Edexcel A-Level Mathematics Formula Booklet Complete Guide — 爱德思A-Level数学公式手册完全指南

一、爱德思A-Level数学公式手册是什么：考场上的合法“锦囊” | What Is the Edexcel Formula Booklet: Your Legally Allowed Exam Companion

爱德思（Edexcel）A-Level 数学考试的每一场试卷都会随卷下发一份官方公式手册，全称为《Mathematical Formulae and Statistical Tables》，也就是同学们口中的“公式本”。这份手册并非试卷的一部分，而是考试委员会为了让考生专注于理解与应用、而非死记硬背公式而提供的标准参考材料。

Every Edexcel A-Level Mathematics paper is accompanied by an official booklet titled “Mathematical Formulae and Statistical Tables”. This is the formula booklet that students refer to throughout the exam. It is not part of the question paper itself; the exam board supplies it so that candidates can focus on understanding and applying mathematical ideas rather than on memorising every formula.

手册分为三大板块：纯数（Pure Mathematics）、统计（Statistics）与力学（Mechanics），覆盖了 AS 与 A-Level 两个阶段课程中所有会用到、但不要求背诵的标准公式。理解手册的编排结构，是高效利用它的第一步。

The booklet is organised into three main sections: Pure Mathematics, Statistics and Mechanics. It covers every standard formula used across both the AS and full A-Level courses that you are not expected to memorise. Understanding how the booklet is laid out is the first step towards using it efficiently.

二、纯数部分·代数与函数：指数、对数与二项展开的核心公式 | Pure Maths: Algebra and Functions—Indices, Logarithms and Binomial Expansion

纯数部分的第一组公式围绕代数与函数展开。指数的基本法则 – 同底数幂相乘指数相加、幂的幂指数相乘 – 是后续所有内容的基础。对数部分则给出了换底公式以及自然对数与常用对数的换算关系，这些公式在解指数方程时不可或缺。

The first block of Pure Maths formulae centres on algebra and functions. The laws of indices – multiplying powers with the same base adds their exponents, and raising a power to a power multiplies them – underpin everything that follows. The logarithms section gives the change-of-base formula and the relationship between natural and common logarithms, which are essential when solving exponential equations.

二项式展开公式是本板块的高频考点。对于正整数指数，二项式定理给出了 (a + b)^n 的完整展开式；对于非正整数或分数指数，则需要用到 (1 + x)^n 的级数展开，并注意其收敛条件 |x| 小于 1。考试中常考的是求某一项的系数，这要求你对通项公式十分熟悉。

The binomial expansion is a frequent exam topic in this section. For a positive integer power, the binomial theorem gives the full expansion of (a + b)^n. For non-integer or fractional powers, you need the series expansion of (1 + x)^n and must remember its validity condition, namely that |x| is less than 1. A common exam question asks for a specific coefficient, which demands solid familiarity with the general term.

此外，二次方程求根公式与判别式也收录于此。判别式 b^2 – 4ac 的符号决定了方程有两个实根、一个重根还是没有实根，这一结论在涉及根的数量与函数图像的题目中反复出现。

The quadratic formula and its discriminant also appear here. The sign of the discriminant b^2 – 4ac determines whether the equation has two distinct real roots, one repeated root, or no real roots. This fact recurs constantly in questions about the number of roots and the shape of function graphs.

三、纯数部分·三角函数：弧度制、恒等式与三角方程 | Pure Maths: Trigonometry—Radians, Identities and Trigonometric Equations

三角函数板块首先强调弧度制。A-Level 中角度默认以弧度为单位，扇形的弧长公式 l = rθ 与扇形面积公式 A = (1/2) r^2 θ 只有在使用弧度时才成立。很多同学在计算扇形相关问题时因为忘记切换到弧度而丢分。

The trigonometry section begins by emphasising radians. In A-Level, angles are given in radians by default. The arc length formula l = rθ and the sector area formula A = (1/2) r^2 θ only hold when angles are measured in radians. Many students lose marks on sector problems simply because they forget to switch to radians.

