Questions — Edexcel (10514 questions)

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Edexcel S1 Q5
Moderate -0.3
5. The following grouped frequency distribution summarises the number of minutes, to the nearest minute, that a random sample of 200 motorists were delayed by roadworks on a stretch of motorway.
Delay (mins)Number of motorists
\(4 - 6\)15
\(7 - 8\)28
949
1053
\(11 - 12\)30
\(13 - 15\)15
\(16 - 20\)10
  1. Using graph paper represent these data by a histogram.
  2. Give a reason to justify the use of a histogram to represent these data.
  3. Use interpolation to estimate the median of this distribution.
  4. Calculate an estimate of the mean and an estimate of the standard deviation of these data. One coefficient of skewness is given by $$\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } } .$$
  5. Evaluate this coefficient for the above data.
  6. Explain why the normal distribution may not be suitable to model the number of minutes that motorists are delayed by these roadworks.
Edexcel S1 Q7
Moderate -0.8
7. A music teacher monitored the sight-reading ability of one of her pupils over a 10 week period. At the end of each week, the pupil was given a new piece to sight-read and the teacher noted the number of errors \(y\). She also recorded the number of hours \(x\) that the pupil had practised each week. The data are shown in the table below.
  1. Given that \(\mathrm { E } ( X ) = - 0.2\), find the value of \(\alpha\) and the value of \(\beta\).
  2. Write down \(\mathrm { F } ( 0.8 )\).
    1. Evaluate \(\operatorname { Var } ( X )\).
Edexcel S1 Q4
Easy -1.2
4. Aeroplanes fly from City \(A\) to City \(B\). Over a long period of time the number of minutes delay in take-off from City \(A\) was recorded. The minimum delay was 5 minutes and the maximum delay was 63 minutes. A quarter of all delays were at most 12 minutes, half were at most 17 minutes and \(75 \%\) were at most 28 minutes. Only one of the delays was longer than 45 minutes. An outlier is an observation that falls either \(1.5 \times\) (interquartile range) above the upper quartile or \(1.5 \times\) (interquartile range) below the lower quartile.
  1. On the graph paper opposite draw a box plot to represent these data.
  2. Comment on the distribution of delays. Justify your answer.
  3. Suggest how the distribution might be interpreted by a passenger who frequently flies from City \(A\) to City \(B\). \includegraphics[max width=\textwidth, alt={}, center]{3d4f7bfb-b235-418a-9411-a4d0b3188254-008_1190_1487_278_223}
Edexcel S1 Q7
Easy -1.8
7. In a school there are 148 students in Years 12 and 13 studying Science, Humanities or Arts subjects. Of these students, 89 wear glasses and the others do not. There are 30 Science students of whom 18 wear glasses. The corresponding figures for the Humanities students are 68 and 44 respectively. A student is chosen at random. Find the probability that this student
  1. is studying Arts subjects,
  2. does not wear glasses, given that the student is studying Arts subjects. Amongst the Science students, \(80 \%\) are right-handed. Corresponding percentages for Humanities and Arts students are 75\% and 70\% respectively. A student is again chosen at random.
  3. Find the probability that this student is right-handed.
  4. Given that this student is right-handed, find the probability that the student is studying Science subjects.
  1. Describe the main features and uses of a box plot.
    Children from schools \(A\) and \(B\) took part in a fun run for charity. The times, to the nearest minute, taken by the children from school \(A\) are summarised in Figure 1. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{3d4f7bfb-b235-418a-9411-a4d0b3188254-015_398_1045_946_461}
    \end{figure}
    1. Write down the time by which \(75 \%\) of the children in school \(A\) had completed the run.
    2. State the name given to this value.
    (c) Explain what you understand by the two crosses ( X ) on Figure 1.
Edexcel S1 Q8
Moderate -0.8
8. The lifetimes of bulbs used in a lamp are normally distributed. A company \(X\) sells bulbs with a mean lifetime of 850 hours and a standard deviation of 50 hours.
