📚 Year 11 Cambridge Maths: Formula & Theorem Quick Reference Handbook | 剑桥 Year 11 数学：公式定理速查手册
This quick reference handbook covers essential formulas, identities, and theorems required for Cambridge Year 11 Mathematics. Whether you are revising for mocks or final assessments, keep these relationships at your fingertips. The content spans algebra, geometry, trigonometry, statistics, and probability, aligning closely with the Cambridge IGCSE and O Level syllabuses.
本速查手册涵盖剑桥 Year 11 数学所必需的核心公式、恒等式与定理。无论你正在准备模拟考试还是最终评测，都可以随时查阅这些关系式。内容涉及代数、几何、三角学、统计与概率，与剑桥 IGCSE 和 O Level 教学大纲高度吻合。
1. Solving Linear Equations | 解一次方程
ng inverse operations. Always perform the same operation on both sides: subtract b, then divide by a (if a ≠ 0).
对于形如 ax + b = c 的一次方程，通过逆运算分离变量。始终对方程两边执行相同操作：先减去 b，再除以 a（若 a ≠ 0）。
ebraic condition for two lines a₁x + b₁y = c₁ and a₂x + b₂y = c₂ is: unique solution if a₁b₂ − a₂b₁ ≠ 0; no solution if the lines are parallel; infinitely many solutions if they coincide.
二元一次联立方程组的解可通过代入法、消元法或图像交点求得。两条直线 a₁x + b₁y = c₁ 和 a₂x + b₂y = c₂ 的代数条件为：若 a₁b₂ − a₂b₁ ≠ 0，有唯一解；若两线平行则无解；若重合则有无穷多解。
2. Quadratic Equations and the Discriminant | 二次方程与判别式
rd form is ax² + bx + c = 0, with a ≠ 0. The solutions are given by the quadratic formula:
标准形式为 ax² + bx + c = 0，其中 a ≠ 0。解由二次公式给出：
x = (−b ± √b² − 4ac) / 2a
there is one repeated real root. When Δ < 0, there are no real roots (two complex roots).
判别式 Δ = b² − 4ac 决定根的性质。当 Δ > 0 时，有两个不等实根；当 Δ = 0 时，有一个重根；当 Δ < 0 时，没有实根（两个复数根）。
q, from which the turning point (−p, q) and axis of symmetry x = −p can be read directly.
配方法将二次式改写为 a(x + p)² + q 的形式，由此可直接读出顶点 (−p, q) 和对称轴 x = −p。
3. Indices and Surds | 指数与根式
Key index laws for all positive bases a, b and real exponents m, n:
对于所有正底数 a, b 及实数指数 m, n，重要指数法则如下：
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- a⁰ = 1 (a ≠ 0)
- a⁻ⁿ = 1 / aⁿ
- (ab)ⁿ = aⁿbⁿ
Surds are irrational roots. Simplify √ab = √a × √b and √a/b = √a / √b. Rationalise denominators such as 1/√a by multiplying numerator and denominator by √a, and for expressions like 1/(a+√b) multiply by the conjugate a−√b.
根式是无理根。化简 √ab = √a × √b 以及 √a/b = √a / √b。化分母如 1/√a 时，分子分母同乘 √a；对于 1/(a+√b)，则乘以其共轭式 a−√b。
4. Algebraic Fractions and Factorisation | 代数分式与因式分解
squares a² − b² = (a − b)(a + b), and trinomials of the form x² + bx + c or ax² + bx + c.
通过提取公因式、运用平方差公式 a² − b² = (a − b)(a + b) 以及对形如 x² + bx + c 或 ax² + bx + c 的三项式进行因式分解来完成完全因式分解。
Simplify algebraic fractions by factorising numerators and denominators first, then cancel common factors. Remember that division by zero is undefined; always state restrictions for denominators.
