2026 Prediction Examination
Level 2 Algebra
91261 Apply algebraic methods in solving problems
Credits: Four
| Achievement | Achievement with Merit | Achievement with Excellence |
|---|---|---|
| Apply algebraic methods in solving problems. | Apply algebraic methods, using relational thinking, in solving problems. | Apply algebraic 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)
Simplify fully, giving your answer with positive indices:
$$\frac{\left(2x^3y^{-1}\right)^4}{\sqrt{64x^{10}y^4}}$$(b)
Write as a single fraction in its simplest form:
$$\frac{3}{x-2}-\frac{12}{x^2-4}$$(c)
Aroha puts $\$6000$ into a savings account. The account pays interest of 4.5% per year, compounded monthly.
How many months will it take for Aroha’s savings to first be more than $\$8000$?
(d)
Solve the equation
$$\left(\log_3x\right)^2-\log_3\left(x^4\right)=5$$(e)
Kiri is watering her garden with a hose. She holds the nozzle 1.2 m above flat ground. The water follows a parabolic path. It is highest when it is 2 m horizontally from the nozzle, and it lands on the ground 6 m horizontally from the nozzle.
The path of the water can be modelled by $y=a(x-h)^2+k$, where $x$ is the horizontal distance from the nozzle and $y$ is the height above the ground, both in metres.
Kiri then stands on a deck that is 0.9 m higher than the garden. She holds the hose in the same way, so the whole path of the water is 0.9 m higher than before.
How much further away, horizontally, does the water now land?
QUESTION TWO
(a)
Use the discriminant to find the values of $k$ for which the equation $9x^2+kx+4=0$ has exactly one solution.
(b)
Write $3x^2-12x+7$ in the form $a(x-h)^2+k$.
Hence, or otherwise, solve $3x^2-12x+7=0$. Give your answers in exact form.
(c)
Find the coordinates of the points where the line $y=x+1$ meets the circle $x^2+y^2=25$.
(d)
Two students are investigating quadratic equations that contain a constant, $k$.
Ella says: “Whatever the value of $k$, the equation $x^2+(k-3)x-k=0$ has two different real roots.”
Josh says: “Whatever the value of $k$, the equation $x^2+(k-3)x+k=0$ has two different real roots.”
Use the discriminant to decide whether each student is correct.
Justify your answers.
(e)
The equation $x^2-px+q=0$, where $p$ and $q$ are constants, has two different real roots. One of the roots is the square of the other.
Given that $q=64$, find the value of $p$.
Find an expression for $p$ in terms of $q$.
QUESTION THREE
(a)
Solve $\log_4x=-\dfrac32$.
(b)
$x=\dfrac53$ is one solution of the equation $6x^2+kx-10=0$, where $k$ is a constant.
Find the value of $k$ and the other solution of the equation.
(c)
Write $2\log x+\log(x+3)-\log\left(x^2-9\right)$ as a single logarithm in its simplest form.
(d)
Wilding pines are spreading across a high-country station. The area they cover, $A$ hectares, $t$ years after the start of 2020, is modelled by
$$A=A_0\times b^t$$where $A_0$ and $b$ are constants.
At the start of 2022 the pines covered 1600 hectares. At the start of 2025 they covered 3125 hectares.
The regional council will begin aerial control when the pines cover more than 10 000 hectares.
According to the model, in which year will this happen?
(e)
A rectangular park is $x$ m wide and $y$ m long, where $x\lt y$. A straight path runs diagonally across the park from one corner, P, to the opposite corner, Q.
Walking from P to Q along the path, instead of along two edges of the park, saves a distance equal to one third of the length of the park.
Find the ratio of the width of the park to its length, $x:y$, in its simplest form.
Pythagoras’ Theorem: $a^2+b^2=c^2$
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 Algebra 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 specificationThe specification names compound interest, growth and decay, completing a model from a real situation, and the meaning of fractional roots. Compound interest hasn’t been examined since 2021, or growth and decay since 2023.
- It comes up every yearEvery paper since 2021 has had index laws, algebraic fractions, the discriminant and logarithms, and most have had a quadratic in a real situation and a question about the roots of a quadratic.
- 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 | Index laws with a bracket, a square root and a negative power | Index laws are in every paper. The 2025 report said candidates who didn’t pass often expanded (2y)³ as 2y³. |
| 1(b) M | Two algebraic fractions as one, then simplified | Simplifying algebraic fractions is in every paper, and the 2025 report said weak algebra showed most when candidates worked with fractions. |
| 1(c) M | Compound interest, compounded monthly | The 2026 specification names compound interest, and NZQA last examined it in 2021. |
| 1(d) E | A quadratic disguised with logarithms | Disguised quadratics were in the 2023 and 2025 papers, made with powers, and the 2023 report listed solving one as an Excellence skill. Making one with logarithms is new. |
| 1(e) E | Completing a parabola model from two facts, then moving it | Parabolic models were examined in 2023 and 2025, and the specification says candidates may have to complete a model from the information given. Raising the whole path is new. |
| 2(a) A | Using the discriminant to find a constant | The discriminant is in every paper. The 2023 report said candidates often missed the negative value when taking a square root. |
| 2(b) M | Completing the square, then exact solutions | Completing the square was examined in 2024, and exact answers with square roots in 2025. |
| 2(c) M | Where a straight line meets a circle | A line or curve meeting another curve was examined in 2021, 2023 and 2025. |
| 2(d) E | Two claims about every value of a constant, decided with the discriminant | The discriminant is in every paper, and the 2025 report said Excellence candidates linked it to a problem. Proving a claim for every value of the constant, and disproving another with one counterexample, is new. |
| 2(e) E | Two roots, one the square of the other | A question about the roots of a quadratic was in the 2021, 2022, 2024 and 2025 papers. This relationship between the roots is new. |
| 3(a) A | A log equation with a negative fraction | Log equations were in the 2023, 2024 and 2025 papers, and the 2025 report said candidates who didn’t pass often struggled with negative numbers. |
| 3(b) M | A fractional root: find the constant and the other root | The specification mentions the meaning of fractional roots, and 2021, 2022 and 2025 asked for an equation from its fractional roots. This asks it the other way round. |
| 3(c) M | Log laws, then factorising | Log laws were examined in 2022, 2023 and 2025, and the 2025 report said candidates who didn’t pass often couldn’t work with logs. |
| 3(d) E | Growth: completing a model from two measurements, then finding when it passes a limit | The specification names growth and decay, last examined in 2023 with one unknown. Here there are two unknowns, and the question leaves it to candidates to see that the model must be completed first. |
| 3(e) E | Pythagoras: a shortcut across a park | The 2025 report said Excellence candidates applied Pythagoras and worked with the result. Finding a ratio this way is new. |
None of the six Excellence questions, 1(d), 1(e), 2(d), 2(e), 3(d) and 3(e), 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 Algebra prediction paper, with a worked answer for every question.