If the domain of the function $f(x)=\sec ^{-1}\left(\frac{2 x}{5 x+3}\right)$ is $[\alpha, \beta) \cup(\gamma, \delta]$, then $|3 \alpha+10(\beta+\gamma)+21 \delta|$ is equal to $…….$.

If a function $g(x)$ is defined in $[-1, 1]$ and two vertices of an equilateral triangle are $(0, 0)$ and $(x, g(x))$ and its area is $\frac{\sqrt 3}{4}$ , then $g(x)$ equals :-

Product of all the solution of the equation  ${x^{1 + {{\log }_{10}}x}} = 100000x$ is

The graph of the function $f(x)=x+\frac{1}{8} \sin (2 \pi x), 0 \leq x \leq 1$ is shown below. Define $f_1(x)=f(x), f_{n+1}(x)=f\left(f_n(x)\right)$, for $n \geq 1$.

Which of the following statements are true?

$I.$ There are infinitely many $x \in[0,1]$ for which $\lim _{n \rightarrow \infty} f_n(x)=0$

$II.$ There are infinitely many $x \in[0,1]$ for which $\lim _{n \rightarrow \infty} f_n(x)=\frac{1}{2}$

$III.$ There are infinitely many $x \in[0,1]$ for which $\lim _{n \rightarrow \infty} f_n(x)=1$

$IV.$ There are infinitely many $x \in[0,1]$ for which $\lim _{n \rightarrow \infty} f_n(x)$ does not exist.

If the function $f\,:\,R – \,\{ 1, – 1\}  \to A$ defined by $f\,(x)\, = \frac{{{x^2}}}{{1 – {x^2}}},$ is surjective, then $A$ is equal to