2 In this question you must show detailed reasoning.
The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by \(z _ { 1 } = 2 - 3 \mathrm { i }\) and \(z _ { 2 } = a + 4 \mathrm { i }\) where \(a\) is a real number.

1. Express \(z _ { 1 }\) in modulus-argument form, giving the modulus in exact form and the argument correct to 3 significant figures.
2. Find \(z _ { 1 } z _ { 2 }\) in terms of \(a\), writing your answer in the form \(c + \mathrm { i } d\).
3. The real and imaginary parts of a complex number on an Argand diagram are \(x\) and \(y\) respectively. Given that the point representing \(z _ { 1 } z _ { 2 }\) lies on the line \(y = x\), find the value of \(a\).
4. Given instead that \(z _ { 1 } z _ { 2 } = \left( z _ { 1 } z _ { 2 } \right) ^ { * }\) find the value of \(a\). In this question you must show detailed reasoning.
5. Express \(( 2 + 3 \mathrm { i } ) ^ { 3 }\) in the form \(a + \mathrm { i } b\).
6. Hence verify that \(2 + 3 \mathrm { i }\) is a root of the equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 23 z + 52 = 0\).
7. Express \(3 z ^ { 3 } - 8 z ^ { 2 } + 23 z + 52\) as the product of a linear factor and a quadratic factor with real coefficients.