📚 GCSE OCR Further Maths: Quick Reference Formula & Theorem Handbook | GCSE OCR 进阶数学：公式定理速查手册
This handbook brings together every essential formula and theorem you need for the OCR GCSE Further Mathematics specification. Each section pairs an English explanation with a Chinese translation to help you memorise, understand, and apply the content efficiently. Use it for last‑minute revision or as a daily drill companion.
本手册汇集了 OCR GCSE 进阶数学考试大纲中所有必备的公式与定理。每个知识点均配有中英文解释，帮助你高效记忆、理解和应用。既可作为考前速查，也可作为日常训练搭档。
1. Algebraic Essentials | 代数基础
The quadratic formula gives the roots of ax² + bx + c = 0: x = [–b ± √(b² – 4ac)] / (2a). The discriminant Δ = b² – 4ac determines the nature of the roots.
二次方程 ax² + bx + c = 0 的求根公式：x = [–b ± √(b² – 4ac)] / (2a)。判别式 Δ = b² – 4ac 决定了根的性质。
Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, a^(1/n) = ⁿ√a.
指数定律：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ，a^(1/n) = ⁿ√a。
Completing the square: x² + bx + c = (x + b/2)² – (b/2)² + c. This form reveals the vertex of a parabola.
配方法：x² + bx + c = (x + b/2)² – (b/2)² + c。该形式可揭示抛物线的顶点。
Factor theorem: (x – a) is a factor of polynomial f(x) if and only if f(a) = 0. Remainder theorem: when f(x) is divided by (x – a), the remainder is f(a).
因式定理：(x – a) 是多项式 f(x) 的因式当且仅当 f(a) = 0。余数定理：f(x) 除以 (x – a) 的余数为 f(a)。
2. Functions and Graphs | 函数与图像
A function f(x) is even if f(–x) = f(x) and odd if f(–x) = –f(x). Composite function: (f ∘ g)(x) = f(g(x)). Inverse function f⁻¹(x) satisfies f(f⁻¹(x)) = x.
若 f(–x) = f(x)，则函数为偶函数；若 f(–x) = –f(x)，则为奇函数。复合函数：(f ∘ g)(x) = f(g(x))。反函数 f⁻¹(x) 满足 f(f⁻¹(x)) = x。
Transformations of y = f(x): y = f(x) + a shifts vertically by a; y = f(x + a) shifts horizontally by –a; y = af(x) stretches vertically by factor a; y = f(ax) stretches horizontally by factor 1/a. Reflection in x‑axis: y = –f(x); reflection in y‑axis: y = f(–x).
y = f(x) 的图像变换：y = f(x) + a 垂直平移 a；y = f(x + a) 水平平移 –a；y = af(x) 垂直拉伸 a 倍；y = f(ax) 水平拉伸 1/a 倍。关于 x 轴反射：y = –f(x)；关于 y 轴反射：y = f(–x)。
The modulus function |x| gives the distance from zero. Graphs involving |f(x)| reflect negative parts in the x‑axis.
绝对值函数 |x| 表示到零的距离。包含 |f(x)| 的图像会将负值部分沿 x 轴向上翻折。
3. Differentiation | 微分
The derivative f'(x) gives the gradient of the tangent. For y = xⁿ, dy/dx = nxⁿ⁻¹. The rule extends to constant multiples and sums: d[k·f(x)]/dx = k·f'(x).
导数 f'(x) 表示切线的斜率。对于 y = xⁿ，有 dy/dx = nxⁿ⁻¹。该法则适用于常数倍与和：d[k·f(x)]/dx = k·f'(x)。
Chain rule: if y = f(g(x)), then dy/dx = f'(g(x))·g'(x). Product rule: if y = u(x)v(x), then dy/dx = u’v + uv’. Quotient rule: if y = u/v, then dy/dx = (u’v – uv’) / v².
链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x))·g'(x)。乘积法则：y = u(x)v(x) 时 dy/dx = u’v + uv’。商法则：y = u/v 时 dy/dx = (u’v – uv’) / v²。
Stationary points occur where f'(x) = 0. Use the second derivative f”(x) to classify: f”(x) > 0 → local minimum, f”(x) < 0 → local maximum, f''(x) = 0 requires further investigation.
