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- Mathematical statementsSuggested nextRead conditionals and quantifiers precisely, and form converses, contrapositives and negations.Shown in a check1 core lessonLesson progress hasn’t loaded yet
- Proofs and reasoning principlesSuggested nextWrite direct, contrapositive, contradiction, case, induction, existence and uniqueness proofs, and use the pigeonhole and symmetry principles and combinatorial arguments.Shown in a check5 core lessonsLesson progress hasn’t loaded yet
- Inequalities, limits, series and number theorySuggested nextUse the H3 additional results: classical inequalities, congruences, the method of differences, reduction formulae, improper integrals, limits, bijections and inclusion–exclusion.Shown in a check4 core lessonsLesson progress hasn’t loaded yet
- Problem solving and investigationSuggested nextUse problem-solving heuristics, investigate conjectures and read, complete or critique mathematical texts.Shown in a check1 core lessonLesson progress hasn’t loaded yet
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Reference
Course info
- GCE A-Level H3 Mathematics 2027
- 4 topics
Course guidance
Lessons for H3 Mathematics cover every topic in its syllabus. Practice questions for this course are not available yet.
Learning goals
- Identify the hypotheses, conclusions, quantifiers and connectives of a mathematical statement
- Form the converse, inverse and contrapositive and decide which are equivalent
- Negate statements with quantifiers and connectives
- Distinguish necessary, sufficient and equivalent conditions
- Construct a direct proof from definitions
- Disprove a universal statement with a verified counterexample
- Prove a statement by contradiction or by its contrapositive
- Prove existence, uniqueness or a construction
- Prove a statement by an exhaustive case analysis
- Prove statements by ordinary or strong mathematical induction
- Apply the pigeonhole principle
- Use symmetry to reduce or complete an argument
- Prove an identity or count by a combinatorial argument
- Sum a series by the method of differences
- Derive and use reduction formulae for integrals
- Decide whether an improper integral converges and evaluate it
- Count with bijections and the inclusion–exclusion principle
- Use and prove the AM–GM, Cauchy–Schwarz and triangle inequalities
- Compare growth rates and combine limits
- Reason with congruences and modular arithmetic
- Choose and use heuristics such as working backwards, simpler cases and restating the problem
- Formulate, test, extend and generalise conjectures
- Read a mathematical text and complete or critique a solution
Syllabus statements covered
- Include: Definition, Proposition and Theorem; Conditionals: if … then …, … if and only if …, Necessary, Sufficient; Quantifiers such as There exists (a unique) …, For all …; Logical connectives ‘and’, ‘or’, ‘not’, ‘implies’; Converse; Inverse; Contrapositive; Negation.
- Include: Direct proof; Disproof by counterexample; Proof by contradiction; Proof of existence; Proof of uniqueness; Proof by construction; Proof by cases; Proof by mathematical induction; Pigeonhole principle; Symmetry principle; Combinatorial arguments and proofs.
- Include: Functions and Graphs concepts from H2 Mathematics; Sequences and Series concepts from H2 Mathematics, with the following addition: Summation of series by the method of differences; Complex Numbers concepts from H2 Mathematics; Calculus concepts from H2 Mathematics, with the following addition: Reduction formulae, Improper integrals; Probability (including counting) concepts from H2 Mathematics, with the following addition: Bijection principle (include the case of distributing indistinguishable objects into distinguishable boxes), Inclusion-Exclusion principle; Additional inequalities: AM-GM inequality, Cauchy-Schwarz inequality, Triangle inequality; Introduction to limits: Comparing polynomial, exponential, and logarithmic growth rates, Operations involving limits e.g., limit of a sum is the sum of the limits, if exist; Concepts of congruence and modular arithmetic.
- Include: Working backwards; Uncovering pattern and structure; Solving a simpler/similar problem; Considering cases; Restating the problem (e.g., contrapositive).
- Include: Formulating a conjecture; Extension, generalisation, special cases; Complete or critique a solution.