📚 Year 13 Cambridge Mathematics: Quick Reference Handbook of Formulas and Theorems | Year 13 Cambridge 数学：公式定理速查手册
This handbook gathers the essential formulas, theorems, and identities that every Cambridge Year 13 Mathematics learner needs at their fingertips. From pure mathematics to mechanics and statistics, the following sections provide a structured revision aid designed to reinforce understanding and boost exam confidence.
本手册汇集了每位剑桥 Year 13 数学学习者必须熟练掌握的核心公式、定理和恒等式。从纯数学到力学与统计，以下章节提供了一份结构化的复习辅助资料，旨在加深理解并提升考试信心。
1. Binomial Expansion for Rational Powers | 有理数指数的二项式展开
For any rational n and |x| t; 1, the expansion of (1 + x)ⁿ is given by the infinite series: (1 + x)ⁿ = 1 + n x + [n(n – 1)/2!] x² + [n(n – 1)(n – 2)/3!] x³ + … . This expansion is valid only when |x| xpression is of the form (a + b x)ⁿ, factor out aⁿ to write it as aⁿ (1 + (b/a) x)ⁿ first.
对于任意有理数 n 且 |x| < 1，(1 + x)ⁿ 的展开由无穷级数给出：(1 + x)ⁿ = 1 + n x + [n(n – 1)/2!] x² + [n(n – 1)(n – 2)/3!] x³ + … 。此展开仅在 |x| < 1 时有效。如果表达式为 (a + b x)ⁿ，应先提取 aⁿ，写成 aⁿ (1 + (b/a) x)ⁿ 的形式。
- Range of validity: |x| < 1, or equivalently |b x / a| < 1 for (a + b x)ⁿ.
- 有效性范围：|x| < 1，对于 (a + b x)ⁿ 等价于 |b x / a| < 1。
- When n is not a positive integer, the series is infinite; for positive integer n it reduces to the finite binomial sum.
- 当 n 不是正整数时，级数为无穷级数；当 n 为正整数时，退化为有限二项式求和。
2. Trigonometric Identities and Compound Angle Formulas | 三角恒等式与复角公式
trigonometric functions. sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; tan(A ± B) = (tan A ± tan B) / (1 ∓ B). From these, the double-angle identities follow: sin 2A = 2 sin A cos A; cos 2A = cos²A – sin²A = 2 cos²A – 1 = 1 – 2 sin²A; tan 2A = 2 tan A / (1 – tan²A).
复角公式是求解三角方程以及对三角函数求导和积分的基础。sin(A ± B) = sin A cos B ± cos A sin B；cos(A ± B) = cos A cos B ∓ sin A sin B；tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)。由此可推导二倍角恒等式：sin 2A = 2 sin A cos A；cos 2A = cos²A – sin²A = 2 cos²A – 1 = 1 – 2 sin²A；tan 2A = 2 tan A / (1 – tan²A)。
c θ = 1/sin θ, cot θ = 1/tan θ; sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ.
倒数恒等式与毕达哥拉斯恒等式：sec θ = 1/cos θ，cosec θ = 1/sin θ，cot θ = 1/tan θ；sin²θ + cos²θ = 1，1 + tan²θ = sec²θ，1 + cot²θ = cosec²θ。
3. Differentiation Rules and Standard Derivatives | 微分法则与标准导数
differentiating composite, product, and quotient functions. If y = f(g(x)), then dy/dx = f '(g(x)) · g'(x). For u and v functions of x, d(uv)/dx = u' v + u v'; d(u/v)/dx = (u' v – u v') / v².
链式法则、乘法法则和除法法则是对复合函数、乘积函数和分式函数求导的关键。若 y = f(g(x))，则 dy/dx = f '(g(x)) · g'(x)。对于 x 的函数 u 和 v，d(uv)/dx = u' v + u v'；d(u/v)/dx = (u' v – u v') / v²。
Standard derivatives to memorise: d/dx (sin x) = cos x, d/dx (cos x) = –sin x, d/dx (tan x) = sec² x, d/dx (sec x) = sec x tan x, d/dx (cosec x) = –cosec x cot x, d/dx (cot x) = –cosec² x. For exponentials and logarithms: d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x, d/dx (aˣ) = aˣ ln a. The derivative of the inverse trigonometric functions: d/dx (sin⁻¹ x) = 1/√1 – x², d/dx (cos⁻¹ x) = –1/√1 – x², d/dx (tan⁻¹ x) = 1/(1 + x²).
