📚 IGCSE CIE Maths Last-Minute Revision Notes | IGCSE CIE 数学：考前冲刺笔记
This last‑minute guide condenses the core IGCSE CIE Mathematics syllabus into sharp, memorable points. Focus on key formulas, typical pitfalls, and efficient problem‑solving strategies.
这份考前冲刺笔记浓缩了 IGCSE CIE 数学核心考点，提炼关键公式、常见失分点与高效解题策略。
1. Number | 数
Prime factorisation: break any integer into a product of prime factors using a factor tree; every number has a unique prime factorisation.
质因数分解：用因数树将任一整数分解为质因数乘积；每个数的质因数分解形式唯一。
HCF and LCM: write numbers as products of primes, then take the lowest powers of common primes for HCF and the highest powers for LCM.
最大公因数（HCF）与最小公倍数（LCM）：将数字写成质数乘积，HCF 取相同质数的最低次幂，LCM 取所有质数的最高次幂。
Standard form: a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. Useful for very large or very small measurements.
标准形式：a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。用于表示极大或极小的度量。
Rounding to significant figures: start counting non‑zero digits from the first non‑zero digit; trailing zeros after a decimal point are significant.
有效数字四舍五入：从第一个非零数字开始计数非零数字；小数点后的尾随零为有效数字。
Upper and lower bounds: if a value is rounded to the nearest unit, the absolute error is half of that unit; use these bounds in compound calculations.
上下界：若一值四舍五入到某单位，绝对误差为该单位的一半；在复合运算中注意使用上下界。
Estimated range: measured value ± ½ × smallest unit
估计范围：测量值 ± ½ × 最小单位
2. Algebra and Equations | 代数与方程
Expanding brackets: multiply each term inside the bracket by the term outside. For two brackets, use FOIL (First, Outer, Inner, Last).
去括号：用括号外的项乘以括号内的每一项。两个括号相乘时使用 FOIL 法则（首、外、内、尾）。
Factorising: always look for a common factor first. For quadratics x² + bx + c, find two numbers that multiply to c and add to b.
因式分解：始终先提取公因数。对于二次式 x² + bx + c，找两个数使其乘积为 c、和为 b。
Solving quadratics: factorise if possible; otherwise use the quadratic formula. Check answers by substitution.
解二次方程：尽可能因式分解；否则使用二次公式。务必代入验根。
x = [ –b ± √(b² – 4ac) ] / (2a)
x = [ –b ± √(b² – 4ac) ] / (2a)
Simultaneous equations: elimination (add/subtract to remove one variable) or substitution. Always state the solution as x = …, y = … .
联立方程：消元法（加减消去一个变量）或代入法。最终答案务必写成 x = … , y = … 的形式。
Inequalities: treat like equations but reverse the inequality sign when multiplying or dividing by a negative number. Show solution on a number line.
不等式：解法类似方程，但当乘以或除以负数时不等号方向逆转。在数轴上表示解集。
Changing the subject: isolate the required variable using inverse operations; factorise if the variable appears in more than one term.
变换公式主项：利用逆运算隔离目标变量；若变量出现在多项中，先提取公因式。
3. Functions | 函数
A function maps each input (x) to exactly one output (f(x)). Use function notation f(x) = … .
函数将每一个输入 (x) 对应到唯一输出 (f(x))。牢记函数记号 f(x) = … 。
Composite function fg(x) means apply g first, then f. Work from right to left.
复合函数 fg(x) 意为先施加 g，再施加 f。从右向左计算。
Inverse function f⁻¹(x): swap x and y, then solve for y. The domain of f⁻¹ is the range of f.
反函数 f⁻¹(x)：交换 x 与 y 然后解出 y。f⁻¹ 的定义域是 f 的值域。
Graphs of functions: learn shapes of linear, quadratic, cubic, reciprocal and exponential graphs. Identify key points: intercepts, vertex, asymptotes.
