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AQA D1 2007 June Q3

11 marks Moderate -0.5

3 [Figure 1, printed on the insert, is provided for use in this question.]
The following network represents the footpaths connecting 12 buildings on a university campus. The number on each edge represents the time taken, in minutes, to walk along a footpath. \includegraphics[max width=\textwidth, alt={}, center]{eb305e75-0e85-4f99-8f04-27046a153532-03_899_1317_589_356}

1. Use Dijkstra's algorithm on Figure 1 to find the minimum time to walk from $A$ to $L$.
2. State the corresponding route.
A new footpath is to be constructed. There are two possibilities:
from $A$ to $D$, with a walking time of 30 minutes; or from $A$ to $I$, with a walking time of 20 minutes. Determine which of the two alternative new footpaths would reduce the walking time from $A$ to $L$ by the greater amount.
(3 marks)

AQA D1 2007 June Q4

17 marks Moderate -0.5

4 The diagram shows the various ski-runs at a ski resort. There is a shop at $S$. The manager of the ski resort intends to install a floodlighting system by placing a floodlight at each of the 12 points $A , B , \ldots , L$ and at the shop at $S$. The number on each edge represents the distance, in metres, between two points. \includegraphics[max width=\textwidth, alt={}, center]{eb305e75-0e85-4f99-8f04-27046a153532-04_842_830_577_607} Total of all edges $= 1135$

The manager wishes to use the minimum amount of cabling, which must be laid along the ski-runs, to connect the 12 points $A , B , \ldots , L$ and the shop at $S$.
1. Starting from the shop, and showing your working at each stage, use Prim's algorithm to find the minimum amount of cabling needed to connect the shop and the 12 points.
2. State the length of your minimum spanning tree.
3. Draw your minimum spanning tree.
4. The manager used Kruskal's algorithm to find the same minimum spanning tree. Find the seventh and the eighth edges that the manager added to his spanning tree.
At the end of each day a snow plough has to drive at least once along each edge shown in the diagram in preparation for the following day's skiing. The snow plough must start and finish at the point $L$. Use the Chinese Postman algorithm to find the minimum distance that the snow plough must travel.
(6 marks)

AQA D1 2007 June Q5

16 marks Moderate -0.8

5 [Figure 2, printed on the insert, is provided for use in this question.]
The Jolly Company sells two types of party pack: excellent and luxury.
Each excellent pack has five balloons and each luxury pack has ten balloons.
Each excellent pack has 32 sweets and each luxury pack has 8 sweets.
The company has 1500 balloons and 4000 sweets available.
The company sells at least 50 of each type of pack and at least 140 packs in total.
The company sells $x$ excellent packs and $y$ luxury packs.

Show that the above information can be modelled by the following inequalities. $$x + 2 y \leqslant 300 , \quad 4 x + y \leqslant 500 , \quad x \geqslant 50 , \quad y \geqslant 50 , \quad x + y \geqslant 140$$ (4 marks)
The company sells each excellent pack for 80 p and each luxury pack for $\pounds 1.20$. The company needs to find its minimum and maximum total income.
1. On Figure 2, draw a suitable diagram to enable this linear programming problem to be solved graphically, indicating the feasible region and an objective line.
2. Find the company's maximum total income and state the corresponding number of each type of pack that needs to be sold.
3. Find the company's minimum total income and state the corresponding number of each type of pack that needs to be sold.

AQA D1 2007 June Q6

14 marks Easy -1.2

Mark is staying at the Grand Hotel ( $G$ ) in Oslo. He is going to visit four famous places in Oslo: Aker Brygge ( $A$ ), the National Theatre ( $N$ ), Parliament House ( $P$ ) and the Royal Palace ( $R$ ). The figures in the table represent the walking times, in seconds, between the places.

	Grand Hotel ( $G$ )	Aker Brygge (A)	National Theatre ( $N$ )	Parliament House (P)	Royal Palace (R)
Grand Hotel ( $G$ )	-	165	185	65	160
Aker Brygge (A)	165	-	155	115	275
National Theatre ( $N$ )	185	155	-	205	125
Parliament House (P)	65	115	205	-	225
Royal Palace (R)	160	275	125	225	-

Mark is to start his tour from the Grand Hotel, visiting each place once before returning to the Grand Hotel. Mark wishes to keep his walking time to a minimum.

Use the nearest neighbour algorithm, starting from the Grand Hotel, to find an upper bound for the walking time for Mark's tour.
By deleting the Grand Hotel, find a lower bound for the walking time for Mark's tour.
The walking time for an optimal tour is $T$ seconds. Use your answers to parts (a)(i) and (a)(ii) to write down a conclusion about $T$.

