Specialist Maths — Question 12

QCAA 2022, Paper 1 · 6 marks

Q12 · 2022 · Technology-freeSimple familiar6 marks

QUESTION 12 (6 marks)

Given

z_1=a+bi

z_2=c+di

for all

a,b,c,d\in\mathbb R

and

z_2\ne0

, prove the identity

\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}.

WORKED SOLUTION

6 marks

ANSWER

The modulus identity is proved.

Worked solution

Multiply the quotient by the conjugate of the denominator:

\frac{z_1}{z_2}=\frac{(a+bi)(c-di)}{c^2+d^2}.

Taking the modulus and simplifying gives

\left|\frac{z_1}{z_2}\right|=\frac{\sqrt{(a^2+b^2)(c^2+d^2)}}{c^2+d^2}=\frac{\sqrt{a^2+b^2}}{\sqrt{c^2+d^2}}=\frac{|z_1|}{|z_2|}.

correctly multiplies the numerator and denominator by the complex conjugate of

z_2

[1 mark]

realises denominator in simplest form and expands numerator

[1 mark]

determines modulus of the expression

[1 mark]

simplifies numerator

[1 mark]

factorises numerator

[1 mark]

completes the proof using mathematical reasoning

[1 mark]

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

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