📚 CIE A Level Pure Mathematics 1 Coursebook: Core Topics and Revision Guide | CIE A Level 纯数学 1 课程书：核心主题与复习指南
The CIE A Level Pure Mathematics 1 (Paper 1) coursebook, aligned to syllabus 9709, builds the essential algebraic, trigonometric, and calculus skills needed for further study. This article breaks down the key topics, common pitfalls, and exam strategies so you can use the coursebook effectively and aim for a top grade.
CIE A Level 纯数学 1（试卷 1）课程书对应 9709 大纲，为后续学习打下代数、三角和微积分的关键基础。本文梳理核心主题、常见错误与考试策略，帮助你高效使用课程书并冲击高分。
1. Syllabus Overview and Assessment | 大纲概览与考核方式
Pure Mathematics 1 is examined by a single 1-hour-50-minute paper worth 75 marks, covering roughly ten content areas: quadratics, functions, coordinate geometry, circular measure, trigonometry, series, differentiation, and integration. Questions range from short, direct skill checks to longer, structured problem-solving tasks.
纯数学 1 由一份时长 1 小时 50 分钟、满分 75 分的试卷考查，涵盖约十个内容领域：二次函数、函数、坐标几何、弧度制、三角学、级数、微分和积分等。题目既有简短直接的技能检测，也有较长的结构化问题求解。
The coursebook mirrors this structure by presenting each topic with worked examples, key formulae, and past-paper-style exercises. Using the chapter summaries and checklists helps you track which syllabus objectives you have mastered.
课程书通过每章的例题、关键公式和真题风格练习来对应这一结构。利用章末总结和检查清单，可以帮助你追踪自己已掌握哪些大纲目标。
| Component | Details |
| Paper 1 Pure Mathematics 1 | 1 h 50 min, 75 marks, 60% of AS Level |
2. Quadratics and Polynomials | 二次函数与多项式
parabola, and proving inequalities. For ax² + bx + c = 0, the roots are given by x = (-b ± √b² - 4ac) / 2a, and the discriminant Δ = b² - 4ac determines the nature of the roots.
配方法是求解二次方程、求抛物线顶点和证明不等式的核心技巧。对于 ax² + bx + c = 0，根由公式 x = (-b ± √b² - 4ac) / 2a 给出，判别式 Δ = b² - 4ac 决定根的性质。
ves two distinct real roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots.
你必须能够画出显示截距和转折点的图像，并能求解一个方程为二次的联立方程组。课程书强调判别式的解读：Δ > 0 有两个不同实根，Δ = 0 有一个重根，Δ < 0 没有实根。
3. Functions and Transformations | 函数与图像变换
d inverse functions f⁻¹(x) are examined regularly, with the condition that f⁻¹ exists only if f is one-to-one.
函数将每个输入 x 唯一映射到一个输出 f(x)。定义域是允许的输入集合，值域是可能输出的集合。复合函数如 fg(x) = f(g(x)) 和反函数 f⁻¹(x) 经常考查，反函数存在的条件是 f 必须是一一对应的。
rizontal stretch. Combining two or more transformations must be done in the correct order.
y = f(x) 的图像变换遵循严格规则：y = f(x) + a 是沿 y 轴平移 a 个单位，y = f(x + a) 是沿 x 轴平移 -a 个单位，y = a f(x) 是垂直伸缩，y = f(ax) 是水平伸缩。组合多个变换时必须按正确顺序进行。
4. Coordinate Geometry | 坐标几何
(x₂ - x₁). Parallel lines have equal gradients, and perpendicular lines satisfy m₁m₂ = -1.
直线方程可以写成 y = mx + c，其中 m 是斜率，c 是 y 轴截距。给定两点 (x₁, y₁) 和 (x₂, y₂)，斜率为 m = (y₂ - y₁)/(x₂ - x₁)。平行线斜率相等，垂直线满足 m₁m₂ = -1。
tangents and normals to circles using the centre and radius from (x - a)² + (y - b)² = r².
课程书还涵盖中点公式 ((x₁ + x₂)/2, (y₁ + y₂)/2) 和距离公式 √((x₂ - x₁)² + (y₂ - y₁)²)。你应该熟练利用圆的方程 (x - a)² + (y - b)² = r² 中的圆心和半径来求圆的切线和法线方程。
5. Circular Measure | 弧度制与圆几何
= 180°. Arc length is given by s = rθ and sector area by A = ½ r²θ, where θ is in radians.
弧度制对于三角函数的微积分至关重要。一弧度是指圆心角所对的弧长等于半径时的角度，且 π 弧度 = 180°。弧长公式为 s = rθ，扇形面积公式为 A = ½ r²θ，其中 θ 以弧度为单位。
Typical exam questions ask you to find shaded areas by subtracting a triangle from a sector, or to solve problems involving chord length and segment area. The coursebook provides many diagrams to help you visualise these relationships.
典型考题要求通过扇形减去三角形来求阴影面积，或解决涉及弦长和弓形面积的问题。课程书提供大量图示帮助你直观理解这些关系。
6. Trigonometry | 三角学
The three trigonometric ratios are defined for angles in right-angled triangles, but the unit circle extends them to all real angles. The exact values
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图