y ax b P P 1
ba ba 100% , 100% a a
180o n 2
1 S L p a , S L Ra 2 1 1 V SB h , V R 2h 3 3
Pt Po 1 r
a
4 S 4 R 2 , V R 3 3
d1 / / d 2 a a
a 3 d a 2 , h 2
x
x2 x1 y2 y1 2
x1 f1 x2 f 2 ... xn f n f1 f 2 ... f
y2 y1 x2 x1
y y A yB y A x x A xB x A
4 S 4 R 2 , V R 3 3
AB
t
d1 d 2 a a 1 2
a 3 b3 a b a 2
ab b 2
:
I. :
II.
:
III.
:
IV. :
V.
:
VI. VII.
:
៩។
(C.U) ២០១២
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´EdlCaGñkeroberogsgÇwmfa elakGñkmitþGñkGannwgeRbIR)as;esovePAenH eFVICaÉksarCMnYykñúg karsikSaKNitviTüa minEmneRbIR)as;esovePAenH edIm,IebIkcmøgeBlRbLgeLIy. sUmmitþGñkGanRbwgERbg bnþeTot edIm,IeFVI[KNitviTüakm<úCamanPaBrIkceRmIn edaysarsm,ÚrGñkmancMeNHdwgEpñkKNitviTüa. sUmmitþGñkGan mansuxPaBl¥ suvtßiPaBRKb;eBl ehIyeronsURt)anBUEk nigTTYl)aneCaKC½ykñgú dMeNIrCIvit . sVayerog> éf¶TI 12 Ex FñÚ qñaM 2012 yuvnisSit ÉkeTsKNitviTüa
esckþIEføgGMNrKuN sUmGrKuNCUndl;elakk«Buk Ca cMerIn nigGñkmþay eBRC esaP½NÐ EdlelakTaMgBIr)anpþl;kMeNIt dl;kUn ehIybI)ac;EfrkSaciBa©wmkUn CYy[kUn)ansikSarhUt)ankøayCaRKÚbeRgónmñak;. sUmCUnBrelakTaMgBIr manRBHCnµynW yUr RbkbedaysuxPaBl¥ edIm,ICYypþl;PaBkk;ekþAdl;kUnecACMnan;eRkay. GrKuNdl;bgb¥Ún RbusRsITaMg 4 nak; Edl)anCYypÁt;pÁg; nigeRCamERCgkareronsURtrbs;´ TaMgfvikar nigsmÖar³ CaBiessKW kMuBüÚT½r eFVI[´GacBRgwg nigBRgIkcMeNHdwgEpñkviTüasaRsþ)anRbesIrCagmun. CUnBrdl;dl;bgb¥ÚnTaMg 4 man suxPaBl¥ nigRbkbrbrkak;kbekIncMNUl. sUmGrKuNdl;elakRKU Kg; Narin sBVéf¶Canaykén GnuviTüal½y eKakRBIg EdlelakRKÚ )anxitxM RbwgERbgminsþaykmøaMgkñúgkarbgðat;beRgón´ k¾dUcCasisSdéTeTotenAkñúgfñak;. ´Kitfa cMeNHdwgKNitviTüa kRmitbzmPUmirbs;´CaeRcIn )anedaysarkarRbwgERbg nigBüayamsþab;karbgðat;beRgónrbs;elakRKU. dUcenH´sUmTTYlsÁal;KuNd¾FMeFgenH Edl´minGacbMePøc)aneTAéf¶GnaKt. sUmCUnBrdl;elakRKU nigRkum RKÜsar sUmmansuxPaBl¥ nigsuvtßiPaBRKb;eBlevla . sUmGrRBHKuNdl;RBHetCKuN Pwm suPNÐ CaRKÚecAGFikarvtþéRBrMdYl EdlRBHGgÁ)anGnuBaØati[´ RBHkruNa sñak;GaRs½yenAkñúgvtþ nigeRbIR)as;GKÁisnI kñúgkarerobcMemeron RBmTaMg)anCYypÁt;pÁg;xagPsþúPa b¤smÖa³epSg². sUmRbeKnBrdl;RBHetCKuNmanRBHCnµyWnyUr. sUmGrKuNdl;GñkRKÚ Niko CaGñksµ½RKcitþCnCatiCb:un EdlGñkRKÚ)anpþl;kmøaMgcitþ nigelIkTwkcitþdl; rUb´kñúgkarRbkbrbrCaRKÚbeRgón nigCYy]btßmdl;rbU ´nUvfvikamYycMnYnFM. sUmCUnBrGñkRKÚmansuxPaBl¥ CaBiessenHeBlGñkRKÚeFVIdMeNIrmatuPUminivtþn_eTAkan;RbeTsCb:unvij sUmmansuvtßiPaBx
elak Gan suxKn§a 2> kBaØa pl RsIFa 3> kBaØa qay suKn§amMu 4> kBaØa pat; suvNÑarI 5> kBaØa sinu suKn§a 6> elak suxa kuslü 7> elak Ém suxsuvNÑ 8> kBaØa Kg; r½tñFI 9> elak kun Git 10 elak yn; m:ar:U 11 elak Gul eGn 12 elak Kwm pan;Na 13 kBaØa v:an; cnßa 14 kBaØa søkw y:at 15> kBaØa sumw sIuNat 16 elak BUk suxsanþ 17> elak etg sarI 18 elak Cwm sar:at; 19 elak raC vutßa 20> elak vn lINa 21 elak mas Parmü 22 kBaØa ey:ak bu)aö 23 elak hYn siuj 24 KrusisS eRbIs dar:a .
ត
CakarBitkarsikSaKNitviTüa vaBMuEmnCakargayRsYlenaHeT CaBiessKWkarsikSaedIm,I[køayxøÜnCasisS BUEkEtmþg. vaTamTar[eyIgmankarts‘U Büayam GMNt; Gt;Fµt; CaeRcInTaMgkay nigcitþ EtBMuEmnmann½yfa eyIgminGaceFVI)anenaHeT. eBlsikSaeyIgRtUvEck[dac;fa {etIeyIgeronykecH b¤eronykCab; ?} . kalNa eyIgcg;køayxøÜneTACasisSBUEkKNitviTüa eyIgRtUveronykecH KWmann½yfaeyIgRtUvyl;[c,as; GMBIemeron rUbmnþ b¤lMhat;EdleyIg)aneronrYc minEmneronbnøMbgáÚvxøÜnÉgenaHeT. kareronbnøMxøÜnÉg nwgeFV[I eyIg køayeTACa mnusSsÞak;esÞrI rhUtdl;maneBlxøH minh‘anfaxøÜnÉgxus b¤RtUv k¾man. skmµPaBEbbenH nwgeFVI[eyIgTTYl)an braC½ykñúgkarsikSa naM[eyIgmanGnaKtminl¥. kñúgnamCaGñkFøab;qøgkat;karsikSa ehIyk¾Føab;TTYl)anTaMgbraC½y nigeCaKC½y kñúgkarsikSaEpñk KNitviTüa ´sUmbgðajGMBbI TBiesaFn_braC½yrbs;´dUcteTA ³ -minyl;BIrebobénkarRbLg b¤rebobénkarsresrviBaØasa. -min)anBRgwgsmtßPaBxøÜnÉg elIlMhat;nImYy²[)anc,as;las; ¬emIlgaylMhat;¦. -caMEtTTYlkarbgðat;beRgónBIRKU min)ansV½ysikSa min)aneFVIkarRsavRCavelIlMhat;epSg². -manemaTnPaBRCulelIxøÜnÉg >>>. skmµPaBTaMgGs;xagelIenHehIy EdleFVI[´TTYl)anbraC½ykñúgkarsikSaKNitviTüa . sUmkMue)aHbg;ecalkartaMgcitþ enAeBlEdleyIgcab;epþImdMbUg ehIyeyIgCYblMhat;sisSBUEksuT§EtlM)ak enaHvaCaerOgFmµtaeT eRBaHeyIgminTan;manbTBiesaFn_RKab;RKan;edIm,IedaHRsayva. eyIgRtUvKitkñúgcitþfa {lMhat;EdleFVIecj KWCalMhat;EdlFøab;CYb } . XøaenHKWcg;mann½yfa sisSBUEkKNitviTüaesÞIrEtTaMgGs;suT§Et CaGñkmanbTBiesaFn_Føab;CYblMhat;eRcIn dUcenHeFVI[eKGacdwgGMBIviFI l,ic b¤bec©keTsedIm,IedaHRsaylMhat; TaMgenaH . kalNaeyIgedaHRsaylMhat;sisSBUEk)ankan;EteRcIn enaHeFVI[eyIgdwgBIviFIedaHRsaykan;EteRcIn Edr. edIm,IkøayxøÜneTACasisSBUEkKNitviTüa GñkminRtUvmanKMnitdUcxageRkam ³ -eKeronecHmkBIeKCakUnGñkman manluyeronKY b¤CYlRKUbeRgón . -lMhat;eKCYbl¥² TajecjBIGIunF½reNt rIÉeyIgKµanlT§PaBdUceK . -eKCamnusSmandugtaMgBIkMeNIt . -suxPaBeKl¥ >>>. sUmkMumanKMnitEbbenH vaKµanRbeyaCn_GVIsRmab;eyIgeT eRkABIeFVI[eyIgtUccitþelIxøÜnÉg. cab;BIeBlenHteTAeyIgRtUvxMeronsURtbEnßmeTot RtUvRsavRCav edaHRsaylMhat;[)aneRcIn ecosvag karx¢il. bisacx¢ilxøackareFVIkargarCaTmøab;Nas; dUcenHsUmkM[u bisacx¢ilcUlxøÜnrbs;eyIg KWeyIgRtUvman EpnkareronsURt[)anc,as;las; ehIyGnuvtþkareronsURtTaMgenaH[køayeTACaTmøab;. CacugeRkaysUmCUnBrdl;mti þGñkGanTaMgGs; mansuxPaBl¥ edIm,IeronsURtkøayeTACamnusSmansmtßPaB nigTTYl)aneCaKC½y edIm,IGPivDÆRbeTsCatieyIg[manPaBrIkceRmIn.
CIvRbvtþisegçbrbs;Gñkeroberog eRbIs dar:a ³ 04 ³
.
PROEUS DARA
1988
24
1989 )។ ។ ។ ។
5
2
³ 090 250 667 ។
4។
³ 011 50 70 65 / 0977 65 70 50 ។ ³ E-mail : [email protected] ។ ³
៩
2009 ។
³
។ ។
³ ³
។
។
³ 2003-2009 ³ ³ 2009-2011 ³ 2011³ 1997-2003
99999999999
3
(RUPP)
(KU)
³
I
> cMnYnGsniTan ............................................001 2> smamaRt ..................................................002 3> kenSamBICKNit .........................................003 4> smIkardWeRkTI 1manmYyGBaØat ..................004 5> vismIkardWeRkTI 1manmYyGBaØat ..................005 6> bMENgEckeRbkg; ......................................006 7> mFümsßiti ..................................................007 8> RbU)ab ........................................................008 9> cm¶ayrvagBIrcMNuc ....................................008
> smIkarénbnÞat; ....................................... 008 11> RbB½n§smIkardWeRkTI 1manBIrGBaØat ......... 009 12> RTwsþIbTBItaK½r ........................................ 009 13> rgVg; nigbnÞat; .......................................... 010 14> lkçN³mMuénrgVg; ...................................... 011 15> RTwsþIbTtaEls ....................................... 013 16> RtIekaNdUcKña ......................................... 014 17> BhuekaN ............................................... 016 18> sUlIt ..................................................... 017
1
II
10
³
-ផ្ផែក ប្បធានលំហាត់ប្បតប ិ តតិ
-ផ្ផែក កំផ្ែលំហាត់ប្បតប ិ តតិ
1
1
> cMnYnGsniTan ............................................018 2> smamaRt ..................................................019 3> kenSamBICKNit .........................................020 4> smIkardWeRkTI 1manmYyGBaØat ..................021 5> vismIkardWeRkTI 1manmYyGBaØat ..................022 6> bMENgEckeRbkg; ......................................023 7> mFümsßiti ..................................................024 8> RbU)ab ........................................................025 9> cm¶ayrvagBIrcMNuc ....................................026 10> smIkarénbnÞat; ........................................027 11> RbB½n§smIkardWeRkTI 1manBIrGBaØat ..........028 12> RTwsþIbTBItaK½r .........................................029 13> rgVg; nigbnÞat; ...........................................030 14> lkçN³mMuénrgVg; .......................................031 15> RTwsþIbTtaEls .......................................033 16> RtIekaNdUcKña ..........................................034 17> BhuekaN ................................................035 18> sUlIt ......................................................036
> cMnYnGsniTan ........................................... 037 2> smamaRt ................................................. 039 3> kenSamBICKNit ........................................ 040 4> smIkardWeRkTI 1manmYyGBaØat ................. 042 5> vismIkardWeRkTI 1manmYyGBaØat.................. 044 6> bMENgEckeRbkg; ..................................... 046 7> mFümsßiti ................................................. 048 8> RbU)ab ........................................................ 050 9> cm¶ayrvagBIrcMNuc ................................... 051 10> smIkarénbnÞat; ....................................... 052 11> RbB½n§smIkardWeRkTI 1manBIrGBaØat ......... 055 12> RTwsþIbTBItaK½r ........................................ 057 13> rgVg; nigbnÞat; .......................................... 058 14> lkçN³mMuénrgVg; ...................................... 060 15> RTwsþIbTtaEls ....................................... 062 16> RtIekaNdUcKña ......................................... 063 17> BhuekaN ............................................... 065 18> sUlIt ..................................................... 066 i
III
³
> cMnYnGsniTan ............................................068 2> smamaRt ..................................................082 3> kenSamBICKNit .........................................085 4> smIkardWeRkTI 1manmYyGBaØat ..................097 5> vismIkardWeRkTI 1manmYyGBaØat ..................106 6> bMENgEckeRbkg; ......................................115 7> mFümsßiti ..................................................122 8> RbU)ab ........................................................129 9> cm¶ayrvagBIrcMNuc ....................................137
> smIkarénbnÞat; ....................................... 149 11> RbB½n§smIkardWeRkTI 1manBIrGBaØat ......... 162 12> RTwsþIbTBItaK½r ........................................ 176 13> rgVg; nigbnÞat; .......................................... 181 14> lkçN³mMuénrgVg; ...................................... 189 15> RTwsþIbTtaEls ....................................... 196 16> RtIekaNdUcKña ......................................... 203 17> BhuekaN ............................................... 217 18> sUlIt ...................................................... 225
1
IV
10
³
ផ្ផែកលំ
ផ្ផែកចម្លយ ើ ននលំ
....238
V
ផ្ែលបាន
............. 309
³
> DIsbøÚmqñaM 1981 ........................................352 2> DIsbøÚmqñaM 1982 ........................................356 3> DIsbøÚmqñaM 1983 .......................................360 4> DIsbøÚmqñaM 1984 ........................................362 5> DIsbøÚmqñaM 1985 ........................................366 6> DIsbøÚmqñaM 1986 .......................................370 7> DIsbøÚmqñaM 1987 .......................................374 8> DIsbøÚmqñaM 1988 ........................................378 9> DIsbøÚmqñaM 1989 ........................................382 10> DIsbøÚmqñaM 1990 ......................................386 11> DIsbøÚmqñaM 1991 ......................................390 12> DIsbøÚmqñaM 1992 ......................................394
> DIsbøÚmqñaM 1993 ..................................... 398 14> DIsbøÚmqñaM 1994 ...................................... 400 15> DIsbøÚmqñaM 1995 (មលើកទ១ ី ) ...................... 404 16> DIsbøÚmqñaM 1995 (មលើកទ២ ី ) ..................... 408 17> DIsbøÚmqñaM 1996 (មលើកទី១) ..................... 412 18> DIsbøÚmqñaM 1996 (មលើកទី២) .................... 416 19> DIsbøÚmqñaM 1997 (មលើកទី១)...................... 420 20> DIsbøÚmqñaM 1997 (មលើកទី២) ..................... 420 21> DIsbøÚmqñaM 1998 (មលើកទី១) ...................... 422 22> DIsbøÚmqñaM 1998 (មលើកទី២) ..................... 424 23> DIsbøÚmqñaM 1999 (មលើកទី១) .................... 428 24> DIsbøÚmqñaM 1999 (មលើកទ២ ី ) ...................... 432
1
13
ii
> DIsbøÚmqñaM 2000 .....................................436 26> DIsbøÚmqñaM 2001 .....................................440 27> DIsbøÚmqñaM 2002 .....................................444 28> DIsbøÚmqñaM 2003 ....................................448 29> DIsbøÚmqñaM 2004 ......................................454 30> DIsbøÚmqñaM 2005 ......................................458 31> DIsbøÚmqñaM 2006 ......................................462
> DIsbøÚmqñaM 2007 ..................................... 466 33> DIsbøÚmqñaM 2008 ..................................... 470 34> DIsbøÚmqñaM 2009 ..................................... 474 35> DIsbøÚmqñaM 2010 ..................................... 478 36> DIsbøÚmqñaM 2011 .................................... 482 37> DIsbøÚmqñaM 2012 ..................................... 486
25
VI
32
³
> sisSBUqñaM 1986 ¬elIkTI1¦ ......................490 2> sisSBUqñaM 1986 ¬elIkTI2¦ ........................491 3> sisSBUqñaM 1987 ¬elIkTI1¦ .......................492 4> sisSBUqñaM 1987 ¬elIkTI2¦ .......................493 5> sisSBUqñaM 1988 ¬elIkTI1¦........................494 6> sisSBUqñaM 1988 ¬elIkTI2¦ ........................495 7> sisSBUqñaM 1989 ¬elIkTI1¦ ........................496 8> sisSBUqñaM 1989 ¬elIkTI2¦ ........................497 9> sisSBUqñaM 1990 ¬elIkTI1¦ ........................498 10> sisSBUqñaM 1990 ¬elIkTI2¦ ......................499 11> sisSBUqñaM 1991 ¬elIkTI1¦ .....................500 12> sisSBUqñaM 1991 ¬elIkTI2¦ .....................501 13> sisSBUqñaM 1992 ¬elIkTI1¦......................502 14> sisSBUqñaM 1992 ¬elIkTI2¦......................503 15> sisSBUqñaM 1993 ¬elIkTI1¦ .....................504 16> sisSBUqñaM 1993 ¬elIkTI2¦ .....................505 17> sisSBUqñaM 1994 ¬elIkTI1¦ ......................506 18> sisSBUqñaM 1994 ¬elIkTI2¦ ......................507
> sisSBUqñaM 1995 ¬elIkTI1¦ ..................... 508 20> sisSBUqñaM 1995 ¬elIkTI2¦ ..................... 509 21> sisSBUqñaM 1996 ¬elIkTI1¦ ..................... 510 22> sisSBUqñaM 1996 ¬elIkTI2¦ ..................... 511 23> sisSBUqñaM 1997 ¬elIkTI1¦..................... 512 24> sisSBUqñaM 1997 ¬elIkTI2¦ ..................... 513 25> sisSBUqñaM 1998 ¬elIkTI1¦ ..................... 514 26> sisSBUqñaM 1998 ¬elIkTI2¦ ..................... 515 27> sisSBUqñaM 1999 ¬elIkTI1¦ ..................... 516 28> sisSBUqñaM 1999 ¬elIkTI2¦ ..................... 517 29> sisSBUqñaM 2000 ¬elIkTI1¦ ..................... 518 30> sisSBUqñaM 2000 ¬elIkTI2¦ ..................... 519 31> sisSBUqñaM 2001 ¬elIkTI1¦ .................... 520 32> sisSBUqñaM 2001 ¬elIkTI2¦ .................... 521 33> sisSBUqñaM 2002 ¬elIkTI1¦ .................... 522 34> sisSBUqñaM 2002 ¬elIkTI2¦ .................... 523 35> sisSBUqñaM 2003 ¬elIkTI1¦ .................... 524 36> sisSBUqñaM 2003 ¬elIkTI2¦ .................... 525
1
19
iii
> sisSBUqñaM 2004 ¬elIkTI1¦......................526 38> sisSBUqñaM 2004 ¬elIkTI2¦ ......................527 39> sisSBUqñaM 2005 ¬elIkTI1¦ ......................528 40> sisSBUqñaM 2005 ¬elIkTI2¦ ......................529 41> sisSBUqñaM 2006 ¬elIkTI1¦......................530 42> sisSBUqñaM 2006 ¬elIkTI2¦......................531 43> sisSBUqñaM 2007 ¬elIkTI1¦ .....................532 44> sisSBUqñaM 2007 ¬elIkTI2¦ .....................533 45> sisSBUqñaM 2008 ¬elIkTI1¦......................534 46> sisSBUqñaM 2008 ¬elIkTI2¦......................535
> sisSBUqñaM 2009 ¬elIkTI1¦ ..................... 536 48> sisSBUqñaM 2009 ¬elIkTI2¦ ..................... 537 49> sisSBUqñaM 2010 ¬elIkTI1¦ .................... 538 50> sisSBUqñaM 2010 ¬elIkTI2¦ .................... 539 51> sisSBUqñaM 2011 ¬elIkTI1¦ .................... 540 52> sisSBUqñaM 2011 ¬elIkTI2¦ .................... 541 53> sisSBUqñaM 2012 ¬elIkTI1¦ .................... 542 54> sisSBUqñaM 2012 ¬elIkTI2¦ .................... 544 55> វញ្ញ ិ ា សាម្ែើប្បឡង្ិ្សពូផ្ក........................ 545 56> វញ្ញ ិ ា សាធាលប់ប្បឡង្ិ្សពូផ្កតា្មេតត .......... 625
37
VII 1
47
³
> EpñkRbFanlMhat;l¥² .................................645 VIII
2
> EpñkcemøIyénlMhat;l¥²
............................ 669
³
> 2> ចម្លយ ..........................787 ិ ើ ននក្ានតគែិតវទា 3> បរមាប្ត នផៃប្កឡា នង ិ ិ មាឌរូបធរែីមាប្ត .........793
> 5> តារាងតន្លផលមធៀបប្តមី ោែមាប្ត ............... 798 6> ខ្នែតននរង្វវ្់រង្វវល់មផសងៗ ........................... 799
1 ក្ានតគែិតវទា ........................................781 ិ
4 តារាង្វ័យគុែ និងឫ្ទី n ....................... 796
3 iv
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1 mile
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1 inch
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1 feet
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1 lb
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147 :
ង
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ងង។ ច
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148 :
150 : ST
ច
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152 :
ង
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155 : ច ច
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153 :
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160 :
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161:
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162:
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164:
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ACB 34o ង CAD 45o ។ ច
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167:
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169 :
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171:
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173:
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174: ង
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175:
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260o S
។
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F 40o
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182 : P
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184 :
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186 :
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188 :
X> g>
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193 : ច
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197 : ច
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PMN
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195 :
y
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B
24 m C
32 m
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-
200 :
x
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20cm
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214 :
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217 : ច
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o
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220 :
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12o
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35
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-
226 : ច
ង ង
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។
10 cm 13 cm
5 cm 55cm cm
2 cm
5 cm
a) -
228 :
-
231 : ច
b)
28cm ង
ង
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12 mm
c)
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13cm ។ ច ង
dm3 ។ ។
3 mm o
o
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DE / / BC , AD 3cm , AB 5cm ។
232 :
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.ច ខ. ច
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tag t Cary³eBlEdlTTYlsBaØaBIPBRBHcnÞ tambRmab; eyIg)an ³ -cm¶ay 186 000 mile eRbIry³eBl 1 s - cm¶ay 240 000 mile eRbIry³eBl t s 000 1 naM[ 186 240 000 t t
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x 2 2 x x 3 2 x 1 x 2 x 2 2 x 2 x 6 2 x 1 x 2 x 2 2 x 2 x 6 2 x 4 x x 2
A 2 x 2 x 2 x 3 2 x 1 x 2
-cMeBaH
2
eday
2
2
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2
eday
2
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2
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eday
A 6 x 2 x 1 2 x x 1
x 1 6 x 2 2 x
dUcenH -cMeBaH
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2
eday dUcenH -cMeBaH
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eday
x 1 x 2 x 1 x 2 x 2
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. tamtaragplKuNExVg
x 5 5 x 3x 2 x
. tamtaragplKuNExVg
x 2 2 x 15 x 3x 5 x 2 11x 30
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.
tamtaragplKuNExVg
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2
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x 4
x 1x 23 x
2
x 1 x 5 x 4 x
x 3
B x 1 x 2 xx 1
tamtaragplKuNExVg
x 5
2
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dUcenH -cMeBaH
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3 x 22 x 12 x 1 2 x 22 x 1 x 22 x 132 x 1 22 x 1 x 22 x 16 x 3 4 x 2 x 22 x 12 x 5
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1
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2 x 2 5 x 3 x 12 x 3
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-TMB½rTI 35 ³ dak;kenSamCaplKuNénktþa
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x 2 6x 7 x 2 6x 9 9 7
A
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b2
.
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-TMB½rTI 36 ³ KNna 1
1 2 x x 1
manPaKEbgrYmKW xx 1
b¤
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a 2 0 2 a 0 a 1 0
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1 b
EdlktþasRmÜl
xx 1 x 1 2 x xx 1
a 2 a 1 a 2a 1
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x 2 x x3 x 2 2 x 2 x x x 1
x 2 x 3x 2 2 x 2 x 2 x x x 1 2x 2 2x x x 1 x 0 x 0 2 x x 1 , x x 1 x 1 0 x 1 2
41
-
o
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-TMB½rTI 45 ³ edaHRsaysmIkarxageRkam ³
6x 2 3x 8
x 2 2x 4x 8 0 x x 2 4 x 2 0 x 2x 4 0
2 x 2 x 8 2x 4 x 8 2 x x 8 4 x 4
x 2 0 x 4 0 x 2 x 4
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2 x 7 3x 27 3x 2 x 7 27 x 34
epÞógpÞat; ³
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43
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2
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x
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0
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1 3
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-TMB½rTI 58 ³ k> rktémø x EdlRtUvyk
epÞógpÞat; ³
tambRmab;RbFaneyIgcg)anRbB½n§vismIkar ³
638 639 640 1918 638 639 640 1914
2x 5 18 2 x 10 18 5 x 50 5 x 50 2x 8 x 4 1 5 x 50 x 10 2
1917 1918 1917 1914
dUcenH cMnYnKt;tKñaTaMgbIEdlRtUvrkKW
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Bit Bit
638 , 639 , 640
.
0 4
cMeBaHvismIkar 2 : x 10
x
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25 5 18 20 18 5 5 50 25 50
Bit Bit
dUcenH vismIkarmancemøIyKW 4 x 10 . x> rkcMnYnKt;TaMgbIenaH tag x CacMnYnKt;TI 1 naM[ x 1 CacMnYnKt;TI 2 nig x 2 CacMnYnKt;TI 3 tambRmab;RbFan eyIg)an RbB½n§vismIkar³ x x 1 x 2 1918 x x 1 x 2 1914
3 x 3 1918 3 x 3 1914 3 x 1915 3 x 1911 3 x 1915 3 x 1911 x 638.33 x 637
cMnYnKt;EdlenAcenøaH 637 x 638.33 manEtmYy Kt;KW x 638 naM[ cMnYnTI2KW x 1 639 / cMnYnTI3 x 2 640
45
-
៦
-TMB½rTI 64 ³ sg;Rkabssr ³
cMnYnkUn cMnYnRKYsar eRbkg;fy eRbkg;eFob eRbkg;eFob fy f % f% f f x
eyIgmanTinñn½ycMnYnéf¶EdlbuKÁlikmin)anbMeBj kargarkñúgry³eBj 20 éf¶ dUcxageRkam ³
0 1 2 3 4
1 2 0 0 1 2 2 1 0 0 4 0 1 1 3 2 1 3 0 1
eyIg)antaragbMENgEckeRbkg;dUcxageRkam ³ cMnYnbuKÁlik eRbkg; f eRbkg;eFob f % 0 1 2 3 4
6 7 4 2 1
4 16 18 6 6
8 32 36 12 12
100 92 60 24 12
k> rkcMnYnRKYsarEdlmankUn 2nak;y:ageRcIn tamtarageRbkg;ekIn cMnYnRKYsarEdlmankUn 2nak; y:ageRcInKWman 38 RKYsar. x> rkcMnYnRKYsarEdlmankUn 2nak;y:agtic tamtarageRbkg;fy cMnYnRKYsarEdlmankUn 2nak; y:agticKWman 30 RKYsar.
30 35 20 10 5
20 100 srub tamtaragbMENgEckeRbkg; eyIgsg;)anRkabssr ³ eRbkg; f
-TMB½rTI 71 ³ sßitim:assisS 200nak;énviTüal½ymYy
7 6 5 4
m:as kg eRbkg;
30-40
40-50
50-60
60-70
35
45
55
65
k> sg;tarageRbkg;ekIn nigp©itfñak;
3 2 1
1
0
2
3
4
fñak;énm:as eRbkg; eRbkg;ekIn p©itfñak;
cMnYnbuKÁlik
30-40 40-50 50-60 60-70
-TMB½rTI 67 ³ sg;tarageRbkg;ekIn nigeRbkg;fy eyIgmanTinñn½y Edl x CacMnYnkUn nig f CacMnYnRKYsar x f
0 1 2 3 4 4 16 18 6 6
4 20 38 44 50
35 80 135 200
35 45 55 65
eRbkg;ekIn
cMnYnkUn cMnYnRKYsar eRbkg;ekIn eRbkg;eFob eRbkg;eFob ekIn f % f% f f x 4 16 18 6 6
35 45 55 65
x> sg;BhuekaNeRbkg;ekIn
eyIgsg;)antarageRbkg;ekIn ³ 0 1 2 3 4
50 46 30 12 6
8 32 36 12 12
200
8 40 76 88 100
150
135
100
80
50 35
eyIgsg;)antarageRbkg;fy ³
30 40 50 60 70
46
m:as -
K> rkcMnYnsisS Edlmanm:aseRkam 50 kg tamtarageRbkg;ekIn cMnYnsisSEdlmanm:aseRkam 50 kg mancMnYn 35 45 80 nak; . dUcenH cMnYnsisSEdlmanm:aseRkam 50 kg man cMnYn 80 nak; .
47
-
៧
-TMB½rTI 77 ³ KNnamFüméntaragTinñn½yxageRkam ³
-rkemdüan m ³ x 0 1 2 3 4 eyIgerobTinñn½ytamlMdab; 7 8 10 12 13 y 3 6 5 2 2 Tinñn½ymancMnYntYsrubKW n 5 KNnamFüm naM[ TItaMgénemdüanKWtY 5 2 1 3 tamrUbmnþ ³ x x f fx f f ......fx f / f y tamTinñn½yerobtamlMdab;rYc tYTI 3 RtUvnwgelx 10 tamtaragTinñn½yxagelIKW ³ 0 3 1 6 2 5 3 2 4 2 dUcenH rk)anemdüanKW m 10 . x 365 2 2 -rkm:Ut m ³ 0 6 10 6 8 30 1.67 18 18 edaym:Ut CatémøEdlmaneRbkg;eRcInCageK . dUcenH mFüménTinñn½yKW x 1.67 . EtTinñn½yxagelImaneRbkg;esµI1 dUcKñaTaMgGs; -TMB½rTI 78 ³ rkemdüanénR)ak;ebovtSn_¬KitCaBan;erol¦ dUcenH Tinñny½ enHKµanm:UteT . R)ak;ebovtSn_ 200 300 350 700 840 950 -TMB½rTI 82 ³ KNnamFüm emdüan nigm:UténTinñn½y ³ 6 2 2 1 1 1 eRbkg; fñak; eyIgsg;)antarageRbkg;dUcxageRkam ³ 1 1
2
1
2
n
2
e
n
n
e
o
R)ak;ebovtSn_ x eRbkg; f eRbkg;ekIn 200 300 350 700 840 950
6 2 2 1 1 1 13
srub
6 8 10 11 12 13
21-28
28-35
35-42
42-49
49-56
56-63
63-70
3
7
12
15
12
7
3
eRbkg;
tamtaragTinñn½y eyIgsg;)antarageRbkg;ekIn ³
xf 1200 600 700 700 840 950 4990
fñak;
edayTinñn½ymancMnYntYsrub n 13 CacMnYness naM[ TItaMgén m KWtY 132 1 142 7 tamtarageRbkg; tYTI 7 RtUvnwgTwkR)ak; 300 Ban;erol dUcenH emdüanénR)ak;ebovtSn_KW m 300 Ban;erol. -TMB½rTI 79 ³ rkmFüm emdüan nigm:Ut énTinñn½y eyIgmanTinñny½ ³ 10 8 13 12 7 -rkmFüm x ³ tamTinñn½y x 10 8 135 12 7 505 10 dUcenH rk)anmFümKW x 10 . e
21-28 28-35 35-42 42-49 49-56 56-63 63-70
3 7 12 15 12 7 3
srub
59
24.5 31.5 38.5 45.5 52.5 59.5 66.5
xf
eRb>ekIn
73.5 220.5 462 682.5 630 416.5 199.5
3 10 22 37 49 56 59
2684.5
-KNnamFüm x ³ .5 45.5 eyIg)an x 2684 59
e
eRbkg; f p©itfñak; x
dUcenH KNna)anmFüm x 45.5 . -KNnaemdüan m ³ emdüan CatémøéntYTI 592 29.5 tamtarageRbkg;ekIntYTI 30 sßitenAkñúgfñak; 42-49 eyIgKNnaemdüantamGaMgETb:ULasüúg e
48
-
naM[ m
e
42
49 4229.5 22 45.5 37 22
dUcenH KNna)anemdüan
me 45 .5
.
-KNnam:Ut m ³ m:Ut Catémøp©itfñak;EdlmaneRbkg;eRcInCageK fñak; 42-49 maneRbkg; 15 eRcInCageK ehIyman p©itfñak;esµI 45.5 o
dUcenH KNna)an
x me mo 45 .5
.
49
-
៨
-TMB½rTI 89 ³ fg;mYymanXøIBN’exµA 4 nigXøIBN’s 2
-TMB½rTI 92 ³
rkRbU)abEdleQµaH k cab;)anXøIBN’exµA -XøIsrubman 6 naM[ krNIGac 6 -XøIBN’exµAman 4 naM[ krNIRsb 4 R sb tamrUbmnþ P cMcMnnYnYnkrNI krNIG ac
rkRbU)abEdlecjelxdUcKña -RKab;LúkLak;manmux 6 enaHeKGacpÁÜbKUlT§pl rbs;va)an TaMgGs;cMnYn 6 6 36 rebob naM[ krNIGac 36 -KUEdlmanelxdUcKñaGacCa 1,1 2 , 2 3, 3 4 , 4 5 , 5 nig 6 , 6 man 6krNI naM[ krNIRsb 6
1.
P
1.
XøIBN’exµA cMnnY krNIRsb 4 2 66 .67 % cMnnY krNIG ac
6
3
dUcenH RbU)ab k cab;)anXøIBN’exµAKW P 66.67% . 2. rkRbU)abEdleQµaH x cab;)anXøIBN’exµA edayeQµaH k cab;)anXøIBN’exµAehIymindak;cUlfg; vij naM[b:HBal;dl;karcab;elIkeRkayrbs;Gñk x -XøITaMgGs;sl;Et 5 naM[ krNIGac 5 -XøIBN’exµAsl;Et 3 naM[ krNIRsb 3 P
dUcenH
P
R sb 6 1 . elxdUcKña cMcMnnYnYnkrNI krNIG ac 36 6
rkRbU)abEdlplbUkRKab;TaMgBIresµInwg 10 KUénRKab;TaMgBIrmanplbUkesµI 10 GacCa 4 , 6 , 5 , 5 , 6 , 4 manbIkrNI naM[ krNIRsb 3
2.
cMnYnkrNIR sb 3 60 % XøIBN’exµA cMnYnkrNIG ac
5
dUcenH
dUcenH RbU)ab x cab;)anXøIBN’exµAKW P 0.6 .
P
cMnYnkrNIR sb plbUkesµI!0 cMnYnkrNIG ac
3 1 36 12
-TMB½rTI 90 ³ rkRbU)abEdleFVIdMeNIredaymFü)aydéT tag P CaRbU)abGñkeFVIdMeNIredayrfynþpÞal;xøÜn -Rkumh‘unmanbuKÁlik 250nak; naM[ krNIGac 250 -GñkCiHrfynþpÞal;xøÜn 50nak; naM[ krNIRsb 50 R sb 50 1 20 % naM[ P cMcMnnYnYnkrNI krNIG ac 250 5 tag P CaRbU)abénGñkeFVIdMeNIredaymeFüa)aydéT Edl P CaRBwtþikarN_bMeBjKñaCamYyRBwtþikarN_ P naM[ P 1 P 1 15 54 80% dUcenH RbU)abénGñkeFVIdMeNIredaymeFüa)aydéT EdlminCiHrfynþpÞal;xøÜnKW P 80% .
50
-
.
៩
-TMB½rTI 99 ³ KNnacm¶ayrvagBIrcMNuc eyIgGacKNnacm¶ayrvagBIrcMNuc edayeRbIrUbmnþ ³ xB x A 2 yB y A 2 A0 , 0 & B8 , 15
AB
k> naM[ AB
8 02 15 02
64 225 289
17
ÉktaRbEvg .
dUcenH KNna)an
AB 17
ÉktaRbEvg .
x> A 2 , 6 & B3 , 6 naM[ AB 3 2 6 6 2
2
25 144 169
ÉktaRbEvg . dUcenH KNna)an AB 13 ÉktaRbEvg . K> A 3 , 5 & B1 , 8 naM[ AB 1 3 8 5 13
2
2
49 13
ÉktaRbEvg .
dUcenH KNna)an
ÉktaRbEvg .
AB 13
X> A5 , 3 & B11 , 11 naM[ AB 11 5 11 3 2
2
36 64 100
10
ÉktaRbEvg .
dUcenH KNna)an
AB 10
ÉktaRbEvg .
51
-
១០
-TMB½rTI 110 ³ sg;bnÞat;TaMgbIkñúgbøg;EtmYy
-TMB½rTI 108 ³ !> sg;bnÞat;EdlmansmIkar y 2x cMeBaH x 0 naM[
0 y 0 2
-cMeBaHbnÞat;
Edr CabnÞat;kat;Kl;
y
x4 2
b£
1 y x2 2 x 0 4 1 y x2 y 2 0 2
taragtémøelxén KW G½kS . bnÞat;enHmanemKuNR)ab;Tis a 12 mann½yfa -cMeBaHbnÞat; y 2 CabnÞat;edk kat;G½kS yy Rtg; 2 ebI x ekInBIrÉkta enaH y fycuHmYyÉkta. -cMeBaHbnÞat; x 4 CabnÞat;Qr kat;G½kS xx Rtg; 4 eyIgsg;bnÞat;)an dUcxageRkam ³ sg;bnÞat;TaMgbIkñúgbøg;EtmYydUcxageRkam ³ y y
x 2
y 1 y x2 2
0
1
x4
x
2 0
4
y2
x
sg;bnÞat;kat;tamcMNuc A0 , 1 nigmanemKuN KNnaRkLaépÞxNÐedaybnÞat;TaMgbIenH R)ab;TisesµI 13 bnÞat;TaMgbIpÁúM)anCaRtIekaNEkgEdlman )atesµI 4 eyIg)anbnÞat;kat;tam A0 , 1 ehIymanemKuNR)ab; ÉktaRbEvg nigkm
¬RbtibtþienHerobcMmin)anl¥ ³ eK[rksmIkarbnÞat;kat;tamBIrcMNuc Et eBl[cMNuc manEtmYycMNuceTAvij KW 2 , 7 dUcKña¦
x
eKGacsg;bnÞat;eRcInrab;minGs;kat;tammYycMNuc dUcenH minGackMNt;)ansmIkarbnÞat;kat; 2 , 7 .
0
-TMB½rTI 113 ³ rksmIkarbnÞat;kat;tammYycMNuc ehIy
KNnaemKuNR)ab;TisénbnÞat;kat;tamBIrcMNuc A 2 , 0 nig B 1 , 4 naM[bnÞat;manemKuNR)ab;Tis y y 40 4 a 4 . x x 1 2 1 3.
B
A
B
A
RsbeTAnwgbnÞat;mYyepSgeTot k> kat;tam 0 , 0 ehIyRsbnwgbnÞat; y 3x 1 tag M x , y enAelIbnÞat;RtUvrkkat;tam A0 , 0 ehIyRsbnwgbnÞat; y 3x 1 enaHemKuNR)ab;TisesµIKña 52
-
eyIg)an
yM y A 3 xM x A y0 3 x0
eyIg)an naM[
y 3x
dUcenH y 3x CasmIkarbnÞat;EdlRtUvrk . karsg;bnÞat;edIm,IepÞógpÞat;
yM y A 1 1 xM x A 2 y 1 1 1 x2 2 y 1 2x 2 y 2 x 3
dUcenH smIkarbnÞat;EdlRtUvrkKW karsg;RkabedIm,IepÞógpÞat;
y 3x
y 2 x 3
.
y 3x 1 2 y x 1 0
x> kat;tam 6 , 3 ehIyRsbnwgbnÞat; x 3 y 3 y 2 x 3 tag M x , y enAelIbnÞat;RtUvrkkat;tam B6 , 3 ehIyRsbnwgbnÞat; x 3 y 3 b¤bnÞat; y 13 x 1 x> tat;tam A 21 , 0 ehIyEkgnwgbnÞat; y 2x enaHemKuNR)ab;TisesµIKña tag M x , y enAelIbnÞat;RtUvrkkat;tam eyIg)an yx xy 13 ehIyEkgnwgbnÞat; y 2x naM[plKuNemKuNR)ab; y 3 1 TisesµI 1 x6 3 y y 1 1 eyI g )an 2 1 y 3 x 6 naM[ y x 1 x x
1
A
M
B
M
B
2
3
3
M
A
M
A
, 0
y0 2 1 1 x 2
dUcenH y 13 x 1CasmIkarbnÞat;EdlRtUvrk . karsg;bnÞat;edIm,IepÞógpÞat;
1 2 1 1 y x 2 4
2 y x y
1 x 1 3 y
1 x 1 3
dUcenH smIkarbnÞat;EdlRtUvrkKW y 12 x 14 . karsg;RkabedIm,IepÞógpÞat;
-TMB½rTI 115 ³ rksmIkarbnÞat;kat;tammYycMNuc ehIy EkgeTAnwgbnÞat;mYyepSgeTot k> tat;tam A2 , 1 ehIyEkgnwgbnÞat; 2 y x 1 0 tag M x , y enAelIbnÞat;RtUvrkkat;tam A2 , 1 ehIyEkgnwgbnÞat; 2 y x 1 0 b¤ y 12 x 12 naM[ plKuNemKuNR)ab;TisesµI 1
y 2x 1 1 y x 2 4
53
-
-TMB½rTI 117 ³ rkeBlevlaEdlsItuNðPaBesµInwg 26
o
tagG½kS xx CaeBlevlaEdlRtUvekIneLIg nigG½kS yy CasItuNðPaBEdlnwgRtUvfycuH edaysItuNðPaBfycuHkñúgGRta 2 / h mann½yfa ebIeBlekIn 1h enaH sItuNðPaBfycuH 2 eyIg)anbnÞat;kat;tamcMNuc 12 , 40 manemKuNR)ab; esµInwg 2 . ebI M x , y CacMNucmYyenAelIbnÞat;enH enaHeyIg)anbERmbRmYlemKuNR)ab;TisKW ³ o
o
y 40 2 x 12
y 40 2 x 24 y 2 x 64
cMeBaHsItuNðPaB y 26 eyIg)an 26 2x 64 o
2 x 38 x 19
CaeBl
dUcenH sItuNðPaB 26 RtUvnwgem:ag 19 7 yb; . o
54
-
១១
-TMB½rTI 123 ³ edaHRsayRbB½n§smIkartamRkab eyIgmanRbB½n§smIkar
4 x 3 y 4 3x y 2
x> 34xx 4yy178
i ii
-eyIgedaHRsayedayeRbIviFICMnYs tam ii : 4x y 17 y 17 4x yk iii CMnYscUlkñúg i eyIg)an ³
eyIgsg;bnÞat; i & ii enAkñúgbøg;EtmYy -taragtémøelxén 4x 3 y 4 KW yx 10 24 -taragtémøelxén 3x y 2 KW
i :
x 1 0 y 1 2
sg;RkabTaMgBIrenAkñúgbøg;EtmYy eyIg)an ³
3x 417 4 x 8 3x 68 16 x 8 19 x 76 x4
iii : y 17 4 4 1
dUcenH RbB½n§smIkarmanKUcemøIy x , y 4 , 1 . -TMB½rTI 128 ³ edaHRsayRbB½n§smIkar k> 43xx35yy97 12
4x 3 y 4
tamtaragbnÞat;RbsBVKñaRtg;cMNuc 2 , 4 dUcenH RbB½n§smIkarmanKUcemøIy x 2 , y 4 . -TMB½rTI 125 ³ edaHRsayRbB½n§smIkar k> 9yx62y 3x3 12
eyIgedaHRsayedayeRbIviFIbUkbM)at; enaHeyIg)an 4x 3y 9 3 x 5 y 7
-eyIgedaHRsaytamviFICMnYs tam 1 : y 6 3x y 3x 6 yk 3 CYscUleTAkñúg 2 eyIg)an ³
3 4
12 x 9 y 27 12 x 20 y 28 11y 55
cMeBaH
3
y5
1 :
y5
ykCMnYskúñg 1 eyIg)an ³
4x 3 5 9 4 x 24 x6
9 x 23 x 6 3 9 x 6 x 12 3 3 x 9 x 3
dUcenH RbB½n§smIkarmanKUcemøIy x 6 , y 5 . x> 77xx31y63y 55xx yy
yk x 3 CMnYscUlkñúg 3 3 :
iii
yk x 4 CMnYscUlkñúg iii eyIg)an ³
3x y 2
2 :
i ii
y 3 3 6 15
eyIg)an
dUcenH RbB½n§smIkarmanKUcemøIy xy 153 .
1 2
7 x 3 y 9 5x 5 y 7 x 7 6 y 5 x 5 y
55
-
7 x 3 y 9 5 x 5 y 7 x 7 6 y 5 x 5 y 9 y 2 10 y y 2
yk
CMnYnskñúg
eyIg)an
1 y 2 1 : 7 x 3 2 3 5x 2 7 x 15 5 x 10 2x 5 x 5/ 2
dUcenH RbB½n§smIkarmanKUcemøIy x 5 / 2 , y 2 .
-TMB½rTI 131 ³ k> rkcMnYnTaMgBIrenaH tag x CacMnYnTI! nig y CacMnYnTI@ tambRmab;RbFaneyIgcg)anRbB½n§smIkar x y 50 1 eyIgedaHRsaytamviFIbUkbM)at; 3x 2 y 60 2
x y 50 3 x 2 y 60
2
2 x 2 y 100 3 x 2 y 60 5 x 160 x 32
yk x 32 cMnYnkñúg 1 edIm,Irk y eyIg)an ³ 1 :
32 y 50 y 18
dUcenH cMnYnTaMgBIrenaHKW cMnYnTI!= 32 nigcMnYnTI@=18 . x> rkRbEvgTTwg nigbeNþayénctuekaNEkg tag x CaTTwg nig y CabeNþay 0 x y xñatEm:t tambRmab;RbFaneyIgcg)anRbB½n§smIkar 2x y 158 edaHRsayedaybUkbM)at; yx5
x y 79 yx5 2 y 84 y 42
cMeBaH y 42 x 42 5 37 dUcenH ctuekaNEkgman TTwg 37m /beNþay 42m .
56
-
១២
-TMB½rTI 138 ³ KNnargVas; x KitCa cm énrUb tag ycm CaGIub:UetnusénRtIekaNEkg i -kúñg i ³ tamRTwsþIbTBItaK½r y 2 12 12 2
ii
x
-kñúg ii ³ tamRTwsþIbTBItaK½r
1
2
1
i
x y 1 2 1 3 2
2
1
x 3 cm
dUcenH rgVas;KNna)anKW x
3 cm
.
-TMB½rTI 139 ³ KNnargVas;Ggát;RTUgénRbelBIEb:tEkg tag y CaRbEvgGgát;RTUgEdlRtUvrk KitCa m cMeBaH ABC ³ RTwsþIbTBItaK½r x 2 22 12 5
cMeBaH ACD ³ tamRTwsþIbTBItaK½r
D
y
2
y 2 x 2 22 5 4 9
naM[
C
x
A
1
2
y 9 3m
dUcenH RbEvgGgát;RtUvKNna)anKW 3 m .
-TMB½rTI 140 ³ rkkm
h 2 64 4
h 60 2 15 m
dUcenH km
h 2 15 m 7.75 m
.
57
-
១៣
-TMB½rTI 145 ³ KNnaRbEvg AP nig BC eK[ AB 25 cm
-TMB½rTI 150 ³ KNnakaMénrgVg; OQ 20 cm
C
Q
BQ 6.5 cm
P
dUcrUbxagsþaM
-eday AB OP Rtg; P naM[ P kNþal AB 25 cm enaH AP AB 12.5 cm 2 2 -eday BC OQ Rtg; Q naM[ Q kNþal BC enaH BC 2BQ 2 6.5 cm 13 cm
o
OT 2 OS 2 ST 2
OQ TQ2 OS 2 ST 2 r 102 r 2 202 r 2 20r 100 r 2 400 20r 300 r 15
dUcenH KNna)an AP 12.5 cm nig BC 13 cm .
-TMB½rTI 147 ³ rkRbEvg NP
dUcenH kaMrgVg;KNna)anKW OQ r 15 cm .
OA
-TMB½rTI 152 ³ RsaybMPøWfa
M
BR RC
eyIgman ABC CaRtIekaNsm)at naM[ AB AC tamRTwsþIbTbnÞat;b:HEdl KUsecjBIcMNucrYmenAeRkArgVg; eyIg)an AP AQ , BP BR , CQ CR cMeBaH AB AC enaH AP BP AQ CQ eday AP AQ eyIg)an BP CQ P
A o
N
o
Q
dUcenH rk)anRbEvg NP 9 cm .
-TMB½rTI 148 ³ KNnargVas; AC
C
eyIgmanrgVg;p©ti O kaM 2 cm A CacMNucenAeRkArgVg; B CacMNucb:HrgVg; AB OB OC 2 cm ¬eRBaH OB CakaMrgVg;¦ kñúg ABOman OA CaGIub:Uetnus tamRTwsþIbTBItaK½r o 2 cm 2 cm
A 2 cm B
l
tamTMnak;TMng
BP BR BP CQ BR CR
dUcenH eyIgbMPøW)anfa BR RC .
-TMB½rTI 153 ³ KNnaRbEvg
AC OA OC 2 2 2 4.83 cm
AB
¬KitCa cm ¦
eyIgman r 5 cm , r 3 cm eyIgKNnaRbEvg AB tamkrNIdUcxageRkam ³
dUcenH RbEvgKNna)anKW AC 4.83 cm .
.
CQ CR
OA AB2 OB2 22 22 2 2 cm
58
B
A
B P
naM[
r
10 cm
o
A
Cacm¶ayBIp©it O eTAGgát; MN OB Cacm¶ayBIp©it O eTAGgát; NP eday OA OB 5.5 cm naM[ MN NP 9 cm .
S
eyIgman ST RT 20 cm r T ehIy TQ 10 cm Q 20 cm tagkaMrgVg; OS OQ r R kñúgRtIekaNEkg TSO man OT CaGIub:Uetnus tamRTwsþIbTBItaK½r eyIg)an ³
B
-
R
C
-krNIrgVg;KµancMNucrYm eyIg)an ³
R r 4 R r 1 2R 5
OO AB r r AB OO r r OO 5 3 OO 8
-krNIrgVg;b:HKñaxageRkA naM[ A RtYtelI B enaH OO r r nig AB 0 -krNIrgVg;kat;Kña)anBIrcMNuc eyIg)an AB 2r b¤ AB 6 cm
A
O
r
O
r
O
-krNIrgVg;b:HKñaxagkñúg eyIg)an AB 2r b¤ AB 6 cm
A B
B
R 2.5 cm
r O
cMeBaH R 2.5 enaH r 4 R 4 2.5 1.5 cm dUcenH KNna)an kaMrgVg;TaMgBIrKW
r O
2.5 cm , 1.5 cm
.
B A r r O
B
Or Or
-krNIrgVg;KµancMNucrYmenAkñúgKña eyIg)an AB r r b£ AB 2 cm .
A
r B A O O r
-TMB½rTI 155 ³ KNnakaMénrgVg;TaMgBIr tag R CakaMrgVg;FM nig r CakaMrgVg;tUc -cMeBaHcm¶ayrvagp©itTaMgBIresµI 4 cm O
O 4 cm
krNIenH eyIg)an R r 4 -cMeBaHcm¶ayrvagp©itTaMgBIresµI 1 cm
1
1 cm
O O
krNIenH eyIg)an R r 1 2 tam 1 nig 2 eyIg)anRbB½n§smIkar
59
-
(
)
១៤
-TMB½rTI 164 ³ KNna AIB
-TMB½rTI 160 ³ bgðajfa AOB COB eyIgman rgVg;p©it o nig B CacMNuckNþalFñÚ AC mMup©it AOB maFñÚsáat; AB mMup©it COB maFñÚsáat; BC eday B CacMNuckNþalFñÚ AC naM[ FñÚ AB FñÚ BC eyIg)an mMup©it AOB COB Edr ¬mMup©itEdlmanFñÚsáat;esµIKña vaCamMub:unKña¦
B
eyIgmanmMu ACB 34 nig CAD 45 . kñúgRtIekaN ADI man ³ CAD ADB AIB¬plbUkmMukñúg@esµImMueRkA!¦ Et ACB ADB 34 ¬manFñÚsáat;rYm AB ¦ naM[ CAD ACB AIB o
C
B
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A
I
o
O
D
A
o
45o 34o AIB 79o AIB
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-TMB½rTI 167 ³ KNna ACB , ADB , AEB eyIgman AOB 60 naM[
P
o
B
AEB
.
D
ACB ADB o
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C
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C
60 o
A
B
¬mMucarwkkñúgrgVg; nigmMup©itmanFñÚsáat;rYm AB ¦
eyIgman ABC CamMucarwk kñúgrgVg;p©it o . A o tamlkçN³én]TahrN_1 C eyIgbgðaj)anfa ³ D 1 1 mMu ABD 2 AD nig mMu CBD 2 CD edaydkGgÁ nigGgÁeyIg)an B
dUcenH
.
ACB ADB AEB 30o
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D
x, y,z
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z
o
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, y z 45 o -
C
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eyIg)an CMD 12 AB CD ¬mMukñúgrgVg;¦
92o
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112o
o
o
o
b
a
o
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b 92 o 180 o b 88 o
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o
P
o
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1 260o 100o 2 80o
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76o
27 o
ACE 27 o , ADE 76 o
k> KNnargVas;mMu
1 150o 120o 135o 2
E
D
260 S
o
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R
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o
o
naM[ ABC ADE 76 kúñgRtIekaN ABC man AB BC CaRtIekaNsm)at enaHmMu)at ACB 180 2ABC
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FET 40o
o
k> KNnamMu EOF mMu EOF CamMup©itmanFñÚsáat; EF naM[ EOF EF 180 76 52 2 edaymMu FET CamMucarwkBiessmanFñÚsáat; EF eRBaH plbUkmMukñúgénRtIekaNesµI 180 . enaH EF 2 FET 2 40 80 dUcenH KNna)anrgVas;mMu ACB 52 . eyIg)an EOF EF 80 x> KNnargVas;mMu BAD dUcenH rgVas;mMuKNna)anKW EOF 80 . edaymMu BCD ACE ACB 27 52 79 x> KNnargVas;mMu DES ehIy BCD BAD 180 ¬mMuQmctu>carwkkñúgrgVg;¦ kñúgRtIekaN EDS man DE DF naM[ BAD 180 BCD 180 79 101 naM[ RtIekaN EDS CaRtIekaNsm)at Edlman mMu EDF FET 40 ¬mMucarwkmanFñÚsáat;rYm EF ¦ dUcenH KNna)anrgVas;mMu BAD 101 . nigmMu)at DEF 12 180 40 70 -TMB½rTI 174³ KNnargVas;mMu CMD A eyIgman CAD 60 nigmMu B eday DE DF enaH DE DF ¬Ggát;FñÚb:unKñaFñÚesµIKña¦ 60 M S
E
T
o
o
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o
o
o
o
o
o
o
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o
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ADB 75o
mMu CAD manFñÚsáat; CD naM[ CD 2 CAD 2 60 mMu ADBmanFñÚsáat; AB naM[ AB 2 ADB 2 75
75o
o
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D
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DF 2 DES DEF 70 o DE DES 2 DEF
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150o
o
61
-
2
2
១៥
-TMB½rTI 182 ³ sg;cMNuc P
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A
sg;Ggát; AB rYcsg; knøHbnÞat; Ax . RkitknøHbnÞat; Ax [)an AC 5 ÉktaRbEvg. P¢ab;BI C eTA B ehIyKUsbnÞat;kat; cMNucÉktaepSg²[Rsbnwg CB . RbsBVénbnÞat;ÉktaTI2 nigGgát; AB Ca cMNuc P
kñúgRtIekaN ABC man DE AB eyIg)an pleFobsmamaRt AD DC AC BC b¤ DC EC BE EC eyIgbMeBjtaragedayeRbIsmamaRtxagelI eyIg)an ³ C
P
D
A
C
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x
dUcenH eyIg)an
k> x>
.
AP 2 AB 5
K>
-TMB½rTI 184 ³ KNnatémøén x nig y eyIgman , , , CabnÞat;RsbKña enaHeyIg)anGgát;smamaRtKñaKW ³ 1
2
3
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y
X> g>
4
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x
y
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6
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3
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5
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18
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45 2
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2
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.
-TMB½rTI 186 ³ KNnakm
A
AF CF BE CE BE CF AF CE
B 1m
b¤ AF CF CE Edl CF CaRbEvgRsemaledImeQI nig CE CaRbEvg BIbegÁaleTAcugRsemal . F
C
E
62
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-TMB½rTI 193 ³ KNna x , y , z
-TMB½rTI 197 ³ KNnaRbEvgénTTwgTenø AB
eyIgman ABC dUcnwgRtIekaN ABC manpleFob dMNUcesµInwg 14 eyIg)an smamaRtpleFob
edaybRmab;RbFan CBA
A
4
3
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C
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AB AC BC 1 AB AC BC 4
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naM[
x
AB 1 3 1 x 12 B AB 4 x 4 AC 1 4 1 y 16 AC 4 y 4
b¤
y
CDE AB CB B 24 m C ED CD AB 24m 40m 32m 24m 40m AB 30m 32m
32 m
D 40 m
E
enaH
C
z
A
dUcenH RbEvgTTwgTenøKNna)anKW AB 30m .
BC 1 7 1 z 28 BC 4 z 4
-TMB½rTI 200 ³ KNnatémø x nig y -tamrUb AEB ADC tamlkçxNÐ m>m BE AE eday AEB ADC CD AD
dUcenH KNna)an x 12 , y 16 , z 28 .
A
-TMB½rTI 195 ³ k> bgðajfa ABC dUc PMN RtIekaN ABC man A 80 , B 54 naM[mMu C 180 A B o
6 4 y 10 6 10 y 15 cm 4
o
o
180 o 80 o 54 o 46 o
dUcenH KNna)an
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ABC
NPM
JLK
tamlkçxNÐ m>m .
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2m J
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AB AC BC NP NM PM
dUcenH eyIg)anpleFobdMNUc AB AC BC NP NM PM
.
63
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-tamrUb RQV RTS tamlkçxNÐ m>m QV QR eday RQV sUmEk ST BI RTS TS TR
20m 20cm
x 16 20 10 16 20 x 32 m 10
dUcenH KNna)an x 32 m . x
Q
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V
R S
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20cm
BAC
DAE DAE BAC
tamlkçxNÐ m>m
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x 220 40 100 220 40 x 88 m 100
dUcenH KNna)an x 88 m 120m
D
.
100m
B
A
40m x
C
E
-TMB½rTI 207 ³ KNna z nigsg;rUbeLIgvij cMeBaH z x y naM[ z x y b¤ x y z 1 tamTMnak;TMng 1 bBa¢ak;fa RtIekaNEdlmanrgVas; RCug x , y , z CaRtIekaNEkg ehIyman x CaRbEvg GIub:Uetnus nig y , z CaRCugCab;mMuEkg . 2
2
2
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2
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dUcenH RbEvgRCug z x , y x . sg;rUb ³
x
y
z x2 y 2
z
64
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១៧
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348o 12 29o
dUcenH eyIgKNna)anmMu 29 . o
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n 17
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o
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n 2 19 n 21
dUcenH BhuekaNenHmancMnYnRCug n 21 .
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2
2 3
2
12o
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65
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១៨
-TMB½rTI 226 ³ KNnaépÞRkLaxag nigépÞRkLaTaMgGs;
épÞRkLaTaMgGs;
ST S L S B
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30 cm 2 6 3 cm 2
1 pa 2 1 5 4 13 2 130 cm 2 S B 5 5 25 cm 2
SL
épÞRkLa)at épÞRkLaTaMgGs;
-TMB½rTI 228 ³ KNnamaDekaNKitCa dm
13 cm
ekaNenHman km
5 cm
2
ST S L S B
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3
5 m 5 cm
R 13 cm
3
3
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3
3
157 cm 2
10 cm
épÞRkLa)at S B R 2 3.14 5 2
5 cm
12 mm
78 .5 cm 2 ST S L S B
157 cm 2 78.5 cm 2 235.5 cm 2
P
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3 mm o
o
3m
dUcenH pleFobbrimaRtKW
P 4 P
.
pleFobépÞRkLa ebI S CaépÞRkLarUbFM S CaépÞRkLarUbtUc
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épÞRkLa)at
h 28 cm
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S L Ra
épÞRkLaTaMgGs;
3
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155 cm 2
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40.39 cm 2
5 cm
2 cm
eyIg)an
1 S B 6 2 2 2 11 2
2
S 12 mm 4 2 16 S 3 mm
dUcenH pleFobépÞRkLaKW
6 3 cm 2 2 cm
66
S 16 S
. -
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x> rképÞRkLaénctuekaN DECB épÞctuekaN DECB S S
pleFobbrimaRt ebI P CabrimaRtrUbFM P CabrimaRtrUbtUc eyIg)an PP 32
ABC
ADE
S S 50 50 18 6 3 3 32 cm 2 3
dUcenH pleFobbrimaRtKW
.
P 3 P 2
dUcenH épÞRkLaénctuekaN DECB KW 323 cm . 2
pleFobépÞRkLa ebI S CaépÞRkLarUbFM S CaépÞRkLarUbtUc eyIg)an SS 32 94
(Y)
2
dUcenH pleFobépÞRkLaKW
.
S 9 S 4
smÁal; ³ eKGaceRbIpleFobrUbtUcelIrUbFMk¾)an .
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A
AD 3 cm , AB 5cm D
nigépÞRkLaRtIekaN ADE 6cm k> rképÞRkLaRtIekaN ABC B tag S CaépÞRkLaénRtIekaN ABC S CaépÞRkLaénRtIekaN ADE naM[eyIg)anpleFobKW ³ S AB S 5 smmUl S AD 6 3 2
2
25 6 50 S cm 2 9 3
dUcenH épÞRkLaRtIekaN
ABC
E C
2
25 9
KW S 503 cm . 2
67
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sYsþI¡ elakGñkmitþGñkGanCaTIRsLaj;rab;Gan enAkñúgEpñkenHelakGñknwg)aneXIj dMeNaHRsay lMhat;tamemeronnImYy² EdlmanenAkñúgesovePAKNitviTüafñak;TI 9 rbs;RksYgGb;rMyuvCn nigkILa e)aHBum< elIkTI 1 qñaM 2011 . enAkñúgEpñkenHmanbgðajTaMgcemøIy nigTaMgRbFanlMhat; eTAtamemeronnImYy²kñgú esovePArbs;RksYg. eTaHbICa´ )aneXIjesovePAkMENKNitviTüa Edldak;lk; enAelITIpSarCaeRcInehIy k¾eday k¾´enAEtbnþbegáItEpñkenHeLIg edIm,IbMeBjlkçN³rbs;esovePAenH [kan; Etsm,ÚrEbb saksmCa esovePAKNitviTüasRmab;RKU nigsisSfñak;TI 9 . RbsinebIelakGñkmanbBaða b¤cm¶l;Rtg;cMNucNa EdlmanenAkñúgEpñkenH elakGñkGacTak;Tg;eTAkan; RKU b¤mitþPkþirbs;GñkEdlmansmtßPaB b¤GñkeroberogesovePAenHkñúgeBlevlasmKYr . …
iii
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k> 9 x> Q> 8 j> 3
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16 3
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3
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625
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X>
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3
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g>
512 343
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216 1000
3.
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x4
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q>
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18
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54
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3
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X>
8x3
C>
64m3 ។
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x 3
y6
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3
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3
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K> q>
4 a
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3
3
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6.
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m
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3
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។
p
X> 2
23 y3 68
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k> K> g>
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3 2 4 2 5 2 3 2 3 15 4 3 3 15 6 3 23 2 83 3 3 2 33 3
5 2 3 3 6 2 5 3 4 3 2 17 3 17 3 3 2 3 83 2 33 3 53 2 23 3 ។
10.
k> 23 X> 5
27 3
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3 48 4
128 3 3 250
3
K> 53 c> 4
3
75 3
2 27 3
54 3 3 128 ។
11.
k> 2 8 3 98 2 200 K> 3 175 2 28 3 63 112 g> 2 16 3 54 2 128 q> 4 54 6 81 4 16 3 24 3
3
3
3
3
3
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k> K>
x> 3 50 32 5 200 X> 108 2 27 40 5 160 c> 3 81 12 128 3 192 4 54 C> 2 40 3 135 5 320 8 5 ។ 3
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3
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2 27 4 50 4 18 3 48 3 5 3
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a, b, x, y
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3
3
3
3
3
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135 4 3 320 3 40
3
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23 135 33 40 ។ 3 2
។
z
A 3 32 x 6 8 x
B 2 125x2 z 8x 80z
C 7a b3 b 4a2b 4b
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E 3xy x 2 y 2 x4 y3
F 3a a3b5 2b a5b3 5 a3b3
G 8a 3 54a 6 3 16a4
H 3 3 x4 y 6 x 3 xy 4 2 3 x4 y 4 ។
14.
a5,b3
A
A ab ab3 9a3b3 a3b ។ a 3 , b 2។
A 4a a a2b b2a b 9b
15. 16.
k> 2 3 3 2 X> 6 2 12 3 q> 2 3 12 3
3
2
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3
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3
17.
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k> 3 5 2 18 3 48 X> 32 2 2 18 3 48 q> 3 3 3 8 2 18 3
3
3
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3
3
3
3
3
3
3
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18.
x> 3 5 2 10 50 2 80 X> 125 75 80 48 c> 80 2 27 3 20 3 12
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3
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a 3 5 2 10 , b 5 7 2 10 , c 3 18 3 27
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k>
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3
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21.
b2 b2 14b 49
A
49 x 2 56 x 16 36 x 2
B
25b2 10ab a 2 ។ 16b2 24ab 9a 2
D
22.
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x> 3 333
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3
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3
23.
3 5 3 20 3
3
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3
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C>
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។
24.
k> c>
3 x
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n 160
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។
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10.
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។
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11.
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A
B
5
។
។ ។
។
72 ។
12. -
80 000
-
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។
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BC
។
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11 31 5 x 2 x 20 7 x 20 x 7
Kµantémø x NaEdlepÞógpÞat; 0x 5 enaHeLIy dUcenH vismIkarKµancemøIy .
cemøIyCaRkab
20 7
)
C1>
dUcenH vismIkarmancemøIy x 207 . c1>
4 x 9 8x 3
cemøIyCaRkab
9 3 8x 4 x
1
[
12 12 x x 1
c2>
2x 2x 1 1 4x 2 1 x 2
q1>
2 x 9 5 x 81 9 81 5 x 2 x 90 3x x 30
[
x 1 1 3 2x 5 2 2 2 x 1 5 53 2 x 10 10 10 2 x 1 5 53 2 x
C2>
1 2
]
2 x 2 5 15 10 x 2 x 3 15 10 x
dUcenH vismIkarmancemøIy x 12 . 2 5 x 3 x 27 3 3
18
dUcenH vismIkarmancemøIy x 187 .
cemøIyCaRkab
1 2 x 1 3x 4 3 12 x 31 2 x 41 3 x 12 12 12 12 x 31 2 x 41 3x 12 x 3 6 x 4 12 x 6 x 12 x 4 3 7 x 18 7 x
cemøIyCaRkab
dUcenH vismIkarmancemøIy x 1 . 1 2 x 1 2 x 2 2x 1 2x 1
0
2 x 2 x 3 2 0 x 5
dUcenH vismIkarmancemøIy x 6 . g2> 2x 11 5x 31
cemøIyCaRkab
2 x 10 x 15 3 8 x 18
cemøIyCaRkab
x
30
)
dUcenH vismIkarmancemøIy x 30 . 109
9 4
9 4
cemøIyCaRkab
[
dUcenH vismIkarmancemøIy
x
9 4
.
2.
edaHRsayRbB½n§vismIkar ³ k1> 22xx 33 95 b¤ 22xx 12 b¤ 2 cemøIyCaRkab -cMeBaH x 6 ³ [ -cMeBaH x 1 ³ -cemøIyénRbB½n§ ³ [
x2> 2x x4335xx21 b¤
x 6 x 1
b¤ b¤
6
]
1 1
cemøIyCaRkab -cMeBaH x 23 ³ -cMeBaH x 3 ³ -cemøIyénRbB½n§ ³
6
]
dUcenH RbB½n§vismIkarmancemøIy 1 x 6 . k2> 33xx 44 87 b¤ cemøIyCaRkab -cMeBaH x 4 ³ -cMeBaH x 1 ³ -cemøIyénRbB½n§ ³
3x 12 3x 3
b¤
x 4 x 1 4
1
K1>
[ 4
[
]
3x 2 4 x 3 4 3 2 2 x 1 3 x 4 4
2 x 5 5 x 4 x 7 2x 3
b¤ b¤ b£
cemøIyCaRkab -cMeBaH x 3 ³ -cMeBaH x 4 ³ -cemøIyénRbB½n§ ³
2 3
3
)
b¤
b¤ b¤ cemøIyCaRkab -cMeBaH x 23 ³ -cMeBaH x 0 ³ -cemøIyénRbB½n§ ³
3
) 4
( (
3
]
b¤
5 4 5 x 2 x 7 3 2 x x 9 3x 4 x x 3 x 4
4
)
b¤
dUcenH RbB½n§vismIkarmancemøIy 1 x 4 . x1>
2 3
dUcenH RbB½n§vismIkarmancemøIy x 23 .
] 1
3 1 5 x 2 x 4 2 3x x 2 3x 6 2 x 2 x 3 x 3
3
)
dUcenH RbB½n§vismIkarmancemøIy 4 x 3 .
9 x 8 64 x 3 12 12 12 42 x 1 3 x 4 4 4 9 x 8 24x 18 8x 4 3x 4 8 18 24x 9 x 8x 3x 4 4 10 15x 5x 0
2 x 3 x 0
2 3
[ 0
( 0
2 3
[
dUcenH RbB½n§vismIkarmancemøIy x 23 . 110
K2>
2 x 1 0 x 2 2 0
b¤
2 x 1 x 4 0
cemøIyCaRkab -cMeBaH x 12 ³ -cMeBaH x 4 ³ -cemøIyénRbB½n§ ³
b¤
1 x 2 x 4
4.
1 2
) 4
(
1 2
4
dUcenH RbB½n§vismIkarKµancemøIy . 3.
rképÞRkLaEdlGacmanFMbMput ³ tag x CaRbEvgbeNþay nig y CaRbEvgTTwg éncmáar Edl x nig y KitCa m ehIy x 0, y 0 eKdaMkUneQI 40 edImB½T§CMuvji cmáar EdlkUneQI nImYy²XøatBIKña 2 m naM[ RbEvgbrimaRt P 2 40 80 m eyIg)an x y P2 802 b¤ x y 40 ¬elIkGgÁTaMgBIrCakaer¦ x 2 2 xy y 2 1600
cMeBaH x 0, y 0 enaH
5.
1
x y 2 0
x 2 2 xy y 2 0
2
2 xy x 2 y 2
-cMeBaH
b ac
b¤
-cMeBaH
c ab
b¤
2a a b c P 1 a 2
bb a c b 2b a b c P 2 b 2 cc abc 2c a b c P 3 c 2
tam 1 , 2 nig 3 eXIjfaRbEvgRCugnImYy² énRtIekaN tUcCagknøHbrimaRt )anbgðajrYc dUcenH kñúgRtIekaNmYy RbEvgRCugnImyY ² RtUvtUcCagknøHbrimaRtrbs;va .
x y 2 402 x 2 y 2 1600 2 xy
bgðajfaRCugRtIekaNnImYy²xøICagknøHbrimaRt tag a , b nig c CaRbEvgRCugénRtIekaN naM[ brimaRt P a b c tamvismPaBénRCugrbs;RtIekaN eyIg)an ³ -cMeBaH a b c b¤ a a b c a
rkcMnYnenaH ³ tag x CacMnYnEdlRtUvrkenaH ³ tambRmab;RbFan 2x 5 3 x 3 5 2 x 8 2
eyIgyk 1 CMnYskñúg 2 eyIg)an ³ 2 xy 1600 2 xy 2 xy 2 xy 1600 4 xy 1600 xy 400
x 2 8 2 2 x 16
eXIjfa épÞRkLacmáarKW xy Edl xy 400
dUcenH cMnYnenaH CacMnYnEdlFMCag b¤esµI 16 .
dUcenH cmáarmanépÞRkLaFMbMputKW 400 m . 2
111
6.
rkcenøaHqñaMeTAmuxeTot ³ tag t CacenøaHqñaMeTAmuxeTot Edl t 0 smµtikmµ ³ «BukmanGayu 32 qñaM kUnmanGayu 12 qñaM tambRmab;RbFan eyIg)anRbB½n§vismIkar ³ 32 t 312 t 32 t 36 3t b¤ 32 t 24 2t 32 t 212 t
8.
32 36 3t t 32 24 2t t
84 93 78 87 89 70 81 x 90 8 582 x 90 8 582 x 720 x 720 582 x 138
4 2t t 2 8 t t 8
eday t 0 enaHeyIg)an 0 t 8 dUcenH cab;BIbc©úb,nñrhUtdl;ticCag 8 qñaMeTot eFVI[Gayu«Buk ticCagbIdg b:uEnþeRcIn CagBIrdgénGayurbs;kUn . 7.
k> etIvisalTTYl)annieTÞs A Edr b¤eT ? tag x CaBinÞúénmuxviC¢acugeRkaymYyeTotEdl visalnwgTTYl)an ehIy 0 x 100 . visal)anRbLg 7 muxviC¢aehIyTTYl)anBinÞú 84 , 93 , 78 , 87 , 89 , 70 nig 81 ]bmafa visalRbLg)annieTÞs A ³ eyIg)an
EttémøénBinÞú 0 x 100 dUcenH visalminGacTTYl)annieTÞs A eT . x> etIvisalTTYl)annieTÞs B Edr b¤eT ? ]bmafa visalRbLg)annieTÞs B ³ eyIg)an
rkcMnYnRkdasR)ak; 500 ` EdlsuxGacman ³ tag x CacMnYnRkdasR)ak; 500 ` EdlsuxhUt)an naM[ suxhUt)anRkdas 1000 ` cMnYn 25 x Edl x KitCasnøwk suxcg;hUt[)anR)ak; 17 500 ` y:agtic nigR)ak; 20 000 ` y:ageRcIn enaHeyIg)anTMnak;TMng ³
84 93 78 87 89 70 81 x 80 8 582 x 80 8 582 x 640 x 640 582 x 58
17500 500x 100025 x 20000 17500 500x 25000 1000x 20000 17500 25000 500 x 20000 17500 25000 500 x 20000 25000 7500 500 x 5000 7500 500x 5000 500 500 500 15 x 10
lT§plenHbBa¢ak;fa visalGacTTYl)annieTÞs B eday 0 x 100 niglT§pl x 58 naM[ 58 x 100 dUcenH visalGacTTYl)annieTÞs B ehIy edIm,I[TTYl)annieT§s B visal caM)ac;RtUvyk[)anBinÞúmxu viC¢a cugeRkayticbMputKW 58 .
dUcenH RkdasR)ak; 500 ` EdlsuxGac hUt)an mancMnYneRcInCag 10 snøwk nigticCag 15 snøwk . 112
kMNt;témøFbM MputéncMnYnessTaMgBIrtKña ³ tag x CacMnYnessTI 1 naM[ cMnYnessbnÞab;KW x 2 eday plbUkéncMnYnessTaMgBIrFMCag 72 eyIg)an x x 2 72 2 x 3 x 15 94 2x 70 b¤ x 35 4 x 15 47 edaymancMnYnKt;ess x eRcInrab;minGs;EdlFM 4 x 32 x8 Cag cMnYn 35 enaHcMnYnKt;ess x kMNt;min)an eday x CacMnYnKt;mantémøGtibrma enaH x 8 dUcenH BIrcMnYnKt;esstKñaFMCageK nigman naM[ beNþay 3x 15 3 8 15 39 m plbUkFMCag 72 KWminGackMNt;)an . dUcenH vimaRtGtibrmaénsYnenaHKW TTwg 12. rkcm¶aypøÚv edIm,IkMu[QñÜlCYlelIsBIR)ak;kk; ³ manRbEvg 8m nigbeNþayman tag x Cacm¶aypøÚvEdlRtUvebIkbr ¬KitCa km ¦ manRbEvg 39m . edayR)ak;QñÜl 80 000 ` kñúgmYyéf¶ nig 400` kñúgcm¶aypøÚv 1km enaHeyIg)an ³ 10. kMNt;RbePTsMbuRtEdlGaclk;)aneRcInbMput ³ 80 000 400x 400 000 tag x CacMnYnRbePTsMbuRt A ¬KitCasnøwk¦ 400x 320 000 naM[ x 5 CacMnYnRbePTsMbuRt B eRBaH x 800 cMnYnRbePTsMbuRt A elIsRbePT B cMnYn 5snøkw dUcenH edIm,IkMu[QñÜlCYlelIsBIR)ak;kk; cMnYnsMbuRtlk;)ay:ageRcInKW 37 snøkw eyIg)an ³ kñúgmYyéf¶GñkCYlminRtUvebIbrelI 9.
rkvimaRtGtibrmaénsYnenaH ³ tag x CaRbEvgTTwgénsYnenaH ¬KitCa m¦ naM[ RbEvgbeNþayKW 3x 15 eRBaH beNþay manrgVas;elIs 3dgénTTwgRbEvg 15m edaysYnmanbrimaRtEvgbMputKW 94 m eyIg)an ³
x x 5 37 2 x 42 x 21
naM[ cMnYnsMbuRtRbePT A lk;)aneRcInbMputKW x 21 snøwk nigRbePT B lk;)aneRcIn bMputKW x 5 21 5 16 snøwk epÞógpÞat; ³ 2116 37 b¤ 37 37 Bit dUcenH cMnYnsMbuRtlk;)aneRcInbMputKW RbePT A cMnYn 21snøkw nig RbePT B cMnYn 16 snøwk
11.
cm¶aypøÚvelIsBI 800 km eLIy . 13. rkRbEvgRCug BC RtIekaN ABC man AB 12cm , AC 3cm tamvismPaB cMeBaHRtIekaN ABC eyIg)an ³ AB AC BC AB AC 12 3 BC 12 3 9 BC 15
C ?
3
A
9
eday BC CacMnYnKt; nigCaBhuKuNén 3 enaHnaM[ BC 12 cm dUcenH rk)anRbEvgRCug BC 12 cm .
. 113
B
rkcMnYnKt;enaH ³ tag x CacMnYnKt;enaH tambRmab;RbFan 2x 8 b¤ x 4 eXIjfa x CacMnYnKt;FMCagb¤esµI 4 naM[ x 4 , 5 , 6 , 7 , 8 , ... dUcenH cMnnY Kt;Edlrk)anKW
14.
x 4 , 5 , 6 , 7 , 8 , ...
.
rkRbEvgbeNþayEdlbUNa RtUveRCIserIs ³ tag x CaRbEvgbeNþay ¬KitCa cm ¦ TTwgmanRbEvg 3 cm ehIyRkLaépÞsßitenA cenøaHBI 15 cm eTA 24 cm tambRmab;RbFan eyIg)anTMnak;TMng ³
15.
2
2
15 3 x 24 15 3 x 24 3 3 3 5 x8
eday x CacMnYnKt; enaH
x6, x7
dUcenH RbEvgbeNþayEdlbUNaRtUv eRCIserIsKW 6 cm , 7 cm .
114
៦
។
1.
3 2 1 6 1 7 9 7 10 10 10 3 5 7 2 5 1 6 4 8 1 2 7 1 10 3 5 5 3 7 6 2 10 6 4 3 4 5 5 7 ។
10 5? ឃ
9? ។ ។
2. x f
0
1
2
3
4
5
6
7
8
144
195
130
80
58
45
24
6
3
។
5 7 ។
3. x f
1
2
3
4
5
6
78
1270
680
320
150
133 ។
x 3 ។ 4.
25 18 16 32
22 30 25 36
30 15 28 21
45 20 31 18
28 24 41 34
51 17 28 41
30 24 26 53
32 41 15 25
34 38 25 42
33 27 19 41
8
5
20-25 , …។ ។ ។ ឃ
25 ។ 115
15-20 ,
5.
45
0-5
5-10
10-15
15-20
20-25
25-30
30-35
35-40
40-45
។
4
5
9
12
15
14
10
8
6
45 15 ។
6.
2 ។
5 5
5-10
10-15
15-20
20-25
25-30
30-35
35-40
40-45
1
1
3
2
6
8
12
12
20
30
។ ។
7. 0-5
5-15
15-35
35-45
45-55
55-70
70-85
85-100
1
104
65
45
70
135
195
210
12
3010
3645
1700
1175
1800
2400
1170
។ ។ ។ ឃ
35
DDCEE
3 116
៦ 1.
eyIgmanTinñn½ykMhus GkçraviruT§énsisSmYyfñak; ³ 3 10 1 6
2 1 3 5 2 7 2 10
6 1 7 7 2 5 1 10 3 6 4 3
9 1 5 4
7 10 10 6 4 8 5 3 7 5 5 7
f % 100%
100% 87.5%
k> pþµúTinñn½yCataragbgðajBI eRbkg; eRbkg;ekIn nigeRbkg;fy eRbkg;eFob eRbkg;eFobekInnigfy kMhus 1 2 3 4 5 6 7 8 9 10
srub
f 5 4 5 3 6 4 6 1 1 5 40
f 5 9 14 17 23 27 33 34 35 40
f 40 35 31 26 23 17 13 7 6 5
g> sg;RkabssréneRbkg;eFobfyCaPaKry ³
f%
f %
f %
12.5 10 12.5 7.5 15 10 15 2.5 2.5 12.5
12.5 22.5 35.0 42.5 57.5 67.5 82.5 85.0 87.5 100
100 87.5 77.5 65.0 57.5 42.5 32.5 17.5 15.0 12.5
80%
77.5% 65%
60%
57% 42.5%
40%
32.5% 17.5%
20%
15% 12.5%
1
2.
2
3
4
5
6
7
8
9
10
kMhus
eyIgmantaragsßiti éncMnYnkUnkñúgRKYsar ³
cMnYnkUn x cMnYnRKYsar f
0
1
2
3
4
5
6
7
8
144
195
130
80
58
45
24
6
3
k> begáIttaragbgðajBIeRbkg; eRbkg;ekInnigfy
x> etIfñak;enH mansisSb:unµannak; ? cMnYnkUn x eRbkg; f eRbkg;ekIn f eRbkg;fy f 0 144 144 685 edayeRbkg;srubKW f 40 1 195 339 541 2 130 469 346 dUcenH fñak;eronenHmancMnYnsisS 40 nak; . 3 80 549 216 K> -sisSb:unµannak;mankMhsu 10 ? 4 58 607 136 5 45 652 78 edaysisSmankMhus 10 mancMnYneRbkg; f 5 6 24 676 33 7 6 682 9 dUcenH cMnYnsisSmankMhsu 10 man 5 nak; . 8 3 685 3 685 -ehIytamtarageRbkg;ekIn nigeRkg;ekIneFob srub x> -edayeRbkg;srub f 685 mansisS 23 nak;RtUvCa 57.5% EdlmankMhus naM[ kñúgtMbn;enHman 685 RKYsar . y:ageRcIncMnnY 5 . -tamtarageRbkg;ekIn man 652 RKYsar Edlman X> tamtarageRbkg;fy nigeRbkg;eFobfy kUny:ageRcIn 5 nak; nigtamtarageRbkg;fy mansisS 6 nak; RtUvCa 15% EdlmankMhus man 9 RKYsarEdlmankUny:agtic 7 nak; . y:agticcMnYn 9 .
117
K> sg;RkabssréneRbkg;ekIn ³
-tagTinñn½yxagelICaRkabssréneRbkg;eFob³ f%
f
700
652
50%
685
682
676
48.27%
607
600
40%
549
500
469
30%
400
25.85% 339
300 200
20% 12.16%
144
10%
100
0
3.
5.70%
5.06%
5
6
2.94%
1
3
2
4
5
6
8
7
taraglT§pléncMnYnkUnsøab;kñúgsRgÁam ³ cMnYnkUnsøab; x cMnYnRKYsar f
1
2
3
4
5
6
78
1270
680
320
150
133
eyIgsg;taragbMENgEckeRbkg; eRbkg;eFob ³ cMnYnkUnsøab; x cMnYnRKYsar f eRbkg;eFob 1 2 3 4 5 6
78 1270 680 320 150 133 2631
srub
1
cMnYnkUn
f%
2.94 48.27 25.85 12.16 5.70 5.06
25 18 16 32
f
1200 1100 1000 900 800 680
600 500 320
200 150
133
78
2
30 15 28 21
45 20 31 18
28 24 41 34
51 17 28 41
30 24 26 53
32 41 15 25
34 38 25 42
33 27 19 41
3
4
eRbkg; eRbkg;ekIn eRbkg;eFob eRbkg;eFobekIn
15-20 20-25 25-30 30-35 35-40 40-45 45-50 50-55
7 5 9 9 2 5 1 2 40
5
6
7 12 21 30 32 37 38 40
17.5 12.5 22.5 22.5 5 12.5 2.5 5
17.5 30 52.5 75 80 92.5 95 100
x> sg;GIusþRkam ³ ¬rMlwk ³ GIusþÚRkamCaRkabssrenACab;²Kña¦ tamtaragbMENgEckeRbkg; eyIgsg;GIusþÚRkam³
300
1
22 30 25 36
Gayu
srub
400
100
cMnYnkUnsab;
k> sresrTinnñ ½yCafñak; mancenøaHfñak;esµI 5 ³
1270
700
4
x> RsaybBa¢ak;elIRkabTaMgBIr Rtg;ssrEdl manGab;sIus x 3 ³ -Rtg;cMNucénRkabssrEdlman x 3 mann½y fa man 680 RKYsarEdlmankUnsøab; 3 nak;enA kñúgsRgÁam EdlRtUvCa 25.85% éncMnYnRKYsar TaMgGs;EdlmankUnsøab;kñúgsRgÁam . 4. eyIgmanTinñn½yGayukmµkrenAkñúgshRKasmYy ³
k> -tagTinñn½yxagelICaRkabssréneRbkg; ³ 1300
3
2
cMnYnkUnsab; 118
tamtaraglT§pl eyIgsg;)antaragbMENkEck eRbkg;dUcxageRkam ³
f 10 9
9
cMnYn)al;
eRbkg;ekIn
eRbkg;fy
eRbkg;eFob
eRbkg;eFobekIn
eRbkg;eFobfy
8
0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40 40-45
4 5 9 12 15 14 10 8 6
4 9 18 30 45 59 69 77 83
83 79 74 65 53 38 24 14 6
4.82 6.02 10.84 14.46 18.07 16.87 12.05 9.64 7.23
4.82 10.84 21.68 36.14 54.21 71.08 83.13 92.77 100
100 95.18 89.16 78.31 63.85 45.78 28.92 16.88 7.23
srub
83
6
5
ry³eBl
7 5
4 2
2
2 1
Gayu
15 20 25 30 35 40 45 50 55
K> -eyIgbMeBjtarageRbkg;ekIn nigeRbkg;eFob ekIn dUckñúgtaragbMENgEckeRbkg;énsMNYr k> -KUsRkabrbs;va ³ f%
k> edayeRbkg;srubKW f 83 dUcenH kñúgry³eBl 45 naTI RkumkILakr)an e)aH)al;cUlTI cMnYn 83 RKab; . x> tamtarageRbkg;ekIn bgðajfakñúgry³eBl 15 naTIdMbUg RkumkILakr)ane)aH cUlTIcMnYn 18 RKab; ehIyenAenAknøHem:ag cugeRkayRkum kILakr)ane)aHcUlTI)ancMnYn 83-18 = 65 RKab; EdlRtUvnwg 78.31% . K> KUsRkabénBhuekaNeRbkg;ekIn ³
f
100%
40
75%
30
50%
20
40 37 38 100% 82.5% 95% 30 32 75% 80%
21 52.5%
12 7 30% 17.5%
10
25%
15 20 25 30 35 40 45 50 55 Gayu
X> bkRsayelIRkabRtg;cMNucEdlman x 25 tamRkabtageRbkg;ekIn nigeRbkg;eFobekInKW Rtg;cMNucEdlmanGab;sIusesµI 25 mann½yfa mankmµkry:ageRcIncMnYn 12 nak; EdlmanGayu 25 qñaM EdlRtUvnwg 30% éncMnYnkmµkrsrub . 5.
f
100
83 80
77 69 59
60
eyIgmantaraglT§ple)aH)al;bBa©ÚlTIeTAtam ry³eBlnImYy² kñúgefrevla 45 naTI ³
45 40
30 18
20
10-15
15-20
20-25
25-30
30-35
35-40
40-45
4
5-10
cMnYn)al;
0-5
ry³eBl
9
5
9
12
15
14
10
8
6
4
5 10 15
119
20 25 30 35 40 45
naTI
cMnYndg
eRbkg;ekIn
eRbkg;fy
ry³eBlpSay
5 1 1 3 2 6 8 12 12
5 6 7 10 12 18 26 38 50
50 45 44 43 40 38 32 24 12
5-10 10-15 15-20 20-25 25-30 30-35 35-40 40-45
srub
70-85
85-100
104
65
45
70
135
195
210
12
3010
3645
1700
1175
1800
2400
1170
Gayu
Rb>C søab; km Rb>C rbYs km
0-5 5-15 15-35 35-45 45-55 55-70 70-85 85-100
1 104 65 45 70 135 195 210
srub
825
0.2 10.4 3.25 4.5 7 9 13 14
12 3010 3645 1700 1175 1800 2400 1170
2.4 301 182.25 170 117.5 120 160 78
h
15
14
13
12 10.4
f
7
6
4.5 3.25
3 0.2
0 5
85
100
Gayu
100
Gayu
301
250
200
182.25
38
170
160
150
32
30
117.5
26
f
24
20
70
h
43 38
35 45 55
-sg;GIusþÚRkaménTin½yRbCaCnsøab; ³
50
40
15
50
40
9
9
300
f
44
f A
14912
50
45
h
-sg;GIusþÚRkaménTin½yRbCaCnsøab; ³
k> tamtaragbMENgEckeRbkg; eXIjfa eRbkg; srubKW f 50 dUcenH kñúgeBll¶acenH eKpSay BaNiC¢kmµ)an 50 dg eRbIeBlGs; 45 naTI . x> -tamtarageRbkg;fy KWry³eBl 20 naTIcug eRkay eKpSayBaNiC¢kmµ)an 38 dg . -tamtarageRbkg;ekIn ry³eBlticCag 30 naTIdMbUg eKpSayBaNiC¢kmµ)an 18 dg . K> KUsRkabénBhuekaNekIn nigfy ³ 50
1
k> eyIgrkkmCsøab; nigrbYs
eyIgsg;)antaragbMENgEckeRbkg;dUcxageRkam³
eRkam 5
55-70
Gayu
12
45-55
12
40-45
35-40
8
35-45
6
Rb>C søab; Rb>C rbYs
15-35
2
30-35
25-30
20-25
3
taraglT§plRbCaCnsøab; nigrgrbYstamGayu ³ 5-15
1
7.
0-5
1
15-20
eRkam 5 5
5-10
ry³eBlpSay ¬naTI¦ cMnYndg
10-15
eyIgmantaragbgðajBIcMnYndgénkarpSayBaNiC¢kmµ
6.
120
100
78
18
50 10
10 5
6
12
12
2.4
7
5 10 15
20 25
30
35 40 45
0 5
naTI 120
15
35 45 55
70
85
x> begáIttarageRbkg; eRbkg;ekIn ³ Gayu
Rb>C søab; eRbkg;ekIn Rb>C rbYs eRbkg;ekIn
0-5 5-15 15-35 35-45 45-55 55-70 70-85 85-100
1 104 65 45 70 135 195 210
srub
825
1 105 170 215 285 420 615 825
12 3010 3645 1700 1175 1800 2400 1170
12 3022 6667 8367 9542 11342 13742 14912
14912
K> rkcMnYnRbCaCnsøab; nigrbYs ³ tamtarageRbkg; RbCaCnsøab; = 825 nak; / RbCaCnrgrbYs = 14912 nak; naM[ RbCaCnsøab; nigrgrbYs mancMnYn 825 + 14912 = 15737 nak; . X> tamtarageRbkg;ekIn RbCaCnEdlmanGayu ticCag 35 qñaM søab;cMnYn 170 nak; nigmanrbYs cMnYn 6667 nak; .
aaa
121
៧ 1. 22 22 20 22 27 20 23
1 2 3 4 5 6
22 23 19 23 27 19 24
21 22 21 23 26 22 24
21 23 20 23 20 24 ។ ។
kg
2.
4 6 2 10 8 5 4 4 8 10 5 4 3 7 2 4 6 8 7 4 5 3 6 4 ។
3. x 16 18 19 f 1 4 9
20 3
21 30 2 1 ។
4.
0
2
4
6
8
10
11
12
34
36
0
2
4
6
8
10
11
12
34
36
5
10
15
20
25
30
35
5.
0 3 1 0
0 2 3 2
0 2 3 4
1 4 4 1
3 1 1 3
3 6 1 4
1 5 0 1
1 6 3 5
1 0 1 6
3 0 0 4 3 3 1 1 0 5 6 2 2 1 1 2 3 5 1 3 2 1 1 1 1 2 1 1 3 3 6 0 0 3 4 1 3 2 3 3 ។
។
122
។
6.
o
C
14o
23o
14o
k>
32o
35o
30o
30o
29o
25o
18o
22o
។ ។
7.
1
89 ។
8
2
90 ។ 8. ។
mm)
0-5 5-10 10-15 15-20 20-25 25-30 30-35 3
15
72
15
91
35
8
។
20 mm
35 mm ។
។
។ ។
45-50
50-55
55-60
60-65
65-70
70-75
75-80
80-85
85-90
g
40-45
9.
8
11
31
61
54
58
43
25
17
7
។ ។ ឃ
។
10.
។
280
ឃ
។
4
។ ។
150-155
155-160
160-165
165-170
170-175
175-180
cm
145-150
11.
31
95
131
272
120
77
48
។ ។ ។
DDCEE
3 123
15o
៧ 1.
eyIgmantaragsikSacMnYnsisSdUcxageRkam ³ fñak; cMnYnsisS fñak;metþyü fñak;TI 1 fñak;TI 2 fñak;TI 3 fñak;TI 4 fñak;TI 5 fñak;TI 6
-cMeBaH fñak;TI 2 ³ eyIgmanTinñn½y 19 20 20 21 ÷mFümKW x 19 20 4 20 21 20 ÷tYén Me KW 4 2 1 2.5 enaHnaM[ Me 20 2 20 20 ÷témøm:Ut KW Mo 20 ¬ maneRbkg;elIseK¦ -cMeBaH fñak;TI 3 ³ eyIgmanTinñn½y 22 23 23 23 ÷mFümKW x 22 23 4 23 23 22.75 ÷tYén Me KW 4 2 1 2.5 enaHnaM[ Me 23 2 23 23 ÷témøm:Ut KW Mo 23 ¬ maneRbkg;elIseK¦ -cMeBaH fñak;TI 4 ³ eyIgmanTinñn½y 26 27 27 ÷mFümKW x 26 273 27 26.67 ÷tYén Me KW 3 2 1 2 naM[ Me 27 ÷témøm:Ut KW Mo 27 ¬ maneRbkg;elIseK¦ -cMeBaH fñak;TI 5 ³ eyIgmanTinnñ ½y 19 20 20 22 ÷mFümKW x 19 20 4 20 22 20.25 ÷tYén Me KW 4 2 1 2.5 enaHnaM[ Me 20 2 20 20 ÷témøm:Ut KW Mo 20 ¬ maneRbkg;elIseK¦
22 22 21 21 22 23 22 23 20 19 21 20 22 23 23 23 27 27 26 20 19 22 20
20
23 24 24 24
rkmFüm emdüan nigm:UténcMnYnsisSkñúgmYyfñak;³ -cMeBaH fñak;metþyü ³ eyIgmanTinñn½y 20 20 21 21 ÷mFümKW x 20 20 4 21 21 20.5 ÷tYén Me KW 4 2 1 2.5 enaHnaM[ Me 20 2 21 20.5 ÷Kµanm:UteT eRBaH 20 nig 21 maneRbkg;dUcKña -cMeBaH fñak;TI 1 ³ eyIgmanTinñn½y 21 22 23 23 ÷mFümKW x 22 22 4 23 23 22.5 ÷tYén Me KW 4 2 1 2.5 enaHnaM[ Me 22 2 23 22.5 ÷Kµanm:UteT eRBaH 22 nig 23 maneRbkg;dUcKña
smÁal; ³
23
27
-mFüm tageday x -emdüan tageday Me -m:Ut tageday Mo
20
124
-cMeBaH fñak;TI 6 ³ eyIgmanTinñn½y 23 24 24 24 ÷mFümKW x 23 24 3 24 24 23.75 ÷tYén Me KW 4 2 1 2.5 enaHnaM[ Me 24 2 24 24 ÷témøm:Ut KW Mo 24 ¬ maneRbkg;elIseK¦
3.
x 16 18 19 f 1 4 9
16 1 18 4 19 9 20 3 21 2 30 1 1 4 9 3 2 1 391 19.55 20
-KNnaemdüan ³ eday tYénemdüanKW 202 1 10.5 ehIytYTI 10.5 sßitkñúgTinñn½y 19 dUcenH Me 19 . -KNnam:Ut ³ edayTinñn½y 19 maneRbkg;elIseK dUcenH Mo 19 .
4 6 2 10 8 5 4 4 8 10 5 4 3 7 2 4 6 8 7 4 5 3 6 4
1 2 3 4 5 6 7 8 10
1 2 2 6 3 3 2 3 2
24 srub -KNnamFüm ³
1 4 6 24 15 18 14 24 20
21 30 2 1
x
eyIgmanlT§plbBa©úHm:as ¬kg¦ dUcxageRkam ³
eyIgGacsresrsRmÜlCatarag dUcxageRkam ³ m:as¬kg¦ eRbkg; f x f f
20 3
-KNnamFüm ³
24
2.
eyIgmanTinñn½y dUcxageRkam ³
4.
1 3 5 11 14 17 19 22 24
KNnamFüm emdüan nigm:Ut ³ k> tamRkaPic eyIgBRgay)anTinñn½y 2 , 4 , 4 , 6 , 6 , 6 , 6 , 6 , 6 , 8 , 8 , 10 2 4 2 6 6 8 2 10 72 x 12 12
-mFüm dUcenH KNna)an x 6 . -KNnaemdüan ³ tYénemdüanKW 122 1 6.5 naM[ Me 6 2 6 6 dUcenH Me 6 . -KNnam:Ut ³ eday 6 maneRbkg;eRcInCageK dUcenH Mo 6 . x> tamRkaPic eyIgBRgay)anTinñn½y
126
11 2 2 3 2 4 6 5 3 6 3 7 2 8 3 10 2 1 2 2 6 3 3 2 3 2 126 5.25 kg 24
x
-KNemdüan ³ tYénemdüanKW 242 1 13 ehIytYTI 13 sßitenA kñúgfñak;m:as 5 kg dUcenH Me 5 kg . -KNnam:Ut ³ eday m:as 4 kg mancMnYneRbkg;eRcInCageK dUcenH Mo 4 kg .
2 , 4 , 4 , 4 , 6 , 8 , 10 , 10 , 10 , 10 , 34 , 34
-mFüm x 2 4 3 6 812 10 4 34 2 136 12 dUcenH KNna)an x 11.33 . -KNnaemdüan ³ tYénemdüanKW 122 1 6.5 naM[ Me 8 210 9 dUcenH Me 9 . 125
-KNnam:Ut ³ eday 10 maneRbkg;eRcInCageK dUcenH Mo 10 . K> tamRkaPic eyIgBRgay)anTinñn½y
6.
o
5 , 5 , 10 , 10 , 10 , 10 , 15 , 15 , 25 , 25 , 30 , 30 , 30 , 30 , 35 , 35
o
14 14 23 32 35 30 30 29 25 22 18 15 12 287 o 23.92 C 12
5 2 10 4 15 2 25 2 30 4 35 2 16
dUcenH KNna)an x 20 . -KNnaemdüan ³ tYénemdüanKW 162 1 8.5 naM[ Me 15 2 25 20 dUcenH Me 20 .
o
o
-KNnamFüm ³ x
k> KNnam:Ut ³ eday sItuNPð aB 14 C nig 30 C maneRbkg; esµI 2 eRcInCageK dUcenH Mo 14 C nig Mo 30 C . x> KNnasItuNðPaBmFüm ³ x
rkcMnYnBinÞúEdl sux RtUvbegáIneEfmeTot ³ tambRmab;RbFan ³ -BinÞú sux enAqmasTI1 KW 89 8 712 BinÞú -ehIyBinÞúedIm,I[)anmFümPaK 90 KW -KNnam:Ut ³ eday 10 nig 30 maneRbkg;eRcIn 90 8 720 BinÞú CageK dUcenH Mo 10 nig Mo 30 . -naM[ cMnYnBinÞú EdlsuxRtUvbegáInKW 5. k> begáIttarageRbkg; eRbkg;ekIneFobCaPaKry 720 712 8 BinÞú cMnYnkUn eRbkg; eRbkg;ekIn eRbkg;eFob eRbkg;ekIneFob dUcenH sux RtUvbegáIncMnYn 8 BinÞúeTot . 0 11 11 14.47 14.47 1 23 34 30.26 44.74 8. k> begáIttarag eRbkg; eRbkg;ekIn nigfy ³ 2 9 43 11.84 56.85 3 4 5 6
18 6 4 5
61 67 71 76
23.68 7.89 5.26 6.58
7.
km
80.26 88.16 93.42 100
srub 76 x> KNnaemdüan ³ eday eRbkg;srub f 76 naM[ tYénemdüanKW 762 1 38.5 tamtarageRbkg;ekIn tYTI 38.5 sßitenATinñn½y cMnYnkUn 2 dUcenH Me 2 .
mm
0-5 5-10 10-15 15-20 20-25 25-30 30-35
srub
f
3 15 72 15 91 35 8 239
f
f
3 18 90 105 196 231 239
239 236 221 149 134 43 8
p©itfñak; x 2.5 7.5 12.5 17.5 22.5 27.5 32.5
x f 7.5 112.5 900 262.5 2047.5 962.5 260 4552.5
x> -tamtarageRbkg;fy cMnYnkUneb:ge)aH Edlman km
K> -KNnam:Ut ³ edayfñak;km
tamtarageRbkg;srubKW f 315 dUcenH cMnYnTarke)ATwkedaHeKaKW 315 nak; . x> rkm:asTwkedaHeKaEdlTarke)ACamFümkñúg1éf¶ 20357.5 x 64.63 g dUcenH x 64.63 g 315 K> KNnam:Ut ³ edayfñak; 55-60 mancMnYneRbkg;eRcInCageK naM[ m:UtCap©ti fñak;enH dUcenH Mo 57.5 . X> KNnaemdüan ³ tYénemdüanKW 315 157.5 2 eday eRbkg; 157.5 sßitenAkñúgfñak; 60-65 157.5 111 64.31 naM[ Me 60 65 60165 111
25 20119.5 105 20.8 196 105
dUcenH emdüanKNna)an -KNnamFüm ³
x
Me 20.8
.
dUcenH KNna)an Me 64.31 . -bkRsaytamRkab ³
4552.5 19.05 239
dUcenH KNna)an x 19.05 . -eRbóbeFobemdüan nigmFüm ³ eday Me 20.8 nig x 19.05 dUcenH eRbóbeFob)an Me x . 9. k> rkcMnYnTarkEdle)ATwkedaHeKa kñúgmYyéf¶ ³ km
40-45 45-50 50-55 55-60 60-65 65-70 70-75 75-80 80-85 85-90
srub
f
8 11 31 61 54 58 43 25 17 7 315
f
8 19 50 111 165 223 266 291 308 315
f
315 307 296 265 204 150 92 49 24 7
p©itfñak; x 42.5 47.5 52.5 57.5 62.5 67.5 72.5 77.5 82.5 87.5
x f 340 522.5 1627.5 3507.5 3375 3915 3117.5 1937.5 1402.5 612.5 20357.5
350 300 250 200 150 100 50 40 45 50 55 60 65 70 75 80 85 90 TwkedaHeKa (g )
tamRkabénBhuekaNeRbkg;ekIn nigBhuekaN eRbkg;fy RkaPicTaMgBIrRbsBVKñaRtg;cMNuc Edl man Gab;sIusesµI 64.31 nigGredaenesµI 157.5 man n½yfa Tinñn½ymanemdüan Me 64.31 EdlRtUvnwg eRbkg; f 157.5 .
127
k> begáIttarageRbkg; eRbkg;ekIn ³ km
10.
145-150 150-155 155-160 160-165 165-170 170-175 175-180
31 95 131 272 120 77 48
srub
774
-bkRsayelIRkab ³ eyIgKUsbnÞat;RsbG½kS Gab;sIus edaykat;G½kSGredaenRtg;eRbkg; 387 rYceFVIcMeNalEkg éncMNucEdlCaRbsBVrvag bnÞat; nigBhuekaNeRbkg;ekIn mkelIGk½ SGab; sIus enaHeyIg)antémøkm
31 126 257 529 649 726 774
x> sg;RkabénBhuekaNeRbkg;ekIn ³ f
800
774
726 700
649
600
529 500
400
387
300
257
200
126 100
31
162.39
145 150 155 160 165 170 175 180
kMBs;sisS
K> KNnaemdüan ³ 387 tYénemdüanKW 774 2 tamtarageRbkg;ekIn tYénemdüansßitenAkñúgfñak; 160-165 mann½yfafñak; 160-165 Cafñak;emdüan eyIgKNna Me tamviFIGaMgETb:LU asüúg ³ Me 160
165 160387 257 162.39 529 257
dUcenH KNna)an
Me 162.39 cm
. 128
៨ 1.
PHNOM PENH P
OឬM
H
2.
1 1
PឬN 21 ឬ
21
3.
1
4
//
//
4. ឬ
5.
182
52
234
254
338
592
1200
760
440 6.
3
A B C
A, B, C
1420 846 570
74 26 41
182 122 60
1676 994 671
2836
141
364
3341 A
BឬC A BឬC A 7.
4
HHHH , HTHH , THHH , TTHH HHHT , HTHT , THHT , TTHT HHTH , HTTH , THTH , TTTH HHTT , HTTT , THTT , TTTT 129
BឬC
H
T
H
H
T 8.
2 5 9. 0
1
2
3
4
5
6
7
8
6
1
15
12
24
18
16
10
7
4
6 6
10.
6
4
2 ABCD
X
11.
Y
6
A, B, C, D, E
F
3
4
12.
H
13.
12
1 3 14.
3
2
A1
15. A1 , A2 , B
A2
A 16.
B
15 3 5
2 3 130
៨ 1.
cMnYnGkSrTaMgGs;énBakü PHNOM PENH man 9 GkSr naM[ cMnYnkrNIGac 9 eKcab;ykGkSmYyBIBaküenHedayécdnü ³ k> rkRbU)abEdlcab;)anGkSr P ³ eday GkSr P mancMnYn 2 naM[ cMnYnkrNIRsbcab;)anGkSr P KW 2 R sb 2 dUcenH P(P) cMcMnnYnYnkrNI . krNIG ac 9 x> rkRbU)abEdlcab;)anGkSr H ³ eday GkSr H mancMnYn 2 naM[ cMnYnkrNIRsbcab;)anGkSr H KW 2 R sb 2 dUcenH P(H) cMcMnnYnYnkrNI . krNIG ac 9 K> rkRbU)abEdlcab;)anGkSr O b¤ M ³ eday GkSr O mancMnYn 1 naM[ cMnYnkrNIRsbcab;)anGkSr O KW 1 ehIy GkSr M mancMnYn 1 naM[ cMnYnkrNIRsbcab;)anGkSr O KW 1 eK)an P(H b¤ M) PO PM 19 19 dUcenH P(H b¤ M) 92 X> rkRbU)abEdlcab;)anGkSr P b¤ N ³ eday GkSr N mancMnYn 2 naM[ cMnYnkrNIRsbcab;)anGkSr N KW 2 eK)an P(P b¤ N) PP PN 92 92 dUcenH P(P b¤ N) 94 .
2.
etIRbU)abéncMnYnEdlsux nigesArkesµIKña b¤eT ? cMnYnenAcenøaHBI 1 dl; 21 KWman 21 cMnYn naM[ cMnYnkrNIGac 21 -sux kMBugrkcMnYnessenAcenøaHBI 1 dl; 21 cMnYnessenAcenøaHBI 1 dl; 21 KWman 11 cMnYn naM[ P(cMnYness) 11 21 -esA kMBugrkcMnYnKUenAcenøaHBI 1 dl; 21 cMnYnKUenAcenøaHBI 1 dl; 21 KWman 10 cMnYn naM[ P(cMnYnKU) 10 21
dUcenH RbU)abéncMnYnEdlsux nigesA kMBugnwgrk KWminesµIKñaeT . 3. rkRbU)abénRKab;LúkLak;ecjBIelx 1dl;elx 4 eday RKab;Lkú Lak;manmux 6 enaHkrNIGac 6 ehIymuxBIelx 1dl; 4 man 4 muxCakrNIRsb naM[ P(1dl;elx 4) 64 23 dUcenH 4.
dl;elx 4)
P(1
2 3
.
etIRbU)abKb;RBYjelIépÞqUt nigminqUtesµIKñab¤eT ? tag a nig b CavimaRtrbs;ctuekaNEkg // // -épÞqUt 12 ab a b 1 b 1 b 1 a a ab 2 2 2 2 2
-épÞminqUt eday épÞqUt nigépÞminqUtesµIKña dUcenH RbU)abKb; RBYjelIépÞqUt nigminqUtk¾esµIKñaEdr . 131
eyIgmanTinñn½ydUcxageRkam ³ GñkenAkñúgRkug GñkenACayRkug srub 182 52 234 rfynþ 254 338 592 m:UtU k> rkRbU)abénGñkeFVIdMeNIredaym:UtU edaym:UtU nigrfynþmansrub 234 592 826 naM[ cMnYnkrNIGacesµInwg 826 ehIy m:UtUmancMnYn 592 CacMnYnkrNIRsb 592 296 eyIg)an P(GñkCiHm:UtU) 826 0.72 413
K> rkcMnYncMNtrfynþEdleKRtUveRtómTukCamun -rkcMnYnGñkenAkñúgRkugCiHrfynþ RbU)abénGñkrs;enAkñúgRkugCiHrfynþKW
5.
P
182 91 182 254 218
edayePJóvenAkñúgRkugGeBa©IjcUlrYm 760 nak; naM[ GñkenAkñúgRkugCiHrfynþmancMnYn 91 760 318 nak; 218 -rkcMnYnGñkenACayRkugCiHrfynþ RbU)abénGñkrs;enACayRkugCiHrfynþKW P
dUcenH RbU)anénGñkeFVIdMeNIredaym:UtUKW 296 P(GñkCiHm:Ut)U . 0.72 413
52 2 52 338 15
edayePJóvenACayRkugGeBa©IjcUlrYm 440 nak; naM[ GñkenACayRkugCiHrfynþmancMnYn 2 x> ¬rebobTI1¦rkRbU)abénGñkeFVIdMeNIredayrynþ 440 59 nak; 15 rfynþmancMnYn 234 CacMnYnkrNIRsb -eyIg)an cMnYnGñkeRbIR)as;rfynþTaMgGs;KW ehIy cMnYnkrNIGacesµInwg 826 ¬rkxagelI¦ 318 59 377 nak; 234 117 eyIg)an P(GñkCiHrfynþ) 826 413 0.28 naM[ rfynþman 377 eRKÓg ehIy RtUvkarcMNt 377 kEnøg dUcenH RbU)anénGñkeFVIdMeNIredayrfynþKW 117 dUcenH eKRtUveRtómcMNtrfynþ 377 kEnøg 0.28 P(GñkCiHrfynþ) . 413 6. eyIgmantaraglT§plénkarsÞabsÞg;sMeLg ³ ¬rebobTI2¦rkRbU)abénGñkeFVIdMeNIredayrynþ KaMRT minKaMRT Kµaneyabl; srub edayRbU)abénGñkeFVIdMeNIredaym:UtUCaRbU)ab A 1420 74 182 1676 B 846 26 122 994 bMeBjnwgRbU)abénGñkeFVIdMeNIredayrfynþ C 570 41 60 671 364 3341 srub 2836 141 naM[ P(GñkCiHm:UtU) + P(GñkCiHrfynþ) = 1 k> rkRbU)abénsMeLgEdlminKaMRTebkçCn A b¤ P(GñkCiHrfynþ) =1 P(GñkCiHm:UtU) 296 sMeLgminKaMRTsrubmancMnnY 3341 CakrNIGac 1 413 sMeLgminKaMRT A mancMnYn 74 CakrNIRsb 117 0.28 74 413 naM[ P(sMeLgminKaMRTA) 3341 0.0221 0.28 . dUcenH P(GñkCiHrfynþ) 117 dUcenH P(sMeLgminKaMRTA) 0.0221 . 413 132
x> rkRbU)abénsMeLgEdlminKaMRTebkçCn B b¤ C tamtaraglT§plénkaresÞabsÞg; eyIg)an ³ 26 41 P(sMeLgminKaMRT B b¤ C) 3341 3341
7.
eyIgmanRBwtþikarN_Gacénkare)aHkak; 4 dgKW ³ HHHH , HTHH , THHH , TTHH HHHT , HTHT , THHT , TTHT HHTH , HTTH , THTH , TTTH HHTT , HTTT , THTT , TTTT
67 0.0201 3341
naM[ cMnYnkrNIGacesµInwg 16 dUcenH P(sMeLgminKaMRT B b¤ C) 0.0201 . k> rkRbU)abEdle)aH)anGkSr H BIr krNIe)aH)anGkSr H BIrman ³ K> rkRbU)abénkarKµaneyabl;cMeBaHebkçCn A TTHH , HTHT , THHT man 6 CakrNIRsb tamtaraglT§plénkaresÞabsÞg; eyIg)an ³ HTTH , THTH , HHTT 182 P(sMeLgKµaneyabl; A) 0.0545 naM[ P( H BIr) 166 83 0.375 3341 dUcenH
sMeLgKµaneyabl; A) 0.0545 .
dUcenH
P(
P( H
BIr) 83 0.375
X> rkRbU)abénkarKµaneyabl;cMeBaHebkçCn B b¤ C x> rkRbU)abEdle)aH)anGkSr T BIrmuneK tamtaraglT§plénkaresÞabsÞg; eyIg)an ³ krNIe)aH)anGkSr T BIrmuneKman ³ 122 60 P(sMeLgKµaneyabl; B b¤ C) TTHH , TTHT , TTTH , TTTT man 4 krNI 3341 3341 4 1 182 naM [ P( T BIrmuneK) 0.25 0.0545 16 4 3341 dUcenH P(sMeLgKµaneyabl; B b¤ C) 0.0545
dUcenH
P( T
BIrmuneK) 14 0.25
g> eRbóbeFobRbU)abKaMRT A nigRbU)abKaMRT B b¤ C K> rkRbU)abEdle)aH)anGkSr H mYymuneK tamtaraglT§plénkaresÞabsÞg; eyIg)an ³ krNIe)aH)anGkSr T BIrmuneKman ³ 1420 0.4250 P(sMeLgKaMRT A) HHHH , HTHH , HHHT , HTHT 3341 man 8 krNI HHTH , HTTH , HHTT , HTTT 846 570 P(sMeLgKaMRT B b¤ C) 3341 3341 naM[ P( H mYymuneK) 168 12 0.5
1416 0.4238 3341
dUcenH
enaH P(sMeLgKaMRT A) P(sMeLgKaMRT B b¤ C) dUcenH eyIgeRbóbeFob)an RbU)abénsMeLg KaMRTebkçCn A FMCagRbU)abénsMeLg ebkçCn B b¤ C
P( H
mYymuneK) 12 0.5
X> rkRbU)abEdle)aH)anGkSr H TaMgbYn ³ e)aH)anGkSr H TaMgbYn man 1 krNI KW HHHH naM[ P( H TaMgbYn) 161 0.0625 dUcenH P( H TaMgbYn) 161 0.0625 133
g> rkRbU)abEdle)aH)anGkSr T mYyy:agtic ³ RbU)abe)aH)anGkSr T mYyy:agtic CaRbU)ab bMeBjnwgRbU)abe)aH)anGkSr H TaMgbYn naM[ P( H TaMgbYn) + P( T mYyy:agtic) = 1 enaH P( T mYyy:agtic) = 1 P( H TaMgbYn) 1
dUcenH
P( T
1 15 0.9375 16 16
x> rkRbU)abEdlbuKÁlikenaH )anQb;y:agticmYyéf¶ edaycMnYnbuKÁlik )anQb;y:agticmYyéf¶ KW ³ 1 15 12 24 18 16 10 7 103 nak; ¬GacKNnaedayxøIKW 109 6 103 nak;¦ 103 eyIg)an P(Qb;y:agticmYyéf¶) 109 dUcenH
15 mYyy:agtic) 16 0.9375 .
103 Qb;y:agticmYyéf¶) 109
P(
K> rkRbU)abEdlbuKÁlikenaH)anQb;cenøaHBI 4eTA 6éf¶ 8. rkRbU)abEdlcab;)anXøIBN’ s ³ edaycMnYnbuKÁlik )anQb;cenøaHBI 4 eTA 6éf¶KW ³ edayRbU)abEdlcab;)anXøBI N’s CaRbU)abbMeBj 24 18 16 58 nak; CacMnYnkrNIRsb 58 nwgRbU)abEdlcab;)anXøBI N’exµA eyIg)an P(Qb;cenøaHBI 4eTA 6éf¶) 109 naM[ P(XøIexµA) + P(XøIs) = 1 58 dU c enH P(Qb;cenøaHBI 4eTA 6éf¶) Taj)an P(XøIs) = 1 P(XøIexµA) 109 tambRmab; P(XøIexµA) 52 K> rkRbU)abEdlbuKÁlikenaH)anQb;eRcInCag 6éf¶ eyIg)an P(XøIs) = 1 52 53 edaycMnYnbuKÁlik )anQb;eRcInCag 6éf¶KW ³ 10 7 17 nak; CacMnYnkrNIRsb dUcenH eyIgrk)an P(XøIs) = 53 . 17 eyIg)an P(Qb;eRcInCag 6éf¶) 109 9. eyIgmantaragéncMnYnGvtþmanrbs;buKl Á ik ³ 17 dUcenH P(Qb;eRcInCag 6éf¶) 109 cM>éf¶Gvtþman cM>buKÁlik
0
1
2
3
4
5
6
7
8
eyIgmanXøIexµA 4 nigXøIs 2 buKÁliksrub 6 115 12 24 18 16 10 7 109 k> rkRbU)abEdlcab;)anXøBI N’sTaMgBIr nak; CacMnYnkrNIGac cMnYnkrNIGacKW 4 2 6 k> rkRbU)abEdlbuKÁlikenaH )anQb;y:ageRcInbIéf¶ eyIg)an P(ss) = P(s)×P(s¼s) 2 1 2 1 edaycMnYnbuKÁlik )anQb;y:ageRcInbIéf¶ KW ³ 6 5 30 15 6 1 15 12 34 nak; CacMnYnkrNIRsb 34 dUcenH RbU)abEdlcab;)anXøIBN’s eyIg)an P(Qb;y:ageRcInbIéf¶) 109 TaMgBIrKW P(ss) 151 34 dUcenH P(Qb;y:ageRcInbIéf¶) 109 6
1
15
12
24
18
16
10
7
10.
134
x> rkRbU)abEdlcab;)anXøIBN’exµAmYyy:agtic edayRbU)abEdlcab;)anXøBI N’exµAmYyy:agtic CaRbU)abbMeBjnwgRbU)abcab;)anXøIsTaMgBIr eyIg)an P(ss) + P(exµAmYyy:agtic) = 1 naM[ P(exµAmYyy:agtic) = 1 P(ss)
rkRbU)abEdle)aH)anmux H TaMgbI kak;manmux H nig T enaHlT§plGac)anKW ³ kak;TI1 kak;TI2 kak;TI3 lT§pl
12.
H T
1 14 1 15 15
dUcenH 11.
P(
exµAmYyy:agtic
H T H T H T H T
H
H T
14 ) 15
T
HHH HHT HTH HTT THH THT TTH TTT
lT§plsrubmancMnYn 8 CacMnYnkrNIGac mux HHH manEt 1 Kt; CacMnYnkrNIRsb
eyIgmanXøIexµA 4 nigXøIs 2
k>
dUcenH
P( H
TaMgbI) 18
k> rkcMnnY énXøIBN’exµA XøITaMgGs;mancMnYn 12 RKab; CacMnnY krNIGac tag x CacMnYnXøIBN’exµA EdlCakrNIRsb naM[ RbU)abcab;)anXøIBN’exµAKW P(exµA) = 12x Et RbU)abEdlcab;)anXøIBN’exµAesµInwg 13
13.
eyIg)an
x 1 12 3
b¤
3x 12
naM[ x 4
dUcenH cMnYnXøIBN’exµAmancMnYn 4 RKab; x> -¬rebobTI1¦ rkRbU)abEdlcab;)anXøBI N’s cMnYnXøIBN’s 12 4 8 CacMnYnkrNIRsb dUcenH
s
P( )
8 2 12 3
-¬rebobTI2¦ rkRbU)abEdlcab;)anXøIBN’s eday P(s) CaRbU)abbMeBjnwg P(exµA) naM[ P(s) = 1 P(exµA) 1 124 128 dUcenH P(s) 128 23 135
rkRbU)abEdleKerIs)annarITaMgBIrnak; bursmancMnYn 3 nak; nignarImancMnYn 2 nak; naM[ cMnYnkrNIGacKW 3+2=5 eyIg)an P(narIBInak;) P(narI) P(narI¼narI)
14.
2 1 1 5 4 10
dUcenH
P(
narIBInak;) 101
rkRbU)abEdleKKb;)anGkSr A TaMgBIelIk eKKb;RBYjBIrdgepSgKña minTak;TgKña fasEckCabIEpñk enaHcMnYnkrNIGacesµInwg 3 GkSr A man 2 KW A nig A CacMnYnkrNIRsb eyIg)an P(ATaMgBIrelIk) P(A) × P(A)
15.
1
x> rkcMnYnXøIRkhmEdlRtUvbEnßmcUlkñúgfg; tag y CacMnYnXøIRkhmEdlRtUvbEnßmcUlkñúgfg; naM[ cMnYnXøsI rubKW 15 x y 25 y cMnYnxøIRkhmekIn)an 15 y naM[ RbU)abcab;)anXøIRkhmfµIKW P 1525 yy EtRbU)abEdlcab;)anXøIRkhmfµIesµInwg 23 eyIg)an 1525 yy 23 225 y 315 y 50 2 y 45 3 y 50 45 3 y 2 y 5 y
2
dUcenH cMnYnXøIBN’RkhmEdlRtUvbEnßm cUlkñúgfg;KWmancMnnY 4 RKab; .
2 2 4 3 3 9
dUcenH RbU)abEdlKb;)anGkSr A TaMgBIelIk KW P(ATaMgBIrelIk) 94 k> KNnacMnnY XøIBN’exovkñúgfg;enaH tag x CacMnYnXøIBN’exovEdlRtUvrk enaHcMnYnXøITaMgGs;KW 15 x CacMnYnkrNIGac naM[ P(Rkhm) 1515 x Et RbU)abEdlcab;)anXøIBN’RkhmesµInwg 53 eyIg)an 1515 x 53
16.
15 5 3 x 25 15 x 10
15 x
dUcenH cMnnY XøIBN’exovmancMnYn 10 RKab;
136
៩ ចម្ងាយរវាងពីរចំណុច 1.
គណនាចម្ងាយរវាងពរី ចំណុច ដែលម្ងនកូអរដោដណែូចខាងដរោម។
k> 9 , 2 nig 6 , 8 K> 2 , 5 nig 0 , 0 g> 8 , 7 nig 8 , 6 q> 100 , 203 nig 97 , 200 2. 3. 4. 5.
ចុងទាំងពីរននអងកតផ្ ់ ចិតរងវងម ់ យ ួ ម្ងនកូអរដោដនដរៀងគ្នា 3 , 1 និង 2 , 5 ។ ចូរគណនាោំរងវងរ់ ច ួ សង់រងវងដ់ នះ។
ចូរគណនាររដវងអងកតរ់ រូងននចតុដោណ PQRS ដែល P 2 , 3 , Q5 , 5 , R6 , 6 នង ិ S 3 , 3 ។
KLM ដែល K 2 , 8 , L10 , 11 នង ិ M 5 , 0 ជារតដី ោណសមបាត ។ ចូរគណនាររម្ងរតននរត ិ ីដោណ ABC ដែល A3 , 7 , B5 , 2 និង C 7 , 3 ។ ចូររង្ហាញថារតដី ោណ ចូររង្ហាញថា
6. 7.
ABC ជារតីដោណដកង រួចគណនានផ្ៃរកឡាររស់វា ។
y ដែើមបឱ្ ី យចម្ងាយរវាងពរី ចំណុចដែលម្ងនកូអរដោដន 4 , 2 នង ិ 4 , y ដសមើនង ឹ 5 ឯកតា ។ ចូររកចំណុច C x , 1 កាុងោរែង់រី 1 ដែើមបឱ្ ី យរតីដោណដែលម្ងនកំពូល A1 , 1 , B4 , 7 និង C x , 1 ចូររកតនមល
ជារតីដោណសមបាតដែល
8.
x> 5 , 3 nig 2 , 4 X> 6 , 3 nig 2 , 1 c> 3 , 12 nig 7 , 12 C> 3 , 8 nig 4 , 7
AB AC ។
កំពូលននរតដី ោណមួយម្ងនកូអរដោដន 3 , 2 , 9 , 2 នង ិ 3 , 10 ។ ក. ចូរគណនាររដវងររុងទាំងរីននរតីដោណ។ ខ. ចូររបារ់ររដេររតីដោណដនះ។
9.
កំពូលននរតដី ោណមួយម្ងនកូអរដោដន 1 , 2 , 5 , 2 , 1 , 4 នង ិ 5 , 4 ។ ក. ចូរគណនាររដវងររុង និងររដវងអងកតរ់ រូងននចតុដោណ។ ខ. ដតើដគសនាិោានែូចដមេចចំដបាះចតុដោណដនះ ?
10.
កូអរដោដនកំពូលររស់ចតុដោណមួយម្ងន 2 , 1 , 6 , 4 , 2 , 9 នង ិ 2 , 4 ។ ក. ចូរគណនាររដវងររុង និងររដវងអងកតរ់ រូងននចតុដោណ។ ខ. ចូររបារ់ររដេរននចតុដោណដនះ។
11. 12.
ចតុដោណមួយម្ងនកំពូល 2 , 3 , 3 , 1 , 5 , 1 នង ិ 4 , 3 ។ ដតច ើ តុដោណដនះម្ងនរាងជាអវ? ី ក. A5 , 8 និង B 5 , 12
13.
ចូរគណនាកូអរដោដនននចំណុចកណ្ត េ លនន AB រួចដៅចំណុចកាុងតរមុយអរតូណរដមតាមករណីនម ួ ៗែូចខាងដរោម ី យ ខ. A 5 , 2 និង B 5 , 14 ។
ដរររើ រ ូ មនេកូអរដោដនននចំណុចកណ្ត េ លអងកត។ ់ ចូរគណនាកូអរដោដនចំណុចកណ្ត េ លននអងកត់ AB កាុងករណី នម ួ ៗែូចខាងដរោម ។ ី យ
ក. A 3 , 5 និង B7 , 5
ខ. A 8 , 6 និង B4 , 2
គ. A3t 5 , 7 និង Bt 7 , 7 , t 1
14.
ចូរគណនាកូអរដោដនននចំណុចកណ្ត េ ល ចូរគណនាររដវង
15.
ឃ. Aa , b នង ិ Ba , c ។
M ននអងកតរ់ វាងពរី ចំណុច A 3 , 1 នង ិ B5 , 7 ។
AM និង BM ។ ដគឱ្យ M 1 , 2 ជាចំណុចកណ្ត េ លននអងកត់ AB ដែល A 2 , 2 ។ ចូរគណនាកូអរដោដនននចំណុច B ។
137
16. 17.
រងវងម ់ យ ួ ម្ងនផ្ចត ់ ចិតរងវងដ់ នះម្ងនចំណុចចុងមួយរតង់ V 3 , 8 ។ គណនាកូអរដោដនននចំណុច ិ F 2 , 6 ។ អងកតផ្
T ដែលជាចំណុចចុងម្ងាងដរៀតននអងកត់ TV រួចសង់រងវងដ់ នះ។ កូអរដោដនននកំពូលររដលឡូរោមមួយដសមើ 1 , 1 , 2 , 3 , 3 , 1 និង 2 , 5 ។ ចូរគណនាកូអរ ដោដនននចំណុចកណ្ត េ លននអងកតរ់ រូងទាំងពីរ។ ដតើដគសនាិោានបានយ៉ាងែូចដមេច?
18.
ចតុដោណមួយម្ងនកំពូល E 5 , 3 , F 5 , 5 , G 3 , 3 នង ិ H 1 , 5 ។ ចូរគណនាកូអរដោដននន ចំណុចកណ្ត េ ល A , B , C និង រាងជាអវី ?
19.
D ននររុង EF , FG , GH និង EH ។ ដតើចតុដោណ ABCD ម្ងន
ចតុដោណដកងមួយកំណត់ដោយចំណុច A3 , 2 , B9 , 2 , C 9 , 6 នង ិ D3 , 6 ។ ចូររាយរញ្ជាក់ថា ចតុដោណដែលកំណត់ដោយ ចំណុចកណ្ត េ លននររុងទាំងរួន ររស់ចតុដោណដកងខាងដលើជាចតុដោណដសមើ។
20.
ចតុដោណមួយម្ងនកំពូល P 3 , 2 , Q 1 , 5 , R3 , 1 និង S 5 , 4 ។ ដរើ A , B , C និង
D
ជាចំណុចកណ្ត េ លដរៀងគ្នាននររុង PQ , QR , RS នង ិ SP ។ ចូររង្ហាញថា
21.
ABCD ជាររដលឡូរោម ។
ដគម្ងនអងកត់ P1 P2 ដែល P1 x1 , y1 និង P2 x2 , y 2 ។ ចូរគណនាកូអរដោដនននចំណុច
ចំណុចដែលដចកអងកត់ P1 P2 ជារីដផ្ាកដសមើៗគ្នា ។
22.
23.
x x2 y1 y 2 P1 x1 , y1 , Pm 1 , នង ិ P2 x2 , y 2 ។ 2 2 ចូររង្ហាញថា P1 Pm Pm P2 P1 P2 ។
A និង B ជា
ដគម្ងនចំណុច
ដគឱ្យ A1 , 0 និង B3 , 2 រួចសង់រងវងដ់ ែលម្ងនអងកតផ្ ់ ចិត ក. ចូររកកូអរដោដនននផ្ចិតរងវងដ់ ែលតាងដោយចំណុច ខ.
5 M ជាចំណុចមួយដែលម្ងនកូអរដោដន , 2 ដែើមបឱ្ ិ ដៅដលើរងវង់ រួចរញ្ជាក់ដតើ M ី យ M ឋត
y
AB ។
I។
y ចូរកំណត់ y ម្ងនរីតាង ំ រានា ុ ម ន?
x
O
A
y
DDCEE
3
138
B
x
៩
KNnacm¶ayrvagBIrcMNuc EdlmankUGredaen ³ k> 9 , 2 nig 6 , 8 ebI d Cacm¶ayrvagBIrcMNucenH eyIg)an ³ 1.
d
X> 6 , 3 nig 2 , 1 ebI d Cacm¶ayrvagBIrcMNucenH eyIg)an ³ d
6 92 8 22 32 6 2
82 4 2 64 16 80 16 5 4 5
9 36
dUcenH KNna)ancm¶ay d 4 5 ÉktaRbEvg
45 95
g> 8 , 7 nig 8 , 6 ebI d Cacm¶ayrvagBIrcMNucenH eyIg)an ³ 5 ÉktaRbEvg
3 5
dUcenH KNna)ancm¶ay d 3
d
x> 5 , 3 nig 2 , 4 ebI d Cacm¶ayrvagBIrcMNucenH eyIg)an ³ d
dUcenH KNna)ancm¶ay d 13 ÉktaRbEvg c> 3 , 12 nig 7 , 12 ebI d Cacm¶ayrvagBIrcMNucenH eyIg)an ³
2 49 7 2
d
dUcenH KNna)ancm¶ay d 7 2 ÉktaRbEvg K> 2 , 5 nig 0 , 0 ebI d Cacm¶ayrvagBIrcMNucenH eyIg)an ³
0 5 0 2 52 22 2
8 82 6 72 2 0 2 13
132 13
2 52 4 32 7 2 7 2
49 49
d
2 62 1 32
2
42 02 42 4
dUcenH KNna)ancm¶ay d 4 ÉktaRbEvg q> 100 , 203 nig 97 , 200 ebI d Cacm¶ayrvagBIrcMNucenH eyIg)an ³ d
25 4
29
dUcenH KNna)ancm¶ay d
29
7 32 12 122
97 100 2 200 203 2 32 32
9 9 18 3 2
ÉktaRbEvg
dUcenH KNna)ancm¶ay d 3 2 ÉktaRbEvg 139
C> 3 , 8 nig 4 , 7 ebI d Cacm¶ayrvagBIrcMNucenH eyIg)an ³
3.
KNnaRbEvgGgát;RTUgénctuekaN PQRS eyIgman kUGredaenéncMNcu dUcxageRkam ³
4 32 7 82 12 12
d
2
6
2
-KNnaRbEvgkaMrgVg; ³ edaycugTaMgBIrénGgát;p©itmankUGredaen 3 , 1 nig 2 , 5 ebI d CaRbEvgGgát;p©it enaH
3
3
dUcenH KNna)ancm¶ay d 2 ÉktaRbEvg
d
Q
P
11
2.
5
S
5
3
6
-KNnaRbEvgGgát;RTUg PR ³ PR
2 32 5 12 52 4 2
6 22 6 32 2 8 2 9
64 81 145
25 16
dUcenH RbEvgGgát;RTUg PR
41
naM[ kaMrgVg; R d2
41 2
dUcenH kaMrgVg;manRbEvg R
ÉktaRbEvg 41 2
145
ÉktaRbEvg
-KNnaRbEvgGgát;RTUg QS ³
ÉktaRbEvg
-sg;rgVg;enH ³ edIm,I[gayRsYlsg;rgVg;enH eyIgrkp©itrbs;rgVg; p©it I 3 2 2 , 1 2 5 b¤ I 12 , 3
QS
3 52 3 52 82 82
64 64 8 2
dUcenH RbEvgGgát;RTUg QS 8 2 ÉktaRbEvg 4.
2 , 5
bgðajfa RtIekaN KLM CaRtIekaNsm)at ³ eyIgman K 2 , 8 , L10 , 11 nig M 5 , 0 L10 , 11
11
R
1 I , 3 2
8
O
2
K 2 , 8
3 , 1
140
5
M 5 , 0
10
-KNnaRbEvg KL ³ KL
naM[ brimaRténRtIekaN ABC KW ³ P AB AC BC
10 22 11 82
29 2 29 145
8 2 32
3 29 145
ÉktaRbEvg
64 9 73
dUcenH
-KNnaRbEvg KM ³ KM
5 22 0 82 2 32 8
-bgðajfa ABC CaRtIekaNEkg eday AB AC 29 2 29 2
ÉktaRbEvg
9 64 73
C 7 , 3
7
3
2
3
AC
AB AC 2 29 2 29 29 2
dUcenH KNna)an S
ÉktaRbEvg
4 29 2 29
BC
6.
7 32 3 72 102 42
ABC
2
29
ÉktaépÞRkLa ÉktaépÞRkLa
2
y 22 5 y 22 25
ÉktaRbEvg
7 52 3 22 122 12
144 1 145
2
rktémøén y EdleFVI[cm¶ayénBIcMNucesµI 5 ³ eyIg)an 4 4 y 2 5
>
100 16 116
2
S ABC
5
5 32 2 72 2 2 2 5
4 25 29
2
-KNnaépÞRkLaénRtIekaN ABC ³ eday ABC CaRtIekaNEkgRtg; A eyIg)an
-KNnabrimaRténRtIekaN ABC ³ AB
2
2
A3 , 7
>
2
2
B5 , 2
O
2
ehIy BC 145 145 enaHeyIg)an AB AC BC tamRTWsþIbTBItaK½r AB AC BC enaH ABC CaRtIekaNEkgRtg; A dUcenH bgðaj)anfa ABC CaRtIekaNEkg
eyIgman A3 , 7 , B5 , 2 nig C 7 , 3 7
2
2
29 4 29 145
edayRtIekaN KLM man KL KM dUcenH KLM CaRtIekaNsm)at 5.
ÉktaRbEvg
P 3 29 145
naM[
y 2 25 y 2 5 y 2 5 y 5 2 y 7 y 2 5 y 5 2 y 3
dUcenH témørk)anKW
ÉktaRbEvg 141
y 7 , y 3
.
7.
rkcMNuc C x , 1 kñúgkaRdg;TI 1 ³ eyIgman A1 , 1 nig B4 , 7 ehIyman AB
4 12 7 12 4 12 7 12
32 6 2 9 36 45 AC
eday
x 12 1 12
AB AC
x 12
x> R)ab;RbePTénRtIekaNenH ³ edayRtIekaN ABC enHman AC BC 10 ÉktaRbEvg dUcKña dUcenH ABC CaRtIekaNsm)aTmankMBUl A . 9. eyIgmankMBUlctuekaNtagdUcxageRkam ³
enaHeyIg)an ³
C 1 , 4
x 12 45 x 12 45
D5 , 4
x 1 45 x 3 5 1
O
cMeBaH x 3 5 1 0 minyk eRBaH C x , 1 enAkñúgkaRdg;TI 1 enaH x 0 dUcenH cMNcu C 3 5 1 , 1 . 8.
A 1 , 2
k> -KNnaRbEvgRCug ³ É>RbEvg AC 1 1 4 2 6 6 É>RbEvg CD 5 1 4 4 6 6 É>RbEvg BD 5 5 4 2 6 6 É>RbEvg dUcenH RbEvgRCugKNna)an ¬ÉktaRbEvg¦ AB 6 / AC 6 / CD 6 / BD 6 -KNnaRbEvgGgát;RTUg ³ AB
k> KNnaRbEvgRCugTaMgbIénRtIekaN ABC ³ eyIgtagkMBUlTaMgbIénRtIekaNenaHeday ³ A 3 , 2 , B9 , 2 nig C 3 , 10 eyIg)an AB
AC
ÉktaRbEvg
3 32 10 22
62 82 100 10 BC
5 12 2 22 2
9 32 2 22
12 2 12
B5 , 2
ÉktaRbEvg
3 92 10 22 62 82 100 10 ÉktaRbEvg
62 6
2
2
2
2
2
2
2
2
AD
5 12 4 22
BC
1 52 4 22 62 62 6
62 62 6 2 2
dUcenH RbEvgGgát;RTUgKNna)anKW AD 6 2 ÉktaRbEvg / BC 6 2 ÉktaRbEvg x> snñdi æancMeBaHctuekaNenH ³ edayRbEvgRCug AB AC BD CD 6 nigRbEvgGgát;RTUg AD BC 6 2 dUcenH ctuekaNenHKWCakaery:agR)akd .
dUcenH KNna)anRbEvgRCugTaMbIénRtIekaNKW ÉktaRbEvg AC 10 ÉktaRbEvg BC 10 ÉktaRbEvg AB 12
142
10.
eyIgtagkUGredaenénkMBUlrbs;ctuekaNeday³ A2 , 1 , B6 , 4 , C 2 , 9 nig D 2 , 4
11.
C 2 , 9
etIctuekaNenH manragCaGVI ? eyIgtagkUGredaenénkMBUlrbs;ctuekaNeday³ A 2 , 3 , B 3 , 1 , C 5 , 1 nig D4 , 3
B6 , 4
B 3 , 1
4
D 2 , 4
2
2
2
O
2 2
O
2
4
6
BC CD
AB
6 22 4 12 4 2 52 41 2 22 4 12 42 52 41 2 62 9 42 42 52 41 2 22 4 92 42 52 41
BD AC BD AD BD
-KNnaRbEvgGgát;RTUgénctuekaN ³ 2 22 9 12 10 2 10 2 62 4 42 82 8
dUcenH KNna)anRbEvgGgát;RTUgKW ³ AC 10 , BD 8 ¬ÉktaRbEvg¦ x> R)ab;RbePTénctuekaNenH ³ edayRbEvgRCug AB AD BC CD 41 RbEvgGgát;RTUg AC BD tamRTwsþI ³ ctuekaNEdlmanRCugTaMgbYnesµIKña ehIyGgát;RTUgminesµKI ña vaCactuekaNesµI . dUcenH ctuekaNenHKWCa ctuekaNesµI .
C 5 , 1
3 22 1 32 12 4 2 17 4 52 3 12 12 4 2 17 5 22 1 32 7 2 2 2 53 4 32 3 12 7 2 2 2 53 4 22 3 32 6 2 6 2 6 2 5 32 1 12 82 22 2 17
edayctuekaNenHman ³ -RbEvgRCug AB BD 17 , AC BD 53 -RbEvgGgát;RTUg AD BD tamRTwsþI ³ ctuekaNEdlmanRCugesµKI ñaBIr² ehIy manGgát;RTUgminesµKI ña vaCaRbelLÚRkam . dUcenH ctuekaNenHmanragCa RbelLÚRkam .
AB 41 , AD 41 , BC 41 , CD 41
BD
eyIgKNnaRbEvgRCug nigRbEvgGgát;RTUg ³
dUcenH KNna)anRbEvgRCug ¬ÉktaRbEvg¦KW
AC
4
k> -KNnaRbEvgRCug ³ AD
2
A 2 , 3
A2 , 1
AB
D4 , 3
4
6
KNnakUGredaencMNuckNþalén AB rYcedA kñúgtRmúyGrtUNrem tamkrNInImYy² ³ k> A5 , 8 nig B5 , 12 tag I CacMNuckNþalén A nig B eyIg)an ³ 5 5 8 12 I , b¤ I 5 , 10 2 2 12.
dUcenH cMNcu kNþal A nig B KW I 5 , 10 . 143
-edAcMNuckñúgtRmúyGetUNrem ³
x> eyIgman A 8 , 6 nig B4 , 2 tamrUbmnþ M x 2 x , y 2 y
B5 , 12
A
I 5 , 10
10
naM[
A5 , 8
8
B
A
8 4 6 2 M , 2 2
B
b¤ M 2 , 2
dUcenH kUGredaencMNuckNþalKW
M 2 , 2
6
K> eyIgman A3t 5 , 7 nig Bt 7 , 7 tamrUbmnþ M x 2 x , y 2 y naM[ M 3t 52 t 7 , 72 7 b¤ M 2t 6 , 0
4
A
2
O
2
6
4
dUcenH cMNcu kNþalKW
x> A 5 , 2 nig B 5 , 14 tag J CacMNuckNþalén A nig B eyIg)an ³ 5 5 2 14 J , b¤ J 5 , 8 2 2 dUcenH cMNcu kNþal A nig B KW J 5 , 8 -edAcMNuckñúgtRmúyGetUNrem ³ B 5 , 14
A
8
6
4
O
KNnakUGredaencMNuckNþalén AB³ k> eyIgman A 3 , 5 nig B7 , 5 tamrUbmnþ M x 2 x , y 2 y naM[
3 7 55 M , 2 2
A
B
-KNnaRbEvg AM nig BM ³ AM 1 3 3 1 4 2 4 2 2 2
13.
B
B
dUcenH kUGredaencMNuckNþalKW M 1 , 3 .
2
A
M 2t 6 , 0
14.
10
6 4 2
B
X> eyIgman Aa , b nig Ba , c tamrUbmnþ M x 2 x , y 2 y a a b c b¤ b c naM [ M , M a , . 2 2 2 dUcenH cMNcu kNþalKW M a , b 2 c
12
A 5 , 2
A
-rkkUGredaenéncMNuckNþal M ³ eyIgman A 3 , 1 nig B5 , 7 naM[ M 32 5 , 12 7 b¤ M 1 , 3
14
J 5 , 8
B
A
2
2
BM 1 5 3 7 2
2
42 42
dUcenH RbEvgKNna)anKW ³ AM 2 2 ÉktaRbEvg BM 2 2 ÉktaRbEvg
B
b¤ M 2 , 5
dUcenH kUGredaencMNuckNþalKW M 2 , 5 144
2 2
KNnakUGredaenéncMNuc B ³ tag Bx , y CacMNucEdlRtUvrk eyIgmancMNuc A 2 , 2 nigcMNuc M 1 , 2 eday M CacMNuckNþalénGgát; AB
15.
B
eyIg)an ³ naM[
KNnakUGredaencMNuckNþalGgát;RTUgTaMgBIr ³ tagkMBUlTaMgbYnenaHénRbelLÚRkameday A1 , 1 , B2 , 3 , C 3 , 1 nig D 2 , 5
17.
B
2 xB 1 2 2 y B 2 2 xB 2 2 y B 4 2
enaH
2
2 x B 2 2 y B 4
b¤
C 3 , 1
4
.
-tag M CacMNuckNþalénGgát;RTUg AD naM[ M 1 2 2 , 1 2 5 b¤ M 12 , 2
naM[
T
enaH b¤
dUcenH cMNcu kNþal AD KW M 12 , 2 -tag N CacMNuckNþalénGgát;RTUg BC naM[ N 2 2 3 , 12 3 b¤ N 12 , 2
xT 3 4 yT 8 12
xT 7 yT 4
dUcenH cMNcu kNþal BC KW N 12 , 2
dUcenH kUGredaencMNuc T 7 , 4
.
-edaycMNuc M 12 , 2 dUcnwg N 12 , 2
-sg;rgVg;enH ³
dUcenH eyIgsnñidæan)anfa cMNucTaMgBIrRtYtsIuKña V 3 , 8
F 2 , 6
T 7 , 4
KNnakUGredaenéncMNuckNþal A , B , C , D ³ eyIgmankMBUlénctuekaN E5 , 3 ; F 5 , 5 ; G 3 , 3 nig H 1 , 5 ³ eday A , B , C , D CacMNuckNþalerogKñaén EF ; FG , GH nig EH eyIg)an ³ 55 35 A , b£ A5 , 1 2 2
18.
8
6
4
2
6
B2 , 3
D 2 , 5
KNnakUGredaenéncMNuc T ³ tag T x , y CacMNucEdlRtUvrk eyIgmancMNuc V 3 , 8 nigp©it F 2 , 6 eday F CacMNuckNþalénGgát; TV eyIg)an ³
2
16.
xT 3 2 2 yT 8 6 2 xT 4 3 yT 12 8
2
O
xB 4 y B 6
dUcenH kUGredaencMNuc B4 , 6
T
2
4
A1 , 1
4
2
O
2
145
b¤ B1 , 4 b¤ C 2 , 1 b¤ D2 , 4 dUcenH kUGredaenKNna)anKW ³ cMNuckNþal EF KW A5 , 1 cMNuckNþal FG KW B1 , 4 cMNuckNþal GH KW C 2 , 1 cMNuckNþal EH KW D2 , 4 53 53 B , 2 2 3 1 3 5 C , 2 2 5 1 3 5 D , 2 2
Rsayfa ctuekaNEdlkMNt;)an CactuekaNEkg eyIgman A3 , 2 ; B9 , 2 ; C9 , 6 ; D3 , 6 tag K , L , M , N CacMNuckNþalerogKñaén RCug AB , BC , CD , AD
19.
8 D3 , 6
6
4
N 3 , 4
-bBa¢ak;BIragctuekaN ABCD ³ O
H 1 , 5
4
4
2
O
4
A5 , 1
4
B1 , 4
KL
F 5 , 5
eyIgKNnaRbEvgRCugénctuekaN ABCD ³ AB AD CD BC
B9 , 2
4
8
6
b¤
K 6 , 2
b¤
L9 , 4
b¤
M 6 , 6
b¤ N 3 , 4 eyIgKNnaRbEvgRCugénctuekaN KLMN ³
2
G 3 , 3
K 6 , 2
39 2 2 K , 2 2 99 26 L , 2 2 93 66 M , 2 2 33 2 6 N , 2 2
E 5 , 3
2
2
L9 , 4
A3 , 2
2
enaHeyIg)an ³
D2 , 4
C 2 , 1
2
C 9 , 6
M 6 , 6
LM MN
1 52 4 12 42 32 5 2 52 4 12 32 52 34 2 22 4 12 4 2 32 5 2 12 1 42 32 52 34
KN
9 62 4 22 32 2 2 13 6 92 6 42 32 2 2 13 3 62 4 62 32 22 13 3 62 4 22 32 2 2 13
eday ctuekaN KLMN manRbEvgRCug ³ KL LM MN KN 13 dUcKña tamRTwsþI ³ ctuekaNEdlmanRCugTaMgbYnesµI²Kña enaHvaKWCactuekaNesµI dUcenH ctuekaNEdlkMNt;)anBIcMNuckNþal énRCugctuekaNEkgvaCa ctuekaNesµI
edayRbEvg AB CD 5 nig AD BC 34 tamRTwsþI ³ ctuekaNEdlmanRCugesµIKñaBIr² vaCa RbelLÚRkam dUcenH ctuekaN ABCD CaRbelLÚRkam . 146
bgðajfa ABCD CaRbelLÚRkam ³ eyIgman kMBUlénctuekaN P 3 , 2 ; Q1 , 5 R 3 , 1 nig S 5 , 4 Edlman A , B , C , D CacMNuckNþalerogKñaRCug PQ ; QR ; RS nig SP enaHeyIg)ankUGredayen ³ 3 1 2 5 b¤ A 2 , 3 A , 2 2 2 1 3 5 1 b¤ B1 , 3 B , 2 2
20.
KNnakUGredaenéncMNuc A nig B ³ eyIgman P x , y nig P x , y
21.
1
1
//
A
O
A
B
1
P 3 , 2
R3 , 1
S 5 , 4
AD
1 22 3 3
2
BC
4 12 3 3 1 42 3 3
2
CD
2
9
2
2
A
2
9
81 117 4 2
9
81 117 4 2
9
9 3 5 4 2
2
B
enaH
9 3 5 4 2
2
2
B
x1 xB x2 x A x2 x x B 2 x2 2 x 1 B 2 2 4 y1 y B y2 y y B 2 y2 y y A y2 2 1 B 2 2 4 x1 2 x2 x 4 xB x1 xB 2 x2 B 3 3 4 y y y 2 y y 2 B 1 B y 1 2 y2 B 3
6 3 C 4 , 2
4
2
1
A
A
4 D1 , 3
B
2
2
B
B
1 22 3 3
P2 x2 , y2
1
B1 , 3
AB
1
A
2
//
1 2
2
B ?
tag Ax , y nig Bx , y CacMNucRtUvrk eday A nig B Eck P P CabIEpñkesµIKña eyIg)an P A AB BP naM;[ A CacMNuckNþalén P B eyIg)an x x 2 x , y y 2 y 1 ehIy B CacMNuckNþalén AP eyIg)an x x 2 x , y y 2 y 2 yk 1 CMnYskñúg 2 enaHeyIg)an ³
4
//
O
Q 1 , 5
2
2
A ?
1
3 A 2 , 2
2
b¤ b¤ D1 , 3 -eyIgKNnaRbEvgRCugénctuekaN ABCD ³ 4
2
P1 x1 , y1
3 C 4 , 2
3 5 1 4 C , 2 2 53 42 D , 2 2
1
b¤
yk 3 CMnYskñúg 1 ³ x 2 x2 x1 1 4 x 2 x2 2 x1 x2 3 1 xA 2 3 2 3 y 2 y2 y 1 4 x 2 x2 2 y1 y 2 y 1 3 1 A 2 3 2 3
eday AB CD 3 25 , AD BC 117 2 ctuekaNenHmanRCugesµIKñaBIr vaCaRbelLÚRkam dUcenH ctuekaN ABCD CaRbelLÚRkam .
dUcenH kUGredaen A 2 x 3 x
2 y1 y 2 3 x 2 x2 y1 2 y 2 B 1 , 3 3 1
kUGredaen 147
2
,
bgðajfa P P P P P P ³ eyIgman P x , y / P x , y nig x x y y P , naM[eyIg)anRbEvg ³ 2 2
22.
1 m
1
1
m
1
2
2
1
1
rkkUGredaenénp©itrgVg;tageday I ³ eyIgman A1 , 0 nig B3 , 2 sg;rgVg;EdlmanGgát;p©it AB
23.
1 2
2
2
2
2
m
x x2 y y2 P1 Pm 1 x1 1 y1 2 2 2
y
2
x x2 2 x1 y y 2 2 y1 1 1 2 2 2
2
x
x x2 y y2 Pm P2 x2 1 y2 1 2 2 2
2
x2 x1 2 y 2 y1 2
eyIgKNnaplbUk P P
1 m
P1 Pm Pm P2
P1 Pm Pm P2
1 2 1 2
³
Pm P2
x2 x1 2 y2 y1 2 x2 x1
y 2 y1
AI
Et P P x x y y dUcenH bgðaj)anfa P P P P P P . -GñkGaceFVItamrebobmü:ageTotKW ³ eday P x 2 x , y 2 y KWCacMNuckNþal énBIrcMNuc P x , y nig P x , y naM[ P P P P 12 P P enaH P P P P 12 P P 12 P P P P 2
1
eyIg)an
2
1
2
1 m
2
m
1
2
1 2
1
2
1 m
m 2
1 2
1 m
m 2
1 2
dUcenH bgðaj)anfa P P
1 m
2
1 2
1 2 1 y 2 4 1 2 1 y 2 4
y 1
2
7 2
dUcenH kMNt;)an y 1 27 , y 1 27 ehIy M manTItaMgBIrenAelIrgVg;KW
1 2
Pm P2 P1 P2
11 2
4 7 1 y 2
2
1
2 12 1 02
2
1 y 2 2 1
m
1
x 5 7 M 1 ,1 2 2
2
2
2
1
2
2
x2 x1 2 y2 y1 2
1 2
2
edayp©it I CacMNuckNþalénGgát;p©it AB eyIg)an I 1 2 3 , 0 2 2 b¤ I 2 , 1 dUcenH KNna)ankUGredaenp©it I 2 , 1 . x> kMNt; y edIm,I[ M 52 , y zitenAelIrgVg; edIm,I[ M zitenAelIrgVg; luHRtaEt MI AI Et MI 2 52 1 y 14 1 y
1 x2 x1 2 1 y 2 y1 2 4 4 1 2
A1
y
2
O
B3 , 2
I 2 , 1
2
2 x x1 x2 2 y 2 y1 y 2 2 2 2
1
x2 x1 2 y 2 y1 2
1 2
2
1 x2 x1 2 1 y 2 y1 2 4 4
5 7 M 2 ,1 2 2
.
5 5 7 7 , M , 1 M , 1 2 2 2 2
148
.
១០ 1.
k> a 2 , 4 , 1
x>
k> 3 , 1 , 5 , 4 K> 2 , 1 , 4 , 4
x> 3 , 0 , 3 , 6 X> 1 , 4 , 4 , 2
k> 6,1 , 6 , 2 K> 2,0 , 1 , 8 g> 2.5, 1 , 0.5 ,1
x> 3, 4 , 9 , 2 X> 4.2, 4 , 2.2 ,6 c> 0,5 , 3 , 2.5
a
2 , 1 , 5 3
2.
3.
4.
0.8% 2000
80%
2008 5.
1992 1992
15.5%
6.
38%
2007 m
P
k> m 1 , P0 , 2 X> m 53 , P 1 , 4
x> m 1 , P1 , 0 g> m 0 , P2 , 1
K> m 2 , P3 , 1 c> m 32 , P0 , 3
k> 2x 3 y 11
x> 4x 8 y 1 0
K> y 23 x 1
7.
8.
k> y x 4 , x y 5 0 K> 3x y 4 0 , y 2 3x 1
x> y 2x 3 , x 2 y 1 0 X> 2 y 5x 6 , 5x 2 y 1 0
9.
k> y 3x 1 x> y 2 x 5 K> y 5x 4
1 , y x 3 , x 2y 6 0 , x 5y 1 0 149
10.
k> y 3x 4 x> 3x 2 y 5 0 K> 2x 2 y 9 0 X> x 5 y 6 0
5 , 1 2 , 4 5 , 0 3
0 , 0
11.
k> y 12 x 4 x> x y 5 0 K> 8x 3y 1 0 X> y x 6
5 , 0 0 , 0 1 , 4 2 4 , 3 A 3 , 5
d1
12. .
m
d2: y m 1 x 2 d1 ឬ
d2
2 d2
m
d1
m
6 , 0
B
A
B
yy
A 0 , 6 , B 3 , 0
13.
