CBSE Questions for Class 11 Engineering Maths Principle Of Mathematical Induction Quiz 2 - MCQExams.com

$$\forall n\in N; 10^{2n - 1}+1$$ is divisible by
  • $$2$$
  • $$3$$
  • $$7$$
  • $$11$$
$$\forall n\in N; 3n^{5} + 5n^3 + 7n\>$$ is divisible by
  • $$3$$
  • $$5$$
  • $$10$$
  • $$15$$
If $$x^n - 1$$ is divisible by $$x - k$$, then the least positive integral value of $$k$$ is
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
$$\forall n\in N, n^4$$ is less than
  • $$10^n$$
  • $$4n$$
  • $$4^n$$
  • $$10^{10}$$
If  $$n$$ $$ \in$$ N, then $$x^{2n - 1} + y^{2n - 1}$$ is divisible by
  • $$x + y$$
  • $$x - y$$
  • $$x^2 + y^2$$
  • none of these
For all $$n\in N, \sum n$$ is
  • $$\displaystyle < \frac{(2n + 1)^2}{8}$$
  • $$\displaystyle > \frac{(2n + 1)^2}{8}$$
  • $$\displaystyle = \frac{(2n + 1)^2}{8}$$
  • none of these
If $$49^{n} + 16 n + \lambda$$ is divisible by $$64$$ for all $$n\in N$$, then the least negative integral value of $$\lambda$$ is
  • $$-12$$
  • $$-1$$
  • $$-3$$
  • $$-4$$
Let $$P(m)$$ be the statement $$m^{2}> 100$$, the statement $$P(k + 1)$$ will be true if
  • $$P(1)$$ is true
  • $$P(2)$$ is true
  • $$P(k)$$ is true
  • none of these
The product of three consecutive natural numbers is divisible by
  • $$3$$
  • $$8$$
  • $$6$$
  • $$11$$
Let $$P(n)$$ be a statement such that truth of $$P\left ( n \right )\Rightarrow $$ the truth of $$P\left ( n+1 \right )$$ for all $$n\epsilon N$$, then $$P(n)$$ is true
  • $$\forall n> 1$$
  • $$\forall n$$
  • nothing can be said
  • $$\forall n> k$$ (k is some fixed positive integer)
$$\displaystyle x^{3^{n}}+y^{3^{n}}$$ is divisible by $$x+y$$, if 
  • $$n$$ is any integer $$\geq0$$
  • $$n$$ is an odd positive integer
  • $$n$$ is an even positive integer
  • $$n$$ is a rational number
Statement  1 : For each natural number $$n, (n + 1)^7 - n^7 - 1$$ is divisible by 7.
Statement  2 : For each natural $$n$$, $$n^7 - n$$ is divisible by 7.
  • A) Statement - 1 is True, Statement - 2 is True; Statement -2 is a correct explanation for Statement - 1
  • B) Statement - 1 is True, Statement - 2 is True; Statement -2 is NOT a correct explanation for Statement - 1
  • C) Statement - 1 is True, Statement-2 is False
  • D) Statement - 1 is False, Statement-2 is True
For $$n \in N, x^{n + 1} + (x + 1)^{2n - 1}$$ is divisible by
  • $$x$$
  • $$x + 1$$
  • $$x^2 + x + 1$$
  • $$x^2 - x + 1$$
Let $$P\left ( n \right )=n\left ( n+1 \right )$$ is an even number, then which of the following satisfy $$P(n)$$
  • $$P(3)$$
  • $$P(100)$$
  • $$P(50)$$
  • All of these
For each natural number, the statement $$P\left ( n \right )=2^{3n}-1$$ is divisible by
  • $$10$$
  • $$6$$
  • $$7$$
  • None of these.
