A-Level CCEA Mathematics: Last-Minute Revision Notes | A-Level CCEA 数学：考前冲刺笔记

📚 A-Level CCEA Mathematics: Last-Minute Revision Notes | A-Level CCEA 数学：考前冲刺笔记

As the exam approaches, a focused revision can make all the difference. This article provides condensed, exam-ready notes covering the core topics of CCEA A-Level Mathematics, including pure mathematics, mechanics, and statistics. Use this guide to refresh key formulas, spot common pitfalls, and build confidence for the big day.

考试临近，集中复习能起到关键作用。本文提供浓缩的应考笔记，涵盖 CCEA A-Level 数学的核心主题，包括纯数学、力学和统计。利用本指南重温关键公式、发现常见陷阱，为考试日建立信心。

---

1. Algebraic Techniques | 代数技巧

indices: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, and a⁻ⁿ = 1/aⁿ. For surds, simplify expressions like √12 = 2√3 and rationalise denominators, e.g. multiply by (√a − √b)/(√a − √b).

掌握代数操作至关重要。复习指数定律：aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ。对于根式，简化如 √12 = 2√3 并有理化分母，例如乘以 (√a − √b)/(√a − √b)。

, Δ = 0 a repeated root, and Δ < 0 no real roots. The roots themselves can be found by factorising, completing the square, or the quadratic formula x = (−b ± √Δ)/(2a).

二次方程无处不在。对于 ax² + bx + c = 0，判别式 Δ = b² − 4ac 决定根的性质：Δ > 0 有两个不等实根，Δ = 0 有重根，Δ < 0 无实根。根本身可通过因式分解、配方法或求根公式 x = (−b ± √Δ)/(2a) 求出。

rtic factorisation. For binomial expansion, the series for (1 + x)ⁿ is 1 + n x + [n(n−1)/2!] x² + ... , valid for |x| < 1 when n is not a positive integer.

因式定理指出若 f(a) = 0，则 (x − a) 是多项式 f(x) 的因式。用于三次或四次多项式的分解。二项展开 (1 + x)ⁿ 的级数为 1 + n x + [n(n−1)/2!] x² + ... ，当 n 不是正整数时仅在 |x| < 1 有效。

---

2. Coordinate Geometry | 坐标几何

lines share the same gradient m; perpendicular lines have gradients satisfying m₁ m₂ = −1.

直线方程主要有三种形式：y = m x + c（斜截式）、y − y₁ = m (x − x₁) 和一般式 a x + b y + c = 0。平行直线斜率相同；垂线的斜率满足 m₁ m₂ = −1。

and radius r has equation (x − a)² + (y − b)² = r². To find tangents, use the fact that the radius to the point of contact is perpendicular to the tangent, or substitute y = m x + c into the circle equation and set the discriminant to zero.

两点 (x₁, y₁) 和 (x₂, y₂) 的中点为 ((x₁+x₂)/2, (y₁+y₂)/2)，距离为 √[(x₂−x₁)² + (y₂−y₁)²]。圆心 (a, b) 半径 r 的圆方程为 (x − a)² + (y − b)² = r²。求切线时，利用半径垂直于切线的性质，或将 y = m x + c 代入圆方程并令判别式为零。

Cartesian form y² = 4x. Differentiation can also be done using dy/dx = (dy/dt) ÷ (dx/dt).

参数方程使用第三个变量（通常为 t）描述曲线。对于抛物线 x = t², y = 2 t，消去 t 得到直角坐标形式 y² = 4x。求导时也可用 dy/dx = (dy/dt) ÷ (dx/dt)。

---

3. Trigonometry | 三角学

Know your exact values for sin, cos, and tan at 0°, 30°, 45°, 60°, 90° and their radian equivalents (0, π/6, π/4, π/3, π/2). Use the two key identities: sin²θ + cos²θ ≡ 1 and tanθ ≡ sinθ/cosθ. These are the foundation for solving equations and proving other identities.

