📚 AQA Maths: High-Frequency Topics Summary | AQA 数学：高频考点总结
Mastering AQA Mathematics requires a strategic focus on the topics that appear most consistently across past papers. This summary highlights the high-frequency areas in pure mathematics, statistics, and mechanics, providing key formulas, common pitfalls, and exam tips to help you maximise your revision efficiency.
要掌握 AQA 数学，需要有针对性地复习历年真题中频繁出现的考点。本文总结纯数学、统计和力学中的高频主题，提供关键公式、常见错误和应试技巧，帮助你提高复习效率。
1. Algebraic Manipulation & Remainder Theorem | 代数运算与余式定理
Algebraic fluency is essential. You must be comfortable with polynomial division, factor theorem, and manipulating rational expressions.
代数运算能力至关重要。你需要熟练掌握多项式除法、因式定理以及有理式的变形。
a polynomial f(x) by (x - a), the remainder is f(a). If f(a)=0, then (x - a) is a factor.
用 (x - a) 去除多项式 f(x) 时，余数为 f(a)。若 f(a)=0，则 (x - a) 为因式。
Use polynomial long division or synthetic division to factorise cubics or simplify improper algebraic fractions.
使用长除法或综合除法分解三次多项式或简化假分数代数式。
Be careful with signs when expanding brackets or factorising by grouping.
在展开括号或分组因式分解时，注意符号。
2. Binomial Expansion | 二项式展开
The binomial expansion appears frequently, either for positive integer powers or for fractional and negative powers using the (1 + x)ⁿ form.
二项式展开经常出现，无论是正整数次幂，还是对分数或负指数使用 (1 + x)ⁿ 形式。
For positive integer n, the expansion: (a + b)ⁿ = Σ (nCk) aⁿ⁻ᵏ bᵏ. nCk = n!/(k!(n-k)!).
对于正整数 n，展开式为 (a + b)ⁿ = Σ (nCk) aⁿ⁻ᵏ bᵏ，nCk = n!/(k!(n-k)!)。
For (1 + bx)ⁿ where n is not a positive integer, the expansion is valid for |bx| < 1, and terms: 1 + n bx + n(n-1)/2! (bx)² + …
对于 n 不是正整数的 (1 + bx)ⁿ，展开式在 |bx| < 1 时有效，项为：1 + n bx + n(n-1)/2! (bx)² + …
State the range of validity explicitly to gain method marks.
明确写出有效范围可获得过程分。
| n | Expansion valid for |
|---|---|
| Positive integer | All x |
| Fraction or negative | |bx| < 1 |
3. Trigonometry | 三角学
Trigonometric identities, solving equations, and radian measure are core AQA topics. Expect questions involving the sine/cosine rules and graphs of trig functions.
三角恒等式、解三角方程和弧度制是 AQA 核心内容。考试会涉及正弦/余弦定理和三角函数图像。
al identity: sin²θ + cos²θ = 1, and derived forms: tan²θ + 1 = sec²θ, 1 + cot²θ = cosec²θ.
牢记基本恒等式：sin²θ + cos²θ = 1，以及推导形式：tan²θ + 1 = sec²θ，1 + cot²θ = cosec²θ。
When solving trig equations, find all solutions in the given interval using CAST diagram or graphical method. Always set calculator to radian mode if required.
解三角方程时，使用 CAST 图或图像法找出给定区间内的所有解。若题目要求弧度，务必把计算器设为弧度模式。
Double-angle formulas: sin2θ = 2sinθcosθ, cos2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ.
倍角公式：sin2θ = 2sinθcosθ，cos2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ。
4. Exponentials & Logarithms | 指数与对数
Be proficient in converting between exponential and logarithmic forms, solving equations, and modelling with ex and ln x.
熟练掌握指数形式与对数形式的互化、解方程以及用 ex 和 ln x 建模。
Key relationship: y = aˣ ⇔ logₐy = x. The natural log, ln, is log base e.
关键关系：y = aˣ ⇔ logₐy = x。自然对数 ln 是以 e 为底的对数。
Laws of logs: logₐ(xy) = logₐx + logₐy; logₐ(x/y) = logₐx - logₐy; logₐ(xᵏ) = k logₐx. Published by TutorHao | Mathematics Revision Series | aleveler.com
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