AB
AC
D D
AB
C
B
AC
A 6 , 2
14.
AB C 2 , 6 C2
B 2 , 2
CAB
AB
C
M 5 , 3
C2
A 6 , 0 , B 6 , 0
15.
C 6 , 0
C 3 , 9
ABC
ABC 16.
a , 6
2 , a
4x 2 y b
a
b
17.
1973 47%
1990
38% x
3 150
x 0
1970
១០ 1.
sg;bnÞat;kat;tammYycMNuc nigmanemKuNR)ab;Tis k> eyIgman a 2 nigcMNuctageday A 4 , 1 eyIgsg;cMNuc A 4 , 1 rYcKUs 1 ÉktaecjBI A RsbG½kS ox tamTisedAviC¢man nigKUs 2 Ékta RsbnwgG½kS oy tamTisedAviC¢man enaHeyIg)an cMNuc B rYcKUsbnÞat;kat;tamcMNuc A nig B ³
2.
rkemKuNR)ab;TisénbnÞat;kat;tamBIrcMNuc ³ emKuNR)ab;TisénbnÞat;kat;tamBIrcMNuc y y Ax , y nig Bx , y KW a x x *
A
A
B
B
A
B
A
B
k> manBIrcMNuctageday A3 , 1 , B5 , 4 emKuNR)ab;TisénbnÞat;Edlkat;tam A nig B kMNt;eday a yx xy 54 31 32 B
A
B
A
dUcenH emKuNR)ab;TisénbnÞat;KW a 32 . x> manBIrcMNcu tageday C3 , 0 , D 3 , 6 emKuNR)ab;TisénbnÞat;Edlkat;tam C nig D kMNt;eday a yx xy 63 30 66 1
B A
x> eyIgman nigcMNuctageday eyIgsg;cMNuc A1 , 5 rYcKUs 3 ÉktaecjBI A RsbG½kS ox tamTisedAviC¢man nigKUs 2 Ékta RsbG½kS oy tamTisedAGviCm¢ an enaHeyIg)an cMNuc B rYcKUsbnÞat;kat;tamcMNuc A nig B ³
A1 , 5
2 a 3
D
C
D
C
dUcenH emKuNR)ab;TisénbnÞat;KW a 1 . K> BIrcMNuctageday I 2 , 1 , J 4 , 4 emKuNR)ab;TisénbnÞat;Edlkat;tam I nig J kMNt;eday a yx xy 44 12 23 23 J
I
J
I
dUcenH emKuNR)ab;TisénbnÞat;KW a 23 .
X> manBIrcMNuctageday M 1 , 4 , N 4 , 2 emKuNR)ab;TisénbnÞat;Edlkat;tam M nig M kMNt;eday a yx xy 4214 36 2
A
B
N
M
N
M
dUcenH emKuNR)ab;TisénbnÞat;KW a 2 . 151
3.
rksmIkarénbnÞat;Edlkat;tamBIrcMNcu ³ k>manBIrcMNuctageday A 6 ,1 , B6 , 2 naM[ emKuNR)ab;TisénbnÞat; AB KW a
K>manBIrcMNuctageday A2 , 0 , B1 , 8 tag M x , y CacMNucenAlIbnÞat; AB naM[ emKuNR)ab;TisénbnÞat; AM nig AB esµIKña eyIg)an yx xy yx xy
2 1 3 1 6 6 12 4
tag M x , y CacMNucenAlIbnÞat; AB naM[ emKuNR)ab;TisénbnÞat; AM nig AB esµIKña eyIg)an x y 16 14
B
A
A
B
A
y 0 80 x2 1 2 y 8 x 2 1 y 8x 2 y 8 x 16
1 x 6 4 1 3 y x 1 4 2 1 1 y x 4 2
y 1
dUcenH smIkarbnÞat;rk)anKW y 8x 16 . X>BIrcMNuctageday A 4.2, 4 , B 2.2 , 6 tag M x , y CacMNucenAlIbnÞat; AB naM[ emKuNR)ab;TisénbnÞat; AM nig AB esµIKña eyIg)an yx xy yx xy
dUcenH smIkarbnÞat;rk)anKW y 14 x 12 . x>manBIrcMNuctageday A3 , 4 , B 9 , 2 naM[ emKuNR)ab;TisénbnÞat; AB KW a
A
A
B
A
A
B
A
y 4 6 4 x 4.2 2.2 4.2 y 4 10 x 4. 2 2 y 4 5 x 4.2 y 5 x 21 4 y 5 x 17
2 4 6 1 9 3 12 2
tag M x , y CacMNucenAlIbnÞat; AB naM[ emKuNR)ab;TisénbnÞat; AM nig AB esµIKña eyIg)an y x 34 12
dUcenH smIkarbnÞat;rk)anKW y 5x 17 .
1 y 4 x 3 2 1 3 y x 4 2 2 1 5 y x 2 2
g> manBIrcMNuctageday A2.5, 1 , B0.5 ,1 tag M x , y CacMNucenAlIbnÞat; AB naM[ emKuNR)ab;TisénbnÞat; AM nig AB esµIKña eyIg)an yx xy yx xy
dUcenH smIkarbnÞat;rk)anKW y 12 x 52 .
A
B
A
A
B
A
y 1 1 1 x 2 .5 0 .5 2 .5 y 1 2 x 2 .5 2 y 1 x 2 .5
smÁal; ³ bnÞat;BIrRtYtelIKña enaHbnÞat;TaMgBIr manemKuNR)ab;TisesµIKña . 152
y x 2.5 1 y x 1 .5
5.
dUcenH smIkarbnÞat;rk)anKW y x 1.5 . c> manBIrcMNcu tageday A0 , 5 , B3 , 2.5 tag M x , y CacMNucenAlIbnÞat; AB naM[ emKuNR)ab;TisénbnÞat; AM nig AB esµIKña eyIg)an yx xy yx xy A
B
A
A
B
A
y 5 2.5 5 x0 30 y 5 7.5 x 3 y 4 2.5 x y 2.5 x 4
b¤
dUcenH smIkarbnÞat;rk)anKW y 2.5x 4 . 4. -sresrsmIkarTak;Tg;ng w karfycuHcMnnY ksikr ³ smIkarEdlRtUvsresrmanrag y ax b ÷eday cMnYnksikrfycuHCalMdab;eTAtamGRta 0.8% kñúgmYyqñaM enaHeyIg)an a 0.8% ÷ehIyenAqñaM 2000 ksikrmancMnYn 80% ebItag x CaqñaM nig y CacMnYnksikr enaHeyIg)an cMeBaH x 2000 naM[ y 80% -smIkar y ax b GacsresrCa ³ 80% 0.8% 2000 b 0.8 0.008 2000 b 0.8 16 b b 16.8
rksmIkarEdlTak;ngw GñkCk;)arI nigcMnYnqñaM ³ tag y ax b CasmIkarbnÞat;RtUvrk Edl x CacMnYnqñaM nig y CacMnYnGñkCk;)arI ÷GñkCk;)arImancMnYn 38% enAqñaM 1992 ³ naM[ y 38% nig x 1992 eyIg)an y ax b kat;cMNuc A1992 , 0.38 ÷GñkCk;)arImancMnYn 15.5% enAqñaM 2007 ³ naM[ y 15.5% nig x 2007 eyIg)an y ax b kat;cMNuc B2007 , 0.155 tag M x , y CacMNucenAlIbnÞat; y ax b naM[ emKuNR)ab;TisénbnÞat; AM nig AB esµIKña eyIg)an yx xy yx xy A
B
A
A
B
A
y 0.38 0.155 0.38 x 1992 2007 1992 y 0.38 0.225 x 1992 15 y 0.38 0.015 x 1992 y 0.38 0.015 x 1992 y 0.015x 29.88 0.38 y 0.015x 30.26
dUcenH smIkarrk)anKW y 0.015x 30.26 6.
dUcenH sresr)ansmIkar y 0.008x 16.8 . -eRbIsmIkar):an;sµancMnnY ksikrenAqñaM 2008 ³ enAqñaM 2008 mann½yfa x 2008 enaHeyIg)an ³ y 0.008 2008 16.8 0.736 b¤ y 73.6% dUcenH enAqñaM 2008 ksikrmancMnYn 73.6% .
sresrsmIkarbnÞat; sg;bnÞat; rYcrkcMNucRbsBV³ k> -eyIgman m 1 , P0 , 2 naM[ a m 1 , x 0 , y 2 bnÞat;RtUvrkmanrag y ax b eyIg)an 2 1 0 b naM[ b 2 dUcenH smIkarbnÞat;RtUvrkKW y x 2 . -sg;bnÞat; y x 2 tamrebobemIl m 1 , P0 , 2
153
¬elOnCag eRbItaragtémøelx¦
-sg;bnÞat; y 2x 5 tamrebobemIl emKuNR)ab;Tis m 2 , P3 , 1
y x2
y 2 x 5
tamRkabcMNucRbsBVrvagbnÞat; y x 2 nig ³ -G½kSGab;sIusKW 2 , 0 -G½kSGredaenKW P0 , 2 .
x> -eyIgman m 1 , P1 , 0 naM[ a m 1 , x 1 , y 0 bnÞat;RtUvrkmanrag y ax b eyIg)an 0 11 b naM[ b 1 dUcenH smIkarbnÞat;RtUvrkKW y x 1 . -sg;bnÞat; y x 1 tamrebobemIl emKuNR)ab;Tis m 1 , P1 , 0
tamRkabcMNucRbsBVrvagbnÞat; y 2x 5 nig ³ -G½kSGab;sIusKW 2.5 , 0 -G½kSGredaenKW 0 , 5 . X> -eyIgman m 53 , P 1 , 4 naM[ a m 53 , x 1 , y 4 bnÞat;RtUvrkmanrag y ax b eyIg)an 4 53 1 b naM[ b 175
y x 1
dUcenH smIkarbnÞat;RtUvrkKW
y
3 17 x 5 5
.
-sg;bnÞat; y 53 x 175 tamrebobemIl emKuNR)ab;Tis m 53 , P 1 , 4
tamRkabcMNucRbsBVrvagbnÞat; y x 1 nig ³ -G½kSGab;sIusKW P1 , 0 -G½kSGredaenKW 0 , 1 .
y
3 17 x 5 5
K> -eyIgman m 2 , P3 , 1 naM[ a m 2 , x 3 , y 1 bnÞat;RtUvrkmanrag y ax b eyIg)an 1 2 3 b naM[ b 5 dUcenH smIkarbnÞat;RtUvrkKW y 2x 5 .
tamRkabcMNucRbsBVrvagbnÞat; y x 1 nig ³ -G½kSGab;sIusKW 17 / 3 , 0 -G½kSGredaenKW 0 , 17 / 5 . 154
g> -eyIgman m 0 , P2 , 1 edayemKuNR)ab;Tis m 0 nig y 1 enaHbnÞat;EdlRtUvrk KWRsbnwgG½kSGab;sIus ehIymansmIkar y 1 dUcenH smIkarbnÞat;RtUvrkKW y 1 . -sg;bnÞat; y 1 RsbnwgG½kSGab;suIs
7.
y 1
8.
tamRkabcMNucRbsBVrvagbnÞat; y 1 nig ³ -G½kSGab;sIusKW Kµan eRBaH y 1 RsbG½kS ox -G½kSGredaenKW 0 , 1 . c> -eyIgman m 32 , P0 , 3 naM[ a m 32 , x 0 , y 3 bnÞat;RtUvrkmanrag y ax b eyIg)an 3 32 0 b naM[ b 3 dUcenH smIkarbnÞat;RtUvrkKW
y
3 x3 2
.
-sg;bnÞat; y 32 x 3 RsbnwgG½kSGab;suIs 3 y x3 2
tamRkabcMNucRbsBVrvagbnÞat; y 32 x 3 nig ³ -G½kSGab;sIusKW 2 , 0 -G½kSGredaenKW 0 , 3 . 155
kñúgcMeNambnÞat;bI etIbnÞat;NaRsbKña ? k> 2x 3 y 11 b¤ y 23 x 113 x> 4x 8 y 1 0 b¤ y 12 x 18 K> y 23 x 1 edabnÞat; k> y 23 x 113 nig K> y 23 x 1 manemKuNR)ab;TisesµIKña naM[vaRsbKña dUcenH k> 2x 3y 11 Rsbnwg K> y 23 x 1 . rkemKuNR)ab;TisénbnÞat; etIvaRsbKña b¤eT ? k> y x 4 manemKuNR)ab;Tis a 1 x y5 0 manemKuNR)ab;Tsi a 1 edaybnÞat;TaMgBIrmanemKuNR)ab;TisesµIKña dUcenH KUsmIkarbnÞat;TaMgBIrRsbKña . x> y 2x 3 manemKuNR)ab;Tis a 2 1 x 2 y 1 0 manemKuNR)ab;Tis a 2 edaybnÞat;TaMgBIrmanemKuNR)ab;TisminesµIKña dUcenH KUsmIkarbnÞat;TaMgBIrminRsbKñaeT . K> 3x y 4 0 manemKuNR)ab;Tis a 3 y 2 3x 1 manemKuNR)ab;Tis a 3 edaybnÞat;TaMgBIrmanemKuNR)ab;TisesµIKña dUcenH KUsmIkarbnÞat;TaMgBIrRsbKña . X> 2 y 5x 6 manemKuNR)ab;Tis a 5 / 2 5x 2 y 1 0 manemKuNR)ab;Tis a 5 / 2 edaybnÞat;TaMgBIrmanemKuNR)ab;TisminesµIKña dUcenH KUsmIkarbnÞat;TaMgBIrminRsbKñaeT .
9.
etIKUsmIkarNaxøHEdlmanbnÞat;EkgKña ? k> y 3x 1 manemKuNR)ab;Tis a 3 1 1 y x manemKuNR)ab;Tis a 3 3 edaybnÞat;TaMgBIrman a a 3 13 1
ehIy y ax b kat;tamcMNuc 2 , 4 enaH x 2 , y 4 eyIg)an 4 32 2 b naM[ b 7
dUcenH KUénsmIkarenH manbnÞat;EkgKña . x> y 2 x 5 manemKuNR)ab;Tis a 2 1 x 2 y 6 0 manemKuNR)ab;Tis a 2 edaybnÞat;TaMgBIrman a a 2 12 1 1
K> eyIgman 2x 2 y 9 0 nigcMNuc 53 , 0 bnÞat;RtUvrkmanrag y ax b EdlRsbnwg bnÞat; 2x 2 y 9 0 enaH a 1 ehIy y ax b kat;tamcMNuc 53 , 0 enaH x 53 , y 0 eyIg)an 0 1 53 b naM[ b 53
dUcenH smIkarbnÞat;RtUvrkKW y 32 x 7 .
dUcenH KUénsmIkarenH manbnÞat;minEkgKñaeT . K> y 5x 4 manemKuNR)ab;Tis a 5 1 x 5 y 1 0 manemKuNR)ab;Tis a 5 edaybnÞat;TaMgBIrman a a 5 15 1
dUcenH smIkarbnÞat;RtUvrkKW y x 53 . X> eyIgman x 5 y 6 0 nigcMNuc 0 , 0 bnÞat;RtUvrkmanrag y ax eRBaHvakat; 0 , 0 ehIyRsbnwgbnÞat; x 5 y 6 0 enaH a 15
dUcenH KUénsmIkarenH manbnÞat;EkgKña . kMNt;smIkarbnÞat;EdlRsbnwg ³ k> eyIgman y 3x 4 nigcMNuc 5 , 1 bnÞat;RtUvrkmanrag y ax b eday y ax b Rsbnwg y 3x 4 enaH a 3 ehIy y ax b kat;tamcMNuc 5 , 1 enaH x 5 , y 1 eyIg)an 1 3 5 b naM[ b 14 dUcenH smIkarbnÞat;RtUvrkKW y 3x 14 .
10.
dUcenH smIkarbnÞat;RtUvrkKW y 15 x . kMNt;smIkarbnÞat;EdlEkgnwg ³ k> eyIgman y 12 x 4 nigcMNuc 5 , 0 bnÞat;RtUvrkmanrag y ax b EdlEkgnwg bnÞat; y 12 x 4 naM[ a 12 1 b¤ a 2 ehIy y ax b kat;tamcMNuc 5 , 0 enaH x 5 , y 0 eyIg)an 0 2 5 b naM[ b 10 dUcenH smIkarbnÞat;RtUvrkKW y 2x 10 .
11.
x> eyIgman 3x 2 y 5 0 nigcMNuc 2 , 4 bnÞat;RtUvrkmanrag y ax b EdlRsbnwg bnÞat; 3x 2 y 5 0 enaH a 32 156
x> eyIgman x y 5 0 nigcMNuc 0 , 0 bnÞat;RtUvrkmanrag y ax eRBaHvakat; 0 , 0 ehIy y ax Ekgnwg x y 5 0 man a 1 naM[ a 1 1 b¤ a 1 dUcenH smIkarbnÞat;RtUvrkKW y x .
k> kMNt;smIkarénbnÞat; d ³ bnÞat;RtUvkMNt;manrag d : y ax b eday d kat;tamcMNuc A 3 , 5 naM[ x 3 , y 5 ehIy d manemKuNR)ab;Tis a 2 eyIg)an 5 2 3 b naM[ b 1 dUcenH smIkarbnÞat; d : y 2x 1 . -sg;bnÞat; d : y 2x 1 ³
12.
1
1
1
1
smÁal; ³ bnÞat;Edlkat;tamKl;tRmúy 0 , 0 CabnÞat;EdlmansmIkarrag y ax .
1
1
K> eyIgman 8x 3y 1 0 nigcMNuc bnÞat;RtUvrkmanrag y ax b EdlEkgnwgbnÞat; 8 8x 3 y 1 0 manemKuNR)ab;Tis a 3 8 3 naM[ a 3 1 b¤ a 8 ehIy y ax b kat;tamcMNuc 1 , 4 enaH x 1 , y 4 eyIg)an 4 83 1 b naM[ b 358 1 , 4
A 3 , 5
d1 : y 2 x 1
dUcenH smIkarbnÞat;RtUvrkKW y 83 x 358 . X> eyIgman y x 6 nigcMNuc 4 , 23 bnÞat;RtUvrkmanrag y ax b EdlEkgnwgbnÞat; y x 6 manemKuNR)ab;Tis a 1 naM[ a 1 1 b¤ a 1 ehIy y ax b kat;tamcMNuc 4 , 23 enaH x 4 , y 23 eyIg)an 23 1 4 b naM[ b 103
x> kMNt;témø m edIm,I[ d // d eyIgman d : y m 1x 2 nigbnÞat; d : y 2 x 1 > edIm,I[bnÞat;TaMgBIr d nig d RsbKñaluHRtaEt ³ m 1 2 naM[ m 1 dUcenH témøkMNt;)anKW m 1 . -etImantémø m EdleFVI[ d d b¤eT ? bnÞat; d d luHRtaEt m 1 2 1 b¤ 2m 2 1 naM[ m 32
dUcenH smIkarbnÞat;RtUvrkKW y x 103 .
dUcenH mantémø m 32 EdleFV[I d
2
1
2
1
2
1
2
2
157
1
1
2
d1
.
-sresrsmIkarkarénbnÞat;RtUvnwgtémø m nImYy² cMeBaH m 1 enaH d : y 2x 2 cMeBaH m 32 enaH d : y 12 x 2 -sg;bnÞat;TaMgenaH ¬emIlsmIkarbnÞat; rYcsg;Etmþg¦
13.
2
kñúgtRmútGrtUNrem eyIgmanbIcMNucKW ³ A0 , 6 , B 3 , 0 nig C 6 , 0 AB : y 2 x 6
2
d 2 : y
A0 , 6
1 D : y x3 2
1 x2 2
AC : y x 6
d1 : y 2 x 1
C 6 , 0
B 3 , 0
d 2 : y 2 x 2
D : y x 3
k> -rksmIkarénbnÞat; AB ³ tag M x , y CacMNucenAlIbnÞat; AB naM[ emKuNR)ab;TisénbnÞat; AM nig AB esµIKña eyIg)an yx xy yx xy
K> sresrsmIkarbnÞat;kat;tamcMNuc A nig B ³ eyIgman A 3 , 5 nig B6 , 0 bnÞat;RtUvrkmanrag AB : y ax b tag M x , y CacMNucenAlIbnÞat; AB naM[ emKuNR)ab;TisénbnÞat; AM nig AB esµIKña eyIg)an yx xy yx xy A
B
A
A
B
A
y 5 05 x 3 6 3 y 5 5 x3 9 5 y 5 x 3 9 5 5 y x 5 9 3 5 10 y x 9 3 5 10 y x x0 9 3 10 yy 0 , 3 5 10 AB : y x 9 3 10 yy 0 , 3
ebI
enaHbnÞat; Rtg;cMNuc
dUcenH bnÞat; nwgG½kS
B
A
A
B
A
y 6 06 x 0 30 y 6 2x y 2x 6
dUcenH smIkarbnÞat;
AB : y 2 x 6
.
- rksmIkarénbnÞat; AC ³ tag N x , y CacMNucenAlIbnÞat; AC naM[ emKuNR)ab;TisénbnÞat; AN nig AC esµIKña eyIg)an yx xy yx xy
CYbnwgG½kS
A
C
A
A
C
A
y6 06 x0 60 y 6 x y x 6
ehIyCYb
Rtg;cMNuc
A
dUcenH smIkarbnÞat;
. 158
AC : y x 6
.
x> rksmIkarénbnÞat; D ³ smIkarbnÞat; D RtUvrkmanrag D : y ax b eday D kat;tam C6 , 0 enaH x 6 , y 0 ehIy D EkgnwgbnÞat; AB : y 2x 6 naM[ a 2 1 enaH a 12 eyIg)an 0 12 6 b naM[ b 3
-rksmIkarénExSemdüaT½rrbs; AB ³ tag d : y ax b CasmIkarExSemdüaT½r AB naM[ d RtUvkat;tam I Edl I kNþal AB enaH I 6 2 2 , 2 2 2 b¤ I 4 , 0 ehIy d : y ax b RtUvEkgnwg AB : y x 4 naM[ a 1 1 enaH a 1 eyIg)an d : 0 1 4 b enaH b 4 dUcenH ExSemdüaT½rén AB KW d : y x 4 .
dUcenH bnÞat;rk)anKW D : y 12 x 3 . K> rksmIkarénbnÞat; D ³ smIkarbnÞat; D RtUvrkmanrag D : y ax b eday D kat;tamcMNuc B 3 , 0 naM[ x 3 , y 0 ehIy D RsbnwgbnÞat; AC : y x 6 naM[ a 1 eyIg)an 0 1 3 b enaH b 3 dUcenH bnÞat;rk)anKW D : y x 3 .
x> bgðajfa CAB CaRtIekaNsm)at ³ eyIgman C 2 , 6 , A6 , 2 nig B2 , 2 eyIgKNnargVas;RCugénRtIekaN CAB ³ AB BC
k-rksmIkarbnÞat;kat;tamcMNuc A nig B ³ eyIgmanBIrcMNuc A6 , 2 nig B2 , 2 tag M x , y CacMNucenAlIbnÞat; AB naM[ emKuNR)ab;TisénbnÞat; AM nig AB esµIKña eyIg)an yx xy yx xy B
A
A
B
A
16 64 80
2
2
2
2
2
2
2
2
y x4 AB : y x 4
16 16 32
K> rksmIkarénbnÞat; C ³ tag C : y ax b CabnÞat;RtUvrk eday C kat; C 2 , 6 enaH x 2 , y 6 ehIy C Rsbnwg AB : y x 4 enaH a 1 eyIg)an C : 6 1 2 b enaH b 8 dUcenH bnÞat;rk)anKW C : y x 8 . X> bgðajfa C kat;tam M 5 , 3 ³ ebIcMNuc M epÞógpÞat;smIkar C enaH C kat; M cMeBaH M 5 , 3 enaH 3 5 8 b¤ 3 3 Bit dUcenH bnÞat; C kat;tam M 5 , 3 .
y2 22 x6 26 y2 1 x6 y 2 x6
dUcenH smIkarbnÞat;
64 16 80
edayRtIekaN CAB man AC BC 80 dUcenH CAB CaRtIekaNsm)atkMBUl C .
14.
A
2 62 6 22 2 62 2 22 2 22 6 22
AC
.
2
159
2
k> rksmIkarénRCugRtIekaN ABC ³ eyIgman A 6 , 0 , B6 , 0 nig C 3 , 9 -rksmIkarénRCug AB ³ tag M x , y CacMNucenAlIbnÞat; AB naM[ emKuNR)ab;TisénbnÞat; AM nig AB esµIKña eyIg)an yx xy yx xy
x> rksmIkarénExSemdüaT½rrbs;RCugRtIekaN ABC -smIkarénExSemdüaT½rRCug AB ³ tag M : y ax b CaExSemdüaT½rRCug AB ³ eday M RtUvkatcMNuc I kNþalRCug AB naM[ I 62 6 , 0 2 0 b¤ I 0 , 0 ¬Kl;>¦ enaHsmIkarRtUvrkbþÚrmkCarag M : y ax ehIy M : y ax Ekgnwg AB: y 0 naM[ 0 ax enaH 0 x dUcenH smIkaremdüaT½r M : x 0 CaG½kS yy
15.
A
B
A
A
B
A
AB
AB
AB
AB
dUcenH smIkarénRCug AB: y 0 CaG½kS xx
AB
-smIkarénExSemdüaT½rRCug AC ³ tag M : y ax b CaExSemdüaT½rRCug AC ³ eday M RtUvkatcMNuc I kNþalRCug AC naM[ I 62 3 , 0 2 9 b¤ I 23 , 92 ehIy M Ekgnwg AC : y x 6 naM[ a 1 1 enaH a 1 eyIg)an 92 1 23 b enaH b 3 dUcenH smIkaremdüaT½r M : y x 3 . -smIkarénExSemdüaT½rRCug BC ³ tag M : y ax b CaExSemdüaT½rRCug BC ³ eday M RtUvkatcMNuc I kNþalRCug BC naM[ I 6 2 3 , 0 2 9 b¤ I 92 , 92 ehIy M Ekgnwg BC : y 3x 18 naM[ a 3 1 enaH a 1 / 3 eyIg)an 92 13 92 b enaH b 3
-rksmIkarénRCug AC ³ tag N x , y CacMNucenAlIbnÞat; AC naM[ emKuNR)ab;TisénbnÞat; AN nig AC esµIKña eyIg)an yx xy yx xy C
A
A
C
A
AC
AC
AC
-rksmIkarénRCug BC ³ tag K x , y CacMNucenAlIbnÞat; BC naM[ emKuNR)ab;TisénbnÞat; BK nig BC esµIKña eyIg)an yx xy yx xy B
B
C
B
AC
AC
dUcenH smIkarénRCug AC : y x 6 .
C
AC
AC
y0 90 x 6 3 6 y x6
B
AB
AB
y0 00 x 6 6 6 y0
A
AB
BC
BC
BC
BC
y0 90 x 6 36 y 3 x6 y 3 x 18
BC
BC
dUcenH smIkarénRCug BC : y 3x 18 .
dUcenH smIkaremdüaT½r M 160
BC
1 : y x3 3
.
kMNt;témø a nig b ³ eyIgmanbnÞat; 4x 2 y b eday cMNuc a , 6 enAelIbnÞat; 4x 2 y b eyIg)an 4a 2 6 b b¤ 4a 12 b 1 ehIy cMNuc 2 , a enAelIbnÞat; 4x 2 y b eyIg)an 4 2 2a b b¤ 2a 8 b 2 eyIgpÞwmtémø b énsmIkar 1 nig 2 eyIg)an³
16.
kMNt;smIkarEdlTak;TgnwgkareRbIfamBl BIeRbgeTAtamqñaM ³ tag y ax b CasmIkarEdlRtUvkMNt; Edl x CaqñaM nig y CafamBleRbgEdleRbIR)as; edayeKkMNt;yk x 0 RtUvnwgqñaM 1970 enaH qñaM 1973 kMNt;eday x 3 qñaM 1990 kMNt;eday x 20 eday enAqñaM 1973 famBleRbgeRbIR)as;KW 47% enAqñaM 1990 famBleRbgeRbIR)as;KW 38% naM[ x 3 RtUvnwgfamBleRbIR)as; y 47% x 20 RtUvnwgfamBleRbIR)as; y 38% edayemKuNR)ab;TisénbnÞat; CapleFobén bERmbRmYlén y nig bERmbRmYlén x eyIg)an a 38%20473 %
17.
4a 12 2a 8 4a 2a 8 12 2a 4 a 2
cMeBaH 1 :
a 2
ykeTACMnYskñgú 1 eyIg)an ³
4a 12 b 4 2 12 b b4
dUcenH témøkMNt;)anKW
a 2 , b 4
.
0.38 0.47 0.09 17 17 0.09 47% 3 b 17 0.09 0.47 3 b 17 0.27 b 0.47 17 8.26 b 17 0.09 8.26 y x 17 17 9 826 y x 1700 1700 a
-sg;bnÞat;enaH ³ cMeBaH b 4 enaHeyIg)an 4x 2 y 4 b¤ 2x y 2 naM[ y 2x 2 eyIgsg;bnÞat; edayemIl y 2x 2 ¬mincaM)ac;eRbItaragtémøelx ¦
naM[
smIkarGacsresr b¤
dUcenH smIkarTMnak;TMngkMNt;)anKW y
y 2 x 2
161
9 826 x 1700 1700
.
១១
១ ឬ
1.
k>
x y 8 x y 2
3 , 5
x>
K>
y 3x x y 3
0 , 0
X>
2 x y 3 x 2 y 4 x y 3 2 x 2 y 6
2 , 1
5 , 2
2.
k> xx 4yy14
x> xy 32 00
2 x y 6 y 8 2x
x y 1 x y 2 2 2
g>
c>
K> 2x x23yy813
X> xy y3x 4
x y 3 7 2x y 2
C> xy 7y 3x
q>
3.
k> 52xx 22yy 92
x> 22xx 2y y58
K> 34xx 33yy 916
y 1 g> 37xx 10 2 y 13
c> 84xx 23 yy 1510
x 8 y 1 5 x 3 y 9 0 q> 210 C> x 4 y 16 3x 4 y 17 0
X> 8x7 x 3 y6 y 17 2
Q> 6x2x552yx67 y j> 43x x35y y97
d> 37xx 49 yy 01
19 z> 46x x23yy1.5
k> 4x x2 yy 16 1
x> 4x x23yy64
K> 2yx34yx135
23 X> 53xx yy 15
g> 22x x3 yy 62
c> 4x x5 yy 112
11 q> 32xx 33yy 18
C> 54xx 32yy 14
3x 6 y 6 2 x 2 y
x 10 y 1 2 x 3 y 2
4.
Q> 5.
5 x 7 y 3 2 x 14 y 2
j>
d>
m
4 x 6 y 12 2 x my 5 m
mx ny 10 4x 3 y 5
12x 3my 1 4 x 2 y 3 n 6 x y 30 n 1x 1 y n m 3 162
z>
x 7 y 1 x 1 y 4
6.
2 3
19 000 ៛
2
8 000 ៛
7. 500 ៛
21 000 ៛
3
11
1 000 ៛
8. 7m 1m 9.
110 m
5m
2m
2
150 m 10.
20 $
H
P
H6
P4
1.5 $
H2
P5
1$
H4
P4
11.
22
246
12.
490
13.
5 4
14.
ABC
3 7 324 192 m
2.5 , 2 15.
BC , CA , AB
1.5
25 81
36
3 5
16. Ax By 9
17.
2 , 1
64 m
3 , 3
A
B
2 2 1 x y 2 1 5 3 x y 4
18.
19.
A
A B
2008 2.3 2.9
B
2009 2.7 3.3
2010 3.1 3.7
2011 3.5 4.1
A
DDCEE
3 163
B ឬ
១១ 1.
1
kUGredaenEdl[CacemøIyrbs;RbB½n§smIkarb¤eT? k> xx yy 82 eyIgmankUGredaen 3 , 5 mann½yfa cMeBaH x 3 , y 5 eyIg)an ³ 3 5 8 3 5 8 8 8 b¤ b¤ 3 5 2 3 5 2 2 2 eXIjfa kUGredaenHvaepÞógpÞat;RbB½n§smIkar dUcenH 3 , 5 CakUGredaenrbs;RbB½n§smIkar
X> 2x xy2y3 6 b¤ xx yy 33 b¤ x y 3 eyIgmankUGredaen 5 , 2 mann½yfa cMeBaH x 5 , y 2 eyIg)an ³ 5 2 3 b¤ 3 3 eXIjfa kUGredaenHvaepÞógpÞat;RbB½n§smIkar dUcenH 5 , 2 CakUGredaenrbs;RbB½n§smIkar 2.
2 x y 3 x 2 y 4
k>
x> eyIgmankUGredaen 2 , 1 mann½yfa cMeBaH x 2 , y 1 eyIg)an ³ 3 3 2 2 1 3 4 1 3 b¤ b¤ 04 2 2 1 4 22 4
edaHRsayRbB½n§smIkartamRkab ³ x 4 y 1 x y 4
b¤
4 y x 1 y x 4
b¤
x 1 y 4 y x 4
eyIgeRbItaragtémøelxedIm,Isg;Rkab ³ -cMeBaH y x 4 1 ³ xy 13 15 - cMeBaH y x 4 ³ xy 22 eyIgsg;RkabénRbB½n§smIkar)an ³
eXIjfa kUGredaenHminvaepÞógpÞat;RbB½n§smIkar dUcenH 2 , 1 minCakUGredaenrbs;Rb>smIkar
0 4
y
K> xy y3x 3 eyIgmankUGredaen 0 , 0 mann½yfa cMeBaH x 0 , y 0 eyIg)an ³ 0 3 0 0 0 b¤ 00 3 03
x 1 4
y x 4
edayRkabTaMgBIrkat;KñaRtg;cMNuc 3 , 1 mann½yfa x 3 , y 1 CacemøIyénRbB½n§
eXIjfa kUGredaenHminvaepÞógpÞat;RbB½n§smIkar dUcenH 0 , 0 minCakUGredaenrbs;Rb>smIkar
dUcenH RbB½n§smIkarmanKUcemøIy xy 31 .
164
edayRkabTaMgBIrkat;KñaRtg;cMNuc 2 , 3 mann½yfa x 2 , y 3 CacemøIyénRbB½n§
x> xy 32 00 b¤ xy 23 y 2 CabnÞat;edkkat;G½kSGredaenRtg; 2 x 3 CabnÞat;Qr kat;G½kSGab;sIusRtg; 3 eyIgsg;RkabénRbB½n§smIkar)an ³
dUcenH RbB½n§smIkarmanKUcemøIy xy 23 .
X> xy y3x 4 b¤ yy 3xx 4 tagragCMnYyedIm,Isg;Rkab ³ x 0 1 -cMeBaH y 3x y 0 3
y2
-cMeBaH y x 4 eyIgsg;Rkab)an ³
x 3
x 0 1 y 4 3
edayRkabTaMgBIrkat;KñaRtg;cMNuc 3 , 2 mann½yfa x 3 , y 2 CacemøIyénRbB½n§ dUcenH RbB½n§smIkarmanKUcemøIy
x 3 y 2
y 3x
.
y x 4
K> 2x x23yy813 b¤
2 x 13 y 3 y x 8 2
edayRkabTaMgBIrkat;KñaRtg;cMNuc 1 , 3 mann½yfa x 1 , y 3 CacemøIyénRbB½n§
eyIgeRbItaragtémøelxedIm,Isg;RkabénbnÞat; -cMeBaH y 2 x 3 13 xy 23 51 -cMeBaH
y
x 8 2
dUcenH RbB½n§smIkarmanKUcemøIy xy 13 .
x 2 4 y 3 2 y
2 x 13 3
y
g> 2yx8y2x6 b¤ yy 86 22xx tagragCMnYyedIm,Isg;Rkab ³ -cMeBaH y 6 2 x xy 22 03
x 8 2
-cMeBaH y 8 2 x eyIgsg;Rkab)an ³ 165
x 3 4 y 2 0
q>
y 6 2x
c>
x y 1 x y 2 2 2
b¤
y x 1 y x 4
tamRkab bnÞat;TaMgBIrRsbKña ¬eRBaHmanemKuN R)ab;TisesµI -1 dUcKña¦ dUcenH RbB½n§smIkarKµanKUcemøIy .
taragtémøelxRtUvKñaén x nig y edIm,Isg;Rkab -cMeBaH y x 1 xy 10 10 -cMeBaH y x 4
b¤
y x 3 7 y x 2
eRbItaragtémøelxedIm,Isg;Rkab ³ -cMeBaH y x 3 xy 03 12 -cMeBaH y x 72 xy 70/ 2 51/ 2
y 8 2x
tamRkabbnÞat;TaMgBIrRsbKña ¬eRBaHemKuNR)ab; Tis énbnÞat;TaMgBIrKWesµI -2 dUcKña ¦ dUcenH RbB½n§smIkarKµanKUcemøIy .
x y 3 7 2x y 2
C> xy 7y 3x b¤ yy x x3 7 taragtémøelxRtUvKñaén x nig y ³ -cMeBaH y x 7 xy 34 43
x 0 2 y 4 2
-cMeBaH
y x3
x 0 1 y 3 4
y x3 y x 1
y x 4
y x 7
tamRkab bnÞat;TaMgBIrRsbKña ¬eRBaHmanemKuN R)ab;TisesµI -1 dUcKña¦
tamRkabbnÞat;TaMgBIRbsBVKñaRtg;cMNuc dUcenH x 2 , y 5 CaKUcemøIyénRbB½n§smIkar
dUcenH RbB½n§smIkarKµanKUcemøIy . 166
3.
edaHRsayRbB½n§smIkartamviFIbUkbM)at; ³ k> 52xx 22yy 92 12
X> 8x7x 3y6y 17 2 16 x 6 y 34 7 x 6 y 2 9 x 36
naM[ x 77 b¤ x 1 yk x 1 CMnYskñúgsmIkar 1 eyIg)an ³
6 y 2 28 30 6 y 5 y
dUcenH RbB½n§smIkarmanKUcemøIy xy 45
dUcenH RbB½n§smIkarmanKUcemøIy xy 12 .
b¤
5 1 7 x 10 y 1 15x 10 y 65 2 22 x 66
naM[ x 2266 3 yk x 3 CMnYskñúgsmIkar 2 eyIg)an ³
x y 4 2 x y 5 3x 9
naM[ x 93 3 yk x 3 CMnYskñúgsmIkar 1 eyIg)an ³
15 3 10 y 65 10 y 65 45
3 y 4
20 10 y 2
y 43
y
y 1
dUcenH RbB½n§smIkarmanKUcemøIy xy 13 .
dUcenH RbB½n§smIkarmanKUcemøIy xy 32 .
1 2
c> 84xx 23 yy 1510
2
8 x 6 y 30 1 8 x 2 y 10 2 8 y 40
3x 3 y 9 4 x 3 y 16 7 x 7
naM[ x 77 1 yk x 1 CMnYskñúgsmIkar 1 eyIg)an ³ 3 1 3 y 9 b¤ 3 y 9 3 y
.
y 1 g> 37xx 10 2 y 13
1 2
x y 4 2 x y 5
K> 34xx 33yy 916
1 2
7 4 6 y 2
2 1 2 y 2 2 y 2 2 4 y 2 y 2
x>
eyIg)an ³
naM[ x 369 4 yk x 4 CMnYskñúgsmIkar 2 eyIg)an ³
2 x 2 y 2 5 x 2 y 9 7x 7
2 x 2 y 8 2 x y 5
2
naM[ y 840 5 yk y 5 CMnYskñúgsmIkar 2 eyIg)an ³ 8 x 2 5 10
8 x 10 10 x0
12 4 3
dUCenH RbB½n§smIkarmanKUcemøIy xy 05 .
dUcenH RbB½n§smIkarmanKUcemøIy xy 41 .
167
q> 2x10x8y 4y116 5
10 x 40 y 5 1 10 x 4 y 16 2 44 y 11 1 y 4 1 10 x 4 16 4 10 x 16 1
yk
j> 4x3x 3 y5 y 9 7
4 1 12 x 9 y 27 12 x 20 y 28 2 11y 55
1 naM[ y 11 44 4
55 naM[ y 11 5 yk y 5 CMnYskñúgsmIkar 1 eyIg)an ³
CMnYskñúgsmIkar 1 eyIg)an ³
x
15 10
12 x 9 5 27 12 x 27 45 72 x 12
b¤
b¤
x
dUcenH RbB½n§smIkarmanKUcemøIy
3 2
x 3 / 2 y 1/ 4
90 5 x 3 y 9 C> 53xx 34 yy 17 b¤ 3x 4 y 17 0 20 x 12 y 36 1 9 x 12 y 51 2 29 x 87
.
3 21x 28 y 7 1 21x 27 y 0 2 y7
4 3
naM[ y 71 7 yk y 7 CMnYskñúgsmIkar 2 eyIg)an ³
naM[ x 2987 3
b¤
dUcenH RbB½n§smIkarmanKUcemøIy Q>
6 x 5 2 x 7 y 2x 5 y 6
b¤
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x 3 y 2
b¤
x9
.
z> 4x6x 2 y3 y 119.5
3
2 12x 6 y 57 1 60 5 12x 6 y 3 2 x 24 2 24x 60 5 2 x 2
4 x 7 y 5 1 4 x 10 y 12 2 17 y 17
yk
1 naM[ y 17 17 yk y 1 CMnYskñúgsmIkar 1 eyIg)an ³ 2 1 4x 7 1 5 b¤ 4x 5 7 enaH x 4 2 1 x 2 y 1
189 21
dUcenH RbB½n§smIkarmanKUcemøIy xy 97 .
y2
4x 7 y 5 2 x 5 y 6 2
dUcenH RbB½n§smIkarmanKUcemøIy
7
d> 37xx 49 yy 01
20 3 12 y 36 12 y 36 60 24 12
x6
dUcenH RbB½n§smIkarmanKUcemøIy xy 65 .
yk x 3 CMnYskñúgsmIkar 1 eyIg)an ³ y
3
naM[ CMnYskñúgsmIkar eyIg)an ³
5 12 6 y 3 2 6 y 3 30 9 y 2
.
dUcenH RbB½n§smIkarmanKUcemøIy xy 95/ 2/ 2 .
168
4.
23 1 X> 53xx yy 15 2 tam 1 : 5x y 23 enaH y 5x 23 yk y 5x 23 CMnYs 2 eyIg)an ³
edaHRsayRbB½n§smIkartamviFICMnYs ³ 1 k> 4x x2 yy 16 1 2 tam 1 : x 2 y 16 b¤ x 16 2 y yk x 16 2 y CMnYskñúg 2 eyIg)an ³
2 : 3x 5x 23 15
3x 5 x 23 15 2 x 23 15 x 4
2 : 416 2 y y 1
64 8 y y 1 9 y 63
b¤
cMeBaH x 4 enaH
y 7
y 5x 23
cMeBaH y 7 enaH x 16 2 y
5 4 23 3
16 2 7
dUcenH RbB½n§smIkarmanKUcemøIy xy 34 .
2
dUcenH RbB½n§smIkarmanKUcemøIy
x 2 y 7
.
g> 2x2x 3yy62 12 tam 1 : 2x y 2 b¤ y 2 2x yk y 2 2x CMnYskñúg 2 eyIg)an ³
x> 4x x23yy64 12 tam 1 : x 2 y 6 b¤ x 6 2 y yk x 6 2 y CMnYskñúg 2 eyIg)an ³
2 :
2 : 46 2 y 3 y 4
24 8 y 3 y 4 5 y 20
cMeBaH y 4 enaH
b¤
cMeBaH
y4
x 2 y 4
x0
enaH
x0
y 2 2x 2 2 0 2
.
c> 4x x5 yy 112 12 tam 1 : x 5 y 11 enaH x 11 5 y yk x 11 5 y CMnYskñúg 2 enaHeyIg)an ³
K> 2yx34yx135 12 yk 1 CMnYskñúg 2 enaHeyIg)an ³
2 : 411 5 y y 2
2 : 2 x 3 4 x 5 13
44 20 y y 2
2 x 12 x 15 13 14 x 28
2 x 32 2 x 6 2x 6 6x 6 8x 0
dUcenH RbB½n§smIkarmanKUcemøIy xy 02 .
x 6 2 y 6 2 4 2
dUcenH RbB½n§smIkarmanKUcemøIy
>
21y 42 y
28 x 2 14
cMeBaH
cMeBaH x 2 enaH y 4x 5
dUcenH RbB½n§smIkarmanKUcemøIy
x 2 y 3
enaH
x 11 5 y 11 52 1
>
42 5 3
y2
42 2 21
>
dUcenH RbB½n§smIkarmanKUcemøIy xy 12 .
.
169
11 1 q> 32xx 33yy 18 2 tam 1 : 2 x 3 y 11 enaH 3 y 11 2x yk 3 y 11 2x CMnYskñúg 2 enaHeyIg)an ³
cMeBaH x 1 enaH 7 y 1 x
7 y 1 1 7y 2 y
2 : 3x 11 2 x 18
dUcenH RbB½n§smIkarmanKUcemøIy xy 21/ 7 .
x 18 11 x 7
cMeBaH x 7 enaH 3 y 11 2x y
11 2 7 1 3
j> 32xx 62yy 6 b¤ xx 2yy 2 iii yk ii : x y CMnYskñúg i enaHeyIg)an ³
dUcenH RbB½n§smIkarmanKUcemøIy xy 71 .
C> 54xx 32yy 14
i :
1 2
tam 1 : 5x 3 y 4 enaH x 4 53 y yk 3 CMnYskñúg 2 enaHeyIg)an ³
3
d>
y 1
tam * : x 10 y 10 1 10 dUcenH RbB½n§smIkarmanKUcemøIy xy 10 . 1
1 2 83 1 5 5 2 2
4 3
z>
x 7 y 1 x 1 y 4
b¤ xx 4y 47y
1 2
yk 1 : x y 7 CMnYskñúg 2 enaHeyIg)an ³
5 x 7 y 3 1 x 7 y 1 2
Q> b¤ tam 2 : x 7 y 1 enaH 7 y 1 x yk 7 y 1 x CMnYskñúg 1 enaHeyIg)an ³
2 : y 7 4 4 y 3 3y y 1
tam 1 : x y 7 1 7 8
: 5 x 1 x 3 4 x 4 x
x 10 y 1 x 3 y 2 * * 2
2 5y 3y 2
dUcenH RbB½n§smIkarmanKUcemøIy x y 12 .
1
2 3
* * : 1 10 y 3 y 2
yk y 12 CMnYskñúg 3 eyIg)an ³
5 x 7 y 3 2 x 14 y 2
y
yk * : x 10 y CMnYskñúg * * enaHeyIg)an ³
11 1 22 y 11 y 22 2
x
y 2y 2
dUcenH RbB½n§smIkarmanKUcemøIy x y 23 .
2 : 4 4 3 y 2 y 1 5 16 12 y 2y 1 5 16 12 y 10 y 5
3 :
2 7
dUcenH RbB½n§smIkarmanKUcemøIy xy 18 .
4 1 4 170
5.
k> rktémø m edIm,I[RbB½n§smIkarKµancemøyI ³ -eyIgman 42xx 6myy 125
naM[ m 8 nig n 6 dUcenH rk)antémø m 8 nig
rMlwk ³ lkçN³énRbB½n§smIkardWeRkTI!manBIrGBaØat -ebI aa bb cc RbB½n§smIkarKµanKUcemøIy -ebI aa bb cc RbB½n§>manKUcemøIyeRcInrab;minGs;
dUcenH témøkMNt;)anKW m 3 . x 3my 1 -eyIgman 12 4 x 2 y 3 edIm,I[RbB½ns§ mIkarKµancemøIyluHRtaEt ³ 12 3m 1 3m 1 b¤ 3 4 2 3 2 3 3m eyIg)an 3 2 b¤ 3m 6 b¤ m 2
dUcenH témøkMNt;)anKW m 2 . x> rktémø m nig n edIm,I[RbB½n§smIkar manKUcemøIyeRcInrab;minGs; ³ -eyIgman mx4xny3 y 105 edIm,I[RbB½ns§ mIkarmanKUcemøIyeRcInrab;luHRta ³ m n 10 m n 2 b¤ 4 3 5 4 3 eyIg)an m4 2 nig n3 2
3
3
eyIg)an n 6 1 3 nig n30 m 3 naM[ n 1 2 b¤ n 3 ehIy m n 10 enaH m 10 n 10 3 7 dUcenH témørk)anKW m 7 nig n 3 .
RbB½n§>manKUcemøIy EtmYyKt;.
edIm,I[RbB½ns§ mIkarKµanKUcemøIyluHRtaEt ³ 4 6 12 enaHeyIgTaj)an ³ 2 m 5 4 6 12 b¤ 4m 12 naM[ m 3 2 m 4
edIm,I[RbB½ns§ mIkarmanKUcemøIyeRcInrab;luHRta ³ 6 30 6 1 30 b¤ 3 n 1 nm n 1 1 n m
-ebI
6 x y 30
.
-eyIgman n 1x 1 y n m
RbB½n§smIkardWeRkTI!manBIrGBaØatmanrag aaxx bbyy cc
a b a b
n 6
rktémøesovePATaMgBIrRbePT ³ -tag x CatémøesovePAesþIg ¬KitCaerol¦ y CatémøesovePARkas; ¬KitCaerol¦ -tambRmab;RbFan ³ ÷suxTijesovePAesþIg 2k, nigRkas; 3k, Gs; R)ak;cMnYn 21 000 ` naM[ 2x 3 y 21000 1 ÷esATijesovePAesþIg 3k, nigRkas; 2k, Gs; R)ak;cMnYn 19 000 ` naM[ 3x 2 y 19000 2 -tam 1 nig 2 eyIg)anRbB½n§smIkar ³ 2 x 3 y 21000 2 eRbIviFIbUkbM)at; 3x 2 y 19000 3 6.
4 x 6 y 42000 9 x 6 y 57000 5 x 15000
3000 naM[ x 15000 5 cMeBaH x 3000 enaH 2 3000 3 y 21000 5000 naM[ 3y 15000 enaH y 15000 3 dUcenH esoePAesþIgtémø 3000` nigRkas; 5000`
171
7.
rkcMnYnRkdas;R)ak;RbePTnImYy² ³ -tag x CacMnYnRkdas;R)ak; 500` ¬KitCasnøwk¦ y CacMnYnRkdas;R)ak; 1000` ¬KitCasnøwk¦ -tambRmab;RbFan ³ ÷kñúgkarbUbmanRkdas;R)ak;BIrRbePTcMnYn 11 s> eyIg)an ³ x y 11 i ÷R)ak;srubkñgú kabUbmancMnYn 8000 ` eyIg)an ³ 500 x 1000 y 8000 ii -tam i nig ii eyIg)anRbB½n§smIkar ³ i x y 11 eRbIviFICMnYs 500x 1000 y 8000 ii tam i : x y 11 enaH x 11 y yk x 11 y CMnYskñúgsmIkar ii eyIg)an ³
eyIg)an 2x 1 y 2 -tam 1 nig 2 eyIg)anRbB½n§smIkar ³ x 1 y 7 1 x 2 2 1 y
b¤
x y 6 i x 2 2 y ii
yk i CMnYskñúg ii eyIg)an ³ ii : y 6 2 2 y
cMeBaH
y 8
CMnYskñúg i eyIg)an ³ i : x y 6 8 6 14 > dUcenH ExSBYrmanRbEvg 14 m nig edImetñatmankm
rkRbEvgbeNþay nigTTwgéndIenaH ³ -tag x CaRbEvgbeNþay ¬KitCa Em:Rt¦ ii : 50011 y 1000 y 8000 5500 500 y 1000 y 8000 y CaRbEvgTTwg ¬KitCa Em:Rt¦ 500 y 8000 5500 -smµtikmµ ³ dImanragCactuekaNEkg 2500 y 5 -tambRmab;RbFan ³ 500 cMeBaH y 5 enaH x 11 y 11 5 6 ÷dImanbrimaRt 110 m eyIg)an 2x y 110 b¤ x y 55 1 dUcenH Rkdas;R)ak; 500 ` mancMnYn 6 snøwk ehIymanépÞ S xy Rkdas;R)ak; 1000 ` mancMnYn 5 snøwk . ÷ebIbEnßm 5m elIbeNþay nig 2m elITTwg enaH 8. rkRbEvgExSBYr nigkm -tambRmab;RbFan ³ -tam 1 nig 2 eyIg)anRbB½n§smIkar ³ ÷ebIKat;minbt;ExSCaBIr enaHKat;bgðÚtbMBg;mkdl; 2 x y 55 eRbIviFIbUkbM)at; dI ehIyenAsl;ExS 7m eTot 2 x 5 y 140 2 x 2 y 110 eyIg)an ³ x 1 y 7 1 2 x 5 y 140 naM[ y 10 ÷ebIKat;bt;ExSCaBIr Kat;bgðÚtmkdl;dlI µm 3 y 30 9.
2
172
yk y 10 CMnYskñúg 1 eyIg)an ³
rkBIrcMnYnenaH ³ -tag x nig y CaBIrcMnYnenaH Edl x y -bRmab;RbFan ³ ÷pldkénBIrcMnYnesµnI wg 22 eyIg)an x y 22 1 ÷BIrdgéncMnYnFMbUkbIdgéncMnYntUcesµInwg 246 eyIg)an 2x 3 y 246 2 -tam 1 nig 2 eyIg)anRbB½n§smIkar ³ 3 x y 22 eRbIviFIbUkbM)at; 2 x 3 y 246
11.
> dUcenH RbEvgbeNþayKW 45m , TTwg 10m . 1
: x 10 55
x 55 10 45
ebITijnM H 4 nignM P 4 etIeKRtUvGab;R)ak;b:unµan? edIm,IrkR)ak;Gab; eyIgRtUvdwgtémønMnImyY ²sin -tag x CatémønM H ¬KitCa $¦ y CatémønM P ¬KitCa $¦ -smµtikmµ ³ eKmanR)ak; 20$ -tambRmab;RbFan ³ ÷ebITijnM H 6 nignM P 4 eKGab;vij 1.5$ eyIg)an ³ 6x 4 y 20 1.5 b¤ 6x 4 y 18.5 1 ÷ebITijnM H 2 nignM P 5 eKGab;vij 1$ eyIg)an ³ 2x 5 y 20 1 b¤ 2x 5 y 19 2 -tam 1 nig 2 eyIg)anRbB½n§smIkar ³ 6 x 4 y 18.5 1 eRbIviFIbUkbM)at; 2 x 5 y 19 3
10.
3x 3 y 66 2 x 3 y 246 5 x 312
naM[ x 312 62.4 5 yk x 62.4 CMnYskñúg 1 1 : 62.4 y 22
y 62.4 22 y 40.4
dUcenH BIrcMnYnEdlRtUvrkKW 62.4 nig 40.4 . KNnacMnYnTaMgBIrenaH ³ -tag x nig y CaBIrcMnYnEdlRtUvKNnaenaH -bRmab;RbFan ³ ÷BIrcMnYnmanplbUkesµI 490 KW x y 490 1 ÷BIrcMnYnmanpleFobesµInwg 73 KW xy 73 2 -tam 1 nig 2 eyIg)anRbB½n§smIkar ³
12.
6 x 4 y 18.5 6 x 15 y 57 11y 38.5
naM[ y 3811.5 3.5 cMeBaH y 3.5 CMnYskñúg 2 eyIg)an ³ 2 : 2 x 53.5 19
2 x 19 17.5
x y 490 x y 490 x 3 x 9 y 7 3 7 x y x y 490 49 3 7 3 7 10
1.5 2 x 0.75 x
b¤
ebITijnM H 4 nignM P 4 enaHeKRtUvGab;R)ak;KW ³ 20 4 0.75 4 3.5 20 3 14 3 > dUcenH eKnwgGab;R)ak;[eyIgvijcMnYn 3$ .
tamlkçN³
smamaRt naM[ x 3 49 147 nig y 7 49 343 dUcenH cMnYnTaMgBIrenaHKW 147 nig 343 . 173
KNnacMnYnnImYy² ³ -tag x nig y CaBIrcMnYnEdlRtUvKNnaenaH -bRmab;RbFan ³ ÷pleFobénBIrcMnYnesµInwg 54 KW ³ xy 54 b¤Gacsresr 5x 4y 1 ÷kaerénpldkrbs;vaesµInwg 324 ³ eyIg)an x y 324 b¤ x y 324 naM[ x y 18 2 b¤ x y 18 3 -tam 1 nig 2 eyIg)anRbB½n§smIkar ³
-tam 1 nig 2 eyIg)anRbB½n§smIkar ³
13.
BC CA AB 192 BC CA AB 2 1.5 2.5
tamlkçN³énsmamaRt eyIg)an ³ BC CA AB BC CA AB 192 32 2.5 2 1.5 2.5 2 1.5 6 BC 32 BC 2.5 32 80 2.5 CA 32 CA 2 32 64 2 AB 32 AB 1.5 32 48 1.5
enaH
2
x y 5 4 x y 18 x y x y 18 18 5 4 54 1
dUcenH KNna)anRbEvgRCugénRtIekaN ABC KW BC 80m , CA 64m , AB 48m .
tamlkçN³énsmamaRt eyIg)an ³
KNnacMnYnnImYy² ³ -tag x nig y CaBIrcMnYnEdlRtUvKNnaenaH -bRmab;RbFan ³ ÷pldkénBIrcMnYnesµnI wg 36 KW x y 36 1 x 25 ÷kaerénpleFobesµInwg 25 KW y 81 81
15.
naM[ x 5 18 90 nig y 4 18 72 -tam 1 nig 3 eyIg)anRbB½n§smIkar ³ x y 5 4 x y 18 x y x y 18 18 5 4 54 1
tamlkçN³énsmamaRt eyIg)an ³
2
x 5 x 5 x 25 y 9 y 9 y 81 x y x y 2 3 5 9 5 9
enaH naM[ b¤ b¤Gacsresr b¤ -tam 1 nig 2 eyIg)anRbB½n§smIkar ³
naM[ x 5 18 90 nig y 4 18 72 dUcenH cMnYnnImYy²KW 90 nig 72 b¤cMnYnnImYy²KW -90 nig -72 .
x y 36 x y 5 9 x y x y 36 9 5 9 59 4
tamlkçN³énsmamaRt eyIg)an³
KNnaRbEvgRCug BC , CA nig AB ³ -bRmab;RbFan ³ ÷RtIekaN ABC manbrimaRt 192m ³ eyIg)an BC CA AB 192 1 ÷smamaRtRCugTaMgbIerogKñaKW 2.5 2 nig 1.5 ³ CA AB 2 eyIg)an BC 2.5 2 1.5
14.
naM[ x 5 9 45 nig y 9 9 81 -tam 1 nig 3 eyIg)anRbB½n§smIkar ³ x y 36 x y 5 9 174
tamlkçN³énsmamaRt eyIg)an³
x y x y 36 18 5 9 5 9 14 7 18 90 18 162 x 5 y 9 7 7 7 7
naM[
yk
CMnYskñúg 2 eyIg)an ³ 2 : 2 B 3 enaH B 5 dUcenH kMNt;)an A 2 nig B 5 .
nig
dUcenH cMnYnnImYy²KW -45 nig -81 b¤cMnYnnImYy²KW 907 nig 162 . 7
18.
A2
edaHRsayRbB½n§smIkar ³ 2 x 1 x
2 1 y 2 5 3 y 4
KNnaRbEvgén)atnImYy² énctuekaNBñay ³ -tag B CaRbEvg)atFM nig b CaRbEvg)attUc -bRmab;RbFan ³ ÷pleFob)atctuekaNBñayesµI 53 KW Bb 53 b¤Gacsresr)an b3 B5 1 ÷plbUk)atrbs;vaesµInwg 64 KW b B 64 2 -tam 1 nig 2 eyIg)anRbB½n§smIkar ³
eyIgman
b B 3 5 b B 64 b B b B 64 8 3 5 35 8
cMeBaH Y 121 eyIgman X 5Y 34 enaH X 34 5Y 34 125 912 5 124 13 eday X 1x nig Y 1y enaHeyIg)an ³ 1 1 1 1 y 12 x 3 nig y 12 x 3
16.
tag X 1x nig Y 1y enaHeyIg)anRbB½n§fµIKW ³ 1 2 X 2Y 4 X 4Y 1 2 4 X 20Y 3 X 5Y 3 4 4 X 4Y 1 2 4 X 20Y 3 Y 24 24Y 2
b¤
naM[
tamlkçN³énsmamaRt eyIg)an ³
enaH b 3 8 24m nig B 4 8 40m dUcenH RbEvg)attUc 24m nig)atFM 40m . kMNt; A nig B ³ eyIgmanbnÞat; Ax By 9 -ebIbnÞat; Ax By 9 kat;tamcMNuc 2 , 1 eyIg)an 2 A B 9 1 -ebIbnÞat; Ax By 9 kat;tamcMNuc 3 , 3 eyIg)an 3A 3B 9 b¤ A B 3 2 -yk 1 - 2 eyIg)an ³
17.
2 A B 9 A B 3 3A 6
naM[
A
6 3
b¤ A 2
b¤
1 1 1 2 x 2 y 2 1 5 1 3 y 4 x
1
b¤ Y 121
3 . dUcenH RbB½n§smIkarmanKUrcemøIy xy 12 19. etIplitpl A ekInCagplitpl B b¤eT? -kMeNInplitpl A tamqñaMnImYy²KW ³ 2.7 2.3 3.1 2.7 3.5 3.1 0.4 ¬efr¦ -kMeNInplitpl B tamqñaMnImYy²KW ³ 3.3 2.9 3.7 3.3 4.1 3.7 0.4 ¬efr¦ dUcenH plitpl A minGaclk;ekInCaplit pl B eT eRBaH GRtakMeNInénka lk;tamqñaMesµIKña . 175
១២ 1.
x 10 m
x
6m
2.
ឬ
ABC
k> AB 7 x> AB 52.8 K> AB 83
,
BC 9
, AC 12
,
BC 45.5
, AC 69.7
,
BC 49
, AC 67
3.
A
ABC
41cm
15 cm
B 12 cm
4.
C
D
5 cm
O
AB
O
8 cm OM
A
//
M
//
B
3 cm
5.
h 6.
x
y
m x
y 12 6
50
38
x
8
16
7. 18cm
12 cm
176
8.
1 cm
9.
A
cm AB
16
AH BH
HC
B
C
H 20
10.
O
Ox
x
o
Oy
O 100 km
100 km
y
DDCEE
3
177
១២ 1.
KNna x kñúgrUbxageRkam ³ eyIgmanRtIekaNEkg Edlman x CaRCugCab; mMuEkg tamRTwsþIbTBItaK½r eyIg)an ³ x 6 10 2
2
67 2 49 2 832 4489 2401 6889
¬minBit¦ dUcenH ABC minEmnCaRtIekaNEkgeT . 6890 6889
2
x 2 100 36
10 m
x 64 x 8 m
x
3. 6m
dUcenH KNna)an 2.
AC 2 BC 2 AB2
x 8 m
.
KNnaépÞRkLaénRtIekaN ABC ³ eyIgmanRtIekaN ABC dUcrUbxageRkam A
bBa¢ak;fa ABC CaRtIekaNEkgb¤eT ? k> AB 7 , BC 9 , AC 12 cMeBaHkrNIenH ebI ABC CaRtIekaNEkgluHRta ³
B
7 9 12 2
eyIgdwgfa S 12 BC AD -cMeBaHRtIekaNEkg ADBEkgRtg; D eyIg)an ³ AD DB AB b¤ AD AB DB ABC
2
49 81 144
¬minBit¦ dUcenH ABC minEmnCaRtIekaNEkgeT .
2
130 144
2
2
2
2
2
2
2
2
2
DC 2 412 9 2 1681 81 1600
52.82 45.52 69.7 2 2787.84 2070.25 4858.09
ABC
2
naM[ AD 81 9 cm -cMeBaHRtIekaNEkg ADC EkgRtg; D eyIg)an ³ DC DA AC b¤ DC AC DA
AB2 BC 2 AC 2
dUcenH
2
AD2 152 122 225 144 81
x> AB 52.8 , BC 45.5 , AC 69.7 cMeBaHkrNIenH ebI ABC CaRtIekaNEkgluHRta ³
4858.09 4858.09
C
D
12 cm
AB 2 BC 2 AC 2 2
41cm
15 cm
naM[ DC 1600 40 cm enaH BC BC DC 12 40 52 cm eyIg)an S 12 52 9 234cm
¬Bit¦
KWCaRtIekaNEkg .
ABC
K> AB 83 , BC 49 , AC 67 cMeBaHkrNIenH ebI ABC CaRtIekaNEkgluHRta ³
dUcenH KNna)an 178
2
S ABC 234 cm 2
.
4.
KNnaRbEvg OM kñúgrUbxageRkam ³ -eyIgmanrgVg;p©it O mankaMRbEvg 5 cm nig AB manRbEvg 8 cm -eday MA MB naM[ M CacMNuckNþal AB M eyIg)an OM AM Rtg; M ¬eRBaH kaMrgVg;EkgnwgGgát;FñÚRtg;cMNuckNþal¦ -kñúgRtIekaN OMA EkgRtg; M tamRTwsþIbTBItaK½r OM AM OA naM[ OM OA AM 8 eday OA 5 cm nig AM AB 4 cm 2 2 eyIg)an OM 5 4 25 16 9 enaH OM 9 3cm dUcenH KNna)anRbEvg OM 3cm .
6.
x 2 6 2 82 36 64 100
O
A
2
2
2
2
5.
2
//
//
2
x
2
2
2
2
2
16
x 2 CD 2 CE 2
CD 2 BE BC
2
KNnaépÞRkLaénRtIekaNsm½gS ³ -ebIRtIekaNsm½gSmYyman rgVas;RCug a enaHkm
16 2 50 38 256 144 400 2
h
Taj)an a 2h3 b¤ a 2 33h -épÞRkLaénRtIekaNsm½gSKW ³ 1 2 3h 3 1 h h S ah b¤ S 2 3 3 2 eday h 3 cm naM[ S 33 3 33 3 3 cm 2
6
naM[ x 100 10 8 ehIy y 12 x cMeBaH x 10 enaH y 144 100 44 b¤ y 44 4 10 2 10 dUcenH KNna)an x 10 nig y 2 10 . E -eyIgmanrUb ³ x eyIgKUsbEnßm CD C D 16 50 Edl CD // AB 38 tamRTwsþIbTBItaK½r A B
B
2
2
kMNt;témø x nig y énrUbxageRkamKitCa m ³ -eyIgmanrUb ³ y tamRTwsþIbTBItaK½r 12
naM[ x 400 20 20 dUcenH KNna)an x 20 . 7. KNnakm
2
h 2 182 122 324 144 180
2
naM[ h 180 6 5 cm ehIy V 13 S h 13 r h b¤ V 13 12 6 5 288
18cm
12 cm
2
b
2
dUcenH KNna)anépÞRkLaénRtIekaN sm½gSKW S 3 cm .
5 cm3
dUcenH KNna)an km
2
3
179
.
8.
KNnaRbEvgGgát;RTUgénKUbenaH ³ A 1cm B eyIgmanKUbEdlman D C rgVas;RCugesµI 1cm F E RbEvgGgát;RTUgKW AG G H tamRTwsþIbTBItaK½r -cMeBaH RtIekaN EHG EkgRtg; H eyIg)an ³
K> KNna BH nig HC ³ -kñúgRtIekaNEkg AHB EkgRtg; H eyIg)an ³ BH 2 AB2 AH 2 2
2304 1296 48 12 2 144 25 25 5
naM[
EG2 GH 2 EH 2
dUcenH KNna)an
1 1 2 2
2
-cMeBaH RtIekaN AEG EkgRtg; G eyIg)an ³ AG EG AE naM[ AG 3 cm 2 1 3 2
2
dUcenH KUbEdlmanRCugesµI 1cm vamanGgát;RTUg 3 cm
2
.
naM[
BC
64 cm 12.8 cm 5
.
¬manviFIRsYlCagenH Et´cg;[GñkGnuvtþRTwsþIbTBItaK½r¦ C
KNnacm¶aypøÚvEdlynþehaHnImYy² ehaH)an ³ edayynþehaHmanel,ÓnesµIKña ehIyehaHecjBI cMNucEtmYyenAem:agEtmYy naM[ cm¶aypøÚvesµIKña -tag x Cacm¶aypøÚvénynþehaHnImYy² ¬km¦ tamRTwsþIbTBItaK½r eyIg)an ³
10.
k> KNna AB ³ 20 kñúgRtIekaN ABC EkgRtg; A eyIg)an AB2 BC 2 AC 2 202 162 400 256 144
naM[ AB 144 12 12cm dUcenH KNna)an AB 12cm x> KNna AH ³ tamlkçN³énRtIekaNEkg ABC mankm
x 2 x 2 1002
2
AH
2
4096 64 64 HC cm 25 5 5
dUcenH KNna)an
16
dUcenH KNna)an
.
2304 4096 48 16 2 256 25 25 5
A
H
36 cm 7.2 cm 5
HC 2 AC 2 AH 2
KNnargVas;RCugénrUbKitCa cm ³
B
BH
-kñúgRtIekaNEkg AHC EkgRtg; H eyIg)an ³
2
2
9.
2
1296 36 36 BH cm 25 5 5
2 x 2 10000
x
o
x 2 5000
100 km
x 5000 x 50 2 km
y
x 70.71km
dUcenH ynþehaHTaMgBIrehaH)ancm¶ay esµIKñaRbEvg 70.71 km .
48 cm 9.6cm 5
180
កំណែលំហាត់តាមមមមរៀន
គែិតវិ ទ្យា អាន. គិត. យល់
១៣ 1.
O
T៖
PT
P T
O
ao bo
50
o
O
O
T
ao
T
P a
a
O
bo
ao 35
ao
P
S o
b
b A
O o
T
P
40 o b o
65 co
B
bo
a
130o
o
P bo
O
T
T
P
a
a,b
b
AOT
c
BOP a
2.
PT T
( )
O
PV
O
c
a
( )
P
42o
O b
P
a
c
V
V T
b
T
25o
a b
V
T
22o
b
( )
T
O
(
P
)
c
X
21o
a
P
O
c
V
3.
O
PA r
PB
d
A cm
A
A 12 17
O r
P
P
O
15
B
O
PA A
cm
5
d
B
4.
B
A
PB
2
b
B
O
៖
c
P
9
181
B
b
កំណែលំហាត់តាមមមមរៀន
គែិតវិ ទ្យា អាន. គិត. យល់
b
c OAPB
OAPB A
5.
AB , BC D,E
10
AC
F
F
ABC
D
cm
5
C
B
6.
PA
PB
O
AP 82.4 cm
3 cm
E
15
A
AOB 140o
82.4
3
៖
P
o O 140
APOB 22 7
AOB
B
A
7.
O
AT
TQO
BT
20 cm
TQ = 10 cm
10
T
8.
O TA
B
20
4.5 cm
TB
O
Q
A
A
B
9
4.5 O
TA = 9 cm
T
OATB B
OATB
9.
A 4 cm CB
B B
10.
TA
DA
C
D
A
B
DC
C
ABCD
TB
5 cm
D
A
CD A
B
A
O ៖
ATB 60o
T
AOB AOB
22 7
182
O
60 o
5
B
កំណែលំហាត់តាមមមមរៀន
គែិតវិ ទ្យា អាន. គិត. យល់
A
11.
O
8 cm
10 cm
AB
T AB
(
:
O
T
OB )
OA
B M
12.
MA , MB
MC
O’
O
C
A
MA = MC
O
13.
EF
EG
H
JEF 22
o
B
O
F
៖
EGH
E
HGF
22 o
J
H
5
FGE
G
DDCEE
3
183
កំណែលំហាត់តាមមមមរៀន
គែិតវិ ទ្យា អាន. គិត. យល់
មេម ៀនទី១៣ 1.
ង្វង្ន ់ ិង្បន្ទាត់
kñúgrUbnImYy²énrgVg;p©it O man PT CabnÞat;b:H ³ k> rktémøénmMu a ³ eday OT TP O 50 naM[ OTP 90 T a ¬eRBaH bnÞat;b:HRtUvEkg P nwgkaMrgVg;Rtg;cMNucb:H¦ eyIg)an a 180 50 90 40 dUcenH rk)antémømMu a 40 . o
X> rktémøénmMu a nig b ³ eday OT TP 90 naM[ OTP a S T 35 55 enaH a90 P ehIy b 2STP ¬mMup©itesµIBIrdgmMuBiess¦ enaH b 23570 dUcenH rk)antémømMu a 55 nig b 70 . g> rktémøénmMu a , b nig c ³ eday OTTP a 90 P naM[ OTP T 65 25 enaH a90 65 32 . 5 32 3 0 ehIy c 2 ¬eRBaH mMucarwkesµIknøHmMup©itEdlmanFñÚsáat;rmY ¦ OTP 90 eday cb naM[ b 90 c 90 32.5 57.5 dUcenH a 25 , b 57.5 , c 32.5 . O
o
o
o
o
o
o
o
o
o
35o
o
o
o
bo
o
o
o
o
o
o
o
o
O
P x> rktémøénmMu a nig b ³ T a eday OT TP 90 O naM[ aOTP smµtikmµ ³ PT OT naM[ PTO CaRtIekaNEkgsm)at enaH mMu)atnImYy²manrgVas;esµIKñaesµInwg 45 eyIg)an b 45 dUcenH rk)antémømMu a 90 nig b 45 .
o
o
o
bo
bo
o
o
o
o
o
o
o
o
o
o
o
o
o
o
c> rktémøénmMu a nig b ³ A mMu a 1302 65 P a ¬eRBaH mMucarwkesµIknøH B T mMup©itEdlmanFñÚsáat;rYm¦ ehIymMu b 180 OTP TOP Edl OTP 90 nig TOP 180 130 50 naM[ b 180 90 50 40 dUcenH rk)an a 65 nig b 40 . o
K> rktémøénmMu b ³ eday OT TP naM[ OTP 90 eyIg)an b 90 40 b¤ b 50
o
o
130o
o
bo
O
O
o
o
o
o
o
o
o
o
o
o
o
c
o
o
o
o
o
65o
o
o
40 o b o
P
o
T
o
o
o
dUcenH rk)antémømMu b 50 . o
o
o
o
o
o
184
o
o
o
o
o
កំណែលំហាត់តាមមមមរៀន
1.
eyIgmanrgVg;p©it O ehIy PT nig PV CabnÞat;b:H k> rkrgVas;mMuEdlminsÁal;KW a , b nig c ³ T
eday PT b:HrgVg;Rtg; T naM[ kñúgRtIekaNEkg OTP man
OTP 90o
b 180 o 90 o 25 o 65 o
22o
a
O
គែិតវិ ទ្យា អាន. គិត. យល់
RtIekaN OTV CaRtIekaNsm)at ¬ OT OV ¦ man TOV 2b 2 65 130 naM[ a c 180 130 2 25 dUcenH a 25 , b 65 , c 25 .
P
b
o
c
V
o
eday PT nig PV CaknøHbnÞat;BIrKUs ecjBIcMNuc P EtmYy eTAb:HnwgrgVg;erogKñaRtg; T nig V enaH OP CaknøHbnÞat;BuHmMu TPV nig TOV naM[ c 22 nigmMu OTP 90 eyIg)an a b 180 90 22 68 dUcenH a 68 , b 68 , c 22 . o
o
o
o
o
o
o
X> rkrgVas;mMuEdlminsÁal;KW
a,b
nig c ³
b
T
o
o
o
o
o
c
o
X
o
21o
a
P
O
o
kñúgRtIekaNEkg OTP man OTP 90 naM[ a 180 90 21 69 ehIy TOX CaRtIekaNsm)at ¬ OT OX ¦ naM[ mMu)at b c cMeBaH RtIekaN TOX man b c a ¬plbUkmMukñúgBIr esµInwgmMueRkAmYyminCab;nwgva¦ naM[ b b a b¤ 2b a enaH b a2 692 o
x> rkrgVas;mMuEdlminsÁal;KW
a,b
nig c ³
o
T c
42o
O b
P
a
V
eday PT nig PV CaknøHbnÞat;BIrKUsecjBIcMNuc P EtmYyeTAb:HnwgrgVg;erog KñaRtg; T nig V naM[ a c 90 cMeBaHctuekaN OTPV manplbUkmMukñúgesµI 360 enaH b 360 90 90 42 138 dUcenH a 90 , b 138 , c 90 . K> rkrgVas;mMuEdlminsÁal;KW a , b nig c ³
o
o
o
o
dUcenH
o
a 69 , b 34 .5o , c 34 o 30
.
o
o
o
o
o
o
o
T a b
o
O
2.
o
-KNnargVas;kaM r KitCa cm ³ A eday PB b:HrgVg;Rtg; B naM[ OBP CaRtIekaN O 17 15 EkgRtg; B B tamRtwsþIbTBItaK½r cMeBaHRtIekaNEkg OBP r 17 15 b¤ r 64 enaH r 8 cm dUcenH KNna)anrgVas;kaM r 8 cm .
r
25o
2
P
c
V
185
2
2
2
P
កំណែលំហាត់តាមមមមរៀន
គែិតវិ ទ្យា អាន. គិត. យល់
-KNnargVas; d KitCa cm ³
K> KNnabrimaRténctuekaN OAPB ³ POAPB OA AP PB OB
A 12 P
2 b 9 c 2 9 9 2 22 cm
5
d
O
dUcenH KNna)anbrimaRt P
OAPB
.
22 cm
B
eday PA b:HrgVg;Rtg; A naM[ OAP CaRtIekaN EkgRtg; A tamRTwsþIbTBItaK½r cMeBaHRtIekaNEkg OAP eyIg)an OP 12 5 144 25 169 naM[ OP 169 13 cm eday OP d OA ¬eRBaH OA CakaMrgVg;¦ eyIg)an d OP OA 13 5 8 cm dUcenH KNna)an d 8 cm . 2
2
X> KNnaépÞRkLaénctuekaN OAPB ³ eday OAP nig OBP CaRtIekaNEkgBIb:unKña ehIy S S S naM[ S 2S 2 12 9 2 18 cm OAPB
OAP
OBP
2
2
OAPB
OAP
dUcenH KNna)anépÞRkLa S 4.
OAPB
.
18 cm 2
KNnaépÞRkLaén RtIekaN ABC KitCa cm ³ 2
A 10
3.
F
eyIgmanrUbdUcxageRkam manrgVas;KitCa cm ³ D
A
5
2
b
B
O
C
E
15
c
P
9
B
eday ABC CaRtIekaNEkgRtg; B ³ naM[ S 12 AB BC ehIy BE BD 5cm ¬ D nig E CacMNucb:H¦ nig AD AF 10cm ¬ D nig F CacMNucb:H¦ enaH AB AD BD 10 5 15 cm BC BE EC 5 15 20 cm > eyIg)an S 12 15 20 150 cm
k> KNnargVas; b ³ eday PA nig PB CaknøHbnÞat;BIrKUsecjBIcMNuc P EtmYy eTAb:HnwgrgVg;erogKñaRtg; A nig B naM[ PA PB b¤ b 9 cm dUcenH KNna)an
b 9 cm
ABC
.
x> KNnargVas; c ³ eday OA OC ¬kaMrgVg;Etp©it O mYy¦ naM[ c 2 cm dUcenH KNna)an
c 2 cm
2
ABC
dUcenH KNna)anépÞRkLa S
. 186
ABC
150 cm 2
.
កំណែលំហាត់តាមមមមរៀន
5.
k> KNnaépÞRkLactuekaN APBO KitCa cm ³ 2
A
គែិតវិ ទ្យា អាន. គិត. យល់
7.
k> rkbrimaRténctuekaN OATB ³ A
82.4
3 o O 140
OAP S APBO
S OAPB 2S OAP 2 2
brimaRt P OA AT TB OB eday OA OB 4.5 cm ¬kaMénrgVg;EtmYy¦ ehIy TA TB 9 cm ¬eRBaH A nig B Ca cMNucb:HrgVg;KUsecjBIcMNuc T EtmYy ¦ eyIg)an P 4.5 9 9 4.5 27 cm dUcenH KNna)anbrimaRt P 27 cm .
nig OBP CaRtIekaNEkgBIb:unKña S S naM[ OAP
T B
B
eday ehIy
9
4.5 O
P
OATB
OBP
1 AO AP 2
1 3 82.4 247.2 cm 2 2
OATB
dUcenH épÞRkLa S 247 .2 cm . x> KNnaépÞRkLaceRmókfastUc AOB ³ tamrUbmnþ épÞRkLaceRmókfasEdlmanmMu nigrgVas;kaM R KW S R 360 eday 140 , R 3cm ehIyyk 227 naM[ S 227 3 140 11 cm 360 2
APBO
OATB
x> rképÞRkLaénctuekaN OATB ³ eday OAT nig OBT CaRtIekaNEkgBIb:unKña ehIy S S S naM[
2
AOB
o
OAT
OATB
o
S OATB 2 S OAT 2
o
2
AOB
2
2
o
dUcenH épÞRkLaceRmókfas S
AOB
11 cm 2
.
OBT
1 OA AT 2
1 4.5 9 40.5 cm 2 2
dUcenH épÞRkLa S 40 .5 cm 8. rképÞRkLaénctuekaN ABCD ³
.
2
OATB
6.
rkrgVas;énkaMrgVg; ³ A eday TB b:HrgVg;Rtg; O Q 10 B enaH TBO CaRtI T B 20 ekaNEkgRtg; B tamRTwsþIbTBItaK½r cMeBaH RtIekaNEkg TBO
D
C
ctuekaN ABCD B A man CB CD eRBaH B nig D CacMNucb:HénrgVg;p©it A nig DA DC eRBaH A nig C CacMNucb:H énrgVg;p©it B ehIy mMu ADC 90 ¬eRBaHbnÞat;b:HEkgnwgkaMrgVg;¦ rgVg;p©it A nigrgVg;p©it B manrgVas;kaMesµIKña 4 cm naM[ ABCD CakaerEdlmanRCugesµI 4 cm dUcenH épÞRkLa S 4 16 cm .
TO 2 TB 2 BO2
10 r 2 202 r 2
o
100 20r r 2 20 2 r 2 20r 400 100 r
300 15 20
dUcenH KNna)anrgVas;kaMrgVg; r 15 cm .
2
ABCD
187
2
កំណែលំហាត់តាមមមមរៀន
k> KNna
9.
AOB
គែិតវិ ទ្យា អាន. គិត. យល់
³
bgðajfa MA MC ³ M -eyIgman MA MB ¬eRBaH MA nig MB A C CabnÞat; KUsecjBI B O cMNucEtmYyehIy b:HrgVg;p©it O Rtg; A nig B -mü:ageTot MB MC ¬eRBaH MB nig MC Ca bnÞat;KUsecjBIcMNucEtmYy ehIyb:HrgVg;p©it O Rtg; B nig C eXIjfa MA MB nig MB MC dUcenH MA MC RtUv)anbgðaj .
11. A O
60 o
T
5
B
edayctuekaN OATB manmMu OAT 90 nig OBT 90 ¬eRBaH TA nig TB CabnÞat;ehIy EkgnwgkaMrgVg;Rtg; A nig B ¦ naM[ AOB 360 90 90 60 120 dUcenH KNna)an AOB 120 . o
o
o
o
o
o
o
o
x> KNnaépÞRkLaénceRmókfatUc AOB ³ tamrUbmnþ épÞRkLaceRmókfasEdlmanmMu nigrgVas;kaM R KW S R 360 eday 120 , R 5 cm ehIyyk 227 naM[ S 227 5 120 26 .19 cm 360
12.
k> KNna EGH ³
2
AOB
o
E
22 o
F J
G
o
2
2
eday EG CaknøHbnÞat;b:HnwgrgVg;p©it H enaH EG GH naM[ EGH 90 dUcenH KNna)an EGH 90 . x> KNna HGF ³ eday EF EG nig HF HG ¬kaMrgVg;EtmYy¦ naM[ EH CaemdüaT½rén FG enaH EH GF eyIg)an HGF GEH 22 ¬eRBaH mMuman RCugEkgerogKñaKW EG GH nig EH GF ¦ dUcenH KNna)an HGF 22 . K> KNna FGE ³ FGE EGH HGF 90 22 68 > dUcenH KNna)an FGE 68 .
o
dUcenH épÞRkLaceRmókfas S
AOB
26 .19 cm 2
o
KNnargVas; AB ³ eyIgKUsP¢ab; OA nig OB bEnßm A eday AB b:HrgVg;tUc naM[ AT TO > T tamRTwsþIbTBItaK½r B AT OA OT > 10 8 36 enaH AT 36 6 cm mü:ageTot AB 2 AT 2 6 12 cm ¬eRBaHGgát;FñÚEkgnwgkaMrgVg;Rtg;cMNuckNþal¦ dUcenH KNna)anrgVas; AB 12 cm .
o
10.
O
2
2
2
H
5
o
AOB
O
o
2
2
o
o
o
o
188
o
១៤ 1.
O
ABC 65o
ACB
B
A
AB = AC
65 o
ABO
o
D
C A
D
ACD , BAE
2.
E
BDC
130o
45 o
C
B
A
B 38 o
3.
O
AB
CD
BCD , ADC
ABC D
C
A C 60 o
y
4.
y
x
x
34 o
D
B A
5.
O
AC
CAB 40 , AIB 60 o
.
I
40 o 60 o
CAD C
D
6.
A, B, C
O
AOC 120o
o
ABC
120o
C
A
B C 25 o 45 o
7.
O
AD
BC
D
o
ACB 45 , ACD 25 o
.
ADC
o
ខ.
B
I
ខ.
ACD
BD
o
OAB
A
189
B
P
A
8.
A, B, C AB CAP 70 .
ACD
CD
ខ.
APC
O
AC
O
70 o
B
CBD 35o ,
P
DBP 75
o
D
35
o
.
o
75 o
P
DAB
D
C
C
9. CD .
T
ATD 20o
AT = AC
ខ.
ADB
BA
BOC
20 o
T
BDC
o
D
B
A Q
10.
O
QS
M
134o
POQ 134o
PR . PRQ
PR
ខ. RPQ
o
RSQ
P
M
R
S S
11.
O
PQ
R
ROQ 48o
RS . OQR
ខ.
o 48 o
TRQ
RST
T
P
Q
R
A
D
110o
12.
O C
AE
ABC
BC
x, y
D
z
x 40 o
B
y
o
A
B
13.
ABCD
AB = BC
ACE 25o
CD .
ABC
E ខ.
ADE 80o
80 o
BAD
25
o
C
TKXM
T J
SNXL
40 o
TKJ 56o
JKL .
MLN
ខ.
NJK 40o .
LMN
KLN
3 190
X M
E
D
K 56 o
L
14.
z
C
N S
E
១៤ 1.
KNnargVas;mMu ACB nig ABO ³
3.
KNnargVas;mMu BCD , ADC nig ABC ³ A
B 38 o
B
A
65 o
o
D
D
C
C
-eyIgman AC AB nigmMu ABC 65 naM[ ABC CaRtIekaNsm)atEdlmanmMu)at ACB ABC 65 > ehIymMu ACB ADB 65 ¬FñÚsáat;rYm AB ¦ -eday OB OD ¬kaMrgVg;EtmYy¦ enaH OBD CaRtIekaNsm)at Edlman ³ OBD ADB 65 ¬mMu)atRtIekaNsm)at¦ nig ABD 90 ¬mMucarwkknøHrgVg;Ggát;p©it AD ¦ naM[ ABO ABD OBD 90 65 25 dUcenH rgVas;mMu ACB 65 nig ABO 25 2. rkrgVas;mMu ACD , BAE nig BDC ³
mMu BCD BAD 38 ¬manFñÚsáat;rYm BD¦ mMu ADC BAD 38 ¬mMuqøas;kñúgeRBaH AB// CD ¦ mMu ABC ADC 38 ¬manFñÚsáat;rYm AC ¦ dUcenH BCD ADC ABC 38 o
o
o
o
o
o
o
4.
o
A
o
o
o
o
C
o
x
eyIgman mMu x y ¬eRBaHmanFñÚsáat;rYmKña CD ¦ eday mMueRkArgVg; 12 ( FñÚsáat;FM FñÚsáat;tUc ) b¤Gacsresr P 12 AB CD naM[ CD AB 2 P eday AB 2 ACB 2 60 120 ehIy P 34 eyIg)an CD 120 2 34 52 naM[ x y 12 CD 12 52 26
C
B
-mMu ACD ABD 45 ¬mMumanFñÚsáat;rYm AD ¦ -mMu BAE 130 45 85 ¬eRBaH RtIekaN ABE manplbUkmMukñúgBIresµImMueRkAmYy¦ -mMu BDC BAC BAE 85 ¬mMumanFñÚsáat;rYm BC ¦ dUcenH ACD 45 , BAE BDC 85
o
o
o
o
o
o
o
o
o
D
B
D
o
P
34 o
o
E 130o
60 o
y
A
45 o
rkrgVas;mMu x nig y ³ rebobTI1 ¬rktameKalbMNgénemeron¦
o
o
dUcenH KNna)anrgVas;mMu
o
191
o
o
x y 26 o
.
rebobTI2 ¬rktamcMeNHdwgmanRsab;¦
6.
KNnargVas;mMu ABC ³
A C 60 o
y
P
34 o
o
120o
x
D
C
A
B
B
kñúg BCP manmMu x 60 34 26 ¬eRBaH plbUkmMukñúgBIrénRtIekaNesµImMueRkAmYy¦ eyIgman mMu x y ¬eRBaHmanFñÚsáat;rYmKña CD ¦ dUcenH x y 26 . o
o
o
eyIgman Edl CA COA Et mMuqk COA 360 120 240 naM[ ABC 12 CA 12 240 120 1 ABC CA 2 o
o
o
o
5.
o
dUcenH KNna)an
k> KNnargVas;mMu ACD ³ 7.
A
o
.
ABC 120o
k> KNnargVas;mMu ADC ³ C
o
40 60
B
o
25 o 45 o
I
D
o
C
D
kñúgRtIekaN ABI man IAB 40 , AIB 60 naM[ ABI ABD 180 40 60 80 enaHmMu ACD ABD 80 ¬FñÚsáat;rYm AD ¦ dUcenH KNna)anrgVas;mMu ACD 80 . o
o
o
o
eday AD// BC naM[ ADC BCD 180 Et BCD BCA ACD 45 25 70 enaHeyIg)an ADC 180 BCD 180 70 110 > dUcenH KNnargVas;mMu ADC 110 .
o
o
o
o
o
x> KNnargVas;mMu CAD ³ tamsmµtikmµ AB BC enaH ABC sm)at vi)ak mMu)at ACB CAB 40 ehIy ADB ACB 40 ¬FñÚsáat;rYm AB ¦ kñúgRtIekaN ADI manplbUkmMu ADB CAD AIB ¬mMueRkArgVg;¦ naM[ CAD AIB ADB 60 40 20 > dUcenH KNna)an CAD 20 .
o
o
o
o
x> KNnargVas;mMu OAB ³ eday OA OB ¬kaMrgVg;EtmYy¦ enaH OAB Ca RtIekaNsm)at Edlman OAB OBA ehIy AOB 2ACB 2 45 90 ¬eRBaH mMup©itesµIBIrdgénmMucarwkmanFñÚsáat;rYm¦ eyIg)an OAB OBA 180 2 90 45
o
o
o
o
o
o
o
B
A
o
o
o
o
o
o
o
dUcenH KNna)an
o
192
OAB 45o
.
8.
k> KNnargVas;mMu ACD ³
9.
k> KNnargVas;mMu ADB ³ C
A 70 o
o
D
B 35 o 75 o
T
20 o
B
A
P D
C
eday AT AC enaH ATC CaRtIekaNsm)at vi)ak ATD ACT 20 ehIy DBA ACD 20 ¬FñÚsáat;rYm AD ¦ mü:ageTot ADC 90 ¬mMucarwkknøHrgVg;¦ enaHmMu ADT ADC 90 ¬mMubEnßmKña¦ cMeBaH ADT ³ TAD 180 ATD ADT b¤ TAD 180 20 90 70 Et ADB DBA TAD ¬mMueRkA ABD¦ b¤ ADB TAD DBA 70 20 50 dUcenH KNna)an ADB 50 .
edaymMu ABD CamMuCab;bEnßmnwgmMu DBP naM[ ABD 180 DBP 180 75 105 ehIy ABCDCactuekaNcarwkkñgú rgVg;enaHvaman plbUkmMuQmKñaesµIngw 180 eyIg)an ACD ABD 180 enaH ACD 180 ABD 180 105 75 dUcenH KNna)an ACD 75 . o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
x> KNnargVas;mMu APC ³ eday CAB CDB 180 ¬mMuQm ABCD¦ naM[ CDB 180 CAB 180 70 110 > kñúg BDP man DBP APC CDB ¬eRBaHplbUkmMukñúgBIrénRtIekaNesµInwgmMueRkAmYy¦ naM[ APC CDB DBP 110 75 35 > dUcenH KNna)anrgVas;mMu APC 35 .
x> KNnargVas;mMu BOC ³ eyIgman BOC sm)at eRBaH OB OC enaHmMu)at BCA OBC ADB 50 ¬eRBaH BCA nig ADB manFñÚsáat;rYm AB ¦ kñúg BOC man BOC 180 BCA OBC naM[ BOC 180 50 50 80 dUcenH KNna)anrgVas;mMu BOC 80 .
o
o
o
o
o
o
o
o
o
o
o
o
K> KNnargVas;mMu BDC ³ eday BDC ADC ADB naM[ BDC 90 50 40 dUcenH KNna)anrgVas;mMu BDC 40 .
o
o
o
o
K> KNnargVas;mMu DAB ³ man CAD CBD 35 ¬manFñÚsáat;rYm CD ¦ ehIy DAB CAB CAD 70 35 35 dUcenH KNna)anrgVas;mMu DAB 35 . o
o
o
o
o
o
o
o
o
o
o
o
193
10.
k> KNnargVas;mMu PRQ ³
x> KNnargVas;mMu RST ³ eday RST 12 ROT 12 48 24 ¬eRBaH mMucarwkesµIknøHmMup©itEdlmanFñÚsáat;rmY TR ¦ dUcenH KNna)anrgVas;mMu RST 24 .
Q
o
134o
o
o
o
P
M
R
S
K> KNnargVas;mMu TRQ ³ eday TRQ RST 24 ¬mMucarwk nigmMu Biess EdlmanFñÚsáat;rYm TR ¦ dUcenH KNna)anrgVas;mMu TRQ 24 .
eday ¬eRBaH mMucarwkesµInwgknøHmMup©itEdlmanFñÚsáat;rYm¦ dUcenH KNna)anrgVas;mMu PRQ 67 . 1 1 PRQ POQ 134o 67o 2 2
o
o
o
x> KNnargVas;mMu RPQ ³ eday QS RbsBVnwg PR Rtg;cMNuckNþal M naM[ QS CaemdüaT½rén PR eFVI[ QR QP eyIg)an PQR CaRtIekaNsm)at kMBUl Q vi)akmMu)atTaMgBIr RPQ PRQ 67 dUcenH KNna)anrgVas;mMu RPQ 67 .
12.
KNnargVas;mMu x , y nig z ³ A 110o
x
B
o
o
40
o
y
o
z
C
E
eday ABCD CactuekaNe):ag carwkkñúgrgVg; enaHplbUkmMuQmrbs;va RtUvEtesµI 180 naM[
K> KNnargVas;mMu RSQ ³ eday RSQ RPQ 67 ¬manFñÚsáat;rYm QR ¦ dUcenH KNna)anrgVas;mMu RSQ 67 .
o
B ADC 180o
o
ADC 180o B 180o 40o 140o
ehIy A BCD 180
o
11.
D
o
BCD 180 o A 180 o 110 o 70 o
k> KNnargVas;mMu OQR ³
naM[
S
>
x 180o ADC 180o 140o 40o y 180 o BCD 180 o 70 o 110 o
cMeBaHRtIekaN naM[ z 180
o 48 o
T
P
o
o
x y
o
180o 150o 30o
eday PQ b:HrgVg;Rtg; R enaH ORQ Ca naM[ OQR 180 90 48 42 dUcenH KNna)anrgVas;mMu OQR 42 . o
o
man x y z 180
180o 40o 110o
Q
R
CDE
dUcenH KNna)anrgVas;mMudUcteTA ³
o
x 40 o , y 110 o , z 30 o
o
194
.
>
13.
k> KNnargVas;mMu ABC ³
eyIgman MKN NJK 40 ¬eRBaH mMu Biess nigmMucarwkmanFñÚsáat;rYm KN énrgVg;FM¦ ehIy MLN MKN 40 ¬eRBaH mMucarwk manFñÚsáat;rYm MN énrgVg;tUc¦ dUcenH KNna)anrgVas;mMu MLN 40 . o
A
B
o
E
80 o 25
D
o
C
o
eday ADC CamMuCab;bEnßmnwgmMu ADE naM[ ADC 180 ADE 180 80 100 ehIy ABCD CactuekaNe):ag carwkkñúgrgVg; enaHplbUkmMuQmrbs;va RtUvEtesµI 180 naM[ ABC ADC 180 b¤ ABC 180 ADC 180 100 80 dUcenH KNnargVas;mMu ABC 80 . o
o
o
x> KNnargVas;mMu LMN ³ eday JKL enAelIbnÞat;EtmYy nig TKXM Ca bnÞat;b:H enaHeyIg)anmMuTl;kMBUl LKM TKJ 56 > naM[ LKN LKM MKN b¤ LKN 56 40 96 mü:ageTot LMNK CactuekaNcarwkkñúgrgVg;tUc naM[ plbUkmMuQm LMN LKN 180
o
o
o
o
o
o
o
o
o
o
x> KNnargVas;mMu BAD ³ eday AB BC enaH ABC CaRtIekaNsm)at vi)ak ACB BAC 12 180 ABC b¤ ACB 12 180 80 50 naM[ BCD ACB ACD b¤ BCD 50 25 75 eday BCD BAD 180 ¬mMuQmKña¦ b¤ BAD 180 BCD 180 75 105 dUcenH KNnargVas;mMu BAD 105 . o
o
LMN 180o LKN 180o 96o 84o
dUcenH KNna)an
o
o
o
o
o
o
14.
o
KLN 180o LKN LNK 180o 96o 40o 44o
T K 56 o
dUcenH KNna)anrgVas;mMu KLN 44 .
J
o
40 o
X M
o
o
k> KNnargVas;mMu MLN ³ L
. o
o
o
LMN 84o
K> KNnargVas;mMu KLN ³ eyIgman MKN LNK NJK 40 ¬ mMuBiess nigmMucarwkmanFñÚsáat;rYm KN énrgVg;FM¦ ehIy LKN LKM MKN b¤ LKN 56 40 96 kñúg LKN manplbUkmMukñúgesµIngw 180 enaH
o
o
o
o
o
o
o
N S
195
១៥ 1.
AB
AB 6 cm
2.
AB
AB 11 cm
3 4 5
3.
AB
AB 12 cm
3 5 7
4.
AB 7 cm
5.
AB 9 mm
6.
5
C
3 5
AB
C
D
5 2
AB
AB / /CD / / EF / /GH .
DF
EG 4cm ខ.
AC 6 cm , CE 12cm
FH
BD 9cm
BD , DF
FH
CE 12cm , EG 4cm .
DF
CG
DB 8cm
B
C
AC 6 cm ,
HB 33cm
E
AC 6 cm , CE 12cm
G
D F
H
FH 9cm
AB / / MN
7.
A
x A
A 9
8
M
N
B
M
M 3
x
16
A
x
8
C
N 4
6 4 N
B
C
B
C 12
A
8.
BE / /CD ED
4
8
E
12
CD
B
D
2
cm
C
9.
a
b
6 a
b
cm
4
2 8
196
x
10.
13 2.40 m
3.40 m
.
kaMTI13
A
ABC
B
1 20
muxQrénkaM
AB
BC ខ.
kaMTI1
AB BC
B
ខ
.
AB
11.
OX I’ , J’
1 20
BC OY
1 cm
OX
OI’ = 3 cm
C
3.40m
I
J
I’J’ = 4.5 cm
20 cm
OI = 2 cm II’
IJ = 3 cm
OY
JJ’
12.
1.65 m
6.50 m 10 m
. ខ.
13.
ABCD AB
F
. ខ.
E
E
E
CD AGE
ADC
FG // BD
DDCEE
3
197
BC
AD
G
AFE
ABC
១៥
eyIgKUsbnÞat; d rYcRkit[)an AC 3 / CD 5 nig DE 7 rYcKUsP¢ab; EB bnÞab;mk A B CM , DN [Rsbnwg EB enaH AB 12 cm RtUv)anEckCaGgát;smamaRtnwg 3 5 7 . x eyIgKUsknøHbnÞat; Ax rYcedARbeLaHRkit 5Ékta 4. rkcMNuc C EdlEckGgát; AB tampleFob 53 ³ esµI²KñaelI Ax rYcKUsP¢ab;BIcMNcu 5 ÉktaeTAkan; C A B cMNuc B ehIyKUsGgát;bnþbnÞab;[RsbKña. M tamRTwsþIbTtaEls AB 6 cm RtUv)anEck x N Ca 5 cMENkesµI²Kña . eyIgKUsknøHbnÞat; Ax rYcRkit 8 RbeLaHesµI²Kña 2. EckGgát; AB 11 cm CaGgát;smamaRtnwg Edl AM 3 , MN 5 ehIyKUsP¢ab; NB nig 3 4 5 ³ N M KUs MC [Rsbnwg NB enaHeyIg)ancMNuc C B A C EdlEckGgát; AB tampleFob 53 . D d 5 E 5. kMNt;cMNuc C nig D EdlEck AB tampleFob 2 M l eyIgKUsbnÞat; d rYcRkit[)an AC 3 / N CD 4 nig DE 5 rYcKUsP¢ab; EB bnÞab;mk B C A D CM , DN [Rsbnwg EB enaH AB 11 cm P d Q RtUv)anEckCaGgát;smamaRtnwg 3 4 5 . 1.
EckGgát; AB 6 cm Ca 5 cMENkesµIKña ³
/
/
/
/
/
/
/
/
/
/ / / / / / / /
/
/
/ / / /
/ / / / / / /
M
/ / / / C / / / /
D
N
/ / / / / / /
E
-KUs d edA[)an AP 2 , PQ 2 rYcP¢ab; QB nig PC [RsbKña enaHeyIg)an C Eck Ggát; AB tampleFob 52 . -KUs l edA[)an AN 9, AM 6 rYcP¢ab; NB nig MD [RsbKña enaHeyIg)an D Eck Ggát; AB tampleFob 156 52 .
EckGgát; AB 12 cm CaGgát;smamaRtnwg 3 5 7 ³ A
/
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
3.
/
B
d
198
6.
eyIgmanrUb Edl AB// CD // EF // GH ³ A
C
E G
K> KNna DF nig CG ³ eday AC 6 cm , CE 12cm , DB 8cm nig FH 6 cm tamRTwsþIbTtaElseyIg)an ³ BD DF 8 DF b¤ 6 12 AC CE naM[ DF 8 612 16cm CG CE CE DH ehIy DH b¤ CG DF DF Et DH DF FH FH naM[ CG CE DF DF
B
D F
H
k> KNna DF nig FH ³ tamRTwsþIbTtaElseyIg)an BD DF FH eday AC 6 cm , AC CE EG CE 12 cm , EG 4cm nig BD 9 cm FH naM[ 96 DF 12 4 9 DF eyIg)an 6 12 enaH DF 9 612 18cm ehIy 96 FH4 enaH FH 9 6 4 6cm dUcenH KNna)an DF 18cm / FH 6cm
CG
dUcenH DF 16cm nig CG 16.5 cm . 7. KNnaRbEvg x énrUbnImYy²xageRkam ³ AN eday AB //MN enaH AM MB NC A -cMeBaHrUb 9
x> KNna BD / DF nig FH ³ tamRTwsþIbTtaElseyIg)an BD DF FH eday AC 6 cm , AC CE EG CE 12 cm , EG 4cm nig HB 33 cm tamlkçN³smamaRteyIg)an ³
dUcenH
8
M
N x
16
B
naM[
C
naM[
9 8 16 x
x
dUcenH KNna)an
BD DF FH BD DF FH AC CE EG AC CE EG BH 33 3 6 12 4 22 2 BD 3 3 AC 3 6 BD 9cm 2 2 AC 2 DF 3 3CE 3 12 DF 18cm CE 2 2 2 FH 3 3EG 3 4 FH 6cm EG 2 2 2
naM[
12 16 6 16.5 cm 16
x
-cMeBaHrUb
N 4
C
B
naM[
x
8 4 32 10.67 3 3
dUcenH KNna)an
x
32 10.67 3
naM[
199
.
x
M 3
BD 9cm , DF 18cm , FH 6cm
128 14.22 9 A
8
b¤ b¤ b¤
8 16 128 14.22 9 9
8 x 3 4
.
-cMeBaHrUb
eyIg)an a6 24 naM[ a 2 4 6 3 cm ehIy b8 6 6 4 naM[ b 810 6 4.8 cm
A x
M
6 4 N
B
dUcenH
C 12
AC MC eday AB //MN enaH BC NC Et AC x , MC 6 , NC 12 4 8 naM[ 12x 86 enaH x 6 812 728 9
dUcenH KNna)an 8.
10.
nig
kaMTI13
muxQrénkaM
.
x 9
KNnaRbEvg ED nig CD KitCa cm ³
kaMTI1 B
4
8
B
D
2
C
eday BE// CD tamRTwsþIbTtaElseyIg)an ³ AB AE Ggát;smamaRt BC ED 8 4 b¤ 2 ED naM[ ED 4 8 2 1 cm AB BE mü:ageTot AC CD 12 b¤ 108 CD naM[ CD 12 810 15 cm dUcenH
ED 1 cm
nig
CD 15 cm
AB
C
2.40m 0.12m 12 cm 20
ehIy RCugedk BC 3.40m RtUvnwgkñúgbøg; BC
3.40m 0.17m 17 cm 20
sg;rUb RtIekaN ABC EkgRtg; B ³ A
.
0.12m
KNnaRbEvg a nig b KitCa cm ³
B
0.17m
C
x> etIeKRtUvEck AB nig BC Cab:unµanEpñk ? -eKRtUvEck AB Ca 13 Epñk edIm,IkMNt;TItaMg km
6 a
b
4
2
3.40m
k> sg;RtIekaN ABC EkgRtg; B ³ eyIgmanbøgm; anmaRtdæan 201 naM[ RCugQr AB 2.40m RtUvnwgkñúgbøg;
E
12
.
b 4.8 cm
eyIgmanrUbdUcxageRkam ³ A
A
9.
a 3 cm
8
tamrUbxagelI eyIgmanGgát;edkTaMgBIrRsbKña 200
K> eFVIviFIEck AB nig BC ³
12.
eyIgmanrUbdUcxageRkam ³
y A
N
k> bkRsaysßanPaBenH edayrUbFrNImaRt ³
C
B
A x
M
-eyIgEck AB 0.12m 12 cm eday ³ KUsknøHbnÞat; By rYcedA[)an 13 RbeLaHRkit esµI²Kña ehIyP¢ab; NA nigP¢ab;Ggát;bnþbnÞab;[ Rsbnwg NA enaH AB RtUvEckCa 13 EpñkesµIKña. -eyIgEck BC 0.17m 17 cm eday ³ KUsknøHbnÞat; Bx rYcedA[)an 13 RbeLaHRkit esµI²Kña ehIyP¢ab; MC nigP¢ab;Ggát;bnþbnÞab;[ Rsbnwg MC enaH BC RtUvEckCa 13 EpñkesµIKña. 11. etIbnÞat; II nig JJ RsbKñab¤eT ?
P B
tag AB Cakm rkkm
X
O
/ /
/
I
/
/
/
dUcenH r)armankm
/
J
C
10m
J
I
1.65m 6.50m Q
/
4.714m
.
Y
cm 2 IJ 3cm 2 eday OOII 23cm nig 3 I J 4.5cm 3 OI IJ naM[ OI I J ¬manGgát;smamaRtKña¦ tamRTwsþIbTRcastaEls enaH II Rsbnwg JJ
13.
tambRmab;RbFaneyIgKUsrUb)an ³ B F A G
dUcenH bnÞat; II Rsbnwg JJ .
D 201
E
C
¬edIm,IgayRsYl ´sUmcmøgrUbdEdlkmvij¦ B F
E
C
A G
D
k> sresrRTwsþIbTtaElskñúgRtIekaN ³ -cMeBaH AGE nig ADC ³ eday AGE nig ADC man GE // DC dUcenH eyIg)an
AG AE GE AD AC DC
-cMeBaH AFE nig ABC ³ eday AFE nig ABCman dUcenH eyIg)an
.
FE // BC
AF AE FE AB AC BC
.
x> bgðajfa FG// BD ³ tamsMeNrénRTwsþIbTtaElsTaMgBIrxagelI AG AE AF AE ni g AB AC AD AC AF naM[ AG AD AB tamRTwsþIbTRcastaEls enaH FG// BD dUcenH FG// BD RtUv)anbgðajrYcral; .
¤ 202
១៦ 1. N F P
A
M
T
G
E
B
S
R
C
2. B
A 51o
R
H
L 34o
82o
40 o
C
57 o
D
ABC
3.
70
P
T
o
E
F
70 o
80 o
ខ
ABC , PMN
G
S
64o
M
N
PM N
A B
K
C
N
N
M M
C A
B
P
P
(A)
(B)
4. 14
16
16
8
32
7
6
30 36
12
(A)
5.
(B)
ABC
27
40
10 20
(C)
GEF
A G
B
C
G
C
A
E
E F
B
AB BC ...? GE ...? FG
F
AB ...? BC EG EF ...? 203
6.
A N
77 o
348m
290m
156.6m
58o
B
400m
7.
ABCD
AD
FB 2.25cm
45o
58o
M
C
130.5 m
BC
FD : FA 8: 5
F
BC
8.
35 dm
56 dm
10 dm
R
A
9.
P
180m
C
BAC 90o
ABC
RS BC
ABC
S
SRC B
10.
C .
D
MN 11 dm ,
a b
MN
a 3 b 5
ខ.
MN 6 dm ,
N
a 1 b 3
C
M D
11.
AB , AC , BC
ABC BC
I
BC
.
J
ABI
ខ.
AJC
AI AJ
.
BAJ CBJ .
JB 2 JA JI
12.
A
B
D AC
AC
DE // AB
13.
CD
D
ខ.
E
CA = 1.8 km ,
DE = 150 m
C
AB
ABC M
D
DE AB
CD = 90 m
B
A
C
.
BAC
A
N ABC
AMN
AB AM AC AN 204
AB
AC
14.
CD .
AB
AD = 4 dm , DB = 9 dm
ខ.
CD
AB = 29 dm , CD = 10 cm
AD
DB aha bhb chc
15. 16.
A
ABC
E
BC
ADB
17.
AB2 AD AE
AEB
ខ
10 m
P
3m 18.
D
P A
AB
A E
AE = 3 cm BA’
EA’=9dm
EB 19.
b
B
c
h a
h
b D
20.
ABCD
IB AD
AD .
C
I I
12 cm
IB
ខ.
A
B
10 cm
P
21.
NMQ
PNQ N
.
MQ
ខ.
3 cm
5 cm
NP Q
M
B 22.
CD I
AB
I
AI 15 cm , IC 12 cm ,
C
ID 4 cm , IB 5 cm , BD 3 cm . ខ.
IAC
IBD
AC
A
205
D
A
23.
P
A C
.
PAC
ខ.
D
PAD
P
AP2 PC PD
D
C
24.
10 cm 5 cm
4 cm
8 cm
25. . 3
v h r V H R ខ.
3
v
V H
h
v 6.28 cm3
h 2 cm , H 4 cm
r
V
DDCEE
3
206
R
១៦ 1.
sresreQµaH nigRCugRtUvKña ³ -cMeBaHKURtIekaNdUcKña ³
-cMeBaH rUb¬K¦ P
L 34o
82o
N
K 64o
M
P
eday PKLman L 180 34 dUcenH PKL MKN eRBaH vamanmMBu Ib:unerogKña . o
B C
÷sresreQµaH ³ NPM ACB NM PM ÷manRCugRtUvKña ³ NP . AC AB CB -cMeBaHKURtIekaNdUcKña ³
3.
F
o
82 o 64 o
eRbóbeFobRtIekaN rYcsresrpleFobdMNUc ³ -cMeBaH rUb(A) A
T
G
B
E
S
B
R
÷sresreQµaH ³ ÷manRCugRtUvKña ³ 2.
N
A
M
.
KURtIekaNxageRkamdUcKñab¤eT? ehtuGVI ? -cMeBaH rUb¬k¦ B
eday ABC naM[ ABC -cMeBaH rUb(B)
A
o
51
C
57 o
D
E
P
70
G
P
tamsmµtikmµénrUb PMN nig PM N man ³ M M nig P P dUcenH PMN PM N tamlkçxNÐ m>m
40 o 80 o
N
M
70 o
F
AB AC BC AB AC BC
N
M
RtIekaNTaMgBIrKW ABCnig ADE mindUcKñaeT eRBaH vamanEtmMumYyb:unKña KWxuslkçxNÐTaMgbI . -cMeBaH rUb¬x¦ R T H o
C A
tamsmµtikmµénrUb ABC nig ABC man ³ A A nig C C dUcenH ABC ABC tamlkçxNÐ m>m
EFG TRS EF FG EG TR RS TS
C
S
RtIekaNTaMgBIrKW HFG nig TSR mindUcKñaeT eRBaH vamanEtmMumYyb:unKña KWxuslkçxNÐTaMgbI .
eday 207
PMN PM N
PN MN naM[ PPM M PN M N
4.
KURtIekaNxageRkamdUcKñab¤eT? ehtuGVI ? -cMeBaH rUb(A)
5.
-bMeBjpleFobdMNUc ³ A B G
C
E
16
8
14
F
7
6
edayRtIekaN ABC dUcnwgRtIekaN GEF AB BC CA eyIgbMeBj)an ³ GE . EF FG -bMeBjpleFobdMNUc ³
12
RtIekaNTaMgBIrCa RtIekaN dUcKñay:agR)akd eRBaH RtIekaNTaMgBIrmanpleFobRCug ³ 6 1 7 1 8 1 6 7 8 , , enaH 12 2 14 2 16 2 12 14 16 mann½yfa vabMeBj[lkçxNÐdMNUc C>C>C . -cMeBaH rUb(B) 16 32
10 20
G C
A B
F
edayRtIekaN ABC dUcnwgRtIekaN GEF AB AC BC eyIgbMeBj)an ³ EG . EF GF 6. RsaybBa¢ak;fa RtIekaNdUcKña rYcTajrkvi)ak ³
RtIekaNTaMgBIrCa RtIekaN dUcKña eRBaH RtIekaNTaMgBIrmanpleFobRCug ³ 16 1 10 1 16 10 ehIyRCug , enaH 32 2 20 2 32 20 smamaRterogKñaTaMgBIrenHenAGmnwgmMub:unKñamYy KWmMuTl;kMBUl vabMeBj[lkçxNÐdMNUc C>m>C . -cMeBaH rUb(C) 40
E
A 348m
B
77 o
58o
400m
N 290m
156.6m
C
M
130.5 m 45o
58o
P
180m
edayRtIekaNTaMgBIrxagelImanpleFobRCug ³
27
naM[
30 36
NM 156.6m 0.45 AB 348m NP 130.5m 0.45 AC 290m MP 180m 0.45 BC 400m NM NP MP AB AC BC
dUcenH ABC NMP tamlkçxNÐ C>C>C -Tajvi)ak ³ A N 77 eday ABC nam[ NMP C P 45
RtIekaNTaMgBIrCa RtIekaN dUcKña eRBaH RtIekaNTaMgBIrmanpleFobRCug ³ 40 4 36 4 40 36 , enaH ehIyRCug 30 3 27 3 30 27 smamaRterogKñaTaMgBIrenHenAGmnwgmMub:unKñamYy KWmMuTl;kMBUl vabMeBj[lkçxNÐdMNUc C>m>C .
o
dUcenH 208
N 77 o , C 45 o
o
.
7.
KNnaRbEvg BC ³ tambRmab;RbFan eyIgKUsrUb)andUcxageRkam ³
AB AC BC AB AC BC
eyIgeRbIlkçN³smamaRt enaHeyIg)an ³ AB AC BC AB AC BC AB AC BC AB AC BC
F
A
Et
B
AB AC BC P 56 dm
C
D
AB AC BC P 35 dm
AC BC 35 5 1 naM[ AAB B AC BC 56 8 mü:ageTot AMB AM B tamlkçxNÐ m>m ¬eRBaH B B nig M M 90 ¦ AB AM vi)ak AMB AM B AB AM
eday ABCD CactuekaNBñayenaH AB// CD naM[ FAB FDA ¬mMuRtUvKña¦ ehIy FBA FCD ¬mMuRtUvKña¦ eyIg)an FAB FDC tamlkçxNÐ m>m FC FD vi)ak FDC FAB FB FA
o
h 2 b¤ AAB B H tam 1 nig 2 eyIg)an Hh 85 ehIybRmab;km
8 Et FD: FA 8 : 5 mann½yfa FD FA 5 nig FB 2.25 m ehIy FC FB BC BC 8 eyIg)an FBFB 5 2.25 BC 8 b¤ 2.25 5
dUcenH km
8 2.25 BC 2.25 5 BC 3.6 2.25 BC 1.35
9.
RsaybBa¢ak;fa ABC SRC ³ A
dUcenH KNna)anRbEvg BC 1.35 m . 8.
R
C
S
rkRbEvgkm
B
eday ABC nig SRC man ³ -mMu ACB SCR ¬mMurYm¦ -mMu BAC RSC 90 CamMuEkgdUcKña dUcenH ABC SRC tamlkçxNÐ m>m
A
A
o
h
B
M
C
B
H M
>
C
edayRtIekaNTaMgBIrdUcKña eyIg)ansmamaRt ³ 209
edAcMNuc C nig D edIm,IEckGgát; MN xagkñúg nigxageRkA tampleFob ba ³ k> eyIgman MN 11 dm , ba 53 ³
x> eyIgman MN 6 dm , ba 13 ³
10.
N Q
N Q
M
C
dUcenH
.
D
dUcenH
D
.
Eck MN xageRkAtampleFob
DM 1 DN 3
Eck MN xageRkAtampleFob
DM 3 DN 5
Eck MN xagkñúgtampleFob
-enAelIbnÞat; l edA[)ancMNuc ³ MQ 1 , MP 2 Edl M enAcenøaH QP KUsP¢ab; PN rYcKUs QD [Rsbnwg PN Edl D CaRbsBVrvag QD nigbnøayén MN enaH DMQ NMP tamlkçxNÐ m>m MQ DM 1 vi)ak DM b¤ MN 2 MN MP MN DM MN DN b¤ DM 1 2 1 2 3 DN DM 1 b¤ DM eyI g )an 1 3 DN 3
-enAelIbnÞat; l edA[)ancMNuc ³ MQ 3 , MP 2 Edl M enAcenøaH QP KUsP¢ab; PN rYcKUs QD [Rsbnwg PN Edl D CaRbsBVrvag QD nigbnøayén MN enaH DMQ NMP MQ DM 3 vi)ak DM b¤ MN 2 MN MP MN DM MN DN b¤ DM 3 2 3 2 5 DN DM 3 b¤ DM eyI g )an 3 5 DN 5 dUcenH
C
CM 1 CN 3
Eck MN xagkñúgtampleFob
CM 3 CN 5
l
¬eyIgedaHRsaydUclMnaM lMhat; k> Edr ¦ eyIgKUsbnÞat; l kat;tam M rYcedA[)an ³ MA 1, AB 3 Edl A nig B enAmçagén M KUsP¢ab; BN rYcKUs AC [Rsbnwg BN Edl C CaRbsBVrvag AC nig BN tamRTwsþIbTtaElseyIg)anpleFob CM AM CM 1 b¤ CN 3 CN AB
l
B
¬eXIjrUbdUclM)ak Ettamdan[c,as;KWFmµta¦ eyIgKUsbnÞat; l kat;tam M rYcedA[)an ³ MA 3 , AB 5 Edl A nig B enAmçagén M KUsP¢ab; BN rYcKUs AC [Rsbnwg BN Edl C CaRbsBVrvag AC nig BN tamRTwsþIbTtaElseyIg)anpleFob CM AM CM 3 b¤ CN 5 CN AB dUcenH
P B
PA D
A
D
C
C
M
.
.
¬´bkRsayEvg EbbBnül;mUlehtudl;GñkGan>>>¦ 210
11.
tambRmab;RbFaneyIgKUsrUb)an ³ A
naM[ JB JA JI dUcenH JB JA JI RtUv)anRsaybBa¢ak; . 12. KNnacm¶ay AB ³ 2
2
I C
B
k> RsaybBa¢ak;fa ABI dUcnwg AJC ³ eday ABI nig AJC man ³ -mMu ABC AJC ¬mMumanFñÚsáat;rmY AC ¦ -mMu BAI JAC ¬eRBaH AJ CaknøHbnÞat;BuH¦ dUcenH ABI AJC tamlkçxNÐ m>m . x> KNnaplKuN AI AJ ³ AI AB eday ABI AJC AC AJ naM[ AI AJ AB AC Et AB , AC enaH AI AJ dUcenH KNna)anplKuN AI AJ . K> bgðajfa BAJ CBJ ³ eday BAJ JAC ¬eRBaH AJ CaknøHbnÞat;BuH¦ ehIy JAC CBJ ¬mMucarwkFñÚsáat;rYm JC ¦ naM[ BAJ CBJ dUcenH BAJ CBJ RtUv)anbgðaj . X> RsaybBa¢ak;fa JB JA JI eday ABJ nig BIJ man ³ -mMu BAJ CBJ IBJ ¬bgðajxagelI¦ -mMu BJA IJB ¬mMurYmEtmYy¦ naM[ ABJ BIJ tamlkçxNÐ m>m JB AJ vi)ak ABJ BIJ JI JB
B
A
J
D
E
C
eday DE // AB enaHeyIg)anmMuRtUvKña ³ CDE CAB nig CED CBA naM[ CDE CAB CD DE vi)ak CDE CAB CA AB naM[ AB DECD CA 150 m901m.8 km KNna)an AB 3 km dUcenH RbEvgKNna)an AB 3 km . 13. tambRmab;RbFaneyIgKUsrUb)an ³ A
N M B
C
k> RbdUc ABCnig AMN ³ eday ABCnig AMN man -mMu ABC PAC ANM eRBaH ABC PAC mMucarwkmanFñÚsáat;rYm AC nig PAC ANM mMuqøas;kñúg -mMu ACB QAB AMN eRBaH ACB QAB mMucarwkmanFñÚsáat;rYm AB ehIy QAB AMN CamMuqøas;kñúg dUcenH ABC AMN tamlkçxNÐ m>m .
2
211
x> RsaybBa¢ak;fa AB AM AC AN ³ AB AC eday ABC ANM AN AM Taj)an AB AM AC AN dUcenH Rsay)anfa AB AM AC AN . 14.
-eyabl;´ ³ RbsinebIeyIgEkRbFanmkCa AB 29 dm , CD 10 dm vijenaHeyIg)an ³ CD 2 AD AB AD 10 2 AD 29 AD 100 29 AD AD2 AD2 29 AD 100 0
eyIgedaHRsayedayeRbIviFIKuNExVg ³
tambRmab;RbFaneyIgKUsrUb)an ³
AD 25 AD2 100 AD 4 4 AD 25 AD 29 AD
C
A
eyIg)an AD 4AD 25 0 naM[ AD 4 0 enaH AD 4 dm AD 25 0 enaH AD 25 dm -cMeBaH AD 4 dm ³ DB AB AD
B
D
eday C CacMNucénknøHrgVg;Ggát;p©it AB enaH ABC CaRtIekaNEkgRtg; C mankm KNna CD ebI AD 4 dm , DB 9 dm ³ naM[ CD 4dm 9dm 36dm 6 dm dUcenH KNna)an CD 6 dm . 2
29 4 25dm
-cMeBaH AD 25 dm ³ DB 29 25 4dm dUcenH AD 4 dm nig DB 25dm b¤ AD 25 dm nig DB 4dm .
2
x> KNna AD nig DB ³ eyIgman AB 29 dm , CD 10 cm 1dm eday CD AD DB
>
15.
RsaybMPøW[eXIjfa ah
a
B
c C
CD AD AB AD 2
³
>
A
2
bhb chc
b
hb h a h c
1 AD 29 AD 2
A
B
1 29 AD AD2
a
C
kñúgRtIekaN AAB nigRtIekaN CCB man ³ -mMu A C 90 ¬ A nig C CaeCIgkmm AB vi)ak ACACBB CACA CB b¤ hh ac Taj)an ah ch ¬k¦
AD2 29 AD 1 0
¬CasmIkardWeRkTI2manb£sminEmnCacMnYnKt; dUcenH sisSBMuGacedaHRsay)aneLIy . ´eXIjmanesovkMEN CaeRcIndak;lk;enAelITIpSar ecHEt edaHRsayy:agRsYl edaymin)anKitBIxñat dm nig cm eFVI[cemøIyrbs;eK xusKW 25dm b¤ 4dm rebobedaHRsayenAfñak;TI 10 ehIycemøIyEdlRtUvKW AD 29 2 837 dm ¦.
o
a c
a
212
c
kñúgRtIekaN AAC nigRtIekaN BBC man ³ -mMu A B 90 ¬ A nig B CaeCIgkmm AC vi)ak ABABCC BABA BC b¤ hh ba Taj)an ah bh ¬x¦ tamTMnak;TMng ¬k¦ nig ¬x¦ eyIg)an ah bh ch dUcenH ah bh ch RtUv)anRsaybMPøW .
17.
KNnaedImb£sSIBIKl;eTAcMNuc)ak; P ³ P
o
10 x
x
B
3m
A
a
tag x CaRbEvgb£sSIBIKl; A eTAcMNuc)ak; P tamRTwsþIbTBItaK½rcMeBaHRtIekaNEkg PAB ³
b
a
b
a
b
a
16.
32 x 2 10 x 2
9 x 100 20 x x 2
c
b
x 2 x 2 20 x 100 9 20 x 91
c
A
dUcenH BIKl;eTAcMNcu )ak;manRbEvg 4.55 m . 18.
KNnaCeRmATwk EB ³
D C A
E
A
RtIekaN ADB nigRtIekaN AEB man ³ -mMu BAD EAB ¬mMurYm¦ -mMu ABC AEB eRBaHedaysarTMnak;TMng ABC ACB mMu)aténRtIekaNsm)at ABC ACB AEB mMucarwkmanFñÚsáat;rYm AB dUcenH ADB AEB tamlkçxNÐ m>m .
E
B
edayépÞTwkEkgnwgpáaQUkQr naM[ EAB EkgRtg; E ehIytamBItaK½r ³ Et AB AB AE EB ¬edImQUkEtmYy¦ eyIg)an AE EB AE EB eday AE 3dm , AE 9 dm 3 EB 9 EB enaH 2
2
2
2
2
2
2
9 6 EB EB2 81 EB2 6 EB 81 9
2
AB2 AD AE
>
AB 2 AE 2 EB2
-rYcTajbBa¢ak;fa AB AD AE ³ AB AD eday ADB ABE AE AB Taj)an AB AB AD AE b¤ AB AD AE dUcenH
91 m 20
x
eRbóbeFobRtIekaN ADB nigRtIekaN AEB³
B
2
EB
RtUv)anbBa¢ak; .
dUcenH CeRmATwkKW 213
72 12 dm 6
EB 12 dm
.
19.
KNna h eTAtamtémøén b nig c ³
x> KNnaépÞRkLaénRbelLÚRkam ³ eday ABCD CaRbelLÚRkamman)at AD nig mankm
a
h b
tamRTwsþIbTBItaK½r RbEvgGIub:UetnusKW tamlkçN³énRtIekaNEkg eyIg)an ³ h a b a b 2
a 2 b2
³
2
h
S ABCD AD IB 12 8 96 cm 2
dUcenH KNna)an 21.
eyIgmanrUbdUcxageRkam ³
ab
P
a b 2
2
CaFmµta eKminEdlTukr:aDIkal;enAPaKEbg naM[ h ab aba a b b 2
a2 b2
2
2
k> KNnargVas; MQ ³ tamRTwsþIbTBItaK½rcMeBaHRtIekaNEkg MNQ ³ eyIg)an MN MQ NQ
2
2
2
D
2
2
MQ2 NQ 2 MN 2
C
MQ NQ 2 MN 2
I
12 cm
A
52 32 16 4 cm
dUcenH KNna)an
B
10 cm
k> KNna IB ³ eyIgman ABCDCaRbelLÚRkam nig I CacMNuc kNþal AD ehIy IB AD enaHeyIg)an ³ AD BC 12 cm 6 cm 2 2 2
2
2
2
IB
Taj)an
2
AB2 IA2
10 2 6 2 100 36 64 8 cm
KNna)an
IB 8 cm
.
o
2
2
MQ 4 cm
x> KNnargVas; NP ³ tamsmµtikmµénrUb MNQ nig NPQ man ³ NMQ PNQ 90 nig PQN NQM naM[ MNQ NPQ tamlkçxNÐ m>m MN MQ vi)ak MNQ NPQ NP NQ
tamRTwsþIbTBItaK½rcMeBaH RtIekaNEkg AIB eyIg)an IA IB AB naM[ IB AB IA
dUcenH
Q
M
eyIgmanrUbdUcxageRkam ³
IA
5 cm
3 cm
2
KNna)antémø h aba a b b .
dUcenH 20.
N
2
2
.
S ABCD 96 cm 2
MN NQ MQ 3 5 15 NP 3.75 cm 4 4
NP
dUcenH KNna)an
. 214
NP 3.75 cm
.
22.
eyIgmanrUbdUcxageRkam Edlman ³ AI 15cm , IC 12cm , ID 4cm , IB 5cm
nig BD 3 cm
x> Taj[eXIjfa AP PC PD PA PC eday PAC PDA PD PA Taj)an PA PC PD dUcenH Taj)anfa PA PC PD . 2
>
B
2
D
I
2
C
24.
A
k> eRbóbeFobRtIekaN IAC nig IBD ³ eday IAC nig IBD man ³ IB 5 cm 1 ID 4 cm 1 nig IA 15cm 3 IC 12cm 3 ehIy BID CIA ¬mMuTl;kMBUl¦ dUcenH IBD IAC tamkrNI C>m>C . x> KNnargVas; AC ³ BD 1 eday IBD IAC enaH IB IA AC 3 BD 1 b¤ naM[ AC 3BD AC 3 eyIg)an AC 3 3cm 9 cm dUcenH KNna)an AC 9 cm . 23.
etIekaNTaMgBIrdUcKñab¤eT ? 10 cm 5 cm
4 cm
8 cm
ekaNTaMgBIrCaekaNdUcKña luHRtaEtpleFob FatuRtUvKñaesµIKña cm 1 h 5cm 1 eday Dd 84cm ehIy 2 H 10cm 2 eyIg)an pleFob Dd Hh 12 dUcenH ekaNTaMgBIrCaekaNdUcKña . 25.
eyIgmanekaNBIrdUcKña dUcrUbxageRkam ³
eyIgmanrUbdUcxageRkam ³ A
H
h
P
C
R
r
D
k> eRbóbeFobRtIekaN PAC nigRtIekaN PAD³ eday PACnig PAD ³ -mMu P CamMurYm -mMu PAC ADP mMucarwkmanFñÚsáat;rYm AC dUcenH PAC PAD tamlkçxNÐ m>m .
3
v h r V H R 1 v r 2 h 3 1 V R 2 H 3
3
k> bgðajfa ³ -maDekaNtUc -manekaNFM eyIgeFVIpleFobmaDekaNTaMgBIr enaHeyIg)an ³ 215
1 2 r h v r 2h 3 2 V 1 2 R H R H 3
edayekaNTaMgBIrdUcKña enaHpleFob Rr Hh naM[ Vv Rr rR b¤ Vv Hh hH eyIg)an Vv Rr b¤ Vv Hh 2
2
2
2
3
3
3
3
Gacsresr Vv Rr
3
h H
3
3
v r h V R H
dUcenH bgðaj)anfa
3
.
x> KNnamaD V énekaNFM ³ ¬esovePABum
3
v h V H
3
eyIgman eday v 6.28 cm , h 2cm , H 4cm 6.28 2 naM[ V 4 3
3
3
6.28 1 V 2 6.28 1 V 8 V 6.28 8
V 50.24 cm3
dUcenH KNna)an
V 50 .24 cm 3
.
216
១៧ 1.
7 8 9
10
12 4860o
1800o , 2 700o
2. 51300
3.
540o
4.
5.
x x
100o 80 o
3x
x
x
2x
50 o
2x
2x x
x o
2x
2x
160
ខ
y
6.
2y
2y
y
2y
y
80o
y
3y
70 o
2y
3y
2y
y
y
2y 2y
ខ
150o , 160o
7.
175o
3240o
8.
900o
9. 10.
168o
n
n
11. .
n6
ខ.
n5
.
n 8 . .
12 900o
12. 13.
12 27
14.
o
n
n 217
15.
n n 3 2
n :
16. 17.
n
30o
n
DDCEE
3
218
១៧ 1.
KNnaplbUkrgVas;mMukñúgénBhuekaN ³ -krNIBhuekaNmancMnYnRCugesµI 7 ³ eday plbUkmMukñúg n 2180 nig n 7 naM[ plbUkmMukñúg 7 2 180 900 dUcenH BhuekaNEdlmancMnYnRCgesµI 7 vamanplbUkmMukñúgesµI 900 .
-krNIBhuekaNmancMnYnRCugesµI 12 ³ eday plbUkmMukñúg n 2180 nig n 12 naM[ plbUkmMukñúg 12 2180 1800 dUcenH BhuekaNEdlmancMnYnRCgesµI 12 vamanplbUkmMukñúgesµI 1800 . o
o
o
o
o
o
o
o
-krNIBhuekaNmancMnYnRCugesµI 8 ³ eday plbUkmMukñúg n 2180 nig n 8 naM[ plbUkmMukñúg 8 2180 1080 dUcenH BhuekaNEdlmancMnYnRCgesµI 8 vamanplbUkmMukñúgesµI 1080 .
2.
o
o
o
o
n 2180o 1800o 1800o n2 180o n 10 2
o
n 12
-krNIBhuekaNmancMnYnRCugesµI 9 ³ eday plbUkmMukñúg n 2180 nig n 9 naM[ plbUkmMukñúg 9 2180 1260 dUcenH BhuekaNEdlmancMnYnRCgesµI 9 vamanplbUkmMukñúgesµI 1260 .
dUcenH BhuekaNEdlmanplbUkmMukñúg esµI 1800 vamancMnYnRCugesµI 12 .
o
o
o
o
-krNIplbUkmMukñúgesµI 2700 ebIBhuekaNenHmancMnYnRCugesµInwg n eyIg)an ³ o
o
n 2180o 2700o
-krNIBhuekaNmancMnYnRCugesµI 10 ³ eday plbUkmMukñúg n 2180 nig n 10 naM[ plbUkmMukñúg 10 2180 1440 dUcenH BhuekaNEdlmancMnYnRCgesµI 10 vamanplbUkmMukñúgesµI 1440 .
2700o n2 180o n 15 2
o
o
rkcMnYnRCugénBhuekaN edaysÁal;plbUkmMukñúg ³ -krNIplbUkmMukñúgesµI 1800 ebIBhuekaNenHmancMnYnRCugesµInwg n eyIg)an ³
o
n 17
dUcenH BhuekaNEdlmanplbUkmMukñúg esµI 2700 vamancMnYnRCugesµI 17 .
o
o
219
-krNIplbUkmMukñúgesµI 4860 ebIBhuekaNenHmancMnYnRCugesµInwg n eyIg)an ³
eyIg)anplbUkmMukñúg
o
x 3x x 80o 4 2180o 5 x 80o 360o
n 2180o 4860o
5 x 360o 80o
4860o n2 180o n 27 2 29
280o x 56o 5
dUcenH témøKNna)anKW
dUcenH BhuekaNEdlmanplbUkmMukñúg esµI 4860 vamancMnYnRCugesµI 29 . o
3.
.
-cMeBaHrUb ¬x¦ ³ CaBhuekaNmancMnYnRCug 5 ³
rkBhuekaNEdlmanplbUkmMukñúgesµI 5130 ebI n CacMnYnRCugénBhuekaN enaH n CacMnYnKt; eyIg)an n 2180 5130 o
o
x 56o
100o x
50 o
2x
o
2x
eyIg)anplbUkmMukñúg
5130 o n2 180 o n 28 .5 2 30 .5
100o x 2 x 2 x 50o 5 2180o 5 x 150o 540o
tamlT§plbgðajfa cMnYnRCugminEmnCacMnYnKt; dUcenH KµanBhuekaNNaEdlman plbUkmMukñúgesµI 5130 eLIy .
5 x 540o 150o x
390o 78o 5
o
4.
dUcenH témøKNna)anKW
rkcMnYnRCugénBhuekaNe):agenaH ³ ebI n CacMnYnRCugénBhuekaNenaH eyIg)an ³
2x
2x x
x
540o n2 180o n 3 2 5
o
2x
160
eyIg)anplbUkmMukñúg
dUcenH BhuekaNe):agenaHmancMnYnRCugesµI 5 .
160o x 2 x 2 x x 2 x 6 2180o 8 x 160o 720o
rktémø x énrUbBhuekaNnImyY ² ³ -cMeBaHrUb ¬k¦ ³ CaBhuekaNmancMnYnRCug 4 ³
8 x 720o 160o x
x
dUcenH témøKNna)anKW
80 o
3x
.
-cMeBaHrUb ¬K¦ ³ CaBhuekaNmancMnYnRCug 6 ³
n 2180o 540o
5.
x 78o
x
220
560o 70o 8
x 70 o
.
6.
KNnatémø y tamrUbnImYy²xageRkam ³ -cMeBaHrUb ¬k¦ ³ manplbUkmMueRkAesµI 360
7.
o
o
2y
y
n 2180o
150o n 180o n 360o 150o n
y 2y
eyIg)anplbUkmMueRkA
180o n 150o n 360o
y 2 y y 2 y 360 o
30 o n 360o
6 y 360 o
n
y 60 o
dUcenH témøKNna)an
.
y 60 o
-BhuekaNniy½tEdlmanrgVas;mMukñgú esµI 160 ebI n CacMnYnRCugénBhuekaNniy½t eyIg)an
o
o
y 80o
eyIg)anplbUkmMueRkA
n 2180o
y
o
360o 12 30o
dUcenH BhuekaNniy½tenaHmancMnYnRCug 12 .
-cMeBaHrUb ¬x¦ ³ manplbUkmMueRkAesµI 360 70
KNnacMnYnRCugénBhuekaNniy½t ³ -BhuekaNniy½tEdlmanrgVas;mMukñgú esµI 150 ebI n CacMnYnRCugénBhuekaNniy½t eyIg)an
160o n 180o n 360o 160o n
y
70o 80o y y y 360o
180o n 160o n 360o
3 y 150o 360o
20 o n 360o
360o 150o y 70o 3
n
360o 18 20 o
dUcenH BhuekaNniy½tenaHmancMnYnRCug 18 .
dUcenH témøKNna)an y 70 . -cMeBaHrUb ¬K¦ ³ manplbUkmMueRkAesµI 360 o
o
-BhuekaNniy½tEdlmanrgVas;mMukñgú esµI 175 ebI n CacMnYnRCugénBhuekaNniy½t eyIg)an
o
2y
2y
3y
2y
3y
n 2180o
175o n 180o n 360o 175o n
2y 2y
eyIg)anplbUkrgVas;mMueRkA
180o n 175o n 360o
3 y 2 y 2 y 3 y 2 y 2 y 2 y 360o
5 o n 360o
16 y 360o y
dUcenH témøKNna)an
n
360o 22.5 o 16
y 22 .5o
360o 72 5o
dUcenH BhuekaNniy½tenaHmancMnYnRCug 72 .
. 221
8.
KNnargVas;mMueRkAnImYy²énBhuekaNniy½t ³ plbUkrgVas;mMukñúgesµInwg 3240 ebI n CacMnYnRCugénBhuekaNniy½t eyIg)an
rkcMnYn n ³ eday BhuekaNniy½tEdlman n RCug nigman rgVas;mMukñúgmYyesµI 168 enaHeyIg)an ³
10.
o
o
n 2180o
n 2180o 3240o
168o n 180o n 360o 168o n
o
3240 180o n 18 2
n2
180o n 168o n 360o
n 20
12o n 360o
eyIgdwgfa cMnYnRCug = cMnYnmMueRkA naM[ cMnYnmMueRkA n 20 edaymMueRkAnImYy² énBhuekaNniy½tmanrgVas; esµI²Kña nigmanplbUkesµI 360 eyIg)an rgVas;mMueRkAnImYy² 36020 18 o
n
360o 30 12o
dUcenH cMnYnEdlRtUvrkKW n 30
.
KNnargVas;mMueRkAnImYy²énBhuekaNniy½t ³ eyIgdwgfa mMeu RkAnImYy²énBhuekaNniy½t kMNt;eday 360n Edl n CacMnYnRCug k> qekaNniy½t n 6 eyIg)an 360n 3606 60
11.
o
o
o
dUcenH rgVas;mMueRkAnImYy² énBhuekaN niy½tenaHKW 18 . o
o
o
o
9.
rkcMnYnRCug énBhuekaNniy½tenaH ³ ebI n CacMnYnRCugénBhuekaNniy½tenaH eyIg)an -plbUkrgVas;mMukñúgesµInwg n 2180 -plbUkrgVas;mMueRkA 360 eday plbUkrgVas;mMukñúgrYmnwgplbUkrgVas;mMueRkA esµInwg 900 naM[eyIgGacsresr)an ³
dUcenH mMueRkAnImYy²esµInwg 60 . o
x> bBa©ekaNniy½t n 5 eyIg)an 360n 3605
o
o
o
o
72 o
dUcenH mMueRkAnImYy²esµInwg 72 . o
o
K> GdæekaNniy½t n 8 eyIg)an 360n 3608
n 2180o 360o 900o n 2180o 900o 360o
o
o
540 180o n 3 2
n2
o
45 o
dUcenH mMueRkAnImYy²esµInwg 45 . o
n5
X> BhuekaNniy½tEdlmanRCug 12 eyIg)an 360n 360 30 12
dUcenH cMnYnRCugénBhuekaNniy½tenaHKW 5 .
o
o
o
dUcenH mMueRkAnImYy²esµInwg 30 . o
222
rkcMnYnRCug énBhuekaNenaH ³ ebI n CacMnYnRCugénBhuekaNenaH eyIg)an -plbUkrgVas;mMukñúgesµInwg n 2180 -plbUkrgVas;mMueRkA 360 eday plbUkrgVas;mMukñúg nigmMueRkAesµIngw 900 naM[ eyIgGacsresr)an ³
eXIjfa BhuekaN ABCDEFGHJKLM man³ -mMu ABC ABI IBC 60 90 150 naM[ mMukñúgnImYy²esµIKña esµInwg 150 -RCug AB BC CD ... MA eyIgrkcMnYnRCugéBhuekaNtamTMnak;TMng ³
12.
o
o
o
o
o
o
n 2180o
150o n 180o n 360o 150o n
n 2180o 360o 900o n 2180o 900o 360o
180o n 150o n 360o
540o 180o n 3 2 5
30 o n 360o
n2
n
dUcenH cMnYnRCugénBhuekaNenaHKW
dUcenH BhuekaNEdlekItCaBhuekaN niy½tmancMnYnRCugesµI 12 .
.
5
360o 12 30o
RsaybBa¢ak;farUbEdlekItCaBhuekaNniy½t EdlmancMnYnRCugesµInwg 12
13.
A
RsayfaBhuekaNEdlman n RCugenaHcMnYn Ggát;RTUgrbs;vaesµInwg nn2 3 ³ eKGacKUsBhuekaN)anluHRtaEtcMnYnRCugrbs; vaRtUvFMCagb¤esµI 3 -cMeBaH n 3 vaCaRtIekaN cMnYnGgát;RTUg 332 3 0 -cMeBaH n 4 vaCaRtIekaN cMnYnGgát;RTUg 442 3 2 -cMeBaH n 5 vaCaRtIekaN cMnYnGgát;RTUg 552 3 5 -cMeBaH n 6 vaCaRtIekaN cMnYnGgát;RTUg 662 3 9 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> dUcenH BhuekaNman n RCugenaHcMnnY Ggát;RTUgesµI nn2 3 , n 3 .
14.
M L
B
I
C
K 120o
D
J
E
H F
G
eyIgP¢ab;kMBUlkaerenACab;²Kña enaHeyIgnwg)an BhuekaN ABCDEFGHJKLM eyIgman qekaNniy½tEdlrgVas;mMukñúgnImYy² esµInwg 6 26180 120 naM[ AIB 360 120 90 90 o
o
o
o
o
o
360o 300o 60o
kñúgRtIekaN AIB man IA IB nig AIB 60 naM[ AIB CaRtIekaNsm½gS vi)ak AIB IBA IAB 60
o
o
223
o
KNnamMukñúg nigmMueRkAénqekaNniy½t ³ -plbUkmMukñúgénqekaN KWesµInwg
15.
6 2180 o 4 180 o 720 o
edaymMunImYy²manrgVas;esµIKña naM[ mMukñúgnImYy²esµInwg 7206 120 -plbUkmMueRkAénqekaN KWesµInwg 360 naM[ mMueRkAnImYy²esµInwg 3606 60 o
o
o
o
o
dUcenH kñúgqekaNniy½t KNna)an ³ mMukñúgesµInwg 120 / mMueRkAesµInwg 60 . o
o
KNnacMnYnRCug n énBhuekaNniy½t ³ -plbUkmMueRkAénqekaN KWesµInwg 360 naM[ mMueRkAmYyénqekaNniy½tKW 3606 60 -eKmanmMueRkAmYyénBhuekaNniy½t manrgVas; 30 eRcInCag rgVas;mMueRkAmYyénqekaNniy½t naM[ mMueRkAénBhuekaNniy½tenaHesµInwg 60 30 90 > eyIg)an cMnYnRCugénBhuekaNniy½tenaHesµInwg
16.
o
o
o
o
o
n
o
o
360 o 4 90 o
dUcenH BhuekaNniy½tenaHmancMnYnRCug n 4 ¬BhuekaNniy½tenaHCakaer¦ .
224
១៨ 1. S S
S
14 cm
9 cm 14cm
o
15 cm
8 cm
8 cm
(a)
(b)
(c)
2.
k> R 5cm , h 12cm X> R 5.2 cm , h 11cm
x> R 3.4 cm , h 8.9 cm g> R 5 mm , h 13.4 mm
K> R 3.7 cm , h 7 cm c> R 7 dm , h 16 dm
3.
k> R 3 cm
x> D 18 dm
k>
46 x> S 16 m
K> R 3.7 cm
X> D 23.2 cm
K>
X>
4.
S
36 cm 2 25
2
S
12 dm 2 5
5. . V 288 cm ខ. V
S 1764 cm
S 3
32 m3 16
6 cm
6. .
o
4.5 cm
ខ.
A
7.
AB // CD
CED
B
BE 1 CE 2 16 cm 2
E
C
AEB
D A
8.
DEFG
ABCD
B
E
F
D
G
C
M N
225
9.
ABCD BD
I
CI
A
AB
o
BIO
O
DOC
C
D
10.
9 cm 2
3 cm
m
6m
2
144 cm 2
11.
256 cm 2
1
18 cm
1
9 600 m 2
12.
B
I
cm
2 1 2 500
cm 2 x
13.
6
ខ
6 cm
.
10 cm
4
10
x
ខ.
14.
2
2
SABCD 80 cm
OO’= x
EFGK
. ខ.
SO
60 cm EFGK
x ខ
x 30 cm
ABCD
BA 6 , SA 8 , SA '
15.
EFGK
2 SA 3
S
cm .
A’B’
ខ.
.
ABCD
.
2 SA’B’C’D’ SABCD 3 1 SM ' SM , SO 10 cm , OM 6 cm 2 O'M '
O '
1 2
ខ. .
2
SO '
H
B
D
C O
3
16.
O
A
2
2 A’B’C’D’ 3 2 SO ' SO 3
C
D
A’B’C’D’
A
H B
S
o
O 1 SO 2
M
M
o O’ 226
S
O
3
1 2
17.
1 4
18.
9 10
-
7 10 19.
R R
h
h h
.
ខ.
R S
20.
R
h R 2
A
o
h
B
. ខ.
C
o
R 2
21.
D R
2a a
2a
.
2a
ខ.
2a
22.
r
2 cm
r
3
23.
2 cm
1 cm 1 cm 1 cm
DDCEE
3 227
1 cm
r
១៨ 1.
rképÞRkLaxag nigépÞRkLaTaMgGs; ³ -cMeBaHrUb (a) S
÷KNnaépÞRkLaTaMgGs; ³ tamrUbmnþ S S S EtépÞRkLa)atmanragCaqekaNniy½t eyIg)an ³ T
1 S B 6 8 82 4 2 2 24 48 96 3 cm 2
15 cm
÷KNnaépÞRkLaxag ³ tamrUbmnþ S 12 pa 12 4 15cm 9 cm KNna)an S 270 cm dUcenH épÞRkLaxagKW S 270 cm .
8 cm
naM[ S 336 cm 96 3 cm 502 .28 cm dUcenH RkLaépÞTaMgGs;KW S 502 .28 cm . 2
L
2
2
2
T
2
-cMeBaHrUb (c)
L
÷KNnaépÞRkLaTaMgGs; ³ tamrUbmnþ S S S EtépÞ)at S 15 cm 15 cm 225 cm naM[ S 270 cm 225 cm 495 cm dUcenH épÞRkLaTaMgGs;KW S 495 cm .
S
14 cm
B
2
2
2
8 cm
o
B
2
÷KNnaépÞRkLaxag ³ tamrUbmnþ S Ra 3.14 8 cm 14 cm KNna)an S 351 .68 cm dUcenH épÞRkLaxagKW S 351 .68 cm .
T
2
T
-cMeBaHrUb (b)
2
T
L
L
B
9 cm
T
L
L
2
S
L
2
L
÷KNnaépÞRkLaTaMgGs; ³ tamrUbmnþ S S S EtépÞ)at S R 3.14 8 cm 200 .96 cm naM[ S 351 .68 cm 200 .96 cm KNna)an S 552 .64 cm dUcenH épÞRkLaTaMgGs;KW S 552 .64 cm .
14cm
T
8 cm
÷KNnaépÞRkLaxag ³ tamrUbmnþ S 12 pa 12 6 8 cm 14cm KNna)an S 336 cm dUcenH épÞRkLaxagKW S 336 cm .
L
B
2
2
2
B
2
2
T
L
2
2
T
L
2
2
T
L
228
2.
KNnamaDekaNtamkrNIdUcxageRkam ³ k> eyIgman R 5cm , h 12cm tamrUbmnþ V 13 S h b¤ V 13 R h naM[ V 13 3.14 5 cm 12 cm KNna)an V 314 cm dUcenH manDekaNKW V 314 cm .
g> eyIgman R 5 mm , h 13.4 mm tamrUbmnþ V 13 S h b¤ V 13 R h naM[ V 13 3.14 5 mm 13.4 mm KNna)an V 70.127 mm dUcenH manDekaNKW V 70.127 mm . 2
B
2
2
B
2
3
3
3
3
x> eyIgman R 3.4 cm , h 8.9 cm tamrUbmnþ V 13 S h b¤ V 13 R h naM[ V 13 3.14 3.4 cm 8.9 cm KNna)an V 107 .685 cm dUcenH manDekaNKW V 107 .685 cm .
c> eyIgman R 7 dm , h 16 dm tamrUbmnþ V 13 S h b¤ V 13 R h naM[ V 13 3.14 7 dm 16 dm KNna)an V 820 .59 dm dUcenH manDekaNKW V 820 .59 dm . 2
B
2
2
B
3
2
3
3
3
3.
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2
2
2
2
B
2
2
3
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3
3
3
3
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B
2
3
2
2
3
2
2
229
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3
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3
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2
2
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1 R 2
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.
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2
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.
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“No part of this test may be reproduced in any form without permission”
RbsinebIelakGñkmanbBaða b¤cm¶l;Rtg;cMNucNa EdlmanenAkñúgEpñkenH elakGñkGacTak;Tg;eTAkan; RKU b¤mitþPkþirbs;GñkEdlmansmtßPaB b¤GñkeroberogesovePAenHkñúgeBlevlasmKYr . …
iv
1) Calculate and simplify. x 2 6 x 8 x 2 2 x 8
x
2) Solve the following inequality. 3 x 2 5x 2
3 x 2 5x 2
3) Express a in terms of 2 a b
2
6 x 8 x 2 2 x 8
and b .
a
b
2 a b
0, 3
4) Find the equation of the straight line that passed through point 0, 3 with a slope
7។
of 7.
l m n
x
5) Find x when l m n as shown in the figure at right.
l
l 3cm
3cm
4cm
m
4cm
m
5cm
xcm
5cm
n
n
6) Expand and simplify. 2 2x 1 2 x 3 2 x 5
2x 1 2 x 3 2 x 5
7) Solve the following equation. 2 x 1 0
x 1
2
238
2
0
xcm
8) Calculate and simplify. 10 5 7 5 5 5
9) Find the range of y when the domain is 2 x 1 regarding the quadratic 1 function y x 2 . 2
5 7
5 5
10 5
y 2 x 1
10) The width and diagonal of a rectangle are 5 cm and 35 cm, respectively. Find its length.
11) Factorize the following expression. xyz xy xz x
y
1 2 x ។ 2
35 cm។
5 cm ។
xyz xy xz x
5x2 19 x 4 0 ។
12) Given the quadratic inequality 5x2 19 x 4 0 . ① Solve the quadratic inequality. ② Draw the range of x found in ① on a number line.
13) Find cos A given that sin A
។
①
②
3 for ABC 5
ABC
cos A 3 sin A 5
with C 90o .
14) How many different ways are there to create a team consisting of one male and one female from among 4 boys and 5 girls?
15) Given that the geometric sequence below. 125 , 25 , A , 1 , B , … ① Find A . ② Find B .
239
①។
x
C 90o ។
4
125 , 25 , ① រក A ។
② រក
B ។
A
5
,1,
B
,…
។
16) Expand and simplify the following expression. x y 4x y 2x y 2x y
x y 4x y 2x y 2x y
17) Express 2010 as the product of prime numbers.
2010
18) Find x satisfying the following equation. 4 x2 5x 8 0
x
19) Calculate expression.
and
simplify
52 2
។
4 x2 5x 8 0
the
following 2 5 5 2 2
20) Find the value of a when the parabola y ax 2 passes through point 2 , 8 .
)
52 2
5 2
y ax 2
a
2 , 8 ។
21) In the diagram on the right, find x when DE BC . C
E
2 5 2
DE BC :
x
C
E
6
6
4 A
x
D
4 A
B
3
22) The diagram on the right shows a right-angled triangle with BC=3cm , AC = 7cm and C=90o . Find the length, in cm, of side AB.
D
x
cm ។
A
A
7cm
B
3cm
7cm
B
C
240
B
C = 90o ។
BC = 3cm , AC = 7cm
AB
3
3cm
C
23) Expand and simplify the following expression. x 6 y x2 6 xy 36 y 2
x 6 y x2 6xy 36 y 2
24) Factor the following expression. x8 1
x8 1
25) Calculate the following expression. Rationalize the denominator if your answer is a fraction. 2 2 2
26) Assume that the universal set consists of integers from 1 to 9. When A = {1 , 3 , 7 , 8 } and B = {1 , 4 , 6 , 7 } find the elements of A B .
2
)
1 A B ។
B = {1 , 4 , 6 , 7 }
①
x
②
x
①
A 3B B x2 x 1 ។
A 3x 2 x
sin cos
0 180 ។ o
)
10 C8
241
2 x 2 3x 9 0 ។
2
for
30) Calculate the following combinations. ① 10 C2
9។
A = {1 , 3 , 7 , 8 }
28) Calculate A 3B when A 3x 2 2 x and B x2 x 1 .
②
22
:
27) Consider the quadratic inequality 2 x 2 3x 9 0 . ① Solve for x in the quadratic inequality. ② Show the range of x in ① on a number line.
29) Find that satisfies sin cos 0o 180o .
①
10
②
10
C2 C8
o
។
31) Expand and simplify the following 2 expression. x 3 y x y x 5 y
៣១)
32) Factor the following expression. 4 x2 225
៣២)
33) Solve for x in the following equation. x 2 10 x 6 0
៣៣)
34) Simplify the following expression. 15 5 3 2 5 5
៣៤)
35) y is directly proportional to the square of x and y 45 when x 3 . Express y in terms of x .
៣៥)
36) In the diagram on the right, find x when m. x
៣៦)
x 3 y x y x 5 y 2
4 x2 225
x
x 2 10 x 6 0
5 3 2 5
15 5
y x 3។
y
x។
m
x
x
4 m
y 45
x
4
1
m
2
1 2
37) For a right-angled triangle, Find the length of the hypotenuse when the other two sides are 5 cm and 12 cm.
242
)
5 cm 12 cm ។
38) Expand and simplify the following expression. x 2 y x 2 2 xy 4 y 2
៣៨)
39) Factor the following expression. x3 6 x 2 y 12 xy 2 8 y 3
៣៩)
40) Solve the following linear inequality. 3x 12 9x 18
៤០)
41) Find the coordinates of the vertex of the parabola y x 2 8 x 15 .
៤១)
42) ① Solve the quadratic inequality 5 x 2 14 x 3 0 . ② Show the range of x in ① on the number line.
