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CIE A-Level Maths

Study Notes

1. Pure Mathematics 1

1.1.1 Completing the Square 1.1.2 The Discriminant 1.1.3 Solving Quadratic Equations 1.1.4 Quadratic Inequalities 1.1.5 Simultaneous Equations: Linear and Quadratic 1.1.6 Equations Reducible to Quadratics

1.2.1 Understanding Functions 1.2.2 Types of Functions 1.2.3 Range and Composition 1.2.4 Inverse Functions 1.2.5 Graphical Interpretation of Inverses 1.2.6 Transformations of Graphs

1.3.1 Equations of a Line 1.3.2 Forms of a Linear Equation 1.3.3 Gradients of Parallel and Perpendicular Lines 1.3.4 Equation of a Circle 1.3.5 Geometrical Properties of Circles 1.3.6 Intersections of Graphs

1.4.1 Understanding Radians 1.4.2 Arc Length and Sector Area 1.4.3 Applications to Triangles

1.5.1 Graphs of Trigonometric Functions 1.5.2 Exact Values of Trigonometric Ratios 1.5.3 Inverse Trigonometric Functions 1.5.4 Trigonometric Identities 1.5.5 Solving Trigonometric Equations

1.6.1 Binomial Expansion 1.6.2 Recognising Sequences 1.6.3 Arithmetic Progressions 1.6.4 Geometric Progressions 1.6.5 Convergence and Sum to Infinity

1.7.1 Understanding the Derivative 1.7.2 Differentiation Techniques 1.7.3 Applications of Differentiation 1.7.4 Stationary Points

1.8.1 Fundamental Concept of Integration 1.8.2 Finding the Constant of Integration 1.8.3 Evaluating Definite Integrals 1.8.4 Areas under Curves 1.8.5 Volumes of Revolution

1.1.1 Completing the Square 1.1.2 The Discriminant 1.1.3 Solving Quadratic Equations 1.1.4 Quadratic Inequalities 1.1.5 Simultaneous Equations: Linear and Quadratic 1.1.6 Equations Reducible to Quadratics

1.2.1 Understanding Functions 1.2.2 Types of Functions 1.2.3 Range and Composition 1.2.4 Inverse Functions 1.2.5 Graphical Interpretation of Inverses 1.2.6 Transformations of Graphs

1.3.1 Equations of a Line 1.3.2 Forms of a Linear Equation 1.3.3 Gradients of Parallel and Perpendicular Lines 1.3.4 Equation of a Circle 1.3.5 Geometrical Properties of Circles 1.3.6 Intersections of Graphs

1.4.1 Understanding Radians 1.4.2 Arc Length and Sector Area 1.4.3 Applications to Triangles

1.5.1 Graphs of Trigonometric Functions 1.5.2 Exact Values of Trigonometric Ratios 1.5.3 Inverse Trigonometric Functions 1.5.4 Trigonometric Identities 1.5.5 Solving Trigonometric Equations

1.6.1 Binomial Expansion 1.6.2 Recognising Sequences 1.6.3 Arithmetic Progressions 1.6.4 Geometric Progressions 1.6.5 Convergence and Sum to Infinity

1.7.1 Understanding the Derivative 1.7.2 Differentiation Techniques 1.7.3 Applications of Differentiation 1.7.4 Stationary Points

1.8.1 Fundamental Concept of Integration 1.8.2 Finding the Constant of Integration 1.8.3 Evaluating Definite Integrals 1.8.4 Areas under Curves 1.8.5 Volumes of Revolution

2. Pure Mathematics 2 & 3
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2.1.1. Understanding Absolute Value (|x|)2.1.2. Polynomial Division 2.1.3. Factor Theorem and Remainder Theorem 2.1.4. Partial Fractions 2.1.5 Expansion of (1 + x)^n

2.2.1 Fundamentals of Logarithms and Indices 2.2.2 Properties and Graphs of e^x and ln(x)2.2.3 Solving Equations with Logarithms 2.2.4 Logarithmic Transformation and Linearization

2.3.1 Extended Trigonometric Functions 2.3.2 Trigonometric Identities and Simplification 2.3.3 Solving Trigonometric Equations

2.4.1 Advanced Differentiation Techniques 2.4.2 Differentiation of Products and Quotients 2.4.3 Parametric and Implicit Differentiation

2.5.1 Advanced Integration Techniques 2.5.2 Trigonometric Integrations 2.5.3 Partial Fractions in Integration 2.5.4 Integrating Special Functions 2.5.5 Integration by Parts

