7.
$$z = 2 - \mathrm { i } \sqrt { } 3$$
- Calculate \(\arg z\), giving your answer in radians to 2 decimal places.
Use algebra to express
- \(z + z ^ { 2 }\) in the form \(a + b \mathrm { i } \sqrt { } 3\), where \(a\) and \(b\) are integers,
- \(\frac { z + 7 } { z - 1 }\) in the form \(c + d \mathrm { i } \sqrt { } 3\), where \(c\) and \(d\) are integers.
Given that
$$w = \lambda - 3 \mathrm { i }$$
where \(\lambda\) is a real constant, and \(\arg ( 4 - 5 \mathrm { i } + 3 w ) = - \frac { \pi } { 2 }\),
- find the value of \(\lambda\).