៤២) ①
43) Find the value of
10
x 2 y x2 2xy 4 y 2
x3 6 x 2 y 12 xy 2 8 y 3
3x 12 9x 18
y x 2 8 x 15 ។
២
5 x 2 14 x 3 0 x ②
៤៣)
P3 .
44) A bag contains6 white balls and 2 red balls. If three balls are chosen at random without replacement, what is the probability that only one ball is red.
10
។
P3 ។
៤៤)
6 2។ ។ ។
45) If is an acute angle and cos ① find the value of sin . ② find the value of tan .
5 , 7
៤៥)
243
cos
①
sin ។
②
tan ។
5 ។ 7
46) Expand and simplify the following expression. x x 2 x 1 x 3
៤៦)
47) Factor the following expression. 9x 2 y 2
៤៧)
48) Solve for x in the following equation. x2 5x 6 0
៤៨)
49) Simplify the following expression.
៤៩)
1 5
2
9x 2 y 2
50) y is directly proportional to the square of x and y 1 when x 2 . Express y in terms of x .
1 5
2
20
y
y 1
x
x 2។
y
x។
51) Find the length of the side of a square whose length of the diagonal is 4 cm.
៥១)
52) In the diagram on the right, find the value m. of x when x 2
៥២)
4 cm ។
m
x
x
6
4
m
53) Expand and simplify the following expression. x y x2 xy y 2
x
x2 5x 6 0
20
6
x x 2 x 1 x 3
m ៥៣)
244
x y x2 xy y 2
2 4
54) Factor the following expression. x3 3x 2 y 3xy 2 y 3
៥៤)
55) Simplify the following expression. 4 1 3 3 3 3 1
៥៥)
56) For the parabola y x 2 ax b , find the value of a and b such that the coordinate of the vertex is 1 , 2 .
៥៦)
57) How many different three-member teams can be formed fromed nine students?
៥៧)
58) Solve for x in the inequality 2 x2 5x 3 0 .
៥៨)
x3 3x 2 y 3xy 2 y 3
59) Answer the following when sin
4 1 3 3 3 3 1
y x 2 ax b a
1 , 2 ។
b
។
x
2 x 5x 3 0 ។ 2
1 for 2
0o 90o . ① find the value of cos . ② find the value of tan .
60) Answer the following for two sets, A = { 2 , 3 , 5 , 7} and B = { 1 , 2 , 3 , 4}. ① Find the elements of set A B and list them. ② Find the number of elements in set A B .
sin
៥៩)
1 2
0o 90o
①
cos ។
②
tan ។
៦០)
A = { 2 , 3 , 5 , 7}
B = { 1 , 2 , 3 , 4} ។
A B
① ។
②
245
A B ។
៦១)
61) Simplify the following expression. a) 8 4 10
b) 56 8 3 c) 1 2 2 4
7 7 1 1 d) 8 9 16 2 e) 6 2
3 2
a) 8 4 10 b) 56 8 3
3
c) 1 2 2 4
7 7 1 1 d) 8 9 16 2
f ) 10 2 5 10 g ) 8x 2 5x 7
e) 6 2 1 10
5 10
3 2
f ) 10 2 5 10 g ) 8x 2 5x 7
h) 6 0.6 x 0.4 0.3 2 x 5
1 10
5 10
h) 6 0.6 x 0.4 0.3 2 x 5
i) 2 3x 4 y 5 4 x 3 y
i) 2 3x 4 y 5 4 x 3 y
3x 5 y x 5 y 8 6 3 2 2 k ) 32 x y 4 x y 8 x
3x 5 y x 5 y 8 6 3 2 2 k ) 32 x y 4 x y 8 x
j)
j)
2
3 6 3 l ) x3 y 2 x 4 y 2 5 5 2
2
62) Expand the following expression. a) 4 x 3 3 x 2 b) x 3 y x 5 y x 4 y
3
2
3 6 3 l ) x3 y 2 x 4 y 2 5 5 2
៦២)
2
63) Factorize the following expression. a) 16 x 2 25
b) x 3 y x 5 y x 4 y
៦៣)
a) 16 x 2 25
b) 9 x 2 24 xy 16 y 2
64) Solve the following equations. a) 4 x 9 x 6
a) 4 x 3 3 x 2
b) 9 x 2 24 xy 16 y 2
៦៤)
b) 0.5 x 3.2 0.8 x 1.6 c) x 2 14 x 45 0
a) 4 x 9 x 6 b) 0.5 x 3.2 0.8 x 1.6 c) x 2 14 x 45 0
d ) x 8x 9 0 2
d ) x2 8x 9 0
246
2
2
៦៥)
65) Solve the following simultaneous equations. 7 x 5 y 1 a) 3x 2 y 1
(
)
7 x 5 y 1 a) 3x 2 y 1 3 1 x y 1 b) 4 2 0.5 x 0.5 y 0.1
3 1 x y 1 b) 4 2 0.5 x 0.5 y 0.1
66) Find the value of a3 b2 when a 2 and b 3.
៦៦)
a3 b 2
a 2
b 3។
67) Solve the following in equation. 4x 11 2x 15
៦៧)
68) y is directly proportional to x and y 4 when x 8 . Find the value of y when x 4 .
៦៨)
69) y is a linear function of x . Express y as a function of x when the graph passes through point 2 , 4 with slope of 3.
៦៩)
70) Is 221 a prime number?
៧០)
71) Find degree measure of each exterior angle of a regular nonagon (regular 9-sided, 9angled, polygon).
៧១)
4x 11 2x 15
y
y 4
x
x 8។
y
x 4 ។
x។
y
y
x
2 , 4
247
(
3។
?
221
9
,
9
)។
៧២)
72) In the accompanying diagram, find x when m
1cm xcm
8cm
1cm xcm
4cm
8cm
m
4cm
m
73) In the rectangle ABCD with 6 cm length and 3 cm width, find the length of the diagonal AC.
6cm
A
៧៣)
ABCD
6 cm
3 cm ។
AC ។
6cm
A
D
3cm
D
3cm C
B
74) Calculate. a ) 13 15 4
C
B
៧៤)
a ) 13 15 4
b) 8 5 20 5
b) 8 5 20 5
c) 2 42
c) 2 42
5 1 d ) 0.75 0.2 6 4
5 1 d ) 0.75 0.2 6 4
e) 18 72 2
e) 18 72 2
3
3
3 2 5 5 4 5 5 g ) 7 3x 2 9 2 x 5
3 2 5 5 4 5 5 g ) 7 3x 2 9 2 x 5
4x 1 5x 3 6 8 i) 3 9 x y 5 x 2 y
4x 1 5x 3 6 8 i) 3 9 x y 5 x 2 y
f)
m
x
f)
h)
h)
j ) 0.3 x 8 y 8 0.4 x 0.2 y
j ) 0.3 x 8 y 8 0.4 x 0.2 y
k ) 18 x y 2 xy 6 x y 2
k ) 18 x 2 y 2 xy 6 x 3 y
3
2
2
5 1 18 l ) xy xy 2 y 9 6 5
2
2
5 1 18 l ) xy xy 2 y 9 6 5
248
75) Expand and simplify the following expressions. a ) 7 x 5 y 3 x 4 y 2 y 10 y
៧៥)
b) x 6 x 5 x 8 2
b) x 6 x 5 x 8 2
76) Factor the following expressions. a) x 2 3x 54
៧៦)
a) x 2 3x 54
b) x 2 y 4 y 3
77) Solve for x in the following equations. a ) 6 x 16 9 x 11 7 x 3 3x 1 b) 8 4 2 c) x 4 x 21 0
b) x 2 y 4 y 3
៧៧)
7 x 3 3x 1 8 4 2 c) x 4 x 21 0 d ) x2 8x 2 0
៧៨)
8 x 3 y 30 a) 5 x 9 y 3 0.8 x 0.3 y 0.8 b) 2 3 9 5 x 2 y 5
0.8 x 0.3 y 0.8 b) 2 3 9 5 x 2 y 5
79) Find the value of 9 x 2 2 xy when x 3 and y 1 .
x a ) 6 x 16 9 x 11
b)
d ) x2 8x 2 0
78) Solve the following system of equations. 8 x 3 y 30 a) 5 x 9 y 3
a ) 7 x 5 y 3 x 4 y 2 y 10 y
៧៩)
9 x 2 2 xy
x 3
249
y 1។
80) A large dice and a small dice are both numbered 1 to 6. When they are rolled, what is the probability that the product of the two numbers on the top faces is 9?
៨០)
6។
1 9។
៨១)
81) Make y the subject of the formula 8x 3 y 7 .
8x 3 y 7 ។
y (
82) y is inversely proportional to x and y 8 when x 3 . Find the value of y when x 2 .
y
85) In the diagram on the right, find x when l m. l 41o
y
x 2 ។
y x 4។
y
x ។
) ។
8។
l m
x
l
41o
95o
m
x
250
y 32
x
95o
m
y 8
x
x 3។
83) y is directly proportional to the square of x and y 32 when x 4 . Express y in terms of x .
84) Find the measure of each exterior angle of a regular octagon in degrees. An octagon is an 8-sided polygon.
8x 3 y 7 )
y
x
86) In the diagram, point A, B, C and D lie on the circumference of circle O. Find x when BD is the diameter of the circle and DBC 38o .
A, B, C
x
BD
DBC 38o ។
D
D
A
A O
x
38o
C
B
B
87) Calculate. a ) 12 13 4
b) 2 5 12 6 c) 4 24
4
2
5 5 d ) 0.2 0.5 8 16 e) 2 8 18
5 5 d ) 0.2 0.5 8 16 e) 2 8 18
f ) 4 3x 5 8 2 x 3
f ) 4 3x 5 8 2 x 3
5x 3 x 1 g) 2 3 h) 4 6 x 7 y 8 2 x 4 y
5x 3 x 1 2 3 h) 4 6 x 7 y 8 2 x 4 y g)
i ) 0.2 5 x 10 y 2 2 x 0.5 y j ) 28 x y 5 xy 35 xy 2
2
C
a ) 12 13 4
b) 2 5 12 6 c) 4 2
O
x
38o
2
O។
D
i ) 0.2 5 x 10 y 2 2 x 0.5 y j ) 28 x 2 y 2 5 xy 35 xy 3
3
2
5 5 2 k ) x 2 y xy 3 y 3 . 6 9 15
2
5 5 2 k ) x 2 y xy 3 y 3 6 9 15
)
88) Expand and simplify the following expressions. a ) x 7 y x 7 y
a ) x 7 y x 7 y b) x 4 x 2 x 8 2
b) x 4 x 2 x 8 2
251
89) Factor the following expressions. a) x 2 8 x 12 . b) ax 2 4axy 5ay 2
b) ax 2 4axy 5ay 2
90) Solve for x in the following equations. a ) 9 x 19 11x 13
x a ) 9 x 19 11x 13
a) x 2 8 x 12
7 x 4 5x 4 8 6 1 c) x 2 0 4 2 d ) x 2x 1 0
7 x 4 5x 4 8 6 1 c) x 2 0 4 2 d ) x 2x 1 0
b)
b)
91) Solve the following systems of equations. x y 1 a) 3 x 2 y 18
x y 1 a) 3 x 2 y 18 2 x y 3 b) 1 1 0.5 x 8 y 2
2 x y 3 b) 1 1 0.5 x 8 y 2
92) Find the value of 3 xy 6 xy 2 when x 5 and y 2 .
3 xy 6 xy 2
x 5
93) Express 2010 as the product of prime numbers.
252
y 2។
2010
។
1 94) Solve for h in the equation mgh mv 2 , 2 where m 0 and g 0 .
)
g 0 ។
m0
95) y is directly proportional to x and y 5 when x 2 . Find the value of y when x 4.
1 mgh mv 2 2
h
y x 2 ។
y
x4 ។
96) y is directly proportional to the square of x and y 3 when x 3 . Express y in terms of x .
y
y 3
x
x 3។
y
x ។
97) Find the measure of each interior angle of a regular octagon in the degrees. An octagon is 8-sided polygon.
)
98) In the diagram on the right, find x when m. x
)
។
8។
m៖
x
x
2
6
y5
x
6
4
m
2
4
m
99) In the diagram on the right, points A, B, C and D lie on the circumference of a circle. Find x when BC : CD 1: 2 and CAD 46o .
A, B, C ។
D
x CAD 46o ។
BC : CD 1: 2
A
A
o x 46
o x 46
D
D
B
B C
C
253
100)
Simplify. a) 5 2 8 4
a) 5 2 8 4
b) 18 30 6
b) 18 30 6
c) 5 32 4 2
2 5 1 d) 3 6 3 e) 2
c) 5 32 4 2
2
2 5 1 d) 3 6 3
3 2 2 24
e) 2
10 5 g ) 2 8 x 5 6 3x 2 f)
2
5 1
3 2 2 24
10 5 g ) 2 8 x 5 6 3x 2 f)
h) 0.8 0.6 x 5 0.5 0.9 x 3
2
5 1
h) 0.8 0.6 x 5 0.5 0.9 x 3
i) 3 4 x 8 y 7 2 x 6 y
i) 3 4 x 8 y 7 2 x 6 y
3x 5 y 4 x y j) 6 9 3 k ) 28 xy 7 xy 2
3x 5 y 4 x y 6 9 3 k ) 28 xy 7 xy 2 j)
2
5 5 4 l ) x3 y x 2 y x 6 3 3
2
5 5 4 l ) x3 y x 2 y x 6 3 3
101) Expand and simplify the following expressions. a ) 3 x 4 y 3x 4 y b) x 6 4 x 3 2 x 5
2
a ) 3 x 4 y 3x 4 y b) x 6 4 x 3 2 x 5
2
102) Factor the following expressions. a) x 2 4 x 12
a) x 2 4 x 12
b) x y 12 x y 36 2
b) x y 12 x y 36 2
254
2
103) Solve for x in the following equations. a ) 7 x 4 4 x 10 x 5 3x 7 b) 1 4 2 c ) 5 x 2 40 0
x a ) 7 x 4 4 x 10
x 5 3x 7 1 4 2 c ) 5 x 2 40 0
b)
d ) x2 8x 4 0
d ) x2 8x 4 0
104) Solve the following systems of equations. 2 x 5 y 2 a) x 3 y 12
2 x 5 y 2 a) x 3 y 12 0.3x 0.4 y 0.1 b) 6 4 7 5 x 3 y 15
0.3x 0.4 y 0.1 b) 6 4 7 5 x 3 y 15
105) Find the value of 2ab b2 when a 4 and b 2 .
2ab b2
a4
b 2 ។
106) Two dice, A and B, are both numbered 1 to 6. When they are rolled, find the probability that the sum of the numbers on the top faces is 4.
107) Solve for y in the equation 5x 2 y 3 .
A 1
6។ 4។
y
255
B
5x 2 y 3 ។
108) y is inversely proportional to x and y 9 when x 4 . Find the value of y when x 6 .
y x 4។
)
110) Find the sum of the measure of interior angle of a dodecagon. A dodecagon is a 12-sided polygon.
)
x
y 8
y
y x 4។
y
x។
។
12។
111) In the diagram on the right, find x when m.
x
62o
62o
x
m
x 46o
46o
m
112) In the diagram on the right, four points A, B, C and D lie on the circumference of circle O. Find x when AB = AC and ADB 64o .
A, B, C O។ AB = AC
A
x
O
x
ADB 64o ។
A
B
y9
x 6 ។
109) y is directly proportional to the square of x and y 8 when x 4 . Express y in terms of x .
m
x
x
64
o
D
O
C
B
256
64o
D C
D
113) Simplify. a ) 8 10 16
a ) 8 10 16
b) 16 24 8 c ) 3 2 2
b) 16 24 8
4
5 1 d) 4 9 3
c ) 3 2 2
2
5 1 d) 4 9 3
e) 18 98 2 2 12 f ) 6 2 6 g ) 7 4 x 3 5 5 x 4
h) 0.6 3 x 0.7 2 0.3 x 0.2
h) 0.6 3 x 0.7 2 0.3 x 0.2
i) 4 9 x 6 y 9 x 2 y
i) 4 9 x 6 y 9 x 2 y
5x 3 y 2x 5 y 12 9 2 3 k ) 63 x y 9 xy 2
5x 3 y 2x 5 y 12 9 2 3 k ) 63 x y 9 xy 2
j)
2
2
e) 18 98 2 2 12 f ) 6 2 6 g ) 7 4 x 3 5 5 x 4
7 2 5 35 l ) xy x 2 y 3 y 20 6 48
4
j)
.
2
7 2 5 35 l ) xy x 2 y 3 y 20 6 48
114) Expand and simplify the following expressions. a ) 5 x y 6 x y
a ) 5 x y 6 x y b) x 7 x 8 x 6 2
b) x 7 x 8 x 6 2
115) Factorize the following expressions. a) x 2 100
a) x 2 100
b) ax 2 6ax 9a
b) ax 2 6ax 9a
257
116) Solve for x in the following equations. a) 9 x 5 6 x 7
x a) 9 x 5 6 x 7
b) 0.2 x 2 0.8 2 x 3
b) 0.2 x 2 0.8 2 x 3
c) x 3x 54 0 2
c) x 2 3x 54 0
d ) x2 8x 9 0
d ) x2 8x 9 0
117) Solve the following systems of equations. 2 x 7 y 11 a) 5 x 2 y 8 1.2 x 3.4 y 1.5 b) 2 4 1 3 x 9 y 3
2 x 7 y 11 a) 5 x 2 y 8 1.2 x 3.4 y 1.5 b) 2 4 1 3 x 9 y 3
118) Find the value of 9a 2 ab2 when a 3 and b 4 .
119) Two dice, A and B, are both numbered 1 to 6. When they are rolled, find the probability the both number facing up even.
120) Which number is the greatest among 3 3 , 2 7 and 5.
258
9a 2 ab2
b 4 ។
a 3
A 1
B
6។ ។
3 3,2 7
5។
121) y is directly proportional to x and y 18 when x 9 . Find the value of y when x 2 .
)
122) y is proportional to the square of x and y 4 when x 4 . Express y in terms of x .
)
123) Find the measure of each interior angle of a regular pentagon. A pentagon is a 5-sided polygon.
)
y x 9 ។
85o
y
A
x 4។
C 68
។
5។
x
m
107o
85o
x
A, B O។
x
AB AC
BAC 68 ។
C o
68
O
x
x
B
B
259
C
o
A
O
y
x។
m
o
y 4
x
x
125) In the figure on the right, three points A, B and C lie on the circumference of circle O. Find x when AB AC and BAC 68o .
y
x 2 ។
124) In the figure on the right, find x when m. 107o
m
y 18
x
126) For each equation, y x 1 , y x 1 , y 2 x 2 , y 3x 3 , . Its slope is equal to its y-intercept. These straight lines pass through a certain point. Find the coordinates of this point.
y x 1 , y x 1 ,
១២៦)
y 2 x 2 , y 3x 3 , x 0។ ។ ។
127) The figure shows trapezoid ABCD with AD||BC and AE = DE where E is the intersection point of AC and BD.
A
១២៧)
ABCD AD||BC
AE = DE AC
D
A
BD ។
E
C
B
a) Prove that EBC is an isosceles triangle. b) Prove that AB = DC.
។
EBC AB = DC ។
128) Solve x 4 289 . Note that x is a positive real number. Write only the answer.
១២៨)
129) Three points A, B and C lie on circle O. Each point is connected to the others. Given that AB : BC : CA 3 : 4 : 5 .
១២៩)
x 4 289 ។
x
។
A
B
D
E C
B
E
A, B
។
O។
C ។
AB : BC : CA 3 : 4 : 5 ។
A
O
B
C
O
C
a) Find CAB , ABC and BCA . Write only the answer. b) Express AB : BC in the simplest ratio.
CAB , ABC
។
AB : BC
260
BCA ។ ។
1 2 3 x x 1 . Find 2 2 the vertex, intersection points with coordinate axis, and then draw its graph.
១៣០)
131) Each surface of cube ABCD-EFGH, as shown in the figure at right, has a distinct label starting from 1 to 6. Each vertex, from A to H, is labeled with a number that is the sum of numbers on the surfaces sharing the vertex. For example, vertex C is labeled with a number that is the sum of numbers on surfaces ABCD, BFGC and CGHD. It is possible to label the surfaces so that the numbers labeled on the vertices occur as eight consecutive integers. Find the eight consecutive integers.
១៣១)
130)
Given a parabola y
A B
F
។
ABCD-EFGH 6។
1
CGHD។ ។
8 ។
A B
D C
E
H F
G
H G
១៣២)
5x 2 y 5x 2 y 5x y
261
H
C ABCD, BFGC
C
132) Simplify. 2 5x 2 y 5x 2 y 5x y
A
។
D
E
1 2 3 x x 1។ 2 2
y
2
133)
១៣៣)
Solve the following equation. x2 1 5x
x2 1 5x
134) Let y be directly proportional to the square of x and y 20 when x 2 . Express y as a function of x .
១៣៤)
135) Find the length of one side of a rhombus whose diagonals measure 10 cm and 6 cm.
១៣៥)
136) Factorize the following expression. xz x 2 y 2 yz
១៣៦)
137) In the accompanying figure, how many different shortest paths are there from point A to point B.
១៣៧)
y
x
x 2 ។
y 20 x។
y
10 cm
xz x 2 y 2 yz
B។
A
B
B
A
A
262
6 cm ។
138) Find that satisfies sin
3 where 2
១៣៨)
0o 180o ។
0o 180o .
139) Find the coordinates of the center of mass for ABC that is created by connecting three points A(-2 , 6) , B(7 , -9) and C(1 , 0).
១៣៩)
140) In the diagram shown on the right, quadrilateral ABCD is inscribed in a circle. Prove that AB = DC when AD||BC.
១៤០)
A
B
3 2
sin
ABC A(-2 , 6) , C(1 , 0) ។
B(7 , -9)
ABCD ។
AB = DC
AD||BC ។
D
D
A
C
C
B
141) Using the equality 8051 72 902 , express 8051 as the product of prime number. Write only your answer.
១៤១)
142) The equality m2 n2 is satisfied for three positive integers , m and n . Prove that can be factored as the product of prime number when n m 1.
១៤២)
8051 72 902 ។
8051 ។
263
m2 n 2
,m
n m 1 ។
n។
143) The diagram on the right shows rectangular prism ABCD-EFGH. Answer the following when AE = 19 cm, EF = 32 cm and FG = 25 cm. Find the length of diagonal AG.
D
ABCD-EFGH ។ AE = 19 cm, EF = 32 cm FG = 25 cm. AG ។
C
D
A 19 cm
១៤៣)
B
H
A
B
G 32 cm
H
19 cm
25 cm
E
C
G
F
25 cm
E
144) Find the remainder when the polynomial x3 2 x2 4 x 8 is divided by x 2 .
១៤៤)
145) Find the value of sin135o .
១៤៥)
sin135o ។
146) Find the value of tan 1 2 when
១៤៦)
tan 1 2
tan 1 2 and tan 2 3 .
147) When the quadratic equation 3x 2 4 x 8 0 has two roots, and , find the value of .
x3 2 x 2 4 x 8
tan 1 2
x2 ។
tan 2 3 ។
១៤៦)
២ 3x 4 x 8 0 2
264
F
32 cm
។
148) The diagram on the right shows a semicircle whose center is O and diameter is line segment AB. Point P lies on arc AB and point H lies on AB such that PH is perpendicular to AB, AH = 3 cm and BH = 7 cm.
១៤៨)
AB ។
O P
AB
AB
3cm H
AB ។
PH BH = 7 cm ។
AH = 3 cm
P
A
H
P
O
A
B
7cm
O
3cm H
PH ។
OP
a) Find the lengths of line segments OP and PH. Write only your answer. b) Find the ratio of the lengths of line segment PA to line segment PB, PA:PB.
B
7cm
។
PA
PB
PA:PB ។
149) Consider the quadratic equation x 2 x a 0 where a is a constant. If one of the roots of the quadratic equation is 5, find the value of a and the other root. Write only your answer.
១៤៩)
150) For two consecutive odd numbers, prove that the square of the smaller odd number subtracted from the square of the larger odd number is equal to 4 times the even number between the two odd number.
១៥០)
២ x xa 0 2
។
a
5
។
a ។
265
4 ។
151) Let x cm be the radius of a sector whose perimeter is 20 cm. a) Express the length of the arc using x . Write only your answer. b) Find the radius of the sector such that the area of the sector has the maximum value. Also find the maximum area.
១៥១)
x cm
20 cm ។ x។ ។ ) ។
152) A die numbered 1 to 6 is rolled 3 times. Find the probability that the same number is on the top face on the 1st and 2nd rolls and a different number is on the top face on the 3rd roll.
១៥២)
153) For ABC, AB=2 , AC=3 and
១៥៣)
1 3 ១
។
២ ៣។
15 for 90o A 180o . 4 a) Find the area of ABC . Write only your answer. b) Find the length of side BC. sin A
ABC, AB=2 , AC=3 sin A
15 90o A 180o ។ 4 ABC ។ ។
BC ។
154) Expand and simplify the following express expression. x 3 y x2 3xy 9 y 2
១៥៤)
155) Factor the following expression. x3 3x 2 y 3xy 2 y 3
១៥៥)
x 3 y x2 3xy 9 y 2
x3 3x 2 y 3xy 2 y 3
266
6
១៥៦)
156) Simplify the following expression.
52 6
52 6
157) Find the value of cos when tan 3 for 90o 180o .
១៥៧)
158) Six students are divided into 2 rooms, A and B, containing 3 students each. How many different ways can the students be divided ?
១៥៨)
159) For two sets, A = { 1 , 4 , 7 , 10 } and B = { 1 , 3 , 5 , 7 , 9 }, find the elements of set A B and list them.
១៥៩)
160) Find the value of a such that the polynomial x3 3x2 ax 4 is divisible by x 2 .
១៦០)
tan 3
cos 90o 180o ។
6
A ។ ។
A = { 1 , 4 , 7 , 10 } B = { 1 , 3 , 5 , 7 , 9 }។ ។
A B
x3 3x2 ax 4
a
x 2។
161) Find the shortest distance between point (0 , 1) and the straight line x 3 y 7 0 .
B
១៦១)
267
(0 , 1) x 3y 7 0 ។
162) Find all sets of integer, x , y , z , satisfying the following equality. Write only your answer. x 6 y 6 z 6 3xyz
១៦២)
163) A sector has a perimeter 20 cm and its central angle is less than 360o . Find the range of the radius of the sector such that the range of the area, A , is given by 16 cm 2 A 24 cm 2 .
១៦៣)
164) Let P and Q be the points of intersection between the circle x 2 y 2 5 and the straight line x y 1 , where the value of the x -coordinate of point P is less than of point Q. a) Find the coordinate of points P and Q. b) Let R be the point of intersection between the tangent line to the circle at P and the tangent line to the circle at Q. Find the coordinates of point R.
១៦៤)
165) Find all sets of integer, x, y, z , satisfying the following equality. Write only your answer. x 4 y 4 z 4 3xyz
១៦៥)
x , y , z ។ ។
x 6 y 6 z 6 3xyz
20 cm 360 ។ o
A។ 16 cm A 24 cm ។ 2
2
P
Q
x y 1
x y 5 2
2
P Q។ P R Q។
P R។
x, y, z ។ ។
x 4 y 4 z 4 3xyz
268
Q។
166) Consider the following proposition for integer n . “If n2 2n is an odd number, then n is an odd number.” a) Give the contrapositive of the proposition. Write only your answer. b) Prove that the proposition is true using your answer in (a).
១៦៦)
n។
n 2n 2
n
។” ។ ។ ។
167) When polynomial P x is divided by
x 3 the remainder is 3. When P x is divided by x 4 the remainder is 1. Find the remainder when P x is divided by
x 3
P x
3។
x4
1។
P x
x 3 x 4 .
168) A rectangular piece of land has length 100m and width 50m. This land is drawn 1 on a scale map. 1000 Find the area, in cm 2 , of the piece of land drawn on the map. You don’t need to write your steps. Write only your answer.
P x
១៦៧)
x 3 x 4 ។
១៦៨)
50m ។
100m
1 ។ 1000 ។
cm 2
។ ។
169) The surface area of a sphere of radius r cm can be express as 4 r 2 cm2 , where is the ratio of the circumference of a circle to its diameter. a) Find the surface area of a sphere when the radius is 5 cm. You don’t need to write your steps. Write only your answer. b) Find the radius of a sphere when the surface area is 81 cm 2 .
១៦៩)
269
r cm
4 r cm 2
2
។
5 cm។ ។
81 cm 2 ។
។
170) Expand and simplify the expression. x 4 y x2 4xy 16 y 2
following
១៧០)
171) Factor the following expression. 125x3 75x 2 15x 1
១៧១)
172) Find the volume of a sphere with radius of 6.
១៧២)
173) A is an acute angle. Find the value of 2 tan A when cos A . 3
១៧៣)
174) How many six-digit integers can be formed using all of the following six number 0, 1, 2, 3, 4 and 5 ?
១៧៤)
175) Find the coordinates of the vertex of the parabola y x 2 8 x 5 .
១៧៥)
x 4 y x2 4xy 16 y 2
125x3 75x 2 15x 1
6។
។
A cos A
2 ។ 3
0, 1, 2, 3, 4
270
tan A
5?
y x2 8x 5 ។
176) Given the parabola y 2 x 2 4kx 3k 9 . ① Find the range of k such that the parabola and the x -axis intersect at two distinct points. ② Show the range of k in ① on the number line.
១៧៦)
177) On the x - y plane, find the equation of the circle that has its center at the origin and passes through point (5, 12).
១៧៧)
178) Solve for x in the following equation. 4 x3 8x2 11x 3 0
១៧៨)
179) Find the range of that satisfies the 1 inequality cos , where 0o 360o . 2
១៧៩)
180) Given the two points A(-3) and B(3) on a number line. a) Find the coordinate of point P which divides internally the line segment AB such that the ratio of AP to PB is 1:2. b) Find the coordinate of point Q which divides externally the line segment AB such that the ratio of AQ to QB is 1:2.
១៨០)
y 2 x 2 4kx 3k 9 ។
k ។ ។
k
x-y
(5, 12) ។
4 x3 8x2 11x 3 0
cos
271
1 2
0o 360o ។
A(-3)
B(3)
។
P
AB
AP:PB=1:2 ។
Q
AB
AQ:QB=1:2 ។
181) How many pages are there in x percent of a 120-page book?
១៨១)
182) Express y as a function of x , given that y is directly proportional to x and y 8 when x 2 .
១៨២)
183) Find the value of y when x 2 , given that y is inversely proportional to x and y 2 when x 4 .
១៨៣)
184) Square tiles with 30 cm sides are arranged without spaces on the floor of Robert’s class. a) What is the area of one tile in cm 2 ? Include units in your answer. b) The area of Robert’s class is 54 m 2 . How many tiles are used in total.
១៨៤)
185) There is 20 g of sugar dissolved in a 160 gram beverage. a) In decimal, what is the weight ratio of sugar to the beverage? b) What is the percentage of sugar in the beverage?
១៨៥)
x ។
120
y
y
x
y 8
x
x 2 ។
x2
y x
y
y2
x 4។
30 cm Robert ។ cm 2 ។
Robert
272
54 m 2 ។
160 g។
20 g ។
១៨៦)
186) In the accompanying diagram of quadrilateral ABCD, answer the following with units. You may use a ruler.
ABCD ។ ។ ។
D
A
D
A
C
B
C
B
a) What is the area of ACD in cm 2 ? Answer this question after measuring the length of sides AD and CD. Round off your answer to one decimal place. b) Find the area of quadrilateral ABCD in cm 2 . Round off your answer to one decimal place.
187) There are 50 identical nails. The total weight is 80 g. a) In grams, what is the weight of 100 nails? b) Some nails are put in a 150 g box such that the total weight is 790 g. How many nails must be in this box? Explain your steps leading to the answer.
ACD
cm 2
AD CD ។ ។
cm 2 ។
ABCD
។
១៨៧)
50 80 g ។ 100 ។
150 g ។ 790 g។ ។
188) Answer the following. a) How many m is 5 km? b) How many m2 is 30 000 cm 2 ? c) How many cm3 is 10 ?
១៨៨)
m
m
5 km? 2
cm3
273
30 000 cm 2 ? 10 ?
189) In parallelogram ABCD shown on the right, let the intersection of the diagonals be O and the intersection points of sides AB and CD with the straight line that passes through point O be M and N, respectively. Answer the following to prove BM = DN.
A
១៨៩)
ABCD O AB O
CD M
។
N
BM = DN ។
A
D
D
N
N
O
M
C
B
O
M
C
B
a) To prove BM = DN, which triangles should be shown to be congruent in the simplest way ? b) Explain in words the congruence condition in your answer for a). There is no need to prove this.
190) Find the volume of the solids below and answer with units. Note that (a) is a rectangular prism and (b) is a solid of combined rectangular prism. (a).
BM = DN ? ។
។
១៩០)
។ )
) ។
)
6cm
6cm 8cm
8cm
10cm
10cm 12cm
(b).
6cm
)
3cm
10cm
12cm 6cm
3cm
10cm 3cm
3cm
6cm
6cm
6cm
6cm
274
191) The table shows the results of mathematics examination for five students A, B, C, D and E. The table shows the difference of each student’s score to 72, which is the average score for the five students. When the score exceeds 72, it is express as a positive number and when it is less than 72, it is expresses as a negative number. Name
A
B
C
Difference to the average score
+4
-1
-8
D
១៩១)
E។
A, B, C, D 72 ។
72 72 ។
E +5
a) Find the score for A. b) Find the difference between B’s score and C’s score as a positive number. c) Find the score for D.
A
B
C
+4
-1
-8
D
A
B
C
។
១៩២)
ABCD BC = b cm ។
AB = a cm
។
D
A
a cm
D
a cm C
B
C
B
b cm
b cm
a) What is the length of the perimeter of the rectangle ABCD in term of a and b? b) What is the area of rectangle ABCD in terms of a and b ?
ABCD
b។
a
ABCD
b។
a
193) Find the area, in cm 2 , of a circle with diameter d cm . Express your answer in terms of d and , the ratio of the circumference of a circle to its diameter.
+5
A។
D។
192) The figure shows the rectangle ABCD with AB = a cm and BC = b cm. Include units in your answer.
E
១៩៣)
275
cm 2 d cm ។
d
។
194) Isosceles triangle ABC with AB = AC is rotated clockwise about vertex A comes to the extension of side BC. Let the rotated vertices A and B be A’ and B’, respectively. Consider a special case when three points A, B’ and A’ are on the same straight line.
១៩៤)
ABC
A BC ។ A’
B
។
A, B’
A’
។
A
B
C
A
B’
A
B
AB=AC
B A'
B
ABC ?
a) How many angles are there that are equal to ABC ? b) Indicate all the angles that are equal to BAC . c) Find the size of ABC and answer with unit.
195) Linda bought 8 cakes consisting of some 90 pesos lemon cakes and some 120 pesos fruitcakes. She paid 870 pesos for the cakes. Let x and y be the number of lemon cakes and fruitcakes she bought respectively. a) Write simultaneous equation in terms of x and y . b) Find the number of lemon cakes and fruitcakes she bought.
A'
C
BAC ។ ។
ABC
១៩៥)
8 90 120 ។
870 ។
x
-
y ។
x
y។ ។
១៩៦)
196) Find the greatest common factor (GCF) of each set of numbers. a) (14, 35) b) (26, 39, 52)
276
(PGCD) (14, 35) (26, 39, 52)
197) The diagram shows rectangular piece of paper ABCD with AB = 4 cm and AD = 5 cm. Point P lies on side AB and the paper is folded along DP so that vertex A comes to side BC. Let A’ be the image of vertex A with DP as the reflection axis. Include units in your answer.
១៩៧)
ABCD AD = 5 cm។
P
AB DP
BC។ A
A A’ ។
DP
5cm
A
AB = 4 cm
។
D
5cm
A
4cm P
B
A'
D
4cm P
C
B a) What is the length in cm of segment A’C? b) What is the length in cm of segment PB?
A’C
១៩៨)
199) One year ago, the price of a digital camera was 8400 pesos. Today, the price is 5040 pesos. By what percent has the price of the digital camera decreased?
១៩៩)
cm ? cm ?
PB
198) There is one melon and one apple. The melon weighs 1.05 kg and the apple weighs 0.3 kg. a) How many kilograms heavier is the melon than the apple? b) How many times more does the melon weigh than the apple?
C
A'
។
0.3 kg ។
1.05 kg
?
277
8400 5040
។
។
200) Kevin and Linda are talking about the properties of their favorite quadrilaterals. Kevin: “I like quadrilateral in which the four side are the same length and the four angles are all 90o .” Linda: “I like quadrilateral in which the two opposite sides are the same length and the opposite angles are equal in measure. But the lengths of the adjacent sides are not equal and all angle are not 90o . a) What kind of quadrilaterals does Kevin like? Write your answer in the most appropriate name. b) What kind of quadrilaterals does Linda like? Write your answer in the most appropriate name.
២០០)
។ :
90o ។ :”
90o ។ ។ ។
201) Lucy bought a bag of snacks. It contained 15% more than the normal bag and weighed 276 g. What is the weight, in grams, of the normal bag of snacks? Include units in your answer.
២០១)
202) Consider the integers from 50 to 150. a) How many integers are a multiple of 6? b) How many integers are multiples of both 6 and 8?
២០២)
។
15% 276 g ។ ។
50
6? 6
278
150 ។
8?
203) There is a triangle in which the ratio of the base to the height is 5:8. Answer the following questions and include units in your answer? a) Find the height, in cm, of the triangle when the base is 40 cm. b) Find the base and height, in cm, of the triangle when the area is 320 cm 2 .
២០៣)
5:8 ។ ។
cm 40 cm ។ cm 320 cm 2 ។
1 liter bottle of juice. 2 2 Yesterday, she drank liters of juice. 5 2 Today, she drank of the rest of the juice. 5 a) How many liters of juice did she drink today? Include units in your answer. b) How many times more juice did she drink today than she did yesterday? c) How many liters of juice are left? Include units in your answer.
204) Ellie bought a 1
២០៤)
1
1 2
។
2 5
2 5
។ ។
។
។
២០៥)
205) Answer the following.
3 of an hour? 4 b) How many mm is 8 cm? c) How many m2 is 400 cm 2 ?
mm
3 4 8 cm ?
m2
400 cm 2 ?
a) How many minutes are
206) Find the values of following expression when x 1and y 5 . a) 7 x 9 y 4 b) 3x 2 2 xy
២០៦)
279
x 1 y5 7x 9 y 4 3x 2 2 xy
207) The table below contains information of the heights of five mountains. The numbers show the difference in meters between 500 m and the actual height. Positive numbers indicate heights higher than 500 m. Negative numbers indicate heights lower than 500 m. Mountain Height
A
B
C
D
E
(difference between 500 m)
99
258
-339
-106
-92
២០៧) ។ ។
500 m
500 m។ 500 m។
(
A
B
C
D
E
99
258
-339
-106
-92
500 m)
E
a) How many meters higher is mountain E than mountain D? b) Find the difference of the height between the highest mountain and the lowest mountain. Write your answer as a positive number. Write the steps leading to your answer.
208) There is a cylinder-shaped log with base diameter 20 cm and height 45 cm. Answer the following question and include units in your answer. Use 3.14 for the ratio of the circumference of a circle to its diameter.
D ។
។
។
២០៨)
45 cm ។
20 cm ។
3.14 ។
20cm 20cm
45cm 45cm 2
a) Find the total surface area, in cm , of the log. b) There is paint that can 2000 cm 2 per liter. How many liters of this paint are required to paint the whole surface? You only need to apply one coat of paint.
280
។
cm 2 2000 cm ។ ។
2
209) Mae’s father gave her x pesos for her one 3 day trip. She thought she would spend 5 of the money for transportation fees and 2 food and of the money for souvenirs but 5 she actually spent 120 pesos more for souvenirs than she expected. Therefore, she had 120 pesos less to spend for transportation fees and food than she originally planned. Answer the following questions and include units in your answer. a) Express the actual cost for souvenirs in terms of x . b) Express the actual cost for transportation fees and food in terms of x . c) Let’s assume that the actual cost for the souvenirs was equal to the actual cost for the transportation fees and food. Find the value of x . Write the steps leading to your answer.
២០៩)
-
x
3 5
។
2 5 120 ។
120 ។ ។
x។ x។ ។
x។ ។
210) Solve for a in the equation s
abc . 3
២១០)
211) Find the value of y when x 4 in the linear function y 2 x 5 .
២១១)
212) Find the measure of each interior angle, in degrees, of a regular decagon. A decagon is a 10-sided polygon.
២១២)
a
s
x4
y
y 2x 5 ។
។
10
281
។
abc ។ 3
213) In the diagram, find x when
២១៣)
m.
58o
m
m។
x 58o
m
32o
32o
x
x
214) The table on the right shows the ingredients needed to make chocolate chip cookies. Ingredient Amount Butter 120 g Sugar 50 g Egg 1 Cake flour 150 g Chocolate chips 60 g
២១៤) ។
<
> 120 g 50 g 1 150 g
a) Express the weight ratio of the sugar to cake flour in simplest form. b) Tracy has 40 g of chocolate chips in the kitchen. If she uses all the chocolate chips to make chocolate chip cookies, how much butter in g does she need? Include units in your answer.
60 g
។
40 g ។ ។ ។
215) Two different number are selected from 1, 2, 3, 4 and 5 to make two-digit integers. a) How many two-digit integers can be created when 1 is in the tens place? b) How many two-digit integers can be created in total?
២១៤)
282
1, 2, 3, 4 ។
5 1
។
216) The figure on the right has the axis of symmetry .
A
H
C
D
B
២១៦)
។
A
G
C
E
F
E
D
B
F A
a) Find the vertex that is symmetric to vertex A with respect to line . b) Find the side that is symmetric to side EF with respect to line .
។
EF ។
217) Shelly bought three sheets of drawing paper and one color pen set. The total cost was 340 pesos. One sheet of drawing paper cost x pesos and one color pen set cost 100 pesos. a) Write the total cost in terms of x . b) Find the price, in pesos, of one sheet of drawing paper. Include units in your answer.
២១៧)
218) The figure shows square ABCD with a side 8 cm. A semicircle with a diameter of AB is in the square. Include units in your answer. Use for the ratio of the circumference of a circle to its diameter. A D
២១៨)
។
។
340 x ។
100
x។ ។
។
8 cm ។
ABCD
AB ។
។
។
A
8 cm
B
G
H
D
8 cm
C
a) Find the perimeter, in cm, of the shaded region. b) Find the area, in cm 2 , of the shaded region. Write the steps leading to your answer.
283
C
B
។
cm ។
cm2 ។
219) The figure shown on the right is rhombus ABCD with acute angle A . Points E and F are located such that CE = CF. Let G be the intersection point between BF and DE. BF = DE will be proved by using congruent triangle in the simplest way.
២១៩)
A ។ CE = CF ។
ABCD F
F
G
DE ។ BF = DE
BF ។
A
A G
B
D
F
E
G
B
D
F
E
C a) Which two triangles should be shown to be congruent? b) Explain in words the condition for congruent triangles in your answer for a). c) Find EGF when ABC 117o and CBF 46o . Include units in your answer.
220) The figure on the right shows right-angled triangle ABC with AB = 8 cm, BC = 6 cm and B 90o . Point P moves on side AB. Let x cm be the length of AP and let y cm 2 be the area of PBC .
C
។
ABC 117o
EGF CBF 46o ។
។
២២០)
ABC B 90o ។
AB = 8 cm, BC = 6 cm AB ។
P
y cm
AP
A
x cm
PBC ។
2
A
x cm
x cm
8 cm P
B
8 cm P
6 cm
C
B
a) Find the length, in cm, of PB in terms of x . Include units in your answer. b) Express y in terms of x . c) When the area of PBC is a cm 2 , express x in terms of a . Write the steps leading to your answer.
cm
6 cm
x។
PB ។
x។
y PBC x
a។ ។
284
C
a cm 2 ។
221) In the diagram on the right, the straight line y ax and the retangular hyperbola b y intersect at point A(4, 3). Point B x lies on the rectangular hyperbola and its x -coordinate is -2. The straight line passing through points O and B is drawn.
y b y x
២២១)
y ax A(4, 3) ។ O
-2 ។ ។
B
y y
b x
A
y ax A
x
O x
b x
B
x
y ax
O
y
B
B a។
b។
a) Find the value of a . b) Find the value of b . c) Find the equation of the line that passes through points O and B.
222) The cone shown on the right has base radius 6 cm and height 12 cm. Answer the following questions and include units in your answer. Express the answer in terms of which is the ratio of circumference of a circle to its diameter.
O
២២២)
12 cm
6 cm 12 cm។ ។
។
12 cm
6 cm
6 cm
a) Find the area, in cm 2 , of the base. b) Find the volume, in cm3 , of the cone.
cm 2 ។ cm3 ។
285
B។
223) Frank went to a store to buy foods for a party. One rice meal set and one burger set cost 220 pesos altogether. When he paid 3000 pesos for 10 rice meal sets 15 burger sets, the change was 200 pesos. Let x and y pesos be the costs of one rice meal set and that of one burger set, respectively. a) Write the system of equations in terms of x and y . b) Find the cost of one rice meal set and the cost of one burger set. Include units in your answer.
២២៣) ។
220 ។
3000 10 ។
200
15
y
x
។
x
។ ។
224) The figure on the right shows a rightangled triangle ABC with BAC 90o and ABC 60o . AH is perpendicular to side BC and BH = 2 cm. Answer the following questions and include units in your answer. A
២២៤)
ABC
2 cm
ABC 60o ។ AH
BAC 90
o
BH = 2 cm ។
BC ។
60o
B
A
60o
H
C
B
a) Find the length, in cm, of line segment AH. Write the steps leading to your answer. b) Find the length, in cm, of line segment CH. c) Find the area, in cm 2 , of ABC .
225) Find the volume, in cm3 , of a cube with edges of length 10 cm.
y។
2 cm
C
H AH ។
cm
។
CH ។
cm cm
២២៥)
286
cm3
10 cm ។
2
ABC ។
226) Find the value of the following expression when x 5 and y 2 . a) 2 x 3 y b) xy 2 y 2
២២៦)
227) y is inversely proportional to x and y 2 when x 6 . Find the value of y when x 3 .
២២៧)
228) Solve for x in the following equations. a) 3x 13 11x 3 4 x 5 3x 2 b) 3 4
២២៨)
229) Find the greatest common (GCF) of each set of numbers. a) (14, 21) b) (10, 15, 20)
២២៩)
230) For triangle ABC, the measures of the three angle A , B and C are x o , 2 x o and 3xo , respectively. a) Find the value of x ? b) What kind of triangle ABC ? Write the most appropriate name.
២៣០)
x 5 y 2 ។ 2x 3 y xy 2 y 2
y
y 2
x
x 6 ។
y
x 3 ។
x
3x 13 11x 3 4 x 5 3x 2 3 4
(GCD) (14, 21) (10, 15, 20)
287
A , B
ABC
C
o
x , 2x x។
ABC ។
o
3x o ។
231) Kevin saw an advertisement for an apartment with the following information written on it: “15 minutes walk from station A.” The distance from station A to the apartment is 1200 m. a) Find the average speed, in meters per minute, of the “walk” written on the advertisement. b) It took 20 minutes for Kevin to walk form station A to the apartment. Find Kevin’s average speed, in meters per minute.
២៣១)
232) When computing 3.14 × 25, some people do it using the way shown in figure 1. Figure 2 displays the detailed version of Figure 1 which shows the decimal position. Only look at figure 2 and answer the following questions.
២៣២)
3.14 25 1570 628 78.50
Figure 1
15
A 1200 m ។
A
” ។
20 ។
A
។
3.14 25 15.70 62.80 78.50
3.14 × 25 1។ 1 ។
2 ។
3.14 25 15.70 62.80 78.50
3.14 25 1570 628 78.50
rbUTI 2
rbUTI 1
Figure 2
a) What number is multiplied by 3.14 to give 15.70 in ①? b) What number is multiplied by 3.14 to give 62.80 in ②? c) Express 3.14×25 as a single expression in terms of 3.14 and the two numbers from questions a) and b).
3.14
15.70
3.14
62.80
①? ②? 3.14×25 3.14 ។
288
2
233) Find area, in cm 2 , of each the following figures. Include units in your answer. a) Triangle
២៣៣)
។
cm 2 ។
3 cm
3 cm
6 cm
6 cm
b) Parallelogram
5 cm
4 cm
4 cm
5 cm 4 cm
4 cm
234) Consider the integers form 1 to a given integer added together like such, 1 2, 1 2 3 , 1 2 3 4 Let’s take the ratio of their sums. For example, the ratio of 1 2 to 1 2 3 is 3: 6 1: 2 . a) Express the ratio of 1 2 3 to 1 2 3 4 in simplest form. b) The ratio of 1 2 3 4 5 6 to a certain sum is 3: 4 . Find the sum and write it in addition form.
២៣៤)
1 1 2, 1 2 3 , 1 2 3 4 ។
1 2
1 2 3
3: 6 1: 2 ។
1 2 3 ។
1 2 3 4
1 2 3 4 5 6 3: 4 ។ ។
235) The figure on the right has axes of symmetry and a point of symmetry.
២៣៥)
a) Construct the point of symmetry using only a ruler. Label the point O. b) Find the total number of axes of symmetry.
289
។
។
O។ ។
236) Charlie collected 400 plastic bottle caps that weighed 1000 g in total. A recycle company buys 400 plastic bottle caps for $1.00. Another company sends polio vaccines to countries in need. Vaccine for one person is sent for every $2.00 donation from the recycle company. Answe the following questions. a) How many plastic bottle caps does Chalie need to collect to send a polio vaccine for one person? b) Suppose each plastic bottle cap has the same weight. Find the weight, in gram, of one plastic bottle cap. Include units in your answer. c) When burning 400 plastic bottle caps, 3150 g of carbon dioxide ( CO 2 ) are emitted. How many times greater is the amount of CO 2 emitted when burning 400 plastic bottle caps than the weight of 400 plastic bottle caps?
២៣៦)
400 1000 g។ $1.00 ។
400
។
$2.00 ។
។ ។ ។ ។
400 ។
( CO 2 ) 3150 g
CO 2
400 400
។
២៣៧)
237) The product of integers from 1 to n is express as n ! . For example, 4! 1 2 3 4 24 and 5! 1 2 3 4 5 120 . a) Calculate 6! b) Calculate 6! 5! 6! 4! 6! 3! .
1
n !។
n
4! 1 2 3 4 24 5! 1 2 3 4 5 120 6! ។
6! 5! 6! 4! 6! 3! ។
238) Brian wants to make a formula to find the area of a circle with circumference a cm in terms of a . Answer the following questions and use for the ratio of the circumference of a circle to its diameter. a) Find the radius, in cm, of the circle. Include units in your answer. b) Let S cm 2 be the area of the circle. Express S in terms of and a . Write the steps leading to your answer.
២៣៨)
a។
a cm
។ ។
cm ។
S cm S
។
2
a។ ។
290
២៣៩)
239) Answer the following. a) How many hours is 20 minutes? b) How many g is 0.3 kg? c) How many m2 is 20000 cm2 ?
20
m
240) Solve the following systems of equations. 3x 2 y 11 a) x y 5 y 6x 3 b) y 4x 7
២៤០)
241) Solve for x in the following equations. a) 2 x 4 x 6 b) 0.4x 5 0.9 x 5 2 x 1 3x 2 c) 3 4
២៤១)
2
20000 cm 2 ?
3x 2 y 11 x y 5 y 6x 3 y 4x 7
x
2x 4 x 6 0.4x 5 0.9 x 5 2 x 1 3x 2 3 4
242) Find the measure of each exterior angle of a regular 18-sided polygon in degrees.
243) In the diagram below, find x when
0.3 kg?
g
m.
២៤២) ។
18
២៤៣)
52o
52o
x
m
m។
x
x
43o
m
291
43o
244) Patrick and Kim play a game using a spinner labeled from A to D as shown on the right. Both players start the game with 10 points. A player spins the spinner and computes their score according to the following rules. When spinning the spinner more than once, each operation is done after computing the previous score.
២៤៤)
A
D
។
10 ។ ។ ។
A: Add 3 B: Subtract 3 C: Multiply by 2 D: Multiply by -2
a) When Patrick spun the spinner, it landed on B. Find Patrick’s score. b) When Kim spun the spinner two times, she got the letters A and D in this order. Find Kim’s score. c) When spinning the spinner two times, find the lowest possible score.
A:
3
B:
3
C:
2
D:
-2 B។
។
A
។
D
។ ។
245) For parallelogram ABCD shown in the figure, diagonals AC and BD intersect at O.
A
31o
ABCD AC A
D O
B
២៤៥)
D O
79 o
C
31o
B
a) AOB and COD are a pair of angles with a certain name. What is this pair of angles called? b) ABO and CDO are a pair of angles with a certain name. What is this pair of angles called? c) Find CAD when DBC 31o and DOC 79o . Include units in your answer.
292
AOB
O។
BD
79 o
C
COD ។
ABO
CDO ។
DBC 31o
CAD DOC 79o ។ ។
246) The product of integers from 1 up to n is expressed as n ! . For example, 3! 1 2 3 6 and 4! 1 2 3 4 24 . a) Compute 5!. b) Find the value of integer n satisfying the following equality. n! 5! n! 4! n! 3! 2010
២៤៦)
1
n
n !។ 3! 1 2 3 6
4! 1 2 3 4 24
5!។ n
n! 5! n! 4! n! 3! 2010 247) Linda does not know why the addition method is true when solving systems of equations. Explain the reason why a c b d is satisfied when a b and cd. ab
២៤៧) ។
ac bd
cd។
ab
ab
) c d ac bd
) c d ac bd
248) For triangle ABC, the measures of the three angles A, B and C are x o , 2 x o and 3xo , respectively. Find the value of x .
២៤៨)
249) For quadrilateral ABCD, the measures of the four angles A, B, C and D are y o , 2 y o , 3 y o and 4 y o , respectively. Find the value of y .
២៤៩)
250) For pentagon ABCDE, the measures of the five exterior angles A, B, C, D and E are z o , 2 z o , 3z o , 4 z o and 5z o , respectively. Find the value of z .
២៥០)
A, B
ABC
C
o
x , 2x
o
3x
។
o
x ។
293
A, B,
ABCD
C
D
4 yo
។
o
o
y , 2 y , 3 yo
y ។
ABCDE
E
A, B, C, D
z o , 2 z o , 3z o , 4 z o z ។
5z o
។
251) For parallelogram ABCD shown in the figure, diagonals AC and BD intersect at O.
A
២៥១)
AC
D
A
D
O
B
O។
BD
O
C
C
B
a) AOB and COD are a pair of angles with a certain name. What is this pair of angles called? b) ABO and CDO are a pair of angles with a certain name. What is this pair of angles called? c) Find the measure of DAB ABC in degrees. Include units in your answer.
AOB
COD ។
ABO
CDO ។ ។
DAB ABC ។
252) A triangle has base 6a cm and height 5 cm. Express the area, in cm2, using a .
២៥២)
253) Kevin weighs 52.5 kg. a) Kevin’s brother weighs 1.2 times more than Kevin. Find Kevin’s brother’s weight, in kg. Include units in your answer. b) Kevin’s sister weighs 42 kg. How many times heavier is Kevin’s sister than Kevin?
២៥៣)
5 cm។
6a cm
a ។
2
cm
52.5 kg. 1.2 ។
kg។ ។
42 kg ។ ។
254) There are 80 students in Jim’s grade. a) 36 students in Jim’s grade are girls. What percent of students in Jim’s grade are girls? b) In Jim’s grade, 15% of students belong to a soccer club. How many students belong to the soccer club?
២៥៤)
294
។
80
។
36
។ , 15% ។ ។
255) In the following figures, find angles X and Y by calculating. Include units in your answer. a) Quadrilateral
២៥៥)
X
Y
។
។
72o
72o
X
X 116o
116o
AD
b) Line segments AD and BC cross at E.
A 74o
A
C
E
E
Y 43o
35o
D
B
C
74o
Y 43o
35o
E។
BC
D
B
256) Find the volume, in cm3, of each of the following figures. Include units in your answer. a) Cube
២៥៦)
8 cm
។
cm3 ។
8 cm
b) Solid made up of rectangular prisms. 10 cm
10 cm
2 cm
2 cm
6 cm
6 cm 5 cm
5 cm
9 cm
9 cm
295
257) There are several identical nails in a box. 50 nails weigh 140 g. a) Find the weight, in g, of 200 nails. Include units in your answer. b) The total weight of all nails in the box is 2800 g. How many nails are in the box? Write the steps leading to your answer.
២៥៧) ។
140 g។
50 g
។
200 ។
2800g។ ។
258) Charlie gives candies to his friends. When he gives 8 candies to each of them, 11 candies are left over. Let x be the number of friends he has. a) Express the total number of candies using x . b) When he gives 8 candies to each of his friends, 11 candies are left over. When he gives 10 candies to each of his friends, he is 5 candies short. Write the equation using x . c) Find the number of friends he has.
២៥៨)
259) In the diagram on the right, each number from 1 to 16 is assigned to each of the 16 squares so that the sum of the 4 number in any row, column or diagonal is the same. (This kind of square is called a magic square) Seven numbers are already assigned. a 14 15 b
២៥៩)
12
c
6
d
e
f
10
g
h
i
3
16
។
8 11 ។
។
x
x ។
8 11 ។ 10
5 x ។ ។
a) Find the sum of four numbers in a row, column or diagonal. b) Find the numbers for a and h .
1 16 ។ 4 ។ ) ។
a
14 15
b
12
c
6
d
e
f
10
g
h
i
3
16
។
a
296
16
h។
។
260) In the diagram on the right, point C lies on segment AB such that AB = 12 cm and AC = 8 cm. There are three semicircles whose diameters are segments AB, AC and BC, respectively. Include units in your answer and use for the ratio of the circumference of a circle to its diameter.
២៦០)
C AC = 8 cm ។
AB = 12 cm
AB, AC
។
12 cm
C
8 cm
B
A
261) In the diagram below, find x when m.
B
C
8 cm
a) Find the area, in cm2, of the semicircle of diameter 12 cm. b) Find the area, in cm2, of the shaded part. Write the steps leading to your answer.
cm2 12 cm ។ ។
cm2
។
២៦១)
m។
x
63o
63o
x
x o
m
BC
។
12 cm
A
AB
38
38o
m
262) There are two fish tanks, A and B. Tank A can hold 27 L of water and tank B can hold 36 L of water. a) Write the ratio of the volume of tank A to the volume of tank B in simplest form. b) 35 fish are divided into the two tanks such that the ratio of the number of fish in tank A to tank B is equal to the ratio of their volumes. Find the number of fish in tank A and in tank B.
២៦២)
A 27 L
A
B
36 L
។
A ។
B 35 A
។
B A
297
B។
B។
263) Becky walks at a speed of 80 m per minute. Answer the following and include units in your answer. a) It takes 8 minutes for Becky to go to school from her home. Find the distance, in m, to school from her home. b) The distance to a station form her home is 2 km. How long, in minutes, does it take for Becky to go to the station from her home?
២៦៣)
264) The figure on the right has axis of symmetry .
២៦៤)
។
8 ។
E
m ។
2 km។ ។
។
A B
។
80 m
A
C
H
D
G
I
B
C
H
D
G
E
F
F ។
C
a) Find the vertex that is symmetric to vertex C with respect to . b) Find the side that is symmetric to side GF with respect to .
265) 17 added to a number is equal to 15 subtracted from that number then multiplied by 5. Let that number be x and answer the following. a) Write the equation in terms of x . b) Find the value of x .
I
GF ។
២៦៥) 17
298
15 5។
x x ។
x ។
266) In the diagram on the right, There is a semicircle of diameter 8 cm in a sector whose radius is 8 cm and its central angle is 90o . Include units in your answer and use for the ratio of the circumference of a circle to its diameter.
២៦៦)
8 cm 8 cm
90o ។
។
8 cm
8 cm
a) Find the area, in cm2, of the sector whose radius is 8 cm and its central angle is 90o . b) Find the area, in cm2, of the shaded part. Write the steps leading to your answer.
267) In the diagram on the right, the straight line y 2 x 6 , denoted by , passes through the y -axis and x -axis at A and B, respectively. The coordinates of point C is (2, 1). Let m be the straight line that passes through points A and C. y m
B
90o ។
8 cm
។
cm2
។
២៦៧)
y 2x 6
y C
m
a) Find the coordinates of point B. b) Find the equation of straight line m . Write the steps leading to your answer.
299
(2, 1) ។
m
C។
y A
B
x
B
A
C
A
x
A
O
cm2
x
O
B។ m ។ ។
C
។
268) The table below contains information on the highest and lowest temperatures for each day from July 20th to July 24th in Osaka city. The numbers show the difference in temperature between 30 o C and the actual temperature. Positive numbers indicate temperatures higher than 30 oC . Negative numbers indicate temperatures lower than 30 oC .
២៦៨) ២០ ២៤ ។ ។
30 oC
30 oC 30 oC ។ ២០
July 20th 21st 22nd 23rd 24th Highest +3.9 +4.2 +5.3 +6.0 +5.6 Lowest -3.6 -3.8 -3.8 -2.5 -2.7
២១
២២
+3.9 +4.2 -3.6 -3.8
+5.3 -3.8
a) Find the actual lowest temperature on the 20th. Include units in your answer. b) Which day has the greatest difference between the highest and lowest temperatures?
269) The diagram on the right shows a rectangular prism of length 5 cm, width 5 cm and height a cm. Include units in your answer.
២៤
+6.0 +5.6 -2.5 -2.7 ២០។ ។ ។
២៦៩)
5 cm
a cm
២៣
5 cm
a cm ។
5 cm ។
5 cm 5 cm
a cm
a) Find the volume, in cm3, of the rectangular prism. b) Find the surface area, in cm2, of the rectangular prism.
300
5 cm
។
cm3 2
cm
។
270) There are two numbers, x and y . The sum of x and y is 30. The sum of 7 times x and 2 times y is 95. a) Write a system of equation using x and y . b) Find the value of x and y .
២៧០)
271) In the graph shown on the right, the parabola y ax 2 has its vertex at O and passes through point A(1, 2). Point B lies on y ax 2 and its x -coordinate is 2. Point H lies on the y -axis and line BH is perpendicular to the y-axis. Point C lies on line BH such that HB = BC. The parabola y bx 2 has its vertex at O and passes through point C.
២៧១)
30 ។
x
y ។
7
x
x
y។
y។
y ax 2
A(1, 2) ។
O
y ax 2
B 2។
H ។
BH y bx
C
HB = BC ។
BH 2
O
C។
y y ax 2 y bx 2
B
y ax 2 y bx 2
C
H
A
B
C
A
O
y 95 ។
2
x
y
H
y
x
x
x
O
a) Find the value of a . b) Find the y -coordinate of point B. c) Find the value of b . Write the steps leading to your answer.
a ។
B។
b ។ ។
301
២៧២)
272) The figure on the right shows trapezoid ABCD with AD || BC and AD DC . Point H lies on side BC and line AH is perpendicular to side BC. Answer the following when AB = 6 cm, BC = 7 cm and AD = 5 cm. 5 cm A D
AD DC ។
AD || BC BC
H BC ។
AH
AB = 6 AD = 5 cm ។
cm, BC = 7 cm
5 cm
A
D
6 cm 6 cm
B
H 7 cm
C
B
C
H 7 cm
a) Find the length, in cm, of AH. b) Find the area, in cm2, of trapezoid ABCD.
AH ។
cm 2
cm ABCD ។
273) Consider the following six integers. 6, 5, 1, 2, 4, 5 a) If you choose two integers and add them, find the least possible sum of the two integers. b) If you choose two integers and multiply them, find the least possible product of the two integers. c) Three difference integers are assigned to each of the letters, A, B and C in the expression below. Find the set of integers, A, B and C such that the expression takes the minimum value. A C B
២៧៣)
1 2 x . 2 a) Find the value of x when y 4 . b) For the problem “Find the range of y in the interval 2 x 4 ” Nick’s answer was “ 2 y 8 ”. Is his answer correct? If it is correct, write “Correct” on your answer sheet. If it is not correct, give your reason.
២៧៤)
274) Consider the function y
6, 5, 1, 2, 4, 5 ។ ។
A, B
C
។
A, B
C ។
A C B
302
y
1 2 x ។ 2
y 4។
x
y
2 x 4 ”
“2 y 8” ។ ” ។
។
275) Find the volume, in cm3, of each of the following solids. Include units in your answer and use for the ratio of the circumference of a circle to its diameter. a) Cylinder
២៧៥)
។
2r cm
r cm
។
cm3
2r cm
r cm
b) Cone 2r cm
2r cm
r cm
r cm
276) 2 students are selected form 4 students, A, B, C and D, at random. a) How many different ways can 2 students be selected? b) What is the probability that student A will be selected? c) After selecting 2 students, a president and a vice president will be selected from the 2 students at random. What is the probability that student A will be president?
២៧៦)
2 B, C
4 ។
D 2
។
A 2 ។
2
277) Circles are arranged according to a rule as shown below. 1st 2nd 3rd 4th …
A,
A
២៧៧) ១
២
៣
៤
…
...
Find the number of circles in the 8th arrangement.
៨
303
។
278) The table on the right shows the number of doctors per 1000 people for the top four countries. Japan is added to the table for comparison. Country Cuba Greece Belarus Russia Japan
២៧៨) ។
1000 ។
Number of doctor per 1000 people 6.4 5.4 4.9 4.3 2.1
1000 6.4 5.4 4.9 4.3
a) How many times more is the number of doctors per 1000 people in Cuba than in Russia? Round your answer off to one decimal place. b) To find the number of doctor per 1000 people in a country, we need two values, “the total number of doctors” and “population” of the country. Let x be the total number of doctors and let y be the population of a country. Express the number of doctors per 1000 people using x and y . c) Suppose that the population of Japan is 130,000,000. Find the total number of doctors in Japan.
2.1 1000 ។
1000 ”
”
y
x ។
1000 x
y ។
130,000,000 ។
279) The figure on the right shows two rightangled triangles ABC and CDE, where point E is the midpoint of side AC. AB = 2 cm, BC = 4 cm and DE = 5 cm. Include units in your answer. D 5 cm A 2 cm
B
។
២៧៩)
ABC CDE
E
AC ។ AB = 2 cm, BC = 4 cm
។
B
a) Find the length, in cm, of side AC. b) Find the length, in cm, of side CD. c) Find the sum of the areas, in cm2, of ABC and CDE .
304
E C
4 cm
cm
AC ។
cm
CD ។ cm2
CDE ។
D
5 cm
A 2 cm
C
DE = 5
cm ។
E
4 cm
។
ABC
280) The diagram on the right shows equilateral triangle ABC. Points D and E lie on side AB and AC, respectively such that AD = CE. ACD CBE will be proven in the simplest way using congruent triangles. A
២៨០) រូបខាងស្តាំបង្ហាញពីររីកោណសម័ងស ABC ។
ចំណុច D នង ិ E កៅកលើរជុង AB នង ិ AC ករៀងគ្នា ដែល AD = CE ។
ACD CBE នឹងររូវ
បង្ហាញកោយោរករបើលក្ខខណឌប៉ន ុ គ្នាននររីកោណ។
A D
D
E
E B
C
B
C
១) ករគ ុ គ្នា? ើ ូររកី ោណណា ដែលគួបង្ហាញថាវាប៉ន
1) Which two triangles should be shown to be congruent? 2) Which conditions are required to prove that the two triangles in your answer for 1) are congruent? Choose three conditions from the following and write the corresponding letters. (a) AD = CE (b) AC = CB (c) CD = BE (d) ADC CEB (e) ADC CEB (f) CAD BCE 3) Explain in words the condition for congruent triangles in your answer for 1).
២) ករល ើ ក្ខខណឌណា ដែលររូវយក្មក្បញ្ជាក្់ចំកោះ
ររីកោណទាំងពីរ ដែលថាវាប៉ន ុ គ្នាក្ាុងចកមលយ ើ របស់
អ្ាក្សរាប់សំណួរទី ១)។ ចូរករជស ខណឌ ើ ករសលក្ខ ើ បនី នបណា ា លក្ខខណឌខាងករោម ក
យ ើ សរកសរ
អ្ក្សរដែលររូវគ្នានឹងលក្ខខណឌកោះ។
ក្. AD = CE
ខ. AC = CB គ. CD = BE ឃ.
ADC CEB
ង.
ACD CBE
ច.
CAD BCE
៣) ពនយល់ជាោក្យ នូវលក្ខខណឌប៉ន ុ គ្នាននររីកោណ ក្ាុងចកមលយ ើ របស់អ្ាក្ សរាប់សំណួរទី ១)។
281) In the figure on the right, a square with sides of length 4 cm is equally divided into 4 4 small squares, each with a side of length 1 cm.
២៨១) ក្ាុងរូបខាងស្តាំ ោករមួយានរង្ហវស់រជុងរបដវង 4 cm
In this figure, construct a line segment of length 5 cm using only a ruler. You can only use the ruler for drawing lines.
305
គរឺ រូវដចក្កចញជាោកររូច
ានរង្ហវស់របដវង 1 cm ។
4 4 កសមើៗគ្នា ដែលរជុង
ក្ាុងរូបកនះ ចូរែឹក្បោារ់ដែលានរបដវង 5 cm
កោយោរករបរើ ាស់ដរបោារ់បកុ៉ ណា ោ ះ។ អ្ាក្អាចករបើ ានដរបោារ់បកុ៉ ណា ោ ះកែើមបគ ី ូសបោារ់។
282)
ំ ស ២៨២) ក្ាុងរូបនម ួ ៗបង្ហាញពអ្ ួ ឱ្យកលខ។ ី យ ី ំពូល៥ ដែលជន
In the accompanying figures, each shows 5 bulbs to represent a number. Each electric bulb can be either on or off. Figure 1 shows all electric bulbs are off and it represents 0. When the electric bulbs are on as in Figure 2 to Figure 6, they represent 1, 2, 4, 8 and 16, respectively. Some othe numbers can also be represented by a combination of these 5 bulbs. For example, Figure 7 is a combination of Figure 2, 5 and 6. Thus Figure 7 represents 1 + 8 + 16 = 25. Answer the following.
អ្ំពូលអ្គគស ួ ៗអាចកឆះភលឺ ឬមន រូបទ១ ិ នន ី ម ី យ ិ កឆះភល។ ឺ ី បង្ហាញពីអ្ំពូលអ្គគិសនីទាង ំ អ្ស់មន ួ ឱ្យ ិ កឆះដែលជំនស
កលខ 0 ។ កៅកពលដែលអ្ំពូលអ្គគិសនីកឆះភលឺ ែូចក្ាុង រូបទ២ ួ ឱ្យកលខ 1, 2, 4, 8 ី ែល់ទី៦ ដែលពួក្វាជំនស
និង 16 ករៀងគ្នា។ កលខកផសងកទៀរអាចររូវានជំនស ួ
កោយោរផសំចូលគ្នាននអ្ំពូលទាំង៥កនះ។ ជាឧទា រណ៍ រូបទី ៧ ជាបនសំននរូបទី 2, 5 នង ិ 6។ ែូចកនះរូបទី ៧ ជំនស ួ ឱ្យកលខ 1 + 8 + 16 = 25 ។ ចូរកឆលើយនឹងសំ ណួរខាងករោម។
Figure 1
រូបទី 1
Figure 2
រូបទី 2
Figure 3
រូបទី 3
រូបទី 4
Figure 4
រូបទី 5
Figure 5
រូបទី 6
Figure 6
រូបទី 7
Figure 7 a) What number is represented when all bulbs are on? b) Configure these bulbs to represent 10. Draw your figure using and just like the figure above. 283) In the rectangle shown on the right, points M and N are the midpoints of sides AD and BC, respectively. BD is the diagonal. MBD NDB will be proved in the simplest way by using congruent triangles.
A
M
D
B
N
C
ក្. ករកើ ៅកពលអ្ំពូលទាំងអ្ស់កឆះភលឺ វាជំនស ួ ឱ្យកលខអ្វ? ី ខ. រូបននអ្ំពូលទាំងកនះជំនស ួ ឱ្យកលខ 10។ ចូរគូសរូប របស់អ្ក្ ា កោយករបរើ ាស់
និង
ែូចរូបខាងកលើ។
២៨៣) ក្ាុងចរុកោណដក្ងបង្ហាញក្ាុងរូបខាងស្តាំ, ចំណុច M និង N គឺជាចំណុចក្ណា ា លករៀងគ្នាននរជុង AD និង
BC ។ BD គជា ឺ អ្ងកររ់ ទូង។ MBD NDB
នឹងររូវ បង្ហាញកោយោរករបើលក្ខខណឌប៉ន ុ គ្នានន ររីកោណ។
1) Which two triangles should be shown to be congruent? 2) Which conditions are required to prove congruency of the two triangles in your answer for 1)? Choose three conditions from the following and write the corresponding letters. a) AM CN b) DM BN c) BD DB d) MBD NDB e) A C f) MDB NBD 3) Explain in words the condition for congruent triangles in your answer for 1). 306
A
M
D
B
N
C
១) ករើគូររីកោណណា ដែលគួបង្ហាញថាវាប៉ន ុ គ្នា?
២) ករើលក្ខខណឌណា ដែលររូវយក្មក្បញ្ជាក្់ចំកោះ
ររីកោណទាំងពីរ ដែលថាវាប៉ន ុ គ្នាក្ាុងចកមលយ ើ របស់
អ្ាក្សរាប់សំណួរទី ១)។ ចូរករជស ខណឌ ើ ករសលក្ខ ើ បនី នបណា ា លក្ខខណឌខាងករោម ក
អ្ក្សរដែលររូវគ្នា។ ក្. AM CN គ. BD DB ង. A C
យ ើ សរកសរ
ខ. DM BN ឃ. MBD NDB ច. MDB NBD
៣) ពនយល់ជាោក្យ នូវលក្ខខណឌប៉ន ុ គ្នាននររកី ោណ ក្ាុងចកមលយ ើ របស់អ្ាក្ សរាប់សំណួរទី ១)។
284) The diagram shows rectangle ABCD. Let M be the midpoint of side AB. When connecting point M to vertices C and D, AMD BMC will be proven in the simplest way using congruent triangles.
A
២៨៤) រូបបង្ហាញពីចរុកោណដក្ង ABCD ។ កគឱ្យ M ជា
ចំណុចក្ណា ា លននរជុង AB ។ កៅកពលដែលកយង ើ ភ្ជាប់ ចំណុច M កៅក្ំពូល C និង D,
នឹងររូវបង្ហាញកោយោរករបើលក្ខខណឌប៉ន ុ គ្នាននររីកោណ
A
D
M B
AMD BMC
D
M B
C
១) ករើគូររីកោណណា ដែលគួបង្ហាញថាវាប៉ន ុ គ្នា?
1) Which two triangles should be shown to be congruent? 2) Which conditions are required to prove that the two triangles in your answer for 1) are congruent? Choose three conditions from the following and write the corresponding letters. a) AD BC b) MD MC c) AM BM d) AMD BMC e) MAD MBC f) ADM BCM 3) Explain in words the condition for congruent triangles in your answer for 1).
285) In the figure on the right, a square with sides of length 8 cm is equally divided into 8 8 small squares, each with a side of length 1 cm.
C
២) ករើលក្ខខណឌណា ដែលររូវយក្មក្បញ្ជាក្់ចំកោះ
ររកី ោណទាំងពរី ដែលថាវាប៉ន ុ គ្នាក្ាុងចកមលយ ើ របស់
អ្ាក្សរាប់សំណួរទី ១)។ ចូរករជស ខណឌ ើ ករសលក្ខ ើ បីននបណា ា លក្ខខណឌខាងករោម ក
ើយសរកសរ
អ្ក្សរដែលររូវគ្នា។ ក្. AD BC ខ. MD MC គ. AM BM ឃ. AMD BMC ង. MAD MBC ច. ADM BCM
៣) ពនយល់ជាោក្យ នូវលក្ខខណឌប៉ន ុ គ្នាននររកី ោណ ក្ាុងចកមលយ ើ របស់អ្ាក្ សរាប់សំណួរទី ១)។
២៨៥) ក្ាុងរូបខាងស្តាំ ោករមួយានរង្ហវស់រជុងរបដវង 8 cm
In this figure, construct a line segment of length 10 cm using only a ruler. You can only use the ruler for drawing lines. You don’t need to write your steps leading to your answer.
307
គឺររូវដចក្កចញជាោកររូច
ានរង្ហវស់របដវង 1 cm ។
8 8 កសមើៗគ្នា ដែលរជុង
ក្ាុងរូបកនះ ចូរែឹក្បោារ់ដែលានរបដវង 10 cm
កោយោរករបរើ ាស់ដរបោារ់បកុ៉ ណា ោ ះ។ អ្ាក្អាចករបើ ានដរបោារ់បកុ៉ ណា ោ ះកែើមបគ ី ូសបោារ់។ អ្ាក្មន ិ ចាំ ាច់សរកសរជំហានននចកមលយ ើ របស់អ្ាក្កទ។
286) Three marbles are picked from a jar containing 6 white marbles and 3 red marbles. Let A be the event “three marbles picked are red”. Three marbles are picked from a jar containing 7 white marbles and 3 red marbles. Let B be the event “three marbles picked are red”. How many time more is the probability of occurring event A than that of event B?
២៨៦) ក្ូនឃលីបីគឺររូវានករជស ា យ ដែល ើ ករសព ើ ីក្ុ ងរក្ឡមួ
287) Cube A has a side of length a . Cube B is a cube that is obtained by extending each side of cube A by length b . a) Write the volume of cube B in terms of a and b . Write only your answer in expanded form. b) How many times more is the volume of cube B than that of cube A? Express your answer in term of t , b where t . a
២៨៧) គូប A ានរទនុង a ។ គូប B គជា ឺ គូបដែលាន
288) Prove x y z a when x y z 3a and x 2 y 2 z 2 3a 2 , where a, x, y and z are real numbers.
២៨៨) បង្ហាញថា
ានផាុក្ក្ូនឃលស ី 6 នង ិ ក្ូនឃលរី ក្
ម 3 ។ តាង A ជា
រពឹរាិោរណ៍ “ក្ូនឃលីទាង ំ បីានករជើសករសគ ើ ឺរក្
ម”។
ក្ូនឃលីបីគឺររូវានករជើសករសព ា យ ដែល ើ ីក្ុ ងរក្ឡមួ
ានផាុក្ក្ូនឃលស ី 7 នង ិ ក្ូនឃលរី ក្
ម 3 ។ តាង B ជា
រពឹរាិោរណ៍ “ក្ូនឃលីទាង ំ បីានករជើសករសគ ើ ឺរក្ ករើបោ ុ៉ ម នែង
ម”។
ដែលរបូាបកក្ើរានកឡង ើ ននរពឹរាិ-
ោរណ៍ A ករចន ាិ រណ៍ B ? ើ ជាងរបូាបននរពរ ឹ ោ
កោយក្វើោរពរងីក្រទនុងនីមយ ួ ៗននគូប A របដវង
ក្. ចូរសរកសរាឌននគូប B ជាអ្នុគមន៍នន a និង
b។
b។
សរកសរចកមលយ ើ របស់អ្ាក្ជាទរមង់ពោលរ។
ខ. ករើបោ ុ៉ ម នែងននាឌគូប B ករចើនជាងាឌននគូប A? បញ្ជាក្់ចកមលយ ើ របស់អ្ាក្ជាអ្នុគមន៍នន ដែល t
b ។ a
x y z a កៅកពលដែល
2 2 2 2 x y z 3a នង ិ x y z 3a ដែល
ួ ពិរ។ a, x, y និង z គឺជាចំនន
លំហាត់ត្តឹមលលខ ២៨៨ លេះ ខញំបា ុ េបកប្ត្បជាភាសាជាតិ េិងលោះត្សាយរកចលមលើយ ជូេដល់មិតអ្ ត នកអាេរួចរាល់ល ើយ។ សូមមមើលដំមោះស្រាយបន្ទាប់ពីទំព័រមេះ មដើមបីម្ាៀងផ្ទាត់ជាមួយចមមលើយរបស់អ្នកមស្រោយពីមោះស្រាយរួច។
3 308
t ,
1)
x
2
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6 x 8 x 2 x 8
2 x 1 2 x 3 2 x 5 2
2
4 x 2 4 x 1 4 x 2 10 x 6 x 15
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4 x 2 4 x 1 4 x 2 4 x 15 8 x 16
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3x 6 5 x 2
x 1
3 x 5 x 2 6
2
x 1 0
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a
b 2 a b
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0 y2 0 y2
3 7 0 b
។
b 3 y 7x 3 ។
5)
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x
l m n 3 cm 4 cm 3 4 ឬ 5 cm x cm 5 x
x
20 3
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30
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5 cm
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។
30 cm ។
309
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4
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f x 0
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4 x 2 xy 4 xy y 2 4 x 2 y 2
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4 x 2 3 xy y 2 4 x 2 y 2 3 xy
( 01 5
x
cos A ABC
2010 2010 2 3 5 67
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x
4 x2 5x 8 0
C
5 4 4 8 25 128 153 2
B
cos A 1 sin A cos A 0 A
C
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sin A
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A
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B
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310
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a
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f x 2x2 3x 9
C AB BC AC 2
2 x 2 3x 9 0 ២
① 2
f x 0
f x 0 2x2 3x 9 0 32 4 2 9 9 72 81
BC = 3cm , AC = 7cm 2
AB2 32 7 9 7 16cm2
AB 16cm 4cm 2
AB 0
22
B = {1 , 4 , 6 , 7 }
C
2
A = {1 , 3 , 7 , 8 }
7cm
ABC
2 2
A B
26)
cm
3cm
24
A
B
2
2 2 2 2 2 2
B
EAD CAB x 4 ឬ x3 6
DE BC AD ED AB CB
2 22
22
AB AB 4 cm ។
311
x1
3 81 3 9 3 2 2 4 2
x1
3 81 3 9 3 2 2 4
៩
x
f x 0
3
+
0
x 3 ឬ x ②
-
3 2 0
33)
x
x 2 10 x 6 0
' 5 6 25 6 31
+
2
3 2
x
x
x 3 ឬ x
x
) 3
34)
3 2
0
15 5
5 3 2 5
( 3 2
3 5 10 3 5
x
10
A 3B
28)
A 3x 2 x
B x2 x 1
2
y
35)
A 3B 3x 2 2 x 3 x 2 x 1
x
y ax
2
y 45
3x 2 2 x 3x 2 3x 3 5x 3 29)
5 31 5 31 1
x 3
45 a 32
a5 y 5x
។
2
sin cos
sin cos
cos
36)
x
tan 1 o
0o 180o
45o
x
l
45 180 k , k o
4
។
1
m
2 30)
10! 10 9 8! ① 10 C2 C 10, 2 45 8! 2! 8! 2 10! 10 9 8! ② 10 C8 C 10,8 45 2! 8! 2 8!
2 1 24 x
m ឬ
x 3 ។
2x 6
37)
31)
5 cm
x 3 y x y x 5 y 2
x 2 6 xy 9 y 2 x 2 5 xy xy y 2
2
12 cm
5cm 12cm 2
x 2 6 xy 9 y 2 x 2 6 xy y 2
25cm2 144cm2
8 y2
169cm2
2
169cm2 13 cm 32) 13 cm ។
4 x 2 225 2 x 152 2
2 x 15 2 x 15 312
៩
38)
x 2 y x 2 2 xy 4 y 2 x 2 y 3
x 8y 3
x
3
+
0
f x 0
3
1 5 -
3
0
+
1 x3 5
39) x3 6 x 2 y 12 xy 2 8 y 3 x 2y
②
x
3
1 x3 5
40)
[ 10 5
x
3x 12 9 x 18 3 x 9 x 18 12 6 x 6 6 x 6 6 6 x 1
43)
10
] 3
x
P3
P P 10,3
10 3
10! 10 9 8 7! 720 7! 7!
44) 6
41) y ax 2 bx c -
C 8 ,3
b b 4ac S , 4a 2a b x 2a 2
2 8 8 4 115 , 2 1 4 1
។
45) ①
5 x 2 14 x 3 0 f x 5x2 14x 3
30 15 56 28
sin sin 2 cos2 1
sin 1 cos2 , 5 cos 7
២
42) ①
8! 56 5! 3!
C 6 , 2 C 2 ,1 6! 2! 30 4! 2! 1!1!
y x 2 8 x 15
4 , 1
2
f x 0
49 25 2 6 5 sin 1 49 7 7 2
f x 0 5x2 14x 3 0 ' 7 5 3 49 15 64
②
2
7 64 7 8 1 5 5 5 7 64 7 8 x2 3 5 5 x1
tan sin tan cos
313
2 6 7 2 6 5 5 7
៩
46)
53)
x x 2 x 1 x 3
x 2 x x 3 x x 3 2
x y x2 xy y 2
។
x3 y 3
2
x2 2 x x2 2x 3 3
54)
x3 3x2 y 3xy 2 y3 x y ។ 3
47)
9 x 2 y 2 3x y 2
3x y 3x y
2
48)
55)
x
x2 5x 6 0 S 5 2 3 P 6 2 3
x2 , x3
1 5
4
3 1
3 1
3 1
3 3 3 3 3
2 3 22 3 2
49) 2
4 1 3 3 3 3 1
56)
a
b
20
y x 2 ax b
1 , 2
1 2 5 5 2 5 6
y ax 2 bx c
y
50)
b b2 4ac , 4a 2a
x
y ax 2
y 1
b 2a 1 b 2 a 2 2 b 4ac 8a b 4ac 2 4a a 2 1 a 2 2 2 a 4 1 b 8 1 a 4b 8
x2
1 a 22
a y
1 4
1 2 x ។ 4
51) a
a
d a 2
d 2
57)
d 4 cm 4 cm a 2 2 cm 2
52)
m x 6 2 4 x 3
9 3 9
3
C 9,3
m
x
b 1 ។
a2
d
x
6
9! 6! 3! 9 8 7 6! 84 6! 3 2
C 9,3
2 4
84 ។
m
314
៩
58)
x
a ) 8 4 10 2
2 x2 5x 3 0
f x 2x2 5x 3
f x 0
b) 56 8 3 10
f x 0 2 x 2 5x 3 0
c) 1 2 2 10 4
' 5 4 2 3 25 24 49 2
7 7 1 1 d ) 1 8 9 16 2
5 49 5 7 1 2 2 4 2 5 49 5 7 x2 3 2 2 4 x1
x
f x 0
+
x 59) ①
1 2 0 -
3
e) 6 2
3 2 2
f ) 10 2 5 10
1 10
5 10
3
21 2 11 2 g ) 8 x 2 5 x 7 3x 5
0
+
h) 6 0.6 x 0.4 0.3 2 x 5 3 x 0.9
i ) 2 3 x 4 y 5 4 x 3 y 14 x 7 y
1 ឬ 3 x 2
3x 5 y x 5 y 13 x 35 y 8 6 24 3 2 2 k ) 32 x y 4 x y 8 x y j)
cos 1 0o 90o 2 2 sin cos2 1
sin
2
2
2 3 6 3 l ) x3 y 2 x 4 y 2 x 2 3 5 5 2
cos 1 sin 2 , 0o 90o 4 1 3 1 cos 1 4 2 2 2
②
12 x 2 8 x 9 x 6 12 x 2 x 6
3 3
x 3 y x 5 y x 4 y
b)
2
x 2 5 xy 3 xy 15 y 2 x 2 8 xy 16 y 2 x 2 8 xy 15 y 2 x 2 8 xy 16 y 2
A B
y2
A = { 2 , 3 , 5 , 7}
B = { 1 , 2 , 3 , 4}
A B 1, 2,3, 4,5,7 ②
4 x 3 3x 2
a)
tan 1 sin 1 tan 2 cos 3 3 2
60) ①
62)
63)
។
16 x 2 25
a)
4 x 52 2
A B
A B 2 , 3
A B
4 x 5 4 x 5
9 x 2 24 xy 16 y 2
b)
3x 2 3x 4 y 4 y 2
n A B 2
។
61) 315
3x 4 y
2
2
៩
64)
a3 b 2
66) a) 4 x 9 x 6
a 2
5 x 15
b3
a3 b2 2 32 8 9 1 3
x 3 b) 0.5 x 3.2 0.8 x 1.6
67)
5 x 32 8 x 16
4 x 11 2 x 15
3 x 48
2x 4
x 16
x2
c) x 14 x 45 0 2
S 14 5 9 P 45 5 9
y
68)
y ax y 4
x5 , x9 ។ d ) x 8x 9 0
' 4 9 16 9 7 2
1 y 4 2 2
x 4
65)
y
69)
7 x 5 y 1 a) 3x 2 y 1
x
y ax b
2 , 4
D 14 15 1 Dx 2 5 3
4 3 2 b
b 2
y 3x 2 ។ 70)
ឬ
221
?
221 1317
3 1 x 6 y 4 1 x y 1 ឬ b) 4 2 5 x 5 y 1 2 0.5 x 0.5 y 0.1
។
221 71)
x 6 y 4 3 1 3 2 2 : 5 6 y 4 5 y 1 35 y 21 y
x2, y4
a 3
Dx 3 3 D 1 D 4 y y 4 D 1
x
x 3 , y 4 ។
3
1 2
1 y x 2
4 7 4 7 1
Dy 7 3 4
a
4 a 8
2
x
x 8
n
n 2 180o n 3 5
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n9 9 2 180o
140o
9 180 140o 40o
2 3 x 6 4 5 5
o
40o ។
2 3 x ,y ។ 5 5
316
៩
72)
x
m
xcm
8cm
1 8 x 4 x ឬ
76)
1cm
a) x 2 3 x 54 x 2 9 x 6 x 54 x x 9 6 x 9
4cm
b) x 2 y 4 y 3 y x 2 4 y 2
32 x 5
x 32 4 x
73)
x 9 x 6
m
y x 2 y x 2 y
AC A
AC 2 AB 2 BC 2
6cm
D
77)
3cm
AC AB 2 BC 2
B
x a) 6 x 16 9 x 11
3x 27 x 9 7 x 3 3x 1 b) 8 4 7x 3 6x 2 x5
C
AB 3 cm , BC AD 6 cm
AC 32 62 45 3 5 cm 74) a) 13 15 4 2
c) x 2 4 x 21 0
b) 8 5 20 5 44
S 4 7 3 P 21 7 3
c) 2 4 24 3
2
5 1 15 d ) 0.75 0.2 6 4 8
x 7 , 3 d ) x2 8x 2 0
e) 18 72 2 4 2
' 4 2 16 2 18 2
3 f) 2 5 5 4 5 6 5 5 g ) 7 3 x 2 9 2 x 5 3 x 59 4x 1 5x 3 x 13 6 8 24 i ) 3 9 x y 5 x 2 y 22 x 13 y h)
x
24 x 9 y 90 5x 9 y 3 29 x 87
2
40 5 1 18 l ) xy xy 2 y y 9 6 5
x 3
75)
8 3 3 y 30 x 3, y 2 ។
21x 2 28 xy 15 xy 20 y 2 20 y 2
0.8 x 0.3 y 0.8 8x 3 y 8 ឬ b) 2 3 9 x y 4 x 15 y 18 2 5 5 1 4 x ,y ។ 2 3
21x 13xy 2
x 6 x 5 x 8 2
x 2 12 x 36 x 2 8 x 5 x 40
x 3
3 y 30 24 y 2
7 x 5 y 3x 4 y 2 y 10 y
b)
។
8x 3 y 30 a) 5x 9 y 3
k ) 18 x 2 y 2 xy 6 x 3 y 6 y
a)
4 18 43 2 1
78)
j ) 0.3 x 8 y 8 0.4 x 0.2 y 3.5 x 0.8 y 2
។
9 x 76 317
៩
9 x 2 2 xy
79)
85)
x
y 1
x 3
l m
9 x2 2 xy 9 3 2 31 81 6 87
m
1
A x
BC ) BDC 90 38 52 o
1 ។ 36
o
O 38o
o
B
C
x BDC 52o ។
y 8x 3 y 7 3 y 7 8x 7 8x y 3
82)
D
x BDC
3
81)
x
x
86)
6 6 36 9
95o
54o
9 6
41o
x 95o 41o
2
80)
l
87) a ) 12 13 4 3 b) 2 5 12 6 12 c ) 4 2 4 0 2
y
5 5 1 d ) 0.2 0.5 8 16 2 e) 2 8 18 0
a y x y 8 x 3 a 24 8 a 24 y 3 x 24 y 12 ។ x 2 2
f ) 4 3x 5 8 2 x 3 4 x 4 5 x 3 x 1 13 x 11 2 3 6 h) 4 6 x 7 y 8 2 x 4 y 8 x 4 y g)
i ) 0.2 5 x 10 y 2 2 x 0.5 y 5 x y
83)
y
j ) 28 x 2 y 2 5 xy 35 xy 3 4 x 2
x
y ax 2
2
y 32 32 a 4
5 5 2 1 k ) x 2 y xy 3 y 3 x 3 y 2 6 9 15 6
x4 a 2
2
y 2 x 2 ។
88) a)
84)
b)
n
n 2 180o n
8 2 180o
x 2 8 x 16 x 2 8 x 2 x 16
)
n 8
x 2 8 x 16 x 2 10 x 16 18 x
135o
8 180 135o 45o o
x 7 y x 7 y x 2 49 y 2 2 x 4 x 2 x 8
89)
x 2 x 6 4axy 5ay 2 a x y x 5 y
a) x 2 8 x 12
45o ។
b) ax 318
2
៩
90)
93)
x a ) 9 x 19 11x 13
2010
2010 2 3 5 67
2 x 32 x 16 7 x 4 5x 4 b) 8 6 21x 12 20 x 16 x 4 1 c) x 2 0 4 1 x2 4 1 x 2 2 d ) x 2x 1 0
94)
h 1 mgh mv 2 2 1 2 mv h 2 mg
y
95)
y ax y5
x 2 5 5 5 a 2 a y x 2 2 5 y 4 10 ។ x4 2
' 12 1 1 1 2 x
1 2 1
1 2
y
96)
91)
x
y ax
x y 1 a) 3x 2 y 18 2 x 2 y 2 3x 2 y 18 5 x 20
2
y 3
x 3 1 3 a 32 a 3 1 2 y x ។ 3
x4
x 4 : 4 y 1 y 3 x 4 , y 3 ។
2 x y 3 2 x y 3 b) 1 1 ឬ 4 x y 4 0.5 x 8 y 2 2 x y 3 7 4 x y 4 x 6 6x 7 7 7 2 x : 2 y 3 y 6 6 3 7 2 x , y 6 3
97) n
n 2 180
o
n 8
n
98)
8 2 180o 8
x
។
3 xy 6 xy 2
x 5
)
135o ។
x
6 92)
v2 2g
2
4
m
y2
3xy 6 xy 2 3 5 2 6 5 22 90
m 319
x 6 2 4
x 3 ។
៩
x
99)
BC : CD 1: 2 CAD BAC 2 o 46 23o ឬ x 2
103)
A
x a ) 7 x 4 4 x 10
3 x 6 x 2
o x 46
D B
b) C
100)
a) 5 2 8 4 7 b) 18 30 6 23
x 5 3x 7 1 4 2 x 5 6 x 14 4 5 x 15 x3
c) 5 x 2 40 0 5 x 2 40
c) 5 32 4 11 2
x2 8
2
2 5 1 2 d) 3 6 3 3 e) 2
x 2 2
d ) x 8x 4 0 2
3 2 2 24 4 6
' 4 4 16 4 20 2
10 6 5 g ) 2 8 x 5 6 3 x 2 34 x 2 f)
2
5 1
h) 0.8 0.6 x 5 0.5 0.9 x 3 0.03 x 2.5 i ) 3 4 x 8 y 7 2 x 6 y 26 x 66 y
x
4 20 1
104)
2 x 5 y 2 a) x 3 y 12
3 x 5 y 4 x y x 17 y j) 6 9 18 3 2 k ) 28 xy 7 xy 4 y
D 6 5 11
2
5 2 5 4 l ) x3 y x 2 y x 6 5y 3 3
Dx 66 6 D 11 D 22 y y 2 D 11
Dx 6 60 66 x Dy 24 2 22
101)
x 6 , y 2 ។
a ) 3 x 4 y 3 x 4 y 9 x 2 16 y 2 b)
x 6 4 x 3 2 x 5
42 5
2
0.3x 0.4 y 0.1 b) 6 4 7 ឬ x y 3 15 5
4 x 2 3 x 24 x 18 4 x 2 20 x 25 4 x 2 21x 18 4 x 2 20 x 25 x 43
3x 4 y 1 18x 20 y 7
D 60 72 12 Dx 8 2 D 12 3 D 3 1 Dy 21 18 3 y y D 12 4 Dx 20 28 8 x
102)
a) x 2 4 x 12 x 2 2 x 6 x 12
x 2 x 6
2 1 x ,y ។ 3 4
x y 12 x y 36 2 x y 6
b)
2
320
៩
2ab b2
105)
a4
x
111)
b 2
62o
m
2ab b2 2 4 2 2 20 2
x
x 62 46 o
106)
A
B
6 6 36 4
x
x
112)
O
AB = AC
3
ABC ACB
1 , 3 ; 2 , 2 ; 3 , 1
46o
m
108o
4 A
o
3 1 36 12
x 180o 2ACB
។
64o
B
ACB ADB 64o
D C
AD
x 180o 2 64o 52o
y
107)
5x 2 y 3 2 y 3 5x 3 5x y 2
108)
113) a) 8 10 16 2 b) 16 24 8 13 c) 3 2 7 2
4
2
5 1 d ) 4 1 9 3
y y
a x
y9
e) 18 98 2 5 2 2 12 f ) 6 2 10 2 6 6 g ) 7 4 x 3 5 5 x 4 3 x 1
x4 a a 36 4
9
y y
x 6
36 6 ។ 6
y
109)
36 x
h) 0.6 3 x 0.7 2 0.3 x 0.2 2.4 x 0.02 i ) 4 9 x 6 y 9 x 2 y 45 x 6 y 5 x 3 y 2 x 5 y 23 x 29 y 12 9 36 2 3 2 k ) 63 x y 9 xy 7 xy
x
j)
y ax 2
y 8
x4
2
7 2 y 5 35 l ) xy x 2 y 3 y 20 3 6 48
x2 y 2
1 8 a 42 a 2
114) 110)
a) n
30 x 2 5 xy 6 xy y 2
n 2 180
o
30 x 2 xy y 2
n 12
12 2 180o 10 180o
5 x y 6 x y
b)
x 7 x 8 x 6 2
x 2 14 x 49 x 2 6 x 8 x 48
1800o
x 2 14 x 49 x 2 14 x 48 1 321
៩
12 x 34 y 15 12 x 8 y 6 42 y 21 1 y 2
115) a ) x 100 2
x 10 x 10
1 2 1 6 x 4 3 2 1 ឬ 6 x 1 x 6 1 1 x , y ។ 6 2
b) ax 2 6ax 9a a x 2 6 x 9
116)
a x 3
2
x a) 9 x 5 6 x 7
3 x 12 x 4
a 3
x 2 8 x 12 7 x 14
b 4
9a 2 ab 2 9 32 3 4
2
81 48 33
x2 x 2 3 x 54 0
c)
9a 2 ab2
118)
b) 0.2 x 2 0.8 2 x 3
y
119)
x 2 6 x 9 x 54 0
A
x x 6 9 x 6 0
B
6 6 36
x 6 x 9 0
9
x 6 0 x 6 x 9 0 ឬ x 9 ឬ x 6 , 9 ។
2, 2 ; 2, 4 ; 2 , 6 ; 4, 2 ; 4, 4 ; 4 , 6 ; 6, 2 ; 6, 4 ; 6 , 6
d ) x 8x 9 0 2
' 4 9 16 9 7
9 1 36 4
។
2
4 7 x 1
?
120)
4 7
3 3,2 7
3 3 32 3 27
117)
2 7 2 2 7 28
2 x 7 y 11 a) 5 x 2 y 8 D 4 35 39
5 52 25 27 27 28
78 2 39 39 y 1 39 x 2 , y 1 ។
2 7 ។
Dx 22 56 78 x Dy 16 55 39
5
121)
y y ax y 18
x 9
18 a 9 a 2
1.2 x 3.4 y 1.5 12 x 34 y 15 b) 2 4 1 ឬ x y 6 x 4 y 3 9 3 3
x 2
y 2 2 4 y4 ។
322
y 2 x
៩
122)
y
x
y ax
EBC
127)
2
A
D
y 4
x4 1 4 a 42 a 4 1 y x2 ។ 4
E C
B
BCA CAD
AD||BC
CBD ADB
CAD ADB
AE = DE 123)
BCA CBD
n
E
n 2 180o n5
124)
BD ។
EBC
n
AC
5 2 180 5
AB = DC
o
EBC
108o ។
x
EC = EB
AE DE EC EB AC DB
107o
m
ABCD
x 180o 107 o
85o
m
AC DB
x
AB = DC ។
73
o
128) 125)
x
A
x 4 289
C o
68
x 2 289
O
x 2 17
x
x 2 17
B
180o 68o ABC 56o 2 180o 2 68o OBC 22o 2
x 17
129)
x
x 17 )
x ។
CAB , ABC
BCA A
AB : BC : CA 3 : 4 : 5
126)
AB BC CA 3 4 5
y x 1 , y x 1 , y 2 x 2 , y 3x 3 ,
x 1
x 2 17
x 17
x ABC OBC 56o 22o 34o
2x 2 3x 3
B
O
C
AB BC CA AB BC CA 360o 30o 3 4 5 3 45 12
y0
x 1, y 0 ។ 323
៩
AB 30o 3
AB 90o
BC 30o 4
BC 120o
CA 30o 5
CA 150o
BCA
AB 90o 45o 2 2
CAB
BC 120o 60o 2 2
ABC
CA 150o 75o 2 2
-
BCA 45o , CAB 60o , ABC 75o
AB : BC AB : BC 3: 4
AB : BC 3 : 4 ។
131) 1
6
130) -
A
ABCD = 1 , ABFE = 2 EFGH = 5 , CDHG = 3 ADHE = 4 , BCGF = 6
1 3 y x2 x 1 2 2 b b2 4ac S , 4a 2a
B
D C
E F
2 3 1 3 4 1 2 2 S 2 , 1 1 4 2 2 2 9 3 4 2 3 1 S , S , 2 2 8 2
G
A = ABCD+ ABFE +ADHE =1+2+4 = 7 B = ABCD+ ABFE+ BCGF = 1+2+6 = 9 C = ABCD+ CDHG+BCGF = 1+3+6 = 10 D = ABCD+ CDHG+ADHE = 1+3+4 = 8 E = ABFE + EFGH+ ADHE = 2+5+4=11 F = ABFE + EFGH+ BCGF = 2+5+6 = 13 G = EFGH +CDHG+ BCGF = 5+3+6 = 14 H = EFGH +CDHG+ ADHE = 5+3+4 = 12
3 1 , 2 8
7, 8, 9, 10, 11, 12, 13, 14 -
y0
H
132)
1 2 3 x x 1 0 ឬ x 2 3x 2 0 2 2 a bc 0 c 2 x1 1 , x2 2 a 1 1, 0 ; 2, 0 ។
5 x 2 y 5 x 2 y 5 x y
2
25 x 2 4 y 2 25 x 2 10 xy y 2 25 x 2 4 y 2 25 x 2 10 xy y 2 5 y 2 10 xy
324
៩
133)
138) x 1 5x ឬ x 5x 1 0 2
2
3 2 o 60 360o k ឬ 120o 360o k sin
5 4 1 1 25 4 29 2
x
5 29 5 29 2 1 2
y
134)
។
k 0o 180o
x
60o , 120o
y ax 2
y 20
x 2 y 5x
2
ABC
139)
20 a 2 a 5 2
G
A xA , yA ; B xB , yB ; C xC , yC
។
y y y x x x G A B C , A B C 3 3 ABC
135) 10 cm
។
6 cm
A(-2 , 6) , B(7 , -9)
6 cm
C(1 , 0)
2 7 1 6 9 0 G , ឬ G 2 , 1 ។ 3 3
10 cm 140)
a 2
6 cm 10 cm a 2 2 a 2 9 cm2 25 cm 2
AB = DC
D
A
AD||BC
2
2
AB DC
C
B
AB DC
a 2 34 cm2 a 34 cm
141)
8051
8051 72 902
34 cm ។
8051 902 7 2 8051 90 7 90 7
136)
8051 83 97
xz x 2 y 2 yz
83
x z 1 2 y 1 z
97
8051 83 97
1 z x 2 y 142)
m2 n 2 ឬ
137)
B
n 2 m2
n m n m
n m 1
A A
B។ 325
(
។
៩
143)
AG
tan 1 2
146)
D
tan 1 2
C
A
tan 1 2
tan 2 3 23 5 tan 1 2 1 1 2 3 5
B
H
19 cm
G 25 cm
E
F
32 cm
tan 1 tan 2 1 tan 1 tan 2
147)
EG 2
3x 2 4 x 8 0
EG 2 EF 2 FG 2 32 cm 25 cm 2
2
b a
S
1024 cm 2 625cm 2 1649 cm 2
AG 2 EG 2 AE 2 1649 cm 19 cm 2
148)
OP
2
4 4 ។ 3 3
PH
P
1649 cm 2 361cm 2 2010 cm 2 AG 2010 cm , AG 0
A
AG 2010 cm ។
O
3cm H
B
7cm
AB 3cm 7 cm 5cm 2 2 OP OA 5 cm
OP OA 144)
f x x
OH OA AH 5 cm 3 cm 2cm
R f
R (
PH 2 OP2 OH 2
១០)
x3 3x 2 4 x 8
PH OP2 OH 2 , PH 0 PH 52 22 cm
x2
PH 21 cm
R 2 3 2 4 2 8 R 8 12 8 8 3
2
PA
AP 2 AH 2 PH 2 , AP 0
R4 145)
PB
AP AH 2 PH 2 32 212 30 cm
o
sin135
sin 180o sin
PB
sin135o sin 180o 45o sin 45
AB PH 10 21 2100 70 cm PA 30 30
PA 30 3 PB 70 7
o
2 2
PA : PB 3 : 7 326
។
៩
149)
a
a 30 , x2 6 x xa 0
360
x1 5
2
20 2 x 2 x
20 2x )
5 5 a 0 a 30 2
S R2
x2 x 30 0
a 30
o
360 20 2 x S x2 2 x 10 x x
x1 x2 P 30 30 30 x2 6 ។ x1 5
x 2 10 x x 2 10 x 25 25
150)
2n 1
2n 1
x 5 25 2
2n
x 5
n
2n 1 2n 1 2
x 5
8n 4 2n 2
2
x 5 25 25 2
S
2
2n 1 2n 1
0 x 5 0 ឬ
2
4n 4n 1 4n 4n 1 2
2
2
*
0 ឬ x 5
25 cm 2 2
4 2n
x 5cm ។
-
2x 20
6
3
6 6 6 216
x
20 cm
២ ៣
។
151)
១
152)
4
-
១
២
6
x cm
-
៣
5
20 2 x cm
-
6 5 30 30 5 ។ 216 36
ABC
153)
0 20 2x 2 x 0 20 2 x x 10 10 20 2 x 2 x x 1 10 x 10 * 1
3 15 4 1 SABC AB AC sin A 2 1 15 3 15 2 3 2 4 4 SABC
S ABC 327
៩
BC
158) BC
6
BC AB AC 2 AB AC cos A sin 2 A cos2 A 1 2
2
A
B
2
cos A 1 sin A 2
, 90 A 180
2
o
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6
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159)
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160)
a
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155) x 3 x y 3 xy y x y 3
2
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156)
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157)
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sin 2 9 9sin 2
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12 3
2
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10
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162)
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1 0 3 1 7
x 6 y 6 z 6 3xyz
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, sin 0
x ,
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1,1, 1 , 1, 1,1
328
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163) x cm
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165)
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164)
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1 y x 2 y a'x b'
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1
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cm
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166)
cm 2
168)
n 2 2n
50 cm 2
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n
1 1000 1000 cm
-
pq qp
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n 2k
n
10cm 5cm 50cm2
k n2 2n 2k 2 2k 2
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169)
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n 2n
2
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167)
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170)
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171) 125 x 3 75 x 2 15 x 1 5 x 1
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172)
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173)
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' 0
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A
sin A 1 cos2 A
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94 5 2 sin A 1 9 3 3 2
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f k 2k 2 3k 9
f k 0 2k 3k 9 0
។
32 4 2 9 9 72 81 3 81 3 9 3 2 2 4 3 81 3 9 3 k2 2 2 4 2
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174) 0, 1, 2, 3, 4
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3
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175)
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177)
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178)
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2
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179)
182)
cos
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45o
y
183)
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o
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x
y ax
0o 360o 2 2
cos
y
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P
30 cm 30 cm 30 cm 900 cm 2
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x
m:n
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P x
-
54 m 2 540 000 cm 2
540000 cm2 600 900 cm2
mb na ។ x mn
m:n
A(-3)
B(3)
P
185)
AP:PB=1:2
AB
20 g
160 g
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1 3 2 3 3 1 1 2 3
20 g 0.125 160 g
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186) 181)
ACD
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120
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184)
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y
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x 120 6 x ។ 100 5
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332
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187)
100 50
80 g
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1
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100
1.6 g 100 160 g
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ABD CDB
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150 g
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190)
790 g 150 g 640 g 6cm
640 g 400 1.6 g
188)
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8cm 10cm
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V 10 cm 8 cm 6 cm 480 cm3
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191)
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A
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192)
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A
195) x y 8 90 x 120 y 870
D
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x y 8 y 8 x 3
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193)
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3 : y 8 x 8 3 5
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d cm
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3 5 8 90 3 120 5 870
cm2
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ABC
194)
196)
A
(PGCD) (14, 35)
B
17 2 7
35 5 7
PGCD(14, 35) = 7 ។
B
A'
C
(26, 39, 52)
ABC
4
26 2 13
BAA '
ACB , CB ' A ' , B ' CA '
39 3 13 52 22 13
BAC BAC
PGCD( 26, 39, 52) = 13 ។
3
B ' A ' C , CAB ' , ACB ' ។
A ' C A ' D CD 2
ABC ABC BAC ABC BAA ' 2
xx
ABC
PB
x 5 180o x 180o x 72o 2 2
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2
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o
2
cm
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D
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197)
PA PA'
។
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cm
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PB PA 4 PA 4 PB 334
C
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202)
PBA '
6
PB 2 BA '2 PA '2
50
PB 2 BA '2 PA2
6
PB 2 22 4 PB
9
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PB 4 16 8 PB PB 8 PB 12
2
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3 cm 2
8
50 24
8
24
150
3
24
198)
24 1.05 kg
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6 3 1
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203)
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b
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199) 8400
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200)
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204) ។
2 1 2 23 2 1 5 2 5 52 5 2 15 4 2 11 5 10 5 10 11 25
201) x
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276 240 g 1.15 335
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208)
11 25 11 5 11 1.1 2 25 2 10 5
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75 42 33 50 50
d2 d S 2S B S L 2 2 h 2 4
205) 1
60
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202
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dh 20 45 2 2 200 900 1100
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2000 cm 2
1
3454 1.727 2000
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209)
x
400 0.04 m 2 10000
400 cm 2
។
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206)
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207)
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D
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500 106 394 m
E
500 92 408 m
D
E
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x
3 x 5
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B
3 x 120 5
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258 339 597 m 336
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x
60 g
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215) 1
210)
1
a
4
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a 3s b c
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211)
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216)
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213)
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217)
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218)
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220)
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C
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219) A
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221)
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224)
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222)
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2
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223)
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225)
cm3 V a3
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a
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226)
ABC y 2
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227)
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231)
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228)
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d
d 1 200 m t 15 1 200 vm 80 15
x
229)
d t
t
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3x 13 11x 3 8x 16
21 3 7
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GCD 14, 21 7
3.14 25 3.14 5 3.14 20
(10, 15, 20) 10 2 5 , 15 3 5 , 21 22 5
GCD 10, 15 , 20 5
230)
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ABC
cm 2
233)
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1 S 6cm 3cm 2
x
ABC
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5 cm
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2
4 cm
340
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234)
237)
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600 696 714 2010
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235)
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m2 400
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1 m2 20000 1 2 cm2 10000
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3 x 2 y 11 3x 2 y 11 3x 3 y 15 ឬ y4 x y 5
1000 2.5 400 CO 2 CO 2 3150 3.15 1000
x y 5 4 5 1
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341
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y 6x 3 y 4x 7 6x 3 4x 7 2 x 4 x 2
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241)
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242)
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243)
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246)
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ac bd
247)
251)
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cd
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248)
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252)
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253)
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52.5 1.2 63 kg
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52.5 kg
z 24o
42 kg
o
360
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។
343
។
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254)
cm3
256) 80 36
8 cm
36 100% 45% 80 V a3 8 cm3 512 cm3
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15% 80
15 80 12 100
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255)
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9 cm
V 10 4 6 5 10 2
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X
240 100 340 cm3
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257) X 116 90 72 360 o
o
o
o
X 278 360 o
X 82
AD
BC
200
50
o
140 g
200 140 g 200 560 g 50
o
E
A
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258)
x
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74 35 Y 43 o
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344
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261)
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2 x 16 x8
x ។
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m
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259)
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m
a
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b
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12
c
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180o 101o 79o
e
f
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g
h
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262) 27 L 3 ឬ 36 L 4
3: 4 ។
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15 6 10 3 34
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a b 35 1 a 3 2 b 4 1 : b 35 a
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C
3
2
a 3 35 a 4 4a 105 3a
7a 105
12 cm
A
B
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12 cm
b 35 15 20
1 122 1 2 36 18 cm ។ 2 4 2
A
15
B
20
345
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263)
m
cm2
266)
d t v 80 m / mn v
d vt t 8 mn
d 80 8 640 m
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។
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267)
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265)
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269)
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347
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273)
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65
A
30
AB, AC, AD A, B A C B
C
A
5 6 11 1 A 5 , B 1 , C 6 ។ 274)
1 2
A
1 2
A
x
1 2 y4 x 2 1 4 x 2 ឬ x 2 8 ឬ x 2 2 ។ 2 y
A
y
“2 y 8”
1 y x2 2 x0 1 y 02 0 ។ 2 “2 y 8” 275)
3 1 6 2
A
1 1 1 25% ។ 2 2 4 277)
។
១
2 x 4
២
៣
៤ ...
x0
។
cm3
1
1 02 12
2
5 12 22
3
13 22 32
4
25 32 42
......................................
7 2 82 49 64 113
8 2r cm
r cm
។
278)
6.4 1.5 4.3
V S B h r 2 2r 2 r 3 cm3
។
1000 x
2r cm
y
x r cm
y
1000
1 1 2 V S B h r 2 2r r 3 cm3 3 3 3 348
x 1000 ។ y
៩
280) ១) តរើគូររីតោណណា ដែលគួបង្ហាញថាវាប៉ន ុ គ្នា?
A y 130 000 000 x 1000 2.1 y x 1000 2.1 130 000 000
D E
។
x 273000
279)
cm
គូររតី ោណដែលគួបង្ហាញថាវាប៉ន ុ គ្នាគៈឺ
ACD នង ិ CBE
AC
D
5 cm
A
អក្សរដែលររូវនឹងលក្ខខណឌទាំងបីររឹមររូវគឺៈ ក្, ខ និង ច
B
។
២) តរជើសតរសលក្ខ ខណឌទាំងបដី ែលររម ើ ឹ ររូវៈ
E
2 cm
C
B
៣) ពនយល់ជាពាក្យ នូវលក្ខខណឌប៉ន ុ គ្នាននររីតោណ
C
4 cm
។
ររីតោណទាំងពីរប៉ន ុ គ្នាតាមលក្ខខណឌទី២ ជ.ម.ជ គឺៈ មុម ំ យ ួ ប៉ន ុ គ្នា អមតោយរជុងពរី ប៉ន ុ តរៀងគ្នា។ (តរពាះ
ABC
ររីតោណ
AC 2 AB 2 BC 2
-រជុង AD = CE បរមាប់របធាន
AC 2 22 42 20 AC 2 5 cm cm
ACD និង CBE មាន៖
-មុំ
, AC 0
CAD BCE មុនំ នររតី ោណសម័ងស
-រជុង AC = CB រជុងននររីតោណសម័ងស)
281) គូសអងករដ់ ែលមានរបដវង 5 cm ក្ាុងរូប៖
CD CDE
CD ED CE 2
CE
2
2
AC 2 5 5 cm 2 2
តរពាះតាមរទស ត ទពតា ឹ ីប ី គ័រ 3 4 5 2
2
CD 5 5 20 2
2
2
282) ក្. តពលអំពូលទាំងអស់តឆះភលឺ វាជំនស ួ ឱ្យតលខអវី?
CD 20 , CD 0
តពលអំពូលទាំងអស់តឆះភលឺ វាជាបនសំននអំពូលចាប់ពី
CD 2 5 cm
ABC
រូបទ២ ួ គឺ៖ ី ែលទី៦ តនះតលខដែលវាររូវជំនស
CDE
1 2 4 8 16 31
S SABC SCDE
ំ ស ខ. គូសរូបននអំពូលទាំងតនះជន ួ ឱ្យតលខ 10
1 1 AB BC CE CD 2 2 1 1 2 4 52 5 2 2
2
ក្ាុងចំតណាមចំនន ួ ពី 1, 2, 4, 8, 16 មានជតរមស ើ ដរមួយ
គរ់ ដែលបូក្តសមើ 10 គឺ
2 8 10 ររូវជាបនសំននអំពូល
ក្ាុងរូបទី 3 និងរូបទី 5 ែូចតនះតយើងគូសរូបបាន៖
4 5 9 cm 2
349
៩
283) ១) តរើគូររីតោណណា ដែលគួបង្ហាញថាវាប៉ន ុ គ្នា? A M D
B
N
285) គូសបន្ទារ់ដែលមានរបដវង 10 cm ក្ាុងរូប
C
គូររតី ោណដែលគួបង្ហាញថាវាប៉ន ុ គ្នាគៈឺ
MBD នង ិ NDB
។
២) តរជស ខណឌទាំងបដី ែលររម ើ តរសលក្ខ ើ ឹ ររូវៈ
អក្សរដែលររូវនឹងលក្ខខណឌទាំងបរី រម ឹ ររូវគៈឺ ខ, គ និង ច
តរពាះតាមរទឹសី តបទពីតាគ័រ 6 8 10 2
។
A តរចន ើ ជាងរបូបាបននរពឹរតិោរណ៍ B -របូបាបននរពឹរតិោរណ៍ A គឺ
C 3 , 3
ររីតោណទាំងពីរប៉ន ុ គ្នាតាមលក្ខខណឌទី២ ជ.ម.ជ គឺៈ
P A
ររតី ោណ
-របូបាបននរពរ តិ រណ៍ B គឺ ឹ ោ
មុម ំ យ ួ ប៉ន ុ គ្នា អមតោយរជុងពីរប៉ន ុ តរៀងគ្នា។ (តរពាះ
-មុំ
MBD នង ិ NDB មាន៖
ត លរជុងរសបគ្នា DM BN ជាពាក្់ក្ណា
P B
ំ ល ស់ក្ុ ង ា MDB NBD មុឆ្ល
-រជុង
A
D
M
1 120
តោយមាឌននគូប B មានរបដវងរទនុងគឺ
C
គូររតី ោណដែលគួបង្ហាញថាវាប៉ន ុ គ្នាគៈឺ
AMD នង ិ BMC
ន្ទំឱ្យ មាឌ ែូចតនះ
។
VB a b
VB a3 3a 2b 3ab 2 b3 ។
តោយគូប A មានមាឌគឺ VA a
អក្សរដែលររូវនឹងលក្ខខណឌទាំងបីររឹមររូវគឺៈ ក្, គ និង ង
a b
3
ខ. រក្ចំនន ួ ែងននមាឌគូប B តរចន ើ ជាងគូប A
២) តរជើសតរសលក្ខ ខណឌទាំងបដី ែលររម ើ ឹ ររូវៈ
3
តយើងបានផលត ៀបរវាងមាឌគូប B និងគូប A គឺៈ
។
VB a 3 3a 2b 3ab 2 b3 VA a3
៣) ពនយល់ជាពាក្យ នូវលក្ខខណឌប៉ន ុ គ្នាននររីតោណ
2
b b b 1 3 3 a a a b តយង ួ t ន្ទំឱ្យតយង ើ ជំនស ើ បានៈ a VB 1 3t 3t 2 t 3 VA
ររីតោណទាំងពីរប៉ន ុ គ្នាតាមលក្ខខណឌទី២ ជ.ម.ជ គឺៈ មុម ំ យ ួ ប៉ន ុ គ្នា អមតោយរជុងពរី ប៉ន ុ តរៀងគ្នា។ (តរពាះ
MBD និង NDB មាន៖
AD BC រជុងរសបគ្នាតសមើគ្នា
MAD MBC 90o មុដំ ក្ងែូចគ្នា
-រជុង
C 10 , 3
P B ដែលៈ
287) ក្. សរតសរមាឌននគូប B ជាអនុគមន៍នន a នង ិ b
B
-មុំ
C 3 , 3
1 84
P A
284) ១) តរើគូររីតោណណា ដែលគួបង្ហាញថាវាប៉ន ុ គ្នា?
-រជុង
C 9 , 3
P A ដែលៈ
1 120 10 -ន្ទំឱ្យចំនន ួ ែង ែង 84 1 P B 84 7 120
BD DB ជារជុងរួម)
ររីតោណ
2
286) រក្ចំនន ួ ែង ដែលរបូបាបតក្រ តិ រណ៍ ើ មានត ង ើ ននរពរ ឹ ោ
៣) ពនយល់ជាពាក្យ នូវលក្ខខណឌប៉ន ុ គ្នាននររតី ោណ
-រជុង
2
ត លរជុង AB AM BM , M ក្ណា
350
3
៩
288) បង្ហាញថា x y z a ពិនិរយ
x a y a z a 2
2
តរពាះ x y z 3a និង x y z 3a 2
2
2
2
ចំតពាះ
2
x 2 y 2 z 2 2a x y z 3a 2
x a 0 តយើងបាន y a 0 z a 0
3a 2 2a 3a 3a 2
ែូចតនះ តយង ើ បញ្ជាក្់បានថា
x 2 2ax a 2 y 2 2ay a 2 z 2 2az a 2 x 2 y 2 z 2 2ax 2ay 2az 3a 2
0
351
x a ន្ទំឱ្យ y a z a
x yza ។
sYsþI¡ elakGñkmitþGñkGanCaTIRsLaj;rab;Gan enAkñúgEpñkenHelakGñknwg)aneXIj BIvBi aØasaFøab; ecjRbLgDIsbøÚm P¢ab;CamYycemøIyRKb;qñaM KWcab;BIqñaM 1981 rhUtdl;qñaM 2012 . CaFmµtaviBaØasa nig cemøIyrbs;va énqñaMnImYy² RtUv)anerobcMCa 4 TMBr½ dUcenHebIviBaØasaxøHxøI eFVI[sl;TMB½rTMenr. elakGñkGaceFVIkarkMNt;TuknUvlkçN³énviBaØasaEdl)anecjRbLgrYcehIyenH edIm,IeFVIkar):an;sµan nUvviBaØasaEdlnwgecjCabnþbnÞab;enAqñaMxagmux. RbsinebIelakGñkmanbBaða b¤cm¶l;Rtg;cMNucNa EdlmanenAkñúgEpñkenH elakGñkGacTak;Tg;eTAkan; RKU b¤mitþPkþirbs;GñkEdlmansmtßPaB b¤GñkeroberogesovePAenHkñúgeBlevlasmKYr . …
v
9
sm½yRbLg ³ 06 kkáda 1981 viBaØasa ³ KNitviTüa ry³eBl ³ 60 naTI BinÞú ³ 10 I. BICKNit
1> rktémø a edIm,I[RbPaK F 3a3a 0.213a2a 26 ³ k> esµInwgsUnü . x> Kµann½y . 2> edaHRsayRbB½n§smIkartamRkaPic yy 2xx4500 12 . 3> mnusSmñak;Gayu 51 qñaM ehIykUnrbs;Kat;manGayu 21 qñaM . etIb:unµanqñaMeTAmuxeTot eTIbGayu«Buk esµInwg 2 dgGayukUn .
II. FrNImaRt
1> rgVg;BIrmanp©ti O nig O kat;KñaRtg; A nig B . eKKUsGgát;p©it AC kat;tam O nigGgát;p©it AD kat;tam O . RsaybMPøW[eXIjfa cMNucTaMgbI C , B nig D sßitenAelIbnÞat;EtmYy ehIybnÞat; AB CD . N 2> MN CargVas;BI)atdIdl;páaQUk nigEpñkput A B BITwk AN 10 cm . eRkamGMNacxül;bk; mYy enaHedImQUkeRTtRbkan;yksßanPaB MB ¬emIlrUb¦ eday AB 30 cm M rkCeRmATwk MA . ENnaM ³ kñúgRtIekaNsm)at MNB KUskm
8
352
9
cemøIy I. BICKNit
y 2x 5 0
1> rktémø a edIm,I[RbPaK F ³ k> esµInwgsUnü eyIgman F 3a3a 0.213a2a 26 ³ 3a 0.07 2a 3 3a 3a 2 2a 0.07a 3 a 3a 2
y x40
tamRkaPic ³ cMNucRbsBVénbnÞat;TaMBIrKW 3,1 dUcenH RbB½n§smIkarmanKUcemøIy x 3, y 1 . 3> rkcMnYnqñaMeTAmuxeTot tag x CacMnYnqñaMeTAmuxeTot EdlRtUvrk tambRmab;RbFan eyIgsresr)ansmIkar ³
eyIgGacKNna F )ankalNavamann½y ehIyvamann½yluHRtaEtPaKEbgxusBIsUnü ³ naM[ aa 32 00 b¤ aa 23 -edIm,I[ F 0 luHRtaEt PaKykrbs;vaesµIsnU ü eyIg)an 2a 0.07 a 3 0 naM[ aa 30.070 0 b¤ aa 30.07
51 x 221 x 51 x 42 2 x 51 42 2 x x x9
dUcenH ebI F 0 enaHtémø a 0.07 , a 3 . x> Kµann½y RbPaK F Kµann½ykalNa PaKEbgrbs;vaesµI 0 naM[ aa 32 00 b¤ aa 23
epÞógpÞat; ³ 51 9 221 9 60 60 Bit dUcenH cMnYnqñaMteTAmuxKW ³ 9 qñaMeTot . II. FrNImaRt
1> RsayfacMNuc C , B nig D enAelIbnÞat;EtmYy
A
dUcenH ebI F Kµann½y enaH a 3 , a 2 . 2> edaHRsayRbB½n§smIkartamRkaPic ³ eyIgman yy 2xx4500 12 eyIgeRbItaragtémøelxedIm,Isg;bnÞat;TaMgenH ³ y 2x 5 0 y x40
O
O C
B
D
eday ABˆC 90 ¬mMucarwkknøHrgVg;Ggát;p©it AC ¦ ABˆ D 90 ¬mMucarwkknøHrgVg;Ggát;p©it AD ¦ naM[ ABˆC ABˆD CBˆD 90 90 180 eXIjfa CBˆ D 180 CamMurab dUcenH cMNcu C , B nig D enAelIbnÞat; EtmYy ehIy AB CD . o
o
o
x 1 2 y 3 1 x 2 3 y 2 1
o
eyIgsg;RkaPic)an ³ 353
o
o
9
2> rkCeRmATwk MA tamENnaMeyIgKUsrUbbEnßm)an ³ N A
smÁal; ebIGñkmineFVItamkarENnaMRsYlCagKW ³ edayépÞTwkEkgnwgpáaQUkQr naM[ ABM EkgRtg; A ehIytamBItaK½r ³
H B
>
MB2 AB2 AM 2
Et MB MN AM AN ¬edImQUkEtmYy¦ eyIg)an AM AN AB AM eday AN 10cm , AB 30cm AM 10 30 AM enaH 2
M
eyIgeRbóbeFob ABN nig HMN eday ABN nig HMN man ³ -mMu BAˆ N MHˆ N 90 eRBaH épÞTwkEkgnwg páaQUkQr nig MH Cakmm AN BN vi)ak ABN HMN HN MN
2
dUcenH CeRmATwkKW
BN 2 AN 2 AB2 , AB 30 cm 102 302 100 900 1000 BN 1000 10 10 cm
-km
50cm 10cm 40cm
2
2
20 AM 800 800 AM 40 cm 20
Taj)an MN BNAN HN 1 eday AN 10 cm -tamRTwsþIbTBItaK½rcMeBaH ABN EkgRtg; A
10
2
AM 2 20 AM 100 900 AM 2
o
tam 1 eyIg)an MN 10 1010 5 naM[ CeRmATwk AM MN AN
2
>
dUcenH TwkmanCeRmA AM 40cm . 354
AM 40cm
.
9
355
9
sm½yRbLg ³ 29 mifuna 1982 viBaØasa ³ KNitviTüa ry³eBl ³ 60 naTI BinÞú ³ 10 I. BICKNit
1> dak;CaplKuNktþa abx y xya b . 2> RkumsamKÁIbgábegáInplmYy lk;RsUvCUnrdæTaMgGs;cMnYn 0.825 t . RsUvenaHmanbIRbePTKW páaxJI nagkuk nigRsUvBeRgaH. témø 1 kg RsUvnagkukesµIBak;kNþaltémøsrubén 1 kg RsUvpáaxJI nig 1 kg RsUvBeRgaH ehIyéføelIstémø 1 kg énRsUvBeRgaHcMnYn 0.50 erol . k> témø 1 kg RsUvpáaxJI 1 kg RsUvnagkuk nig 1 kg RsUvBeRgaHrYmKñaéfø 3.00 erol . KNnatémø 1 kg énRbePTRsUvnImYy² . x> eKdwgfaRkumsamKÁIlk;RsUvnagkuk 250 kg nig RsUvpáaxJI RBmTaMgRsUvBeRgaHTaMgGs;)anR)ak; 837.50 erol. KNnaTm¶n;RsUvpáaxJI nigRsUvBeRgaH . II. FrNImaRt eK[RtIekaN ABC carwkkñúgrgVg;. knøHbnÞat;BuH BAC kat;RCug BC Rtg; I nigFñÚ BC Rtg; J . 1> RsaybBa¢ak;fa AIB nig AJC dUcKña . 2> KNnaplKuN AI AJ . 3> bgðajfa BAJ nig CBJ b:unKña . 4> RsaybBa¢ak;fa JB JA IJ . 2
2
2
2
2
8
356
9
cemøIy I. BICKNit
1> dak;CaplKuNktþa ³
ab x 2 y 2 xy a 2 b 2
abx aby a xy b xy 2
2
2
2
abx 2 a 2 xy b 2 xy aby 2 axbx ay bybx ay bx ay ax by
dUcenH dak;CaplKuNktþa)an
dUcenH témøRsUvkñúg 1kg Edlrk)anKW ³ RtUvpáaxJI mantémø 1.50 erol RtUvnagkuk mantémø 1.00 erol RtUvBeRgaH mantémø 0.50 erol x> KNnaTm¶n;RsUvpáaxJI nigRsUvBeRgaH tag x CaTm¶n;RsUvpáaxJI ¬KitCa kg ¦ y CaTm¶n;RsUvBeRgaH ¬KitCa kg ¦ tambRmab;RbFaneyIg)an ³
ab x 2 y 2 xy a 2 b 2
bx ay ax by
x 250 z 825 1.5x 1 250 0.5z 837.5
2> k> KNnatémø 1 kg énRbePTRsUvnImYy² ³ tag a CatémøRsUvpáaxJI 1kg ¬KitCaerol¦ b CatémøRsUvnagkuk 1kg ¬KitCaerol¦ c CatémøRsUvBeRgaH 1kg ¬KitCaerol¦ - témø 1 kg RsUvnagkukesµIBak;kNþaltémøsrub én 1 kg RsUvpáaxJI nig 1 kg RsUvBeRgaH naM[ b 12 a c b¤ a c 2b 1 - ehIytémø 1 kg RsUvnagkuk éføelIstémø 1 kg énRsUvBeRgaHcMnYn 0.50 erol naM[ b c 0.5 2 - témø 1 kg RsUvpáaxJI 1 kg RsUvnagkuk nig 1 kg RsUvBeRgaHrYmKñaéfø 3.00 erol naM[ a b c 3 3 tam 1 , 2 nig 3 eK)anRbB½n§smIkar ³ a c 2b b c 0.5 a b c 3 a c 2b b c 0.5 3b 3
a c 2b b c 0.5 a c b 3
a c 2 1 1 c 0.5 b 1
¬eRBaHTm¶n;RsUv 0.825 t 825kg ¦
x z 825 250 1.5 x 0.5 z 837.5 250 x z 575 1.5 x 0.5 z 587.5
1 x z 575 3 x z 1175 2
eyIgdkGgÁngi GgÁénsmIkar 2 1 ³ 3x z 1175 x z 575 2 x 600
naM[ x 300 cMeBaH x 300 CMnYskñúg 1 ³ 1 : x z 575
z 575 x 575 300 275
epÞógpÞat;
a c 2b b c 0.5 2b b 3
300 250 275 825 825 825
Bit
dUcenH RsUvpáaxJI manTm¶n; 300 kg RsUvBeRgaH manTm¶n; 275 kg .
a 0.5 2 c 0.5 b 1
357
9
II. FrNImaRt B
JB JA vi)ak ABJ BIJ JI JB Taj)anBIpleFob JB JB JA JI dUcenH RsaybBa¢ak;)anfa JB JA IJ .
J
I
2
C
A
1> RsaybBa¢ak;fa AIB nig AJC dUcKña eday AIB nig AJC man ³ -mMu ABˆ I AJˆC ¬mMucarwkmanFñÚsáat;rYm AC ¦ -mMu BAI JAC ¬ AJ knøHbnÞat;BuH BAC ¦ dUcenH AIB AJC tamlkçxNÐ m>m . 2> KNnaplKuN AI AJ eday AIB AJC ¬sRmayxagelI¦ AI AB eK)an AIB ACJ AC AJ Taj)an AI AJ AB AC dUcenH KNna)anplKuN AI AJ AB AC . 3> bgðajfa BAJ nig CBJ b:unKña eday BAˆ J CAˆ J ¬ AJ knøHbnÞat;BuH BAC ¦ Et CAˆ J CBˆ J ¬mMucarwkmanFñÚsáat;rYm CJ ¦ naM[ BAˆ J CAˆ J CBˆ J 1 dUcenH bgðaj)anfa BAJ CBJ . 4> RsaybBa¢ak;fa JB JA IJ eyIgeRbóbeFob ABJ nig BIJ eday ABJ nig BIJ man ³ -mMu BAˆ J IBˆ J eRBaH tam 1 : BAˆ J CAˆ J CBˆ J IBˆ J -mMu AJˆB BJˆI ¬mMurYm¦ dUcenH ABJ BIJ tamlkçxNÐ m>m 2
358
9
359
9
RksYgGb;rM yuvCn nigkILa RbLgsBaØabRtmFümsikSabzmPUmi cMeNHTUeTA nigbMeBjviC¢a eQµaH nightßelxaGnurkS sm½yRbLg ³ >>>>>>>> >>>>>>>>> 1993
elxbnÞb; ³ >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> elxtu ³ >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> mNÐlRbLg ³ >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
1> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> namRtkUl nignamxøÜn ³ >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 2> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> éf¶ExqñaMkMeNIt ³ >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> GkSrsm¶at; htßelxa ³ >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> ebkçCnminRtUveFVIsBaØasmÁal;Gm VI YyelIsnøwkRbLgeLIy. snøwkRbLgNaEdlmansBaØasmÁal;RtUv)anBinÞúsUnü . --------------------------------------------------------------------------------------------------------------------------
viBaØasa ³ KNitviTüa ry³eBl ³ 120 naTI BinÞú ³ 100 esckþIENnaM ³ GkSrsm¶at; 1> ebkçCnRtUvbt;RkdasenHCaBIr rYcKUsExVgEpñkxagelIénTMB½rTI2 [b:unRbGb;EpñkxagelI énTMB½rTI1 EdlRtUvkat;ecal. hamsresrcemøIyelIkEnøgKUsExVgenaH . 2> ebkçCnRtUvKUsbnÞat;bBaÄr[cMBak;kNþalTMB½rTI2 nigTMB½rTI3 sRmab;sresrcemøIybnþ. I. ¬10 BinÞú¦
360
9
cemøIy
361
9
sm½yRbLg ³ 03 kkáda 1984 viBaØasa ³ KNitviTüa ry³eBl ³ 60 naTI BinÞú ³ 10 I. BICKNit
1> KNna a b ca b c ab ac bc . 2> edaHRsaysmIkar 2x 3x 3x 2 4xx 1 x 11 x . 3> sRmÜlkenSam aabb cc . rYcKNnatémøelxénkenSamenH kñúgkrNIEdl ³ a 35.4 , b 48.6 nig c 29.6 . II. FrNImaRt 1> eK[BIcMNuc O nig O . eKKUsrgVg;p©it O nig O EdlmankaM OO ehIykat;KñaRtg; A nig B . bnÞat;mYykat;tam A kat;rgVg; O Rtg; I nigkat;rgVg; O Rtg; J . k> bgðajfaRtIekaN AOO nig BOO CaRtIekaNsm½gS . x> R)ab;RbePTRtIekaN BIJ . 2> eK[RbelLÚRkam ABCD BC //AD . eKKUsrgVg;mYykat;tam A nig B rgVg;enHkat; RCug AD Rtg; P nig BC Rtg; Q . RsaybMPøWfa ctuekaN CDPQ carwkkñúgrgVg;EtmYy . 3> eK[rgVg;p©it O nigGgát;FñÚ AB. C CacMNucmYyelIFñÚtUc AB . eKbnøay AC [)anGgát; CM Edl CM CB . rkcMNuc M kalNa C rt;elIFñÚtUc AB . smÁal; ³ sMNYrenH minTak;TgKñaeT . 2
2
2
2
2
8
362
9
cemøIy I.
BICKNit 1> KNna
II.
I
a b c a
2
b c ab ac bc 2
2
2
2
2
A J
O
a ab ac a b a c abc 3
FrNImaRt
2
O
a b b bc ab abc b c 2
3
2
2
2
a 2 c b 2 c c 3 abc ac 2 bc 2
B
a 3 b 3 c 3 3abc
2> edaHRsaysmIkar ³
2 x 3 x 3x 2 4 xx 1 x 11 x
2 x 3x 2 6 x 3x 6 4 x 2 4 x 1 x 2 x2 7x 6 1 x2 x2 x2 7x 1 6 7 x 5 x
5 7
dUcenH smIkarman x 75 Cab£s . a b 2 c 2
3> sRmÜlkenSam
abc a b c a b c a b c abc abc abc 2
2
Edl kenSam
1>k> bgðajfa AOO nig BOO Ca sm½gS rgVg;TaMgBIrmankaM OO EtmYy enaHrgVg;TaMgBIr CargVg;b:unKña eday AOO man OA OO OA ¬CakaMrgVg;BIrb:unKña¦ dUcenH AOO CaRtIekaNsm½gS . ehIy BOO man OB OO OB ¬CakaMrgVg;BIrb:unKña¦ dUcenH BOO CaRtIekaNsm½gS . x> R)ab;RbePTRtIekaN BIJ eday AOO nig BOO CaRtIekaNsm½gS naM[ AOˆ B AOˆ O BOˆ O 60 60 120 o
a bc 0
dUcenH sRmYl)an
a b 2 c 2 abc
abc
RtIekaN BIJ man ³ ˆ -mMu IJB AJB AO2B 1202
.
o
o
42.8 2
42 .8
60 o
o
35.4 48.6 29.6
dUcenH témøKNna)an aabb cc
o
- mMu JIB AIB AO2ˆ B 1202 60 ¬mMucarwkelIrgVg;esµkI nøHmMup©ti EdlmanFñÚsáat;rYm¦ eXIjfa BIJ manmMuBIr IJB JIB 60 naM[ BIJ CaRtIekaNsm)at EdlmMu)atesµI 60
2
2
o
AOˆ B AOˆ O BOˆ O 60 o 60 o 120 o
-KNnatémøelxénkenSamenH cMeBaH a 35.4 , b 48.6 nig c 29.6 eday aabb cc a b c 2
o
.
dUcenH RtIekaN BIJ CaRtIekaNsm½gS . 363
o
9
2> RsaybMPøWfa ctuekaN CDPQ carwkkñúg rgVg;EtmYy P
A
naM[ CMB CaRtIekaNsm)at vi)ak CBˆ M CMˆ B ¬mMu)atRtIekaNsm)at¦ ehIy CBˆ M CMˆ B ACˆB ¬mMueRkA CMB ¦ b¤ 2CMˆ B ACˆB naM[ CMˆ B AC2ˆB
D
B
C
Q
Et
-bRmab; ABCD CaRbelLÚRkam BC //AD naM[ ABˆ Q PDˆ C 1 ¬mMuQménRbelLÚRkam¦ ehIy BC //AD man PQ Caxñat; enaH APˆ Q PQˆ C 2 ¬mMukñúgrYmxag¦ -rgVg;mYykat; A nig B ehIyrgVg;enHkat;RCug AD Rtg; P nig BC Rtg; Q mann½yfa A , B , Q, P enAelIrgVg;EtmYy naM[ ctuekaN ABQP carwkkñgú rgVg;mYy vi)ak plbUkmMuQm ABˆ Q APˆ Q 180 3 tam 1, 2 , 3 eyIg)an ³
¬mMucarwkFñÚsáat;FM AB ¦
eyIg)an
AB ˆB A C AB 2 ˆ CMB 2 2 4
b¤Gacsresr eday AB efr enaH FñÚFM AB k¾efrEdr ehIy C rt;elIFñÚtUc AB enaH M k¾rt;elIFñÚén AB CMˆ B AMˆ B 4
rgVg;mYyRbkbedaymMuefrKW -ebI C RtYtelI A enaH M RtYtelI M Edl AB AM -ebI C RtYtelI B enaH M RtYtelI B Edr mann½yfa M rt;elIFñÚ BM énrgVg;mYy dUcenH kalNa C rt;elIFñÚtUc AB enaH M CasMNMucMNucrt;elIFñÚ BM énrgVg;mYy AB AMˆ B 4
o
ABˆ Q PDˆ C APˆ Q PQˆ C
AB ACˆ B 2
PDˆ C PQˆ C 180 o
ABˆ Q APˆ Q 180 o
eday ctuekaN CDPQ manplbUkmMuQm PDˆ C PQˆ C 180 enaHvacarwkkñúgrgVg; dUcenH ctuekaN CDPQ carwkkñgú rgVg;EtmYy . 3> rkcMNuc M kalNa C rt;elIFñÚtUc AB o
RbkbedaymMuefr
M
C
A
//
M B
O
tambRmab;RbFan
CM CB 364
AB AMˆ B 4
.
9
365
9
sm½yRbLg ³ 25 mifuna 1985 viBaØasa ³ KNitviTüa ry³eBl ³ 60 naTI BinÞú ³ 10 I. BICKNit
1> k> dak;kenSamxageRkamCaplKuNktþadWeRkTI1 én x ³ A x 3x 2 9x 3 2
B 2 x 4 x 1 2
2
x> KNnatémøelxén C BA cMeBaHtémø x 3 , x 1 nig x 53 . K> etI D x 31xx5 5 nig C smmUlKña b¤eT ? 2> k> tagRkaPicénTMnak;TMngxageRkam kñúgtRmúyGrtUNremedayyk 1 cm CaÉkta ³ y 2 x 4 nig y x 1 . x> edaHRsayRbB½n§smIkar yy 2x x1 4 . yklT§plxagelIedIm,IedaHRsayRbB½n§smIkar
2 1 y 1 2 x 5 4 1 1 1 y 1 2 x 5
3> k> KNna
.
A 10 48 6 108 4 12
nig
B
1 1 11 3 11 11 3 11
.
x> edaHRsaysmIkar 2 x3 7 x 1 . II. FrNImaRt manrgVg; O p©it O Ggát;p©it AB. O CacMNuckNþalén AB nigCap©itrgVg; O Edl kat;tam A nig B . M nig N CacMNucénFñÚFM AB rbs;rgVg; O . bnÞat; MA , MB CYbrgVg; O Rtg; C nig D ehIy NA nig NB CYbrgVg; O Rtg; E nig F . 1> RbdUcRtIekaN MAD nig NBE . 2> bgðaj[eXIjfa MAD nig NBE CaRtIekaNEkgsm)at . 3> cMNuc M cl½telIFñÚFM AB énrgVg; O bgðajfa CD manrgVas;efr ehIyEdlnwgRtUv KNnaCaGnuKmn_én R CargVas;kaMrgVg;Ggát;p©it AB. rksMNMucMNuckNþal I én CD .
8 366
9
cemøIy K> etI D x 31xx5 5 nig C smmUlKña b¤eT ? eyIgman C x x33x35xx5 1 eXIjfa ebIsRmYl C nwg x 3 enaH C dUc D dUcenH -ebI x 3 enaH C nig D smmUlKña -ebI x 3 enaH C nig D minsmmUlKña . 2> k> tagRkaPicénTMnak;TMngxageRkam ³ eyIgeRbItaragtémøelxedIm,Isg;bnÞat; y 2x 4 y x 1 nig
I. BICKNit
1> k> dak;kenSamCaplKuNktþadWeRkTI1 én x A x 3x 2 9x 3 2
x 3x 2 9 2
x 3x 2 3x 2 3 x 3x 5x 1
dUcenH
.
A x 3x 5x 1
B 2 x 4 x 1 2 x 4 x 12 x 4 x 1 x 33x 5 2
dUcenH
2
B x 33x 5
.
x
0 1
x
y 4 2
0 1
y 1 2
x> KNnatémøelxén C BA eK)an C BA x x33x35xx5 1 -cMeBaH x 3 C
y x 1
3 33 53 1 0 2 4 0 3 33 3 5 04 0
dUcenH C mancemøIyeRcInrab;minGs; . -cMeBaH x 1 C
y 2x 4
x> edaHRsayRbB½n§smIkar yy 2x x1 4 eyIgedaHRsayeday dkGgÁnigGgÁ
1 3 1 5 1 1 4 60 0 1 3 1 3 5 4 8
dUcenH
C
mancemøIyEtmYyKt;KW 0
y 2x 4 y x 1 3x 3
.
eyIg)an cMeBaH x 1 naM[ y x 1
-cMeBaH
5 x 3 5 5 5 4 10 8 3 5 1 3 3 3 3 3 3 C 5 5 4 3 3 5 0 3 3 3
C
x 1
y 1 x 11 2
dUcenH RbB½n§smIkarmanKUcemøIy
manPaKyk 0 ÉPaKEbg 0 . dUcenH
naM[
x 1 , y 2
Kµann½y kñúgkarKNnatémøelx .
¬ebIepÞógpÞat;tamRkaPicKWRtUvKña ¦ 367
.
9
K> yklT§plxagelIedIm,IedaHRsayRbB½n§ 2 1 y 1 2 x 5 4 1 1 1 y 1 2 x 5
smIkar ³
2 x 7 x 1 3 2 x 7 3 x 3
4 x
1 1 y 1 2 2 x 5 4 1 1 1 y 1 2 x 5
Gacsresr
tag Y y 1 1 nig X 2x1 5 Edl y 1 , x 52 eyIg)anRbB½n§fµI YY2XX 14 tamlT§plxagelIKW X 1 , Y 2 naM[ Y y 1 1 2 y 1 1 y 12 X
x> edaHRsaysmIkar ³
edayRKb;cMnYnBit x enaH x 0 Et 4 0 dUcenH smIkarKµanb£sCacMnYnBiteT . II. FrNImaRt
E C
.
40 3 36 3 8 3
A 12 3
.
1 1 11 3 11 11 3 11
11 3 11 11 3 11 11 3 1111 3 11
22 11 9 11 22 1 22
o
2
dUcenH KNna)an
I
o
12 3
B
O
1> RbdUcRtIekaN MAD nigRtIekaN NBE edayRtIekaN MAD nigRtIekaN NBE man ³ -mMu AMˆ B ANˆB ¬mMucarwkmanFñÚsáat;rYm AB énrgVg;p©it O ¦ -mMu ADˆ M BEˆN 90 eRBaH CamMuCab;bEnßm énmMu ADˆ B , AEˆB EdlCamMucarwkknøHrgVg;p©it O dUcenH MAD NBE tamlkçxNÐ m>m . 2> bgðaj[eXIjfa MAD nig NBE Ca RtIekaNEkgsm)at kñúgRtIekaN MAD nig NBE man ³ -mMu ADˆ M BEˆN 90 ¬manbBa¢ak;xagelI¦ ˆ -mMu AMˆ B ANˆ B AO2B 902 45 eRBaH kñúgrgVg;p©it O man AMˆ B , ANˆ B CamMucarwk ehIy
A 10 48 6 108 4 12
B
F
O
3> k> KNna
dUcenH KNna)an
I
I
RbB½n§smIkarmanKUcemøIy 1 x3 , y 2
O D
A
1 1 1 x3 2x 5 2x 5
dUcenH
N
M
o
o
B 1
. 368
9
AOˆ B AMˆ B ANˆ B 2
2
R 2 R2 R2 2 OI R R 2 2 2
mMu CamMup©it enaH mü:ageTot AOˆ B 90 edaysarvaCamMucarwk knøHrgVg;EdlmanGgát;p©it AB . dUcenH MAD nig NBE CaRtIekaNEkgsm)at 3> bgðajfa CD manrgVas;efr edaymMu AMˆ B 45 ehIyCamMueRkArgVg;p©it O AOˆ B
2
o
OI
naM[
AB CD AMˆ B 2
o
2
2
2
o
2
rUbsRmÜlkaremIl N
M
2
CD 2 R 2 R 2
/
dUcenH KNna)an CD R 2 . - rksMNMucMNuckNþal I én CD eday I CacMNuckNþal CD naM[ OI CD ¬kaMrgVg;EkgnwgGgát;FñÚRtg; cMNuckNþal ¦ ehIy CD R 2 efr enaH OI k¾efrEdr Edl OI OC CD 2
E C
O D
I
F
A
B
O I
I
O
2
2
o
o
o
o
2
o
o
o
eday CD 90 efr enaHnaM[ CD k¾efrEdr dUcenH bgðaj)anfa CD manrgVas;efr . -KNna CD CaGnuKmn_én R eday CD 90 enaH mMup©it COˆ D 90 Edr naM[ RtIekaN COD CaRtIekaNEkgRtg; O tamRTwsþIbTBItaK½r CD OC OD eday OC OD R CakaMrgVg;Ggát;p©it AB naM[ CD R R 2R 2
o
180o 2 45o 90o
o
CD AB 2 AMˆ B
R2 R R 2 2 2 2
eday O CacMNucnwg enaH sMNuMcMNuc I rt;enA elIrgVg;kaM OI Et M cl½tenAelIFñÚFM AB Edl FñÚFM AB 360 FñÚtUc AB ¬ FñÚtUc AB mMup©it AOˆ B 90 ¦ naM[ FñÚFM AB 360 90 270 eday M cl½tenAelIFñÚFM 270 enaH I k¾cl½t enAelIFñÚ I II 270 Edr -ebI M RtYtelI A enaH I RtYtelI I -ebI M RtYtelI B enaH I RtYtelI I dUcenH sMNcuM MNuc I rt;enAelIFñÚ I II 270 énrgVg; O mankaM OI R 2 2 .
o
2
2
369
9
sm½yRbLg ³ 07 kkáda 1986 viBaØasa ³ KNitviTüa ry³eBl ³ 60 naTI BinÞú ³ 10 I. BICKNit 1 ab
a b
1> sRmÜlr:aDIkal;
1
1
Edl
0ba
.
a2 b2
2> eK[kenSam A 5x 4 x 4x 4 6 3xx 3 . k> sresr A CaBhuFabRgÜm nigerobtamsV½yKuNcuHén x . x> sresr A CaplKuNktþadWeRkTI 1 . K> edaHRsaysmIkar A 0 nig A 2 . X> eK[RbPaK F 3x 1A x 4 . rktémø x EdlnaM[RbPaKmann½y rYcsRmÜlRbPaK. 3> kñúgtRmúyGrtUNrem ¬Éktþa sg;TIEm:t¦ k> sg;bnÞat;rbs;GnuKmn_ y x 2 D nig y x 4 D . x> kMNt;kUGredaenéncMnucRbsBV D nig D tamkarKNna rYctamRkaPic . K> bgðajfa D nig D EkgKña . II. FrNImaRt eK[rgVg;p©it O Ggát;p©it AB. elIbnÞat;Edlb:HrgVg;Rtg; A eKedAcMNuc C mYy . bnÞat; CB kat;rgVg; O Rtg; D . 1> eRbóbeFobRtIekaN CAD nigRtIekaN CBA rYcbgðajfa CA CB CD . 2> H CaeCIgbnÞat;EkgKUsecjBI A eTAbnÞat; OC . eRbóbeFobRtIekaN CAO nigRtIekaN CHA rYcbgðajfa CH CO CB CD . 3> eRbóbeFobRtIekaN CDH nig COB rYcbBa¢ak;fa O , B , D , H sßitenAelIrgVg;EtmYy. 4> ]bmafa AB a , AC a2 KNna BC , CD , BD , AD CaGnuKmn_én a . 5> bnÞat;Edlb:HrgVg; O Rtg; D nig B . RbsBVKñaRtg; M . bgðajfa M sßitenAelIrgVg; C bBa¢ak;p©itrbs;va rYcKUsrgVg;enH . bgðajfacMNuc A , M , H rt;Rtg;Kña . 2
2
1
2
1
1
2
2
2
8 370
9
cemøIy I. BICKNit
2
1> sRmÜlr:aDIkal; ³
1
a b 1
1
a b
ab
a2 b2 ab
a b 2
ab
ab
1
1
F
a b
a b a b
.
x 23x 1 A 3x 1 x 4 3x 1 x 4
a2 b2
dUcenH F mann½ykalNa x 13 , x 4 .
2
sRmÜl F ³ F 3xx 123xx14 4x 2x Edl ktþasRmÜl 3x 1 0 b¤ x 13
A 5 x 2 4 x 2 4 x 4 6 3 x x 3 5 x 2 20 x 2 4 x 4 6 x 18 3 x 2 9 x 3x 2 7 x 2
dUcenH sRmÜl)an
dUcenH KNna)an A 3x 7 x 2 . x> sresr A CaplKuNktþadWeRkTI 1
F
2
.
edIm,I[ F luHRtaEtPaKEbgrbs;vaminEmnsUnü naM[ 3xx1400 3xx41 xx 14/ 3
2> eyIgman A 5x 4 x 4x 4 6 3xx 3 k> bRgÜmBhuFa A nigerobtam sV½yKuNcuHén x 2
1 3 7 x0, x 3 x2, x
K> rktémø x EdlnaM[RbPaKmann½y ³
1
a b
dUcenH sRmÜl)an
x3x 7 0
x 0 x 0 3x 7 x 7 / 3
2
a2 b2 1 a b
dUcenH cMeBaH A 0 enaH cMeBaH A 2 enaH
a2 b2
a b 1 ab
a b 1
ab a b2 1
a b2 2
2
2
2
b¤ 3x 7 x 0 naM[ 3xx07 0
x2 4 x
.
