Domain and Range of Basic Inverse Trigonometric Functions

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. Which among the following has the largest domain.

1. $y=cosec\left(cose{c}^{-1}\right(x\left)\right)$
2. $y=cot\left(co{t}^{-1}\right(x\left)\right)$
3. $y=sec\left(se{c}^{-1}\right(x\left)\right)$
4. $y=cos\left(co{s}^{-1}\right(x\left)\right)$

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. 3
2. 0
3. 1
4. 2

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. $\frac{4}{3}$
2. 0
3. 4
4. 2

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

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

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. 1
2. 0
3. 3
4. 4

Q. The domain of $y=si{n}^{-1}(1+3x+2x^2)$ is:

1. $(-\infty ,\infty )$
2. $(-\infty ,-\frac{3}{2})\cup (0,\infty )$
3. $[-\frac{3}{2},0]$
4. $(-\infty ,-\frac{1}{2})\cup (2,\infty )$

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. 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. 3
3. 2
4. 4

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. 2
2. 3
3. None
4. 1