📚 Cambridge IGCSE Mathematics Core & Extended: Essential Revision Guide | 剑桥 IGCSE 数学核心与扩展：核心考点精讲
This revision guide follows the Cambridge IGCSE Mathematics Core and Extended coursebook, focusing on the key topics that regularly appear in both Paper 1 and Paper 3. Use the notes below to refresh your understanding of essential methods, notation and problem-solving strategies before your examination.
本复习指南依据剑桥 IGCSE 数学核心与扩展教材编写，聚焦 Paper 1 和 Paper 3 中高频出现的核心主题。使用以下要点帮助你巩固基本方法、数学符号和解题策略，为考试做好准备。
1. Number and Set Language | 数与集合语言
Number work underpins every IGCSE Mathematics question, so you must be confident with integers, fractions, decimals, percentages, powers and roots. Know the symbols for natural numbers (ℕ), integers (ℤ), rational numbers (ℚ) and real numbers (ℝ), and understand that irrational numbers cannot be written as a fraction.
数的运算是每道 IGCSE 数学题的基础，因此你必须熟练掌握整数、分数、小数、百分数、幂与根。熟悉自然数 (ℕ)、整数 (ℤ)、有理数 (ℚ) 和实数 (ℝ) 的符号，并理解无理数不能用分数表示。
Set language uses symbols such as ∪ (union), ∩ (intersection), ∈ (is an element of), ⊂ (subset) and ∅ (empty set). Prime factors, highest common factor (HCF) and lowest common multiple (LCM) are frequently tested, especially in word problems involving cycles or shared quantities.
集合语言使用 ∪（并集）、∩（交集）、∈（属于）、⊂（子集）和 ∅（空集）等符号。质因数、最大公因数 (HCF) 和最小公倍数 (LCM) 常被考查，尤其是在涉及周期或共享数量的应用题中。
aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ
2. Algebra: Expressions and Equations | 代数：表达式与方程
Algebraic manipulation includes expanding brackets, factorising by common factors, difference of two squares, and factorising quadratic trinomials. Always simplify expressions fully before attempting to solve an equation.
代数运算包括去括号、提取公因式、平方差公式以及二次三项式因式分解。在解方程之前，务必先把表达式化简到最简形式。
For linear equations, isolate the unknown using inverse operations. For quadratic equations, set one side to zero and factorise where possible; otherwise apply the quadratic formula. Remember the discriminant b² − 4ac tells you the nature of the roots.
对于线性方程，使用逆运算分离未知数。对于二次方程，先将一边设为零，尽可能因式分解；否则使用求根公式。记住判别式 b² − 4ac 可以判断根的性质。
x = (−b ± √b² − 4ac) ÷ 2a
3. Sequences and Series | 数列与级数
Arithmetic sequences have a constant difference d. The nth term is found using a + (n − 1)d and the sum of the first n terms is given by n/2 [2a + (n − 1)d]. Always check the value of d by subtracting consecutive terms.
等差数列具有固定的公差 d。第 n 项公式为 a + (n − 1)d，前 n 项和公式为 n/2 [2a + (n − 1)d]。务必通过相邻两项相减来检验 d 的值。
Geometric sequences have a constant ratio r. The nth term is arⁿ⁻¹ and the sum of the first n terms is a(1 − rⁿ)/(1 − r) for r ≠ 1. Recognise patterns such as 2, 4, 8, 16 or 81, 27, 9, 3 quickly in examinations.
等比数列具有固定的公比 r。第 n 项公式为 arⁿ⁻¹，前 n 项和公式为 a(1 − rⁿ)/(1 − r)（r ≠ 1）。在考试中要能快速识别 2、4、8、16 或 81、27、9、3 等规律。
uₙ = a + (n − 1)d or uₙ = arⁿ⁻¹
4. Functions and Graphs | 函数与图像
. The domain is the set of allowed inputs, while the range is the set of possible outputs.
函数将每个输入 x 对应到唯一的输出 f(x)。你应当能够根据公式求 f(a)、构造复合函数 f(g(x)) 并解方程 f(x) = k。定义域是允许输入值的集合，值域是可能输出值的集合。
re parabolas; the sign of a determines whether they open upwards or downwards. Use symmetry and the vertex to sketch graphs quickly.
线性图像的形式为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。二次函数图像 y = ax² + bx + c 为抛物线；a 的正负决定开口方向。利用对称性和顶点快速绘制图像。
y = mx + c, f(x) = ax² + bx + c
5. Ratio, Proportion and Rates | 比、比例与速率
Ratios compare quantities in the same units. To divide a quantity in a given ratio, add the total parts and multiply each part by the value of one share. Always simplify ratios to their lowest terms using common factors.
