📚 Pre-U CCEA Mathematics Formula & Theorem Quick Reference Handbook | Pre-U CCEA 数学：公式定理速查手册
Welcome to the ultimate quick-reference guide for Pre-U CCEA Mathematics. This handbook compiles the essential formulae, theorems and identities you will need across Pure Mathematics, Mechanics, Statistics and Probability. Every entry is presented in clear English with a parallel Chinese translation, ensuring you can memorise and apply each piece of knowledge with confidence. Use this resource to reinforce your revision, check a forgotten formula, or consolidate your understanding before the exam.
欢迎使用这份 Pre-U CCEA 数学速查手册。本手册汇集了纯数学、力学、统计与概率部分不可或缺的公式、定理与恒等式。每条内容均以清晰的英文搭配对应的中文呈现，帮助你自信地记忆并运用每一个知识点。无论是强化复习、查阅遗忘的公式，还是在考前巩固理解，都可以借助这份资料。
1. Algebraic Identities and Polynomial Theorems | 代数恒等式与多项式定理
Algebraic manipulation is the foundation of the whole syllabus. The binomial expansion for rational index (|x| < 1, n is rational) is essential for series work and approximations.
代数运算是整个课程的基础。有理指数下的二项式展开（|x| < 1，n 为有理数）在级数运算与近似计算中至关重要。
(1 + x)ⁿ = 1 + nx + [n(n−1)/2!]x² + [n(n−1)(n−2)/3!]x³ + …
The factor theorem states that (x − a) is a factor of polynomial f(x) if and only if f(a) = 0. The remainder theorem gives the remainder when f(x) is divided by (x − a) as f(a).
因式定理指出，(x − a) 是多项式 f(x) 的因式当且仅当 f(a) = 0。余数定理则指明，f(x) 除以 (x − a) 的余数为 f(a)。
Completing the square transforms a quadratic ax² + bx + c into the form a(x + p)² + q, which reveals the vertex and helps find the inverse function.
配方法将二次式 ax² + bx + c 转化为 a(x + p)² + q 的形式，从而揭示顶点并有助于求反函数。
The discriminant Δ = b² − 4ac determines the nature of the roots of ax² + bx + c = 0: Δ > 0 gives two distinct real roots; Δ = 0 gives a repeated real root; Δ < 0 gives two complex conjugate roots.
判别式 Δ = b² − 4ac 决定了二次方程 ax² + bx + c = 0 根的性质：Δ > 0 有两个相异实根；Δ = 0 有一个重根；Δ < 0 有一对共轭复根。
2. Sequences, Series and the Binomial Theorem | 数列、级数与二项式定理
Arithmetic sequences and series appear frequently. The sum of the first n terms of an arithmetic progression with first term a and common difference d is a staple.
等差数列与等差级数频繁出现。首项为 a、公差为 d 的等差数列前 n 项和是一个基础公式。
Sₙ = n/2 [2a + (n − 1)d]
For geometric progressions, the sum to infinity exists only when |r| < 1 and is given by S∞ = a/(1 − r). The sum of the first n terms is Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1.
对于等比数列，无穷和仅在 |r| < 1 时存在，公式为 S∞ = a/(1 − r)。前 n 项和为 Sₙ = a(1 − rⁿ)/(1 − r)，r ≠ 1。
The Maclaurin series expands a function about zero as f(x) = f(0) + f'(0)x + f”(0)x²/2! + … . This underpins series expansions for eˣ, sin x, cos x and ln(1 + x).
麦克劳林级数在零点附近展开函数：f(x) = f(0) + f'(0)x + f”(0)x²/2! + … 。eˣ、sin x、cos x 和 ln(1 + x) 的级数展开均源于此。
Summation notation Σₙᵣ₌₁ r = n(n+1)/2, Σ r² = n(n+1)(2n+1)/6, Σ r³ = [n(n+1)/2]². These are frequently used in proof by induction and integration.
求和记号 Σₙᵣ₌₁ r = n(n+1)/2，Σ r² = n(n+1)(2n+1)/6，Σ r³ = [n(n+1)/2]²。它们常用于数学归纳法证明与积分计算。
3. Trigonometry and Circular Functions | 三角学与圆函数
The definitions of sec, cosec and cot in terms of sin, cos and tan are crucial. The Pythagorean identities form the backbone of trigonometric simplification.
用 sin、cos 和 tan 定义 sec、cosec 和 cot 至关重要。勾股恒等式是三角化简的支柱。
sin² θ + cos² θ ≡ 1 1 + tan² θ ≡ sec² θ 1 + cot² θ ≡ cosec² θ
Compound angle formulae, double angle formulae and factor formulae are used to solve equations and integrate trigonometric expressions.
