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 :-

Domain of function $f(x) = {\sin ^{ – 1}}5x$ 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.

Let $f(x)$ be a quadratic polynomial such that $f(-2)$ $+f(3)=0$. If one of the roots of $f(x)=0$ is $-1$, then the sum of the roots of $f(x)=0$ is equal to

The number of functions $f$, from the set$A=\left\{x \in N: x^{2}-10 x+9 \leq 0\right\}$ to the set $B=\left\{n^{2}: n \in N\right\}$ such that $f(x) \leq(x-3)^{2}+1$, for every $x \in A$, is.