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NCERT Solutions for class 11 Maths | Chapter 4 - Principles of Mathematical Induction

Questions
1 FFor any natural number n, 7n - 2n is divisible by
A 3
B 4
C 5
D 7

Answer:5
2 The sum of the series 13 + 23 + 33 + ...........n3 is
A {(n + 1)/2}2
B {n/2}2
C n(n + 1)/2
D {n(n + 1)/2}2

Answer:{n(n + 1)/2}2
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3 (n² + n) is ____ for all n ∈ N.
A Even
B odd
C Either even or odd
D None of these

Answer:Even
4 The sum of n terms of the series 12 + 32 + 52 +......... is
A n(4n2 - 1)/3
B n2(2n2+1)/6
C none of these.
D n2(n2+1)/3

Answer:n(4n2 - 1)/3
5 (1 + x)n ≥ ____ for all n ∈ N,where x > - 1
A 1 + nx
B 1 - nx
C 1 + nx/2
D 1 - nx/2

Answer:1 + nx
6 (12 + 22 + …... + n2 ) _____ for all values of n ∈ N
A = n3 /3
B < n3 /3
C > n3 /3
D None of these

Answer:> n3 /3
7 The nth term of the series 1 + 4 + 10 + 22 + ….. is
A 2n2+2
B 3*2n-1 - 2
C 2n2+2n
D 5n-1

Answer:3*2n-1 - 2
8 n < 2n for all
A n ∈ R
B n ∈ N
C n may be any number
D None of these

Answer:n ∈ N
9 Find the number of shots arranged in a complete pyramid the base of which is an equilateral triangle, each side containing n shots.
A n(n+1)(n+2)/3
B n(n+1)(n+2)/6
C n(n+2)/6
D (n+1)(n+2)/6

Answer:n(n+1)(n+2)/6
10 (2 ∙ 7N + 3 ∙ 5N - 5) is Divisible by ___ for all N ∈ N
A 6
B 12
C 18
D 24

Answer:24
11 The nth term of the series 1 + 4 + 10 + 22 + ….. is
A 2n2+2
B 3*2n-1 - 2
C 2n2+2n
D 5n-1

Answer:3*2n-1 - 2
12 A + AR + AR2 + ....... + ARN – 1 = (ARN – 1)/(R - 1) for
A r ≥ 1 and n ∈ N
B r ≤ 1 and n ∈ N
C r > 1 and n ∈ N
D r < 1 and n ∈ N

Answer:r > 1 and n ∈ N
13 The sum of n terms of the series 12 + 32 + 52 +......... is
A n(4n2 - 1)/3
B n2(2n2+1)/6
C none of these.
D n2(n2+1)/3

Answer:n(4n2 - 1)/3
14 For all n∈N, 52n − 1 is divisible by :
A 26
B 24
C 11
D 25

Answer:24
15 What is the base case for the inequality 7n > n3, where n = 3?
A 652 > 189
B 42 < 132
C 343 > 27
D 42 <= 431

Answer:343 > 27
16 In the principle of mathematical induction, which of the following steps is mandatory?
A induction hypothesis
B inductive reference
C induction set assumption
D minimal set representation

Answer:induction hypothesis
17 For m = 1, 2, …, 4m+2 is a multiple of ________
A 3
B 5
C 6
D 2

Answer:2
18 For any integer m>=3, the series 2+4+6+…+(4m) can be equivalent to ________
A m2+3
B m+1
C mm
D 3m2+4

Answer:m2+3
19 Which of the following is the base case for 4n+1 > (n+1)2 where n = 2?
A 64 > 9
B 16 > 2
C 27 < 91
D 54 > 8

Answer:64 > 9
20 For any positive integer m ______ is divisible by 4.
A 5m2 + 2
B 3m + 1
C m2 + 3
D m3 + 3m

Answer:m3 + 3m

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