Complex numbers 2 — Question 15

QCAA 2022, Paper 1 · 4 marks

Q15 · 2022 · Technology-freeSimple familiar4 marks

QUESTION 15 (4 marks)

Consider the equation

z^3=1

, where

z\in\mathbb C

Sketch the solutions to

z^3=1

on the Argand diagram.

[2 marks]

The solutions to

z^3=1

can be expressed in the form

z=a+bi

, where

a,b\in\mathbb R

. Determine the largest possible positive value of

ab

[2 marks]

WORKED SOLUTION

4 marks

ANSWER

z=1,\operatorname{cis}(2\pi/3),\operatorname{cis}(4\pi/3)

; largest positive

ab=\sqrt3/4

Worked solution

a) The roots lie on the unit circle at arguments

0

2\pi/3

and

4\pi/3

b) The Cartesian roots are

1

-1/2+(\sqrt3/2)i

, and

-1/2-(\sqrt3/2)i

. The largest positive product

ab

(-1/2)(-\sqrt3/2)=\sqrt3/4

correctly sketches 3 solutions, each with a modulus of 1

[1 mark]

correctly sketches 3 solutions with arguments

0

and

\pm2\pi/3

[1 mark]

recognises a solution from which its

a

and

b

values produce the largest possible positive product

[1 mark]

determines a positive product of

ab

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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