2.6.1 Root Approximation Techniques 2.6.2 Convergence of Approximations 2.6.3 Iterative Solutions and Accuracy

2.7.1 Vector Notation and Fundamentals 2.7.2 Vector Operations and Geometric Interpretations 2.7.3 Vector Magnitudes and Direction 2.7.4 Equation of a Line in Vector Terms 2.7.5 Parallel, Intersecting, and Skew Lines 2.7.6 Scalar Product and Its Applications

2.8.1 Formulating Differential Equations 2.8.2 Solving Separable Differential Equations 2.8.3 Applying Initial Conditions 2.8.4 Interpreting Solutions in Context

2.9.1 Fundamentals of Complex Numbers 2.9.2 Arithmetic Operations with Complex Numbers 2.9.3 Conjugate Pairs in Polynomial Equations 2.9.4 Argand Diagram Representation 2.9.5 Operations in Polar Form 2.9.6 Square Roots of Complex Numbers 2.9.7 Geometrical Interpretations 2.9.8 Complex Loci on Argand Diagram

2.1.1. Understanding Absolute Value (|x|)2.1.2. Polynomial Division 2.1.3. Factor Theorem and Remainder Theorem 2.1.4. Partial Fractions 2.1.5 Expansion of (1 + x)^n

2.2.1 Fundamentals of Logarithms and Indices 2.2.2 Properties and Graphs of e^x and ln(x)2.2.3 Solving Equations with Logarithms 2.2.4 Logarithmic Transformation and Linearization

2.3.1 Extended Trigonometric Functions 2.3.2 Trigonometric Identities and Simplification 2.3.3 Solving Trigonometric Equations

2.4.1 Advanced Differentiation Techniques 2.4.2 Differentiation of Products and Quotients 2.4.3 Parametric and Implicit Differentiation

2.5.1 Advanced Integration Techniques 2.5.2 Trigonometric Integrations 2.5.3 Partial Fractions in Integration 2.5.4 Integrating Special Functions 2.5.5 Integration by Parts

2.6.1 Root Approximation Techniques 2.6.2 Convergence of Approximations 2.6.3 Iterative Solutions and Accuracy

2.7.1 Vector Notation and Fundamentals 2.7.2 Vector Operations and Geometric Interpretations 2.7.3 Vector Magnitudes and Direction 2.7.4 Equation of a Line in Vector Terms 2.7.5 Parallel, Intersecting, and Skew Lines 2.7.6 Scalar Product and Its Applications

2.8.1 Formulating Differential Equations 2.8.2 Solving Separable Differential Equations 2.8.3 Applying Initial Conditions 2.8.4 Interpreting Solutions in Context

2.9.1 Fundamentals of Complex Numbers 2.9.2 Arithmetic Operations with Complex Numbers 2.9.3 Conjugate Pairs in Polynomial Equations 2.9.4 Argand Diagram Representation 2.9.5 Operations in Polar Form 2.9.6 Square Roots of Complex Numbers 2.9.7 Geometrical Interpretations 2.9.8 Complex Loci on Argand Diagram

3. Mechanics
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3.1.1 Force Diagrams 3.1.2 Vector Analysis of Forces 3.1.3 Equilibrium Conditions 3.1.4 Modeling Contact Forces 3.1.5 Frictional Forces and Limiting Equilibrium

3.2.1 Fundamental Kinematic Concepts 3.2.2 Graphical Analysis of Motion 3.2.3 Calculus in Kinematics 3.2.4 Equations of Motion for Constant Acceleration

3.3 Momentum3.4 Newton’s Laws of Motion

3.5.1 Concept of Work Done 3.5.2 Gravitational Potential and Kinetic Energy 3.5.3 Energy Changes and Work Done 3.5.4 Power and Its Calculations 3.5.5 Energy and Power in Motion Problems

3.1.1 Force Diagrams 3.1.2 Vector Analysis of Forces 3.1.3 Equilibrium Conditions 3.1.4 Modeling Contact Forces 3.1.5 Frictional Forces and Limiting Equilibrium

3.2.1 Fundamental Kinematic Concepts 3.2.2 Graphical Analysis of Motion 3.2.3 Calculus in Kinematics 3.2.4 Equations of Motion for Constant Acceleration

3.3 Momentum3.4 Newton’s Laws of Motion

3.5.1 Concept of Work Done 3.5.2 Gravitational Potential and Kinetic Energy 3.5.3 Energy Changes and Work Done 3.5.4 Power and Its Calculations 3.5.5 Energy and Power in Motion Problems

