CBSE Questions for Class 11 Engineering Maths Principle Of Mathematical Induction Quiz 6 - MCQExams.com

If $$n$$ is an odd positive integer, then $$a^n + b^n$$ is divisible by
  • $$a - b$$
  • $$a + b$$
  • $$a^2 + b^2$$
  • none of these
The value of $$\displaystyle \frac{1^2}{1.3} + \frac{2^2}{3 . 5}+\dots+ \frac{n^2}{(2n - 1)(2 n + 1)}$$ is
  • $$\displaystyle \frac{n (n + 1)}{2 (2n + 1)}$$
  • $$\displaystyle \frac{n (n - 1)}{2 (2n - 1)}$$
  • $$\displaystyle \frac{n^2 (n - 1)^2}{2 (2n + 1)}$$
  • none of these
The value of $$\displaystyle \frac{1}{1.2.3} + \frac{1}{2.3.4}+ ....+ \frac{1}{n (n + 1) (n + 2)}$$ is 
  • $$\displaystyle \frac{n(n + 3)}{4(n + 1) (n + 2)}$$
  • $$\displaystyle \frac{n}{(n + 1)(n + 2)}$$
  • $$\displaystyle \frac{n (n + 2)}{ (n + 1)(n + 3)}$$
  • All of these
The value of $$\displaystyle \tan^{-1} \left ( \frac{1}{3} \right ) + \tan^{-1} \left ( \frac{1}{7} \right ) +\dots+ \tan^{-1} \left ( \frac{1}{n^2 + n + 1} \right )$$ is
  • $$\displaystyle \tan^{-1} \left ( \frac{n}{n + 2} \right )$$
  • $$\displaystyle \tan^{-1} \left ( \frac{n+1}{n - 1} \right )$$
  • $$\displaystyle \tan^{-1} \left ( \frac{n-1}{n + 2} \right )$$
  • $$\displaystyle \tan^{-1} \left ( \frac{n+2}{n - 2} \right )$$
By mathematical induction $$p^{n+1} + (p+1)^{2n -1}$$ is divisible by
  • $$p^2 + p + 1$$
  • $$p^2 + 1$$
  • $$p+1$$
  • None of these
If $$a_1 = 1, a_{n+1} = \dfrac{1}{n+1}a_n,\forall n \geq 1$$, then $$a_n \ =$$
  • $$\displaystyle \frac{1}{n !}$$
  • $$\displaystyle \frac{1}{(n + 2)!}$$
  • $$\displaystyle \frac{1}{(n + 1)!}$$
  • none of these
$$3 + 13 + 29 + 51 + 79 + ...$$ to $$n$$ terms $$=$$
  • $$2n^2 + 7n^3$$
  • $$n^2 + 5n^3$$
  • $$n^3 + 2n^2$$
  • none of these
Using the principle of mathematical induction, find $$tan  \alpha  + 2  tan   2 \alpha   + 2^2  tan  2^2  \alpha + ....$$ to $$n$$ terms:
  • $$tan \alpha - 2^n \cdot tan (2^n \cdot \alpha)$$
  • $$cot \alpha - 2^n \cdot cot (2^n \cdot \alpha)$$
  • $$sec \alpha - 2^n \cdot sec (2^n \cdot \alpha)$$
  • None of these
If $$p$$ is a prime number, then $$n^p -n$$ is divisible by $$p$$ for all $$n$$, where
  • $$n\in N$$.
  • $$n$$ is odd natural number.
  • $$n$$ is even natural number.
  • $$n$$ is not a composite number.
For each $$n\, \epsilon\, N$$, then $$3^{2n\, +\, 1}\, +\, 1$$ is divisible by -
  • $$2$$
  • $$3$$
  • $$7$$
  • None of these
For positive integer n, $$3^n < n!$$ when
  • $$n\geq 6$$
  • $$n > 7$$
  • $$n\geq 7$$
  • $$n \leq 7$$
$$\forall$$ $$n\in N$$, $$\displaystyle 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + ...... + \frac{1}{\sqrt{n}}$$ is
  • $$\sqrt{n}$$
  • $$\leq \sqrt{n}$$
  • $$\gt\sqrt{n}$$
  • none of these
Let $$\displaystyle p(n)=x\left(x^{n-1}-n\cdot a^{n-1}+a^{n}(n-1)\right)\>$$ is divisible by $$\left ( x-a \right )^{2}$$ for
  • $$n> 1$$
  • $$n> 2$$
  • $$\forall \;n\;\in \;N$$
  • None of these
If $$10^n + 3\cdot 4^{n +2} + \lambda$$ is exactly divisible by $$9$$ for all $$n\in N$$, then the least positive integral value of $$\lambda$$ is
  • $$5$$
  • $$3$$
  • $$7$$
  • $$1$$
For all $$n \in N$$, $$10^n + 3.4^{n+2} + 5$$ is divisible  by
  • $$23$$
  • $$3$$
  • $$9$$
  • $$207$$
If $$^{4n}{C_{{\text{2n}}}}{:^{2n}}{{\text{C}}_{\text{n}}} = \{ 1.3,.5,...(4{\text{n}} - 1)\} :{\{ 1,3,5....(2{\text{n}} - 1)\} ^\lambda },then\;\lambda  = $$
  • 1
  • 2
  • 3
  • none of these
If $$2^{83}+k $$ is divisible by 127, then the smallest positive integral value of k is:
  • 63
  • 31
  • 15
  • 64
 Using principle of mathematics induct or for all
$$n\ \in \ N:1+2+3+....+n < \dfrac {1}{8}(2n+1)^{2}$$
  • True
  • False
For all $$n \in N ,$$ 
$$1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + 4 ^ { 2 } + \cdots + n ^ { 2 } = \dfrac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }$$
  • True
  • False
 $$\displaystyle{\frac { 1 }{ { 1 }^{ 2 } } +\frac { 1 }{ { 2 }^{ 2 } } +\frac { 3 }{ { 3 }^{ 2 } } +......+\frac { 1 }{ { n }^{ 2 } } \le \frac { 3n+1 }{ 2n+2 } }$$ for every natural number $$n$$
  • True
  • False
For any natural number $$n,$$ $$x^n y^n$$ is divisible by $$x y,$$ where $$x$$ and $$y$$ are any integers with $$x \neq y.$$
  • True
  • False
0:0:1


Answered Not Answered Not Visited Correct : 0 Incorrect : 0

Practice Class 11 Engineering Maths Quiz Questions and Answers