Pre-U Edexcel Chinese: Quick Reference to Key Formulas & Theorems | Pre-U Edexcel 中文：公式定理速查手册

📚 Pre-U Edexcel Chinese: Quick Reference to Key Formulas & Theorems | Pre-U Edexcel 中文：公式定理速查手册

This bilingual handbook presents the essential formulas, theorems, and identities commonly examined under the Edexcel Pre-U (and equivalent Advanced Level) subject in Chinese. It is designed for Chinese-speaking learners who need a systematic revision tool with side-by-side English and Chinese explanations. All content covers Pure Mathematics, Mechanics, Statistics, and selected topics from Physics that frequently appear in integrated science papers.

这本中英双语速查手册整理了 Edexcel Pre-U 及同等 A-Level 中文考试中经常考查的核心公式、定理与恒等式。手册面向母语为中文的学生，采用中英对照的方式提供系统化复习，涵盖纯数学、力学、统计以及综合科学试卷中高频出现的物理公式。

1. Basic Algebra & Functions | 基础代数与函数

Mastering algebraic manipulation is fundamental. The quadratic formula solves ax² + bx + c = 0: x = [ -b ± √(b² – 4ac) ] / (2a). The discriminant Δ = b² – 4ac determines the nature of roots: Δ > 0 (two distinct real roots), Δ = 0 (one repeated real root), Δ < 0 (no real roots).

代数运算是基础。一元二次方程 ax² + bx + c = 0 的求根公式为 x = [ -b ± √(b² – 4ac) ] / (2a)。判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。

For functions f(x) and g(x), the composite function (f ∘ g)(x) = f(g(x)) is defined only when the range of g is contained in the domain of f. The inverse function f⁻¹(x) satisfies f(f⁻¹(x)) = x, and its graph is a reflection in the line y = x.

对于函数 f(x) 与 g(x)，复合函数 (f ∘ g)(x) = f(g(x)) 仅在 g 的值域包含于 f 的定义域时有意义。反函数 f⁻¹(x) 满足 f(f⁻¹(x)) = x，其图像关于直线 y = x 对称。

English	中文
Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, aᵐ / aⁿ = aᵐ⁻ⁿ, a⁰ = 1	指数定律：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，aᵐ / aⁿ = aᵐ⁻ⁿ，a⁰ = 1

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2. Coordinate Geometry | 坐标几何

The distance between two points (x₁, y₁) and (x₂, y₂) is d = √[(x₂ – x₁)² + (y₂ – y₁)²]. The midpoint M has coordinates ((x₁+x₂)/2, (y₁+y₂)/2). The gradient of a line through these points is m = (y₂ – y₁) / (x₂ – x₁).

两点 (x₁, y₁) 与 (x₂, y₂) 间的距离公式为 d = √[(x₂ – x₁)² + (y₂ – y₁)²]。中点 M 坐标为 ((x₁+x₂)/2, (y₁+y₂)/2)，过这两点直线的斜率 m = (y₂ – y₁) / (x₂ – x₁)。

Equation of a straight line: y – y₁ = m(x – x₁) (point-slope form) or y = mx + c (slope-intercept form). For perpendicular lines, the product of their gradients equals -1: m₁ × m₂ = -1. The angle θ between two lines satisfies tan θ = |(m₁ – m₂) / (1 + m₁m₂)|.

直线方程：点斜式 y – y₁ = m(x – x₁)，斜截式 y = mx + c。两条直线垂直时斜率乘积为 -1：m₁ × m₂ = -1。两直线夹角 θ 满足 tan θ = |(m₁ – m₂) / (1 + m₁m₂)|。

The equation of a circle with centre (a, b) and radius r is (x – a)² + (y – b)² = r². The general form is x² + y² + 2gx + 2fy + c = 0, centre (-g, -f), radius √(g² + f² – c).

圆心为 (a, b)、半径为 r 的圆的方程为 (x – a)² + (y – b)² = r²。一般式 x² + y² + 2gx + 2fy + c = 0 中，圆心 (-g, -f)，半径 √(g² + f² – c)。

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3. Trigonometry | 三角学

Basic definitions in a right-angled triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. The key identity is sin²θ + cos²θ = 1. Derived forms: 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.

