AQA D1 (Decision Mathematics 1) 2016 June

1 Alfred has bought six different chocolate bars. He wants to give a chocolate bar to each of his six friends. The table gives the names of the friends and indicates which of Alfred's six chocolate bars they like.

2

1. Use a shuttle sort to rearrange into alphabetical order the following list of names:
   Rob, Eve, Meg, lan, Xavi
   Show the list at the end of each pass.
2. A list of ten numbers is sorted into ascending order, using a shuttle sort.
   1. How many passes are needed?
   2. Give the maximum number of comparisons needed in the sixth pass.
   3. Given that the list is initially in descending order, find the total number of swaps needed.
      [0pt] [4 marks]

3 The network below shows vertices \(A , B , C , D\) and \(E\). The number on each edge shows the distance between vertices.
\includegraphics[max width=\textwidth, alt={}, center]{fb95068f-f76d-492a-b385-bce17b26ae30-06_563_736_402_651}

1. 1. In the case where \(x = 8\), use Kruskal's algorithm to find a minimum spanning tree for the network. Write down the order in which you add edges to your minimum spanning tree.
   2. Draw your minimum spanning tree.
   3. Write down the length of your minimum spanning tree.
2. Alice draws the same network but changes the value of \(x\). She correctly uses Kruskal's algorithm and edge \(C D\) is included in her minimum spanning tree.
   1. Explain why \(x\) cannot be equal to 7 .
   2. Write down an inequality for \(x\).

4 Amal delivers free advertiser magazines to all the houses in his village. The network shows the roads in his village. The number on each road shows the time, in minutes, that Amal takes to walk along that road.
\includegraphics[max width=\textwidth, alt={}, center]{fb95068f-f76d-492a-b385-bce17b26ae30-08_846_1264_445_388}

1. Amal starts his delivery round from his house, at vertex \(A\). He must walk along each road at least once.
   1. Find the length of an optimal Chinese postman route around the village, starting and finishing at Amal's house.
   2. State the number of times that Amal passes his friend Dipak's house, at vertex \(D\).
2. Dipak offers to deliver the magazines while Amal is away on holiday. Dipak must walk along each road at least once. Assume that Dipak takes the same length of time as Amal to walk along each road.
   1. Dipak can start his journey at any vertex and finish his journey at any vertex. Find the length of time for an optimal route for Dipak.
   2. State the vertices at which Dipak could finish, in order to achieve his optimal route.
3. 1. Find the length of time for an optimal route for Dipak, if, instead, he wants to finish at his house, at vertex \(D\), and can start his journey at any other vertex.
   2. State the start vertex.

5 A fair comes to town one year and sets up its rides in two large fields that are separated by a river. The diagram shows the ten main rides, at \(A , B , C , \ldots , J\). The numbers on the edges represent the times, in minutes, it takes to walk between pairs of rides. A footbridge connects the rides at \(D\) and \(F\).

1. 1. Use Dijkstra's algorithm on the diagram below to find the minimum time to walk from \(A\) to each of the other main rides.
   2. Write down the route corresponding to the minimum time to walk from \(A\) to \(G\).
2. The following year, the fair returns. In addition to the information shown on the diagram, another footbridge has been built to connect the rides at \(E\) and \(G\). This reduces the time taken to travel from \(A\) to \(G\), but the time taken to travel from \(A\) to \(J\) is not reduced. The time to walk across the footbridge from \(E\) to \(G\) is \(x\) minutes, where \(x\) is an integer. Find two inequalities for \(x\) and hence state the value of \(x\). \section*{Answer space for question 5}
3. 1. 
      \includegraphics[max width=\textwidth, alt={}, center]{fb95068f-f76d-492a-b385-bce17b26ae30-12_659_1591_1692_223}

6 A connected graph is semi-Eulerian if exactly two of its vertices are of odd degree.

1. A graph is drawn with 4 vertices and 7 edges. What is the sum of the degrees of the vertices?
2. Draw a simple semi-Eulerian graph with exactly 5 vertices and 5 edges, in which exactly one of the vertices has degree 4 .
3. Draw a simple semi-Eulerian graph with exactly 5 vertices that is also a tree.
4. A simple graph has 6 vertices. The graph has two vertices of degree 5 . Explain why the graph can have no vertex of degree 1.

