📚 Edexcel A-Level Mathematics Core Revision: pdfjoiner_(4)-241 | Edexcel A-Level数学核心复习：pdfjoiner_(4)-241
This revision guide covers the key topics in the Edexcel A-Level Mathematics specification, organised as a compact review for the file reference pdfjoiner_(4)-241. It summarises essential definitions, standard results and common exam techniques across pure mathematics, statistics and mechanics.
本复习指南涵盖了 Edexcel A-Level 数学大纲中的核心主题，按照文件编号 pdfjoiner_(4)-241 整理成紧凑的复习讲义。它概述了纯数学、统计学和力学中的基本定义、标准结果和常见考试技巧。
1. Algebraic Expressions, Indices and Surds | 代数表达式、指数与根式
Simplify expressions involving surds and indices. For example, √a × √b = √(ab) and aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Always check whether a denominator needs to be rationalised before giving a final answer.
化简包含根式和指数的表达式。例如，√a × √b = √(ab)，且 aᵐ ÷ aⁿ = aᵐ⁻ⁿ。在给出最终答案前，始终检查分母是否需要有理化。
Rationalise denominators such as 1/√2 by multiplying numerator and denominator by √2. For a binomial denominator like a + √b, multiply by its conjugate a − √b to remove the surd.
对于 1/√2 这类分母，通过将分子和分母同乘 √2 进行有理化。对于 a + √b 这样的二项式分母，乘以其共轭式 a − √b 以消去根号。
- √a × √b = √(ab) only if a, b ≥ 0.
- √a ÷ √b = √(a/b) only if a ≥ 0, b > 0.
- (aᵐ)ⁿ = aᵐⁿ; aᵐ × aⁿ = aᵐ⁺ⁿ.
aᵐ × aⁿ = aᵐ⁺ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁻ⁿ = 1/aⁿ
2. Quadratics, Equations and Inequalities | 二次函数、方程与不等式
Complete the square to find the vertex of a quadratic and to solve equations. The discriminant Δ = b² − 4ac determines the nature of the roots: two distinct real roots if Δ > 0, one repeated real root if Δ = 0, and no real roots if Δ < 0.
通过完成平方求二次函数的顶点并解方程。判别式 Δ = b² − 4ac 决定根的性质：若 Δ > 0，则有两个相异实根；若 Δ = 0，则有一个重根；若 Δ < 0，则无实根。
The factor theorem states that (x − a) is a factor of a polynomial f(x) if and only if f(a) = 0. Use this to factorise cubic and quartic polynomials, then solve polynomial equations by setting each factor to zero.
因式定理指出，当且仅当 f(a) = 0 时，(x − a) 是多项式 f(x) 的一个因式。利用这一定理对三次和四次多项式进行因式分解，然后令每个因式为零来解多项式方程。
x = (−b ± √(b² − 4ac)) / 2a; |x − a| < b ⇔ −b < x − a < b
3. Graphs and Transformations | 函数图像与变换
Recognise how a graph changes under transformations of the form y = af(x), y = f(bx), y = f(x + c) and y = f(x) + d. Stretches multiply coordinates, while translations add or subtract from coordinates.
识别函数图像在 y = af(x)、y = f(bx)、y = f(x + c) 和 y = f(x) + d 等形式变换下的变化。伸缩会乘以坐标，而平移会对坐标进行加减。
For composite transformations, apply horizontal changes inside the bracket first, then vertical changes outside the bracket. Sketching transformed graphs requires careful attention to asymptotes and intercepts.
对于复合变换，先应用括号内的水平变化，再应用括号外的垂直变化。绘制变换后的图像需要仔细关注渐近线和截距。
- y = f(x + c): translation by −c in the x-direction.
- y = f(x) + d: translation by +d in the y-direction.
- y = f(2x): horizontal stretch by scale factor 1/2.
- y = 3f(x): vertical stretch by scale factor 3.
4. Trigonometry | 三角学
Know the exact trigonometric values for 0°, 30°, 45°, 60° and 90°. Use the unit circle and sine, cosine and tangent graphs to solve equations and identify symmetries such as sin(180° − θ) = sin θ.
掌握 0°、30°、45°、60° 和 90° 的精确三角值。使用单位圆以及正弦、余弦和正切图像来解方程，并识别诸如 sin(180° − θ) = sin θ 的对称性。
Key identities include sin²θ + cos²θ = 1, tan θ = sin θ / cos θ and the cosine rule a² = b² + c² − 2bc cos A. These are essential for simplifying expressions and solving triangles.
