📚 Year 13 AQA Mathematics: Quick Reference Formula & Theorem Handbook | Year 13 AQA 数学：公式定理速查手册
This comprehensive reference is designed for Year 13 students following the AQA A-level Mathematics specification. It summarises essential formulae, theorems, and key results across pure mathematics, statistics, and mechanics, serving as a quick revision aid for exam preparation.
本速查手册专为学习AQA A-level数学课程的Year 13学生设计，汇总了纯数学、统计和力学中的核心公式、定理与关键结论，是备考复习的便捷参考资料。
1. Algebra and Functions | 代数与函数
Laws of Indices: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, a¹/ⁿ = ⁿ√a.
指数法则： aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁻ⁿ = 1/aⁿ，a¹/ⁿ = ⁿ√a。
Completing the Square: ax² + bx + c = a(x + b/(2a))² + (c – b²/(4a)).
配方法： ax² + bx + c = a(x + b/(2a))² + (c – b²/(4a))。
Discriminant of ax² + bx + c = 0: Δ = b² – 4ac. If Δ > 0, two distinct real roots; Δ = 0, repeated root; Δ < 0, no real roots.
二次方程判别式： Δ = b² – 4ac。若 Δ > 0，有两个不等实根；Δ = 0，有重根；Δ < 0，无实根。
Factor Theorem: For polynomial f(x), if f(a) = 0 then (x – a) is a factor.
因式定理： 对多项式 f(x)，若 f(a) = 0，则 (x – a) 是其因式。
Modulus Inequalities: |f(x)| < a ⇔ –a < f(x) < a; |f(x)| > a ⇔ f(x) < –a or f(x) > a.
绝对值不等式： |f(x)| < a ⇔ –a < f(x) < a；|f(x)| > a ⇔ f(x) < –a 或 f(x) > a。
Partial Fractions (distinct linear factors): (px+q)/((x+a)(x+b)) = A/(x+a) + B/(x+b).
部分分式（不同线性因式）： (px+q)/((x+a)(x+b)) = A/(x+a) + B/(x+b)。
Inverse Functions: f(f⁻¹(x)) = f⁻¹(f(x)) = x; graph of y = f⁻¹(x) is the reflection of y = f(x) in y = x.
反函数： f(f⁻¹(x)) = f⁻¹(f(x)) = x；y = f⁻¹(x) 的图像是 y = f(x) 关于直线 y = x 的反射。
2. Coordinate Geometry | 坐标几何
Distance between (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 of line through (x₁, y₁) and (x₂, y₂): m = (y₂ – y₁)/(x₂ – x₁).
过两点的直线斜率： 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: (x – a)² + (y – b)² = r², centre (a, b).
圆的方程： (x – a)² + (y – b)² = r²，圆心 (a, b)。
Tangent to a circle: The radius is perpendicular to the tangent at the point of contact. Use discriminant = 0 to find tangents from a line.
圆的切线： 半径与切点处的切线垂直。利用判别式为零可求切线方程。
Parametric Equations: For a curve x = f(t), y = g(t), the gradient dy/dx = (dy/dt) ÷ (dx/dt).
参数方程： 对于曲线 x = f(t), y = g(t)，斜率 dy/dx = (dy/dt) ÷ (dx/dt)。
3. Sequences and Series | 数列与级数
Arithmetic series: nth term uₙ = a + (n – 1)d. Sum to n terms Sₙ = n/2 [2a + (n – 1)d] or Sₙ = n/2 (a + l).
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)。
Geometric series: 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。
Sum to infinity (geometric): S∞ = a/(1 – r), valid for |r| < 1.
无穷等比级数和： S∞ = a/(1 – r)，成立条件 |r| < 1。
Binomial expansion (1 + x)ⁿ: (1 + x)ⁿ = 1 + nx + [n(n–1)/2!]x² + [n(n–1)(n–2)/3!]x³ + … , valid for |x| < 1, n rational.
二项展开式 (1 + x)ⁿ： (1 + x)ⁿ = 1 + nx + [n(n–1)/2!]x² + [n(n–1)(n–2)/3!]x³ + … ，收敛域 |x| < 1，n 为有理数。
Sigma notation and series: Σr = n(n+1)/2, Σr² = n(n+1)(2n+1)/6, Σr³ = [n(n+1)/2]².
