§-Past paper

QLDMath Methods2024Paper 1

QCE Math Methods 2024 Paper 1

Walkthrough of the 2024 QCE math methods exam: what it assessed, strategy tips, and the common errors flagged in the official marker report.

Marks: 55
Time: 90 min
Authority: QCAA
Updated: 2026-10-07

Open the official PDF Official marking notes

Key fact

This paper was set on the 2019 QCAA syllabus. From the 2026 external assessments the 2025 syllabus applies, so some content and the exam format have changed. It is still useful practice, but check the current QCAA Mathematical Methods syllabus and QCAA's 2026 key changes before relying on it. The mark and timing figures shown on this page are approximate study guides, not QCAA's official paper structure.

What this paper assessed

This is Paper 1 of the QCAA Mathematical Methods external assessment: a 55-mark technology-free (no calculator) paper sat over 90 minutes (plus 5 minutes perusal). It assesses Unit 3 (Further calculus) and Unit 4 (Further functions and statistics):

Differential calculus - the product, quotient and chain rules; derivatives of exponential, logarithmic and trigonometric functions; and applications (rates of change, optimisation).
Integral calculus - antidifferentiation, definite integrals, and areas under and between curves.
Exponential and logarithmic functions - including $\frac{d}{dx}(e^{kx})$ and $\frac{d}{dx}(\ln x)$ .
Statistics - discrete and continuous random variables, the binomial distribution, the normal distribution, and sample proportions / confidence intervals.

Because Paper 1 is technology-free, it rewards clean by-hand technique and exact (surd/fraction) answers. The 2024 paper leaned harder on application items where the surface context was unfamiliar but the underlying skill was core.

Structure and timing

55 marks in 90 minutes is 1.6 minutes per mark. The paper has a short-response section (often with some multiple choice) and longer items:

Quick/multiple-choice items: ~1-1.5 min each to bank marks early.
Short and extended response (the bulk of the 55 marks): ~1.6 min/mark. A 5-mark optimisation item deserves ~7-8 minutes including setup.
Reserve the final ~8-10 minutes to check exact answers, signs and that each result addresses the context.

Always write the rule or formula being applied before substituting.

Worked practice questions (exam-style)

Worked example

Question 1 (4 marks): Differentiate $f(x) = (2x^2 + 1)^4$ using the chain rule, and evaluate $f'(1)$ .

This is a composite function: a power applied to a quadratic. We use the chain rule, which requires identifying the inner and outer parts separately before combining.

Step 1: Identify the inner and outer functions

Let $u = 2x^2 + 1$ so that $f = u^4$ . Now differentiate each part:

\frac{df}{du} = 4u^3, \qquad \frac{du}{dx} = 4x.

Step 2: Apply the chain rule

Multiply the two derivatives and substitute $u$ back in terms of $x$ :

f'(x) = \frac{df}{du}\cdot\frac{du}{dx} = 4u^3 \cdot 4x = 16x(2x^2 + 1)^3.

Step 3: Evaluate at $x = 1$

First find the inner value: $2(1)^2 + 1 = 3$ . Then substitute:

f'(1) = 16(1)(3)^3 = 16 \times 27 = 432.

Final answer: $f'(x) = 16x(2x^2+1)^3$ and $f'(1) = 432$ .

Worked example

Question 2 (5 marks): A rectangle has perimeter 20 cm. Find the dimensions that maximise its area, working without a calculator, and justify that it is a maximum.

This is an optimisation problem. We introduce a variable, use the constraint (fixed perimeter) to reduce to one variable, then differentiate to locate the stationary point.

Step 1: Set up the area as a function of one variable

Let the width be $x$ and the length be $y$ . The perimeter constraint gives:

2x + 2y = 20 \quad\Rightarrow\quad y = 10 - x.

Substituting into the area formula:

A = xy = x(10 - x) = 10x - x^2.

Step 2: Differentiate and find the stationary point

Differentiate with respect to $x$ and set equal to zero:

\frac{dA}{dx} = 10 - 2x = 0 \quad\Rightarrow\quad x = 5.

