OCR FP3 Specimen — Question 4

Exam BoardOCR
ModuleFP3 (Further Pure Mathematics 3)
SessionSpecimen
TopicComplex numbers 2

4 In this question, give your answers exactly in polar form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
  1. Express \(4 ( ( \sqrt { } 3 ) - \mathrm { i } )\) in polar form.
  2. Find the cube roots of \(4 ( ( \sqrt { } 3 ) - \mathrm { i } )\) in polar form.
  3. Sketch an Argand diagram showing the positions of the cube roots found in part (ii). Hence, or otherwise, prove that the sum of these cube roots is zero.