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

(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 1+ 3+ 5+......... 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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