Mark then intends to start from the Grand Hotel ( $G$ ), visit three museums, Ibsen ( $I$ ), Munch ( $M$ ) and Viking ( $V$ ), and return to the Grand Hotel. He uses public transport. The table shows the minimum travelling times, in minutes, between the places.

\backslashbox{From}{To}	Grand Hotel ( $G$ )	Ibsen (I)	Munch ( $M$ )	Viking ( $\boldsymbol { V }$ )
Grand Hotel ( $\boldsymbol { G }$ )	-	20	17	30
Ibsen (I)	15	-	32	16
Munch (M)	26	18	-	21
Viking ( $\boldsymbol { V }$ )	19	27	24	-

Find the length of the tour $G I M V G$.
Find the length of the tour GVMIG.
Find the number of different possible tours for Mark.
Write down the number of different possible tours for Mark if he were to visit $n$ museums, starting and finishing at the Grand Hotel.

AQA D1 2014 June Q1

6 marks Moderate -0.8

1 Five people, $A , B , C , D$ and $E$, are to be allocated to five tasks, 1, 2, 3, 4 and 5 . The following bipartite graph shows the tasks that each person is able to undertake. \includegraphics[max width=\textwidth, alt={}, center]{5ee6bc88-6343-4ee6-8ecd-c13868d77049-02_441_437_699_797}

Represent this information in an adjacency matrix.
Initially, $A$ is allocated to task 3, $B$ to task 2 and $C$ to task 4.
1. Demonstrate, by using an alternating-path algorithm from this initial matching, how each person can be allocated to a different task.
2. Find a different allocation of people to tasks.

AQA D1 2014 June Q2

11 marks Moderate -0.8

2 A document which is currently written in English is to be translated into six other European Union languages. The cost of translating a document varies, as it is harder to find translators for some languages. The costs, in euros, are shown in the table below.

1. On the table below, showing the order in which you select the edges, use Prim's algorithm, starting from $E$, to find a minimum spanning tree for the graph connecting $D , E , F , G , H , I$ and $S$.
2. Find the length of your minimum spanning tree.
3. Draw your minimum spanning tree.

It is given that the graph has a unique minimum spanning tree. State the final two edges that would be added to complete the minimum spanning tree in the case where:

Prim's algorithm starting from $H$ is used;

Kruskal's algorithm is used. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Answer space for question 2}

	Danish ( $\boldsymbol { D }$ )	English ( $\boldsymbol { E }$ )	French (F)	German ( $G$ )	Hungarian (H)	Italian (I)	Spanish $\boldsymbol { ( } \boldsymbol { S } \boldsymbol { ) }$
Danish (D)	-	120	140	80	170	140	140
English ( $\boldsymbol { E }$ )	120	-	70	80	130	130	110
French (F)	140	70	-	90	190	85	90
German ( $G$ )	80	80	90	-	110	100	100
Hungarian (H)	170	130	190	110	-	140	150
Italian (I)	140	130	85	100	140	-	60
Spanish ( $\boldsymbol { S }$ )	140	110	90	100	150	60	-

\end{table}

AQA D1 2014 June Q3

10 marks Moderate -0.3

3 The network below shows 11 towns, $A , B , \ldots , K$. The number on each edge shows the time, in minutes, to travel between a pair of towns.

1. Use Dijkstra's algorithm on the diagram below to find the minimum time to travel from $A$ to $K$.
2. State the corresponding route.
On a particular day, Jenny travels from $A$ to $K$ but visits her friend at $D$ on her way. Find Jenny's minimum travelling time.
On a different day, all roads connected to $I$ are closed due to flooding. Jenny does not visit her friend at $D$. Find her minimum time to travel from $A$ to $K$. State the route corresponding to this minimum time.
[0pt] [2 marks] \section*{Answer space for question 3}
\includegraphics[max width=\textwidth, alt={}]{5ee6bc88-6343-4ee6-8ecd-c13868d77049-06_1478_1548_1213_239}

AQA D1 2014 June Q4

10 marks Moderate -0.3

4 Paulo sells vegetables from his van. He drives around the streets of a small village. The network shows the streets in the village. The number on each edge shows the time, in minutes, to drive along that street. Paulo starts from his house located at vertex $A$ and drives along all the streets at least once before returning to his house. \includegraphics[max width=\textwidth, alt={}, center]{5ee6bc88-6343-4ee6-8ecd-c13868d77049-10_1518_1605_598_198} The total of all the times in the diagram is 79.5 minutes.

