Specification points
OCR A-level Further Mathematics A H245
343 points in 6 topics, with OCR's codes and titles.
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Codes and titles are read by a script from OCR's published specification and have not been checked by a person. What each point covers is in the specification itself: OCR’s specification.
4 Pure Core
- 4.01 Proof
- 4.01a Mathematical induction
- 4.01b Demanding proofs and conjecture
- 4.02 Complex Numbers
- 4.02a The language of complex numbers
- 4.02b Cartesian and modulus-argument forms
- 4.02c Complex number notation
- 4.02d Exponential form of a complex number
- 4.02e Basic operations
- 4.02f Converting forms of complex numbers
- 4.02g Solution of equations
- 4.02h Square roots of a complex number
- 4.02i Quadratics with complex roots
- 4.02j Cubic and quartic equations with complex roots
- 4.02k Argand diagrams
- 4.02l Geometry of conjugates, sums and differences
- 4.02m Geometry of multiplying and dividing complex numbers
- 4.02n Euler’s formula
- 4.02o Loci
- 4.02p Set notation for loci
- 4.02q De Moivre’s theorem
- 4.02r nth roots
- 4.02s Roots of unity
- 4.03 Matrices
- 4.03a The language of matrices
- 4.03b Matrix addition and multiplication
- 4.03c Matrix multiplication: associative, not commutative
- 4.03d Linear transformations
- 4.03e Matrices for successive transformations
- 4.03f Matrices for 3-D transformations
- 4.03g Invariance
- 4.03h Determinants
- 4.03i Determinant as area scale factor
- 4.03j Determinant of a 3 × 3 matrix
- 4.03k Determinant as volume scale factor
- 4.03l Singular and non-singular matrices
- 4.03m Determinant of a product
- 4.03n Inverses
- 4.03o Inverse of a 3 × 3 matrix
- 4.03p Properties of inverse matrices
- 4.03q Inverse matrices and inverse transformations
- 4.03r Solution of simultaneous equations
- 4.03s Simultaneous equations with no unique solution
- 4.03t Intersection of planes
- 4.04 Further Vectors
- 4.04a Equation of a straight line
- 4.04b Equation of a plane
- 4.04c Scalar product
- 4.04d Angles between planes and between lines and planes
- 4.04e Intersections
- 4.04f Intersection of a line and a plane
- 4.04g Vector product
- 4.04h Shortest distances
- 4.04i Shortest distance from a point to a line
- 4.04j Shortest distance from a point to a plane
- 4.05 Further Algebra
- 4.05a Roots of equations
- 4.05b Transformation of equations
- 4.05c Partial fractions
- 4.06 Series
- 4.06a Summation of series
- 4.06b Method of differences
- 4.07 Hyperbolic Functions
- 4.07a Definitions
- 4.07b Graphs of hyperbolic functions
- 4.07c Hyperbolic identity cosh² − sinh² = 1
- 4.07d Differentiation and integration
- 4.07e Inverse hyperbolic functions
- 4.07f Inverse hyperbolic functions as logarithms
- 4.08 Further Calculus
- 4.08a Maclaurin series
- 4.08b Standard Maclaurin series
- 4.08c Improper integrals
- 4.08d Volumes of solids of revolution
- 4.08e Mean values
- 4.08f Partial fractions
- 4.08g Inverse trigonometric and hyperbolic functions
- 4.08h Further integration
- 4.09 Polar Coordinates
- 4.09a Polar coordinates
- 4.09b Sketching curves
- 4.09c Area
- 4.10 Differential Equations
- 4.10a General and particular solutions
- 4.10b Modelling
- 4.10c Integrating factor method for first order differential equations
- 4.10d Second order homogeneous differential equations
- 4.10e Second order non-homogeneous differential equations
- 4.10f Simple harmonic motion
- 4.10g Damped oscillations
- 4.10h Linear systems
5 Statistics
- 5.01 Probability
- 5.01a Probability
- 5.01b Probabilities from selections and arrangements
- 5.02 Discrete Random Variables
- 5.02a Probability distributions for general discrete random variables
- 5.02b Expectation and variance of a discrete variable
- 5.02c Linear coding of mean and variance
- 5.02d The binomial distribution
- 5.02e The discrete uniform distribution
- 5.02f The geometric distribution
- 5.02g Geometric distribution probabilities
- 5.02h Mean and variance of a geometric distribution
- 5.02i The Poisson distribution
- 5.02j Poisson probability formula
- 5.02k Poisson probabilities with a calculator
- 5.02l Conditions for a Poisson distribution
- 5.02m Mean and variance of a Poisson distribution