化简代数分式时，先对分子分母进行因式分解，再约去公因式。切记分母为零无意义，应始终注明分母的限制条件。
5. Coordinate Geometry and Straight Lines | 坐标几何与直线
((x₁ + x₂)/2, (y₁ + y₂)/2). The distance between the points is √((x₂ − x₁)² + (y₂ − y₁)²).
两点 (x₁, y₁) 与 (x₂, y₂) 之间的斜率（梯度）为 m = (y₂ − y₁) / (x₂ − x₁)。中点坐标为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。两点间距离为 √((x₂ − x₁)² + (y₂ − y₁)²)。
0. Parallel lines have equal gradients: m₁ = m₂. Perpendicular lines satisfy m₁ × m₂ = −1.
直线方程可写为 y = mx + c（斜率为 m，y 轴截距为 c）或 ax + by + c = 0。平行线斜率相等：m₁ = m₂。垂直线满足 m₁ × m₂ = −1。
6. Geometry: Angles, Polygons and Circles | 几何：角度、多边形与圆
Angle facts: angles on a straight line sum to 180°; angles around a point sum to 360°; vertically opposite angles are equal. In a triangle, the sum of interior angles is 180°. An exterior angle of a triangle equals the sum of the two opposite interior angles.
角度性质：平角为 180°；周角为 360°；对顶角相等。三角形内角和为 180°。三角形的一个外角等于与它不相邻的两个内角之和。
= 360° (one per vertex). For a regular polygon, each interior angle = (n − 2) × 180° / n.
多边形：内角和 = (n − 2) × 180°，外角和 = 360°（每个顶点一个外角）。正多边形每个内角为 (n − 2) × 180° / n。
Circle theorems: angle at centre is twice angle at circumference; angle in a semicircle is 90°; angles in the same segment are equal; opposite angles in a cyclic quadrilateral sum to 180°; the tangent is perpendicular to the radius at the point of contact.
圆定理：圆心角是圆周角的两倍；半圆上的圆周角是 90°；同弧上的圆周角相等；圆内接四边形对角互补；切线垂直于过切点的半径。
7. Perimeter, Area and Volume | 周长、面积与体积
| Shape / 图形 | Formula / 公式 |
|---|---|
| Triangle area | ½ × base × height |
| Parallelogram area | base × height |
| Trapezium area | ½(a + b) × height |
| Circle circumference | 2πr or πd |
| Circle area | πr² |
| Sector arc length | (θ/360) × 2πr |
| Sector area | (θ/360) × πr² |
| Cuboid volume | length × width × height |
| Prism volume | area of cross-section × length |
| Cylinder volume | πr²h |
| Cylinder surface area | 2πr² + 2πrh |
| Sphere volume | 4/3 πr³ |
| Sphere surface area | 4πr² |
| Cone volume | 1/3 πr²h |
| Cone curved surface area | πrl (l = slant height) |
8. Pythagoras’ Theorem and Trigonometry | 勾股定理与三角学
states a² + b² = c². This relationship is fundamental for distance and altitude problems.
在直角边为 a、b，斜边为 c 的直角三角形中，勾股定理指出 a² + b² = c²。这一关系是求解距离和高度问题的基础。
The three trigonometric ratios for an acute angle θ in a right triangle:
直角三角形中锐角 θ 的三个三角比：
- sin θ = opposite / hypotenuse (对边/斜边)
- cos θ = adjacent / hypotenuse (邻边/斜边)
- tan θ = opposite / adjacent (对边/邻边)
Exact values for special angles should be memorised: sin 30° = ½, sin 45° = 1/√2, sin 60° = √3/2; cos 30° = √3/2, cos 45° = 1/√2, cos 60° = ½; tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.
