AQA Further AS Paper 2 Discrete (Further AS Paper 2 Discrete) 2020 June

1 The network represents a system of pipes.
The number on each arc represents the upper capacity for each pipe in \(\mathrm { cm } ^ { 3 } \mathrm {~s} ^ { - 1 }\)
\includegraphics[max width=\textwidth, alt={}, center]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-02_793_1255_731_395} The value of the cut \(\{ S , A , B \} \{ C , D , E , F , T \}\) is \(60 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\)
The maximum flow through the system is \(M \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\)
What does the value of the cut imply about \(M\) ? Circle your answer.
\(M < 60 \quad M \leq 60 \quad M \geq 60 \quad M > 60\)

2 The graph \(G\) has 5 vertices and 6 edges, as shown below.
\includegraphics[max width=\textwidth, alt={}, center]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-03_547_547_360_749} Which of the following statements describes the properties of \(G\) ?
Tick ( \(\checkmark\) ) one box.
\(G\) is Eulerian and Hamiltonian. □
\(G\) is Eulerian but not Hamiltonian. □
\(G\) is semi-Eulerian and Hamiltonian. □
\(G\) is semi-Eulerian but not Hamiltonian. □

3 Summer and Haf play a zero-sum game. The pay-off matrix for the game is shown below. Haf 3

1. Show that the game has a stable solution.
   3
2. 1. State the value of the game for Summer. 3
3. (ii) State the play-safe strategy for each player.

4 The connected planar graph \(P\) is Eulerian and has at least one vertex of degree \(x\). Some of the properties of \(P\) are shown in the table below. Deduce the value of \(x\).
Fully justify your answer.
\includegraphics[max width=\textwidth, alt={}, center]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-07_2488_1716_219_153}

5 A restoration project is divided into a number of activities. The duration and predecessor(s) of each activity are shown in the table below. 5

1. On the opposite page, construct an activity network for the project and fill in the earliest start time and latest finish time for each activity.
   [0pt] [4 marks] \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-09_533_289_2124_1548} \captionsetup{labelformat=empty} \caption{Turn over -}
   \end{figure} 5
2. Due to a change of materials during the project, the duration of activity \(C\) is extended by 3 weeks. Determine the new minimum completion time of the project.
   \includegraphics[max width=\textwidth, alt={}, center]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-11_2488_1716_219_153}

6 A garden has seven statues \(A , B , C , D , E , F\) and \(G\), with paths connecting each pair of statues, either directly or indirectly. To provide better access to all the statues, some of the paths are being made wider.
6

1. State why six is the minimum number of paths that need to be made wider. 6
2. The table below shows the number of trees that need to be removed to make the path between adjacent statues wider. A dash in the table means that there is no direct path between the two statues.

   Statue	\(\boldsymbol { A }\)	\(\boldsymbol { B }\)	C	\(\boldsymbol { D }\)	\(E\)	\(F\)	\(G\)
   \(\boldsymbol { A }\)	-	4	7	-	-	-	-
   B	4	-	6	2	3	-	-
   C	7	6	-	-	3	-	4
   \(D\)	-	2	-	-	4	5	-
   \(E\)	-	3	3	4	-	3	7
   \(F\)	-	-	-	5	3	-	6
   G	-	-	4	-	7	6	-

   Find the minimum number of trees that need to be removed. Fully justify your answer.
   6
3. A landscaper identifies that two new wide paths could be constructed without removing any trees. However, there are only enough resources to build one new wide path. The new wide path could be between \(A\) and \(D\) or between \(A\) and \(F\).
   Explain clearly how the solution to part (b) can be adapted to find the new minimum number of trees that need to be removed.
   [0pt] [2 marks]

7 Robyn manages a bakery. Each day the bakery bakes 900 rolls, 600 teacakes and 450 croissants.
The bakery sells two types of bakery box which contain rolls, teacakes and croissants, as shown in the table below. Robyn formulates a linear programming problem to find the maximum profit the bakery can make from selling the bakery boxes. 7

1. Part of a graphical method to solve this linear programming problem is shown on Figure 1. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-14_1283_1196_1267_424}
   \end{figure} 7
2. 1. Explain how the line shown on Figure 1 relates to the linear programming problem. Clearly define any variables that you introduce.
      [0pt] [3 marks]
      7
3. (ii) Use Figure 1 to find the maximum profit that the bakery can make from selling bakery boxes.
   7
4. State an assumption that you have made in part (a)(ii).
   [0pt] [1 mark]
   \includegraphics[max width=\textwidth, alt={}, center]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-17_2493_1732_214_139}

8 The set \(S\) is defined as $$S = \{ a , b , c , d \}$$ Figure 2 shows a Cayley table for \(S\) under the commutative binary operation \begin{table}[h] \end{table} 8

1. 1. Prove that there exists an identity element for \(S\) under the binary operation
      [0pt] [2 marks]
      8
2. (ii) State the inverse of \(b\) under the binary operation
   8
3. Figure 3 shows a Cayley table for multiplication modulo 4 \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3}

   \(\times _ { 4 }\)	0	1	2	3
   0	0	0	0	0
   1	0	1	2	3
   2	0	2	0	2
   3	0	3	2	1

   \end{table} Mali says that, by substituting suitable distinct values for \(a , b , c\) and \(d\), the Cayley table in Figure 2 could represent multiplication modulo 4 Use your answers to part (a) to show that Mali's statement is incorrect.
   \includegraphics[max width=\textwidth, alt={}, center]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-20_2491_1736_219_139}