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

Question 1(a) · Polynomial Algebra Polynomial division and using the remainder to find \(p\). Question 1(b) · Completing the square and solving in terms Completing the square and solving in terms of \(k\). Question 2(a) · Combining complex numbers from an Argand Combining complex numbers from an Argand diagram. Question 2(d) · Matching real and imaginary parts after 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 and simplifying a complex Squaring and simplifying a complex expression. Question 3(c) · Radical Substitution Solving a radical equation with the substitution \(\sqrt{x}=u\). Question 3(d) · Polynomial Algebra Factorising a cubic and matching coefficients. Question 3(e) · Simultaneous equations with complex Solving simultaneous equations with complex numbers.

NCEA Level 3 · AS91577

2024 Complex Numbers

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) · A complex quotient cannot lie on y=x Showing a complex quotient cannot lie on \(y=x\). Question 2(a) · A complex fraction as a+bi Rewriting a complex fraction as \(a+bi\). Question 2(b) · Equal roots and the discriminant 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) · Squaring a plotted complex number on an Squaring a plotted complex number on an Argand diagram.

NCEA Level 3 · AS91577

2023 Complex Numbers

Question 1(a) · Radicals and Surds Expanding a surd binomial into the form \(a+bp+c\sqrt{p}\). Question 1(b) · Quadratic Algebra Using the discriminant to find when a quadratic has no real roots. Question 1(c) · Rationalising a quotient and reading its 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) · Multiplying by a complex number to solve 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) · Reciprocal complex expressions to find x Simplifying reciprocal complex expressions to find \(x\) and \(y\).

NCEA Level 3 · AS91577

2022 Complex Numbers

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) · An argument condition to match real and 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) · Rationalising first, then finding the Rationalising first, then finding the principal argument. Question 3(b) · Completing the square to solve in terms Completing the square to solve in terms of \(k\). Question 3(d) · That a complex equation has no solution Proving that a complex equation has no solution.

NCEA Level 3 · AS91577

2021 Complex Numbers

Question 1(a) · Multiplying complex numbers and matching Multiplying complex numbers and matching real or imaginary parts. Question 1(b) · Rationalising fraction 26z and plotting Rationalising \(\frac{26}{z}\) 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) · Dividing complex numbers in cis form Dividing complex numbers in \(\operatorname{cis}\) form. Question 2(b) · Quadratic Algebra 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 \(a+b\sqrt{3}\). 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) · A complex equation by equating real and Solving a complex equation by equating real and imaginary parts.

NCEA Level 3 · AS91577

2020 Complex Numbers

Question 1(a) · Multiplying complex numbers and matching Multiplying complex numbers and matching corresponding parts. Question 1(b) · A zero discriminant to find a repeated 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) · Rationalising and simplifying complex Rationalising and simplifying complex fractions. Question 1(e) · An identity involving a complex quotient 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

Question 1(a) · Quadratic Algebra Completing the square to solve a complex quadratic. Question 1(b) · Remainder Theorem Applying the Remainder Theorem to find p. Question 1(c) · Rationalising a complex quotient and 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 after 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) · Making a rationalised complex quotient Making a rationalised complex quotient purely real. Question 3(b) · For real x and y by matching 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) · A discriminant is always positive Proving a discriminant is always positive.

NCEA Level 3 · AS91577

2018 Complex Numbers

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) · Conjugating v, adding complex numbers, Conjugating \(v\), adding complex numbers, and plotting \(w\) on an Argand diagram. Question 2(b) · Radicals and Surds Rationalising a surd denominator into \(a+b\sqrt7\). Question 2(c) · Complex-number Algebra Using conjugate roots and cubic factorisation to find \(A\) and the remaining roots. Question 2(d) · Rationalising a complex fraction and Rationalising a complex fraction and finding its modulus. Question 3(a) · Expanding a complex product and matching Expanding a complex product and matching real and imaginary parts. Question 3(b) · Quadratic Algebra 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

Question 1(a) · Conjugating a complex number, 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) · Rationalising a complex quotient and Rationalising a complex quotient and making it purely real. Question 3(b) · Quadratic Algebra Completing the square to find complex quadratic roots. Question 3(c) · Polynomial Algebra Complete polynomial long division with quotient and remainder. Question 3(d) · And rationalising an equation with square Solving and rationalising an equation with \(\sqrt3\). Question 3(e) · A complex fraction identity step by step Proving a complex fraction identity step by step.

Keep practising

Polar form and De Moivre

Page updated 19 July 2026.