需熟记特殊角的精确值：sin 30° = ½，sin 45° = 1/√2，sin 60° = √3/2；cos 30° = √3/2，cos 45° = 1/√2，cos 60° = ½；tan 30° = 1/√3，tan 45° = 1，tan 60° = √3。
9. Sine and Cosine Rules | 正弦定理与余弦定理
For any triangle ABC with sides a opposite A, b opposite B, c opposite C:
对于任意三角形 ABC，边 a 对角 A，边 b 对角 B，边 c 对角 C：
sin A = b / sin B = c / sin C (also useful in the form sin A / a = sin B / b = sin C / c).
正弦定理：a / sin A = b / sin B = c / sin C（也可使用 sin A / a = sin B / b = sin C / c 的形式）。
c² − a²) / 2bc. This is used to find a side when two sides and the included angle are known (SAS), or to find an angle when all three sides are known (SSS).
余弦定理：a² = b² + c² − 2bc cos A，可变形为 cos A = (b² + c² − a²) / 2bc。当已知两边及其夹角（SAS）时用于求第三边；当已知三边（SSS）时用于求角度。
The area of a triangle can be found using ½ ab sin C (two sides and included angle).
三角形面积可用公式 ½ ab sin C（两边及其夹角）求得。
10. Statistics and Data Handling | 统计与数据处理
For a data set x₁, x₂, …, xₙ, the mean (average) is x̄ when data are ordered; the mode is the most frequent value. The range = maximum − minimum.
对于数据集 x₁, x₂, …, xₙ，平均数 x̄ = (Σx) / n。中位数是排序后中间的值；众数是出现频率最高的值。极差 = 最大值 − 最小值。
For grouped data, estimate the mean using mid-interval values: x̄ ass midpoint and f the frequency. The modal class is the class with the highest frequency.
对于分组数据，用组中值估算平均数：x̄ ≈ (Σfx) / Σf，其中 x 为组中值，f 为频数。众数组是频数最高的组。
). Probability distributions can be summarised with expected value E(X) = Σ[x · P(X = x)].
累积频率曲线有助于估算中位数、四分位数和四分位距（IQR = Q₃ − Q₁）。概率分布可通过期望值 E(X) = Σ[x · P(X = x)] 进行总结。
11. Probability Rules | 概率法则
The probability of an event A, P(A), lies between 0 and 1 inclusive. The sum of probabilities of all mutually exclusive outcomes in a sample space is 1.
事件 A 的概率 P(A) 介于 0 到 1 之间（含 0 和 1）。样本空间中所有互斥结果的概率之和等于 1。
lly exclusive events, use the general addition rule: P(A or B) = P(A) + P(B) − P(A and B).
对于独立事件 A 和 B：P(A 且 B) = P(A) × P(B)。对于互斥事件：P(A 或 B) = P(A) + P(B)。对于非互斥事件，使用一般加法法则：P(A 或 B) = P(A) + P(B) − P(A 且 B)。
agrams are powerful tools to calculate compound probabilities. Conditional probability P(A|B) = P(A and B) / P(B).
树状图和维恩图是计算复合概率的有力工具。条件概率公式为 P(A|B) = P(A 且 B) / P(B)。
12. Transformations of Graphs | 图象变换
while preserving key features. The four basic transformations for any graph y = f(x) are:
给定函数 f(x)，变换会改变其图象的形状和位置，同时保留关键特征。对于任意 y = f(x) 的图象，四种基本变换如下：
- y = f(x) + a : vertical translation by a units (up if a > 0, down if a < 0) (纵向平移 a 单位)
- y = f(x + a) : horizontal translation by −a units (left if a > 0, right if a < 0) (横向平移 −a 单位)
- y = −f(x) : reflection in the x‑axis (关于 x 轴反射)
- y = f(−x) : reflection in the y‑axis (关于 y 轴反射)
ntally, y = f(kx) compresses the graph by factor 1/k if k > 1, and stretches if 0 < k < 1.
纵向伸缩通过 y = k f(x) 实现，k > 1 为拉伸，0 < k < 1 为压缩。横向伸缩 y = f(kx) 中，k > 1 时图象横向压缩为原来的 1/k，0 < k < 1 时横向拉伸。
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