D1 June 2017 Q4
4.
[The total weight of the network is 85]
Figure 4 represents a network of roads. The number on each edge represents the length, in miles, of the corresponding road. Robyn wishes to travel from A to H. She wishes to minimise the distance she travels.
On a particular day, Robyn needs to check each road. She must travel along each road at least once. Robyn must start and finish at vertex A.
The roads BD and BE become damaged and cannot be used. Robyn needs to travel along all the remaining roads to check that there is no damage to any of them. The inspection route must still start and finish at vertex A.
| Scheme | Marks |
|---|---|
| M1 A1 (DCBE) A1 (FG) A1ft (JH) | |
| Shortest path A to H: ADCBEFJH | A1 |
| Length of shortest path: 20 (miles) | A1ft |
| (6) |
Notes
In (a) 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 G the working values must be 20 18 17 in that order (20 17 18 is incorrect) and with no additional working values.It is also important that the order of labelling is checked carefully. 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 labellinga1M1: A larger value replaced by a smaller value at least once in the working values at either B or F or G or H
a1A1: All values at D, C, B and E correct and the working values in the correct order at B. Condone lack of 0 in A’s working value – please check carefully for a 5 in the working values at D
a2A1: All values at F and G correct and the working values in the correct order. Penalise order of labelling only once per question (F and G must be labelled in that order and F labelled after A, D, C, B and E)
a3A1ft: All values in J and H correct on the follow through and the working values in the correct order. Penalise order of labelling only once per question (J and H must be labelled in that order and J labelled after all other nodes (excluding H)). To follow through J check that the working value at J follows from the candidate’s final value from F and that the final value, and order of labelling, follows through correctly. Repeat this process for H (which will have working values from E and J)
a4A1: CAO for the path (from either A to H or H to A)
a5A1ft: If their answer is not 20 follow through their final value at H (ignore units)
| Scheme | Marks |
|---|---|
| AD + E(F)G = 5 + 5 = 10* A(DCB)E + D(CBEF)G = 12 + 12 = 24 A(DCBEF)G + D(CB)E = 17 + 7 = 24 | M1 A1 A1 |
| Repeat edges: AD, EF, FG | A1 |
| Length = 85 + 10 = 95 (miles) | A1ft |
| (5) |
Notes
b1M1: Three distinct pairings of A, D, E and G
b1A1: Any two rows correct including pairings and totals
b2A1: All three rows correct including pairings and totals
b3A1: CAO correct edges clearly (not just in their working) stated as AD, EF, FG. Do not accept EG, EFG or or EG via F
b4A1ft: 85 + their smallest repeat out of a choice of at least two totals seen (the correct answer of 95 must come from the correct pairing of AD and EG)
| Scheme | Marks |
|---|---|
| (i) AD, CD, BC, BG | B1 |
| (ii) Route: e.g. ADABCDCBFEHJFGBGA | B1 |
| Length = 85 + 18 – 6 – 2 = 95 (miles) | M1 A1 |
| (4) | |
| (15 marks) |
Notes
c1B1: CAO (AD, CD, BC, BG)
c2B1: Any correct route (the route may be given in terms of either vertices (ADA…) or arcs (AD, DA, AB,…)) – checks: start and finish at A, 17 vertices (repeats AD, CD, BC, BG, and nodes A, B appear three times, C, D, F, G appear twice and E, H and J appear once)
c1M1: 85 – 6 – 2 + ‘18’ (where ‘18’ is the sum of their repeated arcs from A to G in c1B1 (this may need to be checked) or a clear attempt at their shortest path from A to G) or 85 – 8 + ‘18’ or 95 or for adding the weights of all the arcs from their route (so ft their answers from c1B1 and c2B1)
c1A1: CAO of 95 including as a minimum the working 85 + 18 – 8 or explicitly adding the weights of all the arcs from a correct route given in c2B1. Note that 85 + 10 or 77 + 18 alone is not sufficient for this mark (ignore units)