CBSE Class 11 Mathematics HOTs Principle of Mathematical Induction

Question. If P(n) = 2 + 4 + 6 + .....+ 2n, n∈ N , then P(k) =k(k +1) + 2
⇒ P(k +1) = (k +1)(k + 2) + 2 for all k ∈ N . So we can conclude that P(n) = n(n +1) + 2 for
(a) all n ∈ N
(b) n > 1
(c) n > 2
(d) nothing can be said
Answer : D

Question. For a positive integer n, 
Let a(n) = 1 + 1/2 + 1/3 + 1/4 + .......  + 1/(2ⁿ) -1 Then
(a) a(100) ≤ 100
(b) a(100) > 100
(c) a(200) ≤ 100
(d) a(200) < 100
Answer : A

Question. If n ∈ N , then the result
1/n + 1/n+1 + 1/n+2 + ...... + 1/2n-1 
= 1- 1/2 + 1/3 - 1/4 + ...... + 1/2n-1 holds for
(a) all n ∈ N
(b) for even values of n
(c) for odd values of n
(d) not true for any n
Answer : A

Question. For all n ≥ 1, find 
1/1.2 + 1/2.3 + 1/3.4 + ...... + 1/n(n+1)
(a) n/n+1
(b) 1/n+1
(c) 1/n(n +1)
(d) None of these
Answer : A

Question. For all natural numbers n, find
(1+3/1) (1+5/4) (1+7/9) (1+2n+1/n²)
(a) (n + 1)²
(b) (n – 1)²
(c) n(n + 1)
(d) None of these
Answer : A

Question. 2ⁿ > n² when n ∈ N such that
(a) n > 2
(b) n > 3
(c) n < 5
(d) n ≥ 5
Answer : D

Question. The greatest positive integer, which divides n(n +1)(n + 2)(n + 3) for all n∈ N , is
(a) 2
(b) 6
(c) 24
(d) 120
Answer : C

Question. For any n∈ N , the value of the expression

Answer : A

Question. If 49ⁿ + 16n + λ is divisible by 64 for all n ∈ N, then the least negative value of λ is
(a) –2
(b) –1
(c) –3
(d) – 4
Answer : B

Question. By mathematical induction,
1/1 • 2 • 3 + 1/2 • 3 • 4 +..... 1 + 1/n(n+1)(n+2) is equal to
(a) n(n+1)/4(n+2) (n+3)
(b) n(n+3)4(n+1)(n+2)
(c) n(n+2)/4(n+1)(n+3)
(d) None of these
Answer : B

Question. If n is a positive integer, then 2 . 4²ⁿ⁺¹ + 3³ⁿ⁺¹ is divisible by :
(a) 2
(b) 7
(c) 11
(d) 27
Answer : C

Question. If 4ⁿ/n+1 < (2n)!/(n!)², then P(n) is true for
(a) n ≥ 1
(b) n > 0
(c) n < 0
(d) n ≥ 2
Answer : D

Question. If P(n) : 3ⁿ < n!, n ∈ N, then P(n) is true
(a) for n ≥ 6
(b) for n ≥ 7
(c) for n ≥ 3
(d) for all n
Answer : B

Question. If p is a prime number, then n p – n is divisible by p when n is a
(a) Natural number greater than 1
(b) Irrational number
(c) Complex number
(d) Odd number
Answer : A

Question. When 2³⁰¹ is divided by 5, the least positive remainder is
(a) 4
(b) 8
(c) 2
(d) 6
Answer : C

Question. By the principle of induction ∀ n ∈ N, 32n when divided by 8, leaves remainder
(a) 2
(b) 3
(c) 7
(d) 1
Answer : D

Question. For all n ∈ N, 3.5^{2n + 1} + 2^{3n + 1} is divisible by
(a) 19
(b) 17
(c) 23
(d) 25
Answer : B

Question. For every natural number n, n(n²–1) is divisible by
(a) 4
(b) 6
(c) 10
(d) None of these
Answer : B

Question. For all n ∈ N, 1 + 1/1+2 + 1/1+2+3 +  ..... + 1/1+2+3+ ..... +n  is equal to
(a) 3n/n+1
(b) n/n+1
(c) 2n/n–1
(d) 2n/n+1
Answer : D

Question. For all n ∈ N, 1.3 + 2.3² + 3.3³ + ..... + n.3ⁿ is equal to
(a) (2n+1) 3ⁿ⁺¹+3/4
(b) (2n –1) 3ⁿ⁺¹+3/4
(c) (2n+1)3ⁿ+3/4
(d) (2n–1)3ⁿ⁺¹+1/4
Answer : B

Numeric Value Answer

Question. The remainder when 5⁹⁹ is divided by 13, is ___________.
Answer : 8

Question. For all n ∈ N, 41ⁿ – 14ⁿ is a multiple of ___________.
Answer : 27

Question. If n ∈ N, then 1^{1n + 2} + 12²ⁿ⁺¹ is divisible by ___________.
Answer : 133

Question. For every natural number n, 32^{n + 2} – 8^{n – 9} is divisible by ___________.
Answer : 16

Question. If m, n are any two odd positive integers with n < m, then the largest positive integer which divides all the numbers of the type m² – n² is
___________.
Answer : 8