Let $f(x)=x^6-2 x^3+x^3+x^2-x-1$ and $g(x)=x^4-x^3-x^2-1$ be two polynomials. Let $a, b, c$ and $d$ be the roots of $g(x)=0$. Then, the value of $f(a)+f(b)+f(c)+f(d)$ is

If $f(x)$ and $g(x)$ are functions satisfying $f(g(x))$ = $x^3 + 3x^2 + 3x + 4$  $f(x)$ = $log^3x + 3$, then slope of the tangent to the curve $y = g(x)$ at $x =  \ -1$ is 

Which pair $(s)$ of function $(s)$ is/are equal ?

where $\{x\}$ and $[x]$ denotes the fractional part $\&$ integral part functions.

Let $f (x) = a^x (a > 0)$ be written as $f( x) = f_1( x) + f_2( x)$ , where $f_1( x)$ is an even function and $f_2( x)$ is an odd function. Then $f_1( x + y) + f_1( x - y )$ equals

Let $N$ be the set of positive integers. For all $n \in N$, let $f_n=(n+1)^{1 / 3}-n^{1 / 3} \text { and }$ $A=\left\{n \in N: f_{n+1}<\frac{1}{3(n+1)^{2 / 3}} < f_n\right\}$ Then,

Domain of the function $f(x) = {\sin ^{ - 1}}\left( {\frac{{2 - |x|}}{4}} \right) + {\cos ^{ - 1}}\left( {\frac{{2 - |x|}}{4}} \right) + {\tan ^{ - 1}}\left( {\frac{{2 - |x|}}{4}} \right)$ is