- 5.02n Sum of independent Poisson variables
- 5.03 Continuous Random Variables
- 5.03a Continuous random variables
- 5.03b Probability density functions
- 5.03c Mean and variance of a continuous distribution
- 5.03d Expectation of a function of a continuous variable
- 5.03e Cumulative distribution functions
- 5.03f Probability density and cumulative distribution functions
- 5.03g Cumulative distributions of related variables
- 5.04 Linear Combinations of Random Variables
- 5.04a Linear combinations of any random variables
- 5.04b Linear combinations of normal random variables
- 5.05 Hypothesis Tests and Confidence Intervals
- 5.05a The distribution of X and the central limit theorem
- 5.05b Unbiased estimates of population mean and variance
- 5.05c Using the normal distribution in hypothesis tests
- 5.05d Confidence intervals
- 5.06 Chi-squared Tests
- 5.06a Contingency tables
- 5.06b Fitting a theoretical distribution
- 5.06c Fitting other distributions
- 5.06d Goodness of fit test
- 5.07 Non-parametric tests
- 5.07a Non-parametric tests
- 5.07b The basis of non-parametric tests
- 5.07c Single-sample hypothesis tests
- 5.07d Paired-sample and two-sample hypothesis tests
- 5.07e Sign test and Wilcoxon tests
- 5.07f Normal approximations
- 5.08 Correlation
- 5.08a Pearson’s product-moment correlation coefficient
- 5.08b Correlation coefficient unaffected by linear coding
- 5.08c Pearson's correlation as closeness to a line
- 5.08d Hypothesis tests using Pearson’s product-moment correlation coefficient
- 5.08e Spearman’s rank correlation coefficient
- 5.08f Hypothesis tests using Spearman’s coefficient
- 5.08g Comparison of coefficients
- 5.09 Linear Regression
- 5.09a Dependent and independent variables
- 5.09b Calculation of the equation of the regression line
- 5.09c Regression line of y on x
- 5.09d Linear coding and regression lines
- 5.09e Use of the regression line
6 Mechanics
- 6.01 Dimensional Analysis
- 6.01a Dimensional analysis
- 6.01b Units and dimensions
- 6.01c Dimensional analysis as an error check
- 6.01d Finding unknown indices by dimensional analysis
- 6.01e Models from dimensional arguments
- 6.02 Work, Energy and Power
- 6.02a Work
- 6.02b Work done by a constant force
- 6.02c Work done by forces in two dimensions
- 6.02d Energy
- 6.02e Gravitational potential and kinetic energy
- 6.02f Kinetic energy using the scalar product
- 6.02g Hooke’s law
- 6.02h Elastic potential energy
- 6.02i Conservation of energy
- 6.02j Conservation of mechanical energy and work-energy principle
- 6.02k Power
- 6.02l Power, tractive force and velocity
- 6.02m Power of a variable force using scalar product
- 6.03 Impulse and Momentum
- 6.03a Linear momentum
- 6.03b Conservation of momentum in one dimension
- 6.03c Momentum in two dimensions
- 6.03d Conservation of momentum in two dimensions
- 6.03e Impulse
- 6.03f Impulse and change in momentum
- 6.03g Impulse–momentum principle in two dimensions
- 6.03h Impulse of a constant force
- 6.03i Restitution
- 6.03j Perfectly elastic and inelastic collisions
- 6.03k Newton's experimental law in one dimension
- 6.03l Newton's experimental law in two dimensions
- 6.04 Centre of Mass
- 6.04a Centre of mass
- 6.04b Centre of mass by symmetry
- 6.04c Centre of mass of particles and composite bodies
- 6.04d Centre of mass by integration
- 6.04e Rigid bodies
- 6.05 Motion in a circle
- 6.05a Uniform motion in a circle
- 6.05b Circular motion with constant speed
- 6.05c Motion in a horizontal circle
- 6.05d Motion in a vertical circle
- 6.05e Circular motion with variable speed
- 6.05f Vertical circle problems, including leaving the circle
- 6.06 Further Dynamics and Kinematics
- 6.06a Linear motion under a variable force
7 Discrete Mathematics
- 7.01 Mathematical Preliminaries
- 7.01a Types of problem
- 7.01b Set notation
- 7.01c The pigeonhole principle
- 7.01d Arrangement and selection problems
- 7.01e Permutations
- 7.01f Combinations
- 7.01g Arrangements in a line
- 7.01h Arranging some of a group
- 7.01i Selections with constraints
- 7.01j Problems with several constraints
- 7.01k The inclusion-exclusion principle
- 7.01l Inclusion-exclusion for more than two sets
