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TMUA Past Paper Walkthrough
A question-by-question breakdown of TMUA 2023 Paper 1. Topics, traps, and the approach that gets the mark.
TL;DR
Cambridge Assessment Admissions Testing did not publish a Technical Report for the 2023 TMUA sitting, and Cambridge stopped releasing detailed test statistics after that cycle. The estimates above are drawn from candidate forum reports, school feedback, and aggregated diagnostic data from TMUA.co.uk users who sat the paper. They should be treated as informed estimates rather than official figures.
TMUA 2023 Paper 1 was the last sitting on the old 1.0–9.0 scale and the last sitting before the test moved fully to its current computer-based format. The questions reflect a slightly more leisurely paper than the 2024 recalibration, with a few setups that gave generous time for clean algebra.
What makes this paper genuinely useful for revision is not nostalgia, it is the consistency of the testable skills. The 2023 paper rewards exactly the same core fluencies as 2024: clean algebra, fast graphical reasoning, and pattern recognition in sequences. If you can solve this paper comfortably, you have the technical baseline for the recalibrated test.
The single most striking pattern in 2023 Paper 1 is how often the examiners reward students who can avoid algebra entirely. At least four questions had a graphical or substitution shortcut that turned a 90-second algebraic argument into a 20-second visual check. Students who treated TMUA as a pure algebra paper lost time they did not need to lose.
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. An expression combining a surd and a fractional power must be simplified to a single tidy form.
What it tests. Whether you can apply index laws confidently under exam conditions.
Approach. Rewrite each factor as a power of the same base, combine exponents, simplify.
Setup. A pair of linear equations with one parameter must be solved for a derived quantity in terms of the parameter.
What it tests. Comfort with parameter-laden simultaneous equations.
Approach. Eliminate one variable, solve for the other in terms of the parameter, then compute the derived quantity.
Setup. A quadratic depends on a parameter. The candidate must determine the range of parameter values for which the graph cuts the x-axis at two distinct points.
What it tests. Discriminant reasoning, set up as an inequality in the parameter.
Approach. Write the discriminant as a function of the parameter, set it strictly positive, solve the inequality.
Setup. A trig equation in a restricted domain must be solved exactly. The equation reduces to a quadratic in a single trig function.
What it tests. Substitution u = sin x (or cos x), reduction to a quadratic, then back-substitution.
Approach. Substitute, solve the quadratic for u, then find the angles. Discard solutions outside the given domain or where the substitution is out of range.
Setup. A geometric progression is partially specified by two terms. The candidate must find the sum to infinity (or a related total).
What it tests. Setting up the GP from given terms and applying the sum formula correctly.
Approach. Find the common ratio from the ratio of the two given terms, then use the sum-to-infinity formula. Check that |r| < 1 before applying.
Setup. A log equation reduces to a quadratic in log x. The candidate must find the product of the solutions.
What it tests. Substitution u = log x and use of Vieta's formulas.
Approach. Substitute, write down the quadratic, use sum-of-roots or product-of-roots directly. The product of the solutions in x is the exponential of the sum of u roots, by log properties.
Setup. A circle and a line are given. The candidate must determine the value of a parameter so that the line is tangent to the circle.
What it tests. Whether you reach for perpendicular distance from centre to line, rather than discriminant of a substituted quadratic.
Approach. Compute the perpendicular distance from the centre of the circle to the line as a function of the parameter, set equal to the radius, solve.
Setup. A function of x is given. The candidate must find the x-coordinate of the stationary point, expressed as a simple expression.
What it tests. Differentiation of a product or composite, then solving the resulting equation.
Approach. Apply chain or product rule, set the derivative to zero, factorise to find x.
Setup. Two functions f and g are defined. The candidate must identify which composition is invertible on a given domain.
What it tests. Whether you can check injectivity by reasoning about monotonicity, rather than computing the inverse.
Approach. Test each composition for monotonicity on the given domain. The composition that is strictly monotonic is invertible.
Setup. A definite integral involves a polynomial multiplied by a trig function. The candidate must evaluate exactly.
What it tests. Integration by parts at A Level Year 2 standard.
Approach. Apply integration by parts with u = polynomial, dv = trig dx. Evaluate at both limits.
Setup. A modelling question where a population grows according to an exponential rule. The candidate must find the time at which the population reaches a specified threshold.
What it tests. Translating between exponential and log forms, and isolating the time variable.
Approach. Write the population equation, set equal to the threshold, take natural logs of both sides, solve for time.
Setup. A polynomial inequality must be solved over a specified domain. The polynomial factorises into three real factors.
What it tests. Whether you can build a sign table and read off the solution set without arithmetic errors.
Approach. Factorise, mark the roots on a number line, determine the sign on each interval, take the union of intervals where the inequality holds.
