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Official summer 2024 OCR GCE Further Mathematics A Y544/01: Discrete Mathematics A Level Merged Question Paper + Mark Scheme + Answer Booklet$7.99
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OCR
GCE
Further Mathematics A
Y544/01: Discrete Mathematics
A Level
Merged Question Paper + Mark Scheme + Answer
Booklet
Ace your Mocks!!!
, Oxford Cambridge and RSA
Wednesday 19 June 2024 – Afternoon
A Level Further Mathematics A
Y544/01 Discrete Mathematics
Time allowed: 1 hour 30 minutes
* 1 4 2 2 4 5 0 3 8 9 *
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.
, 2
1 At the end of each year the workers at an office take part in a gift exchange.
Each worker randomly chooses the name of one other worker and buys a small gift for that
person.
Each worker’s name is chosen by exactly one of the others.
A worker cannot choose their own name.
In the first year there were four workers, A, B, C and D.
There are 9 ways in which A, B, C and D can choose the names for the gift exchange.
One of these is already given in the table in the Printed Answer Booklet.
(a) Complete the table in the Printed Answer Booklet to show the remaining 8 ways in which the
names can be chosen. [2]
During the second year, worker D left and was replaced with worker E.
The organiser of the gift exchange wants to know whether it is possible for the event to happen
for another 3 years (starting with the second year) with none of the workers choosing a name they
have chosen before, assuming that there are no further changes in the workers.
(b) Classify the organiser’s problem as an existence, construction, enumeration or optimisation
problem. [1]
After the second year, the organiser drew a graph showing who each worker chose in the first two
years of the gift exchange.
None of the workers chose the same name in the first and second years.
The vertices of the graph represented the workers, A, B, C, D and E, and the arcs showed who had
been chosen by each worker.
(c) Explain why the graph must be a digraph. [1]
(d) State the number of arcs in the digraph that shows the choices for the first two years. [1]
(e) Assuming that the digraph created in part (d) is planar, use Euler’s formula to calculate how
many regions it has. [2]
3x - 4y - z G 30
x-y G6
x - 3y + 2z H - 2
and x H 0, y H 0, z H 0
(a) Complete the table in the Printed Answer Booklet to represent the problem as an initial
simplex tableau. [3]
(b) Carry out one iteration of the simplex algorithm. [3]
(c) State the values of x, y and z that result from your iteration. [1]
After two iterations the resulting tableau is
P x y z s t u RHS
1 0 0 -2 0 2.5 0.5 16
0 0 0 -2 1 -2.5 0.5 16
0 1 0 -1 0 1.5 0.5 10
0 0 1 -1 0 0.5 0.5 4
The boundaries of the feasible region are planes, with edges each defined by two of x, y, z, s, t, u
being zero.
At each vertex of the feasible region there are three basic variables and three non‑basic variables.
(d) Interpret the second iteration geometrically by stating which edge of the feasible region is
being moved along. As part of your geometrical interpretation, you should state the beginning
vertex and end vertex of the second iteration. [2]
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