Cambridge AS Level Maths Revision Guide: Pure Maths Complete

Pure Mathematics forms the foundation of Cambridge AS Level Mathematics. This complete revision guide covers all pure maths topics with key concepts, essential formulas, worked examples, and exam tips to help you excel in your AS Level exams.

Why Focus on Pure Maths?

Pure Mathematics is essential because:

• Foundation - Forms the basis for statistics and mechanics
• High Weighting - Significant portion of AS Level exams
• University Preparation - Essential for many degree courses
• Problem-Solving - Develops logical thinking skills

How to Use This Revision Guide

1. Review Each Topic

• Understand key concepts
• Learn essential formulas
• Practice worked examples
• Test your understanding

2. Practice Regularly

• Do topic-specific questions
• Use past papers
• Identify weak areas
• Focus revision accordingly

3. Exam Preparation

• Review formulas regularly
• Practice under timed conditions
• Learn exam technique
• Build confidence

Topic 1: Algebra and Functions

Quadratic Functions

• Standard form: ax² + bx + c = 0
• Factorizing, completing the square, quadratic formula
• Discriminant: b² - 4ac
• Nature of roots based on discriminant

Polynomials

• Degree and leading coefficient
• Factor theorem
• Remainder theorem
• Polynomial division

Functions

• Domain and range
• Composite functions: f(g(x))
• Inverse functions: f⁻¹(x)
• Transformations of graphs

Key Formulas:

• Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
• Discriminant: Δ = b² - 4ac

Topic 2: Coordinate Geometry

Straight Lines

• Gradient: m = (y₂ - y₁) / (x₂ - x₁)
• Equation: y = mx + c or y - y₁ = m(x - x₁)
• Parallel lines: same gradient
• Perpendicular lines: m₁ × m₂ = -1

Circles

• Equation: (x - a)² + (y - b)² = r²
• Center: (a, b), radius: r
• Tangent to circle
• Intersection of line and circle

Parametric Equations

• x = f(t), y = g(t)
• Converting to Cartesian form
• Finding gradients
• Applications

Topic 3: Trigonometry

Trigonometric Functions

• sin, cos, tan definitions
• Unit circle
• Graphs of trigonometric functions
• Period and amplitude

Trigonometric Identities

• sin²θ + cos²θ = 1
• tanθ = sinθ / cosθ
• Compound angle formulae
• Double angle formulae

Trigonometric Equations

• Solving sin x = k, cos x = k, tan x = k
• General solutions
• Using identities
• Applications

Topic 4: Differentiation

Basic Differentiation

• Power rule: d/dx(xⁿ) = nxⁿ⁻¹
• Sum/difference rule
• Product rule: d/dx(uv) = u’v + uv’
• Quotient rule: d/dx(u/v) = (u’v - uv’) / v²

Applications

• Tangents and normals
• Stationary points
• Maximum and minimum values
• Points of inflection

Second Derivatives

• Finding f”(x)
• Concavity
• Points of inflection
• Curve sketching

Topic 5: Integration

Basic Integration

• Reverse of differentiation
• Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C
• Definite integrals
• Area under curves

Integration Techniques

• Substitution
• Integration by parts
• Partial fractions
• Applications

Applications

• Area between curves
• Volume of revolution
• Average value
• Differential equations

Topic 6: Vectors

2D Vectors

• Component form: (x, y)
• Magnitude: |v| = √(x² + y²)
• Unit vectors
• Position vectors

3D Vectors

• Component form: (x, y, z)
• Magnitude: |v| = √(x² + y² + z²)
• Dot product: a·b = |a||b|cosθ
• Cross product

Vector Equations

• Lines in 2D and 3D
• Parametric form
• Scalar product
• Applications

Exam Tips for AS Level Pure Maths

1. Know Your Formulas

• Learn all formulas
• Practice using them
• Check your work
• Use formula sheet effectively

2. Show Working

• Method marks are crucial
• Show each step
• Check your algebra
• Verify answers

3. Practice Regularly

• Do questions daily
• Use past papers
• Time yourself
• Learn from mistakes

4. Understand Concepts

• Don’t just memorize
• Understand why
• Apply to new problems
• Link related topics

Related Resources

• Cambridge AS Level Maths Past Papers
• AS Level Maths Tutor
• AS Level Maths Exam Tips

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Use this revision guide as part of your comprehensive exam preparation. Combine with past paper practice and expert tutoring for best results.