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

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

This article compiles all the essential formulae, identities, and theorems you need for Year 12 Edexcel Mathematics. Keep this handbook handy for revision and exam preparation, covering Pure, Statistics, and Mechanics topics.

本文汇编了 Year 12 Edexcel 数学所有必备的公式、恒等式和定理。随身携带这本手册，便于复习备考，涵盖纯数学、统计学和力学各主题。

1. Algebraic Rules and Indices | 代数规则与指数

Index Laws: aᵐ × aⁿ = aᵐ⁺ⁿ , aᵐ ÷ aⁿ = aᵐ⁻ⁿ , (aᵐ)ⁿ = aᵐⁿ

指数律：aᵐ × aⁿ = aᵐ⁺ⁿ , aᵐ ÷ aⁿ = aᵐ⁻ⁿ , (aᵐ)ⁿ = aᵐⁿ

Negative and fractional indices: a⁻ⁿ = 1/aⁿ , a¹ᐟⁿ = √[n]{a} , aᵐᐟⁿ = (√[n]{a})ᵐ = √[n]{aᵐ}

负指数与分数指数：a⁻ⁿ = 1/aⁿ , a¹ᐟⁿ = √[n]{a} , aᵐᐟⁿ = (√[n]{a})ᵐ = √[n]{aᵐ}

Expanding brackets: (a + b)(c + d) = ac + ad + bc + bd. Special case: (a + b)² = a² + 2ab + b² , (a − b)² = a² − 2ab + b² , (a + b)(a − b) = a² − b²

展开括号：(a + b)(c + d) = ac + ad + bc + bd。特殊情况：(a + b)² = a² + 2ab + b² , (a − b)² = a² − 2ab + b² , (a + b)(a − b) = a² − b²

Factorising quadratic expressions: x² + bx + c = (x + p)(x + q) where p + q = b and pq = c.

因式分解二次式：x² + bx + c = (x + p)(x + q)，其中 p + q = b，pq = c。

Surds: √(ab) = √a √b , √(a/b) = √a / √b , rationalising denominator: 1/√a = √a / a.

根式：√(ab) = √a √b , √(a/b) = √a / √b，分母有理化：1/√a = √a / a。

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2. Quadratic Functions & the Discriminant | 二次函数与判别式

General form: f(x) = ax² + bx + c, (a ≠ 0). Completed square form: a(x + p)² + q, where vertex is (−p, q).

一般式：f(x) = ax² + bx + c (a ≠ 0)。配方式：a(x + p)² + q，顶点坐标为 (−p, q)。

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

二次公式：x = [−b ± √(b² − 4ac)] / 2a

Discriminant Δ = b² − 4ac. Two distinct real roots if Δ > 0, one repeated real root if Δ = 0, no real roots if Δ < 0.

判别式 Δ = b² − 4ac。Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。

Sum and product of roots: if α and β are roots of ax² + bx + c = 0, then α + β = −b/a , αβ = c/a.

根的和与积：若 α 和 β 是 ax² + bx + c = 0 的根，则 α + β = −b/a，αβ = c/a。

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3. Equations and Inequalities | 方程与不等式

Linear equations: ax + b = 0 → x = −b/a. Simultaneous linear equations can be solved by elimination or substitution.

线性方程：ax + b = 0 → x = −b/a。联立线性方程可通过消元法或代入法求解。

Solving quadratic inequalities: factorise, find critical values, then test intervals. e.g., x² − x − 6 > 0 → (x+2)(x−3) > 0 → x < −2 or x > 3.

解二次不等式：因式分解，找出临界值，然后检验区间。例如 x² − x − 6 > 0 → (x+2)(x−3) > 0 → x < −2 或 x > 3。

Linear inequalities: If a > b then a + c > b + c; if a > b and c > 0 then ac > bc; if a > b and c < 0 then ac < bc. Represent on number line with open/closed circles.

