TMUA Questions by Topic: Practise Every Topic Free

Doing whole past papers is the right way to finish your TMUA preparation, but it is a poor way to improve in the middle of it. When you sit a full paper, your weakest topic gets three or four questions and then disappears, which is nowhere near enough to fix it. The faster route, the one strong candidates rely on, is targeted practice: pick the topic that is costing you marks and drill real TMUA questions on it, one after another, until the method is automatic. This guide is a directory for exactly that. It links every topic to real questions with full worked solutions, and, below the tables, breaks down what the TMUA actually asks in each topic and the trap that recurs in it.

Why practising by topic works

Deliberate practice beats general practice because it concentrates your effort where the return is highest. A student who does ten mixed papers spreads their attention thinly across everything, including the topics they already have. A student who spends the same time drilling the three topics they keep getting wrong closes the exact gaps that are holding their score down. The maths is stable and the question styles repeat, so once a topic clicks, it tends to stay fixed.

Topic practice also fits the TMUA's structure. Paper 1 is applications of standard maths, so its topics map cleanly onto the A-level content you already know. Paper 2 is reasoning, so its "topics" are as much about logical technique as about content. Filtering to one at a time lets you feel that difference and train each properly. Here is a representative Paper 1 question to show the level; try it, then use the directory below to drill whichever topic you need:

Paper 1 topics: applications of maths

These are the pure-maths topics that make up Paper 1 and appear on Paper 2 too. Click "Practise" to jump straight to real questions on that topic in the free bank, and "Learn" for the full in-depth guide.

Algebra and surds is the single most common topic across the papers, so it is usually the highest-value place to start. If your foundations there are shaky, everything else gets harder, because algebra sits underneath every other topic.

What each Paper 1 topic actually asks

The tables get you to the questions; this section tells you what to expect when you arrive. For each topic: what the TMUA really tests, and the trap that keeps recurring in real papers.

Algebra, surds and indices

The bread and butter of Paper 1: rearranging, factorising, completing the square, and manipulating surds and fractional or negative indices exactly, because with no calculator every simplification is done by hand. Questions rarely say "simplify"; they hide an algebraic step inside a longer problem and punish anyone who is slow at it. The recurring trap is the plausible-but-false simplification: a surd "rationalised" with a sign slip, or an index law applied to a sum where it only holds for a product. Full breakdown in the surds and indices guide.

Logarithms and exponentials

Expect equation-solving that needs the log laws cold, work across bases, and exponential equations that reduce to quadratics via a substitution. The classic trap is inventing a law that does not exist: the log of a sum does not split, and candidates under time pressure "expand" it anyway. Watch domains too, since an algebraically valid solution can make a log argument negative. The logarithms guide covers the standard moves.

Sequences and series

Arithmetic and geometric sequences, their sums, and convergence conditions for geometric series, often dressed in wordy setups or combined with logs and inequalities. The recurring trap is the off-by-one: mixing up the number of terms with the last term's index, or summing from the wrong starting point. Writing the first few terms out by hand is slow-looking but reliably faster than misapplying a formula. See the sequences and series guide.

Coordinate geometry

Lines and circles: completing the square to read off a centre and radius, tangency conditions, distances and intersections. The TMUA loves tangency via the discriminant, setting a line against a circle and asking for the condition that they touch. The recurring trap is a sign error reading the centre from the completed-square form, and mixing up the radius with the radius squared. The coordinate geometry guide works through the patterns.

Trigonometry

Exact values, the core identities, and solving equations over a stated interval. The recurring trap is lost solutions: dividing both sides by a trig factor that could be zero, or stopping after the first quadrant when the interval contains a second solution. Sketching the graph for two seconds beats memorising rules about quadrants. The trigonometry guide sets out the toolkit.

Differentiation

Differentiating powers of x, then using it: tangents and normals, stationary points, and increasing or decreasing behaviour. Questions usually require a rewrite first, roots and quotients turned into powers, and the trap is differentiating the unrewritten form. Classifying stationary points by sign change rather than blind second-derivative habits also pays off on the trickier items. See the differentiation guide.

Integration

Definite integrals and areas under curves, frequently combined with a sketch you must infer rather than being given. The single most reliable trap in the whole syllabus lives here: when the curve dips below the axis, the signed integral is not the area, and at least one wrong option always prices in that exact mistake. Split the integral at the roots. The integration guide drills it properly.

Functions, graphs and transformations

Recognising graphs from equations, applying translations and stretches, and reasoning about compositions. The recurring trap is the order of operations inside the argument: a transformation like replacing x with 2x + 1 applies the shift and stretch in the opposite order to most people's instinct. Checking one well-chosen point beats trusting a memorised rule. See the graphs and transformations guide.

Inequalities

Quadratic and rational inequalities, often as the constraint inside a larger problem. The recurring trap is multiplying or dividing both sides by a quantity that might be negative, which silently flips the inequality for some values and not others. The safe method, moving everything to one side and analysing signs, is the one the inequalities guide builds.

