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A tangent is drawn at $(3\sqrt{3}\cos\theta, \sin \theta)\left(0< \theta < \frac{\pi}{2}\right)$ to the ellipse $\frac{x^{2}}{27}+\frac{y^{2}}{1}=1$ . The value of $\theta$ for which the sum of the intercepts on the coordinate axes made by this tangent attains the minimum , is

A) $\frac{\pi}{6}$

B) $\frac{\pi}{3}$

C) $\frac{2\pi}{3}$

D) $\frac{2\pi}{4}$

Consider the function $f(x)=2x^{3}-3x^{2}-x+1$ and the intervals $I_{1}$=[-1,0],$I_{2}$= [0,1], $I_{3}$ =[1,2], $I_{4}$=[-2,-1]

Then,

A) f(x) =0 has a root in the intervals $I_{1}$ and $I_{4}$ only

B) f(x) =0 has a root in the intervals $I_{1}$ and $I_{2}$ only

C) f(x) =0 has a root in every interval except in $I_{4}$

D) f(x)=0 has a root in all the four given intervals

Area of the region (in sq units) bounded by the curve y =$\sqrt{x}$, x= $\sqrt{y}$ and the lines x=1, x=4 , is

A) $\frac{8}{3}$

B) $\frac{49}{3}$

C) $\frac{16}{3}$

D) $\frac{14}{3}$

The solution of the differential equation (2x-3y+5)dx+(9y+6x-7) dy =0 , is

A) 3x-3y+8 log |6x-9y-1|=c

B) 3x-9y+8 log |6x-9y-1|=c

C) 3x-9y+8 log |2x-3y-1|=c

D) 3x-9y+4 log |2x-3y-1|=c

If the point $\left(\frac{k-1}{k},\frac{k-2}{k}\right)$ lies on the locus of z satisfying the inequality $|\frac{z+3i}{3z+i}|$ <1, then the interval in which k lies is

A) $(-\infty ,2) \cup (3, \infty)$

B) [2,3]

C) [1,5]

D) $(-\infty ,1) \cup (5, \infty)$

If OA= $\hat{i}+2\hat{j}+3\hat{k}$ and OB= $4 \hat{i}+\hat{k}$ are the position vectors of the points A and B , then the position vector of a point on the line passing through B and parallel to the vector OA x OB which is at a distance of $\sqrt{189}$ units from B is

A) $6 \hat{i}+11\hat{j}-7 \hat{k}$

B) $4 \hat{i}+11\hat{j}-8 \hat{k}$

C) $2 \hat{i}-11\hat{j}+8 \hat{k}$

D) $-2\hat{i}-11\hat{j}+8\hat{k}$

If f(x) = $\begin{cases}ax+b, & if x\leq 1\\ax^{2}+c ,& if 1<x\leq2 \\ \frac{dx^{2}+1}{x},&if x\geq 2\end{cases}$

is differentiable on R, then ad-bc=

A) 0

B) 1

C) -1

D) 2

The derivative of $\cos h^{-1} x$ with respect to log x at x=5 is

A) $\frac{5}{\sqrt{26}}$

B) $\frac{1}{\sqrt{26}}$

C) $\frac{1}{2\sqrt{6}}$

D) $\frac{5}{2\sqrt{6}}$

If a and b respectively represent the lengths of a side and a diagonal of a regular pentagon that is inscribed in a circle , then $\frac{b}{a}$=

A) $2 \sin \frac{\pi}{5}$

B) $2 \cos \frac{\pi}{5}$

C) $ \cos \frac{\pi}{5}$

D) $\sin \frac{\pi}{5}$

If the position vectors of the points A, B, C, D given by $\hat{i}+2\hat{j}+3\hat{k}$ , $2\hat{i}-\hat{j}+2\hat{k}$,

$\frac{1}{4}(7 \hat{i}+15\hat{j}+15 \hat{k})$ and $\frac{1}{3}[7\hat{i}+2\hat{j}+(5+3a)\hat{k}]$ respectively are such that |AC| =|BD| , then $16(3a-1)^{2}$=

A) 143

B) 139

C) 189

D) 187

If a die is rolled twice and the sum of the numbers appearing on them is observed to be 6 , then the probability that the number 1 appears atleast once on them is

A) $\frac{5}{36}$

B) $\frac{2}{5}$

C) $\frac{11}{36}$

D) $\frac{1}{3}$

The solution of the differential equation $\frac{dy}{dx}=1- \cos (y-x) \cot (y-x) $ is

A) x tan (y-x)=c

B) x= tan(y-x)+c

C) x= sec(y-x)+c

D) x+ sec (y-x)=c

Let $f:R\rightarrow R $ and $g:R\rightarrow R$ be the functions defined by $f(x)= \frac{x}{1+x^{2}}$,

$x \in R, g(x)=\frac{x^{2}}{1+x^{2}},x\in R$ Then, the correct statement (s) among the following is/are

(a) both f.g are one-one

(b) both f.g are onto

(d) f and g are onto but not one-one

A) A

B) A.B

C) D

D) C

If $z=\sqrt{2}\sqrt{1+\sqrt{3i}}$ repesents a point P in the argand plane and P lies in the third quadrant , then the polar form of z is

A) $2\left[ \cos \left(\frac{-4 \pi}{3}\right)+i \sin \left(\frac{-4 \pi}{3}\right)\right]$

B) $2\left[ \cos \left(\frac{-5 \pi}{6}\right)+i \sin \left(\frac{-5 \pi}{6}\right)\right]$

C) $2\left[ \cos \left(\frac{- \pi}{6}\right)+i \sin \left(\frac{- \pi}{6}\right)\right]$

D) $2\left[ \cos \left(\frac{- 2\pi}{3}\right)+i \sin \left(\frac{- 2\pi}{3}\right)\right]$

If $\sqrt{\frac{y}{x}}+4\sqrt{\frac{x}{y}}=4$ then $\frac{dy}{dx}$=

A) xy

B) x/y

C) -4

D) 4

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