3> sg;bnÞat;rbs;GnuKmn_ ³ eyIgman y x 2 D nig y x 4 D eyIgeRbItaragtémøelxedIm,Isg;bnÞat;TaMgBIr
A 5 x 2 4 x 2 4 x 4 6 3x x 3
1
5x 2x 2 x 2 3x 2x 3 x 25x 2 x 2 3x 3 x 25 x 10 x 2 3x 9 x 23x 1 2
D1 : y x 2 D2 : y x 4
dUcenH dak;CaplKuNktþa A x 23x 1 K> > edaHRsaysmIkar ³ cMeBaH A 0 smmUl x 23x 1 0 naM[ 3xx2100 3xx21 xx 12/ 3 cMeBaH A 2 smmUl 3x 7 x 2 2
x y x y
2
0 2 0 4
2 0 4 0
y x 2 D1
3, 1 y x 4 D2
2
371
9
x> kMNt;kUGredaenéncMnucRbsBV D nig D eyIgman y x 2 D nig y x 4 D eyIgpÞwmsmIkarGab;sIusénbnÞat;TaMgBIr eyIg)an ³ x 2 x 4 2x 6 x 3 / cMeBaH x 3 : y x 2 3 2 1 dUcenH kUGredaenéncMNucRbsBVKW 3 , 1 . -tamRkaPiceXIjfa bnÞat;TaMgBIrRbsBVKñaRtg; cMNucmankUGredaen 3 , 1 edaykareFVIcMeNal EkgBIcMNucRbsBVmkelIG½kSTaMgBIr dUcenH kUGredaenéncMNucRbsBVKW 3 , 1 . K> bgðajfa D nig D EkgKña bnÞat;BIrEkgKñaluHRtaplKuNemKuNR)ab;TisKW ³ > a a 1 eday y x 2 D nig y x 4 D manplKuNemKuNR)ab;Tis a a 1 1 1 dUcenH bnÞat; D EkgKñanwgbnÞat; D . II. FrNImaRt tambRmab;RbFaneyIgsg;rbU )an ³ 1
1
1
-mMu ADˆ C BAˆ C 90 eRBaH bnÞat;b:H AC AB nig AD BC edaysar ADˆ B 90 CamMu carwkknøHrgVg;EdlmanGgát;p©it AB dUcenH CAD CBA tamlkçxNÐ m>m o
2
o
2
CA CD vi)ak CAD CBA CB CA Taj)an CA CB CD dUcenH bgðaj)anfa CA CB CD 1 . 2> eRbóbeFobRtIekaN CAO nigRtIekaN CHA eday RtIekaN CAO nigRtIekaN CHA man ³ -mMu OCˆA ACˆH ¬mMurYm¦ -mMu OAˆ C AHˆC 90 ¬eRBaH H CaeCIgkmm kñúg CAO EkgRtg; A nigman AH Cakm eRbóbeFobRtIekaN CDH nigRtIekaN COB eday RtIekaN CDH nigRtIekaN COB man ³ -mMu DCˆH OCˆB ¬mMurYm¦ -tam 3 : CH CO CB CD CD Taj)anpleFobRCug CH CB CO 2
2
o
2
1
2
1
2
2
M
C
D
C
H
A
B
O
dUcenH CDH COB tamlkçxNÐ C>m>C vi)ak DHˆC OBˆC mü:ageTot DHˆ C DHˆ O 180 ¬mMuCab;bEnßm¦ naM[ OBˆ C DHˆ O 180
O
1> eRbóbeFobRtIekaN CAD nigRtIekaN CBA eday RtIekaN CAD nigRtIekaN CBA man ³ -mMu ACˆD BCˆA ¬mMurYm¦
o
o
372
9
eXIjfakñúgctuekaN OBDH manplbUkmMuQm KW OBˆ C DHˆ O 180 enaHvacarwkkñúgrgVg;mYy dUcenH O , B , D , H sßitenAelIrgVg;EtmYy 4> KNna BC , CD , BD , AD CaGnuKmn_én a eyIgmankar]bma AB a , AC a2 -kñúg ABC EkgRtg; A ³ tamRTwsbþI TBItaK½r
A
2
CB CD
a 2 CA a2 2 2 CD CB 4 a 5 a 5 2
-eday
enaH ctuekaN MBOD carwkkñgú rgVg; C EdlmanGgátp; ©it OM dUcenH bgðaj)anfa M sßitenAelIrgVg; C ehIyp©itrgVg; C CacMNuckNþal OM . -bgðajfacMNuc A , M , H rt;Rtg;Kña tamlT§pl O , B , D , H sßitenAelIrgVg;EtmYy nig ctuekaN MBOD carwkkñúgrgVg; C naM[ BhuekaN MBOHD carwkkñgú rgVg; C EdlmanGgátp; ©it OM EtmYy enaH MHˆO 90 ¬mMucarwkknøHrgVg;Ggát;p©it OM ¦ ehIy AH CO naM[ OHˆ A 90 edayplbUkmMu MHˆO OHˆA 90 90 180 Et MHˆ O OHˆ A MHˆ A mann½yfa mMu MHˆ A 180 vaCamMurab dUcenH bgðaj)anfacMNuc A , M , H rt;Rtg;Kña .
a 5 10
2 5 a 5 a 5 BD BC CD 2 10 5a 5 a 5 2 a 5 10 5
-kñúg ACD EkgRtg; D ³ tamRTwsbþI TBItaK½r AC 2 AD2 CD 2 AD2 AC 2 CD 2 2
enaH
2 a 2 5a 2 a a 5 4 100 2 10 a 2 a 2 5a 2 a 2 4a 2 a 2 4 20 20 20 5 2 a a a 5 AD 5 5 5
o
o
o
dUcenH KNnaCaGnuKmn_én a )andUcteTA ³ a 5 a 5 ; CD 2 10 2a 5 a 5 BD ; AD 5 5
>
MBˆ O MDˆ O 90 o 90 o 180 o
2
a
B
o
5a 2 a 5 4 2
O
5> bgðajfa M sßitenAelIrgVg; C bRmab; MB nig MD CabnÞat;b:HrgVg; naM[ MBˆ O MDˆ O 90 ¬eRBaH bnÞat;b:HEkgnwgkaMrgVg;¦ edaykñúgctuekaN MBOD manplbUkmMuQmKW
a 2 5a 2 a a a2 4 4 2
naM[
C
O
2
2
D H
BC 2 AB2 AC 2
enaH BC -tam 1 ³ CA
M
C
o
o
BC
. 373
o
o
9
sm½yRbLg ³ 07 kkáda 1987 viBaØasa ³ KNitviTüa ry³eBl ³ 60 naTI BinÞú ³ 10 I. BICKNit
1> k> eK[kenSam ³ F 2x 1x 3x 3 nig G 3x 4x 3x 3 . KNna ³ A F G nig B F G . x> eK[ C BA . sRmÜlrYcKNnatémøelxénkenSamenH cMeBaHtémø x dUcteTA ³ 3 x , x5 , x 2 . 5 2> edaHRsayvismIkar 3x2 4 2x3 5 x 4 8 rYcbkRsaytamRkaPic . 3> cUrsg;bnÞat;tamTMnak;TMngTaMgbIxageRkamenAkñúgtRmúyEtmYy ³ 2 y x 4 , y x 1 , y 4 x rYcKNnakUGredaenéncMNucRbsBVTaMgbI . 3 II. FrNImaRt
Ggát; AB nig CD CaGgát;p©itEkgKñaénrgVg; O Edlman O Cap©it nigkaMmanrgVas; R . I CacMNuckNþalén OA . CI CYbrgVg; O Rtg; E . tam E eKKUsbnÞat;Rsbnwg AB Edl CYbnwgrgVg; O Rtg; F . CF CYb AB Rtg; J . H CacMeNalEkgén E elI CF . 1> R)ab;RbePTRtIekaN CEF nigctuekaN CIDJ . 2> RbdUcRtIekaN COI nig CED rYcKNna CI , CE nig IE CaGnuKmn_én R . 3> RbdUcRtIekaN CEH nig JDF rYcKNnapleFob CH . CJ
8
374
9
cemøIy I. BICKNit
2> edaHRsayvismIkar ³ 3x 4 2 x 5 x 8 2 3 4 63 x 4 42 x 5 3x 8 18x 24 8 x 20 3 x 24
1> k> KNna ³ A F G nig B F G eyIgman F 2x 1x 3x 3 nig G 3x 4x 3x 3 A F G
10 x 3 x 24 4
/
7 x 28
2 x 1x 3x 3 3x 4x 3x 3 x 3x 32 x 1 3x 4 x 3x 3 x 5
dUcenH KNna)an
x4
dUcenH vismIkarmancemøIy x 4 0 -bkRsaytamRkaPic ³ x EpñkminqUtCacemøIyénvismIkar .
A x 3x 3 x 5
B F G
/
2 x 1x 3x 3 3x 4x 3x 3 x 3x 32 x 1 3x 4 x 3x 35 x 3
Edl
y x4
A x 3x 3 x 5 x 5 B x 3 x 35 x 3 5 x 3 x 3 0 x 3 x 3 0 x 3
2 y x 1 3
b¤
dUcenH sRmÜl)an
C
A x5 B 5x 3
y 4 x
.
B
2 y x 1 3
-rkkUGredaenéncMNucRbsBVTaMgbI tag A CaRbsBVén y x 4 , y 32 x 1 pÞwmsmIkarGab;sIusénbnÞat;TaMgBIr ³
2 5 2 5 5 2 3 C 5 2 3 5 2 3 5 2 3
y x4
A
³
4 0 0 4 3 0 1 1 0 1 0 4
C
Kµann½y
³
x y x y x y
y 4 x
3 22 22 5 C 5 25 25 3 33 0 5 3 5 55 0 0 x5 C 5 5 3 22 x 2
x
eyIgsg;bnÞat;TaMgbI)andUcxageRkam ³
-KNnatémøelxénkenSam C ³ cMeBaH x 53 ³
cMeBaH cMeBaH
3> sg;bnÞat;TaMgbIkñúgtRmúyEtmYy ³ eyIgeRbItaragtémøelxedIm,Isg;bnÞat;TaMgbI ³
dUcenH KNna)an B x 3x 35x 3 x> sRmÜlcMeBaH C BA ³ C
4
10 3 2 25 2 15 25 2 9
5 22 2 41
375
9
2 x 4 x 1 3
b¤
naM[ CD CaemdüaT½rén AB EtGgát;FñÚ EF //AB enaH CD k¾CaemdüaT½r énGgát;FñÚ EF Edr vi)ak CE CF dUcnH RtIekaN CEF CaRtIekaNsm)at . -R)ab;RbePTctuekaN CIDJ ³ eyIgman I nig J enAelI AB enaH IJ // EF naM[ CEˆF CIˆJ nig CFˆE CJˆI ¬mMuRtUvKña¦ Et CEˆF CFˆE ¬mMu)atRtIekaNsm)at CEF ¦ eyIg)an CIˆJ CJˆI enaH CIJ Ca sm)atEdr¦ ebI CD CaemdüaT½rén EF enaH CD k¾Ca emdüaT½rén IJ Edr enaH OI OJ edayctuekaN CIDJ manGgát;RTUg CD IJ Rtg;cMNuckNþal O enaHvaCactuekaNesµI dUcenH ctuekaN CIDJ CactuekaNesµI . 2> RbdUcRtIekaN COI nig CED eday RtIekaN COI nig CED man ³ -mMu COˆ I CEˆD 90 eRBaH CD IJ Rtg; O nig CEˆ D CamMucarwkknøHrgVg;Ggát;p©it CD -mMu OCˆI ECˆD ¬mMurYm¦ dUcenH COI CED tamlkçxNÐ m>m CO CI OI vi)ak COI 1 CED CE CD ED -KNna CI , CE nig IE CaGnuKmn_én R eday CO OA R ¬eRBaH CakaMrgVg;dUcKña¦ OA R OI ¬eRBaH I kNþal OA ¦ 2 2 kñúgRtIekaNEkg COI ³ tamRTwsþIbTBItaK½r
3x 12 2 x 3
3x 2 x 3 12 5 x 15 x 3
cMeBaH x 3 : y x 4 3 4 1 dUcenH kUGredaenéncMNucRbsBV A 3 , 1 . tag B CaRbsBVén y 23 x 1 , y 4x pÞwmsmIkarGab;sIus ³ 23 x 1 4x b¤ 2x 3 12x 10x 3 x 103 cMeBaH x 103 : y 4x 4 103 65 dUcenH kUGredaenéncMNucRbsBV B 103 , 65 tag C CaRbsBVén y x 4 , y 4x pÞwmsmIkarGab;sIus ³ x 4 4x b¤ 5x 4 x 54 cMeBaH x 54 : y 4 x 4 54 165
o
4 16 C , 5 5
dUcenH kUGredaenéncMNucRbsBV II. FrNImaRt
tambRmab;RbFaneyIgsg;rbU )an ³ C
I
A
O H J
E D
O
B F
1> R)ab;RbePTRtIekaN CEF eday AB CD Rtg;cMNuckNþal O
CI 2 CO 2 OI 2 2
CI
376
CO 2 OI 2
R 5 R R2 2 2
9
CE JF JD 4 5R 3 5R 4 5R 3 5R 10 5 10 CH 5 R 5 R 5 2 2 2 2 12 5 R 6R 5 10 5 R 5 R 5 2 2 2 6R 2 12 R 12 5 R 5 25 R 5 5 5
CI tamvi)ak 1 xagelI CO CE CD Taj)an CE COCI CD eday CO R , CD 2R , CI R 2 5 naM[ CE R 2R 4R 4 5R
eyIg)an
2
R 5 2
5
R 5
eday IE CE CI 4 5R R 5 5 2 8 5 R 5 R 5 3R 5 10 10
naM[pleFob CH KW ³ CJ
dUcenH eyIgKNnaCaGnuKmn_én R )an ³ CI
12 5 R CH 12 5 R 2 24 25 CJ 25 R 5 R 5 25 2 CH 24 CJ 25
R 5 4 5R 3 5R , CE , IE 2 5 10
3> RbdUcRtIekaN CEH nig JDF eday RtIekaN CEH nig JDF man ³ -mMu CHˆ E JFˆD 90 eRBaH EH Cakm
dUcenH KNna)anpleFob
sUmKUsrUbEtmYy)anehIy
o
EH CF EH // DF DF CF
.
dUcenH
CEH
tamlkçxNÐ m>m .
vi)ak
CEH
JDF
C
I
A
O H J
E D
CE CH JDF JD JF CE JF CH JD
Taj)an eday CE 4 55R , JD CJ CI R 2 5 eRBaH JD , CI CaRCugénctuekaNesµI CF ehIy CE CI CJ
CH
JF IE
3 5R 10
377
O
B F
.
9
sm½yRbLg ³ 07 kkáda 1988 viBaØasa ³ KNitviTüa ry³eBl ³ 60 naTI BinÞú ³ 10 I. BICKNit
1> sRmÜlkenSam ³ E a b a aa b bb . 2> kMNt;témø m nig n edIm,I[smIkar ³ mx m 4x 3n 2 0 epÞógpÞat;cMeBaH x 2 nig x 5 . 3> k> sg;bnÞat; D tagsmIkar 3x 32 y 12 nigbnÞat; D tagsmIkar 5x 2 y 6 kñúgtRmúyGrtUNrem xoy . x> rkkUGredaenéncMNcu RbsBV I rvag D nig D . K> bMPøWfabnÞat; tagsmIkar m 1x 2m 1y 12 6m kat;tamcMNuc I ¬ m Ca):ar:aEm:Rt ¦. II. FrNImaRt 1> eKmanrgVg;myY p©it O nigcMNuc M mYyenAeRkArgVg; . tam M KUsbnÞat;b:H MA nig MB eTAnwgrgVg;Edl A nig B CacMNucb:H. bnÞat;Ekgnwg OA Rtg; O kat; MB Rtg; E . RsaybMPøWfa EO EM . 2> eKmanctuekaNBñay ABCD EdlmanmMu A nigmMu D CamMuEkg ehIyGgát;RTUg AC Ekgnwg RCugeRTt BC . k> eRbóbeFobRtIekaN ABC nigRtIekaN CAD nigbBa¢ak;fa AC AB CD . x> eKKUsrgVg;EdlmanGgátp; ©it BC ehIykat; AB Rtg; E . R)ab;eQµaHctuekaN AECD. K> ehA AI CaemdüanénRtIekaN ABC . P CacMNucmYyén BI . tam P KUsbnÞat; Rsbnwg AI . kat; AB Rtg; M ehIykat;knøHbnÞat; CA Rtg; N . eRbóbeFob AN pleFob AM ni g . AB AC smÁal; ³ sMNYrFrNImaRt k> x> K> minGaRs½yKñaeT . 2
2
8 378
9
cemøIy I. BICKNit
3> k> sg;bnÞat; D nig D kñúgtRmúy xoy eyIgeRbItaragtémøelxedIm,Isg;bnÞat;TaMgenH x 3 4 D ³ 3x 2 y 12 y 9 0 3
1> sRmÜlkenSam ³ E a b
a b
a a b b a b
a b a a b b a b
D ³
a a a b b a b b a a b b a b
a b b a a b
eyIgsg;bnÞat;TaMgBIr)andUcxageRkam ³
D : 5 x 2 y 6
a b ab ab a b
4.5
EdlktþaEdlsRmÜl a b 0 dUcenH sRmÜlkenSam)an E ab . 2> kMNt;témø m nig n eyIgman mx m 4x 3n 2 0 -cMeBaH x 2 ³ eyIg)an
D : 3x 2 y 12 3
3
2
x> rkkUGredaenéncMNucRbsBV I 2 eyIgmansmIkarbnÞat;TaMgBIr 3x 3 y 12
/
m 2 2 m 42 3n 2 0
5 x 2 y 6
4m 2m 8 3n 2 0
9 x 2 y 36 5 x 2 y 6 42 x 3 14 x 42 14
6m 3n 6 0
6m 3n 6 1
-cMeBaH
x 5
5x 2 y 6
2 x 0 2 y 3 2
³ eyIg)an
2
cMeBaH x 3 ³ 5 3 2 y 6
25m 5m 20 3n 2 0 20m 3n 22 0
dUcenH kUGredaenéncMNucRbsBV I 3 , 92 .
m 5 m 4 5 3n 2 0
20m 3n 22
2
20m 3n 22 6m 3n 6 m 2 14m 28
m 1x 2m 1y 12 6m
yk m 2 CMnYskñúg 1 eyIg)an ³
m 1 3 2m 1 9 12 6m
6 2 3n 6
dUcenH kMNt;)antémø
2 3m 3 9m 9 12 6m
18 6 3
m 2 , n 6
9 2
K> bMPøWfabnÞat; kat;tam I eyIgman ³ m 1x 2m 1y 12 6m ebI kat;tam I 3 , 92 enaHvaepÞógpÞat;smIkar
edayyk 2 1 edIm,IbM)at; n enaHeyIg)an ³
3n 6 12 n
y
Bit dUcenH bnÞat; BitCakat;tam I Emn . 12 6m 12 6m
. 379
9
II. FrNImaRt
vi)ak
M
1> RsaybMPøWfa EO EM
O
eday MA nig MB CabnÞat;b:HKUsecjBI M naM[ MO CaknøHbnÞat;BuHmMu AMˆ B enaHeyIg)an AMˆ O BMˆ O 1 mü:ageTot OA AM ¬kaMrgVg;EkgnwgbnÞat;b:H¦ OA OE ¬smµtikmµ¦ naM[ AM // OE eyIg)an AMˆ O MOˆ E 2 ¬mMuqøas;kñúg¦ tam 1 nig 2 enaH BMˆ O MOˆ E naM[ RtIekaN EMO CaRtIekaNsm)at vi)ak EO EM dUcenH RsaybMPøW)anfa EO EM . 2> tambRmab;RbFaneyIgKUsrUb)an ³
A N
ME
I
¬ CA AC ¦
o
o
o
o
o
AN K> eRbóbeFobpleFob AM ni g AB AC -kñúgRtIekaN ABI man AI // MP IP 1 tamRTwsþIbTtaEls AM AB IB -kñúgRtIekaN CNP man AI // NP IP AN IP tamRTwsþIbTtaEls AN b¤ 2 AC IC AC IB eRBaH AI CaemdüanénRtIekaN ABC Edl naM[ IB IC enaHeKGacCMnYs IC eday IB . AN tamTMnak;TMng 1 nig 2 naM[ AM AB AC
C
AC AB CD AC 2
B
D
CAD
dUcenH Taj)an AC AB CD . x> R)ab;eQµaHctuekaN AECD edayctuekaN AECD man ³ -mMu EAˆ D ADˆ C 90 ¬mMuEkgctuekaNBñay¦ -mMu AEˆC 90 eRBaH CE AB Rtg; E edaysar E enAelIrgVg;Ggát;p©it BC Edlman mMu CEˆB 90 CamMucarwkknøHrgVg; . eXIjfa ctuekaN AECD manmMuEkgcMnYnbIKW EAˆ D ADˆ C 90 nig AEˆ C 90 dUcenH ctuekaN AECD CactuekaNEkg .
E
A
ABC
P B
dUcenH eRbóbeFob)an
k> eRbóbeFobRtIekaN ABC nigRtIekaN CAD eday ABC niig CAD man ³ -mMu ACˆB CDˆ A 90 ¬smµtikmµ CamMuEkg¦ -mMu ACˆD BAˆ C ¬CamMuqøas;kñúg eRBaH AB// CD )atrbs;ctuekaNBñay ABCD¦ dUcenH ABC CAD tamlkçxNÐ m>m o
380
AM AN AB AC
.
9
381
9
sm½yRbLg ³ 07 kkáda 1989 viBaØasa ³ KNitviTüa ry³eBl ³ 60 naTI BinÞú ³ 10 I. BICKNit
1> dak;kenSamxageRkamCaplKuNktþa ³ A x2 x 2 B x 22 x 1 x 4x 2 x 2
2
2> sRmYlkenSam
.
2 1 2 b a b a b a b 1 3x 2 y 0 3x y 7 0 2
3> k> edaHRsayRbB½n§smIkartamRkaPic . x> rYcepÞógpÞat;tamEbbKNna . II. FrNImaRt 1> eKmanrgVg;BIrkat;KñaRtg; A nig B . tam A eKKUsbnÞat;b:HeTAnwgrgVg;TaMgBIr Edlkat;rgVg; TaMgBIrRtg; C nig D . RsaybMPøWfa CBˆ A DBˆ A . 2> eKmanbnÞat;nwg D mYy. enAelIbnÞat;enHeKedAcMNcu nwg A , B , C tamlMdab;enH . tam A nig B eKKUsbnÞat;ERbRbYlBIr nig EkgKñaRtg; E rYceKKUs D Ekgnwg D Rtg; C . bnÞat; D kat; nig erogKñaRtg;cMNuc M nig N . k> bMPøWfabnÞat; BM nig AN EkgKñaRtg; F . rkGrtUsg;énRtIekaN BMN . x> B CacMNucqøHú én B cMeBaH MN . bMPøWfacMNuc B enAelIrgVg;carwkeRkARtIekaN AMN . rksMNMucMNucp©it O énrgVg;enH . K> bMPøWfa CM CN CA CB .
8
382
9
cemøIy I. BICKNit
eyIgsg;RkaPic)an ³
1> dak;kenSamxageRkamCaplKuNktþa ³
3x 2 y 0
A x x2 2
x 2 x 2x 2 xx 1 2x 1
7 3
x 1x 2
3x y 7 0
dUcenH CaplKuNktþa A x 1x 2 . B x 22 x 1 x 4x 2 x 2 x 22 x 1 x 4 x 2 x 22 x 1
dUcenH CaplKuNktþa B x 22x 1 . 2> sRmYlkenSam
1 3x 2 y 0 3x y 7 0 2 3y 7 0
2 1 2 b a b a b a b
manPaKEbgrYmKW a b a b a b eK)an ³ a 2 b a 1 b a2 bb 2 a b a b 2 b a2 b2
cMeBaH
2 y 3 2 7 14 x . 3 3 9
x
2 a 2 b a b 2 b a b a b a b
dUcenH cMNucRbsBVKW 149 , 73 BitEmn . II. FrNImaRt
1> RsaybMPøWfa CBˆ A DBˆ A ¬sUmemIlkarbkRsayenATMB½rbnþ¦
3> k> edaHRsayRbB½n§smIkartamRkaPic ³ eyIgman 33xx 2y y70 0 12 eyIgeRbItaragtémøelxedIm,Isg;RkaPic ³ x y x 2 : 3x y 7 0 : y
1 :
3x 2 y 0
:
7 3
CMnYskñúgsmIkar 1 ³
a b
2 1 2 b a b a b a b a b a b
y
7 3 1 : 3x 2 y 0 y
dUcenH
14 9
tamRkaPic cMNucRbsBVrvagbnÞat;KW 149 , 73 x> rYcepÞógpÞat;tamEbbKNna ³ eyIgman 33xx 2y y70 0 12 eyIgdkGgÁngi GgÁedIm,IbM)at; x eyIg)an ³
2
a2 b2
A D
0 2 0 3 3 2 2 1
B C
383
9
RsaybMPøWfa
km bMPøWfacMNcu B enAelIrgVg;carwkeRkA AMN eday M nig N enAelI D Edl D D ehIy BC BC ¬eRBaH B CacMNucqøúHén B ¦ naM[ M nig N sßitenAelIemdüaT½rén BB naM[ MB MB nig NB NB enaH BMB nig BNB CaRtIekaNsm)at naM[ emdüaT½r MN k¾mannaTICaknøHbnÞat;BuHmMuEdr eK)an BNˆ C BNˆ C nig BMˆ C BMˆ C mü:ageTot BNˆ C MAˆ C nig BMˆ C NAˆ C ¬eRBaH BYkvamanRCugRtUvKñaEkgerogKña¦ ˆ ˆ eXIjfa BNˆ C BNˆ C BNˆ C MAˆ C 1
CBˆ A DBˆ A A
D
O
O
B
C
tag O nig O Cap©iténrgVg;TaMgBIr edaykñúg ACB nig DAB man ³ -mMu ACˆB DAˆ B ¬mMucarwk nigmMuBiessmanFñÚ sþat;rmY AB énrgVg;p©it O ¦ -mMu CAˆ B ADˆ B ¬mMucarwk nigmMuBiessmanFñÚ sþat;rmY AB énrgVg;p©it O ¦ dUcenH ACB DAB tamlkçxNÐ m>m vi)ak mMuTIbIKW CBˆ A DBˆ A dUcenH RsaybMPøW)anfa CBˆ A DBˆ A . 2> lMhat;enH eKykeTARbLgqñaM 1991 mþgeTot tambRmab;RbFaneyIgsg;rbU )an ³
M E
D
O
B
B A F
C N
BNC MAC BMˆ C B Mˆ C
D
BMˆ C NAˆ C
k> bMPøWfabnÞat; BM nig AN EkgKñaRtg; F kñúg AMN man ³ EN AM ¬eRBaH nig EkgKñaRtg; E ¦ AC MN ¬eRBaH D nig D EkgKñaRtg; C ¦ naM[ EN nig AC Cakm
B Mˆ C NAˆ C 2
eyIgbUkGgÁnigGgÁén 1 nig 2 ³ BNˆ C MAˆ C BMˆ C NAˆ C BNˆ C BMˆ C MAˆ N
edaykñúg NMB manplbUkmMukñúgKW ³ ˆ BMˆ C MBˆ N 180o BN C MAˆ N
MAˆ N MBˆ N 180o
9
eXIjfactuekaN AMBN manplbUkmMuQm ³ MAˆ N MBˆ N 180 enaH AMBN CactuekaN carwkkñúgrgVg; mann½yfa B enAelIrgVg;carwkeRkA AMN dUcenH B enAelIrgVg;carwkeRkA AMN . - rksMNMucMNucp©it O énrgVge; nH eday A, B , C CacMNucnwg ¬smµtikmµ¦ enaH B CacMNucqøúHén B k¾CacMNucnwgEdr naM[ AB CaGgát;FñÚEdlmanRbEvgefr ehIyrgVg;p©it O kat;tam A nig B naM[ OA OB ¬CakaMrgVg;EtmYy¦ eK)an O CacMNuccl½t EdleFVI[ OA OB mann½yfa O RtUvsßitelIemdüaT½rén AB dUcenH sMNMucMNuc O sßitelIemdüaT½rén AB K> bMPøWfa CM CN CA CB eyIgeRbóbeFob ACN nig MCB eday ACN nig MCB man ³ -mMu ACˆN MCB ¬eRBaH D D Rtg; C ¦ -mMu NAˆ C BMˆ C ¬mMumanRCugRtUvKñaEkgerogKña¦ dUcenH ACN MCB tamlkçxNÐ m>m CN CA vi)ak ACN MCB CB CM Taj)an CM CN CA CB Et CB CB ¬eRBaH B CacMNucqøúHén B ¦ eK)an CM CN CA CB dUcenH bMPøW)anfa CM CN CA CB . o
385
9
sm½yRbLg ³ 06 kkáda 1990 viBaØasa ³ KNitviTüa ry³eBl ³ 60 naTI BinÞú ³ 10 I. BICKNit
1> eK[kenSam ³ A x 16 3x 1x 4 nig B x k> dak;knSam A nig B CaplKuNktþadWeRkTImYyén x . x> sRmÜlRbPaKsniTan F BA . 2
2
8x 16
.
2> KNnakenSam P 5 5 3 5 3 3 ; Q a aa b bb nig R 2 2 3 . 3> kñúgtRmúyGrtUNrem xoy sg;bnÞat; D rbs;smIkar y 2x 3 nig D : y 5x 6 . k> rkkUGredaenéncMNucRbsBV I rvagbnÞat;TaMgBIr rYcepÞógpÞat;edayKNna . x> rksmIkarbnÞat; Edlkat;tamcMNuc A2 ; 3 ehIyRsbnwgbnÞat; D . II. FrNImaRt
1> eK[ctuekaNBñay ABCD )attUc AB. bnøayRCugeRTt AD nig BC Edlkat;KñaRtg; E . k> eK[ ED a ; EA b ; EB c . KNna BC . x> Ax CaknøHbnÞat;BuH DAB nig Dy CaknøHbnÞat;BuH ADC . knøHbnÞat;TaMgBIrCYbKñaRtg; J . R)ab;eQµaHRtIekaN AJD . 2> rgVg;BIrmanp©it O nig O kat;KñaRtg; A nig B . KUsGgát;p©ti AD nig AE . k> RsaybMPøWfabIcMNcu D , B , E sßitenAelIbnÞat;EtmYy . x> M CacMNuccl½tenAelIrgVg; O . tam M KUsbnÞat; AM kat;rgVg; O Rtg; N . RsaybMPøWfa MBˆ N mantémøefrkalNa M rt;CMuvij A . K> I CacMNuckNþal AM rksMNMucMNuc I kalNa M rt;CMuvji A elIrgVg; O .
8 386
9
cemøIy I. BICKNit
Q
1> mankenSam A x 16 3x 1x 4 nig B x 8x 16 k> dak; A nig B CaplKuNktþadWeRkTImYyén x
a a
2
2
A x 2 16 3 x 1x 4 x 4 x 4 3 x 1x 4 x 4 5 2 x
A x 45 2 x
.
x 2 2 x 4 42 x 4 x 4x 4 2
B x 4x 4
x> sRmÜlRbPaKsniTan F
F
.
A 5 2x B x4
5 5 3 3 5 3 5 3 5 3
Q a b ab
42 3
2
3 2 3 12
2
3> sg;bnÞat; D nig D x 0 D ³ y 2 x 3 ³ y 3
.
1 5
³ xy 11 eyIgsg;)andUcxageRkam ³
D :
2 4
y 2x 3
I 3 , 9
5 2 32
D : y 5x 6
8 52 3 4 5 3 2 P4 5 3
.
3 1 3 1 dUcenH KNna)an R 3 1 .
5 5 5 3 3 5 3 3
dUcenH KNna)an
D : y 5 x 6
5 3 P 5 3 5 3
R 2 2 3
A F B
2> KNnakenSam ³
a b
a2 b2
dUcenH KNna)an
A x 4 5 2 x B x 4 x 4 5 2x , x4 0 x 4 x4
dUcenH sRmÜl)an
a b
B x 2 8 x 16
dUcenH kenSam
b
a b b a b a b a b
a 2 b ab a ab b 2 a b 2 2 a b a ab b ab a b a b a b a b ab a b a b a b ab a b a b ab , a b 0 a b
x 4 x 4 3x 1
dUcenH kenSam
a a b b a b
. 387
9
EA Et AD ED EA enaH BC EB ED EA ehIy ED a ; EA b ; EB c ¬smµtikmµ¦
x> rkkUGredaenéncMNucRbsBV I tamRkaPic ³ ebIeyIgeFVIcMeNalEkgBIrcMNuc RbsBVeTAelIG½kSTaMgBIr enaHeyIg)an I 3, 9 . -epÞógpÞat;tamkarKNna ³ eyIgman D ³ y 2x 3 nig D : y 5x 6 eyIgpÞwmsmIkarGab;sIusénbnÞat;TaMgBIr ³
BC
dUcenH KNna)an
x3
cMeBaH x 3 enaH y 2x 3 2 3 3 9 dUcenH KNna)an I 3, 9 . x> rksmIkarbnÞat; ³ smIkarbnÞat;Edlrkmanrag : y ax b eday : y ax b kat;tam A2 ; 3 eyIg)an 3 a 2 b b 3 2a ehIy RsbnwgbnÞat; D enaH a a 2 cMeBaH a 2 ³ b 3 2a 3 2 2 7 dUcenH smIkarbnÞat; : y 2x 7 .
ac bc b
.
o
o
o
DAˆ J ADˆ J 90o
eXIjfakñúg AJD man DAˆ J ADˆ J 90 naM[ AJˆD 90 ¬CamMuEkg¦ dUcenH RtIekaN AJD CaRtIekaNEkgRtg; J . 2> tambRmab;RbFaneyIgsg;rUb)an ³ o
o
N
II. FrNImaRt
A
1> tambRmab;RbFaneyIgsg;rUb)an ³
I
M B
A
k> RsaybMPøfW a D , B , E sßitenAelIbnÞat;EtmYy edayrgVg;Ggát;p©it AD nig AE RbsBVKñaRtg; cMNuc A , B enaHcMNuc A , B enAelIrgVg;TaMgBIr -rgVg;Ggát;p©it AD man B enAelIrgVg; naM[ ABˆD 90 ¬mMucarwkknøHrgVg;Ggát;p©it AD ¦ -rgVg;Ggát;p©it AE man B enAelIrgVg; naM[ ABˆE 90 ¬mMucarwkknøHrgVg;Ggát;p©it AE ¦
C x
k> KNna BC eday)atctuekaNBñay ABCD man AB//CD tamRTwsþIbTtaEls eyIg)anGgát;smamaRtKW ³
E
BC
B D
y
J
D
O
O
E
EA EB AD BC
BC
x> R)ab;eQµaHRtIekaN AJD eday AB//CD ¬)atTaMgBIrctuekaNBñay¦ naM[ DAˆ B CDˆ A 180 ¬mMubEnßmKña¦ Et DAˆ B 2DAˆ J ¬eRBaH Ax BuHmMu DAB¦ CDˆ A 2 ADˆ J ¬ eRBaH Dy BuHmMu ADC ¦ naM[ 2DAˆ J 2 ADˆ J 180 2DAˆ J ADˆ J 180
2 x 3 5x 6 3x 9
EB ED EA ca b ac bc EA b b
o
EB AD EA
o
388
9
eday ABˆD ABˆE 90 90 180 Et ABˆD ABˆE DBˆE eXIjfa ABˆˆ D ABˆˆ E 180ˆ DBˆ E 180 o
o
o
N
A
o
o
I
ABD ABE DBE
O
E
O
mann½yfa DBˆ E CamMurab b¤ bnÞat;Rtg; dUcenH D , B , E sßitenAelIbnÞat;EtmYy . x> RsaybMPøWfa MBˆ N mantémøefrkalNa M rt;CMuvij A kñúg MNB eTaHbI M rt;CMuvij A y:agNak¾eday k¾eyIgenAEtTTYl)an ³ -cMeBaHrgVg;p©ti O ³ AMˆ B 12 AB efr -cMeBaHrgVg;p©ti O ³ ANˆ B 12 AB efr naM[ MBˆ N 180 AMˆ B ANˆ B k¾efrEdr dUcenH MBˆ N mantémøefrkalNa M rt;CMuvij A K> rksMNMucN M uc I kalNa M rt;CMuvij A elIrgVg; O eyIgman IM IA ¬eRBaH I kNþal AM ¦ enaHeyIg)an OI AM ¬eRBaHkaMrgVgE; kgnwgGgát;FñÚRtg;cMNuckNþal¦ eday cMNuc O , A minERbRbYl eyIg)an OIˆA 90 mantémøefr edIm,I[ OIˆA 90 efr luHRta I sßitenAelI rgVg;EdlmanGgát;p©it OA -ebI M RtYtelI D enaH I RtYtelI O -ebI M RtYtelI A enaH I RtYtelI A dUcenH sMNcMu MNuc I enAelIrgVg;Ggát;p©it OA .
M
B D
o
o
o
389
9
sm½yRbLg ³ 09 mifuna 1991 viBaØasa ³ KNitviTüa ry³eBl ³ 60 naTI BinÞú ³ 10 I. BICKNit
1> dak;kenSamxageRkamCaplKuNktþa ³ A 3 x 5 16 2
B x 32 x 1 x 3 x 3 1 3 0 ; A0 ; B0 B A 2
rYcedaHRsaysmIkar . 2> k> BnøatkenSam P 4 5 3 10 . x> KNnakenSam Q 6 4 2 . 3> tamTMnak;TMng x 2 y 2 KNna xy rYcKNna x nig y edaydwgfa x y 1 . 4> kñúgtRmúyGrtUNrem xoy manÉktaCasg;TIEm:Rt cm . k> sg;bnÞat; : y 2x 3 nig : y 3x 7 . x> bnÞat;TaMgBIrRbsBVKñaRtg;cMNuc A . KNnakUGredaenéncMNuc A . K> sresrsmIkarbnÞat; D Edl D ehIykat;tamcMNuc I 4 , 3 . 1
2
1
II. FrNImaRt
eKmanbnÞat;nwg D mYy. enAelIbnÞat;enHeKedAcMNucnwg A , B , C tamlMdab;enH . tam A nig B eKKUsbnÞat;ERbRbÜlBIr nig EkgKñaRtg; E rYceKKUs D Ekgnwg D Rtg; C . bnÞat; D kat; nig erogKñaRtg;cMNuc M nig N . k> bMPøWfabnÞat; BM nig AN EkgKñaRtg; F . rkGrtUsg;énRtIekaN BMN . x> B CacMNucqøHú én B cMeBaH MN . bMPøWfacMNuc B enAelIrgVg;carwkeRkARtIekaN AMN . rksMNMucMNucp©it O énrgVg;enH . K> bMPøWfa CM CN CA CB .
8 390
9
cemøIy I. BICKNit
-edaHRsaysmIkar cMeBaH B 0 eK)an x 3x 5 0 naM[ xx 53 00 xx 53
1> dak;kenSamxageRkamCaplKuNktþa ³ A 3 x 5 16 2
3 x 5 4 2 3 x 5 43 x 5 4 3 x 13 x 9 33 x 1 x 3
2
dUcenH ebI B 0 enaHb£s x 3 , x 5 . 2> k> BnøatkenSam P ³ P 4 5 3 10
B x 32 x 1 x 3 x 3 2
x 32 x 1 x 3 x 3 2
12 4 10 3 5 50
x 32 x 1 x 3 1 x 3x 5
dUcenH
A 33x 1x 3
12 4 10 3 5 5 2
dUcenH Bnøat)an P 12 4 x> KNnakenSam Q ³
nig B x 3x 5
-edaHRsaysmIkar ³ cMeBaH B1 A3 0 eK)an x 31 x 5 33x 13x 3 0
dUcenH KNna)an
tRmÚvPaKEbgrYm x 3x 53x 1 rYclubecal
3x 1 x 5 0 2 x 6 0 x 3
/
1 3 0 B A
KµancemøyI .
2 2
.
Q 2 2
2 y2
x 2 y
x y x y 1 2 2 2 2 2 2 2 2 2 2 2 22 22 x 2 2 x 2 2 2 2
-edaHRsaysmIkar cMeBaH A 0 eK)an 33x 1x 3 0 naM[ 3xx3100 xx 13/ 3
2
dUcenH KNna)an xy 2 . -KNna x nig y cMeBaH x y 1 x y x 2 y 2 enaH ¬tamsmamaRt¦ 2 2
cMeBaH x 3 minyk eRBaHvaCalkçxNÐ dUcenH smIkar
2 2
3> KNna xy eyIgman TMnak;TMng x Taj)an xy 22
x 3 0 x 3 x 5 0 x 5 3x 1 0 x 1 / 3
eK)an
10 3 5 5 2
Q 6 4 2 4 2 2 2 22
1 1 0 x 3x 5 3x 1x 3
Edl
naM[
y
2
dUcenH ebI A 0 enaHb£s x 13 , x 3
2 2 2 2 22 y 2 1 2 2
dUcenH KNna)an x 2 2 / 391
y 2 1
.
9
4> k> sg;bnÞat; nigbnÞat; eyIgeRbItaragtémøelxedIm,Isg;bnÞat;TaMgenH x 0 1 : y 2 x 3 ³ y 3 1 1
2
II. FrNImaRt
tambRmab;RbFaneyIgsg;rbU )an ³
M
1
E
³ y 4 1 eyIgsg;bnÞat;TaMgBIr)andUcxageRkam ³ 2 : y 3x 7
1 : y 2 x 3
x 1 2
D
2 : y 3x 7
B
B A F
C N
D
k> bMPøWfabnÞat; BM nig AN EkgKñaRtg; F kñúg AMN man ³ EN AM ¬eRBaH nig EkgKñaRtg; E ¦ AC MN ¬eRBaH D nig D EkgKñaRtg; C ¦ naM[ EN nig AC Cakm bMPøWfacMNcu B enAelIrgVg;carwkeRkA AMN eday M nig N enAelI D Edl D D ehIy BC BC ¬eRBaH B CacMNucqøúHén B ¦ naM[ M nig N sßitenAelIemdüaT½rén BB naM[ MB MB nig NB NB enaH BMB nig BNB CaRtIekaNsm)at
D : y 2 x 11
A
x> KNnakUGredaenéncMNuc A eyIgman : y 2x 3 nig : y 3x 7 eyIgpÞwmsmIkarGab;sIusénbnÞat;TaMgBIr ³ eyIg)an 2x 3 3x 7 / 3x 2 x 7 3 x 4 cMeBaH x 4 : y 2x 3 2 4 3 5 dUcenH kUGredaenéncMNucRbsBVKW A4 , 5 . K> sresrsmIkarbnÞat; D smIkarbnÞat;EdlRtUvrkmanrag D : y ax b eday D naM[ a a 2 ehIykat;tamcMNuc I 4 , 3 naM[ 3 2 4 b b 3 8 11 dUcenH smIkarbnÞat; D : y 2x 11 . 1
O
2
1
392
9
naM[ emdüaT½r MN k¾mannaTICaknøHbnÞat;BuHmMuEdr eK)an BNˆ C BNˆ C nig BMˆ C BMˆ C mü:ageTot BNˆ C MAˆ C nig BMˆ C NAˆ C ¬eRBaH BYkvamanRCugRtUvKñaEkgerogKña¦ ˆ ˆ eXIjfa BNˆ C BNˆ C BNˆ C MAˆ C 1
K> bMPøWfa CM CN CA CB eyIgeRbóbeFob ACN nig MCB eday ACN nig MCB man ³ -mMu ACˆN MCB ¬eRBaH D D Rtg; C ¦ -mMu NAˆ C BMˆ C ¬mMumanRCugRtUvKñaEkgerogKña¦ dUcenH ACN MCB tamlkçxNÐ m>m CN CA vi)ak ACN MCB CB CM Taj)an CM CN CA CB Et CB CB ¬eRBaH B CacMNucqøúHén B ¦ eK)an CM CN CA CB dUcenH bMPøW)anfa CM CN CA CB .
BNC MAC
BMˆ C B Mˆ C B Mˆ C NAˆ C 2 ˆ ˆ BMC NAC
eyIgbUkGgÁnigGgÁén 1 nig 2 ³ BNˆ C MAˆ C BMˆ C NAˆ C BNˆ C BMˆ C MAˆ N
edaykñúg NMB manplbUkmMukñúgKW ³ ˆ BMˆ C MBˆ N 180o BN C MAˆ N
rUbdEdl edIm,I[RsYlemIlkñúgkarbkRsay
MAˆ N MBˆ N 180o
eXIjfactuekaN AMBN manplbUkmMuQm ³ MAˆ N MBˆ N 180 enaH AMBN CactuekaN carwkkñúgrgVg; mann½yfa B enAelIrgVg;carwkeRkA AMN dUcenH B enAelIrgVg;carwkeRkA AMN . - rksMNMucMNucp©it O énrgVge; nH eday A, B , C CacMNucnwg ¬smµtikmµ¦ enaH B CacMNucqøúHén B k¾CacMNucnwgEdr naM[ AB CaGgát;FñÚEdlmanRbEvgefr ehIyrgVg;p©it O kat;tam A nig B naM[ OA OB ¬CakaMrgVg;EtmYy¦ eK)an O CacMNuccl½t EdleFVI[ OA OB mann½yfa O RtUvsßitelIemdüaT½rén AB dUcenH sMNMucMNuc O sßitelIemdüaT½rén AB
M
o
E
D
O
B
B A F
C N
D
393
9
sm½yRbLg ³ 14 kkáda 1992 viBaØasa ³ KNitviTüa ry³eBl ³ 60 naTI BinÞú ³ 10 I. BICKNit
1> k> bMEbkkenSam A CaplKuNktþaEdl A 2 x 3x 4 4 x 9 2 x 3 . x> etI x RtUvmantémøesµIb:unµan edIm,I[ A 0 ? K> KNnatémøelxénknSam A cMeBaH x 1 2 . 2> sRmYlkenSam F 2xx32x7x 49 4x6x972xx23 . 3> kñúgtRmúyGrtUNrem xoy EdlmanÉktþaCa sg;TIEm:Rt ¬ cm ¦ . k> sg;bnÞat; D : y 2x 4 nigbnÞat; D : y x 1 . x> bnÞat; D nig D RbsBVKñaRtg;cMNuc I . KNnakUGredaenéncMNuc I . K> sresrsmIkarbnÞat; D Ekgnwg D nigkat;tamcMNuc A0 ,1 . 2
2
2
1
1
2
2
2
2
II. FrNImaRt
eK[rgVg; C p©it O viCÄmaRt AB Edl AB 4R nigrgVg; C p©it O viCÄmaRt OA . tam B KUsbnÞat;b:H BM eTAnwgrgVg; C Rtg; M . BM kat;rgVg; C Rtg; E . 1> bgðajfa AE Rsbnwg OM . rYcTajbBa¢ak;fa AM CaknøHbnÞat;BuHén BAE. 2> AM kat;rgVg; C Rtg; N . bgðajfa M CacMNuckNþalén AN . 3> bnÞat; ON kat; BM Rtg; G . etI G CaGVIcMeBaHRtIekaN ANB . KNna OG CaGnuKmn_én R . 4> bnÞat; AE kat; BN Rtg; P . bgðajfa MP AB .
8
394
9
cemøIy I. BICKNit
3> k> sg;bnÞat; D nigbnÞat; D eyIgeRbItaragtémøelxedIm,Isg;bnÞat;TaMgBIr x 0 2 D : y 2 x 4 ³ y 4 0 1
1> k> bMEbkkenSam A CaplKuNktþa
A 2 x 3x 4 4 x 2 9 2 x 3
2
2 x 3x 4 2 x 32 x 3 2 x 3 2 x 3x 4 2 x 3 2 x 3 2 x 3x 4 2 x 3 2 x 3 2 x 3x 2
1
2
³ xy 10 12 eyIgsg;bnÞat;TaMgBIr)andUcxageRkam ³ D2 : y x 1
dUcenH bMEbk)an A 2x 3x 2 . x> rktémø x edIm,I[ A 0 eyIgman A 2x 3x 2 ebI A 0 enaH 2x 3x 2 0 naM[
D2 : y x 1
D1 : y 2 x 4
3 x 2 x 3 0 2 x 3 2 x 2 0 x 2 x 2
x> KNnakUGredaenéncMNcu I eyIgman D : y 2x 4 b¤ y 2x 4 D : y x 1 b¤ y x 1 eyIgpÞwmsmIkarGab;sIusénbnÞat;TaMgBIr ³ eyIg)an 2x 4 x 1
dUcenH ebI A 0 enaH x 32 , x 2 .
1
K> KNnatémøelxénknSam A cMeBaH x 1 2 eyIgman A 2x 3x 2 cMeBaH x 1 2 eyIg)an A 21 2 31 2 2
2 2 1 3 2
2
3x 3
5 2 1
F
A 5 2 1
.
2x 3x 4 4 x 2 9 2x 32 x 27 x 9 6 x 7 x 2
2
A x 27 x 9 6 x 7 x 2 2 x 3x 2 x 27 x 9 6 x 7 2x 3 , x 2 0 x 2 x2
dUcenH sRmÜl)an
F
2x 3 x2
x 1
/
cMeBaH x 1 enaH y x 1 1 1 2 dUcenH kUGredaenéncMNucRbsBV I 1, 2 . K> sresrsmIkarbnÞat; D smIkarbnÞat; D EdlRtUvrkmanrag y ax b eday D D a a 1 Et D : y x 1 man a 1 naM[ a 1 ehIy D kat;tamcMNuc A0 ,1 eyIg)an 1 a 0 b b 1 dUcenH smIkarbnÞat; D ³ y x 1 .
6 2 43 2
dUcenH cMeBaH x 1 2 enaH 2> sRmYlkenSam F ³
2
2
. 395
9
II. FrNImaRt
rgVg;p©it O man AN CaGgát;FñÚ nig AN MO naM[ MA MN eRBaH kaMEkgnwgGgát;FñÚRtg;cMNuckNþalCanic© dUcenH M CacMNuckNþalén AN . 3> etI G CaGVIceM BaHRtIekaN ANB kñúg ANB mancMNuc M kNþal AN nig O kNþal AB naM[ Ggát; BM nig ON Caemdüan TaMgBIrénRtIekaNenH . edayemdüan BM nig ON RbsBVKñaRtg; G enaH emdüanTIbI k¾kat;tamcMNuc G enHEdr . dUcenH G CaTIRbCMuTm¶n;énRtIekaN ANB . -KNna OG CaGnuKmn_én R tamlkçN³emdüan OG 13 ON 4R Et ON OA AB 2 R ¬kaMrgVg;dUcKña¦ 2 2 naM[ OG 13 2R 23 R
tambRmab;RbFaneyIgsg;rbU )an ³ P N
E
M
G
A
O
B
O
C
C
1> bgðajfa AE Rsbnwg OM rgVg;viCÄmaRt AB man E enAelIrgVg;enH ¬eRBaH BM kat;rgVg; C Rtg; E ¦ naM[ ABE EkgRtg; E Edl AE EB ehIy BM CabnÞat;b:HrgVg;p©it O Rtg; M naM[ OM MB b¤ OM EB eXIjfa OAEMEBEB AE//OM dUcenH bgðaj)anfa AE Rsbnwg OM . -TajbBa¢ak;fa AM CaknøHbnÞat;BuHén BAE eday AE //OM ¬sRmayxagelI ¦ naM[ OMˆ A EAˆ M 1 ¬mMuqøas;kñúg¦ ehIy OAM CaRtIekaNsm)at ¬eRBaH OA OM kaMrgVg;p©it O dUcKña¦ naM[ OMˆ A OAˆ M 2 ¬mMu)at sm)at¦ pÞwm 1 nig 2 eyIg)an ³ EAˆ M OAˆ M b¤ EAˆ M BAˆ M dUcenH AM CaknøHbnÞat;BuHén BAE . 2> bgðajfa M CacMNuckNþalén AN rgVg;viCÄmaRt OA man M enAelIrgVg;enH naM[ mMu OMˆ A 90 enaH AM MO b¤ AN MO ¬eRBaH N Cabnøayén AM ¦
dUcenH KNna)an
OG
2 R 3
.
4> bgðajfa MP AB eK)an AP BE nig BP AN eRBaH E nig N enAelIrgVg;viCÄmaRt AB vaCamMcu arwkknøHrgVg; kñúg APB man AP BE nig BP AN enaH BE nig AN Cakm
o
396
9
397
9
sm½yRbLg ³ 14 kkáda 1993 viBaØasa ³ KNitviTüa ry³eBl ³ 60 naTI BinÞú ³ 10 I. BICKNit
1> eK[kenSam Ax x 2x 3 2 x5 x 4 x . k> BnøatkenSamxagelIenH rYcsresrtamlMdab;sV½yKuNcuHén x . x> sresr Ax CaragplKuNktþa . K> sRmYlRbPaKsniTan F x 5x 2A xxx 2 . kMNt;témø x edIm,I[ F x 0 . 2> sg;bnÞat; D nig D tagGnuKmn_ y x 4 nig y 5 x . rkkUGredaenéncMNucRbsBVrvagbnÞat;TaMgBIr . 2
1
2
II. FrNImaRt
eK[rgVg;p©it O kaM R . KUsbnÞat; D b:HnwgrgVg;Rtg; P . A CacMNucmYyén D . rgVg;Ggát; p©it OA kat;knøHbnÞat;BuH OAP Rtg; H . bnøayén OH kat;bnÞat; D Rtg; M . 1> bgðajfargVgG; gát;p©it OA kat;tamcMNuc P . 2> R)ab;RbePTRtIekaN AOM . 3> KUskm
8 cemøIy I. BICKNit
x> sresr Ax CaragplKuNktþa Ax x 2x 3 2 x 5 x 4 x 2
1> eKman Ax x 2x 3 2 x5 x 4 x k> BnøatkenSam nigsresrtamlMdab;sV½yKuNcuH 2
Ax x 2x 3 2 x 5 x 4 x 2 x 2 3x 2 x 6 10 2 x 5 x x 2 4 x 2 x 2 5 x 6 10 7 x x 2 4 x 2
dUcenH dak; Ax CaplKuNktþa)anKW ³
x 2 2x 8
dUcenH BnøatkenSam)an Ax x
2
2x 8
x 2x 3 x 25 x x 2 4 x 2x 3 x 25 x x 2x 2 x 2x 3 5 x x 2 x 2x 4
Ax x 2x 4
. 398
.
9
K> sRmYlRbPaKsniTan F x ³ eKman F x 5x 2A xxx 2 b¤ F x xx 225x 4x 5x 4x Edl x 2 0 x 2
II. FrNImaRt
tambRmab;RbFaneyIgsg;rbU )andUcxageRkam ³ K
dUcenH sRmÜl)an F x 5x 4x .
A
dUcenH ebI F x 0 kMNt;)antémø x 4 . 2> sg;bnÞat; D nig D eyIgeRbItaragtémøelxedIm,Isg;bnÞat;TaMgBIr x 4 0 D ³ y x 4 y 0 4 2
1
D2 ³
y 5 x
x
H
P M
o
1
O
D
1> bgðajfargVg;Ggát;p©it OA kat;tamcMNuc P bnÞat;b:H D EkgnwgkaMrgVg; OP Rtg;cMNucb:H P naM[ mMu OPA 90 ¬eRBaH A enAelI D ¦ ehIy OA Ggát;p©itrgVg; enaHmann½yfa OPA CamMucarwkknøHrgVg; Ggát;p©it OA dUcenH rgVg;Ggát;p©it OA kat;tamcMNuc P . 2> R)ab;RbePTRtIekaN AOM eday H CacMNucenAelIrgVg;Ggát;p©it OA naM[ AH OH b¤ AH OM ¬eRBaH M enAelIbnøay OH ¦ eday AOM man AH CaknøHbnÞat;BuHpg nig Cakm RsaybMPøWfa MK R eday OPA 90 enaH OP Cakm
- kMNt;témø x edIm,I[ F x 0 ebI F x 0 enaH 5x 4x 0 eK)an 5x x4 00 xx 54
0 5
y 5 0
D1 : y x 4 D2 : y 5 x
-rkkUGredaenéncMNucRbsBVrvagbnÞat;TaMgBIr eyIgpÞwmsmIkarGab;sIusénbnÞat;TaMgBIr eyIg)an x 4 5 x 1 2x 1 x b¤ 2 1 1 9 cMeBaH x 2 : y 5 x 5 2 2
o
dUcenH cMNcu RbsBVénbnÞat;TaMgBIrKW 12 , 92 . 399
9
sm½yRbLg ³ 12 kkáda 1994 viBaØasa ³ KNitviTüa ry³eBl ³ 60 naTI BinÞú ³ 10 I. BICKNit
1> k> eRbóbeFobcMnYn x 3 5 nig y 2 11 . x> sRmYlkenSam E 5 5 3 5 3 3 . 2> eK[kenSam ³ Ax x 4x 4 x 36 3x 5x 4 . k> BnøatkenSam Ax rYcerobtamlMdab;sV½yKuNcuHén x . x> dak; Ax CaplKuNktþadWeRkTI1 . K> edaHRsaysmIkar Ax 0 / Ax 2 . X> eK[RbPaK F x 3x A1x4 x rktémø x EdleFVI[ F x mann½y rYcsRmYl F x . 3> k> kñúgtRmúyGrtUNrem xoy sg;bnÞat; D : y x 2 nig D : y 4 x . x> rkkUGredaenéncMNucRbsBV I rvagbnÞat;TaMgBIr . 2
2
1
2
II. FrNImaRt
eK[rgVg;p©it O EdlmanGgátp; ©itBIrKW AB nig CD EkgKña . E CacMNucmYyén OA . 1> R)ab;eQµaHénRtIekaN CED . 2> tamcMNuc C eKKUsbnÞat;Ekgnwg CE ehIytam D eKKUsbnÞat;Ekgnwg DE . bnÞat;TaMg BIrenHRbsBVKñaRtg;cMNuc F . k> eRbóbeFobRtIekaN CEF nigRtIekaN DEF . x> bgðajfa F enAelIbnÞat; AB . 3> H CacMNucqøúHéncMNuc E eFobnwgcMNuc O . k> R)ab;eQµaHctuekaN CEDH . x> RsaybBa¢ak;fa H CaGrtUsg;énRtIekaN CDF .
8 400
9
cemøIy Ax 3x 2 7 x 2
I. BICKNit
1> k> eRbóbeFobcMnYn x 3 5 nig y 2 eyIgman x 3 5 nig y 2 11 naM[ x 3 5 3 5 9 5 45
3x 2 x 6 x 2 x3x 1 23x 1
11
3x 1x 2
dUcenH plKuNktþa Ax 3x 1x 2 . K> edaHRsaysmIkar ³ eyIgman Ax 3x 1x 2 -cMeBaH Ax 0 enaH 3x 1x 2 0 1 naM[ 3xx2100 3xx21 x 3
2
y 2 11 22 11 4 11 44
eday 45 44 x y dUcenH eRbóbeFob)an x y . x> sRmYlkenSam E ³ eKman E 5 5 3 5 3 3 5 3 5 3 5 3
5 5 3 3 5 3 5 3 5 3
2
dUcenH sRmÜl)an 2> eK[kenSam ³
dUcenH ebI
k> Bnøat Ax rYcerobtamlMdab;sV½yKuNcuHén x
Ax x 2 4 x 4 x 36 3 x 5 x 2 4
Ax 2
enaH
x 0
x
7 , x0 3
.
X> rktémø x EdleFVI[ F x mann½y eyIgman F x 3x A1x4 x RbPaK F x mann½y luHRtaEtPaKEbgxusBI 0
.
x 4 x 4 3 x 3 x 18 5 x 20 2
.
2
Ax x 2 4 x 4 x 36 3x 5 x 2 4
2
1 , x2 3
2
2
E4 5 3
x
-cMeBaH Ax 2 enaH 3x 7x 2 2 b¤ 3x 7 x 0 3x 7x 0 7 naM[ 3xx07 0 3xx07 x 3
8 52 3 53 24 5 3 4 5 3 2
x 2
dUcenH ebI Ax 0 enaH
5 5 5 3 3 5 3 3 5 3
2
3x 2 7 x 2
eK)an
1 x 3x 1 0 3x 1 3 4 x 0 x 4 x 4
dUcenH
F x
mann½y kalNa x 13 , x 4 .
sRmÜl F x 3x A1x4 x
dUcenH Bnøat)an Ax 3x 7 x 2 . x> dak; Ax CaplKuNktþadWeRkTI1 2
3x 1x 2 x 2 3x 14 x 4 x
dUcenH sRmÜl)an 401
F x
x2 4 x
.
9
2> k> eRbóbeFob CEF nig DEF RtIekaNTaMgBIrman EC CF nig ED DF enaHmMu ECˆF EDˆ F 90 naM[ CEF nig DEF CaRtIekaNEkgTaMgBIr Edlman ³ - RCug EC ED ¬bBa¢ak;xagelI¦ -GIub:UetnusrYm EF dUcenH CEF DEF tamkrNI G>C . vi)ak CF DF x> bgðajfa F enAelIbnÞat; AB tamvi)akxagelI CF DF mann½yfa F RtUv sßitenAelIemdüaT½r énGgát; CD ehIyemdüaT½rén CD KW AB dUcenH eXIjfa F enAelIbnÞat; AB . 3> k> R)ab;eQµaHctuekaN CEDH . H CacMNucqøúHéncMNuc E eFobnwgcMNuc O naM[ OE OH Edl E nig H enAelI AB ehIy OC OD ¬kaMrgVg;p©it O dUcKña¦ nigman EH CD Rtg;cMNuckNþal O enaH CEDH CactuekaNmanGgát;RTUgEkgKña Rtg;cMNuckNþal ehIyvaKµanmMuEkg dUcenH ctuekaN CEDH CactuekaNesµI . vi)ak ³ CE // DH nig ED// CH x> RsaybBa¢ak;fa H CaGrtUsg;én CDF // DH eday CE DH CF CE CF
3> k> sg;bnÞat; D nig D kñúgtRmúy xoy eyIgeRbItaragtémøelxedIm,Isg;bnÞat;TaMgBIr 1
D1 : y x 2
D2 : y 4 x
2
x 0 2 y 2 0 x 0 4
o
y 4 0
D1 : y x 2
D2 : y 4 x
x> rkkUGredaencMNucRbsBV I énbnÞat;TaMgBIr eyIgman D : y x 2 nig D : y 4 x edaypÞwmsmIkarGab;sIusénbnÞat;TaMgBIr eK)an x 2 4 x 2x 6 x 3 cMeBaH x 3 enaH y x 2 3 2 1 dUcenH cMNucRbsBVrvagbnÞat;TaMgBIr I 3,1 . 1
2
II. FrNImaRt
tambRmab;RbFaneyIgsg;rbU )an ³ C
H
A
E
O
B
F
D
1> R)ab;eQµaHénRtIekaN CED edayGgát;p©it AB nig CD EkgKña ehIy E CacMNucmYyén OA enaH E sßitenA elIemdüaT½r AB énGgát; CD naM[ EC ED dUcenH RtIekaN CED CaRtIekaNsm)at .
ED // CH ED DF
CH DF
ehIy FH CD eRBaH F nig H enAelI AB naM[ CDF man DH , CH , FH Cakm
9
403
9
sm½yRbLg ³ 13 kkáda 1995 viBaØasa ³ KNitviTüa ¬elIkTI1¦ ry³eBl ³ 60 naTI BinÞú ³ 10 I. BICKNit
1> edaHRsaysmIkar ³ 1 x2x 2 xx 3 . 2> edaHRsayvismIkar ³ 2 1 3x 2x 3 rYcbkRsaycemøIyenAelIG½kS . 3> RsaybBa¢ak;fa ³ x 5 3 xx 29 2xx16 x 2x4 x937 . 2
2
4> sRmYlRbPaK ³ A x
2
2
2
2x 6 x 2 2x 2 x2 4 x 2 3x 2 F x2 1
. 2
5> KNnatémøelxénRbPaK ³ cMeBaH x 1 ; x 3 . 6> enAkñúgtRmúyGrtUNrem xoy sg;bnÞat; D : y 1 2x ; D : y 2x 3 rYcrkkUGredaenéncMNucRbsBVrvagbnÞat;TaMgBIr . 1
1
2
2
II. FrNImaRt
1> tamkMBUlénRtIekaNsm)at DEF eKKUsbnÞat;mYyEdlkat;)at EF Rtg; G nigkat;rgVg; C carwkeRkARtIekaN DEF Rtg; H . eRbóbeFobRtIekaN DEG nigRtIekaN DEH rYcbgðajfa DE DG DH . 2> eK[RbelLÚRkam MNPQ EdlmMu Nˆ 120 nigknøHbnÞat;BuHmMu Mˆ kat; NP Rtg;cMNuckNþal I . k> kMNt;ragRtIekaN NMI rYceRbóbeFob MN nig NP . x>tag J CacMeNalEkgén I elI MQ . KNnargVas;mMuénRtIekaN MIJ . K>tag I CacMNucqøúHén I eFobnwg MQ . kMNt;ragRtIekaN MII rYceRbóbeFob IJ nig MI . 2
o
8 404
9
cemøIy I. BICKNit
1> edaHRsaysmIkar ³ eyIgman 1 x2x 2 xx 3 smIkarmann½yluHRtaEt x 0 eyIg)an 1 x2x 2 xx 3 1 2x 2x 3 ¬lubPaKEbgecal¦
10 x 30 2 x 4 x 2 4 x 3 2 x2 9
x 2 4 x 37 2 x2 9
5 x2 x 1 x 2 4 x 37 2 x 3 x 9 2x 6 2 x2 9
x
2
2
2
2
dUcenH vismIkarmancemøIy x 4 . -bkRsayelIG½kS x 4 0 cMeBaH kmµviFIcas; EpñkminqUtCacemøIyénvismIkar . 0
x
2
2
2
x
2
2
dUcenH cMeBaH x 1 enaH F 0 . -cMeBaH x 3 eK)an F 3 3 3 2 5 43 3
5 x2 x 1 x 2 4 x 37 x 3 x 2 9 2x 6 2 x2 9
.
Edl x 4 0 x 4 x 2 dUcenH sRmÜl)an A 8x 1 . 5> KNnatémøelxénRbPaK F cMeBaHtémø x ³ eyIgman F x x 3x1 2 -cMeBaH x 1 eK)an F 1 1 311 2 02 0
x 4
cMeBaH kmµviFfI µI EpñkKUsDit CacemøIyénvismIkar . 3> RsaybBa¢ak;fa ³
2x 6 x 2 2x 2 x2 4 x 2 2 x 6 x 2 2 x 2x 2 2 x 6 x 2 2 x 2 x2 4 4 x 42 x 2 8 x2 4 4 x 1 2x 2 4 x2 4 8 x 1 A
4
4> sRmYlRbPaK ³
1 3x 2 x 3
2
BinitüGgÁTI1 ³ ¬erobcM[dUcGgÁTI2¦
2
3 1
5 x2 x 1 2 x 3 x 9 2x 6 x 1x 3 2 5 x 3 2x 2 2 2x 3x 3 2 x 9 2x 3x 3
dUcenH
dUcenH smIkarmanb£s x 1 . 2> edaHRsayvismIkar ³ eyIgman 2 1 3x 2x 3
10 x 30 2x 4 x 2 4x 3 2 x2 9 2 x2 9 2 x2 9
eXIjfalT§plénGgÁTI1 dUcGgÁTI2CaR)akd
1 2x 2x 3 2 x 2 x 3 1 4 x 4 x 1
x
dUcenH cMeBaH x
405
3
enaH
F
53 3 4
.
9
6> sg;bnÞat; D nig D kñúgtRmúy xoy eyIgeRbItaragtémøelxedIm,Isg;bnÞat; 1
D1 : y1 1 2 x
DE DG -vi)ak DEG DHE DH DE Taj)anBIsmamaRt DE DG DH dUcenH eyIgbgðaj)anfa DE DG DH . 2> tambRmab;RbFaneyIgKUsrUb)an ³
2
:
D2 : y 2 2 x 3 :
2
x 0 1 y 1 1 x 1 2
2
y 1 1
D2 : y2 2 x 3
N
I
// o
P
//
120
M
Q
J
D1 : y1 1 2 x
I
k-kMNt;ragRtIekaN NMI -rkkUGredaenéncMNucRbsBVrvagbnÞat;TaMgBIr ³ eday D : y 1 2x nig D : y 2x 3 bRmab; Nˆ 120 enaH NMˆ Q 60 ¬eRBaH plbUkmMuCab;RCugEtmYyénRbelLÚRkamesµI 180 ¦ eyIgpÞwmsmIkarGab;sIusénbnÞat;TaMgBIr ³ 1 2x 2x 3 ehIy MI BuHmMu NMˆ Q 60 4x 4 x 1 naM[)an NMˆ I IMˆ J 30 cMeBaH x 1 enaH y 1 2x 1 2 1 1 edaykñúg NMI man Nˆ 120 nig NMˆ I 30 dUcenH kUGredaenéncMNucRbsBVKW 1 , 1 . naM[ NIˆM 30 ¬plbUkmMukñúgén esµI 180 ¦ II. FrNImaRt dUcenH RtIekaN NMI CaRtIekaNsm)at . 1> tambRmab;RbFaneyIgKUsrUb)an ³ vi)ak MN IN D ehIy I CacMNuckNþalén NP enaH IN 12 NP o
o
1
1
2
2
o
o
o
o
o
o
E
G
o
dUcenH eyIgeRbóbeFob)an MN 12 NP .
F
H
eRbóbeFobRtIekaN DEG nigRtIekaN DEH edayRtIekaN DEG nigRtIekaN DEH man ³ -mMu EDG EDH ¬mMurYm¦ -mMu DEG DHE eRBaH mMu DEG DFE ¬mMu)atén sm)at¦ Et DFE DHE ¬mMumanFñÚsáat;rYm DE ¦ dUcenH DEG DEH tamlkçxNÐ m>m .
x- KNnargVas;mMuénRtIekaN MIJ eday J CacMeNalEkgén I elI MQ naM[ IJˆM 90 ehIy IMˆ J 30 ¬sRmaybBa¢ak;xagelI¦ eK)an MIˆJ 60 ¬plbUkmMukñúgén esµI 180 ¦ dUcenH rgVas;mMuénRtIekaN MIJ KW ³ IJˆM 90 , IMˆ J 30 , MIˆJ 60 . o
o
o
o
o
406
o
o
9
K- kMNt;ragRtIekaN MII eday J CacMeNalEkgén I elI MQ naM[ II MJ Rtg; J ehIy I CacMNucqøúHén I eFobnwg MQ naM[ JI JI eK)an M CacMNucenAelIemdüaT½r MJ enaHnaM[ MI MI MI -edaykñúg MII man MI MIˆI MIˆJ 60
o
mann½yfa vaCaRtIekaNsm)atmanmMumYyesµI 60 dUcenH RtIekaN MII CaRtIekaNsm½gS . vi)ak MI II ehIy JI JI mann½yfa J kNþalGgát; II enaH IJ 12 II b¤ IJ 12 MI
o
dUcenH eRbóbeFob)an IJ 12 MI .
407
9
sm½yRbLg ³ 22 sIha 1995 viBaØasa ³ KNitviTüa ¬elIkTI2¦ ry³eBl ³ 60 naTI BinÞú ³ 10 I. BICKNit
1> eK[ ³ A x 1 3x 2 x 1 3x 2 . k> dak;kenSam A CaplKuNktþadWeRkTI 1 én x . x> sRmYlkenSam B 2x A 2 . K> KNnatémø x kalNa B 3 . 2> edaHRsayvismIkar ³ 5x5 4 6x 2 4 x 3 5 . rYcbkRsaytamRkaPic . 3> k> edaHRsayRbB½n§smIkartamRkaPic ³ 33xx 2y y7 0 0 . x> rYcepÞógpÞat;cemøIytamkarKNna . 2
2
2
2
II. FrNImaRt
eK[RtIekaNEkgsm)at ABC EkgRtg; A ehIy AB AC a . DAC k¾CaRtIekaNEkg sm)at EkgRtg; D . RtIekaNTaMgBIrenHsßitenAsgxagRCugrYm AC . 1> kMNt;RbePTctuekaN ABCD . 2> KNna BC ; AD ; BD CaGnuKmn_én a . 3> Ggát;RTUg AC nig BD RbsBVKñaRtg; O . KNna OA , OC , OB , OD CaGnuKmn_én a . 4> bgðajfa AOD nig BOC CaRtIekaNdUcKña . rYcKNnapleFobdMNcU .
8
408
9
cemøIy I. BICKNit
2> edaHRsayvismIkar ³ eyIgman 5x5 4 6x 2 4 x 3 5 eyIgtRmÚvPaKEbgrYm 15 rYclubPaKEbgrYmecal
1> man A x 1 3x 2 x 1 3x 2 k> dak;kenSam A CaplKuNktþadWeRkTI 1 én x 2
2
2
A x 1 3x 2 x 1 3x 2 2
2
2
x 1 2x 13x 2 3x 2 2
15x 12 90 x 30 60 5 x 25 75x 42 5 x 85 70 x 43 43 x 70 43 x 70
2
x 1 3x 2 2x 13x 2 2
2
dUcenH A 2x 13x 2 CaplKuNktþa . x> sRmYlkenSam B 2x A 2 eday A 2x 13x 2 eK)an B 2x2x132x 2
dUcenH vismIkarmancemøyI - bkRsaytamRkaPic ³
2
2x 13 x 2 2 x2 1 2x 13 x 2 2x 1x 1 3 x 2 , x 1 0 x 1 x 1 3x 2 B x 1
x
43 70
3 x 3 3 x 2 3x 3x 2 3
eyIgsg;RkaPicdUcrUbxageRkam ³
3 3 x 2 3
x x
2 3 3 3
x
43 70
3 x 2 y 0 3 x y 7 0
2 3 x 3 3
0
x 0 cMeBaH kmµviFIfµI EpñkKUsDitCacemøIyénvismIkar . 3> k> edaHRsayRbB½n§smIkartamRkaPic ³ eyIgman ³ 33xx 2y y7 0 0 eRbItaragtémøelxedIm,Isg;RkaPicénbnÞat;TaMgBIr
dUcenH smRmÜl)an . K> KNnatémø x kalNa B 3 smIkar B mann½ykalNa x 1 0 b¤ x 1 eK)an 3 3xx1 2
cMeBaH kmµviFIcas; EpñkminqUtCacemøIyénvismIkar
2
x
.
x 0
2
y 0 3 2 1
x y
1
4
3x 2 y 0 7 3
3 3 3 3
3x y 7 0
2 3 633 3 95 3 39 6
dUcenH KNna)antémø x 9 65
3
14 9
tamRkaPic dUcenH bnÞat;TaMgBIrRbsBVKñaRtg;
. 409
14 7 , 9 3
.
9
x> epÞógpÞat;cemøIytamkarKNna ³ eyIgman 33xx 2y y7 0 0 b¤ 33xx 2y y07 edaydkGgÁnigGgÁ énsmIkarTaMgBIr eK)an cMeBaH
2> KNna BC ; AD ; BD CaGnuKmn_én a eday ABC CaRtIekaNEkgsm)at EkgRtg; A man AB AC a tamRTwsþIbTBItaK½r eK)an³ BC 2 AB2 AC 2
3 x 2 y 0 7 3 x y 7 y 3 3y 7 7 3x y 7 y 3 7 3x 7 3 14 14 3x x 3 9
BC AB2 AC 2
naM[
a 2 a 2 2a 2 a 2
enaH
2
dUcenH x 149 , y 73 dUcKNnatamRkaPic . II. FrNImaRt tambRmab;RbFaneyIgsg;rbU )andUcxageRkam ³ B
2
2
/
AC 2 2AD2
Taj)an
AD 2
AC 2 AC AD 2 2
Et AC a naM[
a a 2 2 2
AD
dUcenH KNna)an AD a 2 2 . kñúgRtIekaNEkg BCD EkgRtg; C ¬eRBaH BCD man DCˆB 90 bBa¢ak;xagelI¦ tamRTwsþIbTBItaK½r eK)an ³
O
C
A
dUcenH KNna)an BC a 2 . eday DAC CaRtIekaNEkgsm)at EkgRtg; D naM[ AD CD tamRTwsþIbTBItaK½r AC AD CD
o
D
1> kMNt;RbePTctuekaN ABCD eday ABC CaRtIekaNEkgsm)at EkgRtg; A naM[ mMu)at ABˆ C ACˆB 45 ehIy DAC CaRtIekaNEkgsm)at EkgRtg; D naM[ mMu)at DAˆ C DCˆA 45 eK)an DCˆB DCˆA ACˆB 45 45 90 naM[ BC CD Rtg; C -kñúgctuekaN ABCD manmMuEkg DCˆB 90 AD CD nig BC AD // BC CD dUcenH ABCD CactuekaNBñayEkg .
BD 2 BC 2 CD 2 BD BC 2 CD 2
o
BD
BD 2a 2
o
o
o
a a 2 2
o
2
2
a2 5a 2 10a 2 a 10 2 2 4 2
dUcenH KNna)an
BD
a 10 2
.
3> KNna OA , OC , OB , OD CaGnuKmn_én a eday AC nig BD RbsBVKñaRtg; O ehIy AD// BC ¬sRmayxagelI¦
o
410
9
tamRTwsþIbTtaEls eyIg)anGgát;smamaRtKña
vi)ak
OA OD AD OC OB CB
Et
tamlkçN³smamaRt eK)an ³ OA OC OA OC AC AD CB AD CB AD CB a a a 2 3 a 2 a 2 2a 2 3a 2 a 2 2 2 2 a 2 2 OA 2 2 AD 2 a OA AD 3 3 3 3 OC 2 2 BC 2 a 2 2a OC CB 3 3 3 3
OAD
OCB a OA a 3 1 3 OC 2a 3 2a 2 3 OA OD AD 1 OC OB CB 2
naM[
dUcenH pleFobdMNUcKNna)anKW 12 .
naM[ ehIy
rUbdEdl RKan;EtRsYsemIlb:ueNÑaH ¡¡¡¡ B
dUcenH KNna)an OA a3 nig OC 2a3 . -kñúgRtIekaNEkg ABO EkgRtg; A tamRTwsþIbTBItaK½r OB AB OA 2
OB AB OA 2
2
O
2
D
2
2
a2 10a 2 a 10 a a a2 9 9 3 3
- eday OD BD OB a 210 a 310 3a 10 2a 10 6 a 10 6
OB
a 10 3
nig OD a 610 .
4> bgðajfa AOD nig BOC CaRtIekaNdUcKña eday AOD nig BOC man³ -mMu OAˆ D OCˆB 45 ¬mMu)atRtIekaNsm)at¦ -mMu AOˆ D COˆ B ¬mMuTl;kMBUl¦ o
dUcenH
AOD
BOC
C
A
2
dUcenH
OA OD AD OC OB CB
tamlkçxNÐ m>m . 411
9
sm½yRbLg ³ 01 kkáda 1996 viBaØasa ³ KNitviTüa ¬elIkTI1¦ ry³eBl ³ 60 naTI BinÞú ³ 10 I. BICKNit
1> KNnakenSam E 3 2 5 . 2> sresrBhuFaxageRkamCaplKuNktþa ³ k> x 18x 18 x> 4x 25 . 3> eK[smamaRt ba dc bgðajfa ac 22ca 33db . 4> eK[bnÞat;BIr EdlmansmIkar D : y x 5 nig D : 2 y 3x . k> sg;bnÞat; D nig D kñúgtRmúyGrtUNrem xoy . x> R)ab;emKuNR)ab;TisénbnÞat;nImYy² . K> KNnakUGredaencMNucRbsBV P rvag D nig D . 5> edaHRsaysmIkar 2 x3 7 x 1 . II. FrNImaRt eK[RtIekaN ABC EkgRtg; A . 1> I CacMNuckNþalén BC . R)ab;eQµaHRtIekaN IAC . 2> O CacMNuckNþalén AB . R)ab;eQµaHénctuekaN OICA . 3> KNna BC ebI AB 3cm ; AC 4cm . 4> rgVg;Ggát;p©it AB kat;rgVg;Ggát;p©it AC Rtg;cMNucmYyeTot M . k> bBa¢ak;fa M sßitenAelI BC . x> bgðajfa AB BM BC . 3
4
2
2
2
2
8 412
9
cemøIy I. BICKNit
-eyIgsg; D nig
1> KNnakenSam E ³ eyIgman E 3 2 3
4
D )andUcrUbxageRkam ³ D : 2 y 3 x
5
2
27 16 25 36
D : y x 5
dUcenH KNna)an E 36 . 2> sresrBhuFaxageRkamCaplKuNktþa ³ k> x 18x 81 x 2 x 9 9 2
2
x> R)ab;emKuNR)ab;TisénbnÞat;nImYy² bnÞat; D : y x 5 b¤ D : y x 5 dUcenH emKuNR)ab;TisénbnÞat; D KW a 1
2
x 2 2 x 9 92 x 9 x 9x 9 2
dUcenH x> 4 x
2
2
4 x 2 25 2 x 52 x 5
/ .
3 xx5 2 3 1 x 5 2 5 x5 2
a b 2a 3b 2a 3b c d 2c 3d 2c 3d a 2a 3b c 2c 3d
Taj)an
a 2a 3b c 2c 3d
³
x
D : y 3 x 2
x2
cMeBaH x 2 enaH y x 5 2 5 3 dUcenH kUGredaencMNucRbsBV P2 , 3 .
.
4> k> sg;bnÞat; D nig D kñúgtRmúy xoy eyIgeRbItaragtémøelxedIm,Isg;bnÞat;TaMgenH x 1 2 D : y x 5 ³ y 4 3 D : 2 y 3 x
b¤
K> KNnakUGredaencMNucRbsBV P eyIgman D : y x 5 nig D : y 32 x edaypÞwmsmIkarGab;sIusénbnÞat;TaMgBIr ³ eyIg)an x 5 32 x
3> bgðajfa ac 22ca 33db eyIgmansmamaRt ba dc enaH ac db tamlkçN³smamaRt eyIgGacsesr)an ³
dUcenH eyIgbgðaj)anfa
D : 2 y 3 x
dUcenH emKuNR)ab;TisénbnÞat; D KW a 32 .
25 2 x 5 2
2 x 52 x 5
dUcenH
bnÞat;
.
x 18 x 81 x 9 x 9 2
5> edaHRsaysmIkar 2 x3 7 x 1 smIkar enHmanGBaØat x enAeRkamr:aDIkal; smIkarmann½yluHRtaEt x 0 eK)an ³
0 2
y 0 3
413
9
3> KNna BC ebI AB 3cm ; AC 4cm eday RtIekaN ABC CaRtIekaNEkgRtg; A tamRTwsþIbTBItaK½r eK)an ³
2 x 7 x 1 3 2 x 7 3 x 3 3 x 2 x 7 3
BC 2 AB2 AC 2
x 4 0
BC AB2 AC 2
dUcenH smIkarKµanb£s .
32 4 2 25 5 cm
II. FrNImaRt
tambRmab;RbFaneyIgsg;rbU )an ³ B // M 3 cmO
I
o
//
A
dUcenH KNna)an BC 5 cm . 4> k> bBa¢ak;fa M sßitenAelI BC -edayrgVg;Ggát;p©it AB man M enAelIrgVg; naM[ mMu AMB 90 ¬CamMucarwkknøHrgVg;¦ -edayrgVg;Ggát;p©it AC man M enAelIrgVg; naM[ mMu AMC 90 ¬CamMucarwkknøHrgVg;¦ -edaymMu BMˆ C AMˆ B AMˆ C 90 90 naM[ BMˆ C 180 CamMurab mann½yfa M sßitenAelI BC dUcenH eyIgbBa¢ak;)anfa M sßitenAelI BC . x> bgðajfa AB BM BC eday AMB 90 naM[ BM CacMeNalEkgénGgát; ABelI BC tamTMnak;TMngkñúgRtIekaNEkg ABC eK)an ³ AB BM BC / dUcenH eyIgbgðaj)anfa AB BM BC .
4 cm
C
o
o
1> R)ab;eQµaHRtIekaN IAC ³ -eday I CacMNuckNþalén BC enaH AI Ca emdüan én Ekg ABC RtUvnwgGIub:Uetnus BC tamRTwsþI eK)an IA IB IC -edayRtIekaN IAC man IA IC dUcenH RtIekaN IAC CaRtIekaNsm)at . 2> R)ab;eQµaHénctuekaN OICA ³ -eday I CacMNuckNþalén BC nig O CacMNuckNþalén AB naM[ OI KWCa)atmFüménRtIekaNEkg ABC EkgRtg; A vi)ak AC// OI -eXIjfa ctuekaN OICA man AC// OI nigmMu BAˆ C CamMuEkg dUcenH ctuekaN OICA CactuekaNBñayEkg . 414
o
2
o
2
2
o
9
415
9
sm½yRbLg ³ 19 sIha 1996 viBaØasa ³ KNitviTüa ¬elIkTI2¦ ry³eBl ³ 60 naTI BinÞú ³ 10 I. BICKNit
1> BnøatplKuN ³ 2x 3x 2 . 2> sresrkenSamxageRkamCaplKuNktþa ³ k> 5x 2 42 x x> 2 x 5 4 x 10 x 2 4 x 25 . 3> KNna 20 80 245 . 4> k> edaHRsaysmIkar 1x x 2 1 x 3 x . x> edaHRsayvismIkar 42 x 3 2x 1 4 . 5> edaHRsayRbB½n§smIkar 2yx 2 yx 6 . II. FrNImaRt eK[rgVg;p©it O manGgát;p©it AB nigcMNuc P enAeRkArgVg;enH . bnÞat; PA nig PB kat;rgVg; O erogKñaRtg;cMNuc M nig N enAmçagénbnÞat; AB . 1> RsaybMPøWfa AN nig BM Cakm H CacMNucRbsBVrvag AN nig BM . RsaybMPøWfa PH AB . 3> bgðajfactuekaN HMPN carwkkñúgrgVg;mYy . 4> RsaybBa¢ak;fa AN BP BM AP . 5> cMNuc I cl½tenAelIFñÚtUc AM ehIy E CacMNuckNþalén AI . rksMNMcu MNuc E kalNa I ERbRbYl . 2
2
2
2
2
8
416
9
cemøIy I. BICKNit
eyIg)an
1> BnøatplKuN ³ eyIgman 2 x 3x 2 2 x
2
4 x 3x 6
2x 2 x 6
dUcenH Bnøat)an 2x 3x 2 2x x 6 2> sresrkenSamxageRkamCaplKuNktþa ³ k> 5x 2 42 x 2
2
/ .
dUcenH smIkarmanb£s x 2 . 5> edaHRsayRbB½n§smIkar ³ eyIgman 2yx 2 yx 6 smmUl xx yy 32
2
5 x 2 22 x 5 x 2 22 x 5 x 2 22 x 5 x 2 4 2 x 5 x 2 4 2 x 7 x 6 3 x 2 2
2
2
2
dUcenH RbB½n§smIkarKµanKUcemøIy .
2
2
2
II. FrNImaRt
2
2 x 5 22 x 5x 2 2 x 52 x 5 2 x 52 x 5 2 x 4 2 x 5 2 x 52 x 14 22 x 5x 7 2
dUcenH
2 x 5 4 x 10 x 2 4 x 2
2
3> KNna
tambRmab;RbFaneyIgsg;rbU )an ³ P
I M E
25
20 80 245
A
H
O
B
1> bMPøWfa AN nig BM Cakm
4 5 16 5 49 5 2 5 4 5 7 5 5 5
dUcenH KNna)an 20 80 245 5 5 . 4> k> edaHRsaysmIkar 1x x 2 1 x 3 x
o
2
smIkarmann½yluHRtaEt
N
E
.
22 x 5x 7
a b c a b c
RbB½n§smIkarmanpleFobemKuN KW 11 11 32 mann½yfa vaKµanKUcemøIy
dUcenH 5x 2 42 x 7 x 63x 2 x> 2 x 5 4 x 10 x 2 4 x 25 2 x 5 22 x 5x 2 2 x 5 2
1 2 3 2 x x 1 x x 1 2 3 x x 1 x x 1 x 1 2x 3 x2 x 2
x 0 x 0 x 1 0 x 1 x 2 x 0
417
o
9
2> RsaybMPøWfa PH AB bRmab; H CacMNucRbsBVrvag AN nig BM mann½yfa H sßitenAelIkm bgðajfactuekaN HMPN carwkkñúgrgVg;mYy tamsRmayxagelI ANˆB 90 nig AMˆ B 90 naM[ ANˆP 90 ¬mMubEnßmCamYymMu ANˆB 90 ¦ PMˆ B 90 ¬mMubEnßmCamYymMu AMˆ B 90 ¦ edayctuekaN HMPN manmMuQm ³ -mMu HNˆP ANˆP 90 -mMu PMˆ H PMˆ B 90 naM[plbUkmMQ u m HNˆP PMˆ H 90 90 180 dUcenH ctuekaN HMPN carwkkñúgrgVg;mYy . 4> RsaybBa¢ak;fa AN BP BM AP kñúgRtIekaN PAB man ³ -km
5> rksMNMucN M uc E kalNa I ERbRbYl eday E CacMNuckNþalén AI naM[ EA EI eyIg)an OE AI ¬eRBaH Ggát;FÚñEkgnwgkaMrgVg;Rtg;cMNuckNþal¦ edaycMNuc A , O CacMNucnwg enaHnaM[mMu AEˆO 90 efr edIm,I[ AEˆO 90 efr luHRtaEtcMNuc E RtUv sßitenAelIknøHrgVg;Ggát;p©it AO -ebI I RtYtelI M enaH E RtYtelI E Edl E CacMNuckNþalén AM -ebI I RtYtelI A enaH E k¾RtYtelI A Edr dUcenH sMNMucMNuc E Ca AE énrgVg; Ggát;p©it AO . o
o
o
o
o
o
o
o
o
o
o
o
o
PAB
PAB
1 1 AN BP BM AP 2 2
AN BP BM AP
dUcenH Rsay)anfa
AN BP BM AP
. 418
9
419
9
RksYgGb;rM yuvCn nigkILa RbLgsBaØabRtmFümsikSabzmPUmi cMeNHTUeTA nigbMeBjviC¢a eQµaH nightßelxaGnurkS sm½yRbLg ³ >>>> >>>>>>>>>> 1997
elxbnÞb; ³ >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> elxtu ³ >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> mNÐlRbLg ³ >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
1> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> namRtkUl nignamxøÜn ³ >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 2> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> éf¶ExqñaMkMeNIt ³ >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> GkSrsm¶at; htßelxa ³ >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> ebkçCnminRtUveFVIsBaØasmÁal;Gm VI YyelIsnøwkRbLgeLIy. snøwkRbLgNaEdlmansBaØasmÁal;RtUv)anBinÞúsUnü . --------------------------------------------------------------------------------------------------------------------------
viBaØasa ³ KNitviTüa elIkTI1 ry³eBl ³ 60 naTI BinÞú ³ 100 esckþIENnaM ³ GkSrsm¶at; 1> ebkçCnRtUvbt;RkdasenHCaBIr rYcKUsExVgEpñkxagelIénTMB½rTI2 [b:unRbGb;EpñkxagelI énTMB½rTI1 EdlRtUvkat;ecal. hamsresrcemøIyelIkEnøgKUsExVgenaH . 2> ebkçCnRtUvKUsbnÞat;bBaÄr[cMBak;kNþalTMB½rTI2 nigTMB½rTI3 sRmab;sresrcemøIybnþ.
RbeTsmanPaBvwkvr mankar)aj;KñaenAraCFanIPñMeBj kalBIéf¶TI 05 nig 06 Ex kkáda qñaM 1997 . dUcenH kñúgqñaM 1997 enHBMmu ankarRbLgeLIy .
420
9
cemøIy
421
9
sm½yRbLg ³ 24 sIha 1998 viBaØasa ³ KNitviTüa ¬elIkTI1¦ ry³eBl ³ 60 naTI BinÞú ³ 100
RsaybMPøWfa E 3 2 1 2 1 2 1 CacMnYnKt; . eK[KUbBIrRbePTcMnYnsrub 11. KUbmYyRbePTmanRTnugesµI 3 cm nigKUbmYyRbePTeTotmanRTnug esµI 5 cm . eKmindak;KUbRtYtKñaeT . eKtMerobKUbTaMg 11 CamYyCYrEdlmanRbEvg 47 cm . rkcMnYnKUbénRbePTnImYy² . eK[tRmúyGrtUNrem xoy cUrsg;bnÞat;EdlmansmIkar y 2x 1 nig y 2x 2 . rYcKNna cMNucRbsBVrvagkUGredaenéncMNuc EdlCaRbsBVrvagbnÞat;TaMgBIr . eK[knøHrgVg;p©it O Ggát;p©it AB. cMNuc M cl½tenAelIknøHrgVg;enH . 1> R)ab;RbePTRtIekaN AMB . 2> KNna AM ebI OA 2cm , BM 7cm . 3> rksMNMucMNuc P CacMNuckNþalén AM . 2
I. II.
III.
IV.
8 cemøIy RsaybBa¢ak;fa E CacMnYnKt; ³ eyIgman E 3 2 1 2 1 2 1 2 13 2 1 2 1 2 12 2 2 2 2 1 2 1 22 1 2 dUcenH E 2 CacMnYnKt; . II. rkcMnYnKUbénRbePTnImy Y ²³ tag x CacMnYnKUbEdlmanRbEvgRTnugesµI 3 cm y CacMnYnKUbEdlmanRbEvgRTnugesµI 5 cm tambRmab;RbFaneyIg)anRbB½n§smIkar ³ x y 11 eyIgedaHRsaytamedETmINg; ³ 3x 5 y 47
naM[
I.
D 53 2 Dx 8 4 D 2 D y 14 D y 47 33 14 y 7 D 2
2
Dx 55 47 8 x
dUcenH KUbmanRbEvgRTnugesµI 3cm mancMnYn 4 KUbmanRbEvgRTnugesµI 5cm mancMnYn 7 . III. sg;bnÞat;kñúgtRmúyGrtUNrem xoy eyIgeRbItaragtémøelxedIm,Isg;bnÞat;TaMgenH -cMeBaH y 2x 1 ³ xy 10 22 -cMeBaH y 2x 2 ³
422
x
0 2
y 2 1
9
-eyIgsg;bnÞat;TaMgBIr)andUcxageRkam ³ y
2> KNna AM ebI OA 2cm , BM eday AMB CaRtIekaNEkg Rtg; M tamRTwsþIbTBItaK½r
x 1 2
7cm
AB2 AM 2 BM 2 AM 2 AB2 BM 2 AM AB2 BM 2
x y 2 2
-KNnakUGredaenéncMnucRbsBVrvagbnÞat;TaMgBIr eyIgman y 2x 1 nig y 2x 2 edaypÞwmsmIkarGab;sIus enaHeK)an ³
dUcenH cMNcu RbsBVrvagbnÞat;TaMgBIrKW mankUGredaen x 1 , y 2 .
o
M P
O
B
1> R)ab;RbePTRtIekaN AMB edayRtIekaN AMB man AB CaGgát;p©it nig M CacMNuccl½telIknøHrgVg; enaH AMB Ca RtIekaNcarwkknøHrgVg; Edl AMˆ B efr 90 dUcenH AMB CaRtIekaNEkg .
2 22
o
tambRmab;RbFaneyIgsg;rUb)an ³
AM
7
, AB 2OA
2
dUcenH KNna)an AM 3 cm . 3> rksMNMucN M uc P CacMNuckNþalén AM eday P CacMNuckNþalén AM naM[ PA PM eyIg)an OP AM ¬eRBaH Ggát;FÚñEkgnwgkaMrgVg;Rtg;cMNuckNþal¦ edaycMNuc A , O , B CacMNucnwg enaHnaM[mMu APˆ O 90 efr edIm,I[ APˆ O 90 efr luHRtaEtcMNuc P RtUv sßitenAelIknøHrgVg;Ggát;p©it AO -ebI M RtYtelI B enaH P RtYtelI O -ebI M RtYtelI A enaH P RtYtelI A Edr dUcenH sMNMucMNuc P CaknøHrgVg;Ggát;p©it AO .
cMeBaH x 1 enaH y 2x 1 12 1 2
A
2OA2 BM 2
AM 16 7 9 3 cm
x x 1 2 2 2 x x 2 1 2 2 x 1
IV.
AM
o
423
9
sm½yRbLg ³ 09 tula 1998 viBaØasa ³ KNitviTüa ¬elIkTI2¦ ry³eBl ³ 60 naTI BinÞú ³ 100 I.
II.
eKEckR)ak;[mnusS 3 nak;. GñkTImYy)an 40% énR)ak;srub ehIyGñkTIBrI TTYl)an 15 énR)ak;srub ÉGñkTIbITTYl)anR)ak; 12 000 erol . rkcMENkR)ak;rbs;GñkTImyY nigcMENkR)ak;rbs;GñkTIBIr . bnÞat;b:HrgVg;p©it O Rtg; M kat;bnøayénGgát;p©it CD Rtg; P . bgðajfaRtIekaN PMC nig RtIekaN PDM dUcKña rYcTajbBa¢ak;fa PM PC PD . cUrsresrBakü {xus} b¤ {RtUv} kñúgRbGb;enAxagmuxGMNHGMNagnImYy²xageRkamenH ³ 2
III.
□ □
a 2 b 2 2ab 2 1
1 2 1
□ bnÞat; y 2x 1 nig
IV.
RbsBVKñaRtg;mYycMNcu . cUrKUssBaØa kñúgRbGb;enAxagmuxcemøIyEdlRtwmRtUvmanEtmYyKt; ³ 1> etIkenSamNaEdlesµInwg x 2 ³ k> □ x 2x 4 x> □ x 4x 4 K> □ x 4x 2 X> □ x 4x 4 2> etIcMnYnNa Cab£sénsmIkar ³ x x 2 x 0 . k> □ 0 x> □ 2 K> □ 2 X> □ 0 nig 2 . ABC CaRtIekaNEkgRtg; A ehIy x 0 . eKdwgfa BC 3x 2 nig AB 2 x 1 . KNna AC CaGnuKmn_én x . eK[cMNuc A1 , 2 nig B3 , 1 . rksmIkarbnÞat; AB . y 2x 2
2
2
2
2
2
2
V.
2
VI.
8 424
9
cemøIy I.
rkcMENkR)ak;rbs;GñkTImYy nigGñkTIBIr ³ tag x CacMnYnR)ak;srubTaMgGs; naM[ R)ak;rbs;GñkTI1 KW 40% x R)ak;rbs;GñkTI2 KW 15 x R)ak;srubCaplbUkR)ak;GñkTaMgbI eyIg)an ³ 1 x 40% x x 12000 5 5 x 200% x x 60000 5 x 2 x x 60000 2 x 60000
PM PC vi)ak PMC PDM PD PM Taj)an PM PM PC PD b¤ PM PC PD dUcenH TajbBa¢ak;)anfa PM PC PD . III. sresrBakü {xus} b¤ {RtUv} kñúgRbGb; ³ RtUv a b 2ab eRBaH 2
2
2
a 2 b 2 2ab a 2 2ab b 2 0
x 30000
a b2 0
naM[ -cMENkR)ak;rbs;GñkTI1 KW 40 40% x 30000 12 000 100
erol
RtUv
-cMENkR)ak;rbs;GñkTI2 KW 1 1 x 30000 6 000 5 5
2
BitcMeBaHRKb;cMnYnBit a nig b . 1 2 1 eRBaH 2 1 2 1
erol
dUcenH cMENkR)ak;GñkTI1 KW 12 000 erol cMENkR)ak;GñkTI2 KW 6 000 erol . II. tambRmab;RbFaneyIgKUsrUb)andUcxageRkam³
2 1
1 2 1
2 1 1 2
2 11 1
11
Bit
bnÞat; y 2x 1 nig y 2x 2 RbsBVKñaRtg;mYycMNuc . eRBaH M bnÞat;TaMgBIrmanemKuNR)ab;TisesµI 2 dUcKña naM[ bnÞat;TaMgBIrRsbKña KµancMNucRbsBVmYy . C P D O IV. KUssBaØa kñúgRbGb;muxcemøIyEdlRtwmRtUv ³ bgðajfa PMC nig PDM dUcKña 1> etIkenSamNaEdlesµInwg x 2 ³ eday PMC nig PDM man ³ x> ☑ x 4x 4 -mMu PMC PDM ¬mMucarwkFñÚsáat;rYm MC ¦ eRBaH x 2 x 2 x 2 2 x 4x 4 -mMu MPC DPM ¬mMu P rYmKña ¦ 2> etIcMnYnNa Cab£sénsmIkar ³ x x 2 x 0 eXIjfa PMC nig PDM manmMuBIrb:unerogKña K> ☑ 2 eRBaH cMeBaH x 2 2 22 44 dUcenH PMC PDM tamlkçxNÐ m>m . 0 0 0 0 Bit 2 2 xus
2
2
2
2
2
2
2
2
425
9
V.
tambRmab;RbFaneyIgKUsrUb ³
C
3x 2
KNna AC CaGnuKmn_én x A 2x 1 B tamRTwsþIbTBItaK½r cMeBaH ABC EkgRtg; A eK)an BC AB AC naM[ AC BC AB 2
2
2
2
2
2
2
AC 2 BC 2 AB 2 3 x 2 2 x 1 3 x 2 2 x 13 x 2 2 x 1 2
2
x 15 x 3 5x 2 8x 3
dUcenH KNna)an V.
.
AC 2 5x 2 8x 3
rksmIkarbnÞat; AB eyIgmancMNuc A1 , 2 nig B3 , 1 smIkarbnÞat;EdlRtUvrkmanrag AB : y ax b -eday AB : y ax b kat;tam A1 , 2 eK)an 2 a 1 b a b 2 1 -eday AB : y ax b kat;tam B3 , 1 eK)an 1 a 3 b 3a b 1 2 -edayyk smIkar 2 1 eK)an ³ 3a b 1 a b 2
1 a naM [ 2 2a 1 1 cMeBaH a 2 enaHtam 1 a b 2 naM[ b 2 a 2 12 2 12 52
dUcenH smIkarbnÞat;
AB : y 1 x 5 2
2
.
426
9
427
9
sm½yRbLg ³ 07 sIha 1999 viBaØasa ³ KNitviTüa ¬elIkTI1¦ ry³eBl ³ 60 naTI BinÞú ³ 100 I.
II.
C cUrsresrBakü {xus} b£ {RtUv} kñúgRbGb;xagmuxGMNHGMNagnImYy²xageRkam ³ 6 1> □ x 6 Cab£sénsmIkar 2x 6 3x 19 . C 2> □ BC 5 ebIeK[ ³ 2 1.5 B A . BC B C , AB 1 , AC 2 , CC 6 , BC 1.5 1 B cUrKUssBaØa kñúgRbGb;enAxagmuxcemøIyEdlRtwmRtUv ³ 1> eK[ A 1 , 1 nig B3 , 4 enaHcm¶ay AB manRbEvg ³ k> □ AB 13 x> □ AB 6 K> □ AB 5 X> □ AB 4 . 2> KNnakenSam E 5 12 4 3 48 KW ³ k> □ E 8 3 x> □ E 9 3 K> □ E 10 3 X> □ E 11 3 . 3> eKRkLúkRKab;LúkLak;BIr rkRbU)abEdlGac[eKRkLúk)anBinÞúsrub 7 BinÞú . k> □ P 365 x> □ P 56 K> □ P 16 X> □ P 13 . sYnbEnømYyragctuekaNEkg manbeNþayelIsTTwg 12 m ehIydIenaHmanépÞRkLa 160 m . rkbrimaRténsYnbEnøenaH . enAkñúgtRmúyGrtUNrem xoy . k> sg;bnÞat; D EdlmansmIkar y 2x 3 . x> sresrsmIkarbnÞat; D Edlkat;tamcMnuc P5 , 1 ehIyRsbnwg D . rYcsg; D kñúg tRmúyxagelI . kñúgrgVg;p©it O Ggát;p©it BC Edl BC 5cm . eKKUsGgát;FñÚ BA Edl BA 3cm . k> R)ab;RbePTRtIekaN ABC . x> KNnaRbEvg AC . 7
7
7
III.
IV.
7
2
1
1
2
V.
428
2
9
K> KNna cos ABˆ C nig sin ABˆ C . X> eKKUsemdüan BM énRtIekaN ABC Edlbnøayrbs;vakat;rgVg; O Rtg; D . bgðajfa MB MD MA . 2
8 cemøIy I.
sresrBakü {xus} b£ {RtUv} kñúgRbGb; ³ 1> xus x 6 eRBaH cMeBaH x 6 eK)an 26 6 3 6 19 0 1 minBit 2> xus BC 5 eRBaHebIeK[ ³ BC BC tamlkçN³bnÞat;RsbeK)an ³
1, 6 ; 2 , 5 ; 3, 4 ; 4 , 3 ; 5 , 2 ; 6 ,1
7
C
6
AC BC AC B C 2 AC BC A B C 1 AC 2 6 1.5 12 6 2 2
II.
C 1.5
2
x x 12 160
B
B
x
x 12 x 160 0
160 m 2
2
x 12
man 6 160 196 naM[ x 6 1 196 6 14 0 minyk 2
KUssBaØa kñúgRbGb;enAxagmuxcemøIyRtwmRtUv ³ 1> enaHcm¶ay AB manRbEvg ³ K> ☑ AB 5 eRBaH eyIgman A 1 , 1 nig B3 , 4 naM[ AB 3 1 4 1 4 3 25 5 . 2> KNnakenSam E 5 12 4 3 48 KW ³ K> ☑ E 10 3 eRBaH E 5 12 4 3 48 10 3 4 3 4 3 10 3 . 3> rkRbU)abEdleKRkLúk)anBinÞúsrub 7 BinÞú ³ K> ☑ P 16 eRBaH RKab;LúkLak;nImYy²man mux 6 dUcKña naM[cMnnY krNIGacKW 6 6 36 ehIyplbUkRKab;LúkLak;)anBinÞú 7 BinÞú rYmman ³ 2
mancMnYn 6 krNI enaHcMnYnkrNIRsb 6 Rsb 6 1 naM[ P krNI . krNIGac 36 6 III. rkbrimaRténsYnbEnøenaH tag x CaRbEvgTTwg Edl x 0 KitCa m naM[ x 12 CaRbEvgbeNþay sYn tambRmab;RbFan eK)an ³
1
x2
6 196 6 14 8 m 1
naM[RbEvgbeNþay x 12 8 12 20 m -ebItag P CabrimaRténsYnbEnøenaH eK)an P 28 20 56 m
2
2
dUcenH brimaRténsYnbEnøenaHKW P 56 m . k> sg;bnÞat; D EdlmansmIkar y 2x 3 eyIgeRbItaragtémøelxedIm,I sg;bnÞat;enH cMeBaH D ³ y 2x 3 xy 03 24 eyIgsg;bnÞat; D kñúgtRmúyGrtUNrem xoy )andUcrUbxageRkam ³
IV.
1
1
7
1
429
9
x> KNnaRbEvg AC eday ABC CaRtIekaNEkgRtg; A tamRTwsþIbTBItaK½r AC BC AB eday BC 5 cm , AB 3 cm naM[ AC 5 3 16 4 cm dUcenH KNna)an AC 4 cm . K> KNna cos ABˆ C nig sin ABˆ C kñúgRtIekaNEkg ABC manRbEvgRCug ³ BC 5 cm , AB 3 cm nig AC 4 cm AB 3 naM[ cos ABˆ C BC 5 AC 4 ehIy sin ABˆ C BC 5
D1 : y x 3 2
2
D2 : y 1 x 3 2
2
2
x> sresrsmIkarbnÞat; D smIkarbnÞat;EdlRtUvsresrKW D : y ax b -eday D : y ax b kat;tam P5 , 1 eK)an 1 a 5 b b 1 5a 1 -eday D //D enaH a a Et D ³ y 2x 3 naM[ a a 12 -edayyk a 12 CMnYskñúg 1 ³ eK)an b 1 5a 1 5 12 2 2 5 32 2
2
2
2
1
1
dUcenH
dUcenH sresr)anbnÞat; D : y 12 x 32 . 1 3 1 0
A D
//
B
O
2
cos ABˆ C
3 5
nig sin ABˆ C 54 . 2
2
2
2
X> bgðajfa MB MD MA eyIgeFVIkareRbóbeFob ABM nig DCM edayRtIekaN ABM nig DCM man ³ -mMu ABˆ M DCˆM ¬mMucarwkmanFñÚsáat;rYm AD ¦ -mMu AMˆ B DMˆ C ¬mMuTl;kBM Ul¦ dUcenH ABM DCM tamlkçxNÐ m>m . MB MA vi)ak ABM DCM MC MD Taj)an MB MD MA MC eday BM CaemdüanénRtIekaN ABC naM[ MA MC eK)an MB MD MA MA b¤ MB MD MA dUcenH eyIgbgðaj)an MB MD MA .
2
-sg; D kñúgtRmúyxagelI taragtémøelx D : y 12 x 32 ³ xy V. tambRmab;RbFaneyIgsg;rUb)an ³
2
M //
C
k> R)ab;RbePTRtIekaN ABC eday ABC man BC CaGgát;p©it nig A CacMNucenAelIrgVg; enaH ABC CaRtIekaN carwkknøHrgVg; dUcenH ABC CaRtIekaNEkgRtg; A .
2
2
430
9
431
9
sm½yRbLg ³ 09 sIha 1999 viBaØasa ³ KNitviTüa ¬elIkTI2¦ ry³eBl ³ 60 naTI BinÞú ³ 100 I.
II. III.
cUrKUssBaØa kñúgRbGb;enAxagmuxcemøIyEdlRtwmRtUvmanEtmYyKt; ³ 1> KNna 18 8 ³ k> □ 10 x> □ 5 2 K> □ 2 X> □ 5 g> □ 3 . 2> kñúgfg;mYymanb‘ícRkhm 2 edIm nigb‘ícexov 4 edIm. eKlUkcab;b‘íc 1 edImBIkñúgfg;edayécdnü. rkRbÚ)abénRBwtþikarN_cab;)anb‘ícexovmYyedIm ³ k> □ 16 x> □ 12 K> □ 13 X> □ 23 g> □ 1 . kMNt;témø x edIm,I[RbPaK F x 51x 6 Kµann½y . rksmIkarbnÞat;kñúgrUbxageRkam ³ 2
y
2 B
A x
O
1
x
y
IV.
lT§plénkarRbLgKNitviTüa)an[dwgfa ³ sisS 1 nak;)anBinÞú 100 sisS 4 nak;)anBinÞú 80 sisS 10nak;)anBinÞú 70 sisS 30 nak;)anBinÞú 50 nigsisS 13 nak;)anBinÞú 40 . cUrbMeBjtaragenH . BinÞú eRbkg; eRbkg;ekIn 100 80 70 50 40
V.
M
b£sSImYyedImQrRtg;EkgnwgépÞdI ehIyRtUvxül;vay)ak;Rtg; cMNuc M dUcrUbxagsþaMenH. rkRbEvgTaMgGs;énedImb£sSI . 432
60 o
A
3m
S
9
VI.
ABC
CaRtIekaNsm½gS . cMNuc D sßitenAelIbnÞat; AB . A Cap©itrgVg; . C
F
E
D
60 o
B
G
A
1> KNna GDˆ E rYcbgðajfa FBD CaRtIekaNEkgRtg; F . 2> eRbóbeFob DEG nig EFC .
8 cemøIy KUssBaØa kñúgRbGb;xagmuxcemøIyEdlRtwmRtUv³ 1> KNna 18 8 ³ K> ☑ 2 eRBaH 18 8 3 2 2 2 2 . 2> rkRbU)abénRBwtþikarN_cab;)anb‘ícexov 1edIm ³ X> ☑ 23 ³ eRBaH kñúgfg;manb‘ícRkhm 2 edIm nigb‘ícexov 4 edIm enaHkrNIGac 2 4 6 nigkrNIRsb 4 Rsb 4 2 naM[ P(exov) krNI . krNIGac 6 3 II. kMNt;témø x edIm,I[RbPaK F Kµann½y ³ eyIgman F x 51x 6 edIm,I[RbPaK F Kµann½yluHRtaEt PaKEbgesµI 0 x 5x 6 0 eK)an I.
III.
y
2 B
A
x
smIkarbnÞat;EdlRtUvrkmanTRmg; y ax b -edaybnÞat;kat;G½kS yy Rtg; B0 , 2 naM[ b y 2 -ehIypleFobénbERmbRmYl y nig x BIcMNuc A1, 0 eTAcMNuc B 0 , 2 CaemKuNR)ab;Ts i én bnÞat;kat;tam A nig B eK)an a yx xy 20 10 21 2 dUcenH rk)ansmIkarbnÞat; y 2x 2 .
x 2 2 x 3x 6 0 xx 2 3x 2 0
B
A
B
A
eKGaceRbIviFImü:ageTot
x 2x 3 0
-edaybnÞat;kat;tam A1, 0 nig B0 , 2 eK)anRbB½n§smIkar 02 aa 10bb ba 22 dUcenH smIkarbnÞat;rk)anKW y 2x 2 .
x 2 0 x 2 x 3 0 x 3 x2, x3
1
y
2
dUcenH kMNt;témø)an
O
x
2
naM[
rksmIkarbnÞat;kñúgrUbxageRkam ³
. 433
9
IV.
tambRmab;RbFaneyIgGacbMeBjtarag)an ³ BinÞú eRbkg; eRbkg;ekIn 100 80 70 50 40
1 4 10 30 13
1 5 15 45 58
o
o
o
rkRbEvgedImb£sSITaMgGs; ³ RbEvgedImb¤sSITaMgGs;KW AM MS EdlRtUvrk ³ edaykñúg AMS EkgRtg; A man tan 60 AM AS naM[ AM AS tan 60 3 ehIy tamRTwsþIbTBItaK½r ³ eK)an MS AM AS
IV.
MS
BFD 180o BDF DBF
180o 30o 60o 90 A
3m
S
3 3 3 m
2
AM 2 AS 2
3 3 3 2
o
2
27 9 36 6 m
enaH AM MS 3 3 6 5.2 6 11 .2 m dUcenH RbEvgedImb¤sSITaMgGs;KW 11.2 m . V. eyIgmanrUb ³
o
o
o
F
E
D
60 o
B
G
A
o
o
C
o
dUcenH FBD CaRtIekaNEkgRtg; F . 2> eRbóbeFob DEG nig EFC -eday DEG man DG CaGgát;p©it nig E enA elIrgVg; enaH DEG carwkknøHrgVg; naM[ DEG EkgRtg; E vi)ak DEG 90 nig DGE 60 -eday DEG nig EFC man ³ mMu DEG EFC 90 ¬eRBaH EFC CamMubEnßmCamMu BFD 90 ¦ mMu DGE ECF 60 ¬eRBaHmMu ECF 60 CamMu sm½gS ABC ¦ eXIjfa DEG nig EFC manmMuBIrb:unerogKña dUcenH DEG EFC tamlkçxNÐ m>m .
60 o
o
2
BDF DBF BFD 180o
M
o
2
-bgðajfa FBD CaRtIekaNEkgRtg; F eday FBD man ³ -mMu BDF GDE 30 TajBIxagelI -mMu DBF ABC 60 ¬mMuRtIekaNsm½gS¦ EtplbUkmMukñgú én FBD esµI 180 KW ³
1> KNna GDˆ E tamTMnak;TMngmMup©it nigmMucarwkEdlmanFñÚsáat;rYm ˆ naM[ GDˆ E EA2G 602 30 eRBaH GDˆ E nig EAˆ G manFñÚsáat;rYm EG dUcenH KNna)anmMu GDˆ E 30 . o
o
o
434
9
435
9
sm½yRbLg ³ 24 kkáda 2000 viBaØasa ³ KNitviTüa ry³eBl ³ 120 naTI BinÞú ³ 100 I.
II.
cUrKUssBaØa kñúgRbGb;enAxagmuxcemøIyEdlRtwmRtUvmanEtmYyKt; ³ rktémø m EdlnaM[bnÞat; y m 2x 3 RsbnwgbnÞat; y 3x 1 . k> □ m 5 x> □ m 73 K> □ m 1 X> □ m 53 . D tamrUbxagsþaM cUrbMeBjcenøaHxageRkam[)anRtwmRtUv ³ 1 OA ............ cm OB ............ cm OC ............ cm OD ............ cm
III.
IV.
V.
VI.
O
C
2 A 1 B
1 1
I
eday AB 1 cm ; BI 1 cm ; IC 1 cm ; CD 1 cm nig OI 2 cm . ¬sresrcemøIyCab£skaer¦ RtIekaN ABC mYymanrgVas;RCug AB x , AC x 1 nig BC x 2 Edl x 0 . rktémø x edIm,I[RtIekaN ABC CaRtIekaNEkgRtg; A . kñúgtRmúyGrtUNrem xoy eKmancMNuc A1,1 nig B5 , 1 . k> cUrkMNt;kGU redaenéncMNuckNþal M rbs; AB . x> cUrsresrsmIkarbnÞat; AB . kñúgfg;mYymanXøIelOg 6 nigXøIexov 5 . eKlUkcab;ykXøImþgbI . rkRbU)abénRBwtþikarN_ ³ k> cab;)anXøIexovmYyy:agtic . x> cab;)anXøIelOgBIr nigXøIexovmYy. eK[RtIekaNEkgsm)at ABC Edlman Aˆ 90 , AB AC a nig AH Cakm KNna AH CaGnuKmn_én a . x> O Cap©itrgVgc; arwkkñúgRtIekaN ABC . bgðajfa O sßitenAelI AH . rYcKNna AO CaGnuKmn_én R ¬kaMrgVg; O ¦ . K> KNna R CaGnuKn_én a . o
8 436
9
cemøIy I. KUssBaØa kñúgRbGb;enAmuxcemøIyEdlRtwmRtUv ³ IV. k> kMNt;kUGredaenéncMNuckNþal M én AB rktémø m ³ K> ☑ m 1 eyIgmancMNuc A1,1 nig B5 , 1 eRBaH y m 2x 3 Rsbnwg y 3x 1 naM[ M x 2 x , y 2 y kalNa a a m 2 3 m 1 . 1 5 1 1 M , 2 2 II. bMeBjcenøaHedayKNna OC , OB , OA, OD D M 2 , 0 1 tamRTwsþIbTBItaK½r O C OC OI IC dUcenH kUGredaenéncMNuckNþal M 2 , 0 . 2 1 A
2
2
2 1 5 cm 2
2
OB OI IB 2
A 1 B
1
I
B
x> sresrsmIkarbnÞat; AB ³ smIkarbnÞat;EdlRtUvrkmanrag AB : y ax b -eday y ax b kat;tam A1,1 eK)an 1 a 1 b a b 1 1 -eday y ax b kat;tam B5 , 1 eK)an 1 a 5 b 5a b 1 2 -edayyk 2 1 eK)an ³
2
OB OI 2 IB 2 2 2 12 3 cm OA 2 OB 2 AB 2 OA OB 2 AB 2 2
A
2
OC OI 2 IC 2 2
B
3 12 2 cm
OD 2 OC 2 CD 2 OD OC 2 CD 2 2
5a b 1 a b 1 6a 2 1 a 3
5 12 6 cm
dUcenH eyIgbMeBj)an
OA 2 cm OB 3 cm
-yk
OC 5 cm OD 6 cm
rktémø x edIm,I[ ABC EkgRtg; A ³ eKman AB x , AC x 1 nig BC x 2 B tamRTwsþIbTBItaK½r ³ x 22 x 2 x 12
x
x2 4x 4 x2 x2 2x 1 x2 2x 3 0
A
x 1
krNIBiess ³ a b c 1 2 3 0 x 1 minykeRBaH x 0 naM[ 1
c 3 x2 3 a 1
dUcenH kMNt;)antémø
x 3
.
AB : y 1 x 2 3
3
.
kñúgfg;myY manXøIelOg 6 nigXøIexov 5 naM[ cMnYnXøITaMgGs;KW 6 5 11 k> cab;)anXøIexovmYyy:agtic RBwtþikarN_cab;)anXøIexovmYyy:agtic bMeBjCa mYyRBwtþikarN_ cab;)anXøeI lOgTaMgGs; eK)an ³ P(exovmYyy:agtic) = 1-P(lll) V.
C
2 1 6 3
1 2 b a 1 1 3 3
dUcenH smIkarbnÞat;
x2
a
CMnYskñúg 1 :
1 : a b 1
III.
naM[
437
9
x> bgðajfa O sßitenAelI AH eday AH Cakm cab;)anXøIelOgBIr nigXøIexovmYy naM[ p©itrgVg;carwkkñgú ABCRtUvsßitenAelI AH RBwtþikarN_ cab;)anXøIelOgBIr nigXøIexovmYyKW ¬l2.x1¦ GacCa ³ ¬llx¦ b¤ ¬lxl¦ b¤ ¬xll¦ dUcenH bgðaj)anfa O sßitenAelI AH . naM[RbU)abcab;)anXøIelOgBIr nigXøIexovmYyKW ³ - rYcKNna AO CaGnuKmn_én R ¬kaMrgVg; O ¦ P¬l2.x1¦ = P¬llx¦ + P ¬lxl¦ + P ¬xll¦ tag P nig Q CacMNucb:Hrvag ABC nigrgVg; 6 5 5 6 5 5 5 6 5 ( )( )( ) naM[ APOQ Cakaer BIeRBaHvaman ³ 11 10 9 11 10 9 11 10 9 5 5 5 OP OQ R nig PAˆ Q APˆ O AQˆ O 90 33 33 33 naM[ OA CaGgát;RTUgkaerEdlmanRCug R 5 5 3 0.45 33 11 tamRTwsþI CaGgát;RTUgkaer OA R 2 dUcenH P¬l2.x1¦ 115 0.45 . dUcenH KNna)an OA R 2 . K> KNna R CaGnuKn_én a VI. tambRmab;RbFaneyIgsg;rUb)an ³ A eyIgman OA OH AH Q P a a 2 R OA R 2 / OH R nig AH eday O 2 naM[ P(exovmYyy:agtic) = 1 116 105 94
o
B
eK)an
C
H
k> KNna AH CaGnuKmn_én a eday ABC CaRtIekaNEkgsm)atmankm
AH a sin 45 o a
dUcenH KNna)an
AH
R R
2 a 2 2 2
a 2 2
o
naM[
a 2 2 a 2 R 2 1 2 a 2 R 2 2 1 R 2R
438
a 2 2 1 2 2 1 2 1
2 2 a 1 22 1
dUcenH KNna)an
.
2 a 2
2 a R 1 2
.
9
439
9
sm½yRbLg ³ 13 sIha 2001 viBaØasa ³ KNitviTüa ry³eBl ³ 120 naTI BinÞú ³ 100 I.
cUrKUssBaØa kñúgRbGb;enAxagmuxcemøIyEdlRtwmRtUvmanEtmYyKt; ³ 1> cUrkMNt;témø m EdlnaM[bnÞat; y 2 mx RsbnwgbnÞat; y 3x 1 . k> □ m 3 x> □ m 1 K> □ m 1 X> □ m 73 . 2> tamrUbenH eKman BH 2 , HC 8 , AH x . cUrkMNt;témø x ³ A x k> □ x 10 x> □ x 4 B 2 H K> □ x 4 X> □ x 16
II.
KNna A 2 2 5 5 88 2020 .
III.
edaHRsayRbB½n§smIkartamedETmINg; ³ 2x x2 yy 89
2
IV.
V.
VI.
8
C
2
1 2
.
kñúgfg;mYymanXøIcMnYn 12 RKab; EdlkñúgenaHmanEtXøIBN’s nigXøIBN’exµA . rkcMnYnXøIBN’exµA edIm,I[)anRbU)abénXøIBN’s esµInwg 32 . b£sSI 9 edImmanRbEvg ¬KitCa m ¦ 4 , 12 , 10 , 8 , 2 , 4 , x , 8 , 4 . k> cUrkMNt;témø x edIm,I[ x CamFüménRbEvgb£sSITaMgenaH . x> Tajrkemdüan énRbEvgb£sSITaMgenaH . eK[bIcMNuc A 1 , 1 ; B3 , 2 nig C0 , 4 . k> edAcMNuc A ; B nig C enAkñúgtRmúyGrtUNremEtmYy . x> sresrsmIkarbnÞat; AB . K> KUsemdüan CM énRtIekaN ABC . rkkUGredaenéncMNuc M . X> KUskm
440
9
rgVg;mYymanp©it O kaMRbEvg 4 cm nigmanGgátp; ©it AB . H CacMNuckNþalén OB . bnÞat; mYyEkgnwg AB Rtg; H ehIyCYbrgVg; O Rtg;cMNuc M nig N . bnÞat; AM nig NB CYbKñaRtg;cMNuc I . k> cUrKNnargVas; MA nig MB . x> cUreRbóbeFob ABI nig NMI .
VII.
8 cemøIy I.
KUssBaØa kñúgRbGb;enAmuxcemøIyEdlRtwmRtUv³ 1> kMNt;témø m ³ K> ☑ m 1 . eRBaH bnÞat; y 2 mx Rsbnwg y 3x 1 kalNa a a 2 m 3 m 1 .
III.
eyIgman 2x x2 yy 89
D 1 4 5
Dx 9 16 25
2> kMNt;témø x ³ x> ☑ x 4 . eRBaH x 2 8 x 16 4 EtRbEvgRCugminGacGviC¢man KW x 0 dUcenH x 4
D y 8 18 10
2
x
II.
KNna A ³ 2 5 8 20 A 2 5 8 20 2
2
H
2 2 10 5 8 2 160 20 2 5 2 2 2 5
35 2 10 2 16 10 2 2 5 2 5
35 2 10 8 10 2 2 5 2 5
8
C
36 3 x 24 3 x 36 24 3 x 12 x4
35 6 10 22 5
35 6 10 35 10 6 6
dUcenH KNna)an
A 10
Dx 25 5 D 5 D y 10 y 2 D 5 x
rkcMnYnXøIBN’exµA ³ tag x CacMnnY XøIBN’exµA naM[ cMnYnXøIBN’sKW 12 x eRBaHXøITaMgGs;12 RKab; ebItag P(XøIBN’s) CaRbU)abcab;)anXøIBN’s eK)an P(XøBI N’s) = 1212 x tammRmab;RbFaneK)an ³ P(XøIBN’s) 32 naM[ 1212 x 32
tamedETmINg; ³
IV.
2
1 2
dUcenH RbB½n§smIkarmanKUcemøIy x 5, y 2 .
A
B
edaHRsayRbB½n§smIkartamedETmINg; ³
35 6
dUcenH cMnYnXøIexµAKW 4 RKab; .
. 441
9
V.
k> cUrkMNt;témø x edIm,I[ x CamFüm eyIgmanTinñn½y 4 , 12 , 10 , 8 , 2 , 4 , x , 8 , 4 eK)an x 4 12 10 8 92 4 x 8 4 enaH x 529 x eRBaH x x ¬ x CamFüm¦ 9 x 52 x
eK)an 1 a 1 b a b 1 1 -eday AB : y ax b kat;tam B3 , 2 eK)an 2 a 3 b 3a b 2 2 yk 2 1 enaHeK)an ³ 3a b 2 a b 1 4a 1
8x 52 x 6.5
dUcenH témøkMNt;)anKW
.
x 6.5 m
VI.
Me 6.5
dUcenH smIkarbnÞat;
1 4
AB : y 1 x 5 4
4
.
K> KUsemdüan CM énRtIekaN ABC ¬ emIlrUb cMeBaHkarKUs emdüan CM ¦ -rkkUGredaenéncMNuc M eday M CacMNuckNþalénRCug AB eK)an ³ x xB y A yB M A , 2 2 1 3 1 2 M , 2 2
.
3 M 1 , 2
k> edAcMNuc A ; B nig C enAkñúgtRmúyEtmYy eyIgman A 1 , 1 ; B3 , 2 nig C0 , 4
dUcenH kUGredaenrk)anKW
3 M 1 , 2
.
X> KUskm
C 0 , 4
B3 , 2 A 1 , 1
a
tam 1 : a b 1 b 1 a 1 14 54
x> Tajrkemdüan énRbEvgb£sSITaMgenaH cMeBaH x 6.5 m nigeyIgerobTinñn½ytamlMdab; eyIg)an 2 , 4 , 4 , 4 , 6.5 , 8 , 8 , 10 , 12 mancMnYntYénTinñn½yKW n 9 ¬CacMnYness¦ naM[ Me CatYTI 9 2 1 5 tamTinñn½yeRkayBIerobtamlMdab; tYTI 5 RtUvnwg cMnYn 6.5 enaHnaM[ Me 6.5 dUcenH Taj)anemdüan
naM[
HM
x> sresrsmIkarbnÞat; AB bnÞat; AB manrag y ax b -eday AB : y ax b kat;tam A 1 , 1 442
9
VII.
k> KNnargVas; MA nig MB I M
A
H 4 cm O //
//
B
N
-RtIekaN AMB man M enAelIrgVg; nig AB CaGgát;p©it enaHvaCaRtIekaNcarwkkñúgrgVg; dUcenH RtIekaN AMB CaRtIekaNEkgRtg; M . -eday AB MN Rtg; H naM[ MH Ca km
dUcenH RbEvg
MA 4 3 cm , MB 4 cm
.
x> eRbóbeFob ABI nig NMI edayRtIekaN ABI nig NMI man ³ -mMu ABI NMI ¬mMucarwkmanFñÚsáat;rYm MB ¦ -mMu AIB NIM ¬CamMurYmEtmYy¦ dUcenH
ABI
NMI
lkçxNÐdMNUc m>m .
443
9
sm½yRbLg ³ 19 sIha 2002 viBaØasa ³ KNitviTüa ry³eBl ³ 120 naTI BinÞú ³ 100 I.
cUrKUssBaØa kñúgRbGb;enAxagmuxcemøIyEdlRtwmRtUvmanEtmYyKt; ³ 1> etI 2 Cab£srbs;smIkarNamYy ? k> □ 2x x 2 0 x> □ x 3x 3 2 2 0 K> □ x 2x 3 0 X> □ x x 2 1 0 . 2> cMNuc A1 , 1 sßitenAkñúgtMbn;cemøIyénvismIkarNamYy ? k> □ y x 1 x> □ x y 5 K> □ x y 2 X> □ y x 3 . cUrpÁÚpÁgrvagpleFobRtIekaNmaRténmMu nigtémørbs;va ³ pleFobRtIekaNmaRténmMu témøpleFobRtIekaNmaRt cemøIy 1> cot g k> 0.8 1> g C 2> cos x> 1.25 2> 10 6 3> tan K> 0.6 3> B A 8 4> sin X> 0.75 4> g> 34 2
2
2
II.
III. IV.
V. VI.
cUrsRmYlkenSam ³
2
F
3 18 2 27 45 3 8 2 12 20
.
hwbmYydak;Gavburs nigGav®sþI. GavbursmancMnYn 8 BN’Rkhm nig 2 eTotBN’s. Gav®sþIman cMnYn 6 BN’Rkhm nig 4 eTotmanBN’s . eKhUtykGavmYyBIkñúghwb . k> kMNt;RbU)abEdleKhUt)anGav®sþI . x> kMNt;RbU)abEdleKhUt)anGavBN’Rkhm . rkBIcMnYn x nig y Edl x y 9 nig xy 14 . eK[bnÞat; D : y x 1 enAkñúgtRmúyGrtUNrem xoy . 444
9
VII.
1> sg;bnÞat; D . 2> rksmIkarbnÞat; L Edlkat;tamcMNuc A0 , 3 ehIyEkgnwgbnÞat; D . sg;bnÞat; L enAkñúgtRmúyCamYy D . 3> KNnakUGredaenéncMNucRbsBV M rvagbnÞat; L nig D ehIyepÞógpÞat;lT§pltamRkaPic . M CacMNucmYyenAelIrgVg; C Edlmanp©it O nigkaM R . rgVg; C Edlmanp©it M nigkaM r Edl R r CYbrgVg; C Rtg; A nig B . . bnÞat; OM CYbrgVg; C Rtg; D ehIyCYbrgVg; C Rtg; E nig F Edl E DM . bnÞat; Ekgnwg DM Rtg; F ehIyCYb DA nig DB erogKñaRtg; I nig J . 1>KNnargVas;mMu DAˆ M nig DBˆM . TajbBa¢ak;fa DI nig DJ b:HnwgrgVg; C Rtg; A nig B . 2>bgðajfactuekaN AMFI carwkkñúgrgVg;mYy . 3>eRbobeFob ADM nig FDI . AD DM AM 4>TajbBa¢ak;fa FD ehIy AD DI 2R2R r . DI FI
8 cemøIy
KUssBaØa kñúgRbGb;enAmuxcemøIyEdlRtwmRtUv³ 1> etI 2 Cab£srbs;smIkarNamYy ? x> ☑ x 3x 3 2 2 0 eRBaH 2 3 2 3 2 2 0 0 0 Bit 2> cMNuc A1 , 1 enAkñúgtMbn;cemøIyénvismIkar ³ K> ☑ x y 2 eRBaH A1 , 1 enaH 1 1 2 0 2 Bit . II. pÁÚpÁgrvagpleFobRtIekaNmaRténmMu nigtémø³ I.
III.
sRmYlkenSam ³ F
2
2
6 0.6 10 8 cos 0.8 10
sin
dUcenH 2> k 3> X 4> K
9 2 6 3 3 5 6 2 4 32 5
23
3 5 2
33 2 2 3 5 2 2 3
F
3 2
.
Gavbursman 8 BN’Rkhm nig 2 BN’s Gav®sþIman 6 BN’Rkhm nig 4 BN’s naM[cMnYnGavTaMgGs; 8 2 6 4 20 k> kMNt;RbU)abEdleKhUt)anGav®sþI 6 4 10 0.5 P(hUt)anGavRsþI) 20 20
C 10
8
3 8 2 12 20
IV.
sin 0.6 0.75 cos 0.8 cos 0.8 cot 1.33 sin 0.6
A
3 18 2 27 45
dUcenH sRmÜlkenSam)an
tan
6
F
dUcenH
B
445
P(
hUt)anGavRsþI) 0.5 .
9
x> kMNt;RbU)abEdleKhUt)anGavBN’Rkhm 8 6 14 7 P(hUt)anGavRkhm) 0.7 20 20 10
2> rksmIkarbnÞat; L smIkarbnÞat;EdlRtUvrkmanrag L : y ax b -eday L : y ax b kat;tamcMNuc A0 , 3 eK)an 3 a 0 b b 3 -ehIy L : y ax b EkgnwgbnÞat; D eK)an a a 1 a a1 Et D : y x 1 manemKuNR)ab;Tis a 1 naM[ a 11 1
dUcenH P(hUt)anGavRkhm) 0.7 . V. rkBIrcMnYn x nig y eyIgman x y 9 1 nig xy 14 2 tam 1 : x y 9 x y 9 3 yk 3 CMnYskñúg 2 eK)an ³ 2 :
y 9y 14
y 2 9 y 14 y 2 9 y 14 0
man 9 4 114 81 56 25 naM[ y 921 25 9 2 5 42 2 2
1
y2
9 25 9 5 14 7 2 1 2 2
cMeBaH y 2 ³ 3 : x y 9 2 9 7 cMeBaH y 7 ³ 3 : x y 9 7 9 2 dUcenH rk)ancemøIyBIrKU x 7 , y 2 1
2
x 2 , y 7
1> sg;bnÞat; D eyIgman D : y x 1 eyIgsg;RkaPic)an ³
.
VI.
x 1 2 y 0 1
D : y x 1
L : y x 3
446
dUcenH smIkarbnÞat; L : y x 3 . - sg;bnÞat; L enAkñúgtRmúyCamYy D edayeRbItaragtémøelx edIm,Isg;bnÞat; L x 1 0 L : y x 3 ³ y 2 3 enAkñúgtRmúyCamYy D . 3> KNnakUGredaenéncMNucRbsBV M eyIgman D : y x 1 nig L : y x 3 edaypÞwmsmIkarGab;sIusrvagbnÞat; D nig L eK)an ³ x 1 x 3 2x 4 x 2 / naM[ y x 1 2 1 1 dUcenH kUGredaenéncMNucRbsBV M 2 , 1 . - epÞógpÞat;lT§pltamRkaPic tamRkaPiceyIgeXIjfa ebIeyIgeFVIcMeNalEkg BIcMNucRbsBV eTAelIG½kSTaMgBIrenaHeyIg)an kUGredaenéncMNucRbsBVKW M x 2 , y 1 dUcCamYykarKNnaxagelIBitEmn . dUcenH karKNnaepÞógpÞat;edayRkaPic .
9
VII.
tambRmab;RbFaneyIgsg;rbU )an ³
naM[ctuekaN AMFI manplbUkmMuQm ³ / dUcenH ctuekaN AMFI carwkkñúgrgVg;mYy . 3> eRbóbeFob ADM nig FDI eday ADM nig FDI man ³ -mMu ADM FDI ¬CamMurYm¦ -mMu DAM DFI 90 ¬mMuEkgdUcKña¦ dUcenH ADM FDI tamlkçxNÐ m>m . MFˆI MAˆ I 90 o 90 o 180 o
J
F
B
M
I
C
E O
A
C
D
o
1> KNnargVas;mMu DAˆ M nig DBˆM man A CacMNucenAelIrgVg; C ehIy MD Ca Ggát;p©iténrgVg;p©it C naM[mMu DAˆ M CamMucarwk knøHrgVg; dUcenH mMu DAˆ M 90 . vi)ak RtIekaN DAM CaRtIekaNEkgRtg; A
AD DM AM 4> TajbBa¢ak;fa FD DI FI tamry³sRmayxagelI
ADM
AD DM AM FDI FD DI FI AD DM FD DI
o
eday
A DI A C DI
cMeBaH enaHTaj)an AD DI DM FD eday DM 2R nig FD 2R r
b:HnwgrgVg; C Rtg; A
DI AM
dUcenH
dUcKñaEdr B enAelIrgVg; C ehIy MD Ca Ggát;p©iténrgVg;p©it C naM[mMu DBˆM CamMucarwk knøHrgVg; dUcenH mMu DBˆ M 90 . vi)ak RtIekaN DBM CaRtIekaNEkgRtg; B o
eday
B DJ B C
DJ
b:HnwgrgVg; C Rtg; B
DJ BM
2> bgðajfactuekaN AMFI carwkkñúgrgVg;mYy³ edayctuekaN AMFI man ³ -mMu MFˆI 90 eRBaH Ekgnwg DM Rtg; F -mMu MAˆ I 90 eRBaH DI b:HnwgrgVg; C Rtg; A o
o
447
AD DI 2 R 2 R r
.
9
sm½yRbLg ³ 09 kBaØa 2003 viBaØasa ³ KNitviTüa ry³eBl ³ 120 naTI BinÞú ³ 100 I.
cUrKUssBaØa kñúgRbGb;enAxagmuxcemøIyEdlRtwmRtUv ³ 1> etIsmIkarNamYymanb£sBIrepSgKña ? k> □ x 2x 3 0 x> □ x 4x 4 0 K> □ 2x 5x 3 0 X> □ x 12 x 0 . 2> EFG CaRtIekaNEkgRtg; E ehIyman FG 6 cm nig EFˆG 60 . KNnaRbEvg EF KW ³ k> □ EF 6 3 cm x> □ EF 6 33 cm K> □ EF 3 3 cm X> □ EF 3 cm . emGMe)AmYyehIrcuHTMelIpáamYyTgenAkñgú sYnc,armYy EdlmanpáaBN’s 1Tg páaBN’Rkhm 2 Tg páaBN’sVay 3 Tg nigpáaBN’elOg 4 Tg . cUrpÁÚpÁgrvagRBwtþikarN_ nigRbU)abrbs;vaEdlRtUvKña ³ RBwtþikarN_ RbU)abénRBwtþikarN_ cemøIy 1> {emGMe)AcuHTMelIpáaBN’s } k> 0.4 1> x 2> {emGMe)AcuHTMelIpáaBN’Rkhm } x> 0.1 2> 3> {emGMe)AcuHTMelIpáaBN’sVay } K> 0.5 3> 4> {emGMe)AcuHTMelIpáaBN’elOg } X> 0.2 4> g> 0.3 kMNt;témø m edIm,I[bnÞat; L EdlmansmIkar y m 2m 4x RsbnwgbnÞat; L Edlman smIkar y m 2x 1. KNna A 2 98 2 54 200 3 16 32 2 5 1 5 B . 1 2 5 1 2 5 2
2
2
2
o
II.
III.
IV.
2
1
3
V.
2
3
taragxageRkamenHbgðajBIcMnYnkumar enAkñúgPUmiEdlekItCm¶WxVak;man;tamfñak;Gayu ³ fñak;énGayu ¬qñaM¦ 0-2 2-4 4-6 6-8 4 3 6 7 cMnYnkumar ¬nak;¦ 1> sg;tarageRbkg;ekIn énTinñn½yxagelI . 2> KNnam:Ut nigfñak;énGayuEdlCaemdüan énTinnñ ½yxagelI . 448
9
VI.
ekan K manmaD V 32cm nigmankm kMNt;smIkarbnÞat; BC . 2> kMNt;kUGredaenrbs;cMNuckNþal I én AC . kMNt;smIkarbnÞat; BI . 3
3
1
1
2
1
1
2
2
1
2
1
VII.
3> edaHRsayRbB½n§vismIkartamRkabPic VIII.
y y
2 x2 3 3 x2 2
.
eKman ABC CaRtIekaNEkgRtg;kMBUl A ehIyman AB 3cm , AC 4cm . rgVg; C man Ggát;p©it AB ehIyCYbRCug BC Rtg; H . 1>bgðajfa AH CakmrgVg; C mYyeTotcarwkeRkARtIekaN AHC . M CacMNucmYyenAelIFñÚtUc CH . bnÞat; CM CYbbnÞat; AH Rtg; N . eRbobeFob MAN nig HCN . TajbBa¢ak;fa AM HN MN CH . 1
2
8 cemøIy I.
KUssBaØa kñúgRbGb;enAmuxcemøIyEdlRtwmRtUv³ 1> smIkarEdlmanb£sBIrepSgKñaKW ³ K> ☑ 2x 5x 3 0 eRBaH tamkrNIBiess a b c 2 5 3 0 eK)anb£s x 1 nig x ac 52 . 2> KNnaRbEvg EF KW ³ F 60 X> ☑ EF 3 cm eRBaH G tampleFobRtIekaNmaRt E
II.
2
1
2
o
6 cm
EF cos EFˆG EF EF cos EFˆG EF 6 cos60o 1 6 3 cm 2 449
pÁÚpÁgrvagRBwtþikarN_ nigRbU)abrbs;vaEdlRtUvKña³ páaBN’s 1Tg BN’Rkhm 2 Tg BN’sVay 3 Tg nigBN’elOg 4 Tg enaHsrub 1 2 3 4 10 -RbU)abTMelIpáaBN’ s P(s) 101 0.1 -RbU)abTMelIpáaBN’ Rkhm P(Rkhm) 102 0.2 -RbU)abTMelIpáaBN’ sVay P(sVay) 103 0.3 -RbU)abTMelIpáaBN’ elOg P(elOg) 104 0.4 dUcenH eyIgpÁÚpÁg)an 2> X 3> g 4> k .
9
kMNt;témø m edIm,I[bnÞat; L //L eyIgmanbnÞat;BIrKW L : y m 2m 4x
III.
1
1> sg;tarageRbkg;ekIn énTinñn½y Gayu ¬qñaM¦ cMnYnkumar f eRbkg;ekIn
V.
2
2
1
0-2 2-4 4-6 6-8
L2 : y m 2x 1
bnÞat;BIrRsbKñaluHRtaEtemKuNR)ab;TisesµIKña eK)an ³ m 2m 4 m 2 2
m2 m 6 0
man 1 4 1 6 1 24 25 enaH m 1 2125 12 5 26 3 2
1
m2
1 25 1 5 4 2 2 1 2 2
dUcenH témøkMNt;)an IV.
m1 3 , m2 2
.
dUcenH m:UtKNna)anKW Mo 7 . fñak;GayuEdlCaemdüan Cafñak;GayuEdlman eRbkg;témøkNþalénTinnñ ½y naM[ Me CatémøéntYTI 202 10 tamtarageRbkg;ekIntYTI 10 enAkñúgfñak;Gayu 4-6 dUcenH fñak;EdlCaemdüanKWfñak;Gayu 4-6 .
A 2 98 23 54 200 33 16 32 2 2 49 23 27 2 2 100 33 8 2 16 2 14 2 63 2 10 2 63 2 4 2 14 2 63 2 10 2 63 2 4 2 0
dUcenH KNna)anKW A 0 .
KNnakm
VI.
2 5 1 5 B 1 2 5 1 2 5
1
2
3
1
22 5 5 5 1 20 3 5 19 3 5 19 B
1
1
2 5 1 5 1 2 5 1 2 5
dUcenH KNna)an
4 7 13 20
20 srub / 2> KNnam:Ut nigfñak;énGayuEdlCaemdüan edaym:UtKWCa p©itfñak;énGayuEdlmaneRbkg;eRcIn CageK ehIyfñak;Gayu 6-8 maneRbkg; 7 eRcInelIs eK mann½yfam:UtKWCa p©itfñak;Gayu 6-8 eK)an Mo 6 2 8 7
KNna
4 3 6 7
f
eday naM[
2
1
1
2
V1 32cm 3
h1 5 3
3
2
/
1
2
V2 4cm 3
nig h
2
5cm
32 4
5 3 8 5 2 10 cm
3 5 19
dUcenH RbEvgkm
.
1
450
/ .
9
eyIgmancMNuc A1 , 2 / B0 , 2 / C3 , 0 1> kMNt;smIkarbnÞat; BC bnÞat;RtUvrkmanrag BC : y ax b -eday BC : y ax b kat;tam B0 , 2 naM[ 2 a 0 b b 2 1 -eday BC : y ax b kat;tam C3 , 0 naM[ 0 a 3 b a b3 2 -yk 1 CMnYskñúg 2 eK)an ³
3> edaHRsayRbB½n§vismIkartamRkabPic
VII.
2 :
a
b 3
a
dUcenH smIkarbnÞat;
eyIgmanRbB½n§vismIkar
eyIgsg;bnÞat;RBMEdnénsmIkar
3
2 x2 3 3 x2 2 2 y 3 x 2 3 y x 2 2
edayeRbItaragtémøelxedIm,Isg;bnÞat;RBMEdn 2 3 ni g y x2 y x2 3 2
2 2 3 3
BC : y 2 x 2
y y
x 3 3 y 4 0
x 2 2 y 5 1
eyIgsg;RkaPic)an ³
.
y
3 x2 2
2> kMNt;kUGredaencMNuckNþal I én AC kUGredaenéncMNuckNþalGgát; AC kMNt;eday
y
2 x2 3
x xc y A y c I A , 2 2 1 3 2 0 I , 2 2 I 2 , 1
dUcenH kUGredaencMNuckNþal I 2 , 1 . -kMNt;smIkarbnÞat; BI bnÞat;RtUvrkmanrag BI : y cx d -eday BI : y cx d kat;tam B0 , 2 naM[ 2 c 0 d d 2 -eday BI : y cx d kat;tam I 2 , 1 naM[ 1 c 2 d c 1 2d 2 -yk 1 CMnYskñúg 2 eK)an ³ 2 : c 1 d 2
1 2 3 2 2
dUcenH smIkarbnÞat;
BI : y 3 x 2 2
.
EpñkminqUtCacemøIyénRbB½n§vismIkar . VIII. tambRmab;RbFaneyIgsg;rb U )an ³ C
M C 2
N
4cm
A
H
3cm
B
C1
¬edIm,IkMu[mankarlM)akkñgú kareFVIdMeNaHRsay edaysarrUb nigdMeNaHRsayenATMB½repSg dUcenH ´sUmykrUbeTATMB½rfµIedIm,IgayRsYl ¦ 451
9
2> eRbobeFob MAN nig HCN mMu AMˆ C AHˆ C 90 ¬mMumanFñsÚ áat;rYm AC ¦ naM[ AMˆ N 90 ¬CamMubEnßmCamYymMu AMˆ C ¦ ehIy CHˆN 90 ¬CamMubEnßmCamYymMu AHˆ C ¦ eday MAN nig HCN man ³ -mMu AMN CHN 90 ¬mMuEkgdUcKña¦ -mMu MAˆ N HCˆN ¬mMucarwkmanFñÚsáat;rYm MH ¦ dUcenH MAN HCN tamlkçxNÐ m>m .
C
M C 2
o
N
4cm
o
H
o
3cm
A
B
C1
o
1> bgðajfa AH Cakm
1
vi)ak ³ naM[
2
2
BC 32 4 2 BC 5 cm
-tamTMnak;TMngkñúgRtIekaNEkg ABC eK)an ³ BC AH AB AC AB AC naM[ AH BC Edlman AB 3cm , AC 4cm , BC 5 cm eK)an AH 3 5 4 125 2.4 cm dUcenH eyIgKNna)anRbEvg BC 5 cm , AH 2.4 cm
HCN
. 452
AM MN CH HN
AM HN MN CH
dUcenH Taj)an
o
2
MAN
AM HN MN CH
.
9
453
9
sm½yRbLg ³ 10 kkáda 2004 viBaØasa ³ KNitviTüa ry³eBl ³ 120 naTI BinÞú ³ 100 I.
cUrKUssBaØa kñúgRbGb;enAxagmuxcemøIyEdlRtwmRtUvmanEtmYyKt; ³ 1> etIsmPaBNamYyEdlRtwmRtUv ³ k> □ 81 3 x> □ 16 4 K> □ 1.69 1.3 X> □ 5 5 . 2> EFGH CactuekaNEkgmYyEdlman EF 1.5 cm nig EG 3 cm . KNna GEˆH ³ k> □ GEˆH 60 x> □ GEˆH 35 K> □ GEˆH 45 X> □ GEˆH 30 . ñ ½y. Mo Cam:UtTinñn½y. Me CaemdüanénTinñn½y. cUrpÁÚpÁgEpñk A nig B EdlRtUvKña ³ x CamFümTinn A ¬Tinñn½y¦ B ¬mFümsßiti¦ cemøIy 1> 2 , 3 , 4 , 1 , 6 k> Me 4 1> X 2> 2 , 4 , 5 , 6 , 1 x> Mo 1 2> 3> 4 , 6 , 3 , 7 , 5 K> x 3 3> 4> 1 , 2 , 5 , 4 , 1 X> Me 3 4> g> x 5 enAkñúgfñak;eronmYyeKe)aHeqñateRCIserIssisS 3 nak; kñúgcMeNamsisSQreQµaH 7 nak; EdlkñúgenaH mansisSRbus 5 nak; nigsisSRsI 2 nak;. rkRbU)abEdleKeRCIserIs)ansisSRsI 1 nak;y:agtic. eKykesovePA 200 k,aleTAEck[sisSkñúgfñak;myY . ebIeKEck 4 k,aldUcKña enaHenAsl;esovePA 20 k,al. ebIEck[sisSRbusmñak; 4 k,al nigsisSRsImñak; 5 k,al enaHxVHesovePAcMnYn 5 k,al. rkcMnYnsisSenAkñúgfñak;eronenaH . 1> edaHRsayRbB½n§vismIkartamRkabPic yy 2x 2 . 2> ABC manRCug AB x 1 , BC x 4 nig AC x 2 Edl x 1 . kMNt;témøén x edIm,I[ ABC EkgRtg;kMBUl A . 1> enAkñúgtRmúyGrtUNrem xoy sg;bnÞat; D : y 2x 2 . 2> bnÞat; L kat;tamcMNuc A 1 , 0 ehIyEkgnwg D . 3
4
2
II.
III.
IV.
V.
VI.
o
o
o
o
454
9
VII.
k> rkemKuNR)ab;TisénbnÞat; L . x> sresrsmIkarbnÞat; L . K> sg;bnÞat; L enAkñúgtRmúyCamYy D . kñúgrgVg;p©it O mYymanGgátF; ñÚBIrminb:unKña AB nig CD EdlEkgKñaRtg;cMNuc I . M CacMNuc kNþalén BD . bnÞat; MI CYbGgát; AC Rtg; N . 1> bgðajfa MBI nig MDI CaRtIekaNsm)at . 2> eRbóbeFob IBD nig NCI . TajbBa¢ak;fa MN AC . 3> eKyk IA a nig IC b . KNna AC nig IN [Cab;Tak;TgeTAnwg a nig b .
8 cemøIy I.
KUssBaØa kñúgRbGb;muxcemøIyEdlRtwmRtUv ³ 1> etIsmPaBNamYyEdlRtwmRtUv ³ K> ☑ 1.69 1.3 eRBaH 1.69 1.3 1.3 2> KNna GEˆH ³ X> ☑ GEˆH 30 eRBaH sin GEˆH 13.5 0.5 GEˆH 30
Et P(bbb) = P(b) P(b¼b) P(b¼bb)
eK)an P(s1y:agtic) = 1 72 75 0.71
2
o
dUcenH
o
H
E 1.5cm
F
3cm
5 4 3 2 7 6 5 7
s y:agtic) = 75 0.71 .
P( 1
rkcMnYnsisSenAkñúgfñak;eronenaH tag x CacMnYnsisSRbus ¬KitCa nak;¦ y CacMnYnsisSRsI ¬KitCa nak;¦ tambRmab;RbFan eyIg)anRbB½n§smIkar ³
IV.
G
pÁÚpÁgEpñk A nig B EdlRtUvKña ³ 2> k eRBaH 2 , 4 , 5 , 6 , 1 man Me 4 3> g eRBaH 4 , 6 , 3 , 7 , 5 man x 5 4> x eRBaH 1 , 2 , 5 , 4 , 1 man Mo 1 III. rkRbU)abeRCIserIs)ansisSRsI 1 nak;y:agtic sisSQreQµaH 7 nak; manRbus 5 nak; RsI 2 nak; -RBwtþikarN_eRCIserIs)ansisSRsI 1 nak;y:agtic CaRBwtþikarN_bMeBjCamYyRBwtþikarN_eRCIserIsKµan )ansisSRsI mann½yfaerIs)ansisSRbusTaMgGs; naM[ P(s1y:agtic bbb) II.
4 x 4 y 200 20 4 x 4 y 180 1 4 x 5 y 200 5 4 x 5 y 205 2
edaHRsayedayyk 2 1 ³ 4 x 5 y 205 4 x 4 y 180 y 25
cMeBaH y 25 ³ 1 : 4x 4 y 180 b¤ x y 45 nak; CacMnnY sisSsrubenAkñúgfñak; dUcenH cMnYnsisSsrubenAkñúgfñak;man 45 nak; . 455
9
V.
1> edaHRsayRbB½n§vismIkartamRkabPic
VI.
y x2 y 2
1> sg; D : y 2x 2 kñúgtRmúyGrtUNrem L : y 2 x 2
y x2 y 2
eyIgsg;bnÞat;RBMEdn edayeRbItaragtémøelx ³ y x2 nig x
1
D : y x 2
y2
2
x
y 1 0
2
1 2
y 2 2
2> k> rkemKuNR)ab;TisénbnÞat; L eday L D naM[ a a 1 eK)an 12 a 1 a 2
y2
dUcenH emKuNR)ab;Tisén L KW a 2 . y x2
EpñkminqUtCatMbn;cemøIyénRbB½n§vismIkar . 2> kMNt;témøén x edIm,I[ ABC EkgRtg; A eyIgman AB x 1 , BC x 4 / AC x 2 tamRTwsþIbTBItaK½r ³ ABC EkgRtg;kMBUl A luHRtaEt BC AB AC 2
2
2
x 42 x 12 x 22 x 2 8 x 16 x 2 2 x 1 x 2 4 x 4 x 2 8 x 16 2 x 2 2 x 5 x 2 6 x 11 0
x> sresrsmIkarbnÞat; L bnÞat;RtUvrkmanrag L : y ax b EdlmanemKuNR)ab;Tis a 2 ¬rk)anxagelI¦ -eday L kat;tam A 1 , 0 eK)an 0 2 1 b b 2 dUcenH bnÞat;rk)anKW L : y 2 x 2 . K> sg;bnÞat; L enAkñúgtRmúyCamYy D ¬sUmemIlrUbxagelI Edl)ansg;bnÞat; D ¦ VI. tamRmab;RbFaneyIgsg;rUb)an ³
man 3 1 11 9 11 20 naM[ x 31 20 3 2 5 0 minyk 2
C
A
1
3 20 x2 3 2 5 1
dUcenH témøkMNt;)anKW
x 3 2 5
N I
O
.
B
M D
¬´sUmykrUbeTAdak;TMB½rfµIedIm,IgayRsYlbkRsay¦ 456
9
mann½yfa INC CaRtIekaNEkgRtg; N ehIyman N enAelI AC dUcenH eyIgTaj)an MN AC . 3> KNna AC nig IN [Cab;Tak;Tg a nig b eyIgman IA a nig IC b -cMeBaH RtIekaNEkg AIC EkgRtg; I tamBItaK½r AC IA IC
C
A
N I
O
B
M
D
k> bgðajfa MBI nig MDI Ca sm)at eyIgman AB nig CD EdlEkgKñaRtg;cMNuc I nigman M CacMNuckNþal BD enaH BDI CaRtIekaNEkgRtg; I Edlman IM Caemdüan RtUvnwgGIub:Uetnus BD eK)an MI MB MD -eday MBI man MI MB dUcenH MBI CaRtIekaNsm)at . -eday MDI man MI MD dUcnH MDI CaRtIekaNsm)at . vi)ak ³ mMu)at MDI MID 2> eRbóbeFob IBD nig NCI eday IBD nig NCI man ³ -mMu NCI IBD ¬mMucarwkmanFñÚsáat;rYm AD ¦ -mMu NIC IDB eRBaH NIC MID ¬CamMuTl;kMBUl¦ IDB MDI MID ¬vi)akxagelI¦ dUcenH IBD NCI tamlkçxNÐ m>m . vi)ak ³ INC DIB Et BDI CaRtIekaNEkgRtg; I enaH DIB 90 naM[ INC DIB 90
2
2
2
AC IA2 IC 2 a 2 b 2
tamTMnak;TMngkñúgRtIekaNEkg AIC EkgRtg; I eK)an ³ IN AC IA IC IC naM[ IN IAAC
ab a2 b2
ab a 2 b 2 a2 b2
dUcenH eyIgKNna[Cab;Tak;Tg a nig b )an AC a 2 b 2 IN
o
o
457
ab a 2 b 2 a2 b2
ÉktaRbEvg ÉktaRbEvg .
9
sm½yRbLg ³ 11 kkáda 2005 viBaØasa ³ KNitviTüa ry³eBl ³ 120 naTI BinÞú ³ 100 I.
II.
III. IV. V.
cUrKUssBaØa kñúgRbGb;enAxagmuxcemøIyEdlRtwmRtUvmanEtmYyKt; ³ 1> eKmanTinñn½y 1 , 2 , 3 , a , 8 , 7 , 3a 8 . kMNt;témø a edIm,I[mFümTinñn½yKW x 7 . k> □ a 4 x> □ a 5 K> □ a 6 X> □ a 7 . 2> RtIekaN ABC CaRtIekaNsm)atman AB AC 5cm nig BC 6cm nigkm □ cos ABˆ H 56 x> □ cos ABˆ H 54 K> □ cos ABˆH 53 X> □ cos ABˆ H 12 . fñak;eronmYymansisSGayu 12 qñaM 5 nak; sisSGayu 13 qñaM 8 nak; nigGayu 14 qñaM 7 nak;. RKU)an ehAsisSmñak;edayécdnü . cUrpÁÚpgÁ Epñk A nig B [)ancemøIyRtwmRtUv ³ A B cemøIy 1> RbU)abEdlRKUehAsisSmanGayu 12 qñaM k> 0.65 1> 2> RbU)abEdlRKUehAsisSmanGayu 13 qñaM x> 0.6 2> 3> RbU)abEdlRKUehAsisSmanGayueRkam 14 qñaM K> 0.4 3> 4> RbU)abEdlRKUehAsisSmanGayu 14 qñaM X> 0.35 4> g> 0.25 edaHRsaysmIkar 28 1x 12 x 2 3 . edaHRsayRbB½n§smIkartamRkabPic xy 2x 2 . sYnc,armYymanragCactuekaNEkg manbeNþayelIsTTwg 9 m . eKsg;rbgB½T§CMuvijsYnenaH . rkRbEvgrbgebIvamanépÞRkLa 112 m . kñúgtRmúyGrtUNrem xoy eKmancMNuc A0 , 1 nig B4 , 3 . 1> kMNt;smIkarbnÞat; AB . 2> kMNt;kUGredaenéncMNuckNþal M én AB. kMNt;smIkarbnÞat; D Ekgnwg AB nigkat;tamcMNuc M . 3> sg;bnÞat; AB nig D enAkñúgtRmúyEtmYy . 2
VI.
