AQA Mathematics A-Level : Pure Maths

Outlined below are the pure maths / core topics covered for AQA Mathematics A-Level

It is advisable to check the official AQA Mathematics A-Level specification for any changes.

Contents for Pure Maths AQA

Prior Knowledge
Algebra and Functions 1
Coordinate Geometry
Algebra and Functions 2
Sequences and Series
Trigonometry
Logarithmic and Exponential Functions
Differentiation
Integration
Numerical Methods
Vectors
Proof
Past Papers

Prior Knowledge

Algebra Basics

  1. Expanding a single bracket
  2. Expanding two or more brackets
  3. Squaring a bracket
  4. Terms in expressions and equations
  5. Identity or equation - what is the difference?
  6. Linear Equations with a positive x term
  7. Linear equations with a negative x-term
  8. Linear equations with two x-terms
  9. Linear equations with brackets
  10. Fractional linear equations
  11. f(x) notation
  12. Polynomials

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Polynomials

  1. Polynomials

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Pythagoras’ Theorem

  1. Pythagoras' theorem

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Trigonometry Introduction

  1. Introduction to trigonometry
  2. Finding the length of a side
  3. Finding an angle
  4. Mixed Exercise - Pythagoras and Trigonometry

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Algebra and Functions : 1

Indices

  1. Introduction to indices (exponents)
  2. Multiplication rules for indices
  3. Division rule for indices
  4. Negative indices
  5. Fractions raised to a negative index
  6. Rational (fractional) indices
  7. Simplifying terms with negative powers
  8. Expressing terms in the form axn
  9. Equations in which the power has to be found
  10. Summary of indices
  11. Exam Questions - Indices

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Surds

  1. Surds - introduction & simplifying
  2. Addition and subtraction of surds
  3. Multiplying surds
  4. Dividing surds
  5. Rationalising surds
  6. Exam Questions - Surds

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Functions

  1. f(x) notation
  2. Mappings
  3. Mappings - More examples
  4. Mappings, functions or both?

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Factorising

  1. Introduction to factorising
  2. Highest common factor (HCF)
  3. Factorising by grouping
  4. Factorising quadratic expressions
  5. Mixed Exercise - Factorising
  6. Exam Questions - Factorising

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Completing the Square

  1. Completing the square
  2. Applications of completing the square
  3. Exam Questions - Completing the square

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Quadratic Equations

  1. Solve by factorising
  2. Solve by completing the square
  3. Solve by the quadratic formula
  4. Exam Questions - Solved by the quadratic formula
  5. Solving quadratic equations in some function of x

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Quadratic Equations – Roots and Discriminant

  1. Roots and discriminant of a quadratic equation
  2. Exam Questions - Roots and discriminant

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Quadratic Graphs

  1. Sketching quadratic graphs

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Simultaneous Equations

  1. Elimination method for linear equations
  2. Substitution method for linear and non-linear equations
  3. Exam Questions - Simultaneous equations

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Inequalities

  1. Linear inequalities
  2. Rules for reversing the inequality sign
  3. Solving a linear type
  4. Solving a double inequality
  5. Exam Questions - Solving a double inequality
  6. Shading regions for a linear inequality
  7. Quadratic inequalities
  8. Exam Questions - Quadratic inequalities
  9. Exam Questions - Simultaneous inequalities

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Algebraic Long Division

  1. Algebraic long division
  2. Converting Improper algebraic fractions to mixed fractions

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Factor Theorem

  1. The factor theorem
  2. How to solve a cubic equation
  3. Exam Questions - Factor theorem

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Rational Expressions – Simplifying

  1. Simplifying algebraic fractions
  2. Exam Questions - Simplifying a rational expression
  3. Addition and subtraction of algebraic fractions
  4. Exam Questions - Addition & subtraction
  5. Multiplication of algebraic fractions
  6. Further simplifying of 'stacked fractions'
  7. Exam Questions - Algebraic long division

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Coordinate Geometry

Gradient of Straight Lines

  1. Line segment
  2. Parallel lines
  3. Perpendicular lines

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Straight Lines

  1. Equation of a straight line: y=mx+c
  2. Equation of a line given the gradient and point
  3. Distance between two points
  4. Mid-point of a line segment
  5. Equation of a parallel line
  6. Equation of a perpendicular bisector

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Intersection of Graphs

  1. Intersection of two straight lines
  2. Intersection of a straight line and a parabola
  3. Intersection of a straight line and a hyperbola
  4. Nature of the intersection

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Exam Questions – Straight Lines

  1. Exam Questions - Straight lines

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Circles

  1. Equation of a circle
  2. Finding the centre and radius
  3. Circle properties
  4. Equation of a tangent to a circle
  5. Equation of a circle through 3 points
  6. Exam Questions - Circles

