# Pure Maths Contents

Pure maths tutorials are listed here to help with your course of study

## Prior Knowledge

## Algebra Basics

- Identity or equation - what is the difference?
- Expanding a single bracket
- Expanding two or more brackets
- Squaring a bracket
- Terms in expressions and equations
- Linear equations with a positive x-term
- Linear equations with a negative x-term
- Linear equations with two x-terms
- Linear equations with brackets
- Fractional linear equations

## Trigonometry Introduction

## Indices

- Introduction to indices (exponents)
- Multiplication rules for indices
- Division rule for indices
- Negative indices
- Fractions raised to a negative index
- Rational (fractional) indices
- Simplifying terms with negative powers
- Expressing terms in the form ax
^{n} - Equations in which the power has to be found
- Summary of indices
- Exam Questions - Indices

## Algebra and Functions

## Identity or Equation?

## Solving Linear Equations

## Indices

- Introduction to indices (exponents)
- Multiplication rules for indices
- Division rule for indices
- Negative indices
- Fractions raised to a negative index
- Rational (fractional) indices
- Simplifying terms with negative powers
- Expressing terms in the form ax
^{n} - Equations in which the power has to be found
- Summary of indices
- Exam Questions - Indices

## Functions

## Factorising

## Completing the Square

## Polynomials

## Inequalities 1

## Graph Transformations

## Sequences and Series

## Working with Sequences and Series

## Arithmetic Progressions

## Geometric Series

## Binomial Expansion

- Binomial expansion
- Binomial expansion formula
- Exam Questions - Binomial expansion, basic expansions
- Exam Questions - Binomial expansion, comparing coefficients
- Exam Questions - Binomial expansion, estimating a value
- Exam Questions - Binomial expansion, other
- Binomial expansion for rational powers
- Validity
- Exam Questions - Binomial expansion
- Using partial fractions with the binomial expansion
- Exam Questions - Partial fractions with the binomial expansion

## Algebra and Functions 2

## Algebraic Long Division

## Factor and Remainder Theorems

## Working with Functions

## Graph Transformations and Asymptotes

## Even and Odd and Periodic Functions

## Logarithmic and Exponential Functions

## Exponential Functions and Logarithms

## The Exponential Function e^{x} and Natural Log Functions

## Modelling Curves of the form y=kx^{n} and y=ka^{x}

## Trigonometry

## Pythagoras’ Theorem

## Trigonometry: Right-Angled Triangles

## Trigonometric Ratios

## Trigonometric Graphs and Transformations

## Applications of Trigonometry

## Trigonometric Equations

## Trigonometric Identities

## Sec θ, Cosec θ and Cot θ

## Inverse trigonometric functions

## Identities & Equations – Pythagorean Type

## Identities – Addition type

## Identities & Equations – Double angle type

## Identities – Half angle type

## Identities – Triple angle type

## Identities – Factor Formulae

## Identities & Equations – Harmonic Formulae

## Small-angle Approximations

## Exam Questions – Mixed trigonometry

## Differentiation 1

## Differentiation – Introduction

## Tangents and Normals

## Stationary Points

## Increasing and Decreasing functions

## Standard Differentials

## The Chain Rule

## The Product and Quotient Rules

## More Standard Differentials

## The Reciprocal Function of dy/dx

## Exam Questions – Differentiation

## Exponential Functions

## Parametric Functions

## Implicit Functions

## Integration 1

## Integration – Introduction

## Equations of Curves

## Definite Integration

## Integration – Common Functions

- Integration:(ax+b)
^{n}types - Exam Questions - Integration:(ax+b)
^{n}types - Integrating exponential functions e
^{x}, e^{ax}and e^{(ax+b)} - Exam Questions - Integrating exponential functions e
^{x}, e^{ax}and e^{(ax+b)} - Integrating reciprocal functions 1/x and 1/(ax+b)
- Exam Questions - Integrating reciprocal functions 1/x and 1/(ax+b)
- Integrals of the form : f '(x)/f(x)
- Integrals of the form : f'(x)e
^{f(x)}

## Integrals of Trigonometric Functions

## Integrals involving Partial fractions

## Integration by Substitution

## Integrating Products of the form f[g(x)]g'(x) by inspection

## Integration by Parts

## Mixed Examples & Exam Question – Integration

## Applications of Integration – Area bound by a curve

## Applications of Integration – Volumes of revolution

- Volume of revolution about the x-axis
- Exam Questions - Volume of revolution about the x-axis
- Volume of revolution about the x-axis generated between curves
- Volume of Revolution about the y-axis
- Exam Questions - Volume of Revolution about the y-axis
- Volume of Revolution about the y-axis generated between curves
- Volume of revolution for a curve given in parametric form
- Exam Questions - Volume of revolution: parametric form

## Numerical Solution of Equations

## Solution of Equations by Numerical methods

- Graphical methods
- Change of sign
- Bisection method for locating a root in a given interval
- Exam Questions - Bisection Method
- Linear interpolation method for locating a root in a given interval
- Exam Questions - Linear Interpolation
- Iteration
- Exam Questions - Iteration
- Newton-Raphson method for locating a root in a given interval
- Exam Questions - Newton-Raphson

## Rational Expressions

## Rational Expressions – Simplifying

- Simplifying algebraic fractions
- Exam Questions - Simplifying a rational expression
- Addition and subtraction of algebraic fractions
- Exam Questions - Addition & subtraction
- Multiplication of algebraic fractions
- Further simplifying of 'stacked fractions'
- Division of algebraic fractions
- Exam Questions - Algebraic long division