  1. Find the probability of a bulb, from company \(X\), having a lifetime of less than 830 hours.
  2. In a box of 500 bulbs, from company \(X\), find the expected number having a lifetime of less than 830 hours. A rival company \(Y\) sells bulbs with a mean lifetime of 860 hours and \(20 \%\) of these bulbs have a lifetime of less than 818 hours.
  3. Find the standard deviation of the lifetimes of bulbs from company \(Y\). Both companies sell the bulbs for the same price.
  4. State which company you would recommend. Give reasons for your answer.
    \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{}
    \end{table}
Edexcel S1 2003 June Q1
5 marks Easy -1.8
  1. In a particular week, a dentist treats 100 patients. The length of time, to the nearest minute, for each patient's treatment is summarised in the table below.
Time
(minutes)
\(4 - 7\)8\(9 - 10\)11\(12 - 16\)\(17 - 20\)
Number
of
patients
122018221513
Draw a histogram to illustrate these data.
Edexcel S1 2003 June Q2
6 marks Moderate -0.5
2. The lifetimes of batteries used for a computer game have a mean of 12 hours and a standard deviation of 3 hours. Battery lifetimes may be assumed to be normally distributed. Find the lifetime, \(t\) hours, of a battery such that 1 battery in 5 will have a lifetime longer than \(t\).
Edexcel S1 2003 June Q3
10 marks Moderate -0.8
3. A company owns two petrol stations \(P\) and \(Q\) along a main road. Total daily sales in the same week for \(P ( \pounds p )\) and for \(Q ( \pounds q )\) are summarised in the table below.
\(p\)\(q\)
Monday47605380
Tuesday53954460
Wednesday58404640
Thursday46505450
Friday53654340
Saturday49905550
Sunday43655840
When these data are coded using \(x = \frac { p - 4365 } { 100 }\) and \(y = \frac { q - 4340 } { 100 }\), $$\Sigma x = 48.1 , \Sigma y = 52.8 , \Sigma x ^ { 2 } = 486.44 , \Sigma y ^ { 2 } = 613.22 \text { and } \Sigma x y = 204.95 .$$
  1. Calculate \(S _ { x y } , S _ { x x }\) and \(S _ { y y }\).
  2. Calculate, to 3 significant figures, the value of the product moment correlation coefficient between \(x\) and \(y\).
    1. Write down the value of the product moment correlation coefficient between \(p\) and \(q\).
    2. Give an interpretation of this value.
Edexcel S1 2003 June Q4
11 marks Moderate -0.3
4. The discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \begin{array} { l l } k \left( x ^ { 2 } - 9 \right) , & x = 4,5,6 \\ 0 , & \text { otherwise } \end{array}$$ where \(k\) is a positive constant.
  1. Show that \(k = \frac { 1 } { 50 }\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  3. Find \(\operatorname { Var } ( 2 X - 3 )\).
Edexcel S1 2003 June Q5
12 marks Easy -1.2
5. The random variable \(X\) represents the number on the uppermost face when a fair die is thrown.
  1. Write down the name of the probability distribution of \(X\).
  2. Calculate the mean and the variance of \(X\). Three fair dice are thrown and the numbers on the uppermost faces are recorded.
  3. Find the probability that all three numbers are 6 .
  4. Write down all the different ways of scoring a total of 16 when the three numbers are added together.
  5. Find the probability of scoring a total of 16 .
Edexcel S1 2003 June Q6
16 marks Moderate -0.8
6. The number of bags of potato crisps sold per day in a bar was recorded over a two-week period. The results are shown below. $$20,15,10,30,33,40,5,11,13,20,25,42,31,17$$
  1. Calculate the mean of these data.
  2. Draw a stem and leaf diagram to represent these data.
  3. Find the median and the quartiles of these data. An outlier is an observation that falls either \(1.5 \times\) (interquartile range) above the upper quartile or \(1.5 \times\) (interquartile range) below the lower quartile.
