OCR A-level Mathematics A H240 and Further Mathematics A H245: the route map

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Mathematics A H240

Checked against OCR’s own H240 specification, version 3 (October 2025), on 19 August 2026. That copy was taken from the board itself on the day of the check rather than from a landing page. The H240 document remains the controlling text. The reference codes below are OCR’s; the short labels beside them are InkMaths summaries rather than OCR’s statement text. Each row links to the relevant lessons and records full, partial or missing coverage.

This page is the content. For the examination itself, what each paper is worth and where OCR publishes them: OCR H240 past papers.

A-level Mathematics content is prescribed nationally. Use the table to check how each statement in this qualification maps to InkMaths. Every numbered statement is below, with the lesson that teaches it.

1.01 Proof · 1.02 Algebra and functions · 1.03 Coordinate geometry in the x-y plane · 1.04 Sequences and series · 1.05 Trigonometry · 1.06 Exponentials and logarithms · 1.07 Differentiation · 1.08 Integration · 1.09 Numerical methods · 1.10 Vectors · 2.01 Statistical sampling · 2.02 Data presentation and interpretation · 2.03 Probability · 2.04 Statistical distributions · 2.05 Statistical hypothesis testing · 3.01 Quantities and units in mechanics · 3.02 Kinematics · 3.03 Forces and Newton's laws · 3.04 Moments

1.01 Proof

4 of 4 covered in full.

Ref	Specification statement	Taught in	Coverage
1.01a	Structure of proof, deduction and exhaustion	The structure of proof: deduction and exhaustion	Covered
1.01b	Logical connectives	The structure of proof: deduction and exhaustion, Polynomials and the factor theorem, Logarithms and their laws	Covered
1.01c	Disproof by counter example	Disproof and proof by contradiction, Indices and surds, The general binomial expansion	Covered
1.01d	Proof by contradiction	Disproof and proof by contradiction	Covered

1.02 Algebra and functions

26 of 26 covered in full.

Ref	Specification statement	Taught in	Coverage
1.02a	Laws of indices for all rational exponents	Indices and surds	Covered
1.02b	Surds and rationalising the denominator	Indices and surds	Covered
1.02c	Simultaneous equations by elimination and substitution	Simultaneous equations and inequalities	Covered
1.02d	Quadratic functions, graphs and the discriminant	Quadratic functions	Covered
1.02e	Completing the square	Quadratic functions	Covered
1.02f	Quadratic equations, including in a function of the unknown	Quadratic functions, Indices and surds	Covered
1.02g	Linear and quadratic inequalities	Simultaneous equations and inequalities	Covered
1.02h	Solutions through 'and' and 'or', and set notation	Simultaneous equations and inequalities, Quadratic functions	Covered
1.02i	Representing inequalities graphically	Simultaneous equations and inequalities	Covered
1.02j	Manipulating polynomials and the factor theorem	Polynomials and the factor theorem	Covered
1.02k	Simplifying rational expressions	Polynomials and the factor theorem, Integrating rational functions	Covered
1.02l	The modulus function	Functions, inverses and the modulus	Covered
1.02m	Graphs of functions	Graphs, proportion and transformations, Quadratic functions, Tangents, turning points and curve behaviour	Covered
1.02n	Sketching curves defined by simple equations	Graphs, proportion and transformations, Tangents, turning points and curve behaviour	Covered
1.02o	Sketching reciprocal curves and their asymptotes	Graphs, proportion and transformations	Covered
1.02p	Interpreting algebraic solutions graphically	Simultaneous equations and inequalities, Quadratic functions, Graphs, proportion and transformations	Covered
1.02q	Intersection points of graphs to solve equations	Simultaneous equations and inequalities, Graphs, proportion and transformations, Trigonometric graphs and equations, Functions, inverses and the modulus	Covered
1.02r	Proportional relationships and their graphs	Graphs, proportion and transformations	Covered
1.02s	Sketching the modulus of a linear function	Functions, inverses and the modulus	Covered
1.02t	Solving modulus equations and inequalities graphically	Functions, inverses and the modulus	Covered
1.02u	The definition of a function	Functions, inverses and the modulus	Covered
1.02v	Inverse and composite functions	Functions, inverses and the modulus	Covered
1.02w	Single transformations of y = f(x)	Graphs, proportion and transformations	Covered
1.02x	Combinations of transformations	Graphs, proportion and transformations	Covered
1.02y	Partial fractions	Partial fractions, The general binomial expansion, Integrating rational functions	Covered
1.02z	Functions in modelling	Functions in modelling, Log graphs and exponential models, Trigonometric modelling	Covered

1.03 Coordinate geometry in the x-y plane

8 of 8 covered in full.

Ref	Specification statement	Taught in	Coverage
1.03a	Equation of a straight line	Straight lines, Circles	Covered
1.03b	Gradient conditions for parallel and perpendicular	Straight lines	Covered
1.03c	Straight line models in context	Straight lines	Covered
1.03d	Coordinate geometry of the circle	Circles	Covered
1.03e	Completing the square to find centre and radius	Circles	Covered
1.03f	Circle properties in coordinate problems	Circles	Covered
1.03g	Parametric equations and conversion	Parametric equations	Covered
1.03h	Parametric equations in modelling	Parametric equations, Implicit and parametric differentiation	Covered

1.04 Sequences and series

11 of 11 covered in full.

Ref	Specification statement	Taught in	Coverage
1.04a	Binomial expansion for positive integer n	The binomial expansion	Covered
1.04b	Link to binomial probabilities	The binomial expansion, The binomial distribution	Covered
1.04c	Binomial expansion for rational n	The general binomial expansion	Covered
1.04d	Validity of the expansion	The general binomial expansion	Covered
1.04e	Sequences by formula and by recurrence	Sequences and sigma notation	Covered
1.04f	Increasing, decreasing and periodic sequences	Sequences and sigma notation, Geometric series	Covered
1.04g	Sigma notation	Sequences and sigma notation, Arithmetic series	Covered
1.04h	Arithmetic sequences and series	Arithmetic series	Covered
1.04i	Geometric sequences and finite series	Geometric series	Covered
1.04j	Sum to infinity of a convergent geometric series	Geometric series	Covered
1.04k	Sequences and series in modelling	Arithmetic series, Geometric series, Logarithms and their laws	Covered

1.05 Trigonometry

17 of 17 covered in full.

