📚 Edexcel AS Further Maths Unit 2 (Decision 1) Jan 2019 Question Breakdown | AS进阶数学单元2 2019年1月试卷题型剖析
The January 2019 Edexcel AS Further Mathematics Unit 2 (Decision Mathematics 1) paper is a classic benchmark for students aiming to master algorithmic thinking, graph theory, and optimisation. This guide breaks down every question style that appeared in that exam, giving you bilingual step‑by‑step strategies, exam‑ready techniques, and common pitfalls to avoid.
2019年1月Edexcel考试局AS进阶数学单元2（决策数学1）真题，是检验学生算法思维、图论与优化能力的经典标杆。本文逐一拆解该试卷中出现的各题型，提供中英双语的分步策略、应试技巧和常见失分点剖析，助你稳扎稳打拿高分。
1. Sorting Algorithms – Bubble & Quick Sort | 排序算法——冒泡排序与快速排序
In Question 1 of the Jan 2019 paper, you were given a list of numbers and asked to perform both a bubble sort and a quick sort. For bubble sort, you needed to record the number of comparisons and swaps after each pass, and state the final sorted list. Remember to clearly mark the end of each pass; a common mistake is forgetting to reduce the number of comparisons as elements sink to their correct positions.
2019年1月卷第一题给出一组数字，要求分别执行冒泡排序和快速排序。冒泡排序必须记录每一轮比较次数和交换次数，并写出最终有序列表。别忘了在每轮结束后标注状态，常见错误是未随着元素归位而减少后续轮次的比较次数。
Quick sort required choosing a pivot (usually the first or middle element) and partitioning the list into two sublists, then recursively sorting. The mark scheme rewarded clear sublisting and proper pivot notation. Practice both pivot selection by first item and by middle item, because the exam can specify either.
快速排序需要选取基准值（pivot），分划子表并递归排序。评分方案看重清晰的子表分划和基准值标注。务必熟练以首元素或中间元素为基准的两种做法，因为考题可能指定方法。
2. Bin Packing – First‑Fit & First‑Fit Decreasing | 装箱问题——首次适应与降序首次适应
A bin packing problem appeared with a set of item sizes and bin capacity. You had to apply the first‑fit algorithm and then the first‑fit decreasing algorithm, showing the contents of each bin. For first‑fit decreasing, you must first sort the items into descending order before packing.
试卷出现了一组物品尺寸和固定箱容量的装箱问题。要求先用首次适应算法，再用降序首次适应算法，展示每个箱子里的物品。降序首次适应必须先将物品从大到小排序再装箱，遗漏排序直接失分。
The Jan 2019 paper also asked you to compare the efficiency of the two bin‑packing heuristics. You should always comment on the number of bins used and whether the decreasing variant achieves a better fill. A quick lower bound check using ∑(item size)/bin capacity can strengthen your answer.
2019年1月卷还比较了两种启发式算法的效率。一定要指出所用箱子数量，以及降序变体是否提高了装满率。用 ∑(物品体积)/箱容量 快速估算下界，能让你的回答更有说服力。
3. Minimum Spanning Trees – Kruskal & Prim | 最小生成树——Kruskal与Prim算法
One question provided a network table and asked for a minimum spanning tree using both Kruskal’s and Prim’s algorithms. For Kruskal, you must list edges in ascending order of weight, only adding an edge if it does not create a cycle, and state the total weight.
有一题给出网络表格，要求分别用Kruskal和Prim算法求最小生成树。Kruskal算法需按权值升序列出所有边，选择时避开成圈，最后给出总权值。考试中要清晰写出拒绝某条边的理由，否则容易丢分。
Prim’s algorithm was required in both graph form and matrix form. When using the matrix, highlight the row of the newly joined vertex and mark the crossing‑out of previous columns. The examiners expect a neat sequence of choices; your final answer must include the edges added and total weight.
