TMUA Proof Techniques for Paper 2 (Logic & Reasoning)

Paper 2 is the half of the TMUA that surprises everyone. It looks like a maths paper, but there is barely any calculation. Instead it hands you statements, definitions and arguments and asks what logically follows. Almost no school teaches this, so most candidates walk in underprepared, which means it is also where the easiest marks are hiding for anyone who trains it. This guide covers the core techniques Paper 2 rewards and, just as importantly, how to recognise which one a question is asking for.

The techniques that cover almost everything

1. Counterexample

To disprove a statement of the form "for all x, something is true", you only need one x where it fails. This is the single most common Paper 2 move. When a question says "which of the following is false" or "is this claim true?", your first instinct should be to hunt for a small case that breaks it (often 0, 1, a negative, or a boundary value), before you try to prove anything.

Here is a real Paper 2 question in exactly this spirit. Read it, decide whether the statement holds, and if not, find the case that kills it:

The move here is systematic, not lucky. When you are hunting a counterexample, you do not try random values; you try the values most likely to break a claim: the smallest allowed number, a boundary of the given range, an even versus an odd, a prime versus a composite, zero, one, or a negative if they are permitted. One of those almost always exposes a false "for all" statement, and the moment it does, you are finished: a single counterexample is a complete disproof, so you can stop and move on. Training this habit until it is automatic is worth more marks per hour than almost anything else on Paper 2, because so many questions are secretly counterexample hunts wearing a disguise.

2. Direct deduction

Sometimes there is nothing to disprove: you are given facts and asked what must follow. Here you chain the given information step by step, keeping careful track of what you actually know versus what you are assuming. The trap is to import a "reasonable" assumption that was never stated. On the TMUA, only use what you are given.

3. Proof by contradiction

To prove a statement is true, assume it is false, then follow the logic until you reach something impossible. The impossibility means your assumption was wrong, so the original statement holds. This is the standard tool for classic results (there are infinitely many primes, the square root of two is irrational) and it appears on Paper 2 both directly and hidden inside answer options.

4. "Must be true" over every case

Many questions give a setup with some freedom in it (a variable, an unknown arrangement) and ask which statements must be true. A statement only qualifies if it holds in every allowed configuration; a single exception disqualifies it. So your job is to test the extremes and awkward cases, not just the obvious one. This is where careful, systematic candidates pull ahead of intuitive guessers.

5. Necessary versus sufficient

A huge share of Paper 2 turns on the direction of an implication. "A is sufficient for B" means A implies B. "A is necessary for B" means B implies A. They are not the same, and questions are built to catch you swapping them. Whenever you see "necessary", "sufficient", "if and only if", or "only if", slow down and write out which way the arrow points.

6. Spot the flaw

Some questions show you a full "proof" and ask where it goes wrong. The error is usually a single specific line: dividing by something that could be zero, taking a square root and dropping the negative case, or misapplying a rule outside its conditions. Read the argument line by line and check that each step is genuinely justified, rather than skimming for the general shape.

The specification names two invalid deductions outright as examples of what to watch for: claiming that if $ab = ac$ then $b = c$, and assuming that if $\sin A = \sin B$ then $A = B$. Neither follows. The first collapses when $a = 0$, which makes $ab = ac$ true whatever $b$ and $c$ are. The second fails at $A = 30°$ and $B = 150°$, where both sines equal $\frac{1}{2}$. Both are worth memorising as shapes, because a printed "proof" that quietly cancels a factor or strips a trig function off both sides is a stock Paper 2 construction.

The inequality errors that a third of the flaw questions are built on

Inequality manipulation deserves its own heading, because it is a whole third of the board's own error taxonomy for faulty proofs and it is the family most candidates never think to check. The rule students carry over from equations, "whatever you do to one side you do to the other", is simply not true for an inequality sign.

The board's own three examples are as short as they are convincing:

• Squaring both sides. $-5 < 4$ is true. Square it and you get $25 < 16$, which is false.
• Multiplying by a negative number. $1 < 2$ is true. Multiply by $-1$ and you get $-1 < -2$, which is false.
• Applying a function to both sides. $\frac{\pi}{4} < \frac{\pi}{3}$ is true. Take the cosine of both sides and you get $\cos\frac{\pi}{4} < \cos\frac{\pi}{3}$, which is false, because cosine is decreasing there. Only an increasing function preserves the direction.

