CBSE Questions for Class 11 Engineering Maths Principle Of Mathematical Induction Quiz 4 - MCQExams.com

A student was asked to prove a statement by induction.
(i) $$P(5)$$ is true and
(ii) Truth of $$P(n) \Rightarrow$$ truth of $$P(n + 1)$$, $$n\epsilon N$$
On the basis of this, he could conclude that $$P(n)$$ is true for
  • no $$n\epsilon N$$
  • all $$n\epsilon N$$
  • all $$n\geq 5$$
  • none of these
If $$n \in N$$, then $$x^{2n+1} + y^{2n+1}$$ is divisible by
  • $$x+y$$
  • $$x-y$$
  • $$x^2+y^2$$
  • $$x^2+xy$$
The inequality $$n! > 2^{n-1}$$ is true
  • for all $$n > 1$$
  • for all $$n > 2$$
  • for all $$n \epsilon N$$
  • none of these
For positive integer n, $$10^{n-2} > 81n$$ when
  • $$n < 5$$
  • $$n > 5$$
  • $$n\geq 5$$
  • $$n > 6$$
State which of the following statements is true $$ \displaystyle 2^{16}-1 $$   is divisible by 
  • 11
  • 13
  • 17
  • 19
$$7$$ is a factor of $$2^{3n}-1$$ for all natural numbers n.
  • True
  • False
Let $$S(K) =1+ 3 + 5...+ (2K -1) =3 + K^2$$. Then which of the following is true
  • Principle of mathematical induction can be used to prove the formula
  • $$S(K) \Rightarrow S(K + 1)$$
  • $$S(K) \not{\Rightarrow} S(K + 1)$$
  • S(1) is correct
The statement P(n) $$"1 \times 1! + 2\times 2! + 3\times 3! + ... + n \times n! =(n + 1)! 1"$$ is
  • True for all n > 1
  • Not true for any n
  • True for all $$n\epsilon N$$
  • None of these
If $$a, b$$ and $$n$$ are natural numbers then $$a^{2n-1}+b^{2n-1}$$ is divisible by
  • $$a + b$$
  • $$a - b$$
  • $$a^3+b^3$$
  • $$a^2+b^2$$
Let $$a, b, c$$ and $$d$$ be any four real numbers. Then, $$a^n+b^n=c^n+d^n$$ holds for any natural number $$n$$, if

(This question has some ambiguity, but appeared in WBJEE 2015 exam).
  • $$a+b = c+d$$
  • $$a-b=c-d$$
  • $$a+b=c+d, a^2+b^2=c^2+d^2$$
  • $$a-b=c-d, a^2-b^2=c^2-d^2$$
For any integer $$n\ge 1$$, the sum $$\displaystyle\sum _{ k=1 }^{ n }{ k\left( k+2 \right)  } $$ is equal to
  • $$\dfrac { n\left( n+1 \right) \left( n+2 \right) }{ 6 } $$
  • $$\dfrac { n\left( n+1 \right) \left( 2n+1 \right) }{ 6 } $$
  • $$\dfrac { n\left( n+1 \right) \left( 2n+7 \right) }{ 6 } $$
  • $$\dfrac { n\left( n+1 \right) \left( 2n+9 \right) }{ 6 } $$
For any $$+$$ve integer $$n, n^3 + 2n$$ is always divisible by
  • $$3$$
  • $$7$$
  • $$5$$
  • $$6$$
The last digit in $$\displaystyle { 7 }^{ 300 }$$ is:
  • 7
  • 9
  • 1
  • 3
Let $$A = \begin{pmatrix}1 & 1 & 1\\ 0 & 1 & 1\\ 0 & 0 & 1\end{pmatrix}$$. Then for positive integer $$n, A^{n}$$ is
  • $$\begin{pmatrix}1 & n & n^{2}\\ 0 & n^{2} & n\\ 0 & 0 & n\end{pmatrix}$$
  • $$\begin{pmatrix}1 & n & n\left (\dfrac {n + 1}{2}\right )\\ 0 & 1 & n\\ 0 & 0 & 1\end{pmatrix}$$
  • $$\begin{pmatrix}1 & n^{2} & n\\ 0 & n & n^{2}\\ 0 & 0 & n^{2}\end{pmatrix}$$
  • $$\begin{pmatrix}1 & n & 2n - 1\\ 0 & \dfrac {n + 1}{2} & n^{2}\\ 0 & 0 & \dfrac {n + 1}{2}\end{pmatrix}$$
The number $$4^n \, + \, 15n \,-\, 1$$ is a multiple of $$9$$ for any natural $$n$$.
  • True
  • False
State true or false 
$$1 +$$ $$\dfrac{x}{a_1} \, + \, \dfrac{x(x \, + \, a_1)}{a_1 \, a_2} \, + \, ... \, + \, \dfrac{x(x \, + \, a_1) \, (x \, + \, a_2) \, .... \, (x \, + \, a_{n \, -  \, 1})}{a_1 \, a_2 \, a_3 \,... \, a_n}$$ $$= \, \dfrac{(x \, + \, a_1) \, (x \, + \, a_2) \, .... \, (x \, + \, a_n)}{a_1 \, a_2 \, a_3 \,... \, a_n}$$
  • True
  • False
Using the principle of mathematical induction ,  $$\forall  \, n\epsilon \, N$$ if 
$$y = \cot$$$$^{-1} \,$$ x then     
$$y_n \, = \, (-1)^n\, (n \, - \, 1)!\, \sin^n$$ $$y \sin ny$$.   
  • True
  • False
Prove by mathematical induction that 
n. 1 + ( n - 1) 2 + (n - 2) 3 + ..... + 2 (n -1)  + 1.n = $$\dfrac{n}{6} $$ (n + 1)(n + 2) 
  • True
  • False
State the whether the statement id True/False

