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 Candidate number
Surname
Forename(s)
Candidate signature
I declare this is my own work.
A-level
FURTHER MATHEMATICS
Paper 3 Discrete
Friday 7 June 2024 Afternoon Time allowed: 2 hours
Materials
You must have the AQA Formulae and statistical tables booklet for For Examiner’s Use
A-level Mathematics and A-level Further Mathematics. Question Mark
You should have a graphical or scientific calculator that meets the
requirements of the specification. 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
2 hours to complete both papers. 3
Instructions 4
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
Answer all questions.
You must answer each question in the space provided for that question. 6
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). 7
Do not write outside the box around each page or on blank pages.
Show all necessary working; otherwise marks for method may be lost. 8
Do all rough work in this book. Cross through any work that you do not want
to be marked. 9
Information 10
The marks for questions are shown in brackets.
TOTAL
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.
,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.
G/LM/Jun24/G4006/V9 7367/3D
, 2
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 () 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
G/Jun24/7367/3D
, 3
Do not write
outside the
box
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
Turn over U
G/Jun24/7367/3D
, 4
Do not write
outside the
box
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
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]
G/Jun24/7367/3D
, 5
Do not write
outside the
box
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
Turn over U
G/Jun24/7367/3D
(Merged Question Paper and Marking Scheme)
(Friday 7 June 2024)
Please write clearly in block capitals.
Centre number Candidate number
Surname
Forename(s)
Candidate signature
I declare this is my own work.
A-level
FURTHER MATHEMATICS
Paper 3 Discrete
Friday 7 June 2024 Afternoon Time allowed: 2 hours
Materials
You must have the AQA Formulae and statistical tables booklet for For Examiner’s Use
A-level Mathematics and A-level Further Mathematics. Question Mark
You should have a graphical or scientific calculator that meets the
requirements of the specification. 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
2 hours to complete both papers. 3
Instructions 4
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
Answer all questions.
You must answer each question in the space provided for that question. 6
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). 7
Do not write outside the box around each page or on blank pages.
Show all necessary working; otherwise marks for method may be lost. 8
Do all rough work in this book. Cross through any work that you do not want
to be marked. 9
Information 10
The marks for questions are shown in brackets.
TOTAL
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.
,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.
G/LM/Jun24/G4006/V9 7367/3D
, 2
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 () 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
G/Jun24/7367/3D
, 3
Do not write
outside the
box
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
Turn over U
G/Jun24/7367/3D
, 4
Do not write
outside the
box
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
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]
G/Jun24/7367/3D
, 5
Do not write
outside the
box
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
Turn over U
G/Jun24/7367/3D
