📚 CIE A-Level Pure Math 1: Core Concepts Explained | CIE A-Level 纯数学1：核心知识点精讲
The CIE A-Level Pure Mathematics 1 syllabus builds the essential groundwork for all further study in mathematics. It covers algebraic manipulation, functions, coordinate geometry, sequences, trigonometry, and introductory calculus. Mastering these topics not only prepares you for the AS examination but also develops the precision and logical reasoning required for Pure Math 3 and beyond. This guide distills the core concepts, emphasizes key formulas, and provides bilingual explanations to support deep understanding.
CIE A-Level 纯数学1课程为后续所有数学学习奠定必要基础，涵盖代数运算、函数、坐标几何、数列、三角学以及微积分入门。掌握这些专题不仅能帮助你在AS考试中取得好成绩，还能培养纯数学3及更高内容所需的严谨性与逻辑推理能力。本指南提炼核心概念，突出关键公式，并通过中英双语解释帮助巩固理解。
1. Quadratics and Inequalities | 二次函数与不等式
The standard form of a quadratic equation is ax² + bx + c = 0, with a ≠ 0. The discriminant Δ = b² − 4ac reveals the nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 means no real roots. Completing the square rewrites the expression as a(x + p)² + q, where the vertex of the parabola y = ax² + bx + c is at (−p, q).
二次方程的标准形式为 ax² + bx + c = 0，其中 a ≠ 0。判别式 Δ = b² − 4ac 显示根的性质：Δ > 0 时有两个不等实根，Δ = 0 时有一个重根，Δ < 0 时无实根。配方法将表达式改写成 a(x + p)² + q 的形式，此时抛物线 y = ax² + bx + c 的顶点坐标为 (−p, q)。
When solving quadratic inequalities, first find the real roots (if any) and then consider the graph’s sign pattern. For example, ax² + bx + c > 0 is satisfied outside the interval of roots when a > 0. Always illustrate the solution on a number line and write it using interval notation.
在解二次不等式时，先求出实根（如果存在），再考虑图像的正负分布。例如，当 a > 0 时，ax² + bx + c > 0 的解集在两根区间之外。务必用数轴表示解集，并使用区间记号书写答案。
The quadratic formula supplies the exact roots:
x = [−b ± √(b² − 4ac)] / 2a
二次求根公式给出准确根：x = [−b ± √(b² − 4ac)] / 2a。在无法因式分解或需要保留根式时，该公式至关重要。
Hidden quadratics arise when a substitution such as y = x² or y = √x transforms an equation into a standard quadratic. Always check the validity of solutions in the original variable to avoid extraneous answers.
隐二次方程通过代换（如 y = x² 或 y = √x）可化为标准二次方程。切记在原变量中检验解的合理性，避免引入增根。
2. Functions and Their Graphs | 函数及其图像
A function f maps each input x (from the domain) to exactly one output f(x) (in the range). The notation f : x → f(x) describes the mapping, and we often write y = f(x). One-to-one functions have different inputs giving different outputs; they pass both the vertical and horizontal line tests. The inverse function f⁻¹(x) exists only if f is one-to-one, and its graph is the reflection of y = f(x) in the line y = x.
函数 f 将定义域中的每个输入 x 映射到值域中唯一的输出 f(x)。我们常用 f : x → f(x) 表示对应关系，并写成 y = f(x)。一一函数要求不同输入对应不同输出，它同时通过垂直线和水平线检验。反函数 f⁻¹(x) 仅在一一映射下存在，其图像是 y = f(x) 关于直线 y = x 的对称图形。
Composite functions are formed by applying two functions successively: gf(x) = g(f(x)). The domain of gf is those x in the domain of f for which f(x) belongs to the domain of g. Always work from the inside out and pay close attention to domain restrictions, especially when square roots or denominators appear.
复合函数由两个函数依次作用得到：gf(x) = g(f(x))。gf 的定义域是那些既属于 f 的定义域，又使得 f(x) 属于 g 的定义域的 x。务必由内向外计算，并严格注意定义域限制，尤其是出现平方根或分母时。
Transformations of graphs include translations, stretches, and reflections. The map y = f(x) + a translates the graph vertically by a; y = f(x + a) translates horizontally by −a. y = a f(x) stretches vertically by factor a, while y = f(ax) stretches horizontally by factor 1/a. Reflections: y = −f(x) reflects in the x‑axis, y = f(−x) reflects in the y‑axis. Apply transformations in the correct order: first horizontal, then stretches/reflections, then vertical shifts.
图像变换包括平移、伸缩和反射。y = f(x) + a 将图像垂直平移 a 个单位；y = f(x + a) 水平平移 −a 个单位。y = a f(x) 垂直伸缩 a 倍，y = f(ax) 水平伸缩 1/a 倍。反射：y = −f(x) 关于 x 轴，y = f(−x) 关于 y 轴。变换顺序为：先水平方向，再伸缩与反射，最后垂直平移。
3. Coordinate Geometry: Lines and Circles | 坐标几何：直线与圆
The distance between two points A(x₁, y₁) and B(x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²], the midpoint is ((x₁+x₂)/2, (y₁+y₂)/2), and the gradient of AB is (y₂ − y₁)/(x₂ − x₁). Straight line equations can be expressed in the forms y = mx + c or y − y₁ = m(x − x₁). Two lines are parallel if their gradients are equal; they are perpendicular if m₁ × m₂ = −1.
