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TMUA Past Paper Walkthrough
A question-by-question breakdown of TMUA 2024 Paper 1. Topics, traps, and the approach that gets the mark.
TL;DR
Score distribution and band thresholds taken from the UAT-UK TMUA Technical Report 2024-25 (published September 2025). The report confirms 2024 was the first sitting on the recalibrated 1.0–7.0 scale and gives modal performance bands, though it does not publish individual question-level statistics.
Paper 1 in 2024 was the first sitting on the new 1.0–7.0 scale, and the cohort treated it as a fresh exam rather than a continuation of the old TMUA. That matters because the difficulty mix shifted slightly: pure algebra and quadratics felt familiar, but the sequences and combined function questions punished candidates who had memorised template solutions from 2016–2022 papers.
If you only take one lesson from 2024 Paper 1, take this: the test rewards students who can translate between representations. The hardest questions sat algebraic statements next to graph features, log identities, or coordinate geometry, and asked you to commute between them. Students who could only work in one direction lost time on three or four questions, which is the difference between a 5.5 and a 6.2.
The other lesson is that 2024 Paper 1 punished partial fluency in differentiation. Candidates comfortable with polynomial differentiation but rusty on differentiating products of trig and exponentials gave up easy marks. The TMUA never asks for product rule manipulations beyond A Level Year 2 standard, but it does expect you to apply them confidently inside a larger argument.
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.
Setup. A short manipulation of an expression involving fractional indices and a single surd. The candidate is asked to find the value of a numerical expression in a tidy form.
What it tests. Whether you can apply standard index laws without writing intermediate working. The paper opens with a confidence question.
Approach. Rewrite each term as a power of the same base, combine, then re-express. Do not be tempted to evaluate a square root numerically; the answer choices will look like a surd.
Setup. An inequality involving a parameter where the candidate must determine the range of the parameter for which the inequality has integer solutions.
What it tests. Comfort with parameterised inequalities and the discipline to test boundary cases.
Approach. Solve the inequality for the variable in terms of the parameter, then ask which parameter values make the resulting interval contain an integer. Sketching the boundary line beats algebraic manipulation here.
Setup. A quadratic with a varying constant term is set equal to a linear expression. The question asks for the number of intersection points as the parameter varies.
What it tests. Whether you instinctively reach for the discriminant when a question asks about the number of solutions.
Approach. Set up the quadratic in standard form, then write the discriminant as a function of the parameter. The answer follows from where the discriminant changes sign.
Setup. A trig equation in a restricted domain where the candidate must count the number of solutions. The equation mixes a sin and a cos term with different arguments.
What it tests. Whether you can move between identities and graphs to count solutions, rather than solving algebraically.
Approach. Use the identity to combine the two terms into a single trig expression, then read the number of intersections from the period structure. A quick sketch is faster than algebra.
Setup. A recurrence relation is defined with two seeded terms. The candidate must determine the value of the tenth term or recognise the eventual periodic pattern.
What it tests. Pattern recognition under time pressure. Most students brute-force the first several terms and lose minutes.
Approach. Compute three or four terms by hand, look for a pattern, and verify it algebraically. If the recurrence depends on differences or ratios, that usually signals a closed-form trick.
Setup. An equation involves logs to different bases and asks for the product of the solutions, expressed in terms of natural logs.
What it tests. Change-of-base fluency and willingness to use Vieta's formulas instead of solving directly.
Approach. Substitute u = log of the variable, reduce to a quadratic in u, and use the sum and product of roots to skip the explicit factoring.
Setup. A circle and a line are defined parametrically. The candidate must identify the range of parameters for which the line cuts the circle in exactly two distinct points.
What it tests. Whether you reach for the perpendicular-distance shortcut rather than solving a quadratic discriminant.
Approach. Compute the perpendicular distance from the centre to the line as a function of the parameter, then compare with the radius. This avoids a four-line discriminant expansion.
Setup. A function combines an exponential and a polynomial. The candidate must find the x-coordinate of the stationary point in terms of a constant.
What it tests. Product rule fluency and recognition that the stationary condition reduces to a tidy expression.
Approach. Apply product rule, factorise out the exponential, and solve the polynomial factor. The exponential is never zero, so the answer comes from the polynomial root.
Setup. Two functions f and g are defined. The candidate must determine which composition of f and g produces an even function.
What it tests. Whether you can reason about symmetry without computing a single value, plus comfort with f(-x) reasoning.
Approach. Test each composition against the definition that h(-x) = h(x). Eliminate compositions that fail at a simple test value before checking algebraically.
Setup. A definite integral of a rational function over a closed interval. The integrand simplifies after long division or partial fractions.
What it tests. Algebraic preparation before integration, and whether you spot when a rational function is improper.
Approach. Perform long division to reduce the integrand to a polynomial plus a remainder term. Integrate each piece, then evaluate.
Setup. A modelling question where a quantity grows exponentially. The candidate must find the time for the quantity to halve relative to a reference value.
What it tests. Translation between log statements and exponential statements; comfort with manipulating natural logs.
Approach. Set up the half-life equation, take logs of both sides, and isolate the time variable. Keep the answer in terms of ln to match the answer choices.
Setup. A rational inequality where the numerator and denominator each contain a parameter. The candidate must find the range of the variable satisfying the inequality.
What it tests. Sign analysis under a sign-changing denominator. A standard trap.
Approach. Bring everything to one side, factorise numerator and denominator, then build a sign table. Resist the urge to multiply across, since the denominator changes sign.
