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

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

This quick reference handbook gathers all the essential formulas and theorems you need to master Year 11 CAIE IGCSE Mathematics. Whether you are revising for your final exams or working through past papers, having these results at your fingertips saves time and helps you avoid common mistakes. Use this guide alongside your textbook and class notes to reinforce understanding.

本速查手册汇总了 Year 11 CAIE IGCSE 数学所需的所有核心公式与定理。无论你在备战大考还是练习历年真题，把关键结论信手拈来都能节省时间并减少常见错误。结合教材和课堂笔记一起使用，将有效巩固理解。

1. Algebraic Identities & Equations | 代数恒等式与方程

Expanding a perfect square: (a + b)² = a² + 2ab + b²

完全平方展开： (a + b)² = a² + 2ab + b²

Difference of squares: a² – b² = (a – b)(a + b)

平方差公式： a² – b² = (a – b)(a + b)

Quadratic formula: For ax² + bx + c = 0, the solutions are x = [-b ± √(b² – 4ac)] / 2a

二次方程求根公式：若 ax² + bx + c = 0，则解为 x = [-b ± √(b² – 4ac)] / 2a

Discriminant Δ = b² – 4ac. When Δ > 0 there are two distinct real roots; Δ = 0 gives one repeated root; Δ < 0 means no real roots.

判别式 Δ = b² – 4ac。Δ > 0 时有两个不等实根；Δ = 0 有一个重根；Δ < 0 则无实根。

Completing the square: x² + bx = (x + b/2)² – (b/2)²

配方法： x² + bx = (x + b/2)² – (b/2)²

2. Surds & Indices | 根式与指数

Product of powers: aᵐ × aⁿ = aᵐ⁺ⁿ

同底数幂的乘法： aᵐ × aⁿ = aᵐ⁺ⁿ

Power of a power: (aᵐ)ⁿ = aᵐⁿ

幂的乘方： (aᵐ)ⁿ = aᵐⁿ

Negative index: a⁻ⁿ = 1 / aⁿ

负指数： a⁻ⁿ = 1 / aⁿ

Fractional index: a^(1/n) = ⁿ√a (the nth root of a)

分数指数： a^(1/n) = ⁿ√a（a 的 n 次方根）

Simplifying surds: √(ab) = √a × √b, and √(a/b) = √a / √b

根式化简： √(ab) = √a × √b，且 √(a/b) = √a / √b

3. Sequences & Series | 数列与级数

Arithmetic sequence nth term: uₙ = a + (n – 1)d, where a is the first term and d is the common difference.

等差数列第 n 项： uₙ = a + (n – 1)d，其中 a 为首项，d 为公差。

Arithmetic sum: Sₙ = n/2 [2a + (n – 1)d] or Sₙ = n/2 (a + l), where l is the last term.

等差数列求和： Sₙ = n/2 [2a + (n – 1)d] 或 Sₙ = n/2 (a + l)，其中 l 为末项。

Geometric sequence nth term: uₙ = ar^(n – 1), where r is the common ratio.

等比数列第 n 项： uₙ = ar^(n – 1)，其中 r 为公比。

Geometric sum (r ≠ 1): Sₙ = a(1 – rⁿ) / (1 – r)

等比数列求和（r ≠ 1）： Sₙ = a(1 – rⁿ) / (1 – r)

4. Coordinate Geometry | 坐标几何

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

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

Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

中点： M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Gradient of a line: m = (y₂ – y₁) / (x₂ – x₁)

直线的斜率： m = (y₂ – y₁) / (x₂ – x₁)

Equation of a straight line: y = mx + c (slope-intercept form) or y – y₁ = m(x – x₁) (point-slope form).

直线方程： y = mx + c（斜截式）或 y – y₁ = m(x – x₁)（点斜式）。

Perpendicular lines: If two lines are perpendicular, the product of their gradients is –1, i.e. m₁ × m₂ = –1.

垂直直线：若两直线垂直，则其斜率乘积为 –1，即 m₁ × m₂ = –1。

5. Geometry & Mensuration | 几何与测量

Key area, surface area and volume formulas are listed below. In the formulas, b stands for base, h for height, r for radius and l for slant height.

