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

Let $f: R \rightarrow R$ be defined by $f(x)=5^{-|x|}+\operatorname{sgn}\left(5^{-x}\right)$, where sgn $x$ denotes signum function of $x$. Then $f$ is

If the range of the real valued function $f(x)=\frac{x^2+x+k}{x^2-x+k}$ is $\left[\frac{1}{3}, 3\right]$, then $k=$

For a real number ' $a$ ', if a real valued function $f(x)=4 x^3+a x^2+3 x-2$ is monotonic in its domain, then the range of ' $a$ ' is