• # Mathematics Revision Notes List

This page is a 'working page' - we are using it to help audit our revision notes and to audit the exam syllabuses. By doing this we hope to work out which topic areas we cover and which we are missing. We also hope to identify which exams each revision note page is suitable for in order to better direct you to the notes you need. It's gonna be a long project, but once there we'll have an amazing, userfriendly collection of maths revision notes..

# Exam Level

Revision notes for mathematics are currently spread across 5 different exam levels or types:

## Key Stage 3

Mathematics at KS3 can be split into four broad topics and further sub-divided into smaller areas:

### Data handling and probability

• Handling data: Specify, plan, collect and represent data
• Handling data: processing, interpreting and discussing data
• Probability

### Numbers and calculations

• Written methods
• Fractions, decimals and percentages
• Property of numbers
• Ratio and proportion
• Measures

### Algebra

• Sequences and functions
• Equations and formulae
• Functions and graphs

### Shape, space and measure

• 2D and 3D shapes
• Transformations
• Angles and lines
• Coordinates and geometrical reasoning
• Constructions an loci

## GCSE

Mathematics at GCSE can be split into 4 main topic areas:

## A Level

Mathematics at A Level is based around the 4 main areas of core mathematics (or 'pure maths'), statistics, mechanics and decision (also known as discrete) mathematics. A Level courses are split into numerous modules and our revision notes are split up approximately in to the modules the topics appear in. Please note there are some slight differences between exam boards as to which topic falls into which module.

### Core 1

#### Algebra and Functions

• Indices (Edexcel, OCR)
• Surds (Edexcel, OCR, AQA)
• Quadratic functions (Edexcel, OCR, AQA)
• Graphs of quadratic functions (Edexcel, OCR, AQA)
• Completing the square (Edexcel, OCR, AQA)
• Quadratic equations (Edexcel, OCR, AQA)
• Solution by factorisation (Edexcel, OCR, AQA)
• Solution by completing the square (Edexcel, OCR, AQA)
• Solution by general formula (Edexcel, OCR, AQA)
• The discriminant (Edexcel, OCR, AQA)
• Simultaneous equations (Edexcel, OCR, AQA)
• Analytical solution (Edexcel, OCR, AQA)
• Graphical solution (Edexcel, OCR, AQA)
• inequalities (Edexcel, OCR, AQA)
• Solution of linear inequalities (Edexcel, OCR, AQA)
• Solution of quadratic inequalities (Edexcel, OCR, AQA)
• Polynomials (Edexcel, OCR, AQA)
• Expanding brackets (Edexcel, OCR, AQA)
• Simplifying (Edexcel, OCR, AQA)
• Algebraic long division (AQA)
• Factor theorem (AQA)
• Remainder theorem (AQA)
• Graphs (Edexcel, OCR, AQA)
• Graphs of cubic functions (Edexcel, OCR, AQA)
• Graphs of (Edexcel)
• Graphs of for any n (OCR)
• Translations of quadratic functions and circles only (AQA)
• Graph transformations (Edexcel, OCR)
• Graphical solutions to equations (Edexcel, OCR, AQA)

#### Coordinate Geometry

• Straight lines (Edexcel, OCR, AQA)
• Midpoints of lines (OCR)
• Gradients (Edexcel, OCR, AQA)
• General equation (Edexcel, OCR, AQA)
• Finding a line's equation (Edexcel, OCR, AQA)
• Perpendicular lines (Edexcel, OCR, AQA)
• Parallel lines (Edexcel, OCR, AQA)
• Circles (OCR, AQA)
• Drawing circles (OCR, AQA)
• Geometric properties (OCR, AQA)
• Tangents and normals to circles (AQA, maybe OCR)
• Cartesian equation (Pythagorean form) (OCR, AQA)
• Cartesian equation (Expanded form) (OCR)

#### Sequences and Series

• Sequences and sequence notation (Edexcel)
• Recurrence relations (Edexcel)
• Series and sigma notation (Edexcel)
• Arithmetic series (Edexcel)
• Sum of an arithmetic series (Edexcel)

#### Differentiation

• Gradients of curves (Edexcel, OCR, AQA)
• Differentiation and the derivative (Edexcel, OCR, AQA)
• Tangents and normals to curves (Edexcel, OCR, AQA)
• Rates of change (Edexcel, OCR, AQA)
• Derivatives of polynomials with positive integer exponents only (AQA)
• Derivatives of polynomials (Edexcel, OCR)
• Higher order derivatives (OCR, AQA)
• Stationary points (OCR, AQA)

#### Integration

• Indefinite integration (Edexcel, AQA)
• Finding constants of integration (Edexcel, AQA)
• Integrating polynomials with positive integer exponents only (AQA)
• Integrating polynomials (Edexcel)
• Definite integration (AQA)
• Areas under curves (AQA)

