📚 A-Level Maths D1 Exam Review and Revision Guide | A-Level 数学 D1 考情回顾与备考指导
Decision Mathematics 1 (D1) is a unique module in A-Level Mathematics that focuses on algorithms, optimisation, and discrete structures. Recent exam papers have shown a consistent emphasis on applying standard algorithms to unseen scenarios, with questions often combining multiple topics. Understanding the common question types and refining your problem-solving approach are key to securing top marks.
决策数学 1 (D1) 是 A-Level 数学中一个独特的模块，主要研究算法、最优化和离散结构。近年的试卷显示，考试持续强调将标准算法应用到新情景中，且题目常常综合多个知识点。掌握常见题型并完善解题方法，是取得高分的关键。
1. Overview of D1 and Recent Exam Trends | D1 模块概览与近年考情趋势
D1 covers topics including algorithms, graph theory, linear programming, critical path analysis, and matchings. Examiners frequently test the ability to execute algorithms precisely, interpret results, and adapt methods to slightly modified conditions. Recent papers show a growing tendency to include wordy, real-world context problems, particularly in linear programming and critical path analysis.
D1 涵盖算法、图论、线性规划、关键路径分析和匹配等主题。考官经常测试精确执行算法、解释结果以及根据稍有变化的场景调整方法的能力。近年试卷越来越倾向于包括文字量较大的实际情境题，尤其是在线性规划和关键路径分析中。
A notable trend is the increased weighting on tracing algorithms, such as the shuttle sort or Dijkstra's algorithm, where candidates must show clear working in a table. Examiners' reports highlight common errors like incorrect tracing of bubble sort passes or forgetting to update temporary labels in Dijkstra. Understanding these pitfalls can help you avoid losing marks.
一个显著的趋势是，算法追踪题（如穿梭排序或 Dijkstra 算法）的权重有所增加，考生必须在表格中清晰展示过程。考官报告指出了常见错误，例如错误地追踪冒泡排序的趟数，或在 Dijkstra 算法中忘记更新临时标签。了解这些易错点有助于避免失分。
2. Algorithms: Sorting and Searching | 算法：排序与搜索
Sorting algorithms (bubble sort, shuttle sort, quick sort) and searching algorithms (binary search) are core to D1. You must be able to carry out each algorithm step by step, recording comparisons and swaps as requested. Practice writing out passes efficiently to meet time constraints.
排序算法（冒泡排序、穿梭排序、快速排序）和搜索算法（二分搜索）是 D1 的核心。你必须能够逐步执行每个算法，并根据要求记录比较和交换次数。练习高效写出各趟过程，以满足时间要求。
ublist from the left. For quick sort, you must clearly indicate the pivot and sublists. Binary search requires a sorted list and a midpoint calculation: midpoint = floor((low+high)/2). Use consistent notation to avoid confusion.
对于冒泡排序，记住第一趟之后最大元素已处于最终位置。在穿梭排序中，将列表视为从左向右增长已排序子列表。对于快速排序，必须清楚地指出枢轴和子列表。二分搜索要求列表已排序，并计算中点：中点 = floor((low+high)/2)。使用一致的符号以避免混淆。
Examiners often ask for the number of comparisons or swaps. Ensure you count correctly and show reasoning. Diagrams or tables can help. For example, a bubble sort table might list each pass and label the final sorted elements.
考官常要求写出比较次数或交换次数。确保计数正确并展示推理过程。图表或表格会有所帮助。例如，冒泡排序表可以列出每趟并标注最终有序的元素。
3. Graph Theory Fundamentals | 图论基础
Key definitions include vertices, edges, degree, walks, paths, cycles, and trees. A complete graph Kn has n vertices and every pair connected; a bipartite graph has two disjoint sets of vertices with edges only between sets. You may be asked to draw a graph from an adjacency matrix or list.
关键定义包括顶点、边、度、途径、路径、回路和树。完全图 Kn 有 n 个顶点且每对顶点都相连；二部图有两组不相交的顶点集，边只存在于两集之间。你可能会被要求根据邻接矩阵或列表画出图形。
Understanding Eulerian and Hamiltonian paths is important. A graph is Eulerian if all vertices have even degree and it is connected (excluding isolated vertices). A Hamiltonian cycle visits each vertex exactly once and returns to start. Identify these properties quickly from degree sequences.
理解欧拉路径和哈密顿路径很重要。如果所有顶点都是偶数度且图连通（忽略孤立点），则图是欧拉图。哈密顿回路恰好访问每个顶点一次并返回起点。根据度序列快速确定这些性质。
Recent exams have included questions on isomorphic graphs and complementary graphs. Recall that the sum of degrees is twice the number of edges. Use this to find missing edges or to verify graph properties.
近期试题包含了同构图和补图的问题。记住度之和是边数的两倍。利用此性质来找缺失边或验证图的性质。
4. Minimum Spanning Trees and Shortest Paths | 最小生成树与最短路径
Prim's and Kruskal's algorithms are used to find a minimum spanning tree (MST) in a weighted network. Both can be tested, and you may be asked to apply them in tabular form. For Prim's, start from a given vertex and at each step add the cheapest available edge that connects a visited vertex to an unvisited one.
