1 Let S= {$x\in R:x\geq0$ and $2\mid \sqrt{x}-3\mid +\sqrt{x}(\sqrt{x}-6)+6=0$. Then , S A) is an empty set B) contains exactly one element C) contains exactly two element D) contains exactly four element
2 If α,β $\in$ C are the distinct roots of the equation $x^{2}-x+1=0, $ , then $\alpha^{101}+\beta^{107}$ is equal to A) -1 B) 0 C) 1 D) 2
3 The sum of the coefficients of all odd degree terms in the expansion is $(x+\sqrt{x^{3}-1})^{5}+(x-\sqrt{x^{3}-1} )^{5},(x>1)is$ A) -1 B) 0 C) 1 D) 2
4 If the system of linear equation x+ky+3z=0;3x+ky-2z=0; 2x+4y-3z=0 has a non-zero solution ( x,y,z), then $\frac{xz}{y^{2}}$ is equal to A) -10 B) 10 C) -30 D) 30
5 Two sets A and B are as under A= { (a,b)} $\in$ RxR: $\mid a-5\mid<1$ and $\mid b-5\mid<1$; B={(a,b) $\in$ RxR: $4(a-6)^{2}+9(b-5)^{2}\leq 36$ } , Then A) $B\subset A$ B) $A\subset B$ C) $A\cap B =\phi $ (an empty set) D) neither $A\subset B $ nor $B\subset A $
6 The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\sin ^{2}x}{1+2^{x}}$ is A) $\frac{\pi}{8}$ B) $\frac{\pi}{2}$ C) $\frac{\pi}{4}$ D) $4\pi$
7 For each t ε R, let [t] be the greatest integer less than or equal to t. Then, $\lim_{x \rightarrow 0}x([\frac{1}{x}+[\frac{2}{x}]+......+[\frac{15}{x}])$ A) is equal to 0 B) is equal to 15 C) is equal to 120 D) does not exist (R)
8 In a Δ PQR , Let $\angle PQR=30^{0}$ and the side PQ and QR have lengths 10√3 and 10, respectively . Then , which of the following statement (s) is (are) TRUE? A) $\angle QPR=45^{0}$ B) The area of the $\triangle PQR$ is $25\sqrt{3}$ and $\angle QRP=120^{0}$ C) The radius of the incircle of the $\triangle PQR$ is $10\sqrt{3}-15$ D) The area of the circumcircle of the $\triangle PQR$ is $100\pi$
9 Let P1: 2x+y-z=3 and P2 : x+2y+z=2 be two planes. Then, which of the following statements (s) is (are) TRUE ?1 A) The line of intersection of $P_{1}$ and $P_{2}$ has direction ratio 1,2,-1 B) The line $\frac{3x-4}{9}=\frac{1-3y}{9}=\frac{z}{3}$ is perpendicular to the line of intersection of $P_{1}$ and $P_{2}$ C) The acute angle between $P_{1}$ and $P_{2}$ is $60^{0}$ D) If $P_{3}$ is the plane passing through the point (4,2,-2) and perpendicular to the line of intersection of $P_{1}$ and $P_{2}$ , then the distance of the point (2,1,1) from the plane $P_{3}$ is $\frac{2}{\sqrt{3}}$
10 Let a and b be two unit vectors such that a.b=0. For some x,y $\in$ R, let c=xa+yb+ (a× b) . If $\mid c\mid=2$ and the vector c is inclined at the same angle $\alpha$ to both a and b, then the value of $8\cos^{2}\alpha$ is A) 2 B) 3 C) 1 D) 5
11 Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect the coordinate axes at the points M and N. Then the mid -point of the line segment MN must lie on the curve A) $(x+y)^{2}=3xy$ B) $x^{\frac{2}{3}}+y^{\frac{2}{3}}=2^{\frac{4}{3}}$ C) $x^{2}+y^{2}=2xy$ D) $x^{2}+y^{2}=x^{2}y^{2}$
12 Let T be the line passing through the points P(-2,7) and Q(2,-5). Let F1 be the set of all pairs of circles (S1, S2) such that T is tangent to S1 at P and tangent to S2 at Q , and also such that S1 and S2 touch each other at a point, say M. Let E1 be the set representing the locus of M as the pair (S1 , S2) varies in F1. Let the set of all straight line segments joining a pair of distinct points of E1 and passing through the point R(1,1) be F2. Let E2 be the set of the mid-points of the line segments in the set F2. Then , which of the following statement (s) is (are) TRUE ? A) The point (-2,7) lies in $E_{1}$ B) The point $(\frac{4}{5},\frac{7}{5})$ does NOT lie in $E_{2}$ C) The point $(\frac{1}{2},1)$ lies in $E_{2}$ D) The point $(0,\frac{3}{2})$ does not lie in $E_{1}$
13 Let s,t, r be non zero complex numbers and L be the set of solutions z=$ x + iy (x, y \in R$ ,i = $\sqrt{-1}$ ) of the equation $sz+i\bar{z}+r=0$ , where $\bar{z}=x-iy$ , Then , which of the following statement (s) is (are) TRUE ? A) If L has exactly one element, then $\mid s\mid \neq \mid t\mid$ B) If $\mid s\mid = \mid t\mid$ , then L has infinitely many elements C) The number of elements in $L \cap ({z:\mid z-1+i\mid=5})$ is at most 2 D) If L has more than one element then L has infinitely many elements.
14 The value of integral $\int_{0}^{1/2} \frac{1+\sqrt{3}}{((x+1)^{2}(1-x)^{6})^{1/4}}dx$ is ............ A) 3 B) 4 C) 1 D) 2
15 Let P be a point in the first octant , whose image Q in the plane x+y= 3 (that is, the line segment PQ is perpendicular to the plane x+y=3 and the mid- point of PQ lies in the plane x+y=3) lies on the Z-axis. Let the distance of P from the X-axis be 5. If R is the image of P in the XY-plane, then the length of PR is ..... A) 4 B) 6 C) 12 D) 8
