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
3 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) $\frac{2\pi}{4}$
4 If $e_{1}$ is the eccentricity of the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{25}=1$ and $e_{2}$ is the eccentricity of a hyperbola passing through the foci of the given ellipse and $e_{1}e_{2}=1$ , then the equation of such a hyperbola among the following is A) $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$ B) $\frac{y^{2}}{9}-\frac{x^{2}}{16}=1$ C) $\frac{x^{2}}{9}-\frac{y^{2}}{25}=1$ D) $\frac{x^{2}}{25}-\frac{y^{2}}{9}=1$
5 The differential equation representing the family of circles of constant radius r is A) $ r^{2} y"=[1+(y')^{2}]^{2}$ B) $ r^{2} (y")^{2}=[1+(y')^{2}]^{2}$ C) $r^{2} (y")^{2}=[1+(y')^{2}]^{3}$ D) $ (y")^{2}=r^{2}[1+(y')^{2}]^{2}$
6 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
7 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) $-2\hat{i}-11\hat{j}+8\hat{k}$
8 In a communication network , ninety eight precent of messages are transmitted with no error.If a random variable X denotes the number of incorrectly transmitted messages , then the probability that atmost one message is transmitted incorrectly out of 500 messages sent, is A) $\frac{11}{e^{10}}$ B) $\frac{e^{10}-1}{e^{10}}$ C) $\frac{10}{e^{10}}$ D) $\frac{98}{e^{10}}$
9 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
10 If a triangle ABC with two vertices A(5,4,6) and B(1,-1,3) has its centroid at ($\frac{10}{3},2,\frac{11}{3})$ then the third vertex C is A) (4,2,3) B) (-4,-3,2) C) (4,3,2) D) (2,4,3)
11 $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot^{9} x dx=$ 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) $\frac{1}{24}+2 log 2$
12 $\sum_{r=1}^{16} \left( \sin \frac{2r\pi}{17}+i\cos \frac{2r \pi}{17}\right)=$ A) 1 B) -1 C) i D) -i
13 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 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)
