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## Algebra and Functions

### Indices

- Multiplication
- Division
- Negative indices
- Fractions with negative indices
- Rational indices
- Simplifying negative powers
- Expressions terms in the form ax
^{n} - Equation types
- Summary
- Practice questions
- All grades | A*/A | B | C | D/E

### Surds

- Simplifying
- Addition and subtraction
- Multiplication
- Division
- Rationalising
- Practice questions
- All grades | A*/A | B | C | D/E

### Factorising

- Introduction
- HCF types
- Grouping types
- Quadratic expressions
- HCF types
- Difference of two square types
- Trinomials by grouping
- Trinomials by inspection 1 | 2 | 3 | 4 | 5 | 6 | 7
- Misc exercise 1 | 2 | 3
- Practice questions
- All grades | A*/A | B | C | D/E

### Completing the Square

- Completing the square for x
^{2}terms - Completing the square for 2x
^{2}, 3x^{2}, ... terms - Completing the square for negative terms
- Sketching positive graphs
- Sketching negative graphs
- Practice questions
- All grades | A*/A | B | C | D/E

### Quadratic Equations

- Solve by factorising
- Solve by completing the square
- Solve by the quadratic formula
- Solving in some function of x
- Roots and discriminant
- Common types of discriminant questions
- Practice questions
- All grades | A*/A | B | C | D/E

### Simultaneous Equations

- Elimination method 1 | 2 | 3 | 4 | 5 | 6
- Substitution method
- Substitution method example
- Practice questions
- All grades | A*/A | B | C | D/E

### Inequalities

- Introduction
- Rules for reversing the inequality sign
- Solving linear inequalities
- Solving double inequalities
- Linear inequalities in two variables
- Solving quadratic inequalities
- Practice questions
- All grades | A*/A | B | C | D/E

### Polynomials

### Algebraic Long Division

### Factor Theorem

- The factor theorem
- Showing that x-1 is a factor of a cubic polynomial
- Factorising a cubic polynomial (Method 1)
- Factorising a cubic polynomial (Method 2)
- Solving a cubic equation
- Finding constants in a polynomial given the factors

### Rational Expressions

- Simplifying algebraic fractions
- Addition and subtraction of algebraic fractions 1
- Addition and subtraction of algebraic fractions 2
- Multiplication of algebraic fractions
- Simplifying "stacked" fractions
- Division of algebraic fractions

## Coordinate Geometry

### Gradient of Straight Lines

### Straight Lines

- Equation of a line in the form y = mx + c
- Equation of a line in the form y - y
_{1}= m(x - x_{1}) - Distance between two points
- Mid-point of a line segment
- Equation of a parallel line
- Equation of a perpendicular bisector

### Intersection of Graphs

- Two straight lines
- Parabola and a straight line
- Nature of intersection
- Tangent to a curve
- Hyperbola and straight line

### Straight Lines Practice Questions

- All grades | A*/A | B | C | D/E

### Circles

- Equation of a circle
- Finding the centre and radius
- Equation of a tangent
- Equation of a circle through 3 points
- Circle properties

### Parametric Equations

## Algebra and Functions 2

### Sketching Curves

### Modulus Functions

- The Modulus function: |x|
- Graphing y = |f(x)|
- Graphing y = f(|x|)
- Modulus Equations
- Example: How to solve |x + 1| = -2x - 5
- Example: How to solve |x - 2| = 3
- Example: How to solve |3x + 9| = |2x + 1|
- Modulus Inequalities

### Working with Functions

- f(x) notation
- Domain and range 1
- Domain and range 2
- Combination of functions | Example 1 | Example 2
- The inverse of a function | Example 1 | Example 2 | Example 3
- Graphical relationship between f(x) and its inverse | Example 1 | Example 2

### Graph Transformations

- Baisc graphs used in transformations
- Translations
- Reflections
- Stretches
- y = af(x) and y = f(ax)
- How y = af(x) stretches y = f(x) by scale factor a parallel to the y-axis
- How y = f(ax) stretches y = f(x) by scale factor 1/a parallel to the x-axis
- Asymptotes

### Partial Fractions

## Sequences and Series

### Binomial Expansion

- What is nCr?
- Binomial Expansion using the nCr method
- Finding a certain term or coefficient in a Binomial expansion
- Binomial expansion using Pascal's triangle method
- Binomial expansion formula as an alternative to the nCr method
- Rational powers
- Validity
- Partial fractions

