A-Level MathematicsYear 2023Q7
12 P72799A 7. A publisher plans to produce three versions of the same book: a paperback, a hardcover, and a deluxe edition. • Each paperback takes 4 minutes to print and 1 minute to bind • Each hardcover takes 8 minutes to print and 5 minutes to bind • Each deluxe edition takes 15 minutes to print and 12 minutes to bind The printing machine is available for at most 150 hours and the binding machine must be used for at least 60 hours. The publisher decides to produce • at least 1600 books in total • at least three times as many paperbacks as hardcovers The profit on each paperback sold is £8, the profit on each hardcover sold is £20 and the profit on each deluxe edition sold is £40 Let x, y and z represent the number of paperbacks, hardcovers and deluxe editions produced. (a) Formulate this as a linear programming problem, stating the objective and listing the constraints as simplified inequalities with integer coefficients. (5) The publisher decides to solve this linear programming problem by using the two-stage simplex method. (b) Set up an initial tableau for solving this problem using the two-stage simplex method. As part of your solution, you must show how • the constraints have been made into equations by using slack variables, exactly two surplus variables and exactly two artificial variables • the rows for the two objective functions are formed (6) 13 Turn over P72799A The following tableau is obtained after two iterations of the first stage of the two-stage simplex method. b.v. x y z s1 s2 s3 s4 a1 a2 Value s1 0 0 0 1 1 3 0 –1 –3 600 z 0 4 11 1 0 −1 11 1 11 0 1 11 −1 11 2000 11 x 1 7 11 0 0 1 11 −12 11 0 −1 11 12 11 15600 11 s4 0 40 11 0 0 1 11 −12 11 1 −1 11 12 11 15600 11 P 0 −4 11 0 0 −32 11 −56 11 0 32 11 56 11 204800 11 I 0 0 0 0 0 0 0 1 1 0 (c) Taking the most negative number in the profit row to indicate the pivot column, perform one complete iteration of the second stage of the two-stage simplex method to obtain a new tableau. Make your method clear by stating the row operations you use. (5) 14 P72799A After three iterations of the second stage of the two-stage simplex method, the following tableau is obtained. b.v. x y z s1 s2 s3 s4 Value s2 0 0 0 1 1 3 0 600 z 0 0 1 1 10 0 1 2 −1 10 100 x 1 0 0 −3 40 0 −9 8 −7 40 1125 y 0 1 0 −1 40 0 −3 8 11 40 375 P 0 0 0 29 10 0 7 2 1 10 20500 Given that the publisher produces the optimal number of each version of the book, (d) (i) state the maximum profit the publisher can earn, (ii) find the number of hours the binding machine will be used. (2) (e) Give a reason why the publisher may not earn the profit stated in (d)(i). (1) (Total for Question 7 is 19 marks) TOTAL FOR PAPER IS 75 MARKS 15 P72799A BLANK PAGE 16 P72799A BLANK PAGE Turn over P72799A ©2023 Pearson Education Ltd. N:1/1/1/ Please check the examination details below before entering your candidate information Candidate surname Other names Centre Number Candidate Number Pearson Edexcel Level 3 GCE Friday 23 June 2023 Afternoon (Time: 1 hour 30 minutes) 9FM0/3D Paper reference Total Marks Further Mathematics Advanced PAPER 3D: Decision Mathematics 1 Answer Book Do not return the question paper with the answer book. DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 2 1. A B D C E Figure 1 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over 3 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (e) Initial time matrix A B C D E A – B – C – D – E – (f) Final iteration A B C D E A – B – C – D – E – (Total for Question 1 is 7 marks) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 4 2. Key: Early event time Late event time A(3) D(7) G(4) B(5) E(1) H(1) C(4) F(2) Diagram 1 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over 5 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 2 continued 8 7 6 5 4 3 2 1 1 O 2 3 4 5 6 7 8 9 10 11 12 13 14 Number of workers Time (hours) Grid 1 