### Core 2

#### Algebra and Functions

• Indices (AQA)
• Polynomial Division (Edexcel, OCR)
• Algebraic long division (Edexcel, OCR)
• Factor theorem (Edexcel, OCR)
• Remainder theorem (Edexcel, OCR)
• Graphs (Edexcel, OCR, AQA)
• Graph transformations (AQA)
• General graph sketching with calculus (Edexcel, presumably OCR and AQA)

#### Coordinate Geometry

• Circles (Edexcel)
• Drawing circles (Edexcel)
• Geometric properties (Edexcel)
• Tangents and normals to circles (Edexcel)
• Cartesian equation (Pythagorean form) (Edexcel)

#### Sequences and Series

• Sequences and sequence notation (AQA, OCR)
• Recurrence relations (AQA, OCR)
• Series and sigma notation (AQA, OCR)
• Limits and convergence (Edexcel, AQA, OCR)
• Arithmetic series (AQA, OCR)
• Sum of an arithmetic series (AQA, OCR)
• Geometric series (Edexcel, AQA, OCR)
• Sum of a convergent geometric series (Edexcel, AQA, OCR)
• Binomial expansions (Edexcel, AQA, OCR)
• Factorials (Edexcel, AQA, OCR)
• Combinatorics (Edexcel, AQA, OCR)
• Expansion of (integer case) (Edexcel, AQA, OCR)

#### Differentiation

• Derivatives of polynomials (AQA)
• Higher order derivatives (Edexcel)
• Stationary points (Edexcel)

#### Integration

• Indefinite integration (OCR)
• Finding constants of integration (OCR)
• Integrating polynomials (OCR, AQA)
• Definite integration (OCR, Edexcel)
• Areas under curves (OCR, Edexcel)
• The trapezium rule (OCR, Edexcel, AQA)

#### Exponentials and Logarithms

• Exponential functions (AQA, OCR, Edexcel)
• Graphs of exponential functions (AQA, OCR, Edexcel)
• Logarithms (AQA, OCR, Edexcel)
• Graphs of logarithms (AQA, OCR, Edexcel)
• Solution of (AQA, OCR, Edexcel)

#### Trigonometry

• Sine, cosine and tangent (AQA, OCR, Edexcel)
• Exact values (AQA, OCR, Edexcel)
• Graphs of sine, cosine and tangent (AQA, OCR, Edexcel)
• Sine rule (AQA, OCR, Edexcel)
• Cosine rule (AQA, OCR, Edexcel)
• Area of a triangle (AQA, OCR, Edexcel)
• Simple identities (AQA, OCR, Edexcel)
• Radians (AQA, OCR, Edexcel)
• Arc lengths (AQA, OCR, Edexcel)
• Sector areas (AQA, OCR, Edexcel)
• Trigonometric equations (AQA, OCR, Edexcel)

### Core 3

#### Algebra and Functions

• Simplification of rational expressions (Edexcel)
• Functions (Edexcel)
• Domains and ranges (Edexcel)
• Function composition (Edexcel)
• Inverse functions and their graphs (Edexcel)
• The modulus function (Edexcel)
• Graphs involving the modulus function (Edexcel)

#### Trigonometry

• Secant, cosecant and cotangent (Edexcel)
• Graphs of secant, cosecant and cotangent (Edexcel)
• Arcsin, arccos and arctan (Edexcel)
• Graphs of arcsin, arccos and arctan (Edexcel)
• Basic identities involving secant, cosecant and cotangent (Edexcel)
• Double angle formulae (Edexcel)
• Addition formulae (Edexcel)
• Simplifying (Edexcel)

#### Exponentials and Logarithms

• Euler's number (Edexcel)
• The natural logarithm (Edexcel)

#### Differentiation

• Derivative of (Edexcel)
• Derivative of the natural logarithm (Edexcel)
• Derivative of sine (Edexcel)
• Derivative of cosine (Edexcel)
• Derivative of tangent (Edexcel)
• Product rule (Edexcel)
• Quotient rule (Edexcel)
• Chain rule (Edexcel)

#### Numerical Methods

• Solving equations numerically (Edexcel)
• Considering changes of sign (Edexcel)
• Using given iterative procedures (Edexcel)

#### Proof

• Disproof by counter-example (Edexcel)
• Proof by deduction (Edexcel)
• Proof by contradiction (Edexcel)

### Core 4

#### Algebra and Functions

• "Rational functions" (which could mean anything) (Edexcel)
• Partial fractions (Edexcel)

