Maths Methods Paper 1: what to expect in 2026

Maths Methods Paper 1 and Paper 2 are both on Friday 13 November, and I know a week can feel like no time at all! At this point, I have found it helps to look at what each paper actually asks, rather than working through the whole syllabus again. This post covers Paper 1, and what to expect on Paper 2 is the other half.

I went back through every past QCAA Mathematical Methods Paper 1 from 2020 to 2025, which is 60 multiple choice questions and 56 short response questions, and sorted each one by topic and syllabus dot point. A handful of question shapes have turned up on Paper 1 every single year, while a few others have dropped off.

First, here is how the paper is set out. Paper 1 is technology-free. Questions 1 to 10 are multiple choice, worth a mark each, and the short response questions are worth 45 marks (50 in 2020). Since 2022 there have been nine short response questions. The first four or five are simple familiar, the next two or three are complex familiar, and the last two, Questions 18 and 19, are complex unfamiliar. 2020 and 2021 had a tenth question, so their complex unfamiliar questions were 19 and 20.

Please keep in mind that past papers can only ever be a guide to what comes next. 2026 is also the first external exam on the new 2025 syllabus, so every pattern below comes from papers written for the old one. I will point out where that matters.

The question shapes that keep coming back

Here they are, year by year. Every question number below opens the worked solution, so you can go straight to any of them.

You will notice that Paper 1 is all about calculus and probability by hand. Everything on it is built to work without a calculator, so expect to leave your answers as fractions, surds or logs.

Probability by hand

A Bernoulli or binomial question has been on Paper 1 in every one of the last six years. In 2023 there were three in the short response alone. They tend to come in one of four forms:

• Binomial with fractions. 2024 Q12 spins a pointer with three equal sections over a long weekend and wants the probabilities as fully simplified fractions. 2021 Q17 does the same with someone who catches the bus on three days in five, on average. My advice is to write out \(\binom{n}{r}p^r(1-p)^{n-r}\) every time, and then simplify.
• Bernoulli mean and variance. 2022 Q12 gives \(p=\frac{4}{5}\) and asks for the mean, variance and standard deviation. 2023 Q11 works from two samples recorded as 1s and 0s. You will want \(E(X)=p\) and \(\operatorname{Var}(X)=p(1-p)\) off by heart.
• Conditions. 2023 Q13 asks you to state two conditions that make a situation binomial. Have all four ready: a fixed number of trials, two outcomes, the same probability each time, and independent trials.
• The complex familiar version. 2025 Q17 is a fishing game: three casts in either a blue pond or a red pond, at least 30 points to win, and a claim about which pond gives the better chance. Before any binomial calculation, you have to work out how many fish each pond needs.

Confidence intervals by hand

Every Paper 1 has had a confidence interval or sample proportion question, done by hand. Usually the \(z\)-value is given, so use the value they give, but 2022 MC8 expected you to know \(z\approx1.96\) for 95%. 2025 Q12 said \(z=1\), not 2, and wanted the interval for 5 students out of 25 as simplified fractions. 2024 Q15 runs it backwards: from the interval, find the margin of error, then the number of people surveyed. 2021 Q19 finds a sample size with no past data, using \(\hat{p}=0.5\). You need to justify that mathematically, by showing that \(\hat{p}(1-\hat{p})\) is greatest at \(\hat{p}=0.5\), so it gives the largest, safest sample size. The subject report says many students lost that mark by giving a reason in words only.

New this year: general discrete random variables

General discrete random variables now sit in Unit 3 of the 2025 syllabus. Past papers have only touched them lightly (2025 MC10 reads a probability off a frequency graph), and none has asked for \(E(X)\) or \(\operatorname{Var}(X)\) from a general probability table. The arithmetic is fractions and short sums, so I would expect one on Paper 1. The likeliest form is a probability table with an unknown, where the probabilities add to 1. Then find \(E(X)=\sum x\,p(x)\), the variance and the standard deviation, perhaps finishing with a decision about whether a game is fair. The closest past questions are the ones above: 2023 Q15 gives the probabilities for one water-balloon throw, then asks for the binomial mean and variance for 20 throws.

Calculus by hand: the core of Paper 1

Antiderivatives, definite integrals and areas

There has been an exact definite integral or area on every Paper 1, and a bare antiderivative on five of the six. 2025 Q11 is the friendliest version: three integrals, worth 1, 1 and 3 marks. The area questions each add one more step:

• 2021 Q13: find where two graphs meet, then use those \(x\)-values as the limits for the area between them.
• 2023 Q12: estimate the area of a garden bed with rectangles, then find it exactly with a definite integral.
• 2022 Q17: two definite integrals are given with unknown limits, and you work backwards to find \(b\).
• 2021 Q18: two shaded regions with known areas, and you find the constants in the function.

