1 Let AX=D be a system of three linear non-homogeneous equations, If |A| =0 and rank(A) =rank ([AD])= $\alpha$ , then A) AX=D will have infinite number of solutions when $\alpha$=3 B) AX=D will have unique solution when $\alpha$ <3 C) AX=D will have infinite number of solutions when $\alpha$ < 3 D) AX=D will have no solution when $\alpha$ <3
2 If a,b and c are three non-collinear points and ka+2b+3c is a point in the plane of a,b, c then k= A) 4 B) 5 C) -5 D) -4
3 Consider the following system of equations in matrix form $\begin{bmatrix}1 \\2 \\\lambda \end{bmatrix}$ (1 2 $\lambda$) $\begin{bmatrix}x \\y \\z\end{bmatrix} =0 $ Then which one of the following statements is ture? A) $\forall \lambda\epsilon(-\infty,\infty)$ , the given system has non trivial solution B) $\forall \lambda\epsilon(-\infty,\infty)$ , the given system has only trivial solution C) For $\lambda\neq0$ , the given system does not have any solution D) For $\lambda =0$ , the given system is inconsistent
4 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) $(-\infty ,1) \cup (5, \infty)$
5 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) -$2^{13}$
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 If $\int\frac{2x^{2}}{(2x^{2}+\alpha)(x^{2}+5)}dx=\frac{\sqrt{5}}{3} \tan ^{-1}\frac{x}{\sqrt{5}}-\frac{\sqrt{2}}{3} \tan ^{-1}\frac{x}{\sqrt{2}}+c,$ then $\alpha$ = A) 1 B) 2 C) 3 D) 4
8 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
9 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) $\frac{1}{3}$
10 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) $x^{2}+y^{2}-6x+12y+15=0$
11 $\int\frac{dx}{(1+x)\sqrt{8+7x-x^{2}}}$= 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) $\frac{2}{9}\sqrt{\frac{8+x}{1+x}}+c$
12 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) $(-\infty,-1)\cup (3, \infty)$
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 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) $\cos ^{-1}\left(\frac{19}{\sqrt{350}}\right)$
15 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)