函数图像：熟记一次、二次、三次、反比例及指数函数的形状。识别关键点：截距、顶点、渐近线。
4. Coordinate Geometry | 坐标几何
Mid‑point: ((x₁ + x₂)/2, (y₁ + y₂)/2)
中点公式：((x₁ + x₂)/2, (y₁ + y₂)/2)
Distance between two points: √[(x₂ – x₁)² + (y₂ – y₁)²]
两点间距离：√[(x₂ – x₁)² + (y₂ – y₁)²]
Gradient: m = (y₂ – y₁) / (x₂ – x₁). Positive gradient slopes upward; negative slopes downward.
斜率：m = (y₂ – y₁) / (x₂ – x₁)。正斜率向上倾斜；负斜率向下倾斜。
Equation of a straight line: y = mx + c, where m is the gradient and c is the y‑intercept. Can also use y – y₁ = m(x – x₁).
直线方程：y = mx + c，其中 m 为斜率，c 为 y 轴截距。也可使用点斜式 y – y₁ = m(x – x₁)。
Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = –1.
平行直线斜率相等；垂直直线斜率乘积为 –1。
5. Mensuration | 测量
Area: triangle (½bh), parallelogram (bh), trapezium [½(a + b)h], circle (πr²), sector ((θ/360)×πr²).
面积：三角形 ½bh，平行四边形 bh，梯形 ½(a+b)h，圆 πr²，扇形 (θ/360)×πr²。
Arc length: (θ/360) × 2πr, where θ is in degrees. Always check whether the angle is given in degrees or radians (if CIE requires, but IGCSE uses degrees).
弧长：(θ/360)×2πr，θ 以度为单位。务必确认角度单位是度。
Volume: prism (area of cross‑section × length), cylinder (πr²h), pyramid (⅓ × base area × height), cone (⅓πr²h), sphere (4/3 πr³).
体积：棱柱（横截面积 × 长），圆柱 πr²h，棱锥 ⅓×底面积×高，圆锥 ⅓πr²h，球体 4/3 πr³。
Surface area: cylinder (2πrh + 2πr²), cone (πrl + πr², where l is slant height), sphere (4πr²). Know how to combine shapes.
表面积：圆柱 2πrh+2πr²，圆锥 πrl+πr²（l 为斜高），球体 4πr²。掌握组合体的表面积求法。
6. Trigonometry | 三角学
SOH CAH TOA: sin = opp/hyp, cos = adj/hyp, tan = opp/adj. Use this only for right‑angled triangles.
SOH CAH TOA：sin = 对/斜，cos = 邻/斜，tan = 对/邻。仅适用于直角三角形。
Exact values: memorise sin, cos, tan of 0°, 30°, 45°, 60°, 90°. Often appear in non‑calculator papers.
精确值：熟记 0°、30°、45°、60°、90° 的 sin、cos、tan 值。非计算器试卷常考。
Sine rule: a/sin A = b/sin B = c/sin C, or sin A / a = sin B / b = sin C / c. Use when given two sides and a non‑included angle (SSA) or two angles and any side.
正弦定理：a/sin A = b/sin B = c/sin C。已知两边及一对角（SSA）或两角任一边时使用。
Cosine rule: a² = b² + c² – 2bc cos A. Use for SAS or SSS cases. Rearrange to find an angle: cos A = (b² + c² – a²) / 2bc.
余弦定理：a² = b² + c² – 2bc cos A。用于 SAS 或 SSS 情形。求角时变形为 cos A = (b² + c² – a²) / 2bc。
Area of any triangle: ½ ab sin C, where a and b are sides and C is the included angle.
任意三角形面积：½ ab sin C，a、b 为边，C 为夹角。
Trigonometric graphs: y = sin x, y = cos x, y = tan x. Know their amplitude, period, and asymptotes.
三角函数图像：y = sin x, y = cos x, y = tan x。熟悉振幅、周期及渐近线。
7. Vectors | 向量
Vectors have magnitude and direction; they can be represented as column vectors [x, y] or as a letter with an arrow.
