📚 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。
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When dividing 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。
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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。
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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。
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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)!)。
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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 + …
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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。
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Know the fundamental 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。
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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。
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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 e^x 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 e^x 和 ln x 建模。
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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。
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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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