External exams 2020–2025 · 24 questions · 148 marks
Past QCAA complex unfamiliar questions
Every complex unfamiliar question from the 2020–2025 Specialist Mathematics external exams, newest first. Each one has an animation and an interactive to work through first, then the marking guide, one mark at a time.
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4 questions · 24 marks- Paper 1, Question 18 Find the launch angle of a projectile fired below the horizontal Unit 3 · Topic 4 · Vector calculus
- Paper 1, Question 19 Prove a result for any non-real fifth root of unity Unit 3 · Topic 1 · Further complex numbers
- Paper 2, Question 17 Find the chance that two confidence intervals overlap Unit 4 · Topic 5 · Statistical inference
- Paper 2, Question 18 Find the length of a polar curve above the $x$-axis Unit 4 · Topic 1 · Integration techniques
2024
4 questions · 24 marks- Paper 1, Question 18 Find $k$ so that $k\sin^{-1}(x)$ is a probability density function Unit 4 · Topic 2 · Applications of integral calculus
- Paper 1, Question 19 Find every positive $x$ that makes $\operatorname{Re}\big((x + i)^7\big)$ zero Unit 3 · Topic 1 · Further complex numbers
- Paper 2, Question 18 Use two shaded normal curves to find $P(4 \le X \le 6)$ Unit 4 · Topic 5 · Statistical inference
- Paper 2, Question 19 Build a Leslie matrix for an insect and predict week 4 Unit 3 · Topic 5 · Further matrices
2023
4 questions · 24 marks- Paper 1, Question 18 Solve a separable differential equation through the origin Unit 4 · Topic 3 · Rates of change and differential equations
- Paper 1, Question 19 Find how much later object B was released before the collision Unit 3 · Topic 4 · Vector calculus
- Paper 2, Question 18 Show that $w_1^4 \div w_2$ is an integer for roots of a degree 15 equation Unit 3 · Topic 1 · Further complex numbers
- Paper 2, Question 19 Find the confidence level at which two intervals just touch Unit 4 · Topic 5 · Statistical inference
2022
4 questions · 24 marks- Paper 1, Question 18 Prove by induction that $1 + \operatorname{cis}\theta$ divides a sum of $\operatorname{cis}$ terms Unit 3 · Topic 2 · Mathematical induction and trigonometric proofs
- Paper 1, Question 19 Find $f(x)$ from a gradient of $e^x\sin^{-1}(e^x)$ through the origin Unit 4 · Topic 1 · Integration techniques
- Paper 2, Question 18 Use a remainder of $a - bi$ to test whether $z - (a - bi)$ is a factor Unit 3 · Topic 1 · Further complex numbers
- Paper 2, Question 19 Find the drone heights that give a safe impact speed with air resistance Unit 4 · Topic 4 · Modelling motion
2021
4 questions · 26 marks- Paper 1, Question 18 Show that the current in a circuit can never exceed $V \div R$ Unit 4 · Topic 3 · Rates of change and differential equations
- Paper 1, Question 19 Find where object A is when it is first 4 m from object B Unit 3 · Topic 4 · Vector calculus
- Paper 2, Question 18 Build a cubic that shares two roots with $z^4 + z^3 + z^2 + z + 1$ Unit 3 · Topic 1 · Further complex numbers
- Paper 2, Question 19 Find the chance a sample mean waiting time is over 16 minutes Unit 4 · Topic 5 · Statistical inference
2020
4 questions · 26 marks- Paper 1, Question 18 Find $a$ and $b$ so every other root of $P(z)$ has an imaginary part Unit 3 · Topic 1 · Further complex numbers
- Paper 1, Question 19 Find how fast the depth rises in a bowl made by rotating a curve Unit 4 · Topic 3 · Rates of change and differential equations
- Paper 2, Question 18 Find the sample sizes whose confidence interval misses $\mu$ Unit 4 · Topic 5 · Statistical inference
- Paper 2, Question 19 Find the acceleration of a vehicle hit by a projectile from a swinging string Unit 3 · Topic 4 · Vector calculus