Ref	Specification statement	Taught in	Coverage
1.05a	Sine, cosine and tangent for all arguments	Trigonometric graphs and equations	Covered
1.05b	The sine and cosine rules	Triangles and the sine and cosine rules, Vectors in two dimensions	Covered
1.05c	Area of a triangle	Triangles and the sine and cosine rules	Covered
1.05d	Radian measure, arc length and sector area	Radians, arcs and small angles	Covered
1.05e	Small angle approximations	Radians, arcs and small angles	Covered
1.05f	The trigonometric functions, their graphs and periodicity	Trigonometric graphs and equations	Covered
1.05g	Exact values in radians	Radians, arcs and small angles, Trigonometric graphs and equations, Reciprocal and inverse trigonometric functions	Covered
1.05h	Secant, cosecant, cotangent and the inverse functions	Reciprocal and inverse trigonometric functions	Covered
1.05i	Graphs, ranges and domains of those functions	Reciprocal and inverse trigonometric functions	Covered
1.05j	The quotient and Pythagorean identities	Trigonometric graphs and equations	Covered
1.05k	The secant and cosecant identities	Reciprocal and inverse trigonometric functions, Integrating standard functions	Covered
1.05l	Compound and double angle formulae	Compound angles and the harmonic form, Integrating standard functions, The product, quotient and chain rules	Covered
1.05m	Geometrical proofs of those formulae	Compound angles and the harmonic form	Covered
1.05n	The harmonic form	Compound angles and the harmonic form, Trigonometric modelling	Covered
1.05o	Solving trigonometric equations in an interval	Trigonometric graphs and equations, Reciprocal and inverse trigonometric functions, Compound angles and the harmonic form	Covered
1.05p	Constructing trigonometric proofs	Compound angles and the harmonic form, Reciprocal and inverse trigonometric functions	Covered
1.05q	Trigonometric functions in context	Trigonometric modelling, Vectors in two dimensions, Projectiles, Forces and Newton's laws, Friction and inclined planes, Statics of a particle	Covered

1.06 Exponentials and logarithms

9 of 9 covered in full.

Ref	Specification statement	Taught in	Coverage
1.06a	The functions a^x and e^x and their graphs	Exponential functions and e	Covered
1.06b	The gradient of e^kx and why the model suits	Exponential functions and e	Covered
1.06c	Logarithms as the inverse of a^x	Logarithms and their laws	Covered
1.06d	The natural logarithm and its graph	Logarithms and their laws	Covered
1.06e	The natural logarithm as the inverse of e^x	Logarithms and their laws, Solving differential equations	Covered
1.06f	The laws of logarithms	Logarithms and their laws	Covered
1.06g	Solving equations of the form a^x = b	Logarithms and their laws	Covered
1.06h	Logarithmic graphs to estimate parameters	Log graphs and exponential models, Correlation and regression	Covered
1.06i	Exponential growth and decay in modelling	Log graphs and exponential models, Exponential functions and e, Functions in modelling, Solving differential equations	Covered

1.07 Differentiation

20 of 20 covered in full.

Ref	Specification statement	Taught in	Coverage
1.07a	The derivative as the gradient of the tangent	The derivative from first principles	Covered
1.07b	The tangent gradient as a limit and as a rate of change	The derivative from first principles, Rates of change and building differential equations	Covered
1.07c	Sketching the gradient function	The derivative from first principles	Covered
1.07d	Second derivatives	The derivative from first principles	Covered
1.07e	The second derivative as the rate of change of gradient	The derivative from first principles, Tangents, turning points and curve behaviour	Covered
1.07f	Convex and concave sections, and points of inflection	Tangents, turning points and curve behaviour	Covered
1.07g	First principles for small integer powers	The derivative from first principles	Covered
1.07h	First principles for sin x and cos x	Differentiating trig, exponentials and logs	Covered
1.07i	Differentiating powers of x	Differentiating powers of x	Covered
1.07j	Differentiating exponentials	Differentiating trig, exponentials and logs	Covered
1.07k	Differentiating trigonometric functions	Differentiating trig, exponentials and logs	Covered
1.07l	The derivative of ln x	Differentiating trig, exponentials and logs, The product, quotient and chain rules	Covered
1.07m	Gradients, tangents and normals	Tangents, turning points and curve behaviour, Differentiating trig, exponentials and logs, Implicit and parametric differentiation	Covered
1.07n	Finding and classifying stationary points	Tangents, turning points and curve behaviour	Covered
1.07o	Increasing and decreasing functions	Tangents, turning points and curve behaviour	Covered
1.07p	Points of inflection	Tangents, turning points and curve behaviour, The derivative from first principles	Covered
1.07q	Product and quotient rules	The product, quotient and chain rules	Covered
1.07r	The chain rule, connected rates and inverse functions	The product, quotient and chain rules, Rates of change and building differential equations, Implicit and parametric differentiation	Covered
1.07s	Implicit and parametric differentiation	Implicit and parametric differentiation	Covered
1.07t	Constructing differential equations	Rates of change and building differential equations	Covered

1.08 Integration

12 of 12 covered in full.

Ref	Specification statement	Taught in	Coverage
1.08a	The fundamental theorem of calculus	Integration as antidifferentiation, Definite integrals and areas	Covered
1.08b	Integrating powers of x	Integration as antidifferentiation	Covered
1.08c	Integrating standard functions	Integrating standard functions	Covered
1.08d	Evaluating definite integrals	Definite integrals and areas	Covered
1.08e	Area between a curve and the x-axis	Definite integrals and areas	Covered
1.08f	Area between two curves	Areas, parametric curves and the limit of a sum, Definite integrals and areas	Covered
1.08g	Integration as the limit of a sum	Areas, parametric curves and the limit of a sum	Covered
1.08h	Integration by substitution	Integration by substitution and by parts	Covered
1.08i	Integration by parts	Integration by substitution and by parts	Covered
1.08j	Integrating with partial fractions	Integrating rational functions	Covered
1.08k	Separable first order differential equations	Solving differential equations	Covered
1.08l	Interpreting the solution of a differential equation	Solving differential equations	Covered

1.09 Numerical methods

7 of 7 covered in full.

Ref	Specification statement	Taught in	Coverage
1.09a	Locating roots by change of sign	Locating roots and iteration	Covered
1.09b	How change of sign methods fail	Locating roots and iteration	Covered
1.09c	Simple iterative methods, cobweb and staircase diagrams	Locating roots and iteration	Covered
1.09d	Newton-Raphson and other recurrence relations	The Newton-Raphson method, Locating roots and iteration	Covered
1.09e	How iterative methods fail	Locating roots and iteration, The Newton-Raphson method	Covered
1.09f	Numerical integration and bounds	The trapezium rule, Areas, parametric curves and the limit of a sum	Covered
1.09g	Numerical methods in context	The trapezium rule, The Newton-Raphson method, Locating roots and iteration	Covered

1.10 Vectors

8 of 8 covered in full.

