Q. If $y=\frac{x{\sin }^{-1}x}{\sqrt{1-x^2}}$, prove that: $(1-x^2)\frac{dy}{dx}=x+\frac{y}{x}$

Q. $P,Q$ and $R$ represent switches in on position and $P^{\prime },Q^{\prime }$ and $R^{\prime }$ represent switches in off position. Construct a switching circuit representing the polynomial $PR+Q(Q^{\prime }+R)(P+QR)$. Using Boolean Algebra, simplify the polynomial expression and construct the simplified circuit

Q. If ${\cos }^{-1}x+{\cos }^{-1}y+{\cos }^{-1}z=\pi$, prove that $x^2+y^2+z^2+2xyz=1$

Q. A psychologist selected a random sample of $22$ students. He grouped them in $11$ pairs so that the students in each pair have nearly equal scores in an intelligence test. In each pair, one student was taught by method $A$ and the other by method $B$ and examined after the course. The marks obtained by them after the course are as follows:

Pairs	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
Method $A$	$24$	$29$	$19$	$14$	$30$	$19$	$27$	$30$	$20$	$28$	$11$
Method $B$	$37$	$35$	$16$	$26$	$23$	$27$	$19$	$20$	$16$	$11$	$21$

Calculate Spearman's Rank correlation

Q. A wire of length $50m$ is cut into two pieces. One piece of the wire is bent in the shape of a square and the other in the shape of a circle. What should be the length of each piece so that the combined area of the two is minimum?

Q. For the given lines of regression, $3x-2y=5$ and $x-4y=7$, find:
coefficient of correlation $r(x,y)$

Q. Solve the differential equation:
$(x+1)dy-2xydx=0$

Q. A bag contains $20$ balls numbered from $1$ to $20$. One ball is drawn at random from the bag. What is the probability that the ball drawn is marked with a number which is multiple of $3$ or $4$?

Q. In a college, $70$% students pass in Physics, $75$% pass in Mathematics and $10$% students fail in both. One student is chosen at random. What is the probability that:
He passes in Physics given that he passes in Mathematics

Q. A company manufactures two types of toys $A$ and $B$. A toy of type $A$ requires $5$ minutes for cutting and $10$ minutes for assembling. A toy of type $B$ requires $8$ minutes for cutting and $8$ minutes for assembling. There are $3$ hours available for cutting and $4$ hours available for assembling the toy in a day. The profit is $Rs.50$ each on a toy of type $A$ and $Rs.60$ each on a toy of type $B$. How many toys of each type should the company manufacture in a day to maximize the profit? Use linear programming to find the solution

Q. If $A=\left[\begin{array}{cc}3& 1\\ 7& 5\end{array}\right]$, find the values of $x$ and $y$ such that $A^2+x{I}_2=yA$

Q. Express the complex number $\frac{(1+\sqrt{3}i{)}^2}{\sqrt{3}-i}$ in the form of $a+ib$. Hence, find the modulus and argument of the complex number

Q. Evaluate: $\tan [2{\tan }^{-2}\frac{1}{2}-{\cot }^{-1}3]$

Q. Evaluate the following:
$\underset{x\to 0}{lim}(1+\sin x{)}^{\cot x}$

Q. For the given lines of regression, $3x-2y=5$ and $x-4y=7$, find:
regression coefficients $b_{yx}$ and $b_{xy}$

Q. In a college, $70$% students pass in Physics, $75$% pass in Mathematics and $10$% students fail in both. One student is chosen at random. What is the probability that:
He passes in Mathematics given that he passes in Physics

Q. Find the equation of the parabola with latus - rectum joining points $(4,6)$ and $(4,-2)$

Q. Evaluate: $\int \frac{x+\sin x}{1+\cos x}dx$

Q. A bag contains $5$ white and $4$ black balls and another bag contains $7$ white and $9$ black balls. A ball is drawn from the first bag and two balls drawn from the seconds bag. What is the probability of drawing one white and two black balls?

Q. In a college, $70$% students pass in Physics, $75$% pass in Mathematics and $10$% students fail in both. One student is chosen at random. What is the probability that:
(i) He passes in Physics and Mathematics.
(ii) He passes in Mathematics given that he passes in Physics.
(iii) He passes in Physics given that he passes in Mathematics.

Q. Find the eccentricity and the coordinates of foci of the hyperbola $25x^2-9y^2=225$

Q. Evaluate: $\int e^x\frac{(2+\sin 2x)}{{\cos }^{2}x}dx$

Q. The coefficient of correlation between the values denoted by $X$ and $Y$ is $0.5$. The mean of $X$ is $3$ and that of $Y$ is $5$. Their standard deviations are $5$ and $4$ respectively. Find:
The expected value of $X$, when $Y$ is given $9$

Q. The coefficient of correlation between the values denoted by $X$ and $Y$ is $0.5$. The mean of $X$ is $3$ and that of $Y$ is $5$. Their standard deviations are $5$ and $4$ respectively. Find:
The two lines of regression

Q. The coefficient of correlation between the values denoted by $X$ and $Y$ is $0.5$. The mean of $X$ is $3$ and that of $Y$ is $5$. Their standard deviations are $5$ and $4$ respectively. Find:
The expected value of $Y$, when $X$ is given $14$

Q. Sketch the graphs of the curves $y^2=x$ and $y^2=4-3x$ and find the area enclosed between them

Q. Using matrix method, solve the following system of equation:
$x-2y=10,2x+y+3z=8$ and $-2y+z=7$

Q. Using properties of determinants, prove that:
$\left|\begin{array}{ccc}{a}^2+1& ab& ac\\ ba& b^2+1& bc\\ ca& cb& c^2+1\end{array}\right|=a^2+b^2+c^2+1$

Q. Using properties of definite integrals, evaluate ${\int }_0^{\frac{\pi }{2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$

Q. Verify Rolle's Theorem for the function $f\left(x\right)=e^x(\sin x-\cos x)$ on $[\frac{\pi }{4},\frac{5\pi }{4}]$