📚 Edexcel A Level Maths Year 2 Pure: Complete Revision Guide | Edexcel A Level 数学 Year 2 Pure 完整复习指南
This revision guide covers the core content of Edexcel A Level Mathematics Year 2 Pure. It focuses on proof, algebraic methods, sequences, trigonometry, calculus, parametric equations, numerical methods and vectors. Use the worked ideas and paired explanations to check your understanding and to prepare for examination-style questions.
本复习指南涵盖 Edexcel A Level 数学 Year 2 Pure 的核心内容。重点包括证明、代数方法、数列、三角学、微积分、参数方程、数值方法和向量。请使用其中的解题思路与英中对照解释来检查理解情况，并为考试题型做好准备。
1. Proof by Contradiction and Partial Fractions | 反证法与部分分式
Proof by contradiction begins by assuming the opposite of the required statement. You then use valid logical steps until you reach an impossible conclusion. This contradiction proves that the original statement must be true.
反证法从假设命题的反面成立开始。然后通过正确的逻辑推理，直到推出一个不可能的结论。这个矛盾就证明原命题一定成立。
A classic example is proving that √2 is irrational. Suppose √2 = p/q where p and q are integers with no common factor. Squaring gives 2q² = p², so p must be even. Write p = 2k, which gives 2q² = 4k² and therefore q² = 2k². This means q is also even, contradicting the assumption that p and q have no common factor.
一个经典的例子是证明 √2 是无理数。假设 √2 = p/q，其中 p 和 q 是没有公因数的整数。平方后得到 2q² = p²，所以 p 必为偶数。令 p = 2k，则 2q² = 4k²，因此 q² = 2k²。这说明 q 也是偶数，从而与 p、q 没有公因数的假设矛盾。
For partial fractions, a rational function with distinct linear denominators can be split into simpler fractions. For example, (5x + 1)/((x − 1)(x + 2)) can be written as A/(x − 1) + B/(x + 2). Multiplying through and comparing coefficients gives A = 2 and B = 3, so the decomposition is 2/(x − 1) + 3/(x + 2).
对于部分分式，分母为互不相同的一次因式的有理函数可以拆分成更简单的分式。例如，(5x + 1)/((x − 1)(x + 2)) 可以写成 A/(x − 1) + B/(x + 2)。通分并比较系数可得 A = 2、B = 3，因此分解结果为 2/(x − 1) + 3/(x + 2)。
If a denominator has a repeated factor such as (x + 1)², you must include both A/(x + 1) and B/(x + 1)². If the fraction is improper, carry out algebraic division first before applying partial fractions.
如果分母含有重复因式，例如 (x + 1)²，则必须同时写出 A/(x + 1) 和 B/(x + 1)²。若分式为假分式，应先进行多项式除法，再使用部分分式。
2. Sequences and Series | 数列与级数
For an arithmetic sequence, the nth term is uₙ = a + (n − 1)d, where a is the first term and d is the common difference. The sum of the first n terms is Sₙ = n/2 [2a + (n − 1)d] or Sₙ = n/2 (a + l), where l is the last term.
对于等差数列，第 n 项为 uₙ = a + (n − 1)d，其中 a 为首项，d 为公差。前 n 项和为 Sₙ = n/2 [2a + (n − 1)d] 或 Sₙ = n/2 (a + l)，其中 l 为末项。
For a geometric sequence, the nth term is uₙ = arⁿ⁻¹, where a is the first term and r is the common ratio. The sum of the first n terms is Sₙ = a(1 − rⁿ)/(1 − r) when r ≠ 1. If |r| < 1, the infinite series converges with sum to infinity S∞ = a/(1 − r).
对于等比数列，第 n 项为 uₙ = arⁿ⁻¹，其中 a 为首项，r 为公比。前 n 项和为 Sₙ = a(1 − rⁿ)/(1 − r)，其中 r ≠ 1。如果 |r| < 1，无穷级数收敛，其无穷和为 S∞ = a/(1 − r)。
Sigma notation is used to write series compactly. You may need to change the index, split a sum, or use standard results. Recurrence relations of the form uₙ₊₁ = f(uₙ) are often tested by substituting an initial value and generating successive terms.
级数常用 Σ 符号简洁表示。你可能需要变换下标、拆分和式，或使用标准结果。形如 uₙ₊₁ = f(uₙ) 的递推关系常通过代入初值并逐项生成后项来考查。
3. Binomial Expansion for Rational Powers | 有理数幂的二项式展开
The binomial expansion for rational n is (1 + x)ⁿ = 1 + nx + n(n − 1)/2! x² + n(n − 1)(n − 2)/3! x³ + … . When n is not a positive integer, the expansion is infinite and is valid only for |x| < 1.
