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Two free courses: the logic of TMUA Paper 2, and the useful techniques that sit just outside a standard A-level course.
Logic for the TMUA
A self-contained course on the logic and proof topics tested in TMUA Paper 2. Work through the sections in order: each one ends with worksheets and full solutions, so you can practise and check your understanding as you go.
- 1
An introduction to P ⇒ Q: what it means for one statement to imply another, and why ‘deduced’ and ‘implied’ are two words for the same idea.
2 worksheets - 2
The many ways mathematicians express P ⇒ Q — if/then, only if, sufficient and necessary conditions — and why they all say the same thing.
2 worksheets - 3
How to negate mathematical statements — including ‘and’/‘or’ and quantifiers, simple and nested.
3 worksheets - 4
What counterexamples are, how they disprove general claims, and how to disprove if-then statements by finding examples where the if part is true and the then part is false, including statements with quantifiers and nested implications.
1 worksheet - 5
The converse of an implication and why it need not be true, and what happens when an implication and its converse both hold: equivalence, P if and only if Q, ‘if and only if’.
1 worksheet - 6
How to rewrite P ⇒ Q as not Q ⇒ not P, why this follows naturally from necessary conditions, and why the original implication and its contrapositive are equivalent.
1 worksheet - 7
How proof by contradiction works by assuming the opposite, reaching an impossible consequence, and connecting the method to the contrapositive.
1 worksheet - 8
Truth tables, why P ⇒ Q means ‘P and not-Q can’t both hold’, vacuous truth, and a few other equivalent faces of the conditional.
1 worksheet - 9
Three TMUA-style logic questions with full worked solutions, pulling together implication, necessary and sufficient conditions, negation, quantifiers and the contrapositive.
1 worksheet
Additional Topics
Topics I think are useful for TMUA preparation, because the ideas are unfamiliar, overlooked or not covered in much depth in the mainstream curriculum. They are not a full list of TMUA topics, and many are not high priority for the exam.
Each topic is self-contained, so you can work through them in any order.
- 1
Vertical and horizontal graph transformations, the order in which they combine, reflections in and , and a first-principles method for transforming the equation of any curve.
- 2
The modulus transformations and , a direct shortcut for mapping a point to its image without drawing intermediate graphs, and rotating a curve through 90° about any point.
- 3
A collection of useful numerical examples involving arithmetic and geometric sequences, recurrence relations, arithmetic–geometric sums and the method of differences.
- 4
Use the circle definition to establish the basic properties of sine, derive its general solutions, and apply the same reasoning to equations involving variable angles and cosine.
- 5
Learn how to factorise multivariate polynomials using the factor theorem, including homogeneous polynomials and hidden factors, with techniques highly relevant to TMUA.
- 6
An introduction to equivalence modulo an integer and its basic arithmetic properties, followed by five TMUA-style examples showing useful applications.
- 7
Learn the Cauchy-Schwarz and AM-GM inequalities, with simple proofs, general forms, and examples showing when they can provide useful shortcuts in TMUA problem solving.