📚 A-Level CIE Mathematics: Last-Minute Revision Notes | A-Level CIE 数学：考前冲刺笔记
This revision guide distills the most critical concepts, formulas, and problem-solving strategies for the CIE A-Level Mathematics syllabus. It is designed to help you quickly locate key information, avoid common pitfalls, and reinforce your understanding before the exam. Keep it by your side as you work through past papers.
这份考前冲刺笔记浓缩了 CIE A-Level 数学大纲中最核心的概念、公式和解题策略，旨在帮助你快速定位关键信息、避开常见陷阱、在考前巩固理解。建议你在刷历年真题时随身参考。
1. Algebra Basics and Polynomials | 代数基础与多项式
Mastering algebraic manipulation is essential across all topics. Pay special attention to factorising, completing the square, and the Remainder Theorem.
掌握代数运算是所有专题的基础，尤其要注意因式分解、配方法和余数定理。
The factor theorem states that (x − a) is a factor of polynomial f(x) if and only if f(a) = 0. If a polynomial f(x) is divided by (px + q), the remainder is f(−q/p). Completing the square rewrites ax² + bx + c as a(x + b/(2a))² + (c − b²/(4a)).
因式定理指出，(x − a) 是多项式 f(x) 的因式当且仅当 f(a) = 0。若多项式 f(x) 除以 (px + q)，余数为 f(−q/p)。配方法将 ax² + bx + c 化为 a(x + b/(2a))² + (c − b²/(4a))。
- Always check for common factors first before applying the quadratic formula.
- 使用求根公式前务必先提取公因式。
- Remember the discriminant Δ = b² − 4ac: Δ > 0 gives two distinct real roots, Δ = 0 one repeated root, Δ < 0 no real roots.
- 牢记判别式 Δ = b² − 4ac：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。
2. Functions and Graph Transformations | 函数与图形变换
Be comfortable with domain, range, composite functions, and inverse functions. A function must be one‑one to have an inverse, which can be found by swapping x and y and rearranging.
务必熟悉定义域、值域、复合函数和反函数。只有一一对应的函数才有反函数，求反函数可通过交换 x 和 y 并整理得出。
Transformations of the graph y = f(x):
y = f(x) + a → translation by (0, a)
y = f(x + a) → translation by (−a, 0)
y = af(x) → vertical stretch scale factor a
y = f(ax) → horizontal stretch scale factor 1/a
y = −f(x) → reflection in the x‑axis
y = f(−x) → reflection in the y‑axis
函数 y = f(x) 的图像变换规律：
y = f(x) + a → 沿向量 (0, a) 平移
y = f(x + a) → 沿向量 (−a, 0) 平移
y = af(x) → 垂直方向伸缩，比例因子为 a
y = f(ax) → 水平方向伸缩，比例因子为 1/a
y = −f(x) → 关于 x 轴反射
y = f(−x) → 关于 y 轴反射
For composite functions fg(x) = f(g(x)), the domain is determined by the domain of g such that the output of g lies in the domain of f. When sketching modulus functions |f(x)|, reflect any parts below the x‑axis above it.
对于复合函数 fg(x) = f(g(x))，其定义域由 g 的定义域和 f 对 g 的输出要求共同决定。绘制绝对值函数 y = |f(x)| 的图像时，将 x 轴下方的部分翻折至上方。
3. Trigonometry | 三角学
Know your exact values for sin, cos, tan at 0°, 30°, 45°, 60°, 90° and their radian equivalents. Radians: 180° = π rad.
熟记 0°、30°、45°、60°、90° 的正弦、余弦、正切精确值及其弧度制对应。弧度转换：180° = π rad。
| Angle / 角度 | 0° | 30° (π/6) | 45° (π/4) | 60° (π/3) | 90° (π/2) |
|---|---|---|---|---|---|
| sin | 0 | ½ | 1/√2 or √2/2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | ½ | 0 |
| tan | 0 | 1/√3 | 1 | √3 | undefined |
Key identities: sin²θ + cos²θ ≡ 1, tanθ ≡ sinθ/cosθ. For solving equations like sinθ = k, find the principal value and use the CAST diagram or general solutions: for sinθ = k, θ = nπ + (−1)ⁿα; for cosθ = k, θ = 2nπ ± α; for tanθ = k, θ = nπ + α (α in radians, n ∈ Z).
