📚 AQA International AS Unit PSM1 Complete Revision Guide | AQA 国际AS单元PSM1 完整复习指南
Welcome to this complete revision guide for AQA International AS Unit PSM1 (Pure Mathematics, Statistics and Mechanics). This unit combines all three branches of A-level mathematics at the AS level. To secure top marks, you need to master algebraic fluency, interpret statistical data with confidence, and apply Newton’s laws to physical situations. This guide condenses the entire specification into a structured revision companion.
欢迎阅读 AQA 国际AS单元 PSM1（纯数学、统计与力学）完整复习指南。本单元涵盖 AS 阶段数学的三大分支。要想斩获高分，你需要精通代数运算、从容解读统计数据，并熟练运用牛顿定律解决实际问题。本指南将全部考纲浓缩为一份结构清晰的复习手册。
1. Pure Maths: Algebra & Functions | 纯数学：代数与函数
Algebra is the engine of the entire PSM1 unit. Quadratic equations, inequalities and simultaneous equations reappear constantly in statistics and mechanics questions, so mastering them first is essential.
代数是整个 PSM1 单元的核心引擎。二次方程、不等式与联立方程在统计和力学题目中反复出现，因此优先掌握它们是重中之重。
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Solving quadratics: three methods. Factorising works when roots are rational, completing the square reveals the vertex, and the quadratic formula always works. For x² + 5x + 6 = 0, factorising gives (x + 2)(x + 3) = 0, so x = -2 or x = -3.
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求解二次方程：三种方法。根为有理数时用因式分解；配方法可揭示顶点坐标；求根公式则万无一失。例如 x² + 5x + 6 = 0，因式分解得 (x + 2)(x + 3) = 0，故 x = -2 或 x = -3。
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Quadratic formula and discriminant. For ax² + bx + c = 0, the solutions are given by the formula below. The discriminant Δ = b² – 4ac determines the nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots.
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求根公式与判别式。对于 ax² + bx + c = 0，解由下式给出。判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。
x = (-b ± √(b² – 4ac)) / 2a, Δ = b² – 4ac
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Completing the square. Writing ax² + bx + c in the form a(x + p)² + q makes it easy to identify the turning point (-p, q). For example, x² + 6x + 5 = (x + 3)² – 4, so the vertex is (-3, -4).
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配方法。将 ax² + bx + c 写成 a(x + p)² + q 的形式，可快速确定顶点 (-p, q)。例如 x² + 6x + 5 = (x + 3)² – 4，顶点为 (-3, -4)。
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Quadratic inequalities. Solve the related equation first, then sketch the parabola. For (x – 1)(x – 3) > 0, the solution is x < 1 or x > 3; for (x – 1)(x – 3) < 0, the solution is 1 < x < 3.
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二次不等式。先解对应方程，再画出抛物线草图。对于 (x – 1)(x – 3) > 0，解为 x < 1 或 x > 3；对于 (x – 1)(x – 3) < 0，解为 1 < x < 3。
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Simultaneous equations. When solving one linear and one quadratic equation, substitute the linear expression into the quadratic, solve for x, then substitute back to find y. Discard any solutions that do not satisfy both equations.
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联立方程。当一个线性方程与一个二次方程联立时，将线性表达式代入二次方程，求出 x 后再代回求 y。丢弃不满足两个方程的解。
2. Pure Maths: Coordinate Geometry & Circles | 纯数学：坐标几何与圆
Coordinate geometry connects algebra with shapes. You must be confident with straight-line gradients, perpendicularity, circle equations, and finding tangents to circles.
坐标几何将代数与图形联系起来。你必须熟练运用直线斜率、垂直条件、圆的方程以及圆的切线求法。
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Straight-line essentials. The gradient between two points is m = (y₂ – y₁) / (x₂ – x₁). Midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2) and distance is √((x₂ – x₁)² + (y₂ – y₁)²). Line equation: y – y₁ = m(x – x₁).
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直线基础。两点间斜率为 m = (y₂ – y₁) / (x₂ – x₁)。中点坐标为 ((x₁ + x₂)/2, (y₁ + y₂)/2)，距离为 √((x₂ – x₁)² + (y₂ – y₁)²)。直线方程：y – y₁ = m(x – x₁)。
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Parallel and perpendicular lines. Parallel lines share the same gradient m₁ = m₂. Perpendicular lines satisfy m₁ × m₂ = -1. For example, if a line has gradient 2, a perpendicular line has
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