Q. If $|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}-\overrightarrow{b}|$, prove that $\overrightarrow{a}$ and $\overrightarrow{b}$ are perpendicular.

Q. Find the value of $\cos ({\sec }^{-1}x+{\text{cosec}}^{-1}x),$ for $|x|\ge 1$.

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

Q. If $y=a^{\frac{1}{2}{\log }_a\cos x}$, find $\frac{dy}{dx}$

Q. If $x^y=a^x$, prove that $\frac{dy}{dx}=\frac{x{\log }_{e}a-y}{x{\log }_{e}x}$.

Q. If $P\left(A\right)=0.8$ and $P\left(B|A\right)=0.4$, then find $P(A\cap B)$.

Q. Verify Mean Value Theorem if $f\left(x\right)=x^3-5x^2-3x$ in the interval $[1,-3]$

Q. Find $\int \frac{x}{(x-1)(x-2)}dx$.

Q. Using differentials find the approximate value of $(25{)}^{\frac{1}{3}}$.

Q. If function $f:R\to R$ and $g:R\to R$ are given by $f\left(x\right)=|x|$ and $g\left(x\right)=\left[x\right]$, (where $\left[x\right]$ is greatest integer function) find $f\circ g(-\frac{1}{2})$ and $g\circ f(-\frac{1}{2})$.

Q. The random variable x has a probability distribution $P\left(x\right)$ of the following form where $k$ is some number:
$P\left(x\right)=\{\begin{array}{cc}k& ifx=0\\ 2k& ifx=1\\ 3k& ifx=2\\ 0& otherwise\end{array}$
Determine the value of $k$ and $P(X\le 2).$

Q. Find order and degree (if defined) of the differential equation $\frac{d^{4}y}{d{x}^4}+sin\left(\frac{d^{3}y}{d{x}^3}\right)=0$.

Q. If $\left[\begin{array}{cc}x+2& y-3\\ 0& 4\end{array}\right]$ is a scalar matrix, find $x$ and $y$.

Q. Find the distance of the point $(-6,0,0)$ from the plane $2x-3y+6y=2$

Q. Show that $ta{n}^{-1}\frac{1}{2}+ta{n}^{-1}\frac{2}{11}+ta{n}^{-1}\frac{4}{3}=\frac{\pi }{2}$.

Q. Find a value of $x$ if $\left|\begin{array}{cc}x& 2\\ 18& x\end{array}\right|=\left|\begin{array}{cc}6& 2\\ 18& 6\end{array}\right|$

Q. Using determinants show that points $A(a,b+c),B(b,c+a)$ and $C(c,a+b)$ are collinear.

Q. Find $\int \text{cosec}x(\text{cosec}x+\cot x)dx$.

Q. If vector $\overrightarrow{AB}=2\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{OB}=3\hat{i}-4\hat{j}+4\hat{k}$, find the position vector $\overrightarrow{OA}$

Q. Show that the reaction R in the $setA=\{x:x\in z,0\le x\le 12\}$ given by $R=\left\{\right(a,b):|a-b|\text{is a multiple of 4}\}$ is an equivalence relation.

Q. If $x=aco{s}^3\theta$ and $y=asi{n}^3\theta$, prove that $\frac{dy}{dx}=-\sqrt[3]{3\frac{y}{x}}$.

Q. Evalue: ${\int }_0^{\pi }(si{n}^2\left(\frac{x}{2}\right)-co{s}^2\left(\frac{x}{2}\right))dx$.

Q. Write the simplest form of ${\tan }^{-1}\left[\frac{a\cos x-b\sin x}{b\cos x+a\sin x}\right]$, if $\frac{a}{b}\tan x>-1$.

Q. Box-I contains 2 gold coins, while another Box-II contains 1 gold and 1 silver coin. A person choose a box at random and takes out a coin. If the coin is of gold, what is the probability that the order coin in the box is also of gold?

Q. Find $\int \frac{1}{sinxco{s}^{3}x}dx$

Q. Find the Cartesian equation of the line parallel to y-axis and passing through the point (1, 1, 1).

Q. Integrate $\frac{2x}{(x^2+1)(x^2+2)}$ with respect to x.

Q. An equation $\ast$ on $Z^{+}$ (the set of all non-negative integers) is defined as $a\ast b=a-b,\forall a,b\in Z^{+}$. Is $\ast$ a binary operation on $Z^{+}$ ?

Q. Find $\frac{dy}{dx}$ if $y=se{c}^{-1}\left[\frac{1}{2x^2-1}\right]$, $0<x>\frac{1}{\sqrt{2}}$

Q. Find angle between the vectors $\overrightarrow{a}=\hat{i}+\hat{j}-\hat{k}$ and $\hat{b}=\hat{i}+\hat{j}+\hat{k}$.