| Exam Board | OCR |
|---|---|
| Module | Further Discrete (Further Discrete) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Graph Theory Fundamentals |
| Type | Game and interaction modeling |
| Difficulty | Standard +0.8 This question requires understanding of Venn diagrams, graph construction, and careful application of inclusion-exclusion principle with multiple sets. Part (c) involves complex set arithmetic with overlapping conditions across 5 sets, requiring systematic tracking of intersections. While the techniques are standard Further Maths content, the multi-step reasoning and potential for arithmetic errors elevate it above average difficulty. |
| Spec | 7.01k Inclusion-exclusion: for two sets7.01l Inclusion-exclusion: extended to more than two sets7.02a Graphs: vertices (nodes) and arcs (edges) |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (a) | A 12 B |
| Answer | Marks |
|---|---|
| D 5 C | M1 |
| Answer | Marks |
|---|---|
| [3] | 3.3 |
| Answer | Marks |
|---|---|
| 1.1 | Graph K with vertices labelled A, B, C, D (in any order) |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (b) | e.g. no parties attended by only one these people |
| [1] | 3.5b | An appropriate reference to the number of parties attended by 0, 1, 3 |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (c) | (8 + 7 + 5 + 8) – (5 + 2 + 3 + 3 + 4 ) + (1 + 1) |
| Answer | Marks |
|---|---|
| = 13 | M1 |
| A1 | 1.1 |
| 1.1 | Evidence of using inclusion-exclusion with E |
| Answer | Marks | Guidance |
|---|---|---|
| 23 = (16+15+6+13+n(E)) – (12+2+6+6+5+8+7 | M1 | Using inclusion-exclusion for all 5 sets |
| Answer | Marks |
|---|---|
| n(E) = 13 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| SC B1 | SC B1 | 13 from a Venn diagram showing how many attended with E, but |
| Answer | Marks |
|---|---|
| 13 from valid working | 13 from a Venn diagram showing how many attended with E, but |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | -2 | 1 |
| 1 | 0 | 5 |
| 3 | 8 |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | 2 | |
| 3 | 0 | 0 |
Question 1:
1 | (a) | A 12 B
2 6
6 0
D 5 C | M1
M1
A1
[3] | 3.3
3.4
1.1 | Graph K with vertices labelled A, B, C, D (in any order)
4
K with any correct non-zero arc weight
4
Fully correct graph with all six arc weights correct
For reference
1 | (b) | e.g. no parties attended by only one these people | B1
[1] | 3.5b | An appropriate reference to the number of parties attended by 0, 1, 3
or 4 of A, B, C, D OR the total number of parties
1 | (c) | (8 + 7 + 5 + 8) – (5 + 2 + 3 + 3 + 4 ) + (1 + 1)
= 28 – 17 + 2
= 13 | M1
A1 | 1.1
1.1 | Evidence of using inclusion-exclusion with E
May also show 0’s
From correct working
Alternative method 1
23 = (16+15+6+13+n(E)) – (12+2+6+6+5+8+7 | M1 | Using inclusion-exclusion for all 5 sets | Using inclusion-exclusion for all 5 sets
+5+8) + (3+1+5+2+3+3+4) – (1+1)
23 = (50+n(E)) – 59 + 21 – 2
n(E) = 13 | A1
Alternative method 2
SC B1 | SC B1 | 13 from a Venn diagram showing how many attended with E, but
without evidence of using inclusion-exclusion
Or 4 + 1 + 1 + 1 + 1 + 2 + 3 = 13
13 from valid working | 13 from a Venn diagram showing how many attended with E, but
without evidence of using inclusion-exclusion
Or 4 + 1 + 1 + 1 + 1 + 2 + 3 = 13
13 from valid working
[2]
1 | -2 | 1 | 0 | 0 | 0 | 0 | 0
1 | 0 | 5
3 | 8
−
3 | 2
3 | 0 | 0 | 161 Four children, A, B, C and D, discuss how many of the 23 birthday parties held by their classmates they had gone to. Each party was attended by at least one of the four children.
The results are shown in the Venn diagram below.\\
\includegraphics[max width=\textwidth, alt={}, center]{50697293-6cdc-475f-981f-71a351b0ff5a-2_387_618_589_246}
\begin{enumerate}[label=(\alph*)]
\item Construct a complete graph $\mathrm { K } _ { 4 }$, with vertices representing the four children and arcs weighted to show the number of parties each pair of children went to.
\item State a piece of information about the number of parties the children went to that is shown in the Venn diagram but is not shown in the graph.
A fifth child, E, also went to some of the 23 parties shown in the Venn diagram.\\
Every party that E went to was also attended by at least one of $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D .
\begin{itemize}
\item A was at 8 of these parties, B at 7, C at 5 and D at 8 .
\item These include 5 parties attended by both A and $\mathrm { B } , 2$ by both A and $\mathrm { C } , 3$ by both A and $\mathrm { D } , 3$ by both B and D and 4 by both C and D .
\item These include 1 party attended by $\mathrm { A } , \mathrm { B }$ and D and 1 party attended by $\mathrm { A } , \mathrm { C }$ and D .
\item Use the inclusion-exclusion principle to determine the number of parties that E went to.
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR Further Discrete 2022 Q1 [6]}}