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 From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf, so that the dictionary is always in the middle. The number of such arrangement is A) atleast 1000 B) less than 500 C) atleast 500 but less than 750 D) atleast 750 but less than 1000
3 A straight line through a fixed point ( 2,3) intersects the coordinate axes at distinct point P and Q, If O is the origin and the rectangle OPRQ is completed, then the locus of R is A) $3x+2y=6$ B) $2x+3y=xy$ C) $3x+2y=xy$ D) $3x+2y=6xy$
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 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 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) $\frac{3}{4}$
8 If sum of all the solutions of the equation $8\cos x.( \cos ( \frac{\pi}{6}+x).\cos ( \frac{\pi}{6}-x)-\frac{1}{2})=1$ in [0,π ] is kπ , then k is equal to A) $\frac{2}{3}$ B) $\frac{13}{9}$ C) $\frac{8}{9}$ D) $\frac{20}{9}$
9 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,$\pi$}
10 Let A be the sum of the first 20 terms and B be the sum of first 40 terms of the series 12+2.22+32+2.42+52+2.62+........ If B-2A=100λ then λ is equal to A) 232 B) 248 C) 464 D) 496
11 The value of $((\log_{2}{9})^{2})^{\frac{1}{\log_{2}{}(\log_{2}{9})}\times\frac{1}{(\sqrt{7})^{\log_{4}{7}}}}$ is............ A) 6 B) 8 C) 10 D) 4
12 For each positive integer n, let $y_{n}=\frac{1}{n}((n+1)(n+2)...(n+n))^{\frac{1}{n}}. $. For x ε R, let [x] be the greatest integer less than or equal to x. If $ \lim_{n \rightarrow\infty} y_{n}=L$, then the value of [L] is ........... A) 2 B) 5 C) 3 D) 1
13 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
14 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) $x^{2}+y^{2}=x^{2}y^{2}$
15 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