NCEA Level 2 and Level 3 Maths

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Guided walkthroughs for Level 2 and Level 3 questions that build Achieved, Merit, and Excellence reasoning.

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Achieved to Excellence thinking built through reasoning, not just final answers.

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From picking a paper to understanding the thinking behind the marks.

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What you get

Guided hints Exam-style questions Reasoning-first feedback

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Step 1: Choose a level

Choose Level 2 or Level 3 to browse the standards and papers available.

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Level 2

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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.

Integration

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Select the Level 3 Integration paper you want to open.

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

Question 1

Question 1(a) Focus: recognising a reverse derivative with a sec-tan pattern. Question 1(b) Focus: integrating a derivative and using an initial condition. Question 1(c) Focus: linking two definite integrals to solve for constants. Question 1(d) Focus: using product-to-sum before integrating a shaded trig area. Question 1(e) Focus: separating variables and evaluating the resulting model.

Question 2

Question 2(a) Focus: reverse chain rule integration of a linear power. Question 2(b) Focus: integrating a rate function and fitting the constant. Question 2(c) Focus: using a definite integral to determine a constant. Question 2(d) Focus: building velocity and displacement from acceleration with initial conditions. Question 2(e) Focus: substitution in a shaded trigonometric area problem.

Question 3

Question 3(a) Focus: estimating cross-sectional area with the trapezium rule. Question 3(b) Focus: logarithmic antiderivatives with a linear inside. Question 3(c) Focus: solving a separable differential equation with a radical. Question 3(d) Focus: proving a cubic relation from a shaded area. Question 3(e) Focus: using weighted-average integrals to find a balance point.

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

Question 1

Question 1(a) Focus: recognising a reverse sec-tan derivative pattern. Question 1(b) Focus: setting up and evaluating a shaded polynomial area. Question 1(c) Focus: integrating velocity and using a position condition. Question 1(d) Focus: product-to-sum before integrating a differential equation. Question 1(e) Focus: finding area between sec² and tan² curves.

Question 2

Question 2(a) Focus: expanding a square before integrating term by term. Question 2(b) Focus: solving for a lower limit from a definite integral. Question 2(c) Focus: separating variables with an exponential model. Question 2(d) Focus: using trig identities in a shaded \(\sin^2x\) area. Question 2(e) Focus: logarithmic substitution in a spherical mass model.

Question 3

Question 3(a) Focus: integrating exponential terms with different inside coefficients. Question 3(b) Focus: solving a logarithmic differential equation using a condition. Question 3(c) Focus: decomposing a rational integrand before solving for a parameter. Question 3(d) Focus: splitting a cosine area into two equal parts. Question 3(e) Focus: building and solving a Newton's law of cooling model.

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

Question 1

Question 1(a) Focus: splitting an integral into three familiar antiderivatives. Question 1(b) Focus: integrating velocity to get position, then using the starting distance. Question 1(c) Focus: finding intersections and the area between \(\sqrt{x}\) and \(\frac{x^2}{8}\). Question 1(d) Focus: separating variables and using a condition to keep the correct solution branch. Question 1(e) Focus: combining curve intersections with a subtract-a-cap area strategy.

Question 2

Question 2(a) Focus: reversing a linear exponential chain rule. Question 2(b) Focus: integrating a radical differential equation and fitting the constant. Question 2(c) Focus: rewriting a rational integrand before solving for \(k\). Question 2(d) Focus: spotting a logarithmic quotient by differentiating the denominator. Question 2(e) Focus: building and solving a quadratic decay model for volume.

Question 3

Question 3(a) Focus: using Simpson’s Rule with measured garden ordinates. Question 3(b) Focus: simplifying radicals before integrating term by term. Question 3(c) Focus: using product-to-sum in a shaded trigonometric area. Question 3(d) Focus: integrating acceleration to velocity with an initial condition. Question 3(e) Focus: factor cancellation in a separable differential equation.

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

Question 1

Question 1(a) Focus: splitting an integral into logarithm and trig parts. Question 1(b) Focus: using symmetry and signed area on a graph with no formula. Question 1(c) Focus: rewriting \(\sin^2(2x)\) before integrating exactly. Question 1(d) Focus: setting up a shaded-area equation to solve for \(k\). Question 1(e) Focus: finding exact area between exponential curves.

Question 2

Question 2(a) Focus: integrating an exponential term and a square-root term. Question 2(b) Focus: solving for an upper limit from a definite integral. Question 2(c) Focus: separating variables and using a condition in a log model. Question 2(d) Focus: turning acceleration into velocity, then distance over one second. Question 2(e) Focus: solving a separable leakage model for the required time.

Question 3

Question 3(a) Focus: reverse chain rule on a linear power. Question 3(b) Focus: applying the Trapezium Rule from tabulated values. Question 3(c) Focus: simplifying a rational integrand before evaluating it. Question 3(d) Focus: identifying top-minus-bottom for an area between a curve and a line. Question 3(e) Focus: expressing a shaded region as rectangle minus logarithmic curve area.

Work through the 2021 Integration walkthroughs question by question.

Topic: Integration. Year: 2021. Standard: NCEA Level 3 Calculus. Difficulty: mixed / Excellence-style.

