Roadmap
Core Pure
Complex numbers
Pj1, Pj2 - Complex number terms and quadratic roots
Pj4, Pj5 - Complex arithmetic and equating parts
Pj3 - Solving cubics and quartics
Pj6–Pj8 - Modulus-argument form
Pj9, Pj10 - Using the Argand diagram
Pj11 - Loci on the Argand diagram
Pj11 - Regions on the Argand diagram
Pj12, Pj13 - De Moivre's theorem and multiple angles
Pj13 - Summing trigonometric series
Pj14–Pj17 - Exponential form and nth roots
Pj18–Pj20 - Geometry on the Argand diagram
Matrices and transformations
Pm1–Pm3 - Adding and multiplying matrices
Pm4 - Matrices of 2-D transformations
Pm4, Pm5 - Successive and 3-D transformations
Pm6 - Invariant points and lines
Pm7–Pm10 - Determinants and scale factors
Pm11–Pm14 - Inverse matrices
Pm15 - Determinants and inverses by hand
Proof
Pp4 - Induction for sequences, series and matrices
Pp5 - Induction for divisibility and other results
Vectors and 3-D space
Pv1 - Scalar product and angles between vectors
Pv2, Pv3 - Equations of planes
Pv4–Pv6 - Intersections and angles of planes
Pv7, Pv8 - The vector product
Pv9, Pv10 - Lines and the angle between them
Pv11, Pv12 - Intersecting, parallel or skew lines
Pv14, Pv15 - Line–plane intersections and angles
Pv16, Pv17 - Point-to-line and point-to-plane distances
Pv13 - Distances between parallel or skew lines
Algebra
Pa1 - Roots and coefficients of polynomials
Pa2 - Equations with transformed roots
Series
Ps1 - Standard summation formulae
Ps1 - The method of differences
Ps2 - Summing series with partial fractions
Ps3, Ps4 - Maclaurin series and their validity
Ps5 - Standard Maclaurin series
Calculus
Pc1 - Improper integrals
Pc2, Pc3 - Volumes of revolution and mean values
Pc5 - Differentiating inverse trigonometric functions
Pc6 - Integrals giving inverse trigonometric functions
Locked
Pc4 - Integrating with partial fractions
Locked
Polar coordinates
PP1, PP2 - Polar conversions and curve sketching
Locked
PP3 - Areas enclosed by polar curves
Locked
Hyperbolic functions
Pa3, Pa4 - Sinh, cosh and tanh
Locked
Pa5 - Calculus of sinh, cosh and tanh
Locked
Pa6–Pa8 - Inverse hyperbolic functions and integrals
Locked
Differential equations
Pp19–Pp22 - Forming differential equation models
Locked
Pc7–Pc10 - First order equations and integrating factors
Locked
Pc11, Pc12 - Second order homogeneous equations
Locked
Pc13, Pc14 - Second order non-homogeneous equations
Locked
Pc15 - Simple harmonic motion
Locked
Pc16, Pc17 - Damped oscillations
Locked
Pc18 - Coupled first order equations
Locked
Mechanics (Major and Minor)
Dimensional analysis
Mq1–Mq5 - Dimensions, units and consistency
Locked
Mq6, Mq7 - Building models from dimensions
Locked
Forces
Md1–Md5 - Modelling friction
Locked
Md6–Md10 - Resolving forces and particle equilibrium
Locked
Md11–Md14 - Moments and couples
Locked
Md15, Md16 - Equilibrium, sliding or toppling
Locked
Work, energy and power
Mw1–Mw5 - Work done and mechanical energy
Locked
Mw6, Mw7 - Energy conservation and the work-energy principle
Locked
Mw8 - Power as rate of working
Locked
Momentum and impulse
Mi1–Mi3 - Impulse and momentum as vectors
Locked
Mi4–Mi6 - Conservation of momentum in direct impacts
Locked
Mi7–Mi12 - Restitution and energy loss in impacts
Locked
Centre of mass
MG1–MG3 - Centres of mass of particles and shapes
Locked
MG4 - Centres of mass of composite bodies
Locked
MG5 - Centre of mass and equilibrium
Locked
Momentum and impulse (Major only)
Mi13–Mi15 - Oblique impact with a smooth plane
Locked
Mi16, Mi17 - Oblique impact between two bodies
Locked
Circular motion (Major only)
Mr1–Mr3 - Angular speed, forces and acceleration
Locked
Mr4 - Horizontal circles at constant speed
Locked
Mr5–Mr8 - Varying speed and vertical circles
Locked
Hooke's law (Major only)
Mh1–Mh5 - Elastic tension and equilibrium
Locked
Mh6–Mh8 - Elastic energy and Hooke's law limits
Locked
Centre of mass (Major only)
MG6–MG8 - Centres of mass of solids of revolution
Locked
MG9, MG10 - Lamina centres of mass and equilibrium
Locked
Vectors and variable forces (Major only)
Mk1, Mk2, Mv1, Mv2 - Position and velocity vectors
Locked
Mv3–Mv5 - Equation of a particle's path
Locked
Mv6, Mv7 - Projectiles on an inclined plane
Locked
Mv8–Mv10 - Differential equations for variable forces
Locked
Mv11, Mv12 - Modelling with simple harmonic motion
Locked
Statistics (Major and Minor)
Sampling
Sx1–Sx3 - Sample size and random samples
