📚 AQA AS Mathematics 9660 MA02 Pure, Statistics and Mechanics Unit 1 Revision | AQA AS数学9660 MA02纯数、统计与力学第一单元复习
This revision guide covers the core content of AQA International AS Mathematics 9660, Unit MA02 Pure, Statistics and Mechanics. It summarises the key methods, formulas and examiner tips for the first AS unit, helping you build fluency across all three strands: Pure Mathematics, Statistics and Mechanics.
本复习指南覆盖AQA国际AS数学9660 MA02纯数、统计与力学第一单元的核心内容。它总结了该AS单元的关键方法、公式与考官提示，帮助你在纯数、统计和力学三个模块中建立熟练度。
1. Algebraic Expressions and Index Laws | 代数表达式与指数定律
In AS Pure Mathematics, fluent manipulation of algebraic expressions is essential. You must be able to simplify surds, expand brackets, factorise quadratics and cubics, and use the laws of indices confidently. The index laws are particularly important because they underpin differentiation, integration and binomial expansions.
在AS纯数学中，熟练处理代数表达式至关重要。你必须能够化简根式、展开括号、因式分解二次和三次多项式，并熟练运用指数定律。指数定律尤其重要，因为它们是微分、积分和二项式展开的基础。
| Rule | 中文 |
|---|---|
| aᵐ × aⁿ = aᵐ⁺ⁿ | 同底数幂相乘，指数相加 |
| aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 同底数幂相除，指数相减 |
| (aᵐ)ⁿ = aᵐⁿ | 幂的乘方，指数相乘 |
| a⁰ = 1 | 任何非零数的零次方等于1 |
| a⁻ⁿ = 1 / aⁿ | 负指数表示倒数 |
| a¹⁄ⁿ = ⁿ√a | 分数指数表示开方 |
| aᵐ⁄ⁿ = ⁿ√aᵐ | 分数指数表示幂与开方结合 |
A common exam question asks you to write 3√5 + √20 in the form k√5, rationalise denominators such as 1/(2+√3 e 2x³y² × 3x⁻²y⁵. Rationalising the denominator uses the difference of two squares: (a + b√c)(a - b√c) = a² - b²c.
常见考题要求把 3√5 + √20 写成 k√5 的形式，对 1/(2+√3) 这样的分母进行有理化，或者化简 2x³y² × 3x⁻²y⁵ 这类表达式。有理化分母时利用平方差公式：(a + b√c)(a - b√c) = a² - b²c。
2. Quadratics and Simultaneous Equations | 二次函数与联立方程
no real roots. Hidden quadratics, such as x⁴ - 5x² + 4 = 0, can be solved by substitution.
二次函数是AS纯数的核心内容。三种主要工具为因式分解、配方法和二次方程求根公式。判别式 Δ = b² - 4ac 决定实根的个数：Δ > 0 有两个不同实根，Δ = 0 有一个重根，Δ < 0 没有实根。隐藏二次方程，如 x⁴ - 5x² + 4 = 0，可以通过换元来求解。
x = (-b ± √b² - 4ac) / (2a)
For simultaneous equations, one linear and one quadratic usually appear. Substitute the linear expression into the quadratic to form a single quadratic equation, solve it, then substitute back to find the other variable. Always check both solutions in the original equations.
联立方程通常出现一个一次方程和一个二次方程。把一次表达式代入二次方程，形成一个一元二次方程并求解，然后代回求出另一个变量。务必把两个解代回原方程检验。
3. Coordinate Geometry and Linear Graphs | 坐标几何与直线图像
two points is √[(x₂ - x₁)² + (y₂ - y₁)²], and the midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).
坐标几何将代数与几何联系起来。两点之间的斜率为 m = (y₂ - y₁)/(x₂ - x₁)。直线方程为 y - y₁ = m(x - x₁)。平行直线斜率相等；垂直直线的斜率满足 m₁m₂ = -1。两点之间的距离为 √[(x₂ - x₁)² + (y₂ - y₁)²]，中点为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
². If a circle is given in expanded form, complete the square on both x and y to find its centre and radius.
圆心为 (a, b)、半径为 r 的圆的方程为 (x - a)² + (y - b)² = r²。如果圆以展开形式给出，需要分别对 x 和 y 配方，求出圆心和半径。
4. Graphs and Transformations | 图像与变换
rizontal translation, y = a f(x), a vertical stretch, and y = f(ax), a horizontal stretch.
你必须熟悉三次函数 y = ax³ + bx² + cx + d、四次函数、反比例函数 y = a/x 和 y = a/x² 的图像。y = f(x) 的变换包括 y = f(x) + a 表示竖直平移，y = f(x + a)
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