驻点出现在 f'(x) = 0 处。用二阶导数 f”(x) 分类：f”(x) > 0 → 局部极小值，f”(x) < 0 → 局部极大值，f''(x) = 0 需进一步判断。
Increasing and decreasing functions: f'(x) > 0 implies increasing; f'(x) < 0 implies decreasing.
函数的增减性：f'(x) > 0 时函数递增；f'(x) < 0 时函数递减。
4. Integration | 积分
Integration reverses differentiation. The indefinite integral ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ –1. The constant of integration C must be included.
积分是微分的逆运算。不定积分 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，n ≠ –1。必须包含积分常数 C。
Definite integral ∫ₐᵇ f(x) dx gives the signed area between the curve and the x‑axis from x = a to x = b. Area = ∫ₐᵇ f(x) dx when f(x) ≥ 0.
定积分 ∫ₐᵇ f(x) dx 表示曲线与 x 轴在 x = a 到 x = b 之间的带符号面积。当 f(x) ≥ 0 时，面积 = ∫ₐᵇ f(x) dx。
Area between two curves y = f(x) and y = g(x) from a to b: ∫ₐᵇ [f(x) – g(x)] dx, assuming f(x) ≥ g(x). Remember to find intersection points for limits.
两条曲线 y = f(x) 与 y = g(x) 在 a 到 b 之间的面积：∫ₐᵇ [f(x) – g(x)] dx，需保证 f(x) ≥ g(x)。注意先求交点确定积分限。
Trapezium rule (numerical integration): Area ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ], where h = (b – a)/n.
梯形法则（数值积分）：面积 ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ]，其中 h = (b – a)/n。
5. Matrices | 矩阵
A matrix is a rectangular array of numbers. Order is rows × columns. Two matrices can be added or subtracted if they have the same order. Multiplication AB is defined if the number of columns of A equals the number of rows of B.
矩阵是一个矩形数字阵列，阶数为 行数 × 列数。同型矩阵可进行加减运算。乘法 AB 有定义的条件是 A 的列数等于 B 的行数。
Matrix multiplication is not commutative (AB ≠ BA in general). The identity matrix I satisfies AI = IA = A. For 2×2 matrices, I = [[1,0],[0,1]].
矩阵乘法不满足交换律（通常 AB ≠ BA）。单位矩阵 I 满足 AI = IA = A。对于 2×2 矩阵，I = [[1,0],[0,1]]。
The determinant of a 2×2 matrix A = [[a,b],[c,d]] is det(A) = ad – bc. If det(A) = 0, the matrix is singular (no inverse).
2×2 矩阵 A = [[a,b],[c,d]] 的行列式为 det(A) = ad – bc。若 det(A) = 0，矩阵为奇异矩阵（无逆矩阵）。
The inverse of A: A⁻¹ = 1/(ad – bc) [[d, –b],[–c, a]], provided det(A) ≠ 0. Matrix equations AX = B are solved by X = A⁻¹B.
A 的逆矩阵：A⁻¹ = 1/(ad – bc) [[d, –b],[–c, a]]，前提是 det(A) ≠ 0。矩阵方程 AX = B 的解为 X = A⁻¹B。
Transformations can be represented by 2×2 matrices: reflection, rotation, enlargement, shear. Rotation by θ anticlockwise: [[cos θ, –sin θ],[sin θ, cos θ]].
几何变换可用 2×2 矩阵表示：反射、旋转、放大、剪切。逆时针旋转 θ：[[cos θ, –sin θ],[sin θ, cos θ]]。
6. Vectors | 向量
A vector has magnitude and direction. Notation: a = (x, y) or xi + yj. Magnitude |a| = √(x² + y²). Unit vector in the direction of a: â = a/|a|.
向量具有大小和方向。记作 a = (x, y) 或 xi + yj。模长 |a| = √(x² + y²)。与 a 同方向的单位向量：â = a/|a|。
Vector addition follows the triangle or parallelogram law. Scalar multiplication stretches the vector.
向量加法遵循三角形或平行四边形法则。标量乘法拉伸向量。
Scalar (dot) product: a · b = |a||b| cos θ, where θ is the angle between them. For a = (x₁, y₁) and b = (x₂, y₂), a · b = x₁x₂ + y₁y₂. If a · b = 0, the vectors are perpendicular.