熟记特殊角 0°, 30°, 45°, 60°, 90° 及其弧度值 (0, π/6, π/4, π/3, π/2) 的 sin、cos、tan 精确值。使用两个基本恒等式：sin²θ + cos²θ ≡ 1 和 tanθ ≡ sinθ/cosθ。这是解方程和证明其他恒等式的基础。

ves θ = arcsin k and θ = π − arcsin k; cosθ = k gives θ = ± arccos k; tanθ = k gives θ = arctan k + nπ.

解三角方程时，务必画图或使用 CAST 图求出给定区间内的所有解。牢记：sinθ = k 的解为 θ = arcsin k 和 θ = π − arcsin k；cosθ = k 的解为 θ = ± arccos k；tanθ = k 的解为 θ = arctan k + nπ。

A. Area of triangle = ½ ab sin C. Radian measure: arc length s = rθ, sector area = ½ r²θ.

正弦和余弦定理对非直角三角形至关重要。正弦定理：a/sin A = b/sin B = c/sin C。余弦定理：a² = b² + c² − 2bc cos A。三角形面积 = ½ ab sin C。弧度制下：弧长 s = rθ，扇形面积 = ½ r²θ。

---

4. Differentiation | 微分

Differentiation from first principles gives the derivative f'(x) = limₕ→₀ [f(x+h) − f(x)] / h. For standard functions: d/dx (xⁿ) = n xⁿ⁻¹; 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.

根据第一原理求导得到 f'(x) = limₕ→₀ [f(x+h) − f(x)] / h。标准函数的导数：d/dx (xⁿ) = n xⁿ⁻¹；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。

Apply the chain rule for composite functions: if y = g(f(x)), then dy/dx = g'(f(x)) · f'(x). Product rule: d/dx (u v) = u' v + u v'. Quotient rule: d/dx (u/v) = (u' v − u v' sides of an equation with respect to x and applying the chain rule to terms containing y.

复合函数使用链式法则：若 y = g(f(x))，则 dy/dx = g'(f(x)) · f'(x)。乘法法则：d/dx (u v) = u' v + u v'。除法法则：d/dx (u/v) = (u' v − u v') / v²。参数方程中 dy/dx = (dy/dt) / (dx/dt)。隐函数求导是对方程两边关于 x 求导，并对含 y 的项使用链式法则。

ange table for dy/dx. Connected rates of change use the chain rule: dV/dt = dV/dr · dr/dt.

驻点出现在 dy/dx = 0 处。利用二阶导数进行分类：d²y/dx² > 0 为极小值点，< 0 为极大值点，= 0 时需用一阶导数符号变化表判断。相关变化率使用链式法则：dV/dt = dV/dr · dr/dt。

---

5. Integration | 积分

is the reverse of differentiation. Basic integrals: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1); ∫ eˣ dx = eˣ + C; ∫ 1/x dx = ln|x| sec² x dx = tan x + C. Always add the constant of integration C for indefinite integrals.

积分是微分的逆运算。基本积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)；∫ eˣ dx = eˣ + C；∫ 1/x dx = ln|x| + C；∫ cos x dx = sin x + C；∫ sin x dx = −cos x + C；∫ sec² x dx = tan x + C。不定积分务必加上积分常数 C。

For more complex integrals, use substitution: let u = g(x), then ∫ f(g(x)) g' rivative appears. For definite integrals, remember to change the limits when substituting.

更复杂的积分使用换元法：令 u = g(x)，则 ∫ f(g(x)) g'(x) dx = ∫ f(u) du。选择 u 以简化函数，通常是内层函数或其导数已出现。定积分换元时记得更换积分限。

Inverse trig, Algebraic, Trigonometric, Exponential) rule to choose u. For example, ∫ x eˣ dx, set u = x, dv = eˣ dx. It can also be applied repeatedly when necessary.

分部积分源自乘法法则：∫ u dv = uv − ∫ v du。用 LIATE（对数、反三角、代数、三角、指数）规则选择 u。例如 ∫ x eˣ dx，令 u = x，dv = eˣ dx。必要时可重复使用。

)]² dx. Use these to solve applied problems, being careful with regions above and below the axis.