7 A company operates a steam railway between six stations. The minimum cost (in euros) of travelling between pairs of stations is shown in the table below.

1. On Figure 1 below, use Prim's algorithm, starting from \(P\), to find a minimum spanning tree for the graph connecting \(P , Q , R , S , T\) and \(U\). State clearly the order in which you select the vertices and draw your minimum spanning tree. \section*{Question 7 continues on page 20} \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1}

   	\(\boldsymbol { P }\)	\(\boldsymbol { Q }\)	\(\boldsymbol { R }\)	\(\boldsymbol { S }\)	\(\boldsymbol { T }\)	\(\boldsymbol { U }\)
   \(\boldsymbol { P }\)	-	14	7	11	6	12
   \(\boldsymbol { Q }\)	14	-	8	10	9	10
   \(\boldsymbol { R }\)	7	8	-	12	13	15
   \(\boldsymbol { S }\)	11	10	12	-	5	11
   \(\boldsymbol { T }\)	6	9	13	5	-	10
   \(\boldsymbol { U }\)	12	10	15	11	10	-

   \end{table}
2. Another station, \(V\), is opened. The minimum costs (in euros) of travelling to and from \(V\) to each of the other stations are added to the table in part (a), as shown in Figure 2(i) below. Further copies of this table are shown in Figure 2(ii). Arjen is on holiday and he plans to visit each station. He intends to board a train at \(V\) and visit all the other stations, once only, before returning to \(V\).
   1. By first removing \(V\), obtain a lower bound for the minimum travelling cost of Arjen's tour. (You may use Figure 2(i) for your working.)
   2. Use the nearest neighbour algorithm twice, starting each time from \(V\), to find two different upper bounds for the minimum cost of Arjen's tour. State, with a reason, which of your two answers gives the better upper bound. (You may use Figure 2(ii) for your working.)
   3. Hence find an optimal tour of the seven stations. Explain how you know that it is optimal. Answer space for question 7(b) \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Figure 2(ii)}

      	\(\boldsymbol { P }\)	\(Q\)	\(\boldsymbol { R }\)	\(\boldsymbol { S }\)	\(T\)	\(\boldsymbol { U }\)	\(V\)
      \(\boldsymbol { P }\)	-	14	7	11	6	12	15
      \(Q\)	14	-	8	10	9	10	18
      \(\boldsymbol { R }\)	7	8	-	12	13	15	14
      \(\boldsymbol { S }\)	11	10	12	-	5	11	14
      \(T\)	6	9	13	5	-	10	17
      \(\boldsymbol { U }\)	12	10	15	11	10	-	12
      \(V\)	15	18	14	14	17	12	-

      \end{table}

8 Nerys runs a cake shop. In November and December she sells Christmas hampers. She makes up the hampers herself, in two sizes: Luxury and Special. Each day, Nerys prepares \(x\) Luxury hampers and \(y\) Special hampers.
It takes Nerys 10 minutes to prepare a Luxury hamper and 15 minutes to prepare a Special hamper. She has 6 hours available, each day, for preparing hampers. From past experience, Nerys knows that each day:

• she will need to prepare at least 5 hampers of each size
• she will prepare at most a total of 32 hampers
• she will prepare at least twice as many Luxury hampers as Special hampers.

Each Luxury hamper that Nerys prepares makes her a profit of \(\pounds 15\); each Special hamper makes a profit of \(\pounds 20\). Nerys wishes to maximise her daily profit, \(\pounds P\).

1. Show that \(x\) and \(y\) must satisfy the inequality \(2 x + 3 y \leqslant 72\).
2. In addition to \(x \geqslant 5\) and \(y \geqslant 5\), write down two more inequalities that model the constraints above.
3. On the grid opposite draw a suitable diagram to enable this problem to be solved graphically. Indicate a feasible region and the direction of an objective line.
4. 1. Use your diagram to find the number of each type of hamper that Nerys should prepare each day to achieve a maximum profit.
   2. Calculate this profit.
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