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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 axn
- 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 x2 terms
- Completing the square for 2x2, 3x2, ... 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
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 - y1 = m(x - x1)
- 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
Algebra and Functions 2
Sketching Curves
Graph Transformations
Sequences and Series
Binomial Expansion
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
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 sin2θ+cos2θ = 1
- Proving trig. identities 1 | 2 | 3
- cos(θ) = cos(-θ) and sin(θ) = -sin(-θ)
- Solving trig. equations using identities 1 | 2 | 3
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 ex and Natural Log Functions
- Exponential Functions
- Transformations of exponential graphs
- The natural log function: ln(x)
Modelling Curves of the form y=kxn and y=kax
Differentiation
Introduction
- The gradient function dy/dx
- Differentiation from 1st principles
- Terms of the form axn
- 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
Integration
Introduction
Equation of Curves
Definite Integration
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
For worked papers, see the top of this section.