Sets, relations and functions
- Representations of sets, subsets, universal sets, finite and infinite sets, and real intervals
- Set operations, their properties, Venn diagrams, ordered pairs and Cartesian products
- Types of relations and equivalence relations
- Types of functions, composition and inverses
- Operations and graphs of real functions, including polynomial, rational, modulus and signum functions
Counting and the binomial theorem
- Counting principles and factorial notation
- Deriving and applying permutation and combination formulae
- Binomial expansion for positive integral powers and Pascal's triangle
- General terms, middle terms and applications of binomial expansions
Complex numbers and equations
- Complex numbers as ordered pairs and in a + ib form
- Argand diagrams, polar form, modulus, conjugation and complex arithmetic
- Statement of the fundamental theorem of algebra
- Solving quadratics and connecting roots with coefficients
- Algebraic and graphical solutions of linear inequalities in one variable
Trigonometry
- Angle measures and conversion between units
- Trigonometric functions, identities and geometric meaning
- Trigonometric equations and applications
- Inverse trigonometric functions, their properties and graphs
Vectors
- Scalars, vectors, magnitudes, directions, direction cosines and ratios
- Position vectors, components, negative vectors and vector types
- Addition and scalar multiplication
- Vectors for internal and external division of a segment
- Scalar and vector products and projection onto a vector or line
Matrices and determinants
- Matrix notation, order, equality, types and transposes
- Addition, scalar multiplication, matrix products and their properties
- Elementary row and column operations
- Invertibility, uniqueness of an inverse and applications
- Determinants up to order three, minors, cofactors and determinant properties
- Triangle areas, adjoints, inverse matrices and applications
Coordinate geometry
- Cartesian coordinates and shifting the origin
- Line slopes, angles, equations in different forms and point-to-line distance
- Sections of a cone and the standard equations of circles, parabolas, ellipses and hyperbolas
- Basic conic properties and the general circle equation
- Degenerate conics: a point, a line and a pair of intersecting lines
Three-dimensional geometry
- Coordinate axes and planes, point distances and section formulae
- Line direction ratios and cosines, Cartesian and vector line equations
- Angles between lines and between planes
- Coplanar and skew lines and shortest distances
- Cartesian and vector equations of planes and point-to-plane distance
Sequences and series
- Arithmetic and geometric progressions
- Finite arithmetic sums and finite or infinite geometric sums
- Arithmetic and geometric means and the relation between them
- Sums of the first n natural numbers, their squares and cubes
Limits and continuity
- Limits and the algebra of limits
- Continuity and operations on continuous functions
- Limits and continuity of polynomial, rational, trigonometric, exponential and inverse functions
Differentiation and applications
- Derivative definition and tangent slope
- Sum, difference, product and quotient rules
- Derivatives of polynomial, trigonometric, inverse trigonometric, exponential and logarithmic functions
- Chain rule, implicit and parametric differentiation, logarithmic differentiation and second derivatives
- Rates of change, monotonicity, tangents, normals, maxima and minima
Integration and applications
- Antidifferentiation of algebraic, trigonometric, exponential and logarithmic functions
- Substitution, partial fractions, integration by parts and trigonometric identities
- Standard integral forms, the fundamental theorem of calculus and definite-integral properties
- Areas under curves and between two curves
Differential equations
- Definition, order, degree and formation of differential equations
- Separation of variables
- First-order homogeneous equations
- First-order linear equations of the form dy/dx + Py = Q, with P and Q depending on x
- The corresponding linear form dx/dy + Px = Q, with coefficients depending on y
Statistics and probability
- Dispersion, mean deviation, variance and standard deviation for grouped and ungrouped data
- Random experiments, outcomes, sample spaces and events
- Complementary, combined, exhaustive and mutually exclusive events
- Set-based probability axioms and probabilities of unions, intersections and complements
- Multiplication rule, conditional probability and independent events
- Total probability and Bayes' theorem
Linear programming: the 2026 document includes it in the final unit heading but supplies no separate subtopic list. Treat graphing a feasible region and optimising a linear objective as useful preparation, rather than an invented detailed syllabus requirement. A linear-programming question appears in our 2023 paper.
Put the topics into practice
For each topic, explain the concept, solve a question without notes, then return to it after a gap. Older papers may include material outside the 2026 scope.
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