Prim算法既要在图上演示，也要在表格矩阵上操作。矩阵操作时，务必突出新加入顶点的行，并划掉已选列。阅卷人看重步骤清晰，最终答案必须包含所加边以及总权值。
4. Shortest Path – Dijkstra’s Algorithm | 最短路径——Dijkstra算法
The paper contained a network with weighted edges and required the shortest path from a start node to several destinations using Dijkstra’s algorithm. You had to produce a labelling table with permanent labels boxed, and clearly state the route and its length.
试卷给出了带权网络，要求用Dijkstra算法求起点到多个终点的最短路径。必须制作标签表格，将永久标签用方框标出，并明确指出路径及长度。务必将每次迭代的顺序编号，方便检查。
Besides the actual shortest path, a ‘trace back’ was needed to show how the route was derived from the permanent labels. Many students lose marks by not writing the full vertex sequence, e.g. A → C → E → G. Practise drawing the working values and replacing temporary labels systematically.
除最短路径外，还需通过永久标签回溯路线。很多考生只写距离却未列出完整顶点序列，如A → C → E → G，白白失分。平时要多练习系统地更新临时标签并绘制工作值表。
5. Critical Path Analysis – Activity Networks & Floats | 关键路径分析——活动网络与浮动时间
An activity‑on‑node network had to be drawn from a precedence table, with clear node numbers and duration indications. Jan 2019 asked for early and late event times, the critical path, and the total float of specific activities. Always double‑check your forward pass and backward pass calculations.
试卷要求根据前驱表画出节点活动网络，清楚标注节点编号和工期。随后计算最早与最晚事件时间，找出关键路径和指定活动的总浮动。正向传递和反向传递计算务必反复核对，避免一错皆错。
Total float = LFT – EST – Duration. The exam also tested whether you could identify a delay’s effect on the project completion time. If total float is zero, any delay to that activity will push back the whole project. This conceptual link is heavily examined.
总浮动 = 最晚完成时间 − 最早开始时间 − 工期。题目还考查了某项活动延误对项目工期的影响。若总浮动为零，任何延误都会推迟整个项目。这个逻辑考点极受命题者青睐。
6. Linear Programming – Formulation & Graphical Solution | 线性规划——建模与图解法
A production‑planning scenario required writing down inequalities, identifying the feasible region, and using an objective line to find the optimal integer solution. The Jan 2019 constraints involved resource limits and non‑negativity conditions.
一个生产计划场景要求列出不等式组，确定可行域，并用目标函数线寻找最优整数解。2019年1月卷的约束条件包含资源上限和非负条件。建模时注意表述“不超过”、“至少”等关键词的符号方向。
After graphing the constraints, you had to use a ruler to draw an objective line and slide it parallel until leaving the feasible region. The optimal vertex often has fractional coordinates, so you must test surrounding integer points to confirm the maximum profit or minimum cost. A table of integer candidates is always expected.
画出约束图形后，需用直尺画出目标函数线并平行推移至离开可行域。最优顶点常有小数坐标，因此必须检验周边整数点，以确认最大利润或最小成本。答案中通常应包含一张整数候选点表格。
7. Transportation Problem – Initial Solution & MODI | 运输问题——初始解与位势法
The paper featured a classic transportation problem with supply and demand values. You had to find an initial feasible solution using the north‑west corner rule or the least‑cost method, then test optimality using the modified distribution (MODI) method and find improvement indices.
试卷上有一道经典的运输问题，给出了供应量与需求量。需用西北角法或最小成本法求得初始可行解，然后运用位势法（MODI）检验最优性，并计算改进指数。确保所有分配数之和与供需匹配。
Stepping‑stone paths were needed to re‑allocate units if a negative improvement index appeared. Always write the closed loop clearly, indicating + and – signs on the cells, and recalculate the new cost. The Jan 2019 paper wanted you to continue iterating until all indices were non‑negative.
若出现负改进指数，则需用闭回路法重分配运量。闭回路务必清晰画出，并在格子上标注+、‑号，再计算新成本。2019年1月卷要求不断迭代，直到所有改进指数非负。
8. Assignment Problem – Hungarian Algorithm | 指派问题——匈牙利算法
A minimisation assignment problem appeared where you had to allocate tasks to workers using the Hungarian algorithm. The initial matrix needed row and column reductions, followed by finding the minimum number of lines to cover all zeros. If the number of lines is less than the order of the matrix, you must proceed with additional adjustments.
