Syllabus
What TMUA covers
Algebra and Functions
- Quadratics, the discriminant and completing the square
- Polynomials, the factor theorem and simultaneous equations
- Linear and quadratic inequalities
- Graphs of functions and their transformations
Sequences, Geometry and Trigonometry
- Arithmetic and geometric sequences and series
- Binomial expansion for positive integer powers
- Coordinate geometry of lines and circles
- Trigonometric graphs, identities and equations
Calculus, Exponentials and Logarithms
- Exponential functions and the laws of logarithms
- Differentiation, tangents, normals and stationary points
- Integration and areas under curves
- Using calculus to reason about the shape of a graph
Logic and Proof
- Implication, converse, contrapositive and negation
- Necessary and sufficient conditions
- Direct proof, proof by contradiction and counterexamples
- Identifying the error in a flawed argument
Study Notes
TMUA notes from our tutors
TMUA
16 sections · 18 notes
- Algebraic Expressions and Identities
- Rational Expressions and Restrictions
- Linear Equations and Formula Rearrangement
- Quadratic Equations and Completing the Square
- Discriminants and Parameter Conditions
- Quadratic Graphs and Intersections
- Modulus Equations and Inequalities · Locked, notes not yet available
- Quadratics and Inequalities Review · Locked, notes not yet available
- Function Notation Domain and Range · Locked, notes not yet available
- Composite Functions · Locked, notes not yet available
- Inverse Functions · Locked, notes not yet available
- Graph Transformations · Locked, notes not yet available
- Interpreting and Sketching Graphs · Locked, notes not yet available
- Functions and Graph Transformations Review · Locked, notes not yet available
- Arithmetic Sequences and Series · Locked, notes not yet available
- Geometric Sequences and Series · Locked, notes not yet available
- Recurrence Relations · Locked, notes not yet available
- Binomial Expansion · Locked, notes not yet available
- Sequences and Binomial Expansion Review · Locked, notes not yet available
- Exponential Functions · Locked, notes not yet available
- Laws of Logarithms · Locked, notes not yet available
- Exponential and Logarithmic Equations · Locked, notes not yet available
- Exponential Models · Locked, notes not yet available
- Exponentials and Logarithms Review · Locked, notes not yet available
- Coordinates Distance and Midpoint · Locked, notes not yet available
- Straight Lines and Gradients · Locked, notes not yet available
- Parallel and Perpendicular Lines · Locked, notes not yet available
- Equations of Circles · Locked, notes not yet available
- Line and Circle Intersections · Locked, notes not yet available
- Coordinate Geometry Review · Locked, notes not yet available
- Angles Polygons and Geometric Reasoning · Locked, notes not yet available
- Congruence and Similarity · Locked, notes not yet available
- Circle Theorems · Locked, notes not yet available
- Perimeter Area and Volume · Locked, notes not yet available
- Bearings and Spatial Problems · Locked, notes not yet available
- Vectors and Geometric Proofs · Locked, notes not yet available
- Geometry Measures and Vectors Review · Locked, notes not yet available
- Right Triangle Trigonometry · Locked, notes not yet available
- Radians Arcs and Sectors · Locked, notes not yet available
- Trigonometric Functions and Graphs · Locked, notes not yet available
- Trigonometric Identities and Equations · Locked, notes not yet available
- Sine and Cosine Rules · Locked, notes not yet available
- Trigonometry Review · Locked, notes not yet available
- Gradients and Rates of Change · Locked, notes not yet available
- Differentiating Powers and Sums · Locked, notes not yet available
- Tangents and Normals · Locked, notes not yet available
- Stationary Points and Curve Sketching · Locked, notes not yet available
- Optimisation and Motion · Locked, notes not yet available
- Differentiation and Applications Review · Locked, notes not yet available
- Integration as Reverse Differentiation · Locked, notes not yet available
- Constants of Integration and Initial Conditions · Locked, notes not yet available