需记忆的标准导数：d/dx (sin x) = cos x，d/dx (cos x) = –sin x，d/dx (tan x) = sec² x，d/dx (sec x) = sec x tan x，d/dx (cosec x) = –cosec x cot x，d/dx (cot x) = –cosec² x。指数与对数：d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x，d/dx (aˣ) = aˣ ln a。反三角函数的导数：d/dx (sin⁻¹ x) = 1/√1 – x²，d/dx (cos⁻¹ x) = –1/√1 – x²，d/dx (tan⁻¹ x) = 1/(1 + x²)。
4. Integration Techniques and Standard Integrals | 积分方法与标准积分
the standard integrals follow directly from derivatives. ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c, for n ≠ –1; ∫ 1/x dx = ln|x| + c; ∫ eˣ dx = eˣ + c; ∫ aˣ dx = aˣ/ln a + c; ∫ cos x dx = sin x + c; ∫ sin x dx = –cos x + c; ∫ sec² x dx = tan x + c; ∫ cosec² x dx = –cot x + c; ∫ sec x tan x dx = sec x + c; ∫ cosec x cot x dx = –cosec x + c.
积分是微分的逆运算，标准积分可直接由导数得出。∫ xⁿ dx = xⁿ⁺¹/(n+1) + c，n ≠ –1；∫ 1/x dx = ln|x| + c；∫ eˣ dx = eˣ + c；∫ aˣ dx = aˣ/ln a + c；∫ cos x dx = sin x + c；∫ sin x dx = –cos x + c；∫ sec² x dx = tan x + c；∫ cosec² x dx = –cot x + c；∫ sec x tan x dx = sec x + c；∫ cosec x cot x dx = –cosec x + c。
integration methods: substitution (reverse chain rule), integration by parts: ∫ u dv = u v – s. For definite integrals, the fundamental theorem of calculus applies: ∫ₐᵇ f(x) dx = F(b) – F(a) where F'(x) = f(x).
核心积分方法：换元积分法（链式法则的逆用）、分部积分法：∫ u dv = u v – ∫ v du，以及用于有理函数的部分分式分解。对于定积分，应用微积分基本定理：∫ₐᵇ f(x) dx = F(b) – F(a)，其中 F'(x) = f(x)。
5. Logarithmic and Exponential Functions | 对数与指数函数
The natural logarithm ln x is the inverse of eˣ. Properties: ln(ab) = ln a + ln b; ln(a/b) = ln a – ln b; ln(aᵇ) = b ln a. The exponential function satisfies eˣ⁺ʸ = eˣ eʸ, eˣ⁻ʸ = eˣ / eʸ, (eˣ)ʸ = eˣʸ. For solving equations, if eᵏˣ = c then x = (1/k) ln c.
自然对数 ln x 是 eˣ 的反函数。性质：ln(ab) = ln a + ln b；ln(a/b) = ln a – ln b；ln(aᵇ) = b ln a。指数函数满足 eˣ⁺ʸ = eˣ eʸ，eˣ⁻ʸ = eˣ / eʸ，(eˣ)ʸ = eˣʸ。解方程时，若 eᵏˣ = c，则 x = (1/k) ln c。
k > 0 for growth and k < 0 for decay. The half-life T₁/₂ for decay satisfies T₁/₂ = ln 2 / |k|.
指数增长和衰减模型由 dN/dt = k N 描述，得到 N = N₀ eᵏᵗ，其中对于增长 k > 0，对于衰减 k < 0。衰减的半衰期 T₁/₂ 满足 T₁/₂ = ln 2 / |k|。
6. Sequences and Series: Arithmetic and Geometric | 数列与级数：等差与等比
Arithmetic progression: nth term uₙ = a + (n – 1)d; sum of first n terms Sₙ = n/2 [2a + (n – 1)d]. Geometric progression: nth term uₙ = a rⁿ⁻¹; sum of first n terms Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1; sum to infinity S∞ = a/(1 – r) exists only when |r| < 1.
