A-Level CCEA Mathematics Formula Handbook | A-Level CCEA 数学公式汇总手册

📚 A-Level CCEA Mathematics Formula Handbook | A-Level CCEA 数学公式汇总手册

This handbook provides a concise summary of the essential formulas required for the CCEA A-Level Mathematics syllabus. Covering pure mathematics, mechanics, and statistics, these formulas serve as a quick reference for exam preparation and problem-solving.

本手册汇总了 CCEA A-Level 数学课程大纲所需的核心公式，涵盖纯数学、力学和统计，可作为备考与解题的快速参考。

1. Algebra and Polynomials | 代数与多项式

Key algebraic formulas underpin almost every topic in A-Level mathematics. Memorising these will speed up equation solving, factorisation, and series expansions.

关键代数公式是 A-Level 数学几乎所有主题的基础。熟记它们能加快解方程、因式分解和级数展开的速度。

For a quadratic equation ax² + bx + c = 0, the roots are given by x = [−b ± √(b² − 4ac)] / 2a

对于二次方程 ax² + bx + c = 0，求根公式为 x = [−b ± √(b² − 4ac)] / 2a

The discriminant Δ = b² − 4ac determines the nature of the roots.

判别式 Δ = b² − 4ac 决定根的性质。

Factor theorem: (x − a) is a factor of f(x) if and only if f(a) = 0.

因式定理： (x − a) 是 f(x) 的因式当且仅当 f(a) = 0。

Remainder theorem: When f(x) is divided by (x − a), the remainder is f(a).

余式定理：当 f(x) 除以 (x − a) 时，余式为 f(a)。

Binomial expansion ( |x| < 1 ): (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + …

二项式展开 ( |x| < 1 )： (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + …

General binomial coefficient: ⁿCᵣ = n! / [r!(n−r)!]

通用二项式系数： ⁿCᵣ = n! / [r!(n−r)!]

2. Coordinate Geometry | 坐标几何

Coordinate geometry formulas describe lines, circles, and their intersections. Mastering gradient, midpoint, and distance relations is crucial for graph sketching and algebraic geometry problems.

坐标几何公式描述直线、圆及其交点。掌握斜率、中点和距离关系对于图形绘制和代数几何问题至关重要。

Distance between two points (x₁,y₁) and (x₂,y₂): d = √[(x₂−x₁)² + (y₂−y₁)²]

两点 (x₁,y₁) 与 (x₂,y₂) 的距离： d = √[(x₂−x₁)² + (y₂−y₁)²]

Midpoint: M = ((x₁+x₂)/2 , (y₁+y₂)/2)

中点： M = ((x₁+x₂)/2 , (y₁+y₂)/2)

Gradient of line through (x₁,y₁) and (x₂,y₂): m = (y₂ − y₁) / (x₂ − x₁)

过两点直线的斜率： m = (y₂ − y₁) / (x₂ − x₁)

Equation of a straight line: y − y₁ = m(x − x₁) or y = mx + c

直线方程： y − y₁ = m(x − x₁) 或 y = mx + c

Equation of a circle centre (a,b) radius r: (x − a)² + (y − b)² = r²

圆心 (a,b)、半径 r 的圆的方程： (x − a)² + (y − b)² = r²

Parametric equations: for a circle, x = a + r cos θ, y = b + r sin θ

参数方程：对于圆， x = a + r cos θ, y = b + r sin θ

3. Trigonometry | 三角学

Trigonometric identities and radian measure are fundamental for calculus and solving geometric problems. The following relationships allow simplification of expressions and equations.

三角恒等式与弧度制是微积分和几何问题求解的基础。以下关系可用于简化表达式和方程。

Basic identities: tan θ = sin θ / cos θ ; sin²θ + cos²θ = 1

基本恒等式： tan θ = sin θ / cos θ ； sin²θ + cos²θ = 1

Compound angle formulae: sin(A ± B) = sin A cos B ± cos A sin 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

倍角公式： sin 2θ = 2 sin θ cos θ ； cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ

Radian measure: 180° = π rad ; sector area = ½ r²θ ; arc length = rθ

弧度制： 180° = π rad ；扇形面积 = ½ r²θ ；弧长 = rθ

4. Exponentials and Logarithms | 指数与对数

Exponential and logarithmic functions model growth and decay; their manipulation rules are essential for solving equations and calculus operations.

指数函数和对数函数用于描述增长与衰减；其运算法则是解方程与微积分运算的关键。

Index laws: aᵐ × aⁿ = aᵐ⁺ⁿ ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ ; (aᵐ)ⁿ = aᵐⁿ ; a⁻ⁿ = 1/aⁿ

指数法则： aᵐ × aⁿ = aᵐ⁺ⁿ ； aᵐ ÷ aⁿ = aᵐ⁻ⁿ ； (aᵐ)ⁿ = aᵐⁿ ； a⁻ⁿ = 1/aⁿ

Logarithm laws: logₐ(xy) = logₐx + logₐy ; logₐ(x/y) = logₐx − logₐy ; logₐ(xᵏ) = k logₐx

对数法则： logₐ(xy) = logₐx + logₐy ； logₐ(x/y) = logₐx − logₐy ； logₐ(xᵏ) = k logₐx

Change of base: logₐb = log_c b / log_c a

换底公式： logₐb = log_c b / log_c a

Natural logarithm: ln x means log base e; eˣ and ln x are inverse functions.

