Pure Maths Help

Identity or Equation?

1. Identity or equation - what is the difference?

Expanding Brackets

1. Expanding a single bracket
2. Expanding two or more brackets
3. Squaring a bracket

Solving Linear Equations

1. Terms in expressions and equations
2. Linear Equations with a positive x term
3. Linear equations with a negative x-term
4. Linear equations with two x-terms
5. Linear equations with brackets
6. Fractional linear equations

Indices (Exponents)

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

Surds

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

Functions

1. f(x) notation

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

Completing the Square

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

Polynomials

1. Polynomials

Simultaneous Equations

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

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

Quadratic Equations – Roots and Discriminant

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

Quadratic Graphs

1. Sketching quadratic graphs

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. Quadratic inequalities
7. Exam Questions - Quadratic inequalities
8. Exam Questions - Simultaneous inequalities

Sketching Cubic and Reciprocal Curves

1. Sketching cubic curves
2. Sketching reciprocal curves of the form y = k/x
3. The graph of f=1/f(x) given the graph of y=f(x)

Asymptotes

1. Asymptotes - horizontal and vertical types

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

Working with Sequences and Series

1. Definition and finding the nth term
2. Recurrence relationships
3. Sigma notation
4. Exam Questions - Recurrence relationships

Arithmetic Progressions

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

Geometric Series

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

Gradient

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

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

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

Useful Methods

1. Shortest distance from a point to a line
2. Acute angle between two intersecting lines

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

Parametric Equations

1. Parametric equations
2. Converting to Cartesian form
3. Exam Questions - Parametric to Cartesian equations
4. Sketching parametric graphs
5. Exam Questions - Parametric equations

Algebraic Long Division

1. Algebraic long division

Factor and Remainder Theorems

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

Working with Functions

1. f(x) notation
2. Domain and range
3. Exam Questions - Domain and range
4. Combination of functions
5. The inverse of a function
6. Graphical relationship between f(x) and its inverse
7. Exam Questions - Inverse functions
8. Exam Questions - Functions
9. Injections, surjections, bijections
10. Composition of functions and inverse functions

Graph Transformations and Asymptotes

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

Even and Odd and Periodic Functions

1. Even and odd and periodic functions

Modulus Functions

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

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. Modelling exponential growth and decay
7. Exponential and log equations
8. Simultaneous equations
9. Solving inequalities
10. Exam Questions - Logarithms

The Exponential Function e^x and Natural Log Functions

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

Modelling Curves of the form y=kxⁿ and y=ka^x

1. Modelling curves - Converting to linear form

Pythagoras’ Theorem

1. Pythagoras' theorem

Trigonometry: Right-Angled Triangles

1. Introduction to trigonometry
2. Finding the length of a side
3. Finding an angle

Trigonometric Ratios

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

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

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

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

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

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

Inverse trigonometric functions

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

Identities & Equations – Pythagorean Type

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

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

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

Identities – Half angle type

1. Examples using half angle identities

Identities – Triple angle type

1. Identity for cos 3θ and sin 3θ

Identities – Factor Formulae

1. Proving identities using the factor formulae

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

Small-angle Approximations

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

Differentiation – Introduction

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

Tangents and Normals

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

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

Increasing and Decreasing functions

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

Standard Differentials

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

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

The Product and Quotient Rules

1. The product rule
2. The quotient rule

More Standard Differentials

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

The Reciprocal Function of dy/dx

1. The reciprocal function of dy/dx

Exponential Functions

1. Differentiation: Exponential functions of the form y=a^x

Parametric Functions

1. Differentiation: Parametric functions
2. Tangents and normals
3. Stationary points
4. Exam Questions - Parametric functions

Implicit Functions

1. Implicit functions
2. Tangents and normals
3. Stationary points
4. Exam Questions - Implicit functions

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

Integration – Introduction

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

Equations of Curves

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

Definite Integration

1. Definite integration
2. Exam Questions - Definite Integration
3. Convergent and Divergent Integrals

Integration – Common Functions

1. Integration:(ax+b)ⁿ types
2. Exam Questions - Integration:(ax+b)ⁿ types
3. Integrating exponential functions e^x, e^ax and e^(ax+b)
4. Exam Questions - Integrating exponential functions e^x, e^ax 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)e^f(x)

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

Integrals involving Partial fractions

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

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

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

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

Integration by Parts

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

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 bound by a curve and y-axis
4. Exam Questions - Area bound by a curve and y-axis
5. Area under a graph : parametric type

Applications of Integration – Volumes of revolution

1. Volume of revolution about the x-axis
2. Exam Questions - Volume of revolution about the x-axis
3. Volume of revolution about the x-axis generated between curves
4. Volume of Revolution about the y-axis
5. Exam Questions - Volume of Revolution about the y-axis
6. Volume of Revolution about the y-axis generated between curves
7. Volume of revolution for a curve given in parametric form
8. Exam Questions - Volume of revolution: parametric form

Numerical Integration

1. Trapezium rule
2. Exam Questions - Trapezium rule
3. Simpson's Rule
4. Exam Questions - Simpson's Rule
5. Mid-ordinate Rule

Solution of Equations by Numerical methods

1. Graphical methods
2. Change of sign
3. Bisection method for locating a root in a given interval
4. Exam Questions - Bisection Method
5. Linear interpolation method for locating a root in a given interval
6. Exam Questions - Linear Interpolation
7. Iteration
8. Exam Questions - Iteration
9. Newton-Raphson Method NEW!!
10. Newton-Raphson method for locating a root in a given interval
11. Exam Questions - Newton-Raphson

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

Partial Fractions

1. Partial fractions
2. Denominator contains 2 or 3 linear factors
3. Denominator contains repeated factors
4. Exam Questions - Partial fractions
5. Denominator contains 1 linear and 1 quadratic factor
6. Improper types

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
4. Separating the variables and sketching a family of curves
5. Geometric interpretation using slope fields, including identification of isoclines

Forming differential equations

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

Vectors

1. What is a vector and a scalar quantity?
2. Vector notation
3. Position vectors
4. Equal and negative vectors
5. Multiplying a vector by a scalar
6. Addition and subtraction of vectors
7. Magnitude of a 2 dimensional vector
8. Magnitude of a 3 dimensional vector
9. Unit vectors

Scalar Product (Dot Product)

1. Scalar product
2. Perpendicular vectors
3. Exam Questions - Scalar product

Vector Equations of Lines

1. Vector equation of a line
2. Angle between two lines
3. Parallel lines
4. Intersecting and skew lines
5. Exam Questions - Parallel intersecting and skew lines
6. Closest point to a line and shortest distance from the origin
7. Shortest distance of a point to a line

Permutations and Combinations

The Language of Mathematics

1. Problem solving
2. Types of numbers
3. Writing mathematics
4. The converse of a theorem
5. Proof

Venn Diagrams

1. Algebra of sets
2. De' Morgan's Laws
3. Distributive, associative and commutative laws
4. 3 set problems

Cartesian Product

1. Cartesian Product

Relations

Logic

1. Introduction to logic
2. Compound statements and symbols
3. Truth tables : negation
4. Truth tables : conjunction (and)
5. Truth tables : resolving an ambiguity - the 'or' connective
6. Logical equivalence, tautologies and contradictions
7. Compound statements made up from three simple statements
8. Arguments