Setup. A function is transformed by a combination of shifts and scalings. The candidate must identify the resulting expression.
What it tests. Whether you apply transformations in the correct order (inside-the-function first, then outside).
Approach. Apply the inside-the-function transformations first, then the outside-the-function ones. Test against a known point if you are unsure of order.
Setup. Two events are partially dependent. The candidate must compute a conditional probability from given marginal and joint probabilities.
What it tests. Conditional probability as a ratio of joint over marginal, not as a Bayes plug-in.
Approach. Write P(A given B) = P(A and B) / P(B). Substitute the given numbers.
Setup. Two polynomial expressions are claimed to be equal for all values of x. The candidate must find the coefficients on both sides.
What it tests. Comparison of coefficients on both sides.
Approach. Expand both sides, equate coefficients of equal powers, solve the resulting linear system.
Setup. An expression involves sin and cos of multiple angles. The candidate must simplify to a single trig function or numerical value.
What it tests. Pattern recognition for double-angle, factor, or sum-to-product identities.
Approach. Identify the structural pattern. If unclear, convert everything to sin x and cos x, simplify, then reidentify.
Setup. An arithmetic series sums to a given total. The candidate must find the first term and the common difference given a structural condition on the terms.
What it tests. Setting up simultaneous equations from sum and term conditions.
Approach. Write the sum formula and the term condition as two linear equations in first term and common difference. Solve.
Setup. A cubic polynomial with a parameter has a specified relationship between its roots (for example, two roots equal, or roots in arithmetic progression). The candidate must find the parameter.
What it tests. Vieta's formulas in disguise; ability to translate root conditions into coefficient conditions.
Approach. Use sum, product, and pairwise products of roots to set up equations in the parameter. Resist the temptation to solve the cubic explicitly.
Setup. A function involving a logarithm of a polynomial must have its domain identified. The polynomial inside the log changes sign over the real line.
What it tests. Domain reasoning combined with polynomial sign analysis.
Approach. Set the polynomial inside the log strictly positive, factorise, build a sign table, identify the intervals where positivity holds.
Setup. An optimisation question where a geometric shape has dimensions constrained by a fixed quantity. The candidate must minimise (or maximise) a derived quantity.
What it tests. Setting up the objective function from a geometric description, then differentiating cleanly.
Approach. Express the constraint, write the objective as a function of one variable, differentiate, solve, confirm with a second-derivative check or boundary inspection.
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.
Discriminants, Vieta's formulas, and sign tables drove four of the five hardest questions. Drill each one independently.
Composition, invertibility, and transformations all appeared. Practise reasoning about monotonicity without computing explicit inverses.
Integration by parts and chain-rule differentiation both featured. The standard A Level Year 2 toolkit is enough; comfort with it is essential.
Tangency was the recurring theme. Perpendicular-distance arguments beat substitution-and-discriminant nearly every time.
Both an arithmetic and a geometric progression featured. Memorise the sum formulas cold; do not derive them under exam pressure.
Change of base, log equations as quadratics, and log domains all appeared. Treat logs as functions with a domain, not as algebraic placeholders.
Identity recognition decided the harder trig questions. If you cannot name the structural identity within five seconds, the algebra will not save you.
Conditional probability as a ratio of counts. A single hard question, but it rewarded clean setup over formula recall.
Not by Cambridge Assessment. The 2023 sitting was the last under the old 1.0–9.0 scale, and Cambridge did not publish a per-question or per-paper statistics document for that year. Most published estimates come from candidate forums, school internal data, and platforms like TMUA.co.uk that aggregate diagnostic results from candidates who sat the paper.
Yes, with caveats. The topics tested have not changed materially. The new 1.0–7.0 scaling and the computer-based format are different, so do not interpret 2023 raw marks as equivalent to 2024 scaled scores. Use 2023 for breadth and topic practice, 2024 for calibration.
Most students find 2023 Paper 1 slightly more leisurely on pure algebra and slightly harder on coordinate geometry. The 2024 paper leans harder on sequences and combined functions. They are similar overall, but a student who scored 70% raw on one would not necessarily score the same on the other.
Official TMUA 2023 papers are still hosted in the UAT-UK question-bank library. We do not republish them on this site due to Cambridge Assessment's copyright. Use UAT-UK's release alongside this walkthrough.
Spending too long on Question 18 (the cubic with parameter-related roots) when Question 20 (the optimisation question) is more straightforward and worth the same. Bank the optimisation mark before sinking time into the cubic.
Roughly 3.5 minutes per question across 20 questions, with five minutes left for a sweep. Questions 1 through 6 should each take under two minutes; this banks time for the harder questions in the back half.
The TMUA Pro question bank contains Paper 1 sets that mirror the 2023 difficulty profile, plus standalone mock papers with worked solutions. The free diagnostic is a 10-question sampler at TMUA-style difficulty.
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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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