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TMUA Past Paper Walkthrough
A question-by-question breakdown of TMUA 2024 Paper 2. Topics, traps, and the approach that gets the mark.
TL;DR
Score-band figures inferred from the UAT-UK TMUA Technical Report 2024-25. The report confirms the wider Paper 2 distribution and the lower modal band relative to Paper 1, though it stops short of publishing item-level statistics.
Paper 2 in 2024 doubled down on language precision. The questions still tested logical reasoning, but the wording was sharpened so that students who paraphrased a statement into 'rough English' before evaluating it were systematically wrong. The examiners are explicitly checking whether you read the statement as written, not as you would have written it.
The other big lesson is that the paper now treats quantifier order as a first-class skill. At least three questions hinge on whether a statement says 'for every x there exists a y' or 'there exists a y such that for every x', and students who treat those as interchangeable lost marks they did not need to lose.
If you want one heuristic that would have helped on this paper, it is this: when a statement contains both 'all' and 'some', annotate which quantifier governs which variable before you start evaluating. Half the wrong answers in this paper come from getting that wrong on the first read.
Questions are paraphrased to respect Cambridge Assessment's copyright. For the verbatim wording, work through the official paper from the UAT-UK question bank, then return to this guide for the strategy notes.
Setup. A pair of conditional statements is presented, and the candidate must determine which is logically equivalent to the contrapositive of the first.
What it tests. Whether you reliably know the contrapositive of 'if P then Q' is 'if not Q then not P', not 'if not P then not Q'.
Approach. Write out the four logically related forms — converse, inverse, contrapositive, original — and match by symbol.
Setup. A statement of the form 'every member of the set has property X' is given, and the candidate must select the correct negation.
What it tests. Whether you know that the negation of a universal claim is an existential claim about the opposite property.
Approach. Apply the rule: not(for every x, P(x)) is the same as (there exists x with not P(x)). Reject paraphrases that talk about 'no member' rather than 'at least one member'.
Setup. Four statements about pairs of conditions are presented. The candidate must identify which correctly describes the relationship as necessary, sufficient, both, or neither.
What it tests. Whether you can isolate the direction of an implication and check both directions independently.
Approach. For each candidate pair, test the implication left-to-right and right-to-left with a counterexample. If both directions hold, it is necessary and sufficient; if only one, it is just one of the two.
Setup. An argument is presented with two premises and a conclusion. The candidate must identify whether the conclusion follows from the premises.
What it tests. Whether you separate validity (does the conclusion follow if the premises are true) from soundness (are the premises actually true).
Approach. Treat the premises as definitely true, then check whether the conclusion has to follow. Look for a counterexample where the premises are true but the conclusion is false.
Setup. A general claim about integers or sets is made. The candidate must identify the smallest set or value that disproves the claim.
What it tests. Whether you can construct minimal counterexamples by working with edge cases (zero, negative numbers, the empty set, single-element sets).
Approach. Test the claim against small, edgy cases first: 0, 1, -1, the empty set, a single-element set. The counterexample is almost always one of these.
Setup. A set is defined by a logical predicate. The candidate must determine which of the answer choices correctly describes the set.
What it tests. Whether you can translate a logical predicate into a concrete set description without losing edge cases.
Approach. Construct the set element by element from the smallest values upward. Check each answer choice against your concrete construction.
Setup. A conditional chain of three statements is given. The candidate must identify which conclusion follows.
What it tests. Chained modus ponens or modus tollens, plus the ability to spot when a chain is broken.
Approach. Map each statement to a symbol (P implies Q, Q implies R, etc.), then chain the implications. The conclusion has to follow strictly by valid inference rules.
Setup. A claim is made about functions or sequences with a particular property. The candidate must select the function that serves as a counterexample.
What it tests. Whether you carry a small mental library of well-known counterexamples (constant functions, sin, characteristic functions, piecewise constructions).
Approach. Test each candidate function against the claim. The counterexample is usually a function with a discontinuity or a degenerate case (constant, sign function, characteristic of rationals).
Setup. Two compound statements are presented. The candidate must determine whether they are logically equivalent.
What it tests. Whether you can verify equivalence with a small truth table or by manipulating logical connectives algebraically.
Approach. Build the truth table for both statements over the four possible truth assignments. They are equivalent iff the final columns match.
Setup. A short verbal argument is given with a hidden assumption. The candidate must identify the unstated premise that makes the argument valid.
What it tests. Whether you can find the missing premise that, if added, would make the conclusion follow.
Approach. Identify the gap between the stated premises and the conclusion. The missing premise must close exactly that gap, no more and no less.
Setup. Three sets are defined by their elements. The candidate must determine which expression correctly describes a derived set.
What it tests. Understanding of unions, intersections, and complements applied to small explicit sets.
Approach. Compute the derived set element by element, then test each answer choice. Mistakes here are arithmetic, not logical.
Setup. Two statements differ only in quantifier order. The candidate must determine which is stronger or which implies the other.
What it tests. Whether you understand that 'for every x there exists a y' is weaker than 'there exists a y such that for every x'.
Approach. Construct a small example where the weaker statement holds but the stronger does not. The order matters because the y in the weaker form can depend on x.
Setup. A short proof is presented with one logically invalid step. The candidate must identify the step that does not follow.
What it tests. Close reading of a chain of inferences, and recognition of common fallacies (affirming the consequent, denying the antecedent, dividing by zero in disguise).