数量积（点积）：a · b = |a||b| cos θ，θ 为夹角。对于 a = (x₁, y₁)，b = (x₂, y₂)，a · b = x₁x₂ + y₁y₂。如果 a · b = 0，则两向量垂直。
Vector equation of a line: r = a + t b, where a is a point on the line, b the direction vector, t a scalar parameter.
直线的向量方程：r = a + t b，其中 a 为直线上一点，b 为方向向量，t 为标量参数。
7. Complex Numbers | 复数
A complex number is of the form z = a + bi, where i² = –1. a is the real part, b the imaginary part. The complex conjugate is z̄ = a – bi.
复数形如 z = a + bi，其中 i² = –1。a 为实部，b 为虚部。共轭复数为 z̄ = a – bi。
Addition and subtraction: (a + bi) ± (c + di) = (a ± c) + (b ± d)i. Multiplication uses i² = –1. Division: multiply numerator and denominator by the conjugate of the denominator.
加减法：(a + bi) ± (c + di) = (a ± c) + (b ± d)i。乘法利用 i² = –1。除法：分子分母同乘分母的共轭复数。
Modulus |z| = √(a² + b²). Argument arg(z) is the angle θ with the positive real axis, usually given in radians. Argand diagram shows z as a point or vector.
模长 |z| = √(a² + b²)。辐角 arg(z) 是与正实轴的夹角 θ，通常以弧度表示。Argand 图将复数表示为点或向量。
Complex numbers can be used to solve quadratic equations with negative discriminant. The roots occur in conjugate pairs.
复数可用于求解判别式为负的二次方程。根以共轭对形式出现。
8. Trigonometry | 三角学
Radian measure: π rad = 180°. Arc length s = rθ, sector area A = ½r²θ (θ in rad).
弧度制：π 弧度 = 180°。弧长 s = rθ，扇形面积 A = ½r²θ（θ 以弧度计）。
Basic identities: tan θ = sin θ / cos θ; sin²θ + cos²θ = 1. These hold for all θ.
基本恒等式：tan θ = sin θ / cos θ；sin²θ + cos²θ = 1。对所有 θ 成立。
Sine rule: a/sin A = b/sin B = c/sin C. Cosine rule: a² = b² + c² – 2bc cos A. Area of triangle: ½ab sin C.
正弦定理：a/sin A = b/sin B = c/sin C。余弦定理：a² = b² + c² – 2bc cos A。三角形面积：½ab sin C。
Compound angle formulas: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B).
和角公式：sin(A ± B) = sin A cos B ± cos A sin B；cos(A ± B) = cos A cos B ∓ sin A sin B；tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)。
Double angle formulas: sin 2A = 2 sin A cos A; cos 2A = cos²A – sin²A = 2 cos²A – 1 = 1 – 2 sin²A; tan 2A = 2 tan A/(1 – tan²A).
倍角公式：sin 2A = 2 sin A cos A；cos 2A = cos²A – sin²A = 2 cos²A – 1 = 1 – 2 sin²A；tan 2A = 2 tan A/(1 – tan²A)。
9. Sequences and Series | 数列与级数
Arithmetic sequence: nth term uₙ = a + (n – 1)d, where a is first term, d common difference. Sum of first n terms: Sₙ = n/2 [2a + (n – 1)d] = n/2 (a + l), l is last term.
等差数列：第 n 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] = n/2 (a + l)，l 为末项。
Geometric sequence: uₙ = arⁿ⁻¹, where a is first term, r common ratio. Sum of first n terms: Sₙ = a(1 – rⁿ)/(1 – r) for |r| < 1. Infinite sum convergent when |r| < 1: S∞ = a/(1 – r).
等比数列：uₙ = arⁿ⁻¹，a 为首项，r 为公比。前 n 项和：Sₙ = a(1 – rⁿ)/(1 – r)（|r| < 1 时适用）。当 |r| < 1 时无穷级数收敛：S∞ = a/(1 – r)。
Sigma notation: Σᵢ₌₁ⁿ f(i) represents the sum of terms f(i) from i = 1 to n. Standard results: Σ k = kn, Σ i = n(n+1)/2, Σ i² = n(n+1)(2n+1)/6.
