Year 12 OCR Mathematics: Formula & Theorem Quick Reference Guide | Year 12 OCR 数学：公式定理速查手册

📚 Year 12 OCR Mathematics: Formula & Theorem Quick Reference Guide | Year 12 OCR 数学：公式定理速查手册

Welcome to the ultimate quick reference guide for Year 12 OCR Mathematics. This handbook consolidates the essential formulas, theorems, and key concepts from Pure Mathematics, Mechanics, and Statistics topics covered in the AS-level syllabus. Use it for rapid revision, homework support, and exam preparation. Each section presents the core facts in clear English followed by the Chinese translation, ensuring you master both terminology and application.

欢迎使用 Year 12 OCR 数学终极速查手册。本手册汇集了纯数学、力学和统计学 AS 阶段大纲中的核心公式、定理和关键概念，适合快速复习、完成作业和备考。每个部分先以清晰的英文呈现核心内容，再给出中文翻译，帮助你同时掌握术语与应用。

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1. Algebraic Laws and Indices | 代数法则与指数

Algebraic laws: For all real numbers a, b, c:

代数法则：对所有实数 a, b, c：

Commutative law: a + b = b + a, ab = ba.

交换律：a + b = b + a，ab = ba。

Associative law: (a + b) + c = a + (b + c), (ab)c = a(bc).

结合律：(a + b) + c = a + (b + c)，(ab)c = a(bc)。

Distributive law: a(b + c) = ab + ac.

分配律：a(b + c) = ab + ac。

Index laws for a > 0, m, n real:

指数法则（a > 0，m, n 为实数）：

aᵐ × aⁿ = a⁽ᵐ⁺ⁿ⁾

aᵐ × aⁿ = a⁽ᵐ⁺ⁿ⁾

aᵐ ÷ aⁿ = a⁽ᵐ⁻ⁿ⁾

aᵐ ÷ aⁿ = a⁽ᵐ⁻ⁿ⁾

(aᵐ)ⁿ = a⁽ᵐⁿ⁾

(aᵐ)ⁿ = a⁽ᵐⁿ⁾

a⁰ = 1 (a ≠ 0)

a⁰ = 1 (a ≠ 0)

a⁻ⁿ = 1 / aⁿ

a⁻ⁿ = 1 / aⁿ

a^(1/n) = ⁿ√a (the nth root of a)

a^(1/n) = ⁿ√a（a 的 n 次方根）

To rationalise a denominator like 1/√a, multiply numerator and denominator by √a. For 1/(√a + √b), use the conjugate (√a – √b).

有理化分母，例如 1/√a，分子分母同乘 √a；对于 1/(√a + √b)，用共轭根式 (√a – √b)。

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

Standard quadratic equation: ax² + bx + c = 0, a ≠ 0.

标准二次方程：ax² + bx + c = 0，a ≠ 0。

The solutions are given by the quadratic formula:

解由二次公式给出：

x = [-b ± √(b² – 4ac)] / (2a)

x = [-b ± √(b² – 4ac)] / (2a)

Discriminant Δ = b² – 4ac determines the nature of roots:

判别式 Δ = b² – 4ac 决定根的性质：

If Δ > 0: two distinct real roots. If Δ = 0: one repeated real root. If Δ < 0: no real roots (two complex roots).

若 Δ > 0，有两个相异实根；若 Δ = 0，有一个重根（相等实根）；若 Δ < 0，无实根。

Completing the square transforms ax²+bx+c into a(x + p)² + q, where p = b/(2a) and q = c – (b²/(4a)). The vertex is (-p, q).

配方法将 ax²+bx+c 化为 a(x + p)² + q，其中 p = b/(2a)，q = c – (b²/(4a))，顶点坐标为 (-p, q)。

Sum of roots: α + β = -b/a; product of roots: αβ = c/a.

根的和：α + β = -b/a；根的积：αβ = c/a。

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3. Coordinate Geometry & Straight Lines | 坐标几何与直线

Distance between two points (x₁, y₁) and (x₂, y₂):

两点 (x₁, y₁) 与 (x₂, y₂) 间距离：

d = √[(x₂ – x₁)² + (y₂ – y₁)²]

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 forms: y = mx + c, point-slope y – y₁ = m(x – x₁), general form ax + by + c = 0.

直线方程形式：斜截式 y = mx + c，点斜式 y – y₁ = m(x – x₁)，一般式 ax + by + c = 0。

Parallel lines have equal gradients: m₁ = m₂.

平行线斜率相等：m₁ = m₂。

Perpendicular lines satisfy m₁ × m₂ = -1 (unless one is vertical).

垂直线满足 m₁ × m₂ = -1（除非其中一条为垂直线）。

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4. Trigonometry Basics & Identities | 三角学基础与恒等式

In a right-angled triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent (SOH CAH TOA).

于直角三角形中：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边（SOH CAH TOA）。

Radians: π rad = 180°. Arc length s = rθ, sector area A = ½ r²θ (θ in radians).

弧度制：π rad = 180°。弧长 s = rθ，扇形面积 A = ½ r²θ（θ 以弧度为单位）。

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