2026 Prediction Examination
Level 2 Calculus
91262 Apply calculus methods in solving problems
Credits: Five
| Achievement | Achievement with Merit | Achievement with Excellence |
|---|---|---|
| Apply calculus methods in solving problems. | Apply calculus methods, using relational thinking, in solving problems. | Apply calculus methods, using extended abstract thinking, in solving problems. |
Instructions
- Suggested time: 60 minutes.
- You should attempt ALL the questions in this paper.
- Show ALL relevant working.
- A scientific calculator is permitted.
- The formula sheet opens from the margin beside any question.
- Use the margin beside each question for its timer and worked answer.
- Start the timer below when you are ready to begin.
Main Exam Timer
Our 2026 prediction paper, written from the 2023 to 2025 papers and this year's assessment specification. Every question has a worked answer.
A Maths Study Hub practice paper. Not an official NZQA examination.
QUESTION ONE
(a)
Find the gradient of the curve $y=4-3x-\dfrac{x^3}{6}$ at the point where $x=-2$.
(b)
The curve $y=x^3-6x^2+9x+1$ has two tangents that are parallel to the line $y=9x-2$.
Find the equations of both tangents.
(c)
The graph of $y=2x^3+3x^2-12x+k$, where $k$ is a constant, has two stationary points. The local minimum of the graph lies on the $x$-axis.
Find the value of $k$ and the coordinates of both stationary points. Use calculus to determine the nature of each.
(d)
A solid wooden cone has a base radius of 6 cm and a vertical height of 12 cm.
A cylinder is to be carved from the cone. The cylinder stands on the base of the cone, and the circle around its top touches the sloping surface of the cone. The diagrams show the cylinder inside the cone, and a vertical cross-section through the centre of the cone.
The radius of the cylinder is $x$ cm and its height is $h$ cm.
Volume of a cylinder: $V=\pi r^2h$
Show that $h=12-2x$, and hence find the largest possible volume of the cylinder.
Use calculus to justify that your answer is a maximum.
QUESTION TWO
(a)
The graph of the function $y=f(x)$ is shown below.
On the axes below, sketch the graph of the gradient function $y=f'(x)$.
(b)
The graph of the gradient function $y=f'(x)$ is shown below. It is a parabola.
The graph of $y=f(x)$ passes through the point $(0,\,2)$.
Find the equation of $f(x)$.
On the axes below, sketch the graph of $y=f(x)$. Show the $x$-coordinates of its stationary points.
(c)
A remote-controlled boat moves along a straight canal that runs north–south. A positive velocity means the boat is moving north.
The acceleration of the boat, $a$ m s$^{-2}$, $t$ seconds after it starts moving is modelled by
$$a(t)=0.6t-2.4\qquad 0\le t\le8$$When $t=0$, the boat is 10 m north of a footbridge and is moving north at 3.6 m s$^{-1}$.
Find the velocity of the boat when $t=5$, and state what it tells you about the boat at that moment.
Use calculus to find how far the boat is from the footbridge when $t=8$.
Find the total distance the boat travels during the 8 seconds.
Show full working and justify your answer.
(d)
The cross-section of a hill is modelled by
$$y=20-0.05x^2\qquad -20\le x\le20$$where $x$ and $y$ are in metres. The ground on either side of the hill is flat, along $y=0$.
A wildlife camera sits on the ground at $C(25,\,0)$. It can see an object only if the straight line from the camera to the object does not pass through the hill.
A drone hovers directly above the point $D(-30,\,0)$, on the other side of the hill.
Use calculus to find the lowest height at which the drone can hover and still be seen by the camera.
Show full working and justify your answer.
QUESTION THREE
(a)
Find the coordinates of the point on the curve $y=3+4x-\tfrac12x^2$ where the gradient is $-2$.
(b)
A spherical balloon is being inflated.
The volume of a sphere is $V=\tfrac43\pi r^3$, where $r$ is the radius in cm.
Find the rate of change of the volume with respect to the radius at the moment when the volume of the balloon is $288\pi$ cm$^3$.
(c)
The side view of a children's slide is modelled by
$$\begin{gathered}h(x)=ax^3+bx^2+cx+d\\0\le x\le4\end{gathered}$$where $h$ is the height of the slide above the ground and $x$ is the horizontal distance from the top of the slide, both in metres.
