📚 IGCSE CIE Additional Mathematics (0606): A Complete Syllabus Breakdown | IGCSE CIE 附加数学(0606)：课程大纲完全解析
IGCSE Additional Mathematics (0606) is designed for learners who have a strong foundation in basic mathematics and wish to deepen their understanding and problem-solving skills before progressing to AS and A Level courses. The syllabus extends the content of IGCSE Mathematics (0580) by introducing more advanced concepts such as calculus, logarithms, vectors, and permutations and combinations. This article provides a thorough breakdown of the entire syllabus, helping students and teachers plan an effective revision strategy and understand exactly what is expected in the final examinations.
IGCSE 附加数学(0606)是为那些已经具备扎实基础数学功底的学习者设计的，旨在帮助他们深入发展理解和解题技能，为衔接 AS 和 A Level 课程做好准备。该大纲在 IGCSE 普通数学(0580)的基础上，引入了微积分、对数、向量、排列组合等更进阶的概念。本文将对整个课程大纲进行全面解析，帮助学生和教师规划高效的复习策略，并清晰了解最终考试的具体要求。
1. Exam Overview & Paper Structure | 考试概览与试卷结构
All candidates take two written papers, Paper 1 and Paper 2. Each paper lasts 2 hours, contains about 11-13 questions of varying lengths, and contributes 80 marks. The total raw mark of 160 forms the basis for the final grade, which is reported on an A* to G scale. There is no coursework component. A calculator is required for both papers; knowledge of algebraic manipulation and numerical accuracy is essential.
所有考生都需要参加两份笔试，即试卷一和试卷二。每份试卷时长 2 小时，包含约 11 至 13 道长短不一的题目，满分 80 分。两份试卷原始总分 160 分，最终成绩以 A* 到 G 的等级呈现。该科目没有课程作业部分，两份试卷均需使用计算器，但同时要求学生具备熟练的代数运算与数值精度控制能力。
| Component | Duration | Marks | Weighting |
|---|---|---|---|
| Paper 1 | 2 hours | 80 | 50% |
| Paper 2 | 2 hours | 80 | 50% |
The questions cover the full syllabus and may require candidates to combine techniques from multiple topic areas. It is crucial to practise past papers under timed conditions to become familiar with the command words such as ‘evaluate’, ‘solve’, ‘sketch’, and ‘prove’.
试题覆盖整个大纲范围，可能要求考生综合运用不同主题的技巧。因此，在计时条件下练习历年真题，熟悉“求值”、“求解”、“草图绘制”和“证明”等指令词，是备考的关键。
2. Functions | 函数
The concept of a function, domain, range, and the notation f(x) are fundamental. You must be able to form composite functions (gf(x)) and find inverse functions f⁻¹(x). The condition for an inverse to exist is that the function must be one-to-one over its given domain. Graphical representation of functions and their inverses, reflecting in the line y = x, is also expected.
函数的概念、定义域、值域以及记号 f(x) 是基础中的基础。你必须掌握复合函数 gf(x) 的构造以及反函数 f⁻¹(x) 的求法。反函数存在的条件是该函数在其定义域上是一一映射。此外，还需要理解函数及其反函数的图像关系，即关于直线 y = x 对称。
Understanding transformations of graphs is a key skill. You will need to apply translations, stretches, and reflections to given graphs, interpreting the effect of changes such as y = f(x) + a, y = f(ax), y = |f(x)|, and y = f(|x|).
图像变换是一项核心技能。你需要对给定图形进行平移、伸缩和翻折操作，并理解诸如 y = f(x) + a、y = f(ax)、y = |f(x)| 以及 y = f(|x|) 等变化带来的影响。
3. Quadratic Functions & Inequalities | 二次函数与不等式
Quadratic functions of the form f(x) = ax² + bx + c are central. Candidates must be able to find the maximum or minimum value by completing the square, and hence determine the vertex of the parabola and its axis of symmetry. The discriminant Δ = b² – 4ac is used to identify the nature of the roots: two distinct real roots, one repeated real root, or no real roots.
形如 f(x) = ax² + bx + c 的二次函数是重点内容。考生必须能够通过配方法求最大值或最小值，进而确定抛物线的顶点和对称轴。判别式 Δ = b² – 4ac 用来判断根的性质：有两个不等实根、有一个重根，还是没有实数根。
Solving quadratic inequalities such as ax² + bx + c > 0 is often tackled by sketching the parabola and identifying the required intervals. You are also expected to solve equations reducible to quadratics, for example, those involving exponential terms like 3²ˣ – 10(3ˣ) + 9 = 0.
