Inverse Function

Q. The number of positive integral solutions of ${\tan }^{-1}x+{\cos }^{-1}\frac{y}{\sqrt{1+y^2}}={\sin }^{-1}\frac{3}{\sqrt{10}}$ is

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

Q. If $\int \frac{x^2-x+1}{x^2+1}{e}^{{\cot }^{-1}x}dx=f\left(x\right)\cdot e^{{\cot }^{-1}x}+C$, then $f\left(x\right)$ is equal to:
(where $C$ is integration constant)

Q. The value of $2{\tan }^{-1}\left[\sqrt{\frac{a-b}{a+b}}\tan \frac{\theta }{2}\right]$ is equal to

1. ${\cos }^{-1}\left(\frac{a+b\cos \theta }{a\cos \theta +b}\right)$
2. ${\cos }^{-1}\left(\frac{a\cos \theta +b}{a+b\cos \theta }\right)$
3. ${\cos }^{-1}\left(\frac{a\cos \theta }{a+b\cos \theta }\right)$
4. ${\cos }^{-1}\left(\frac{b\cos \theta }{a+b\cos \theta }\right)$

Q. The graph of inverse function can be obtained from the graph of its corresponding function by interchanging x and y axes.

1. True
2. False

Q. If ${\cot }^{-1}\frac{n^2-10n+21.6}{\pi }>\frac{\pi }{6},n\in N$, then $n$ can be

1. $3$
2. $2$
3. $4$
4. $8$

Q. $\mathrm{If}{\cos }^{-1}x+{\cos }^{-1}y+{\cos }^{-1}z=3\pi ,$then the value of $xy+yz+zx$ =

Q. The range of the function $f\left(x\right)={\sin }^{-1}\left(\frac{x^2}{1+x^2}\right),x\in R$ is equal to :

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