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
- Suggested time: 75 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 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 $n$.
Use an algebraic method to find 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 90 mm long.
Calculate the volume of glass in the paperweight, in cm$^3$.
(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.
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.