和角公式、倍角公式与和差化积公式可用于解方程和积分三角表达式。
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)
sin 2A = 2 sin A cos A
cos 2A = cos² A − sin² A = 2 cos² A − 1 = 1 − 2 sin² A
sin P + sin Q = 2 sin[(P+Q)/2] cos[(P−Q)/2]
The expression a sin θ + b cos θ can be written as R sin(θ + α) or R cos(θ − α), where R = √(a² + b²) and tan α = b/a (after suitable adjustment). This form is vital for solving equations and finding maximum/minimum values.
表达式 a sin θ + b cos θ 可写成 R sin(θ + α) 或 R cos(θ − α)，其中 R = √(a² + b²)，tan α = b/a（适当调整）。该形式对解方程和求最值至关重要。
4. Exponentials, Logarithms and Their Calculus | 指数、对数及其微积分
The key relationship y = aˣ ⇔ x = logₐ y connects exponentials and logarithms. The natural logarithm ln x is the inverse of eˣ.
关键关系 y = aˣ ⇔ x = logₐ y 将指数与对数联系起来。自然对数 ln x 是 eˣ 的反函数。
eˡⁿˣ = x lneˣ = x
logₐ x = ln x / ln a
Differentiation and integration of exponential and logarithmic functions appear throughout the syllabus.
指数函数与对数函数的求导与积分贯穿整个课程。
d/dx (eˣ) = eˣ ∫ eˣ dx = eˣ + C
d/dx (ln x) = 1/x ∫ 1/x dx = ln|x| + C
d/dx (aˣ) = aˣ ln a
Exponential growth and decay models follow the form dx/dt = kx, giving x = x₀ eᵏᵗ. The half‑life or doubling time is found using ln 2 / k.
指数增长与衰减模型遵循 dx/dt = kx，其解为 x = x₀ eᵏᵗ。半衰期或加倍时间可用 ln 2 / k 求出。
5. Differentiation: Rules, Techniques and Applications | 微分：法则、技巧与应用
The product rule, quotient rule and chain rule are the workhorses of differentiation. Remember them accurately.
乘法法则、除法法则和链式法则是微分的三大主力。必须准确记忆。
d/dx (uv) = u’v + uv’
d/dx (u/v) = (u’v − uv’) / v²
dy/dx = dy/du × du/dx (chain rule)
Implicit differentiation is used when y is not given explicitly as a function of x. Differentiate term‑by‑term, treating y as a function and adding dy/dx each time.
隐函数微分在 y 未显式表达为 x 的函数时使用。逐项求导，将 y 视为函数并每次乘上 dy/dx。
Parametric differentiation: if x = f(t) and y = g(t), then dy/dx = (dy/dt) ÷ (dx/dt). Second derivative: d²y/dx² = d/dt (dy/dx) ÷ dx/dt.
参数微分：若 x = f(t)，y = g(t)，则 dy/dx = (dy/dt) ÷ (dx/dt)。二阶导数：d²y/dx² = d/dt (dy/dx) ÷ dx/dt。
Stationary points occur where dy/dx = 0. The second derivative test or sign change of the first derivative determines whether a point is a maximum, minimum or point of inflection.
驻点出现在 dy/dx = 0 处。通过二阶导数检验或一阶导数符号变化可判断该点为极大值点、极小值点还是拐点。
6. Integration: Techniques and Definite Integrals | 积分：技巧与定积分
Integration by inspection (reverse chain rule) and integration by parts are essential for functions that are products or involve ln x.
观察积分法（逆链式法则）和分部积分法对于乘积函数或含 ln x 的被积函数至关重要。
∫ u dv = uv − ∫ v du (integration by parts)
∫ f'(g(x)) g'(x) dx = f(g(x)) + C (reverse chain rule)
Integration using partial fractions is used when the integrand is a rational function where the denominator factorises. Each linear factor contributes a form A/(ax + b).
当被积函数为有理函数且分母可分解时，使用部分分式积分。每个线性因式贡献 A/(ax + b) 的形式。
Trapezium rule for numerical integration: ∫ₐᵇ y dx ≈ h/2 [y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)], where h = (b − a)/n.
数值积分中的梯形法则：∫ₐᵇ y dx ≈ h/2 [y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)]，其中 h = (b − a)/n。
Areas between curves are given by ∫ (top − bottom) dx or ∫ (right − left) dy. Volumes of revolution about the x‑axis: V = π ∫ y² dx; about the y‑axis: V = π ∫ x² dy.
