TMUA PAPER 2 PROOF
A proof can contain several true statements and still fail because one implication is invalid. Train yourself to identify the first unsupported step and to distinguish a proof of the claim from a proof of its converse.
A universal statement such as ‘for every integer’ or ‘for all real \(x\)’ is disproved by one valid counterexample. You do not need to classify every failure unless the question asks for that classification.
When testing a claim, choose examples that target the weakest assumption: boundary values, repeated roots, zero, negative values, parity or a residue class that the proposed pattern ignores.
A chain of equivalences requires every step to work in both directions. Squaring, cancelling, taking square roots, multiplying inequalities and substituting parameterisations can all lose information or introduce extra possibilities.
Write the target implication at the top of your rough work. Then ask whether the displayed proof actually starts from the hypothesis and reaches the conclusion, or whether it has silently started from the conclusion and proved the converse instead.
FREE PRACTICE
These questions now draw from the current reviewed CF Maths banks, including image and diagram items where appropriate. Open the solution only after you have committed to a route.
