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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

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

p₁,p₂,p₃ arte the altitudes of a triangle ABC drawn from the vertices A,B and C respecively . If $\triangle$ is the area of the triangle and 2s is the sum of its sides a,b and c , then $\frac{1}{p_{1}}+\frac{1}{p_{2}}-\frac{1}{p_{3}}$=

A) $\frac{s-a}{\triangle}$

B) $\frac{s-b}{\triangle}$

C) $\frac{s-c}{\triangle}$

D) $\frac{s}{\triangle}$

If two events , E₁ ,E₂ are such that

$P(E_{1}\cup E_{2})=\frac{5}{8},P(\overline{E_{1}})=\frac{3}{4}, P(E_{2})=\frac{1}{2} $ then $E_{1}$ and $E_{2}$ are

A) independent s events

B) mutually exclusive events

C) exhaustive events

D) not independent events

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 equation of a circle concentric with the circle $x^{2}+y^{2}-6x+12y+15=0$ and having area that is twice the area of the given circle is

A) $x^{2}+y^{2}-6x+12y-15=0$

B) $x^{2}+y^{2}-6x+12y-30=0$

C) $x^{2}+y^{2}-6x+12y-60=0$

D) $x^{2}+y^{2}-6x+12y+15=0$

$\int\frac{dx}{(1+x)\sqrt{8+7x-x^{2}}}$=

A) $-\frac{2}{9}\sqrt{\frac{8-x}{1+x}}+c$

B) $-\frac{1}{9}\sqrt{\frac{1+x}{8-x}}+c$

C) $-\frac{2}{9}\sqrt{\frac{1+x}{8-x}}+c$

D) $\frac{2}{9}\sqrt{\frac{8+x}{1+x}}+c$

$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot^{9} x dx=$

A) $\frac{-7}{42}+\frac{1}{2} log 2$

B) $\frac{7}{24}-\frac{1}{2} log 2$

C) $\frac{25}{24}+\frac{1}{2} log 2$

D) $\frac{1}{24}+2 log 2$

if the function $f:[a,b]\rightarrow \left[-\frac{\sqrt{3}}{4},\frac{1}{2}\right]$ defined by

$f(x)=\begin{bmatrix}1 & 1&1 \\1 & 1+\sin_{}x&1\\1+\cos x&1&1 \end{bmatrix}$

is one-one and onto , then

A) $a=\frac{-\pi}{4},b=\frac{\pi}{6}$

B) $a=\frac{-\pi}{2},b=\frac{\pi}{2}$

C) $a=\frac{-\pi}{6},b=\frac{\pi}{4}$

D) $a=-\pi, b=\pi$

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

The solution of the differential equation $(x+2y^{3})\frac{dy}{dx}=y$ is

A) $x=y^{3}+c$

B) $x=y^{3}+cy$

C) $y=x^{3}+c$

D) $y=x^{3}+cx+d$

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