Find the length of an optimal Chinese postman route around the village, starting and finishing at $A$. (Shortest routes between vertices may be found by inspection.)
For an optimal Chinese postman route, state:
1. the number of times the vertex $F$ would occur;
2. the number of times the vertex $D$ would occur.
Toto is standing for the position of Mayor in the local elections. He intends to travel along all the roads at least once. He can start his journey at any vertex and can finish his journey at any vertex.
1. Find the length of an optimal route for Toto.
  [0pt]
2. State the vertices from which Toto could start in order to achieve this optimal route. [3 marks]

AQA D1 2014 June Q5

11 marks Moderate -0.8

5 The feasible region of a linear programming problem is determined by the following: $$\begin{aligned} x & \geqslant 1 \\ y & \geqslant 3 \\ x + y & \geqslant 5 \\ x + y & \leqslant 12 \\ 3 x + 8 y & \leqslant 64 \end{aligned}$$

On the grid below, draw a suitable diagram to represent the inequalities and indicate the feasible region.
Use your diagram to find, on the feasible region:
1. the maximum value of $3 x + y$;
2. the maximum value of $2 x + 3 y$;
3. the minimum value of $- 2 x + y$. In each case, state the coordinates of the point corresponding to your answer.
  [0pt] [6 marks]

AQA D1 2014 June Q6

10 marks Easy -1.8

Sarah is solving a travelling-salesman problem.
1. She finds the following upper bounds: $32,33,32,32,30,32,32$. Write down the best upper bound.
2. She finds the following lower bounds: 17, 18, 17, 20, 18, 17, 20. Write down the best lower bound.

Rob is travelling by train to a number of cities. He is to start at $M$ and visit each other city at least once before returning to $M$. The diagram shows the travelling times, in minutes, between cities. Where no time is shown, there is no direct journey available. \includegraphics[max width=\textwidth, alt={}, center]{5ee6bc88-6343-4ee6-8ecd-c13868d77049-16_959_1122_1059_443} The table below shows the minimum travelling times between all pairs of cities.

\cline { 2 - 6 } \multicolumn{1}{c\|}{}	$\boldsymbol { B }$	$\boldsymbol { E }$	$\boldsymbol { L }$	$\boldsymbol { M }$	$\boldsymbol { N }$
$\boldsymbol { B }$	-	230	82	102	192
$\boldsymbol { E }$	230	-	148	244	258
$\boldsymbol { L }$	82	148	-	126	110
$\boldsymbol { M }$	102	244	126	-	236
$\boldsymbol { N }$	192	258	110	236	-

Explain why the minimum travelling time from $M$ to $N$ is not 283 .
Find an upper bound for the minimum travelling time by using the tour $M N B E L M$.
Write down the actual route corresponding to the tour $M N B E L M$.
Use the nearest-neighbour algorithm, starting from $M$, to find another upper bound for the minimum travelling time of Rob's tour.
[0pt] [4 marks] QUESTION

1. The best upper bound is $\_\_\_\_$
2. The best lower bound is $\_\_\_\_$

AQA D1 2014 June Q7

8 marks Standard +0.3

AQA D1 2014 June Q8

9 marks Standard +0.3

8 In this question you may use the fact that any simple graph must have an even number of vertices of odd degree.

A simple graph has five vertices and their degrees are $$x , x + 1 , x + 1 , x + 2 \text { and } x + 3$$
1. Show that $x$ must be odd.
2. Find the value of $x$ and draw a graph with vertices having the given degrees.
A simple graph has 10 vertices.
1. State the minimum possible degree and maximum possible degree of a vertex.
2. Show that the degrees of the vertices cannot all be different.
  □
  □

AQA D1 2015 June Q1

5 marks Moderate -0.8

1 A quiz team must answer questions from six different topics, numbered 1 to 6. The team has six players, $A , B , C , D , E$ and $F$. Each player can only answer questions on one of the topics. The players list their preferred topics. The bipartite graph shows their choices. \includegraphics[max width=\textwidth, alt={}, center]{f5890e58-38c3-413c-8762-6f80ce6dcec7-02_711_499_781_760} Initially, $A$ is allocated topic 2, $B$ is allocated topic $3 , C$ is allocated topic 1 and $F$ is allocated topic 4. By using an alternating path algorithm from this initial matching, find a complete matching.
[0pt] [5 marks]

AQA D1 2015 June Q2

9 marks Moderate -0.8

2 The network below shows 8 towns, $A , B , \ldots , H$. The number on each edge shows the length of the road, in miles, between towns. During the winter, the council treats some of the roads with salt so that each town can be safely reached on treated roads from any other town. It costs $\pounds 30$ to treat a mile of road. \includegraphics[max width=\textwidth, alt={}, center]{f5890e58-38c3-413c-8762-6f80ce6dcec7-04_876_1611_497_210}

