The five Specialist multiple choice questions that caught Queensland out

Multiple choice questions should be the marks you bank. One mark each, four options, and the answer is right there in front of you. So what happens when most of Queensland gets one wrong?

The QCAA subject reports tell us exactly how many students chose each option for every multiple choice question. I went through every Specialist Mathematics multiple choice question from 2021 to 2025 and found the five with the fewest correct answers.

Here is the interesting part. In all five, a wrong answer was more popular than the right one. These were not careless slips. Each question had a trap built into it, and the trap worked because it targeted a game plan that we knew and tried to get us to use it in the wrong context.

For each one, I will look at the trap, why it got us, and what we can learn from it. Every question is in the free Specialist simple familiar question bank, with the full solution, animation and interactive.

5. The plane that was not really new

2023 Paper 1, Question 9 · 28.4% correct

You are given two planes, \(x+y-z=0.5\) and \(x-y-z=0.5\), and a picture of a system with no solution. Which equation could be the third plane?

The trap: The most popular answer was \(2x+2y-2z=1\), chosen by 33.7% of students. It has the same normal as the first plane, so it looks parallel. But divide it by 2 and you get \(x+y-z=0.5\). That is not a parallel plane. It is the same plane!

Why it got us: We learn that parallel normals mean parallel planes, and that is true. But a plane is also parallel to itself. Two equations with the same normal are either parallel and separate, or exactly the same plane, and only the constant tells you which.

What we can learn: Before you compare two planes, scale them so they match. Same normal and same constant means the same plane. Same normal and a different constant means parallel. The answer was \(2x+2y-2z=3\), which divides to \(x+y-z=1.5\): parallel to the first plane, but never touching it.

Try 2023 Paper 1, Question 9

4. The conjugate root trap

2022 Paper 2, Question 1 · 26.5% correct

One solution of \(z^2=ai\) is \(z=-2-2i\). What is the other solution?

The trap: Half the state, 49.9%, chose the conjugate, \(-2+2i\). It feels right. We have all heard “complex roots come in conjugate pairs” so many times that it has become a reflex.

Why it got us: That rule only works when the polynomial has real coefficients. Square the given root and you get \(8i\), so the equation is \(z^2-8i=0\). That \(-8i\) is not real, so the conjugate pair rule does not apply. Square \(-2+2i\) and you get \(-8i\), the wrong sign. The QCAA love to test the conjugate root theorem, so if we practise with past papers, we are primed to look for it!

What we can learn: Every rule comes with conditions, so check them before you use it. We should be in the habit of checking whether the conjugate root theorem actually applies!

Try 2022 Paper 2, Question 1

3. The zero that was really a value

2021 Paper 1, Question 5 · 26.0% correct

You are given an augmented matrix in row-echelon form, with a last row of \(0\ \ 0\ \ 1\ |\ 0\). Which picture of three planes matches the solution?

The trap: Students split almost evenly across all four options. More chose the prism, which means no solution, than the right answer.

Why it got us: We train our eyes to look for zeros in the last row, because a row of zeros tells us something special. But this row is not all zeros. It reads \(0x+0y+1z=0\), which simply says \(z=0\). A zero on the right-hand side is still a value!

What we can learn: Read the last row as an equation, out loud if you need to. \(0=0\) means a free variable and infinitely many solutions. \(0=6\) is impossible, so there is no solution. \(z=0\) is a value, so back-substitute: \(y=6\) and \(x=-2\). One solution, \((-2,6,0)\), so the three planes meet at a single point.

Try 2021 Paper 1, Question 5

2. The modulus that would not split

2025 Paper 1, Question 10 · 21.1% correct

For complex numbers \(z_1\) and \(z_2\), which statement about \(|z_1+z_2|\) and \(|z_1|+|z_2|\) is always true?

The trap: The most popular answer, chosen by 42.1% of students, was that they are equal. That is twice as many students as chose the right answer.

Why it got us: We get very comfortable with the modulus rules for products, where \(|z_1z_2|=|z_1||z_2|\) always works. It is tempting to split the modulus across a sum in the same way. But it does not work for sums.

What we can learn: A modulus is a length, so draw it! Put \(z_1\) and \(z_2\) head to tail on the Argand diagram and the three moduli make a triangle. One side of a triangle can never be longer than the other two put together, so \(|z_1+z_2|\le|z_1|+|z_2|\). They are only equal when both arrows point the same way. Whenever a question says “always true”, hunt for one example that breaks each option. One counterexample is all you need.

Try 2025 Paper 1, Question 10

1. The logistic equation in disguise

2023 Paper 1, Question 7 · 15.5% correct

Which differential equation has a solution of the form \(y=\dfrac{a}{b+Ce^{-at}}\)?

The trap: Only 15.5% of students chose the right answer. That is fewer than one in six, and fewer than you would expect from guessing! Two thirds of the state chose one of the two options where the right-hand side depended only on \(t\).

Why it got us: Most students knew what a logistic solution looks like. Far fewer knew the differential equation it comes from. The options in \(t\) look like something you could just integrate, so they felt safe. This is also a small part of the syllabus, and this exact form only appeared in the syllabus glossary, so it is very possible students just missed it!

What we can learn: Learn each model as a pair: the differential equation and its solution. The logistic differential equation is \(\dfrac{dy}{dt}=y(a-by)\). It depends on \(y\), not \(t\), and it has a factor of \(y\) out the front. The answer was \(\dfrac{dy}{dt}=0.25y(1-0.01y)\), which expands to \(y(0.25-0.0025y)\). The graph shows it best: only that curve makes the S shape, rising slowly, then quickly, then levelling off at 100.

Try 2023 Paper 1, Question 7

What these five have in common

Looking at all five together, a pattern jumps out. In every question, a reflex beat a check:

• Same normal, so parallel.
• Complex roots, so conjugates.
• A zero in the last row, so something special.
• Modulus of a sum, so split it.
• Something to integrate, so integrate it.

Each wrong answer was only one condition away from being right. That is what makes a good trap!

Four of the five were also on Paper 1, which is technology-free. There is no calculator to save you there, so these checks need to be habits.

My advice for every multiple choice question is a simple two-step routine. Before you choose, say what condition makes your answer true. Then check one option you are about to reject, and make sure you know why it is wrong. It takes thirty seconds, and it would have picked up every one of these traps.

If you want to practise spotting traps like these, every one of these questions is in the free Specialist simple familiar question bank, along with every other past QCAA multiple choice question. From 2021 on, each one shows how Queensland students answered, so you can see which traps caught the most people. Try the “Pick one at random” button for some interleaved practice.

If a topic keeps tripping you up, an hour of tutoring might be worthwhile. We can work through the traps together and build some habits that hold up in the exam room.

Good luck, and remember to check the condition before you pick the answer!