AQA_2024: A-level Further Mathematics - Paper 3 Discrete (Merged Question Paper and Marking Scheme) (Friday 7 June 2024)

AQA_2024: A-level Further Mathematics - Paper 3 Discrete (Merged Question Paper and Marking Scheme) (Friday 7 June 2024) Please write clearly in block capitals. Centre number Surname Forename(s) Candidate signature Candidate number I declare this is my own work. A-level FURTHER MATHEMATICS Paper 3 Discrete Friday 7 June 2024 Materials Afternoon  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. Time allowed: 2 hours For Examiner’s Use Question Mark 1  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  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 50. Advice  Unless stated otherwise, you may quote formulae, without proof, from the booklet.  You do not necessarily need to use all the space provided. 2 3 4 5 6 7 8 9 10 TOTAL A-Level Further Mathematics: Paper 3 Discrete (Friday 7 June 2024) Exam Preview Areas This paper focuses on Discrete Mathematics, which involves mathematical structures and techniques that are fundamentally distinct from continuous mathematics. Key topics usually covered include: 1. Combinatorics:  Counting Principles: Basic principles of counting, including the addition and multiplication rules for counting possible outcomes.  Permutations and Combinations: Calculating the number of ways to arrange or select items (with or without repetition).  Binomial Coefficients: Understanding the binomial theorem and using combinations to solve problems involving subsets and arrangements. 2. Graph Theory:  Graphs and Networks: Studying vertices, edges, and their relationships in graphs. Concepts include directed and undirected graphs, bipartite graphs, and weighted graphs.  Euler and Hamiltonian Paths: Finding paths and circuits in graphs, including Euler’s path and circuit, and Hamiltonian paths and cycles.  Planar Graphs: Understanding properties of graphs that can be drawn on a plane without edges crossing, and applying Euler's formula for planar graphs. 3. Algorithms:  Graph Algorithms: Solving problems using algorithms like Dijkstra’s Algorithm for shortest paths and Kruskal’s Algorithm for minimum spanning trees.  Recursion: Solving problems using recursive methods, including analyzing the efficiency of recursive algorithms.  Searching and Sorting: Understanding and applying algorithms like binary search, quicksort, and merge sort. 4. Linear Programming:  Formulating Problems: Setting up linear inequalities to represent constraints in optimization problems.  Simplex Method: Solving linear programming problems to find the maximum or minimum values of an objective function under given constraints. 5. Boolean Algebra:  Logic Gates: Using Boolean logic to simplify expressions, and applying it to digital circuits.  Simplification: Simplifying Boolean expressions using laws of Boolean algebra and Karnaugh maps. This paper assesses your ability to apply discrete mathematics concepts to solve problems related to graphs, counting, optimization, algorithms, and logic. You’ll need to demonstrate strong problem-solving skills and a solid understanding of key discrete structures and techniques. 7367/3D G/LM/Jun24/G4006/V9 2 Do not write outside the Answer all questions in the spaces provided. 1 2 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 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. Best Lower Bound 15 Best Upper Bound 48 15 51 19 48 19 51 [1 mark] G/Jun24/7367/3D box 3 Do not write outside the G/Jun24/7367/3D 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] box G is a tree G is complete G is Eulerian G is planar Turn over for the next question Turn over U 4 4 Do not write outside the Daniel and Jackson play a zero-sum game. The game is represented by the following pay-off matrix for Daniel. Jackson Strategy W X A 3 Y Z – 2 B Daniel 5 1 1 – 4 4 1 C 2 – 1 D – 3 1 2 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. box [3 marks] G/Jun24/7367/3D 5 Do not write outside the G/Jun24/7367/3D 4 (b) Determine the play-safe strategy for each player. [1 mark] box Play-safe strategy for Daniel Play-safe strategy for Jackson Turn over for the next question Turn over U 5 6 Do not write outside the 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 X 100 A 135 125 B 115 75 D 100 95 105 65 E 165 H 75 150 135 125 145 95 145 F 45 105 G I C 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. box G/Jun24/7367/3D 7 Do not write outside the G/Jun24/7367/3D 5 (a) (i) Determine the electrical connections that should be installed. [2 marks] box 5 (a) (ii) Find the minimum possib