📚 Edexcel AS and A level Decision Mathematics 1: Textbook and E-Book Question Type Analysis | Edexcel AS和A Level进阶数学决策数学1教材与电子书题型解析
The Edexcel Decision Mathematics 1 (D1) textbook and its accompanying electronic resources provide a structured introduction to algorithms and discrete mathematical modelling. This article dissects the most common question types found in the AS and A level Further Mathematics exams, offering step-by-step breakdowns and revision strategies that mirror the official textbook approach.
Edexcel决策数学1（D1）教材及其配套电子资源为算法和离散数学建模提供了系统的入门学习。本文拆解了AS和A Level进阶数学考试中最常见的题型，结合教材思路进行逐步解析，并给出复习策略。
1. Understanding Sorting Algorithms | 排序算法题型解析
Exam questions often require you to trace the bubble sort and quick sort algorithms on a list of numbers. You must show each pass or pivot selection clearly, counting comparisons and swaps. The textbook emphasises that for bubble sort, after the first pass the largest element is in its final position.
考试题常要求对一列数字跟踪冒泡排序和快速排序算法。你需要清晰地展示每一趟或每次基准选择，并统计比较和交换次数。教材强调，对于冒泡排序，第一趟之后最大元素会位于最终位置。
A typical question might ask: 'Use bubble sort to arrange these numbers into ascending order. Show the result after each pass.' You should present a table or list, indicating the state after each pass, for example: Pass 1: 5, 3, 8, 1, 9 -> 3, 5, 1, 8, 9. Then continue until no swaps are needed.
典型题目可能要求：“用冒泡排序将下列数字排列为升序。展示每一趟后的结果。”你应当以表格或列表呈现，标明每趟后的状态，例如：第1趟：5, 3, 8, 1, 9 -> 3, 5, 1, 8, 9。然后继续直到无需交换。
2. Bin Packing Strategies | 装箱问题题型解析
Bin packing questions test first-fit, first-fit decreasing, and full-bin strategies. The textbook highlights that first-fit decreasing requires sorting items into descending order before applying first-fit. You must justify the number of bins used and state whether a solution is optimal.
装箱问题考察首次适应、降序首次适应和完全装箱策略。教材指出，降序首次适应需要先将物品按降序排列后再应用首次适应。你必须说明所用箱子的数量，并判断解是否最优。
In an exam, you might be given items of lengths 4, 8, 3, 5, 7, 2 and bin capacity 10. Using first-fit decreasing, list the bins and their contents. The answer would show bins: Bin1: 8, 2; Bin2: 7, 3; Bin3: 5, 4. This uses exactly the lower bound ceil(total length / bin capacity), so it is optimal.
考试中，可能给定物品长度4, 8, 3, 5, 7, 2及箱容量10。使用降序首次适应，列出各箱及其内容。答案将展示箱子：箱1: 8, 2；箱2: 7, 3；箱3: 5, 4。这恰好达到下限值ceil(总长度/箱容量)，因此是最优解。
3. Graph Representation and Traversal | 图的表示与遍历题型
Questions on graph theory commonly ask you to represent a network using an adjacency matrix or adjacency list, and to perform breadth-first or depth-first traversal. The textbook stresses that breadth-first uses a queue, while depth-first uses a stack. You must keep track of visited vertices and the order of discovery.
图论相关题常要求用邻接矩阵或邻接表表示网络，并执行广度优先或深度优先遍历。教材强调广度优先用队列，深度优先用栈。你必须跟踪已访问顶点和发现顺序。
For example, starting from vertex A, perform a breadth-first search on the given graph. List the vertices in the order they are visited. You would initialise the queue with A, then dequeue A, enqueue its unvisited neighbours B and C, then next dequeue B, enqueue its unvisited neighbour D, and so on. The visit order might be A, B, C, D, E, F.
例如，从顶点A开始，对给定图执行广度优先搜索。按访问顺序列出顶点。你将先把A入队，然后A出队，将其未访问邻居B和C入队，之后B出队，其未访问邻居D入队，依此类推。访问顺序可能为A, B, C, D, E, F。
4. Minimum Spanning Trees – Kruskal’s and Prim’s | 最小生成树 – Kruskal与Prim算法题型
Finding a minimum spanning tree is a core D1 topic. You need to apply Kruskal's algorithm (select edges in increasing weight order, avoiding cycles) or Prim's algorithm (growing a tree from a start vertex by always adding the cheapest connecting edge). The textbook provides matrix and list forms for Prim's.
