Questions D1 (932 questions)

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Edexcel D1 Q4
11 marks Standard +0.3
4. The following matrix gives the capacities of the pipes in a system.
To FromS\(T\)A\(B\)\(C\)D
S--1626--
T------
A----135
B-16---11
C-11----
D-11----
  1. Represent this information as a digraph.
  2. Find the minimum cut, expressing it in the form \(\{ \} \mid \{ \}\), and state its value.
  3. Starting from having no flow in the system, use the labelling procedure to find a maximal flow through the system. You should list each flow-augmenting route you use, together with its flow.
  4. Explain how you know that this flow is maximal.
Edexcel D1 Q5
11 marks Moderate -0.5
5. This question should be answered on the sheet provided. An algorithm is described by the flow chart shown in Figure 1 below. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1e518ab0-9852-4d1d-a4c9-344a5edf9547-05_1337_937_388_404} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Complete the table on the answer sheet recording the results of each instruction as the algorithm is applied and state the final output.
  2. Explain what the algorithm achieves.
  3. Attempt to apply the algorithm again, with the initial value of \(a\), as specified in Box 2, changed to 5 . Explain what happens.
    (2 mark)
  4. Find the set of positive initial values of \(a\) for which the algorithm will work.
    (2 marks)
Edexcel D1 Q6
14 marks Moderate -0.5
6. The manager of a new leisure complex needs to maximise the Revenue \(( \pounds R )\) from providing the following two weekend programmes.
\(\frac { \text { Participants } } { \text { Children } }\)7 hours windsurfing, 2 hours sailing\(\frac { \text { Revenue } } { \pounds 50 }\)
Adults5 hours windsurfing, 6 hours sailing\(\pounds 100\)
The following restrictions apply to each weekend.
No more than 90 participants can be accommodated.
There must be at most 40 adults.
A maximum of 600 person-hours of windsurfing can be offered.
A maximum of 300 person-hours of sailing can be offered.
  1. Formulate the above information as a linear programming problem, listing the constraints as inequalities and stating the objective function \(R\).
  2. On graph paper, illustrate the constraints, indicating clearly the feasible region.
  3. Solve the problem graphically, stating how many adults and how many children should be accepted each weekend and what the revenue will be. The manager is considering buying more windsurfing equipment at a cost of \(\pounds 2000\). This would increase windsurfing provision by \(10 \%\).
  4. State, with a reason, whether such a purchase would be cost effective.
Edexcel D1 Q7
14 marks Standard +0.3
7. This question should be answered on the sheet provided. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1e518ab0-9852-4d1d-a4c9-344a5edf9547-07_576_1360_331_278} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows an activity network modelling the tasks involved in widening a bridge over the B451. The arcs represent the tasks and the numbers in brackets gives the time, in days, to complete each task.
  1. Find the early and late times for each event.
  2. Determine those activities which lie on the critical path and list them in order.
  3. State the minimum length of time needed to widen the bridge. Each task needs a single worker.
  4. Show that two men would not be sufficient to widen the bridge in the shortest time.
    (2 marks)
  5. Draw up a schedule showing how 3 men could complete the project in the shortest time. \section*{Please hand this sheet in for marking}
    1. Complete matching:
      \(P\)\(\bullet\)\(\bullet\)\(D\)
      \(Q\)\(\bullet\)\(\bullet\)\(G\)
      \(R\)\(\bullet\)\(\bullet\)\(E\)
      \(S\)\(\bullet\)\(\bullet\)\(L ( H )\)
      \(T\)\(\bullet\)\(\bullet\)\(L\)
      \section*{Please hand this sheet in for marking}
    2. \(x\)\(a\)\(b\)\(( a - b ) < 0.01\) ?
      1005026No
      -2614.923No
      Final output
    3. \(\_\_\_\_\)
    4. \(x\)\(a\)\(b\)\(( a - b ) < 0.01 ?\)
      100
    5. \(\_\_\_\_\) \section*{Please hand this sheet in for marking}
    6. \includegraphics[max width=\textwidth, alt={}, center]{1e518ab0-9852-4d1d-a4c9-344a5edf9547-11_768_1689_427_221}
    7. \(\_\_\_\_\)
    8. \(\_\_\_\_\)
    9. 051015202530354045505560
      Worker 1
      Worker 2
    10. 051015202530354045505560
      Worker 1
      Worker 2
      Worker 3
Edexcel D1 2015 January Q1
6 marks Moderate -0.8
1.
ABCDEFGH
A-981317111210
B9-11211524137
C811-2023171715
D132120-15281118
E17152315-312330
F1124172831-1315
G121317112313-23
H1071518301523-
The table represents a network that shows the time taken, in minutes, to travel by car between eight villages, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G }\) and H .
