Partial preview of the text
Download 2024 AQA A-level FURTHER MATHEMATICS 7367/3D Paper 3 Discrete Question Paper + Mark Schem and more Exams Mathematics in PDF only on Docsity!
2024 AQA A-level FURTHER MATHEMATICS 7367/3D Paper 3 Discrete Question Paper + Mark Scheme AQA va Please write clearly in block capitals. Centre number Candidate number Surname Forename(s) Candidate signature | declare this is my own work. A-level FURTHER MATHEMATICS Paper 3 Discrete Friday 7 June 2024 Afternoon Time allowed: 2 hours Materials — e You must have the AQA Formulae and statistical tables booklet for Gal lesclul. asst: A-level Mathematics and A-level Further Mathematics. Question Mark e You should have a graphical or scientific calculator that meets the requirements of the specification. e 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. 1 2 3 Instructions 4 e Use black ink or black ball-point pen. Pencil should only be used for drawing. Fill in the boxes at the top of this page. 5 6 7 8 ° e Answer all questions. e 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). e Do not write outside the box around each page or on blank pages. e Show all necessary working; otherwise marks for method may be lost. Do all rough work in this book. Cross through any work that you do not want to be marked. 9 Information 10 e The marks for questions are shown in brackets. TOTAL e The maximum mark for this paper is 50. Advice e Unless stated otherwise, you may quote formulae, without proof, from the booklet. e You do not necessarily need to use all the space provided. TL | Do not write outside the box Answer all questions in the spaces provided. 1 Which one of the following sets forms a group under the given binary operation? Tick (Vv) one box. [1 mark] Set Binary Operation {1, 2, 3} Addition modulo 4 {1, 2, 3} Multiplication modulo 4 {0, 1, 2, 3} Addition modulo 4 {0, 1, 2, 3} Multiplication modulo 4 2 A student is trying to find the solution to the travelling salesperson problem for a network. They correctly find two lower bounds for the solution: 15 and 19 They also correctly find two upper bounds for the solution: 48 and 51 Based on the above information only, which of the following pairs give the best lower bound and best upper bound for the solution of this problem? Tick (Vv) one box. [1 mark] Best Lower Bound Best Upper Bound 15 48 15 51 19 48 19 51 IML G/Jun24/7367/3D 2 Do not write outside the 4 Daniel and Jackson play a zero-sum game. box The game is represented by the following pay-off matrix for Daniel. Jackson Strategy Ww x Y Zz A 3 —2 1 4 B 5 1 -4 1 Daniel c 2 1 1 2 D -3 0 2 -1 Neither player has any strategies which can be ignored due to dominance. 4 (a) Prove that the game does not have a stable solution. Fully justify your answer. [3 marks] IML G/Jun24/7367/3D, 4 Do not write outside the 4 (b) Determine the play-safe strategy for each player. box [1 mark] Play-safe strategy for Daniel Play-safe strategy for Jackson Turn over for the next question Turn over > AMM G/Jun24/7367/3D, 5 Do not write outside the 5 (a) (i) Determine the electrical connections that should be installed. box [2 marks] 5 (a) (ii) Find the minimum possible total time needed to install the required electrical connections. [1 mark] 5 (b) Following the installation of the electrical connections, some of the car parks have an indirect connection to the stadium's main electricity power supply. Give one limitation of this installation. [1 mark] Turn over > IML G/Jun24/7367/3D, 7 AMM Do not write outside the A company delivers parcels to houses in a village, using a van. box The network below shows the roads in the village. Each node represents a road junction and the weight of each arc represents the length, in miles, of the road between the junctions. A The total length of all of the roads in the village is 31.4 miles. On one particular day, the driver is due to make deliveries to at least one house on each road, so the van must travel