Name: ESAT Maths Course Book: M1 and M2
Author: VerityPrep

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Prepare for ESAT maths with chapters on circle theorems, conditional probability, differentiation and integration, backed by 241 questions. Explore the book.

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About this book

This ESAT maths book is written for students preparing for the Engineering and Science Admissions Test, and it follows the two maths parts of the exam. The Maths 1 chapters build the core: units and compound measures, numbers, ratio and proportion, algebra and functions, geometry, statistics and probability.

The Maths 2 chapters move into more advanced territory, from indices, surds, polynomials and graph transformations to trigonometric identities and equations, exponentials and logarithms, sequences and the binomial expansion, modulus and inverse functions, differentiation with optimisation and related rates, and integration by substitution and by parts, ending with volumes of revolution.

The opening chapter on units includes worked examples, and each chapter closes with a revision section. Across the book there are 241 questions to test your understanding, so you can move from reading a topic to practising it on VerityPrep without switching resources.

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14 chapters · 98 topics

M1-01-Units and Compound Measures
1. 01-The Core Volume-Capacity Bridges Sample
2. 02-Metric Conversion Reference
3. 01-The dimensional "Power Rule"
4. 02-Quick Conversion Reference
5. 03-Precision Under Calculator-Free Conditions
6. 04-Converting Compound Formats: The Split-Fraction Method
7. 05-Watch the Trap!
8. 01-Dimensional Variable Shifting
9. 02-Kinematic Graph Analysis
10. Chapter 1 Comprehensive Practice Exam
M1-02-Numbers
1. 01-Rounding: decimal places vs significant figures
2. 02-Estimation strategy
3. 03-Common traps and error intervals
4. 01-Working with fractions
5. 02-Percentages and the multiplier method
6. 03-The reverse-percentage trap
7. 02-Standard form
8. 03-Two standard-form traps
M1-03-Ratio and Proportion
1. 3.1 Ratio
2. 3.2 Direct and Inverse Proportion
3. 3.3 Proportional Reasoning in Context
4. Chapter 3 Revision Exam
M1-04-Algebra and Functions
1. 4.1 Manipulating Algebraic Expressions
2. 4.2 Equations and Inequalities
3. 4.3 Sequences and Graphs
4. Chapter 4 Revision Exam
M1-05-Geometry
1. 5.1 Angles and Polygons
2. 5.2 Pythagoras and Trigonometry
3. 5.3 Circle Theorems
4. 5.4 Mensuration: Area, Surface Area and Volume
5. Chapter 5 Revision Exam
M1-06-Statistics
1. 6.1 Collecting and Representing Data
2. 6.2 Averages: Mean, Median and Mode
3. 6.3 Measures of Spread
4. 6.4 Scatter Graphs and Correlation
5. Chapter 6 Revision Exam
M1-07-Probability
1. 7.1 Sample Spaces and Single Events
2. 7.2 Combined Events, the Addition Rule and Venn Diagrams
3. 7.3 Independent Events and Tree Diagrams
4. 7.4 Conditional Probability
5. 7.5 Expectation
6. Chapter 7 Revision Exam
M2-01-Algebra and Functions
1. 01-Index & surd rules
2. 02-Common traps with surds
3. 01-The discriminant
4. 02-Equal roots condition
5. 01-Reading inequalities
6. 02-Sign outside vs inside roots
7. 01-Theorems for polynomials
8. 02-Complete the factorisation
9. 01-Function facts
10. 02-Composition is not commutative
11. 01-Transformations of y = f(x)
12. 02-Inside the bracket is backwards
13. REVISION
M2-02-Sequences and Series
1. Chapter 2 Overview
2. A2.1 Sequences and Recurrence Relations
3. A2.2 Arithmetic Sequences and Series
4. A2.3 Geometric Sequences and Series
5. A2.4 Sum to Infinity and Convergence
6. A2.5 Sigma Notation
7. A2.6 The Binomial Expansion
M2-03-Exponentials and Logarithms
1. Chapter 10 Overview
2. 10.1 Exponential Functions and Their Graphs
3. 10.2 Logarithms and the Laws of Logs
4. 10.3 The Exponential Function \(e^{x}\) and Natural Logarithm \(\ln x\)
5. 10.4 Solving Exponential and Logarithmic Equations
6. 10.5 Transformations of Exponential and Log Graphs
7. 10.6 Exponential Models and Linearising with Logarithms
M2-03-Trigonometry
1. Chapter 9 Overview
2. 9.1 Radian Measure, Arc and Sector
3. 9.2 Trigonometric Graphs and Their Properties
4. 9.3 Transformations of Trigonometric Graphs
5. 9.4 Trigonometric Identities
6. 9.5 Solving Trigonometric Equations
7. 9.6 The Sine and Cosine Rules; Triangle Area
M2-04-Functions and Graphs
1. Chapter 8 Overview
2. 8.1 Modulus Functions and Graphs
3. 8.2 Functions, Mappings, Domain and Range
4. 8.3 Composite Functions
5. 8.4 Inverse Functions
6. 8.5 The Graphs of y=|f(x)| and y=f(|x|)
M2-05-Differentiation
1. Chapter 6 Overview
2. A6.1 The Derivative: First Principles and the Power Rule
3. A6.2 The Product, Quotient and Chain Rules
4. A6.3 Derivatives of Exponential, Logarithmic and Trigonometric Functions
5. A6.4 Tangents, Normals and Stationary Points
6. A6.5 Optimization: Applied Maxima and Minima
7. A6.6 Related Rates of Change
8. A6.7 Higher Derivatives and the Graph of the Gradient Function
M2-07-Integration
1. Chapter 7 Overview
2. A7.1 Integration as the Reverse of Differentiation
3. A7.2 Integrating Standard Functions
4. A7.3 The Definite Integral and the Area Under a Curve
5. A7.4 Area Between Curves and Regions Below the Axis
6. A7.5 Integration by Substitution
7. A7.6 Integration by Parts
8. A7.7 Applications: Volumes of Revolution

Sample · M1-01-Units and Compound Measures

01-The Core Volume-Capacity Bridges

Memorize these exact conversions to bypass step-by-step logic during the exam:

1 ml = 1 cm³
1 l = 1 dm³ = 1000 cm³
1000 l = 1 m³

02-Metric Conversion Reference

Quantity	Key metric conversions
Length	1 cm = 10 mm \| 1 m = 100 cm \| 1 km = 1,000 m
Area	1 cm² = 100 mm² \| 1 m² = 10,000 cm² \| 1 km² = 1,000,000 m²
Volume	1 cm³ = 1,000 mm³ \| 1 m³ = 1,000,000 cm³
Capacity	1 ml = 1 cm³ \| 1 l = 1,000 cm³ \| 1 m³ = 1,000 l
Mass	1 g = 1,000 mg \| 1 kg = 1,000 g \| 1 t = 1,000 kg

01-The dimensional "Power Rule"

When converting an underlying unit of length, you must raise the scale factor to the power of that dimension (squared for area, cubed for volume).

Linear Scale: 1 m = 100 cm
Area Scale: 1 m² = (100)² cm² = 10,000 cm²
Volume Scale: 1 m³ = (100)³ cm³ = 1,000,000 cm³

Continue reading — 14 chapters, worked examples and practice.

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