Year 11 Cambridge Maths: Formula & Theorem Quick Reference | 剑桥11年级数学：公式定理速查手册

📚 Year 11 Cambridge Maths: Formula & Theorem Quick Reference | 剑桥11年级数学：公式定理速查手册

This quick reference handbook is compiled for Year 11 Cambridge Mathematics students (IGCSE / O Level) to provide a concise yet comprehensive summary of all the essential formulas, identities and theorems needed for revision and examinations. Use it to refresh your memory and strengthen your problem-solving toolkit.

本速查手册专为剑桥11年级数学（IGCSE / O Level）学生编写，精炼总结所有必备公式、恒等式与定理，方便复习与考试前快速查阅。使用本手册巩固记忆，完善你的解题工具箱。

1. Algebraic Identities & Factorisation | 代数恒等式与因式分解

Difference of squares: a² – b² = (a – b)(a + b). Essential for factorising expressions with two squared terms.

平方差公式：a² – b² = (a – b)(a + b)。分解两个平方项之差的核心方法。

Perfect squares: (a + b)² = a² + 2ab + b² and (a – b)² = a² – 2ab + b². Recognising these patterns helps when completing the square or expanding binomials quickly.

完全平方公式：(a + b)² = a² + 2ab + b² 以及 (a – b)² = a² – 2ab + b²。识别此模式有利于配方法和快速展开二项式。

Sum and difference of cubes: a³ + b³ = (a + b)(a² – ab + b²); a³ – b³ = (a – b)(a² + ab + b²). Sometimes required for higher-level factorisation.

立方和与立方差：a³ + b³ = (a + b)(a² – ab + b²)；a³ – b³ = (a – b)(a² + ab + b²)。高阶因式分解偶有涉及。

Factorising by grouping: Group terms to reveal common factors, e.g., ax + ay + bx + by = (a + b)(x + y).

分组分解法：将项进行分组以提取公因式，如 ax + ay + bx + by = (a + b)(x + y)。

Cross method for quadratics: For ax² + bx + c, find two numbers that multiply to ac and sum to b.

十字相乘法：对于二次三项式 ax² + bx + c，寻找两个数使其乘积为 ac 且和为 b。

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2. Linear Equations & Coordinate Geometry | 线性方程与坐标几何

Gradient (slope): m = (y₂ – y₁) / (x₂ – x₁). The gradient shows the steepness of a straight line.

斜率：m = (y₂ – y₁) / (x₂ – x₁)，表示直线的倾斜程度。

Equation of a line: y = mx + c (slope-intercept form), where c is the y-intercept. Also point-slope form: y – y₁ = m(x – x₁).

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

Parallel lines: Two lines are parallel if and only if m₁ = m₂.

平行线：两直线平行当且仅当 m₁ = m₂。

Perpendicular lines: Two lines are perpendicular if and only if m₁ × m₂ = –1.

垂直线：两直线垂直当且仅当 m₁ × m₂ = –1。

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

两点中点：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。

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

两点间距离：d = √[(x₂ – x₁)² + (y₂ – y₁)²]，由勾股定理导出。

Intercepts: To find x-intercept, set y = 0; to find y-intercept, set x = 0.

截距：求 x 轴截距时令 y=0；求 y 轴截距时令 x=0。

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3. Quadratic Equations & Functions | 二次方程与函数

Standard form: ax² + bx + c = 0, where a ≠ 0.

标准形式：ax² + bx + c = 0，a ≠ 0。

Quadratic formula: x = [–b ± √b² – 4ac] / (2a). This solves any quadratic equation.

求根公式：x = [–b ± √b² – 4ac] / (2a)，可解任意二次方程。

Discriminant: D = b² – 4ac. D > 0 → two distinct real roots; D = 0 → one repeated root; D < 0 → no real roots.

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

Sum and product of roots: For α and β, α+β = –b/a, αβ = c/a.

根的和与积：若根为 α 与 β，则 α+β = –b/a，αβ = c/a。

Completing the square: x² + bx = (x + b/2)² – (b/2)². Used to find the vertex form.

配方法：x² + bx = (x + b/2)² – (b/2)²。用于化为顶点式。

Vertex of a parabola: For y = ax² + bx + c, vertex at (–b/(2a), f(–b/(2a))).

