IB Mathematics AA HL
Analysis & Approaches - Higher Level
Welcome, students! This specialized section contains syllabus breakdowns, core study notes, targeted practice question banks, and actual exam past papers organized dynamically to optimize your learning and help you achieve a Grade 7.
1. Course Introduction
Mathematics: Analysis and Approaches (AA) Higher Level (HL) focuses on developing strong analytical and rational thinking skills. The course emphasizes rigorous mathematical proofs, algebraic methods, and deep concepts of calculus and algebraic analysis. It is highly recommended for students aiming to study Computer Science, Engineering, Physics, or Mathematics at premier universities worldwide.
Assessment Outline
The IB Mathematics AA HL course is assessed through final written examinations (80%) and an independent mathematical exploration task (20%). A graphic display calculator (GDC) is required only for Paper 2 and Paper 3:
| Assessment Component | Duration | Calculator Policy | Marks | Weighting |
|---|---|---|---|---|
| Paper 1 (Section A & B) | 120 minutes (2 hours) | Non-calculator (Non-GDC) | 110 marks | 30% |
| Paper 2 (Section A & B) | 120 minutes (2 hours) | Calculator allowed (GDC required) | 110 marks | 30% |
| Paper 3 (Problem Solving) | 60 minutes (1 hour) | Calculator allowed (GDC required) | 55 marks | 20% |
| Internal Assessment (IA) - Exploration | Independent research (12-20 pages) | No restrictions | 20 marks | 20% |
Syllabus Structure
The syllabus is organized into five core topics consisting of both standard core content and advanced high-level extension topics (SL Core & HL Extensions):
1.1: Operations in standard form
1.2: Arithmetic sequences and series
1.3: Geometric sequences and series
1.4: Financial applications
1.5: Laws of exponents and logarithms
1.6: Simple deductive proofs
1.7: Rational exponents
1.8: Infinite geometric series
1.9: Binomial theorem
1.10: Permutations and combinations [AHL]
1.11: Partial fractions [AHL]
1.12: Complex numbers in Cartesian form [AHL]
1.13: Polar and Euler forms of complex numbers [AHL]
1.14: Conjugate roots of polynomials [AHL]
1.15: Proof by mathematical induction [AHL]
1.16: Systems of linear equations [AHL]
2.1: Straight lines: equations, gradient, intercepts
2.2: Concept of a function: domain, range, notation
2.3: Graphs of functions
2.4: Key features of graphs
2.5: Composite and inverse functions
2.6: Quadratic functions and their graphs
2.7: Quadratic equations and discriminant
2.8: Reciprocal functions and graphs
2.9: Exponential and logarithmic functions
2.10: Solving equations graphically and analytically
2.11: Transformations of graphs
2.12: Factor and remainder theorems [AHL]
2.13: Rational functions [AHL]
2.14: Odd and even functions [AHL]
2.15: Solving inequalities [AHL]
2.16: Advanced graph transformations and modulus functions [AHL]
3.1: Distance and midpoint in 3D space
3.2: Right-angled trigonometry
3.3: Sine and cosine rules, area of a triangle
3.4: Radian measure, arc length, sector area
3.5: Unit circle trig definitions
3.6: Pythagorean identity and double angle formulae
3.7: Trigonometric graphs
3.8: Solving trigonometric equations
3.9: Reciprocal trigonometric functions [AHL]
3.10: Compound and double angle identities [AHL]
3.11: Vectors: position, displacement, magnitude [AHL]
3.12: Scalar product of vectors [AHL]
3.13: Vector product of vectors [AHL]
