📚 GCSE CCEA Further Mathematics: Case Study Practical Drill | CCEA GCSE 进阶数学：案例分析实战演练
This article works through a set of case studies that cover the core of the CCEA GCSE Further Mathematics specification. Each case is broken into numbered steps, with English and Chinese explanations placed side by side, so you can practise the method and the vocabulary at the same time. Every worked example is written the way a marker expects to see it: data, decision, algebra, conclusion.
本文通过一系列案例分析，覆盖 CCEA GCSE 进阶数学考试的核心内容。每个案例都拆解为编号步骤，中英双语同步讲解，让你在练习解题方法的同时掌握术语表达。每个例题都按阅卷老师期待的方式书写：已知数据、判断思路、代数运算、最终结论。
1. Why Case Studies Beat Passive Revision | 为什么案例分析胜过被动复习
Further Mathematics is not a memory test. The same three or four techniques - differentiating, solving equations, manipulating matrices - reappear in different disguises, and the marks come from applying them in the right order rather than recognising a familiar page.
进阶数学不是记忆型考试。同样的三四种技巧——求导、解方程、矩阵运算——会以不同面貌反复出现，分数来自把它们按正确顺序使用，而不是"认出一页熟悉的题目"。
A case study forces that order into the open. You see the data, the decision point, the algebra and the closing sentence in one place, and you can copy the pattern under time pressure.
案例分析把解题顺序摊开给你看：数据、判断点、代数运算与收尾结论一次呈现，考试时间紧张时照着这个模式写即可。
Use each case in three passes: first read the solution, then redo it from a blank page, then change one number and redo it again. The third pass is where real fluency is built.
每个案例建议分三遍使用：第一遍读解答，第二遍在空白纸上重做，第三遍改动其中一个数字再做一次。第三遍才是真正练出熟练度的地方。
2. Specification Map: Where the Marks Sit | 大纲地图：分数藏在哪里
The qualification is built around a pure mathematics core - algebra, trigonometry, sequences and series, coordinate geometry, calculus and matrices - supported by applied material that draws on mechanics and statistics. Unit titles, weightings and grade arrangements are revised from time to time, so always download the current specification and the most recent past papers from the CCEA website before you build a revision plan.
该资格考试以纯数学为核心——代数、三角、数列与级数、坐标几何、微积分和矩阵——并辅以力学与统计的应用内容。单元名称、权重与评分安排会不定期调整，因此制定复习计划前，务必先从 CCEA 官网下载最新大纲与近年真题。
Mechanics supplies the kinematics case in this article; matrices, trigonometry, series and calculus supply the rest. The table below maps each case to the skill it drills, so you can jump straight to a weak topic.
本文的运动学案例来自力学部分，其余案例分别来自矩阵、三角、数列与微积分。下表把每个案例对应到它训练的技能，方便你直接跳到薄弱环节。
| Case | 案例 | Topic | 主题 | Core skill | 核心技能 |
| 3-7 | Kinematics | 运动学 | Differentiation, interpretation | 求导与结果解读 |
| 8 | Matrices | 矩阵 | Inverse matrix, simultaneous equations | 逆矩阵与联立方程 |
| 9 | Trigonometry | 三角 | Quadratic in sin θ, full solution set | 关于 sin θ 的二次方程与完整解集 |
| 10 | Sequences and series | 数列与级数 | AP and GP formulae | 等差与等比公式 |
| 11 | Calculus | 微积分 | Stationary points, second derivative test | 驻点与二阶导数判别 |
3. Case Study 1: Setting Up the Kinematics Problem | 案例一：运动学审题与建模
sions for the velocity and the acceleration; (b) the velocity at t = 4 s; (c) the time at which the acceleration is zero.
一质点沿直线运动。t 秒后相对于固定点 O 的位移 s（单位：米）由 s(t) = t³ − 5t² + 3t + 2 给出。求：(a) 速度与加速度的表达式；(b) t = 4 s 时的速度；(c) 加速度为零的时刻。
The first decision is what each symbol means. Displacement is signed distance from O, so a negative value simply means the particle is on the other side of O. Velocity is the rate of change of displacement, and acceleration is the rate of change of velocity, which gives the standard chain below.
第一步是明确符号含义。位移是相对 O 的有向距离，负值只表示质点在 O 的另一侧。速度是位移的变化率，加速度是速度的变化率，于是得到下面这条标准链条。
s(t) → differentiate → v(t) = ds/dt → differentiate → a(t) = dv/dt = d²s/dt²
ng also restricts the domain to t ≥ 0, which matters when you decide which of several solutions to keep.
单位会自动跟着走：s 用米、t 用秒，所以速度单位是 m/s（m s⁻¹），加速度单位是 m/s²（m s⁻²）。物理背景还限定了定义域 t ≥ 0，这在后面取舍多个解时非常关键。
4. Case Study 1(a): Velocity and Acceleration by Differentiation | 案例一(a)：用求导求速度与加速度
Differentiate term by term. The derivative of t³ is 3t², of −5t² is −10t, of 3t is 3 and of the constant 2 is 0. So the velocity is the quadratic below.
逐项求导：t³ 的导数是 3t²，−5t² 的导数是 −10t，3t 的导数是 3，常数 2 的导数是 0。于是速度是一个二次函数。
v(t) = ds/dt = 3t² − 10t + 3
Differentiating again gives the acceleration, which is linear in t.
再求一次导即得加速度，它是关于 t 的一次函数。
a(t) = dv/dt = 6t − 10
s, and that the sign of the velocity changes at those two instants. Examiners often award a mark for this observation because it demonstrates understanding rather than mechanical differentiation.
把速度因式分解能得到额外信息：v(t) = (3t − 1)(t − 3)。这说明质点在 t = 1/3 s 和 t = 3 s 时瞬时静止，且速度符号在这两个时刻发生改变。阅卷时这种观察常常能得分，因为它体现的是理解而非机械求导。
5. Case Study 1(b): The Velocity at t = 4 s | 案例一(b)：求 t = 4 s 时的速度
stitute t = 4 into the velocity expression and work line by line so the method is visible.
把 t = 4 代入速度
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