A2 June 2022 Q2
2.
[The total weight of the network is 299]
Figure 1 represents a network of cycle tracks between 10 landmarks, A, B, C, D, E, F, G, H, J and K. The number on each edge represents the length, in kilometres, of the corresponding track.
One day, Blanche wishes to cycle from A to K. She wishes to minimise the distance she travels.
The cycle tracks between the landmarks now need to be inspected. Blanche must travel along each track at least once. She wishes to minimise the length of her inspection route. Blanche will start her inspection route at D and finish at E.
It is now decided to start the inspection route at A and finish at K. Blanche must minimise the length of her route and travel along each track at least once.
| Scheme | Marks | AO |
|---|---|---|
(i) | M1 A1 A1 A1ft | 1.1b 1.1b 1.1b 1.1b |
| Path from A to K: ABEHGK | A1 | 2.2a |
| (ii) Length of shortest path from A to K is 59 (km) | A1ft | 2.2a |
| (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, for example at F the working values must be 29 28 25 in that order (so 29 25 28 is incorrect)
It is also important that the order of labelling is checked carefully – some candidates start with a 0 label 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. Condone crossed out working values if they are still legible
(a) M1: A larger value replaced by a smaller value in at least two different Working Value boxes at either C or F or G or H or J or K
A1: All values at A, B, E, D and C correct. Condone lack of 0 in A’s working value
A1: All values at F, H and G correct and the working values in the correct order. Penalise order of labelling only once per question (F, H and G must be labelled in that order and F must be labelled after A, B, E, D and C)
A1ft: All values in J and K correct on the follow through and the working values in the correct order. To follow through J check that the working values at J follow from the candidate’s final values for the nodes that are directly attached to J (which are G and H). For example, if correct then the order of labelling of nodes G and H are 8 and 7 respectively so the working values at J should come from H and G in that order. The first working value at J should be their 28 (the Final value at H) + 23 (the weight of the arc HJ), the second working value at J should be their 34 (the Final value at G) + 10 (the weight of the arc GJ). Repeat the process for K (which will have working values from G and H with the order of these nodes determined by the candidate’s order of labelling at G and H)
A1: cao (correct path from A to K – ABEHGK). Do not accept KGHEBA. Condone the length of the shortest path and the path interchanged on the answer lines in the answer book. Allow the paths written in terms of arcs, e.g. AB, BE, EH, HG, GK
A1ft: ft their final value at K only (if 59 stated and 59 is not the final value at K then A0)
| Scheme | Marks | AO |
|---|---|---|
| (i) Repeated edges: AB, BE, EH | B1 | 2.2a |
| (ii) Length of route = 299 + 28 = 327 (km) | B1ft | 2.2a |
| (2) |
Notes
(b)(i) B1: cao (AB, BE, EH only but in any order) – must be stated as edges so B0 for ABEH
(ii) B1ft: 327 or follow through their final value at H from (a) + 299
| Scheme | Marks | AO |
|---|---|---|
| Route must start at A and finish at K therefore need to consider pairings of the nodes D, E, H and K | M1 | 3.1b |
| D(AB)E + H(G)K = 26 + 31 = 57 D(FG)H + E(HG)K = 30 + 47 = 77 D(FG)K + EH = 49 + 16 = 65 | A1 A1 A1 | 1.1b 1.1b 1.1b |
| Length of new route = 299 + 57 = 356 (km) | A1 | 2.2a |
| (5) | ||
| (13 marks) |
Notes
(c) M1: The correct three pairings of the correct four nodes (D, E, H and K)
A1: One row correct including pairing and total
A1: Two rows correct including pairings and totals
A1: All three rows correct including pairings and totals
A1: cao (356) – dependent on all previous marks in this part
Condone lack of units throughout