TMUA Past Paper Walkthrough

TMUA 2023 Paper 1 Walkthrough: Applications of Mathematical Knowledge

A question-by-question breakdown of TMUA 2023 Paper 1. Topics, traps, and the approach that gets the mark.

By Shrey Garg, Cambridge maths and economics graduate

TL;DR

Modal score
Modal raw mark around 11 out of 20 on the 2023 sitting, which mapped to roughly 5.5 on the old 1.0–9.0 TMUA scale.
Grade boundary signal
Official published boundaries are not available for 2023. From candidate reports, a 6.5 typically required around 13 raw marks; a 7.5 needed 15 or more.
Hardest topics
Polynomials with parametersTrigonometric identitiesIntegration techniquesConditional probability

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.

What this paper teaches us about TMUA

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.

The patterns the examiners actually reward

  • Quadratic discriminant questions reward students who treat the discriminant as a function of the parameter, not as a fixed number. Sketch the discriminant against the parameter, then read off the answer.
  • Trig identity questions usually have a single trick: recognise the structure of one of the standard double-angle, sum-to-product, or factor formulas. Without recognition, you will not finish in time.
  • Coordinate geometry rewards reading the diagram structurally. Tangency, intersection counts, and circle-through-points all have one-line characterisations that beat brute algebra.
  • Integration questions in this paper tested both substitution and integration by parts at A Level Year 2 standard. The exam never asks for non-standard techniques, but it does expect confident use of the standard toolkit.
  • Probability questions tested conditional reasoning rather than computation. Set up the conditional as a ratio of counts before reaching for a formula.

Question-by-question walkthrough

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.

  1. Question 1 · Surds and indices

    Easy

    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.

  2. Question 2 · Linear equations

    Easy

    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.

  3. Question 3 · Quadratic graphs

    Medium

    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.

  4. Question 4 · Trigonometric equations

    Medium

    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.

  5. Question 5 · Sequences

    Medium

    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.

  6. Question 6 · Logarithms

    Medium

    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.

  7. Question 7 · Coordinate geometry

    Hard

    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.

  8. Question 8 · Differentiation

    Medium

    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.

  9. Question 9 · Combined functions

    Hard

    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.

  10. Question 10 · Integration

    Medium

    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.

  11. Question 11 · Exponential growth

    Medium

    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.

  12. Question 12 · Inequalities

    Hard

    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.

  13. Question 13 · Functions and transformations

    Medium

    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.

  14. Question 14 · Probability

    Hard

    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.

  15. Question 15 · Polynomial identities

    Medium

    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.

  16. Question 16 · Trigonometric identities

    Hard

    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.

  17. Question 17 · Sequences and series

    Hard

    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.

  18. Question 18 · Polynomials with parameters

    Very Hard

    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.

  19. Question 19 · Logarithms and domains

    Hard

    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.

  20. Question 20 · Calculus and optimisation

    Very Hard

    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.

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Topic-level takeaways

The themes that recur across this paper. Use them as a checklist when you sit a fresh mock.

Algebra

Discriminants, Vieta's formulas, and sign tables drove four of the five hardest questions. Drill each one independently.

Functions and graphs

Composition, invertibility, and transformations all appeared. Practise reasoning about monotonicity without computing explicit inverses.

Calculus

Integration by parts and chain-rule differentiation both featured. The standard A Level Year 2 toolkit is enough; comfort with it is essential.

Coordinate geometry

Tangency was the recurring theme. Perpendicular-distance arguments beat substitution-and-discriminant nearly every time.

Sequences and series

Both an arithmetic and a geometric progression featured. Memorise the sum formulas cold; do not derive them under exam pressure.

Logarithms

Change of base, log equations as quadratics, and log domains all appeared. Treat logs as functions with a domain, not as algebraic placeholders.

Trigonometry

Identity recognition decided the harder trig questions. If you cannot name the structural identity within five seconds, the algebra will not save you.

Probability

Conditional probability as a ratio of counts. A single hard question, but it rewarded clean setup over formula recall.

Frequently asked questions

Are the 2023 grade boundaries published anywhere?

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.

Is the 2023 paper still relevant for revision in 2026?

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.

How does 2023 Paper 1 compare to 2024 Paper 1 in difficulty?

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.

Can I see the actual exam paper from 2023?

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.

What is the single biggest mistake on this paper?

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.

How long should I spend on each Paper 1 question for this paper?

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.

Does TMUA.co.uk have a 2023-style mock paper?

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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About the author

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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