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AQA Further Paper 3 Discrete 2020 June Q4
7 marks Standard +0.3
4 Joe, a courier, is required to deliver parcels to six different locations, \(A , B , C , D , E\) and \(F\). Joe needs to start and finish his journey at the depot.
The distances, in miles, between the depot and the six different locations are shown in the table below.
Depot\(\boldsymbol { A }\)\(\boldsymbol { B }\)C\(\boldsymbol { D }\)\(E\)\(F\)
Depot-181715161930
\(\boldsymbol { A }\)18-2920253521
B1729-26301614
C152026-283127
D16253028-3424
E1935163134-28
F302114272428-
The minimum total distance that Joe can travel in order to make all six deliveries, starting and finishing at the depot, is \(L\) miles. 4
  1. Using the nearest neighbour algorithm starting from the depot, find an upper bound for \(L\).
    4
  2. By deleting the depot, find a lower bound for \(L\).
    4
  3. Joe starts from the depot, delivers parcels to all six different locations and arrives back at the depot, covering 134 miles in the process. Joe claims that this is the minimum total distance that is possible for the journey. Comment on Joe's claim.
AQA Further Paper 3 Discrete 2020 June Q5
4 marks Moderate -0.5
5 The planar graph \(P\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-08_410_406_360_817} 5
  1. Determine the number of faces of \(P\).
    5
  2. Akwasi claims that \(P\) is semi-Eulerian as it is connected and it has exactly two vertices with even degree. Comment on the validity of Akwasi's claim. \includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-09_2488_1716_219_153}
AQA Further Paper 3 Discrete 2020 June Q6
8 marks Challenging +1.2
6 The group \(( G , \boldsymbol { A } )\) has the elements \(e , r , r ^ { 2 } , q , q r\) and \(q r ^ { 2 }\), where \(r ^ { 2 } = r \boldsymbol { \Delta } r , q r = q \boldsymbol { \Delta } r , q r ^ { 2 } = q \boldsymbol { \Delta } r ^ { 2 }\) and \(e\) is the identity element of \(G\). The elements \(q\) and \(r\) have the following properties: $$\begin{aligned} & r \boldsymbol { \Delta } r \boldsymbol { \Delta } r = e \\ & q \boldsymbol { \Delta } q = e \\ & r ^ { 2 } \boldsymbol { \Delta } q = q \boldsymbol { \Delta } r \end{aligned}$$ 6
    1. State the order of \(G\). 6
      1. (ii) Prove that the inverse of \(q r\) is \(q r\).
        6
    2. Complete the Cayley table for elements of \(G\). 6
    3. Complete the Cayley table for elements of \(G\).
      A\(e\)\(r\)\(r ^ { 2 }\)\(q\)\(q r\)\(q r ^ { 2 }\)
      \(e\)\(e\)\(r\)\(r ^ { 2 }\)\(q\)\(q r\)\(q r ^ { 2 }\)
      \(r\)\(r\)\(r ^ { 2 }\)\(e\)
      \(r ^ { 2 }\)\(r ^ { 2 }\)\(e\)\(r\)
      \(q\)\(q\)\(q r\)\(q r ^ { 2 }\)\(e\)
      \(q r\)\(q r\)\(q r ^ { 2 }\)\(q\)\(r ^ { 2 }\)
      \(q r ^ { 2 }\)\(q r ^ { 2 }\)\(q\)\(q r\)\(r\)\(r ^ { 2 }\)\(e\)
      6
    4. State the name of a group which is isomorphic to \(G\).
AQA Further Paper 3 Discrete 2020 June Q7
11 marks Moderate -0.5
7 An engineering company makes brake kits and clutch kits to sell to motorsport teams. The table below summarises the time taken and costs involved in making the two different types of kit.
