📚 AS AQA Further Maths Unit 2 Mark Scheme January 2021: A Full Walkthrough | AS AQA 进阶数学第二单元 2021年1月评分标准详解
The January 2021 AQA AS Further Mathematics Unit 2 paper (Paper 2: Discrete) tested candidates on algorithms, graph theory, critical path analysis, linear programming, and binary operations. Understanding the mark scheme is just as important as being able to solve the questions, because AQA rewards method, accuracy, and clear communication in specific ways.
2021年1月AQA AS进阶数学第二单元（离散数学卷）考查了算法、图论、关键路径分析、线性规划和二元运算。理解评分标准与能够解题同样重要，因为AQA对方法、准确性和清晰的表达方式有明确的给分要求。
1. Sorting Algorithms: Bubble Sort and Quick Sort | 排序算法：冒泡排序与快速排序
The first question typically required candidates to perform a bubble sort on a small list of numbers and then apply a quick sort, showing each pass clearly. The mark scheme awarded one method mark for each complete pass, and one accuracy mark for the final sorted list.
第一题通常要求考生对一组较小的数字执行冒泡排序，然后再执行快速排序，并清晰展示每一趟的结果。评分标准对每一趟完整的过程给一个方法分，对最终排序正确的列表给一个准确分。
Bubble sort: you must compare adjacent pairs and swap if out of order. A full pass goes from first to last pair. The largest element “bubbles” to the end.
冒泡排序：必须比较相邻元素对，若顺序错误则交换。一趟完整的排序从第一对比较到最后一对。最大的元素会“冒泡”到末尾。
Quick sort: choose a pivot (usually the middle value), partition into less-than and greater-than groups, then repeat on each sublist. The mark scheme accepts either “first item” or “middle item” pivot choices as long as the method is consistent.
快速排序：选择一个基准（通常取中间值），将列表分为小于基准和大于基准两组，然后对每个子列表重复此过程。评分标准接受“取首项”或“取中间项”作为基准，只要方法保持一致。
Common error: in bubble sort, many candidates forget that after each pass the last element is fixed and need not be compared again. Failing to reduce the comparison range loses a method mark.
常见错误：在冒泡排序中，许多考生忘记每一趟后最后一个元素已固定，无需再参与比较。若未减少比较范围会丢失方法分。
In quick sort, tracing each step is essential. Write down the pivot, the new sublists, and the next pivot clearly.
在快速排序中，逐步追踪非常重要。请明确写出基准、新的子列表以及下一步的基准。
| Mark awarded | What must be shown |
| M1 | One complete pass of bubble sort or quick sort shown |
| A1 | Final list correct and full method clearly documented |
| B1 | Correct statement of number of comparisons (e.g., for n items, maximum n−1 passes) |
2. Binary Operations and Tables | 二元运算与运算表
Unit 2 often contains a short question on binary operations defined on finite sets, using operation tables. The mark scheme expects you to check closure, commutativity, identity, and inverses directly from the table.
第二单元通常包含一个关于有限集合上二元运算的简答题，并使用运算表。评分标准要求你直接从表中检查封闭性、交换律、单位元和逆元。
For a binary operation * on a set S, closure means that for all a, b ∈ S, a * b ∈ S. In a table, every entry must belong to S.
对于集合S上的二元运算*，封闭性意味着对所有a, b ∈ S，都有 a * b ∈ S。在运算表中，每一个元素都必须属于S。
Identity e satisfies e * x = x * e = x for all x ∈ S
单位元 e 满足对所有 x ∈ S，e * x = x * e = x
To find the identity, look for a row and column identical to the original set order. That element is the identity.
要寻找单位元，请找出一行和一列与原始集合顺序完全相同的元素，该元素就是单位元。
To check commutativity, verify that the table is symmetric about the leading diagonal.
要检查交换律，请验证运算表关于主对角线是否对称。
t x * y = e and y * x = e. If any element lacks an inverse, the set does not form a group.
