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CBSE Questions for Class 11 Engineering Maths Principle Of Mathematical Induction Quiz 3 - MCQExams.com

Let f(n) equals to the sum of the cubes of three consecutive natural numbers.
f(n) leaves the remainder zero when divided by
  • 11
  • 9
  • 99
  • none of these
For all nI+, the statement P(n)=n77+n55+23n3n105 is a natural number is true, if
  • P(1) is true
  • P(2) is true
  • P(3) is true
  • True nN
The number of values of n, for which p(n)=1!+2!+3!+4!++n! is the square of a natural number, is equal to
  • 0
  • 1
  • 2
  • 3
If nN, then n3+2n is divisible by
  • 2
  • 3
  • 5
  • 6
If P(n)=n55+n33+7n15, then statement "P(n) is a natural number" is true,
  • only for n>1.
  • only for n is an odd positive integer.
  • only for n is an even positive integer.
  • nN
Let P\left ( n \right )= n^{3}-n, the largest number by which P\left ( n \right ) is divisible \forall possible integral values of n is
  • 2
  • 3
  • 5
  • 6
For every natural number n -
  • n\, >\, 2^{n}
  • n\, <\, 2^{n}
  • n\, / 2^{n}
  • n\, / 2^{2n}
If \displaystyle a_{n}=\sqrt{7+\sqrt{7+\sqrt{7+.....}}} having n radical signs, then by method of mathematical induction which of the following is true?
  • \displaystyle a_{n}>6,\forall n> 1
  • \displaystyle a_{n}>3,\forall n> 1
  • \displaystyle a_{n}>4,\forall n> 1
  • \displaystyle a_{n}<2,\forall n> 1
  • Both (A) & (R) are individually true & (R) is correct explanation of (A).
  • Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
  • (A) is true but (R) is false.
  • (A) is false but (R) is true.
Let P(n)\, :\, n^{2}\, +\, n is an odd integer. It is seen that truth of P(n)\, \Rightarrow the truth of P(n + 1). Therefore, P(n) is true for all -
  • n > 1
  • n
  • n > 2
  • None of these
The inequality n!\, >\, 2^{n\, -\, 1} is true - 
  • For all n > 1
  • For all n > 2
  • For all n\, \epsilon\, N
  • None of these
Let P\left ( n \right )= x^{2n-1}+y^{2n-1} is divisible by x+y as P\left ( 1 \right ) is true, then truth of P\left ( k+1 \right ) indicates
  • P\left ( n \right ) s divisible by x+y only for odd positive integer
  • P\left ( n \right ) is divisible by x+y only for positive even integer
  • P\left ( n \right ) is true \forall n\in N
  • P\left ( k \right ) is true
The sequence \displaystyle \left ( x_{n}n\geq 1 \right ) is defined by \displaystyle x_{1}=0 and \displaystyle x_{n+1}=5x_{n}+\sqrt{24x^{2}_{n}+1}  for all \displaystyle n\geq 1. Then all \displaystyle x_{n} are 
  • Negative integers
  • Positive integers
  • Rational numbers
  • None of these
Let P\left ( n \right )=11^{n+2}+12^{2n+1}, then the least value of the following which P\left ( n \right ) is divisible by is
  • 19
  • 7
  • 133
  • none of these
For positive integer n, 3^{n} < n! when
  • n \geq 6
  • n > 7
  • n \geq 7
  • n \leq 7
Let P(n) : n^2 + n is an odd integer. It is seen that truth of P(n)\Rightarrow the truth of P(n + 1). Therefore, P(n) is true for all
  • n > 1
  • n
  • n > 2
  • none of these
For natural number n, 2^{n}\, (n - 1) ! <n^{n}, if.
  • n < 2
  • n > 2
  • n \geq 2
  • Never
For every positive integer
n, \displaystyle \frac{n^{7}}{7} + \frac{n^{5}}{5} + \frac {2n^{3}}{3} - \frac{n}{105} is
  • an integer
  • a rational number
  • a negative real number
  • an odd integer
If P is a prime number then n^{p} - n is divisible by p when n is a 
  • natural number greater than 1
  • odd number
  • even number
  • None of these
The difference between an +ve integer and its cube, is divisible by
  • 4
  • 6
  • 9
  • None of these
If n is a natural number then \left( \displaystyle \frac{n + 1}{2} \right)^{n}\, \geq n! is true when.
  • n > 1
  • n \geq 1
  • n > 2
  • Never
For  every natural number n, n(n + 3) is always :
  • multiple of 4
  • multiple of 5
  • even
  • odd
For all n \in N, n^{4} is less than
  • 10^{n}
  • 4^{n}
  • 10^{10}
  • None of these
A student was asked to prove a statement by induction. He proved
(i) P(5) is true and
(ii) Trutyh of P(n) \Rightarrow truth of p(n + 1), n\inN
On the basis of this, he could conclude that P(n) is true for
  • no n \in N
  • all n \in N
  • all n \geq 5
  • None of these
For every positive integral value of n, 3^n > n^3 when
  • n > 2
  • n\geq 3
  • n\geq 4
  • n < 4
For every positive integer n, \dfrac {n^7}{7}+\dfrac {n^5}{5}+\dfrac {2n^3}{3}-\dfrac {n}{105} is
  • an integer
  • a rational number
  • a negative real number
  • an odd integer
P(n) : 3^{2n+2} -8n -9 is divisible by 64, is true for
  • all n\epsilon N \cup \left \{0\right \}
  • n\geq 2, n\epsilon N
  • n\epsilon N, n > 2
  • none of these
The smallest positive integer for which the statement 3^{n+1} < 4^n holds is
  • 1
  • 2
  • 3
  • 4
If x > 1, then the statement P(n) : (1 + x)^n > 1 + nx is true for
  • all n\epsilon N
  • all n > 1
  • all n > 1 and x\neq 0
  • None of these
If n is a natural number then \left (\frac {n+1}{2}\right )^n \geq n! is true when
  • n > 1
  • n\geq 1
  • n > 2
  • never
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