Complex Loci and Modulus NCEA Practice Questions | Calc.nz

Skill overview

What these questions practise

Choose a question, attempt it independently, then compare the reasoning and restrictions in its walkthrough.

Relevant standards: NCEA Level 3 Complex Numbers — AS91577

This collection contains 31 matching walkthroughs across 9 examination years.

The grouping is based on the method recorded for each walkthrough. Use the linked NZQA standard and paper material for authoritative wording, diagrams, and assessment information.

Before you start

Common mistakes in these methods

A modulus is a nonnegative distance. Translate the original distance statement carefully and check any sign restriction introduced by squaring.

Related practice

Complex Loci and Modulus questions from different years

Open a question for hints and a progressive worked solution, or use the standard and paper links to return to its assessment context.

NCEA Level 3 · AS91577 2025 Complex Numbers 3 questions

Open a question or return to its assessment context.

Question 1(e) — Complex-number Loci Converting a modulus locus into a Cartesian equation. Question 2(a) — Argand Diagram Algebra Combining complex numbers from an Argand diagram. Question 2(c) — Modulus Equations Solving a modulus equation with a real parameter.

NCEA Level 3 · AS91577 2024 Complex Numbers 4 questions

Open a question or return to its assessment context.

Question 1(d) — Complex-number Loci Turning a modulus locus into a line. Question 2(c) — Complex Modulus Solving for w before finding |w|. Question 2(e) — Complex-number Loci Finding a circle locus and the matching complex numbers. Question 3(b) — Argand Diagram Algebra Squaring a plotted complex number on an Argand diagram.

NCEA Level 3 · AS91577 2023 Complex Numbers 3 questions

Open a question or return to its assessment context.

Question 1(e) — Modulus Equations Simplifying u divided by v and solving a modulus equation in k. Question 2(b) — Complex Modulus Finding a modulus in terms of a real parameter. Question 2(e) — Complex-number Loci Turning a locus into a circle and using tangency.

NCEA Level 3 · AS91577 2022 Complex Numbers 3 questions

Open a question or return to its assessment context.

Question 1(e) — Complex-number Algebra Proving an identity involving moduli and real parts. Question 2(e) — Complex-number Loci Turning a modulus locus into a circle equation. Question 3(a) — Argand Diagram Algebra Combining Argand-diagram points and plotting the result.

NCEA Level 3 · AS91577 2021 Complex Numbers 4 questions

Open a question or return to its assessment context.

Question 1(b) — Argand Diagram Algebra Rationalising 26 divided by z and plotting 4 - 6i on an Argand diagram. Question 1(e) — Complex-number Loci Proving a circular locus in Cartesian form. Question 2(e) — Modulus Equations Turning a modulus equation into the circle |z| = 4. Question 3(a) — Polar Form Using modulus and argument to choose the correct value of m.

NCEA Level 3 · AS91577 2020 Complex Numbers 4 questions

Open a question or return to its assessment context.

Question 2(b) — Modulus Equations Solving a modulus equation for all real values of k. Question 2(e) — Complex-number Loci Converting a modulus locus into a Cartesian circle. Question 3(b) — Complex Modulus Comparing squared moduli directly. Question 3(e) — Complex-number Algebra Proving a quotient is purely imaginary from equal moduli.

NCEA Level 3 · AS91577 2019 Complex Numbers 3 questions

Open a question or return to its assessment context.

Question 1(c) — Complex-number Algebra Rationalising a complex quotient and applying its modulus. Question 2(e) — Complex-number Algebra Proving a unit-circle quotient is purely imaginary. Question 3(a) — Polar Form Dividing moduli and subtracting arguments in polar form.

NCEA Level 3 · AS91577 2018 Complex Numbers 5 questions

Open a question or return to its assessment context.

Question 1(b) — Arguments and Quadrants Multiplying moduli and adding arguments in cis form. Question 1(e) — Complex-number Algebra Rationalising z/ overline z and proving its modulus is 1. Question 2(a) — Argand Diagram Algebra Conjugating v, adding complex numbers, and plotting w on an Argand diagram. Question 2(d) — Complex-number Algebra Rationalising a complex fraction and finding its modulus. Question 2(e) — Complex-number Loci Using an equidistant locus to find the imaginary coordinate m.

NCEA Level 3 · AS91577 2017 Complex Numbers 2 questions

Open a question or return to its assessment context.

Question 1(e) — Complex-number Loci Converting a modulus locus into the Cartesian equation of a circle. Question 3(a) — Polar Form Dividing moduli and subtracting arguments in polar form.

Keep practising

Page updated 24 September 2026.