Domain and Range of Basic Inverse Trigonometric Functions

Q.

Find the value of $\tan 480°$.

Q. If $\left[si{n}^{-1}x\right]>\left[co{s}^{-1}x\right]$ where [.] denotes the greatest integer function, then the set of values of x is:

1. [cos 1, 1]
2. [cos 1, sin 1]
3. [sin 1, 1]
4. None of these

Q. Value of $si{n}^{-1}\left(sin\frac{3\pi }{4}\right)$

1. $\frac{\pi }{4}$
2. $\frac{-\pi }{4}$
3. $\frac{3\pi }{4}$
4. None

Q.

The greatest and least value of ${\left({\sin }^{-1}x\right)}^2+{\left({\cos }^{-1}x\right)}^2$are respectively

1. $\frac{{\mathrm{\pi }}^2}{4}$ and $0$

2. $\frac{{\mathrm{\pi }}^2}{4}$ and $-\frac{\mathrm{\pi }}{2}$

3. $\frac{5{\mathrm{\pi }}^2}{4}$ and $\frac{{\mathrm{\pi }}^2}{8}$

4. $\frac{{\mathrm{\pi }}^2}{4}$ and $-\frac{{\mathrm{\pi }}^2}{4}$

Q. If $f:[0,4\pi ]\to [0,\pi ]$ be defined by $f\left(x\right)=co{s}^{-1}\left(cosx\right)$Then, the number of points $xϵ[0,4\pi ]$ satisfying the equation $f\left(x\right)=\frac{10-x}{10}$, is ___

Q. The number of solutions of the equation $ta{n}^{-1}\left(\frac{x}{1-x^2}\right)+ta{n}^{-1}\left(\frac{1}{x^3}\right)=\frac{3\pi }{4}$ belonging to the interval (0, 1) is

1. \N
2. 1
3. 2
4. 3

Q. Find maximum value of x for which $2ta{n}^{-1}x+co{s}^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ is independent of x.

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

Q. Which of the following is the principal value of ${\text{cosec}}^{-1}x$

1. $(0,\pi )-\left\{\frac{\pi }{2}\right\}$
2. $(-\frac{\pi }{2},\frac{\pi }{2})$
3. $[-\frac{\pi }{2},\frac{\pi }{2}]$
4. $[-\frac{\pi }{2},\frac{\pi }{2}]-\left\{0\right\}$

Q.

The greatest and least values of $(si{n}^{-1}x{)}^3+(co{s}^{-1}x{)}^3$ are

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

2. $\frac{-{\pi }^3}{2},\frac{{\pi }^3}{2}$

3. $\frac{{\pi }^3}{32},\frac{7{\pi }^3}{8}$

4. None

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

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

Q. The complete solution set of the inequality $[co{s}^{-1}x{]}^2-6[co{t}^{-1}x]+\le 0$, where [.] denotes the greatest integer function, is :

1. $(-\infty ,cot3]$
2. $[cot3,cot2]$
3. $[cot3,\infty ]$
4. $Noneofthese$

Q.

The value of $co{s}^{-1}x+co{s}^{-1}(\frac{x}{2}+\frac{1}{2}\sqrt{3-3x^2})$ where $(\frac{1}{2}\le x\le 1)$ is equal to

1. $\frac{\pi }{6}$
2. $\frac{\pi }{3}$
3. $\pi$
4. $0$

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. 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. If $xϵ[-1,1],thenrangeofta{n}^{-1}(-x)$ is

1. $[\frac{3\pi }{4},\frac{7\pi }{4}]$
2. $[\pi ,0]$
3. $[\frac{3\pi }{4},\frac{5\pi }{4}]$
4. $[\frac{-\pi }{4},\frac{\pi }{4}]$

Q. If $y=si{n}^{-1}x+ta{n}^{-1}x+se{c}^{-1}x$, then set of values of y is

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

Q. The exhaustive domain of the function ${\sin }^{-1}\left[{\log }_2\right(x^2/2\left)\right]$ is

1. $[0,2]$
2. $[-2,-1]\cup [1,2]$
3. $[1,2]$
4. $[-2,2]$

Q. Match the following by appropriately matching the lists based on the information given in $\text{Column I}$ and $\text{Column II}$.

