8. (a) It is given that \(z = x + y \mathrm { i }\), where \(x\) and \(y\) are real numbers.
- Write down, in terms of \(x\) and \(y\), an expression for \(( z - 2 \mathrm { i } ) ^ { * }\).
(1 mark) - Solve the equation
$$( z - 2 i ) ^ { * } = 4 i z + 3$$
giving your answer in the form \(a + b \mathrm { i }\).
(b) It is given that \(p + q \mathrm { i }\), where \(p\) and \(q\) are real numbers, is a root of the equation \(z ^ { 2 } + 10 \mathrm { i } z - 29 = 0\).
Without finding the values of \(p\) and \(q\), state why \(p - q \mathrm { i }\) is not a root of the equation \(z ^ { 2 } + 10 \mathrm { i } z - 29 = 0\).
[0pt]
[BLANK PAGE]