📚 Mastering AQA 9660 MA03 Pure Mathematics Unit 2 | 精通 AQA 9660 MA03 纯数学第二单元
This revision guide covers the core content of AQA 9660 Mathematics International A level MA03 Pure Mathematics Unit 2. The unit extends your algebra, trigonometry and calculus skills and places strong emphasis on rigorous proof, modelling and problem solving. Work through each section in order, and use the paired bilingual notes to check your understanding.
本复习指南涵盖 AQA 9660 数学国际 A level MA03 纯数学第二单元的核心内容。该单元拓展你的代数、三角和微积分技能，并强调严谨证明、建模与问题解决。请按顺序学习各节，并利用中英双语对照检查理解。
1. Proof and Mathematical Argument | 证明与数学论证
MA03 rewards clear logical structure rather than a single 'correct' answer. You need to prove statements by deduction, disprove them by counterexample, and sometimes use exhaustion for a small finite set of cases.
MA03 注重清晰的逻辑结构，而不是单一的 "正确" 答案。你需要通过演绎法证明命题，通过反例法反驳命题，有时还要对有限的小范围情况使用穷举法。
A direct proof of an identity often starts from one side and transforms it using algebra or known identities until it matches the other side. Never begin by writing the identity as an assumption.
直接证明恒等式通常从等式的一侧出发，利用代数或已知恒等式逐步变形，直到与另一侧相同。切勿一开始就把待证恒等式当作假设写下来。
- Deduction: from general rules to a specific conclusion | 演绎法：从一般规则推出具体结论
- Counterexample: one example that disproves a statement | 反例：一个能推翻命题的例子
- Exhaustion: check every possible case in a finite set | 穷举法：检查有限集合中的每一种可能情况
2. Algebra and Functions: Polynomial Division and Factor Theorem | 代数与函数：多项式除法与因式定理
a factor. The remainder theorem gives f(a) as the remainder when f(x) is divided by (x − a).
多项式除法和因式定理是核心工具。如果 f(x) 是多项式且 f(a) = 0，则 (x − a) 是一个因式。余数定理指出，f(x) 除以 (x − a) 的余数就是 f(a)。
You must also manipulate rational expressions, including decomposing improper fractions into a polynomial plus a proper fraction. This skill often supports integration and curve sketching later.
你还必须会处理有理式，包括将假分式分解为一个多项式加一个真分式。这一技能常常为后续的积分和曲线作图提供支持。
f(x) = (x − a)q(x) + f(a)
3. Coordinate Geometry: Circles and Parametric Equations | 坐标几何：圆与参数方程
) is the centre and r is the radius. Completing the square is needed when a circle is given in expanded form.
圆的标准方程为 (x − a)² + (y − b)² = r²，其中 (a, b) 为圆心，r 为半径。当圆以一般式给出时，需要用配方法化为标准式。
Parametric equations define x and y separately in terms of a third variable, usually t. To find a Cartesian equation, eliminate the parameter, but always consider any domain restrictions from the original parametric definitions.
参数方程用第三个变量（通常是 t）分别定义 x 和 y。求直角坐标方程时需要消去参数，但必须始终考虑原参数定义带来的定义域限制。
(x − a)² + (y − b)² = r²
4. Sequences and Series: Arithmetic and Geometric Progressions | 数列与级数：等差与等比数列
n/2 [2a + (n − 1)d]. Use these when the difference between consecutive terms is constant.
对于等差数列，第 n 项为 aₙ = a + (n − 1)d，前 n 项和为 Sₙ = n/2 [2a + (n − 1)d]。当相邻项之差为常数时使用这些公式。
n, uₙ = arⁿ⁻¹ and Sₙ = a(1 − rⁿ) / (1 − r) for r ≠ 1. The sum to infinity exists only when |r| < 1 and is given by S∞ = a / (1 − r).
对于等比数列，uₙ = arⁿ⁻¹，且当
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