TMUA Paper 2: mathematical reasoning, logic and proof
Paper 2 asks for everything Paper 1 asks for, and adds Section 2 on top. About 38% of it tests reasoning that no A-level maths course formally teaches, which is exactly why it is the best place to spend your preparation.
Written by the specialists behind our practice bank — over a decade writing, teaching and tracking the UK admissions tests.
What Paper 2 actually is
| Paper 2: Mathematical Reasoning | |
|---|---|
| Time | 75 minutes |
| Questions | 20 multiple choice |
| Specification required | Sections 1 and 2 |
| Calculator | Not permitted |
| Formulae booklet | None |
| Negative marking | None. Equal weight per question |
| Score | No separate Paper 2 score. Combined with Paper 1 |
Paper 2 is sat straight after Paper 1, in the same 2 hour 30 minute session, with no break between them.
For the test as a whole, including dates, fees and which courses require it, see the complete TMUA guide.
The specification says its job is to test your ability to apply conceptual knowledge to constructing and analysing mathematical arguments.
The structural difference from Paper 1 is one line in the specification, and it is the whole story.
As with Paper 1, there is no separate Paper 2 score. The two papers are equated and scaled together into one result on the 1.0 to 9.0 scale.
That has a consequence for this page worth stating up front: nobody can tell you how candidates perform on Paper 2 specifically, because that number is never produced.
Anyone who quotes you a Paper 2 average is quoting something that does not exist. The mechanism is on the scoring guide.
Why this is the paper to prepare for
Paper 2 gets called "the harder paper", which is a mood rather than advice.
The sharper claim is about return on preparation:
Paper 2 is where a few weeks of work moves your score furthest, and the reason is specific.
Everywhere else, the TMUA tests material you have been studying for two years. Preparation buys polish on the familiar.
Paper 2 contains a block of content most candidates have never formally studied. Preparation there buys competence where you had none.
But the usual version of this argument overstates it, so here is the accurate one.
What A-level does not teach is the language of logic.
Converse and contrapositive. Necessary and sufficient conditions.
Quantifiers, and how to negate a statement containing one. Finding the flawed step in an argument someone else wrote.
A-level asks you to use logical connectives. Paper 2 asks you to reason about them, which is a different skill and is examined directly.
We classified every Paper 2 question in the official archive by its primary reasoning demand. Here is how the split falls.
Here's how that breaks down:
| Demand | Share | On A-level maths? |
|---|---|---|
| Necessary and sufficient conditions | 17% | No |
| Constructing or following a proof | 15% | Yes |
| Deducing implications from given statements | 11% | Partly |
| Identifying errors in a supplied proof | 9% | No |
| Sophisticated multi-step chains of reasoning | 5% | Partly |
| Quantifiers: for all, for some, there exists | 4% | No |
| Logic language: if-then, only if, converse, contrapositive | 4% | No |
| Conjecture from small cases, then justify | 3% | Partly |
| Negating a quantified statement | 2% | No |
| Common named errors in proofs | 2% | No |
| Rearranging statements into a correct proof | 1% | Partly |
| Comprehension of a definition invented in the question | 1% | No |
| Computational, with reasoning content light | 26% | Yes |
Add the rows A-level does not cover and you get roughly 38% of Paper 2 sitting in genuinely unfamiliar territory.
That is the argument, with a number on it. Not "Paper 2 is all new", but "about two questions in five test something your school course never formally taught you".
Two further findings support spending your time here.
Paper 2 is measurably harder. On our difficulty assessment, 31% of Paper 2 questions rate as hard against 24% of Paper 1's.
Paper 2 rewards insight over machinery. Only 19 of 180 Paper 2 questions can be solved by standard procedure alone, against 37 on Paper 1.
Roughly a third turn on spotting a way of looking at the problem that is not obvious.
One honest limit on all of this.
We are reasoning from what the specification requires against what A-level covers, and from our own classification of the archive. We are not reasoning from per-paper score data, because none exists and none can, given the papers are scaled together.
What Section 2 contains, and how Paper 2 examines it
Section 2 of the specification is short. What follows is each part of it, introduced by what the paper does with it rather than as a lesson.
1. Implication, and the traps in "if" and "only if"
What the paper does with it: gives you a statement in one of several grammatical forms and asks which other statements must follow.
The specification lists the forms explicitly: if-then, A if B, A only if B, and A if and only if B.
They are not interchangeable, and the wording is the whole question.