  1. Use Prim's algorithm, starting at A , to find a minimum spanning tree for this network. You must list the arcs that form your tree in the order in which you select them.
  2. Draw your minimum spanning tree using the vertices given in Diagram 1 in the answer book and state the weight of the tree.
  3. State whether your minimum spanning tree is unique. Justify your answer.
Edexcel D1 2015 January Q2
8 marks Moderate -0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00abfcc0-63b3-4784-a4b5-06aba234068c-3_616_561_278_356} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00abfcc0-63b3-4784-a4b5-06aba234068c-3_606_552_285_1146} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 1 shows the possible allocations of six workers, Amrit (A), Bernard (B), Cameron (C), David (D), Emily (E) and Francis (F), to six tasks, 1, 2, 3, 4, 5 and 6
  1. Explain why it is not possible to find a complete matching. Figure 2 shows an initial matching. Starting from this initial matching,
  2. find the two alternating paths that start at C .
  3. List the two improved matchings generated by using the two alternating paths found in (b). After training, task 5 is added to Bernard's possible allocation.
    Starting from either of the two improved matchings found in (c),
  4. use the maximum matching algorithm to obtain a complete matching. You must list the additional alternating path that you use, and state the complete matching.
    (3)
Edexcel D1 2015 January Q3
14 marks Easy -1.2
3. $$\begin{array} { l l l l l l l l l l } 1.1 & 0.7 & 1.9 & 0.9 & 2.1 & 0.2 & 2.3 & 0.4 & 0.5 & 1.7 \end{array}$$
  1. Use the first-fit bin packing algorithm to determine how the numbers listed above can be packed into bins of size 3 The list is to be sorted into descending order.
    1. Starting at the left-hand end of the list, perform one pass through the list using a bubble sort. Write down the list that results at the end of your first pass.
    2. Write down the number of comparisons and the number of swaps performed during your first pass. After a second pass using this bubble sort, the updated list is $$\begin{array} { l l l l l l l l l l } 1.9 & 1.1 & 2.1 & 0.9 & 2.3 & 0.7 & 0.5 & 1.7 & 0.4 & 0.2 \end{array}$$
  3. Use a quick sort on this updated list to obtain the fully sorted list. You must make your pivots clear.
  4. Apply the first-fit decreasing bin packing algorithm to your fully sorted list to pack the numbers into bins of size 3
Edexcel D1 2015 January Q4
16 marks Challenging +1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00abfcc0-63b3-4784-a4b5-06aba234068c-5_889_1591_258_239} \captionsetup{labelformat=empty} \caption{Figure 3
[0pt] [The total weight of the network is 100]}
\end{figure} Figure 3 represents a network of pipes in a building. The number on each arc represents the length, in metres, of the corresponding pipe.
  1. Use Dijkstra's algorithm to find the shortest path from A to J . State your path and its length. On a particular day Kim needs to check each pipe. A route of minimum length, which traverses each pipe at least once and starts and finishes at A, needs to be found.
  2. Use an appropriate algorithm to find the arcs that will need to be traversed twice. You must make your method and working clear.
  3. Write down a possible route, giving its length. All the pipes directly attached to B are removed. Kim needs to check all the remaining pipes and may now start at any vertex and finish at any vertex. A route is required that excludes all those pipes directly attached to B .
  4. State all possible combinations of starting and finishing points so that the length of Kim's route is minimised. State the length of Kim's route.
Edexcel D1 2015 January Q5
7 marks Moderate -0.8
5.
ActivityImmediately preceding activities
A-
B-
CA
DA
EA, B
FC
GC, D
HE
IE
JH, I
KF, G
  1. Draw the activity network described in the precedence table, using activity on arc. Your activity network must contain only the minimum number of dummies.
    (5)
  2. Explain why, in general, dummies may be required in an activity network.
Edexcel D1 2015 January Q6
12 marks Moderate -0.8
6. Jonathan is going to make hats to sell at a fete. He can make red hats and green hats. Jonathan can use linear programming to determine the number of each colour of hat that he should make. Let \(x\) be the number of red hats he makes and \(y\) be the number of green hats he makes.
One of the constraints is that there must be at least 30 hats.
  1. Write down an inequality, in terms of \(x\) and \(y\), to model this constraint. Two further constraints are $$\begin{aligned} & 2 y + x \geqslant 40 \\ & 2 y - x \geqslant - 30 \end{aligned}$$
  2. Write down two more constraints which apply.
  3. Represent all these constraints on Diagram 1 in the answer book. Hence determine, and label, the feasible region R . The cost of making a green hat is three times the cost of making a red hat. Jonathan wishes to minimise the total cost.
