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GCSE/i-GCSE
GCSE/i-GCSE Course Info
1. Numbers
1. Basic Stuff
2. Top & Bottom Decks of Fractions
3. Fractions & Decimals 1
4. Fractions & Decimals 2.
5. Large & small Numbers
6. Percentages: An Introduction
7. Percentages, Fractions & Decimals
8. HCFs & LCMs, Factors & Primes
9. HCFs & LCMs, Factors & Primes Examples
10. Squares & Cubes
11. BODMAS
12. Decimal Places & Significant Figures
13. Truncated Numbers & Error Ranges
14. Estimation
15. Powers: An Introduction
16. More on Positive Powers
17. Fractional & Negative Powers
18. Powers in Equations
19. Powers: Examples 1
20. Powers: Examples 2
21. Standard Form
22. Recurring Decimals
23. Surds
24. Surds: Examples 1
25. Removing Surds: Rationalising the Denominator
26. Rationalising the Denominator: Harder Examples
27. Bounds: An Introduction
28. Bounds: Examples
Worked Ex's: 1. Numbers
Key Facts: 1. Numbers
2. Ratio & Proportion
1. What is Ratio & Proportion
2. Ratios, Fractions & Equations
3. Working with Ratios
4. Ratios & Scaling Factors
5. Proportionality
6. Direct Linear Proportionality: Examples
7. Gradients & Rates of Change
8. Rates of Change: Examples
9. Distance-Time Graphs
10. Variable Speeds & Distance-Time Graphs
11. Velocity-Time Graphs
12. Ratios, Percentages & Proportions
13. Single Percentage Problems
14. Percentage Multipliers
15. Populations & Compound Growth or Decay
16. Simple Interest
17. Compound Interest
Worked Ex's: 2. Ratio & Proportion
Key Facts: 2. Ratio & Proportion
3. Algebra & Equations
1. Solving & Simplifying
2. Expanding Brackets
3. Equations
4. Equations & Algebraic Families
5. Changing the Subject
6. Finding Common Factors
7. Cancelling Common Factors
8. Linear Equations: Worked Examples
9. Simultaneous Linear Equations: Substitution
10. Simultaneous Linear Equations: Elimination
11. Quadratics: Expanding & Factorising
12. Factorising Quadratics: The 4 Methods
13. Factorising Quads 1: By Inspection
14. Factorising Quads 1: Plotting the Results
15. Factorising Quads: 2. Quadratic Formula
16. Factorising Quads 2: The Discriminant
17. Factorising Quads 3: Completing the Square
18. Factorising Quads 3: Roots & Turning Points
19. Factorising Quads 3: Completing the Square: Example
20. Factorising Quads 4: Iteration
21. Factorising Quads 4: Iteration: Using a Calculator
22. Difference of Two Squares Quadratics
23. Simultaneous Equations: Linear & Quadratic
24. Algebraic Fractions
25. Inequalities: An Introduction
26. Inequalities: Number Lines
27. Multiplying an Inequality by a Negaitve: The proof
28. Quadratic Inequalities
29. Quadratic Inequalities: Examples
30. Real Life Formulas: Speed, Density & Pressure
Worked Ex's: 3. Algebra & Equations
Key Facts: 3. Algebra & Equations
4. Sequences & Series
1. Introduction to Sequences & Series
2. Types of Sequences
3. Arithmetic Sequences
4. Geometric Sequences
5. Geometric Sequences: Examples
6. Quadratic Sequences
7. Quadratic Sequences: A Proof
8. Quadratic Sequences: Examples
9. Number Sequences
Worked Ex's: 4. Sequences & Series
Key Facts: 4. Sequences & Series
5. Proofs
1. Proof Types
2. Algebraic Proofs
3. Algebraic Proofs: Examples
Worked Ex's: 5. Proofs
Key Facts: 5. Proofs
6. Functions & Graphs
1. What is a Function?
2. Sketching Functions
3. Sketching Functions: An Example
4. Roots: What & Where are They?
5. Graphical Solutions
6. Gradient & Straight Line Equations
7. More on Straight Line Gradients
8. Three Straight Lines You Must Know
9. Straight Lines: Where they Meet
10. Parallel & Perpendicular Lines
11. Shapes of Quadratics
12. Advantages of Factorising Quadratics
13. Completing the Square & Transformations
14. Shape of 'Difference of 2 Squares' Quadratics
15. Simultaneous Equations and Quadratics
16. Circles Straight Lines & Tangents
17. Other Common Functions
18. Exponentials
19. Areas Under Curves
20. Areas Under Curves: Triangles & Trapesiums
21. Transformations of Functions: An Introduction
22. Function Machines
23. Function Notation: f(x) g(x) h(x)
24. Composite Function Notation: fg(x) gf(x)
25. Inverse Functions
26. Inverse Functions: Example
27. Reflections of Functions
28. When Functions Misbehave: Dividing by Zero
29. When Functions Misbehave: Tan x
30. More on Domain & Range
31. Restrictions of Domain & Range
32. Domain & Range and Inverse Functions
33. More on Domain & Range and Inverse Functions
34. Domain & Range Restricted by Choice
35. Sectional Functions Defined by Domain
36. Domain & Range: Further Examples
Worked Ex's: 6. Functions & Graphs
Key Facts: 6. Functions & Graphs
7. Geometry
1. Basic Angle Stuff
2. Polygons: Interior & Exterior Angles
3. Geometric Constructions 1
4. Geometric Constructions 2
5. Loci
6. Angles of Depression & Elevation
7. Pythagoras
8. Congruent Triangles
9. Similar Triangles
10. Plans & Elevations
11. Circles: An Introduction
12. Circle Areas: Sectors & Arcs
13. Areas of Common Shapes
14. Cylinders, Spheres, Cones and Pyramids
15. Unit Conversions: Lengths, Areas & Volumes
16. Length, Area & Volume Scale Factors
17. Length, Area, Volume Scale Factors: Examples
18. Complex & Compound Areas
19. Coordinate Geometry: An Introduction
20. Coordinate Geometry: Mid-points
21. Bounded Areas & Graphical Area Inequalities
22. Bounded Areas: Examples
23. Circles, Lines & Tangents
24. Circle Theorems:An Introduction
25. Circle Theorems 1
26. Circle Theorems 2
27. Circle Theorems 3
28. Circle Theorems 4 & Examples
29. Circle Theorems 5 Intersecting Chords
30. Pythagoras in 3D
31. Translations & Reflections
32. Rotations & Enlargements
33. Combining Transformations
34. Bearings
35. Triangle Areas: Acute & Obtuse Angles
36. Triangle Area using a Calculotor
Worked Ex's: 7. Geometry
Key Facts: 7. Geometry
8. Trigonometry
1. Trigonometry: An Introduction
2. SOH CAH TOA
3. Finding Sides from Trig Ratios: sin, cos & tan
4. Finding Angles from Trig Ratios: sin^-1, cos^-1 & tan^-1
5. Pythagoras & SOH CAH TOA
6. Unit Circles & Trig Functions
7. Shape of Trig Functions: An Introduction
8. Standard Trig Values
9. Shape of sin x & cos x
10. Shape of tan x
11. Complete cycles of sin x cos x tan x
12. Standard Trig Values on the Waveforms
13. Trig Functions: y-Transformations
14. Trig Functions: x-Transformations
15. Sine & Cosine Rules
16. Sine or Cosine Rule?
17. Sine & Cosine Rules: With Bearings
18. Triangle Area using Sine Formula
19. Sine Area Formula & Circle Sectors
20. Sinusoidal Waves in Nature
Worked Ex's: 8. Trigonometry
Key Facts: 8. Trigonometry
9. Vectors
1. Vector Addition
2. Negative Vectors
3. Vector Addition: Examples
4. Vector Subtraction: Examples
5. Unit Vectors
6. Parallel Vectors
7. Parallel Split Straight-Line Vectors
8. Vector Geometry: Example 1
9. Vector Geometry: Example 2
10. Magnitude of a Vector (i-GCSE)
Worked Ex's: 9. Vectors
Key Facts: 9. Vectors
10. Data & Statistics
1. Frequency Tables
2. Mean Mode Median & Range
3. Quartiles
4. Finding Quartiles by a Rule
5. Quartiles with Grouped Data
6. Average & Range Examples
7. Populations
8. Frequency Tables: An Example
9. Grouped Frequency Tables
10. Grouped Frequency Tables: An Example
11. Frequency Polygons
12. Cumulative Frequency Tables
13. Cumulative Frequency Graphs
14. Cumulative Frequency Graphs: Another Example
15. Box Plots
16. Box Plot Examples
17. Comparing Box Plots
18. Combinations: The Product Rule
19. Combinations: The Product Rule: Examples
20. Bar Charts
21. Frequency Density
22. Histograms
23. Histograms: Examples
24. Histograms: Finding the Median
25. Stem & Leaf Diagrams
26. Line Graphs
27. Pie Charts
28. Scatter Graphs & Correlation
29. Venn Diagrams
30. Venn Diagrams: Examples
31: Probability
32. Venn Diagrams & Probability
33. Venn Diagrams: The Dangers of the OR Condition
34. Dependent and Independent Events
35. Dependent & Independent events: Examples 1
36. Dependent & Independent Events: Examples 2
37. Probability Tree Diagrams
38. Probability Tree Diagrams: A Useful Shortcut
39. Algebraic Probability Trees: Example 1
40. Algebraic Probability Trees: Example 1
Worked Ex's: 10. Data & Statistics
Key Facts: 10. Data & Statistics
11. Differentiation (i-GCSE)
1. Differentiation: An Introduction
2. From First Principles 1
3. From First Principles 2
4. How to Differentiate
5. Gradients of Tangents & Normals
6. Stationary Points
7. Finding Turning Points: Example 1
8. Finding Turning Points: Example 2
9. Optimisation Problems Using Differetiation
10. Optimisation Problems Using Differentiation: An Example
11. Optimisation Problems Using Differentiation: Another Example
Worked Ex's: 11. Differentiation (i-GCSE)
Key Facts: 11. Differentiation (i-GCSE)