直角三角形中的基本定义：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。核心恒等式为 sin²θ + cos²θ = 1。由此可得 1 + tan²θ = sec²θ 与 1 + cot²θ = cosec²θ。

Sine and cosine rules apply to any triangle. Sine rule: a / sin A = b / sin B = c / sin C = 2R, where R is the circumradius. Cosine rule: a² = b² + c² – 2bc cos A. Area of triangle = ½ab sin C.

正弦定理与余弦定理适用于任意三角形。正弦定理：a / sin A = b / sin B = c / sin C = 2R（R 为外接圆半径）。余弦定理：a² = b² + c² – 2bc cos A。三角形面积 = ½ab sin C。

Radian measure: π radians = 180°. Arc length s = rθ, sector area A = ½r²θ. Compound angle formulas: sin(A±B) = sinA cosB ± cosA sinB; cos(A±B) = cosA cosB ∓ sinA sinB; tan(A±B) = (tanA ± tanB)/(1 ∓ tanA tanB).

弧度制：π 弧度 = 180°。弧长 s = rθ，扇形面积 A = ½r²θ。两角和差公式：sin(A±B) = sinA cosB ± cosA sinB；cos(A±B) = cosA cosB ∓ sinA sinB；tan(A±B) = (tanA ± tanB)/(1 ∓ tanA tanB)。

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4. Calculus — Differentiation | 微积分 — 微分

The derivative of a function f(x) is defined as f'(x) = lim_h→0 [f(x+h) – f(x)] / h. 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) = sec²x.

函数 f(x) 的导数定义为 f'(x) = lim_h→0 [f(x+h) – f(x)] / h。基本导数公式：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) = sec²x。

Product rule: d/dx (uv) = u’v + uv’. Quotient rule: d/dx (u/v) = (u’v – uv’) / v². Chain rule: If y = f(u) and u = g(x), then dy/dx = dy/du × du/dx.

乘法法则：d/dx (uv) = u’v + uv’。除法法则：d/dx (u/v) = (u’v – uv’) / v²。链式法则：若 y = f(u)，u = g(x)，则 dy/dx = dy/du × du/dx。

Second derivative d²y/dx² gives concavity. At a stationary point f'(x)=0: if f”(x) > 0, it is a minimum; if f”(x) < 0, a maximum; if f''(x)=0, investigate further.

二阶导数 d²y/dx² 可判断凹凸性。在驻点 f'(x)=0 处：若 f”(x) > 0 为极小值；f”(x) < 0 为极大值；f''(x)=0 需进一步检验。

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5. Calculus — Integration | 微积分 — 积分

Indefinite integration reverses differentiation. Basic integrals: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n≠-1); ∫ eˣ dx = eˣ + C; ∫ 1/x dx = ln|x| + C; ∫ sin x dx = -cos x + C; ∫ cos x dx = sin x + C; ∫ sec²x dx = tan x + C.

不定积分是微分的逆运算。基本积分公式：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n≠-1)；∫ eˣ dx = eˣ + C；∫ 1/x dx = ln|x| + C；∫ sin x dx = -cos x + C；∫ cos x dx = sin x + C；∫ sec²x dx = tan x + C。

The definite integral ∫ₐᵇ f(x) dx represents the net area between the curve and the x‑axis from x=a to x=b. The Fundamental Theorem of Calculus links differentiation and integration: ∫ₐᵇ f(x) dx = F(b) – F(a), where F'(x)=f(x).

定积分 ∫ₐᵇ f(x) dx 表示曲线与 x 轴在 x=a 到 x=b 之间的代数和面积。微积分基本定理将微分与积分联系起来：∫ₐᵇ f(x) dx = F(b) – F(a)，其中 F'(x)=f(x)。

Integration by substitution: ∫ f(g(x))g'(x) dx = ∫ f(u) du, where u=g(x). Integration by parts: ∫ u dv = uv – ∫ v du. This is particularly useful for products of functions like x cos x or x eˣ.

