Q. Differentiate ${\left(\begin{array}{c}x+\frac{1}{x}\end{array}\right)}^x$ with respect to $x$.

Q. Find the angle between the pair of lines given by
$\overrightarrow{r}=3\hat{i}+2\hat{j}-4\hat{k}+\lambda (\hat{i}+2\hat{j}+2\hat{k})$ and $\overrightarrow{r}=5\hat{i}-2\hat{j}+\mu (3\hat{i}+2\hat{j}+6\hat{k})$.

Q. Verify Rolles theorem for the function $f\left(x\right)=x^2+2x-8,x\in [-4,2]$.

Q. Show that ${\sin }^{-1}\left(2x\sqrt{1-x^2}\right)=2{\sin }^{-1}x$ for $\frac{-1}{\sqrt{2}}\le x\le \frac{1}{\sqrt{2}}$.

Q. If ${\tan }^{-1}\frac{x-1}{x-2}+{\tan }^{-1}\frac{x+1}{x+2}=\frac{\pi }{4}$, then find the values of $x$.

Q. Let $X$ denote the number of hours you study during a randomly selected school day. The probability that $X$ can take the values of $x$, has the following form, where $k$ is some constant
$P(X=x)=\begin{array}{c}\{\begin{array}{cc}0,1& \text{if}x=0\\ Kx& \text{if}x=1\text{or}2\\ K(5-x)& \text{if}x=3\text{or}4\\ 0& \text{otherwise}\end{array}\end{array}$
find the value of $K$.

Q. If $P\left(A\right)=\frac{3}{5}$ and $P\left(B\right)=\frac{1}{5}$, then find $P(A\cap B)$, if $A$ and $B$ are independent events.

Q. Define feasible region in LPP.

Q. If $A$ and $B$ are invertible matrices of the same order, then prove that
$(AB{)}^{-1}=B^{-1}{A}^{-1}$.

Q. If the area of the triangle with vertices $(-2,0),(0,4)$ and $(0,k)$ is $4$ square units, find the values of $k$ using determinants.

Q. Construct a $2\times 2$ matrix $A=\left[a_{ij}\right]$ whose elements are given by $\frac{1}{2}|-3i+j|$.

Q. Find the area of the region bounded by the curve $y^2=4x$ and the line $x=3$.

Q. Write the direction cosines of $x$-axis.

Q. Write the values of $x$ for which $2{\tan }^{-1}x={\cos }^{-1}\frac{1-x^2}{1+x^2}$, holds.

Q. Determine whether the relation $R$ in the set $A=\{1,2,3,.....,13,14\}$ defined as $R=\left\{\right(x,y):3x-y=0\}$, is reflexive, symmetric and transitive.

Q. Evaluate: $\int \frac{\cos 2x-\cos 2\alpha }{\cos x-\cos \alpha }dx$.

Q. Show that $2{\tan }^{-1}\frac{1}{2}+{\tan }^{-1}\frac{1}{7}={\tan }^{-1}\frac{31}{17}$.

Q. Find the order and degree, if defined of the differential equation
${\left(\begin{array}{c}\frac{d^{2}y}{d{x}^2}\end{array}\right)}^3+{\left(\begin{array}{c}\frac{dy}{dx}\end{array}\right)}^2+\sin \frac{dy}{dx}+1=0$.

Q. Evaluate: $\int e^x\left(\begin{array}{c}\frac{x-1}{x^2}\end{array}\right)dx$.

Q. Find the values of $x$ for which $\left|\begin{array}{cc}3& x\\ x& 1\end{array}\right|=\left|\begin{array}{cc}3& 2\\ 4& 1\end{array}\right|$.

Q. Find $\frac{dy}{dx}$, if $y=\sin (x^2+5)$.

Q. Show that if $f:A\to B$ and $g:B\to C$ are one-one, then $g\circ f:A\to C$ is also one-one.

Q. If $x=\sqrt{a^{{\sin }^{-1}t}}$ and $y=\sqrt{a^{{\cos }^{-1}t}}$ then prove that $\frac{dy}{dx}=\frac{-y}{x}$.

Q. Find the slope of the tangent to the curve $y=\frac{x-1}{x-2},x\ne 2$ at $x=10$.

Q. Find $|\overrightarrow{b}|$, if $(\overrightarrow{a}+\overrightarrow{b}).(\overrightarrow{a}-\overrightarrow{b})=8$ and $|\overrightarrow{a}|=8|\overrightarrow{b}|$.

Q. Let $\ast$ be a operation defined on the set of rational numbers by $a\ast b=\frac{ab}{4}$, find the identity element.

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

Q. Define negative of a vector.

Q. Find $\frac{dy}{dx}$, if $x^2+xy+y^2=100$.

Q. Find the area of the parallelogram whose adjacent sides are determined by the vectors $\overrightarrow{a}=\hat{i}-\hat{j}+3\hat{k}$ and $\overrightarrow{b}=2\hat{i}-7\hat{j}+\hat{k}$.