核心恒等式：sin²θ + cos²θ ≡ 1，tanθ ≡ sinθ/cosθ。求解形如 sinθ = k 的方程时，先求主值，再利用 CAST 图或通解公式：sinθ = k 时 θ = nπ + (−1)ⁿα；cosθ = k 时 θ = 2nπ ± α；tanθ = k 时 θ = nπ + α（α 为弧度，n 为整数）。
Remember the double‑angle formulas: sin2θ = 2sinθcosθ, cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ. Use them to integrate sin²x and cos²x: cos²x = (1 + cos2x)/2, sin²x = (1 − cos2x)/2.
牢记二倍角公式：sin2θ = 2sinθcosθ，cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ。利用它们可积分 sin²x 和 cos²x：cos²x = (1 + cos2x)/2，sin²x = (1 − cos2x)/2。
4. Exponentials and Logarithms | 指数与对数
The function y = eˣ and its inverse y = ln x appear frequently in differentiation, integration, and growth/decay models. Remember that ln x is only defined for x > 0.
函数 y = eˣ 及其反函数 y = ln x 在微积分和增长/衰减模型中频繁出现。注意 ln x 仅当 x > 0 时有定义。
Key laws: ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, ln aᵏ = k ln a. To solve an equation like e²ˣ = 5, take natural logs: 2x = ln5, so x = ½ ln5. When differentiating, d/dx (eᵏˣ) = k eᵏˣ and d/dx (ln x) = 1/x. By the chain rule, d/dx ln(f(x)) = f'(x)/f(x).
核心运算法则：ln(ab) = ln a + ln b，ln(a/b) = ln a − ln b，ln aᵏ = k ln a。求解 e²ˣ = 5 时，两边取自然对数：2x = ln5，故 x = ½ ln5。求导时，d/dx (eᵏˣ) = k eᵏˣ，d/dx (ln x) = 1/x。根据链式法则，d/dx ln(f(x)) = f'(x)/f(x)。
Integration: ∫ eᵏˣ dx = (1/k)eᵏˣ + c, ∫ 1/x dx = ln|x| + c. Watch out for definite integrals crossing the vertical asymptote of 1/x.
积分公式：∫ eᵏˣ dx = (1/k)eᵏˣ + c，∫ 1/x dx = ln|x| + c。注意当定积分区间跨越 1/x 的竖直渐近线时，积分无效。
5. Calculus: Differentiation | 微积分：微分
Differentiation gives the gradient of a curve. The derivative of xⁿ is nxⁿ⁻¹. Know the chain, product, and quotient rules.
微分用于求曲线的斜率。xⁿ 的导数为 nxⁿ⁻¹。需熟练掌握链式法则、乘积法则和商法则。
Product rule: d/dx (uv) = u’v + uv’
Quotient rule: d/dx (u/v) = (u’v − uv’)/v²
Chain rule: dy/dx = dy/du × du/dx
Implicit differentiation: differentiate both sides with respect to x, treating y as a function of x. For example, d/dx (y²) = 2y dy/dx. Parametric differentiation: if x = f(t), y = g(t), then dy/dx = (dy/dt) / (dx/dt).
隐函数求导：方程两边同时对 x 求导，将 y 视为 x 的函数。例如 d/dx (y²) = 2y dy/dx。参数方程求导：若 x = f(t), y = g(t)，则 dy/dx = (dy/dt) / (dx/dt)。
Connected rates of change: if three variables are related, use dA/dt = dA/dr × dr/dt etc. Stationary points occur where dy/dx = 0; determine their nature using the second derivative or a sign test.
相关变化率：若三个变量相关，则使用 dA/dt = dA/dr × dr/dt 等公式。驻点出现在 dy/dx = 0 处；用二阶导数或符号表判断其性质（极大、极小或拐点）。
6. Calculus: Integration | 微积分：积分
Integration is the reverse of differentiation. The basic rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c, provided n ≠ −1.
