Derivative of Standard Inverse Trigonometric Functions

Q.

$\int \frac{(x^4+x^2+1)}{(x^2-x+1)}dx=?$

1. $\frac{x^3}{3}-\frac{x^2}{2}+x+c$

2. $\frac{x^3}{3}+\frac{x^2}{2}+x+c$

3. $\frac{x^3}{3}-\frac{x^2}{2}-x+c$

4. $\frac{x^3}{3}+\frac{x^2}{2}-x+c$

Q.

The domain of the function defined by $f\left(x\right)=si{n}^{-1}\sqrt{x-1}$ is

(a) $[1,2]$ (b) $[-1,1]$ (c) $[0,1]$ (d) None of these

Q.

If ${\cos }^{-1}x-{\cos }^{-1}\left(\frac{y}{2}\right)=\alpha$, then $4x^2-4xy\cos \alpha +y^2=$

1. $4{\sin }^2\alpha$

2. $-4{\sin }^2\alpha$

3. $2\sin 2\alpha$

4. $4$

Q. Let $f\left(x\right)=\frac{\sin x}{\left[\frac{x}{\pi }\right]+\frac{1}{2}}$, where $\left[x\right]$ denotes greatest integer function, then $f\left(x\right)$ is

1. odd function
2. even function
3. neither odd or even
4. both odd and even function

Q.

Evaluate:$\int (x+1){(x+2)}^7(x+3)dx$

1. $\left[\frac{{(x+2)}^{10}}{10}\right]–\left[\frac{{(x+2)}^8}{8}\right]+c$

2. $\left[\frac{{(x+2)}^2}{2}\right]–\left[\frac{{(x+2)}^8}{8}\right]-\left[\frac{{(x+3)}^2}{2}\right]+c$

3. $\left[\frac{{(x+1)}^{10}}{10}\right]+c$

4. $\left[\frac{{(x+1)}^2}{2}\right]+\left[\frac{{(x+2)}^8}{8}\right]+\left[\frac{{(x+3)}^2}{2}\right]+c$

5. $\left[\frac{{(x+2)}^9}{9}\right]–\left[\frac{{(x+2)}^7}{7}\right]+c$

Q.

$\sin \left(4{\tan }^{-1}\left(\frac{1}{3}\right)\right)=$

1. $\frac{12}{25}$

2. $\frac{24}{25}$

3. $\frac{1}{5}$

4. None of these

Q. If $\underset{x\to a}{lim}\left[{\sin }^{-1}\frac{2x}{1+x^2}\right]$ doesn't exist, then the number of possible value(s) of $a$ is
(Here, $[.]$ denotes the greatest integer function)

Q. If $f\left(x\right)=\frac{2{\sin }^{2}x+2\sin x+3}{{\sin }^{2}x+\sin x+1}$, then the number of integers in the range of $f\left(x\right)$ is

Q.

Let $f\left(x\right)={\int }_0^x{e}^{t}f\left(t\right)dt+e^x$ be a differentiable function for all $x\in R$. Then $f\left(x\right)$ equals:

1. $2e^{e^{x-1}}-1$

2. $e^{e^{x-1}}-1$

3. $2e^{e^x}-1$

4. $e^{e^x}-1$

Q.

$\int \left(1+x-\frac{1}{x}\right)e\left({}^{x+\frac{1}{x}}\right)dx=$

1. $(x-1)e^{\left(x+\frac{1}{x}\right)}+C$

2. $x{e}^{\left(x+\frac{1}{x}\right)}+C$

3. $(x+1)e^{\left(x+\frac{1}{x}\right)}+C$

4. $(-x)e^{\left(x+\frac{1}{x}\right)}+C$

Q.

If $\int \left[\log \left(\log \left(x\right)\right)+\frac{1}{{\left(\log \left(x\right)\right)}^2}\right]dx=x\left[f\left(x\right)-g\left(x\right)\right]+c$then

1. $f\left(x\right)=\log \left(\log \left(x\right)\right);g\left(x\right)=\frac{1}{\log \left(x\right)}$

2. $f\left(x\right)=\log \left(x\right);g\left(x\right)=\frac{1}{\log \left(x\right)}$

3. $f\left(x\right)=\frac{1}{x\log \left(x\right)};g\left(x\right)=\log \left(\log \left(x\right)\right)$

4. $f\left(x\right)=\frac{1}{x\log \left(x\right)};g\left(x\right)=\frac{1}{\log \left(x\right)}$

Q.

Find imaginary part of ${\sin }^{-1}(\cos ec\theta )$

1. $\log \left(cot\frac{\theta }{2}\right)$

2. $\frac{\mathrm{\pi }}{2}$

3. $\frac{1}{2}\log \left(cot\frac{\theta }{2}\right)$

4. None of these

Q. The value of $\underset{n\to \infty }{\text{lim}}6\tan \left\{\sum _{r=1}^n{\tan }^{-1}\left(\frac{1}{r^2+3r+3}\right)\right\}$ is equal to :

1. $2$
2. $6$
3. $1$
4. $3$

Q. $\underset{0}{\overset{102}{\int }}\left[{\tan }^{-1}x\right]dx$ is equal to
(where $[\cdot ]$ denotes the greatest integer function)

1. $102-\tan 1$
2. $101$
3. $102+\tan 1$
4. $102-\frac{\pi }{4}$

Q. In $△ABC$ with usual notations, if $2a^2+4b^2+c^2-4ab-2ac=0$, then $\cos A+\cos B+\cos C$ is equal to:

1. $\frac{1}{4}$
2. $\frac{1}{2}$
3. $\frac{7}{8}$
4. $\frac{11}{8}$

Q.