12. Exam Tips & Traps
1. Exam Tips & Traps 1
2. Exam Tips & Traps 2
3. Exam Tips & Traps 3
4. Exam Tips & Traps 4
5. Exam Tips & Traps 5
13. GCSE Exam Papers
Nov 2022 Higher P1 Edexcel GCSE Worked Soln's
Nov 2022 Higher P2 Edexcel GCSE Worked Soln's
Nov 2022 Higher P3 Edexcel GCSE Worked Soln's
Tough Questions Higher P1 Edexcel GCSE Worked Soln's
Tough Questions Higher P2 Edexcel GCSE Worked Soln's
May 2023 P1 Edexcel GCSE Worked Soln's
June 2023 P2 Edexcel GCSE Worked Soln's
June 2023 P3 Edexcel GCSE Worked Soln's
Nov 2023 P1 Edexcel GCSE Worked Soln's
Nov 2023 P2 Edexcel GCSE Worked Soln's
Nov 2023 P3 Edexcel GCSE Worked Soln's
14. i-GCSE Exam Papers
Nov 2021 Paper 1 Edexcel i-GCSE Worked Soln's
Nov 2021 Paper 2 Edexcel i-GCSE Worked Soln's
A-LEVEL
AS PURE: Course Topics
1. Algebra & Equations
1. A Reminder of the Power Rules
2. Power Rules: Ex’s 1
3. Power Rules: Ex’s 2
4. Rationalisation
5. Dealing With Surds: More Ex's
6. Expanding & Factorising
7. Finding Common Factors
8. Cancelling & Simplification
9. Algebraic Fractions
Worked Ex's: 1. Algebra & Equations
Key Facts: 1. Algebra & Equations
2. Quadratics
1. ‘u’ or n’ Shapes
2. Factorising & Solving Quadratics: The 4 Methods
3. Factorising Quadratics: 1: By Inspection
4. Factorising Quadratics: 1: Plotting the Results
5. Factorising Quadratics: 1: By Inspection: Ex’s
6. Factorising Quadratics: 2: The Quadratic Formula
7. Factorising Quadratics: 2: Quadratic Formula Discriminant
8. Factorising Quadratics: 2: Testing the Discriminant
9. Factorising Quadratics: 3: Completing the Square
10. Factorising Quadratics: 3: Completing the Square: Finding Roots
11. Factorising Quadratics: 3: Completing the Square: Ex
12. Factorising Quadratics: 3: Completing the Square & Transformations
13. Completing the Square & the Turning Point
14. Completing the Square & the Quadratic Formula
15. Factorising Quadratics: 4: Iteration
16. Factorising Quadratics: 4: Iteration & Using a Calculator
17. Factorising Quadratics: The Difference of Two Squares
18. A Typical Exam Question
Worked Ex's: 2. Quadratics
Key Facts: 2. Quadratics
3. Equations & Inequalities
1. Simultaneous Linear Eqns: 1: Substitution Method
2. Simultaneous Linear Eqns: 2: Elimination Method
3. Simultaneous Eqns: Linear & Quadratic
4. Inequalities
5. Quadratic Inequalities: Ex’s 1
6. Quadratic Inequalities: Ex’s 2
7. Quadratic Inequalities: Ex’s 3
Worked Ex's: 3. Equations & Inequalities
Key Facts: 3. Equations & Inequalities
4. Functions & Transformations
1. Functions & Graphs
2. Standard Function Shapes 1
3. Standard Function Shapes 2
4. Parabolas, Cubics, Reciprocals & Exponentials
5. Exponential Functions
6. Sketching Functions
7. Sketching Functions: Ex 1
8. Sketching Functions: Ex 2
9. Roots: What & Where Are They?
10. Function Transformations: An Introduction
11. The Four Transformations
12. Reflections
13. Y Direction ↓ ↑ Transformations
14. X Direction → ← Transformations
Worked Ex's: 4. Functions & Transformations
Key Facts: 4. Functions & Transformations
5. Straight Lines
1. Straight Lines & Gradient 1
2. Straight Lines & Gradient 2
3. Parallel & Perpendicular Lines
Worked Ex's: 5. Straight Lines
Key Facts: 5. Straight Lines
6. Circles
1. Circles Centre (0, 0)
2. Circles Centre (a, b)
3. Radians, Arcs, Sectors & Segments
4. Circle Properties & Tangents
5. Circles Meeting Lines: 1
6. Circles Meeting Lines: 2
7. Circles Meeting Circles: Ex 1
8. Circles Meeting Circles: Ex 2
9. Circles Meeting Circles: Ex 3
Worked Ex's: 6. Circles
Key Facts: 6. Circles
7. Algebraic Methods (AS)
1. Algebraic Long Division: An Introduction
2. Algebraic Long Division: Examples
3. Remainder Theorem
4. Factor Theorem
5. Factorising Polynomials A Shortcut
6. Factorising Polynomials : Examples
7. Algebraic Fractions
8. Algebraic Fractions: Examples
Worked Ex's: 7. Algebraic Methods (AS)
Key Facts: 7. Algebraic Methods (AS)
8. Proofs (AS)
1. Proofs by Deduction
2. Proofs by Deduction: Examples 1
3. Proofs by Deduction: Examples 2
4. Proofs by Exhaustion
5. Proofs by Exhaustion: Examples
6. Proofs by Counter Example
7. Proofs by Counter Example: Examples 1
8. Proofs by Counter Example: Examples 2
9. Proofs by Counter Example: Examples 3
Worked Ex's: 8. Proofs (AS)
Key Facts: 8. Proofs (AS)
9. Binomial Expansions (AS)
1. Permutations & Combinations 1
2. Permutations & Combinations 2
3. Binomial Expansion of (a+b)^n
4. More about Binomial Expansion of (a+b)^n
5. Binomial Expansion of (a+b)^n: Ex's 1
6. Binomial Expansion of (a+b)^n: Ex's 2
Worked Ex's: 9. Binomial Expansions (AS)
Key Facts: 9. Binomial Expansions (AS)
10. Trigonometry (AS)
1. Sin Cos & Tan Definitions
2. Shape of Trig Functions
3. Standard Trig Values
4. Shape of Sin x & Cos x 0 to 90
5. Shape of Tan x 0 to 90
6. Complete Cycles of Sin Cos and Tan
7. Transforming Sin X
8. Sin & Cosine Rules
9. Area of a Triangle using Trig
Worked Ex's: 10. Trigonometry (AS)
Key Facts: 10. Trigonometry
11. Trig Identities
1. Pythagorean Trig Identities
2. Pythagorean Trig Identities: Ex's
3. Trig Equations: Graphical Solutions
4. Trig Equations: CAST Solutions
5. Trig Equations: CAST Solutions: Ex's 1
6. Trig Equations: CAST Solutions: Ex's 2
7. Trig Equations: CAST Solutions: Ex's 3
Worked Ex's: 11. Trig Identities
Key Facts: 11. Trig Identities
12. Vectors (AS)
1. Notation Magnitudes & Unit Vectors
2. Vector Addition
3. Vector Subtraction
4. Parallel Vectors
5. Position & Direction Vectors
6. Polar Form Notation for Vectors
7. Vector Magnitudes: Example
8. Distance between Two Points
9. Parallel & Split Straight-Line Vectors
10. Straight Line Vectors: Example
11. Vectors & Geometry: Harder Example 1
12. Vectors & Geometry: Harder Example 2
Worked Ex's: 12. Vectors (AS)
Key Facts: 12. Vectors (AS)
13. Differentiation
1. Differentiation: An Introduction
2. From First Principles 1
3. From First Principles 2
4. How to Differentiate
5. Gradients of Tangents & Normals
6. Stationary Points
7. Concave & Convex Curves
8. Finding Turning Points: Example 1
9. Finding Turning Points: Example 2
10. Finding Turning Points: Example 3
11. More on Function Inflections 1
12. More on Function Inflections
13. Optimisation Problems Using Differetiation
14. Optimisation Problems Using Differentiation: An Example
15. Optimisation Problems Using Differentiation: Another Example
16. Chain Rule
17. Chain Rule: An Example
Worked Ex's: 13. Differentiation (AS)
Key Facts: 13. Differentiation (AS)
14. Integration (AS)
1. Integration: An Introduction 1
2. Integration: An Introduction 2
3. Finding the Area Under a Curve
4. Integration Examples 1
5. Integration Examples 2
6. Integration Examples 3
7. Integration Examples 4
Worked Ex's: 14. Integration (AS)
Key Facts: 14. Integration (AS)
15. Logs & Exponentials
1. Exponential Functions
2. Log Functions
3. Log Laws & Common Log Values
4. Log Laws: Ex's
5. Changing the Base of Logs
6. Exponential & Log Functions
7. Exponential & Log Functions: Ex's
8. A Special Exponential e^x & Natural Logs
9. Exponentials & Natural Logs: Ex's 1
10. Exponentials & Natural Logs: Ex's 2
11. Exponentials & Natural Logs: Ex's 3
12. Exponential Functions Involving Time
13. Exponential Functions Involving Time: Ex's 1
14. Exponential Functions Involving Time: Ex's 2
15. Modelling Functions Using Logs
16. Modelling Functions Using Logs: Ex's
Worked Ex's: 15. Logs & Exponentials
Key Facts: 15. Logs & Exponentials
Exam Papers: AS Pure
Edexcel AS 2023 Pure
AS MECHANICS: Course Topics
1. Units & Model Simplification
1. Units of Measurements
2. Mathematical Model Simplifications
3. Weight & Mass, Speed & Velocity
Worked Ex's: 1. Unit & Simplifications
Key Facts: 1. Model Simplification
2. Constant Acceleration
1. Displacement-Time Graphs
2. Velocity-Time Graphs
3. Acceleration & Decelleration
4. Constant Acceleration Motion Eqns
5. SUVAT: The 5 Motion Eqns
6. SUVAT: The 5 Motion Eqns: Ex's
7. Velocity-Time Graphs Ex's
8. SUVAT: Harder Ex's 1
9. SUVAT: Harder Ex's 2
10. Acceleration due to Gravity
11. Acceleration due to Gravity: Ex's 1
12. Acceleration due to Gravity: Ex's 2
13. Acceleration due to Gravity: Ex's 3
14. Acceleration due to Gravity: Ex's 4
15. Vectors: Column Vectors & Magnitudes
16. Vectors: Polar & Component Form
17. Distance, Velocity & Acceleration Vectors
18. SUVAT Vectors in 2 Dimensions
19. SUVAT Vectors in 2 Dimensions: Ex's
Worked Ex's: 2. Constant Acceleration
Key Facts: 2. Constant Acceleration
3. Static Forces
1. Forces…What Are They?
2. Weight & Normal Reaction Forces
3. Tension & Thrust
4. Friction
5. Splitting Forces into Components
6. Forces in Equilibrium
7. Forces & Vectors
8. Forces & Vectors: Ex's 1
9. Forces & Vectors: Ex's 2
Worked Ex's: 3. Static Forces
Key Facts: 3. Static Forces
4. Forces & Motion
1. Force Diagrams
2. Newton’s Laws of Motion
3. Motion Alone a Line: Ex's
4. Motion in 2D: Ex 1
5. Motion in 2D: Ex 2
6. Motion in 2D: Ex 3
7. Particles Connected by Pulleys 1
8. Particles Connected by Pulleys 2
9. Connected Bodies in Motion Ex 1
10. Connected Bodies in Motion Ex 2
11. Connected Bodies in Motion Ex 3
12. Connected Bodies in Motion Ex 3 (Cont)
Worked Ex's: 4. Forces & Motion
Key Facts: 4. Forces & Motion
5. Variable Acceleration
1. Acceleration-Time Graphs
2. Differentiation & Displacement, Velocity & Acceleration
3. Displacement, Velocity & Acceleration Varying with Time
4. Motion Varying with Time: Ex's 1g with Time: Ex's 1
5. Motion Varying with Time: Ex's 2
Worked Ex's: 5. Variable Acceleration
Key Facts: 5. Variable Acceleration
Exam Papers: AS Mechanics
AS STATISTICS: Course Topics
1. Data Collection & Sampling
1. Large Data Set: An Introduction
2. Collection of Data: Planning the Collection
3. Types of Data: Discrete vs. Continuous Variables
4. Types of Data: Quantitative vs. Qualitative
5. Population & Samples
6. Sampling Methods
7. Stratified Sampling: The Maths
8. Sampling Techniques: Examples
Worked Ex's: 1. Data Collection
Key Facts: 1. Data Collection
2. Data Representation
1. Mean, Median, Mode, Range, Quartiles & Percentiles
2. Mean from Ungrouped Data
3. Median & Quartiles from Ungrouped Data
4. Finding the Median & the Quartiles: A Rule
5. Quartiles & Mid-Range Values in Grouped Data
6. Finding the Mean: Ex's
7. Quartiles from Grouped Data
8. Quartiles, Percentiles & Grouped Data: Ex's
9. Quartiles, Percentiles & Grouped Data: Ex 2
10. Box Plots: Quartiles & Interquartile Range
11. Outliers in Box Plots
12. Cleaning the Data: Outliers
13. Cleaning the Data: Outliers: Ex
14. Outliers & Comparison of Box Plots: Ex
15. Histograms
16. Scatter Graphs: Correlation & Association
17. Scatter Graphs: to Correlate or Not to Correlate
Worked Ex's: 2. Data Representation
Key Facts: 2. Data Representation
3. Location & Spread
1. Location & Spread: An Introduction
2. The Maths of Location & Spread
3. Variance: An Introduction
4. Variance & Standard Deviation: Longhand Ex
5. Variance & Standard Deviation: Calculator Ex
6. Standard Deviation: Well, not that Standard!
7. Variance: A Simplified Formula
8. Standard Deviation & Grouped Data
9. Standard Deviation & Grouped Data: Ex
10. Finding Outliers Using Mean & Standard Deviation
11. Cleaning Data Using Mean & Standard Deviation
12. Coding
13. Coding: How it works
14. Coding: Ex's
Worked Ex's: 3. Location & Spread
Key Facts: 3. Location & Spread
4. Correlation
1. Bivariate Data & Scatter Graphs
2. Linear Regression: An Introduction
3. Correlation Coefficient: r
4. Linear Correlation: Ex
Worked Ex's: 4. Correlation
Key Facts: 4. Correlation
5. Probability
1. Probability: Definitions & Notation
2. Probability Tree Diagrams
3. Venn Diagrams: Ind't & Dependent Events
4. Venn Diagrams: Dangers of the 'OR' Condition
5. Venn Diagrams: Ind't & Dependent Events: Ex's
6. Venn Diagrams: Harder Ex
7. Multiple Probability Outcomes
8. Probability Space Diagrams
9. Two-Way Probability Tables: Ex 1
10. Two-Way Probability Tables: Ex 2
Worked Ex's: 5. Probability
Key Facts: 5. Probability
6. Binomial Distributions
1. Discrete Random Variables: An Introduction
2. Discrete Random Variables: Real Life Examples
3. Probability Mass Functions
4. Discrete Random Variables: Examples
5. Probability Distribution Notation
6. Binomial Distributions: A Formula
7. Binomial Distributions: Finding the Formula: 1
8. Binomial Distributions: Finding the Formula: 2
9. Binomial Distributions: Finding the Formula: 3
10. Binomial Distributions: Finding the Formula: 4
11. Binomial Probability Density: Using the Formula
12. Binomial Probabilities on a Calculator: Example
13. Binomial Cumulative Probabilities on a Calculator
14. Binomial Cumulative Probabilities on a Calculator: Example
15. Binomial Probabilities: Tabular Probabilities on a Calculator
16. Binomial Probabilities: Using Tables
17. Binomial Probabilities: Using Tables: Example
Worked Ex's: 6. Binomial Distributions
Key Facts: 6. Binomial Distributions
7. Hypothesis Testing
1. Hypothesis Testing: An Introduction
2. Significance Probability Levels & Critical Values
3. Significance Probability Levels & Binomial Distributions
4. Significance Probability Levels Notation
5. Beware of the Upper Tail
6. Lower Tail Tests: Ex 1
7. Lower Tail Tests: Ex 2
8. Lower Tail Tests: Ex 3
9. Upper Tail Tests: Ex 1
10. Upper Tail Tests: Ex 2
11. Two Tail Tests: Ex 1
12. Two Tail Tests: Ex 2
13. Finding the Critical Region Using a List
14. Finding the Critical Region Using a List: Ex's
Worked Ex's: 7. Hypothesis Testing
Key Facts: 7. Hypothesis Testing
Exam Papers: AS Stats
May 2023 Stats Edexcel
A2 PURE: Course Topics
1. Algebraic Methods (A2)
1. Algebraic Fractions: 1
2. Algebraic Fractions: 2
3. Proper & Improper Rational Expressions
4. Decomposing Rules for Partial Fractions
5. Partial Fractions: Ex's 1
6. Partial Fractions: Ex's 2
7. Partial Fractions: Ex's 3
8. Numerators & Denominators of Equal Degree
9. Top-Heavy Improper Rational Expressions
10. Binomial Expansions & Partial Fractions
Worked Ex's: 1. Algebraic Methods (A2)
Key Facts: 1. Algebraic Methods (A2)