两点 A(x₁, y₁) 与 B(x₂, y₂) 的距离为 √[(x₂ − x₁)² + (y₂ − y₁)²]，中点坐标为 ((x₁+x₂)/2, (y₁+y₂)/2)，AB 的斜率为 (y₂ − y₁)/(x₂ − x₁)。直线方程可写为 y = mx + c 或点斜式 y − y₁ = m(x − x₁)。两直线平行当且仅当斜率相等；垂直条件为 m₁ × m₂ = −1。
The equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r². To find the centre and radius from an expanded form x² + y² + 2gx + 2fy + c = 0, complete the square: centre = (−g, −f), radius = √(g² + f² − c). When a line and a circle intersect, substitute the line equation into the circle to obtain a quadratic; the discriminant indicates whether the line cuts the circle (Δ > 0), is tangent (Δ = 0), or misses it (Δ < 0).
以 (a, b) 为圆心、半径为 r 的圆方程为 (x − a)² + (y − b)² = r²。若给定一般式 x² + y² + 2gx + 2fy + c = 0，通过配方法可得圆心 (−g, −f)，半径 r = √(g² + f² − c)。直线与圆的交点问题通过代入直线方程得到二次方程，其判别式表明位置关系：Δ > 0 相交，Δ = 0 相切，Δ < 0 相离。
4. Sequences and Series | 数列与级数
A sequence is an ordered list of numbers. An arithmetic progression (AP) has a common difference d: the nth term is uₙ = a + (n − 1)d, where a is the first term. The sum of the first n terms, Sₙ, is given by Sₙ = n/2 [2a + (n − 1)d] or Sₙ = n/2 (a + l), where l is the last term.
数列是一组有序排列的数。等差数列具有公差 d：第 n 项 uₙ = a + (n − 1)d，a 为首项。前 n 项和 Sₙ = n/2 [2a + (n − 1)d] 或 Sₙ = n/2 (a + l)，l 为末项。
A geometric progression (GP) has a common ratio r: uₙ = a rⁿ⁻¹. The sum of the first n terms is Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. A GP converges to an infinite sum only when |r| < 1, and then S∞ = a/(1 − r). Be careful with conditions: the formula for S∞ is valid solely for convergent series.
等比数列具有公比 r：uₙ = a rⁿ⁻¹。前 n 项和 Sₙ = a(1 − rⁿ)/(1 − r)（r ≠ 1）。仅当 |r| < 1 时，无穷等比级数收敛，其和为 S∞ = a/(1 − r)。使用无穷和公式必须确保收敛条件成立。
Sigma notation Σ represents a sum systematically. Always understand the index, its starting value, and the general term. Exam questions often ask to derive a sum from a given sigma expression or to express a word problem using sequences.
求和符号 Σ 系统地表示累加。务必明确下标、起始值和通项。考试常要求根据 Σ 表达式求和，或将文字题转化为数列模型。
5. Binomial Expansion | 二项式展开
The binomial theorem expands (a + b)ⁿ for positive integer n: the general term is ⁿCᵣ aⁿ⁻ʳ bʳ, where ⁿCᵣ = n! / [r!(n − r)!]. The coefficients can be read from Pascal’s triangle or computed using combinations. Pure Math 1 also covers the expansion of (1 + x)ⁿ for rational n when |x| < 1; in this case the series is infinite and the general term is n(n−1)...(n−r+1) / r! xʳ.
对于正整数 n，(a + b)ⁿ 的二项展开通项为 ⁿCᵣ aⁿ⁻ʳ bʳ，其中 ⁿCᵣ = n! / [r!(n − r)!]。系数可从杨辉三角获得或用组合数计算。纯数学1还涉及 |x| < 1 时 (1 + x)ⁿ 的展开，其中 n 为有理数，此时级数为无穷级数，通项为 n(n−1)...(n−r+1) / r! xʳ。
You must be able to find a specific coefficient or the constant term without fully expanding. Often a substitution or equating powers renders the term. When n is not a positive integer, the expansion is valid only for |x| < 1, and you typically need to write the expression in the form (1 + ax)ⁿ before expanding.
你需要能直接求某一特定系数或常数项而不必完全展开。通常通过代换或比较指数得到。当 n 不是正整数时，展开仅在 |x| < 1 下有效，且通常需要将表达式先写成 (1 + ax)ⁿ 形式。
6. Trigonometry and Circular Measure | 三角学与弧度法
Angles can be measured in degrees or radians. The radian is the angle subtended at the centre of a circle by an arc equal in length to the radius: π rad = 180°. Sector area = ½ r² θ and arc length = r θ, where θ is in radians. Convert fluently between the two units, as calculus of trigonometric functions works only in radians.