Type of kitTime taken to make a kit (hours)Cost to engineering company per kit (£)Profit to engineering company per kit (£)
Brake kit55002000
Clutch kit32001000
The workers at the engineering company have a combined 2500 hours available to make the kits every month. The engineering company has \(\pounds 200000\) available to cover the costs of making the kits every month. To meet the minimum demands of the motorsport teams, the engineering company must make at least 100 of each type of kit every month. 7
  1. Using a graphical method on the grid opposite, find the number of each type of kit that the engineering company should make every month, in order to maximise its total monthly profit. Show clearly how you obtain your answer. \includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-13_2486_1709_221_153} Do not write outside the box 7
  2. Give a reason why the engineering company may not be able to make the number of each kit that you found in part (a). 7
  3. During one particular month the engineering company removes the need to make at least 100 of each type of kit. Explain whether or not this has an effect on your answer to part (a).
AQA Further Paper 3 Discrete 2020 June Q8
10 marks Challenging +1.2
8 Daryl and Clare play a zero-sum game. The game is represented by the following pay-off matrix for Daryl. Clare
AQA Further Paper 3 Discrete 2021 June Q1
1 marks Easy -1.8
1 Which of the following statements about critical path analysis is always true? Tick ( \(\checkmark\) ) one box. All activity networks have exactly one critical path. □ All critical activities have a non-zero float. □ The first activity in a critical path has an earliest start time of zero. □ A delay on a critical activity may not delay the project. □
AQA Further Paper 3 Discrete 2021 June Q2
1 marks Standard +0.3
2 The network below represents a system of pipes. The numbers on each arc represent the lower and upper capacity for each pipe. \includegraphics[max width=\textwidth, alt={}, center]{59347089-ea4a-4ee6-b40e-1ab78aa7cdc3-03_616_1415_447_310} Find the value of the cut \(\{ A , B , C , D , E \} \{ F , G , H , I \}\).
Circle your answer. 56586370
AQA Further Paper 3 Discrete 2021 June Q3
8 marks Standard +0.3
3 A mining company wants to open a new mine in an area where the ground contains a precious metal. The mining company has carried out a survey of the area. The network below shows nodes which represent the entrance to the new mine, \(X\), and the 8 ventilation shafts, \(A , B , \ldots , H\), which have been installed to prevent the build up of dangerous gases underground. \includegraphics[max width=\textwidth, alt={}, center]{59347089-ea4a-4ee6-b40e-1ab78aa7cdc3-04_846_1228_623_404} Each arc represents a possible underground tunnel which could be mined.
The weight on each arc represents the estimated amount of precious metal in that possible underground tunnel in tonnes. Due to geological reasons, the mining company can only create 8 underground tunnels. All 8 ventilation shafts must be accessible from the entrance of the mine. 3
    1. The mining company wants to maximise the amount of precious metal it can extract from the new mine. Determine the tunnels the mining company should use.
      3
      1. (ii) Estimate the maximum amount of precious metal the mining company can extract from the new mine. 3
    2. Comment on why the maximum amount of precious metal the mining company can extract from the new mine may be different from your answer to part (a)(ii).
      [0pt] [2 marks]
      3
    3. Before the mining company begins work on the new mine, a government survey prevents the mining company drilling the tunnel represented by \(C F\). Determine the effect, if any, the government survey has on your answers to part (a)(i) and part (a)(ii).
AQA Further Paper 3 Discrete 2021 June Q4
7 marks Moderate -0.5
4 Derrick, a tanker driver, is required to deliver fuel to 6 different service stations \(A , B\), \(C , D , E\) and \(F\). Derrick needs to begin and finish his delivery journey at the refinery \(O\).
The distances, in miles, between the 7 locations which have a direct road between them are shown in the network below. \includegraphics[max width=\textwidth, alt={}, center]{59347089-ea4a-4ee6-b40e-1ab78aa7cdc3-06_921_1440_628_303} Derrick spends 30 minutes at each service station to complete the fuel delivery.
When driving, the tanker travels at an average speed of 40 miles per hour.