出现一道最小化指派问题，要求用匈牙利算法将任务分配给工人。初始矩阵需进行行削减和列削减，然后用最少直线覆盖所有零。若直线数量小于矩阵阶数，必须进一步调整矩阵——即寻找未被覆盖的最小元素进行加减。
In the Jan 2019 question, you had to clearly state the final optimal assignment and the total cost. Always present the assignment as a one‑to‑one matching, e.g. Worker A → Task 3, Worker B → Task 1, etc. Full marks require a logical progression of the reduced matrices.
2019年1月考题要求清晰写出最终最优分配及总成本。指派必须以一一对应形式呈现，如工人A → 任务3，工人B → 任务1等。要拿满分，必须展示每次约化矩阵的逻辑演变过程。
9. Matchings – Bipartite Graph & Maximal Matching | 匹配——二分图与最大匹配
The paper tested bipartite matching: you were given an initial matching and had to find a maximal matching using an alternating path starting from an unmatched vertex on the left set. It is crucial to label the alternating path correctly and change the matching status of each edge along that path.
试卷考了二分图匹配：给定一个初始匹配，要求从左部未饱和点出发，寻找交替路径，从而得到最大匹配。准确标记交替路径并翻转路径上的匹配边是解题关键，画图时务必整洁清晰。
After each augmentation, you needed to redraw the bipartite graph showing the new matching. The final answer had to state the maximal matching explicitly and explain why no further augmentation is possible (e.g. all left vertices are matched or no alternating path exists).
每次增广后，需重新绘制二分图展示新匹配。最终答案须明确写出最大匹配，并解释为何无法进一步增广（如所有左部顶点都已匹配，或不存在交替路径）。
10. Scheduling – Gantt Charts & Worker Allocation | 调度——甘特图与工人分配
Based on the activity network from a previous part, a scheduling question required you to draw a Gantt chart and schedule the activities using a given number of workers. You had to respect precedence constraints and resource limits, and then calculate the minimum completion time.
调度题基于前面得出的活动网络，要求绘制甘特图并在给定工人数下安排活动。必须严格遵守紧前约束和资源限制，进而计算最短完工时间。绘制甘特图时，每个活动的横条需对齐时间轴并标注工人编号。
The Jan 2019 paper also asked you to determine the minimum number of workers needed to complete the project in a specified time, or to find the shortest possible duration given a fixed workforce. Such questions rely on careful packing of activities into time slots; use a scheduling table to avoid missing overlaps.
2019年1月卷还探讨了在指定时间内完成项目所需的最少工人数，或给定工人数下的最短工期。这类题目相当于把活动“打包”进时间槽，使用调度表格可有效避免忽略重叠冲突。
11. Common Pitfalls & Exam Strategy | 常见失分点与备考策略
Throughout the Jan 2019 paper, examiners penalised sloppy working, missing units, and failing to state all steps explicitly. When performing algorithms, always show intermediate stages – even if the final answer is correct, omitted working loses method marks. Get into the habit of labelling every iteration.
整份2019年1月卷中，阅卷人对手草、遗漏单位、省去关键步骤的现象扣分严厉。执行算法时必须展示中间过程——即使最终答案正确，缺少步骤也会丢掉方法分。养成每次迭代都加标注的习惯。
For graph‑based questions, use a ruler for straight edges and clearly indicate permanent labels. In linear programming, shade the feasible region neatly and label all vertices with their coordinates. Time management is also crucial: allocate roughly 1 minute per mark and leave time for checking arithmetic.
图论题用直尺画边，永久标签用方框标识；线性规划题均匀涂绘可行域，并标出所有顶点坐标。时间分配同样重要：约1分钟/分，并留出检查运算的时间。善用计算器验证加减绝非丢脸，实为明智之举。
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