📚 AQA AS Further Maths Unit 2 (June 2022) – Key Concepts Review | AQA AS 进阶数学单元二 2022年6月卷知识点精讲
The June 2022 AQA AS Further Mathematics Paper 2 (Unit 2) covers both Statistics and Discrete Mathematics. This paper tests a wide range of topics including discrete probability distributions, chi-squared tests, algorithms, graph theory, linear programming and critical path analysis. In this article, we will review the essential knowledge points and typical question styles, helping you strengthen your understanding and prepare effectively for similar assessments.
2022年6月AQA AS进阶数学试卷二（单元二）涵盖统计与离散数学两大板块。试卷涉及离散概率分布、卡方检验、算法、图论、线性规划及关键路径分析等多个主题。本文将对核心知识点与典型题型进行梳理，帮助考生巩固理解，高效备考。
1. Discrete Random Variables | 离散随机变量
value E(X) is the long-run average, calculated as Σ x·P(X=x). The variance Var(X) = E(X²) – ad. Transformations follow simple rules: E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X).
离散随机变量 X 取可数个值，并对应各自的概率。你需要建立概率分布表并验证总概率为1。期望值 E(X) 是长期平均值，计算公式为 Σ x·P(X=x)。方差 Var(X) = E(X²) – [E(X)]² 衡量离散程度。线性变换遵循 E(aX + b) = aE(X) + b 和 Var(aX + b) = a²Var(X)。
E(X) = Σ xᵢ pᵢ, Var(X) = Σ(xᵢ² pᵢ) – μ²
In the June 2022 paper, a typical question asked to find unknown probabilities and calculate the mean and standard deviation of a discrete distribution. Remember to present your working clearly and to interpret the context, e.g. 'the expected number of heads'.
在2022年6月试卷中，典型题目要求求出未知概率并计算离散分布的均值与标准差。答题时注意清晰展示计算过程，并结合题目背景进行解释，如“正面向上的期望次数”。
2. Poisson Distribution | 泊松分布
d a constant average rate λ. The probability of exactly r events is given by P(X = r) = (e–λ λr) / r! o approximate a binomial distribution B(n, p) when n is large and p is small, with λ = np.
泊松分布用于描述在固定时间或空间间隔内、事件以恒定平均速率 λ 且相互独立发生的次数。恰好发生 r 次的概率为 P(X = r) = e–λ λr / r!。均值与方差都等于 λ。当二项分布 B(n, p) 的 n 很大、p 很小时，可用 λ = np 的泊松分布进行近似。
P(X = r) = e–λ λr / r!, r = 0, 1, 2, …
Cumulative probabilities from tables or calculators are often required. The June 2022 paper included a context-based question where you had to justify the Poisson model and then calculate probabilities such as 'at least 3 events' or use the distribution of a sum of independent Poisson variables.
试卷经常要求使用累积概率表或计算器。2022年6月试题中有一道应用题，需要先说明选取泊松模型的理由，再计算“至少发生3次”等概率，或利用独立泊松变量之和的性质。
3. Geometric Distribution | 几何分布
success p. There are two main conventions: one where r starts at 1, giving P(X = r) = p(1 – p)r–1, and another starting at 0. For AQA AS Further Maths, the r = 1, 2, … definition is standard. Then E(X) = 1/p and Var(X) = (1 – p)/p².
几何分布描述在一系列独立的伯努利试验中，首次成功所需的试验次数（包括成功的那一次）。AQA AS进阶数学采用从 r = 1 开始的定义，概率函数为 P(X = r) = p(1 – p)r–1。期望 E(X) = 1/p，方差 Var(X) = (1 – p)/p²。
E(X) = 1/p, Var(X) = (1 – p) / p²
Questions often ask for 'the probability that the first success occurs on or after the 5th trial', which links to the cumulative property P(X > k) = (1 – p)k. In the 2022 paper, candidates needed to recognise the memoryless nature of the distribution and apply it in a quality-control context.
试题常考“首次成功发生在第5次或之后”的概率，这利用性质 P(X > k) = (1 – p)k。在2022年试卷中，考生需要识别几何分布的无记忆性，并将其应用到质量控制的情境中。
4. Chi-Squared Goodness of Fit Test | 卡方拟合优度检验
H₀ states that the data follow the specified distribution. The test statistic is X² = Σ (O – to be combined to achieve this. The degrees of freedom ν = number of classes after pooling – 1 – number of estimated parameters.
