Year 12 Edexcel Maths: Formula & Theorem Quick-Reference Handbook | Year 12 Edexcel 数学：公式定理速查手册

📚 Year 12 Edexcel Maths: Formula & Theorem Quick-Reference Handbook | Year 12 Edexcel 数学：公式定理速查手册

This quick-reference handbook gathers every essential formula, identity and theorem from the Year 12 Edexcel Pure Mathematics syllabus. Designed for last-minute revision and daily problem-solving, the guide covers algebra, coordinate geometry, trigonometry, calculus, sequences, vectors and more. Each entry is paired with a simplified Chinese translation to support bilingual learners.

本速查手册汇集了 Year 12 Edexcel 纯数学课程中的所有核心公式、恒等式与定理。它既适合考前冲刺，也适合日常练习参考，内容覆盖代数、坐标几何、三角学、微积分、数列、向量等模块。每条公式均配有简体中文解释，方便双语学习者对照记忆。

1. Algebraic Methods and Surds | 代数方法与根式

Manipulating surds and algebraic expressions is the foundation of Pure Mathematics. Below are the key identities and operations you must be fluent in.

根式运算与代数变形是纯数学的基础。以下是你必须熟练掌握的关键恒等式与运算法则。

Surd multiplication and division: √a × √b = √(ab) and √a / √b = √(a/b), provided a, b ≥ 0.

根式乘除：√a × √b = √(ab)，√a / √b = √(a/b)，其中 a, b ≥ 0。

Rationalising denominators: 1/√a = √a / a, and 1/(a + √b) = (a − √b)/(a² − b).

分母有理化：1/√a = √a / a，1/(a + √b) = (a − √b)/(a² − b)。

Expanding brackets: (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², (a + b)(a − b) = a² − b².

括号展开：(a + b)² = a² + 2ab + b²，(a − b)² = a² − 2ab + b²，(a + b)(a − b) = a² − b²。

Factorising: always check for common factors and recognise difference of squares a² − b² = (a − b)(a + b).

因式分解：先提取公因式，并识别平方差 a² − b² = (a − b)(a + b)。

2. Quadratic Equations and the Discriminant | 二次方程与判别式

The quadratic formula and discriminant determine the nature of roots and form the core of equation solving in Year 12.

二次公式与判别式决定了根的性质，是 Year 12 方程求解的核心工具。

Standard form: ax² + bx + c = 0, where a ≠ 0.

标准形式：ax² + bx + c = 0，其中 a ≠ 0。

Quadratic formula: x = [−b ± √(b² − 4ac)] / (2a).

求根公式：x = [−b ± √(b² − 4ac)] / (2a)。

Discriminant: Δ = b² − 4ac. If Δ > 0, two distinct real roots; if Δ = 0, two equal real roots (one repeated root); if Δ < 0, no real roots.

判别式：Δ = b² − 4ac。若 Δ > 0，有两个不等实根；若 Δ = 0，有两个相等实根（重根）；若 Δ < 0，无实根。

Completing the square: x² + bx = (x + b/2)² − (b/2)². This leads to the vertex form a(x − h)² + k.

配方法：x² + bx = (x + b/2)² − (b/2)²。由此可得顶点式 a(x − h)² + k。

3. Laws of Indices and Logarithms | 指数与对数定律

Indices and logarithms are inverse operations. Mastery of their laws is essential for exponentials, growth models and calculus.

指数与对数互为逆运算。掌握它们的定律对于理解指数函数、增长模型以及微积分至关重要。

Index laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ / aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, a^{1/n} = ⁿ√a.

指数定律：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ / aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ，a^{1/n} = ⁿ√a。

Logarithm laws: logₐ(xy) = logₐx + logₐy; logₐ(x/y) = logₐx − logₐy; logₐ(xᵏ) = k logₐx; logₐ1 = 0; logₐa = 1.

对数定律：logₐ(xy) = logₐx + logₐy；logₐ(x/y) = logₐx − logₐy；logₐ(xᵏ) = k logₐx；logₐ1 = 0；logₐa = 1。

Change of base: logₐb = logₓb / logₓa. This is particularly useful with log₁₀ or ln.

换底公式：logₐb = logₓb / logₓa。通常选用常用对数 log₁₀ 或自然对数 ln 进行计算。

4. Coordinate Geometry: Straight Lines and Circles | 坐标几何：直线与圆

Coordinate geometry lets you describe lines and circles algebraically. These formulas are the toolkit for intersection and tangent problems.

坐标几何用代数描述直线与圆。以下公式是解决交点与切线问题的基本工具箱。

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

两点间距离：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 (slope) of a line: m = (y₂ − y₁)/(x₂ − x₁).

直线斜率：m = (y₂ − y₁)/(x₂ − x₁)。

Equation of a line: y − y₁ = m(x − x₁) or y = mx + c, where c is the y-intercept. Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = −1.

直线方程：y − y₁ = m(x − x₁) 或 y = mx + c，其中 c 为 y 轴截距。平行线斜率相等；垂直线满足 m₁ × m₂ = −1。

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

圆方程：圆心 (a, b)，半径 r，则 (x − a)² + (y − b)² = r²。

General circle form: x² + y² + 2gx + 2fy + c = 0, centre (−g, −f), radius √(g² + f² − c).

圆的一般方程：x² + y² + 2gx + 2fy + c = 0，圆心 (−g, −f)，半径 √(g² + f² − c)。

5. Functions and Graphs | 函数与图像

Functions map inputs to outputs. Understanding domain, range, composition and inverse functions is critical for modelling and calculus.

