AQA A-level FURTHER MATHEMATICS 7367/3D Paper 3 Discrete Version: 1.0 Final PB/KL/Jun23/E4 7367/3D A-level FURTHER MATHEMATICS Paper 3 Discrete// QUESTION PAPER & MARKING SCHEME/ [MERGED] Marl( scheme June 2023

AQA A-level FURTHER MATHEMATICS 7367/3D Paper 3 Discrete Version: 1.0 Final PB/KL/Jun23/E4 7367/3D A-level FURTHER MATHEMATICS Paper 3 Discrete Time allowed: 2 hours Materials l You must have the AQA Formulae and statistical tables booklet for A‑level Mathematics and A‑level Further Mathematics. l You should have a graphical or scientific calculator that meets the requirements of the specification. l You must ensure you have the other optional Question Paper/Answer Book for which you are entered (either Mechanics or Statistics). You will have 2 hours to complete both papers. Instructions l Use black ink or black ball‑point pen. Pencil should only be used for drawing. l Fill in the boxes at the top of this page. l Answer all questions. l You must answer each question in the space provided for that question. If you require extra space for your answer(s), use the lined pages at the end of this book. Write the question number against your answer(s). l Do not write outside the box around each page or on blank pages. l Show all necessary working; otherwise marks for method may be lost. l Do all rough work in this book. Cross through any work that you do not want to be marked. Information l The marks for questions are shown in brackets. l The maximum mark for this paper is 50. Advice l Unless stated otherwise, you may quote formulae, without proof, from the booklet. l You do not necessarily need to use all the space provided. Please write clearly in block capitals. Centre number Candidate number Surname ________________________________________________________________________ Forename(s) ________________________________________________________________________ Candidate signature ________________________________________________________________________ For Examiner’s Use Question Mark 1 2 3 4 5 6 7 8 9 TOTAL I declare this is my own work. 2 Answer all questions in the spaces provided. 1 The simple-connected graph G is shown below. The graph G has n faces. State the value of n Circle your answer. [1 mark] 2 34 5 2 Jonathan and Hoshi play a zero-sum game. The game is represented by the following pay-off matrix for Jonathan. Hoshi Strategy H1 H2 H3 Jonathan J1 23 2 J2 320 J3 4 1 3 J4 310 The game does not have a stable solution. Which strategy should Jonathan never play? Circle your answer. [1 mark] J1 J2 J3 J4 Jun23/7367/3D Do not write outside the box (02) 3 3 A student is solving a maximising linear programming problem. The graph below shows the constraints, feasible region and objective line for the student’s linear programming problem. 0 50 100 150 200 250 y 25 75 125 175 225 50 Objective line 250 x A D C Feasible region B Which vertex is the optimal vertex? Circle your answer. [1 mark] A BCD Turn over for the next question Do not write outside the box Jun23/7367/3D Turn over s (03) 4 4 The network below represents a system of water pipes in a geothermal power station. The numbers on each arc represent the lower and upper capacity for each pipe in gallons per second. B D C F I 7, 23 3, 28 3, 10 7, 15 2, 15 8, 22 4, 15 A J H E G 1, 30 8, 17 5, 18 7, 11 2, 21 6, 12 3, 9 0, 14 6, 9 5, 24 The water is taken from a nearby river at node A The water is then pumped through the system of pipes and passes through one of three treatment facilities at nodes H, I and J before returning to the river. 4 (a) The senior management at the power station want all of the water to undergo a final quality control check at a new facility before it returns to the river. Using the language of networks, explain how the network above could be modified to include the new facility. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 4 (b) Find the value of the cut {A, B, C, D, E} {F, G, H, I, J } [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Do not write outside the box Jun23/7367/3D (04) 5 4 (c) Tim, a trainee engineer at the power station, correctly calculates the value of the cut {A, B, C, D, E, F } {G, H, I, J } to be 106 gallons per second. Tim then claims that the maximum flow through the network of pipes is 106 gallons per second. Comment on the validity of Tim’s claim. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over for the next question Do not write outside the box Jun23/7367/3D Turn over s (05) 6 5 A student is solving the following linear programming problem. Minimise Q ¼ 4x  3y subject to x þ y  520 2x  3y  570 and x  0, y  0 5 (a) The student wants to use the simplex algorithm to solve the linear programming problem. They modify the linear programming problem by introducing the objective function P ¼ 4x þ 3y and the slack variables r and s State one further modification that must be made to the linear programming problem so that it can be solved using the simplex algorithm. [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 5 (b) (i) Complete the initial simplex tableau for the modified linear programming problem. [2 marks] Pxy r s value Do not write outside the box Jun23/7367/3D (06) 7 5 (b) (ii) Hence, perform one iteration of the simplex algorithm. [2 marks] Pxy r s value 5 (c) The student performs one further iteration of the simplex algorithm, which results in the following correct simplex tableau. Pxy r s value 100 18 5 1 5 1986 001 2 5  1 5 94 010 3 5 1 5 426 5 (c) (i) Explain how the student can tell that the optimal solution to the modified linear programming problem can be determined from the above simplex tableau. [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ 5 (c) (ii) Find the optimal solution of the original linear programming problem