Ref	Specification statement	Taught in	Coverage
1.10a	Vectors in two dimensions	Vectors in two dimensions	Covered
1.10b	Vectors in three dimensions	Vectors in three dimensions	Covered
1.10c	Magnitude, direction and conversion between forms	Vectors in two dimensions	Covered
1.10d	Vector addition and multiplication by scalars	Vectors in two dimensions	Covered
1.10e	Position vectors	Vectors in two dimensions, Vectors in three dimensions	Covered
1.10f	Distance between two points	Vectors in two dimensions, Vectors in three dimensions	Covered
1.10g	Vectors in pure problems and with forces	Vectors in two dimensions, Vectors in three dimensions, Forces and Newton's laws, Statics of a particle	Covered
1.10h	Vectors in kinematics	Kinematics with variable acceleration, Vectors in two dimensions	Covered

2.01 Statistical sampling

4 of 4 covered in full.

Ref	Specification statement	Taught in	Coverage
2.01a	Population and sample	Sampling and the large data set	Covered
2.01b	Informal inferences about the population	Sampling and the large data set, Hypothesis testing with the binomial	Covered
2.01c	Sampling techniques	Sampling and the large data set	Covered
2.01d	Selecting and critiquing sampling techniques	Sampling and the large data set	Covered

2.02 Data presentation and interpretation

10 of 10 covered in full.

Ref	Specification statement	Taught in	Coverage
2.02a	Tables and diagrams for single-variable data	Representing and interpreting data, Measures of location and spread	Covered
2.02b	Area in a histogram represents frequency	Representing and interpreting data, The normal distribution	Covered
2.02c	Scatter diagrams and regression lines	Correlation and regression	Covered
2.02d	Informal interpretation of correlation	Correlation and regression	Covered
2.02e	Correlation does not imply causation	Correlation and regression	Covered
2.02f	Measures of central tendency and variation	Measures of location and spread	Covered
2.02g	Calculating mean and standard deviation	Measures of location and spread	Covered
2.02h	Recognising and interpreting outliers	Representing and interpreting data	Covered
2.02i	Selecting and critiquing data presentation	Representing and interpreting data	Covered
2.02j	Cleaning data	Representing and interpreting data, Sampling and the large data set	Covered

2.03 Probability

5 of 5 covered in full.

Ref	Specification statement	Taught in	Coverage
2.03a	Mutually exclusive and independent events	Probability and Venn diagrams, The binomial distribution, The normal distribution	Covered
2.03b	Diagrams to assist probability calculations	Probability and Venn diagrams, Conditional probability	Covered
2.03c	Conditional probability with diagrams and tables	Conditional probability, Probability and Venn diagrams	Covered
2.03d	Conditional probability from first principles	Conditional probability	Covered
2.03e	Modelling with probability	Conditional probability, The binomial distribution	Covered

2.04 Statistical distributions

8 of 8 covered in full.

Ref	Specification statement	Taught in	Coverage
2.04a	Discrete probability distributions	The binomial distribution	Covered
2.04b	The binomial distribution as a model	The binomial distribution	Covered
2.04c	Calculating binomial probabilities	The binomial distribution	Covered
2.04d	Mean and variance for the normal approximation	The normal distribution	Covered
2.04e	The normal distribution as a model	The normal distribution	Covered
2.04f	Finding normal probabilities	The normal distribution	Covered
2.04g	Links to histograms, mean and standard deviation	The normal distribution	Covered
2.04h	Selecting an appropriate distribution	The normal distribution, The binomial distribution	Covered

2.05 Statistical hypothesis testing

7 of 7 covered in full.

Ref	Specification statement	Taught in	Coverage
2.05a	The language of hypothesis testing	Hypothesis testing with the binomial	Covered
2.05b	Testing a binomial proportion	Hypothesis testing with the binomial	Covered
2.05c	Inference, significance level and critical values	Hypothesis testing with the binomial	Covered
2.05d	The sample mean as a random variable	Hypothesis testing: correlation and the normal	Covered
2.05e	Testing a normal mean with known variance	Hypothesis testing: correlation and the normal	Covered
2.05f	Pearson's correlation coefficient as a measure	Correlation and regression	Covered
2.05g	Correlation coefficients in hypothesis tests	Hypothesis testing: correlation and the normal	Covered

3.01 Quantities and units in mechanics

3 of 3 covered in full.

Ref	Specification statement	Taught in	Coverage
3.01a	Fundamental quantities and units	Modelling, quantities and units	Covered
3.01b	Derived quantities and units	Modelling, quantities and units, Forces and Newton's laws	Covered
3.01c	The unit for moment	Moments	Covered

3.02 Kinematics

9 of 9 covered in full.

Ref	Specification statement	Taught in	Coverage
3.02a	The language of kinematics	Modelling, quantities and units, Kinematics with constant acceleration, Kinematics with variable acceleration	Covered
3.02b	Graphs in kinematics	Kinematics with constant acceleration	Covered
3.02c	Interpreting displacement-time and velocity-time graphs	Kinematics with constant acceleration	Covered
3.02d	The constant acceleration formulae	Kinematics with constant acceleration, Projectiles	Covered
3.02e	Constant acceleration in two dimensions using vectors	Kinematics with constant acceleration, Projectiles	Covered
3.02f	Calculus in kinematics in one dimension	Kinematics with variable acceleration	Covered
3.02g	Calculus in kinematics in two dimensions	Kinematics with variable acceleration	Covered
3.02h	Motion under gravity in a vertical plane using vectors	Projectiles, Kinematics with constant acceleration	Covered
3.02i	Projectiles	Projectiles, Parametric equations	Covered

3.03 Forces and Newton's laws

22 of 22 covered in full.

