30 VCE Math Methods practice questions for 2026

How to use this question bank

VCE Math Methods is examined in two papers, Paper 1 (technology-free, 1 hour, 40 marks) and Paper 2 (technology-active, 2 hours, 80 marks). These 30 practice questions are split across both modes and the three content areas (functions, calculus, probability and statistics).

Three rules for VCE Math Methods practice.

1. Practice Paper 1 without any calculator at all. No scientific, no CAS, nothing. The point of Paper 1 is by-hand fluency, and any calculator use during practice undermines that.
2. Practice Paper 2 with your CAS calculator beside you and use it. Build the reflexes of which CAS commands solve which problems. Save the by-hand work for what is faster by hand.
3. Always state exact answers where possible. $\pi/4$ not $0.7854$. $\sqrt{3}/2$ not $0.866$. Approximations cost marks.

Paper 1 (Technology-free, 1-15)

Allocate 1.5 minutes per mark. Total Paper 1 budget if done in full: 60 minutes for the 40-mark paper.

Functions and algebra (1-5)

1. Factorise $f(x) = 2x^3 - x^2 - 7x + 6$ completely over the rationals. (3 marks)

2. Solve for $x \in \mathbb{R}$: $\ln(2x + 1) = \ln(x) + \ln(3)$. (3 marks)

3. Let $f(x) = 2\sin(2x - \pi/3) + 1$. State the amplitude, period, and the maximum value of $f$. (3 marks)

4. Let $g(x) = \sqrt{x - 2}$, $x \geq 2$. Find $g^{-1}(x)$ and state its domain and range. (4 marks)

5. Solve $\cos(2x) = -1/2$ for $x \in [0, 2\pi]$. (3 marks)

Calculus (6-11)

6. Find $f'(x)$ from first principles for $f(x) = 3x^2 - x$. (3 marks)

7. Differentiate $h(x) = x^2 \ln(x)$. (2 marks)

8. Differentiate $y = \frac{\sin(x)}{e^x}$. (3 marks)

9. Find the equation of the tangent to $y = x e^{x}$ at the point $x = 1$. (4 marks)

10. Evaluate $\int_0^{\pi/4} \sec^2(x)\,dx$ exactly. (2 marks)

11. The acceleration of a particle is $a(t) = 4 - 2t$, with $v(0) = 0$ and $x(0) = 0$. Find $x(t)$ and the time at which the particle is momentarily at rest. (5 marks)

Probability and statistics (12-15)

12. A discrete random variable $X$ has the distribution $P(X = 1) = 0.2$, $P(X = 2) = 0.3$, $P(X = 3) = 0.5$. Find $E(X)$ and $\text{Var}(X)$. (3 marks)

13. IMATH_31 . Find $P(X = 3)$ as an exact fraction. (2 marks)

14. The continuous random variable $X$ has PDF $f(x) = kx$ for $0 \leq x \leq 2$, zero elsewhere. Find $k$ and $E(X)$. (3 marks)

15. IMATH_38 . Approximately what proportion of values lie in the interval $[\mu - 2\sigma, \mu + \sigma]$? (2 marks)

Paper 2 Section A (Technology-active multi-choice, 16-22)

For each question, select the single best answer A-D. CAS permitted.

16. The function $f: \mathbb{R} \to \mathbb{R}$, $f(x) = x^3 - 6x^2 + 9x + 1$ has stationary points at $x =$

    • A. 1 only
    • B. 3 only
    • C. 1 and 3
    • D. 0 and 2

17. The maximum value of $f(x) = -2x^2 + 8x - 3$ is

    • A. 3
    • B. 5
    • C. 8
    • D. IMATH_44

18. For $f(x) = e^{2x}$, $\int_0^1 f(x)\,dx =$

    • A. $\frac{e^2 - 1}{2}$
    • B. $e^2 - 1$
    • C. $2e^2$
    • D. IMATH_50

19. The inverse function of $f: [0, \infty) \to \mathbb{R}$, $f(x) = x^2 + 1$ has rule

    • A. $f^{-1}(x) = \sqrt{x} - 1$
    • B. $f^{-1}(x) = \sqrt{x - 1}$
    • C. $f^{-1}(x) = (x - 1)^2$
    • D. IMATH_56

20. IMATH_57 . $P(X = 4)$ to four decimal places is approximately
    - A. 0.1029
    - B. 0.2001
    - C. 0.0917
    - D. 0.2335

21. Heights are distributed $N(170, 64)$. The proportion of heights between 160 cm and 180 cm is approximately

    • A. 0.683
    • B. 0.789
    • C. 0.500
    • D. 0.954

22. A 95% confidence interval for a population proportion based on $n = 200$ with $\hat{p} = 0.6$ has approximate width

    • A. 0.068
    • B. 0.136
    • C. 0.034
    • D. 0.272

Paper 2 Section B (Technology-active extended response, 23-30)

Use your CAS calculator. Show working where appropriate.

