2025
2025
2025
OCR
OCROCR
ASAS
Level
AS
Level
Level
Further
Further
Further
Mathematics
Mathematics
Mathematics
B (MEI)
B (MEI)
B (MEI)
Y413-01
Y413-01
Y413-01
Modelling
Modelling
Modelling
with
with
Algorithms
with
Algorithms
Algorithms
Verified
Verified
Verified
Question
Question
Question
paper
paper
paper
with
with
Marking
with
Marking
Marking
Scheme
Scheme
Scheme
Attached.pdf
Attached.pdf
Attached.pdf
2025 OCR AS Level Further Mathematics B (MEI)
Y413/01 Modelling with Algorithms
Verified Question paper with Marking Scheme Attached
Oxford Cambridge and RSA
Tuesday 3 June 2025 – Afternoon AS Level
Further Mathematics B (MEI) Y413/01 Modelling
with Algorithms
Time allowed: 1 hour 15 minutes
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for Further Mathematics B
QP
(MEI)
• a scientific or graphical calculator
INSTRUCTIONS
• Use black ink. You can use an HB pencil, but only for graphs and diagrams.
• Write your answer to each question in the space provided in the Printed Answer Booklet. If
you need extra space use the lined page at the end of the Printed Answer Booklet. The
question numbers must be clearly shown.
• Fill in the boxes on the front of the Printed Answer Booklet.
• Answer all the questions.
• Where appropriate, your answer should be supported with working. Marks might be given
for using a correct method, even if your answer is wrong.
• Give your final answers to a degree of accuracy that is appropriate to the context.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 60.
• The marks for each question are shown in brackets [ ].
• This document has 8 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2025 [L/508/5556] OCR is an exempt Charity
DC (DE/SW) 358206/4 Turn over
2025
2025
OCR
OCR
ASAS
Level
Level
Further
Further
Mathematics
Mathematics
B (MEI)
B (MEI)
Y413-01
Y413-01
Modelling
Modelling
with
with
Algorithms
Algorithms
Verified
Verified
Question
Question
paper
paper
with
with
Marking
Marking
Scheme
Scheme
Attached.pdf
Attached.pdf
4/28/2026
,2025
2025
2025
OCR
OCROCR
ASAS
Level
AS
Level
Level
Further
Further
Further
Mathematics
Mathematics
Mathematics
B (MEI)
B (MEI)
B (MEI)
Y413-01
Y413-01
Y413-01
Modelling
Modelling
Modelling
with
with
Algorithms
with
Algorithms
Algorithms
Verified
Verified
Verified
Question
Question
Question
paper
paper
paper
with
with
Marking
with
Marking
Marking
Scheme
Scheme
Scheme
Attached.pdf
Attached.pdf
Attached.pdf
2
1 A network has ten vertices, A to J. The table shows the distances between each pair of vertices
for which there is a connecting arc.
A B C D E F G H I J
A 12 6 7 8 2
B 12 5 14 9
C 6 5 10 7
D 14 3 8
E 7 6 4 9
F 9 6
G 8 10 5
H 3 11
I 4 5
J 2 7 8 9 11
Apply the tabular form of Prim’s algorithm to the network, starting at A, to find a minimum
spanning tree for the network.
Your solution should contain the following.
• The order in which the arcs are selected
• The total length of the arcs in the minimum spanning tree [4]
2 The list below shows the sizes of eleven items.
28 25 19 32 18 22 3 12 20 7 5
(a) (i) Show the result of applying the first fit algorithm to pack items with the sizes listed
above into bins that have a capacity of 50. [2]
(ii) Show the result of applying the first fit decreasing algorithm to pack items with the
sizes listed above into bins that have a capacity of 50. [2]
(b) A computer takes 4.7 # 10-8 seconds to pack 50 items with sizes 100, 99, 98, … , 53, 52, 51
into bins that have a capacity of 100 using the first fit decreasing algorithm.