- 7.01m Derangements
- 7.02 Graphs and Networks
- 7.02a Terminology and notation
- 7.02b Trees, simple and connected graphs
- 7.02c Walks, trails, paths, cycles and routes
- 7.02d Complete graphs
- 7.02e Bipartite graphs
- 7.02f Colouring arguments for bipartite graphs
- 7.02g Eulerian graphs
- 7.02h Hamiltonian graphs
- 7.02i Ore's theorem
- 7.02j Isomorphism
- 7.02k Digraphs
- 7.02l Planar graphs
- 7.02m Euler's formula for planar graphs
- 7.02n Kuratowski's theorem
- 7.02o Thickness of a graph
- 7.02p Using graphs and networks
- 7.02q Adjacency and weighted matrices
- 7.02r Modelling with graphs and networks
- 7.03 Algorithms
- 7.03a Definition of an algorithm
- 7.03b Awareness of the uses and practical limitations of algorithms
- 7.03c Working with algorithms
- 7.03d The order of an algorithm
- 7.03e Efficiency and complexity
- 7.03f Worst case time complexity
- 7.03g Orders of complexity O(nᵏ)
- 7.03h Best, worst and average case run-times
- 7.03i Exponential and logarithmic orders of complexity
- 7.03j Strategies for sorting
- 7.03k Quick sort
- 7.03l Strategies for packing
- 7.03m Extend their knowledge of packing methods
- 7.04 Network Algorithms
- 7.04a Least weight path between two vertices
- 7.04b Least weight set of arcs connecting all vertices
- 7.04c Least weight cycle through all vertices
- 7.04d Lower bound for the travelling salesperson problem
- 7.04e Route inspection problem
- 7.04f Network problems
- 7.05 Decision Making in Project Management
- 7.05a Critical path analysis
- 7.05b Forward pass and earliest start times
- 7.05c Total float
- 7.05d Latest start times and independent float
- 7.05e Cascade charts and delays
- 7.06 Graphical Linear Programming
- 7.06a Formulating LP problems
- 7.06b Slack variables
- 7.06c Working with constraints
- 7.06d Graphical solutions
- 7.06e Changing objective function coefficients
- 7.06f Integer programming and branch-and-bound
- 7.07 The Simplex Algorithm
- 7.07a Use a simplex tableau
- 7.07b Simplex tableau iterations
- 7.07c Interpreting a simplex tableau
- 7.07d Terminology
- 7.07e Graphical and algebraic interpretation of iterations
- 7.07f Algebra behind the simplex algorithm
- 7.08 Game Theory
- 7.08a Pay-off matrix
- 7.08b Reducing a matrix by dominance
- 7.08c Pure strategies
- 7.08d Nash equilibrium
- 7.08e Mixed strategies
- 7.08f Optimal mixed strategies
8 Additional Pure Mathematics
- 8.01 Sequences and Series
- 8.01a Recurrence relations
- 8.01b Induction for sequences and series
- 8.01c Properties of sequences
- 8.01d Limits of sequences and steady states
- 8.01e Fibonacci and related numbers
- 8.01f Solving recurrence systems
- 8.01g Second-order linear recurrence relations
- 8.01h Modelling
- 8.01i Modelling with second order recurrences
- 8.02 Number Theory
- 8.02a Number bases
- 8.02b Divisibility tests
- 8.02c Divisibility tests for primes under 50
- 8.02d The division algorithm
- 8.02e Finite (modular) arithmetics
- 8.02f Single linear congruences
- 8.02g Quadratic residues
- 8.02h Simultaneous linear congruences
- 8.02i Prime numbers
- 8.02j Divisibility of linear combinations
- 8.02k Euclid’s lemma
- 8.02l Fermat’s little theorem
- 8.02m The order of a modulo p
- 8.02n The order of a modulo p divides p−1
- 8.02o Binomial theorem
- 8.03 Groups
- 8.03a Binary operations
- 8.03b Cayley tables
- 8.03c Definition of a group
- 8.03d Latin square property
- 8.03e Orders of elements and groups
- 8.03f Subgroups
- 8.03g Cyclic groups
- 8.03h Generators
- 8.03i Properties of groups
- 8.03j Groups of higher or infinite order
- 8.03k Lagrange’s theorem
- 8.03l Isomorphism
- 8.03m Abstract groups
- 8.04 Further Vectors
- 8.04a Vector product
- 8.04b Properties of the vector product
- 8.04c Areas using the vector product
- 8.04d Significance of a zero vector product
- 8.04e Scalar triple product
- 8.05 Surfaces and Partial Differentiation
- 8.05a 3-D surfaces
- 8.05b Surfaces of more than two variables
- 8.05c Sections and contours
- 8.05d Partial differentiation
- 8.05e Stationary points
- 8.05f Maxima, minima and saddle points of surfaces
- 8.05g Tangent planes
- 8.06 Further Calculus
- 8.06a Reduction formulae
- 8.06b Arc lengths and surface areas