2 The number of even numbers greater than 1000000 that can be formed using all the digits 1,2,0,2,4,2 and 4 is A) 120 B) 240 C) 310 D) 480
3 If one of the diameters of the circle $x^{2}+y^{2}-2x-6y+6=0$ is a chord to the circle with centre (2,1), then the radius of the bigger circle is A) 6 B) 4 C) 2 D) 3
4 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) $\frac{2\pi}{4}$
5 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
6 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) $-2\hat{i}-11\hat{j}+8\hat{k}$
7 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 distance between the tangents to the hyperbola $\frac{x^{2}}{20}- \frac{3y^{2}}{4}$=1 which are parallel to the line x+3y=7 is A) $4\sqrt{5}$ B) $\frac{4}{\sqrt{5}}$ C) $\frac{2}{\sqrt{5}}$ D) $2\sqrt{5}$
10 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
11 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
12 The rank of the matrix $\begin{bmatrix}3 & 5&-1&4 \\2 & 1&3&-2\\8&11&1&6\\-7&-14&6&-14 \end{bmatrix}$ is A) 1 B) 2 C) 3 D) 4
13 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) $\left(-\frac{1}{2},\frac{1}{2}\right)$
14 The volume of the tetrahedron (in cubix units) formed by the plane 2x+y+z=K and the coordinate planes is $\frac {2V^{3}}{3}$ , then K:V= A) 1:2 B) 1:6 C) 4:3 D) 2:1
15 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) $\cos ^{-1}\left(\frac{19}{\sqrt{350}}\right)$