Let ${{z}_{1}}=\sqrt{2}-\left( \sqrt{2} \right)\mathrm{i}$ and ${{z}_{2}}=1+\left( \sqrt{3} \right)\

Question

Let ${{z}_{1}}=\sqrt{2}-\left( \sqrt{2} ight)\mathrm{i}$ and ${{z}_{2}}=1+\left( \sqrt{3} ight)\mathrm{i}$. Find ${{z}_{3}}=-\frac{{{\left( {{z}_{2}} ight)}^{2}}}{{{z}_{1}}^{*}}$ exactly in the form $r{{\mathrm{e}}^{\mathrm{i}	heta }}$, where $r>0$ and $-\pi [4] On a single Argand diagram, sketch the points representing the complex numbers ${{z}_{1}}$, ${{z}_{2}}$ and ${{z}_{3}}$. Label each point and show clearly any relationships between the moduli and arguments of the complex numbers. [3] Given that ${{z}_{3}}^{n}$ is purely imaginary, find the possible values of $n$ and $\left| {{z}_{3}}^{n} ight|$. [3] Given that $w=\frac{1}{{{\mathrm{e}}^{\mathrm{i}4	heta }}-1}$, show that $\operatorname{Re}\left( w ight)~=-\frac{1}{2}$ and find $\operatorname{Im}\left( w ight)$ in terms of $	heta $. [4]

Accepted Answer

(a)(i) z₃ = 2e^(-i7π/12); (b) Re(w) = -1/2, Im(w) = -(1/2)cot2θ

2015 NJC P1 Q12 [Modified]

2015 NJC P1 Q12 [Modified]

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