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ADDTTTONAL Kertasl
rvrArr@arr
cs
Septernber
2 jam
MAJLIS PENGETUA SEKOLAH MALAYSIA CAWANGAI\ PULAU PINANG
MODUL LATIHAN BERFOKUS SPM 2OI4 MARKSCHEME
ADDITIONAT MATHEMATIC Paper
I
Two hours
S
347211
2
ADDITIONAL MATHEMATICS PAPER 1 2AI4 Solution and Mark Scheme
Question
I
2
Marhs
Mark
3, g))
g), (2, 4), t(3 ,
(- 2, 4), (-
(b)
.f :x->*2
oR f(x)=i2
(a)
L7
1
(b)
-14
2
(a)
(b)
m2+6m+11
-2 2
J-
(3)2
2
+ pQ)+
18 = 0
x:6 :
4
11
p:-9
81
3
2
S(r):i2 +2
Bl : (b)
2
1
-3
Bl : g(Y): (V -3)'+ 6U- 3) + (a)
1
f-t (x): +
Bl:
4
Total
(a)
81:
3
Sub
(r-3Xr-6)-0
or 3a:18 or
m1-6, m> 6 B2: (- m)2 -
3+o=
2
4
3
3
(-9)
4QXe) >
Bl :x2 -mx+ 9 = Q
6
(a)
x=3
I
(b)
(3,2)
1
(b)
f(x)= (r- 3)'+2
3472n
I
3
34721r
3
7
/l
3
2
3
3
4
4
3
3
rr,s
2
B2:2x -3
Bl:5b"51 or 1
I
83
: R2 =26P oR R2 =26 P
B2
:
log2 R2
= lo1z26 +log2 P OR rogz
81
,
l=og'P. log24
or 3log2
,R2 T=
24
9
B2:
-1 37
+(n-
0
1X6)
: d-6
B1
10
or fi-}JF
gP2
R
53
(a)
4
I
(b)
20
3
4
3
3
B2: (2n + 4t)(n-2q BI
11
;va)*b-1)4I
-2r 55l-+Ll-4,2 -7
B.2:
1-
2 3
B1
34721I
,-23
0
= 820
3472tI
tz
(a)
1 L*'+4 v
(b)
I
2
p:2,q-7(both)
3
4
3
3
4
4
B2:p-2 or q-7 Bt
: 5 =+p+4 or stz--G) +4 1
r =14,t
13
-+
0oth)
5
82: r =14 or
5
: 8 _z(r)+t(a)
Bt
t7
+-
or
, _20)*3(t)
o-
5
t4
5
2100, 3300
83 : 210o or
3300
B2: (2sinr + l)(sin )c +2)=
Bl :
ls
(a)
(1 -2sin2x)-5sin
0
x=3
t6
2
Bl 1 /. PoR:0.45 (b)
344.576 /344.59/344.6
Bl
16
3472tr
4
2
2
, te6)2(z.6sz)
Ji5 /3.6a6 81
2
:
22
+32
3472n
t7
(a)
-5 a *4b
(b)
2
)=z or 4( L)=t' \3+n) \3+n)
Bz: 5 -s( -L
-6 81 : -3(2x+ t-4 Q)
2t
(a)
-4
(b)
16
81
347211
:
or 5
3472n
Br
23
(a)
: p=ry--($Ef
144
Bl:4!
x3!
72
Bl:31 x2! x3!
24
(a)
(b)
0.1
4AI
Bl : 1.08
u72lt
0.
I3l4
3:47212
ADDITIONAL MATFIEh4ATIC
S
Kertas 2 Septernber Al
z- lam 2" .