"A if B" means B implies A. "A only if B" means A implies B, which is the reverse, and reading it as the first is the most reliably exploited misreading in Section 2.
The technique: rewrite every conditional into one standard form before you do anything else. Pick "P implies Q" and convert everything into it.
Do not reason in the sentence's own grammar. Convert first, then reason.
2. Converse, and why it is usually the distractor
What the paper does with it: gives you a true implication and offers its converse as an option.
The converse of "P implies Q" is "Q implies P", and the two are independent. One being true says nothing about the other.
This is the single most productive distractor in Section 2, because assuming the converse feels like reasoning rather than like a mistake.
The technique: when an option reverses the direction of a given implication, treat it as false until you have separately proved it. Look for a case satisfying Q but not P.
3. Contrapositive, and using it as a tool
What the paper does with it: offers the contrapositive as an option alongside the converse, to see whether you can tell them apart.
The contrapositive of "P implies Q" is "not Q implies not P". Unlike the converse, it is always logically equivalent to the original.
The technique: this is not just something to identify, it is something to use. When a statement is awkward to prove directly, proving the contrapositive proves the original.
Questions about odd and even, or about divisibility, are frequently far easier from the contrapositive direction.
4. Necessary and sufficient conditions
The single most examined idea in Section 2, at roughly one Paper 2 question in six where reasoning is the main demand.
What the paper does with it: gives a condition and asks whether it is necessary, sufficient, both, or neither. Often as a fixed four-way menu.
Most candidates guess at this vocabulary, which is why it is worth ten minutes of proper attention.
Sufficient means the condition being true is enough to force the conclusion. P is sufficient for Q when P implies Q.
Necessary means the conclusion cannot hold without it. P is necessary for Q when Q implies P.
The technique: test the two directions separately and in a fixed order. Ask "does P force Q" for sufficiency, then "does Q force P" for necessity.
Answer both, then read off the label. Never try to feel which word fits.
Here's how that breaks down:
| Answer | P implies Q? | Q implies P? | Typical shape |
|---|---|---|---|
| Sufficient but not necessary | Yes | No | A condition stronger than needed |
| Necessary but not sufficient | No | Yes | A condition weaker than needed |
| Both | Yes | Yes | An equivalent restatement |
| Neither | No | No | A plausible but unrelated condition |
5. Quantifiers, and negating them
What the paper does with it: asks you to select the correct negation of a statement containing "all", "some" or "there exists".
Negation is where quantifiers stop being vocabulary and start being a technique, and it is the one rule worth memorising outright.
The negation of "all x satisfy P" is "there exists an x that does not satisfy P". Not "no x satisfies P".
The same rule the other way round: the negation of "there exists an x satisfying P" is "no x satisfies P", or equally "every x fails P".
The technique: negation flips the quantifier and negates the inside. Both, every time.
Doing only one is precisely the option the question is built around.
6. Proof by contradiction
What the paper does with it: supplies a proof and asks what the opening assumption should be, or asks which contradiction closes it.
This is one of the constructs A-level does teach, so you may well have met it.
What differs is the direction of the task. A-level asks you to write one.
Paper 2 mostly asks you to audit one somebody else wrote.
The technique: check the assumption first. The most common flaw in a supplied contradiction proof is that it assumes the wrong negation, and everything after that is irrelevant.
7. Disproof by counterexample
What the paper does with it: offers several candidate counterexamples and asks which actually disproves the claim.
Also familiar from A-level, and also inverted: the work is judging a counterexample, not producing one.
The technique: a valid counterexample must satisfy every hypothesis of the claim and fail its conclusion. Candidates that fail a hypothesis are not counterexamples at all, and the option list will contain some.
One counterexample disproves a universal claim completely. No number of supporting examples proves one.
8. Finding the flawed step
Roughly 11% of Paper 2 combining the two error-spotting codes, and one of the most distinctive things the test does.
What the paper does with it: presents a proof in numbered steps, with a conclusion you can see is false, and asks where it first goes wrong.
The specification names two errors as exemplars: that ab = ac does not imply b = c, and that sin A = sin B does not imply A = B.
Both are the same underlying move. Cancelling something that might be zero, or inverting a function that is not one-to-one.
The technique: work forwards from step one and validate each line in isolation, rather than working back from the false conclusion.
The question asks where the argument first fails. Later steps are often invalid too, as a consequence, and those wrong options are always available.
