📚 AQA International AS Unit FPSM1: Pure Maths, Statistics and Mechanics | AQA国际AS单元FPSM1：纯数学、统计与力学
Welcome to this revision guide for the AQA International AS Unit FPSM1, which blends pure mathematics, statistics and mechanics into a single demanding paper. This guide covers the essential ideas you need to master for the AS exam.
欢迎阅读这份 AQA 国际 AS 单元 FPSM1 复习指南，该单元将纯数学、统计和力学融合在一张高要求的试卷中。本指南涵盖 AS 考试必须掌握的核心知识。
1. Algebra and Functions | 代数与函数
Core algebraic skills include factorising quadratics, completing the square, using the discriminant, and solving simultaneous equations. You must be able to manipulate expressions accurately under time pressure.
核心代数技能包括二次式因式分解、配方法、使用判别式以及解联立方程。你必须能够在时间压力下准确处理各种表达式。
The discriminant Δ = b² – 4ac determines the nature of the roots of ax² + bx + c = 0: if Δ > 0 there are two distinct real roots, if Δ = 0 there is one repeated root, and if Δ < 0 there are no real roots.
判别式 Δ = b² – 4ac 决定 ax² + bx + c = 0 的根的性质：若 Δ > 0，有两个不同实根；若 Δ = 0，有一个重根；若 Δ < 0，没有实根。
For functions, understand domain, range, inverse functions, and composite functions. A function f(x) has an inverse only if it is one-to-one on its domain, and the inverse is usually denoted by f⁻¹(x).
对于函数，要理解定义域、值域、反函数和复合函数。函数 f(x) 仅在其定义域上是一一对应时才有反函数，反函数通常记作 f⁻¹(x)。
2. Trigonometry | 三角学
Work in radians as well as degrees. Recall that π rad = 180°, so 1 rad ≈ 57.3°. Many AS questions require exact answers in terms of π, so avoid reaching for a calculator too early.
弧度制和角度制都要会使用。记住 π 弧度 = 180°，因此 1 弧度 ≈ 57.3°。许多 AS 题目要求以 π 表示精确答案，所以不要过早依赖计算器。
Key identities include sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ. Use them to simplify expressions and solve trigonometric equations within a given interval.
关键恒等式包括 sin²θ + cos²θ = 1 和 tan θ = sin θ / cos θ。利用它们化简表达式，并在给定区间内解三角方程。
Exact values such as sin(π/6) = 1/2, cos(π/3) = 1/2 and tan(π/4) = 1 must be memorised. They often appear without warning in later pure and mechanics problems.
必须记住精确值，例如 sin(π/6) = 1/2，cos(π/3) = 1/2，tan(π/4) = 1。它们经常突然出现在后续的纯数学和力学问题中。
3. Coordinate Geometry | 坐标几何
The gradient of a straight line through (x₁, y₁) and (x₂, y₂) is m = (y₂ – y₁)/(x₂ – x₁). A positive gradient means the line slopes upwards from left to right.
经过点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率为 m = (y₂ – y₁)/(x₂ – x₁)。正斜率表示直线从左到右向上倾斜。
The equation of a line can be written as y – y₁ = m(x – x₁). Parallel lines have equal gradients; perpendicular lines have gradients whose product is -1.
直线方程可写为 y – y₁ = m(x – x₁)。平行直线斜率相等；垂直直线的斜率乘积为 -1。
For circles, the equation (x – a)² + (y – b)² = r² has centre (a, b) and radius r. Be ready to find the centre and radius by completing the square when the equation is given in expanded form.
对于圆，方程 (x – a)² + (y – b)² = r² 的圆心为 (a, b)，半径为 r。当题目给出展开式时，要会通过配方求圆心和半径。
4. Sequences and Series | 数列与级数
An arithmetic sequence has a constant difference d: uₙ = a + (n – 1)d. The sum of the first n terms is Sₙ = n/2 [2a + (n – 1)d] or Sₙ = n/2 (a + l), where l is the last term.
等差数列具有常数公差 d：uₙ = a + (n – 1)d。前 n 项和为 Sₙ = n/2 [2a + (n – 1)d] 或 Sₙ = n/2 (a + l)，其中 l 是末项。
A geometric sequence has a constant ratio r: uₙ = arⁿ⁻¹. The sum of the first n terms is Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1; if |r| < 1, the sum to infinity is S∞ = a/(1 – r).
等比数列具有常数公比 r：uₙ = arⁿ⁻¹。前 n 项和为 Sₙ = a(1 – rⁿ)/(1 – r)（r ≠ 1）；若 |r| < 1，无穷和为 S∞ = a/(1 – r)。
Pay attention to the difference between the nth term and the sum of the first n terms. Many errors come from substituting values into the wrong formula.
注意区分第 n 项和前 n 项和。许多错误来自于将数值代入错误的公式。
5. Differentiation | 微分
For y = xⁿ, the derivative is dy/dx = nxⁿ⁻¹. Key rules include the product rule, quotient rule and chain rule. The chain rule states that if y = f(g(x)), then dy/dx = f'(g(x)) × g'(x).
对于 y = xⁿ，导数为 dy/dx = nxⁿ⁻¹。关键法则包括乘法法则、除法法则和链式法则。链式法则指出，若 y = f(g(x))，则 dy/dx = f'(g(x)) × g'(x)。
Use differentiation to find tangents and normals, and to solve optimisation problems where a maximum or minimum value is required. Always justify whether a stationary point is a maximum or minimum.
利用微分求切线和法线，并解决需要最大值或最小值的优化问题。始终要说明驻点是极大值还是极小值。
6. Integration | 积分
Integration is the reverse of differentiation. For xⁿ with n ≠ -1, ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C. Never forget the constant of integration in an indefinite integral.
积分是微分的逆运算。对于 n ≠ -1 的 xⁿ，∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C。不定积分中千万不要漏掉积分常数。
Use definite integrals to find the area between a curve and the x-axis: A = ∫ₐᵇ f(x) dx if f(x) ≥ 0 on [a, b]. Remember that areas below the x-axis give a negative integral, so use absolute values or split the interval.
使用定积分求曲线与 x 轴之间的面积：若 f(x) ≥ 0 在 [a, b] 上，A = ∫ₐᵇ f(x) dx。记住 x 轴下方的面积给出的积分值为负，因此使用绝对值或拆分区间。
7. Statistical Measures and Probability | 统计度量与概率
For a data set, the mean is x̄ = Σx/n. The variance is σ² = Σ(x – x̄)²/n or σ² = Σx²/n – x̄². The standard deviation is the square root of the variance.
对于一个数据集，平均数为 x̄ = Σx/n。方差为 σ² = Σ(x – x̄)²/n 或 σ² = Σx²/n – x̄²。标准差是方差的平方根。
Probability rules include P(A ∪ B)
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