2026 Prediction Examination
Level 1 Mathematical Reasoning
91947 Demonstrate mathematical reasoning
Credits: Five
| Achievement | Achievement with Merit | Achievement with Excellence |
|---|---|---|
| Demonstrate mathematical reasoning. | Demonstrate mathematical reasoning with relational thinking. | Demonstrate mathematical reasoning with extended abstract thinking. |
Instructions
- 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 suggested time and worked answer.
- Check your answers with the marking schedule at the end of the paper.
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)
The diagram below shows a pentagon ABCDE.
Find the value of $x$.
Clearly show all steps of your working.
(b)
A rectangular garden measures $(5x+1)$ metres by $(2x+1)$ metres. A square patio, with sides of $2x$ metres, is built in one corner of the garden. The rest of the garden is lawn.
The area of the lawn is $25\text{ m}^2$.
Form a quadratic equation, then factorise AND solve it to find the value of $x$.
Clearly show all steps of your working.
(c)
The diagrams below show the first three patterns in a sequence.
To make the next pattern, every shaded triangle is divided into four smaller triangles, and the middle one of the four is left unshaded.
Find a rule for the number of shaded triangles, $S$, in Pattern $n$.
Which pattern has exactly 729 shaded triangles?
The graph of your rule from part (i) is drawn for all values of $n$.
Identify TWO features of this graph.
(d)
The diagram below shows the lines $x+2y=10$ and $2x-y=k$, where $k$ is a constant. In the diagram, $k=4$.
When $k=4$, find the coordinates of the point where the two lines intersect, using an algebraic method.
For some values of $k$, the two lines intersect at a point where both coordinates are positive.
Find all the values of $k$ for which this happens.
Show full working and justify your answer.
QUESTION TWO
(a)
The diagram below shows a solid cone with a base radius of 8 and a vertical height of 15.
Find the total surface area of the cone.
(b)
A boat leaves harbour H and sails 12 km on a bearing of 040° to a buoy B. It then sails 9 km due east to a reef C.
Find the direct distance from H to C, and the bearing of C from H.
Clearly show all steps of your working.
(c)
A glass paperweight is a square-based pyramid VABCD.
The base ABCD is a square with sides of 8 cm, and each of the four sloping edges is 9 cm long. Its top, V, is directly above the centre of the base.
It is known that 1 cm$^3$ of the glass has a mass of 2.5 g.
Calculate the mass of the paperweight.
(d)
A vertical radio mast TM stands on flat ground, with its base at T and its top at M.
Point A is due south of T, and point B is due east of T.
From A, the angle of elevation of the top of the mast is 30°.
From B, the angle of elevation of the top of the mast is 40°.
A and B are 120 m apart.
Let the height of the mast be $h$ metres.
Show that the length TA $=1.7321h$ metres.
Show your working clearly.
Find the height of the mast.
Show full working and justify your answer.
QUESTION THREE
(a)
Two candles, A and B, are lit at the same time. Each candle burns down at a constant rate.
The graph shows the height, $h$ cm, of candle A, $t$ hours after it is lit.
The table shows the height of candle B.
| Time, $t$ (hours) | 0 | 2 | 4 | 6 |
|---|---|---|---|---|
| Height, $h$ (cm) | 30 | 25 | 20 | 15 |
Find the gradient of the graph for candle A, and explain what it tells you about the candle. Hence write an equation for the height of candle A.
Form and solve an inequation to find when candle A is taller than candle B.
Explain your answer in context.
(b)
The graph below shows a straight line L and a parabola M. The parabola passes through the point P(3, 6).
Find the equation of the straight line L.
Find the equation of the parabola M.
Justify your working with appropriate reasoning.
(c)
Mere has a right-angled triangular offcut of plywood, ABC, with AB $=60$ cm, BC $=80$ cm and angle ABC $=90^\circ$.
She wants to cut a rectangular shelf BRQP from it, as shown. P lies on AB, R lies on BC, and the corner Q lies on the long edge AC.
Let BR $=x$ cm and RQ $=y$ cm.
Use similar triangles to show that $y=60-\tfrac34x$.
Find the dimensions of the largest shelf Mere can cut in this way.
Mere thinks that this largest shelf uses exactly half of the plywood. Is she correct?
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 Mathematical Reasoning examinations and marking schedules, the examiners’ reports on those years, and the 2026 assessment specification.
Every question is here for at least one of three reasons
- It’s in the 2026 specificationInequations, similar shapes, cones, the volume of a pyramid, optimal solutions, and rate of change as a gradient are all listed for this standard, but none has been examined in it yet.
- It comes up most yearsSome question types are set almost every year, such as finding a rule from a table of values, and trigonometry in a real situation.
- It’s where marks are lostThe examiners’ 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 | Angles in a polygon, solved as an equation | A steady opener: angle questions were in the 2023 and 2025 papers. |
| 1(b) M | Forming and factorising a quadratic from an area | In 2025, factorising a quadratic with a number bigger than 1 in front of x² “highlighted gaps in many candidates’ knowledge”, the examiners said. |
| 1(c) M | A picture pattern with an exponential rule | Every paper for this standard has asked for a rule from a table of values, and the 2025 report flagged the features of exponential graphs. |
| 1(d) E | Two lines with an unknown constant | Inequations are in the specification but haven’t been examined in this standard yet, and solving simultaneous equations by algebra was a weak spot in 2025. |
| 2(a) A | Surface area of a cone | Cones are in the specification but haven’t been examined yet. In 2025, candidates who didn’t pass often confused surface area with volume, or put the “squared” unit in the wrong place. |
| 2(b) M | Bearings and trigonometry | Bearings were examined in 2023, and trigonometry in a real situation comes up most years. |
| 2(c) M | Volume and mass of a right square-based pyramid | The 2024 paper had a pyramid with its top above one corner of its base, and asked for the length of an edge. This one is a right pyramid, with its top directly above the centre of the base, and it needs the volume, which hasn’t been examined yet. The volume then gives the mass, as the 2025 paper turned a volume of earth into a mass. |
| 2(d) E | Height of a mast from two angles of elevation | Pythagoras and trigonometry in three dimensions were examined in 2024 and 2025. This set-up, with two angles of elevation, is new. |
| 3(a) M | Rate of change as a gradient | The specification asks candidates to relate rate of change to the gradient of a graph; so far it has only been examined as speeds (2023). |
| 3(b) M | Equations of a line and a parabola from their graphs | The 2025 report named the equations of lines and parabolas as a weak spot, and lines with a negative gradient caused trouble in both 2024 and 2025. |
| 3(c) E | Similar triangles and the largest shelf | Similar shapes and optimal solutions are in the specification but haven’t been examined in this standard yet. |
The three Excellence questions, 1(d), 2(d) and 3(c), are not repeats of NZQA’s 2023–2025 questions: each is a new set-up.
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 Mathematical Reasoning prediction paper, with a worked answer for every question.