458
9
VII.
CaRtIekaNcarwkkñgú rgVg;p©it O Edlman AB AC BC 2 cm . eKKUsGgát;p©it AD Edlkat; BC Rtg; H . 1> KNna AD nig BD ebI sin 30 12 , cos 30 23 . 2> eRbóeFob BHD nig AHC . TajrkpleFobdMNUc . 3> yk K CacMNuckNþal BD . bnÞat; KH kat;RCug AC Rtg; I . bgðajfa KI AC . ABC
o
o
8 cemøIy I.
KUssBaØa kñúgRbGb;enAxagmuxcemøIyRtwmRtUv 1> kMNt;témø a ³ x> ☑ a 5 eRBaH X 1 2 3 a 78 7 3a 8
III.
edaHRsaysmIkar ³
28 1x 12 x 2 7 1x 2 3 x
3 cos ABˆ H 5
2 7x 2 3
A 5cm
5cm
x
2 3 2 7
x
17 21 14
edaHRsayRbB½n§smIkartamRkaPic ³ eyIgmanRbB½n§vismIkar xy 2x 2 sg;bnÞat;RBMEdnénsmIkar xy 2x 2
IV. B
H
C
6cm
II.
2 3
2 7x x x 2 3 2 3
4a 29 7 4a 29 49 a 5 7 3 cos ABˆ H ☑ cos ABˆ H 5
2> KW ³ K> eRBaH ABH man
2 3
pÁÚpÁgEpñk A nig B [)ancemøIyRtwmRtUv ³ sisSGayu 12 qñaM 5 nak; sisSGayu 13 qñaM 8 nak; nigGayu 14qñaM 7nak; naM[krNIGac 5 8 7 20 naM[ 1> P(12qñaM) 205 0.25 2> P(13qña)M 208 0.4 3> P(eRkam14qñaM) 520 8 0.65 4> P(14qña)M 207 0.35 dUcenH eyIgpÁÚpÁg)an ³ 1> g 2> K 3> k 4> X .
taragtémøelxRtUvKña
y x2 x2
y x2
x 0 2 y 2 0 x 2 2 y 2 0
x2
EpñkminqUtCacemøIyénRbB½n§vismIkar . 459
9
V.
rkRbEvgrbgénsYnc,arenaH tag x CaRbEvgTTwg ¬Edl x 0 KitCa m ¦ naM[ RbEvgbeNþayKW x 9 tambRmab;RbFaneK)an ³
-kMNt;smIkarbnÞat; D smIkarbnÞat;RtUvkMNt;manrag D : y ax b eday D : y ax b kat;tam M 2 , 1 eK)an 1 a 2 b b 2a 1 eday D AB a a 1 naM[ a a1 11 1 ¬eRBaH a 1¦ eK)an b 2a 1 2 11 3
x x 9 112
x 2 9 x 112 0
man 9 4 1 112 81 448 529 naM[ x 9 2 1529 9 2 23 0 minyk 2
1
x2
9 529 9 23 7 2 1 2
dUcenH smIkarbnÞat; D : y x 3
cMeBaH x 7 enaH x 9 7 9 16 eK)an P 2( TTwg + beNþay )
3> sg;bnÞat; AB nig D kñúgtRmúyEtmYy taragtémøelxRtUvKña AB : y x 1 nig D : y x 3
/
27 16 46 m
edayRbEvgrbgCaRbEvgbrimaRtrbs;sYn dUcenH RbEvgrbgsYnc,arKW P 46m .
x
0 1
x
VI.
0 4 1 3 M , M 2 , 1 2 2
dUcenH kUGredaencMNuckNþalKW M 2 , 1
1
2
y 2 1
y 1 0
eKmancMNuc A0 , 1 nig B4 , 3 1> kMNt;smIkarbnÞat; AB bnÞat;EdlRtUvrkmanrag AB : y ax b -eday AB : y ax b kat;tam A0 , 1 eK)an 1 a 0 b b 1 -eday AB : y ax b kat;tam B4 , 3 eK)an 3 a 4 b 4a b 3 cMeBaH b 1 enaH 4a b 3 4a 3 b eK)an 4a 3 1 a 1 dUcenH smIkarbnÞat; AB : y x 1 . 2> kMNt;kUGredaenéncMNuc M kNþal AB
.
D : y x 3
AB : y x 1
VII.
KNna AD nig BD
A
2 cm
B
//
K
//
O H
I C
D
¬ebIrUb nigsRmayenATMB½repSgBIKña enaHvaBi)akdl; GñkGan dUcenH´sUmdak;rUb nigsRmayenATMB½rfµI . edIm,IkMu[GñkGanBi)ak ¦ 460
9
-TajrkpleFobdMNcU
A
eday
2 cm
B
//
//
K
O
I C
H
2 3 BHD BH HD BD 3 3 AHC AH HC AC 2 3
dUcenH pleFobdMNUcEdlTaj)anKW
D
BH HD BD 3 . -eday ABC man AB AC BC 2 cm AH HC AC 3 naM[ ABC CaRtIekaNsm½gS 3> bgðajfa KI AC vi)ak ABC ACB BAC 60 -eyIgman BAD 30 enaH DAC 30 -eday ABD man B enAelIrgVg; nig AD Ca naM[ AH Cakm eRbóeFob BHD nig AHC o
o
o
o
o
o
o
o
o
o
o
o
o
o
eday BHD nig AHC man ³ -mMu BHD AHC ¬CamMuTl;kMBUl¦ -mMu ADB ACB 60 ¬sRmayxagelI¦ dUcenH BHD AHC lkçxNÐdMNUc m>m . o
461
9
sm½yRbLg ³ 10 kkáda 2006 viBaØasa ³ KNitviTüa ry³eBl ³ 120 naTI BinÞú ³ 100 I.
cUrbMeBjcenøaHkñúgtaragxageRkam[)anRtwmRtUv edayKNna 8
x
cMeBaH x 8 .
8
x2 3
x 2 , 3 x , x 17
8
x
x 17
II.
III.
taragxageRkamenH CakarbgðajBIcMnYnsisS énGnuviTüal½ymYy Edl)anTTYl)anC½ylaPI sisSBUEkRbcaMqñaM ³ 10-12 12-14 14-16 16-18 fñak;énGayu ¬qñaM¦ 3 4 6 3 cMnYnsisS ¬nak;¦ 1> sg;tarageRbkg;ekIn énTinñn½yxagelI . 2> KNnam:Ut nigrkfñak;emdüan énTinñn½yxagelI . k> edaHRsaysmIkar x 2 8 x . x> rk m nig p edaydwgfasmIkar x mx p manb£s x 2 nig x 5 . KNna A 5 5 2 5 2 2 2
2
1
IV.
B 4 2 4 5 3 3 4 10 1 4 1000 4 2 3 3
V. VI.
VII.
edaHRsayRbB½n§vismIkar
1 y x 1 2 y x
2
.
.
kñúgkabUbmYymanRkdasR)ak; 15 snøkw . enAkñúgenaHmanRkdasR)ak;BIrRbePTKW 1000` nig 5000`. TwkR)ak;srubman 39000 ` . cUrrkcMnnY RkdasR)ak;RbePTnImYy ² . fg;mYymanGkSrCaBakü HAPPY . eKlUkykGkSrkñúgfg;mgþ mYyedaymindak;cUlvijeT . rkRbU)abénRBwtþikarN_ ³ 1> lUkyk)anGkSr P . 2> lUkyk)anGkSr AY tamlMdab;enH . 462
9
3> lUkyk)anGkSr HPP tamlMdab;enH . rgVg;p©it O manGgát;p©it AB Edl AB 5 cm . bnÞat; L mYyb:HnwgrgVg; O Rtg;cMNuc C ehIy AC 4 cm . bnÞat; AD CYbrgVg;mþgeTotRtg; D ehIyEkgeTAnwg L Rtg; E . M CacMNuc RbsBVrvag AC nig BD . N CacMNuckNþalGgát; AD . 1> kMNt;RbePT ABC nig ABD. KNnabrimaRtén ABC . 2> bgðajfactuekaN CENO CactuekaNEkg . 3> eRbóbeFob MAB nig MDC . Tajfa MA MC MB MD .
VIII.
8 cemøIy I.
bMeBjcenøaHkñúgtarag[)anRtwmRtUv x
8
8
x2
8
8
x
2
2
x 17
3
5
3
-cMeBaH
x 8
x2
2> KNnam:Ut nigrkfñak;emdüan tamtaragTinñn½yfñak;Gayu 14-16 maneRbkg;esµI 6 eRcInCageK naM[fñak;Gayu 14-16 Cafñak;m:Ut Etm:UtCatémøénp©itfñak; enaH Mo 14 2 16 15 dUcenH m:UtKNna)anKW Mo 15 . -emdüanCatémøkNþal edayTinñn½ymaneRbkg;srub 16 naM[témøkNþal éneRbkg;KW 162 8 . tamtarageRbkg;ekIn ³ eRbkg; 8 sßitenAkñúgfñak; 14-16 dUcenH fñak;Gayu 14-16 Cafñak;énemdüan .
enaH
82
8 8
x 3 8 2
3
x 17 8 17 9 3
-cMeBaH
x 8
x2 3
enaH
82
8 8
x 3 82 x 17 8 17 25 5
II.
1> sg;tarageRbkg;ekIn
III.
k> edaHRsaysmIkar x 22 8 x
fñak;énGayu eRbkg; f eRbkg;ekIn f 10-12 12-14 14-16 16-18
3 4 6 3
srub
16
x 2 4x 4 8 x
x 2 3x 4 0
3 7 13 16
tamkrNIBiess a b c 1 3 4 0 naM[ x 1 , x ac 14 4 1
2
dUcenH smIkarmanb£s x
1
463
1 , x2 4
.
9
x> rk m nig p eyIgmansmIkar x mx p manb£s x 2 nig x 5 cMeBaHtémøbs£ TaMgBIr eK)an ³
V.
edaHRsayRbB½n§vismIkar
2
1
eyIgsg;bnÞat;RBMEdn
2
taragtémøelxRtUvKña
2 2 m 2 p 0 2 5 m 5 p 0
y
4 2 m p 0 25 5m p 0 p 2m 4 p 5m 25
1 x 1 2 x y
2 4
yx x
y
0 1
0 1
-sg;RkaPic
5m 25 2m 4 3m 21
yx
m 7 : p 2m 4 2 7 4 10
dUcenH témøEdlrk)anKW IV.
nig
2 3
m7
cMeBaH
1 y x 1 2 y x 1 y x 1 2 y x
m 7 , p 10
y
1 x 1 2
.
KNna 5 2 A 5 2 5 2
5 5 2 2 5 2 5 2 5 2
5 10 10 2 52 7 3
dUcenH KNna)an
A
7 3
.
EpñkminqUtCacemøIyénRbB½n§vismIkar . VI. rkcMnYnRkdasR)ak;RbePTnImYy² tag x CacMnYnRkdasR)ak; 1000` y CacMnYnRkdasR)ak; 5000` tambRmab;RbFaneK)an ³ x y 15 x y 15 1000x 5000 y 39000 x 5 y 39
tamedETmINg;
B 4 2 4 5 3 3 4 10 1 4 1000 4 2 3 3
Dx 36 9 D 4 D y 24 D y 39 15 24 y 6 D 4
Dx 75 39 36 x
4 10 4 2 3 3 4 10 4 10000 4 2 3 3 4 10000 4 10 4 10
dUcenH KNna)an
B 10
D 5 1 4
dUcenH RkdasR)ak; 1000` man 9 snøwk RkdasR)ak; 5000` man 6 snøkw .
. 464
9
fg;manGkSr HAPPY man 5 GkSr naM[ krNIGac 5 rkRbU)abénRBwtþikarN_ ³ 1> lUkyk)anGkSr P edayGkSr P mancMnYn 2 enaHkrNIRsbesµI 2 naM[ P(P) = 52 0.4
-KNnabrimaRtén ABC eday ABC CaRtIekaNmanGuIb:Uetnus AB tamRTwsþIbTBItaK½r AB AC BC eday AB 5 cm nig AC 4 cm naM[ BC AB AC
VII.
dUcenH RbU)ablUk)anGkSr P KW
2 P(P) = 5
2
2
2
2
2
52 42 9 3 cm
eK)an
.
PABC AB AC BC
5 4 3 12 cm
2> lUkyk)anGkSr AY tamlMdab;enH
dUcenH brimaRt ABC KW P 12 cm . P(AY) P(A) P(Y/A) 1 1 1 2> bgðajfactuekaN CENO CactuekaNEkg 0.05 5 4 20 edayctuekaN CENO man ³ 1 dUcenH RbU)ablUk)an AY KW P(AY) = 20 . -mMu OCˆE 90 ¬bnÞat;b:HEkgnwgkaMrgVg;¦ -mMu NEˆC 90 ¬eRBaH AD L Rtg; E ¦ 3> lUkyk)anGkSr HPP tamlMdab;enH -mMu ONˆE 90 ¬kaMrgVg;EkgGgát;FñÚRtg;cMNuckNþal¦ P(HPP) P(H) P(P/H) P(P/HP) 1 2 1 1 ctuekaNEdlmanmMuTaMgbICamMuEkg enaHvaCa 0.033 5 4 3 30 ctuekaNEkg ¬eRBaHmMuTI4 k¾CamMuEkgEdr¦ 1 dUcenH RbU)ablUk)an HPP KW P(HPP) = 30 . dUcenH ctuekaN CENO CactuekaNEkg . VIII. tambRmab;RbFaneyIgKUsrUb)an ³ 3> eRbóbeFob MAB nig MDC E eday MAB nig MDC man ³ C L D -mMu MAB MDC ¬mMucarwkmanFñÚsáat;rYm BC ¦ N M -mMu AMB DMC ¬mMuTl;kBM Ul¦ B A ABC
o
o
o
O
dUcenH
1> kMNt;RbePT ABC nig ABD ³ eday ABCmankMBUl C enAelIrgVg;manGgát;p©it AB enaHvaCaRtIekaNcarwkknøHrgVg; dUcenH ABC CaRtIekaNEkgRtg; C .
MAB
MDC
tamlkçxNÐ m>m .
MA MB vi)ak MAB MDC MD MC Taj)anBIsmamaRt MA MC MB MD
dUcenH Taj)an 465
MA MC MB MD
.
9
sm½yRbLg ³ 09 kkáda 2007 viBaØasa ³ KNitviTüa ry³eBl ³ 120 naTI BinÞú ³ 100 I.
II.
cUrKUssBaØa kñúgRbGb;enAxagmuxcemøIyEdlRtwmRtUvmanEtmYyKt; ³ 1> smIkar 2x 1 2 manb£s ³ k> □ x 1 22 x> □ x 2
K> □ x 2 2 X> □ x 12 2 . 2> RtIekaN ABC CaRtIekaNEkgRtg;kMBUl A . AH Ekgnwg BC Rtg; H Edl BH 4.5cm nig HC 2 cm . rkRbEvg AH ³ k> □ AH 9 cm x> □ AH 6.5 cm K> □ AH 3 cm X> □ AH 3 cm . 1> edaHRsayRbB½n§smIkar 2x xy3y3 11 . 2> edaHRsayvismIkar 2 x 1 4 x x rYcbkRsaycemøyI elIG½kSéncMnYnBit . enAkñúgRKYsarmYy «BukmanGayueRcInCagkUnc,gcMnYn 25 qñaM ehIykUnc,gmanGayueRcInCagkUnb¥Ún cMnYn 5 qñaM . rkGayumñak;² ebIdwgfaplbUkénGayuGñkTaMgbIesµIngw 65 qñaM . sisSRbus 3 nak; nigRsI 3 nak; )anQreQµaHe)aHeqñat edIm,IeRCIserIseFVICaRbFanfñak; nigbnÞab;mk eRCIserIsGnuRbFanfñak;mñak;eTot . rkRbU)abénRBwtþki aN_ ³ 1> eRCIserIs)anRbFanfñak;CasisSRbus . 2> eRCIserIs)anRbFanfñak;CasisSRsI . 3> eRCIserIs)anRbFanfñak;CasisSRsI nigGnuRbFanfñak;CasisSRbus . enAkñúgtRmúyGrtUNrem xoy eKmanbnÞat; L : y x 2 nigmancMNuc A1 , 2 . 1> rksmIkarbnÞat; L Edlkat;tamcMNuc A1 , 2 ehIyEkgCamYybnÞat; L . 2> rkkUGredaencMNucRbsBVrvagbnÞat; L nig L ehIyepÞógpÞat;lT§pltamRkabPic . rgVg;p©it O manGgát;p©it BC Edl BC 4 cm . A CacMNucmYyenAelIrgVg; O Edl ABˆC 30 . bnÞat; AE CYbCamYy BC Rtg; D ehIyCYbrgVg; O mþgeTotRtg; E . 1> kMNt;RbePTRtIekaN ABC nigRtIekaN AOC . 2
III.
IV.
V.
2
1
2
1
2
1
VI.
2
o
466
9
2> KNna AB , AC , AEˆB . 3> eRbóbeFobRtIekaN ACD nigRtIekaN BED. rYcRsaybBa¢ak;fa DA DE DB DC .
8 cemøIy I.
KUssBaØa kñúgRbGb;enAmuxcemøIyEdlRtwmRtUv 1> smIkar 2x 1 2 manb£s ³ k> ☑ x 1 22 eRBaH 2 x 1 2 2 x 2 1 enaH x 1 2 2 x 12 1 x 1 22 2> rkRbEvg AH ³ X> ☑ AH 3 cm tamTMnak;TMngkñúgRtIekaNEkg ABC km
AH 2 HB HC
2 cm
H
AH HB HC
x
A
Dx 2 2 D 1 Dy 5 y 5 D 1
x
D y 11 6 5
1 3
0
3 x 20 65 3 x 45 x 15
B
naM[ Gayu«BukKW x 25 15 25 35 qñaM nig GayuknU b¥ÚnKW x 5 15 5 10 qñaM / /
dUcenH
dUcenH RbB½n§manKUcemøIy x 2 , y 5 .
«BukmanGayu 35 qñaM / kUnc,gmanGayu 15 qñaM kUnb¥ÚnmanGayu 10 qñaM
.
sisSRbus 3 nak; nigRsI 3 nak; naM[ cMnYnkrNIGac 3 3 6 . rkRbU)ab ³ 1> eRCIserIs)anRbFanfñak;CasisSRbus RbusCaRbFan GacRbusCaGnu> b¤GacRsICaGnu> P(b>CaRbFan) = P(b>b) + P(b>s) = P(b) P(b¼b) + P(b) P(s¼b)
IV.
2> edaHRsayvismIkar ³ 2 x 12 4 x 2 x 4x 2 4x 1 4x 2 x 3x 1 x
x
x 25 x x 5 65
1> edaHRsayRbB½n§smIkar ³ eyIgmansmIkar 2x xy3y3 11 tamedETmINg; eK)an D 3 2 1 Dx 9 11 2
EpñkminqUtCacemøIyénvismIkar . III. rkGayurbs;mñak;² tag x CaGayukUnc,g ¬Edl x 0 KitCaqñaM¦ naM[ Gayu«BukKW x 25 nig GayukUnb¥ÚnKW x 5 tambRmab;RbFan eK)ansmIkar ³
4.5 cm
4.5 2 3 cm
II.
-bkRsaycemøIyvismIkarelIG½kScMnYnBit ³
1 3
dUcenH vismIkarmancemøIy x 13 .
3 2 3 3 15 1 6 5 6 5 30 2
467
9
dUcenH P(b>CaRbFan
naM[
.
1 ) = 0.5 2
y x2
5 1 2 2 2
dUcenH cMNucRbsBVrvag L nig L KW ³
2> eRCIserIs)anRbFanfñak;CasisSRsI sisSRsICaRbFan GacRbusCaGnu> b¤GacRsICaGnu> P(s>CaRbFan) = P(s>b) + P(s>s) = P(s) P(b¼s) + P(s) P(s¼s)
2
1
5 1 , y x 2 2
.
-epÞógpÞat;tamRkaPic L2 : y x 3
3 3 3 2 15 1 6 5 6 5 30 2
dUcenH P(b>CaRbFan) = 12 0.5 . 3> eRCIserIs)anRbFanCaRsI nigGnu>CaRbus P(RbFanRsI>GnuRbus) = P(s) P(b)
L1 : y x 2
3 3 3 0.3 6 5 10
dUcenH P(RbFanRsI>GnuRbus) = 0.3 . V.
tamRkaPiceXIjfa cMNucRbsBVénbnÞat;TaMgBIr mankUGredaen x 52 , y 12 nigEkgKñaBitEmn . VI. 1> kMNt;RbePT ABC nigRtIekaN AOC
1> rksmIkarbnÞat; L eyIgman L : y x 2 nigcMNuc A1 , 2 smIkarbnÞat;RtUvrkmanrag L : y ax b -eday L kat;tam A1 , 2 eK)an 2 a 1 b b 2 a -eday L L eK)an a 1 1 a 1 naM[ b 2 a 2 1 3 2
1
A
2
2
2
B
30 o
O D
1
C
E
-edayRtIekaN ABC manGgát;p©it BC nig A enA elIrgVg; ehIy ABˆC 30 enaHvaCaRtIekaNcarwkknøHrgVg; dUcenH smIkarbnÞat;RtUvrkKW L : y x 3 . dUcenH ABC CaRtIekaNEkgknøHsm½gS . vi)ak ACB 60 2> rkkUGredaencMNucRbsBVrvag L nig L -eday AOC man OA OC ¬kaMrgVg;EtmYy¦ eyIgman L : y x 2 nig L : y x 3 pÞwmsmIkarGab;sIus énbnÞat;TaMgBIr L nig L enaHvaCaRtIekaNsm)at EdlmanmMu)at ACˆB 60 eK)an x 2 x 3 x 52 dUcenH AOC CaRtIekaNsm½gS . o
2
o
2
1
1
2
o
1
2
468
9
KUsrUbEtmYy)anehIy ´RKan;Etdak;[RsYlemIl A B
30 o
C
O D
E
2> KNna AB , AC , AEˆB kñúg ABCman ABˆC 30 nig BC 4 cm AB naM[ cos ABˆ C BC AB BC cos ABˆ C o
eK)an AB 4 cos 30 AC ehIy sin ABˆC BC eK)an AC 4 sin 30
o
o
dUcenH RbEvg AB 2
3 2 3 cm 2
4
AC BC sin ABˆ C
4
1 2 cm 2
3 cm
nig AC 2 cm .
cMeBaH AEˆB ACˆB 60 eRBaH AEˆB nigmMu ACˆ B CamMucarwkelIrgVg;EtmYy nigmanFñÚsáat;rYm AB dUcenH AEˆB 60 . o
o
3> eRbóbeFob ACD nig BED eday ACD nig BED man ³ -mMu AEˆB ACˆB 60 ¬manbBa¢ak;xagelI¦ -mMu EDˆ B CDˆ A ¬mMuTl;kMBUl¦ o
dUcenH vi)ak Taj)an
ACD BED ACD BED
tamlkçxNÐ m>m .
DA DC DB DE
DA DE DB DC
. 469
9
sm½yRbLg ³ 19 kBaØa 2008 viBaØasa ³ KNitviTüa ry³eBl ³ 120 naTI BinÞú ³ 100 I.
cUrKUssBaØa kñúgRbGb;enAxagmuxcemøIyEdlRtwmRtUvmanEtmYyKt; ³ 1> etIsmIkarmYyNa Edlmanb£sBIrepSgKña ? k> x 2 x 1 0 x> x 9 0 K> 5 x 7 x 1 0 X> x 5x 8 0 2> ABC CaRtIekaNEkgRtg;kMBUl A EdlmanGIub:Uetnus BC a nig ACˆB 60 . RCug ABmanRbEvg ³ a 2 a 3 a 3 a k> AB x> AB K> AB X> AB 2 3 2 2 eKføwgGgár 7 fg;CabnþbnÞab;ehIy)anlT§pl ³ 3 kg , 5 kg , 4 kg , 9 kg , 10 kg , 4 kg nig 7 kg . cUrpÁÚpÁgEpñk A nig Epñk B rYcsresrcemøIyenAkñúgEpñk C [)anRtwmRtUv ³ Epñk A Epñk B C ¬cemøIy¦ 1> m:UténTinñn½yxagelI k> 5 kg 1> 2> mFüménTinñny½ xagelI x> 7 kg 2> 3> emdüanénTinnñ ½yxagelI K> 6 kg 3> X> 4 kg 1> KNna A 2 8 nig B 12 3 1 2> KNna C 2 32 12 6 1 . kñúgRbGb;mYymanesovePAlMhat;FrNImaRtcMnYn 5 k,al nigesovePAlMhat;BICKNitcMnYn7 k,al. suPaB cab;ykesovePA 2 k,alecjBIRbGb;enaHedayécdnü . 1> rkRbU)ab EdlsuPaB cab;)anesovePAlMhat;FrNImaRtTaMgBIrk,al . 2> rkRbU)ab EdlsuPaB cab;)anesovePAlMhat;BICKNity:agticmYyk,al. 1> cUrsg;cMNuc A 1 , 2 nig B2 , 0 enAkñúgtRmúyGrtUNrem xOy . 2> rksmIkarbnÞat; AB . 3> sg;bnÞat; D mansmIkar y 32 x enAkñúgtRmúyxagelI. bgðajfabnÞat; D nigbnÞat; AB EkgKña . 2
2
2
2
o
II.
III.
IV.
V.
2
470
9
VI.
VII.
bUNa manGayueRcInCag cinþa . plbUkGayuGñkTaMgBIresµI 33 qñaM . ehIypldkGayuGñkTaMgBIresµI 3 qñaM. rkGayurbs; bUNa nigGayurbs; cinþa . eK[RtIekaNsm½gS ABC EdlmanRCugRbEvg 4 cm . eKKUsknøHbnÞat; At RsbnwgRCug BC . H CacMeNalEkgén C elI At . 1> KNna CH nig AH edaydwgfa cos60 12 nig sin 60 23 . 2> KNna BH . 3> KNnaépÞRkLaénRtIekaN ABH . o
o
8 cemøIy I.
KUssBaØa kñúgRbGb;xagmuxcemøIyEdlRtwmRtUv³ 1> smIkarEdlmanb£sBIrepSgKñaKW ³ ☑ K> 5x 7 x 1 0 eRBaHvaman 0 Edl 7 4 5 1 49 20 29 0 . 2> RCug ABmanRbEvg ³ ☑ X> AB a 2 3 AB C eRBaH sin 60 BC enaH AB BC sin 60 60 a
III.
1> KNna ³ A
2
22 8 22 22 2 2 2
dUcenH KNna)an
2
2
A 22 2
B 12 3 1 2 3 3 1
o
3 1
o
o
AB a
II.
3 2
A
dUcenH KNna)an B 3 1 . 2> KNna 22 2 C 6 1 3 1 21 2 3 1 3 1 3 1 6 1 21 2 3 1 6 1 3 1
B
pÁÚpÁgEpñk A nig Epñk B [)anRtwmRtUv ³ pÁÚpÁg)an ³ 1> X / 2> K / 3> k edaysareyIgmanTinñn½ydUcxageRkam ³ 3 kg , 5 kg , 4 kg , 9 kg , 10 kg , 4 kg nig 7 kg enaHeyIgGacKNna)annUv ³ -m:Ut KW 4kg eRBaH 4kg maneRbkg;eRcInCageK -mFüm 3 5 4 97 10 4 7 427 6 -emdüanCatYTI n 2 1 7 2 1 4 énTinñn½yerob tamlMdab;KW 3 , 4 , 4 , 5 , 7 , 9 , 10 KitCa kg .
3 1 6 2 6 1 3 2
dUcenH KNna)an
471
A 3 2
9
kñúgRbGb;manesovePAFrNImaRt 5 k,al nig esovePA BICKNit 7 k,al naM[ cMnYnkrNIGac 5 7 12 1>rkRbU)absuPaBcab;)anesovePAFrNImaRtTaMg 2 eK)an P(F>F) = P(F) P(F¼F)
-eday AB kat;tam A 1 , 2 eyIg)an
IV.
ab 2
-eday AB kat;tam B2 , 0 eyIg)an tam 1 nig 2 eyIg)anRbB½n§smIkar ³ a b 2 edayeRbIviFIedETmINg; ³ 2a b 0 enaH D 1 2 3
dUcenH RbU)ab P(F>F) 335 0.15 . 2> rkRbU)abEdlsuPaB cab;)anesovePABICKNit y:agticmYyk,al RBwtþikarN_Edlcab;)anesovePABICKNit 1 k,al y:agtic CaRBwtþikarN_bMeBjCamYyRBwtþikarN_cab; Kµan)anesovePABICKNitmYyk,alesaH eK)an P(F>F) + P(B>1y:agtic ) = 1 naM[ P(B>1y:agtic ) = 1 P(F>F) dUcenH P(B>1y:agtic V.
Da 2 2 D 3 3 D 4 4 Db 0 4 4 b b D 3 3 Da 2 0 2
dUcenH
a
smIkarbnÞat;EdlRtUvrkKW AB : y 2 x 4 3
3
3> sg;bnÞat; D : y 32 x enAkñúgtRmúyxagelI eyIgsg;bnÞat; D edayeRbItaragtémøelx
5 28 0.85 33 33
5 28 )=1 0.85 33 33
2
2a b 0
5 4 5 12 11 33
=1
1
D : y 3 x
.
2
x
0 2
y 0 3
-bgðajfabnÞat; D nigbnÞat; AB EkgKña edaybnÞat; AB : y 23 x 43 manemKuN R)ab;Tis a 23 nigbnÞat; D : y 32 x manemKuNR)ab;Tis a 32 ehIyplKuNemKuN R)ab;TisKW 23 32 1 enHbBa¢ak;fa bnÞat; D nig bnÞat; AB EkgKñaBitEmn
sg;cMNuc A 1 , 2 nig B2 , 0 enAkñúgtRmúy GrtUNrem xOy D : y 2 x 3
A B
dUcenH
2 4 y x 3 3
2> rksmIkarbnÞat; AB smIkarbnÞat;EdlRtUvrkmanrag AB : y ax b 472
D AB RtUv)anbgðaj
.
9
rkGayurbs; bUNa nigGayurbs; cinþa ³ tag x CaGayurbs; bUNa ¬KitCaqñaM¦ y CaGayurbs; cinþa ¬KitCaqñaM¦ Edl x y 0 bRmab; ³ plbUkGayuGkñ TaMgBIrKW 33 qñaM nigpl dkGayuGñkTaMgBIrKW 3 qñaM tambRmab;enHeyIgsresr)anRbB½n§smIkar ³ x y 33 eyIgedaHRsayedaybUkbM)at; x y 3
-kUsIunus ³ cosCAˆ H AH AC naM[ AH AC cos CAˆ H
VI.
1 2 cm 2 1 cos60 o 2 CH sin CAˆ H AC CH AC sin CAˆ H 4 cos60o 4
eRBaH
-sIunus ³ naM[
3 4 cos60o 4 2 3 cm 2 3 sin 60 o 2
eRBaH
x y 33 x y 3 2 x 36 x 18
eyIg)an ³ cMeBaH x y 33 nig x 18 naM[ y 33 x b¤ y 3318 enaH y 15 dUcenH bUNa manGayu 18 qñaM nig cinþa manGayu 15 qñaM . VII.
dUcenH AH 2 cm nig CH 2 3 cm . 2> KNna BH //BC eday At CH BC Rtg; C At CH enaH BCH CaRtIekaNEkgmanGIub:Uetnus BH tamRTwsþIbTBItaK½r BH BC CH 2
H
2
>
naM[ BH 28 2 7 2 7 cm dUcenH KNna)an BH 2 7 cm . 3> KNnaépÞRkLaénRtIekaN ABH tamrUb S S S Edl ABCH CactuekaNBñayEkg CH naM[ S AH BC 2 2
t
4 cm
C
B
2
4 2 2 3 28
tambRmab;RbFaneyIgsg;rbU )an ³ A
2
ABH
1> KNna CH nig AH ³ eyIgman ABC CaRtIekaNsm½gS enaH ACˆB 60 knøHbnÞat; At RsbnwgRCug BC naM[ CAˆ H ACˆB 60 ¬mMuqøas;kñúgxñat; AC ¦ RtIekaN AHC CaRtIekaNEkgRtg; H eRBaH H CacMeNalEkgén C elI At -tamTMnak;TMngRtIekaNmaRtkñúgRtIekaNEkg AHC man GIub:Uetnus AC 4 cm eyIg)an ³
o
ABCH
BCH
ABCH
o
ehIy naM[ dUcenH 473
BCH
2 4 2 2
3
6 3 cm 2
CaRtIekaNEkgRtg; C
S BCH
BC CH 4 2 3 4 3 cm 2 2 2
S ABH 6 3 4 3
2 3 cm 2
.
9
sm½yRbLg ³ 06 kkáda 2009 viBaØasa ³ KNitviTüa ry³eBl ³ 120 naTI BinÞú ³ 100 I.
II.
III.
IV.
cUrKUssBaØa kñúgRbGb;enAxagmuxcemøIyEdlRtwmRtUvmanEtmYyKt; ³ 1> sisSmYyRkummankm □ X 14 dm x> □ X 13dm K> □ X 11dm X> □ X 12 dm . 2> AI Cakm □ AI 2 2 cm x> □ AI 12 cm K> □ AI 2 3 cm X> □ AI 3 2 cm . kñúgRbGb;mYymanesovePAlMhat;FrNImaRtcMnYn 5 k,al nigesovePAlMhat;BICKNitcMnYn 7 k,al. suPaB cab;ykesovePAmþgmYyk,alcMnnY 2 dg CabnþbnÞab;Kña ecjBIRbGb;enaHedayécdnü nigmin dak;cUlvijeT. 1> rkRbU)ab EdlsuPaB cab;)anesovePAlMhat;FrNImaRtTaMgBIrk,al . 2> rkRbU)ab EdlsuPaB cab;)anesovePAlMhat;BICKNity:agticmYyk,al. BUsuxepJIR)ak; 2 000 000 erol enAFnaKar A TTYl)anGRtakarR)ak; 5% kñúg 1 qñaM nigepJIR)ak;cMnYn 3 000 000 erol enAFnaKar B )anTTYlGRtakarR)ak; 6% kñúg 1 qñaM . etIBUsuxTTYl)ankarR)ak; srubb:unµankñúg 1 qñaM BIFnaKarTaMgBIr A nig B . sYnc,armYymanragCactuekaNEkg EdlmanknøHbrimaRtesµI 14 m . eKdwgfa 3 dgénTTwg esµInwgBak; kNþalénbeNþay. 1> rkRbEvgTTwg nigbeNþayrbs;sYn . 2> rkcMnYnedImpáakñúgsYn ebIeKdwgfaépÞrbs;sYn 1 m manedImpáacMnYn 4 edIm. eK[bnÞat; L : y x 1 nig L : y x 3 enAkñúgtRmúyGrtUNrem xoy mYy . 1> sg;bnÞat; L nig L enAkñúgtRmúy xoy . 2
V.
1
2
1
2
474
9
2> rkkUGredaenéncMNucRbsBVrvag L nig L tamkarKNna ehIyeFVIkarepÞógpÞat;lT§pltam RkaPic . 3> TajrkcemøIyénvismIkartamRkabPic yy x x1 3 . rgVg;p©it O mYycarwkeRkARtIekaN ABC Edlman BAˆ C 60 , AB 6cm nigmMuBIreTotCamMuRsYc. eKKUs BD Ekgnwg AC Rtg; I ehIyCYbrgVg;p©it O Rtg; D . eKedAcMNuc E enAelI BD [)an BAˆ E CAˆ D . 1> KNna AI nig BI . 2> bgðajfa ACD nig ABE dUcKña . TajbBa¢ak;fa AB CD AC BE . 3> eRbóbeFob ABC nig AED . TajbBa¢ak;fa BC AD AC ED ehIy AB CD BC AD AC BD . 1
VI.
2
o
8 cemøIy I.
2> rkRbU)abEdlsuPaB cab;)anesovePABICKNit y:agticmYyk,al RBwtþikarN_Edlcab;)anesovePABICKNit 1 k,al y:agtic CaRBwtþikarN_bMeBjCamYyRBwtþikarN_cab; Kµan)anesovePABICKNitmYyk,alesaH mann½yfa cab;)anesovePAFrNImaRtTaMgBIrk,al eK)an P(F>F) + P(B>1y:agtic ) = 1 naM[ P(B>1y:agtic ) = 1 P(F>F) 28 dUcenH P(B>1y:agtic ) = 1 335 33 0.85 .
KUssBaØa kñúgRbGb;xagmuxcemøIyEdlRtwmRtUv ³ 1> rkmFüménTinñn½y ³ sisSTTYl)anBinÞú ¬eRBaH RbGb;TaMg 4 KµancemøIyRtwmRtUv¦ 2> KNna AI manRbEvg ³ K> ☑ AI 2 3 cm eRBaH tamTMnak;TMngkñúg ABC km
C
I 4cm
3 4 2 3 cm
II.
3cm
A
B
kñúgRbGb;manesovePAFrNImaRt 5 k,al nig esovePA BICKNit 7 k,al naM[ cMnYnkrNIGac 5 7 12 1>rkRbU)absuPaBcab;)anesovePAFrNImaRtTaMg 2 eK)an P(F>F) = P(F) P(F¼F)
rkkarR)ak;EdlBUsxu TTYl)anBIFnaKarTaMgBIr -karR)ak;EdlKat;TTYl)anBIFnaKar A KW ³ 2 000 000 5% 100 000 erol -karR)ak;EdlKat;TTYl)anBIFnaKar B KW ³ 3 000 000 6% 180 000 erol
III.
5 4 5 12 11 33
dUcenH RbU)ab P(F>F) 335 0.15 . 475
9
-karR)ak;EdlTTY)anBIFnaKarTaMgBIr A nig B KW 100 000 180 000 280 000 erol dUcenH BUsux TTYl)ankarR)ak; 280 000 erol. IV. 1> rkRbEvgTTwg nigbeNþayrbs;sn Y c,ar tag x CaRbEvgTTwgrbs;snY ¬KitCa m ¦ Y ¬KitCa m ¦ y CaRbEvgbeNþayrbs;sn tambRmab; RbFaneyIg)anRbB½n§smIkar ³
L1 : y x 1
L2 : y x 3
2> rkkUGredaenéncMNcu RbsBVrvag L nig L eyIgman L : y x 1 nig L : y x 3 eyIgpÞwmsmIkarGab;sIusénbnÞat;TaMgBIr 1
x y 14 x y 14 x y 14 y 3x 6 x y 6 x y 0 2 x y 14 6 x y 0 7 x 14 x 2 m
1
2
2
x 1 x 3 2x 2
bUkGgÁnigGgÁ naM[ y 6x 6 2 12 m dUcenH RbEvgTTwg x 2 m nig RbEvgbeNþay y 12 m .
x 1
cMeBaH x 1 enaH y x 1 1 1 2 dUcenH cMNucRbsBVvvag L nig L KW 1 , 2 . tamRkab cMNucRbsBVrvag L nig L eday kareFVIcMeNalEkgelIG½kSTaMgBIrKW x 1 nig y 2 2> rkcMnYnedImpáaEdlmanenAkñúgsYn ³ dUcKñaCamYycemøIytamEbbKNnaR)akdEmn . épÞsYnTaMgGs; TTwg beNþay 3> TajrkcemøIyénvismIkartamRkabPic S x y 2 12 24 m > eyIgmanRbB½n§vismIkar yy x x1 3 edayépÞsYn 1 m manedImpáacMnYn 4 edIm enaH tamRkab EpñkminqUtCacemøIyénRbB½n§vismIkar . cMnYnedImpáakñúgsYn 4 S 4 24 96 edIm VI. 1> KNna AI nig BI C D dUcenH cMnYnedImpáamanenAkñúgsYnKW 96 edIm . kñúgRtIekaNEkg ABI E I V. 1> sg;bnÞat; L nig L enAkñúgtRmúy xoy EkgRtg; I man A B 6 cm eyIgman L : y x 1 nig L : y x 3 BAˆ C 60 , AB 6cm AI eyIgeRbItaragtémøelx edIm,Isg;bnÞat; naM[ cosBAˆ C AB AI AB cos BAˆ C L : y x 1 nig L : y x 3 1 AI 6 3cm dUcenH AI 3cm . eK)an x y x y 2 BI 0 1 0 3 BI AB sin BAˆ C ehIy sin BAˆ C AB 1 2 1 2 eK)an BI 6 23 3 3 cm enaH BI 3 3 cm -sg;Rkab L nig L 1
1
2
2
1
2
1
o
2
1
2
1
2
2
476
2
9
2> bgðajfa ACD nig ABE dUcKña Binitü ACD nig ABE man ³ -mMu BAˆ E CAˆ D ¬smµtki mµ¦ -mMu ACˆ D ABˆ E ¬mMucarwkmanFñÚsáat;rYm AD ¦ dUcenH ACD ABE lkçxNÐdMNUc m>m . vi)ak
ACD ABE
AB BE AC CD
dUcenH eyIgTaj)an AB CD AC BE 1 3> eRbóbeFob ABC nig AED BinitüKUén ABC nig AED man ³ -mMu ADˆ E ACˆB ¬mMucarwkmanFñÚsáat;rYm AB ¦ -mMu DAˆ E CAˆ B eRBaH CAˆ D CAˆ E DAˆ E BAˆ E CAˆ E CAˆ B DAˆ E CAˆ B BAˆ E CAˆ D
dUcenH
ABC
vi)ak
ABC AED
AED
lkçxNÐdMNUc m>m .
BC AC ED AD
dUcenH eyIgTaj)an BC AD AC ED 2 -edaybUkGgÁnigGgÁén 1 nig 2 eK)an ³ AB CD AC BE BC AD AC ED AB CD BC AD AC BE AC ED AB CD BC AD ACBE ED
eday BE ED BD eRBaH E enAelI BD naM[ AB CD BC AD AC BD . dUcenH AB CD BC AD AC BD . 477
9
sm½yRbLg ³ 05 kkáda 2010 viBaØasa ³ KNitviTüa ry³eBl ³ 120 naTI BinÞú ³ 100 I.
cUrKUssBaØa kñúgRbGb;enAxagmuxcemøIyEdlRtwmRtUvmanEtmYyKt; ³ 1> RtIekaNEkg EFG manGIub:Uetnus FG 5 nig EFˆG 60 . EF manRbEvg ³ k> □ EF 5 33 x> □ EF 5 23 K> □ EF 52 X> □ EF 5 3 . 2> rfynþ 5 eRKÓg)andwgkgT½B 8 nak; / 11 nak; / 11 nak; / 8 nak; nig 12 nak;erogKña . mFümén Tinñn½yxagelIenHKW ³ k> □ X 11 nak; x> □ X 12 nak; K> □ X 8 nak; X> □ X 10 nak; . o
II.
KNna
A
1 1 7 3 7 3
B3 2
III.
3
V.
x y z 2 3 4 2 x 3 y 4 z 33 3x 2m 2010 x m 4 2
.
1> edaHRsayRbB½n§smIkar 2> edaHRsaysmIkar
IV.
4 3 3 12 3 2 45 4 5 8
manGBaØat x .
fñak;eronmYymansisSRsIcMnYn 17 nak; nigsisSRbuscMnYn 23 nak;. RKU)anehAsisSBIrnak;Ca bnþbnÞab;edayécdnü edIm,I[eLIgedaHRsaylMhat;RbU)ab RbLgRbNaMgKña . 1> rkRbU)abEdlRKUehA)ansisSRsITaMgBIrnak; . 2> rkRbU)abEdlRKUehA)ansisSRbus 1 nak;y:agtic . 3> rkRbU)abEdlRKUehA)ansisSTaMgBIrePT . kñúgtRmúyGtUNrem xoy eyIgmancMNuc Ax , 2 ; B9 , 2 nig C3 , y . 1> rkGab;sIuséncMNuc A edaydwgfa AB 12 nigGredaenéncMNuc C edaydwgfa BC 10 . 2> rksmIkarbnÞat; d : y ax b Edlkat;tamcMNuc B9 , 2 nig D3 , 10 . 478
9
rgVg;mYymanp©it O nigGgát;p©it AB Edl AB 8cm . M CacMNucmYyénrgVg;p©it O ehIyEdl BM 3cm . eKbnøay AB xag B [)an BE 3cm . bnÞat; L mYyEkgnwg AE Rtg; E ehIy L CYb AM Rtg; N . 1>KNna AE nig AM . 2>bgðajfa MAB nig EAN dUcKña . Tajrk AN nig EN . 3>bgðajfactuekaN BENM carwkkñúgrgVg;mYy EdleKnwgbBa¢ak;TItaMgp©it I nigRbEvgkaM r rbs;va.
VI.
8 cemøIy I.
KUssBaØa kñúgRbGb;enAxagmuxcemøIyRtwmRtUv ³ 1> EF manRbEvg ³ K> ☑ EF 52 eRBaH ³ 60
5 G
E
b¤
EF cos EFˆG FG EF FG cos60 o 1 5 5 2 2
F o
tamlkçN³smamaRteK)an ³
naM[
2> mFüménTinñn½yKW ³ X> ☑ X 10 nak; eRBaH 8 11 11 8 12 50 X 10 nak; 5 5 II. KNna
dUcenH KUcemøIyénRbB½n§smIkarKW ³
1 1 7 3 7 3
A
x
7 3 7 3 2 7 7 3 7 3 73
B3 2
3
4 3 3 12 3 2 45 4 5 8
1> edaHRsayRbB½n§smIkar
.
23x 2m 2010 4x m
3 8 3 2 3 36 3 2 3 45 32 2 68 0 III.
22 33 44 , y , z 7 7 7
2> edaHRsaysmIkar eyIgman 3x 2m4 2010 x 2 m eyIgTajecjBIsmamaRtenH)an ³
7 2
2x 3 y 4z 2x 3 y 4z 33 4 9 16 4 9 16 21 x y z 33 11 2 3 4 21 7 22 x 11 2 7 x 7 33 y 11 y 7 3 7 z 11 44 4 7 z 7
3x 2m 2010 2x m 3x 2m 2010 2 x 2m
3x 2 x 2m 2m 2010
x y z 2 3 4 2 x 3 y 4 z 33 2x 3 y 4z 4 9 16
x 2010
dUcenH
eKman 2x 3y 4z b¤esµnI wg
479
x 2010
Cab£sénsmIkar .
9
IV.
kñúgfñak;mansisS ³ RsI 17 nak; nigRbus 23 nak; naM[ krNIGac 17 23 40 1> rkRbU)abEdlRKUehA)ansisSRsITaMgBIrnak; eK)an P(s>s) = P(s) P(s¼s)
9 x 2 2 22 2 AB 9 x 2 12 2 9 x AB
b¤ 9 x 144 enaH 9 x 144 12 naM[ 99 xx 1212 xx 213 2
17 16 34 0.17 40 39 195
dUcenH RbU)abRKUehAsisSRsITaMgBIrnak;KW ³ 34 P(s>s) 0.17 . 195
dUcenH Gab;sIuskMNt;)an x 3 , x 21 . -rkGredaen éncMNuc C ebI BC 10
2> RbU)abEdlRKUehA)ansisSRbusmñak;y:agtic RBwtþikarN_EdlRKUehA)ansisSRbusmñak;y:agtic CaRBwtþikarN_bMeBjCamYyRBwtþikarN_KµanRbusesaH mann½yfa ehA)ansisSRsITaMgBIrnak; eK)an P(s>s) + P(b>1y:agtic ) = 1 naM[ P(b>1y:agtic ) = 1 P(s>s)
3 92 y 22 2 BC 36 y 2 2 10 2 36 y 2 BC
b¤ y 2 64 enaH y 2 8 naM[ yy 22 88 yy 10 6 2
dUcenH GredaenkMNt;)an y 6 , y 10 .
34 161 dUcenH P(b>1y:agtic ) = 1 195 . 195
2> rksmIkarbnÞat; d : y ax b -eday d : y ax b kat;tam B9 , 2 eK)an 2 a 9 b 9a b 2 1 -eday d : y ax b kat;tam D3 , 10 eK)an 10 a 3 b 3a b 10 2 -edayyk 1 - 2
3> rkRbU)abEdlRKUehA)ansisSTaMgBIrePT ³ sisSTaMgBIrePTGac RbusnigRsI b¤ RsInigRbus P(BIrePT) = P(bs) + P(sb) = P(b) P(s¼b) + P(s) P(b¼s) 23 17 17 23 40 39 40 39 391 391 391 1560 1560 780
eK)an
391 0.50 . dUcenH P(BIrePT) 780 V.
tam
eyIgmancMNuc Ax , 2 ; B9 , 2 nig C3 , y 1> -rkGab;sIuséncMNuc A ebI AB 12
9a b 2 3a b 10 4 a 6a 8 3
2 ³ 3a b 10
dUcenH smIkarbnÞat; 480
4 b 10 3 14 3
d : y 4 x 14 3
.
9
VI.
KNna AE nig AM
3> bgðajfactuekaN BENM carwkkñúgrgVg;mYy ³ eday AMˆ B 90 enaH NMˆ B 90 ¬mMubEnßm¦ ctuekaN BENM manplbUkmMuQmKña NMˆ B BEˆ N 90 90 180 /
N
o
M I
A
o
3cm
8cm
O
o
E
B
o
o
dUcenH ctuekaN BENM carwkkñúgrgVg;mYy .
L
-bBa¢ak;TItaMgp©it I nigRbEvgkaM r rbs;va -KNna AE AB BE 8 3 11cm ctuekaN BENM carwkñúgrgVg;man NMˆ B 90 -KNna AM ³ ABM CaRtIekaNcarwkknøH mann½yfa BN CaGgát;p©iténrgVg;enH rgVg;manGgát;p©it AB enaH ABM CaRtIekaNEkg dUcenH p©it I CacMNuckNþalén BN . tamBItaK½r AM AB BM 2
2
2
AM 2 8 2 32 AM 2 55
-rkRbEvgkaM r rbs;va kñúg BEN ³ tamRTwsþIbTBItaK½r
AM 55 cm
2> bgðajfa MAB nig EAN dUcKña eday MAB nig EAN man -mMu AMˆ B AEˆN 90 ¬RtIekaNEkgTaMgBIr¦ -mMu MAˆ B EAˆ N ¬mMumankMBUlrYm A dUcKña¦
BN 2 BE 2 EN 2 BN BE 2 EN 2 33 3 55
o
dUcenH
MAB MAB
vi)ak ³
EAN
EAN
tamlkçxNÐTI m>m .
AB MA AN EA
ehIy
AN
88 55 cm 55
9
99 5
45 99 5 12 12 5 cm 5 5
eday I kNþal BN naM[
enaH EN 1155 3 335555 cm EN
1089 55
MB MA EA MB EN EN EA MA
dUcenH
9
naM[ AN EAMA AB enaH AN 1155 8 885555 cm dUcenH
2
2
12 5 BN 6 5 r 5 cm 2 2 5
dUcenH kaMrgVg;manRbEvg r 6 5 5 cm .
33 55 cm 55
481
o
9
sm½yRbLg ³ 04 kkáda 2011 viBaØasa ³ KNitviTüa ry³eBl ³ 120 naTI BinÞú ³ 100 I.
II. III.
IV.
RtIekaN ABC mYymanrgVas;RCug AB x 3 , AC x 1 nig BC x 3 Edl x 3 . rktémø x edIm,I[RtIekaN ABC CaRtIekaNEkgRtg; A . edaHRsayRbB½n§vismIkartamRkabPic yy 22 x . kñúges,agmYymanXøIBN’Rkhm 4 RKab; BN’exov 3 RKab; nigBN’exµA 5 RKab;. sisSBIrnak;)anykXøI mñak;mYyRKab; CabnþbnÞab;ecjBIes,agedayécdnü ehIymindak;cUlvijeT . 1> rkRbU)abEdlsisSTI1 yk)anXøIBN’Rkhm ehIysisSTI2 yk)anXøIBN’exµA . 2> rkRbU)abEdl sisSTaMgBIrnak; yk)anXøIBN’exovmYyRKab;mñak; . BinÞúeFVIetsþsisSmYyRkum TTYl)anlT§pldUcxageRkam ³ BinÞú cMnYnsisS
V.
0
1
2
3
4
1
2
5
1
x
1> rktémø x ebIBinÞú 3 KWCaemdüanénTinñn½y . 2> rkcMnYnsisS)aneFVIetsþ . 3> rkcMnYnsisS Edl)anBinÞúticCag b¤esµI 3 . eKmanRtIekaN EFG EkgRtg; E ehIyrgVas;mMu EGˆ F 30 . K CacMNucmYyenAelIRCug EG EdlmanrgVas;mMu EKˆF 45 nigrgVas; EK 4cm . 1> KNnargVas;mMu GFˆK nigRbEvg EF nig FG . 2> KNnaépÞRkLaénRtIekaN EFG . kñúgtRmúyGrtUNrem xoy mYyeKmancMNuc A1 , 0 nig B3 , 2 . 1> rksmIkarbnÞat; AB . 2> rksmIkarénbnÞat; D EdlEkgnwg AB Rtg;cMNuckNþal E rbs;Ggát;enH . o
o
VI.
8 482
9
cemøIy I.
rktémø x edIm,I[ ABC CaRtIekaNEkgRtg; A eyIgmanRtIekaNEkg ABC B x 3 x 3 EdlcMeBaH x 3 mandUcrUb ³ A x 1 C
III.
tamRTwsþIbTBItaK½r ³ edIm,I[ ABC CaRtIekaN EkgRtg; A luHRtaEt ³ AB AC BC eK)an x 3 x 1 x 3 2
2
2
2
2
2
x2 6x 9 x2 2x 1 x2 6x 9 x 10 x 1 0
man 5 1 24 naM[ x 51 24 5 2
2> rkRbU)abEdlsisSyk)anXøIBN’exovdUcKña naM[ P¬exov exov¦ P¬exov¦ P¬exov¼exov¦
2
6 3
1
minyk
5 24 x2 5 2 6 1
IV.
edaHRsayRbB½n§vismIkartamRkabPic eyIgman yy 22 x eyIgsg;bnÞat;RBMEdn ³ D2 : y 2 x
y
0 2
1> rktémø x ebIBinÞú 3 KWCaemdüanénTinñn½y eyIgmantaragTinñn½y BinÞú 0 1 2 3 4 cMnYnsisS 1 2 5 1 x eyIgGacerobBRgayTinñn½ytamlMdab; 0 ,1,1, 2 , 2 , 2 , 2 , 2 , 3 , 4 , 4 , ...
1 2
8
2 2
2
tY
x
tY
eday 3 CaemdüanKWCatYEdlenAkNþaleK naM[cMnYntYEdlenAsgxag 3 RtUvEtesµIKña dUcenH x 8 CatémøEdlRtUvrk . 2> rkcMnYnsisSEdl)aneFVIetsþ cMnYnsisS)aneFVIetsþ 1 2 5 1 x , x 8 9 8 17 nak;
4
4
3 2 1 12 11 22
dUcenH P¬exov exov ¦ 221 0.045 .
dUcenH témørk)anKW x 5 2 6 ÉktaRbEvg .
D1 : y 2 x x y 0 2 1 1
4 5 5 12 11 33
dUcenH P¬Rkhm exµA¦ 335 0.15 .
2
II.
kñúges,agmanXøI Rkhm 4 exov 3 nigexµA 5 naM[cMnYnkrNIGac 4 3 5 12 1> rkRbU)abEdlsisSTI1yk)anXøIRkhm nig sisSTI2 yk)anXøIBN’exµA naM[ P¬Rkhm exµA¦ P¬Rk¦ P¬xµ¼Rk¦
4
2
4
dUcenH tamRkaPicEpñkEdlminqUtCacemøIy rbs;RbB½n§vismIkar .
dUcenH sisS)aneFVIetsþmancMnYn 17 nak; . 483
9
3> rkcMnYnsisSEdl)anBinÞútci Cag b¤esµI 3 cMnYnsisS)anBinÞúticCag b¤esµIbI 1 2 5 1 dUcenH cM>sisS)anBinÞúticCag b¤esµIbIKW 9 nak; . V. 1> KNnargVas;mMu GFˆK nigRbEvg EF nig FG
naM[
1 S EFG EF EG 2 1 4 4 3 8 3 cm 2 2
dUcenH KNna)an S 8 3 cm . VI. 1> rksmIkarbnÞat; AB F eKmancMNuc A1 , 0 nig B3 , 2 -kñúgRtIekaN smIkarbnÞat; EdlRtUvrkmanrag y ax b G E K FKG man ³ -ebI y ax b kat;tam A1 , 0 KGˆ F GFˆK EKˆ F ¬ EKˆ F mMueRkARtIekaN ¦ eK)an 0 a 1 b a b 1 Taj)an GFˆK EKˆF KGˆ F -ebI y ax b kat;tam B3 , 2 eK)an GFˆK 45 30 15 eK)an 2 a 3 b 3a b 2 2 dUcenH KNna)anrgVas;mMu GFˆK 15 . -edayyk 1 CMnYskñúg 2 eK)an 3 b b 2 -RtIekaN FEK CaRtIekaNEkgsm)at 2b 2 b 1 / eRBaH FEK CaRtIekaNEkgRtg; E nigman -cMeBaH b 1 CMnYskñúg 1 mMu)at EKˆF 45 1 : a b 1 1 vi)ak RCugCab;mMu)at EF EK 4cm dUcenH smIkarbnÞat;EdlrkKW y x 1 . dUcenH KNna)an EF 4 cm . 2> rksmIkarénbnÞat; D -kñúgRtIekaNEkg EFG EF kUGredaenéncMNuc E kNþalGgát; AB KW tamrUbmnþRtIekaNmaRt sin EGˆ F FG 1 3 0 2 E , E 2 , 1 EF 2 2 Taj)an FG sin EGˆ F bnÞat;EdlRtUvrkmansmIkar D : y ax b eday EF 4 cm nig sin EGˆ F sin 30 0.5 -eday D : y ax b kat;tam E2 , 1 4 eK)an FG 0.5 8 cm eK)an 1 2a b 3 -ehIy D AB b¤ D AB dUcenH KNna)an FG 8 cm . naM[ a a 1 a a1 11 1 2> KNnaépÞRkLaénRtIekaN EFG -enaH 3 : 1 2 1 b b 3 kñúgRtIekaNEkg EFG tamBItaK½r dUcenH smIkarbnÞat; D : y x 3 . EG FG EF EG FG EF enaH EG 8 4 48 4 3 cm 45 o
EFG
30 o
4 cm
o
o
o
o
o
o
2
2
2
2
2
2
2
484
2
9
485
9
sm½yRbLg ³ 16 kkáda 2012 viBaØasa ³ KNitviTüa ry³eBl ³ 120 naTI BinÞú ³ 100 I.
cUrKUssBaØa kñúgRbGb;enAxagmuxcemøIyEdlRtwmRtUv manEtmYyKt; ³ RtIekaN ABC mYyman BAˆ C 90 , AB 3 , AC a nig BC 4 . rktémø a . k> a 5 x> a 7 K> a 7 X> a 1 cmáarragctuekaNEkgmYymanvimaRt 2x 1 nig 20 KitCaEm:Rt . eKdaMeBatenAelIépÞdI 160 m ehIyeKdaMl¶enAelIEpñkdIenAsl; EdlmanragCactuekaNEkgmanvimaRt x nig 5 KitCaEm:Rt . rktémø x . sBVéf¶enHsuxmanGayu 35 qñaM ehIyesAmanGayu 14 qñaM. eKdwgfa t qñaMeTAmuxeTot suxnwgman GayutUcCag 3 dg EtFMCag 2 dgénGayurbs;esA . rkRKb;témøKt;viC¢manén t EdlGacman . kñúgfg;mYymanXøIs XøIexµA nigXøIRkhm TaMgGs;cMnYn 24 RKab; . 1> eKcab;ykXøI 1RKab;edayécdnü. eKdwgfaRbU)abEdleKcab;)anXøIsesµInwg 13 nigRbU)abEdlcab; )anXøRI khmesµI 14 . rkcMnYnXøIexµA . 2> eKcab;ykXøImþg 3 RKab;edayécdnü. rkRbU)abEdleKcab;yk)an XøITaMgbImanBN’dUcKña . cMNuc M 1 , 1 nig N 3 , 1 enAkñúgtRmúyGrtUNrem xOy mYy. 1> rkRbEvg MN nigkUGredaenéncMNuckNþal P énGgát; MN . 2> rksmIkarbnÞat; d Edlkat;tamcMNuc P nig R2 , 2 . sg;Ggát; MN nigbnÞat; d kñúgtRmúy xOy EtmYy . rgVg;p©it O mYycarwkeRkARtIekaN ABC mYyEdlmankm bgðajfa ABˆC ADˆ C nig ACˆB ADˆ B x> eRbóbeFob HAB nig CAD rYcehIy HAC nig BAD . K> bgðajfa AB AC AD AH ehIy BAˆ D HAˆ C . o
II.
III.
IV.
V.
VI.
2
8 486
9
cemøIy I. KUssBaØa kñúgRbGb;xagmuxcemøIyEdlRtwmRtUv³ eyIgGacsresr)an 0 t 7 eRBaH t 0 rktémø a ³ ☑ K> a 7 eRBaH tamRTwsþIbT edaycMnYnKt;viC¢man enAcenøaH 0 nig 7 man ³ 1,2,3,4,5,6 BItaK½r BC AB AC B dUcenH témøcMnYnKt; t EdlGacmanKW ³ naM[ AC BC AB 4 3 1 , 2 , 3 , 4 , 5 nig 6 . enaH AC BC AB A a? C 2
2
2
2
2
2
2
2
42 32 7 II.
20 rktémøén x ³ 160 m eyIgGactagcmáarenaH eBat l¶ )andUcrUbxagsþaM 5 eXIjfa épÞdIsrub = épÞddI aMeBat + épÞdIdaMl¶ eyIg)an 202x 1 160 5x 2
x
7 5 1 1 1 1 12 12 3 4 1 1 P( ) P( ) 3 4
40 x 20 160 5 x 40 x 5 x 160 20 35x 140 140 x 4 35
dUcenH témørk)anKW
x4m
kñúgfg;manXøI s exµA Rkhm TaMgGs; 24 RKab; 1> rkcMnnY XøIexµA ¬rebobTI1¦ edayplbUkRbU)abénRBwtþki arN_TaMgGs;kñúg viBaØasaEtmYyesµI 1 eyIg)an P(s) +P(Rkhm) + P(exµA) = 1 Taj)an P(exµA) = 1- [ P(s) +P(Rkhm) ]
IV.
eRBaHsmµtikmµ s nig Rkhm mü:ageTot tag n CacMnYnXøIBN’exµA enaHeyIg)an P(exµA) 24n naM[eyIgpÞwm)an ³ 24n 125 b¤ n 5 1224 10
.
rkRKb;témøKt;viC¢manén t EdlGacman ³ smµtikmµ ³ sBVéf¶suxGayu 35 qñaM nig esAGayu 14 qñaM ehIy t qñaMeTot suxmanGayutc U Cag 2 dg EtFMCagBIrdgénGayuesA enaHeyIgGacsresr)an RbB½n§vismIkar ³ 35 t 314 t Edl t CacMnYnKt;viC¢manRtUvrk 35 t 214 t
III.
dUcenH cMnYnXøIBN’exµAmancMnYn 10 RKab; . ÷rkcMnYnXøIexµA ¬rebobTI2¦ -tag x CacMnYnXøIs kñgú cMeNamXøITaMg 24 RKab; eyIg)an P(s) 24x Etsmµtikmµ P(s) 13 enaHeyIg)an 24x 13 enaH x 243 8 RKab; -tag y CacMnYnXIøRkhménXøITaMg 24 RKab; eyIg)an P(Rkhm) 24y Et P(Rkhm) 14 enaHeyIg)an 24y 14 enaH y 244 6 RKab;
35 t 42 3t 35 t 28 2t 35 42 3t t 35 28 2t t 7 2t 7 t
487
9
eday XøITaMgGs;mancMnYn 24 RKab; naM[ cMnYnXøIexµA 24 ¬cMnYnXøIs+cMnYnXøIRkhm¦ 24 8 6 10 RKab; dUcenH XøIBN’exµAmancMnYn 10 RKab; .
2> rksmIkarbnÞat; d smIkarbnÞat;EdlRtUvrkmanrag d : y ax b -eday d kat;tamcMNuc P1 , 1 enaHeyIg)an a b 1 1 -eday d kat;tam R2 , 2 enaHeyIg)an 2a b 2 2 -eyIgyksmIkar 2 1 eyIg)an ³
¬GñkKYreFVItamrebobTI2 eRBaHvaTak;TgsMNYrbnþ¦
2> rkRbU)abcab;)anXøITaMgbImanBN’dcU Kña eKcab;ykXøImþgbI cat;Tku CaviBaØasahUtehIymin dak;vij enaHvaTak;Tgdl;karcab;bnþbnÞab;eTot. eyIgmanXøI s 8RKab; / Rkhm 6RKab; / exµA 10RKab; -XøIbI BN’dUcKña GacCa sTaMgbI b¤ RkhmTaMgbI b¤ exµATaMgbI enaHeyIg)an ³ P(XøIbIBN’dUcKña) = P(sss) +P(RkRkRk)+P(xxx)
2 a b 2 a b 1 a 3
yk a 3 CMnYskñúg 1 1 : 3 b 1 naM[ b 4 dUcenH smIkarEdlRtUvrkKW d : y 3x 4 . - sg;Ggát; MN nigbnÞat; d kñúgtRmúy xOy EtmYy eyIgman M 1 , 1 nig N 3 , 1 ehIy d : y 3x 4 xy 04 11 eyIgsg;bnÞat; d nigGgát; MN dUcxageRkam ³
8 7 6 6 5 4 10 9 8 24 23 22 24 23 22 24 23 22 4 7 3 3 5 2 5 9 4 12 23 11 12 23 11 12 23 11 84 30 180 3036 3036 3036 294 49 0.097 3036 506
dUcenH V.
294 49 XøIbIBN’dUcKña) 3036 0.097 506
P(
1> rkRbEvg MN eyIgman cMNuc M 1 , 1 nig N 3 , 1 naM[ MN 3 1 1 1 4 4 dUcenH rk)an MN 4 ÉktaRbEvg . 2
2
d : y 3x 4
2
M 1 , 1
-rkkUGredaencMNuckNþal P énGgát; MN ³ eyIg)an P 12 3 , 1 2 1 b¤ P1 , 1 dUcenH rk)ancMNuckNþal MN KW P1 , 1 . 488
P1 , 1
N 3 , 1
9
VI.
tambRmab;RbFaneyIgKUsrUb)an ³
K> bgðajfa AB AC AD AH AB AH eday HAB CAD AD AC naM[eyIg)an AB AC AD AH dUcenH AB AC AD AH )anbgðajrYc .
A
O
H
C
B D
k> bgðajfa ABˆC ADˆ C nig ACˆB ADˆ B -eday ABˆ C nig ADˆ C CamMucarwkEdlman FñÚsáat;rYm AC naM[ ABˆC ADˆ C dUcenH ABˆC ADˆ C RtUv)anbgðajrYc . -eday ACˆ B nig ADˆ B CamMucarwkEdlman FñÚsáat;rYm AB naM[ ACˆB ADˆ B dUcenH ACˆB ADˆ B RtUv)anbgðajrYc . x> ÷ eRbóbeFob HAB nig CAD eday HAB nig CAD man ³ -mMu AHˆ B ACˆD 90 CamMuEkgdUcKña eRBaH AH Cakmm . ÷ eRbóbeFob HAC nig BAD eday HAC nig BAD man ³ -mMu AHˆ C ABˆ D 90 CamMuEkgdUcKña eRBaH AH Cakmm .
-bgðajfa ehIy BAˆ D HAˆ C eday HAC HAˆ C BAˆ D BAD b¤GacsresrCa BAˆ D HAˆ C dUcenH BAˆ D HAˆ C RtUv)anRsaybBa¢ak; .
o
o
489
sYsþI¡ elakGñkmitþGñkGanCaTIRsLaj;rab;Gan enAkñúgEpñkenHelakGñknwg)aneXIj GMBIviBaØasaFøab; ecjRbLgsisSBUEkfñak;TI 9 RKb;qñaM KWcab;BIqñaM 1986 rhUtdl;qñaM 2012 . ´sUmGP½yeTasdl;elakGñk mitþGñkGan cMeBaHEpñkenHmanEtviBaØasab:ueNÑaHKµancemøIyeT eRBaHlMhat;FrNImaRtxøHenAkñúgsm½yGtItkal ´xVHÉksarBieRKaH nigGñksYreyabl;bEnßm edaysarsisSBUEkCMnan;enaH eRcInCaGñkeFVIkar EdlKat;KµaneBl evlaRKb;RKan; edIm,I[´)anRbwkSaeyabl;CamYyKat; . EteTaHCay:agNak¾eday ´)andkRsg;lMhat;kñúg viBaØasaTaMgenHmYycMnYn eTAbkRsaykñúgEpñk lMhat;l¥² nigcemøyI . eyIgKYrdwgfa viBaØasaénqñaMnImYy²man BIr KWRbLgelIkTI 1 nigRbLgelIkTI 2 . ´nwgBüamerobcMcemøIyrbs;viBaØasaTaMgenH enAeBleRkay . RbsinebIelakGñkmanbBaða b¤cm¶l;Rtg;cMNucNa EdlmanenAkñúgEpñkenH elakGñkGacTak;Tg;eTAkan; RKU b¤mitþPkþirbs;GñkEdlmansmtßPaB b¤GñkeroberogesovePAenHkñúgeBlevlaTMng . …
vi
៩
: ០៤ :
៧(
១៩៨៦ ១
២
៣០
7 4 2 x x2 7
I. II.
III.
IV.
V.
edaHRsaykñúg Q smIkar ³ . sYnc,arrbs;ksikrsux manragctuekaNEkg nigbrimaRtmanrgVas; 296m . edIm,IerobcMpøÚvPUmi[Rtg; KN³kmµkarPUmiseRmckat;beNþaydIKat;Gs; 20m EtedIm,I[RkLaépÞénsYnenAdEdl KN³kmµkarPUmi seRmcbEnßmTTwgdI[Kat; 12m . cUrrkvimaRtrbs;sYnmunerobcMpÚøvPUmi . «BuknarImanGayu 48 qñaM / narImanGayu 13 qñaM nigb¥ÚnRbusnag Gayu 6 qñaM. etIry³eBlb:unµanqñaM eTIb Gayu«BuknarIesµI ³ k> 73 énGayunarI . x> 15 dgénGayukUnRbusKat; . K> esµIplbUkGayuknU Kat;TaMgBIr . X> esµIBak;kNþalénplbUkGayukUnKat;TaMgBIr . ABC CaRtIekaNsm)atkMBUl A . P CacMNucmYyenAelI BC . I nig J CacMeNalEkgén P elIbnÞat; AB nig AC . k> CabnÞat;EdlRsbeTAnwg AB kat;tam C . J CacMeNalEkgén P elIbnÞat; . bgðajfa J qøúHnwg J eFobnwgbnÞat; BC . x> bgðajfa plbUk PI PJ efr kalNa P rt;elI BC . K> P CacMNucmYyenAelIbnÞat; BC eRkA BC . bgðajfa PI PJ efr viC¢man b¤mYyefr GviC¢man. C CargVg;Ggát;p©it AB nig M CacMNucenAelI C xusBI A nig B . M CacMNucqøúHén M eFobnwg AB . k> I , J , K , L CacMNuckNþalerogKñaén MB , BM , M A , AM . Rsayfa IJKL CactuekaNEkg . x> kMNt;TItaMgén M edIm,I[ctuekaNEkg IJKL Cakaer . K> etImanb¤eTTItaMgén M EdleFVI[ctuekaNEkg IJKL CactuekaNesµI EtminEmnCakaer ?
8
490
៩
: ០៤ :
៧(
១៩៨៦ ២
២
៣០
I.
1> rk x , y , z CaFaturbs; Q sMNMuéncMnnY sniTan Edl x y z 3 nig 2x 5y 8z . 2> edaHRsaykñúg Q smIkar 12 xx 0 , 12 xx 1 . 1> RsaybBa¢afa ebI a nig b CacMnYnsniTanBIrxusKña enaHeK)an a 2 b ab . 2
II.
2
1 999 2 2 001 2
III.
IV.
3 999 999 . 2> Gnuvtþn_rUbmnþxagelIbgðajfa 2 yuvCnbInak;KW k , x nig K cUlrYmkILart;RbNaMgcm¶ay 100 m . eBlEdl {K} dl;eKaledA eKeXIj {k} enAxVH 10 m eTot . eBlEdl {k} dl;eKaledA eKeXIj {x} enAxVH 10 m eTot . eKsnµt;faGñkTaMgbIrt;kñgú el,Ónefr. 1> rkpleFobénel,Ónrbs; k nigel,Ónrbs; K . 2> eBl K eTAdl;eKaledA etIvaenAXøatBI x b:unµanEm:Rt . eKmanRtIekaN ABC , A CacMNuckNþalGgát; BC nig D CacMNucqøúHén C eFobnwg A . bnÞat; AB nig DA CYbKñaRtg; E . 1> RsaybBa¢ak;fa AB 3 AE . 2> O CacMNucminenAelIbnÞat; AB nigminenAelI AC . kMNt;cN M uc M elI AB nig M elI AC Edl O CacMNuckNþalén M M . 3> P CacMNucmYyelI BC . eKKUs PP Rsbnwg AC nig PP Rsbnwg AB Edl P enAelI AB nig P enAelI AC . rksMNMucMNuckNþalénGgát; P P kalNa P rt;elI BC . 4> H CaeCIgkm RsaybBa¢ak;facMNcu B , C , H , B , C rt;Rtg;Kña . x> kMNt;enAelIrUbenH Ggát;BImanemdüaT½rrYmKña . kMNt;smÁal; ³ sMNYrTI1 , TI2 , TI3 nigTI4 énlMhat;FrNImaRtminTak;TgKñaeT . 1
2
1
2
1
2
1 2
2
8
491
1
៩
: ០២ :
៨(
១៩៨៧ ១
២
៣០
I.
eK[ a b c 1 , 1a b1 1c 0 a 0 , b 0 , c 0 . bMPøWfa a b c 1 . eKmanbIcMnYnxusKña a , b , c . KNnaplbUk S a baa c b cbb a c a c c b . 2
II.
III.
2
2
edaHRsayRbB½n§smIkar
2 x y 3 x y 3
smÁal; ³ témødac;xatén a KW IV.
V.
.
a a a
ebI a 0 . ebI a 0
eKmanRtIFa A x 4x 72 . rktémøén x EdlnaM[RtIFa A mantémøGb,brma rYcR)ab;témø Gb,brmaenH . eK[cMNuc B nig C enAelIrgVg; O kaM R ¬ O BC ¦ . A CacMNucERbRbÜlelIFñÚFM BC . cMNuc H CaGrtUsg;énRtIekaN ABC . 1> bMPøWfa k> BAˆ O HAˆ C . x> cMNuc H , G , O rt;Rtg;Kña ¬ G CaTIRbCMuTm¶n;énRtIekaN ABC ¦ 2> rksMNMuéncMNuc H ¬bBa¢ak;lImIténcMNucenHpg¦ . 2
8
492
៩
: ០២ :
៨(
១៩៨៧ ២
២
៣០
I.
II.
edaHRsaysmIkarxageRkamkñúgsMNMucMnYnKt;rWuLaTIb ³ 4 x 3 x 2 x 1 x 4 . 1986 1987 1988 1989 1> sresrBhuFaxageRkamCaplKuNktþa ³ Ax x 4 x 3 x 1 B x x 4 x 3 2 x 2 x 1
III. IV.
2> sRmÜlRbPaKsniTan ³ Ax E x . Bx 3> bgðajfa Ex mantémøCacMnYnviC¢man b¤sUnü cMeBaHRKb;témøén x . KNnatémøelxénkenSam ³ aa bb ebIeKdwgfa 2a 2b 5ab b a 0 . rgVg;p©it O nigrgVg;p©it O kat;KñaRtg; A nig B . xñat;ERbRbYlmYyKUsecjBI B kat;rgVg;p©it O Rtg; E nigkat;rgVg;p©it O Rtg; F . 1> bnÞat;b:HrgVg;p©it O Rtg; E nigbnÞat;b:HrgVg;p©it O Rtg; F RbsBVKñaRtg;cMNuc P . bMPøWfa EPF manrgVas;efr . 2> bnÞat; EO nigbnÞat; FO RbsBVKñaRtg; M . bMPøWfa cMNuc E , M , A , F , P enAelIrgVg;mYy . 3> bMPøWfa emdüaT½rén EF kat;tamcMNucnwgmYy . 2
2
8
493
៩
: ០២ :
១៩៨៨
៨(
១
២
៣០
I. II.
III.
edaHRsaysmIkar x 3x 3x 4 0 . eKmanRbB½n§smIkar ³ 52xx 3yym1 ¬ m CacMnYnKt;sÁal;¦ . 3
2
kMNt;sMNMutémøKt;ngi GviC¢manén m EdlnaM[KUcemøIy x nig y énRbB½n§epÞógpÞat;lkçxNÐ x 0 y . eKmanbIcMnYn a , b , c Edl a b c 0 nig abc 0 . KNna E b a5 c b a5 c a b5 c . Ggát; PQ enAkñúgrgVg;p©it O ehIyEkgnwgkaM OS Rtg;cMNuc I . bnÞat; SP nig SQ kat;rgVg;erogKñaRtg;cMNcu M nig N . k> bMPøWfa rgVg;carwkeRkARtIekaN PQN kat;tamcMNuc M . x> K CacMNucqøHú éncMNuc S cMeBaHcMNuc O . bMPøWfa SMˆ I PKˆ Q SNˆ I . K> knøHbnÞat;BuHkñúgénmMu PIK kat;Ggát; PK Rtg;cMNuc E . H CacMeNalEkgéncMNuc E elIGgát; IK . bMPøWfa ³ IP IPIK IK HE 1 . 2
IV.
2
2
2
2
2
2
2
2
2
8
494
៩
: ០២ :
៨(
១៩៨៨ ២
២
៣០
I. II.
etIcMnYn A 2 5 ¬ n IN ¦ bBa©b;edaycMnYnsUnücMnYnb:unµan ? bMPøWfa cMeBaHRKb;témø a nig b eK)an ³ a b 5a 9b 6ab 30 a 45 . rktémøGb,brmaénkenSamRbPaK ³ F 2xx 513 . eKmanmMuRsÜc xOy . cMNuc C cl½tenAelIRCug Ox nigcMNuc D cl½telIRCug Oy edaybMeBj lkçxNÐ ³ OC OD 2a ¬ a CaRbEvgefr¦. rksMNMucMNuckNþal M én CD . eKdwgfakñúgRtIekaNEkg plbUkkaerénrgVas;RCugmMuEkgTaMgBIr esµIngw kaerénrgVas;GIub:Uetnus . cUrGñksg;Ggát;FñÚ AB énrgVg;p©it O kaMRbEvg 5 cm Edl AB 8 cm ehIy AB kat;tam cMNuc P ¬ P enAeRkArgVg;p©it O ¦ . n
2
2
2
2 n10
2
2
2
III. IV.
V.
2
8
495
៩
: ០២ :
៨(
១៩៨៩ ១
២
៣០
I. II. III.
IV.
V.
sresrcMnYn N 3a 1 4a 6a 9 CaplKuNktþadWeRkTI1 . KNna ³ E x xbaa b x xabb a a axbb x ¬ x ; a ; b CabIcMnYnxusKña ¦ . eK[smIkar 2x 3m 1 m 2 5 3 2 3x 3 ¬ m CacMnYnsÁal; ¦ . kMNt;sMNMutémøKt;GviCm¢ anén m edIm,I[smIkar manb¤sCacMnYnviC¢man . AM CaemdüanénRtIekaN ABC . R)ab;eQµaH BAC tamkrNInImYy²dUcxageRkam ³ k> ebI AM BC2 . x> ebI AM BC2 . cMNuc A cl½telIknøHrgVg;Ggát;p©it BC . enAeRkARtIekaN ABC eKsg;RtIekaNEkgsm)at ABE Bˆ 90 nigRtIekaNEkgsm)at ACF Cˆ 90 . I nig K CacMeNalEkgerogKña éncMNuc E nig F elIbnÞat; BC . bMPøWfa plbUk EI FK manrgVas;efr . 2
2
o
o
8
496
៩
: ០២ :
៨(
១៩៨៩ ២
២
៣០
bMPøWfa RbPaK F 2ba 51 mantémøelxCacMnYnGviC¢mancMeBaHRKb;témø a nig b . edaHRsaysmIkar x 1 5 x 2 x x 21410 2 x 1 . rkBIrcMnYnKt; a nig b a b edaydwgfaplbUk énBIrcMnYnenaH CaBhuKuNén 15 ehIypldk kaeréncMnYnTaMgBIresµI 45 . eK[rgVg;p©it O nigGgát;FñÚ AB . cMNuc I enAkñúgrgVg; cMNuc M enAelIrgVg; nigcMNuc E enAeRkArgVg; . eRbóbeFob AMB , AIB nig AEB . G CaTIRbCMuTm¶né; nRtIekaN ABC . CabnÞat;mYykat;tamcMNuc G . H , K , K CacMeNalEkgerogKñaénkMBUl A , B , C elIbnÞat; . bMPøWfa AH BK CK . 2
I. II. III.
IV.
V.
2
8
497
៩
: ០២ :
៨(
១៩៩០ ១
២
៣០
I.
II. III.
eK[BhuFa ³ E 4a x 20a 9x 45 12ax 60a . 1> sresr E CaplKuNktþa . 2> kMNt;témø a nig x EdlnaM[ E mantémøGb,brma . rkmYycMnYnedaydwgfa bIdgéncMnYnenaH nigkaeréncMnYndEdlenaH CacMnYnpÞúyKña . eK[RtIekaN ABC Edl AB BC2 ehIy BC2 AC BC . rgVg;Ggát;p©it BC nigrgVg;Ggátp; ©it AB kat;KñaRtg;cMNuc B nig E . rgVg;Ggát;p©it BC nigrgVg;Ggátp; ©it AC kat;KñaRtg;cMNuc C nig F . rgVg;Ggát;p©it AB nigrgVg;Ggátp; ©it AC kat;KñaRtg;cMNuc A nig K . 1> bnÞat; BE nig CF RbsBVKñaRtg;cN M uc M . etIcMNuc A tagGVIcMeBaHRtIekaN MBC . 2> bnÞat; EF nig BC RbsBVKñaRtg;cN M uc P . bMPøWfa EPˆ B FAˆ C 2 AMˆ F . 3> cMNuc N cl½telIknøHrgVg;Ggát;p©it BC EdlKµancMNuc E nig F . rksMNMucMNuckNþal I én MN . 2
2
2
2
2
8
498
៩
: ០២ :
៨(
១៩៩០ ២
២
៣០
I. II. III.
IV.
5x 3 2 x 4 3
3x 4 3 3
x
edaHRsaysmmIkar . eK[ ³ a b 1 . KNnatémøelxén P 2a b 3a b 1 . eK[ctuekaNBñay ABCD )attUc AB. knøHbnÞat;BuHén A nig D RbsBVKñaRtg;cMNuc O . H nig K CacMeNalEkgén O elI AB nig DC . bMPøWfa AD AH DK . eK[RtIekaN ABC . eKedAcMNuc P enAkñúgRtIekaN ABC Edl CBˆ P CAˆ P . M nig N CacMeNalEkgén P elIRCug BC nig AC . I CacMNuckNþalén AB . bMPøWfa IM IN . 3
3
8
499
2
2
៩
: ០២ :
៨(
១៩៩១ ១
២
៣០
I. II. III. IV.
V.
2b sRmÜlRbPaKsniTan ³ E aa 35ab . ab 6b kMNt;sMNMutémøén x x EdlepÞógpÞat;lkçxNÐ ³ x 12 x 3 3x 1 4 x 5 . edaHRsaysmIkar x 3x 7 x 9x 30 0 . eK[ctuekaNBñay ABCD )attUc AB Edl DC 2 AB . M nig N CacMNuckNþalerogKña én BD nig AC . RsaybMPøWfa AB 2 MN . eK[cMNuc E enAeRkArgVg;p©ti O . tam E eKKUsbnÞat;BIr kat;rgVg; O ³ bnÞatt;TI1 kat;rgVg;Rtg; cMNuc A nig B ehIybnÞat;TI2 kat;rgVg;Rtg;cMNuc C nig D . BC AD I . RsaybMPøWfa 2AEˆC AIˆC BCˆD 3BOˆ D . 4
3
2
2
2
2
2
8
500
៩
: ០២ :
៨(
១៩៩១ ២
២
៣០
I.
II. III. IV.
eK[smIkar ³ mx 2x m x 1 ¬m CacMnYnKt;sÁal;¦ . kMNt;témø m EdlnaM[smIkarminGacman . R)ab;témøtUcbMputénRtIFa ³ x 43 x 409 . RsaybMPøWfa ³ ebIRtIekaN ABC nig ABC b:unKña enaHrgVg;carwkeRkAénRtIekaNTaMgBIr RtUvb:unKña. eK[RtIekaN ABC . cMNuc M cl½tenAelIRCug AB . eKsg;RbelLÚRkam BMCN Edlman Ggát;RTUg BC . rksMNMukMBUl N . 2
2
8
501
៩
: ០៤ :
៨(
១៩៩២ ១
២
៣០
I.
II.
III.
IV.
pldkrvagRkLaépÞénkaerBIresµInwg 1152 m ehIypldkrvagRbEvgRCugénkaerTaMgBIresµI 16 m . KNnaRbEvgRCugénkaernImYy² . mYycMnYnKt;pSMedaybYnelx elxxÞg;rayKW b elxxÞg;db;KW 5 elxxÞg;ryKW a ehIyelxxÞg;Ban;KW 3 . KNna a nig b edaydwgfa cMnYnenaHEckdac;nwg 5 pg nig 9 pg . RtIekaN ABC nigRtIekaN ABC man BC BC nigmMu Aˆ Aˆ . RsaybMPøWfa rgVg;carwkeRkA énRtIekaNTaMgBIr CargVg;b:unKña . cMNuc B nig C enAelIrgVg;p©it O . cMNuc A cl½telIrgVg;enH . H CaGrtUsg;énRtIekaN ABC . RsaybMPøWfa AH b:un ehIyRsbnwgGgát;nwgmYy kalNa A ERbRbÜl . 2
smÁal; ³ sisSGaceFVIlMhat;NamYymunk¾)an .
8
502
៩
: ០៤ :
៨(
១៩៩២ ២
២
៣០
I.
k> sresrRtIFa A 841a 870ab 225b CakaeréneTVFa . x> sresrBhuFa B 196a 841a b 870ab 225b CaplKuNbYnktþa . eK[bIcMnYnKt; a , b , c xusBIsnU ü Edl a b c . bc ca RsaybMPøWfa 1a ab 3abc . eKedAcMNuc M enAkñúgRtIekaN ABC . k> bgðajfa BMˆ C BAˆ C . x> eKKUskm
2
4
II.
III.
IV.
2
2
3
4
smÁal; ³ sisSGaceFVIlMhat;NamYymunk¾)an .
8
503
៩
: ០៣ :
៨(
១៩៩៣ ១
២
៣០
I. II. III.
IV.
edaHRsaysmIkarkñúgsMNMucMnYnKt;rWuLaTIb ³ 6x 3x 8x 4 0 . sRmÜlkenSam ³ E 22aa 11 22aa 11 1 1 1a 41a . eK[RtIekaNsm)at OAB OA OB . eKedAcMNuc C enAelIRCug OA rYcekbnøayRCug OB [)an BD Edl BD AC . CD kat; AB Rtg;cMNuc M . RsaybBa¢ak;fa cMNuc C nig D qøúHKñaeFobnwgcMNuc M . eK[RtIekaN ABC Edl AB AC . knøHbnÞat;BuHénmMu BAC kat;emdüaT½rén BC Rtg;cMNuc I . H CacMeNalEkgéncMNuc I elI AB . RsaybBa¢ak;fa AB AC 2 AH . 3
2
2
smÁal; ³ sisSeFVIlMhat;NamYymunk¾)an .
8
504
៩
: ០៣ :
៨(
១៩៩៣ ២
២
៣០
I. II. III. IV.
RsaybBa¢ak;fa RbPaK x 1993 mann½yCanic© . 4x 7 KNna A 5 3 8 60 . RsaybBa¢ak;fa cMnYn N 44a 1 100 Eckdac;ngw 32 . eK[ctuekaNBñay ABCD )attUc AB. Ggát;RTUg AC nig BD kat;KñaRtg;cMNcu O . bnÞat;Rsbnwg DC EdlKUsecjBI O kat;RCugeRTt AD nig BC erogKñaRtg;cMNcu M nig N . 1> RsaybBa¢ak;fa O CacMNuckNþalén MN . 2> ebI AB a , CD b KNna MN CaGnuKmn_én a nig b . 2 1 1 3> RsaybBa¢ak;fa MN . AB DC 2
2
smÁal; ³ sisSGaceFVIlMhat;NamYymunk¾)an .
8
505
៩
: ០៣ :
៨(
១៩៩៤ ១
២
៣០
I.
II.
III. IV.
V.
RbGb;katugmYyragRbelBIEb:tEkg EdlmanvimaRt 180 mm , 60 mm , 90 mm . rkcMnYnKUbtic bMputEdlGacerobbMeBjkñúgRbGb;enH . rkBIrcMnYnKt; a nig b a b edaydwgfa plbUkénBIcMnYnenH CaBhuKuNén 9 ehIypldkkaer éncMnYnTaMgBIresµI 63 . etIcMnYn A 2 5 n IN bBa©b;edaysUnücMnYnb:unµan ? G CaTIRbCMuTm¶g;énRtIekaN ABC . eKKUs GD // AB rYcKUs GE // AC eday D nig E CacMNucénGgát; BC . RsaybBa¢ak;fa BD DE EC . ABC CaRtIekaNsm)atkMBUl A . P CacMNucmYyenAelI BC . I nig J CacMeNalEkgerogKña én P elIbnÞat; AB nig AC . 1> CabnÞat;EdlRsbnwg AB kat;tam C . J CacMeNalEkgén P elIbnÞat; . bgðajfa J qøúHnwg J eFobnwgbnÞat; BC . 2> bgðajfaplbUk PI PJ efr kalNa P rt;elI BC . 3> P CacMNucmYyelIbnÞat; BC eRkA BC . bgðajfa PI PJ efr . 2n
3n 5
8
506
៩
: ០៣ :
១៩៩៤
៨(
២
២
I.
KNnatémøén edaydwgfa ³
F
x y 1 x y 1
eFobnwgtémø a 0
a x bx 1
. edaHRsaysmIkarxageRkam kñúgsMNMucnM YnKt;rWuLaTIb ³ 5 x 4 x 3 x 2 x 1 x 5 . 1989 1990 1991 1992 1993 rkbIcMnYnKt;rLWu aTIb a , b , c edaydwgfacMeBaHRKb;témø x eK)an ³ x a x 10 1 x bx c . eK[Ggát; AB enAkñúgrgVg;p©it O ehIyEkgnwgkaM OC Rtg;cMNcu D . bnÞat; AC nig BC kat;rgVg;erogKña Rtg;cMNuc E nig F . k> bMPøWfargVgc; arwkeRkARtIekaN ABE kat;tamcMNuc F . x> G CacMNucqøHú éncMNuc C cMeBaHcMNuc O . bMPøWfa CED AGˆ E CFˆD . K> knøHbnÞat;BuHkñúgénmMu ADG kat;Ggát; AG Rtg;cMNuc H . I CacMeNalEkgéncMNuc H DG IH elIGgát; DG . bMPøWfa ³ DADA 1 . DG b 1y a b
II.
III.
IV.
b 1
8
507
៩
: ០២ :
១៩៩៥
៨(
១
២
I. II. III. IV.
KNnakenSam ³ A x 11x 2 2 x23 x 1 x3x 3 . RsaybBa¢ak;fa cMnYn N 4n 3 25 Eckdac;ngw 8 cMeBaHRKb;témøén n . edaHRsaysmIkarkñúgsMNMucMnYnKt;rWuLaTIb ³ 2x 5x x 5x 3 0 . eKedAcMNuc B elIGgát; AC . O Cap©itrgVg;ERbRbYl Edlkat;tamcMNuc B nig C . 1> BM CaGgát;p©ti énrgVg;p©it O . rksMNMuéncMNuc M . 2> bnÞat; AM kat;rgVg;p©it O Rtg;cMNuc P . rksMNMuéncMNuc P . 3> AT CabnÞat;b:HrgVg;p©it O Rtg;cMNuc T . rksMNMéu ncMNuc T . 2
4
3
smÁal; ³ sisSGaceFVIlMhat;NamYymunk¾)an .
8
508
2
៩
: ០២ :
១៩៩៥
៨(
២
២
I.
dak;kenSam A nig B CaplKuNktþa ³ 3 x 53x 32 3 x 12 5 x 2 2 2 2 2 B x 2 x 7 x 2 x 7 A
II.
III. IV.
edaHRsayRbB½n§smIkar ³ 2 x 2 y 3xy . 6 x y 4 xy KNna 7 2 12 4 12 . RtIekaN ABC carwkkñúgrgVg;p©it O kaM R . eKKUskm RsaybBa¢ak;fa ABAH AC 2R . 2> bnÞat;b:HrgVg; O Rtg;cMNuc A kat;bnÞat; BC Rtg;cMNuc I . AB RsaybBa¢ak;fa ICIB AC . 2
2
smÁal; ³ sisSGaceFVIlMhat;NamYymunk¾)an .
8
509
៩
: :
៨(
១
8
510
២
៩
: :
៨(
២
8
511
២
៩
: ០៧ :
១៩៩៧
៩
១
៣
I.
sresrkenSam A nig B CaplKuNktþa ³
A a x2 1 x a2 1
70 cm
B 2 x 3 9x 5 2
II.
III.
2
2 .1 m BinitürUb ³ kñúgbnÞb;mYymankm etIrUbenHRtwmRtUvEdrb¤eT ? O 13 x> cUrKUsrUbenHeLIgvij[)anRtwmRtUv . kñúgtRmúyGrtUNrem xOy eK[cMNuc A1 , 8 ; B 2 , 1 nig C 23 , 7 . RsaybBa¢ak;fa ³ cMNcu A , B nig C rt;Rtg;Kña . RtIekaN ABC carwkkñúgrgVg;p©it O . H CaGrtUsg;énRtIekaN ABC ehIy M CacMNuc kNþalén BC . RsaybBa¢ak;fa AH 2 OM . RsaybBa¢ak;fa RbPaK 2xx 53 mantémøelxCacMnYnviC¢man b¤sUnücMeBaHRKb;témø x .
IV.
V.
2
VI.
4
8
512
2.2 m
៩
: ០៧ :
១៩៩៧
៩
២
៣
I. II.
KNna ³ E 11 6 2 11 6 2 . edaHRsaysmIkar ³ x 3x 10 0 . F KNna ³ S a baa c b cbb a c acc b . 2m xügehonmYyenAcm¶ay 1.50 m BICBa¢aMgEdlman B km RsaybBa¢ak;fa MC nig ND EkgKñaRtg; I . x> eRbóbeFob AI nig BC . 4
2
III. IV.
V.
VI.
2
2
2
8
513
៩
: ១៥ :
១៩៩៨
៩
១
៣
I. II. III. IV. V. VI.
KNnaplbUk ³ S 3 5 7 2 3 . bBa¢ak;dWeRkénsmIkar ³ x 2x 5 x 4 . KNnaplEck ³ D 9x 27 2 4 x 12 ¬eday x 0 ¦ dak;kenSam E 4 x 9 53 2 x CaplKuNktþa . eK[RbPaK F BA 2xx14 . kMNt;sMNMutémøén x EdlnaM[ A 0 nig B 0 . eK[Ggát; AB nigcMNuc C cl½tkñúgbøg;EdlxNÐedaybnÞat; AB . rgVg;Ggát;p©it AC p©it O kat;bnÞat; AB Rtg;cMNuc E ehIykat;rgVg;Ggát;p©it AB Rtg;cMNucmYyeTot D . 1> RsaybBa¢ak;fa cMNuc B , C nig D rt;Rtg;Kña . 2> M CacMNucqøúHén E eFobnwgcMNuc O . rksMNMucMNuc M . 2
2
2
2
2
2
smÁal; ³ ebkçCnGaceFVIlMhat;NamYymunk¾)an .
8
514
៩
: ១៥ :
១៩៩៨
៩
២
៣
I.
II.
III.
IV.
V.
VI.
eK[ ³
x2 2 y2 x2 2 y2 306 294
edaHRsaysmIkar ³
x2 y2
.
x 2 1 1 x 2 x 2 x 2 2 x 2 x 2
.
. KNna
eKEcknMeTA[ekµgbInak; . GñkTImYyTTYl)anBak;kNþal éncMnYnnMsrub nignMmyY kMNat;eTot. bnÞab;mkGñkTIBIrTTYl)an Bak;kNþaléncMnYnnMEdlenAsl; ¬eRkayeBlGñkTImYyTTYlykehIy¦ nignMmYykMNat;eTot . TIbBa©b;GñkTIbTI TYl)an Bak;kNþaléncMnYnnMEdlenAsl; ¬eRkayeBlGñkTIBIr TTYlykehIy ¦ nignMmYykMNat;Tot ehIynMBuMmanenAsl;eToteT . cUrrkcMnYnnMEdleKman . ABC CaRtIekaNEkgRtg; C ehIy CH Cakm rksMNMucMNuc I kalNa AB vilCMuvij P . x> bBa¢ak;TItaMgén AB edIm,I[Ggát;FñÚ AB manrgVas;xøIbMput . smÁal; ³ ebkçCneFVIlMhat;NamYymunk¾)an .
8
515
៩
: ២១ :
១៩៩៩
៩
១
៣
I. II.
III. IV.
edaHRsaysmIkar 2 2 2 2 2 . etImancMnYnKt;b:unµanEdlsßitenAcenøaH 99 nig 9999 edaydwgfacMnYnKt;TaMgenaHCakaerR)akdpg ehIyEckdac;nwg 11 pg . edaHRsayRbB½n§smIkar xxyyz 4 6z 13 . 18
17
15
x
2
rkRkLaépÞénRtIekaN ABC tamrUbxageRkam ³ 5
B 5
4
C
2 A
V.
14
KNnaplbUkmMutamrUbxageRkam ³ S Aˆ Bˆ Cˆ Dˆ Eˆ Fˆ F
C
A
B
D
E
8
516
៩
: ២១ :
១៩៩៩
៩
២
៣
I. II. III.
IV.
cMeBaHRKb;cMnYn x bgðajfa ³ 0 x 61x 10 1 . plbUkénBIrcMnYnesµInwg 1 . bgðajfa plKuNvatUcCag b¤esµInwg 14 . ebI abc 10 KNnatémøénplKuN P Edl ³ 1 1 1 1 1 1 1 1 P . a b c a b c ab bc ac ab bc ac eK[ MNPQ CakaermYyEdlmanRkLaépÞesµInwg 9x ehIy ABCD CactuekaNEkgEdl AM BM CP DP 1 . KNnaépÞRkLaén ABCD . AQ BN CN DQ 2 2
2
A
Q
M
B D P
V.
N
C
nig Q zitenAelIrgVg;mYyEdlmanp©it O . bnÞat;mYyb:HrgVg;Rtg; P ehIyeKedAcMNuc R eday ély:agNa[rgVas; PR esµInwgRbEvgFñÚ PQ . bgðajfa EpñkqUtTaMgBIrmanépÞRkLaesµIKña . P
O
Q M
P
R
8
517
៩
: ១១ :
២០០០
៩
១
៣
I. II.
edaHRsaysmIkar x 6 x 2 3 x . a nig b CacMnYnKt;rWuLaTIb . cUrkMNt; a nig b edIm,I[smIkarxageRkamenHepÞógpÞat;Canic© ³ a 2 b 44 24 2 . enAkñúgtRmúyGrtUNrem eK[cMNucEdlmankUGredaen ³ 0, 0 ; 0,1 ; 0, 2 ; 1, 0 ; 1,1 ; 2, 0 . rkrgVas;RCugénRKb;RtIekaN minb:unKñaEdlmankMBUlCa 3 cMNuc kñúgcMeNamcMNucTaMg 6 Edl[ . ABCD nig DEFG CakaerEdlmanRCugerogKña a nig x . AOˆ B . KNna x CaGnuKmn_én a nig tg . 2000
1998
1999
2
III.
IV.
B a
A
V.
C
F x
E
D
O
G
knøHbnÞat;BuHkñúgénmMu B énRtIekaN ABC mYykat;knøHbnÞat;BuHkñúg nigeRkA énmMu A Rtg; I nig E nigkat;rgVg;carwkeRkARtIekaN ABC Rtg; D . RsaybMPøWfa ID DE .
8
518
៩
: ១១ :
២០០០
៩
២
៣
I.
eK[kenSam ³ A 100 2 99 2 98 2 97 2 96 2 ... 1 B 100 99 98 97 96 ... 1
k> cUreRbóbeFob A nig B . x> etI 2 2 Eckdac;nwg 5 b¤eT ? cUrbgðaj . 20
18
1
1 27 3 A 3 54 3 4 4
II.
K> sresr edaHRsayRbB½n§smIkar ³ x y 35 . xy 6 3
CasV½yKuNén 2 .
3
III.
RkLaépÞctuekaNEkgmYynAdEdl kalNaeKbEnßm 2.5 dm elIbeNþay nigdk 23 dm BITTwg b¤kalNaeKdk 2.5 dm BIbeNþay nigEfm 43 dm elITTwg . KNnaRkLaépÞctuekaNEkgKitCa dm . eragcRkmYymankmµkrelIsBI 600 nak; nigticCag 700 nak; . ebIkmµkrTaMgGs;eFVIkarCaRkumEdl manKña 5 nak; b¤ 7 nak; b¤ 9 nak; enaHKµansl;kmµkreT . rkcMnYnkmµkrTaMgGs; . rgVg;p©it O kaM R nigrgVg;p©it I kaM r R r b:HKñaRtg; A nigb:HbnÞat;EtmYyRtg; B nig C . KNnaRkLaépÞénRtIekaN ABC CaGnuKmn_én R nig r . 2
IV.
V.
8
519
៩
: ០៣ :
២០០១
៩
១
៣
I. II.
cUrRsaybMPøWfa x 13 2 1 2 1 CacMnYnKt;viC¢man . eK[ a , b , c CabIcMnYnEdlepÞógpÞat; TMnak;TMngTaMgbIxageRkamenH ³ a 2b 1 0 ; b 2c 1 0 ; c 2a 1 0 . cUrKNna S a b c . 3
2
IV.
3
3
2
2002
III.
3
2
2002
cUredaHRsayRbB½n§smIkar
2002
x y z xyz x y z xyz x y z xyz
.
tamcMNuc M enAkñúgRtIekaN ABC mYy eKsg;Ggát; EF , GH nig IJ RsberogKñanwg AB , AC nig BC edaycMNuc G nig I enAelI AB . E nig J enAelI AC . F nig H enAelI BC . RtIekaN MEJ; MFH nig MGI manépÞRkLaerogKña S , S nig S . RsaybMPøWfa RtIekaN ABC manépÞRkLaesµInwg S S S . eK[RbelLÚRkam ABCD mYy . enAxageRkA ABCD eKsg;kaer ABMN , BCPQ , CDRS nig W a ctuekaN I I I I Cakaer . ADXY Edlmanp©iterogKña I , I , I nig I . cUrRsaybMPøf 1
2
2
1
V.
1
2
3
3
1 2 3 4
4
8
2
520
3
៩
: ០៣ :
២០០១
៩
២
៣
I.
1> cUreRbóbeFob 1 2000 nig 2001 2 2000 . 2> edaymineRbI 2000 4 000 000 nig 2001 4 004 001 2000 cUrKNna A 1 2000 2000 . 2001 2001 2
2
2
2
2
2
II.
III.
eK[ a b c 1 ; a b c 1 nig ax by cz m . cUrKNna P xy yz zx . eKmansmIkardWeRkTI2 ³ ax 2bx c 0 EdlmanDIsRKImINg; bx 2cx a 0 EdlmanDIsRKImINg; cx 2ax b 0 EdlmanDIsRKImINg; cUrKNna rYcbgðajfa y:agticNas; k¾mansmIkarmYy kñúgcMeNamsmIkarTaMgbIenaH CasmIkarmanb£sEdr . eK[ AA d ; BB d ; AA a ; BB b . A B A nig B sßitenAEtmçagénbnÞat; d . M CacMNuc a b A B A B RbsBVrvagbnÞat; nig . d cUrRsaybMPøWfa cm¶ayBI M eTAbnÞat; d mantémøefr A B kalNacMNuc A nig B cl½telI d ¬cMNuc A epSgBI B ¦. rgVg;p©it O kaM r nigrgVg;p©it O kaM r b:HKñaxageRkARtg; M . rgVg;p©it O kaM r mYyeTotb:H xageRkAnwgrgVg;p©it O pg ehIyb:HxageRkAnwgrgVg;p©it O pg . bnÞat;bH: rYmRtg; M énrgVg; O nig O CYbrgVg; O Rtg; A nig B . I CacMNuckNþalénGgát; AB . cUrRsaybMPøWfa AB 4rr rrr . 2
2
2
2
1
2
2
2
3
1
IV.
2
3
V.
1
2
1
3
2
2
1
2
3
3
1 2
1
2
8
521
3
1
៩
: ២៣ :
២០០២
៩
១
៣
I. II. III.
IV. V.
cUrRsaybMPøWfa ³ 2001 2003 2002 12002 12002 12002 1 2002 1 . cUredaHRsaysmIkar ³ x 2x 4x 2 x 3 x 2 x 7 . eK[ a ; b ; c CabIcMnYnEdlepÞógpÞat;TMnak;TMng abc 1 . cUrRsaybMPøWfa ³ 1 a1 ab 1 b1 bc 1 c1 ca 1 . cUrRsaybMPøWfa ³ 49 20 6 49 20 6 2 3 . eK[ctuekaNEkg ABCD mYy . I CacMNucmYyenAxagkñúgctuekaNEkg ABCD ehIyEdl IA a ; IB b ; IC c ; ID x . cUrRsaybMPøWfa ³ x a b c . ABC CaRtIekaNEkgRtg; A ehIyman ABˆ C 2 . rgVg;p©it M kaM R carwkkñúgRtIekaN ABC . rgVg;p©it N kaM r b:HrgVg;p©it M kaM R ehIyrgVg;p©ti N kaM r enaHb:HnwgRCug AC nig BC rbs; RtIekaN ABC eTotpg . 1> cUrRsaybMPøWfa ³ MN sin R45 r . 2> cUrKNnapleFob Rr . 2
2
4
4
2
8
32
2
4
2
VI.
16
o
8
522
2
2
2
៩
: ២៣ :
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៩
២
៣
I.
II.
III.
ctuekaNe):ag ABCD carwkkñúgrgVg;p©it O mYy. bnÞat; AB CYb CD Rtg; I ehIybnÞat; AD CYb BC Rtg; J Edl IAˆ J 3 ; AJˆB 2 ; AIˆD . cUrKNna KitCadWeRk . eK[ x y 1 nig bx ay Edl a 0 , b 0 . cUrRsaybMPøWfa ³ xa by a 2b . cUredaHRsayRbB½n§smIkar ³ 2
2
2
2
2002
2002
1001
1001
1001
x y y z 187 x; y; z y z z x 154 z x x y 238 1 1 1 2 a b c abc a b c 1 1 1 2 2 2 2 a b c
¬Edl
IV.
V.
VI.
CacMnYnviC¢man¦ .
eK[ nig . cUrRsaybMPøWfa ³ . EFGH CasYnc,arragctuekaNBñay Edlman)atFM EF a nig)attUc GH b . eKEcksYnc,ar enaHCaBIrEpñk EdlmanépÞRkLaesµIKña edayeFVIrbgtam MN Rsb)atTaMgBIrehIy M EH / N FG . cUrKNnaRbEvg MN [Cab;Tak;Tg a nig b . P CacMNucmYyenAelIrgVg;p©it O kaM R Ggát;p©it AB . 1> cUrKNnaépÞRkLaGtibrmarbs;RtIekaN PAB . 2> RsaybMPøWfa PA PB 2R ehIy PA PB 2R 2 . 2
8
523
៩
: ២៨ :
២០០៣
៩
១
៣
I. II.
III. IV. V.
VI.
1 KNnaplbUkedaymineRbI]bkrN_Kitelx ³ S 13 151 351 ... 9999 . cMnYnmYymanelxbIxÞg; Edlman 4 CaelxxÞg;ray. ebIeKelIkRtLb;elxxÞg;ray mkdak;xagmux énelxBIrxÞg;eTot enaHeKnwg)ancMnYnTIBIrfµImYyeTot EdlmanelxbIxÞg; ehIyman 4 CaelxxÞg;ry. ebIeKdwgfa {cMnYnTIBIr} elIs {cMnYn 400 dkcMnYnTImYy} 400 . rkcMnYnTImYyenaH . rkRKb;cMnYnKt;sniTan EdlepÞógpÞat;smIkar 2x 2xy y 25 . 2
bgðajfa
4 5 3 5 48 10 7 4 3 4
2
2003
CacMnYnKt;sniTan .
rgVg;BIrmanp©ti O nig O ehIyenAeRkAKña . bnÞat;b:HrYmeRkAmYyb:HrgVg;TaMgBIrerogKñaRtg; M nig N Edl MN a . bnÞat;b:HrYmkñgú mYyb:HrgVg;TaMgBIrerogKñaRtg; P nig Q Edl PQ b / a b 0 . KNnaplKuNrvagkaMénrgVg;TaMgBIrenaH[Cab;Tak;Tg a nig b . M CacMNucmYyenAxagkñúgRtIekaN ABC mYy. bnÞat; BM CYbRCug AC Rtg; N . bnÞat; CM CYbRCug AB Rtg; L . BLM , CMN nig MBC manépÞRkLaerogKña 5 cm , 8 cm nig 10 cm . KNnaépÞRkLaénctuekaN ALMN . 2
1
2
2
2
8
524
៩
: ២៨ :
២០០៣
៩
២
៣
I.
II.
III.
IV.
A
1 2 3 2002 ... 1 2 1 2 3 1 2 3 4 1 2 3 ... 2003 1 A 1 1 2 3 ... 2003
bgðajfa . eKdwgfa a b a 3b 4a b 10a 3b 29 0 . KNnatémøelxén 2a 4b . kñúgkarRkal)atbnÞb;mYy EdlmanragCactuekaNEkg eK)aneRbI\dækRmalBN’Rkhm nigBN’s EdlmanragCakaer ehIymanRCugRbEvg 5dm . eK)aneRbI\dækRmalBN’Rkhm nig\dækRmalBN’s Gs;cMnYnesµIKañ . bnÞab;enaHmanTTwg 5x nigbeNþay 5 y ¬KitCa dm ¦. eK)anRkal\dæBN’Rkhm TaMgGs;enAtamekonCBa¢aMg EtmYyCYrB½TC§ MuvijbnÞb; ehIyeRkABIenaH eK)anRkal\dækRmalBN’s TaMgGs; . kMNt;cMnYnKt; x nig y TaMgGs;EdlGacman . a , b , c CabIcMnYnKt;EdlepÞógpÞat; a b c 1 ; a b c 1 nig a b c 1 . KNnatémøelxén P a b c . M CacMNucmYyenAkñúgkaer EFGH mYy ehIyEdl MEˆ G MGˆ H 0 45 . KNnargVgs;mMu EFˆM [Cab;Tak;Tgnwg . rgVg;bImanp©iterogKña I , I nig I ehIymankaM R dUcKña . rgVg; I CYbrgVg; I Rtg;cMNuc O nig A . rgVg; I CYbrgVg; I Rtg;cMNuc O nig B . rgVg; I CYbrgVg; I Rtg;cMNuc O nig C . bgðajfacMNuc A , B nig C sßitenAelIrgVg;EtmYy EdlmankaM R Edr . 2
2
2
2002
V.
VI.
.
2003
2
2
3
3
3
2004
o
1
2
3
2
1
3
3
8
525
o
2
1
៩
: ១១ :
២០០៤
៩
១
៣
I. II.
rk x edIm,I[ 3x 6x 9x 12x 1 12 13 14 . KUb 7 b:un²Kña pÁúMCab;KñadUcrUbxagerkamenH ³
maDénsUlItEdlpÁúM)anesµI 189 cm . rképÞRkLaTaMgGs;énsUlItenH . rkRKb;témø m m 0 edIm,I[smIkar mx m 1 0 manb£sviC¢man . ksikrmñak;mandMLÚg 5 l¥IEdlmanTm¶n;erogKña 7kg , 10kg , 14kg , 18kg nig 19kg . Kat;)anlk;dMLÚg 4 l¥I[mnusS 2 nak;. GñkTI1 Tij)andMLÚgEdlmanTm¶n;esµI 2 dgénTm¶n; dMLÚgrbs;GñkTI2 . rkTm¶n;dMLÚgEdlenAsl; . RtIekaN ABC mYymanrgVas;RCug x , x a nig x 2a Edl a CacMnYnKt;viCm¢ an . rkRKb;rgVas;RCugénRtIekaN ABC EdltUcCagb¤esµI 10 edIm,I[RtIekaN ABC CaRtIekaNEkg . RtIekaNmYymanrgVas;RCug 3 , 4 nig 5 . RtIekaNenHcarwkkñúgrgVg;mYy. tag A , B nig C CaépÞ RkLaénEpñkbøg; EdlenAkñúgrgVg; nigenAeRkARtIekaN ehIy C CaépÞRkLaEdlFMCageK . KNna A + B [Cab;Tak;Tgnwg C . 3
III. IV.
V.
VI.
8
526
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: ១១ :
២០០៤
៩
២
៣
I. II. III.
IV.
ebI
etI x 1 3 esµIb:unµan ? a nig b CacMnYnKt;viCm ¢ an . edaydwgfa a b 13 cUrKNna a b . eK[kaer ABCD EdlRCugmanrgVas; 1 Ékta . enAelIRCug BC eKdak;cMNuc E eday BE 2 1 . RsaybMPøWfa AE CaknøHbnÞat;BuHénmMu BAC . eK[cMnYn t 1 , t 3 , t 6 , t 10 , t 15 , t 21 EdlcMnYnnImYy²CaplbUkén cMnYnKt;tKña . edaysegátKRmUEdl[ cUrsresrkenSam t . KNna t t rYcTajrkrUbmnþ sRmab;KNna t . KNna t . mnusScas;mñak;eFVIkargarmYycb;kñúgry³eBl 15h . ekµgmñak;eFVkI argardEdlcb;kñúgry³eBl 20h . k> etImnusScas;mñak; minekµgmñak;rYmKñaeFVIkargarenHcb;kñúgry³eBlb:unµanem:ag ? x> etImnusScas;b:unµannak; nigekµgb:unµannak;rYmKñaeFVIkargarenHcb;kñgú ry³eBl 6h ? ABC CaRtIekaNsm½gS . enAelIbnøayénRCug AB eKdak;cMNuc P ¬ A enAcenøaH P nig B ¦. tag a CargVas;RCugénRtIekaNsm½gS ABC r CakaMrgVg;carwkkñúgRtIekaN PAC r CakaMrgVg;Edl b:HRCugénmMu P ehIyb:HRCug BC xageRkARtIekaN ABC . rgVg;kaM r b:H PA Rtg; D b:H CA Rtg; E nigb:H PC Rtg; H . rgVg;kaM r b:H BC Rtg; F b:H PB Rtg; G nigb:H PC Rtg; I . k> bgðajfa DG HI 3a2 . x> KNna r r eday[Cab;Tak;TgEtnwg a . 1 5 x2
2
1
2
3
4
2
5
6
n
n
V.
VI.
n
100
2
1
1
2
1
n
2
8
527
៩
: ០៥ :
២០០៥
៩
១
៣
I. II. III.
IV.
V.
VI.
ebI A 1 12 14 18 161 nig B 1 12 A . etI B elIs A b:unµan ? sresrcMnYn 93 CaplbUkénsV½yKuNeKal 3 . karRbLgmYyman 25 sMNYr. cemøIyRtUvmYy)anBinÞú 4 ehIycemøIyxusmYy)anBinÞú 1 . sisSmñak; eqøIy)anRKb;sMNYr ehIy)anBinÞúsrub 70 . etIsisSenaHeqøIyRtUv)anb:unµansMNYr ? enAkñúgrUbxageRkammanRtIekaNEdlmanqUt ¬ ¦ kaertUcBN’s ¬ ¦ nigkaerFMenAkNþal BN’exµA ¬ ¦ . tag A CaépÞRkLasrubénRtIekaNqUtTaMgGs; B CaépÞRkLasrubénkaerBN’s TaMgGs; nig C CaépÞRkLaénkaerexµAenAkNþal. bNþacemøIyxageRkamenH etIcemøIyNaxøHCacemøIyRtwmRtUv ? eRBaHGVI ? k> A B x> B C K> 2 A 3C X> 3B 2C . RtIekaN ABC mYymanmMu A 90 nig AB AC 1 . eKbnøay AC eTAxag C [)an CD CB . rkRkLaépÞénRtIekaN BCD KitCaÉktaépÞRkLa . kILakrbInak;Bak;Gav Edlmanpøakelxxus²Kña . kILakrTI1 Bak;GavEdlmanpøakelx1 / kILakrTI2 Bak;GavEdlmanpøakelx2 nigkILakrTI3 Bak;GavEdlmanpøakelx3 . muneBlRbkYtkILakrTaMgbI )anTukGavcUlKñaenAkñgú es,agEtmYy . eBlerobRbkYtkILakrTaMgbInak; lUkykGavBIkñúges,ag mkBak; . Ca]TahrN_kILakrTI1 TI2 TI3 lUkyk)anerogKñaGavEdlmanelx 3 / 2 / 1 . ehIy lT§plsresrCa 3 , 2 , 1 . k> cUrsresrlT§plTaMgGs;EdlGacekItman . x> rkRbU)abénRBwtþikarN_ A ³ {kILakrlUkyk)anGav EdlmanpøakelxminEmnCaelxrbs;xøÜn TaMgbInak; } . o
8
528
៩
: ០៥ :
២០០៥
៩
២
៣
I. II.
III.
IV.
V.
rkcMnYnKt;tUcbMput C C 1 Edl C a b / a nig b CacMnYnKt; . k> sresr 4 2 3 Carag a b . x> KNna 2 2 2 2 4 2 3 . mFümPaKén n cMnYnKt;xusBI 0 a , a , a , ... , a esµI 22 . ebIeKEfm 29 eTAelI n cMnYnKt;TaMg enaH enaHmFümPaKfµIKW 23 . etIeKmancMnYnKt;TaMgGs;b:unµan ? mnusSmñak;rt;edayel,Ón 10 km/h enAelIcm¶aypøÚv 1 km rYcedIrbnþedayel,Ón 5 km/h enAelI cm¶aypøÚv 3 km . rkel,ÓnmFümrbs;mnusSenaH elIcm¶aypøÚvTaMgGs; . enAkñúgkaer ABCD EdlmanrgVas;RCug 4 cm eKKUsRtIekaNsm½gS AEF , E enAelIRCug BC nig F enAelIRCug CD . k> rkrgVas;RCugRtIekaNsm½gS AEF [Cab;Tak;Tgnwg cos15 . 2a 1 x> edaydwgfa cos a 1 cos ni g cUrKNna cos15 . cos30 2 2 K> TajrkrgVas;RCugénRtIekaNsm½gS AEF edayKittémøCacMnYnTsPaKykRtwm 101 én cm . 2
1
2
3
3
n
o
2
o
o
8
529
៩
: ០៥ :
២០០៦
៩
១
៣
I. II. III. IV. V. VI.
smIkar x bx 1 0 manb£smYyesµI 4 dgénm£smYyeTot . etI b esµIb:unµan ? rktémø x EdlepÞógpÞat;smIkar x x 2 0 . KNna X a 1b 1c 1 1 edaydwgfa a b 4 , ab 3 nig ab ac bc 31 . KNna x y z edaydwgfa 2x y z 7 , x 2 y z 8 nig x y 2z 3 . enAkñúgkaer ABCD eKsg;RtIekaNsm½gS AID. KNnargVas;mMu BIˆC KitCadWeRk . rkbIcMnYnKt;vCi ¢manxusKña x , y nig z edaydwgfa xy yz 106 . 2
2 3
1 3
3
8
530
៩
: ០៥ :
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៩
២
៣
KNna S 32 64 96 128 ......200 . 300 KNnaépÞRkLaénrUbxageRkamedaydwgfa ABC nig PQR CaRtIekaNsm½gS EdlmanrgVas;RCug 3 cm ehIyRCugBIrénRtIekaNmYy kat;RCugmYyénRtIekaNmYyeTot)an 3 Ggát;b:un²Kña . 2
I. II.
2
2
2
2
2
2
2
2
2
A P
Q
B
C
R
III.
IV.
kumar 3 nak; A , B , C manXøIerog²xøÜn . ebIsinCa A [XøI 1RKab;eTA B ehIy B [XøI 1 RKab;eTA C enaHGñkTaMgbImancMnYnXøIesµIKña . rkcMnYnXøI Edlmñak;²man edaydwgfamñak;²manXøIticCag 10 RKab; ehIycMnYnXøIrbs; A nig B CacMnYnbzm . enAkñúgrUbxageRkam ABCD CactuekaNBñayEkg . AD DI , M CacMNucrt;enAelI CD . kMNt;TItaMgén M enAelI CD edIm,I[ AM MB manRbEvgtUcbMput . C
B
M A V.
D
I
k> sresrcMnYnbzmEdltUcCag 100 . x> 5 cMnYnbzmtUcCag 100 dUcCa 2 , 41 , 53 , 67 , 89 RtUv)ansresredayeRbIRKb;elxBI 1 dl; 9 ehIyelxnImYy² eRbIEtmþg. cUrrksMNMuén 5 cMnYnbzmdUcenHepSgeTot[)an 5 sMNMu .
8
531
៩
: ១៩ :
២០០៧
៩
១
៣
I.
bursmñak;RtUvkarpwkTwkmþg 700 cm b:uEnþKat;mankaTwkEtBIrKW kamYymancMNuH 500 cm nigka mYyeTotmancMNuH 300 cm . etIbursenaHeFVIdUcemþc eTIbGacpwkTwk[Gs; 700 cm . cUrbMeBjcenøaHenAkñúgRbmaNviFIxageRkam edayeRbIelxxus²KñaBI 1 dl; 5 ³ 2 ..... ..... ..... ...... 6 . ]bmafa a a 1 . cUrKNnatémøénkenSam A a 2a 4a 3a 2 . enAkñúgkareFVIEtsþKNitviTüaelIkTI2 sisSmñak;)anBinÞúelIsBinÞúEdl)ankalBIeFIVEtsþelIkTI1 cMnYn 16 BinÞú. sisSenaHsgÇwmfaxøÜnnwg)anBinÞú 88 kñúgkareFVIEtsþelIkTI3 ehIyBinÞúenHnwgdMeLIgmFümPaK énBinÞúTaMgbIelIkdl; 80 BinÞú. etIsisSenaH)anBinÞúb:unµanxøH enAkñúgkareFVIEtsþelIkTI2 nigenAelIkTI2 ? rkRKb;KUéncMnYnKt;rWuLaTIb x , y EdlepÞógpÞat; 1x 1y 13 . kñúgrgg;p©it O kaM R eKKUsGgát;FñÚ AB Edl AB R rYcKUsGgát;FñÚ AC Edl BAˆ C 45 . k> KNnargVas;mMu ACˆ B nig ABˆ C . x> KNna BC nig AC CaGnuKmn_én R . ¬eK[ cos 45 22 nig cos 30 23 ¦ 3
3
3
II.
III. IV.
V. VI.
3
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4
3
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eday[témøeTAcMnYnKt; n BI 0 , 1 , 2 , 3 , ... cUrbBa¢ak;témø n EdlnaM[vismPaB 2 n Bit . []TahrN_témø X mYy ¬ X CacMnYnBitEdlnaM[smPaB X 1 X 1 Bit¦ nig[témø X mYy EdlsmPaBenHminBit . etItémø X NaxøH EdleFVI[smPaBenHBit ? rkBIrcMnYnKt;viC¢man a nig b EdlmanplbUkesµInwg 92 nig a 1 CaBhuKuNén b . cUrrkcMnYnKt;minGviC¢mant²Kña EdlmanplbUkesµInwg 10 . ¬ GacCaplbUkén 2 cMnYnKt; 3 cMnYnKt; 4 cMnYnKt; >>> ¦. ABC CaRtIekaNsamBaØ EdlmanmMu A CamMuTal . eKKUskm
II.
III. IV.
V.
2
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KNnakenSam A eday[lT§plCabIktþa ³ A 12 2823 6 . cUrbMEbkRbPaK 30 CaplbUkénbIRbPaK EdlRbPaKnImyY ²manPaKEbg CacMnYnbzm . 31 eK[smIkar ax bx c 0 a 0 Edlmanb¤sBIrepSgKñaKW x nig x . k> bgðajfa 2ax b 0 nig 2ax b 0 . x> KNna y 2axx b 2axx b . yuvtInak;manR)ak; 2500 erol ykeTATijEtmbiTsMbuRt[)anBIrRbePT EdlRbePTTI1 mantémø 300 erol nigRbePTTI2 mantémø 500 erol. etIyuvtIenaHTijEtm)anb:unµansnøwk ? etIkñúgenaHman RbePTEtm 300 erol b:unµansnøwk ? nigRbePTEtm 500 erol b:unµansnøkw ? eK[kaer ABCD. eKbnøayRCug AB eTAxag B [)an BE AB . M CacMNucenAkñúgkaereday MAC MCD x . k> KNnargVas;mMu AMˆ C KitCadWeRk . x> bgðajfactuekaN AMCE CactuekaNcarwkñúgrgVg;mYy. bBa¢ak;p©iténrgVg;enH ? K> bgðajfa MC CaknøHbnÞat;BuHmMu EdlpÁúMedaybnÞat; AM nig ME . 2
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V.
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2
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534
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I. II. III. IV.
V.
VI.
eK[BIrcMnYnBit a nig b Edl 2 a 5 nig 3 b 7 . cUrkMNt;témøFMbpM uténRbPaK ba . mnusSmYyRkummanKña 13 nak;. cUrBnül;famanmnusS 2 nak;y:agtic ekItenAkñúgExEtmYy . cMeBaHcMnYnBitxusKña a , b , c cUrKNna ³ A a baa c b abb c c a c c b . sresrcMnYnEdlmanelxbIxÞg; edayeRbInitþsBaØa abc ehIy abc 100 a 10b c ¬]TahrN_ 432 100 4 10 3 2 ¦ . cUrrk a nig b edaydwgfa 5ab 500 500 ab5 . RtIekaNmYymanrgVas;RCug a , b , c EdlepÞógpÞat; 24a 18b 12c . k> KNnargVas;RCugénRtIekaN edaydwgfa b c 10 cm . x> KNnargVas;km
B 5cm
A
10cm 8cm
4cm
K
H 14cm
D
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I. II.
III.
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etImancMnYnKt; n b:unµanxøH EdlepÞógpÞat; 72 13n 54 . 24 B E RtIekaN ABC RtUv)ankat;ecjBIRkdasragCactuekaNEkg 12 dUcrUbxagsþaM . etIRkdasEdlenAsl;manb:unµanPaK ? A C 10 D 1> ebI x 1 2 2 etI x 1 3 esµIb:unµan ? 2> ebI x 1 3 3 etI x 1 4 esµIb:unµan ? 3> Tajrk x 1n 1 ebIeKdwgfa x 1 n n . A r)arEdkRtg;mYymanRbEvg 25m RtUv)aneKdak;Ep¥kcugxagelI C 25 m eTAelICBa¢aMgQrmYy Edlcm¶ayBICBa¢aMeTAcugr)arxageRkam manRbEvg 20 m . ebIeKbgçitcugr)arxageRkamecjBICBa¢aMg O 20 m Efm 4 m eTot etIcugr)ar)anFøak;cuHcMnYnb:nu µan m ?¬emIlrUb¦ D 4 m B Rcvak;EdlmankgmUlb:un²Kña 2 manRbEvg 12 cm . Rcvak;EdlmankgmUlb:un²Kña 5 ¬b:nu nwgkgRcvak; mun¦ manRbEvg 27 cm . etIRcvak;kgmUlb:un²Kña 40 ¬b:unnwgkgRcvak;mun¦ manRbEvgb:unµan ?
12 cm
VI.
27 cm
RtIekaN ABC mYyEkgRtg; A ehIymanbrimaRt 60 cm nigépÞRkLa 120 cm . cUrrkrgVas;RCugnImYy²rbs;RtIekaN ABC . 2
8
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I. II.
bgðajfaRtIekaNEdlmanrgVas;mMu ; 2 ; 3 CaRtIekaNEkg . cUrrktémø x nig y enAkñúgrUbxageRkamenH ³ 40 o 40 x
III. IV.
V.
VI.
o
3x
o
yo
o
80 o
eKdwgfa f x f x 1 9 nig f 3 81 . cUrrk f 9 . ABC CaRtIekaNEkgRtg; A . eKKUskm
nig z CabIcMnYnKt;viC¢mantUcCag 9 . rkRKb;cemøIyrbs;smIkar x y z 20 .
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eK)ansaksYrsisSmkRbLgsisSBUEkcMnYn 240 nak; GMBImeFüa)ayeFVIdMeNIrmkkan;raCFanIPñMeBj. GMBImeFüa)ayeFVIdMeNIrTaMgenaHrYmman ³ A: rfynþRkug B: rfynþtak;sIu B 135 C: rfynþpÞal;xøÜn nig D: eTacRkyanynþ . RkabpøitenHbgðajBIlT§pl A 120 énkarsaksYr ¬dUcrUb¦. cUrrkcMnYnsisSEdleFVIdMeNIrtammeFüa)aynImYy² . II. sisSmYyRkum)anRBmeRBógcUlluyKñaedIm,IeFVIBiFICb;elogmYy. luyEdlRbmUl)ansrub 770 000 erol edaydwgfasisSRbusmñak;RtUvbg; 30 000 erol nigsisSRsImñak;RtUvbg; 20 000 erol. cUrkcMnYnsisSRbus nigcMnYnsisSRsIénRkumenH . III. cUrrkcMnYnKt; x Edl 700 x 800 edaydwgfaplbUkRKb;xÞg;én x esµI 17 nigebIeKbþÚrelxxÞg;ryeTA xagsþaMéncMnYnenaH eK)ancMnYnfµImYyticCagcMnYnedImc 441 . 1 1 1 1 IV. cUrKNnakenSam ³ A 1 1 1 ... 1 2 3 4 n 1 B 0.001 0.02 0.0001 10 10 . 3x 7 y m V. 1> eK[RbB½n§smIkar . cUrkMNt;témø m edIm,I[RbB½n§smIkarmanKUcemøIyviC¢man. 2 x 5 y 20 2> cUredaHRsaysmIkar a b x b x a x a b 0 ¬ x CaGBaØat ehIy a , b 0 ¦ . 3> eK[smIkar 2x 1 5 ax . cUrrktémø a edIm,I[smIkarmanb£stUcCag b¤esµI 1 . VI. TIFøamYymanépÞRkLaesµI 864 m eKEckTIFøaenHCabIcMENk. eKdwgfacMENkGñkTIbI esµInwgplbUk cMENkGñkTImYy nigcMENkGñkTIBIr. pleFobcMENkGñkTImYy nigcMENkGñkTIBIresµInwg 11/ 5 . rképÞRkLaéncMENknImYy² . VII. eK[RtIekaNsm½gS ABC EdlRCugmanrgVas;esµI 2a . I CacMNuckNþal AB . B enAxageRkARtIekaN ABC sg;knøHrgVg;BIrEdlman AI nig BI CaGgát;p©itrYcsg; A FñÚ énrgVg;Edlmanp©it A kaM AB EdlFñÚénrgVg;p©it A kaM AB x½NÐedaycMNuc B nig i B kaM AB Edlx½NÐedaycMNuc A nig C . eK)an C nigsg;FñÚénrgVg;Edlmanp©t C rUbmYy ragebHdUg ¬dUcrUbxagsþaM ¦. KNnaépÞRkLaénrUbebHdUgenH . VIII. eK[RtIekaNEkg ABC EkgRtg; A EdlRCugénmMuEkgmanrgVas;esµI b nig c . AI CaknøHbnÞat;BuHénmMu BAC . cUrKNnargVas; AD . I.
o
D 45o
C 60
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2
2
2
2
4
5
4
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2
2
2
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538
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1> cUrKNnakenSam ³ A 40 2 57 40 2 57 . 2> eK[smIkar x 2ax k 0 manb£smYyesµI a b 0 . cUrkMNt;témø k . 3> cUredaHRsayRbB½n§smIkar EdlmansmIkarnImYy²KW ³ 12xx y ; 12yy z ; 12zz x . eK[RtIekaN ABC Edlman AB c ; AC b nig BC a . D ; M nig N CaeCIgénknøHbnÞat;BuH mMukñúgTaMgbIrbs;RtIekaN edaydwgfa AD x ; BM y nig CN z . cUrRsaybMPøWfa 1x 1y 1z 1a b1 1c . f x CaGnuKmn_kMNt;cMeBaHRKb;cMnYnBit x nigepÞógpÞat; xf x 2 x 9 f x . cUrRsaybBa¢ak;fasmIkar f x 0 manb£sbIy:agtic . cUrrkcMnYnEdlRtUvdak;bnþenAral;esrILÚsIuk kñúgcMeNamcemøIyEdl)anesñIeLIgTaMgR)aM . ]TahrN_ ³ 3 6 12 24 ? k> 72 x> 18 K> 37 X> 48 g> 11 cemøIyRtwmRtUvKW X> 48 ¬eRBaHcMnYnEdl)anRtUvKuNnwg 2 ral;elIk¦ . 1> 1 5 9 13 ? k> 15 x> 16 K> 17 X> 18 g> 19 . 2> 5 2 4 1 3 ? k> 0 x> 1 K> 2 X> 3 g> 4 . 3> 6 9 13 18 ? k> 23 x> 24 K> 25 X> 26 g> 27 . 4> 18 54 45 135 126 ? k> 136 x> 257 K> 378 X> 499 g> 620 . P ; Q ; R nig S CacMNuckNþalerogKñaénRCug AB ; BC ; CD nig DA rbs;RbelLÚRkam ABCD . cUrKNnaépÞRkLaénctuekaNEdlpÁúMedaybnÞat; AQ ; BR ; CS nig DP edaydwgfa épÞRkLarbs;RbelLÚRkam ABCD esµI a . eK[ctuekaNBñaysm)at ABCD mankm
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VI.
2
2
2
8
539
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1> eKmanbnÞHelah³mYysnøwkEdlmankRmas; 3mm ehIymanm:as 264kg . cUrrképÞRkLaénbnÞH elah³enHKitCa m edaydwgfam:asmaDénelah³enH 8kg/dm . 2> eKmancMnnY Kt;tKña 91 , 92 , 93 , 94 , 95 , 96 , 97 , 98 nig 99 . cUrdak;cMnYnKt;xagelIkñúgkaertUc²énrUbxagsþaM eFVIy:agNa[)an ³ plbUkCYredknImYy² esµInwgplbUkCYrQrnImYy² esµInwgplbUkCYr Ggát;RTUgnImYy² esµnI wg 285 . ¬smÁal; ³ cMnnY nImYy²eRbI)anEtmþg¦ 1> cUrKNnakenSam D 243 81 4 16 . 2> x nig x Cab£sénsmIkar ³ x 2m 1x m 2m 0 . cUrbgðajfa x x CacMnYnefr. 3> cUrrktémø m edIm,I[RbB½n§smIkar ³ 32xx 75 yy m20 manKUcemøIyviC¢man . RkumsamKÁImYymandIERsBIrkEnøgmanragCakaerdUcKña EdlmanépÞRkLaxusKña 75m ehIyplbUk brimaRtdIERsTaMgBIrkEnøg manrgVas;esµI 100 m . cUrKNnaplRsUvtamkEnøgnImYy² edaydwgfa ERsTaMgBIrkEnøg TTYlpl)an 0.35kg/m . cUrrkKUéncMnYnKt;rWuLaTIb x , y EdlepÞógpÞat; x 249 xy 250 y 6033 . A B eK[kaer ABCD EdlmanrgVas;RCugesµI a . enAkñúgkaerenH M sg;mYyPaKbYnénrgVg; EdlmankaMman rgVas;esµI a ehIyp©it Q N rbs;vaCakMBUlTaMgbYnrbs;kaer ABCD . P C D cUrKNnabrimaRtrUbpáakUlab AMBNCPDQA Edlsg;)anenH . Ggát;RTUgénctuekaNBñay ABCD EckctuekaNBñayCaRtIekaN 4 . s nig s CaépÞRkLaén RtIekaNEdlmanRCugmYyCa)at rbs;ctuekaNBñay . cUrrképÞRkLa S énctuekaNBñay CaGnuKmn_eTAnwg s nig s . 3
2
II.
0. 2
0.25
0. 5
2
1
III.
0.75
2
1
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2
2
IV. V.
VI.
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2
1
1
2
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540
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1> cUrRsaybBa¢ak;fa 3 2> cUredaHRsaysmIkar
2
2
3
.
x 4 x 3 x 2 x 1 0 2008 2009 2010 2011 x y 5 x y0 x 6 y x y 5
.
3> edaHRsayRbB½nsmIkar II. III.
cMeBaH
.
cUrrkcMnYnmYyEdlmanelx 4 xÞg; abca edaydwgfa abca 5c 1 . eK[kenSam M x y 2 z t cMeBaH x , y , z nig t CacMnYnKt;minGviC¢man . cUrrktémø tUcbMputén M nigbNþatémøRtUvKñaén x , y , z nig t edaydwgfa x y t 21 nig x 3 y 4 z 101 . a , b nig c CargVas;RCugrbs;RtIekaN . cUrRsaybBa¢ak;fa ³ 3ab bc ca a b c 4ab bc ca . kaermYyRCugmanrgVas;esµInwg 1 ÉktaRbEvg. tamcMNucRbsBVénGgát;RTUgkaer eKKUsbnÞat;cl½tmYy. cUrKNnaplbUkkaeréncm¶ay BIkMBUlTaMgbYnrbs;kaer eTAbnÞat;cl½t . eK[RtIekaNsm)at ABC kMBUl A manépÞRkLaesµI S . MN Ca)atmFümRtUvnwg)at BC ehIy O CacMNucRbsBVén MN nigkm
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¬10BinÞú¦ KNnaplbUk ³ 1> A a a1 1 a 11 a 2 ... a 20111 a 2012
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a0
2> B 2011 2012 2012 ... 2012 2012 2012 1 1 . ¬10BinÞú¦ 1> cUrrktémøénb£skaer ³ 2012 2011 2012 2011 2012 2011 ... . 2> M nig N CaBIrcMnYnénkaerR)akdtUcCag 100 . ebI M N 27 cUrKNnatémøén M N . ¬10BinÞú¦ cUreRbóbeFobBIrcMnYnxageRkam ³ 1> A 2013 nig B 2010 2012 2014 2016 . 2> P 2 3 4 nig Q 3 24 . ¬10BinÞú¦ haglk;smÖar³kILamYy )aneFVIkmµviFIlk;BiessdUcteTA ³ ral;GtifiCnTI 25 RtUvTTYl)an rgVan;)al;mYy nigral;GtifCi nTI 35 RtUvTTYl)anrgVan;GavyWtmYy . vIr³ CaGtifiCnTImYy EdlTTYl )anrgVan;TaMg)al; nigGavyWt. 1> etImanGtifiCncMnYnb:nu µannak; Edl)ancUlTijhagenHmunvIr³ ? 2> dar:a CaGtifiCnEdlTTYlrgVan;mun vIr³ ehIybnÞab;mkKWvIr³ . etIdar:aCaGtifiCnTIb:unµan ? ehIyTTYl)anrgVan;GVI ? ¬10BinÞú¦ kñúgkarsikSaRsavRCavsisS 8000 nak;)an[dwgfa sisSEdlcUrcitþeronKNitviTüa mancMnYn esµInwgbIdgéncMnYnsisSEdlcUlcitþeronrUbviTüa nigmancMnYn 48 nak; cUlcitþeronTaMgBIrmuxviC¢a . cUrrkcMnYnsisSEdlcitþeronrUbviTüa nigcMnYnsisScUlcitþeronKNitviTüa . ¬10 BinÞú¦ xYbkMeNItTI n rbs; davId nwgRbRBwtþeTAenAqñaM n énstvtSTI 21 enH. etIxYbkMeNItelIkTI 32 rbs;edvIdeFVIenAqñaMNa ? ¬20 BinÞú¦ ABCD CactuekaNEkgmYy Edl AB a nig AD b , E CacMNucmYysßitenAelIRCug CD . RtIekaN AED manRkLaépÞesµInwg 15 énépÞRkLactuekaNe):ag ABCE . cUrKNnaRbEvgGgát; DE [Cab;Tak;TgeTAnwg a . 9
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¬20 BinÞú¦ PQRS CactuekaNe):agEdlmancMNuc A , B , C nig D CacMNuckNþalerogKñaénGgát; PQ , QR , RS nig SP ehIy M CacMNuckNþalénGgát; CD nig H CacMNucmYyenAelIGgát; AM Edl HC = BC . bgðajfa BHM 90 . o
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¬15 BinÞú¦ 1> eK[ x
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.
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1 n
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. n
.
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A
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.
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IV.
bgðajfa ³ 6 2 5 13 48 3 1 . 3 x 2 x x 1 edaHRsaysmIkar ³ 1997 3 . 1998 1999 bIcMnYn x , y , z smamaRtnwg 5 , 7 , 11 . 1> KNnacMnYnTaMgbIenH kalNa ³ k> x 2 y 3z 48 x> x my z 72 ¬ m CacMnYnKt;EdleK[¦ rktémø m minRtUvman . 2> kñúglT§plénsMNYr 1> ¬x¦ eKcg;[ x , y , z CacMnYnKt;viC¢man. etIcMnYn 16 7m RtUvbMeBj lkçxNÐy:agNa ? etImantémø m b:unµanEdlbMeBjlkçxNÐenH . 3> eKcg;[ m CacMnYnKt; . KNna m ¬mancemøIyeRcIn¦ nigtémørbs; x , y , z . eK[kaer ABCD manRCugRbEvg a . eKcg;sg;RtIekaNsm½gS EdlmanGgát;RTUg AC CaRCug ehIy mankMBUl E enAxag B . O CacMNucenAkNþalén AC . k> bgðajfacMNuc E , B , O nig D rt;Rtg;Kña . x> KNna BE CaGnuKmn_én a . K> bgðajfa EB ED AB . X> BI E eKKUsbnÞat; EP b:HrgVg;carwkeRkAkaer ABCD Rtg; P . bgðajfa EP a . 2
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545
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KNnakenSamxageRkam ³ k> Y 169 14 52 2 1 16 5 4 19 27 3
II.
x> Z 11 2 2 1 3 3 1 4 49 1 50 50 1 51 . RsaybBa¢ak;faebI a , b , c CargVas;RCugénRtIekaNmYy enaHeK)an ³ a b c 2ab bc ca . edaHRsayRbB½n§smIkar xxy 6y 35 . 2
2
2
3
III. IV.
V.
3
sYnc,arragctuekaNEkgmYy manépÞRkLa 720 m . ebIeKbEnßmbeNþay 6 m nigbnßyTTwg 4 m enaHépÞRkLasYnc,arminpøas;bþÚreT . KNnavimaRténctuekaNEkgenH . eK[RtIekaN ABC mYy carwkkñgú rgVg;p©it O kaM R . bnÞat;KUsBI B RsbnwgbnÞat;b:HrgVg;Rtg; A kat;bnÞat; AC Rtg; D . 1> bgðajfa ABC~ ADB rYcbBa¢ak;fa AB AC AD . 2> bnÞat;mYyKUsBI B RsbnwgbnÞat;b:HrgVg;Rtg; C kat; AC Rtg; E nigrgVg;Rtg; F . cUreRbobeFob ABC nig BEC . 3> bgðajfa BED nig BFC CaRtIekaNsm)at . 4> eKsnµt;fa BAC 60 . R)ab;eQµaHRtIekaN BFC rYcKNna BF CaGnuKmn_én R . 2
2
o
PñMeBj> éf¶TI 28 Ex kumÖ³ qñaM 2003 GñkeFVIviBaØasa eday Tk; yUGUn
3
546
៩
៩ :
I. II. III. IV.
V.
bM)at;r:aDIkal;BIPaKEbg edaHRsaysmIkar ³ 32 x
3
11 2m 3 12n 1 2 2 x 0 3 3 1 2 4 2 y x4 y 2x 7 3
. .
begáItsmIkaredaysÁal;bs£ nig 1 4 2 . 1> edaHRsayRbB½n§smIkar . 2> sg;bnÞat;EdlmansmIkar TaMgBIrelIRkabPicEtmYy . bBa¢ak;cemøIyelIRkaPic ebI M CacMNuc RbsBVénbnÞat;TaMgBIr . 3> sresrsmIkarénbnÞat;Edlkat;tamcMNuc P 2 , 5 nig Q4 , 2 . bgðajfabnÞat;enHkat;tam M . 4> bnÞat;TI1 kat;G½kS xx Rtg; A ehIybnÞat;TI2 kat;G½kS yy Rtg; B . sresrsmIkarbnÞat; AB . 5> bgðajfa AM BM rYcKNnaRkLaépÞRtIekaN AMB . eK[rgVg; C O , R nigrgVg; C O , R b:HKñaxageRkARtg; A eday OA 4 cm nig OA 2 cm . tam A eKTaj ExSFñÚEkgKña eday AB C O , R nig AC C O , R . k> bgðajfa OB // OC . x> BC OO I . bgðajfa I CacMNucnwg ehIybBa¢ak;TItaMgva edayKNna IO nig IO . K> eKKUs OH Ekgnwg BC Rtg; H . rksMNMucMNuc H kalNa B rt;elI C O , R . eFVIenAERBklab> éf¶TI 10 Ex mIna qñaM 2003 GñkeFVIviBaØasa eday hIug sm,tþi
3
547
៩
៩ :
I. II.
dak; E CaplKuNktþa E ab c ba c ca b . eK[cMnYnviC¢man a , b , c , m , n , p Edl a b c 1 . eKdwgfacMnYn a , b nig c RcassmamaRt erogKñaeTAnwgcMnYn m , n nig p . bgðajfaeK)an am bn cp 1 11 1 . 2
2
2
2
2
2
m
III. IV.
V. VI.
n
p
eK[kenSam A xx 13 . kMNt;cMnYnKt; x viC¢man edIm,I[kenSam A CacMnYnKt;viCm¢ an . RsaybBa¢ak;fa smIkarxagsþaMenHmanb£sCanic© ³ x ax b x bx c x cx a 0 cMeBaH x CaGBaØat nig a , b , c CacMnYnBit . RsaybMPøWfa 49 20 6 49 20 6 2 3 . eK[RtIekaN ABC EdlRCugnigkm
4
1
1
VII.
2
3
1
2
3
2
3
eK[RtIekaNsm)at OAB Edlman)at AB nig AOB 30 . bnÞat;Ekgnwg OA Rtg; O kat; bnøayén)at ABRtg; D . bnÞat;Ekgnwg OB Rtg; B kat; OD Rtg; E . RsaybMPøWfa RtIekaN BED CaRtIekaNsm)at . o
GUrEbkk¥m> éf¶TI 18 Ex mIna qñaM 2004 GñkeFVIviBaØasa eday Kwm b‘unFI
3
548
៩
៩ :
I.
cUrKUssBaØa kñúgRbGb; cMeBaHcemøIyEdlRtwmRtUv ³ eK[bIcMNuc A , B , C mankUGredaenerogKñaKW 3 , 1 ; 4 , 4 nig 2 , 5 . k> kUGredaenéncMNuc I kNþal AC KW ³ I 12 , 23 I 3 , 92
. x> kUGredaencMNuc D qøúHénc B eFobnwg I KW ³ D 3 , 2 D 1 , 2 I 2 , 1 . 5 I , 3 2
K> tamrUbxageRkamenH eK[ d // d . KNnatémø x ³ 1
2
10
cemøIy ³
x 6
12
x 66
x 36
x 11
.
x
II.
k> KNna A 16 16 Edl x CacMnYnkt;viC¢man . x> edaHRsaysmIkar x x 2 1 A . K> rktémø a EdlnaM[bnÞat; y a 2 x 3 EkgnwgbnÞat; y 12 x 1 . rYcsg;bnÞat;TaMgBIrenH kñúgtRmúyGrtUNremEtmYy . X> rktémø a nig b EdlnaM[RbB½n§smIkarxageRkam manKUcemøIyeRcInrab;minGs; ³ 4 x 1
1> III. IV.
x
x
ax by 10 4 x 3 y 5
2>
6 x y 30 1 b 1 x y b a 3
.
eK[BIrcMnYn tUc b¤FMCagKñabIÉkta ehIyplbUkkaeréncMnYnTaMgBIrenH 89 . rkcMnYnTaMgBIrenH . eK[RtIekaN ABC carwkkñúgrgVg; . knøHbnÞat;BuHén BAC kat;RCug BC Rtg; I nigFñÚ BC Rtg; J . 1> RsaybMPøWfa ABI AJC rYcbBa¢ak;fa AI AJ AB AC . 2> Bgðajfa BAJ CBJ . 3> RsaybMPøWfa JB JA IJ . PñMeBj> éf¶TI 28 Ex kumÖ³ qñaM 2003 GñkeFVIviBaØasa ³ R)ak; kaNaDI 2
3
549
៩
៩ :
I.
RsaybBa¢ak;fa ³ k> 11a aa a 11 aa
x>
2 3 2 3 6
2
1
.
.
2 2 x x 1 1 1 A 2 2 15 x 1 2 x 1 2 x 1 1 1 3 3
II.
KNnatémøénkenSam ³
III.
KNnatémø x ; y Edl y 0 BIsmIkar ³
IV.
V.
eK[ x ; y ; z Edl
x2 2 y 1 0 2 y 2z 1 0 z2 2x 1 0
x2 4x y 6 y 13 0
. KNnakenSam ³ 2
2
VI.
2
2
3
550
.
A x 2000 y 2000 z 2000
eK[RtIekaN ABC mYyman AM Caemdüan nigkm
.
.
៩
៩ :
I.
BICKNit 1> edaHRsaysmIkar ³ 4x 3x x . 2> eK[ M 1 , 2 CacMNuckNþalénGgát; AB Edl A 2 , 2 . KNnakUGredaenéncMNuc B . 3> KNna A x 1 x x 1 x x2x 1 . 4
2
4> sRmÜlkenSam ³
2
3
2
2
xa xa 2 xa F xa xa 1 xa
.
5> eTscrN_mñak;ecjdMeNIrenAem:ag 7 BIeCIgPñMeTAkMBUlPñMkñúgel,ÓnmFüm 300 m/h . luHcuHBI elIPñMmkvijedayel,ÓnmFüm 450 m/h mkdl;eCIgPñMvijenAkEnøgecjdMeNIr. enAevlaem:ag 18h 45mn . kñúgry³eBleFVIdMeNIr eTscrN_enaHsRmakGs; 4h 15mn . rkcm¶aypøÚvBI eCIgPñMeTAkMBUlPñM . 6> edaHRsayRbB½n§smIkar ³ II.
2 x y 2 x y x 24 2 x y 2 x y 2 x y 11 3 y 2 x 1
1 2
.
FrNImaRt 1> CD Cakm eK[rgVg;p©it O BIcMNuc A mYyelIrgVg; eKKUsGgát;FñÚBrI AB , AC enAmçagénGgát;p©itKUs ecjBI A . bnÞat;b:HRtg; A nig C énrgVg;Rtg; S . Ggát;RTUgmYyEkgnwg AB kat; BC Rtg; T RsaybMPøWfa ³ k> SAT nig TAB sm)at ehIymMu)atrbs;vamanrgVas;esµIKña . x> R)aMcMNuc A , S , C , T nig O sßitenAelIrgVg;EtmYy . K> ST // AB .
3
551
៩
៩ :
I. II. III.
IV.
KNna A 19781979 1979 1979 ... 1980 1 . eK[ x 3 2 3 2 KNna f x x 3x . eK[smIkar x a b x b c x a ca b c abcx . edaHRsaysmIkarebI a 2 , b 3 , c 4 . rktémøKt;én x , y nig z EdlepÞógpÞat;smIkar x 4 y 1 z 25 80xyz . edaHRsayRbB½n§smIkar ³ xy 73xx 37 yy 00 . 9
3
8
7
3
3
2
2
2
3
V.
3
rktémøKt;én x , y nig z EdlepÞógpÞat;smIkar x y z 4 2 x 2 4 y 3 6 z 5 . VII. eK[RtIekaN ABC manemdüan AM , BN , CP . 1> RsaybMPøWfa AM BN CP AB BC AC . 2> RsaybMPøWfa AM BN CP AB BC2 AC . VIII. eK[mMuEkg xoy enAelI ox eKedAcMNuc B nigenAelI oy eKedAcMNucbIKW C , D nig E Edl OC DE CD OB . RsaybMPøWfa ODB OEB 45 . VI.
o
¬viTüal½ysn§rm:uk¦ PñMeBj> 03 mIna 2004 GñkeFVIviBaØasa eday Ect lI
3
552
៩
៩ :
I.
cUrKUssBaØa kñúgRbGb;xagmuxcemøIyEdlRtwmRtUv ³ 1> kenSam A 53 95 13 45 5 15 mantémø ³ k> A 33 x> A 5 K> A 3 5 X> A 55 . 2> ABC CaRtIekaNEkgRtg; A Edl BC 5 3 cm nig AC 5 2 cm . AB manRbEvg ³ B
II. III.
IV.
V. VI.
k> AB 2 5 cm x> AB 4 cm K> AB 5 cm X> AB 5 cm A eRbóbeFobcMnYn 3 3 nig 6 2 3 . plbUkBIrcMnYnesµI 84 ÉplEckvaesµInwg 3 . rkcMnYnTaMgBIrenaH . edaHRsayRbB½n§vismIkar
8 3x x 3 3 4 x 1 x 2 5
5 3 cm
5 2 cm
C
.
KNna ¬KitCa cm¦ RbEvgRCugTaMgbI x , x 2 , x 4 énRtIekaNEkgmYy . edaHRsaysmIkar ³ k> 2 x 2 35 4x 43 12 x 0
x> 4x 2x 1 19x 3 . VII. ABC CaRtIekaNEkgRtg; A Edl BC 2a nig Bˆ 60 . KNnargVas;RCug AB nig AC CaGnuKmn_én a . VIII. kñúgRtIekaN ABC eKKUskm eRbóbeFob AC ni g rY c ni g . AE AF AD AG x> bgðajfa AC AF AB AG . K> bgðajfa BC // GF . ¬GnuvTi üal½y h‘un Esn burI 100 xñg ¦ 2
o
3
553
៩
៩ :
I.
KNna
P
III.
IV.
V. VI.
1 1 2a 4a 3 8a 7 2 a b a b a b 2 a 4 b 4 a 8 b8
. eK[smIkar 3a 3b 10ab nig b a 0 . rktémøénRbPaK F aa bb . cMBYyTwkbI EdlkñúgenaHmanBIrbBa©ÚlTwkdak;kñúgGag nigmYyeTotbeBa©jTwkBIGag. edIm,IbMeBjGag cMBYyTI1 RtUvkarry³eBl 1h 30mn . cMENkÉcMByY TI2 vijvaGacbMeBjGagkñúgry³eBlminsÁal;. edIm,IbeBa©jTwk[Gs;BIGag cMBYyTI3 RtUveRbIry³eBl 2h . ebIeKebIkcMBYyTaMgbI[hUrTwkRBmKña eK segáteXIjfa GagenaHeBjkñúgry³eBl 40mn . etIcMBYyTI2 RtUveRbIry³eBlb:unµanedIm,IbMeBjGag? E
II.
1 1 1 2 2 2 2 b c a ac b bc c a b ab c ac a b c bc a 2 ab 2
2
eK[
2
a b c 1 2 2 2 a b c 1 x y z a b c
1 2
. RsaybBa¢ak;fa
xy yz xz 0
.
3
eK[RtIekaN ABC carwkkñúgrgVg;p©it O sg;km
¬TYlsVayéRB ¦
3
554
៩
៩ :
I. II. III. IV. V.
VI.
RsaybMPøWfa 4 2 3 4 2 3 2 3 . KNnatémøelxénkenSam A 1 1 2 2 1 edaHRsayRbB½n§smIkar
x 2 y 2 130 xy x y 47
3
1 1 ... 3 4 99 100
.
.
eKman a b c 0 nig ab bc ca 4010 . cUrKNna a b c 0 . 2 eK[ctuekaNBñaysm)at ABCD man BC C B Ca)attUc Edl BC 16cm . Ggát;RTUg BD nigRCugeRTt AB EkgKñaRtg; B . A D O H eK[ BD 40 cm . KNna AD x . eK[RtIekaNsm½gS ABC Edl AB AB BC a ehIyman AH Cakm KNna CM nig BM CaGnuKmn_én a . 2> KNnaRkLaépÞctuekaN AHCM CaGnuKmn_én a . 4
4
4
¬vi> vtþekaH ¦
3
555
៩
៩ :
I. II. III.
IV.
V.