  4. Determine whether or not any items of data are outliers.
  5. On graph paper draw a box plot to represent these data. Show your scale clearly.
  6. Comment on the skewness of the distribution of bags of crisps sold per day. Justify your answer.
Edexcel S1 2003 June Q7
16 marks Moderate -0.8
  1. Eight students took tests in mathematics and physics. The marks for each student are given in the table below where \(m\) represents the mathematics mark and \(p\) the physics mark.
\multirow{2}{*}{}Student
\(A\)B\(C\)D\(E\)\(F\)G\(H\)
\multirow{2}{*}{Mark}\(m\)9141310782017
\(p\)1123211519103126
A science teacher believes that students' marks in physics depend upon their mathematical ability. The teacher decides to investigate this relationship using the test marks.
  1. Write down which is the explanatory variable in this investigation.
  2. Draw a scatter diagram to illustrate these data.
  3. Showing your working, find the equation of the regression line of \(p\) on \(m\).
  4. Draw the regression line on your scatter diagram. A ninth student was absent for the physics test, but she sat the mathematics test and scored 15 .
  5. Using this model, estimate the mark she would have scored in the physics test.
Edexcel D1 2014 January Q1
8 marks Easy -1.3
1. 11
17
10
14
8
13
6
4
15
7
  1. Use the bubble sort algorithm to perform ONE complete pass towards sorting these numbers into ascending order. The original list is now to be sorted into descending order.
  2. Use a quick sort to obtain the sorted list, giving the state of the list after each complete pass. You must make your pivots clear. The numbers are to be packed into bins of size 26
  3. Calculate a lower bound for the minimum number of bins required. You must show your working.
Edexcel D1 2014 January Q2
10 marks Moderate -0.5
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e3ac0632-9560-4cb8-99dd-8f4bf28315f4-3_549_1175_260_443} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 represents nine buildings, A, B, C, D, E, F, G, H and I, recently bought by Newberry Enterprises. The company wishes to connect the alarm systems between the buildings to form a single network. The number on each arc represents the cost, in pounds, of connecting the alarm systems between the buildings.
  1. Use Prim's algorithm, starting at A , to find the minimum spanning tree for this network. You must list the arcs that form your tree in the order that you select them.
  2. State the minimum cost of connecting the alarm systems in the nine buildings. It is discovered that some alarm systems are already connected. There are connections along BC and EF, as shown in bold in Diagram 1 in the answer book. Since these already exist, it is decided to use these arcs as part of the spanning tree.
    1. Use Kruskal's algorithm to find the minimum spanning tree that includes arcs BC and EF . You must list the arcs in the order that you consider them. In each case, state whether you are adding the arc to your spanning tree.
    2. Explain why Kruskal's algorithm is a better choice than Prim's algorithm in this case. Since arcs BC and EF already exist, there is no cost for these connections.
  4. State the new minimum cost of connecting the nine buildings.
Edexcel D1 2014 January Q4
8 marks Moderate -0.8
4
15
7
  1. Use the bubble sort algorithm to perform ONE complete pass towards sorting these numbers into ascending order. The original list is now to be sorted into descending order.
  2. Use a quick sort to obtain the sorted list, giving the state of the list after each complete pass. You must make your pivots clear. The numbers are to be packed into bins of size 26
  3. Calculate a lower bound for the minimum number of bins required. You must show your working.
    2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e3ac0632-9560-4cb8-99dd-8f4bf28315f4-3_549_1175_260_443} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 represents nine buildings, A, B, C, D, E, F, G, H and I, recently bought by Newberry Enterprises. The company wishes to connect the alarm systems between the buildings to form a single network. The number on each arc represents the cost, in pounds, of connecting the alarm systems between the buildings.
    1. Use Prim's algorithm, starting at A , to find the minimum spanning tree for this network. You must list the arcs that form your tree in the order that you select them.
    2. State the minimum cost of connecting the alarm systems in the nine buildings. It is discovered that some alarm systems are already connected. There are connections along BC and EF, as shown in bold in Diagram 1 in the answer book. Since these already exist, it is decided to use these arcs as part of the spanning tree.
      1. Use Kruskal's algorithm to find the minimum spanning tree that includes arcs BC and EF . You must list the arcs in the order that you consider them. In each case, state whether you are adding the arc to your spanning tree.
