Year 11 OCR Further Mathematics: Quick Reference Formulas and Theorems Handbook | Year 11 OCR 进阶数学：公式定理速查手册

📚 Year 11 OCR Further Mathematics: Quick Reference Formulas and Theorems Handbook | Year 11 OCR 进阶数学：公式定理速查手册

This handbook provides a concise summary of essential formulas, identities and theorems required for the OCR Year 11 Further Mathematics course. Use it as a quick recall tool during revision and before assessments.

本手册为 OCR Year 11 进阶数学课程提供了必需公式、恒等式与定理的简明总结，可作为复习与考前快速回顾的工具。

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1. Algebraic Manipulation and Identities | 代数运算与恒等式

Expand the product of two binomials: (a + b)(c + d) = ac + ad + bc + bd.

展开两个二项式的乘积：(a + b)(c + d) = ac + ad + bc + bd。

Perfect square expansions: (a + b)² = a² + 2ab + b² and (a – b)² = a² – 2ab + b².

完全平方展开：(a + b)² = a² + 2ab + b² 以及 (a – b)² = a² – 2ab + b²。

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

平方差公式：a² – b² = (a + b)(a – b)。

Sum and difference of cubes: a³ + b³ = (a + b)(a² – ab + b²); a³ – b³ = (a – b)(a² + ab + b²).

立方和与立方差：a³ + b³ = (a + b)(a² – ab + b²)；a³ – b³ = (a – b)(a² + ab + b²)。

Factorising by grouping: ac + ad + bc + bd = (a + b)(c + d).

分组分解法：ac + ad + bc + bd = (a + b)(c + d)。

Completing the square for a monic quadratic: x² + bx = (x + b/2)² – (b/2)².

首一二次式的配方法：x² + bx = (x + b/2)² – (b/2)²。

General completing the square: ax² + bx + c = a(x + b/(2a))² + (c – b²/(4a)).

一般二次式的配方法：ax² + bx + c = a(x + b/(2a))² + (c – b²/(4a))。

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2. Quadratic Functions and the Discriminant | 二次函数与判别式

Standard quadratic function: f(x) = ax² + bx + c (a ≠ 0).

标准二次函数：f(x) = ax² + bx + c (a ≠ 0)。

Quadratic formula for roots of ax² + bx + c = 0:

二次方程 ax² + bx + c = 0 的求根公式：

x = (–b ± √(b² – 4ac)) / (2a)

The discriminant Δ = b² – 4ac determines the nature of the roots.

判别式 Δ = b² – 4ac 决定根的性质。

If Δ > 0: two distinct real roots. If Δ = 0: one repeated real root. If Δ < 0: no real roots.

若 Δ > 0，有两个相异实根；若 Δ = 0，有一个重根；若 Δ < 0，无实根。

Sum of roots = –b/a; product of roots = c/a.

根之和 = –b/a；根之积 = c/a。

Vertex (turning point) of parabola y = ax² + bx + c: x-coordinate = –b/(2a); y-coordinate = f(–b/(2a)).

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

If a > 0 the parabola opens upwards (minimum); if a < 0 it opens downwards (maximum).

若 a > 0，抛物线开口向上（极小值）；若 a < 0，开口向下（极大值）。

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3. Coordinate Geometry and Straight Lines | 坐标几何与直线

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

两点 (x₁, y₁) 与 (x₂, y₂) 的距离：d = √[(x₂ – x₁)² + (y₂ – y₁)²]。

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

两点中点：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。

Gradient of a line through the points: m = (y₂ – y₁) / (x₂ – x₁).

过两点的直线斜率：m = (y₂ – y₁) / (x₂ – x₁)。

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

直线方程：y – y₁ = m(x – x₁) 或 y = mx + c（c 为 y 轴截距）。

Parallel lines have equal gradients: m₁ = m₂.

平行直线斜率相等：m₁ = m₂。

Perpendicular lines satisfy: m₁ × m₂ = –1 (provided lines are not vertical/horizontal).

垂直直线满足：m₁ × m₂ = –1（非水平或垂直线）。

General equation of a circle with centre (a, b) and radius r: (x – a)² + (y – b)² = r².