求解二次不等式（如 ax² + bx + c > 0）时，通常会先画出抛物线草图，再确定满足条件的区间。此外，你还需会解可化为二次方程的方程，例如包含指数形式的方程 3²ˣ – 10(3ˣ) + 9 = 0。
4. Indices, Surds & Logarithmic and Exponential Functions | 指数、根式、对数与指数函数
Laws of indices for rational exponents are extended, including negative and fractional powers. You must be able to simplify expressions involving surds and rationalise denominators. These algebraic skills underpin many other parts of the syllabus.
有理指数幂的运算法则将扩展到负指数和分数指数。你必须能够化简含有根式的表达式并进行分母有理化。这些代数技能是学习大纲许多其他部分的基础。
Logarithmic functions are introduced as inverses of exponential functions. Key relationships, such as logₐ(mn) = logₐ m + logₐ n, and the change of base formula, are tested. Solving equations like 2²ˣ = 5 by taking logarithms, and solving logarithmic equations using the properties of logs, are typical exam questions. You will also study the graphs of functions of the form y = keⁿˣ + c and y = k ln(ax + b).
对数函数被引入为指数函数的反函数。核心运算规则如 logₐ(mn) = logₐ m + logₐ n，以及换底公式都是考查内容。通过取对数解方程 2²ˣ = 5，以及利用对数性质解对数方程，是考试中常见的题型。你还将学习 y = keⁿˣ + c 及 y = k ln(ax + b) 这类函数的图像。
5. Polynomials & Simultaneous Equations | 多项式与联立方程组
The remainder theorem and factor theorem are essential tools for factorising cubic polynomials. You should be able to find the remainder when a polynomial P(x) is divided by (ax + b), and to use the factor theorem to solve polynomial equations. Full factorisation of cubic expressions and solving cubic equations with integer coefficients are required.
余式定理和因式定理是因式分解三次多项式的关键工具。你需要能求出多项式 P(x) 除以 (ax + b) 的余式，并利用因式定理求解多项式方程。大纲要求能对整系数三次式进行完全因式分解并求解三次方程。
Solving simultaneous equations extends to cases where one equation is linear and the other is quadratic or of a form that can be reduced to a pair of linear equations. Substitution is the main method, and you must be able to interpret the solutions as the intersection points of the two graphs.
解联立方程组延伸至一元一次与一元二次联立的情形，或可化为两个线性方程的形式。代入法是主要解题手段，你还要能够将方程组的解解释为两条图像曲线的交点。
6. Straight Line Graphs & Coordinate Geometry | 直线图形与坐标几何
You will work extensively with the equation of a straight line in various forms: y = mx + c, y – y₁ = m(x – x₁), and the general form ax + by + c = 0. Finding the distance between two points, the midpoint, and the gradient of perpendicular and parallel lines are fundamental skills.
你将大量使用不同形式的直线方程：y = mx + c、点斜式 y – y₁ = m(x – x₁)，以及一般式 ax + by + c = 0。计算两点间距离、中点坐标、以及平行线和垂直线的斜率，都是基本技能。
An important application is converting non-linear relationships into linear form to determine constants graphically. For instance, if given y = abˣ, you would rewrite it as log y = log a + x log b, then plot log y against x to find a and b from the intercept and gradient.
一个重要的应用是将非线性关系转化为线性形式，从而通过图形确定常数。例如，给定 y = abˣ，可改写为 log y = log a + x log b，然后以 log y 对 x 作图，从截距和斜率计算出 a 和 b。
7. Circular Measure & Trigonometry | 弧度制与三角学
Radian measure replaces degrees for most advanced work. You need to convert between degrees and radians, and use the arc length formula s = rθ and area of sector A = ½r²θ. Problems often combine these formulas with basic trigonometry to find areas of segments and perimeters of shaded regions.
在大部分进阶内容中，弧度制取代了角度制。你需要能在度与弧度间进行转换，并熟练运用弧长公式 s = rθ 和扇形面积公式 A = ½r²θ。实际问题常将这些公式与基础三角学结合，以求解弓形面积和阴影区域周长。
The syllabus covers the three basic trigonometric ratios for angles of any magnitude, graphs of sin, cos, and tan, and simple trigonometric identities such as tanθ = sinθ/cosθ and sin²θ + cos²θ = 1. Solving trigonometric equations within a specified interval, including those using double-angle formulas like sin2θ = 2sinθcosθ, is a regular feature of the exam.
大纲涵盖任意角下的三个基本三角比、正弦、余弦和正切函数的图像，以及简单的三角恒等式，如 tanθ = sinθ/cosθ 和 sin²θ + cos²θ = 1。在指定区间内求解三角方程（包括使用诸如 sin2θ = 2sinθcosθ 等倍角公式）是考试的常客。
8. Permutations & Combinations | 排列与组合
You will learn to distinguish between arrangements where order matters (permutations) and selections where order does not matter (combinations). The notation nPr and nCr is used, and you must be able to evaluate expressions such as 10P3 and 8C5 as well as solve equations involving factorials.
