Two loci in the Argand diagram are given by the equations \[|z-3-\mathrm i|=1\quad\text{and}\quad\arg z=\alpha,\quad\text{where}\hspace{0.5em}\tan\alpha=0.4.\] The complex numbers \(z_1\) and \(z_2\), where \(|z_1|<|z_2|\), correspond to the points of intersection of these loci.
Draw an Argand diagram to show both loci, and mark the points represented by \(z_1\) and \(z_2\).[2]
Find the two values of \(z\) which represent points on \(|z-3-\mathrm i|=1\) such that \(|z-z_1|=|z-z_2|\).[4]
The complex number \(2-2\mathrm i\) is denoted by \(w\). By writing \(w\) in polar form \(r\mathrm e^{\mathrm i\theta}\), where \(r>0\) and \(-\pi<\theta\le\pi\), find exactly all the cube roots of \(w\) in polar form.[3]
Find the smallest positive whole number value of \(n\) such that \(\arg(w^*w^n)=\frac12\pi\).[3]