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AQA D1 2005 January Q4

9 marks Easy -1.2

4 The Head Teacher of a school is allocating five teachers, Amy ( $A$ ), Ben ( $B$ ), Celia ( $C$ ), Duncan ( $D$ ), and Erica ( $E$ ), to the five posts of Head of Year 7, 8, 9, 10 and 11. The five teachers are asked which year(s) they would be willing to take. This information is shown below. Amy is willing to take Year 7 or Year 8 . Ben is willing to take Year 7, Year 8 or Year 10. Celia is willing to take Year 8, Year 9 or Year 11. Duncan will take only Year 9.
Erica will take only Year 11.

Show this information on a bipartite graph.
Initially the Head Teacher assigns Amy to Year 8, Ben to Year 10, Celia to Year 9 and Erica to Year 11. Demonstrate, by using an alternating path from this initial matching, how each teacher can be matched to a year that they are willing to take.

AQA D1 2005 January Q5

10 marks Moderate -0.8

5 The network shows the lengths, in miles, of roads connecting eleven villages. \includegraphics[max width=\textwidth, alt={}, center]{76bccb26-f2ec-4798-bb6b-89c922f9651a-04_1100_1575_406_251}

Use Prim's algorithm, starting from $A$, to find the minimum spanning tree for the network.
State the length of your minimum spanning tree.
Draw your minimum spanning tree.
A student used Kruskal's algorithm to find the same minimum spanning tree. Find the seventh and eighth edges that the student added to his spanning tree.

AQA D1 2005 January Q6

8 marks Standard +0.8

6 [Figure 1, printed on a separate sheet, is provided for use in this question.]
A theme park is built on two levels. The levels are connected by a staircase. There are five rides on each of the levels. The diagram shows the ten rides: $A , B , \ldots \ldots J$. The numbers on the edges represent the times, in minutes, taken to travel between pairs of rides. \includegraphics[max width=\textwidth, alt={}, center]{76bccb26-f2ec-4798-bb6b-89c922f9651a-05_984_1593_584_226}

Use Dijkstra's algorithm on Figure 1 to find the minimum time to travel from $A$ to $J$.
A new staircase is built connecting rides $B$ and $G$. The time taken to travel from $B$ to $G$ using this staircase is $x$ minutes, where $x$ is an integer. The time taken to travel from $A$ to $G$ is reduced, but the time taken to travel from $A$ to $J$ is not reduced. Find two inequalities for $x$ and hence state the value of $x$.
(4 marks)

AQA D1 2005 January Q7

16 marks Moderate -0.8

7 Rob delivers bread to six shops $A , B , C , D , E$ and $F$. Each day, Rob starts at shop $A$, travels to each of the other shops before returning to shop $A$. The table shows the distances, in miles, between the shops.

	$\boldsymbol { A }$	$\boldsymbol { B }$	$\boldsymbol { C }$	$\boldsymbol { D }$	$\boldsymbol { E }$	$\boldsymbol { F }$
$\boldsymbol { A }$	-	8	6	9	12	7
$\boldsymbol { B }$	8	-	10	14	13	8
$\boldsymbol { C }$	6	10	-	7	16	10
$\boldsymbol { D }$	9	14	7	-	15	5
$\boldsymbol { E }$	12	13	16	15	-	11
$\boldsymbol { F }$	7	8	10	5	11	-

1. Find the length of the tour $A B C D E F A$.
2. Find the length of the tour obtained by using the nearest neighbour algorithm starting from $A$.
By deleting $A$, find a lower bound for the length of a minimum tour.
By deleting $F$, another lower bound of 45 miles is obtained for the length of a minimum tour. The length of a minimum tour is $T$ miles. Write down the smallest interval for $T$ which can be obtained from your answers to parts (a) and (b), and the information given in this part.
(3 marks)

AQA D1 2005 January Q8

18 marks Moderate -0.8

8 [Figure 2, printed on a separate sheet, is provided for use in this question.]
A bakery makes two types of pizza, large and medium.
Every day the bakery must make at least 40 of each type.
Every day the bakery must make at least 120 in total but not more than 400 pizzas in total.
Each large pizza takes 4 minutes to make, and each medium pizza takes 2 minutes to make. There are four workers available, each for five hours a day, to make the pizzas. The bakery makes a profit of $\pounds 3$ on each large pizza sold and $\pounds 1$ on each medium pizza sold.
Each day, the bakery makes and sells $x$ large pizzas and $y$ medium pizzas.
The bakery wishes to maximise its profit, $\pounds P$.

