If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $P ( x )=f\left( x ^{3}\right)+ xg \left( x ^{3}\right)$ is divisible by $x^{2}+x+1,$ then $P(1)$ is equal to ....... .

The domain of ${\sin ^{ - 1}}\left[ {{{\log }_3}\left( {\frac{x}{3}} \right)} \right]$ is

Which of the following is true 

Let ${f_k}\left( x \right) = \frac{1}{k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)\;,x \in R$ and $k \ge 1$, then ${f_4}\left( x \right) - {f_6}\left( x \right)$ is equal to

Which one of the following best represent the graph of $y = \frac{|x-x^2|}{x^2-x}$ ? 

The domain of the derivative of the function $f(x) = \left\{ \begin{array}{l}{\tan ^{ - 1}}x\;\;\;\;\;,\;|x|\; \le 1\\\frac{1}{2}(|x|\; - 1)\;,\;|x|\; > 1\end{array} \right.$ is