Roadmap
Core Pure Mathematics
Complex numbers
CP 2.2, 2.3 - Complex number arithmetic and the conjugate
CP 2.1, 2.3 - Complex roots of polynomial equations
CP 2.4 - Argand diagrams
CP 2.5 - Modulus-argument form
CP 2.6 - Multiplying and dividing in modulus-argument form
CP 2.7 - Loci in the Argand diagram
CP 2.7 - Regions in the Argand diagram
CP 2.8 - De Moivre's theorem
CP 2.8 - Multiple-angle formulae
CP 2.8 - Powers of sine and cosine
CP 2.8 - Sums of series with de Moivre
CP 2.9 - Exponential form of a complex number
CP 2.10 - nth roots of a complex number
CP 2.11 - Roots of unity in geometric problems
Matrices
CP 3.1, 3.2 - Matrix arithmetic, zero and identity matrices
CP 3.3 - Matrices for 2-D transformations
CP 3.3 - Successive and 3-D transformations
CP 3.4 - Invariant points and lines
CP 3.5 - Determinants and scale factors
CP 3.6 - Singular and inverse matrices
CP 3.7 - Solving three equations with matrices
CP 3.8 - How three planes intersect
Proof
CP 1.1 - Induction for sums of series
CP 1.1 - Induction for divisibility
CP 1.1 - Induction for powers of matrices
Further algebra and functions
CP 4.1 - Roots and coefficients of polynomials
CP 4.2 - Equations with transformed roots
CP 4.3 - Sums of integers, squares and cubes
CP 4.4 - The method of differences
CP 4.5 - Maclaurin series and the general term
CP 4.6 - Standard and compound Maclaurin series
Further calculus
CP 5.1 - Volumes of revolution from Cartesian equations
CP 5.1 - Volumes of revolution from parametric equations
CP 5.2 - Improper integrals
CP 5.3 - Mean value of a function
CP 5.5 - Differentiating inverse trigonometric functions
CP 5.6 - Integrals giving inverse trigonometric functions
CP 5.4 - Partial fractions with quadratic factors
Further vectors
CP 6.1 - Equations of lines in 3-D
CP 6.2 - Equations of a plane
CP 6.3, 6.4 - The scalar product and normal vectors
CP 6.3 - Angles between lines and planes
CP 6.5 - Where a line meets a plane
CP 6.5 - Perpendicular distances to lines and planes
Polar coordinates
CP 7.1 - Converting between polar and Cartesian
CP 7.2 - Sketching polar curves
CP 7.3 - Areas enclosed by polar curves
CP 7.3 - Tangents to polar curves
Hyperbolic functions
CP 8.1 - The functions sinh, cosh and tanh
Locked
CP 8.2 - Hyperbolic derivatives and integrals
Locked
CP 8.3, 8.4 - Inverse hyperbolic functions as logarithms
Locked
CP 8.5 - Integrals giving inverse hyperbolic functions
Locked
Differential equations
CP 9.1 - The integrating factor method
Locked
CP 9.2 - General and particular solutions
Locked
CP 9.3 - Modelling with differential equations
Locked
CP 9.4, 9.6 - Second-order homogeneous equations
Locked
CP 9.5 - Complementary functions and particular integrals
Locked
CP 9.7 - Simple harmonic motion
Locked
CP 9.8 - Damped and forced oscillations
Locked
CP 9.9 - Coupled first-order equations
Locked
Further Pure Mathematics 1
Further trigonometry
FP1 1.1 - The t-formulae
Locked
FP1 1.2, 1.3 - Identities and equations with t-formulae
Locked
Further calculus
FP1 2.1 - Taylor series
Locked
FP1 2.2 - Limits using series expansions
Locked
FP1 2.3 - Leibniz's theorem
Locked
FP1 2.4 - L'Hospital's rule
Locked
FP1 2.5 - The Weierstrass substitution
Locked
Further differential equations
FP1 3.1 - Series solutions of differential equations
Locked
FP1 3.2 - Solving differential equations by substitution
Locked
Coordinate systems
FP1 4.1 - Parabolas, ellipses and hyperbolas
Locked
FP1 4.2 - Foci, directrices and eccentricity
Locked
FP1 4.3 - Tangents and normals to conics
Locked
FP1 4.4 - Loci problems with conics
Locked
Further vectors
FP1 5.1 - The vector product
Locked