3.14: Vector equations of lines in 2D and 3D [AHL]
3.15: Line intersections in 3D [AHL]
3.16: Kinematics applications of vectors [AHL]
3.17: Vector equations of planes [AHL]
3.18: Intersections of lines and planes, intersections of planes [AHL]
4.1: Population, sampling, discrete and continuous data
4.2: Data presentation: frequency tables, histograms, box plots
4.3: Measures of central tendency and dispersion
4.4: Linear correlation of bivariate data
4.5: Probability of events, Venn and tree diagrams
4.6: Independent and mutually exclusive events
4.7: Discrete random variables and expectation
4.8: Binomial distribution
4.9: Normal distribution
4.10: Regression line and predictions
4.11: Conditional probability
4.12: Standardized normal variables (z-scores)
4.13: Bayes’ theorem [AHL]
4.14: Probability density functions for continuous random variables [AHL]
5.1: Concept of limits
5.2: Increasing and decreasing functions
5.3: Derivative of power functions and basic rules
5.4: Tangents and normals equations
5.5: Indefinite integration as anti-differentiation
5.6: Derivative rules: product, quotient, and chain rules
5.7: Second derivative, concavity, points of inflection
5.8: Local maximum and minimum points
5.9: Kinematic applications
5.10: Indefinite integrals of standard functions
5.11: Definite integrals and area under a curve
5.12: Continuity and differentiability at a point [AHL]
5.13: Limits of indeterminate forms and L’Hopital’s rule [AHL]
5.14: Implicit differentiation and related rates [AHL]
5.15: Derivatives of reciprocal and inverse trig functions [AHL]
5.16: Integration by substitution and parts [AHL]
5.17: Volumes of revolution [AHL]
5.18: First-order differential equations [AHL]
5.19: Maclaurin series expansions [AHL]
Official Academic Resources
Download official IB resources and curriculum guides below (Google Drive links will be loaded from metadata):
2. Study Notes
Access curated study notes summarizing core formulas and key theorems across all topics:
| Topic | Notes | Questions |
|---|---|---|
| 1. Number & Algebra | ||
| 1.1 Partial Fractions, Indices & Standard Form | Notes | Questions |
| 1.2 Exponentials & Logs | Notes | Questions |
| 1.3 Sequences & Series | Notes | Questions |
| 1.4 Simple Proof & Reasoning | Notes | Questions |
| 1.5 Proof by Induction & Contradiction | Notes | Questions |
| 1.6 Binomial Theorem | Notes | Questions |
| 1.7 Permutations & Combinations | Notes | Questions |
| 1.8 Complex Numbers | Notes | Questions |
| 1.9 Further Complex Numbers | Notes | Questions |
| 1.10 Systems of Linear Equations | Notes | Questions |
| 2. Functions | ||
| 2.1 Linear Functions & Graphs | Notes | Questions |
| 2.2 Quadratic Functions & Graphs | Notes | Questions |
| 2.3 Functions Toolkit | Notes | Questions |
| 2.4 Other Functions & Graphs | Notes | Questions |
| 2.5 Reciprocal & Rational Functions | Notes | Questions |
| 2.6 Transformations of Graphs | Notes | Questions |
| 2.7 Polynomial Functions | Notes | Questions |
| 2.8 Inequalities | Notes | Questions |
| 2.9 Modulus Functions & Further Transformations | Notes | Questions |
| 3. Geometry & Trigonometry | ||
| 3.1 Geometry Toolkit | Notes | Questions |
| 3.2 Geometry of 3D Shapes | Notes | Questions |
| 3.3 Trigonometry | Notes | Questions |
| 3.4 The Unit Circle & Exact Values | Notes | Questions |
| 3.5 Trigonometric Functions & Graphs | Notes | Questions |