Let $$P\left ( n \right ):n^{2}+n$$ is an odd integer
$$P\left ( k \right )\Rightarrow P\left ( k+1 \right )$$ is true
Then $$P\left ( n \right )$$ is true for all
  • $$n> 2$$
  • $$n> 1$$
  • $$n$$
  • none of these
If $$P(n)$$ be the statement $$n(n+1)+1$$ is odd, then which of the following is false?
  • $$P(2)$$
  • $$P(3)$$
  • $$P(4)$$
  • none of these
Let $$P\left ( n \right )=2^{3n}-7n-1$$ then $$P(n)$$ is divisible by
  • $$63$$
  • $$36$$
  • $$49$$
  • $$25$$
Let $$P(n)$$ be the statement representing the sum of next three successive natural numbers of $$n$$, $$\forall n\in N$$, then the smallest value of $$n$$ to which $$P(n)$$ is divisible by $$9$$ is
  • $$1$$
  • $$3!$$
  • $$3$$
  • $$9!$$
Let $$\displaystyle P\left ( n \right )=1+\frac{1}{4}+\frac{1}{9}+...+\frac{1}{n^{2}}< 2-\frac{1}{n}$$ is true for
  • $$\forall n$$
  • for $$n=1$$
  • $$n=2$$
  • none of these
$$P\left ( n \right ):2^{n+2}< 3^{n}$$, is true for
  • $$n\in N$$
  • $$n>3, n\in N$$
  • $$n>2, \forall n\in N$$
  • none of these.
Let $$P\left ( k \right ):2+4+6+...+2k=k\left ( k+1 \right )+2$$, then the statement $$P(m+1)$$ will be true if
  • $$P(1)$$ is true
  • $$P(2)$$ is true
  • $$P(m)$$ is true
  • none of these
Let $$P(n)$$ be the statement that $$n^{2}-n+41$$ is prime, then which of the following is not true ?
  • $$P(2)$$
  • $$P(3)$$
  • $$P(41)$$
  • none of these
Let $$P\left ( n \right ):2^{n}> n, \forall n\in N$$ and $$2^{k}> k, \forall n=k$$, then which of the following is true $$\forall k\geq 2$$?
  • $$2^{k}> 5k> 1$$
  • $$2^{k+1}> 2k> k+1$$
  • $$2^{k}> 2\left ( k+1 \right )> k$$
  • None of these.
The inequality, $$P(n)=n!> 2^{n}$$ is true for
  • $$n\geq 4$$
  • $$n> 1$$
  • $$n> 2$$
  • $$\forall n\in N$$
Let $$P\left ( n \right ):a^{n}+b^{n}$$ such that $$a, b$$ are even, then $$p(n)$$ will be divisible by $$a + b$$ if
  • $$n> 1$$
  • $$n$$ is odd
  • $$n$$ is even
  • none of these
Let $$f\left ( n \right ) = 8^{n}-3^{n}$$, if $$n$$ is odd natural number then $$f\left ( n \right )$$ is divisible by
  • $$2$$
  • $$3$$
  • $$5$$
  • none of these
If $$\displaystyle P\left ( n \right )= 1^{2}+3^{2}+5^{2}+...+\left ( 2n-1 \right )^{2}=  \frac{n\left ( 4n^{2} -1\right )}{3}$$, then which of the following does NOT hold good?
  • $$P\left ( 1 \right )$$
  • $$P\left ( 2 \right )$$
  • $$P\left ( 3 \right )$$
  • None of these.
If $$P\left ( n \right )= 1+2+3+\dots+n$$ is a perfect square, $$N$$ is less than $$100$$, then possible values of $$n$$ is/are
  • only $$1$$
  • $$1$$ and $$8$$
  • only $$8$$
  • $$1, 8, 49$$
Let the statement $$P\left ( n \right ):2^{n}\geq 3n$$, the truth of $$P\left( k \right), \forall   k\in N  \Rightarrow$$ truth of,
  • $$P\left ( 1 \right )$$
  • $$P\left ( 2 \right )$$
  • $$P\left ( 3 \right )$$
  • $$P\left ( k+1 \right )$$
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