Pearson Edexcel Level 3 GCE
Monday 22 June 2026
Afternoon (Time: 1 hour 30 minutes)
Paper
reference 9FM0/4D
Further Mathematics
Advanced
PAPER 4D: Decision Mathematics 2 Mark Scheme (Results)
You must have:
Mathematical Formulae and Statistical Tables (Green), calculator,
Decision Mathematics Answer Book (enclosed)
Summer 2026
Candidates may use any calculator permitted by Pearson regulations. Calculators
must not have the facility for symbolic algebra manipulation, differentiation and
integration, or have retrievable mathematical formulae stored in them.
Instructions
•• Use black ink or ball-point pen.
If pencil is used for diagrams/sketches/graphs it must be dark (HB or B).
• Write your answers for this paper in the Decision Mathematics answer
book provided.
Pearson Edexcel GCE
• Fill in the boxes at the top of the answer book with your name, centre number
and candidate number. In Further Mathematics (9FM0)
•• Answer
Do not return the question paper with the answer book.
all questions and ensure your answers to parts of questions are Paper 4D Decision Mathematics 2
clearly labelled.
• Answer the questions in the answer book provided
– there may be more space than you need.
• You should show sufficient working to make your methods clear. Answers without
working may not gain full credit.
• Inexact
stated.
answers should be given to three significant figures unless otherwise
Information
•• AThere
booklet ‘Mathematical Formulae and Statistical Tables’ is provided.
are 8 questions in this question paper. The total mark for this paper is 75.
• – use this asfora guide
The marks each question are shown in brackets
as to how much time to spend on each question.
Advice
•• Read each question carefully before you start to answer it.
Try to answer every question.
• Check your answers if you have time at the end. Turn over
P79406APearson Education Ltd.
P:1/1/1/1/1/1/
*P79406A* for more:
for more:
, Edexcel and BTEC Qualifications
1. Five workers, A, B, C, D and E, are to be assigned to four tasks, P, Q, R and S. Each
task must be assigned to just one worker and each worker may do at most one task.
• Worker A cannot do task R
• Worker C cannot do task Q Edexcel and BTEC qualifications are awarded by Pearson, the UK’s
largest awarding body. We provide a wide range of qualifications
• Worker E cannot do task S including academic, vocational, occupational and specific
The cost, in pounds, of assigning each worker to each task is shown in the table. programmes for employers. For further information visit our
qualifications websites at or .
The total cost is to be minimised. Alternatively, you can get in touch with us using the details on our
contact us page at
P Q R S
A 121 143 - 132
B
C 128 - 151 150
Pearson: helping people progress, everywhere
D
Pearson aspires to be the world’s leading learning company. Our aim is to help
E - everyone progress in their lives through education. We believe in every kind of
learning, for all kinds of people, wherever they are in the world. We’ve been involved
(a) Modify the table so that the Hungarian Algorithm can be used to obtain an allocation in education for over 150 years, and by working across 70 countries, in 100
with minimum total cost. languages, we have built an international reputation for our commitment to high
(2)
standards and raising achievement through innovation in education. Find out more
(b) Use the Hungarian Algorithm to obtain an allocation with minimum total cost. about how we can help you and your students at:
(5)
State the minimum total cost.
(1)
(Total for Question 1 is 8 marks)
Summer 2026
Question Paper Log Number P79406A
Publication Code 9FM0_4D_2606_MS
All the material in this publication is
Pearson Education Ltd
for more:
2
for more:
9406A
,2.
General Marking Guidance
Player Y
• All candidates must receive the same treatment. Examiners must mark the
P Q R first candidate in exactly the same way as they mark the last.
A 4 1 −3 • Mark schemes should be applied positively. Candidates must be rewarded for
what they have shown they can do rather than penalised for omissions.
B −3 −4 4
• Examiners should mark according to the mark scheme not according to their
Player X
C 3 −1 2 perception of where the grade boundaries may lie.
• There is no ceiling on achievement. All marks on the mark scheme should be
D −1 −2 3
used appropriately.
Two players, X and Y, are playing a zero-sum game which does not have a stable • All the marks on the mark scheme are designed to be awarded. Examiners
solution. The table shows the pay-off matrix for Player X. should always award full marks if deserved, i.e. if the answer matches the mark
(a) Explain why Player Y should never play option P. scheme. Examiners should also be prepared to award zero marks if the
(2) candidate’s response is not worthy of credit according to the mark scheme.
(b) Rewrite the pay-off matrix for the reduced game for Player Y. • Where some judgement is required, mark schemes will provide the principles
(1)
by which marks will be awarded and exemplification may be limited.
Use a graphical method to determine the optimum strategy for Player Y, defining • When examiners are in doubt regarding the application of the mark scheme to
any variables that you use.
(6) a candidate’s response, the team leader must be consulted.
• Crossed out work should be marked UNLESS the candidate has replaced it
(d) State the value of the game to Player X.
(1) with an alternative response.