      2. Explain why Kruskal's algorithm is a better choice than Prim's algorithm in this case. Since arcs BC and EF already exist, there is no cost for these connections.
      3. State the new minimum cost of connecting the nine buildings.
        3. \begin{figure}[h]
        \includegraphics[alt={},max width=\textwidth]{e3ac0632-9560-4cb8-99dd-8f4bf28315f4-4_547_413_260_504} \captionsetup{labelformat=empty} \caption{Figure 2}
        \end{figure} \begin{figure}[h]
        \includegraphics[alt={},max width=\textwidth]{e3ac0632-9560-4cb8-99dd-8f4bf28315f4-4_549_412_258_1146} \captionsetup{labelformat=empty} \caption{Figure 3}
        \end{figure} Figure 2 shows the possible allocations of six people, Beth (B), Charlie (C), Harry (H), Karam (K), Sam (S) and Theresa (T), to six tasks 1, 2, 3, 4, 5 and 6. Figure 3 shows an initial matching.
    4. Define the term 'matching'.
      (2)
    5. Starting from the given initial matching, use the maximum matching algorithm to find an improved matching. You should list the alternating path that you use, and state the improved matching.
      (3) After training, a possible allocation for Harry is task 6, and an additional possible allocation for Karam is task 1.
    6. Starting from the matching found in (b), use the maximum matching algorithm to find a complete matching. You should list the alternating path that you use, and state your complete matching.
      (3)
      4. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e3ac0632-9560-4cb8-99dd-8f4bf28315f4-5_814_1303_251_390} \captionsetup{labelformat=empty} \caption{Figure 4
      [0pt] [The total weight of the network is 367 metres]}
      \end{figure} Figure 4 represents a network of water pipes. The number on each arc represents the length, in metres, of that water pipe. A robot will travel along each pipe to check that the pipe is in good repair.
      The robot will travel along each pipe at least once. It will start and finish at A and the total distance travelled must be minimised.
    7. Use the route inspection algorithm to find the pipes that will need to be traversed twice. You must make your method and working clear.
    8. Write down the length of a shortest inspection route. A new pipe, IJ, of length 35 m is added to the network. This pipe must now be included in a new minimum inspection route starting and finishing at A .
    9. Determine if the addition of this pipe will increase or decrease the distance the robot must travel. You must give a reason for your answer.
Edexcel D1 2014 January Q11
Moderate -0.5
11
17
10
14
8
Edexcel D1 2014 January Q14
Moderate -0.5
14
8
13
6
4
Edexcel D1 2014 January Q17
Easy -1.8
17
10
14
8
13
6
4
15
7
  1. Use the bubble sort algorithm to perform ONE complete pass towards sorting these numbers into ascending order. The original list is now to be sorted into descending order.
  2. Use a quick sort to obtain the sorted list, giving the state of the list after each complete pass. You must make your pivots clear. The numbers are to be packed into bins of size 26
  3. Calculate a lower bound for the minimum number of bins required. You must show your working.
    2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e3ac0632-9560-4cb8-99dd-8f4bf28315f4-3_549_1175_260_443} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 represents nine buildings, A, B, C, D, E, F, G, H and I, recently bought by Newberry Enterprises. The company wishes to connect the alarm systems between the buildings to form a single network. The number on each arc represents the cost, in pounds, of connecting the alarm systems between the buildings.
    1. Use Prim's algorithm, starting at A , to find the minimum spanning tree for this network. You must list the arcs that form your tree in the order that you select them.
    2. State the minimum cost of connecting the alarm systems in the nine buildings. It is discovered that some alarm systems are already connected. There are connections along BC and EF, as shown in bold in Diagram 1 in the answer book. Since these already exist, it is decided to use these arcs as part of the spanning tree.
      1. Use Kruskal's algorithm to find the minimum spanning tree that includes arcs BC and EF . You must list the arcs in the order that you consider them. In each case, state whether you are adding the arc to your spanning tree.