核心恒等式是必须熟练掌握的内容。sin²θ + cos²θ = 1、tanθ = sinθ / cosθ，以及二倍角公式 sin2θ、cos2θ 与 tan2θ 的各种形式，是化简三角表达式、证明恒等式与解三角方程的主力工具。手册中把这些恒等式集中列出，方便你在证明题中快速对照。

The core identities are the tools you must master. The identities sin²θ + cos²θ = 1 and tanθ = sinθ / cosθ, together with the double-angle formulae for sin2θ, cos2θ and tan2θ in all their forms, are the workhorses for simplifying expressions, proving identities and solving trigonometric equations. The booklet lists these together so you can quickly cross-check them during proof questions.

三角函数与反三角函数、以及正弦定理、余弦定理也收录其中。余弦定理 a² = b² + c² – 2bc cosA 在解非直角三角形时尤其关键，它能与三角形面积公式 (1/2)bc sinA 配合，处理一大类几何与测量问题。

The section also includes inverse trigonometric functions and the sine and cosine rules. The cosine rule, a² = b² + c² – 2bc cosA, is especially important for solving non-right-angled triangles, and it pairs with the area formula (1/2)bc sinA to handle a whole family of geometry and measurement problems.

四、纯数部分·微积分：微分与积分的核心公式 | Pure Maths: Calculus—Core Differentiation and Integration Rules

微积分是纯数部分篇幅最大的板块。微分部分给出多项式、指数函数、对数函数与三角函数的导数表，以及乘积法则、商法则与链式法则。链式法则 dy/dx = (dy/du)(du/dx) 是处理复合函数求导的核心，几乎所有稍复杂的求导题都会用到它。

Calculus is the largest block in the Pure Maths section. The differentiation part gives the derivative table for polynomials, exponentials, logarithms and trigonometric functions, along with the product rule, the quotient rule and the chain rule. The chain rule, dy/dx = (dy/du)(du/dx), is the key to differentiating composite functions and appears in almost every slightly more involved differentiation question.

积分部分则给出与微分对应的不定积分公式，以及定积分的基本性质。分部积分法（integration by parts）与换元积分法（integration by substitution）是 A-Level 阶段的两大积分技巧，手册中给出的标准积分公式表是你在考试中快速完成积分步骤的底气所在。

The integration part provides the indefinite integrals that correspond to the derivatives above, together with the basic properties of definite integrals. Integration by parts and integration by substitution are the two main techniques at A-Level, and the standard table of integrals in the booklet is what lets you complete integration steps quickly and confidently in the exam.

此外，参数方程与隐函数求导也属于微积分板块。对于 x = f(t), y = g(t) 形式的参数方程，dy/dx 由 dy/dt 除以 dx/dt 得到，这一公式在涉及曲线的切线与法线的问题中十分重要。

In addition, parametric equations and implicit differentiation belong to the calculus section. For parametric equations of the form x = f(t), y = g(t), the derivative dy/dx is found by dividing dy/dt by dx/dt. This formula is crucial in problems involving tangents and normals to curves.

五、纯数部分·数列与级数：等差、等比与二项级数 | Pure Maths: Sequences and Series—Arithmetic, Geometric and Binomial Series

数列与级数板块给出了等差数列与等比数列的通项公式与求和公式。等差数列的第 n 项为 a + (n-1)d，前 n 项和为 n/2 [2a + (n-1)d]；等比数列的第 n 项为 ar^(n-1)，当公比 r 的绝对值小于 1 时，无穷等比级数收敛到 a / (1 – r)。

The sequences and series section provides the nth term and sum formulae for arithmetic and geometric sequences. For an arithmetic sequence the nth term is a + (n-1)d and the sum of the first n terms is n/2 [2a + (n-1)d]. For a geometric sequence the nth term is ar^(n-1), and when the common ratio r has absolute value less than 1, the infinite geometric series converges to a / (1 – r).

无穷级数的求和是考试的常见难点。判断一个无穷等比级数是否收敛、以及求出其收敛值，是这一板块的核心技能。许多同学会把“收敛到某值”误当成“恰好等于某值”，导致在证明题中表述不严谨而失分。

Summing infinite series is a common exam difficulty. Deciding whether an infinite geometric series converges, and finding the value it converges to, is the core skill of this section. Many students mistake “converges to a value” for “equals exactly that value”, which costs them precision in proof questions.