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 7 marks) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 6 3. Key: Vertex Order of labelling Final value Working values A B E H C F J D G 34 22 28 42 50 32 32 35 37 30 21 10 11 43 45 7 41 23 8 7 Shortest path from A to J: ___________________________________________________________________________________________________ Length of shortest path from A to J: ___________________________________________________________________________________________ Turn over 7 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 8 marks) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 8 4. 15 22 3 9 23 x 5 4 18 20 13 _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 10 Question 4 continued 15 22 3 9 23 x 5 4 18 20 13 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over 11 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 4 continued 15 22 3 9 23 x 5 4 18 20 13 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 12 marks) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 12 5. A B C D E F G H J A – 34 51 31 79 20 8 55 61 B 34 – 17 65 45 54 42 21 27 C 51 17 – 82 28 71 59 22 10 D 31 65 82 – 87 22 23 86 92 E 79 45 28 87 – 65 87 30 18 F 20 54 71 22 65 – 28 75 81 G 8 42 59 23 87 28 – 63 69 H 55 21 22 86 30 75 63 – 12 J 61 27 10 92 18 81 69 12 – _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over 13 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 5 continued A B C D E F G H J A – 34 51 31 79 20 8 55 61 B 34 – 17 65 45 54 42 21 27 C 51 17 – 82 28 71 59 22 10 D 31 65 82 – 87 22 23 86 92 E 79 45 28 87 – 65 87 30 18 F 20 54 71 22 65 – 28 75 81 G 8 42 59 23 87 28 – 63 69 H 55 21 22 86 30 75 63 – 12 J 61 27 10 92 18 81 69 12 – _____________________________________________________________________________________ _____________________________________________________________________________________ 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NOT WRITE IN THIS AREA 14 Question 5 continued A B C D E F G H J A – 34 51 31 79 20 8 55 61 B 34 – 17 65 45 54 42 21 27 C 51 17 – 82 28 71 59 22 10 D 31 65 82 – 87 22 23 86 92 E 79 45 28 87 – 65 87 30 18 F 20 54 71 22 65 – 28 75 81 G 8 42 59 23 87 28 – 63 69 H 55 21 22 86 30 75 63 – 12 J 61 27 10 92 18 81 69 12 – _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over 15 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 5 continued A B C D E F G H J A – 34 51 31 79 20 8 55 61 B 34 – 17 65 45 54 42 21 27 C 51 17 – 82 28 71 59 22 10 D 31 65 82 – 87 22 23 86 92 E 79 45 28 87 – 65 87 30 18 F 20 54 71 22 65 – 28 75 81 G 8 42 59 23 87 28 – 63 69 H 55 21 22 86 30 75 63 – 12 J 61 27 10 92 18 81 69 12 – _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 13 marks) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 16 6. Turn over 17 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 6 is 9 marks) DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 18 7. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over 19 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ b.v. x y z Value P I Parts (c), (d) and (e) to be answered on the next page. DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 20 Question 7 continued b.v. x y z s1 s2 s3 s4 Value s1 0 0 0 1 1 3 0 600 z 0 4 11 1 0 −1 11 1 11 0 2000 11 x 1 7 11 0 0 1 11 −12 11 0 15600 11 s4 0 40 11 0 0 1 11 −12 11 1 15600 11 P 0 −4 11 0 0 −32 11 −56 11 0 204800 11 b.v. x y z s1 s2 s3 s4 Value Row Ops P _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 7 is 19 marks) TOTAL FOR PAPER IS 75 MARKS
Paper Source:EAMF3119fm0-3d-que-20230624.pdf
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Exam Specification Info
This question is part of the UK A-Level Mathematics syllabus. In the actual exam, structured questions typically require linking specific keywords to gain full marks. Applaa helps you drill these topics.
Syllabus levelAdvanced Level (A-Level)
SubjectMathematics
Official MarksVariable (2–6 marks)