IPMAT Indore 2023 (SA) PYQs

Suppose that a real-valued function $f(x)$ of real numbers satisfies $f(x + xy) = f(x) + f(xy$) for all real $x, y,$ and that $f(2020) = 1$. Compute $f(2021)$.

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 _________.

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