AQA Further AS Paper 2 Discrete (Further AS Paper 2 Discrete) 2022 June

1 The connected graph \(G\) is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{ecbeedf5-148e-40ad-b8a2-a7aa3db4a115-02_542_834_630_603} The graphs \(A\) and \(B\) are subgraphs of \(G\)
Both \(A\) and \(B\) have four vertices. 1

1. The graph \(A\) is a tree with \(x\) edges.
   State the value of \(x\) Circle your answer. 3459 1
2. The graph \(B\) is simple-connected with \(y\) edges.
   Find the maximum possible value of \(y\)
   Circle your answer. 3459

2 The diagram shows a network of pipes. Each pipe is labelled with its upper capacity in \(\mathrm { m } ^ { 3 } \mathrm {~s} ^ { - 1 }\)
\includegraphics[max width=\textwidth, alt={}, center]{ecbeedf5-148e-40ad-b8a2-a7aa3db4a115-03_424_1262_445_388} 2

1. Find the value of the cut \(\{ A , C , D , G , H \} \{ B , E , F , I \}\) 2
2. Write down a cut with a value of \(300 \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 1 }\) 2
3. Using the values from part (a) and part (b), state what can be deduced about the maximum flow through the network. Fully justify your answer.

3 A project consists of 11 activities \(A , B , \ldots , K\) A completed activity network for the project is shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{ecbeedf5-148e-40ad-b8a2-a7aa3db4a115-04_972_1604_445_219} All times on the activity network are given in days.
3

1. Write down the critical path.
   [0pt] [1 mark] 3
2. Due to an issue with the supply of materials, the duration of activity \(G\) is doubled. Deduce the effect, if any, that this change will have on the earliest start time and latest finish time for each of the activities \(I , J\) and \(K\)

4 Alun, a baker, delivers bread to community shops located in Aber, Bangor, Conwy, and E'bach. Alun starts and finishes his journey at the bakery, which is located in Deganwy.
The distances, in miles, between the five locations are given in the table below. The minimum total distance that Alun can travel in order to make all four deliveries, starting and finishing at the bakery in Deganwy is \(x\) miles. 4

1. Using the nearest neighbour algorithm starting from Deganwy, find an upper bound for \(x\)

5

1. A connected planar graph has 9 vertices, 20 edges and \(f\) faces. Use Euler's formula for connected planar graphs to find \(f\) 5
2. The graph \(J\), shown in Figure 1, has 9 vertices and 20 edges. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{ecbeedf5-148e-40ad-b8a2-a7aa3db4a115-09_778_760_440_641}
   \end{figure} By redrawing the graph \(J\) using Figure 2, show that \(J\) is planar. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2}

   \(A\)	\(B\)	\(C\)
   \(\bullet\)	\(\bullet\)	\(\bullet\)
   \(D \bullet\)	\(E \bullet\)	\(\bullet F\)
   \(\bullet\)	\(\stackrel { \theta } { H }\)	\(\bullet\)

   \end{table}

6 The set \(S\) is given by \(S = \{ \mathbf { A } , \mathbf { B } , \mathbf { C } , \mathbf { D } \}\) where
\(\mathbf { A } = \left[ \begin{array} { l l } 1 & 0
0 & 0 \end{array} \right]\)
\(\mathbf { B } = \left[ \begin{array} { l l } 1 & 0
0 & 1 \end{array} \right]\)
\(\mathbf { C } = \left[ \begin{array} { l l } 0 & 0
0 & 1 \end{array} \right]\)
\(\mathbf { D } = \left[ \begin{array} { l l } 0 & 0
0 & 0 \end{array} \right]\) 6

1. Complete the Cayley table for \(S\) under matrix multiplication.

   	A	B	C	D
   A	A		D
   B		B
   C			C
   D				D

   6
2. Using the Cayley table above, explain why \(\mathbf { B }\) is the identity element of \(S\) under matrix multiplication.
   [0pt] [1 mark] 6
3. Sam states that the Cayley table in part (a) shows that matrix multiplication is commutative. Comment on the validity of Sam's statement.

7 Kez and Lui play a zero-sum game. The game does not have a stable solution. The game is represented by the following pay-off matrix for Kez. 7

1. State, with a reason, why Kez should never play strategy \(\mathbf { K } _ { \mathbf { 2 } }\) 7
2. \(\quad\) Kez and Lui play the game 20 times.
   Kez plays their optimal mixed strategy.
   Find the expected number of times that Kez will play strategy \(\mathbf { K } _ { \mathbf { 3 } }\)
   Fully justify your answer.

8 Alli is planting garlic cloves and leek seedlings in a garden. The planting density is the number of plants that are planted per \(\mathrm { m } ^ { 2 }\)
The planting densities and costs are shown in the table below.