📚 A-Level OCR Maths: Last-Minute Revision Notes | A-Level OCR 数学：考前冲刺笔记
As the exam approaches, effective revision becomes the key to unlocking your full potential. This guide distils the essential topics from the OCR A-Level Mathematics specification into concise, high-yield notes. It covers Pure Mathematics, Statistics, and Mechanics, highlighting common pitfalls and key formulas you must have at your fingertips. Each section reinforces understanding through clear explanations paired with Chinese translations, ensuring you grasp the concepts deeply and quickly.
随着考试临近，高效复习是发挥你全部潜能的关键。本指南将 OCR A-Level 数学大纲中的核心主题提炼成简洁、高收益的冲刺笔记，涵盖纯数、统计和力学，重点标出常见易错点和必须烂熟于心的关键公式。每个部分都通过清晰的解释配合中文翻译来强化理解，帮助你深入快速地掌握概念。
1. Algebraic Manipulation & Functions | 代数运算与函数
Master factorising by inspection and completing the square. For a quadratic ax² + bx + c, the discriminant is Δ = b² – 4ac. If Δ > 0, two distinct real roots exist; Δ = 0 gives a repeated root; Δ < 0 yields no real roots. Always state the domain and range of functions, including modulus functions f(x) = |x| and composite functions. Watch out for restrictions, such as the denominator cannot be zero or the argument of a square root must be non-negative.
熟练掌握因式分解和配方法。对于二次式 ax² + bx + c，判别式为 Δ = b² – 4ac。若 Δ > 0，有两个不等实根；Δ = 0 给出重根；Δ < 0 则无实根。务必指出函数的定义域和值域，包括绝对值函数 f(x) = |x| 与复合函数。注意限制条件，例如分母不能为零，平方根的被开方数必须非负。
When transforming graphs, remember f(x + a) shifts left by a, f(x) + a shifts up, f(ax) is a horizontal stretch by scale factor 1/a, and af(x) is a vertical stretch by scale factor a. Sketching modulus graphs often involves reflecting negative parts of f(x) in the x-axis.
进行图像变换时，记住 f(x + a) 向左平移 a 个单位，f(x) + a 向上平移，f(ax) 是水平方向拉伸 1/a 倍，af(x) 是竖直方向拉伸 a 倍。绘制绝对值图像通常需将 f(x) 的负值部分关于 x 轴反射。
2. Trigonometry & Identities | 三角学与恒等式
Know exact values for sin, cos, tan of 0°, 30°, 45°, 60°, 90° and radian equivalents. The Pythagorean identity sin²θ + cos²θ = 1 is fundamental; derive tan²θ + 1 = sec²θ and 1 + cot²θ = csc²θ when solving equations. Use double-angle formulas sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ, and tan 2θ = (2 tan θ) / (1 – tan²θ).
牢记 0°、30°、45°、60°、90° 以及对应弧度的 sin、cos、tan 精确值。基本恒等式 sin²θ + cos²θ = 1 至关重要；解方程时推导 tan²θ + 1 = sec²θ 和 1 + cot²θ = csc²θ。使用倍角公式：sin 2θ = 2 sin θ cos θ，cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ，以及 tan 2θ = (2 tan θ) / (1 – tan²θ)。
For solving trigonometric equations, always consider the given interval and use the CAST diagram or graph to find all solutions. Beware of extraneous roots when squaring both sides. R cos(θ ± α) and R sin(θ ± α) forms help solve equations like a sin θ + b cos θ = c.
解三角方程时，始终考虑给定区间，利用 CAST 图或图像求出所有解。两边平方时警惕增根。将 a sin θ + b cos θ = c 型方程化为 R cos(θ ± α) 或 R sin(θ ± α) 形式有助于求解。
3. Differentiation Techniques | 微分技巧
Differentiate powers of x: d/dx (xⁿ) = n xⁿ⁻¹, extending to rational and negative n. The chain rule dy/dx = dy/du × du/dx is essential for composites. Product rule: d/dx (uv) = u’v + uv’. Quotient rule: d/dx (u/v) = (u’v – uv’) / v². For exponentials, d/dx (eᵏˣ) = k eᵏˣ; for natural logs, d/dx (ln x) = 1/x.
对 x 的幂函数求导：d/dx (xⁿ) = n xⁿ⁻¹，可推广到有理数和负数 n。链式法则 dy/dx = dy/du × du/dx 是复合函数求导的核心。乘法法则：d/dx (uv) = u’v + uv’。除法法则：d/dx (u/v) = (u’v – uv’) / v²。对于指数函数，d/dx (eᵏˣ) = k eᵏˣ；对于自然对数，d/dx (ln x) = 1/x。
Differentiate trigonometric functions: d/dx (sin x) = cos x, d/dx (cos x) = -sin x, d/dx (tan x) = sec² x. Use implicit differentiation when y cannot be expressed explicitly. For parametric equations x = f(t), y = g(t), dy/dx = (dy/dt) / (dx/dt). Find stationary points by setting dy/dx = 0 and classify using second derivative or sign change.
