Official June 2024 AQA A-level FURTHER MATHEMATICS 7367/3D Paper 3 Discrete Merged Question Paper + Mark Scheme Ace your Mocks!!! G/LM/Jun24/G4006/V9 7367/3D (JUN2473673D01) A-level FURTHER MATHEMATICS Paper 3 Discrete Friday 7 June 2024 Afternoon 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. For Examiner’s Use Question Mark 1 2 3 4 5 6 7 8 9 10 TOTAL Please write clearly in block capitals. Centre number Candidate number Surname _________________________________________________________________________ Forename(s) _________________________________________________________________________ Candidate signature _________________________________________________________________________ I declare this is my own work. 2 Do not write outside the box (02) G/Jun24/7367/3D Answer all questions in the spaces provided. 1 Which one of the following sets forms a group under the given binary operation? Tick () 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 () one box. [1 mark] Best Lower Bound Best Upper Bound 15 48 15 51 19 48 19 51 3 Do not write outside the box (03) G/Jun24/7367/3D Turn over U 3 The simple-connected graph G has the adjacency matrix A B C D A 0 1 1 1 B 1 0 1 0 C 1 1 0 1 D 1 0 1 0 Which one of the following statements about G is true? Tick () one box. [1 mark] G is a tree G is complete G is Eulerian G is planar Turn over for the next question 4 Do not write outside the box (04) G/Jun24/7367/3D 4 Daniel and Jackson play a zero-sum game. The game is represented by the following pay-off matrix for Daniel. Jackson Strategy W X Y Z Daniel A 3 –2 1 4 B 5 1 –4 1 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] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 5 Do not write outside the box (05) G/Jun24/7367/3D Turn over U 4 (b) Determine the play-safe strategy for each player. [1 mark] Play-safe strategy for Daniel __________________________________________________________ Play-safe strategy for Jackson _________________________________________________________ Turn over for the next question 6 Do not write outside the box (06) G/Jun24/7367/3D 5 The owners of a sports stadium want to install electric car charging points in each of the stadium’s nine car parks. An engineer creates a plan which requires installing electrical connections so that each car park is connected, directly or indirectly, to the stadium’s main electricity power supply. The engineer produces the network shown below, where the nodes represent the stadium’s main electricity power supply X and the nine car parks A, B, …, I 100 150 135 135 125 65 165 75 75 95 105 95 105 45 145 100 145 115 125 I E F G H A B X C D Each arc represents a possible electrical connection which could be installed. The weight on each arc represents the time, in hours, it would take to install the electrical connection. The electrical connections can only be installed one at a time. To reduce disruption, the owners of the sports stadium want the required electrical connections to be installed in the minimum possible total time. 7 Do not write outside the box (07) G/Jun24/7367/3D Turn over U 5 (a) (i) Determine the electrical connections that should be installed. [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] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 8 Do not write outside the box (08) G/Jun24/7367/3D 6 A company delivers parcels to houses in a village, using a van. 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. 1.2 1.2 1.0 1.0 1.0 1.0 1.1 1.1 1.1 1.0 1.0 1.4 0.7 0.9 0.9 0.9 0.9 0.8 0.8 0.8 0.6 0.8 0.6 0.9 I J K L P O N M E F G H A B C D 1.4 1.5 1.5 1.1 1.6 1.6 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 turns 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] 9 Do not write outside the box (09) G/Jun24/7367/3D Turn over U _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 10 Do not write outside the box G/Jun24/7367/3D (10) 7 (a) By considering associativity, show that the set of integers does not form a group under the binary operation of subtraction. 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] ×19 1 7 8 11 12 18 1 1 7 8 11 12 18 7 7 11 8 8 7 11 11 7 12 12 11 18 18 1 7 (b) (ii) State the inverse of 11 in G [1 mark] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 11 Do not write outside the box G/Jun24/7367/3D Turn over U (11) 7 (c) (i) State, with a reason, the possible orders of the proper subgroups of G [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 7 (c) (ii) Find all the proper subgroups of G Give your answers in the form (< g>, ×19) where g ∈ G [3 marks]