NCEA 91262: calculus methods (Level 2 Maths)

What NCEA 91262 assesses

NZQA says you are assessed on applying calculus methods in solving problems, set in real-life or mathematical contexts. The standard (version 3) lists the methods:

• derivatives and anti-derivatives of polynomials given in expanded form
• gradient functions and the gradient at a point
• the equation of a tangent
• turning points, where the gradient is zero, and their nature
• finding a function from a derived function
• rate of change problems, such as kinematics

The three grades read the same way in the standard, with a different kind of thinking. Achieved applies calculus methods in solving problems. Merit uses relational thinking: selecting and carrying out a logical sequence of steps, connecting concepts or representations, and relating findings to a context. Excellence uses extended abstract thinking: devising a strategy, developing a chain of logical reasoning or proof, or forming a generalisation.

The NCEA 91262 exam in 2026 and 2027

NCEA 91262 is one of three Level 2 Mathematics and Statistics externals, with 91261 (algebra) and 91267 (probability). The 2026 exam is on Thursday 19 November, in the morning session. NZQA's specifications for both years say the exam is on printed paper and 91262 is a 5-credit external.

• Each exam has three questions made up of multiple parts, some of them resource-based and some knowledge-based. Parts are not ordered by difficulty, so attempt every part.
• Calculator: you must bring an approved calculator.
• Graphs: you may be asked to draw the graph of the gradient of a function from the graph of the function, or the other way round.
• Justify turning points: you may be asked to justify whether a stationary point is a maximum or a minimum, for example with the second derivative, by testing gradients either side, or with an annotated sketch.
• Modelling: you may need to form a polynomial from a context, and you should know common 2D and 3D shapes and their formulae.

How NCEA 91262 is marked

Three questions, each scored N0 to E8, give a total out of 24. In calculus, the units of evidence usually come from a sequence of correct steps, so one slip in differentiating can cost a unit at one level without losing the rest. NZQA's report says kinematics answers need the "+ c" and how it was found for the higher grades.

NCEA 91262 cut scores

Calculus has held fairly still: Achieved started at 7 or 8, Merit at 14, and Excellence at 19 or 20 in all five years. Scores below are where each grade begins, out of 24.

Cut scores 2025

2025 schedule: Achieved from 8, Merit from 14, Excellence from 20. The same as 2024.

Cut scores 2024

2024 schedule: Achieved from 8, Merit from 14, Excellence from 20. The same as 2023.

Cut scores 2023

2023 schedule: Achieved from 8, Merit from 14, Excellence from 20. Compared with 2022, Excellence went up by 1.

Cut scores 2022

2022 schedule: Achieved from 8, Merit from 14, Excellence from 19. Compared with 2021, Achieved went up by 1 and Excellence came down by 1.

Cut scores 2021

2021 schedule: Achieved from 7, Merit from 14, Excellence from 20.

Worked examples at each grade

Achieved: gradient and tangent

For f(x) = 2x³ − 9x² + 12x + 1, find the gradient at x = 3 and the equation of the tangent there.

Check the gradient in the Derivative Calculator.

Merit: turning points and their nature

Find and classify the turning points of the same function. Set the gradient to zero:

The second derivative is negative at x = 1, so (1, 6) is a local maximum, and positive at x = 2, so (2, 5) is a local minimum. Merit wants the working in a logical order and the nature of each point justified. See f″(1) in the Derivative Calculator.

Excellence: a kinematics problem

A particle moves in a straight line with velocity v = 3t² − 12t + 9 m/s, and starts 2 m from the origin. Find its position at time t, when it is at rest, and the distance it travels in the first 4 seconds.

The position is s(1) = 6 m, s(3) = 2 m and s(4) = 6 m. The particle moves forward from 2 m to 6 m, back to 2 m, then forward to 6 m again, so it travels 4 + 4 + 4 = 12 m, not the 4 m that the final displacement would suggest. NZQA's 2025 report says showing how the constant of integration is found is needed for the higher grades.

Now classify some stationary points in our Calculus course, then press Mark to check your answer in the course:

Exam answers

The answers and a worked mark scheme come with today's paper once it is marked. We do not publish official or past paper answers. Every question on the daily paper is our own.

Use the daily paper as a practice exam

The track shown above is a short timed mock of 8 questions, not a full exam and not written by NZQA (NCEA Level 2 Maths). It practises algebra and functions, not calculus, so it supports the function and graph skills behind 91262 but not the differentiation itself. For calculus practice use the NCEA Level 3 Calculus track, where tangents and normals, stationary points and rates of change match 91262, and treat its chain rule, parametric and integration-by-parts questions as extension.

Is NCEA 91262 changing?

NZQA's 2027 specification keeps the same three standards for Level 2 maths and the same calculus requirements. For Year 12 students the first year of the new NZ Certificate of Education is 2029.

Common mistakes in NCEA 91262

• Substituting the gradient into the original function. NZQA's 2025 report says some candidates did this instead of setting the derivative equal to the gradient.
• Forgetting + c. The report lists leaving out the constant of integration as a common Not Achieved error, and says showing how it is found is needed for higher grades.
• Sloppy gradient graphs. Draw the graph of a derived function as one smooth curve with the right x-intercepts.
• Negative numbers and fractions. Take care when substituting them, and when differentiating a term with a fraction.
• No justification. Use the second derivative, or check the gradient either side of a turning point.

Read next

New Zealand · NCEA Level 2NCEA Level 2 Maths: standards, exam date and practiceNCEA Level 2 Maths explained: the algebra, calculus and probability externals, the internals, the 2026 exam date, 2027 dates and worked examples for each.