2022 NCEA Level 3 Complex Numbers

2022 Complex Numbers AS91577 worked answers

Guided, step-by-step worked solutions for Apply the algebra of complex numbers in solving problems.

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Paper overview

Work through the 2022 questions one step at a time

This page contains 15 independent worked question walkthroughs for NCEA Level 3 Complex Numbers (AS91577). Each walkthrough offers hints, reveals the full working in a logical sequence, and focuses on the method described below.

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Official 2022 NZQA Complex Numbers material

Calc.nz is independent of NZQA. Use the official NZQA AS91577 assessment resources for the original paper, diagrams, assessment schedule, and authoritative standard information.

Worked question index

2022 Complex Numbers questions

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Question 1

Question 1 walkthroughs

Question 1(a) worked solution Rationalising a surd denominator into the required exact form. Question 1(b) worked solution Dividing complex numbers in polar form. Question 1(c) worked solution Using an argument condition to match real and imaginary parts. Question 1(d) worked solution Isolating a surd and using the discriminant to find \(p\). Question 1(e) worked solution Proving an identity involving moduli and real parts.

Question 2

Question 2 walkthroughs

Question 2(a) worked solution Using the remainder theorem to find \(b\). Question 2(b) worked solution Solving for \(z\) using \(z\) and \(\overline{z}\). Question 2(c) worked solution Factorising a cubic with a known root and finding \(h\). Question 2(d) worked solution Rationalising first, then finding the principal argument. Question 2(e) worked solution Turning a modulus locus into a circle equation.

Question 3

Question 3 walkthroughs

Question 3(a) worked solution Combining Argand-diagram points and plotting the result. Question 3(b) worked solution Completing the square to solve in terms of \(k\). Question 3(c) worked solution Finding the three cube roots in polar form. Question 3(d) worked solution Proving that a complex equation has no solution. Question 3(e) worked solution Solving for \(a\) and \(b\) from \(z\) and \(\overline{z}\).

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Complex-number algebra Polar form and De Moivre

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