along each road at least once. However, the driver has forgotten to add fuel to the van and it only has 4.5 litres of fuel to use to make its deliveries. The van uses, on average, 1 litre of fuel to travel 7.8 miles along the roads of this village. Whilst making each delivery, the driver tums off the van’s engine so it does not use any fuel. Determine whether the van has enough fuel for the driver to make all of the deliveries to houses on each road of the village, starting and finishing at the same junction. Fully justify your answer. [6 marks] G/Jun24/7367/3D 8 10 Do not write outside the By considering associativity, show that the set of integers does not form a group under box the binary operation of subtraction. 7 (a) Fully justify your answer. [2 marks] 7 (b) The group G is formed by the set {1, 7, 8, 11, 12, 18} under the operation of multiplication modulo 19 7 (b)(i) Complete the Cayley table for G [3 marks] xi | 1 | 7 | 8 | 11) 12 | 18 1 1 7 8 11 12 | 18 7 7 11 8 8 6 11 11 7 12 | 12 11 18 | 18 1 7 (b) (ii) State the inverse of 11 in G [1 mark] IMI G/Jun24/7367/3D 10 11 Do not write outside the 7 (c) (i) State, with a reason, the possible orders of the proper subgroups of G box [2 marks] 7 (c) (ii) Find all the proper subgroups of G Give your answers in the form ((z): x19) where g € G [3 marks] 7 (ce) (iii) The group His such that G=H State a possible name for H [1 mark] Turn over > 11 G/Jun24/7367/3D 11 13 Do not write outside the Figure 2 box 8 (c) While the flow through the network is at its maximum value, the pipe EG develops a leak To repair the leak, an engineer turns off the flow of water through EG The engineer claims that the maximum flow of water through the network will reduce by 31 litres per second. Comment on the validity of the engineer’s claim. [2 marks] Turn over > IM G/Jun24/7367/3D 13 14 Do not write outside the 9 Janet and Samantha play a zero-sum game. box The game is represented by the following pay-off matrix for Janet. Samantha Strategy Ss, So S3 dy 2 7 6 Jo 5 5 1 Janet Jy 4 3 8 Jy 1 6 4 9 (a) Explain why Janet should never play strategy J4 [1 mark] 9 (b) Janet wants to maximise her winnings from the game. She defines the following variables. Pi = the probability of Janet playing strategy J, Py = the probability of Janet playing strategy J. P3 = the probability of Janet playing strategy J3 vy = the value of the game for Janet Janet then formulates her situation as the following linear programming problem. Maximise P=v subject to 2p, + Sp, + 4p, 2v P, + Sp, +3p,2V 6p, +P, +8p,2¥ and P,+P,+P3S1 Py Py P32 9 G/Jun24/7367/3D 14 16 Do not write outside the 9 (c) Further iterations of the Simplex algorithm are performed until an optimal solution box is reached. The grid below shows part of the final Simplex tableau. P, Py value 1 1 0 72 1 0 1 2 Find the probability of Janet playing strategy J3 when she is playing to maximise her winnings from the game. [1 mark] IMI G/Jun24/7367/3D 16 17 Do not write outside the box Turn over for the next question DO NOT WRITE/ON THIS PAGE ANSWER IN THE/SPACES PROVIDED Turn over > G/Jun24/7367/3D 17 19 Do not write outside the ‘box 10 (b) Using Figure 4 below, draw a resource histogram for the project to show how the project can be completed in the minimum possible time. Assume that each activity is to start as early as possible. [3 marks] Figure 4 8x 7 p 6 : g 55 = 364 3 2 3 5 ae HH Ht aa aa re 1 FCEH 22 1 0 > 0 10 20 30 40 50 60 70 80 90 100 Days 10 (c) Higton Engineering Ltd only has four workers available to work on the project. Find the minimum completion time for the project. Use Figure 5 below in your answer. [3 marks] Figure 5 on | t 4 = { $3 o 5 2 2 coo aan Soot S| 2 0 | | 0 10 20 30 40 50 60 70 80 90 100 Days Minimum completion time END OF QUESTIONS ll . 9 G/dun24/7367/3D 1 20 Do not write outside the box There are no questions printed on this page DO NOT WRITE/ON THIS PAGE ANSWER IN THE/SPACES PROVIDED IMU G/Jun24/7367/3D 20 