Question 1

Question 1(a) Focus: integrating a linear term and a reciprocal term. Question 1(b)(i) Focus: integrating a gradient function and fitting the constant. Question 1(b)(ii) Focus: using the curve equation to find the area between \(x=1\) and \(x=2\). Question 1(c) Focus: integrating acceleration twice with initial conditions. Question 1(d) Focus: solving a square-root leakage model from two volume readings.

Question 2

Question 2(a) Focus: integrating exponential and radical terms. Question 2(b) Focus: adding a constant inside a definite integral. Question 2(c) Focus: using product-to-sum before integrating a trig product. Question 2(d) Focus: setting a shaded logarithmic area equal to \(4\) and solving for \(k\). Question 2(e) Focus: solving a separable differential equation to find a vertical distance.

Question 3

Question 3(a) Focus: expanding \((x+\sqrt{x})^2\) before integrating. Question 3(b) Focus: applying the Trapezium Rule with a width of \(0.25\). Question 3(c) Focus: separating a trigonometric differential equation and using a condition. Question 3(d) Focus: finding graph limits and integrating a rational curve. Question 3(e) Focus: proving an exponential shaded-area formula in terms of \(k\).

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 \(x^3+\frac{1}{x^3}\).

Work through the 2023 Level 3 Complex Numbers walkthroughs question by question.

Question 1

Question 1(a) Focus: expanding a surd binomial into the form \(a+bp+c\sqrt{p}\). Question 1(b) Focus: using the discriminant to find when a quadratic has no real roots. Question 1(c) Focus: rationalising a quotient and reading its real part. Question 1(d) Focus: conjugate roots and factorising a cubic with a real constant. Question 1(e) Focus: simplifying \(\frac{u}{v}\) and solving a modulus equation in \(k\).

Question 2

Question 2(a) Focus: dividing complex numbers in polar form. Question 2(b) Focus: finding a modulus in terms of a real parameter. Question 2(c) Focus: solving for \(z\) before finding its argument. Question 2(d) Focus: writing all cube roots in polar form. Question 2(e) Focus: turning a locus into a circle and using tangency.

Question 3

Question 3(a) Focus: using the remainder theorem to find \(p\). Question 3(b) Focus: multiplying by a complex number to solve for \(n\). Question 3(c) Focus: solving a surd equation for \(x\) in terms of \(w\). Question 3(d) Focus: simplifying reciprocal complex expressions to find \(x\) and \(y\). Question 3(e) Focus: finding \(z\), then \(w\), and using \(\operatorname{Im}(w)\).

Work through the 2022 Level 3 Complex Numbers walkthroughs question by question.

Question 1

Question 1(a) Focus: rationalising a surd denominator into the required exact form. Question 1(b) Focus: dividing complex numbers in polar form. Question 1(c) Focus: using an argument condition to match real and imaginary parts. Question 1(d) Focus: isolating a surd and using the discriminant to find \(p\). Question 1(e) Focus: proving an identity involving moduli and real parts.

Question 2

Question 2(a) Focus: using the remainder theorem to find \(b\). Question 2(b) Focus: solving for \(z\) using \(z\) and \(\overline{z}\). Question 2(c) Focus: factorising a cubic with a known root and finding \(h\). Question 2(d) Focus: rationalising first, then finding the principal argument. Question 2(e) Focus: turning a modulus locus into a circle equation.

Question 3

Question 3(a) Focus: combining Argand-diagram points and plotting the result. Question 3(b) Focus: completing the square to solve in terms of \(k\). Question 3(c) Focus: finding the three cube roots in polar form. Question 3(d) Focus: proving that a complex equation has no solution. Question 3(e) Focus: solving for \(a\) and \(b\) from \(z\) and \(\overline{z}\).

Work through the 2021 Complex Numbers walkthroughs question by question.

Topic: Complex Numbers. Year: 2021. Standard: NCEA Level 3 Calculus. Difficulty: mixed / Excellence-style.

Question 1

Question 1(a) Focus: multiplying complex numbers and matching real or imaginary parts. Question 1(b) Focus: rationalising \(\frac{26}{z}\) and plotting \(4-6i\) on an Argand diagram. Question 1(c) Focus: using the remainder theorem and factor theorem to find \(a\) and \(b\). Question 1(d) Focus: converting a quotient to \(a+bi\) and reading its argument. Question 1(e) Focus: proving a circular locus in Cartesian form.

Question 2

Question 2(a) Focus: dividing complex numbers in \(\operatorname{cis}\) form. Question 2(b) Focus: completing the square to solve a quadratic in terms of \(q\). Question 2(c) Focus: using conjugate multiplication to prove a quotient is purely imaginary. Question 2(d) Focus: finding cube roots in \(\operatorname{cis}\) form. Question 2(e) Focus: turning a modulus equation into the circle \(|z|=4\).

Question 3

Question 3(a) Focus: using modulus and argument to choose the correct value of \(m\). Question 3(b) Focus: rationalising a surd denominator into \(a+b\sqrt{3}\). Question 3(c) Focus: conjugate roots, cubic factorisation, and finding \(A\). Question 3(d) Focus: solving a radical equation in terms of \(m\). Question 3(e) Focus: solving a complex equation by equating real and imaginary parts.