How to Improve Your HSC Maths Advanced Score: 2025 Examiner Report Breakdown

HSC Mathematics Advanced rewards students who can do more than reproduce a method. The 2025 exam tested whether students could read a question carefully, choose the right approach and set out clear, logical working. NESA's marking feedback shows exactly where students did this well, and where marks slipped away.

This guide breaks down the 2025 HSC Mathematics Advanced exam question by question. Every question is classified by syllabus topic, so you can see which areas carried the most marks. It also covers the final answers, what better responses did and the areas NESA's markers said students need to improve. For the previous year's analysis, see our 2024 HSC Maths Advanced examiner report breakdown.

Key Takeaways

• Calculus was the biggest topic, worth 30 of the 100 marks under NESA's own mapping grid. Statistical Analysis and Trigonometric Functions were close behind, at 22 marks each.
• Several questions looked like calculus but didn't need it. In Question 26(b), the maximum came from the endpoints of the domain, not a stationary point.
• "Hence" questions (15(c), 25(b), 28(b)) and multi-part questions (17, 27, 29) rewarded students who used their earlier answers.
• Markers repeatedly flagged notation and precision: interval notation, inequality direction, rounding only at the final step and checking the calculator is in radians.
• In "show" and "prove" questions (17(a), 22, 26(a), 29(a)), every step needed to be written out clearly.

HSC Mathematics Advanced Examination Structure

The exam has 10 minutes of reading time and 3 hours of working time, and is worth 100 marks. NESA-approved calculators are allowed and a reference sheet is provided. The HSC exam mark makes up 50% of the final HSC mark. The other 50% comes from school-based assessment.

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Marks by Topic in the 2025 Exam

Every question below is classified using the mapping grid in NESA's 2025 marking guidelines, which links each question to a syllabus topic. Sequences and series sit within Financial Mathematics (MA-M1) in the syllabus. Where NESA mapped a question to more than one topic, we've counted it under its main topic.

Source: NESA 2025 HSC Mathematics Advanced marking guidelines (mapping grid). NESA also mapped Q17(c), Q21(b) and Q27(c) to Logarithms and Exponentials, alongside their main topic.

Section I: Multiple-Choice Breakdown

Source: NESA 2025 HSC Mathematics Advanced marking guidelines (answer key and mapping grid). Descriptions of what each question tested are our own.

Multiple-Choice Questions to Watch

• Question 4: Rewrite −5x(x − 2)(3 − x) as 5x(x − 2)(x − 3). The leading coefficient is positive and the roots are 0, 2 and 3. Watch the sign change hidden in the (3 − x) factor.
• Question 6: y = −f(−x) reflects the graph in both axes, so every feature moves to the opposite side. The asymptotes become x = 2 and y = −1, and the y-intercept becomes −3.
• Question 9: f′(1) ≈ 2, so f(1.1) ≈ 6 + 0.1 × 2 = 6.2. Because f′ is decreasing just after x = 1, the true value is slightly less than 6.2, which puts it in [6.0, 6.2).
• Question 10: By the chain rule, dy/dx = eˣ f′(eˣ). Since eˣ is always positive, only the stationary points of f with positive x-values count. That's 2 of the 3.

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Section II: Question-by-Question Breakdown

Answers match the sample answers in NESA's 2025 marking guidelines. Topics follow NESA's mapping grid.

What the Markers Said: Calculus (30 marks)

Question 12 – Equation of a tangent

• Better responses differentiated the negative index term correctly (−2/x² = −2x⁻²) and substituted the point and gradient into y − y₁ = m(x − x₁).
• Students need to know the difference between finding the gradient at a point, f′(a) = m, and solving f′(x) = a to find an x-value. Some confused finding a tangent with locating a stationary point.

Question 16 – Stationary points of f(x) = x²/eˣ

• Better responses found both the x- and y-values of each stationary point and determined their nature using a table of values or the second derivative.
• Markers recommended practising the product or quotient rule, and factorising the common eˣ correctly to simplify the derivative.
• In 16(b), students needed to use their results from part (a) to draw a smooth curve through (0, 0) and (2, 0.54).

Question 24 – Common tangent

• Better responses recognised that y = x is a tangent to both curves, so each derivative equals 1 at the point of intersection.
• Common errors included differentiating logarithmic functions incorrectly, equating a derivative to a function, and treating e as a variable instead of a number.

