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June 2023 p33 q11
1983
The complex number \(z\) is defined by \(z = \frac{5a - 2i}{3 + ai}\), where \(a\) is an integer. It is given that \(\arg z = -\frac{1}{4}\pi\).
(a) Find the value of \(a\) and hence express \(z\) in the form \(x + iy\), where \(x\) and \(y\) are real. [6]
(b) Express \(z^3\) in the form \(re^{i\theta}\), where \(r > 0\) and \(-\pi < \theta \leq \pi\). Give the simplified exact values of \(r\) and \(\theta\). [3]
Solution
(a) Multiply numerator and denominator by \(3 - ai\):