9. Definitions invented in the question
A species the specification does not name but the paper uses: a notation or operation defined in the question itself, with the work being comprehension under the new rules.
The technique: nothing you know about the ordinary meaning of the symbol applies. Compute the first small case by hand, strictly from the definition given, before forming any intuition.
The official Notes on Logic and Proof
UAT-UK publish a document covering Section 2 directly. It is free, and it is the right place to start on this content.
The thing to get right is which version you are holding.
Use it for the definitions and the vocabulary, which it sets out precisely and which most candidates are otherwise guessing at.
What it will not give you is speed. Recognising which of the constructs above a question is testing, in ten seconds, under a clock, comes from working through enough Paper 2 questions that the shapes become familiar.
One other free resource is worth knowing about:
Warwick's mathematics department publishes a logic worksheet aimed squarely at this content.
It is a foundation to build on rather than complete preparation, but it costs nothing.
The Paper 2 question styles
Reasoning has to become a multiple-choice question somehow, and the ways it does are limited and learnable.
Before the styles, the finding that surprised us most, and that we have not seen written anywhere.
That matters because a structured option set has to be read differently, and candidates who have only practiced Paper 1 have never had to.
Here's how that breaks down:
| Format | Paper 1 | Paper 2 |
|---|---|---|
| Plain list of independent options | 98% | 58% |
| Combination options: "I and III only", "II only", "none of them" | 1% | 23% |
| Step-error: "the first error occurs at step ..." | 0% | 8% |
| Classification menu: necessary, sufficient, both, neither | 0% | 4% |
| Truth-table grid: classify each row | 1% | 3% |
| Graph selection or matching | 0% | 4% |
Paper 2 also runs longer option lists. 38% of its questions offer eight options against 16% on Paper 1, which is what combination formats do to a list.
The seven styles, in the order they are covered below:
Its wrong answers are also more carefully built. We traced 93% of Paper 2's wrong options to a specific reasoning error, against 74% on Paper 1.
On Paper 2, a wrong answer almost always names the exact mistake you made.
1. Which of these must be true
What it looks like: a statement, then two or three numbered claims, then options that are combinations of them.
How to work it: evaluate each numbered claim independently and write down true or false for each before looking at the options at all.
Then match your pattern to the list. Reading the options first invites you to reason towards one that looks plausible.
The distractor: a claim that is usually true but not necessarily true. The question asks what must follow, and one supporting instance is not enough.
2. Which of these does not follow
What it looks like: the same shape, with the polarity flipped.
How to work it: the same way, but read the question twice. The most expensive error on this style is answering the version you expected instead of the one printed.
The distractor: the answer to the un-flipped question, which is always present.
3. Find the first invalid step
What it looks like: a numbered proof of something false, and options naming each step, usually with "the proof is correct" among them.
How to work it: forwards, one line at a time, asking only whether that line follows from what precedes it.
The distractor: a later step that is also invalid, or the step where the falsity becomes visible rather than where the reasoning broke. Those are rarely the same line.
4. Select the correct negation
What it looks like: a quantified statement and several candidate negations.
How to work it: construct the negation yourself before reading the options, flipping the quantifier and negating the inside.
The distractor: the version that flips the quantifier without negating the inside, or negates the inside while leaving "all" as "all".
5. Select the valid counterexample
What it looks like: a universal claim and several candidate objects.
How to work it: check each candidate against every hypothesis first, then against the conclusion. Discard anything failing a hypothesis immediately.
The distractor: an object that fails the conclusion but also fails a hypothesis, so it never had standing to be a counterexample.
6. Necessary, sufficient, both or neither
What it looks like: a fixed four-way menu, sometimes across several statements at once.
How to work it: the two-direction test above, in a fixed order, written down. This style punishes doing it in your head.
The distractor:
"both", offered whenever one direction happens to hold, and "neither" whenever the relationship is real but runs the other way.
7. Order the steps of a proof
What it looks like: scrambled lines of a valid argument, options giving possible orderings.
How to work it: find the line that depends on nothing, and the line that is the conclusion. Anchoring both ends usually collapses the option set to one or two.
The distractor: an ordering that reads naturally in English but uses a result before establishing it.
The taxonomy above is GhostPrep's own, built from the archive. UAT-UK publish no question-style classification.