1. Use Prim's algorithm starting from $A$, showing the order in which you select the edges, to find a minimum spanning tree for the network.
2. Draw your minimum spanning tree.
3. Calculate the minimum cost to the council of making it possible for each town to be safely reached on treated roads from any other town.
On one occasion, the road from $C$ to $E$ is impassable because of flooding. Find the minimum cost of treating sufficient roads for safe travel in this case.
[0pt] [2 marks]

AQA D1 2015 June Q3

5 marks Easy -1.2

3 Four students, $A , B , C$ and $D$, are using different algorithms to sort 16 numbers into ascending order.

Student $A$ uses the quicksort algorithm. State the number of comparisons on the first pass.
Student $B$ uses the Shell sort algorithm. State the number of comparisons on the first pass.
Student $C$ uses the shuttle sort algorithm. State the minimum number of comparisons on the final pass.
Student $D$ uses the bubble sort algorithm. Find the maximum total number of comparisons.

AQA D1 2015 June Q4

8 marks Easy -1.2

4 The network opposite shows roads connecting 10 villages, $A , B , \ldots , J$. The time taken to drive along a road is not proportional to the length of the road. The number on each edge shows the average time, in minutes, to drive along each road.

A commuter wishes to drive from village $A$ to the railway station at $J$.
1. Use Dijkstra's algorithm, on the diagram opposite, to find the shortest driving time from $A$ to $J$.
2. State the corresponding route.
A taxi driver is in village $D$ at 10.30 am when she receives a radio call asking her to pick up a passenger at village $A$ and take him to the station at $J$. Assuming that it takes her 3 minutes to load the passenger and his luggage, at what time should she expect to arrive at the station?
[0pt] [2 marks]
\includegraphics[max width=\textwidth, alt={}]{f5890e58-38c3-413c-8762-6f80ce6dcec7-09_2484_1717_223_150}

AQA D1 2015 June Q5

7 marks Moderate -0.5

5 The network shows the paths mown through a wildflower meadow so that visitors can use these paths to enjoy the flowers. The lengths of the paths are shown in metres. \includegraphics[max width=\textwidth, alt={}, center]{f5890e58-38c3-413c-8762-6f80ce6dcec7-10_1097_1603_413_214} The total length of all the paths is 1400 m .
The mower is kept in a shed at $A$. The groundskeeper must mow all the paths and return the mower to its shed.

Find the length of an optimal Chinese postman route starting and finishing at $A$.
State the number of times that the mower, following the optimal route, will pass through:
1. $C$;
2. $D$.

AQA D1 2015 June Q6

12 marks Easy -1.2

6 The network shows the roads linking a warehouse at $A$ and five shops, $B , C , D , E$ and $F$. The numbers on the edges show the lengths, in miles, of the roads. A delivery van leaves the warehouse, delivers to each of the shops and returns to the warehouse. \includegraphics[max width=\textwidth, alt={}, center]{f5890e58-38c3-413c-8762-6f80ce6dcec7-12_1241_1239_484_402}

Complete the table, on the page opposite, showing the shortest distances between the vertices.
1. Find the total distance travelled if the van follows the cycle $A E F B C D A$.
2. Explain why your answer to part (b)(i) provides an upper bound for the minimum journey length.
Use the nearest neighbour algorithm starting from $D$ to find a second upper bound.
By deleting $A$, find a lower bound for the minimum journey length.
Given that the minimum journey length is $T$, write down the best inequality for $T$ that can be obtained from your answers to parts (b), (c) and (d).
[0pt] [1 mark] \section*{Answer space for question 6} REFERENCE

	$\boldsymbol { A }$	$\boldsymbol { B }$	$\boldsymbol { C }$	$\boldsymbol { D }$	$\boldsymbol { E }$	$\boldsymbol { F }$
$\boldsymbol { A }$	-	7
$\boldsymbol { B }$	7	-	5
$\boldsymbol { C }$		5	-	4
$\boldsymbol { D }$			4	-	6
$\boldsymbol { E }$				6	-	10
$\boldsymbol { F }$					10	-

AQA D1 2015 June Q7

6 marks Moderate -0.8

A simple connected graph has 4 edges and $m$ vertices. State the possible values of $m$.
A simple connected graph has $n$ edges and 4 vertices. State the possible values of $n$.
A simple connected graph, $G$, has 5 vertices and is Eulerian but not Hamiltonian. Draw a possible graph $G$.
[0pt] [2 marks]

AQA D1 2015 June Q8

6 marks Easy -1.2

8 A student is tracing the following algorithm. \includegraphics[max width=\textwidth, alt={}, center]{f5890e58-38c3-413c-8762-6f80ce6dcec7-18_1431_955_372_539}

Trace the algorithm illustrated in the flowchart for the case where the input value of $N$ is 5 .
Explain the role of $N$ in the algorithm.

