Pure Maths

Outlined below are the topics covered for OCR Pure maths AS course

Pure Maths

Pure Maths topics

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

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

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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 equations with equal terms on both sides (Common error)
  6. 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

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Factor Theorem

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

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

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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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

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

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

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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
  5. Solving equations with equal terms on both sides (Common error)

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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

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

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

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

Vectors

Vectors

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

Proof

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