23. The function $f: [0, 6] \to \mathbb{R}$, $f(x) = -\frac{1}{6}x^3 + x^2$, models a hill profile.

    a. Find the coordinates of the highest point of the hill. (3 marks)

    b. Find the area enclosed between the curve and the x-axis. (3 marks)

    c. The hill is to be approximated by a rectangle of the same base width and the same area. Find the height of this rectangle. (2 marks)

24. A particle moves in a straight line with velocity $v(t) = 3t^2 - 12t + 9$ for $t \geq 0$, in metres per second.

    a. Find the times when the particle is at rest. (2 marks)

    b. Find the displacement of the particle from $t = 0$ to $t = 4$. (3 marks)

    c. Find the total distance travelled by the particle from $t = 0$ to $t = 4$. (4 marks)

25. Let $f(x) = 2\sin(3x) + 1$.

    a. State the period and the range of $f$. (2 marks)

    b. Find all solutions of $f(x) = 2$ for $x \in [0, 2\pi]$. (4 marks)

26. The volume of a solid is given by $V = \pi \int_0^4 [g(x)]^2 dx$ where $g(x) = e^{-x/2}$.

    a. Find $V$ in exact form. (3 marks)

    b. Find $V$ correct to 2 decimal places. (1 mark)

27. A factory produces light bulbs. The lifetime $X$ in hours is normally distributed with mean 1200 and standard deviation 150.

    a. Find $P(X > 1500)$. (2 marks)

    b. Find the lifetime that is exceeded by 5% of bulbs. (2 marks)

    c. Three bulbs are tested independently. Find the probability that all three last more than 1500 hours. (3 marks)

28. A pollster surveys $n = 800$ voters and finds 480 support a policy.

    a. State the sample proportion $\hat{p}$. (1 mark)

    b. Construct a 95% confidence interval for the true population proportion $p$. (3 marks)

    c. The pollster wishes to halve the margin of error. What sample size is required (assuming $\hat{p}$ stays the same)? (3 marks)

29. The function $f: \mathbb{R} \to \mathbb{R}$, $f(x) = x e^{-x^2}$.

    a. Find $f'(x)$. (3 marks)

    b. Find the coordinates of the stationary points. (3 marks)

    c. Sketch the graph of $f$, showing all key features. (3 marks)

30. A box has dimensions $x$ by $x$ by $h$ (a square base of side $x$ and height $h$). The total surface area is fixed at 600 cm$^2$ (open top).

    a. Express $h$ in terms of $x$. (2 marks)

    b. Express the volume $V$ in terms of $x$ alone. (1 mark)

    c. Find the value of $x$ that maximises $V$, and the maximum volume. (5 marks)

Marking your own work

For each VCAA-style question.

• Paper 1 short answer. Final answer is exact (no decimals), with correct notation, and clean working that a marker can follow.
• Paper 2 Section A multi-choice. Pure right or wrong. Re-do any wrong answer until you know why.
• Paper 2 Section B extended response. Marks are awarded for: setup (defining variables, stating constraints), method (correct differentiation/integration/CAS command), and final answer (with units where required). Partial credit is generous if the method is sound.

A useful self-mark question for Section B. Did I show all working in a way the marker can follow even when the answer is wrong? If yes, you likely banked partial marks.

Past papers

These practice questions complement past VCAA exam papers; they do not replace them. VCAA publishes papers, assessment reports, and answer guides for VCE Math Methods at vcaa.vic.edu.au. Papers from 2023 onwards use the current study design. Aim for 6 to 8 full timed pairs (Paper 1 plus Paper 2) in the final term.

Related guides

• VCE Math Methods functions, graphs and transformations
• VCE Math Methods calculus
• VCE Math Methods probability and statistics
• VCE Math Methods hub

These questions are written by ExamExplained for practice purposes only. They are not endorsed by VCAA.