Q. Box I contains two white and three black balls. Box II contains four white and one black balls and box III contains three white and four black balls. A dice having three red, two yellow and one green face, is thrown to select the box. If red face turns up, we pick up box I, if a yellow face turns up we pick up box II, otherwise, we pick up box III. Then, we draw a ball from the selected box. If the ball drawn is white, what is the probability that the dice had turned up with a red face?

Q. Evaluate : $\int \frac{2y^2}{y^2+4}dy.$

Q. The two lines of regressions are $4x+2y-3=0$ and $3x+6y+5=0$. Find the correlation coefficient between $x$ and $y$.

Q. Three persons $A,B$ and $C$ shoot to hit a target. If $A$ hits the target four times in five trials, $B$ hits it three times in four trials and $C$ hits it two times in three trials, find the probability that:
i. Exactly two persons hit the target.
ii. At least two persons hit the target.
iii. None hit the target.

Q. Evaluate : ${\int }_0^{3}f\left(x\right)dx$, where $f\left(x\right)=\{\begin{array}{c}\cos 2x0\le x\le \frac{\pi }{2}\\ 3\frac{\pi }{2}\le x\le 3\end{array}$

Q. Find the volume of a parallelopiped whose edges are represented by the vectors :
$\overrightarrow{a}=2\hat{i}-3\hat{j}-4\hat{k},\overrightarrow{b}=\hat{i}+2\hat{j}-\hat{k},$ and $\overrightarrow{c}=3\hat{i}+\hat{j}+2\hat{k}$.

Q. Solve the equation for $x:{\sin }^{-1}\frac{5}{x}+{\sin }^{-1}\frac{12}{x}=\frac{\pi }{2},x\ne 0$.

Q. Using vectors, prove that angle in a semicircle is a right angle.

Q. A card is drawn from a well shuffled pack of playing cards. What is the probability that it is either a spade or an ace or both?

Q. Find the equation of the hyperbola whose foci are $(0,\pm \sqrt{10})$ and passing through the point $(2,3)$.

Q. $A,B$ and $C$ represents switches in 'on' position and $A^{\prime },B^{\prime }$ and $C^{\prime }$ represent them in 'off' position. Construct a switching circuit representing the polynomial $ABC+A{B}^{\prime }C+A^{\prime }{B}^{\prime }C$. Using Boolean Algebra, prove that the given polynomial can be simplified to $C(A+B^{\prime })$. Construct an equivalent switching circuit.

Q. Show that the rectangle of maximum perimeter which can be inscribed in a circle of radius $10cm$ is square of side $10\sqrt{2}cm$.

Q. Find the equation of an ellipse whose latus rectum is $8$ and eccentricity is $\frac{1}{3}$.

Q. Given that the observations are:
$(9,-4),(10,-3),(11,-1),(12,0),(13,1),(14,3),(15,5),(16,8)$.
Find the tow lines of regression and estimate the value of $y$ when $x=\mathrm{13.5.}$

Q. Using L' Hospital's rule, evaluate: $\underset{x\to 0}{lim}\frac{x-\sin x}{x^2\sin x}$.

Q. If $y=e^{m.{\cos }^{-1}x}$, prove that :
$(1-x^2)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}=m^{2}y$

Q. Solve : ${\cos }^{-1}\left(\sin \right({\cos }^{-1}x\left)\right)=\frac{\pi }{6}$.

Q. Find the smaller area enclosed by the circle $x^2+y^2=4$ and the line $x+y=2$.

Q. Find the equation of the plane passing through the intersection of the planes :
$x+y+z+1=0$ and $2x-3y+5z-2=0$ and the point $(-1,2,1)$.

Q. If $1,\omega$ and ${\omega }^2$ are the cube roots of unity, prove that $\frac{a+b\omega +c{\omega }^2}{c+a\omega +b{\omega }^2}={\omega }^2$

Q. Verify Lagrange's Mean Value Theorem for the following function:
$f\left(x\right)=2\sin x+\sin 2x$ on $[0,\pi ]$

Q. Find the shortest distance between the lines $\overrightarrow{r}=\hat{i}+2\hat{j}+3\hat{k}+\lambda (2\hat{i}+3\hat{j}+4\hat{k})$ and $\overrightarrow{r}=2\hat{i}+4\hat{j}+5\hat{k}+\mu (4\hat{i}+6\hat{j}+8\hat{k})$.

Q. If $z=x+iy,\omega =\frac{2-iz}{2z-i}$ and $|\omega |=1$, find the locus of $z$ and illustrate it in the Argand Plane.

Q. Solve the differential equation :
$e^{\frac{x}{y}}\left(\begin{array}{c}1-\frac{x}{y}\end{array}\right)+(1+e^{\frac{x}{y}})\frac{dx}{dy}=0$ when $x=0,y=1$

Q. An urn contains $2$ white and $2$ black balls. A ball is drawn at random. If it is white, it is not replaced into the urn. Otherwise, it is replaced with another ball of the same colour. The process is repeated. Find the probability that the third ball drawn is black.

Q. Evaluate : $\int \frac{\sec x}{1+\text{cosec}x}dx$

Q. Using properties of determinants, prove that:

Q. In a contest the competitors are awarded marks out of $20$ by two judges. The scores of the $10$ competitors are given below. Calculate Spearman's rank correlation.

Competitors	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$	$I$	$J$
Judge $A$	$2$	$11$	$11$	$18$	$6$	$5$	$8$	$16$	$13$	$15$
Judge $B$	$6$	$11$	$16$	$9$	$14$	$2$	$4$	$3$	$13$	$17$

Q. Given two matrices $A$ and $B$
$A=\left[\begin{array}{ccc}1& -2& 3\\ 1& 4& 1\\ 1& -3& 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}11& -5& -14\\ -1& -1& 2\\ -7& 1& 6\end{array}\right]$

Q. Solve the differential equation :