Year 11 CAIE Math: Formula & Theorem Quick Reference Handbook | Year 11 CAIE 数学：公式定理速查手册

📚 Year 11 CAIE Math: Formula & Theorem Quick Reference Handbook | Year 11 CAIE 数学：公式定理速查手册

This short reference compiles the most important formulas, theorems, and relationships you need to master for the CAIE IGCSE Mathematics (0580/0980) syllabus at Year 11. Each section pairs an English explanation with a precise Chinese translation so you can check meanings instantly. Bookmark this page for revision, homework checks, and last‑minute exam prep.

本文是 Year 11 CAIE IGCSE 数学（0580/0980）最核心公式、定理与关系的速查手册。每一部分都采用英文与中文一对一对照讲解，方便你随时查阅、核对概念，也适合期末考试前的快速复习。

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1. Algebraic Essentials | 代数基础

Expanding brackets: a(b + c) = ab + ac. The same rule applies when signs are mixed, e.g. a(b – c) = ab – ac.

去括号法则：a(b + c) = ab + ac。包含负号时同理，例如 a(b – c) = ab – ac。

Factorising reverses expansion: look for common factors or use the difference of two squares, a² – b² = (a + b)(a – b).

因式分解是去括号的逆运算：先提取公因式，或利用平方差公式 a² – b² = (a + b)(a – b)。

For a quadratic equation ax² + bx + c = 0, the solutions are given by the quadratic formula:

x = ( –b ± √b² – 4ac ) ÷ (2a)

对于二次方程 ax² + bx + c = 0，求根公式为：

x = ( –b ± √b² – 4ac ) ÷ (2a)

The discriminant Δ = b² – 4ac es two real distinct roots; Δ = 0 gives one repeated real root; Δ < 0 gives no real roots.

判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 有两个不等实根；Δ = 0 有两个相等实根；Δ < 0 无实数根。

Simultaneous linear equations can be solved by substitution or elimination. For example, solve 2x + y = 7 and x – y = 2 by adding the two equations to eliminate y.

联立一次方程组可用代入法或消元法求解。例如解 2x + y = 7 与 x – y = 2，可将两式相加消去 y。

Direct proportion: y = kx; inverse proportion: y = k/x. The constant k can be found from a given pair of values.

正比例：y = kx；反比例：y = k/x。常数 k 可由已知数值对求出。

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2. Number & Exponents | 数系与指数

Laws of indices for any real base a (a ≠ 0):

• aᵐ × aⁿ = aᵐ⁺ⁿ
• aᵐ ÷ aⁿ = aᵐ⁻ⁿ
• (aᵐ)ⁿ = aᵐⁿ
• a⁰ = 1
• a⁻ⁿ = 1 / aⁿ
• a^1/n = ⁿ√a, e.g. a^1/2 = √a

指数运算律（底数 a ≠ 0）：

• aᵐ × aⁿ = aᵐ⁺ⁿ
• aᵐ ÷ aⁿ = aᵐ⁻ⁿ
• (aᵐ)ⁿ = aᵐⁿ
• a⁰ = 1
• a⁻ⁿ = 1 / aⁿ
• a^1/n = ⁿ√a，例如 a^1/2 = √a

Surds: simplify √ab = √a × √b, and rationalise denominators like 1/√a by multiplying numerator and denominator by √a.

根式化简：√ab = √a × √b，分母有理化如 1/√a 可分子分母同乘 √a。

Percentages: percentage change = (change ÷ original) × 100%. For repeated percentage increase/decrease use the multiplier (1 ± r%)ⁿ.

百分比：变化率 = (变化量 ÷ 原值) × 100%。重复增减用乘数 (1 ± r%)ⁿ。

Standard form: a number is written as A × 10ⁿ here 1 ≤ A < 10 and n is an integer. It makes very large or very small numbers manageable.

科学记数法：数字写成 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数，适合表示极大或极小的数。

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3. Coordinate Geometry | 坐标几何

The gradient (slope) of a straight line through points (x₁, y₁) and (x₂, y₂) is m = (y₂ – y₁) ÷ (x₂ – x₁).

过两点 (x₁, y₁) 和 (x₂, y₂) 直线的斜率（梯度）为 m = (y₂ – y₁) ÷ (x₂ – x₁)。

The equation of a straight line can be written as y = mx + c (slope-intercept form) or y – y₁ = m(x – x₁) (point-slope form). m is the gradient and c the y‑intercept.