### Working with Sequences and Series

### Arithmetic Progressions

- Arithmetic sequences and series
- Sum of the first n terms
- Finding a and d given two terms
- Working with consecutive terms

### Geometric Series

## Trigonometry

### Trigonometric Ratios

- Simple way to learn the trig. ratios for 30, 45 and 60 degrees
- Simple way to work out the trig. ratios for multiples of 30, 45 and 60 degrees

### Graphs and Transformations

- Trigonometric graphs
- Translations parallel to the y-axis
- Translations parallel to the x-axis
- Reflection in the x-axis
- Reflection in the y-axis
- Stretch parallel to the y-axis: y = kf(x)
- Stretch parallel to the x-axis: y = f(kx)
- Combining transformations

### Applications

- Finding the area of a triangle with two sides and an included angle | Proof
- Sine Rule
- Finding the length of a side of a non-right triangle
- Finding an angle of a non-right triangle
- The Ambiguous Case
- Cosine Rule
- Arcs, Sectors and Segments

### Trig. Equations and the Quadrant Rule

- What is the Quadrant Rule/CAST diagram?
- Using the Quadrant Rule to solve trig. equations
- Solving trig. equations in various ranges
- Solving trig. equations with multiple angles
- Solving trig. equations that can be factorised 1 | 2 | 3 | A common mistake

### Identities

- tanθ = sinθ/cosθ and sin
^{2}θ+cos^{2}θ = 1 - Proving trig. identities 1 | 2 | 3
- cos(θ) = cos(-θ) and sin(θ) = -sin(-θ)
- Solving trig. equations using identities 1 | 2 | 3

### Secθ, Cosecθ and Cotθ

### Inverse Trigonometric Functions

- arcsin(x) or sin
^{-1}(x) - arccos(x) or cos
^{-1}(x) - arctan(x) or tan
^{-1}(x) - Examples using inverse trigonometric functions

### Indentities and Equations: Pythagorean Type

- sin
^{2}x + cos^{2}x = 1, 1 + tan^{2}x = sec^{2}x, 1 + cot^{2}x = cosec^{2}x - Proving Pythagorean identities
- Solving equations using Pythagorean identities

### Small-angle Approximations

### Indentities and Equations: Addition Type

- sin(A + B), cos(A + B) and tan(A + B) addition formulae
- Finding exact trig. ratios
- Exact values of sin(A + B) etc
- Proving identitites using the Addition Formulae
- Equations that use the Addition Formulae

### Indentities and Equations: Double Angle Type

- Identities for sin(2A), cos(2A) and tan(2A)
- Proving identities using the Double Angle identities
- Solving equations using Double Angle identities

### Indentities and Equations: Half Angle Type

### Indentities and Equations: Triple Angle Type

### Indentities and Equations: Factor Formulae

### Indentities and Equations: Harmonic Formulae

- Asin(x) + Bcos(x) = Rsin(x + a)
- Asin(x) - Bcos(x) = Rsin(x - a)
- Acos(x) + Bsin(x) = Rcos(x - a)
- Acos(x) - Bsin(x) = Rcos(x + a)
- Solving equations using Harmonic identities
- Max and min values
- Max and min values - fractional type

## Logarthmic and Exponential Functions

### Logarthmic and Exponential Functions

- What is an exponential function?
- What is a log?
- Logarithm Rules
- Equations
- Logarithms - change of base
- Logarithms - change of base (examples)
- Simplifying and expanding equations
- Solving equations where x is in the power
- Solving equations that contain log terms
- Solving equations that contain logs with different bases
- Solving equations that contain exponential functions
- Simultaneous equations 1 | 2 | 3
- Solving inequalities

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

- Exponential Functions
- Transformations of exponential graphs
- The natural log function: ln(x)

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

## Differentiation

### Introduction

- The gradient function dy/dx
- Differentiation from 1st principles
- Terms of the form ax
^{n} - Extending to root types
- Extending to fractional types
- The second derivative

### Tangents and Normals

### Stationary Points

- What are stationary points?
- An example of finding a stationary point
- Nature of a stationary point using 1st differential
- Nature of a stationary point using 2nd differential
- An example of finding stationary points and their nature
- Applications of stationary points
- Increasing and decreasing functions

### Standard Differentials

- Exponential function: e
^{x} - Natural log function: ln(x)
- Trig. functions: sin(x), cos(x) and tan(x)
- How to differentiate sinθ from first principles
- How to differentiate cosθ from first principles

