CAIE P3 2006 November — Question 9

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2006
SessionNovember
TopicComplex Numbers Argand & Loci

9 The complex number \(u\) is given by $$u = \frac { 3 + \mathrm { i } } { 2 - \mathrm { i } }$$
  1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. Find the modulus and argument of \(u\).
  3. Sketch an Argand diagram showing the point representing the complex number \(u\). Show on the same diagram the locus of the point representing the complex number \(z\) such that \(| z - u | = 1\).
  4. Using your diagram, calculate the least value of \(| z |\) for points on this locus.