六、统计学部分·概率分布与假设检验 | Statistics: Probability Distributions and Hypothesis Testing

统计板块首先介绍二项分布与泊松分布，以及它们各自的条件。二项分布 B(n, p) 适用于固定次数独立试验，其概率质量函数 P(X = r) = nCr p^r (1-p)^(n-r)；泊松分布适用于单位时间内随机事件发生的次数，其参数 λ 同时等于期望与方差。

The statistics section first introduces the binomial and Poisson distributions and their respective conditions. The binomial distribution B(n, p) applies to a fixed number of independent trials, with probability mass function P(X = r) = nCr p^r (1-p)^(n-r). The Poisson distribution models the number of random events in a fixed interval, and its parameter λ equals both the mean and the variance.

正态分布与标准正态分布是本板块的另一重点。手册给出了标准正态分布表，用于将一般正态变量标准化为 Z = (X – μ) / σ 后查表求概率。连续性校正与逆查表操作是常考但易错的地方，务必在理解原理的基础上练习。

The normal distribution and the standard normal distribution are another key part of this section. The booklet provides the standard normal table, used after standardising a general normal variable with Z = (X – μ) / σ. Continuity corrections and inverse table look-ups are frequent but error-prone; practise them on a firm understanding of the underlying principle.

假设检验是统计板块的压轴内容。你需要写出原假设 H0 与备择假设 H1，计算检验统计量或 p 值，再与显著性水平比较，最后给出“拒绝”或“不拒绝”原假设的结论。手册中正态分布表与检验统计量公式共同支撑起这一整套流程。

Hypothesis testing rounds off the statistics section. You must state the null hypothesis H0 and the alternative hypothesis H1, compute the test statistic or p-value, compare it with the significance level, and conclude whether to reject or not reject H0. The normal table and the test-statistic formulae in the booklet together support this entire workflow.

七、力学部分·运动学与牛顿定律 | Mechanics: Kinematics and Newton’s Laws

力学板块以匀速直线运动与匀变速直线运动的五大公式（SUVAT）为核心。这组公式把初速度 u、末速度 v、加速度 a、位移 s 与时间 t 联系起来，只要已知其中三个量，就能求出其余两个。熟练掌握并能正确挑选公式，是力学得分的基础。

The mechanics section is built around the SUVAT equations for uniform acceleration. These five equations link initial velocity u, final velocity v, acceleration a, displacement s and time t; given any three quantities you can find the other two. Mastering these equations and choosing the right one is the foundation of scoring well in mechanics.

牛顿第二定律 F = ma 是力学中最重要的一条公式，它把作用在物体上的合力与加速度联系起来。配合摩擦系数、斜面分解与连接体分析，F = ma 能处理从简单质点运动到复杂多物体系统的各类问题。

Newton’s second law, F = ma, is the single most important formula in mechanics, linking the resultant force on an object to its acceleration. Combined with friction coefficients, resolved components on slopes and connected-particle analysis, F = ma handles everything from simple particle motion to complex multi-body systems.

此外，动量与冲量公式也收录在手册中。动量 p = mv 与冲量-动量定理 FΔt = Δ(mv) 在碰撞与受力分析问题中频繁出现，而功、能与功率的公式则把力学与能量观点统一起来。

The booklet also includes momentum and impulse. The momentum formula p = mv and the impulse-momentum principle FΔt = Δ(mv) appear frequently in collision and force-analysis problems, while the work, energy and power formulae unify mechanics with an energy-based viewpoint.

八、公式手册的正确使用姿势：考前、考中、考后 | Using the Booklet Correctly: Before, During and After the Exam

考前的关键不是把手册“背下来”，而是做到“指哪打哪”。你应该对每一页的内容心中有数：打开目录能立刻定位到所需公式所在的位置。与其背公式，不如通过大量练习熟悉公式的结构与适用条件，这样在考场上才能在几秒内找到并正确使用它们。

The key before the exam is not to memorise the booklet but to know exactly where everything lives. You should be able to open the contents page and locate any formula you need in seconds. Rather than memorising formulae, practise enough that you know their structure and validity conditions, so that in the exam you can find and apply them correctly within seconds.