函数将输入映射为输出。理解定义域、值域、复合函数与反函数对建模和微积分至关重要。

Notation: f(x) means ‘the value of the function f at x’. The domain is the set of allowed x-values; the range is the set of possible f(x)-values.

记号：f(x) 表示“函数 f 在 x 处的值”。定义域是允许输入的 x 值集合；值域是所有可能的 f(x) 值集合。

Composite function: (g ∘ f)(x) = g(f(x)). Apply f first, then g.

复合函数：(g ∘ f)(x) = g(f(x))。先作用 f，再作用 g。

Inverse function: f⁻¹(x) reverses the effect of f. Graphically, y = f⁻¹(x) is the reflection of y = f(x) in the line y = x. To find it, write y = f(x), swap x and y, and solve for y.

反函数：f⁻¹(x) 逆转 f 的作用。y = f⁻¹(x) 的图像是 y = f(x) 关于直线 y = x 的对称图形。求法：设 y = f(x)，交换 x 与 y，解出 y。

Transformations: f(x) + a translates a units up; f(x + a) translates −a units left; a f(x) stretches vertically by factor a; f(ax) stretches horizontally by factor 1/a.

图像变换：f(x) + a 向上平移 a 单位；f(x + a) 向左平移 a 单位；a f(x) 纵向伸缩 a 倍；f(ax) 横向伸缩 1/a 倍。

6. Trigonometry Basics and Identities | 三角学基础与恒等式

Trigonometric ratios, exact values, identities and the sine/cosine rules are vital for solving triangles and wave problems.

三角比、特殊角精确值、恒等式以及正弦定理与余弦定理是解三角形与波动问题的关键。

Right-angled definitions: sinθ = opp/hyp, cosθ = adj/hyp, tanθ = opp/adj. For any angle, use the unit circle.

直角三角形定义：sinθ = 对边/斜边，cosθ = 邻边/斜边，tanθ = 对边/邻边。对于任意角，使用单位圆定义。

Exact values to memorise: sin30° = 1/2, sin45° = √2/2, sin60° = √3/2; cos30° = √3/2, cos45° = √2/2, cos60° = 1/2; tan30° = 1/√3, tan45° = 1, tan60° = √3. Equivalent radian measures are π/6, π/4, π/3.

需要熟记的精确值：sin30° = 1/2，sin45° = √2/2，sin60° = √3/2；cos30° = √3/2，cos45° = √2/2，cos60° = 1/2；tan30° = 1/√3，tan45° = 1，tan60° = √3。对应的弧度值为 π/6、π/4、π/3。

Fundamental identity: sin²θ + cos²θ = 1. It leads to 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ, but the basic form is sufficient for Year 12.

基本恒等式：sin²θ + cos²θ = 1。它可导出 1 + tan²θ = sec²θ 等，但 Year 12 阶段主要使用基本形式。

Sine rule: a/sinA = b/sinB = c/sinC, used when you know two angles and a side, or two sides and a non-included angle.

正弦定理：a/sinA = b/sinB = c/sinC，适用于已知两角一边或两边一对角的情况。

Cosine rule: a² = b² + c² − 2bc cosA. It is used for two sides and the included angle, or all three sides.

余弦定理：a² = b² + c² − 2bc cosA，适用于已知两边及其夹角或三边全知的情形。

Area of a triangle: ½ ab sinC.

三角形面积公式：½ ab sinC。

7. Binomial Expansion | 二项展开

The binomial theorem expands expressions of the form (a + b)ⁿ. In Year 12, the focus is on positive integer powers and the use of Pascal’s triangle or nCr.

二项式定理将 (a + b)ⁿ 展开为多项式。Year 12 重点关注正整数次幂以及杨辉三角或 nCr 组合数的运用。

For positive integer n: (a + b)ⁿ = Σ from k=0 to n [ⁿCₖ aⁿ⁻ᵏ bᵏ], where ⁿCₖ = n! / [k!(n − k)!].

对正整数 n：(a + b)ⁿ = Σ{k=0 到 n} [ⁿCₖ aⁿ⁻ᵏ bᵏ]，其中 ⁿCₖ = n! / [k!(n − k)!]。

General term: the (r+1)th term is ⁿCᵣ aⁿ⁻ʳ bʳ, with r starting at 0.

通项公式：第 (r+1) 项为 ⁿCᵣ aⁿ⁻ʳ bʳ，r 从 0 开始计数。

Pascal’s triangle gives the coefficients quickly for small n. Each entry is the sum of the two above it.

对于较小的 n，可用杨辉三角快速得到系数。每个数等于上方两数之和。

The expansion of (1 + x)ⁿ is often used: (1 + x)ⁿ = 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + …

常用 (1 + x)ⁿ 展开：(1 + x)ⁿ = 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + …

8. Exponentials and Logarithms (Natural) | 自然指数与自然对数

Exponential functions with base e and the natural logarithm ln x appear in growth, decay and calculus. Their properties are directly linked to differentiation and integration.

以 e 为底的指数函数与自然对数 ln x 在增长、衰减和微积分中大量出现。它们的性质与微分、积分直接相关。

Definition: e is the unique number such that the derivative of eˣ is eˣ. e ≈ 2.71828.

定义：e 是使得 eˣ 的导数仍为 eˣ 的唯一常数，e ≈ 2.71828。

Natural

Published by TutorHao | Year 12 Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）