Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 4F / i / Solution

The pumping lemma for regular languages says that sufficiently long accepted words split as $xyz$ with $|xy|\le P$, $|y|>0$, and $x{y}^{k}z$ accepted for all $k\ge 0$. Applying it to $0^P{1}^{P}01$ pumps only zeros, so language (i) is not regular.