Noteful Mathematics

• Structure of Mathematical Proof
• Logical Connectives
• Disproof by Counter Example
• Proof by Deduction
• Proof by Exhaustion
• Proof by Contradiction

GCSE Recap

• Indices
• Surds
• Simultaneous Equations
• Quadratic Functions
• Inequalities
• Polynomials

The Modulus Function

• Understand and Use the Modulus Function
• Solve Equations and Inequalities involving the Modulus
• Solve Equations and Inequalities involving the Modulus Graphically
• Sketch the Modulus and the Modulus of Linear Functions

Functions

• Understand and be able to use the definiton of a function
• Vocabulary and associated notation i.e. many-one, one-many, one-one, mapping, image, range, domain
• Definitions of range and domain
• Understand and be able to use inverse functions and their graphs
• Know the conditions for the inverse function to exist
• Be able to find the inverse of a function either graphically, by reflection in the line y = x, or algebraically
• Composite functions

Graph Transformations

Partial Fractions

• Decompose rational functions into partial fractions
• Denominator will be up to squared terms and maximum three terms
• Numerator will be numerial or linear
• Use of factor theorem and polynomial division to form rational functions
• For Integration see later

Straight Lines Summary

• Equations of Straight lines
• Draw graphs of straight lines
• Find the midpoint of a line segment
• Find the distance between two points
• Find the intersection of two lines
• Gradient conditions of parallel and perpendicular lines
• Use straight line models in a variety of contexts

Circles

• Understand and be able to use the coordinate geometry of a circle including using the equation of a circle
• Be able to draw a circle given its euation or form the equation given its centre and radius
• Be able to complete the square to find the centre and radius of a circle
• Be able to investigate wheter a line and a circle or two circles intersect

• Circle Theorems

Parametric Equations

• Understand and use parametric equations of curves
• Convert between cartesian and parametric forms
• Sketch simple parametric curves
• Use parametric curves in modelling in a variety of contexts

Calculus with Parametric Equations

• Be able to differentiate simple functions defined parametrically
• Find the gradient of a point on a curve and use to find the equations of tangents and normals, and solve associated problems

Binomial Expansion

• Understand and be able to use the binomial expansion for positive integers of n
• Coefficient notations
• Calculate binomial coefficents
• Relationship to Pascal's traingle
• Know that 0! = 1
• Understand and know the link to binomial probabilities (see probability section)

• Extend the binomal expansion for any rational n
• Write (a+bx)ⁿ as aⁿ(1 + ^bx⁄_a)ⁿ
• Know when the expansion is valid

Arithmetic Sequences

Definitions and Introduction

• Understand and use definitions for sin x, cos x and tan x
• Understand the functions, their graphs, symmetries and periodicities
• Know and use exact values of sin x, cos x, tan x for x = 0, 30, 45, 60, 90 (not for tan), 180 degrees and multiples thereof
• Know and be able to use simple trigonometric identities
• Be able to solve trigonometric equations in a given interval including quadratic equations in sin x, cos x and tan x and equations involving multiples of the known angle

Triangle Geometry

• Understand and use the sine and cosine rules
• Understand and be able to use the area of a triangle

Radians

• Be able to work with radian measure
• Use the relationship between degrees and radians
• Know and able to use exact values of sin x, cos x, tan x (table in notes)
• Extend knowledge of trigonmetric equations and identities to include radians

• Be able to work with radian measure and use for arc length and sector

Inverse Functions

• Understand and use definitions of arcsin x, arccos x and arctan x and their relationship to sin x, cos x and tan x respectively
• Know their graphs and relate them to the graphs of sin x, cos x and tan x

Modelling

• Be able to use trigonometric functions to solve problems in context

a cos x + b sin x

Small Angle Approximations

• Understand and be able to use the standard small angle approximations

Compound & Double Angle Formulae

• Understand and be able to use the double angle formulae and compound angle formulae
• Understand the geometric proofs of these formulae
• Be able to prove the double angle formulae
• Be able to construct proofs involving trigonometric functions and identities
• Use the formulae to solve trigonometric equations, prove identities or evaluate integrals

Reciprocal Functions

• Understand and be able to use the definitions of secant (sec x), cosecant (cosec x) and contangent (cot x)
• Understand and be able to use their relationship with sin x, cos x and tan x
• Understand the graphs of the functions, their ranges and domains
• Understand and be able to use the identities (see notes)
• Use the identities to solve trigonometric equations, prove identities or evaluate integrals

Indefinite Integration

• Know and be able to use the fundamental theorem of calculus
• Understand and be able to use the terms indefinite and definite when applied to integrals
• Be able to integrate x_n (except n = -1) and related sums, differences and multiples
• Solve problems including evaluating a constant of integration

Definite Integration and Complex Functions

• Be able to integrate e_kx,¹⁄_x, sin kx, cos kx, and related sums, differences and constant multiples
• Integrals of arcsin, arccos and arctan will be given if required
• This includes using trigonmetric relations (eg double angle formulae) to integrate functions such as cos₂x
• Be able to evaluate definite integrals
• Be able to use a definite integral to find the area between a curve and x-axis

Partial Fractions and Integration

• Be able to integrate functions using partial fractions that have linear terms in their denominator

Complicated Areas

Integration as the limit of a sum

Integration by Substitution

Integration by Parts

Separable Differential Equations

Vectors in 2D

• Be able to use vectors in two dimensions
• Column vectors and i,j notation
• Difference between a scalar and vector and distinguish between them when writing
• Calculate magnitude and direction of a vector
• Convert between magnitude/direction form and component form
• Calculate the modulus of a vector and interpret as magnitude
• Be able to add vectors diagrammatically
• Perform vector addition
• Multiply vectors by scalars
• Understand and be able to use position vectors
• Understand the meaning of displacement vector, component vector, resultant vector, parallel vector, equal vector and unit vector
• Calculate the distance between two points represented by position vectors
• Use vectors to solve problems in pure mathematics and in context, including forces (see forces notes)
• Use vectors to solve problems in kinematics (see kinematics notes)

Vectors in 3D

• Be able to use vectors in three dimensions
• Column vectors and i,j notation
• Extend points above to 3D (excluding the direction of a 3D vector)