Q. Verify Rolle's theorem for each of the following functions on the indicated intervals:
$f\left(x\right)=e^x\sin x$ on $[0,\pi ]$.

Q. If $A=\left(\begin{array}{cc}5& a\\ b& 0\end{array}\right)$ and A is symmetric matrix, show that a$=$b.

Q. Using matrices, solve the following system of equation.
$2x-3y+5z=11$

Q. If the function $f\left(x\right)=\sqrt{2x-3}$ is invertible then find its inverse. Hence prove that $\left(fo{f}^{-1}\right)\left(x\right)=x$.

Q. Evaluate: ${\int }_0^{\pi /2}\frac{{\cos }^{2}x}{1+\sin x\cos x}dx$.

Q. Water is dripping out from a conical funnel of semi-vertical angle $\frac{\pi }{4}$ at the uniform rate of $2$ $c{m}^2$/sec in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is $4$ cm, find the rate of decrease of the slant height of the water.

Q. Find the approximate changes in the volume 'V' of a cube of side x metres caused by decreasing the side by $1\%$.

Q. Use properties of determinants to solve for x: $\left|\begin{array}{ccc}x+a& b& x\\ c& x+b& a\\ a& b& x+c\end{array}\right|=0$ and $x\ne 0$.

Q. Find the differential equation of the family of concentric circles $x^2+y^2=a^2$.

Q. Evaluate: $\int \frac{x^3+5x^2+4x+1}{x^2}dx$.

Q. Without expanding at any stage, find the value of $\left|\begin{array}{ccc}a& b& c\\ a+2x& b+2y& c+2z\\ x& y& z\end{array}\right|$.

Q. A cone is inscribed in a sphere of radius $12$cm. If the volume of the cone is maximum, find its height.

Q. If A and B are events such that $P\left(A\right)=\frac{1}{2}$, $P\left(B\right)=\frac{1}{3}$ and $P(A\cap B)=\frac{1}{4}$, then find $P\left(A/B\right)$.

Q. If $x=\tan \left(\frac{1}{a}logy\right)$, prove that $(1+x^2)\frac{d^{2}y}{d{x}^2}+2x\frac{dy}{dx}-a0$.

Q. The population of a town grows at the rate of $10\%$ per year. Using differential equation, find how long will it take for the population to grow $4$ times.

Q. In a race, the probabilities of A and B winning the race are $\frac{1}{3}$ and $\frac{1}{6}$ respectively. find the probability of neither of them winning the race.

Q. If ${\tan }^{-1}a+{\tan }^{-1}b+{\tan }^{-1}c=\pi$, prove that $a+b+c=abc$.

Q. Solve: $\sin x\frac{dy}{dx}-y=\sin x\cdot \tan \frac{x}{2}$.

Q. From a lot of $6$ items containing $2$ defective items, a sample of $4$ items are drawn at random. Let the random variable X denote the number of defective items in the sample. If the sample is drawn without replacement, find Variance of X.

Q. Find the value of constant 'k' so that the function f(x) defined as; $f\left(x\right)=\{\begin{array}{cc}\frac{x^2-2x-3}{x+3},& x\ne -1\\ k,& x=-1\end{array}$ is continuous at $x=-1$.

Q. Evaluate: $\int {\tan }^{-1}\sqrt{x}dx$.

Q. Find the points on the curve $y=4x^3-3x+5$ at which the equation of the tangent is parallel to the x-axis.

Q. From a lot of $6$ items containing $2$ defective items, a sample of $4$ items are drawn at random. Let the random variable X denote the number of defective items in the sample. If the sample is drawn without replacement, find the probability distribution of X.

Q. Solve: $3{\tan }^{-1}x+{\cot }^{-1}x=\pi$.

Q. Evaluate: $\int \frac{x-1}{\sqrt{x^2-x}}dx$.

Q. Find $\lambda$ if the scalar projection of $\overrightarrow{a}=\lambda \hat{i}+\hat{j}+4\hat{k}$ on $\overrightarrow{b}=2\hat{i}+6\hat{j}+3\hat{k}$ is $4$ units.

Q. Show that the function $f\left(x\right)=\{\begin{array}{cc}{x}^2,& x\le 1\\ \frac{1}{x},& x>1\end{array}$ is continuous at $x=1$ but not differentiable.

Q. From a lot of $6$ items containing $2$ defective items, a sample of $4$ items are drawn at random. Let the random variable X denote the number of defective items in the sample. If the sample is drawn without replacement, find Mean of X.

Q. The binary operation $\ast$ : R $\times$ R $\to$ R is defined as a $a\ast b=2a+b$. Find $(2\ast 3)\ast 4$.

Q. If A and B are events such that $P\left(A\right)=\frac{1}{2}$, $P\left(B\right)=\frac{1}{3}$ and $P(A\cap B)=\frac{1}{4}$, then find $P\left(B/A\right)$.