Oxford Cambridge and
RSA Examinations GCE
Further Mathematics
AY544/01: Discrete
Mathematics
A Level
Question paper with
marking scheme
(merged)
, Oxford Cambridge and RSA
Thursday 22 June 2023 – Afternoon
A Level Further Mathematics A
Y544/01 Discrete Mathematics
Time allowed: 1 hour 30 minutes
* 9 9 7 8 0 7 8 2 3 8 *
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for A Level Further
QP
Mathematics A
• 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 pages 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 non-exact numerical answers correct to 3 significant figures unless a different
degree of accuracy is specified in the question.
• The acceleration due to gravity is denoted by g m s–2. When a numerical value is
needed use g = 9.8 unless a different value is specified in the question.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 75.
• 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 2023 [Y/508/5513] OCR is an exempt Charity
DC (PQ/SG) 329617/3 Turn over
, 2
1 The table below shows the activities involved in a project together with the immediate
predecessors and the duration of each activity.
Activity Immediate predecessors Duration (hours)
A – 2
B A 3
C – 4
D C 2
E B, C 2
F D, E 3
G E 2
H F, G 1
(a) Model the project using an activity network. [3]
(b) Determine the minimum project completion time. [2]
The start of activity C is delayed by 2 hours.
(c) Determine the minimum project completion time with this delay. [2]
2 A graph is shown below.
A B C
D E F
(a) Write down a cycle through all six vertices. [1]
(b) Write down a continuous route that uses every arc exactly once. [2]
(c) Use Kuratowski’s theorem to show that the graph is not planar. [2]
(d) Show that the graph has thickness 2. [3]
© OCR 2023 Y544/01 Jun23
, 3
3 An initial simplex tableau is given below.
P x y z s t RHS
1 -2 3 -1 0 0 0
0 5 -4 1 1 0 20
0 2 -1 0 0 1 6
(a) Carry out two iterations of the simplex algorithm, choosing the first pivot from the x column.
[4]
After three iterations the resulting tableau is as follows.
P x y z s t RHS
1 3 -1 0 1 0 20
0 5 -4 1 1 0 20
0 2 -1 0 0 1 6
(b) State the values of P, x, y, z, s and t that result from these three iterations. [2]
(c) Explain why no further iterations are possible. [2]
The initial simplex tableau is changed to the following, where k is a positive real value.
P x y z s t RHS
1 2 -3 1 0 0 0
0 5 k 1 1 0 20
0 2 -1 0 0 1 6
After one iteration of the simplex algorithm the value of P is 500.
(d) Deduce the value of k. [4]
© OCR 2023 Y544/01 Jun23 Turn over
RSA Examinations GCE
Further Mathematics
AY544/01: Discrete
Mathematics
A Level
Question paper with
marking scheme
(merged)
, Oxford Cambridge and RSA
Thursday 22 June 2023 – Afternoon
A Level Further Mathematics A
Y544/01 Discrete Mathematics
Time allowed: 1 hour 30 minutes
* 9 9 7 8 0 7 8 2 3 8 *
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for A Level Further
QP
Mathematics A
• 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 pages 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 non-exact numerical answers correct to 3 significant figures unless a different
degree of accuracy is specified in the question.
• The acceleration due to gravity is denoted by g m s–2. When a numerical value is
needed use g = 9.8 unless a different value is specified in the question.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 75.
• 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 2023 [Y/508/5513] OCR is an exempt Charity
DC (PQ/SG) 329617/3 Turn over
, 2
1 The table below shows the activities involved in a project together with the immediate
predecessors and the duration of each activity.
Activity Immediate predecessors Duration (hours)
A – 2
B A 3
C – 4
D C 2
E B, C 2
F D, E 3
G E 2
H F, G 1
(a) Model the project using an activity network. [3]
(b) Determine the minimum project completion time. [2]
The start of activity C is delayed by 2 hours.
(c) Determine the minimum project completion time with this delay. [2]
2 A graph is shown below.
A B C
D E F
(a) Write down a cycle through all six vertices. [1]
(b) Write down a continuous route that uses every arc exactly once. [2]
(c) Use Kuratowski’s theorem to show that the graph is not planar. [2]
(d) Show that the graph has thickness 2. [3]
© OCR 2023 Y544/01 Jun23
, 3
3 An initial simplex tableau is given below.
P x y z s t RHS
1 -2 3 -1 0 0 0
0 5 -4 1 1 0 20
0 2 -1 0 0 1 6
(a) Carry out two iterations of the simplex algorithm, choosing the first pivot from the x column.
[4]
After three iterations the resulting tableau is as follows.
P x y z s t RHS
1 3 -1 0 1 0 20
0 5 -4 1 1 0 20
0 2 -1 0 0 1 6
(b) State the values of P, x, y, z, s and t that result from these three iterations. [2]
(c) Explain why no further iterations are possible. [2]
The initial simplex tableau is changed to the following, where k is a positive real value.
P x y z s t RHS
1 2 -3 1 0 0 0
0 5 k 1 1 0 20
0 2 -1 0 0 1 6
After one iteration of the simplex algorithm the value of P is 500.
(d) Deduce the value of k. [4]
© OCR 2023 Y544/01 Jun23 Turn over
