AQA Further AS Paper 2 Discrete (Further AS Paper 2 Discrete) 2019 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]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-03_691_1067_721_482} The value of the cut \(\{ S , A , B \} \{ C , D , E , T \}\) is \(V \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\)
Find \(V\). Circle your answer.
[0pt] [1 mark]
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2 Part of an activity network is shown in the diagram below.
\(A B C\) is part of the critical path of the activity network.
\includegraphics[max width=\textwidth, alt={}, center]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-04_264_908_447_566} The duration of activity \(B\) is \(d\).
Which of the following statements about \(d\) is correct? Circle your answer. $$0 < d < 10 \quad d = 10 \quad 10 < d < 20 \quad d = 20$$

3 Manon makes apple cakes and banana cakes. Each apple cake is made with 3 eggs and 100 grams of flour. Each banana cake is made with 2 eggs and 150 grams of flour. Manon has 36 eggs and 1500 grams of flour.
Manon wants to make as many cakes as possible.
Formulate Manon's situation as a linear programming problem, clearly defining any variables you introduce.

4

1. State the definition of a bipartite graph. 4
2. A jazz quintet has five musical instruments: bassoon, clarinet, flute, oboe and violin. Jay, Kay, Lee, Mel and Nish are musicians and each plays a musical instrument in the jazz quintet. Jay knows how to play the bassoon and the clarinet.
   Kay knows how to play the bassoon, the oboe and the violin.
   Lee knows how to play the clarinet and the flute.
   Mel knows how to play the clarinet, the oboe and the violin.
   Nish knows how to play the flute, the oboe and the violin. 4
3. 1. Draw a graph to show which musicians know how to play which instruments. 4
4. (ii) Nish arrives late to a jazz quintet rehearsal. Each of the other four musicians is already playing an instrument: \begin{displayquote} Jay is playing the clarinet
   Kay is playing the oboe
   Lee is playing the flute
   Mel is playing the violin. \end{displayquote} Explain how the graph in part (b)(i) shows that there is no instrument available that Nish knows how to play. 4
5. (iii) When Nish arrives the rehearsal stops. When they restart the rehearsal, Nish is playing the flute. Draw all possible subgraphs of the graph in part (b)(i) that show how Jay, Kay, Lee and Mel can each be assigned a unique musical instrument they know how to play.
   [0pt] [2 marks]

5

1. Complete the Cayley table in Figure 1 for multiplication modulo 4 \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-08_761_1017_434_493}
   \end{figure} 5
2. The set \(S\) is defined as $$S = \{ a , b , c , d \}$$ Figure 2 shows an incomplete Cayley table for \(S\) under the commutative binary operation • \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2}

   •	\(a\)	\(b\)	\(c\)	\(d\)
   \(a\)	\(b\)	\(a\)	\(a\)	\(c\)
   \(b\)		\(c\)		\(c\)
   \(c\)		\(d\)	\(d\)
   \(d\)			\(d\)	\(d\)

   \end{table} 5
3. 1. Complete the Cayley table in Figure 2. 5
4. (ii) Determine whether the binary operation • is associative when acting on the elements of \(S\). Fully justify your answer.

6 The diagram shows a nature reserve which has its entrance at \(A\), eight information signs at \(B , C , \ldots , I\), and fifteen grass paths. The length of each grass path is given in metres.
The total length of the grass paths is 1465 metres.
\includegraphics[max width=\textwidth, alt={}, center]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-10_812_1192_584_424} To cut the grass, Ashley starts at the entrance and drives a mower along every grass path in the nature reserve. The mower moves at 7 kilometres per hour. 6

1. Find the least possible time that it takes for Ashley to cut the grass on all fifteen paths in the nature reserve and return to the entrance. Fully justify your answer.
   6
2. Brook visits every information sign in the nature reserve to update them, starting and finishing at the entrance. For the eight information signs, the minimum connecting distance of the grass paths is 510 metres. 6
3. 1. Determine a lower bound for the distance Brook walks to visit every information sign.
      Fully justify your answer.
      [0pt] [2 marks]
      6
4. (ii) Using the nearest neighbour algorithm starting from the entrance, determine an upper bound for the distance Brook walks to visit every information sign.
   [0pt] [2 marks]
   6
5. Brook takes one minute to update the information at one information sign. Brook walks on the grass paths at an average speed of 5 kilometres per hour. Ashley and Brook start from the entrance at the same time. 6
6. 1. Use your answers from parts (a) and (b) to show that Ashley and Brook will return to the entrance at approximately the same time. Fully justify your answer.
      6
7. (ii) State an assumption that you have used in part (c)(i).
   \includegraphics[max width=\textwidth, alt={}, center]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-13_2488_1716_219_153}
   \(7 \quad\) Ali and Bex play a zero-sum game. The game is represented by the following pay-off matrix for Ali.

   \multirow{2}{*}{}	Bex
   	Strategy	\(\mathbf { B } _ { \mathbf { 1 } }\)	\(\mathbf { B } _ { \mathbf { 2 } }\)	\(\mathbf { B } _ { \mathbf { 3 } }\)
   \multirow{4}{*}{Ali}	\(\mathbf { A } _ { \mathbf { 1 } }\)	2	-1	3
   	\(\mathbf { A } _ { \mathbf { 2 } }\)	-4	-2	2
   	\(\mathbf { A } _ { \mathbf { 3 } }\)	0	1	1
   	\(\mathrm { A } _ { 4 }\)	-3	2	-2

7

1. 1. Write down the pay-off matrix for Bex. 7
2. (ii) Explain why the pay-off matrix for Bex can be written as