AQA D1 2015 June Q9

17 marks Moderate -0.8

9 A company producing chicken food makes three products, Basic, Premium and Supreme, from wheat, maize and barley. A tonne $( 1000 \mathrm {~kg} )$ of Basic uses 400 kg of wheat, 200 kg of maize and 400 kg of barley.
A tonne of Premium uses 400 kg of wheat, 500 kg of maize and 100 kg of barley.
A tonne of Supreme uses 600 kg of wheat, 200 kg of maize and 200 kg of barley.
The company has 130 tonnes of wheat, 70 tonnes of maize and 72 tonnes of barley available. The company must make at least 75 tonnes of Supreme.
The company makes $\pounds 50$ profit per tonne of Basic, $\pounds 100$ per tonne of Premium and $\pounds 150$ per tonne of Supreme. They plan to make $x$ tonnes of Basic, $y$ tonnes of Premium and $z$ tonnes of Supreme.

Write down four inequalities representing the constraints (in addition to $x , y \geqslant 0$ ).
[0pt] [4 marks]
The company want exactly half the production to be Supreme. Show that the constraints in part (a) become $$\begin{aligned} x + y & \leqslant 130 \\ 4 x + 7 y & \leqslant 700 \\ 2 x + y & \leqslant 240 \\ x + y & \geqslant 75 \\ x & \geqslant 0 \\ y & \geqslant 0 \end{aligned}$$
On the grid opposite, illustrate all the constraints and label the feasible region.
Write an expression for $P$, the profit for the whole production, in terms of $x$ and $y$ only.
[0pt] [2 marks]
1. By drawing an objective line on your graph, or otherwise, find the values of $x$ and $y$ which give the maximum profit.
  [0pt] [2 marks]
2. State the maximum profit and the amount of each product that must be made.
  [0pt] [2 marks] \section*{Answer space for question 9}
  \includegraphics[max width=\textwidth, alt={}]{f5890e58-38c3-413c-8762-6f80ce6dcec7-21_1349_1728_310_148}
  QUESTION
  PART
  REFERENCE \includegraphics[max width=\textwidth, alt={}, center]{f5890e58-38c3-413c-8762-6f80ce6dcec7-24_2488_1728_219_141}

AQA D1 2016 June Q1

5 marks Easy -1.2

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.

AQA D1 2016 June Q2

7 marks Easy -1.8

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.
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]

AQA D1 2016 June Q3

7 marks Moderate -0.3

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. 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.
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 $x$.

AQA D1 2016 June Q4

11 marks Moderate -0.3

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}

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 $D$.
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.
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.

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	\(\boldsymbol { A }\)	\(\boldsymbol { B }\)	\(\boldsymbol { C }\)	\(\boldsymbol { D }\)	\(\boldsymbol { E }\)	\(\boldsymbol { F }\)
\(\boldsymbol { A }\)	-	7
\(\boldsymbol { B }\)	7	-	5
\(\boldsymbol { C }\)		5	-	4
\(\boldsymbol { D }\)			4	-	6
\(\boldsymbol { E }\)				6	-	10
\(\boldsymbol { F }\)					10	-

	Grand Hotel ( \(G\) )	Aker Brygge (A)	National Theatre ( \(N\) )	Parliament House (P)	Royal Palace (R)
Grand Hotel ( \(G\) )	-	165	185	65	160
Aker Brygge (A)	165	-	155	115	275
National Theatre ( \(N\) )	185	155	-	205	125
Parliament House (P)	65	115	205	-	225
Royal Palace (R)	160	275	125	225	-

	Danish ( \(\boldsymbol { D }\) )	English ( \(\boldsymbol { E }\) )	French (F)	German ( \(G\) )	Hungarian (H)	Italian (I)	Spanish \(\boldsymbol { ( } \boldsymbol { S } \boldsymbol { ) }\)
Danish (D)	-	120	140	80	170	140	140
English ( \(\boldsymbol { E }\) )	120	-	70	80	130	130	110
French (F)	140	70	-	90	190	85	90
German ( \(G\) )	80	80	90	-	110	100	100
Hungarian (H)	170	130	190	110	-	140	150
Italian (I)	140	130	85	100	140	-	60
Spanish ( \(\boldsymbol { S }\) )	140	110	90	100	150	60	-

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