📚 AS Eduqas Further Mathematics: Formula & Theorem Quick-Reference Handbook | AS Eduqas 进阶数学：公式定理速查手册
This article provides a concise, exam-focused collection of essential formulae and theorems for the AS Level Eduqas Further Mathematics specification. It covers core pure mathematics topics, including algebra, trigonometry, calculus and vectors. Use this handbook as a quick revision aid or to reinforce your understanding ahead of assessments.
本文为 AS 阶段 Eduqas 进阶数学考试整理了一份简明、聚焦考点的公式定理速查手册。内容涵盖纯数核心主题，包括代数、三角、微积分和向量。在备考复习时，可用作快速查阅工具，也可用于巩固理解。
1. Algebraic Fundamentals & Proof | 代数基础与证明
× aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ. Also recall that a⁰ = 1 (a ≠ 0) and a⁻ⁿ = 1/aⁿ.
掌握有理指数幂的运算法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ。同时记住 a⁰ = 1（a ≠ 0）以及 a⁻ⁿ = 1/aⁿ。
Surds can be simplified using √ab = √a × √b and rationalised by multiplying numerator and denominator by the conjugate, e.g. 1/(√a + √b) = (√a − √b)/(a − b).
根式可借助 √ab = √a × √b 化简，并通过乘以共轭式进行分母有理化，如：1/(√a + √b) = (√a − √b)/(a − b)。
For proof, be familiar with direct proof, proof by exhaustion and disproof by counter‑example. In algebraic proof, express even numbers as 2k and odd numbers as 2k + 1.
在证明中，需熟悉直接证明、穷举法和反例证伪。在代数证明中，偶数表示为 2k，奇数表示为 2k + 1。
2. Quadratic Functions & the Discriminant | 二次函数与判别式
ic equation is ax² + bx + c = 0 (a ≠ 0). Its solutions are given by the quadratic formula:
标准二次方程为 ax² + bx + c = 0（a ≠ 0）。其解由求根公式给出：
x = (−b ± √b² − 4ac) / 2a
distinct real roots if Δ > 0, one repeated real root if Δ = 0, and no real roots if Δ < 0.
判别式 Δ = b² − 4ac 决定了根的性质：若 Δ > 0，有两个不等实根；若 Δ = 0，有一个重根；若 Δ < 0，无实根。
Completing the square rewrites ax² + bx + c as a(x + p)² + q, where the vertex of the parabola is at (−p, q). For a quadratic in the form x² + bx, add (b/2)² to complete the square.
配方法将 ax² + bx + c 写成 a(x + p)² + q，其中抛物线顶点为 (−p, q)。对于 x² + bx 形式的二次式，加上 (b/2)² 即可完成配方。
3. Polynomials & the Factor Theorem | 多项式与因式定理
ainder is f(a). If f(a) = 0, then (x − a) is a factor of f(x). This is the factor theorem.
对于多项式 f(x)，余式定理指出：f(x) 除以 (x − a) 的余式为 f(a)。若 f(a) = 0，则 (x − a) 是 f(x) 的一个因式，即因式定理。
When dividing a polynomial, equate coefficients or use algebraic long division. The factor theorem is useful for factorising cubic and higher-degree polynomials. Always check for common factors first.
进行多项式除法时，可采用待定系数法或代数长除法。因式定理对于分解三次及更高次多项式十分有用。务必首先检查是否存在公因式。
4. Binomial Expansion | 二项展开
For any rational n and |x| < 1 (for non‑positive integer n), the binomial expansion is:
对于任何有理数 n 及 |x| < 1（当 n 不是正整数时），二项展开为：
(1 + x)ⁿ = 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + …
he binomial theorem with nCr coefficients: (a + b)ⁿ = Σₖ₌₀ⁿ ⁿCₖ aⁿ⁻ᵏ bᵏ. Note that ⁿCₖ = n! / [k!(n−k)!].