要寻找逆元，对于每个x，寻找y使得 x * y = e 且 y * x = e。若任一元素缺少逆元，则该集合不构成群。
Mark scheme note: A conclusion without a table-based justification earns zero marks. Always cite specific entries.
评分标准提示：没有表格证据的结论不得分。务必引用具体的表格条目。
3. Graph Theory: Eulerian and Hamiltonian Paths | 图论：欧拉路径与哈密顿路径
The graph theory question usually asks whether a graph has an Eulerian trail or circuit, and whether it contains a Hamiltonian path. The mark scheme values correct use of vertex degrees.
图论问题通常要求判断一个图是否具有欧拉回路或欧拉路径，以及是否包含哈密顿路径。评分标准非常看重对顶点度数的正确运用。
Eulerian circuit: every vertex has even degree. Eulerian trail: exactly two vertices have odd degree.
欧拉回路：每个顶点度数均为偶数。欧拉路径：恰好有两个顶点度数为奇数。
If you write “all degrees are even” without listing them, you may earn the method mark but not the accuracy mark. Listing degrees is expected.
若只写“所有度数为偶数”而不列出各顶点度数，可能只能得到方法分而得不到准确分。列出度数才是标准做法。
For Hamiltonian paths, there is no simple degree theorem in AS Further Maths. You must show a valid path by tracing vertices, or argue via a general theorem if applicable.
对于哈密顿路径，AS进阶数学中没有简单的度数定理可直接使用。你必须通过标出一条经过所有顶点的路径来证明，或在适用时使用一般定理论证。
Be careful with graph terminology: a cycle must return to the start and have no repeated vertices except the start/end.
注意图论术语：回路必须回到起点，并且除起点/终点外不重复经过其他顶点。
The mark scheme often has a final “conclusion” mark. Even if your reasoning is correct, forgetting the explicit statement “therefore it has an Eulerian circuit” loses one mark.
评分标准通常设有“结论”分。即使推理正确，若忘记明确写出“因此它存在欧拉回路”，也会丢失一分。
4. Critical Path Analysis: Earliest and Latest Times | 关键路径分析：最早与最迟时间
Critical path analysis questions in this paper tested forward and backward passes on an activity network. The mark scheme clearly separates the forward pass (earliest event times) and backward pass (latest event times).
本次试卷中的关键路径分析题考查了活动网络上的正向扫描（最早事件时间）和反向扫描（最迟事件时间）。评分标准明确区分了正向扫描和反向扫描。
Forward pass: start at the source, add durations along each path, and take the maximum at each event.
正向扫描：从源点开始，沿每条路径累加工期，在每个事件处取最大值。
The source event is always given earliest time 0.
源事件的最早时间始终设为0。
When two activities meet at an event, choose the larger total time.
当两个活动在同一事件汇合时，选择较大的总时间。
Backward pass: start at the sink with the final earliest time, subtract durations, and take the minimum at each event.
反向扫描：从汇点开始，以最终最早时间作为初始值，沿路径反向减去工期，并在每个事件处取最小值。
rliest time for each activity. The critical path is the set of activities with zero float.
时差（浮动时间）= 最迟时间 − 最早时间。关键路径即所有时差为零的活动组成的路径。
| Event | Earliest time | Latest time | Float |
| A | 0 | 0 | 0 |
| B | 3 | 4 | 1 |
| C | 5 | 5 | 0 |
Note: the source and sink always have zero float. If your final critical path does not start at the source and end at the sink, check your backward pass.
注意：源点和汇点的时差始终为零。如果你的关键路径没有从源点开始并到达汇点，请检查反向扫描的计算。
5. Linear Programming: Formulation | 线性规划：模型建立
Linear programming questions in January 2021 required translating a word problem into constraints and an objective function. The mark scheme awards marks for each correct inequality and for correctly identifying the objective.