#### Co-ordinate Geometry

• Parametric equations (Edexcel)
• Converting from cartesian to parametric form (Edexcel)
• Converting from parametric to cartesian form (Edexcel)
• Areas under curves given by parametric equations (Edexcel)

#### Sequences and Series

• Series expansion of for any rational n (Edexcel)

#### Differentiation

• Implicit differentiation (Edexcel)
• Differentiating parametric curves (Edexcel)
• Differentiating (Edexcel)
• Formation of differential equations (Edexcel)

#### Integration

• Integration of exponential functions (Edexcel)
• Integration of trigonometric functions (Edexcel)
• Integration by substitution (Edexcel)
• Integration by parts (Edexcel)
• Integration using partial fractions (Edexcel)
• General procedure for evaluating unfamiliar integrals (Edexcel)
• Volumes of revolution (Edexcel)

#### Differential Equations

• Separating variables (Edexcel)

#### Vectors

• Definition of a vector (Edexcel)
• Vector addition (Edexcel)
• Multiplication by scalars (Edexcel)
• Magnitude (Edexcel)
• Unit vectors (Edexcel)
• , and (Edexcel)
• Position vectors (Edexcel)
• Distance between two points (Edexcel)
• Vector equations of lines (Edexcel)
• Intersections between lines (Edexcel)
• Skew lines (Edexcel)
• The dot product (Edexcel)
• Expression in terms of components (Edexcel)
• Angles between lines (Edexcel)
• Perpendicular lines (Edexcel)

### Further Pure Mathematics 1

#### Inequalities

• Solution of inequalities involving fractions (Edexcel)
• Solution of inequalities involving the modulus function (Edexcel)

#### Series

• Summation identities (Edexcel)
• Summing unknown series (Edexcel)
• The method of differences (Edexcel)

#### Numerical Solution of Equations

• Interval bisection (Edexcel)
• Linear interpolation (Edexcel)
• The Newton-Raphson process (Edexcel)

#### First-Order Differential Equations

• Solving linear differential equations (Edexcel)
• Using given substitutions (Edexcel)
• Homogeneous equations (Comes up often on Edexcel, but not on the syllabus)
• Bernoulli equations (Comes up often on Edexcel, but not on the syllabus)

#### Second-Order Differential Equations

• Solving linear second-order differential equations with constant coefficients (Edexcel)
• General solution to (Edexcel)
• General solution to (Edexcel)
• Finding the constants (Edexcel)
• Using given substitutions (Edexcel)
• Euler-Cauchy equations (Comes up often on Edexcel, but not on the syllabus)

#### Polar Coordinates

• Definition of polar coordinates (Edexcel)
• Sketching curves in polar co-ordinates (Edexcel)
• Common curves (Edexcel)
• Tangents of curves in polar co-ordinates (Edexcel)
• Areas of curves in polar co-ordinates (Edexcel)

### Further Pure Mathematics 2

#### Conic Sections

• Parabolas (Edexcel)
• Cartesian equation (Edexcel)
• Parametric equation (Edexcel)
• Focus-directrix properties (Edexcel)
• Tangents and normals (Edexcel)
• Ellipses (Edexcel)
• Cartesian equation (Edexcel)
• Parametric equation (Edexcel)
• Focus-directrix properties (Edexcel)
• Tangents and normals (Edexcel)
• Hyperbolas (Edexcel)
• Cartesian equation (Edexcel)
• Parametric equation (Edexcel)
• Focus-directrix properties (Edexcel)
• Tangents and normals (Edexcel)
• Loci problems (Edexcel)

#### Intrinsic Coordinates

• Definition of intrinsic coordinates (Edexcel)
• Intrinsic coordinate identities (Edexcel)
• Deriving intrinsic equations from cartesian equations (Edexcel)
• Deriving intrinsic equations from parametric equations (Edexcel)
• The radius of curvature (Edexcel)
• Curves with cartesian equations (Edexcel)
• Curves with parametric equations (Edexcel)
• Curves with intrinsic equations (Edexcel)

### Further Pure Mathematics 3

#### Complex Numbers

• Euler's formula (Edexcel)
• Relation between hyperbolic functions and trigonometric functions (Edexcel)
• De Moivre's theorem (Edexcel)
• Deriving trigonometric identities (Edexcel)
• Roots of a complex number (Edexcel)
• Loci in the Argand diagram (Edexcel)
• Transformations between planes (Edexcel)

#### Maclaurin and Taylor Series

• Maclaurin series (Edexcel)
• Taylor series (Edexcel)
• Series solutions of differential equations (Edexcel)

#### Numerical Solution of Differential Equations

• Approximation identities (Edexcel)

#### Proof by induction

• Definition of proof by induction (Edexcel)
• Common types of proof by induction (Edexcel)
• Series (Edexcel)
• Divisibility (Edexcel)
• Inequalities (Edexcel)
• Sequences (Edexcel)

