Edexcel A-Level Maths Core Revision (pdfjoiner_(4)-199) | 爱德思A-Level数学核心复习（pdfjoiner_(4)-199）

📚 Edexcel A-Level Maths Core Revision (pdfjoiner_(4)-199) | 爱德思A-Level数学核心复习（pdfjoiner_(4)-199）

This article provides a structured revision overview for Edexcel A-Level Mathematics. It focuses on core techniques and key formulae that often appear across Pure, Statistics and Mechanics papers. Use each section to test recall before attempting past-paper questions. The file tag pdfjoiner_(4)-199 identifies this support sheet within the TutorHao series.

本文为爱德思 A-Level 数学提供结构化复习概览，重点梳理纯数、统计与力学试卷中常见的核心技巧与关键公式。建议在练习历年真题前，先利用每一小节检验自己的记忆与理解。文件标识 pdfjoiner_(4)-199 对应 TutorHao 系列中的本页复习资料。

1. Algebraic Expressions | 代数表达式

In Edexcel A-Level Maths, simplifying algebraic expressions including surds, indices and rational functions is a fundamental skill. You must be confident using the laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ and aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Rationalising denominators such as 1/√2 also appears frequently in Paper 1 and Paper 2. Factorising and cancelling common terms in algebraic fractions are essential before attempting harder problems.

在爱德思 A-Level 数学中，化简代数表达式（包括根式、指数与有理函数）是一项基础技能。你必须熟练使用指数律：aᵐ × aⁿ = aᵐ⁺ⁿ，以及 aᵐ ÷ aⁿ = aᵐ⁻ⁿ。将分母有理化（例如 1/√2）在试卷一和试卷二中也经常出现。在处理更复杂的问题前，必须先掌握代数分式的因式分解与约分。

aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ

2. Quadratic Functions | 二次函数

Quadratic functions of the form ax² + bx + c are central to Edexcel Pure Mathematics. The discriminant Δ = b² – 4ac determines the nature of the roots: if Δ > 0 there are two distinct real roots, if Δ = 0 there is one repeated real root, and if Δ < 0 there are no real roots. The quadratic formula gives the roots explicitly, and completing the square helps identify the vertex of the parabola.

形如 ax² + bx + c 的二次函数是爱德思纯数部分的核心内容。判别式 Δ = b² – 4ac 决定根的性质：若 Δ > 0，则有两个不同实根；若 Δ = 0，则有一个重根；若 Δ < 0，则没有实根。求根公式可显式给出根，而配方法有助于确定抛物线的顶点。

x = (-b ± √(b² – 4ac)) / (2a)

3. Differentiation | 微分

Differentiation measures the rate of change of a function. The standard rule for powers states that if y = xⁿ, then dy/dx = n xⁿ⁻¹. This rule is extended to polynomials, and the chain, product and quotient rules handle composite functions. Differentiation is used to find gradients, tangents, normals and stationary points. A stationary point is a maximum if the second derivative is negative and a minimum if it is positive.

微分用于衡量函数的变化率。幂函数的标准求导法则为：若 y = xⁿ，则 dy/dx = n xⁿ⁻¹。该法则可推广到多项式，链式法则、乘法法则和商法则用于处理复合函数。微分可用于求梯度、切线、法线以及驻点。若二阶导数为负，则驻点为极大值；若二阶导数为正，则驻点为极小值。

d/dx (xⁿ) = n xⁿ⁻¹

4. Integration | 积分

Integration is the reverse process of differentiation. The fundamental rule for powers is ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c, where c is the constant of integration and n ≠ -1. Definite integrals calculate the area between a curve and the x-axis, while indefinite integrals return a family of functions. Integration by substitution and integration by parts are also required for more advanced Edexcel questions.

积分是微分的逆运算。幂函数的基本积分法则为 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c，其中 c 为积分常数，且 n ≠ -1。定积分用于计算曲线与 x 轴之间的面积，而不定积分返回一族函数。更高级的爱德思题目还要求掌握换元积分法与分部积分法。

∫ xⁿ dx = xⁿ⁺¹/(n+1) + c, n ≠ -1

5. Trigonometric Ratios and Identities | 三角比与恒等式

Trigonometry links angles and side lengths in right-angled triangles, but A-Level work extends to all angles using the unit circle. The key identity sin²θ + cos²θ = 1 is used to simplify expressions and solve equations. The tangent ratio satisfies tanθ = sinθ/cosθ, and you also need exact values for 30°, 45° and 60°. Radian measure is introduced, and the graphs of sine, cosine and tangent must be recognised.

三角学将直角三角形中的角度与边长联系起来，但 A-Level 内容通过单位圆扩展到任意角度。关键恒等式 sin²θ + cos²θ = 1 常用于化简表达式与解方程。正切比满足 tanθ = sinθ/cosθ，你还需要掌握 30°、45° 和 60° 的精确值。引入弧度制后，必须能够识别正弦、余弦和正切函数的图像。

sin²θ + cos²θ = 1, tanθ = sinθ/cosθ

6. Exponentials and Logarithms | 指数与对数

Exponential functions of the form aˣ and eˣ model growth and decay. The natural logarithm ln x is the inverse of eˣ,

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