直线方程可以写成 y = mx + c（斜截式）或 y – y₁ = m(x – x₁)（点斜式）。m 为斜率，c 为 y 轴截距。

Midpoint of two points: M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 ).

两点中点坐标：M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )。

Distance between two points: d = √[ (x₂ – x₁)² + (y₂ – y₁)² ].

两点间距离公式：d = √[ (x₂ – x₁)² + (y₂ – y₁)² ]。

Parallel lines have the same gradient m₁ = m₂; perpendicular lines satisfy m₁ × m₂ = –1.

两直线平行则斜率相等 m₁ = m₂；两直线垂直则斜率乘积 m₁ × m₂ = –1。

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4. Geometry & Angle Properties | 几何与角性质

Angles on a straight line add to 180°. Angles around a point sum to 360°.

直线上的邻角和为 180°；绕一点一周的角之和为 360°。

Vertically opposite angles are equal. Alternate angles on parallel lines are equal; corresponding angles are equal; interior (co‑interior) angles sum to 180°.

对顶角相等。平行线中的内错角相等；同位角相等；同旁内角和为 180°。

For any polygon: sum of interior angles = (n – 2) × 180°, where n is the number of sides. Sum of exterior angles always equals 360°.

任意多边形内角和 = (n – 2) × 180°，其中 n 为边数。外角和恒为 360°。

In a triangle, the interior angles add to 180°. The exterior angle equals the sum of the two opposite interior angles.

三角形内角和为 180°；一个外角等于与其不相邻的两个内角之和。

Symmetry: a regular polygon has equal sides and equal angles; lines of symmetry and order of rotational symmetry each equal n.

对称性：正多边形各边等长、各角相等；对称轴的条数和旋转对称的阶数都等于边数 n。

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5. Circle Theorems | 圆的定理

Angle at the centre is twice the angle at the circumference standing on the same arc: ∠AOB = 2 × ∠APB.

圆心角等于同弧所对圆周角的两倍：∠AOB = 2 × ∠APB。

Angle in a semicircle is a right angle (90°).

半圆上的圆周角是直角（90°）。

Angles in the same segment are equal.

同弧上的圆周角相等。

Opposite angles of a cyclic quadrilateral add to 180°: ∠A + ∠C = 180°, ∠B + ∠D = 180°.

圆内接四边形的对角互补：∠A + ∠C = 180°，∠B + ∠D = 180°。

The perpendicular from the centre to a chord bisects the chord. The tangent to a circle is perpendicular to the radius at the point of contact.

圆心到弦的垂线平分该弦。圆的切线与过切点的半径垂直。

Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.

弦切角定理：切线与弦的夹角等于该弦所对的另一侧圆周角。

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6. Trigonometry | 三角学

In a right‑angled triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

在直角三角形中：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。

Exact trigonometric values to learn:

θ	sin θ	cos θ	tan θ
0°	0	1	0
30°	½	√3 / 2	1 / √3
45°	1 / √2	1 / √2	1
60°	√3 / 2	½	√3
90°	1	0	undefined

必记的精确三角值见上表。

For non‑right triangles use the sine rule: a / sin A = b / sin B = c / sin C.

非直角三角形使用正弦定理：a / sin A = b / sin B = c / sin C。

Cosine rule: a² = b² + c² – 2bc cos A (and the rearranged form to find an angle: cos A = (b² + c² – a²) / (2bc) ).

余弦定理：a² = b² + c² – 2bc cos A（求角时的变形为 cos A = (b² + c² – a²) / (2bc)）。

Area of a triangle using sine: Area = ½ ab sin C.

三角形面积公式（已知两边夹角）：面积 = ½ ab sin C。

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7. Mensuration | 测量

= ½ × base × height, parallelogram = base × height, trapezium = ½ (a + b)h, circle = πr².

常用面积公式：矩形 = 长 × 宽，三角形 = ½ × 底 × 高，平行四边形 = 底 × 高，梯形 = ½ (a + b)h，圆 = πr²。

Circumference of a circle: C = 2πr = πd. Length of an arc = (θ/360) × 2πr. Area of a sector = (θ/360) × πr².

圆周长：C = 2πr = πd。弧长 = (θ/360) × 2πr。扇形面积 = (θ/360) × πr²。

Volume and surface area for prisms: volume = area of cross‑section × length. For a cylinder, V = πr²h, curved surface area = 2πrh, total surface area = 2πr(r + h). Cone: V = ⅓πr²h, slant height l, curved surface area = πrl. Sphere: V = ⁴⁄₃πr³, surface area = 4πr².

