CAIE A-Level Mathematics A2 3.9.5 Complex Numbers Questions

Practise using modulus–argument and exponential forms to multiply, divide and raise complex numbers to powers.

Exam points

convert complex numbers between Cartesian, modulus–argument and exponential forms
use arguments and moduli in products, quotients and powers
solve trigonometric and algebraic results from exponential complex forms

[Maximum number: 3]

The complex numbers z and $\omega$ are defined by z=1-i and $\omega=-3+3 \sqrt{3} \mathrm{i}$ .

Using your answers to part (b), prove that $\tan \frac{5}{12} \pi=\frac{\sqrt{3}+1}{\sqrt{3}-1}$ .

$\arg z \omega=\arg z+\arg \omega\left(=\frac{2 \pi}{3}-\frac{\pi}{4}=\frac{5 \pi}{12}\right)$
For showing correct use of their angles from part (b).
Must demonstrate where $\frac{5 \pi}{12}$ comes from.

$\arg z \omega=\tan ^{-1} \frac{3+3 \sqrt{3}}{-3+3 \sqrt{3}}$
Correct method for their z $\omega$ from part (a). Must link to point B on diagram or to $\arg z \omega$ .

Need to see $\tan ^{-1} \frac{3+3 \sqrt{3}}{-3+3 \sqrt{3}}$ or $\tan \theta=\frac{3+3 \sqrt{3}}{-3+3 \sqrt{3}}$ and not just $\tan ^{-1} \frac{1+\sqrt{3}}{-1+\sqrt{3}}$ .

$\Rightarrow \tan \left(\frac{5}{12} \pi\right)=\frac{\sqrt{3}+1}{\sqrt{3}-1}$
Obtain given answer from full and correct working.

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CAIE A-Level Mathematics A2 3.9.5 Complex Numbers Questions question 1