CAIE Further Paper 2 2021 June — Question 1

Exam BoardCAIE
ModuleFurther Paper 2 (Further Paper 2)
Year2021
SessionJune
TopicComplex numbers 2

1
  1. Find \(a\) and \(b\) such that $$z ^ { 8 } - i z ^ { 5 } - z ^ { 3 } + i = \left( z ^ { 5 } - a \right) \left( z ^ { 3 } - b \right) .$$
  2. Hence find the roots of $$z ^ { 8 } - i z ^ { 5 } - z ^ { 3 } + i = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).