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) = $

Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = 2x + |x|$,then $f(2x) + f(-x) - f(x) = $

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 $A = \{1, 4, 7\}$ and $B = \{2, 3, 8\}$. Then the number of elements in the relation $R = \{((a_1, b_1), (a_2, b_2)) \in ((A \times B) \times (A \times B)) : a_1 + a_2 \text{ divides } b_2 + b_1\}$ is . . . . . . .

Let $f(x) = \frac{\sqrt{x - 2\sqrt{x - 1}}}{\sqrt{x - 1} - 1}$. Then:

$A$ function $f(x)$ is given by $f(x) = \frac{5^{x}}{5^{x} + \sqrt{5}}$. Then the sum of the series $f\left(\frac{1}{20}\right) + f\left(\frac{2}{20}\right) + f\left(\frac{3}{20}\right) + \ldots + f\left(\frac{39}{20}\right)$ is equal to ....... .