📚 IB & CIE Mathematics: High-Frequency Exam Topics Summary | IB与CIE数学高频考点总结
Whether you are sitting IB Mathematics Analysis & Approaches or CIE A-Level Mathematics, certain topics appear year after year and carry significant weight. Mastering these high-frequency areas not only boosts your grade but also builds a solid foundation for the entire syllabus. This bilingual summary revisits the most commonly tested concepts, key formulas, and typical question styles found in both IB and CIE exams, helping you focus your revision efficiently.
无论你参加的是IB数学分析与方法还是CIE A-Level数学，一些知识点年复一年出现且占分很重。掌握这些高频考点不仅能提升成绩，还能为整个课程打下坚实基础。这篇中英双语总结回顾了IB和CIE考试中最常考的概念、重要公式和典型题型，帮助你高效复习。
1. Algebraic Techniques | 代数运算技巧
Fluency in algebraic manipulation is assumed in almost every question. Common tasks include expanding brackets, factorising quadratic and cubic expressions, simplifying rational functions, and using the laws of indices.
几乎所有题目都默认你能熟练进行代数变形。常见任务包括展开括号、分解二次和三次式、化简有理函数以及运用指数法则。
For example, factorising x² – 7x + 12 quickly as (x – 3)(x – 4) is essential for solving equations and sketching graphs.
比如，将 x² – 7x + 12 快速分解为 (x – 3)(x – 4) 对解方程和画图至关重要。
(a + b)² = a² + 2ab + b² a² – b² = (a – b)(a + b)
The difference of two squares and perfect square expansions are heavily tested in both IB and CIE, often hidden inside larger problems.
平方差和完全平方展开在IB和CIE中都大量出现，常常隐藏在大题内部。
2. Functions and Graphs | 函数与图像
Understanding domain, range, composite functions, and inverse functions is a recurring theme. You must interpret function notation f(x), find f(g(x)), and determine whether a function is one-to-one.
理解定义域、值域、复合函数与反函数是反复出现的主题。你必须解读函数记号 f(x)，求 f(g(x))，并判断函数是否是一一映射。
Graphical skills such as sketching parabolas, hyperbolas, exponentials, and trigonometric curves are fundamental. Transformations—translations, reflections, stretches—appear in both IB and CIE papers.
绘制抛物线、双曲线、指数曲线和三角函数曲线等图形技能是基本功。平移、对称、伸缩变换在IB和CIE试卷中都会出现。
For f(x) = a f(b(x – c)) + d, the parameter a affects vertical stretch, b horizontal stretch, c horizontal shift, and d vertical shift.
对于 f(x) = a f(b(x – c)) + d，参数 a 影响垂直伸缩，b 影响水平伸缩，c 水平平移，d 垂直平移。
3. Quadratics and Polynomials | 二次函数与多项式
Quadratic equations and their discriminants are a cornerstone. The discriminant Δ = b² – 4ac determines the nature of roots: two real distinct roots if Δ > 0, one repeated root if Δ = 0, and no real roots if Δ < 0.
二次方程及其判别式是基石。判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。
Sum and product of roots (Vieta’s formulas) are frequently examined: for ax² + bx + c = 0, sum = –b/a, product = c/a.
根的和与积（韦达定理）经常考查：对于 ax² + bx + c = 0，和为 –b/a，积为 c/a。
Polynomial division, the factor theorem, and the remainder theorem allow you to factorise cubic expressions like x³ – 3x + 2 into (x – 1)(x² + x – 2).
多项式除法、因式定理和余式定理能让你将 x³ – 3x + 2 这样的三次式分解为 (x – 1)(x² + x – 2)。
4. Inequalities | 不等式
Solving linear, quadratic, and rational inequalities is a high-frequency skill. Always remember to reverse the inequality sign when multiplying or dividing by a negative number.
求解一次、二次和分式不等式是高频考点。当乘以或除以负数时，务必记得反转不等号。
For a quadratic inequality such as x² – 4x – 5 ≤ 0, factorise to (x – 5)(x + 1) ≤ 0, then sketch the sign diagram to obtain the solution –1 ≤ x ≤ 5.
对于 x² – 4x – 5 ≤ 0 这样的二次不等式，先分解为 (x – 5)(x + 1) ≤ 0，再画符号图得到解 –1 ≤ x ≤ 5。
Absolute value inequalities like |2x – 3| > 5 are solved by splitting into two linear cases. IB and CIE both test the graphical interpretation of |f(x)| and f(|x|).