VI.
x x 1 x 2 x 3 KNna x Edl 1999 4 . 2000 2001 2002 eRbóbeFobcMnYn 14 48 nig 7 4 3 7 4 3 edaHRsayRbB½n§smIkar nigBiPakSatémø m xageRkam ³ x my 1 . mx 3my 2m 3 fñalsMNabmYymanragCactuekaNEkgmanRkLaépÞ 400 m nigbeNþaymanRbEvgelIsTTwg 23 m . cUrrkvimaRtfñalsMNabenaH . cUrsg;bnÞat; D : y 2x 5 nigeK[bnÞat; : m 1 x my m 2 , m Ca):ar:aEm:t . k> KNnakUGredaencMNucRbsBV I rvag D nigbnÞat; cMeBaH m 13 . x> KNna m edIm,I[bnÞat; D . K> KNnaépÞRkLaénRtIekaN EdlpÁúMedaybnÞat; D nigG½kSkUGredaen . cUrsresrCaRbB½n§bIvismIkar ¬kñúgrUb¦ ³ D 4
4
2
2
4
I D3 tbMnc;eMly 1 2
VII.
o
4
D1
eK[rgVg;BIr (O) nig (O’) manp©it O nig O’ mankaMRbEvg R < R’ . rgVg;TaMgBIrb:HKñaxageRkARtg; A . eKKUsbnÞat;b:HrYmkñúg nigbnÞat;rYmeRkA Edlb:HrgVg; (O) Rtg; B nigb:HrgVg; (O’) Rtg; C . bnÞat;b:HTaMg BIrkat;KñaRtg; M . 1> bgðajfa MA MB MC ehIyfaRtIekaN ABC EkgRtg; A . 2> bnÞat; BA nigbnÞat; CA kat;rgVg; (O’) nig (O) Rtg; D nig E . bgðajfa BE CaGgát; p©itrgVg; (O) ehIy CD CaGgát;p©iténrgVg; (O’) . R)ab;eQµaHctuekaN BCDE . 3> bgðajfa BEA ABC eRbóbeFobRtIekaN BCD nig BCE rYcbgðajfa BC BE CD . 4> eK[ R 4.5cm nig R 2cm . KNna BC nigRkLaépÞctuekaN BCDE . 2
3
556
៩
៩ :
I.
eK[ 2a 2b 5ab Edl b a 0 . cUrKNnatémøelxén F aa bb . eKman ax by 0 . bgðajfa a a b x x y 1 . 2
2
2
II. III.
2
2
cUreRbóbeFobpleFob
x y
2
2
nig
x2 y 2
2
2
x2 y 2
x y
2
ebI x y 0 .
RsaybBa¢ak;fa 3 3 2 3 4 24 Eckdac;nwgcMnYn 27 . V. eK[bIcMnYnKt;rWuLaTIbtKña a , b , c Edl a b c . k> KNnaplbUk S a b c CaGnuKmn_én b . x> Tajrktémø a , b , c edaydwgfa S 333 . VI. edaHRsaysmIkar y 2 x 6 2 y x 4 0 . VII. dImYykEnøgmanragCactuekaNEkgmanépÞRkLa S Edl 2hm < S < 3hm eKEckdIenaHCaLÚt_² man épÞRkLa 150m , 180m nig 210m . k> KNnaRkLaépÞéndIragCactuekaNEkgenaH . x> KNnaRbEvgbeNþay nigTTwgctuekaNEkgenaH ebIdwgfaTTwgesµInwg 74 énbeNþay . VIII. eK[RtIekaNEkg ABC EkgRtg; A Edl Bˆ 2Cˆ nig BC a . enAeRkARtIekaN ABC sg;RtIekaN sm½gS ABF nig ACM . k> eRbóbeFobRkLaépÞénRtIekaN ABF nigRkLaépÞénRtIekaN AFM . x> KNnaRkLaépÞctuekaN BCMF CaGnuKmn_én a . IX. eK[RtIekaNsm)at ABC man)at BC . D CacMeNalEkgéncMNuc B elI AC . bgðajfa BC 2 AC CD . 2005
IV.
2006
2005
2006
2
2005
2005
2
2
2
2
2
2
2
¬vi> )ak;TUk ¦
3
557
៩
៩ :
I. II.
III.
IV.
3 4
edaHRsaysmIkar nigepÞógpÞat;cemøIy ³ x x 4 8 . RsaybBa¢ak;fa 49 20 6 49 20 6 2 3 . 2
4
KNnakenSam
4
1 ab
a b A 1
1
Edl a b 0 .
a 2 b2
eK[cMNuc A 1 , 0 ; B 1 , 0 ; C 11 xx
2 2
,
2x 1 x2
enAkñúgtRmúyGrtUNremEtmYy. bgðajfa
CaRtIekaNEkgRtg; C . ABC CaRtIekaNEkgRtg; A kMNt;edayrgVas;RCug AB c , AC b . knøHbnÞat;BuHkñúgénmMu A kat; GIub:Uetnus BC Rtg; D . bnÞat;Rsbnwg (AD) EdlKUsecjBI B CYbbnøayénRCug CA Rtg; E . k> RsaybMPøWfa AE AB rYcKNna BE . x> KNna AD . K> KNnaépÞctuekaN ADBE . ABC
V.
¬Gnu> TYlGMBil ¦
3
558
៩
៩ :
KNnakenSam A 3273 , B 2 16 16 . x 18 x 21 x 15 x 12 x 555 edaHRsaysmIkar ³ k> 241979 0 . 1985 1982 1988 1991 111 x> 84 16 . 3
I. II.
4 x 1
x
x
6
x 1
x
x 1
III.
IV.
edaHRsayRbB½n§smIkar ³
1 1 1 1 a b c 12 1 1 1 7 c b a 12 1 1 1 5 a c b 12
eK[bnÞat;BIrKW D : y 2m 5m 4 x 2m 5m 5 nig kMNt;témø m edIm,I[bnÞat; D nig RsbKña . 2
2
m
m
V.
.
edaHRsayRbB½n§vismIkar ³
m
: y m2 m 1 x m2 m 1
m
x 0 y 0 x y 0 3 x 4 y 36
tamRkaPic .
rktémø x edIm,I[cm¶ayrvagBIrcMNuc A 2 , 9 nig B x , 9 esµInwg 2004 Ékta . VII. eK[ A 2 , 2 ; B 3 , 3 nig C 6 , 6 . bgðajfa A , B , C sßitenAelIbnÞat;EtmYy . VIII. ABC CaRtIekaNEkgRtg; C ehIy CH Cakm
vi> b£sSIEkv> éf¶TI 09 mIna 2005 eday cab e):as‘ag
3
559
៩
៩ :
I.
KNnakenSam
II.
. eK[ a , b , c CargVas;RCugTaMgbIénRtIekaNmYy. cUrbgðajfasmIkar b x b Kµanb£s .
III.
bgðajfa X 2
IV.
edaHRsayRbB½n§smIkar ³
V.
VI.
1 1 1 1 ... 1 2 2 3 3 4 1999 2000 n 1 P 2 n 4 x 2 4 x 1 2x 1 S
2 2
3 5 13 48
c2 a2 x c2 0
CacMnYnKt;viC¢man .
6 2 x xy y 1 y yz z 4 z zx x 9
.
sisS 6 nak; A , B , C , D , E nig F RbLgsisSBUEkKNitviTüaRbcaMsala. KNemRbeyaKRbkasfa mansisSBIrnak;)anRbLgCab;edayKat;niyayfa ³ 1> A nig C Cab; 2> B nig E Cab; 3> F nig A Cab; 4> B nig F Cab; 5> D nig A Cab; . eKdwgfa XøaTaMgR)aMmanXøamYyxussuT§saF ehIybYnXøaeTotmanmYyXøa²RtUvBak;kNþal. etIsisS BIrnak;NaEdl)anRbLgCab; . cUrRsaybBa¢ak;fa A 11...11 22...22 CacMnYnmYymankaerR)akd . 2n
VII.
2
tYén 1
n
tYén 2
manRtIekaN ABC EkgRtg; A nigkm
1
2
2
3
3
vi> c,arGMeBA> éf¶TI 13 mIna 2005 eday nk kn
3
560
៩
៩ :
I. II.
eRbóbeFobcMnYn edaymin)ac;eRbIm:asIunKitelx ³ 2000 nig 3000 . KNnatémøelxénkenSam A x x x 2x 1 cMeBaH x 1 2 3000
4
3
2000
2005
2
2 1 1
III. IV.
VI.
.
2 1 1
rktémøGb,brmaénBhuFa ³ W a b 5a 9b 6ab 30a 45 . RsaybMPøWfa a b c d 1 a b c d RKb;témøén a , b , c , d . eK[kenSam P x x2x2 5 . rkRKb;témøKt;viC¢manén x edIm,I[ P mantémøelxCacMnYnKt; . 2 2
2
2
2
4
V.
1
2
2
2
2
3
1> edaHRsayRbB½n§smIkar ³
xy 64 1 1 1 x y 4
.
2> KNna N 4 15 4 15 2 3 5 . 3> RsaybBa¢ak;fa RKb;cMnYnBit a , b , c enaHsmIkar x 1 a x 1 b c1 manb£sCanic© . VII. eK[RtIekaN ABC . km
ABC
MNP
vi> bwgRtEbk> éf¶TI 10 mIna 2005 eday kuy EkvLúg
3
561
៩
៩ :
I.
KNna A 12 19 6 10 3 2 2 5 , B 3 5 3 5 2 2 3 5 2 2 3 2. rkcMnYnEdlmanelxBIrxÞg; AB ^epÞógpÞat; ³ AB BA 1980 . 1. edaHRsaysmIkar 2 3 3 x 3 y 3 Edl x , y CacMnYnsniTan . 2 x 12 x 17 x 2 2. KNnatémøelxénkenSam ³ A x2 x 13 169 27 ebI a b a c , . c b 2a b c 1.
2
II.
3
a c
III.
2
2
eKman x 0 , y 0 , z 0 ehIyEdl x y z xyz , x xy . RsaybBa¢ak;fa x 3 . 2. smIkar a 1 x 2 2bx c 1 x 0 Edl a , b , c CaRCugénRtIekaNEkg ABC Edl Cˆ 90 ³ k> RsaybBa¢ak;fa smIkarmanb£sBIr x , x . x> ebI x x 12 cUrkMNt;témøelxén M a b c . 2
1.
2 1
1.
2
o
2
1
2
2 2
edaHRsayRbB½n§smIkar
2 2 3x 2 xy y 0 2 x 5 y 6
.
enAkñúgRtIekaN ABC man Bˆ 120 , BA 8 cm , BC 12 cm . cMNuc P rt;elI AB edayepþIm BIcMNcu A eTA B nigcMNuc Q rt;elI BC ecjBI C eTA B . ebI P rt;kñúgel,Ón 1cm/s ehIy Q rt; kñgú el,Ón 2cm/s ecjdMeNIrkñúgeBlEtmYy. k> etIb:unµanvinaTIeRkay eTIbépÞRkLaRtIekaN PBQ esµInwgBak;kNþalépÞRkLaRtIekaN ABC ? x> kñúgeBlCamYyKñaenaH etI PQ manRbEvgb:unµan ? 1. ctuekaNesµI ABCD EdlmanRCugesµI a . R nig r CakaMénrgVg;carwkeRkARtIekaN ABD nigRtIekaN 1 1 4 ABC . RsaybBa¢ak;fa . R r a 2. kñúgRtIekaN ABC man BAC 60 . AD CaknøHbnÞat;BuHmMu BAC 60 kat; BC Rtg; D . ebI SS 85 ehIyrgVg;p©it I carwkkñúgRtIekaN ABC manépÞesµI 12 . AB , BC nig AC b:HrgVg;p©it I Rtg;cMNuc E , N , M . KNnaRbEvg AD ? o
2.
V.
5
2
2
IV.
.
5
10
2
2
2
o
o
ABC
ADC
3
562
៩
៩ :
I. II. III.
rktémø x nig y edaydwgfa x y 120 nig PGCD x , y 30 . eK[ ba dc bgðajfa 44ba 33dc xbxa ydyc rYcrktémø x nig y . eKmanbIcMnYn a 0 , b 0 , c 0 Edl a b c 1 nig 1a b1 1c 0 . RsaybMPøWfa a b c 1 . edaHRsay nigBiPakSa ³ ax 2x 1 0 , a Cab)a:r:aEm:t . ebkçCnRbLgmñak;cab;eqñat 3 snøwkkñúgsMNYr 22 EdlerobcMeLIgedayKN³emRbeyaK. kñúgsMNYr 22 enHman ³ 10 sMNYrBICKNit , 7 sMNYrRtIekaNmaRt , 5 sMNYrnBVnþsaRsþ . k> KNnaRb)ab edIm,I[sMNYrTaMgbICasMNYrBICKNit . x> KNnaRbU)ab edIm,I[sMnYrTaMgbI CaBIKNit RtIekaNmaRt nignBVnþ . 2
IV. V.
VI.
2
2
2
1> edaHRsayRbB½n§smIkar ³
xy 64 1 1 1 x y 4
.
2> KNna N 4 15 4 15 2 3 5 . 3> RsaybBa¢ak;fa RKb;cMnYnBit a , b , c enaHsmIkar x 1 a x 1 b c1 manb£sCanic© . eK[RtIekaN ABC EkgRtg; A nig AH Cakm KNna CH , BH , ABH . x> R)ab;eQµaHRtIekaN IAC ( I CacMNuckNþal BC ) . K> rgVg;Ggát;p©it AB nigrgVg;Ggát;p©it AC kat;KñaRtg;cMNcu mYyeTot M . RsaybMPøWfa M RtYtelI H enAelI BC . 2
VII.
o
Gnu> BgTwk> éf¶TI 13 mIna 2005 eday RkUc suvNÑara
3
563
៩
៩ :
I. II.
rktémøKt; x , y , z EdlepÞógpÞat;smIkar x 4 y 1 z 25 80xyz . eK[ a , b , c CaRCugénRtIekaN ABC . KNnaRkLaépÞénRtIekaN ABC edaydwgfa ³ 2a 2b 2c . b , c ,a 1 a 1 b 1 c edaHRsaysmIkar xx 11 xx 44 xx 22 xx 33 . eK[ x y x z y z . RsaybBa¢ak;fa 1x 1y 1z 0 . 2
2
2
2
III. IV. V.
2
2
2
2
edaHRsayRbB½n§smIkar
2
x 2 y 2 z 2 xy yz xz 2005 2005 2005 2006 x y z 3
.
KNnatémøtUcbMputénBhuFa 2x 4xy 5 y 12 y 13 . 3 2 . VII. edaHRsaysmIkar 2 VIII. eK[RtIekaN ABC EkgRtg; A nigman AD Cakm
VI.
1 x 3 y
2
2 x 4 y 1
o
3
3
2
vi> sn§rm:uk> éf¶TI 08 mIna 2005 eday Ect lI
3
564
៩
៩ :
I. II.
eK[ a b c 1 , a b c 1 , ax by cz m . cUrKNna KNna A 1 12 1 13 1 14 ... 1 991 2
2
2
B 6 4 x 1 81x 2 81
P xy yz zx
.
x
C 4 74 3 2 3
S x 1 x 2 x 3 ... x 100 III.
edaHRsaysmIkar ³ k> 2 x 5 x> 2x x K> 3 9 x 2005
3x 5 2
2003
V.
VI.
edaHRsayRbB½n§smIkar ³ k>
.
2 2 x 2004 2
2 x2 3 x
IV.
rYcbBa¢ak;BItémøelx S ebI x 1 , 2 , 3 .
x 1 x x2003 x2002
x xy y 2 3 2 2 2 x y 6
x>
1 x x y z t 2 3 4 6 zyt xzt xyt xyz 14625 y z t x
.
.
eK[smIkarbnÞat; y 32m mx . 1 k> sresrsmIkarbnÞat;Edlkat;tamcMNuc A0 , 1 . x> K CacMNucEdlsßitelIbnÞat;tagRkaPicRKb; m . KNnakUGredaenén K’ qøúHnwg K eFobnwg A . ctuekaNEkg ABCD EdlTTwg AB a tamcMNuckNþalén AB eKKUsmMu KOH 90 EdlcMNuc H enAelI AD nigcMNuc K enAelI BC . k> eRbóbeFobRtIekaN AOH nigRtIekaN OBK rYcKNna AH × BK . x> eK[ AH = x nig BK = y KNnaépÞGb,brmaénRtIekaN KOH . o
Gnu> Twkl¥k;> éf¶TI 01 mIna 2005 eday hYt sIN u n
3
565
៩
៩ :
I. II. III. IV.
V.
VI. VII.
KNnakenSam ³ A 2 2 edaHRsaysmIkar ³ k> 5 6
KNnakenSam RsaybBa¢ak;fa
1 x 4 y 4
ebI
2
3 7 2 3 6 7 2 3 6 7 2 4
5 5
x
5 8
4
1 x 1296 y
52003 52004 52005
edaHRsayRbB½n§smIkar ³ k>
x> nig
x 4 y
2 4x x 1 8 3 x 1
3
.
.
.
Eckdac;nwg 31 .
1 1 1 x y z 4 2 1 8 xy z 2
x>
xy 18 2 2 x y 24
.
KNnacm¶ayrvagcMNuc A6 , 6 eTAbnÞat; D : y x 4 kñúgtRmúyGrtUNrem . eKmansmIkar x mx 4 0 KNnatémø m nigb£smYyeTot edaysÁal;b£sesµI 2 . 2
D VIII.
tamrUbxagsþaM AB // EF // DC . RsaybBa¢ak;fa 1x 1y 1z .
B E x
A IX.
z
y F
C
eK[rgVg; C(O , R) nigGgát;p©it AB nigcMNuc M rbs;rgVg; C ehIybnÞat; D b:HrgVg; C Rtg;cMNcu A . P CacMeNalEkgén M eTAelI (AB) ehIy Q CacMeNalEkgén M eTAelI (PQ) . k> bgðajfa AIO CaRtIekaNEkg . x> bnÞat;Edlb:HrgVg;Rtg; M kat;bnÞat; D Rtg; T . bgðajfa AM CaknøHbnÞat;BuHénmMu QMO nig TMP . K> bgðajfa AIO ATM ehIy AIP AOM . X> KNna AQ , AI , AP ebI AT = x , OA = R .
vi> b‘unr:anI h‘un Esn vtþPñM> éf¶TI 03 mIna 2005 eday Eg:t can;sux
3
566
៩
៩ :
I.
rktémø x edIm,I[ sRmYl xx 11 .
x x x x 1 1 1 1 5 10 15 20 2 4 5
.
2
II. III. IV. V.
VI.
rkRKb;témø m Edl m 0 edIm,I[smIkar m 2 x m 5 0 manb£sviC¢man . RsaybMPøWsmPaB 2 3 2 3 2 . 2 2 3
2 2 3
GtþBlkrBIrnak;rt;edayclnaesµIenAelIpøvÚ mYymanragCargVg; Edlmanrg; 480 m. ebIGtþBlkrTaMgBIr rt;tamTisedApÞúyKña enaHGñkTaMgBIrCYbKña 45 vinaTImþg. ebIGtþBlkrTaMgBIrrt;tamTisedAEtmYy enaH Gñkrt;elOnECgGñkrt;yWt ral;bInaTI. k> KNnael,ÓnénGtþBlkrTaMgBIr . x> ]bmafa A CacMNucecjdMeNIr etIGtþBlkrmñak;² rt;b:unµanCMu)anCYbKñaRtg; A . eK[RtIekaN ABC mYycarwkkñúgrgVg; (C) Edl Bˆ Cˆ 90 . k> bgðajfa Ggát;p©it AA’= 2R énrgVg; (C) Rsbnwg BC . x> bgðajfa AB AC 4R . K> eK[ AH Cakm
2
2
2
2
b£sSIEkv> éf¶TI 28 kumÖ³ 2005 eday pn sar:at
3
567
៩
៩ :
I.
II.
3 A 4
2000
9 16
1
1> KNna . 2> cUrkMNt;TtI aMgvg;Rkck edIm,I[)ansmPaB ³ B 8 4 2 2 9 3 3 3 4 0 . 1> cUrdak;kenSam A CaplKuNktþa Edl A a b c b a c c a b . 2> bgðajfa ³ ebi a b c enaH a 2bab c b 2cbc a 0 Edl a 0 , b 0 , c 0 . x x 1 x 2 x 3 1> edaHRsaysmIkar ³ 2004 4 . 2005 2006 2007 1
1
1
2
2
2
III.
2
x 3 2 y 1 2
2> edaHRsaysmIkar
2 x 3 y 1 4
IV.
V.
2
2
2
1 2
2
2
2
2
2
2
.
1> cUrrkbIcMnnY Kt;esstKña edaydwgfa plbUkcMnYnTaMgbIesµInwg 909 . 2> cUrrkbYncMnYnKt;KU edaydwgfaplbUkcMnYnTaMgbYnesµInwg 1028 . eKmanrgVg;p©ti O mYycarwkeRkARtIekaNsm)at ABC kMBUl B nigkm cUrRsaybBa¢ak;fa BCK CBF BCH nig FAE EAK . x> bgðajfa FK 2MP . K> cUrRsaybMPøWfabIcMNuc F , M , H sßitenAelIbnÞat;EtmYy . X> cUreRbobeFob OM nig AH .
GUEbkk¥m> éf¶TI 13 mIna 2005 eday Xun Bisidæ
3
568
៩
៩ :
I. II. III.
bgðajfa ³ 6 2 5 13 48 3 1 . 3 x 2 x x 1 edaHRsaysmIkar 2002 3 . 2003 2004 eK[bIcMnYn a , b , x epÞógpÞat;tMnak;TMngTaMgbIxageRkam ³ a 2b 1 0 , b 2 x 1 0 , x 2a 1 0 . cUrKNnaplbUk S a b x . eK[smIkar x 1 x 4 x 5 x 6 m . k> kMNt;témø m edIm,I[smIkarKµanb£s . x> kMNt;témø m edIm,I[smIkarmanb£sDub . K> edaHRsaysmIkarebI m = 8 . kñúgtRmúyGrtUNrem mancMNuc A 3 , 2 ; B 9 , 2 nig C 3 , 10 . KNnabrimaRt nigRkLaépÞén RtIekaN ABC . eK[bnÞat; D : 2 3x y 0 nig D : 4 m 1 m x my 0 . rktémø m EdlnaM[ ³ k> bnÞat; D nig D kat;KñaelIGk½ SGab;sIus . x> bnÞat; D nig D EkgKña . RCugénkaer ABCD manrgVas;esµI 1 . enAelIRCug AB nigRCug AD edAcMNuc E nig F EdlbrimaRt RtIekaN AEF esµI 2 . RsaybBa¢ak;fa ECF 45 . 2
2
2
2005
IV.
V.
VI.
VII.
1
2005
2005
2
1
2
1
2
o
Gnu> TYlR)asaTEsnsux> éf¶TI 09 mIna 2005 eday BuT§ v½nþ
3
569
៩
៩ :
I.
1> dak;CaplKuNktþadWeRkTI 1 nUvBhuFa ³ P x 2x 5x 6 . 2> dak;CaplKuNktþadWeRkTI 2 nUveTVFa ³ A x 64 . 3> bgðajfa plbUk S 39 51 Eckdac;nwg 45 . 3
2
4
51
ax by cz dt 0 bx ay dz ct 0 cx dy az bt 0 dx cy bz at 0
II.
eKman
III.
edaHRsayRbB½n§smIkar ³ k> xx 1y y 651 18 2
IV.
V.
2
39
. bgðajfa
x>
x y z t 0
1 1 5 x y 6 1 1 13 x 2 y 2 36
b¤
a bc d 0
K>
yz 10 x 3 zx 15 y 2 xy 6 z 5
.
.
1> brimaRténRtIekaN ABC manrgVas;esµInwg 80 cm . eyIgsnµt;fa BC a , AC b , AB c . eK[ a , b , c smamaRtnwgcMnYn 5 , 7 , 4 . KNna a , b , c . 2> tamcMNuc M enAelI BC eKKUs ME // AB Edl E AC . eK[ MB x ehIy y Ca brimaRténctuekaN ABME . KNna y CaGnuKmn_én x . 3> sg;bnÞat; D tagGnuKmn_én y EdlrkeXIjkñúgsMNYrTI 2 . 4> KNnatémø x ebI y = 56 . ABC CaRtIekaNcarwkkñúgrgVg;p©it O kaM R ehIy [AH] Cakm
vi> vtþekaH ³ eday )an fn
3
570
៩
៩ :
I.
sRmYlkenSam ³ A xx 11 22xx 11 2xx11 2xx11 . m 1 1 m 4m2 1 1 m B 2 2 3 2 1 2 m 1 m 1 m m 1 m m
II.
1> edaHRsayRbB½n§smIkar Edlman x CaGBaØat nig m Ca):ar:aEm:t ³ 5x m 2x m m 5 5 x 1 . 6 5 5 20 2> rktémø m Kt;viC¢man edIm,I[ 0 x 10 . KNnakenSam ³ A 2 3 2 3 . 5 2 6 52 6 B 2 25 . 52 6 52 6 eK[rgVg;p©it O kaM R Ggát;p©it AB . sg;kaM OI Ekgnwg AB . J CacMNucmYyelIFñÚtUc AI . BJ kat; OI Rtg; K ehIy AJ kat; OI Rtg; L nig AK kat;rgVg;Rtg; P . 1> RsaybMPøWfa B , P , L enAelIbnÞat;EtmYy . 2> bgðajfactuekaN AOKJ nig JKPL carwkkñúgrgVg; . 3> RbdUc BOK nig ALO rYcbgðajfa OK OL R . 4> RsaybBa¢ak;fa OJ b:HrgVg;carwkeRkActuekaN JKPL . 2004
III.
.
2005
2
IV.
2
PñMeBj ³ éf¶TI 14 mIna 2005 eday Qwm esg
3
571
៩
៩ :
I. II. III.
RsaybBa¢ak;fa 5 5 5 Eckdac;nwg 31 . KNnatémøénkenSam A 1 1 2 2 1 3 3 1 4 ... 99 1 100 . eK[ a 0 , b 0 nig a b c 0 . k> RsaybBa¢ak;fa a b c 3abc . x> eKtag c 2n . RsaybBa¢ak;fa aa bb cc 3n . cabmYyhVÚgehIreTATMelIpáaQUkkñúgRsHmYy. ebIcabmYyTMelIpáaQUkmYy enaHmancabmYyKµanpáaQUkTM EtebIcabBIrTMpáaQUkmYy enaHenAsl;páaQUkmYyTMenr. rkcMnYnpáaQUk nigcMnYncab . edaHRsayRbB½n§smIkar xy 33xy 66 22yx 21 . 2001
2002
3
IV.
3
2003
3
3
3
3
2
2
2
2
V. VI.
2
Ggát;RTUgénctuekaNBñaysm)at EckctuekaNBñayenH)anRtIekaNcMnnY 4 EdlkñúgenaHRkLaépÞ RtIekaNBIr Cab;nwg)atmanrgVas; S nig S . KNnaRkLaépÞénctuekaNBñayxagelI . eK[RtIekaN ABC . M CacMNuckNþalén [AB] , D CacMNucmYyenAelI [MB] Edl 2MD = DB , MCD BCD . cUrbgðajfa ACD 90 . 1
VII.
2
o
vi> h‘un Esn b‘unr:anIvtþPñM ³ éf¶TI 09 mIna 2005 eday erol rdæa
3
572
៩
៩ :
I.
RsaybBa¢ak;fa plbUk 385a 1001b Eckdac;nwg 77 .
II.
x2 2 y 1 0 2 y 2z 1 0 z2 2x 1 0
III.
IV.
V.
3
eK[ x , y , z Edl
5
. KNna A x
2000
y 2000 z 2000
.
eK[smIkar 4 m x 4x m 0 . manb£sBIrKW x , x . kMNt;témø m edIm,I[smIkarman plbUkkaerénb£sesµI 10 . kukmYyehIrCYbRksarmYyhVÚgk¾ERsksYrfa {sYsþI¡ mitþTaMg 100 { eBlenaHemxül;énhVÚgRksareqøIyfa {eT¡ cMnYnBYkeyIgminRKb; 100 eT} cMnYnBYkeyIgbUknwgcMnBYkeyIg bEnßmBak;kNþalcMnYnBYkeyIg ehIy Efm 14 éncMnYnBYkeyIg RBmTaMgmitþÉgeToteTIbRKb;cMnnY 100 . KNnacMnnY RksarenAkñúghVÚg . eK[bIcMNuc A , B , C sßitenAelI (xy) . KUsRtIekaNsm½gS ABD nigRtIekaNsm½gS BCE enAEt mçagénbnÞat;enH . 1> RsaybMPøWfa AE = CD . 2> eK[ M nig N kNþalRCug [AE] nig [CD] . bgðajfaRCug BM = BN . 3> RsaybMPøWfa RtIekaN BMN CaRtIekaNsm½gS . 2
1
2
vi> h‘un Esn b‘unr:anIvtþPñM ³ éf¶TI 07 mIna 2005 eday em:A TUc
3
573
៩
៩ :
I. II. III.
KNna
S
a
b
a b a c b c b a c a c b
eK[bIcMnYn x , y , z xusBIsUnü Edl x y z 1 nig 1x 1y 1z 0 . bgðajfa x dak;CaplKuNktþanUvkenSam A ab x y xy a b . RsaybBa¢ak;smPaB 1 2axaxa ax x 1x 1 a 1 2axax x 1 1a . 2
2
2
V.
VI.
2 2
y2 z2 1
.
2
2
2
1> kñúgtRmúyGrtUNrem rksmIkarénbnÞat;kat;tamcMNuc A6 , 2 nig B 2 , 2 rYcrksmIkarExS emdüaT½rrbs;Ggát; [AB] . 2> eK[cMNuc C 2 , 6 . bgðajfaRtIekaN CAB CaRtIekaNsm)at . rgVg;mYykaMmanrgVas; R . eKKUsGgát;FñÚBrI EkgKña [BC] nig [DE] ehIyRbsBVKñaRtg; A ehIy [BB’] CaGgát;p©iténrgVg;enaH . RsaybMPøWfa AB AC AD AE 4R . RtIekaNsm)at BAC nig ADC EkgRtg; A nig D manRCugrYm [AC] ehIytaMgenAsgxagRCugrYmenH. eK[ AB AC a . 1> bgðajfactuekaN ABCD CactuekaNBñayEkg . 2> KNna BC , AD , BD . 3> Ggát;RTUgTaMgBIrrbs;ctuekaNkat;KñaRtg; O . bgðajfa OAD COB rYcrkpleFob dMNUc RBmTaMgKNna OA , OB , OC , OD . 2
VII.
2
2
2
IV.
.
c
2
2
2
2
PñMeBj ³ éf¶TI 23 kumÖ³ 2005 eday R)ak; kaNaDI
3
574
៩
៩ :
I.
1> bgðajfa
2 3 2 2 3
2> KNnatémøelxénkenSam II. III.
.
5 3 2 27 2 3 3 4 3
.
9 16
2
2
b2 2ab
x> ba ba 2 .
2
2
y 1 x a 2 a 2 a 2 x y 1 a 2 a 2 a 2
.
1> KNnabIcnM Yn x , y , z edaydwgfaplbUkvaesµInwg 292 ehIy x nig y smamaRtnwgcMnYn 150 nig 90 ehIy y nig z smamaRtnwgcMnYn 24 nig 21 . 2> eK[knSam E a 1 x 2 b 4 x 3c 15 . cUrkMNt;témø a , b nig c edim,I[BhuFa E = 0 cMeBaHRKb;témøén x . 2
VI.
2
eK[ a nig b CaBIrcMnYnmansBaØadUcKña . bgðajfa k> a 1> dak;CaplKuNktþa E 12 x 3 27 x 5 . 2> edaHRsaysmIkar 4x 5x 1 0 . 3> edaHRsayvismIkar 1x 21 1x 12 x 4 . 4> edaHRsayRbB½n§vismIkar
V.
2 2 3 A
4
IV.
2 3
edaHRsayRbB½n§smIkar
2
9 x 6 y 10 z 1 6 x 4 y 7 z 0 x2 y 2 z 2 9
.
eK[rgVg; C (O , R) Ggát;p©it [AB] EdlbnÞat; AB EkgnwgbnÞat; D Rtg; H . (B enAcenøaH O nig H). M C (O , R) bnÞat; AM , BM nigbnÞat;b:HrgVg;Rtg; M kat;bnÞat; T er[gKñaRtg; D , C nig I . bnÞat; AC kat;rgVg;Rtg; E . 1> bgðajfa I CacMNuckNþalén [DC] rYcTajbBa¢ak;fa (IE) CabnÞat;b:HénrgVg; C (O , R) . 2> K CacMNucRbsBVrvag (OI) nig (ME) , J CacMNucRbsBVrvag (ME) nig (AB) . RsaybBa¢ak;fa OJ OH OI OK R rYcbgðajfabnÞat;(ME) kat;tamcMNucnwgmYykalNa M rt;elIrgVg; (C). 2
3
575
៩
៩ :
I. II.
RbsinebI a bc 2 nig ab 2 . KNnaplKuN P abc . rkBIrtYbnÞab;énsVIútcMnYn 1002 , 2005 , 4011 , ....... , ......... . 2004
2005
2004
2004
2 3 1 x 1 y 2 z 3 x 2 y 3z 56
x 2 y 2 2 xy 8 2
x>
III.
edaHRsayRbB½n§smIkar k>
IV.
KNnaplbUk S edayRsaybBa¢ak;rUbmnþTUeTACamunsinKW ³
VI.
1 ... n n n 1
n 1
.
rYceTIbKNna ³
. eK[ [AM] CaemdüanénRtIekaN ABC EkgRtg; A . ehIyman AB = 4cm , AC = 5cm . KNna épÞRkLaénRtIekaN AMB nigRtIekaN AMC . man [Sx) , [Sy) nig [Sz) tamlMdab;enH Edl xSy 60 , ySz 90 . cMNuc A , B nig C enAelIknøH bnÞat;TaMgbIerogKña eday SA = SB = SC = a . 1> KNna AB , BC nig AC CaGnuKmn_én a . rYcR)ab;RbePTRtIekaN ABC . 2> H CacMNuckNþalén [AC] . bgðajfa [SH] EkgnwgépÞénRtIekaN ABC . bEnßm ³ xSz 120 . S
V.
x y 4
1 1 1 ... 2 1 1 2 3 2 2 3 100 99 99 100
o
o
o
vi> b‘unr:anI hs vtþPñM> éf¶TI 14 kumÖ³ 2005 eday hYt riT§I
3
576
៩
៩ :
125 4 2 4 5 4 625 10
II.
sRmÜlr:aDIkal; ³ eK[ x y 0 nig
III.
edaHRsayRbB½n§smIkar
I.
IV.
V. VI. VII.
4
.
2 x 2 2 y 2 5 xy
. KNna
x 2 y 2 13 xy 6
E
eK[smIkar x 4a 34 x 4a a 2 x 4 a 1 0 . k> edaHRsaysmIkarcMeBaH a 12 . x> edaHRsaysmIkarxagelI[Cab;GnuKmn_én a . cMkarmYyragCactuekaNEkg. RkLaépÞvaesµIBIrdgénplbUkbeNþay nigTTwg. KNnavimaRtcMkar . bgðajfa 39 51 Eckdac;ngw 45 . KNna B 10110001100000001 ...1 00...00 1 . 3
51
2
2
39
énelx
IX.
.
.
2n 1
VIII.
x y x y
0
eKmanRbGb;bYndUcKña nigb:unKña. kñúgRbGb;TI1manXøIs 1 Rkhm 1. kñúgRbGb;TI2manXøIs 2 Rkhm 3 kñúgRbGb;TI3manXøIs 3 Rkhm 5 . kñúgRbGb;TI4manXøIs 4 Rkhm 7 . eKeRCIserIsRbGb;mYy rYceK ykXøImYyedayécdnü ecjBIRbGb;enaH . RbU)abedIm,I[eKerIsykRbGb;TI1 esµInwg 101 . KNna RbU)abedIm,I[eKerIsyk)anXøIBN’s . eK[ctuekaN ABCD mYycarwkkúñgknøHrgVg;Ggát;p©it [AD] nigp©it O ehIy AB 2 5 cm , BC 2 5 cm nig CD 6cm . KNnakaMrgVg;carwkeRkActuekaNenH .
PñMeBj> éf¶TI 05 kumÖ³ 2004 eday taMg RTI
3
577
៩
៩ :
I.
KNnatémøelxénkenSam ³ A 2005 02005 12005 2 ... 2005 3005 15 2 27 8 3 4 9 16 5 3 4 . B 3
3
4
II.
1> eK[1 x
2
0
3
3
33 4
9 3 16
2
1 1 x 1 1 x x2 1 A x 1 2 1 x 1 x 1 x 1 x 4 4 4 4 f a 1 1 1 ... 1 2 1 9 25 2a 1
cUrsRmÜlkenSam
2> eK[ a 1 . bgðajfa III.
4
.
CacMnYnGviC¢man .
1> edaHRsaysmIkar 12 x 14 x 2 . 2> edaHRsayRbB½n§smIkar xy 11 22yx 12 . 3
3
3
IV.
3
1> RtIekaN ABC sm)atkMBUl A , [AH] Cakm H CaGetUsg;énRtIekaN ABC . k> bgðajfa H’ CacMNucqøúHén H eFobnwg (BC) sßitenAelIrgVg;carwkeRkAénRtIekaN ABC . x> I , M nig Q Cap©itrgVg;carwkeRkAerogKñaénRtIekaN AHB , AHC nig BHC . bgðajfa (IM)//(BC) nig (MQ)//(AB) . 2
Gnu hs bUrI100xñg> éf¶TI 25 kumÖ³ 2005 eday C½y sMGun
3
578
៩
៩ :
I.
II.
III. IV. V. VI.
eK[smPaB x y y z z x x y 2z y z 2x x z 2 y . bgðajfa x y z . 2
edaHRsayRbB½n§smIkar
2
2
1 1 1 x y z 2 2 1 4 xy z 2
2
2
2
.
edaHRsaysmIkar 2 x 2 x 1 4 x 1 . eK[kenSam A x y x 2 y x 3y x 4y y . bgðajfakenSam A CakaerR)akd . KNnakenSam ³ B 1 1 2 1 21 3 1 2 1 3 4 ... 1 2 3 1 ... 100 . RsaybBa¢ak;fa TIRbCMuTm¶n; p©itrgVg;carwkeRkA nigGrtUsg; énRtIekaNmYy CabIcMNucrt;Rtg;Kña . 2
4
Gnu hs bUrI100xñg> éf¶TI 01 mIna 2005 eday Tk; yUGUn
3
579
៩
៩ :
I. II. III.
IV.
bgðajfa ³ 6 2 5 13 48 3 1 . 3 x 2 x x 1 edaHRsaysmIkar 1997 3 . 1998 1999 eK[bIcMnYn x , y , z smamaRtnwgcMnYn 5 , 7 , 11 . 1> KNnacMnYnTaMgbIkalNa ³ k> x 2 y 3z 48 . x> x my z 72 , m CacMnYnKt;EdleK[ . rktémøén m minRtUvman . 2> kñúglT§plénsMNYr 1>¬x¦ eKcg;[ x , y , z CacMnYnKt;viC¢man . etIcMnYn 16-7m RtUvbMeBj lkçxNÐy:agNa ? etImantémø m b:unµanEdlbMeBjlkçxNÐenH ? 3> eKcg;[ m CacMcMnYnKt; . KNna m ¬mancemøIyeRcIn¦ nigtémørbs; x , y , z . eK[kaer ABCD manRCugRbEvg a . eKsg;RtIekaNsm½gS EdlmanGgát;RTUg [AC] CaRCug ehIyman kMBUl E enAxag B . O CacMNuckNþalénRCug [AC] . k> bgðajfacMNuc E , B , O nig D rt;Rtg;Kña . x> KNna BE CaGnuKmn_én a . K> bgðajfa EB ED AB . X> BI E eKKUsbnÞat; (EP) b:HrgVg;carwkeRkAkaerRtg; P . bgðajfa EP = a . 2
eday hak; sm,tþi
3
580
៩
៩ :
rkcMnYnEdlmanelxBIrxÞg; ab EdlepÞógpÞat; ab ba 1980 . x 5 x 3 x 2000 x 2002 II. edaHRsaysmIkar . 2000 2002 5 3 2 III. bM)at;r:aDIkal;BIPaKEbg ³ A . 4 16 64 256 IV. KNnabIcMnYnviC¢man a , b , c edaydwgfa ab 30 , ac 40 , bc 75 . V. eKmanR)ak;my Y cMnYnEck[mnusSbInak; A , B , C . R)ak;cMENk A nig B elIscMENk C cMnYn 500 ` R)ak;cMENk A nig C elIscMENk B cMnYn 330 ` nig R)ak;cMENk B nig C elIscMENk A cMnYn 60 `. 1> rkcMnYnR)ak;EdlRtUvEck[mnusSTaMgbI . 2> KNnaR)ak;EdlmnusSmñak;RtUvTTYl)an . VI. edaHRsaysmIkar ³ 0.17 2.3 x 0.3 . VII. eK[kaer ABCD nig M mYyenAelI BC . (AM) kat; (DC) Rtg; N . RsaybBa¢ak;fa AB1 AM1 AN1 . VIII. kMNt;RbePTRtIekaN ABC edaydwgfakm
I.
8
2
2
8
8
2
8
2
Gnu> pSaredImfáÚv> 01 mIna 2005 eday eqg Kwmetg
3
581
៩
៩ :
I. II. III. IV.
V.
sRmÜlkenSam E 625 2 80 . eKmanBIcMnYn a b 0 bgðajfa RKb;cMnYn n eK)an edaHRsayRbB½n§smIkar a a b 8b 3 . 3
4 3
*
3
a n bn
.
3
hIugmYyenAkñúgGNþÚgmYymanCeRmA 7m . stVhIugenHxMetageLIgmkelI Edlkñgú mYyéf¶etageLIg)an 3m EtFøak;cuHvij 1m . etIb:unµanéf¶eTIbhuIgenH eLIgputBIGNþÚg ? eK[RtIekaNsm)at COB Edlman)at [OB] nigkm
2
2
2
vi hs Cm<Úv½n > 28 kumÖ³ 2005 eday Rs‘un saerOn
3
582
៩
៩ :
I.
1> etIcMnYn A 2 5 , n bBa©b;edayelxsunücMnYnb:unµan ? 2> eRbóbeFobcMnYn 200 nig 300 . x 4 x 3 x 2 x 1 edaHRsaysmIkar 2001 4 . 2002 2003 2004 edaHRsayvismIkar 1x 21 1x 12 x 4 . enAkñúgtRmúyGrtUNrem edAcMNuc A 6 , 0 ; B 6 , 0 nig C 3 , 9 . 1> sresrsmIkarRCugénRtIekaN ABC . 2> sresrsmIkaremdüaT½rrbs;RCugRtIekaN ABC . eK[RtIekaNEkgsm)at PMN EkgRtg; M . kñúgmMu MNP eKKUsknøHbnÞat; [Nx) Edlkat; [MP] Rtg; D . tamcMNuc P eKKUsbnÞat;Ekgnwg [Nx) Rtg; E ehIyknøHbnÞat;enHkat; (MN) Rtg; F . 1> bgðajfabnÞat; (FD) Ekgnwg [NP] rYcKNnargVas;mMu NFD . 2> bgðajfa M , D , E , F sßitenAelIrgVg;EtmYy rYcbBa¢ak;p©iténrgVg;enH . 3> bgðajfa MD MF . 4> kñúgkrNI MNx 30 nig NP a . KNna MN nig DM CaGnuKmn_én a . n
2 n 1
*
300
II. III. IV.
V.
200
o
vi samKÁI ¬PñMeBj¦ > 20 kumÖ³ 2005 eday vn vNÑa
3
583
៩
៩ :
I. II.
III.
KNnakenSam
x xy y x y E xy x y x y
V.
.
eK[RtIekaN ABC EkgRtg; A manbrimaRt 12cm nigépÞRkLa 6cm . KNnargVas;RCugénRtIekaN ABC enH . 2
edaHRsayRbB½n§smIkar
x y xy 1 yz 2 yz x z 5 xz
.
eK[kenSam A xx 3xx 24 . 1> sRmÜlRbPaK A . 2> rktémø x EdlnaM[ A = 4 . eK[rgVg;nwg C(O , R) Edlmanp©it O nigkaM R nigGgát;p©it [AB]. H CacMNucmYyEdlEckGgát; [AB] tampleFob 13 eday H enAcenøaH A nig B ehIy HA < HB . eKKUsGgát;FñÚ [CD] Ekgnwg [AB] Rtg; H . 1> KNna HA nig HB CaGnuKmn_én R ehIyTajbBa¢ak;fa H CacMNuckNþalén [OA] . 2> KNna CH , CA nig CB CaGnuKmn_én R . 3> P CacMNucGefrenAelI [CD]. bnÞat; (AP) kat; C(O , R) Rtg; M’ CacMeNalEkgén M elI [AB] k> rksMNMucMNucénp©itrgVg; carwkeRkActuekaN HPMB kalNa P rt;elI [CD] . x> eRbóbeFobRtIekaN AHP ; AMB nig BMM’ ehIyTajrktémøénplKuN AP AM CaGnuKmn_én R . 4> eKsnµt;fa PAH 30 . kñúgkrNIenHKNna AP , AM nig BM CaGnuKmn_én R . 8
IV.
2
4
4
2
o
Gnuvi> TYlsVayéRB
3
584
៩
៩ :
I. II. III. IV.
V.
eK[ a b c 1 , a b c 1 nig ax by cz . RsaybBa¢ak;fa xy yz zx 0 . edaHRsaysmIkar 3x 11 3x 2 3x 21 3x 3 3x 31 3x 4 3x2 4 . 2
2
2
KNnatémøelxénknSam A x 12x 12x 12x 12x ... 12x 12x 1 cMeBaH x 11 . rkcMnYnKt;EdlenAcenøaH 40 nig 50 edaydwgfa ebIbrþÚ lMdab;elx eK)anCYbcMnYnfµImYyeTot esµInwg 3223 éncMnYnenaH . eKmanRtIekaNsm½gS ABC nigrgVg;carwkeRkARtIekaNenH mankaM R . eKP¢ab;cMNuckNþal D énFñÚ AC eTAcMNuckNþal H énRCug [BC] nigbnøay [DH] [RbsBVrgVg;Rtg;cMNuc M . 1> R)ab;RbePTRtIekaN DCH . 2> KNna DH CaGnuKmn_én R . 3> RsaybBa¢ak;fa HD HM HC HB . 4> rktémø HM nig DH CaGnuKmn_én R . 17
16
15
14
13
2
vi> samKÁI ¬PñMeBj¦
3
585
៩
៩ :
I. II. III.
IV.
V.
VI.
KNnatémøelxénknSam A 1 1 2 2 1 3 3 1 cUrRsaybBa¢ak;fa 4 2 3 4 2 3 2 3 . x 4 x 3 x 2 x 1 edaHRsaysmIkar 2001 4 . 2002 2003 2004 edaHRsayRbB½n§smIkar
2 x y z 18 x 2 y z 1 x y 2z 2
4
...
1 99 100
.
.
tamcMNuc O enAkñúgRtIekaNsm½gS ABC eKKUsGgát;EkgeTAnwgRCugTaMgbIénRtIekaNenaH . bgðajfa plbUkcm¶ayBIcMNuc O eTARCugTaMgbI esµInwgrgVas;km KNna CM nig MB CaGnuKmn_én a . 2> KNnaRkLaépÞctuekaN AHCM CaGnuKmn_én a .
vi> samKÁI éf¶TI 25 kumÖ³ 2005 eday haM sarun
3
586
៩
៩ :
I. II.
III. IV.
V.
edaHRsaysmIkar ³ k> 5x 3 x> 2x 3 5x eK[kenSam A 5x 7 x 2x nig B 2x 2 x 5x 6x . k> dak;kenSam A nig B CaplKuNénktþa . x> sRmÜlRbPaK E BA . K> KNnatémøelxén E cMeBaH x 2 . kMNt;témø m nig n edIm,I[smIkarCaÉlkçN³PaB ³ x13 x m 4 x 3n x 4 . 2
2
3
2
eK[RbB½n§smIkar 2y y2xx 74
3
D D '
2
2
.
1> k> sg;bnÞat;TaMgBIrenAkñúgtRmúyGrtUNremEtmYy . x> KNnakUGredaenéncMNuc M CacMNucRbsBVénbnÞat;TaMgBIr . 2> sresrsmIkarbnÞat;Edlkat;tamcMNuc P 2 , 5 nig Q 4 , 2 . rYcbgðajfabnÞat;enHkat;tam cMNuc M Edlrk)anxagelI . 3> bnÞat;TI1kat;G½kS x’x Rtg;cMNuc A nigbnÞat;TI2kat;G½kS y’y Rtg; B . sresrsmIkar bnÞat; AB . 4> bgðajfa AM Ekgnwg BM rYcKNnaRkLaépÞénRtIekaN AMB . eKmanRtIekaNEkg ABC Edl AB = 4cm ; AC = 3cm . eKP¢ab;cMNuckNþalerogKña A’ , C’ , B’ énRCug BC , AB , AC . 1> eRbógeFobRtIekaN ABC nigRtIekaN A’B’C’ . 2> eRbóbeFobrgVas;km
eday S.B
3
587
៩
៩ :
I.
bM)at;r:aDIkal;BIPaKEbg ³
II.
edaHRsayRbB½n§smIkar
III. IV.
V.
A
20
3 5 2 2 5 x y 3 x z 2 xy yz zx 2
.
.
eK[ a b 1 . cUreRbóbeFob ba 11 , ba nig ba 11 . eK[RtIekaN ABC manrgVas;RCug a , b , c nigRtIekaN A’B’C’ manrgVas;RCug a’ , b’ , c’ . ]bmafa ABC ABC cUrbgðajfa aa bb cc a b c a b c . knøHbnÞat;BuHénmMu A rbs;RtIekaN ABC kat;rgVg;carwkeRkARtIekaNenHRtg; D . RsaybMPøWvismIkar AD AB 2 AC .
vi> RBHsIusuvtßi> éf¶TI 05 kumÖ³ 2005 eday Gwug Kwmh‘ag
3
588
៩
៩ :
I. II.
III.
IV.
V.
eRbóbeFobBIrcMnYn x 5 2 7 5 2 7 nig eK[ M 2 40 12 2 75 3 5 48 nig N 3
VII.
.
y 51 10 2 51 10 2
2 8 2 15
1 52 6
3 7 2 10
.
bgðajfa M = N . bgðajfa k> 1 1 2 2 1 3 3 1 4 ... 9999 1 10000 99 . 1 100 . x> 11 12 13 ... 10000 edaHRsaysmIkar ³ k> x 2004 x 0 . x> 1 x x 2 x x 3 x x ... 2004x x 2005x x . k> sresrCaplKuNktþanUvsmPaB ³ 1a b1 1c a 1b c . x> bgðajfa ebIbIcMnYnminsUnü a , b , c ehIymanBIrcMnYnpÞúyKñay:agtic enaHeK)anTMnak;TMng ³ 1 1 1 1 . a b c abc 2004
K> edaHRsayRbB½n§smIkar VI.
3
x , y
2004
³
x y x y 20 2 x y 15 3
.
enAem:ag 6h30mn narImYyRkumcab;epþImsÞÚgRsUvBIPøWmçageTAPømW çageTotEdlmancm¶ay 100m eday el,Ón 30m/h ehIyyuvCnmYyRkumeTotcab;epþImsÞgÚ BIPøWmçagedayel,Ón 20m/h . k> etIRkumTaMgBIrsÞÚgCYbKñaenAem:agb:unµan ? x> etIRkumnImYy²sÞÚg)anRbEvgb:unµanEm:Rt ? eK[kaer OACB p©it I . sg;rgVg;p©it P (P enAelI AB) kat;tam O kat;bnÞat; (OA) Rtg; M nig (OB) Rtg; N . k> RbdUcRtIekaN CMA , CNB nig CPI . KNna AM rYcbgðajfa CP MN . PI x> sg; C’EdlCacugTI2énGgát; [CC’] énrgVg; (P) bnÞat;TajBI C’Ekgnwg [OC’] kat; [MN] Rtg; Q . bgðajfa [CC’) BuHmMu OCQ .
3 vi> RBHsIusuvtßi> éf¶TI 13 mIna 2005
589
៩
៩ :
I. II. III.
IV.
V.
VI.
edaHRsaysmIkar 3 3 30 . RsaybBa¢ak;fa ebI abc 1 nig a 36 enaHeK)an sRmÜlRbPaK A x3x14xx 1x eday x 0 . 2 x
2 x
3
a2 b 2 c 2 ab bc ca 3
.
2
edaHRsayRbB½n§smIkar ³
x y 1 xyz 2 yz 5 xyz 6 x z 2 xyz 3
.
RCugBIrénctuekaNEkg ABCD manrgVas; 20cm nig 30cm . kMNt;TItaMgrbs;kBM UlRbelLÚRkam MNPQ EdlcMNuc M , N , P , Q enAelIRCugerogKña [BC] , [AB] , [AD] , [DC] nig MB BN QD DP edIm,I[épÞRbelLÚRkamFMbMput rYcKNnaépÞFMbMputenaH . kñúgRtIekaN ABC emdüan [AA’] , [BB’] nig [CC’] kat;KñaRtg; G . F CacMNuckNþal [BG] . BI F eKTajbnÞat;Rsbnwg (AB) ehIyBI G eKTajbnÞat;Rsbnwg (AC) ehIybnÞat;TaMgBIrkat;KñaRtg; E . bgðajfa cMNuc B’, E , C’ rt;Rtg;CYr .
vi> TYlTMBUg> éf¶TI 09 mIna 2005 eday G‘uk sam:aNa
3
590
៩
៩ :
I. II.
CacMnYnKt;FmµCati n . rkcMnYnKt;FmµCati n edIm,I[BIcnM Yn n 12 nig n 77 CakaerR)akd. k> eRbóbeFob 1 2012 nig 2013 2 2012 . x> edaymineRbI 2012 4048144 nig 2013 4052169 cUrrKNna ³ 2012 2012 . A 1 2012 2013 2013 n
2
2
2
2
2
2
III. IV.
bgðajfa N 2013 5 3 29 12 5 CacMnYnKt; . eK[bIcMnYn a , b , c EdlepÞógpÞat;lkçxNÐ a 2b 1 0 , b 2c 1 0 , c 2a 1 0 cUrKNnakenSam F a b c 2012 . kñúgrgVg;p©it O Ggát;p©it [BC] Edl BC = 5cm . eKKUsGgát;FñÚ [BA] Edl BA = 3cm . k> KNnargVas;RCug AC . x> RsaybMPøWfa cos ABC sin ABC 1 K> eKKUsemdüan [BM] énRtIekaN ABC Edlbnøayrbs;va kat;rgVg;p©it Rtg;cMNuc D . Bgðajfa MB MD MA . 2
14
V.
2
2
2
2
2013
2
2
Gnu> s éf¶TI 24 mkra 2013 eday Gan suxKn§a
3
591
៩
៩ :
I.
eRbóbeFobcMnYn 2012 2014 nig 2 2013 . zx edaHRsayRbB½n§smIkar xx y y z zxy yz2013 . 2
II.
III.
KNnaplbUk S 2 1 1 1
IV.
bgðajfa
V.
2 3 2 2 3
eK[ ax by cz 0 nig cUrKNnakenSam A
2
2013
2
2
2013
2013
2013
1 1 1 ... 3 2 2 3 4 3 3 4 n 1 n n
2 3
.
.
2 2 2 3 1 abc 2013 ax 2 by 2 cz 2
.
bc y z ac x z ab x z
VI.
n 1
2
2
2
.
CaRtIekaNcarwkkñúgrgVg;p©it O kaM R . ehIy [AH] Cakm
ABC
a
, AC b ,
Gnu> s éf¶TI 24 mkra 2013 eday Gan suxKn§a
3
592
៩
៩ :
I.
A
B
A a x 2 1 x a 2 1 B 2 x 3 9 x 5 2
2
2 A 1 , 8 , B 2 , 1 , C , 7 ។ 3 1 1 1 0 a b c 1 a b c
II. III.
a,b,c K
IV. V.
3x 2 14 x2 4
O [AD]
VI.
ABC
a 2 b2 c 2 1
E
[AC]
[BC]
M
[AM]
E
[BD] ME EN
N [AB]
BE 2EA CE
[AM]
>
២
O O
[AM]
CE 4OE
3
593
៤
៩
៩ :
I.
E 2 x2 y 2 2x x y x y
II.
A a2b2 b a b2c2 c b a2c2 c a
III.
a c b d
ad bc 2cd 2ab ad bc
IV.
110 cm
7cm
4cm
V.
ABC E
H
O
[AC]
ABH
[BC] ,
D
G DOE
AH 2DO ២
AHG
DOG
H ,G,O
GH
GO
>
3
594
២
៤
៩
៩ :
I.
A 11 6 2 11 6 2 x4 3x2 10 0
II. y
III. IV.
1 3
3
2 1
3
2 1
3
a,b,c a 2 2b 1 0 , b 2 2c 1 0 , c 2 2a 1 0
V.
S a 2004 b2004 c 2004 30o
O
>
3
595
២
៤
៩
៩ :
x 4 x 3 x 2 x 1 4 2001 2002 2003 2004
I.
4
II. 4
III.
8
2 1
2 1 4
8
2 1
1 2
3 9 15 6n 3 75 ... 2 2 2 2 2
n ២
8
A
6 12 18 ... 96
12 24 36 ... 192 S
IV. V.
2 2
1 1 1 1 ... 2 1 1 2 3 2 2 3 4 3 3 4 100 99 99 100 ˆ ˆ 20o Aˆ 20o ABC CBx ABC [Bx)
AB3 BC 3 3 AB2 BC
VI.
ABCD
O
ˆ OBA ˆ 15o OAB
ˆ OCD
២
3
596
៩
៩ :
24x 112x 18x 16x 1 330 x, y A x y x 2 y x 3y x 4 y y 4
I. II. III.
72002 72003 72004
IV.
220 1 5 1 20 6 2 5
D1 : 2 3x y 0 D1 D2 D1 D2
V.
VI.
ABCD 2
57
D2 : 4 m 1 m x my 0
1
[AB]
[AD]
m
E
AEF
F
ˆ 45o ECF
២
3
597
៤
៩
៩ :
I. 6 5
22004 3 2
21
1 2004 11 72
ax3 by 3 cz 3 1 1 1 x y z 1
II.
xa 2 by 2 cz 2 3 a 3 b 3 c
1 1 1 1 ... 1 1 2 2 3 3 4 n n 1
n 1
III.
3
2 xy y 4 x 2 2 x 1
x, y
A 2004 xy 2003 y3
f x x17 13x16 13x15 13x4 ... 13x2 13x 1
x 12
15x 9 y z 300 (x , y , z x y z 100
IV. V.
48 cm 2
ABCD
5,7
ABCD VI.
ABCD H
[AB] [AE]
[BC]
E
D
[AC]
DFH
២
3
598
F
៤
៩
៩ :
D2 : y
D1 : y 2x 4
I.
D1 D2 D1 D2
1 x4 2
T A
TO2 OA OB
B
TAB II.
x 5m
y
110 m 150 m 2
2m
III.
TAB
ABCD
A
D
[AB]
[AC] IV.
[BC] O
[AB]
O’
AB
O’ (MA) , (MB) MAD
(O)
C
D
O’
A
(NA) , (NB)
B
M
(O)
E
F
NBE
២
3
599
២
N
៤
៩
៩ :
A 1002 992 982 972 ... 32 22 1
I.
B 100 99 98 97 .. 3 2 1
A
B
1
1 27 3 E 54 3 4 4
២
3
2
3x3 x 2 8 x 4 0 x 1 x 2 x 3 3 2000 1999 1998 ៤
3951 5139
45
II.
18 cm , 24 cm
III.
A(-3 , 2) ; B(9 , 2)
36 cm
C(3 , 10)
ABC IV.
ABC
O
R
R V.
ABCD [AC]
E
O A
[BC]
B
[AD] [BD]
F
OA OB OC OF OD OE ២
EF || CD
២
3
600
២
៤
៩
៩ :
1999 2001
I. ២
S
2 2000
1 1 1 1 ... 2 1 1 2 3 2 2 3 1999 1998 1998 1999 2000 1999 1999 2000
A 3 9 80 3 9 80
II.
២
B A 3 26 15 3 2 3
III.
២
x 4 2 x3 ax 2 bx 1
a,b ២
x 2 y 2 24
x,y a,b,c
ab bc cd a2 b2 c2 2 ab bc cd x xy y 1 y yz z 3 z zx x 7
IV. ២
a b c abc
a,b,c
a b2 1 c2 1 b a 2 1 c2 1 c a 2 1b2 1 4abc
x2 2m 3 x m2 3m 0 1 x1 x2 6
m V.
ABC ABC , AHB ២
(O)
(O’)
R1 R2 R3 AH
AHC (O)
[AH] , R1 , R2 , R3
A
(O’) (CA)
B
A (O’)
B
[AC]
[AD]
E
EBD
AC CE 2BE 2
២
3
601
៤
៩
៩ :
P 2 3 3
S 1 3 3
I.
F
II.
A 2x 4 B x 1
4x
A0
x
B0
2 4 x 1 x 1 4 A 2 2 x 6 x 9 x 1 x x 1
III.
2
x
A
1 2
2 x 3 y z 4 4 x 2 y 3z 1 ។ 6 x y 2 z 7
IV.
2 dm 3
2.5 dm
V.
4 dm 3
2.5 dm
VI.
A
dm 2
B
AAB ២
A
B A
VII.
C(O , R)
[CD]
(DC)
T
[CD]
H
[AH]
I
M
BC
M
CMD
[MA) ២
[AB ]
TIM ACM
IMD
MA MI MC MD ៤
HIMB AMB
ˆ " 30o MAB
AH
MB
R
២
3
602
៤
៩
៩ :
385a3 1001b5
I.
K
II. III.
77 3x 2 14 x2 4
a,b,c
ab bc ac a2 b2 c2 2 ab bc ac
IV.
x2 2 y 1 0 2 y 2z 1 0 z2 2x 1 0
x,y,z
A x 2000 y 2000 z 2000
4 m x 2 4 x m 0
V.
x1 , x2
m
VI.
.
10 ABC
G
H , K , K’
A,B,C
AH BK CK '
G
[KK’] , [AG] VII.
(C)
C
R
OI
I
[AG] R 3
[MN]
I
[MN] [MN] J
[MN]
J
[MN]
I
២
3
603
២
៤
៩
៩ :
A
I.
C 1 x 4 y 4
II. III.
4 x 1
16 x 16 x
B
2 1 3 3 9
6
4
V.
ABC
OD BC
O
OF AB ABC
3
x 1 4 x 1296 y y x 12 x 11 x 74 x 73 77 78 15 16
A 4 15 4 15 2 3 5
VI.
3 3 27
6
ឃ D 4 2 3 1 3
3
IV.
C
3
A
D
OE AC
BD2 CE 2 AF 2 DC 2 AE 2 FB2
F [AH]
H
AB 3 BD AC EC
3
604
[AC]
E
AB
D
៩
៩ :
A 1 3 3
I.
B 3 2 27 48 3
26 2 26 2 28 6 3 1
A
B A
II. III.
1 1 1 1 ... 1 2 2 3 3 4 99 100 x 2 px 1
a,b
x 2 qx 1
c,d
a c b c a d c d q2 p2 x3 3ax2 3 a 2 bc x a3 b3 c3 3abc 0
IV. V.
[BC]
R
[DE]
ឃ
A
AB AC AD AE 4R2 2
2
VI.
2
2
O
a
AB
[BC]
ABC
a
AC
AB CD
D
F
AO
BFC
OF AG
E AF
2 3
AC a
CF
ABC
BC OF BF
BC H
3
AB
E
1 2
AC
F
a
605
H
G (G
A ABC
៩
៩ :
I. II. III.
ab bc ac a2 b2 c2 2 ab bc ac
a,b,c
A
1 1 1 1 ... 1 2 2 3 3 4 99 100
x2 x 1 0
x14
1 x14
x2 x 1 0
1 1930
x1945
1 1945
x1987
1 1987
x x x x 1 x6 x2 x5 2 2 2 2 x 2 x x 12 x 35 x 4 x 3 x 10 x 24 x 2 y 3 z 4v 31 x y z v 3 x 3 y 4 z v 8 1 3 4 5 2 x y 5 x 3v 13 xyz xyv yzv xzv 20 2 x 2 y z v 17
IV.
V.
VI.
x1930
AB CD
O
OB
M
OC
N
AN || DM
ˆ OBD ˆ ˆ , OBN ˆ OCA OCM
3
606
ឃ
២
.
៩
៩ :
3 x 3 y 4 xy 27
I.
E 12 6 3 3 729 4 9
II. III.
x2 2 x 1 0
t 2
3
0
IV.
8 37.50%
V.
230
150
75 1 cm
300
50
y x 2 1 x 2 y 2 2
VI. 1 I 0 , 2
2
PQ
1
Q
yy '
VII.
AB AD
ABCD I
P
BH , J
CD
BD
A ˆ AIJ
២
3
607
H
៩
៩ :
A 4 15 4 15 2 3 5
I.
35 278 2 911
II. III. IV.
15 816 12 319
x y0
2 x 2 2 y 2 5 xy
x y x y
1 1 1 1 42 56 72 90 1 1 1 1 B 15 35 63 99 A
y 3 x 3 4 5 xy yz zx 21
V. ២
VI. VII.
E
ABC
1 10 72
x
AM ; BN MNP
CP
1 10 6 2
MA ; NB
H
២
3
608
PC
៩
៩ :
1 1 1 1 (x ab 0) a b x a b x 1 1 5 x y 6 , x 0 , y 0 1 1 13 x 2 y 2 36
I.
II.
x xy y
III.
x y K
IV.
x y
2 y x y
xy 1 , x y
3a 2 14 a2 4
VI.
n n 1 , n 2
9A
40
VII.
9B
A,B,C (AC) AP = 4cm
[Ax)
[Cy)
BC CM
AB = 5cm, BC = 12 cm
C
[BM]
36
[Ax)
[Cy)
P
M
[BP]
BAP
MBC
PBM H
៤
A
CM = 15cm AP AB
២
, x y
K 1 2 3 ... n
V.
x 0 , y 0
B
[MP]
PABH
BCMH
AHC
២
3
609
៤
៩
៩ :
I.
II.
a5 b5 a3 b3 a 2 b2 a b
ab 1
x2 2 y 1 0 2 y 2z 1 0 z2 2x 1 0
x, y,z
III.
ABC
IV.
0 a 1 2 1 1 x2
V.
AH
AM
AB 2 AC 2 2 AM 2
BC 2 2
x 1 a 2 a
1
A x 2000 y 2000 z 2000
2 1 1 x2
a 1 1 a a 1
x 1 y 2002 2003 x 2002 x 1 2003
VI.
២
3
610
២
៤
៩
៩ :
I.
x 2003 3x 2002 2 x 2001
II.
a,b,x
III.
a 2 2b 1 0 , b 2 2 x 1 0 , x 2 2a 1 0
IV.
x y 2 y x
x 2 y 2 2 xy
y
x
a a a 2 b2 b b A 2 2 a a b b 2ab
a,b
V.
S a 2003 b2003 c 2003
a
1
a b 1
b
a b a2 b2 a4 b4 a8 b8 0 1 y x a 2 a 2 a 2 x 1 y a 2 a 2 a 2
VI.
VII. P
ABC
AA , BB
CC
2 AA BB CC 3P
xoy
VIII.
120o , [oz ) [ox)
M
២
B
[oy )
[oz )
xoy M A
B
AM AM MM OA AB OB 1 1 1 OM OA OB
[oy )
M
[oz )
[ox)
[ox)
B
OA AM OM AB AB BB
៤
3
M
611
២
៩
៩ :
២
I.
3x 2 2 x 5 0
២ x 2 x 4
x2 0.4 x 0.16 0
៤ x 7 x 15
2 x 3 x 4 1 3x 7 3 5 3x 3 2 4 6 2
២
3 14 x 3 3 x 2 4 x 4 3 x 14
5 6 x 7 4 x 7 ៤ 8 x 3 2 x 25 2
2
2
II.
720 m 2
III.
IV.
V.
x 5 x 3 x 1 x 2 2 4 3 6
ABC
(3 , 0) ; (10 , 3)
6m
4m
(3 , 6)
x
B \
9
M A
x?
N 3
6
\
P
8
C
24
៤
3
612
២
៩
៩ :
1 1 1 1 ... 1 2 2 3 3 4 99 100 1 1 1 1 B ... 1 2 2 3 3 4 899 900 A
I.
x 3 x 4 x 4 x 6 m
II.
m m
m8 x yz 0
x, y,z
III.
x2 y 2 z 2 2
x3 y 3 z 3 3xyz
S xy yz xz
IV.
[Ox)
[ Ny)
MON [Ox)
V.
[ Ny)
A
B
AB 4 dm
ABCD
BD
MNO
BCD
S ABD 6 dm 2
3
613
MON
AB AD 3.5 dm
M
NO BC ABCD
៩
៩ :
4
I. x y0
II.
125 4 2 4 5 4 625 10
E
2 x 2 2 y 2 5 xy
x y x y
x 2 y 2 13 xy 6
III.
x3 4a 34 x2 4a a 2 x 4 a 2 1 0
IV.
a
1 2 a
V. 3951 5139
VI.
45
B 101100011000001 ... 100...001
VII.
2n 1
elx 0 ឃ
VIII. ឃ
ឃ
3
3
ឃ
1
ឃ
ឃ
1 ឃ
5
4
ឃ
2
7
ឃ
1 10 IX.
ឃ
AD
ABCD AB 2 5 cm , BC 2 5 cm
O
CD 6 cm
២
3
614
២
៤
៩
៩ :
I.
A
II.
B
4
53 5 54 2
511
2 3
74 3
x
III.
2
7 x 2 x 2 7 x 48 2
1
1
1
3x 1 3x 2 3x 2 3x 3 3x 3 3x 4 IV.
2
2 3x 4
2 3 3 5
2 19 5 15
2 x 2 y 3xy y 6 x 4 xy
V. VI.
3x 2 14 K 2 x 4
VII.
ABC
VIII.
Aˆ 2 Bˆ
A
ABC H
D
R
[BC]
[DH]
AC
M
DCH DH
R
HD HM HC HB
>
3
615
២
២
៤
៩
៩ :
1
2 3 2 A 2 3 6 2 4 3
I.
B 2
2 3 2 2 3
3
2
1
2 3
2 2 3
4 x 163 x1 8
II.
mx2 m n 1 x 2m n 8 0
III.
m
S 4
n
P2 a,b,c
IV. 7
a
V.
Bˆ 20o
ABC
70
b
c
a 2b 3c 90
a,b,c
9
50
b
[AB]
BM AC
M
ˆ AMC
VI.
O
[AB]
(AM)
d
(BM)
ANI
d
R N
[OB]
I
M
P
PBI
IN IP IA IB
3 2 R 4
A,I,P,M
BN AP
>
3
616
៩
៩ :
I.
20022003 20021979
II. III.
6
a , 36
a < 36
1 3
a 2 b2 c 2
a b c 1
IV.
PPCM a , 36 72
a
x
2
x 3 x 2 x 4
2 x y 2 , (x 0 ; y 0 2 x y 2
V. VI.
x y 1)
A 2 , 3 ; B 4 , 4
[AB] ឃ
PGCD a , 36 12
2
4
ឃ
6
ឃ
VII.
4 2
VIII.
4
R ABC
IX.
ABC
R
O
ABC
B
(OA)
ABC
ABM X.
BC 20 cm
ABC
[AK] [AC]
E
[BD]
H
[BC]
F
ABC
H,G,O
3
(AC)
N
ABN
AH 2OF G
M
617
O
ABC
៩
៩ :
I.
a
a 2 b2 2ab
b
1999 2001 2
2
2 II.
a
3999999
b
x2 x 1 0
1 1 x14
x14
a b c d
a b c d a b c d a b c d a b c d
III.
C
6 2
32
32
A a b c a b c a b c a b c 2
IV.
2
2
1 1 1 2 2 0 2 2 2 2 b c a c a b a b2 c 2
a bc 0
2
x 2 y 2 xy 37 1 2 2 x z xz 28 2 2 2 y z yz 19 3
V.
2
1 2 x
VI.
2
2 1 2 x 1 3 2 0
x x x 1 abc x a b c ab bc ac x x x 2 a b a c b a b c c a c b VII.
O
a
[BC]
AC
a
D (AC)
[AB]
[AB] F
ABC [CD]
F
[AO]
E
E
1 2
[AC]
2 3
BFC (OF) (AG)
a
[BC] (BF)
CF
a
ABC (OF) H
[BC] H
3
AF
(AB)
618
G(G ABC ៕
A
៩
៩ :
A 2 32 5 2 32 5
I.
3
2
3 5 3 3 5
2a 2b 2a 2b 3a b 3a b F 2
2
9b2 6ab a 2
a bc 0
a,b,c
III.
E
3 3 2 1 2 1 3
abc 0
5 5 5 2 2 2 2 2 2 2 2 b a c b a c a b c 2
a n , bn , cn , d n
a,b,c,d
IV.
3
a bb c c a abc 0
ab bc ca 0
a,b,c
II.
E3
(n
a , b , c
a,b,c
a
2
b2 c 2 a2 b2 c2 aa bb cc
a b c 1
a,b,c
1 1 1 0 a b c
a b c 1
a,b,c
V.
2
ab bc ca 0
a 2 b2 c 2 1 K
a 2 b2 c2 2003
A 6 , 3 ; B 2 , 5 ; C 4 , 8
VI. A ABC
a2 x2 1 a 2 b2 x2 y 2
ax by 0
VII.
x xy y
x y VIII.
x y
ABC [AC]
2 y x y
xy x y
x0
x, y
[AH]
A
M
E
D
H
[AB]
[BC]
ED AM K
[AM] M
BK || ED
[HD] [EH]
[AB]
L
3
619
BEDK
ML AH
[BK]
៩
៩ :
3 2 3 5
M
I.
1 ab
a b
II. 1
2
18
III.
1 a b2 2
217 215 214 2x x y 4 1 2 xy z 6 z 13 2
IV.
1 1 1 1 1 1 1 1 P a b c a b c ab bc ac ab ac bc
V. VI.
9x 2
MNPQ
ABCD
AM BM CP DP 1 AQ BN CN DQ 2
A
Q
abc 10
ABCD ?
M
B D ២
P
C
N
3
620
៤
៩
៩ :
A ab a b bc b c ca c a
I.
A 2 , 2 ; B 3 , 3 ; C 6 , 6
II.
x y z 5 3 4 xy yz zx 21
III.
AB 1
A, B,C
IV. x
5 1 2
BC x
BC AC AB BC
V.
10 m
M
3m VI. VII.
AB 9cm
C
O
[AB] [AB]
D
AB
[CD] (AB > CD)
P
PH
H
[CD] I
ˆ HPO
5 2
[OP]
O,K,P,H
ˆ KPO
PK
3
621
K
៩
៩ :
I.
220 1 5 1 20 6 2 5
3
II.
1 1 1 x y z 2 2 1 4 xy z 2
x xy y 2 3 2 2 2 x y 6
x 2 px 1 0
III.
a,b
3 3 3 3 3 3 3 23 3
x 2 qx 2 0
b,c
b a b c pq 6 IV.
10
20
1.70 m
1.61 m V.
ABCD
AC BD ,
AC 20a , BD 12a
AM 1 BN 2 AP 3 , , AB 2 BC 3 AD 4
[AB] , N BC , P AD
MBNP
S ABD 2S BCD
3
622
M
៩
៩ :
A
I.
3
1 3
3
2 1
2 1
3
3
3 2 2 2 1 a b 1
1 ឃ
1 ab ,
a 2 b2
B 6 6 6 ... ២
II.
ax 2 2bx c 0
1
bx 2 2cx a 0
2
cx 2 2ax b 0
3
S
abc 1
a,b,c
III.
a b 0
1 2 3
1 1 1 1 a ab 1 b bc 1 c ca
.
IV. V.
O [AB]
xy
[AB] D
E
A
C
[AB]
xy
F
xy
C M
[DE]
AMO M
C O
OCE
[CA) ឃ
C
AME
AFC
២
3
623
២
៤
៩
៩ :
M a 2 b2 2ab 1
I.
N a 2 b2 1 2ab
P 4a 2b 2 a 2 b 2 1
2
M
P N
4 3x 6 12 x 1 3 3x 2 3x 2 3 x 5
II.
x x2 4
x 4 4 x 2 32 0 x y z 1
a,b,c
III. IV.
1 1 1 0 x y z
M’N’P’
MNP
MI
M’I’
A
D
x2 y 2 z 2 1
MI M I V.
O
D
R
(AC)
O
OA
B I
OIH
OEA
C
OE
D
E
(AB) BC
OE
H
OE OI OH OA
OB2 OA OH
២
3
624
៩
៩ ២០១២-២០១៣ :
១២០
១០០
EE o DD 31 ។
52013 52014 52015
I.
។
, 2 n 1 n
II.
3
។
n n 1 ។
A 2012 2013 2013 2013 2014 4026 2013 ។
6 4 3 2 5 15 ។ 23 3 5 a b c 0 ។ ab bc ca 0 14 2013 a 1 b2 c 1 2011 ។ N
III.
IV.
a,b
c
1 20122
20132 2 2012 ។
20122 4048144
20132 4052169 2
2012 2012 A 1 2012 ។ 2013 2013 ABCD O ។ AC BD AB CD AD BC ។ ˆ 45o ។ BAC O AC R AB AB R 2
V. VI.
ˆ ACB
BC cos 45o
AC
ˆ ។ ABC
R។
2 3 , cos 30o ) 2 2
7
625
៩
៩ ២០១២-២០១៣ :
១២០
១០០
EE o DD I.
N
5 3 2 27 2 3 3 4 4
9 3 16
1 1 x y x y 2 3 4 7 x y x y
II.
III.
,
M 22 288
A 2012
។
x y z ។ 5 3 4 xy yz zx 21 ។
5 3 29 12 5
a 2a 3b 3b 4a ។ ។ b2 3 2 P x 1 x 2 x 3 x 6 2
IV.
។
V.
។
AC BC ។
AC ។
ABC
PBA
P
D ( AB
A
ABCD
x
B
PBC ។
PB 2 PA PC ។ BA2 PA ។ BC 2 PC
7
626
AC
P។
៩
៩ ២០១២-២០១៣ ១:
១ ០
១០០
EE o DD I.
1)
A 20133 2013 2014 2012
2)
B 1 2 2 4 5 6 7 8 9 ... 2014 2015 2016
3) 4) II.
x x 20132013
2014
x
44 4
128
22
2n
។ 2009
9
1)
a,b
2)
9
11 ។
92011 92012 92013
abc
c
A
III.
n ។ 2010
a b c 1 ; ab bc ac 4
20131111 1 20132222 1
20132222 1 ។ 20133333 1
B
13
IV.
a b S 8 8 b c
a b c ab ac bc ។ 2
2
2
x 2 3x 1 0
V.
x P
2 1
2x
x1 1
a,b,c
VI.
2
2000
c 8 a
2013
។
x2 ។
x1
x
abc 2 ។
2 2
2 x2
x2 1
។
2
a 2 2b 1 0 ; b 2 2c 1 0 ; c 2 2a 1 0 ។
S a 2013 b2013 c 2013 ។ VII.
5cm ។
ABCD
x2 2m 1 x 4 m 1 0 ។
OB
O។
OA
m ។
ABCD ។ VIII.
O1
C ។ PC
O2
P។
O2
D។ APB ។
PD
PC AC BC PA PB ។ 2
7
627
AB
O2
O1
៩
៩ ២០១២-២០១៣ (១៤/០២/២០១៣) ២:
១ ០
១០០
EE o DD
x
I.
2.63 2 ។
x ។
x ។
x 20132 2014
N 2010 2011 2012 2013 1 ។
II.
N
។
IV.
A
B f
V. VI. (
A 3 2 5 3 2 5 ។
VII. (
ABCD
B
x ។
E ។
C
F E D
7
628
a ។
B A ។
42 cm 2 ។
3 AE FC 4EF ។
A
។
x4 ay 2 2ax y 1 x6 A B ។ 2 x 2x x x 2 f x 2 f 5 x x
III.
F
f 1 ។ AC ។
៩
៩ (០៣/០៣/២០០ ) ១:
១ ០
១០០
EE o DD x x 4 4m 0 ។
I. 2 2 6
II.
2 2 3
2 2 6 2 2 3
3
III.
1 x x 1 x 3 x
2 ។
x 1 ។
a 3 2 17 ; b 4 19 ; c 5
IV. V.
32 ។
cba ។
។
N X
X X
។
34
X
។
X+1 VI.
ABC ABC
ACD ។
។
15 [AD] ។
A
O1O2
AB
ABC
AKL ។
S1 2S2 ។
7
629
O2
O1
AC
X
K
L។
S1
S2
N។
៩
៩ (១៧/០២/២០១០) ១:
១២០
១០០
EE o DD 2010 ។ x 4x 7 1 2 x x A ។ x 1 x xx x x 14 x 5 3 ។ 3 x 5
I.
2
II.
III. IV.
a
a 2 b2 ab ។ 2
។
b
9992 1 0012 1 999 998 ។
V.
O
AB
R
CD
I។
IA2 IB2 IC 2 ID2 2R2 ។ VI.
ABC ។
G AC
BC
P
Q។
BP , PQ
7
G
630
AB QC ។
៩
៩ (១ /០២/២០០៩) ២:
១ ០
១០០
EE o DD I.
a,b,c
ab cd II.
a
2
a c b d
។
d
a 2 c 2 b2 d 2 ។
b
III.
a 2 b2 41 ។
។
a b ។
A 2 x 2 9 y 2 6 xy 6 x 12 y 30
y
x
a c b d
ab cd
។
។
m2 x 2 m2 x 2 2 2 2 x m 1 m 1 1 ។ 8 8 BC a , CA b , AB c ។ A BC 2
IV. (
m
។
V.
ABC D។I
2
។
IA ID
DB , DC I
O
VI.
I
I
BC
AB
O
E
AB = 2R
O
EF ។
OC
AB ។
J
OB ។
I
C
F។
AC
J
OA
c ។
a,b
M
O
J
OC
N។
C x ។
OI
R
x ។
x
MN ។ S
O ។
OI
O
I
7
631
។
S
S ។
៩
៩ (១៣/០៣/២០០ ) ១:
១ ០
១០០
EE o DD I. 1 2 3 6 2 35 12 2 2 6 3 4 7 12 2 2 32 6 ...............................
12
។
S
II.
13 1 23 1 33 1 20083 1 ... ។ 11 2 1 3 1 2008 1 ។
p
។
2008o ។
III. IV. ( V.
p
។
1 1 1 1 ... 43 ។ 4 5 5 6 6 7 n n 1
n
: 4 x 4 3 4 x 3n 24 0 2
n ។
n ។
VI.
។
ABC ABC ។
VII.
n។
1 2008 ។
។ ។
7
632
៩
៩ (១៣/០៣/២០០ ) ២:
១ ០
១០០
EE o DD I.
a0 , b0
II.
x , y
III.
ABC
bc ។ ab
b c 2008 ។ a b
x 2 20082 y 2 2007 2 ។
P
ABC ។
PAB , PAC , PBA
P ។
IV. (
A A។
100 km/h A។
A។
A
B។
B
V.
3 8
B
A, B, C ។
A
B 16 $
៣។
C
។
។
VI.
12 ។
5
។
9
។
7
633
10
3
៩
៩ (១ /០២/២០១០) ១:
១ ០
១០០
EE o DD A
I. II.
2a3 326 5b9
2a3
246912 ។ 123457 123456 123458 2
5b9
5b9
9
a b ។ F
III. IV. (
1666...66 6666...64 a * b ab
*
a *b * c a *b * c
a * bn a * n * c
a *b
n
a * bn
( a, b, c, n
9។
V. ។
VI.
10
x ។
19 96
1
d
a
b
។
1
9
។
9 1។
i
ABCD ។
7
c
10
ABCD c, d, e, f, g, h
។
x
។
3
b ។
a
a *b b * a
VII.
។
2010
634
a, b,
៩
៩ (១ /០២/២០១០) ២:
១ ០
១០០
EE o DD a 2 a 1 ។
I.
A a 4 2a3 4a 2 3a 3 ។
III. IV. (
a
92
b
HAO ACB
b ។
a 1
។
A O ។
n
។
ABC B
VI.
0, 1, 2, 3, …
n
2n n 2
V.
1 1 1 ។ x y 3
x , y
II.
[AO] ( H
[AH]
BC = 2a ។
AH
AO
a។ ABCD ។
AB
B
BE = AB ។ M
MAC MCD ។ ។
AMC
។
AMCE
7
635
។
៩
៩ (១ /០២/២០០៩) ១:
១២០
១០០
EE o DD ។
I.
8 ។
9 II.
A
។
1 3 2 2 3 ។ 2 3 3 2 2 3
xy 100 1 1 1 ។ x y 5
III. IV. (
4
O
។
M
M ។
M
។
V.
2 cm ។
12
។
7
636
។
៩
៩ (១ /០២/២០០៩) ២:
១២០
១០០
EE o DD I.
2 2 2 A x 1 3x 2 x 1 3x 2 ។
x ។
A
B
A ។ 2x 2 B 2 ។
x
II.
x 2 x 2 x 2 x 2 2 ។ x 2 1 x 2
III.
1 2 y 1 2x 1 1 4 2 3 2 x 1 y 1
1 2
។
5x 4 x6 ។ 6x 2 4 5 3
IV. ( V.
BAC ។
ADC
A
D
AC
AB AC a ។ ។
ABCD
BC, AD
O។ ។
OA, OB, OC [BC]
[BD]
DK
7
637
a ។
K។
BCD ។
[CK]
AOD OD
a ។
BD
OK ។
COB
៩
៩ (២០០៩-២០១០) ១:
១២០
១០០
EE o DD 4 40 ។ f x x2 x 3 9
I.
P 2 a3 b3 3 a 2 b2 1 ។
a b 1 ។
2 x y 16 3xy ។
II.
A 3 7 5 2 3 7 5 2 ។
III.
333 IV. (
b a b
a
9
63 ។ V. K VI.
[AB] ។
ABCD O
a, b, c
[AB]
O
VII.
D
[CD] ។
ABC ABC ។
A
A
O
AD = AH + DK ។ O
R
[AH]
S
abc 4RS ។ [AB] ។ C
។M
C។D
B
C
CD2 AM BN ។
7
638
H
N
AB ។
៩
៩ (២០០៩-២០១០) ១:
១២០
១០០
EE o DD I.
a
។
b
B a b 2 ab ។
A a b
A
m, n, p
II.
ab 2
a 2 b2 ។ 2
B
q
m 2 2n 1 0 , n 2 2 p 1 0 p 2q 1 0 , q 2 m 1 0 2
2
។
I m 2010 n 2010 p 2010 q 2010 ។
III.
x 9 x 8 x 7 x 6 x 5 5 ។ 2009 2008 2007 2006 2005
IV. (
x1 , x2
a 2 pa 1 0
y1 , y2
x1 y1 x2 y1 x1 y2 x2 y2 q2 p2
b 2 qb 1 0
។
a, b, c
V.
a2 b2 c2 2 ab bc ca ។ VI.
218 220
2001
1 2000
2
2
VII. VIII.
20
ABC
2 2000 ។
។
100 1
IX.
។
5
។
100
BAC ។
[AD)
។
D។
BC
AB DB ។ AC DC ។
X.
A
។
B ។
។
PmUi A
7
639
PmUi B
៩
៩ (១ /០២/២០១០) ១:
១ ០
១០០
EE o DD 1 2 x 5
I.
4
II.
1 x2
49 5 24 4 49 5 24 2 3 ។ B 2 2009 ។
A 2008 2010
III.
1 2 3 2009 ។ ... 1 2 1 2 3 1 2 3 4 1 2 3 ... 2010 1 ។ A 1 1 2 3 ... 2010
IV. (
A
V.
x2 y 2 1
a0 , b0 ។
bx 2 ay 2
x 2010 y 2010 2 ។ 1005 1005 1005 a b a b
VI. (
a,b
c
a b c 1 , a 2 b 2 c 2 1 , a 3 b3 c 3 1 ។
A a 2008 b2009 c2010 ។ VII. (
O1
O2
r1
M។
r2
O2
O1 O2
O3
AB
A
B។I
។
[AB] ។
4r3 r1r2 ។ r1 r2
7
O3
640
r3
M
O1
៩
៩ (១ /០២/២០១០) ១:
១ ០
១០០
EE o DD I. II. III.
A
A x2 x 6
n
7h30mn
n2
A
2 2 ។ 12 8 3 6 ។
x
5 km/h ។
B
1h
20 km/h ។
A 17 km ។
1 a b c a 1
V.
ABC
bc ។
VI. (
CD CA ។
[BN] ។
A
BD 2BN ។
7
641
C
។
BCD ABC
AC
AC
-
13h6mn ។
ba ។ AB AC 1 ។
A 90o
CD BC ។
A
B ។
A IV. (
B
D
៩
៩ (២២/០៥/២០០៩) ១:
១ ០
១៥០
EE o DD 2 2 x1 4x 4x
I. II.
3
a bc ; b ac
a, b, c
1 3
3
2 1
3
2 1
3
។
a 2 b2 c 2 2 ។
c a b
a 2 b2 c2 2abc 2 ។
x 1 x 2 x 3 x 4 3 ។
III. IV. (
72016 ។
a, b
V.
c
a 2 2b 1 0 ; b 2 2c 1 0 ; c 2 2a 1 0 ។
VI. (
10 m
M ។
3m VII. ( VIII.
M។
A។
ABC E។
A
BD = 20 , DC = 16 ។ D B
[BC]
DA
[AC]
[BD]
BC
D
Y។
[BD]
X។ XY = 2
AYC ។
AXC IX.
AB AC2 ។
។
ABCD
S a 2009 b2009 c 2009 ។
X, Y, Z Z X
X Z
X។
Z
X។
Z Z។
X
1 4
Y
7
642
។
៩
៩ (២៧/០២/២០០៩) ១:
១២០
១០០
EE o DD 13 x 3
I.
1 2 x 3 x 20 ។ x
II.
12 cm
8 cm។
x cm ។
x
64 cm 2 ។
a, b, c
III. IV. ( V.
a b 1 ។
B a 2 b2 ab ។
a, b, c
a3 b3 c3 3abc 0 ។ ។
x, y, z
1 1 1 1 1 1 ។ x y z a b c
VI. (
A។
M
MB2 MC 2 2MA2 ។
7
643
[BC] ។
៩
៩ (២៧/០២/២០០៩) ២:
១២០
១០០
EE o DD 1 3 2 x y x ។ 2 y 1 3 x y
I.
f n 32n 7 , n
II. III.
E
n
f n 1 f n
។
IV. (
n n 2 n3 3 2 6 1 8
8។
។ ។
20% ។
។
V.
13 cm
17 cm ។
។
VI. (
ABC
AA ' CA ។
។
AB , BC
CA
S A ' B 'C ' 7 S ABC ។
7
644
BB ' AB , CC ' BC
sYsþI¡ elakGñkmitþGñkGanCaTIRsLaj;rab;Gan enAkñúgEpñkenHelakGñknwg)aneXIj BIlMhat;l¥² nig cemøIyrbs;vaCaeRcIn Edlbgáb;eTAedayviFIsaRsþedaHRsayl¥² RbkbedayKMnitEbøk² . ´Føab;erobcMEpñkenH rYcehIykalBImun EtmanlMhat;cRmúHfñak;rhUtdl;fñak;TI 12 ៩
critlkçN³enAkñúgEpñkenH KWEckecjCaBIrRkumrYmmman RkumRbFanlMhat;l¥² nigRkumcemøIyén lMhat;l¥² EdlpÁÚpgÁ KñaeTAedayelxerogBIxagmuxcab;BI 1 rhUtdl; 400 . sUmkMuBüayamedaHRsaylMhat; TaMgenHtamlMdab;lMeday ebIcg;eFVIlMhat;Na cUreFVlI Mhat;enaHEtmþgeTA . RbsinebIelakGñkmanbBaða b¤cm¶l;Rtg;cMNucNa EdlmanenAkñúgEpñkenH elakGñkGacTak;Tg;eTAkan; RKU b¤mitþPkþirbs;GñkEdlmansmtßPaB b¤GñkeroberogesovePAenHkñúgeBlevlaGñkKitfaKYr . …
vii
1> 2> 3> 4>
lMhat;l¥² rkRKb;KUéncMnYnKt;viCm¢ anén a , b EdlepÞógpÞat; TMnak;TMng ³ a b a b 8 . RsaybBa¢ak;fa E CakaerR)akd Edl E x 1000 x 2000 x 3000 x 4000 1000 . rkelxenAxÞgr; ayéncMnYn ³ A 3 7 3 7 . KNnakenSam ³ P 3 2 2 3 2 2 . 2
2
4
2011
2012
17 12 2
2013
2014
17 12 2
5> KNnatémø x edIm,I[ F mantémøGb,brma ¬tUcbMput¦ Edl ³ F x 1x 2x 3x 6 . 6> KNna x y edaydwgfa x y 8100 nig x y 30 . 7> RsaybMPøWfa ³ 11 2 21 3 31 4 ... n n1 1 1 Edl n CacMnYnKt;viCm¢ an ehIyFMCag 1 . 2
3
2
3
8> eK[ n CacMnYnKt;viC¢man Edl n Ecknwg 7 [sMNl; 5 ehIy n Ecknwg 8 [sMNl; 3 .
k> etI n Ecknwg 56 [sMNl;b:unµan ? 9> eK[sVIútcMnYnBit a 1 1 1n 2
n
14> 15> 16>
2
bgðajfa A a1 a1 a1 ... a1 CacMnYnKt; . KNnatémøelxénkenSamelx A 182 33125 182 33125 . eK[BIrcMnYnKt;viC¢man x , y Edlman 2 xy y 4 x 2 x 1 . KNna A x y . eK[ f x x 13 x 13 x 13 x ... 13 x 13 x 1 . KNna f 12 . brimaRténRbelLÚRkam ABCDmanrgVas;esµInwg 48cm . rgVas;km
10> 11> 12> 13>
x> rkcMnYn n enaHedaydwgfa 5616 n 5626 . 1 1 1 Edl n 1 n
2
3
20
3
3
2
2011
2010
2009
2011
2008
2011
2
2
2
17> eK[ x y 0 nig 2 x
2
KNna E xx yy .
2 y 2 5 xy
18> sRmYlkenSam E 2x y 2x x y x y x y x y Edl x y 0 . 19> edaydwgfa sin a cos a 2 sin 45 a nig 3 cos a sin a 2 cos30 a . 2
2
o
o
cUrKNnaplKuN ³ k> A 1 cot1 1 cot 2 1 cot 3 1 cot 44 x> B 3 tan1 3 tan 2 3 tan 3 3 tan 29 . o
o
o
o
o
645
o
o
o
-
20> 21> 22> 23> 24>
25> 26> 27> 28> 29> 30> 31> 32> 33> 34>
ebIeK[bIcMnYnBitviC¢man a , b , c . cUrbgðajfa ³ a bb cc a 8abc . eK[bIcMnYnBitviC¢man a , b , c . bgðajfa ³ a b c ab bc ca . rkb¤sKt;viCm¢ anénsmIkar ³ 2x y 16 3xy . RsaybBa¢ak;fasmIkar ³ x ax b x bx c x cx a 0 manb¤sCanic© EdlenAkñúgsmIkarenHman x CaGBaØat nig a , b , c CacMnYnBit. eK[TMnak;TMng ³ f x 4 4 2 . k> KNna f x f y edaydwgfa x y 1 . 1 2 3 2010 x> TajrkplbUk S f 2011 f f ... f . 2011 2011 2011 KNnalImIt ³ lim x . KNna ³ A 2010 2011 2011 2011 ... 2011 2012 1 . eK[BIrcMnYn a nig b EdlepÞógpÞat; a b 1 . bgðajfa a ba b a b a b 0 . 1 KNna A 1 12 1 13 1 14 ... 1 2011 . edaHRsaysmIkar ³ 4x 5x 1 0 . ebI abc 1 sUmbgðaj[eCOfa ab 1a 1 bc 1b 1 ca 1c 1 1 . KNna S 2 1 1 1 2 3 2 1 2 3 4 3 1 3 4 ... 100 99 1 99 100 . sUmbgðaj[eKeCOfa ³ 2002 2002 Eckdac;nwg 6 . RKb;cMnYnBit a , b , c EdlepÞógpÞat; a b c 1 . bgðaj[eKeCOfa a b c 13 . RKb;cMnYnBit a , b , c ,..., x , y viC¢man nigxusBI 1. sUmbgðaj[eKeCOfa ³ log b log c log d .... log y log a 1 . eyIgman f x log 11 xx . bgðajfa f x f y f 1xxyy . 2
2
x
x
x
x0
2010
2009
2008
2
2
4
2
c
4
8
8
1979
2
b
4
2
2003
a
35>
2
x
2
2
y
36> edaHRsaysmIkar 7 48 7 48 14 . 37> RKb; a nig b CacMnYnBitviC¢man nigxusBI 1. sUmbgðajfa ³ x
x
log a b log b a 2log a b log ab b log b a 1 log a b
.
cosa 1 cosa 4 cot a 38> sUmbgðajfa ³ 11 cos . a 1 cosa sin a 39> edaHRsaysmIkar ³ 5 50 x . log x
log 5
646
-
40> eKe)aHRKab;LúkLak;mYyEtmþgKt; . KNnaRbU)abEdle)aH)anelxess b¤ elxFMCag 5 . 41> edaHRsaysmIkar x 4x 16x ... 42> cMeBaHRKb;cMnYnKt;viC¢man n eK[ f n 3
4n x 3 x 1 1
n 2n 1 n 2 1 3 n 2 2n 1 2
3
f 1 f 3 f 5 ... f 999997 f 999999
43> edaHRsayRbB½n§smIkar ³
44> eK[ a nig n CacMnYnviC¢man ehIy n Kt; EdlplKuN a a 1
45> edaHRsayRbB½n§smIkar
. KNnaplbUk
. 1 x y 4 5 400 b/ 1 5 x 6 y 900
5log y x log x y 26 a/ xy 64
1 a1 1 a2 1 a3 ... 1 an 2 n
.
2
a3 ... an 1
.
. sUmbgðajfa
.
7 x 3 3x 2 y 21xy 2 26 y 3 342 a/ 3 2 2 3 9 x 21x y 33xy 28 y 344
4 x 3 y 1 27 y 171 b/ x x 1 2 y 172 8 2 3
.
46> cMnYnmYymanelxR)aMxgÞ ;EdlelxxÞg;vaerobtamlMdab;KW x , x 1 , x 2 , 3x , x 3 ehIyeKdwgfa 47> 48> 49> 50> 51> 52> 53> 54> 55> 56> 57> 58>
cMnYnenaHCakaerR)akd . cUrrkcMnYnenaH . KNnaplbUk S 1 1 2 2 1 3 3 1 4 ... 2024 1 2025 . RsaybBa¢ak;fa ³ f n 3 7 Eckdac;nwg 8 Edl n CacMnYnKt; . man f x 2x 1 nig g f x x 3x 1 . sUmKNna g 3 . RKb;cMnYnKt;viC¢man n RsaybMPøWfa n3 n2 n6 k¾CacMnYnKt;viC¢manEdr . KNnaplbUk S 11 2 21 3 31 4 ... 20101 2011 . eK]bmafa x x 1 0 . cUrKNna x x 1 . cMnYnmYyCakaerR)akdmanrag abcd . edaydwgfa ab cd 1 cUrrkcMnYn abcd enaH . sUmbgðajfa n! 3 cMeBaHRKb;cMnYnBitviC¢man n 7 . GnuKmn_ f kMNt;eday f x a 12 f x f x cMeBaHRKb; x 0 nig a 0 . sUmbgðaj[eXIjfa f CaGnuKmn_xYbelIcenøaH 0 , . cMeBaH n! n n 1 n 2 ... 3 2 1 ¬ n! Ganfa n hVak;tUErül¦. bgðajfa 0 ! 1 . 21n 4 bgðajfa F 14 CaRbPaKsRmYlmin)an cMeBaHRKb;cMnYnKt;FmµCati n . n3 rkcMnYnKt;FmµCatitUcbMput Edlman 6 CaelxenAxÞg;Ékta ¬xÞg;ray¦ ehIyebIbþÚrelx 6 eTAxagmuxeK bMputvij enaHeKnwg)ancMnYnfµI EdlesµInwg 4 dgéncMnYnedIm . 2n
2
2
2
3
2011
2011
n
2
647
-
59> eK[lkçxNÐbI EdlbursbInak;
eTAsþIdNþwgnarIbInak; 1, 2 , 3 CaKUGnaKt dUcxageRkam ³ k> ebIburs A CaKUnwgnarI 1 enaH burs B CaKUnwgnarI 2 x> ebIburs A CaKUnwgnarI 3 enaH burs C CaKUnwgnarI 1 K> ebIburs B minCaKUnwgnarI 3 enaH burs C CaKUnwgnarI 1 . sMNYrsYrfa etIbursNa CaKUnwgnarINa ? 60> rk 11 cMnYnminGviC¢man EdlcMnYnnImYy² esµInwgkaerénplbUkén 10 cMnYnepSgeTot . abc 0 61> eK[ a , b , c CacMnYnBit epÞógpÞat; ab . KNna A a 1 b 1 a 1 . bc ca 0 A, B , C
2010
2011
2012
62> KNnaplbUk
S 1
1 1 1 1 1 1 2 1 2 2 ... 1 2 2 1 2 2 3 2010 2011 2
63> edaHRsaysmIkarman x CacMnYnKt;³ k> x
.
x 2011 x 2012 x 2013 4
2010
x> x x x x 0 . 64> edaHRsaysmIkar ³ 2 x9 y 2 9 . ¬eKsresr 2 x9 y bB¢ak;faCaelx4xÞg;mann½yfa 0 x 9 , 0 y 9 ¦ . 21 65> rkRKb;cMnYnKt;FmµCati N a a a ...a EdlnaM[ 2a a a ...a 1 12 . 1a a a ...a 2 2010
x
2011
2012
2013
y
1 2 3
1 2 3
1 2 3
66> KNnaplbUk ³
n
n
n
1 1 1 1 S ... 1 2 3 2 3 4 3 4 5 nn 1n 2
.
67> sRmYlkenSam ³ 2 3 2 3 3 . 68> sUmbgðaj[eXIjfa x 11 x 1 1 x 2 1 x 4 1 x 81 . 2011
2012
2
4
8
69> cMeBaH n! n n 1 n 2 ... 3 2 1 . KNnaplbUk S 21! 32! 43! ... n n 1! . 70> 71> 72> 73> 74>
kMNt;elxxagcugénplbUk ³ 1 2 3 ... 2010 2011 . kMNt;elxxagcugénplbUk ³ 1!2!3!... 2010!2011! . eK[ a a 1b b 1 1 . KNnatémø A a ab . 4022 . edaHRsaysmIkar ³ 2011 x eK[ ab 1 . RsaybBa¢ak;[eXIjfa ³ a b a b a b a b . 2
2
2
log 2012 x
a
a c b d
. bgðajfa
2
2012
2011
log 2012 2011
5
76> BIsmamaRt
2
2
75> eK[ x , y nig z EdlepÞógpÞat;
77>
2
5
3
3
2
2
x 2 2 y 1 0 2 A x 2010 y 2011 x 2012 y 2z 1 0 z 2 2x 1 0 ad bc 2cd 2ab ad bc 1 a a a 2 b 2 b b A 2 2 a a b b 2011 ab
. KNnakenSam ³
.
.
nig b CaBIrcMnYnKt;viC¢man. cUrsRmYlkenSam ³
648
-
.
78> eK[ a b c 1 , a b c 1 nig ax by cz m . cUrKNna 79> eK[RtIekaNsm)aTmYy EdlmanmMumYymanrgVas;esµI 2
2
P xy yz zx
2
.
a nigRCugBIrEdlGmnwgmMuenH manrgVas;esµIKña ehIyesµI a . KNnaRkLaépÞRtIekaNenH CaGnuKmn_én nig a . 80> eRbóbeFobcMnYn k> 6 5 nig 21 x> 1 11 72 nig 2 3 2 . 81> rktémøKt;én x , y nig z EdlepÞógpÞat;smIkar ³ x y z 4 2 x 2 4 y 3 6 z 5 . 2011
2011
82> eK[
ax 3 by 3 cz 3 1 1 1 1 x y z
. RsaybMPøWfa
3
ax 2 by 2 cz 2 3 a 3 b 3 c
.
83> RsaybBa¢ak;fa TIRbCMuTm¶n; G p©itrgVg;carwkeRkARtIekaN O nigGrtUsg;énRtIekaN H énRtIekaN
mYy CabIcN M ucrt;Rtg;Kña . 84> eK[ P a b 5a 9b 6ab 30a 45 . cMeBaHRKb;témøéncMnYnBit a nig b bgðajfa P 0 . 85> sUmbgðaj[eXIjfa 39 51 Eckdac;nwg 45 . 86> KNna A 10110001100000001 ...1000...001 . 2 2
2
2
2
51
39
man 2
87> KNnatémø n EdlnaM[ 88> edaHRsayRbB½n§smIkar
3 9 15 6n 3 300 ... 2011 2011 2011 2011 2011 x2 2x y 2 y 2y z z 2 2z x
k>
4 4
8
8
2 1
91> sRmYlkenSam B 2
xx x x 1
92> eK[bIcMnYnrYmman ³
1 dgénelx 0
x 4 x 3 x 2 x 1 4 2007 2008 2009 2010
89> edaHRsaysmIkar³ 90> RsaybB¢aak;fa
n
A 888...88
man n tYénelx *
2 1 4
8
2 1
1 2
. x>
.
1 1 1 1 a b c 12 1 1 1 7 c b a 12 111 5 a c b 12
.
.
1 x 2 1 x 1 x x 1
B 222...222
. nig C 444...444
man n 1 tYénelx @
man 2n tYénelx $
RsaybB¢ak;[eXIjfa A B C 7 CakaerR)akd . 93> kalNa)aténRtIekaNmYyekIneLIg 10% etIkm eK[ a , b nig c CargVas;RCugénRtIekaNmYy.Rsayfa ³ ab bc ac a b c 2ab bc ac . 2
649
2
2
-
95> eK[bIcMnYn a , b nig c Edl a b c 0 nig abc 0 . KNna E Edl E
2011 2011 2011 2 2 2 2 2 2 b a c b a c a b2 c2 2
.
96> bgðajfa 2010 2012 2011 12011 12011 12011 1 2011 1 . 97> eK[ ABC CaRtIekaNmYy Edlman AM Caemdüan . bMPøWfa 2 AM AB AC . 98> RtIekaNmYymanrgVas;RCug x , x a nig x 2a Edl x 0 nig a CacMnYnKt;viC¢man . rkRKb;rgVas; 2
4
8
16
32
RCugénRtIekaNenH EdltUcCag b¤esµI 10 edIm,I[vaCaRtIekaNEkg . 99> RtIekaNmYymanrgVas;RCug 3 , 4 , 5 . RtIekaNenHCaRtIekaNcarwkkñúgrgVgm; Yy . tag A , B , C CaRkLaépÞEpñkbøg; EdlenAkñúgrgVg; nigeRkARtIekaN ehIy C CaEpñkRkLaépÞEdlFMCageK . KNna A B CaGnuKmn_nwg C . 6 cm 100> eK[ctuekaN ABCD CactuekaNBñay nigman MN Ca)atmFümén A B N ctuekaNBñayenH. KNnaRbEvg)atmFümenAcenøaHGgát;RTUg x dUcrUb? M x C D ebIeKdwgfa )attUc AB 6 cm nig)atFM CD 10cm . 10 cm 1000 101> cUrKNnaplbUk S 11 3 32 5 53 7 ... 1999 . 2001 2
102> RsaybBa¢ak;[eCOfa
2
2
2
6 6 6 ... 6 30 30 30 ... 30 9
man n r:aDIkal;
man n r:aDIkal;
103> k> cUrKNnaplbUk S 1 2 3 2 3 4 3 4 5 ... nn 1n 2
x> RsaybBa¢ak;fa 4S 1 CakaerR)akd cMeBaHRKb;cMnYnKt;viC¢man n . 104> eK[ 0 x x x ... x . RsaybBa¢ak;fa x xx xx ...x x x xx xx ...x x 105> ebI x , y CacMnYnBitEdlepÞógpÞat; x 3xy y 60 . rktémøFMbMputénplKuN xy . F 106> rkplbUkrgVas;mMu ¬KitCadWeRk¦énmMu A, B , C , D , E nig F C A B enAkñúgrUbxagsþaM . 1
1
2
3
2
3
3
6
9
1
2
3
4
8
12
12
2
9
7
.
12
2
D
E
107> RsaybBa¢ak;facMnYn A 111...11222...22 EdlmancMnYntYénelx! nigelx@ esµIKñaKW 2011tY.
RsaybBa¢ak;fa A CaplKuNénBIrcMnYnKt;tKña . 108> ]bmafa x , x Cab¤sénsmIkar ³ 2011 x t 2011 x 2011 0 . rktémøtcU bMputénkenSam ³ x x 1 1 H x x 4 . 2 x x 2
1
2
2
2
2
1
109> KNnatémøelxén
1
2
1
2
A 3 5 3 5 3...
.
650
-
110> eKmanBhuFa
. eKdwgfasMNl;énviFIEckrvag px nig x esµI 1 ehIysMNl;énviFIEckrvag px nig x 1esµI 2 . rksMNl;énviFIEckrvag px nig xx 1 . ¬GñkGaceRbobeFoCamYylMhat;TI8 TMBr½ TI!¦ 111> RsaybBa¢ak;facMnYn 5 5 5 Eckdac;nwg 31 . 112> rkelxéntYcugeRkaybg¥s;éncMnYn 2 . 113> rkelxéntYcugeRkaybg¥s;éncMnYn 123456789 114> RsaybBa¢ak;facMnYn A 13 2 1 2 1 CacMnYnKt;viC¢man . 115> bBa¢ak;témøéncMnYnBit x edayeRbIsBaØavismPaB. edaydwgfa x CacMnYnEdlmanEpñkKt;manelx BIrxÞg; kalNaeKsresrvaCacMnYnTsPaK. 116> k> eRbóbeFobcMnYn 1 2000 nig 2001 2 2000 2000 x> edaymineRbI 2000 4000000nig 2001 4004001KNna A 1 2000 2000 . 2001 2001 p x
2011
2012
2013
34
2011
3
3
3
3
2
2
2
2
2
2
2 3
117> eK[TMnak;TMng x x y y x y a . RsaybBa¢ak;fa a x 118> eRbóbeFobcMnYn 2001 2002 nig 2 2002 . 119> RsaybBa¢ak;fa ³ A 111...111 444...44 1 CakaerR)akdéncMnYnKt; . 2
4
3
2
2
1
1 1 p q
p q 121> KNnatémøelxénkenSam ³ A 3 2
3
122> RsaybBa¢ak;fa ³
3 3
2 1 3
4
9
1 1 3 q p q p 2
125 3 125 3 9 27 27
1 3 2 3 4 9 9 9
x 1 x 5 x 7 x 11 4 . 125> edaHRsaysmIkar ³ 1991 1987 1999 1981 126> edaHRsaysmIkar ³ x 6 x 2 3 x . 127> edaHRsaysmIkar ³ log log log x 2 . 128> KNnaplbUk A 111111 ...111...11 . 1998
2
y
2 3
.
3
Edl p , q 0 .
.
.
123> KNnatémøelxénkenSam S 6 6 6 6 ... . 124> ]bmafa a 0 , b 0 nig ba bc 2011 . KNnatémøén 2000
2 3
n énelx$
2n énelx!
120> sRmYlkenSam ³ E
2
3
bc ab
.
1999
4
n énelx!
129> edaHRsaysmIkar ³k> x
x
xx
x> x
x x
x x x
651
. -
130> eK[ a , b , c CacMnYnBitEdlepÞógpÞat;RbB½n§smIkar
a b 2 2ac 29 2 b c 2ab 18 c a 2 2bc 25
. cUrKNna a b c .
131> KNnaplbUk S 1 2 3 ... 2011 . ¬edaHRsayBMutamrUbmnþsVúIt¦. 132> fñak;TI 9B mansisSsrub 28 nak;. sisSRbusmankm
. ebIdwgfakm KNnakenSam A a b c a b c a b c a b c . 2011 2011 134> eK[ a b c 0 . bgðajfa b 2011 0 . c a c a b a b c 135> kMNt;témø a nig b edIm,I[BhuFa x 2x ax bx 1 CakaerénRtIFadWeRkTI@ x px q . 136> cUrRsabMPWøfa 49 20 6 49 20 6 2 3 . 137> KNna E 3 5 3 5 3 5 3 5 . 138> kMNt;témø m nig n edIm,I[smIkar x 1 3 x m 4 x 3n x 4 manb¤seRcInrab;minGs; . 1.60 m
2
2
2
2
2
4
4
2
2
3
2
2
2
2
2
2
2
2
4
2
2
139> kMNt;témø a edIm,I[RbB½n§smIkar 2xxyya3 manb¤sepÞógpÞat; x y .
140> eK[ a , b , c epÞógpÞat; a b c 0 nig ab bc ca 0 . KNna D a 1 b 1 c 1 . 141> KNna S 1 3 2 4 3 5 ... 2010 2012 1 2 3 ... 2010 . 142> bgðajfa A 9 8 7 6 5 1 2 3 4 Eckdac;ngw 5 . 143> eK[GnuKmn_BIr f nig g kMNt;eday ³ f : x f x px 2 Edl p Ca):ar:aEm:RtBit nig 2010
2
2011
2011
2011
2011
2011
2
2
2011
2011
2012
2
2011
2011
2011
. nigKNna g f .
g : x g x 4 x 3
k> KNna f g x> kMNt;témø p edIm,I[eK)an f g g f . 144> eKman P cos 2x cos 2x cos 2x ... cos 2x . bgðajfa P 21 sin xx . n
2
3
n
n
n
sin
145> edaHRsayRbB½n§smIkarkñúg énRbB½n§smIkar x y 2
2
5440
PGCDx, y 8
2n
.
146> edaHRsaysmIkarkñúgsMNMucMnYKt;FmµCati énsmIkar xy 3x 2 y 37 . 147> bMPøWfacMnYn N 4n 3 25 Eckdac;nwg * . 148> RsaybMPøWfa ebIeKmansmPaB a b c d ad bc enaHeKRtUv)an a , b , c nig d CatYén 2
2
smamaRtmYy. 149> eK[ ax by 0 . bgðajfa
2
2
2
2
a2 x2 1 a2 b2 x2 y 2 652
. -
b c Edl a 0 , b 0 , c 0 . KNnatémøén K a 2011 . 151> bgðajfa A 2011 5 3 29 12 5 CacMnYnKt; . 152> edaHRsaysmIkar x 4x 4 x 2x 1 . 153> KNnabIcMnYnKt;tKña ebIeKdwgfa plEckénplKuNcMnYnTaMgbI nwgBak;kNþalkaerénplbUkcMnYnTaMgbI esµI 130 . 21 154> ebI x yz 7 nig xy 7 . cUrKNnatémøelxén xyz . 155> rktémøelxén 22 22 . 156> cMeBaH a b c KNnakenSam H a baa c b cbb a c acc b . 157> eKdaMkUneQI 40edIm B½T§CMuvijcmáarmYymanragCactuekaNEkg. etIcmáarenaHGacmanRkLaépÞFMbMput b:unµan ? ebIeKdwgfa KmøatBIedImeQImYyeTAedImeQImYyeTotmanRbEvg 2 m . 158> bgðajfacMnYn 3 BMuEmnCacMnYnsniTan. ¬cMnYnsniTan CacMnYnmanTRmg;CaRbPaK ba Edl >>> ¦ . 159> ebI x nig y CacMnnY Bit cUredaHRsaysmIkar x y 0 . 3 3 3 3 160> KNna A 2 34 . B 2 C 2 E 2 D 2 4 4 4 4 C2 161> KNna A 2 . D2 E 2 2 B 2 B 0.12 C 0.12 D 1.20 162> sresrCaTRmg;RbPaKén A 1.2 . 163> eKmankaermYymanRCug a . KNnaGgát;RTUg nigcm¶ayxøIbMputBIcMNucRbsBVrvagGgát;RTUgeTARCugkaer. 164> eKmanRtIekaN ABC EkgRtg;kMBUl B ehIymanRbEvgRCug AB 1. A 1 2 2 D CacMNucmYyenAelIRCug BC Edl AD CD 2 . KNna AC . C B D 165> edaHRsaysmIkar 2 3 2 2 3 2 2 3 x 2011 0 . 166> edaHRsaysmIkar x 2 x 4x 2 x 3 x 2 x 7 . 167> ]bmafasmIkar ax bx c 0 manb¤s x nig x . tag S x x Edl n CacMnYnKt;FmµCati. RsaybBa¢ak;fa aS bS cS 0 . 2y x 2y x 168> eK[ x 306 . KNna . 294 y 169> KNna A x 2 2x 2x 8 x 8 cMeBaH x 2011 . 170> eK[ a , b nig c CabIcMnYnxusBIsUnü Edl a b c 1 nig 1a b1 1c 0 . RsaybMPøWfa a b c 1 .
150> eK[ a b c 0 nig
2
2
3
2
1 1 1 0 a b c
3
2
9
2012
2011
2012
2011
2
2
22
22
2
2
2 2
2
1
n 1
n
2
2
2
2
2 2
2
2
2
2
2
2
2
2
n
n 1
2
n 2
n 1
2
2
2
2
2
2
2012
2
653
-
171> rkBIrcMnYnBitEdlmanplbUkesµI 13 ehIyplbUkcRmasrbs;vaesµInwg 13 . 40 172> eK[smIkar x 1x 2x 3x 4 k .
k> edaHRsaysmIkar cMeBaH k 3 . x> kMNt;témø k edIm,I[smIkarmanb¤s . 173> eRbóbeFobBIrcMnYn ³ 200 nig 300 . 174> eRbóbeFobcMnYn 31 nig 17 . 175> ebI a Cab¤sénsmIkar x 1 ehIy a 1 . cUrKNna 1 a a a a . 176> KNnaplbUk S 3 3 3 ... 3 . ¬BMuedaHRsaytamrUbmnþsVIútFrNImaRt¦. 177> KNna A 1 21 1 31 1 41 ... 1 n1 . 300
200
14
11
2011
2
2
3
2
2
2
2
nig y 0 . 1 2 3 ... 2009 2010 2011 2010 2009 ... 3 2 1 .
E
180> edaHRsayRbB½n§smIkar 181> bgðajfa 182> eK[BhuFa
2010
n
178> kMNt;témø m énRbB½n§smIkar 23xx 4yym7 edIm,I[vamanb¤s 179> KNna
3
a b c 3 b a d 4 a c d 5 b c d 6
x0
.
ehIy 4 4 4 4 . f x x 4 x 1 Edl n CacMnYnKt;FmµCati ¬ n ¦. kMNt;témø n edIm,I[viFIEckén BhuFaxaelI nigBhuFa x 3 mansMNl;esµI 46 . x 183> edaHRsaysmIkar ³ 1 . x 7 3 7 4 7 7 n
3
4
2
2
x
2
x
2 2
x 1 1 x
184> kMNt;RKb;cMnYnKt; n EdleFVI[ B 4 4 4 CakaerR)akdénmYycMnYnKt; . 185> edaHRsaysmIkar x xx 1 x xx 1 . 27
4
2
1016
n
2
2
186> edaHRsayRbB½n§smIkar ³ k>
x y z 6 xy yz zx 12 2 2 2 x y z 3
x>
1 1 1 x y z 3 1 1 1 3 xy yz zx 1 1 xyz
.
187> eK[ 4 4 23 . KNna 2 2 . 188> rkcMnYnlT§PaBTaMgGs; kñúgkarbegáItelxTUrs½BÞ Edlmank,alRbB½n§ ³ k> 011 x> 097 . a
a
a
a
654
-
189> edayeRbIelxTaMgbYnman ³ 1 , 2 , 4 nig 9 . etIeKGacbegáIt)anelxbYnxÞg; b:unµanrebobxusKña? 190> ]bmafasmIkar x px 1 0 manb¤s a , b nigsmIkar x qx 2 0 manb¤s b , c . 2
2
RsaybBa¢ak;fa b ab c pq 6 . 191> RsaybMPøWfa ebIeKman a b c abc nig 1a b1 1c 2 enaHeyIg)an a1 b1 c1 2 . 192> rkBIrcMnYnedaydwgfa plbUk plKuN nigplEckrvagcMnYnTaMgBIr esµIKña . 193> KNnaplKuN P 1 x1 x 1 x ... 1 x . 194> KNnaplbUk S 1 1 2 1 21 3 1 2 1 3 4 ... 1 2 3 1... 2011 . 195> KNnaplbUk S x 1 x x 13x 2 x 15x 6 x 71x 12 x 91x 20 cMeBaH x 95 . 196> ebI abc 1 bgðajfa ab aa 1 bc bb 1 ca cc 1 1 a1 ab 1 b1 bc 1 c1 ca . 197> rktémø x ebI ³ xxxx 1 x 1 Edl x CacMnYnKt;FmµCati . 198> sRmYlkenSam A log 2 log 3 log 4 log 5 log 6 log 7 . 199> KNnaplbUk S log1 x log1 x log1 x ... log1 x Edl k CacMnnY Kt;FmµCati . 2
2
2
2
2
2n
4
2
2
2
2
x2
3
4
a2
a
200> edaHRsayRbB½n§smIkar ³
5
6
7
a3
8
ak
x1 x2 x3 x2011 0 2 2 x1 x22 x32 x2011 0 3 3 3 3 x1 x2 x3 x2011 0 ......................................................... 2011 2011 2011 2011 x1 x2 x3 x2011 0
201> edaHRsayRbB½n§smIkartamedETmINg; ³ 3
x 3 y 2
2 3 x 3 y 1
.
.
202> eK[BIcMnYnBit x nig y mansBaØadUcKña. bgðajfa ³ k> x y 2 xy x> xy xy 2 . 203> sRmYlkenSam ³ E a b 4 ab nig F a b b a Edl a 0 , b 0 . 2
2
2
a b
ab
rYcKNna ³ E F nig E F . 204> cMeBaH x 100 KNnatémøénkenSam ³ A x 100x x 100x x 100x . 205> KNnatémøénkenSam A 22 30 20 27 8 . 206> eK[ a nig b CaBIrcMnYnKt;tKña . edaydwgfa a b 321 cUrKNnatémøén a nig b . 207> dUcemþcEdlehAfamMuTl;kBM Ul ? bgðajfamMuTl;kMBUlCamMub:unKña . 208> rkcMnYnKt;viCm¢ an n edIm,I[ x x 6 esµInwg n Edl x CacMnYnKt; . 209> eRbóbeFobkenSam ³ aa bb nig a b . a b n1
n
3
n3
n 4
n5
3
2
2
2
2
n 2
2
2
2
2
2
2
655
-
210> eKmansmIkar m 1x
. k> kMNt;témø m edIm,I[ x 1 Cab¤sénsmIkarxagelI rYcKNnab¤smYyeTot . x> kMNt;témø m edIm,I[smIkarmanb¤sDub rYcKNnab¤sDubTaMgenaH . 211> rkBIrcMnYn x nig y edaysÁal;plbUk nigplKuN ³ x y 25 . xy 12 2
m 3x 3 m 0
2
2
212> edaHRsayRbB½n§smIkar
x y x y 8 2 2 x y xy 7 2
2
213> dUcemþcEdlehAfa)atmFüménRtIekaN ? bgðajfa)atmFüménRtIekaN esµInwgBak;kNþal)atmYyén
RtIekaNenaH . 214> etImancMnYnKt; n b:unµanxøHEdlepÞógpÞat; 72 12n 54 ? 24 B E 215> RtIekaN ABC RtUv)ankat;ecjBIRkdasragCactuekaNEkg 12 dUcrUbxagsþaM . etIRkdasEdlenAsl;manb:unµanPaK ? A 1 1 C 10 D 216> Tajrk x 2012 ? ebIdwgfa x 2011 2011 . A C 217> r)arEdkmYymanRbEvg 25m RtUv)anEp¥kcugxagelI eTAelI 25 m CBa¢aMgQrmYy Edlcm¶ayBICBa¢aMeTAcugr)arxageRkamman RbEvg 20 m . ebIeKbgçitr)aecjBICBa¢aMgEfm 4 m eTot O D 4m B 20 m etIcugr)ar)anFøak;cuHcMnYnb:unµan m ?¬sUmemIlrUbxagsþaM ¦ 218> RtIekaN ABC mYyEkgRtg; A ehIymanbrimaRt 60 cm nigmanépÞRkLa 120 cm . cUeKNna rgVas;RCugnImYy²énRtIekaN ABC . 219> bgðajfaRtIekaNEdlmanrgVas;mMuTaMgbI , 2 nig 3 CaRtIekaNEkg . 220> ebIeKdwgfa f x f x 1 9 nig f 3 81 . cUrrk f 9 . 221> suxeFVIdMeNIredaymeFüa)aybIRbePTKW rfynþ)an 83 ehIym:UtU)an 53 éncm¶aypøÚvTaMgGs;. k> rkRbPaKtagcm¶aypøÚvEdlsuxRtUvedIr . x> rkcm¶aypøÚvTaMgGs;ebIsuxedIr)an 2 km . 222> k> sresr 45m CaPaKryén 1km . x> sresr 1kg CaPaKryén 800 g . 223> Ggát; MN manrgVas; 18 cm . I CacMNucmYyenAelI MN ehIy MI NI . A nig B CacMNuckNþal erogKñaén MI nig NI . KNna AB . 224> RsaybBa¢ak;fa plbUkmMukñúgBIrénRtIekaN esµImMueRkAmYyénRtIekaN EdlminCab;mMuTaMgBIrenaH . 2
225> KNnatémøénkenSam
A 4 5 3 5 48 10 7 4 3 4
656
2011 2012
. -
226> edaHRsayRbB½n§smIkar ³
2 x 2 y 2 z 7 x 7 y z 2 2 2 4 x y z 3
. 2011
. cUrKNna 2ba . 228> eKdwgfa log 2 0.3010.... cUrrkcMnYnxÞg;énEpñkKt;viC¢manén A 2 . 229> edaHRsaysmIkar 3 x2 5 2 x3 7 x 1 . 230> eRbóbeFobcMnYn ab nig PGCD a, b PPCM a, b cMeBaH a 90 , b 280 . 231> epÞógpÞat;ÉklkçN³PaB ³ 1 sin x cos x 21 sin x 1 cos x . 232> edaHRsayRbB½n§smIkar ³ xyx yz 4 6z 13 . 233> eKmannimitþsBaØaelakarIt ³ Logx , log x , lg x , ln x , log x , log x . etInimitþsBaØaNaxøHCa elakarItTsSPaK ¬elakarIteKal 10¦ ehIyelakarItNaxøHCaelakarItenEB ¬elakarIteKal e ¦ ? 234> RtIekaNmYymanrgVas;RCug 7 , 8 nig 11 . rkRkLaépÞrgVg;carwkkñúgRtIekaNenH . 235> eKmanbIcMnYn a , b , c epÞógpÞat; a b c 1 , a b c 1 nig a b c 1 . KNnatémøelxénkenSam P a b c . 236> bgðajfa ³ a b c d 1 a b c d . 237> Gayu«BukticCagplbUkGayukUnTaMgbI 3qñaM. ebIdwgfa ³ Gayu«Buk nigkUnTaMgbIsmamaRtnwgcMnYn ³ 15 , 7 , 5 , 4 . rkGayumñak;² . 225 238> eRbóbeFobBIrcMnYn ³ 259 nig 625 . 227> a nig b CaBIrcMnYnKt;viC¢manEdl a
b 15 216
2011
2
2
10
2
2010
2
2
2
2
3
3
3
2012
2
60
239> eK[smamaRt
2011
2
e
50
ab a b cd c d 2 2
RsaybMPøWfaeK)an ³ . 240> rkBIrcMnYnKt;viC¢mantKña edaydwgfapldkkaeréncMnYnTaMgBIresµInwg 321 . 241> ekµgelgXøIbInak;KW A , B nig C manXøIsmamaRtnwg 3 , 4 , 5 . eRkayeBlQb;elgXøIGkñ TaMgbIman smamaRt 15 , 16 , 17 . etIekµgNaQñH ekµgNacaj; ? 242> edaHRsaysmIkar 3x 6x 9x 12x 1 12 13 14 . a c b d
243> edaHRsayRbB½n§smIkar ³ 244> eRbóbeFobcMnYn 2
2
22
x xy y 1 y yz z 4 z zx x 9
manelx 2 cMnYn 1001dg nig 657
3 3
33
manelx 3 cMnYn 1000dg . -
245> edaHRsaysmIkar ³ x 3 x 4 x 4 x 5 x 5 x 3 x . 246> bursmñak;eFVIdMeNIrBIraCFanIPñMeBjeTAkan;extþsVayerog. eRkayqøgsaLagGñkelOgKat;eFVIdMeNIrbnþ
dl;Rtg;cMNucmYy Kat;)anecalEPñkeXIjbegÁalR)ab;cm¶aydak;fa {sVayRCM 17 km } nigbegÁalbnÞab; dak;fa {sVayerog 25 km }. sMNYrsYfa etIcm¶ayBI sVayRCM eTAsVayerogmanRbEvgb:unµan km ? 247> cabmYyhVÚgehIrTMelIpáaQUkénRsHmYy. ebIcabmYyTMelIpáaQUkmYy enaHmancabmYyKµanpáaQUkTM . EtebIcabBIrTMpáaQUkmYy enaHmanpáaQUkmYyKµancabTM. cUrrkcMnYncab nigcMnYnpáaQUk . 248> stVExVkTMelIEmkxVav )ak;EmkR)avgab;bIrs;BIr cuHebIstV 120 gab;b:unµan ? rs;b:unµan ? 249> davI manGayutageday x ebIKitKU[xÞic x CaBhuKuNén8 edaydwgfa x FMCag10 nigtUcCag 20 . etIsBVéf¶ davIRsImanGayub:unµan ? 250> ksikmñak;mancMNIsRmab;pÁt;pÁg;eKarbs;Kat; 40 k,al)an 35 éf¶. ebIKat;TijeKa 10 k,albEnßmeTot etIkat;GacpÁt;pÁg;cMNIdEdl[eKa)anb:unµanéf¶ ? 251> eK[bIcMnYnBit a , a nig a Edl a sin a , a cos a sin a nig a cos a cos a . bgðajfa a a a 1 . 252> KNnaplKuN P 1 x x 1 x x 1 x x 1 x x . 253> ebI a 0 , b 0 nig c 0 bgðajfa a 3b c a 1 b b 1 c c 1 a . 254> KNnaplbUk S 11!2 2!3 3! n n! Edl n! nn 1n 2 3 2 1 . 1 255> KNnaplbUk n n 1 n n 1 . 1
2 1
3
2
2 2
1
1
2
1
3
2
1
2
2 3
2
2
4
4
2n 1
2n
8
n
n
9999 n 1
256> sRmYlkenSam ³
A
4
a b c 3abc a b 2 c 2 ab bc ca 3
3
3
.
2
257> RsaybBa¢ak;fa ebIeKman 258> eKman
4
a c b d
enaHeK)an ³
a4 b4 a b 4 c d4 cd 4
.
6 2 52 11 , 56 2 45 2 1111 , 556 2 445 2 111111 , 5556 2 4445 2 11111111 , ...
BIkarbgðaj]TahrN_xagelIcUrrkrUbmnþTUeTA nigRsaybBa¢ak;rUbmnþenaHpg. 259> KNnaplbUk ³ S cos 1 cos 2 cos 3 cos 89 . 260> KNnaplbUk ³ S sin 0 sin 1 sin 2 sin 90 . 261> cUrkMNt;témø a nig b edIm,I[cMnYn abba CaKUbR)akdénmYycMnYnKt; . 262> k> kMNt;témøelxénGBaØat a , b , c nig d éncMnYn abcd edaydwgfa abcd 9 dcba . x> bBa¢ak;fa abcd nig dcba suT§EtCakaerR)akdénmYycMnYnKt;. 263> BIrcMnYnEdlCakaerR)akd edaydwgfaplKuNva elIsplbUkvacMnYn 4844 . 2 o
2
o
2
o
2
o
2
2
o
2
o
2
o
o
658
-
.
264> RtIekaNEkgmYymanépÞRkLa 24 cm nigmanbrimaRt 24 cm . KNnargVas;RCugnImYy²rbs;RtIekaN. 265> manBIrcMnYnKt;tKña EdlcMnYnTImYyCacMnYnbzm nigcMnYnFMbnÞab;CakaerR)akd . cUrrkcMnYnTaMgBIenaH . 266> eKmanrgVg;b:un²KñaehIyRtUv)an 2
eKerobdUcrUbxagsþaM Edlman 2m km Rsay[eXIjfa RtIekaNcarwkknøHrgVg;CaRtIekaNEkgCanic© . 268> BhuekaNmYymanplbUkmMukñúgminelIsBI 2011 . etIBhuekaNenHGacmancMnYnRCugeRcInbMputb:unµan ? 269> cUrRsayfa rgVas;RCugnImYy²rbs;RtIekaNRtUvEttUcCagknøHbrimaRtrbs;vaCanic© . 270> bursmñak;man)arImUledayéd 10 edIm. Kat;Ck;)arIedayrkSaknÞúy)arI ebICk;Gs;bIedIm enaHKat;Gac ykbnÞúy)arIEdlsl;mUl)anmYyedImfµvI ij . etIKat;Ck;b:unµandgeTIbGs;)arITaMg 10 edImKµansl; ? 271> edaHRsaysmIkar k> 3 4 5 x> 3 4 5 6 . 272> RsaybBa¢ak; BIsmPaB y z z x x y y z 2 x z x 2 y x y 2 z enaHeK)an x y z . 273> cUrKNna S ab cd edaydwgfa a b c d 2011 nig ac bd 0 . 274> k> RsaybBa¢ak;fa sin12a cot a cot 2a . x> cUrKNnaplbUk S sin1 a 1 a 1 a 1 a 1 a . o
x
x
x
2
x
2
2
2
2
2
x
2
x
x
2
2
2
n
sin
sin 2 sin 3 sin n 2 2 2 2 x cot 1 2 1 cos x cot x 1 1 1 1 1 1 Pn 1 1 cos x cos x cos x cos x 2 22 2n yzt ztx txy xyz x y z t 14625 x, y, z t x y z t 2 3 4 6
275> k> RsaybBa¢ak;fa
.
x> cUrKNnaplKuN 276> eK[bYncMnYn
.
nig viC¢man edaydwgfa nig . KNna x , y , z nig t . 277> eKmanBhuFa P x x 1 nig Q 2 x x . k> kMNt;témø x edIm,I[ P mantémøGb,brma. x> kMNt;témø x edIm,I[ Q mantémøGtibrma. 278> rkcMnYnKt;viCm¢ an n tUcbMput Edl n mansMNl; 1 , 2 , 3 , 4 , 5 eBlEcknwg 2 , 3, 4 , 5 , 6 erogKña . 279> rkcMnYnKt; n tUcbMput Edl n Eckdac;nwg 7 EtebIEcknwg 2 , 3 , 4 , 5 , 6 [sMMNl;esµI 1 Canic© . 280> RsaybBa¢ak;fakenSam E cos x sin x 3sin x cos x mantémøefr RKb;témørbs; x . 2
2
6
6
2
659
2
-
281> edaHRsayRbB½n§smIkar ³ x
3
y3 9
xy 2
.
282> edaHRsaysmIkar 3 3 3 3 4 3 . 283> sresr N CaplKuNktþadWeRkTI1 Edl N 3a 1 4a 6a 9 . 284> RsaybBa¢ak;fa cMnnY N 44a 1 100 Eckdac;nwg 32 . 285> eK[ a b 1. cUrKNnatémøelxénkenSam P 2a b 3a b 1 . 286> rkBIrcMnYnKt;viC¢man a nig b edaydwgfa a b 24 . 287> kaerBIrmanpldkRkLaépÞesµI 1152 m nigmanpldkRbEvgRCugesµI 16 m . KNnaRCugkaernImYy² . 288> KNnakenSam A x 11x 2 2 x23 x 1 x 3x 3 . 2011
2010
2009
2008
x
2
2
2
3
2
3
2
2
2
2
289> rkmYycMnYn edaydwgfa bIdgéncMnYnenH nigkaeréncMnYndEdlenH CacMnYnpÞúyKña . 290> eKmancMnYn N 12345678910111213...998999EdlcMnYnenHsresrBIelx 1 dl;elx 999 .
etIelxb:unµanenARtg;TItaMgtYTI 2011 rab;BIxageqVgéd ? 291> eK[GnuKmn_ f kMNt; x 1 , 0 eday x2 x 1 f x f 1x x 1 . cUrKNna plbUk S f 1 f 2 f 3 f 2011 . 292> RbGb;mYymanragCaRbelBIEb:tEkg manvimaRt 180mm , 600mm , 90 mm . rkcMnYnKUbticbMputEdl GacerobbMeBjkñgú RbGb;enH. 293> edaHRsayRbB½n§smIkar 26xx2yy43xyxy . 294> rkBIrcMnYnKt; a nig b Edl a b . ebIeKdwgfa plbUkrbs;vaCaBhuKuNén 15 ehIypldkkaerrbs; vaesµInwg 45 . 295> x, y , z CacMnYnsniTanxusBIsUnü. ebI A yz zy , B xz xz , C xy xy . bgðajfa témøén A B C ABC minGaRs½ynwg x , y , z . 296> yuvCnbInak;KW A , B , C eFVIkarrt;RbNaMgKñacm¶ay 100 m . eBlrt;RbNaMgeKsegáteXIjfa ³ xN³eBl A rt;dl;TI B enAxVH 10 m eTot nigxN³eBl B rt;dl;TI C enAxVH 10 m eTot . eKsnµt; faGñkTaMgbIrt;kñúgel,Ónefr. sYrfa xN³eBlEdl A rt;dl;TI etI C enAcm¶ayb:unµanEm:Rt BI A ? 2
2
2
297> edaHRsayRbB½n§smIkar ³
x y z xyz x y z xyz x y z xyz
.
298> eKman a , b , c CabIcnM YnxusKña. KNnaplbUk S a baa c b c bb a c a c c b . 660
-
299> rkRKb;bNþaKUéncMnYnKt;viC¢man x , y epÞógpÞat;smIkar x x 13 y . 300> eKmancMnYn A 2 5 Edl n CacMnYnKt;viC¢man. etIcMnYn A bBa©b;edayelxsUnü ¬0¦b:unµan ? 301> eK[bIcMnYnviCm¢ an a , b , c . edaHRsaysmIkar xbc a xac b xab c 2 1a b1 1c . 2
n
2
2 n 1
302> rkb¤sKUbén
Z 8 , 4
. 303> eKmanKUbcMnnY 7 manTMhMb:un²Kña² ehIypÁMúP¢ab;Kña)andUcrUbsUlItxagsþaM ³ edaydwgfasUlItenHmanmaD 448cm cUrrképÞRkLaTaMgGs;rbs;slU ItenH. 304> eKmanRtIekaN ABC mYy nigman AM Caemdüan . bgðaj[eXIjfa 2 AM AB AC . 305> GñkebIkbrmñak;ecjBITIRkug A eTATIRkug B edayel,ÓnefrCaragral;éf¶. Kat;cab;GarmµN_eXIjfa ³ ebIKat;bEnßmel,Ón 3 km/ h enaHKat;eTAdl;eKaledA muneBlkMNt; 1em:ag. EtebIKat;bnßyel,Ón 2 km/ h enaHKat;eTAdl;eKaledA eRkayeBlkMNt; 1 em:agEdr. KNna ry³eBlFmµta el,ÓnFmµta nigcm¶ayBITIRkug A eTATIRkug B . 306> mYycMnYnmanplKuNxøÜnÉgesµInwgplbUkxøÜnÉg. KNnamYycMnYnenaH . 307> kñúgkic©RbCMumYymanmnusS 10 nak; )ancUlrYm ehIyGñkTaMgenaH)ancab;édsVaKmn_KñaeTAvijeTAmk. ebIdwgfaGñkTaMgGs;)ancab;édKña RKb;²Kña. cUrrkcMnYnénkarcab;édKñaTaMgGs;. 308> RsaybBa¢ak;facMnYn 2 3 Eckdac;nwg 35 . 2b 309> sRmYlRbPaKsniTan ³ E aa 35ab . ab 6b 310> RtIekaNmYymanbrimaRt 24 cm . ebIRbEvgRCugTaMgbIsmamaRterogKñaKW 3 : 4 : 5 cUrKNnargVas; RCugnImYy²rbs;va. 311> cMnYnKt;mYymanelxBIrxÞg; EdlmanplbUkelxtamxÞg;esµInwg 9 . ebIeKdUrxÞg;rayeTACaxÞg;db;vji enaHeK)ancMnnY fµIeRcInCagcMnYncas; 63 . KNnacMnnY Kt;enaH. 312> brimaRténRbelLÚRkam ABCD manrgVas;esµI 48 cm . rgVas;km rktémø x EdleFVI[kenSam P x 1x 2x 3x 6 mantémøtUcbMput ? rkémøtUcbMputenaH. 314> mnusSmYyRkumeLIgCiHLan. ebImñak;GgÁúyekAGImYy enaHenAsl;mnusS 4 nak;KµankEnøgGgÁúy. EtebI mnusSBIrnak; GgÁúyekAGImYy enaHenAsl;ekAGI 4 KµanmnusSGgÁúy. rkcMnYnmnusS nigcMnYnekAGI. 315> plbUkénBIrcMnYnesµInwg 1. bgðajfaplKuNvatUcCag b¤esµInwg 14 . 316> kaermYymanRkLaépÞ 100 cm carwkkñúgknøHrgVg;mYy. rkRkLaépÞénkaerEdlcarwkkñúgrgVg;TaMgmUl. 3
9
9
2
2
2
2
2
661
-
317> kMNt;témøéncMnYnBit a nig b edaydwgfa a
2
x y z 2 3 4 xy yz zx 26
318> edaHRsayRbB½n§smIkar ³
.
b2 0
.
319> KNnatémøKt;kenSamelx ³ E 2 5 2 5 . 320> BIrcMnYnmanplbUkesµI12 nigmanplKuNesµI 4 . KNnaplbUkcRmascMnYnTaMgBIrtamBIrrebobepSgKña. 321> ebI x 3x 8 CaktþamYyénkenSam x rx s cUrKNnark r nig s . 322> KNna A 2 4 6 ... 2n . 323> eKman S 1 21 2 1 2 1 2 ... 1 2 . cUrKNna S 1 . 2012 324> edaymineRbIm:asIunKitelx cUrKNnarktémøén A 1234568 1234567 . 12345679 325> rkcMnYnKt;viCm¢ an n edIm,I[ 4 n CacMnYnbzm . 326> rkelxxagcugénplKuN 7 2013 7 2013 . 327> KNnarkelxxagcugén A Edl A 2012 . 328> rkRKb;cMnYnKt;viC¢man n EdleFVI[ 2 1 Eckdac;nwg 7 . 329> bgðajfa 12 2 Eckdac;ngw 10 . 330> eKmancMnYn A Edl A n n . bgðajfa cMeBaHRKb;cMnnY Kt;viC¢man n enaH A Eckdac;nwg 30 . 331> eK[ f CaGnuKmn_BhuFakMNt;eday f x x 3x . cUrKNna f x cMeBaH 3
3
2
4
2
4
2
8
1024
1024
2
4
2012
2010
2000
2012
2013
n
2012
2008
5
3
. 332> rkBIrcMnYnKt;viC¢man a nig b EdlmanplbUkesµI 92 nig a 1 CaBhuKuNén b . 333> cMeBaHcMnYnKt;viC¢man n / eK[ P n 1n 2n 3 ... n n nig P 1 3 5 ... 2n 1 . bgðajfa P Eckdac;ngw P nigrkplEckrbs;vapg . 334> eK[ 2a3 326 5b9 Edl 2a3 nig 5b9 CacMnYnmanelxbIxÞg;. ebI 5b9 Eckdac;nwg 9 KNna a b . 335> ]bmafa a a 1 cUrKNnatémøelxénkenSam A a 2a 4a 3a 3 . 2 2 336> KNnakenSam A eday[lT§plCaplKuNbIktþa Edl A . 12 8 3 6 x3
3 2 3
3 2
2
4
3
2
200 337> KNnatémøelxén S 23 64 96 ...... 300 . 338> RtIekaNEkgmYymanGIub:UetnusesµI 13 cm nigplbUkRCugBIreTotesµI 17 cm . rkrgVas;RCugmMuEkgTaMgBIr. 339> RsaybBa¢ak;fa ³ 3 2 . 340> bgðajfa A 6 Edl A 6 6 6 ... 6 24 24 24 ... 24 . 2
2
2
2
2
2
2
2
2
3
3
662
3
3
3
-
341> edaHRsaysmIkar ³
x 4 x 3 x 2 x 1 0 2008 2009 2010 2011
. 342> rkbIcMnYnKt;vCi ¢manxusKña x , y nig z EdlepÞógpÞat;RbB½n§smIkar yx zy106 . 343> edaHRsaysmIkar ³ x 1 x 1 . d 344> eKmanbnÞat;BrI RsbKñaKW d l . x l KNnamMu ³ x énrUbxagsþaM . 345> eKmanRtIekaNsm½gSBIr KW ABC nig DEF EdlmanrgVas;RCug esµI 3 cm dUcKña. RtIekaNTaMgBIrpÁúMKñadUcrUbxagsþaM )anRtIekaNsm½gS tUc²cMnYn 6 EdlmanrgVas;RCugesµI 1 cm dUcKña. KNnaRkLaépÞénrUbxagsþaM . 346> eKmancMnYn 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . cUrykelxTaMgenHeTAbMeBjkñúgRbGb; kaerxagsþaM edIm,I[plbUkCYredk esµIplbUkCYrQr esµIplbUkGgÁt;RTUg esµIngw 15 . 347> RtIekaNmYymanrgVas;RCug a , b nig c EdlepÞógpÞat; 24a 18b 12c . k> KNnargVas;RCugénRtIekaN ebIeKdwgfa b c 10 cm . x> KNnargVas;km kñúgfñak;eronmYyeKerobsisSCaRkum. ebIerobsisS 8 nak;kñúgmYyRkum enAsl;sisS 4 nak;eTot. EtebI erobsisS 9 nak;kñúgmYyRkum enaHenAxVHsisS 2 nak;eTot. rkcMnYnsisSkñúgfñak;eronenaH . 349> rkmYycMnYnebIdwgfa cMnYnenaHminFMCagBIr ehIyk¾mintUcCagBIrEdr. 350> rkmYycMnYn EdlmanplbUkesµIplKuNénxønÜ Ég. 351> mFüménBIrcMnYnesµI 2012 nigmFüménbIcMnYnk¾esµInwg 2012 Edr. cUrrkcMnYnTI 3 . 352> x CacMnYnKt;viCm¢ an manelxBIrxÞg;. cUrKNna x ebIdwgfa 44 Eckdac;nwg x sl;sMNl; 10 . 353> edaHRsaysmIkar HE SHE . 354> kMNt;témø a nig b edIm,I[kenSam ax bx 54x 27 GacsresrCaKUbéneTVFamYy)an . rYckMNt;eTVFaenaH . 355> RtIekaN ABC EkgRtg; A manGIub:Uetnus BC 17 cm nig AB AC 23cm . KNna AB nig AC . m 356> eK[RbB½n§smIkar 23xx 57yy 20 . kMNt;témø m edIm,I[KUcemøIyénRbB½n§smIkarviC¢man . 30 o
110o
A
D
F
B
C
E
2
3
357> KNnakaMrgVg;énrUbxagsþaM ³
2
1
1
1
1 1
1
358> KNnacMnYnKt;FmµCati n EdleFVI[cMnYn n 13 nigcMnYn n 76 CakaerR)akd . 663
-
5x
359> KNnatémømMu x KitCadWeRk énrUbxagsþaM ³
3x
7x 4x
360> edaHRsayRbB½n§smIkar EdlmansmIkarnImYy²KW
6x 2
2x 2y2 2z 2 y 1 , z 2 , x 3 1 x2 1 y2 1 z2
.
361> RsaybBa¢ak;[eXIjfa plbUkmMukñúgTaMgbIénRtIekaNesµInwg 180 . 362> cUrRsaybBa¢ak;BIRTwsþIbTBItaK½r EdlfakñúgRtIekaNEkg a b c man c CaGIub:Uetnus. 363> cUrrkcMnYnmYyEdlmanelxbYnxÞg;KW abca edaydwgfa abca 5c 1 . 364> eKmanBIrcMnYn a b 0 . bgðajfa RKb;cMnYnKt;FmµCati n enaHeK)an a b . 9900 ... 365> KNnaplKuN P 22 36 12 . 4 100 366> dak;kenSam P x y y z z x CaplKuNktþa . 367> eK[ a b 0 EdlepÞógpÞat;lkçxNÐ 3a 3b 10ab . KNnatémøénkenSam P aa bb . o
2
2
2
2
n
3
3
n
3
2
2
368> eK[kenSam A xx 13 . kMNt;témø x viC¢man edIm,I[ A CacMnYnKt;viC¢man . 369> bgðajfa ebI a b c 0 ehIy a 0 , b 0 , c 0 enaHeK)an a b c 3abc . 370> edaHRsaysmIkar x 2012 x 0 . 371> KNnatémøénkenSam A x x x 2x 1 cMeBaH x 1 2 1 . 3
2012
4
3
3
2012
3
2012
2
2 1 1
2 1 1
372> sYnc,armYymanragCactuekaNEkg EdlmanépÞRkLa 720 m . ebIeKEnßmbeNþay 6m nigbnßyTTwg 2
enaHépÞRkLarbs;sYn minmankarpøas;bþÚreT . KNnavimaRténsYnc,arenaH. 373> eK[BIrcMnYnEdltUc b¤FMCagKña 3 Ékta. ebIdwgfaplbUkkaeréncMnYnTaMgBIresµI 89 cUrrkcMnYnTaMgBIr. 374> KNnatémø x , y Edl y 0 BIsmIkar ³ x 4x y 6 y 13 0 . 375> k> eRbóbeFob 3 3 nig 62 3 . x> eRbóbeFob 14 48 nig 7 4 3 7 4 3 . 376> edaHRsaysmIkar x 2 x 3 1 . 377> rkcMnYnEdlmanelxBIrxÞg; AB EdlepÞógpÞat; AB BA 1980 . 378> ebI ab 4 , ac 5 , bc 20 . KNna a , b , c . 379> KNna A 2025 1 2025 2 2025 3 ... 2025 50 . 380> etIcMnYn P 1 2 3...100 bBa©b;edayelxsUnücMnYnb:unµanxÞg; ? 4m
2
4
4
4
4
2
2
2
2
664
2
2
-
381> etIcMnYn 2012! manelxsUnüenAxagcugb:unµan ? 382> KNna N 4 15 4 15 2 3 5 . 383> kñúgfg;mYymanXøIBN’s nigXøIBN’exµAcMnYn 12 RKab; . rkcMnYnXøI s edIm,I[RbU)ab énXøIexµAesµI 13 . 384> eK[ ABC Edlmanemdüan AM , BN nig CP .
k> RsaybMPøWfa AM BN CP AB BC AC . x> RsaybMPøWfa AM BN CP AB BC2 AC . 385> eK[bIcMnYnKt;ruWLaTIbtKña a , b , c Edl a b c . k> KNnaplbUk S a b c CaGnuKmn_én b . x> Tajrktémøén a , b , c edaydwgfa S 333 . 386> kukmYyehIrCYbRksamYyhVÚgk¾ERsksYrfa {sYsþImitþTaMg 100 }. emxül;énhVÚgRksa)aneqøIytbvijfa {eTcMnYnBYkeyIgminRKb; 100 eT} . cMnYnBYkeyIgbUkcMnnY BYkeyIg bEnßmBak;kNþaléncMnYnBYkeyIg ehIyEfm 14 énBYkeyIg RBmTaMgmitþÉeToteTIbRKb; 100 . cUrrkcMnYnRksakñúghVÚgTaMgGs; . 2 2 2 x y z xy yz xz 2011 2011 2011 32012 x y z
387> edaHRsayRbB½n§smIkar 388> brimaRténRtIekaN
.
mYymanrgVas;esµI 80 cm . ebIRCugTaMgbImansmamaRterogKña 5 , 7 , 4 . cUrKNnargVas;RCugnImYy² . bc ac 389> eK[ 1a b1 1c 0 . KNnatémøénkenSam P ab . c a b ABC
2
x 3 2 y 1 2
390> edaHRsayRbB½n§smIkar
2 x 3 y 1 4
2
2
.
391> eKman a 1 , b 1 nig a b . cUreRbóbeFob ba 11 , ba nig ba 11 . 392> eRbobeFob A 5 2 7 5 2 7 nig B 5110 2 5110 393> edaHRsaysmIkar 1 x x 2 x x 3 x x ... 2013x x 0 . 3
3
394> k> dak;CaplKuNktþanUvsmPaB
2
.
. x> bgðajfabIcMnYnBitminsUnü a , b , c ehIymanBIrcMnYnpÞúyKñamYyy:agtickñgú cMeNam a , b , c enaH eK)anTMnak;TMng 1a b1 1c a 1b c . 395> edaHRsaysmIkar 3 3 30 . 396> sRmYlRbPaK A x3x 14xx 1x eday x 0 . 2 x
1 1 1 1 a b c abc
2 x
2
665
-
397> edaHRsayRbB½n§smIkar
x y 1 xyz 2 yz 5 xyz 6 x z 2 xyz 3
.
398> RtIekaNEkgmYymanrgVas;RCug CabIcnM YnKUtKña . KNnabrimaRt nigRkLaépÞrbs;RtIekaNenH. 399> sRmYlkenSam F xx 11 . 2
400> KNnacm¶ayBIcMNuc A6 , 6 eTAbnÞat; D : y x 4 kñúgtRmúyGrtUNrem. 401> 402> dak;kenSam x 1x 3x 5x 7 24 CaplKuNktþa . 403> eKmanbIcMnYn a , b , c epÞógpÞat;TMnak;TMng a b c 1 nig a b c 1 . 2
2
2
3
3
3
RsaybBa¢ak;[eXIjfa ³ a b c 1 . 1 404> eK[ ax by cz 0 nig a b c 2005 . KNna A bc y z ax acxby z cz abx y . 405> cUrkMNt;témø x nig y edIm,I[cMnYn N 3x82 y Eckdac;nwg 3pg nigEckdac;nwg 11pg . 406> eKmancMnYn A 2 . cUrrkelx ³ k> mYyxÞg;xagcug x> BIrxÞg;xagcug K> bIxÞg;xagcug . 407> KNnaplbUk S 101 102 103 ... 10n . 2
3
2
2
2
2
2
2
2004
n
2
3
408> edaHRsayRbB½n§smIkar ³ 3x y
x y
n
x y
9 324 18x 2 12 xy 2 y 2
.
409> eK[ a CacMnYnKt;FmµCati . bgðajfa n , a 1 an 1 1 CaBhuKuNén a . 410> dak;kenSamxagsþaMCaplKuNktþadWeRkTI! ³ a 4b 2 a 2b x a b 0 . 411> RsaybBa¢ak;fa 4a 4a 5a 4a 1 0 , a CacMnYnBit . 412> eK[smIkar x a bx b cx c aa b c abcx . edaHRsaysmIkarebI a 2, b 3, c 4 . 413> eK[ x , y , z CabIcMnYnviC¢manminsUnü . cUrbgðajfa x1 y1 z1 x y9 z . n 1
2
4
3
2
2
3
3
4
4
2
2
414> enAkñúgtRmuyGrtUNemeK[bIcMNuc
2
2
2
2
2
. bnÞat; D nig L kat;tam C . k> sresrsmIkabnÞat; AB x> rksmIkarbnÞat; D Rsbnwg AB K> rkbnÞat; L Ekgnwg AB . 415> edaHRsaysmIkar x x 21x x x12x6 35 x x 4x2 3 x x10x5 24 . 416> edaHRsaysmIkar 24 x 112 x 18x 16x 1 330 . 417> edaHRsaysmIkar x 3x 4 2x 5x 3 3x 2x 1 . 2
2
2
3
A1, 2, B2 , 3, C 3, 4
2
2
2
3
666
2
3
-
99 1 418> bgðajfa 151 12 34 56 ... 100 . 10 419> dak;CaplKuNktþanUvsmIkar 16x 20x 5x 1 0 ebIeKdwgfa 1 Cab¤sénsmIkarenH . bc ca 420> ebI a 0 , b 0 , c 0 Edl a b c . cUrRsaybMPøWfa 1a ab 3abc . 421> rkbIcMnYnKt;rL Wu aTIb a , b , c edaydwgfa cMeBaHRKb;témø x eK)an ³ x ax 10 1 x bx c . 5
422> edaHRsayRbB½n§smIkar
3
x1 x2 x3 ... x2000 1 x x x ... x 4 2000 2 1 3 x1 x2 x4 ... x2000 3 ....................................... x1 x2 x3 ... x1999 2000
EdlmancMnYn 2000 smIkar .
