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QLDMath Methods2022Paper 1

QCE Math Methods 2022 Paper 1

Walkthrough of the 2022 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 the QCAA Mathematical Methods external assessment Paper 1 (technology-free), covering Units 3 and 4. With no calculator, it rewards fluent by-hand technique and exact-value work across:

  • Calculus: differentiation (chain, product and quotient rules), applications of derivatives (rates of change, optimisation, motion), and integration including the fundamental theorem and areas.
  • Functions and their graphs: exponential, logarithmic and trigonometric functions and transformations.
  • Probability and statistics: discrete and continuous random variables, the binomial distribution, and the normal distribution (interpreted via standardisation).
  • Algebra: logarithmic and exponential manipulation, and solving equations exactly.

Items mix short-answer technique with applied problems where students set up a model, apply a technique cleanly, and interpret the result in context - all without technology, so accurate hand computation and exact values matter.

Structure and timing

55 marks in 90 minutes (about 1.6 min/mark), technology-free, with perusal time and the QCAA formula sheet:

  • Section 1 is ten multiple-choice questions (10 marks) and Section 2 is nine short-response questions (45 marks), with the short-response items of increasing difficulty.
  • A workable pace is ~1.6 minutes per mark: a 3-mark item ~5 minutes, a 6-mark item ~10 minutes. Bank the early routine marks quickly, then spend the saved time on the harder applied items, leaving ~5 minutes to check.

Because there is no calculator, write the rule or formula you are applying before substituting, keep answers exact (surds, fractions, ln⁡\ln, π\pi) unless asked to round, and show every algebraic step so method marks are visible.

Worked practice questions (exam-style)

Worked example

Question 1 (4 marks): Differentiate y=(3x2−5)4y = (3x^2 - 5)^4 and evaluate the gradient at x=1x = 1.

This is a composite function: a power of 4 applied to an inner polynomial. The chain rule differentiates the outer function and multiplies by the derivative of the inner function.

Step 1: Identify the inner and outer functions

Let u=3x2−5u = 3x^2 - 5 so that y=u4y = u^4. Differentiating each piece separately:

dydu=4u3,dudx=6x\frac{dy}{du} = 4u^3, \qquad \frac{du}{dx} = 6x

(2 marks.)

Step 2: Multiply to get dydx\dfrac{dy}{dx}

By the chain rule dydx=dydu⋅dudx\dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx}. Substitute back to express the result in terms of xx.

dydx=4(3x2−5)3×6x=24x(3x2−5)3\frac{dy}{dx} = 4(3x^2 - 5)^3 \times 6x = 24x(3x^2 - 5)^3

(1 mark.)

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

First compute the inner function at x=1x = 1: 3(1)2−5=−23(1)^2 - 5 = -2. Then substitute into the derivative.

dydx∣x=1=24(1)(−2)3=24×(−8)=−192\frac{dy}{dx}\bigg|_{x=1} = 24(1)(-2)^3 = 24 \times (-8) = -192

Final answer: the gradient at x=1x = 1 is −192-192 (1 mark).

Worked example

Question 2 (5 marks): Definite integral and area

Evaluate ∫12(3x2−2x2) dx\displaystyle\int_{1}^{2} \left(3x^2 - \frac{2}{x^2}\right)\,dx, giving an exact answer.

Solution. Rewrite 2x2=2x−2\dfrac{2}{x^2} = 2x^{-2} and integrate term by term:

∫(3x2−2x−2)dx=x3+2x−1=x3+2x\int \left(3x^2 - 2x^{-2}\right)dx = x^3 + 2x^{-1} = x^3 + \frac{2}{x}

(2 marks.) Evaluate between the limits:

[x3+2x]12=(8+1)−(1+2)\left[x^3 + \frac{2}{x}\right]_1^2 = \left(8 + 1\right) - \left(1 + 2\right)

(2 marks.) So the value is 9−3=69 - 3 = 6 (1 mark). Keeping the answer exact is essential in the technology-free paper.

Worked example

Question 3 (5 marks): Binomial probability

A fair six-sided die is rolled 4 times. Find the exact probability of obtaining exactly two sixes.

Solution. Each roll is independent with P(six)=16P(\text{six}) = \dfrac{1}{6}, so the number of sixes X∼Binomial(n=4,p=16)X \sim \text{Binomial}(n=4, p=\tfrac{1}{6}) (1 mark). Using P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k}p^k(1-p)^{n-k} with k=2k = 2:

P(X=2)=(42)(16)2(56)2P(X = 2) = \binom{4}{2}\left(\frac{1}{6}\right)^2\left(\frac{5}{6}\right)^2

(2 marks.) Now (42)=6\binom{4}{2} = 6, so

P(X=2)=6×136×2536=6×251296=1501296=25216P(X=2) = 6 \times \frac{1}{36} \times \frac{25}{36} = \frac{6 \times 25}{1296} = \frac{150}{1296} = \frac{25}{216}

(2 marks for evaluating to the exact simplified fraction.)

Common errors students made

The 2022 QCAA marker report flagged chain-rule slips on composite functions, sign errors when integrating, skipping domain/restriction statements, and probability questions answered with the wrong conditional setup. Add these recurring traps:

  • Forgetting the "inner derivative" in the chain rule (omitting the ×6x\times 6x), or mis-applying the product/quotient rule.
  • Premature rounding / non-exact answers - using decimals in the technology-free paper where an exact surd or fraction is required.
  • Integration sign and power errors - mishandling x−2x^{-2} integrating to −x−1-x^{-1}, or dropping the negative.
  • Binomial vs other models - applying the binomial formula without checking independence and fixed nn, or forgetting the (nk)\binom{n}{k} coefficient.

How to use this paper

Sit the full paper in 90 minutes, technology-free, with only the formula sheet. Mark against the official QCAA marking guide and marker report (linked at the top of this page), which show exactly where method marks are awarded. Re-attempt any whole question scoring below half after a 24-hour gap, writing one line of method per step and keeping answers exact. Maintain an error log by topic (chain rule, integration signs, binomial setup) and drill those specific by-hand techniques until they are automatic before the next sitting.

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