曲线 y = f(x) 与 x 轴在 x = a 到 x = b 之间的面积为 ∫ₐᵇ f(x) dx。绕 x 轴旋转的体积为 V = π ∫ₐᵇ [f(x)]² dx。解应用题时注意位于轴上下的区域。

---

广告 / Sponsored

6. Sequences and Series | 数列与级数

An arithmetic sequence has a common difference d: terms are a, a+d, a+2d, ... The nᵗʰ um of the first n terms is Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l), where l is the last term.

等差数列有公差 d：各项为 a, a+d, a+2d, ...。第 n 项为 a + (n−1)d。前 n 项和为 Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l)，其中 l 为末项。

A geometric sequence has a common ratio r: a, ar, ar², ... The nᵗʰ − rⁿ)/(1 − r) for r ≠ 1. An infinite geometric series converges to S∞ = a/(1 − r) provided |r| < 1. Watch out for the sum to infinity formula – it only works for convergent series.

等比数列有公比 r：a, ar, ar², ...。第 n 项为 a rⁿ⁻¹。前 n 项和为 Sₙ = a(1 − rⁿ)/(1 − r)，r ≠ 1。无穷等比级数当 |r| < 1 时收敛，其和为 S∞ = a/(1 − r)。注意无穷和公式仅适用于收敛级数。

vₖ. The binomial expansion can also be written as a series: (a + b)ⁿ = Σₖ₌₀ⁿ (ⁿₖ) aⁿ⁻ᵏ bᵏ.

使用西格玛符号 Σₖ₌₁ⁿ uₖ 表示级数。要能用线性性质处理求和：Σ (a uₖ + b vₖ) = a Σ uₖ + b Σ vₖ。二项式展开也可写成级数：(a + b)ⁿ = Σₖ₌₀ⁿ (ⁿₖ) aⁿ⁻ᵏ bᵏ。

---

7. Vectors | 向量

s column vectors or using i, j, k notation. The magnitude of vector a = x i + y j + z k is |a| = √x² + y² + z². A unit vector in the direction of a is â = a / |a|.

二维和三维向量表示具有大小和方向的量，可写作列向量或使用 i, j, k 记法。向量 a = x i + y j + z k 的模为 |a| = √x² + y² + z²。沿 a 方向的单位向量为 â = a / |a|。

The scalar (dot) product a · b = x₁ x₂ + y₁ y₂ + z₁ z₂ = |a| |b| on of a line is r = a + t b, where a is a point on the line and b is the direction vector.

标量积（点积）a · b = x₁ x₂ + y₁ y₂ + z₁ z₂ = |a| |b| cos θ，其中 θ 为两向量夹角。用于判断垂直（a · b = 0）和求角度。直线的向量方程为 r = a + t b，其中 a 为直线上一点，b 为方向向量。

For two lines to intersect, the parameters must satisfy all coordinates simultaneously. The shortest distance from a point to a line can be found using the perpendicular condition or vector cross product in some specifications, but in CCEA pure, dot product methods often suffice. Remind yourself of position vectors and displacement vectors.

两直线相交需参数同时满足所有坐标。点到直线的最短距离可用垂直条件或向量叉积（部分考纲），CCEA 纯数中通常使用点积方法即可。回顾位置向量和位移向量的概念。

---

8. Mechanics – Kinematics | 力学 – 运动学

) t / 2, s = u t + ½ a t², s = v t − ½ a t², v² = u² + 2 a s. Identify each symbol: s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time.

运动学描述运动而不考虑力。匀加速直线运动使用 SUVAT 方程：v = u + a t, s = (u + v) t / 2, s = u t + ½ a t², s = v t − ½ a t², v² = u² + 2 a s。明确各符号：s = 位移，u = 初速度，v = 末速度，a = 加速度，t = 时间。

ant velocity) and vertical motion (constant acceleration g). Always use sₓ = u cosθ t and sᵧ = u sinθ t − ½ g t².

重力作用下的运动取 a = 9.8 m s⁻²（或 9.81）向下。明确正方向。抛体运动将水平方向（匀速）与竖直方向（匀加速 g）分开处理。始终使用 sₓ = u cosθ t 和 sᵧ = u sinθ t − ½ g t²。

graph = displacement. With calculus, velocity v = ds/dt and acceleration a = dv/dt = d²s/dt². Conversely, s = ∫ v dt and v = ∫ a dt.

