CBSE Questions for Class 11 Engineering Maths Principle Of Mathematical Induction Quiz 5 - MCQExams.com

The integer next above $$(\sqrt{3}+1)^{2n}$$ contains
  • $$2^{n+1}$$ as a factor
  • $$2^{n+2}$$ as a factor
  • $$2^{n+3}$$ as a factor
  • $$2^{n}$$ as a factor
If $${a_{1,}}{a_{2,}}{a_{3,}}........{a_{2n + 1}}$$ are in A.P. then$$\frac{{{a_{2n + 1}} - {a_1}}}{{{a_{2n + 1}} + {a_1}}} + \frac{{{a_{2n + 1}} - {a_2}}}{{{a_{2n + 1}} + {a_2}}} + ...... + \frac{{{a_{2n + 1}} - {a_n}}}{{{a_{2n + 1}} + {a_n}}}$$ 
  • $$\frac{{n\left( {n + 1} \right)}}{2}\,\,\frac{{{a_2} - {a_1}}}{{{a_{n + 1}}}}$$
  • $$\frac{{n\left( {n + 1} \right)}}{2}$$
  • $$\left( {n + 1} \right)\left( {{a_2} - {a_1}} \right)$$
  • $$\frac{{n\left( {n + 1} \right)}}{2}\,\,\frac{{{a_{n + 1}} - {a_n}}}{{{a_n}}}$$
State the whether given statement is true or false
Prove that by P.M.I.
$$1^2+3^2+........+(2n-1)^2=\frac{n(4n^2-1)}{3}$$
  • True
  • False
 16 divides $${n^4} + 4{n^2} + 11$$ , if n is an odd integer.
  • True
  • False
The product of two consecutive positive integers is divisible by 22
  • True
  • False
$$^{\text{1}}{{\text{p}}_{\text{1}}}{\text{ + 2}}{{\text{.}}^{\text{2}}}{{\text{p}}_{\text{2}}}{\text{ + 3}}{{\text{.}}^{\text{3}}}{{\text{p}}_{\text{3}}}{\text{ + }}....{\text{ + n}}{{\text{.}}^{\text{n}}}{{\text{p}}_{\text{n}}}{{\text{ = }}^{{\text{n + 1}}}}{{\text{p}}_{{\text{n + 1}}}}{\text{ -2}}$$
  • True
  • False
$$33!$$ is divisble by $${2^n}$$, then $$n$$=......
  • $$16$$
  • $$31$$
  • $$17$$
  • $$19$$
By induction it can be proved $$n ^ { 7 } - 7 n ^ { 5 } + 14 n ^ { 3 } - 8 n$$ is divisible by 8.
  • True
  • False
Using P. M. 1, whether  $$41^n-14^n$$ is a multiple of $$27$$.
  • True
  • False
Use Principle of Mathematical Induction it can be proved $$1 ^ { 3 } + 2 ^ { 3 } + 3 ^ { 3 } + \cdots + n ^ { 3 } = \left( \frac { n ( n + 1 ) } { 2 } \right) ^ { 2 }$$ for all $$n \in N$$.
  • True
  • False
Prove by method of induction, for all $$n \in $$
$$3 + 7 + 11 + ...$$ to $$n$$ terms $$= n(2n + 1)$$
State True or False
  • True
  • False
Let $$P(n)$$ be the statement $$2^{n}<n!$$ where $$n$$ is a natural number, then $$P(n)$$ is true for:
  • all $$n$$
  • all $$n>2$$
  • all $$n>3$$
  • all $$n<3$$
Using principle of mathematic inductor for all
$$n\ \epsilon\ N:1+2+3+....+n < \dfrac {1}{8}(2n+1)^{2}$$
  • True
  • False
$$\sin(\alpha )+\sin\left( \alpha +\dfrac { \pi  }{ 6 }  \right) +\sin\left( \alpha +\dfrac { 2\pi  }{ 6 }  \right) +...$$ $$+\sin\left( \alpha +\dfrac { (n-1)\pi  }{ 6 }  \right)=\frac { \sin\left( \alpha +\frac { (n-1)\pi  }{ 6 }  \right) \times \sin\left( \dfrac { n\pi  }{ 12 }  \right)  }{ \sin\left( \dfrac { \pi  }{ 12 }  \right)  } $$ 
  • True
  • False
Using mathematical induction, for all $$n\epsilon N$$
$${ 1 }^{ 2 }+{ 2 }^{ 2 }+{ 3 }^{ 2 }+...+{ n }^{ 2 }=\dfrac { n\left( n+1 \right) \left( 2n+1 \right)  }{ 6 } $$
  • True
  • False
By method of Induction for all $$n\in N$$
$$\dfrac { 1 }{ 3.3 } +\dfrac { 1 }{ 3.5 } +\dfrac { 1 }{ 5.7 } +....+\dfrac { 1 }{ (2n-1)(2n+1) } =\dfrac { n }{ 2n+1 } $$
  • True
  • False
Consider the statement: $$"P(n):n^2-n+41$$ is prime". Then which one of the following is true?
  • P$$(5)$$ is false but P$$(3)$$ is true
  • Both P$$(3)$$ and P$$(5)$$ are false
  • P$$(3)$$ is false but P$$(5)$$ is true
  • Both P$$(3)$$ and P$$(5)$$ are true
For all positive integrals $$10^{n}+3^{4n+2}+8$$ is divisible by
  • $$8$$
  • $$9$$
  • $$7$$
  • $$6$$
Is $$n(n^2+5)$$ divisible by $$6$$, $$\forall n\in Z$$?
  • True
  • False
Is $$4^n-1$$ is divisible by $$3$$?
  • True
  • False
Is $$2n<(n+2)!$$ for all natural number $$n?$$
  • True
  • False
Is $$n^3-7n+3$$ divisible by $$3$$?
  • True
  • False
Is $$2+4+6+......2n=n^2+n$$ true for all natural numbers $$n$$?
  • True
  • False
Is $$n^2<2^n$$ for all natural numbers $$n\geq 5?$$
  • True
  • False
Is $$3^{2n}-1$$ is divisible by $$8?$$
  • True
  • False
State true or false.
Is $$1+5+9+........+(4n-3)=n(2n-1)$$ for all natural numbers $$n$$?
  • True
  • False
Let P(n) be a statement and let $$P(k) \Rightarrow P(k + 1),$$ for some natural number $$k,$$then $$P(n)$$ is true for all $$n \in N.$$
  • True
  • False
State true or false.
For any natural number n, $$7^n – 2^n$$ is divisible by $$5.$$
  • True
  • False
Is $$1+2+2^2+......+2^n=2^{n+1}-1$$ for all natural numbers n?
  • True
  • False
By mathamatical induction $$n(n^2 - 1)$$ is divisible by
  • $$19$$
  • $$23$$
  • $$24$$
  • $$29$$
0:0:1


Answered Not Answered Not Visited Correct : 0 Incorrect : 0

Practice Class 11 Engineering Maths Quiz Questions and Answers