2021年1月的线性规划问题要求将文字题转化为约束条件和目标函数。评分标准对每个正确的不等式和正确识别目标函数分别给分。
ust be linear, and non-negativity constraints x ≥ 0 and y ≥ 0 usually earn separate marks.
令x和y为决策变量。每个约束都必须是线性的，并且非负约束 x ≥ 0 和 y ≥ 0 通常单独给分。
ify the quantity to maximise or minimise (e.g., profit, cost) and write it as P = ax + by.
确定要最大化或最小化的量（如利润、成本），并将其写为 P = ax + by 的形式。
Convert phrases like “at most”, “no more than”, “at least” into ≤ or ≥ signs. One mark per correct inequality.
将“至多”“不超过”“至少”等措辞转换为 ≤ 或 ≥ 符号。每个正确的不等式得一分。
Do not forget to state “x, y ≥ 0” icitly. Many candidates assume it is implied, but the mark scheme gives a clear B1 for it.
不要忘记明确写出“x, y ≥ 0”。许多考生认为这是默认条件，但评分标准为此单独给了B1分。
If the problem involves integer quantities (e.g., number of items), mention integrality if required, but in AS AQA usually continuous variables are accepted unless stated.
若问题涉及整数数量（如物品数量），必要时需说明整数性；但在AS AQA中，除非特别说明，通常接受连续变量。
6. Graphical Solution and Integer Points | 图解与整数点
Once constraints are plotted, the feasible region is found. The mark scheme requires correct shading of the rejected region, not the feasible region. This is a common point of confusion.
在绘制约束条件后，需要确定可行域。评分标准要求对“拒绝区域”进行正确着色，而不是对可行域着色。这是常见的混淆点。
For each inequality, shade the side that does not satisfy it. The remaining unshaded region is the feasible region.
对每个不等式，将不满足该不等式的一侧染色。剩余未染色的区域即为可行域。
Use a straight edge and label axes clearly. A wrong scale can cause loss of all graph marks.
使用直尺并清楚标注坐标轴。比例错误可能导致整个图解部分失分。
The objective line should be drawn last. Use the gradient from the objective function, and slide it across the feasible region.
目标函数线应最后绘制。使用目标函数的斜率，并沿可行域平移。
For integer programming, test integer points close to the continuous optimum. The mark scheme gives one mark for identifying candidate points, one for testing them, and one for the final optimal integer solution.
对于整数规划，测试连续最优解附近的整数点。评分标准对识别候选点、代入检验和最终最优整数解各给一分。
If you find the optimal point by drawing, you must clearly mark the point on your graph. Coordinates without a visible mark may not earn the final accuracy mark.
若你通过作图找到最优点，必须在图上明确标记该点。仅有坐标而没有图中标记可能无法获得最后的准确分。
7. Simplex Algorithm (Standard Maximisation) | 单纯形算法（标准最大化）
The simplex tableau question tested the standard maximisation procedure. The mark scheme gives method marks for choosing the pivot column, pivot row, and performing row operations correctly.
单纯形表问题考查了标准最大化流程。评分标准对选择主元列、主元行以及正确执行行运算给予方法分。
Pivot column: the most negative entry in the objective row (for maximisation). Pivot row: the smallest non-negative ratio of RHS to pivot column entry.
主元列：目标行中最负的项（对于最大化问题）。主元行：右端项与主元列对应项之比中最小的非负比值所对应的行。
Show the ratios clearly. For each row, write the quotient, then decide the minimum. This earns a method mark even if the arithmetic is wrong.
清楚地写出比值。对每一行写出商，然后确定最小值。即使计算有误，也能获得方法分。
Perform pivot row division first: divide the entire pivot row by the pivot element.
先执行主元行归一化：将整个主元行除以主元元素。
tries in the pivot column zero. Write down each operation explicitly, e.g., R₂ = R₂ − 2R₁.
