30 HSC Mathematics Advanced practice questions for 2026 (calculus, trig, statistics)

How to use this question bank

HSC Mathematics Advanced is a 3-hour exam covering calculus, trigonometric functions, statistical analysis, financial mathematics, and (for the new 2024 syllabus) vectors.

These 30 practice questions are grouped by topic and difficulty. For each session: pick 4-6 questions, set a timer (about 1.5 minutes per mark), work through them, then mark yourself.

Three rules for practice:

1. Show full working. HSC markers award method marks even when the final answer is wrong. Sloppy working that gets the right answer scores lower than careful working that gets a slightly wrong answer.
2. Check units and contexts. A worded problem asking for "the dimensions of the box in cm" loses marks if you give the answer in unlabelled numerical form.
3. Read the question twice. The single highest-yield habit. Most lost marks come from misreading what was asked.

Calculus: differentiation (1-8)

1. Differentiate $f(x) = (2x^2 + 3)^5$.

2. Find $\frac{dy}{dx}$ for $y = e^{3x} \sin x$.

3. Find the equation of the tangent to $y = \ln x$ at $x = e$.

4. The position of a particle is $s(t) = t^3 - 6t^2 + 9t$. Find the times at which the particle is stationary.

5. A rectangular box has a square base with side $x$ cm and height $h$ cm. Its volume is fixed at 1000 cm³. Express the surface area in terms of $x$ and find the value of $x$ that minimises it.

6. Find $\frac{dy}{dx}$ if $y = \tan(2x + 1)$.

7. A particle moves along the x-axis with velocity $v(t) = 6t - t^2$ m/s. Find the maximum displacement of the particle.

8. The function $f(x) = x^4 - 8x^3 + 18x^2$ has stationary points. Find them and classify each as a maximum, minimum, or point of inflection.

Calculus: integration (9-15)

9. Evaluate $\int_0^{\pi/2} \sin(2x) \, dx$.

10. Find $\int \frac{2x}{x^2 + 1} \, dx$.

11. Find the area enclosed between $y = x^2$ and $y = 4$.

12. The region bounded by $y = \sin x$, $y = 0$, $x = 0$, and $x = \pi$ is rotated about the x-axis. Find the volume of revolution.

13. Evaluate $\int_1^e \frac{\ln x}{x} \, dx$ using substitution $u = \ln x$.

14. Find the area between $y = e^x$ and $y = x + 1$ from $x = 0$ to $x = 1$.

15. A function $f(x)$ satisfies f'(x) = 3x^2 - 4x + 1 and $f(1) = 5$. Find $f(x)$.

Trigonometric functions (16-22)

16. Solve $2\sin\theta + 1 = 0$ for $0 \leq \theta \leq 2\pi$.

17. Prove that $\frac{1 - \cos 2\theta}{\sin 2\theta} = \tan\theta$.

18. Find the period and amplitude of $y = 4\sin\left(\frac{x}{3} - \frac{\pi}{4}\right) + 1$.

19. Solve $\cos 2\theta = \sin\theta$ for $0 \leq \theta \leq 2\pi$.

20. The temperature in a city varies sinusoidally during the day, with a maximum of 28°C at 3pm and a minimum of 12°C at 3am. Write an equation for the temperature $T$ in terms of time $t$ (hours after midnight).

21. Evaluate $\sin\left(\frac{5\pi}{6}\right)$ using exact values.

22. A sector of a circle has radius 10 cm and central angle $\frac{\pi}{3}$ radians. Find the area of the sector and the length of its arc.

Statistical analysis (23-28)

23. The marks in a test are normally distributed with mean 65 and standard deviation 12. Find the probability that a randomly chosen student scores above 80.

24. A factory produces resistors with a mean resistance of 100 ohms and standard deviation 2 ohms. Resistors outside the range 96 to 104 ohms are rejected. What percentage of resistors are accepted?

25. A discrete random variable $X$ takes values 1, 2, 3, 4 with probabilities 0.1, 0.3, 0.4, 0.2 respectively. Find $E(X)$ and the standard deviation of $X$.

26. The heights of Year 12 boys at a school are normally distributed with mean 178 cm and SD 7 cm. A sample of 49 boys is taken. What is the probability the sample mean is above 180 cm?

27. A class of 30 students has scores with mean 70 and standard deviation 10. Assuming approximately normal distribution, how many students would you expect to score above 90?

28. The 68-95-99.7 rule for a normal distribution gives confidence in three standard-deviation intervals. The IQ test is calibrated to be normal with mean 100 and standard deviation 15. What range of IQ scores contains the middle 95% of the population?

Financial mathematics (29-30)

29. A loan of $200,000 is repaid by monthly instalments over 25 years at an interest rate of 5% per annum compounded monthly. Find the monthly repayment.

30. $5,000 is invested at 4% per annum compounded annually. Find the amount in the account after 10 years. How long would it take for the investment to double in value?

Marking your own work

For each question:

• 2-3 marks: short answer. Aim for correct final answer with concise working.
• 4-6 marks: extended response. Show full working. Method marks are awarded for correct setup and identifiable steps even if the final answer is wrong.
• 7-10 marks: multi-part. Each part typically builds on the previous; if you cannot do part (a), assume a plausible answer and move on to part (b). Markers reward continuation.

A useful habit: rate your confidence on each question (1-5) before checking the answer. Calibrate your gut against your actual performance. Strong students learn to trust their gut, which saves checking time on the real exam.

Past papers are the gold standard

These practice questions complement past NESA exam papers; they do not replace them. Aim for 8-10 full past papers under timed 3-hour conditions in the final term. NESA publishes papers and marking guides at educationstandards.nsw.edu.au.

Where to go next

For topic-specific theory and worked examples, see:

• HSC Maths Advanced: calculus guide
• HSC Maths Advanced: trigonometric functions guide
• HSC Maths Advanced: statistical analysis guide
• HSC Maths Advanced: vectors guide (new 2024 syllabus)

These prompts are written by ExamExplained for practice purposes only. For official HSC past papers and marking guides, refer to NESA.