AQA A Level Further Maths Paper 3 Discrete June 2025 question paper

The group A is of order 9

The group B is a proper, non-trivial subgroup of A

Find the order of B

Circle your answer.

1 3 9 18

The group (X, ) is an abelian group of order 4

The elements of X are e, a, b and c

where e is the identity element of (X, )

The following is a partially completed Cayley table for (X, )

Which one of the following fully completed Cayley tables for (X, ) is correct?

Tick (✓) one box.

A water company is planning to install a new sewer system between 5 districts of a city.

The lengths, in miles, of possible sewer pipes between districts is shown in the table.

The water company wants to create a network connecting, directly or indirectly, each of the 5 districts using the minimum possible total length of sewer pipes.

Find the minimum possible total length of sewer pipes required.

As part of their planning, the water company carries out a geological survey of the city.

As a result, the water company cannot connect Hoole directly to Upton in the plans for the new sewer system.

Explain the effect, if any, this has on your answer to part (a).

A project is being planned by Lepton Electricity Installations.

The project is broken down into 10 separate activities A, B, …, J and the activity network for the project is shown in Figure 1

The duration of each activity is given in days.

Find the earliest start time and the latest finish time for each activity and write these values on the activity network in Figure 1

Determine the float of activity F

On Figure 2 draw a cascade diagram (Gantt chart) for the project, assuming that each activity starts as early as possible.

The network below represents a system of pipes that transport chemicals at a factory.

The numbers on each arc represent the lower and upper capacity of each pipe in litres per second.

The management team at the factory plan to connect a supersource and a supersink to the network.

Determine which node(s) should connect to the supersource and which node(s) should connect to the supersink.

Find the value of the cut { A, B, C, E, F, H } { D, G, I, J }

Andrew, a manager at the factory, finds a flow of 79 litres per second through the network. He claims this is the maximum possible flow through the network.

Explain whether this claim is correct.

Fully justify your answer.

Andrew also claims that introducing a supersource and a supersink to the network will increase the maximum flow through the network.

Explain why this claim is incorrect.

A large bakery makes and sells two different types of loaf: malt and rye.

Each day, the bakery makes x malt loaves and y rye loaves.

The bakery wants to maximise the total profit, in £s, it generates each day from these two types of loaf.

The bakery formulates their situation as a linear programming problem by considering: the total profit, £P, generated on all the malt loaves and rye loaves made each day

the total time available to make the loaves each day

the ingredients available to make the loaves each day

a minimum number of each type of loaf made each day.

The bakery’s linear programming problem is shown below.

Maximise
P = 2 x + 3 y

and
x, y are integers

The graph shows the line 3 x + 2 y = 210

On the graph, draw the line x + 2y = 150

On the graph, identify the feasible region of the linear programming problem.

Hence, find the number of malt loaves and the number of rye loaves the bakery should make each day.

Find the maximum total profit, in £s, the bakery will generate each day.

Give a reason why your answer to part (b)(ii) may not be accurate.

The connected planar graph G has n² vertices, 2n edges and 3n – 4 faces, where n is an integer.

Show that n = 2

Hence, write down the number of vertices, edges and faces for G

In the case when G is Eulerian, draw G

Alexie claims that in the case when G is Eulerian, G is also a bipartite graph.

Comment on the validity of Alexie’s claim.

In the case when G is semi-Eulerian, draw G

Zackary claims that in the case when G is semi-Eulerian, G is also a tree.

Comment on the validity of Zackary’s claim.

State, with a reason, which strategy John should never play.

Find the optimal mixed strategy for John.

Fully justify your answer.

The set S contains the elements (a, b) where a, b ∈ ℝ except (0, 0)

The group G is formed by S under the binary operation ◊ where

(a, b) ◊ (c, d) = (a × d + b × c, b × d – a × c)

Prove that (0, 1) is the identity element of G

It is given that (a, b) ∈ G

Find, in terms of a and b, the inverse of (a, b) in G