Formula & Theorem Quick Reference Handbook | 公式定理速查手册

📚 Formula & Theorem Quick Reference Handbook | 公式定理速查手册

This handbook provides a concise summary of the essential formulae, theorems and notation you need to master for the WJEC GCSE Further Mathematics course. It is designed for quick revision and as a ready reference when you are practising past papers or checking your understanding. Each section pairs a key topic from algebra, geometry, trigonometry, calculus, matrices, vectors, sequences and probability with clear statements of the results you must recall, together with short notes on their use. Keep it to hand and build confidence by testing yourself on each formula.

本手册简要总结了 WJEC GCSE 进阶数学课程必须掌握的核心公式、定理和符号。它既可用于快速复习，也适合在做历年真题或检查理解时随手查阅。每一节涵盖代数、几何、三角、微积分、矩阵、向量、数列和概率中的一个关键主题，列出你必须记住的结论，并附上简要的使用说明。请将本手册放在手边，通过自测每个公式来建立信心。

1. Algebraic Expressions and Laws of Indices | 代数表达式与指数法则

When simplifying algebraic products, you combine coefficients and apply the laws of indices to identical letter bases. For any non-zero number a and integers m, n:

化简代数乘积时，要合并系数并对相同字母底数运用指数法则。对于任意非零数 a 和整数 m, n：

a^m × aⁿ = a^m+n

a^m ÷ aⁿ = a^m−n

(a^m)ⁿ = a^mn

The rule a⁰ = 1 (provided a ≠ 0) and a⁻ⁿ = 1/aⁿ are essential when simplifying expressions or writing answers with positive indices. Fractional indices link roots and powers: a^1/n = ⁿ√a and a^m/n = (ⁿ√a)^m.

法则 a⁰ = 1（a ≠ 0）和 a⁻ⁿ = 1/aⁿ 在化简表达式或用正指数书写答案时至关重要。分数指数将根式与幂联系起来：a^1/n = ⁿ√a，a^m/n = (ⁿ√a)^m。

2. Quadratic Equations and the Discriminant | 二次方程与判别式

A quadratic equation written as ax² + bx + c = 0, where a ≠ 0, can be solved by factorising, completing the square or using the quadratic formula. The formula itself is given in the examination and should be memorised:

形如 ax² + bx + c = 0（a ≠ 0）的二次方程可通过因式分解、配方法或求根公式求解。公式本身在考试中提供，但应熟记：

x = [−b ± √(b² − 4ac)] / 2a

The expression inside the square root, Δ = b² − 4ac, is called the discriminant. It determines the nature of the roots: Δ > 0 gives two distinct real roots; Δ = 0 gives one repeated real root; Δ < 0 gives no real roots. This is often tested in conjunction with sketching graphs of y = ax² + bx + c. Remember that the sum of roots is −b/a and the product is c/a; these relationships can help you form equations from given roots.

根号内的表达式 Δ = b² − 4ac 称为判别式。它决定根的性质：Δ > 0 有两个不等实根；Δ = 0 有一个重根；Δ < 0 无实根。这一知识点经常与绘制 y = ax² + bx + c 图像结合考查。记住两根之和为 −b/a，两根之积为 c/a；这些关系有助于从给定根构造方程。

3. Factor and Remainder Theorems | 因式定理与余数定理

For a polynomial p(x), the remainder theorem states that when p(x) is divided by (x − a), the remainder is p(a). The factor theorem is a special case: (x − a) is a factor of p(x) if and only if p(a) = 0. Together they allow you to factorise cubics and higher-degree polynomials by finding one factor through trial values (usually ±1, ±2, ±3) and then performing long division or comparing coefficients.

对于多项式 p(x)，余数定理指出：当 p(x) 除以 (x − a) 时，余数为 p(a)。因式定理是其特例：(x − a) 是 p(x) 的因式当且仅当 p(a) = 0。两者结合可以让你通过尝试小整数（通常为 ±1, ±2, ±3）找到一个因式，然后用长除法或比较系数法将三次或更高次多项式进行因式分解。

When dividing a polynomial by a quadratic divisor, the remainder can be of degree 1, i.e. of the form ax + b. You can find a and b by equating coefficients or by using the remainder theorem with carefully chosen values.

当多项式除以一个二次除式时，余式可以为一次式，即 ax + b 的形式。你可以通过比较系数或选用恰当的值代入余数定理来求出 a 和 b。

4. Binomial Expansion | 二项式展开

The binomial expansion for a positive integer n is:

(1 + x)ⁿ = 1 + n·x + [n(n−1)/2!]·x² + [n(n−1)(n−2)/3!]·x³ + … + xⁿ

For small values of n, you can also use Pascal’s triangle to find the coefficients quickly. In WJEC Further Mathematics, you must be able to expand expressions of the form (a + bx)ⁿ, treating a + bx as a[1 + (b/a)x] and then expanding. Always state the range of validity when the expansion is infinite (not required for positive integer n). Questions often ask for the first few terms and then evaluation of a power or an approximation.

