Roadmap
Pure Core
Complex numbers
4.02a–4.02c - The language of complex numbers
4.02e - Arithmetic in cartesian form
4.02g–4.02j - Solving equations with complex roots
4.02k, 4.02l - Argand diagrams
4.02e, 4.02f, 4.02m - Working in modulus-argument form
4.02n - Euler's formula
4.02d - Using exponential form
4.02o, 4.02p - Drawing loci and regions
4.02q - De Moivre's theorem for multiple angles
4.02q - Rewriting powers of sine and cosine
4.02q - Summing trigonometric series
4.02r - nth roots of complex numbers
4.02s - Using roots of unity
Matrices
4.03a - The language of matrices
4.03b, 4.03c - Adding and multiplying matrices
4.03d - Matrices for 2D transformations
4.03e, 4.03f - Successive and 3D transformations
4.03g - Invariant points and lines
4.03h–4.03k - Calculating and interpreting determinants
4.03l, 4.03m - Singular matrices and det(AB)
4.03n–4.03q - Finding and using inverse matrices
4.03r, 4.03s - Solving simultaneous equations
Proof
4.01a - Proof by induction
4.01b - Harder proofs by induction
Further vectors
4.04a - Equations of straight lines
4.04c - Scalar product and angles between lines
4.04e - Intersecting, parallel or skew lines
4.04g - Finding a common perpendicular
4.04b - Equations of planes
4.04d - Angles with planes
4.04f - Where a line meets a plane
4.04h–4.04j - Finding shortest distances
Intersection of planes
4.03t - Arrangements of three planes
Further algebra
4.05a - Roots and coefficients of polynomials
4.05b - Transforming polynomial equations
4.05c - Quadratic and improper partial fractions
Series
4.06a - Sums of integers, squares and cubes
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4.06b - The method of differences
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Hyperbolic functions
4.07a–4.07c - Hyperbolic definitions, graphs and identities
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4.07d - Differentiating and integrating hyperbolic functions
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4.07e, 4.07f - Inverse hyperbolic functions as logarithms
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Further calculus
4.08a, 4.08b - Maclaurin series
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4.08c - Improper integrals
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4.08d - Volumes of revolution
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4.08e - Mean value of a function
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4.08g - Inverse trig and hyperbolic derivatives
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4.08h - Inverse trig and hyperbolic integrals
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4.08f - Integrating with partial fractions
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Polar coordinates
4.09a - Using polar coordinates
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4.09b - Sketching polar curves
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4.09c - Areas enclosed by polar curves
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Differential equations
4.10a - General and particular solutions
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4.10c - The integrating factor method
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4.10b - Modelling with differential equations
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4.10d - Second order homogeneous equations
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4.10e - Second order non-homogeneous equations
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4.10f - Simple harmonic motion
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4.10g - Damped oscillations
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4.10h - Coupled first-order systems
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Statistics
Probability
5.01a, 5.01b - Probability with permutations and combinations
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Discrete random variables
5.02a–5.02c - Discrete distributions, expectation and variance
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5.02d, 5.02e - Binomial and discrete uniform models
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5.02f–5.02h - The geometric distribution
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5.02i–5.02l - The Poisson distribution as a model
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5.02m, 5.02n - Poisson mean, variance and sums
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Continuous random variables
5.03a–5.03d - Probability density, mean and variance
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5.03e–5.03g - Cumulative distribution functions
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5.03a - Continuous uniform and exponential distributions
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Linear combinations of random variables
5.04a, 5.04b - Combining random variables
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Hypothesis tests and confidence intervals
5.05a, 5.05b - Sample means and unbiased estimates
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5.05c - Hypothesis tests for a population mean
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5.05d - Confidence intervals for a population mean
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Chi-squared tests
5.06a - Chi-squared tests for independence
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5.06b, 5.06c - Fitting a theoretical distribution
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5.06d - Chi-squared goodness of fit tests
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Non-parametric tests
5.07a–5.07c - Single-sample non-parametric tests
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5.07d, 5.07e - Paired-sample and two-sample tests
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5.07f - Normal approximations for Wilcoxon tests
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Correlation
5.08a–5.08d - Pearson's correlation coefficient and its test
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5.08e–5.08g - Rank correlation and choosing a coefficient
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Linear regression
5.09a–5.09e - Least squares regression lines
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Mechanics
Dimensional analysis
6.01a–6.01e - Finding and using dimensions
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Work, energy and power
6.02a–6.02c - Work done by a force
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6.02d–6.02f - Kinetic and potential energy
