Domain and Range of Basic Inverse Trigonometric Functions

Q.

If $\sin \theta -\cos \theta =\frac{1}{2}$then find the value of $\sin \theta +\cos \theta$.

Q. The number of integral value(s) of $k$ for which ${\sin }^{2}x+(\cos x-1)\sin x-\cos x-k\sin x+k=0$ have exactly three real roots, where $x\in (0,2\pi )$, is

Q.

Prove that $\frac{(\tan A+\sin A)}{(\tan A-\sin A)}=\frac{(secA+1)}{(secA-1)}$.

Q.

${\cos }^{-1}\left(\frac{15}{17}\right)+2{\tan }^{-1}\left(\frac{1}{5}\right)=$

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

2. ${\cos }^{-1}\left(\frac{171}{221}\right)$

3. $\frac{\pi }{4}$

4. none of these

Q.

Determine the principal value of ${\cos }^{-1}\left(-\frac{1}{2}\right)$.

Q. The domain of ${\text{cosec}}^{-1}x$ is

1. $R-[-1,1]$
2. $R-(-1,1)$
3. $[-1,1]$
4. $R$

Q.

Evaluate:$\sin \frac{\mathrm{\pi }}{4}$

Q.

If $\frac{{\sin }^{4}A}{a}+\frac{{\cos }^{4}A}{b}=\frac{1}{(a+b)}$, then the value of $\left(\frac{{\sin }^{8}A}{a^3}\right)+\left(\frac{{\cos }^{8}A}{b^3}\right)$ is equal to

1. $\frac{1}{{\left(a+b\right)}^3}$

2. $\frac{a^3{b}^3}{(a+b{)}^3}$

3. $\frac{a^2{b}^2}{(a+b{)}^2}$

4. None of these

Q.

If $\mathrm{f}\left(\mathrm{x}\right)=\frac{(1-\mathrm{x})}{(1+\mathrm{x})}$then $\mathrm{f}\left(\mathrm{f}\right(\cos 2\mathrm{\theta }\left)\right)=$

1. $\tan \left(2\mathrm{\theta }\right)$

2. $\sec \left(2\mathrm{\theta }\right)$

3. $\cos \left(2\mathrm{\theta }\right)$

4. $\cot \left(2\mathrm{\theta }\right)$

Q. $\cos [2{\sin }^{-1}\frac{3}{4}+{\cos }^{-1}\frac{3}{4}]=$

1. $\frac{3}{5}$
2. $\frac{-3}{4}$
3. does not exist
4. $\frac{3}{4}$

Q.

$cotx-\tan x=$

1. $cot2x$

2. $2cot2x$

3. $2cot2x$

4. $co{t}^2\left(2x\right)$

Q. Considering only the principal values of the inverse trigonometric functions, the domain of the function $f\left(x\right)={\cos }^{-1}\left(\frac{x^2-4x+2}{x^2+3}\right)$ is

1. $[-\frac{1}{4},\infty )$
2. $(\frac{-1}{3},\infty )$
3. $(-\infty ,\frac{1}{3}]$
4. $(-\infty ,\frac{1}{4}]$

Q.

If $L={\sin }^2\frac{\mathrm{\pi }}{16}-{\sin }^2\frac{\mathrm{\pi }}{8}$ and $M={\cos }^2\frac{\mathrm{\pi }}{16}-{\sin }^2\frac{\mathrm{\pi }}{8}$, then

1. $M=\frac{1}{2\sqrt{2}}+\frac{1}{2}\cos \frac{\mathrm{\pi }}{8}$

2. $M=\frac{1}{4\sqrt{2}}+\frac{1}{4}\cos \frac{\mathrm{\pi }}{8}$

3. $L=-\frac{1}{2\sqrt{2}}+\frac{1}{2}\cos \frac{\mathrm{\pi }}{8}$

4. $L=\frac{1}{4\sqrt{2}}-\frac{1}{4}\cos \frac{\mathrm{\pi }}{8}$

Q.

If $\sin A+\sin B+\sin C=3$, then $\cos A+\cos B+\cos c$ is equal to

1. $3$

2. $2$

3. $1$

4. $0$

Q.

The range of ${\tan }^{-1}x$ is:

1. $\left(\mathrm{\pi },\frac{\mathrm{\pi }}{2}\right)$

2. $\left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$

3. $(-\pi ,\pi )$

4. $(0,\pi )$

Q.

If $\begin{array}{rcl}{\tan }^{-1}\left(\frac{1}{3}\right)+{\tan }^{-1}\left(\frac{3}{4}\right)-{\tan }^{-1}\left(\frac{x}{3}\right)& =& 0\end{array}$, then $x$ is equal

1. $\frac{7}{3}$

2. $3$

3. $\frac{11}{3}$

4. $\frac{13}{3}$

Q. Let $z_k=\cos \left(\frac{2k\pi }{10}\right)+i\sin \left(\frac{2k\pi }{10}\right);k=1,2,\cdots ,9.$

$\begin{array}{cccc}& List\left(I\right)& & List\left(II\right)\\ \text{P}.& \text{For each}{z}_k\text{there exist a}{z}_j& \left(1\right)& \text{True}\\ & \text{such that}{z}_k\cdot z_j=1& & \\ \text{Q}.& \text{There exists a}k\in \{1,2,\cdots ,9\}& & \\ & \text{such that}{z}_1\cdot z=z_k\text{has no solution}& \left(2\right)& \text{False}\\ & z\text{in the set of complex numbers.}& & \\ \text{R}.& \frac{|1-z_1||1-z_2|\cdots |1-z_9|}{10}\text{equals}& \left(3\right)& 1\\ \text{S}.& 1-\sum _{k=1}^9\cos \left(\frac{2k\pi }{10}\right)\text{equals}& \left(4\right)& 2\end{array}$

Which of the following option is correct?