换元积分法：∫ f(g(x))g'(x) dx = ∫ f(u) du，令 u=g(x)。分部积分法：∫ u dv = uv – ∫ v du，特别适用于形如 x cos x 或 x eˣ 的函数乘积。

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6. Sequences, Series & Binomial Expansion | 数列、级数与二项展开

Arithmetic progression: nth term aₙ = a₁ + (n-1)d; sum of first n terms Sₙ = n/2 [2a₁ + (n-1)d] = n/2 (a₁ + aₙ). Geometric progression: nth term aₙ = a₁ rⁿ⁻¹; sum Sₙ = a₁(1 – rⁿ)/(1 – r) for r≠1. Sum to infinity exists when |r| < 1: S∞ = a₁/(1 - r).

等差数列：第 n 项 aₙ = a₁ + (n-1)d；前 n 项和 Sₙ = n/2 [2a₁ + (n-1)d] = n/2 (a₁ + aₙ)。等比数列：第 n 项 aₙ = a₁ rⁿ⁻¹；当 r≠1 时和 Sₙ = a₁(1 – rⁿ)/(1 – r)。当 |r| < 1 时无穷和存在：S∞ = a₁/(1 - r)。

The binomial theorem for (1 + x)ⁿ where n is a rational number and |x| < 1: (1 + x)ⁿ = 1 + nx + [n(n-1)/2!] x² + [n(n-1)(n-2)/3!] x³ + … . For positive integer n, the expansion is finite and the coefficient of xʳ is ⁿCᵣ = n!/(r!(n-r)!).

当 n 为有理数且 |x| < 1 时，(1 + x)ⁿ 的二项展开为：(1 + x)ⁿ = 1 + nx + [n(n-1)/2!] x² + [n(n-1)(n-2)/3!] x³ + …。若 n 为正整数，展开式有限，xʳ 的系数为 ⁿCᵣ = n!/(r!(n-r)!)。

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7. Exponentials & Logarithms | 指数与对数

The exponential function y = eˣ is its own derivative. Natural logarithm ln x is the inverse: eˡⁿ ˣ = x. Key properties: ln(ab) = ln a + ln b; ln(a/b) = ln a – ln b; ln(aᵏ) = k ln a. Change of base: logₐ b = ln b / ln a.

指数函数 y = eˣ 的导数等于自身。自然对数 ln x 是其反函数，满足 eˡⁿ ˣ = x。主要性质：ln(ab) = ln a + ln b；ln(a/b) = ln a – ln b；ln(aᵏ) = k ln a。换底公式：logₐ b = ln b / ln a。

Exponential growth/decay models: N = N₀ e⁽ᵏᵗ⁾. If k > 0 it is growth; if k < 0 it is decay. Half-life formula: t₁/₂ = ln 2 / |k|. Doubling time: t₂ = ln 2 / k.

指数增长/衰减模型：N = N₀ e⁽ᵏᵗ⁾。k > 0 为增长，k < 0 为衰减。半衰期公式：t₁/₂ = ln 2 / |k|；倍增时间：t₂ = ln 2 / k。

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8. Vectors | 向量

A vector v can be expressed as xi + yj + zk. Magnitude: |v| = √(x² + y² + z²). The dot product a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃ is used to find the angle between vectors. Two vectors are perpendicular if a·b = 0.

向量 v 可表示为 xi + yj + zk。模长：|v| = √(x² + y² + z²)。点积 a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃，用于计算向量间夹角。若 a·b = 0，则两向量垂直。

Parametric equation of a line through point A with direction vector d: r = A + λd. The angle between two lines is the angle between their direction vectors. The shortest distance from a point to a line can be found using cross product in 3D.

过点 A、方向向量为 d 的直线参数方程：r = A + λd。两直线夹角等于方向向量夹角。点到直线的最短距离在三维空间中可用叉积计算。

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9. Probability & Statistics | 概率与统计

P(A ∪ B) = P(A) + P(B) – P(A ∩ B). For independent events, P(A ∩ B) = P(A)P(B). Conditional probability: P(A|B) = P(A ∩ B) / P(B). Bayes’ theorem: P(A|B) = P(B|A)P(A) / P(B).

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。对于独立事件，P(A ∩ B) = P(A)P(B)。条件概率：P(A|B) = P(A ∩ B) / P(B)。贝叶斯定理：P(A|B) = P(B|A)P(A) / P(B)。

For a discrete random variable X with probabilities p(x), the expected value E(X) = Σ x p(x), and variance Var(X) = E(X²) – [E(X)]². For the Binomial distribution B(n, p), mean μ = np, variance σ² = np(1-p).