Ref	Specification statement	Taught in	Coverage
3.03a	The concept and vector nature of a force	Forces and Newton's laws	Covered
3.03b	Newton's first law	Forces and Newton's laws, Statics of a particle	Covered
3.03c	Newton's second law in a straight line	Forces and Newton's laws	Covered
3.03d	Newton's second law with forces as 2D vectors	Forces and Newton's laws	Covered
3.03e	Newton's second law with forces resolved	Forces and Newton's laws, Friction and inclined planes	Covered
3.03f	Weight and motion under gravity	Modelling, quantities and units, Kinematics with constant acceleration, Forces and Newton's laws	Covered
3.03g	Gravitational acceleration	Modelling, quantities and units, Kinematics with constant acceleration	Covered
3.03h	Newton's third law	Forces and Newton's laws, Connected particles and pulleys	Covered
3.03i	The normal reaction force	Forces and Newton's laws, Friction and inclined planes, Moments	Covered
3.03j	The smooth contact model	Modelling, quantities and units, Connected particles and pulleys, Friction and inclined planes	Covered
3.03k	Equilibrium with connected particles and pulleys	Connected particles and pulleys	Covered
3.03l	Newton's third law with forces resolved	Connected particles and pulleys, Friction and inclined planes, Statics of a particle	Covered
3.03m	Equilibrium as zero resolved parts	Statics of a particle, Friction and inclined planes	Covered
3.03n	Simple equilibrium in two dimensions using vectors	Forces and Newton's laws, Statics of a particle, Connected particles and pulleys	Covered
3.03o	Resolving forces in advanced pulley problems	Connected particles and pulleys, Friction and inclined planes, Statics of a particle	Covered
3.03p	Resultants and components of forces	Forces and Newton's laws, Statics of a particle, Projectiles	Covered
3.03q	Dynamics of motion in a plane	Forces and Newton's laws, Kinematics with variable acceleration	Covered
3.03r	The concept of a frictional force	Friction and inclined planes, Statics of a particle	Covered
3.03s	The contact force and its two components	Friction and inclined planes, Forces and Newton's laws	Covered
3.03t	The coefficient of friction and the friction model	Friction and inclined planes, Statics of a particle, Connected particles and pulleys	Covered
3.03u	Static equilibrium on a rough surface	Statics of a particle, Friction and inclined planes	Covered
3.03v	Motion of a body on a rough surface	Friction and inclined planes	Covered

3.04 Moments

3 of 3 covered in full.

Ref	Specification statement	Taught in	Coverage
3.04a	The moment of a force about a point	Moments	Covered
3.04b	Equilibrium of a rigid body	Moments	Covered
3.04c	Moments in simple static contexts	Moments	Covered

Further Mathematics A H245

Checked against OCR’s own H245 specification, version 2, October 2025, on 21 August 2026. That copy was taken from the board itself on the day of the check rather than from a landing page. The H245 document remains the controlling text. The reference codes below are OCR’s; the short labels beside them are InkMaths summaries rather than OCR’s statement text. Each row links to the relevant lessons and records full, partial or missing coverage.

This page is the content. For the examination itself, what each paper is worth and where OCR publishes them: OCR H245 past papers.

This qualification includes compulsory content and two of four optional applications. Use the section links to find the papers you take. Option content differs between awarding organisations, so treat every recorded gap as qualification-specific.

Pure Core: 4 Pure Core
Statistics: 5 Statistics
Mechanics: 6 Mechanics
Discrete Mathematics: 7 Discrete Mathematics
Additional Pure Mathematics: 8 Additional Pure Mathematics

4 Pure Core

81 of 81 covered in full.

Ref	Specification statement	Taught in	Coverage
4.01 Proof
4.01a	Mathematical induction	Proof by induction: sums and series, Induction: divisibility and matrices	Covered
4.01b	Mathematical induction, stage 2	Proof by induction: sums and series, Induction: divisibility and matrices, Leibniz's theorem and the Weierstrass substitution	Covered
4.02 Complex Numbers
4.02a	The language of complex numbers	Complex arithmetic and the Argand diagram, Modulus, argument and loci	Covered
4.02b	Cartesian and modulus-argument form	Complex arithmetic and the Argand diagram, Modulus, argument and loci	Covered
4.02c	Notation for a complex number	Complex arithmetic and the Argand diagram, Modulus, argument and loci, Further loci and regions in the Argand diagram, Transformations of the complex plane	Covered
4.02d	Exponential form	De Moivre's theorem and trigonometric identities, Roots of unity and complex roots	Covered
4.02e	Basic operations	Complex arithmetic and the Argand diagram, Modulus, argument and loci	Covered
4.02f	Converting between forms	Modulus, argument and loci	Covered
4.02g	Conjugate pairs of roots	Complex arithmetic and the Argand diagram, Roots of polynomials	Covered
4.02h	Square roots of a complex number	Complex arithmetic and the Argand diagram, Roots of unity and complex roots	Covered
4.02i	Quadratics with complex roots	Complex arithmetic and the Argand diagram	Covered
4.02j	Cubics and quartics	Roots of polynomials	Covered
4.02k	Argand diagrams	Complex arithmetic and the Argand diagram	Covered
4.02l	Geometry of conjugates and sums	Complex arithmetic and the Argand diagram	Covered
4.02m	Geometry of products and quotients	Modulus, argument and loci, De Moivre's theorem and trigonometric identities	Covered
4.02n	Euler's formula	De Moivre's theorem and trigonometric identities	Covered
4.02o	Loci and inequalities	Modulus, argument and loci, Further loci and regions in the Argand diagram	Covered
4.02p	Set notation for loci	Further loci and regions in the Argand diagram, Simultaneous equations and inequalities, Quadratic functions	Covered