| 3.6 Trigonometric Equations & Identities | Notes | Questions |
| 3.7 Inverse & Reciprocal Trigonometric Functions | Notes | Questions |
| 3.8 Trigonometric Proof & Equation Strategies | Notes | Questions |
| 3.9 Vector Properties | Notes | Questions |
| 3.10 Vector Equations of Lines | Notes | Questions |
| 3.11 Vector Planes | Notes | Questions |
| 4. Statistics & Probability | ||
| 4.1 Statistics Toolkit | Notes | Questions |
| 4.2 Correlation & Regression | Notes | Questions |
| 4.3 Probability | Notes | Questions |
| 4.4 Discrete Random Variables | Notes | Questions |
| 4.5 Binomial Distribution | Notes | Questions |
| 4.6 Normal Distribution | Notes | Questions |
| 4.7 Continuous Random Variables | Notes | Questions |
| 5. Calculus | ||
| 5.1 Differentiation | Notes | Questions |
| 5.2 Techniques & Applications of Differentiation | Notes | Questions |
| 5.3 Integration | Notes | Questions |
| 5.4 Techniques & Applications of Integration | Notes | Questions |
| 5.5 Optimisation | Notes | Questions |
| 5.6 Kinematics | Notes | Questions |
| 5.7 Basic Limits & Continuity | Notes | Questions |
| 5.8 Further Differentiation | Notes | Questions |
| 5.9 Further Integration | Notes | Questions |
| 5.10 Differential Equations | Notes | Questions |
| 5.11 Maclaurin Series | Notes | Questions |
| 5.12 Limits using l’Hôpital’s Rule & Maclaurin Series | Notes | Questions |
| Syllabus Topic | Summary Guide |
|---|---|
| Topic 1: Number and Algebra | Download PDF |
| Topic 2: Functions | Download PDF |
| Topic 3: Geometry and Trigonometry | Download PDF |
| Topic 4: Statistics and Probability | Download PDF |
| Topic 5: Calculus | Download PDF |
3. Practice Questions
| Subtopic | Worksheet Title | Question Paper | Mark Scheme |
|---|---|---|---|
| Topic 1: Number & Algebra | |||
1.1, 1.2
|
Numbers & Methods of Proof | Questions | Solutions |
1.3, 1.4, 1.5, 1.6
|
Sequences & Series | Questions | Solutions |
1.7
|
Binomial Theorem | Questions | Solutions |
1.8
|
Mathematical Induction | Questions | Solutions |
1.9
|
Systems of Linear Equations | Questions | Solutions |
1.10, 1.11
|
Complex Numbers (Cartesian Form) | Questions | Solutions |
1.12, 1.13, 1.14
|
Complex Numbers (Polar & Euler Forms) | Questions | Solutions |
| Topic 2: Functions | |||
2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.16
|
Functions, Graphs & Modulus Operations | Questions | Solutions |
2.8, 2.9, 2.10
|
Exponents & Logarithms | Questions | Solutions |
2.11, 2.12, 2.13, 2.14, 2.15
|
Polynomials & Rational Functions | Questions | Solutions |
| Topic 3: Geometry & Trigonometry | |||
3.1, 3.2, 3.3, 3.4
|
3D Geometry, Triangles, Arcs & Sectors | Questions | Solutions |
3.5, 3.6, 3.7, 3.8, 3.9
|
Trigonometric Equations & Functions | Questions | Solutions |
3.10, 3.11, 3.12, 3.13, 3.14
|
Vectors & Equations of Lines | Questions | Solutions |
3.15, 3.16, 3.17, 3.18
|
Cross Product & Vector Planes | Questions | Solutions |
| Topic 4: Statistics & Probability | |||
4.1, 4.2, 4.3, 4.4
|
Statistics Toolkit & Regression | Questions | Solutions |
4.5, 4.6, 4.7, 4.8
|
Probability Theory & Expectation | Questions | Solutions |
4.9, 4.10
|
Discrete Random Variables & Binomial Distribution | Questions | Solutions |
4.11, 4.12
|
Continuous Random Variables & Normal Distribution | Questions | Solutions |
4.13
|
Counting & Probability | Questions | Solutions |