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6.02g, 6.02h - Hooke's law and elastic energy
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6.02i, 6.02j - Energy conservation and the work-energy principle
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6.02k–6.02m - Power and driving force
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Impulse and momentum
6.03a–6.03d - Momentum and its conservation
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6.03e–6.03h - Impulse and change in momentum
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6.03i–6.03k - Restitution and direct impact
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6.03l - Oblique impacts
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Centre of mass
6.04a–6.04c - Centres of mass of composite bodies
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6.04d - Centres of mass by integration
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6.04e - Rigid bodies in equilibrium
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Motion in a circle
6.05a–6.05c - Uniform circular motion
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6.05d, 6.05e - Motion in a vertical circle
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6.05f - Leaving a circular path
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Further dynamics and kinematics
6.06a - Motion under a variable force
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Discrete Mathematics
Sets and counting
7.01a–7.01c - Problem types, sets and pigeonholes
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7.01d–7.01f - Permutations and combinations
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7.01g, 7.01h - Counting arrangements of objects
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7.01i, 7.01j - Selections with constraints
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7.01k–7.01m - Inclusion-exclusion and derangements
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Graphs and networks
7.02a–7.02c - Graph terminology and notation
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7.02d–7.02f - Complete and bipartite graphs
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7.02g–7.02i - Eulerian and Hamiltonian graphs
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7.02j, 7.02k - Isomorphic graphs and digraphs
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7.02l–7.02o - Planar graphs
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7.02p–7.02r - Networks, matrices and modelling
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Algorithms
7.03a–7.03d - Defining, tracing and timing algorithms
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7.03e–7.03i - Efficiency and complexity
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7.03j, 7.03k - Sorting algorithms
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7.03l, 7.03m - Packing algorithms
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Network algorithms
7.04a - Dijkstra's algorithm
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7.04b - Minimum spanning trees
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7.04c, 7.04d - Bounds for the travelling salesperson
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7.04e, 7.04f - Route inspection and choosing algorithms
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Project management
7.05a–7.05c - Activity networks and critical paths
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7.05d, 7.05e - Float, cascade charts and scheduling
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Graphical linear programming
7.06a–7.06c - Formulating linear programmes
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7.06d - Solving linear programmes graphically
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7.06e, 7.06f - Changing coefficients and integer programming
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The simplex algorithm
7.07a–7.07d - Using a simplex tableau
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7.07e, 7.07f - Interpreting simplex iterations
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Game theory
7.08a–7.08d - Zero-sum games and pure strategies
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7.08e, 7.08f - Optimal mixed strategies
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Additional Pure Mathematics
Sequences and series
8.01a, 8.01b - Recurrence relations and induction
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8.01c–8.01e - Sequence behaviour and Fibonacci numbers
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8.01f - Solving first-order recurrence relations
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8.01g - Solving second-order recurrence relations
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8.01h, 8.01i - Modelling with recurrence relations
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Number theory
8.02a–8.02d - Bases, divisibility and the division algorithm
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8.02e–8.02h - Modular arithmetic and congruences
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8.02i–8.02k - Primes, factors and Euclid's lemma
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8.02l–8.02o - Fermat's little theorem and orders
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Groups
8.03a–8.03d - Binary operations and group axioms
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8.03e–8.03h - Orders, subgroups and cyclic groups
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8.03i–8.03k - Group structures and Lagrange's theorem
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8.03l, 8.03m - Isomorphism and abstract groups
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Further vectors
8.04a–8.04d - Vector product, areas and lines
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8.04e - The scalar triple product
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Surfaces and partial differentiation
8.05a–8.05c - Surfaces, sections and contours
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8.05d - First and second partial derivatives
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8.05e–8.05g - Stationary points and tangent planes
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Further calculus
8.06a - Reduction formulae
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8.06b - Arc lengths and surface areas
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