MAJLIS PENGETUA SEKOLAH MALAYSIA CAWANGAN PULAU PINAIIG
MODUL LATIHAN BERFOKUS SPM 2OI4
MARKSCHEME
ADI}ITIONAL MATHEMATIC S Paper 2
Two hours and thirty minutes
347212
2
sEcTroN A ( 40 MARKS ) Mark Scheme
Ques
1.
y=2x-6 or x-3+
ry E
or
y=x2-15
_t1Ttiil'_t_:'_: *z
_ *(b_6) -
15
or
.f,
.:r)'-y =
15
Solve the Quadr aticEquation
il;il;;
-(e)*Jt-z)z
-4(rX-9)
2(r)
or
OR
l-e-15)=3 2\
-(8)t
(8)2
- 4(1) (-24)
2(r)
Cqmnletine the sgugfe
(x- i2 - (- D2 -g= Q OR
b, + 4)2
-
42
x = 4.16, -2.16 Qr
y:2,32,-
10.32
OR
y :2.32r-
x:
10.32
or 4.16, -2,16
Notg:
OW-l if
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steps to solve quadratic equation arcnot showq.
-24=
o
Sub
TotaI
Marhs
Mark
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Mark Scheme
Ques
2
(a)
k+(9-lXx):52A
?Po+
Sub
Total
Marks
Mark
(t z-rX')J =474a
Solve simultaneous linear equations
x-
50
or
k: 120 I I
k- 120 or
x(b)
50
* 120 + (n- 1)*(50)
:
n-
4
5
48 + (n
- I)(74) 2
7
Note : Accept complete listing of the terms
3
(a)
9:t-
ytl!?:-.J
Y -(-3) =*2(x-1)
fllyC- -r/2, fllAD-2
or
-3 : * 2(I)+ c
Y: hc-5 (b)
3
Solve simultaneous equations ?x + 4y:20 and *y:2x - 5 2
D(4,3)
(c)
B (t0, o)
use
A
-%lf )-(
)l
;: i-1ro-iib:i;;:i;i +o *r| 22.5
347212
I
3
8
347212
(a)
I
| Use tTtrx ntz: -
1
I
rl I
substitute
l*r:;,mt:-3
gg-e-gY
x-- 1 into 4
?t:19-l(:-3-) ::
n;(
!)3 =_r
(- r)t
(b)
2- !
Integr ate
\ rrt
3
)c
11 y---+-+C 2xn
x
Substitute to find c
I 17 l)=--+-+'2xax2
s
(a)
of
Shape
sine
2cyclesfor0
v
-
3
(- I ,2) into ty
3
6
E
E] EI E)
Accept cosine graph 4
3
2
I
.r 3x y-5-1T
Sketch the straight line
Number of solutions =
347212
3
7
347212 It
Mark
Ques
(a)
| Height of the bars proportional
I to the frequencies I
I ruUel the class bound afies or I midpoints of the classe s or I the class interval
Scheme
Sub
Marks Mark
E E
Method to find mode
26
(b)1"
@
F: 14
@
0)
*14
Iu;
g
22.83
347212
"f* -9
l
4
E
TotaI
347212
347212
347212
7
sEcrroNB(40MARKS)
(a)
x-I
1
2
3
4
5
6
logroy
0.44
0.63
0.82
1.03
1.20
L.40
(b)
Plot logt $/ against ( x
Full Marks Mark Sub
Mark Scheme
Ques
E E
- 1)
& 6 *points plotted correctly
(c)
Line of best fit
log;,oy= logr ok + ( r
- 1) logrch
E Use
;Use
*c
*m: logoh
h
- 1.56
lg-
Note : SS-1 if part of the scale'is not uniform or not using the scales given or not using the graph paper
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log1s fr
1.74
10
347212
logro
347212
347212
Sub
Mark Scheme
Ques
(i) &(ii)
(a)
Marks
Use Triangle Law for
Full Mark
0R or OS
fr:4q-6v OS
9l: _qetsl_11ry:_"_ ${ Or -9u-4v
(b)
- 4u +3v
_9_T_
-3-
se QT = lQR
I
(c)
| (i) &(ii)
I tI
tubstitute u =3iand Y
- -21+ liinto 'r ffi
OS
=
6r_+
lSi
6t+Lsj 4 ?_a- - "llao
3472n
4
10
t0
Mark
347212
Scheme
Solve simultaneous equations
y:(x-3)2andy--x*9.
