2023 SJI P1 Q5

Consider the complex number \(w=a+\mathrm i\), where \(a\) is a non-zero real number and \(-\pi<\arg w\leq\pi\). It is given that \(\dfrac{w^2}{w^*}\) is a real number.

1. Find the possible values of \(a\).[4]
2. Hence find the possible arguments of \(w\).[2]