Do not use a calculator in answering this question. The complex numbers $u$ and $v$ are such that $u

Question

Do not use a calculator in answering this question. The complex numbers $u$ and $v$ are such that $u=5{{\mathrm{e}}^{\frac{7}{12}\pi \mathrm{i}}}$ and $v=6\mathrm{i}{{\mathrm{e}}^{-\frac{1}{3}\pi \mathrm{i}}}$ respectively. Find an exact expression of $\frac{{{u}^{2}}}{{{v}^{*}}}$, giving your answer in the form $r{{\mathrm{e}}^{\mathrm{i}	heta }}$, where $r>0$ and $-\pi [4] Find the three smallest positive integer values of $n$ for which $\frac{{{u}^{n}}}{{{v}^{*}}}$ is purely imaginary. [2]

Accepted Answer

$\dfrac{u^2}{v^*} = \dfrac{25}{6}e^{-i2\pi/3}$; $n = 4, 16, 28$

2018 HCI P1 Q9 (b)

2018 HCI P1 Q9 (b)

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