Let $f(x)$ be a non-constant polynomial with real coefficients such that $f\left(\frac{1}{2}\right)=100$ and $f(x) \leq 100$ for all real $x$. Which of the following statements is NOT necessarily true?

The graph of $y = f(x)$ is shown then number of solutions of the equation $f(f(x)) =2$ is

Let $A=\{0,1,2,3,4,5,6,7\} .$ Then the number of bijective functions $f: A \rightarrow A$such that $f(1)+f(2)=3-f(3)$ is equal to $.....$

Let $f: R \rightarrow R$ be a function defined $f(x)=\frac{2 e^{2 x}}{e^{2 x}+\varepsilon}$. Then $f\left(\frac{1}{100}\right)+f\left(\frac{2}{100}\right)+f\left(\frac{3}{100}\right)+\ldots .+f\left(\frac{99}{100}\right)$ is equal to

Domain of $log\,log\,log\,  ....(x)$ is 

                        $ \leftarrow \,n\,\,times\, \to $

Let $c, k \in R$. If $f(x)=(c+1) x^{2}+\left(1-c^{2}\right) x+2 k$ and $f(x+y)=f(x)+f(y)-x y$, for all $x, y \in R$, then the value of $|2( f (1)+ f (2)+ f (3)+\ldots \ldots+ f (20)) \mid$ is equal to