The area question that caught us out

2025 MC8 asked for the area between \(y=4x^3\) and the \(x\)-axis from \(x=-1\) to \(x=2\). Only 28.7% of Queensland students got it right, and 58.9% chose 15, which is the value of the integral from \(-1\) to 2. However, the graph sits below the axis between \(-1\) and 0, so the area is \(1+16=17\). If a graph crosses the \(x\)-axis inside your interval, split the integral at the crossing and add the sizes of the pieces. The same idea caught 62% of students on Paper 2 the year before (2024 Paper 2 MC8).

A rate given: integrate to the total

Five of the last six Paper 1s gave you a rate and asked for an amount. My advice is to integrate, use the starting value to find the constant, and then make sure you answer the question that was asked. 2023 Q14 starts with \(\frac{dP}{dt}=e^{2t}\) and \(P(0)=60\), then asks for the change during the third hour in terms of \(e\). The third hour runs from \(t=2\) to \(t=3\). 2022 Q14 fills a vessel, and 2024 Q17 grows a social media group to seven times its size, with the answer in the form \(a\ln(b)\). The complex familiar version is 2025 Q16: solar panels charge a battery at a sine-shaped rate while the house drains it at a steady rate, and you find the net change over daylight hours.

Derivatives and what they tell you

From 2020 to 2022, every Paper 1 had a “determine the derivative of the following” question (2020 Q11, 2021 Q11, 2022 Q13). Since 2023 the product, quotient and chain rules have mostly moved into the multiple choice, and the short response asks what a derivative tells you instead. You still need to be fluent, and my chain rule post is a quick refresher.

• Second derivatives. 2024 Q11 finds \(y''\), evaluates it, then finds the point where the rate of change of the first derivative is zero. 2025 Q13 asks you to use \(f''(x)=2x-6\) to demonstrate a point of inflection at \(x=3\). Show that \(f''\) changes sign either side of 3, not just that \(f''(3)=0\).
• The maximum rate. In 2023 Q17 a pool chemical is absorbed as \(A=10t^2-4t^3\) from 10:00 am. The rate is \(A'\), so the maximum rate is where \(A''=0\), and the answer is a time of day. Then verify it is a maximum, with \(A'''\) or a sign test.
• Reading graphs. 2022 Q15 gives \(f'(x)=e^x(x-4)\) and asks where \(f\) is both decreasing and concave up. In the multiple choice, 2022 MC1 asked how many maxima and minima \(f\) has from the graph of \(f'\), and only 34.4% got it right. 2024 MC8 matches a graph of \(f\) to its \(f''\). A rule I would strongly recommend saying out loud is this: where \(f'\) crosses zero, \(f\) has a maximum or minimum; where \(f'\) touches zero without crossing, \(f\) has a stationary point of inflection; where \(f'\) has a turning point, \(f\) has a point of inflection.

Exponential and log equations

Every Paper 1 has had an exact equation to solve, and five of the six included a log to a base other than \(e\) (2020 Q15, 2021 Q12, 2022 Q11, 2024 Q16 and 2025 Q15). Under the 2025 syllabus, logs to other bases are Unit 2 assumed knowledge, and QCAA says assumed knowledge can be drawn on. The log laws also turn up in the multiple choice every year. I would practise base \(e\) first, but keep a multi-step \(\log_2\) equation like 2025 Q15 in your set. The 2025 subject report specifically asked for more practice with equations that need several log laws.

2023 Q16 is a great one to model your working on. It asks you to solve \(4+7e^{-2x}=3e^{2x}\). Multiply through by \(e^{2x}\) and it becomes a quadratic in \(e^{2x}\): \(3\left(e^{2x}\right)^2-4e^{2x}-7=0\). One root is negative, and \(e^{2x}\) can never be negative, so reject it and show that you have. The marking guide gives a mark for rejecting it.

The complex familiar version builds or integrates an exponential model, then solves for a time exactly. Do 2020 Q17 (fit \(V=Ae^{kt}\) from the starting volume and the starting rate, with the answer in the form \(\ln(a)\)) and 2024 Q17 (integrate a growth rate from 1000 members, with the answer in the form \(a\ln(b)\)). When a question asks for a form, the last mark depends on it.