Step 3: Justify the maximum and state the answer

Compute the second derivative: $\dfrac{d^2A}{dx^2} = -2 < 0$ , which confirms $x = 5$ is a local maximum. Then $y = 10 - 5 = 5$ .

Final answer: the rectangle is a $5\ \text{cm}$ by $5\ \text{cm}$ square, with maximum area $A = 25\ \text{cm}^2$ .

Worked example

Question 3 (5 marks): A fair coin is tossed 6 times. Using the binomial distribution, find the exact probability of obtaining exactly 4 heads.

The toss sequence meets all binomial conditions: fixed number of trials, independent outcomes, constant probability. We set up the distribution and apply the formula directly.

Step 1: Define the distribution

Let $X$ be the number of heads. Since the coin is fair, $p = \tfrac{1}{2}$ , and there are $n = 6$ tosses, so $X \sim \text{Bin}(6,\, \tfrac{1}{2})$ .

Step 2: Write the binomial formula for $k = 4$

The general binomial probability is $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ . Substituting $k = 4$ :

P(X = 4) = \binom{6}{4}\left(\tfrac{1}{2}\right)^4 \left(\tfrac{1}{2}\right)^2 = \binom{6}{4}\left(\tfrac{1}{2}\right)^6.

Step 3: Evaluate the binomial coefficient and simplify

Compute $\binom{6}{4} = \dfrac{6!}{4!\,2!} = 15$ , and note that $\left(\tfrac{1}{2}\right)^6 = \dfrac{1}{64}$ :

P(X = 4) = 15 \times \frac{1}{64} = \frac{15}{64}.

Final answer: the exact probability is $\dfrac{15}{64}$ ( $\approx 0.234$ ).

Common errors students made

The 2024 QCAA marker report flagged that students lost marks by:

chain-rule slips on composite functions;
sign errors when integrating (and forgetting the constant);
skipping domain or restriction statements;
using the wrong conditional setup in probability questions.

Further traps to avoid (especially technology-free):

Giving decimal approximations where an exact surd/fraction answer is required.
Forgetting to justify a maximum/minimum with the second derivative or a sign test.
Mixing radians and degrees when differentiating trig functions.
Not changing limits when using substitution, or not addressing the original context in an applied item.

How to use this paper

Sit the full paper in 90 minutes with no calculator, working all answers by hand.
Self-mark against the QCAA marking guide and marker report using the links at the top of this page, awarding method marks line by line.
Re-attempt any item scored below half, writing the relationship first, then working exactly.
Keep an error log of recurring algebra/sign slips and drill technology-free technique before each timed re-sit.

What this paper assessed

Structure and timing

Worked practice questions (exam-style)

Question 1 (4 marks): Differentiate f(x)=(2x2+1)4f(x) = (2x^2 + 1)^4f(x)=(2x2+1)4 using the chain rule, and evaluate f′(1)f'(1)f′(1).

Step 1: Identify the inner and outer functions

Step 2: Apply the chain rule

Step 3: Evaluate at x=1x = 1x=1

Question 2 (5 marks): A rectangle has perimeter 20 cm. Find the dimensions that maximise its area, working without a calculator, and justify that it is a maximum.

Step 1: Set up the area as a function of one variable

Step 2: Differentiate and find the stationary point

Step 3: Justify the maximum and state the answer

Question 3 (5 marks): A fair coin is tossed 6 times. Using the binomial distribution, find the exact probability of obtaining exactly 4 heads.

Step 1: Define the distribution

Step 2: Write the binomial formula for k=4k = 4k=4

Step 3: Evaluate the binomial coefficient and simplify

Common errors students made

How to use this paper

Keep going

Question 1 (4 marks): Differentiate $f(x) = (2x^2 + 1)^4$ using the chain rule, and evaluate $f'(1)$ .

Step 3: Evaluate at $x = 1$

Step 2: Write the binomial formula for $k = 4$