HSC Maths Extension 2 complex numbers

Write z in polar form, r(cos θ + i sin θ), and multiplying multiplies the moduli and adds the arguments. That one idea gives de Moivre's theorem for powers, the n equally spaced nth roots, and multiplication as a rotation and an enlargement in the complex plane.

The paper	One written paper, 100 marks, 3 hours plus 10 minutes reading time
Section I	Objective-response questions worth 10 marks
Section II	90 marks in 37 to 42 items; at least two items worth 4 or 5 marks
Provided	A reference sheet; NESA-approved calculators may be used

The ideas that carry the focus area

Polar form: the modulus |z| and the principal argument Arg(z), the one argument in the interval from minus pi (not included) to pi.
De Moivre's theorem: (cos θ + i sin θ) to the power n is cos nθ + i sin nθ, proved by induction for positive integers and true for every integer n.
The nth roots of a complex number have the same modulus and arguments 2π/n apart, so they sit at the vertices of a regular n-sided polygon.
A polynomial with real coefficients has its non-real roots in conjugate pairs, so a known root gives a quadratic factor.

Try 3 questions

Section I · 1 mark · Complex numbers: arithmetic and polar form
Let $z = 5 - 12 i$ . What is $\mathrm{Im}\left(\dfrac{1}{z}\right)$ ?
Answer it, every step marked
Section II · 7 marks · Complex numbers: arithmetic and polar form
Let $z = 3 + 4 i$ and $w = 1 - 2 i$ .
Answer it, every step marked
Section I · 1 mark · Complex numbers: arithmetic and polar form
What are the two square roots of $-5 + 12 i$ ?
Answer it, every step marked

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Questions people ask

Is the principal argument between 0 and 2π?

No. The principal argument Arg(z) is the value of the argument in the interval from minus pi, not included, to pi, included.

Are complex numbers in both sections?

They can be. Both Section I and Section II are set on the whole Year 12 course.