📚 GCSE CCEA Further Mathematics: Quick Reference Formulas and Theorems | GCSE CCEA 进阶数学：公式定理速查手册
This article provides a concise, organised summary of the essential formulas and theorems you need for the CCEA GCSE Further Mathematics examination. Each section pairs key concepts in English and Chinese, helping bilingual learners quickly locate and review the most critical material. Use this as a last‑minute reference or as a checklist to ensure you have memorised every necessary relationship.
本文为 CCEA GCSE 进阶数学考试整理了一份简洁有序的核心公式与定理摘要。每个部分都采用中英双语要点对照，帮助双语学习者快速查找和复习最关键的内容。可将本文用作考前速查手册，也可用作核对清单，确保你已记住每一条必须掌握的关系式。
1. Algebraic Expressions and Identities | 代数表达式与恒等式
Expand, factorise and manipulate algebraic expressions confidently. Recognise difference of two squares, perfect squares, and the sum/difference of two cubes.
熟练掌握代数式的展开、因式分解和变形。能识别平方差、完全平方以及两数的立方和与立方差。
Common expansions and factorisations
常见展开与因式分解
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
a² − b² = (a − b)(a + b)
To factorise a quadratic ax² + bx + c, find two numbers that multiply to ac and add to b, or use completing the square.
对二次式 ax² + bx + c 进行因式分解时，可寻找两个数使其乘积为 ac 且和为 b，或使用配方法。
x² + bx + c completed square form: (x + b/2)² − (b/2)² + c
x² + bx + c 的配方式： (x + b/2)² − (b/2)² + c
Sum and difference of two cubes:
两数的立方和与立方差：
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)
2. Quadratic Equations and Inequalities | 二次方程与不等式
Quadratic equations of the form ax² + bx + c = 0 can be solved by factorising, completing the square, or using the quadratic formula. Inequalities are handled by considering the shape of the graph.
形如 ax² + bx + c = 0 的二次方程可通过因式分解、配方法或求根公式求解。二次不等式则需结合图像形状来判定解集。
Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a
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 决定方程根的性质：
- Δ > 0: two distinct real roots 两个不等实根
- Δ = 0: one repeated real root 一个重实根
- Δ < 0: no real roots 无实根
For the quadratic inequality ax² + bx + c > 0 or < 0, sketch the graph and identify where it lies above or below the x‑axis.
对于二次不等式 ax² + bx + c > 0 或 < 0，画出草图并判断图像位于 x 轴上方或下方的区间。
3. Coordinate Geometry | 坐标几何
Know how to find the distance between two points, the midpoint, and the equation of a straight line. Apply these to circles and tangents.
要会求两点间的距离、中点坐标以及直线的方程。将这些知识运用到圆和切线问题中。
Distance between (x₁, y₁) and (x₂, y₂):
两点 (x₁, y₁) 与 (x₂, y₂) 间的距离：
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Midpoint M:
中点 M：
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Gradient m of a line through (x₁, y₁) and (x₂, y₂):
过 (x₁, y₁) 与 (x₂, y₂) 的直线斜率 m：
m = (y₂ − y₁) / (x₂ − x₁)
Equation of a straight line: y − y₁ = m(x − x₁) or y = mx + c, where c is the y‑intercept.
直线方程： y − y₁ = m(x − x₁) 或 y = mx + c，其中 c 为 y 轴截距。
For parallel lines, m₁ = m₂. For perpendicular lines, m₁ × m₂ = −1.
平行线满足 m₁ = m₂；垂直线满足 m₁ × m₂ = −1。
The equation of a circle with centre (a, b) and radius r:
圆心为 (a, b)、半径为 r 的圆方程：
(x − a)² + (y − b)² = r²
The tangent to a circle at a point is perpendicular to the radius at that point.
圆上一点的切线与过该点的半径互相垂直。
4. Functions and Graphs | 函数与图像
Understand function notation, domain, range, and the effect of transformations. Be able to sketch linear, quadratic, cubic, reciprocal, exponential and trigonometric graphs.
理解函数记法、定义域、值域以及变换的效果。能够画出一次、二次、三次、倒数、指数和三角函数的草图。
Transformation rules for y = f(x):
对 y = f(x) 的变换规则：
- f(x) + a: vertical translation by a 向上平移 a 个单位
- f(x + a): horizontal translation by −a 向左平移 a 个单位
- a f(x): vertical stretch by factor a 垂直方向拉伸 a 倍
- f(ax): horizontal stretch by factor 1/a 水平方向压缩至 1/a 倍
- −f(x): reflection in the x‑axis 关于 x 轴反射
- f(−x): reflection in the y‑axis 关于 y 轴反射
Inverse function f⁻¹(x) reverses the effect of f: f⁻¹(f(x)) = x. The graph of f⁻¹(x) is a reflection of f(x) in the line y = x.
