📚 AQA AS International Mathematics MA02: Core Revision Guide | AQA 国际数学 AS MA02 核心复习指南
This revision guide breaks down the key skills assessed in the AQA International AS Mathematics MA02 paper. It covers the main pure mathematics topics and common exam pitfalls so you can work through past paper questions with confidence.
本复习指南梳理 AQA 国际数学 AS MA02 试卷考查的核心技能，覆盖纯数学主要专题与常见考试易错点，帮助你自信应对历年真题。
1. Algebraic Manipulation and Surds | 代数运算与根式
Algebraic manipulation is the foundation of the MA02 paper. You must be able to simplify surds such as √18, √50 and expressions like √8 + √2, as well as rationalise denominators in fractions such as 1/(2 + √3). A common error is to combine unlike surds incorrectly: √12 + √27 is not √39, but 2√3 + 3√3 = 5√3.
代数运算是 MA02 试卷的基础。你需要熟练化简根式，如 √18、√50，以及表达式 √8 + √2，并会对分母如 1/(2 + √3) 进行有理化。常见错误是把不同根式直接相加：√12 + √27 不等于 √39，而应化为 2√3 + 3√3 = 5√3。
- √a × √b = √ab for a, b ≥ 0 | 当 a, b ≥ 0 时，√a × √b = √ab
- To rationalise 1/(a + √b), multiply numerator and denominator by the conjugate a − √b | 分母有理化时，分子分母同乘共轭式 a − √b
2. Quadratic Functions and Discriminant | 二次函数与判别式
, there are no real roots. Completing the square gives the vertex form a(x + b/2a)² + (c − b²/4a), which is useful for finding the minimum or maximum point.
对于二次方程 ax² + bx + c = 0，判别式 Δ = b² − 4ac 可以判断实根个数。若 Δ > 0，有两个不同实根；若 Δ = 0，有一个重根；若 Δ < 0，无实根。配方法可得到顶点式 a(x + b/2a)² + (c − b²/4a)，便于求最小或最大值点。
Δ = b² − 4ac
| Condition | Meaning | 中文 |
|---|---|---|
| Δ > 0 | Two distinct real roots | 两个不同实根 |
| Δ = 0 | One repeated real root | 一个重根 |
| Δ < 0 | No real roots | 无实根 |
3. Coordinate Geometry | 坐标几何
m₁ and m₂, parallel lines satisfy m₁ = m₂, while perpendicular lines satisfy m₁ × m₂ = −1.
坐标几何题考查直线与圆的运算。两点 (x₁, y₁) 和 (x₂, y₂) 的距离为 √((x₂−x₁)² + (y₂−y₁)²)，中点坐标为 ((x₁+x₂)/2, (y₁+y₂)/2)。对于斜率分别为 m₁ 和 m₂ 的两条直线，平行时 m₁ = m₂，垂直时 m₁ × m₂ = −1。
². A tangent to a circle is perpendicular to the radius at the point of contact, so use the negative reciprocal gradient to find the tangent equation.
圆心为 (a, b)、半径为 r 的圆的方程为 (x − a)² + (y − b)² = r²。圆的切线在切点处与半径垂直，因此可用负倒数斜率求切线方程。
4. Polynomials and Factor Theorem | 多项式与因式定理
is a factor. The remainder theorem states that when f(x) is divided by (x − a), the remainder is f(a). These results let you factorise cubic and quartic expressions and solve polynomial equations without guessing too many values.
因式定理指出：若多项式 f(x) 满足 f(a) = 0，则 (x − a) 是一个因式。余式定理指出：f(x) 除以 (x − a) 的余数为 f(a)。利用这两个定理，可对三次或四次多项式因式分解并求解方程，无需大量试值。
- To divide polynomials, use long division or compare coefficients | 多项式除法可用长除法或比较系数法
- If x = p is a root, then (x − p) is a factor | 若 x = p 是根，则 (x − p) 为因式
5. Binomial Expansion | 二项式展开
For a positive integer n, the binomial expansion is (a + b)ⁿ = Σ [nCr] aⁿ⁻ʳ bʳ for an unknown constant. When n is not a positive integer, the expansion is valid only for |x| < 1, and it becomes an infinite series.
对于正整数 n，二项式展开为 (a + b)ⁿ = Σ [nCr] aⁿ⁻ʳ bʳ。在 MA02 试卷中，常需展开 (1 + ax)ⁿ 并求特定项系数，或解出未知常数。当 n 不是正整数时，展开仅在 |x| < 1 时成立，且为无穷级数。
(1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + …
Always state the validity condition when the index is fractional or negative. Many candidates lose marks by omitting |x| < 1 or by writing the expansion as if it were finite.
当指数为分数或负数时，务必写出成立条件 |x| < 1。许多考生因漏写该条件或将无穷展开写成有限项而失分。
6. Trigonometry | 三角学
A = b/sin B = c/sin C, the cosine rule a² = b² + c² − 2bc cos A, and the area formula Area
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