If $x$ is a real number, then the number of solutions of $\tan ^{-1}(\sqrt{x(x+1)})+\sin ^{-1}\left(\sqrt{x^2+x+1}\right)=\frac{\pi}{2}$ is

Domain of the real valued function $f(x)=\log \left(x^2-1\right)+x \operatorname{coth}^{-1} x$ is

In a $\triangle A B C$, if $\sin \frac{A}{2}=\frac{1}{4} \sqrt{\frac{3}{5}}, a=2, c=5$ and $b$ is an integer, then the area (in sq. units) of $\triangle A B C$ is

In a $\triangle A B C$ if $a+c=5 b$, then $\cot \frac{A}{2} \cot \frac{C}{2}=$