| Topic 5: Calculus | |||
5.1, 5.2, 5.3, 5.4
|
Derivatives & Rates of Change | Questions | Solutions |
5.2
|
5.2 | Questions | Solutions |
5.5, 5.6, 5.7
|
Monotony, Concavity & Optimisation | Questions | Solutions |
5.8, 5.9, 5.16
|
Indefinite Integrals & Substitution Techniques | Questions | Solutions |
5.10, 5.18
|
Definite Integrals, Areas & Volumes | Questions | Solutions |
5.11
|
Kinematics | Questions | Solutions |
5.12, 5.13
|
Continuity, Differentiability & L’Hôpital’s Rule | Questions | Solutions |
5.14, 5.15
|
Implicit Differentiation & Related Rates | Questions | Solutions |
5.17
|
Integration by Parts | Questions | Solutions |
5.19
|
Differential Equations | Questions | Solutions |
4. Topical past paper questions
Questions taken from actual exam questions organized by the main syllabus topics:
| Chapter | Topic / Concept Area | Paper 1 | Paper 2 |
|---|---|---|---|
| Ch 1 | Algebra, Functions & Roots | Part 1 | Part 1 |
| Ch 2 | Quadratics | Part 1 | Part 1 |
| Ch 3 | Exponents & Logarithms | Part 1 | Part 1 |
| Ch 4 | Graphs | Part 1 | Part 1 |
| Ch 5 | Sequences & Series | Part 1 | Part 1 |
| Ch 6 | Complex Numbers | Part 1 | Part 1 |
| Ch 7 | Permutations & Combinations | Part 1 | Part 1 |
| Ch 8 | Binomial Theorem | Part 1 | Part 1 |
| Ch 9 | Remainder & Factor Theorem | Part 1 | Part 1 |
| Ch 10 | Mathematical Induction | Part 1 | Part 1 |
| Ch 11 | Radians | Part 1 | Part 1 |
| Ch 12 | Trigonometry | Part 1 | Part 1 |
| Ch 13 | Matrices | Part 1 | Part 1 |
| Ch 14 | Vectors, Lines & Planes | Part 1 | Part 1 |
| Ch 15 | Statistics | Part 1 | Part 1 |
| Ch 16 | Probability | Part 1 | Part 1 |
| Ch 17 | Differentiation | Part 1 | Part 1 |
| Ch 18 | Integration | Part 1 | Part 1 |
| Ch 19 | Differential Equations | Part 1 | Part 1 |
| Ch 20 | Kinematics | Part 1 | Part 1 |
| Ch 21 | Geometry | Part 1 | Part 1 |
[!IMPORTANT] Paper 3 Extended Exam Questions: Students preparing for the HL exam can access the Paper 3 extended analysis past paper packs:
| Chapter | Topic / Concept Area | Paper 1 | Paper 2 |
|---|---|---|---|
| Ch 1 | Algebra, Functions & Roots | Part 1 | Part 1 |
| Ch 2 | Quadratics | Part 1 | Part 1 |
| Ch 3 | Exponents & Logarithms | Part 1 | Part 1 |
| Ch 4 | Graphs | Part 1 | Part 1 |
| Ch 5 | Sequences & Series | Part 1 | Part 1 |
| Ch 6 | Complex Numbers | Part 1 | Part 1 |
| Ch 7 | Permutations & Combinations | Part 1 | Part 1 |
| Ch 8 | Binomial Theorem | Part 1 | Part 1 |
| Ch 9 | Remainder & Factor Theorem | Part 1 | Part 1 |
| Ch 10 | Mathematical Induction | Part 1 | Part 1 |
| Ch 11 | Radians | Part 1 | Part 1 |
| Ch 12 | Trigonometry | Part 1 | Part 1 |
| Ch 13 | Matrices | Part 1 | Part 1 |
| Ch 14 | Vectors, Lines & Planes | Part 1 | Part 1 |
| Ch 15 | Statistics | Part 1 | Part 1 |
| Ch 16 | Probability | Part 1 | Part 1 Part 2 |
| Ch 17 | Differentiation | Part 1 | Part 1 |
| Ch 18 | Integration | Part 1 | Part 1 |
| Ch 19 | Differential Equations | Part 1 | Part 1 |
| Ch 20 | Kinematics | Part 1 | Part 1 |
[!IMPORTANT] Paper 3 Extended Exam Questions (Options): Students preparing for the HL exam can access the specialized Paper 3 option papers below:
5. Past Papers Bank
Search and filter actual past exam papers. Use the quick filter bar to locate papers by year, component, or document type:
IB Mathematics AA HL Past Papers
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