TMUA Paper 2: practising conditions, counterexamples and proof
Oxbridge Admissions Tests · Updated
Paper 2 asks you to assess reasoning, not just produce a numerical result. It covers mathematical reasoning and elementary logic, with 20 multiple-choice questions in 75 minutes. Record condition errors separately from calculation errors when reviewing a set.
Separate sufficient, necessary and equivalent conditions
Original example: for real x, if x > 2 then x² > 4. The reverse implication is false: x = −3 also satisfies x² > 4. Thus x > 2 is sufficient, but not necessary, for x² > 4.
Restricting x to positive real numbers changes the conclusion. Mark words such as integer, positive, non-zero, every and exists before calculating. A counterexample must satisfy the stated domain.
Try boundaries and special values with a purpose
For an 'always true' claim, inspect zero, one, negative values, equality cases or equal variables. One valid counterexample disproves a universal claim. Several supporting examples do not establish a proof.
Avoid random substitution. If a claim depends on parity, monotonicity or signs, name the property you are trying to break before choosing a value. Failing to find a counterexample does not prove the claim.
Ask whether each transformation is reversible
Squaring, taking roots and dividing by an expression containing a variable can alter the solution set or omit cases. Mark these steps in your working and check them against the original conditions.
Keep three lines in your review: the original claim, the step that failed and the missing condition. 'I divided without checking the divisor was non-zero' is more useful than copying a full solution without identifying the error.
Build accuracy before practising decisions under time pressure
Start with untimed sets where you can explain each option. Then use timed practice and compare the initial answer with your checked answer to distinguish reasoning errors from execution errors.
The official preparation page links to Notes on Logic and Proof. Use our practice report to plan the next set, not to convert a practice percentage directly into an official score.
Official sources and scope
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