Question 25 – Integration and area

• In 25(b), "hence" meant using the result from part (a). Markers reminded students to check their calculator is in radians.
• Question 25(c) is mapped to Financial Mathematics because it's an arithmetic series question. Better responses used the given result |Aₙ| = (2n − 1)π to recognise an arithmetic series (π, 3π, 5π, …) and summed 2025 terms. The absolute value already deals with regions below the x-axis.

Question 27 – Trapezoidal rule and exact area

• Many students didn't understand what "two applications" means. It's two strips, which needs three function values. Using a table of values helps.
• In 27(c), students needed to recognise that the trapezoidal rule overestimates the area here (the curve is concave up), and write the inequality the right way round.

Question 26 – Optimisation (calculus not required)

• In 26(a), better responses drew a diagram and wrote the circle's radius as r = (100 − 3x)/(2π). Every step of a "show" question needed to be written out neatly.
• In 26(b), calculus wasn't needed. A(x) is a concave-up quadratic, so its stationary point is a minimum. The maximum must occur at an endpoint of the domain 0 ≤ x ≤ 100/3. Comparing both endpoints shows the maximum is at x = 0, when all the wire forms the circle.
• Markers noted that the words "maximum" or "minimum" don't always mean differentiating, even though NESA mapped this question to Applications of Differentiation.

What the Markers Said: Statistical Analysis (22 marks)

Question 14 – Bivariate data

• Use the syllabus terms: the form is linear and the direction is negative.
• In 14(b), better responses interpreted the slope and y-intercept in context: each extra minute of television is associated with 0.7 fewer minutes of exercise, and someone who watches no television is predicted to exercise for about 64.3 minutes. Markers noted that the slope is not the correlation coefficient.
• In 14(d), better responses converted 2 hours to 120 minutes, noted it's outside the dataset and showed that the line predicts a negative amount of exercise.

Question 19 – Conditional probability

• Better responses converted the ratio 1 : 2 : 3 into probabilities of 1/6, 1/3 and 1/2 and drew a two-stage tree diagram.
• The word "given" signals conditional probability, and "respectively" means "in the same order".

Question 21 – Continuous probability

• The derivative −1/x² is never zero, so the mode can't be found from a turning point. Better responses explained that f(x) = 1/x is decreasing on 1 ≤ x ≤ e, so the maximum is at the endpoint x = 1.
• For 21(b), the 25th percentile is found by setting the area under the curve from 1 to x equal to 0.25, so ln x = 0.25.

Question 23 – Normal distribution

• Better responses found that there were 600 male sheep (12 600 ÷ 21). 15 of them is 2.5%, which corresponds to a z-score of 2.
• Common errors included calculating the number of males from the ratio incorrectly and confusing a z-score with a percentage. Markers recommended a labelled sketch of the normal curve.
• In 23(b), 300 of 12 000 females is also 2.5%. Because the females have a smaller mean and standard deviation, fewer than 2.5% would be above x kg, so 300 couldn't be expected.

What the Markers Said: Trigonometric Functions (22 marks)

Question 15 – Modelling a sound wave

• In 15(a), the amplitude is the coefficient k, and a comes from period = 2π/a.
• In 15(b), P₂ has a period of 20 and an amplitude of 4, so only half a cycle appears on 0 ≤ t ≤ 10. Some students struggled because the horizontal axis was in whole numbers rather than radians or degrees.
• In 15(c), the answer could be read straight from the graph. Markers stressed reading the grid scale accurately, excluding stationary points, and using correct inequality or interval notation.

Question 22 – Trigonometric proof

• Better responses worked from one side only, used a common denominator and recognised that (sin²θ + cos²θ)² = sin⁴θ + cos⁴θ + 2sin²θcos²θ.
• A common error was assuming sin⁴θ + cos⁴θ = (sin²θ · cos²θ)², which is false.

Question 28 – Segments, radians and arc length

• In 28(a), better responses linked the segment area (sector minus triangle) to a quarter of the circle's area, using radian formulas from the reference sheet.
• In 28(b), students needed to draw the line y = θ on the given graph to solve θ = sin θ + π/2, read off θ ≈ 2.3 (the value of θ, not y), then use l = rθ.

Question 29 – 3D trigonometry and bearings

• In 29(a), better responses used the right-angled triangle TOY and tan 60° = h/OY with exact values.
• In 29(b), better responses recognised that triangle TOF is isosceles (so OF = h) and that angle OYF is 45° from the bearing. They then applied the cosine rule and solved the resulting quadratic, rejecting the negative solution.
• Markers warned students not to assume right angles unless they're given or can be proved.
• In 29(c), the sine rule gave angle OFY. Students then needed to convert it into a true bearing from F.

