| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Discrete (Further Paper 3 Discrete) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Route Inspection |
| Type | Basic Chinese Postman (closed route) |
| Difficulty | Standard +0.8 This is a standard Chinese Postman Problem application requiring identification of odd-degree vertices, calculation of minimum repeat edges, and a straightforward mean calculation. While it tests graph theory understanding, the solution follows a well-defined algorithm with no novel insights required. The 6-mark total and routine nature place it moderately above average difficulty. |
| Spec | 7.02g Eulerian graphs: vertex degrees and traversability7.04e Route inspection: Chinese postman, pairing odd nodes |
| Answer | Marks |
|---|---|
| 5(a) | Sets up a model by identifying |
| Answer | Marks | Guidance |
|---|---|---|
| degree nodes (PI) | 3.3 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| between the odd nodes | 3.4 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| correct pairing | 1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| of their pairing | 3.2a | A1F |
| Total | 4 | |
| Q | Marking instructions | AO |
| Answer | Marks |
|---|---|
| 5(b) | Evaluates the council regulation |
| Answer | Marks | Guidance |
|---|---|---|
| required | 3.5a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| meet the new council regulation | 3.2a | R1 |
| Total | 2 | |
| Question total | 6 | |
| Q | Marking instructions | AO |
Question 5:
--- 5(a) ---
5(a) | Sets up a model by identifying
the problem as a route
inspection problem by noting
that B, D, H and J are odd-
degree nodes (PI) | 3.3 | M1 | Odd nodes: B, D, H, J
Shortest Distances
B–D: 225 H–J: 275
B–H: 425 D–J: 75
B–J: 250 D–H: 200
Pairings
(B–D)(H–J) = 500
(B–H)(D–J) = 500
(B–J)(D–H) = 450*
Minimum distance the council
worker can walk in order to count
all street lights is
2250 + 450 = 2700 m
Hence x = 2700
Uses the model to find at least
five correct shortest distances
between the odd nodes | 3.4 | M1
Uses the model to find the
correct shortest distance for the
correct pairing | 1.1b | A1
Determines correctly the value
of x using the shortest distance
of their pairing | 3.2a | A1F
Total | 4
Q | Marking instructions | AO | Marks | Typical solution
--- 5(b) ---
5(b) | Evaluates the council regulation
by considering the total distance
of all the streets in the village
OR
Evaluates the council regulation
by finding the minimum number
of street lights that would be
required | 3.5a | M1 | There are 91 street lights spread
over a total distance of 2250 m, so
there is, on average, a street light
every 24.7 m.
Hence, as 24.7 < 25, the village will
meet the new council regulation.
Correctly compares two
comparable values and
concludes that the village will
meet the new council regulation | 3.2a | R1
Total | 2
Question total | 6
Q | Marking instructions | AO | Marks | Typical solutionA council wants to convert all of the street lighting in a village to use LED lighting.
The network below shows each street in the village. Each node represents a junction and the weight of each arc represents the length, in metres, of the street.
The street lights are only positioned on one side of each street in the village.
\includegraphics{figure_6}
The total length of all of the streets in the village is 2250 metres.
In order to determine the total number of street lights in the village, a council worker is required to walk along every street in the village at least once, starting and finishing at the same junction.
The shortest possible distance the council worker can walk in order to determine the total number of street lights in the village is $x$ metres.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $x$
Fully justify your answer.
[4 marks]
\item A new council regulation requires that the mean distance along a street between adjacent LED street lights in a village be less than 25 metres.
The council worker counted 91 different street lights on their journey around the village.
Determine whether or not the village will meet the new council regulation.
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Discrete 2022 Q5 [6]}}