OCR Specification Codes

Browse questions by OCR A-Level Mathematics (H240) and Further Mathematics (H245) specification reference.

1.01 Proof

1.01a Proof: structure of mathematical proof and logical steps

1.01b Logical connectives: congruence, if-then, if and only if

1.01c Disproof by counter example

1.01d Proof by contradiction

1.02 Algebra and Functions

1.02a Indices: laws of indices for rational exponents

1.02b Surds: manipulation and rationalising denominators

1.02c Simultaneous equations: two variables by elimination and substitution

1.02d Quadratic functions: graphs and discriminant conditions

1.02e Complete the square: quadratic polynomials and turning points

1.02f Solve quadratic equations: including in a function of unknown

1.02g Inequalities: linear and quadratic in single variable

1.02h Express solutions: using 'and', 'or', set and interval notation

1.02i Represent inequalities: graphically on coordinate plane

1.02j Manipulate polynomials: expanding, factorising, division, factor theorem

1.02k Simplify rational expressions: factorising, cancelling, algebraic division

1.02l Modulus function: notation, relations, equations and inequalities

1.02m Graphs of functions: difference between plotting and sketching

1.02n Sketch curves: simple equations including polynomials

1.02o Sketch reciprocal curves: y=a/x and y=a/x^2

1.02p Interpret algebraic solutions: graphically

1.02q Use intersection points: of graphs to solve equations

1.02r Proportional relationships: and their graphs

1.02s Modulus graphs: sketch graph of |ax+b|

1.02t Solve modulus equations: graphically with modulus function

1.02u Functions: definition and vocabulary (domain, range, mapping)

1.02v Inverse and composite functions: graphs and conditions for existence

1.02w Graph transformations: simple transformations of f(x)

1.02x Combinations of transformations: multiple transformations

1.02y Partial fractions: decompose rational functions

1.02z Models in context: use functions in modelling

1.03 Coordinate Geometry

1.03a Straight lines: equation forms y=mx+c, ax+by+c=0

1.03b Straight lines: parallel and perpendicular relationships

1.03c Straight line models: in variety of contexts

1.03d Circles: equation (x-a)^2+(y-b)^2=r^2

1.03e Complete the square: find centre and radius of circle

1.03f Circle properties: angles, chords, tangents

1.03g Parametric equations: of curves and conversion to cartesian

1.03h Parametric equations: in modelling contexts

1.04 Sequences and Series

1.04a Binomial expansion: (a+b)^n for positive integer n

1.04b Binomial probabilities: link to binomial expansion

1.04c Extend binomial expansion: rational n, |x|<1

1.04d Binomial expansion validity: convergence conditions

1.04e Sequences: nth term and recurrence relations

1.04f Sequence types: increasing, decreasing, periodic

1.04g Sigma notation: for sums of series

1.04h Arithmetic sequences: nth term and sum formulae

1.04i Geometric sequences: nth term and finite series sum

1.04j Sum to infinity: convergent geometric series |r|<1

1.04k Modelling with sequences: compound interest, growth/decay

1.05 Trigonometry

1.05a Sine, cosine, tangent: definitions for all arguments

1.05b Sine and cosine rules: including ambiguous case

1.05c Area of triangle: using 1/2 ab sin(C)

1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta

1.05e Small angle approximations: sin x ~ x, cos x ~ 1-x^2/2, tan x ~ x

1.05f Trigonometric function graphs: symmetries and periodicities

1.05g Exact trigonometric values: for standard angles

1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs

1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs

1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=1

1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^2

1.05l Double angle formulae: and compound angle formulae

1.05m Geometric proofs: of trig sum and double angle formulae

1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc

1.05o Trigonometric equations: solve in given intervals

1.05p Proof involving trig: functions and identities

1.05q Trig in context: vectors, kinematics, forces

1.06 Exponentials and Logarithms

1.06a Exponential function: a^x and e^x graphs and properties

1.06b Gradient of e^(kx): derivative and exponential model

1.06c Logarithm definition: log_a(x) as inverse of a^x

1.06d Natural logarithm: ln(x) function and properties

1.06e Logarithm as inverse: ln(x) inverse of e^x

1.06f Laws of logarithms: addition, subtraction, power rules

1.06g Equations with exponentials: solve a^x = b

1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form

1.06i Exponential growth/decay: in modelling context

1.07 Differentiation

1.07a Derivative as gradient: of tangent to curve

1.07b Gradient as rate of change: dy/dx notation

1.07c Sketch gradient function: for given curve

1.07d Second derivatives: d^2y/dx^2 notation

1.07e Second derivative: as rate of change of gradient

1.07f Convexity/concavity: points of inflection

1.07g Differentiation from first principles: for small positive integer powers of x

