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

If $f(x) = \sqrt{x^2 + x} + \frac{\tan^2 \alpha}{\sqrt{x^2 + x}}$,where $\alpha \in (0, \pi/2)$ and $x > 0$,then the value of $f(x)$ is greater than or equal to:

If $f: R \rightarrow R$ and $g: R \rightarrow R$ are given by $f(x)=|x|$ and $g(x)=[x]$ for each $x \in R$, then $\{x \in R: g(f(x)) \leq f(g(x))\}$ is equal to

If $N$ denotes the set of all positive integers and if $f: N \rightarrow N$ is defined by $f(n) = \text{the sum of positive divisors of } n$, then $f(2^k \cdot 3)$, where $k$ is a positive integer, is

Let $A = \{2, 3, 4, 5, 6\}$. Let $R$ be a relation on the set $A \times A$ defined by $(x, y) R (z, w)$ if and only if $x$ divides $z$ and $y \le w$. Then the number of elements in $R$ is . . . . . . .

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