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

Number of solutions of the equation $2^x + x = 2^{\sin x} + \sin x$ in $[0, 10\pi]$ is -

Function $f(x) = {\left( {\left\{ x \right\} - \frac{1}{2}} \right)^2}$ is (where $\{.\}$ represents the fractional part function).

If $\alpha_1 < \alpha_2 < \alpha_3 < \alpha_4 < \alpha_5 < \alpha_6$,then the equation $(x-\alpha_1)(x-\alpha_3)(x-\alpha_5) + 3(x-\alpha_2)(x-\alpha_4)(x-\alpha_6) = 0$ has :-

Find the minimum integral value of $\alpha$ for which the graph of $f(x) = ||x - 2| - \alpha| - 5$ has exactly four $x-$intercepts.

Let $f(x) = 2x^{2} - x - 1$ and $S = \{n \in \mathbb{Z} : |f(n)| \leq 800\}$. Then the value of $\sum_{n \in S} f(n)$ is: