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 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) $2\sqrt{2}$
3 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
4 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) $4\pi$
5 The length of the projection of the line segment joining the points (5,-1,4) and (4,-1,3) on the plane , x+y+z=7 is A) $\frac{2}{\sqrt{3}}$ B) $\frac{2}{3}$ C) $\frac{1}{3}$ D) $\sqrt{\frac{2}{3}}$
6 Let y= y(x) be the solution of the different equation $\sin x\frac{dy}{dx}+ycosx=4x,x\epsilon (0,\pi)$. If $y(\frac{\pi}{2})=0$, then $y(\frac{\pi}{6})$ is equal to A) $\frac{4}{9\sqrt{3}}\pi^{2}$ B) $\frac{-8}{9\sqrt{3}}\pi^{2}$ C) $-\frac{8}{9}\pi^{2}$ D) $-\frac{4}{9}\pi^{2}$
7 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,$\pi$}
8 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 $\frac{2}{\sqrt{3}}$
9 Let X be the set consisting of the first 2018 terms of the arithmetic progression 1,6,11,..... and Y be the set consisting of the first 2018 terms of the arithmetic progression 9,16,23,.... Then , the number of elements in the set X υ Y is..... A) 3784 B) 2847 C) 3748 D) 4827
10 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
11 Let a,b,c be three non-zero real numbers such that the equation $\sqrt{3} a \cos x+2b \sin x= c$ , $x\in[-\frac{\pi}{2},\frac{\pi}{2}]$, has two distinct real roots $\alpha$ and $\beta$ with $\alpha$ + $\beta$= $\frac{\pi}{3}$ , Then , the value of $\frac{b}{a}$ is..... A) 0.5 B) 0.75 C) 1.5 D) 1
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 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
14 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
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