- Definite Integrals and Signed Area · Locked, notes not yet available
- Areas Between Curves · Locked, notes not yet available
- Trapezium Estimates · Locked, notes not yet available
- Integration and Applications Review · Locked, notes not yet available
- Data Types and Statistical Diagrams · Locked, notes not yet available
- Averages and Measures of Spread · Locked, notes not yet available
- Histograms and Frequency Density · Locked, notes not yet available
- Cumulative Frequency and Box Plots · Locked, notes not yet available
- Scatter Diagrams and Data Interpretation · Locked, notes not yet available
- Statistics and Data Interpretation Review · Locked, notes not yet available
- Sample Spaces and Counting · Locked, notes not yet available
- Probability Rules and Combined Events · Locked, notes not yet available
- Tree Diagrams and Independent Events · Locked, notes not yet available
- Conditional Probability · Locked, notes not yet available
- Probability and Counting Review · Locked, notes not yet available
- Mathematical Statements and Negations · Locked, notes not yet available
- Implication and Equivalence · Locked, notes not yet available
- Necessary and Sufficient Conditions · Locked, notes not yet available
- Quantifiers and Logical Reasoning · Locked, notes not yet available
- Logic and Mathematical Language Review · Locked, notes not yet available
- Direct Proof and Deduction · Locked, notes not yet available
- Proof by Exhaustion and Cases · Locked, notes not yet available
- Proof by Contradiction · Locked, notes not yet available
- Disproof by Counterexample · Locked, notes not yet available
- Identifying Errors in Arguments · Locked, notes not yet available
- Proof and Errors in Reasoning Review · Locked, notes not yet available
- Mixed Algebra and Functions · Locked, notes not yet available
- Mixed Geometry and Trigonometry · Locked, notes not yet available
- Mixed Calculus and Graphs · Locked, notes not yet available
- Mixed Data and Probability · Locked, notes not yet available
- Mathematical Reasoning Review · Locked, notes not yet available
- Full Course Review · Locked, notes not yet available
Exam Pattern
How TMUA is assessed
Paper 1: Applications of Mathematical Knowledge
Multiple choice. Tests fluent use of GCSE and AS-level pure maths in unfamiliar problems, without a calculator.
Paper 2: Mathematical Reasoning
Multiple choice. Assesses logic, proof and spotting errors in arguments, built on the same maths knowledge as Paper 1.
Scoring
No negative marking. Results are reported on a 1.0 to 9.0 scale. Required scores and accepted sittings vary by university and course, so check each offer carefully.
Required Materials
What you'll need
Official TMUA specification and past papers
The published content specification plus official past papers and worked solutions are the most realistic practice available.
AS-level pure maths textbook
Paper 1 draws on GCSE and AS-level pure content, so a solid AS textbook covers the knowledge base.
Official guide to logic and proof
The official notes on logic and proof explain the reasoning skills Paper 2 assesses and are essential reading.
Try It Yourself
Sample questions, solved step by step
Scroll into view to watch each solution build itself, one step at a time, exactly how our tutors walk students through it.
Sample question
Find all real values of k for which the equation x² + (k − 3)x + k = 0 has no real roots.
Animated solution
State the condition
No real roots means the discriminant is negative: b² − 4ac < 0
Substitute
(k − 3)² − 4k < 0, so k² − 6k + 9 − 4k < 0
Simplify and factorise
k² − 10k + 9 < 0, so (k − 1)(k − 9) < 0
Solve the inequality
A positive quadratic is negative between its roots, so 1 < k < 9
Answer: 1 < k < 9
Sample question
Find a counterexample to the claim: for every positive integer n, n² + n + 41 is prime.
Animated solution
Rewrite the expression
n² + n + 41 = n(n + 1) + 41
Look for a shared factor
If n(n + 1) is a multiple of 41, the whole expression is a multiple of 41
Choose n = 40
40 × 41 + 41 = 41 × (40 + 1) = 41 × 41
Conclude
1681 = 41², which is not prime, so the claim is false
Answer: n = 40 gives 1681 = 41², which is not prime