423> edaHRsaysmIkar x 1x 2x 3x 4 3 . 424> eKmanctuekaNBñay ABCDmYyman)at AB 80 cm , CD 40 cm . KNnaépÞRkLaénctuekaN
BñayenH ebIdgw faRCug AD 68 cm nig AC 84 cm . 425> fñak;TI 9C mansisSbIRbePTKW BUEk mFüm exSay. enAedImqñaMcMnYnsisSsmamaRtnwg 3 , 4 , 7 luHdl; cugqñaMcMnYnsisSsmamaRtnwg 2 , 5 , 7. ebIdwgfacugqñaMsisSexSayfycuH 10 nak; cUrrkcMnYnsisSsrub. 426> eK[ ax by cz k Edl k CacMnYnKt;FmµCati. bgðajfa ax by cz a b cx y z . 427> k> rkbIcMnYnKt;esstKña ebIplbUkcMnYnTaMgbIesµInwg 909 . x> rkbYncMnYnKt;KUtKña ebIdwgfaplbUkcMnYnTaMgbYnesµnI wg 1028 . 428> edaHRsaysmIkar 0.17 2. 3 x 0. 3 . 429> xügmYyenAkñúgGNþÚglU manCeRmA 7m.eBlyb;xügenHvareLIgelI)an 3m EtenAeBléf¶eRkamGMNac énkemþAvaFøak;cuH 1m vij. ]bmafa xügBüayamvareLIgrhUt etIb:unµanéf¶eTIbxügecjrYcBIGNþgÚ . 430> enAem:ag 6:30 mn narImYyRkumcab;epþImsÞgÚ RsUvBIPøWmçag eTAPøWmçageTot Edlmancm¶ay 100m eday el,Ón 30m/h ehIybursmYyRkumeTotcab;epþImsÞÚgBIPøWmçagedayel,Ón 20m/h . k>etIRkumTaMgBIrsÞÚgCYbKñaenAem:agb:unµan ? x> etImYyRkum²sÞÚg)anRbEvgb:unµan m ? 431> KNnakenSam F x 2 3 2x 5 x 2 3 2x 5 . 432> eKmankaer ABCD nigcMNuc K enAelIGgát;RTUg AC . bgðajfa KA KC 2KD . 433> KNnakenSam x y1 ebIdwgfa x 1y 1296 nig xy 4 . 2
2
2
2
4
4
4
434> edaHRsayRbB½n§smIkar
ab 8 3 3 a b 3
.
667
-
435> eK[RtIekaNsm)at COB Edlman)at OB nigkm ABC CaRtIekaNEkgRtg;kMBUl A manbrimaRtesµI 12 cm nigmanRkLaépÞesµI 6 cm . KNnargVas; 2
2
2
2
2
RCugnImYy² énRtIekaNEkg ABC enH . 437> rkcMnYnKt;EdlenAcenøaH 40 nig 50 edaydwgfa ebIeKbþÚrlMdab;elx enaHeKnwg)anCYbcMnYnfµImYyeTot EdlesµInwg 32 éncMnYnenaH . 23 438> tamcMNuc O kñúgRtIekaNsm½gS ABC eKKUsGgát;EkgeTAnwgRCugTaMgbIénRtIekaNenaH. bgðajfaplbUkcm¶ayBIcMNuc O eTARCugTaMgbIesµInwgrgVas;km bM)at;r:aDIkal;BIPaKEbg A . 440> edaHRsayRbB½n§smIkar 441> eK[RtIekaN
3 5 2 2 5 x y 3 x z 2 xy yz zx 2
.
manrgVas;RCug a , b , c nigRtIekaN ABC manrgVas;RCug a , b , c . ]bmafa ABC dUcnwg ABC bgðajfa aa bb cc a b ca b c . 442> ABC CaRtIekaNEkgRtg; A Edlman BC 2a nig B 60 . kMNt;témø a edIm,I[témøbrimaRt esµInwgtémøépÞRkLa . 443> enAmanlMhat;bnþCaeRcInTot>>>. ABC
o
668
-
1> rkRKb;KUéncMnYnKt;viC¢manén a , b eyIgman ³ eyIg)an ³
-ebI A 2 enaHelxcugén 3 -ebI A 3 enaHelxcugén 3 naM[ 3 3
4 k A
4 k A
a 2 b2 a b 8 4a 2 4a 1 4b 2 4b 1 34
32013 345031
2a 12 2b 12 34
eday 34 36 6 ehIy 2a 1 nig 2b 1CacMnYness enaH 34 CaplbUkkaercMnYnesstUcCag 6 . cMnYnesstUcCag 6man 1, 3, 5 manEt 3 5 34 b¤ 5 3 34 eyIg)an ³ 2a 1 3 nig 2b 1 5 enaH a 2 , b 3 b¤ 2a 1 5 nig 2b 1 3 enaH a 3 , b 2 dUcenH KUcMnYnepÞógpÞat;KW ³ aa 32 ,, bb 23 . 2
2
2
2
E x y x 2 y x 3 y x 4 y y 4 x 2 4 xy xy 4 y 2 x 2 3 xy 2 xy 6 y 2 y 4 2
5 xy 5 y 2 y 2 x 2 5 xy 5 y 2 y 2 y 4
2
5 xy 5 y 2 5 xy 5 y
2
5 xy 6 y y
3 3
, 3 9 2
4 k A
4 k A
4 k A
4 k A
32 2
P
CakaerR)akd . 2
2013
, 3 27 3
17 12 2
3 2 2
2 2 2 1
9 12 2 8
17 12 2
2 2 2 1 9 12 2 8
2 2 2 2 12
3 2 12 2 2 2
2014
2 1 3 2 2 2
2
, 3 81 4
35 243 , 36 729 , 37 2187 , 3 6561 , ...
eyIgTaj)anTUeTA sV½yKuNén 3 ³ eyIgman 3 -ebI A 0 enaHelxcugén 3 KW ³ 1 -ebI A 1enaHelxcugén 3 KW ³ 3
4503 2
4
2012
4503 0
2012
2 2
E x 2 5xy 5 y 2
2011
4k A
y4 y4
3> rkelxenAxÞg;rayéncMnYn A ³ eyIgman A 3 7 3 7 BinitüelxcugsV½yKuNén 3 ³ 1
5 xy 4 y
2
dUcenH
x
2
2
7 5 16807 , 7 6 ...9 , 7 7 ...3 , 78 ...1 , ...
4> KNnakenSam P ³ eyIgmankenSam
tag y 1000 eyIg)an ³
, 7 2 49 , 7 3 343 , 7 4 2401
E x 1000 x 2000 x 3000 x 4000 1000 4
2
71 7
2014
2> RsaybBa¢ak;fa E CakaerR)akd eyIgman ³
2
BinitüelxcugsV½yKuNén 7 ³
2
x y x 4 y x 2 y x 3 y y 4
eyIgTaj)anTUeTA sV½yKuNén 7 ³ eyIgman 7 , k N -ebI A 0 enaHelxxagcugén 3 KW ³ 1 -ebI A 1enaHelxxagcugén 3 KW ³ 7 -ebI A 2 enaHelxxagcugén 3 KW ³ 9 -ebI A 3 enaHelxxagcugén 3 KW ³ 3 naM[ 7 7 manelxcug 1 7 7 manelxcug 9 eyIg)an ³ elxcug A =elxcug 7 elxcug1 elxcug 3 elxcug 9 = elxcug 7 elxcug 7 = elxcug 4 dUcenH cMnYn A manelx 4 enAxagcug .
x x x x
45023
2011
4a 2 4b 2 4a 4b 32
KW ³ 9 KW ³ 7 manelxcug 7 manelxcug 3
4k A
,k N
2 1 32 2
2
3 2 12 2 2 2
2 1 3 2 2 2
2
2 1 3 2 2
2 13 2 2 2 2
2 1 3 2 2 32
4 k A
2 2 2 2 12
2
3 2 432 2 3 2 43 2 2 2
4 k A
dUcenH kenSam P 2 . 669
2
5> KNnatémø x edIm,I[ eyIgman ³
F
mantémøGb,brma
tamlMnaMxagelIeKGacbMEbk)an dUcxageRkam ³ 1 1 1 1 2 1 2 1 1 1 23 2 3 1 1 1 3 4 3 4
F x 1 x 2 x 3 x 6 x 1 x 6 x 2 x 3
x x
x 2 6 x x 6 x 2 3x 2 x 6
2
5x 6 x 5x 6
2
5 x 36
2
bUkGgÁnwgGgÁ
>>>>>>>>>>>>>>>>>>>>>>>
2
eday x 5x 0 enaH x 5x naM[ F mantémøGb,brmaKW F 36 eyIg)an x 5x 36 36 x 5x 0 2
2
2
2
36 36
2
2
b¤ cMeBaH n CacMnYnKt;viC¢manehIyFMCag! eK)an n n 1 b¤ n n 1 1
2
2
x 5x 0 xx 5 0 2
naM[ x 0 , x 5 dUcenH témøKNna)anKW
6> KNnatémøén eyIgman ³
x 0 , x 5
x2 y2
.
naM[
³
dUcenH
x y 30
x 3 y 3 3xyx y 27000
eday x y 8100 nig x y 30 eyIg)an ³ 8100 3xy 30 27000 3
90 xy 27000 8100 18900 xy 210 90 x 3 y 3 x y x 2 xy y 2
8100 30 x 210 y
x
2
2
210 y
2
270
2
dkGgÁnwgGgÁ
1
1
2
2
2 8n 56q1 40 7n 56q2 21 n 56q1 q2 19
ebItag q CacMnYnKt;FmµCati Edl q q q eyIg)an n 56q 19 dUcenH témøsMNl;Edl n Ecknwg 56 KW 19 . x> rkcMnYn n ³ eday 5616 n 5626 naM[ 5616 56q 19 5626 1
x 2 y 2 480
x 2 y 2 480
56
1
x 2 y 2 270 210
dUcenH témøKNna)anKW ³
.
smµtikmµ ³ n Ecknwg 7 [sMNl; 5 ebItag q CaplEck enaHeK)an n 7q 5 n Ecknwg 8 [sMNl; 3 ebItag q CaplEck enaHeK)an n 8q 3 eyIg)anRbB½n§smIkar nn 87qq 53 87
x 3 3x 2 y 3xy 2 y 3 27000
Et
1 1 1 1 ... 1 1 2 2 3 3 4 n n 1 1 1 1 1 ... 1 1 2 2 3 3 4 n n 1
8> k> rksMNl;ebI n Ecknwg
x y 3 303
3
1 1 1 n n 1 n n 1 1 1 1 1 1 1 ... 1 2 2 3 3 4 n n 1 1 n 1 1 1 1 1 n ... 1 2 2 3 3 4 n n 1 n 1
.
7> RsaybMPøWfa ³ 11 2 21 3 31 4 ... n n1 1 1 Binitü ³ 1n n 1 1 nnn11n nn1 1
5597 56q 5607 99.946 q 100.125
670
2
eday q CacMnYnKt;FmµCati enaH q 100 eyIg)an n 56100 19 5619 dUcenH cMnYn n Edlrk)anKW ³ n 5619 .
10> KNnatémøelxénkenSamelx A eyIgman ³ A 182 33125 182 33125 tamrUbmnþ a b a 3a b 3ab b a b 3aba b naM[ ³ 3
3
3
3
3
9> bgðajfa A a1 a1 a1 ... a1 CacMnYnKt; 1
eyIgman ³ a 1 an
2
3
2
n
naM[
2
1 1 1 1 1 1 n n 1 1 1 1 1 1 n n 2
1 1 1 1 1 1 n n
2
A3 364 33 1822 331252 A A3 364 33 33124 33125 A
2
A3 364 3 A
2
A3 3 A 364 0
edayemIleXIjb¤sgay A 7 eyIg)an ³
A3 3 A 364 0 A3 7 A 2 7 A 2 49 A 52 A 364 0 A 2 A 7 7 A A 7 52 A 7 0
A 7A2 7 A 52 0 A 7 0
2
Taj)an A
2
7 A 52 0
naM[ A 7 0
A7
eRBaHKµanb¤s edaysar 0 dUcenH témøelxénkenSam A 7 . A2 7 A 52 0
11> KNna A x eyIgman ³ 2 xy y 4 x 2011
1 1 5 1 a1 4 1 1 13 5 a2 4 1 1 25 13 a3 4
y 2011 2
2x 1
y 2 x 1 4 x 2 x 1
2
4x 2 2x 1 y 2x 1 2 x2 x 1 1 y 2x 1 1 y 2x 2x 1
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
3
3
2n 2n 1 2n 2n 1 n2 n2 4 n 2 n 2n 2n 1 2n 2 2n 1 4 n2 n2 1 2n 2 2n 1 2n 2 2n 1 4
eyIg)an ³
3
3 3 182 33125 3 182 33125 A
2 2 1 1 1 1 1 12 2 1 1 12 2 1 n n n n 2 2 1 1 1 1 12 2 1 12 2 1 n n n n 2
3
A3 182 33125 182 33125
1
2
2
2
A3 3 182 33125 3 182 33125
20
1 1 1 1 1 1 n n
2
1 1 841 761 a20 4 1 1 1 1 1 ... 841 1 a1 a2 a3 a 20 4 1 29 1 7 4
smµtikmµ x , y CacMnYnKt; enHmann½yfa 1 1 CacM n Y n Kt; l u H RtaEt CacMnYnKt; y 2x 2x 1 2x 1
dUcenH KNna)an A 7 CacMnYnKt; . 671
ehIy 2 x1 1 CacMnYnKt;luHRtaEt 2x 1 1 naM[ x 1Taj)an y 2 x 2 x1 1 2 1
eyIg)an ³ A x y dUcenH KNna)an A x 2011
12> KNna
f 12
2011
13x
f x x
5 7 2 AD AB AD AB 5 7 12 1
1 2 1 1 2 1 1
2011
2011
smµtikmµ smamaRtkm
12011 12011 2
.
y 2011 2
³ eyIgman ³ 2010
13x
2009
13x
2008
... 13x 13x 1 2
x 2011 12 x 2010 x 2010 12 x 2009 ... 12 x 2 x 2 12 x x 1
x 2010 x 12 x 2009 x 12 x 2008 x 12 x 2007 x 12 ... x x 12 x 1
x 12( x 2010 x 2009 x 2008 x 2007 ... x) x 1
cMeBaH
x 12
CMnYscUl eyIg)an ³
f 12 12 12(122010 122009 122008 122007 ... 12) 12 1 f 12 0 11 11
dUcenH eRkayBIKNna f 12 11 .
13> KNnargVas;RCugénRbelLÚRkam ABCD A
B
h1
eyIgmanrUbmnþRkLaépÞRbelLÚRkam S AB h b¤ S AD h Taj)an AB h AD h )ansmamaRt hh AD AB 1
ABCD
1
2
2
dUcenH rgVas;RCugRbelLÚRkamKW 10cm nig 14cm .
2
AB
smIkaremdüaT½rénGgát; AB CasmIkarbnÞat;Edl kat;tamcMNuckNþalGgát; ABehIyEkgnwgGgát; AB. eyIgman ³ A2 , 3 nig B4 , 4 naM[kUGredaencMNuckNþalénGgát; ABKW ³ x xB y A y B 2 4 3 4 7 I A , , I I 3, 2 2 2 2 2
-bnÞat; AB kat;tam A2 , 3 enaH A AB naM[kUGredaencMNuc A epÞógpÞat;smIkar³ 3 2a b GacsresrCa 2a b 3 1 -bnÞat; AB kat;tam B4 , 4 enaH B AB naM[kUGredaencMNuc B epÞógpÞat;smIkar³ 2 4 4a b GacsresrCa 4a b 4 yksmIkar 2 1 eyIg)an ³ 4a b 4 2a b 3 2a 1
C
1
AB 2 AB 14cm 7
h2
D
ABCD
5 naM[ AD smmUl AB 7 EtRbelLÚRkammanbrimaRt 48cm naM[ 2 AB AD 48 enaH AB AD 24 AB AD AB 24 eyIg)an AD 2 5 7 12 12 TMnak;TMngsmamaRt AD 2 AD 10cm 5
14> rksmIkaremdüaT½rénGgát;
x 2009 12 x 2008 x 2008
2 1
naM[ a 12 -smIkaremdüaT½rRtUvrkmanrag y ax b eday y ax b AB enaH a a 1 b¤ 12 a 1 a 2 ehIy emdüaT½rkat;cMNuckNþal I enaH I y ax b 672
eyIg)an ³ 72 2 3 b
b
17> KNna E xx yy
7 19 6 2 2
naM[smIkaremdüaT½rEdlRtUvrkGacsresr³ y 2x 192 dUcenH smIkaremdüaT½rénGgát; ABKW³ y 2x 192
15> rkRbU)abcab;)anXøI@ minEmnBN’exov
x y E x y
cMeBaH smmUl E . eyIgman ³ 2 x 2 y 5xy naM[ 2 x 2 y 5xy
RBwtiþkarN_cab;min)anXøIBN’exov mann½yfa ³ )anXøIRkhmTaMg@ b¤ elOgTaMg@ b¤ Rkhm!elOg! -RbU)abcab;)anXøIRkhmTaMg@KW P ¬k>k¦= 122 111 661 -RbU)abcab;)anXøIelOgTaMg@KW P ¬l>l¦= -RbU)abcab;)anXøIRkhm!elOg!KW ³ 2 4 4 2 4 P ¬k!>l!¦= P ¬k>l¦÷ P ¬l>k¦= 12 11 12 11 33
4 3 1 12 11 11
2
2
2
2
2
2 x 2 y 2 5 xy 5 x 2 y 2 xy 2
-Efm 2 xy elIGgÁTaMgBIrénsmIkar eyIg)an x 2xy y 52 xy 2xy smmUl x y 92 xy -Efm 2 xy elIGgÁTaMgBIrénsmIkar eyIg)an x 2xy y 52 xy 2xy smmUl x y 12 xy 2
2
2
2
naM[ RbU)abcab;XøI@minEmnBN’exovKW ³ P
2 x y x y 2
2
2
¬minEmnexov¦= P ¬k>k¦÷ P ¬l>l¦÷ P ¬k!>l!¦ 1 1 4 66 11 33 1 6 8 15 5 66 66 22
x y 2 E2 x y 2
eyIg)an ³
9 xy 2 9 1 xy 2
dUcenH RbU)abcab;min)anXøIBN’exovKW P 225 .
Taj)an E 9 3 eRBaH x y 0 enaH E 0 dUcenH eRkayBIKNna E 3 .
16> KNnakenSam
18> sRmYlkenSam
A
³ eyIgman
6 12 18 ... 962 3 A 12 24 36 ... 1922 4 2 6 12 18 ... 96 3 2 2 6 2 12 2 18 ... 2 96 4 6 12 18 ... 962 3 26 12 18 ... 962 4 2 6 12 18 ... 96 3 2 2 2 6 12 18 ... 96 4
1 3 1 4 4
³ eyIgman
E
x y x y
E 2x 2 y 2 2x x y x y
x y 2 x x y x x y x y 2 x x y x x y x y x x y x y x x y x 2
2
2
2
2
2 2
2
2
2
2
2
2
2
2
2
2
2
2
2
dUcenH eRkayBIsRmYlrYc E x .
dUcenH eRkayBIKNna A 1 . 673
2
2
2
2
19> KNnaplKuN iii c a 2 ca k> A 1 cot1 1 cot 2 1 cot 3 1 cot 44 edayKuNGgÁnwgGgÁén i ii iii eyIg)an ³ a a b b c c a 2 ab 2 bc 2 ca tamrUbmnþ cos cot a eyIg)an sin a o
o
o
o
cos1o cos 2 o cos3o cos 44o 1 A 1 1 1 o o o o sin 1 sin 2 sin 3 sin 44 sin 1o cos1o A sin 1o
sin 2 o cos 2 o sin 2 o
sin 3o cos3o sin 3o
edaydwgfa sin a cos a 2 sin 44 A o sin 1
o
2 sin 43 sin 2o
o
sin 44 o cos 44 o sin 44 o
2 sin 45 o a
2 sin 42 sin 3o
o
a b b c c a 8 ab bc ca a b b c c a 8abc dUcenH eyIgeXIjfa a bb cc a 8abc
21> bgðajfa a b c ab bc ca ³ eday a , b , c CacMnYnBitviC¢man eyIg)an ³ 2
2 sin 1 sin 44o o
44 sin 44 o sin 43o sin 42 o sin 1o A 2 o o o o sin 1 sin 2 sin 3 sin 44
Emn .
2
2
a b 0 a 2 2ab b 2 0 a 2 b 2 2ab i 2
b c 0 b 2 2ab c 2 0 b 2 c 2 2bc ii
eRkayBIsRmYl eyIg)an A 2 4 194 304 dUcenH lT§plénplKuNKW A 2 4 194 304 . c a 0 c 2ab a 0 c a 2ca iii x> B 3 tan1 3 tan 2 3 tan 3 3 tan 29 edaybUkGgÁngw GgÁén i ii iii eyIg)an ³ sin a a b b c c a 2ab 2bc 2ca tamrUbmnþ cos tan a eyIg)an a 2
22
2
2
2
2
2
22
o
o
o
o
2
sin 1o B 3 cos1o
sin 2 o 3 cos 2 o
sin 29 o 3 cos 29 o
3 cos a sin a 2 cos 30 o a
dUcenH eXIjfa a
ii
¬Efm 43 ¦
2 3 y 2 48 4 3 3 2 52 x 3 y 2 3 3
2 3 x 3 y 2 52 3 3x 23 y 2 52
2
b c 2 bc
Emn .
x3 y 2
20> cUrbgðajfa ³ a bb cc a 8abc eday a , b , c CacMnYnBitviC¢man eyIg)an ³ a b 0 dUcKñaenHEdr eyIg)an ³
b 2 c 2 ab bc ca
2 x 2 y 16 3 xy 3 xy 2 x 2 y 16 4 4 3xy 2 x 2 y 16 3 3
29
i
2
22> rkb¤sKt;rbs;smIkar eyIgman 2x y 16 3xy
29
ab
2
2
eRkayBIsRmYl eyIg)an B 2 536 870 912 dUcenH lT§plénplKuNKW B 2 536 870 912 .
a b 2
2
a 2 b 2 c 2 ab bc ca
2 cos 29 o 2 cos 28 o 2 cos 27 o 2 cos1o B o o o o cos1 cos 2 cos 3 cos 29 cos 29 o cos 28 o cos 27 o cos1o B 2 29 o o o o cos1 cos 2 cos 3 cos 29
a2 2 a b b2 0
2
2 a 2 b 2 c 2 2ab bc ca
3 cos1o sin 1o 3 cos 2 o sin 2 o 3 cos 29 o sin 29 o B o o cos1 cos 2 cos 29 o
edaydwgfa
2
edaycg;rkEtb¤sKt;én x nig y EdlepÞógpÞat; ehIy 674
1 52 52 2 26 4 13
b¤
52 1 52 26 2 13 4
-cMeBaH 33yx22521 -cMeBaH
3x 2 2 3 y 2 26 3x 2 4 3 y 2 13
24> k> KNna f x f y eyIgmanTMnak;TMng f x 4 4 2 naM[ f y 4 4 2 eyIg)an f x f y 4 4 2 4 4 2
3x 3 x 1 3 y 54 y 18 3x 4 minykeRBaHminKt;¦ 3 y 28 3x 6 x 2 3 y 15 y 5
x
¬
x
y
y
-cMeBaH dUcenH KUb£sKt;EdlrkeXIjrYmman³ x 18 x 5 x 1 x 2 b¤ . , , y 1 y2 y 18 y 5
x
y
x
y
4x 4 y 2 4 y 4x 2 4x 2 4 y 2 4 x y 2 4 x 4 x y 2 4 y x y 4 2 4 x 2 4 y 22
23> RsaybBa¢ak;fasmIkarEdl[manb¤Canic© edaydwgfa x y 1 enaHeyIg)an eyIgman x ax b x bx c x cx a 0 f x f y 4 2 4 4 2 4 4 24 24 2 edayBnøatsmIkarxagelI eyIg)an ³ 8 24 24 x
x 2 bx ax ab x 2 cx bx bc x 2 ax cx ac 0
dUcenH
3x 2 2ax 2bx 2cx ab bc ac 0
y
x
y
x
y
8 2 4x 2 4y
f x f y 1
2
1
tamlkçxNÐEdl[ .
3x 2 2a b c x ab bc ac 0
x> TajrkplbUk S ³ eyIgman eXIjfaeRkayBIbRgYmrYcvaCasmIkardWeRkTI2 1 2 3 2010 edIm,I[smIkardWeRkTI2manb¤sCanic© luHRtaEt 0 S f 2011 f 2011 f 2011 ... f 2011 edayyl;fa f x f y 1 kalNa x y 1 eyIg)an b ac Edl b b2 ehIyeXIjfa a c c 3ab bc ac 1 2010 1 2010 1 enaH f a b c 2ab bc ac 3ab bc ac f 1 2011 2011 2011 2011 2
2
2
2
2
a 2 b 2 c 2 ab bc ac 1 2a 2 2b 2 2c 2 2ab 2bc 2ac 2 1 a 2 2ab b 2 b 2 2bc c 2 c 2 2ac a 2 2 1 2 2 2 a b b c c a 2 a,b,c
eday
2 2009 1 2011 2011
CacMnYnBit enaHeyIg)an ³
enaH
2 f 2011
2009 f 2011
1
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1005 1006 1 2011 2011
enaH
1005 1006 f f 1 2011 2011
edaybUkGgÁngw GgÁ 1 2 2010 f f ... f 1 1 ... 1 2011 2011 2011
a b 0 2
b c 0
man !00% tY
2
c a 0 2 a b b c 2 c a 2 0 2
naM[ 12 a b b c c a 0 Emn dUcenH smIkarmanb¤sCanic© . 2
2
1 2 2010 f f ... f 1 1005 2011 2011 2011
2
dUcenH eyIgTaj)anplbUk S 1005 .
675
25> KNnalImIt lim x 27> bgðajfa tag y x naM[ ln y ln x smmUl ln y x ln x a ba b a b a b 0 eyIgGacbMElg)an ln y ln1x ¬lkçxNÐ x 0 ¦ Binitü a b a b a b x
x0
x
x
2
2
8
8
eyIg)an
4
4
4
8
8
4
4
a2 b2 a2 b2 a4 b4
x
ln x lim ln y lim x 0 x 0 1 x
4
a b a b a b a b 4
¬ragminkMNt;¦
EtbRmab; ³ ; a b 1 eyIg)an a b a ba
2
2
4
naM[ -eyIgGacedaHRsaytamrebobTI! ¬edaHedaytag¦³ a b a b a b a b 0 tag u 1x naM[ x u1 ebI x 0 enaH u a ba b a b a ba b a b 0 00 ¬)anCah‘andak; x 0 eRBaH x 0 xitCitBIxagsþa¦M dUcenH a ba b a b a b 0 . eyIg)an ³ 1 ln 1 ln x 28> KNna A 1 12 1 13 1 14 ... 1 2011 u lim ln u lim ln u 0 lim lim 8
8
2
2
4
4
2
2
4
4
2
2
2
b2 a4 b4
8
8
2
2
4
4
4
4
8
8
1
1 x
x 0
u
u
u
u
u
u
1 1 1 1 A 1 1 1 ... 1 2 3 4 2011 2 1 3 1 4 1 2011 1 ... 2 3 4 2011 1 2 3 2010 1 ... 2 3 4 2011 2011 1 A 2011
¬eRBaHtamrUbmnþ 1a a / log b x log b / lim lnuu 0 ¦ naM[ lim ln y 0 ln lim y 0 lim y e lim x 1 . dUcenH eRkayBIKNna - eyIgGacedaHRsaytamrebobTI@ ¬LÚBItal;¦³ 1
x
a
a
x0
x0
0
x
x0
cMeBaH
u
x 0
1 ln x lim ln y lim lim x lim x 0 x 0 x 0 1 x 0 x 0 1 2 x x
¬eFVIedrIevTaMgPaKyk nigPaKEbg¦
naM[
lim ln y 0 x0
lim y e 0
29> edaHRsaysmIkar ³ 4x 5x eyIgman ³ 4 x 5x 1 0 4
4
x0 x
dUcenH eRkayBIKNnaeXIjfa
lim x x x0
. 1 .
2
A 2011 1(20112010 20112009 20112008
A 2011 2011 12011 1 2011 2011 1 1 2011 2011
dUcenH KNna)an
A 20112011
.
2
plKuNktþaesµI 0 luHRtaktþanImYy²esµI 0 ³ eyIg)an ³ x 1 0 x 1
... 20112 2011 1) 1
1 0
x 1x 12 x 12 x 1 0
A 2010 2011 2010 2011 2009 2011 2008 ... 2011 2 2012 1
2
x 14 x
26> KNna A ³ eyIgman
1 0
4x 2 x 2 1 x 2 1 0
lim x 1 x 0
2
4x 4 4x 2 x 2 1 0
ln lim y 0
x0
.
x 1 0
x 1
2x 1 0
x
2x 1 0
x
1 2 1 2
dUcenH smIkarmanb¤sbYnKW x 1, 12 , 12 ,1 .
676
30> bgðajfa ab 1a 1 bc 1b 1 ca 1c 1 1 eyIg)an ³ BinitüGgÁTImyY ³
1 1 ab a 1 ab a 1 1 1 a a bc b 1 bc b 1 a abc ab a a eRBaH abc 1 1 ab a 1 1 ab ab ca c 1 ca c 1 ab a abc abc ab ab eRBaH abc 1 a 1 ab
eyIg)an ³
dUcenH
1 1 1 ab a 1 bc b 1 ca c 1 1 a ab 1 a ab 1 a ab 1 a ab 1 a ab 1 1 a ab
1 1 1 1 ab a 1 bc b 1 ca c 1
BitEmn .
1 1 1 2 1 1 2 1 2 1 1 1 3 22 3 2 3 1 1 1 4 3 3 4 3 4 ………………………. 1 1 1 100 99 99 100 99 100
1 2 1 1 2
1
...
3 22 3
1 100 99 99 100
1 1
1 100
1 1 1 1 9 ... 1 10 10 2 1 1 2 3 2 2 3 100 99 99 100
dUcenH S 2 1 1 1 2 3 2 1 2 3 ... 100 99 1 99 100 109 . 32> bgðajfa 2002 eyIgman ³ 20022003 20021979
Eckdac;nwg 6
20021979
2003
20021979 200224 1
20021979 2002 1 200223 200222 ... 2002 1
31> KNna S
1 1 1 1 ... 2 1 1 2 3 2 2 3 4 3 3 4 100 99 99 100
2001 2002 2002 ... 2002 1
2 2002
3 667 2002 2002 ... 2002 1
6 2002
667 2002 2002 ... 2002 1
1978
1978
23
22
23
22
23
tamlT§pl cMnYn 2002 2002 CaBhuKuNén 6 dUcenH cMnYn 2002 2002 Eckdac;nwg 6 .
Binitü ³
1979
2003
1 n 1 n n n 1 1 n 1n n 1 n
n 1n
2
n 1 n n 1 n n 1 n
n 1 n 1n
1979
33> bgðajfa a b c 13 eyIgman a b c 1 elIkCakaer eyIg)an 2
2
a 2 b 2 c 2 2ab 2bc 2ca 1
n 1 n n 1 n n 1n n 1 n n 1n
22
2003
2 2002
1978
a 2 b 2 c 2 1 2ab 2bc 2ca
RKb;cMnYnBit a , b , c enaHeyIg)anvismPaB ³ a 2 b 2 2ab b 2 c 2 2bc c 2 a 2 2ca
n 1 1 n 1n n n 1
2a 2 2b 2 c 2 2ab 2bc 2ca 677
ii
i
edayyk i ii enaHeyIg)an ³ a 2 b 2 c 2 1 2ab 2bc 2ca 2 2a 2b 2 2c 2 2ab 2bc 2ca 3a 2 3b 2 3c 2 1
Taj)an
x y 1 1 xy log x y 1 1 xy
i ii
1 3a b c 1 a b c 3 2
2
dUcenH eXIjfa a
2
2
2
b2 c2
1 3
2
2
R)akdEmn .
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b
c
x
y
a 1
a
log a b log b c log c d .... log x y log y a 1 log b log c log d log y log a ... 1 log a log b log c log x log y 11
dUcenH
log a b log b c log c d .... log x y log y x y f x f y f 1 xy 1 x f x log 1 x 1 y f y log 1 y
35> bgðajfa eyIgman ³ naM[)an ³ eyIg)an ³
dUcenH eXIjfa
x y f 1 xy
x y f x f y f 1 xy
36> edaHRsaysmIkar
Emn .
x
x
7 48 7 48 14
eXIjfa ³ 7 48 7 48 49 48 1 Taj)an 7 48 1 7 48
1
7 48
eyIg)ansmIkarfµI ¬CMnYs
7 48
eday
1 7 48
¦
x
x 1 7 48 14 7 48
tag t 7 48 a 1 . smIkareTACa 1t t 14 smmUl t 14t 1 0 man 7 1 48 naM[ t 71 48 7 48 7 48 x
2
2
2
1
t2
1 y 1 x f x f y log log 1 x 1 y 1 x 1 y log 1 x 1 y
1 x y xy log 1 x y xy 1 xy x y log 1 xy x y
7 48 1 7 48 1 7 48
7 48
1
7 48
cMeBaH
t1 7 48
enaH Taj)an
2
x
7 48 7 48
cMeBaH
naM[)an ³
enaH 7 48 7 48 Taj)an x 2 dUcenH smIkarmanb¤s x 2 x 2 .
678
t 2 7 48
2
x
2
x2
edayEckTaMgPaKyk nigPaKEbgnwg 1 xy
1 xy x y 1 xy f x f y log 1 xy x y 1 xy
2
2
37> bgðajfa ³
39> edaHRsaysmIkar ³ 5 50 x Binitü 5 bMBak;elakarItenEBr eK)an log x
log a b log b a 2log a b log ab b log b a 1 log a b
log x
Binitü ³
log a b log b a 2log a b log ab b log b a 1 log b log a log b log b log a 2 1 log a log b log a log ab log b log 2 a 2 log a log b log 2 b log a 1 1 log a log b log ab log a log b 2 log a 1 1 log a log b log a log b log a log b 2 log a log b log a 1 log a log b log a log b log a log b 2 log b 1 log a log b log a log b log a log b 1 1 log a b 1 log a b log a
ehIy
5log x
bMBak;elakarItenEBr eK)an
ln x log 5 log 5 ln x
eXIjfa log x ln 5 log 5 ln x naM[ 5 x CMnYs x eday 5 eyIg)ansmIkar 5 50 5 log x
log 5
log x
log x
log x
2 5log x 50
5log x 25
5log x 25
5log x 5 2
Taj)an log x 2 x 10 100 . dUcenH smIkarmanb¤sKW x 100 . 2
log a b log b a 2log a b log ab b log b a 1 log a b cosa 1 cosa 4 cot a 38> bgðajfa 11 cos a 1 cosa sin a Binitü RBmTaMgKNnaGgÁTImYy eyIg)an ³
1 cosa 1 cosa 1 cosa 1 cosa 2 2 1 cosa 1 cosa 1 cosa 1 cosa 1 cosa 1 cosa 1 cosa 1 cosa 1 cosa 1 cosa 2 cosa 2 4 cosa 4 cosa 4 cot a 2 2 1 cos a sin a sin a sin a sin a 1 cosa 1 cosa 4 cot a 1 cosa 1 cosa sin a
ln 5 log x log x ln 5
log 5
dUcenH eRkayBIbgðajrYceXIjfa
dUcenH
log 5
BitEmn .
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dUcenH RbU)abe)aH)anelxess b¤ elxFMCag5esµInwg 23 .
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679
n
x 4 x 16 x ... 4 n x 3 x 2 x 1 4 x 16 x ... 4 n x 3 2 x 1 4 x 16 x ... 4 n x 3 4 x 4 x 1
16 x ... 4 n x 3 4 x 1 16 x ... 4 n x 3 16 x 8 x 1 ... 4 n x 3 8 x 1 ......................................................
f 1 f 3 f 5 ... f 999997 f 999999
1 3 6 10 0 2 1 100 50 2
4n x 3 2n x 1
dUcenH f 1 f 3 f 5 ... f 999997 f 999999 50 .
4n x 3 4n x 2 2n x 1 2 2 2n x
43> edaHRsayRbB½n§smIkar
2n x 1
5log y x log x y 26 a/ xy 64
2 2n x 1 1 4n
x
1 3 3 f 1 2 2 0 f 3 1 3 4 3 2 2 1 f 5 3 6 3 4 2 ............................ f 999999 1 3 1000000 3 999998 2
x 4 x 16 x ... 4 n x 3 x 1
i ii
¬GñkKYcaMrUbmnþbþÚreKal log x log1 y ¦ -tam i ³ 5log x log y 26 y
dUcenH smIkarmanb¤s
x
.
1 x n 4
y
1 26 log x y log x y 5
42> KNnaplbUk
5 log 2x y 26 log x y 5 0
f 1 f 3 f 5 ... f 999997 f 999999
f n 3
n 2 2n 1 3 n 2 1 3 n 2 2n 1 1
n 1
2
n 1 2
3
3
3
3
2
3
1 2
3
n 1 3 n 1
naM[eyIg)an ³
3
3
2
eyIg)ansmIkarfµIKW ³ 26t 5 0 ¬tamDIsRKImINg; ¦ log x y t
man 13 25 169 25 144 naM[ t 135 12 15 / t 135 12 5 2
n 1
3
3
5t 2
2
n 1 n 1 n 1 n 1 n 1 n 1 1 n 1 n 1 n 1 n 1
3
tag
1 3
x
1
3
n 12
1
1
cMeBaH naM[ cMeBaH t 5 naM[ log y 5 y x -tam ii : xy 64 krNI y x enaH x x 64 x 2 x 2 naM[ y x y 2 y 2 krNI y x enaH x x 64 x 2 x 2 1 log x y y x 5 5
1 t 5
5
x
1 5
1 5
1 5
5
680
6 5
6
1 5 5
5
6
6
5
naM[ y x y 2 y 32 dUcenH RbB½n§smIkarmanKUcemøIyBIrKW ³ x 32 , y 2 b¤ x 2 , y 32 . 5
³
naM[
.
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eday a CacMnYnviC¢man enaHeyIg)an ³ 1 a1 2 a1 1 a2 2 a2 1 a3 2 a3 ................... 1 a 2 a n n 1 a1 1 a2 1 a3 ... 1 an 2 n a1a2 a3 ...an
x ln 4 y ln 5 2ln 4 5
x ln 4 y ln 5 2 ln 4 2 ln 5 i 5x 6 y 1 ln 900 1 ln 5 x 6 y ln 900 ln 5 x ln 6 y ln 1 ln 900
³
¬bMBak; elIGgÁTaMgBIr¦
smµtikmµ plKuN a a a ... a 1 enaHeyIg)an 1 a 1 a 1 a ... 1 a 2 dUcenH 1 a 1 a 1 a ... 1 a 2 .
x ln 5 y ln 6 0 ln 30 2
1
x ln 5 y ln 6 2ln 5 6
1
x ln 5 y ln 6 2 ln 5 2 ln 6 ii
3
n
n
2
3
n
edaybUkGgÁngw GgÁ eyIg)an ³
D ln 4 ln 6 ln 2 5 Dx ln 4 2 ln 5ln 6 2 ln 5 2 ln 6ln 5
7 x 3 3x 2 y 21xy 2 26 y 3 342 3 2 2 3 9 x 21x y 33xy 28 y 344 16 x 3 24 x 2 y 12 xy 2 2 y 3 686
2 ln 4 ln 6 2 ln 5 ln 6 2 ln 2 5 2 ln 5 ln 6 2 ln 4 ln 6 2 ln 2 5
2 ln 4 ln 6 ln 2 5 D y ln 4 2 ln 5 2 ln 6 ln 5 2 ln 4 2 ln 5
8 x 3 12 x 2 y 6 xy 2 y 3 343
2 x 3 3 2 x 2 y 3 2 x y 2 y 3 7 3 2 x y 3 7 3
2 ln 4 ln 5 2 ln 4 ln 6 2 ln 5 ln 4 2 ln 2 5 2 ln 4 ln 6 2 ln 2 5
2
n
7 x 3 3x 2 y 21xy 2 26 y 3 342 a/ 3 2 2 3 9 x 21x y 33xy 28 y 344
edaHRsaytamedETmINg; ³
2 ln 4 ln 6 ln 2 5
3
45> edaHRsayRbB½n§smIkar
x ln 4 y ln 5 2 ln 4 2 ln 5 x ln 5 y ln 6 2 ln 5 2 ln 6
2
n
1
tam i nig ii eyIg)anRbB½n§smIkar ³
x 2 , y 2
x ln 4 y ln 5 0 ln 20 2
tam
dUcenH RbB½n§smIkarmanKUcemøIyKW ³
1 x y 4 5 400 b/ 1 5 x 6 y 900 4 x 5 y 1 ln 400 1 ln 4 x 5 y ln 400 ln 4 x ln 5 y ln 1 ln 400
tam
Dx 2 ln 4 ln 6 ln 2 5 x 2 D ln 4 ln 6 ln 2 5 2 y D y 2 ln 4 ln 6 ln 5 2 D ln 4 ln 6 ln 2 5
5
2x y 7
edaydkGgÁnwgGgÁ eyIg)an ³ 681
i
7 x 3 3 x 2 y 21xy 2 26 y 3 342 3 2 2 3 9 x 21x y 33xy 28 y 344 2 x 3 18x 2 y 54 xy 2 54 y 3 2 x 3 9 x 2 y 27 xy 2 27 y 3 1 x 3 3 x 2 3 y 3 x 3 y 3 y 1 2
3
x 3 y 3 13
ii
x 3y 1
tamry³
i nig ii
eyIg)anRbB½n§smIkarfµIKW³
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2x y 7 2 x 6 y 2 5y 5 y 1 x 1 3y 1 3 4
2 x y 7 x 3y 1
smmUl
naM[ dUcenH RbB½n§smIkarmanKUcemøIy x 4 y 1 . 4 x 3 y 1 27 y 171 b/ x x 1 2 y 172 8 2 3 2 3 2 x 3 3 y 3 y 171 x 3 x y 2 2 2 3 3 172
tamry³lkçxNÐTaMg5xagelI lkçxNÐrYmKW mann½yfa ³ x 1 , 2 , 3
eyIgGacsresrCa
erob[tamlMdab;
2
x 3
2 3
2 3 3 2 x 3 y 3 y 171 x 3 2 3 2 x 3 y 172 2 2 2 3 3 2 x 3 y 3 2 x 3 y 3 y 343
y 3
x
7
cMeBaH x 3 cMnYnenaHKW 34596 186 ¬CakaerR)akd¦ 2
dUcenH cMnnY manelxR)aMxÞg;enaHKW 34596 . 3
2 3 7 x
y
i
edaydkGgÁnwgGgÁ ³ 3
y 3
3 2 1 y 3
y
x 3
3
edayyk i ii eyIg)an ³ 2 3 7 y x 3 2 1 2 3y 2 3 x
3 3
cMeBaH y 1 enaH
2 1
3 2
3 2 1
4
4
3
2025 y 1
S
2 3 7 y
2 1 1
1
3 1
2 1
1 3 2 1 4
3
2 1
3 2
4
3
2024
1 1 2
2025 1 2 3
2024 1
...
1 2025 2024
1 2024 2025
2025 2024
2025 1 45 1 44
2x 7 3 2 x 22
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
y
y
1 1 1 1 ... 1 2 2 3 3 4 2024 2025
Binitü rYcbUkGgÁnigGgÁ énkenSamxageRkam
3 y 2 x 1 ii
x
47> KNnaplbUk S ³ eyIgman S
3 2 3y 3 171 x 3 x y 2 3 2 3 172 2 3 x y 2 x 2 y 3 2 3 3 2 3 2 x 1 x 2
0 x3
cMeBaH x 1 cMnYnenaHKW 12334 minyk ¬eRBaHminCakaerR)akd¦ cMeBaH x 2 cMnYnenaHKW 23465 minyk ¬eRBaHminCakaerR)akd¦
edaybUkGgÁngw GgÁ ³
smmUl
0 x 9 0 x 8 0 x 7 0 x 3 0 x 6
x2
dUcenH RbB½n§smIkarmanKUcemøIy x 2 nig y 1 .
dUcenH plbUkKNna)an S 44 .
682
48> RsaybBa¢ak;fa ³ f n 3 7 Eckdac;nwg 8 eyIgman f n 3 7 cMeBaH n 0 : f 0 3 7 1 7 8 Eckdac;nwg 8 n 1 : f 1 3 7 9 7 16 Eckdac;nwg 8 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> ]bmafa vaBitdl; n k : f k 3 7 Eckdac;nwg 8 eyIgnwgRsaybBa¢ak;fa n k 1 k¾Eckdac;nwg 8 Edr cMeBaH n k 1 : f k 1 3 7 2n
2n
0
2
50> RsaybMPøWfa n3 n2 n6 CacMnYnKt;viC¢man tag A n3 n2 n6 2
2
eyIg)an
3
2n 3n 2 n 3 6 n 2 3n n 2 6 nn 1n 2 6
A
2k
3
ebI A CacMnYnKt; luHRtaEt nn 1n 2 Eckdac;nwg 6 3 7 mann½yfa nn 1n 2 Eckdac;nwg 2 pg nig 3 pg 93 7 -ebI n CacMnYnKt; enaH n , n 1, n 2 CabIcMnYnKt;tKña 8 3 3 7 eday 8 3 Eckdac;nwg 8 ¬eRBaH 8 3 CaBhuKuNén 8 ¦ -bIcMnYnKt;tKñay:agehas; k¾mancMnYnKUmYyEdr naM[ nn 1n 2 Eckdac;nwg 2 niig 3 7 Eckdac;nwg 8 ¬]bmafaBit xagelI ¦ -bIcMnYnKt;tKñay:agehas; k¾mancMnYnmYyCaBhuKuN naM[ f k 1 8 3 3 7 Eckdac;nwg 8 dUcenH f n 3 7 Eckdac;nwg 8 RKb; n CacMnYnKt; . én 3 Edr naM[ nn 1n 2 Eckdac;nwg 2 enaHeyIg)an nn 1n 2 Eckdac;nwg 6 eyIgGacbkRsaytamviFImüa:geTot 9 1 8 ¬mann½yfa 9 Eckdac;nwg 8 [sMNl; 1 ¦ dUcenH RKb;cMnYnKt;viC¢man n enaH n3 n2 n6 9 1 8 ¬GacelIkCasVy ½ KuN)an RKb; n CacMnYnKt;¦ k¾CacMnYnKt;viC¢manEdr . 3 1 8 ¬eRBaH 9 3 ehIy 1 1 ¦ 3 7 1 7 8 smmUl 3 7 8 8 51> KNna S 11 2 21 3 31 4 ... 20101 2011 Et 8 0 8 naM[ 3 7 0 8 Edr 1 1 n 1 n 1 dUcenH f n 3 7 Eckdac;nwg 8 RKb; n CacMnYnKt; . Binitü ³ n n 1 nn 1 nn 1 2 k 1 2k 2
2k
2k
2k
2k
2k
2k
2k
2k
2n
2
n
3
n
2n
2
n
2n
2n
2n
2n
tamlMnaMxagelIeKGacbMEbk)an dUcxageRkam ³
49> KNna g 3 ³ 1 1 1 1 2 1 2 eyIgman f x 2x 1 1 1 1 [ f x 3 smmUl 2x 1 3 x 1 23 2 3 bUkGgÁnwgGgÁ 31 4 13 14 ehIyman g f x x 3x 1 g 3 1 3 1 1 ¬eRBaH f x 3 nig x 1 ¦ naM[ >>>>>>>>>>>>>>>>>>>>>>> 2
2
g 3 5
1 1 1 2010 2011 2010 2011 1 1 1 1 1 1 ... 1 2 2 3 3 4 2010 2011 1 2011
dUcenH eRkayBIKNna g 3 5 . 683
b¤
1 1 1 1 2010 ... 1 2 2 3 3 4 2010 2011 2011
dUcenH eRkayBIKNnaplbUk S 2010 . 2011
52> KNna x x 1 eyIgman x x 1 0 Taj)an 2011
2011
2
x
1 1 x
enaH
1 1 1 1 x x 1 1 x 2 2 2 1 x 2 2 1 x x x x 1 1 1 1 1 x x 2 2 1 1 x 3 3 x 1 x 3 3 2 x x x x x 1 1 1 1 3 1 4 2 4 x x 3 1 2 x 4 x 2 2 x 4 1 x x x x x 1 1 1 1 1 x x 4 4 1 1 x 5 5 x 3 3 1 x 5 5 1 x x x x x 1 1 1 1 5 1 6 4 6 x x 5 1 1 x 6 x 4 1 x 6 2 x x x x x
-yl;fa ³ cd CacMnYnKt; enaHRtUvEtmanktþamYy kñúg cMeNamktþa n 10 nig n 10 EdlRtUvEtEckdac;nwg 101 eRBaH 101CacMnYnbzm . enaHeyIgTaj)an ³ n 10 101b¤ n 10 101 naM[ n 111 b¤ n 91 -cMeBaH n 111 minyk eRBaH n Caelx4xÞg; enaH n 100 2
-cMeBaH n 91 enaH n 8281 yk dUcenH cMnYnenaHKW abcd n 8281 . 2
2
54> bgðajfa n! 3 cMeBaH n 7 eyIgman n! 3 -cMeBaH n 7 enaH 7! 5040 3 2187 ³ Bit -cMeBaH n 8 enaH 8! 40320 3 6561 ³ Bit n
n
7
8
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n
n
k 1
naM[
x 2011
1 x
2011
dUcenH eRkayBIKNna eXIjfa x
2011
1 x
2011
1
53> rkcMnYn abcd enaH tambRmab;RbFan cMnYn abcd CakaerR)akd eyIg)an abcd n Edl n CacMnYnKt; eyIgsresr)an ab00 cd n enaHTaj)an 100 ab cd n 1 bRmab;bEnßm ab cd 1 b¤ ab 1 cd yk 2 CMnYskñúg 1 ³ 100 100cd cd n 2
k
k
k 1
2
eday enaHeyIg)an k 1 k ! 3 3 KWCakarBit dUcenH eXIjfa n! 3 cMeBaHRKb;cMnYnBitviC¢man n 7 . -eRBaHebI k! 3 nig k 1 3 naM[ k 1k! 3 3 b¤ k 1 ! 3 . 55> bgðaj[eXIjfa f CaGnuKmn_xYb GnuKmn_ f manEdnkMNt; D 0 , cMeBaH RKb; x D enaH x 2a D eRBaH a 0 eyIgman ³ f x a 12 f x f x naM[ f x 2a f x a a n
2
100 1 cd cd n 2
.
k
2
k 1 k ! 3 3k k! 3k ehIy k 1 7 3
1
2
2
101cd n 2 100 n 10n 10 cd 101
684
f x a f x a
1 2
1 1 2 2
2
1 2 f x f x 2
2 f x f x
2
1 1 2 1 2 2 f x f x f x f x f x f x 2 2 4
2 3 42n 8 2ad i 42n 9 3bd ii
cgCaRbB½n§
21n 4 ad 14n 3 bd
1 1 1 2 2 2 f x f x f x f x f x f x 2 2 4 1 1 2 f x f x 2 4 1 1 f x 2 2 1 1 f x 2 2 f x
edayyk ii i eyIg)an ³
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n
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2
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>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 1 1 1 1 nn 1n 2 2 nn 1 n 1n 2
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8
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2012
2011
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690
2012
2011
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73> edaHRsaysmIkar ³ eyIgman 2011 x -tag A 2011 naM[ log A log 2011 log 2012 x
b¤ x
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i
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1
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edayyk
dUcenH eXIjfa
83> RsaybBa¢ak;fa H / G nig O CabIcMNucrt;Rtg;Kña
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naM[ eday
1 1 1 a 3 b 3 c 3 ax 2 by 2 cz 2 x y z
¬!¦ ¬@¦ ¬#¦
693
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84> bgðajfa P 0 eyIgman ³
dUcenH
P a b 5a 9b 6ab 30a 45 2
2
2
2
2
a 6a 9 b 5 a 3 b 5 2
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naM[ cMnYn 51 Eckdac;ngw 9 39
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51
man 2
0
2
n
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3
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ebI n CacMnYnKt;ess ebI n CacMnYnKt;KU
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511
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3
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2n
n
1 10 2 1 n
n 1 1 10 2 1 99 1 1000...00 1 99
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39
51
n 1
39
39
39 51 13 3 13 51 351 13 51 3 49 3 2 13 51 3 49 9 51
694
8
n 1
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1 10 4 1 108 1 ... 10 2 1
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n
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>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> n
10 1 10 1
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39
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102 1 104 1 108 1 ... 102 1
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51
1 dgénelx 0
man 2 dgénelx 0
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manTRmg; 9
3951 5139
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A 101 10001 100000001 ...1000...001
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86> KNna A ³ eyIgman
a 2 b 2 6ab 2 9b 2 5a 2 30a 45
b 2 a 2 6a 9 5 a 2 6a 9
51
n
87> KNnatémø n ³ eyIgman
-cMeBaH A 0 eyIg)an³ C A 0 0 ehIy B A
3 9 15 6n 3 300 ... 2011 2011 2011 2011 2011 3 9 15 ... 6n 3 300 2011 2011 3 9 15 ... 6n 3 300
2
naM[
1 3 5 ... 2n 1 100
2
n 2 10 2
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n 10
k>
eyIg)an³
x2 2x y 2 y 2 y z z 2 2z x
x>
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x 1 1 y x 2x 1 1 y 2 2 y 2 y 1 1 z y 1 1 z z 2 2z 1 1 x z 12 1 x 1 x 2 1 y 2 1 y 1 z 1 z 2 1 x
edaytag A 1 x eyIg)an³
, B 1 y , C 1 z
-tam 1 : A C -tam 2 : C B -tam 3 : B A naM[ A A 2
A4 B
2
A8 A
A A7 1 0
enaH
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b¤ A
7
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A 1 x 1 1 x x 0 B 1 y 1 1 y y 0 C 1 z 1 1 z z 0 A, B, C x, y, z
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b¤
x 1 y 1 z 1
.
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1 1 1 7 2 6 2 12 b 4 b b 12 12 b 12 6 1 & 3 1 1 1 5 2 4 2 12 a 6 a a 12 12 a 12 4 1 & 3 1 1 7 5 2 12 2 12 c 2 c c 12 12 c 12 12
-edaybUkGgÁnwgGgÁénsmIkar
eyIg)an ³
-edaybUkGgÁnwgGgÁénsmIkar
eyIg)an ³
x 4 x 3 x 2 x 1 4 2007 2008 2009 2010 x4 x 3 x2 x 1 1 1 1 1 0 2007 2008 2009 2010 x4 x3 x2 x 1 1 1 1 1 0 2007 2008 2009 2010
A8 A 0
89> edaHRsaysmIkar eyIgmansmIkar
2
8
x 1 y 1 z 1
dUcenH RbB½n§smIkarmanKUcemøIy a, b, c 6,4,2 .
1 2 3
A2 C 2 C B B2 A
04 0
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2
2
4
dUcenH KUcemøIyénRbB½n§KW
88> edaHRsayRbB½n§smIkar ³ x2 2x y 2 y 2y z z 2 2z x
2
-ykKUcemøIyéntémø
2
4
A 1 x 0 1 x B 1 y 0 1 y C 1 z 0 1 z
-cMeBaH A 1 eyIg)an³ C A 1 1 ehIy B A
31 3 5 ... 2n 1 3 100
GñkRtUvcaMfa 1 3 5 ... 2n 1 n dUcenH témøKNna)anKW n 10 .
2
1 0 A7 1 A 1
x 4 2007 x 3 2008 x 2 2009 x 1 2010 0 2007 2008 2009 2010 x 2011 x 2011 x 2011 x 2011 0 2007 2008 2009 2010
695
x 2011
1 1 1 1 0 2007 2008 2009 2010
naM[ ehIy
x 2011 0 x 2011
dUcenH 4
1 1 1 1 0 2007 2008 2009 2010
tag n n2
8
4
4
4
8
2 1
8
8
4
8
n 2 24 8 2
8
24 8 2
8
2
4
2
4
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2 1
2
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2 1
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8
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4
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2
8
8
eRBaH n 0 BMuEmn eT
8 4
8
x 1 x x 1 x 1
x 1 x x 1
n
1 9x
8
x 1
x 2
x x 1
x x 1 x 2 x x 1 x 1
x x 1 x x 1 1 ,0 x 1 x 1 x 1
man n tYénelx ! 10 n 9 x 1
¬sUmKit¦
man n tYénelx ! man n tYénelx ! 222...220 2
man n 1 tYénelx @ man n tYénelx @
2 1 2
4
.
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4
8
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8
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dUcenH eRkayBIsRmYl B x 1 1 .
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24 8 2 2 2 2 1 2 1
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2
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enaH
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91> sRmYlkenSam
dUcenH smIkarmanb¤sEtmYyKt;KW x 2011 . 90> RsaybB¢aak;fa
4
2 1
nig C 444 ...444
444 ..44 10 n 444 ..44
man 2n tYénelx $ man n tYénelx $ man n tYénelx $
2 1
1 2 2 1
4 x9 x 1 4 x 36 x 2 4 x 4 x 36 x 2 8 x
696
naM[
A B C 7
b a c b 2 ab bc 5
8 x 20 x 2 36 x 2 8 x 7
c a b c 2 ac bc 6
edayyk 5 6 7 : eyIg)an
36 x 2 36 x 9 6 x 2 6 x 3 3 2
2
6 x 3 666...66 3 666...69 2
2
2
man n tYénelx ^ man n 1 tYénelx ^
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a 2 b 2 c 2 2ab bc ac
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ab bc ac a 2 b 2 c 2 2ab bc ac
dUcenH ab bc ac a
2
b 2 c 2 2ab bc ac
95> KNna E ³ eyIgman
2011 2011 2011 RkLaépÞRtIekaN E b a c b a c a b c b -tag x CaPaKryénkm
1 S bh 2
2
2
2
2
2
2
2
2
2
1 1.11 x
1 1.1 1.1x
1.1 1 0.1 x 0.09091 9.09 % 1.1 1.1
dUcenH km
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2
a b
2
2
E
2
2
2
2
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c
2
2
2
a b c
a 2 2ab b 2 c 2
2
3
2011 2011 2011 2 2 2 2 2 2 b a c b a c a b2 c2 2
eRBaH a b c 0 nig abc 0 dUcenH eRkayBIKNnaeXIjfa E 0 697
2
2
2011 2011 2011 2bc 2ac 2ab 2011 2011 2011 2bc 2ac 2ab 2011a 2011b 2011c 2abc abc 2011 0 2abc
2
2
E
2a 2 b 2 c 2 2ab bc ac
a b c a 2 ab ac 4
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2
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2
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.
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2
.
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2
2
2
2
2
2
2010 2012 2011 2 1 2011 4 1 2011 8 1 2011 16 1 2011 32 1 2
2
2
2
4
8
2
2
2
16
4
4
8
16
8
16
2011 12011 12011 1 2011 12011 1 2011 1
20114 1 20114 1 20118 1 201116 1 8
8
16
5a 10 a 2
16
16
32
dUcenH eRkayBIkarbgðaj eXIjfa
2010 2012 2011 2 1 2011 4 1 2011 8 1 2011 16 1 2011 32 1
ehIy a CacMnYnKt;viCm¢ an enaH a 1 nig a 2 cMeBaH a 1 enaHRtIekaNmanrgVas;RCug 3, 4 , 5 cMeBaH a 2 enaHRtIekaNmanrgVas;RCug 6, 8 , 10 dUcenH rgVas;RCugénRtIekaNEkgtUcCag b¤esµI!0 KW³ 3, 4, 5 nig 6, 8, 10 . 99> KNna A B CaGnuKmn_nwg C BinitülkçN³RCugrbs;RtIekaN -tamBItaK½r eXIjfa³ 3 4 5 mann½yfavaCaRtIekaNEkg -ehIyvaCaRtIekaNCarwkkñúgrgVg;enaHeyIgTajfa RbEvgRCugEvgCageKmanrgVas;esµI 5 CaGgát;[p©it -eyIgKUsrUb)abdUcxageRkam³ rgVg;enHman kaM R 52
97> bMPøWfa 2AM AB AC A -yk N kNþal AC N 1 naM[ AN AC 2 C -bRmab; AM Caemdüan B M naM[ M kNþal BC Taj)an MN Ca)atmFüménRtIekaN ABC 2 ehIy MN AB 2 -tamc,ab;vismPaBkñúgRtIekaN AMN eK)an ³ AM MN AN 3 C A 5 4 -edayyk 1 & 2 CMnYykñúg 3 AC 3 2 AM AB AC eyIg)an AM AB B 2 2 dUcenH eXIjfa 2AM AB AC R)akdEmn . -RkLaépÞRtIekaN S 1 3 4 6 ÉktaépÞ 2 -RkLaépÞrgVg; S R 52 254 ÉktaépÞ 98> rkRKb;rgVas;RCugénRtIekaNEkg -eday a CacMnYnKt;viC¢man -RkLaépÞEpñk C KW C R2 S2 258 ÉktaépÞ enaHeyIg)anrgVas;RCug x x a x 2a naM[ A B S S C -edayvaCaRtIekaNEkg tamBItaK½r 25 25 25 6 6 C 6 eyIg)an x x a x 2a 4 8 8 x x 2ax a x 4ax 4a dUcenH A B C 6 . 2
2
2
2
2
2
2
2
2
2
2
2
2
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x 2 2ax 3a 2 0 698
100 KNnaRbEvg)atmFümenAcenøaHGgát;RTUg x 4S 1 1 2 2 3 ... 1000 1000 1999 2001 1 3 3 5 5 -tag P nig Q CacMNuc A 6 cm B 1 2 2 3 999 1000 1000 4S 1 ... RbsBVrvagGgát;RTUg nig M P x Q N 3 3 5 5 1999 1999 2001 C )atmFüm eyIg)an ³ D 10 cm man 999 tYénKUplbUk - MP Ca)atmFümén DAB eRBaH MP kat;tamcMNuc 4S 1 1 1 1 ... 1 1000 2001 cMNuckNþal AD ehIyRsbnwg AB ¬)atmFümctu>Bñay¦ manelx! cMnYn 999 tY 6 cm naM[ MP AB 3 cm 1000 2001000 1000 2002000 2 2 4S 1000 2001 2001 2001 - MQ Ca)atmFümén ACD eRBaH MQ kat;tamcMNuc 4 500500 500500 cMNuckNþal AD ehIyRsbnwg CD ¬)atmFümctu>Bñay¦ S 4 2001 2001 10 cm naM[ MQ CD 5 cm dUcenH plbUkKNna)anKW S 500500 . 2 2 2001 -eyIg)an x PQ MQ MP 5cm 3cm 2 cm 102 RsaybBa¢ak;fa ³ dUcenH RbEvgcenøaHGgát;RTUgKW x 2 cm . 101 KNnaplbUk S 11 3 32 5 53 7 ... 19991000 2001 6 6 6 ... 6 30 30 30 ... 30 9 eyIgBinitüemIlGgÁTI! dUcbgðajxageRkam ³ -Binitü ragTUeTAéntYnImYy²rbs; S -cMeBaH 6 6 6 ... 6 6 6 6 ... 9 n 4n 2n 12n 1 42n 12n 1 Et 6 6 6 ... 9 3 ¬beBa©jBIr:aDIkal;¦ 2n n 2n n 42n 12n 1 naM[)an ³ 6 6 6 ... 6 3 (i) n 2n 1 n 2n 1 -cMeBaH 42n 12n 1 1 n2n 1 n2n 1 Et 30 30 30 ... 36 6 ¬beBa©jBIr:aDIkal;¦ 4 2n 12n 1 2n 12n 1 naM[)an ³ 1 n n 4 2n 1 2n 1 -edaybUkGgÁnigGgÁ ³ (i) (ii) eyIg)an ³ eyIg[témø n 1, 2, 3,...,1000 eyIg)an ³ (i) 6 6 6 ... 6 3 1 1 1 1 ebI n 1³ 30 30 30 ... 30 6 (ii) 1 3 4 1 3 2 1 2 2 ebI n 2 ³ 6 6 6 ... 6 30 30 30 ... 30 9 35 4 3 5 dUcenH eyIgGacRsaybBa¢ak;)anfa ³ ebI n 3 ³ 53 7 14 15 17 2
2
2
2
2
2
2
2
30 30 30 ... 30 30 30 30 ... 36
30 30 30 ... 30 6
(ii)
2
2
2
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
ebI n 1000³
1000 2 1 1000 1000 1999 2001 4 1999 2001
6 6 6 ... 6 30 30 30 ... 30 9
1 1 1 2 2 3 1000 1000 S ... 4 1 3 3 5 5 1999 2001
bBa¢ak; ³ karedaHRsayxagelIKWman n r:aDIkal;dUcKña. ¬GñkGacsegçbkaredaHRsay[xøICagenHk¾)an¦
699
4 S 1 t t 2 1
103 k> cUrKNnaplbUk
t 2 2t 1
S 1 2 3 2 3 4 3 4 5 ... nn 1n 2
eyIgBinitütYnImYy²én S manragCa Et k k 1k 2
k k 1k 2
1 k k 1k 2 4 4 1 k k 1k 2 4 k k 4 1 k k 1k 2 k 3 k 1 4 1 k 3k k 1k 2 k 1k k 1k 2 4 1 k k 1k 2 k 3 k 1k k 1k 2 4
edaytémø k 1, 2 , 3, ... , n eyIg)an ³ 1 1 2 3 4 1 2 3 4 0 1 2 3 2 3 4 1 2 3 4 5 1 2 3 4 4 1 3 4 5 3 4 5 6 2 3 4 5 4 ................................................... 1 nn 1n 2 nn 1n 2 n 3 4 n 1nn 1n 2 1 2 3 2 3 4 3 4 5 ... nn 1n 2
t 1
2
b¤Gacsresr)an³ 4S 1 n 3n1 dUcenH 4S 1 CakaerR)akd . 104 RsaybBa¢ak;fa ³
x1 x2 x3 ... x9 x1 x2 x3 ... x12 7 x3 x 6 x9 x4 x8 x12
eyIgman ³ 0 x x x ... x naM[ x x x x x x 3x 1
1
x> RsaybBa¢ak;fa 4S 1 CakaerR)akd eyIgman S 14 nn 1n 2n 3 naM[ 4S 1 4 14 nn 1n 2n 3 1 nn 3n 1n 2 1
ebItag
n 2 3n n 2 3n 2 1 t n 2 3n
eyIggayRsYlKuN enaH
2
2
3
3
12
3
3
3
3
x4 x5 x6 x6 x6 x6 3x6
x7 x8 x9 x9 x9 x9 3x9
1 2 3
edayyk 1 + 2 + 3 eyIg)an ³ x1 x 2 x3 3 x3 x 4 x5 x 6 3 x 6 x x x 3x 9 7 8 9
x1 x 2 x3 ... x9 3x3 x6 x9
b¤sresr)an ³ x xx xx ...x x ehIy x x x x x x x 1
1
2
3
2
3
9
3
6
9
4
4
4
4
3
i
x4 4x4 4
x5 x6 x7 x8 x8 x8 x8 x8 4x8 5 x9 x10 x11 x12 x12 x12 x12 x12 4x12 6
edayyk 4 + 5 + 6 eyIg)an ³ x1 x 2 x3 x 4 4 x 4 x 5 x 6 x 7 x8 4 x8 x x x x 4x 12 9 10 11 12
1 nn 1n 2 n 3 4
dUcenH eRkayBIKNna S 14 nn 1n 2n 3
2
2
x1 x 2 x3 ... x12 4x 4 x8 x12 x1 x2 x3 ... x12 4 x 4 x8 x12
ii b¤sresr)an ³ CalT§plRKan;Etyk i ii CakareRsc
dUcenH
x1 x 2 x3 ... x9 3 x3 x 6 x9 x1 x 2 x3 ... x12 4 x 4 x8 x12
x1 x2 x3 ... x9 x1 x2 x3 ... x12 7 x3 x 6 x9 x4 x8 x12
700
105 rktémøFMbMputénplKuN xy x 3 xy y 60 eyIgman GacEfmfy x 2xy y 5xy 60
A 111...11222...22
2
2
111...1110 2011 2111...11
2
2
111...11 10 2011 2
x y 2 5 xy 60 2 60 x y xy
Edlman elx#cMnYn @0!! tY A 111 ... 11222 ... 22 333 ... 33 333 ... 33 1
EdlmancMnYntYénelx! /@nigelx# esµIKñaKW 2011tY .
F 1
M 2 D
1 2 N
dUcenH
106 rkplbUkrgVas;mMu A, B , C , D , E nig F EfmcMNuc M nig N Rtg;RbsBVdUcrUb ³ A
5 x y 2 xy 12 5 2 x y 0 0 5
C
1 999...99 10 2011 2 9 1 999...99 10 2011 1 3 9 1 999...99999...99 3 9 1 1 999...99 999...99 3 3 3 333...33333...33 1
eday x y naM[ xy 12 0 12 CatémøFMbMput cMeBaH x y dUcenH témøplKuNFMbMputénplKuN xy 12 . 2
108 rktémøtUcbMputénkenSam H ³ ebI x , x Cab¤smIkar 2011 x t 2011 x 2011 0 tamrUbmnþ plbUkplKuNb¤s eyIg)an ³
B
E
2
1
2
BinitüplbUkmMukñúgén ³ -RtIekaN ANF ³ A N F 180 ¬!¦ b t 2011 x x a 2011 -RtIekaN BMC ³ B M C 180 ¬@¦ c 2011 xx 1 -ctuekaN DMNE ³ D M N E 360 ¬#¦ a 2011 edaybUk (1)+(2)+(3) eyIg)anplbUkmMu ³ eyIgman ³ H x x 4 x 2 x o
1
1
o
2
1
o
2
1 2
2
2
2
A B C D E F M 1 M 2 N1 N 2 720
EtmMu M M 180 , N N 180 naM[ A B C D E F 180 180 720 b¤ A B C D E F 360 dUcenH plbUkmMu A B C D E F 360 . o
1
2
o
1
2
o
o
1
o
H x 2 x1
2
o
o
o
107 RsaybBa¢ak;fa A CaplKuNénBIrcMnYnKt;tKña eyIgman A 111...11222...22 man @0!!tYdUcKña ¬BinitüedayykcitþTukdak;BImYycMNuceTAmYycMNuc¦
x x 2 x 2 x1 4 1 x1 x 2 2
1 1 x1 x2 2
2
2 1 1 x 2 x1 x 2 x 1 4 2 x1 x 2
701
2
1 2 2 1 x 2 x1 4 x 2 x1 2 x1 x 2 2 1 1 2 x 2 x1 1 4 2 x1 x 2 2 2
2 1
x1 x 2
eyIgGacbMEbk A CaTRmg;gaydUcxageRkam ³
1
2
2 1 1 4 x1 x 2 1 4 2 x1 x 2
2
edayCMnYstémøénplbUknig plKuN eyIg)an ³
eday xx 1 CaBhuFadWeRkTI@ enaH sMNl;CaBhuFa dWeRkTI! manragCa rx ax b . -]bmafa qx CaplEckén px nwg xx 1 eyIg)an ³ px xx 1 qx ax b -cMeBaH p0 1 nig p1 2 eyIg)anRbB½n§smIkar³
2 t 2011 2 1 1 H 4 1 1 4 2 1 2011
t 2011 2 4 10 2011
p0 1 00 1qx a 0 b 1 p1 2 11 1qx a 1 b 2 b 1 b 1 a b 2 a 1 r x ax b x 1
t 2011 10 40 2011 2
2011 eday t 2011
2
0
naM[
t 2011 10 40 40 2011 2
dUcenH témøtUcbMputén H 40 cMeBaH t 2011 . 109 KNnatémøelxén A 3 eyIgman ³ A 3 5 3 5 3...
5 3 5 3...
enaHsMNl;KW
dUcenH sMNl;énviFIEckKW rx x 1 . 111 RsaybBa¢ak; 5 eyIgman 5 5 2011
2011
52012 52013
2012
52013
5 2011 1 5 5 2
A 2 3 5 3 5 3...
Eckdac;nwg 31
31 5 2011
A 9 5 3 5 3 5 3... 4
dUcenH
A 45 3 5 3 5 3... 4
A 4 45 A 4
3
3
3
3
110 rksMNl;énviFIEckrvag
45
p x
. nig xx 1
¬GñkRtUvcaMfa ³ sMNl;énviFIEck f x nig x a KW r f a Edl f x CaBhuFa nig r f a CasMNl; .¦
bRmab;
Ecknwg x )ansMNl; 1 mann½yfa p0 1 ehIy px Ecknwg x 1)ansMNl; 2 mann½yfa p1 2 -dWeRkénsMNl;Eck RtUvtUcCagdWeRkéntYEck p x
Eckdac;nwg 31 .
112 rkelxéntYcugeRkaybg¥s;éncMnYn 2 BinitüsV½yKuNén @ dUcxageRkam ³ 2 2 2 2 2 2 2 4 2 4 2 4 2 8 2 8 2 8 >>> 2 6 2 6 2 6 eXIjfa elxcugsV½yKuNén @ manxYbesµI$vileTAmk ehIy 3 81 naM[ 2 2 2 2 2 2 2 6 2 2 34
Gacsresr A 45 A 0 AA 45 0 naM[)an A 0 minykeRBaH A 0 A 45 0 A 45 yk dUcenH eRkayBIKNna A
52011 52012 52013
1
5
9
2
6
10
3
7
11
4
8
12
4
81
80
420
80
dUcenH cMnYn 2 manelx 2 enAxagcugeKbg¥s; . 34
smÁal; ³ eKkMNt;sresr 2 8 mann½yfa elxxagcugén 2 KWelx 8 . 11
11
702
113 rkelxéntYcugeRkaybg¥s;éncMnYn123456789 -ebItémø x GviCm¢ aneyIg)an 10 x 100 elxxagcugéncMnYn 123456789 KWGaRs½yEt elIsV½yKuNelxxagcugéncMnYnenHb:ueNÑaH mann½yfa dUcenH témøéncMnnY BitbBa¢ak;)anKW ³ . 10 x 100 b¤ 10 x 100 elxxagcug123456789 =elxxagcug 9 -BinitüelxxagcugsV½yKuNénelx ( ³ 116 k>eRbóbeFobcMnYn 1 2000 nig 2001 2 2000 9 9 , 9 1 , 9 9 , 9 1 , ... Binitü ³ 2001 2 2000 9 ebI n ess naM[ 9 1 ebI n KU 1 2000 2 2000 1 2 1 2000 2000 2 2000 eyIg)an 9 9 eRBaH @0!! CacMnYness 1 2000 2011
2011
2011
2011
2
1
2
3
4
2
2
n
2
2
2
2011
2
dUcenH elxxagcugbg¥s;én 123456789 KW 9 . 2011
114 RsaybBa¢ak;facMnYn A CacMnYnKt;viC¢man eyIgman A 13 2 1 2 1 3
3
1 A3 3
1 3
3
1 3
3
3
3
3 3
3
3
3
2 1
3
3
3
2 1
2
2
2
2
2
2
2
2000 2000 2001 2001 2001 2000 2000 2001 2001 2001 2001
2 2 1 2 3 3 2 3 3 2 1
2 2 1 1 3 2 3 2 2
2
dUcenH eRkayBIKNnaeXIjfa A 2001 . 117 RsaybBa¢ak;fa a x y eyIgman ³ x x y y x y a
2 3 2 3 2 1 3 2 3 2
A 1
2
2000 2000 A 20012 2 2000 2001 2001
3 2 2 1 3 2 3 3 2 3 3 2 1
3
2
2
3
2 3
3 2 1 1
dUcenH
dUcenH 1 2000 2001 2 2000 . 2000 x> KNna A 1 2000 2000 2001 2001 2000 eyIgman ³ A 1 2000 2000 2001 2001 eday 1 2000 2001 2 2000 eyIg)an ³
2
CacMnYnKt;viC¢man .
115 bBa¢ak;témøéncMnYnBit x edayeRbIsBaØavismPaB -bRmab; ³ x CacMnYnBit mann½yfa x GacCacMnYnviC¢man b¤GviC¢man -smµtikmµ ³ x CacMnYnEdlmanEpñkKt;manelxBIrxÞg; mann½yfa 10 x 100 ¬TRmg;enHeFVI[ x CacMnYn EdlmanEpñkKt;manelxBIrxÞg;Canic©¦ 703
a
4
3
2
x2 3 x4 y2 3
x6 3 x4 y2
3
x4
3 x2
3
3
3
x2 3
3
x2 3
2
2
3
4
y6 3 x2 y4
3
y y y y y x y y
2
2
2 3
y2 3 x2 y4
x x
2 3
3 y2 2
3
3
2
2
3
2
y4
3
2
3
3
3
3
2
2
2
3 x2 3 x2
cMeBaH a x y eyIgGacTaj)anCa ³ a x b¤ a x tamrUbmnþ ³ a a eyIg)an ³ a x y 2
3
2
3
3
2 3
2
3
2
y
2
3
2
3 y2
3
2
dUcenH cMnYn A CakaerR)akdéncMnYnKt; . 120 sRmYlkenSam E ³ eyIgman ³
3
m n
m
n
3
2 3
E
2 3
dUcenH eyIgRsay)anfa a
2 3
2 3
x y
2 3
.
118 eRbóbeFobcMnYn 2001 2002 nig 2 2002 eyIgdwgfa 2001 2002 EfmGgÁTaMgBIrénvismPaBnwg 2002 eyIg)an ³
p q 1
2
pq pq
p q pq p q 2
p q
p 2 pq q 2 pq pq p q
2
1 1 3 q p q p 2
1
1 1 p q
3
2
1
2001 2002 2 2002
2
2
p q
2 p q 2 pq 2 pq p q p 2 p q 2 pq 1 2 pq pq p q p 1 2 2 2 2 pq p q pq p q pq
2001 2002 2002 2002
smmUl
1
2
q
2 q
2
2
dUcenH eRbóbeFo)an 2001 2002 2 2002 . 1 pq 119 RsaybBa¢ak;fa A CakaerR)akdéncMnYnKt; dUcenH sRmYl)an E pq1 . eyIgman A 111...111 444...44 1 n énelx$
2n énelx!
1 4 A 999...999 999...99 1 9 9 2n énelx(
121 sRmYlkenSam E ³ eyIgman ³ A 3 9 125 27
n énelx(
3
1 2n 4 10 1 10 n 1 1 9 9 1 1 4 4 10 2 n 10 n 1 9 9 9 9 1 4 4 10 2 n 10 n 9 9 9 1 10 2 n 4 10 n 4 9
A
tamrUbmnþ ³
2 1 1 2 10 n 2 10 n 1 3 3 3 2
10 n 1 999...99 1 1 3 3
a 3 b 3 3aba b
125 125 3 9 A3 3 9 27 27
2
125 125 3 9 A 33 3 9 27 27
2 2
125 A 6 33 32 9 27
333...33 1 333...34 2 2
n énelx#
3 9
6 33
n 1 énelx# 704
125 27
¬RsedoglMhat;elx 10 TMB½r!¦ a b 3 a 3 3a 2b 3ab 2 b 3
eyIg)an ³
3
125 A 27
pq
pq
pq
naM[)an ³
A3 6 5 A
dUcenH Rsay)anfa
A 5A 6 0 3
A3 A 6 A 6 0
3 3
2 1 3
1 3 2 3 4 9 9 9
123 KNnatémøelxénkenSam S ³ eyIgman ³ S 6 6 6 6 ... , naM[ S 6 6 6 6 6 ...
A A 2 1 6 A 1 0
A 1A 2 A 6 0
naM[ A 1 0 A 1 ehIy A A 6 0 eRBaH a 11240 0 dUcenH eRkayBIsRmYlEbbbec©keTs A 1 .
.
S 0
2
2
122 RsaybBa¢ak;fa tag A 2 1
3 3
S 2 6 6 6 6 6 ... S2 6 S
man
1 2 4 2 1 3 3 3 9 9 9
3 3
A3 3 2 1
3
A3
ehIy
2 1 1 3 2 3 22
1
3
2 3 22
3
1 3 2 3 4 9 9 9 1 2 4 B 3 3 3 9 9 9 1 3 1 3 2 3 4 9 1 3 1 3 2 1 3 2 3 4 3 9 1 2 1 3 13 3 2 3 3 9 1 2 3 B 3 9 1 3 2 27 3 B3 3 2 1 3 1 3 2 3 1 3 2 2 3 2 3 9 1 3 2 3 1 3 33 2 33 4 1 3 2 3 4 B3
eXIjfa A
3
B3
1 5 S 2 1 21 1 5 S2 3 21
eyIg)an
cMeBaH S 2 minyk eRBaH S 0 dUcenH témøelxKNna)anKW S 3 . 124 KNnatémøén ab bc ³ eyIgman ba bc 2011 Edl a 0 , b 0 Taj)an ba 2011 b 2011a ehIy bc 2011 c 2011b naM[ ab bc 2011aa b2011b 2011a ab b 2011 1
2 13 1 1 3 2 3 4 1 3 2 3 4 3
S2 S 6 0 2 b 2 4ac 1 41 6 25 5 2
dUcenH témøKNna)anKW ab bc 2011 . 125 edaHRsaysmIkar x 1 x 5 x 7 x 11 4 eyIgman 1991 1987 1999 1981
A B
705
x 1 x5 x7 x 11 1 1 1 1 0 1991 1987 1999 1981
x 1 1991 x 5 1987 x 7 1999 x 11 1981 0 1991 1987 1999 1981
x 1992 x 1992 x 1992 x 1992 0 1991 1987 1999 1981 x 1992 1 1 1 1 0 1991 1987 1999 1981
naM[ x 1992 0 x 1992 dUcenH smIkarmanb¤s x 1992 . 126 edaHRsaysmIkar eyIgmansmIkar x 6 x 2 3 x Taj)an x 2 3 x 6 x 0 x 2 3 x 6 x 0 naM[ x 0 x 0 x 2 3 x 6 0 smIkarenHman S 2 3 , P 2 3 6 eyIg)an 2 nig 3 Cab¤énsmIkar dUcenH smIkarmanb¤sbIKW x 0 , 2 , 3 . 127 edaHRsaysmIkar log log log x 2 eyIgman log log log x 2 eyIg)an log log log x 2 log 2 2000
1998
2000
90 A 100 1000 10000 ... 10 n1 10n n 9 A 10 100 1000 ... 10 10 81A 10 n1 9n 10 10 n1 9n 10 A 81
b¤Gacsresr ³
1999
1999
1998
2
dUcenH plbUkKW A 10
n 1
1998
1998 2
2
2
3
4
2
3
4
3
4
x
eyIgman ³ x Gacsresr x naM[)an
xx
x
x 2
x
x 0 x 1 x x 0 2
x 1
x x x2 0 x x 2 2 4 2 2 4 x x 4 x x 0 x4 x 0
b¤ naM[ x 0 , x 4 cMeBaH x 0 minyk eRBaH x 0
log 4 x 81log 4 4 x 481
¬rUbmnþelakarIt
x
x
log 4 x 34
x 481
v x
log 3 log 4 x 4 log 3 3
dUcenH smIkarmanb¤s
x
u x
eyIg)an x 1 0
log 3 log 4 x 4
.
129 edaHRsaysmIkar k> x x GñkRtUvcaM rUbmnþ ³ f x f x naM[)an ff xx 10ux vx 0
2
log 2 log 3 log 4 x log 2 2 2
9n 10 81
.
log a x k x a k
dUcenH smIkarmanb¤s x 1 x 4 .
¦
128 KNnaplbUk A 111111 ...111...11 x> x x ¬lMnaMdUcsMNYr k>¦ eyIgman ³ A 111111 ...111...11 tYcugmanelx eyIg)an ³ x x x x ! cMnYn n dg eyIgGacsresr ³ x 0 naM [ )an 9 A 9 99 999 ... 999...99 ¬tYcugman n dgelx(¦ x 1x x 0 9 A 10 1 100 1 1000 1 ... 10 1 eyIg)an x 1 0 x 1 x x
xx
xx
xx
2
x2
xx
x
n
9 A 10 100 1000 ... 10n n
edayKuN ¬ ¦nwg !0 eyIg)an ³
90 A 100 1000 10000 ... 10 n1 10 n
eyIgyk ¬ ¦-¬ ¦ eyIg)an ³
x2 x x 0 x2 x x x 2
dUcenH smIkarmanb¤s x 1 b¤
706
x2
.
130 KNna a b c eyIgman
bRmab; ³ km
a b 2 2ac 29 2 b c 2ab 18 c a 2 2bc 25
edaybUkGgÁngi GgÁ eyIg)an ³ a 2 b 2 c 2 2ab 2bc 2ac a b c 72
a b c 2 a b c 72 a b c 2 a b c 72 0
tag t a b c eyIg)an ³ smIkarfµIgaysresr niggayKit t man 1 288 289 17
sisSsrubkñúgfñak; ³ x y 28 b¤ 1.68x 1.68y 28 1.68
2
edayyk (2)-(1) eyIg)an ³
t 72 0
1.68x 1.68 y 47.04 1.68x 1.60 y 46.48 0.56 7 0.08 y 0.56 y 0.08
1 17 t 2 9 a b c 9 a b c 8 t 1 17 8 2
dUcenH cMnYnsisSRsIenAkñgú fñak;KW 7 nak; .
dUcenH KNna)an a b c 9 , 8 . 131 KNnaplbUk S 1 2 3 ... 2011 eyIgman S 1 2 3 ... 2011 ¬!¦ b¤srsesr S 2011 2010 2009 ...1 ¬@¦ edayyk (1)+(2) eyIg)an ³ S 1 2 3 ... 2011 S 2011 2010 2009 ... 1 2S 2012 2012 2012 ... 2012 2012 2011
Edlmanelx eyIg)an ³
cMnYn
2
1.68 x 1.68 y 47 .04
2
naM[
1
1.68 x 1.60 y 46 .48
dg
2S 2012 2011 2012 2011 S 1006 2011 2023066 2
133 KNnakenSam A eyIgman ³ A a b c a b c a b c a b c 2
edayeRbIrUbmnþ a
2
2
2
b 2 a b a b
2
eyIg)an ³
A a b c a b c a b c a b c a b c a b c a b c a b c A a b c a b c a b c a b c
a b c a b ca b c a b c A 2a 2b 2c 2b 2c 2a A 2a2b 2c 2b 2c 2a 4b 8ab
dUcenH kenSamEdlKNna)anKW
.
A 8ab
2011 2011 134 bgðajfa b 2011 c a c a b a b c eyIgman a b c 0 naM[ ³ 2
2
2
2
2
2
2
b c a b c a 2
dUcenH plbUkKNna)anKW S 2023066 .
2
b 2 2bc c 2 a 2 b 2 c 2 a 2 2bc
132 rkcMnYnsisSRsIenAkñúgfñak; tag x CacMMnYnsisSRbus y CacMMnYnsisSRsI Edl x 0 , y 0
1
c a b c a b 2
2
c 2 2ca a 2 b 2 c 2 a 2 b 2 2ca 707
2
2
2
0
136 cUrRsabMPWøfa Binitü ³
a b c a b c 2
2
a 2 2ab b 2 c 2 a 2 b 2 c 2 2ab
tamry³ (1) , (2) & (3) eyIg)an ³
3
4
4 5 2
.
135 kMNt;témø a nig b eyIgman ³ x px q
4
2
20 6 2 6
2
2
4
2
2
2
3 2 6 2
3 2 6 2
2
2 3 2
3 2
2
2
3 2 3 2 2 3
dUecñH eyIgbMPøW)anfa
x 4 p 2 x 2 q 2 2 px 3 2qx 2 2 pqx x 4 2 px 3 p 2 2q x 2 2 pqx q 2
4
49 20 6 4 49 20 6 2 3
.
2x 3 ax2 bx 1 x 2 px q
137 KNna E ³ 2
2
x 4 2 px 3 p 2 2q x 2 2 pqx q 2 2 p 2 p 1 p 2 2q a p 2 2q a 2 pq b 2 pq b 2 q 1 q 1 2 p 2q a 2 pq b
Taj)an ³
smmUl
a 3 b 2
naM[ 1 212 11 ba 2
dUcenH témøkMNt;)anKW
4 9 5 4 4 8
naM[ 121211ba p 1 & q 1
4 3 5 3 5
p 1 & q 1 2
2
E 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 2 3 5 2 3 5
x 4 2 x 3 ax 2 bx 1
-krNI
2
2
2
2
cMeBaH -krNI
5 6 5 2 6
5 2 6 52 6
2011 2011 2011 2 2 2 2 2 2 0 2 2 b c a c a b a b c
4
49 20 6 4 49 20 6
4 5 2 20 6 2 6
2
bRmab; ³ x eyIg)an
49 20 6 4 49 20 6 2 3
4 25 20 6 24 4 25 20 6 24
2011 2011 2011 2 2 2 2 2 2 2 b c a c a b a b2 c2 2011 2011 2011 2bc 2ca 2ab 2011a 2011b 2011c 2abc 2011a b c 2abc 0
dUcenH
4
dUcenH KNna)an
a 1 b 2
a 3 , b 2 a 1 , b 2
.
E 8
.
138 kMNt;témø m nig n eyIgman x 1 3 x m 4 x 3n x 4 RbPaKmann½ykalNa x 3 , x 4 tRmUvPaKEbg rYclubPaKEbgecal eyIg)an ³
708
eyIg)andwgehIyfa ebI a , b , c enaH
x 4 m x 3 n x 4 mx 3m n
a2 0 , b2 0 & c2 0
x mx n 3m 4 n 3m 4 x 1 m
2
2
edIm,I[RbPaK x mantémøeRcInrab;minGs; luHRtaEt PaKykesµIsUnü nigPaKEbgesµIsUnü dUcKña -eyIg)an ³ n 3m 4 0 n 3m 4 n 1 1 m 0 m 1 m 1
dUcenH témøkMNt;)anKW
m n 1
139 kMNt;témø a eyIgmanRbB½n§smIkar ³ 2xxyya3 edaybUkGgÁnigGgÁ ³
-manEtmYykrNIKt;EdleFVI[ a b c KWmann½yfa a 0 , b 0 & c 0 naM[ a 0 , b 0 & c 0 eyIg)an ³ D a 1 b 1 c 1
.
2010
2011
2010
1
2010
dUcenH KNna)an
0 1
2011
1
2011
2
0
2012
0 1
2012
1
2012
111 1
.
D 1
S 1 3 2 4 3 5 ... 2010 2012 12 2 2 32 ... 20102
1 2
1 3 2 4 3 5 ... 2010 2012 2 12 1 3 13 1 4 14 1 ... 2011 12011 1
2 2 1 3 2 1 4 2 1 ... 20112 1 2 3 4 ... 2011 2010 2
2
2
2
naM[
S 2 2 3 2 4 2 ... 20112 2010
tamsmIkar edIm,I[epÞógpÞat; x y eyIg)an ³ a 3 3 2a3 3
12 2 2 3 2 ... 20102
2 2 32 4 2 ... 20112 2010 12 2 2 32 ... 20102 20112 2010 12 20112 2011 20112011 1 2011 2010 4042110
a 3 2a 3
dUcenH KNna)anKW S 2011 2010 4042110 .
a6
.
a6
140 KNna D a 1 b 1 c 1 smµtikmµ a b c 0 nig ab bc ca 0 a b c 0 naM[)an 2010
2011
2
a 2 b 2 c 2 2ab 2ac 2bc 0 a b c 2ab ac bc 0 2
2
0 1
1 : y a x a a 3 2a 3 3 3
2
2
141 KNna S ³ eyIgman ³ Binitü
x y a 2 x y 3 a3 3x a 3 x 3
dUcenH témøkMNt;)anKW
2
2
a2 b2 c2 2 0 0 a2 b2 c2 0
2012
142 bgðajfa A Eckdac;nwg 5 ³ eyIgman ³ A 92011 82011 7 2011 62011 52011 12011 22011 32011 42011 A 9 2011 4 2011 8 2011 3 2011 7 2011 2 2011 6 2011 12011 5 2011
tamrUbmnþ ³
a n b n a b a n1b a n2 b 2 ... a 2b n2 ab n1
eyIgeXIjfa ³
9 2011 4 2011 9 4 9 2010 4 9 2009 4 2 ... 9 4 2010
5k1 , k1 9
709
2010
49
2009
4 ... 9 4 2
2010
8 2011 3 2011 8 38 2010 3 8 2009 3 2 ... 8 3 2010
144 bgðajfa
5k 2 , k 2 8 2010 3 8 2009 3 2 ... 8 3 2010
1 sin x 2 n sin x 2n x x x x Pn cos cos 2 cos 3 ... cos n 2 2 2 2 a a sin a 2 sin cos 2 2 x x x x a x , , , , ..., n1 2 4 8 2 x x sin x 2 sin 2 cos 2 sin x 2 sin x cos x 2 4 4 x x x sin 2 sin cos 8 8 4 ................................ sin x 2 sin x cos x 2 n 1 2n 2n x x x x x sin x 2 n sin n cos cos cos ... cos n 2 4 8 2 2
7 2011 2 2011 7 27 2010 2 7 2009 2 2 ... 7 2 2010
5k 3 , k 3 7 2010 2 7 2009 2 2 ... 7 2 2010
eyIgman tamrUbmnþ ³ eyIg[témø
6 2011 12011 6 16 2010 1 6 2009 12 ... 6 12010
5k 4 , k 4 6 2010 1 6 2009 12 ... 6 12010
5
5 5 2010
2011
5k 5 , k 5 5 2010
eyIg)an ³ A 5k 5k 1
2
5k 3 5k 4 5k 5
5k1 k 2 k3 k 4 k5
eXIjfa A CaBhuKuNén 5 mann½yfa A Eckdac;nwg 5 dUcenH bgðaj)anfa A Eckdac;nwg 5 . 143 k> KNna fog nigKNna gof eyIgman f : x f x px 2 g : x g x 4 x 3
naM[
fog f g x
f 4 x 3
p4 x 3 2 4 px 3 p 2
ehIy
x sin x 2 sin n Pn 2 1 sin x Pn n 2 sin x 2n
Taj)an
g px 2 4 px 2 3 4 px 8 3 4 px 5
dUcenH bgðaj)anfa
dUcenH KNna)an fog 4 px 3 p 2 ehIy gof 4 px 5 . x> KNna p edIm,I[ fog gof eyIgman fog 4 px 3 p 2 nig gof 4 px 5 naM[ fog gof enaH 4 px 3 p 2 4 px 5 3 p 5 2
Pn
1 sin x 2 n sin x 2n
.
145 edaHRsayRbB½n§smIkarkñúg x y 5440 1 eyIgman ³ PGCD x, y 8 2 2
2
-tam 2 ³ PGCD x, y 8 mann½yfa x 8a & y 8b Edl a , b CacMny Y bzmrvagKña ehIy a b eRBaH x y 5440 0 -tam 1 ³ x y 5440 2
2
2
2
x y x y 5440 8a 8b 8a 8b 5440
p 1
p 1
Pn
n
gof g f x
dUcenH témøKNna)anKW
Pn
.
8 2 a b a b 5440
710
y 3 43 y 46 x 2 1 x3 y 3 1 y4 x 2 43 x 45
a ba b 85
eday a , b CacMnYnKt;FmµCati nigedaHRsaykñúg b¤ 1 85 eyIg)anplKuNktþaén 85 5 17 eday a b dUcenH smIkarmanb¤s x 3, y 46 a b 1 a b 5 enaHeyIg)an ³ a b 85 i or a b 17 ii b¤ x 35 , y 4 . i
³
a b 1 a b 85 2a 86
147 bMPøWfacMnYn N 4n 3 eyIgman ³ N 4n 3 25
2
a 43
ii
³
4n 3 5 2 4n 3 54n 3 5 2
4n 24n 8 22n 1 4n 2
a 11
82n 1n 2
ehIy b 17 a 17 11 6 -cMeBaH a 43 , b 42 344 naM[ xy 88ab 88 43 42 336 -cMeBaH a 11 , b 6 88 naM[ xy 88ab 88 11 6 48 dUcenH RbB½n§smIkarmanKUcemøIyBIrKW x , y 344 , 336 b¤ x , y 88 , 48 .
eXIjfa N CaBhuKuNén 8 mann½yfa N Eckdac;nwg 8 dUcenH eXIjfa N 4n 3 25 Eckdac;nwg 8 . 2
148 bMPøWfacMnYn a , b , c nig d CatYén smamaRt smµtikmµ a b c d ad bc 2
2
2
edayedaHRsaykñúgsMNcMu MnYnKt;FmµCati mann½yfa y 3 nig x 2 CacMnYnKt;FmµCati ehIyplKuNcMnYnFmµCatiesµI 43 manEtBIrkrNIKW 431 eyIg)anktþanImYy²KW ³ 43 1 43
2
2
a 2 c 2 a 2 d 2 b 2 c 2 b 2 d 2 a 2 d 2 2adbc b 2 c 2 a 2 c 2 2adbc b 2 d 2 0
ac bd 2 0 ac bd 0
146 edaHRsaysmIkarkñúgsMNMucMnYKt;FmµCati eyIgman ³ xy 3x 2 y 37 xy 3x 2 y 37 xy 3x 2 y 6 37 6 x y 3 2 y 3 43 y 3x 2 43
Eckdac;nwg 8
2
ehIy b 85 a 85 43 42 a b 5 a b 17 2a 22
25
ac bd a c b d
dUcenH eyIg)antYénsmmaRt
.
a c b d
a2 x2 1 a2 b2 x2 y 2
149 bgðajfa smµtikmµ ³ ax by 0 naM[ x bya x bay yktémø x bay CMnYscUl eyIg)an ³ 2
2
2
2
2
2
2
2
711
151 bgðajfa eyIgman ³
b2 y 2 a a2 1 a2 b2 b2 y 2 2 y a2 b2 y 2 2 2 a 2 2a 2 2 1 2 2 b y a y a b a2 a2 b2 y 2 a2 1 a2 b2 a2 b2 y 2 a2 y 2 2
2
2
A 2011
a2 b2 1 a2 b2 a2 b2 a2 b2 1 a2 b2
1 1
2011
5 3
2011
5 3
2011
5 32 5 3
2011
5
2011
5
2011
5 5 1 2011 1 2011
dUcenH eXIjfa
Bit
a2 x2 1 a2 b2 x2 y 2
dUcenH eXIjfa
5 3 29 12 5 20 2 12 5 32
2
a2 b2 c2 K 2011
150 KNnatémøén smµtikmµ ³ 1a b1 1c 0
.
5 1
2
A 2011
CacMnYnKt; .
2
2
x2 4x 4 x2 2x 1 2 x 3 x
3 2
2
2
a 2 b 2 c 2 2ab 2bc 2ac 0
9 64 4
a 2 b 2 c 2 2ab bc ac 0
9 2 4
a2 b2 c2 2 0 0 a2 b2 c2 0
eRBaH ab bc ac 0 b c eyIg)an ³ K a 2011 dUcenH témøKNna)anKW
2
5 2 2 5 12
naM[ ab bc ac 0 Edl a 0 , b 0 , c 0 epÞógpÞat; smµtikmµbEnßm ³ a b c 0 3 3 4 4 2 a b c 0 naM[ 2
2
152 edaHRsaysmIkar eyIgman ³ x 4x 4 x 2x 1 elIkGgÁTaMgBIrCakaer enaHeyIg)an ³
bc ac ab 0 abc
2
5 3
2
a b y 2 1 2 a b b a2 y2 2
CacMnYnKt;
A
2
0 0 2011
K 0
.
2
3 3 2 1 2 2
9 3 1 4 9 2 4
1 1 4 4 1 1 2 2
dUenH smIkarmanb¤s
x
¬ebImin[lkçxNÐ RtUvepÞógpÞat;[eK¦ 712
3 2
.
153 KNnabIcMnYnKt;tKña 155 rktémøelxén 22 tag n CacnYnTI2 enaHcMnYnTI1KW n 1 nigTI3KW n 1 eyIgman 22 22 22 Edl n CacMnYnKt;viC¢man -tambRmab;RbFaneyIgsresr)an ³ dUcenH témøén 22 22
2 2011 2012 2 2011 2011 2 1 1 2011 2 1 3 2012
n 1nn 1 2 n 1 n n 1
130 21
2011
2012
2011
2012
2011
2012
2011
1 3
.
2 n 1nn 1 130 2 21 3n 2
156 KNnakenSam H ³ eyIgman H a baa c b cbb a c acc b kenSamenHmanPaKEbgrYm a ba cb c enaH
130 9n 2 nn 1n 1 21 4
H
a2 b2 c2 a b a c b c b a c a c b
a2 b2 c2 a b a c b c a b a c b c
14n
195 n 14 1 195n
n n2 1 2
2
2
14n 2 195n 14 0
man 195 naM[
2012
2
4 14 14
38025 784 38809 197 195 197 1 n1 2 14 14 195 197 n2 14 2 14
2
a 2 b c b 2 a c c 2 a b a b a c b c
a 2 b a 2 c ab 2 b 2 c ac 2 bc 2 a 2 ac ab bc b c
a 2 b a 2 c ab 2 b 2 c ac 2 bc 2 a 2 b a 2 c abc ac 2 ab 2 abc b 2 c bc 2 a 2 b a 2 c ab 2 b 2 c ac 2 bc 2 2 1 a b a 2 c ab 2 b 2 c ac 2 bc 2
ducenH eRkayBIKNna témø H 1 . 157 rkRkLaépÞFMbMputéncmáar tag a CaRbEvgTTwg nig b CaRbEvgbeNþay b bRmab; ³ kUneQIman 40 edIm KmøaténkUneQI 2 m naM[ cmáarmanbrimaRt P 2 40m Et P 2a b eyIg)an 2a b 2 40 m a b 40 m -cMeBaHctuekaNEkgEdlmanbrimaRtdUcKña KWctuekaN EkgmanTTwgesµIbeNþaymanépÞFMbMput . enaH a b a a 40 a 20m a
154 KNnatémøelxén xyz smµtikmµ ³ x yz 7 i nig xy edayyk 1 2 eyIg)an ³ 2
x 2 yz 3 7 3 2 9 xy 7 x 3 y 3 z 3 712
dUcenH témøén
3
3
xyz 7 3
xyz 7 4
12
2
7
79
12 3
74
.
ii
2
minyk
-cMeBaH n 14 eyIg)an cMnYnTI1KW n 1 14 1 13 cMnYnTI3KW n 1 14 1 15 dUcenH cMnYnKt;TaMgbItKñaKW 13 , 14 , 15 .
2
713
naM[ RkLaépÞFMbMputKW ³
160 KNna A , B , C , D nig E ³ 3 2 4 3 8 3 11 A2 4 4 4 4
S Max ab aa a 2 20m 400 m2 2
dUcenH RkLaépÞFMbMputéncmáarKW S 158 bgðajfacMnYn
3
Max
.
400 m 2
B 2
BMuEmnCacMnYnsniTan
3 23 6 3 4 4 4 2
C 2
¬cMnYnsniTanCacMnYnmanTRmg;CaRbPaK a ehIy a nig b Ca b cMnYnbzmrvagKña ¬bzmrvagKñaKWtYEckrYmFMbMputén a nig b esµI!¦¦
3 2 3 23 6 4 1 4 1 4 4
3 3 3 D 2 2 22 2 4
3 2 3 2
-]bmafa 3 CacMnYnsniTan 3 2 4 3 8 3 8 3 11 E 2 4 4 4 4 4 4 4 naM[ 3 manTRmg;CaRbPaKsRmYlmin)an 161 KNna A , B , C , D nig E ³ -tag 3 ba 3 ba a 3b A 2 2 2 2 2 2 16 -eXIjfa a CaBhuKuNén 3 enaH a k¾CaBhuKuN B 2 4 4 4 16 én 3 Edr mann½yfa a 3k , k CacMnYnKt; C 2 2 16 -naM[ a 3b 3k 3b b 3k -eXIjfa b CaBhuKuNén 3 enaH b k¾CaBhuKuN D 2 2 16 E 22 24 8 én 3 Edr mann½yfa b 3 p , p CacMnYnKt; -naM[ 3 ba 33kp kp GacsRmYl)an mann½yfa 162 sresrCaTRmg;RbPaKén A , B , C ³ A 1.2 mann½yfa A 1.2222222222222... vaBMu)aneKarBtamlkçxNÐcMnYnsniTan enaHvaBMuEmnCa naM[ 10A 12.2222222222222... cMnYnsniTaneT ¬GBa©wgmanEt CacMnYnGsniTan¦ sikSapldk 10 A 12.2222222222... dUcenH cMnYn 3 BMuEmnCacMnYnsniTan . 2
2
2
2
22
4
2
2 2
22
2
2
2
2
2
2
4
2
2 2
4
2
2
¬xagelIenHbkRsaytamsMeNIpÞúy tkáviTüa ¦
159 edaHRsaysmIkar x y 0 smµtikmµ³ x nig y CacMnYnBit enaH x 0 nig y 0 manEtmYykrNIKt;EdlepÞógpÞat;smIkar x y 0 KW x 0 & y 0 dUcenH smIkarmanb¤ x 0 & y 0 . ¬lMhat;xagelIRtUvkarcMeNHdwg BMuEmnbec©keTseT¦ 2
A 1.22222222222... 9 A 11
A
11 9
2
2
2
2
2
mann½yfa B 0.121212121212... naM[ 100B 12.121212121212... sikSapldk
714
B 0.12
100B 12.1212121212... B 0.121212121212... 12 4 B 99B 12 99 33
mann½yfa C 0.1222222222222... naM[ 10C 0.1222222222222... sikSapldk C 0.12
10C 1.2222222222222... C 0.12222222222222... 9C 1.1
D 1.20
C
12 10
.
6 5
2
2
dUcenH Ggát;RTUgkaer
d a 2
BD
BD 2 2 12 3
2
ÉktaRbEvg .
2
d a 2 2 a2 a2 2 4
AB2 BD DC
2
x
a 2
32
2
1 3 4 3 4
AC 2 2 3
.
2 3 2 2 3 2 2 3 x 2011 0 2 3 2 2 3 2 2 3 x 2011 0 2 2 3 2 2 2 3 x 2011 0 2 2 3 2 2 2 3 x 2011 0 2 3 4 2 3 x 2011 0 2 3 2 3 x 2011 0 2 3 2 3 x 2011 4 3 2 x 2011 x 2011
2
2 a 2 a 2 2
a2 a 4 2
dUcenH cm¶ayxøIbMputKW
AD2 AB2
165 edaHRsaysmIkar eyIgmansmIkar ³
2
2
2
C
AB2 BC 2
dUcenH RbEvg
-cm¶ayxøIbMputBIcMNucRbsBVrvagGgát;RTUgeTARCugkaer KWCaRbEvgGgát;EdlP¢ab;BIcMNucRbsBVenaHeTAcMNuc kNþalénRCugkaerenaH . ebI x CaRbEvgxøIbMputenaH ¬tamRTwsþIbTBItaK½r¦ eyIg)an ³ x a2 d2 2
2
8 4 3 2 2 3
2
-ebI d CaRbEvgGgát;RTUgkaerEdlmanRCug a ³ tamRTwsþIbTBItaK½r ³ d a a 2a naM[ d 2a d a 2 eRBaH d 0 2
D
BD2 AD2 AB2
AC
cm¶ayxøIbMputBIcMNuc RbsBVeTARCugkaer
a 2
B
12
x
2
AC 2 AB2 BC 2
Ggát;RTUg
a
1
-cMeBaHRtIekaNEkg ABC
163 KNnaGgát;RTUg nigcm¶ayxøIbMputBIcMNuc RbsBVrvagGgát;RTUgeTARCugkaer a
A
AD2 AB2 BD2
1.1 11 9 90
mann½yfa
D 1.2000000... 1.2
b¤ x
164 KNna AC ABC nig ABD CaRtIekaNEkg tamRTwsþIbTBItaK½r ³ -cMeBaHRtIekaNEkg ABD
ÉktaRbEvg .
dUcenH b¤sénsmIkar x 2011 .
715
166 edaHRsaysmIkar eyIgmansmIkar ³
168 KNna eK[
x 2 x 4x 2 x 3 x 2 x 7 x 2 x 4x 2 x 3 x 2 x 3 4 x 2 x 4 x 2 x 4 4 x 2 x 3 x 2 x 4 1 4 x 2 x 3 2 x 2 x 3 2 2
2
2
2
2
2
2
2
2
2
2y2
2 x 2 200 y 2 x 2 200 2 y2
x 2 2x 3 2
x 2 2 x 1 0 x 1 0 x 1 0 x 1 2
-cMeBaH
2y2
49 x 2 51x 2 92 y 2 102 y 2
2
2
-cMeBaH
2
6 51x
49 x 2 92 y 2 51x 2 102 y 2
2
2
x2 2 y2 x2 2y2 306 294 2 2 294 x 2 y 306 x 2 2 y 2
6 49x
2
x2 y2
x2 100 y2
x2 100 y2
dUcenH KNna)an . 169 KNna A ³ eK[ A x 2 2x 2x 8 x 8 tamrUbmnþ a b a 2ab b eyIg)an A x 2 x 8
x 2 2 x 3 2 x 2 2 x 1 4 x 1 2 2
eday x 1 0 naM[ x 1 2 CasmIkarKµanb¤s dUcenH smIkarmanb¤s x 1 . 2
2
2
2
2
2
2
2
x 2 x 8
2
167 RsaybBa¢ak;fa aS bS cS 0 eday x nig x Cab¤sénsmIkar ax bx c 0 enaHvaepÞógpÞat; ³ axax bxbx cc 00 n 1
x 2 x 8
2
n 1
n
10 2 100
2
1
2
2 1 2 2
1
2 2 ax1 bx1 c 0 2 ax2 bx2 c 0 ax12 n1 bx11 n1 cx1n1 0 2 n1 1 n 1 n 1 ax2 bx2 cx2 0
x1n1
eyIgGacKuNEfm eyIg)an
x2n1
n
n 1
n 2
n 1 1
n 1
n 1 1
170 RsaybMPøW;fa a b c 1 eK[ a b c 1 a b c 1 naM[
n
x
n 1
2
n 1 2
0
.
2
a 2 b 2 c 2 2ab 2ac 2bc 1 a 2 b 2 c 2 2ab ac bc 1
1
smµtikmµbEnßm 1a b1 1c 0 bc ac ab 0 abc bc ac ab 0
n 1 2
n 1
2
2
eday S x x enaH S x ehIy S x x dUcenH eyIg)an aS bS cS n 1
2012
2
ax n 1 bx1n cx1n 1 0 1n 1 n n 1 ax1 bx2 cx2 0 a x1n 1 x 2n 1 b x1n x 2n c x1n 1 x 2n 1 0
cMeBaH x 2011 KµanTak;Tgnwgtémørbs; A dUcenH eyIgKNna)an A 100 .
naM[smIkar 1 eTACa ³
716
, abc 0
naM[ktþanImYy²esµIsUnü eyIg)an ³
a2 b2 c2 2 0 1 a2 b2 c2 1
dUcenH eyIgeXIjfa a
2
b2 c2 1
BitEmn .
171 rkBIrcMnYnBit tag a nig b CaBIrcMnYnBitEdlRtUvrkenaH tambRmab; ³ a b 13 1 ehIyplbUkcRmasKW 1a b1 1340 2 tam 2 : 1a b1 1340 aab b 13 40 Taj)anplKuN ab 40 3 tam 1 & 3 eyIg)anRbB½n§smIkar³ aab b4013 edaysÁal;plbUk nigplKuNeyIg)ansmIkardWeRkTI2 Edlman a nig b Cab¤s KWsmIkar ³ x 13x 40 0 man 13 4 1 40 169 160 9 3
x 2 5x 3 0
man 25 12 13 naM[ x b2a 5 2 13
x 2 5x 7 0
0 man a 25 28 3 0 naM[ smIkarKµanb¤s dUcenH cMeBaH k 3 smIkarmanb¤s x 5 2 13 .
x> kMNt;témø k edIm,I[smIkarmanb¤s BIsRmayxagelIBinitüRtg;kenSam ³ x
2
2
x
2
2
naM[ b¤
a b b a
3
13 3 10 5 2 1 2 13 3 16 8 2 1 2 13 3 10 5 2 1 2 13 3 16 8 2 1 2
dUcenH BIrcMnYnBitrk)an
a 5 b 8
b¤
a 8 b 5
x 1x 4x 2x 3 3
x 5x 5 1x 5x 5 1 3 x 5x 5 1 3 x 5 x 5 2 0 x 5 x 5 2x 5x 5 2 0 x 5 x 3x 5 x 7 0 2
5x 4 x 2 5x 6 3
2
2
2
2
2
2
2
2
2
2
5x 5
2
k 1
eday x 5x 5 0 naM[ k 1 0 k 1 dUcenH témøkMNt;)anKW k 1 2
2
.
173 eRbóbeFobBIrcMnYn 200 nig 300 Binitü ³ 200 200 8000000 ehIy 300 300 90000 tamry³lT§plbgðajfa 8000000 90000 dUcenH Taj)anfa 200 300 . 174 eRbóbeFobBIrcMnYn 31 nig 17 Binitü ³ 31 32 2 2 enaH 31 2 ehIy 17 16 2 2 enaH 17 2 eday 31 2 2 17 dUcenH eRbóbeFob)an 31 17 . 300
172 k> edaHRsaysmIkar cMeBaH k 3 x 1x 2x 3x 4 k eK[ x
5 x 5 12 k
2
2
.
300
3 100
200
2 100
200
100
100
100
300
200
14
11
11
14
11
11
14
55
511
55
414
56
56
14
11
2
717
100
14
11
55
14
56
175 KNna 1 a a a a bRmab; ³ a Cab¤sénsmIkar x 1 mann½yfa vaepÞógpÞat; a 1 a eyIg)an ³ 2
3
1 1 1 1 A 1 2 1 2 1 2 ... 1 2 2 3 4 n 2 1 2 1 3 1 3 1 4 1 4 1 2 3 3 4 4 2
2010
2011
2011
2011
n 1 n 1 ... n n 1 3 2 4 3 5 n 1 n 1 ... n 2 2 3 3 4 4 n
1 0
a 1a 2011 a 2011 a 2011 ... a 1 0 b¤Gacsresr ³ a 11 a a 2 a 3 ... a 2010 0
1 1 1 1 smµtikmµ ³ a 1 enaH a 1 0 1 n 1 n 1 1 1 1 ... 2 n 2n dUecñHmanEtktþa 1 a a a ... a 0 n 1 EdlnaM[plKuN a 11 a a a ... a 0 dUcenH eRkayBIKNna A 2n . dUcenH KNna)an 1 a a a a 0 . 178 kMNt;témø m ³ eyIgman ³ 23xx 4yym7 176 KNna S 3 3 3 ... 3 eyIgman ³ S 3 3 3 ... 3 1 edayedaHRsaytamedETmINg; eyIg)an ³ yk 1 3 ³ 3S 3 3 ... 3 2 D ab ab 2 12 14 D cb cb 7 4m edayyk 2 1 eyIg)an ³ 2
3
2010
2
2
2
2010
3
2
2
3
2010
3
3
n
n
n 1
3
x
3S 32 33 34 ... 3n1 2 3 n S 3 3 3 ... 3 2S 3n1 3
dUcenH KNna)an
S
3
n 1
3 2
D x 7 4m D 14 D y ac ac 2m 21 x
S
3n1 3 2
.
x
edIm,I[vamanb¤s
2 n 1 n2 n 1n 1 n2 n 1 n 1 n n
D
x0
2m 21 14
nig y 0 luHRtaEt ³
7 7 4m m 7 4 m 0 14 0 4 2m 21 21 2 m 21 0 m 0 2 14 7 21 m 4 2
177 KNna A ³ BinitütYnImYy²manrag 1 1 2 n
Dy
b¤ Gacsresrsruby:agxøIKW
dUcenH témøEdlkMNt;)anKW
eyIgman ³
1 1 1 1 A 1 2 1 2 1 2 ... 1 2 2 3 4 n
edaybMEbkktþanImYy²eyIg)an ³ 718
7 21 m 4 2
.
179 KNna E ³
3
eyIgman E 1 2 3 ... 2009 2010 2011 2010 2009 ... 3 2 1
tag S 1 2 3 ... 2010 eyIg)an E S 2011 S eday
181 bgðajfa ³ 7 7 7 7 eyIgdwgehIyfa ³ 7 9 3 enaH 7 3 1 7 8 2 enaH 7 2 2 7 16 2 enaH 7 2 3 edayyk ³ (1)+(2)+(3) : eyIg)an ³
2S 2011
man 2011 cMnYn 2010 dg
naM[ 2S 2011 2010 eyIg)an E 2S 2011 2011 2011 20112 2011
dUcenH eRkayBIKNna E 2011
.
180 edaHRsayRbB½n§smIkar ³ edaybUksmIkarTaMgbYnbBa©ÚlKña eyIg)an ³
d 3
3
a b c d 6 acd 5 b 1
.
7 3 7 4 7 7
cMENkÉ karbgðajfa 4 4 4 4 eyIgman ³ 4 2 1 4 1 1 enaH 4 1 2 4 1 1 enaH 4 1 3 edayyk ³ (1)+(2)+(3) : eyIg)an ³ 3
3
3
4
4
4
4
dUcenH eXIjfa 4 4 4 4 . 182 kMNt;témø n ³ tag g x CaplEckén f x x 4 x 1 CamYy x 3 EdlmansMNl;esµI 46 eyIg)an ³ x 4 x 1 x 3 g x 46 ebIeyIg[témø x 3 enaHsmIkareTACa ³ 3
a b c d 6
4
n
a b c d 6 2 b a d 4 c2
a b c d 6 4 b c d 6
n
3
a0
n
2
2
4 32 1 3 3 g 3 46 3 35 0 46 n
3n 46 35
dUcenH RbB½n§smIkarmancemøIyKW a , b , c , d 0 , 1 , 2 , 3
4
4 2 3 4 1 4 4 1 3 4 44 4 4
edayyksmIkar dksmIkarnImYy²bnþbnÞab; a b c d 6 1 a b c 3
4
3
3
20112010 1
18
4
dUcenH eXIjfa
2011 2010 2011
1 2 3 4
3
7 3 3 7 2 4 7 2 3 7 7 4 7 7
S 1 2 3 ... 2010 S 2010 2009 2008 ... 1 2S 2011 2011 2011 ... 2011
a b c 3 b a d 4 a c d 5 b c d 6 3a 3b 3c 3d
3
4
.
3n 81
dUcenH témøkMNt;)anKW 719
3n 34
n4
.
n4
183 edaHRsaysmIkar ³ eyIgmansmIkar ³
x
1
x
2
2
x
2 1 2 1
1 2 1 x x 1 1 1 x
21978 2 n27
27 2
2
2 1 21977 2 n27 2
2
a 2 2ab b 2
luHRtaEt
n
B 4 27 4 n 41016
2 1 2 1
1 x 2
2 2 2 1 2 2
4 27 1 4 n27 4989
1 1 x
2
27 2
2
27 2
2
989 2
2 n 54
2
a 2 2ab b 2
luHRtaEt
2 2 n 55 2 989 2n 55 989 1044 2 n 522 n
x 1 1 1 x x 1 1 x
989 2
2 n 55
edIm,I[bMeBjrag a b
ebIeyIgyklT§plenHeTACMnYs vanwgkøayeTACaTRmg; dEdl²vij. ebICMnYsbnþbnÞab; cugeRkayeyIg)an ³
dUenH témøEdlkMNt;)anKW n 2004 , n 522 .
x 1 1 x
x 12 1 x 2
185 edaHRsaysmIkar ³ eyIgman ³ x xx 1 x
x 2 2x 1 1 x
x 1 x 1 1 x2 1 2 x 1 x x 1 1 x2 2 x x x 1 1 x2 2 2 2 x x x 4
x 3x 0 x x 3 0 2
2
2
2
x 0 x 3
-cMeBaH x 0 minyk eRBaHeBlepÞógpÞat; 0 1 -cMeBaH x 3 yk eRBaHeBlepÞógpÞat; 1 1
2
x 3
-krNI ³ 4 CatYkNþal eyIg)an ³
1 1 x
dUcenH smIkarmanb¤s
2
n 2004
12 2 1 1 x 1 x 1 1 x
2
2 n27 21977 n 27 1977
2 2 1 x x 1 1 x
1
27 2
edIm,I[bMeBjrag a b
n
4 27 1 4 989 4 n27
x 2 1 1 x x 2 1 1 x 1 1 x
1016
B 4 27 41016 4 n
x 2 1 1 x
smIkarmann½yluHRtaEt 1 x 0 x 1 Binitü RTg;RTayknÞúycugeRkayeK ³
27
n
x
2
184 kMNt;RKb;cMnYnKt; n B 4 4 4 CakaerR)akdéncMnYnKt; -krNI ³ 4 CatYcug eyIg)an ³
1 1 x 2 x x x
. 720
,
x0
t2 2 t
eday 2 : xy yz zx 12 naM[ x y z 2 12 36
t2 t 2 0
x 2 y 2 z 2 12
ebItag t x 1x eyIg)ansmIkardWeRkTI@ ³
2
man a b c 1 1 2 0 CakrNIBiess naM[ t 1 , t ac 12 2 -cMeBaH t 1 eyIg)an ³ 1
ehIy 2 : 2xy 2 yz 2zx 24 eyIgyk (*)-(**) enaHeyIg)an ³
2
x y 2 y z 2 z x 2 0 eday x y 2 0 , y z 2 0 , z x 2 0
2
1 3 x 0 2 4
edIm,I[smIkarepÞótpÞat;manEtmYykrNIKt; KWktþa nImYy²esµIsUnü mann½yfa ³
2
1 3 x 0 , 0 2 4
x y 0 y z 0 x y z z x 0 3 : 2 2 2 3 x yz x y z 2 2 2 3 x x x 222 6 3 x x 2 x 3
2
enaH x 12 34 0 naM[smIkarKµanb¤s -cMeBaH t 2 eyIg)an ³
tam eyIg)an
2
1 2 x x 2 2x 1 0 x
x 12 0 x 1 0 x 1
186 edaHRsayRbB½n§smIkar ³ k>
x> 1 2 3
smIkarmann½yluHRtaEt ³ x 0 , y 0 , z 0 x yz 6 tam 1 ³ x y z 2 6 2 x 2 y 2 z 2 2xy yz zx 36
eday
dUcenH RbB½n§smIkarmanb¤s x y z 2
dUcenH eRkayBIedaHRsay smIkarmanb¤s x 1 . x y z 6 xy yz zx 12 2 2 2 x y z 3
2 x 2 2 y 2 2 z 2 24 2 xy 2 yz 2 zx 24 x 2 2 xy y 2 y 2 2 yz z 2 z 2 2 zx x 2 0
1 x 1 x 2 x x 1 0 1 1 3 x2 2 x 0 2 4 4
2
2 x 2 2 y 2 2 z 2 24
1
eday
2
1 1 1 x y z 3 1 1 1 3 xy yz zx 1 1 xyz
.
1 2 3
smIkarmann½yluHRtaEt ³ x 0 , y 0 , z 0 yz zx 3 -tam 1 : 1x 1y 1z 3 smmUl xy xyz naM[ xy yz zx 3 eRBaH 3 : xyz1 1 -tam 2 : xy1 yz1 zx1 3 721
x yz 3 x yz 3 xyz
naM[
k> 011 edayRbB½n§ 011 manelx 6xÞg;P¢ab;CamYyk,alRbB½n§ ehIymanTRmg;Ca ³ 011 xxx xxx -CeRmIsenAtamxÞg;nImYy²man 10elxdUcKñaKW ³
1
x y z 2 32
x 2 y 2 z 2 2 xy yz zx 9 x2 y2 z2 2 3 9 x2 y2 z2 3
2
0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9
edayyk (2)-2×(1) eyIg)an ³
¬tameKalkarN_r)ab; ¦ naM[ cMnYnelxTaMgGs;KW ³
x 2 y 2 z 2 3 2 x 2 y 2 z 6 2 2 x 2 x y 2 y z 2 2 z 3
x
2
10 10 10 10 10 10 10 6 1 000 000
2x 1 y 2 2 y 1 z 2 2z 1 0
x 12 y 12 z 12 0
eday x 1 0 , y 1 0 , z 1 0 edIm,I[smIkarepÞótpÞat;manEtmYykrNIKt; KWktþa nImYy²esµIsUnü mann½yfa ³ 2
2
x 1 0 y 1 0 z 1 0
2
x 1 y 1 z 1
dUcenH RbB½n§smIkarmanb¤s x y z 1 . 187 KNna 2 2 ³ eyIgman 4 4 23 a
a
a
a
2
a 2
23 2 2 2 25
2 2 a
dUcenH cMnYnlT§PaBénkarbegáItelxTUrs½BÞrbs; RbB½n§ 011 KW 1 000 000 elx . x> 097 dUcKñanwgRbB½n§ 011 EdrRKan;EtRbB½n§ 097 man7xÞg; manTRmg; 097 xxx xxxx naM[ cMnYnelxTaMGs;KW ³ 10 10 10 10 10 10 10 10 7 10 000 000
dUcenH cMnYnlT§PaBénkarbegáItelxTUrs½BÞrbs; RbB½n§ 097 KW 10 000 000 elx .
2
189 rkcMnYnrebobénkarbegáItelx4xÞg; 2 2 25 eyIgmanelxTaMgbYnKW 1 , 2 , 4 nig 9 2 2 5 eyIgnwgbegáIelx4xÞg;Edlman4elxxagelI edayKit smÁal; ³ xN³bMBak;r:aDIkal;BMuEmn 25 eT eRBaH tamviFIdUcxageRkam ³ 2 2 0 Canic© ¬CaGnuKmn_Gics,:ÚNg;Esül¦ -xÞg;dMbUgman4CeRmIs xÞg;TIBIrman3CeRmIs xÞg;TIbImanBIrCeRmIs nigxÞg;cugeRkayman1CeRmIs dUcenH eRkayBIKNna 2 2 5 . -tameKalkarN_r)ab; cMnYnrebob= 4 3 21 4! 12 188 rkcMnYnlT§PaBTaMgGs;énkarbegáItelxTUrs½BÞ dUcenH cMnYnrebobkñúgkarbegáItelx4xÞg; eyIg)andwgehIyfa enARbeTskm<úCaeyIgmanRbB½n§ TUrs½BÞCaeRcIn. RbB½n§xøHmanelx 6xÞg; xøHeTotman elx 7xÞg; ehIyP¢ab;CamYyk,al xusKñaman 12 rebob . a 2
a
a
a
a
a
a
a
a
a
RbB½n§. etIcMnYnén karbegáItlxTUrs½BÞ y:agNaEdr ?
722
190 RsaybBa¢ak;fa ³ b ab c pq 6 -smµtikmµ ³ a , b Cab¤sénsmIkar x px 1 0 naM[ a abb 1 p 12 -smµtikmµ ³ b , c Cab¤sénsmIkar x qx 2 0 naM[ b bcc 2q 34 -Taj)an a bb c p q pq ehIy ab bc 1 2 3 b¤ 2ab bc 6 -Binitü pq 6 a bb c 6 2
2
a b b c 2ab bc
192 rkBIrcMnYnenaH ³ tag x , y CaBIrcMnYnEdlRtUvrkenaH tambRmab;RbFan ³ plbUk=plKuN=plEck eyIg)an ³ x y xy xy -BinitüsmPaB ³ xy xy y 1y naM[ y 1 y 1 y 1 -BinitüsmPaB ³ x y xy cMeBaH y 1 enaH x 1 x Kµantémø x epÞógpÞat; cMeBaH y 1 enaH x 1 x 2x 1 x 12 2
dUcenH BIrcMnYnenaHKW 12 nig 1 .
ab ac b 2 bc 2ba 2bc b ba bc ac bb a cb a b a b c 2
193 KNnaplKuN P ³ eyIgman ³ P 1 x1 x 1 x ... 1 x a b a b a b -Binitü ³ naM[ a b aa bb tamrUbmnþxagelIeyIg)an ³ P 1 x 1 x 1 x ... 1 x 2
dUcenH eXIjfa
.
b a b c pq 6
2
191 RsaybMPøWfaeyIg)an a1 b1 c1 2 bc smµtikmµ ³ a b c abc naM[ a abc 1 eyIgman 1a b1 1c 2 tamrUbmnþ a b c a b b 2ab bc ac 1 1 1 naM[ 4 a b c 2
2
2
2
2
2
2
2
2
2
2
2
2
2n
4
4
2
2n
1 x2 1 x 2 1 x 4 1 x 8 ... 2 4 1 x 2n 1 x 1 x 1 x
n 1
n 1
1 x2 1 x
Edl x 1
2
1 1 1 1 1 1 2 4 a b b ab bc ac 1 1 1 cab 2 2 2 4 2 a b c abc 1 1 1 2 2 2 1 4 2 a b c 1 1 1 2 2 2 2 a b c
dUcenH plKuNKNna)anKW
2
.
194 KNnaplbUk S ³ eyIgman ³ S
1 1 1 1 ... 1 2 1 2 3 1 2 3 4 1 2 3 ... 2011
Binitü ³ S
dUcenH eXIjfa a1 b1 c1 2 . 2
n 1
1 x2 P 1 x
2
723
k
1 2 3 ... k
naM[
x 2 9 x 20 x 4x 5
S 1 2 3 ... k k S k k k 1 ... 2 1 2S k k 1 k 1 k 1 ... k 1
enaH S xx1 1 x 11x 2 x 21x 3
1 1 x 3x 4 x 4x 5 x2 x 1 S xx 1x 2 x 2x 3 x5 x3 x 3x 4x 5 2 1 2 S xx 2 x 2x 3 x 3x 5 2x 6 x 2 xx 2x 3 x 3x 5 3 2 xx 3 x 3x 5 3x 15 2 x 5 xx 3x 5 xx 5
man k dgén k 1
Taj)an ³ 2S k k 1 S k k2 1 eyIg)an ³ S S1 S1 S1 ... S 1 k
k
2
eday S
k
k k 1 2
3
4
2011
1 1 1 1 ... 22 1 33 1 44 1 20112011 1 2 2 2 2 2 2 2 2 S ... 22 1 33 1 44 1 20112011 1
S
1 1 1 1 S 2 ... 2011 2012 23 3 4 45
cMeBaH x 95 eyIg)an ³
ehIyBinitütYnImYy²én S manTRmg; ³ 1 n 1 n n 1 n 1 1 nn 1 nn 1 nn 1 nn 1 n n 1 1 1 1 1 1 1 1 1 S 2 ... 2011 2012 2 3 3 4 4 5 1 1 1005 1 2 1 1006 1006 2 2012
a b c 1 1 1 ab a 1 bc b 1 ca c 1 1 a ab 1 b bc 1 c ca
-cMeBaH abc 1 Binitü ³
195 KNnaplbUk S cMeBaH x 95 ³ eyIgman S x 1 x x 13x 2 x 15x 6
eday
5 5 5 1 xx 5 95 95 5 5 19 100 1900
196 bgðaj[eXIjBIsmPaB
1005 dUcenH plbUkKNna)anKW S 1006 .
2
S
1 dUcenH plbUkKNna)anKW S 1900 .
naM[
2
2
1 1 2 x 7 x 12 x 9 x 20 2
x 2 x xx 1
1 1 1 1 a ab 1 b bc 1 c ca 1 c 1 a 1 b 1 a ab c 1 b bc a 1 c ca b c a b c ac abc a ab abc b bc cab c a b c ac 1 a ab 1 b bc 1
x 3x 2 x 1x 2 2
x 2 5 x 6 x 2x 3
a b c ab a 1 bc b 1 ca c 1
dUcenH eyIg)ansmPaBR)akdEmn .
x 7 x 12 x 3x 4 2
724
197 rktémø x ³ 199 KNnaplbUk S ³ ¬rMlkw rUbmnþ log b 1 log b , 1 2 3 ... n nn 1 eyIgman xxxx 1 x 1 n 2 elxénxÞg;nImYy² CacMnYnKt;RtUvEtFMCag0 nigtUcCag10 eyIgman ³ 1 1 1 1 eyIg)an 00 xx 110 10 1 x 10 S ... log x log x log x log x 1 1 1 1 elx x 2 CasV½yKuNRtUvmantémøFMCag0 ... 1 1 1 log x log x log x log x enaH x 2 0 x 2 2 3 k 1 2 3 k ... srublkçxNÐKW 3 x 9 log x log x log x log x 1 mann½yfa x GacmantémøesµI 3 , 4 , 5, 6 , 7 , 8 , 9 1 2 3 ... k log x b:uEnþ xxxx 1 CaelxbYnxÞg; ehIymanEt 6 EtmYy 1 k k 1 k k 1 log x 2 2 log x Kt;EdlmanelxbYnxÞg; naM[eyIgTaj)an ³ x 1 6 k k 1 S dU c enH plbU k KNna)anKW . x7 2 log x x 2 5 enaNeyIg)an ³ xxxx 1 x 1 200 edaHRsayRbB½n§smIkar 777 7 1 7 1 eyIgmanRbB½n§smIkar ³ 7776 6 x2
an
a
a
a2
a
a
a3
a
ak
a
a
a
a
a
a
5
a
a
a
x2
72
5
x1 x2 x3 x2011 0 1 2 2 x1 x22 x32 x2011 0 2 3 3 3 3 x1 x2 x3 x2011 0 3 ......................................................... 2011 x12011 x22011 x32011 x2011 0 2011
7776 7776
dUcenH témøEdlrk)anKW x 7 . 198 sRmYlkenSam
A
³
¬rMlkw rUbmnþelakarIt log a b log b c log a c 1 log a n b log a b , log a a 1 n
2 1
>>> ¦
2 1
eyIgman ³
A log 3 2 log 4 3 log 5 4 log 6 5 log 7 6 log 8 7
log 3 2 log 4 3 log 5 4 log 6 5 log 7 6 log 8 7
log 3 2 log 5 3 log 6 5 log 8 6
log 3 2 log 5 3 log 6 5 log 8 6
2 3
2 3
2 2011
2 2011
0
0
2 x12 x22 x32 x2011 0
cMeBaHsmIkarepSgeTot k¾epÞógpÞat;EdrsRmab;témøenH 2 x12 x22 x32 x2011 0
1 1 1 log 8 2 log 23 2 log 2 2 1 3 3 3
dUcenH eRkayBIsRmYleXIjfa
2 2
2 2
dUcenH RbB½n§smIkarmantémøb¤sesµI²KñaKW ³
log 5 2 log 8 5
1 A 3
tamsmIkar 2 ³ x x x x eday x 0 , x 0 , x 0 , , x naM[smIkar 2 epÞógpÞat;luHRtaEt ³
201 edaHRsayRbB½n§smIkartamedETmINg; eyIgmansmIkar ³ 23 3x x 3 3 y y21
.
725
.
edaHRsaytamedETmINg; eyIg)an ³ D ab a b
203 sRmYlkenSam E & F ³ eyIgman E a ab b4 ab a b 4 ab 2
3 3 2 3 3
3
23
2
1 2 3
Dx cb cb
a b
2 3 1 3
D y cb cb 3 1 2 3 2 3 43
ehIy
3 3
dUcenH
2
EF
a b a b 2 a EF
202 bgðajfa k> x y 2 xy eday x nig y CacMnYnBitenaHeyIg)an ³
a b 2
a b
2
a b a b
dUcenH
x y 2 0 x 2 2 xy y 2 0 x 2 y 2 2 xy
dUcenH eXIjfa x y 2 xy . bgðajfa x> xy xy 2 enAkñúgsMNYr k> eXIjfa x y 2 xy smµtikmµ x nig y mansBaØadUcKña enaHmann½yfa xy 0 Canic© eBlEcknwg xy vismIkar minbþÚrTisedA eyIg)an xxy yxy 2xyxy xy xy 2 2
2
2
x y 2 y x
.
E a b , F a b
2
2
nig E F E F a b a b
.
y 3
, 2
dUcenH eXIjfa
a b a b
a b b a ab ab a b a b ab
KNna
dUcenH RbB½n§smIkarmanKUcemøIy
2
F
D x 3 3 x 3 D 3 D y 3 3 y 3 D 3
2
2
a b
x 3
2
a 2 ab b a b
3 3
naM[
2
a 2 ab b 4 ab a b
2 3 3
2
.
EF 2 a
EF a b
&
.
204 KNnatémøénkenSam A ³ eyIgman ³ A x n 100x n1 x n2 100x n3 x n4 100x n5
x n 100x n1 x n2 100x n3 x n4 100x n5
x n 100 x n 1 x n 2 100 x n 3 x n 4 100 x n 5 x n 1 x 100 x n 3 x 100 x n 5 x 100
x 100 x n 1 x n 3 x n 5
cMeBaH x 100 naM[ A 100 100 100 dUcenH 726
n 1
100 n 3 100 n 5 0
témøKNna)anKW A 0 .
205 KNnatémøénkenSam A ³ eyIgman A 22 30 20 3
180o o 180 0
27 3 8
dUcenH mMuTl;kMBUl . 208 rkcMnYnKt;viC¢man n ³ eyIgman ³ x x 6 n KuNnwg4 ³ 4x 4x 24 4n
22 30 20 27 3 8 3
3 22 30 20 27 2
2
3 22 30 20 5
2
2
3 22 30 5
4 x
3 22 5
2
2
4 x 1 23 4n 2
2n 2 x 12 23 2n 2 x 12n 2 x 1 1 23 2
3
dUcenH KNna)an
A3
.
206 KNnatémøén a nig b bRmab; a nig b CacMnYnKt; eday a b 321 0 mann½yfa a CacMnYnFMCag b ehIysnñidæanfa b a 1 eRBaH vaCacMnYnKt;tKña eyIg)an ³ a a 1 321 2
bRmab; ³ x nig n CacMnYnKt; ehIy n x enaHeyIg)an ³ 2n 2 x 1 1 2n 2 x 1 23 4n 24
2
-cMeBaH
2
x 2 x 30 0
eday 5 6 1 & 5 6 30 tamRTwsþIbTEvüt enaH 5 & 6 Cab¤sKt;énsmIkar dUcenH témøEdlkMNt;)anKW n 6 .
a 2 a 2 2a 1 321 a a 2a 1 321 2
a
n6
x 2 x 6 62
a 2 a 1 321
eyIg)ansmIkar ³
n6
2
2
2
322 161 2
naM[ b a 1 1611 160 a b dUcenH témøKNna)anKW a 161 , b 160 . 209 eRbóbeFobkenSam a b nig 207 mMuTl;kMBUlCaGVI? Binitü ³ aa bb a abba b mMuTl;kMBlU CamMumankMBUlrYmKña ehIyRCugénmMunImYy² a b a b pÁúMKñabegáIt)anbnÞat;BIrkat;Kña nigrgVas;mMuTaMgBIrb:unKña . 2
2
2
2
2
a2 b2 a b2
2
2
a b a b a 2 b 2 a b a b a b 2
eyIgnwgRsayfa mMuTl;kMBUl cMeBaH RKb;cMnYnBit a nig b Edl a b enaH ³ nig CamMub:unKña a b a b a b a b 2b 0 Bit a b a b eday 180 nig 180 eRBaHCamMurab sikSapldk ³ 2
2
2
2
2
2
o
o
727
2
2
2
2
2
eday aabb aa bb 2
2
2 2
a2 b2 a b2
-cMeBaH m 1: dUcmanenAsMNYr k> bgðajrYc enaHsmIkarmanb¤DubKW x x 1 -cMeBaH m 53 smIkar eTACa ³ 1
1
dUcenH eRbóbeFob)an aabb aa bb . 210 k> kMNt;témø m edIm,I[ x 1 Cab¤s eyIgman ³ m 1x m 3x 3 m 0 ebI x 1 Cab¤sénsmIkareyIg)an ³ 2
2
2
2
2
m 112 m 31 3 m 0 m 1 m 3 3 m 0
2
2
3 2 3 3 1 x 3 x 3 0 5 5 5 2 2 12 18 x x 0 5 5 5 2 2 x 6x 9 0 5 x 2 6x 9 0
1 m 0
x 32 0
m 1
naM[ smIkarmanb¤sdubKW x x 3 dUcenH témøkMNt;)an m 1 , m ac 53 nigb¤sDub x x 1 b¤ x x 3 .
cMeBaH m 1 smIkarka eTACa ³
1
1 1x 2 1 3x 3 1 0
2
1
2x 2 4x 2 0
1
2
2
1
2
x 2 2x 1 0
man a b c 1 2 1 0 ¬CakrNIBiess¦ naM[ x 1 , x ac 11 1 1
2
211 rkBIrcMnYn x nig y y 25 eyIgsÁal; ³ xxy 12 eyIg)an 2
x 2 y 2 25 2 xy 24 2 x 2 xy y 2 1
dUcenH témø m 1 nigb¤smYyeTotKW x 1 . x> kMNt;témø m edIm,I[smIkarmanb¤sDub smIkar m 1x m 3x 3 m 0 man b 4ac
x y 2 1
2
x y 1
2
m 3 4m 13 m 2
2
2
5m 2 2m 3
a 0 0 m 1 0 m 1 2 2 5m 2m 3 0 5m 2m 3 0
edIm,I[smIkarmanb¤sDub luHRtaEt
naM[ sRmab; ³ 5m 2m 3 0 man a b c 0 naM[smIkarmanb¤s ³ m 1 , m ac 53 eyIgnwgrk b¤sDubcMeBaHtémø m nImYy²Edlrk)an 2
1
-cMeBaH x y 1 x y 1 naM[ y 1y 12 y y 12 0 man y 1 1 2 414 1 2 49 28 4 2
m 6m 9 12m 4m 12 4m 2
2
2
1
1 12 4 14 1 49 6 y2 3 2 2 2
ebI y 4 : x 124 3 ehIy y 3: x 123 4 -cMeBaH x y 1 x y 1 naM[ y 1y 12 y y 12 0 man y 1 12 4 14 1 2 7 3 2
1
2
2
3
728
ebI
1 12 4 14 1 7 y2 4 2 2 y 3 3: x
12 4 3
ehIy y
4
b¤ x 3 , y 1 -cMeBaH S 3 & P 2 eyIg)an xxy y23 x & y Cab¤sénsmIkar X 3 X 2 0 man a b c 0 enaHsmIkarmanb¤s ³ x 1 , y 2 b¤ x 2 , y 1 dUcenH RbB½n§smIkarmanKUcemøIybYnKU ³ x 1 , y 3 b¤ x 3 , y 1 x 1 , y 2 b¤ x 2 , y 1 . x 1 , y 3
4: x
12 3 4
2
dUcenH BIrcMnYnEdlkMNt;)anKW ³ x 3, y 4 x 4, y 3 b¤ . x 3 , y 4 x 4 , y 3
212 edaHRsayRbB½n§smIkar eyIgman xx yy xyx y7 8
2
2
2
2
1 2
tag S x y nig P xy tam 1 ³ x 2 xy y 2 xy x y 8 2
2
x y 2 2 xy x y 8 S 2 2P S 8
tam 2 ³
3
x 2 y 2 xy 7
A
x 2 2 xy y 2 2 xy xy 7
x y 2 xy 7
M
S2 P 7
4
yk 4 CMnYskñúg 3 eyIg)an ³ S 2 2P S 8
S2 2 S2 7 S 8 S 2 2S 2 14 S 8 S2 S 6 0 S2 S 6 0
naM[ S
1
N
B
P S2 7
213 )atmFüménRtIekaNKWCaGVI? )atmFüménRtIekaN KWCaGgát;EdlKUsP¢ab;BIcMNcu kNþalénRCugBIrrbs;RtIekaN ehIyvaRsbnwgRCugTI# énRtIekaNenH nigmantémøesµIBak;kNþalRCugTI#. eyIgnwgbgðajfa MN BC 2 tamniymn½y AB AC / nigman MN Rsbnwg BC AM AN 2 2 -RtIekaN ³ ABC & AMN man ³ mMu AMN ABC / mMu ANM ACB ¬mMuRtUvKña¦ MN AM naM[ AMN ABC BC AB AM BC AM BC naM[ MN BCAB 2 AM 2 dUcenH )atmFüménRtIekaNekaNesµIBak; kNþalén)atTI# EdlRsbnwgva .
1 1 24 2 2 3 S 2 : P 22 7 3 2 S 3 : P 3 7 2
tam 4 ebI
x y 2 xy 3
-cMeBaH S 2 & P 3 eyIg)an x & y Cab¤sénsmIkar X 2 X 3 0 man a b c 0 enaHsmIkarmanb¤s ³ 2
214 rkcMnYnKt; n EdlepÞógpÞat; 72 12n 54 eyIgman 72 12n 54 b¤Gacsresr 247 n 485 smmUl 3 73 n 9 53 naM[ cMnYnKt;epÞógpÞat;KW n 4 , 5, 6 , 7 ,8, 9 729
C
dUcenH cMnYnKt;EdlepÞógpÞat;KW
AO2 AB2 OB 2
215 rkPaKEdlenAsl; -RtIekaNEkg AEB ³ S AEB
252 20 2 625 400 225
-cMeBaHRtIekaNEkg COD man ³ 24
A
PaKenAsl; S S 144 60 204 PaK dUcenH PaKEdlenAsl;KW 204 PaK . BDC
252 24 2 625 576 49
216 Tajrk eyIgman ³ x 12011 2011 1 1 smmUl x 2011 2011 b¤ x 2011 2011 naM[ x 12012 1 1
eyIg)an AC AO CO 15 m 7 m 8 m dUcenH RbEvgcugr)aFøak;cHu KW AC 8 m .
2
2011 2012 2011 1 1 2011 1 1 2011 2012 1 2011 2011
2
20 m
a b c 60 1 2 bc 240 a 2 b 2 c 2 3
b c 2 60 a 2 b 2 c 2 2bc 3600 120a a 2 A
D 4 m B
2
tam 1 : a b 60 c
217 rkRbEvgEdlcugr)aFøak;cuH 25 m
2
srubmkeyIg)anRbB½n§smIkar
.
1 2011 x 2012 2012
7m
218 KNnargVas;RCugnImYy²énRtIekaN tagrUbRtIekaNenaHdUcrUbxageRkam ³ B a tambRmab;RbFan c -brimaRt a b c 60 cm A b 1 -RkLaépÞ 2 bc 120 bc 240cm -tamBItaK½r a b c
1 ? x 2012
dUcenH Taj)an
CO CD 2 OD 2
C 10 D
1 S BDC 10 12 60 2 AEB
CO 2 CD 2 OD 2
B
12
- RtIekaNEkg BDC ³
AO AB2 OB 2 15 m
E
1 12 24 144 2
.
n 4 , 5, 6 , 7 ,8, 9
b 2 c 2 3600 120a a 2 2bc 4
C
edayyk (2) nig (3) CMnYskñúg (4) a 2 3600 120a a 2 2 240
O
120a 3600 480
RbEvgcugr)aFøak;cuHKW AC AO CO tamRTwsþIbTBItaK½r -cMeBaHRtIekaNEkg AOB man ³
3120 26 120 b c 60 26 34 bc 240
a
naM[ tamEvüt b nig c Cab¤sénsmIkar ³
730
C
x 2 34 x 240 0
man b ac 17 240 289 240 7 naM[ b 171 7 10 , c 171 7 24 b¤Gac c 171 7 10 , b 171 7 24 2
2
dUcenH rgVas;RCugnImYy²rbs;RtIekaNKW .
10cm , 24cm , 26cm
2
221 k> rkRbPaKtagcm¶aypøÚvEdlsuxRtUvedIr -bRmab; ³ suxeFVIdMeNIredayrfynþ)an 83 15 40 suxeFVIdMeNIredaym:UtU)an 53 24 40 15 24 39 naM[ kareFVIdMeNIrtamrfynþ nigm:UtKU W³ 40 40 40 39 1 enaH kareFVIdMeNIredayefµIeCIgKW 40 40 40 40 dUcenH RbPaKtagkareFVIdeM NIredayefµIeCIgKW 401 .
219 bgðajfaRtIekaNmanmMuTaMgbICaRtIekaNEkg x> rkcm¶aypøÚvTaMgGs;ebIsuxedIr)an 2 km bRmab;mMuTaMgbIKW , 2 nig 3 suxedIr)an 2 km RtUvnwgRbPaK 401 edayeyIgdwgfaplbUkmMukgñú énRtIekaNesµI 180 1 mann½ y fa 2 km cm¶aypøÚvTaMGs;KW eyIg)an 2 3 180 40 o
o
6 180 o
40 40 2 km 80 km 40
30 o
mMuTaMgbIénRtIekaNenHKW
dUcenH cm¶aypøÚvTaMgGs;KW 80 km .
30 o , 2 60 o , 3 90 o
RtIekaNenHmanmMumYymanrgVas; 90 vaCaRtIekaNEkg 222 k> sresr 45m CaPaKryén 1km eday 1 km 1000 m dUcenH RtIekaNmanmMuTaMgbICaRtIekaNEkg . 1 45 4.5 eyIg)an 45 1000 4.5% 1000 100 220 rk f 9 dUcenH eyIgsresr)an 4.5% . eyIgman f x f x 1 9 nig f 3 81 naM[ x> sresr 1kg CaPaKryén 800 g 9 9 1 f 3 f 4 9 f 4 f 3 81 9 eday 1kg 1000g 9 9 1 1000 1 125 f 4 f 5 9 f 5 81 125% eyIg)an 1000 800 f 4 1 / 9 8 100 100 o
f 5 f 6 9
f 6 f 7 9
f 7 f 8 9
f 8 f 9 9
dUcenH rk)an
f 9 81
f 6
9 9 1 f 5 81 9 9 9 f 7 81 f 6 1 / 9 9 9 1 f 8 f 7 81 9 9 9 f 9 81 f 8 1 / 9
.
dUcenH eyIgsresr)an 223 KNnaRbEvg bRmab; ³ MN 18cm M
A
AB
I
eyIg)an ³ AB AI 18BIcm
731
.
125%
³ B
N
eday A nig B CacMNuckNþal erogKñaén MI nig NI naM[ AI MI2 nig BI NI2 enaH AB MI2 NI2 MI NI MN 18 cm 9 cm 2 2 2
dUcenH RbEvgKNna)anKW
4 5 3 5 5 2 10 3
o
o
o
x 180o x 180o 0
A 4 5 3 5 48 10 7 4 3 4
2011 2012
4
4
4 5 3 25 5 3 4 4 25 4 45 4 1
naM[ A 1 eday sV½yKuNén 2011CacMnYnmanelx 1 cugCanic© naM[ A 2011 ...1 CacMnYness eyIg)an A 1 1 dUcenH témøKNna)anKW A 1 . 20112012
2012
20112012
rMlwkragsmIkardWeRkTI3 ³ tamRTwsþIbTEvüt smIkar manrag X 3 SX 2 RX P 0 S a b c R ab ac bc P abc
2
4 5 3 5 48 10 2 2 4 3 3 4
2
4
Edl
a ,b,c
Cab¤s ³
. GñkGacepÞógpÞat;ragxagelI
edaykarBnøatkenSam x ax bx c 0 .
eyIgman
4 5 3 5 48 10 7 4 3 4
2
2
226 edaHRsayRbB½n§smIkar
dUcenH pkbUkmMukñúgBIrénRtIekaNesµImMueRkAmYy EdlminCab;ngw va . 225 KNnatémøénkenSam A ³ eyIgman ³
4 5 3 5 48 10 2 3
3
4 5 3 5 5 3 4
224 RsaybBa¢ak;faplbUkmMukñúgBIresµImMueRkAmYy ]bmafa eyIgmanRtIekaN nig x mMu dUcrUbxagsþaM eyIgnwgbgðajfa -plbUkmMukñúgénRtIekaNesµI 180 enaH x 180 -ehIymMu x 180 eRBaHvaCamMurab sikSapldk ³
Binitü ³
4 5 3 5 28 10 3 4
4 5 3 5 5 3
.
AB 9 cm
4 5 3 5 48 10 2 3 4
2 x 2 y 2 z 7 x 7 y z 2 2 2 4 x y z 3
2 x 2 y 2 z 7 1 1 1 7 x y z 4 2 x y z2 32 2 2 732
2x 2 y 2z 7 2 y 2 z 2 x 2 z 2 x 2 y 7 4 2x2 y2z x y z 2 2 2 8
2x 2y 2z 7 2x 2y 2z 7 y z 7 y z x z x y x z x y 2 2 2 2 2 2 8 2 2 2 2 2 2 14 4 2x2y2z 8 2x2y2z 8
cMeBaH
edIm,IgayRsYl eyIgtag a 2
dUcenH KNna)an 2ba 1 . 228 rkcMnYnxÞg;EpñkKt;éncMnYn A 2 eyIgman ³ A 2 edaybMBak;elakarIteKal10 eyIg)an ³ log A log 2
x
, b 2 y , c 2z
abc 7
eyIg)anRbB½n§fµI bc ac ab 14 tamRTwsþIbTEvüt abc 8
a ,b,c
Cab¤sénsmIkar X
7 X 2 14 X 8 0
3
smIkarxagelImanb¤sgay ³ X 1 eRkayBIEckBhuFa X 7 X 14X 8 0 nig X 1 eyIg)an ³ X 1X 6 X 8 0 X 1 X 2 X 4 0
23 6
2011
6 6
2011
12011 1
2011
2011
log A 2011 log 2
bRmab; ³ log 2 0.301 naM[ log A 2011 0.301 log A 605.311
edaycMnYn 605 605.311 606 naM[ 605 log A 606 10605 A 10606
X 1 0 X a 1 X 2 0 X b 2 X 4 0 X c 4
eyIg)an
2011
2011
2
2
naM[
2a b
eyIg)an ³
2011
¬manviFIedaHRsaysmIkardWeRkTI3tam Cadan b:uEnþBuMmankñúgkmµviFIsikSarbs; RksYgGb;rM dUcenHeyIgenAEtedaHRsayedayrebobTajb¤sgay. sUmcgcaMfa vaBMuEmnhYssmtßPaBrbs;eyIgeT eRBaHeKerobcMlMhat;)anl¥Nas; ¦ 3
a3 &b6
eday
CacMnYnmanelx 606xÞg; ehIycMnYn 10 CacMnYnmanelx 607xÞg; Taj)an A CacMnYnEdlmanelx 606xÞg; dUcenH A 2 CacMnYnEdlmanEpñkKt; 606xÞg; . 10605
606
2 x 2 0 a 1 x 0 y 1 b 2 2 2 y 1 c 4 2 z 2 2 z 2
2011
dUcenH RbB½n§smIkarmanb¤s x , y , z 0 ,1, 2 . 229 edaHRsaysmIkar 3 x 5 2 x 7 x 1 2 3 3 x 5 2 x 7 227 KNna 2ba x 1 , x 0 eyI g man 2 3 33 x 5 22 x 7 6 x 1 eK[ a b 15 216 6 6 6 a b 9 36 6 6 33 x 5 22 x 7 6 x 1 2011
32 6 6 6
2
9 x 15 4 x 14 6 x 6
3 6
5 x 1 6 x 6
2
x 5
3 6
x 25 kenSaménGgÁTaMgBIrmanTRmg;dUcKña nig a nig b CaBIr cMnYnKt;viC¢man naM[eyIgTaj)ankrNIEtmYYyKt;KW ³ cMeBaH x 25 eRkayBIepÞógpÞat; 4=4 Bit dUcenH smIkarmanb¤s x 25 . a3 &b6
733
230 eRbóbeFob ab nig PGCD a,b PPCM a,b 232 edaHRsayRbB½n§smIkar x y 4 eyIgman a 90 , b 280 edaybMEbkCaktþabzm ³ eyIgman x y 4 xy z 6 z 13 xy z 6 z 13 b 280 2 a 90 2 edaysÁal;plbUk nigplKuN eyIgbegáIt)ansmIkar 45 3 140 2 15 3 70 2 X 4 X z 6 z 13 0 Edlman x & y Cab¤s 35 5 5 5 man b ac 7 7 1 2
2
2
2
2
1
2
naM[ a 90 2 3 5 nig b 280 2 5 7 eyIg)an ³ ab 90 280 25200 2
2 z 2 6 z 13
3
4 z 2 6 z 13 z 2 6z 9
z 2 6z 9
PGCD a , b 2 5 10
z 3
2
PPCM a , b 23 32 5 7 2520
eday z 3 0 manEtmYykrNIKt; EdlnaM[ smIkarmanb¤sKW z 3 0 enaH z 3 -cMeBaH z 3 eyIg)ansmIkarfµI ³ 2
enaH PGCD a , b PPCM a , b 10 2520 25200 eXIjfa ab PGCD a , b PPCM a , b 25200 dUcenH eyIgeRbobeFob)an ab PGCD a , b PPCM a , b . 231 epÞógpÞat;ÉklkçN³PaB
2
X 2 4 X 32 6 3 13 0 X 2 4X 4 0
X 22 0
smIkarmanb¤sDub X X 2 eyIg)anb¤sénsmIkarKW ³ XX xy 22 1
1 sin x cos x 21 sin x 1 cos x Binitü ³ 1 sin x cos x 2 2
2
1
2
tamrUbmnþ a b c a b c 2ab 2bc 2ac dUcenH RbB½n§smIkarmancemøIy x 2 , y 2 , z 3 . edayBnøattamrUbmnþxagelI eyIg)an ³ 233 EbgEcknimitþsBaØaelakarIt 1 sin x cos x 1 sin x cos x 2 sin x 2 cos x 2 sin x cos x eyIgmannimitsþ BaØaelakarItdUcxageRkam ³ 2
2
2
2
2
2
2
2
12 1 2 sin x 2 cos x 2 sin x cos x 21 sin x cos x sin x cos x
Logx , log x , lg x , ln x , log 10 x , log e x
tamkareRbIR)as; rbs;GñkKNitviTüanimitþsBaØa ³ - elakarItTsSPaK b¤ elakarIteKal 10rYmman ³
21 sin x cos x sin x cos x 21 sin x cos x1 sin x 21 sin x 1 cos x
log x , lg x , log 10 x
eRBaH edaysar sin x cos x 1 naM[ 1 sin x cos x 21 sin x 1 cos x 2
2
2
dUcenH 1 sin x cos x
2
21 sin x 1 cos x
.