你将学习区分有序排列（排列）和无序选择（组合）。需要使用记号 nPr 和 nCr，并能够计算 10P3 和 8C5 这类表达式，以及求解含有阶乘符号的方程。
Typical problems involve arranging letters of a word with repeated characters, selecting a committee from a group with restrictions, or arranging items around a circle. Understanding when to multiply and when to add the count of possibilities is crucial.
典型的题目包括排列含重复字母的单词、从带有约束条件的一组人中选取委员会成员，或将物品排列成一个圆圈。理解何时对可能情况数目进行乘法、何时进行加法，这一点至关重要。
9. Series & Binomial Theorem | 级数与二项式定理
The binomial expansion of (a + b)n for positive integer n uses the formula involving nCr. You must be able to expand expressions such as (2x – 3)5 and find a specific term, for example, the term independent of x. The link between binomial coefficients and Pascal’s triangle is also explored.
二项式 (a + b)n （n 为正整数）的展开用到包含 nCr 的公式。你应能展开如 (2x – 3)5 这类表达式，并找出特定项，例如不含 x 的项。同时还要探究二项式系数与杨辉三角间的联系。
Arithmetic progression (AP) and geometric progression (GP) are covered in detail. You need to find the n-th term, the sum of the first n terms, and the sum to infinity for a convergent GP. Real-world applications, such as compound interest or population growth, often serve as the context for series problems.
等差数列和等比数列是详细考察的内容。你需要能求第 n 项、前 n 项和，以及对收敛的等比数列求无穷项和。复利计算或人口增长等实际应用常常作为数列问题的背景出现。
10. Vectors in Two Dimensions | 平面向量
Vectors are expressed in column form, component form using i and j, and position vectors. Operations include addition, subtraction, multiplication by a scalar, and finding the magnitude of a vector. You must be able to use the dot product to determine the angle between two vectors or to check for perpendicularity.
向量以列形式、用 i 和 j 表示的分量形式，以及位置向量等形式表达。运算包括加法、减法、与标量的乘法，以及求向量的模长。你必须会用点积来求两向量间的夹角，或者验证垂直关系。
Geometry problems using vectors are common: for instance, finding a unit vector in a given direction, expressing a point dividing a line segment in a given ratio, or proving that three points are collinear. Velocity and relative velocity problems round out the vector topic.
利用向量的几何问题也很常见：例如求给定方向的单位向量、用给定比例表示线段分割点，或证明三点共线。速度与相对速度问题使向量这一主题更加完整。
11. Differentiation | 微分
Differentiation from first principles is not required, but you must be proficient with the power rule: if y = xⁿ, then dy/dx = nxⁿ⁻¹. This extends to sums, differences, and constant multiples. The derivative of functions such as sin x, cos x, eˣ, and ln x must be known. The chain rule, product rule, and quotient rule are all assessed.
大纲不要求从第一性原理推导导数，但你必须熟练运用幂法则：若 y = xⁿ，则 dy/dx = nxⁿ⁻¹。这一法则扩展到和、差以及常数倍的情形。sin x、cos x、eˣ 和 ln x 等函数的导数必须牢记。链式法则、乘积法则和商法则均在考查范围之内。
Applications of differentiation include finding gradients of curves, equations of tangents and normals, and determining the nature of stationary points (maximum, minimum, or point of inflection). You will also solve problems involving rates of change and small increments using δy ≈ (dy/dx) δx.
微分的应用包括求曲线的斜率、切线和法线方程，以及判断驻点的性质（极大点、极小点或拐点）。你还将解决涉及变化率的问题，并利用 δy ≈ (dy/dx) δx 处理微小增量。
12. Integration | 积分
Integration is treated as the reverse of differentiation. The fundamental formula is ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, for n ≠ -1. You must also integrate standard functions like eˣ, sin x, and cos x. Definite integrals are used to find the area under a curve between two x-limits or the area enclosed between a curve and a line.
积分被视为微分的逆运算。基本公式为 ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C（n ≠ -1）。你还需要能对 eˣ、sin x 及 cos x 等标准函数进行积分。定积分用来求一条曲线在指定的两个 x 界限之间的面积，或求曲线和某直线所围区域的面积。
Syllabus coverage also includes solving simple differential equations of the form dy/dx = f(x) by separating variables, and finding the equation of a curve given its derivative and a point on the curve. Kinematics-style problems involving displacement, velocity, and acceleration are a common context.
大纲还覆盖了通过分离变量法求解形如 dy/dx = f(x) 的简单微分方程，以及已知导数与曲线上一点求曲线方程。涉及位移、速度和加速度的运动学背景问题也是常见的考查情境。
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