Complex numbers 2 — Question 12

QCAA 2023, Paper 2 · 7 marks

Q12 · 2023 · Technology-activeSimple familiar7 marks

QUESTION 12 (7 marks)

Consider the complex number

z=-3+2i

Determine

z^3

using the binomial theorem. Leave your answer in the form

a+bi

, where

a,b\in\mathbb R

[2 marks]

Convert

z

into the form

r\operatorname{cis}(\theta)

, where

-\pi<\theta\le\pi

[1 mark]

Use the result from Question 12b) to determine

z^3

using De Moivre’s theorem. Leave your answer in the form

r\operatorname{cis}(\theta)

, where

-\pi<\theta\le\pi

[2 marks]

Evaluate the reasonableness of your results from Questions 12a) and 12c), noting that both represent

z^3

[2 marks]

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

7 marks

ANSWER

z^3=9+46i\approx46.87\operatorname{cis}(1.38)

Worked solution

(-3+2i)^3=9+46i

z\approx3.61\operatorname{cis}(2.55)

c) By De Moivre’s theorem,

z^3\approx46.87\operatorname{cis}(1.38)

d) Converting this polar form back to Cartesian form gives approximately

8.89+46.02i

, which agrees with

9+46i

apart from rounding, so the results are reasonable.

uses the binomial theorem

[1 mark]

expresses

z^3

in Cartesian form

[1 mark]

converts

z

to polar form

[1 mark]

uses De Moivre’s theorem

[1 mark]

expresses

z^3

in the required polar form

[1 mark]

converts the polar result to Cartesian form

[1 mark]

evaluates the reasonableness of the results

[1 mark]

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Complex numbers 2 — Question 12

QUESTION 12 (7 marks)

QCAA guide · typeset solution

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