📚 AQA International AS FPSM1 Complete Revision Guide | AQA 国际AS FPSM1 全面复习指南
This revision guide covers the essential topics from the AQA International AS Unit FPSM1, which combines Pure Mathematics, Statistics, and Mechanics into a single modular examination. Whether you are preparing for your first sitting or retaking the paper, this guide distils the core techniques, formulas, and exam strategies you need.
本复习指南涵盖 AQA 国际 AS 单元 FPSM1 的核心内容，该单元将纯数学、统计学与力学整合为一次模块化考试。无论您是首次应考还是补考，本指南都将提炼您所需的核⼼技巧、公式与应试策略。
1. Algebra: Quadratics and Inequalities | 代数：二次函数与不等式
The quadratic function is the cornerstone of AS Pure Mathematics. You must be able to solve quadratic equations by factorisation, completing the square, and using the quadratic formula. The discriminant Δ = b² − 4ac tells you the nature of the roots: two distinct real roots if Δ > 0, one repeated root if Δ = 0, and no real roots if Δ < 0.
二次函数是 AS 纯数学的基石。您必须掌握因式分解、配方法和求根公式来求解二次方程。判别式 Δ = b² − 4ac 告诉您根的性质：Δ > 0 时有两个不同的实根，Δ = 0 时有一个重根，Δ < 0 时无实根。
Completing the square is particularly useful for finding the vertex: a(x − h)² + k has its turning point at (h, k). For inequalities, always sketch the parabola first. Solve x² − 5x + 6 > 0 by factorising to (x − 2)(x − 3) > 0; the solution is x < 2 or x > 3.
配方法在寻找顶点时尤为有用：a(x − h)² + k 的顶点在 (h, k) 处。解不等式时，务必先画出抛物线草图。解 x² − 5x + 6 > 0，可因式分解为 (x − 2)(x − 3) > 0，其解为 x < 2 或 x > 3。
x = (−b ± √(b² − 4ac)) / 2a, Δ = b² − 4ac
Simultaneous equations involving one linear and one quadratic can be solved by substitution. Substitute the linear expression into the quadratic, solve the resulting quadratic, then back-substitute to find the corresponding y-values.
含一个线性方程与一个二次方程的联立方程组可通过代入法求解。将线性表达式代入二次方程，解所得二次方程，再回代求出对应的 y 值。
2. Coordinate Geometry: Lines and Circles | 坐标几何：直线与圆
The equation of a straight line can be written in the forms y = mx + c, y − y₁ = m(x − x₁), or ax + by + c = 0. The gradient m between two points (x₁, y₁) and (x₂, y₂) is given by (y₂ − y₁)/(x₂ − x₁). Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = −1.
直线方程可以写成 y = mx + c、y − y₁ = m(x − x₁) 或 ax + by + c = 0 等形式。两点 (x₁, y₁) 与 (x₂, y₂) 之间的斜率 m 由 (y₂ − y₁)/(x₂ − x₁) 给出。平行线斜率相等；垂直线满足 m₁ × m₂ = −1。
The equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r². To find whether a point lies on, inside, or outside the circle, compare its distance from the centre with r. A tangent to a circle at a given point is perpendicular to the radius at that point.
圆心为 (a, b)、半径为 r 的圆方程为 (x − a)² + (y − b)² = r²。要判断点与圆的位置关系，可将该点到圆心的距离与 r 比较。圆在某点的切线垂直于该点处的半径。
For the midpoint of two points: ((x₁ + x₂)/2, (y₁ + y₂)/2). The distance between two points is √((x₂ − x₁)² + (y₂ − y₁)²). These formulae appear repeatedly across both the pure and mechanics sections of the paper.
两点的中点为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。两点间距离为 √((x₂ − x₁)² + (y₂ − y₁)²)。这些公式在试卷的纯数学与力学部分反复出现。
3. Trigonometry Essentials | 三角学基础
At AS level, you must work confidently with sine, cosine, and tangent. For any right-angled triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. The sine and cosine rules apply to non-right-angled triangles:
在 AS 阶段，您必须熟练运用正弦、余弦和正切。对任意直角三角形：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。正弦定理和余弦定理适用于非直角三角形：
a/sin A = b/sin B = c/sin C, a² = b² + c² − 2bc cos A
The identity sin²θ + cos²θ = 1 and the ratio identity tan θ = sin θ/cos θ are essential for simplifying expressions and solving equations. Know the exact values for 0°, 30°, 45°, 60°, and 90° — they appear regularly.
恒等式 sin²θ + cos²θ = 1 以及比例恒等式 tan θ = sin θ/cos θ 对于化简表达式和求解方程至关重要。请记住 0°、30°、45°、60° 和 90° 的精确值——它们在考卷中经常出现。
When solving trigonometric equations over a given interval, sketch the relevant graph to determine all solutions. For example, solving sin θ = 0.5 for 0° ≤ θ < 360° gives θ = 30° and θ = 150° due to symmetry.
在给定区间内求解三角方程时，画出相应函数图以确定所有解。例如，在 0° ≤ θ < 360° 内解 sin θ = 0.5，根据对称性可得 θ = 30° 和 θ = 150°。
4. Differentiation Fundamentals | 微分基础
Differentiation measures the rate of change. The derivative f'(x) of f(x) = xⁿ is nxⁿ⁻¹. For a sum of terms, differentiate term by term. The gradient of a curve at a point is found by substituting the x-coordinate into f'(x).
微分衡量变化率。f(x) = xⁿ 的导数 f'(x) 为 nxⁿ⁻¹。对于多项之和，逐项求导即可。曲线在某点的斜率通过将该点横坐标代入 f'(x) 得到。
Stationary points occur where f'(x) = 0. Use the second derivative or a sign table to classify them: f”(x) > 0 indicates a local minimum, f”(x) < 0 a local maximum, and f''(x) = 0 requires further investigation.
驻点出现在 f'(x) = 0 处。用二阶导数或符号表进行分类：f”(x) > 0 表示局部极小值，f”(x) < 0 表示局部极大值，f''(x) = 0 则需要进一步判断。
Practical optimisation problems — maximising area or minimising cost — follow a routine: write the quantity to be optimised as a function of one variable, differentiate, set the derivative to zero, and verify the nature of the stationary point.
实际优化问题——最大化面积或最小化成本——遵循固定流程：将待优化量写成单变量函数，求导，令导数为零，并验证驻点的性质。
d/dx (xⁿ) = nxⁿ⁻¹, d/dx (axⁿ) = anxⁿ⁻¹
5. Integration Basics | 积分基础
Integration is the reverse process of differentiation. For xⁿ, the indefinite integral is xⁿ⁺¹/(n+1) + C, provided n ≠ −1. The constant of integration C represents the family of curves with the same gradient function.
积分是微分的逆过程。对于 xⁿ，其不定积分为 xⁿ⁺¹/(n+1) + C，前提是 n ≠ −1。积分常数 C 代表具有相同导函数的一族曲线。
Definite integrals are evaluated by finding the antiderivative and substituting the limits:
定积分通过求原函数并代入上下限来求值：
∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a)
The area under a curve y = f(x) between x = a and x = b is given by the definite integral ∫ₐᵇ f(x) dx, as long as the curve lies entirely above the x-axis. If the curve dips below the axis, split the integral at the x-intercepts and take absolute values of negative areas.
曲线 y = f(x) 与 x 轴在 x = a 和
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