YEAR 12 · COMPLEX NUMBERS

Topic review.

Connect the key ideas, then choose a practice paper.

01 / QUICK RECAP

Connect the ideas.

For nonzero $z = x + i y$ , $∣ z ∣ = \sqrt{x^{2} + y^{2}}$ and $z = ∣ z ∣ cis θ$ . Our principal-argument convention is $- π < Arg z \leq π$ .
Add in Cartesian form; multiply or divide conveniently in polar form. Multiply moduli and add arguments, or divide moduli and subtract arguments.
Multiplication by $ρ cis ϕ$ scales distances from the origin by $ρ$ and rotates anticlockwise through $ϕ$ .
De Moivre applies to integer powers. The nth roots of a nonzero number have equal modulus and angles separated by $2 π / n$ .
An nth root of unity is $cis (2 k π / n)$ . Their regular polygon links exact sums, products and geometry.
$∣ z - a ∣$ is the distance from a. A fixed distance gives a circle, equal distances give a perpendicular bisector, and a fixed argument gives a ray with its starting point excluded.
Complete factorisation over $ℂ$ uses linear factors and counts multiplicities. Real coefficients force non-real roots to occur in conjugate pairs.

02 / PRACTICE TESTS

Put it together.

Choose a paper and show your working.

Complex Numbers — Paper 1
A balanced first paper on exact forms, arithmetic, powers, roots and simple geometry.
10 questions · 51 marks · 70 minutes · Scientific calculator
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Complex Numbers — Paper 2
A second balanced paper with signals, conjugates, roots, loci and reconstruction of polynomials.
10 questions · 55 marks · 70 minutes · Scientific calculator
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Complex Numbers — Paper 3
Reasoning and proof: inspect claims, derive identities and connect algebra with geometry.
10 questions · 56 marks · 70 minutes · By hand
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Complex Numbers — Paper 4
Roots and factorisations, from recognising a square to exact root geometry and real coefficient constraints.
10 questions · 59 marks · 70 minutes · Scientific calculator
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Complex Numbers — Paper 5
Applications and geometry: transformations, distance constraints, phase relations and coefficient models.
10 questions · 56 marks · 70 minutes · Scientific calculator
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Complex Numbers — Paper 6
A challenging synthesis paper with exact roots, geometric restrictions, proof and polynomial reconstruction.
10 questions · 68 marks · 75 minutes · Scientific calculator
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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Specialist Mathematics complex numbers: Cartesian and polar representations, de Moivre, roots and roots of unity, Argand geometry and loci, complex factorisation and conjugate roots of real polynomials. Real-polynomial techniques are taught in the separate Real Polynomials chapter.

Official curriculum source ↗

Connect the ideas.

Put it together.

Complex Numbers — Paper 1

Complex Numbers — Paper 2

Complex Numbers — Paper 3

Complex Numbers — Paper 4

Complex Numbers — Paper 5

Complex Numbers — Paper 6