Q. What is the value of ${\int }_0^{\frac{\pi }{2}}|\cos x|dx$ ?

Q. $n^{th}$ derivative of $e^{ax}$ is ________.

Q. In perfect correlation both regression lines be _________.

Q. Fraction $\frac{2x+3}{(x+1)(x-3)}=\frac{a}{x+1}+\frac{b}{x-3}$, then $a+b=$.

1. $0$
2. $1$
3. $2$
4. $3$

Q. Find the vector equation of the sphere concentric with the sphere $|\overrightarrow{r}+(\hat{i}-2\hat{j}-3\hat{k})|=5$ and its radius is two times of that sphere.

Q. Match the column.

Q. Write the value of $\int \frac{\cot x}{\log \left(\sin x\right)}dx$.

Q. Write the value of ${\int }_0^{\frac{\pi }{2}}\sin xdx$.

Q. Resolve the following fraction into partial fraction: $\frac{x^2+7x}{x^2+7x-8}$.

Q. If $\frac{1}{x^3-2x^2-x+2}=\frac{A}{x-1}+\frac{B}{x+1}+\frac{C}{x-2}$, then find the value of $A+B+C$.

Q. If $xy=6$, then minimum value of $2x+3y$ is __________.

Q. Evaluate: $\int {\log }_{e}xdx$.

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

Q. Solve: $\int \frac{\sec x}{(\sec x-\tan x)}dx$.

Q. Write the value of $\int e^x(\sin x+\cos x)dx$.

Q. The work done by a force $\overrightarrow{F}=3\hat{i}-4\hat{j}+5\hat{k}$ displaced the body from a point $(3,4,6)$ to a point $(7,2,5)$ is

Q. What is the value of ${\int }_0^{\frac{\pi }{2}}\log \left(\sin x\right)dx$ ?

Q. Direction cosines of $Y$-axis are _______.

Q. Projection of $\overrightarrow{b}$ in the direction of $\overrightarrow{a}$ ___________.

Q. If $y=\log \left[\log \right(\log x\left)\right]$, then value of $\frac{dy}{dx}$ is.

1. $\frac{1}{x\log x}$
2. $\frac{1}{x\log x\log \left(\log x\right)}$
3. $\frac{1}{x\log \left(\log x\right)}$
4. $\frac{1}{x}$

Q. Coordinates of centre of sphere $6x^2+6y^2+6z^2-16x+9z-6=0$ are.

1. $\left(\frac{3}{4},0,\frac{4}{3}\right)$
2. $(-16,0,9)$
3. $\left(-8,0,\frac{9}{2}\right)$
4. $\left(\frac{4}{3},0,\frac{-3}{4}\right)$

Q. $\sin \left[{\sin }^{-1}\frac{1}{2}+{\cos }^{-1}\frac{1}{2}\right]$ is equal to.

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

Q. Evaluate: $\int \sqrt{\frac{1-x}{1+x}}dx$.

Q. Evaluate: $\int \frac{1}{1-\sin x}dx$.

Q. Solve the equation: ${\tan }^{-1}x+2{\cot }^{-1}x=\frac{2\pi }{3}$.

Q. Equation of plane parallel to z-axis.

1. $ax+by+d=0$
2. $by+cz+d=0$
3. $bx+cz+d=0$
4. None of these

Q. If $\overrightarrow{a}=2\hat{i}-5\hat{j}+8\hat{k},\overrightarrow{b}=\hat{i}-3\hat{j}-\hat{k},\overrightarrow{c}=-3\hat{i}-2\hat{j}-\hat{k}$, then find $|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}|$.

Q. $AC$ and $BD$ are the diagonals of a quadrilateral $ABCD$, prove that: $\overrightarrow{AB}+\overrightarrow{DC}=\overrightarrow{AC}+\overrightarrow{DB}$.

Q. If $\overrightarrow{a}=2\hat{i}+5\hat{j}+8\hat{k},\overrightarrow{b}=\hat{i}+3\hat{j}+7\hat{k}$ and $\overrightarrow{c}=3\hat{i}+2\hat{j}+\hat{k}$ then find the value of $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})$.

Q. Find the Vector and Cartesian equation of the sphere whose centre $(2,-3,4)$ and radius is $5$.