曲线间的面积由 ∫ (上方曲线 − 下方曲线) dx 或 ∫ (右边曲线 − 左边曲线) dy 给出。绕 x 轴旋转体积：V = π ∫ y² dx；绕 y 轴旋转体积：V = π ∫ x² dy。
7. Differential Equations | 微分方程
First‑order separable differential equations are solved by separating variables and integrating both sides. The constant of integration is determined by initial conditions.
一阶可分离变量的微分方程通过分离变量并两边积分求解。积分常数由初始条件确定。
dy/dx = g(x)h(y) ⇒ ∫ 1/h(y) dy = ∫ g(x) dx
The integrating factor method is used for first‑order linear differential equations of the form dy/dx + P(x)y = Q(x). The integrating factor is I = e^(∫ P dx). Multiply through and rewrite as d/dx (Iy) = I Q.
积分因子法用于形如 dy/dx + P(x)y = Q(x) 的一阶线性微分方程。积分因子为 I = e^(∫ P dx)。两边乘以 I 后可写成 d/dx (Iy) = I Q。
Second‑order homogeneous linear differential equations with constant coefficients: ay” + by’ + cy = 0. The auxiliary equation is am² + bm + c = 0. The form of the general solution depends on whether roots are real and distinct, repeated or complex.
常系数二阶齐次线性微分方程：ay” + by’ + cy = 0。辅助方程为 am² + bm + c = 0。通解形式取决于根是相异实根、重根还是复根。
| Roots of auxiliary equation | General solution |
|---|---|
| Real distinct m₁, m₂ | y = A e^(m₁x) + B e^(m₂x) |
| Repeated root m | y = (A + Bx) e^(mx) |
| Complex α ± iβ | y = e^(αx) (C cos βx + D sin βx) |
辅助方程的根 通解（中文对照）：相异实根 m₁, m₂ 对应 y = A e^(m₁x) + B e^(m₂x)；重根 m 对应 y = (A + Bx) e^(mx)；共轭复根 α ± iβ 对应 y = e^(αx) (C cos βx + D sin βx)。
8. Vectors and 3‑D Geometry | 向量与三维几何
The scalar product (dot product) a·b = |a||b| cos θ is used to calculate angles and to test perpendicularity (a·b = 0). In component form, if a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k, then a·b = a₁b₁ + a₂b₂ + a₃b₃.
标量积（点乘）a·b = |a||b| cos θ 用于计算角度和判断垂直（a·b = 0）。分量形式下，若 a = a₁i + a₂j + a₃k，b = b₁i + b₂j + b₃k，则 a·b = a₁b₁ + a₂b₂ + a₃b₃。
The vector equation of a line is r = a + t d, where a is a point on the line and d is the direction vector. The vector product (cross product) a × b gives a vector perpendicular to both a and b, with magnitude |a||b| sin θ.
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。向量积（叉乘）a × b 给出一个垂直于 a 和 b 的向量，其模长为 |a||b| sin θ。
a × b = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k
The shortest distance from a point to a line is found by projecting the point‑to‑line vector onto the normal to the line direction. The area of a triangle formed by two vectors a and b is ½|a × b|.
点到直线的最短距离可通过将点线向量投影到与直线方向垂直的方向上求得。由两向量 a 与 b 构成的三角形面积为 ½|a × b|。
9. Probability and Discrete Random Variables | 概率与离散随机变量
The addition rule for mutually exclusive events: P(A ∪ B) = P(A) + P(B). For non‑mutually exclusive events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
互斥事件的加法法则：P(A ∪ B) = P(A) + P(B)。非互斥事件：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。
Conditional probability: P(A|B) = P(A ∩ B)/P(B). Events A and B are independent if P(A ∩ B) = P(A)P(B) or P(A|B) = P(A).
条件概率：P(A|B) = P(A ∩ B)/P(B)。若 P(A ∩ B) = P(A)P(B) 或 P(A|B) = P(A)，则事件 A 与 B 相互独立。
For a discrete random variable X, the expectation E(X) = Σ x P(X = x) and variance Var(X) = E(X²) − [E(X)]². The sum of probabilities is always 1.
对于离散随机变量 X，期望 E(X) = Σ x P(X = x)，方差 Var(X) = E(X²) − [E(X)]²。所有概率之和恒为 1。
The binomial distribution X ~ B(n, p) has probability function P(X = r) = ⁿCᵣ p^r (1−p)^(n−r). Mean = np, variance = np(1−p). The Poisson distribution X ~ Po(λ) has P(X = r) = e^(−λ) λ^r / r!, with mean and variance both λ.