      2. Explain why Kruskal's algorithm is a better choice than Prim's algorithm in this case. Since arcs BC and EF already exist, there is no cost for these connections.
      3. State the new minimum cost of connecting the nine buildings.
        3. \begin{figure}[h]
        \includegraphics[alt={},max width=\textwidth]{e3ac0632-9560-4cb8-99dd-8f4bf28315f4-4_547_413_260_504} \captionsetup{labelformat=empty} \caption{Figure 2}
        \end{figure} \begin{figure}[h]
        \includegraphics[alt={},max width=\textwidth]{e3ac0632-9560-4cb8-99dd-8f4bf28315f4-4_549_412_258_1146} \captionsetup{labelformat=empty} \caption{Figure 3}
        \end{figure} Figure 2 shows the possible allocations of six people, Beth (B), Charlie (C), Harry (H), Karam (K), Sam (S) and Theresa (T), to six tasks 1, 2, 3, 4, 5 and 6. Figure 3 shows an initial matching.
    4. Define the term 'matching'.
      (2)
    5. Starting from the given initial matching, use the maximum matching algorithm to find an improved matching. You should list the alternating path that you use, and state the improved matching.
      (3) After training, a possible allocation for Harry is task 6, and an additional possible allocation for Karam is task 1.
    6. Starting from the matching found in (b), use the maximum matching algorithm to find a complete matching. You should list the alternating path that you use, and state your complete matching.
      (3)
      4. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e3ac0632-9560-4cb8-99dd-8f4bf28315f4-5_814_1303_251_390} \captionsetup{labelformat=empty} \caption{Figure 4
      [0pt] [The total weight of the network is 367 metres]}
      \end{figure} Figure 4 represents a network of water pipes. The number on each arc represents the length, in metres, of that water pipe. A robot will travel along each pipe to check that the pipe is in good repair.
      The robot will travel along each pipe at least once. It will start and finish at A and the total distance travelled must be minimised.
    7. Use the route inspection algorithm to find the pipes that will need to be traversed twice. You must make your method and working clear.
    8. Write down the length of a shortest inspection route. A new pipe, IJ, of length 35 m is added to the network. This pipe must now be included in a new minimum inspection route starting and finishing at A .
    9. Determine if the addition of this pipe will increase or decrease the distance the robot must travel. You must give a reason for your answer.
      5. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e3ac0632-9560-4cb8-99dd-8f4bf28315f4-6_560_1134_251_470} \captionsetup{labelformat=empty} \caption{Figure 5}
      \end{figure} Figure 5 represents a network of roads. The number on each arc represents the length, in km, of the corresponding road.
    10. Use Dijkstra's algorithm to find the shortest route from S to T . State your route and its length. The road represented by arc CE is now closed for repairs.
    11. Find two shortest routes from S to T that do not include arc CE . State the length of these routes.
      (3)
      6. A linear programming problem in \(x\) and \(y\) is described as follows. Minimise \(\quad C = 2 x + 5 y\) subject to $$\begin{aligned} x + y & \geqslant 500 \\ 5 x + 4 y & \geqslant 4000 \\ y & \leqslant 2 x \\ y & \geqslant x - 250 \\ x , y & \geqslant 0 \end{aligned}$$
    12. Add lines and shading to Diagram 1 in the answer book to represent these constraints. Hence determine the feasible region and label it R .
    13. Use point testing to determine the exact coordinates of the optimal point, P. You must show your working. The first constraint is changed to \(x + y \geqslant k\) for some value of \(k\).
    14. Determine the greatest value of \(k\) for which P is still the optimal point.
      7. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e3ac0632-9560-4cb8-99dd-8f4bf28315f4-8_582_1226_248_422} \captionsetup{labelformat=empty} \caption{Figure 6}
      \end{figure} A project is modelled by the activity network shown in Figure 6. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, to complete the activity. Each activity requires one worker. The project is to be completed in the shortest possible time.
    15. Complete Diagram 1 in the answer book to show the early event times and late event times.
    16. Draw a cascade (Gantt) chart for this project on Grid 1 in the answer book.
    17. Use your cascade chart to determine a lower bound for the number of workers needed to complete the project in the shortest possible time. You must make specific reference to times and activities. The project is to be completed in the minimum time using as few workers as possible.