(Total for Question 2 is 10 marks)
for more:
P79406A for more:
3
Turn over
, EDEXCEL GCE MATHEMATICS
3. General Instructions for Marking
C1 18 (5, 20) 1. The total number of marks for the paper is 75.
A F
4 (3, 18) (4, 17) 6 2. The Edexcel Mathematics mark schemes use the following types of marks:
(12, 34)
(8, 30) 26 4 (4, 14) • M marks: method marks are awarded for ‘knowing a method and attempting
(6, 12)
D 10 (3, 10) to apply it’, unless otherwise indicated.
• A marks: Accuracy marks can only be awarded if the relevant method (M)
12 (5, 12)
B (10, 22) marks have been earned.
S G
6 (2, 7) T • B marks are unconditional accuracy marks (independent of M marks)
(6, 24) 18 (5, 11) 11 • Marks should not be subdivided.
(4, 13)
(10, 25) 25 3. Abbreviations
4 (4, 16) E 3 (3, 10) (8, 26)
2 (2, 14) These are some of the traditional marking abbreviations that will appear in the mark
schemes.
C H • bod – benefit of doubt
(16, 27)
• ft – follow through
C1
• the symbol will be used for correct ft
The diagram shows a directed capacitated network. • cao – correct answer only
The numbers in brackets on each arc show the lower and upper capacities of the • cso - correct solution only. There must be no errors in this part of the
corresponding arc. question to obtain this mark
The numbers in circles represent a feasible flow from S to T. • isw – ignore subsequent working
• awrt – answers which round to
(a) State the value of the feasible flow.
(1) • SC: special case
• oe – or equivalent (and appropriate)
(b) Explain why the flow in HT can never be greater than 20 • dep – dependent
(2)
• indep – independent
Calculate the value of the cut C1 • dp – decimal places
(1) • sf – significant figures
• The answer is printed on the paper
(d) By inspection, increase the flow by exactly 4 units. You must state the
flow-augmenting routes used and their values. • The second mark is dependent on gaining the first mark
(1)
4. For misreading which does not alter the character of a question or materially
(e) Prove that the flow found in part (d) is maximal.
simplify it, deduct two from any A or B marks gained, in that part of the question
(3)
affected.
(Total for Question 3 is 8 marks)
5. Where a candidate has made multiple responses and indicates which response
they wish to submit, examiners should mark this response.
If there are several attempts at a question which have not been crossed out,
examiners should mark the final answer which is the answer that is the most
complete.
for more:
4
for more:
9406A
Monday 22 June 2026
Afternoon (Time: 1 hour 30 minutes)
Paper
reference 9FM0/4D
Further Mathematics
Advanced
PAPER 4D: Decision Mathematics 2 Mark Scheme (Results)
You must have:
Mathematical Formulae and Statistical Tables (Green), calculator,
Decision Mathematics Answer Book (enclosed)
Summer 2026
Candidates may use any calculator permitted by Pearson regulations. Calculators
must not have the facility for symbolic algebra manipulation, differentiation and
integration, or have retrievable mathematical formulae stored in them.
Instructions
•• Use black ink or ball-point pen.
If pencil is used for diagrams/sketches/graphs it must be dark (HB or B).
• Write your answers for this paper in the Decision Mathematics answer
book provided.
Pearson Edexcel GCE
• Fill in the boxes at the top of the answer book with your name, centre number
and candidate number. In Further Mathematics (9FM0)
•• Answer
Do not return the question paper with the answer book.
all questions and ensure your answers to parts of questions are Paper 4D Decision Mathematics 2
clearly labelled.
• Answer the questions in the answer book provided
– there may be more space than you need.
• You should show sufficient working to make your methods clear. Answers without
working may not gain full credit.
• Inexact
stated.
answers should be given to three significant figures unless otherwise
Information
•• AThere
booklet ‘Mathematical Formulae and Statistical Tables’ is provided.
are 8 questions in this question paper. The total mark for this paper is 75.
• – use this asfora guide
The marks each question are shown in brackets
as to how much time to spend on each question.
Advice
•• Read each question carefully before you start to answer it.
Try to answer every question.
• Check your answers if you have time at the end. Turn over
P79406APearson Education Ltd.
P:1/1/1/1/1/1/
*P79406A* for more:
for more:
, Edexcel and BTEC Qualifications
1. Five workers, A, B, C, D and E, are to be assigned to four tasks, P, Q, R and S. Each
task must be assigned to just one worker and each worker may do at most one task.
• Worker A cannot do task R
• Worker C cannot do task Q Edexcel and BTEC qualifications are awarded by Pearson, the UK’s
largest awarding body. We provide a wide range of qualifications
• Worker E cannot do task S including academic, vocational, occupational and specific
The cost, in pounds, of assigning each worker to each task is shown in the table. programmes for employers. For further information visit our
qualifications websites at or .
