Roadmap
Core Pure
Complex arithmetic
B2 - Introducing complex numbers
B3 - The complex conjugate
B2, B3 - Dividing complex numbers
B1, B3 - Quadratics with non-real roots
B1, B3 - Complex roots of cubic and quartic equations
Complex geometry
B4 - Points on the Argand diagram
B5 - Modulus and argument
B6 - Multiplication as rotation and scaling
B7 - Loci and regions in the complex plane
Complex powers and roots
B9 - Exponential form
B8 - De Moivre’s theorem
B8 - Sums of trigonometric series
B10 - Finding all complex roots
B11 - Roots of unity in geometry
Matrix arithmetic
C1, C2 - Calculating with matrices
C1, C2 - Matrix multiplication
C5 - Determinants in two and three dimensions
C9 - Factorising determinants
C6 - Inverse matrices
C7, C8 - Three simultaneous equations
Matrix transformations
C3 - Transformations in two dimensions
C3, C5 - Combining transformations
C3 - Transformations in three dimensions
C4 - Invariant points and lines
C10 - Eigenvalues and eigenvectors
C11 - Diagonalisation and matrix powers
Polynomials and inequalities
D1 - Roots and coefficients
D2 - Transforming polynomial roots
D8 - Polynomial inequalities
D9 - Rational inequalities
D10 - Modulus functions and inequalities
Graphs and curves
D11 - Modulus and reciprocal graphs
D12 - Linear rational functions
D13 - Quadratic rational functions
D14 - Ranges of rational functions
D15 - Parabolas, ellipses and hyperbolas
D16 - Transforming curves
Series and induction
D3 - Sums of powers
D4 - The method of differences
A1 - Proof by induction
A1 - Induction for divisibility
Locked
A1 - Induction for matrix powers
Locked
Power series and limits
D5 - Building Maclaurin series
Locked
D6 - Using standard Maclaurin series
Locked
D7 - Limits from power series
Locked
D7 - L’Hôpital’s rule
Locked
Vectors in space
F1 - Equations of lines in space
Locked
F3, F4 - Scalar products and angles
Locked
F2, F3 - Equations of planes
Locked
F3 - Angles involving planes
Locked
F5 - The vector product
Locked
F6 - Intersections in space
Locked
F6 - Perpendicular distances
Locked
Hyperbolic functions
H1 - Hyperbolic functions from exponentials
Locked
H6, H7 - Hyperbolic identities
Locked
H2 - Differentiating and integrating hyperbolic functions
Locked
H3, H4 - Inverse hyperbolic functions
Locked
H5 - Hyperbolic substitutions
Locked
Further integration
E5, E6 - Inverse trigonometric functions
Locked
E6 - Trigonometric substitutions
Locked
E4 - Partial fractions with quadratic factors
Locked
E1, E9 - Improper integrals
Locked
E3 - Mean value of a function
Locked
E8 - Reduction formulae
Locked
Calculus in geometry
E2 - Volumes of revolution
Locked
E7 - Arc length
Locked
E7 - Surface area of revolution
Locked
G1 - Polar coordinates
Locked
G2 - Sketching polar curves
Locked
G3 - Areas inside polar curves
Locked
First-order differential equations
I1, I2 - Integrating factors
Locked
I2, I3 - Initial conditions and models
Locked
I9 - Coupled differential equations
Locked
Second-order differential equations
I2, I4, I6 - Auxiliary equations with real roots
Locked
I2, I4, I6 - Oscillatory solutions
Locked
I5 - Forced differential equations
Locked
I9 - Solving coupled models by elimination
Locked
I3, I7, I10 - Simple harmonic motion
Locked
I8, I11 - Damped oscillations
Locked
Numerical methods
J1 - Mid-ordinate and Simpson’s rules
Locked
J2 - Euler’s method
Locked
J3 - Improved Euler’s method
Locked
Mechanics
Dimensions and momentum
MA1 - Dimensions and consistency
Locked