  4. Use the objective line (ruler) method to determine the number of red hats and number of green hats that Jonathan should make. You must clearly draw and label your objective line. Given that the minimum total cost of making the hats is \(\pounds 107.50\)
  5. determine the cost of making one green hat and the cost of making one red hat. You must make your method clear.
Edexcel D1 2015 January Q7
12 marks Moderate -0.5
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00abfcc0-63b3-4784-a4b5-06aba234068c-8_980_1577_229_268} \captionsetup{labelformat=empty} \caption{Figure 4
[0pt] [The sum of all the activity durations is 99 days]}
\end{figure} The network in Figure 4 shows the activities that need to be undertaken to complete a project. Each activity is represented by an arc and the duration of the activity, in days, is shown in brackets. The early event times and late event times are to be shown at each vertex and some have been completed for you. Given that activity F is a critical activity and that the total float on activity G is 2 days,
  1. write down the value of \(x\) and the value of \(y\),
  2. calculate the missing early event times and late event times and hence complete Diagram 1 in your answer book. Each activity requires one worker and the project must be completed in the shortest possible time.
  3. Calculate a lower bound for the number of workers needed to complete the project in the shortest possible time.
  4. Draw a cascade (Gantt) chart for this project on Grid 1 in the answer book.
  5. Use your cascade chart to determine the minimum number of workers needed to complete the project in the shortest possible time. You must make specific reference to times and activities. (You do not need to provide a schedule of the activities.)
Edexcel D1 2016 January Q1
8 marks Moderate -0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a3ca2743-2311-4225-8b78-dcd5eb592704-2_741_558_370_372} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a3ca2743-2311-4225-8b78-dcd5eb592704-2_734_561_374_1114} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 1 shows the possible allocations of six workers, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\) and F , to six tasks, 1, 2, 3, 4, 5 and 6. Figure 2 shows an initial matching.
  1. Starting from this initial matching, use the maximum matching algorithm to find an improved matching. You must list the alternating path used and state your improved matching.
  2. Explain why it is not possible to find a complete matching. Now, exactly one worker may be trained so that a complete matching becomes possible.
    Either worker A can be trained to do task 1 or worker E can be trained to do task 4.
  3. Decide which worker, A or E , should be trained. Give a reason for your answer. You may now assume that the worker you identified in (c) has been trained.
  4. Starting from the improved matching found in (a), use the maximum matching algorithm to find a complete matching. You must list the alternating path used and state your complete matching.
Edexcel D1 2016 January Q2
10 marks Easy -1.2
2. Kruskal's algorithm finds a minimum spanning tree for a connected graph with \(n\) vertices.
  1. Explain the terms
    1. connected graph,
    2. tree,
    3. spanning tree.
  2. Write down, in terms of \(n\), the number of arcs in the minimum spanning tree. The table below shows the lengths, in km, of a network of roads between seven villages, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F }\) and G .
    ABCDEFG
    A-17-1930--
    B17-2123---
    C-21-27293122
    D192327--40-
    E30-29--3325
    F--314033-39
    G--22-2539-
  3. Complete the drawing of the network on Diagram 1 in the answer book by adding the necessary arcs from vertex C together with their weights.
  4. Use Kruskal's algorithm to find a minimum spanning tree for the network. You should list the arcs in the order that you consider them. In each case, state whether you are adding the arc to your minimum spanning tree.
  5. State the weight of the minimum spanning tree.
Edexcel D1 2016 January Q3
15 marks Easy -1.2
3.
6.4
7.9
8.1
12.19 .3
14.0
15.7
17.4
20.1
  1. Use the first-fit bin packing algorithm to determine how the numbers listed above can be packed into bins of size 33 The list is to be sorted into descending order.
    1. Starting at the left-hand end of the list, perform two passes through the list using a bubble sort. Write down the state of the list that results at the end of each pass.
    2. Write down the total number of comparisons and the total number of swaps performed during your two passes.
  3. Use a quick sort on the original list to obtain a fully sorted list in descending order. You must make your pivots clear.
  4. Use the first-fit decreasing bin packing algorithm to determine how the numbers listed can be packed into bins of size 33
  5. Determine whether your answer to (d) uses the minimum number of bins. You must justify your answer.
Edexcel D1 2016 January Q4
15 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a3ca2743-2311-4225-8b78-dcd5eb592704-5_926_1479_219_296} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} [The total weight of the network is 196]
Figure 3 models a network of roads. The number on each edge gives the time, in minutes, taken to travel along that road. Oliver wishes to travel by road from A to K as quickly as possible.
