LEVEL-II MQ Objective questions ( One correct Answer) 1.

In a G.P., T2 + T5 = 216 and T4 : T6 = 1:4 and all terms are integers, then its first term is (A) 16

(B) 14

(C) 12

(D) none

If a1,a2 ,a3 ,..an be an A.P. of non-zero terms then

2.

1 1 1   ...  a1a2 a2a3 an1an =

4.

n 1 (C) a a 1 n

(D) none

(A) 2m – 1 : 2n – 1

(B) M : n

(C) 2m + 1 : 2n + 1

(D) none

The ratio between the sum of n terms of two A.P.’s is 3n + 8 : 7 n + 15. Then the ratio between their 12th terms is (A) 5 : 7

(B) 7 : 16

(C) 12 : 11

(D) none

If A.M. between two numbers is 5 and their G.M. is 4, then the H.M. will be: a)

6.

n (B) a a 1 n

If the ratio of sum of m terms and n terms of an A.P. be m2: n2, then the ratio of its mth and nth terms will be

3.

5.

a (A) a a 1 n

16 5

c)

11 5

d) none of these

b) 2

c) 3

d) 4

Between 1 and 31 are inserted m arithmetic means, so that the ratio of the 7th and (m – 1)th means is 5 : 9. Then the value of m is a) 12

8.

14 5

The A.M. between two numbers b and c is a and the two G.M.s between them are g1 and g2. If g13 + g23 = k abc, then k is equal to a) 1

7.

b)

b) 13

c) 14

Sum to n terms of the series

d) 15

1 1 1    .......... is 1  x 1  2x  1  2x 1  3x  1  3x 1  4x  nx (A) 1  x 1  nx   

n (B) 1  x  1  n  1 x   

x (C) 1  x  1   n  1 x 

(D) none of these



9.

1

 (n  1) (n  2) (n  3)....(n  k) is equal to n 1

10.

(A)

1 (k  1) k  1

(B)

(C)

1 (k  1) k

(D)

k k 1 k

Sum of the series S  1

11.

1

1 1 1 1  2   1  2  3  1  2  3  4   ......... upto 20 terms is 2 3 4

(A) 110

(B) 111

(C) 115

(D) 116

The sum of first n terms of the series 12 + 2 × 22 + 32 + 2 × 42 + 52 + 2 × 62 + ........... is n (n + 1)2/2 when n is even. When n is odd the sum of the series is 2 (B) n

(C) n3 (n – 1)/2

(D) none of these n

12.

 n  1

(A) n2 (3n + 1)/4

Sum of the series

(A) 1 

1 n!

r

 n  1! r 1

is

1 (B) 1  n  1 !  

2

1 (C) 2  n  1 !  

(D) none of these

13. If the A.M. and G.M. of two numbers are 13 and 12 respectively then the two numbers are a) 8, 12 14.

15

b) 8, 18

c) 10, 18 d) 12, 18

If 2p + 3q + 4r = 15, the maximum value of p3q5 r7 will be (A) 2180

54.35 (B) 15 2

55.7 7 (C) 17 2 .9

(D) 2285

If a1, a2, ... an are positive real numbers whose product is a fixed number c, then the minimum value of a1 + a2 + ... an–1 + 2an is (A) n(2c)1/n

(B) (n + 1)c1/n

(C) 2nc1/n

(D) (n + 1) (2c)1/n

Multiple Choice Questions with one or more than one correct Answers: n

16.

If a1, a2, a3, ............... are in A.P. with common difference d, then

1  a n  1  a 1   (A) tan   1  an1a1 

r 1

1 

 d   equals  1  ar ar 1 

 nd 1   (B) tan   1  an1a1 

(C) tan–1 an+1 – tan–1 (a1) 17.

 tan

(D) / 2

Let a1, a2, a3, ........ be terms of an A.P. a  a  .............  a

p2

a

1 2 p 6 If a  a  ..........  a  q 2 , p  q then a must be 1 2 q 21

(A) less than 1

(C) 18.