The minimum total time that it takes Derrick to travel to and deliver fuel to all 6 service stations, starting and finishing at the refinery, is \(T\) minutes. 4
  1. Using the nearest neighbour algorithm starting from the refinery, find an upper bound for \(T\) 4
  2. Before setting off to make his fuel deliveries, Derrick is notified that, due to a low bridge, the road represented by CE is not suitable for tankers to travel along. State, with a reason, the effect this new information has on your answer to part (a).
    [0pt] [2 marks]
AQA Further Paper 3 Discrete 2021 June Q5
11 marks Standard +0.3
5
  1. Describe the conditions necessary for a set of elements, \(S\), under a binary operation * to form a group.
    5
  2. In the multiplicative group of integers modulo 13, the group \(G\) is defined as $$G = \left( \langle 10 \rangle , \times _ { 13 } \right)$$ 5 (b) (i) Explain why \(G\) is an abelian group.
    5 (b) (ii) Find the order of \(G\).
    5
  3. State the identity element of \(G\) and prove it is an identity element. Fully justify your answer.
    5
  4. Find all the proper non-trivial subgroups of \(G\), giving your answers in the form \(\left( \langle g \rangle , \times _ { 13 } \right)\), where \(g\) is an integer less than 13
AQA Further Paper 3 Discrete 2021 June Q6
8 marks Challenging +1.8
6
  1. A connected planar graph has \(( x + 1 ) ^ { 2 }\) vertices, \(( 25 + 2 x - 2 y )\) edges and \(( y - 1 ) ^ { 2 }\) faces, where \(x > 0\) and \(y > 0\) Find the possible values for the number of vertices, edges and faces for the graph.
    [0pt] [6 marks]
    LL
    6
  2. Explain why \(K _ { 6 }\), the complete graph with 6 vertices, is not planar. Fully justify your answer.
AQA Further Paper 3 Discrete 2021 June Q7
14 marks Standard +0.3
7 Avon and Roj play a zero-sum game. The game is represented by the following pay-off matrix for Avon. 7 (c)
  1. Find the optimal mixed strategy for Avon.
    7
  2. Find the value of the game for Avon.
7 (d) Roj thinks that his best outcome from the game is to play strategy \(\mathbf { R } _ { \mathbf { 2 } }\) each time. Avon notices that Roj always plays strategy \(\mathbf { R } _ { \mathbf { 2 } }\) and Avon wants to use this knowledge to maximise his expected pay-off from the game. Explain how your answer to part (c)(i) should change and find Avon's maximum expected pay-off from the game. \includegraphics[max width=\textwidth, alt={}, center]{59347089-ea4a-4ee6-b40e-1ab78aa7cdc3-16_2490_1735_219_139}
AQA Further Paper 3 Discrete 2023 June Q1
1 marks Easy -1.8
1 The simple-connected graph \(G\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-02_271_515_632_762} The graph \(G\) has \(n\) faces. State the value of \(n\) Circle your answer. 2345
AQA Further Paper 3 Discrete 2023 June Q2
1 marks Moderate -0.5
2 Jonathan and Hoshi play a zero-sum game.
The game is represented by the following pay-off matrix for Jonathan.
\multirow{6}{*}{Jonathan}Hoshi
Strategy\(\mathbf { H } _ { \mathbf { 1 } }\)\(\mathbf { H } _ { \mathbf { 2 } }\)\(\mathbf { H } _ { \mathbf { 3 } }\)
\(\mathbf { J } _ { \mathbf { 1 } }\)-232
\(\mathbf { J } _ { \mathbf { 2 } }\)320
\(\mathbf { J } _ { \mathbf { 3 } }\)4-13
\(\mathbf { J } _ { \mathbf { 4 } }\)310
The game does not have a stable solution.
Which strategy should Jonathan never play?
Circle your answer.