Show that one of the constraints leads to the inequality $$2 x + y \leqslant 600$$
Formulate this situation as a linear programming problem.
On Figure 2, draw a suitable diagram to enable the problem to be solved graphically, indicating the feasible region and an objective line.
Use your diagram to find the maximum daily profit.
The bakery introduces a new pricing structure in which the profit is $\pounds 2$ on each large pizza sold and $\pounds 2$ on each medium pizza sold.
1. Find the new maximum daily profit for the bakery.
2. Write down the number of different combinations that would give the new maximum daily profit.

AQA D1 2012 January Q1

5 marks Easy -1.2

1 Use a Shell sort to rearrange the following numbers into ascending order, showing the new arrangement after each pass. $\begin{array} { l l l l l l l l } 37 & 25 & 16 & 12 & 36 & 24 & 13 & 11 \end{array}$ (5 marks) PART REFERENCE REFERENCE

AQA D1 2012 January Q2

5 marks Easy -1.8

Draw a bipartite graph representing the following adjacency matrix.

\cline { 2 - 7 } \multicolumn{1}{c\|}{}	$\mathbf { 1 }$	$\mathbf { 2 }$	$\mathbf { 3 }$	$\mathbf { 4 }$	$\mathbf { 5 }$	$\mathbf { 6 }$
$\boldsymbol { A }$	1	1	0	0	1	1
$\boldsymbol { B }$	0	1	0	0	1	1
$\boldsymbol { C }$	1	0	0	0	1	1
$\boldsymbol { D }$	0	1	1	1	0	1
$\boldsymbol { E }$	0	0	0	0	1	1
$\boldsymbol { F }$	0	0	0	0	0	1

Given that $A , B , C , D , E$ and $F$ represent six people and that 1, 2, 3, 4, 5 and 6 represent six tasks to which they may be assigned, explain why a complete matching is impossible. \begin{verbatim} QUESTION PART REFERENCE \end{verbatim}

AQA D1 2012 January Q3

9 marks Moderate -0.8

3 The following network shows the roads connecting seven villages, $A , B , C , \ldots , G$. The number on each edge represents the length, in miles, between a pair of villages. \includegraphics[max width=\textwidth, alt={}, center]{5a414265-6273-41c5-ad5f-f6316bd774d0-06_978_1108_443_466}

Use Kruskal's algorithm to find a minimum spanning tree for the network. (5 marks)
State the length of your minimum spanning tree.
There are two minimum spanning trees for this network. Draw both of these minimum spanning trees.

AQA D1 2012 January Q4

6 marks Standard +0.3

4 The following network shows the times, in minutes, taken by a policeman to walk along roads connecting 12 places, $A , B , \ldots , L$, on his beat. Each day, the policeman has to walk along each road at least once, starting and finishing at $A$. \includegraphics[max width=\textwidth, alt={}, center]{5a414265-6273-41c5-ad5f-f6316bd774d0-08_1141_1313_461_360} The total of all the times in the network is 224 minutes.

Find the length of an optimal Chinese postman route for the policeman.
State the number of times that the vertex $J$ would appear in a route corresponding to the length found in part (a).

AQA D1 2012 January Q5

10 marks Standard +0.3

5 The feasible region of a linear programming problem is determined by the following: $$\begin{aligned} y & \geqslant 20 \\ x + y & \geqslant 25 \\ 5 x + 2 y & \leqslant 100 \\ y & \leqslant 4 x \\ y & \geqslant 2 x \end{aligned}$$

On Figure 1 opposite, draw a suitable diagram to represent the inequalities and indicate the feasible region.
Use your diagram to find the minimum value of $P$, on the feasible region, in the case where:
1. $P = x + 2 y$;
2. $P = - x + y$. In each case, state the corresponding values of $x$ and $y$.

AQA D1 2012 January Q6

11 marks Standard +0.3

6 The network below shows the lengths of roads, in miles, connecting 10 towns, $A , B , \ldots , J$.

Use Dijkstra's algorithm on the network to find the shortest distance from $A$ to $J$. Show all your working at each vertex.
Write down the corresponding route.
A new road is to be constructed connecting $B$ to $G$. Find the length of this new road if the shortest distance from $A$ to $J$ is reduced by 10 miles. State the new route.
(3 marks) \includegraphics[max width=\textwidth, alt={}, center]{5a414265-6273-41c5-ad5f-f6316bd774d0-12_1846_1719_861_150}

AQA D1 2012 January Q7

19 marks Easy -1.2

7 The diagram shows the locations of some schools. The number on each edge shows the distance, in miles, between pairs of schools. \includegraphics[max width=\textwidth, alt={}, center]{5a414265-6273-41c5-ad5f-f6316bd774d0-14_1031_1231_428_392} Sam, an adviser, intends to travel from one school to the next until he has visited all of the schools, before returning to his starting school. The shortest distances for Sam to travel between some of the schools are shown in Table 1 opposite.

