Class 11 Mathematics Sequences and Series MCQs Set 11

Question. If \( a_1, a_2, a_3, \dots, a_{n+1} \) are in A.P., of non zero terms then \( \frac{1}{a_1 a_2} + \frac{1}{a_2 a_3} + \dots + \frac{1}{a_n a_{n+1}} \), is
(a) \( \frac{n-1}{a_1 a_{n+1}} \)
(b) \( \frac{1}{a_1 a_{n+1}} \)
(c) \( \frac{n+1}{a_1 a_{n+1}} \)
(d) \( \frac{n}{a_1 a_{n+1}} \)
Answer: (d) \( \frac{n}{a_1 a_{n+1}} \)

Question. If \( a_1, a_2, a_3, \dots, a_n \) is an A.P of non zero terms such that \( \frac{1}{a_1 a_n} + \frac{1}{a_2 a_{n-1}} + \frac{1}{a_3 a_{n-2}} + \dots + \frac{1}{a_n a_1} = \lambda \left\{ \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n} \right\} \) then \( \lambda = \)
(a) 2
(b) \( a_1 + a_n \)
(c) \( 2(a_1 + a_n) \)
(d) \( \frac{2}{a_1 + a_n} \)
Answer: (d) \( \frac{2}{a_1 + a_n} \)

Question. If \( A_1, A_2, A_3, \dots, A_n \) are 'n' numbers inserted between a,b to form A.P. then \( A_1 + A_2 + A_3 + \dots + A_n \)
(a) \( \frac{a+b}{2} \)
(b) \( \frac{n}{2}(a+b) \)
(c) \( \frac{n}{4}(a+b) \)
(d) \( \frac{n}{3}(a+b) \)
Answer: (b) \( \frac{n}{2}(a+b) \)

Question. If the arthimetic mean between a and b is \( \frac{a^{n+1} + b^{n+1}}{a^n + b^n} \), then n =
(a) 0
(b) 1
(c) –1
(d) 1/2
Answer: (a) 0

Question. If the number of terms in a G.P. is odd, then product of terms =
(a) \( (middle\ term)^{no\ of\ terms} \)
(b) \( (middle\ term)^{no\ of\ terms - 1} \)
(c) \( (middle\ term)^{no\ of\ even\ terms + 2} \)
(d) \( (middle\ term)^{no\ of\ terms + 1} \)
Answer: (a) \( (middle\ term)^{no\ of\ terms} \)

Question. If the number of terms in a G.P. is even, then product of terms =
(a) \( (G.M.\ of\ middle\ terms)^{no\ of\ terms} \)
(b) \( (A.M.\ of\ middle\ terms)^{no\ of\ terms} \)
(c) \( (H.M.\ of\ middle\ terms)^{no\ of\ terms} \)
(d) \( (G.M.\ of\ middle\ terms)^{no\ of\ terms - 1} \)
Answer: (a) \( (G.M.\ of\ middle\ terms)^{no\ of\ terms} \)

Question. If \( G_1, G_2, G_3 \dots G_n \) are n numbers inserted between a,b of G.P. then common ratio is
(a) \( \left( \frac{a}{b} \right)^{\frac{1}{n+1}} \)
(b) \( \left( \frac{b}{a} \right)^{\frac{1}{n+1}} \)
(c) \( \left( \frac{b}{a} \right)^{n+1} \)
(d) \( \left( \frac{a}{b} \right)^{n+1} \)
Answer: (b) \( \left( \frac{b}{a} \right)^{\frac{1}{n+1}} \)

Question. If \( G_1, G_2, G_3 \dots G_n \) are n numbers inserted between a,b to form a G.P. then \( G_1 G_2 G_3 \dots G_n = \)
(a) \( \sqrt{ab} \)
(b) \( (\sqrt{ab})^n \)
(c) \( (\sqrt{ab})^{n+1} \)
(d) \( \left( \frac{a+b}{2} \right)^n \)
Answer: (b) \( (\sqrt{ab})^n \)

Question. If the geometric mean between a and b is \( \frac{a^{n+1} + b^{n+1}}{a^n + b^n} \) then n =
(a) 0
(b) 1/2
(c) – 1/2
(d) 1/4
Answer: (c) – 1/2

Question. If the harmonic mean between a and b is \( \frac{a^{n+1} + b^{n+1}}{a^n + b^n} \), then n=
(a) 0
(b) –1
(c) –1/2
(d) 1
Answer: (b) –1

Question. In an A.P., if first term is 4, 9th term is 20, then 15th term is
(a) 16
(b) 32
(c) 18
(d) 36
Answer: (b) 32

Question. The number of numbers that are divisible by 9 between 1 and 1000 is
(a) 101
(b) 110
(c) 111
(d) 100
Answer: (c) 111

Question. Let \( T_r \) be the \( r^{th} \) term of an A.P. whose first term is a and common difference is d. If for some positive integers m,n, \( m \neq n \), \( T_m = \frac{1}{n} \) and \( T_n = \frac{1}{m} \), then a-d=
(a) 0
(b) \( \frac{1}{m} + \frac{1}{n} \)
(c) \( \frac{1}{mn} \)
(d) 1
Answer: (a) 0

Question. If \( p^{th}, q^{th}, r^{th} \) terms of an A.P are a, b, c then a(q – r) + b(r – p) + c(p – q) =
(a) 0
(b) 1
(c) a + b + c
(d) abc
Answer: (a) 0