Calculate approximately how long it will take the same computer to pack 1 000 000 items
with sizes 2 000 000, 1 999 999, 1 999 998, … , 1 000 003, 1 000 002, 1 000 001 into bins that
have a capacity of 2 000 000 using the first fit decreasing algorithm. [2]
© OCR 2025 Y413/01 Jun25
2025
2025
OCR
OCR
ASAS
Level
Level
Further
Further
Mathematics
Mathematics
B (MEI)
B (MEI)
Y413-01
Y413-01
Modelling
Modelling
with
with
Algorithms
Algorithms
Verified
Verified
Question
Question
paper
paper
with
with
Marking
Marking
Scheme
Scheme
Attached.pdf
Attached.pdf
4/28/2026
,2025
2025
2025
OCR
OCROCR
ASAS
Level
AS
Level
Level
Further
Further
Further
Mathematics
Mathematics
Mathematics
B (MEI)
B (MEI)
B (MEI)
Y413-01
Y413-01
Y413-01
Modelling
Modelling
Modelling
with
with
Algorithms
with
Algorithms
Algorithms
Verified
Verified
Verified
Question
Question
Question
paper
paper
paper
with
with
Marking
with
Marking
Marking
Scheme
Scheme
Scheme
Attached.pdf
Attached.pdf
Attached.pdf
3
3 The diagram shows a network of roads. The number on each arc represents the length, in miles,
of the corresponding road.
E 77 F
14 24
11 10 12
65
29 34
A D G
50 10
28 8
34
C 40 H
5
B
Finley, who lives at F, needs to drive to a friend’s house at A before they travel together to a
campsite situated at C.
Finley decides to use Dijkstra’s algorithm once to find the shortest route from F to C via A.
(a) Explain why vertex A should be chosen as the starting vertex for the algorithm. [1]
(b) (i) On the copy of the network in the Printed Answer Booklet, apply Dijkstra’s algorithm
once to find the length of the shortest route from F to C via A. [5]
(ii) State the corresponding shortest route from F to C via A. [1]
4 (a) Determine the number of nodes in the complete graph with 1431 arcs. [2]
(b) In this question you must show detailed reasoning.
A simply connected graph G has eight nodes of orders 1, 1, 2, 3, 3, 4, 4 and x.
• Determine the three possible values of x.
• Use the nodes in the Printed Answer Booklet to draw an example of G for each of these
three values of x. [6]
© OCR 2025 Y413/01 Jun25 Turn over
2025
2025
OCR
OCR
ASAS
Level
Level
Further
Further
Mathematics
Mathematics
B (MEI)
B (MEI)
Y413-01
Y413-01
Modelling
Modelling
with
with
Algorithms
Algorithms
Verified
Verified
Question
Question
paper
paper
with
with
Marking
Marking
Scheme
Scheme
Attached.pdf
Attached.pdf
4/28/2026
, 2025
2025
2025
OCR
OCROCR
ASAS
Level
AS
Level
Level
Further
Further
Further
Mathematics
Mathematics
Mathematics
B (MEI)
B (MEI)
B (MEI)
Y413-01
Y413-01
Y413-01
Modelling
Modelling
Modelling
with
with
Algorithms
with
Algorithms
Algorithms
Verified
Verified
Verified
Question
Question
Question
paper
paper
paper
with
with
Marking
with
Marking
Marking
Scheme
Scheme
Scheme
Attached.pdf
Attached.pdf
Attached.pdf
4
5 The diagram represents a system of pipes through which water flows continuously from a
source S to a sink T. The weights on the arcs show the capacities of the pipes in litres per
minute.
A 17 D
43
9 E
19
32 12
5 53
11
C 17
S 28 T
15 12
23 25 18
B 7 F
(a) Explain why the maximum possible flow along DE must be less than 43 litres per minute. [1]
(b) The cut a partitions the vertices into the sets {S, A, B, C, D, F}, {E, T}.
Calculate the capacity of cut a. [1]
(c) Explain why partitioning the vertices into sets {S, A, D, E, T}, {B, C, F} does not give a cut.
[1]
An LP formulation is set up to find the maximum flow through the system.