向量既有大小又有方向；可用列向量 [x, y] 或带箭头的字母表示。
Addition and subtraction: add or subtract corresponding components. Scalar multiplication multiplies each component.
加减法：对应分量相加或相减。数乘将每个分量乘以该标量。
Magnitude: |a| = √(x² + y²). This is the length of the vector.
向量的模：|a| = √(x² + y²)，即向量的长度。
Position vectors: the vector from the origin to a point A is written as OA, having the same components as the coordinates of A.
位置向量：从原点指向点 A 的向量记作 OA，其分量与 A 的坐标相同。
Parallel vectors: vectors are parallel if one is a scalar multiple of the other. Collinear points share a common line when connecting vectors are multiples.
平行向量：若一向量是另一向量的标量倍数，则两者平行。当连接向量成倍数关系时，点共线。
8. Transformations | 变换
Reflection: mirror image in a given line. Matrices for reflections in x‑axis, y‑axis, y = x, y = –x are worth memorising.
反射：关于给定直线作镜像。记住关于 x 轴、y 轴、y=x、y=–x 反射的变换矩阵。
Rotation: turn around a centre by a given angle. Use tracing paper if needed.
旋转：绕一点转动给定角度。考试时可借助描图纸。
Translation: described by a column vector [x, y]; moves every point by the same vector.
平移：用列向量 [x, y] 描述，每个点都移动相同的向量。
Enlargement: scale factor k about a centre. If k > 1, shape enlarges; if 0 < k < 1, shape shrinks. Area scales by k².
放大：以某点为中心、比例因子 k。k>1 放大，0 Combined transformations: apply in the correct order. A transformation followed by another can be represented by matrix multiplication. 组合变换：按正确顺序施加。两个变换依次作用可用矩阵乘法表示。 Probability of an event not happening = 1 – P(event). Always check that probabilities sum to 1. 事件不发生的概率 = 1 – P(事件)。务必检查所有可能结果的概率之和为 1。 Tree diagrams: multiply along branches for combined events. For dependent events, probabilities change on second branches. 树状图：沿分支相乘求联合概率。若事件不独立，第二层分支的概率会变化。 Addition rule: P(A or B) = P(A) + P(B) – P(A and B). For mutually exclusive events, P(A and B) = 0. 加法法则：P(A 或 B) = P(A) + P(B) – P(A 且 B)。互斥事件时 P(A 且 B)=0。 Conditional probability: P(A|B) = P(A and B) / P(B). Read questions carefully to identify the reduced sample space. 条件概率：P(A|B) = P(A 且 B) / P(B)。仔细审题，识别缩小后的样本空间。 Mean = (sum of values) ÷ number of values. For grouped data, use midpoints × frequencies. 平均数 = 数值总和 ÷ 数据个数。分组数据使用组中值×频数。 Median: middle value when data are ordered. For cumulative frequency graphs, the median is at the 50th percentile. 中位数：数据排序后的中间值。在累积频率图中，中位数对应 50% 百分位数。 Interquartile range (IQR) = upper quartile – lower quartile. Box plots display minimum, Q₁, median, Q₃, maximum. 四分位距 (IQR) = 上四分位数 – 下四分位数。箱线图展示最小值、Q₁、中位数、Q₃、最大值。 Histograms: area of each bar represents frequency; frequency density = frequency ÷ class width. Use frequency density on the vertical axis. 直方图：每个柱的面积代表频数；频数密度 = 频数 ÷ 组距。纵轴为频数密度。 Scatter graphs: draw a line of best fit to estimate values. Correlation does not imply causation. 散点图：绘制最佳拟合线进行估值。相关性不等于因果性。 Published by TutorHao | Mathematics Revision Series | aleveler.com 更多咨询请联系16621398022（同微信）
9. Probability | 概率
10. Statistics | 统计学
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