Principle Of Mathematical Induction(11th And 12th > Mathematics ) Questions and Answers

Question 1.


Let P(n) denote the statement that n2 + n is odd. It


is seem that P(n)  ⇒ P(n + 1), Pn is true for all


  1.     n > 1
  2.     n
  3.     n > 2
  4.     None of these
Explanation:-
Answer: Option D. -> None of these
:
D

P(n) = n2 + n. It is always odd (statement) but


square of any odd number is always odd and 


also, sum of odd number is always even. So


for no any 'n' for which this statement is true.


Question 2.


For natural number n, (n!)2 > nn, if


  1.     n > 3
  2.     n > 4
  3.     n≥ 4
  4.     n≥ 3
Explanation:-
Answer: Option D. -> n≥ 3
:
D

Check through option, condition (n!)2 > nn is


true when n ≥ 3.


Question 3.


For every positive integral value of n, 3n > n3 when


  1.     n > 2
  2.     n≥3
  3.     n≥ 4
  4.     n < 4
Explanation:-
Answer: Option C. -> n≥ 4
:
C

Check through option, the condition 3n > n3 is


true when n ≥ 4.


Question 4.


If n is a natural number then (n+12)n ≥ n ! is true


when


  1.     n > 1
  2.     n≥ 1
  3.     n > 2 
  4.     n≥2
Explanation:-
Answer: Option B. -> n≥ 1
:
B

Check through option, the condition


 (n+12)n ≥ n ! is true for n  ≥ 1.


Question 5.


For positive integer n, 10n−2 > 81n, if


  1.     n > 5
  2.     n≥ 5
  3.     n < 5
  4.     n > 6
Explanation:-
Answer: Option B. -> n≥ 5
:
B

Check through option, the condition


10n−2 > 81n is satisfied if n ≥ 5.


Question 6.


For every positive integer n, 2n < n! when


  1.     n < 4
  2.     n≥ 4
  3.     n < 3
  4.     None of these
Explanation:-
Answer: Option B. -> n≥ 4
:
B

Check through option, the condition 2n < n! is


true when n ≥ 4.


Question 7.


For every natural number n, n(n2−1) is divisible by


 


  1.     4
  2.     6
  3.     10
  4.     None of these
Explanation:-
Answer: Option B. -> 6
:
B

n(n2−1) = (n - 1)(n)(n + 1)


It is product of three consecutive natural


numbers, so according to Langrange's theorem


it is divisible by 3 ! i.e., 6.


Question 8.


For every natural number n


  1.     n>2n
  2.     n<2n
  3.     n≥2n
  4.     Can't be determined.
Explanation:-
Answer: Option B. -> n<2n
:
B

Let n = 1 then option (a) and (d) is eliminated.


Equality can't be attained for any value of n so,


option (b) satisfied.


Question 9.


If n ∈ N, then 72n + 23n−3.3n−1 is always divisible by


 


  1.     25
  2.     35
  3.     45
  4.     None of these
Explanation:-
Answer: Option A. -> 25
:
A

Putting n = 1 in 72n+23n−3.3n−1


=50, divisible by 25


Question 10.


If n ∈ N, then x2n−1+y2n−1 is divisible by 


  1.     x + y
  2.     x - y
  3.     x2 + y2
  4.     x2+xy
Explanation:-
Answer: Option A. -> x + y
:
A

x2n−1+y2n−1 is always contain equal odd power.


So it is always divisible by x + y.


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