3 In this question you must show detailed reasoning.
The complex number \(7 - 4 \mathrm { i }\) is denoted by \(z\).
- Giving your answers in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are rational numbers, find the following.
- \(3 z - 4 z ^ { * }\)
- \(( z + 1 - 3 i ) ^ { 2 }\)
- \(\frac { z + 1 } { z - 1 }\)
- Express \(z\) in modulus-argument form giving the modulus exactly and the argument correct to 3 significant figures.
- The complex number \(\omega\) is such that \(z \omega = \sqrt { 585 } ( \cos ( 0.5 ) + \mathrm { i } \sin ( 0.5 ) )\).
Find the following.
- \(| \omega |\)
- \(\arg ( \omega )\), giving your answer correct to 3 significant figures