The live version, and the one that shows up disguised as algebra rather than as a logic puzzle, is multiplying by a quantity whose sign you do not know. Faced with $\frac{2x + 5}{x + 3} > 1$, the obvious move is to multiply both sides by $x + 3$. The board declines, precisely because $x + 3$ is negative for some $x$ and the inequality sign would then be pointing the wrong way for those values. It multiplies by $(x + 3)^2$ instead, which is never negative. Multiply by $x + 3$ carelessly and you conclude $x > -2$, having silently thrown away the entire branch $x < -3$.

Two questions catch every one of these. Before a step on an inequality, ask: could this quantity be negative? and is this function increasing over the range I was given? If either answer is no or unknown, the step is not justified, and on a spot-the-flaw question that is the line you are looking for. Our inequalities guide works the algebra, and TMUA maths mistakes collects the rest of the errors the examiners name.

The mistakes that cost the most marks on Paper 2

Knowing the techniques is only half of it. The other half is avoiding the specific errors Paper 2 is built to punish. These five account for the majority of dropped marks:

• Importing unstated assumptions. The single most common error. A question tells you "n is a whole number"; you quietly assume it is positive, or non-zero, and walk into the trap. On the TMUA, use only what is written. If a case is not ruled out, it is allowed, and it is usually exactly the case the examiner is testing.
• Confusing "true for some" with "true for all". A statement that holds for a few examples you tried is not proved; a statement that fails for one example you did not try is not true. Always ask which quantifier the claim actually uses, then test accordingly: examples can only disprove a universal claim, never prove it.
• Swapping necessary and sufficient. "A only if B" means A implies B, not the reverse. Under time pressure it is dangerously easy to reverse the arrow. Whenever those words appear, write the implication out with an actual arrow before you evaluate the options.
• Not negating quantifiers correctly. The negation of "every x satisfies P" is "some x does not satisfy P", and the negation of "some x satisfies P" is "no x satisfies P". Getting this backwards turns a right answer into a wrong one, and it comes up constantly in "which of these is the negation" questions.
• Negating a compound statement term by term. The negation of "A and B" is "not A or not B", not "not A and not B": the connective flips. This is why the contrapositive of "if $a$ and $b$ are odd then $ab$ is odd" is "if $ab$ is even then $a$ and $b$ are not both odd", and not the tempting "then $a$ and $b$ are even", which is false at $a = 2$, $b = 3$. The board flags this careless "not" explicitly, and it is worked through in full in Paper 2: logic and proof.
• Reasoning from intuition instead of logic. Paper 2 rewards the answer that is forced, not the one that feels likely. The distractors are written to be the intuitive-but-wrong choice, so if an option feels obviously right without a chain of reasoning behind it, treat that as a warning, not a green light.

If you want a broader list of avoidable errors across both papers, see common TMUA mistakes.

How to know which technique a question wants

The question stem almost always tells you:

Reading for these signals before you dive in is half the battle. It stops you grinding algebra on a question that only wanted one clean counterexample.

A simple plan to get good at Paper 2

Paper 2 responds to deliberate practice better than almost anything on the TMUA, precisely because it is unfamiliar: you are not fighting years of ingrained habit, you are learning a small toolkit from a low base, so progress is fast. Here is a sequence that works:

1. Learn the six moves first, in isolation. Before doing timed questions, make sure you can define and use each technique above. You cannot pick the right tool from a set you do not know. Read Paper 2: logic and proof alongside this to see each one in its natural habitat.
2. Do Paper 2 questions untimed at first. Early on, correctness and reasoning matter more than speed. Work slowly enough to actually follow the logic, because a fast wrong habit is harder to unlearn than a slow right one.
3. After every question, name the technique that solved it. This is the highest-leverage habit on the page. Do not just check the answer; say "that was a counterexample" or "that was necessary-versus-sufficient". Labelling is what makes the next similar question instant, and it is why every CrackTMUA solution names the key idea rather than only listing steps.
4. Then add the clock. Once the moves are reliable, practise under time pressure, because Paper 2 gives you under four minutes a question and hesitation is expensive.
5. Space your review. Reasoning skills fade without revisiting, so review older Paper 2 questions on a schedule instead of cramming. This is exactly what spaced repetition for the TMUA automates.

To see these techniques worked against real questions of every difficulty, from an accessible starter to a genuine 8.0, go through the TMUA example questions. Then put it into practice: the CrackTMUA bank gives you Paper 2 questions with a worked solution that names the technique on every one, free at 10 a day, so the reasoning stops feeling like a foreign language and starts feeling like a checklist.

Frequently asked questions