    18. Schedule the activities, using Grid 2 in the answer book.
      8. A charity produces mixed packs of posters and flyers to send out to sponsors. Pack A contains 40 posters and 20 flyers.
      Pack B contains 30 posters and 50 flyers.
      The charity must send out at least 15000 flyers.
      The charity wants between \(40 \%\) and \(60 \%\) of the total packs produced to be Pack As.
      Posters cost 15p each and flyers cost 3p each.
      The charity wishes to minimise its costs.
      Let \(x\) represent the number of Pack As produced, and \(y\) represent the number of Pack Bs produced.
      Formulate this as a linear programming problem, stating the objective and listing the constraints as simplified inequalities with integer coefficients.
      You should not attempt to solve the problem.
      (Total 6 marks)
Edexcel D1 Q4
Moderate -0.5
4. This question should be answered on the sheet provided in the answer booklet. A manager has five workers, Mr. Ahmed, Miss Brown, Ms. Clough, Mr. Dingle and Mrs. Evans. To finish an urgent order he needs each of them to work overtime, one on each evening, in the next week. The workers are only available on the following evenings: Mr. Ahmed \(( A )\) - Monday and Wednesday;
Miss Brown ( \(B\) ) - Monday, Wednesday and Friday;
Ms. Clough ( \(C\) ) - Monday;
Mr. Dingle ( \(D\) ) - Tuesday, Wednesday and Thursday;
Mrs. Evans \(( E )\) - Wednesday and Thursday.
The manager initially suggests that \(A\) might work on Monday, \(B\) on Wednesday and \(D\) on Thursday.
  1. Using the nodes printed on the answer sheet, draw a bipartite graph to model the availability of the five workers. Indicate, in a distinctive way, the manager's initial suggestion.
    (2 marks)
  2. Obtain an alternating path, starting at \(C\), and use this to improve the initial matching.
    (3 marks)
  3. Find another alternating path and hence obtain a complete matching.
    (3 marks)
Edexcel D1 Q5
Standard +0.3
5. This question should be answered on the sheet provided in the answer booklet. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3147dad8-2d3c-42fd-b288-7017ff1fce16-003_352_904_450_287} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows the activity network used to model a small building project. The activities are represented by the edges and the number in brackets on each edge represents the time, in hours, taken to complete that activity.
  1. Calculate the early time and the late time for each event. Write your answers in the boxes on the answer sheet.
  2. Hence determine the critical activities and the length of the critical path. Each activity requires one worker. The project is to be completed in the minimum time.
  3. Schedule the activities for the minimum number of workers using the time line on the answer sheet. Ensure that you make clear the order in which each worker undertakes his activities.
    (5 marks)
Edexcel D1 Q6
Moderate -0.3
6. This question should be answered on the sheet provided in the answer booklet. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3147dad8-2d3c-42fd-b288-7017ff1fce16-003_469_844_422_1731} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 shows a capacitated, directed network. The number on each arc indicates the capacity of that arc.
  1. State the maximum flow along
    1. SAET,
    2. SBDT,
    3. SCFT.
      (3 marks)
  2. Show these maximum flows on Diagram 1 on the answer sheet.
    (1 mark)
  3. Taking your answer to part (b) as the initial flow pattern, use the labelling procedure to find a maximum flow from \(S\) to \(T\). Your working should be shown on Diagram 2. List each flow augmenting route you find, together with its flow.
    (6 marks)
  4. Indicate a maximum flow on Diagram 3.
    (2 marks)
  5. Prove that your flow is maximal.
    (2 marks)
Edexcel D1 Q7
Moderate -0.8
7. A tailor makes two types of garment, \(A\) and \(B\). He has available \(70 \mathrm {~m} ^ { 2 }\) of cotton fabric and \(90 \mathrm {~m} ^ { 2 }\) of woollen fabric. Garment \(A\) requires \(1 \mathrm {~m} ^ { 2 }\) of cotton fabric and \(3 \mathrm {~m} ^ { 2 }\) of woollen fabric. Garment \(B\) requires \(2 \mathrm {~m} ^ { 2 }\) of each fabric. The tailor makes \(x\) garments of type \(A\) and \(y\) garments of type \(B\).
