📚 Formulae | 公式
Formulae are the backbone of A-Level Mathematics. They give you a compact, reliable way to model problems, justify reasoning, and move from given information to a final answer. Edexcel papers expect you not only to recall key formulae but also to select, rearrange, and combine them under exam pressure. This revision guide collects the most important formulae across Pure Mathematics, Statistics, and Mechanics, with practical notes on when and how to use each one.
公式是 A-Level 数学的支柱。它们为你提供了一种紧凑而可靠的方式来建立问题模型、论证推理过程，并从已知信息推导出最终答案。Edexcel 试卷不仅要求你记住关键公式，还要求你在考试压力下能够选择、变形并综合运用这些公式。本复习指南汇集了纯数学、统计学和力学中最主要的公式，并附有使用时机和方法的实用说明。
1. Quadratic Formula | 二次公式
n by the quadratic formula. It should be used when factorisation is not obvious, or when the question asks for answers to a given degree of accuracy.
方程 ax² + bx + c = 0 的解由二次公式给出。当因式分解不明显，或题目要求将答案保留到指定精度时，就应当使用该公式。
x = (-b ± √b² - 4ac) ÷ 2a
real roots; if Δ = 0 there is one repeated real root; if Δ < 0 there are no real roots.
判别式为 Δ = b² - 4ac。若 Δ > 0，则有两个不同的实根；若 Δ = 0，则有一个重根；若 Δ < 0，则没有实根。
- Always rearrange the equation into the form ax² + bx + c = 0 before substituting.
- 代入前务必先把方程整理成 ax² + bx + c = 0 的形式。
- Watch for negative coefficients, as they change the sign of -4ac.
- 注意负系数，它们会改变 -4ac 的符号。
2. Arithmetic Series | 等差数列
An arithmetic sequence has a constant difference d between consecutive terms. Its nth term and sum formulae appear frequently in both Pure and applied contexts, such as savings plans or simple linear patterns.
等差数列相邻两项之间具有固定的公差 d。它的第 n 项和求和公式经常出现在纯数学和应用题中，例如储蓄计划或简单线性规律。
uₙ = a + (n - 1)d
Sₙ = n/2 [2a + (n - 1)d] = n/2 (a + l)
Here a is the first term, d is the common difference, n is the term number, and l is the last term. Use the version with a + l when the last term is already known.
其中 a 是首项，d 是公差，n 是项数，l 是末项。当末项已知时，使用含有 a + l 的求和形式。
- Check whether the problem gives you a, d, n, or l, and choose the most direct formula.
- 先判断题目给出的是 a、d、n 还是 l，再选择最直接的公式。
- Proof questions often ask you to derive Sₙ by pairing terms from both ends.
- 证明题常要求你通过首尾配对的方法推导 Sₙ。
3. Geometric Series | 等比数列
A geometric sequence has a constant ratio r between consecutive terms. Its sum formula is essential when r ≠ 1, while the sum to infinity only exists when |r| < 1.
等比数列相邻两项之间具有固定的公比 r。当 r ≠ 1 时，其求和公式至关重要；而无穷级数求和仅当 |r| < 1 时存在。
uₙ = arⁿ⁻¹
Sₙ = a(1 - rⁿ) ÷ (1 - r)
S∞ = a ÷ (1 - r) for |r| < 1
If |r| ess all terms are zero. Edexcel questions often ask you to find the range of values of r for which convergence occurs.
如果 |r| ≥ 1，除非所有项都为零，否则无穷级数发散。Edexcel 的题目常要求你求出使级数收敛的 r 的取值范围。
- Check that the ratio is constant before applying the geometric formulae.
- 在应用等比公式之前，先验证公比是否为常数。
- Sum to infinity models real contexts such as bouncing balls or repeating decimals.
- 无穷级数求和可以模拟弹跳球或循环小数等实际背景。
4. Binomial Expansion | 二项式展开
The binomial expansion expresses (a + b)ⁿ as a sum of terms involving powers of a and b. For positive integer n, the expansion is finite and the coefficients are given by the binomial coefficients.