卡方拟合优度检验用于判断观测频数分布是否符合某一理论分布。原假设 H₀ 声明数据服从特定分布。检验统计量为 X² = Σ (O – E)² / E，O 为观测频数，E 为期望频数。期望频数应不小于5，必要时可合并类别。自由度 ν = 合并后的类别数 – 1 – 估计的参数个数。
X² = Σ (O – E)² / E
In the June 2022 paper, a question gave observed counts for each day of the week and asked to test whether sales were uniformly distributed. You needed to calculate expected frequencies, combine categories if necessary, compute the test statistic and compare it against a critical value from statistical tables.
2022年6月试卷中有一题给出了一周中各天的销售观测次数，要求检验销售是否服从均匀分布。你需要计算期望频数、必要时合并类别、算出检验统计量并与临界值比较。
5. Chi-Squared Test for Association | 卡方独立性检验
cy for each cell is (row total × column total) / grand total. The test statistic X² = Σ (O – E)² / E is compared with a chi-squared distribution with (r–1)(c– and c are the numbers of rows and columns. Again, expected frequencies must be at least 5.
卡方独立性检验用于判断两个分类变量是否独立。数据以列联表呈现。在独立性原假设下，每个单元格的期望频数为 (行合计 × 列合计) / 总合计。检验统计量 X² = Σ (O – E)² / E，与自由度为 (r–1)(c–1) 的卡方分布比较，其中 r、c 为行数与列数。同样要求期望频数不小于5。
ν = (r – 1)(c – 1)
r there is an association between, for instance, gender and preference for a product. Candidates needed to set up hypotheses, compute expected values, find X² and draw a conclusion at a 5% significance level.
2022年夏季的一道典型题给出了2×2或3×2的列联表，问例如性别与产品偏好之间是否存在关联。考生需设立假设、计算期望值、求出 X² 并在5%显著性水平下得出结论。
6. Sorting and Searching Algorithms | 排序与搜索算法
In the discrete part of Unit 2, you must be able to trace and apply standard algorithms. Bubble sort repeatedly compares adjacent items and swaps them if they are in the wrong order, passing through the list until no swaps are needed. Quick sort selects a pivot (usually the first item) and partitions the list into items less than or greater than the pivot, then recursively sorts sublists. For search, the binary search algorithm works on a sorted list by repeatedly dividing the interval in half and discarding the half that cannot contain the target.
在单元二的离散部分，你需要掌握标准算法的跟踪与应用。冒泡排序反复比较相邻元素并在顺序错误时交换，遍历列表直至无需再交换。快速排序选取一个枢轴（通常为首项），将列表划分为小于和大于枢轴的两部分，然后递归排序子列表。二分搜索算法用于已排序列表，通过反复将区间对半分并丢弃不可能包含目标的一半来查找。
A typical exam question provides a short list and asks to show each pass of bubble sort, or to perform a quick sort using the first element as the pivot, writing down the result after each partition. For binary search, you might be asked to count the number of comparisons needed to locate an item. The June 2022 paper included such an algorithmic trace.
典型的试卷题会给出一个简短列表，要求展示冒泡排序的每一趟结果，或以首元素为枢轴执行快速排序，写出每次划分后的结果。对于二分搜索，可能要求统计查找某一项所需的比较次数。2022年6月试卷包含了此类算法跟踪题。
7. Minimum Spanning Trees | 最小生成树
A minimum spanning tree (MST) connects all vertices of a network with the minimum total weight, without forming cycles. Kruskal’s algorithm lists all edges in ascending order of weight and selects an edge if it does not create a cycle, until all vertices are connected. Prim’s algorithm grows a tree from a start vertex by repeatedly adding the shortest edge that connects a tree vertex to a non-tree vertex. Both algorithms appear in the exam; you may need to use a distance matrix or list version.
最小生成树（MST）以最小总权重连接网络中的所有顶点，且不形成回路。克鲁斯卡尔算法将所有边按权重升序排列，依次选取不构成回路的边，直至连通所有顶点。普里姆算法从起始顶点开始，重复添加连接已选树与未选顶点的最短边，逐步扩展树。两种算法都可能考到，可使用距离矩阵或列表形式作答。
Total weight of MST = sum of chosen edges
Questions often ask to apply both algorithms to a small network and compare the resulting MST. In the 2022 paper, there was a network with vertices labelled A, B, C, … and you had to list the edges in order of selection using Kruskal and Prim from a specified start vertex, then state the total weight.