寻找最小生成树是D1的核心课题。你需要应用Kruskal算法（按权值递增顺序选边，避免回路）或Prim算法（从起始顶点不断添加最便宜的连接边来生长树）。教材为Prim算法提供了矩阵和列表形式。
A question might give a table of distances between six towns and ask you to use Prim's algorithm starting from town P to find the minimum spanning tree. You should draw the tree step by step, clearly stating the order in which edges are added and their weights. The total weight is then calculated.
题目可能给出六个城镇之间的距离表，要求从城镇P开始用Prim算法求最小生成树。你应按步骤画出树形，清晰说明边添加的顺序及其权重，最后计算总权重。
5. Shortest Path – Dijkstra’s Algorithm | 最短路径 – Dijkstra算法题型
Dijkstra's algorithm appears in practically every D1 exam. The textbook teaches you to use boxes at each vertex to record the working values: order of labelling, final distance, and temporary distances. You must update labels systematically and extract the shortest path by tracing backwards.
Dijkstra算法几乎出现在每份D1试卷中。教材教你使用各顶点上的方框记录工作值：标注顺序、最终距离和临时距离。你必须系统地更新标签，并通过反向追踪提取最短路径。
A typical problem gives a network with weighted edges and asks for the shortest path from S to T. Your solution should include a complete labelling box diagram and a clear statement of the path, such as S -> A -> C -> T with length 17. Remember to check for alternative paths when distances are equal.
典型问题给出一个加权边网络，要求找出S到T的最短路径。你的解答应包括完整的标签框图和清晰的路径陈述，例如S -> A -> C -> T，长度17。当距离相等时，记得检查备选路径。
6. Route Inspection (Chinese Postman) | 邮路问题（中国邮递员）题型
Route inspection problems require you to find the shortest route that traverses every edge at least once, starting and ending at the same vertex. The textbook approach identifies odd vertices, considers all possible pairings to find the minimum extra distance, and adds those extra edges to make the graph Eulerian.
邮路问题要求找出遍历每条边至少一次且起点和终点相同的最短路线。教材方法先找出奇度顶点，考虑所有可能配对以找到最小额外距离，然后添加这些额外边使图成为欧拉图。
An exam question might present a network with four odd vertices, A, B, C, D. You must list the possible pairings, calculate the sum of shortest distances for each pairing, and select the minimum. The route length is then total weight of all edges plus this minimum extra distance. You should describe a possible route.
考试题可能给出一个具有四个奇度顶点A、B、C、D的网络。你必须列出可能的配对，计算每种配对的最短距离之和，选择最小值。路线长度则为所有边权重之和加上这个最小额外距离。你还需描述一条可行路线。
7. Travelling Salesperson Problem – Upper and Lower Bounds | 旅行商问题 – 上界与下界题型
The travelling salesperson problem (TSP) asks for the shortest Hamiltonian cycle. The textbook teaches the nearest neighbour algorithm to find an upper bound, and a minimum spanning tree approach for a lower bound. You must find the smallest upper bound by trying different start vertices, and refine the lower bound by deleting a vertex and finding the MST of the remainder.
旅行商问题（TSP）要求寻找最短哈密顿圈。教材教授用最近邻算法求上界，用最小生成树方法求下界。你需要通过尝试不同起始顶点找到最小上界，并通过删除一个顶点并对其余图求MST来优化下界。
In a typical question, a table of distances between cities is given. Find an upper bound using the nearest neighbour algorithm starting from each city, and state the best upper bound. Then find a lower bound by deleting city E and applying Prim’s algorithm on the reduced network, remembering to add back the two shortest edges from E. Interpret the bounds in an inequality.
典型题目给出城市间距离表。用最近邻算法从每个城市出发求上界，并给出最佳上界。然后删除城市E并在缩小后的网络上应用Prim算法求下界，别忘了将E的两条最短边加回。将上下界写成不等式解释。
8. Critical Path Analysis – Activity Networks | 关键路径分析 – 活动网络题型
Critical path analysis is a mainstay of D1. You must construct an activity-on-arc network from a precedence table, perform forward and backward passes to calculate earliest and latest start times, identify the critical path(s), and calculate float times. The textbook uses clearly defined conventions for node numbering and boxed times.
关键路径分析是D1的重要内容。你必须根据先后关系表构建箭线式网络，执行前推和后推以计算最早和最迟开始时间，识别关键路径，并计算浮动时间。教材对节点编号和时间方框有明确定义。
A question might provide a list of activities with durations and predecessors. Draw the network, find the overall project duration, and list the critical activities. Then, for a given activity, calculate its total float. The critical path is the sequence with zero total float. Your diagram must show EST and LFT clearly.