11 41

(B)

2 7

(D)

7 2

If a1, a2, ............ an are in H.P. and d be the common difference of the corresponding A.P. then the expression a1a2 + a2a3 + ......... + an–1an is equal to

(A)

a1  a n d

(B) (n – 1) (a1 – an)

(C) n (a1 – an) 19.

20.

For 0   

(D) (n – 1)a1an

    , if x   cos 2n , y   sin 2n , z   cos 2n  sin 2n , then 2 n0 n 0 n 0

(A) xyz = xz + y

(B) xyz = xy + z

(C) xyz = x + y + z

(D) xyz = yz + x

If a, b, c are in H.P. , then the expression

 1 1 1 1 1 1 E          equals  b c c  c a b 

21.

22.

23.

24.

25.

1 3 2 1   2  2 4c ca a 

(A)

2 1  2 bc b

(B)

(C)

3 2  2 b ab

(D) none of these

Three positive numbers form a GP. If the middle number is increased by 8, the three numbers form an AP. If the last number is also increased by 64 along with the previous increase in the middle number, the resulting numbers form a GP again. Then (A) common ratio = 3

(B) first number = 4/9

(C) common ratio = –5

(D) first number = 4

Let the harmonic mean and the geometric mean of two positive numbers be in the ratio 4 : 5. Then the two numbers are in the ratio (A) 1 : 4

(B) 4 : 1

(C) 3 : 4

(D) 4 : 3

Between two unequal numbers, if a1, a2 are two AMs; g1, g2 are two GMs and h1, h2 are two HMs then g1.g2 is equal to (A) a1h1

(B) a1h2

(C) a2h2

(D) a2h1

The sum of the series 1 + 2.2 + 3.22 + 4.23 + 4.23 + .......... + 100. 299 (A) is more than 2106

(B) is 99.2100 + 1

(C) is 100.2100 + 1

(D) is 99.299 + 1

Let the sum of the series

1 1 2 1  2  .........  n   ...........  3 upto n terms be 13 13  23 1  23  ..........  n 3

Sn ,n = 1, 2, 3, ........... Then Sn cannot be greater than

(A) 1/2

(B) 1

(C) 2

(D) 4

(Assertion – Reason Type) Each question contains STATEMENT – 1 (Assertion) and STATEMENT – 2 (Reason). Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct. 26.

Let a1, a2, a3, ........... be a sequence of real numbers such that sum to n terms of the sequence is  n  2  2n 1 n  N STATEMENT-1 : If tr = ar/2r, then t1, t2, t3 ........... forms an A.P. STATEMENT-2 : If Tr = r/ar, then T1, T2, T3, ........... forms a G.P.

27.

Let a, b, c be three positive real numbers which are in H.P. STATEMENT-1 :

ab cb  4 2a  b 2c  b

because STATEMENT-2 : If x > 0, then x  28.

1 4 x

STATEMENT-1 : If a, b, c are three positive real numbers such that a  c  b and 1 1 1 1     0 then a, b, c are in H.P.. a ab c cb

because STATEMENT-2 : If a, b, c are distinct positive real numbers such that a (b – c)x2 + b (c – a) xy + c (a – b)y2 is a perfect square, then a, b, c are in H.P. Linked Comprehension Type This section contains 3 paragraphs. Based upon each paragraph, 3 multiple choice questions have to be answered. Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

P29–31 : Paragraph for Question Nos. 29 to 31 Suppose x1, x2 be the roots of ax2 + bx + c = 0 and x3, x4 be the roots of px2 + qx + r = 0. 29.

If x1 , x 2 ,

1 1 b 2  4ac , are in A.P. , then q 2  4pr equals x3 x 4

(A) a2/r2

(B) b2/q2

(C) c2/p2

(D) a2/p2

30.

31.

If a, b, c are in G.P. as well as x1, x2, x3, x4 are in G.P. then p, q, r are in (A) A.P.

(B) G.P.

(C) H.P.

(D) A.G.P.