求和公式： Σr = n(n+1)/2，Σr² = n(n+1)(2n+1)/6，Σr³ = [n(n+1)/2]²。
4. Trigonometry | 三角学
Fundamental Identities: sin²θ + cos²θ = 1; tanθ = sinθ/cosθ; 1 + cot²θ = csc²θ; 1 + tan²θ = sec²θ.
基本恒等式： sin²θ + cos²θ = 1；tanθ = sinθ/cosθ；1 + cot²θ = csc²θ；1 + tan²θ = sec²θ。
Compound Angle Formulae: sin(A ± B) = sinA cosB ± cosA sinB; cos(A ± B) = cosA cosB ∓ sinA sinB; tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB).
和角公式： sin(A ± B) = sinA cosB ± cosA sinB；cos(A ± B) = cosA cosB ∓ sinA sinB；tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB)。
Double Angle Formulae: sin2θ = 2sinθ cosθ; cos2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ; tan2θ = 2tanθ/(1 – tan²θ).
倍角公式： sin2θ = 2sinθ cosθ；cos2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ；tan2θ = 2tanθ/(1 – tan²θ)。
Rsin(θ ± α) and Rcos(θ ± α): a sinθ ± b cosθ = R sin(θ ± α), where R = √(a² + b²), tanα = b/a.
辅助角形式： a sinθ ± b cosθ = R sin(θ ± α)，其中 R = √(a² + b²)，tanα = b/a。
Sine and Cosine Rules: a/sinA = b/sinB = c/sinC; a² = b² + c² – 2bc cosA. Area = ½ ab sinC.
正弦定理与余弦定理： a/sinA = b/sinB = c/sinC；a² = b² + c² – 2bc cosA。面积 = ½ ab sinC。
Radian Measure: π rad = 180°; arc length s = rθ; area of sector = ½ r²θ.
弧度制： π rad = 180°；弧长 s = rθ；扇形面积 = ½ r²θ。
Small Angle Approximations: sinθ ≈ θ, cosθ ≈ 1 – θ²/2, tanθ ≈ θ (θ in radians, small).
小角近似： sinθ ≈ θ，cosθ ≈ 1 – θ²/2，tanθ ≈ θ（θ 以弧度为单位且很小）。
5. Exponentials and Logarithms | 指数与对数
The exponential function: y = eˣ, where e ≈ 2.71828. Its derivative is d/dx (eˣ) = eˣ.
指数函数： y = eˣ，其中 e ≈ 2.71828。其导数为 d/dx (eˣ) = eˣ。
Natural logarithm: y = ln x is the inverse of y = eˣ, defined for x > 0. d/dx (ln x) = 1/x.
自然对数： y = ln x 是 y = eˣ 的反函数，定义域 x > 0。d/dx (ln x) = 1/x。
Laws of Logarithms: ln(ab) = ln a + ln b; ln(a/b) = ln a – ln b; ln aⁿ = n ln a.
对数法则： ln(ab) = ln a + ln b；ln(a/b) = ln a – ln b；ln aⁿ = n ln a。
Change of base and exponential equations: aˣ = e^{x ln a}; logₐ x = ln x / ln a.
换底及其应用： aˣ = e^{x ln a}；logₐ x = ln x / ln a。
Exponential growth and decay: Models of the form y = A e^{kt} or y = A aˣ; k > 0 growth, k < 0 decay.
指数增长与衰减： 模型形式为 y = A e^{kt} 或 y = A aˣ；k > 0 增长，k < 0 衰减。
6. Differentiation | 微分
Power Rule: d/dx (xⁿ) = nxⁿ⁻¹, for any real n.
幂函数求导： d/dx (xⁿ) = nxⁿ⁻¹，n 为任意实数。
Standard Derivatives: d/dx (eˣ) = eˣ; d/dx (ln x) = 1/x; d/dx (sin x) = cos x; d/dx (cos x) = –sin x; d/dx (tan x) = sec² x.
基本导数公式： d/dx (eˣ) = eˣ；d/dx (ln x) = 1/x；d/dx (sin x) = cos x；d/dx (cos x) = –sin x；d/dx (tan x) = sec² x。
Chain Rule: If y = f(g(x)), then dy/dx = f'(g(x)) · g'(x).
链式法则： 若 y = f(g(x))，则 dy/dx = f'(g(x)) · g'(x)。
Product Rule: d/dx (uv) = u dv/dx + v du/dx.
