📚 Analysis of Further Maths Decision Maths 1 Exam Question Types | Further Maths决策数学1试卷题型解析
The Decision Mathematics 1 (D1) module for A-level Further Mathematics tests your ability to apply algorithms to optimisation, networks, and scheduling problems. Exam papers consistently feature a core set of question types that reward procedural accuracy and clear reasoning. This revision guide identifies the most common D1 topics, highlights typical question formats, and provides targeted advice to help you secure top marks.
A-level Further Mathematics中的决策数学1（D1）模块考查你将算法应用于优化、网络和调度问题的能力。历年试卷反复出现一些核心题型，强调步骤准确和推理清晰。本复习指南归纳最常见的D1专题，指出典型命题方式，并提供针对性建议，助你稳取高分。
1. Sorting Algorithms and Order of Complexity | 排序算法与复杂度
Expect a question that asks you to perform bubble sort, shuttle sort (insertion sort) or quicksort on a small list of numbers. You need to show every pass clearly and count comparisons and swaps.
考题通常会要求你对一组较小的数字列表执行冒泡排序、穿梭排序（插入排序）或快速排序。你需要清晰地展示每一趟，并计数比较次数和交换次数。
For bubble sort, after the first pass the largest number is guaranteed to be at the end. The maximum number of comparisons is ½n(n – 1) and the maximum number of swaps is the same.
冒泡排序中，第一趟后最大的数必定在末尾。最大比较次数为 ½n(n – 1)，最大交换次数与之相同。
In shuttle sort, treat the first item as sorted; repeatedly take the next item and insert it into the correct position among those already sorted, shifting larger items to the right. You must record each intermediate list.
在穿梭排序中，将第一项视为已排序；反复取出下一项并将其插入已排序部分的正确位置，将较大的项向右移位。你必须记录每一个中间列表。
Quick sort uses a pivot. Partition the list into numbers less than the pivot and numbers greater than the pivot, keeping their relative order. Show the two sublists and recurse. Average time complexity is O(n log n); worst case is O(n²).
快速排序使用枢轴。将列表划分为小于枢轴的数和大于枢轴的数，保持相对顺序。展示两个子列表并进行递归。平均时间复杂度为 O(n log n)，最坏情况为 O(n²)。
2. Bin Packing and Scheduling | 装箱与调度
First-fit, first-fit decreasing, and full-bin packing are standard. For first-fit decreasing, sort the items into decreasing order, then place each item into the first bin that has enough space.
首次适应、首次适应递减和满箱组合是标准算法。首次适应递减先将物品按大小降序排列，再依次放入第一个有足够剩余空间的箱子。
Lower bounds for the number of bins are often required. Use ⌈(sum of item sizes) ÷ (bin capacity)⌉, and also consider the bound based on the largest item: at least that many bins.
常要求确定箱数的下界。使用 ⌈(物品大小总和) ÷ (箱子容量)⌉，还要考虑基于最大物品的下界：至少需要其个数那么多箱子。
Scheduling questions involve drawing Gantt charts to display the completion times of activities on processors. Be ready to identify idle time and the schedule length when using a given number of machines.
调度问题涉及绘制甘特图来展示各活动在处理器上的完成时间。要能够识别空闲时间并使用给定数量的机器时的总调度时长。
3. Graph Terminology and Minimum Spanning Trees | 图论术语与最小生成树
You must be comfortable with basic terms: graph, vertex, edge, degree, path, cycle, tree, spanning tree, and complete graph. A tree on n vertices has exactly n – 1 edges.
你必须熟悉基本术语：图、顶点、边、度、路径、回路、树、生成树、完全图。n 个顶点的树恰好有 n – 1 条边。
Kruskal’s algorithm: list all edges in order of increasing weight, and select the next smallest edge that does not form a cycle. Stop when n – 1 edges have been added. Prim’s algorithm: start at any vertex, then repeatedly add the cheapest edge connecting the current tree to a new vertex. Know how to apply Prim using a distance matrix.
Kruskal算法：将所有边按权重递增顺序排列，选择不构成回路的最小边，直到加入 n – 1 条边。Prim算法：从任意顶点开始，反复加入连接当前树与新顶点的最便宜边。要会使用距离矩阵执行Prim算法。
Common errors include forgetting to check for cycles in Kruskal and misreading rows/columns in a matrix-based Prim. Always list the edges in the order selected.
常见错误包括Kruskal中忘记检查回路，以及基于矩阵的Prim中读错行/列。务必依选择顺序列出每条边。
4. Dijkstra’s Algorithm and Shortest Path | 迪杰斯特拉算法与最短路径
Questions typically provide a weighted graph or network and ask for the shortest path from a given source to one or all vertices. Set up boxes with order of labelling, working values, and final label.
题目通常会给出一个带权图或网络，要求从给定源点到一个或所有顶点的最短路径。需设置标签框，包含标记顺序、工作值和最终标签。
At each vertex, update working values if a smaller distance is found through the currently permanently labelled vertex. When the target is permanently labelled, you can stop. Backtrack to state the actual shortest path.
