TMUA Example Questions (With Worked Solutions)

The single best way to understand the TMUA is to see real questions. Descriptions only get you so far; the moment you actually try one, you understand why it feels different from an A-level paper. This guide walks through three genuine past-paper questions, one at a time, with the full worked solution revealed when you are ready, so you can see the exact style, difficulty and traps before you sit down to a full session.

What a TMUA question actually looks like

Every question is multiple choice, usually with five to eight options (most often six). There is no method mark and no working credit: you either land on the correct option or you do not. That changes how you should work. The fastest route is rarely the full, formal method you would write in an exam; it is the shortest valid path to eliminating options. Keep that in mind as you try the examples below.

Example 1 — Paper 1: a calculation with a twist

This is a very typical Paper 1 opener. Everything in it (expanding a bracket, matching coefficients, working with surds) is first-year A-level content. The difficulty comes from doing it cleanly by hand and not slipping on the surd:

Notice what happened there: no single step was hard, but there were four or five of them, they had to be exact, and a calculator would not have helped. That is the essence of Paper 1. Speed and accuracy under no-calculator conditions, not exotic knowledge.

Example 2 — Paper 1: combining two topics

TMUA questions love to make you join two ideas that your textbook keeps in separate chapters. This one fuses coordinate geometry with differentiation: you differentiate to get a gradient, turn it into the gradient of a normal, form a line, and then chase where that line meets the axes.

If you found yourself thinking "which topic even is this?", that is the intended friction. On the TMUA, recognising what to do is often harder than doing it. This is exactly the skill that drilling lots of varied questions builds, and why passive revision (re-reading notes) does so little for your score.

Example 3 — Paper 2: reasoning, not computation

Paper 2 is what surprises most people, and it is where CrackTMUA puts its deepest solutions, because almost no school teaches it. There is barely any calculation here. Instead you are handed a set of statements and asked what must logically follow. It rewards careful deduction and punishes the intuitive-but-wrong leap:

That is a completely different muscle from Example 1. You cannot grind through it with algebra; you have to reason. Paper 2 makes up half your TMUA and is the half most candidates neglect, which means it is also where the easiest marks are hiding if you start early. For a full breakdown of this side of the test, see Paper 2: logic and proof.

What these three examples tell you

Put side by side, the pattern is clear:

None of it is beyond A-level Maths. If a question ever feels like it needs content you have not met, you have missed a simpler route, because by design there always is one. For an honest look at the overall difficulty, see is the TMUA hard? and what is a good TMUA score?.

And now the hard end: two genuine 8.0s

If those three felt manageable, good, that is most of the paper. But the TMUA is scored all the way to 9.0, and the gap between a 6.5 and an 8.0 lives in a small handful of questions like the two below. Only about 15 of the 360 real past-paper questions reach this level. Do not expect to crack them in four minutes on a first read; they are here so you know what the top of the scale actually looks like, and so the hard questions you meet in practice feel less alien.

First, a Paper 1 question that looks impossible and hides an elegant collapse. The phrase "length of the curve" is the whole clue: work out what the equation genuinely forces (combine the logs, then respect the domains) and the "curve" turns out to be far simpler than it looks.

And finally, the kind of pure-reasoning puzzle that makes Paper 2 famous. There is no formula to reach for and nothing to compute. You have to hold several statements in your head at once and follow the logic without a single wrong turn:

If you solved either of those cold, you are already operating near the top of the scale. If you did not, that is completely normal, and it is exactly what deliberate practice fixes: seeing the one insight that unlocks a hard question is how you learn to spot the same move next time. That is why every CrackTMUA solution names the key idea rather than just listing steps.

The traps that separate a 6 from an 8

Across these five questions, the same handful of failure modes recur, and they are what actually decide your band. Learn to watch for them:

• The calculator reflex. Half of TMUA errors are arithmetic slips that a calculator would have caught, so your mental and by-hand arithmetic has to be genuinely reliable, not just "good enough with a calculator". Drill it until it stops being the weak link.
• The dropped case. Squaring an equation, taking a square root, or dividing by a variable can quietly create or destroy solutions. Strong candidates always ask "could this be negative? could this be zero?" before committing to an answer, especially on the surd and inequality questions.
• Solving forwards when backwards is faster. Because it is multiple choice with no method marks, the quickest route is often to test the options or reason about the shape of the answer, not to grind out a full derivation. Spotting when to work backwards is a skill in itself.
• Misreading the quantifier. On Paper 2, "must be true" is a completely different demand from "could be true" or "is true in this example". One misread word turns a correct chain of reasoning into the wrong option. Read the exact wording twice.
• Finishing the maths but not the question. A question might ask for a range of values, the number of solutions, or the largest option, not the value you just computed. Re-read what was actually asked before you pick. For a fuller list, see common TMUA mistakes.

Every one of these is trainable, and none of them is about knowing harder content. That is the core message of the TMUA: the marks are won on precision, method-choice and reading, not on syllabus.

How to practise with questions like these

Three examples show you the style. Getting a good score takes volume: enough varied questions that method-choice becomes automatic and no-calculator arithmetic stops slowing you down. A focused practice bank is far more useful than a handful of loose PDFs, because you get instant worked solutions, you can filter by topic and difficulty, and you can drill your weak areas rather than re-doing what you already know.

CrackTMUA gives you interactive past papers from every recent sitting, plus original questions and full mock exams, with a proper worked solution and the specific trap called out on every single one. It is free at 10 questions a day, and premium is a one-time £37 for 12 months if you want the entire library with no daily cap. The best next step after reading examples is simply to start answering them: open the practice bank and do ten right now, or read how to prepare for the TMUA for the full study plan.

Frequently asked questions