§-Past paper

VICMath Methods2023

VCE Math Methods 2023

Walkthrough of the 2023 VCE math methods exam: what it assessed, strategy tips, and the common errors flagged in the official marker report.

Marks: 80
Time: 120 min
Authority: VCAA
Updated: 2026-05-29

Open the official PDF ->Official marking notes

What this paper assessed

VCE Mathematical Methods is assessed by two examinations; this Exam 2 is technology-active (CAS permitted), combining multiple choice with extended-response questions. It draws across the VCAA study design:

Functions and graphs: polynomial, power, exponential, logarithmic and circular functions; transformations, domain and range, and composite/inverse functions.
Calculus: differentiation and its applications (rates, tangents, optimisation), antidifferentiation and definite integrals (area, average value).
Probability and statistics: discrete and continuous random variables, the binomial and normal distributions, and sample proportions/confidence intervals.

Several items required students to interpret a context, set up a model, then execute the technique - often using CAS for computation but still showing reasoning. The paper rewarded stating the rule or relationship before substituting and answering the exact command word.

Structure and timing

The paper is 80 marks in 120 minutes, plus 15 minutes reading time.

Section A - 20 multiple choice (20 marks): target ~30 minutes (about 1.5 min/question).
Section B - extended response (60 marks): target ~90 minutes at roughly 1.5 minutes per mark.

A 4-mark item deserves ~6 minutes; on "show that" items, write the reasoning even though CAS gives the answer. Use reading time to plan the extended-response questions. Reserve ~8 minutes to check that final answers address the original context.

Worked practice questions (exam-style)

Worked example

Question 1 (4 marks): For $f(x) = x^3 - 6x^2 + 9x$ , find the coordinates of the stationary points and classify them.

Stationary points are where the derivative equals zero. We differentiate and factor, then substitute back to find the coordinates, and use the second derivative to classify each point.

Step 1: Differentiate and factor to find the stationary $x$ -values

Differentiating term by term and factoring:

f'(x) = 3x^2 - 12x + 9 = 3(x^2 - 4x + 3) = 3(x-1)(x-3).

Setting $f'(x) = 0$ gives $x = 1$ and $x = 3$ .

Step 2: Find the $y$ -coordinates

Substitute each $x$ -value into the original function:

f(1) = 1 - 6 + 9 = 4, \qquad f(3) = 27 - 54 + 27 = 0.

The stationary points are $(1, 4)$ and $(3, 0)$ .

Step 3: Classify using the second derivative

Compute $f''(x) = 6x - 12$ and evaluate at each stationary point:

At $x = 1$ : $f''(1) = 6 - 12 = -6 < 0$ , so $(1, 4)$ is a local maximum.
At $x = 3$ : $f''(3) = 18 - 12 = 6 > 0$ , so $(3, 0)$ is a local minimum.

Final answer: local maximum at $(1, 4)$ and local minimum at $(3, 0)$ .

Worked example

Question 2 (5 marks): The continuous random variable $X$ has probability density function $f(x) = kx$ for $0 \le x \le 4$ , and $0$ otherwise. Find $k$ , then find $P(X \le 2)$ .

The constant $k$ is found by requiring the pdf to integrate to 1 over its support. Once $k$ is known, any probability is found by integrating the pdf over the required interval.

Step 1: Find $k$ from the normalisation condition

For a valid pdf, the integral over the entire support must equal 1. Evaluating over $[0, 4]$ :

\int_0^4 kx\, dx = k\left[\frac{x^2}{2}\right]_0^4 = k \cdot \frac{16}{2} = 8k = 1 \implies k = \frac{1}{8}.

Step 2: Compute $P(X \le 2)$

Integrate the pdf (with $k = \dfrac{1}{8}$ ) from the lower bound of the support to $x = 2$ :

P(X \le 2) = \int_0^2 \frac{x}{8}\, dx = \frac{1}{8}\left[\frac{x^2}{2}\right]_0^2 = \frac{1}{8}\cdot\frac{4}{2} = \frac{1}{8}\times 2 = \frac{1}{4}.

Final answer: $k = \dfrac{1}{8}$ and $P(X \le 2) = \dfrac{1}{4}$ .

Worked example

Question 3 (5 marks): A factory's daily output (in tonnes) follows a normal distribution with mean $\mu = 50$ and standard deviation $\sigma = 4$ . Using CAS, the probability that output is below 45 tonnes is to be found. Show the setup and state the result, then interpret it.

Standardise to make the reasoning explicit:

z = \frac{x - \mu}{\sigma} = \frac{45 - 50}{4} = -1.25

So $P(X < 45) = P(Z < -1.25)$ . Evaluating with CAS (or standard normal tables):

P(Z < -1.25) \approx 0.1056

Interpretation: on about 10.6% of days the factory produces less than 45 tonnes. State the relationship and the standardisation before quoting the CAS value so method marks are secured even if a syntax slip occurs.

Common errors students made

Recurring errors in VCE Methods Exam 2 responses include:

Chain-rule slips on composite functions and sign errors when integrating.
Skipping domain or restriction statements (e.g. for inverse functions or restricted intervals).
Setting up probability questions with the wrong distribution or conditional.
Relying on CAS output without showing the reasoning that earns method marks.

Add these subject-specific traps:

Forgetting to check the validity condition for a probability density function (total area = 1).
CAS syntax errors (wrong brackets, degree vs radian mode) producing a plausible but wrong value.
Not answering the exact command word ("show that" vs "find" vs "hence").
Final answers that ignore the original context (units, sensible range).

How to use this paper

Sit Section A in 30 minutes and Section B in 90 minutes under timed conditions with an approved CAS and the VCAA formula sheet. Mark against the VCAA examination report (linked in the frontmatter), which lists the most common errors and the average marks per question. For every CAS-active item, practise writing the relationship before the computation so you keep method marks. Keep a personal error log distinguishing conceptual, algebraic, and CAS-syntax slips, and re-attempt every multi-step modelling item from a clean page after a 48-hour gap.

Use this paper well

Sit the paper under exam conditions (120 minutes, 80 marks).
Mark yourself against the official VCAA marking notes.
Compare against the Math Methods hub to find the syllabus dot points this paper tested.

What this paper assessed

Structure and timing

Worked practice questions (exam-style)

Common errors students made

How to use this paper

Use this paper well

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