线性不等式：若 a > b 则 a + c > b + c；若 a > b 且 c > 0 则 ac > bc；若 a > b 且 c < 0 则 ac < bc。在数轴上用空/实心圆表示。

Polynomial division and factor theorem: f(a) = 0 ⇔ (x − a) is a factor of f(x).

多项式除法与因式定理：f(a) = 0 ⇔ (x − a) 是 f(x) 的因式。

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4. Straight Lines and Circles | 直线与圆

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

过点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率：m = (y₂ − y₁) / (x₂ − x₁)。

Equation of a straight line: y − y₁ = m (x − x₁), or y = mx + c where m = gradient, c = y-intercept.

直线方程：y − y₁ = m (x − x₁)，或 y = mx + c，其中 m 为斜率，c 为 y 轴截距。

Parallel lines have equal gradients: m₁ = m₂. Perpendicular lines: m₁ × m₂ = −1.

平行线斜率相等：m₁ = m₂。垂直线斜率满足：m₁ × m₂ = −1。

Midpoint of (x₁, y₁) and (x₂, y₂): ((x₁+x₂)/2, (y₁+y₂)/2). Distance between them: √[(x₂−x₁)² + (y₂−y₁)²].

中点坐标：((x₁+x₂)/2, (y₁+y₂)/2)。两点间距：√[(x₂−x₁)² + (y₂−y₁)²]。

Equation of a circle with centre (a, b) and radius r: (x − a)² + (y − b)² = r². The angle in a semicircle is a right angle.

以 (a, b) 为圆心、半径为 r 的圆方程：(x − a)² + (y − b)² = r²。直径所对圆周角为直角。

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5. Trigonometry | 三角学

Exact values for 0°, 30°, 45°, 60°, 90°: sin 30° = ½, cos 60° = ½, tan 45° = 1, etc. Use triangles to recall.

0°、30°、45°、60°、90° 的精确值：sin 30° = ½，cos 60° = ½，tan 45° = 1，利用特殊三角形记忆。

Radian measure: π radians = 180°. To convert degrees to radians multiply by π/180. Arc length s = rθ, sector area = ½ r² θ.

弧度制：π 弧度 = 180°。度转弧度乘以 π/180。弧长 s = rθ，扇形面积 = ½ r² θ。

Trigonometric identities: tan θ = sin θ / cos θ . Pythagorean identity: sin² θ + cos² θ = 1.

三角恒等式：tan θ = sin θ / cos θ，毕达哥拉斯恒等式：sin² θ + cos² θ = 1。

Sine rule: a/sin A = b/sin B = c/sin C (for any triangle). Cosine rule: a² = b² + c² − 2bc cos A.

正弦定理：a/sin A = b/sin B = c/sin C（任意三角形）。余弦定理：a² = b² + c² − 2bc cos A。

Area of triangle: ½ ab sin C. For right‑angled triangle: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj.

三角形面积：½ ab sin C。直角三角形中：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。

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6. Exponentials and Logarithms | 指数与对数

Exponential function: y = aˣ, with a > 0, a ≠ 1. The natural exponential function is y = eˣ where e ≈ 2.718.

指数函数：y = aˣ (a > 0, a ≠ 1)。自然指数函数为 y = eˣ，e ≈ 2.718。

Logarithm definition: if aˣ = b then logₐ b = x. Natural log: ln x = logₑ x.

对数定义：若 aˣ = b，则 logₐ b = x。自然对数：ln x = logₑ x。

Laws of logarithms: logₐ (mn) = logₐ m + logₐ n, logₐ (m/n) = logₐ m − logₐ n, logₐ mᵏ = k logₐ m.

对数律：logₐ (mn) = logₐ m + logₐ n，logₐ (m/n) = logₐ m − logₐ n，logₐ mᵏ = k logₐ m。

Change of base: logₐ b = logₓ b / logₓ a. Special cases: ln(eˣ) = x, e^(ln x) = x.