考中要警惕“抄错公式”这个隐形杀手。抄写时把变量代错、把幂次看错、把加减号抄反，是最常见的无谓失分。一个有效的习惯是：抄完公式后，先在心里用一个小例子验证一下，再代入具体数据计算。宁可多花五秒钟检查，也不要因为一个笔误丢掉整道题的分。

During the exam, watch out for the silent killer of miscopying formulae. Substituting the wrong variable, misreading an exponent, or flipping a plus into a minus are the most common avoidable errors. A useful habit is to verify a copied formula with a tiny example in your head before plugging in the actual numbers. Spending five extra seconds checking is far better than losing a whole question’s marks to a slip.

考后要复盘：哪道题因为找不到公式而卡住，哪道题因为抄错而丢分，把这些都记下来。针对薄弱环节做专项练习，下一次考试时你对手册的依赖就会更少、更精准。手册只是工具，真正的能力在于你对数学结构的理解。

After the exam, reflect: which question stalled because you could not find a formula, and which one lost marks because you copied it wrong. Record these and drill the weak spots. Next time your reliance on the booklet will be lighter and more precise. The booklet is only a tool; your real strength lies in understanding mathematical structure.

九、常见失分陷阱：这些公式用法最容易出错 | Common Mistakes: The Formula Misuses That Cost Marks

第一个陷阱是弧度与角度的混用。三角函数求导公式只有在角度为弧度时才成立，如果你在求导前忘记把角度换算成弧度，结果会完全错误。养成“见到三角函数先确认单位”的习惯，能避免一整类低级错误。

The first trap is mixing radians and degrees. The differentiation formulae for trigonometric functions only hold when the angle is in radians. If you forget to convert degrees to radians before differentiating, the result is completely wrong. Make a habit of confirming the unit the moment you see a trigonometric function, and you will dodge a whole class of basic errors.

第二个陷阱是二项展开的收敛条件。对 (1 + x)^n 的非整数指数展开，只有当 |x| 小于 1 时级数才有效。很多同学在得到展开式后直接代入超出范围的 x 值，得到一个看似“算出来”实则错误的结果。始终先检查 x 的取值范围。

The second trap is the validity condition of binomial expansions. For the expansion of (1 + x)^n with non-integer powers, the series is valid only when |x| is less than 1. Many students substitute an out-of-range x value into the expansion and obtain a result that looks computed but is actually wrong. Always check the allowed range of x first.

第三个陷阱是假设检验的结论表述。统计中的结论只能说“在显著性水平下，有足够证据拒绝 H0”，而不能说“证明了 H1 正确”或“H0 错误”。用词不严谨会直接丢掉结论分，务必使用“拒绝/不拒绝原假设”的标准表述。

The third trap is wording the conclusion of a hypothesis test. In statistics you may only say “at this significance level there is sufficient evidence to reject H0”, never “H1 is proven” or “H0 is wrong”. Loose wording loses conclusion marks directly, so always use the standard phrasing of rejecting or not rejecting the null hypothesis.

第四个陷阱是力学中的方向与符号。速度、加速度和力都是矢量，在建立方程时必须统一正方向。方向弄反会让 F = ma 或 SUVAT 的结果出现符号错误，而这类错误往往在最后一步才暴露出来。

The fourth trap is direction and sign in mechanics. Velocity, acceleration and force are all vectors, so you must fix a consistent positive direction when setting up equations. Getting the direction wrong introduces sign errors into F = ma or SUVAT results, and these often only surface at the final step.

十、备考策略：如何把公式手册变成你的提分工具 | Study Strategy: Turn the Booklet into a Scoring Tool

第一步是“结构化整理”。把手册内容按你自己的理解重新梳理成一张思维导图或一页速查表，标注每类公式的适用条件与典型题型。这个整理过程本身就是最好的复习，它迫使你把零散的公式组织成有逻辑的知识网络。

The first step is structured organisation. Reorganise the booklet’s contents into a mind map or a one-page quick-reference sheet in your own words, marking each formula’s validity conditions and typical question types. The act of organising is itself excellent revision, because it forces you to turn scattered formulae into a logical knowledge network.

第二步是“刻意练习”。每学完一类公式，立刻做对应的真题，尤其是需要你从手册中查公式并正确代入的题目。通过反复练习，你会逐渐记住常用公式，把查手册的时间省下来用于思考和检查，这才是手册带来的真正优势。

The second step is deliberate practice. As soon as you finish one category of formulae, immediately work on matching past-paper questions, especially those that require you to look up a formula and substitute correctly. Through repetition you will gradually remember the common formulae, freeing up time to think and check – and that is the real advantage the booklet gives you.