使用行运算将主元列中的其他元素化为零。明确写出每次运算，例如 R₂ = R₂ − 2R₁。
The final tableau is optimal when every entry in the objective row is non-negative. State “optimal solution found” as the final conclusion.
当目标行中所有元素均为非负时，最终单纯形表达到最优。最后必须写出“已找到最优解”的结论。
8. Game Theory: Pure and Mixed Strategies | 博弈论：纯策略与混合策略
The January 2021 paper included a two-player zero-sum game. The mark scheme expected identification of a stable solution via a saddle point, or computation of mixed strategy probabilities.
2021年1月试卷包含一个两人零和博弈。评分标准期望考生通过鞍点识别稳定解，或计算混合策略概率。
A saddle point is the entry that is both the minimum of its row and the maximum of its column. If such a point exists, the game has a pure strategy solution and the value of the game is that entry.
鞍点是同时满足“行中最小值”和“列中最大值”的条目。若存在这样的点，则博弈具有纯策略解，博弈值即为该条目。
For pure strategies, simply circle the saddle point and write the value of the game. One mark for the saddle point, one for the value.
对于纯策略，只需圈出鞍点并写出博弈值。鞍点一分，博弈值一分。
er choose strategy 1 with probability p. Solve for p such that the column player’s expected payoffs are equal.
若不存在鞍点，则建立方程。对于2×2博弈，设行玩家以概率p选择策略1。求解p使列玩家的期望收益相等。
Write the expected value equations clearly. Do not skip the algebra, because method marks are awarded for setting up the correct equations.
清楚地写出期望值方程。不要跳过代数步骤，因为正确建立方程即可获得方法分。
State the value of the game at the end, even if you found it during the calculation. AQA examiners expect a final boxed answer.
在最后写出博弈值，即使计算过程中已经得出。AQA考官期望看到最终框出的答案。
9. Network Flows: Maximum Flow / Minimum Cut | 网络流：最大流/最小割
The network flow question tested the max-flow min-cut theorem. The mark scheme awards marks for augmenting paths, checking the final flow, and identifying a minimum cut.
网络流问题考查了最大流最小割定理。评分标准对增广路径、检验最终流量和识别最小割分别给分。
Max flow = Min cut capacity
最大流 = 最小割容量
Use the labelling algorithm: label each vertex with (incoming flow, outgoing capacity) and find an augmenting path from source to sink.
使用标记算法：为每个顶点标记（入流量，出容量），并寻找从源点到汇点的增广路径。
After each augmentation, update the residual capacities and draw a new diagram if possible. The mark scheme often gives one mark per successful augmentation.
每次增广后更新剩余容量，如有可能就绘制新图。评分标准通常对每次成功的增广给一分。
To prove maximality, use the min-cut. Draw a cut separating the source from the sink, calculate its capacity, and show it equals the flow.
为了证明最大性，使用最小割。画一条将源点与汇点分离的割线，计算其容量，并证明其等于流量。
Be careful with edge labels: flow cannot exceed capacity. Many candidates lose marks by writing a flow greater than capacity in a single edge.
注意边的标注：流量不能超过容量。许多考生因在单条边上写出超过容量的流量而失分。
10. Recurrence Relations and Closed Forms | 递推关系与闭式解
The final part of the paper often introduced a recurrence relation. The mark scheme looked for substitution, iteration, and pattern recognition, plus a proof by induction if required.
试卷的最后部分通常引入一个递推关系。评分标准关注代入、迭代和模式识别，若题目要求则还需使用归纳法证明。
first-order recurrence xₙ = axₙ₋₁ + b, try iterative substitution until a pattern emerges.
对于一阶递推关系 xₙ = axₙ₋₁ + b，请尝试迭代代入，直到出现规律。
Write the first few terms explicitly: x₁, x₂, x₃, then guess the general form.
明确写出前几项：x₁, x₂, x₃，然后猜测通项形式。
If the question says “using induction” u must state the base case, assume the formula for n = k, prove for n = k+1, and conclude.