对于小的 n，你也可以用帕斯卡三角形快速找到系数。在 WJEC 进阶数学中，你必须能够展开形如 (a + bx)ⁿ 的表达式，将其写成 a[1 + (b/a)x] 的形式再展开。当展开为无穷级数时需要说明有效性范围（正整数 n 无需此步）。题目常要求写出前几项并据此计算某次幂或近似值。

5. Coordinate Geometry | 坐标几何

Given two points A(x₁, y₁) and B(x₂, y₂), the key results you must know are:

给定两点 A(x₁, y₁) 和 B(x₂, y₂)，必须掌握以下关键结论：

Midpoint of AB: ((x₁ + x₂)/2, (y₁ + y₂)/2)

AB 的中点：((x₁ + x₂)/2, (y₁ + y₂)/2)
Gradient of AB: m = (y₂ − y₁) / (x₂ − x₁)

AB 的斜率：m = (y₂ − y₁) / (x₂ − x₁)
Distance AB: √[(x₂ − x₁)² + (y₂ − y₁)²]

AB 的距离：√[(x₂ − x₁)² + (y₂ − y₁)²]

For a straight line, the equation y = mx + c gives intercept c on the y-axis. An alternative form y − y₁ = m(x − x₁) is useful when you know a point and the gradient. Parallel lines have equal gradients; two lines are perpendicular if and only if the product of their gradients is −1. The equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r².

对于直线，方程 y = mx + c 中的 c 为 y 轴截距。当已知一点和斜率时，使用 y − y₁ = m(x − x₁) 更方便。平行线斜率相等；两直线垂直当且仅当它们斜率的乘积为 −1。以 (a, b) 为圆心、r 为半径的圆方程为 (x − a)² + (y − b)² = r²。

6. Trigonometry and Trigonometric Identities | 三角学与三角恒等式

In any right-angled triangle, the sine, cosine and tangent ratios are defined by SOH CAH TOA. For general triangles you will use the sine rule and cosine rule as listed in the formula booklet:

在任意直角三角形中，正弦、余弦和正切比由 SOH CAH TOA 定义。对于一般三角形，可使用公式表中提供的正弦定理和余弦定理：

Sine rule: a/sin A = b/sin B = c/sin C or sin A/a = sin B/b = sin C/c

正弦定理：a/sin A = b/sin B = c/sin C 或 sin A/a = sin B/b = sin C/c

Cosine rule: a² = b² + c² − 2bc cos A

余弦定理：a² = b² + c² − 2bc cos A

The area of a triangle is given by ½ab sin C. You are also expected to know the fundamental identity sin²θ + cos²θ = 1 and be able to solve simple trigonometric equations within a given interval, often using the graphs of y = sin x, y = cos x and y = tan x.

三角形面积公式为 ½ab sin C。还需掌握基本恒等式 sin²θ + cos²θ = 1，并会利用 y = sin x、y = cos x 和 y = tan x 的图像在给定区间内解简单三角方程。

7. Matrices: Operations and Determinants | 矩阵：运算与行列式

A matrix is a rectangular array of numbers. Two matrices can be added or subtracted only if they have the same dimensions. Multiplication of matrices A (m × n) and B (n × p) yields an m × p matrix; the number of columns of A must equal the number of rows of B. Matrix multiplication is not commutative: in general AB ≠ BA.

矩阵是一个矩形的数字阵列。两个矩阵只有在维数相同时才能相加或相减。矩阵 A (m × n) 与 B (n × p) 相乘得到 m × p 矩阵；A 的列数必须等于 B 的行数。矩阵乘法不满足交换律：一般而言 AB ≠ BA。

For a 2 × 2 matrix M = [a b; c d], the determinant is det(M) = ad − bc. The inverse of M exists only if det(M) ≠ 0 and is given by 1/(ad−bc) [d −b; −c a]. Matrices are used to represent linear transformations such as reflections, rotations and enlargements; you should be able to identify the transformation from its matrix and vice versa.

对于 2 × 2 矩阵 M = [a b; c d]，行列式为 det(M) = ad − bc。M 的逆矩阵仅在 det(M) ≠ 0 时存在，公式为 1/(ad−bc) [d −b; −c a]。矩阵用于表示线性变换，如反射、旋转和放大；你应能由矩阵识别变换，反之亦然。

8. Vector Geometry | 向量几何

A vector is a quantity with both magnitude and direction. In two dimensions it can be written as a column vector (x y) or using i, j notation: xi + yj. Position vectors locate a point relative to an origin. The vector from A to B is found by subtracting the position vector of A from that of B: AB = b − a.

向量是既有大小又有方向的量。在二维情形下，它可以写成列向量 (x y) 或使用 i, j 记号：xi + yj。位置向量表示点相对于原点的位置。从 A 到 B 的向量可由 B 的位置向量减去 A 的位置向量得到：AB = b − a。

The magnitude (length) of vector v = xi + yj is |v| = √(x² + y²). A unit vector has magnitude 1. Vectors are parallel if one is a scalar multiple of the other. To prove points are collinear, show that the vectors between them share a common direction (are scalar multiples). Geometric problems are often solved by expressing one vector as a combination of known vectors.

