If $f(x) = \log_e \left( \frac{1-x}{1+x} \right)$,$|x| < 1$,then $f\left( \frac{2x}{1+x^2} \right)$ is equal to

Let $A = \{a, b, c\}$ and $B = \{1, 2, 3, 4\}$. Then the number of elements in the set $C = \{ f : A \rightarrow B \mid 2 \in f(A) \text{ and } f \text{ is not one-one} \}$ is

The number of solutions of the equation $2{e^{\left| x \right|}}{\tan ^{ - 1}}\left| x \right| = 1$ is

Let $f: R \rightarrow R$ be a function defined by $f(x) = \frac{2e^{2x}}{e^{2x} + e}$. Then $f\left(\frac{1}{100}\right) + f\left(\frac{2}{100}\right) + f\left(\frac{3}{100}\right) + \dots + f\left(\frac{99}{100}\right)$ is equal to

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

If $f(x) = \frac{1}{\sqrt{x + 2\sqrt{2x - 4}}} + \frac{1}{\sqrt{x - 2\sqrt{2x - 4}}}$ for $x > 2$,then $f(11) = $