[0pt] [1 mark] \(\mathbf { J } _ { \mathbf { 1 } }\) \(\mathbf { J } _ { \mathbf { 2 } }\) \(\mathbf { J } _ { \mathbf { 3 } }\) \(\mathbf { J } _ { \mathbf { 4 } }\)
AQA Further Paper 3 Discrete 2023 June Q3
1 marks Easy -1.8
3 A student is solving a maximising linear programming problem. The graph below shows the constraints, feasible region and objective line for the student's linear programming problem. \includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-03_1248_1184_502_427} Which vertex is the optimal vertex? Circle your answer. \(A\) B
C
D
AQA Further Paper 3 Discrete 2023 June Q4
5 marks Standard +0.3
4 The network below represents a system of water pipes in a geothermal power station. The numbers on each arc represent the lower and upper capacity for each pipe in gallons per second. \includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-04_837_1413_493_312} The water is taken from a nearby river at node \(A\) The water is then pumped through the system of pipes and passes through one of three treatment facilities at nodes \(H , I\) and \(J\) before returning to the river. 4
  1. The senior management at the power station want all of the water to undergo a final quality control check at a new facility before it returns to the river. Using the language of networks, explain how the network above could be modified to include the new facility. 4
  2. Find the value of the cut \(\{ A , B , C , D , E \} \{ F , G , H , I , J \}\) 4
  3. Tim, a trainee engineer at the power station, correctly calculates the value of the cut \(\{ A , B , C , D , E , F \} \{ G , H , I , J \}\) to be 106 gallons per second. Tim then claims that the maximum flow through the network of pipes is 106 gallons per second. Comment on the validity of Tim's claim.
AQA Further Paper 3 Discrete 2023 June Q5
8 marks Standard +0.3
5 A student is solving the following linear programming problem. $$\begin{array} { l r } \text { Minimise } & Q = - 4 x - 3 y \\ \text { subject to } & x + y \leq 520 \\ & 2 x - 3 y \leq 570 \\ \text { and } & x \geq 0 , y \geq 0 \end{array}$$ 5
  1. The student wants to use the simplex algorithm to solve the linear programming problem. They modify the linear programming problem by introducing the objective function $$P = 4 x + 3 y$$ and the slack variables \(r\) and \(s\) State one further modification that must be made to the linear programming problem so that it can be solved using the simplex algorithm. 5
  2. (i) Complete the initial simplex tableau for the modified linear programming problem.
    [0pt] [2 marks]
    \(P\)\(x\)\(y\)\(r\)\(S\)value
    5 (b) (ii) Hence, perform one iteration of the simplex algorithm.
    \(P\)\(x\)\(y\)\(r\)\(s\)value
    5
  3. The student performs one further iteration of the simplex algorithm, which results in the following correct simplex tableau.
    \(P\)\(x\)\(y\)\(r\)\(s\)value
    100\(\frac { 18 } { 5 }\)\(\frac { 1 } { 5 }\)1986
    001\(\frac { 2 } { 5 }\)\(- \frac { 1 } { 5 }\)94
    010\(\frac { 3 } { 5 }\)\(\frac { 1 } { 5 }\)426
    5 (c) (i) Explain how the student can tell that the optimal solution to the modified linear programming problem can be determined from the above simplex tableau.
    5 (c) (ii) Find the optimal solution of the original linear programming problem.
AQA Further Paper 3 Discrete 2023 June Q6
8 marks Standard +0.3
6 A council wants to grit all of the roads on a housing estate. The network shows the roads on a housing estate. Each node represents a junction between two or more roads and the weight of each arc represents the length, in metres, of the road. \includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-08_1145_1458_539_292} The total length of all of the roads on the housing estate is 9175 metres.
In order to grit all of the roads, the council requires a gritter truck to travel along each road at least once. The gritter truck starts and finishes at the same junction. 6
  1. The gritter truck starts gritting the roads at 7:00 pm and moves with an average speed of 5 metres per second during its journey. Find the earliest time for the gritter truck to have gritted each road at least once and arrived back at the junction it started from, giving your answer to the nearest minute. Fully justify your answer.