二项式展开将 (a + b)ⁿ 表示为包含 a 和 b 的幂的项之和。当 n 为正整数时，展开式是有限的，其系数由二项式系数给出。
(a + b)ⁿ = Σₖ₌₀ⁿ C(n,k) aⁿ⁻ᵏ bᵏ
where C(n,k) = n! ÷ [k!(n-k)!]. For rational n, the expansion of (1 + x)ⁿ is valid for |x| < 1.
其中 C(n,k) = n! ÷ [k!(n-k)!]。对于有理数 n，(1 + x)ⁿ 的展开式在 |x| < 1 时成立。
(1 + x)ⁿ = 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + ...
You must state the validity range in Edexcel answers, especially when approximating expressions such as (4 + 3x)⁻¹ after rewriting it as 4⁻¹(1 + 3x/4)⁻¹.
在 Edexcel 答案中必须注明有效范围，尤其是在将 (4 + 3x)⁻¹ 改写为 4⁻¹(1 + 3x/4)⁻¹ 后进行近似计算时。
- Write the expression in the form (1 + something) first, then expand.
- 首先将表达式写成 (1 + 某个量) 的形式，然后展开。
- For approximations, use the first few terms only and clearly state the range of validity.
- 进行近似计算时，只使用前几项，并明确写出有效范围。
5. Trigonometric Identities | 三角恒等式
Trigonometric identities allow you to simplify expressions, solve equations, and prove relationships. The two most fundamental identities form the basis for nearly all trigonometric manipulation.
三角恒等式可用于化简表达式、解方程和证明关系。两个最基本的恒等式构成了几乎所有三角变换的基础。
tan θ = sin θ ÷ cos θ
sin² θ + cos² θ = 1
From these, the squared identities follow directly after dividing by cos² θ or sin² θ.
由这两个公式，分别除以 cos² θ 或 sin² θ 即可直接得到平方恒等式。
1 + tan² θ = sec² θ
1 + cot² θ = cosec² θ
The addition formulae and double-angle formulae are also essential for Edexcel Pure Mathematics.
和角公式和二倍角公式也是 Edexcel 纯数学的重要内容。
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
sin 2A = 2 sin A cos A
cos 2A = cos² A - sin² A = 2 cos² A - 1 = 1 - 2 sin² A
- sin² θ + cos² θ = 1 to turn a quadratic in sin θ into a quadratic in cos θ, or vice versa.
- 利用 sin² θ + cos² θ = 1 可将 sin θ 的二次式转化为 cos θ 的二次式，反之亦然。
- Choose the correct form of cos 2A depending on whether the expression contains sin, cos, or both.
- 根据表达式中含有 sin、cos 还是两者都有，选择 cos 2A 的适当形式。
6. Exponentials and Logarithms | 指数与对数
Exponential and logarithmic functions are inverses of each other. The natural logarithm ln x has base e, and its properties mirror the laws of indices.
指数函数与对数函数互为反函数。自然对数 ln x 以 e 为底，其性质与指数法则相对应。
eln x = x and ln(eˣ) = x
ln a + ln b = ln(ab)
ln a - ln b = ln(a/b)
ln aⁿ = n ln a
To solve an equation such as 3ˣ s: x ln 3 = ln 7, so x = ln 7 ÷ ln 3. This method works for any base and avoids guesswork.
解形如 3ˣ = 7 的方程时，对方程两边取对数：x ln 3 = ln 7，因此 x = ln 7 ÷ ln 3。这种方法适用于任何底数，避免猜测。
- Always isolate the exponential or logarithmic term before taking logs or exponentiating.
- 在取对数或指数化之前，务必先分离出指数项或对数项。
- Remember that ln 1 = 0 and ln e = 1.
- 记住 ln 1 = 0 以及 ln e = 1。
7. Differentiation Rules | 微分法则
Differentiation gives the gradient of a curve, the rate of change, or the marginal value in applied contexts. The power rule is the starting point, but Edexcel questions often require the product, quotient, and chain rules.
