AS June 2024 Q3
3.
[The total weight of the network is \(139 + x + y\)]
Figure 3 represents a network of walkways in a warehouse.
The arcs represent the walkways and the nodes represent junctions between them.
The number on each arc represents the length, in metres, of the corresponding walkway.
The values \(x\) and \(y\) are unknown, however it is known that \(x\) and \(y\) are integers and that
\[9 \lt x \lt y \lt 14\]The warehouse manager wants to check that all of the walkways are in good condition.
Their inspection route starts at B and finishes at C.
The inspection route must traverse each walkway at least once and be as short as possible.
The warehouse manager finds that the total length of the inspection route is 172 metres.
| Scheme | Marks | AO |
|---|---|---|
| A tree is a connected graph with no cycles | B1 | 1.2 |
| (1) |
Notes
B1: Must state connected graph and must state no cycles. Paths and/or loops are NOT correct, ignore any extras unless incorrect.
| Scheme | Marks | AO |
|---|---|---|
(i) | M1 A1 (A,B,C, E,F,K) A1 (D,J,H) A1ft (G,L,M) | 1.1b 1.1b 1.1b 1.1b |
| Shortest route from A to M is AEJHLM | A1 | 2.2a |
| (ii) Length of shortest route is \(x + 15\) | A1ft | 2.2a |
| (6) |
Notes
In (b) it is important that all values at each node are checked very carefully – the order of the working values must be correct for the corresponding A mark to be awarded e.g. at H the working values must be 22 21 \(x+7\) in that order (so \(x+7\) 22 21 is incorrect).
It is also important that the order of labelling is checked carefully – some candidates start with a label of 0 at A (rather than 1) – which is fine. Also the order of labelling must be a strictly increasing sequence – so 1, 2, 3, 3, 4, … will be penalised once (see notes below) but 1, 2, 3, 5, 6, … is fine. Errors in the final values and working values are penalised before errors in the order of labelling.
M1: A working value in at least three of the boxes and a larger numerical value replaced by a smaller numerical value in at least two of C, F, G and H.
A1: All values in A, B, C, E, F and K correct. Condone lack of 0 in A’s working value. Penalise order of labelling only once per question. (A, B, C, E, F and K must be labelled in that order).
A1: All values D, J, H correct and the working values in the correct order. (For the labelling, check that D, J and H are labelled in that order and D is labelled before K).
A1ft: All values/expressions in G, L and M correct on the follow through and the working values in the correct order. To follow through M check that the working value at M follows from the candidate’s final values from their feeds into M (which will come from nodes G, K and L (in the order in which the candidate has labelled them)) and that the final value, and order of labelling, follows through correctly. Note that an additional working value of \(x+23\) after the \(x+15\) is not an error so \(x+15\) \(x+23\) is fine, however, any other number or \(x+23\) \(x+15\) in this order is incorrect and scores A0 in this part
Do not award ft if candidate has used numerical values, the final answer must be an expression.
A1: CAO - correct path from A to M (AEJHLM)
A1ft: ft their final value at M only (must be an expression).
| Scheme | Marks | AO |
|---|---|---|
| Arcs HL and LM need to be traversed twice | B1 | 1.1b |
| (1) |
Notes
B1: CAO (arcs HL and LM NOT H(L)M) do NOT accept if extra arcs included.
| Scheme | Marks | AO |
|---|---|---|
| H would appear 3 times | B1 | 2.2a |
| (1) |
Notes
B1: CAO (3 times) NOT dependent on the correct repeated arcs in part (c)
| Scheme | Marks | AO |
|---|---|---|
| \(139 + x + y + 5 + 3 = 172\) | M1 | 1.1b |
| \(x + y = 25\) \(x = 12,\ y = 13\) | A1 | 2.2a |
| (2) | ||
| (11 marks) |
Notes
M1: setting the total weight of the network (\(139 + x + y\)) plus 8 (or their repeated arcs) equal to 172
A1: CAO both values clearly assigned; \(x\)=12, \(y\)=13. Dependent on the correct repeated arcs [accept as a minimum sight of H and M and 8 (5+3) here] in part (c)
If you just see \(x\)=12, \(y\)=13 stated with no working, M0A0.