1.07h Differentiation from first principles: for sin(x) and cos(x)

1.07i Differentiate x^n: for rational n and sums

1.07j Differentiate exponentials: e^(kx) and a^(kx)

1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)

1.07l Derivative of ln(x): and related functions

1.07m Tangents and normals: gradient and equations

1.07n Stationary points: find maxima, minima using derivatives

1.07o Increasing/decreasing: functions using sign of dy/dx

1.07p Points of inflection: using second derivative

1.07q Product and quotient rules: differentiation

1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates

1.07s Parametric and implicit differentiation

1.07t Construct differential equations: in context

1.08 Integration

1.08a Fundamental theorem of calculus: integration as reverse of differentiation

1.08b Integrate x^n: where n != -1 and sums

1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)

1.08d Evaluate definite integrals: between limits

1.08e Area between curve and x-axis: using definite integrals

1.08f Area between two curves: using integration

1.08g Integration as limit of sum: Riemann sums

1.08h Integration by substitution

1.08i Integration by parts

1.08j Integration using partial fractions

1.08k Separable differential equations: dy/dx = f(x)g(y)

1.08l Interpret differential equation solutions: in context

1.09 Numerical Methods

1.09a Sign change methods: locate roots

1.09b Sign change methods: understand failure cases

1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams

1.09d Newton-Raphson method

1.09e Iterative method failure: convergence conditions

1.09f Trapezium rule: numerical integration

1.09g Numerical methods in context

1.10 Vectors

1.10a Vectors in 2D: i,j notation and column vectors

1.10b Vectors in 3D: i,j,k notation

1.10c Magnitude and direction: of vectors

1.10d Vector operations: addition and scalar multiplication

1.10e Position vectors: and displacement

1.10f Distance between points: using position vectors

1.10g Problem solving with vectors: in geometry

1.10h Vectors in kinematics: uniform acceleration in vector form

2.01 Statistical Sampling

2.01a Population and sample: terminology

2.01b Informal inferences: from samples

2.01c Sampling techniques: simple random, opportunity, etc

2.01d Select/critique sampling: in context

2.02 Data Presentation and Interpretation

2.02a Interpret single variable data: tables and diagrams

2.02b Histogram: area represents frequency

2.02c Scatter diagrams and regression lines

2.02d Informal interpretation of correlation

2.02e Correlation does not imply causation

2.02f Measures of average and spread

2.02g Calculate mean and standard deviation

2.02h Recognize outliers

2.02i Select/critique data presentation

2.02j Clean data: missing data, errors

2.03 Probability

2.03a Mutually exclusive and independent events

2.03b Probability diagrams: tree, Venn, sample space

2.03c Conditional probability: using diagrams/tables

2.03d Calculate conditional probability: from first principles

2.03e Model with probability: critiquing assumptions

2.04 Statistical Distributions

2.04a Discrete probability distributions

2.04b Binomial distribution: as model B(n,p)

2.04c Calculate binomial probabilities

2.04d Normal approximation to binomial

2.04e Normal distribution: as model N(mu, sigma^2)

2.04f Find normal probabilities: Z transformation

2.04g Normal distribution properties: empirical rule (68-95-99.7), points of inflection

