A-Level MathematicsYear 2020Q7
9 P62678A Turn over 7. A maximisation linear programming problem in x, y and z is to be solved using the two-stage simplex method. The partially completed initial tableau is shown below. Basic variable x y z s1 s2 s3 a1 a2 Value s1 1 2 3 1 0 0 0 0 45 a1 3 2 0 0 –1 0 1 0 9 a2 –1 0 4 0 0 –1 0 1 4 P –2 –1 –3 0 0 0 0 0 0 A (a) Using the information in the above tableau, formulate the linear programming problem. State the objective and list the constraints as inequalities. (4) (b) Complete the bottom row of Table 1 in the answer book. You should make your method and working clear. (2) The following tableau is obtained after two iterations of the first stage of the two-stage simplex method. Basic variable x y z s1 s2 s3 a1 a2 Value s1 0 5 6 0 1 7 12 3 4 −7 12 −3 4 147 4 x 1 2 3 0 0 −1 3 0 1 3 0 3 z 0 1 6 1 0 −1 12 −1 4 1 12 1 4 7 4 P 0 5 6 0 0 −11 12 −3 4 11 12 3 4 45 4 A 0 0 0 0 0 0 1 1 0 (c) (i) Explain how the above tableau shows that a basic feasible solution has been found for the original linear programming problem. (ii) Write down the basic feasible solution for the second stage. (3) (d) 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, T. Make your method clear by stating the row operations you use. (5) 10 P62678A (e) (i) Explain, using T, whether or not an optimal solution to the original linear programming problem has been found. (ii) Write down the value of the objective function. (iii) State the values of the basic variables. (3) (Total for Question 7 is 17 marks) TOTAL FOR PAPER IS 75 MARKS 11 P62678A BLANK PAGE 12 P62678A BLANK PAGE Please check the examination details below before entering your candidate information Candidate surname Other names Total Marks Centre Number Candidate Number Pearson Edexcel Level 3 GCE P62678A ©2020 Pearson Education Ltd. 1/1/1/1/1/1/1/ Turn over Answer book Do not return the question paper with the answer book. Paper Reference 9FM0/3D Afternoon (Time: 1 hour 30 minutes) Wednesday 17 June 2020 Turn over Further Mathematics Advanced Paper 3D: Decision Mathematics 1 2 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 1. A B C D E F G A – 24 – 22 35 – – B 24 – 25 27 – – – C – 25 – 33 31 36 26 D 22 27 33 – – 42 – E 35 – 31 – – 37 29 F – – 36 42 37 – 40 G – – 26 – 29 40 – C G E D B A F 24 25 35 29 26 31 40 37 36 Diagram 1 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 3 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 6 marks) 4 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 2. (a) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (b) Activity Immediately preceding activities Activity Immediately preceding activities Activity Immediately preceding activities A E I B F J C G K D H L Key: Early event time Late event time A(6) B(7) E(5) I(4) J(6) G(4) F(3) H(3) D(4) C(5) K(5) L(6) Diagram 1 5 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ time (hours) O 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 number of workers Grid 1 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 15 marks) 6 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 3. (a) Initial time matrix Initial route matrix A B C D E A B C D E A B C D E A B C D E (b) Time matrix Route matrix A B C D E A B C D E A B C D E A B C D E Time matrix Route matrix A B C D E A B C D E A B C D E A B C D E If you make an error there are spare copies of these matrices on Page 9. 7 Turn over 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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you have made an error in your working. (a) Spare copy of initial time matrix Spare copy of initial route matrix A B C D E A B C D E A B C D E A B C D E (b) Spare copy of time matrix Spare copy of route matrix A B C D E A B C D E A B C D E A B C D E Spare copy of time matrix Spare copy of route matrix A B C D E A B C D E A B C D E A B C D E (Total for Question 3 is 9 marks) 10 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 4. y O 7 – 6 – 5 – 4 – 3 – 2 – 1 – 1 2 3 4 5 x 2y = 5x 6x + 5y = 30 R y = x + 1 – – – – – – – Figure 3 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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(a) ________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________ Vertex Order of labelling Final value Working values Key: D F E B C A G 24 y 42 15 65 21 34 31 24 x 19 45 17 Turn over 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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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Basic variable x y z s1 s2 s3 a1 a2 Value s1 1 2 3 1 0 0 0 0 45 a1 3 2 0 0 –1 0 1 0 9 a2 –1 0 4 0 0 –1 0 1 4 P –2 –1 –3 0 0 0 0 0 0 A Table 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_____________________________________________________________________________________ b.v. x y z s1 s2 s3 Value s1 0 5 6 0 1 7 12 3 4 147 4 x 1 2 3 0 0 −1 3 0 3 z 0 1 6 1 0 −1 12 −1 4 7 4 P 0 5 6 0 0 −11 12 −3 4 45 4 b.v. x y z s1 s2 s3 Value Row Ops P Spare copy b.v. x y z s1 s2 s3 Value Row Ops P _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 20 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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Paper Source:9FM0_3D_que_20201022.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)