试题常要求对一个小型网络分别应用两种算法并比较生成的MST。2022年试卷中出现了一个标有字母 A, B, C, … 的网络，需要按顺序列出克鲁斯卡尔算法以及从给定起点开始的普里姆算法所选取的边，并给出总权重。
8. Shortest Path Problems | 最短路径问题
Dijkstra’s algorithm finds the shortest path from a source node to all other nodes in a weighted graph with non-negative weights. It assigns a temporary distance label to every vertex, updates the smallest temporary label to permanent, and then uses it to update neighbours' labels. The process repeats until all vertices have permanent labels. You must record the working values in boxes and be prepared to show how the path is traced back.
迪杰斯特拉算法用于在非负权重的图中寻找从源点到其他所有节点的最短路径。它给每个顶点赋予临时距离标记，将其中最小的改为永久标记，然后用它来更新相邻顶点的标记。重复直至所有顶点都获得永久标记。你需要在表格中记录计算过程，并能沿标记回溯展示具体路径。
Dijkstra: temporary labels → permanent, trace precedences
In the June 2022 paper, a network was given with distances, and you had to use Dijkstra’s algorithm to find the shortest route from the home node to the destination node, clearly stating the route and its length. Remember that when two vertices have the same smallest temporary label, either may be chosen.
2022年6月试卷中给出了一个带距离的网络，要求使用迪杰斯特拉算法求出从起点到终点的最短路线，并明确写出路线与长度。注意当两个顶点具有相同的最小临时标记时，可任意选择。
9. Linear Programming | 线性规划
om a word problem, using non-negativity restrictions (x ≥ 0, y ≥ 0) and other inequalities. To find the optimum, test the coordinates of all vertices of the feasible region in the objective function, or use a ruler method.
线性规划（LP）是在一组线性不等式约束下最大化或最小化一个线性目标函数。对于两个变量，可在坐标系中画出可行域，最优解出现在可行域的某个顶点（或沿一条边）。你需要根据文字题建立约束条件，包括非负限制 x ≥ 0, y ≥ 0，以及其他不等式。求最优解时，可将所有顶点坐标代入目标函数检验，或采用平行线移动法。
Maximise P = ax + by, subject to constraints
A typical summer 2022 question described a manufacturing scenario with constraints on machine hours and materials, asking to formulate the inequalities, graph the feasible region, label vertices and determine the maximum profit. Integer solutions may be required if variables must be whole numbers.
2022年夏季的一道典型题描述了一个生产场景，有机器时间和原料等限制，要求写出不等式、画出可行域、标出顶点并确定最大利润。若变量必须为整数，则可能需要求整数解。
10. Critical Path Analysis | 关键路径分析
Critical path analysis is used to schedule project activities. Activities have durations and dependencies. An activity-on-node network shows each activity as a vertex, with directed edges representing precedence. The forward pass calculates the earliest start time (EST) and earliest finish time (EFT) for each activity using maximum times from predecessors. The backward pass determines the latest finish time (LFT) and latest start time (LST) by working backwards. The total float is LFT – EFT (or LST – EST). Activities with zero total float are critical and form a critical path, which determines the minimum project duration.
关键路径分析用于项目活动的调度。活动具有时长和依赖关系。节点表示活动的网络（AON）中，每个活动是一个顶点，有向边代表先后关系。前向推断根据前置活动的最早完成时间计算每个活动的最早开始时间（EST）和最早完成时间（EFT）。后向推断反向工作，确定最晚完成时间（LFT）和最晚开始时间（LST）。总时差为 LFT – EFT（或 LST – EST）。总时差为零的活动为关键活动，它们构成关键路径，决定了项目的最短工期。
Total Float = LFT – EFT = LST – EST
In the 2022 exam, a question presented a table of activities and their immediate predecessors, asking to draw the network, perform forward and backward passes, and state the critical path and project duration. Candidates also had to comment on the effect of a delay in a non-critical activity.
在2022年试卷中，一道题给出了活动及其紧前活动的表格，要求画出网络、进行前向和后向推断，并指出关键路径和项目工期。考生还需评论非关键活动发生延迟所产生的影响。
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