All roots of $A_3$ have the same length, so $|\alpha_2|/|\alpha_3|=1$. In $B_3$, $\alpha_2$ is long and $\alpha_3$ short, giving $\sqrt2$. In $C_3$ their roles are reversed, giving $1/\sqrt2$. These ratios also follow from $A_{23}/A_{32}=|\alpha_2|^2/|\alpha_3|^2$.


Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 4 / b / ii / Solution

 Solution (source code)