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Parametric Equations

  1. Parametric equations
  2. Converting to Cartesian form
  3. Exam Questions - Parametric to Cartesian equations
  4. Sketching parametric graphs
  5. Parametric equations of circle, ellipse, parabola and hyperbola
  6. Finding Points of Intersection between a Parametric and Cartesian Equation
  7. Exam Questions - Parametric equations

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Algebra and Functions : 2

Sketching Cubic and Reciprocal Curves

  1. Sketching cubic curves
  2. Sketching reciprocal curves of the form y = k/x

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Modulus Functions, Equations and Inequalities

  1. The modulus function
  2. Graphing y=|f(x)|
  3. Modulus equations
  4. Exam Questions - Modulus equations
  5. Modulus inequalities
  6. Exam Questions - Modulus inequalities

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Working with Functions

  1. Mappings
  2. Mappings - More examples
  3. Mappings, functions or both?
  4. f(x) notation
  5. Domain and range
  6. Exam Questions - Domain and range
  7. Combination of functions
  8. The inverse of a function
  9. Graphical relationship between f(x) and its inverse
  10. Exam Questions - Inverse functions
  11. Exam Questions - Functions

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Graph Transformations

  1. Basic graphs used in transformations
  2. Translations of graphs
  3. Reflections of graphs
  4. Stretches of graphs
  5. Exam Questions - Graph transformations

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Asymptotes

  1. Asymptotes - horizontal and vertical types

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Partial Fractions

  1. Partial fractions
  2. Denominator contains 2 or 3 linear factors
  3. Denominator contains repeated factors
  4. Exam Questions - Partial fractions

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Sequences and Series

Binomial Expansion

  1. Binomial expansion
  2. Exam Questions - Binomial expansion for positive integer powers
  3. Exam Questions - Binomial expansion, comparing coefficients
  4. Exam Questions - Binomial expansion, estimating a value
  5. Exam Questions - Binomial expansion, other
  6. Exam Questions - Binomial expansion for rational and negative powers
  7. Exam Questions - Partial fractions with the binomial expansion

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Working with Sequences and Series

  1. Definition and finding the nth term
  2. Increasing and decreasing sequences
  3. Recurrence relationships
  4. Sigma notation
  5. Exam Questions - Recurrence relationships
  6. Periodic Sequences

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Arithmetic Sequence and Series

  1. Arithmetic progressions
  2. Sum of the first n terms
  3. Finding a and d given two terms
  4. Working with consecutive terms
  5. Exam Questions - Arithmetic sequences and series
  6. Examsolutions Beastie - Arithmetic progressions

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Geometric Sequence and Series

  1. Geometric series
  2. Proof of sum of first n terms, Sn
  3. Sum to infinity
  4. Exam Style Questions - Geometric Series and Progressions
  5. Exam Questions - Geometric series

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Trigonometry

Trigonometric Ratios

  1. Trigonometric ratios for 30°, 45° and 60°
  2. Trig ratios for multiples of 30°, 45° and 60°

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Trigonometric Graphs and Transformations

  1. Trigonometric graphs
  2. Translations of trig graphs
  3. Reflections of trig graphs
  4. Stretches of trig graphs
  5. Combining transformations
  6. Exam Questions - Trigonometric graphs and transformations

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Applications of Trigonometry

  1. Area of a triangle - Given two sides and an included angle
  2. Sine rule
  3. Exam Questions - Sine rule
  4. Cosine rule
  5. Radians
  6. Arcs, sectors and segments
  7. Exam Questions - Arcs, sectors and segments

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Trigonometric Equations

  1. Quadrant rule to solve trig equations
  2. Trig equations with different ranges
  3. Trig equations with multiple angles
  4. Trig equations that factorise

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Trigonometric Identities

  1. Using the identities: tanθ ≡sinθ/cosθ and sin²θ+cos²θ ≡1
  2. Using the identities: cos(θ) ≡ cos(-θ), sin(θ) ≡ -sin(-θ)
  3. Solving equations using identities
  4. Exam Questions - Trigonometric identities

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Sec θ, Cosec θ and Cot θ

  1. Trig functions sec θ, cosec θ and cot θ
  2. Graphs of sec θ, cosec θ and cot θ

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Inverse trigonometric functions

  1. Inverse trigonometric functions - arcsin x, arccos x, arctan x
  2. Examples using Inverse trigonometric functions

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Identities & Equations – Pythagorean Type

  1. sin²x + cos²x ≡1 , 1 + tan²x ≡ sec²x , 1 + cot²x ≡ cosec²x
  2. Solving equations using Pythagorean identities

