📚 A-Level CCEA Mathematics: End-of-Term Revision Checklist | A-Level CCEA 数学：期末复习提纲
This comprehensive revision guide covers the essential topics for the CCEA A-Level Mathematics examinations, including both AS and A2 units. Key areas from Pure Mathematics, Mechanics and Statistics are summarised below so that you can focus your end-of-term revision efficiently, check off each skill and walk into the exam hall with a clear plan.
这份全面的复习指南涵盖了 CCEA A-Level 数学考试的核心主题，包括 AS 和 A2 单元。下面总结了纯数学、力学和统计学的关键领域，帮助你高效聚焦期末复习，逐项核对每项技能，带着清晰的计划走进考场。
1. Algebraic Techniques: Indices, Surds and Quadratics | 1. 代数技巧：指数、根式与二次方程
Master the manipulation of indices and surds, including simplifying expressions such as √8 ÷ √2 ng denominators. The core index laws are aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ and am/n = ⁿ√aᵐ. For surds, remember that √ab = √a × √b and that rationalising means multiplying top and bottom by the conjugate.
掌握指数和根式的运算，包括化简诸如 √8 ÷ √2 这类表达式以及分母有理化。核心指数律为 aᵐ × aⁿ = aᵐ⁺ⁿ、aᵐ ÷ aⁿ = aᵐ⁻ⁿ、(aᵐ)ⁿ = aᵐⁿ 以及 am/n = ⁿ√aᵐ。对于根式，要记住 √ab = √a × √b，而有理化就是把分子分母同乘共轭式。
− 4ac) to determine the nature of the roots. The quadratic formula is written as x = (−b ± √b² − 4ac) ÷ 2a, and completing the square turns x² − 6x + 4 into (x − 3)² − 5.
二次方程应通过因式分解、配方法或求根公式求解；务必检查判别式 (b² − 4ac) 以确定根的性质。求根公式写作 x = (−b ± √b² − 4ac) ÷ 2a，而配方则把 x² − 6x + 4 化为 (x − 3)² − 5。
4ac > 0 → two distinct real roots; Δ = 0 → one repeated root; Δ < 0 → no real roots
You should also be able to work with the completed-square form to find the vertex of a parabola and to prove that a quadratic is always positive.
你还应能利用配方形式求抛物线的顶点，并证明某个二次式恒为正。
2. Polynomials, Factor Theorem and Algebraic Division | 2. 多项式、因式定理与代数除法
Algebraic fractions and polynomial division (up to cubics) are tested regularly; remember the Factor Theorem: if f(a) = 0, then (x − a) is a factor. The Remainder Theorem states that when f(x) is divided by (x − a), the remainder is f(a).
代数分式和多项式除法（最高三次）经常出现；记住因式定理：若 f(a) = 0，则 (x − a) 为一个因式。余数定理指出：当 f(x) 除以 (x − a) 时，余数为 f(a)。
For division you may use long division or the inspection method of comparing coefficients; both are acceptable in CCEA papers, but show every line of working so that method marks are secure. Once one factor is found, factorise the remaining quadratic to obtain all three roots of a cubic.
除法可以使用长除法，也可以使用比较系数的待定系数法；在 CCEA 试卷中两者均可接受，但要写出每一步过程，以确保拿到方法分。找到一个因式后，再把剩余的二次式分解，从而得到三次方程的三个根。
A typical exam instruction reads "show that (x − 2) is a factor of f(x)", followed by "hence solve f(x) = 0". Always answer the "hence" part using your factorised form rather than restarting the calculation.
典型的考题表述是"证明 (x − 2) 是 f(x) 的因式"，接着是"由此解 f(x) = 0"。一定要用你分解后的形式来回答"由此"部分，而不要重新开始计算。
3. Simultaneous Equations and Inequalities | 3. 联立方程与不等式
Simultaneous equations (including two linear equations, or one linear and one quadratic) require an analytical solution. Substitute the linear equation into the quadratic one, rearrange to zero and solve; geometrically, the number of solutions equals the number of intersection points between a line and a curve.
联立方程组（包括两个线性方程，或一个线性一个二次方程）需要解析求解。把线性方程代入二次方程，整理为零后求解；从几何上看，解的个数等于直线与曲线的交点个数。
ons using set notation or interval notation, for example x < −2 or x > 3 as (−∞, −2) ∪ (3, ∞).
不等式需通过图像或代数方法求解，注意乘以负数时不等号的方向变化。用集合符号或区间符号表示解集，例如 x < −2 或 x > 3 写作 (−∞, −2) ∪ (3, ∞)。
For quadratic inequalities, always find the critical values first, sketch the parabola and then read off the regions that satisfy the inequality. For rational inequalities, multiply through carefully and state that the denominator cannot be zero.