位移-时间和速度-时间图提供直观信息。s-t 图斜率 = 速度；v-t 图斜率 = 加速度；v-t 图下方面积 = 位移。通过微积分，速度 v = ds/dt，加速度 a = dv/dt = d²s/dt²。反向地，s = ∫ v dt，v = ∫ a dt。

---

9. Mechanics – Forces and Motion | 力学 – 力与运动

Newton' mponents when dealing with inclined planes or angled forces. Draw clear free-body diagrams.

牛顿定律支配力与运动的关系。第二定律：F = m a（合力 = 质量 × 加速度）。重量 W = m g。处理斜面或有角度的力时，将力分解为分量。绘制清晰的受力图。

cient of friction. On the point of sliding, F = μ R. For connected particles, apply Newton' nd law to each mass or to the whole system, being careful with the direction of tension T.

摩擦力阻碍运动，模型为 F ≤ μ R，R 为法向反作用力，μ 为摩擦系数。即将滑动时 F = μ R。对于连接物体问题，对每个质量或整体系统应用牛顿第二定律，注意张力 T 的方向。

= sum of anticlockwise moments, and the net force in any direction must be zero. Use these principles to find unknown forces or distances in static systems.

力矩度量力对支点的转动效应：力矩 = 力 × 垂直距离。平衡状态下，顺时针力矩之和等反时针力矩之和，且任何方向上的合力为零。利用这些原理求静态系统中的未知力或距离。

---

10. Statistics – Probability and Distributions | 统计 – 概率与分布

Probability revision: P(A∪B) = P(A) + P(B) − P(A∩B). For conditional probability, P(A|B) = P(A∩B) / P(B). Independent events satisfy P(A∩B) = P(A) P(B) or P(A|B) = P(A). Tree diagrams and Venn diagrams are helpful for multi-stage problems.

概率复习：P(A∪B) = P(A) + P(B) − P(A∩B)。条件概率 P(A|B) = P(A∩B) / P(B)。独立事件满足 P(A∩B) = P(A) P(B) 或 P(A|B) = P(A)。树形图和维恩图有助于多阶段问题。

(X = x) − μ². Expectation is linear: E(a X + b) = a E(X) + b, and Var(a X + b) = a² Var(X).

离散随机变量有概率分布且满足 Σ P(X = x) = 1。期望 E(X) = Σ x P(X = x)，方差 Var(X) = E(X²) − [E(X)]² = Σ x² P(X = x) − μ²。期望具有线性性质：E(a X + b) = a E(X) + b，Var(a X + b) = a² Var(X)。

The binomial distribution X ~ B(n, p) models the number of successes in n independent trials: P(X = r) = (ⁿᵣ) pʳ (1−p)ⁿ⁻ʳ. Mean = np, variance = np(1−p). The normal distribution X ~ . Standardise to Z = (X − μ)/σ and use tables to find probabilities. For the sample mean X̄ ion is N(μ, σ²/n) if the population is normal or by the Central Limit Theorem for large n.

二项分布 X ~ B(n, p) 模拟 n 次独立试验的成功次数：P(X = r) = (ⁿᵣ) pʳ (1−p)ⁿ⁻ʳ。均值为 np，方差为 np(1−p)。正态分布 X ~ N(μ, σ²) 是连续分布。标准化为 Z = (X − μ)/σ 并使用表格求概率。样本均值 X̄ 的分布，若总体正态或由中心极限定理对大样本成立，则为 N(μ, σ²/n)。

Hypothesis testing involves a null hypothesis H₀ and alternative H₁. For a binomial test, calculate the p-value (probability of observed result or more extreme under H₀) and compare with significance level α. For normal tests, use Z-tests or t-tests as appropriate, remembering critical regions and one-tailed vs two-tailed alternatives.

假设检验涉及原假设 H₀ 和备择假设 H₁。二项检验计算 p-值（在 H₀ 下得到观察结果或更极端结果的概率）并与显著性水平 α 比较。正态检验使用 Z 检验或 t 检验，注意临界域以及单尾与双尾的区别。

---

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）