📚 A-Level Maths: Last-Minute Revision Notes | A-Level 数学：考前冲刺笔记
This article distils the essential concepts across Pure Mathematics, Statistics, and Mechanics that you need to have at your fingertips for the A-Level Maths exam. Use these bilingual paired notes to rapidly reinforce key formulas, methods, and common pitfalls. Each section presents English explanations followed immediately by a Chinese translation, so you can revise effectively in either language.
本文浓缩了 A-Level 数学纯数、统计与力学三大模块的核心考点，助你快速巩固关键公式、方法与常见陷阱。每个要点均采用先英文后中文的配对形式，方便你高效切换双语复习，查漏补缺。
1. Core Algebra and Functions | 核心代数与函数
Factorising quadratics: for ax²+bx+c, find two numbers multiplying to ac and adding to b, then split the middle term and factor by grouping. Always extract any common factor first.
二次式因式分解：对 ax²+bx+c，找到乘积为 ac 且和为 b 的两数，拆中项后分组提取公因式。切记先提取任何公共因子。
Completing the square transforms x²+bx into (x + b/2)² – (b/2)². This technique reveals the vertex of a parabola and is the foundation for the quadratic formula.
配方法将 x²+bx 化为 (x + b/2)² – (b/2)²。这个技巧能直接给出抛物线的顶点，也是求根公式的基础。
x = [–b ± √(b² – 4ac)] / 2a
The modulus function |x| behaves like x when x ≥ 0 and –x when x < 0. Solving |f(x)| = k gives f(x) = ±k, but always check for extraneous solutions.
绝对值函数 |x| 在 x ≥ 0 时等于 x，x < 0 时等于 –x。解 |f(x)| = k 得到 f(x) = ±k，但务必检验增根。
For composite functions fg(x) = f(g(x)), the domain is restricted by g(x) and then by the domain of f. Inverse functions f⁻¹(x) reflect the graph in y = x, provided f is one-to-one.
复合函数 fg(x) = f(g(x)) 的定义域先受限于 g(x)，再受限于 f 的定义域。反函数 f⁻¹(x) 的图像关于 y = x 对称，前提是 f 为一一映射。
2. Coordinate Geometry | 坐标几何
The gradient of a line through (x₁,y₁) and (x₂,y₂) is (y₂ – y₁)/(x₂ – x₁). The equation y – y₁ = m(x – x₁) is often the quickest to use.
过 (x₁,y₁) 和 (x₂,y₂) 的直线斜率是 (y₂ – y₁)/(x₂ – x₁)。点斜式 y – y₁ = m(x – x₁) 通常最快捷。
For perpendicular lines, m₁m₂ = –1. The distance between two points is √[(x₂ – x₁)² + (y₂ – y₁)²] and the midpoint is ((x₁+x₂)/2, (y₁+y₂)/2).
垂直直线满足 m₁m₂ = –1。两点间距离为 √[(x₂ – x₁)² + (y₂ – y₁)²]，中点坐标为 ((x₁+x₂)/2, (y₁+y₂)/2)。
The circle equation (x – a)² + (y – b)² = r² has centre (a,b) and radius r. To find tangents, use the discriminant Δ = 0 after substituting the line equation, or use the perpendicular radius property.
圆方程 (x – a)² + (y – b)² = r² 的圆心为 (a,b)，半径为 r。求切线可代入直线方程后用判别式 Δ = 0，或利用半径与切线垂直的性质。
Parametric equations: for a curve given by x = f(t), y = g(t), eliminate t to find the Cartesian equation. The gradient dy/dx = (dy/dt) / (dx/dt).
参数方程：给定 x = f(t), y = g(t)，可消去 t 得到直角坐标方程。斜率 dy/dx = (dy/dt) / (dx/dt)。
3. Trigonometry | 三角学
Radians: π rad = 180°. Arc length s = rθ, area of sector A = ½r²θ. Use radians in calculus, as the derivatives of sin x and cos x only hold when x is in radians.
弧度制：π 弧度 = 180°。弧长 s = rθ，扇形面积 A = ½r²θ。微积分中务必使用弧度，因为 sin x 和 cos x 的导数仅在 x 为弧度时成立。
Fundamental identities: sin²θ + cos²θ ≡ 1, 1 + tan²θ ≡ sec²θ, 1 + cot²θ ≡ cosec²θ. The double-angle formulae: sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ – sin²θ = 2 cos²θ – 1 = 1 – 2 sin²θ.