向量 v = xi + yj 的模（长度）为 |v| = √(x² + y²)。单位向量的模为 1。若两个向量成标量倍数，则它们平行。证明三点共线的方法是证明它们之间的向量具有相同方向（即互为标量倍数）。几何问题通常通过将某个向量表示为已知向量的组合来解决。

9. Introduction to Differentiation | 微分入门

Differentiation gives the gradient of a curve at a point. For y = xⁿ, the derivative is dy/dx = nxⁿ⁻¹. This rule applies for any rational n. For a sum of terms, differentiate term by term. The derivative tells you the rate of change of y with respect to x.

微分给出曲线在某点处的斜率。对于 y = xⁿ，导数为 dy/dx = nxⁿ⁻¹。此法则对任意有理数 n 适用。对多项式逐项求导即可。导数表示 y 关于 x 的变化率。

For a function f(x), the derivative f ‘(x) is zero at stationary points. You can identify a maximum or minimum point by examining the sign of the derivative either side (first derivative test) or by using the second derivative if required. The tangent to a curve at x = a has gradient f ‘(a) and equation y − f(a) = f ‘(a)(x − a). The normal is perpendicular to the tangent.

对于函数 f(x)，在驻点处导数 f ‘(x) = 0。可通过检查导数两侧的符号变化（一阶导数检验法）或在需要时使用二阶导数来判断极大值或极小值。曲线在 x = a 处的切线斜率为 f ‘(a)，方程为 y − f(a) = f ‘(a)(x − a)。法线与切线垂直。

10. Introduction to Integration | 积分入门

Integration is the reverse process of differentiation. The indefinite integral of xⁿ (n ≠ −1) is ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c, where c is the constant of integration. The integral of a sum is the sum of the integrals.

积分是微分的逆运算。xⁿ（n ≠ −1）的不定积分为 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c，其中 c 为积分常数。多项式的积分等于各项积分之和。

A definite integral ∫ₐᵇ f(x) dx gives the exact area between the curve y = f(x) and the x-axis from x = a to x = b. Areas below the axis are negative; you may need to split the interval to find the total area enclosed. The fundamental theorem of calculus links differentiation and integration.

定积分 ∫ₐᵇ f(x) dx 给出曲线 y = f(x) 与 x 轴之间从 x = a 到 x = b 的精确面积。x 轴下方的面积为负值；求总面积时可能需要分割区间。微积分基本定理将微分与积分联系起来。

11. Sequences and Series | 数列与级数

An arithmetic sequence has a constant difference d between consecutive terms. The n-th term is uₙ = a + (n−1)d, and the sum of the first n terms is Sₙ = n/2 [2a + (n−1)d] or Sₙ = n/2 (a + l) where l is the last term. These formulae appear in the formula booklet.

等差数列相邻项的差 d 为常数。第 n 项为 uₙ = a + (n−1)d，前 n 项和为 Sₙ = n/2 [2a + (n−1)d] 或 Sₙ = n/2 (a + l)，其中 l 为末项。这些公式在公式表中提供。

A geometric sequence has a constant ratio r. The n-th term is uₙ = arⁿ⁻¹. The sum of the first n terms is Sₙ = a(1 − rⁿ)/(1 − r), valid for r ≠ 1. You may be asked to prove this formula or to use it in context. For a sequence that converges, |r| < 1; the sum to infinity is S∞ = a/(1 − r).

等比数列的相邻项比值 r 为常数。第 n 项为 uₙ = arⁿ⁻¹。前 n 项和为 Sₙ = a(1 − rⁿ)/(1 − r)（r ≠ 1）。你可能会被要求证明该公式或在实际问题中运用它。收敛数列需满足 |r| < 1；无穷项和为 S∞ = a/(1 − r)。

12. Probability and Statistics Essentials | 概率与统计基础

Probability is measured on a scale from 0 (impossible) to 1 (certain). The probability that two independent events both occur is P(A and B) = P(A) × P(B). For mutually exclusive events, P(A or B) = P(A) + P(B). Tree diagrams help you handle conditional probability: P(A|B) = P(A and B) / P(B). In WJEC Further Mathematics, you may be expected to use Venn diagrams and two-way tables to organise outcomes.

概率的度量范围从 0（不可能）到 1（必然）。两个独立事件同时发生的概率为 P(A and B) = P(A) × P(B)。对于互斥事件，P(A or B) = P(A) + P(B)。树状图有助于处理条件概率：P(A|B) = P(A and B) / P(B)。在 WJEC 进阶数学中，你可能需要运用维恩图和双向表格来整理结果。

For statistics, the mean of a set of data is Σx/n; the median is the middle value when data are ordered; the mode is the most frequent value. The range measures spread. You should be able to construct and interpret cumulative frequency diagrams for grouped data and find quartiles and the interquartile range.

在统计中，一组数据的平均数为 Σx/n；中位数是排序后位于中间的值；众数是出现次数最多的值。极差衡量数据的离散程度。你应能构造并能解读分组数据的累积频率图，并找出四分位数和四分位距。

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Formula & Theorem Quick Reference Handbook | 公式定理速查手册