MHT CET 2018 :: Mathematics :: General questions

• Mathematics - General questions

If  $X= \left\{4^{n}-3n-1:n\epsilon N\right\}$  and  $Y= \left\{9(n-1):n\epsilon N\right\}$  then $X\cap Y$= 

Answer:

Explanation:

We have , 

   $X= \left\{4^{n}-3n-1:n\epsilon N\right\}$ 

   = {0,9,54,........}

and   $Y= \left\{9(n-1):n\epsilon N\right\}$

      = {0,9,18,27,6,45,54.........}

 $\therefore$    $X\cap Y$=   X

If $f(x)=\frac{x}{x^{2}+1}$   is increasing function , then the value of x lies in 

Answer:

Explanation:

 Given, $f(x)=\frac{x}{x^{2}+1}$

 $\Rightarrow$    $  f'(x)=\frac{(x^{2}+1)\times 1-x(2x)}{(x^{2}+1)^{2}}$

$=\frac{(x^{2}+1 -2x^{2})}{(x^{2}+1)^{2}}=\frac{1-x^{2}}{(x^{2}+1)^{2}}$

since, f(x)  is increasing function,

$\therefore$     f '(x)  >0

$\Rightarrow$    $\frac{1-x^{2}}{(x^{2}+1)^{2}}>0$

$\Rightarrow$  $1- x^{2} >0$

$\Rightarrow$    $x^{2} <1 $

$\Rightarrow$   -1 < x< 1

 $\therefore$  x ε  (-1,1)

  Hence, option (d)  is correct.

The contrapositive of the statement:"If the weather is fine then my friends will come and we go for a picnic".

Answer:

Explanation:

Given statement is 

" If the weather is fine then my friends will come and we go for a picnic".

 Here, p= the weather  is fine

q= my friends will come

r= we go for a picnic

 Symbolic form is  $p\rightarrow (q \wedge r)$

 contrapositive of   $p\rightarrow q$  is $\sim q\wedge \sim p$

 $\therefore$     Contrapositive  of  $p\rightarrow (q   \wedge r)\sim(q\wedge r)\sim p$

 $\Rightarrow$    $( \sim q \vee\sim r)\rightarrow\sim p$

$\therefore$ Contrapositive of the given statement is, if my friends will not come  or we do not go for a picnic then weather will not fine.

Matrix A=$\begin{bmatrix}1 & 2&3 \\1 & 1&5 \\ 2& 4& 7 \end{bmatrix}$, then   the value of a₃₁A₃₁+a₃₂A₃₂+a₃₃A₃₃ is 

Answer:

Explanation:

 We have

   $A=\begin{bmatrix}1 & 2&3 \\1 & 1&5 \\ 2& 4& 7 \end{bmatrix}$

 a₃₁=2, a₃₂=4, a₃₃= 7

 and   $A_{31}=\begin{bmatrix}2 & 3 \\1 & 5 \end{bmatrix}=10-3=7$

 $A_{32}=-\begin{bmatrix}1 & 3 \\1 & 5 \end{bmatrix}=-(5-3)=-2$

$A_{33}=\begin{bmatrix}1 & 2 \\1 & 1 \end{bmatrix}=1-2=-1$

$\therefore$    $a_{31}A_{31}+a_{32}A_{32}+a_{33}A_{33}$

  = 2(7)+4(-2)+7(-1)

   =14-8-7=-1

The number  of solution of $\sin x$ +sin 3x+sin 5x=0  in the interval  $\left[ \frac{\pi}{2},3\frac{\pi}{2}\right]$  is 

Answer:

Explanation:

 We have,

 $\sin x$ +sin 3x+sin 5x=0

$\Rightarrow$     $\sin x+\sin 5x+\sin x$=0

$\Rightarrow$  2$\sin 3x \cos 2x+\sin 3x$=0

$\Rightarrow$    $\sin 3x (2\cos 2x +1)$=0

$\Rightarrow$   $\sin 3x=0$  or $\cos 2x=-\frac{1}{2}=\cos \frac{2 \pi}{3}$

$\Rightarrow$ $3x=n\pi $  or   $2n=2n\pi\pm\frac{2\pi}{3}$

$\Rightarrow$    $x=\frac{n\pi}{3}$   or   $x=n\pi\pm\frac{\pi}{3}$  

  But, it is given that

$x \epsilon \left[\frac{\pi}{2},\frac{3\pi}{2}\right]$

$\therefore$      $x=\frac{2\pi}{3},\pi,\frac{4\pi}{3}$

$\therefore$    Number of solutions is 3.

• Mathematics - General questions