2. Functions & Graphs (A2)
1. Functions…What are they?
2. Domain & Range
3. Domain & Range: Ex’s
4. Inverse Functions
5. Inverse Functions: Domain & Range: Ex 1
6. Inverse Functions: Further Restrictions to Domain & Range
7. Modulus Functions: Ex’s 1
8. Modulus Functions: Ex’s 2
9. Modulus Functions: Ex’s 3
Worked Ex's: 2. Functions & Graphs (A2)
Key Facts: 2. Functions & Graphs (A2)
3. Sequences & Series (A2)
1. Series by Definition & Σ Notation
2. Series by Definition: Ex’s
3. Arithmetic Series
4. Arithmetic Series: Ex’s
5. Geometric Series
6. Geometric Series: Ex’s 1
7. Geometric Series: Ex’s 2
8. Geometric Series: Ex’s 3
Worked Ex's: 3. Sequences & Series (A2)
Key Facts: 3. Sequences & Series (A2)
4. Binomial Expansions (A2)
1. Binomial Expansions: The (1 +“x”)^n Expansion
2. (1 +“x”)^n Expansion: Ex’s
3. Binomial Estimation
4. Binomial Estimation: Ex’s 1
5. Binomial Estimation: Ex’s 2
6. Binomial Expansions & Partial Fractions
Worked Ex's: 4. Binomial Expansions (A2)
Key Facts: 4. Binomial Expansions (A2)
5. Radians
1. Radians, Arcs, Sectors & Segments
2. Radians, Arcs, Sectors & Segments Ex's
Worked Ex's: 5. Radians
Key Facts: 5. Radians
6. Trigonometric Functions
1. Reciprocal Functions: sec, cosec & cot
2. Inverse Trigonometric Functions: arcsin, arccos, arctan 1
3. Inverse Trigonometric Functions: arcsin, arccos, arctan 2
Worked Ex's: 6. Trigonometric Functions
Key Facts: 6. Trigonometric Functions
7. Further Trigonometry
1. Compound Angle Formula Proofs: 1
2. Compound Angle Formula Proofs: 2
3. Double Angle Formulas
4. Double Angle Formulas: Ex’s 1
5. Double Angle Formulas: Ex’s 2
6. Triple-Angle Formulas
7. Triple-Angle Formulas: Ex’s
8. Trig Equation Solutions: Using Trig Identities
9. Trig Equation Solutions: LHS/RHS Proofs
10. Harmonic Form: R sin (Θ ± ∝) & R cos (Θ ± ∝)
11. Harmonic Form: Ex’s 1
12. Harmonic Form: Ex’s 2
13. Harmonic Form: Ex’s 3
14. Harmonic Form: Ex’s 4
15. Factor Formulas
16. Small Angle Approximations
17. Small Angle Approximations: Ex’s 1
18. Small Angle Approximations: Ex’s 2
Worked Ex's: 7. Further Trigonometry
Key Facts: 7. Further Trigonometry
8. Parametric Equations
1. Parametric Equations
2. Parametric Equations: Ex's
3. Parametric Equations: Domain & Range
4. Parametric Equations: Ellipses
5. Parametric Equations: Ellipses: Ex's
6. Parametric to Cartesian Conversions Ex's
Worked Ex's: 8. Parametric Equations
Key Facts: 8. Parametric Equations
9. Differentiation (A2)
1. The Product Rule for Differentiation
2. The Product Rule: A Proof
3. The Quotient Rule for Differentiation
4. Related Rates of Change
5. Implicit Differentiation
6. Implicit Differentiation: Ex
7. Differentiating Trig Functions: sin x
8. Standard Trig Functions and their Differentials
9. Trig Functions & Powers
10. Differentiating Log Functions
11. Differentiating Exponential Functions
12. Differentiating Exponential & Log Functions: Ex’s
13. Parametric Differentiation
14. Parametric Differentiation: Ex’s 1
15. Parametric Differentiation: Ex’s 2
16. Differential Equations
17. Modelling With Differential Equations: Ex 1
18. Modelling With Differential Equations: Ex 2
Worked Ex's: 9. Differentiation (A2)
Key Facts: 9. Differentiation (A2)
10. Numerical Methods: Areas & Roots
1. Trapezium Rule for Finding Areas
2. Mid-Ordinate Rule for Finding Areas
3. Simpson’s Rule for Finding Areas
4. Finding Roots Using Sign Changes
5. Finding Roots Using Iteration
6. Finding Roots Using Newton-Raphson
7. Finding Turning Points Using Newton-Raphson
Worked Ex's: 10. Numerical Methods: Areas & Roots
Key Facts: Key Facts: 10. Numerical Methods: Areas & Roots
11. Integration (A2)
1. Integrating Simple Terms
2. Integral Simplification Techniques
3. Standard Trig Integrals
4. Standard Trig Integrals: Example
5. Integrating Even Powers of sin x & cos x
6. Integrating Even Powers of sin x & cos x: Ex’s
7. Integrating Trig Products using the Double Angle Formulas
8. Integrating Trig Products using the Factor Formulae
9. Integration by Substitution: Spotting the Derivative 1
10. Integration by Substitution: Spotting the Derivative 2
11. Integration by Substitution: Ex’s 1
12. Integration by Substitution: Ex’s 2
13. Integration by Substitution: Ex’s 3
14. Integration by Substitution: Ex’s 4
15. Integrating 1 / x
16. Integrating 1 / x: Ex
17. Integrating tan x & cot x
18. Integrating Odd Powers of sin x & cos x
19. Integrating Odd Powers of tan x
20. Integrating cosec x: A Tricky Trig Integral
21. Integrating Functions With: e
22. Integration by Parts
23. Integration by Parts: Ex 1
24. Integration by Parts: Ex 2
25. Repeated Use of Integration by Parts 1
26. Repeated Use of Integration by Parts 2
27. Repeated Use of Integration by Parts 3
28. Inverse Trig Integrals
29. Partial Fractions & Integration
30. Partial Fractions & Integration: Ex
31. Exponential Functions Involving Time
32. Parametric Integration
33. Parametric Integration: Ex
34. Volumes of Revolution
35. Volumes of Revolution of Parametric Functions
Worked Ex's: 11. Integration (A2)
Key Facts: 11. Integration (A2)
12. Integration Revision
1. Integration Revision: 1
2. Integration Revision: 2
3. Integration Revision: 3
4. Integration Revision: 4
5. Integration Results Table
6. Practice Integrals 1
7. Practice Integrals 2
8. Practice Integrals 3
9. Practice Integrals 4
10. Practice Integrals 5
11. Practice Integrals 6
Worked Ex's: 12. Integration Revision
Key Facts: 12. Integration Revision
13. Vectors (A2)
1. 3D Vectors
2. Distance Between Two 3D Points
3. Vector Between Two 3D Points
4. Unit Vectors
5. Angles Between Axes & 3D Vectors 1
6. Angles Between Axes & 3D Vectors 2
7. Angles Between Axes & 3D Vectors Ex
8. Collinearity of Vectors
9. Solving Geometric Problems Using Vectors 1
10. Solving Geometric Problems Using Vectors 2
11. Solving Geometric Problems Using Vectors 3
12. Solving Geometric Problems Using Vectors 4
13.Solving Geometric Problems Using Vectors 5
14. Solving Geometric Problems Using Vectors 6
15. THE FINAL PROBLEM
Worked Ex's: 13. Vectors (A2)
Key Facts: 13. Vectors (A2)
Exam Papers: A2 Pure
A2 MECHANICS: Course Topics
1. Moments
1. Moment of a Force: The Definition
2. Moments About a Point
3. Moments & Equilibrium
4. Moments & Non-Uniform Rods
5. Moments & Tilting Rods
6. Moments & Tilting Rods: Example
7. Laminas & Moments
8. Laminas & Moments: Example
9. Laminas: Finding the Centre of Mass
Worked Ex's: 1. Moments
Key Facts: 1. Moments
2. Forces & Friction
1. Forces & Friction: Force Diagrams
2. Friction Forces 1
3. Friction Forces 2
4. Angles of Slopes
5. Resolution of Forces on a Plane
6. More on the Coefficient of Friction & Limiting Equilibrium
Worked Ex's: 2. Forces & Friction
Key Facts: 2. Forces & Friction
3. Projectiles
1. Projectiles
2. Velocity Components of Projectiles
3. Projectiles: Displacement is the Integration of Velocity
4. Solving Projectile Problems Using SUVAT
5. Solving Projectile Problems Using the Displacement Equation
6. Horizontal Projectile Distances Using SUVAT
7. Horizontal Projectile Distances Using the Displacement Eqn.
Worked Ex's: 3. Projectiles
Key Facts: 3. Projectiles
4. Application of Forces
1. Masses on Slopes
2. Masses on Slopes & Pullies 1
3. Masses on Slopes & Pullies 2
4. Forces on a Ladder
5. Forces on a Ladder: Example
6. Motion in Two Dimensions 1
7. Motion in Two Dimensions 2
Worked Ex's: 4. Application of Forces
Key Facts: 4. Application of Forces
5. Further Kinematics
1. Displacement, Velocity & Acceleration: A Reminder
2. SUVAT In 2 Dimensions
3. Compass Points & Unit Vectors
4. SUVAT In 2 Dimensions: Vector Motion Equations 1
5. SUVAT in 2 Dimensions: Vector Motion Equations 2
6. Displacement, Velocity & Acceleration Varying with Time
7. Time Dependent Motion: Examples 1
8. Time Dependent Motion: Examples 2
9. Time Dependent Motion: Examples 3
10. Time Dependent Motion: Examples 4
11. Time Dependent Motion: Examples 5
12. Newton’s 2nd Law as a Differential Equation
Worked Ex's: 5. Further Kinematics
Key Facts: 5. Further Kinematics
Exam Papers: A2 Mechanics
A2 STATISTICS: Course Topics
1. Correlation & Regression
1. Bivariate Data & Scatter Graphsr
2. Linear Regression & Correlation: An Introduction
3. Linear Regression Line & Correlation: Using a Calculator
4. Calculating Linear Regression: Summing the Squares 1
5. Calculating Linear Regression: Summing the Squares 2
6. Calculating Linear Regression: Summing the Squares 3
7. Regression Line Gradient
8. Using the Regression Line Gradient Formula 1
9. Using the Regression Line Gradient Formula 2
10. Product Moment Correlation Coefficient: r
11. Non-Linear Relationships: Exponential Models
12. Non-Linear Relationships: Exponential Models: Ex 1
13. Non-Linear Relationships: Exponential Models: Ex 2
Worked Ex's: 1. Correlation & Regression
Key Facts: 1. Correlation & Regression
2. Conditional Probability
1. Probability: Independent & Dependent Events
2. Conditional Probabilities & Venn Diagrams
3. Conditional Probability & Venn Diagrams: Exs 1
4. Conditional Probability & Venn Diagrams: Exs 2
5. Conditional Probability & Venn Diagrams: Exs 3
6. Conditional Probability & Venn Diagrams: Exs 4
7. Conditional Probability & Tree Diagrams
Worked Ex's: 2. Conditional Probability
Key Facts: 2. Conditional Probability
3. Normal Distributions
1. Normal Distributions: An Introduction
2. Normal Distributions: Mean & Standard Deviation
3. Standard Normal Distributions
4. Standard Normal Distributions: Converting to Z-Scores
5. Converting X to Z to P Using a Calculator
6. Z-Score Tables
7. Normal Distributions: Ex’s 1
8. Normal Distributions: Ex's 2
9. Normal Distributions: Ex's 3
10. Normal Distributions: Ex's 4
11. Inverse Normal Function
12. Normal & Inverse Normal Ex's 1
13. Normal & Inverse Normal Ex's 2
14. Normal & Inverse Normal Ex's 3
15. Normal & Inverse Normal Ex's 4
16. Simultaneous Eqns & Normal Distributions: Ex's 1
17. Simultaneous Eqns & Normal Distributions: Ex's 2
18. Modelling Probability & Normal Distributions
19. Normal & Binomial Distributions: A Revision Comparison
20. Normal & Binomial Distribution Ex's
21. Normal Distributions & Histograms
Worked Ex's: 3. Normal Distributions
Key Facts: 3. Normal Distributions
4. Statistical Approximations
1. Normal Approximation for a Binomial Distribution
2. Binomial Distributions: Mean & Variance
3. Normal Distribution Approximations: Continuity Correction
4. Binomial to Normal Distribution Approximations: Ex’s 1
5. Binomial to Normal Distribution Approximations: Ex’s 2
6. Binomial to Normal Distribution Approximations: Ex’s 3
7. Choosing & Approximating Distributions
8. Distribution of the Sample Mean
9. Distribution of the Sample Mean: Ex’s
Worked Ex's: 4. Statistical Approximations
Key Facts: 4. Statistical Approximations
5. Hypothesis Testing & Distributions
1. Hypothesis Testing for Correlation: 1
2. Hypothesis Testing for Correlation: 2
3. Hypothesis Testing for Correlation: Ex’s 1
4. Hypothesis Testing for Correlation: Ex’s 2
5. Hypothesis Testing for Correlation: Ex’s 3
6. Rank Correlation (OCR Exam Board Only)
7. Rank Correlation (OCR Exam Board Only): Ex
8. Hypothesis Testing & Normal Distributions: Changes to the Mean
9. Sample Means
10. Sample Means: Finding the Critical Region
11. Sample Means: Ex’s 1
12. Sample Means: Ex’s 2
13. Sample Means: Ex’s 3
Worked Ex's: 5. Hypothesis Testing & Distributions
Key Facts: 5. Hypothesis Testing & Distributions
Exam Papers: A2 Statistics
AS & A2: APPENDIX & FORMULA SHEETS
1. Introduction to the Large Data Set
2. Large Data Set: Dataset Variables & Explanatory Notes 1
3. Large Data Set: Dataset Variables & Explanatory Notes 2
4. PMCC Table
5. Variance: A Simplification of the Formula
6. Standard Deviation: A Proof
7. Linear Regression: Derivation of the Formula
8. Proof of: μ=np for a Binomial Distribution
9. Proof of Variance: σ² = npq for a Binomial Distribution
10. Pure Maths Formula Sheet
11. Mechanics & Statistics Formula Sheet
SAT (USA)
