📚 AQA A-level Maths Formula List: Complete Revision Guide | AQA A-level 数学公式表完整速览
This revision guide summarises the most important formulae in the AQA A-level Mathematics data and formula booklet, with practical notes on how to use them in the exam. AQA provides a clean formula list covering pure mathematics, statistics and mechanics, but many expressions still need to be memorised or derived from first principles, and knowing exactly which is which saves you time and marks.
本复习指南总结 AQA A-level 数学公式册中最重要的公式，并附上考试实战用法。AQA 为纯数学、统计和力学提供了一份简洁的公式表，但许多式子仍需你记忆或自行推导。清楚区分"给出的"与"要背的"，能帮你在考场上省下大量时间与分数。
1. What the AQA Formula List Covers | AQA 公式表涵盖内容
The AQA A-level Mathematics formulae booklet is organised into three strands: pure mathematics, statistics and mechanics. The pure section contains the core identities, series results and standard calculus results that you are expected to quote; the statistics section lists probability rules plus the binomial and normal distributions; the mechanics section gives the constant-acceleration equations, momentum and impulse relations, and the standard modelling assumptions.
AQA A-level 数学公式册分为三条主线：纯数学、统计与力学。纯数部分包含核心恒等式、级数结果和标准微积分结果；统计部分列出概率法则以及二项分布与正态分布；力学部分给出匀加速公式、动量与冲量关系，以及标准建模假设。
Always download the current version from the AQA website before you start revising. The booklet is revised occasionally, so a result that appeared in an older edition may not be printed in the present one, and vice versa.
开始复习前，请务必从 AQA 官网下载现行版本。公式册会不定期修订，旧版中印出的结果在现行版本里可能不再出现，反之亦然。
Key point: the booklet gives you the quadratic formula and the standard trigonometric identities, but it does not give every rearrangement, every derivation step or every modelling assumption. Manipulating and rearranging expressions remains entirely your responsibility.
关键点：公式册提供二次公式和标准三角恒等式，但并非所有变形、推导步骤或建模假设都会给出。式子的变形与整理仍然完全由你负责。
| Topic area | 板块 | Usually provided | 通常给出 | Usually memorised | 通常需记忆 |
|---|---|---|
| Algebra | 代数 | Quadratic formula, laws of logarithms | 二次公式、对数法则 | Discriminant, completing the square, index and surd laws | 判别式、配方法、指数与根式法则 |
| Series | 数列 | Arithmetic and geometric sums, binomial expansion | 等差与等比求和、二项展开 | Convergence conditions, sigma notation manipulation | 收敛条件、求和号运算 |
| Trigonometry | 三角 | Pythagorean identities, compound and double angle formulae | 平方恒等式、和角与倍角公式 | Exact values, radian conversions, small-angle approximations | 精确值、弧度换算、小角近似 |
| Calculus | 微积分 | Product, quotient and chain rules, integration by parts | 乘积、商、链式法则，分部积分 | Standard derivatives and integrals, limits, areas and volumes | 标准导数与积分、极限、面积与体积 |
| Statistics | 统计 | Binomial and normal distribution formulae | 二项与正态分布公式 | E(X) and Var(X) of a discrete variable, hypothesis test logic | 离散型随机变量的 E(X) 与 Var(X)、假设检验逻辑 |
| Mechanics | 力学 | Constant-acceleration equations, F = ma | 匀加速公式、F = ma | Energy and work relations, friction, resolving forces | 能量与功的关系、摩擦力、力的分解 |
Treat this table as a revision checklist rather than a fixed rule: verify each item against your own copy of the booklet, since exam boards can and do change what is printed.
把这张表当作复习清单，而不是死规定：请逐条对照你手上那份公式册核实，因为考试局确实会更改印刷内容。
2. Quadratics and Inequalities | 二次方程与不等式
For any quadratic equation ax² + bx + c = 0, the AQA booklet gives the solutions as:
对于任意二次方程 ax² + bx + c = 0，AQA 公式册给出的解为：
x = (-b ± √b² - 4ac) / 2a
gives two distinct roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots.
判别式 Δ = b² - 4ac 决定实根个数：Δ > 0 有两个不同实根，Δ = 0 有一个重根，Δ < 0 无实根。
Completing the square turns ax² + bx + c into a(x + p)² + q, which immediately reveals the vertex at (-p, q) and the minimum or maximum value. This form is not printed in the booklet in general, so practise it until it is automatic.
配方把 ax² + bx + c 化为 a(x + p)² + q，可立刻读出顶点 (-p, q) 以及最小或最大值。这个形式一般不会印在公式册上，所以要练到能条件反射般写出。
For inequalities, solve the associated equation first, then test intervals or sketch the curve. When a > 0 with roots α < β, you get ax² + bx + c > 0 for x < α or x > β, and the inequality is reversed between the roots. Remember that the sign flips when you multiply or divide by a negative quantity, and that you may not multiply an inequality by an expression whose sign
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