<<345678>>
26.

Tangents are drawn to the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$ parallel to the straight line 2x-y=1.The points of contacts of the tangents on the hyperbola are


A) $\left(\frac{9}{2\sqrt{2}},\frac{1}{\sqrt{2}}\right)$

B) $\left(-\frac{9}{2\sqrt{2}},-\frac{1}{\sqrt{2}}\right)$

C) $(3\sqrt{3},-2\sqrt{2})$

D) $(-3\sqrt{3},2\sqrt{2})$



27.

If S be the area of the region enclosed   by $ y^{e^{-x^{2}}}$, y=0,x=0 and x=1 , then


A) $ S \geq \frac{1}{e}$

B) $ S \geq 1-\frac{1}{e}$

C) $S \leq \frac{1}{4} \left(1+\frac{1}{\sqrt{e}}\right)$

D) $S \leq \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{e}} \left(1-\frac{1}{\sqrt{2}}\right)$



28.

Let $\theta$ , $\phi \epsilon [0,2\pi]$ be such that $2 \cos \theta(1-\sin \phi)=\sin^{2} \theta$

$\left( \tan \frac{\theta}{2}+\cot \frac{\theta}{2}\right)\cos \phi-1$,

 $\tan (2\pi-\theta)>0$

  and $-1 < \sin \theta < -\frac{-\sqrt{3}}{2}$ then , $\phi$ cannot satisfy 


A) $0&lt; \phi &lt; \frac{\pi}{2}$

B) $\frac{\pi}{2} &lt; \phi &lt; \frac{4 \pi}{3}$

C) $\frac{4 \pi}{3} &lt; \phi &lt; \frac{3 \pi}{2}$

D) $\frac{3 \pi}{2} &lt; \phi &lt; 2 \pi$



29.

A ship is fitted with with three engines $E_{1},E_{2}$ and $E_{3}$. The engines function independently of each other with respective probabilities 1/2,1/4 and 1/4. For the ship to be operational at least two of its engines must function. Let X denote the event that the ship is operational and let $X_{1},X_{2}$ and $X_{3}$ denotes,respectively the  events that engines $E_{1},E_{2}$ and $E_{3}$ are functioning .Which of the following is/are true?

 


A) $P[X_{1}^{c} X]$

B) P[Exactly two engines of the ship are functioning x]7/8

C) $P[X|X_{2}]=\frac{5}{16}$

D) $P[X|X_{1}]=\frac{7}{16}$



30.

 If y(x) satisfies the differential equation $y'-y \tan x=2 x \sec x$ and y(0)  , then 


A) $y\left(\frac{\pi}{4}\right)= \frac{\pi^{2}}{8 \sqrt{2}}$

B) $y' \left(\frac{\pi}{4}\right)= \frac{\pi^{2}}{18}$

C) $y\left(\frac{\pi}{3}\right)= \frac{\pi^{2}}{9}$

D) $y'\left(\frac{\pi}{3}\right)= \frac{4 \pi^{}}{3}+\frac{2\pi^{2}}{3 \sqrt{3}}$



<<345678>>