Locked
Discrete random variables
SR1–SR3 - Discrete distributions, mean and variance
Locked
SR4–SR6 - Expectation and variance of combinations
Locked
SR7–SR9 - The discrete uniform distribution
Locked
SR10 - The binomial distribution
Locked
SR11–SR15 - The Poisson distribution
Locked
SR16–SR18 - The geometric distribution
Locked
Bivariate data
Sb1–Sb3 - Scatter diagrams
Locked
Sb4–Sb7 - Pearson's product moment correlation
Locked
Sb8–Sb10 - Spearman's rank and choosing a test
Locked
Sb11, Sb12 - Regression on a non-random variable
Locked
Sb13–Sb15 - Regression lines for two random variables
Locked
Chi-squared tests
Sb16, SH1, SH2 - Testing contingency tables for association
Locked
SH3, SH4 - Goodness of fit tests
Locked
Continuous random variables (Major only)
SR19–SR21 - Probability density functions
Locked
SR22, SR23 - Averages and spread from a pdf
Locked
SR24–SR26 - Cumulative distribution functions
Locked
SR27–SR29 - Combining Normal random variables
Locked
SR30, SR31 - Checking a Normal model
Locked
Inference (Major only)
SI1–SI6 - Sample means and the Central Limit Theorem
Locked
SI7–SI9, SR32 - Confidence intervals for a mean
Locked
SI10, SI11 - Confidence intervals for paired samples
Locked
SI12, SI13 - Confidence intervals from software
Locked
SH5 - The Wilcoxon signed rank test
Locked
SH6 - Hypothesis tests for a mean
Locked
Simulation (Major only)
SZ1, SZ2 - Simulating random variables
Locked
Modelling with Algorithms
Algorithms
AA1–AA3 - Using and adapting algorithms
Locked
AA4–AA6 - Efficiency, correctness and heuristics
Locked
AA7–AA9 - Sorting algorithms
Locked
AA10, AA11 - Bin packing algorithms
Locked
Networks
AN1–AN4 - Graphs and networks as models
Locked
AN5 - Minimum connectors by Kruskal and Prim
Locked
AN6, AN7 - Dijkstra's algorithm and complexity
Locked
AN8, AN9 - Critical path analysis
Locked
AN10–AN12 - Network flows and cuts
Locked
Linear programming
AL1–AL5 - Formulating linear programs
Locked
AL6, AN13 - Network problems as linear programs
Locked
AL7–AL11 - Graphical solution of linear programs
Locked
AL12, AL13 - The simplex method
Locked
AL14–AL16 - Simplex for non-standard problems
Locked
AL17, AL18 - Solving linear programs with software
Locked
Numerical Methods
Use of technology
NQ1, NQ2 - Spreadsheets and calculator iteration
Locked
Errors
NU1–NU3 - Absolute, relative and propagated errors
Locked
NU4–NU6 - Rounding, chopping and precision
Locked
Solution of equations
Ne1, Ne2 - Bisection, false position and secant methods
Locked
Ne1, Ne2 - Fixed point iteration and Newton-Raphson
Locked
Ne3, Ne4, NU7 - Order of convergence and failure
Locked
Ne5 - Relaxation of fixed point iteration
Locked
Numerical differentiation
Nc1, Nc2 - Forward and central difference methods
Locked
Numerical integration
Nc3, Nc4 - Midpoint, trapezium and Simpson's rules
Locked
Improving a solution
NU8 - Extrapolation from a sequence of estimates
Locked
Approximations to functions
Nf1 - Newton's forward difference interpolation
Locked
Nf2 - Lagrange's interpolating polynomial
Locked
Extra Pure
Recurrence relations
Xs1–Xs3 - Recurrence models and their behaviour
Locked
Xs4–Xs6 - Solving first order recurrences
Locked
Xs7 - Second order homogeneous recurrences
Locked
Xs8, Xs9 - Second order non-homogeneous recurrences
Locked
Groups
XS1, XS2, Xa1, Xa2 - Sets, group axioms and examples
Locked
Xa3–Xa6 - Cyclic groups, subgroups and Lagrange's theorem
Locked
Xa7, Xa8 - Isomorphic groups
Locked
Matrices
Xm1, Xm2 - Eigenvalues and eigenvectors
Locked
Xm3, Xm4 - Diagonalisation and matrix powers
Locked
Xm5, Xm6 - Cayley-Hamilton and transformation eigenvectors
Locked
Multivariable calculus
Xc1–Xc4 - Surfaces, partial derivatives and stationary points
Locked
Xc5–Xc7 - Normal lines and tangent planes
Locked
Further Pure with Technology
Investigation of curves
TC1–TC5 - Families of curves with software
Locked
TC6, TC7 - Tangents, normals and chords
Locked
TC8, TC9 - Arc length and envelopes
Locked
TC10–TC12 - Limits, asymptotes and cusps
Locked
Exploring differential equations
Tc1, Tc2, Tc4–Tc6 - Software solutions and tangent fields
Locked
Tc3, Tc7, Tc8 - Euler methods in a spreadsheet
Locked
Tc9, Tc10 - Runge-Kutta methods
Locked
Number theory
TT1–TT3 - Programs and prime factorisation
Locked
TT4–TT7 - Theorems of Fermat, Euler and Wilson
Locked
TT8–TT10 - Diophantine and Pell's equations
Locked