4.02q	De Moivre's theorem	De Moivre's theorem and trigonometric identities	Covered
4.02r	nth roots	Roots of unity and complex roots	Covered
4.02s	Roots of unity in geometry	Roots of unity and complex roots	Covered
4.03 Matrices
4.03a	The language of matrices	Matrix algebra and transformations, Determinants and inverses	Covered
4.03b	Matrix arithmetic	Matrix algebra and transformations, Induction: divisibility and matrices	Covered
4.03c	Associativity and commutativity	Matrix algebra and transformations, Groups and their axioms	Covered
4.03d	Linear transformations in 2-D	Matrix algebra and transformations	Covered
4.03e	Successive transformations	Matrix algebra and transformations	Covered
4.03f	Single transformations in 3-D	Matrix algebra and transformations	Covered
4.03g	Invariant points and lines	Systems of equations and invariance	Covered
4.03h	Determinant of a 2 x 2 matrix	Determinants and inverses, Matrix algebra and transformations	Covered
4.03i	Determinant as an area scale factor	Determinants and inverses, Matrix algebra and transformations	Covered
4.03j	Determinant of a 3 x 3 matrix	Determinants and inverses	Covered
4.03k	Determinant as a volume scale factor	Determinants and inverses	Covered
4.03l	Singular and non-singular matrices	Determinants and inverses	Covered
4.03m	Determinant of a product	Determinants and inverses	Covered
4.03n	Inverse of a 2 x 2 matrix	Determinants and inverses	Covered
4.03o	Inverse of a 3 x 3 matrix	Determinants and inverses	Covered
4.03p	Properties of inverse matrices	Determinants and inverses, Matrix algebra and transformations	Covered
4.03q	Inverse matrices and inverse transformations	Determinants and inverses, Matrix algebra and transformations	Covered
4.03r	Solving by inverse matrix	Systems of equations and invariance	Covered
4.03s	Consistency when no unique solution exists	Systems of equations and invariance	Covered
4.03t	Three planes, geometrically	Systems of equations and invariance, Lines and planes in three dimensions	Covered
4.04 Further Vectors
4.04a	Equation of a straight line	Lines and planes in three dimensions, Straight lines, Vectors in two dimensions	Covered
4.04b	Equation of a plane	Lines and planes in three dimensions	Covered
4.04c	The scalar product	Vectors in three dimensions, Intersections, angles and distances, Lines and planes in three dimensions	Covered
4.04d	Angles with planes	Intersections, angles and distances	Covered
4.04e	Intersection of two lines	Lines and planes in three dimensions	Covered
4.04f	Intersection of a line and a plane	Intersections, angles and distances	Covered
4.04g	The vector product	The vector product and the scalar triple product	Covered
4.04h	Distances between lines	Intersections, angles and distances, The vector product and the scalar triple product	Covered
4.04i	Distance from a point to a line	Intersections, angles and distances, The vector product and the scalar triple product	Covered
4.04j	Distance from a point to a plane	Intersections, angles and distances	Covered
4.05 Further Algebra
4.05a	Roots and coefficients	Roots of polynomials	Covered
4.05b	Substitution to transform roots	Roots of polynomials	Covered
4.05c	Further partial fractions	Partial fractions, Calculus with inverse trigonometric functions, Polynomials and the factor theorem	Covered
4.06 Series
4.06a	Summation of series	Summing series and the method of differences, Proof by induction: sums and series, Sequences and sigma notation	Covered
4.06b	Method of differences	Summing series and the method of differences, Partial fractions, Geometric series	Covered
4.07 Hyperbolic Functions
4.07a	Definitions	Hyperbolic functions and identities	Covered
4.07b	Graphs	Hyperbolic functions and identities	Covered
4.07c	The fundamental identity	Hyperbolic functions and identities	Covered
4.07d	Differentiation and integration	Calculus with hyperbolic functions	Covered
4.07e	Inverse hyperbolic functions	Hyperbolic functions and identities	Covered
4.07f	Logarithmic forms	Hyperbolic functions and identities	Covered
4.08 Further Calculus
4.08a	Maclaurin series	Maclaurin series	Covered
4.08b	Standard series	Maclaurin series, The general binomial expansion	Covered
4.08c	Improper integrals	Mean values and improper integrals	Covered
4.08d	Volumes of revolution	Volumes of revolution	Covered
4.08e	Mean value of a function	Mean values and improper integrals	Covered
4.08f	Integration by partial fractions	Integrating rational functions, Calculus with inverse trigonometric functions	Covered
4.08g	Derivatives of inverse functions	Calculus with inverse trigonometric functions, Calculus with hyperbolic functions, Hyperbolic functions and identities	Covered
4.08h	Standard integrals and substitutions	Calculus with inverse trigonometric functions, Calculus with hyperbolic functions	Covered
4.09 Polar Coordinates
4.09a	Polar coordinates	Polar curves	Covered
4.09b	Sketching polar curves	Polar curves	Covered
4.09c	Area enclosed by a polar curve	Areas with polar coordinates	Covered
4.10 Differential Equations
4.10a	General and particular solutions	First order equations and integrating factors, Solving differential equations, Second order equations	Covered
4.10b	Modelling with differential equations	Modelling with differential equations, Rates of change and building differential equations, Newton's laws with a variable force, Kinematics with variable acceleration	Covered
4.10c	Integrating factor	First order equations and integrating factors	Covered
4.10d	Second order, homogeneous	Second order equations	Covered
4.10e	Second order, non-homogeneous	Second order equations	Covered
4.10f	Simple harmonic motion	Modelling with differential equations, Simple harmonic motion, Compound angles and the harmonic form	Covered
4.10g	Damped oscillations	Modelling with differential equations	Covered
4.10h	Coupled first order systems	Modelling with differential equations	Covered