当 n 为正整数时，(a + b)ⁿ 的展开使用组合数系数： (a + b)ⁿ = Σₖ₌₀ⁿ ⁿCₖ aⁿ⁻ᵏ bᵏ，其中 ⁿCₖ = n! / [k!(n−k)!]。
Be able to find the range of validity for infinite expansions: for (a + bx)ⁿ, the expansion is valid when |bx/a| < 1.
需能确定无穷展开的有效范围：对于 (a + bx)ⁿ，当 |bx/a| < 1 时展开有效。
5. Trigonometry: Ratios, Graphs & Identities | 三角学：比值、图像与恒等式
Exact trigonometric values you must remember:
| Angle θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° (π/6) | 1/2 | √3/2 | 1/√3 |
| 45° (π/4) | 1/√2 | 1/√2 | 1 |
| 60° (π/3) | √3/2 | 1/2 | √3 |
| 90° (π/2) | 1 | 0 | undefined |
必须熟记以下三角函数精确值：
hagorean identity sin²θ + cos²θ = 1. From this, dividing by cos²θ gives 1 + tan²θ = sec²θ.
核心恒等式包括 tan θ = sin θ / cos θ，以及毕达哥拉斯恒等式 sin²θ + cos²θ = 1。将其除以 cos²θ 可得 1 + tan²θ = sec²θ。
For solving trigonometric equations, use the quadrant rule (CAST diagram) to find all solutions within a given interval. Always express angles in the correct unit – degrees or radians.
求解三角方程时，利用象限法则（CAST 图）求出给定区间内的所有解。务必使用正确的角度单位——度或弧度。
6. Exponentials & Logarithms | 指数与对数
The natural exponential function y = eˣ has derivative f '(x) = eˣ and inverse y = ln x, where ln x = logₑ x. The logarithmic laws mirror index laws:
自然指数函数 y = eˣ 的导数为 f '(x) = eˣ，其反函数为 y = ln x，其中 ln x = logₑ x。对数运算法则与指数法则对应：
- Product rule: ln(ab) = ln a + ln b
- 商法则： ln(a/b) = ln a − ln b
- Power rule: ln(aⁿ) = n ln a
- 幂法则： ln(aⁿ) = n ln a
Also recall that ln e = 1 and ln 1 = 0. To solve equations like e²ˣ = 5, take natural logs of both sides: 2x = ln 5.
同时记住 ln e = 1 且 ln 1 = 0。为求解形如 e²ˣ = 5 的方程，两边取自然对数：2x = ln 5。
The function aˣ can be expressed as eˣˡⁿᵃ. Its derivative can be found using the chain rule.
函数 aˣ 可表示为 eˣˡⁿᵃ。其导数可利用链式法则求得。
7. Coordinate Geometry & Parametric Equations | 坐标几何与参数方程
(x₂ − x₁). The equation of a line can be written as y − y₁ = m(x − x₁) or ax + by + c = 0.
过 (x₁, y₁) 和 (x₂, y₂) 的直线斜率为 m = (y₂ − y₁)/(x₂ − x₁)。直线方程可写为 y − y₁ = m(x − x₁) 或 ax + by + c = 0。
To find tangents or intersections, use the discriminant condition for a line touching the circle.
圆心为 (a, b)、半径为 r 的圆方程为 (x − a)² + (y − b)² = r²。求切线或交点时，利用直线与圆相切的判别式条件。
d the Cartesian equation, eliminate t. The gradient is given by dy/dx = (dy/dt) / (dx/dt).
参数方程用第三个变量（通常为 t）定义 x 和 y。要找出笛卡尔方程，需消去 t。斜率为 dy/dx = (dy/dt) / (dx/dt)。
8. Differentiation | 微分
erivative is dy/dx = nxⁿ⁻¹. This rule works for any real n. For y = f(x) ± g(x), dy/dx = f '(x) ± g'(x).