The errors that cost marks
Six mistakes account for a large share of Paper 2's mapped distractors. Each one is a wrong option somewhere in the archive, put there deliberately.
| The error | What it looks like | The fix |
|---|---|---|
| Assuming the converse | Given "P implies Q", concluding "Q implies P" | Treat a reversed implication as false until separately proved |
| Confusing converse with contrapositive | Both reverse something, so they feel interchangeable | The contrapositive negates as well as reverses, and is always equivalent |
| Negating "all" as "none" | "Not all swans are white" read as "no swans are white" | Flip the quantifier and negate the inside |
| Reading "only if" as "if" | "A only if B" taken to mean B implies A | "A only if B" means A implies B. Rewrite it in one standard form first |
| Treating an example as a proof | One case satisfies the claim, so the claim is accepted | Examples disprove universal claims. They never prove them |
| Assuming what is being proved | A supplied proof uses its conclusion partway through | Validate each line against only what precedes it |
The named exemplars in the specification are versions of one deeper error: cancelling something that might be zero, and inverting something that is not one-to-one.
Both feel like ordinary algebra, which is exactly why they work as traps.
How to practice Paper 2
Every official paper from 2016 to 2023 is free with UAT-UK's own worked answers. The full archive is here.
Paper 2 rewards a different practice method from Paper 1, and this is the part most candidates get wrong.
Mark by reasoning, not by letter. On a reasoning question, arriving at the right option through faulty logic is not a mark you can rely on repeating.
Read the official worked answer line by line even when you were right, and check that your route matches. Given 93% of wrong options map to a specific reasoning error, the letter you chose diagnoses your thinking precisely.
Sit it straight after Paper 1, timed. Paper 2 is the second 75 minutes of a two and a half hour session, and reasoning under fatigue is what it actually tests.
Sitting it fresh on its own flatters you in a way that does not transfer.
No calculator, no formulae booklet, and a whiteboard rather than the page. Structured option formats in particular are harder on screen, where you cannot annotate a numbered proof beside the text.
Practice the structured formats deliberately. If 42% of Paper 2 uses combination, classification or step-error options, those need dedicated attention rather than being met for the first time under a clock.
The archive is the only free source of them, which is a reason to ration it rather than burn through it early.
UAT-UK also note that performance on a practice test "will not necessarily be indicative of performance under exam conditions", and their practice tests deliberately award no score.
Frequently asked questions
How long is TMUA Paper 2?
75 minutes for 20 multiple-choice questions, sat straight after Paper 1 in the same 2 hour 30 minute session.
What is the difference between TMUA Paper 1 and Paper 2?
Paper 1 requires Section 1 of the specification. Paper 2 requires Sections 1 and 2, so it needs everything Paper 1 needs plus logic and proof. It is additive, not a different subject.
Is Paper 2 harder than Paper 1?
On our analysis of the archive, yes: 31% of Paper 2 questions rate as hard against 24% of Paper 1's, and Paper 2 relies far more on non-obvious insight. No official per-paper difficulty data exists, because the papers are scored together.
Do I get a separate score for Paper 2?
No. Both papers are equated and scaled together into one score from 1.0 to 9.0. Anyone quoting a Paper 2 average is quoting a number that is never produced.
Is logic and proof on A-level maths?
Partly. A-level requires proof by deduction, exhaustion, contradiction and disproof by counterexample. It does not formally teach converse, contrapositive, necessary and sufficient conditions, quantifier negation or error-spotting in supplied proofs, and those account for roughly 38% of Paper 2.
What is the difference between necessary and sufficient?
P is sufficient for Q when P implies Q: having P is enough. P is necessary for Q when Q implies P: you cannot have Q without it. Test the two directions separately rather than trying to feel which word fits.
Do I need to know formal logic notation?
No. The specification explicitly excludes formal truth tables and symbolic logic notation, and formal set notation. Section 2 is examined in words and mathematics.
Which edition of the Notes on Logic and Proof should I use?
The June 2025 revision, taken from UAT-UK directly. Revision sites frequently host the 2020 or 2021 edition, so check the date on the document rather than where you found it.
How many answer options does a Paper 2 question have?
It varies, from four to eight. 38% of Paper 2 questions offer eight, which is more than double Paper 1's rate, largely because combination formats produce longer lists.
Is there negative marking on Paper 2?
No. Every question carries equal weight and nothing is deducted for a wrong answer, so a blank is always worse than a guess.
Work the part of the test that moves your score
Section 2 is where preparation pays most, and the official archive is the only free source of questions in the formats Paper 2 actually uses.
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