有理数 n 的二项展开式为 (1 + x)ⁿ = 1 + nx + n(n − 1)/2! x² + n(n − 1)(n − 2)/3! x³ + … 。当 n 不是正整数时，该展开式为无穷级数，并且仅在 |x| < 1 时有效。
To expand (a + x)ⁿ, first factor out aⁿ to obtain aⁿ(1 + x/a)ⁿ. For example, (2 + x)⁻¹ = 1/2 (1 + x/2)⁻¹, and the expansion is valid for |x/2| < 1, so |x| < 2.
要展开 (a + x)ⁿ，可先提取 aⁿ，得到 aⁿ(1 + x/a)ⁿ。例如，(2 + x)⁻¹ = 1/2 (1 + x/2)⁻¹，展开式的有效范围为 |x/2| < 1，即 |x| < 2。
Questions often ask you to find the first few terms or to use the expansion for approximations. Always state the range of validity when required and check whether a term is negligible for a given small value of x.
考试题常要求写出前几项，或利用展开式进行近似计算。务必在需要时说明有效范围，并判断对于给定的较小 x 值某项是否可以忽略。
4. Functions, Modulus and Transformations | 函数、模与变换
The domain is the set of possible input values, and the range is the set of possible output values. The inverse function f⁻¹(x) exists only if f is one-to-one on its domain. To find an inverse, write y = f(x), rearrange to make x the subject, and then swap x and y.
定义域是可能输入值的集合，值域是可能输出值的集合。反函数 f⁻¹(x) 只有在 f 在其定义域上是一一对应时才存在。求反函数时，写出 y = f(x)，整理成 x 关于 y 的表达式，然后交换 x 和 y。
The composite function fg(x) means f(g(x)). The output of g must be a valid input for f. For modulus equations, |ax + b| = c is equivalent to ax + b = c or ax + b = −c. For inequalities, |ax + b| < c gives −c < ax + b < c, while |ax + b| > c gives ax + b < −c or ax + b > c.
复合函数 fg(x) 表示 f(g(x))。g 的输出必须是 f 的有效输入。对于模方程，|ax + b| = c 等价于 ax + b = c 或 ax + b = −c。对于不等式，|ax + b| < c 得到 −c < ax + b < c，而 |ax + b| > c 得到 ax + b < −c 或 ax + b > c。
Transformations include y = af(x), which is a vertical stretch by scale factor a; y = f(bx), which is a horizontal stretch by scale factor 1/b; y = f(x + c), which translates the graph left by c; and y = f(x) + d, which translates the graph up by d.
函数变换包括 y = af(x)，表示纵向拉伸，比例因子为 a；y = f(bx)，表示横向拉伸，比例因子为 1/b；y = f(x + c)，表示将图像向左平移 c 个单位；y = f(x) + d，表示将图像向上平移 d 个单位。
5. Radians, Arc Length and Sector Area | 弧度制、弧长与扇形面积
Radians are essential in Year 2 trigonometry. The conversion is π radians = 180°. To convert degrees to radians, multiply by π/180; to convert radians to degrees, multiply by 180/π.
弧度制在 Year 2 三角学中非常重要。换算关系为 π 弧度 = 180°。将角度转换为弧度时乘以 π/180；将弧度转换为角度时乘以 180/π。
For a circle of radius r, the arc length is s = rθ and the area of a sector is A = ½r²θ, where θ must be measured in radians. The area of a segment is the sector area minus the triangle area: A = ½r²θ − ½r² sinθ = ½r²(θ − sinθ).
对于半径为 r 的圆，弧长为 s = rθ，扇形面积为 A = ½r²θ，其中 θ 必须以弧度为单位。弓形面积等于扇形面积减去三角形面积：A = ½r²θ − ½r² sinθ = ½r²(θ − sinθ)。
When solving trigonometric equations, answers may be required in radians within a given interval, often 0 ≤ θ < 2π or −π < θ ≤ π. Use a sketch or CAST diagram to find all solutions.
解三角方程时，答案可能要求以弧度表示，并限定在某一区间内，通常为 0 ≤ θ < 2π 或 −π < θ ≤ π。可使用图像或 CAST 图找出所有解。
6. Further Trigonometry and Identities | 进阶三角学与恒等式
The reciprocal trigonometric functions are defined by secθ = 1/cosθ, cosecθ = 1/sinθ and cotθ = 1/tanθ = cosθ/sinθ. Two important identities are 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.
倒数三角函数定义为 secθ = 1/cosθ，cosecθ = 1/sinθ，cotθ = 1/t
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