The last two questions: models built from words

About half of the complex unfamiliar questions on Paper 1 make you build the model from a description, and the rest give you a function but no parts to guide you. Two shapes have repeated:

• A sinusoid built from a description. 2023 Q18 is a Ferris wheel: a six-metre diameter, the bottom carriage one metre off the ground, and three seconds from the lowest point to the highest. You build the height function, differentiate twice, and compare the largest acceleration with half of \(9.8\) m s\(^{-2}\). 2024 Q19 does the same with a glacier that moves between 270 m and 280 m from a car park over a year. The amplitude is half the range, the centre line is the middle, and lowest to highest is only half a cycle.
• Optimisation on a figure. 2022 Q19 uses similar triangles and \(\frac{1}{2}ab\sin(C)\) to write the area between two parallel walls in terms of one length, then minimises it. 2025 Q19 finds the shortest fence through a gate in the corner of a paddock. Draw it, name one variable (a length or an angle), write the quantity to optimise in terms of it, then differentiate and set the derivative to zero. 2025 said that verifying the minimum was not required. Normally it is, so do it unless you are told otherwise.

The others were one-offs that used the same moves, for example 2025 Q18 (two objects launched towards each other), 2024 Q18 (a justified decision about the height of a storage container), 2023 Q19 (two log models), 2022 Q18 (a percentile from a probability density function) and 2021 Q20 (when a rabbit population is increasing).

On these, write a short plan before you start: what the function is, what you will differentiate, and what you will compare at the end. A complex unfamiliar question has no parts to guide you, so your plan has to do that job instead.

What Paper 1 rarely asks

Knowing what Paper 1 avoids is just as useful when time is short.

• Normal distribution calculations. Most normal probabilities need a calculator, so they live on Paper 2. On Paper 1 the normal distribution has only appeared in the multiple choice: the 68–95–99.7 rule, symmetry, or combining given tail areas (2021 MC10, 2022 MC4). There has been none since 2022.
• Probability density functions. Four of the six Paper 1s had one, but there were none in 2024 or 2025. The last were 2023 MC7 and 2023 MC10, and 2022 asked for a percentile (2022 Q18). Please do not skip them, but they are more likely on Paper 2.
• The trapezoidal rule. It has been on Paper 1 in only one year out of six, but twice that year: 2022 MC9, which only 29.4% got right, and 2022 Q14, which applies it to a table of rates. Know where the 2s go in the rule: the end heights once, the middle heights twice.
• Integrals outside the 2025 list. The 2025 syllabus integrates \(x^n\), \(e^x\), \(\frac{1}{x}\), sine, cosine and \(f(ax+b)\). Integrals with a non-linear inside function, like 2020 MC9, are outside it, so skip them in older papers. “Recognise anti-differentiation as the reverse of differentiation” is still a dot point, though, so a “differentiate, then integrate” question could still appear.

Triangles with exact values sit in the middle. They appeared in five of six years, as a multiple choice question or a short simple familiar question like 2020 Q14. Make sure you know \(\sin(30^\circ)\), \(\cos(60^\circ)\) and the other exact values, as well as the area formula \(\frac{1}{2}ab\sin(C)\).

A revision plan for Paper 1

Both papers are on the same day, so split your last few evenings between them. Here is how I would spend two sessions on Paper 1. Do every question by hand, without notes, and give yourself about a minute and a half per mark, which is the pace the paper sets. Then check your work against the QCAA marking guide, one mark at a time.

Session 1: probability and statistics by hand

• Binomial with fractions: 2024 Q12.
• Binomial conditions, then a binomial mean and variance: 2023 Q13, then 2023 Q15 (use 2022 Q12 for a Bernoulli mean and variance).
• A confidence interval forwards, then backwards: 2025 Q12, then 2024 Q15.
• The complex familiar version: 2025 Q17.

Session 2: calculus and the last two questions

• A rate to a total: 2023 Q14, then 2025 Q16.
• The area trap: 2025 MC8. It only takes a minute, so take another minute to explain why 15 is wrong.
• A base-\(e\) equation: 2023 Q16.
• The maximum rate: 2023 Q17.
• One sinusoid built from words (2023 Q18 or 2024 Q19) and one optimisation (2025 Q19).

If there is time left over, press “Pick one at random” in the simple familiar bank and give whatever comes up a go! Mixing topics is closer to what the exam feels like than doing one topic at a time.

The Methods game plans and topic guides are free until the end of 13 November, if you would like a structured run through any of these.

Before Friday

None of this guarantees what is on Friday’s paper, as 2026 is the first year of a new syllabus and QCAA could break any of these patterns. Even so, I think six years of past papers are the best guide we have to where your last few hours of study should go.

You have put in two years of hard work, and your best effort will be good enough. Be kind to yourself this week, and best of luck on Friday!

Paper 2 is the same day. Here is what to expect on Methods Paper 2.

If you are a Year 11 student starting Units 3 and 4 and want help building these skills earlier, that is what tutoring is for.