下面的表格列出了重要的面积、表面积和体积公式。公式中 b 表示底边，h 表示高，r 表示半径，l 表示斜高。

Shape / 形状	Area / 面积	Volume / 体积
Rectangle / 矩形	A = b × h	—
Parallelogram / 平行四边形	A = b × h	—
Triangle / 三角形	A = ½ × b × h	—
Trapezium / 梯形	A = ½ (a + b)h	—
Circle / 圆	A = πr², C = 2πr	—
Cylinder / 圆柱	Curved surface = 2πrh	V = πr²h
Cone / 圆锥	Curved surface = πrl	V = ⅓πr²h
Sphere / 球体	Surface area = 4πr²	V = ⁴⁄₃πr³

Pythagoras’ theorem: In a right‑angled triangle, a² + b² = c², where c is the hypotenuse.

勾股定理：在直角三角形中，a² + b² = c²，其中 c 是斜边。

6. Trigonometry | 三角学

Right‑angled triangle ratios: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent.

直角三角形的三角比： sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。

Exact values for key angles (in degrees): sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2; cos 30° = √3/2, cos 45° = √2/2, cos 60° = ½; tan 30° = √3/3, tan 45° = 1, tan 60° = √3.

特殊角的精确值： sin 30° = ½，sin 45° = √2/2，sin 60° = √3/2；cos 30° = √3/2，cos 45° = √2/2，cos 60° = ½；tan 30° = √3/3，tan 45° = 1，tan 60° = √3。

Area of a triangle using sine: Area = ½ ab sin C, where C is the included angle between sides a and b.

用正弦求三角形面积：面积 = ½ ab sin C，其中 C 是边 a 与边 b 的夹角。

Sine rule: a / sin A = b / sin B = c / sin C, used for non‑right‑angled triangles.

正弦定理： a / sin A = b / sin B = c / sin C，适用于非直角三角形。

Cosine rule: a² = b² + c² – 2bc cos A, can be rearranged to find an angle: cos A = (b² + c² – a²) / 2bc.

余弦定理： a² = b² + c² – 2bc cos A，也可变形求角： cos A = (b² + c² – a²) / 2bc。

7. Circle Theorems | 圆定理

Angle at the centre is twice the angle at the circumference: ∠AOB = 2 × ∠APB, where O is the centre and P is on the circumference.

圆心角是圆周角的两倍： ∠AOB = 2 × ∠APB，其中 O 是圆心，P 在圆周上。

Angle in a semicircle: The angle subtended by a diameter at the circumference is always 90°.

半圆上的圆周角：直径所对的圆周角总是 90°。

Angles in the same segment are equal: If two angles are subtended by the same chord on the same side, they are equal.

同弦上的圆周角相等：同一段弧所对的圆周角相等。

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

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

Tangent–radius property: A tangent is perpendicular to the radius at the point of contact.

切线与半径：圆的切线垂直于过切点的半径。

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的圆周角。

8. Vectors | 向量

A vector is usually written as a column vector [x, y] or in component form xi + yj. The magnitude of vector v = xi + yj is |v| = √(x² + y²).

向量通常写成列向量 [x, y] 或分量形式 xi + yj。向量 v = xi + yj 的模为 |v| = √(x² + y²)。

Addition and subtraction: (a, b) + (c, d) = (a + c, b + d).

向量的加法与减法： (a, b) + (c, d) = (a + c, b + d)。

Scalar multiplication: k(a, b) = (ka, kb).

标量乘法： k(a, b) = (ka, kb)。

Position vector of a point P(x, y) relative to the origin O is OP = (x, y). The vector AB from point A to B is OB – OA.

点 P(x, y) 相对于原点 O 的位置向量为 OP = (x, y)。从点 A 到点 B 的向量 AB = OB – OA。

Parallel vectors: Two vectors are parallel if one is a scalar multiple of the other, i.e. v = k w.

平行向量：若一个向量是另一个的标量倍数，即 v = k w，则两向量平行。

9. Statistics | 统计

Measures of central tendency: Mean = (Σx) / n, Median = middle value when ordered, Mode = most frequent value.

集中趋势的度量：平均数 = (Σx) / n，中位数 = 排序后的中间值，众数 = 出现频率最高的值。

Range = maximum value – minimum value.

极差 = 最大值 – 最小值。

Interquartile range (IQR) = Q₃ – Q₁, where Q₁ is the lower quartile and Q₃ is the upper quartile.

四分位距 (IQR) = Q₃ – Q₁，其中 Q₁ 是下四分位数，Q₃ 是上四分位数。

Estimated mean from a frequency table: Mean ≈ Σ(f · m) / Σf, where m is the class midpoint.

由频数表估算平均数：平均数 ≈ Σ(f · m

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