对于 y = xⁿ，导数为 dy/dx = nxⁿ⁻¹。该法则对任意实数 n 均适用。对于 y = f(x) ± g(x)，dy/dx = f '(x) ± g'(x)。
The chain rule: if y = f(g(x)), then dy/dx = f '(g(x)) × g'(x). It is often written as dy/dx = (dy/du) × (du/dx).
链式法则：若 y = f(g(x))，则 dy/dx = f '(g(x)) × g'(x)。常记作 dy/dx = (dy/du) × (du/dx)。
= u(dv/dx) + v(du/dx). The quotient rule: for y = u/v, dy/dx = [v(du/dx) − u(dv/dx)] / v².
乘积法则：对于 y = uv，dy/dx = u(dv/dx) + v(du/dx)。商法则：对于 y = u/v，dy/dx = [v(du/dx) − u(dv/dx)] / v²。
Standard derivatives to memorise:
- d/dx (eˣ) = eˣ
- d/dx (ln x) = 1/x
- d/dx (sin x) = cos x
- d/dx (cos x) = −sin x
- d/dx (tan x) = sec² x
需记忆的标准导数：
- d/dx (eˣ) = eˣ
- d/dx (ln x) = 1/x
- d/dx (sin x) = cos x
- d/dx (cos x) = −sin x
- d/dx (tan x) = sec² x
9. Integration | 积分
fferentiation. The indefinite integral of xⁿ (n ≠ −1) is ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C. For eˣ, ∫ eˣ dx = eˣ + C, and ∫ 1/x dx = ln|x| + C.
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。对于 eˣ，∫ eˣ dx = eˣ + C；∫ 1/x dx = ln|x| + C。
rals include ∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C, and ∫ sec² x dx = tan x + C.
标准三角积分包括 ∫ sin x dx = −cos x + C、∫ cos x dx = sin x + C 以及 ∫ sec² x dx = tan x + C。
e definite integral ∫ₐᵇ f(x) dx gives the signed area between the curve y = f(x) and the x‑ mental theorem of calculus: ∫ₐᵇ f(x) dx = F(b) − F(a), where F is any antiderivative of f.
定积分 ∫ₐᵇ f(x) dx 表示曲线 y = f(x) 和 x 轴之间从 x = a 到 x = b 的带号面积。微积分基本定理：∫ₐᵇ f(x) dx = F(b) − F(a)，其中 F 是 f 的任一原函数。
Integration by 'reverse chain rule' (or inspection) works for integrals of the form ∫ f '(g(x)) g'(x) dx. For example, ∫ 2x eˣ² dx = eˣ² + C.
反向链式法则（或观察法）适用于形如 ∫ f '(g(x)) g'(x) dx 的积分。例如 ∫ 2x eˣ² dx = eˣ² + C。
10. Vectors | 向量
A vector has both magnitude and direction. In 2D, a vector can be written as v = xi + yj or as a column vector. The magnitude is |v| = √x² + y².
向量既有大小又有方向。在二维中，向量可表示为 v = xi + yj 或列向量形式。其模长为 |v| = √x² + y²。
If a = a₁i + a₂j and b = b₁i + b₂j, then a + b = (a₁+b₁)i + (a₂+b₂)j. The scalar product (dot product) is a · b = a₁b₁ + a₂b₂ = |a||b| cos θ, where θ is the angle between the vectors.
若 a = a₁i + a₂j，b = b₁i + b₂j，则 a + b = (a₁+b₁)i + (a₂+b₂)j。数量积（点积）为 a · b = a₁b₁ + a₂b₂ = |a||b| cos θ，其中 θ 为两向量间的夹角。
Vectors are parallel if one is a scalar multiple of the other. They are perpendicular (orthogonal) if their dot product is zero.
若一个向量是另一向量的标量倍数，则二者平行。若点积为零，则二者垂直（正交）。
To find the angle between two vectors, use cos θ = ( a · b ) / (|a||b|). Position vectors, displacement vectors and vector equations of lines are also examinable.
求两向量间的夹角可使用 cos θ = ( a · b ) / (|a||b|)。位置向量、位移向量以及直线的向量方程也是考查内容。
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