绝对值不等式如 |2x – 3| > 5 可通过拆分成两个一次不等式求解。IB和CIE都会考查 |f(x)| 和 f(|x|) 的图形解释。
5. Sequences and Series | 数列与级数
Arithmetic and geometric sequences appear frequently. Key formulas: for an arithmetic sequence, nth term uₙ = a + (n – 1)d, sum Sₙ = n/2 (2a + (n – 1)d). For a geometric sequence, uₙ = arⁿ⁻¹, sum Sₙ = a(1 – rⁿ)/(1 – r) for |r| < 1.
等差数列和等比数列经常出现。关键公式：等差数列第n项 uₙ = a + (n – 1)d，和 Sₙ = n/2 (2a + (n – 1)d)。等比数列 uₙ = arⁿ⁻¹，和 Sₙ = a(1 – rⁿ)/(1 – r)（|r| < 1）。
Infinite geometric series converge to a/(1 – r) when |r| < 1, a concept tested in both basic and applied contexts, such as recurring decimals.
无穷等比级数当 |r| < 1 时收敛于 a/(1 – r)，这一概念在基础和实际应用题（如循环小数）中都会出现。
Binomial expansion for rational powers, given by (1 + x)ⁿ = 1 + nx + n(n – 1)/2! x² + … , is valid for |x| < 1 and is common in IB HL and CIE P3.
有理次幂的二项展开式 (1 + x)ⁿ = 1 + nx + n(n – 1)/2! x² + … 在 |x| < 1 时成立，常见于IB HL和CIE P3。
6. Exponents and Logarithms | 指数与对数
The rules: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, and a⁰ = 1 (a ≠ 0) are used constantly. Converting between exponential form and logarithmic form—b = aˣ ↔ x = logₐ b—is crucial.
指数法则 aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ 和 a⁰ = 1（a ≠ 0）被频繁使用。指数式与对数式的互化 b = aˣ ↔ x = logₐ b 至关重要。
Logarithm laws: logₐ (xy) = logₐ x + logₐ y, logₐ (x/y) = logₐ x – logₐ y, logₐ xⁿ = n logₐ x. Always check the domain when solving log equations, as arguments must be positive.
对数法则：logₐ (xy) = logₐ x + logₐ y, logₐ (x/y) = logₐ x – logₐ y, logₐ xⁿ = n logₐ x。解对数方程时务必检查定义域，因为真数必须为正。
Exponential growth and decay models, P = P₀ eᵏᵗ, appear in both pure and applied sections, particularly in IB Paper 2 and CIE Mechanics.
指数增长和衰减模型 P = P₀ eᵏᵗ 在纯数和应用题中都会出现，尤其在IB卷二和CIE力学中。
7. Trigonometry | 三角学
Exact values of sin, cos, tan for 0°, 30°, 45°, 60°, 90° must be memorised. Radian measure is used throughout calculus; you must convert freely between degrees and radians (π rad = 180°).
必须熟记 0°, 30°, 45°, 60°, 90° 的 sin, cos, tan 精确值。微积分中统一使用弧度制，你需要熟练转换角度和弧度（π rad = 180°）。
Key identities: sin²θ + cos²θ = 1, tanθ = sinθ/cosθ, and double angle formulas sin2θ = 2sinθcosθ, cos2θ = cos²θ – sin²θ = 1 – 2sin²θ = 2cos²θ – 1. These are tested in solving trigonometric equations and simplifying expressions.
关键恒等式：sin²θ + cos²θ = 1, tanθ = sinθ/cosθ，二倍角公式 sin2θ = 2sinθcosθ, cos2θ = cos²θ – sin²θ = 1 – 2sin²θ = 2cos²θ – 1。这些恒等式用于解三角方程和化简表达式。
Graphs of sinx, cosx, tanx, their amplitude, period, and phase shift are examined. For y = a sin(bx + c) + d, period = 2π/|b|.
sinx, cosx, tanx 的图像及其振幅、周期和相位平移常考。对于 y = a sin(bx + c) + d，周期 = 2π/|b|。
8. Vectors | 向量
Vectors in 2D and 3D are essential in both IB and CIE. You must be able to add, subtract, and multiply vectors by scalars, and find magnitude: |v| = √(x² + y² + z²).
二维和三维向量在IB和CIE中都很关键。你必须掌握向量的加减、数乘以及求模长：|v| = √(x² + y² + z²)。
The dot product a · b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃ is used to find angles between vectors and test perpendicularity. The cross product (CIE and IB HL) gives a vector perpendicular to two vectors.
点积 a · b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃ 用于求向量夹角和检验垂直。叉积（CIE和IB HL）给出垂直于两个向量的向量。
Vector equations of lines: r = a + λb, and planes (IB HL, CIE P3) r · n = a · n. Finding intersections and distances between points and lines is a high-frequency written question.
