If $2^{4 n+3}+3^{3 n+1}$ is divisible by $P$ for all natural numbers $n$, then $P$ is

Consider the following statements

Statement $\mathrm{I} \cosh ^{-1} x=\tanh ^{-1} x$ has no solution

Statement II $\cosh ^{-1} x=\operatorname{coth}^{-1} x$ has only one solution

The correct answer is

The domain of the real valued function $f(x)=\log _{\sqrt{2}}\left(\sqrt{x^2+x}+\sqrt{x^2-x}\right)$ is

If $\frac{x+1}{x^3(x-1)}=\frac{a}{x}+\frac{b}{x^2}+\frac{c}{x^3}+\frac{d}{x-1}$, then