| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2022 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Route Inspection |
| Type | Basic Chinese Postman (closed route) |
| Difficulty | Moderate -0.8 This is a straightforward Chinese Postman Problem application requiring identification of odd vertices, pairing them optimally, and adding the shortest path between them to the total weight. The network is small, the odd vertices are easily identified, and the algorithm is mechanical with minimal problem-solving required—well below average difficulty for A-level. |
| Spec | 7.04e Route inspection: Chinese postman, pairing odd nodes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Pair the odd nodes: C, D or repeated arcs are CF, FG, DG | B1 | Must state C and D are odd or state arcs CF, FG, DG; B0 if only stating C and D or CD without mention of 'odd' |
| Time \(= 82 + 7 = 89\) | B1 | cao (89) |
| e.g. route GDGJHEADCABEFBCFCGFG | B1 | Correct route: starts and finishes at G, 20 nodes, CF, FG and DG repeated, A(2), B(2), C(3), D(2), E(2), F(3), G(4), H(1), J(1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(BC + DG = B(F)C + DG = 6 + 3 = 9\)* | M1 A1 | Correct three distinct pairings of correct four odd nodes B, C, D and G |
| \(BD + CG = B(FG)D + C(F)G = 11 + 4 = 15\) | A1 | Any two rows correct including pairings and totals |
| \(BG + CD = B(F)G + C(FG)D = 8 + 7 = 15\) | A1 | All three rows correct including pairings and totals |
| Repeat arcs: BF, CF, DG | A1 | cao – correct arcs clearly stated; do not accept BC, BFC or BC via F |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Route starting from G is quicker | B1 | cao (oe e.g. B to G is slower) – dependent on correct repeat arcs in (a) and (b) or clearly implied in (c) |
| e.g. difference \(= (82 + 9) - 89 = 2\) or \(9 - 7 = 2\) | B1 | cao (difference of 2 or comparing 89 and 91 or comparing 7 with 9) |
# Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Pair the odd nodes: C, D **or** repeated arcs are CF, FG, DG | B1 | Must state C and D are odd **or** state arcs CF, FG, DG; B0 if only stating C and D or CD without mention of 'odd' |
| Time $= 82 + 7 = 89$ | B1 | cao (89) |
| e.g. route GDGJHEADCABEFBCFCGFG | B1 | Correct route: starts and finishes at G, 20 nodes, CF, FG and DG repeated, A(2), B(2), C(3), D(2), E(2), F(3), G(4), H(1), J(1) |
# Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $BC + DG = B(F)C + DG = 6 + 3 = 9$* | M1 A1 | Correct three distinct pairings of correct four odd nodes B, C, D and G |
| $BD + CG = B(FG)D + C(F)G = 11 + 4 = 15$ | A1 | Any two rows correct including pairings **and** totals |
| $BG + CD = B(F)G + C(FG)D = 8 + 7 = 15$ | A1 | All three rows correct including pairings **and** totals |
| Repeat arcs: BF, CF, DG | A1 | cao – correct arcs clearly stated; do not accept BC, BFC or BC via F |
# Question 5(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Route starting from G is quicker | B1 | cao (oe e.g. B to G is slower) – dependent on correct repeat arcs in (a) and (b) **or** clearly implied in (c) |
| e.g. difference $= (82 + 9) - 89 = 2$ or $9 - 7 = 2$ | B1 | cao (difference of 2 **or** comparing 89 and 91 **or** comparing 7 with 9) |
---5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{765ea64e-d4b8-4f0f-9a43-2619f9db0c18-12_705_1237_239_411}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}
[The total weight of the network is 82]
\section*{Question 5 continued}
\hfill \mbox{\textit{Edexcel D1 2022 Q5 [10]}}