MHT CET 2018 :: Mathematics :: General questions

• Mathematics - General questions

The cartesian  coordinates of the point on the parabola y²=-16x, whose parameter is $\frac{1}{2}$ , are 

Answer:

Explanation:

We know that parametric form y²=4ax is

 x= at²   and y=2at

 Here, y²  =-16x

$\therefore$      x= -4t²   and y= -8t

 Given,  t= $\frac{1}{2}$

 $\therefore$    $x=-4(\frac{1}{2})^{2}$ and  $ y=-8(\frac{1}{2})$

  x=-1 and y=-4

 Hence, cartesian coordinates are (-1,-4).

If x= $e^{\theta}(\sin\theta-\cos\theta),y=e^{\theta}(\sin\theta+\cos\theta)$ then $\frac{dy}{dx}$at$ \theta=\frac{\pi}{4}$   is 

Answer:

Explanation:

Given

   x= $e^{\theta}(\sin\theta-\cos\theta)$

and   $y=e^{\theta}(\sin\theta+\cos\theta)$ 

 $\Rightarrow$   $\frac{dx}{d\theta}=e^{\theta} (\sin \theta-\cos\theta+\cos\theta+\sin\theta)=2e^{\theta}\sin\theta$

 and  $\frac{dy}{d\theta}=e^{\theta} (\sin \theta+\cos\theta+\cos\theta-\sin\theta)=2e^{\theta}\cos \theta$

 $\therefore$     $\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}=\frac{2e^{\theta}\cos \theta}{2e^{\theta}\sin \theta}=\cot \theta$

 $\Rightarrow$   $\left(\frac{dy}{dx}\right)_{\theta=\frac{\pi}{4}}=\cot \frac{\pi}{4}=1$

$\int_{0}^{\pi/4} x.\sec^{2}x dx=$

Answer:

Explanation:

 Let   l=$\int_{0}^{\pi/4} x.\sec^{2}x dx$

 $\Rightarrow$      $l=[x\int \sec^{2}x dx]^{\pi/4}_{0}-\int_{0}^{\pi/4} \left(\frac{dy}{dx}\int \sec^{2}x dx\right) dx$

                                                                                             [integration by parts]

$\Rightarrow$    $l=[x\tan^{}x ]^{\pi/4}_{0}-\int_{0}^{\pi/4} \tan x dx$

$\Rightarrow$      $l=[\frac{\pi}{4}\tan^{}\frac{\pi}{4}-0 ]-[\log \sec x]^{\pi/4}_{0}$

$\Rightarrow$      $l=\frac{\pi}{4}-\left( \log \sec\frac{\pi}{4}-\log \sec 0\right)$

                   $l=\frac{\pi}{4}-\left( \log\sqrt{2}-\log 1\right)$

$\Rightarrow$ l=  $\frac{\pi}{4}-\log \sqrt{2}$

A die is rolled  .If X denotes the number of positive divisors of the outcome, then the range of the random  variable X is

Answer:

Explanation:

 A die is rolled

 $\therefore$ Simple space = {1,2,3,4,5,6}

Given X denotes the number of positive divisors of the outcomes

$\therefore$      X= {1,2,3,4,5,6}

cos 1°. cos 2°. cos 3° ............. cos 179°=

Answer:

Explanation:

cos 1°. cos 2°. cos 3° .......

 cos 90°    cos 91°...... cos 179°= 0   [$\because$  cos 90° =0]

• Mathematics - General questions