2024 RI P1 Q8

Do not use a calculator in answering this question.

1. The complex number \(w\) is such that \(w=a+ib\), where \(a\) and \(b\) are non-zero real numbers. The complex conjugate of \(w\) is denoted by \({{w}^{*}}\). Given that
   \(w{{w}^{*}}=4-2i+2i{{w}^{*}}\),
   find the two possible values of \(w\).[4]
2. The complex number \(z\) is given by \(z=\frac{1-\sqrt{3}i}{-1+i}\).
   1. Find \(\arg (z)\).[3]
   2. Find \(z\) in cartesian form \(x+iy\).[2]
   3. Hence find the value of \(\tan \frac{\pi }{12}\) in the form \(c+d\sqrt{3}\), where \(c\) and \(d\) are integers to be found.[3]