对于二次不等式，先求出临界值，画出抛物线草图，再读出满足不等式的区间。对于分式不等式，要仔细通分，并注明分母不能为零。
4. Functions: Domain, Range and Mappings | 4. 函数：定义域、值域与映射
Understand the definitions of domain, range and injective or many-to-one mappings. A one-to-one (injective) function gives a different output for every input, whereas a many-to-one function maps several inputs to the same output.
理解定义域、值域以及单射或多对一映射的定义。单射函数对每个输入给出不同的输出，而多对一函数会把多个输入映射到同一个输出。
Typical questions ask for the largest possible domain of a function such as f(x) = √x − 2, which requires x ≥ 2, or g x ≠ 3. The range then follows from testing the end points and the behaviour of the graph.
典型题目会要求写出如 f(x) = √x − 2 这类函数的最大定义域，它要求 x ≥ 2；或 g(x) = 1 ÷ (x − 3)，它要求 x ≠ 3。值域则通过检验端点以及图像的变化趋势得出。
If a function is not one-to-one, you may need to restrict its domain before an inverse can exist; for example, restricting f(x) = x² to x ≥ 0 makes the inverse f⁻¹(x) = √x valid.
如果函数不是单射，你可能需要先限制其定义域，反函数才存在；例如把 f(x) = x² 限制在 x ≥ 0 上，反函数 f⁻¹(x) = √x 才成立。
5. Composite and Inverse Functions | 5. 复合函数与反函数
For composite functions fg(x), apply the inner function first: fg(x) means f(g(x)). Be careful with the order, because fg(x) is generally not equal to gf(x), and always state the domain of the composite carefully.
对于复合函数 fg(x)，先应用内层函数：fg(x) 表示 f(g(x))。注意顺序，因为 fg(x) 通常不等于 gf(x)，并且务必仔细写出复合函数的定义域。
Make sure you can find inverse functions f⁻¹(x) by swapping x and y and rearranging, and know that the graph of f⁻¹ is a reflection in y = x. The domain of f⁻¹ equals the range of f, and the range of f⁻¹ equals the domain of f.
确保能通过交换 x 和 y 再重排找到反函数 f⁻¹(x)，并知道反函数的图像是关于 y = x 的反射。反函数的定义域等于原函数的值域，反函数的值域等于原函数的定义域。
Remember the self-inverse property: ff⁻¹(x) = f⁻¹f y x in the appropriate domain. This is often used to verify an answer quickly in the exam.
记住自反性质：对相应定义域内的每个 x，都有 ff⁻¹(x) = f⁻¹f(x) = x。考试中常用它来快速验证答案。
6. Graph Transformations and Modulus Functions | 6. 图像变换与模函数
You must know the transformations: translations (y = f(x) + a moves the curve up by a, y = f(x + a) moves it left by a), stretches (y = af(x) is a vertical stretch of scale factor a, y = f(ax) is a horizontal stretch of scale factor 1 ÷ a) and reflections (y = −f(x), y = f(−x)).
你必须掌握这些变换：平移（y = f(x) + a 使曲线向上平移 a，y = f(x + a) 使其向左平移 a）、伸缩（y = af(x) 为纵向伸缩，比例因子 a；y = f(ax) 为横向伸缩，比例因子 1 ÷ a）以及反射（y = −f(x)、y = f(−x)）。
You should be able to sketch modulus functions y = |f(x)| and y = f(|x|). For y = |f(x)|, draw y = f(x) and reflect any part below the x-axis upwards; for y = f(|x|), delete the part for x < 0 and mirror the right-hand branch into the left.
你应该能够绘制模函数 y = |f(x)| 和 y = f(|x|) 的图像。对于 y = |f(x)|，先画 y = f(x)，再把 x 轴下方的部分向上翻折；对于 y = f(|x|)，删去 x < 0 的部分，并把右侧分支镜像到左侧。
Modulus equations such as |2x − 5| ses, 2x − 5 = 7 and 2x − 5 = −7, and each solution checked against the original statement.
解 |2x − 5| = 7 这类模方程时应分两种情况：2x − 5 = 7 与 2x − 5 = −7，并把每个解代回原式检验。
7. Standard Graphs, Asymptotes and Curve Sketching | 7. 标准函数图像与渐近线
Recognise standard functions: quadratic, cubic, reciprocal, exponential, logarithmic and trigonometric graphs, including their asymptotes and key features. A sketch must show intercepts, turning points, asymptotes and the general shape.
识别标准函数：二次、三次、倒数、指数、对数和三角函数图像，包括它们的渐近线和关键特征。草图必须体现截距、转折点、渐近线以及整体形状。
The reciprocal curve y = 1 ÷ x has asymptotes x = 0 and y = 0; the exponential curve y = eˣ has asymptote y = 0 and passes through (0, 1); the logarithmic curve y = ln x has asymptote x = 0 and passes through (1, 0).