SAT (USA) Practice Questions
SAT (USA) Practice Paper 1 Module 1
SAT (USA) Subject Questions
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GCSE/i-GCSE
GCSE/i-GCSE Course Info
1. Numbers
1. Basic Stuff
2. Top & Bottom Decks of Fractions
3. Fractions & Decimals 1
4. Fractions & Decimals 2.
5. Large & small Numbers
6. Percentages: An Introduction
7. Percentages, Fractions & Decimals
8. HCFs & LCMs, Factors & Primes
9. HCFs & LCMs, Factors & Primes Examples
10. Squares & Cubes
11. BODMAS
12. Decimal Places & Significant Figures
13. Truncated Numbers & Error Ranges
14. Estimation
15. Powers: An Introduction
16. More on Positive Powers
17. Fractional & Negative Powers
18. Powers in Equations
19. Powers: Examples 1
20. Powers: Examples 2
21. Standard Form
22. Recurring Decimals
23. Surds
24. Surds: Examples 1
25. Removing Surds: Rationalising the Denominator
26. Rationalising the Denominator: Harder Examples
27. Bounds: An Introduction
28. Bounds: Examples
Worked Ex's: 1. Numbers
Key Facts: 1. Numbers
2. Ratio & Proportion
1. What is Ratio & Proportion
2. Ratios, Fractions & Equations
3. Working with Ratios
4. Ratios & Scaling Factors
5. Proportionality
6. Direct Linear Proportionality: Examples
7. Gradients & Rates of Change
8. Rates of Change: Examples
9. Distance-Time Graphs
10. Variable Speeds & Distance-Time Graphs
11. Velocity-Time Graphs
12. Ratios, Percentages & Proportions
13. Single Percentage Problems
14. Percentage Multipliers
15. Populations & Compound Growth or Decay
16. Simple Interest
17. Compound Interest
Worked Ex's: 2. Ratio & Proportion
Key Facts: 2. Ratio & Proportion
3. Algebra & Equations
1. Solving & Simplifying
2. Expanding Brackets
3. Equations
4. Equations & Algebraic Families
5. Changing the Subject
6. Finding Common Factors
7. Cancelling Common Factors
8. Linear Equations: Worked Examples
9. Simultaneous Linear Equations: Substitution
10. Simultaneous Linear Equations: Elimination
11. Quadratics: Expanding & Factorising
12. Factorising Quadratics: The 4 Methods
13. Factorising Quads 1: By Inspection
14. Factorising Quads 1: Plotting the Results
15. Factorising Quads: 2. Quadratic Formula
16. Factorising Quads 2: The Discriminant
17. Factorising Quads 3: Completing the Square
18. Factorising Quads 3: Roots & Turning Points
19. Factorising Quads 3: Completing the Square: Example
20. Factorising Quads 4: Iteration
21. Factorising Quads 4: Iteration: Using a Calculator
22. Difference of Two Squares Quadratics
23. Simultaneous Equations: Linear & Quadratic
24. Algebraic Fractions
25. Inequalities: An Introduction
26. Inequalities: Number Lines
27. Multiplying an Inequality by a Negaitve: The proof
28. Quadratic Inequalities
29. Quadratic Inequalities: Examples
30. Real Life Formulas: Speed, Density & Pressure
Worked Ex's: 3. Algebra & Equations
Key Facts: 3. Algebra & Equations
4. Sequences & Series
1. Introduction to Sequences & Series
2. Types of Sequences
3. Arithmetic Sequences
4. Geometric Sequences
5. Geometric Sequences: Examples
6. Quadratic Sequences
7. Quadratic Sequences: A Proof
8. Quadratic Sequences: Examples
9. Number Sequences
Worked Ex's: 4. Sequences & Series
Key Facts: 4. Sequences & Series
5. Proofs
1. Proof Types
2. Algebraic Proofs
3. Algebraic Proofs: Examples
Worked Ex's: 5. Proofs
Key Facts: 5. Proofs
6. Functions & Graphs
1. What is a Function?
2. Sketching Functions
3. Sketching Functions: An Example
4. Roots: What & Where are They?
5. Graphical Solutions
6. Gradient & Straight Line Equations
7. More on Straight Line Gradients
8. Three Straight Lines You Must Know
9. Straight Lines: Where they Meet
10. Parallel & Perpendicular Lines
11. Shapes of Quadratics
12. Advantages of Factorising Quadratics
13. Completing the Square & Transformations
14. Shape of 'Difference of 2 Squares' Quadratics
15. Simultaneous Equations and Quadratics
16. Circles Straight Lines & Tangents
17. Other Common Functions
18. Exponentials
19. Areas Under Curves
20. Areas Under Curves: Triangles & Trapesiums
21. Transformations of Functions: An Introduction
22. Function Machines
23. Function Notation: f(x) g(x) h(x)
24. Composite Function Notation: fg(x) gf(x)
25. Inverse Functions
26. Inverse Functions: Example
27. Reflections of Functions
28. When Functions Misbehave: Dividing by Zero
29. When Functions Misbehave: Tan x
30. More on Domain & Range
31. Restrictions of Domain & Range
32. Domain & Range and Inverse Functions
33. More on Domain & Range and Inverse Functions
34. Domain & Range Restricted by Choice
35. Sectional Functions Defined by Domain
36. Domain & Range: Further Examples
Worked Ex's: 6. Functions & Graphs
Key Facts: 6. Functions & Graphs
7. Geometry
1. Basic Angle Stuff
2. Polygons: Interior & Exterior Angles
3. Geometric Constructions 1
4. Geometric Constructions 2
5. Loci
6. Angles of Depression & Elevation
7. Pythagoras
8. Congruent Triangles
9. Similar Triangles
10. Plans & Elevations
11. Circles: An Introduction
12. Circle Areas: Sectors & Arcs
13. Areas of Common Shapes
14. Cylinders, Spheres, Cones and Pyramids
15. Unit Conversions: Lengths, Areas & Volumes
16. Length, Area & Volume Scale Factors
17. Length, Area, Volume Scale Factors: Examples
18. Complex & Compound Areas
19. Coordinate Geometry: An Introduction
20. Coordinate Geometry: Mid-points
21. Bounded Areas & Graphical Area Inequalities
22. Bounded Areas: Examples
23. Circles, Lines & Tangents
24. Circle Theorems:An Introduction
25. Circle Theorems 1
26. Circle Theorems 2
27. Circle Theorems 3
28. Circle Theorems 4 & Examples
29. Circle Theorems 5 Intersecting Chords
30. Pythagoras in 3D
31. Translations & Reflections
32. Rotations & Enlargements
33. Combining Transformations
34. Bearings
35. Triangle Areas: Acute & Obtuse Angles
36. Triangle Area using a Calculotor
Worked Ex's: 7. Geometry
Key Facts: 7. Geometry
8. Trigonometry
1. Trigonometry: An Introduction
2. SOH CAH TOA
3. Finding Sides from Trig Ratios: sin, cos & tan
4. Finding Angles from Trig Ratios: sin^-1, cos^-1 & tan^-1
5. Pythagoras & SOH CAH TOA
6. Unit Circles & Trig Functions
7. Shape of Trig Functions: An Introduction
8. Standard Trig Values
9. Shape of sin x & cos x
10. Shape of tan x
11. Complete cycles of sin x cos x tan x
12. Standard Trig Values on the Waveforms
13. Trig Functions: y-Transformations
14. Trig Functions: x-Transformations
15. Sine & Cosine Rules
16. Sine or Cosine Rule?
17. Sine & Cosine Rules: With Bearings
18. Triangle Area using Sine Formula
19. Sine Area Formula & Circle Sectors
20. Sinusoidal Waves in Nature
Worked Ex's: 8. Trigonometry
Key Facts: 8. Trigonometry
9. Vectors
1. Vector Addition
2. Negative Vectors
3. Vector Addition: Examples
4. Vector Subtraction: Examples
5. Unit Vectors
6. Parallel Vectors
7. Parallel Split Straight-Line Vectors
8. Vector Geometry: Example 1
9. Vector Geometry: Example 2
10. Magnitude of a Vector (i-GCSE)
Worked Ex's: 9. Vectors
Key Facts: 9. Vectors
10. Data & Statistics
1. Frequency Tables
2. Mean Mode Median & Range
3. Quartiles
4. Finding Quartiles by a Rule
5. Quartiles with Grouped Data
6. Average & Range Examples
7. Populations
8. Frequency Tables: An Example
9. Grouped Frequency Tables
10. Grouped Frequency Tables: An Example
11. Frequency Polygons
12. Cumulative Frequency Tables
13. Cumulative Frequency Graphs
14. Cumulative Frequency Graphs: Another Example
15. Box Plots
16. Box Plot Examples
17. Comparing Box Plots
18. Combinations: The Product Rule
19. Combinations: The Product Rule: Examples
20. Bar Charts
21. Frequency Density
22. Histograms
23. Histograms: Examples
24. Histograms: Finding the Median
25. Stem & Leaf Diagrams
26. Line Graphs
27. Pie Charts
28. Scatter Graphs & Correlation
29. Venn Diagrams
30. Venn Diagrams: Examples
31: Probability
32. Venn Diagrams & Probability
33. Venn Diagrams: The Dangers of the OR Condition
34. Dependent and Independent Events
35. Dependent & Independent events: Examples 1
36. Dependent & Independent Events: Examples 2
37. Probability Tree Diagrams
38. Probability Tree Diagrams: A Useful Shortcut
39. Algebraic Probability Trees: Example 1
40. Algebraic Probability Trees: Example 1
Worked Ex's: 10. Data & Statistics
Key Facts: 10. Data & Statistics
11. Differentiation (i-GCSE)
1. Differentiation: An Introduction
2. From First Principles 1
3. From First Principles 2
4. How to Differentiate
5. Gradients of Tangents & Normals
6. Stationary Points
7. Finding Turning Points: Example 1
8. Finding Turning Points: Example 2
9. Optimisation Problems Using Differetiation
10. Optimisation Problems Using Differentiation: An Example
11. Optimisation Problems Using Differentiation: Another Example
Worked Ex's: 11. Differentiation (i-GCSE)
Key Facts: 11. Differentiation (i-GCSE)
12. Exam Tips & Traps
1. Exam Tips & Traps 1
2. Exam Tips & Traps 2
3. Exam Tips & Traps 3
4. Exam Tips & Traps 4
5. Exam Tips & Traps 5
13. GCSE Exam Papers
Nov 2022 Higher P1 Edexcel GCSE Worked Soln's
Nov 2022 Higher P2 Edexcel GCSE Worked Soln's
Nov 2022 Higher P3 Edexcel GCSE Worked Soln's
Tough Questions Higher P1 Edexcel GCSE Worked Soln's
Tough Questions Higher P2 Edexcel GCSE Worked Soln's
May 2023 P1 Edexcel GCSE Worked Soln's
June 2023 P2 Edexcel GCSE Worked Soln's
June 2023 P3 Edexcel GCSE Worked Soln's
Nov 2023 P1 Edexcel GCSE Worked Soln's
Nov 2023 P2 Edexcel GCSE Worked Soln's
Nov 2023 P3 Edexcel GCSE Worked Soln's
14. i-GCSE Exam Papers
Nov 2021 Paper 1 Edexcel i-GCSE Worked Soln's
Nov 2021 Paper 2 Edexcel i-GCSE Worked Soln's
A-LEVEL
AS PURE: Course Topics
1. Algebra & Equations
1. A Reminder of the Power Rules
2. Power Rules: Ex’s 1
3. Power Rules: Ex’s 2
4. Rationalisation
5. Dealing With Surds: More Ex's
6. Expanding & Factorising
7. Finding Common Factors
8. Cancelling & Simplification
9. Algebraic Fractions
Worked Ex's: 1. Algebra & Equations
Key Facts: 1. Algebra & Equations
2. Quadratics
1. ‘u’ or n’ Shapes
2. Factorising & Solving Quadratics: The 4 Methods
3. Factorising Quadratics: 1: By Inspection
4. Factorising Quadratics: 1: Plotting the Results
5. Factorising Quadratics: 1: By Inspection: Ex’s
6. Factorising Quadratics: 2: The Quadratic Formula
7. Factorising Quadratics: 2: Quadratic Formula Discriminant
8. Factorising Quadratics: 2: Testing the Discriminant
9. Factorising Quadratics: 3: Completing the Square
10. Factorising Quadratics: 3: Completing the Square: Finding Roots
11. Factorising Quadratics: 3: Completing the Square: Ex
12. Factorising Quadratics: 3: Completing the Square & Transformations
13. Completing the Square & the Turning Point
14. Completing the Square & the Quadratic Formula
15. Factorising Quadratics: 4: Iteration
16. Factorising Quadratics: 4: Iteration & Using a Calculator
17. Factorising Quadratics: The Difference of Two Squares
18. A Typical Exam Question
Worked Ex's: 2. Quadratics
Key Facts: 2. Quadratics
3. Equations & Inequalities
1. Simultaneous Linear Eqns: 1: Substitution Method
2. Simultaneous Linear Eqns: 2: Elimination Method
3. Simultaneous Eqns: Linear & Quadratic
4. Inequalities
5. Quadratic Inequalities: Ex’s 1
6. Quadratic Inequalities: Ex’s 2
7. Quadratic Inequalities: Ex’s 3
Worked Ex's: 3. Equations & Inequalities
Key Facts: 3. Equations & Inequalities
4. Functions & Transformations
1. Functions & Graphs
2. Standard Function Shapes 1
3. Standard Function Shapes 2
4. Parabolas, Cubics, Reciprocals & Exponentials
5. Exponential Functions
6. Sketching Functions
7. Sketching Functions: Ex 1
8. Sketching Functions: Ex 2
9. Roots: What & Where Are They?
10. Function Transformations: An Introduction
11. The Four Transformations
12. Reflections
13. Y Direction ↓ ↑ Transformations
14. X Direction → ← Transformations
Worked Ex's: 4. Functions & Transformations
Key Facts: 4. Functions & Transformations
5. Straight Lines
1. Straight Lines & Gradient 1
2. Straight Lines & Gradient 2
3. Parallel & Perpendicular Lines
Worked Ex's: 5. Straight Lines
Key Facts: 5. Straight Lines
6. Circles
1. Circles Centre (0, 0)
2. Circles Centre (a, b)
3. Radians, Arcs, Sectors & Segments
4. Circle Properties & Tangents
5. Circles Meeting Lines: 1
6. Circles Meeting Lines: 2
7. Circles Meeting Circles: Ex 1
8. Circles Meeting Circles: Ex 2
9. Circles Meeting Circles: Ex 3
Worked Ex's: 6. Circles
Key Facts: 6. Circles
7. Algebraic Methods (AS)
1. Algebraic Long Division: An Introduction
2. Algebraic Long Division: Examples
3. Remainder Theorem
4. Factor Theorem
5. Factorising Polynomials A Shortcut
6. Factorising Polynomials : Examples
7. Algebraic Fractions
8. Algebraic Fractions: Examples
Worked Ex's: 7. Algebraic Methods (AS)
Key Facts: 7. Algebraic Methods (AS)
8. Proofs (AS)
1. Proofs by Deduction
2. Proofs by Deduction: Examples 1
3. Proofs by Deduction: Examples 2
4. Proofs by Exhaustion
5. Proofs by Exhaustion: Examples
6. Proofs by Counter Example
7. Proofs by Counter Example: Examples 1
8. Proofs by Counter Example: Examples 2
9. Proofs by Counter Example: Examples 3
Worked Ex's: 8. Proofs (AS)
Key Facts: 8. Proofs (AS)
9. Binomial Expansions (AS)
1. Permutations & Combinations 1
2. Permutations & Combinations 2
3. Binomial Expansion of (a+b)^n
4. More about Binomial Expansion of (a+b)^n
5. Binomial Expansion of (a+b)^n: Ex's 1
6. Binomial Expansion of (a+b)^n: Ex's 2
Worked Ex's: 9. Binomial Expansions (AS)
Key Facts: 9. Binomial Expansions (AS)
10. Trigonometry (AS)
1. Sin Cos & Tan Definitions
2. Shape of Trig Functions
3. Standard Trig Values
4. Shape of Sin x & Cos x 0 to 90
5. Shape of Tan x 0 to 90
6. Complete Cycles of Sin Cos and Tan
7. Transforming Sin X
8. Sin & Cosine Rules
9. Area of a Triangle using Trig
Worked Ex's: 10. Trigonometry (AS)
Key Facts: 10. Trigonometry
11. Trig Identities
1. Pythagorean Trig Identities
2. Pythagorean Trig Identities: Ex's
3. Trig Equations: Graphical Solutions
4. Trig Equations: CAST Solutions
5. Trig Equations: CAST Solutions: Ex's 1
6. Trig Equations: CAST Solutions: Ex's 2
7. Trig Equations: CAST Solutions: Ex's 3
Worked Ex's: 11. Trig Identities
Key Facts: 11. Trig Identities
12. Vectors (AS)
1. Notation Magnitudes & Unit Vectors
2. Vector Addition
3. Vector Subtraction
4. Parallel Vectors
5. Position & Direction Vectors
6. Polar Form Notation for Vectors
7. Vector Magnitudes: Example
8. Distance between Two Points
9. Parallel & Split Straight-Line Vectors
10. Straight Line Vectors: Example
11. Vectors & Geometry: Harder Example 1
12. Vectors & Geometry: Harder Example 2
Worked Ex's: 12. Vectors (AS)
Key Facts: 12. Vectors (AS)
13. Differentiation
1. Differentiation: An Introduction
2. From First Principles 1
3. From First Principles 2
4. How to Differentiate
5. Gradients of Tangents & Normals
6. Stationary Points
7. Concave & Convex Curves
8. Finding Turning Points: Example 1
9. Finding Turning Points: Example 2
10. Finding Turning Points: Example 3
11. More on Function Inflections 1
12. More on Function Inflections
13. Optimisation Problems Using Differetiation
14. Optimisation Problems Using Differentiation: An Example
15. Optimisation Problems Using Differentiation: Another Example
16. Chain Rule
17. Chain Rule: An Example
Worked Ex's: 13. Differentiation (AS)
Key Facts: 13. Differentiation (AS)
14. Integration (AS)
1. Integration: An Introduction 1
2. Integration: An Introduction 2
3. Finding the Area Under a Curve
4. Integration Examples 1
5. Integration Examples 2
6. Integration Examples 3
7. Integration Examples 4
Worked Ex's: 14. Integration (AS)
Key Facts: 14. Integration (AS)
