NCEA Maths Walkthroughs

Maths Exam Walkthroughs

Step through NCEA maths questions and build the reasoning behind strong exam answers.

Each walkthrough breaks a question into manageable steps, with feedback that explains why each move works.

Use the papers to practise the thinking needed for Achieved, Merit, and Excellence responses, not just to check the final answer.

Start with a level below, then choose the standard and paper you want to practise.

Start practising

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Step 1: Choose a level

Open Level 2 or Level 3 to see the standards and papers available.

Select a level to show its standards, or select it again to hide them.

Level 2

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Choose the Level 2 standard you want to work on first.

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Calculus

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Select the Level 2 Calculus paper you want to open.

Work through the 2025 Level 2 Calculus walkthroughs question by question.

Question 1

Question 1(a) Focus: using the power rule term by term and evaluating the gradient at a point. Question 1(b) Focus: finding the point of tangency, the gradient, and the tangent equation. Question 1(c) Focus: finding critical points, identifying the minimum, and solving for the constant. Question 1(d) Focus: writing displacement equations, solving when they are equal, and answering in context.

Question 2

Question 2(a) Focus: sketching the derivative from stationary points, signs, and changes in concavity. Question 2(b) Focus: antidifferentiating and using a point on the graph to find the constant. Question 2(c) Focus: differentiating, evaluating a rate, and using derivatives to justify the minimum turning point. Question 2(d) Focus: using a stationary point and an x-intercept to find constants, then classifying the other stationary point.

Question 3

Question 3(a) Focus: differentiating, solving for the given gradient, and finding the coordinate on the curve. Question 3(b) Focus: forming an area expression, finding the height from the given area, and evaluating the rate of change. Question 3(c) Focus: using a surface-area constraint, writing volume in one variable, maximizing it, and proving the maximum. Question 3(d) Focus: differentiating, factorising, classifying critical points, and finding where the function is decreasing.

Algebra

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Select the Level 2 Algebra paper you want to open.

Work through the 2025 Level 2 Algebra walkthroughs question by question.

Question 1

Question 1(a) Focus: simplifying a radical expression. Question 1(b) Focus: rearranging an equation to make x the subject. Question 1(c) Focus: building a quadratic from a root and a point on the graph. Question 1(d) Focus: using Pythagoras to write area in terms of the radius. Question 1(e) Focus: solving an exponential equation with a substitution.

Question 2

Question 2(a) Focus: identifying the missing term in a perfect square. Question 2(b) Focus: factorising and simplifying an algebraic fraction. Question 2(c) Focus: solving a quadratic and matching it to a surd form. Question 2(d) Focus: combining logarithms and solving the resulting equation. Question 2(e) Focus: using the discriminant in simultaneous equations.

Question 3

Question 3(a) Focus: solving a logarithm with a variable base. Question 3(b) Focus: forming a quadratic equation from its roots. Question 3(c) Focus: rewriting a logarithm in terms of given values. Question 3(d) Focus: using root relationships to rewrite an expression. Question 3(e) Focus: modelling a golf shot with a quadratic and interpreting the result.

Level 3

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Differentiation

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Select the Level 3 Differentiation paper you want to open.

Work through the 2025 Level 3 Differentiation walkthroughs question by question.

Question 1

Question 1(a) Focus: chain rule differentiation of a composite polynomial. Question 1(b) Focus: rewriting in power form and finding a rate of change. Question 1(c) Focus: product rule and the gradient of a normal. Question 1(d) Focus: finding a tangent and its x-intercept. Question 1(e) Focus: parametric differentiation with stationary inflection points.

Question 2

Question 2(a) Focus: differentiating reciprocal and logarithmic terms. Question 2(b) Focus: product rule and deciding whether a model is increasing or decreasing. Question 2(c) Focus: quotient rule and tangents with a given gradient. Question 2(d) Focus: parametric gradients and tangent equations. Question 2(e) Focus: first and second derivatives in a radius of curvature problem.

Question 3

Question 3(a) Focus: reading differentiability, stationary points, and a limit from a graph. Question 3(b) Focus: stationary points on a logarithmic curve. Question 3(c) Focus: related rates for the volume of a sphere. Question 3(d) Focus: solving a parametric gradient condition. Question 3(e) Focus: maximising the area of a rectangle inside a curve.

Work through the 2024 Level 3 Differentiation walkthroughs question by question.

Question 1

Question 1(a) Focus: rewriting a square root as a power and applying the chain rule. Question 1(b) Focus: product rule differentiation and evaluating the gradient at \(x=0\). Question 1(c) Focus: differentiating, solving \(f'(x)=0\), and finding where a function is decreasing. Question 1(d) Focus: stationary points, the second derivative test, and classifying the point. Question 1(e) Focus: locating a point of inflection and finding the tangent there.

Question 2

Question 2(a) Focus: parametric differentiation using \(\frac{dy/dt}{dx/dt}\). Question 2(b) Focus: chain rule differentiation of a logarithmic displacement function. Question 2(c) Focus: first and second derivatives in a proof-style differential-equation question. Question 2(d) Focus: finding a point of inflection on \(\frac{\ln x}{x}\). Question 2(e) Focus: using a discriminant condition to force a single turning point.

