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Free Test of Mathematics for University Admission (TMUA) Practice Questions

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Sample Practice Questions

Try these 10 sample questions from the Test of Mathematics for University Admission (TMUA) practice bank. Each includes the answer and explanation. Continue with the interactive practice above to review the full question set.

  1. 1A positive real number t satisfies t^(3/2) = 8. What is t?

    1. A.2
    2. B.4Correct answer: B
    3. C.8
    4. D.64

    Explanation: The notes state that "we must interpret 𝑎𝑛 as the 𝑛th root of 𝑎" in their fractional-power discussion. Thus t^(3/2) = (sqrt(t))^3. Taking the cube root gives sqrt(t) = 2, so t = 4. The value 2 misses the final square; 8 ignores the exponent; 64 squares 8 without first taking its cube root.

    Reference: UAT-UK Notes on Mathematics — pp. 7–8, MM1.1, rational exponents

  2. 2A positive integer is selected when it is divisible by 3 or divisible by 5, using the mathematical meaning of 'or'. How does this rule apply to 15?

    1. A.Select it: the rule allows both conditions to hold.Correct answer: A
    2. B.Reject it: the rule requires exactly one condition to hold.
    3. C.Reject it: both conditions must instead be linked by 'and'.
    4. D.Select it: the rule requires one condition to be false.

    Explanation: The guide explicitly includes "or both are true" in the meaning of mathematical or. Since 15 is divisible by both 3 and 5, it is selected. B and C wrongly exclude the both-true case; D gives a justification that contradicts the two true divisibility conditions.

    Reference: UAT-UK Notes on Logic and Proof — p. 18, inclusive or

  3. 3Which expression is equal to 1/(sqrt(3) - 1)?

    1. A.(sqrt(3) minus 1) divided by 2
    2. B.(sqrt(3) plus 1) divided by 1
    3. C.(sqrt(3) plus 1) divided by 2Correct answer: C
    4. D.(sqrt(3) plus 1) divided by 4

    Explanation: The guide describes the equivalent rationalised expression as one "but which has a surd in the numerator [top] of the fraction and no surds in the denominator [bottom]." Multiply numerator and denominator by sqrt(3) + 1: the denominator becomes 3 - 1 = 2, so the value is (sqrt(3) + 1)/2. A uses the wrong conjugate sign; B omits division by 2; D incorrectly squares the resulting denominator.

    Reference: UAT-UK Notes on Mathematics — p. 13, rationalising denominators

  4. 4What is the converse of the statement: If a quadrilateral is a square, then it is a rectangle?

    1. A.If it is not a rectangle, then it is not a square.
    2. B.If it is a rectangle, then it is a square.Correct answer: B
    3. C.If it is not a square, then it is not a rectangle.
    4. D.It is a square if and only if it is a rectangle.

    Explanation: The guide pairs "if A then B and its converse if B then A". Interchanging the original hypothesis and conclusion gives: if a quadrilateral is a rectangle, then it is a square. Identifying the converse does not assert that it is true. A is the contrapositive; C is the inverse; D asserts both directions.

    Reference: UAT-UK Notes on Logic and Proof — p. 45, converses

  5. 5For which values of k does x^2 - 2kx + k + 2 = 0 have a repeated real root?

    1. A.k = -1 or k = 2Correct answer: A
    2. B.k = -2 or k = 1
    3. C.k = -1 or k = 1
    4. D.k = -2 or k = 2

    Explanation: The discriminant rule is "𝑏 2 − 4𝑎𝑐 = 0 quadratic has one repeated root [touches 𝑥-axis]". Here 4k^2 - 4(k + 2) = 0, so k^2 - k - 2 = (k - 2)(k + 1) = 0. Hence k = 2 or -1. B reverses the linear-term sign; C and D incorrectly impose symmetric roots for the parameter equation.

    Reference: UAT-UK Notes on Mathematics — p. 19, discriminant

  6. 6Which statement is the contrapositive of: If an integer n is divisible by 12, then n is divisible by 3?

    1. A.If n is divisible by 3, then n is divisible by 12.
    2. B.If n is not divisible by 12, then n is not divisible by 3.
    3. C.If n is divisible by 12, then n is not divisible by 3.
    4. D.If n is not divisible by 3, then n is not divisible by 12.Correct answer: D

    Explanation: The guide states that "the contrapositive of if A then B is if not B then not A". Negate the conclusion and use it as the new hypothesis, then negate the original hypothesis as the new conclusion. This gives D. A is the converse; B is the inverse; C negates only the conclusion without reversing the implication.

    Reference: UAT-UK Notes on Logic and Proof — p. 69, proof by contrapositive

  7. 7The sequence has u_1 = 2 and u_(n+1) = 3 - u_n for every positive integer n. What is u_2027?

    1. A.Negative two
    2. B.Three halves
    3. C.One
    4. D.Positive twoCorrect answer: D

    Explanation: The guide notes that "the number of terms you need to write out to spot any patterns that might emerge is dictated by the structure of the recurrence relationship." Here u_2 = 1 and u_3 = 2. Since 3 - (3 - u_n) = u_n, odd-indexed terms are 2 and even-indexed terms are 1. Thus u_2027 = 2. A discards the constant and alternates signs; B solves the fixed-point equation u = 3 - u instead of following this periodic sequence; C reverses the index parity.

    Reference: UAT-UK Notes on Mathematics — p. 48, MM2.1, recurrence patterns

  8. 8A conditional statement says: If P is true, then Q is true. What circumstances make this conditional false?

    1. A.P is false and Q is false.
    2. B.P is false and Q is true.
    3. C.P is true and Q is false.Correct answer: C
    4. D.P is true and Q is true.

    Explanation: The guide states that "the only way that if A then B can be false is if we can find an example of statement A that is true and an example of statement B that is false." Hence P must hold while Q fails. A and B have a false hypothesis and do not falsify the conditional; D satisfies the promised conclusion.

    Reference: UAT-UK Notes on Logic and Proof — p. 70, counterexamples to conditionals

  9. 9An arithmetic sequence begins 7, 12, 17. What is the sum of its first 20 terms?

    1. A.1,040
    2. B.1,090Correct answer: B
    3. C.1,140
    4. D.2,180

    Explanation: The sum formula is described as "𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑒𝑟𝑚𝑠 × 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡𝑒𝑟𝑚𝑠". The common difference is 5, so the twentieth term is 7 + 19 × 5 = 102. The average of the first and last terms is (7 + 102)/2 = 54.5. Thus the sum is 20 × 54.5 = 1,090. A and C use 18 or 20 increments instead of 19; D omits division by 2.

    Reference: UAT-UK Notes on Mathematics — p. 49, arithmetic series

  10. 10Two statements A and B satisfy A if and only if B. If A is false, what follows about B?

    1. A.B must be false.Correct answer: A
    2. B.B must be true.
    3. C.B may have either truth value.
    4. D.B cannot be assigned a truth value.

    Explanation: The guide states that in a necessary-and-sufficient relationship, "when A is false then B is false, and vice versa." A biconditional requires matching truth values, so B is false. B incorrectly requires the opposite truth value; C treats equivalence as only a one-way implication; D mistakes a false component for an undefined component.

    Reference: UAT-UK Notes on Logic and Proof — p. 52, necessary and sufficient equivalence

Exam Review

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

What is the exam 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.

How many questions are on the exam?

Total questions: 40.

How long is the exam?

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

What is the 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.

How much does the exam cost?

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.

Official sources and references

Confirm current exam arrangements with the exam provider.