TMUA Past Paper Walkthrough
TMUA 2024 Paper 2 Walkthrough: Mathematical Reasoning
A question-by-question breakdown of TMUA 2024 Paper 2. Topics, traps, and the approach that gets the mark.
TL;DR
- Modal score
- 4.7 on the new 1.0–7.0 scale, slightly below Paper 1, with a noticeably wider spread.
- Grade boundary signal
- About 60% raw to reach a 6.0; closer to 75% raw for a 7.0 in this sitting.
- Hardest topics
- Quantifiers and negationNecessary vs sufficientCounterexample selectionMulti-step deduction chains
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.
What this paper teaches us about TMUA
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.
The patterns the examiners actually reward
- Most negation questions look easy until you check edge cases. The trick is to negate exactly the structure of the original statement, not a paraphrase of it.
- Counterexample questions reward students who can construct a small concrete instance fast. The set is usually tiny — two or three elements — and the rule is usually broken by a corner case.
- Necessary vs sufficient questions in 2024 were almost evenly split. Train both directions with the same effort; you cannot guess your way around them.
- Deduction-chain questions reward students who track which statements are facts, which are assumptions, and which are conclusions. A two-column scratchpad pays for itself.
- The hardest questions test whether you can resist plausible-sounding distractors. If an answer 'feels right' but you cannot defend the inference step that produced it, it is usually a trap.
Question-by-question walkthrough
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.
Question 1 · Logical equivalence
EasySetup. 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.
Question 2 · Quantifier negation
EasySetup. 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'.
Question 3 · Necessary vs sufficient
MediumSetup. 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.
Question 4 · Argument validity
MediumSetup. 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.
Question 5 · Counterexample selection
HardSetup. 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.
Question 6 · Set theory and logic
MediumSetup. 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.
Question 7 · Conditional reasoning
MediumSetup. 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.
Question 8 · Counterexamples in real analysis
HardSetup. 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).
Question 9 · Truth tables and equivalence
MediumSetup. 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.
Question 10 · Argument analysis
HardSetup. 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.
Question 11 · Set operations
MediumSetup. 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.
Question 12 · Existential and universal quantifiers
HardSetup. 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.
Question 13 · Proof analysis
HardSetup. 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.
Question 14 · Counterexample construction
Very HardSetup. 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.
Question 15 · Combinations of logical statements
HardSetup. 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.
Question 16 · Inductive reasoning
MediumSetup. 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.
Question 17 · Negation of compound statements
HardSetup. 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.
Question 18 · Implication chains
HardSetup. 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.
Question 19 · Sufficient conditions
Very HardSetup. 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.
Question 20 · Mixed quantifiers and deduction
Very HardSetup. 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.
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Topic-level takeaways
The themes that recur across this paper. Use them as a checklist when you sit a fresh mock.
Quantifiers
Order matters. Train yourself to read 'for every…there exists' and 'there exists…for every' as structurally different statements before you start evaluating.
Negation
Apply De Morgan inside quantifier scope deliberately. Flip the quantifier first, then negate the predicate.
Necessary and sufficient
Always check both directions of an implication. Necessary-only and sufficient-only are equally common answer choices.
Counterexamples
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.
Conditional reasoning
Affirming the consequent and denying the antecedent are the two most common fallacies in TMUA distractors. Recognise them by shape, not by meaning.
Set theory
Set notation in TMUA is shorthand for logic. Expanding to element-wise checks beats trying to manipulate the set expression.
Argument analysis
Hidden-premise questions reward finding the smallest premise that closes the gap. Bigger or stronger premises are usually wrong distractors.
Proof verification
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.
Frequently asked questions
Is TMUA Paper 2 really just logic, or is there maths in it?
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.
How is the modal score for Paper 2 different from Paper 1?
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.
What is the single biggest mistake on TMUA 2024 Paper 2?
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.
Should I learn formal logic notation for Paper 2?
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.
How long should I spend on each Paper 2 question?
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.
Is the Logic Decoder PDF enough preparation for Paper 2?
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.
Did the 2024 paper change the style of Paper 2 questions?
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.
Related TMUA resources
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Free PDF guides
Logic Decoder, Top 30 Mistakes, and the Score Reality Check guide.
About the author
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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