15. Logs & Exponentials
1. Exponential Functions
2. Log Functions
3. Log Laws & Common Log Values
4. Log Laws: Ex's
5. Changing the Base of Logs
6. Exponential & Log Functions
7. Exponential & Log Functions: Ex's
8. A Special Exponential e^x & Natural Logs
9. Exponentials & Natural Logs: Ex's 1
10. Exponentials & Natural Logs: Ex's 2
11. Exponentials & Natural Logs: Ex's 3
12. Exponential Functions Involving Time
13. Exponential Functions Involving Time: Ex's 1
14. Exponential Functions Involving Time: Ex's 2
15. Modelling Functions Using Logs
16. Modelling Functions Using Logs: Ex's
Worked Ex's: 15. Logs & Exponentials
Key Facts: 15. Logs & Exponentials
Exam Papers: AS Pure
Edexcel AS 2023 Pure
AS MECHANICS: Course Topics
1. Units & Model Simplification
1. Units of Measurements
2. Mathematical Model Simplifications
3. Weight & Mass, Speed & Velocity
Worked Ex's: 1. Unit & Simplifications
Key Facts: 1. Model Simplification
2. Constant Acceleration
1. Displacement-Time Graphs
2. Velocity-Time Graphs
3. Acceleration & Decelleration
4. Constant Acceleration Motion Eqns
5. SUVAT: The 5 Motion Eqns
6. SUVAT: The 5 Motion Eqns: Ex's
7. Velocity-Time Graphs Ex's
8. SUVAT: Harder Ex's 1
9. SUVAT: Harder Ex's 2
10. Acceleration due to Gravity
11. Acceleration due to Gravity: Ex's 1
12. Acceleration due to Gravity: Ex's 2
13. Acceleration due to Gravity: Ex's 3
14. Acceleration due to Gravity: Ex's 4
15. Vectors: Column Vectors & Magnitudes
16. Vectors: Polar & Component Form
17. Distance, Velocity & Acceleration Vectors
18. SUVAT Vectors in 2 Dimensions
19. SUVAT Vectors in 2 Dimensions: Ex's
Worked Ex's: 2. Constant Acceleration
Key Facts: 2. Constant Acceleration
3. Static Forces
1. Forces…What Are They?
2. Weight & Normal Reaction Forces
3. Tension & Thrust
4. Friction
5. Splitting Forces into Components
6. Forces in Equilibrium
7. Forces & Vectors
8. Forces & Vectors: Ex's 1
9. Forces & Vectors: Ex's 2
Worked Ex's: 3. Static Forces
Key Facts: 3. Static Forces
4. Forces & Motion
1. Force Diagrams
2. Newton’s Laws of Motion
3. Motion Alone a Line: Ex's
4. Motion in 2D: Ex 1
5. Motion in 2D: Ex 2
6. Motion in 2D: Ex 3
7. Particles Connected by Pulleys 1
8. Particles Connected by Pulleys 2
9. Connected Bodies in Motion Ex 1
10. Connected Bodies in Motion Ex 2
11. Connected Bodies in Motion Ex 3
12. Connected Bodies in Motion Ex 3 (Cont)
Worked Ex's: 4. Forces & Motion
Key Facts: 4. Forces & Motion
5. Variable Acceleration
1. Acceleration-Time Graphs
2. Differentiation & Displacement, Velocity & Acceleration
3. Displacement, Velocity & Acceleration Varying with Time
4. Motion Varying with Time: Ex's 1g with Time: Ex's 1
5. Motion Varying with Time: Ex's 2
Worked Ex's: 5. Variable Acceleration
Key Facts: 5. Variable Acceleration
Exam Papers: AS Mechanics
AS STATISTICS: Course Topics
1. Data Collection & Sampling
1. Large Data Set: An Introduction
2. Collection of Data: Planning the Collection
3. Types of Data: Discrete vs. Continuous Variables
4. Types of Data: Quantitative vs. Qualitative
5. Population & Samples
6. Sampling Methods
7. Stratified Sampling: The Maths
8. Sampling Techniques: Examples
Worked Ex's: 1. Data Collection
Key Facts: 1. Data Collection
2. Data Representation
1. Mean, Median, Mode, Range, Quartiles & Percentiles
2. Mean from Ungrouped Data
3. Median & Quartiles from Ungrouped Data
4. Finding the Median & the Quartiles: A Rule
5. Quartiles & Mid-Range Values in Grouped Data
6. Finding the Mean: Ex's
7. Quartiles from Grouped Data
8. Quartiles, Percentiles & Grouped Data: Ex's
9. Quartiles, Percentiles & Grouped Data: Ex 2
10. Box Plots: Quartiles & Interquartile Range
11. Outliers in Box Plots
12. Cleaning the Data: Outliers
13. Cleaning the Data: Outliers: Ex
14. Outliers & Comparison of Box Plots: Ex
15. Histograms
16. Scatter Graphs: Correlation & Association
17. Scatter Graphs: to Correlate or Not to Correlate
Worked Ex's: 2. Data Representation
Key Facts: 2. Data Representation
3. Location & Spread
1. Location & Spread: An Introduction
2. The Maths of Location & Spread
3. Variance: An Introduction
4. Variance & Standard Deviation: Longhand Ex
5. Variance & Standard Deviation: Calculator Ex
6. Standard Deviation: Well, not that Standard!
7. Variance: A Simplified Formula
8. Standard Deviation & Grouped Data
9. Standard Deviation & Grouped Data: Ex
10. Finding Outliers Using Mean & Standard Deviation
11. Cleaning Data Using Mean & Standard Deviation
12. Coding
13. Coding: How it works
14. Coding: Ex's
Worked Ex's: 3. Location & Spread
Key Facts: 3. Location & Spread
4. Correlation
1. Bivariate Data & Scatter Graphs
2. Linear Regression: An Introduction
3. Correlation Coefficient: r
4. Linear Correlation: Ex
Worked Ex's: 4. Correlation
Key Facts: 4. Correlation
5. Probability
1. Probability: Definitions & Notation
2. Probability Tree Diagrams
3. Venn Diagrams: Ind't & Dependent Events
4. Venn Diagrams: Dangers of the 'OR' Condition
5. Venn Diagrams: Ind't & Dependent Events: Ex's
6. Venn Diagrams: Harder Ex
7. Multiple Probability Outcomes
8. Probability Space Diagrams
9. Two-Way Probability Tables: Ex 1
10. Two-Way Probability Tables: Ex 2
Worked Ex's: 5. Probability
Key Facts: 5. Probability
6. Binomial Distributions
1. Discrete Random Variables: An Introduction
2. Discrete Random Variables: Real Life Examples
3. Probability Mass Functions
4. Discrete Random Variables: Examples
5. Probability Distribution Notation
6. Binomial Distributions: A Formula
7. Binomial Distributions: Finding the Formula: 1
8. Binomial Distributions: Finding the Formula: 2
9. Binomial Distributions: Finding the Formula: 3
10. Binomial Distributions: Finding the Formula: 4
11. Binomial Probability Density: Using the Formula
12. Binomial Probabilities on a Calculator: Example
13. Binomial Cumulative Probabilities on a Calculator
14. Binomial Cumulative Probabilities on a Calculator: Example
15. Binomial Probabilities: Tabular Probabilities on a Calculator
16. Binomial Probabilities: Using Tables
17. Binomial Probabilities: Using Tables: Example
Worked Ex's: 6. Binomial Distributions
Key Facts: 6. Binomial Distributions
7. Hypothesis Testing
1. Hypothesis Testing: An Introduction
2. Significance Probability Levels & Critical Values
3. Significance Probability Levels & Binomial Distributions
4. Significance Probability Levels Notation
5. Beware of the Upper Tail
6. Lower Tail Tests: Ex 1
7. Lower Tail Tests: Ex 2
8. Lower Tail Tests: Ex 3
9. Upper Tail Tests: Ex 1
10. Upper Tail Tests: Ex 2
11. Two Tail Tests: Ex 1
12. Two Tail Tests: Ex 2
13. Finding the Critical Region Using a List
14. Finding the Critical Region Using a List: Ex's
Worked Ex's: 7. Hypothesis Testing
Key Facts: 7. Hypothesis Testing
Exam Papers: AS Stats
May 2023 Stats Edexcel
A2 PURE: Course Topics
1. Algebraic Methods (A2)
1. Algebraic Fractions: 1
2. Algebraic Fractions: 2
3. Proper & Improper Rational Expressions
4. Decomposing Rules for Partial Fractions
5. Partial Fractions: Ex's 1
6. Partial Fractions: Ex's 2
7. Partial Fractions: Ex's 3
8. Numerators & Denominators of Equal Degree
9. Top-Heavy Improper Rational Expressions
10. Binomial Expansions & Partial Fractions
Worked Ex's: 1. Algebraic Methods (A2)
Key Facts: 1. Algebraic Methods (A2)
2. Functions & Graphs (A2)
1. Functions…What are they?
2. Domain & Range
3. Domain & Range: Ex’s
4. Inverse Functions
5. Inverse Functions: Domain & Range: Ex 1
6. Inverse Functions: Further Restrictions to Domain & Range
7. Modulus Functions: Ex’s 1
8. Modulus Functions: Ex’s 2
9. Modulus Functions: Ex’s 3
Worked Ex's: 2. Functions & Graphs (A2)
Key Facts: 2. Functions & Graphs (A2)
3. Sequences & Series (A2)
1. Series by Definition & Σ Notation
2. Series by Definition: Ex’s
3. Arithmetic Series
4. Arithmetic Series: Ex’s
5. Geometric Series
6. Geometric Series: Ex’s 1
7. Geometric Series: Ex’s 2
8. Geometric Series: Ex’s 3
Worked Ex's: 3. Sequences & Series (A2)
Key Facts: 3. Sequences & Series (A2)
4. Binomial Expansions (A2)
1. Binomial Expansions: The (1 +“x”)^n Expansion
2. (1 +“x”)^n Expansion: Ex’s
3. Binomial Estimation
4. Binomial Estimation: Ex’s 1
5. Binomial Estimation: Ex’s 2
6. Binomial Expansions & Partial Fractions
Worked Ex's: 4. Binomial Expansions (A2)
Key Facts: 4. Binomial Expansions (A2)
5. Radians
1. Radians, Arcs, Sectors & Segments
2. Radians, Arcs, Sectors & Segments Ex's
Worked Ex's: 5. Radians
Key Facts: 5. Radians
6. Trigonometric Functions
1. Reciprocal Functions: sec, cosec & cot
2. Inverse Trigonometric Functions: arcsin, arccos, arctan 1
3. Inverse Trigonometric Functions: arcsin, arccos, arctan 2
Worked Ex's: 6. Trigonometric Functions
Key Facts: 6. Trigonometric Functions
7. Further Trigonometry
1. Compound Angle Formula Proofs: 1
2. Compound Angle Formula Proofs: 2
3. Double Angle Formulas
4. Double Angle Formulas: Ex’s 1
5. Double Angle Formulas: Ex’s 2
6. Triple-Angle Formulas
7. Triple-Angle Formulas: Ex’s
8. Trig Equation Solutions: Using Trig Identities
9. Trig Equation Solutions: LHS/RHS Proofs
10. Harmonic Form: R sin (Θ ± ∝) & R cos (Θ ± ∝)
11. Harmonic Form: Ex’s 1
12. Harmonic Form: Ex’s 2
13. Harmonic Form: Ex’s 3
14. Harmonic Form: Ex’s 4
15. Factor Formulas
16. Small Angle Approximations
17. Small Angle Approximations: Ex’s 1
18. Small Angle Approximations: Ex’s 2
Worked Ex's: 7. Further Trigonometry
Key Facts: 7. Further Trigonometry
8. Parametric Equations
1. Parametric Equations
2. Parametric Equations: Ex's
3. Parametric Equations: Domain & Range
4. Parametric Equations: Ellipses
5. Parametric Equations: Ellipses: Ex's
6. Parametric to Cartesian Conversions Ex's
Worked Ex's: 8. Parametric Equations
Key Facts: 8. Parametric Equations
9. Differentiation (A2)
1. The Product Rule for Differentiation
2. The Product Rule: A Proof
3. The Quotient Rule for Differentiation
4. Related Rates of Change
5. Implicit Differentiation
6. Implicit Differentiation: Ex
7. Differentiating Trig Functions: sin x
8. Standard Trig Functions and their Differentials
9. Trig Functions & Powers
10. Differentiating Log Functions
11. Differentiating Exponential Functions
12. Differentiating Exponential & Log Functions: Ex’s
13. Parametric Differentiation
14. Parametric Differentiation: Ex’s 1
15. Parametric Differentiation: Ex’s 2
16. Differential Equations
17. Modelling With Differential Equations: Ex 1
18. Modelling With Differential Equations: Ex 2
Worked Ex's: 9. Differentiation (A2)
Key Facts: 9. Differentiation (A2)
10. Numerical Methods: Areas & Roots
1. Trapezium Rule for Finding Areas
2. Mid-Ordinate Rule for Finding Areas
3. Simpson’s Rule for Finding Areas
4. Finding Roots Using Sign Changes
5. Finding Roots Using Iteration
6. Finding Roots Using Newton-Raphson
7. Finding Turning Points Using Newton-Raphson
Worked Ex's: 10. Numerical Methods: Areas & Roots
Key Facts: Key Facts: 10. Numerical Methods: Areas & Roots
11. Integration (A2)
1. Integrating Simple Terms
2. Integral Simplification Techniques
3. Standard Trig Integrals
4. Standard Trig Integrals: Example
5. Integrating Even Powers of sin x & cos x
6. Integrating Even Powers of sin x & cos x: Ex’s
7. Integrating Trig Products using the Double Angle Formulas
8. Integrating Trig Products using the Factor Formulae
9. Integration by Substitution: Spotting the Derivative 1
10. Integration by Substitution: Spotting the Derivative 2
11. Integration by Substitution: Ex’s 1
12. Integration by Substitution: Ex’s 2
13. Integration by Substitution: Ex’s 3
14. Integration by Substitution: Ex’s 4
15. Integrating 1 / x
16. Integrating 1 / x: Ex
17. Integrating tan x & cot x
18. Integrating Odd Powers of sin x & cos x
19. Integrating Odd Powers of tan x
20. Integrating cosec x: A Tricky Trig Integral
21. Integrating Functions With: e
22. Integration by Parts
23. Integration by Parts: Ex 1
24. Integration by Parts: Ex 2
25. Repeated Use of Integration by Parts 1
26. Repeated Use of Integration by Parts 2
27. Repeated Use of Integration by Parts 3
28. Inverse Trig Integrals
29. Partial Fractions & Integration
30. Partial Fractions & Integration: Ex
31. Exponential Functions Involving Time
32. Parametric Integration
33. Parametric Integration: Ex
34. Volumes of Revolution
35. Volumes of Revolution of Parametric Functions
Worked Ex's: 11. Integration (A2)
Key Facts: 11. Integration (A2)
12. Integration Revision
1. Integration Revision: 1
2. Integration Revision: 2
3. Integration Revision: 3
4. Integration Revision: 4
5. Integration Results Table
6. Practice Integrals 1
7. Practice Integrals 2
8. Practice Integrals 3
9. Practice Integrals 4
10. Practice Integrals 5
11. Practice Integrals 6
Worked Ex's: 12. Integration Revision
Key Facts: 12. Integration Revision
13. Vectors (A2)
1. 3D Vectors
2. Distance Between Two 3D Points
3. Vector Between Two 3D Points
4. Unit Vectors
5. Angles Between Axes & 3D Vectors 1
6. Angles Between Axes & 3D Vectors 2
7. Angles Between Axes & 3D Vectors Ex
8. Collinearity of Vectors
9. Solving Geometric Problems Using Vectors 1
10. Solving Geometric Problems Using Vectors 2
11. Solving Geometric Problems Using Vectors 3
12. Solving Geometric Problems Using Vectors 4
13.Solving Geometric Problems Using Vectors 5
14. Solving Geometric Problems Using Vectors 6
15. THE FINAL PROBLEM
Worked Ex's: 13. Vectors (A2)
Key Facts: 13. Vectors (A2)