第三步是“限时模拟”。在完整真题模拟中练习翻手册的动作，让它成为考试流程中自然的一环。模拟时记录每次查手册的位置与时间，找出自己最不熟悉的板块并重点攻克。把翻手册练成肌肉记忆，考场上就不会因为它而慌乱。

The third step is timed mock exams. Practise the action of flicking through the booklet during full past-paper mocks, so it becomes a natural part of your exam routine. Record where and how long you look things up each time, identify your least familiar sections, and target them. Turning booklet navigation into muscle memory means it will never unsettle you on exam day.

十一、向量板块：直线、平面与数量积 | Vectors: Lines, Planes and the Scalar Product

向量板块在 A-Level 纯数中占有一席之地，也是进阶数学（Further Maths）的重要内容。手册给出了二维与三维向量的基本运算规则：向量加减、数乘、模长以及单位向量的求法。向量的模长公式 |a| = √(x² + y² + z²) 与单位向量公式 a / |a| 是后续所有向量几何问题的基础。

Vectors occupy a solid place in A-Level Pure Maths and are also central to Further Maths. The booklet gives the basic operations on two- and three-dimensional vectors: addition, scalar multiplication, magnitude, and unit vectors. The magnitude formula |a| = √(x² + y² + z²) and the unit vector formula a / |a| underpin every vector geometry problem that follows.

向量的数量积（点积）是本板块的核心工具。a · b = |a||b|cosθ 把两个向量的大小与夹角联系起来，而分量形式 a · b = x1x2 + y1y2 + z1z2 则给出了计算上的便捷路径。通过数量积可以判断两向量是否垂直（数量积为零）、求夹角，以及计算向量在某一方向上的投影。

The scalar product (dot product) is the core tool of this section. The formula a · b = |a||b|cosθ links the magnitudes of two vectors to the angle between them, while the component form a · b = x1x2 + y1y2 + z1z2 offers a convenient computational route. Using the scalar product you can test for perpendicularity (dot product equals zero), find angles, and compute the projection of one vector onto another.

直线的向量方程 r = a + tb 是描述三维空间直线的标准形式，其中 a 是直线上一点的位矢，b 是方向向量。两条直线平行、相交还是异面，可以通过比较方向向量与解联立方程来判断，这类题型在进阶数学的向量几何中反复出现。

The vector equation of a line, r = a + tb, is the standard way to describe a line in three dimensions, where a is the position vector of a point on the line and b is the direction vector. Whether two lines are parallel, intersecting, or skew can be determined by comparing direction vectors and solving simultaneous equations – a question type that recurs throughout Further Maths vector geometry.

十二、AS与A-Level、数学与进阶数学：手册的适用范围 | AS vs A-Level, Maths vs Further Maths: The Booklet’s Coverage

同一份手册同时服务于 AS 与完整的 A-Level 两个阶段。AS 阶段只考察纯数、统计与力学的前半部分内容，因此你只需用到手册中的一部分公式；到了 A-Level 第二年，新增的微积分技巧、进阶统计与力学内容会调用手册中更多的公式。理解“当前阶段用得到哪一部分”，可以避免在考场上翻到无关内容而浪费时间。

The same booklet serves both the AS and the full A-Level stages. AS only examines the first half of Pure Maths, Statistics and Mechanics, so you will only need part of the booklet. By the second year of A-Level, the new calculus techniques and advanced statistics and mechanics material call on more of its formulae. Knowing which part is relevant to your current stage saves you from wasting time flipping to irrelevant content in the exam.

进阶数学（Further Maths）则使用同一本手册的扩展部分，或者单独的手册。进阶数学会用到双曲函数、复数、矩阵、极坐标、更深的微积分与微分方程等内容，其中相当一部分公式并不出现在普通数学的手册主表里。选读进阶数学的同学需要确认自己用的是哪一份公式材料，并清楚哪些公式需要额外记忆。

Further Maths uses the extended sections of the same booklet, or a separate booklet altogether. Further Maths draws on hyperbolic functions, complex numbers, matrices, polar coordinates, deeper calculus and differential equations, many of whose formulae do not appear in the main tables of the ordinary Mathematics booklet. Students taking Further Maths should confirm which formula materials they are given and which formulae they must memorise on top.

