| Exam Board | OCR MEI |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Graph Theory Fundamentals |
| Type | Abstract shape and algorithm modeling |
| Difficulty | Moderate -0.3 This D1 question tests understanding of how set relationships translate to graph representations. Part (i) requires straightforward translation from Venn diagram to graphs (routine application), while part (ii) requires reverse translation from graphs to Venn diagram (slightly more challenging but still mechanical). The concepts are foundational and the question follows a standard pattern with no novel problem-solving required, making it slightly easier than average. |
| Spec | 7.01b Set notation: basic language and notation of sets, partitions7.02a Graphs: vertices (nodes) and arcs (edges)7.02b Graph terminology: tree, simple, connected, simply connected |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Directed graph on 3 vertices: "is a subset of": \(X \rightarrow Y\), arrow to Z | M1 | Directed graph on 3 vertices |
| All correct | A1 | |
| Undirected graph on 3 vertices: "shares at least one element with": \(X \leftrightarrow Y\), edge to Z | M1 | Arcs must either have an arrow at each end, or no arrows. |
| All correct | A1 | |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| R subset of Q, no other subsets (Venn diagram with R inside Q, P separate) | M1, A1 | Allow area split in two, with third area. If P and R shown intersecting then can score M1 A1 B0 B0. |
| \(P \cap Q\) | B1 | |
| \(P \cap Q'\) | B1 | |
| [4] |
# Question 3:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Directed graph on 3 vertices: "is a subset of": $X \rightarrow Y$, arrow to Z | M1 | Directed graph on 3 vertices |
| All correct | A1 | |
| Undirected graph on 3 vertices: "shares at least one element with": $X \leftrightarrow Y$, edge to Z | M1 | Arcs must either have an arrow at each end, or no arrows. |
| All correct | A1 | |
| **[4]** | | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| R subset of Q, no other subsets (Venn diagram with R inside Q, P separate) | M1, A1 | Allow area split in two, with third area. If P and R shown intersecting then can score M1 A1 B0 B0. |
| $P \cap Q$ | B1 | |
| $P \cap Q'$ | B1 | |
| **[4]** | | |
---3 The diagram shows three sets, A, B and C. Each region of the diagram contains at least one element. The diagram shows that B is a subset of $\mathrm { A } , \mathrm { C }$ is a subset of A , and that B shares at least one element with C .\\
\includegraphics[max width=\textwidth, alt={}, center]{7330fba5-720f-47d5-a2ac-ad24e0cf097b-3_410_615_342_726}
The two graphs below give information about the three sets $\mathrm { A } , \mathrm { B }$ and C . The first graph shows the relation 'is a subset of' and the second graph shows the relation 'shares at least one element with'.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{7330fba5-720f-47d5-a2ac-ad24e0cf097b-3_195_261_977_621}
\captionsetup{labelformat=empty}
\caption{'is a subset of'}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{7330fba5-720f-47d5-a2ac-ad24e0cf097b-3_195_257_977_1155}
\captionsetup{labelformat=empty}
\caption{'shares at least one element with'}
\end{center}
\end{figure}
(i) Draw two graphs to represent the sets $\mathrm { X } , \mathrm { Y }$ and Z shown in the following diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{7330fba5-720f-47d5-a2ac-ad24e0cf097b-3_415_613_1388_731}\\
(ii) Draw a diagram to represent the sets $\mathrm { P } , \mathrm { Q }$ and R for which both of the following graphs apply.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{7330fba5-720f-47d5-a2ac-ad24e0cf097b-3_202_264_1980_621}
\captionsetup{labelformat=empty}
\caption{'is a subset of'}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{7330fba5-720f-47d5-a2ac-ad24e0cf097b-3_200_260_1982_1155}
\captionsetup{labelformat=empty}
\caption{'shares at least one element with'}
\end{center}
\end{figure}
\hfill \mbox{\textit{OCR MEI D1 2012 Q3 [8]}}