Setup. A piecewise function is defined with a parameter chosen to make the function continuous. The candidate must find the value of the parameter and a derived quantity.
What it tests. Continuity reasoning and willingness to evaluate two pieces at the same point.
Approach. Set the two pieces equal at the join point, solve for the parameter, then compute the requested quantity using that value.
Setup. A counting problem with conditional structure where the candidate must determine the probability of a specific arrangement, given a constraint.
What it tests. Whether you can structure conditional probability as a ratio of counts rather than reaching for Bayes.
Approach. Count the favourable arrangements directly, then count the constrained arrangements. Take the ratio. Avoid plugging into Bayes if a counting argument works.
Setup. An identity in two variables holds for all values of one variable. The candidate must determine specific coefficients in the other.
What it tests. Comparison of coefficients on both sides and willingness to substitute particular values.
Approach. Either expand and compare coefficients, or substitute three specific values to get a small linear system. The substitution method is usually faster.
Setup. An expression involves multiple trig functions of different angles. The candidate must simplify to a single value or recognise a structural identity.
What it tests. Familiarity with double-angle and sum-to-product identities, plus pattern recognition.
Approach. Identify whether the expression is a recognisable expansion of a single identity. If not, convert everything to sin and cos and simplify.
Setup. An arithmetic progression has terms in a specified ratio. The candidate must find the common difference and a derived sum.
What it tests. Setting up simultaneous equations from ratio statements and managing the resulting linear system.
Approach. Express the three terms in terms of the first term and the common difference, write the ratio as an equation, and solve.
Setup. A cubic has known roots related by a structural condition. The candidate must determine a coefficient.
What it tests. Vieta's formulas in disguise and willingness to avoid expansion.
Approach. Use sum and product of roots to set up two equations, then solve for the coefficient. Expansion works but takes twice as long.
Setup. A function is defined as a logarithm of a quadratic. The candidate must determine the range of values for which the function is defined.
What it tests. Domain reasoning combined with quadratic inequalities.
Approach. Set up the quadratic inside the log as strictly positive, then solve the inequality. The domain is the union of intervals where the inequality holds.
Setup. An optimisation question where a shape's dimensions depend on a parameter. The candidate must find the parameter that minimises the area, expressed in a non-obvious form.
What it tests. Setting up the right objective function, then differentiating cleanly. The setup is harder than the calculus.
Approach. Draw the shape, write the area as a function of the parameter, differentiate, set to zero, and confirm with a second-derivative or boundary check.
Stop reading. Start practising.
Reading walkthroughs is half the job. The free diagnostic shows which topics from this paper you would actually drop marks on, with a TMUA-style grade at the end.
The themes that recur across this paper. Use them as a checklist when you sit a fresh mock.
Expect at least four questions that reward algebraic prep before the main argument. Long division, Vieta, partial fractions, and clean factoring save minutes.
Symmetry, continuity, and composition arguments featured heavily. Practise reasoning about f(-x), f ∘ g, and piecewise joins without computing values.
Differentiation of products and quotients was standard. The trap was identifying which factor matters at the stationary point. Always factor before solving.
Two questions used perpendicular distance from a point to a line. If you only know the discriminant approach, learn the alternative now.
Recurrences and arithmetic progressions appeared. The recurrence rewarded pattern spotting; the AP rewarded a clean simultaneous-equation setup.
Change of base, log inequalities, and log domains all appeared. Treat logs as just another function with a domain; the domain question alone was worth a mark.
Identity recognition mattered more than algebraic manipulation. Sketch first, manipulate second.
A single hard counting question. Conditional probability framed as a ratio of counts beats Bayes notation for TMUA-style problems.
The UAT-UK Technical Report 2024-25 does not publish raw-to-scaled conversion tables. Based on the published modal-band distribution, candidates around the 6.0 mark generally needed roughly two-thirds of the raw marks. A 7.0 sat closer to 80% raw. Treat these as informed estimates from the report's band shape, not official numbers.
Most students who sat both reported similar difficulty in pure manipulation, with 2024 Paper 1 leaning harder on sequences and combined functions. The change of scale to 1.0–7.0 also makes direct comparison awkward. The Technical Report describes the 2024 cohort distribution as broadly in line with prior expectations once recalibrated.
Yes. The 2024 paper reflects the recalibrated TMUA most closely, so it is the best signal for what the next sitting will look like. Use older papers for breadth and the 2024 paper for calibration.
On average 3.5 minutes per question across 20 questions, leaving five minutes for a sweep. The 2024 paper has at least three questions where finishing in 2 minutes is essential to bank time for the harder sequences and optimisation questions later.
Official TMUA papers are released by UAT-UK each year and sit in their question-bank library. We do not republish them on this site because Cambridge Assessment owns the copyright. Use UAT-UK's release alongside this walkthrough.
Spending too long on Question 9 (the combined-functions question) and Question 14 (the conditional probability question). Both are solvable in under three minutes with the right framing, but they look like they need longer, so students sink time and shortchange the optimisation question at the end.
Yes. The TMUA Pro question bank includes Paper 1 sets calibrated against the 2024 paper's difficulty profile, with worked solutions. The free diagnostic is a 10-question version drawn from the same content.
10-question diagnostic with a TMUA-style score and weak-spot report.
Convert historic TMUA scores to the new 2026 scale and see a percentile band.
Timed mock exams with worked solutions and a TMUA grade. Free account required.
Logic Decoder, Top 30 Mistakes, and the Score Reality Check guide.
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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