MA2 - Predicting formulae
Locked
MB1 - Conservation of momentum
Locked
MB2 - Direct collisions
Locked
MB1, MB2 - Oblique impacts
Locked
MB3, MB4 - Impulse
Locked
Work, energy and power
MC1–MC3 - Work and mechanical energy
Locked
MC4 - Hooke’s law
Locked
MC5, MC6 - Variable forces and elastic energy
Locked
MC7 - Power
Locked
Circular motion
MD1–MD3 - Speed in circular motion
Locked
MD4 - Circular motion with vectors
Locked
MD5 - Conical pendulums
Locked
MD6 - Vertical circles
Locked
Centres of mass and equilibrium
ME1 - Centres of mass of particles
Locked
ME2 - Composite bodies
Locked
ME3 - Laminas by integration
Locked
ME4 - Solids of revolution
Locked
ME6 - Moments and couples
Locked
ME5 - Sliding, toppling and suspension
Locked
Statistics
Discrete probability
SA1, SA2 - Discrete probability distributions
Locked
SA3, SA4 - Averages and spread
Locked
SA5 - Functions of random variables
Locked
SA6, SA7 - The discrete uniform model
Locked
Poisson models and testing
SB1–SB3 - The Poisson distribution
Locked
SB4 - Combining Poisson variables
Locked
SB5 - Testing a Poisson mean
Locked
SC1 - Type I errors
Locked
SC2 - Type II errors and power
Locked
Continuous probability
SD1, SD2 - Probability density functions
Locked
SD6 - Cumulative distribution functions
Locked
SD3 - Medians and quartiles
Locked
SD4, SD5 - Continuous expectation and variance
Locked
SD1 - Mixed distributions
Locked
SD7 - The rectangular distribution
Locked
SD8 - Sums of independent variables
Locked
Exponential waiting times
SF1, SF2, SF4 - Exponential distributions
Locked
SF3 - Mean and variance of waiting times
Locked
Association and inference
SE1–SE3 - Chi-squared tests for association
Locked
SE4, SE5 - Interpreting association
Locked
SG1 - One-sample t-tests
Locked
SH1, SH3 - Confidence intervals with known variance
Locked
SH2–SH4 - Confidence intervals with estimated variance
Locked
Discrete Mathematics
Graphs and planarity
DA1, DA6 - The language of graphs
Locked
DA5 - Complete and bipartite graphs
Locked
DA7 - Graph isomorphism
Locked
DA2 - Eulerian and Hamiltonian graphs
Locked
DA3 - Euler’s formula for planar graphs
Locked
DA4 - Proving non-planarity
Locked
Networks and routes
DB1, DB2 - Minimum spanning trees
Locked
DB3 - Route inspection
Locked
DB4 - Travelling salesperson bounds
Locked
DB5 - Refining network models
Locked
Network flows
DC1, DC2 - Flows and cuts
Locked
DC3, DC5 - Maximum flow and minimum cut
Locked
DC4, DC7 - Multiple sources and restricted nodes
Locked
DC6 - Lower capacity constraints
Locked
Linear programming
DD1 - Formulating linear programmes
Locked
DD2 - Graphical optimisation
Locked
DD3, DD4 - The simplex algorithm
Locked
DD3, DD4 - Minimisation and tableau interpretation
Locked
Project scheduling
DE1 - Activity-on-node networks
Locked
DE2, DE3 - Critical paths and float
Locked
DE4 - Changing project schedules
Locked
DE5 - Gantt charts and resource histograms
Locked
DE6 - Scheduling with limited resources
Locked
Zero-sum games
DF1–DF3 - Pay-off matrices and stable games
Locked
DF4 - Dominated strategies
Locked
DF5 - Mixed strategies
Locked
DF6 - Games as linear programmes
Locked
Binary operations and groups
DG1, DG4 - Binary operations and Cayley tables
Locked
DG2, DG3 - Commutativity and associativity
Locked
DG5, DG6 - Identity and inverses
Locked
DG7, DG8 - The group axioms
Locked
DG7, DG9 - Groups and subgroups
Locked
DG10 - Lagrange’s theorem
Locked
DG11 - Generators of groups
Locked
DG12 - Isomorphism of finite groups
Locked