  1. Use Dijkstra's algorithm to find the shortest time needed to travel from A to K . State the quickest route.
    (6) On a particular day Oliver must travel from B to K via A.
  2. Find a route of minimal time from B to K that includes A , and state its length.
    (2) Oliver needs to travel along each road to check that it is in good repair. He wishes to minimise the total time required to traverse the network.
  3. Use the route inspection algorithm to find the shortest time needed. You must state all combinations of edges that Oliver could repeat, making your method and working clear.
Edexcel D1 2016 January Q5
11 marks Moderate -0.8
5. A linear programming problem in \(x\) and \(y\) is described as follows. $$\begin{array} { l l } \text { Maximise } & \mathrm { P } = 5 x + 3 y \\ \text { subject to: } & x \geqslant 3 \\ & x + y \leqslant 9 \\ & 15 x + 22 y \leqslant 165 \\ & 26 x - 50 y \leqslant 325 \end{array}$$
  1. Add lines and shading to Diagram 1 in the answer book to represent these constraints. Hence determine the feasible region and label it R .
  2. Use the objective line method to find the optimal vertex, V, of the feasible region. You must draw and label your objective line and label vertex V clearly.
  3. Calculate the exact coordinates of vertex V and hence calculate the corresponding value of P at V . The objective is now to minimise \(5 x + 3 y\), where \(x\) and \(y\) are integers.
  4. Write down the minimum value of \(5 x + 3 y\) and the corresponding value of \(x\) and corresponding value of \(y\).
Edexcel D1 2016 January Q6
16 marks Moderate -0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a3ca2743-2311-4225-8b78-dcd5eb592704-7_664_1520_239_276} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A project is modelled by the activity network shown in Figure 4. The activities are represented by the arcs. The number in brackets on each arc gives the time required, in hours, to complete the activity. The numbers in circles are the event numbers. Each activity requires one worker.
  1. Explain the significance of the dummy activity
    1. from event 5 to event 6
    2. from event 7 to event 9
  2. Complete Diagram 1 in the answer book to show the early event times and the late event times.
  3. State the minimum project completion time.
  4. Calculate a lower bound for the minimum number of workers required to complete the project in the minimum time. You must show your working.
  5. On Grid 1 in your answer book, draw a cascade (Gantt) chart for this project.
  6. On Grid 2 in your answer book, construct a scheduling diagram to show that this project can be completed with three workers in just one more hour than the minimum project completion time.
    (3)
Edexcel D1 2017 January Q1
4 marks Easy -1.8
  1. Use the binary search algorithm to try to locate the name Hilbert in the following alphabetical list. Clearly indicate how you chose your pivots and which part of the list is being rejected at each stage.
Edexcel D1 2017 January Q2
5 marks Moderate -0.8
2.
ABCDEFGH
A-27513229234740
B27-243520423328
C5124-3743312634
D323537-39454430
E29204339-384555
F2342314538-5345
G473326444553-39
H40283430554539-
The table represents a network that shows the average journey time, in minutes, between eight towns, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G }\) and H .
  1. Use Prim's algorithm, starting at A , to find the minimum spanning tree for this network. You must clearly state the order in which you select the edges of your tree.
  2. Draw the minimum spanning tree using the vertices given in Diagram 1 in the answer book.
  3. State the weight of the minimum spanning tree.
Edexcel D1 2017 January Q3
7 marks Moderate -0.5
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8c9bce2c-4156-4bf6-8d02-9e01d6f11948-04_608_511_242_358} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8c9bce2c-4156-4bf6-8d02-9e01d6f11948-04_611_510_242_1201} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 1 shows the possible allocations of six workers, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\) and F , to six tasks, \(1,2,3\), 4, 5 and 6. Each task must be assigned to only one worker and each worker must be assigned to exactly one task. Figure 2 shows an initial matching.
  1. Starting from the given initial matching, use the maximum matching algorithm to find an alternating path from A to 4 . Hence find an improved matching. You should list the alternating path you use, and state your improved matching.
  2. Explain why it is not possible to find a complete matching. After training, task 1 is added to worker A's possible allocations.
  3. Starting from the improved matching found in (a), use the maximum matching algorithm to find a complete matching. You should list the alternating path you use, and state your complete matching.
    (3)
Edexcel D1 2017 January Q4
11 marks Easy -1.2
4. \(\begin{array} { l l l l l l l l l } 23 & 18 & 27 & 9 & 25 & 10 & 12 & 30 & 24 \end{array}\) The numbers in the list represent the weights, in kilograms, of nine suitcases. The suitcases are to be transported in containers that will each hold a maximum weight of 45 kilograms.
