Edexcel FD1 AS (Further Decision 1 AS) 2024 June

1. $$\begin{array} { l l l l l l l l l l l } 4 & 6.5 & 7 & 1.3 & 2 & 5 & 1.5 & 6 & 4.5 & 6 & 1 \end{array}$$ The list of eleven numbers shown above is to be sorted into descending order.

1. Carry out a quick sort to produce the sorted list. You should show the result of each pass and identify the pivots clearly.
2. Use the first-fit decreasing bin packing algorithm to pack the numbers into bins of size 10
3. Determine whether your answer to part (b) uses the minimum number of bins. You must justify your answer. A different list of eleven numbers is to be sorted into descending order using a bubble sort. The list after the second pass is
   1.6
   1.7
   1.5
   3.8
   3.3
   4.5
   4.8
   5.6
   5.4
   6.7
   9.1
4. Explain how you know that at least one of the first two passes of the bubble sort was not carried out correctly.

2. A company manages an awards evening. The table below lists the activities required to set up the room for the evening, and their immediately preceding activities. Each activity requires exactly one person. Figure 1 shows a partially completed activity network used to model the project. Each activity is represented by an arc. \begin{figure}[h] \end{figure}

1. Add the remaining five activities to Diagram 1 in the answer book to complete the activity network, using exactly two dummies. In addition to setting up the room, the company must prepare the meals for the guests. Figure 2 shows the activity network for preparing the main courses. The numbers in brackets represent the time, in minutes, to complete each task. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ca57c64b-0b33-4179-be7f-684bd6ea2162-05_793_1515_451_373} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure}
2. Complete Diagram 2 in the answer book to show the early event times and the late event times for the activity network shown in Figure 2.
3. State the critical activities.
4. Given that the main courses need to be ready to be served (with all activities completed) at 8 pm , state the latest time that activity \(R\) can start.

3. \begin{figure}[h] \end{figure} [The total weight of the network is \(139 + x + y\) ]

1. Explain what is meant by the term "tree". Figure 3 represents a network of walkways in a warehouse.
   The arcs represent the walkways and the nodes represent junctions between them.
   The number on each arc represents the length, in metres, of the corresponding walkway.
   The values \(x\) and \(y\) are unknown, however it is known that \(x\) and \(y\) are integers and that $$9 < x < y < 14$$
2. 1. Use Dijkstra's algorithm to find the shortest route from A to M.
   2. State an expression for the length of the shortest route from A to M . The warehouse manager wants to check that all of the walkways are in good condition.
      Their inspection route starts at B and finishes at C .
      The inspection route must traverse each walkway at least once and be as short as possible.
3. State the arcs that are traversed twice.
4. State the number of times that H appears in the inspection route. The warehouse manager finds that the total length of the inspection route is 172 metres.
5. Determine the value of \(x\) and the value of \(y\)

4. \begin{figure}[h] \end{figure} Figure 4 shows three of the six constraints for a linear programming problem in \(x\) and \(y\) The unshaded region and its boundaries satisfy these three constraints.

1. State these three constraints as simplified inequalities with integer coefficients. The variables \(x\) and \(y\) represent the number of orange fish and the number of blue fish, respectively, that are to be kept in an aquarium. The number of fish in the aquarium is subject to these three further constraints
   • there must be at least one blue fish
   • the orange fish must not outnumber the blue fish by more than ten
   • there must be no more than five blue fish for every orange fish
   • Write each of these three constraints as a simplified inequality with integer coefficients.
   • Represent these three constraints by adding lines and shading to Diagram 1 in the answer book, labelling the feasible region, \(R\)
   The total value (in pounds) of the fish in the aquarium is given by the objective function $$\text { Maximise } P = 3 x + 5 y$$
2. 1. Use the objective line method to determine the optimal point of the feasible region, giving its coordinates as exact fractions.
   2. Hence find the maximum total value of the fish in the aquarium, stating the optimal number of orange fish and the optimal number of blue fish. \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Please check the examination details below before entering your candidate information}

      Candidate surname								Other names
      Centre Number				Candidate Number
      □			□

      \end{table} \section*{Pearson Edexcel Level 3 GCE} \section*{Friday 17 May 2024} Afternoon \section*{Further Mathematics} Advanced Subsidiary
      Further Mathematics options
      27: Decision Mathematics 1
      (Part of options D, F, H and K) \section*{D1 Answer Book} Do not return the question paper with the answer book.
      1.
      \(\begin{array} { l l l l l l l l l l l } 4 & 6.5 & 7 & 1.3 & 2 & 5 & 1.5 & 6 & 4.5 & 6 & 1 \end{array}\) 2.
      \includegraphics[max width=\textwidth, alt={}]{ca57c64b-0b33-4179-be7f-684bd6ea2162-12_435_815_392_463}
      \section*{Diagram 1} Use this diagram only if you need to redraw your activity network.
      \includegraphics[max width=\textwidth, alt={}, center]{ca57c64b-0b33-4179-be7f-684bd6ea2162-12_442_820_2043_458} Copy of Diagram 1

      VJYV SIHI NI JIIYM ION OC

      V346 SIHI NI JLIYM ION OC

      V34V SIHI NI IIIIM ION OC

      Key: \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{ca57c64b-0b33-4179-be7f-684bd6ea2162-13_1217_1783_451_236} \captionsetup{labelformat=empty} \caption{Diagram 2}
      \end{figure} 3.
      \includegraphics[max width=\textwidth, alt={}, center]{ca57c64b-0b33-4179-be7f-684bd6ea2162-14_2463_1240_339_465}
      Shortest route from A to M:
      Length of shortest route from A to M:
      
      \includegraphics[max width=\textwidth, alt={}]{ca57c64b-0b33-4179-be7f-684bd6ea2162-16_3038_2264_0_0}
      \includegraphics[max width=\textwidth, alt={}]{ca57c64b-0b33-4179-be7f-684bd6ea2162-17_1103_1247_397_512}
      \section*{Diagram 1} \section*{There is a copy of Diagram 1 on page 11 if you need to redraw your graph.}

      VJYV SIHI NI JIIIM ION OC

      V341 S1H1 NI JLIYM ION OA

      V34V SIHI NI IIIVM ION OC

      Use this diagram only if you need to redraw your graph.
      \includegraphics[max width=\textwidth, alt={}, center]{ca57c64b-0b33-4179-be7f-684bd6ea2162-19_1108_1252_1606_509} Copy of Diagram 1