Approach. Read each line of the proof in isolation and check whether it follows from the previous lines by a valid rule. The wrong line is usually a converse error or an unjustified case split.
Setup. A claim is made about sequences of integers. The candidate must select the answer that minimally disproves the claim.
What it tests. Constructing the smallest possible counterexample under a constraint. The answers differ by sequence length and element values.
Approach. Try the smallest length first. If the claim allows a counterexample at that length, the smaller answer is correct. Verify by direct computation.
Setup. Several statements about properties of numbers are presented as a system. The candidate must determine which combination is internally consistent.
What it tests. Whether you can detect logical contradictions across a set of statements without solving each in isolation.
Approach. Pair statements off and check for direct contradiction. If no two statements clash pairwise, check the system as a whole using a small witness.
Setup. A pattern is presented and the candidate must determine which of the answer choices correctly continues or generalises the pattern.
What it tests. Whether you can extract a closed-form description of a pattern from a few terms.
Approach. Write the first three or four terms in a normalised form, then look for the structural rule. The wrong answers usually fail at the fourth or fifth term.
Setup. A statement combines a conjunction inside a quantifier. The candidate must select the correct negation.
What it tests. Whether you can apply De Morgan's laws inside the scope of a quantifier without dropping a connective.
Approach. Negate the quantifier first (universal becomes existential, and vice versa), then apply De Morgan to the inside. Check the result against a small test case.
Setup. Several conditional statements connect five variables. The candidate must determine which inference is valid.
What it tests. Whether you can build a graph of implications and read off valid conclusions without circular reasoning.
Approach. Draw an arrow diagram with the variables as nodes and implications as arrows. The valid inference is the one that follows a path in the diagram.
Setup. A target condition is given, with four possible additional conditions. The candidate must identify the weakest condition that guarantees the target.
What it tests. Whether you can rank conditions by strength and identify the minimal sufficient one.
Approach. Order the candidate conditions from weakest to strongest. Test the weakest first; if it implies the target, it is the answer.
Setup. A statement with mixed universal and existential quantifiers, plus a conditional structure. The candidate must determine which of the offered conclusions follows.
What it tests. Combined fluency in quantifier handling, conditional inference, and small witness construction.
Approach. Translate the statement into formal notation, then test each conclusion against a small witness model. The correct conclusion follows on every witness; the others fail on at least one.
Stop reading. Start practising.
Reading walkthroughs is half the job. The free diagnostic shows which topics from this paper you would actually drop marks on, with a TMUA-style grade at the end.
The themes that recur across this paper. Use them as a checklist when you sit a fresh mock.
Order matters. Train yourself to read 'for every…there exists' and 'there exists…for every' as structurally different statements before you start evaluating.
Apply De Morgan inside quantifier scope deliberately. Flip the quantifier first, then negate the predicate.
Always check both directions of an implication. Necessary-only and sufficient-only are equally common answer choices.
Maintain a mental library of small edge cases: 0, 1, -1, the empty set, a singleton, constant functions. Most TMUA Paper 2 counterexamples come from this set.
Affirming the consequent and denying the antecedent are the two most common fallacies in TMUA distractors. Recognise them by shape, not by meaning.
Set notation in TMUA is shorthand for logic. Expanding to element-wise checks beats trying to manipulate the set expression.
Hidden-premise questions reward finding the smallest premise that closes the gap. Bigger or stronger premises are usually wrong distractors.
TMUA proof questions reward reading each line as if you have never seen the conclusion. Each line must follow from the previous by a named rule.
Paper 2 is logic and reasoning over mathematical content. The objects are still numbers, functions, and sets, but the questions ask about validity, negation, and counterexamples rather than computation. You will not be asked to integrate or differentiate, but you do need to know what a continuous function or an injective map is.
In 2024, Paper 2 had a modal score of roughly 4.7, slightly lower than Paper 1's 5.0, with a wider spread. The wider spread reflects that Paper 2 separates students by reading discipline, which is harder to improve through practice than computational fluency.
Paraphrasing statements before evaluating them. Students who rewrite a logical statement in 'their own words' before checking it almost always change the quantifier scope or drop a connective. Read the statement verbatim, then evaluate.
Not formally, but knowing the symbols (for-all, there-exists, implies, and the contrapositive) helps you write a clean scratch pad. The exam does not use formal notation, but your working will be tidier if you do.
Roughly 3.5 minutes per question, the same as Paper 1. Paper 2 questions are usually shorter to read but heavier to evaluate. Bank time on the easier negation questions to spend on the quantifier-order and proof-analysis questions later.
The Logic Decoder covers every Paper 2 topic with worked examples and is the right starting point. For exam-day timing and pressure, supplement with the TMUA Pro question bank or the free diagnostic to test under timed conditions.
The style was consistent with prior Paper 2 sittings, but the wording was tightened. Quantifier-order questions were more prominent, and the proof-analysis questions had cleaner traps. The reasoning skills you need have not changed; the precision required has.
10-question diagnostic with a TMUA-style score and weak-spot report.
Convert historic TMUA scores to the new 2026 scale and see a percentile band.
Timed mock exams with worked solutions and a TMUA grade. Free account required.
Logic Decoder, Top 30 Mistakes, and the Score Reality Check guide.
Shrey Garg read maths and economics at Cambridge. He co-built TMUA.co.uk after sitting the test himself and has since written the Logic Decoder, Top 30 Mistakes, and Score Reality Check guides. Walkthroughs reflect what he teaches one-to-one, not generic exam advice.
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