The slide is horizontal at its top, 3 m above the ground. At its bottom end, where $x=4$, it is 1 m above the ground and horizontal again.
Use calculus to show that $c=0$, and find the values of $a$, $b$ and $d$.
A safety guideline says the gradient of a slide must never be steeper than $-0.8$; that is, it must never drop more than 0.8 m for each metre across.
Does this slide meet the guideline? Use calculus to justify your answer.
(d)
A function is given by $f(x)=x^3+kx^2+3x-2$, where $k$ is a constant.
When $k=-5$, use calculus to find the values of $x$ for which $f(x)$ is decreasing.
Find all the values of $k$ for which the graph of $y=f(x)$ has no stationary points.
Show full working and justify your answer.
How we chose these questions
A prediction paper is our best judgement of what the 2026 examination is likely to ask. It isn’t a copy of any past paper, and nobody can know the real questions in advance. We built this one from NZQA’s 2023, 2024 and 2025 Calculus examinations and marking schedules, the examiners’ reports on 2024 and 2025, and the 2026 assessment specification.
Every question is here for at least one of three reasons
- It’s in the 2026 specificationThe specification names three things for this standard: drawing a gradient graph from a function’s graph (or the reverse), justifying whether a turning point is a maximum or a minimum, and forming polynomials from real situations. All three are in this paper.
- It comes up every yearEvery paper since 2021 has had a gradient at a point, a tangent, a gradient graph to sketch and an optimisation problem.
- It’s where marks are lostThe 2024 and 2025 reports name the mistakes candidates make most often. We built several of them into these questions.
Question by question
| Question | What it tests | Why it’s here |
|---|---|---|
| 1(a) A | The gradient at a point | Every paper has asked for one. The 2025 report warned about substituting negative numbers and working with fractions, which this question has. |
| 1(b) M | Tangents parallel to a given line | Tangents are in every paper. In 2025, candidates who didn’t pass often put the gradient into the function instead of the derivative. |
| 1(c) E | Turning points and their nature, with the minimum on the x-axis | The 2026 specification names justifying a maximum or minimum, and NZQA awarded Excellence for that justification in 2024 and 2025. |
| 1(d) E | The largest cylinder that can be carved from a cone | Optimisation is in every paper, and in 2025 NZQA gave Excellence for proving that a volume is a maximum. A cone and a cylinder is new. |
| 2(a) M | Sketching a gradient graph | The specification names it, and both the 2024 and 2025 reports asked for one smooth, symmetrical curve with the right x-intercepts. |
| 2(b) M | Sketching a function from its gradient graph | The specification says “or vice versa”. 2024 did this from a straight-line gradient graph; this one is a parabola. |
| 2(c) E | Motion, starting from an acceleration | Both 2024 and 2025 started from an acceleration, and both reports flagged the “+ c” and the starting position. |
| 2(d) E | A line of sight over a hill: a tangent from a point off the curve | Tangents are in every paper. In 2023 the point where the tangent touched was given; here it has to be found. |
| 3(a) A | The point with a given gradient | Asked in 2023, 2024 and 2025. In 2025, candidates who didn’t pass often put the gradient into the function. |
| 3(b) M | Rate of change for a sphere | Rates of change come up most years, and the 2024 report gave not understanding rate of change as a reason for Not Achieved. |
| 3(c) E | Building a cubic from a description, then where it is steepest | The specification names forming polynomials from contexts. 2022 asked when a model was growing fastest, the same idea on a function that was given. |
| 3(d) E | Where a function decreases, and when it has no turning points | Where a function increases or decreases was examined in 2022, 2023 and 2025. Using the discriminant to decide is new. |
None of the Excellence questions, 1(c), 1(d), 2(c), 2(d), 3(c) and 3(d), repeats an NZQA question from 2021–2025.
NZQA sources: 2026 assessment specification, 2025 assessment report, 2025 examination, 2025 assessment schedule. The 2023 and 2024 papers, schedules and reports are on the same NZQA pages.
Maths Study Hub is independent and is not affiliated with or endorsed by NZQA. Written October 2026.
More practice: our 2025 Calculus prediction paper, with a worked answer for every question.