CAIE P3 2015 June — Question 8

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2015
SessionJune
TopicComplex Numbers Argand & Loci

8 The complex number 1 - i is denoted by \(u\).
  1. Showing your working and without using a calculator, express $$\frac { \mathrm { i } } { u }$$ in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. On an Argand diagram, sketch the loci representing complex numbers \(z\) satisfying the equations \(| z - u | = | z |\) and \(| z - \mathrm { i } | = 2\).
  3. Find the argument of each of the complex numbers represented by the points of intersection of the two loci in part (ii).