Paper 2 topics: reasoning

Paper 2 layers logic and proof on top of the content above, and these "topics" are really techniques. They are the most under-practised part of the TMUA and, because the toolkit is finite, the most trainable.

Logic and quantifiers

The heart of Paper 2: necessary versus sufficient conditions, negating statements correctly, converses and contrapositives, and claims quantified with "all", "some" or "exactly one". The recurring trap is direction: reading "P is necessary for Q" as "P is sufficient for Q", or proving a converse and believing the original is done. This is by far the largest Paper 2 category and where the most gettable marks hide, precisely because so many candidates neglect it. Here is a real one; be precise about which direction each implication runs before you reveal:

Proof and argument analysis

You are given a complete argument, sometimes correct and sometimes not, and asked to judge it or find the first invalid line. The recurring trap is being seduced by a true conclusion: a proof can arrive at a correct statement through an invalid step, and the step is still wrong. Check each line against the previous one, not against the destination. The proof techniques guide covers the standard flaws to hunt for.

Number theory and divisibility

Parity arguments, primes, factors and remainders, usually as the raw material for a reasoning question rather than heavy theory. The recurring trap is the unchecked edge case: forgetting that 1 is not prime, that zero and negatives exist, or testing two small cases and calling it a proof. Counterexample-hunting with small numbers is the core skill, done systematically rather than hopefully.

Sets and counting

Combinatorial reasoning dressed as logic: counting arrangements or possibilities under constraints, and set relationships between conditions. The recurring trap is overcounting, treating ordered selections as unordered or double-counting an overlap. Listing small cases in a fixed order is the reliable antidote. The counting and probability guide builds the method.

Topic is only half of it: the same topic gets asked in several different shapes, and the shape changes what a wrong option means. Our guide to the TMUA question types walks through all six formats with a real example of each.

How much practice each topic has

One of the reasons topic practice works so well on CrackTMUA is depth: because the bank combines every official past paper with a large set of original questions, most topics have enough questions to drill properly, not just the handful you would get from a single paper. The pools are not equal, and knowing that helps you plan.

The deepest pools are the ones that appear most often on the real test. Logic and quantifiers is the single largest, since it underpins the whole of Paper 2, followed closely by algebra and surds, which turns up everywhere on Paper 1. Functions and graphs, trigonometry, sequences and series, coordinate geometry, inequalities, differentiation, integration and logarithms all have substantial pools of their own, comfortably enough for focused, back-to-back practice. Number theory and the counting topics are smaller but concentrated on Paper 2, and statistics is the smallest, reflecting how lightly the TMUA tests it.

The practical takeaway is to spend your time in proportion to both your weakness and the topic's weight. There is little value in over-drilling statistics, which might give you one question on the day, and enormous value in getting fluent at algebra and Paper 2 reasoning, which shape a large share of your score. Filter to a topic, set a difficulty range you can handle, and work upward.

How to run topic-first practice properly

Topic practice works best as a loop, not a one-off, and each step has a tool built for it:

1. Diagnose first. Do a paper or two, or a diagnostic, and note which topics you consistently miss. If you practise on CrackTMUA, the dashboard keeps a topic-by-topic accuracy map so you are measuring your weak spots rather than guessing them.
2. Drill in blocks. Take one topic and do a run of questions on it back to back in the practice bank with the topic filter on, easy to hard, reading the worked solution each time. Doing them in a cluster builds the pattern recognition that scattered practice never quite delivers.
3. Re-test timed. Once the topic feels solid untimed, prove it under pressure: build a timed quiz on the same topic and check the accuracy survives the clock. A topic is only fixed when it holds at exam pace, and this step is the one most students skip.
4. Let review keep it fixed. The questions you missed resurface in spaced review over the following days, which is what stops a drilled topic quietly decaying while you work on the next one.
5. Mix back in. Once a topic stops costing you marks, fold it back into mixed practice and full papers so you can still switch between topics on demand, which is what the real exam requires.

This is the middle phase of the three-phase past-paper method: familiarise, then drill by topic, then sit timed papers. For the full plan around it, see how to prepare for the TMUA, and for the complete topic list with what each one contains, the TMUA syllabus guide.

Practise any topic free

Every topic above links straight into the CrackTMUA question bank, filtered to that topic, with a full worked solution on every question that names the trap and the fastest method. You can narrow further by paper and difficulty, and your attempts, flags and weak topics are tracked so you always know what to drill next.

It is free at 10 questions a day, and premium is a one-time £49 for 12 months if you want the whole library, every official paper plus 330+ original questions and 26+ full mocks, with no daily cap. Pick your weakest topic from the tables above and start there, or open the full practice bank and set your own filters. To see what a good bank should offer, read what a TMUA question bank needs.

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