A-Level Maths Year 2 Pure: Comprehensive Revision Notes | A-Level数学 Year 2 Pure 知识点精讲

📚 A-Level Maths Year 2 Pure: Comprehensive Revision Notes | A-Level数学 Year 2 Pure 知识点精讲

The second year of A-Level Pure Mathematics deepens your algebraic, trigonometric, and calculus skills, introducing more advanced differentiation, integration techniques, vectors in 3D, and differential equations. This guide covers key topics with paired English and Chinese explanations to support bilingual revision.

A-Level纯数学第二年深化了代数、三角和微积分技能，引入了更高级的微分、积分技巧、三维向量以及微分方程。本指南以中英文对照讲解覆盖各核心知识点，助力双语复习。

1. Algebra, Partial Fractions and Proof | 代数、部分分式与证明

In Year 2, you extend algebraic methods to include dividing polynomials, splitting rational expressions into partial fractions (with linear, repeated, and quadratic factors), and using proof by contradiction.

第二年，代数方法扩展到多项式除法、将有理式拆分为部分分式（含线性、重复和二次因式），以及反证法证明。

For partial fractions with a denominator like (x+1)(x−2)(x+3), you write A/(x+1) + B/(x−2) + C/(x+3) and solve for constants. For repeated factors such as (x+2)², include terms A/(x+2) and B/(x+2)².

对于分母如 (x+1)(x−2)(x+3)，可写成 A/(x+1) + B/(x−2) + C/(x+3) 求解待定系数。对于重因式如 (x+2)²，需包含 A/(x+2) 和 B/(x+2)² 两项。

Proof by contradiction starts by assuming the opposite of what you need to prove, then logically reaching an impossibility. For example, prove √2 is irrational.

反证法先假设需证结论的反面，然后逻辑推导出矛盾。例如证明√2是无理数。

2. Functions, Modulus and Transformations | 函数、模函数与变换

The modulus function |x| and solving equations like |f(x)| = a or |f(x)| = |g(x)| require considering positive and negative branches. Composite and inverse functions are extended with domain and range restrictions.

模函数 |x| 以及解形如 |f(x)| = a 或 |f(x)| = |g(x)| 的方程需要考虑正负分支。复合函数和反函数在定义域和值域限制下得到深化。

Transformations of graphs can be combined: y = a f(bx + c) + d involves stretches, reflections and translations. Always apply horizontal transformations first.

图像变换可组合：y = a f(bx + c) + d 涉及伸缩、反射和平移。务必先进行水平方向变换。

3. Sequences, Series and Binomial Expansion | 数列、级数与二项展开式

The binomial expansion (1 + x)ⁿ for rational n is valid for |x| < 1 and yields an infinite series. Use the formula n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + … .

对有理数 n 的二项展开式 (1 + x)ⁿ 在 |x| < 1 时有效，可展开为无穷级数。使用公式 n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + … 。

Sequence behaviour is analysed using sigma notation, arithmetic and geometric progressions. For a geometric series with |r| < 1, sum to infinity is a/(1−r).

利用∑符号、等差与等比数列分析数列行为。对于 |r| < 1 的等比级数，无穷和为 a/(1−r)。

You also learn to find limits of sequences and apply recurrence relations.

还将学习求数列极限并应用递推关系。

4. Trigonometry: Radians, Identities and Harmonic Form | 三角学：弧度、恒等式与谐波形式

Radian measure is essential for calculus with trig functions. Arc length s = rθ, sector area A = ½ r²θ. Reciprocal functions sec, csc, cot and inverse functions sin^-1, cos^-1, tan^-1 are introduced.

弧度制是三角微积分的基础。弧长 s = rθ，扇形面积 A = ½ r²θ。引入倒数函数 sec, csc, cot 和反三角函数 sin^-1, cos^-1, tan^-1。

Key identities include compound angle formulas sin(A±B), cos(A±B), tan(A±B), and double-angle formulas sin2θ = 2sinθcosθ, cos2θ = cos²θ − sin²θ = 2cos²θ − 1 =

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