1. $\left(P\right)\to \left(1\right),\left(Q\right)\to \left(2\right)\left(R\right)\to \left(4\right),\left(S\right)\to \left(3\right)$
2. $\left(P\right)\to \left(2\right),\left(Q\right)\to \left(1\right)\left(R\right)\to \left(3\right),\left(S\right)\to \left(4\right)$
3. $\left(P\right)\to \left(1\right),\left(Q\right)\to \left(2\right)\left(R\right)\to \left(3\right),\left(S\right)\to \left(4\right)$
4. $\left(P\right)\to \left(2\right),\left(Q\right)\to \left(1\right)\left(R\right)\to \left(4\right),\left(S\right)\to \left(3\right)$

Q.

The solution set of the equation ${\sin }^{-1}x=2{\tan }^{-1}x$ is

1. $\left\{1,2\right\}$

2. $\left\{-1,2\right\}$

3. $\left\{-1,1,0\right\}$

4. $\left\{1,\frac{1}{2},0\right\}$

Q. Evaluate the limit:

$\underset{x\to 0}{lim}\frac{x^3\cot x}{1-\cos x}$

Q.

If $\alpha ,\beta$are roots of the equation $6x^2+11x+3=0$ then

1. Both ${\cos }^{-1}\alpha$ and ${\cos }^{-1}\beta$ are real

2. Both ${\text{cosec}}^{-1}\alpha$ and ${\text{cosec}}^{-1}\beta$ are real

3. Both ${\cot }^{-1}\alpha$ and ${\cot }^{-1}\beta$ are real

4. None of the above

Q.

What is the domain and range of $y=\mathrm{cosec}x$?

Q.

$si{n}^{-1}x+si{n}^{-1}y+si{n}^{-1}z=\frac{3\pi }{2},$ then the value of $x^{100}+y^{100}+z^{100}-\frac{9}{x^{101}+y^{101}+z^{101}}=$

1. 0
   

2. 1
   

3. 2
   

4. 13

Q.

If $f\left(x\right)={\cos }^{-1}x+{\cos }^{-1}\left\{\frac{x}{2}+\frac{1}{2}\sqrt{3-3x^2}\right\}$, then

1. $f\left(\frac{2}{3}\right)=\frac{\mathrm{\pi }}{3}$

2. $f\left(\frac{2}{3}\right)=2{\cos }^{-1}\left(\frac{2}{3}\right)-\frac{\mathrm{\pi }}{3}$

3. $f\left(\frac{1}{3}\right)=\frac{\mathrm{\pi }}{3}$

4. $f\left(\frac{1}{3}\right)=\frac{\mathrm{\pi }}{3}+2{\cos }^{-1}\left(\frac{1}{3}\right)$

Q.

The differential coefficient of $a^x+\log x\sin x$ is:

1. $a^x\log a+\frac{\sin x}{x}+\log x\cos x$

2. $a^x+\frac{\sin x}{x}+\log x\cos x$

3. $a^x\log a+\frac{\cos x}{x}+\log x\sin x$

4. None of these

Q.

Show that $3{\tan }^{-1}\mathrm{x}=\frac{{\tan }^{-1}(3\mathrm{x}-{\mathrm{x}}^3)}{(1-3{\mathrm{x}}^2)}$

Q. If $f\left(x\right)=\left|\begin{array}{ccc}\sin x& \cos x& \tan x\\ x^3& x^2& x\\ 2x& 1& 1\end{array}\right|,$ then the value of $\underset{x\to 0}{lim}\left(\frac{f\left(x\right)}{x^2}\right)$

Q.

The minimum value of $2^{\sin x}+2^{\cos x}$ is

1. $2^{\left(1-\frac{1}{\sqrt{2}}\right)}$

2. $2^{\left(1+\frac{1}{\sqrt{2}}\right)}$

3. $2^{\sqrt{2}}$

4. $2$

Q.

What is the maximum value of the function $sinx+cosx$?

Q. Total number of positive integral value of `n' such that the equations
$co{s}^{-1}x+(si{n}^{-1}y{)}^2=\frac{n{\pi }^2}{4}$ and
$(si{n}^{-1}y{)}^2-co{s}^{-1}x=\frac{{\pi }^2}{16}$
are consistent, is equal to :

1. 1
2. 4
3. 3
4. 2

Q.

Which of the following is the principal value branch of $cose{c}^{-1}x?$

(a) $(\frac{-\pi }{2},\frac{\pi }{2})$ (b) $[0,\pi ]-\left\{\frac{\pi }{2}\right\}$ (c) $[\frac{\pi }{2},\frac{\pi }{2}]$ (d) $[\frac{-\pi }{2},\frac{\pi }{2}]-\left[0\right]$