微分可求得曲线的斜率、变化率或应用背景中的边际值。幂法则是起点，但 Edexcel 题目往往需要乘积法则、商法则和链式法则。
d/dx (xⁿ) = nxⁿ⁻¹
d/dx (eˣ) = eˣ
d/dx (ln x) = 1/x
d/dx (sin x) = cos x
d/dx (cos x) = -sin x
d/dx (tan x) = sec² x
⁵ or ln(sin x). In Leibniz notation, if y = f(u) and u = g(x), then dy/dx = dy/du × du/dx.
链式法则用于复合函数求导，例如 (3x² + 1)⁵ 或 ln(sin x)。用莱布尼茨记号表示，若 y = f(u) 且 u = g(x)，则 dy/dx = dy/du × du/dx。
dy/dx = dy/du × du/dx
Product and quotient rules are listed below.
乘积法则和商法则如下所示。
d/dx (uv) = u dv/dx + v du/dx
d/dx (u/v) = (v du/dx - u dv/dx) ÷ v²
- For the chain rule, differentiate the outer function first, then multiply by the derivative of the inner function.
- 使用链式法则时，先对外层函数求导，再乘以内层函数的导数。
- With quotient rule, keep the denominator as v² and watch the minus sign in the numerator.
- 使用商法则时，分母保持为 v²，并注意分子中的负号。
8. Integration Rules | 积分法则
Integration reverses differentiation and is used to find areas, volumes, and general solutions to differential equations. The power rule for integration requires special care when the power is -1.
积分是微分的逆运算，用于求面积、体积以及微分方程的通解。当幂指数为 -1 时，积分幂法则需要特别处理。
∫ xⁿ dx = xⁿ⁺¹ ÷ (n + 1) + C for n ≠ -1
∫ eˣ dx = eˣ + C
∫ 1/x dx = ln |x| + C
∫ sin x dx = -cos x + C
∫ cos x dx = sin x + C
For definite integrals, remember to evaluate the antiderivative at both limits and subtract: F(b) - F(a). The constant C cancels out.
对于定积分，要记住在两个积分限处分别计算原函数并相减：F(b) - F(a)。常数 C 会相互抵消。
Integration by substitution and integration by parts are the two main advanced techniques in Edexcel A-Level.
换元积分法和分部积分法是 Edexcel A-Level 中两种主要的高级积分方法。
∫ u dv = uv - ∫ v du
For substitution, replace the original variable, its derivative, and the limits if it is a definite integral. Integration by parts is typically used for products such as x sin x or x eˣ, where one part becomes simpler when differentiated.
使用换元法时，要替换原变量、其导数，以及定积分中的积分限。分部积分法通常用于 x sin x 或 x eˣ 这类乘积，其中一部分求导后会变得更简单。
- Add the constant of integration + C for every indefinite integral.
- 每个不定积分都要加上积分常数 + C。
- Check your answer by differentiating it back to the original integrand where possible.
- 尽可能通过对答案求导来检验是否得到原被积函数。
9. Coordinate Geometry and Circles | 坐标几何与圆
Coordinate geometry formulae connect algebraic equations with geometric shapes. For straight lines, the gradient and distance measures are fundamental building blocks.
坐标几何公式将代数方程与几何图形联系起来。对于直线而言，斜率和距离度量是最基本的组成部分。
Gradient: m = (y₂ - y₁) ÷ (x₂ - x₁)
Distance: d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
For a circle with centre (a, b) and radius r, the standard equation is:
对于圆心为 (a, b)、半径为 r 的圆，其标准方程为：
(x - a)² + (y - b)² = r²
The general form x² + y² + 2gx + 2fy + c = 0 has centre (-g, -f) and radius √g² + f² - c. Completing the square is the usual first step.
一般式 x² + y² + 2gx + 2fy + c = 0 的圆心为 (-g, -f)，半径为 √g² + f² - c。通常第一步是配方。
A tangent to a circle is perpendicular to the radius at the point of contact. Use the negative reciprocal of the radius gradient to find the tangent gradient.
圆上一点的切线垂直于该点处的半径。利用半径斜率的负倒数即可求出切线斜率。
- Complete the square to convert a circle equation into standard form.
- 通过配方将圆的方程化为标准形式。
- For tangents, find the radius gradient first, then use m_tangent = -1 ÷ mradius.
