Polar Form and De Moivre’s Theorem NCEA Practice Questions | Calc.nz

Skill overview

What these questions practise

The matched AS91577 methods explicitly mention polar form, cis form, arguments, or De Moivre's theorem. This excludes generic modulus and Argand-diagram questions that do not identify a polar method.

Relevant standards: NCEA Level 3 Complex Numbers — AS91577

This collection contains 27 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

Choose the argument from the correct quadrant and within the range required by the question.
When dividing in polar form, divide the moduli and subtract the arguments in a consistent order.
When multiplying in polar form, multiply the moduli and add the arguments before normalising the angle if required.

Related practice

Polar Form and De Moivre’s Theorem 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

Question 1(d) · Polar Form Writing cube roots in polar form. Question 2(b) · De Moivre’s Theorem De Moivre’s Theorem and converting to \(a+bi\). Question 2(e) · An argument condition to solve for a Using an argument condition to solve for a real constant.

NCEA Level 3 · AS91577

2024 Complex Numbers

Question 1(b) · De Moivre’s Theorem De Moivre’s Theorem in polar form. Question 3(d) · Polar Form Fourth roots in polar form.

NCEA Level 3 · AS91577

2023 Complex Numbers

Question 2(a) · Polar Form Dividing complex numbers in polar form. Question 2(c) · For z before finding its argument Solving for \(z\) before finding its argument. Question 2(d) · Polar Form Writing all cube roots in polar form.

NCEA Level 3 · AS91577

2022 Complex Numbers

Question 1(b) · Polar Form Dividing complex numbers in polar form. Question 1(c) · An argument condition to match real and Using an argument condition to match real and imaginary parts. Question 2(d) · Rationalising first, then finding the Rationalising first, then finding the principal argument. Question 3(c) · Polar Form Finding the three cube roots in polar form.

NCEA Level 3 · AS91577

2021 Complex Numbers

Question 2(a) · Dividing complex numbers in cis form Dividing complex numbers in \(\operatorname{cis}\) form. Question 2(d) · Complex Roots Finding cube roots in \(\operatorname{cis}\) form. Question 3(a) · Polar Form Using modulus and argument to choose the correct value of \(m\).

NCEA Level 3 · AS91577

2020 Complex Numbers

Question 2(d) · The principal argument of a quotient Finding the principal argument of a quotient. Question 3(a) · Polar Form Dividing complex numbers in polar form. Question 3(d) · Polar Form Finding all fourth roots in polar form.

NCEA Level 3 · AS91577

2019 Complex Numbers

Question 2(c) · Polar Form Finding fourth roots in polar form for every real p. Question 3(a) · Polar Form Dividing moduli and subtracting arguments in polar form. Question 3(d) · An argument condition and verifying the Using an argument condition and verifying the quadrant.

NCEA Level 3 · AS91577

2018 Complex Numbers

Question 1(b) · Multiplying moduli and adding arguments Multiplying moduli and adding arguments in cis form. Question 3(c) · De Moivre’s Theorem Finding all cube roots in polar form with De Moivre's theorem. Question 3(d) · Complex-number Algebra Combining an argument condition with \(w\overline w=|w|^2\).

NCEA Level 3 · AS91577

2017 Complex Numbers

Question 2(c) · Complex-number Algebra Exact complex division and choosing the principal argument. Question 2(d) · Polar Form Finding and listing all four fourth roots in polar form. Question 3(a) · Polar Form Dividing moduli and subtracting arguments in polar form.

Keep practising

Complex-number algebra

Page updated 19 July 2026.