If x1, x2, x3, x4 are in G.P., then its common ratio is 1/ 4

1/ 3

 ar  (A)    cp 

(C)

 cr  (B)    ap 

cr ap

(D)

ap bq

P32–34 : Paragraph for Question Nos. 32 to 34 For k,n  N , we define B (k, n) = 1.2.3 ...... k + 2.3.4 ...... (k + 1) + ........ + n (n + 1) ........ (n + k – 1), S0 (n) = n and Sk (n) = 1k + 2k + ...... + nk To obtain value of B (k, n), we rewrite B (k, n) as follows:  k   k  1  k  2   n  k  1   n  k  n  n  1 ........  n  k  B  k,n   k !        .........     k !   k 1  k   k  1  k   k   k 

32.

S3(n) + 3S2(n) equals (A) B (3, n)

(B) B (3, n) – 2B (2, n)

(C) B (3, n) – 2B (1, n)

33.

 k  1  k  1  k  1  k  1   Sk n     Sk 1 n   ...........    S1  n     S0  n  equals  1   2   k   k  1

(A) (n + 1)k

(B) (n + 1)k – 1

(C) nk – (n – 1)k n

34.

(D) B (3, n) + 2B (1, n)

 k k  1k  2

(D) (n + 1)k – (n – 1)k 2

k 1

(A)

1 n (n + 1) (n + 2) (n + 3) (4n + 15) 20

(B)

1 n (n + 1) (n + 2) (n + 3) (2n + 12) 20

(C)

1 n (n + 1) (n + 2) (n + 3) (2n + 13) 20

(D)

1 n (n + 1) (n + 2) (n + 3) (n + 14) 20

Matrix–Match Type This section contains 2questions. Each question contains statements given in two column which have to be matched. Statements (A, B, C, D) in Column I have to be matched with statements (p, q, r, s) in Column II. The answers to these questions have to be appropriately bubbles as illustrated in the following example. If the correct matches are A–p, A–s, B–q, B–r, C–p, C–q and D–s, then the correctly bubbled 4 × 4 matrix should be as follows : 35.

If n > 1 Sum of all the terms in the nth row of the triangle Column I (A)

(p) n3

1 2 4

7

3 5

6

8

(B)

9

10

1 1 1

1

(q)

1

3

3

1 (r) 2n–1

1 3 7

13

5 9

15

11A 17

19

12

(D) 12 12

(s) 12

22 32

1 n  n 2  1 2

1 2

(C)

12

Column II

12 32

12

2n–2

Cn–2

36.

Match the conditions for the equation ax3 + bx2 + cx + d = 0 having roots in Column I

Column II

(A) AP (P) b3d = ac3 (B) GP (Q) 27ad3 = abcd2 – 2c3 d (C) HP (R) 2b3 – 9abc + 27a2d = 0 (D) 3 =  +

(S) 4ad -bc = 0 LEVEL-II MQ (ANS KEY) Objective questions ( One correct Answer)

1-C

2-C

3-A

4-B

5-A

11-B

12-B

13-B

14-C

15-A

16-A,B 21-A,D 26-C 29-A 3536.

17-A,B

18-A,D

6-B

7-C

19-B,C

8-B

9-C

10-C

20-A,B,C

22-A,B 23-B,D 24-A,B 25-C,D 27-C 28-C 30-B 31-A 32-B 33-B 34-A A Q BR CP DS A P BQ C R DS

LEVEL-II PQ Objective questions ( One correct Answer) 1.

If the pth, qth and rth terms of an A.P. be a, b,c respectively, then a  q  r   b  r  p   c p  q  

(A) 0

(B) 2

(C) p + q + r

(D) pqr



2.





n n n If x   a ,y   b ,z   c where a, b, c are in A.P. such that |a| <1, |b| n0

n 0

<

1and

n0

\|c| < 1, then x, y, z are in

3.

4.

5.

(A) A.P.

(B) G.P.

(C) H.P.

(D) None of these

log3 2, log6 2, log122 are in (A) A.P.