自然对数： ln x 表示以 e 为底的对数； eˣ 与 ln x 互为反函数。

Growth/decay model: N = N₀ eᵏᵗ

增长/衰减模型： N = N₀ eᵏᵗ

5. Sequences and Series | 数列与级数

Sequences and series appear in pure mathematics and applied contexts. Understanding arithmetic and geometric progressions, along with binomial series, is essential for the CCEA syllabus.

数列与级数在纯数学和应用情境中均有出现。理解等差数列、等比数列以及二项式级数对 CCEA 考纲至关重要。

Arithmetic progression: nth term uₙ = a + (n−1)d ; sum to n terms Sₙ = n/2 [2a + (n−1)d]

等差数列：第 n 项 uₙ = a + (n−1)d ；前 n 项和 Sₙ = n/2 [2a + (n−1)d]

Geometric progression: nth term uₙ = arⁿ⁻¹ ; sum to n terms Sₙ = a(1−rⁿ) / (1−r) for |r|≠1

等比数列：第 n 项 uₙ = arⁿ⁻¹ ；前 n 项和 Sₙ = a(1−rⁿ) / (1−r) (|r|≠1)

Infinite sum for |r|<1: S∞ = a / (1−r)

|r|<1 时的无穷和： S∞ = a / (1−r)

Binomial series valid for |x|<1, |x|<|b/a|: (a + bx)ⁿ = aⁿ[1 + n(bx/a) + n(n−1)(bx/a)²/2! + …]

适用于 |x|<1，|x|<|b/a| 的二项式级数： (a + bx)ⁿ = aⁿ[1 + n(bx/a) + n(n−1)(bx/a)²/2! + …]

6. Differentiation | 微分

Differentiation gives the gradient of a curve and rates of change. CCEA requires fluency with standard derivatives, rules, and parametric forms.

微分给出曲线的斜率和变化率。CCEA 要求熟练掌握基本导数、运算法则和参数形式。

Standard derivatives: d/dx (xⁿ) = n xⁿ⁻¹ ; d/dx (sin x) = cos x ; d/dx (cos x) = −sin x ; d/dx (eˣ) = eˣ ; d/dx (ln x) = 1/x

基本导数： d/dx (xⁿ) = n xⁿ⁻¹ ； d/dx (sin x) = cos x ； d/dx (cos x) = −sin x ； d/dx (eˣ) = eˣ ； d/dx (ln x) = 1/x

Product rule: d/dx (uv) = u’v + uv’

乘积法则： d/dx (uv) = u’v + uv’

Quotient rule: d/dx (u/v) = (u’v − uv’) / v²

商法则： 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

链式法则：若 y = f(u) 且 u = g(x)，则 dy/dx = dy/du × du/dx

Parametric differentiation: given x = f(t), y = g(t), dy/dx = (dy/dt) / (dx/dt)

参数微分：给定 x = f(t), y = g(t)， dy/dx = (dy/dt) / (dx/dt)

Second derivative: d²y/dx² = d/dx (dy/dx)

二阶导数： d²y/dx² = d/dx (dy/dx)

7. Integration | 积分

Integration reverses differentiation and is used to find areas, volumes, and solve differential equations. Memorising standard integrals and techniques is vital.

积分是微分的逆运算，用于计算面积、体积以及求解微分方程。记忆基本积分和技巧至关重要。

Standard 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

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

Integration by parts: ∫ u dv = uv − ∫ v du or ∫ u v’ dx = uv − ∫ u’ v dx

分部积分： ∫ u dv = uv − ∫ v du 或 ∫ u v’ dx = uv − ∫ u’ v dx

Substitution (reverse chain rule): ∫ f(g(x)) g'(x) dx = ∫ f(u) du with u = g(x)

代换积分（逆链式法则）： ∫ f(g(x)) g'(x) dx = ∫ f(u) du，其中 u = g(x)

Definite integral and area: Area between curve y = f(x) and x‑axis from a to b is ∫[a,b] |f(x)| dx

定积分与面积：曲线 y = f(x) 与 x 轴在 a 到 b 之间的面积为 ∫[a,b] |f(x)| dx

Volume of revolution about x‑axis: V = π ∫[a,b] y² dx

绕 x 轴旋转体积： V = π ∫[a,b] y² dx

8. Vectors in 2D and 3D | 二维与三维向量

Vectors describe both magnitude and direction, forming the backbone of geometry in three dimensions and mechanics. Core operations include addition, scalar product, and finding magnitudes.

向量兼具大小与方向，是三维几何和力学的基础。核心运算包括加法、点乘与求模。

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A-Level CCEA Mathematics Formula Handbook | A-Level CCEA 数学公式汇总手册