  1. Explain why this can be modelled by the inequalities $$\begin{aligned} & x + 2 y \leq 70 \\ & 3 x + 2 y \leq 90 \\ & x \geq 0 , y \geq 0 \end{aligned}$$ (2 marks)
    The tailor sells type \(A\) for \(\pounds 30\) and type \(B\) for \(\pounds 40\). All garments made are sold. The tailor wishes to maximise his total income.
  2. Set up an initial Simplex tableau for this problem.
    (3 marks)
  3. Solve the problem using the Simplex algorithm.
    (8 marks) Figure 4 shows a graphical representation of the feasible region for this problem. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3147dad8-2d3c-42fd-b288-7017ff1fce16-004_452_828_995_356} \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{figure}
  4. Obtain the coordinates of the points A, \(C\) and \(D\).
  5. Relate each stage of the Simplex algorithm to the corresponding point in Fig. 4.
    (3 marks) Answer Book (AB12)
    Graph Paper (ASG2) Items included with question papers Answer booklet
Edexcel D1 Q4
Moderate -0.5
4. This question should be answered on the sheet provided in the answer booklet. A manager has five workers, Mr. Ahmed, Miss Brown, Ms. Clough, Mr. Dingle and Mrs. Evans. To finish an urgent order he needs each of them to work overtime, one on each evening, in the next week. The workers are only available on the following evenings: $$\begin{aligned} & \text { Mr. Ahmed } ( A ) \text { - Monday and Wednesday; } \\ & \text { Miss Brown } ( B ) \text { - Monday, Wednesday and Friday; } \\ & \text { Ms. Clough } ( C ) \text { - Monday; } \\ & \text { Mr. Dingle } ( D ) \text { - Tuesday, Wednesday and Thursday; } \\ & \text { Mrs. Evans } ( E ) \text { - Wednesday and Thursday. } \end{aligned}$$ The manager initially suggests that \(A\) might work on Monday, \(B\) on Wednesday and \(D\) on Thursday.
  1. Using the nodes printed on the answer sheet, draw a bipartite graph to model the availability of the five workers. Indicate, in a distinctive way, the manager's initial suggestion.
    (2 marks)
  2. Obtain an alternating path, starting at \(C\), and use this to improve the initial matching.
  3. Find another alternating path and hence obtain a complete matching.
    (3 marks)
Edexcel D1 Q5
Standard +0.3
5. This question should be answered on the sheet provided in the answer booklet. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-006_542_1389_483_352} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows the activity network used to model a small building project. The activities are represented by the edges and the number in brackets on each edge represents the time, in hours, taken to complete that activity.
  1. Calculate the early time and the late time for each event. Write your answers in the boxes on the answer sheet.
    (6 marks)
  2. Hence determine the critical activities and the length of the critical path.
    (2 marks)
    Each activity requires one worker. The project is to be completed in the minimum time.
  3. Schedule the activities for the minimum number of workers using the time line on the answer sheet. Ensure that you make clear the order in which each worker undertakes his activities.
    (5 marks)
Edexcel D1 Q6
Standard +0.3
6. This question should be answered on the sheet provided in the answer booklet. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-007_732_1308_433_388} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 shows a capacitated, directed network. The number on each arc indicates the capacity of that arc.
  1. State the maximum flow along
    1. SAET,
    2. SBDT,
    3. SCFT.
      (3 marks)
  2. Show these maximum flows on Diagram 1 on the answer sheet.
  3. Taking your answer to part (b) as the initial flow pattern, use the labelling procedure to find a maximum flow from \(S\) to \(T\). Your working should be shown on Diagram 2. List each flow augmenting route you find, together with its flow.
  4. Indicate a maximum flow on Diagram 3.
  5. Prove that your flow is maximal.