📚 Edexcel AS & A Level Mathematics: Pure Mathematics Year 1/AS Textbook Guide | Edexcel AS 与 A Level 数学：纯数学第一册教材指南
The Edexcel AS and A Level Mathematics Pure Mathematics Year 1/AS textbook is the core student book for the first year of the A Level Mathematics course. It covers all pure mathematics content required for the AS qualification and forms the foundation for the full A Level. This guide summarises the key chapters, essential formulas, common exam pitfalls and effective study strategies to help you master the material.
Edexcel AS 与 A Level 数学纯数学第一册教材是 A Level 数学课程第一年的核心学生用书。它覆盖 AS 资格所需的所有纯数学内容，并为完整 A Level 打下基础。本指南总结主要章节、关键公式、常见考试陷阱和高效学习策略，帮助你掌握这些内容。
1. Overview of the Textbook | 教材概览
This textbook is published by Pearson Edexcel and follows the 2017 specification. It is divided into 14 chapters, each matching a topic in the pure mathematics syllabus. Worked examples, exercises and mixed practice sections are included. The book assumes GCSE Higher Tier knowledge and builds from algebraic manipulation to introductory calculus.
该教材由 Pearson Edexcel 出版，遵循 2017 年考试大纲。全书分为 14 章，每章对应纯数学大纲中的一个主题。书中包含例题、练习和混合复习部分。教材以 GCSE 高等级知识为基础，从代数运算逐步过渡到微积分入门。
Each chapter begins with a list of objectives and ends with a summary of key points, followed by exam-style questions. The worked examples show step-by-step reasoning, which is especially useful for understanding how marks are awarded in Edexcel examinations.
每章开头列出学习目标，结尾总结关键知识点，并附有考试风格练习。书中例题展示分步推理过程，这对于理解 Edexcel 考试中如何给分特别有帮助。
The textbook also includes mixed exercises that draw together content from several chapters, encouraging you to identify which technique to use without prompts. This mirrors the real exam, where questions often combine topics such as algebra and coordinate geometry.
教材还包含混合练习，综合多个章节的内容，训练你在没有提示的情况下识别该用哪种方法。这与真实考试相似，因为考题常常结合代数与坐标几何等不同主题。
2. Structure and Chapter Breakdown | 结构与章节划分
Chapters 1-4 cover algebraic expressions, quadratics, equations and inequalities, and graphs and transformations. Chapters 5-8 develop coordinate geometry, trigonometry, sequences and series, and the binomial expansion. Chapters 9-14 introduce differentiation, integration, exponentials and logarithms, vectors, and proof. Each chapter ends with a summary and exam-style questions.
第1-4章涵盖代数表达式、二次函数、方程与不等式以及图像与变换。第5-8章逐步展开坐标几何、三角学、数列与级数以及二项式展开。第9-14章引入微分、积分、指数与对数、向量和证明。每章末尾附有总结和考试风格练习。
The table below gives a concise map of the main topics and where they appear in the book. Knowing this structure helps you plan a revision timetable and focus on weaker areas.
下表简要列出主要主题及其在书中的对应章节。了解这一结构有助于你制定复习时间表并集中攻克薄弱环节。
| Chapter | 章节 | Topic | 主题 |
|---|---|
| 1-4 | Algebra, quadratics, inequalities, graphs | 代数、二次函数、不等式、图像 |
| 5-8 | Coordinate geometry, trigonometry, sequences, binomial expansion | 坐标几何、三角学、数列、二项式展开 |
| 9-14 | Differentiation, integration, exponentials, vectors, proof | 微分、积分、指数与对数、向量、证明 |
When working through the book, do not skip the proof chapter even if it appears at the end. Proof questions are increasingly common in Edexcel papers and require clear logical argument rather than just calculation.
学习本书时，不要跳过最后的证明章节。证明题在 Edexcel 试卷中出现频率越来越高，需要清晰的逻辑论证，而不只是计算。
3. Key Topic: Algebra and Functions | 核心主题：代数与函数
You must be confident simplifying surds, rationalising denominators, and using index laws. Functions include domain, range, composite functions fg(x) and inverse functions f⁻¹(x). The discriminant Δ = b² – 4ac determines the nature of roots of a quadratic equation ax² + bx + c = 0.