Exam Papers: A2 Pure
A2 MECHANICS: Course Topics
1. Moments
1. Moment of a Force: The Definition
2. Moments About a Point
3. Moments & Equilibrium
4. Moments & Non-Uniform Rods
5. Moments & Tilting Rods
6. Moments & Tilting Rods: Example
7. Laminas & Moments
8. Laminas & Moments: Example
9. Laminas: Finding the Centre of Mass
Worked Ex's: 1. Moments
Key Facts: 1. Moments
2. Forces & Friction
1. Forces & Friction: Force Diagrams
2. Friction Forces 1
3. Friction Forces 2
4. Angles of Slopes
5. Resolution of Forces on a Plane
6. More on the Coefficient of Friction & Limiting Equilibrium
Worked Ex's: 2. Forces & Friction
Key Facts: 2. Forces & Friction
3. Projectiles
1. Projectiles
2. Velocity Components of Projectiles
3. Projectiles: Displacement is the Integration of Velocity
4. Solving Projectile Problems Using SUVAT
5. Solving Projectile Problems Using the Displacement Equation
6. Horizontal Projectile Distances Using SUVAT
7. Horizontal Projectile Distances Using the Displacement Eqn.
Worked Ex's: 3. Projectiles
Key Facts: 3. Projectiles
4. Application of Forces
1. Masses on Slopes
2. Masses on Slopes & Pullies 1
3. Masses on Slopes & Pullies 2
4. Forces on a Ladder
5. Forces on a Ladder: Example
6. Motion in Two Dimensions 1
7. Motion in Two Dimensions 2
Worked Ex's: 4. Application of Forces
Key Facts: 4. Application of Forces
5. Further Kinematics
1. Displacement, Velocity & Acceleration: A Reminder
2. SUVAT In 2 Dimensions
3. Compass Points & Unit Vectors
4. SUVAT In 2 Dimensions: Vector Motion Equations 1
5. SUVAT in 2 Dimensions: Vector Motion Equations 2
6. Displacement, Velocity & Acceleration Varying with Time
7. Time Dependent Motion: Examples 1
8. Time Dependent Motion: Examples 2
9. Time Dependent Motion: Examples 3
10. Time Dependent Motion: Examples 4
11. Time Dependent Motion: Examples 5
12. Newton’s 2nd Law as a Differential Equation
Worked Ex's: 5. Further Kinematics
Key Facts: 5. Further Kinematics
Exam Papers: A2 Mechanics
A2 STATISTICS: Course Topics
1. Correlation & Regression
1. Bivariate Data & Scatter Graphsr
2. Linear Regression & Correlation: An Introduction
3. Linear Regression Line & Correlation: Using a Calculator
4. Calculating Linear Regression: Summing the Squares 1
5. Calculating Linear Regression: Summing the Squares 2
6. Calculating Linear Regression: Summing the Squares 3
7. Regression Line Gradient
8. Using the Regression Line Gradient Formula 1
9. Using the Regression Line Gradient Formula 2
10. Product Moment Correlation Coefficient: r
11. Non-Linear Relationships: Exponential Models
12. Non-Linear Relationships: Exponential Models: Ex 1
13. Non-Linear Relationships: Exponential Models: Ex 2
Worked Ex's: 1. Correlation & Regression
Key Facts: 1. Correlation & Regression
2. Conditional Probability
1. Probability: Independent & Dependent Events
2. Conditional Probabilities & Venn Diagrams
3. Conditional Probability & Venn Diagrams: Exs 1
4. Conditional Probability & Venn Diagrams: Exs 2
5. Conditional Probability & Venn Diagrams: Exs 3
6. Conditional Probability & Venn Diagrams: Exs 4
7. Conditional Probability & Tree Diagrams
Worked Ex's: 2. Conditional Probability
Key Facts: 2. Conditional Probability
3. Normal Distributions
1. Normal Distributions: An Introduction
2. Normal Distributions: Mean & Standard Deviation
3. Standard Normal Distributions
4. Standard Normal Distributions: Converting to Z-Scores
5. Converting X to Z to P Using a Calculator
6. Z-Score Tables
7. Normal Distributions: Ex’s 1
8. Normal Distributions: Ex's 2
9. Normal Distributions: Ex's 3
10. Normal Distributions: Ex's 4
11. Inverse Normal Function
12. Normal & Inverse Normal Ex's 1
13. Normal & Inverse Normal Ex's 2
14. Normal & Inverse Normal Ex's 3
15. Normal & Inverse Normal Ex's 4
16. Simultaneous Eqns & Normal Distributions: Ex's 1
17. Simultaneous Eqns & Normal Distributions: Ex's 2
18. Modelling Probability & Normal Distributions
19. Normal & Binomial Distributions: A Revision Comparison
20. Normal & Binomial Distribution Ex's
21. Normal Distributions & Histograms
Worked Ex's: 3. Normal Distributions
Key Facts: 3. Normal Distributions
4. Statistical Approximations
1. Normal Approximation for a Binomial Distribution
2. Binomial Distributions: Mean & Variance
3. Normal Distribution Approximations: Continuity Correction
4. Binomial to Normal Distribution Approximations: Ex’s 1
5. Binomial to Normal Distribution Approximations: Ex’s 2
6. Binomial to Normal Distribution Approximations: Ex’s 3
7. Choosing & Approximating Distributions
8. Distribution of the Sample Mean
9. Distribution of the Sample Mean: Ex’s
Worked Ex's: 4. Statistical Approximations
Key Facts: 4. Statistical Approximations
5. Hypothesis Testing & Distributions
1. Hypothesis Testing for Correlation: 1
2. Hypothesis Testing for Correlation: 2
3. Hypothesis Testing for Correlation: Ex’s 1
4. Hypothesis Testing for Correlation: Ex’s 2
5. Hypothesis Testing for Correlation: Ex’s 3
6. Rank Correlation (OCR Exam Board Only)
7. Rank Correlation (OCR Exam Board Only): Ex
8. Hypothesis Testing & Normal Distributions: Changes to the Mean
9. Sample Means
10. Sample Means: Finding the Critical Region
11. Sample Means: Ex’s 1
12. Sample Means: Ex’s 2
13. Sample Means: Ex’s 3
Worked Ex's: 5. Hypothesis Testing & Distributions
Key Facts: 5. Hypothesis Testing & Distributions
Exam Papers: A2 Statistics
AS & A2: APPENDIX & FORMULA SHEETS
1. Introduction to the Large Data Set
2. Large Data Set: Dataset Variables & Explanatory Notes 1
3. Large Data Set: Dataset Variables & Explanatory Notes 2
4. PMCC Table
5. Variance: A Simplification of the Formula
6. Standard Deviation: A Proof
7. Linear Regression: Derivation of the Formula
8. Proof of: μ=np for a Binomial Distribution
9. Proof of Variance: σ² = npq for a Binomial Distribution
10. Pure Maths Formula Sheet
11. Mechanics & Statistics Formula Sheet
SAT (USA)
SAT (USA) Practice Questions
SAT (USA) Practice Paper 1 Module 1
SAT (USA) Subject Questions
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GCSE/i-GCSE
GCSE/i-GCSE Course Info
1. Numbers
1. Basic Stuff
2. Top & Bottom Decks of Fractions
3. Fractions & Decimals 1
4. Fractions & Decimals 2.
5. Large & small Numbers
6. Percentages: An Introduction
7. Percentages, Fractions & Decimals
8. HCFs & LCMs, Factors & Primes
9. HCFs & LCMs, Factors & Primes Examples
10. Squares & Cubes
11. BODMAS
12. Decimal Places & Significant Figures
13. Truncated Numbers & Error Ranges
14. Estimation
15. Powers: An Introduction
16. More on Positive Powers
17. Fractional & Negative Powers
18. Powers in Equations
19. Powers: Examples 1
20. Powers: Examples 2
21. Standard Form
22. Recurring Decimals
23. Surds
24. Surds: Examples 1
25. Removing Surds: Rationalising the Denominator
26. Rationalising the Denominator: Harder Examples
27. Bounds: An Introduction
28. Bounds: Examples
Worked Ex's: 1. Numbers
Key Facts: 1. Numbers
2. Ratio & Proportion
1. What is Ratio & Proportion
2. Ratios, Fractions & Equations
3. Working with Ratios
4. Ratios & Scaling Factors
5. Proportionality
6. Direct Linear Proportionality: Examples
7. Gradients & Rates of Change
8. Rates of Change: Examples
9. Distance-Time Graphs
10. Variable Speeds & Distance-Time Graphs
11. Velocity-Time Graphs
12. Ratios, Percentages & Proportions
13. Single Percentage Problems
14. Percentage Multipliers
15. Populations & Compound Growth or Decay
16. Simple Interest
17. Compound Interest
Worked Ex's: 2. Ratio & Proportion
Key Facts: 2. Ratio & Proportion
3. Algebra & Equations
1. Solving & Simplifying
2. Expanding Brackets
3. Equations
4. Equations & Algebraic Families
5. Changing the Subject
6. Finding Common Factors
7. Cancelling Common Factors
8. Linear Equations: Worked Examples
9. Simultaneous Linear Equations: Substitution
10. Simultaneous Linear Equations: Elimination
11. Quadratics: Expanding & Factorising
12. Factorising Quadratics: The 4 Methods
13. Factorising Quads 1: By Inspection
14. Factorising Quads 1: Plotting the Results
15. Factorising Quads: 2. Quadratic Formula
16. Factorising Quads 2: The Discriminant
17. Factorising Quads 3: Completing the Square
18. Factorising Quads 3: Roots & Turning Points
19. Factorising Quads 3: Completing the Square: Example
20. Factorising Quads 4: Iteration
21. Factorising Quads 4: Iteration: Using a Calculator
22. Difference of Two Squares Quadratics
23. Simultaneous Equations: Linear & Quadratic
24. Algebraic Fractions
25. Inequalities: An Introduction
26. Inequalities: Number Lines
27. Multiplying an Inequality by a Negaitve: The proof
28. Quadratic Inequalities
29. Quadratic Inequalities: Examples
30. Real Life Formulas: Speed, Density & Pressure
Worked Ex's: 3. Algebra & Equations
Key Facts: 3. Algebra & Equations
4. Sequences & Series
1. Introduction to Sequences & Series
2. Types of Sequences
3. Arithmetic Sequences
4. Geometric Sequences
5. Geometric Sequences: Examples
6. Quadratic Sequences
7. Quadratic Sequences: A Proof
8. Quadratic Sequences: Examples
9. Number Sequences
Worked Ex's: 4. Sequences & Series
Key Facts: 4. Sequences & Series
5. Proofs
1. Proof Types
2. Algebraic Proofs
3. Algebraic Proofs: Examples
Worked Ex's: 5. Proofs
Key Facts: 5. Proofs
6. Functions & Graphs
1. What is a Function?
2. Sketching Functions
3. Sketching Functions: An Example
4. Roots: What & Where are They?
5. Graphical Solutions
6. Gradient & Straight Line Equations
7. More on Straight Line Gradients
8. Three Straight Lines You Must Know
9. Straight Lines: Where they Meet
10. Parallel & Perpendicular Lines
11. Shapes of Quadratics
12. Advantages of Factorising Quadratics
13. Completing the Square & Transformations
14. Shape of 'Difference of 2 Squares' Quadratics
15. Simultaneous Equations and Quadratics
16. Circles Straight Lines & Tangents
17. Other Common Functions
18. Exponentials
19. Areas Under Curves
20. Areas Under Curves: Triangles & Trapesiums
21. Transformations of Functions: An Introduction
22. Function Machines
23. Function Notation: f(x) g(x) h(x)
24. Composite Function Notation: fg(x) gf(x)
25. Inverse Functions
26. Inverse Functions: Example
27. Reflections of Functions
28. When Functions Misbehave: Dividing by Zero
29. When Functions Misbehave: Tan x
30. More on Domain & Range
31. Restrictions of Domain & Range
32. Domain & Range and Inverse Functions
33. More on Domain & Range and Inverse Functions
34. Domain & Range Restricted by Choice
35. Sectional Functions Defined by Domain
36. Domain & Range: Further Examples
Worked Ex's: 6. Functions & Graphs
Key Facts: 6. Functions & Graphs
7. Geometry
1. Basic Angle Stuff
2. Polygons: Interior & Exterior Angles
3. Geometric Constructions 1
4. Geometric Constructions 2
5. Loci
6. Angles of Depression & Elevation
7. Pythagoras
8. Congruent Triangles
9. Similar Triangles
10. Plans & Elevations
11. Circles: An Introduction
12. Circle Areas: Sectors & Arcs
13. Areas of Common Shapes
14. Cylinders, Spheres, Cones and Pyramids
15. Unit Conversions: Lengths, Areas & Volumes
16. Length, Area & Volume Scale Factors
17. Length, Area, Volume Scale Factors: Examples
18. Complex & Compound Areas
19. Coordinate Geometry: An Introduction
20. Coordinate Geometry: Mid-points
21. Bounded Areas & Graphical Area Inequalities
22. Bounded Areas: Examples
23. Circles, Lines & Tangents
24. Circle Theorems:An Introduction
25. Circle Theorems 1
26. Circle Theorems 2
27. Circle Theorems 3
28. Circle Theorems 4 & Examples
29. Circle Theorems 5 Intersecting Chords
30. Pythagoras in 3D
31. Translations & Reflections
32. Rotations & Enlargements
33. Combining Transformations
34. Bearings
35. Triangle Areas: Acute & Obtuse Angles
36. Triangle Area using a Calculotor
Worked Ex's: 7. Geometry
Key Facts: 7. Geometry
8. Trigonometry
1. Trigonometry: An Introduction
2. SOH CAH TOA
3. Finding Sides from Trig Ratios: sin, cos & tan
4. Finding Angles from Trig Ratios: sin^-1, cos^-1 & tan^-1
5. Pythagoras & SOH CAH TOA
6. Unit Circles & Trig Functions
7. Shape of Trig Functions: An Introduction
8. Standard Trig Values
9. Shape of sin x & cos x
10. Shape of tan x
11. Complete cycles of sin x cos x tan x
12. Standard Trig Values on the Waveforms
13. Trig Functions: y-Transformations
14. Trig Functions: x-Transformations
15. Sine & Cosine Rules
16. Sine or Cosine Rule?
17. Sine & Cosine Rules: With Bearings
18. Triangle Area using Sine Formula
19. Sine Area Formula & Circle Sectors
20. Sinusoidal Waves in Nature
Worked Ex's: 8. Trigonometry
Key Facts: 8. Trigonometry
9. Vectors
1. Vector Addition
2. Negative Vectors
3. Vector Addition: Examples
4. Vector Subtraction: Examples
5. Unit Vectors
6. Parallel Vectors
7. Parallel Split Straight-Line Vectors
8. Vector Geometry: Example 1
9. Vector Geometry: Example 2
10. Magnitude of a Vector (i-GCSE)
Worked Ex's: 9. Vectors
Key Facts: 9. Vectors
10. Data & Statistics
1. Frequency Tables
2. Mean Mode Median & Range
3. Quartiles
4. Finding Quartiles by a Rule
5. Quartiles with Grouped Data
6. Average & Range Examples
7. Populations
8. Frequency Tables: An Example
9. Grouped Frequency Tables
10. Grouped Frequency Tables: An Example
11. Frequency Polygons
12. Cumulative Frequency Tables
13. Cumulative Frequency Graphs
14. Cumulative Frequency Graphs: Another Example
15. Box Plots
16. Box Plot Examples
17. Comparing Box Plots
18. Combinations: The Product Rule
19. Combinations: The Product Rule: Examples
20. Bar Charts
21. Frequency Density
22. Histograms
23. Histograms: Examples
24. Histograms: Finding the Median
25. Stem & Leaf Diagrams
26. Line Graphs
27. Pie Charts
28. Scatter Graphs & Correlation
29. Venn Diagrams
30. Venn Diagrams: Examples
31: Probability
32. Venn Diagrams & Probability
33. Venn Diagrams: The Dangers of the OR Condition
34. Dependent and Independent Events
35. Dependent & Independent events: Examples 1
36. Dependent & Independent Events: Examples 2
37. Probability Tree Diagrams
38. Probability Tree Diagrams: A Useful Shortcut
39. Algebraic Probability Trees: Example 1
40. Algebraic Probability Trees: Example 1
Worked Ex's: 10. Data & Statistics
Key Facts: 10. Data & Statistics
11. Differentiation (i-GCSE)
1. Differentiation: An Introduction
2. From First Principles 1
3. From First Principles 2
4. How to Differentiate
5. Gradients of Tangents & Normals
6. Stationary Points
7. Finding Turning Points: Example 1
8. Finding Turning Points: Example 2
9. Optimisation Problems Using Differetiation
10. Optimisation Problems Using Differentiation: An Example
11. Optimisation Problems Using Differentiation: Another Example
Worked Ex's: 11. Differentiation (i-GCSE)
Key Facts: 11. Differentiation (i-GCSE)