5 Statistics (Y542, an optional paper)

51 of 51 covered in full.

Ref	Specification statement	Taught in	Coverage
5.01 Probability
5.01a	Permutations and combinations	Combinatorics, Probability and Venn diagrams, Conditional probability	Covered
5.01b	Selections and arrangements	Combinatorics	Covered
5.02 Discrete Random Variables
5.02a	Discrete probability distributions	The binomial distribution, Discrete random variables and expectation	Covered
5.02b	Expectation and variance	Discrete random variables and expectation	Covered
5.02c	Linear coding	Discrete random variables and expectation, Measures of location and spread	Covered
5.02d	Mean and variance of a binomial	The Poisson distribution, The binomial distribution, Discrete random variables and expectation	Covered
5.02e	Discrete uniform distribution	The binomial distribution, Discrete random variables and expectation, Goodness-of-fit tests	Covered
5.02f	Conditions for a geometric distribution	Geometric and negative binomial distributions	Covered
5.02g	Geometric probabilities	Geometric and negative binomial distributions	Covered
5.02h	Mean and variance of a geometric	Geometric and negative binomial distributions	Covered
5.02i	The Poisson model	The Poisson distribution	Covered
5.02j	The Poisson formula	The Poisson distribution	Covered
5.02k	Poisson probabilities	The Poisson distribution	Covered
5.02l	Conditions for a Poisson distribution	The Poisson distribution	Covered
5.02m	Mean and variance equal	The Poisson distribution, Discrete random variables and expectation	Covered
5.02n	Sums of Poisson variables	The Poisson distribution	Covered
5.03 Continuous Random Variables
5.03a	Continuous random variables	Continuous random variables: density and distribution functions, The continuous uniform distribution, The normal distribution, Mean, variance and skewness of continuous variables	Covered
5.03b	Using a probability density function	Continuous random variables: density and distribution functions	Covered
5.03c	Mean and variance by integration	Mean, variance and skewness of continuous variables	Covered
5.03d	Expectation of a function	Mean, variance and skewness of continuous variables, Discrete random variables and expectation	Covered
5.03e	Cumulative distribution function	Continuous random variables: density and distribution functions	Covered
5.03f	Median, quartiles and percentiles	Continuous random variables: density and distribution functions, Mean, variance and skewness of continuous variables	Covered
5.03g	Distributions of related variables	Continuous random variables: density and distribution functions, Mean, variance and skewness of continuous variables	Covered
5.04 Linear Combinations of Random Variables
5.04a	Expectation and variance of a combination	Combinations of normal random variables, Discrete random variables and expectation	Covered
5.04b	Combinations of normal variables	Combinations of normal random variables	Covered
5.05 Hypothesis Tests and Confidence Intervals
5.05a	The distribution of the sample mean	The Central Limit Theorem, Estimators, standard error and confidence intervals, Hypothesis testing: correlation and the normal	Covered
5.05b	Unbiased estimates	Estimators, standard error and confidence intervals	Covered
5.05c	Hypothesis test for a population mean	Hypothesis testing: correlation and the normal, The Central Limit Theorem, Comparing two normal means	Covered
5.05d	Confidence interval for a population mean	Estimators, standard error and confidence intervals, Comparing two normal means	Covered
5.06 Chi-squared Tests
5.06a	Contingency tables	Contingency tables, Goodness-of-fit tests	Covered
5.06b	Fitting a theoretical distribution	Goodness-of-fit tests	Covered
5.06c	Fitting other distributions	Goodness-of-fit tests	Covered
5.06d	Goodness of fit test	Goodness-of-fit tests	Covered
5.07 Non-parametric tests
5.07a	Non-parametric tests	Non-parametric tests, Testing a correlation coefficient, Type I and Type II errors, test size and power, Goodness-of-fit tests	Covered
5.07b	The basis of the tests	Non-parametric tests	Covered
5.07c	Single-sample tests for a median	Non-parametric tests	Covered
5.07d	Paired and two-sample tests	Non-parametric tests, Confidence intervals and tests with the t-distribution, Comparing two normal means	Covered
5.07e	Tests for medians or identity of population	Non-parametric tests	Covered
5.07f	Large-sample approximations	Non-parametric tests	Covered
5.08 Correlation
5.08a	The product-moment correlation coefficient	Correlation coefficients: product moment and Spearman, Correlation and regression	Covered
5.08b	Coding and correlation	Correlation coefficients: product moment and Spearman	Covered
5.08c	What the coefficient measures	Correlation coefficients: product moment and Spearman, Correlation and regression	Covered
5.08d	Testing the product-moment coefficient	Testing a correlation coefficient, Hypothesis testing: correlation and the normal, Hypothesis testing with the binomial	Covered
5.08e	Spearman's rank correlation coefficient	Correlation coefficients: product moment and Spearman	Covered
5.08f	Testing for association	Testing a correlation coefficient	Covered
5.08g	Choosing between the two coefficients	Correlation coefficients: product moment and Spearman, Testing a correlation coefficient, Correlation and regression	Covered
5.09 Linear Regression
5.09a	Independent and dependent variables	Least squares regression and residuals, Correlation and regression	Covered
5.09b	Least squares and regression lines	Least squares regression and residuals	Covered
5.09c	The regression line of y on x	Least squares regression and residuals, Correlation and regression	Covered
5.09d	Coding and the regression line	Least squares regression and residuals, Correlation coefficients: product moment and Spearman, Measures of location and spread	Covered
5.09e	Estimating from the regression line	Correlation and regression, Least squares regression and residuals	Covered

6 Mechanics (Y543, an optional paper)

42 of 42 covered in full.

Ref	Specification statement	Taught in	Coverage
6.01 Dimensional Analysis
6.01a	Dimensions of a quantity	Modelling, quantities and units	Covered
6.01b	Units and dimensions	Modelling, quantities and units	Covered
6.01c	Dimensional analysis as an error check	Modelling, quantities and units	Covered
6.01d	Finding unknown indices	Modelling, quantities and units	Covered
6.01e	Modelling by a dimensional argument	Modelling, quantities and units	Covered
6.02 Work, Energy and Power
6.02a	The concept of work	Work, energy and power	Covered
6.02b	Work done by a constant force	Work, energy and power, Statics of a particle, Friction and inclined planes	Covered
6.02c	Work in two dimensions and by a variable force	Work, energy and power, Newton's laws with a variable force, Vectors in three dimensions	Covered
6.02d	Mechanical energy	Work, energy and power	Covered
6.02e	Kinetic and potential energy	Work, energy and power	Covered
6.02f	Kinetic energy from the scalar product	Work, energy and power, Impulse and momentum as vectors, Vectors in three dimensions	Covered
6.02g	Hooke's law	Hooke's law and elastic strings	Covered
6.02h	Elastic potential energy	Elastic potential energy	Covered
6.02i	Conservation of mechanical energy	Work, energy and power	Covered
6.02j	Energy with elastic strings and springs	Elastic potential energy, Oscillations on strings and springs	Covered
6.02k	Power	Work, energy and power	Covered
6.02l	Power, tractive force and velocity	Work, energy and power, Friction and inclined planes	Covered
6.02m	Power from the scalar product	Work, energy and power, Vectors in three dimensions	Covered
6.03 Impulse and Momentum
6.03a	Linear momentum in one dimension	Momentum and impulse	Covered
6.03b	Conservation of momentum in one dimension	Momentum and impulse	Covered
6.03c	Momentum in two dimensions	Impulse and momentum as vectors	Covered
6.03d	Conservation of momentum in two dimensions	Impulse and momentum as vectors	Covered
6.03e	Impulse	Momentum and impulse	Covered
6.03f	Instantaneous impulse in one dimension	Momentum and impulse, Direct impact and Newton's law of restitution, Successive impacts and impacts with a wall	Covered
6.03g	Impulse and momentum in two dimensions	Impulse and momentum as vectors, Oblique impact and impact with a smooth surface	Covered
6.03h	Impulse of a constant or variable force	Momentum and impulse	Covered
6.03i	The coefficient of restitution	Direct impact and Newton's law of restitution	Covered
6.03j	Perfectly elastic and inelastic collisions	Direct impact and Newton's law of restitution	Covered
6.03k	Newton's experimental law in one dimension	Direct impact and Newton's law of restitution, Successive impacts and impacts with a wall	Covered
6.03l	Newton's experimental law in two dimensions	Oblique impact and impact with a smooth surface	Covered
6.04 Centre of Mass
6.04a	The centre of mass and gravity	Centre of mass of a discrete distribution, Modelling, quantities and units, Centres of mass of plane figures and frameworks	Covered
6.04b	Centre of mass by symmetry	Centres of mass of plane figures and frameworks, Centres of mass by integration	Covered
6.04c	Systems of particles and composite bodies	Centre of mass of a discrete distribution, Centres of mass of plane figures and frameworks	Covered
6.04d	Centre of mass by integration	Centres of mass by integration	Covered
6.04e	Equilibrium of a rigid body	Equilibrium, suspension, toppling and sliding, Moments, Centres of mass of plane figures and frameworks	Covered
6.05 Motion in a Circle
6.05a	Angular velocity and circular motion	Angular speed and horizontal circular motion	Covered
6.05b	Relations for constant speed	Angular speed and horizontal circular motion	Covered
6.05c	Motion in a horizontal circle	Angular speed and horizontal circular motion	Covered
6.05d	Variable speed by energy	Motion in a vertical circle	Covered
6.05e	Radial and tangential acceleration	Motion in a vertical circle	Covered
6.05f	Motion in a vertical circle	Motion in a vertical circle, Projectiles	Covered
6.06 Further Dynamics and Kinematics
6.06a	Linear motion under a variable force	Newton's laws with a variable force, Further kinematics: acceleration as a function of x, t or v, Solving differential equations, First order equations and integrating factors	Covered

7 Discrete Mathematics (Y544, an optional paper)

73 of 73 covered in full.

