If $f(x+2y, x-2y) = xy$, then $f(x, y)$ is equal to

If $f(a) = a^2 + a + 1$,then the number of solutions of the equation $f(a^2) = 3f(a)$ is:

If $f: R \rightarrow R$ is defined as $f(x)=\frac{3^x+3^{-x}}{2}, \forall x \in R$ and it satisfies $f(x+y)+f(x-y)=a f(x) f(y)$, then $a=$

If $f : \mathbb{Z} \rightarrow \mathbb{Z}$ is defined by $f(x) = x^{9} - 11 x^{8} - 2 x^{7} + 22 x^{6} + x^{4} - 12 x^{3} + 11 x^{2} + x - 3, \forall x \in \mathbb{Z}$,then $f(11) = $

If $f(f(0)) = 0$,where $f(x) = x^2 + ax + b$ and $b \neq 0$,then $a + b =$

If $f(x)$ is a polynomial function satisfying $f(x) \cdot f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)$ and $f(4)=257$,then $f(3)=$