### Differentiating with the Chain Rule

- Polynomial to a rational power
- Exponential types
- Natural log types
- Trigonometric types
- Trigonometric functions to a power
- Trigonometric functions to a power (part 2)

### Differentiating with the Product Rule

### Differentiating with the Quotient Rule

### More Standard Differentials

### Reciprocal Function of dy/dx

### Exponential Functions

### Parametric Functions

- Differentiating Parametric functions
- Second Differential
- Tangents and normals
- Finding stationary points

### Implicit Functions

- Differentiating Implicit equations
- Using the Product Rule with Implicit equations
- Tangents to Implicit curves
- Finding stationary points
- Nature of stationary points

### Connected Rates of Change

## Integration

### Introduction

### Equation of Curves

### Definite Integration

### Common Functions

- (ax + b)
^{n}type functions - Summary Exercise: (ax + b)
^{n}type functions - Exponential functions: e
^{x}, e^{ax}and e^{(ax + b)} - Reciprocal functions 1/x and 1/(ax + b)
- Integrals of the form: f'(x)/f(x)
- Integrals of the form: f'(x)/f(x) - Example
- Why the modulus sign?
- Why the modulus sign? - example
- Integrals of the form: f'(x)e
^{f(x)}

### Trigonometric Functions

- Integrals of sin(x), cos(x) and sec
^{2}(x) - Integrals of the form sin(ax + b), cos(ax + b) and sec
^{2}(ax + b) - Identity types: 1 | 2 | 3
- sin
^{2}types - cos
^{2}types

### Partial Fractions

### Integration by Substitution

- Examples: 1 | 2
- Square root types (Method 1)
- Square root types (Method 2)
- Integrating trig. functions
- Integrating exponential types
- Integration by substitution with limits
- Integrating trig. functions with limits

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

### Integration by Parts

- Introduction
- Applying integration by parts twice over
- Worked Example
- Natural Log types: ln(x)
- e
^{ax}sin(bx) and e^{ax}cos(bx) types - How to integrate e
^{ax}sin(bx) and e^{ax}cos(bx) - Example - Integrate e
^{x}cos(x) - Example - Integrate e
^{2x}sin(3x) - Example - Integrate e
^{-3x}cos(5x) - Integration by parts with limits
- Proof of the formula

### General Methods of Integration

- Step 1 - Is it a standard Integral?
- Step 2a - Is it a product of the form f[g(x)]g'(x)?
- Step 2b - If not, try integration by parts or substitution
- Step 3 - Is it a fraction?
- Examples: 1 | 2 | 3 | 4 | 5 | 6

### Area Bound by a Curve

### Differential Equations - Seperating the Variables

- What do we mean by seperating the variables?
- Where does the Constant C go?
- Handling the constant in log types
- Exponential and trig. types

### Differential Equations - Forming Differential Equations

## Numerical Methods

### Solving Equations by Numerical Methods

- Graphical method of finding roots
- Change of sign method
- Iteration
- How does Iteration work?
- Newton-Raphson method

### Vectors

## Vectors

### Vectors

- What is a vector and a scalar quantity?
- 2D vector notation
- Vector notation
- 2D position vectors
- Position vectors
- Equal and negative vectors
- 2D multiplying a vector by a scalar
- Multiplying a vector by a scalar
- 2D addition and subtraction
- Addition and subtraction of vectors
- Magnitude of a 2 dimensional vector
- Distance between 2 points (2D)
- Magnitude of a 3 dimensional vector
- Unit vectors

## Worked Papers

Pure papers are for the new specification, C1-C4 are from the old but are still mostly applicable.

Pure Paper 1

Pure Paper 2

C1

- June 2015
- June 2014
- June 2013
- January 2013
- June 2012
- January 2012
- June 2011
- January 2011
- June 2010
- January 2010
- June 2009
- January 2009
- June 2008
- January 2008
- June 2007

C2

- June 2015
- June 2014
- June 2013
- January 2013
- June 2012
- January 2012
- June 2011
- January 2011
- June 2010
- January 2010
- June 2009
- January 2009
- June 2008
- January 2008
- June 2007

C3

- June 2015
- June 2014
- June 2013
- January 2013
- June 2012
- January 2012
- June 2011
- January 2011
- June 2010
- January 2010
- June 2009
- January 2009
- June 2008

C4

- June 2015
- June 2014
- June 2013
- June 2012
- January 2012
- June 2011
- January 2011
- June 2010
- January 2010
- June 2009
- January 2009
- June 2008
- January 2008
- June 2007

For worked papers, see the top of this section.