Ref	Specification statement	Taught in	Coverage
7.01 Mathematical Preliminaries
7.01a	Classifying problems	Algorithms, sorting and bin packing	Covered
7.01b	Sets and partitions	Combinatorics, Probability and Venn diagrams, Simultaneous equations and inequalities	Covered
7.01c	The pigeonhole principle	Combinatorics	Covered
7.01d	The multiplicative principle	Combinatorics	Covered
7.01e	Ordered subsets	Combinatorics	Covered
7.01f	Unordered subsets	Combinatorics, The binomial expansion	Covered
7.01g	Arrangements in a line	Combinatorics	Covered
7.01h	Arrangements of some of a group	Combinatorics	Covered
7.01i	Selections with constraints	Combinatorics	Covered
7.01j	Selections with several constraints	Combinatorics, Graphs: order, Eulerian paths and planarity	Covered
7.01k	Inclusion-exclusion for two sets	Probability and Venn diagrams, Combinatorics	Covered
7.01l	Inclusion-exclusion for more sets	Probability and Venn diagrams, Combinatorics	Covered
7.01m	Derangements	Combinatorics	Covered
7.02 Graphs and Networks
7.02a	Vertices, arcs and degree	Graphs: order, Eulerian paths and planarity, Shortest paths: Dijkstra and Floyd	Covered
7.02b	Trees and connectedness	Minimum spanning trees: Prim and Kruskal, Graphs: order, Eulerian paths and planarity	Covered
7.02c	Walks, trails, paths and cycles	Graphs: order, Eulerian paths and planarity, Route inspection, Shortest paths: Dijkstra and Floyd	Covered
7.02d	Complete graphs	Graphs: order, Eulerian paths and planarity, The travelling salesman problem	Covered
7.02e	Bipartite graphs	Graphs: order, Eulerian paths and planarity	Covered
7.02f	Colouring arguments	Graphs: order, Eulerian paths and planarity	Covered
7.02g	Eulerian and semi-Eulerian graphs	Graphs: order, Eulerian paths and planarity, Route inspection	Covered
7.02h	Hamiltonian paths and cycles	Graphs: order, Eulerian paths and planarity, The travelling salesman problem	Covered
7.02i	Ore's theorem	Graphs: order, Eulerian paths and planarity	Covered
7.02j	Isomorphic graphs	Graphs: order, Eulerian paths and planarity	Covered
7.02k	Digraphs	Graphs: order, Eulerian paths and planarity, Maximum flow and the labelling procedure, Critical path analysis, Flows in networks: cuts and capacity, Minimum spanning trees: Prim and Kruskal	Covered
7.02l	Planarity, subdivision and contraction	Graphs: order, Eulerian paths and planarity	Covered
7.02m	Euler's formula	Graphs: order, Eulerian paths and planarity	Covered
7.02n	Kuratowski's theorem	Graphs: order, Eulerian paths and planarity	Covered
7.02o	Thickness	Graphs: order, Eulerian paths and planarity	Covered
7.02p	Networks as weighted graphs	Minimum spanning trees: Prim and Kruskal, Shortest paths: Dijkstra and Floyd, Flows in networks: cuts and capacity, Critical path analysis	Covered
7.02q	Matrix representations	Minimum spanning trees: Prim and Kruskal, Shortest paths: Dijkstra and Floyd	Covered
7.03 Algorithms
7.03a	What an algorithm is	Algorithms, sorting and bin packing, Shortest paths: Dijkstra and Floyd, The travelling salesman problem	Covered
7.03b	Why algorithms	Algorithms, sorting and bin packing	Covered
7.03c	Tracing an algorithm	Algorithms, sorting and bin packing	Covered
7.03d	Order and run-time	Algorithms, sorting and bin packing, Summing series and the method of differences	Covered
7.03e	Comparing two algorithms	Algorithms, sorting and bin packing, The travelling salesman problem	Covered
7.03f	Worst case time complexity	Algorithms, sorting and bin packing	Covered
7.03g	Polynomial orders	Algorithms, sorting and bin packing	Covered
7.03h	Best, worst and typical cases	Algorithms, sorting and bin packing	Covered
7.03i	The hierarchy of orders	Algorithms, sorting and bin packing	Covered
7.03j	Bubble sort and shuttle sort	Algorithms, sorting and bin packing	Covered
7.03k	Quick sort	Algorithms, sorting and bin packing	Covered
7.03l	Bin packing	Algorithms, sorting and bin packing	Covered
7.03m	Further packing methods	Further packing and the knapsack problem	Covered
7.02 Graphs and Networks
7.02r	Modelling with graphs and networks	Critical path analysis, Route inspection, The travelling salesman problem, Flows in networks: cuts and capacity, Minimum spanning trees: Prim and Kruskal	Covered
7.04 Network Algorithms
7.04a	Dijkstra's algorithm	Shortest paths: Dijkstra and Floyd	Covered
7.04b	Prim's and Kruskal's algorithms	Minimum spanning trees: Prim and Kruskal	Covered
7.04c	Upper bounds for the travelling salesperson problem	The travelling salesman problem, Minimum spanning trees: Prim and Kruskal	Covered
7.04d	Lower bounds for the travelling salesperson problem	The travelling salesman problem	Covered
7.04e	Route inspection	Route inspection, Shortest paths: Dijkstra and Floyd	Covered
7.04f	Choosing an algorithm	Route inspection, The travelling salesman problem, Minimum spanning trees: Prim and Kruskal, Shortest paths: Dijkstra and Floyd, Float, Gantt charts and scheduling	Covered
7.05 Decision Making in Project Management
7.05a	Activity networks	Critical path analysis	Covered
7.05b	Forward and backward passes	Critical path analysis	Covered
7.05c	Total float	Float, Gantt charts and scheduling	Covered
7.05d	Independent and interfering float	Float, Gantt charts and scheduling, Critical path analysis	Covered
7.05e	Cascade charts and scheduling	Float, Gantt charts and scheduling	Covered
7.06 Graphical Linear Programming
7.06a	Formulating a linear programme	Linear programming: formulation and graphical solution	Covered
7.06b	Slack variables	Linear programming: formulation and graphical solution, The Simplex algorithm	Covered
7.06c	Investigating constraints and objectives	Linear programming: formulation and graphical solution	Covered
7.06d	Graphical solution	Linear programming: formulation and graphical solution	Covered
7.06e	Changing the coefficients	Linear programming: formulation and graphical solution	Covered
7.06f	Integer programming	Linear programming: formulation and graphical solution	Covered
7.07 The Simplex Algorithm
7.07a	The initial tableau	The Simplex algorithm, Linear programming: formulation and graphical solution	Covered
7.07b	Performing an iteration	The Simplex algorithm	Covered
7.07c	Interpreting the tableau	The Simplex algorithm	Covered
7.07d	Simplex terminology	The Simplex algorithm, Transportation problems and the north-west corner method	Covered
7.07e	What an iteration achieves	The Simplex algorithm, Linear programming: formulation and graphical solution	Covered
7.07f	Explaining the calculations	The Simplex algorithm	Covered
7.08 Game Theory
7.08a	Zero-sum games and pay-off matrices	Game theory: play safe and stable solutions	Covered
7.08b	Dominance	Game theory: play safe and stable solutions	Covered
7.08c	Play-safe strategies and stable solutions	Game theory: play safe and stable solutions	Covered
7.08d	Nash equilibrium	Game theory: play safe and stable solutions	Covered
7.08e	Optimal mixed strategies	Mixed strategies in game theory, Game theory: play safe and stable solutions	Covered
7.08f	Games as linear programmes	Mixed strategies in game theory, The Simplex algorithm	Covered