Question. If the numbers \( a, b, c, d, e \) form an A.P., then the value of \( a - 4b + 6c - 4d + e \) is
(a) 1
(b) 2
(c) 0
(d) None of the options
Answer: (c) 0

Question. If \( S_n = nP + \frac{n}{2}(n-1)Q \), where \( S_n \) denotes the sum of the first n terms of an A.P., then the common difference is
(a) \( P + Q \)
(b) \( 2P + 3Q \)
(c) 2Q
(d) Q
Answer: (d) Q

Question. In an A.P., if common difference is 2, sum to n terms is 49, 7th term is 13, then n=
(a) 0
(b) 5
(c) 7
(d) 13
Answer: (c) 7

Question. Consider an A.P. with first term \( a \) and common difference \( d \). Let \( S_k \) denote the sum of the first \( k \) terms. If \( \frac{S_{kx}}{S_x} \) is independent of \( x \), then
(a) \( a = 2d \)
(b) \( a = d \)
(c) \( 2a = d \)
(d) \( 2a = 3d \)
Answer: (c) \( 2a = d \)

Question. If \( a_1, a_2, a_3, \dots \) are in A.P. such that \( a_1 + a_5 + a_{10} + a_{15} + a_{20} + a_{24} = 225 \), then \( a_1 + a_2 + a_3 + \dots + a_{23} + a_{24} = \)
(a) 909
(b) 75
(c) 750
(d) 900
Answer: (d) 900

Question. The degree of the expression \( (1+x)(1+x^6)(1+x^{11}) \dots (1+x^{101}) \) is
(a) 1081
(b) 1061
(c) 1071
(d) 1091
Answer: (c) 1071

Question. If the sum of three numbers which are in A.P is 27 and the product of first and last is 77, then the numbers are
(a) 7, 9, 11
(b) 6, 9, 12
(c) 7, 10, 11
(d) 7, 11, 9
Answer: (a) 7, 9, 11

Question. If the ratio of \( n^{th} \) terms of two A.P.'s is \( (2n+8):(5n-3) \), then the ratio of the sums of their n terms is
(a) \( (2n+18):(5n+1) \)
(b) \( (5n-1):(2n+18) \)
(c) \( (2n+18):(5n-1) \)
(d) \( (3n+18):(4n+1) \)
Answer: (c) \( (2n+18):(5n-1) \)

Question. 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
(b) 13
(c) 14
(d) 15
Answer: (c) 14

Question. If \( S_1, S_2, S_3 \) are the sums of first n natural numbers, their squares and their cubes respectively, then \( S_3(1+8S_1)= \)
(a) \( S_2^2 \)
(b) \( 9S_2 \)
(c) \( 9S_2^2 \)
(d) \( 3S_2^2 \)
Answer: (c) \( 9S_2^2 \)

Question. If 6th term of a G.P. is –1/32 and 9th term is 1/256, then 11th term =
(a) 1024
(b) 1/1024
(c) 1/256
(d) 1/512
Answer: (b) 1/1024

Question. If \( (1-y)(1+2x+4x^2+8x^3+16x^4+32x^5)=1-y^6 \), (\( y \neq 1 \)), then a value of y/x is
(a) 1/2
(b) 2
(c) 25/24
(d) 24/25
Answer: (b) 2

Question. If the sum of three numbers in a G.P. is 26 and the sum of products taken two at a time is 156, then the numbers are
(a) 2,6,18
(b) 1,8,64
(c) 1,5,25
(d) 1,4,1
Answer: (a) 2,6,18

Question. One of the 5 geometric means between \( \frac{1}{3} \) and 243 is
(a) 79
(b) 80
(c) 81
(d) 82
Answer: (c) 81

Question. If the fourth term of a H.P is 1/3 and 7th term is 1/4, then 16th term is
(a) 1/5
(b) 1/6
(c) 1/7
(d) 1/8
Answer: (c) 1/7

Question. If a, b, c are in H.P and ab + bc + ca = 15, then ca =
(a) 5
(b) 7
(c) 9
(d) 10
Answer: (a) 5

Question. If \( \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots \text{upto } \infty = \frac{\pi^2}{6} \), then value of \( \frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + \dots \text{up to } \infty \) is
(a) \( \frac{\pi^2}{4} \)
(b) \( \frac{\pi^2}{6} \)
(c) \( \frac{\pi^2}{8} \)
(d) \( \frac{\pi^2}{12} \)
Answer: (c) \( \frac{\pi^2}{8} \)

Question. The sum of \( \frac{3}{1.2} \left( \frac{1}{2} \right) + \frac{4}{2.3} \left( \frac{1}{2} \right)^2 + \frac{5}{3.4} \left( \frac{1}{2} \right)^3 + \dots \) to n terms is equal to
(a) \( 1 - \frac{1}{(n+1)2^n} \)
(b) \( 1 - \frac{1}{n \cdot 2^{n-1}} \)
(c) \( 1 + \frac{1}{(n+1) \cdot 2^n} \)
(d) \( 1 + \frac{1}{n \cdot 2^{n-1}} \)
Answer: (a) \( 1 - \frac{1}{(n+1)2^n} \)

Question. If \( \sum_{r=1}^n (r^2+1)r! = 200 \times 201! \), then n =
(a) 200
(b) 201
(c) 199
(d) 202
Answer: (a) 200