TMUA 2024 Paper 1 Walkthrough: Applications of Mathematical Knowledge

What this paper teaches us about TMUA

The patterns the examiners actually reward

• Setting up a clean equation matters more than the final manipulation. Roughly a third of the paper hinges on translating worded conditions into algebra without losing a sign or a constraint.
• Multiple-choice distractors target sign errors, off-by-one indices, and misplaced inequalities. Almost every wrong answer in the multiple-choice set corresponds to a specific predictable slip.
• Examiners reward students who can rule out two or three answer choices using a quick numerical check before doing the full algebraic work. Pure substitution beats pure manipulation under time pressure.
• Coordinate geometry questions in this paper assume you can spot when a line is tangent to a curve from the structure of the simultaneous equations, not from a discriminant calculation. The shortcut saves a minute on every geometry question.
• Sequences questions test pattern recognition first and algebraic technique second. If you cannot describe what the sequence is doing in one sentence, the algebra will not save you.

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 · Indices and surds

   Easy

   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.

2. Question 2 · Linear equations and inequalities

   Easy

   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.

3. Question 3 · Quadratic functions

   Medium

   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.

4. Question 4 · Trigonometry

   Medium

   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.

5. Question 5 · Sequences

   Hard

   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.

6. Question 6 · Logarithms

   Medium

   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.

7. Question 7 · Coordinate geometry

   Hard

   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.

8. Question 8 · Differentiation

   Medium

   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.

9. Question 9 · Combined functions

   Very Hard

   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.

10. Question 10 · Integration

    Medium

    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.

11. Question 11 · Exponentials

    Medium

    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.

12. Question 12 · Inequalities

    Hard

    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.

13. Question 13 · Functions and graphs

    Medium

    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.

14. Question 14 · Probability

    Hard

    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.

15. Question 15 · Algebraic identities

    Medium

    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.

16. Question 16 · Trigonometric identities

    Hard

    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.

17. Question 17 · Sequences and series

    Hard

    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.

18. Question 18 · Polynomials

    Medium

    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.

19. Question 19 · Logarithmic functions

    Hard

    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.

20. Question 20 · Calculus and optimisation

    Very Hard

    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.

Topic-level takeaways

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

Frequently asked questions

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