求和符号：Σᵢ₌₁ⁿ f(i) 表示 f(i) 从 i = 1 到 n 的和。常用结果：Σ k = kn，Σ i = n(n+1)/2，Σ i² = n(n+1)(2n+1)/6。
10. Binomial Expansion | 二项展开式
Binomial theorem for positive integer n: (a + b)ⁿ = Σᵣ₌₀ⁿ (ⁿCᵣ) aⁿ⁻ʳ bʳ, where ⁿCᵣ = n!/(r!(n – r)!). Pascal’s triangle gives coefficients.
正整数指数的二项式定理：(a + b)ⁿ = Σᵣ₌₀ⁿ (ⁿCᵣ) aⁿ⁻ʳ bʳ，其中 ⁿCᵣ = n!/(r!(n – r)!)。杨辉三角给出系数。
For (1 + x)ⁿ where n is rational and |x| < 1, the expansion is (1 + x)ⁿ = 1 + nx + n(n–1)/2! x² + n(n–1)(n–2)/3! x³ + … . This is an infinite series.
对于 (1 + x)ⁿ，当 n 为有理数且 |x| < 1 时，展开式为 (1 + x)ⁿ = 1 + nx + n(n–1)/2! x² + n(n–1)(n–2)/3! x³ + …。这是一个无穷级数。
The expansion is valid only within the interval of convergence |x| < 1 unless n is a positive integer. Use partial fractions to expand rational expressions in combination with binomial series.
该展开式仅在收敛区间 |x| < 1 内有效，除非 n 为正整数。可结合部分分式与二项级数展开有理式。
11. Inequalities | 不等式
Solving linear inequalities follows similar rules to equations, but multiplying or dividing by a negative number reverses the inequality sign.
解线性不等式与解方程规则类似，但当两边同乘或同除以负数时，不等式方向需要反转。
Quadratic inequalities: solve ax² + bx + c > 0 by finding critical values where the expression equals zero, then test intervals. Use a sign diagram or sketch.
二次不等式：求解 ax² + bx + c > 0 时，先求出使表达式等于零的临界值，然后检验各区间。可借助符号表或草图。
Linear programming: represent constraints as linear inequalities on a graph, identify the feasible region, then evaluate the objective function at vertices to find maximum or minimum.
线性规划：将约束条件表示为图上的线性不等式，确定可行域，然后在各顶点处计算目标函数值，以求得最大值或最小值。
12. Coordinate Geometry | 坐标几何
The distance between points (x₁, y₁) and (x₂, y₂): √[(x₂ – x₁)² + (y₂ – y₁)²]. Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2).
两点 (x₁, y₁) 与 (x₂, y₂) 间距离：√[(x₂ – x₁)² + (y₂ – y₁)²]。中点坐标：((x₁ + x₂)/2, (y₁ + y₂)/2)。
Gradient of line through (x₁, y₁) and (x₂, y₂): m = (y₂ – y₁)/(x₂ – x₁). Equation of a straight line: y – y₁ = m(x – x₁), or y = mx + c, or ax + by + c = 0.
过 (x₁, y₁) 与 (x₂, y₂) 的直线斜率：m = (y₂ – y₁)/(x₂ – x₁)。直线方程：y – y₁ = m(x – x₁)，或 y = mx + c，或 ax + by + c = 0。
Parallel lines have equal gradients; perpendicular lines have gradients product –1 (m₁m₂ = –1).
平行线斜率相等；垂直线斜率乘积为 –1（m₁m₂ = –1）。
Circle with centre (a, b) and radius r: (x – a)² + (y – b)² = r². General form: x² + y² + 2gx + 2fy + c = 0, centre (–g, –f), radius √(g² + f² – c).
圆心 (a, b)、半径 r 的圆方程：(x – a)² + (y – b)² = r²。一般式：x² + y² + 2gx + 2fy + c = 0，圆心 (–g, –f)，半径 √(g² + f² – c)。
Parametric equations: x = f(t), y = g(t). To convert to Cartesian form, eliminate the parameter t. Differentiation: dy/dx = (dy/dt)/(dx/dt).
参数方程：x = f(t)，y = g(t)。化为笛卡尔形式需消去参数 t。求导：dy/dx = (dy/dt)/(dx/dt)。
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