The total cost is to be minimised. Alternatively, you can get in touch with us using the details on our
contact us page at
P Q R S
A 121 143 - 132
B
C 128 - 151 150
Pearson: helping people progress, everywhere
D
Pearson aspires to be the world’s leading learning company. Our aim is to help
E - everyone progress in their lives through education. We believe in every kind of
learning, for all kinds of people, wherever they are in the world. We’ve been involved
(a) Modify the table so that the Hungarian Algorithm can be used to obtain an allocation in education for over 150 years, and by working across 70 countries, in 100
with minimum total cost. languages, we have built an international reputation for our commitment to high
(2)
standards and raising achievement through innovation in education. Find out more
(b) Use the Hungarian Algorithm to obtain an allocation with minimum total cost. about how we can help you and your students at:
(5)
State the minimum total cost.
(1)
(Total for Question 1 is 8 marks)
Summer 2026
Question Paper Log Number P79406A
Publication Code 9FM0_4D_2606_MS
All the material in this publication is
Pearson Education Ltd
for more:
2
for more:
9406A
,2.
General Marking Guidance
Player Y
• All candidates must receive the same treatment. Examiners must mark the
P Q R first candidate in exactly the same way as they mark the last.
A 4 1 −3 • Mark schemes should be applied positively. Candidates must be rewarded for
what they have shown they can do rather than penalised for omissions.
B −3 −4 4
• Examiners should mark according to the mark scheme not according to their
Player X
C 3 −1 2 perception of where the grade boundaries may lie.
• There is no ceiling on achievement. All marks on the mark scheme should be
D −1 −2 3
used appropriately.
Two players, X and Y, are playing a zero-sum game which does not have a stable • All the marks on the mark scheme are designed to be awarded. Examiners
solution. The table shows the pay-off matrix for Player X. should always award full marks if deserved, i.e. if the answer matches the mark
(a) Explain why Player Y should never play option P. scheme. Examiners should also be prepared to award zero marks if the
(2) candidate’s response is not worthy of credit according to the mark scheme.
(b) Rewrite the pay-off matrix for the reduced game for Player Y. • Where some judgement is required, mark schemes will provide the principles
(1)
by which marks will be awarded and exemplification may be limited.
Use a graphical method to determine the optimum strategy for Player Y, defining • When examiners are in doubt regarding the application of the mark scheme to
any variables that you use.
(6) a candidate’s response, the team leader must be consulted.
• Crossed out work should be marked UNLESS the candidate has replaced it
(d) State the value of the game to Player X.
(1) with an alternative response.
(Total for Question 2 is 10 marks)
for more:
P79406A for more:
3
Turn over
, EDEXCEL GCE MATHEMATICS
3. General Instructions for Marking
C1 18 (5, 20) 1. The total number of marks for the paper is 75.
A F
4 (3, 18) (4, 17) 6 2. The Edexcel Mathematics mark schemes use the following types of marks:
(12, 34)
(8, 30) 26 4 (4, 14) • M marks: method marks are awarded for ‘knowing a method and attempting
(6, 12)
D 10 (3, 10) to apply it’, unless otherwise indicated.
• A marks: Accuracy marks can only be awarded if the relevant method (M)
12 (5, 12)
B (10, 22) marks have been earned.
S G
6 (2, 7) T • B marks are unconditional accuracy marks (independent of M marks)
(6, 24) 18 (5, 11) 11 • Marks should not be subdivided.
(4, 13)
(10, 25) 25 3. Abbreviations
4 (4, 16) E 3 (3, 10) (8, 26)
2 (2, 14) These are some of the traditional marking abbreviations that will appear in the mark
schemes.
C H • bod – benefit of doubt
(16, 27)
• ft – follow through
C1
• the symbol will be used for correct ft
The diagram shows a directed capacitated network. • cao – correct answer only
The numbers in brackets on each arc show the lower and upper capacities of the • cso - correct solution only. There must be no errors in this part of the
corresponding arc. question to obtain this mark
The numbers in circles represent a feasible flow from S to T. • isw – ignore subsequent working
• awrt – answers which round to
(a) State the value of the feasible flow.
(1) • SC: special case
• oe – or equivalent (and appropriate)
(b) Explain why the flow in HT can never be greater than 20 • dep – dependent
(2)
• indep – independent
Calculate the value of the cut C1 • dp – decimal places
(1) • sf – significant figures
• The answer is printed on the paper
(d) By inspection, increase the flow by exactly 4 units. You must state the
flow-augmenting routes used and their values. • The second mark is dependent on gaining the first mark
(1)
4. For misreading which does not alter the character of a question or materially
(e) Prove that the flow found in part (d) is maximal.
simplify it, deduct two from any A or B marks gained, in that part of the question
(3)
affected.
(Total for Question 3 is 8 marks)
5. Where a candidate has made multiple responses and indicates which response
they wish to submit, examiners should mark this response.
If there are several attempts at a question which have not been crossed out,
examiners should mark the final answer which is the answer that is the most
complete.
for more:
4
for more:
9406A