(d) (i) Write down the required constraint in the LP formulation regarding the flow through
vertex D. [1]
(ii) Write down the required constraint in the LP formulation regarding the flow along arc
AD. [1]
© OCR 2025 Y413/01 Jun25
2025
2025
OCR
OCR
ASAS
Level
Level
Further
Further
Mathematics
Mathematics
B (MEI)
B (MEI)
Y413-01
Y413-01
Modelling
Modelling
with
with
Algorithms
Algorithms
Verified
Verified
Question
Question
paper
paper
with
with
Marking
Marking
Scheme
Scheme
Attached.pdf
Attached.pdf
4/28/2026
2025
2025
OCR
OCROCR
ASAS
Level
AS
Level
Level
Further
Further
Further
Mathematics
Mathematics
Mathematics
B (MEI)
B (MEI)
B (MEI)
Y413-01
Y413-01
Y413-01
Modelling
Modelling
Modelling
with
with
Algorithms
with
Algorithms
Algorithms
Verified
Verified
Verified
Question
Question
Question
paper
paper
paper
with
with
Marking
with
Marking
Marking
Scheme
Scheme
Scheme
Attached.pdf
Attached.pdf
Attached.pdf
2025 OCR AS Level Further Mathematics B (MEI)
Y413/01 Modelling with Algorithms
Verified Question paper with Marking Scheme Attached
Oxford Cambridge and RSA
Tuesday 3 June 2025 – Afternoon AS Level
Further Mathematics B (MEI) Y413/01 Modelling
with Algorithms
Time allowed: 1 hour 15 minutes
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for Further Mathematics B
QP
(MEI)
• a scientific or graphical calculator
INSTRUCTIONS
• Use black ink. You can use an HB pencil, but only for graphs and diagrams.
• Write your answer to each question in the space provided in the Printed Answer Booklet. If
you need extra space use the lined page at the end of the Printed Answer Booklet. The
question numbers must be clearly shown.
• Fill in the boxes on the front of the Printed Answer Booklet.
• Answer all the questions.
• Where appropriate, your answer should be supported with working. Marks might be given
for using a correct method, even if your answer is wrong.
• Give your final answers to a degree of accuracy that is appropriate to the context.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 60.
• The marks for each question are shown in brackets [ ].
• This document has 8 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2025 [L/508/5556] OCR is an exempt Charity
DC (DE/SW) 358206/4 Turn over
2025
2025
OCR
OCR
ASAS
Level
Level
Further
Further
Mathematics
Mathematics
B (MEI)
B (MEI)
Y413-01
Y413-01
Modelling
Modelling
with
with
Algorithms
Algorithms
Verified
Verified
Question
Question
paper
paper
with
with
Marking
Marking
Scheme
Scheme
Attached.pdf
Attached.pdf
4/28/2026
,2025
2025
2025
OCR
OCROCR
ASAS
Level
AS
Level
Level
Further
Further
Further
Mathematics
Mathematics
Mathematics
B (MEI)
B (MEI)
B (MEI)
Y413-01
Y413-01
Y413-01
Modelling
Modelling
Modelling
with
with
Algorithms
with
Algorithms
Algorithms
Verified
Verified
Verified
Question
Question
Question
paper
paper
paper
with
with
Marking
with
Marking
Marking
Scheme
Scheme
Scheme
Attached.pdf
Attached.pdf
Attached.pdf
2
1 A network has ten vertices, A to J. The table shows the distances between each pair of vertices
for which there is a connecting arc.
A B C D E F G H I J
A 12 6 7 8 2
B 12 5 14 9
C 6 5 10 7
D 14 3 8
E 7 6 4 9
F 9 6
G 8 10 5
H 3 11
I 4 5
J 2 7 8 9 11
Apply the tabular form of Prim’s algorithm to the network, starting at A, to find a minimum
spanning tree for the network.
Your solution should contain the following.
• The order in which the arcs are selected
• The total length of the arcs in the minimum spanning tree [4]
2 The list below shows the sizes of eleven items.
28 25 19 32 18 22 3 12 20 7 5
(a) (i) Show the result of applying the first fit algorithm to pack items with the sizes listed
above into bins that have a capacity of 50. [2]
(ii) Show the result of applying the first fit decreasing algorithm to pack items with the
sizes listed above into bins that have a capacity of 50. [2]
(b) A computer takes 4.7 # 10-8 seconds to pack 50 items with sizes 100, 99, 98, … , 53, 52, 51
into bins that have a capacity of 100 using the first fit decreasing algorithm.
