2 If the cubic equation $x^{3}-a x^{2}+ax-1$=0 is identical with the cubic equation whose roots are the squares of the roots of the given cubic equation , then the non-zero real value of 'a' is A) $\frac{1}{2}$ B) 2 C) 3 D) $\frac{7}{2}$
3 Consider the function $f(x)=2x^{3}-3x^{2}-x+1$ and the intervals $I_{1}$=[-1,0],$I_{2}$= [0,1], $I_{3}$ =[1,2], $I_{4}$=[-2,-1] Then, A) f(x) =0 has a root in the intervals $I_{1}$ and $I_{4}$ only B) f(x) =0 has a root in the intervals $I_{1}$ and $I_{2}$ only C) f(x) =0 has a root in every interval except in $I_{4}$ D) f(x)=0 has a root in all the four given intervals
4 Area of the region (in sq units) bounded by the curve y =$\sqrt{x}$, x= $\sqrt{y}$ and the lines x=1, x=4 , is A) $\frac{8}{3}$ B) $\frac{49}{3}$ C) $\frac{16}{3}$ D) $\frac{14}{3}$
5 If the point $\left(\frac{k-1}{k},\frac{k-2}{k}\right)$ lies on the locus of z satisfying the inequality $|\frac{z+3i}{3z+i}|$ <1, then the interval in which k lies is A) $(-\infty ,2) \cup (3, \infty)$ B) [2,3] C) [1,5] D) $(-\infty ,1) \cup (5, \infty)$
6 Bag I contains 3 red and 4 black balls , Bag II contains 5 red and 6 black balls .If one ball is drawn at random from one of the bags and it is found to be red , then the probability that it was drawn from Bag II, is A) $\frac{33}{68}$ B) $\frac{35}{68}$ C) $\frac{37}{68}$ D) $\frac{41}{68}$
7 The derivative of $\cos h^{-1} x$ with respect to log x at x=5 is A) $\frac{5}{\sqrt{26}}$ B) $\frac{1}{\sqrt{26}}$ C) $\frac{1}{2\sqrt{6}}$ D) $\frac{5}{2\sqrt{6}}$
8 If $\int\frac{2x^{2}}{(2x^{2}+\alpha)(x^{2}+5)}dx=\frac{\sqrt{5}}{3} \tan ^{-1}\frac{x}{\sqrt{5}}-\frac{\sqrt{2}}{3} \tan ^{-1}\frac{x}{\sqrt{2}}+c,$ then $\alpha$ = A) 1 B) 2 C) 3 D) 4
9 If a die is rolled twice and the sum of the numbers appearing on them is observed to be 6 , then the probability that the number 1 appears atleast once on them is A) $\frac{5}{36}$ B) $\frac{2}{5}$ C) $\frac{11}{36}$ D) $\frac{1}{3}$
10 If the function f:R→ r defined by $f(x)=\begin{cases}a\left(\frac{1-\cos 2x}{x^{2}}\right), & for x < 0\\b ,& x = 0\\\frac{\sqrt{x}}{\sqrt{4+\sqrt{x}}-2}&for x>0\end{cases}$ is continuous at x=0 , then a+b= A) 2 B) 4 C) 6 D) 8
11 $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot^{9} x dx=$ A) $\frac{-7}{42}+\frac{1}{2} log 2$ B) $\frac{7}{24}-\frac{1}{2} log 2$ C) $\frac{25}{24}+\frac{1}{2} log 2$ D) $\frac{1}{24}+2 log 2$
12 The value of $\theta$ for which the following system of equations has a non-trivial solution is $(4 \sin \theta)x-3y-z=0 , x-(6 \cos 2\theta )y+z=0$, 3x-12y+4z=0 A) $\tan^{-1} (\frac{1}{2})$ B) $\frac{\pi}{4}$ C) $\sin^{-1} (\frac{3}{16})$ D) $\frac{\pi}{12}$
13 If $z=\sqrt{2}\sqrt{1+\sqrt{3i}}$ repesents a point P in the argand plane and P lies in the third quadrant , then the polar form of z is A) $2\left[ \cos \left(\frac{-4 \pi}{3}\right)+i \sin \left(\frac{-4 \pi}{3}\right)\right]$ B) $2\left[ \cos \left(\frac{-5 \pi}{6}\right)+i \sin \left(\frac{-5 \pi}{6}\right)\right]$ C) $2\left[ \cos \left(\frac{- \pi}{6}\right)+i \sin \left(\frac{- \pi}{6}\right)\right]$ D) $2\left[ \cos \left(\frac{- 2\pi}{3}\right)+i \sin \left(\frac{- 2\pi}{3}\right)\right]$
14 If a variable circle S=0 touches the line y=x and passes through the point (0,0) , then the fixed point that lies on the common chord of the circles $x^{2}+y^{2}+6x+8y-7=0$ and S=0 is A) $\left(\frac{1}{2},\frac{1}{2}\right)$ B) $\left(-\frac{1}{2},-\frac{1}{2}\right)$ C) $\left(\frac{1}{2},-\frac{1}{2}\right)$ D) $\left(-\frac{1}{2},\frac{1}{2}\right)$
15 Electric current is measured by tangent galvanometer, the current being proportional to the tangent of the angle $\theta$ of deflection. If the deflection is read as $45^{0}$ and an error of 1% is made in reading it.then the percentage error in the current is A) $\pi$ B) $\frac{\pi}{2}$ C) $\frac{\pi}{3}$ D) $\frac{\pi}{4}$
