Test of Mathematics for University Admission (TMUA) — Information & Free Practice

About Test of Mathematics for University Admission (TMUA)

Test of Mathematics for University Admission (TMUA) · GB

Key exam facts

Provider

UAT-UK owns the TMUA; UAT-UK is a joint venture of Imperial College London and the University of Cambridge. Pearson delivers it at professional test centres.

Format

Computer-based multiple-choice examination at Pearson test centres. Two papers taken consecutively: Applications of Mathematical Knowledge and Mathematical Reasoning. No calculator or formula booklet.

Exam questions

Total questions: 40.

Time limit

150 minutes of standard answer time: two separately timed 75-minute papers, taken one after the other. Approved access arrangements may alter individual timing.

Passing score

No pass/fail score or UAT-UK cut score. One overall TMUA score from 1.0 (low) to 9.0 (high), reported to one decimal place; universities determine how to use it.

Exam fee

GBP £78 when sitting in the UK or Republic of Ireland; GBP £133 elsewhere. An approved bursary for eligible UK candidates covers the full test cost and must be arranged before payment.

Topics covered

• Paper 1: Applications of Mathematical Knowledge (50%)

  • Section 1: mathematical knowledge (also required by Paper 2)
  • Part 1: AS pure mathematics content
  • Part 2: Higher Level GCSE content
  • MM1. Algebra and functions
  • MM1.1: Laws of indices for rational exponents
  • MM1.2: Surds and rationalising denominators
  • MM1.3: Quadratics, discriminant, completing the square and equations
  • MM1.4: Simultaneous linear and quadratic equations
  • MM1.5: Linear and quadratic inequalities
  • MM1.6: Polynomial manipulation, algebraic division, Factor and Remainder Theorems
  • MM1.7: Functions as mappings; square root and modulus
  • MM2. Sequences and series
  • MM2.1: Explicit and recurrent sequences
  • MM2.2: Arithmetic series and sums of natural numbers
  • MM2.3: Finite and convergent infinite geometric series
  • MM2.4: Positive integer binomial expansions; factorials and combinations
  • MM3. Coordinate geometry in the (x, y)-plane
  • MM3.1: Equations of lines; parallel and perpendicular gradients
  • MM3.2: Equations of circles
  • MM3.3: Circle properties: chords, tangents, arcs, semicircles, segments and cyclic quadrilaterals
  • MM4. Trigonometry
  • MM4.1: Sine and cosine rules; triangle area; ambiguous case; 2D and 3D
  • MM4.2: Radians; arc length, sectors and segments
  • MM4.3: Exact trigonometric values
  • MM4.4: Trigonometric graphs, symmetries and periodicity
  • MM4.5: Tangent quotient and Pythagorean identities
  • MM4.6: Trigonometric equations in specified intervals
  • MM5. Exponentials and logarithms
  • MM5.1: Exponential functions and graphs
  • MM5.2: Laws of logarithms; change of base excluded
  • MM5.3: Exponential equations including prior algebraic manipulation
  • MM6. Differentiation
  • MM6.1: Tangent gradient, rate of change, second derivatives; first principles excluded
  • MM6.2: Differentiation of rational powers and sums after simplification
  • MM6.3: Tangents, normals, maxima, minima and monotonicity; inflexion constraints
  • MM7. Integration
  • MM7.1: Definite integrals versus geometric area
  • MM7.2: Rational-power indefinite and definite integration, excluding power -1
  • MM7.3: Fundamental Theorem of Calculus
  • MM7.4: Combining integrals with equal or contiguous ranges
  • MM7.5: Trapezium approximation and over/underestimation
  • MM7.6: Differential equations of the form dy/dx = f(x)
  • MM8. Graphs of functions
  • MM8.1: Graphs of common functions in the specification
  • MM8.2: Graph transformations and composition
  • MM8.3: Effect of line parameters
  • MM8.4: Effect of quadratic parameters
  • MM8.5: Differentiation and graph shape, excluding stationary inflexions
  • MM8.6: Intercepts and possible numbers of real polynomial roots
  • MM8.7: Intersections and simultaneous equations
  • M1. Units
  • M1.1: Standard and compound units
  • M1.2: Conversions of standard and compound units
  • M2. Number
  • M2.1: Ordering integers, decimals and fractions; inequality symbols
  • M2.2: Four operations and place value
  • M2.3: Primes, factors, HCF, LCM and unique factorisation
  • M2.4: Inverse operations, cancellation and priority of operations
  • M2.5: Systematic listing and multiplication principle
  • M2.6: Squares, square roots, cubes and cube roots
  • M2.7: Integer, fractional and negative index laws
  • M2.8: Standard index form
  • M2.9: Terminating/recurring decimals, percentages and fractions
  • M2.10: Interchangeable fractions, decimals and percentages
  • M2.11: Exact calculations with fractions, surds and pi
  • M2.12: Upper and lower bounds
  • M2.13: Rounding and error intervals
  • M2.14: Approximation and estimation
  • M3. Ratio and proportion
  • M3.1: Scale factors, diagrams and maps
  • M3.2: Expressing one quantity as a fraction of another
  • M3.3: Ratio notation
  • M3.4: Division in a given ratio
  • M3.5: Ratio in conversions, scaling, mixing and concentration
  • M3.6: Proportion, ratios, fractions and linear functions
  • M3.7: Fractions in ratio problems
  • M3.8: Percentages, percentage changes, reverse percentages and simple interest
  • M3.9: Direct and inverse proportion, including powers
  • M3.10: Lengths, areas, volumes and similarity ratios
  • M3.11: Growth, decay, compound interest and iteration
  • M4. Algebra
  • M4.1: Algebraic notation and index laws
  • M4.2: Substitution into formulae and expressions
  • M4.3: Vocabulary of expressions, equations, formulae, identities and inequalities
  • M4.4: Expanding and collecting terms
  • M4.5: Quadratic factorisation and difference of squares
  • M4.6: Rational expressions and algebraic simplification
  • M4.7: Rearranging formulae
  • M4.8: Equations versus identities and equivalent expressions
  • M4.9: Coordinates in all quadrants
  • M4.10: Line gradients, intercepts and equations
  • M4.11: Quadratic roots, intercepts and turning points
  • M4.12: Linear, quadratic, cubic, reciprocal, exponential and trigonometric graphs
  • M4.13: Graph interpretation in real contexts
  • M4.14: Graph gradients, areas and contextual meaning
  • M4.15: Setting up and solving simultaneous equations
  • M4.16: Quadratic equations and the quadratic formula
  • M4.17: Linear inequalities and representations
  • M4.18: Term-to-term and position-to-term sequences
  • M4.19: Linear and quadratic nth terms
  • M5. Geometry
  • M5.1: Geometric language, polygons and symmetries
  • M5.2: Angle facts and polygon angle sums
  • M5.3: Properties of special quadrilaterals and triangles
  • M5.4: Triangle congruence criteria
  • M5.5: Applying angles, congruence, similarity and quadrilateral properties
  • M5.6: Transformations and invariance; translation vectors
  • M5.7: Pythagoras in 2D and 3D
  • M5.8: Circle terminology
  • M5.9: Circle theorems and related proofs
  • M5.10: Geometry on coordinate axes
  • M5.11: Solid geometry terminology
  • M5.12: Plans and elevations
  • M5.13: Maps, scales and bearings
  • M5.14: Areas of triangles, parallelograms, trapezia; volumes of prisms
  • M5.15: Circle/cylinder formulae and composite shapes; other solid formulae supplied
  • M5.16: Circle arcs and sectors
  • M5.17: Similarity: lengths, areas and volumes
  • M5.18: Trigonometric ratios and exact values in 2D and 3D
  • M5.19: Vector operations and geometric proofs
  • M6. Statistics
  • M6.1: Tables, charts and discrete/time-series representations
  • M6.2: Histograms, cumulative frequency and frequency density
  • M6.3: Averages, spread, grouped estimates and distribution comparisons
  • M6.4: Scatter graphs, correlation, best fit and extrapolation
  • M7. Probability
  • M7.1: Frequencies and frequency trees
  • M7.2: Randomness, fairness and expected outcomes
  • M7.3: Expected relative frequency and probability scale
  • M7.4: Exhaustive and mutually exclusive outcomes
  • M7.5: Systematic sets and combinations; no formal set notation required
  • M7.6: Equally likely sample spaces
  • M7.7: Addition, multiplication, conditional and dependent probabilities

