2021 RI J2 BT Q5

The complex number \(w\) can be expressed as \(2\left( \cos \theta +\mathrm{i}\sin \theta \right)\).

1. Show that \(w-\frac{4}{w}\) is a purely imaginary number.[2]
2. Show that \(\frac{w+2}{w-2}\) can be expressed as \(k\cot \frac{1}{2}\theta \), where \(k\) is a complex number to be found.[4]
3. Find the modulus of the complex number \(\frac{w+6\mathrm{i}}{2-3\mathrm{i}w}\).[4]