二项分布 X ~ B(n, p) 的概率函数为 P(X = r) = ⁿCᵣ p^r (1−p)^(n−r)，均值 = np，方差 = np(1−p)。泊松分布 X ~ Po(λ) 有 P(X = r) = e^(−λ) λ^r / r!，均值与方差均为 λ。
10. Continuous Random Variables and the Normal Distribution | 连续随机变量与正态分布
A continuous random variable has probability density function (p.d.f.) f(x) satisfying ∫ f(x) dx = 1 over the domain. Probabilities are areas: P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx.
连续随机变量具有概率密度函数 f(x)，在其定义域上满足 ∫ f(x) dx = 1。概率为面积：P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx。
The cumulative distribution function (c.d.f.) is F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt. The mean is μ = ∫ x f(x) dx, variance σ² = ∫ (x−μ)² f(x) dx = ∫ x² f(x) dx − μ².
累积分布函数为 F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt。均值为 μ = ∫ x f(x) dx，方差 σ² = ∫ (x−μ)² f(x) dx = ∫ x² f(x) dx − μ²。
The normal distribution X ~ N(μ, σ²) has p.d.f. f(x) = (1/√(2πσ²)) e^[−(x−μ)²/(2σ²)]. Standardisation: Z = (X − μ)/σ ~ N(0, 1). Use tables for Φ(z) = P(Z ≤ z).
正态分布 X ~ N(μ, σ²) 的密度函数为 f(x) = (1/√(2πσ²)) e^[−(x−μ)²/(2σ²)]。标准化：Z = (X − μ)/σ ~ N(0, 1)。使用表格查 Φ(z) = P(Z ≤ z)。
The central limit theorem states that for a sufficiently large sample size n, the distribution of the sample mean X̄ tends to N(μ, σ²/n) regardless of the original population distribution.
中心极限定理指出，当样本量 n 足够大时，无论总体分布如何，样本均值 X̄ 的分布趋于 N(μ, σ²/n)。
11. Mechanics: Kinematics, Forces and Newton’s Laws | 力学：运动学、力与牛顿定律
The SUVAT equations describe uniform acceleration motion in a straight line. They are valid only when acceleration a is constant.
SUVAT 方程描述匀加速直线运动，仅在加速度 a 恒定时有效。
v = u + at s = ut + ½at² v² = u² + 2as s = ½(u + v)t
Newton’s second law: F = ma. When forces act on a particle, resolve along chosen directions. The resultant force equals mass times acceleration vectorially.
牛顿第二定律：F = ma。当力作用于质点时，需沿选定方向分解。合力在矢量上等于质量乘以加速度。
For connected particles, write equations of motion for each body and solve simultaneously. Include tension and consider inextensible light strings where tension is constant and acceleration is the same for connected bodies.
对于连接体，需分别写出每个物体的运动方程并联立求解。包含张力，并注意不可伸长的轻绳中张力处处相等，连接体加速度相同。
Momentum p = mv. Impulse = change in momentum = FΔt = mv − mu. The principle of conservation of momentum applies when no external forces act.
动量 p = mv。冲量等于动量变化 = FΔt = mv − mu。当无外力作用时，动量守恒定律适用。
12. Moments, Equilibrium and Work–Energy | 力矩、平衡与功—能
For a body in static equilibrium, the vector sum of forces is zero and the sum of moments about any point is zero. The moment of a force = force × perpendicular distance from the pivot.
对于静态平衡的物体，力的矢量和为零，且对任意点的力矩之和为零。力矩 = 力 × 到支点的垂直距离。
Work done by a constant force is Fd cos θ, where θ is the angle between force and displacement. Kinetic energy KE = ½mv², gravitational potential energy GPE = mgh.
恒力做功为 Fd cos θ，θ 是力与位移的夹角。动能 KE = ½mv²，重力势能 GPE = mgh。
The work–energy principle: total work done by all forces equals change in kinetic energy. For conservative systems, mechanical energy (KE + PE) is conserved when no external work is done.
功—能原理：所有力做的总功等于动能的变化量。对于保守系统，当无外部做功时，机械能（动能 + 势能）守恒。
Power = work done / time = Fv, where v is velocity in the direction of the force. Unit: watt (W). This is useful for problems involving vehicles and constant tractive forces.
功率 = 做功 / 时间 = Fv，v 为沿力方向的速度，单位为瓦特 (W)。这在涉及车辆与恒定牵引力的题目中非常实用。
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