• Paper 2: Mathematical Reasoning (50%)

  • Section 2: mathematical arguments, drawing on all Section 1 mathematical knowledge
  • The Logic of Arguments
  • Arg1: Mathematical logic, connectives, conditional statements, converse and contrapositive
  • Arg2: Necessary and sufficient conditions
  • Arg3: Universal and existential statements
  • Arg4: Negation of mathematical statements
  • Mathematical Proof
  • Prf1: Direct, cases, contradiction and counterexample proofs
  • Prf2: Deduction of implications
  • Prf3: Conjectures from small cases and their justification
  • Prf4: Ordering statements to form a proof
  • Prf5: Sophisticated chains of reasoning
  • Identifying Errors in Proofs
  • Err1: Identifying errors in purported proofs
  • Err2: Common invalid deductions and their missing conditions

How to prepare

Work through the topics, read the explanation after each answer, and revisit the areas where you miss questions. Use the official outline and reference materials below to guide your study.

These independent practice questions are not actual exam items and do not guarantee a passing result.

Frequently asked questions

Official sources and references

• TMUA Content Specification

  p. 1

• TMUA Content Specification

  p. 2, Structure of the Test

• TMUA Content Specification

  p. 2, Structure of the Test, equal weighting and incorrect answers

• About UAT-UK

  About UAT-UK

• UAT-UK TMUA overview

  Test Format

• UAT-UK Test Results

  Your Test Score

• UAT-UK TMUA overview

  How is the TMUA scored?

• UAT-UK TMUA overview

  Booking and Sitting the TMUA

• UAT-UK TMUA overview

  Test Format, both paper rows

• UAT-UK Candidate Handbook 2027 Entry

  p. 13, Access arrangements

• UAT-UK TMUA overview

  Candidate Support, Bursary Scheme

Confirm current exam arrangements with the exam provider.