2.04h Select appropriate distribution

2.05 Statistical Hypothesis Testing

2.05a Hypothesis testing language: null, alternative, p-value, significance

2.05b Hypothesis test for binomial proportion

2.05c Significance levels: one-tail and two-tail

2.05d Sample mean as random variable

2.05e Hypothesis test for normal mean: known variance

2.05f Pearson correlation coefficient

2.05g Hypothesis test using Pearson's r

3.01 Quantities and Units in Mechanics

3.01b Derived quantities and units

3.01c Moment unit: N m

3.02 Kinematics

3.02a Kinematics language: position, displacement, velocity, acceleration

3.02b Kinematic graphs: displacement-time and velocity-time

3.02c Interpret kinematic graphs: gradient and area

3.02d Constant acceleration: SUVAT formulae

3.02e Two-dimensional constant acceleration: with vectors

3.02f Non-uniform acceleration: using differentiation and integration

3.02g Two-dimensional variable acceleration

3.02h Motion under gravity: vector form

3.02i Projectile motion: constant acceleration model

3.03 Forces and Newton's Laws

3.03a Force: vector nature and diagrams

3.03b Newton's first law: equilibrium

3.03c Newton's second law: F=ma one dimension

3.03d Newton's second law: 2D vectors

3.03e Resolve forces: two dimensions

3.03f Weight: W=mg

3.03g Gravitational acceleration

3.03h Newton's third law: action-reaction pairs

3.03i Normal reaction force

3.03j Smooth contact model: and limitations

3.03k Connected particles: pulleys and equilibrium

3.03l Newton's third law: extend to situations requiring force resolution

3.03m Equilibrium: sum of resolved forces = 0

3.03n Equilibrium in 2D: particle under forces

3.03o Advanced connected particles: and pulleys

3.03p Resultant forces: using vectors

3.03q Dynamics: motion under force in plane

3.03r Friction: concept and vector form

3.03s Contact force components: normal and frictional

3.03t Coefficient of friction: F <= mu*R model

3.03u Static equilibrium: on rough surfaces

3.03v Motion on rough surface: including inclined planes

3.04 Moments

3.04a Calculate moments: about a point

3.04b Equilibrium: zero resultant moment and force

3.04c Use moments: beams, ladders, static problems

4.01 Proof

4.01a Mathematical induction: construct proofs

4.01b Complex proofs: conjecture and demanding proofs

4.02 Complex Numbers

4.02a Complex numbers: real/imaginary parts, modulus, argument

4.02b Express complex numbers: cartesian and modulus-argument forms

4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)

4.02d Exponential form: re^(i*theta)

4.02e Arithmetic of complex numbers: add, subtract, multiply, divide

4.02f Convert between forms: cartesian and modulus-argument

4.02g Conjugate pairs: real coefficient polynomials

4.02h Square roots: of complex numbers

4.02i Quadratic equations: with complex roots

4.02j Cubic/quartic equations: conjugate pairs and factor theorem

4.02k Argand diagrams: geometric interpretation

4.02l Geometrical effects: conjugate, addition, subtraction

4.02m Geometrical effects: multiplication and division

4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)

4.02o Loci in Argand diagram: circles, half-lines

4.02p Set notation: for loci

4.02q De Moivre's theorem: multiple angle formulae

4.02r nth roots: of complex numbers

4.02s Roots of unity: geometric applications

4.03 Matrices

4.03a Matrix language: terminology and notation

4.03b Matrix operations: addition, multiplication, scalar

4.03c Matrix multiplication: properties (associative, not commutative)

4.03d Linear transformations 2D: reflection, rotation, enlargement, shear

4.03e Successive transformations: matrix products

4.03f Linear transformations 3D: reflections and rotations about axes

4.03g Invariant points and lines

4.03h Determinant 2x2: calculation

4.03i Determinant: area scale factor and orientation

4.03j Determinant 3x3: calculation

4.03k Determinant 3x3: volume scale factor

4.03l Singular/non-singular matrices

4.03m det(AB) = det(A)*det(B)

4.03n Inverse 2x2 matrix

4.03o Inverse 3x3 matrix

4.03p Inverse properties: (AB)^(-1) = B^(-1)*A^(-1)

4.03q Inverse transformations

4.03r Solve simultaneous equations: using inverse matrix

4.03s Consistent/inconsistent: systems of equations

4.03t Plane intersection: geometric interpretation

4.04 Further Vectors

4.04a Line equations: 2D and 3D, cartesian and vector forms

4.04b Plane equations: cartesian and vector forms

4.04c Scalar product: calculate and use for angles

4.04d Angles: between planes and between line and plane

4.04e Line intersections: parallel, skew, or intersecting

4.04f Line-plane intersection: find point

4.04g Vector product: a x b perpendicular vector

4.04h Shortest distances: between parallel lines and between skew lines

4.04i Shortest distance: between a point and a line

4.04j Shortest distance: between a point and a plane

4.05 Further Algebra

4.05a Roots and coefficients: symmetric functions

4.05b Transform equations: substitution for new roots

4.05c Partial fractions: extended to quadratic denominators

4.06 Series

4.06a Summation formulae: sum of r, r^2, r^3

4.06b Method of differences: telescoping series

4.07 Hyperbolic Functions

4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials

4.07b Hyperbolic graphs: sketch and properties

4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 1

4.07d Differentiate/integrate: hyperbolic functions

4.07e Inverse hyperbolic: definitions, domains, ranges

4.07f Inverse hyperbolic: logarithmic forms

4.08 Further Calculus

4.08a Maclaurin series: find series for function

4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n

4.08c Improper integrals: infinite limits or discontinuous integrands

4.08d Volumes of revolution: about x and y axes

4.08e Mean value of function: using integral

4.08f Integrate using partial fractions

4.08g Derivatives: inverse trig and hyperbolic functions

4.08h Integration: inverse trig/hyperbolic substitutions

4.09 Polar Coordinates

4.09a Polar coordinates: convert to/from cartesian

4.09b Sketch polar curves: r = f(theta)

