Q. If $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],$ find $|2A|$.

Q. Write the domain of $f\left(x\right)={\cos }^{-1}x$.

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

Q. Verify Rolle's theorem for the function $y=x^2+2$. $xϵ|-2,2|$.

Q. Write the simplest form of ${\tan }^{-1}\left(\begin{array}{c}\sqrt{\frac{1-\cos x}{1+\cos x}}\end{array}\right)0<x<\pi .$.

Q. Evaluate $\sin \left[\begin{array}{c}\frac{\pi }{3}-{\sin }^{-1}\left(\begin{array}{c}-\frac{1}{2}\end{array}\right)\end{array}\right].$

Q. Evaluate $\int (\sin x+\cos x)dx$.

Q. Find the equation of line joining (1, 2) and (3, 6) using determinants.

Q. Find the equation of the plane with intercept 4 on Z-axis and parallel to $XY$ plane.

Q. A relation R on A = {1, 2, 3} defined by R = { (1, 1) (1, 2) (3, 3) } is not symmetric. Why?

Q. Define a scalar matrix.

Q. Find the approximate change in the volume $V$ of a cube of side $x$ meters caused by increasing the side by $2\%$.

Q. Find the direction cosines of the vector $\hat{i}+2\hat{j}+3\hat{k}$.

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

Q. If $y=\log \left(\sin x\right),$ find $\frac{dy}{dx}$.

Q. Find the probability distribution of number of heads in two tosses of a coin.

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

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

Q. If $\overrightarrow{a}$ is a unit vector such that $(\overrightarrow{x}-\overrightarrow{a})(\overrightarrow{x}+\overrightarrow{a})=8$, find $|\overrightarrow{x}|$.

Q. Verify whether the operation $\ast$ defined on $\varrho$ by $a\ast b=\frac{ab}{2}$ is associative or not.

Q. Define feasible region.

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

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

Q. Prove that $3{\cos }^{-1}x={\cos }^{-1}(4x^3-3x),xϵ\left[\begin{array}{c}\frac{1}{2},1\end{array}\right]$.

Q. Find the general solution of the differential equation
$e^x\tan ydx+(1-e^x){\sec }^{2}ydy=0$.

Q. If $P\left(A\right)=\frac{4}{5}$ and $P\left(B/A\right)=\frac{2}{5}$, find $P(A\cap B).$

Q. Find $\frac{dy}{dx}$ if $x=a(\cos \theta +\theta \sin \theta )$. and $y=a(\sin \theta -\theta \cos \theta )$.

Q. Evaluate : $\stackrel{e}{\underset{1}{\int }}\frac{1}{x}dx$.

Q. If $y+\sin y=\cos x$, find $\frac{dy}{dx}$.

Q. Find g o f and f o g if $f:R\to R$ and $g:R\to R$ are given by $f\left(x\right)=\cos x$ and $g\left(x\right)=3x^2$. Show that g o f $\ne$ f o g.