Class 11 Mathematics Principle of Mathematical Induction Functions MCQs

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

Answer :  D
 

Question. Let P(n) be statement 2n < n!. Where n is a natural number, then P(n) is true for: 
(a) all n
(b) all n > 2
(c) all n > 3
(d) none of these

Answer :  C
 

Question. What is the sum of 12 + 22 + 32 + ... + n2? 
(a) n(n+1)(2n+1) /6
(b) n(n+1)/6
(c) n(n+1+2n+1) /6
(d) n(n+1)(n+2) /3

Answer :  A
 

Question. Let T(k) be the statement 1 + 3 + 5 + .... + (2k – 1) = k² +10 
Which of the following is correct?
(a) T(1) is true
(b) T(k) is true ⇒ T(k + 1) is true
(c) T(n) is true for all n ÎN
(d) All above are correct

Answer :  B
 

Question. Let P(n) : n2 + n + 1 is an even integer. If P(k) is assumed true then P(k + 1) is true. Therefore P(n) is true. 
(a) for n > 1
(b) for all n Î N
(c) for n > 2
(d) none of these

Answer :  D


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 P(n) : 2 + 4 + 6 +... + (2n), n Î N, then P(k) =k (k + 1) + 2 implies P (k + 1) = (k + 1) (k + 2) + 2 is true for all k Î N. 
So statement P(n) = n (n + 1) + 2 is true for:
(a) n ≥ 1
(b) n ≥ 2
(c) n ≥ 3
(d) none of these

Answer :  D
 

Question. The smallest +ve integer n for which n! (n+1/2) holds is 
(a) 1
(b) 2
(c) 3
(d) 4

Answer :  B
 

Question. Let S(K) = 1+ 3+ 5...+ (2K -1) = 3 + K2 then which of the following is true? 
(a) Principle of mathematical induction can be used to prove the formula
(b) S(K)ÞS(K +1)
(c) S(K)Þ/ S(K +1)
(d) S(1) is correct

Answer :  B


Question. Let P(n) : “2n < (1 × 2 × 3 × ... × n)”. Then the smallest positive integer for which P(n) is true is 
(a) 1
(b) 2
(c) 3
(d) 4

Answer :  D


Question. For all n Î N, 3.52n +1 + 23n + 1 is divisible by 
(a) 19
(b) 17
(c) 23
(d) 25

Answer :  B
 

Question. What is the sum of 13 + 23 + 33 + ........ + n3 ? 
(a) [n(n+1)/3]²
(b) [n(n+1)/2]²
(c) [n(n+2)/3]²
(d) [n(n+1)/2]³

Answer :  B
 

Question. What is the sum of 1 + 2 + 3 + ... n ? 
(a) n+1/2
(b) n/2
(c) n(n+1)/2
(d) n(n+2)/2

Answer :  C
 

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

Answer :  D
 

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

Answer :  C
 

Question. If P(n) : “46ⁿ + 16ⁿ + k is divisible by 64 for n Î N” is true, then the least negative integral value of k is. 
(a) – 1
(b) 1
(c) 2
(d) – 2

Answer :  A
 

Question. If an = √7 + √7 + √7 +... ... having n radical signs then by methods of mathematical induction which is true 
(a) an > 7 " n ≥ 1
(b) an < 7 " n ≥ 1
(c) an < 4 " n ≥ 1
(d) an < 3 " n ≥ 1

Answer :  B
 

Question. Let T(k) be the statement 1 + 3 + 5 + .... + (2k – 1)= k² +10 
Which of the following is correct
(a) T(1) is true
(b) T(k) is true ⇒ T(k + 1) is true
(c) T(n) is true for all nεN
(d) All above are correct 

Answer :  B

Question. If xn – 1 is divisible by x – k, then the least positive integral value of k is 
(a) 1
(b) 2
(c) 3
(d) 4

Answer :  A
 

Question. A student was asked to prove a statement P(n) by induction. He proved that P(k + 1) is true whenever P(k) is true for all k > 5 Î N and also that P (5) is true. On the basis of this he could conclude that P(n) is true 
(a) for all n Î N
(b) for all n > 5
(c) for all n ³ 5
(d) for all n < 5

Answer :  C


Question. If 10n + 3.4n+2 k is divisible by 9 for all n ÎN, then the least positive integral value of k is 
(a) 5
(b) 3
(c) 7
(d) 1

Answer :  A
 

Question. What is the sum of 2 + 4 + 6 + 8 + ....+ 2n ? A
(a) n (n + 1)
(b) n(n + 2)
(c) n (n + 3)
(d) n(n + 4)

Answer :  A