基本恒等式：sin²θ + cos²θ ≡ 1, 1 + tan²θ ≡ sec²θ, 1 + cot²θ ≡ cosec²θ。倍角公式：sin 2θ = 2 sin θ cos θ；cos 2θ = cos²θ – sin²θ = 2 cos²θ – 1 = 1 – 2 sin²θ。
Solving trig equations: sketch the graph or use the CAST diagram to find all solutions within the given interval. Always adjust the range first if the angle is transformed, e.g., for sin(2x – 30°), set 2x – 30° in the required range.
解三角方程：画图或使用 CAST 图找出给定区间内的所有解。若角度有变换，如 sin(2x – 30°)，应先将 2x – 30° 置于所需范围。
The sine rule: a/sin A = b/sin B = c/sin C; cosine rule: a² = b² + c² – 2bc cos A. Use the cosine rule when given two sides and the included angle or three sides.
正弦定理：a/sin A = b/sin B = c/sin C；余弦定理：a² = b² + c² – 2bc cos A。当已知两边及其夹角或已知三边时，使用余弦定理。
4. Sequences and Series | 数列与级数
An arithmetic sequence has common difference d: uₙ = a + (n – 1)d. The sum of the first n terms: Sₙ = n/2 [2a + (n – 1)d] or Sₙ = n/2 (a + l).
等差数列公差为 d，通项 uₙ = a + (n – 1)d。前 n 项和 Sₙ = n/2 [2a + (n – 1)d] 或 Sₙ = n/2 (a + l)。
A geometric sequence has common ratio r: uₙ = arⁿ⁻¹. The sum Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1. An infinite geometric series converges to a/(1 – r) if |r| < 1.
等比数列公比为 r，通项 uₙ = arⁿ⁻¹。前 n 项和 Sₙ = a(1 – rⁿ)/(1 – r) (r ≠ 1)。若 |r| < 1，无穷等比级数收敛于 a/(1 – r)。
The binomial expansion for (1 + x)ⁿ is 1 + nx + n(n–1)/2! x² + … + nCr xʳ + …, valid for |x| < 1 when n is not a positive integer. Always state the validity range.
(1 + x)ⁿ 的二项展开式为 1 + nx + n(n–1)/2! x² + … + nCr xʳ + …，当 n 不是正整数时，要求 |x| < 1。务必注明适用范围。
Sigma notation: ∑ⁿᵣ₌₁ r = n(n+1)/2; ∑ⁿᵣ₌₁ r² = n(n+1)(2n+1)/6; ∑ⁿᵣ₌₁ r³ = [n(n+1)/2]². These standard results often appear in summation questions.
求和符号：∑ⁿᵣ₌₁ r = n(n+1)/2；∑ⁿᵣ₌₁ r² = n(n+1)(2n+1)/6；∑ⁿᵣ₌₁ r³ = [n(n+1)/2]²。这些标准结果常出现在求和题中。
5. Differentiation | 微分
| f(x) | f'(x) |
| xⁿ | nxⁿ⁻¹ |
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | –sin x |
| tan x | sec² x |
The chain rule: if y = f(u) and u = g(x), then dy/dx = (dy/du) × (du/dx). The product rule: if y = uv, then dy/dx = u dv/dx + v du/dx. The quotient rule: if y = u/v, then dy/dx = (v du/dx – u dv/dx) / v².
链式法则：若 y = f(u), u = g(x)，则 dy/dx = (dy/du) × (du/dx)。乘积法则：若 y = uv，dy/dx = u dv/dx + v du/dx。商法则：若 y = u/v，dy/dx = (v du/dx – u dv/dx) / v²。
For stationary points, solve f'(x) = 0. Classify using the second derivative: f”(x) > 0 gives a minimum, f”(x) < 0 gives a maximum, and f''(x) = 0 requires the first derivative test. Concave up: f''(x) > 0; concave down: f”(x) < 0; points of inflection occur where concavity changes.
驻点解 f'(x) = 0。用二阶导数分类：f”(x) > 0 为极小值，f”(x) < 0 为极大值，f''(x) = 0 时需用一阶导数检验。凹向上：f''(x) > 0；凹向下：f”(x) < 0；拐点即凹凸性发生变化的点。
6. Integration | 积分
Integration is the reverse of differentiation. The general power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ –1. Also ∫ 1/x dx = ln|x| + C, and ∫ eˣ dx = eˣ + C.