普林姆算法和克鲁斯卡尔算法用于在赋权网络中寻找最小生成树（MST）。两者都可能考到，你可能需要以表格形式应用它们。对于普林姆算法，从给定顶点开始，每一步添加连接已访问顶点与未访问顶点的最便宜边。
Kruskal's algorithm lists edges in ascending order of weight, adding an edge if it does not form a cycle. When drawing the MST, mark the order of edge selection. For Dijkstra's algorithm for shortest path, use a table with columns: vertex, visited, shortest distance, and previous vertex. Carefully update working values and final values.
克鲁斯卡尔算法按权重升序列出边，若添加边不构成回路则添加。在画 MST 时，标记边选择的顺序。对于求最短路径的 Dijkstra 算法，使用包含列：顶点、已访问、最短距离和前驱顶点的表格。仔细更新工作值和最终值。
A common mistake is misinterpreting 'shortest path' when multiple paths exist. Always double-check by tracing back from the destination. Also, be aware that the examination may ask you to apply the route inspection algorithm (Chinese Postman) or to determine whether a network is traversable.
常见错误是当存在多条路径时误解"最短路径"。务必从终点回溯复核。另外，注意考试可能要求应用路线检查算法（中国邮路）或判断网络是否可遍历。
5. Linear Programming | 线性规划
x. Use decision variables x and y clearly. Inequations are typically in the form ax + by ≤ c, with x, y ≥ 0. Plot the boundary lines and shade the unwanted region, or label the feasible region.
线性规划问题涉及从文字情境中列出约束条件、画出可行域并找到最优顶点。清晰地使用决策变量 x 和 y。不等式通常为 ax + by ≤ c 的形式，且 x, y ≥ 0。画出边界线并遮蔽不需要的区域，或标记可行域。
ax + by. Use an objective line method or test all vertices of the feasible region. If the problem is to minimise, you can maximise the negative or directly test vertices. Be careful with strict inequalities and integer solutions; if integer coordinates are required, you may need to use the ruler method and check nearby integer points.
目标函数表示为 P = ax + by。使用目标直线法或检验可行域的所有顶点。如果问题为最小化，可以对目标函数取负后最大化，或直接检验顶点。注意严格不等式和整数解；如果要求整数坐标，可能需要使用平行直线方法并检查附近整数点。
Examiners' fine variables precisely or do not write down the non-negativity constraints. Always write 'let x = ..., y = ...' ≥ 0. Show working for finding intersection points, possibly using simultaneous equations.
考官报告指出，学生往往未能精确定义变量或未写出非负约束。一定要写出"设 x = ...，y = ..."并包含 x ≥ 0，y ≥ 0。展示求交点的工作，可能需用到联立方程。
6. Critical Path Analysis | 关键路径分析
Critical path analysis (CPA) uses activity-on-arc or activity-on-node notation (the latter is common in Edexcel D1). You need to construct a precedence table and draw the network, ensuring correct dummies when necessary. Calculate earliest start times (EST) and latest finish times (LFT) using forward and backward passes.
关键路径分析（CPA）使用箭线图或节点图（后者在爱德思 D1 中常见）。你需要构建前序表并绘制网络，必要时确保正确地使用虚工作。通过向前和向后推算计算最早开始时间（EST）和最迟完成时间（LFT）。
The critical path is the longest path through the network, where total float is zero. Show all floats, including total float and free float. Interpretation questions might ask about the effect of a delay in a specific activity. If the delay is less than or equal to total float, the project completion time is unchanged; otherwise it increases.
关键路径是网络中最长的路径，其上总时差为零。展示所有时差，包括总时差和自由时差。解释性题目可能询问某个活动延迟的影响。若延迟小于或等于总时差，则项目完成时间不变；否则会增加。
A recent exam trend is to provide a partially completed network and ask you to complete it, or to schedule activities with limited resources using a Gantt chart. Practice reading and constructing Gantt charts, and be able to schedule at most the given number of workers.
近期考试的趋势是给出部分完成的网络并要求你补全，或者使用甘特图在有限资源下安排活动。练习阅读和绘制甘特图，并能够安排不超过给定工人数的调度。
7. Matchings and Allocation Problems | 匹配与分配问题
Matchings involve bipartite graphs where you need to find an alternating path to improve an initial matching. The Hungarian algorithm is used for allocation problems (e.g., assigning workers to tasks) to minimise total cost. You must be able to reduce rows and columns, find the minimum number of lines to cover zeros, and augment the matching.
匹配涉及二部图，需要找到交错路径以改进初始匹配。匈牙利算法用于分配问题（例如给工人分配任务）以最小化总成本。你必须能够进行行归约、列归约，找到覆盖所有零的最少直线数，并扩充匹配。
In exam questions, you may be asked to use the algorithmic steps and show all intermediate matrices. A common error is forgetting to add the minimum uncovered element to doubly covered elements after line-drawing, or incorrectly determining the minimal number of lines. Always
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