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 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
3 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) $\frac{9}{2}$
4 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) $2\sqrt{2}$
5 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) $3x+2y=6xy$
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 Let $g(x)=\cos x ^{2}, f(x)=\sqrt{x}$ and $\alpha ,\beta (\alpha <\beta)$ be the roots of the quadratic equation $18x^{2}-9\pi x +\pi^{2}=0$ .Then , the area (in sq units) bounded by the curve $y= (gof) (x)$ and the lines x=α, x=β and y=0, is A) $\frac{1}{2}(\sqrt{3}-1)$ B) $\frac{1}{2}(\sqrt{3}+1)$ C) $\frac{1}{2}(\sqrt{3}-\sqrt{2})$ D) $\frac{1}{2}(\sqrt{2}-1)$
8 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)
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 If $\sum_\left(i=1\right)^9\left(x_{i}-5\right)=9$ and $\sum_\left(i=1\right)^9\left(x_{i}-5\right)^{2}=45$ then the standard deviation of the 9 items x1, x2,........ x9 is A) 9 B) 4 C) 2 D) 3
11 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 $\frac{2}{\sqrt{3}}$
12 A farmer F1 has a land in the shape of a triangle with vertices at P(0,0), Q(1,1), and R (2,0). From this land, a neighbouring farmer F2 takes away the region which lies between the sides PQ and a curve of the form y=xn (n>1). If the area of the region taken away by the farmer F2 is exactly 30% of the area Δ PQR , then the value of n is........ A) 5 B) 4 C) 2 D) 1
13 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) $x^{2}+y^{2}=x^{2}y^{2}$
14 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.
15 Let $f:R\rightarrow R$ be a differentiable function with f(0) =1 and satisfying the equation f(x+y) =f(x) f ' (y)+ f ' (x) f(y) for all x , $y \in R $, Then, the value of loge (f(4)) is.............. A) 2 B) 6 C) 8 D) 4