柱体体积 = 截面面积 × 长。圆柱：V = πr²h，侧面积 = 2πrh，总表面积 = 2πr(r + h)。圆锥：V = ⅓πr²h，母线长 l，侧面积 = πrl。球：V = ⁴⁄₃πr³，表面积 = 4πr²。

Pyramids: V = ⅓ × base area × vertical height. Pythagoras' theorem often helps find slant heights: a² + b² = c².

棱锥：V = ⅓ × 底面积 × 垂直高度。勾股定理常用来求斜高：a² + b² = c²。

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8. Vectors & Transformations | 向量与变换

A vector describes a translation: v = (x, y) where (x,y) are the horizontal and vertical components. The magnitude (length) is |v| = √x² + y².

向量表示平移：v = (x, y)，其中 (x,y) 为水平分量和竖直分量。模（长度）为 |v| = √x² + y²。

Vector addition: a + b = (x₁ + x₂, y₁ + y₂). Scalar multiplication: kv = (kx, ky). The vector AB = position of B − position of A.

向量加法：a + b = (x₁ + x₂, y₁ + y₂)。数乘：kv = (kx, ky)。向量 AB = B 的位置向量 − A 的位置向量。

Transformations in the plane: Reflection in the x‑axis flips the y‑nate sign; reflection in y = x swaps coordinates. Rotation of 90° about origin: (x, y) → (–y, x). Enlargement with scale factor k about the origin: (x, y) → (kx, ky).

平面变换：关于 x 轴的反射改变 y 坐标符号；关于直线 y = x 反射则交换坐标。绕原点转 90°：(x, y) → (–y, x)。以原点为中心、缩放因子 k 的放大：(x, y) → (kx, ky)。

A combination of transformations is applied step by step; the order matters.

复合变换按顺序逐步执行；顺序会影响结果。

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9. Sequences & Sets | 数列与集合

Arithmetic sequence: nth term uₙ = a + (n – 1)d, where a is the first term and d the common difference. Sum of first n terms: Sₙ = n/2 [2a + (n – 1)d] or Sₙ = n/2 (a + l) where l is the last term.

等差数列：通项 uₙ = a + (n – 1)d，a 为首项，d 为公差。前 n 项和：Sₙ = n/2 [2a + (n – 1)d] 或 Sₙ = n/2 (a + l)，l 为末项。

Quadratic sequence recognition: if the second differences are constant, the nth term is of the form an² + bn + c.

二次数列识别：若二阶差分为常数，则通项为 an² + bn + c 的形式。

Set notation: A ∪ B (union – all elements), A ∩ B (intersection – common elements), A' (complement). n(A) means the number of elements in set A. Venn diagrams help solve problems.

集合符号：A ∪ B（并集—所有元素），A ∩ B（交集—公共元素），A'（补集）。n(A) 表示集合 A 的元素个数。文氏图常用来解题。

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10. Probability & Statistics | 概率与统计

Probability of a single event: P(A) = number of favourable outcomes / total number of possible outcomes. For equally likely outcomes, 0 ≤ P(A) ≤ 1.

单事件概率：P(A) = 有利结果数 / 总可能结果数。等可能结果时，0 ≤ P(A) ≤ 1。

Addition rule: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). If A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B).

加法法则：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。若 A、B 互斥，则 P(A ∪ B) = P(A) + P(B)。

n branches from the same point sum to 1. For independent events, P(A and B) = P(A) × P(B).

树状图沿分支相乘概率；从同一点出发的各分支概率之和为 1。独立事件满足 P(A and B) = P(A) × P(B)。

Conditional probability: P(A|B) = P(A ∩ B) / P(B).

条件概率：P(A|B) = P(A ∩ B) / P(B)。

Statistics – an = (Σx) / n; median = middle value when ordered; mode = most frequent value. Range = max – min.

统计—平均数：平均数 = (Σx) / n；中位数 = 排序后中间的值；众数 = 出现次数最多的值。极差 = 最大值 – 最小值。

e highest frequency. Frequency density = frequency ÷ class width, used to draw histograms.

分组数据以组中值 × 频数估算平均数。众数组为频率最高的组。频率密度 = 频数 ÷ 组距，用于绘制直方图。

ive frequency graphs help find quartiles and the interquartile range (IQR = upper quartile – artile). Scatter diagrams show correlation; a line of best fit can be used for estimation.

累积频率图用于求四分位数和四分位距（IQR = 上四分位数 – 下四分位数）。散点图显示相关性；最佳拟合线可用于估计。

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