Questions — Edexcel D2 (231 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Edexcel D2 2019 June Q2
2. Table 1 shows the cost, in pounds, of transporting one unit of stock from each of three supply points, \(\mathrm { A } , \mathrm { B }\) and C , to each of four demand points, 1, 2, 3 and 4 . It also shows the stock held at each supply point and the stock required at each demand point. A minimum cost solution is required. \begin{table}[h]
1234Supply
A1720231425
B1615192229
C1914111532
Demand28172318
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} Table 2 shows an initial solution given by the north-west corner method. \begin{table}[h]
1234
\(A\)25
\(B\)3179
\(C\)1418
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}
  1. Taking A4 as the entering cell, use the stepping-stone method to find an improved solution. Make your route clear.
  2. Taking the most negative improvement index to indicate the entering cell, use the stepping-stone method once to obtain an improved solution. You must make your method clear by stating your shadow costs, improvement indices, route, entering cell and exiting cell.
  3. Determine whether your current solution is optimal, giving a reason for your answer.
  4. State the cost of your current solution.
Edexcel D2 2019 June Q3
3. Five friends have rented a house that has five bedrooms. They each require their own bedroom. The table below shows how each friend rated the five bedrooms, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E , where 0 is low and 10 is high.
ABCDE
Frank50734
Gill538101
Harry43790
Imogen63654
Jiao02732
Reducing rows first, use the Hungarian algorithm to obtain an allocation that maximises the total of all the ratings. You must make your method clear and show the table after each stage.
(Total 8 marks)
Edexcel D2 2019 June Q4
4. Eugene and Stephen play a zero-sum game. The pay-off matrix shows the number of points that Eugene scores for each combination of strategies.
Stephen plays 1Stephen plays 2Stephen plays 3
Eugene plays 1450
Eugene plays 2-211
Eugene plays 3-3-43
  1. Find the play-safe strategies for each of Eugene and Stephen, and hence show that this zero-sum game does not have a stable solution.
  2. Suppose that Eugene knows that Stephen will use his play-safe strategy. Explain why Eugene should change from his play-safe strategy. You should state as part of your answer which strategy Eugene should now play.
  3. Formulate the game as a linear programming problem for Stephen. Define your variables clearly. Write the constraints as equations.
Edexcel D2 2019 June Q5
5. A linear programming problem in \(x , y\) and \(z\) is described as follows. Maximise \(P = 2 x + 3 y + z\)
subject to \(\quad 2 y - 3 z \leqslant 30\) $$\begin{array} { r } - 3 x + y + z \leqslant 60
x + 4 y - z \leqslant 80 \end{array}$$
  1. Complete the initial tableau in the answer book for this linear programming problem.
    (3)
  2. Taking the most negative number in the profit row to indicate the pivot column, perform one complete iteration of the simplex algorithm to obtain a new tableau, T. Make your method clear by stating the row operations you use.
    (5)
  3. Write down the profit equation given by T and state the values of the slack variables given by T . The following tableau is obtained after further iterations.
    Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
    \(r\)02-310030
    \(s\)013-2013300
    \(x\)14-100180
    \(P\)05-3002160
  4. Explain why no optimal solution can be found by applying the simplex algorithm to the above tableau.
Edexcel D2 2019 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0c66144-9e34-42fc-9f40-a87a49331483-07_719_1313_246_376} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a capacitated, directed network. The number on each arc represents the capacity of that arc. The numbers in circles represent an initial flow.
  1. State the value of the initial flow.
    1. Add a supersource, S , and a supersink, T , and corresponding arcs to Diagrams 1 and 2 in the answer book.
    2. Enter the flow value and appropriate capacity on each of the arcs you have added to Diagram 1.
  3. Complete the initialisation of the labelling procedure on Diagram 2 in the answer book by entering values along the new arcs from S to T , and along \(\operatorname { arcs } \mathrm { S } _ { 1 } \mathrm {~B}\) and \(\mathrm { AT } _ { 1 }\)
  4. Hence use the labelling procedure to find a maximum flow through the network. You must list each flow-augmenting route you use, together with its flow.
  5. Draw a maximal flow pattern on Diagram 3 in the answer book.
  6. Prove that your flow is maximal.
Edexcel D2 2019 June Q7
7. A company has purchased a plot of land and has decided to build four holiday homes, A, B, C and D, on the land at the rate of one home per year. The company expects that the construction costs each year will vary, depending on which houses have already been constructed and which house is currently under construction. The expected construction costs, in thousands of pounds, are shown in the table below.
\cline { 2 - 7 } \multicolumn{1}{c|}{}ABCDEF
A-5347393540
B53-32464143
C4732-514737
D394651-3649
E35414736-42
F4043374942-
\begin{table}[h]
1234Supply
A1720231425
B1615192229
C1914111532
Demand28172318
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} 2. You may not need to use all of these tables \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Table 1}
1234Supply
A25
B29
C32
Demand28172318
\end{table}
1234Supply
A25
B29
C32
Demand28172318
1234Supply
A25
B29
C32
Demand28172318
1234Supply
A25
B29
C32
Demand28172318
1234Supply
A25
B29
C32
Demand28172318
1234Supply
A25
B29
C32
Demand28172318
1234Supply
A25
B29
C32
Demand28172318
1234Supply
A25
B29
C32
Demand28172318
1234Supply
A25
B29
C32
Demand28172318
3.
ABCDE
Frank50734
Gill538101
Harry43790
Imogen63654
Jiao02732
You may not need to use all of these tables
\(A\)\(B\)\(C\)\(D\)\(E\)
\(F\)
\(G\)
\(H\)
\(I\)
\(J\)
ABCDE
F
G
H
I
J
\(A\)\(B\)\(C\)\(D\)\(E\)
\(F\)
\(G\)
\(H\)
\(I\)
\(J\)
\(A\)\(B\)\(C\)\(D\)\(E\)
\(F\)
\(G\)
\(H\)
\(I\)
\(J\)
\(A\)\(B\)\(C\)\(D\)\(E\)
\(F\)
\(G\)
\(H\)
\(I\)
\(J\)
\(A\)\(B\)\(C\)\(D\)\(E\)
\(F\)
\(G\)
\(H\)
\(I\)
\(J\)
\(A\)\(B\)\(C\)\(D\)\(E\)
\(F\)
\(G\)
\(H\)
\(I\)
\(J\)
Stephen plays 1Stephen plays 2Stephen plays 3
Eugene plays 1450
Eugene plays 2-211
Eugene plays 3-3-43
4. 5. (a)
b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
30
60
80
0
You may not need to use all of these tableaux
b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow Ops
\(P\)
b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow Ops
\(P\)
b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow Ops
\(P\)
6. (a) Value of initial flow
(b) and (c)
\includegraphics[max width=\textwidth, alt={}, center]{e0c66144-9e34-42fc-9f40-a87a49331483-20_725_1251_404_349} \section*{Diagram 1}
\includegraphics[max width=\textwidth, alt={}]{e0c66144-9e34-42fc-9f40-a87a49331483-20_1070_1264_1322_349}
\section*{Diagram 2} (d)
(e)
\includegraphics[max width=\textwidth, alt={}, center]{e0c66144-9e34-42fc-9f40-a87a49331483-21_714_1385_1306_283} \section*{Diagram 3} (f)
7. (a)
StageStateActionDest.Value
1ABCDABCD65*
StageStateActionDest.Value
\includegraphics[max width=\textwidth, alt={}]{e0c66144-9e34-42fc-9f40-a87a49331483-24_2642_1833_118_118}