反函数 f⁻¹(x) 逆转 f 的作用：f⁻¹(f(x)) = x。f⁻¹(x) 的图像是 f(x) 关于直线 y = x 的反射。
5. Trigonometry | 三角学
Master the sine, cosine and tangent ratios, the sine and cosine rules, and the solution of trigonometric equations within specified ranges. Work in both degrees and radians where applicable.
熟练掌握正弦、余弦和正切比，正弦定理与余弦定理，以及给定区间内三角方程的求解。根据题目要求使用角度制或弧度制。
Right‑angled triangle ratios (SOH CAH TOA):
直角三角形中的三角比（SOH CAH TOA）：
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
Sine rule (for any triangle):
正弦定理（任意三角形）：
a / sin A = b / sin B = c / sin C
Cosine rule:
余弦定理：
a² = b² + c² − 2bc cos A
Area of a triangle:
三角形面积：
Area = ½ ab sin C
Key identities:
核心恒等式：
sin² θ + cos² θ = 1
tan θ = sin θ / cos θ
For solving sin θ = k, cos θ = k, or tan θ = k, use the CAST diagram or graph to identify all solutions within the given interval.
解 sin θ = k、cos θ = k 或 tan θ = k 时，利用 CAST 图或图像找出给定区间内的所有解。
6. Sequences and Series | 数列与级数
Recognise linear and quadratic sequences, and work with arithmetic progressions. Summation notation (Σ) is often used to represent series.
能识别一次和二次数列，并能处理等差数列。级数常用求和符号 Σ 表示。
nth term of a linear sequence: an = a₁ + (n − 1)d
等差数列的第 n 项： an = a₁ + (n − 1)d
Sum of the first n terms of an arithmetic series:
等差数列前 n 项和：
Sₙ = n/2 [2a₁ + (n − 1)d] or Sₙ = n/2 (a₁ + aₙ)
To find the nth term of a quadratic sequence, examine the second differences; if constant 2a, the nth term is of the form an² + bn + c.
求二次数列的第 n 项时，检查二阶差：若二阶差为常数 2a，则第 n 项形如 an² + bn + c。
Sigma notation: Σ from r=1 to n of a_r means sum of the terms a₁, a₂, …, aₙ.
求和符号： Σ (r=1 到 n) a_r 表示 a₁, a₂, …, aₙ 各项的和。
7. Calculus: Differentiation and Integration | 微积分：微分与积分
GCSE Further Mathematics introduces differentiation and integration of polynomial functions. Use these to find gradients, tangents, normals, stationary points, and areas under curves.
GCSE 进阶数学介绍了多项式函数的微分与积分，可用于求梯度、切线、法线、驻点以及曲线下的面积。
Basic differentiation: If y = xⁿ, then dy/dx = n xⁿ⁻¹.
基本微分： 若 y = xⁿ，则 dy/dx = n xⁿ⁻¹。
The derivative f ‘(x) gives the gradient of the curve at any point. The tangent at x = a has gradient f ‘(a), and the normal gradient is −1 / f ‘(a).
导数 f ‘(x) 表示曲线上任意一点的梯度。曲线在 x = a 处的切线斜率为 f ‘(a)，法线斜率为 −1 / f ‘(a)。
Stationary points occur where f ‘(x) = 0. Test using the second derivative or sign change to classify as maximum, minimum, or point of inflection.
驻点出现在 f ‘(x) = 0 处。利用二阶导数或符号变化法判断是极大点、极小点还是拐点。
Basic integration: ∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C, for n ≠ −1.
基本积分： ∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C，其中 n ≠ −1。
The definite integral ∫ₐᵇ f(x) dx gives the area between the curve, the x‑axis, and the lines x = a and x = b.
定积分 ∫ₐᵇ f(x) dx 表示曲线、x 轴以及直线 x = a 与 x = b 所围成的面积。
8. Vectors | 向量
Vectors describe quantities with both magnitude and direction. Use column vectors, unit vectors, and vector algebra to solve geometric problems.
向量用于描述既有大小又有方向的量。使用列向量、单位向量和向量代数解决几何问题。
A vector v from A to B is written as AB = b − a (position vectors).
从 A 到 B 的向量写作 AB = b − a（位置向量）。
If v = (x, y) then its magnitude |v| = √(x² + y²).
若 v = (x, y)，则其模长 |v| = √(x² + y²)。
Scalar multiplication changes the magnitude but not the direction (for k > 0). Parallel vectors are scalar multiples of each other.
数乘改变向量大小而不改变方向（当 k > 0 时）。平行向量互为标量倍数。
The midpoint of AB is given by (a + b)/2. This is the same as the line‑segment midpoint formula.
AB 的中点由 (a + b)/2 给出，这与线段中点公式一致。
Vector addition and subtraction are performed component‑wise. Draw vector diagrams to handle relative position and motion problems.
向量的加法和减法按分量进行。利用向量图处理相对位置和相对运动问题。
9. Matrices and Transformations | 矩阵与变换
Matrices are used to represent linear transformations in the plane. Know how to add, subtract, multiply matrices, and find the determinant and inverse of a 2×2 matrix.
