2020 DHS P2 Q4

Using \({{\mathrm{e}}^{\mathbf{i}\theta }}=\cos \theta +\mathbf{i}\sin \theta \) and the trigonometry formulas, show that \(\left( {{\mathrm{e}}^{\mathbf{i}{{\theta }_{1}}}} \right)\left( {{\mathrm{e}}^{\mathbf{i}{{\theta }_{2}}}} \right)={{\mathrm{e}}^{\mathbf{i}\left( {{\theta }_{1}}+{{\theta }_{2}} \right)}}\).[2]

Hence find the least positive integer value of \(n\) such that the complex number \({{\left( \mathbf{i}{{\mathrm{e}}^{\mathbf{i}\frac{\pi }{5}}} \right)}^{n}}\) is purely imaginary.[3]

Without the use of a graphing calculator, show that the roots of the equation of \(\mathrm{f}\left( x \right)=0\) are \(m\), \({{z}_{0}}\) and \({{z}_{1}}\) where \(m\) is a real constant and \({{z}_{0}}\) and \({{z}_{1}}\) are complex constants to be determined.[4]

The complex number \(w\) is such that \[w={{z}_{0}}+\lambda \left( {{z}_{0}}-{{z}_{1}} \right)\] where \(\lambda \) is a real number. By considering an Argand diagram or otherwise, find the least value of \(\left| w \right|\) when \(\lambda \) varies.[2]