Complex numbers 2 — Question 11

QCAA 2020, Paper 1 · 7 marks

Q11 · 2020 · Technology-freeSimple familiar7 marks

QUESTION 11 (7 marks)

The vertices of a regular hexagon are positioned on the circumference of a unit circle as shown on the Argand plane.

Consider the complex number

w

, as shown on the plane.

Argand diagram of a regular hexagon on the unit circle with vertex w

Determine

w

, expressing your answer in the form

r\,\operatorname{cis}(\theta)

[1 mark]

Convert

w

into Cartesian form.

[2 marks]

Each vertex of the hexagon is a solution of an equation of the form

z^n=a

, where

z\in\mathbb{C}

. State the value of

n

[1 mark]

State the value of

a

[1 mark]

Verify that

w

satisfies the equation

z^n=a

using the results from 11c) and 11d).

[2 marks]

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

7 marks

ANSWER

w=\operatorname{cis}(2\pi/3)=-\frac12+\frac{\sqrt3}{2}i

Worked solution

|w|=1

and

\arg(w)=\frac{2\pi}{3}

, so

w=\operatorname{cis}\left(\frac{2\pi}{3}\right)

w=\cos\left(\frac{2\pi}{3}\right)+i\sin\left(\frac{2\pi}{3}\right)=-\frac12+\frac{\sqrt3}{2}i

n=6

a=1

w^6=\operatorname{cis}(4\pi)=\operatorname{cis}(0)=1

, so the required result is verified.

correctly expresses

w

r\operatorname{cis}(\theta)

form

[1 mark]

converts

w

into

r\cos(\theta)+r\sin(\theta)i

form

[1 mark]

uses exact values for

\operatorname{Re}(w)

and

\operatorname{Im}(w)

[1 mark]

correctly states the value of

n

[1 mark]

correctly states the value of

a

[1 mark]

uses De Moivre’s theorem in the calculation of

w^n

for the value of

n

[1 mark]

verifies the result by showing that the result of the calculation is the value of

a

[1 mark]

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Complex numbers 2 — Question 11

QUESTION 11 (7 marks)

QCAA guide · typeset solution

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