Question 4:
4 | 2 | 7
4 | 6 | 1
Activity | Immediate predecessors | Duration (hours)
A Have skip delivered | – | 3
B Remodel walls | A | 3
C Buy new fittings | – | 2
D Fit electrics | B | 2
E Fit plumbing | B | 2
F Install fittings | C, E | 3
G Plastering | D,E | 2
H Decorating | F, G | 3
4
5 (i) How many arcs does the complete bipartite graph K have? [1]
5,5
A subgraph of K contains 5 arcs joining each of the elements of the set {1, 2, 3, 4, 5} to an element in a
5,5
permutation of the set {1, 2, 3, 4, 5}. Suppose that r is connected to p(r) for r = 1, 2, 3, 4, 5.
(ii) How many permutations would have p(1) ! 1? [2]
(iii) Using the pigeonhole principle, show that for every permutation of {1, 2, 3, 4, 5}, the product
P5 (r-p(r)) is even (i.e. an integer multiple of 2, including 0). [3]
r=1
(iv) Is the result in part (iii) true when the permutation is of the set {1, 2, 3, 4, 5, 6}?
Give a reason for your answer. [2]
6 An online magazine consists of an editorial, articles, reviews and advertisements.
The magazine must have a total of at least 12 pages. The editorial always takes up exactly half a page. There
must be at least 3 pages of articles and at most 1.5 pages of reviews. At least a quarter but fewer than half of
the pages in the magazine must be used for advertisements.
Let x be the number of pages used for articles, y be the number of pages used for reviews and z be the
number of pages used for advertisements.
The constraints on the values of x, y and z are:
x+y+z H 11.5
xH 3
yG 1.5
2x+2y-2z+1 H 0
2x+2y-6z+1 G 0
yH 0
(i) (a) Explain why x+y+z H 11.5. [1]
(b) Explain why only one non-negativity constraint is needed. [1]
(c) Show that the requirement that at least one quarter of the pages in the magazine must be used for
advertisements leads to the constraint 2x+2y-6z+1 G 0. [2]
Advertisements bring in money but are not popular with the subscribers. The editor decides to limit the
number of pages of advertisements to at most four.
(ii) Graph the feasible region in the case when z =4 using the axes in the Printed Answer Booklet. [5]
To be successful the magazine needs to maximise the number of subscribers.
The editor has found that when z G 4 the expected number of subscribers is given by P=300x+400y.
(iii) (a) What is the maximum expected number of subscribers when z =4? [2]
(b) By first considering the feasible region for z =k, where kG 4, find an expression for the
maximum number of subscribers in terms of k. [5]
END OF QUESTION PAPER
© OCR 2018 Practice paper Y534/01
8
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