📚 Pre-U CAIE Mathematics: Formula and Theorem Quick Reference Handbook | Pre-U CAIE 数学：公式定理速查手册
This handbook provides a concise reference of essential formulas and theorems for the Cambridge Pre-U Mathematics syllabus. It covers Pure Mathematics, Mechanics and Probability & Statistics, offering quick access to key results that are frequently used in problem-solving and examinations.
本手册为剑桥 Pre-U 数学课程提供了一份简明扼要的公式定理速查参考，涵盖纯数学、力学和概率与统计，方便快速查阅解题和考试中常用的关键结果。
1. Algebraic Laws and Identities | 代数定律与恒等式
Exponent rules: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, a⁰ = 1 (a ≠ 0).
指数律：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁻ⁿ = 1/aⁿ，a⁰ = 1（a ≠ 0）。
Logarithm rules: logₐ(xy) = logₐ x + logₐ y, logₐ(x/y) = logₐ x − logₐ y, logₐ(xⁿ) = n logₐ x.
对数律：logₐ(xy) = logₐ x + logₐ y，logₐ(x/y) = logₐ x − logₐ y，logₐ(xⁿ) = n logₐ x。
Binomial theorem: (a + b)ⁿ = Σ (ⁿCᵣ) aⁿ⁻ʳ bʳ for r = 0 to n, where ⁿCᵣ = n! / [(n − r)! r!].
二项式定理：(a + b)ⁿ = Σ (ⁿCᵣ) aⁿ⁻ʳ bʳ，r 从 0 到 n，其中 ⁿCᵣ = n! / [(n − r)! r!]。
Quadratic formula: Solutions of ax² + bx + c = 0 are x = [−b ± √(b² − 4ac)] / (2a).
二次公式：方程 ax² + bx + c = 0 的解为 x = [−b ± √(b² − 4ac)] / (2a)。
Partial fractions: For distinct linear factors, 1/[(x − a)(x − b)] = A/(x − a) + B/(x − b).
部分分式：对于相异线性因子，1/[(x − a)(x − b)] = A/(x − a) + B/(x − b)。
Difference of powers: aⁿ − bⁿ = (a − b)(aⁿ⁻¹ + aⁿ⁻²b + … + bⁿ⁻¹).
幂差公式：aⁿ − bⁿ = (a − b)(aⁿ⁻¹ + aⁿ⁻²b + … + bⁿ⁻¹)。
2. Functions and Graphs | 函数与图像
Modulus function: |x| = x for x ≥ 0; |x| = −x for x < 0. Inequality |f(x)| < a ⇔ −a < f(x) < a.
绝对值函数：|x| 当 x ≥ 0 时为 x，当 x < 0 时为 −x。不等式 |f(x)| < a ⇔ −a < f(x) < a。
Inverse function: f⁻¹(x) exists if f is one‑one; domain of f⁻¹ = range of f.
反函数：若 f 是单射则 f⁻¹(x) 存在；f⁻¹ 的定义域等于 f 的值域。
Composite function: (g ∘ f)(x) = g(f(x)); domain is x ∈ domain of f such that f(x) ∈ domain of g.
复合函数：(g ∘ f)(x) = g(f(x))；定义域为使得 f(x) 在 g 的定义域内的所有 x。
Transformation of graphs: y = f(x − a) shifts right by a; y = f(kx) stretches horizontally by factor 1/k.
图像变换：y = f(x − a) 右移 a 单位；y = f(kx) 水平缩放因子 1/k。
Even / odd functions: Even: f(−x) = f(x); odd: f(−x) = −f(x).
奇偶函数：偶函数满足 f(−x) = f(x)；奇函数满足 f(−x) = −f(x)。
3. Sequences and Series | 数列与级数
Arithmetic progression: n-th term uₙ = a + (n − 1)d; sum Sₙ = (n/2)[2a + (n − 1)d] = (n/2)(a + l).
等差数列：第 n 项 uₙ = a + (n − 1)d；前 n 项和 Sₙ = (n/2)[2a + (n − 1)d] = (n/2)(a + l)。
Geometric progression: uₙ = arⁿ⁻¹; sum Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. Infinite sum S∞ = a/(1 − r) for |r| < 1.
