FURTHER MATHEMATICS Paper 2 Discrete Friday 19 May 2023 Afternoon

FURTHER MATHEMATICS Paper 2 Discrete Friday 19 May 2023 Afternoon Time allowed: 1 hour 30 minutes Materials  You must have the AQA Formulae and statistical tables booklet for A‑level Mathematics and A‑level Further Mathematics.  You should have a graphical or scientific calculator that meets the requirements of the specification.  You must ensure you have the other optional Question Paper/Answer Book for which you are entered (either Mechanics or Statistics). You will have 1 hour 30 minutes to complete both papers. Instructions  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.  Answer all questions.  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).  Do not write outside the box around each page or on blank pages.  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. Information  The marks for questions are shown in brackets.  The maximum mark for this paper is 40. Advice  Unless stated otherwise, you may quote formulae, without proof, from the booklet.  You do not necessarily need to use all the space provided. (JUN2373662D01) G/LM/Jun23/E4 7366/2D There are no questions printed on this page DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED box Answer all questions in the spaces provided. 1 The graph G has 8 vertices and 13 edges as shown in the diagram below. Q box P R S T U W V Graph H is a simple‑connected subgraph of graph G Which of the following diagrams could represent graph H ? Tick () one box. P Q [1 mark] W R V S U T P Q W R V S U T P Q W R V S U T P Q W R V S U T Turn over U 2 The diagram below shows a network of pipes with their capacities. box A G 42 42 D 42 42 C 42 F 42 42 I 42 E 42 42 B H A supersource is added to the network. Which nodes are connected to the supersource? Tick () one box. [1 mark] A and B A and G G and H H and I 3 Ben is packing eggs into boxes, labelled Town Box or Country Box. Each Town Box must contain 10 chicken eggs and 2 duck eggs. Each Country Box must contain 4 chicken eggs and 8 duck eggs. Ben has 253 chicken eggs and 151 duck eggs. Ben wants to pack as many boxes as possible. Formulate Ben’s situation as a linear programming problem, defining any variables you introduce. [4 marks] box Turn over for the next question Turn over U 4 A community project consists of 10 activities A, B, . . . , J, as shown in the activity network below. Latest finish time Duration Earliest start time box The duration of each activity is shown in days. 4 (a) (i) Complete the activity network in the diagram above, showing the earliest start time and latest finish time for each activity. [3 marks] 4 (a) (ii) State the minimum completion time for the community project. [1 mark] 4 (b) Write down the critical activities of the network. [1 mark] 4 (c) Glyn claims that a project’s activity network can be used to determine its minimum completion time by adding together the durations of all the project’s critical activities. 4 (c) (i) Show that Glyn’s claim is false for this community project’s activity network. [1 mark] box 4 (c) (ii) Describe a situation in which Glyn’s claim would be true. [1 mark] Turn over for the next question Turn over U (07) 5 (a) The set S is defined as S = {0, 1, 2, 3, 4, 5} 5 (a) (i) State the identity element of S under the operation multiplication modulo 6 [1 mark] box 5 (a) (ii) An element g of a set is said to be self‑inverse under a binary operation * if g *g = e where e is the identity element of the set. Find all the self‑inverse elements in S under the operation multiplication modulo 6 [2 marks] (0 ) 5 (b) The set T is defined as T = {a, b, c} box Figure 1 shows a partially completed Cayley table for T under the commutative binary operation ♦ Figure 1 a b c a a c b b b a c c 5 (b) (i) Complete the Cayley table in Figure 1 [1 mark] 5 (b) (ii) Prove that ♦ is not associative when acting on the elements of T [3 marks] Turn over U 6 Xander and Yvonne are playing a zero‑sum game. The game is represented by the pay‑off matrix for Xander. box Yvonne Xander 6 (a) Show that the game has a stable solution. [3 marks] 6 (b) State the play‑safe strategy for each player. [1 mark] Play‑safe strategy for Xander is Play‑safe strategy for Yvonne is 6 (c) The game that Xander and Yvonne are playing is part of a marbles challenge. The pay‑off matrix values represent the number of marbles gained by Xander in each game. In the challenge, the game is repeated until one player has 24 marbles more than the other player. Explain why Xander and Yvonne must play at least 3 games to complete the challenge. [2 marks] box Turn over for the next question Turn over U