抛物线顶点：对于 y = ax² + bx + c，顶点坐标为 (–b/(2a), f(–b/(2a)))。

Factorised form: If roots are p and q, then ax² + bx + c = a(x – p)(x – q).

因式分解形式：若根为 p 和 q，则 ax² + bx + c = a(x – p)(x – q)。

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4. Inequalities | 不等式

Linear inequalities: Solve similar to equations, but if multiplying or dividing by a negative number, reverse the inequality sign.

线性不等式：解法与等式类似，但乘除负数时需反转不等号方向。

Quadratic inequalities: Solve ax² + bx + c > 0 by first finding roots and testing intervals, or using a sign graph.

二次不等式：解 ax² + bx + c > 0 时先求根并检验区间，或绘制符号图。

Representing on a number line: Open circle for strict inequalities (<, >); closed circle for inclusive (≤, ≥).

数轴表示：严格不等号用空心圆；包含等号用实心圆。

Compound inequalities: e.g., –2 < x ≤ 5, represent intersection of separate inequalities.

复合不等式：如 –2 < x ≤ 5，表示两个不等式区间的交集。

Solving inequalities with fractions: Multiply by the LCM of denominators, careful with sign changes.

分式不等式：乘以各分母的最小公倍数，注意符号变化。

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5. Trigonometry | 三角学

Right-angled triangle ratios (SOH CAH TOA): sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ = opposite/adjacent.

直角三角形比值 (SOH CAH TOA)：sin θ = 对边/斜边；cos θ = 邻边/斜边；tan θ = 对边/邻边。

Exact values for special angles:

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

特殊角的精确值：记忆常用角度的正弦、余弦、正切值，可快速求解。

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。用于非直角三角形。

Cosine rule: a² = b² + c² – ed to find a side or an angle when given two sides and the included angle, or three sides.

余弦定理：a² = b² + c² – 2bc cos A。已知两边及其夹角或三边时求边或角。

Area of triangle: Area = ½ ab sin C. Any two sides and the included angle give the area.

三角形面积公式：面积 = ½ ab sin C。任意两边及其夹角可求面积。

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

勾股定理：直角三角形中，a² + b² = c²，c 为斜边。

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6. Mensuration (Perimeter, Area & Volume) | 测量（周长、面积与体积）

Rectangle: Perimeter = 2(l + w), Area = l × w.

矩形：周长 = 2(长 + 宽)，面积 = 长 × 宽。

Triangle: Area = ½ × base × height. Also Heron's formula where s = semi-perimeter: Area = √[s(s–a)(s–b)(s–c)].

三角形：面积 = ½ × 底 × 高。也可用海伦公式，s = 半周长：面积 = √[s(s–a)(s–b)(s–c)]。

Parallelogram: Area = base × perpendicular height.

平行四边形：面积 = 底 × 垂直高。

Trapezium: Area = ½ (a + b) × h, where a, b are parallel sides.

梯形：面积 = ½ (a + b) × h，a, b 为平行边。

Circle: Circumference = 2πr or πd, Area = πr².

圆：周长 = 2πr 或 πd，面积 = πr²。

Arc length: l = (θ/360) × 2πr. Sector area: A = (θ/360) × πr².

弧长：l = (θ/360) × 2πr。扇形面积：A = (θ/360) × πr²。

Cylinder: Volume = πr²h, Curved surface area = 2πrh, Total surface area = 2πr(r + h).

圆柱：体积 = πr²h，侧面积 = 2πrh，全面积 = 2πr(r + h)。

Cone: Volume = ⅓ πr²h, Curved surface area = πrl (l = slant height).

圆锥：体积 = ⅓ πr²h，侧面积 = πrl（l 为斜高）。

Sphere: Volume = ⁴⁄₃ πr³, Surface area = 4πr².

球体：体积 = ⁴⁄₃ πr³，表面积 = 4πr²。

Prism: Volume = area of cross-section × length.

棱柱：体积 = 横截面积 × 长。

Pyramid: Volume = ⅓ × base area × height.

棱锥：体积 = ⅓ × 底面积 × 高。

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7. Statistics | 统计

Mean (average): x̄ = Σx / n. For grouped data, estimate using midpoints.

平均数：x̄ = Σx / n。分组数据用组中值估算。

Median: : Middle value when data is ordered. For n odd, median = (n+1)/2 th value; for n even, average the two middle values.

中位数：排序后居于中间的值。n 为奇数时取第 (n+1)/2 个；n 为偶数时

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