1 If $\frac{x}{\sqrt{1+x}}+\frac{y}{\sqrt{1+y}}=0, x\neq y,$ then $ (1+x)^{2}\frac{dy}{dx}=$ A) 1 B) $\frac{1}{2}$ C) -1 D) 0
2 The value of $\sin^{-1}\left(\frac{1}{2}\right)+\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right)$ is A) $\cos^{-1}\left(\frac{1}{2}\right)$ B) $\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right)$ C) $\cos^{-1}\left(-\frac{1}{2}\right)$ D) $\sin^{-1}\left(-\frac{1}{2}\right)$
3 The p.d.f of c.r.v X is given by $f(x)=\frac{x+2}{18}$ , if -2 < x < 4=0, otherwise =0, then P[|x| <1]= A) $\frac{2}{9}$ B) $\frac{4}{9}$ C) $\frac{1}{9}$ D) $\frac{1}{18}$
4 If $\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}=4$, then $\frac{dy}{dx}=$ A) $\frac{7y-x}{y-7x}$ B) $\frac{y-7x}{7x-y}$ C) $\frac{y+7x}{7y-x}$ D) $\frac{7x+y}{x-7y}$
5 The symbolic form of the following circuit is (where p,q represents switches S1 and S2 closed respectively ) A) $(p\wedge q)\wedge (\sim p\wedge \sim q)=l$ B) $(p\wedge [q\wedge (\sim p\wedge \sim q)=l$ C) $(p\vee q)\vee (\sim p\wedge \sim q)=l$ D) $(p\vee [q\wedge (\sim p\wedge \sim q)=l$
6 If the equation ax2+2hxy+by2+2gx+2fy=0 has one line as the bisector of the angle between co-ordinate axes, then A) $(a+b)^{2}=4(h^{2}+f^{2})$ B) $(a+b)^{2}=4(h^{2}+g^{2}+f^{2})$ C) $(a+b)^{2}=4h^{2}$ D) $(a+b)^{2}=4(h^{2}+g^{2})$
8 The angle between the lines $\frac{x-1}{4}=\frac{y-3}{1}=\frac{z}{8}$ and $\frac{x-2}{2}=\frac{y+1}{2}=\frac{z-4}{1}$ A) $\cos^{-1}\left(\frac{2}{3}\right)$ B) $\cos^{-1}\left(\frac{1}{2}\right)$ C) $\cos^{-1}\left(\frac{3}{4}\right)$ D) $\cos^{-1}\left(\frac{1}{3}\right)$
9 If the line $r= (\hat{i}-2\hat{j}+3\hat{k})+\lambda (2\hat{i}+\hat{j}+2\hat{k})$ is parallel to the plane $r. (3\hat{i}-2\hat{j}+m\hat{k})=10$ , then the value of m is A) -2 B) 3 C) 2 D) -3
12 The negation of the statement pattern $\sim p \vee (q\rightarrow\sim r)$ is A) $p\wedge (q\wedge r)$ B) $\sim p\wedge (q\wedge r)$ C) $p \vee (q\wedge r)$ D) $p \rightarrow (q\wedge \sim r)$
13 The c.d. f F(x) associated with p.d.f. $f(x) =3(1-2x^{2})$. If 0 < x <1. is $k\left( x-\frac{2x^{3}}{k}\right)$ , then value of k is A) 3 B) $\frac{1}{3}$ C) 1 D) $\frac{1}{6}$
14 The equation of planes parallel to the plane x+2y+2z+8=0 , which are at a distance of 2 units from the point (1,1,2) are A) x+2y+2z-5=0 or x+2y+2z-3=0 B) x+2y+2z-6=0 or x+2y+2z-7=0 C) x+2y+2z-13=0 or x+2y+2z-1=0 D) x+2y+2z+3=0 or x+2y+2z-5=0
15 $\int \frac{dx}{x^{2}+4x+13}=$ A) $\frac{1}{3}\tan^{-1}\left(\frac{x+2}{3}\right)+C$ B) $\frac{1}{6}\tan^{-1}\left(\frac{x-1}{x+5}\right)+C$ C) $3\tan^{-1}\left(\frac{x+2}{3}\right)+C$ D) $\frac{1}{6}\tan^{-1}\left(\frac{x+2}{3}\right)+C$