2018 RI P1 Q7

Do not use a calculator in answering this question.

1. Find the roots of the equation \({{w}^{2}}\left( 1-\mathrm{i} \right)+4w+\left( 10+10\mathrm{i} \right)=0\), giving your answers in cartesian form \(a+\mathrm{i}b\).[3]
2. It is given that \(z=-\sqrt{3}+\mathrm{i}\).
   1. Find an exact expression for \({{z}^{5}}\). Give your answer in the form \(r{{\mathrm{e}}^{\mathrm{i}\theta }}\), where \(r>0\) and \(-\pi <\theta \le \pi \).[2]
   2. Find the three smallest positive whole number values of \(n\) for which \(\frac{{{z}^{n}}}{\mathrm{i}{{z}^{*}}}\) is purely imaginary.[4]
   3. Given that \(\left| 1+\frac{p}{{{z}^{*}}} \right|=\sqrt{7}\), find exactly the possible values of the real number \(p\).[3]