8 Additional Pure Mathematics (Y545, an optional paper)

51 of 51 covered in full.

Ref	Specification statement	Taught in	Coverage
8.01 Sequences and Series
8.01a	General sequences	Sequences and sigma notation, Recurrence relations	Covered
8.01b	Induction on sequences and series	Proof by induction: sums and series, Recurrence relations, Induction: divisibility and matrices	Covered
8.01c	Behaviour of sequences	Sequences and sigma notation, Recurrence relations	Covered
8.01d	Limits of sequences	Sequences and sigma notation, Geometric series, Recurrence relations, First order equations and integrating factors	Covered
8.01e	Fibonacci numbers	Recurrence relations	Covered
8.01f	First-order recurrence relations	Recurrence relations, Second order equations	Covered
8.01g	Second-order recurrence relations	Recurrence relations, Second order equations	Covered
8.01h	Modelling with recurrence relations	Recurrence relations	Covered
8.01i	Second-order modelling	Recurrence relations	Covered
8.02 Number Theory
8.02a	Number bases	The Euclidean algorithm and Bézout's identity	Covered
8.02b	Standard divisibility tests	Modular arithmetic and Fermat's little theorem	Covered
8.02c	Establishing new divisibility tests	Modular arithmetic and Fermat's little theorem	Covered
8.02d	The division theorem	The Euclidean algorithm and Bézout's identity	Covered
8.02e	Arithmetic modulo n	Modular arithmetic and Fermat's little theorem, Groups and their axioms	Covered
8.02f	Single linear congruences	Modular arithmetic and Fermat's little theorem, The Euclidean algorithm and Bézout's identity	Covered
8.02g	Quadratic residues	Modular arithmetic and Fermat's little theorem	Covered
8.02h	Simultaneous linear congruences	Modular arithmetic and Fermat's little theorem, The Euclidean algorithm and Bézout's identity	Covered
8.02i	Primes, factors and coprimality	The Euclidean algorithm and Bézout's identity, Disproof and proof by contradiction, Modular arithmetic and Fermat's little theorem	Covered
8.02j	Divisibility of linear combinations	The Euclidean algorithm and Bézout's identity	Covered
8.02k	Euclid's lemma	The Euclidean algorithm and Bézout's identity, Modular arithmetic and Fermat's little theorem	Covered
8.02l	Fermat's little theorem	Modular arithmetic and Fermat's little theorem	Covered
8.02m	Shorter cycles of powers	Modular arithmetic and Fermat's little theorem	Covered
8.02n	The order of an element modulo p	Modular arithmetic and Fermat's little theorem, Groups and their axioms	Covered
8.02o	The binomial theorem modulo p	The binomial expansion, Modular arithmetic and Fermat's little theorem	Covered
8.03 Groups
8.03a	Binary operations	Groups and their axioms	Covered
8.03b	Cayley tables	Groups and their axioms	Covered
8.03c	The definition of a group	Groups and their axioms	Covered
8.03d	The Latin square property	Groups and their axioms	Covered
8.03e	Order of a group and of an element	Groups and their axioms, Subgroups, Lagrange's theorem and isomorphism	Covered
8.03f	Subgroups	Subgroups, Lagrange's theorem and isomorphism	Covered
8.03g	Cyclic groups	Groups and their axioms	Covered
8.03h	Generators	Groups and their axioms, Subgroups, Lagrange's theorem and isomorphism	Covered
8.03i	Finite groups up to order seven	Subgroups, Lagrange's theorem and isomorphism, Groups and their axioms	Covered
8.03j	Larger and infinite groups	Subgroups, Lagrange's theorem and isomorphism, Groups and their axioms	Covered
8.03k	Lagrange's theorem	Subgroups, Lagrange's theorem and isomorphism	Covered
8.03l	Isomorphism	Subgroups, Lagrange's theorem and isomorphism	Covered
8.03m	Abstract group properties	Groups and their axioms, Subgroups, Lagrange's theorem and isomorphism	Covered
8.04 Further Vectors
8.04a	The vector product	The vector product and the scalar triple product	Covered
8.04b	Properties of the vector product	The vector product and the scalar triple product	Covered
8.04c	Areas from the vector product	The vector product and the scalar triple product	Covered
8.04d	A vanishing vector product	The vector product and the scalar triple product	Covered
8.04e	The scalar triple product	The vector product and the scalar triple product	Covered
8.05 Surfaces and Partial Differentiation
8.05a	Functions of two variables	Functions of two variables	Covered
8.05b	More variables and other functions	Functions of two variables	Covered
8.05c	Sections and contours	Functions of two variables	Covered
8.05d	Partial derivatives	Functions of two variables	Covered
8.05e	Stationary points of a surface	Functions of two variables	Covered
8.05f	The Hessian test	Functions of two variables	Covered
8.05g	Tangent planes	Functions of two variables	Covered
8.06 Further Calculus
8.06a	Reduction formulae	Reduction formulae	Covered
8.06b	Arc length and surface area	Arc length and surface area	Covered

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