§-Past paper

QLDMath Methods2025Paper 1

QCE Math Methods 2025 Paper 1

Walkthrough of the 2025 QCE math methods exam: what it assessed, strategy tips, and the common errors flagged in the official marker report.

Marks: 60
Time: 90 min
Authority: QCAA
Updated: 2026-05-29

Open the official PDF ->Official marking notes

What this paper assessed

QCAA Mathematical Methods external assessment is split across Paper 1 (technology-free, 60 marks, 90 minutes) and Paper 2 (technology-active). This guide covers Paper 1, where no calculator is permitted, so the emphasis is on exact values, algebraic technique and clean by-hand calculus. It draws on Units 3 and 4 of the senior syllabus:

Calculus: further differentiation (chain, product and quotient rules), the second derivative and curve sketching, integration as antidifferentiation, definite integrals and areas, and the fundamental theorem of calculus.
Functions and their derivatives: exponential, logarithmic and trigonometric functions and their rates of change.
Probability and statistics: discrete and continuous random variables, the binomial distribution, the normal distribution, and sample proportions and confidence intervals.

Because it is technology-free, the paper rewards exact answers (leaving $\pi$ , $e$ , surds and fractions unsimplified-where-appropriate) and showing the rule applied before substituting.

Structure and timing

Paper 1 is 60 marks in 90 minutes (plus perusal) - exactly 1.5 minutes per mark.

Multiple choice / short items first: bank the quick marks at under a minute each where possible.
Extended-response items: the remaining time at ~1.5 min/mark, with multi-step questions getting a moment to map the steps.

Use perusal to spot the heaviest calculus or probability question. The most common time loss is over-working a single hard part; if a part stalls beyond ~3 minutes, leave space, move on, and return. Keep ~8 minutes at the end to recheck signs and exact-value simplifications.

Worked practice questions (exam-style)

Worked example

Question 1 (4 marks): Differentiate $f(x) = x^2\ln(3x)$ and evaluate $f'(1)$ . Give an exact answer.

This is a product of two functions, so we apply the product rule. The second factor, $\ln(3x)$ , also requires the chain rule to differentiate correctly.

Step 1: Identify the product and differentiate $\ln(3x)$

Set $u = x^2$ and $v = \ln(3x)$ . The derivative of $u$ is $u' = 2x$ . For $v$ , we apply the chain rule: $\dfrac{d}{dx}\ln(3x) = \dfrac{1}{3x}\cdot 3 = \dfrac{1}{x}$ . The factor of $3$ in the numerator and denominator cancel, leaving $v' = \dfrac{1}{x}$ .

Step 2: Apply the product rule

The product rule gives $f'(x) = u'v + uv'$ :

f'(x) = 2x\ln(3x) + x^2\cdot\frac{1}{x} = 2x\ln(3x) + x.

The second term simplifies because $x^2 \cdot \dfrac{1}{x} = x$ .

Step 3: Substitute $x = 1$ for the exact value

f'(1) = 2(1)\ln 3 + 1 = 2\ln 3 + 1.

This is already in exact form. The common technology-free slip is writing $\dfrac{d}{dx}\ln(3x) = \dfrac{1}{3x}$ (forgetting to multiply by the inner derivative), which gives the wrong answer.

Final answer: $f'(1) = 2\ln 3 + 1$ .

Worked example

Question 2 (5 marks): Evaluate $\displaystyle\int_{0}^{2}\left(3x^2 - 4x + 1\right)\,dx$ and state what the result represents if the integrand is non-negative on the interval.

We apply the fundamental theorem of calculus: antidifferentiate, then evaluate at the upper and lower limits and subtract. Showing each term separately earns the available method marks.

Step 1: Antidifferentiate term by term

Raise each power by one and divide by the new power:

\int (3x^2 - 4x + 1)\,dx = x^3 - 2x^2 + x + C.

The constant of integration drops out when we evaluate a definite integral, because it cancels in the subtraction.

Step 2: Substitute the limits and compute

Evaluate at $x = 2$ first, then at $x = 0$ , and subtract:

\Big[x^3 - 2x^2 + x\Big]_0^2 = \big(8 - 8 + 2\big) - \big(0\big) = 2.

Step 3: Interpret the result geometrically

The definite integral equals $2$ . If the integrand is non-negative everywhere on $[0,2]$ , this value equals the area enclosed between the curve $y = 3x^2 - 4x + 1$ and the $x$ -axis. In fact, the curve dips below the axis between its roots, so the value here is the net signed area rather than the total enclosed area. A point worth checking before writing "area" is whether the integrand changes sign on the interval.

Final answer: $\displaystyle\int_{0}^{2}(3x^2 - 4x + 1)\,dx = 2$ .

Worked example

Question 3 (6 marks): A biased coin lands heads with probability $p = 0.4$ . It is tossed 5 times. Let $X$ be the number of heads. (a) Write the distribution of $X$ . (b) Find $P(X = 2)$ exactly. (c) State the mean $E(X)$ .

This question tests three aspects of the binomial distribution in sequence: naming the distribution, computing a specific probability, and recalling the mean formula.

Step 1: Name the distribution

Because each toss is independent, there are a fixed five tosses, and the probability of heads is the same constant $p = 0.4$ on every toss, $X$ follows a binomial distribution: $X \sim \text{Bin}(5, 0.4)$ .

Step 2: Calculate $P(X = 2)$ exactly

Apply the binomial formula $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$ with $k=2$ , $n=5$ , $p=0.4$ , and $1-p=0.6$ :

P(X=2) = \binom{5}{2}(0.4)^2(0.6)^3 = 10 \times 0.16 \times 0.216.

= 10 \times 0.03456 = 0.3456.

So $P(X=2) = 0.3456$ (or $\tfrac{432}{1250}$ exactly). The recurring error is swapping $p$ and $1-p$ in the exponents; here heads is the "success", so $p=0.4$ carries the power $k=2$ and $0.6$ carries the remaining power $3$ .

Step 3: State the mean

For any binomial distribution, the mean is $E(X) = np$ :

E(X) = 5 \times 0.4 = 2.

Final answer: $X \sim \text{Bin}(5, 0.4)$ ; $P(X=2) = 0.3456$ ; $E(X) = 2$ .

Common errors students made

The 2025 QCAA marker report flagged chain-rule slips on composite functions, sign errors when integrating, skipped domain or restriction statements, and probability questions answered with the wrong conditional setup. Further recurring traps (especially technology-free):

Mishandling derivatives of $\ln$ and $e$ with an inner function, dropping or misplacing the chain-rule factor.
Forgetting the constant of integration for indefinite integrals, and sign errors when subtracting the lower limit.
Claiming "area" for a definite integral that crosses the axis, where the integral gives signed (net) area.
Confusing $p$ with $1-p$ in the binomial, or misreading which outcome is the "success".
Leaving inexact decimals where the technology-free paper expects exact surd, $\pi$ or fraction form.

How to use this paper

Sit Paper 1 in 90 minutes with no calculator, then mark against the official marking guide and marker report at the QCAA links in the frontmatter, watching for where method marks are awarded. Rework every item that lost more than half its marks, writing the rule or formula on its own line before substituting and checking each exact-value simplification. Keep an error log sorted by question class (chain rule, integration sign, binomial setup) and re-test that class technology-free next session.

Use this paper well

Sit the paper under exam conditions (90 minutes, 60 marks).
Mark yourself against the official QCAA 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

Keep going