TMUA Worked Solutions: Every Official Paper Explained

This guide is the solutions companion to our TMUA past papers hub: that page covers which official papers exist and how to schedule them, while this one covers the worked solutions that turn each paper from a score into a lesson, year by year, question by question.

Why worked solutions beat answer keys

The official past papers are distributed with answer keys, and an answer key answers exactly one question: which letter was right. On a test with no method marks, that is close to useless as feedback, because the letter never tells you why you missed it.

There are three completely different ways to get a TMUA question wrong, and each needs a different fix:

• Wrong method. You did not know the intended approach, or picked a slower one and ran out of time. The fix is learning the route, and the key cannot teach it.
• Right method, wrong execution. A sign slip, a dropped domain condition, an arithmetic error under pressure. The fix is knowing which step you habitually fumble, and the key cannot show you the steps at all.
• Right answer, slow method. You got the mark but spent seven minutes on a question the setter designed to take three. On a 75-minute paper this is a silent score-killer, and an answer key literally cannot detect it, because your letter was correct.

A proper worked solution addresses all three at once: it shows the intended fastest route so you can compare it against yours, it walks the execution so you can find the exact step where you slipped, and it exposes the gap between "I got it" and "I got it efficiently".

What a good worked solution contains

Not all worked solutions are equal. Plenty of solution documents floating around are just the algebra written out longhand, which proves the answer without teaching the question. Here is what to demand from any solution you study.

The fastest route, not just a route

TMUA questions allow under four minutes each. A solution that grinds through fifteen lines of correct algebra is a bad lesson even when the maths is right, because it trains the habit that loses you the last five questions of the paper. A good solution leads with the efficient method: the substitution that collapses the mess, the option-testing shortcut when solving forward is slow, the symmetry that halves the work. If a slower textbook route exists, the solution should say so and explain why the fast one is preferred.

Why each distractor tempts

The second thing to demand is trap analysis. When a solution says "the answer is D" and moves on, it leaves the most valuable information on the table: option B is there because students forget the negative branch, option E is there because squaring introduces a false solution. Reading the trap analysis for a question you got right is where a lot of quiet improvement happens, because you find out which cousin of the trap is waiting for you on a future paper.

Difficulty context

Finally, a solution should tell you how hard the question actually is. Missing a question rated 8 on the 1.0 to 9.0 scale is normal, and obsessing over it is a poor use of revision time. Missing a 3 is a red flag that needs attention this week. Without difficulty context you cannot triage your errors, and triage is most of what a good review is. The exam board publishes no per-question data, so on CrackTMUA every question carries our own calibrated difficulty rating, built by rating the whole official archive on one consistent scale.

Here is a real Paper 1 question that shows all three layers working together. It looks like a routine inequality until a substitution changes the variable, and the trap is a sign flip that one of the options is waiting to catch. Attempt it, then read the full solution:

Worked solutions by year

Every released official paper is below, with a one-line read on what the year is like and a link to its full breakdown. Every question from every one of these papers has a complete worked solution on CrackTMUA, with the fastest method, the trap analysis and a difficulty rating.

• Specimen: the gentlest full paper and the standard place to start untimed, though its final Paper 2 logic puzzle is one of the hardest questions anywhere in the archive. Specimen paper guide.
• 2016: the first live sitting, mid difficulty overall, and home of a classic trigonometric sign-analysis question late in Paper 1. 2016 paper guide.
• 2017: gentle to mid, with a late Paper 1 stretch that punishes autopilot algebra. 2017 paper guide.
• 2018: one of the gentler sittings overall, but Paper 2 hides a genuinely hard question about transformed cubics that catches confident students. 2018 paper guide.
• 2019: the most balanced and representative year, and a strong choice for a first serious timed attempt; it closes Paper 1 with a tough cubic-roots problem. 2019 paper guide.
• 2020: mid to tough, with more multi-step questions than the earlier years. 2020 paper guide.
• 2021: the hardest top end of any released paper; its final two Paper 1 questions are two of the hardest ever set. 2021 paper guide.
• 2022: one of the tougher sittings, with demanding closers on both papers. 2022 paper guide.
• 2023: the gentlest recent paper and the best warm-up before you attempt 2021 or 2022. 2023 paper guide.

The 2024 and 2025 sittings have taken place but their papers have not been released, so the specimen plus 2016 to 2023 is the complete official archive you can actually study from.

Jump straight to a question

If you searched for the solution to one specific question, chances are it is one of these. They are the questions people talk about, the hardest and most instructive in the archive, and each link goes straight to the question with its full worked solution:

• TMUA 2021 Paper 1 Q19: an exponential meets a sine and you must count solutions in a range. One of the hardest Paper 1 questions ever set.
• TMUA 2022 Paper 1 Q18: two shaped polynomials, several interacting conditions, and no single hard step, just many careful ones.
• TMUA 2019 Paper 1 Q20: locating the roots of a cubic through its stationary points, the classic hard finish to a balanced paper.
• TMUA 2018 Paper 2 Q15: which statements about a shifted cubic must be true. Harder than the rest of its paper by some distance.
• TMUA 2016 Paper 1 Q17: a product of trigonometric factors and a sign analysis across a union of intervals.
• TMUA 2022 Paper 2 Q19: a necessary-versus-sufficient question about triangles cut from a regular polygon, pure reasoning at the top of the scale.
• TMUA Specimen Paper 2 Q20: the famous five-logicians puzzle. Nothing to compute, everything to track, and it catches very strong mathematicians.

For the full tour of the top of the scale, with several of these embedded and dissected in one place, read the hardest TMUA questions.

How to study from a worked solution

A worked solution is a tool, and there is a right way to hold it. The students who improve fastest do the same five things every time:

1. Attempt the question first, under a timer. A solution read before a genuine attempt teaches almost nothing; the struggle is what makes the resolution stick. Give it the four minutes it would get in the exam.
2. Read the whole solution even when you were right. Check your route against the intended one. If yours was slower, the question goes in your review pile exactly as if you had missed it.
3. Name the trap in your own words. Do not just nod at the trap analysis. Say precisely what error the tempting option encodes, because naming it is what stops you falling for its cousins.
4. Log the one-line insight. For every miss and every slow solve, write the single sentence that would have made it faster or correct. A page of these is worth more than any bought notes.
5. Reattempt cold, days later. The test of whether a solution taught you anything is whether you can reproduce the method from nothing once the memory has faded.

Every solution, in one place

CrackTMUA hosts the entire official archive as an interactive bank: every question from the specimen and the 2016 to 2023 papers, each with an in-depth worked solution that leads with the fastest method, names the trap behind each wrong option, and carries a calibrated difficulty rating. On top of the officials there are 280+ original trap-based questions written to the same standard, 640+ in total, filterable by paper, topic and difficulty, with spaced repetition to resurface your misses and a predicted band built from your real attempts.

The free tier gives you 10 questions a day with their full solutions, which is enough to feel the difference between an answer key and a worked solution today. Premium unlocks everything for £49 one-time, with 12 months of access and no subscription. Start with any question above, or see how to build the papers into a plan.

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