📚 A-Level AQA Maths: Formula Summary Handbook | A-Level AQA 数学：公式汇总手册
This handbook gathers the essential formulae required for the AQA A-Level Mathematics specification. It covers pure mathematics, mechanics and statistics in a clear, bilingual format so that you can check formulas quickly during revision and exam preparation. All formulae are presented using standard notation and are aligned with the requirements of the AQA course.
本手册汇集了 AQA A-Level 数学考试所需的核心公式，涵盖纯数学、力学与统计，以清晰的双语对照形式呈现，便于复习和考前快速查阅。所有公式均采用标准符号，并严格贴合 AQA 课程要求。
1. Algebraic Laws and Indices | 代数运算法则与指数
Multiplying like bases: am × an = am+n.
同底数幂相乘：am × an = am+n。
Dividing like bases: am ÷ an = am−n.
同底数幂相除：am ÷ an = am−n。
Power of a power: (am)n = amn.
幂的乘方：(am)n = amn。
Negative and fractional indices: a−n = 1/an; a1/n = n√a; am/n = (n√a)m.
负指数与分数指数：a−n = 1/an；a1/n = n√a；am/n = (n√a)m。
Laws of surds: √(ab) = √a × √b; √(a/b) = √a / √b (a, b ≥ 0).
根式运算法则：√(ab) = √a × √b；√(a/b) = √a / √b（a, b ≥ 0）。
Completing the square: x² + bx = (x + b/2)² − (b/2)².
配方法：x² + bx = (x + b/2)² − (b/2)²。
2. Coordinate Geometry | 坐标几何
Gradient of a straight line: m = (y₂ − y₁)/(x₂ − x₁).
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。
Midpoint of two points: ((x₁ + x₂)/2, (y₁ + y₂)/2).
两点中点：((x₁ + x₂)/2, (y₁ + y₂)/2)。
Distance between two points: d = √[(x₂ − x₁)² + (y₂ − y₁)²].
两点间距离：d = √[(x₂ − x₁)² + (y₂ − y₁)²]。
Equation of a circle centre (a, b) radius r: (x − a)² + (y − b)² = r².
以 (a, b) 为圆心、半径为 r 的圆方程：(x − a)² + (y − b)² = r²。
Circumference and area of a circle: C = 2πr, A = πr².
圆的周长与面积：C = 2πr，A = πr²。
3. Trigonometry | 三角学
Sine rule: a/sin A = b/sin B = c/sin C.
正弦定理：a/sin A = b/sin B = c/sin C。
Cosine rule: a² = b² + c² − 2bc cos A.
余弦定理：a² = b² + c² − 2bc cos A。
Area of a triangle: Area = ½ ab sin C.
三角形面积：面积 = ½ ab sin C。
Trigonometric identities: sin²θ + cos²θ = 1; tan θ = sin θ / cos θ; sec²θ = 1 + tan²θ; cosec²θ = 1 + cot²θ.
三角恒等式：sin²θ + cos²θ = 1；tan θ = sin θ / cos θ；sec²θ = 1 + tan²θ；cosec²θ = 1 + cot²θ。
Small-angle approximations (θ in radians): sin θ ≈ θ, cos θ ≈ 1 − θ²/2, tan θ ≈ θ.
小角近似（弧度制）：sin θ ≈ θ，cos θ ≈ 1 − θ²/2，tan θ ≈ θ。
Radian measure: 180° = π rad.
弧度制：180° = π 弧度。
4. Exponentials and Logarithms | 指数与对数
Relationship: logₐ x = y ⇔ aʸ = x.
关系：logₐ x = y ⇔ aʸ = x。
Natural log and exponential: ln x = logₑ x; e^(ln x) = x.
自然对数与指数：ln x = logₑ x；e^(ln x) = x。
Laws of logs: logₐ(xy) = logₐ x + logₐ y; logₐ(x/y) = logₐ x − logₐ y; logₐ(xⁿ) = n logₐ x.
对数运算法则：logₐ(xy) = logₐ x + logₐ y；logₐ(x/y) = logₐ x − logₐ y；logₐ(xⁿ) = n logₐ x。
Change of base: logₐ x = log_b x / log_b a.
换底公式：logₐ x = log_b x / log_b a。
Derivative of eˣ: d/dx (eˣ) = eˣ.
eˣ 的导数：d/dx (eˣ) = eˣ。
Derivative of ln x: d/dx (ln x) = 1/x.
ln x 的导数：d/dx (ln x) = 1/x。
5. Differentiation | 微分
Basic derivative: d/dx (xⁿ) = n xⁿ⁻¹.
基本导数：d/dx (xⁿ) = n xⁿ⁻¹。
Product rule: if y = u v, then dy/dx = u dv/dx + v du/dx.
乘法法则：若 y = u v，则 dy/dx = u dv/dx + v du/dx。
Quotient rule: if y = u/v, then dy/dx = (v du/dx − u dv/dx) / v².
除法法则：若 y = u/v，则 dy/dx = (v du/dx − u dv/dx) / v²。
Chain rule: dy/dx = dy/du × du/dx.
链式法则：dy/dx = dy/du × du/dx。
Derivatives of trig functions: d/dx (sin x) = cos x; d/dx (cos x) = −sin x; d/dx (tan x) = sec² x.
三角函数导数：d/dx (sin x) = cos x；d/dx (cos x) = −sin x；d/dx (tan x) = sec² x。
Stationary points: solve dy/dx = 0; nature determined by second derivative d²y/dx².
驻点：解 dy/dx = 0；由二阶导数 d²y/dx² 判断性质。
6. Integration | 积分
Basic indefinite integral: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c (n ≠ −1).
基本不定积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + c（n ≠ −1）。
Integral of 1/x: ∫ 1/x dx = ln|x| + c.