求三角函数的导数：d/dx (sin x) = cos x，d/dx (cos x) = -sin x，d/dx (tan x) = sec² x。当 y 无法显式表达时，用隐函数求导。对于参数方程 x = f(t), y = g(t)，dy/dx = (dy/dt) / (dx/dt)。令 dy/dx = 0 找驻点，用二阶导数或符号变化分类。
4. Integration & Area Under Curves | 积分与曲线下面积
Integration is the reverse of differentiation. Basic integrals: ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C (n ≠ -1), ∫ 1/x dx = ln|x| + C. For exponentials, ∫ eᵏˣ dx = (1/k) eᵏˣ + C. For trig, ∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C, ∫ sec² x dx = tan x + C. Use reverse chain rule or substitution for more complex integrals.
积分是微分的逆运算。基本积分公式：∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C（n ≠ -1），∫ 1/x dx = ln|x| + C。对于指数函数，∫ eᵏˣ dx = (1/k) eᵏˣ + C。三角函数积分：∫ sin x dx = -cos x + C，∫ cos x dx = sin x + C，∫ sec² x dx = tan x + C。复杂积分可用反链式法则或换元法。
Definite integrals give the signed area between the curve and the x-axis. The area between two curves is ∫ (upper – lower) dx over the interval. Integration by parts: ∫ u dv/dx dx = uv – ∫ v du/dx dx. For parametric curves, area = ∫ y dx = ∫ y (dx/dt) dt. Use partial fractions to integrate rational functions.
定积分给出曲线与 x 轴之间的有向面积。两曲线之间的面积为 ∫ (上方曲线 – 下方曲线) dx。分部积分法：∫ u dv/dx dx = uv – ∫ v du/dx dx。对于参数曲线，面积 = ∫ y dx = ∫ y (dx/dt) dt。用部分分式积分有理函数。
5. Sequences & Series | 数列与级数
Arithmetic sequences have a common difference d: n-th term uₙ = a + (n-1)d, sum Sₙ = n/2 [2a + (n-1)d]. Geometric sequences have a common ratio r: uₙ = arⁿ⁻¹; if |r| < 1 the infinite sum converges to a/(1-r). Sigma notation Σ expresses sums succinctly; know how to split sums and extract constants.
等差数列有公差 d：第 n 项 uₙ = a + (n-1)d，求和 Sₙ = n/2 [2a + (n-1)d]。等比数列有公比 r：uₙ = arⁿ⁻¹；若 |r| < 1，无穷级数和收敛于 a/(1-r)。用 Σ 符号简洁表达求和；掌握拆分求和与提取常数的方法。
The binomial expansion (1 + x)ⁿ = 1 + nx + n(n-1)/2! x² + … is valid for any rational n provided |x| < 1. For (a + b)ⁿ, factor out aⁿ. Find the range of validity by ensuring the modulus of the variable part is less than 1. Use expansions to approximate values or integrate term by term.
二项式展开 (1 + x)ⁿ = 1 + nx + n(n-1)/2! x² + … 对任意有理数 n 成立，要求 |x| < 1。对于 (a + b)ⁿ，提取 aⁿ。通过确保变量部分的模小于 1 来确定有效范围。可用展开式进行近似计算或逐项积分。
6. Vectors in 2D & 3D | 二维和三维向量
Vectors have magnitude and direction. In component form, a vector from A to B is given by b – a. Magnitude |a| = √(x² + y² + z²). Unit vector in direction a is a / |a|. Addition, subtraction, and scalar multiplication follow component-wise rules. The dot product a · b = |a||b| cos θ = x₁x₂ + y₁y₂ + z₁z₂ is used to find angles and test perpendicularity (a · b = 0).
向量具有大小和方向。分量形式下，从 A 到 B 的向量为 b – a。模长 |a| = √(x² + y² + z²)。方向 a 的单位向量为 a / |a|。加减法和数乘遵循分量逐项运算。点积 a · b = |a||b| cos θ = x₁x₂ + y₁y₂ + z₁z₂ 用于求夹角和判断垂直（a · b = 0）。
Vector equations of a straight line: r = a + λd, where d is a direction vector. For two lines, check if they intersect by equating components and solving for λ and μ. If no solution, lines are skew or parallel. Use the scalar product to find the angle between lines or between a line and a plane.
直线的向量方程：r = a + λd，其中 d 为方向向量。判断两直线是否相交：令对应分量相等，解 λ 和 μ。若无解，则两直线为异面或平行。利用点积求线线角或线面角。
7. Probability & Statistical Distributions | 概率与统计分布
Recap basic probability: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). Mutually exclusive events: P(A ∩ B) = 0. Independent events: P(A ∩ B) = P(A)P(B). Conditional probability P(A|B) = P(A ∩ B) / P(B). Use Venn diagrams and tree diagrams to organise information. A discrete random variable has a probability distribution summing to 1.
