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 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 $e_{1}$ 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) $\frac{x^{2}}{25}-\frac{y^{2}}{9}=1$
4 Consider the function $f(x)=2x^{3}-3x^{2}-x+1$ and the intervals $I_{1}$=[-1,0],$I_{2}$= [0,1], $I_{3}$ =[1,2], $I_{4}$=[-2,-1] Then, A) f(x) =0 has a root in the intervals $I_{1}$ and $I_{4}$ only B) f(x) =0 has a root in the intervals $I_{1}$ and $I_{2}$ only C) f(x) =0 has a root in every interval except in $I_{4}$ D) f(x)=0 has a root in all the four given intervals
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 p1,p2,p3 arte the altitudes of a triangle ABC drawn from the vertices A,B and C respecively . If $\triangle$ is the area of the triangle and 2s is the sum of its sides a,b and c , then $\frac{1}{p_{1}}+\frac{1}{p_{2}}-\frac{1}{p_{3}}$= A) $\frac{s-a}{\triangle}$ B) $\frac{s-b}{\triangle}$ C) $\frac{s-c}{\triangle}$ D) $\frac{s}{\triangle}$
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 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) $y\left( e^{x^{3}}-(1+e)\right)+x=0$
9 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) $\sin \frac{\pi}{5}$
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 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) $x^{2}+y^{2}-6x+12y+15=0$
12 If $z=\sqrt{2}\sqrt{1+\sqrt{3i}}$ repesents a point P in the argand plane and P lies in the third quadrant , then the polar form of z is A) $2\left[ \cos \left(\frac{-4 \pi}{3}\right)+i \sin \left(\frac{-4 \pi}{3}\right)\right]$ B) $2\left[ \cos \left(\frac{-5 \pi}{6}\right)+i \sin \left(\frac{-5 \pi}{6}\right)\right]$ C) $2\left[ \cos \left(\frac{- \pi}{6}\right)+i \sin \left(\frac{- \pi}{6}\right)\right]$ D) $2\left[ \cos \left(\frac{- 2\pi}{3}\right)+i \sin \left(\frac{- 2\pi}{3}\right)\right]$
13 $\sum_{r=1}^{16} \left( \sin \frac{2r\pi}{17}+i\cos \frac{2r \pi}{17}\right)=$ A) 1 B) -1 C) i D) -i
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 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) $y=x^{3}+cx+d$