矩阵用于表示平面中的线性变换。要掌握矩阵的加减、乘法，以及 2×2 矩阵的行列式和逆矩阵。
2×2 matrix A: [[a, b], [c, d]] has determinant det(A) = ad − bc.
2×2 矩阵 A: [[a, b], [c, d]] 的行列式 det(A) = ad − bc。
Inverse matrix A⁻¹ exists only if det(A) ≠ 0:
逆矩阵 A⁻¹ 仅在 det(A) ≠ 0 时存在：
A⁻¹ = (1/det(A)) [[d, −b], [−c, a]]
Common transformation matrices:
常见变换矩阵：
- Rotation by 90° anticlockwise: [[0, −1], [1, 0]]
- Reflection in x‑axis: [[1, 0], [0, −1]]
- Enlargement scale factor k: [[k, 0], [0, k]]
Multiply matrices to combine transformations; remember that matrix multiplication is not commutative (AB ≠ BA in general).
对多个变换进行组合时，将矩阵相乘；注意矩阵乘法不满足交换律（通常 AB ≠ BA）。
10. Probability and Statistics | 概率与统计
Use tree diagrams for combined events, understand independent and mutually exclusive events, and interpret measures of central tendency and spread.
利用树图处理复合事件，理解独立事件和互斥事件，能够解读集中趋势量和离散程度量。
Probability of event A: P(A) = number of favourable outcomes / total outcomes. 0 ≤ P(A) ≤ 1.
事件 A 的概率： P(A) = 有利结果数 / 总结果数，且 0 ≤ P(A) ≤ 1。
For independent events A and B: P(A and B) = P(A) × P(B).
独立事件 A 和 B：P(A 且 B) = P(A) × P(B)。
For mutually exclusive events: P(A or B) = P(A) + P(B).
互斥事件：P(A 或 B) = P(A) + P(B)。
Conditional probability: P(A|B) = P(A and B) / P(B).
条件概率：P(A|B) = P(A 且 B) / P(B)。
Key statistics: mean = Σx / n; median is the middle value when ordered; range = max − min. For grouped data, estimate the mean using midpoints.
关键统计量：平均数 = Σx / n；中位数为排序后的中间值；极差 = 最大值 − 最小值。对于分组数据，用组中点估计平均数。
11. Logarithms and Exponentials | 对数与指数
Logarithms are the inverses of exponential functions. Use the laws of logarithms to simplify expressions and solve equations where the unknown is in the exponent.
对数是指数函数的逆运算。运用对数法则简化表达式并求解未知量位于指数位置的方程。
Definition: logₐ y = x ⇔ aˣ = y. The natural log ln x = logₑ x.
定义： logₐ y = x ⇔ aˣ = y。自然对数 ln x = logₑ x。
Laws of logarithms (for any base a > 0, a ≠ 1):
对数运算法则（对任意 a > 0, a ≠ 1 成立）：
- logₐ (xy) = logₐ x + logₐ y
- logₐ (x/y) = logₐ x − logₐ y
- logₐ (xⁿ) = n logₐ x
To solve aˣ = b, take logs of both sides: x = log b / log a or x = ln b / ln a.
求解 aˣ = b 时，两边取对数：x = log b / log a 或 x = ln b / ln a。
Exponential growth/decay models often take the form y = A eᵏᵗ. The doubling time or half‑life can be derived using logs.
指数增长或衰减模型常取形式 y = A eᵏᵗ。借助对数可推导出倍增时间或半衰期。
12. Binomial Expansion | 二项式展开
For positive integer powers n, (a + b)ⁿ can be expanded using Pascal’s triangle or the binomial theorem. The expansion allows quick evaluation of powers of binomials and approximations.
对于正整数指数 n，(a + b)ⁿ 可利用帕斯卡三角形或二项式定理展开。利用展开式可快速计算二项式的幂并进行近似。
Binomial theorem:
二项式定理：
(a + b)ⁿ = Σ (k=0 to n) [nCk] aⁿ⁻ᵏ bᵏ
where nCk = n! / (k! (n − k)!) is the binomial coefficient.
其中 nCk = n! / (k! (n − k)!) 为二项式系数。
For (1 + x)ⁿ with small x, the expansion truncated after a few terms gives approximations: (1 + x)ⁿ ≈ 1 + nx (when x is sufficiently small).
当 (1 + x)ⁿ 中的 x 较小时，截取展开式前几项可用于近似： (1 + x)ⁿ ≈ 1 + nx（当 x 充分小时有效）。
Pascal’s triangle gives the coefficients for expansions of (a + b)ⁿ, starting with n=0: row 0 = 1; row 1 = 1 1; row 2 = 1 2 1; row 3 = 1 3 3 1; row 4 = 1 4 6 4 1; and so on.
帕斯卡三角形给出 (a + b)ⁿ 展开式的系数，从 n=0 起：第 0 行 = 1；第 1 行 = 1 1；第 2 行 = 1 2 1；第 3 行 = 1 3 3 1；第 4 行 = 1 4 6 4 1；依此类推。
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