CAIE P3 2022 November — Question 5

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2022
SessionNovember
TopicComplex numbers 2

5 The complex numbers \(u\) and \(w\) are defined by \(u = 2 \mathrm { e } ^ { \frac { 1 } { 4 } \pi \mathrm { i } }\) and \(w = 3 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } }\).
  1. Find \(\frac { u ^ { 2 } } { w }\), giving your answer in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the exact values of \(r\) and \(\theta\).
  2. State the least positive integer \(n\) such that both \(\operatorname { Im } w ^ { n } = 0\) and \(\operatorname { Re } w ^ { n } > 0\).