📚 Year 12 AQA Maths: Formula & Theorem Quick Reference | AQA 数学公式定理速查手册
This quick reference handbook is designed for Year 12 students following the AQA Mathematics specification. It compiles essential formulas, theorems and key concepts from the pure and applied components, providing a concise revision tool for both classroom tests and end‑of‑year assessments. Use this guide to rapidly recall the rules and relationships that underpin your problem‑solving toolkit.
本速查手册专为学习 AQA 数学大纲的 12 年级学生设计。它汇编了纯数及应用部分的关键公式、定理和核心概念，为课堂测验和年终评估提供简洁的复习工具。使用本指南快速回顾构成解题基础的各种规则和关系。
1. Algebraic Fundamentals | 代数基础
overn powers: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, a⁻ⁿ = 1/aⁿ, and a⁰ = 1 (a ≠ 0).
指数法则：aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, a⁻ⁿ = 1/aⁿ, 以及 a⁰ = 1 (a ≠ 0)。
d factorising rely on the distributive law. The quadratic formula solves ax² + bx + c = 0:
展开括号与因式分解依赖分配律。二次公式求解 ax² + bx + c = 0：
x = (−b ± √b² − 4ac) / (2a)
The discriminant Δ = b² − 4ac determines the nature of roots: Δ > 0 ⇒ two distinct real roots, Δ = 0 ⇒ one repeated real root, Δ < 0 ⇒ no real roots.
判别式 Δ = b² − 4ac 决定根的性质：Δ > 0 ⇒ 两个相异实根，Δ = 0 ⇒ 一个重实根，Δ < 0 ⇒ 无实根。
= 0. The remainder theorem says f(p) gives the remainder when f(x) is divided by (x − p).
因式定理指出，(x − p) 是多项式 f(x) 的因式当且仅当 f(p) = 0。余数定理表明，f(p) 等于 f(x) 除以 (x − p) 的余数。
Completing the square rewrites a quadratic as a(x + b/2a)² + (c − b²/4a).
配方法将二次式写为 a(x + b/2a)² + (c − b²/4a)。
2. Coordinate Geometry | 坐标几何
n the points is √[(x₂ − x₁)² + (y₂ − y₁)²] and the midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).
过点 (x₁, y₁) 与 (x₂, y₂) 的直线斜率为 m = (y₂ − y₁)/(x₂ − x₁)。两点距离为 √[(x₂ − x₁)² + (y₂ − y₁)²]，中点为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。
by + c = 0. Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = −1.
直线方程形式：y − y₁ = m(x − x₁), y = mx + c 以及 ax + by + c = 0。平行线斜率相等；垂直线满足 m₁ × m₂ = −1。
y − b)² = r². The expanded form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √g² + f² − c.
圆心 (a, b)、半径 r 的圆方程为 (x − a)² + (y − b)² = r²。展开式 x² + y² + 2gx + 2fy + c = 0 的圆心为 (−g, −f)，半径为 √g² + f² − c。
The tangent to a circle at a point is perpendicular to the radius at that point.
圆上一点处的切线与该点处的半径垂直。
3. Sequences and Series | 数列与级数
e first n terms is Sₙ = n/2 [2a + (n − 1)d] or Sₙ = n/2 (a + l), where l is the last term.
等差数列有首项 a 和公差 d。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。
of the first n terms is Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. The sum to infinity exists when |r| < 1 and is S∞ = a/(1 − r).
等比数列有首项 a 和公比 r。第 n 项为 uₙ = arⁿ⁻¹。前 n 项和为 Sₙ = a(1 − rⁿ)/(1 − r)，r ≠ 1。无穷级数求和当 |r| < 1 时存在，且 S∞ = a/(1 − r)。
Sigma notation: Σ (from k=1 to n) aₖ represents the sum of terms a₁, a₂, …, aₙ.
Σ 符号：Σ (k=1 到 n) aₖ 表示项 a₁, a₂, …, aₙ 的和。
4. Trigonometry | 三角学
Radians: π rad = 180°. Arc length s = rθ, sector area A = ½ r²θ, where θ is in radians.
弧度制：π 弧度 = 180°。弧长 s = rθ，扇形面积 A = ½ r²θ，其中 θ 以弧度为单位。
te/adjacent. In the unit circle, (cos θ, sin θ) are the coordinates of a point at angle θ.
定义：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。在单位圆中，(cos θ, sin θ) 是角度 θ 处点的坐标。
Key identities: sin²θ + cos²θ ≡ 1, tan θ ≡ = c/sin C. The cosine rule: a² = b² + c² − 2bc cos A, and cos A = (b² + c² − a²) / (2bc).
重要恒等式：sin²θ + cos²θ ≡ 1，tan θ ≡ sin θ/cos θ。正弦定理：a/sin A = b/sin B = c/sin C。余弦定理：a² = b² + c² − 2bc cos A，以及 cos A = (b² + c² − a²) / (2bc)。
Area of a triangle using sine: Area = ½ ab sin C.
三角形面积（用正弦）：面积 = ½ ab sin C。
Exact values to memorise: sin 30° = ½, cos 45° = √2/2, tan 60° = √3, etc.
需记忆的精确值：sin 30° = ½，cos 45° = √2/2，tan 60° = √3 等。
5. Exponentials and Logarithms | 指数与对数
The exponential function with base e (e ≈ 2.718) is f(x) = eˣ. Its inverse is the natural logarithm, ln x, so ln(eˣ) = x and eˡⁿˣ = x for x > 0.
