If the graph of a non-constant function $f(x)$ is symmetric about the point $(3,4)$,then the value of $\sum\limits_{r = 0}^6 {f(r) + f(3)}$ is equal to

For $\theta \in [0, \pi]$,let $f(\theta) = \sin(\cos \theta)$ and $g(\theta) = \cos(\sin \theta)$. Let $a = \max_{0 \leq \theta \leq \pi} f(\theta)$,$b = \min_{0 \leq \theta \leq \pi} f(\theta)$,$c = \max_{0 \leq \theta \leq \pi} g(\theta)$,and $d = \min_{0 \leq \theta \leq \pi} g(\theta)$. The correct inequalities satisfied by $a, b, c, d$ are:

Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be two functions defined by $f(x)=\log _{e}(x^{2}+1)-e^{-x}+1$ and $g(x)=\frac{1-2e^{2x}}{e^{x}}$. Then,for which of the following range of $\alpha$,the inequality $f(g(\frac{(\alpha-1)^{2}}{3})) > f(g(\alpha-\frac{5}{3}))$ holds?

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

If $f(x) = 2x$ and $g$ is the identity function,then:

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