IPMAT Indore 2023 (SA) PYQs

A real-valued function $f$ satisfies the relation $f(x)f(y) = f(2xy + 3) + 3f(x + y) - 3f(y) + 6y$, for all real numbers $x$ and $y$, then the value of $f(8)$ is

The function $f(x) = \dfrac{x^3 - 5x^2 - 8x}{3}$ is

If $f\left(x^{2}+f(y)\right)=x f(x)+y$ for all non-negative integers $x$ and $y$, then the value of $[f(0)]^{2}+f(0)$ equals _________.