Specialist Maths — Question 13

QCAA 2023, Paper 1 · 5 marks

Q13 · 2023 · Technology-freeSimple familiar5 marks

QUESTION 13 (5 marks)

Given

z\in\mathbb C

, where

z\ne0

, prove

\left|\frac{|z|}{z\bar z}\right|=|z^{-1}|.

WORKED SOLUTION

5 marks

ANSWER

The identity is proved.

Worked solution

Let

z=a+bi

, so

\bar z=a-bi

|z|=\sqrt{a^2+b^2}

and

z\bar z=a^2+b^2

. Then

\left|\frac{|z|}{z\bar z}\right| =\left|\frac{\sqrt{a^2+b^2}}{a^2+b^2}\right| =\frac1{\sqrt{a^2+b^2}}.

Also

|z^{-1}|=1/|z|=1/\sqrt{a^2+b^2}

, so the two sides are equal.

correctly represents

\bar z

in terms of

a

and

b

[1 mark]

correctly represents

|z|

as a radical in terms of

a

and

b

[1 mark]

correctly simplifies

z\bar z

in terms of

a

and

b

[1 mark]

determines an expression without an index that represents

|z|^{-1}

[1 mark]

correctly shows mathematical reasoning to complete the proof

[1 mark]

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

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