  1. Calculate a lower bound for the number of containers that will be needed to transport the suitcases.
  2. Use the first-fit bin packing algorithm to allocate the suitcases to the containers.
  3. Using the list provided, carry out a bubble sort to produce a list of the weights in descending order. You need only give the state of the list after each complete pass.
  4. Use the first-fit decreasing bin packing algorithm to allocate the suitcases to the containers.
  5. Explain why it is not possible to transport the suitcases using fewer containers than the number used in (d).
Edexcel D1 2017 January Q5
9 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8c9bce2c-4156-4bf6-8d02-9e01d6f11948-06_897_1499_239_283} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} [The total weight of the network is 106.7]
Figure 3 models a network of cycle tracks that have to be inspected. The number on each arc represents the length, in km , of the corresponding track. Angela needs to travel along each cycle track at least once and wishes to minimise the length of her inspection route. She must start and finish at A.
  1. Use an appropriate algorithm to find the tracks that will need to be traversed twice. You should make your method and working clear.
  2. Find a route of minimum length, starting and finishing at A . State the length of your route. A new cycle track, AC, is under construction. It will be 15 km long. Angela will have to include this new track in her inspection route.
  3. State the effect this new track will have on the total length of her route. Justify your answer.
Edexcel D1 2017 January Q6
9 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8c9bce2c-4156-4bf6-8d02-9e01d6f11948-07_1052_1447_212_310} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 represents a network of roads. The number on each arc represents the time taken, in minutes, to drive along the corresponding road. Stieg wishes to minimise the time spent driving from his home at A , to his office at H . The amount of traffic on two of the roads leading into H varies each day, and so the length of time taken to drive along these roads is expressed in terms of \(x\), where \(x > 7\)
  1. Use Dijkstra's algorithm to find the possible routes that minimise the driving time from A to H . State the length of each route, leaving your answer in terms of \(x\) where necessary.
    (7) On a particular day, the quickest route from A to H via G is 2 minutes quicker than the quickest route from A to H via E .
  2. Calculate the value of \(x\). You must make your method and working clear.
Edexcel D1 2017 January Q7
14 marks Moderate -0.5
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8c9bce2c-4156-4bf6-8d02-9e01d6f11948-08_1024_1495_226_276} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A project is modelled by the activity network shown in Figure 5. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, to complete the corresponding activity. Each activity requires exactly one worker. The project is to be completed in the shortest possible time.
  1. Complete Diagram 1 in the answer book to show the early event times and late event times.
  2. Explain what is meant by a critical path.
  3. List the critical path for this network.
  4. For each of the situations below, state the effect that the delay would have on the project completion date.
    1. A 4-day delay during activity J.
    2. A 4-day delay during activity M . The delays mentioned in (d) do not occur.
  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. Schedule the activities using the minimum number of workers so that the project is completed in the minimum time.
Edexcel D1 2017 January Q8
16 marks Moderate -0.3
8. A shop sells three types of pen. These are ballpoint pens, rollerball pens and fountain pens. The shop manager knows that each week she should order
  • at least 50 pens in total
  • at least twice as many rollerball pens as fountain pens
In addition,
  • at most \(60 \%\) of the pens she orders must be ballpoint pens
  • at least a third of the pens she orders must be rollerball pens
Each ballpoint pen costs \(\pounds 2\), each rollerball pen costs \(\pounds 3\) and each fountain pen costs \(\pounds 5\) The shop manager wants to minimise her costs.
Let \(x\) represent the number of ballpoint pens ordered, let \(y\) represent the number of rollerball pens ordered and let \(z\) represent the number of fountain pens ordered.
  1. Formulate this information as a linear programming problem. State the objective and list the constraints as simplified inequalities with integer coefficients. The shop manager decides to order exactly 10 fountain pens. This reduces the problem to the following $$\begin{array} { l r } \text { Minimise } & P = 2 x + 3 y \\ \text { subject to } & x + y \geqslant 40 \\ & 2 x - 3 y \leqslant 30 \\ - x + 2 y \geqslant 10 \\ & y \geqslant 20 \\ & x \geqslant 0 \end{array}$$
  2. Represent these constraints on Diagram 1 in the answer book. Hence determine, and label, the feasible region R .
  3. Use the objective line method to find the optimal vertex, V, of the feasible region. You must make your objective line clear and label the optimal vertex V.
  4. Write down the number of each type of pen that the shop manager should order. Calculate the cost of this order.
    (Total \(\mathbf { 1 6 }\) marks)