4.09c Area enclosed: by polar curve

4.10 Differential Equations

4.10a General/particular solutions: of differential equations

4.10b Model with differential equations: kinematics and other contexts

4.10c Integrating factor: first order equations

4.10d Second order homogeneous: auxiliary equation method

4.10e Second order non-homogeneous: complementary + particular integral

4.10f Simple harmonic motion: x'' = -omega^2 x

4.10g Damped oscillations: model and interpret

4.10h Coupled systems: simultaneous first order DEs

5.01 Probability

5.01a Permutations and combinations: evaluate probabilities

5.01b Selection/arrangement: probability problems

5.02 Discrete Random Variables

5.02a Discrete probability distributions: general

5.02b Expectation and variance: discrete random variables

5.02c Linear coding: effects on mean and variance

5.02d Binomial: mean np and variance np(1-p)

5.02e Discrete uniform distribution

5.02f Geometric distribution: conditions

5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1)

5.02h Geometric: mean 1/p and variance (1-p)/p^2

5.02i Poisson distribution: random events model

5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!

5.02k Calculate Poisson probabilities

5.02l Poisson conditions: for modelling

5.02m Poisson: mean = variance = lambda

5.02n Sum of Poisson variables: is Poisson

5.03 Continuous Random Variables

5.03a Continuous random variables: pdf and cdf

5.03b Solve problems: using pdf

5.03c Calculate mean/variance: by integration

5.03d E(g(X)): general expectation formula

5.03e Find cdf: by integration

5.03f Relate pdf-cdf: medians and percentiles

5.03g Cdf of transformed variables

5.04 Linear Combinations of Random Variables

5.04a Linear combinations: E(aX+bY), Var(aX+bY)

