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
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
5 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) $\frac{41}{68}$
6 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) $\frac{98}{e^{10}}$
7 The solution of the differential equation $ydx-xdy+3x^{2}y^{2}e^{x^{3}}dx=0$ satisfying y=1 when x=1, is A) $y\left( e^{x^{3}}-(1+2e)\right)-x=0$ B) $y\left( e^{x^{3}}+(1-e)\right)+x=0$ C) $y\left( e^{x^{3}}+(1+e)\right)-x=0$ D) $y\left( e^{x^{3}}-(1+e)\right)+x=0$
8 If a and b respectively represent the lengths of a side and a diagonal of a regular pentagon that is inscribed in a circle , then $\frac{b}{a}$= A) $2 \sin \frac{\pi}{5}$ B) $2 \cos \frac{\pi}{5}$ C) $ \cos \frac{\pi}{5}$ D) $\sin \frac{\pi}{5}$
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 If the locus of a point which divides a chord with slope 2 of the parabola $y^{2}=4x$ , internally in the ratio 1:3 is a parabola , then its vertes is A) (2,1) B) ($\frac{3}{16},\frac{3}{2})$ C) ($\frac{3}{4},\frac{3}{16})$ D) ($\frac{3}{16},\frac{3}{4})$
11 Let $f:D\rightarrow R,D\subseteq R, c \in D$ and r be a non zero real number . Consider the following statements: I: c is an extreme point of f $\Rightarrow$ c is an extreme point of rf II. c is an extreme point of f $\Rightarrow$ c is an extreme point of r+f Which of the following is correct? A) Only (i) is true B) Only (ii) is true C) Both (i) and (ii) are true D) Neither (i) nor(ii) is true
12 if the function $f:[a,b]\rightarrow \left[-\frac{\sqrt{3}}{4},\frac{1}{2}\right]$ defined by $f(x)=\begin{bmatrix}1 & 1&1 \\1 & 1+\sin_{}x&1\\1+\cos x&1&1 \end{bmatrix}$ is one-one and onto , then A) $a=\frac{-\pi}{4},b=\frac{\pi}{6}$ B) $a=\frac{-\pi}{2},b=\frac{\pi}{2}$ C) $a=\frac{-\pi}{6},b=\frac{\pi}{4}$ D) $a=-\pi, b=\pi$
13 If P is a complex number whose modulus is one , then the equation $\left(\frac{1+iz}{1-iz}\right)^{4}$ =P has A) real and equal roots B) real and distinct roots C) two real and two complex roots D) all complex roots
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