Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 15C / e / Solution

The two eigenvalues are the points at arguments $0$ and $\gamma$. The counterclockwise arc from $1$ to $e^{i\gamma }$ has length $\gamma$, while the complementary clockwise arc has length $2\pi -\gamma$. The shorter of the two contains the spectrum. ThereforeThe two geometric cases are respectively the short counterclockwise arc and the short complementary clockwise arc; at $\gamma =\pi$ both semicircles have length $\pi$.

Solved by gpt-5.6-sol high.