5.04b Linear combinations: of normal distributions

5.05 Hypothesis Tests and Confidence Intervals

5.05a Sample mean distribution: central limit theorem

5.05b Unbiased estimates: of population mean and variance

5.05c Hypothesis test: normal distribution for population mean

5.05d Confidence intervals: using normal distribution

5.06 Chi-squared Tests

5.06a Chi-squared: contingency tables

5.06b Fit prescribed distribution: chi-squared test

5.06c Fit other distributions: discrete and continuous

5.06d Goodness of fit: chi-squared test

5.07 Non-parametric Tests

5.07a Non-parametric tests: when to use

5.07b Sign test: and Wilcoxon signed-rank

5.07c Single-sample tests

5.07d Paired vs two-sample: selection

5.07e Test medians

5.08 Correlation

5.08a Pearson correlation: calculate pmcc

5.08b Linear coding: effect on pmcc

5.08c Pearson: measure of straight-line fit

5.08d Hypothesis test: Pearson correlation

5.08e Spearman rank correlation

5.08f Hypothesis test: Spearman rank

5.08g Compare: Pearson vs Spearman

5.09 Linear Regression

5.09a Dependent/independent variables

5.09b Least squares regression: concepts

5.09c Calculate regression line

5.09d Linear coding: effect on regression

5.09e Use regression: for estimation in context

6.01 Dimensional Analysis

6.01a Dimensions: M, L, T notation

6.01b Units vs dimensions: relationship

6.01c Dimensional analysis: error checking

6.01d Unknown indices: using dimensions

6.01e Formulate models: dimensional arguments

6.02 Work, Energy and Power

6.02a Work done: concept and definition

6.02b Calculate work: constant force, resolved component

6.02c Work by variable force: using integration

6.02d Mechanical energy: KE and PE concepts

6.02e Calculate KE and PE: using formulae

6.02f KE with vectors: using scalar product

6.02g Hooke's law: T = k*x or T = lambda*x/l

6.02h Elastic PE: 1/2 k x^2

6.02i Conservation of energy: mechanical energy principle

6.02j Conservation with elastics: springs and strings

6.02k Power: rate of doing work

6.02l Power and velocity: P = Fv

6.02m Variable force power: using scalar product

6.03 Impulse and Momentum

6.03a Linear momentum: p = mv

6.03b Conservation of momentum: 1D two particles

6.03c Momentum in 2D: vector form

6.03d Conservation in 2D: vector momentum

6.03e Impulse: by a force

6.03f Impulse-momentum: relation

6.03g Impulse in 2D: vector form

6.03h Variable force impulse: using integration

6.03i Coefficient of restitution: e

6.03j Perfectly elastic/inelastic: collisions

6.03k Newton's experimental law: direct impact

6.03l Newton's law: oblique impacts

6.04 Centre of Mass

6.04a Centre of mass: gravitational effect

6.04b Find centre of mass: using symmetry

6.04c Composite bodies: centre of mass

6.04d Integration: for centre of mass of laminas/solids

6.04e Rigid body equilibrium: coplanar forces

6.05 Motion in a Circle

6.05a Angular velocity: definitions

6.05b Circular motion: v=r*omega and a=v^2/r

6.05c Horizontal circles: conical pendulum, banked tracks

6.05d Variable speed circles: energy methods

6.05e Radial/tangential acceleration

6.05f Vertical circle: motion including free fall

6.06 Further Dynamics and Kinematics

6.06a Variable force: dv/dt or v*dv/dx methods

7.01 Mathematical Preliminaries

7.01a Types of problem: existence, construction, enumeration, optimisation

7.01b Set notation: basic language and notation of sets, partitions

7.01c Pigeonhole principle

7.01d Multiplicative principle: arrangements of n distinct objects

7.01e Permutations: ordered subsets of r from n elements

7.01f Combinations: unordered subsets of r from n elements

7.01g Arrangements in a line: with repetition and restriction

7.01i Selections with constraints

7.01j Selections with several constraints

7.01k Inclusion-exclusion: for two sets

7.01l Inclusion-exclusion: extended to more than two sets

7.01m Derangements: enumeration by ad hoc methods

7.02 Graphs and Networks

7.02a Graphs: vertices (nodes) and arcs (edges)

7.02b Graph terminology: tree, simple, connected, simply connected

7.02c Graph terminology: walk, trail, path, cycle, route

7.02d Complete graphs: K_n and number of arcs

7.02e Bipartite graphs: K_{m,n} notation

7.02f Bipartite test: colouring argument

7.02g Eulerian graphs: vertex degrees and traversability

7.02h Hamiltonian paths: and cycles

7.02i Ore's theorem: sufficient condition for Hamiltonian graphs

7.02j Isomorphism: of graphs, degree sequences

7.02k Digraphs: directed graphs, indegree and outdegree

7.02l Planar graphs: planarity, subdivision, contraction

7.02m Euler's formula: V + R = E + 2

7.02n Kuratowski's theorem: K_5 and K_{3,3} subdivisions

7.02o Thickness: of graphs

7.02p Networks: weighted graphs, modelling connections

7.02q Adjacency matrix: and weighted matrix representation

7.02r Graph modelling: model and solve problems using graphs

7.03 Algorithms

7.03a Algorithm definition: input, output, deterministic, finite

7.03b Algorithm awareness: uses and practical limitations

7.03c Working with algorithms: trace, interpret, adapt

7.03d Order of algorithm: dominant term and scaling run-times

7.03e Efficiency comparison: of two algorithms on specific cases

7.03f Worst case complexity: T(n) as function of problem size

7.03g Order notation: O(n^k) for k = 0,1,2,3,4

7.03h Best/worst/typical case: run-time analysis

7.03i Order hierarchy: O(n^k), O(a^n), O(log n), O(n!)