无论哪个阶段，都要在考前通读一遍手册目录，标记出自己课程所覆盖的部分。这样既能做到心中有数，也能避免在考试时因为翻错区域而产生不必要的紧张。对手册适用范围的清醒认识，本身就是一种考场优势。

Whichever stage you are at, read through the booklet’s contents page before the exam and mark the parts your course actually covers. This gives you a clear mental map and prevents unnecessary panic from flipping to the wrong region mid-exam. A clear awareness of the booklet’s scope is itself an exam advantage.

十三、数值方法：迭代法与梯形法则 | Numerical Methods: Iteration and the Trapezium Rule

数值方法是 A-Level 数学中一个容易被忽视却常考的小板块。迭代法用于求方程的近似根：把方程 f(x) = 0 改写为 x = g(x) 的形式，从初值 x0 出发反复代入 x_{n+1} = g(x_n)，当迭代收敛时，序列会逐渐逼近真实根。判断迭代是否收敛，通常要看 g'(x) 在根附近的绝对值是否小于 1。

Numerical methods form a small but frequently examined section of A-Level Mathematics. Iteration is used to find approximate roots of equations: rewrite f(x) = 0 as x = g(x), then repeatedly substitute x_{n+1} = g(x_n) starting from an initial value x0. When the iteration converges, the sequence approaches the true root. To judge whether an iteration converges, check whether the absolute value of g'(x) near the root is less than 1.

梯形法则（Trapezium Rule）用于近似计算定积分，特别适用于被积函数没有初等原函数的情形。它把积分区间等分为 n 个小梯形，用这些梯形面积之和来逼近曲线下方面积，公式为 ∫ f(x) dx ≈ (h/2)[y0 + 2(y1 + y2 + … + y_{n-1}) + yn]。增加分段数 n 会减小步长 h，从而提高近似的精度。

The trapezium rule approximates a definite integral and is especially useful when the integrand has no elementary antiderivative. It divides the interval into n equal trapeziums and approximates the area under the curve by their total area, with the formula ∫ f(x) dx ≈ (h/2)[y0 + 2(y1 + y2 + … + y_{n-1}) + yn]. Increasing the number of strips n reduces the step size h and improves accuracy.

这类题型常要求你在保留一定小数位的前提下完成若干次迭代，或比较梯形法则近似值与真实值的误差。计算的准确性与格式的规范（保留有效数字、写清每一步的迭代值）同等重要。虽然公式手册未必逐条列出数值方法的公式，但理解其原理后，你就能在考场上快速写出正确步骤。

Such questions often ask you to carry out several iterations to a specified number of decimal places, or to compare the trapezium approximation with the true value. Accuracy of calculation and neatness of presentation (keeping significant figures and writing out each iterative value) matter equally. The booklet may not list every numerical-methods formula, but once you understand the principles you can write out the correct steps quickly in the exam.

Summary | 总结

爱德思 A-Level 数学公式手册是贯穿纯数、统计与力学三大板块的标准参考材料，它的价值在于把考生的注意力从“记公式”转移到“用公式”。掌握手册的编排结构、熟悉每类公式的适用条件，并通过刻意练习与限时模拟把它变成肌肉记忆，你就能在考场上从容、准确地调用每一个公式。

The Edexcel A-Level Mathematics formula booklet is the standard reference spanning Pure Maths, Statistics and Mechanics, and its value lies in shifting a candidate’s attention from memorising formulae to applying them. Master the booklet’s layout, learn the validity conditions of every formula, and turn its navigation into muscle memory through deliberate practice and timed mocks. Then you will be able to call up each formula calmly and accurately in the exam.

真正的数学能力不在于记住了多少公式，而在于知道何时用、怎样用、以及用了之后如何检验结果是否合理。把手册当作朋友而非拐杖，你的 A-Level 数学之路会走得更加稳健。

Real mathematical ability is not about how many formulae you can recite, but about knowing when and how to use them, and how to check afterwards whether a result is reasonable. Treat the booklet as a friend rather than a crutch, and your journey through A-Level Mathematics will be far more assured.

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