1 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
2 If the curves $y^{2}=6x,9x^{2}+by^{2}=16$ intersect each other at right angles, then the value of b is A) 6 B) $\frac{7}{2}$ C) 4 D) $\frac{9}{2}$
3 Let $f( x)=x^{2}+\frac{1}{x^{2}}$ and $g( x)=x^{}-\frac{1}{x^{'}}$ x $\in$ R - {-1,0,1}. If $h( x)=\frac{f( x)}{g( x)}$ , then the local minimum value of h(x) is A) 3 B) -3 C) $-2\sqrt{2}$ D) $2\sqrt{2}$
4 If the tangent at ( 1,7) to the curve $x^{2}=y-6$ touches the circle $x^{2}+y^{2}+16x+12y+c=0$, then the value of c is A) 195 B) 185 C) 85 D) 95
5 Tangents are drawn to the hyperbola 4x2-y2=36 at the points P and Q . If these tangents intersect at the point T ( 0,3), then the area ( in sq units ) of $\triangle PTQ$ is A) $45\sqrt{5}$ B) $54\sqrt{3}$ C) $60\sqrt{3}$ D) $36\sqrt{5}$
6 The boolean expression $\sim (p\vee q) \vee (\sim p \wedge q)$ is equivalent to A) $\sim p$ B) p C) q D) $\sim q$
7 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 $
8 A bag contains 4 red and 6 black balls, a ball is drawn at random from the bag. Its colour is observed and this ball along with two additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is A) $\frac{3}{10}$ B) $\frac{2}{5}$ C) $\frac{1}{5}$ D) $\frac{3}{4}$
9 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)
10 Let $S= ( t\in R:f( x)=\mid x-\pi\mid .e^{\mid x\mid}-1)sin\mid x\mid$ is not differentiable at t). Then, the set S is equal to A) $\phi $ (an empty set) B) { 0} C) {$\pi$} D) {0,$\pi$}
11 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 $100\pi$
12 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 $\frac{2}{\sqrt{3}}$
13 Let E1 E2 and F1 F2 be the chords of S passing through the point P0 (1,1) and parallel to the X-axis and the Y-axis, respectively. Let G1 G2 be the chord of S passing through P0 and having slope -1. Let the tangents to S at E1 and E2 meet at E3, then tangents to S at F1 and F2 meet at F3, and the tangents to S at G1 and G2 meet at G3. Then, the points E3, F3, and G3 lie on the curve A) x+y=4 B) $(x-4)^{2}+(y-4)^{2}=16$ C) (x-4)(y-4)=4 D) xy=4
14 Consider two straight lines, each of which is tangent to both the circle x2+ y2=(1/2) and the parabola y2 =4x. Let these lines intersect at the point Q. Consider the ellipse whose center ia at the origin O(0,0) and whose semi-major axis is OQ. If the length of the minor axis of this ellipse is $\sqrt{2}$ , then which of the following statement (s) is (are) TRUE? A) For the ellipse, the eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is 1 B) For the ellipse, the eccentricty is 1/2 and the length of the latus rectum is 1/2 C) The area of the region bounded by the ellipse between the lines x=$\frac{1}{\sqrt{2}}$ and x=1 is $\frac{1}{4\sqrt{2}}(\pi -2)$ D) The area of the region bounded by the ellipse between the lines x= $\frac{1}{\sqrt{2}}$ and x=1 is $\frac{1}{16}(\pi -2)$
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