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Small-angle Approximations

  1. Small-angle approximations
  2. Exam Questions - Small Angle Approximations

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Identities – Addition type

  1. sin(A±B), cos(A±B) and tan(A±B)
  2. Using the Addition formulae to get exact values
  3. Proving identities using the addition formulae
  4. Identities - Addition type - Equations

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Identities & Equations – Double angle type

  1. Identities for sin2A, cos2A and tan2A
  2. Examples using double angle identities
  3. Solving equations using double angle identities
  4. Exam Questions - Double Angles

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Identities – Half angle type

  1. Examples using half angle identities

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Identities – Triple angle type

  1. Identity for cos 3θ and sin 3θ

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Identities & Equations – Harmonic Formulae

  1. Harmonic Identities Rsin(x ± α), Rcos(x ± α)
  2. Equations using harmonic identities
  3. Harmonic identities - Max and Min
  4. Exam Questions - Harmonic identities and equations

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Exam Questions – Mixed trigonometry

  1. Exam Questions - Mixed trigonometry

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Logarithmic and Exponential Functions

Exponential Functions and Logarithms

  1. Exponential functions: what they are and their graphs
  2. What do we mean by a log?
  3. Rules of logs
  4. Logarithms - Change of Base
  5. Simplifying and expanding
  6. Converting between Logarithmic and Power (Exponential) equations
  7. Modelling exponential growth and decay
  8. Modelling exponential equations | Exam Questions
  9. Exponential and log equations
  10. Simultaneous equations
  11. Solving inequalities
  12. Exam Questions - Logarithms

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The Exponential Function ex and Natural Log Functions

  1. The exponential function ex
  2. Sketching exponential graphs based on transformations
  3. The natural logarithmic function, ln x
  4. Exam Questions - Natural log functions

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Modelling Curves of the form y=kxn and y=kax

  1. Modelling curves - Converting to linear form

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Differentiation

Differentiation – Introduction

  1. The gradient function dy/dx
  2. Differentiation - terms of the form axn
  3. The second derivative
  4. Exam Questions - Differentiation: introduction

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Tangents and Normals

  1. Equations of tangents and normals
  2. Exam Questions - Tangents and normals

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Stationary Points

  1. Stationary points
  2. Nature of a stationary point
  3. Exam Questions - Stationary points
  4. Applications of stationary points
  5. Exam Questions - Applications of stationary points

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Increasing and Decreasing functions

  1. Increasing and decreasing functions
  2. Exam Questions - Increasing and decreasing functions

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Standard Differentials

  1. Exponential function ex
  2. The natural log function, ln(x)
  3. The trig functions sin(x), cos(x) and tan(x)

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The Chain Rule

  1. Chain rule: Polynomial to a rational power
  2. Chain rule: Exponential types
  3. Chain rule: Natural log types
  4. Chain rule: Trigonometric types

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The Product and Quotient Rules

  1. The product rule
  2. The quotient rule

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More Standard Differentials

  1. The trig functions, sec(x), cosec(x) and cot(x)

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The Reciprocal Function of dy/dx

  1. The reciprocal function of dy/dx

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Exam Questions – Differentiation

  1. Exam Questions - Differentiation methods
  2. Exam Questions - Differentiation: tangents, normals and stationary points
  3. Exam Questions - Exponential rates of change

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Exponential Functions

  1. Differentiation: Exponential functions of the form y=ax

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Parametric Functions

  1. Differentiation: Parametric functions
  2. Tangents and normals

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Implicit Functions

  1. Implicit functions
  2. Implicit functions with product rule
  3. Tangents and normals
  4. Exam Questions - Implicit functions

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Connected Rates of Change

  1. Connected rates of change
  2. Using three connected rates of change
  3. Connected rates of change cone type problems
  4. Exam Questions - Connected rates of change

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Integration

Integration – Introduction

  1. Integration - Terms of the form axn
  2. Exam Questions - Integration: introduction

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Equations of Curves

  1. Finding the equation of a curve given the gradient function
  2. Exam Questions - Equations of curves

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Definite Integration

  1. Definite integration
  2. Exam Questions - Definite Integration

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Integration – Common Functions

  1. Integration:(ax+b)n types
  2. Exam Questions - Integration:(ax+b)n types
  3. Integrating exponential functions ex, eax and e(ax+b)
  4. Exam Questions - Integrating exponential functions ex, eax and e(ax+b)
  5. Integrating reciprocal functions 1/x and 1/(ax+b)
  6. Exam Questions - Integrating reciprocal functions 1/x and 1/(ax+b)
  7. Integrals of the form : f '(x)/f(x)
  8. Integrals of the form : f'(x)ef(x)

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Integrals of Trigonometric Functions