(B) G.P.

(C) H.P.

(D) none

In a G.P. if the (m + n)th term be p and (m – n)th term be q, then its mth term is (A)

pq

(B)

p / q 

(C)

q / p

(D) p/q

If the roots of the equation x3 – 12x2 + 39x – 28 = 0 are in A.P., then their common difference will be

6.

7.

(A) 1

(B) 2

(C) 3

(D) 4

If a, b, c, d are nonzero real numbers such that (a2 + b2 + c2) (b2 + c2 + d2)  (ab + bc + cd)2, then a, b, c, d are in (A) AP

(B) GP

(C) HP

(D) none of these 1

1

1



Value of y   0.36 log0.25  3  32  33 ......... upto   is (A) 0.9

(B) 0.8

(C) 0.6

(D) 0.25

8.

9.

10.

Let a1, a2, a3, ... be in AP and ap, aq, ar be in GP. Then aq : ap is equal to (A)

rp qp

(B)

(C)

rq qp

(D) none of these

Three distinct real numbers a, b, c are in G.P. such that a + b + c = x b, then (A) 0 < x < 1

(B) –1 < x < 3

(C) x < –1 or x > 3

(D) –1 < x < 2

If x = 1 + y + y2 + y3 +..... to  , then y is x (A) x  1  

(C) 11.

x (B) 1  x  

x 1 x

(D)

(A)

1 2



cos x  ,0  x  is cos x  sin x 2



3 1

(C) 0

13.

14.

1 x x

If exp. {(sin2 x + sin4 x + sin6 x + .... in f) loge 2} satisfies the equation x2 – 9x + 8 = 0, t hen the value of

12.

qp rq

(B)

1 2





3 1

(D) none of these

If x, y, z are in G.P., ax = by = cz, then (A) logc b = loga c

(B) log1 c = logb c

(C) loga b=logc b

(D) logb a = logc b

The rth, sth and tth terms of a certain G.P. are R, S and T respectively, then the value of Rs –t St–r. Tr – s is . (A) 0

(B) 1

(C) –1

(D) none of these

The greatest value of x2y3z4, (if x + y + z = 1, x, y, z > 0) is (A)

29 35

(B)

210 315

(C) 15.

215 310

(D) none of these

If a, b and c are three positive real numbers, then the minimum value of the expression

bc c a ab   is a b c (A) 1

(B) 2

(C) 3

(D) 6

Multiple Choice Questions with one or more than one correct Answers: 16.

If positive numbers a, b, c, d are in harmonic progression and a  b , then (A) a + d > b + c is always true

(B) a + b > c + d is always true

(C) a + c > b + d always true

(D) ad > bc



17.



n 0

18.

n 0

n 0

(A) are all more than 1

(B) are all less than 1

(C) are in G.P.

(D) are in H.P.

If S 

1 1 1  2  3  ........ upto  , then 3 3 3 log2  S 

(A)  0.25 

(B)  0.008 

4

log5  S 

(C)  0.008  19.



n n n If x   a , y   b , z   c , where a, b, c are in A.P and |a| < 1, |b| < 1, |c| < 1 then x, y, z

(D)  0.25 

4

log5  S 

log2  S 

8

8

The sum of the first n terms of the series 2

2

2

2

2

2

2

1  2.2  3  2.4  5  2.6  ..........is

n n  1 2

,

when n is even. When n is odd, the sum is

(A)

n2  n  1

(B)

2

(C) even, if odd ‘n’ of the type 4 l + 1. n

20.

The value of

 r 1

1 is a  rx  a  (r  1)x

 n  1

2

n2

2

(D) even, if the odd ‘n’ is of the type 4l + 3

(A)

n a  a  nx

(C)

n( a  nx  a) x

(B)

a  nx  a x

(D) none of these

(Assertion – Reason Type) Each question contains STATEMENT – 1 (Assertion) and STATEMENT – 2 (Reason). Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

21.