- 求切线时，先求半径斜率，再使用 m_tangent = -1 ÷ mradius。
10. Statistical Formulae | 统计公式
Statistics formulae summarise data and model random variables. The mean, variance, and standard deviation are basic measures of location and spread.
统计公式用于总结数据并对随机变量建模。均值、方差和标准差是描述位置和离散程度的基本度量。
Mean: x̄ = Σx ÷ n
Variance: s² = Σ(x - x̄)² ÷ (n - 1)
Standard deviation: s = √s²
For a binomial random variable X ~ B(n, p), the key formulae are:
对于二项随机变量 X ~ B(n, p)，关键公式为：
P(X = r) = C(n,r) pʳ (1 - p)ⁿ⁻ʳ
E(X) = np and Var(X) = np(1 - p)
For the normal distribution X ~ standardise using z = (x - μ) ÷ σ. Then use tables or a calculator to find probabilities.
对于正态分布 X ~ N(μ, σ²)，使用 z = (x - μ) ÷ σ 进行标准化。然后查表或使用计算器求概率。
z = (x - μ) ÷ σ
The product moment correlation coefficient (PMCC) measures linear correlation strength. Its formula is provided in the Edexcel formula booklet, but you must be able to interpret values from -1 to 1.
积矩相关系数 (PMCC) 衡量线性相关的强度。其公式在 Edexcel 公式手册中给出，但你必须能够解释 -1 到 1 之间的取值含义。
- Use the correct denominator n - 1 for sample variance, not population variance.
- 样本方差要使用分母 n - 1，而不是总体方差的分母 n。
- For normal probabilities, always sketch the bell curve and shade the required region.
- 计算正态概率时，先画出钟形曲线并标出所需区域。
11. Mechanics Formulae | 力学公式
Mechanics applies mathematics to motion and forces. The SUVAT equations describe constant acceleration motion and are collected here for quick reference.
力学将数学应用于运动和力。SUVAT 方程描述匀加速运动，以下汇总便于快速查阅。
v = u + at
s = ut + ½at²
v² = u² + 2as
s = (u + v)t ÷ 2
Here u is initial velocity, v is final velocity, a is acceleration, t is time, and s is displacement. You must choose the equation that contains the quantity you are trying to find and all the quantities you already know.
其中 u 是初速度，v 是末速度，a 是加速度，t 是时间，s 是位移。你必须选择包含待求量以及所有已知量的方程。
Newton's second law links resultant force, mass, and acceleration.
牛顿第二定律将合力、质量和加速度联系起来。
F = ma
Momentum and impulse are also important in collisions and motion.
动量和冲量在碰撞和运动中也很重要。
Momentum: p = mv
Impulse: I = mv - mu = Ft
- Assign a positive direction before writing SUVAT values, especially when velocity or acceleration is negative.
- 在写出 SUVAT 各量之前，先规定正方向，尤其是当速度或加速度为负时。
- Keep units consistent: metres, seconds, kilograms, newtons.
- 保持单位一致：米、秒、千克、牛顿。
12. Key Exam Tips | 考试要点
Success in Edexcel A-Level Mathematics depends on more than memorising formulae. You also need to read questions carefully, choose the right method, and present clear working.
在 Edexcel A-Level 数学中取得成功，不仅仅依靠记忆公式。你还需要仔细审题、选择正确的方法，并写出清晰的解题过程。
- Write down any formula you use before substituting numbers, as method marks are awarded for correct structure.
- 在代入数值之前先写出所使用的公式，因为正确的公式结构也会获得方法分。
- Check whether your answer is reasonable; for example, a probability should lie between 0 and 1.
- 检查答案是否合理；例如，概率应当介于 0 和 1 之间。
- For trig equations, state all solutions within the required interval, not just the principal value.
- 解三角方程时，要写出所求区间内的所有解，而不仅仅是主值。
- In mechanics, always indicate your positive direction and show units in final answers.
- 在力学中，始终标明正方向，并在最终答案中写出单位。
- Use the Edexcel formula booklet intelligently: know what is inside it, but do not rely on it to replace understanding.
- 合理使用 Edexcel 公式手册：了解其中内容，但不要依赖它来代替理解。
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