6 It is given that $z = \mathrm { e } ^ { \mathrm { i } \theta }$.\\
(a) (i) Show that

$$z + \frac { 1 } { z } = 2 \cos \theta$$

(2 marks)\\
(ii) Find a similar expression for

$$z ^ { 2 } + \frac { 1 } { z ^ { 2 } }$$

(2 marks)\\
(iii) Hence show that

$$z ^ { 2 } - z + 2 - \frac { 1 } { z } + \frac { 1 } { z ^ { 2 } } = 4 \cos ^ { 2 } \theta - 2 \cos \theta$$

(3 marks)\\
(b) Hence solve the quartic equation

$$z ^ { 4 } - z ^ { 3 } + 2 z ^ { 2 } - z + 1 = 0$$

giving the roots in the form $a + \mathrm { i } b$.

\hfill \mbox{\textit{AQA FP2  Q6}}