Complex Numbers
1. Complex Numbers
10. Do not use a calculator in answering this question.
(a). Solve the equation \(z^3-4 z^2+6 z-4=0\).
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(b). Hence solve the equation \(\mathrm{i} w^3+4 w^2-6 \mathrm{i} w-4=0\), giving the roots in the form \(r \mathrm{e}^{\mathrm{i} \theta}\), where \(r>0\) and \(0 \leq \theta<2 \pi\).
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(c). For the value of \(w\) found in part (b) with the largest \(\theta\), find the smallest positive integer \(n\) such that \(w^n\) is a positive real number, showing your working clearly.
[3]