等比数列：uₙ = arⁿ⁻¹；和 Sₙ = a(1 − rⁿ)/(1 − r)（r ≠ 1）。无穷和 S∞ = a/(1 − r) 当 |r| < 1 时成立。
Maclaurin series: f(x) = f(0) + f′(0)x + f″(0)x²/2! + … + f⁽ⁿ⁾(0)xⁿ/n! + …
麦克劳林级数：f(x) = f(0) + f′(0)x + f″(0)x²/2! + … + f⁽ⁿ⁾(0)xⁿ/n! + …
Important expansions: eˣ = 1 + x + x²/2! + x³/3! + …; sin x = x − x³/3! + x⁵/5! − …; cos x = 1 − x²/2! + x⁴/4! − …; ln(1+x) = x − x²/2 + x³/3 − … for −1 < x ≤ 1.
重要展开式：eˣ = 1 + x + x²/2! + x³/3! + …；sin x = x − x³/3! + x⁵/5! − …；cos x = 1 − x²/2! + x⁴/4! − …；ln(1+x) = x − x²/2 + x³/3 − …，−1 < x ≤ 1。
4. Trigonometry | 三角学
Fundamental identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ.
基本恒等式：sin²θ + cos²θ = 1；1 + tan²θ = sec²θ；1 + cot²θ = csc²θ。
Compound angle formulas: 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).
和差角公式：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)。
Double angle: sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ; tan 2θ = 2 tan θ / (1 − tan²θ).
倍角公式：sin 2θ = 2 sin θ cos θ；cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ；tan 2θ = 2 tan θ / (1 − tan²θ)。
Sine and cosine rules: a / sin A = b / sin B = c / sin C = 2R; a² = b² + c² − 2bc cos A.
正弦与余弦定理：a / sin A = b / sin B = c / sin C = 2R；a² = b² + c² − 2bc cos A。
Radian measure: π rad = 180°; arc length s = rθ; sector area A = ½ r²θ.
弧度制：π 弧度 = 180°；弧长 s = rθ；扇形面积 A = ½ r²θ。
Inverse trigonometric functions: sin⁻¹x, cos⁻¹x, tan⁻¹x; ranges typically [−π/2, π/2], [0, π], (−π/2, π/2).
反三角函数：sin⁻¹x，cos⁻¹x，tan⁻¹x；常见值域分别为 [−π/2, π/2]，[0, π]，(−π/2, π/2)。
5. Differentiation | 求导法则
Basic derivatives: 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.
基本导数：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。
Product rule: d/dx [u(x) v(x)] = u′v + uv′.
乘法法则：d/dx [u v] = u′v + u v′。
Quotient rule: d/dx [u/v] = (u′v − uv′) / v².
除法法则：d/dx [u/v] = (u′v − u v′) / v²。
Chain rule: dy/dx = (dy/du) × (du/dx).
链式法则：dy/dx = (dy/du) × (du/dx)。
Implicit differentiation: For F(x, y) = 0, differentiate both sides with respect to x, treating y as a function of x.
隐函数求导：对 F(x, y) = 0 两边关于 x 求导，将 y 视为 x 的函数。
Parametric differentiation: If x = f(t), y = g(t), then dy/dx = (dy/dt) / (dx/dt).
参数方程求导：若 x = f(t), y = g(t)，则 dy/dx = (dy/dt) / (dx/dt)。
6. Integration | 积分法则
Basic integrals: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1); ∫ 1/x dx = ln|x| + C; ∫ eˣ dx = eˣ + C; ∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C; ∫ sec² x dx = tan x + C.
基本积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ −1）；∫ 1/x dx = ln|x| + C；∫ eˣ dx = eˣ + C；∫ sin x dx = −cos x + C；∫ cos x dx = sin x + C；∫ sec² x dx = tan x + C。
Reverse chain rule: ∫ f′(g(x)) g′(x) dx = f(g(x)) + C.
反向链式法则：∫ f′(g(x)) g′(x) dx = f(g(x)) + C。
Integration by substitution: Let u = g(x), then dx = du / g′(x).
换元积分法：令 u = g(x)，则 dx = du / g′(x)。
Integration by parts: ∫ u dv = uv − ∫ v du.
分部积分法：∫ u dv = uv − ∫ v du。
Definite integrals and area: Area under y = f(x) from a to b is ∫ₐᵇ f(x) dx (taking absolute values if below x‑axis).