12. Exam Tips & Traps
1. Exam Tips & Traps 1
2. Exam Tips & Traps 2
3. Exam Tips & Traps 3
4. Exam Tips & Traps 4
5. Exam Tips & Traps 5
13. GCSE Exam Papers
Nov 2022 Higher P1 Edexcel GCSE Worked Soln's
Nov 2022 Higher P2 Edexcel GCSE Worked Soln's
Nov 2022 Higher P3 Edexcel GCSE Worked Soln's
Tough Questions Higher P1 Edexcel GCSE Worked Soln's
Tough Questions Higher P2 Edexcel GCSE Worked Soln's
May 2023 P1 Edexcel GCSE Worked Soln's
June 2023 P2 Edexcel GCSE Worked Soln's
June 2023 P3 Edexcel GCSE Worked Soln's
Nov 2023 P1 Edexcel GCSE Worked Soln's
Nov 2023 P2 Edexcel GCSE Worked Soln's
Nov 2023 P3 Edexcel GCSE Worked Soln's
14. i-GCSE Exam Papers
Nov 2021 Paper 1 Edexcel i-GCSE Worked Soln's
Nov 2021 Paper 2 Edexcel i-GCSE Worked Soln's
A-LEVEL
AS PURE: Course Topics
1. Algebra & Equations
1. A Reminder of the Power Rules
2. Power Rules: Ex’s 1
3. Power Rules: Ex’s 2
4. Rationalisation
5. Dealing With Surds: More Ex's
6. Expanding & Factorising
7. Finding Common Factors
8. Cancelling & Simplification
9. Algebraic Fractions
Worked Ex's: 1. Algebra & Equations
Key Facts: 1. Algebra & Equations
2. Quadratics
1. ‘u’ or n’ Shapes
2. Factorising & Solving Quadratics: The 4 Methods
3. Factorising Quadratics: 1: By Inspection
4. Factorising Quadratics: 1: Plotting the Results
5. Factorising Quadratics: 1: By Inspection: Ex’s
6. Factorising Quadratics: 2: The Quadratic Formula
7. Factorising Quadratics: 2: Quadratic Formula Discriminant
8. Factorising Quadratics: 2: Testing the Discriminant
9. Factorising Quadratics: 3: Completing the Square
10. Factorising Quadratics: 3: Completing the Square: Finding Roots
11. Factorising Quadratics: 3: Completing the Square: Ex
12. Factorising Quadratics: 3: Completing the Square & Transformations
13. Completing the Square & the Turning Point
14. Completing the Square & the Quadratic Formula
15. Factorising Quadratics: 4: Iteration
16. Factorising Quadratics: 4: Iteration & Using a Calculator
17. Factorising Quadratics: The Difference of Two Squares
18. A Typical Exam Question
Worked Ex's: 2. Quadratics
Key Facts: 2. Quadratics
3. Equations & Inequalities
1. Simultaneous Linear Eqns: 1: Substitution Method
2. Simultaneous Linear Eqns: 2: Elimination Method
3. Simultaneous Eqns: Linear & Quadratic
4. Inequalities
5. Quadratic Inequalities: Ex’s 1
6. Quadratic Inequalities: Ex’s 2
7. Quadratic Inequalities: Ex’s 3
Worked Ex's: 3. Equations & Inequalities
Key Facts: 3. Equations & Inequalities
4. Functions & Transformations
1. Functions & Graphs
2. Standard Function Shapes 1
3. Standard Function Shapes 2
4. Parabolas, Cubics, Reciprocals & Exponentials
5. Exponential Functions
6. Sketching Functions
7. Sketching Functions: Ex 1
8. Sketching Functions: Ex 2
9. Roots: What & Where Are They?
10. Function Transformations: An Introduction
11. The Four Transformations
12. Reflections
13. Y Direction ↓ ↑ Transformations
14. X Direction → ← Transformations
Worked Ex's: 4. Functions & Transformations
Key Facts: 4. Functions & Transformations
5. Straight Lines
1. Straight Lines & Gradient 1
2. Straight Lines & Gradient 2
3. Parallel & Perpendicular Lines
Worked Ex's: 5. Straight Lines
Key Facts: 5. Straight Lines
6. Circles
1. Circles Centre (0, 0)
2. Circles Centre (a, b)
3. Radians, Arcs, Sectors & Segments
4. Circle Properties & Tangents
5. Circles Meeting Lines: 1
6. Circles Meeting Lines: 2
7. Circles Meeting Circles: Ex 1
8. Circles Meeting Circles: Ex 2
9. Circles Meeting Circles: Ex 3
Worked Ex's: 6. Circles
Key Facts: 6. Circles
7. Algebraic Methods (AS)
1. Algebraic Long Division: An Introduction
2. Algebraic Long Division: Examples
3. Remainder Theorem
4. Factor Theorem
5. Factorising Polynomials A Shortcut
6. Factorising Polynomials : Examples
7. Algebraic Fractions
8. Algebraic Fractions: Examples
Worked Ex's: 7. Algebraic Methods (AS)
Key Facts: 7. Algebraic Methods (AS)
8. Proofs (AS)
1. Proofs by Deduction
2. Proofs by Deduction: Examples 1
3. Proofs by Deduction: Examples 2
4. Proofs by Exhaustion
5. Proofs by Exhaustion: Examples
6. Proofs by Counter Example
7. Proofs by Counter Example: Examples 1
8. Proofs by Counter Example: Examples 2
9. Proofs by Counter Example: Examples 3
Worked Ex's: 8. Proofs (AS)
Key Facts: 8. Proofs (AS)
9. Binomial Expansions (AS)
1. Permutations & Combinations 1
2. Permutations & Combinations 2
3. Binomial Expansion of (a+b)^n
4. More about Binomial Expansion of (a+b)^n
5. Binomial Expansion of (a+b)^n: Ex's 1
6. Binomial Expansion of (a+b)^n: Ex's 2
Worked Ex's: 9. Binomial Expansions (AS)
Key Facts: 9. Binomial Expansions (AS)
10. Trigonometry (AS)
1. Sin Cos & Tan Definitions
2. Shape of Trig Functions
3. Standard Trig Values
4. Shape of Sin x & Cos x 0 to 90
5. Shape of Tan x 0 to 90
6. Complete Cycles of Sin Cos and Tan
7. Transforming Sin X
8. Sin & Cosine Rules
9. Area of a Triangle using Trig
Worked Ex's: 10. Trigonometry (AS)
Key Facts: 10. Trigonometry
11. Trig Identities
1. Pythagorean Trig Identities
2. Pythagorean Trig Identities: Ex's
3. Trig Equations: Graphical Solutions
4. Trig Equations: CAST Solutions
5. Trig Equations: CAST Solutions: Ex's 1
6. Trig Equations: CAST Solutions: Ex's 2
7. Trig Equations: CAST Solutions: Ex's 3
Worked Ex's: 11. Trig Identities
Key Facts: 11. Trig Identities
12. Vectors (AS)
1. Notation Magnitudes & Unit Vectors
2. Vector Addition
3. Vector Subtraction
4. Parallel Vectors
5. Position & Direction Vectors
6. Polar Form Notation for Vectors
7. Vector Magnitudes: Example
8. Distance between Two Points
9. Parallel & Split Straight-Line Vectors
10. Straight Line Vectors: Example
11. Vectors & Geometry: Harder Example 1
12. Vectors & Geometry: Harder Example 2
Worked Ex's: 12. Vectors (AS)
Key Facts: 12. Vectors (AS)
13. Differentiation
1. Differentiation: An Introduction
2. From First Principles 1
3. From First Principles 2
4. How to Differentiate
5. Gradients of Tangents & Normals
6. Stationary Points
7. Concave & Convex Curves
8. Finding Turning Points: Example 1
9. Finding Turning Points: Example 2
10. Finding Turning Points: Example 3
11. More on Function Inflections 1
12. More on Function Inflections
13. Optimisation Problems Using Differetiation
14. Optimisation Problems Using Differentiation: An Example
15. Optimisation Problems Using Differentiation: Another Example
16. Chain Rule
17. Chain Rule: An Example
Worked Ex's: 13. Differentiation (AS)
Key Facts: 13. Differentiation (AS)
14. Integration (AS)
1. Integration: An Introduction 1
2. Integration: An Introduction 2
3. Finding the Area Under a Curve
4. Integration Examples 1
5. Integration Examples 2
6. Integration Examples 3
7. Integration Examples 4
Worked Ex's: 14. Integration (AS)
Key Facts: 14. Integration (AS)
15. Logs & Exponentials
1. Exponential Functions
2. Log Functions
3. Log Laws & Common Log Values
4. Log Laws: Ex's
5. Changing the Base of Logs
6. Exponential & Log Functions
7. Exponential & Log Functions: Ex's
8. A Special Exponential e^x & Natural Logs
9. Exponentials & Natural Logs: Ex's 1
10. Exponentials & Natural Logs: Ex's 2
11. Exponentials & Natural Logs: Ex's 3
12. Exponential Functions Involving Time
13. Exponential Functions Involving Time: Ex's 1
14. Exponential Functions Involving Time: Ex's 2
15. Modelling Functions Using Logs
16. Modelling Functions Using Logs: Ex's
Worked Ex's: 15. Logs & Exponentials
Key Facts: 15. Logs & Exponentials
Exam Papers: AS Pure
Edexcel AS 2023 Pure
AS MECHANICS: Course Topics
1. Units & Model Simplification
1. Units of Measurements
2. Mathematical Model Simplifications
3. Weight & Mass, Speed & Velocity
Worked Ex's: 1. Unit & Simplifications
Key Facts: 1. Model Simplification
2. Constant Acceleration
1. Displacement-Time Graphs
2. Velocity-Time Graphs
3. Acceleration & Decelleration
4. Constant Acceleration Motion Eqns
5. SUVAT: The 5 Motion Eqns
6. SUVAT: The 5 Motion Eqns: Ex's
7. Velocity-Time Graphs Ex's
8. SUVAT: Harder Ex's 1
9. SUVAT: Harder Ex's 2
10. Acceleration due to Gravity
11. Acceleration due to Gravity: Ex's 1
12. Acceleration due to Gravity: Ex's 2
13. Acceleration due to Gravity: Ex's 3
14. Acceleration due to Gravity: Ex's 4
15. Vectors: Column Vectors & Magnitudes
16. Vectors: Polar & Component Form
17. Distance, Velocity & Acceleration Vectors
18. SUVAT Vectors in 2 Dimensions
19. SUVAT Vectors in 2 Dimensions: Ex's
Worked Ex's: 2. Constant Acceleration
Key Facts: 2. Constant Acceleration
3. Static Forces
1. Forces…What Are They?
2. Weight & Normal Reaction Forces
3. Tension & Thrust
4. Friction
5. Splitting Forces into Components
6. Forces in Equilibrium
7. Forces & Vectors
8. Forces & Vectors: Ex's 1
9. Forces & Vectors: Ex's 2
Worked Ex's: 3. Static Forces
Key Facts: 3. Static Forces
4. Forces & Motion
1. Force Diagrams
2. Newton’s Laws of Motion
3. Motion Alone a Line: Ex's
4. Motion in 2D: Ex 1
5. Motion in 2D: Ex 2
6. Motion in 2D: Ex 3
7. Particles Connected by Pulleys 1
8. Particles Connected by Pulleys 2
9. Connected Bodies in Motion Ex 1
10. Connected Bodies in Motion Ex 2
11. Connected Bodies in Motion Ex 3
12. Connected Bodies in Motion Ex 3 (Cont)
Worked Ex's: 4. Forces & Motion
Key Facts: 4. Forces & Motion
5. Variable Acceleration
1. Acceleration-Time Graphs
2. Differentiation & Displacement, Velocity & Acceleration
3. Displacement, Velocity & Acceleration Varying with Time
4. Motion Varying with Time: Ex's 1g with Time: Ex's 1
5. Motion Varying with Time: Ex's 2
Worked Ex's: 5. Variable Acceleration
Key Facts: 5. Variable Acceleration
Exam Papers: AS Mechanics
AS STATISTICS: Course Topics
1. Data Collection & Sampling
1. Large Data Set: An Introduction
2. Collection of Data: Planning the Collection
3. Types of Data: Discrete vs. Continuous Variables
4. Types of Data: Quantitative vs. Qualitative
5. Population & Samples
6. Sampling Methods
7. Stratified Sampling: The Maths
8. Sampling Techniques: Examples
Worked Ex's: 1. Data Collection
Key Facts: 1. Data Collection
2. Data Representation
1. Mean, Median, Mode, Range, Quartiles & Percentiles
2. Mean from Ungrouped Data
3. Median & Quartiles from Ungrouped Data
4. Finding the Median & the Quartiles: A Rule
5. Quartiles & Mid-Range Values in Grouped Data
6. Finding the Mean: Ex's
7. Quartiles from Grouped Data
8. Quartiles, Percentiles & Grouped Data: Ex's
9. Quartiles, Percentiles & Grouped Data: Ex 2
10. Box Plots: Quartiles & Interquartile Range
11. Outliers in Box Plots
12. Cleaning the Data: Outliers
13. Cleaning the Data: Outliers: Ex
14. Outliers & Comparison of Box Plots: Ex
15. Histograms
16. Scatter Graphs: Correlation & Association
17. Scatter Graphs: to Correlate or Not to Correlate
Worked Ex's: 2. Data Representation
Key Facts: 2. Data Representation
3. Location & Spread
1. Location & Spread: An Introduction
2. The Maths of Location & Spread
3. Variance: An Introduction
4. Variance & Standard Deviation: Longhand Ex
5. Variance & Standard Deviation: Calculator Ex
6. Standard Deviation: Well, not that Standard!
7. Variance: A Simplified Formula
8. Standard Deviation & Grouped Data
9. Standard Deviation & Grouped Data: Ex
10. Finding Outliers Using Mean & Standard Deviation
11. Cleaning Data Using Mean & Standard Deviation
12. Coding
13. Coding: How it works
14. Coding: Ex's
Worked Ex's: 3. Location & Spread
Key Facts: 3. Location & Spread
4. Correlation
1. Bivariate Data & Scatter Graphs
2. Linear Regression: An Introduction
3. Correlation Coefficient: r
4. Linear Correlation: Ex
Worked Ex's: 4. Correlation
Key Facts: 4. Correlation
5. Probability
1. Probability: Definitions & Notation
2. Probability Tree Diagrams
3. Venn Diagrams: Ind't & Dependent Events
4. Venn Diagrams: Dangers of the 'OR' Condition
5. Venn Diagrams: Ind't & Dependent Events: Ex's
6. Venn Diagrams: Harder Ex
7. Multiple Probability Outcomes
8. Probability Space Diagrams
9. Two-Way Probability Tables: Ex 1
10. Two-Way Probability Tables: Ex 2
Worked Ex's: 5. Probability
Key Facts: 5. Probability
6. Binomial Distributions
1. Discrete Random Variables: An Introduction
2. Discrete Random Variables: Real Life Examples
3. Probability Mass Functions
4. Discrete Random Variables: Examples
5. Probability Distribution Notation
6. Binomial Distributions: A Formula
7. Binomial Distributions: Finding the Formula: 1
8. Binomial Distributions: Finding the Formula: 2
9. Binomial Distributions: Finding the Formula: 3
10. Binomial Distributions: Finding the Formula: 4
11. Binomial Probability Density: Using the Formula
12. Binomial Probabilities on a Calculator: Example
13. Binomial Cumulative Probabilities on a Calculator
14. Binomial Cumulative Probabilities on a Calculator: Example
15. Binomial Probabilities: Tabular Probabilities on a Calculator
16. Binomial Probabilities: Using Tables
17. Binomial Probabilities: Using Tables: Example
Worked Ex's: 6. Binomial Distributions
Key Facts: 6. Binomial Distributions
7. Hypothesis Testing
1. Hypothesis Testing: An Introduction
2. Significance Probability Levels & Critical Values
3. Significance Probability Levels & Binomial Distributions
4. Significance Probability Levels Notation
5. Beware of the Upper Tail
6. Lower Tail Tests: Ex 1
7. Lower Tail Tests: Ex 2
8. Lower Tail Tests: Ex 3
9. Upper Tail Tests: Ex 1
10. Upper Tail Tests: Ex 2
11. Two Tail Tests: Ex 1
12. Two Tail Tests: Ex 2
13. Finding the Critical Region Using a List
14. Finding the Critical Region Using a List: Ex's
Worked Ex's: 7. Hypothesis Testing
Key Facts: 7. Hypothesis Testing
Exam Papers: AS Stats
May 2023 Stats Edexcel
A2 PURE: Course Topics
1. Algebraic Methods (A2)
1. Algebraic Fractions: 1
2. Algebraic Fractions: 2
3. Proper & Improper Rational Expressions
4. Decomposing Rules for Partial Fractions
5. Partial Fractions: Ex's 1
6. Partial Fractions: Ex's 2
7. Partial Fractions: Ex's 3
8. Numerators & Denominators of Equal Degree
9. Top-Heavy Improper Rational Expressions
10. Binomial Expansions & Partial Fractions
Worked Ex's: 1. Algebraic Methods (A2)
Key Facts: 1. Algebraic Methods (A2)