### Mechanics 1

#### Mathematical Models in Mechanics

• Particles (Edexcel)
• Laminae (Edexcel)
• Rigid bodies (Edexcel)
• Rods (Edexcel)
• Light rods (Edexcel)
• Uniform rods (Edexcel)
• Inextensible strings (Edexcel)
• Smooth and rough surfaces (Edexcel)
• Light smooth pulleys (Edexcel)
• Beads (Edexcel)
• Wires (Edexcel)
• Pegs (Edexcel)

#### Vectors

• Definition of a vector (Edexcel)
• Vector addition (Edexcel)
• Multiplication by scalars (Edexcel)
• Magnitude (Edexcel)
• Unit vectors (Edexcel)
• and (Edexcel)
• Resolving vectors into two components (Edexcel)
• Velocities, displacements, forces and accelerations as vectors (Edexcel)
• Vector diagrams (Edexcel)

#### Kinematics

• Graphs (Edexcel)
• Displacement-time graphs (Edexcel)
• Velocity-time graphs (Edexcel)
• Constant acceleration formulae (Edexcel)

#### Statics

• Resolving forces (Edexcel)
• Common forces on a particle (Edexcel)
• Weight (Edexcel)
• Normal reaction (Edexcel)
• Tension and thrust (Edexcel)
• Friction (Edexcel)

#### Dynamics

• Forces (Edexcel)
• Newton's First Law (Edexcel)
• Newton's Second Law (Edexcel)
• Newton's Third Law (Edexcel)
• The motion of two connected particles (e.g. by a pulley) (Edexcel)
• Inclined planes (Edexcel)
• Impulse (Edexcel)
• Momentum (Edexcel)
• The impulse-momentum principle (Edexcel)
• Conservation of linear momentum (Edexcel)
• Collisions in one dimension (Edexcel)

#### Moments

• Moment of a force (Edexcel)

### Mechanics 2

#### Centres of Mass

• Definition of the centre of mass (Edexcel)
• Centre of mass of a system of bodies (Edexcel)
• Centres of mass of symmetric bodies (Edexcel)
• Centres of mass of plane lamina (Edexcel)
• Centre of mass of a triangle (Edexcel)
• Centre of mass of a circular arc (Edexcel)
• Centre of mass of a sector of a circle (Edexcel)
• Centre of mass of a composite lamina (Edexcel)
• Equilibrium of a plane lamina (Edexcel)
• When suspended from a fixed point (Edexcel)
• When free to rotate about a horizontal axis (Edexcel)
• When on an inclined plane (Edexcel)

#### Work, Energy and Power

• Mechanical energy (Edexcel)
• Kinetic energy (Edexcel)
• Potential energy (Edexcel)
• Conservation of mechanical energy (Edexcel)
• Work (Edexcel)
• Work-energy principle (Edexcel)
• Using the work-energy principle to solve problems in dynamics (Edexcel)
• Power (Edexcel)
• Motors running at constant power (Edexcel)

#### Collisions and Impulse

• Momentum as a vector (Edexcel)
• Newton's law of restitution (Edexcel)
• Collisions between elastic particles (Edexcel)
• Loss of mechanical energy due to impact (Edexcel)
• Successive collisions (Edexcel)

#### Statics

• Equilibrium of a rigid body (Edexcel)
• Using moments (Edexcel)
• Ladder problems (Edexcel)

(Not on Edexcel)

(Not on Edexcel)

### Mechanics 3

• (Edexcel)

#### Elastic Strings and Springs

• Springs and strings (Edexcel)
• Hooke's Law (Edexcel)
• Energy stored in an elastic spring or string (Edexcel)

#### Dynamics

• Newton's Second Law for variable forces (Edexcel)
• Simple harmonic motion (Edexcel)
• Amplitude (Edexcel)
• Frequency and time period (Edexcel)
• Displacement at a given time (Edexcel)
• Velocity at a given time (Edexcel)
• Proving that something moves with simple harmonic motion (Edexcel)

#### Circular Motion

• Angular velocity (Edexcel)
• Acceleration in circular motion (Edexcel)
• Relations to angular velocity, velocity and radius of circle (Edexcel)
• Circular motion under a varying force (possibly Edexcel)
• Particles moving in horizontal circles (Edexcel)
• Particles moving in vertical circles (Edexcel)
• Condition for particles moving in a full vertical circle (Edexcel)

#### Statics

• Centres of mass of rigid bodies (Edexcel)
• Use of integration (Edexcel)
• Equilibrium of rigid bodies (Edexcel)
• When suspended from a fixed point (Edexcel)
• When on an inclined plane (Edexcel)

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