2020 DHS P2 Q4
(a)
(i) Using ${{\text{e}}^{\mathbf{i}\theta }}=\cos \theta +\mathbf{i}\sin \theta $ and the trigonometry formulas, show that $\left( {{\text{e}}^{\mathbf{i}{{\theta }_{1}}}} \right)\left( {{\text{e}}^{\mathbf{i}{{\theta }_{2}}}} \right)={{\text{e}}^{\mathbf{i}\left( {{\theta }_{1}}+{{\theta }_{2}} \right)}}$.
[2]
(ii) Hence find the least positive integer value of $n$ such that the complex number ${{\left( \mathbf{i}{{\text{e}}^{\mathbf{i}\frac{\pi }{5}}} \right)}^{n}}$ is purely imaginary.
[3]
(b)
The graph of $\text{f}\left( x \right)={{x}^{3}}-7{{x}^{2}}+17x-15$ is shown below. It cuts the $x$-axis at $x=3$.
(i) Without the use of a graphing calculator, show that the roots of the equation of $\text{f}\left( x \right)=0$ 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]
(ii) 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]
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