📚 Year 12 OCR Mathematics: Formula & Theorem Quick Reference Handbook | Year 12 OCR 数学：公式定理速查手册
This comprehensive quick reference handbook covers all essential formulas and theorems from the OCR Year 12 Mathematics syllabus. It includes pure mathematics, statistics, and mechanics. Designed for efficient revision, each section presents key results in a clear bilingual format, helping you to master the concepts required for success in your AS-level exams.
这份全面的速查手册涵盖了 OCR Year 12 数学课程中所有必要的公式和定理，包括纯数学、统计与力学。手册专为高效复习而设计，每个部分都以清晰的中英对照形式呈现关键结论，帮助您掌握 AS 考试所需的概念。
1. Laws of Algebra and Indices | 代数法则与指数律
For any real numbers a, b and integers m, n: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0), (aᵐ)ⁿ = aᵐⁿ, (ab)ⁿ = aⁿbⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ. Surds follow √(ab) = √a × √b and √(a/b) = √a/√b.
对于任意实数 a, b 及整数 m, n：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ（a ≠ 0），(aᵐ)ⁿ = aᵐⁿ，(ab)ⁿ = aⁿbⁿ，a⁰ = 1（a ≠ 0），a⁻ⁿ = 1/aⁿ。根式遵循 √(ab) = √a × √b 及 √(a/b) = √a/√b。
2. Quadratic Functions and Equations | 二次函数与方程
The solutions of ax² + bx + c = 0 are given by the quadratic formula:
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. Completing the square: ax² + bx + c = a(x + b/(2a))² – (b² – 4ac)/(4a).
方程 ax² + bx + c = 0 的解由二次公式给出：
x = (-b ± √(b² – 4ac)) / (2a)
判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 ⇒ 两个相异实根；Δ = 0 ⇒ 一个重实根；Δ < 0 ⇒ 无实根。配方法：ax² + bx + c = a(x + b/(2a))² – (b² – 4ac)/(4a)。
3. Coordinate Geometry and Straight Lines | 坐标几何与直线
The distance between points (x₁, y₁) and (x₂, y₂) is √[(x₂ – x₁)² + (y₂ – y₁)²]. The midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2). The gradient of a line through these points is m = (y₂ – y₁)/(x₂ – x₁). Equation of a line: y – y₁ = m(x – x₁), or y = mx + c, where c is the y-intercept. For two lines with gradients m₁ and m₂: parallel if m₁ = m₂; perpendicular if m₁m₂ = –1.
两点 (x₁, y₁) 和 (x₂, y₂) 之间的距离为 √[(x₂ – x₁)² + (y₂ – y₁)²]。中点为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。过这两点的直线斜率为 m = (y₂ – y₁)/(x₂ – x₁)。直线方程：y – y₁ = m(x – x₁) 或 y = mx + c，其中 c 是 y 轴截距。对于斜率为 m₁ 和 m₂ 的两条直线：若 m₁ = m₂ 则平行；若 m₁m₂ = –1 则垂直。
4. Sequences and Series | 数列与级数
Arithmetic sequence: nth term uₙ = a + (n – 1)d, sum of n terms Sₙ = n/2 [2a + (n – 1)d] or Sₙ = n/2 (a + l), where l is the last term. Geometric sequence: uₙ = arⁿ⁻¹, sum Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1. Sum to infinity for |r| < 1: S∞ = a/(1 – r). Standard sums: ∑ᵢ₌₁ⁿ r = n(n+1)/2, ∑ᵢ₌₁ⁿ r² = n(n+1)(2n+1)/6, ∑ᵢ₌₁ⁿ r³ = [n(n+1)/2]².