  1. Integrals of sin x, cos x, sec² x
  2. Integrals of the form sin(ax+b), cos(ax+b), sec² (ax+b) types
  3. Integrals Using Trigonometric Identities
  4. sin2x and cos2x types
  5. Exam Questions - Trigonometric types

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Integrals involving Partial fractions

  1. Integrals involving partial fractions
  2. Exam Questions - Integrals involving partial fractions

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Integration by Substitution

  1. Integration by substitution
  2. Square root types
  3. Integration of trigonometric functions by substitution
  4. Integration of exponential types by substitution
  5. Integration by substitution using limits
  6. Integration of trigonometric functions by substitution with limits
  7. Exam Questions - Integration by substitution

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Integrals of the form f[g(x)]g'(x) by inspection

  1. Integrating products of the form f[g(x)]g'(x) by inspection

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Integration by Parts

  1. Integration by parts
  2. Integration by parts (ln types)
  3. eaxsin(bx) and eaxcos(bx) types
  4. Integration by parts using limits
  5. Proof of the formula - Integration by parts
  6. Exam Questions - Integration by parts

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General Methods – Integration

  1. Integration - General Methods
  2. Mixed Examples - Integration
  3. Exam Questions - Integration

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Applications of Integration – Area bound by a curve

  1. Area bound by a curve and x-axis
  2. Exam Questions - Area bound by a curve and x-axis
  3. Area under a graph : parametric type

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Differential equations – Separating the variables

  1. Differential Equations - Finding a general and a particular solution
  2. Working with constants in log types
  3. Differential Equations - Exponential and trig type

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Differential equations – Forming differential equations

  1. Direct proportion type
  2. Inverse proportion type
  3. Newton's law of cooling
  4. Exam Questions – Forming differential equations

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Numerical Methods

Solution of Equations by Numerical methods

  1. Graphical methods
  2. Change of sign
  3. Iteration
  4. Exam Questions - Iteration
  5. Newton-Raphson Method NEW!!
  6. Newton-Raphson method for locating a root in a given interval
  7. Exam Questions - Newton-Raphson

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Numerical Integration

  1. Trapezium rule
  2. Exam Questions - Trapezium rule

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Vectors

Vectors

  1. What is a vector and a scalar quantity?
  2. Vector notation 2d
  3. Vector notation
  4. Position vectors 2d
  5. Position vectors
  6. Equal and negative vectors
  7. Multiplying a vector by a scalar 2d
  8. Multiplying a vector by a scalar
  9. Addition and subtraction of vectors 2d
  10. Addition and subtraction of vectors
  11. Magnitude of a 2 dimensional vector
  12. Magnitude of a 3 dimensional vector
  13. Unit vectors
  14. How to solve vector geometry problems
  15. Vector Methods in Geometry - Parallelogram problem
  16. Exam Questions - Vectors

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Proof

  1. Proving Identities
  2. Proof by Exhaustion and Deduction
  3. Using a Counter-Example
  4. Exam Questions - Proof By Counter Example
  5. Proof by Contradiction

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Past Papers

For further revision and practice I have past papers, some with worked solutions – Check them out.

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Index

algebra and functions
arc length
arithmetic sequence/series
asymptote
binomial expansion
circles
chain rule
change of sign
completing the square
connected rates of change
coordinate geometry
cosine rule
cubic graph
decreasing functions
differential equations:
solving
forming
differentiation:
chain rule
connected rates of change
implicit
parametric
product rule
quotient rule
differentials:
ax
ex
ln x
sin, cos, tan(x)
sec, cosec, cot(x)
discriminant
domain – functions
exponential functions
ex
factorising
factor theorem
functions
geometric sequence/series
gradient – straight line
graphs:
cubic
intersection
reciprocal
implicit differentiation
increasing functions
indices
inequalities
integration
by parts
common functions
general methods
partial fractions
substitution
trigonometric
definite
integration applications
area under a curve
iteration
intersection – graphs
inverse functions
logarithms
natural logarithms
long division – algebraic
maximum stationary point
minimum stationary point
modelling curves
modulus function
Newton-Raphson method
normals to curves
numerical methods
integration
solving equations
parametric:
differentiating
equations
partial fractions
product rule
proof
quadrant rule
quadratics:
discriminant
equations
graphs
quotient rule
radian
range – functions
rational expressions
reciprocal graphs
recurrence relationships
sector
segment
sequences and series
sigma notation
simultaneous equations
sine rule
stationary points
straight lines
surds
tangents to curves
transformations:
graphs
trapezium rule
trigonometric:
equations
graphs
ratios 30°, 60°, 45°
trigonometry
cosine rule
cosec, sec and cot θ
inverse functions
sine rule
small angles
trigonometry – identities:
addition formulae
double angle formulae
half angle formulae
harmonic formulae
Pythagorean
triple angle formulae
vectors