Proving that $\{0^n 1^{n^2} \mid n \in \mathbb{N}\}$ is not context-free
If $v x y$ is just 0s or 1s, pumping does unbalance the 0s and 1s.

If $v$ is $0^a$ and $y = 1^b$, pumping once gives $0^{N + a} 1^{N^2 + b}$, now prove that that can't possibly work out as $a + b \le N$ (in particular, $N^2 + b$ can't be a perfect square).