在每个顶点，若经由当前已永久标记的顶点找到更短距离，则更新工作值。当目标顶点被永久标记时可停止。通过回溯给出确切的最短路径。
Watch out for directed edges – the direction must be followed strictly. Exam questions may also ask for the length only, or compare two alternative routes.
注意有向边——必须严格遵循方向。试卷也可能只要求给出最短距离，或比较两条替代路线。
5. Chinese Postman Problem (Route Inspection) | 中国邮递员问题(路径检查)
The task is to find the shortest closed walk that traverses every edge at least once. Start by listing all odd-degree vertices – there will always be an even number of them.
任务是为覆盖每条边至少一次的最短闭合路线。首先列出所有奇度顶点——它们的个数必为偶数。
Pair up the odd vertices, and for each pair find the shortest path between them. The minimum extra distance is the sum of these shortest path lengths. Add this to the total weight of all edges for the length of the optimal Chinese postman route.
将奇度顶点两两配对，并找出每对之间的最短路径。最小额外距离就是这些最短路径长度之和。将其加到所有边总权重上，即为最优中国邮递员路线长度。
If the network is already Eulerian (all degrees even), no extra edges are needed. If it is semi-Eulerian (exactly two odd vertices), the route starts and ends at those two odd vertices.
如果网络已是欧拉图（所有度数为偶数），则不需要额外边。如果是半欧拉图（恰好两个奇度顶点），路线以这两个奇度顶点为起点和终点。
6. Travelling Salesman Problem (Upper and Lower Bounds) | 旅行商问题(上下界)
TSP questions ask you to find a Hamiltonian cycle of minimum weight. Upper bounds are found using the Nearest Neighbour algorithm: start at a chosen vertex, repeatedly go to the nearest unvisited vertex, then return to the start.
TSP问题要求找到最小权重的哈密顿回路。上界可通过最近邻算法求得：从选定顶点出发，反复前往最近的未访问顶点，最后返回起点。
For a better upper bound, run Nearest Neighbour from different starting vertices and take the smallest value. Lower bounds are obtained by deleting a vertex, finding the MST of the remaining network, and adding the two smallest edges incident to the deleted vertex. Repeat for several vertices and choose the maximum of these lower bounds.
为得到更好的上界，从不同起点运行最近邻算法并取最小值。下界通过删除一个顶点、求剩余网络的最小生成树，并加上与该删除顶点关联的两条最小边来获得。对多个顶点重复，取最大的下界。
Present your answer as an inequality: lower bound ≤ optimal tour length ≤ upper bound. Marks are awarded for clear working at each step.
将答案表达为不等式：下界 ≤ 最优巡回路长度 ≤ 上界。每一步清晰的演算都可获分。
7. Critical Path Analysis (CPA) | 关键路径分析
Construct an Activity-on-Arc (AoA) network from a precedence table. Dummy activities are introduced when an activity depends on more than one preceding activity that do not share identical predecessors.
根据前导表构建Activity-on-Arc (AoA) 网络。当一项活动依赖于多个前驱活动而这些前驱活动又不完全相同的时候，就需要引入虚活动。
Calculate earliest start time (EST) and latest start time (LST) – some boards use earliest event time and latest event time. Total float = LST – EST – duration. Activities with zero total float are critical; the longest path through the network is the critical path.
计算最早开始时间 (EST) 和最晚开始时间 (LST) ——某些考试局使用最早事件时间和最晚事件时间。总浮动 = LST – EST – 时长。总浮动为零的活动是关键活动；穿过网络的最长路径即为关键路径。
You may need to draw a cascade (Gantt) chart to show the scheduling of activities, highlighting float times and the critical path.
你可能需要绘制级联（甘特）图来展示活动的排程，突出浮动时间和关键路径。
8. Linear Programming (Graphical Method) | 线性规划(图解法)
Formulate the problem by defining decision variables x and y, then write the objective function and constraints as linear inequalities. Plot the constraint lines, shade the unwanted regions, and identify the feasible region.
通过定义决策变量 x 和 y 来表达问题，然后用线性不等式写出目标函数和约束条件。画出约束线，涂掉不满足约束的区域，确定可行域。
Use the objective line method: draw a line of the form ax + by = k for a chosen k, then slide it parallel until it leaves the feasible region. The last point of contact gives the optimal solution. If integer solutions are required, test the integer points around the optimal vertex.
使用目标直线法：选取一个k画出直线 ax + by = k，然后平行移动直至要离开可行域。最后接触点即为最优解。如需整数解，围绕最优顶点测试整数点。
Never forget non-negativity constraints x ≥ 0, y ≥ 0. Clearly write the maximum or minimum value of the objective function at the end.
绝不要忘记非负约束 x ≥ 0, y ≥ 0。最后务必清楚地写出目标函数的最大值或最小值。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply