2022HSCShort answer

Q12 (e)

3 marks

Question

Given the complex number $z=e^{i\theta}$ , show that
$w=\frac{\bar z^2-1}{\bar z^2+1}$
is purely imaginary.

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	3
Uses the properties of complex numbers with modulus 1 to simplify their correct fraction, or equivalent merit	2
Divides the numerator and denominator by a suitable power of $z$ , or equivalent merit	1

Sample answer:
$z=e^{i\theta}\Rightarrow\frac1z=e^{-i\theta}$
$w=\frac{z^2-1}{z^2+1}=\frac{z\left(z-\frac1z\right)}{z\left(z+\frac1z\right)}$

=\frac{e^{i\theta}-e^{-i\theta}}{e^{i\theta}+e^{-i\theta}} =\frac{\cos\theta+i\sin\theta-\cos\theta+i\sin\theta}{\cos\theta+i\sin\theta+\cos\theta-i\sin\theta} =\frac{2i\sin\theta}{2\cos\theta}=i\tan\theta

which is purely imaginary.

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Syllabus alignment

ProofComplex Numbers

The Nature of ProofUse formal proof language, contradiction, counter-examples and rigorous arguments involving numbers and inequalities.
Geometrical implications of complex numbersInterpret complex-number operations geometrically and solve problems involving roots and loci in the complex plane.

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