FP1 5.2 - Scalar triple product and volumes
Locked
FP1 5.3 - Points, lines and planes in 3-D
Locked
Further numerical methods
FP1 6.1 - Numerical solution of differential equations
Locked
FP1 6.2 - Simpson's rule
Locked
Inequalities
FP1 7.1 - Algebraic and modulus inequalities
Locked
Further Pure Mathematics 2
Number theory
FP2 5.1, 5.2 - The Euclidean algorithm and Bezout's identity
Locked
FP2 5.3 - Modular arithmetic and congruences
Locked
FP2 5.4 - Fermat's little theorem
Locked
FP2 5.5 - Divisibility tests
Locked
FP2 5.6 - Solving congruence equations
Locked
FP2 5.7 - Counting principles and subsets
Locked
FP2 5.7 - Permutations and combinations
Locked
Groups
FP2 1.1 - Binary operations and group axioms
Locked
FP2 1.2 - Cayley tables and cyclic groups
Locked
FP2 1.3 - Orders of elements and subgroups
Locked
FP2 1.4 - Lagrange's theorem
Locked
FP2 1.5 - Isomorphic groups
Locked
Further calculus
FP2 2.1 - Reduction formulae
Locked
FP2 2.2 - Arc length and surface area
Locked
Further matrix algebra
FP2 3.1 - Eigenvalues and eigenvectors
Locked
FP2 3.2 - Diagonalising matrices
Locked
FP2 3.3 - The Cayley-Hamilton theorem
Locked
Further complex numbers
FP2 4.1 - Further loci and regions
Locked
FP2 4.2 - Transformations from z-plane to w-plane
Locked
Further sequences and series
FP2 6.1, 6.2 - First-order recurrence relations
Locked
FP2 6.1, 6.2 - Second-order recurrence relations
Locked
FP2 6.3 - Proving closed forms by induction
Locked
Further Statistics 1
Discrete probability distributions
FS1 1.1 - Mean and variance of discrete distributions
Locked
Poisson and binomial distributions
FS1 2.1 - The Poisson distribution
Locked
FS1 2.1 - Adding independent Poisson variables
Locked
FS1 2.2, 2.3 - Means, variances and the Poisson approximation
Locked
Geometric and negative binomial distributions
FS1 3.1 - Geometric and negative binomial probabilities
Locked
FS1 3.2, 3.3 - Means and variances of both distributions
Locked
Hypothesis testing
FS1 4.1 - Testing a Poisson mean
Locked
FS1 4.2 - Testing a geometric parameter
Locked
Central Limit Theorem
FS1 5.1 - Sample means from other distributions
Locked
Chi-squared tests
FS1 6.1 - Goodness of fit tests
Locked
FS1 6.1 - Contingency table tests
Locked
Probability generating functions
FS1 7.1 - Generating functions of standard distributions
Locked
FS1 7.2 - Mean and variance from generating functions
Locked
FS1 7.3 - Generating functions of sums
Locked
Quality of tests
FS1 8.1 - Type I and Type II errors
Locked
FS1 8.1 - Size and power of a test
Locked
Further Statistics 2
Linear regression
FS2 1.1 - Least squares regression lines
Locked
FS2 1.2 - Residuals and residual sum of squares
Locked
Continuous probability distributions
FS2 2.1, 2.2 - Density and cumulative distribution functions
Locked
FS2 2.3 - Mean and variance of continuous variables
Locked
FS2 2.3 - Mode, median, percentiles and skewness
Locked
FS2 2.4 - The continuous uniform distribution
Locked
Correlation
FS2 3.1 - Product moment correlation coefficient
Locked
FS2 3.2 - Spearman's rank correlation
Locked
FS2 3.3 - Testing for zero correlation
Locked
Combinations of random variables
FS2 4.1 - Linear combinations of normal variables
Locked
Estimation and tests using a normal distribution
FS2 5.1 - Estimators, bias and standard error
Locked
FS2 5.2, 5.3 - Confidence intervals for a normal mean
Locked
FS2 5.4, 5.5 - Testing the difference between two means
Locked
Tests and confidence intervals for variances
FS2 6.1 - Testing and estimating a normal variance
Locked
FS2 6.2 - The F-test for two variances
Locked
Confidence intervals and tests using the t-distribution
FS2 7.1 - A normal mean with unknown variance
