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

The domain of definition of the function $f (x) = {\log _{\left[ {x + \frac{1}{x}} \right]}}|{x^2} - x - 6|+ ^{16-x}C_{2x-1} + ^{20-3x}P_{2x-5}$  is

Where $[x]$ denotes greatest integer function.

If non-zero real numbers $b$ and $c$ are such that $min \,f\left( x \right) > \max \,g\left( x \right)$, where $f\left( x \right) = {x^2} + 2bx + 2{c^2}$  and $g\left( x \right) = {-x^2} - 2cx + {b^2}$$\left( {x \in R} \right)$; then $\left| {\frac{c}{b}} \right|$ lies in the interval

Let $A$ be the set of all $50$ students of Class $X$ in a school. Let $f: A \rightarrow N$ be function defined by $f(x)=$ roll number of the student $x$. Show that $f$ is one-one but not onto.

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 $f(\theta ) = \sin \theta (\sin \theta + \sin 3\theta )$, then $f(\theta )$