等差数列：第 n 项 uₙ = a + (n – 1)d；前 n 项和 Sₙ = n/2 [2a + (n – 1)d]。等比数列：第 n 项 uₙ = a rⁿ⁻¹；前 n 项和当 r ≠ 1 时 Sₙ = a(1 – rⁿ)/(1 – r)；无穷项和 S∞ = a/(1 – r) 仅在 |r| < 1 时存在。
Convergence of a geometric series is a crucial test for infinite series in pure mathematics. The condition |r| < 1 ensures the sum approaches a finite limit. This concept extends to binomial expansions and power series.
等比级数的收敛性是纯数学中无穷级数的一个关键检验。条件 |r| < 1 确保总和趋近于一个有限极限。这一概念延伸至二项式展开和幂级数。
7. Vectors in 3D: Dot Product and Equations of Lines and Planes | 三维向量：点积与直线和平面方程
The dot product of two vectors a and b is a · b = |a||b| ne is r = a + t d, where a is a point on the line and d is a direction vector. The Cartesian equation of a plane is n₁x + n₂y + n₃z = k, or in vector form r · n = a · n, where n is the normal vector.
两向量 a 和 b 的点积为 a · b = |a||b| cos θ，其中 θ 为夹角。分量形式下，a · b = a₁b₁ + a₂b₂ + a₃b₃。若 a · b = 0，则两向量垂直。直线的向量方程为 r = a + t d，其中 a 为直线上一点，d 为方向向量。平面的笛卡尔方程为 n₁x + n₂y + n₃z = k，或向量形式 r · n = a · n，其中 n 为法向量。
The angle between two lines is found using the dot product of their direction vectors. The angle between two planes is the angle between their normals. The shortest distance from a point to a line or a plane can be calculated using vector projection or the distance formula.
两直线间的夹角通过其方向向量的点积求得。两平面间的夹角即为其法向量间的夹角。点到直线或点到平面的最短距离可通过向量投影或距离公式计算。
8. Forces and Newton’s Laws in Mechanics | 力学中的力与牛顿定律
Newton' , m is mass, and a is acceleration. (3) For every action, there is an equal and opposite reaction. Force is a vector measured in newtons (N). Weight is the force due to gravity: W = m g.
牛顿三定律支配着运动：（1）除非受到净外力作用，物体将保持静止或匀速直线运动状态。（2）F = m a，F 为净力，m 为质量，a 为加速度。（3）每一个作用力都有一个大小相等、方向相反的反作用力。力是矢量，单位为牛顿 (N)。重力即由引力产生的力：W = m g。
tic) friction, Ff = μₖ R. Resolving forces into perpendicular components is essential for equilibrium problems.
摩擦力 Ff ≤ μ R，其中 μ 为摩擦系数，R 为法向反作用力。对于静摩擦，最大值为 μₛ R；对于动摩擦，Ff = μₖ R。将力分解为互相垂直的分量是解决平衡问题的关键。
9. Kinematics with Constant and Variable Acceleration | 匀加速与变加速运动学
constant acceleration a, the SUVAT equations apply: v = u + a t; s = u t + ½ a t²; s = v t – velocity, s displacement, t time. These are vector equations when direction is considered.
对于恒定加速度 a，适用 SUVAT 方程：v = u + a t；s = u t + ½ a t²；s = v t – ½ a t²；v² = u² + 2 a s；s = ½ (u + v) t。其中 u 为初速度，v 为末速度，s 为位移，t 为时间。考虑方向时这些均为矢量方程。
a = dv/dt = d²s/dt². Integration yields v = ∫ a dt, s = ∫ v dt. Motion under gravity: a = –g (taking upward as positive) near the Earth's surface.
