Format & Structure
TMUA Paper 1
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- Time: 75 minutes
- Questions: 20 multiple-choice (options A-H)
- Topics: Pure maths only (no mechanics or statistics).
- Difficulty: High, questions test deep understanding and creative application of concepts.
- No calculators or formulae sheets allowed.
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Key Features
✔ No penalty for wrong answers (guess if unsure!).
✔ Questions range from quick tricks to multi-step proofs.
✔ Tests algebra, calculus, functions, and logical reasoning heavily.
TMUA Paper 2
- Duration: 75 minutes
- Questions: 20 multiple-choice (options A–H)
- Content: Advanced pure mathematics with an emphasis on proofs, logic, and abstract reasoning
- Difficulty: High—questions require deep analytical thinking and creative application of concepts
- Restrictions: No calculators or formula sheets allowed
Key Features
✔ No penalty for incorrect answers—candidates should attempt every question.
✔ Heavy focus on mathematical reasoning, including proofs, counterexamples, and logical conditions.
✔ Tests problem-solving versatility rather than rote memorization.
Core Topics to Master
TMUA Paper 1
Based on recent papers, focus on these high-frequency topics
Calculus
• Integration: Definite/indefinite integrals, area under curves.
• Differentiation: Stationary points, optimization, and rates of change.
• Numerical Methods: Approximation techniques like the Trapezium Rule.
Algebra & Functions
• Polynomials & Roots: Solving equations, analyzing graphs.
• Exponentials & Logarithms: Transformations and equation-solving.
• Inequalities: Finding solution ranges and critical values.
Trigonometry
• Identities & Equations: Solving complex trigonometric equations.
• Graphs: Understanding periodicity and transformations.
Geometry
• Circles & Symmetry: Calculating distances and areas.
• Triangles: Properties and coordinate geometry.
Sequences & Series
• Infinite Series: Testing convergence and evaluating sums.
• Binomial Expansion: Analyzing coefficients and divisibility.
Problem-Solving & Proof
• Logical Reasoning: Constructing and interpreting proofs.
• Differential Equations: Solving basic differential equations.
Core Topics to Master
TMUA Paper 2
Advanced Algebra & Functions
• Equations & Inequalities: Solving complex polynomial and exponential equations.
• Function Analysis: Exploring properties and transformations of functions.
• Logarithmic & Exponential Relationships: Understanding log and exponentials equations
Calculus & Integration
• Definite Integrals: Techniques for evaluating integrals, including substitution and symmetry.
• Differential Equations: Basic solving and interpretation of solutions.
Proof & Logical Reasoning
• Necessary vs. Sufficient Conditions: Identifying correct logical relationships.
• Counterexamples: Disproving mathematical claims.
• Contrapositive & Converse Statements: Analyzing implications in theorems.
Number Theory & Sequences
• Prime Numbers & Divisibility: Analyzing properties of primes in sequences.
• Recursive Sequences: Understanding patterns in recurrence relations.
• Base Conversions: Interpreting numbers in different bases (e.g., binary).
Geometry & Graphs
• Line Geometry: Conditions for parallel and perpendicular lines.
• Polynomial Roots: Criteria for distinct real roots in equations.
Abstract Problem-Solving
• Special Functions: Working with ceiling functions and their applications.
• Advanced Sequences: Analyzing behavior of complex recursive sequences.
Pro Tips for Success
Practice Past Papers
Familiarize yourself with the unique problem-solving style.
Master Time Management
Aim for ~3.5 minutes per question—skip and revisit tough ones.
Deep Conceptual Understanding
Many questions combine multiple topics, so focus on underlying principles rather than rote memorization.
Focus on Proofs & Logic
Practice constructing and deconstructing mathematical arguments.
Master Problem-Solving Techniques
Learn to approach unfamiliar problems methodically.
Example Problem Types:
Integration: Solve simultaneous integral equations.
Functions: Find specific values using exponential laws.
Geometry: Maximize distances between points on circles.
Example Skills Tested:
Logical Analysis: Determining whether a condition is necessary, sufficient, or both.
Integration Tricks: Simplifying complex integrals using algebraic manipulation.
Number Theory Insights: Proving properties of sequences or divisibility.