7.03j Sorting: bubble sort and shuttle sort

7.03k Sorting: quick sort

7.03l Bin packing: next-fit, first-fit, first-fit decreasing, full bin

7.03m Packing extensions: 2D/3D packing and knapsack problems

7.04 Network Algorithms

7.04a Shortest path: Dijkstra's algorithm

7.04b Minimum spanning tree: Prim's and Kruskal's algorithms

7.04c Travelling salesman upper bound: nearest neighbour method

7.04d Travelling salesman lower bound: using minimum spanning tree

7.04e Route inspection: Chinese postman, pairing odd nodes

7.04f Network problems: choosing appropriate algorithm

7.05 Decision Making in Project Management

7.05a Critical path analysis: activity on arc networks

7.05b Forward and backward pass: earliest/latest times, critical activities

7.05c Total float: calculation and interpretation

7.05d Latest start and earliest finish: independent and interfering float

7.05e Cascade charts: scheduling and effect of delays

7.06 Graphical Linear Programming

7.06a LP formulation: variables, constraints, objective function

7.06b Slack variables: converting inequalities to equations

7.06c Working with constraints: algebra and ad hoc methods

7.06d Graphical solution: feasible region, two variables

7.06e Sensitivity analysis: effect of changing coefficients

7.06f Integer programming: branch-and-bound method

7.07 The Simplex Algorithm

7.07a Simplex tableau: initial setup in standard format

7.07b Simplex iterations: pivot choice and row operations

7.07c Interpret simplex: values of variables, slack, and objective

7.07d Simplex terminology: basic feasible solution, basic/non-basic variable

7.07e Graphical interpretation: iterations as edges of convex polygon

7.07f Algebraic interpretation: explain simplex calculations

7.08 Game Theory

7.08a Pay-off matrix: zero-sum games

7.08b Dominance: reduce pay-off matrix

7.08c Pure strategies: play-safe strategies and stable solutions

7.08d Nash equilibrium: identification and interpretation

7.08e Mixed strategies: optimal strategy using equations or graphical method

7.08f Mixed strategies via LP: reformulate as linear programming problem

8.01 Sequences and Series

8.01a Recurrence relations: general sequences, closed form and recurrence

8.01b Induction: prove results for sequences and series

8.01c Sequence behaviour: periodic, convergent, divergent, oscillating, monotonic

8.01d Sequence limits: limit of nth term as n tends to infinity, steady-states

8.01e Fibonacci: and related sequences (e.g. Lucas numbers)

8.01f First-order recurrence: solve using auxiliary equation and complementary function

8.01g Second-order recurrence: solve with distinct, repeated, or complex roots

8.01h Modelling with recurrence: birth/death rates, INT function

8.01i Modelling with second-order: recurrence relations

8.02 Number Theory

8.02a Number bases: conversion and arithmetic in base n

8.02b Divisibility tests: standard tests for 2, 3, 4, 5, 8, 9, 11

8.02c Divisibility by primes: algorithmic tests for primes less than 50

8.02d Division algorithm: a = bq + r uniquely

8.02e Finite (modular) arithmetic: integers modulo n

8.02f Single linear congruences: solve ax = b (mod n)

8.02g Quadratic residues: calculate and solve equations involving them

8.02h Simultaneous linear congruences: solve up to three

8.02i Prime numbers: composites, HCF, coprimality

8.02j Divisibility property: a|b and a|c implies a|(bx+cy)

8.02k Euclid's lemma: if a|rs and hcf(r,a)=1 then a|s

8.02l Fermat's little theorem: both forms

8.02m Order of a modulo p: p-1 not necessarily least such n

8.02o Binomial theorem: (a+b)^p = a^p + b^p (mod p) for prime p

8.03 Groups

8.03a Binary operations: and their properties on given sets

8.03b Cayley tables: construct for finite sets under binary operation

8.03c Group definition: recall and use, show structure is/isn't a group

8.03d Latin square property: for group tables

8.03e Order of elements: and order of groups

8.03f Subgroups: definition and tests for proper subgroups

8.03g Cyclic groups: meaning of the term

8.03h Generators: of cyclic and non-cyclic groups

8.03i Properties of groups: structure of finite groups up to order 7

8.03j Properties of groups: higher finite order or infinite order

8.03k Lagrange's theorem: order of subgroup divides order of group

8.03l Isomorphism: determine using informal methods

8.04 Further Vectors

8.04a Vector product: definition, magnitude/direction, component form

8.04b Vector product properties: anti-commutative and distributive

8.04c Areas using vector product: triangles and parallelograms

8.04d Significance of a x b = 0: and line equations using vector product

8.04e Scalar triple product: volumes of tetrahedra and parallelepipeds

8.05 Surfaces and Partial Differentiation

8.05a 3D surfaces: z = f(x,y) and implicit form, partial derivatives

8.05b Extended surfaces: more than two variables, trig/log/exp functions

8.05c Sections and contours: sketch and relate to surface

8.05d Partial differentiation: first and second order, mixed derivatives

8.05e Stationary points: where partial derivatives are zero

8.05f Nature of stationary points: classify using Hessian matrix

8.05g Tangent planes: equation at a given point on surface

8.06 Further Calculus

8.06a Reduction formulae: establish, use, and evaluate recursively

8.06b Arc length and surface area: of revolution, cartesian or parametric