$\begin{array}{cc}ColumnI& ColumnII\\ \text{a. Range of}f\left(x\right)={\sin }^{-1}x+{\cos }^{-1}x+{\cot }^{-1}x\text{is}& \text{p.}[0,\frac{\pi }{2})\cup (\frac{\pi }{2},\pi ]\\ \text{b. Range of}f\left(x\right)={\cot }^{-1}x+{\tan }^{-1}x+{\text{cosec}}^{-1}x\text{is}& \text{q.}[\frac{\pi }{2},\frac{3\pi }{2}]\\ \text{c. Range of}f\left(x\right)={\cot }^{-1}x+{\tan }^{-1}x+{\cos }^{-1}x\text{is}& \text{r.}\{0,\pi \}\\ \text{d. Range of}f\left(x\right)={\sec }^{-1}x+{\text{cosec}}^{-1}x+{\sin }^{-1}x\text{is}& \text{s.}[\frac{3\pi }{4},\frac{5\pi }{4}]\end{array}$

1. $a-p;b-q;c-r;d-s$
2. $a-q;b-s;c-p;d-r$
3. $a-s;b-r;c-q;d-p$
4. $a-s;b-p;c-q;d-r$

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

1. \N
2. 1
3. 2
4. 3

Q. If $si{n}^{-1}x+si{n}^{-1}y+si{n}^{-1}z=\frac{3\pi }{2}$ and $f\left(1\right)=2,f(a+b)=f\left(a\right)f\left(b\right)$ for all $a,b,ϵR$, then
$x^{f\left(1\right)}+y^{f\left(2\right)}+z^{f\left(3\right)}-\frac{x+y+z}{x^{f\left(1\right)}+y^{f\left(2\right)}+z^{f\left(3\right)}}$ is equal to :

1. 10
2. 11
3. 2
4. 3

Q. Number of common points for the curves $y=si{n}^{-1}\left(2x\right)+ta{n}^{-1}\left(\frac{1}{\left[2x\right]}\right)+2$ and $y=co{s}^{-1}(2x+5)+1$ is (where [.] denotes greatest integer function)

1. \N
2. 1
3. 3
4. 4

Q. The domain of the function ${\cos }^{-1}(3x-2)$ is

1. $\left(\frac{1}{3},1\right]$
2. $\left(-1,\frac{1}{3}\right]$
3. $\left(-1,1\right]$
4. $\left(-\frac{1}{3},\frac{1}{3}\right]$

Q. If $co{s}^{-1}x+co{s}^{-1}y+co{s}^{-1}z=3\pi$, then $xy+yz+zx=$

1. \N
2. 1
3. 3
4. -3

Q. Number of integer(s) for which the function
$f\left(x\right)={\sin }^{-1}\left({\log }_2\left(\frac{x}{3}\right)\right)$ is defined is

Q. The number of real solutions of $ta{n}^{-1}\sqrt{x(x+1)}+si{n}^{-1}\sqrt{x^2+x+1}=\frac{\pi }{2}$ is equal to :

1. 1
2. 2
3. 3
4. None

Q. If $si{n}^{-1}x+si{n}^{-1}y+si{n}^{-1}z=\frac{3\pi }{2}$
then $\frac{{\sum }_{r=1}^2(x^{100r}+y^{103r})}{\sum x^{201}{y}^{201}}$ =

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

Q. If $({\sin }^{-1}x+{\sin }^{-1}w)({\sin }^{-1}y+{\sin }^{-1}z)={\pi }^2$, then $D=\left|\begin{array}{cc}{x}^{N_1}& y^{N_2}\\ z^{N_3}& w^{N_4}\end{array}\right|$, $(N_1,N_2,N_3,N_4\in N)$

1. has a maximum value of $2$
2. has a minimum value of $0$
3. $16$ different $D$ are possible
4. has a minimum value of $-2$

Q.

Prove the following identity: $\frac{\cot \theta }{cosec\theta +1}+\frac{cosec\theta +1}{\cot \theta }=2\sec \theta$

Q. Greatest value of $(si{n}^{-1}x{)}^3+(co{s}^{-1}x{)}^3$ is :

1. $\frac{7{\pi }^3}{32}$
2. $\frac{{\pi }^3}{32}$
3. $\frac{7{\pi }^3}{8}$
4. None