积分是微分的逆运算。基本公式：∫ xⁿ dx = xⁿ⁺¹/(n+1) + c，其中 n ≠ −1。
Definite integrals compute the area under a curve between limits a and b. The area between a curve and the x‑axis is ∫ₐᵇ y dx (where y ≥ 0). If the curve is given parametrically, area = ∫ y (dx/dt) dt with appropriate t limits. Volumes of revolution: about x‑axis V = π ∫ y² dx; about y‑axis V = π ∫ x² dy.
定积分用于计算曲线与 x 轴在区间 [a, b] 上围成的面积。面积 = ∫ₐᵇ y dx（要求 y ≥ 0）。若曲线由参数方程给出，面积 = ∫ y (dx/dt) dt，并配上相应的 t 积分限。旋转体体积：绕 x 轴旋转 V = π ∫ y² dx；绕 y 轴旋转 V = π ∫ x² dy。
For integration by substitution, choose u = g(x), then du = g'(x) dx, and replace all x terms with u before integrating. For integration by parts: ∫ u dv = uv − ∫ v du. Use the order of choosing u by the LIATE rule (Logs, Inverse trig, Algebraic, Trig, Exponentials).
换元积分法：令 u = g(x)，则 du = g'(x) dx，将原积分全部化为 u 的表达式后再积分。分部积分法：∫ u dv = uv − ∫ v du。选择 u 时遵循 LIATE 顺序（对数函数、反三角函数、代数函数、三角函数、指数函数）。
7. Numerical Methods | 数值方法
When algebraic solutions are impossible, numerical methods come to the rescue. The sign‑change rule locates roots: if f(a) and f(b) have opposite signs, there is at least one root in (a, b).
当无法用代数方法求解时，数值方法便能派上用场。符号变化法用于定位根：若 f(a) 与 f(b) 异号，则在区间 (a, b) 内至少存在一个根。
The Newton‑Raphson iteration: xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ). It converges rapidly but requires a suitable starting value. An alternative is the iterative rearrangement x = g(x) where xₙ₊₁ = g(xₙ); it converges if |g'(x)| < 1 near the root.
牛顿‑拉弗森迭代公式：xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ)。该方法收敛速度快，但需要合适的初始值。另一种方法是迭代重整 x = g(x)，迭代式为 xₙ₊₁ = g(xₙ)；当根附近满足 |g'(x)| < 1 时收敛。
The trapezium rule approximates ∫ₐᵇ f(x) dx ≈ h/2 [y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)], where h = (b − a)/n. This is an overestimate when the curve is convex and an underestimate when concave.
梯形法则近似计算定积分：∫ₐᵇ f(x) dx ≈ h/2 [y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)]，其中 h = (b − a)/n。当曲线为凸函数时结果为高估，为凹函数时结果为低估。
8. Vectors | 向量
Vectors describe both magnitude and direction. In 3D, a vector can be written as xi + yj + zk or as a column. The magnitude is √(x² + y² + z²).
向量用于描述大小和方向。在三维空间中，向量可表示为 xi + yj + zk 或列向量形式。其模长为 √(x² + y² + z²)。
The scalar (dot) product: a·b = |a||b| cos θ = x₁x₂ + y₁y₂ + z₁z₂. Use it to find the angle between two vectors or to test perpendicularity (a·b = 0). The vector equation of a line: r = a + td, where a is a fixed point on the line and d is the direction vector.
标量积（点乘）：a·b = |a||b| cos θ = x₁x₂ + y₁y₂ + z₁z₂。用于求两向量之间的夹角或判断垂直（a·b = 0）。直线的向量方程：r = a + td，其中 a 为直线上一定点，d 为方向向量。
To find the intersection of two lines, set their position vectors equal and solve for the parameters. To find the shortest distance from a point to a line, use the formula involving the modulus of the vector product.
求两直线交点时，令它们的位矢相等并解参数。求点到直线的最短距离，可使用包含向量积模长的公式。
9. Sequences and Series | 序列与级数
Arithmetic sequences have a common difference d: uₙ = a + (n−1)d. The sum to n terms: Sₙ = n/2 [2a + (n−1)d] or Sₙ = n/2 (a + l).