直线的向量方程：r = a + λb，以及平面的方程（IB HL, CIE P3）r · n = a · n。求交点和点线距离是高频书面题。
9. Differentiation | 微分
Differentiation from first principles and the power rule d/dx (xⁿ) = nxⁿ⁻¹ are the starting points. You must differentiate polynomials, trigonometric, exponential, and logarithmic functions accurately.
从第一原理求导和幂法则 d/dx (xⁿ) = nxⁿ⁻¹ 是起点。你必须准确对多项式、三角函数、指数函数和对数函数求导。
| f(x) | f'(x) |
|---|---|
| sinx | cosx |
| cosx | –sinx |
| eˣ | eˣ |
| lnx | 1/x |
Product rule: (uv)’ = u’v + uv’. Quotient rule: (u/v)’ = (u’v – uv’)/v². Chain rule: dy/dx = dy/du × du/dx. These are combined in problems involving parametric equations.
乘积法则：(uv)’ = u’v + uv’。商法则：(u/v)’ = (u’v – uv’)/v²。链式法则：dy/dx = dy/du × du/dx。这些法则会综合用于参数方程问题。
Applications: finding tangents and normals, stationary points, and optimisation. Set f'(x) = 0 and use the second derivative f”(x) to classify maxima, minima, and points of inflection.
应用：求切线法线、驻点和最优化。令 f'(x) = 0 并用二阶导数 f”(x) 判断极大值、极小值和拐点。
10. Integration | 积分
Integration as the reverse of differentiation: ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ –1, and ∫ 1/x dx = ln|x| + C. Definite integrals find the area between a curve and the x-axis.
积分是微分的逆运算：∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C（n ≠ –1），且 ∫ 1/x dx = ln|x| + C。定积分用于求曲线与 x 轴之间的面积。
Integration by substitution and by parts are essential in IB HL and CIE P3. The formula for integration by parts: ∫ u dv = uv – ∫ v du.
换元积分法和分部积分法是IB HL和CIE P3的必要内容。分部积分公式：∫ u dv = uv – ∫ v du。
Volumes of revolution about the x-axis: V = π ∫ [f(x)]² dx from a to b. Trapezium rule for numerical integration is also examined when analytic integration is not possible.
围绕 x 轴旋转体的体积：V = π ∫ₐᵇ [f(x)]² dx。当解析积分不可行时，也会考察数值积分中的梯形法则。
11. Probability and Statistics | 概率与统计
Basic probability: P(A ∪ B) = P(A) + P(B) – P(A ∩ B), conditional probability P(A|B) = P(A ∩ B)/P(B). Tree diagrams and Venn diagrams help visualise independent and mutually exclusive events.
基础概率：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)，条件概率 P(A|B) = P(A ∩ B)/P(B)。树状图和韦恩图有助于展现独立和互斥事件。
Discrete random variables, expectation E(X) = ∑ x P(X = x), variance Var(X) = E(X²) – [E(X)]². The binomial distribution X ~ B(n, p) gives P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ.
离散随机变量，期望 E(X) = ∑ x P(X = x)，方差 Var(X) = E(X²) – [E(X)]²。二项分布 X ~ B(n, p) 有 P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ。
For continuous data, use the normal distribution X ~ N(μ, σ²). Standardise with Z = (X – μ)/σ then use the standard normal table. In CIE and IB, inverse normal calculations are common in Paper 2.
对于连续数据，使用正态分布 X ~ N(μ, σ²)。用 Z = (X – μ)/σ 标准化后查标准正态表。在CIE和IB的卷二里，反向正态计算很常见。
12. Complex Numbers | 复数
Complex numbers (IB HL, CIE P3) appear in the form z = a + bi where i² = –1. Operations: addition, multiplication, and division (multiplying by the complex conjugate).
复数（IB HL, CIE P3）以 z = a + bi 形式出现，其中 i² = –1。运算包括加法、乘法和除法（乘以共轭复数）。
The Argand diagram plots z as points. Modulus |z| = √(a² + b²), argument arg(z) = θ where tanθ = b/a. Polar form z = r(cosθ + i sinθ) = r cisθ simplifies multiplication and exponentiation.
阿甘图将 z 表示为点。模 |z| = √(a² + b²)，辐角 arg(z) = θ 满足 tanθ = b/a。极式 z = r(cosθ + i sinθ) = r cisθ 可简化乘法和幂运算。
De Moivre’s theorem: (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ) for integer n. It is used to find powers and roots of complex numbers, a high-frequency question in both IB HL and CIE P3.
棣莫弗定理：(cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)（n 为整数）。它用于求复数的幂和根，这是IB HL和CIE P3的高频考题。
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