The <highest root> is
$$
\boxed{\beta=\varepsilon_1+\varepsilon_2
=\alpha_1+2\alpha_2}.
$$
Since $\beta$ is long and the $\varepsilon_i$ are orthonormal, $\beta^\vee=\beta$. The <coroot> pairing gives
$$
\boxed{\langle\alpha_1,\beta^\vee\rangle=0,
\qquad
\langle\alpha_2,\beta^\vee\rangle=1}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 102 / 3 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-102/3/d/solution