倒数曲线 y = 1 ÷ x 的渐近线为 x = 0 和 y = 0；指数曲线 y = eˣ 的渐近线为 y = 0，并经过 (0, 1)；对数曲线 y = ln x 的渐近线为 x = 0，并经过 (1, 0)。
| Function 函数 | Key feature 关键特征 |
| y = x² | U-shape, minimum at the vertex 抛物线，顶点处取最小值 |
| y = x³ | Point of inflection at the origin 原点处为拐点 |
| y = 1 ÷ x | Two branches, asymptotes x = 0, y = 0 两支曲线，渐近线 x = 0、y = 0 |
| y = sin x, y = cos x | Period 360° or 2π 周期为 360° 或 2π |
| y = tan x | Asymptotes at x = 90° + 180°n 渐近线位于 x = 90° + 180°n |
8. Sequences, Series and the Binomial Expansion | 8. 数列、级数与二项式展开
× (2a + (n − 1)d). For geometric sequences, uₙ = arⁿ⁻¹ and Sₙ = a(1 − rⁿ) ÷ (1 − r); when |r| < 1 the series converges and S∞ = a ÷ (1 − r).
对于等差数列，第 n 项为 uₙ = a + (n − 1)d，前 n 项和为 Sₙ = n ÷ 2 × (2a + (n − 1)d)。对于等比数列，uₙ = arⁿ⁻¹，Sₙ = a(1 − rⁿ) ÷ (1 − r)；当 |r| < 1 时级数收敛，且 S∞ = a ÷ (1 − r)。
The binomial expansion of (a + b)ⁿ uses coefficients ⁿCᵣ, with general term ⁿCᵣ aⁿ⁻ʳ bʳ. For positive integer n the series terminates; for fractional or negative n the expansion is infinite and valid only for a restricted range of x.
(a + b)ⁿ 的二项式展开使用系数 ⁿCᵣ，通项为 ⁿCᵣ aⁿ⁻ʳ bʳ。当 n 为正整数时级数有限；当 n 为分数或负数时展开式无限，仅在 x 的有限范围内成立。
ity of an infinite binomial expansion, and check convergence conditions before quoting S∞.
对于无限二项式展开，一定要写出成立范围，并在引用 S∞ 前检查收敛条件。
9. Differentiation and Its Applications | 9. 微分及其应用
Differentiate from first principles, then apply the standard rules: the chain rule dy ÷ dx = dy ÷ du × du ÷ dx, the product rule (uv)′ = u′v + uv′, and the quotient rule (u ÷ v)′ = (u′v − uv′) ÷ v².
先掌握从基本原理求导，再运用标准法则：链式法则 dy ÷ dx = dy ÷ du × du ÷ dx，乘积法则 (uv)′ = u′v + uv′，商法则 (u ÷ v)′ = (u′v − uv′) ÷ v²。
Other essentials include differentiating eˣ, ln x, sin x, cos x and tan x, finding equations of tangents and normals, and locating stationary points where dy ÷ dx = 0. The second derivative decides the nature: d²y ÷ dx² > 0 gives a minimum and d²y ÷ dx² < 0 gives a maximum.
其他要点包括对 eˣ、ln x、sin x、cos x 和 tan x 求导，求切线与法线方程，以及找出 dy ÷ dx = 0 的驻点。二阶导数决定性质：d²y ÷ dx² > 0 为极小值，d²y ÷ dx² < 0 为极大值。
Applied questions on rates of change, connected rates and exponential growth or decay are common in A2 papers; set up the equation first, then differentiate with respect to time.
A2 试卷中常见的应用题涉及变化率、相关变化率以及指数增长或衰减；先建立方程，再对时间求导。
10. Integration and Areas Under Curves | 10. 积分与曲线下面积
e standard result ∫xⁿ dx = xⁿ⁺¹ ÷ (n + 1) + c for n ≠ −1, together with the integrals of eˣ imits and give a signed area, so sketch the curve to see which parts lie below the x-axis.
记住标准结果 ∫xⁿ dx = xⁿ⁺¹ ÷ (n + 1) + c（n ≠ −1），以及 eˣ、1 ÷ x、sin x 和 cos x 的积分。定积分带上下限，得到的是有向面积，因此要画草图以判断哪些部分位于 x 轴下方。
The area between two curves is found by integrating (upper curve − lower curve) between the intersection points, and the trapezium rule estimates an integral when an exact form is impossible.
两曲线之间的面积通过在两交点之间对（上方曲线 − 下方曲线）积分得到；当无法求出精确形式时，可用梯形法则估算积分值。
For A2, be ready to use integration by substitution, integration by parts and volumes of revolution, and always substitute the limits back in terms of the original variable.
在 A2 中，要准备好使用换元积分法、分部积分法和旋转体体积，并记得把上下限换回原变量。
11. Trigonometry, Exponentials and Logarithms | 11. 三角、指数与对数
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