1 If two unbiased dice are rolled simultaneously until a sum of the number appeared on these dice is either 7 or 11, then the probability that 7 comes before 11, is A) $\frac{3}{8}$ B) $\frac{3}{4}$ C) $\frac{5}{6}$ D) $\frac{2}{9}$
2 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
3 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) $\frac{14}{3}$
4 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) -$2^{13}$
5 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) $\frac{s}{\triangle}$
6 Bag I contains 3 red and 4 black balls , Bag II contains 5 red and 6 black balls .If one ball is drawn at random from one of the bags and it is found to be red , then the probability that it was drawn from Bag II, is A) $\frac{33}{68}$ B) $\frac{35}{68}$ C) $\frac{37}{68}$ D) $\frac{41}{68}$
8 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) $x^{2}+y^{2}-6x+12y+15=0$
9 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) $\frac{3}{64}$
10 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
11 The domain of the function $f(x)= \frac{1}{\sqrt{[x]^{2}-[x]^{}-2}}$ is Here [x] denotes the greatest integer not exceeding the value of [x] A) $(-\infty,-2)\cup (1, \infty)$ B) $(-\infty,-2)\cup (0, \infty)$ C) $(-\infty,-2)\cup (2, \infty)$ D) $(-\infty,-1)\cup (3, \infty)$
12 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) $2\left[ \cos \left(\frac{- 2\pi}{3}\right)+i \sin \left(\frac{- 2\pi}{3}\right)\right]$
13 $\sum_{r=1}^{16} \left( \sin \frac{2r\pi}{17}+i\cos \frac{2r \pi}{17}\right)=$ A) 1 B) -1 C) i D) -i
14 If a variable circle S=0 touches the line y=x and passes through the point (0,0) , then the fixed point that lies on the common chord of the circles $x^{2}+y^{2}+6x+8y-7=0$ and S=0 is A) $\left(\frac{1}{2},\frac{1}{2}\right)$ B) $\left(-\frac{1}{2},-\frac{1}{2}\right)$ C) $\left(\frac{1}{2},-\frac{1}{2}\right)$ D) $\left(-\frac{1}{2},\frac{1}{2}\right)$