积分是微分的逆运算。一般幂法则：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ –1)。同时 ∫ 1/x dx = ln|x| + C，∫ eˣ dx = eˣ + C。
Standard integrals: ∫ sin x dx = –cos x + C; ∫ cos x dx = sin x + C; ∫ sec² x dx = tan x + C. Remember the reverse chain rule (inverse chain rule) for linear functions: ∫ f'(ax+b) dx = (1/a) f(ax+b) + C.
标准积分：∫ sin x dx = –cos x + C；∫ cos x dx = sin x + C；∫ sec² x dx = tan x + C。对线性函数的逆链式法则：∫ f'(ax+b) dx = (1/a) f(ax+b) + C。
Integration by substitution: substitute u = g(x), replace dx with du / g'(x), and change limits for definite integrals. Integration by parts: ∫ u dv/dx dx = uv – ∫ v du/dx dx. Use L-I-A-T-E (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) to choose u.
换元积分法：令 u = g(x)，将 dx 替换为 du / g'(x)，定积分需同步更换上下限。分部积分法：∫ u dv/dx dx = uv – ∫ v du/dx dx。按 L-I-A-T-E 顺序（对数、反三角、代数、三角、指数）选择 u。
Definite integrals give the area between the curve and the x‑axis. Area between two curves y = f(x) and y = g(x) is ∫ [f(x) – g(x)] dx over the intersection interval. Volume of revolution about the x‑axis is π ∫ y² dx.
定积分给出曲线与 x 轴之间的面积。两曲线 y = f(x) 与 y = g(x) 间的面积为在交点区间 ∫ [f(x) – g(x)] dx。绕 x 轴旋转体积为 π ∫ y² dx。
7. Differential Equations | 微分方程
A first-order separable differential equation can be written as dy/dx = f(x)g(y). Separate variables: (1/g(y)) dy = f(x) dx, then integrate both sides. Don’t forget the constant of integration.
一阶可分离变量的微分方程可写为 dy/dx = f(x)g(y)。分离变量：(1/g(y)) dy = f(x) dx，然后两边积分。切勿遗漏积分常数。
After integration, use a given boundary condition (x₀, y₀) to find the particular solution. Model real-world contexts like population growth (dP/dt = kP) or Newton’s law of cooling.
积分后，利用给定的边界条件 (x₀, y₀) 求出特解。常应用于现实模型，如种群增长 (dP/dt = kP) 或牛顿冷却定律。
8. Vectors | 向量
A vector is represented as a = (a₁, a₂, a₃) in 3D. The magnitude |a| = √(a₁² + a₂² + a₃²). The unit vector in the direction of a is a / |a|.
三维向量表示为 a = (a₁, a₂, a₃)。模长 |a| = √(a₁² + a₂² + a₃²)。沿 a 方向的单位向量为 a / |a|。
The scalar (dot) product a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ, where θ is the angle between the vectors. Two non-zero vectors are perpendicular iff a·b = 0.
标量积（点积）a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ，θ 为向量间夹角。两非零向量垂直当且仅当 a·b = 0。
The vector equation of a line: r = a + λ d, where a is a point on the line and d is the direction vector. To find the angle between two lines, use the dot product of their direction vectors.
直线的向量方程：r = a + λ d，其中 a 为线上一点，d 为方向向量。求两直线夹角时，用其方向向量的点积。
9. Statistical Distributions | 统计分布
For a discrete probability distribution, ΣP(X = x) = 1. The expectation E(X) = Σ x P(X = x) and variance Var(X) = E(X²) – [E(X)]².
对于离散概率分布，ΣP(X = x) = 1。期望 E(X) = Σ x P(X = x)，方差 Var(X) = E(X²) – [E(X)]²。
The binomial distribution X ~ B(n, p): P(X = r) = nCr pʳ (1–p)ⁿ⁻ʳ. Mean μ = np, variance σ² = np(1–p). Use the tables or calculator for cumulative probabilities.
二项分布 X ~ B(n, p)：P(X = r) = nCr pʳ (1–p)ⁿ⁻ʳ。均值 μ = np，方差 σ² = np(1–p)。使用表格或计算器求累积概率。
The normal distribution X ~ N(μ, σ²). Standardise using Z = (X – μ)/σ. For the sample mean, X̄ ~ N(μ, σ²/n). When using the normal approximation to a binomial, ensure np > 5 and n(1–p) > 5, and apply a continuity correction (e.g., P(X ≤ a) becomes P(Y < a + 0.5)).