    [0pt] [6 marks]
    6
  2. Explain how a refinement to the council's requirement, that the gritter truck must start and finish at the same junction, could reduce the time taken to grit all of the roads at least once.
    [2 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
    The planning involves producing an activity network for the project, which is shown in Figure 1 below. The duration of each activity is given in weeks. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-10_965_1600_559_221}
    \end{figure}
AQA Further Paper 3 Discrete 2023 June Q7
6 marks Moderate -0.8
7
    1. Find the earliest start time and the latest finish time for each activity and write these values on the activity network in Figure 1 7
      1. (ii) Write down the critical path. 7
    2. On Figure 2 below, draw a cascade diagram (Gantt chart) for the planned building project, assuming that each activity starts as early as possible. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-11_1127_1641_539_201}
      \end{figure} 7
    3. During further planning of the building project, Nova Merit Construction find that activity \(F\) is not necessary and they remove it from the project. Explain the effect removing activity \(F\) has on the minimum completion time of the project.
AQA Further Paper 3 Discrete 2023 June Q8
6 marks Moderate -0.3
8 The graph \(G\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-12_301_688_351_676} 8
    1. State, with a reason, whether or not \(G\) is simple. 8
      1. (ii) A student states that \(G\) is Eulerian.
        Explain why the student is correct. 8
    2. The graph \(H\) has 8 vertices with degrees 2, 2, 4, 4, 4, 4, 4 and 4 Comment on whether \(H\) is isomorphic to \(G\) 8
    3. The formula \(v - e + f = 2\), where \(v =\) number of vertices \(e =\) number of edges \(f =\) number of faces
      can be used with graphs which satisfy certain conditions. Prove that \(G\) does not satisfy the conditions for the above formula to apply.
AQA Further Paper 3 Discrete 2023 June Q9
14 marks Standard +0.3
9 The group \(\left( C , + _ { 4 } \right)\) contains the elements \(0,1,2\) and 3 9
    1. Show that \(C\) is a cyclic group.
      9
      1. (ii) State the group of symmetries of a regular polygon that is isomorphic to \(C\) 9
    2. The group ( \(V , \otimes\) ) contains the elements (1, 1), (1, -1), (-1, 1) and (-1, -1) The binary operation \(\otimes\) between elements of \(V\) is defined by $$( a , b ) \otimes ( c , d ) = ( a \times c , b \times d )$$ 9
      1. Find the element in \(V\) that is the inverse of \(( - 1,1 )\) Fully justify your answer.
        [0pt] [2 marks]
        9
    4. (ii) Determine, with a reason, whether or not \(C \cong V\) \(\mathbf { 9 }\) (c) The group \(G\) has order 16
      Rachel claims that as \(1,2,4,8\) and 16 are the only factors of 16 then, by Lagrange's theorem, the group \(G\) will have exactly 5 distinct subgroups, including the trivial subgroup and \(G\) itself. Comment on the validity of Rachel's claim. \includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-16_2493_1721_214_150}
Edexcel FD1 AS 2020 June Q1
6 marks Moderate -0.8
1. \(3.7 \quad 2.5\) \(5.4 \quad 1.9\) 2.7
3.2
3.1
2.7
4.2
2.0
  1. Use the first-fit bin packing algorithm to determine how the numbers listed above can be packed into bins of size 8.5 The first-fit bin packing algorithm is to be used to pack \(n\) numbers into bins. The number of comparisons is used to measure the order of the first-fit bin packing algorithm.
  2. By considering the worst case, determine the order of the first-fit bin packing algorithm in terms of \(n\). You must make your method and working clear.
Edexcel FD1 AS 2020 June Q2
14 marks Moderate -0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2a6e659-aab5-4eec-9af4-ca6ab895f1c8-03_693_1379_233_342} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A project is modelled by the activity network shown in Figure 1. The activities are represented by the arcs. The number in brackets on each arc gives the time, in hours, to complete the corresponding activity. Each activity requires one worker. The project is to be completed in the shortest possible time.