题目可能给出活动列表及其工期和紧前活动。绘制网络图，求出项目总工期，并列出关键活动。然后对某指定活动计算其总浮动时间。关键路径是总浮动为零的活动序列。你的图示必须清晰标明最早开始时间和最迟完成时间。
9. Scheduling and Gantt Charts | 调度与甘特图题型
Once the activity network is completed, scheduling questions ask you to allocate activities to workers, respecting precedence constraints and resource limits. The textbook introduces Gantt (cascade) charts to visualise the schedule. You must be able to construct and interpret such charts from a given priority list or by using heuristic rules.
完成活动网络后，调度题目要求你在遵守先后约束和资源限制的条件下将活动分配给工人。教材引入了甘特图（级联图）来可视化调度方案。你必须能根据给定优先级列表或启发式规则构建并解读甘特图。
For example, given 2 workers and a project duration of 20 days, draw a Gantt chart for the project using the activity times from part (a). Activities are shown as blocks of time; you must ensure an activity does not start before its predecessors finish. The chart makes it easy to spot resource clashes and idle time.
例如，有2名工人，项目工期20天，用(a)部分的活动时间绘制甘特图。活动显示为时间段块；你必须确保活动不早于其紧前活动结束前启动。甘特图便于发现资源冲突和闲置时间。
10. Linear Programming – Graphical Method | 线性规划 – 图解法题型
Linear programming in D1 is solved graphically for two variables. You start by defining decision variables, writing objective function and constraints, then plotting the feasible region. The textbook emphasises accurate graph drawing and testing vertices to find the optimal point. You must interpret integer solutions when required.
D1中的线性规划针对两个变量采用图解法。首先定义决策变量，写出目标函数和约束条件，然后绘制可行域。教材强调精确绘图并通过测试顶点找到最优点。必要时还需解释整数解。
ld plot lines such as 2x + y ≤ 40, shade the unwanted regions, and calculate the profit at each vertex of the feasible region. The answer might be (x=10, y=20) giving profit £380.
典型考题描述制造商生产两种纪念品。建立线性规划模型，绘制图形，找出最大利润。你将绘制像2x + y ≤ 40这样的直线，排除不可行区域，并计算可行域各顶点处的利润。答案可能为(x=10, y=20)，利润380英镑。
11. Matchings – Maximum Matching Algorithm | 匹配问题 – 最大匹配算法题型
Matching problems use bipartite graphs to model assignments. The textbook covers the maximum matching algorithm, starting from an initial matching and using alternating paths to augment it. An exam question expects you to break through a vertex, find an alternating path, and change the matching status until no further augmenting path exists.
匹配问题用二分图来模拟分配。教材介绍了最大匹配算法，从初始匹配出发，利用交替路径增广匹配。考试题要求你突破一个顶点，找到一条交替路径，并改变匹配状态，直到无增广路径为止。
Given a bipartite graph with sets X and Y, construct an initial matching. Then, choosing an unmatched vertex in X, try to find an alternating path. You must show the labelling process and the resulting improved matching. The final matching is maximum, and its size should match the minimum vertex cover.
给定二分图集合X和Y，构造一个初始匹配。然后选择一个X中未匹配的顶点，尝试寻找一条交替路径。你必须展示标注过程和所得改善匹配。最终匹配是最大匹配，其大小应与最小顶点覆盖相等。
12. Exam Strategy and Common Pitfalls | 考试策略与常见陷阱
The D1 paper is known for its methodical mark schemes. Always show your working clearly, especially in algorithms that require iterative steps like Dijkstra or bubble sort. The textbook’s e-book bonus often includes interactive walkthroughs that help you internalise the step-by-step logic needed for full marks.
D1试卷以有条理的评分方案著称。务必清晰展示运算过程，尤其是在需要迭代步骤的算法中，如Dijkstra或冒泡排序。教材的电子书通常包含互动式演示，帮助你将逐步逻辑内化以拿到满分。
Common pitfalls include misreading precedence tables for critical path analysis, forgetting to reset temporary labels in Dijkstra, and not checking the number of bins against the lower bound in bin packing. Practice with the textbook end-of-chapter mixed exercises and the e-book’s self-test questions to build both speed and accuracy.
常见陷阱包括在关键路径分析中误读先后关系表、在Dijkstra算法中忘记重置临时标签，以及在装箱问题中未将箱子数与下界进行比较。通过教材章末混合练习和电子书中的自测题进行练习，可同时提升速度与准确性。
Published by TutorHao | Decision Mathematics 1 Revision Series | aleveler.com
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