Question 1421a

Complex Numbers

2021 Paper 1 Question 4 Variant

Question

Do not use a calculator in answering this question.

The complex number

{z}

is given by

z = \frac{\textstyle \Bigl( \cos \left( \frac{1}{4} \pi \right) + \mathrm{i} \sin \left( \frac{1}{4} \pi \right) \Bigr)^2}{\textstyle \cos \left( \frac{1}{6} \pi \right) + \mathrm{i} \sin \left( \frac{1}{6} \pi \right)}.

(a)

Find

{|z|}

and

{\arg(z).}

Hence find the value of

{z^{3}}

[3]

(bi)

Show that

(\cos \theta + \mathrm{i} \sin \theta)(1 + \cos \theta - \mathrm{i} \sin \theta) = 1 + \cos \theta + \mathrm{i} \sin \theta.

[2]

(bii)

Hence, or otherwise, find the value of

(1+z)^3 + (1+z^*)^3.

[2]

Answer

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