比用于比较同一单位下的数量。按给定比例分配数量时，先求出总份数，再用一份的值乘以各部分对应的份数。务必用公因数将比化为最简形式。
nd pressure are common rate problems: speed = distance ÷ time and density = mass ÷ volume.
正比例表示 y = kx，因此 y 与 x 以相同速率增加。反比例表示 y = k/x，因此 x 增大时 y 减小。速度、密度和压强是常见的速率问题：速度 = 路程 ÷ 时间，密度 = 质量 ÷ 体积。
y = kx (direct) and y = k/x (inverse)
6. Geometry and Mensuration | 几何与测量
les on parallel lines. The sum of interior angles of an n-sided polygon is (n − 2) × 180°.
角的性质包括平角（180°）、周角（360°）、对顶角、平行线中的内错角和同位角。n 边形内角和为 (n − 2) × 180°。
Circle theorems require precise reasoning: angle at the centre is twice the angle at the circumference, angles in the same segment are equal, and the angle in a semicircle is 90°. Mensuration formulas for area and volume must be memorised, including πr² for a circle and 4/3 πr³ for a sphere.
圆定理要求精确推理：圆心角是圆周角的两倍、同弧上的圆周角相等、直径所对的圆周角为 90°。面积和体积公式必须牢记，包括圆面积 πr² 和球体积 4/3 πr³。
Area = πr², Volume of sphere = 4/3 πr³, Volume of cylinder = πr²h
7. Trigonometry | 三角学
nt. Identify the correct sides relative to the angle before substituting into the formula.
在直角三角形中，SOHCAHTOA 给出 sin θ = 对边/斜边、cos θ = 邻边/斜边、tan θ = 对边/邻边。代入公式前，先根据角度正确识别各边。
rea can also be found by ½ ab sin C. Bearings, angles of elevation and depression are common applications in Paper 3.
对于非直角三角形，使用正弦定理 a/sin A = b/sin B 和余弦定理 a² = b² + c² − 2bc cos A。面积也可用 ½ ab sin C 求得。方位角、仰角和俯角是 Paper 3 中常见的应用题。
a/sin A = b/sin B, a² = b² + c² − 2bc cos A
8. Vectors and Transformations | 向量与变换
A vector has both magnitude and direction. Vectors can be written as column vectors or in component form, and added or subtracted by combining corresponding components. Multiplying by a scalar changes the magnitude but not the direction unless the scalar is negative.
向量既有大小又有方向。向量可以写成列向量或分量形式，并通过合并对应分量进行加减。乘以标量会改变大小，但除非标量为负，否则不改变方向。
Transformations include translation, rotation, reflection and enlargement. A translation is described by a vector, an enlargement by a centre and scale factor k, and a rotation by a centre, angle and direction. The inverse of a transformation returns a shape to its original position.
变换包括平移、旋转、反射和放大。平移用向量描述，放大用中心和比例因子 k 描述，旋转用中心、角度和方向描述。逆变换使图形回到原始位置。
Vector addition: (x₁, y₁) + (x₂, y₂) = (x₁ + x₂, y₁ + y₂)
9. Probability | 概率
tcomes ÷ total number of outcomes. The sum of probabilities of all possible outcomes is 1.
概率是介于 0 和 1 之间的度量。对于等可能结果，P(事件) = 有利结果数 ÷ 总结果数。所有可能结果的概率之和为 1。
ee diagrams help to organise conditional probabilities and repeated trials without replacement.
互斥事件不能同时发生，因此 P(A 或 B) = P(A) + P(B)。独立事件互不影响，因此 P(A 与 B) = P(A) × P(B)。树状图有助于整理条件概率和不放回反复试验。
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
10. Statistics | 统计
Measures of central tendency include the mean, median and mode. The mean is the sum of values divided by the number of values, the median is the middle value when data are ordered, and the mode is the most frequent value.
集中趋势的度量包括平均数、中位数和众数。平均数是数据之和除以数据个数；中位数是排序后位于中间的值；众数是最常出现的值。
Measures of spread include range, interquartile range and standard deviation. Cumulative frequency graphs are used to estimate medians and quartiles from grouped data. Histograms display frequency density when class widths are unequal.
离散程度的度量包括极差、四分位距和标准差。累积频率图用于从分组数据中估计中位数和四分位数。当组距不等时，直方图使用频率密度来展示。
Mean = Σfx ÷ Σf, Range = highest value − lowest value
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