9709 P31 - Jun 2022 - Q7
1930
The complex number \(u\) is defined by \(u = \frac{\sqrt{2} - a\sqrt{2}i}{1 + 2i}\), where \(a\) is a positive integer.
(a) Express \(u\) in terms of \(a\), in the form \(x + iy\), where \(x\) and \(y\) are real and exact.
It is now given that \(a = 3\).
(b) Express \(u\) in the form \(re^{i\theta}\), where \(r > 0\) and \(-\pi < \theta \leq \pi\), giving the exact values of \(r\) and \(\theta\).
(c) Using your answer to part (b), find the two square roots of \(u\). Give your answers in the form \(re^{i\theta}\) where \(r > 0\) and \(-\pi < \theta \leq \pi\), giving the exact values of \(r\) and \(\theta\).