4. Statistics and Probability 1

4.1.1 Selection of Data Presentation Methods 4.1.2 Creation and Interpretation of Graphical Representations 4.1.3 Measures of Central Tendency and Variation 4.1.4 Cumulative Frequency Analysis 4.1.5 Advanced Calculations with Mean and Standard Deviation

4.2.1 Understanding Permutations and Combinations 4.2.2 Arrangements with Repetition 4.2.3 Arrangements with Restrictions

4.3.1 Fundamental Probability Concepts 4.3.2 Addition and Multiplication Rules 4.3.3 Exclusive and Independent Events 4.3.4 Conditional Probability

4.4.1 Probability Distributions of Discrete Random Variables 4.4.2 Binomial Distribution 4.4.3 Geometric Distribution 4.4.4 Expectation and Variance of Binomial Distribution 4.4.5 Expectation of Geometric Distribution

4.5.1 Fundamentals of the Normal Distribution 4.5.2 Calculations Involving the Normal Distribution 4.5.3 Normal Approximation to the Binomial

4.1.1 Selection of Data Presentation Methods 4.1.2 Creation and Interpretation of Graphical Representations 4.1.3 Measures of Central Tendency and Variation 4.1.4 Cumulative Frequency Analysis 4.1.5 Advanced Calculations with Mean and Standard Deviation

4.2.1 Understanding Permutations and Combinations 4.2.2 Arrangements with Repetition 4.2.3 Arrangements with Restrictions

4.3.1 Fundamental Probability Concepts 4.3.2 Addition and Multiplication Rules 4.3.3 Exclusive and Independent Events 4.3.4 Conditional Probability

4.4.1 Probability Distributions of Discrete Random Variables 4.4.2 Binomial Distribution 4.4.3 Geometric Distribution 4.4.4 Expectation and Variance of Binomial Distribution 4.4.5 Expectation of Geometric Distribution

4.5.1 Fundamentals of the Normal Distribution 4.5.2 Calculations Involving the Normal Distribution 4.5.3 Normal Approximation to the Binomial

5. Statistics and Probability 2
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5.1.1 Poisson Probability Calculations 5.1.2 Mean and Variance of the Poisson Distribution 5.1.3 Poisson Distribution as a Model for Random Events 5.1.4 Poisson Approximation to the Binomial Distribution 5.1.5 Normal Approximation to the Poisson Distribution

5.2.1 Expectation of Linear Combinations 5.2.2 Variance of Linear Combinations 5.2.3 Normal Distribution of Linear Combinations 5.2.4 Poisson Distribution of Linear Combinations

5.3.1 Fundamentals of Continuous Random Variables 5.3.2 Utilizing Probability Density Functions 5.3.3 Mean and Variance from Density Functions 5.3.4 Determining Medians and Percentiles

5.4.1 Sample vs. Population 5.4.2 Critique of Sampling Methods 5.4.3 Sample Mean as a Random Variable 5.4.4 Distribution of the Sample Mean 5.4.5 Unbiased Estimation 5.4.6 Confidence Intervals for Population Mean 5.4.7 Confidence Interval for Population Proportion

5.5.1 Fundamentals of Hypothesis Testing 5.5.2 Formulating and Conducting Hypothesis Tests 5.5.3 Hypothesis Testing for Population Means 5.5.4 Understanding Type I and Type II Errors 5.5.5 Calculating Error Probabilities

5.1.1 Poisson Probability Calculations 5.1.2 Mean and Variance of the Poisson Distribution 5.1.3 Poisson Distribution as a Model for Random Events 5.1.4 Poisson Approximation to the Binomial Distribution 5.1.5 Normal Approximation to the Poisson Distribution

5.2.1 Expectation of Linear Combinations 5.2.2 Variance of Linear Combinations 5.2.3 Normal Distribution of Linear Combinations 5.2.4 Poisson Distribution of Linear Combinations

5.3.1 Fundamentals of Continuous Random Variables 5.3.2 Utilizing Probability Density Functions 5.3.3 Mean and Variance from Density Functions 5.3.4 Determining Medians and Percentiles

5.4.1 Sample vs. Population 5.4.2 Critique of Sampling Methods 5.4.3 Sample Mean as a Random Variable 5.4.4 Distribution of the Sample Mean 5.4.5 Unbiased Estimation 5.4.6 Confidence Intervals for Population Mean 5.4.7 Confidence Interval for Population Proportion

5.5.1 Fundamentals of Hypothesis Testing 5.5.2 Formulating and Conducting Hypothesis Tests 5.5.3 Hypothesis Testing for Population Means 5.5.4 Understanding Type I and Type II Errors 5.5.5 Calculating Error Probabilities

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