Let $f(x ) = x^3 - 2x + 2$. If real numbers $a$, $b$ and $c$ such that $\left| {f\left( a \right)} \right| + \left| {f\left( b \right)} \right| + \left| {f\left( c \right)} \right| = 0$ then the value of ${f^2}\left( {{a^2} + \frac{2}{a}} \right) + {f^2}\left( {{b^2} + \frac{2}{b}} \right) - {f^2}\left( {{c^2} + \frac{2}{c}} \right)$ equal to

Let $A = \{ {x_1},\,{x_2},\,............,{x_7}\} $ and $B = \{ {y_1},\,{y_2},\,{y_3}\} $ be two sets containing seven and three distinct elements respectively. Then the total number of functions $f : A \to B$ that are onto, if there exist exactly three elements $x$ in $A$ such that $f(x)\, = y_2$, is equal to

If for the function $f(x) = \frac{1}{4}{x^2} + bx + 10$ ; $f\left( {12 - x} \right) = f\left( x \right)\,\forall \,x\, \in \,R$ , then the value of $'b'$ is 

Let $c, k \in R$. If $f(x)=(c+1) x^{2}+\left(1-c^{2}\right) x+2 k$ and $f(x+y)=f(x)+f(y)-x y$, for all $x, y \in R$, then the value of $|2( f (1)+ f (2)+ f (3)+\ldots \ldots+ f (20)) \mid$ is equal to

If $f(x) = \cos (\log x)$, then $f({x^2})f({y^2}) - \frac{1}{2}\left[ {f\,\left( {\frac{{{x^2}}}{2}} \right) + f\left( {\frac{{{x^2}}}{{{y^2}}}} \right)} \right]$ has the value

If $a+\alpha=1, b+\beta=2$ and $\operatorname{af}(x)+\alpha f\left(\frac{1}{x}\right)=b x+\frac{\beta}{x}, x \neq 0,$ then the value of expression $\frac{ f ( x )+ f \left(\frac{1}{ x }\right)}{ x +\frac{1}{ x }}$ is ..... .