CAIE A-Level Further Math A2 2.5.3 De Moivres Theorem Questions

Practise applying de Moivre's theorem to trigonometric identities, complex roots and related series with Further Mathematics Paper 2 questions and mark schemes.

Exam points

Apply de Moivre's theorem to powers and roots of complex numbers.

[Maximum number: 5]

Find the roots of the equation $(z+5 \mathrm{i})^{3}=4+4 \sqrt{3} \mathrm{i}$ , giving your answers in the form $r \cos \theta+\mathrm{i}(r \sin \theta-5)$ , where r>0 and $0<\theta<2 \pi$ .

$(z+5 \mathrm{i})^{3}=4+4 \mathrm{i} \sqrt{3}=8 \mathrm{e}^{\frac{1}{3} \pi}$
Finds modulus and argument of $4+4 \mathrm{i} \sqrt{3}$ .

$Z_{1}=2\left(\cos \frac{1}{9} \pi+i \sin \frac{1}{9} \pi\right)-5 i=2 \cos \frac{1}{9} \pi+i\left(2 \sin \frac{1}{9} \pi-5\right)$
Finds one root.

$z_{2}=2 \cos \frac{7}{9} \pi+\mathrm{i}\left(2 \sin \frac{7}{9} \pi-5\right), z_{3}=2 \cos \frac{13}{9} \pi+\mathrm{i}\left(2 \sin \frac{13}{9} \pi-5\right)$

A1 FT

Finds other two roots. FT on their modulus.

A1 FT

5

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CAIE A-Level Further Math A2 2.5.3 De Moivres Theorem Questions question 1