Domain and Range of Inverse Functions

The number of solutions of the equation;
$1+a^2+2a\sin \left({\cos }^{-1}b\right)=0$ is:

If $a,b$ are the roots of the equation; 

$6x^2+11x+3=0$

The number of real solutions of ${\cos }^{-1}a+{\cos }^{-1}2a=-\pi$ is:

Let $f\left(x\right)={cosec}^{-1}\left[1+{\sin }^{2}x\right],$ where $\left[·\right]$ denotes the greatest integer function. Then $f\left(x\right)$ equals;

Find the number of real solutions: ${\tan }^{-1}\sqrt{x(x+1)}+{\sin }^{-1}\sqrt{x^2+x+1}=\frac{\mathrm{\pi }}{2}$

Find the value of $x$ for which;  ${\sec }^{-1}x+{\tan }^{-1}x$ is real.

Find the value of $x$ for which; ${\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)=2{\tan }^{-1}x$ satisfied.

Find the value of $x$ for which; ${cosec}^{-1}\left(\cos x\right)$ is real.

Find the value of $x$ for which; ${\cos }^{-1}\left(\cos x\right)=x$ is satisfied.