B(5,4)
A(0,9)
Integrate (x
- 3)'wrt x
6xz ,+ Ar:t9x 32
At\/'
(* -3)3 3
,f5
Uselimit
t
in Ar
OR
Az
Find the arcaof tapezium
Az=
or
At
*(e 4)x *(5) l-. 2\ +
l+-'{: r25
/Do1/lzo.B3
Integrate n(x - 3)o wrt x
v:rry 3
Use
Iin
0
243
n
ll48:- n //
48.6n
o(ry-sf 5
341212
11
Mark
Use ^r - r0 to
Scheme
find arc
DC or EC
Arc DC-s
(Z\ \2)
AC:
+72
or arc EC - 5(1.2
AC*arcDC+2
19.45
/l
/
= Y, fg to find area of sector CBE
Use
15.5
Use
A- Yrfe b find
area
of sector DBE
I
Use *
A,: *Of(;-Lz4)
bxh
22
find
lo|sin
Ar:LABC or A3: or A+: L EBC
Az= 17.5', At
A+: Il,82
/-
OR
*15.5
A
Cto
ABE
- 5 .687 ;
+As-At-A+
OR
A- A2-Ar-A+* (*t5.5 -A+)
5.22
347212
ll
5.23
t2
347212
Mark Scheme
8p:2
(i)
DL4
1 ll o.z5
(ii)
Use
* (0.25), (0.75)n-r
P(X -
1
Write P(X>Z): 3) + P(X - 4) +.. .+ P(X: 8) OR
P(X:
0)
-
P(X - 1) - P(X= 2)
0.3015
(i) z- 2.50.8-1.6
0.13029
ll
0,1303
fn -1.6 -''r' =1.175 .
0.8
m=2,54
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13
SECTTON C (20 MARKS)
Mark
Integr
Scheme
ate Izt - 8 dt
v- f
8t +
v-
find Use
t:
Substitute 12 in *
c
0 and
y to
c
c: 12
a -0tofindr
t -4
Solvev.0tofindr
(t-2)(t-6)-0
t:2,
6
!t' -8r+12 dt s=
t- -
4t2
+r2t
3
Substitute t = *2 or t
-
4 into *s
OR
Find total distance travelled
use
fi"at arll:"dtl
t6
347212
I:
or I:
in
*s
347212
L4
Mark
Scheme
use Qt x loo Qo
)c
I=
-
y=
125
* I2s(27) + l2s(34) + 137 .5(10) + 1s0(10) 27 +34+10+10+19
127.8
use Qn 26A0
x loo = *127 .8
RM
118
x *1 27 .B
100
150.8
347212
3322.8
t2o(1e)
l5
3472/2
t.
Mark Scheme
Ques
1,4
(a)
(i)
Yr-'-: :rlt 19 lt ! 4! 9- ly -!$BC
2
= 82
+lz2 -
BC:
(ii) y::
2(8)
B
Sub
FUII
Marks Mark
c
(lz) cos 25'
5.83
:ir :l:11 y:?
::.1i
C sin 25 12 =-5.83
::
::o
sin
60.450
I 19.550
(b) I
G)
(ii)
(iii)
ADB = 60,45o
Z DAfi:59. use
1o
|olez)sin
(*5e.1
47 "75
347212
+ 25)
10
t6
Mark
x + y S 8 or
Scheme
equivalent
x 600x +
2A0y < 3 000
or
equivalent
Draw coffectly at least one straight line from the *inequalities which involves xand y Draw coffectly all the three *straight lines
Note : Accept dashed lines
The correct region shaded
(i)
I
(ii)
Maxirnum
point:
(4, 3)
Use 3bc + 7y
for point in the *region R
Note:
ss-l
if
in (a), the symbol 63-" is not used at all or rnore than three inequalities are given
in (b), does not use the scale given or does not use graph paper or interchange between r-anis and y-axis
347212
347212
E E E
t7
END OF N{ARK SCHEME 347212
347212