等差数列：第 n 项 uₙ = a + (n – 1)d，前 n 项和 Sₙ = n/2 [2a + (n – 1)d] 或 Sₙ = n/2 (a + l)，其中 l 为末项。等比数列：uₙ = arⁿ⁻¹，和 Sₙ = a(1 – rⁿ)/(1 – r)，r ≠ 1。当 |r| < 1 时的无穷和为 S∞ = a/(1 – r)。标准求和：∑ᵢ₌₁ⁿ r = n(n+1)/2，∑ᵢ₌₁ⁿ r² = n(n+1)(2n+1)/6，∑ᵢ₌₁ⁿ r³ = [n(n+1)/2]²。
5. Trigonometry | 三角学
Definitions: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = sin θ / cos θ. Exact values: sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2; cos 30° = √3/2, cos 45° = √2/2, cos 60° = ½; tan 30° = 1/√3, tan 45° = 1, tan 60° = √3. Identities: tan θ = sin θ / cos θ, sin²θ + cos²θ = 1. Sine rule: a/sin A = b/sin B = c/sin C. Cosine rule: a² = b² + c² – 2bc cos A. Area of triangle: ½ab sin C. Radian measure: π rad = 180°, arc length s = rθ, sector area A = ½r²θ.
定义：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = sin θ / cos θ。特殊值：sin 30° = ½，sin 45° = √2/2，sin 60° = √3/2；cos 30° = √3/2，cos 45° = √2/2，cos 60° = ½；tan 30° = 1/√3，tan 45° = 1，tan 60° = √3。恒等式：tan θ = sin θ / cos θ，sin²θ + cos²θ = 1。正弦定理：a/sin A = b/sin B = c/sin C。余弦定理：a² = b² + c² – 2bc cos A。三角形面积：½ab sin C。弧度制：π 弧度 = 180°，弧长 s = rθ，扇形面积 A = ½r²θ。
6. Exponentials and Logarithms | 指数与对数
For a > 0, a ≠ 1: y = aˣ ⇔ x = logₐ y. Natural logarithm: ln x = logₑ x, with e ≈ 2.718. Laws: logₐ (xy) = logₐ x + logₐ y, logₐ (x/y) = logₐ x – logₐ y, logₐ (xᵏ) = k logₐ x. Special cases: logₐ 1 = 0, logₐ a = 1. The function eˣ is its own derivative; ln x differentiates to 1/x. Growth/decay model: N = N₀ e^(kt).
对于 a > 0，a ≠ 1：y = aˣ ⇔ x = logₐ y。自然对数：ln x = logₑ x，e ≈ 2.718。运算律：logₐ (xy) = logₐ x + logₐ y，logₐ (x/y) = logₐ x – logₐ y，logₐ (xᵏ) = k logₐ x。特例：logₐ 1 = 0，logₐ a = 1。函数 eˣ 的导数为其本身；ln x 的导数为 1/x。增长/衰减模型：N = N₀ e^(kt)。
7. Differentiation | 微分
The derivative of f(x) = xⁿ is f'(x) = nxⁿ⁻¹. General rules: (u ± v)’ = u’ ± v’, (ku)’ = k u’. The gradient of a curve at point (x₁, y₁) is f'(x₁). Equation of tangent: y – y₁ = m(x – x₁); normal gradient = –1/m. For exponentials: d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x. For trig: d/dx (sin x) = cos x, d/dx (cos x) = –sin x, d/dx (tan x) = sec² x. Chain rule: dy/dx = dy/du × du/dx. Product rule: (uv)’ = u’v + uv’. Quotient rule: (u/v)’ = (u’v – uv’)/v².
f(x) = xⁿ 的导数为 f'(x) = nxⁿ⁻¹。一般法则：(u ± v)’ = u’ ± v’，(ku)’ = k u’。曲线在点 (x₁, y₁) 处的斜率为 f'(x₁)。切线方程：y – y₁ = m(x – x₁)；法线斜率为 –1/m。指数函数：d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x。三角：d/dx (sin x) = cos x，d/dx (cos x) = –sin x，d/dx (tan x) = sec² x。链式法则：dy/dx = dy/du × du/dx。乘法法则：(uv)’ = u’v + uv’。除法法则：(u/v)’ = (u’v – uv’)/v²。
8. Integration | 积分
Integration is the reverse of differentiation. ∫ 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. Definite integral ∫ₐᵇ f(x) dx gives the area between the curve and the x-axis. Area between two curves: ∫ (upper – lower) dx, within limits of intersection. For equations of the form dy/dx = g(x), integrating yields the general solution.