  1. Complete the precedence table in the answer book.
  2. Complete Diagram 1 in the answer book to show the early event times and the late event times.
    1. State the minimum project completion time.
    2. List the critical activities.
  4. Calculate the maximum number of hours by which activity H could be delayed without affecting the shortest possible completion time of the project. You must make the numbers used in your calculation clear.
  5. Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working.
  6. Draw a cascade chart for this project on Grid 1 in the answer book.
  7. Using the answer to (f), explain why it is not possible to complete the project in the shortest possible time using the number of workers found in (e).
Edexcel FD1 AS 2020 June Q3
11 marks Challenging +1.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2a6e659-aab5-4eec-9af4-ca6ab895f1c8-04_720_1470_233_296} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} [The weight of the network is \(5 x + 246\) ]
  1. Explain why it is not possible to draw a graph with an odd number of vertices of odd valency. Figure 2 represents a network of 14 roads in a town. The expression on each arc gives the time, in minutes, to travel along the corresponding road. Prim's algorithm, starting at A, is applied to the network. The order in which the arcs are selected is \(\mathrm { AD } , \mathrm { DH } , \mathrm { DG } , \mathrm { FG } , \mathrm { EF } , \mathrm { CG } , \mathrm { BD }\). It is given that the order in which the arcs are selected is unique.
  2. Using this information, find the smallest possible range of values for \(x\), showing your working clearly. A route that minimises the total time taken to traverse each road at least once is required. The route must start and finish at the same vertex. Given that the time taken to traverse this route is 318 minutes,
  3. use an appropriate algorithm to determine the value of \(x\), showing your working clearly.
Edexcel FD1 AS 2020 June Q4
9 marks Challenging +1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2a6e659-aab5-4eec-9af4-ca6ab895f1c8-05_1472_1320_233_376} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the constraints of a linear programming problem in \(x\) and \(y\), where \(R\) is the feasible region. Figure 3 also shows an objective line for the problem and the optimal vertex, which is labelled as \(V\). The value of the objective at \(V\) is 556
Express the linear programming problem in algebraic form. List the constraints as simplified inequalities with integer coefficients and determine the objective. Please check the examination details below before entering your candidate information
Candidate surname
Other names Pearson Edexcel
Centre Number
Candidate Number Level 3 GCE \includegraphics[max width=\textwidth, alt={}, center]{a2a6e659-aab5-4eec-9af4-ca6ab895f1c8-09_122_433_356_991}
□
□
□
□ \section*{Thursday 14 May 2020} Afternoon
Paper Reference 8FMO/27 \section*{Further Mathematics} Advanced Subsidiary
Further Mathematics options
27: Decision Mathematics 1
(Part of options D, F, H and K) \section*{Answer Book} Do not return the question paper with the answer book.
1. \(\begin{array} { l l l l l l l l l l } 3.7 & 2.5 & 5.4 & 1.9 & 2.7 & 3.2 & 3.1 & 2.7 & 4.2 & 2.0 \end{array}\)
  1. (a)
Activity
Immediately
preceding
activities
A
B
C
D
Activity
Immediately
preceding
activities
E
F
G
H
Activity
Immediately
preceding
activities
I
J
K
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2a6e659-aab5-4eec-9af4-ca6ab895f1c8-12_734_1646_925_196} \captionsetup{labelformat=empty} \caption{Diagram 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2a6e659-aab5-4eec-9af4-ca6ab895f1c8-13_1116_1475_979_296} \captionsetup{labelformat=empty} \caption{Grid 1}
\end{figure} 3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2a6e659-aab5-4eec-9af4-ca6ab895f1c8-14_716_1467_255_299} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} [The weight of the network is \(5 x + 246\) ] 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2a6e659-aab5-4eec-9af4-ca6ab895f1c8-18_1470_1319_255_388} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure}