AQA Further Paper 3 Discrete (Further Paper 3 Discrete) 2020 June

1 The diagram below shows a network of pipes with their capacities.
\includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-02_734_1275_630_386} A supersource and a supersink will be added to the network.
To which nodes should the supersource and supersink be connected?
Tick \(( \checkmark )\) one box. □
□
□
\includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-02_118_113_2261_1324}

2 Which of the following statements is true about the operation of matrix multiplication on the set of all \(2 \times 2\) real matrices? Tick ( \(\checkmark\) ) one box. Matrix multiplication is associative and commutative.
\includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-03_109_112_552_1599} Matrix multiplication is associative but not commutative. □ Matrix multiplication is commutative but not associative. □ Matrix multiplication is not commutative and not associative. □

3 A company is installing an internal telephone network between the offices in a council building. Each office is required to be connected with telephone cables, either directly or indirectly, to every other office in the building. The lengths of cable, in metres, needed to connect the offices are shown in the table below. The council wants the total length of cable that is used to be as small as possible.
The cost to the council to install one metre of cable is \(\pounds 8\)
3

1. 1. Find the minimum total cost to the council to install the cable required for the internal telephone network.
      [0pt] [4 marks]
      3
2. (ii) Suggest a reason why the total cost to the council for installing the internal telephone network is likely to be different from your answer to part (a)(i). 3
3. Before the company starts installing the cable, it is told that the Education office cannot be connected directly to the Transport office due to issues with the building. Explain the possible impact of this on your answer to part (a)(i).

4 Joe, a courier, is required to deliver parcels to six different locations, \(A , B , C , D , E\) and \(F\). Joe needs to start and finish his journey at the depot.
The distances, in miles, between the depot and the six different locations are shown in the table below. The minimum total distance that Joe can travel in order to make all six deliveries, starting and finishing at the depot, is \(L\) miles. 4

1. Using the nearest neighbour algorithm starting from the depot, find an upper bound for \(L\).
   4
2. By deleting the depot, find a lower bound for \(L\).
   4
3. Joe starts from the depot, delivers parcels to all six different locations and arrives back at the depot, covering 134 miles in the process. Joe claims that this is the minimum total distance that is possible for the journey. Comment on Joe's claim.

5 The planar graph \(P\) is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-08_410_406_360_817} 5

1. Determine the number of faces of \(P\).
   5
2. Akwasi claims that \(P\) is semi-Eulerian as it is connected and it has exactly two vertices with even degree. Comment on the validity of Akwasi's claim.
   \includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-09_2488_1716_219_153}

6 The group \(( G , \boldsymbol { A } )\) has the elements \(e , r , r ^ { 2 } , q , q r\) and \(q r ^ { 2 }\), where \(r ^ { 2 } = r \boldsymbol { \Delta } r , q r = q \boldsymbol { \Delta } r , q r ^ { 2 } = q \boldsymbol { \Delta } r ^ { 2 }\) and \(e\) is the identity element of \(G\). The elements \(q\) and \(r\) have the following properties: $$\begin{aligned} & r \boldsymbol { \Delta } r \boldsymbol { \Delta } r = e
& q \boldsymbol { \Delta } q = e
& r ^ { 2 } \boldsymbol { \Delta } q = q \boldsymbol { \Delta } r \end{aligned}$$ 6

1. 1. State the order of \(G\). 6
2. (ii) Prove that the inverse of \(q r\) is \(q r\).
   6
3. Complete the Cayley table for elements of \(G\). 6
4. Complete the Cayley table for elements of \(G\).

   A	\(e\)	\(r\)	\(r ^ { 2 }\)	\(q\)	\(q r\)	\(q r ^ { 2 }\)
   \(e\)	\(e\)	\(r\)	\(r ^ { 2 }\)	\(q\)	\(q r\)	\(q r ^ { 2 }\)
   \(r\)	\(r\)	\(r ^ { 2 }\)	\(e\)
   \(r ^ { 2 }\)	\(r ^ { 2 }\)	\(e\)	\(r\)
   \(q\)	\(q\)	\(q r\)	\(q r ^ { 2 }\)	\(e\)
   \(q r\)	\(q r\)	\(q r ^ { 2 }\)	\(q\)	\(r ^ { 2 }\)
   \(q r ^ { 2 }\)	\(q r ^ { 2 }\)	\(q\)	\(q r\)	\(r\)	\(r ^ { 2 }\)	\(e\)

   6
5. State the name of a group which is isomorphic to \(G\).

7 An engineering company makes brake kits and clutch kits to sell to motorsport teams. The table below summarises the time taken and costs involved in making the two different types of kit. The workers at the engineering company have a combined 2500 hours available to make the kits every month. The engineering company has \(\pounds 200000\) available to cover the costs of making the kits every month. To meet the minimum demands of the motorsport teams, the engineering company must make at least 100 of each type of kit every month. 7

1. Using a graphical method on the grid opposite, find the number of each type of kit that the engineering company should make every month, in order to maximise its total monthly profit. Show clearly how you obtain your answer.
   \includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-13_2486_1709_221_153} Do not write outside the box 7
2. Give a reason why the engineering company may not be able to make the number of each kit that you found in part (a). 7
3. During one particular month the engineering company removes the need to make at least 100 of each type of kit. Explain whether or not this has an effect on your answer to part (a).

8 Daryl and Clare play a zero-sum game. The game is represented by the following pay-off matrix for Daryl. Clare