Complex-number Algebra NCEA Practice Questions | Calc.nz

Skill overview

What these questions practise

The matched AS91577 methods explicitly identify complex-number operations or the polynomial, discriminant, conjugate, rationalising, surd, and radical techniques used in the standard.

Relevant standards: NCEA Level 3 Complex Numbers — AS91577

This collection contains 88 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

When using a conjugate, change only the sign of the imaginary part and multiply the complete numerator and denominator.
Collect real and imaginary components separately before equating their corresponding coefficients.
Keep the divisor, root, and remainder signs consistent, then verify the factorisation or reconstructed polynomial.

Related practice

Complex-number Algebra 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 9 questions

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Question 1(a) — Polynomial Algebra Polynomial division and using the remainder to find p. Question 1(b) — Completing the Square Completing the square and solving in terms of k. Question 2(a) — Argand Diagram Algebra Combining complex numbers from an Argand diagram. Question 2(d) — Equating Real and Imaginary Parts Matching real and imaginary parts after expansion. Question 3(a) — Quadratic Algebra Conjugate roots and forming a quadratic with real coefficients. Question 3(b) — Squaring a Complex Expression Squaring and simplifying a complex expression. Question 3(c) — Radical Substitution Solving a radical equation with the substitution square root of x = u. Question 3(d) — Polynomial Algebra Factorising a cubic and matching coefficients. Question 3(e) — Complex Simultaneous Equations Solving simultaneous equations with complex numbers.

NCEA Level 3 · AS91577 2024 Complex Numbers 9 questions

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Question 1(a) — Factor Theorem Using the factor theorem to find p. Question 1(c) — Radicals and Surds Solving a surd equation in terms of k. Question 1(e) — Complex-number Algebra Showing a complex quotient cannot lie on y = x. Question 2(a) — Complex-number Algebra Rewriting a complex fraction as a + bi. Question 2(b) — Discriminants and Real Roots Equal roots and the discriminant. Question 2(d) — Complex-number Algebra Conjugate roots, factorisation, and finding d. Question 2(e) — Complex-number Loci Finding a circle locus and the matching complex numbers. Question 3(a) — Radicals and Surds Rationalising a surd denominator. Question 3(b) — Argand Diagram Algebra Squaring a plotted complex number on an Argand diagram.

NCEA Level 3 · AS91577 2023 Complex Numbers 9 questions

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Question 1(a) — Radicals and Surds Expanding a surd binomial into the form a + bp + c square root of p. Question 1(b) — Quadratic Algebra Using the discriminant to find when a quadratic has no real roots. Question 1(c) — Complex-number Algebra Rationalising a quotient and reading its real part. Question 1(d) — Complex-number Algebra Conjugate roots and factorising a cubic with a real constant. Question 2(a) — Polar Form Dividing complex numbers in polar form. Question 3(a) — Remainder Theorem Using the remainder theorem to find p. Question 3(b) — Complex-number Algebra Multiplying by a complex number to solve for n. Question 3(c) — Radicals and Surds Solving a surd equation for x in terms of w. Question 3(d) — Complex-number Algebra Simplifying reciprocal complex expressions to find x and y.

NCEA Level 3 · AS91577 2022 Complex Numbers 9 questions

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Question 1(a) — Radicals and Surds Rationalising a surd denominator into the required exact form. Question 1(b) — Polar Form Dividing complex numbers in polar form. Question 1(c) — Equating Real and Imaginary Parts Using an argument condition to match real and imaginary parts. Question 1(d) — Radicals and Surds Isolating a surd and using the discriminant to find p. Question 2(a) — Remainder Theorem Using the remainder theorem to find b. Question 2(c) — Polynomial Algebra Factorising a cubic with a known root and finding h. Question 2(d) — Complex-number Algebra Rationalising first, then finding the principal argument. Question 3(b) — Completing the Square Completing the square to solve in terms of k. Question 3(d) — Complex-number Algebra Proving that a complex equation has no solution.