对于变加速运动，位移、速度和加速度通过微积分关联：v = ds/dt，a = dv/dt = d²s/dt²。积分得 v = ∫ a dt，s = ∫ v dt。重力作用下的运动：在地球表面附近 a = –g（取向上为正）。
10. Probability, Conditional Probability, and Bayes’ Theorem | 概率、条件概率与贝叶斯定理
For events A and B, P(A ∪ B) = P(A) + P(B) – P(A ∩ B). Conditional probability: P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0. Two events are independent if P(A ∩ B) = P(A) P(B), or equivalently P(A|B) = P(A). Bayes' theorem allows reversing conditional probabilities: P(A|B) = [P(B|A) P(A)] / P(B).
对于事件 A 和 B，P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。条件概率：P(A|B) = P(A ∩ B) / P(B)，前提是 P(B) > 0。若 P(A ∩ B) = P(A) P(B) 或等价地 P(A|B) = P(A)，则两事件独立。贝叶斯定理可逆转条件概率：P(A|B) = [P(B|A) P(A)] / P(B)。
are powerful tools for multi-stage experiments. The law of total probability: P(B) = Σ P(B|Aᵢ) P(Aᵢ) for a partition Aᵢ. This is often used with Bayes' theorem.
树形图和韦恩图是多阶段实验的有力工具。全概率公式：对于一组划分 Aᵢ，P(B) = Σ P(B|Aᵢ) P(Aᵢ)。这常与贝叶斯定理结合使用。
11. Discrete Random Variables and Probability Distributions | 离散随机变量与概率分布
= 1. The expected value (mean) is E(X) = μ = Σ x P(X = x); the variance is Var(X) = E(X²) – [E(X)]² = Σ (x – μ)² P(X = x). The standard deviation is σ = √Var(X).
离散随机变量 X 具有概率质量函数 P(X = x)。总和 Σ P(X = x) = 1。期望值（均值）为 E(X) = μ = Σ x P(X = x)；方差为 Var(X) = E(X²) – [E(X)]² = Σ (x – μ)² P(X = x)。标准差为 σ = √Var(X)。
Common distributions: Binomial B(n, p) with P(X = r) = C(n, r) pʳ qⁿ⁻ʳ, mean = n p, variance = n p q. Poisson distribution Po(λ) with P(X = r) = (e⁻λ λʳ) / r! with P(X = n) = qⁿ⁻¹ p, mean = 1/p. These distributions are used for modelling count data.
常见分布：二项分布 B(n, p)，P(X = r) = C(n, r) pʳ qⁿ⁻ʳ，均值 = n p，方差 = n p q。泊松分布 Po(λ)，P(X = r) = (e⁻λ λʳ) / r!，均值 = 方差 = λ。几何分布 P(X = n) = qⁿ⁻¹ p，均值 = 1/p。这些分布用于计数数据建模。
12. Hypothesis Testing and the Normal Distribution | 假设检验与正态分布
The normal distribution N(μ, σ²) has probability density function f(x) = (1 / σ √2π) e⁻⁽ˣ⁻μ⁾²/⁽²σ²⁾. The standard normal Z ~ N(0, 1) uses the transformation Z = (X – lt; z). For sample means, the central limit theorem states that for large sample size n, X̄ ≈ N(μ, σ²/n).
正态分布 N(μ, σ²) 的概率密度函数为 f(x) = (1 / σ √2π) e⁻⁽ˣ⁻μ⁾²/⁽²σ²⁾。标准正态 Z ~ N(0, 1) 使用变换 Z = (X – μ) / σ。表格给出 Φ(z) = P(Z < z)。对于样本均值，中心极限定理表明对于大样本容量 n，X̄ 近似服从 N(μ, σ²/n)。
A hypothesis test involves stating the null hypothesis H₀ and alternative hypothesis H₁, calculating a test statistic (e.g., Z or t), finding the p-value or critical region, and making a conclusion. Significance level α is the probability of rejecting H₀ when it is true (Type I error). For binomial and Poisson models, exact probabilities are used; for normal, standardised test statistics are compared with critical values.
假设检验包括设立原假设 H₀ 和备择假设 H₁，计算检验统计量（如 Z 或 t），求出 p 值或临界域，并得出结论。显著性水平 α 是当 H₀ 为真时拒绝它的概率（I 类错误）。对于二项与泊松模型，使用精确概率；对于正态，则将标准化检验统计量与临界值进行比较。
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