乘法法则： d/dx (uv) = u dv/dx + v du/dx。
Quotient Rule: d/dx (u/v) = (v du/dx – u dv/dx)/v².
除法法则： d/dx (u/v) = (v du/dx – u dv/dx)/v²。
Parametric Differentiation: dy/dx = (dy/dt) ÷ (dx/dt), provided dx/dt ≠ 0.
参数微分： dy/dx = (dy/dt) ÷ (dx/dt)，要求 dx/dt ≠ 0。
Implicit Differentiation: Differentiate each term with respect to x, treating y as a function of x, and apply chain rule to functions of y.
隐函数微分： 对方程每一项关于 x 求导，将 y 视作 x 的函数，对 y 函数应用链式法则。
Second Derivative: d²y/dx² = d/dx (dy/dx). Used to determine the nature of stationary points.
二阶导数： d²y/dx² = d/dx (dy/dx)。用于判断驻点的性质。
Stationary Points: dy/dx = 0 gives stationary points; d²y/dx² > 0 indicates a minimum, d²y/dx² < 0 a maximum.
驻点与极值： 令 dy/dx = 0 得驻点；若 d²y/dx² > 0 则为极小值点，d²y/dx² < 0 为极大值点。
Connected Rates of Change: If y = f(x) and x = g(t), then dy/dt = (dy/dx) · (dx/dt).
相关变化率： 若 y = f(x) 且 x = g(t)，则 dy/dt = (dy/dx) · (dx/dt)。
7. Integration | 积分
Reverse of Differentiation: ∫ f'(x) dx = f(x) + C.
微分的逆运算： ∫ f'(x) dx = f(x) + C。
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; ∫ sec² x dx = tan x + C.
基本积分公式： ∫ 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；∫ sec² x dx = tan x + C。
Integration by Substitution: Replace the variable and dx, often simplifying the integrand. For definite integrals, change limits accordingly.
换元积分法： 替换变量与 dx，往往化简被积函数。对于定积分，需同时更换积分限。
Integration by Parts: ∫ u dv/dx dx = uv – ∫ v du/dx dx. Often used for products of functions.
分部积分法： ∫ u dv/dx dx = uv – ∫ v du/dx dx。常用于两个函数的乘积积分。
Integrating Partial Fractions: Express rational function as sum of simpler fractions before integrating.
部分分式积分： 先将有理函数拆分为简单分式之和，再逐项积分。
Area Under a Curve: Area = ∫ₐᵇ y dx (y ≥ 0). Area between curves: ∫ (upper – lower) dx.
曲线下方面积： 面积 = ∫ₐᵇ y dx（当 y ≥ 0）。两曲线间面积：∫ (上曲线 – 下曲线) dx。
Volume of Revolution about x-axis: V = π ∫ₐᵇ y² dx.
绕 x 轴旋转体体积： V = π ∫ₐᵇ y² dx。
Solving Differential Equations: Separate variables: ∫ 1/g(y) dy = ∫ f(x) dx. Use initial conditions to find particular solutions.
解微分方程： 可分离变量：∫ 1/g(y) dy = ∫ f(x) dx。利用初始条件求特解。
8. Vectors | 向量
Magnitude: |a| = √(x² + y² + z²) for a = xi + yj + zk.
模长： 若 a = xi + yj + zk，则 |a| = √(x² + y² + z²)。
Scalar (Dot) Product: a·b = |a||b| cosθ = x₁x₂ + y₁y₂ + z₁z₂.
标量积（点积）： a·b = |a||b| cosθ = x₁x₂ + y₁y₂ + z₁z₂。
Angle Between Vectors: cosθ = (a·b) / (|a||b|). Vectors are perpendicular if a·b = 0.
向量间夹角： cosθ = (a·b) / (|a||b|)。若 a·b = 0，则两向量垂直。
Vector Equation of a Line: r = a + tb, where a is a point on the line, b is the direction vector, t a scalar parameter.
直线向量方程： r = a + tb，其中 a 为直线上一点的位置向量，b 为方向向量，t 为标量参数。
Distance from a point to a line: Use the formula d = |(a – p) × b| / |b|, but in AQA using scalar product to find shortest distance via perpendicular construction is common.
点到直线距离： 通常通过构造垂直条件利用点积求最短距离。
9. Statistical Distributions | 统计分布
Binomial Distribution X ~ B(n, p
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