2 If the cubic equation $x^{3}-a x^{2}+ax-1$=0 is identical with the cubic equation whose roots are the squares of the roots of the given cubic equation , then the non-zero real value of 'a' is A) $\frac{1}{2}$ B) 2 C) 3 D) $\frac{7}{2}$
4 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
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 The lines represented by $5x^{2}-xy-5x+y=0$ are normals to a circle S=0 .If this circle touches the circle $S'= x^{2}+y^{2}-2x+2y-7=0$ externally , then the equation of the chord of contact of centre of S'=0 with respect to S=0 is A) 2y-7=0 B) x-1=0 C) 3x+4y-7=0 D) x+y=5
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 Let $\lim_{t \rightarrow 0}(1+5t)^{1/t}=K$ and X be the random variable representing number of successes in 100 independent trails .If the probability of success in each trial is 0.05 ,then the probability of getting at least one success is A) $\frac{1-K}{K}$ B) $\frac{K-1}{K}$ C) $\frac{K+1}{2K}$ D) $\frac{5K+2}{7K}$
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 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) $\frac{\pi}{12}$
13 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) $\cos ^{-1}\left(\frac{19}{\sqrt{350}}\right)$
14 Electric current is measured by tangent galvanometer, the current being proportional to the tangent of the angle $\theta$ of deflection. If the deflection is read as $45^{0}$ and an error of 1% is made in reading it.then the percentage error in the current is A) $\pi$ B) $\frac{\pi}{2}$ C) $\frac{\pi}{3}$ D) $\frac{\pi}{4}$
15 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) $y=x^{3}+cx+d$