Verify the following identity $\tan \left(3x\right)=\frac{3\tan \left(x\right)-{\tan }^3\left(x\right)}{1-3{\tan }^2\left(x\right)}$

Q. Solve:$\underset{x\to 0}{lim}\frac{{\tan }^{-1}x}{x}.$ solve.

Q. Evaluate the following integrals:

${\int }_{-\mathrm{\pi }}^{\mathrm{\pi }}\frac{2x\left(1+\sin x\right)}{1+{\cos }^{2}x}dx$

Q.

$co{s}^{-1}(-\frac{1}{2})-2si{n}^{-1}\left(\frac{1}{2}\right)+3co{s}^{-1}(-\frac{1}{\sqrt{2}})-4ta{n}^{-1}(-1)$ equals to

1. 19π/12

2. 35π/12

3. 47π/12

4. 43π/12

Q. If $\int \frac{x-1}{(x+1)\sqrt{x^3+x^2+x}}dx=K{\tan }^{-1}\left(f\right(x\left)\right)+C,$ then which of the following is/are true?
$($Where $K$ is fixed constant and $C$ is integration constant$)$

1. $K=2$
2. $f\left(x\right)=(x+\frac{1}{x}+1)$
3. $f\left(x\right)=\sqrt{x+\frac{1}{x}+1}$
4. $K=\frac{1}{\sqrt{2}}$

Q. Match the given functions in the first column with their first derivatives.

1. $\frac{1}{\sqrt{1-x^2}}$
2. $\frac{-1}{\sqrt{1-x^2}}$
3. $\frac{1}{1+x^2}$
4. $\frac{-1}{\left(x\right|\sqrt{x^2-1}}$

Q. Evaluate each of the following:

$\left(\mathrm{i}\right){\tan }^{-1}1+{\cos }^{-1}\left(-\frac{1}{2}\right)+{\sin }^{-1}\left(-\frac{1}{2}\right)$
(ii) ${\tan }^{-1}\left(-\frac{1}{\sqrt{3}}\right)+{\tan }^{-1}\left(-\sqrt{3}\right)+{\tan }^{-1}\left(\sin \left(-\frac{\mathrm{\pi }}{2}\right)\right)$

(iii) ${\tan }^{-1}\left(\tan \frac{5\mathrm{\pi }}{6}\right)+{\cos }^{-1}\left\{\cos \left(\frac{13\mathrm{\pi }}{6}\right)\right\}$

Q. Match the following functions to their derivatives?
$\begin{array}{cc}Function& Derivatives\\ a)si{n}^{-1}x& 1)\frac{-1}{|x|\sqrt{x^2-1}}\\ b)co{s}^{-1}x& 2)\frac{-1}{1+x^2}\\ c)ta{n}^{-1}x& 3)\frac{1}{|x|\sqrt{x^2-1}}\\ d)se{c}^{-1}x& 4)\frac{1}{1+x^2}\\ e)co{t}^{-1}x& 5)\frac{-1}{\sqrt{1-x^2}}\\ f)cose{c}^{-1}x& 6)\frac{1}{\sqrt{1-x^2}}\end{array}$

1. a-6, b-5, c-4, d-3, e-2, f-1
2. a-5, b-6, c-4, d-3, e-2, f-1
3. a-6, b-5, c-3, d-4, e-2, f-1
4. a-6, b-5, c-3, d-2, e-1, f-2

Q. $\int {\tan }^2(3x+5)dx$ is equal to (where $C$ is the constant of integration)

Q. If $f\left(x\right)=\sum _{i=1}^{2020}{\cos }^{-1}{\alpha }_i=0$, then $\sum _{i=1}^4{\alpha }_i=$

Q.

How do you simplify the expression $1-se{c}^{2}x$?

Q. If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ be unit vectors such that $\overrightarrow{a}\cdot \overrightarrow{b}=\overrightarrow{a}\cdot \overrightarrow{c}=0$ and the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is $\frac{\pi }{6}$ then $\overrightarrow{a}=$

1. $\pm (\overrightarrow{a}\times \overrightarrow{c})$
2. $\pm 2(\overrightarrow{a}\times \overrightarrow{c})$
3. $\pm (\overrightarrow{b}\times \overrightarrow{c})$
4. $\pm 2(\overrightarrow{b}\times \overrightarrow{c})$

Q. Let [a] denote the greatest integer which is less than or equal to a. Then the value of the integral ${\int }_{\frac{\pi }{2}}^{\frac{\pi }{2}}\left[sinxcosx\right]dx$ is

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

Q. Find the range if $\left[2\sin x\right]+\left[\cos x\right]=-3,$ then the range of the function $f\left(x\right)=\sin x+\sqrt{3}\cos x$ in $[0,2\pi ]$ (where $[.]$ denotes the greatest integer function)

1. $(2,-1)$
2. $(-1,-\frac{1}{2})$
3. $(-2,-1)$
4. None of these

Q. If $A=\sum _{r=0}^n{}^n{C}_r{\cos }^{n}x\cdot {\tan }^{r}x$, then which of the following option(s) is/are correct?

1. Maximum value of $A$ is $(\sqrt{2}{)}^n$
2. Minimum value of $A$ is $(-\sqrt{2}{)}^n$ when $n\in N$.
3. Minimum value of $A$ is $(-\sqrt{2}{)}^n$ when $n$ is odd.
4. Minimum value of $A$ is $0$ when $n$ is even.