Q. If $y=e^{x^3}$, find $\frac{dy}{dx}$.

Q. If $P\left(A\right)=\frac{7}{13},P\left(B\right)=\frac{9}{13}$ and $P(A\cap B)=\frac{4}{13}$, find $P\left(A/B\right)$.

Q. If $A$ is an invertible matrix of order $2$ then find $|A^{-1}|$.

Q. Write the simplest form of ${\tan }^{-1}\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right),0<x<\frac{\pi }{2}$.

Q. Find $\int \frac{x^3-1}{x^2}dx$.

Q. Find $\int e^x\sin xdx$.

Q. A random variable $X$ has the following probability distribution:

$X$	$0$	$1$	$2$	$3$	$4$
$P\left(X\right)$	$0.1$	$k$	$2k$	$2k$	$k$

Determine:
(i) $k$
(ii) $P(X\ge 2)$.

Q. Construct a $2\times 2$ matrix $A=\left[a_{ij}\right]$, whose elements are given by $a_{ij}=\frac{i}{j}$.

Q. Find the unit vector in the direction of the vector $\overrightarrow{a}=\hat{i}+\hat{j}+2\hat{k}$.

Q. Find $\int \frac{xdx}{(x+1)(x+2)}$.

Q. Find the angle, between the planes whose vector equations are $\overrightarrow{r}\cdot (2\hat{i}+2\hat{j}-3\hat{k})=5$ and $\overrightarrow{r}\cdot (3\hat{i}-3\hat{j}+5\hat{j}+5\hat{k})=3$.

Q. Sand is pouring from a pipe at the rate of $12c{m}^3/s$. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is $3cm$?

Q. Let $\ast$ be a binary operation on $Q$, defined by $a\ast b=\frac{ab}{2},\forall a,b\in Q$.
Determine whether $\ast$ is associative or not.

Q. Find the slope of the tangent to the curve $y=x^3-x$ at $x=2$.

Q. Find the projection of the vector $\hat{i}+3\hat{j}+\hat{k}$ on the vector $7\hat{i}-\hat{j}+8\hat{k}$.

Q. Find the area of the parallelogram whose adjacent sides are determined by the vectors $\overrightarrow{a}=3\hat{i}+\hat{j}+4\hat{k}$ and $\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}$.

Q. If $x=\sin t,y=\cos 2t$ then prove that $\frac{dy}{dx}=-4\sin t$.

Q. Find the order and degree of the differential equation:
${\left(\frac{d^{3}y}{d{x}^3}\right)}^2+{\left(\frac{d^{2}y}{d{x}^2}\right)}^3+{\left(\frac{dy}{dx}\right)}^4+y^5=0$.

Q. Find the area of the triangle whose vertices are $(-2,3),(3,2)$ and $(-1,-8)$ by using determinant method.

Q. Verify Rolle's theorem for the function $f\left(x\right)=x^2+2,xϵ[-2,2]$.

Q. Show that the relation $R$ in the set $A=\{1,2,3,4,5\}$ given by $R=\left\{\right(a,b):|a-b|iseven\}$, is an equivalence relation.

Q. Differentiate $x^{\sin x},x>0$ with respect to $x$.

Q. Write the principal value branch of ${\cos }^{-1}x$.

Q. If $\sin ({\sin }^{-1}\frac{1}{5}+{\cos }^{-1}x)=1$ then find the value of $x$.

Q. Evaluate: ${\int }_2^3\frac{xdx}{x^2+1}$.

Q. If a line makes angle ${90}^{\circ },{60}^{\circ }$ and ${30}^{\circ }$ with the positive direction of $x,y$ and $z-axis$ respectively, find its direction cosines.

Q. Integrate $\frac{e^{{\tan }^{-1}}x}{1+x^2}$ with respect to $x$.

Q. Find two number whose sum is $24$ and whose product is as large as possible.

Q. Define bijective function.

Q. By using elementary transformations, find the inverse of the matrix.
$A=\left[\begin{array}{cc}1& 3\\ 2& 7\end{array}\right]$.