STATEMENT-1 : There exists no A.P. whose three terms are

3, 5 and 7

because STATEMENT-2 : If tp, tq and tr are three distinct terms of an tr  tp

A.P., then t  t is a rational number q p 22.

STATEMENT-1 : There exists an A.P. whose three terms are

2, 3, 5

STATEMENT-2 : There exists distinct real number p, q, r satisfying

2  A   p  1 d, 3  A   q  1 d 5  A   r  1 d 23.

STATEMENT-1 : The sum of an infinite A.G.P. a + (a + d) x + (a + 2d) x2 + (a + 3d) x3 + ........... where |x| < 1 always exist

STATEMENT-2 : The sum of the infinite series a + ar + ar2 + ............. converges if |r| < 1. Linked Comprehension Type This section contains 3 paragraphs. Based upon each paragraph, 3 multiple choice questions have to be answered. Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

P24–26 : Paragraph for Question Nos. 24 to 26 Sum of the following three series is given

1

1 1 1 1     .............  log2 2 3 4 5

1

1 1 1 1 1       .............  3 5 7 9 11 2 2

(1)

(2)

1

24.

25.

(3)

 1  1 1 1 2    .............  is   3  5   7  9  1113    

(A)

 2

(B)

 1 2

(C)

 4

(D)

 1 4

Sum of series

1

1 1 1 1    ............upto  is 7 9 15 17

(A)

  1 2 (B) 2 8

(C) 26.

1 1 1 1     .............  3 5 7 9 4



 4 2

1

(D)

 

4 2

Sum of the series

1

1 1 1 1 1 1       .......... upto  is 2 3 4 5 6 7

(A)

   log 2 (B)  log 2 4 4

(C)  – log 2

(D)  + log 2

Matrix–Match Type This section contains 2questions. Each question contains statements given in two column which have to be matched. Statements (A, B, C, D) in Column I have to be matched with statements (p, q, r, s) in Column II. The answers to these questions have to be appropriately bubbles as illustrated in the following example. If the correct matches are A–p, A–s, B–q, B–r, C–p, C–q and D–s, then the correctly bubbled 4 × 4 matrix should be as follows : 27.

Match the following, if f(n) denotes the sum of the series

12 + 2.22 + 32 + 2.42 + 52 + 2.62 + …. Upto n terms, then Column I Column II

28.

1 (49) 2 .50 2

(A) f(49)

(p)

(B) f(50)

(q) 25(51)2

(C) f(51)

(r)

(D) f(52)

(s) 26(53)2

1 (51) 2 52 2

Let a, b, c, p > 1 and q > 0. Suppose a, b, c are in G.P. Column I

Column II

(A) logp a, logp b, logp c are in

(p) G.P.

(B) loga p, logb p, logc p are in

(q) A.G.P.

(C) a logp c, b logp b, c logp a

(r) H.P.

(D) qlogaap, qlogb bp, qlogc cp are in

(s) A.P. LEVEL-II PQ (ANS KEY)

Objective questions ( One correct Answer) 1-A

2-C

3-C

4-A

5-C

11-B

12-D

13-B

14-B

15-D

16-A,D

17-A,D

18-A,B

21-A 24-C 2736.

22-B 26-A

23-A

25-B A P A S

BQ BR

6-B

7-C

8-C

19-B,D

CR C Q

9-C

20-A,B

DS DP

10-C

LEVEL-II MQ Objective questions ( One correct Answer) -

bq. P. 32–34. : Paragraph for Question Nos. 32 to 34. For k,n N. ∈ , we define. B (k, n) = 1.2.3 ...... k + 2.3.4 ...... (k + 1) + ........ + n (n + 1) ........ (n + k – 1), S0 (n) = n and Sk (n) = 1k + 2k + ...... + nk. To obtain value of B (k, n), we rewrite B (k, n) as follows: ( ) k k 1 k 2. n k 1. B k,n k! ......... k k k k. ⌈. ⌉. +. +. + -. ⎛ ⎞ ⎛. ⎞ ⎛. ⎞. ⎛. ⎞. =.

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