换底公式：logₐ b = logₓ b / logₓ a。特殊情形：ln(eˣ) = x，e^(ln x) = x。

Solving exponential equations: 2ˣ = 5 → x = log₂5 = ln5 / ln2. Logarithmic equations: ln(x) + ln(x+3) = …

解指数方程：2ˣ = 5 → x = log₂5 = ln5 / ln2。对数方程：ln(x) + ln(x+3) = …

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7. Differentiation | 微分

For y = xⁿ, dy/dx = n xⁿ⁻¹ (n real). Sum rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x).

对 y = xⁿ，dy/dx = n xⁿ⁻¹（n 为实数）。和差法则：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。

Gradient of a curve at a point is given by dy/dx. Equation of tangent at P(x₁, y₁): y − y₁ = m(x − x₁) where m = dy/dx at P.

曲线上一点处的斜率由 dy/dx 给出。点 P(x₁, y₁) 处切线方程：y − y₁ = m(x − x₁)，其中 m 为该点导数。

Stationary points occur where dy/dx = 0. Nature tested using second derivative d²y/dx²: >0 minimum, <0 maximum, or by sign change of first derivative.

驻点满足 dy/dx = 0。利用二阶导数 d²y/dx² 判断：>0 为极小，<0 为极大，或利用一阶导数符号变化。

Derivatives of standard functions: d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x, d/dx (sin x) = cos x, d/dx (cos x) = −sin x.

标准函数求导：d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x，d/dx (sin x) = cos x，d/dx (cos x) = −sin x。

Chain rule: if y = f(g(x)), then dy/dx = f'(g(x)) g'(x). Product rule: d/dx(uv) = u’v + uv’. Quotient rule: d/dx(u/v) = (u’v − uv’)/v².

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

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8. Integration | 积分

Integration as reverse of differentiation: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1. ∫ 1/x dx = ln|x| + C.

积分是微分的逆运算：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，n ≠ −1。∫ 1/x dx = ln|x| + C。

Definite integral ∫ₐᵇ f(x) dx gives the area between the curve y = f(x) and the x‑axis from x=a to x=b (areas below axis count negative).

定积分 ∫ₐᵇ f(x) dx 表示曲线 y = f(x) 与 x 轴在 x=a 到 x=b 之间的面积（x 轴下方为负）。

Area between a curve and the x‑axis: ∫ₐᵇ f(x) dx, and between two curves: ∫ₐᵇ [f(x) − g(x)] dx.

曲线与 x 轴间的面积：∫ₐᵇ f(x) dx；两曲线间面积：∫ₐᵇ [f(x) − g(x)] dx。

Standard integrals: ∫ eˣ dx = eˣ + C, ∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C.

常用积分：∫ eˣ dx = eˣ + C，∫ sin x dx = −cos x + C，∫ cos x dx = sin x + C。

Using integration to find the equation of a curve when given a derivative and a point: integrate to get general form, then substitute coordinates to find constant C.

已知导数及一点求原函数：积分得一般式，代入坐标确定常数 C。

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9. Sequences, Series and Binomial Expansion | 数列、级数与二项式展开

Arithmetic sequence: nth term uₙ = a + (n−1)d, where a = first term, d = common difference. Sum of first n terms: Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l).

等差数列：第 n 项 uₙ = a + (n−1)d，其中 a 为首项，d 为公差。前 n 项和：Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l)。

Geometric sequence: uₙ = a rⁿ⁻¹. Sum of first n terms: Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. Sum to infinity exists if |r| < 1: S∞ = a/(1 − r).

等比数列：uₙ = a rⁿ⁻¹。前 n 项和：Sₙ = a(1 − rⁿ)/(1 − r) (r ≠ 1)。若 |r| < 1，无穷级数和存在：S∞ = a/(1 − r)。

Binomial expansion for (1 + x)ⁿ when n is rational: (1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + … , valid for |x| < 1.

当 n 为有理数时，(1 + x)ⁿ 的二项式展开：(1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + …，|x| < 1 时成立。

For small x, (a + bx)ⁿ can be written as aⁿ(1 + (b/a)x)ⁿ and expanded.