定积分与面积：曲线 y = f(x) 在 [a, b] 下的面积为 ∫ₐᵇ f(x) dx（若在 x 轴下方需取绝对值）。
Volume of revolution: about x‑axis: V = π ∫ₐᵇ y² dx; about y‑axis: V = π ∫ₐᵇ x² dy.
旋转体体积：绕 x 轴：V = π ∫ₐᵇ y² dx；绕 y 轴：V = π ∫ₐᵇ x² dy。
7. Vectors in 3D | 三维向量
Position vector: r = xi + yj + zk, magnitude |r| = √(x² + y² + z²).
位置向量：r = x i + y j + z k，模长 |r| = √(x² + y² + z²)。
Scalar (dot) product: a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃. Vectors perpendicular if a · b = 0.
数量积（点积）：a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃。若 a · b = 0 则两向量垂直。
Vector (cross) product: a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k; |a × b| = |a||b| sin θ.
向量积（叉积）：a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k；|a × b| = |a||b| sin θ。
Equation of a line: r = a + t d, where a is a point on the line, d is a direction vector, t ∈ ℝ.
直线方程：r = a + t d，a 为直线上一点，d 为方向向量，t ∈ ℝ。
Equation of a plane: r · n = a · n, where n is a normal vector and a is a point on the plane.
平面方程：r · n = a · n，n 为法向量，a 为平面上一点。
Distance from a point to a plane: |(r₀ − a)·n| / |n|.
点到平面的距离：|(r₀ − a)·n| / |n|。
8. Complex Numbers | 复数
Standard form: z = x + iy, where i² = −1. Modulus |z| = √(x² + y²), argument θ = arctan(y/x), adjusted for quadrant.
标准形式：z = x + iy，i² = −1。模 |z| = √(x² + y²)，辐角 θ = arctan(y/x)（按象限调整）。
Polar form: z = r(cos θ + i sin θ), where r = |z|, θ = arg(z).
极坐标形式：z = r(cos θ + i sin θ)，r = |z|，θ = arg(z)。
Euler’s formula: eⁱᶿ = cos θ + i sin θ; thus z = reⁱᶿ.
欧拉公式：eⁱᶿ = cos θ + i sin θ；因此 z = reⁱᶿ。
de Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ).
棣莫弗定理：(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)。
Roots of unity: Solutions of zⁿ = 1 are z = cos(2kπ/n) + i sin(2kπ/n), k = 0, 1, …, n−1.
n次单位根：zⁿ = 1 的解为 z = cos(2kπ/n) + i sin(2kπ/n)，k = 0, 1, …, n−1。
Complex conjugate: z* = x − iy; z z* = |z|².
共轭复数：z* = x − iy；z z* = |z|²。
9. Differential Equations | 微分方程
Separation of variables: For dy/dx = f(x)g(y), write ∫ 1/g(y) dy = ∫ f(x) dx.
分离变量法：对于 dy/dx = f(x)g(y)，化为 ∫ 1/g(y) dy = ∫ f(x) dx。
Integrating factor method: For dy/dx + P(x)y = Q(x), IF = e∫ P dx; then d/dx (y·IF) = Q·IF.
积分因子法：对于 dy/dx + P(x)y = Q(x)，积分因子 IF = e∫ P dx；则 d/dx (y·IF) = Q·IF。
Second-order linear ODE with constant coefficients: a d²y/dx² + b dy/dx + c y = f(x). Auxiliary equation: am² + bm + c = 0. General solution of homogeneous part: y = Ae^(m₁x) + Be^(m₂x) (if distinct real roots); y = (A + Bx)e^(mx) (repeated real); y = e^(αx)(A cos(βx) + B sin(βx)) (complex conjugates m = α ± iβ).
常系数二阶线性微分方程：a d²y/dx² + b dy/dx + c y = f(x)。辅助方程：am² + bm + c = 0。齐次通解：若相异实根 m₁, m₂，y = Ae^(m₁x) + Be^(m₂x)；重根 m，y = (A + Bx)e^(mx)；共轭复根 α ± iβ，y = e^(αx)(A cos(βx) + B sin(βx))。
Particular integral: Assume a trial function based on f(x) (polynomial, exponential, trigonometric) and substitute to find coefficients.
特解：根据 f(x) 设定试函数（多项式、指数、三角），代入求系数。
10. Probability and Combinations | 概率与组合
Permutations: n
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