6. The complex number \(z\) is defined by
$$z = - \lambda + 3 i \quad \text { where } \lambda \text { is a positive real constant }$$
Given that the modulus of \(z\) is 5
- write down the value of \(\lambda\)
- determine the argument of \(z\), giving your answer in radians to one decimal place.
In part (c) you must show detailed reasoning.
Solutions relying on calculator technology are not acceptable. - Express in the form \(a + \mathrm { i } b\) where \(a\) and \(b\) are real,
- \(\frac { z + 3 i } { 2 - 4 i }\)
- \(\mathrm { Z } ^ { 2 }\)
- Show on a single Argand diagram the points \(A\), \(B\), \(C\) and \(D\) that represent the complex numbers
$$z , z ^ { * } , \frac { z + 3 i } { 2 - 4 i } \text { and } z ^ { 2 }$$