正态分布 X ~ N(μ, σ²)。用 Z = (X – μ)/σ 标准化。样本均值 X̄ ~ N(μ, σ²/n)。用正态近似二项分布时，确保 np > 5 且 n(1–p) > 5，并应用连续性校正（如 P(X ≤ a) 变为 P(Y < a + 0.5)）。
10. Hypothesis Testing | 假设检验
State the null hypothesis H₀ and alternative H₁. For a binomial test, find the probability of obtaining the observed result or more extreme under H₀. Compare this p-value with the significance level α (often 0.05).
陈述原假设 H₀ 与备择假设 H₁。对于二项检验，计算在原假设下得到当前结果或更极端情况的概率。将该 p 值与显著性水平 α（通常为 0.05）进行比较。
If p ≤ α, reject H₀; the result is significant. If p > α, do not reject H₀. The critical region is the set of outcomes leading to rejection of H₀. The actual significance level is the total probability in the critical region.
若 p ≤ α，拒绝 H₀，结果显著。若 p > α，不拒绝 H₀。拒绝域是导致拒绝 H₀ 的所有结果构成的集合。实际显著性水平为拒绝域的总概率。
For a normal distribution test, use the Z‑test: Z = (X̄ – μ₀) / (σ/√n). For a two-tailed test, halve the significance level when finding critical Z‑values.
对于正态分布检验，使用 Z 检验：Z = (X̄ – μ₀) / (σ/√n)。双尾检验查找临界 Z 值时，需将显著性水平除以 2。
11. Mechanics: Kinematics | 力学：运动学
Displacement (s), velocity (v), and acceleration (a) are functions of time t. v = ds/dt, a = dv/dt = d²s/dt². Constant acceleration equations (SUVAT) apply when a is constant.
位移 (s)、速度 (v) 和加速度 (a) 均为时间 t 的函数。v = ds/dt，a = dv/dt = d²s/dt²。当加速度恒定时，适用匀加速运动方程（SUVAT）。
v = u + at s = ut + ½at² s = ½(u+v)t v² = u² + 2as
Select the SUVAT equation that includes the four variables you know and the one you need. For projectile motion, resolve into horizontal and vertical components; horizontal velocity is constant (aₓ = 0), vertical acceleration is g downward.
选择包含三个已知量和一个所求量的 SUVAT 方程。对于抛体运动，分解为水平与竖直分量；水平速度恒定 (aₓ = 0)，竖直加速度为向下的 g。
Use calculus for variable acceleration: v = ∫ a dt, s = ∫ v dt. Integrate with initial conditions to find constants. The maximum height in a projectile occurs when vᵧ = 0.
变加速运动用微积分：v = ∫ a dt，s = ∫ v dt。结合初始条件积分求常数。抛体最高点对应 vᵧ = 0。
12. Mechanics: Forces and Newton’s Laws | 力学：力与牛顿定律
Newton’s second law: F = ma, where F is the resultant force. Always draw a clear force diagram showing weight (mg), normal reaction (R), friction (Fᵣ ≤ μR), and tension.
牛顿第二定律：F = ma，F 为合力。务必绘制清晰的受力图，标明重力 (mg)、法向反作用力 (R)、摩擦力 (Fᵣ ≤ μR) 和张力。
Friction: limiting friction Fₘₐₓ = μR. Friction acts opposite to motion or potential motion. On an inclined plane, resolve weight into components mg sin θ along the plane and mg cos θ perpendicular.
摩擦力：极限摩擦力 Fₘₐₓ = μR。摩擦力与运动或运动趋势方向相反。在斜面上，将重力分解为沿斜面的 mg sin θ 和垂直于斜面的 mg cos θ。
Connected particles: treat each particle separately, applying F = ma. Use the same tension throughout an inextensible string passing over a smooth pulley. The acceleration of connected objects has the same magnitude.
连接体：对每个物体单独应用 F = ma。轻绳绕过光滑滑轮时，绳上张力处处相等。连接物体的加速度大小相同。
For equilibrium, the resultant force in any direction is zero. Resolve forces into components and set ΣFₓ = 0, ΣFᵧ = 0. Moments about a point: moment = force × perpendicular distance. An object is in equilibrium when both the resultant force and resultant moment are zero.
平衡状态时，任意方向上的合力为零。将力分解并令 ΣFₓ = 0, ΣFᵧ = 0。力矩 = 力 × 垂直距离。物体平衡需满足合力矩与合力均为零。
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