角度可用度或弧度表示。弧度是弧长等于半径所对的圆心角：π 弧度 = 180°。扇形面积 = ½ r² θ，弧长 = r θ，其中 θ 以弧度为单位。务必灵活换算，因为三角函数的微积分仅在弧度制下成立。
The three basic trigonometric ratios are defined on a right‑angled triangle and extended via the unit circle. The sine, cosine, and tangent graphs are periodic: sin and cos have period 2π, while tan has period π. Key exact values: sin 30° = 1/2, cos 45° = 1/√2, tan 60° = √3 (and their radian equivalents).
三个基本三角函数比由直角三角形定义，并通过单位圆推广。正弦、余弦和正切图像均为周期函数：sin 与 cos 周期为 2π，tan 周期为 π。重要精确值：sin 30° = 1/2，cos 45° = 1/√2，tan 60° = √3（及对应的弧度值）。
Trigonometric equations are solved by finding all solutions within a given interval using the quadrant rule (CAST) or the graphs. For example, if sin θ = k, then the other solutions in [0,2π] are π − θ, etc. Transform equations using identities such as tan θ = sin θ / cos θ and sin² θ + cos² θ = 1. Always check that solutions lie in the required domain.
解三角方程时，借助象限规则（CAST）或图像找出指定区间内的所有解。例如，若 sin θ = k，在 [0,2π] 中的其他解为 π − θ 等。利用恒等式，如 tan θ = sin θ / cos θ 与 sin² θ + cos² θ = 1，对方程进行变形。务必确认解落在要求的定义域内。
7. Differentiation | 微分
Differentiation gives the gradient of a curve. The derivative of xⁿ is n xⁿ⁻¹ for any rational n. The derivative is denoted f'(x) or dy/dx. The gradient of a curve y = f(x) at x = a is f'(a). The tangent line at that point has equation y − f(a) = f'(a)(x − a); the normal has gradient −1/f'(a) if f'(a) ≠ 0.
微分用于求曲线的斜率。对于任意有理数 n，xⁿ 的导数为 n xⁿ⁻¹。导数记作 f'(x) 或 dy/dx。曲线 y = f(x) 在 x = a 处的切线斜率为 f'(a)，切线方程为 y − f(a) = f'(a)(x − a)；当 f'(a) ≠ 0 时，法线斜率为 −1/f'(a)。
Derivatives help determine whether a function is increasing or decreasing: f'(x) > 0 on an interval means the function is increasing there, and f'(x) < 0 means decreasing. Stationary points occur where f'(x) = 0. Use the first derivative test or the shape of the gradient to classify them as maximum, minimum, or point of inflection.
导数可判断函数的增减性：在某区间上 f'(x) > 0，函数在该区间递增；f'(x) < 0 则递减。驻点满足 f'(x) = 0。利用一阶导数符号变化或二阶导数判定驻点是极大值、极小值还是拐点。
Higher derivatives (f”(x) = d²y/dx²) are used to test concavity and classify stationary points more rapidly when f”(x) ≠ 0. Always show clear working when identifying the nature of stationary points.
高阶导数（f”(x) = d²y/dx²）用于判断曲线凹凸性，并在 f”(x) ≠ 0 时快速判定驻点类型。在判定驻点性质时，务必展示清晰的推理过程。
8. Integration | 积分
Integration is the reverse process of differentiation. The indefinite integral ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1. The constant C is essential for indefinite integrals. Integrate sums term by term, and always interpret the result as a family of functions.
积分是微分的逆运算。不定积分 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，n ≠ −1。常数 C 对于不定积分至关重要。积分可逐项进行，其结果表示一族函数。
A definite integral ∫ₐᵇ f(x) dx calculates the signed area between the curve y = f(x), the x‑axis, and the vertical lines x = a and x = b. The fundamental theorem of calculus states that ∫ₐᵇ f(x) dx = F(b) − F(a), where F'(x) = f(x). Areas below the x‑axis give negative contributions; to find a total area, split the interval where f(x) changes sign and add absolute values.
定积分 ∫ₐᵇ f(x) dx 计算曲线 y = f(x)、x 轴以及直线 x = a、x = b 之间的有向面积。微积分基本定理指出 ∫ₐᵇ f(x) dx = F(b) − F(a)，其中 F'(x) = f(x)。x 轴下方的面积贡献为负；求总面积时需根据 f(x) 的符号分段积分，并将各段绝对值相加。
Area between a curve and a line, or between two curves, is found by integrating the difference of the functions. Always sketch the region, identify intersection points, and set up the correct integral limits. Integration is also used to solve simple differential equations dy/dx = f(x) by separating variables.
计算曲线与直线之间或两曲线之间的面积时，需对被积函数的差进行积分。务必画出草图，确定交点，正确设定积分限。积分还可通过分离变量法求解简单的微分方程 dy/dx = f(x)。
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