若题目要求“使用归纳法”，你必须写出基础情形、假设n = k时成立、证明n = k+1时成立，并作出总结论。
Clarity of notation matters: write xₙ₊₁ and xₙ with correct subscripts. A missing subscript can turn a recurrence into nonsense and cost method marks.
符号清晰非常重要：正确书写xₙ₊₁和xₙ。若遗漏下标，递推关系将失去意义，并导致方法分丢失。
After deriving a closed form, test it against the first two given values. If it matches, you can be confident in the final accuracy mark.
推导出闭式解后，请用前两个给定值检验。若吻合，则最后准确分拿到的可能性很大。
11. Mark Scheme Common Traps and Examiner Comments | 评分标准常见陷阱与考官点评
Analysing the official mark scheme from January 2021 reveals several recurring traps. The examiner’s report highlighted four main issues across candidates.
分析2021年1月官方评分标准可以发现几个反复出现的陷阱。考官报告指出了考生普遍存在的四个主要问题。
Trap 1: giving answers without working. In process-heavy questions, the final numerical value is rarely worth more than one mark. Show every step.
陷阱一：只给答案不写过程。在过程为主的问题中，最终数值很少超过一分。请展示每一个步骤。
Trap 2: using LaTeX-style notation or arrow symbols incorrectly. The mathematical notation in your answer must be standard and unambiguous.
陷阱二：错误使用LaTeX式符号或箭头。你答案中的数学符号必须标准且无歧义。
Trap 3: confusing time with float in critical path analysis. Float is the difference between latest and earliest times, not a duration.
陷阱三：在关键路径分析中混淆时间与时差。时差是最迟时间与最早时间之差，而不是持续时间。
Trap 4: in graphical linear programming, drawing the objective line with the wrong gradient because the objective function was not simplified.
陷阱四：在图形化线性规划中，因目标函数未化简而画出错误斜率的等值线。
To avoid these, always read the command words carefully: “show”, “find”, “calculate”, “prove”, “determine”. Each command implies a different level of explanation required.
要避免这些错误，请仔细阅读指令词：“show”（展示）、“find”（求出）、“calculate”（计算）、“prove”（证明）、“determine”（确定）。每个指令词意味着所需的解答详细程度不同。
12. Final Strategy for the Mark Scheme | 评分标准应试策略总结
Success in this paper is not only about knowing the mathematics but about aligning your written solution with what AQA expects. Here is a final checklist drawn directly from the January 2021 mark scheme.
本卷的成功不仅取决于掌握数学知识，还取决于将你的解题书写与AQA的评分期望对齐。以下清单直接来源于2021年1月评分标准。
For every graph algorithm, list your working in a table or clear rows. One row per pass is the standard.
对于每个图算法，用表格或清晰行列出过程。每一趟一行是标准格式。
For every optimization problem, state the objective function and all constraints before drawing anything.
对于每个优化问题，在画图前先写出目标函数和全部约束条件。
For network flow, label every edge with “flow / capacity” and update both after each augmentation.
对于网络流，每条边都标注“流量/容量”，并在每次增广后同时更新。
For proof questions, use the standard induction format and never omit the conclusion sentence.
对于证明题，使用标准归纳法格式，切勿省略结论句。
Use a ruler for all graph drawings. A freehand line may not be accurate enough to earn the mark.
画所有图形时必须使用直尺。手绘直线可能因不够准确而无法得分。
Final answers should be boxed. The mark scheme says “accept unboxed answers if clear”, but a box removes all ambiguity.
最终答案应加框。评分标准说“若清晰则接受未框答案”，但加框可以消除一切歧义。
By internalising these patterns, you not only improve your score on this specific paper, but also build a robust approach for any future discrete mathematics assessment.
通过内化这些模式，你不仅能在本次试卷中获得更高分数，还能为未来任何离散数学评估建立可靠的方法体系。
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