回顾基础概率：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。互斥事件满足 P(A ∩ B) = 0。独立事件满足 P(A ∩ B) = P(A)P(B)。条件概率 P(A|B) = P(A ∩ B) / P(B)。使用韦恩图和树状图整理信息。离散随机变量的概率分布之和为 1。
Binomial distribution X ~ B(n, p) models number of successes in n independent trials. P(X = r) = ⁿCᵣ pʳ (1-p)ⁿ⁻ʳ. The mean is np, variance np(1-p). Normal distribution X ~ N(μ, σ²) is continuous and bell-shaped. Use standardisation Z = (X – μ) / σ ~ N(0,1) to find probabilities from tables. Remember continuity correction when approximating binomial with normal.
二项分布 X ~ B(n, p) 模拟 n 次独立试验的成功次数。P(X = r) = ⁿCᵣ pʳ (1-p)ⁿ⁻ʳ。均值为 np，方差为 np(1-p)。正态分布 X ~ N(μ, σ²) 是连续钟形分布。使用标准化 Z = (X – μ) / σ ~ N(0,1) 查表求概率。用正态分布近似二项分布时注意连续性校正。
8. Hypothesis Testing | 假设检验
A hypothesis test evaluates evidence against a null hypothesis H₀. The alternative H₁ can be one-tailed or two-tailed. Define the test statistic, significance level α (commonly 0.05), and critical region. For binomial tests, find the probability of the observed result or more extreme under H₀; if p-value ≤ α, reject H₀.
假设检验用于评估反对原假设 H₀ 的证据。备择假设 H₁ 可以是单尾或双尾。定义检验统计量、显著性水平 α（常取 0.05）和拒绝域。对于二项检验，计算在原假设下观察到当前结果及更极端结果的概率；若 p 值 ≤ α，则拒绝 H₀。
For normal tests with known variance, use the z-test. For small samples with unknown variance, use the t-distribution. Always state a conclusion in context: ‘there is sufficient evidence to reject H₀ and accept H₁’ or ‘there is insufficient evidence to reject H₀’. Beware of Type I (false positive) and Type II (false negative) errors.
对于已知方差的正态检验，使用 z 检验。对于未知方差的小样本，使用 t 分布。结论始终要联系背景：“有充分证据拒绝 H₀，接受 H₁”或“没有充分证据拒绝 H₀”。警惕第一类错误（拒真）和第二类错误（取伪）。
9. Kinematics in One Dimension | 一维运动学
Master the SUVAT equations for constant acceleration: v = u + at, s = ut + ½at², v² = u² + 2as, s = ½(u + v)t, s = vt – ½at². You must know three variables to find the other two. Choose the direction of positive motion and apply signs consistently. Gravity g is 9.8 m/s² downwards.
掌握匀加速运动的 SUVAT 公式：v = u + at，s = ut + ½at²，v² = u² + 2as，s = ½(u + v)t，s = vt – ½at²。必须已知三个变量才能求另外两个。选定正方向后，符号要一致使用。重力加速度 g 取 9.8 m/s² 向下。
Displacement-time graphs: gradient equals velocity. Velocity-time graphs: gradient gives acceleration, area under graph gives displacement. For variable acceleration, use differentiation and integration: v = ds/dt, a = dv/dt = d²s/dt²; conversely s = ∫ v dt, v = ∫ a dt. Pay attention to initial conditions to find constants of integration.
位移-时间图：斜率等于速度。速度-时间图：斜率给出加速度，图下面积给出位移。对于变加速度，使用微积分：v = ds/dt，a = dv/dt = d²s/dt²；反之 s = ∫ v dt，v = ∫ a dt。注意利用初始条件求出积分常数。
10. Newton’s Laws & Connected Particles | 牛顿定律与连接体
Newton’s second law F = ma links resultant force, mass, and acceleration. Resolve forces in perpendicular directions. For particles on slopes, resolve weight into components parallel (mg sin θ) and perpendicular (mg cos θ). Friction F ≤ μR, where R is normal reaction; limiting equilibrium when F = μR. Use Fmax = μR for sliding.
牛顿第二定律 F = ma 将合力、质量与加速度联系起来。沿垂直方向分解力。对于斜面上的物体，将重力分解为平行于斜面的分量 (mg sin θ) 和垂直于斜面的分量 (mg cos θ)。摩擦力 F ≤ μR，其中 R 为法向反力；极限平衡时 F = μR。滑动时用 Fmax = μR。
Connected particles linked by a light inextensible string share the same speed and acceleration. Draw clear force diagrams for each particle, write equations of motion, and solve simultaneously. For pulleys, the tension is the same on both sides if the string is light and the pulley smooth. Include both linear and rotational contexts only if specified.
由轻质不可伸长绳子连接的物体具有相同的速率和加速度。为每个物体绘制清晰受力图，列出运动方程并联立求解。对于滑轮，若绳轻质且滑轮光滑，则两侧张力相等。除非指定，否则仅考虑直线运动情境。
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