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If 'a' is the point of discontinuity of the function

$f(x)=\begin{cases}\cos 2x & for -\infty<x<0\\e^{3x} & for 0\leq x<3\\ x^{2}-4x+3& ,for 3\leq x\leq6\\\frac{\log(15x-89)}{x-6}&, for x>6\end{cases}$

Then , $\lim_{x \rightarrow a}\frac{x^{2}-9}{x^{3}-5x^{2}+9x-9}$=

A) 1

B) 0

C) 6

D) 3

If $y=\sqrt{x+\sqrt{y+\sqrt{x+\sqrt{y+..........\infty}}}}$, then $\frac{dy}{dx}$=

A) $\frac{y^{3}-x}{2y^{2}-2xy+1}$

B) $\frac{x+y^{3}}{2y^{2}-x}$

C) $\frac{y+x}{y^{2}-2x}$

D) $\frac{y^{2}-x}{2y^{3}-2xy-1}$

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 $\alpha$ and $\beta$ are the roots of the equation

$x^{2}-2x+4=0$ , then $\alpha^{12}+\beta^{12}$=

A) $2^{12}$

B) $2^{10}$

C) $2^{13}$

D) -$2^{13}$

If $\frac{2x+7}{(x^{2}+4)(x^{2}+9)(x^{2}+16)}= \frac{Ax+1}{x^{2}+4}+\frac{Bx+m}{x^{2}+9}+\frac{Cx+n}{x^{2}+16}$ , then

$\frac{1}{A}+\frac{1}{B}+\frac{1}{C}$=........

A) 0

B) 27

C) $\frac{105}{2}$

D) $\frac{109}{2}$

The solution of the differential equation $ydx-xdy+3x^{2}y^{2}e^{x^{3}}dx=0$ satisfying y=1 when x=1, is

A) $y\left( e^{x^{3}}-(1+2e)\right)-x=0$

B) $y\left( e^{x^{3}}+(1-e)\right)+x=0$

C) $y\left( e^{x^{3}}+(1+e)\right)-x=0$

D) $y\left( e^{x^{3}}-(1+e)\right)+x=0$

If the function f:R→ r defined by

$f(x)=\begin{cases}a\left(\frac{1-\cos 2x}{x^{2}}\right), & for x < 0\\b ,& x = 0\\\frac{\sqrt{x}}{\sqrt{4+\sqrt{x}}-2}&for x>0\end{cases}$

is continuous at x=0 , then a+b=

A) 2

B) 4

C) 6

D) 8

If $\frac{dy}{dx}=4$ and $\frac{d^{2}y}{d^{2}x}=-3$ at a point P on the curve y=f(x), then

$\left(\frac{d^{2}x}{dy^{2}}\right)=$

A) 0

B) -$\frac{3}{4}$

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

D) $\frac{3}{64}$

Let $f:D\rightarrow R,D\subseteq R, c \in D$ and r be a non zero real number . Consider the following statements:

I: c is an extreme point of f $\Rightarrow$ c is an extreme point of rf

II. c is an extreme point of f $\Rightarrow$ c is an extreme point of r+f

Which of the following is correct?

A) Only (i) is true

B) Only (ii) is true

C) Both (i) and (ii) are true

D) Neither (i) nor(ii) is true

The value of $\theta$ for which the following system of equations has a non-trivial solution is $(4 \sin \theta)x-3y-z=0 , x-(6 \cos 2\theta )y+z=0$, 3x-12y+4z=0

A) $\tan^{-1} (\frac{1}{2})$

B) $\frac{\pi}{4}$

C) $\sin^{-1} (\frac{3}{16})$

D) $\frac{\pi}{12}$

$\sum_{r=1}^{16} \left( \sin \frac{2r\pi}{17}+i\cos \frac{2r \pi}{17}\right)=$

A) 1

B) -1

C) i

D) -i

Let P(1,-2,5) be the foot of the perpendicular drawn from the origin to the plane $\pi_{1}$ and the same P be the foot of the perpendicular from (1,2,-1) to the plane $\pi_{2}$ . then the acute angle between the planes $\pi_{1}$ and $\pi_{2}$ is

A) $\cos ^{-1}\left(\frac{19}{\sqrt{390}}\right)$

B) $\cos ^{-1}\left(\frac{19}{\sqrt{340}}\right)$

C) $\cos ^{-1}\left(\frac{19}{\sqrt{370}}\right)$

D) $\cos ^{-1}\left(\frac{19}{\sqrt{350}}\right)$

The coordinates of a a point on the curve $x=a(\theta +\sin \theta), y =a(1-\cos \theta)$ where the tangent is inclined at an angle $\frac{\pi}{4}$ to the positive X-axis , are

A) $\left(a\left(\frac{\pi}{2}-1\right)a\right)$

B) $\left(a\left(\frac{\pi}{2}+1\right)a\right)$

C) $\left(a \frac{\pi}{2},a\right)$

D) (a,a)

The area (in square units) of the region bounded by the curve y=|sin 2x| and X-axis in [0, $2 \pi$] is

A) 0

B) 1

C) 3

D) 4

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