| Exam Board | OCR MEI |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2005 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Graph Theory Fundamentals |
| Type | Game and interaction modeling |
| Difficulty | Moderate -0.8 This is a straightforward graph traversal exercise requiring basic tree drawing, path enumeration, and simple logical reasoning. All parts involve direct reading from the given graph with no complex problem-solving, making it easier than average A-level material despite being from Further Maths D1. |
| Spec | 7.02a Graphs: vertices (nodes) and arcs (edges) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Tree diagram drawn | M1 | Method for tree diagram |
| First branch correct | A1 | |
| Second branch correct | A1 | |
| Third branch correct | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 1 (T) 4 (H) 5 (T) 9 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 2; 4; 6; 7; 9 | M1 | At least four different values |
| A1 | Complete and no repeats |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| e.g. \(1 \rightarrow 4 \rightarrow 1 \rightarrow \ldots\) | B1 |
# Question 1:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Tree diagram drawn | M1 | Method for tree diagram |
| First branch correct | A1 | |
| Second branch correct | A1 | |
| Third branch correct | A1 | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| 1 (T) 4 (H) 5 (T) 9 | B1 | |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| 2; 4; 6; 7; 9 | M1 | At least four different values |
| | A1 | Complete and no repeats |
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| e.g. $1 \rightarrow 4 \rightarrow 1 \rightarrow \ldots$ | B1 | |
---1 The bipartite graph in Fig. 1 represents a board game for two players. At each turn a player tosses a coin and moves their counter. The graph shows which square the counter is moved to if the coin shows heads, and which square if it shows tails. Each player starts with their counter on square 1. Play continues until one player gets their counter to square 9 and wins.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{b9ee9306-18ca-42b3-9f2e-b23849374b5e-2_723_1287_569_425}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}
(i) Draw a tree to show all of the possibilities for the player's first three moves.\\
(ii) Show how a player can win in 3 turns.\\
(iii) List all squares which it is possible for a counter to occupy after 3 turns.\\
(iv) Show that a game can continue indefinitely.
\hfill \mbox{\textit{OCR MEI D1 2005 Q1 [8]}}