你必须熟练掌握根式化简、分母有理化和指数法则。函数包括定义域、值域、复合函数 fg(x) 和反函数 f⁻¹(x)。判别式 Δ = b² – 4ac 决定二次方程 ax² + bx + c = 0 根的性质。
Δ = b² – 4ac
If Δ > 0, there are two distinct real roots; if Δ = 0, one repeated real root; if Δ < 0, no real roots. This concept appears both in pure algebra questions and in problems involving intersections of curves and lines.
若 Δ > 0，有两个不同实根；若 Δ = 0，有一个重根；若 Δ < 0，无实根。这一概念既出现在纯代数题中，也出现在曲线与直线交点问题中。
For composite functions, remember that fg(x) means apply g first, then f. The inverse function f⁻¹(x) can be found by writing y = f(x), swapping x and y, and then solving for y. The domain of f⁻¹ is the range of f.
对于复合函数，记住 fg(x) 表示先应用 g，再应用 f。反函数 f⁻¹(x) 可通过写出 y = f(x)、交换 x 与 y 然后解出 y 来求得。f⁻¹ 的定义域就是 f 的值域。
Index laws such as aᵐ × aⁿ = aᵐ⁺ⁿ and (aᵐ)ⁿ = aᵐⁿ are essential tools. Negative and fractional indices, for example x⁻¹ = 1/x and x¹ᐟ² = √x, are frequently tested at AS level.
指数法则如 aᵐ × aⁿ = aᵐ⁺ⁿ 和 (aᵐ)ⁿ = aᵐⁿ 是基本工具。负指数和分数指数，例如 x⁻¹ = 1/x 和 x¹ᐟ² = √x，在 AS 考试中经常出现。
4. Coordinate Geometry | 坐标几何
The gradient of a line through points (x₁, y₁) and (x₂, y₂) is m = (y₂ – y₁)/(x₂ – x₁). The equation of a straight line can be written as y – y₁ = m(x – x₁) or ax + by + c = 0. Parallel lines have equal gradients, and perpendicular lines satisfy m₁ × m₂ = -1.
经过点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率为 m = (y₂ – y₁)/(x₂ – x₁)。直线方程可写为 y – y₁ = m(x – x₁) 或 ax + by + c = 0。平行直线斜率相等，垂直直线满足 m₁ × m₂ = -1。
m = (y₂ – y₁)/(x₂ – x₁), m₁ × m₂ = -1
The equation of a circle with centre (a, b) and radius r is (x – a)² + (y – b)² = r². Questions often ask you to complete the square to find the centre and radius from an expanded equation such as x² + y² + 2x – 6y – 15 = 0.
圆心为 (a, b)、半径为 r 的圆的方程是 (x – a)² + (y – b)² = r²。题目常要求你通过配方法从形如 x² + y² + 2x – 6y – 15 = 0 的展开式中求出圆心和半径。
When finding the intersection of a line and a circle, substitute the line equation into the circle equation to form a quadratic. The discriminant then tells you whether the line cuts the circle twice, touches it once, or misses it entirely.
求直线与圆的交点时，将直线方程代入圆方程得到一个二次方程。判别式随后告诉你直线与圆相交于两点、相切于一点，还是完全没有交点。
In exam problems involving perpendicular bisectors or tangents, always draw a small sketch. This helps you avoid sign errors and makes the geometry of the question clearer.
在涉及垂直平分线或切线的考试题中，务必画一个简图。这有助于避免符号错误，并使题目的几何关系更加清晰。
5. Trigonometry | 三角学
The sine and cosine rules are required for non-right-angled triangles. The area of a triangle is ½ ab sin C. You must also know exact values for sin, cos and tan at 0°, 30°, 45°, 60° and 90°. Trigonometric identities include tan θ = sin θ / cos θ and sin² θ + cos² θ = 1.
正弦定理和余弦定理用于非直角三角形。三角形面积为 ½ ab sin C。你还必须记住 0°、30°、45°、60° 和 90° 时 sin、cos、tan 的精确值。三角恒等式包括 tan θ = sin θ / cos θ 和 sin² θ + cos² θ
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