当 x 很小时，(a + bx)ⁿ 可写为 aⁿ(1 + (b/a)x)ⁿ 并展开。

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10. Vectors | 向量

A vector has magnitude and direction. Written as a = (x, y) or xi + yj. Magnitude |a| = √(x² + y²).

向量具有大小和方向。表示为 a = (x, y) 或 xi + yj。模 |a| = √(x² + y²)。

Vector addition: a + b = (x₁+x₂, y₁+y₂). Scalar multiplication: k a = (k x, k y).

向量加法：a + b = (x₁+x₂, y₁+y₂)。数乘：k a = (k x, k y)。

Position vector of point A relative to origin O: OA. Vector AB = OB − OA.

点 A 相对于原点 O 的位置向量：OA。向量 AB = OB − OA。

Two vectors are parallel if a = k b for some scalar k. Collinear points have position vectors satisfying AB = λ AC.

若 a = k b (k 为标量)，则两向量平行。共线点的位置向量满足 AB = λ AC。

Unit vector in direction of a: â = a / |a|. Dot product: a·b = x₁x₂ + y₁y₂ = |a||b| cos θ.

方向 a 的单位向量：â = a / |a|。点积：a·b = x₁x₂ + y₁y₂ = |a||b| cos θ。

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11. Statistics and Probability Basics | 统计与概率基础

Mean of data: x̄ = Σx / n (raw), or Σfx / Σf for grouped data. Median: middle value or use cumulative frequency. Mode: most frequent value.

数据平均数：x̄ = Σx / n（原始数据），或 Σfx / Σf（分组数据）。中位数：中间值或利用累积频率。众数：频数最高的值。

Variance and standard deviation: s² = Σ(x − x̄)² / (n−1) (sample) or σ² = Σ(x − μ)² / n. sd = √(variance).

方差与标准差：样本方差 s² = Σ(x − x̄)² / (n−1)，总体方差 σ² = Σ(x − μ)² / n。标准差为方差的平方根。

Probability: P(event) = number of favourable outcomes / total outcomes. For mutually exclusive events A and B, P(A ∪ B) = P(A) + P(B). For independent events, P(A ∩ B) = P(A)P(B).

概率：P(事件) = 有利结果数 / 总结果数。互斥事件 A、B：P(A ∪ B) = P(A) + P(B)。独立事件：P(A ∩ B) = P(A)P(B)。

Binomial distribution: X ~ B(n, p). P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ, where q = 1 − p. Mean = np, variance = npq.

二项分布：X ~ B(n, p)。P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ，q = 1 − p。均值 = np，方差 = npq。

Mutually exclusive and independent events: understand difference. Tree diagrams help with conditional probabilities.

互斥事件与独立事件：理解区别。树形图有助于处理条件概率。

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12. Mechanics: Kinematics & Newton’s Laws | 力学：运动学与牛顿定律

Constant acceleration equations (SUVAT): v = u + at, s = ut + ½ at², s = vt − ½ at², v² = u² + 2as, s = ½ (u + v) t.

匀加速度运动方程（SUVAT）：v = u + at, s = ut + ½ at², s = vt − ½ at², v² = u² + 2as, s = ½ (u + v) t。

Displacement-time graphs: gradient = velocity. Velocity-time graphs: gradient = acceleration, area under graph = displacement.

位移-时间图：斜率 = 速度。速度-时间图：斜率 = 加速度，图下方面积 = 位移。

Newton’s Second Law: F = ma, where F is resultant force in newtons, m in kg, a in m/s².

牛顿第二定律：F = ma，合力 F 单位牛顿，质量 m 单位千克，加速度 a 单位米/秒²。

Weight: W = mg (g = 9.8 m/s²). Friction: Fₘₐₓ = μR, where R is normal reaction and μ coefficient of friction.

重力：W = mg (g = 9.8 m/s²)。摩擦力：Fₘₐ

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