AQA D1 2015 June — Question 1 5 marks

Exam BoardAQA
ModuleD1 (Decision Mathematics 1)
Year2015
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicMatchings and Allocation
TypeMaximum matching algorithm application
DifficultyModerate -0.8 This is a standard textbook application of the alternating path algorithm with an initial matching already provided. Students simply need to execute the mechanical procedure to find an augmenting path and improve the matching—no problem-solving insight or novel thinking required, just routine algorithm application.
Spec7.02e Bipartite graphs: K_{m,n} notation7.02r Graph modelling: model and solve problems using graphs

1 A quiz team must answer questions from six different topics, numbered 1 to 6. The team has six players, \(A , B , C , D , E\) and \(F\). Each player can only answer questions on one of the topics. The players list their preferred topics. The bipartite graph shows their choices. \includegraphics[max width=\textwidth, alt={}, center]{f5890e58-38c3-413c-8762-6f80ce6dcec7-02_711_499_781_760} Initially, \(A\) is allocated topic 2, \(B\) is allocated topic \(3 , C\) is allocated topic 1 and \(F\) is allocated topic 4. By using an alternating path algorithm from this initial matching, find a complete matching.
[0pt] [5 marks]

Question 1:
AnswerMarks Guidance
AnswerMarks Guidance
Initial matching: \(A-2, B-3, C-1, F-4\) stated or implied
Unmatched players: \(D\) and \(E\); unmatched topics: \(5\) and \(6\)B1 identifying unmatched vertices
Alternating path from \(D\): e.g. \(D-4-F-6\) (or \(D-3-B-...\))M1 valid alternating path attempted from unmatched player
e.g. \(D-4-F-6\) is an augmenting path; update matching: \(D-4, F-6\)A1 correct augmenting path identified and matching updated
Alternating path from \(E\): e.g. \(E-3-B-2-A-1\) or \(E-5\)M1 second valid alternating path attempted
Complete matching e.g. \(A-1, B-2, C-3, D-4, E-5, F-6\) (or equivalent valid complete matching)A1 all six players matched to distinct topics
# Question 1:

| Answer | Marks | Guidance |
|--------|-------|----------|
| Initial matching: $A-2, B-3, C-1, F-4$ | | stated or implied |
| Unmatched players: $D$ and $E$; unmatched topics: $5$ and $6$ | B1 | identifying unmatched vertices |
| Alternating path from $D$: e.g. $D-4-F-6$ (or $D-3-B-...$) | M1 | valid alternating path attempted from unmatched player |
| e.g. $D-4-F-6$ is an augmenting path; update matching: $D-4, F-6$ | A1 | correct augmenting path identified and matching updated |
| Alternating path from $E$: e.g. $E-3-B-2-A-1$ or $E-5$ | M1 | second valid alternating path attempted |
| Complete matching e.g. $A-1, B-2, C-3, D-4, E-5, F-6$ (or equivalent valid complete matching) | A1 | all six players matched to distinct topics |

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1 A quiz team must answer questions from six different topics, numbered 1 to 6. The team has six players, $A , B , C , D , E$ and $F$. Each player can only answer questions on one of the topics. The players list their preferred topics. The bipartite graph shows their choices.\\
\includegraphics[max width=\textwidth, alt={}, center]{f5890e58-38c3-413c-8762-6f80ce6dcec7-02_711_499_781_760}

Initially, $A$ is allocated topic 2, $B$ is allocated topic $3 , C$ is allocated topic 1 and $F$ is allocated topic 4. By using an alternating path algorithm from this initial matching, find a complete matching.\\[0pt]
[5 marks]

\hfill \mbox{\textit{AQA D1 2015 Q1 [5]}}