积分是微分的逆运算。∫ 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。定积分 ∫ₐᵇ f(x) dx 给出曲线与 x 轴之间的面积。两曲线间面积：∫ (上方函数 – 下方函数) dx，积分限取交点范围。对于形如 dy/dx = g(x) 的方程，积分可得通解。
9. Vectors | 向量
A vector can be expressed in component form (ai + bj) or as a column vector. Magnitude: |ai + bj| = √(a² + b²). Unit vector = v/|v|. Addition: (a₁i + b₁j) + (a₂i + b₂j) = (a₁+a₂)i + (b₁+b₂)j. Scalar product: v·w = |v||w| cos θ, where θ is the angle between them; also v·w = a₁a₂ + b₁b₂. Two vectors are perpendicular if v·w = 0. Position vector of the midpoint of AB: (OA + OB)/2.
向量可用分量形式 (ai + bj) 或列向量表示。模长：|ai + bj| = √(a² + b²)。单位向量 = v/|v|。加法：(a₁i + b₁j) + (a₂i + b₂j) = (a₁+a₂)i + (b₁+b₂)j。数量积：v·w = |v||w| cos θ，其中 θ 为两向量夹角；同时 v·w = a₁a₂ + b₁b₂。若 v·w = 0，则两向量垂直。线段 AB 中点的位置向量为 (OA + OB)/2。
10. Statistical Measures and Probability | 统计量度与概率
Mean of data: x̄ = ∑x/n. For grouped data: x̄ = ∑fx/∑f. Variance σ² = ∑(x – x̄)²/n = ∑x²/n – x̄². Probability: P(A∪B) = P(A) + P(B) – P(A∩B). Mutually exclusive: P(A∩B) = 0. Independent events: P(A∩B) = P(A)P(B). Binomial distribution X ~ B(n, p): P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ, where ⁿCᵣ = n!/(r!(n – r)!). Mean = np, variance = np(1 – p). For hypothesis testing: critical region determined by significance level.
数据均值：x̄ = ∑x/n。分组数据：x̄ = ∑fx/∑f。方差 σ² = ∑(x – x̄)²/n = ∑x²/n – x̄²。概率：P(A∪B) = P(A) + P(B) – P(A∩B)。互斥事件：P(A∩B) = 0。独立事件：P(A∩B) = P(A)P(B)。二项分布 X ~ B(n, p)：P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ，其中 ⁿCᵣ = n!/(r!(n – r)!)。均值 = np，方差 = np(1 – p)。假设检验中，临界域由显著性水平确定。
11. Kinematics and Newton’s Laws | 运动学与牛顿定律
Constant acceleration formulae: v = u + at, s = ut + ½at², v² = u² + 2as, s = ½(u + v)t. Displacement, velocity, acceleration: v = ds/dt, a = dv/dt = d²s/dt². Newton’s second law: F = ma. Weight W = mg (g = 9.8 m/s²). For equilibrium, resultant force = 0. Frictional force F ≤ μR, where μ is the coefficient of friction and R is the normal reaction. Momentum p = mv; impulse = change in momentum.
匀加速运动公式：v = u + at，s = ut + ½at²，v² = u² + 2as，s = ½(u + v)t。位移、速度、加速度关系：v = ds/dt，a = dv/dt = d²s/dt²。牛顿第二定律：F = ma。重力 W = mg（g = 9.8 米/秒²）。平衡时合力为零。摩擦力 F ≤ μR，其中 μ 为摩擦系数，R 为法向反作用力。动量 p = mv；冲量 = 动量的变化。
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