2020 ASRJC P1 Q5(a) | 2024 SAJC P1 Q8b

A graphing calculator is not allowed for this question.

1. Find the square roots of \(2\mathrm{i}\).[2]
2. Solve the equation \(w^2 = 3 - 4\mathrm{i}\). Hence find the roots of the equation \(z^2 - 4\mathrm{i}z + 4\mathrm{i} - 7 = 0\), giving your answers in Cartesian form.[6]
3. Given that \(w = \frac{1}{2 + \mathrm{i}\alpha}\) where \(\alpha \in \mathbb{R}\), and \(w^*\) denotes the complex conjugate of \(w\), find \(ww^*\) and hence prove that \(w + w^* = 4ww^*\) ---(1).
   The complex number \(w\) is represented in an Argand diagram by the point \(A\). By using (1) and considering \(w = x + y\mathrm{i}\), where \(x, y \in \mathbb{R}\), show that, \(A\) lies on a circle as \(\alpha\) varies. Find the Cartesian equation of this circle, stating clearly its centre and radius.[6]