HSC Maths Advanced past papers: what to use for 2027

The 2027 HSC is the first exam for the Mathematics Advanced 11-12 Syllabus (2024). Older past papers still contain relevant questions, but you need to compare them with the new content before using a whole paper for exam practice. NESA syllabus and implementation dates

Begin with the new exam specifications and sample materials. You can then choose older questions on the topics you are studying. Below are three questions from the 2025 paper that match particular skills, followed by two original worked examples.

Where to find official resources

NESA's published sample examination materials illustrate the new format. They do not predict which questions will appear in 2027.

Allow the right time and equipment

The new exam allows three hours of writing after ten minutes of reading. It is worth 100 marks: 10 for objective-response questions and 90 for Section II. NESA provides a reference sheet and permits approved calculators. Year 11 content may also be examined. NESA examination specifications

Follow these requirements when preparing for the new exam. If you shorten an older paper by leaving out questions, record which ones you excluded alongside the paper's year. That will help you interpret your mark, because a result from a shortened paper is not directly comparable with a result from a complete one. Our 2027 Maths Advanced and Extension 1 syllabus guide explains the broader course changes.

Three questions from the 2025 paper

The questions below match specific outcomes in the new syllabus. They cover only part of the course, so check with your teacher that you have learnt the methods needed before attempting them.

These references use the printed page numbers in the 2025 paper and the new Advanced outcomes.

Attempt each question before reading its marking guidelines. If you cannot get started, write down what you recognise and what you cannot work out. A teacher can use those notes to explain the step you are missing.

Worked example 1: a geometric sequence

The following exercises are original Plume examples. They practise relevant methods and are neither NESA questions nor predictions of the 2027 exam.

A geometric sequence has first term 12 and common ratio 1.5. Find its fourth term and the sum of its first four terms.

To reach the fourth term, multiply the first term by the common ratio three times:

$$u_4=12(1.5)^3=40.5.$$

Use the geometric-series formula to find the sum:

$$S_4=\frac{12(1.5^4-1)}{1.5-1}=97.5.$$

You can check that result by adding the four terms directly:

$$12+18+27+40.5=97.5.$$

If you obtained 60.75 for the fourth term, you multiplied by the ratio four times and reached the fifth term. If you found 40.5 but gave it as the sum, check which quantity each part asks for. A term is one member of the sequence; a sum adds the specified members.

Try the calculation again with a different first term and ratio. That will help you check whether you can apply the method without relying on the numbers in this example.

Worked example 2: a stationary point

Find the stationary point of

$$f(x)=(x+1)e^{-x},\qquad x\in\mathbb{R},$$

and determine whether it is a maximum or minimum.

Applying the product and chain rules gives

$$f'(x)=e^{-x}-(x+1)e^{-x}=-xe^{-x}.$$

Since $e^{-x}$ is positive for every real $x$, the derivative is zero only at $x=0$. Substituting this into the original function gives $f(0)=1$, so the stationary point is $(0,1)$. To classify it, check the sign of the derivative on either side:

The function increases before the point and decreases after it, so $(0,1)$ is a maximum.

A complete answer needs both coordinates and a reason for the classification. When reviewing your solution, also check that you can explain why the exponential factor does not produce another stationary point.

For more practice choosing and checking a method, use our Maths Advanced formula-sheet guide. It links the reference sheet for each cohort and works through logarithmic differentiation and total area.

Use marking to decide what to practise

Compare your working with the guidelines and find the first step that needs changing. You may have chosen the wrong method, made an algebra or calculator error, or left out an explanation. Record that step and what you will do on the next attempt:

Keep the first attempt while you correct it, then try again without the solution beside you. If you still need to look at the answer to choose a method, practise a simpler question on the same skill.

As you select more questions, include some you can complete, some that revisit recent mistakes and some that require you to choose between methods. Save the topic and syllabus match with each question. Before an exam, ask your teacher or tutor to check that the set covers the required content; the three questions listed above cover only a small part of it.

If you need help with a method, bring your unfinished working when discussing Maths Advanced tutoring. To fit that practice around other subjects, use our HSC study timetable.