1/x 的积分：∫ 1/x dx = ln|x| + c。
Definite integral: ∫ab f(x) dx = F(b) − F(a) where F'(x) = f(x).
定积分：∫ab f(x) dx = F(b) − F(a)，其中 F'(x) = f(x)。
Area under a curve: area = ∫ab y dx for regions above x-axis.
曲线下方面积：位于 x 轴上方的区域面积 = ∫ab y dx。
Trapezium rule (approximation): ∫ab y dx ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ], where h = (b − a)/n.
梯形法则（近似）：∫ab y dx ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ]，其中 h = (b − a)/n。
Reverse chain rule and integration by substitution methods also required.
也需要掌握反向链式法则与换元积分法。
7. Sequences and Series | 数列与级数
Arithmetic sequence: nth term uₙ = a + (n−1)d; sum Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l).
等差数列：第 n 项 uₙ = a + (n−1)d；和 Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l)。
Geometric sequence: nth term uₙ = a rⁿ⁻¹; sum to n terms Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1.
等比数列：第 n 项 uₙ = a rⁿ⁻¹；前 n 项和 Sₙ = a(1 − rⁿ)/(1 − r)（r ≠ 1）。
Sum to infinity of a geometric series: S∞ = a/(1 − r) provided |r| < 1.
等比级数无穷和：S∞ = a/(1 − r)，当 |r| < 1 时收敛。
Binomial expansion: (1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + … for |x| < 1, n rational.
二项展开式：(1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + …，适用于 |x| < 1，n 为有理数。
8. Vectors | 向量
Magnitude of 2D vector v = xi + yj: |v| = √(x² + y²).
二维向量 v = xi + yj 的模：|v| = √(x² + y²)。
Magnitude of 3D vector: |v| = √(x² + y² + z²).
三维向量的模：|v| = √(x² + y² + z²)。
Dot product: a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃.
点积：a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃。
Angle between vectors: cos θ = (a · b) / (|a||b|).
向量夹角：cos θ = (a · b) / (|a||b|)。
Equation of a straight line in vector form: r = a + t d, where a is a point on the line and d is a direction vector.
直线向量方程：r = a + t d，其中 a 为直线上一点，d 为方向向量。
9. Mechanics – Kinematics | 力学 – 运动学
Constant acceleration formulae (SUVAT): v = u + at; s = ut + ½ at²; s = ½ (u + v)t; v² = u² + 2as; s = vt − ½ at².
匀加速度公式（SUVAT）：v = u + at；s = ut + ½ at²；s = ½ (u + v)t；v² = u² + 2as；s = vt − ½ at²。
Displacement, velocity, acceleration under varying motion: v = ds/dt, a = dv/dt = d²s/dt².
变加速运动中，位移、速度与加速度的关系：v = ds/dt，a = dv/dt = d²s/dt²。
Distance travelled: total area under a velocity–time graph.
运动距离：速度–时间图线下方的总面积。
10. Mechanics – Forces and Newton’s Laws | 力学 – 力与牛顿定律
Newton’s second law: F = m a (resultant force = mass × acceleration).
牛顿第二定律：F = m a（合力 = 质量 × 加速度）。
Weight: W = m g, where g = 9.8 m s⁻² (unless stated otherwise).
重力：W = m g，通常 g = 9.8 m s⁻²。
Friction: F_max = μ R, where μ is the coefficient of friction and R is the normal reaction.
摩擦力：F_max = μ R，μ 为摩擦系数，R 为法向反作用力。
Moment of a force about a point: moment = force × perpendicular distance.
力关于一点的力矩：力矩 = 力 × 垂直距离。
Equilibrium: resultant force = 0 and resultant moment = 0.
平衡条件：合力 = 0 且合力矩 = 0。
11. Statistics – Probability and Distributions | 统计 – 概率与分布
Conditional probability: P(A|B) = P(A ∩ B) / P(B).
条件概率：P(A|B) = P(A ∩ B) / P(B)。
Mutually exclusive events: P(A ∪ B) = P(A) + P(B) if A and B cannot occur together.
互斥事件：若 A 与 B 不能同时发生，则 P(A ∪ B) = P(A) + P(B)。
Independent events: 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，其中 q = 1 − p。均值 = np；方差 = npq。
Normal distribution: X ~ N(μ, σ²). Standardisation: Z = (X − μ)/σ ~ N(0, 1).
正态分布：X ~ N(μ, σ²)。标准化：Z = (X − μ)/σ ~ N(0, 1)。
Approximating binomial with normal: if np > 5 and nq > 5, B(n, p) ≈ N(np, npq). Apply continuity correction.
用正态分布近似二项分布：当 np > 5 且 nq > 5 时，B(n, p) ≈ N(np, npq)，需使用连续性校正。
12. Statistics – Hypothesis Testing | 统计 – 假设检验
Test statistic for a binomial proportion: use the binomial distribution directly or normal approximation as appropriate.
二项比例检验统计量：直接使用二项分布，或在合适时使用正态近似。
Critical region and significance level: reject H₀ if test statistic falls in the critical region based on the chosen α (often 0.05).
拒绝域与显著性水平：若检验统计量落在基于选定 α（常为 0.05）的拒绝域内，则拒绝 H₀。
p-value approach: reject H₀ if p-value ≤ significance level.
P 值法：若 p 值 ≤ 显著性水平，则拒绝 H₀。
One-tailed and two-tailed tests: adjust critical values accordingly.
单尾与双尾检验：相应调整临界值。
For normal distribution tests with known variance: use Z-test; for unknown variance estimated from sample, use t-test if required by the specification.
已知方差的正态分布检验：使用 Z 检验；若方差未知且由样本估计，在课程要求时使用 t 检验。
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