以 e 为底的指数函数（e ≈ 2.718）为 f(x) = eˣ。其反函数为自然对数 ln x，因此有 ln(eˣ) = x 和 eˡⁿˣ = x（x > 0）。
ln a + ln b, ln(a/b) = ln a − ln b, ln(aᵏ) = k ln a. Change of base: logₐ x = ln x / ln a.
对数法则：ln(ab) = ln a + ln b，ln(a/b) = ln a − ln b，ln(aᵏ) = k ln a。换底公式：logₐ x = ln x / ln a。
rowth and decay models take the form y = aeᵏᵗ, where k > 0 for growth and k < 0 for decay.
增长与衰减模型形如 y = aeᵏᵗ，其中 k > 0 为增长，k < 0 为衰减。
6. Differentiation | 微分
The derivative f' (h→0) [f(x+h) − f(x)]/h. It represents the gradient of the tangent to the curve y = f(x).
导数 f'(x) = lim (h→0) [f(x+h) − f(x)]/h。它表示曲线 y = f(x) 切线的斜率。
| f(x) | f'(x) |
|---|---|
| xⁿ | nxⁿ⁻¹ |
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
Rules: product rule d/dx (uv) = u'v + uv', quotient rule d/dx (u/v) = (u'v − uv')/v², chain rule d/dx [f(g(x))] = f'(g(x))·g'(x). For composite powers, d/dx [ (ax+b)ⁿ ] = na(ax+b)ⁿ⁻¹.
法则：乘积法则 d/dx (uv) = u'v + uv'，商法则 d/dx (u/v) = (u'v − uv')/v²，链式法则 d/dx [f(g(x))] = f'(g(x))·g'(x)。对于幂次复合，d/dx [ (ax+b)ⁿ ] = na(ax+b)ⁿ⁻¹。
Stationary points occur where f'(x) = 0. Use the second derivative f''(x): f''(x) > 0 ⇒ minimum, f''(x) < 0 ⇒ maximum. The equation of the tangent at (a, f(a)) is y − f(a) = f'(a)(x − a); the normal has gradient −1/f'(a).
驻点出现在 f'(x) = 0 处。利用二阶导数 f''(x)：f''(x) > 0 ⇒ 极小值，f''(x) < 0 ⇒ 极大值。点 (a, f(a)) 处的切线方程为 y − f(a) = f'(a)(x − a)；法线斜率为 −1/f'(a)。
7. Integration | 积分
n is the reverse of differentiation. The indefinite integral ∫ f(x) dx = F(x) + C, where F'(x) = f(x). Definite integrals give the signed area between the curve and the x‑axis.
积分是微分的逆运算。不定积分 ∫ f(x) dx = F(x) + C，其中 F'(x) = f(x)。定积分给出曲线与 x 轴之间的有号面积。
| f(x) | ∫ f(x) dx |
|---|---|
| xⁿ (n ≠ −1) | xⁿ⁺¹/(n+1) + C |
| 1/x | ln|x| + C |
| eˣ | eˣ + C |
| sin x | −cos x + C |
| cos x | sin x + C |
| sec² x | tan x + C |
the form (ax + b)ⁿ (n ≠ −1), use ∫ (ax + b)ⁿ dx = 1/a · (ax + b)ⁿ⁺¹/(n + 1) + C. For ∫ eᵃˣ⁺ᵇ dx = 1/a · eᵃˣ⁺ᵇ + C.
积分形如 (ax + b)ⁿ (n ≠ −1) 的函数，用 ∫ (ax + b)ⁿ dx = 1/a · (ax + b)ⁿ⁺¹/(n + 1) + C。对于 ∫ eᵃˣ⁺ᵇ dx = 1/a · eᵃˣ⁺ᵇ + C。
Area between a curve and the x‑ ∫ₐᵇ f(x) dx. Area between two curves is ∫ₐᵇ [f(x) − g(x)] dx, where f(x) ≥ g(x) on [a, b].
曲线与 x 轴在 x = a 到 x = b 间的面积为 A = ∫ₐᵇ f(x) dx。两曲线之间的面积为 ∫ₐᵇ [f(x) − g(x)] dx，其中在 [a, b] 上 f(x) ≥ g(x)。
8. Vectors | 向量
A two‑ x, y) or xi + yj, with i and j being unit vectors along the x and y axes. The magnitude is |v| = √x² + y².
二维向量可表示为 v = (x, y) 或 xi + yj，i 和 j 分别为沿 x 轴和 y 轴的单位向量。模为 |v| = √x² + y²。
A unit vector in the direction of v is v̂ = v/|v|. The position vector of a point P is OP. The vector from A to B is AB = OB − OA.
沿 v 方向的单位向量为 v̂ = v/|v|。点 P 的位置向量为 OP。从 A 到 B 的向量为 AB = OB − OA。
The scalar (dot) product is a · b = |a||b| he vectors. In component form, if a = (x₁, y₁) and b = (x₂, y₂), then a · b = x₁x₂ + y₁y₂.
数量积（点积）为 a · b = |a||b| cos θ，θ 为两向量之间的夹角。以分量表示，若 a = (x₁, y₁) 且 b = (x₂, y₂)，则 a · b = x₁x₂ + y₁y₂。
Use the dot product to find the angle: cos θ = (a · b) / (|a||b|). Perpendicular vectors satisfy a · b = 0.
利用点积求角度：cos θ = (a · b) / (|a||b|)。垂直向量满足 a · b = 0。
9. Numerical Methods | 数值方法
ocate a root of f(x) = 0, find an interval [a, b] where f(a) and f(b) have opposite signs,
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