- elakarItenEB b¤ elakarIteKal e rYmman ³ Logx , ln x , log x . e
¬lMhat;xagelI RKan;Etcg;bgðajBInimitþsBaØaelakarIt¦ 734
3a b b c a c 0 a b b c a c 0
234 rkRkLaépÞrgVg;carwkkñúgRtIekaNenH -tamrUbmnþehrug épÞénRtIekaNKW S p p a p b p c
7
Taj)an
8
r Edlman a , b , c CargVas;RCug 11 abc ehIy p 2 CaknøHbrimaRt -smµtikmµ ³ RtIekaNmanrgVas;RCug 7 , 8 nig 11 enaH p 7 82 11 13 eyIg)an ³ S 1313 713 813 11
a b 0 b c 0 a c 0
-cMeBaH a b 0 tambRmab; a b c 1 enaH c 1 ehIy a b c 1enaH a b 0 Taj)an a 0 , b 0 eyIgKNnatémøénkenSam ³ 2
2
2
2
2
P a 2010 b 2011 c 2012 0 2010 0 2011 12012 1
-cMeBaH b c 0 tambRmab; a b c 1 enaH a 1 ehIy a b c 1enaH b c 0 Taj)an 2 13 15 -épÞRtIekaNFM )anBIplbUképÞRtIekaNtUc²bIEdlman b 0 , c 0 eyIgKNnatémøénkenSam ³ Pa b c 1 0 0 1 rgVas;RCugnImYy²Ca)at nigkaMrgVg; r énrgVg;carwkkñúg -cMeBaH a c 0 tambRmab; a b c 1 enaH b 1 RtIekaN Cakm
13 6 5 2
2
2
2010
2
2
2011
2
2012
2
2
2010
2
2011
2
2012
2
2
2
2
13 r
naM[ 13r 2 13 15 -épÞrgVg;carwkkñúgRtIekaNKW S
r
P a 2010 b 2011 c 2012 02010 12011 02012 1
2 13 15 13
srumk eTaHCakrNINak¾eday P 1 Canic© dUcenH témøKNna)anKW P 1 .
r 2
2
2 13 15 S 13 4 13 15 60 13 132
236 bgðajfa a b c d 1 a b c d eyIgman a b c d 1 a b c d 2
2
2
2
2
2
2
a2 a b2 b c2 c d 2 d 1 0
dUcenH épÞRkkLargVg;carwkkñúgRtIekaNKW S
60 13
.
1 2 1 2 1 2 1 2 a a b b c c d d 0 4 4 4 4 2
235 KNnatémøénkenSam P a tamrUbmnþÉklkçN³PaB
2010
2
3
3
1
2
2
dUcenH
3
2
eday a 12 0 , b 12 0 , c 12 0 , d 12 0 naM[ a 12 b 12 c 12 d 12 0 BitCanic© 2
smµtikmµ ³ a b c 1 nig a b c naM[ 1 1 3a bb ca c
2
1 1 1 1 a b c d 0 2 2 2 2
b 2011 c 2012
a b c 3 a 3 b 3 c 3 3a b b c a c 3
2
735
2
2
2
2
2
a 2 b2 c2 d 2 1 a b c d
.
237 rkGayumñak;² tag a CaGayu«Buk nig b , c , d CaGayukUnTaMgbI bRmab; ³ GayuGñkTaMgbYnmansmamaRt 15 , 7 , 5 , 4 ebItag x CapleFobsmamaRtGayuGñkTaMgbYn eyIg)an ³ 15a b7 5c d4 x Taj)an ³ 15a x a 15x b7 x b 7 x
ab a b cd c d 2 a c a b (1) c d b d ab b 2 (2) cd d 2 a 2 ab (3) c 2 cd 2
239 RsaybMPøWfaeK)an
eyIgmansmamaRt ³ smmUl -KuN (1) nwg db eyIg)an ³ -KuN (1) nwg ac eyIg)an ³ tam (2) nig (3) eyIg)anpleFobesµItKñaKW ³ a ab b a 2ab b c d b¤ G acsresr x c 5x x d 4x c cd d 2cd d c 5 4 smµtikmµ³ Gayu«BukticCagplbUkGayukUnTaMgbI 3qñaM eyIg)an a 2ab b a 2ab b c 2cd d 2cd d c eyIg)an a 3 b c d a b 2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
15 x 3 7 x 5 x 4 x x3
naM[ ³ 15a 3 a 45
b 3 b 21 7 d 3 d 12 4
c 3 c 15 5
epÞógpÞat; ³ 45 3 2115 12 48 48 Bit dUcenH «BukmanGayu45qñaM nigkUnTaMgbImanGayu erogKñaKW ³ 21qñaM , 15qñaM , 12qñaM . 60
225 238 eRbóbeFobBIrcMnYn 259 nig 625
Binitü ³ ehIy eday naM[
naM[
2ab a b 2cd c d 2 2
dUcenH bgðaj)anfa
60
120
9 3 25 5
100
dUcenH eRkayBIeRbóbeFob
225 625 60
50
.
320 160 2
eday x 160 CacMnYnTI1 enaHcMnYnTI2 KW x 1 161 dUcenH cMnYnKt;viC¢mantKñaenaHKW 160 nig 161 .
50
9 225 25 625
2
x
eRBaHvamaneKal 53 1 3 5
ab a b cd c d 2
2 x 321 1
50
100
x 2 2 x 1 x 2 321
50 250 100 15 2 225 3 3 625 5 5 25
3 5
2
x 12 x 2 321
50
60
120
ab a b cd c d 2
240 rkBIrcMnYnKt;viC¢mantKña tag x CacMnYnTI1 enaHcMnYnTI2 bnÞab;KW x 1 tambRmab;RbFaneyIg)an ³
60 260 120 3 2 9 3 3 25 5 5 5
3 5
c d 2
.
¬lMhat;xagelImaneRcInnak;Nas; EdlRclMePøcKitfa eKaltUcCag!¦
241 etIekµgNaQñH ehIyekµgNacaj; tag N CacMnYnXøIsrub mann½yfa A B C N -muneBlelg eyIgmansmamaRt ³ A B C 3 4 5 736
tamlkçN³smamaRteyIgGacsresr)an ³
243 edaHRsayRbB½n§smIkar ³
A B C A BC N 3 4 5 3 45 12
x xy y 1 y yz z 4 z zx x 9 x xy y 1 y yz z 4 z zx x 9
eyIgman ³
Taj)an ³ A N 3N 12 N A 3 12 12 48 B N 4 N 16 N B 4 12 12 48
x xy y 1 1 1 y yz z 1 4 1 z zx x 1 9 1
x y 1 y 1 2 y 1 x 1 2 i y z 1 z 1 5 z 1 y 1 5 ii z x 1 x 1 10 x 1 z 1 10 iii
C N 5 N 20 N C 5 12 12 48
-eRkayeBlelg eyIgmansmamaRt ³
edayKuNsmIkar ³ i ii iii bBa¢ÚlKña eyIg)an
A B C 15 16 17
x 12 y 12 z 12 100 x 1 y 1z 1 10 x 1 y 1z 1 10
tamlkçN³smamaRteyIgGacsresr)an ³ A B C A B C N 15 16 17 15 16 17 48
-cMeBaH EcksmIkar nigsmIkar i , ii , iii eyIg)an ³
Taj)an ³
x 1 y 1z 1 10 y 1x 1 2 z 1 5 z 4 x 1 y 1z 1 10 x `1 2 x 1 5 z 1 y 1 y 1 1 y 0 x 1 y 1 z 1 10 x 1z 1 10
A N 15N A 15 48 48 B N 16 N B 16 48 48 C N 17 N C 17 48 48
-eRbobeFobmuneBlelg nigeRkayeBlelg ³ ekµg A ³ 1248N 1548N mann½yfa A QñH ekµg B ³ 1648N 1648N mann½yfa B rYcxøÜn ekµg C ³ 2048N 1748N mann½yfa C caj;
-cMeBaH x 1 y 1z 1 10 EcksmIkar nigsmIkar i , ii , iii eyIg)an ³
dUcenH ekµg ³ A QñH / B rYcxøÜn nig C caj; . 242 edaHRsaysmIkar x x x x 1 1 1 1 3 6 9 12 2 3 4 x 1 1 1 1 1 1 1 1 3 2 3 4 2 3 4 x 1 x 3 3
dUcenH smIkarmanb¤s x 3
eyIg)an
x 1 y 1z 1 10 y 1x 1 2 z 1 5 z 6 x 1 y 1z 1 10 x `1 2 x 3 5 z 1 y 1 y 1 1 y 2 x 1 y 1z 1 10 x 1z 1 10
dUcenH RbB½n§smIkarmancemøIyBIrKW ³ x , y, z 1, 0 , 4 b¤ x , y , z 3, 2 , 6 . 244 eRbóbeFobcMnYn ³ eyIgman 2 manelx 2 cMnYn 1001dg ehIy 3 manelx 3 cMnYn 1000dg Binitü ³ 2 2 16 ehIy 3 27 2
22
3
33
.
22
737
4
3
225 240t 64t 64t 64 eXIjfa 16 27 2 3 240t 289 eyIgdwgfa 2 tUcCag 3 Edlman 998 dg 289 t 240 CasV½yKuNbnþbnÞab;dUcKña 289 960 289 671 naM [ x 4t 4 3 enaHeyIg)an 16 27 Edlmanelx2 nigelx3 240 240 240 671 epÞógpÞat; cMBaH x 240 cMnYn 998 dgCasV½yKuNbnþbnÞab;dcU Kña 3 x 4 x 4 x 5 x 5 x 3 x eday 16 ¬man dgelx ¦ 2 ¬man dgelx ¦ 22
33
2
22
3
33
2
2
998
3 3
273
dUcenH 2
2
22
2
3
2
22
22
2
3
2
22
1001
2
3 3
¬man 998 dgelx 3¦ 33 ¬man 1000 dgelx 3¦ ¬man 1001 dgelx 2¦ 3
3
33
¬man 1000 dgelx 3¦
245 edaHRsaysmIkar x 3 x 4 x 4 x 5 x 5 x 3 x
¬vaCasmIkarGsniTan caM)ac;RtUvykcemøIEdlrkeXIjmkepÞógpÞat; ¦
49 289 289 529 529 49 240 240 240 240 240 240
7 2 17 2 17 2 232 2402 2402 7 17 17 23 23 7 240 240 240 119 391 161 671 240 240 671 671 x 240 240
enaH
232 7 2 2402
smIkarmann½ykalNa ³
eday
3 x 0 x 3 4 x 0 x 4 x 3 5 x 0 x 5
671 dUcenH smIkarmanb¤s x 240
671 240
Bit .
4 t t 2 t t 2 t t 2 1
246 rkcm¶ayBIsVayRCMeTAsVayerog -smµtikmµ ³ begÁalcm¶ayTI1dak;fa {sVayRCM 17 km } mann½yfaeFVIdMeNIr 17 km eTot dl;sVayRCM ehIy begÁalcm¶ayTI2bnÞab;dak;fa {sVayerog 25 km } mann½yfaeFVIdMeNIr 25 km eTot dl;sVayerog . eyIgGacKUsrUbgayemIl
4 t
PñMeBj
-edIm,IkMu[EvgeBk nigBi)akkñúgkarsresr ³ eyIgtag
x 4t 4 x t 3 x t 1 5 x t 1
eyIg)an ³
4 t t 1 t t t 1 t 1 t 1 4t
t 1t t t 1 t 1t 1 t 2 1 t 2 t t 2 t
elIkGgÁTaMgBICakaer eyIg)an ³
sVayRCM
1km
sVayerog
25km
4 t t 2 1 t 2 t t 2 t 4 t 2 24 t t 2 1 t 2 1 2t 2 2t t 2 1 4 t 2 t 2 1 2t 2 2t t 2 1 24 t t 2 1 15 8t 8 t 2 1
17km
?
tamrUbeyIgGacbkRsay)any:aggayfa cm¶ayBIsVayRCM eTAsVayerogKW ³ 25 1 17 26 17 9 km
elIkGgÁTaMgBIrCakaermþgeToteyIg)an ³
dUcenH cm¶ayBIsVayRCMeTAsVayerogKW 9 km .
738
247 rkcMnYncab nigcMnYnpáaQUk tag x CacMnYncab nig y CacMnYnpáaQUk -tamsmµtikmµ ³ cabmYyTMpáaQUkmYy enaHsl;cab mYyKµanpáaQUkTM eyIg)ansmIkar ³
249 rkGayurbs;davIRsIsBVé;f¶ -smµtikmµ ³ Gayurbs;davItageday x ehIy x Ca BhuKuNén mann½yfa x Gacmantémø CaeRcIndUcCa ³ 0 , 8 , 16 , 24 , 32 , 40 , 48 , ...
kñúgcMeNamelxTaMgGs;xagelI manEtelx 16 eTEdl -ehIy ebIcabBIrTMpáaQUkmYy enaHenAsl;páaQUkmYy eKarBtamlkçx½NÐ FMCag10 nigtUcCag 20 KµancabTM eyIg)ansmIkar ³ na[témø x manEtmYyKt;KW 16 x 1 y
i
x y 1 2
ii
-tamsmIkar i & ii eyIgcgCaRbB½n§smIkarKW ³ x 1 y x y 1 x y 1 x y 1 x 2 y 2 x 2 y 2 2
edaybUkGgÁ nigGgÁénsmIkarTaMgBIr eyIg)an ³ x y 1 x 2 y 2 y 3
dUcenH sBVéf¶davIRsImanGayu 16 qñaM . 250 rkcMnYnéf¶EdlKat;pÁt;pÁg;cMNIdEdl[eKa -eyIgsegáteXIjfa kalNacMnYneKakan;EteRcIn enaHry³eBlsIukan;EtfycuH vaCasmamaRtRcas naM[ cMnYnéf¶EdlRtUvpÁt;pÁg;KW ³ 40 35 40 1400 35 28 40 10 40 10 50
x y 1 3 1 4
dUcenH cabmancMnYn 4 k,al / páaQUkmancMnYn 3 Tg .
¬BMuEmn 40 10 35 43.75 éf¶enaHeT ¦ 40
dUcenH cMnYnéf¶EdlRtUvpÁt;pÁg;KW 28 éf¶ 248 rkcMnYnstVExVkEdlgab; nig rs; -eyIgKYKb,IKitfa eRkABIstVExVkgab; nigrs;vaKµanGVI 251 bgðajfa a a a 1 eToteT ehIysmµtikmµniyayfa gab;bIrs;BIr enaH eyIgman ³ eyIgGacsnñidæan)anfa stVTaMgGs;mancMnYn 5k,al . a sin a a sin a 2 1
1
¬eRkABIgab; nigrs;KµanGVIeToteLIy ¦
éf¶
2 2
2 3
2 1 2 2 2 3
1
a2 cos a1 sin a2
.
2
1
a cos a1 sin 2 a2 2
-BicarNafa ³ ebIstV5 gab;3 enaHstV120gab;b:unµan? a cos a cos a a cos a cos a naM[ a a a naM[ cMnYnstVgab; 1205 3 72 k,al sin a cos a sin a cos a cos a ebIstV5 rs;;2 enaHstV120rs;b:unµan? sin a cos a sin a cos a sin a cos a 1 naM[ cMnYnstVrs; 1205 2 48 k,al eRBaH sin cos 1 dUcenH cMnYnstVExVkEdlgab;man 72 k,al ehIycMnYnstVExVkEdlrs;man 48 k,al . dUcenH eXIjfa a a a 1 R)akdEmn . 3
1
2 1
2
2 2
2
1
2
1
2
2
2
2
1
2
1
2
2 1
2 2
2 3
2
1
2
2
2
1
739
2
1
2
2
1
2 3
2
2
2
2
2
252 KNnaplKuN P Binitü ³ 1 a a 1 2a
abc ca
n
2
4
a
a4 a2
2
4
2 2
2
2
2
1 1 a b c a b 1 1 a b c b c 1 1 a b c c a 3 1 1 1 abc ab bc ca
2
2
2
1 a2 a4 1 a a 1 a a2
naM[ ebIbþÚ a Ca x eyIg)an ³ 1 x x 11xx xx eyIg[témø k 0 , 1 , 2 , 3, ... , n ebI k 0 ³ 1 x x 11xx xx ebI k 1 ³ 1 x x 11 xx xx ebI k 2 ³ 1 x x 11 xx xx >>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> ebI k n ³ 1 x x 1 x x 2
2k
2k 1
2k 1
2k
2k 2
2k 1
2k
2
dUcenH eXIjfa a 3b c a 1 b b 1 c c 1 a .
4
2
254 KNnaplbUk S 11!2 2!3 3! n n! Binitü ³ k 1! k 1 k!
2
2
4
2
n
4
8
2
4
8
16
4
8
8
2
2n 1
2n 2
1 x2 x2 n
1 x x 1 x 2
2
x 4 1 x 4 x8 1 x 2 x 2 n 1
b¤Gacsresr
1 x2 x2 1 x x2 n 1
1 x2 x2 Pn 1 x x2
n
2 n 1
n 1
abc bc
eyIg)an ³
1 1! 2!1! 2 2! 3!2! 3 3! 4!3! 4 4! 5!4! ................. n n! n 1!n!
n2
1 1!2 2!3 3!... n n! n 1!1! 2n 2
2
.
1 1 abc ab 1 1 abc bc
k 1 , 2 , 3 , 4 , ... , n
n 1
dUcenH eRkayBIKNnaeXIjfa S
253 bgðajfa a 3b c a 1 b b 1 c c 1 a tambRmab; a 0 , b 0 nig c 0 naM[ abc ab
naM[ ebI[témø
k 1! k k!k! k k! k 1!k!
n2
dUcenH KNna)an P 1 1x x xx n
n
4
n 1
3
edaybUkGgÁngi GgÁénvismPaB ³ 1 2 3 eyIg)an ³
1 a a 1 a a 1 a a 1 a a 1 a a 1 2a a 2
1 1 abc ca
1
2
255 KNnaplbUk n 9999 n 1
n
n 1!1
.
1 n 1 4 n 4 n 1
Binitü ³ n 1 n n 1 n n 1 n Et n 1 n n 1 n 1 naM[ n 1 n n 11 n 4
4
4
4
eyIg)an n 1 n n 1 n n 11 n 4
740
4
4
4
Taj)an n 1 n n 1 n 1 n 1 n 257 RsaybBa¢ak;faeK)an a b a b c d cd eyIg)an ³ eyIgman 1 1 n n 1 n n 1 n 1 n ba dc ac db ac db ca db mü:ageTot ac db ca db ebI[témø n 1 , 2 , 3 , ... , 9999 eyIg)an ³ 4
4
4
4
9999
4
9999
4
4
n 1
4
n 1
4 2 4 1 4 3 4 2 4 4 4 3 ............ 4 10000 4 9999
smmUl naM[
n 1 4 n 4 10000 4 1
n 1 4 n 10 1 9
4
n 1
4
n 1
9999
n 1
1
n n 1
4
9 4 n n 1
.
a b c 3 3aba b c 3
a b c a b a b c c 2 3aba b c
a b c a a b c a
4
4
4
4
4
4
2
4
a4 b4 a b c4 d 4 c d
a4 b4 a b c4 d 4 c d
4
4
a4 b4 a b c4 d 4 cd 4
.
5562 4452 111111 55562 44452 11111111 ..................................................
a 3 3a 2 b 3ab 2 b 3 c 3 3a 2 b 3ab 2 3abc 2
4
4
56 2 452 1111
3
4
4
6 2 5 2 11
256 sRmYlkenSam eday a b c 3abc 3
4
4
258 rkrUbmnþTUeTA eyIgman]TahrN_ ³
a 3 b 3 c 3 3abc A 2 2 2 a b c ab bc ca
3
4
dUcenH Rsay)anfa
9999
4
a b a b c d cd
tam 1& 2
9999
dUcenH
4
4
a b c a 2 2ab b 2 ac bc c 2 3ab
ab bc ac
2
b 2 ac bc c 2 ab
2
b2 c2
eyIgeXIjfa elx1 ekIneLIgeTVdgCanic© dUcenH tamlMnaMKMrUxagelIenHeyIgTaj)anrUbmnþTUeTA 555 ... 56 2
man n xÞg;
naM[
444 ... 45 2 man n xÞg;
111 ... 111 man 2n xÞg;
RsaybBa¢ak;rUbmnþxagelI a b c 3abc a b 2 c 2 ab bc ca a b c a 2 b 2 c 2 ab bc ca a 2 b 2 c 2 ab bc ca abc
A
3
3
2
Edl a b c ab bc ca 0 dUcenH eRkayBIsRmYl A a b c 2
tag
3
2
An 555 ... 56 2 444 ... 45 2
555...55 1 444...44 1
2
2
2
5 4 999...99 1 999...99 1 9 9 2
5 4 10n 1 1 10n 1 1 9 9
2
.
741
2
2
2
5 4 An 10n 1 1 10 n 1 1 9 9 4 5 10n 1 1 10 n 1 1 9 9
2
cos2 1o cos2 89o 2 o 2 o cos 2 cos 88 cos2 3o cos2 87 o ........................ cos2 44o cos2 46o cos2 45o S
4 5 10n 1 1 10n 1 1 9 9 1 10n 1 10n 1 2 9 1 10n 1 10n 1 9 1 102 n 1 9 1 9 999...999 9
enaH
Edlmanelx cMnYn 2n 111...11 Edlmanelx 1 cMnYn 2n dUcenH CakarBit rUbbmnþenHBitCaRtwmRtUvKW 555 ... 56 2
man n xÞg;
444 ... 45 2 man n xÞg;
2
o
2
2
o
o
o
o
o
cos 2 o cos 90 o 88 o sin 88 o
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
cos 44 o cos 90 o 46 o sin 46 o
enaHeyIg)an ³ cos2 1o sin 2 89o
S
89 2
1 2
.
260 KNna S sin 2 0o sin 2 1o sin 2 2o sin 2 90o
111 ... 111 man 2n xÞg; o
44
1 88 1 89 2 2 2
dUcenH KNna)an
259 KNna S cos 1 cos 2 cos 3 cos 89 tamrUbmnþ cos 2 sin b¤ cos90 sin naM[ cos1 cos90 89 sin 89 2 o
S 44
cos2 1o sin 2 1o 1 2 o 2 o cos 2 sin 2 1 cos2 3o sin 2 3o 1 ........................ cos2 44o sin 2 44o 1 2 2 2 o cos 45 2
/
o
¬RsedoglMhat;TI259 ¦ tamrUbmnþ sin 2 cos eyIg)an ³ sin 2 0 o cos2 0 o 1 2 o 2 o sin 1 cos 1 1 sin 2 0 o sin 2 90 o sin 2 2 o cos2 2 o 1 2 o 2 o sin 1 sin 89 sin 2 2 o sin 2 88o ........................ sin 2 44 o cos2 46 o 1 ........................ 2 sin 2 44 o sin 2 46 o 2 2 o sin 45 sin 2 45o 2 1 S 45 2
enaH
cos2 2o sin 2 88o
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
S 45
1 90 1 91 2 2 2
dUcenH KNna)an
cos2 44o sin 2 46o
enaHeyIg)an ³
742
S
91 2
.
261 kMNt;témø a & b eyIgmancMnYn abba Edl 1 a 9 , 0 b 9 eyIg)an abba 1000 a 100 b 10b a
¬eRBaH b 9 minRtUvmanRtaTukeT ¦ -cMeBaH b 0 eyIg)an 10 c9 9 9c01 4 tamTMnak;TMng 4 Taj)an c 8 eRBaH 8 9 72 1001 a 110 b ehIyEfmRtaTuk 8 KW 72 8 80 manelx0enAcug 1191a 10 b smµtikmµ abba CaKUbR)akdéncMnYnKt; enaHryIg)an -cMeBaH c 8 eyIg)an 1089 8 9801 abba k dUcenH témøkMNt;)anKW a 1 , b 0 , c 8 , d 9 3
1191a 10b k 3
x> bBa¢ak;fa abcd nig dcba suT§EtCakaerR)akd eday a & b CacMnYnKt; enaH 91a 10b CacMnYnKt;Edr eday abcd 1089 33 naM[ k Eckdac;nwg 11 enaH k 11n / n Kt;viC¢man ehIy dcba 9801 99 eyIg)an k 11n Edl n 1 , 2 , 3 , ... dUcenH cMnYn abcd nig dcba suT§EtCakarR)akd . EtebI n 2 enaH k 22 10648 abba 263 rkBIrcMnYnCakaerR)akd naM[ cMnYnKt;viC¢man n mantémøEtmYyKt;KW n 1 tag a k CacMnYnTI 1 nig b n CacMnYnTI 2 Edl eyIg)an abba k 11 1 11 1331 k & n CacMnYnKt; ehIy k n b¤ k n Taj)an témø a 1 , b 3 tambRmab;RbFan ab a b 4844 dUcenH témøkMNt;)anKW a 1 , b 3 . eyIg)an ³ ab a b 4844 ab a b 1 4844 1 262 k> kMNt;témøelxénGBaØat a , b , c nig d ab 1 b 1 4845 a 1b 1 4845 eyIgmancMnYn abcd nig dcba tamxÞg;nImYy²enaH eday a k nig b n enaHeyIg)an ³ 48453 1 a 9 , 0 b 9 , 0 c 9 , 1 d 9 1615 5 a 1b 1 4845 eK[ abcd 9 dcba 1 323 17 k 1n 1 4845 19 19 tamTMnak;TMng 1 naM[kMNt;)antémø a 1 EtmYyKt; k 1k 1n 1n 1 4845 1 ¬eRBaHebI a 1 enaHlT§plCaelx R)aMxÞg; ¦ Et 4845 3 51719 CaplKuNénbYncMnYnKt; -cMeBaH a 1 eyIg)an 1bcd 9 dcb1 2 EdlmanEtmYyEbbKt; tamTMnak;TMng 2 naM[kMNt;)antémø d 9 EtmYyKt; eday 3 5 4 14 1 nig 17 19 18 118 1 ¬eRBaHmanEtkrNIenHeTIb 9 9 81 manelx!cug¦ enaH k 1k 1n 1n 1 4 14 118 118 1 -cMeBaH d 9 eyIg)an 1bc9 9 9cb1 3 -cMeBaH k n Taj)an k 18 , n 4 tamTMnak;TMng 3 Taj)an b 0 -cMeBaH k n Taj)an k 4 , n 18 91a 10b
k3 11
2
2
3
3
3
3
3
3
3
2
3
2
2
2
2
743
2
naM[ b¤
2 2 a k 18 324 2 2 b n 4 16 2 2 a k 4 16 2 2 b n 18 324
2
epÞógpÞat; ³ 324 16 324 16 4844 Bit ehIy 16 324 16 324 4844 Bit
2
dUcenH BIrcMnYnEdlCakaerR)akdKW 324 & 16 . 264 KNnargVas;RCugnImYy²rbs;RtIekaN tag a , b , c CargVas;RCugén c a RtIekaNEkg Edl c CaGIubU:etnus b -tamBItaK½r ³ a b c 1 -bRmab;RkLaépÞesµI 24 cm naM[ 12 ab 24 2 -bRmab;brimaRtesµI 24 cm naM[ a b c 24 3 tam 3 ³ a b 24 c elIkCakaer 2
2
265 rkcMnYnTaMgBIrenaH tag x CacMnYnTI ! Edl x CacMnYnbzm enaHcMnYnTI@KW x 1 ehIy x 1 CakaerR)akd enaH x 1 n eyIg)an n 1 x smmUl n 1n 1 x Et x CacMnYnbzm naM[plKuNrbs; x KW x x 1 eyIg)an n 1n 1 1 x Taj)an ³ nn 11 1x nn 12 x 3n x2
2
2
2
266 rkkaMrgVg; tag r CakaMrgVg;EdlRtUvrk 2m edayP¢ab;p©itrgVg;TaMgbIdUcrUb enaHeyIg)anRtIekaNEkg EdlmanTMnak;TMngCamYy r tamBItaK½reyIg)an ³ 2 2r 2 2r 2 4r 2
b 2 2ab 576 48c c 2 4
4 8r 4r 2 4r 2 16r 2
edayyk 1 & 2 CMnYnskñúg 4 eyIg)an ³
8r 2 8r 4 0 2r 2 2r 1 0
c 2 2 48 576 48c c 2 96 576 48c 48c 576 96
dUcenH cMnnY TI!KW 3 nigcMnYnTI@KW 4 .
a 2 2ab b 2 242 48c c 2
a
tam b ac 1 2 3 naM[ r 1 2 3 0 minyk 2
c
480 10 cm 48
cMeBaH c 10 enaH a b 14 nig ab 48 tamEvüteyIg)ansmIkar ³ x 14x 48 0 Edl man a nig b Cab¤sénsmIkar
1
r1
2
b 2 ac 49 48 1 7 1 7 1 a 6 , b 8 1 1
1 3 2
ÉktaRbEvg
dUcenH kaMrgVg;EdlRtUvrkKW ³ r 12
.
3
b¤ a 8 , b 6 267 RsayfaRtIekaNcarwkknøHrgVg;CaRtIekaNEkg ¬munbkRsay GñkRtUvsÁal; mMucarwk C dUcenH rgVas;RCugnImYy²énRtIekaNEkgKW ³ nig mMup©ti Camunsin ³ mMucarwk CamMmu ankMBUl o elI r gV g ; ni g RCu g TaM g BI r kat; r gV g ; ÉmM u p © i t CamM u 6 cm , 8 cm , 10 cm . enaH
EdlmankMBUlelIp©itrgVg; .
744
A
B
-TMnak;TMngrvagmMup©it nigmMucarwkKW ebI CamMup©it nig CamMucarwkEdlmanFñÚsáat;rYm enaHeyIg)anTMnak;TMng ³ 2 . -RtIekaNcarwkknøHrgVg; CaRtIekaNEdlmanRCugmYy CaGgát;p©iténrgVg; EckrgVg;CaBIrEpñkb:unKña -¬emIlrUb¦ mMu nig A manFñsÚ áat;rYm BC naM[ A 2 eday CamMurab enaH 180 eyIg)an ³ A 1802 90 CamMuEkg . naM[ RtIekaNcarwkknøHrgVg;manmMuEkgCanigc© o
o
o
269 RsayfaRCugnImYy²énRtIekaNtUcCagknøH brimaRtrbs;vaCanic© c a ]bmafaeyIgmanRtIekaN EdlmanrgVas;RCug a , b , c b nigmanknøHbrimaRt a 2b c -tamvismPaBénRtIekaN eyIg)an ³ abc 2 abc 2b a b c b 2 abc 2c a b c c 2
a bc
2a a b c a
b ac c ab
dUcenH RtIekaNcarwkknøHrgVg;CaRtIekaNEkgCanic© . dUcenH rgVas;RCugnImYy²énRtIekaNtUcCagknøH brimaRtrbs;vaCanic© . 268 rkcMnYnRCugeRcInbMputénBhuekaN 270 rkcMnYndgénkarCk;)arIrbs;bursenaH -TMnak;TMngrvagcMnYnRCug nigmMuénBhuekaNKW ³ Ck;)an 10 dg ebIBhuekaNmYyman n RCugenaHplbUkmMukñúgTaMgGs; -)arI 10 edIm enAsl;knÞúy)arIcMnYn 10 knÞúy 3 edIm nig 1 knÞúy énBhuekaNKW ³ 180 n 2 ; -)arI 3 edIm nig 1 knÞúy Ck;)an 3 dg tambRmab;eyIg)an ³ enAsl;knÞúy)arIcMnYn 4 knÞúy 1 edIm nig 1 knÞúy 180 n 2 2011 2011 -)arI 1 edIm nig 1 knÞúy Ck;)an 1 dg n2 180 enAsl; knÞúy)arIcMnYn 2 knÞúy x©IknÞúy)arIeK 1 2011 n 2 180 naM[)an 3 knÞúy 1 edIm Ck;)an 1 dg 1980 31 n 2 enAsl;knÞúy)arI 1 yksgeKvij enaH)arIGs;Kµansl; 180 31 naM[cMnYndgénkarCk;)arIKW 10 3 11 15 dg n 11 2 180 dUcenH bursenaHRtUvCk;cMnYn 15 dgeTIbGs;)arIKat; . 31 n 13 o
o
o
o
o
180
271 edaHRsaysmIkar ³ k> 3 4 5 Et n CacMnYnRCug EdlCacMnnY Kt; enaH n 13 epÞógpÞat; 180 13 2 2011 1980 2011 Bit eyIgsegáteXIjfa x 2 Cab¤sénsmIkar eRBaH 3 4 5 9 16 25 25 25 Bit dUcenH BhuekaNenHGacmanRCugeRcInbMputcMnYn 13 . eyIgnwgbgðajfasmIkarmanb¤sEtmYyKt;KW x 2 x
o
o
o
o
2
745
2
2
x
x
smIkarxagelIGacsresr ³ -krNI x 2 eyIg)an
x
x
2
x
2
3 3 5 5
4 4 5 5
ehIy edaybUkGgÁ nigGgÁ
x
x
x
2
3 4 3 4 5 5 5 5
2
edaybUkGgÁ nigGgÁ
x
x
x
2
x
2
4 4 5 5 x
x
x
x
3
3
3
x
smmUl naM[ x 3 minEmnCab¤sénsmIkar . -krNI x 3 eyIg)an 63 63 x
3
x
3
x
3
4 4 6 6
ehIy
x
smmUl 53 54 1 naM[ x 2 minEmnCab¤sénsmIkar . -krNI x 2 eyIg)an 53 53 ehIy
x
3 4 5 1 6 6 6
x
x
x
3 4 5 3 4 5 6 6 6 6 6 6
3 4 1 5 5
5 5 6 6
ehIynig edaybUkGgÁ nigGgÁ enaHeyIg)an ³ x
x
x
3
x
x
x
3
3 4 5 3 4 5 6 6 6 6 6 6 2
3 4 3 4 5 5 5 5
2
3 4 1 5 5
smmUl naM[ x 2 minEmnCab¤sénsmIkar . dUcenH smIkarmanb¤sEtmYyKt;KW x 2 .
3
3 4 5 1 6 6 6
smmUl naM[ x 3 minEmnCab¤sénsmIkar . dUcenH smIkarmanb¤sEtmYyKt;KW x 3 . 272 bgðajfa eKTaj)an x y z eyIgmansmPaB
edaHRsaysmIkar ³ x> 3 4 5 6 Binitü y z 2x z x 2 y x y 2 z eyIgsegáteXIjfa x 3 Cab¤sénsmIkar eRBaH tamrUbmnþ³ a b c a b c 2ab 2ac 2bc 3 4 5 6 216 216 Bit g)anktþanImYy²KW ³ eyIgnwgbgðajfasmIkarmanb¤sEtmYyKt;KW x 3 enaHeyI y z 2 x y z 4 x 2 yz 4 xy 4 xz smIkarxagelIGacsresr ³ 63 64 56 1 z x 2 y z x 4 y 2 xz 4 yz 4 xy yz zx x y 2
x
x
x
x
2
2
y z 2x z x 2 y x y 2z 2
2
2
2
3
3
3
2
2
2
2
2
2
3
x
-krNI x 3 eyIg)an ehIy
x
x
3
x
3
3 3 6 6
4 4 6 6 x
5 5 6 6
ehIynig edaybUkGgÁ nigGgÁ enaHeyIg)an ³
2
2
2
2
2
2
2
2
x
2 2 2 2 x y 2 z x y 4 z 2 xy 4 xz 4 yz
y z 2 x 2 z x 2 y 2 x y 2 z 2 6 x 2 6 y 2 6 z 2 6 xy 6 yz 6 xz
1
Binitü ³ y z 2 z x 2 x y 2
3
y 2 2 yz z 2 z 2 2 xz x 2 x 2 2 xy y 2 2 x 2 2 y 2 2 z 2 2 xy 2 xz 2 yz
edayeFVIkarpÞwm 1 & 2 enaHeyIg)an
746
2
6 x 2 6 y 2 6 z 2 6 xy 6 yz 6 xz
274 k> RsaybBa¢ak;fa sin12a cota cot2a Binitü ³ cota cot2a
2 x 2 2 y 2 2 z 2 2 xy 2 yz 2 xz
eRkayBIeFVIkarTUTat; eyIg)anliT§plKW ³
4 x 2 4 y 2 4 z 2 4 xy 4 yz 4 xz 0 2 x 2 2 y 2 2 z 2 2 xy 2 yz 2 xz 0 x 2 xy y y 2 yz z z 2 xz x 0 2
2
2
2
2
2
x y y z z x 2
2
2
¬rUbmnþ cos 2a 2 cos a 1 , sin 2a 2 sin a cos a ¦ 2
cosa 2 cos2 a 1 sin a 2 sin a cosa 2 cos2 a 2 cos2 a 1 2 sin a cosa 1 sin 2a
0
plbUkxagelIepÞógpÞat;manEtmYykrNIKt;KW ³ x y 2 0 x y 0 x y 2 y z 0 y z 0 y z z x 0 z x 2 z x 0
naM[eyIg)an ³ x y z dUcenH eKTaj)an x y z
dUcenH bgðaj)anfa sin12a cota cot2a .
.
x> KNnaplbUk Sn
273 KNna S ab cd eyIgman ³
2011ab cd S ab cd 2011 2011ab 2011cd 2011 2 c d 2 ab a 2 b 2 cd 2011 2 abc abd 2 a 2 cd b 2 cd 2011 2 2 abc a cd abd 2 b 2 cd 2011 acbc ad bd ad bc 2011 bc ad 0 0 2011
a 1 sin a cot 2 cot a 1 cot a cot a a 4 2 sin 2 a a 1 cot cot a 8 4 sin 4 ................................... 1 cot a cot a a 2 n1 2n sin n 2 a S n cot n1 cot a 2 a S n cot n1 cot a 2
eRBaH smµtikmµ ac bd 0 dUcenH KNna)an S 0
1 1 1 1 1 a a a a sin a sin sin 2 sin 3 sin n 2 2 2 2
tamsRmaybBa¢ak;sMNYr k> eyIg)an ³
cos a cos 2a sin a sin 2a
dUcenH eRkayBIKNna .
x cot 1 2 1 cos x cot x 1 cos x 1 1 cos x cos x
275 k> RsaybBa¢ak;fa Binitü ³
747
.
¬rUbmnþ cos 2a 2 cos
2
a 1 cos 2a 1 2 cos 2 a
x x x 2 cos2 2 cos2 sin 1 cos x 1 2 2 2 1 x cos x cos x cos x cos x sin 2 x x cos sin x cos sin x 2 2 x cos x x cos x sin sin 2 2 x cot x 2 tan x cot 2 cot x
dUcenH Rsay)anfa
¦ 276 KNna x , y , z nig t smµtikmµ ebIeyIg[pleFob 2x 3y 4z 6t a Edl a 0 eRBaH bYncMnYn x , y , z nig t viC¢man Taj)an ³ x 2a , y 3a , z 4a , t 6a naM[ yztx ztxy txyz xyzt 14625 )anCa 72a 3 48a 3 36a 3 24a 3 14625 2a 3a 4a 6a 36a 2 16a 2 9a 2 4a 2 14625 65a 2 14625
x 1 2 1 cos x cot x cot
.
a 2 225
eyIg)an ³
x> KNnaplKuN 1 1 1 1 Pn 1 1 cos x cos x cos x 2 22
1 1 cos x 2n
tamsRmaybBa¢ak;sMNYr k> eyIg)an ³
Pn tan x cot
x 2
n 1
x 2a y 3a z 4a t 6a
x 2 15 x 30 y 3 15 y 45 z 4 15 z 60 t 6 15 t 90
dUcenH KNna)an x 30 , y 45 , z 60 , t 90 .
x cot 1 2 1 cos x cot x x cot 1 4 1 x x cos cot 2 2 x cot 1 8 1 x x cos cot 4 4 ................................ x cot n1 1 2 1 x x cos n cot n 2 2 x cot n1 2 tan x cot x Pn cot x 2 n1
dUcenH eRkayBIKNna
a 15
276 k> kMNt;témø x edIm,I[ P mantémøGb,brma eyIgman ³ P x x 1 ¬edayeFVIkarEfmfytY¦ 2
P x2 x 1 2
2
1 1 1 x2 2 x 1 2 2 2 2
2
1 1 4 1 3 x x 2 4 4 2 4
edIm,I[ P mantémøGb,brma luHRtaEtkenSam 2
1 1 1 x 0 x 0 x 2 2 2 1 3 x P 4 2
dUcenH témø eFVI[ mantémøGb,brma . x> kMNt;témø x edIm,I[ Q mantémøGtibrma eyIgman Q 2 x x Gacsresr ³ 2
.
Q 2 x2 x 2 x2 x
748
n 2a 1 n 3b 1 n 4c 1 n 5d 1 n 6e 1
2 2 2 1 1 1 Q 2 x 2 x 2 2 2 2
1 1 2x 2 4 9 1 x 2 2
2
n 1 2 a n 1 3b n 1 4c n 1 5d n 1 6e
naM[ n 1 RtUvEtEckdac;CamYy 2, 3, 4, 5 , 6 edIm,I[ Q mantémøGtibrma luHRtaEtkenSam mann½yfa n 1 CaBhuKuNén 2, 3, 4, 5 , 6 1 1 1 eday PPCM 2 , 3, 4 , 5 , 6 2 3 5 60 x 0 x 0 x 2 2 2 dUcenH témø x 12 eFVI[ Q 94 mantémøGtibrma . naM[BhuKuNbnþbnÞab;én 2, 3, 4, 5 , 6 KW ³ 60 ,120 ,180 , 240 , 300 , 360 , 420 , ... enaHnaM[ n 1 60 , 120 , 180 , 240 , 300 , 360 , 420 , ... 278 rkcMnYnKt;tUcbMput n n 61 , 121 , 181 , 241 , 301 , 361 , 421 , ... ebI a , b , c , d , e CacMnYnKt; Etsmµtikmµ n Eckdac;nwg 7 tambRmab;eyIg)an ³ ehIy kñúgcMeNamtémø n nImYy² tamlMdab;BItcU eTAFM n 2 a 1 n 2 a 1 manEttémø 301 eT EdltUcCageK nig Eckdac;nwg 7 n 3b 2 n 3b 2 n 4c 3 n 4c 3 dUcenH cMnYnKt;tUcbMputEdlRtUvrkKW n 301 . 2
2
n 5d 4 n 5d 4 n 6e 5 n 6e 5 n 1 2 a 2 n 1 2a 1 n 1 3b 3 n 1 3b 1 n 1 4 c 4 n 1 4c 1 n 1 5d 5 n 1 5d 1 n 1 6e 6 n 1 6e 1
280 RsaybBa¢ak;fakenSam E mantémøefr rMlwk cos x sin x 1 2
2
a 3 b 3 a b a 2 ab b 2
eyIgman
naM[ n 1 RtUvEtEckdac;CamYy 2, 3, 4, 5 , 6 mann½yfa n 1 CaBhuKuNén 2, 3, 4, 5 , 6 smµtikmµ rkEttémø n EdltUcbMput naM[ n 1 PPCM (2 , 3, 4 , 5, 6) 2 3 5 60 Taj)an n 60 1 59 dUcenH cMnYnKt;tUcbMputEdlRtUvrkKW n 59 . 279 rkcMnYnKt;tUcbMput n ebI a , b , c , d , e CacMnYnKt; tambRmab;eyIg)an ³ 2
E cos6 x sin 6 x 3sin 2 x cos2 x
cos x sin x cos 3
3
cos2 x sin 2 x 3 sin 2 x cos2 x 2
2
4
x cos2 x sin 2 x sin 4 x
3 sin 2 x cos2 x cos4 x cos2 x sin 2 x sin 4 x 3 sin 2 x cos2 x cos4 x 2 cos2 x sin 2 x sin 4 x
cos2 x sin 2 x
2
12 1
dUcenH RKb;témørbs; x eyIg)antémø E 1efrCanic© . 749
281 edaHRsayRbB½n§smIkar eyIgmanRbB½n§smIkar x xy y 2 9 3
3
tamsmIkar 2 Taj)an x 2y
283 sresr N CaplKuNktþadWeRkTI1 eyIgman N 3a 1 4a 6a 9
1 2
2
3a 1 2 2 a 3 3a 1 2a 33a 1 2a 3 2
3 CMnYskñúg 1
dUcenH sresr)an N a 75a 5
8 y6 9 y3
tag y t y t eyIg)an t 9t 8 0 smIkarman a b c 1 9 8 0 naM[ t 1 , t 8 cMeBaH t 1 y 1 y 1 enaHtam 3 : x 2y 12 2 cMeBaH t 8 y 8 y 2 enaHtam 3 : x 2y 22 1 6
2
2
2 2 4a 1 10 2 24a 1 1024a 1 10 2
2
8a 2 108a 2 10 8a 88a 12
8a 1 42a 3
3
32a 12a 3
eday N 32a 12a 3 CaBhuKuNén 32
3
dUcenH
N
CacMnYnRtUvEtEckdac;nwg 32
285 KNnatémøelxénkenSam eK[ a b 1 nigkenSam
dUcenH RbB½n§smIkarmanKUcemøIy x 2 , y 1 , x 1, y 2
.
2011
2010
2009
2008
2
2
2
2
2
2
a 2 2ab b 2 1
a 2 2ab b 2 1 a b 1 2
34018 3 x 4018 x
x 4018
2
2a 2 2ab 2b 2 3a 2 3b 2 1
x
2 2 32010 2008 4 3 x
dUcenH smIkarmanb£s
³
2a b a ab b 3a b 1 2a ab b 3a b 1
32010 3 1 32008 3 1 4 3 x
P
.
P 2 a 3 b3 3 a 2 b 2 1 2
282 edaHRsaysmIkar ³ eyIgman 3 3 3 3 4 3
.
284 Rsayfa N Eckdac;nwg 32 ³ eyIgman N 44a 1 100
y6 9 y3 8 0 3
2
3a 1 2a 63a 1 2a 6 a 7 5a 5
3
2 y 3 9 y 8 y3 9 3 y
eyIg)an
2
12 1 1 1
.
0
dUcenH témøelxénkenSamKNna)an 750
P0
.
286 rkBIrcMnYnKt;viC¢man a nig b ³ eyIgman a b 24 eyIg)an a ba b 24 ebI a nig b CacMnnY Kt;enaH a b nig a b k¾Ca cMnYnKt;Edr EdleyIgnwgrkplKuNBIrcMnYnKt;esµI 24 2
eday naM[ b¤ b¤ b¤
2
1 24 2 12 24 3 8 4 6 a b 1 a b 24 a b 2 a b 12 a b 3 a b 8 a b 4 a b 6
dUcenH
nig
288 KNnakenSam A
A
1
2
3
1
2
3
x 1x 2 2 x 3 x 1 x x 3
x 1x 2 x 2x 3 x 1x 3 x 3 2x 1 3x 2 x 1x 2x 3 x 3 2 x 2 3x 6 x 1x 2x 3 1 x 1x 2x 3
a b a b
1 minykeRBaH a nig b minCacMnYnKt; dUcenH kenSamKNna)anKW A . x 1x 2x 3 edaHRsay)anKUcemøIy a 7 , b 5 minykeRBaH a nig b minCacMnYnKt; 289 rkmYycMnYnenaH tag x CamYycMnYnEdlRtUvrkenaH edaHRsay)anKUcemøIy a 5 , b 1 tambRmab;RbFan eyIg)antémø 3x pÞúyBI x mann½yfa 3x x smIikarmanKUcemøIy ba 57 b ba 15 . edaHRsay 3x x 2
2
2
3x x 2 0 x3 x 0
287 KNnaRCugrbs;kaernImYy² tag a CargVas;RCugkaerFM ¬KitCa m ¦ b CargVas;RCugkaertUc ¬KitCa m ¦ tambRmab;RbFan eyIg)anRbB½n§smIkar a 2 b 2 1152 a b 16 16a b 1152 a b 16
x0 3 x 0 x 0 x 3
a b a b 1152 a b 16 a b 72 a b 16
edaHRsaytamviFIbUkbM)at; eyIg)an ³ a b 72 a b 16 2a 88 a 44
enaH b 44 16 28
dUcenH RbEvgRCugkaernImYy²KW 44 m , 28 m .
cMeBaH x 0 minykeRBaH 0 KµancMnYnpÞúyeT dUcenH cMnYnEdlRtUvrkenaHKW
3
.
290 rkelxEdlenARtg;TItaMgtYTI 2011 eyIgmanelxsresrBI 1 dl;elx 999 KW N 12345678910111213...998999
edIm,IrkelxenARtg;TItaMgtYTI 2011 eyIgsikSatamEpñk²dUcbgðajxageRkam ³ 751
-BIelx 1 9 mancMnYntY ³ 91 9 tY -BIelx 10 99 mancMnYntY ³ 90 2 180 tY -BIelx 100 199 mancMnYntY ³ 100 3 300 tY -BIelx 200 299 mancMnYntY ³ 100 3 300 tY -BIelx 300 399 mancMnYntY ³ 100 3 300 tY -BIelx 400 499 mancMnYntY ³ 100 3 300 tY -BIelx 500 599 mancMnYntY ³ 100 3 300 tY -BIelx 600 699 mancMnYntY ³ 100 3 300 tY ebIeyIgsrubtYBIelx 1 699 vamancMnYntYKW 9 180 6 300 1989 tY eyIgRtUvkar 22 tYeTot edIm,IbEnßm[RKb;tYTI 2011 ehIytYbnþbnÞab;eTot bnÞab;BI 699 enaHKW ³ 700701702703704705706707708709800...999
tYTI 1990
tYTI 2000
x2 2 x 2 x x 2
x 2 1 x 1 f f x x 2 x x
x2 x x2 1 f f x 2 x x 2 x
edayyksmIkar 1 3 enaHeyIg)an ³ 1 x2 x 1 f x f x x 1 2 2 f 1 x f x x x x 2 x 2 x 2 x x2 x x2 x 1 f x x 1 2 x 2 x
2 x
4 x 2 x 2 x f x 2 x 2 2 xx 2 x 1 f x 2 x 1 2
3
2
>
2 x 1 2 2 x x 1 1 f x x x 1 1 1 f x x x 1
f x
tYTI 2011
eday[témø x 1 , 2 , 3 , ... , 2011 eyIg)an ³
291 KNnaplbUk S ³ S f 1 f 2 f 3 f 2011 1 x2 x 1 f x f x 1 x 1 x x 1 1 1 1 1 2 1 f f 1 x x x 1/ x x
1 1 f 1 1 2 f 2 1 1 2 3 1 1 f 3 3 4 ..................... f 2011 1 1 2011 2012 1 1 S 1 2012 2012 1 2011 S 2012 2012
1
enaHeyIg)an ³
12 x 1 1 x f f x x x x x 2 x x2
1 x 1 f f x x x
2 2
edayKuNGgÁTaMgBIrénsmIkar 2 nwg 2x x edIm,I [emKuNén f 1x esµInwg 1 eyIg)an ³
x 2 x x 2 x 12 x x x 2 f x 2 x 2 x 2 3 2 2 2 4 x 2 x 2 x x x f x x 3x 2 x x 2 2
dUcenH elxEdlenARtg;TItaMgtYTI 2011 KWelx 7 .
eyIgman ebIeyIg CMnYs eday
3
dUcenH plbUkKNna)an 752
S
2011 2012
.
292 rkcMnYnKUbticbMputEdlRtUverobkñúgRbGb; edIm,I[)ancMnYnKUbticbMptu erobenAkñúgRbGb; enaHmaD rbs;KUbRtUvFMbMput. enaHeyIgRtUvrkRCugKUbFMbMput EdlvimaRtTaMgbIrbs;RbGb;Eckdac; nwgRCugrbs;KUb. -mann½yfa RCugrbs;KUbCatYEckrYmFMbMptu énvimaRt rbs;RbGb; EdleyIgnwgkMNt; -ebItag a CaRCugrbs;KUb ¬KitCa mm ¦ naM[ a PGCD 180 , 60 , 90 2 3 5 30 eRBaH 180 2 3 5 2
2
600 2 3 3 5 2 90 2 32 5
naM[ maDKUbesµI V 30 27 000 mm maDRbGb;esµI V 180 600 90 9720 000 mm eyIg)an cMnYnKUbEdlerob)anKW 9720000 360 KUb 27000 3
3
294 rkBIrcMnYnKt; a nig b tambRmab; ³ eyIg)anRbB½n§smIkar a b 15n Edl n 1 , 2 , 3 , ... eRBaH a b a b 45 2
2
a b 15n a b a b 45 a b 15n na b 3
10 5 y
y2
a b 15n a b 3 n
a b 15 a b 3 2a 18 a 9 a b 45 n3 a b 1 a b 45 a b 1 2a 46 a 23
ehIy b 6
dUcenH cMnYnKUbticbMputEdlGacerob)anKW 360 KUb . -cMeBaH
12 x 2 y 8 xy 2 x 2 y 3xy 10 x 5 xy
a b 15n 15na b 45
ebI a nig b CacMnnY Kt;enaH a b k¾CacMnYnKt;Edr enaH n3 CacMnYnKt; mann½yfa 3 RtUvEtEckdac;nwg n eyIgTaj)antémø n KW n 1 b¤ n 3 -cMeBaH n 1 ³ aabb153 edaHRsaytambUkbM)at;
3
293 edaHRsayRbB½n§smIkar eyIgmanRbB½n§smIkar 26xx2yy43xyxy 12 edaysikSapldksmIkar ³ 2 2 1
³
edaHRsaytambUkbM)at; ehIy b 22
dUcenH RbB½n§smIkarmanKUcemøIyBIrKUKW a 9 , b 6 b¤ a 23 , b 22 . 295 bgðajfa A nwg x, y , z eyIgman
cMeBaH y 2 tamsmIkar 1 eyIg)an ³ 2 x 2 2 3x 2 2x 4 6x 4x 4 x 1
2
B 2 C 2 ABC
2
minGaRs½y
2
y z y z y z A A 2 2 z y z y z y B
dUcenH RbB½n§smIkarmanKUcemøIy x 1 , y 2 .
z x z B2 x z x
2
2
2
2
x z x 2 z x z 2
2
x y x y x y C C 2 2 y x y x y x
753
2
naM[
10VC VB t 100 V 10 B VB VC t 90 VC 9 9
A2 B 2 C 2
2
2
2
2
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y z z x x y 2 2 2 2 2 2 2 2 2 z y x z y x
6
ehIy
y ABC z y x
x x y z y x xy z 2 x x y z 2 xy y y x
V A t 100 t
2
2
2
y2 z2 z2 x2 x2 y2 z2 y2 x2 z2 y2 x2
y2 x2 y2 z2 z2 x2 2 2 2 2 2 2 2 x z z y x y 4
dUcenH A
2
B 2 C 2 ABC 4
B
A
C
B
V A t 100 V 9V 10 A VB A VB t 90 VB 9 10
i
19 m
.
297 edaHRsayRbB½n§smIkar x y z xyz x y z xyz x y z xyz
1 2 3
pÞwm 1 & 2 eyIg)an ³ x yz x yz
i
x y
y z ii
z x iii
pÞwm 2 & 3 eyIg)an ³ x yz x yz
pÞwm 1 & 3 eyIg)an ³ x yz x yz
tam i , ii nig iii eyIg)an x y z edayyk x y z CMnYskñúgsmIkar 1 eyIg)an ³ x x x xxx x x3
x3 x 0
x x2 1 0
C
100 100 VC 81 m 100VC VA 81
dUcenH cm¶ayKmøatBI C eTA A KW
bRmab;bnþ ³ xN³eBl B rt;dl;TI C enAxVH 10 m naM[ V t 100 3 nig V t 90 4 eyIgeFVIpleFob 3 nig 4 eyIg)an ³ B
100 VA
mann½yfa xN³eBl t Edl A rt;dl;TIcm¶ay100 m eXIjfa C rt;)anEtcm¶ay 81 m b:ueNÑaH naM[ KmøatBI C eTA A KW 100 m 81 m 19 m
minGaRs½y x, y , z . eyIgman
296 rkcm¶ayKmøatBI C eTA A xN³eBl A rt;dl;TI tag V , V , V Cael,ÓnerogKñaén A , B , C t Cary³eBlEdl A rt;dl;TI t Cary³eBlEdl B rt;dl;TI bRmab; ³ xN³eBl A rt;dl;TI B enAxVH 10 m naM[ V t 100 1 nig V t 90 2 eyIgeFVIpleFob 1 nig 2 eyIg)an ³ A
d VC t VC
2
A2 B 2 C 2 ABC 6
100VC 81
ebI d Cacm¶ayEdl C rt;)ankñúgry³eBl t enaH
y x y z z x 1 2 2 2 2 2 2 1 x z z y x y y2 x2 y2 z2 z2 x2 2 2 2 2 2 2 2 x z z y x y 2
VA
ebI t Cary³eBlEdl A rt;dl;TI enaH
z z y x
2
eyIg)an
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y2 z2 z2 x2 x2 y2 z2 y2 x2 z2 y2 x2
ii
xx 1x 1 0 754
naM[
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x 0 x 1 x 1
dUcenH RbB½n§smIkarmanRkumcemøIybIKW x y z 0 b£ x y z 1 b£ x y z 1 298 KNnaplbUk S ³ eK[ a b c S
a
b
c
a
b
c
a b a c b c b a c a c b
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y x 1 y x 25 2 y 26
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y x 2 y x 8 2 y 10
dUcenH témøplbUkKNna)anKW
enaH S 0
2
y5
x y 2 52 3
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.
299 rkRKb;bNþaKUéncMnYnKt;viC¢man x , y eyIgman x x 13 y naM[ y x x 13 ¬KuNGgÁTaMgBIrnwg 4 ¦ 2
y 13
2
2
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n
4 y 2 4 x 2 4 x 52
2 n 5 n n 1
4 y 2 4 x 2 4 x 1 52 1
2 n 5 n 5 n 1
10 n 5 n 1
4 y 2 4 x 2 4 x 1 51
eday -sV½yKuNén 5 CacMnYnmanelx 5 xagcugCanic© naM[ 5 CacMnYnEdlmanelx 5 xagcugCanic© -ehIy 10 manelx 0 cMnYn n dg naM[plKuN 10 5 manelx 0 cMnYn n dg
2 y 2 2 x 12 51 2 y 2 x 12 y 2 x 1 51 eday x , y CacMnYnKt;viC¢man
n 1
n
naM[ 2 y 2x 1 2 y 2x 1 ehIyplKuNBIrktþaéncMnYnKt;esµInwg 51 KW ³
n1
n
dUcenH cMnYn A 2 cMnYn n dg
1 51 51 3 17
755
n
52 n1
bBa©b;edayelx 0 .
301 edaHRsaysmIkar xa xb xc 1 1 1 2 bc ac ab a b c ax a bx b cx c bc ac ab 2 abc abc ax a bx b cx c 2ab bc ac ax a 2 bx b 2 cx c 2 2ab 2bc 2ac
a b c x a 2 b 2 c 2 2ab 2bc 2ac a b c x a b c 2 a b c 2 x a b c x a b c
dUcenH smIkarman
x a bc
301 rkRkLaépÞTaMgGs;rbs;sUlIt ³ eyIgmanKUb 7 pÁúMP¢ab;Kña)anCa sUlItmanmaD 448 cm naM[KUbnImYy²manmaD V 4487cm 64 cm Taj)anRTnugrbs;KUb a V 64 cm 4 cm KUbnImYy²manépÞ 6a -Kitfa ³ KUb7 manKUbmYyenAkNþaleK )at;épÞGs; nigKUb6 eTot)at;épÞGs;1dUcKña naM[épÞsUlIt ³ 3
3
3
3
3
3
2
S 7 6a 2 6a 2 6 a 2
42a 2 12a 2
Cab£s .
30a 2
a 4 cm
,
30 4 cm 30 16 cm 2 480 cm 2 2
302 rkb£sKUbén Z ³ eyIgman Z 8 , 4 manTRmg; Z r , Edl r Cam:UDul nig CaGaKuym:g; b£sKUbén Z KW 2k Z r , Edl k 0 , 1 , 2 3 3
k
/ 4 2 0 3 0 8 , 3 2 , 12 / 4 2 1 Z1 3 8 , 3 2 3 2 , 2 , 4 12 3 / 4 2 2 Z2 3 8 , 3 4 17 2 , 2 , 12 12 3
-cMeBaH k 0 ³ Z -cMeBaH k 1 ³ -cMeBaH k 2 ³
dUcenH b£sKUbén Z KNna)anKW ³
dUcenH épÞRkLaTaMgGs;rbs;sUlItKW S 480 cm . 2
304 bgðajfa 2AM AB AC A -eyIgman AM Caemdüan N naM[ cMNuc M kNþal BC -ykcMNuc N kNþal AB B M 1 naM[ AN AB 2 -kñúgRtIekaN ABC man MN Ca)atmFüm 2 naM[ MN AC 2 -edaybUk 1 2 eyIg)an ³ AN MN
AB AC 2 2
-Et kñúgRtIekaN AMN tamvismPaBénRtIekaN eyIg)an AM AN MN enaHeyIgTaj)an ³ AM
3 17 Z 0 2 , , Z1 2 , , Z1 2 , 4 12 12
C
AB AC 2 2
. dUcenH eyIgeXIjfa 756
2 AM AB AC
2 AM AB AC
.
305 KNnael,Ón ry³eBl nigcm¶aycr rMlwkrUbmnþ cm¶aycrKW ³ d v t tambRmab;RbFan ³ -ebIbEnßmel,Ón 3 km/ h eTAdl;muneBlkMNt; 1 h naM[)an ³ v 3t 1 d 1 -ebIbnßyel,Ón 2 km/ h eTAdl;eRkayeBlkMNt; 1 h naM[)an ³ v 2t 1 d 2 tam 1 & 2 eyIg)anRbB½n§smIkar ³ v 3t 1 d v 2t 1 d v 3t 3 d vt v 2t 2 d vt
eRBaH
vt v 3t 3 d vt v 2t 2 d
v 3t 3 0 v 2t 2 0
d vt v 3t 3 0 v 2t 2 0 t 5 0 t 5 h
bUkGgÁnigGgÁ ³ cMeBaH v 2t 2 0 v 2 5 2 0 v 12 0
naM[
307 rkcMnYnénkarcab;édKñaTaMgGs; karcab;édBMumanRcMEdleLIy enaHeyIgKiteXIjfa ³ GñkTI1 cab;)an 9 dg / GñkTI2 cab;)an 8 dg GñkTI3 cab;)an 7 dg / GñkTI4 cab;)an 6 dg >>> rhUtdl;GñkTI10 )an 0 dg naM[ cMnYndgénkarcab;édKñaKW ³ 9 8 7 6 5 4 3 2 1 0 45 dg -GñkGaceFVItamrebobmü:ageTotKW ³ mnusSman 10nak; ehIycab;édKñaBIr²nak; mann½yfa³ vaCabnSM én10Fatu eRCIserIsyk2Fatu naM[ C 10 , 2 10 10 2!!2! 108!92!8 ! 45 dg dUcenH cMnYndgénkarcab;édTaMgGs;KW
308 RsaybBa¢ak;fa 2 3 Eckdab;nwg eyIgman 2 3 2 3 9
9
.
45
9
3 3
9
35
3 3
8 27 2 2 3 3 35 2 2 3 3
v 12 km / h
2 3 33 2 6 2 3 33 3 6
d v t 12 5 60 km
6
6
3
3
3
3
6
6
dUcenH eyIgKNna)an el,Ón v 12km/ h eXIjfalT§plén 2 3 CaBhuKuNén 35 ry³eBl t 5h cm¶aycr d 60km . enaH 2 3 RtUvEtEckdac;nwg 35 9
9
9
9
306 KNnamYycMnYnenaH tag x CamYycMnYnEdlRtUvrkenaH tambRmab;RbFan eyIg)an ³ x x x x b£ x 2x naM[ x 2x 0 x 0 xx 2 0 Taj)an x2
dUcenH cMnYn 2
dUcenH cMnYnEdlRtUvKNnaKW 0 b£
dUcenH eyIgsRmYl)an E aa3bb .
2
2
.
39
Eckdac;nwg 35 .
309 sRmYlRbPaKsniTan E ³ 2b eyIgman E aa 35ab ab 6b
2
9
757
2
2
2
2
a b a 2b a b a 2b a 3b a 3b
310 KNnargVas;RCugnImYy² 312 KNnargVas;RCugénRbelLÚRkam ABCD ³ C D tag x , y , z CargVas;RCugTaMgbIénRtIekaN KitCa cm tag h 5 RtUvnwg)at AB h 7 RtUvnwg)at AD A tambRmab;RbFan eyIg)an 3x 4y 5z B eyIg)an épÞRkLaRbelLÚRlam ABCD KW tamlkçN³smamaRt eyIg)anpleFob S AB h b¤ S AD h x y z x y z 24 2 3 4 5 3 4 5 12 h AD AB h AD h Taj)an naM [ h AB eRBaH brimaRt x y z 24 cm h AB 5 hAD 7
AB
AD
ABCD
AB
ABCD
AD
AB
AB
AD
AD
eyIg)an
x 2 3 y 2 4 z 2 5
x6
cm
y 8
cm
z 10 cm
tambRmab; smamaRtkm
5 AB 7 AD 0 1
bRmab;brimaRt
2 AB AD 48
cm
AB AD 24
2
dUcenH rgVas;RCugénRtIekaNKW 6cm , 8cm , 10cm .
tam 1 & 2 eyIg)anRbB½n§smIkar ³ 5 AB 7 AD 0 bUkbM)at;edayyksmIkar 2 7 AB AD 24
311 KNnacMnYnKt;enaH tag ab CacMnYnKK;EdlmanelxBIrxÞg;enaH naM[eyIg)an ³ 0 a 9 , 0 b 9 tambRmab;RbFan eyIgcg)anRbB½n§smIkar ³ Binitü
ab 9 ba ab 63
h AB 5 h AD 7
5 AB 7 AD 0 7 AB 7 AD 168 12 AB 168 AB 14 5 AB 5 14 5 AB 7 AD AD 10 7 7
eyIg)an
eday dUcenH rgVas;RCugRbelLÚRkameRkayBIKNna)an KW ³ AB 14 cm , AD 10 cm .
ba ab 10 b a 10 a b
10b a 10a b 9a 9b
9 a b
313 rktémø x edIm,I[ P mantémøtUcbMput eyIgman P x 1x 2x 3x 6
enaHeyIg)an RbB½n§smIkar eTACa ³ ab 9 9 a b 63
ab 9 a b 7 ab 9 a b 7 2b 16
x 1x 6 x 2 x 3
x 5 x 6 x 5 x 36 eday x 5x 0 naM[ x 5x 36 36 enaHtémø P x 5x 36 P 36 x 2 5x 6 x 2 5x 6
tamviFIbUkbM)at; b8 Taj)an a 9 b a 9 8 a 1 dUcenH cMnYnKt;enaHKW ab 18 .
2
2
2
2
2
2
2
758
2
2
2
2
mann½yfa P mantémøtUcbMputesµI 36 cMeBaH x 5x 0 xx 5 0 naM[ x 0 , x 5
315 bgðajfaplKuNénBIrcMnYnenaHtUcCag b¤esµI 14
2
dUcenH
CatémøEdleFV[I . P 36 CatémøtUcbMput
x 0 , x 5
tag a nig b CaBIrcMnYnenaH tambRmab; a b 1 ¬elIkGgÁTaMgBIrCakaer¦ eyIg)an a b 1 2
2
a 2 2ab b 2 1
2ab 1 a 2 b 2
314 rkcMnYnmnusS nigcMnYnekAGI -rebobTI1 ³ ¬tamRbB½n§smIkar ¦ tag x CacMnYnmnusS nig y CacNYnekAGI Edl x nig y CacMnYnKt;
1
mü:ageTot ebI a nig b CaBIrcMnYnenaHeyIg)an ³ a b2 0 a 2 2ab b 2 0
x y 4 y 4 x 2 x y 4 x 4 y 1 x 2 y 8 x 2 y 8 2
tambRmab;RbFan eyIg)anRbB½n§smIkar Taj[gay edaypÞwm 1 & 2 enaHeyIg)an ³
2ab a 2 b 2
2
edaybUkGgÁngi GgÁén 1 nig 2 eyIg)an ³
2ab 1 a 2 b 2 2 2 2ab a b 4ab 1
ab
1 4
dUcenH plKuNénBIrcMnYnenaHtUcCag b¤esµI 14 .
2y 8 4 y 2y y 4 8 y 12
316 rképÞkaerEdlcarwkkñúgrgVg;TaMgmUl r tam 1 : x 4 y x 4 12 x 16 tag s nig a CaépÞ nigRCug énkaertUc dUcenH mnusSmancMnYn 16 nak; /ekAGImancMnYn 12 . S nig b CaépÞ nigRCug énkaerFM r tambRmab; ³ s 100 cm -rebobTI2 ³ ¬tamsmIkar¦ naM[ a s 100 enaH a 10 cm tag x CacMnYnmnusS enaHcMnYnekAGIKW x 4 tamRTwsþIbTBIrtaK½r r CaRbEvgGIub:Uetnus ¬kaMrgVg;¦ Edl x CacMnYnKt; a 10 enaH r a 10 125 cm tambRmab;RbFaneyIg)ansmIkar ³ 2 2 x nig 2r b b 4r 2b b 2r x 4 4 2 eyIg)an épÞRkLakaerFMKW S b x 2 x 16 x 16 naM[ S 2r S 2 125 S 225 cm naM[ cMnYnekAGIKW x 4 16 4 12 dUcenH épÞkaercarwkkñúgrgVg;TaMgmUlKW S 225 cm . dUcenH mnusSmancMnYn 16 nak; /ekAGImancMnYn 12 . a
a 2
2
b
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
759
317 kMNt;témøéncMnYnBit a nig b ³ eyIgman a b 0 RKb;témøéncMnYnBit a nig b enaHeyIg)an ³ 2
x 2 y 2 z 2 52 x 2 y 2 z 2 81 29 2 2 2 2 29 x y z 52 81 x y 2 z 2
2
29x y z 29 52 81x y z 81x y z 29x y z 29 52 52x y z 29 52
a 0 , b 0 2
2
2
2
2
2
2
2
2
2
2
2
2
2
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2
2
2
2
2
2
2
2
2
2
2
2
2
2
4 y2 1 9 z2 1 16
318 edaHRsayRbB½n§smIkar eyIgman
x y z 2 3 4 xy yz zx 26
1 2
tamsmamaRtenaHeyIg)an 1 :
x y z x yz x yz 2 3 4 23 4 9 x y z x yz 2 3 4 9 2 2 2 2 x y z x y z 4 9 16 81 2 2 2 2 x y z x y 2 z 2 2xy yz zx 4 9 16 81 2 2 2 2 2 2 x y z x y z 2 26 4 9 16 81 2 2 2 2 2 x y z x y z 2 52 i 4 9 16 81
eRBaHtam 2 : xy yz zx 26 EttamsmamaRt 2
2
2
2
y2 9
y 3
z 2 16
z 4
dUcenH RbB½n§smIkarmanKUcemøIyRtUvKñaKW ³ x , y , z 2 , 3, 4 b¤ x , y , z 2 , 3, 4 . 319 KNnatémøKt; énkenSamelx E ³ eyIgman E 2 5 2 5 naM[ E 2 5 2 5 3
3
3
3
3
3
2
2 5 3 3 2 5 3 2 5 2
3 3 2 5 3 2 5 2 5 4 33 2 5 3 2 5 3 2 5 3 2 5 4 33 2 2 5 2 E E 3 4 3E
x y z x y z 4 9 16 4 9 16 2 2 2 x y z x2 y2 z2 4 9 16 29 2
eyIg)an E
2
3
3E 4 0
E 1E 2 E 4 0
naM[ EE 1E 0 4 0 eRkayBIedaHRsaymanEttémøKt; E 1 b:ueNÑaH
ii
tam i nig ii enaHeyIg)an ³
2
dUcenH KNnatémøKt;)anKW E 1 .
760
320 KNnaplbUkcRmaséncMnYnTaMgBIr tag a nig b CaBIrcMnYnenaH naM[plbUkcRmascMnYnTaMgBIrKW 1a b1 -rebobTI1 ³ eyIgKNna tamtRmUvPaKEbgrYmKW ab naM[ 1a b1 aab b edaybRmab; plbUk a b 12 nigplKuN ab 4 eK)an 1a b1 aab b 124 3 enaH 1a b1 3 -rebobTI2 ³ edayEckGgÁ nigGgÁ eyIgman a b 12 edayEckGgÁTaMgBIrnwg ab eyIg)an ³ a b 12 1 1 12 3 eRBaH ab 4 ab ab a b 4
322 KNna A ³ eyIgman A 2 4 6 ... 2n
2 1 2 2 2 3 ... 2 n 2 2 2 ... 21 2 3 ... n 2 n n!
eRBaH
2 2 2 ... 2 2 n n dgénelx 2
nig
n! 1 2 3 ... n
dUcenH témøEdlKNna)an A 2 323 KNna eyIgman
1024
dUcenH plbUkcRmascMnYnTaMgBIrKW
321 KNnark r nig s ³ ebI x 3x 8 CaktþamYyénkenSam x 2
4
2 11 21 2 1 2 ... 1 2 2 11 2 1 2 1 2 ... 1 2 2 11 2 1 2 ... 1 2 2 11 2 ... 1 2
S 1 2 1 2 2 1 2 4 1 28 ... 1 21024
.
x 4 rx 2 s x 2 3x 8 x 2 ax b
4
2
2
4
4
4
8
8
8
...
enaH
rx 2 s
.
n!
³
S 1
2
1 1 3 a b
n
1024
8
1024
1024
1024
21024 1 1 21024
2 2048 1
x 4 ax 3 bx 2 3x 3 3ax 2 3bx
naM[
1024
8 x 2 8ax 8b
S 1 1024 22048 1 1 1024 2 2048
x 4 a 3x 3 b 3a 8x 2 3b 8a x 8b
1024 2 2
1024
22 4
edaypÞwmelxemKuNRtUvKñaén x enaHeK)an ³
dUcenH KNna)an
a 3 0 a 3 a 3 r b 3a 8 r b98 r7 3b 8a 0 3b 24 0 b8 s 8b s 8b s 64
324 KNna A ³ 2012 man A 1234568 1234567 12345679
eK)an
x 7 x 64 x 3x 8 x 3x 8 4
2
dUcenH témøKNna)anKW
2
r 7
2
1024
2
2012 1234568 1234568 11234568 1 2012 2012 2012 2 2 1 1234568 1234568 1
2
nig s 64 .
dUcenH KNna)an
.
S 1 4
761
A 2012
.
325 rkcMnYnKt;viC¢man n ³ eyIgman 4 n n 4n 4 4n
327 KNnarkelxxagcugén A ³ man A 2012 2010 2 n 2 2n eday 2010 manelxxagcug 0 enaHelxxagcugén n 2 2n n 2 2n 2010 2 GaRs½yelIelxxagcugén 2 n 2n 2n 2n 2 eday n enaHRtUvEt n 2n 2 n 2n 2 -Binitü lkçN³TUeTAcMeBaHsV½yKuNén 2 KW ³ ebI 4 n CacMnYnbzm enaHvaCaplKuNBIrktþa Edl cMeBaH n 1 , 2 , 3, 4 , 5 , 6 , 7 , 8 , ... enaHelxcug ktþaTI1esµInwg 1 nigktþaTI2esµInwgxøÜnÉg eyIg)an ³ én 2 KW 2 , 4 , 8, 6 , 2 , 4 , 8, 6 , ... manTRmg;TUeTA KW ³ 2 manelxxagcug 2 ebI n 4k 1 , k n 2n 2 1 n 2n 1 0 n 2n 2 4 n n n 2n 2 0 2 manelxxagcug 4 ebI n 4k 2 , k n 1 0 n 1 0 2 manelxxagcug 8 ebI n 4k 3 , k n 1 n n 2 0 n n 1 2n 1 0 , k 2 manelxxagcug 6 ebI n 4k n 1 n 1 eday 2013 4 503 1 manTRmg; n 4k 1 n 2 n 2 0 n 1 n 2n 2 0 naM[ 2 2 manelxxagcugKW 2 cMeBaH n 2n 2 0 man ³ 1 2 1 0 naM[smIkarKµanb£s dUcenH elxxagcugén A KW 2 . dUcenH cMnYnKt;viC¢manEdlrk)anKW n 1 . 328 rkRKb;cMnYnKt;viC¢man n ³ cMeBaH n 1 , 2 , 3, 4 , 5 , 6 , 7 , 8 , ... enaHeyIg)an 326 rkelxxagcugénplKuN ³ 2 KW 2 , 4 , 8 , 16 , 32 , 64 , 128, 256 , ... ehIy eyIgman 7 2013 7 2013 sMNl;Eckén 2 nwg 7 KW 2 , 4 , 1, 2 , 4 , 1 , 2, 4 , 1 ,... 7 7 2013 2013 EdlsMNl;énplEckenHmanTRmg;TUeTAKW ³ 7 2013 7 2013 -ebI n 3k 1 , k enaH 2 7 mansMNl; 2 7 2013 -ebI n 3k 2 , k enaH 2 7 mansMNl; 4 14091 -ebI n 3k , k enaH 2 7 mansMNl; 1 ...1 sV½yKuNénmYycMnYnEdlmanelx1xagcug esµInwgmYy cMeBaHkrNI 2 7 mansMNl; 1 Edl n 3k cMnYnEdlmanelx1 enAxagcugCanic©. naM[ 2 1 7 mansMNl; 0 Edl mann½yfa 2 1 Eckdac;nwg 7 kñúgkrNI n 3k dUcenH elxxagcugénplKuNKW 1 . dUcenH 2 1 Eckdac;nwg 7 kalNa n CaBhuKuNén 3 . 4
4
2
2
2
2013
2013
2
2
2
2
2013
2
2013
2
2
2
4
n
2
n
2
2
4
4
2
n
2
2
n
3
2
2
n
2
2
2
45031
2013
2
n
2012
2012
2010
2000
2012 2000 4012
2000
2012
2010
2002
n
2010 2002
4012
n
4012
n
4012
n
n
n
n
n
762
329 bgðajfa 12 2 Eckdac;nwg 10 ³ edIm,I[ 12 2 Eckdac;nwg 10 eyIgRKan;Etbgðaj faplkdén 12 2 manelxsUnüenAxagcug . elxcugén12 2 GaRs½yelIelxcugén 2 2 -sikSasV½yKuNén 2 KW ³ cMeBaH n 1 , 2 , 3, 4 , 5 , 6 , 7 , 8 , ... enaHelxcug én 2 KW 2 , 4 , 8, 6 , 2 , 4 , 8, 6 , ... manTRmg;TUeTA KW ³ 2 manelxxagcug 2 ebI n 4k 1 , k 2 manelxxagcug 4 ebI n 4k 2 , k 2 manelxxagcug 8 ebI n 4k 3 , k , k 2 manelxxagcug 6 ebI n 4k eday 2 2 manTRmg; 2 naM[ 2 CacMnYnmanelx 6 enAxagcug nig 2 2 manTRmg; 2 naM[ 2 CacMnYnmanelx 6 enAxagcugEdr eyIg)anpldkén 12 2 manelx 0 enAxagcug 2012
2012
2008
2008
2012
2012
2008
2008
2012
2008
n
n
n
n
n
4503
2012
4k
ehIy
man n 1nn 1 CabIcMnYn Kt;tKñaenaH 5n 1nn 1 RtUvEtEckdac;nwg 2 , 3 nig 5 b¤cMnYnenHRtUvEtEckdac;nwg 6 nig 5 eday PGCD 6 , 5 1 enaHcMnYnEdlEckdac;nwg 6 nig 5 CaCMnYnEdlEckdac;nwg 30 dUcenH cMnYn A n
4502
2012
dUcenH cMnYn 12
2012
2008
2 2008
3
x 3 3
2
1n
2
1
nn 1n 1n
4 5nn 1n 1
nn 1n 1 n 2 4 5 2
3 2
2 3 33
3 2
3 2
naM[
3
3 2 33
3 2
3 2 x
32 2 2 x
2 3 3x
Taj)an
x 3 3x 2 3
dUcenH KNna)an
enaH
f x x 3 3x 2 3
f x 2 3
.
332 rkBIrcMnYnKt; a nig b ³ bRmab; a b 92 naM[ a 1 b 93 eday a 1 CaBhuKuNén b enaH a 1 kb , k eyIg)an kb b 93 k 1b 93 naM[ b k93 1 eday b CacMnYnKt;viC¢man enaH 93 RtUvEtEckdac;nwg k 1
A n5 n
nn
3 2 3
3
Eckdac;nwg 10 .
330 bgðajfa A Eckdac;nwg 30 ³ cMeBaH RKb;cMnYnKt;viC¢man n eyIg)an ³ n n4 1
Eckdac;nwg 30 .
n
3
4k
2008
5
331 KNna f x ³ eyIgman f x x 3x cMeBaH x 3 2
2012
2008
5n 1nn 1
nn 1n 1n 2 n 2 5nn 1n 1 n 2 n 1nn 1n 2 5n 1nn 1
eday n 2n 1nn 1n 2 CaR)aMcMnYntKña enaHcMnYnenH RtUvEckdak;nwg 2 , 3 nig 5 b¤cMnYnenHRtUvEtEckdac;nwg 6 nig 5
eday 93 1393 enaH 31
-cMeBaH -cMeBaH 763
1 0 93 92 k 1 k 3 2 31 30
enaH b 93 minyk eRBaH a b 92 k 92 enaH b 1 k 0
-cMeBaH -cMeBaH eyIg)an
enaH b 31 k 30 enaH b 3
335 KNnakenSam A ³ eyIgman a a 1 nig
k 2
1 b 31 3
naM[
2
A a 4 2a 3 4a 2 3a 3 a 4 2a 3 a 2 3a 2 3a 3
92 1 91 a 92 b 92 31 61 92 3 89
a
dUcenH cMnYnKt;viC¢manEdlrkeXIj a 91 , b 1 a 61 , b 31 nig a 89 , b 3 . 333 bgðajfa P Eckdac;nwg P ³ man P n 1n 2n 3 ... n n ehIy P 1 3 5 ... 2n 1 enaHeyIg)an ³
2
2
336 KNna A [lT§plCaplKuNbIktþa ³ eyIgman A 12 2823 6
tamlT§plbgðajfa P Eckdac;nwg P )anplEck 2
12 8 3 6
334 KNna a b ³ eyIgman 2a3 326 5b9 Edl 0 a 9 ,0 b 9 bRmab;RbFan ³ 5b9 Eckdac;ngw 9 naM[ 5 b 9 Eckdac;nwg 9 enaHmanEt b 4 eRBaH 5 4 9 18 Eckdac;nwg 9 eyIg)an 2a3 326 549 2a3 223 Taj)an a 2 naM[ a b 2 4 6 dUcenH eyIgKNna)an a b 6 .
2 2
4 3 2 3 3 2
n
.
2 2
P 2 n P
dUcenH P Eckdac;nwg )anplEck 2
a 3 a2 a 3
n
1 3 1 3 1 3 3 1
n
P
2
dUcenH kenSam A 1 RtUv)anKNna .
n!P n!n 1n 2n 3 ... n n 1 2 ... nn 1n 2n 3 ... n n 1 2 ... nn 1n 2n 3 ... 2n 1 3 5 ... 2n 12 4 6 ... 2n P 2 12 22 3 ... 2 n P 1 2 ... n 2 2 2 ... 2 n!P P n!2
a 4 2a 3 a 2 3a 2 3a 3
2 2 3 2
4 3
2 2
3 2 2 2 2 2
3 2
3 2
4 3
4 3
3 2 4 3 3 24 3
3 2
dUcenH KNna)an A
2 2
4 3
4 3
3 2 2 3
2
2
2
2
2
2
2
2
2 2 1 2 2 32 ... 1002 32 1 2 2 32 ... 1002
22 4 32 9
dUcenH témøelxKNna)an E 94 . 764
337 KNnatémøelxénkenSam E ³ 200 eyIgman S 23 64 96 ...... 300
.
338 rkrgVas;RCugmMuEkgTaMgBIr ³ tag x nig y CargVas;RCugmMuEkg x Edl x 0 , y 0 KitCa cm tambRmab;eyIg)an ³
341 edaHRsaysmIkar ³ 13 cm
y
x 2 y 2 132 x 2 y 2 169 x y 17 x y 17
cMeBaH
x y 17
elIkGgÁTaMgTaMgBIrCakaer
x 2 2 xy y 2 289
naM[ x 2012 0 x 2012 1 1 1 1 ehIy 2008 0 2009 2010 2011
2 xy 289 x 2 y 2 289 169 xy 60 2 x y 17 5 12 xy 60 5 12
dUcenH
eday enaH x 5, y 12 b¤ x 12, y 5 dUcenH RCugmMuEkgTaMgBIr
nig 12cm .
5cm
339 RsaybBa¢ak;fa 3 2 ³ Binitü 9 8 3 2 3 2 nig 2 2 2 2 tamTMnak;TMng nig eyIg)an ³ 3 2 Taj)an 3 2 dUcenH 3 2 RtUv)anRsaybBa¢ak; . 2
2
9
2
2
3
3
3
2
6
3
2
2
3
3
2
2
3
2
3
3
3
340 bgðajfa
A6
³
A 6 6 6 ... 6 3 24 3 24 3 24 ... 3 24
eday
6 6 6 ... 6 6 6 6 ... 9 3 3 24 3 24 ... 3 24 3 24 3 24 ... 3 27 3 6 6 6 ... 6 24 24 ... 3 24 6 3
3
A6
dUcenH
A6
x 4 x 3 x 2 x 1 0 2008 2009 2010 2011 x 2012 2008 x 2012 2009 x 2012 2010 x 2012 2011 0 2008 2009 2010 2011 x 2012 x 2012 x 2012 x 2012 1 1 1 1 0 2008 2009 2010 2011 x 2012 x 2012 x 2012 x 2012 0 2008 2009 2010 2011 x 2012 1 1 1 1 0 2008 2009 2010 2011
RtUv)anRsaybBa¢ak;
.
x 2012
Cab£sénsmIkar .
342 rkbIcMnYnKt;viC¢manxusKña x , y nig z ³ eyIgman yx zy106 eyIg[témøén x edIm,IKNnatémørbs; y nig z ³ eday x y 6 naM[ 0 x 6 enaH ³ ebI x 1 eyIg)an y 5 nig z 5 ¬minyk¦ x 2 eyIg)an y 4 nig z 6 ¬yk¦ x 3 eyIg)an y 3 nig z 7 ¬minyk¦ x 4 eyIg)an y 2 nig z 8 ¬yk¦ x 5 eyIg)an y 1 nig z 9 ¬yk¦ dUcenH bIcMnYnKt;viC¢manxusKñaEdlepÞógpÞat;KW ³ x 2 , y 4 , z 6 , x 4 , y 2 , z 8 . x 5 , y 1, z 9 343 edaHRsaysmIkar ³ x 1 x 1 elIkGgÁTaMgBIrCakaer x 1 x 2 2x 1 x 2 3x 0
minyk eRBaHminepÞógpÞat; x 3 yk dUcenH smIkarman x 3 Cab£s . xx 3 0 x 0
765
344 KNnamMu x ³
347 k> KNnargVas;RCugénRtIekaN eyIgman ³ 24a 18b 12c nig b c 10 cm tamlkçN³smamaRteyIg)an ³
d
30 o
x
30
110
o
l
o
b c bc 10 1 18 12 18 12 30 3 a b c 1 24 18 12 3 a 1 24 3 3a 24 a 8 b 1 3b 18 b 6 18 3 3c 12 c 4 c 1 12 3
bnøayRCugénmMu x [kat;bnÞat; l naM[ enaHeyIg)anRtIekaNmYy enAkñúgbnÞat;Rsb Edl;man mMu x CamMueRkAénRtIekaNenH manrgVas;esµpI lbUkmMu kñúgBIrEdlminCab;nwgva . Taj)an eyIg)an ³ x 30 180 110 100 . dUcenH mMu x 100 RtUv)anKNnabgðaj . dUcenH rgVas;RCugénRtIekaNEdlKNna)anKW ³ o
o
o
o
o
a 8 cm , b 6 cm , c 4am
345 KNnaépÞRkLaénrUb ³ rMlkw RtIekaNsm½gSEdlmanRCug a enaHkm
RkLaépÞrUbenH )anBIépÞRtIekaNFMmYynig RtIekaNtUcbI EdlsuT§EtRtIekaNsm½gS.
3 a 2 A
D
F
B
ebI S CaépÞRtIekaNFM ¬ Big =FM¦ nig S CaépÞRtIekaNtUc ¬ Small =tUc¦ naM[ S S 3S B
C
E
x 2 h2 42 2 2 2 8 x h 6 x 2 h 2 16 2 2 64 16 x x h 36
S
B
S
1 1 3 3 3 3 3 1 1 2 2 2 2
x
a 8cm
1 2
yk 1 CMnYskñúg 2 eyIg)an ³
9 3 3 3 4 4 2 3 3 cm
64 16 x 16 36
dUcenH RkLaépÞénrUbKW
S 3 3 cm 2
7 5 3
6 15 1 15 8 15
15 15 15
16 x 80 36 44 11 x x 16 4
.
346 bMeBjelxkñúgRbGb; ³ eyIgGacbMeBj)antambRmab;RbFan dUcxageRkam ³ 15 2 9 4
x> KNnargVas;km
cMeBaH x 114 enaH naM[
h 2 16 x 2
256 121 135 3 15 11 h 16 16 4 4 4 2
dUcenH km
15
766
h
3 15 cm 4
.
abc 2012 3
348 rkcMnYnsisSenAkñúgfñak;eronenaH ehIy tag x CacMnYnsisS enAkñgú fñak;eronenaH ¬KitCa nak;¦ enaH tambRmab;RbFan eyIgsresr)ansmIkar ³ x4 x2 8 9 9 x 36 8 x 16 9 x 8 x 16 36
a b c 3 2012 4024 c 6036 c 6036 4024 c 2012
dUcenH cMnYnTI3 enaHKW
.
2012
x 52
dUcenH cMnYnsisSenAkñgú fñak;KW 52 nak; . 349 rkmYycMnYnenaH tag A CacMnYnenaH tambRmab;eyIg)an ³ AA 44
A4
dUcenH cMnYnEdlRtUvrkenaHKW
352 KNnacMnYnBIrxÞg; x ³ bRmab; ³ 44 Eckdac;nwg x sl;sMNl; 10 ebI k CaplEckrvag 44 nig x Edl k eyIg)an ³ 44 kx 10 kx 34 eday 34 2 17 enaH kx 217 Et x CaelxBIrxÞg; enaHRtUvEt x 17 , k 2
4
.
dUcenH cMnYnBIrxÞg;én x KW
350 rkmYycMnYnenaH tag x CacMnYnEdlRtUvrkenaH tambRmab;RbFan eyIg)an ³
353 edaHRsaysmIkar HE SHE ³ BinitüsmIkar ³ HE SHE kaerénmYycMnYnenArkSaelxxagcugenAEt E dEdl naM[ E 0 ,1, 5 , 6 -ebI E 0 enaH HE RtUvmanelxsUnüBIrenAxagcug Et HE SHE manelxxagcugmindUcKña enaH E 0 -eyIgepÞógpÞat;cMeBaH E 1, 5, 6 tamtémø H Edl lT§plrbs; HE RtwmEtCaelxbIxÞg;b:ueNÑaH eyIg)an³ 21 441 minepÞógpÞat; / 31 961 minepÞógpÞat; 15 225 minepÞógpÞat; / 25 625 epÞógpÞat; 16 256 minepÞógpÞat; / 26 676 minepÞógpÞat; eXIjfa HE SHE epÞógpÞat;eday 25 625 dUcenH E 5 , H 2 , S 6 . 2
x 2 2x
naM[
.
2
x x x x
x 2 2x 0 x x 2 0 x0 x 2 0
x 17
2
enaH
2
x 0 x 2
dUcenH cMnYnEdlRtUvrkenaHKW
x0 , x2
.
2
2
2
351 rkcMnYnTIbIenaH tag a , b , c CacMnYnTI1 / TI2 / TI3 erogKña tambRmab;RbFan ³
2
2
2
2
2
ab 2012 a b 4024 2
767
2
354 kMNt;témø a nig b nigkMNt;eTVFaenaH ³ eyIgman ax bx 54x 27 CaKUbéneTVFa tag x CaeTVFaEdlRtUvkMNt; naM[ x x 3 x 3 x x pÞwmelxemKuN ³ axx 3bxx 354 x 27 3
2
3
3
3
2
3
3
2
2
2
3
eyIg)an ³
356 kMNt;témø m ³ eyIgman 23xx 57yy 20m ¬edaHRsaytamedETmINg;¦
2
D
3
2
3
2 5
Dx
2
3 a 3 a 8 a 2 2 36 b 3 b 3 b 2 3 54 2 2 3 27 3 3
epÞógpÞat; cMeBaH a 8 , b 36 nigeTVFa 2x 3 eyIg)an
3 7
2 x 33 8 x 3 36 x 2 54 x 27
15 14 1
m 7 5m 140 20 5
5m 140 5m 140 1 3 m Dy 60 2m 2 20 x
y
60 2m 60 2m 1
edIm,I[RbB½ns§ mIkarmanKUcemøIyviC¢manluHRtaEt ³ x 0 5m 140 0 m 28 28 m 30 y 0 60 2m 0 m 30
Bit
dUcenH kMNt;)an a 8 , b 36 nigeTVFa 2x 3 . dUcenH kMNt;)anKW 28 m 30 b¤ m 28 , 30 . 355 KNnargVas;RCug AB nig AC ³ eyIgman AB AC 23 cm ¬elIkGgÁTaMgBIrCakaer¦ C
BC 17 cm B
A
1
AB 2 AC 2 2 AB AC 529
tamRTwsþIbTBItaK½r AB AC BC 17 289 naM[ 1 : 289 2AB AC 529 naM[ AB AC 120 AC 23 eyIg)an AB ¬tamRTwsþIbTEvüt¦ AB AC 120 2
eday b¤
AB AC 8 15 AB AC 8 15 AB AC 15 8 AB AC 15 8
2
2
2
enaH AB 8 , AC 15 enaH AB 15 , AC 8
2
x
1
1 1
r
1
2 2 2 1 r 2 x 2 2 xr r 2 3 2 2 x 2 2 xr 3 2 2 x x 2r
, x 2r 1
x 3 2 2
cMeBaH x 3 2 2 naM[ x 2r 1
3 2 2 2r 1
2r 2 2 2
dUcenH kaMrgVg;énrUbKW .
1
2
2
dUcenH rgVas;RCugénRtIekaNEdlKNna)anKW ³ AB 8cm , AC 15cm b¤ AB 15cm , AC 8cm
357 KNnakaMrgVg;énrUb ³ tamrUb eyIg)an ³ 1 x 2r 1 nig tamRTwsþIbTBItaK½r 2 1 r x r
768
r 2 1 r 2 1
ÉktaRbEvg .
1
358 KNnacMnYnKt;FmµCati n ³ bRmab; n 13 nig n 76 CakaerR)akd 0 naM[ nn 13 n 76 76 0
360 edaHRsayRbB½n§smIkar ³ man 12xx y 1 , 12yy z 2 , 12zz eyIgdwgfa 1 x 0 enaH 1 2x x 0 2
2
2
2
2
x 3
2
tag n 13 a 1 nig n 76 b 2 Edl a nig b CacMnYnKt; ehIy a b edayyk 1 2 eyIg)an ³ 2
2
2
2
1 x 2 2x
2x 1 1 x2
2x 2 x 4 1 x2
tamlMnaMdUcKñaenH eyIgk¾)an ³ 2z 2y z 6 y 5 nig 1 z 1 y eyIgCMnYs³ 1 kñúg 4 / 2 kñúg 5 nig 3 kñúg 6
n 13 a 2 2 n 76 b 89 a 2 b 2
2
2
2
2
Gacsresr ³ a ba b 89 mann½yfa 89 CaplKuNénBIrcMnYnKt; ³ edaydwgfa 1 89 89 ¬eRBaH 89 CacMnnY bzm¦ eyIgpÞwm)an aa bb 189 ¬edaybUkGgÁnigGgÁ¦
y x z y x y z x z
eyIg)an
tam 1 ³ 12xx
2
a b 1 a b 89 2a 90
2
y
2x 2 x 1 x2 2x 2 x x 3
eyIg)an ³ x 2x x 0 a 45 xx 2 x 1 0 naM[ n 13 45 enaH n 13 2025 b¤ n 2012 xx 1 0 ¬min)ac;sikSa b eRBaHeKalbMNgcg;rktémø n ¦ naM[ x 0 , x 1 dUcenH cMnYnKt;FmµCatiKNna)anKW n 2012 . eyIg)an ³ x y z 0 , x y z 1 dUcenH x y z 0 , x y z 1 . 359 KNnatémøénmMu x ³ 5x a b tamc,ab;plbUkmMukñúg 6x 361 RsaybBa¢ak;plbUkmMukñúgénRtIekaNesµI 180 énRtIekaN eyIg)an ³ 3 x 4 x a 180 KUsbnÞat;kat;kMBUl A ehIyRsbnwgRCug BC 6 x 7 x b 180 naM[ B A ¬mMuqøas;kñúg¦ 1 20 x a b 180 C A ¬mMuqøas;kñúg¦ ehIy 5x a b 180 a b 180 5x 2 Et A A A 180 ¬mMurab¦ eyIgyk 2 CMnYskñúg 1 enaHeyIgnwg)an ³ enaHeyIg)an A B C 180 ¬CaplbUkmMukñúg ¦ 20x 180 5x 360 b¤ 15x 180 x 12 dUcenH témømMuKNna)anKW x 12 . dUcenH plbUkmMukñúgRtIekaNTaMgGs;esµI 180 . 3
2
2
2
2
3x
7x
4x
o
o o
A
o
1
1
o
2
o
3
C
B
o
1
o
o
o
3
o
o
o
o
769
362 RsaybBa¢ak;BIRTwsþIbTBItaK½r a b c ³ -eyIgmanRtIekaNEkgEdlmanrgVas; a c RCug a , b , c ehIy c CaGIub:Uetnus b -eyIgpÁúMRtIekaNenH)andUcrUbxageRkam ³ -ctuekaNxageRkA Cakaerman a A b a c b c D rgVas;RCug a b naM[ ³ c B épÞkaerFM a b b c a a 2ab b . C a b - ABCD Cakaer eRBaH plbUkmMuRsYcBIrEdlenACab;mMu énctuekaNenH CamMubMeBjKñanaM[ ³ épÞkaertUc c -RtIekaNEkgnImYy²man RkLaépÞ 12 ab -tamrUbEdlpÁúM eyIg)anTMnak;TMngrvagRkLaépÞ ³ épÞkaerFM épÞkaertUc épÞRtIekaNtUcTaMgbYn 2
2
2
2
2
2
2
1 a 2 2ab b 2 c 2 4 ab 2 a 2 2ab b 2 c 2 2ab
cMeBaH c 9 naM[ 5c 1 2116 minyk ¬)anCayk 5c 1 1681 eRBaHelx abca manTRmg;dUcelx 1681 Edl a 1 , b 6 , c 8 dUcenH cMnYnmanelxbYnxÞg;enaHKW 1681 . 2
2
364 bgðajfa n IN enaHeK)an a b ³ Binitü ³ a b a ba a b ... b smµtikmµ ³ a b 0 naM[ a b 0 nig n
n
n 1
n
n
n2
n
a n1 a n2b ... b n 0
naM[plKuN a ba a b ... b 0 eyIg)an a b 0 Taj)an a b dUcenH bgðaj)anfa a b . n 1
n
n2
n
n
n
n
n
n
365 KNnaplKuN P 22 36 124 ... eyIgKNnaplKuN P )andUcxageRkam ³
2 6 12 9900 ... 2 3 4 100 1 2 23 3 4 99 100 ... 2 3 4 100 1 2 2 3 3 4 99 100 ... 2 3 4 100
P
a 2 b 2 c 2 2ab 2ab a2 b2 c2
dUcenH RTwsþIbTBItaK½rRtUv)anRsaybBa¢ak; .
1 2 2 3 2 4 2 ... 99 2 100 2 3 4 ... 100 1 2 3 4 ... 99 10 2 3 4 ... 100 10 1 100 10
363 rkmYycMnYnEdlmanelxbYnxÞg; abca ³ eyIgman abca 5c 1 enaH a , b , c CacMnYnKt; naM[ 0 a 9 , 0 b 9 , 0 c 9 1 ehIy 10000 abca 961
2
1002 5c 1 312 2
dUcenH KNna)anplKuN P 101 .
100 5c 1 31
cMeBaH 5c 1 31 5c 30 c 6 2 tam 1 nig 2 eyIg)an 6 c 9 c 7 , 8 , 9 kargarlM)ak KWCasRtUvénkarBüayam >>> cMeBaH c 7 naM[ 5c 1 1296 minyk cMeBaH c 8 naM[ 5c 1 1681 yk 2
2
9900 100
770
366 dak;kenSam P CaplKuNktþa ³ eyIgman P x y y z z x tag a x y , b y z , c z x eyIg)an P a b c a bc x y y z z x eday 3
3
3
3
368 kMNt; x viC¢man edIm,I[ A CacMnYnKt;viC¢man ³ eyIgman A xx 13
3
A
3
a b 3 c 3
x 3 1 x 1 3 x 3 2 x 2 3
a 2 3a 2 b 3ab 2 b 3 c 3 a 2 b 3 c 3 3a 2 b 3ab 2
eyIg)an ³
369 bgðajfaeK)an a b c 3abc ³ smµtikmµ ³ a b c 0 enaH a b c eyIg)an ³ 3
P 3x y y z x y y z P 3x y y z x z P 3x y y z z x
2 a b a b2
a 2 b 2 2ab 2 a b 2 2ab
P
eyIg)an
3a 2 3b 2 6ab P 2 3a 3b 2 6ab 10ab 6ab 4ab 1 P2 10ab 6ab 16ab 4 1 1 P ¬ P 0¦ 4 2
a 2 b 3 c 3 3a 2 b 3ab 2 a 2 b 3 c 3 3aba b a 2 b 3 c 3 3ab c
dUcenH KNna)an
a 2 b 3 c 3 3abc
dUcenH bgðaj)anfa
2012
1 2
x 2012 x
-cMeBaH -cMeBaH
.
smIkarKµanb£s x 2012 x naM[ x 1006 x 2012 x
dUcenH smIkarmanb£s
771
x 1006
.
x 2012 0
x 2012 2012 x 2012 0 x 2012 2012 x 2012
eRBaH
P
a 2 b 3 c 3 3abc
370 edaHRsaysmIkar x 2012 eyIg)an ³
2
Taj)an
3
a 2 3a 2 b 3ab 2 b 3 c 3
2
naM[
2
3
a b 3 c 3
.
367 KNnatémøénkenSam P aa bb ³ smµtikmµ 3a 3b 10ab eyIgman P aa bb Edl a b 0 enaH P 0 2
x 4 x 5
dUcenH kMNt;)antémø x 4 , x 5 .
a 2 b 3 c 3 3aba b
dUcenH dak;)an
x 3
2 x 3
-edIm,I[ A CacMnYnKt;viCm¢ an luHRtaEt x 2 3 CacMnYn Kt;viC¢manEdr naM[ 2 RtUvEckdac;nwg x 3 -edaytYEckviC¢manén 2 KW 1 b¤ 2 enaHeyIg)an ³
abc 0 a b c
P 3x y y z z x
x 3 2 1
.
371 KNnatémø eyIgman ³ x
A x 4 x 3 x 2 2 x 1 2 1 2 1 1
tag
1
X
2012
2 1 1
naM[ eyIg)an
2 1 1
2 1 1
x 2 x 2 6 x 9 89 2 x 2 6 x 80 0
2012
12012 1
.
374 KNnatémø eyIgmansmIkar x
2
x, y
2
8 , 5
b¤
5,8
.
BIsmIkar ³
4x y 6 y 13 0
x 2 y 3 0 plbUkénBIrkaeresµI 0 luHRtakaerTaMgBIresµI 0 ³ x
2
4x 4 y 6 y 9 0 2
ab 4a 6b 24 ab 4a 6b 24
eyIg)an
2a 3b 12
eyIg)an 2 720 3b 12 b
2
x 22 0 x 2 0 x 2 2 y 3 0 y 9 y 3 0
dUcenH témøKNna)anKW
240 b 4 b 480 b 2 4b
2
x2, y 9
.
cMNaM ³ ebIman A B 0 enaHeyIg)an ³ A 0 nig B 0 eRBaHfa A 0 , B 0 .
b 2 4b 480 0
2
2 480 484 22 2 2
2 22 20 0 1 2 22 b 24 1
naM[ xx 85 00 xx 58 -cMeBaH x 8 naM[ x 3 8 3 5 -cMeBaH x 5 naM[ x 3 5 3 8 dUcenH cMnYnTaMgBIrenaHKW
372 KNnavimaRténsYnc,arenaH ³ tag a CabeNþay nig b CaTTwg énsYn tamsmµtikmµ ³ sYnmanépÞRkLa 720 m eyIg)an ab 720 naM[ a 720 b müa:geTot a 6b 4 ab
b
x 2 3 x 40 0 x 8x 5 0
2012
A 1
720 720 30 b 24
2
4 3 2 A 2 2 2 2 2 1
dUcenH KNna)an
a
x 2 x 3 89
enaH
373 rkcMnYnTaMgBIr ³ tag x nig x 3 CacMnYnTaMgBIrenaH tambRmab;RbFan eyIg)an ³
1
4 2 2 2 2 2 1
man enaH
b 24
dUcenH KNna)anvimaRtrbs;sYnKW 30m nig 24m .
1
2 1 1 2 1 1 2 1 1 2 1 1 2 2 2 2 11 2 2 2 x 2 X 2
cMeBaH
minyk
2
2
772
2
375 k> eRbóbeFob 3 3 nig 62 3 ³ Binitü 62 3 12 6 3
-cMeBaH
dUcenH smIkarmanb£sBIrKW
96 33 32 6 3 3
3 3
2
3 3 6 2 3 4
4
2
3 3
x> eRbóbeFob 14 48 nig 7 4 Binitü 14 48 14 16 3 3 ehIy 7 4 3 7 4 3 7 4 3 7 4 3
3 4 74 3
4 49 48 4 1 1 3 1
48 4 7 4 3 4 7 4 3
.
1 48 4 7 4 3 4 7 4 3 4
376 edaHRsaysmIkar x 2 x 3 eyIgmansmIkar x 2 x 3 1 tag t x 2 naM[ x 3 t 1 eyIg)ansmIkareTACa t t 1 1 4
4
4
1
³
2
2
2
2
2
2
2
2
378 KNna eyIgman
4
t 4 t 1 t 1 1
eXIjfa kaerén A nigkaerén B manelxcugdUcKña eday 0 A 9 , 0 B 9 eyIg)an ³ -ebI A 81 B 61 minyk 61 minCakaerR)akd -ebI A 64 B 44 minyk 44 minCakaerR)akd -ebI A 49 B 29 minyk 29 minCakaerR)akd -ebI A 36 B 16 yk ¬CakaerR)akddUcKña¦ eyIg)an A 6 , B 4 dUcenH cMnYnmanelxBIrxÞg;enaHKW AB 64 .
4
4
2
2
A 2 B 2 20
4
dUcenH
.
10 A B 2 10B A2 1980 10 A B 10B A10 A B 10B A 1980 9 A 9 B 11A 11B 1980 99 A B A B 99 20 A B A B 20
.
4
edaysar naM[ 14
x2, x3
377 rkcMnYnEdlmanelxBIrxÞg; AB ³ AB BA 1980 eyIgman
2
dUcenH eRbóbeFob)an
t 1 : 1 x 2 x 3
2
t 4 t 2 2t 1 t 2 2t 1 1
a,b,c
ab 4 i ac 5 ii bc 20 iii
³ naM[
abc2 400
cMeBaH abc 400 Taj)an abc 20 20 c 5 -yk i : abc ab 4 20 b 4 -yk ii : abc ac 5 20 a 1 -yk iii : abc bc 20 2
t 4 t 4 2t 3 t 2 2t 3 4t 2 2t t 2 2t 1 1 2t 4 4t 3 6t 2 4t 0
2t t 1t t 2 0
2t t 3 2t 2 3t 2 0 2
naM[ tt10 0 tt 10 ehIy t t 2 man 1 8 7 0 Kµanb£s -cMeBaH t 0 : 0 x 2 x 2 2
dUcenH KNna)an 773
a 1 , b 4 , c 5 a 1 , b 4 , c 5
.
379 KNna
³ eyIgman
381 rkcMnYnelxsUnüenAxagcugén 2012! A 2025 1 2025 2 2025 3 ... 2025 50 eday 2012! 1 2 3 ... 2012 eday 2025 45 enaHeyIgsresr)an ³ cMnYnelxsUnüenAxagcugéncMnYn 2012! bgáBIelxEdl A 45 1 45 2 45 3 ... 45 50 CaBhuKuNén 5 , 5 , 5 , 5 kñúgcMeNam ktþaBI 45 1 rhUtdl; 45 50 ¬minyk 5 eRBaH 5 3125 2012 ¦ R)akdCamanktþamYyKW 45 45 0 -tag n CacMnYnBhuKuNén 5 naM[ A CaplKuN Edlman 0 Caktþa enaH A 0 enaH 5n 2012 KNna)an n 402 -tag n CacMnYnBhuKuNén 5 = 25 dUcenH KNna)an A 0 . enaH 25n 2012 KNna)an n 80 380 rkcMnYnelxsUnüenAxagcugrbs; P : -tag n CacMnYnBhuKuNén 5 = 125 eyIgman P 1 2 3...100 enaH 125 n 2012 KNna)an n 16 cMnYnelxsUnüxagcugrbs; P ekItBIcMnYnCaBhuKuNén 5 -tag n CacMnYnBhuKuNén 5 = 625 enaHeyIgRKan;EtrkcMnYnEdlCaBhuKuNén5 kñúgcMeNam enaH 625 n 2012 KNna)an n 3 elxBI 1 rhUtdl; 100 mann½yfaeyIgkMBugrkcMnYn eyIg)an cMnYnelxsUnüTaMgGs;éncMnYn 2012! KW n n n n 402 80 16 3 501 elxenAxagcugTaMgGs;rbs; P elxEdlCaBhuKuN 5 man ³ dUcenH cMnUn 2012! bBa©b;edayelxsUnücMnYn 501 . 5 , 10 ,15 , … , 100 man 20 elx EtcMeBaHelx 25 , 50 , 75 , 100 ekItBIBhuKuNén 5 382 KNna N : eyIgman N 4 15 4 15 2 3 5 dl;eTA 2 dg xusBIcMnYnepSgeTot enaHmanelx 5 8 2 15 8 2 15 62 5 2 cMnYn 4 eTot 2 2 2 srubBhuKuNén 5 KW 20 + 4 = 24 énelx 5 5 3 5 3 2 5 1 2 2 2 dUcenH P manelxsUnüenAxagcMnYn 24 . 5 3 5 3 2 5 1 2 2 2 ***smÁal; ³ GñkGaceFVItamrebobdUcxageRkam ³ 5 3 5 32 5 2 -ebI n CacMnYnBhuKuNén 5 2 2 enaH 5n 100 KNna)an n 20 2 2 -ebI n CacMnYnBhuKuNén 5 = 25 dUcenH KNna)an N 2 . enaH 25n 100 KNna)an n 4 eyIg)an cMnYnelxsUnüTaMgGs; n n 20 4 24 A
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
4
5
2
2
1
1
1
2
2
2
2
3
3
3
3
4
4
4
1
4
2
3
2
1
1
1
2
2
2
2
1
2
774
4
2
2
383 rkcMnYnXøIBN’ s ³ tag x CacMnYnXøIBN’ s enaHcMnYnXøIBN’exµAKW 12 x RbU)abcab;)anXøIBN’exµAKW P(x) 1212 x (1) Etsmµtikmµ RbU)abXøIBN’exµAKW P(x) 13 (2) tam (1) nig (2) eyIg)an ³ 12 x 1 12 3
12 3 x 12 4 x 8
12 x
eyIgbUkGgÁ nigGgÁénvismIkar
1 , 2 nig 3 ³
AB AC AM 2 2 BC AB BN 2 2 BC AC CP 2 2 AM BN CP AB BC AC
dUcenH
AM BN CP AB BC AC
.
x> Rsayfa AM BN CP AB BC2 AC dUcenH XøIBN’ s mancMnYn 8 RKab; . -tamc,ab;vismPaBkñúg ABM ³ AM AB MB BC 384 k> Rsayfa AM BN CP AB BC AC tam Gacsresr AM AB 2 i -tamc,ab;vismPaBkñúg BNC ³ BN BC NC tambRmab;RbFaneyIgKUsrUb)an ³ A ii tam Gacsresr BN BC AC 2 o N -tamc,ab;vismPaBkñúg ACP ³ CP AC PA P o iii tam Gacsresr CP AC AB // // 2 C B M i , ii eday AM , BN nig CP Caemdüanén ABC enaH eyIgbUkGgÁ nigGgÁ énvismIkar BC nig iii ³ AM AB 2 M , N , P CacMNuckNþalerogKñaén BC , AC , AB PA
AB BC AC , MB , NC 2 2 2
AC BN BC 2 AB CP AC 2 AB BC AC AM BN CP 2 2 2
mü:ageTot eyIg)an )atmFüm cMnYnbIKW ³ PM
AC AB BC , MN , NP 2 2 2
Gacsresr AM BN CP AB BC2 AC
-tamc,ab;vismPaBkñúg AMP ³ AM PA PM AC 1 tam nig eyIg)an AM AB 2 2 dUcenH AM BN CP AB BC2 AC . -tamc,ab;vismPaBkñúg BMN ³ BN MB MN AB 2 ***smÁal; ³ vismPaBkñúgRtIekaNmYyKW ³ tam nig eyIg)an BN BC 2 2 -tamc,ab;vismPaBkñúg CPN ³ CP NP NC -plbUkRCugBIrRtUv > RCugmYyeTot AC 3 - pldkRCugBIrRtUv < RCugmYyeTot . tam nig eyIg)an CP BC 2 2
775
385 k> KNna S a b c CaGnuKmn_én b ³ eyIgman a , b , c CabIcMnYnKt;ruWLaTIbtKña naM[ eyIg)an a b 1 nig c b 1 eyIg)an S a b c
387 edaHRsayRbB½n§smIkar eyIgman xx y y z zxy yz3 xz 2
tam 1 :
2011
2011
2012
1 2
x 2 y 2 z 2 xy yz xz
x 2 2 xy y 2 y 2 2 yz z 2 z 2 2 zx x 2 0
3b
S 3b
2011
2
2 x 2 2 y 2 2 z 2 2 xy 2 yz 2 xz
b 1 b b 1
dUcenH KNna)an
2
vaCaGnuKmn_én b .
x y 2 y z 2 z x 2 0
plbUkéneRcInkaeresµIsUnü luHRta kaernImYy²esµIsnU ü x y 2 0 x y 0 2 y z 0 y z 0 x y z z x 0 2 z x 0
x> Tajrktémøén a , b , c edaydwgfa S 333 eyIg)an eyIgman S 3b EteyIgdwgfa S 333 naM[ 3b 333 Taj)an b 111 tam 2 : x enaHeyIg)an a b 1 b¤ a 1111 112 eyIg)an x nig c b 1 b¤ c 111 1 110 dUcenH KNna)an a 112 , b 111 , c 110
2011
386 rkcMnYnRksakñúghVÚgTaMgGs; ³ tag x CacMnYnRksaTaMgGs;enAkñúghVÚg tambRmab; EdlCaBaküsmþIrbs;emxül;énhVÚgRksa eyIgsresr)ansmIkardUcxageRkam ³ 1 1 x x x x 1 100 2 4 4 x 4 x 2 x x 4 400 11x 396 x 36
x 2011 x 2011 32012 3x 2011 32012 x 2011 32011 x3
dUcenH RbB½n§smIkarmanb£s x y z 3 . 388 KNnargVas;RCugnImYy² ³ smµtikmµ ³ RtIekaN ABC manbrimaRt 80 cm tag a , b , c CargVas;RCugTaMgbIénRtIekaN ABC eyIg)an a b c 80 edayRCugTaMgbImansmamaRterogKña 5 , 7 , 4 tamlkçN³smamaRt eyIgsresr)an ³
epÞógpÞat; ³ 36 36 18 9 1 100 100 100
2011
y 2011 z 2011 32012
Bit
dUcenH cMnYnRksaenAkñgú hVÚgTaMgGs;mancMnnY 36 . eyIg)an
a b c a b c 80 5 5 7 4 5 7 4 16 a 5 5 a 25 b 5 b 35 7 c 20 c 4 5
dUcenH RCugTaMgbImanrgVas; 25cm , 35cm , 20cm .
776
bc ac 389 KNnatémøénkenSam P ab : c a b eyIgman 1a b1 1c 0 2
2
2
1 1 1 a b c 3
1 1 1 a b c 1 1 3 1 1 1 3 3 3 ab a b a b c
391 eRbóbeFob ba 11 , ba nig ba 11 eyIgman a 1 , b 1 nig a b eyIg)an ab a ab b a b 1 ba 1
3
a a 1 b b 1
ehIy a b enaH
1 1 1 3 1 1 3 3 3 ab a b a b c 1 1 1 3 1 3 3 3 ab c a b c 1 1 1 3 3 3 3 abc a b c 1 1 1 abc 3 3 3 3 b c a bc ac ab 3 a2 b2 c2 P3
dUcenH KNna)antémøénkenSam
P3
tam 1 nig
.
3 2 2 6 3 2 1 3 2 2 6 3 2 1
.
3
2 3 3 2 2 3 2 13 2 3 3 2 2 3 2 13
3
3
2 1 2 1 2 1 2 3
2 1 3
3
B 51 10 2 51 10 2
x 3 2 y 1 2 4 x 3 2 y 1 8 3 x 3 6
50 10 2 1 50 10 2 1
5 2 2 5 2 1 5 2 2 5 2 1 5 2 1 5 2 1 5 2 1 5 2 1 2 eRkayBIKNna eXIjfa A 2 nig B 2 Edr naM[ eyIg)an A B 2
x 3 4 x 1
yktémø x 1 CMnYskñúgsmIkar 1 eyIg)an 1 3 2 y 1 2 2 2 y 1 2
y 1 0
a 1 a a 1 b 1 b b 1
A 3 5 2 7 3 5 2 7
eyIgyk 1 2 2 enaHeyIg)an ³
2 y 1 0
eyIg)an
392 eRbóbeFob A nig B :
y 1 2 1 2 x 3 y 1 4 2
1 :
a b ab a ab b ab 1 ba 1 a a 1 2 b b 1 a 1 a a 1 2 b 1 b b 1
dUcenH eyIgeRbóbeFob)an
390 edaHRsayRbB½n§smIkar ³ eyIgmanRbB½n§smIkar x 3 2
x3 2
1
y 1 0
y 1
2
2
2
dUcenH RbB½n§smIkarmanKUcemøIy x 1 , y 1 . dUcenH eRbóbeFob)an
777
2
A B
.
393 edaHRsaysmIkar ³ eyIgman 1 x x 2 x x 3 x x ... 2013x x 0 1 2 3 2013 1 0 1 1 1 ... x x x x 1 2 3 2013 ... 2013 0 x x x x 1 2 3 ... 2013 1 2013 x 1 2 3 ... 2013 x 2013 2013 1 2013 1 2 3 ... 2013 2
1 1 1 1 a b b c a c a b c abc
ebI a , b , c mancMnYnpÞúyKñamYyy:agticKW ³ a b a b 0 b c b c 0 c a a c 0
ebImankrNImYyy:agtic kñúgcMeNamkrNIxagelIenaH 1 1 1 1 0 a b c abc 1 1 1 1 a b c abc
Et
20131007
eyIg)an
x
>
2013 1007 1007 2013
dUcenH smIkarmanb£s
x 1007
dUcenH ebImanBIrcMnYnpÞúyKñamYyy:agtickñúgcMeNam a , b , c enaHeyIg)anTMnak;TMng ³
.
395 edaHRsaysmIkar ³ eyIgmansmIkar 3 3
394 k> dak;CaplKuNktþa ³ eyIgman 1a b1 1c a 1b c eyIg)an
2 x
2 x
30
32 3 x 32 3 x 30
bc ac ab 1 abc abc bc ac aba b c abc
9 3 x 3 x 30 10 3 1 10 3x x 3 3
3 x 3 x
a b bc ac ab cbc ac ab abc 0 a b bc ac ab bc 2 ac 2 abc abc 0 a b bc ac ab bc 2 ac 2 0 a b bc ac ab a b c 2 0 a b bc ac ab c 2 0 a b bc ab ac c 2 0 a b bc a a c c 0 a b a c b c 0
tag t 3 Edl t 0 ¬eRBaH 3 0 ¦ eyIg)an t 1t 103 3t 10t 3 0 x
x
2
man 25 9 16 enaH -cMeBaH t 3 -cMeBaH t 3
dUcenH eyIgdak;CaplKuNktþa)an a ba c b c 0
.
1 1 1 1 a b c abc
1
1
.
2
x> mancMnYnpÞúyKñamYyy:agtickñúgcMeNam a , b , c ³ tamsRmayenAkñúgsMNYr k> eyIg)an ³
5 4 1 1 3 t1 3 3 t 5 4 3 2 3
3 x 31 x 1 3x 3
x 1
dUcenH smIkarmanb£sBIrKW x 1 b¤ x 1 .
778
396 sRmÜlRbPaK ³ eyIgman A x3x 14xx 1x eday x 0 naM[ x 1 x 1 ehIy x x eyIg)an A 3xx 14xx1 x 3x 3xx1x 1
398 KNnabrimaRt nigRkLaépÞénRtIekaNenH ³
2
2
2
x 1 x 1 3 x x 1 x 1 x 13 x 1 1 1 3x 1 1 3x
dUcenH sRmÜlRbPaK)an
A
1 1 3x
x2
tag x , x 2 , x 4 CaRCugénRtIekaNEkgenH tamRTwsþIbTBItaK½r eyIg)an ³ x 2 x 2 x 4 2
x 2 4 x 12 0
.
x y 1 xyz 2 1 y z 5 2 6 xyz x z 2 3 3 xyz 2 x y z 1 5 2 xyz 2 6 3
naM[
x 2 6 x 2 x 12 0 x 6x 2x 6 0 x 6x 2 0 x 6 0 x 6 x 2 0 x 2 x 2 6 2 8 x6 x 4 6 4 10
minyk
cMeBaH enaH eXIjfa RtIekaNenHmanRbEvgRCug 6 , 8 , 10 naM[ brimaRt P = 6 + 8 + 10 = 24 ÉktþaRbEvg RkLaépÞ S 12 6 8 24 ÉktþaépÞRkLa
2x y z 3 5 4 xyz 6 x yz 4 1 xyz 2 xyz 36 xyz 6 xyz 6 , z 3 xy 2 xyz 6 xyz 6 x 1 , y 2 yz 6 xz 3
4 1 :
z 1 xy 2 xyz 2 4 2 : x 1 yz 6 xyz 6 4 3 : y 1 xz 3 xyz 3
dUcenH KNna)an brimaRtesµI 24 ÉktþaRbEvg RkLaépÞesµI 24 ÉktþaépÞRkLa . 399 sRmYlkenSam F xx 11 2
-krNI x 1 naM[ x 1 x 1 eyIg)an F xx 11 x x1x1 1 x 1 2
-krNI x 1 naM[
F
Kµann½y .
-krNI x 1 naM[ x 1 x 1 eyIg)an F xx 11 x1xx11
dUcenH RbB½n§smIkarmanKUcemøIy ³ x 1 , y 2 , z 3
2
x 2 x 2 4 x 4 x 2 8 x 16
397 edaHRsaysmIkar ³ eyIgman
x 1 , y 2 , z 3
x4
x
2
b¤ .
779
x 1
.
400 KNnacm¶ayBIcMNuc A eTAbnÞat; (D) ³ eyIgmancMNuc A6 , 6 nigbnÞat; D: y x 4
401 >>>>>>>>>>>>>>>¬enAmanbnþ¦
A6 , 6
D : y x B2 , 2
D : y x 4
-rkbnÞat;Edlkat;tam A6 , 6 nigEkgnwg D bnÞat;RtUvrkmanrag D : y ax b eday D D a a1 11 1 ehIy D kat;tam A6 , 6 enaH x 6 , y 6 eyIg)an D : 6 1 6 b b 0 dUcenH bnÞat; D : y x -rkcMNucEdl D RbsBVnwg D tageday B ³ pÞwmsmIkarGab;sIus eyIg)an ³ x 4 x 2x 4 x 2 > cMeBaH x 2 y 2 naM[cMNuc B2 , 2 -rkcm¶atBIcMNuc A6 , 6 eTAbnÞat; D: y x 4 mann½yfa rkcm¶ayBI A6 , 6 eTAcMNuc B2 , 2 d AB
xB x A 2 y B y A 2 2 62 2 62
16 16 32 4 2
dUcenH cm¶ayrk)anKW
d 4 2
ÉktþaRbEvg .
780
sYsþI¡ elakGñkmitþGñkGanCaTIRsLaj;rab;Gan enAkñúgEpñkenHelakGñknwg)aneXIj BIkarbEnßmCUn Biess² nigl¥² dUcCakarkmSanþKNitviTüa taragsV½yKuN >>>. ral;GVI² EdlsMxan; ehIy Tak;TgnwgcMeNHdwgKNitviTüafñak;TI 9 RtUv)an´erobcMCUnmitþGñkGanenAkñúgEpñkenH (...) . critlkçN³sMxan;enARtg;EpñkenH KWkarKNnaelx[)anrh½s dwgBIcMNucl¥² EdlmanTMnak;TMng CamYyemeronKNitviTüafñak;TI 9 . karkmSanþsb,ayCamYyKNitviTüa eFVI[eyIgmancMNg;cMNUlcitþ edIm,I sikSaemeronKNitviTüa . elIsBIenHeTAeTot EpñkenHk¾manbgðajGMBIkarrIkceRmInrbs;KNitviTüapgEdr. RbsinebIelakGñkmanbBaða b¤cm¶l;Rtg;cMNucNa EdlmanenAkñúgEpñkenH elakGñkGacTak;Tg;eTAkan; RKU b¤mitþPkþirbs;GñkEdlmansmtßPaB b¤GñkeroberogesovePAenHkñúgeBlevlasmRsb . …
viii
:
001
kñúgcMeNamrUbTaMgbYnxageRkam etIrUbNamYyCarUbEdlmanlkçN³xuseK ??? k> kaer
:
002
:
003
:
004
:
005
:
006
:
007
x> RtIekaNsm½gS
K> RtIekaNEkg
X> ctuekaNesµI
GñksYrnrNamñak;nUvsMNYrxageRkam ehIy[eKeqøIyy:agrh½sbMput ³ etI«Bukekµkbgéfø´ RtUvCaGVInwg´ ??? GñksYrnrNamñak;nUvsMNYrxageRkam ehIy[eKeqøIyy:agrh½sbMput ³ etI 2 2 2 b:unµan ??? GñksYrnrNamñak;nUvsMNYrxageRkam ehIy[eKeqøIyy:agrh½sbMput ³ elxenaHminFMCag 2 ehIymintUcCag 2 etIelxenaHb:unµan ??? bursmñak;mankUneQIEt 6 edIm b:uEnþKat;GacdaM)an 4 CYr EdlkñúgmYyCYr²mankUneQI cMnYn 3 edImCanic©. etIKat;daMtamreboby:agNa cUrKUsrUbbBa¢ak;kardaM ??? bursmñak;mankUneQIEt 10 edIm b:uEnþKat;GacdaM)an 5 CYr EdlkñúgmYyCYr²mankUneQI cMnYn 4 edImCanic©. etIKat;daMtamreboby:agNa cUrKUsrUbbBa¢ak;kardaM ??? bursmñak;mankUneQIEt 12 edIm b:uEnþKat;GacdaM)an 6 CYr EdlkñúgmYyCYr²mankUneQI cMnYn 4 edImCanic©. etIKat;daMtamreboby:agNa cUrKUsrUbbBa¢ak;kardaM ???
781
eKmanelx 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . cUrykelxTaMgenH eTAbMeBjkñúgRbGb;kaerxagsþaM edIm,I[plbUkCYredk esµIplbUk CYrQr esµIplbUkGgÁt;RTUg esµInwg 15 .
:
008
:
009
etIelxb:unµan EdlmanplbUkesµIplKuNénxøÜnÉg ???
:
010
cUrKUsbnÞat; 6 edIm,I[)anRtIekaNcMnYn 8 .
:
011
stVExVkTMelIEmkxVav )ak;EmkR)av gab; 3 rs; 2 cuHebIstV 120 gab;b:unµanrs;b:unµan ???
:
012
etImanviFIsaRsþGVI EdleFVI[eKeCOeyIgfa 11 bUkEfm 2 eTotesµInwg 1 ???
:
013
:
014
:
015
:
016
edayeRbIelx 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 cUrykelxTaMgenHdak; kñúgRbGb;TaMg 8 xagsþaM ély:agNa kM[u elxEdlenAbnÞab;Kña sßitenAkñúgRbGb;Cab;Kña b¤enAExVgKñak¾eday . etIeKemIly:agdUcemþcdwgfaqñaMNacUlqñaeM nAéf¶TI 13 ehIyqñaMNacUlqñaMéf¶TI 14 ??? enAelIpøÚvRtg;mYymanLanebIkCaCYr ebIdwgfa LanmuxebIkmuxLanBIr LankNþal ebIkkNþalLanBIr LaneRkayebIkeRkayLanBIr etILanTaMgGs;manb:unµan ??? davImanGayudUc x ebIKit[xÞic x CaBhuKuNén 8 edaydwgfa x FMCag 10 nigtUcCag 20 etIsBVéf¶davIRsImanGayub:unµan ???
782
:
:
017
018
enACMuvijRtIekaNxagsþaMeKeRbIcMnYnBI 2 dl; 10 edIm,IteRmób [cMnYnEdlenAelIRCugnImYy²énRtIekaNmanplbUkesµI 21. etIeKRtUvteRmóbcMnYnenHya:gNa edIm,I[cMnYnelIRCugnImYy² manplbUkesµInwg 24 vijmþg ???
2
:
019
:
020
:
021
6
5 9
10
2
7
8
4
tamry³lMnaMrUbxageRkam etIRtUvykelxb:unµanmkbMeBjenAkñúgRbGb;sBaØa {?} ??? 6
7
8
3
3
4
20
30
9
4
?
8
rkbIelxEdldUcKña éllkeFVIplbUkedIm,I[esµI 24 . bUkya:gNaeRsccitþ [EtesµI 24 ehIybIelxenaHKWCaelxdUcKña ]TahrN_ 8 8 8 24 . eKmanb‘íccMnYn 10 edImerobmYy²enAdac;edayELkBIKña)anmYyCYrdUcrUbxageRkam. cUrelIkb‘ícmþgmYy²eTAcab;KUKña edaykñúgkarelIkRtUvrMlgb‘íccMnYnBIKt; eTaHbICa sßitenAkñúgsßanPaBEbbNak¾edayKWrMlgb‘ícBIrCanic© edIm,I[bí‘cTaMgGs;enACaKU².
enAkñúgrUbctuekaNEkgxageRkammanGkSr A , B , C CaKU². cUrGñkKUsExSP¢ab;KUGkSr EdldUcKña KWBI A eTA A / BI B eTA B nigBI C eTA C edaymin[ExSenaHkat;KñaeLIy ehIyExSEdlKUsP¢ab;minGacenAeRkActuekaNEkg)aneLIy. B A
A
C
C
B
smÁal; ³ GñkRtUvcmøgrUbdak;RkdasepSgeTot minRtUvKUsP¢ab;elIrUbxageqVgenHeLIg edIm,ITuk[GñkeRkaykmSanþbnþeTot .
783
:
022
:
023
:
024
:
025
:
026
bursmñak;cg;cmøg xøa / eKa nig esµA eTAeRtIymçageToténTenømYy. TUkrbs;Kat;tUc EdleFVI[Kat;Gaccmøg)anmþgmYyb:ueNÑaHkñúgcMeNam xøa / eKa nigesµA . eKdwgfaebI bursenaHminenAKW xøasIueKa b¤ eKasIuesµA EtebIbursenaHenAKWKµankarsIuKñaekIteLIgeT. etIKat;KYcmøg xøa / eKa nigesµA eTAeRtIymçageTotedayviFINaeTIbKµankar)at;bg; . GñksYrnrNamñak;nUvsMNYrxageRkam ehIy[eKeqøIyy:agrh½sbMput ³ mþayBUsn mankUnRbusbInak;KW kUnRbusb¥ÚneKmaneQµaHfa sux kUnRbusTI2maneQµaH fa esA etIkUnRbusc,geKmaneQµaHfaGVI ??? kaerFMmYyeKGacEckeTACakaertUcCagcMnYn 7 dUcrUbxagsþaM cUrGñkEckkaerFMenHeTACakaertUcCagcMnYn 6 vijmþg . eKeRbIeQIcak;eFµjedIm,Ierob)ankaercMnYn 4 b:un²Kña. cUrbþÚrTItaMg eQIcak;eFµjcMnYn 3 edIm,I[)ankaercMnYn 3 EdlmanTMhMb:un²Kña. eKmanEmkeQIBIrmanRbEvgRbEhlKña. kac;EmkeQITaMgBIrRtg;cMNuckNþaldUcKña edaykac;mYy[dac; nigmYyeTotkMu[dac;. eKykEmkeQIEdldac;mYyeTAdak;Tl; EmkeQIEdlminTan;dac; [mansßanPaBlMnwg ¬dUcrUbxageRkam¦ . eKnwgykEmk eQIEdldac;mYyeToteTAelIkEmkeQITaMgGs;[eLIgputBIdIkñúgeBlEtmYy . etIeKRtUvelIkdUcemþc ???
784
eKeRbIeQIcak;eFµjedIm,Ierob)ankaercMnnY 5 b:un²Kña. cUrbþÚrTItaMg eQIcak;eFµjcMnYn 2 edIm,I[)ankaercMnYn 4 EdlmanTMhMb:un²Kña.
:
027
:
028
:
029
:
030
cUrbMeBjRbGb;[)anRtwmRtUv ³ c6 30 s
:
031
tamlMnaMKRmUénrUbbIxagedIm cUrrkelxmkCMnYssBaØa {?} [)anRtwmRtUv ³
bursmñak;mandIERsragCakaer . Kat;cg;daMbEnøelIdIERsenH dUcenHKat;RtUveFVIrbg. ebIdwgfaRCugnImYy²RtUvmanbegÁal 10 edImdUcKña rkcMnYnbegÁalTaMgGs;EdlKat; RtUvkar ? cUrKUsGgát;bIbEnßmBIelIrUbxagsþaM edIm,I[vakøayCaRbGb;dIs.
¬sUmcmøgrUbecj kMuKUselIesovePAenH Tuk[GñkeRkaykmSanþbnþ¦
5
4
9
6
2
:
032
5
8
30
4
6
6
20 5
3
.
?
4
2
2
5
3
eK[lkçxNÐbI EdlbursbInak; A , B , C eTAsþIdNþwgnarIbInak; 1, 2 , 3 CaKUGnaKt dUcxageRkam ³ k> ebIburs A CaKUnwgnarI 1 enaH burs B CaKUnwgnarI 2 x> ebIburs A CaKUnwgnarI 3 enaH burs C CaKUnwgnarI 1 K> ebIburs B minCaKUnwgnarI 3 enaH burs C CaKUnwgnarI 1 . sMNYrsYrfa etIbursNa CaKUnwgnarINa ?
785
rkBIrcMnYnedaydwgfa plbUk plKuN nigplEckrvagcMnYnTaMgBIr esµIKña .
:
033
:
034
:
035
:
036
:
037
mFüménBIrcMnYnesµI 2012 nigmFüménbIcMnYnk¾esµInwg 2012 Edr. cUrrkcMnYnTI 3 .
:
038
etIrUbxagsþamM ankaerTaMgGs;cMnYnb:unµan ?
:
039
eKmanGkSrLataMg T ,
:
040
ksikmñak;mancMNIsRmab;pÁt;pÁg;eKarbs;Kat; 40 k,al)an 35 éf¶. ebIKat;TijeKa 10 k,albEnßmeTot etIkat;GacpÁt;pgÁ ;cMNIdEdl[eKa)anb:unµanéf¶ ? bursmñak;man)arImUledayéd 10 edIm. Kat;Ck;)arIedayrkSaknÞúy)arI ebICk;Gs;bIedIm enaHKat;GacykbnÞúy)arIEdlsl;mUl)anmYyedImfµvI ij . etIKat;Ck;b:unµandgeTIbGs;)arITaMg 10 edImKµansl; ? kñúgkic©RbCMumYymanmnusS 6 nak; )ancUlrYm ehIyGñkTaMgenaH)ancab;édsVaKmn_Kña eTAvijeTAmk. ebIdwgfaGñkTaMgGs;)ancab;édKñaRKb;²Kña. cUrrkcMnYnénkarcab;édKña TaMgGs;.
H , L,K , E , F
etIGkSrNamanlkçN³xuseK ???
kñúgfñak;eronmYy eQµaHrbs;sux enAelxerogTI 20 ebIrab;BIxagelI EtebIrab;BIxag eRkamvij k¾enAEtelxerogTI 20 dEdl. etIfñak;eronenaHmansisSb:unµannak; ?
786
005
001
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6 ÂÂCÂÂ
3
3 002
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3 788
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3 040 39 20 20
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3
792
brimaRt épÞRkLa nigmaD énrUbFrNImaRt ctuekaNEkg -CactuekaNmanRCugb:unKñaBIr² nigmanmMuEkg . A -RkLaépÞ ³ S ab -brimaRt ³ P 2a b a -Ggát;RTUg ³ d a b B b
RtIekaNsam½BaØ -RtIekaNsam½BaØ CaRtIekaNKµanlkçN³Biess. B -RkLaépÞ ³ S 12 bh
5)
1)
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a
c
h
P a bc A
2
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2
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c
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C
RbelLÚRkam -CactuekaN EdlmanRCugRsbKñaBIr² . -RkLaépÞ ³ S bh ab sin A -brimaRt ³ P 2a b B h 6)
D
a
C
b
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b
2
2
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b
2)
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D
a
RtIekaNsm½gS
CaRtIekaNmanRCugTaMgbIb:unKña mMuTaMgbIesµIKañ esµInwg 60 .
-brimaRt ³
c
P 4a b
o
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1 a2 3 S ah 2 4
3 h a 2
a
a
A
h
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b
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793
r
2
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o
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15)
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2
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F
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18)
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27)
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r
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b
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795
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1
n n
n2
n3
n4
n5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400 441 484 529 576 625 676 729 784 841 900 961 1024 1089 1156 1225 1296 1369 1444 1521 1600 1681 1764 1849 1936 2025 2116 2209 2304 2401 2500
1 8 27 64 125 216 343 512 729 1000 1331 1728 2197 2744 3375 4096 4913 5832 6859 8000 9261 10648 12167 13824 15625 17576 19683 21952 24389 27000 29791 32768 35937 39304 42875 46656 50653 54872 59319 64000 68921 74088 79507 85184 91125 97336 103823 110592 117649 125000
1 16 81 256 625 1296 2401 4096 6561 10000 14641 20736 28561 38416 50625 65536 83521 104976 130321 160000 194481 234256 279841 331776 390625 456976 531441 614656 707281 810000 923521 1048576 1185921 1336336 1500625 1679616 1874161 2085136 2313441 2560000 2825761 3111696 3418801 3748096 4100625 4477456 4879681 5308416 5764801 6250000
1 32 243 1024 3125 7776 16807 32768 59049 100000 161051 248832 371293 537824 759375 1048576 1419857 1889568 2476099 3200000 4084101 5153632 6436343 7962624 9765625 11881376 14348907 17210368 20511149 24300000 28629151 33554432 39135393 45435424 52521875 60466176 69343957 79235168 90224199 102400000 115856201 130691232 147008443 164916224 184528125 205962976 229345007 254803968 282475249 312500000
n
796
1 1.414214 1.732051 2.000000 2.236068 2.449490 2.645751 2.828427 3.000000 3.162278 3.316625 3.464102 3.605551 3.741657 3.872983 4.000000 4.123106 4.242641 4.358899 4.472136 4.582576 4.690416 4.795832 4.898979 5.000000 5.099020 5.196152 5.291503 5.385165 5.477226 5.567764 5.656854 5.744563 5.830952 5.916080 6.000000 6.082763 6.164414 6.244998 6.324555 6.403124 6.480741 6.557439 6.633250 6.708204 6.782330 6.855655 6.928203 7.000000 7.071068
100 3
n
1 1.259921 1.442250 1.587401 1.709976 1.817121 1.912931 2.000000 2.080084 2.154435 2.223980 2.289428 2.351335 2.410142 2.466212 2.519842 2.571282 2.620741 2.668402 2.714418 2.758924 2.802039 2.843867 2.884499 2.924018 2.962496 3.000000 3.036589 3.072317 3.107233 3.141381 3.174802 3.207534 3.239612 3.271066 3.301927 3.332222 3.361975 3.391211 3.419952 3.448217 3.476027 3.503398 3.530348 3.556893 3.583048 3.608826 3.634241 3.659306 3.684031
4
n
1 1.189207 1.316074 1.414214 1.495349 1.565085 1.626577 1.681793 1.732051 1.778279 1.821160 1.861210 1.898829 1.934336 1.967990 2.000000 2.030543 2.059767 2.087798 2.114743 2.140695 2.165737 2.189939 2.213364 2.236068 2.258101 2.279507 2.300327 2.320596 2.340347 2.359611 2.378414 2.396782 2.414736 2.432299 2.449490 2.466326 2.482824 2.498999 2.514867 2.530440 2.545730 2.560750 2.575510 2.590020 2.604291 2.618330 2.632148 2.645751 2.659148
5
n
1 1.148698 1.245731 1.319508 1.379730 1.430969 1.475773 1.515717 1.551846 1.584893 1.615394 1.643752 1.670278 1.695218 1.718772 1.741101 1.762340 1.782602 1.801983 1.820564 1.838416 1.855601 1.872171 1.888175 1.903654 1.918645 1.933182 1.947294 1.961009 1.974350 1.987341 2.000000 2.012347 2.024397 2.036168 2.047673 2.058924 2.069935 2.080717 2.091279 2.101632 2.111786 2.121747 2.131526 2.141127 2.150560 2.159830 2.168944 2.177906 2.186724 -
n
n2
n3
n4
n5
51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
2601 2704 2809 2916 3025 3136 3249 3364 3481 3600 3721 3844 3969 4096 4225 4356 4489 4624 4761 4900 5041 5184 5329 5476 5625 5776 5929 6084 6241 6400 6561 6724 6889 7056 7225 7396 7569 7744 7921 8100 8281 8464 8649 8836 9025 9216 9409 9604 9801 10000
132651 140608 148877 157464 166375 175616 185193 195112 205379 216000 226981 238328 250047 262144 274625 287496 300763 314432 328509 343000 357911 373248 389017 405224 421875 438976 456533 474552 493039 512000 531441 551368 571787 592704 614125 636056 658503 681472 704969 729000 753571 778688 804357 830584 857375 884736 912673 941192 970299 1000000
6765201 7311616 7890481 8503056 9150625 9834496 10556001 11316496 12117361 12960000 13845841 14776336 15752961 16777216 17850625 18974736 20151121 21381376 22667121 24010000 25411681 26873856 28398241 29986576 31640625 33362176 35153041 37015056 38950081 40960000 43046721 45212176 47458321 49787136 52200625 54700816 57289761 59969536 62742241 65610000 68574961 71639296 74805201 78074896 81450625 84934656 88529281 92236816 96059601 100000000
345025251 380204032 418195493 459165024 503284375 550731776 601692057 656356768 714924299 777600000 844596301 916132832 992436543 1073741824 1160290625 1252332576 1350125107 1453933568 1564031349 1680700000 1804229351 1934917632 2073071593 2219006624 2373046875 2535525376 2706784157 2887174368 3077056399 3276800000 3486784401 3707398432 3939040643 4182119424 4437053125 4704270176 4984209207 5277319168 5584059449 5904900000 6240321451 6590815232 6956883693 7339040224 7737809375 8153726976 8587340257 9039207968 9509900499 10000000000
n 7.141428 7.211103 7.280110 7.348469 7.416198 7.483315 7.549834 7.615773 7.681146 7.745967 7.810250 7.874008 7.937254 8.000000 8.062258 8.124038 8.185353 8.246211 8.306624 8.366600 8.426150 8.485281 8.544004 8.602325 8.660254 8.717798 8.774964 8.831761 8.888194 8.944272 9.000000 9.055385 9.110434 9.165151 9.219544 9.273618 9.327379 9.380832 9.433981 9.486833 9.539392 9.591663 9.643651 9.695360 9.746794 9.797959 9.848858 9.899495 9.949874 10.000000
3
n
3.708430 3.732511 3.756286 3.779763 3.802952 3.825862 3.848501 3.870877 3.892996 3.914868 3.936497 3.957892 3.979057 4.000000 4.020726 4.041240 4.061548 4.081655 4.101566 4.121285 4.140818 4.160168 4.179339 4.198336 4.217163 4.235824 4.254321 4.272659 4.290840 4.308869 4.326749 4.344481 4.362071 4.379519 4.396830 4.414005 4.431048 4.447960 4.464745 4.481405 4.497941 4.514357 4.530655 4.546836 4.562903 4.578857 4.594701 4.610436 4.626065 4.641589
4
n
2.672345 2.685350 2.698168 2.710806 2.723270 2.735565 2.747696 2.759669 2.771488 2.783158 2.794682 2.806066 2.817313 2.828427 2.839412 2.850270 2.861006 2.871622 2.882121 2.892508 2.902783 2.912951 2.923013 2.932972 2.942831 2.952592 2.962257 2.971828 2.981308 2.990698 3.000000 3.009217 3.018349 3.027400 3.036370 3.045262 3.054076 3.062814 3.071479 3.080070 3.088591 3.097041 3.105423 3.113737 3.121986 3.130169 3.138289 3.146346 3.154342 3.162278
5
n
2.195402 2.203945 2.212357 2.220643 2.228807 2.236854 2.244786 2.252608 2.260322 2.267933 2.275443 2.282855 2.290172 2.297397 2.304532 2.311579 2.318542 2.325422 2.332222 2.338943 2.345588 2.352158 2.358656 2.365083 2.371441 2.377731 2.383956 2.390116 2.396213 2.402249 2.408225 2.414142 2.420001 2.425805 2.431553 2.437248 2.442890 2.448480 2.454019 2.459509 2.464951 2.470345 2.475692 2.480993 2.486250 2.491462 2.496631 2.501758 2.506842 2.511886
Computer
797
-
º 1º 2º 3º 4º 5º 6º 7º 8º 9º 10º 11º 12º 13º 14º 15º 16º 17º 18º 19º 20º 21º 22º 23º 24º 25º 26º 27º 28º 29º 30º 31º 32º 33º 34º 35º 36º 37º 38º 39º 40º 41º 42º 43º 44º 45º
0.0000 0.0175 0.0349 0.0524 0.0698 0.0873 0.1047 0.1222 0.1396 0.1571 0.1745 0.1920 0.2094 0.2269 0.2443 0.2618 0.2793 0.2967 0.3142 0.3316 0.3491 0.3665 0.3840 0.4014 0.4189 0.4363 0.4538 0.4712 0.4887 0.5061 0.5236 0.5411 0.5585 0.5760 0.5934 0.6109 0.6283 0.6458 0.6632 0.6807 0.6981 0.7156 0.7330 0.7505 0.7679 0.7854
sin
cos
tan
cot
0.0000 0.0175 0.0349 0.0523 0.0698 0.0872 0.1045 0.1219 0.1392 0.1564 0.1736 0.1908 0.2079 0.2250 0.2419 0.2588 0.2756 0.2924 0.3090 0.3256 0.3420 0.3584 0.3746 0.3907 0.4067 0.4226 0.4384 0.4540 0.4695 0.4848 0.5000 0.5150 0.5299 0.5446 0.5592 0.5736 0.5878 0.6018 0.6157 0.6293 0.6428 0.6561 0.6691 0.6820 0.6947 0.7071
1.0000 0.9998 0.9994 0.9986 0.9976 0.9962 0.9945 0.9925 0.9903 0.9877 0.9848 0.9816 0.9781 0.9744 0.9703 0.9659 0.9613 0.9563 0.9511 0.9455 0.9397 0.9336 0.9272 0.9205 0.9135 0.9063 0.8988 0.8910 0.8829 0.8746 0.8660 0.8572 0.8480 0.8387 0.8290 0.8192 0.8090 0.7986 0.7880 0.7771 0.7660 0.7547 0.7431 0.7314 0.7193 0.7071
0.0000 0.0175 0.0349 0.0524 0.0699 0.0875 0.1051 0.1228 0.1405 0.1584 0.1763 0.1944 0.2126 0.2309 0.2493 0.2679 0.2867 0.3057 0.3249 0.3443 0.3640 0.3839 0.4040 0.4245 0.4452 0.4663 0.4877 0.5095 0.5317 0.5543 0.5774 0.6009 0.6249 0.6494 0.6745 0.7002 0.7265 0.7536 0.7813 0.8098 0.8391 0.8693 0.9004 0.9325 0.9657 1.0000
------57.2900 28.6363 19.0811 14.3007 11.4301 9.5144 8.1443 7.1154 6.3138 5.6713 5.1446 4.7046 4.3315 4.0108 3.7321 3.4874 3.2709 3.0777 2.9042 2.7475 2.6051 2.4751 2.3559 2.2460 2.1445 2.0503 1.9626 1.8807 1.8040 1.7321 1.6643 1.6003 1.5399 1.4826 1.4281 1.3764 1.3270 1.2799 1.2349 1.1918 1.1504 1.1106 1.0724 1.0355 1.0000
798
46º 47º 48º 49º 50º 51º 52º 53º 54º 55º 56º 57º 58º 59º 60º 61º 62º 63º 64º 65º 66º 67º 68º 69º 70º 71º 72º 73º 74º 75º 76º 77º 78º 79º 80º 81º 82º 83º 84º 85º 86º 87º 88º 89º 90º
0.8029 0.8203 0.8378 0.8552 0.8727 0.8901 0.9076 0.9250 0.9425 0.9599 0.9774 0.9948 1.0123 1.0297 1.0472 1.0647 1.0821 1.0996 1.1170 1.1345 1.1519 1.1694 1.1868 1.2043 1.2217 1.2392 1.2566 1.2741 1.2915 1.3090 1.3265 1.3439 1.3614 1.3788 1.3963 1.4137 1.4312 1.4486 1.4661 1.4835 1.5010 1.5184 1.5359 1.5533 1.5708
sin
cos
tan
cot
0.7193 0.7314 0.7431 0.7547 0.7660 0.7771 0.7880 0.7986 0.8090 0.8192 0.8290 0.8387 0.8480 0.8572 0.8660 0.8746 0.8829 0.8910 0.8988 0.9063 0.9135 0.9205 0.9272 0.9336 0.9397 0.9455 0.9511 0.9563 0.9613 0.9659 0.9703 0.9744 0.9781 0.9816 0.9848 0.9877 0.9903 0.9925 0.9945 0.9962 0.9976 0.9986 0.9994 0.9998 1.0000
0.6947 0.6820 0.6691 0.6561 0.6428 0.6293 0.6157 0.6018 0.5878 0.5736 0.5592 0.5446 0.5299 0.5150 0.5000 0.4848 0.4695 0.4540 0.4384 0.4226 0.4067 0.3907 0.3746 0.3584 0.3420 0.3256 0.3090 0.2924 0.2756 0.2588 0.2419 0.2250 0.2079 0.1908 0.1736 0.1564 0.1392 0.1219 0.1045 0.0872 0.0698 0.0523 0.0349 0.0175 0.0000
1.0355 1.0724 1.1106 1.1504 1.1918 1.2349 1.2799 1.3270 1.3764 1.4281 1.4826 1.5399 1.6003 1.6643 1.7321 1.8040 1.8807 1.9626 2.0503 2.1445 2.2460 2.3559 2.4751 2.6051 2.7475 2.9042 3.0777 3.2709 3.4874 3.7321 4.0108 4.3315 4.7046 5.1446 5.6713 6.3138 7.1154 8.1443 9.5144 11.4301 14.3007 19.0811 28.6363 57.2900 --------
0.9657 0.9325 0.9004 0.8693 0.8391 0.8098 0.7813 0.7536 0.7265 0.7002 0.6745 0.6494 0.6249 0.6009 0.5774 0.5543 0.5317 0.5095 0.4877 0.4663 0.4452 0.4245 0.4040 0.3839 0.3640 0.3443 0.3249 0.3057 0.2867 0.2679 0.2493 0.2309 0.2126 0.1944 0.1763 0.1584 0.1405 0.1228 0.1051 0.0875 0.0699 0.0524 0.0349 0.0175 0.0000
-
sBVéf¶kareRbIR)as;xñatmaneRcInsn§wksn§ab;Nas; edIm,IbMeBjeTAnwgkarrIkceRmInénviTüasaRsþ . minEt b:uNÑaHxñatmYycMnYnRtUv)anCMnYsedayxñatfµIepSgeTot edIm,I[RsbeTAnwgsþg;daGnþrCati. enHehIyCa mUlehtuEdl naM[xñatFmµtamYycMnYn)an)at;bg;bnþicmþg² mYyvijeTot xñatmYycMnYnFMenAEteRbIR)as;enAkñúg CIvPaBrs;enAsBVéf¶ b:uEnþKµanÉksarNa )anniyaylm¥it nigrebobeRbIxñatTaMgenaHeLIy. CaBiessedIm,I kMu[sñaédExµr)at;bg; EdlbuBVburs )anbnSl;Tuk eTIb´)anerobcMeLIgvij GMBIxñatCaeRcIn CamYykarRsavRCav Éksarmintic . kñúgEpñkenHnwgmankarbgðajxñatepSg²CaeRcInrYmman ³
1> xñatRbEvg BhuKuNén m
1
=1 =2m
1Tm
1012 m
1
= 16 km
1
1Gm
109 m
1
= 4 km
1 inch
1.54 cm
1Mm
106 m
1
= 40 m
1 foot
30.48 cm
1Km
103 m
1
= 20 m
1 yard
0.9114 m
1
= 0.50 m
1 feet
0.3038 m
1
= 0.20 m
1 mile
GnuBhuKuNén m
1609.344 m
1mm
103 m
1
= 0.10 m
1 yard
= 3 feet (
1μm
106 m
1
= 0.01 m
1 mile
= 1760 yd
1nm
109 m
1
= 100
1 mill , mrin
1pm
1012 m
1
= 400
1 m 1010 Ao (
…
Km
hm
dam
m
dm
cm
1852m
mm
…
-
799
-
2> xñatépÞRkLa -
-
m2
-
m2
1 Tm 2 1024 m 2
1 mm 2 106 m 2
1 ha
= 10 000 m 2
1 Gm 2 1018 m 2
1 μm 2 1012 m 2
1a (
= 100 m 2
1 Mm 2 1012 m 2
1 nm 2 1018 m 2
1 ca
= 1 m2
1 Km 2 106 m 2
1 pm 2 1024 m 2
Km2
…
hm2 ha
dam2 a
m2
dm2
cm2
mm2
…
ca
-
3> xñatmaD 1 pm3 1036 m3
1 Tm3 1036 m3
= 0.001
1 ml 1 000 ml
1
1 dm
1 Mm 10 m
1 hl
100
1 mm3 1 cc
1 Km3 109 m3
1 cl
= 0.01
1 ml 1 cm3
1 dal
= 10
1
1 Gm 10 3
3
27
m
3
18
1 mm3 109 m3 1 μm 10 3
18
m
3
1
1 nm3 1027 m3
…
3
Km3
hm3
dl
dam3
= 30 = 0.1 m 3
1
= 0.1
m3
1
dm3
cm3
mm3 cc
…
-
800
-
4> xñateBl s 1
= 1000
1
j,
h
mn ,
s
mn
1h
= 3600 s (
) = 60 s (
)
1j(
= 1440 mn
)
1j
= 60
1
= 100
1 mn (
1
= 10
1s(
)
= 60 (
1
= 365
1 (
)
= 60
6
h,
= 86400 s
1
= 8766 h
1
= 10
5> xñatm:as 1 kg
= 1 000 g
1 hg
= 100 g
edkaRkam dag = 10 g 1 Rkam g = 0.001 kg 1 edsIuRkam dg = 0.1 g 1
sg;TIRkam cg = 0.01 g 1 mIlIRkam mg = 0.001 g 1
= 60 kg
1
¬Ggár¦
1
= 15 kg
1
= 600 g
1
= 37.5 g
1
= 3.75 g
1
= 0.375 g
1
= 0.0375 g
1
=2
1
=2
1
=4
1
= 30 kg
1
= 16
1
= 30 kg
1
= 10
1
= 10
1 kg ( 1
) = 10
= 100 g = 1000 kg
1 1
= 907.18 kg
1
= 1016.05 kg
1
pound =
1
N 100 g
0.453 kg
6> xñatsItuNðPaB ³ K t 273
o
F
180 oC 32 100
C o
K ,
,
³K ³ F
,
F o
t ,
o
o
C
7> xñatmMu 1o (
= 60
180º = 200 Grad
1
= 60
180º = rad
1 D = 90º
)
1 rad 57 o1745
801
1º = 0.0174533 rad 1º = 1.111111 Grad 1 rad 63.662 Grad rad = 200 Grad -
8> xñatepSg² LÚ = 12 1 ekH ¬Rsaebo¦ = 24 kMb:ug 1 b‘íc = 25 sug ¬)arI¦ 1 sug = 10 kBa©b; ¬)arI¦ 1 kBa©b; = 20 edIm¬)arI¦ 1 sIu = 0.25 L lIt
ry = 10 erol 1 erol = 10 kak; 1 kak; = 10 esn 1 dMbr = 4 ¬b¤5ebIKitk,ac;¦ 1 pøÚn = 10 dMbr 1 søwk = 10 pøÚn
1
1
)aj; = 100 g ¬fñaM¦ 1eb ¬Ggár¦= 50 kg 1 )avRkecA = 100 kg 1 yYr¬TwksuT¦§ = 12 db 1
1
= 0.25 kg
20 10
9> esckþIbEnßm -xñatsemøgKitCa edsIuEbl -xñatrBa¢ÜydIKitCa -xñatfamBlGKÁisnIKitCa kw/h -xñatkMlaMgKitCa jÚtun (N) -xñatem:m:UrI KitCa éb -cMnYnGaKuyKitCa RKab; -xñatKitCa eRKÓg Qut sRmab; kMebø ér: >>> .
802
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