Q. Resolve $\frac{x^2-5x-1}{(x-1{)}^2(x-2)}$ into partial fraction.

Q. The value of ${\tan }^{-1}x+{\cot }^{-1}x$ is.

1. $\pi$
2. Zero
3. $\frac{\pi }{2}$
4. $1$

Q. If vectors $\overrightarrow{a}=\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{i}+\hat{j}+\hat{k}$, then find the value of $\overrightarrow{a}\cdot \overrightarrow{b}$.

Q. If the regression line of $y$ on $x$ is $ax+by+c=0$ and the regression line of $x$ on $y$ is $a_{1}x+b_{1}y+c_1=0$, then prove that: $a{b}_1\le a_{1}b$.

Q. If $y=({\sin }^{-1}x{)}^2$, then prove that $(1-x^2)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}=2$.

Q. The equation of a plane which intercept unit length from the co-ordinate axes is ________.

Q. Match the columns by choosing the correct answer from B for A.

Q. The centre of the sphere $x^2+y^2+z^2-ax-by-cz=0$ is.

1. $(a,b,c)$
2. $\left(\frac{a}{2},\frac{b}{2},\frac{c}{2}\right)$
3. $\left(-\frac{a}{2},-\frac{b}{2},-\frac{c}{2}\right)$
4. $(-a,-b,-c)$

Q. Find the coefficient of correlation from the following data.

$x$	$2$	$3$	$5$	$7$	$3$
$y$	$15$	$17$	$4$	$5$	$4$

Q. Find the maximum value of $\sin x+\cos x$ (using differentiation).

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

Q. If $\overrightarrow{a}=-\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{b}=-\hat{i}+\hat{j}-\hat{k}$, then find the value of $|\overrightarrow{a}\cdot \overrightarrow{b}|$.

Q. Find the coefficient of correlation from the following data.

$x$	$-10$	$-5$	$0$	$5$	$10$
$y$	$5$	$9$	$7$	$11$	$13$

Q. Resolve $\frac{x}{1-x^3}$ into partial fraction.

Q. The nth derivative of $\sin x$ is equal to ________.

Q. In numerical methods, write the formula for Simpson's one-third rule.

Q. The velocity of any particle at maximum height is equal to ________.

Q. Partial fraction of $\frac{2}{x^2-1}$ is.

1. $\frac{1}{2}\left[\frac{1}{x-1}+\frac{1}{x+1}\right]$
2. $\frac{1}{2}\left[\frac{1}{x-1}-\frac{1}{x+1}\right]$
3. $\frac{1}{2}\left[\frac{1}{x+1}-\frac{1}{x-1}\right]$
4. $\left[\frac{1}{x-1}-\frac{1}{x+1}\right]$

Q. Find the equation of the plane passes through the point $(-1,3,2)$ and is perpendicular to planes $x+2y+2z=5$ and $3x+3y+2z=8$.

Q. In trapezoidal rule ${\int }_a^{b}f\left(x\right)dx=\frac{h}{2}[y_0+2(y_1+y_2+.....y_{a-1})+y_a$. Write the value of $h$.

Q. If regression lines are $3x+12y-19=0$ and $9x+3y-46=0$, then find the coefficient of correlation.

Q. The shortest distance between two intersecting lines is always equal to _________.

Q. If $x=a{\sin }^3\theta$ and $y=a{\cos }^3\theta$, then find the value of $\frac{dy}{dx}$.

Q. The equation of X-axis is.

1. $\frac{X}{1}=\frac{Y}{0}=\frac{Z}{0}$
2. $\frac{X}{1}=\frac{Y}{1}=\frac{Z}{1}$
3. $\frac{X}{0}=\frac{Y}{1}=\frac{Z}{1}$
4. None of these

Q. Find the differential coefficient of $x^{\sin x}$.

Q. If $\hat{i},\hat{j},\hat{k}$ are the unit vectors along the axis X, Y and Z respectively, then the value of $\hat{i}\times (\hat{j}\times \hat{k})$ is.

1. Zero
2. $1$
3. $-1$
4. None of these

Q. Write the value of $0.2642E05+0.3781E05$.

Q. A spherical ball of ice melts uniformly. When the radius of the ball is $5$ $cm$, find the rate of change of its volume with respect to its radius.

Q. Prove that: ${\cos }^{-1}\frac{4}{5}+{\sin }^{-1}\frac{5}{13}={\cos }^{-1}\frac{33}{65}$.

Q. If $y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+.......\infty }}}$ then prove that $\frac{dy}{dx}=\frac{-\sin x}{(2y-1)}$.