📚 9550-FM05 Further Mathematics Specimen Paper 2019: Key Concepts | 9550-FM05 国际 A-Level 进阶数学样卷 2019：知识点精讲
The 9550-FM05 specimen paper from 2019 targets the Decision Mathematics 1 unit of the International A-Level Further Mathematics qualification. This paper tests a range of algorithms and graphical techniques used to solve optimisation and modelling problems. In this article, we break down the essential topics that appear regularly, explaining each concept with practical examples and exam-focused insights.
2019 年发布的 9550-FM05 样卷针对国际 A-Level 进阶数学的决策数学 1 单元。这份试卷考查了一系列用于解决优化和建模问题的算法与图形技术。本文将逐一解析常考的核心主题，结合实例和应试要点，帮助考生彻底掌握这些知识点。
1. Introduction to Decision Mathematics | 决策数学导论
Decision Mathematics 1 (D1) is concerned with algorithms that solve real-world problems involving networks, scheduling, and resource allocation. Unlike pure mathematics, it focuses on step-by-step procedures rather than continuous functions. The FM05 specimen paper examines your ability to apply, trace, and adapt these algorithms under exam conditions.
决策数学 1 (D1) 关注解决实际问题的算法，如网络、调度和资源分配。与纯数学不同，它强调逐步执行的过程而非连续函数。FM05 样卷考查你在考试环境下应用、追踪和调整这些算法的能力。
2. Graphs and Networks Terminology | 图与网络的术语
A graph consists of vertices (nodes) connected by edges (arcs). A weighted graph assigns a number to each edge, representing cost, time, or distance. Important definitions include simple graph, connected graph, complete graph, and bipartite graph. In the exam, you must be able to distinguish between a walk, trail, path, and cycle.
图由顶点（节点）和边（弧）连接而成。加权图给每条边赋予一个数值，代表成本、时间或距离。重要定义包括简单图、连通图、完全图和二分图。考试中必须能区分行走、迹、路径和回路等概念。
3. Minimum Spanning Trees – Kruskal’s and Prim’s Algorithms | 最小生成树 – 克鲁斯卡尔与普里姆算法
A minimum spanning tree (MST) connects all vertices with the smallest total weight and no cycles. Kruskal’s algorithm sorts edges by weight and adds the shortest edge that does not form a cycle, while Prim’s algorithm grows a tree from a starting vertex by adding the cheapest outgoing edge. Both may be tested on a network given as a list or matrix. Candidates should show clear steps for full marks.
最小生成树（MST）用最小的总权重连接所有顶点且不含圈。克鲁斯卡尔算法先将边按权重排序，再依次添加不形成圈的最短边；普里姆算法从一个起始顶点出发，逐步添加最廉价的出边。考试中网络可能以列表或矩阵形式给出，需要清晰展示步骤才能获得满分。
4. Shortest Path – Dijkstra’s Algorithm | 最短路径 – 迪杰斯特拉算法
Dijkstra’s algorithm finds the shortest path from a start vertex to all other vertices in a weighted graph with non-negative edges. You must maintain a table of distances, update temporary labels, and record the order of permanent labelling. The final answers often require the route as a sequence of vertices and the total length.
迪杰斯特拉算法用于在边权非负的加权图中寻找从起点到所有其他顶点的最短路径。你需要维护距离表格，更新临时标号，并记录永久标号的顺序。最终答案通常要求给出由顶点序列组成的路径及其总长度。
5. Critical Path Analysis – Activity Networks | 关键路径分析 – 活动网络
Activity-on-arc networks represent a project where each activity is an edge. You need to perform forward and backward scans to calculate earliest start times (EST), latest start times (LST), and total float. The critical path consists of activities with zero float, determining the minimum project completion time. A Gantt chart (cascade chart) may also be required to visualise the schedule.
弧表示活动的网络（AoA）将每项活动作为一条边。需要执行前推和后推扫描，计算最早开始时间（EST）、最迟开始时间（LST）和总时差。关键路径由总时差为零的活动组成，决定了项目的最短完成时间。考试还可能要求画出甘特图（级联图）来可视化进度安排。
6. Linear Programming – Graphical Method | 线性规划 – 图解法
In D1, linear programming (LP) problems with two variables are solved by graphing the feasible region defined by linear inequalities. You must identify the objective function line and use the vertex method or sliding line to find the optimal point. Integer solutions may require additional testing of nearby lattice points.
在 D1 中，涉及两个变量的线性规划问题通过绘制由线性不等式定义的可行域来求解。需要确定目标函数线，并利用顶点法或平移直线找到最优点。整数解可能还需要测试附近的格点。
7. Matchings in Bipartite Graphs | 二分图中的匹配
A matching pairs vertices from two disjoint sets, with no vertex used more than once. The maximum matching algorithm starts from an initial matching and uses alternating paths to improve it until no more augmenting paths exist. The final matching is maximal, and you must show the alternating paths used in each iteration.
匹配是将两个不相交集合中的顶点进行配对，每个顶点最多使用一次。最大匹配算法从初始匹配开始，通过交替路径进行改进，直到不存在增广路径为止。最终匹配是最大的，你需要展示每次迭代使用的交替路径。
8. Sorting Algorithms | 排序算法
Two common sorting algorithms are bubble sort and quick sort. Bubble sort compares adjacent pairs and swaps them if they are out of order, making multiple passes until sorted. Quick sort selects a pivot and partitions the list into smaller and larger sublists, then sorts recursively. You should be able to draw a binary tree to represent the quick sort process.
两种常见的排序算法是冒泡排序和快速排序。冒泡排序比较相邻元素，若顺序错误则交换，经过多轮扫描直至有序。快速排序选择一个枢轴，将列表划分为较小和较大的子列表，再递归排序。你需要能够画出二叉树来表示快速排序过程。
9. Binary Search and Traversal | 二分查找与遍历
Binary search efficiently finds a target in a sorted list by repeatedly halving the search interval. The algorithm requires a clear trace of the lower bound, upper bound, and midpoint at each step. Pre-order, in-order, and post-order traversals of binary trees are also tested, often linked to sorting or expression trees.
二分查找通过在有序列表中反复将搜索区间减半来高效定位目标。算法需要清晰记录每一步的下界、上界和中点。二叉树的前序、中序和后序遍历也常出现在考题中，通常与排序或表达式树结合。
10. Route Inspection (Chinese Postman) Problem | 路由检查（中国邮递员）问题
For networks where every edge must be traversed at least once, the route inspection algorithm finds a closed walk of minimum weight. If the graph is Eulerian, any Eulerian circuit works. If not, you must add repeats by matching odd vertices to make the graph Eulerian, minimising extra weight. This is a classic D1 problem appearing on many specimens.
对于每条边都至少遍历一次的网络，路由检查算法找出最小权重的闭合行走。若图是欧拉图，任意欧拉回路即可。否则，需要通过匹配奇度顶点来添加重复边，使图变为欧拉图，并最小化额外权重。这是 D1 中经典的问题，常在样卷中出现。
11. Simulation and Flow Charts | 模拟与流程图
Simulation uses random numbers to model real-world processes when direct analysis is impractical. Questions may ask you to run a simulation given a flowchart, use random digits to determine outcomes, and estimate quantities such as queue length or waiting time. Understanding the logic of a flow chart is essential.
当直接分析不可行时，模拟利用随机数来建模现实过程。考题可能会给出流程图，要求你运行模拟，使用随机数字确定结果，并估计队列长度或等待时间等指标。理解流程图的逻辑至关重要。
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