Question 3

Question 3(a) Focus: product rule with \(\sqrt{x}\) and a trig chain rule. Question 3(b) Focus: reading continuity, differentiability, and limits from a graph. Question 3(c) Focus: quotient rule differentiation and finding stationary points. Question 3(d) Focus: related rates for the height of a conical pile. Question 3(e) Focus: forming an area function and proving its largest possible value.

Work through the 2023 Level 3 Differentiation walkthroughs question by question.

Question 1

Question 1(a) Focus: rewriting a square root and applying the chain rule. Question 1(b) Focus: product rule differentiation and evaluating a rate of change. Question 1(c) Focus: using derivatives to find a parallel tangent on a rational curve. Question 1(d) Focus: parametric differentiation and building a tangent equation. Question 1(e) Focus: maximising a triangle area on the curve \(y=x(x-2m)^2\).

Question 2

Question 2(a) Focus: rewriting \(\frac{x^2}{\cos x}\) and using the product rule. Question 2(b) Focus: chain rule differentiation of \(y=\cot(2x)\). Question 2(c) Focus: quotient rule differentiation and horizontal tangents. Question 2(d) Focus: second derivatives and finding a logarithmic point of inflection. Question 2(e) Focus: related rates for a helicopter chasing a car.

Question 3

Question 3(a) Focus: the chain rule inside a logarithm. Question 3(b) Focus: reading continuity, concavity, and limits from a graph. Question 3(c) Focus: parametric differentiation for a Ferris wheel model. Question 3(d) Focus: stationary points and classifying them with the second derivative. Question 3(e) Focus: verifying a differential equation from a catenary model.

Work through the 2022 Level 3 Differentiation walkthroughs question by question.

Question 1

Question 1(a) Focus: differentiation of the function. Question 1(b) Focus: finding and interpreting stationary points. Question 1(c) Focus: the normal line and its x-intercept. Question 1(d) Focus: parametric differentiation. Question 1(e) Focus: proving there are no points of inflection.

Question 2

Question 2(a) Focus: product rule with a trig chain rule. Question 2(b) Focus: chain rule and finding a tangent gradient. Question 2(c) Focus: quotient rule and finding when motion is stationary. Question 2(d) Focus: forming an area function and maximising it. Question 2(e) Focus: implicit differentiation and tangent-line distance.

Question 3

Question 3(a) Focus: rewriting radicals and using the chain rule. Question 3(b) Focus: reading differentiability, derivatives, and limits from a graph. Question 3(c) Focus: related rates using volume and height. Question 3(d) Focus: finding stationary points and classifying them. Question 3(e) Focus: building a time function and minimising it.

Complex Numbers

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Select the Level 3 Complex Numbers paper you want to open.

Work through the 2025 Level 3 Complex Numbers walkthroughs question by question.

Question 1

Question 1(a) Focus: polynomial division and using the remainder to find \(p\). Question 1(b) Focus: completing the square and solving in terms of \(k\). Question 1(c) Focus: discriminants and proving real roots for every real \(k\). Question 1(d) Focus: writing cube roots in polar form. Question 1(e) Focus: converting a modulus locus into a Cartesian equation.

Question 2

Question 2(a) Focus: combining complex numbers from an Argand diagram. Question 2(b) Focus: De Moivre’s Theorem and converting to \(a+bi\). Question 2(c) Focus: solving a modulus equation with a real parameter. Question 2(d) Focus: matching real and imaginary parts after expansion. Question 2(e) Focus: using an argument condition to solve for a real constant.

Question 3

Question 3(a) Focus: conjugate roots and forming a quadratic with real coefficients. Question 3(b) Focus: squaring and simplifying a complex expression. Question 3(c) Focus: solving a radical equation with the substitution \(\sqrt{x}=u\). Question 3(d) Focus: factorising a cubic and matching coefficients. Question 3(e) Focus: solving simultaneous equations with complex numbers.

Work through the 2024 Level 3 Complex Numbers walkthroughs question by question.

Question 1

Question 1(a) Focus: using the factor theorem to find \(p\). Question 1(b) Focus: De Moivre’s Theorem in polar form. Question 1(c) Focus: solving a surd equation in terms of \(k\). Question 1(d) Focus: turning a modulus locus into a line. Question 1(e) Focus: showing a complex quotient cannot lie on \(y=x\).

Question 2

Question 2(a) Focus: rewriting a complex fraction as \(a+bi\). Question 2(b) Focus: equal roots and the discriminant. Question 2(c) Focus: solving for \(w\) before finding \(|w|\). Question 2(d) Focus: conjugate roots, factorisation, and finding \(d\). Question 2(e) Focus: finding a circle locus and the matching complex numbers.

Question 3

Question 3(a) Focus: rationalising a surd denominator. Question 3(b) Focus: squaring a plotted complex number on an Argand diagram. Question 3(c) Focus: using \(z\), \(\overline{z}\), and \(z^{-1}\) to solve for \(d\). Question 3(d) Focus: fourth roots in polar form. Question 3(e) Focus: cubing \(x+\frac{1}{x}\) to find \(x^3+\frac{1}{x^3}\).