等差数列的公差为 d，通项公式：uₙ = a + (n−1)d。前 n 项和公式：Sₙ = n/2 [2a + (n−1)d] 或 Sₙ = n/2 (a + l)。
Geometric sequences have a common ratio r: uₙ = arⁿ⁻¹. The sum to n terms: Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. An infinite geometric series converges when |r| < 1, with sum S∞ = a/(1 − r).
等比数列的公比为 r，通项公式：uₙ = arⁿ⁻¹。前 n 项和：Sₙ = a(1 − rⁿ)/(1 − r)（r ≠ 1）。无穷等比级数当 |r| < 1 时收敛，和为 S∞ = a/(1 − r)。
Binomial expansion: (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + … The expansion is valid for |x| < 1 when n is not a positive integer. For (a + b)ⁿ, factor out aⁿ to get aⁿ(1 + b/a)ⁿ.
二项式展开：(1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + … 当 n 不为正整数时，展开式在 |x| < 1 时有效。对于 (a + b)ⁿ，提取 aⁿ 化为 aⁿ(1 + b/a)ⁿ。
10. Probability and Statistics | 概率与统计
Probability basics: P(A∪B) = P(A) + P(B) − P(A∩B). Conditional probability: P(A|B) = P(A∩B)/P(B). Independent events satisfy P(A∩B) = P(A)P(B).
概率基础：P(A∪B) = P(A) + P(B) − P(A∩B)。条件概率：P(A|B) = P(A∩B)/P(B)。若事件独立，则 P(A∩B) = P(A)P(B)。
For discrete random variables, E(X) = Σ x p(x) and Var(X) = E(X²) − [E(X)]². For linear combinations: E(aX + b) = aE(X) + b, Var(aX + b) = a² Var(X). For independent variables X and Y: E(X ± Y) = E(X) ± E(Y), Var(X ± Y) = Var(X) + Var(Y).
对于离散型随机变量，期望 E(X) = Σ x p(x)，方差 Var(X) = E(X²) − [E(X)]²。线性组合的性质：E(aX + b) = aE(X) + b，Var(aX + b) = a² Var(X)。若 X 与 Y 独立，则 E(X ± Y) = E(X) ± E(Y)，Var(X ± Y) = Var(X) + Var(Y)。
The binomial distribution X ~ B(n, p) has P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ, where q = 1 − p. The normal distribution X ~ N(μ, σ²) is standardised using Z = (X − μ)/σ. For sums of independent normal variables, use linear combinations. When approximating binomial with normal, apply continuity correction.
二项分布 X ~ B(n, p) 的概率公式：P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ，其中 q = 1 − p。正态分布 X ~ N(μ, σ²) 通过 Z = (X − μ)/σ 标准化。独立正态变量之和的分布仍为正态。用正态分布近似二项分布时需进行连续性修正。
11. Mechanics | 力学
Mechanics problems often involve constant acceleration formulas (SUVAT). Memorise: v = u + at, s = ut + ½ at², v² = u² + 2as, s = ½ (u + v)t.
力学问题常涉及匀加速运动公式（SUVAT）。牢记：v = u + at，s = ut + ½ at²，v² = u² + 2as，s = ½ (u + v)t。
Newton’s second law: F = ma. Resolve forces along sloping planes, remembering the weight component along the plane is mg sinθ. Friction, when limiting, is F = μR, where R is the normal reaction.
牛顿第二定律：F = ma。分析斜面上的力时需分解，注意沿斜面方向的重力分量为 mg sinθ。极限摩擦力为 F = μR，其中 R 为法向反力。
Momentum = mv. Conservation of momentum: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. For connected particles, treat them as a single system or write equations of motion for each. Power = Fv, and for a car travelling at constant speed, the driving force equals resistance.
动量 = mv。动量守恒：m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂。对于连接体问题，可将其视为一个整体或分别对每个物体列运动方程。功率 = Fv；汽车匀速行驶时，驱动力等于阻力。
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