📚 AS AQA Maths Paper 2 January 2022 Core Skills & Question Walkthrough | AS AQA 数学 2022年1月卷2 核心技能与真题解析
The January 2022 AQA AS Mathematics Paper 2 gives a clear picture of the balance between pure mathematics and statistics. This walkthrough explains the recurring question types, the key formulas, and the most common mistakes so you can use the paper as a focused revision resource.
2022年1月 AQA AS 数学卷二清晰地体现了纯数学与统计学之间的平衡。本解析说明反复出现的题型、核心公式以及最常见错误，让你把这份试卷用作高效的复习资源。
1. Paper Structure and Mark Allocation | 试卷结构与分值分配
Paper 2 is an 80-mark paper lasting 1 hour 30 minutes. It is one of two equally weighted AS papers, so it counts for half of the AS Mathematics qualification. The paper is split into pure mathematics and statistics, with the statistics section usually placed at the end.
卷二满分 80 分，考试时间 1 小时 30 分钟。它是 AS 数学两张等权重试卷之一，因此占 AS 数学成绩的一半。试卷分为纯数学和统计学两部分，统计部分通常放在最后。
The pure content includes algebra, coordinate geometry, differentiation, integration, exponentials and logarithms. The statistics content includes sampling, probability, binomial distribution and hypothesis testing. Marks are printed next to each part, and "show that" questions often guide the method.
纯数学内容包括代数、坐标几何、微分、积分、指数与对数。统计学内容包括抽样、概率、二项分布和假设检验。每道小题旁边标有分值，"show that" 题型通常会提示解题方法。
2. Polynomial Division and the Factor Theorem | 多项式除法与因式定理
t state the result. If f(2) ≠ 0, the value is the remainder when f(x) is divided by (x - 2).
2022 年 1 月常见题型要求你使用因式定理证明某个一次式（如 x - 2）是三次多项式 f(x) 的因式。你必须代入 x = 2 并展示 f(2) = 0，而不能只写出结论。如果 f(2) ≠ 0，该值就是 f(x) 除以 (x - 2) 的余数。
f(x) = 2x³ - 5x² + 4x - 3, f(2) = 2(8) - 5(4) + 4(2) - 3 = 16 - 20 + 8 - 3 = 1
x² - x + 2) + 1 after dividing or comparing coefficients. Many candidates lose accuracy by forgetting the constant remainder.
这里 f(2) = 1，所以 (x - 2) 不是因式；余数为 1。完成多项式除法后，可写成 f(x) = (x - 2)(2x² - x + 2) + 1。很多考生因忘记常余数项而失分。
3. Coordinate Geometry and Circles | 坐标几何与圆
group terms to get (x - 3)² + (y + 2)² = 25, so the centre is (3, -2) and the radius is 5.
圆方程题目通常会给出圆心和半径，或者要求你通过配方法求圆心和半径。例如，由 x² + y² - 6x + 4y - 12 = 0 配方得到 (x - 3)² + (y + 2)² = 25，因此圆心为 (3, -2)，半径为 5。
(x - a)² + (y - b)² = r²
If a line touches a circle, the perpendicular distance from the centre to the line equals the radius. Use the distance formula d = |px₀ + qy₀ + r| / √p² + q² e the distance as an exact value or to three significant figures unless the question says otherwise.
如果一条直线与圆相切，圆心到直线的垂直距离等于半径。对于直线 px + qy + r = 0，使用距离公式 d = |px₀ + qy₀ + r| / √p² + q²。除非题目另有说明，距离要保留精确值或三位有效数字。
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