NCEA Level 3 · AS91577 2021 Complex Numbers 10 questions

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Question 1(a) — Complex-number Algebra Multiplying complex numbers and matching real or imaginary parts. Question 1(b) — Argand Diagram Algebra Rationalising 26 divided by z and plotting 4 - 6i on an Argand diagram. Question 1(c) — Factor Theorem Using the remainder theorem and factor theorem to find a and b. Question 2(a) — Polar Form Dividing complex numbers in cis form. Question 2(b) — Completing the Square Completing the square to solve a quadratic in terms of q. Question 2(c) — Complex-number Algebra Using conjugate multiplication to prove a quotient is purely imaginary. Question 3(b) — Radicals and Surds Rationalising a surd denominator into a + b square root of 3. Question 3(c) — Complex-number Algebra Conjugate roots, cubic factorisation, and finding A. Question 3(d) — Radicals and Surds Solving a radical equation in terms of m. Question 3(e) — Equating Real and Imaginary Parts Solving a complex equation by equating real and imaginary parts.

NCEA Level 3 · AS91577 2020 Complex Numbers 9 questions

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Question 1(a) — Complex-number Algebra Multiplying complex numbers and matching corresponding parts. Question 1(b) — Discriminants and Real Roots Using a zero discriminant to find a repeated root. Question 1(c) — Radicals and Surds Solving a surd equation in terms of g. Question 1(d) — Complex-number Algebra Rationalising and simplifying complex fractions. Question 1(e) — Complex-number Algebra Proving an identity involving a complex quotient. Question 2(a) — Factor Theorem Applying the factor theorem to find q. Question 2(c) — Complex-number Algebra Conjugate roots, cubic factorisation, and coefficient matching. Question 3(a) — Polar Form Dividing complex numbers in polar form. Question 3(c) — Complex-number Algebra Using conjugates to prove a quotient is real.

NCEA Level 3 · AS91577 2019 Complex Numbers 11 questions

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Question 1(a) — Completing the Square Completing the square to solve a complex quadratic. Question 1(b) — Remainder Theorem Applying the Remainder Theorem to find p. Question 1(c) — Complex-number Algebra Rationalising a complex quotient and applying its modulus. Question 1(d) — Complex-number Algebra Conjugate roots, cubic factorisation, and coefficient matching. Question 1(e) — Equating Real and Imaginary Parts Equating real and imaginary parts after clearing denominators. Question 2(a) — Complex-number Algebra Using a conjugate and collecting complex-number terms. Question 2(b) — Radicals and Surds Rationalising a surd denominator into a + b√5. Question 2(d) — Complex-number Algebra Making a rationalised complex quotient purely real. Question 3(b) — Complex-number Algebra Solving for real x and y by matching complex parts. Question 3(c) — Radicals and Surds Solving a radical equation with real-domain checks. Question 3(e) — Discriminants and Real Roots Proving a discriminant is always positive.

NCEA Level 3 · AS91577 2018 Complex Numbers 11 questions

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Question 1(a) — Remainder Theorem Applying the Remainder Theorem to evaluate f(2). Question 1(c) — Radicals and Surds Solving a radical equation in terms of k with real-domain checks. Question 1(d) — Quadratic Algebra Using a zero discriminant to find when a quadratic has a repeated root. 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(b) — Radicals and Surds Rationalising a surd denominator into a + b square root of 7. Question 2(c) — Complex-number Algebra Using conjugate roots and cubic factorisation to find A and the remaining roots. Question 2(d) — Complex-number Algebra Rationalising a complex fraction and finding its modulus. Question 3(a) — Equating Real and Imaginary Parts Expanding a complex product and matching real and imaginary parts. Question 3(b) — Completing the Square Completing the square to solve a quadratic in terms of p. Question 3(e) — Radicals and Surds Rationalising a radical quotient and checking the real domain.

NCEA Level 3 · AS91577 2017 Complex Numbers 11 questions

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Question 1(a) — Complex-number Algebra Conjugating a complex number, distributing a real factor, and collecting parts. Question 1(b) — Radicals and Surds Rationalising a surd denominator with a difference of two squares. Question 1(c) — Radicals and Surds Solving a radical equation with domain and parameter restrictions. Question 1(d) — Complex-number Algebra Conjugate roots, Vieta's formula, and cubic factorisation. Question 2(a) — Remainder Theorem Applying the Remainder Theorem to find a constant. Question 2(c) — Complex-number Algebra Exact complex division and choosing the principal argument. Question 2(e) — Complex-number Algebra Rationalising a complex quotient and making it purely real. Question 3(b) — Completing the Square Completing the square to find complex quadratic roots. Question 3(c) — Polynomial Algebra Complete polynomial long division with quotient and remainder. Question 3(d) — Complex-number Algebra Solving and rationalising an equation with square root of 3. Question 3(e) — Complex-number Algebra Proving a complex fraction identity step by step.

Keep practising

Polar form and De Moivre

Page updated 9 August 2026.