6 The complex number \(z\) is given by
$$z = ( \sqrt { } 3 ) + \mathrm { i } .$$
- Find the modulus and argument of \(z\).
- The complex conjugate of \(z\) is denoted by \(z ^ { * }\). Showing your working, express in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real,
(a) \(2 z + z ^ { * }\),
(b) \(\frac { \mathrm { i } z ^ { * } } { z }\). - On a sketch of an Argand diagram with origin \(O\), show the points \(A\) and \(B\) representing the complex numbers \(z\) and \(\mathrm { i } z ^ { * }\) respectively. Prove that angle \(A O B = \frac { 1 } { 6 } \pi\).