The given relation is  $${ \left( 1+x \right)  }^{ n }\ge \left( 1+nx \right)$$ if $$x\ge -1$$
  • True
  • False

If $$A=\left ( \begin{matrix}cos \theta & isin
\theta \\isin \theta & cos
\theta\end{matrix} \right )$$, where $$ i =
\sqrt { - 1} $$, then by principle of Mathematical Induction then $$ A^n=\left [ \begin{matrix}cos\,n \theta & isin\,n \theta \\isin\,n
\theta & cos\,n \theta\end{matrix} \right ]$$.
  • True
  • False
State whether the statement is true/false

If $$a$$ and $$b$$ are $$2$$ +ve no. such that $$a>b$$ then one of the two numbers $$\dfrac { a+b }{ 2 }$$ and $$\dfrac { a-b }{ 2 }$$is even or odd
  • True
  • False
State true or false.
$${1^3}\, + \,{3^3}\, + \,{5^3}\, + \,...\, + \,{\left( {2n\, - \,1} \right)^3}\, = \,{n^2}\,\left( {2{n^2}\, - \,1} \right)$$ n is a natural number
  • True
  • False
State true or false $$1.3\, + \,{\left( {{{2.3}^{}}} \right)^2}\, + \,{3.3^3}\, + \,.....\, + \,n{.3^n}\, = \,\dfrac{{\left( {2n\, - \,1} \right){3^{n\, + \,1}}\, + \,3}}{4}$$
  • True
  • False
State whether following statement is true or false.
By using the principle of mathematical induction we can proove that $$n\left( {n + 1} \right)\left( {n + 5} \right)$$ is a multiple of $$3$$.
  • True
  • False
For every positive integer $$n$$, $${7^n} - {3^n}$$ is divisible by $$4$$.
  • True
  • False
$$1.3 + {\left( {2.3} \right)^2} + {(3.3)^3} + .... + {(n.3)^n} = \frac{{\left( {2n - 1} \right){3^{n + 1}} + 3}}{4}$$
  • True
  • False
State True or False
$${7^n} - {2^n}$$ is divisible by $$5$$.
  • True
  • False
$${5}^{2n+2}-24n-25$$ is divisible by $$576$$ for all $$n\in N$$ by using principle of mathematical induction.
  • True
  • False
State whether the following statement is true or false.
$${1^3}\, + \,{3^3}\, + \,{5^3}\, + \,...\, + \,{\left( {2n\, - \,1} \right)^3}\, = \,{n^2}\left( {2{n^2}\, - \,1} \right)$$
  • True
  • False
 Using the principle of mathematical induction for all $$n\in N$$ it can be proved $$1+2+3+...+n < \dfrac{1}{8}(2n+1)^2$$.
  • True
  • False
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