关键恒等式包括 sin²θ + cos²θ = 1、tan θ = sin θ / cos θ 以及余弦定理 a² = b² + c² − 2bc cos A。这些等式对于化简表达式和解三角形至关重要。
sin²θ + cos²θ = 1; tan θ = sin θ / cos θ; a² = b² + c² − 2bc cos A
5. Exponentials and Logarithms | 指数与对数
Exponential functions of the form y = aˣ or y = eˣ grow or decay rapidly. The natural logarithm ln x is the inverse of eˣ, so ln eˣ = x and eˡⁿˣ = x for suitable domains.
形如 y = aˣ 或 y = eˣ 的指数函数会迅速增长或衰减。自然对数 ln x 是 eˣ 的反函数，因此在适当的定义域内有 ln eˣ = x 和 eˡⁿˣ = x。
Use logarithm laws to solve exponential equations: logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx − logₐy and logₐ(xⁿ) = n logₐx.
使用对数法则解指数方程：logₐ(xy) = logₐx + logₐy，logₐ(x/y) = logₐx − logₐy，以及 logₐ(xⁿ) = n logₐx。
logₐ x = y ⇔ aʸ = x; ln x = logₑ x; eˡⁿˣ = x
6. Differentiation | 微分
Differentiate powers, exponentials, logarithms and trigonometric functions. For a function y = xⁿ, the derivative is dy/dx = nxⁿ⁻¹. The derivative gives the gradient of the tangent at any point on a curve.
对幂函数、指数函数、对数函数和三角函数求导。对于函数 y = xⁿ，其导数为 dy/dx = nxⁿ⁻¹。导数给出曲线上任意一点切线的斜率。
Use the product rule, quotient rule and chain rule for composite functions. Stationary points occur where dy/dx = 0; determine their nature using the second derivative or by testing the sign of dy/dx on either side.
对复合函数使用乘积法则、商法则和链式法则。驻点出现在 dy/dx = 0 处；通过二阶导数或检验 dy/dx 在两边的符号来确定驻点的性质。
d/dx (xⁿ) = nxⁿ⁻¹; d/dx (eˣ) = eˣ; d/dx (ln x) = 1/x
7. Integration | 积分
Integration reverses differentiation and calculates areas under curves. The indefinite integral of xⁿ is xⁿ⁺¹/(n+1) + C for n ≠ −1, where C is the constant of integration.
积分是微分的逆运算，用于计算曲线下的面积。对于 n ≠ −1，xⁿ 的不定积分为 xⁿ⁺¹/(n+1) + C，其中 C 为积分常数。
Evaluate definite integrals using limits and apply integration techniques such as substitution, integration by parts and partial fractions when integrands are not standard. Remember ∫ 1/x dx = ln |x| + C.
通过代入上下限计算定积分，并在被积函数非标准形式时应用换元法、分部积分法和部分分式法。记住 ∫ 1/x dx = ln |x| + C。
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1); ∫ 1/x dx = ln |x| + C
8. Vectors | 向量
Represent vectors as column vectors or using i, j, k unit vectors. Add and subtract vectors component-wise, and multiply a vector by a scalar to change its magnitude without changing its direction.
将向量表示为列向量或使用 i、j、k 单位向量。按分量进行向量的加法和减法，并将向量乘以标量以改变其大小而不改变其方向。
The scalar product a · b = |a||b| cos θ is used to find angles between vectors and to test perpendicularity: if a · b = 0, the vectors are perpendicular. Vector equations of lines require a position vector and a direction vector.
标量积 a · b = |a||b| cos θ 用于求向量间的夹角并检验垂直性：若 a · b = 0，则两向量垂直。直线的向量方程需要一个位置向量和一个方向向量。
a · b = |a||b| cos θ; a · b = 0 ⇒ a ⟂ b
9. Sequences and Series | 数列与级数
Arithmetic sequences have a constant difference d; the n-th term is a + (n−1)d and the sum of the first n terms is n/2 [2a + (n−1)d]. Geometric sequences have a constant ratio r; the n-th term is arⁿ⁻¹.
等差数列有恒定公差 d；第 n 项为 a + (n−1)d，前 n 项和为 n/2 [2a + (n−1)d]。等比数列有恒定公比 r；第 n 项为 arⁿ⁻¹。
For a geometric series with |r| < 1, the sum to infinity is a/(1 − r). Use the binomial expansion for (1 + x)ⁿ when n is rational or negative, remembering the expansion is valid for |x| < 1.
对于 |r| < 1 的等比级数，无穷和为 a/(1 − r)。当 n 为有理数或负数时，使用 (1 + x)ⁿ 的二项式展开，记住该展开在 |x| < 1 时有效。
等差: Sₙ = n/2 [2a + (n−1)d]; 等比: Sₙ = a(1−rⁿ)/(1−r); S∞ = a/(1−r)
10. Probability and Statistics
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