Calculate approximately how long it will take the same computer to pack 1 000 000 items
with sizes 2 000 000, 1 999 999, 1 999 998, … , 1 000 003, 1 000 002, 1 000 001 into bins that
have a capacity of 2 000 000 using the first fit decreasing algorithm. [2]
© OCR 2025 Y413/01 Jun25
2025
2025
OCR
OCR
ASAS
Level
Level
Further
Further
Mathematics
Mathematics
B (MEI)
B (MEI)
Y413-01
Y413-01
Modelling
Modelling
with
with
Algorithms
Algorithms
Verified
Verified
Question
Question
paper
paper
with
with
Marking
Marking
Scheme
Scheme
Attached.pdf
Attached.pdf
4/28/2026
,2025
2025
2025
OCR
OCROCR
ASAS
Level
AS
Level
Level
Further
Further
Further
Mathematics
Mathematics
Mathematics
B (MEI)
B (MEI)
B (MEI)
Y413-01
Y413-01
Y413-01
Modelling
Modelling
Modelling
with
with
Algorithms
with
Algorithms
Algorithms
Verified
Verified
Verified
Question
Question
Question
paper
paper
paper
with
with
Marking
with
Marking
Marking
Scheme
Scheme
Scheme
Attached.pdf
Attached.pdf
Attached.pdf
3
3 The diagram shows a network of roads. The number on each arc represents the length, in miles,
of the corresponding road.
E 77 F
14 24
11 10 12
65
29 34
A D G
50 10
28 8
34
C 40 H
5
B
Finley, who lives at F, needs to drive to a friend’s house at A before they travel together to a
campsite situated at C.
Finley decides to use Dijkstra’s algorithm once to find the shortest route from F to C via A.
(a) Explain why vertex A should be chosen as the starting vertex for the algorithm. [1]
(b) (i) On the copy of the network in the Printed Answer Booklet, apply Dijkstra’s algorithm
once to find the length of the shortest route from F to C via A. [5]
(ii) State the corresponding shortest route from F to C via A. [1]
4 (a) Determine the number of nodes in the complete graph with 1431 arcs. [2]
(b) In this question you must show detailed reasoning.
A simply connected graph G has eight nodes of orders 1, 1, 2, 3, 3, 4, 4 and x.
• Determine the three possible values of x.
• Use the nodes in the Printed Answer Booklet to draw an example of G for each of these
three values of x. [6]
© OCR 2025 Y413/01 Jun25 Turn over
2025
2025
OCR
OCR
ASAS
Level
Level
Further
Further
Mathematics
Mathematics
B (MEI)
B (MEI)
Y413-01
Y413-01
Modelling
Modelling
with
with
Algorithms
Algorithms
Verified
Verified
Question
Question
paper
paper
with
with
Marking
Marking
Scheme
Scheme
Attached.pdf
Attached.pdf
4/28/2026
, 2025
2025
2025
OCR
OCROCR
ASAS
Level
AS
Level
Level
Further
Further
Further
Mathematics
Mathematics
Mathematics
B (MEI)
B (MEI)
B (MEI)
Y413-01
Y413-01
Y413-01
Modelling
Modelling
Modelling
with
with
Algorithms
with
Algorithms
Algorithms
Verified
Verified
Verified
Question
Question
Question
paper
paper
paper
with
with
Marking
with
Marking
Marking
Scheme
Scheme
Scheme
Attached.pdf
Attached.pdf
Attached.pdf
4
5 The diagram represents a system of pipes through which water flows continuously from a
source S to a sink T. The weights on the arcs show the capacities of the pipes in litres per
minute.
A 17 D
43
9 E
19
32 12
5 53
11
C 17
S 28 T
15 12
23 25 18
B 7 F
(a) Explain why the maximum possible flow along DE must be less than 43 litres per minute. [1]
(b) The cut a partitions the vertices into the sets {S, A, B, C, D, F}, {E, T}.
Calculate the capacity of cut a. [1]
(c) Explain why partitioning the vertices into sets {S, A, D, E, T}, {B, C, F} does not give a cut.
[1]
An LP formulation is set up to find the maximum flow through the system.
(d) (i) Write down the required constraint in the LP formulation regarding the flow through
vertex D. [1]
(ii) Write down the required constraint in the LP formulation regarding the flow along arc
AD. [1]
© OCR 2025 Y413/01 Jun25
2025
2025
OCR
OCR
ASAS
Level
Level
Further
Further
Mathematics
Mathematics
B (MEI)
B (MEI)
Y413-01
Y413-01
Modelling
Modelling
with
with
Algorithms
Algorithms
Verified
Verified
Question
Question
paper
paper
with
with
Marking
Marking
Scheme
Scheme
Attached.pdf
Attached.pdf
4/28/2026
