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QLDMath Methods2021Paper 1

QCE Math Methods 2021 Paper 1

Walkthrough of the 2021 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
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 (55 marks in 90 minutes). Paper 1 is the technology-free paper (Paper 2 is technology-active), so it rewards clean by-hand technique and exact-value work. It examines Unit 4 (further differentiation and applications, integrals, and the binomial distribution) alongside earlier units, covering:

  • Differential calculus: derivatives of polynomial, exponential, logarithmic and trigonometric functions; the chain, product and quotient rules; and applications such as rates of change and optimisation.
  • Integral calculus: antidifferentiation, the fundamental theorem of calculus, definite integrals and areas under curves.
  • Functions and algebra: exponential and logarithmic relationships, and the solution of equations.
  • Probability and statistics: discrete random variables, the binomial distribution, and continuous distributions including the normal.

Items require interpreting a context, setting up a model, and carrying out a technique cleanly, with exact answers expected where calculators are not permitted.

Structure and timing

55 marks in 90 minutes is 1.6 minutes per mark - tighter than the Biology and Physics papers (about 2 minutes per mark), so pace matters.

  • Multiple choice / short items. Bank these quickly at ~1.6 min/mark; do not linger.
  • Short-answer technique items. Show each line of working; method marks are awarded even if the final value slips.
  • Extended-response/modelling items. A 5-mark context item deserves ~7-8 minutes, including checking the answer makes sense in context.

A workable plan: work steadily at 1.6 min/mark and keep a final 8-10 minutes to recheck signs, domains and exact values.

Worked practice questions (exam-style)

Worked example

Question 1 (3 marks): Differentiate f(x)=ln⁡(2x2+1)f(x) = \ln(2x^2 + 1) and evaluate f′(1)f'(1).

This is a composite function where the outer function is ln⁡\ln and the inner function is 2x2+12x^2 + 1. The chain rule handles it.

Step 1: Apply the chain rule for ln⁡(u)\ln(u)

The derivative of ln⁡(u)\ln(u) with respect to xx is u′u\dfrac{u'}{u}. Here u=2x2+1u = 2x^2 + 1, so u′=4xu' = 4x.

f′(x)=4x2x2+1f'(x) = \frac{4x}{2x^2 + 1}

Step 2: Evaluate at x=1x = 1

Substitute x=1x = 1 into the derivative expression.

f′(1)=4(1)2(1)2+1=43f'(1) = \frac{4(1)}{2(1)^2 + 1} = \frac{4}{3}

Final answer: f′(1)=43f'(1) = \dfrac{4}{3}.

Worked example

Question 2 (4 marks): Exact area enclosed between y=x2y = x^2 and y=2xy = 2x.

The area between two curves requires knowing where they intersect, which function is on top, and then integrating the difference.

Step 1: Find the intersection points

Set the two expressions equal and solve exactly.

x2=2x  ⟹  x2−2x=0  ⟹  x(x−2)=0  ⟹  x=0 or x=2x^2 = 2x \implies x^2 - 2x = 0 \implies x(x - 2) = 0 \implies x = 0 \text{ or } x = 2

Step 2: Identify which function is greater on the interval

On [0,2][0, 2], test x=1x = 1: the line gives 2(1)=22(1) = 2 and the parabola gives 12=11^2 = 1. The line y=2xy = 2x lies above the parabola y=x2y = x^2, so integrate (top minus bottom).

Step 3: Set up and evaluate the definite integral

A=∫02(2x−x2) dx=[x2−x33]02=(4−83)−0=123−83=43A = \int_0^2 (2x - x^2)\,dx = \left[x^2 - \frac{x^3}{3}\right]_0^2 = \left(4 - \frac{8}{3}\right) - 0 = \frac{12}{3} - \frac{8}{3} = \frac{4}{3}

Final answer: the exact enclosed area is 43\dfrac{4}{3} square units.

Worked example

Question 3 (5 marks): Binomial probability for a fair coin tossed 6 times.

Each toss is an independent trial with a fixed probability of success. This matches the conditions for a binomial distribution.

Step 1: State the distribution of XX

Each of the 6 tosses is independent, and the probability of a head on each toss is p=12p = \tfrac{1}{2}. The number of heads XX therefore follows a binomial distribution.

X∼Bin ⁣(6, 12)X \sim \text{Bin}\!\left(6,\ \tfrac{1}{2}\right)

Step 2: Apply the binomial probability formula for k=4k = 4

Use P(X=k)=(nk)pk(1−p)n−kP(X = k) = \dbinom{n}{k} p^k (1-p)^{n-k} with n=6n = 6, k=4k = 4, and p=12p = \tfrac{1}{2}.

P(X=4)=(64)(12)4(12)2P(X = 4) = \binom{6}{4}\left(\frac{1}{2}\right)^4\left(\frac{1}{2}\right)^{2}

Step 3: Evaluate the binomial coefficient and simplify

(64)=15\dbinom{6}{4} = 15. All six factors of 12\tfrac{1}{2} combine to give (12)6=164\left(\tfrac{1}{2}\right)^6 = \tfrac{1}{64}.

P(X=4)=15×164=1564P(X = 4) = 15 \times \frac{1}{64} = \frac{15}{64}

Final answer: the exact probability of exactly 4 heads is 1564\dfrac{15}{64}.

Common errors students made

The 2021 QCAA marker report noted chain-rule slips on composite functions, sign errors when integrating, skipped domain or restriction statements, and wrong conditional setups in probability. Adding to those:

  • Decimal instead of exact answers - Paper 1 expects exact values (43\tfrac{4}{3}, ln⁡2\ln 2, surds), and decimals lose accuracy marks.
  • Forgetting the constant of integration in indefinite integrals, or omitting/mis-ordering limits in definite integrals.
  • Area sign errors - not subtracting "top minus bottom", giving a negative area.
  • Binomial setup slips - using the wrong kk, or forgetting the (nk)\dbinom{n}{k} coefficient.

How to use this paper

Sit the full paper in a single timed 90-minute block, technology-free, keeping a final 8-10 minute buffer to recheck signs, exact values and domains. Mark each item against the official QCAA marking guide at the authority page linked at the top of this page, paying attention to where method marks are awarded - they are recoverable with practice. Re-attempt any item that lost more than half its marks on a fresh page, writing the rule or formula before substituting. Keep an error log of recurring slips (chain rule, missing +C+C, area orientation) and re-test those item types after a week. Drill exact-value manipulation so it is automatic without a calculator.

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