Locked
FS2 7.2 - The paired t-test
Locked
FS2 7.3 - Two-sample t-tests with pooled variance
Locked
Further Mechanics 1
Momentum and impulse
FM1 1.1 - Impulse and conservation of momentum
Locked
FM1 1.2 - Momentum as a vector
Locked
Work, energy and power
FM1 2.1 - Work-energy principle and energy conservation
Locked
FM1 2.1 - Power and driving force
Locked
Elastic strings, springs and elastic energy
FM1 3.1 - Hooke's law for strings and springs
Locked
FM1 3.2 - Elastic potential energy
Locked
Elastic collisions in one dimension
FM1 4.1 - Newton's law of restitution
Locked
FM1 4.2 - Successive direct impacts
Locked
Elastic collisions in two dimensions
FM1 5.1 - Oblique impact with a fixed surface
Locked
FM1 5.1 - Oblique impact of two spheres
Locked
FM1 5.2 - Successive oblique impacts
Locked
Further Mechanics 2
Motion in a circle
FM2 1.1 - Angular speed and horizontal circles
Locked
FM2 1.1 - Conical pendulums and banked surfaces
Locked
FM2 1.2 - Motion in a vertical circle
Locked
Centres of mass of plane figures
FM2 2.1 - Centre of mass of particles
Locked
FM2 2.2 - Composite laminas and frameworks
Locked
FM2 2.2 - Equilibrium of laminas and frameworks
Locked
Further centres of mass
FM2 3.1 - Centres of mass by integration
Locked
FM2 3.1 - Composite rigid bodies
Locked
FM2 3.2 - Equilibrium of rigid bodies
Locked
FM2 3.3 - Toppling and sliding
Locked
Further kinematics
FM2 5.1 - Variable acceleration in a straight line
Locked
Further dynamics
FM2 4.1 - Variable forces and gravitation
Locked
FM2 4.2 - Simple harmonic motion of a particle
Locked
FM2 4.2 - Oscillations on strings and springs
Locked
Decision Mathematics 1
Algorithms and graph theory
D1 1.1 - Algorithms and their order
Locked
D1 1.2 - Bubble sort and quick sort
Locked
D1 1.2 - Bin packing algorithms
Locked
D1 1.3 - Graph terminology and Eulerian graphs
Locked
D1 1.4 - The planarity algorithm
Locked
Algorithms on graphs
D1 2.1 - Minimum spanning trees
Locked
D1 2.2 - Dijkstra's algorithm
Locked
D1 2.2 - Floyd's algorithm
Locked
Route inspection and travelling salesman problems
D1 3.1 - The route inspection algorithm
Locked
D1 3.2 - Travelling salesman upper bounds
Locked
D1 3.2 - Travelling salesman lower bounds
Locked
Critical path analysis
D1 4.1, 4.2 - Activity networks and precedence tables
Locked
D1 4.3 - Finding the critical path
Locked
D1 4.4 - Total float and Gantt charts
Locked
D1 4.5, 4.6 - Resource histograms and scheduling
Locked
Linear programming
D1 5.1 - Formulating linear programs
Locked
D1 5.2 - Solving linear programs graphically
Locked
D1 5.3 - The Simplex algorithm
Locked
D1 5.4 - Two-stage Simplex and big-M methods
Locked
Decision Mathematics 2
Transportation problems
D2 1.1 - The north-west corner method
Locked
D2 1.2 - Stepping-stone method and improvement indices
Locked
D2 1.3 - Transportation as a linear program
Locked
Allocation (assignment) problems
D2 2.1 - The Hungarian algorithm
Locked
D2 2.2 - Allocation as a linear program
Locked
Flows in networks
D2 3.1 - Cuts and their capacity
Locked
D2 3.2, 3.3 - Labelling and the max-flow min-cut theorem
Locked
D2 3.4, 3.5 - Multiple sources, sinks and restricted capacities
Locked
Dynamic programming
D2 4.1 - Bellman's principle and tabulation
Locked
Game theory
D2 5.1, 5.2 - Zero-sum games and stable solutions
Locked
D2 5.3, 5.4 - Dominance and graphical mixed strategies
Locked
D2 5.5 - Games as linear programs
Locked
Recurrence relations
D2 6.1, 6.2 - Recurrence models and first-order solutions
Locked
D2 6.2 - Solving second-order recurrence relations
Locked
Decision analysis
D2 7.1, 7.2 - Decision trees, EMVs and utility
Locked