以 (a, b) 为圆心、r 为半径的圆方程：(x – a)² + (y – b)² = r²。

Tangent to a circle at a point: radius is perpendicular to tangent; use gradient relationship to find tangent equation.

圆上一点处的切线：半径垂直于切线；利用斜率关系求切线方程。

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4. Trigonometry Ratios and Identities | 三角比与恒等式

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。

Pythagorean identity: sin²θ + cos²θ = 1.

毕达哥拉斯恒等式：sin²θ + cos²θ = 1。

Tangent identity: tan θ = sin θ / cos θ.

正切恒等式：tan θ = sin θ / cos θ。

Exact trig values for common angles (θ in degrees):

常见角度的精确三角值（角度制）：

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

Sine rule (for any triangle): a/sin A = b/sin B = c/sin C.

正弦定理（适用于任意三角形）：a/sin A = b/sin B = c/sin C。

Cosine rule: a² = b² + c² – 2bc cos A; also cos A = (b² + c² – a²)/(2bc).

余弦定理：a² = b² + c² – 2bc cos A；以及 cos A = (b² + c² – a²)/(2bc)。

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

三角形面积：面积 = ½ ab sin C。

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5. Sequences and Series | 数列与级数

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

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

Sum of first n terms of an AP: Sₙ = n/2 [2a + (n – 1)d] = n/2 (a + l), where l is last term.

等差数列前 n 项和：Sₙ = n/2 [2a + (n – 1)d] = n/2 (a + l)，l 为末项。

Geometric sequence (GP): nth term uₙ = arⁿ⁻¹, where a is first term, r is common ratio.

等比数列：第 n 项 uₙ = arⁿ⁻¹，a 为首项，r 为公比。

Sum of first n terms of a GP (r ≠ 1): Sₙ = a(1 – rⁿ)/(1 – r) = a(rⁿ – 1)/(r – 1).

等比数列前 n 项和（r ≠ 1）：Sₙ = a(1 – rⁿ)/(1 – r) = a(rⁿ – 1)/(r – 1)。

Sum to infinity of a GP (|r| < 1): S∞ = a/(1 – r).

等比数列无穷项和（|r| < 1）：S∞ = a/(1 – r)。

Sigma notation: Σ (from k=1 to n) of uₖ denotes sum of terms u₁ + u₂ + … + uₙ.

西格玛求和符号：Σ (k=1 到 n) uₖ 表示 u₁ + u₂ + … + uₙ 的和。

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6. Binomial Expansion | 二项式展开

For a positive integer n: (a + b)ⁿ = Σ (from r=0 to n) ⁿCᵣ aⁿ⁻ʳ bʳ, where ⁿCᵣ = n! / [r!(n – r)!].

对于正整数 n：(a + b)ⁿ = Σ (r=0 到 n) ⁿCᵣ aⁿ⁻ʳ bʳ，其中 ⁿCᵣ = n! / [r!(n – r)!]。

Pascal’s triangle can be used to find binomial coefficients for small n.

杨辉三角形可用于求较小的 n 的二项式系数。

The general term (r+1)th term: Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ.

一般项，第 (r+1) 项：Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ。

Special case (1 + x)ⁿ = 1 + nx + [n(n – 1)/2!] x² + … + xⁿ.

特殊情况 (1 + x)ⁿ = 1 + nx + [n(n – 1)/2!] x² + … + xⁿ。

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7. Calculus – Differentiation | 微积分 — 微分

The derivative of f(x) is f'(x) or dy/dx, representing the gradient of the curve y = f(x).

f(x) 的导数为 f'(x) 或 dy/dx，表示曲线 y = f(x) 的斜率。

Power rule: If f(x) = axⁿ, then f'(x) = naxⁿ⁻¹.

幂函数法则：若 f(x) = axⁿ，则 f'(x) = naxⁿ⁻¹。

Constant multiple rule: d/dx [k·f(x)] = k·f'(x).

常数倍法则：d/dx [k·f(x)] = k·f'(x)。

Sum/difference rule: d/dx [f(x) ± g(x)]

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