对于概率为 p(x) 的离散随机变量 X，期望 E(X) = Σ x p(x)，方差 Var(X) = E(X²) – [E(X)]²。二项分布 B(n, p) 的均值 μ = np，方差 σ² = np(1-p)。

The Normal distribution N(μ, σ²) is defined by the probability density function f(x) = (1/(σ√(2π))) e⁻⁽ˣ⁻µ⁾²/⁽²σ²⁾. Standardization: Z = (X – μ)/σ ~ N(0,1). Use Z-tables for probability calculations.

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⁾。标准化：Z = (X – μ)/σ 服从 N(0,1)，借助 Z 表计算概率。

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10. Mechanics | 力学

Kinematics equations for constant acceleration a: v = u + at; s = ut + ½at²; v² = u² + 2as; s = ½(u + v)t. These apply in one dimension with uniform acceleration like free fall under gravity (g = 9.8 m/s²).

匀变速直线运动公式：v = u + at；s = ut + ½at²；v² = u² + 2as；s = ½(u + v)t。适用于重力加速度 g = 9.8 m/s² 的自由落体等场景。

Newton’s laws: First – inertia; Second – F = ma (net force = mass × acceleration); Third – action-reaction. Weight W = mg. Friction f ≤ μR, where R is the normal reaction and μ is the coefficient of friction.

牛顿运动定律：第一定律（惯性）；第二定律 F = ma（合外力 = 质量 × 加速度）；第三定律（作用与反作用）。重量 W = mg。摩擦力 f ≤ μR，其中 R 为法向反作用力，μ 为摩擦系数。

For projectile motion launched with speed u at angle θ to horizontal: horizontal range R = (u² sin 2θ)/g; maximum height H = (u² sin²θ)/(2g); time of flight T = (2u sin θ)/g. Momentum p = mv; impulse = change in momentum = FΔt.

斜抛运动初速度 u、仰角 θ：水平射程 R = (u² sin 2θ)/g；最大高度 H = (u² sin²θ)/(2g)；飞行时间 T = (2u sin θ)/g。动量 p = mv；冲量 = 动量变化 = FΔt。

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11. Physics & Applied Formulas | 物理应用公式

Ideal gas law: pV = nRT, where p is pressure, V volume, n number of moles, R = 8.31 J/(mol·K). Kinetic energy: Eₖ = ½mv². Gravitational potential energy: Eₚ = mgh. Work done: W = Fd cos θ. Power: P = W/t = Fv.

理想气体状态方程：pV = nRT，R = 8.31 J/(mol·K)。动能：Eₖ = ½mv²。重力势能：Eₚ = mgh。功：W = Fd cos θ。功率：P = W/t = Fv。

Ohm’s law: V = IR. Resistance in series: Rₜ = R₁ + R₂ + … ; in parallel: 1/Rₜ = 1/R₁ + 1/R₂ + … . Coulomb’s law: F = kQ₁Q₂ / r². These relationships often appear in the mechanics–electricity hybrid problems within the Chinese curriculum.

欧姆定律：V = IR。电阻串联：Rₜ = R₁ + R₂ + …；并联：1/Rₜ = 1/R₁ + 1/R₂ + …。库仑定律：F = kQ₁Q₂ / r²。这些关系常出现在中文课程中的力学-电学综合题里。

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12. Final Review Tips | 复习策略

Pay attention to units: convert all quantities to SI before substituting into formulas. Memorise the standard derivatives and integrals, but always verify with the formula booklet provided in the exam. Practise applying the chain rule and integration by parts in unfamiliar contexts.

注意单位换算：代入公式前先将所有量统一为国际单位制。牢记标准导数与积分公式，但务必结合考试提供的公式手册核对。多练习链式法则与分部积分法在不同背景下的应用。

For statistics, focus on choosing the correct distribution and interpreting Z-values correctly. In mechanics, drawing a clear free‑body diagram is often the key to setting up equations. This quick‑reference is a supplement to, not a replacement for, thorough understanding of concepts.

统计部分要重点区分分布类型并正确解读 Z 值。力学中清晰的受力分析图往往是列对方程的关键。本速查手册是概念理解的补充，不能替代对原理的透彻学习。

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