Q.

Q. Differentiate ${\tan }^{-1}\frac{x}{\sqrt{a^2-x^2}}$, with respect to $x$.

Q. Differential coefficient of $\sin x^3$ with respect to $x^3$ is _____.

Q. Find the value of $\int x\sin xdx$.

Q. Prove that:
${\sin }^{-1}\frac{3}{5}+{\sin }^{-1}\frac{8}{17}={\sin }^{-1}\frac{77}{85}$

Q. Find the angle between two vectors $(\overrightarrow{a}=2\hat{i}+\hat{j}-3\hat{k})$ and $(\overrightarrow{b}=3\hat{i}-2\hat{j}-\hat{k})$.

Q. The partial fraction form of $\frac{1}{x^2+2x}$ is:

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

Q. The value of $2{\cos }^{-1}x$ is:

1. ${\cos }^{-1}(2x^2-1)$
2. ${\cos }^{-1}\frac{2x}{1+x^2}$
3. ${\cos }^{-1}(2x^2+1)$
4. ${\tan }^{-1}\frac{2x}{1-x^2}$

Q. Match the correct pair

Q. Equation of the plane which intercepts unit length from coordinate axes is _____.

Q. The value of ${\int }_a^b\cos xdx$ is:

1. $-\sin b+\sin a$
2. $\sin b+\sin a$
3. $\cos b-\cos a$
4. $\sin b-\sin a$

Q. The value of $\frac{d}{dx}\sin x^2$ is:

1. $\cos x^2$
2. $x\sin x^2$
3. $2x\cos x^2$
4. $x\cos x^2$

Q. If $y=\frac{1-\cos x}{1+\cos x}$, then find the value of $\frac{dy}{dx}$.

Q. Resolve $\frac{11-2x}{x^2+9x+14}$ partial fractions.

Q. Find the value of $\int \frac{2x+1}{\sqrt{x^2+x+1}}dx$

Q. If $\overrightarrow{a}=a_1\hat{i}+a_2\hat{j}-a_3\hat{k}$ and $\overrightarrow{b}=b_1\hat{i}+b_2\hat{j}-b_3\hat{k}$ then find the value of $\overrightarrow{a}\times \overrightarrow{b}$.

Q. Projection of vector $\overrightarrow{a}$ on $\overrightarrow{b}$:

1. $\frac{\overrightarrow{a}.\overrightarrow{b}}{\left|\overrightarrow{b}\right|}$
2. $\frac{\overrightarrow{a}}{\left|\overrightarrow{a}\right|}$
3. $\frac{\overrightarrow{a}\times \overrightarrow{b}}{\left|\overrightarrow{b}\right|}$
4. $\frac{\overrightarrow{a}.\overrightarrow{b}}{\left|\overrightarrow{a}\right|}$

Q. Find the interval in which the root of equation $x^3-2x-5=0$ lies?

Q. Find the vector equation of the sphere with centre $(\hat{i}+2\hat{j}-3\hat{k})$ and radius $5$ units.

Q. Find the direction cosines of the vector $6\hat{i}+2\hat{j}-3\hat{k}$.

Q. Write the conditions for intersecting the two lines $\overrightarrow{r}=\overrightarrow{a_1}+l\overrightarrow{b_1}$ and $\overrightarrow{r}=\overrightarrow{a_2}+l\overrightarrow{b_2}$.

Q. Find the value of $\int \frac{1}{e^x+1}dx$.

Q. Prove that:
${\tan }^{-1}\frac{1}{4}+{\tan }^{-1}\frac{2}{9}=\frac{1}{2}{\cos }^{-1}\frac{3}{5}$

Q. Resolve $\frac{x}{1+x^3}$ into partial fractions.

Q. The acute angle between the planes $2x-y+z=6$ and $x+y+2z=3$ is _____.

Q. On which principle is Simpson's rule based?

Q. Write Simpson's one-third rule formula in numerical methods.

Q. If $\overrightarrow{a}=2\hat{i}+l\hat{j}-3\hat{k}$ and $\overrightarrow{b}=4\hat{i}-3\hat{j}-2\hat{k}$ are perpendicular to each other then find the value of scalar $\lambda$..

Q. The equation of plane parallel to $x$-axis is _____.

Q. The perpendicular distance of point $(18,5,12)$ from $x$-axis is _____.