Question 31 – Trigonometric equations

• Better responses found the solutions of cos x = 1/2 in radians (π/3, 5π/3, 7π/3, …) and recognised that exactly two solutions means 5π/3 must fall inside the domain while 7π/3 falls outside it.
• Markers recommended revising how p affects the period of cos(px).

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What the Markers Said: Functions (11 marks)

Question 11 – Quadratics

• Better responses found the turning point from the axis of symmetry, t = 4, then substituted it to find h. Some students confused the intercepts with the vertex.
• In 11(b), symmetry about t = 4 gives the other value immediately: t = 8.

Question 18 – Composite function

• g(f(x)) = 3/(x − 1) + 5 is a hyperbola translated up 5 units, with asymptotes x = 1 and y = 5.
• Markers wanted accurate notation for the range: (−∞, 5) ∪ (5, ∞), or all real y with y ≠ 5.

Question 30 – Translations

• Translating right by k means replacing x with x − k. Translating up by k means adding k to the function.
• Better responses substituted (6, 11) early to simplify the algebra, solved the quadratic in k and rejected the negative value, since k is positive.
• Translating a single point instead of the whole curve wasn't a complete solution.

What the Markers Said: Financial Mathematics (14 marks)

Question 13 – Geometric sequence

• Better responses used Tₙ = arⁿ⁻¹ from the reference sheet: 2025 = 75r³, so r = 3.
• Some students didn't recognise whether the sequence was arithmetic or geometric.

Question 17 – Reducing balance loan

• In 17(a), interest is charged on the balance for the whole month before the repayment is subtracted. A₂ should be written in terms of A₁ before expanding.
• In 17(b), better responses wrote Aₙ as a series with the correct first and last terms, then used the geometric series formula from the reference sheet.
• In 17(c), students needed to use part (b), make 1.005ⁿ the subject, solve with logarithms and round only at the final step. Checking large numbers have the right number of zeros also mattered.

Question 20 – Annuity tables

• Better responses converted to monthly values (n = 84, r = 0.005), found Lin's future value with A = P(1 + r)ⁿ, then divided by the table factor of 104.07393.
• Markers flagged confusion about when to use the formula and when to use the table, and when to divide or multiply by the factor.

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Common Mistakes Across the 2025 Exam

• Not using earlier parts: "Hence" questions and multi-part questions (15, 17, 25, 27, 28) were designed to build on previous answers.
• Reaching for calculus when it wasn't needed: Questions 15(c), 21(a) and 26(b) were solved by reading a graph, using a function's shape or checking endpoints.
• Imprecise notation: Interval notation, inequality direction, "between" versus "inclusive", and the range notation in Question 18 all cost marks.
• Rounding too early and using the calculator in degrees in calculus questions.
• Assuming right angles in 3D trigonometry and bearings problems.
• Skipping steps in "show" and "prove" questions, where the working is the answer.
• Misreading key words such as "given", "respectively", "extrapolate", "hence" and "two applications".

How to Prepare for HSC Mathematics Advanced

• Know the reference sheet well, including the geometric series, trapezoidal rule, sine and cosine rules, and the normal distribution percentages.
• Practise interpreting results in context, especially for regression lines, probability and financial questions.
• Learn to recognise when calculus isn't the best tool, such as a quadratic's shape, a decreasing function on a closed interval or reading values from a given graph.
• Practise multi-part questions as a whole, asking how each part sets up the next.
• Write full working for "show" questions and check your final line matches the result you were asked to show.
• Complete full past papers under timed conditions to practise managing 3 hours of working time.

For more preparation:

• read how one of our tutors scored a 98 in HSC Maths Advanced
• see everything you need to know for HSC Maths Advanced
• also taking Extension 1? See our 2025 HSC Maths Extension 1 examiner report breakdown
• studying Standard 2 instead? See our 2025 HSC Maths Standard 2 examiner report breakdown
• brush up on key words with our guide to HSC command terms
• understand HSC performance bands
• see the HSC exam timetable 2026 for this year's exam date.

Need Help With HSC Maths Advanced?

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Sources: NSW Education Standards Authority, 2025 HSC Mathematics Advanced examination and HSC marking feedback 2025.