VITEEE 2016 :: Mathematics :: General questions

• Mathematics - General questions

$\int\frac{1}{x^{2}(x^{4}+1)^{3/4}}dx$   is equal to

Answer:

Explanation:

$\int  \frac{dx}{x^{2}(x^{4}-1)^{3/4}}=\int \frac{dx}{x^{5}\left(1+\frac{1}{x^{4}}\right)^{3/4}}$

 Put   $1+\frac{1}{x^{4}}=t\Rightarrow-\frac{4}{x^{5}}dx=dt$

 So, integral is 

$I= -\frac{1}{4}\int\frac{dt}{t^{3/4}}=-t^{\frac{1}{4}}+c=-\left(1+\frac{1}{x^{4}}\right)^{1/4}+c$

If y= 2^x/Inx   then  $\frac{dy}{dx}$    at x=e   is 

Answer:

Explanation:

$\log y= \frac{x}{\log x}\log 2$

 $\Rightarrow\frac{dy}{dx}]_{x=e}=0$

 If $f(x)= (1+x)^{2/x}$  for x≠0 and f(0)= e² is

Answer:

Explanation:

 $\lim_{x \rightarrow 0}f(x)=\lim_{x \rightarrow 0}[(1+x)^{1/x}]^{2}=e^{2}= f(0)$

The solution of the equation  $\cos^{2}\theta+\sin^{2}\theta+1=0$   , lies in the interval

Answer:

Explanation:

We have ,

  $\cos^{2}\theta+\sin^{2}\theta+1=0\Rightarrow 1-\sin^{2}+\sin\theta+1=0$

 $\Rightarrow \sin\theta=-1(\because\sin\theta\neq2)\Rightarrow \theta=3\pi/2$

$\therefore$   $\theta\in \left(\frac{5\pi}{4},\frac{7\pi}{4}\right)$

The number of common tangents to the circles 

$x^{2}+y^{2}-6x-14y+48=0 $  and  $x^{2}+y^{2}-6x=0 $ is 

Answer:

Explanation:

 For the first circle centre  =(3,7)

 Radius   $r_{1}=\sqrt{3^{2}+7^{2}-48}=\sqrt{10}$

 For the second circle , centre (3,0) ;radius r₂=3

 So, r₁+r₂ < d  (distance betweeen the centres)

 $\therefore$  Circle don't  cut and hence the number of common tangents =4

• Mathematics - General questions