2. Functions & Graphs (A2)
1. Functions…What are they?
2. Domain & Range
3. Domain & Range: Ex’s
4. Inverse Functions
5. Inverse Functions: Domain & Range: Ex 1
6. Inverse Functions: Further Restrictions to Domain & Range
7. Modulus Functions: Ex’s 1
8. Modulus Functions: Ex’s 2
9. Modulus Functions: Ex’s 3
Worked Ex's: 2. Functions & Graphs (A2)
Key Facts: 2. Functions & Graphs (A2)
3. Sequences & Series (A2)
1. Series by Definition & Σ Notation
2. Series by Definition: Ex’s
3. Arithmetic Series
4. Arithmetic Series: Ex’s
5. Geometric Series
6. Geometric Series: Ex’s 1
7. Geometric Series: Ex’s 2
8. Geometric Series: Ex’s 3
Worked Ex's: 3. Sequences & Series (A2)
Key Facts: 3. Sequences & Series (A2)
4. Binomial Expansions (A2)
1. Binomial Expansions: The (1 +“x”)^n Expansion
2. (1 +“x”)^n Expansion: Ex’s
3. Binomial Estimation
4. Binomial Estimation: Ex’s 1
5. Binomial Estimation: Ex’s 2
6. Binomial Expansions & Partial Fractions
Worked Ex's: 4. Binomial Expansions (A2)
Key Facts: 4. Binomial Expansions (A2)
5. Radians
1. Radians, Arcs, Sectors & Segments
2. Radians, Arcs, Sectors & Segments Ex's
Worked Ex's: 5. Radians
Key Facts: 5. Radians
6. Trigonometric Functions
1. Reciprocal Functions: sec, cosec & cot
2. Inverse Trigonometric Functions: arcsin, arccos, arctan 1
3. Inverse Trigonometric Functions: arcsin, arccos, arctan 2
Worked Ex's: 6. Trigonometric Functions
Key Facts: 6. Trigonometric Functions
7. Further Trigonometry
1. Compound Angle Formula Proofs: 1
2. Compound Angle Formula Proofs: 2
3. Double Angle Formulas
4. Double Angle Formulas: Ex’s 1
5. Double Angle Formulas: Ex’s 2
6. Triple-Angle Formulas
7. Triple-Angle Formulas: Ex’s
8. Trig Equation Solutions: Using Trig Identities
9. Trig Equation Solutions: LHS/RHS Proofs
10. Harmonic Form: R sin (Θ ± ∝) & R cos (Θ ± ∝)
11. Harmonic Form: Ex’s 1
12. Harmonic Form: Ex’s 2
13. Harmonic Form: Ex’s 3
14. Harmonic Form: Ex’s 4
15. Factor Formulas
16. Small Angle Approximations
17. Small Angle Approximations: Ex’s 1
18. Small Angle Approximations: Ex’s 2
Worked Ex's: 7. Further Trigonometry
Key Facts: 7. Further Trigonometry
8. Parametric Equations
1. Parametric Equations
2. Parametric Equations: Ex's
3. Parametric Equations: Domain & Range
4. Parametric Equations: Ellipses
5. Parametric Equations: Ellipses: Ex's
6. Parametric to Cartesian Conversions Ex's
Worked Ex's: 8. Parametric Equations
Key Facts: 8. Parametric Equations
9. Differentiation (A2)
1. The Product Rule for Differentiation
2. The Product Rule: A Proof
3. The Quotient Rule for Differentiation
4. Related Rates of Change
5. Implicit Differentiation
6. Implicit Differentiation: Ex
7. Differentiating Trig Functions: sin x
8. Standard Trig Functions and their Differentials
9. Trig Functions & Powers
10. Differentiating Log Functions
11. Differentiating Exponential Functions
12. Differentiating Exponential & Log Functions: Ex’s
13. Parametric Differentiation
14. Parametric Differentiation: Ex’s 1
15. Parametric Differentiation: Ex’s 2
16. Differential Equations
17. Modelling With Differential Equations: Ex 1
18. Modelling With Differential Equations: Ex 2
Worked Ex's: 9. Differentiation (A2)
Key Facts: 9. Differentiation (A2)
10. Numerical Methods: Areas & Roots
1. Trapezium Rule for Finding Areas
2. Mid-Ordinate Rule for Finding Areas
3. Simpson’s Rule for Finding Areas
4. Finding Roots Using Sign Changes
5. Finding Roots Using Iteration
6. Finding Roots Using Newton-Raphson
7. Finding Turning Points Using Newton-Raphson
Worked Ex's: 10. Numerical Methods: Areas & Roots
Key Facts: Key Facts: 10. Numerical Methods: Areas & Roots
11. Integration (A2)
1. Integrating Simple Terms
2. Integral Simplification Techniques
3. Standard Trig Integrals
4. Standard Trig Integrals: Example
5. Integrating Even Powers of sin x & cos x
6. Integrating Even Powers of sin x & cos x: Ex’s
7. Integrating Trig Products using the Double Angle Formulas
8. Integrating Trig Products using the Factor Formulae
9. Integration by Substitution: Spotting the Derivative 1
10. Integration by Substitution: Spotting the Derivative 2
11. Integration by Substitution: Ex’s 1
12. Integration by Substitution: Ex’s 2
13. Integration by Substitution: Ex’s 3
14. Integration by Substitution: Ex’s 4
15. Integrating 1 / x
16. Integrating 1 / x: Ex
17. Integrating tan x & cot x
18. Integrating Odd Powers of sin x & cos x
19. Integrating Odd Powers of tan x
20. Integrating cosec x: A Tricky Trig Integral
21. Integrating Functions With: e
22. Integration by Parts
23. Integration by Parts: Ex 1
24. Integration by Parts: Ex 2
25. Repeated Use of Integration by Parts 1
26. Repeated Use of Integration by Parts 2
27. Repeated Use of Integration by Parts 3
28. Inverse Trig Integrals
29. Partial Fractions & Integration
30. Partial Fractions & Integration: Ex
31. Exponential Functions Involving Time
32. Parametric Integration
33. Parametric Integration: Ex
34. Volumes of Revolution
35. Volumes of Revolution of Parametric Functions
Worked Ex's: 11. Integration (A2)
Key Facts: 11. Integration (A2)
12. Integration Revision
1. Integration Revision: 1
2. Integration Revision: 2
3. Integration Revision: 3
4. Integration Revision: 4
5. Integration Results Table
6. Practice Integrals 1
7. Practice Integrals 2
8. Practice Integrals 3
9. Practice Integrals 4
10. Practice Integrals 5
11. Practice Integrals 6
Worked Ex's: 12. Integration Revision
Key Facts: 12. Integration Revision
13. Vectors (A2)
1. 3D Vectors
2. Distance Between Two 3D Points
3. Vector Between Two 3D Points
4. Unit Vectors
5. Angles Between Axes & 3D Vectors 1
6. Angles Between Axes & 3D Vectors 2
7. Angles Between Axes & 3D Vectors Ex
8. Collinearity of Vectors
9. Solving Geometric Problems Using Vectors 1
10. Solving Geometric Problems Using Vectors 2
11. Solving Geometric Problems Using Vectors 3
12. Solving Geometric Problems Using Vectors 4
13.Solving Geometric Problems Using Vectors 5
14. Solving Geometric Problems Using Vectors 6
15. THE FINAL PROBLEM
Worked Ex's: 13. Vectors (A2)
Key Facts: 13. Vectors (A2)
Exam Papers: A2 Pure
A2 MECHANICS: Course Topics
1. Moments
1. Moment of a Force: The Definition
2. Moments About a Point
3. Moments & Equilibrium
4. Moments & Non-Uniform Rods
5. Moments & Tilting Rods
6. Moments & Tilting Rods: Example
7. Laminas & Moments
8. Laminas & Moments: Example
9. Laminas: Finding the Centre of Mass
Worked Ex's: 1. Moments
Key Facts: 1. Moments
2. Forces & Friction
1. Forces & Friction: Force Diagrams
2. Friction Forces 1
3. Friction Forces 2
4. Angles of Slopes
5. Resolution of Forces on a Plane
6. More on the Coefficient of Friction & Limiting Equilibrium
Worked Ex's: 2. Forces & Friction
Key Facts: 2. Forces & Friction
3. Projectiles
1. Projectiles
2. Velocity Components of Projectiles
3. Projectiles: Displacement is the Integration of Velocity
4. Solving Projectile Problems Using SUVAT
5. Solving Projectile Problems Using the Displacement Equation
6. Horizontal Projectile Distances Using SUVAT
7. Horizontal Projectile Distances Using the Displacement Eqn.
Worked Ex's: 3. Projectiles
Key Facts: 3. Projectiles
4. Application of Forces
1. Masses on Slopes
2. Masses on Slopes & Pullies 1
3. Masses on Slopes & Pullies 2
4. Forces on a Ladder
5. Forces on a Ladder: Example
6. Motion in Two Dimensions 1
7. Motion in Two Dimensions 2
Worked Ex's: 4. Application of Forces
Key Facts: 4. Application of Forces
5. Further Kinematics
1. Displacement, Velocity & Acceleration: A Reminder
2. SUVAT In 2 Dimensions
3. Compass Points & Unit Vectors
4. SUVAT In 2 Dimensions: Vector Motion Equations 1
5. SUVAT in 2 Dimensions: Vector Motion Equations 2
6. Displacement, Velocity & Acceleration Varying with Time
7. Time Dependent Motion: Examples 1
8. Time Dependent Motion: Examples 2
9. Time Dependent Motion: Examples 3
10. Time Dependent Motion: Examples 4
11. Time Dependent Motion: Examples 5
12. Newton’s 2nd Law as a Differential Equation
Worked Ex's: 5. Further Kinematics
Key Facts: 5. Further Kinematics
Exam Papers: A2 Mechanics
A2 STATISTICS: Course Topics
1. Correlation & Regression
1. Bivariate Data & Scatter Graphsr
2. Linear Regression & Correlation: An Introduction
3. Linear Regression Line & Correlation: Using a Calculator
4. Calculating Linear Regression: Summing the Squares 1
5. Calculating Linear Regression: Summing the Squares 2
6. Calculating Linear Regression: Summing the Squares 3
7. Regression Line Gradient
8. Using the Regression Line Gradient Formula 1
9. Using the Regression Line Gradient Formula 2
10. Product Moment Correlation Coefficient: r
11. Non-Linear Relationships: Exponential Models
12. Non-Linear Relationships: Exponential Models: Ex 1
13. Non-Linear Relationships: Exponential Models: Ex 2
Worked Ex's: 1. Correlation & Regression
Key Facts: 1. Correlation & Regression
2. Conditional Probability
1. Probability: Independent & Dependent Events
2. Conditional Probabilities & Venn Diagrams
3. Conditional Probability & Venn Diagrams: Exs 1
4. Conditional Probability & Venn Diagrams: Exs 2
5. Conditional Probability & Venn Diagrams: Exs 3
6. Conditional Probability & Venn Diagrams: Exs 4
7. Conditional Probability & Tree Diagrams
Worked Ex's: 2. Conditional Probability
Key Facts: 2. Conditional Probability
3. Normal Distributions
1. Normal Distributions: An Introduction
2. Normal Distributions: Mean & Standard Deviation
3. Standard Normal Distributions
4. Standard Normal Distributions: Converting to Z-Scores
5. Converting X to Z to P Using a Calculator
6. Z-Score Tables
7. Normal Distributions: Ex’s 1
8. Normal Distributions: Ex's 2
9. Normal Distributions: Ex's 3
10. Normal Distributions: Ex's 4
11. Inverse Normal Function
12. Normal & Inverse Normal Ex's 1
13. Normal & Inverse Normal Ex's 2
14. Normal & Inverse Normal Ex's 3
15. Normal & Inverse Normal Ex's 4
16. Simultaneous Eqns & Normal Distributions: Ex's 1
17. Simultaneous Eqns & Normal Distributions: Ex's 2
18. Modelling Probability & Normal Distributions
19. Normal & Binomial Distributions: A Revision Comparison
20. Normal & Binomial Distribution Ex's
21. Normal Distributions & Histograms
Worked Ex's: 3. Normal Distributions
Key Facts: 3. Normal Distributions
4. Statistical Approximations
1. Normal Approximation for a Binomial Distribution
2. Binomial Distributions: Mean & Variance
3. Normal Distribution Approximations: Continuity Correction
4. Binomial to Normal Distribution Approximations: Ex’s 1
5. Binomial to Normal Distribution Approximations: Ex’s 2
6. Binomial to Normal Distribution Approximations: Ex’s 3
7. Choosing & Approximating Distributions
8. Distribution of the Sample Mean
9. Distribution of the Sample Mean: Ex’s
Worked Ex's: 4. Statistical Approximations
Key Facts: 4. Statistical Approximations
5. Hypothesis Testing & Distributions
1. Hypothesis Testing for Correlation: 1
2. Hypothesis Testing for Correlation: 2
3. Hypothesis Testing for Correlation: Ex’s 1
4. Hypothesis Testing for Correlation: Ex’s 2
5. Hypothesis Testing for Correlation: Ex’s 3
6. Rank Correlation (OCR Exam Board Only)
7. Rank Correlation (OCR Exam Board Only): Ex
8. Hypothesis Testing & Normal Distributions: Changes to the Mean
9. Sample Means
10. Sample Means: Finding the Critical Region
11. Sample Means: Ex’s 1
12. Sample Means: Ex’s 2
13. Sample Means: Ex’s 3
Worked Ex's: 5. Hypothesis Testing & Distributions
Key Facts: 5. Hypothesis Testing & Distributions
Exam Papers: A2 Statistics
AS & A2: APPENDIX & FORMULA SHEETS
1. Introduction to the Large Data Set
2. Large Data Set: Dataset Variables & Explanatory Notes 1
3. Large Data Set: Dataset Variables & Explanatory Notes 2
4. PMCC Table
5. Variance: A Simplification of the Formula
6. Standard Deviation: A Proof
7. Linear Regression: Derivation of the Formula
8. Proof of: μ=np for a Binomial Distribution
9. Proof of Variance: σ² = npq for a Binomial Distribution
10. Pure Maths Formula Sheet
11. Mechanics & Statistics Formula Sheet
SAT (USA)
SAT (USA) Practice Questions
SAT (USA) Practice Paper 1 Module 1
SAT (USA) Subject Questions
A2 Pure: Radians
2. Radians, Arcs, Sectors & Segments Ex's
🟢
A2 PURE: Course Topics
⚪ 1.
Algebraic Methods (A2)
⚪ 2.
Functions & Graphs (A2)
⚪ 3.
Sequences & Series (A2)
⚪ 4.
Binomial Expansions (A2)
🟢 5.
Radians
Radians, Arcs, Sectors & Segments
Radians, Arcs, Sectors & Segments: Ex’s
🎞️
Worked Ex's: Radians
🔑
Key Facts: Radians
⚪ 6.
Trigonometric Functions
⚪ 7.
Further Trigonometry
⚪ 8.
Parametric Equations
⚪ 9.
Differentiation (A2)
⚪ 10.
Numerical Methods: Areas & Roots
⚪ 11.
Integration (A2)
⚪ 12.
Integration Revision
⚪ 13.
Vector (A2)
🎞️
Worked Ex's: A2 Pure
📋
AS & A2 Appendix & Formula Sheets
♻️
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