AQA Further AS Paper 2 Discrete (Further AS Paper 2 Discrete) Specimen

1 A graph has 5 vertices and 6 edges.
Find the sum of the degrees of the vertices. Circle your answer. 10111215

2 A connected planar graph has \(x\) vertices and \(2 x - 4\) edges.
Find the number of faces of the planar graph in terms of \(x\).
Circle your answer.
\(x - 6\)
\(x - 2\)
\(6 - x\)
\(2 - x\)

3 The function min \(( a , b )\) is defined by: $$\begin{aligned} \min ( a , b ) & = a , a < b
& = b , \text { otherwise } \end{aligned}$$ For example, \(\min ( 7,2 ) = 2\) and \(\min ( - 4,6 ) = - 4\). Gary claims that the binary operation \(\Delta\), which is defined as $$x \Delta y = \min ( x , y - 3 )$$ where \(x\) and \(y\) are real numbers, is associative as finding the smallest number is not affected by the order of operation. Disprove Gary's claim.
[0pt] [2 marks]

4 A communications company is conducting a feasibility study into the installation of underground television cables between 5 neighbouring districts. The length of the possible pathways for the television cables between each pair of districts, in miles, is shown in the table. The pathways all run alongside cycle tracks. 4

1. Give a possible reason, in context, why some of the table entries are labelled as ***. 4
2. As part of the feasibility study, Sally, an engineer needs to assess each possible pathway between the districts. To do this, Sally decides to travel along every pathway using a bicycle, starting and finishing in the same district. From past experience, Sally knows that she can travel at an average speed of 12 miles per hour on a bicycle. Find the minimum time, in minutes, that it will take Sally to cycle along every pathway.
   [0pt] [4 marks]

5 Charlotte is visiting a city and plans to visit its five monuments: \(A , B , C , D\) and \(E\).
The network shows the time, in minutes, that a typical tourist would take to walk between the monuments on a busy weekday morning.
\includegraphics[max width=\textwidth, alt={}, center]{ba9e9840-ce27-4ca7-ab05-50461d135445-06_902_1134_529_543} Charlotte intends to walk from one monument to another until she has visited them all, before returning to her starting place. 5

1. Use the nearest neighbour algorithm, starting from \(A\), to find an upper bound for the minimum time for Charlotte's tour.
   5
2. By deleting vertex \(B\), find a lower bound for the minimum time for Charlotte's tour.
   [0pt] [3 marks]
   5
3. Charlotte wants to complete the tour in 52 minutes. Use your answers to parts (a) and (b) to comment on whether this could be possible.
   5
4. Charlotte takes 58 minutes to complete the tour. Evaluate your answers to part (a) and part (b) given this information.
   5
5. Explain how this model for a typical tourist's tour may not be applicable if the tourist walked between the monuments during the evening.

6 Victoria and Albert play a zero-sum game. The game is represented by the following pay-off matrix for Victoria. 6

1. Find the play-safe strategies for each player.
   6
2. State, with a reason, the strategy that Albert should never play.
   6
3. 1. Determine an optimal mixed strategy for Victoria.
      [0pt] [5 marks]
      6
4. (ii) Find the value of the game for Victoria.
   6
5. (iii) State an assumption that must made in order that your answer for part (c)(ii) is the maximum expected pay-off that Victoria can achieve.

7 The network shows a system of pipes, where \(S\) is the source and \(T\) is the sink.
The capacity, in litres per second, of each pipe is shown on each arc.
The cut shown in the diagram can be represented as \(\{ S , P , R \} , \{ Q , T \}\).
\includegraphics[max width=\textwidth, alt={}, center]{ba9e9840-ce27-4ca7-ab05-50461d135445-10_629_1168_616_557} 7

1. Complete the table below to give the value of each of the 8 possible cuts.

   Cut		Value
   \{ S \}	\(\{ P , Q , R , T \}\)	31
   \(\{ S , P \}\)	\(\{ Q , R , T \}\)	32
   \(\{ S , Q \}\)	\(\{ P , R , T \}\)
   \(\{ S , R \}\)	\(\{ P , Q , T \}\)
   \(\{ S , P , Q \}\)	\(\{ R , T \}\)	30
   \(\{ S , P , R \}\)	\(\{ Q , T \}\)	37
   \(\{ S , Q , R \}\)	\(\{ P , T \}\)	35
   \(\{ S , P , Q , R \}\)	\(\{ T \}\)	30

   7
2. State the value of the maximum flow through the network. Give a reason for your answer.
   [0pt] [1 mark] 7
3. Indicate on Figure 1 a possible flow along each arc, corresponding to the maximum flow through the network.
   [0pt] [2 marks] \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{ba9e9840-ce27-4ca7-ab05-50461d135445-11_618_1150_1260_557}
   \end{figure}

8 A family business makes and sells two kinds of kitchen table.
Each pine table takes 6 hours to make and the cost of materials is \(\pounds 30\).
Each oak table takes 10 hours to make and the cost of materials is \(\pounds 70\).
Each month, the business has 360 hours available for making the tables and \(\pounds 2100\) available for the materials.
Each month, the business sells all of its tables to a wholesaler.
The wholesaler specifies that it requires at least 10 oak tables per month and at least as many pine tables as oak tables. Each pine table sold gives the business a profit of \(\pounds 40\) and each oak table sold gives the business a profit of \(\pounds 75\). Use a graphical method to find the number of each type of table the business should make each month, in order to maximise its total profit. Show clearly how you obtain your answer.
[0pt] [8 marks]