AQA_2024: A-level Further Mathematics - Paper 3
Discrete.
(Merged Question Paper and Marking Scheme)
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.
,For A-Level Further Mathematics - Paper 3: Discrete, focus on the following key areas:
1. Graph Theory:
Graphs and Networks: Understand the basics of graphs, vertices, edges, and paths. Learn about
directed and undirected graphs, and apply concepts such as Eulerian paths and Hamiltonian
cycles.
Graph Traversal: Master algorithms for traversing graphs, including depth-first search (DFS) and
breadth-first search (BFS).
Planar Graphs: Study the properties of planar graphs and apply Euler’s formula for planar graphs to
solve problems.
Dijkstra’s Algorithm: Use Dijkstra’s algorithm to find the shortest path between two vertices in a
weighted graph.
2. Algorithms:
Algorithm Design: Understand how to design and analyze algorithms for problems like sorting,
searching, and optimization.
Big O Notation: Know how to express the complexity of algorithms using Big O notation and how to
compare the efficiency of algorithms.
Greedy Algorithms: Study the greedy algorithm approach for optimization problems like the
knapsack problem and Huffman coding.
3. Combinatorics:
Permutations and Combinations: Master the concepts of permutations and combinations, and
apply these to solve problems involving counting arrangements, selections, and distributions.
Binomial Coefficients: Understand the use of binomial coefficients (nCr) and their relationship to
Pascal’s triangle.
Inclusion-Exclusion Principle: Use the inclusion-exclusion principle to solve problems involving
overlapping sets.
4. Recurrence Relations:
Solving Recurrence Relations: Understand how to solve linear recurrence relations (both
homogeneous and non-homogeneous), and use iteration or the characteristic equation method.
Applications of Recurrence Relations: Apply recurrence relations to real-world problems like
population growth, and algorithmic complexities.
5. Binary Relations:
Properties of Binary Relations: Study the properties of binary relations such as reflexivity,
symmetry, transitivity, and antisymmetry. Understand how to determine equivalence relations and
partial orders.
Matrices and Binary Relations: Represent binary relations using adjacency matrices and use these
representations to analyze properties of the relations.
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
Discrete.
(Merged Question Paper and Marking Scheme)
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.
,For A-Level Further Mathematics - Paper 3: Discrete, focus on the following key areas:
1. Graph Theory:
Graphs and Networks: Understand the basics of graphs, vertices, edges, and paths. Learn about
directed and undirected graphs, and apply concepts such as Eulerian paths and Hamiltonian
cycles.
Graph Traversal: Master algorithms for traversing graphs, including depth-first search (DFS) and
breadth-first search (BFS).
Planar Graphs: Study the properties of planar graphs and apply Euler’s formula for planar graphs to
solve problems.
Dijkstra’s Algorithm: Use Dijkstra’s algorithm to find the shortest path between two vertices in a
weighted graph.
2. Algorithms:
Algorithm Design: Understand how to design and analyze algorithms for problems like sorting,
searching, and optimization.
Big O Notation: Know how to express the complexity of algorithms using Big O notation and how to
compare the efficiency of algorithms.
Greedy Algorithms: Study the greedy algorithm approach for optimization problems like the
knapsack problem and Huffman coding.
3. Combinatorics:
Permutations and Combinations: Master the concepts of permutations and combinations, and
apply these to solve problems involving counting arrangements, selections, and distributions.
Binomial Coefficients: Understand the use of binomial coefficients (nCr) and their relationship to
Pascal’s triangle.
Inclusion-Exclusion Principle: Use the inclusion-exclusion principle to solve problems involving
overlapping sets.
4. Recurrence Relations:
Solving Recurrence Relations: Understand how to solve linear recurrence relations (both
homogeneous and non-homogeneous), and use iteration or the characteristic equation method.
Applications of Recurrence Relations: Apply recurrence relations to real-world problems like
population growth, and algorithmic complexities.
5. Binary Relations:
Properties of Binary Relations: Study the properties of binary relations such as reflexivity,
symmetry, transitivity, and antisymmetry. Understand how to determine equivalence relations and
partial orders.
Matrices and Binary Relations: Represent binary relations using adjacency matrices and use these
representations to analyze properties of the relations.
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