Complete Table 1.
1. On the completed Table 1, use the nearest neighbour algorithm, starting from $B$, to find an upper bound for the length of Sam's tour.
2. Write down Sam's actual route if he were to follow the tour corresponding to the answer in part (b)(i).
3. Using the nearest neighbour algorithm, starting from each of the other vertices in turn, the following upper bounds for the length of Sam's tour were obtained: 77, 77, 77, 76, 77 and 76. Write down the best upper bound. 7
1. On Table 2 below, showing the order in which you select the edges, use Prim's algorithm, starting from $A$, to find a minimum spanning tree for the schools $A , B , C$, $D , F$ and $G$.
2. Hence find a lower bound for the length of Sam's minimum tour.
3. By deleting each of the other vertices in turn, the following lower bounds for the length of a minimum tour were found: $50,48,52,51,56$ and 64 . Write down the best lower bound.

Given that the length of a minimum tour is $T$ miles, use your answers to parts (b) and (c) to write down the smallest interval within which you know $T$ must lie.
(2 marks) \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Table 2}

	$\boldsymbol { A }$	$\boldsymbol { B }$	$\boldsymbol { C }$	$\boldsymbol { D }$	$\boldsymbol { F }$	G
A	-	2	6	4	16	27
$\boldsymbol { B }$	2	-	8	3	15	26
$\boldsymbol { C }$	6	8	-	10	22	32
$\boldsymbol { D }$	4	3	10	-	12	23
$\boldsymbol { F }$	16	15	22	12	-	20
G	27	26	32	23	20	-

\end{table}

\includegraphics[max width=\textwidth, alt={}]{5a414265-6273-41c5-ad5f-f6316bd774d0-17_2486_1714_221_153}

AQA D1 2012 January Q8

10 marks Standard +0.8

8 Four distinct positive integers are $( 3 x - 5 ) , ( 2 x + 3 ) , ( x + 1 )$ and $( 4 x - 13 )$.

Explain why $x \geqslant 4$.
The four integers are to be sorted into ascending order using a bubble sort. The original list is $\begin{array} { c c c c } ( 3 x - 5 ) & ( 2 x + 3 ) & ( x + 1 ) & ( 4 x - 13 ) \end{array}$ After the first pass, the list is $( 3 x - 5 ) \quad ( x + 1 ) \quad ( 4 x - 13 ) \quad ( 2 x + 3 )$ After the second pass, the list is $( x + 1 )$ $( 4 x - 13 )$ $( 3 x - 5 )$ $( 2 x + 3 )$ After the third pass, the list is $( 4 x - 13 ) \quad ( x + 1 )$ $( 3 x - 5 )$ ( $2 x + 3$ )
1. By considering the list after the first pass, write down three inequalities in terms of $x$.
2. By considering the list after the second pass, write down two further inequalities in terms of $x$.
3. By considering the list after the third pass, write down one further inequality in terms of $x$.
Hence, by considering the results above, find the value of $x$.
\includegraphics[max width=\textwidth, alt={}]{5a414265-6273-41c5-ad5f-f6316bd774d0-19_2486_1714_221_153}

AQA D1 2013 January Q1

4 marks Moderate -0.8

Draw a bipartite graph to represent the following adjacency matrix.
1 2 3 4 5
A 0 0 0 0 1
$\boldsymbol { B }$ 0 0 0 1 1
$\boldsymbol { C }$ 0 1 0 1 1
$\boldsymbol { D }$ 0 0 0 1 1
$\boldsymbol { E }$ 1 1 1 0 1
If $A , B , C , D$ and $E$ represent five people and 1, 2, 3, 4 and 5 represent five tasks to which they are to be assigned, explain why a complete matching is impossible.
(2 marks)

AQA D1 2013 January Q2

5 marks Easy -1.2

Use a Shell sort to arrange the following numbers into ascending order. $$\begin{array} { l l l l l l l l } 7 & 8 & 1 & 6 & 3 & 4 & 5 & 2 \end{array}$$
Write down the number of comparisons on the first pass.

AQA D1 2013 January Q3

7 marks Moderate -0.8

3 The following network shows the lengths, in miles, of roads connecting nine villages, $A , B , \ldots , I$. A delivery man lives in village $A$ and is to drive along all the roads at least once before returning to $A$. \includegraphics[max width=\textwidth, alt={}, center]{d666b2d9-cb14-4d29-a842-8c87f1b25dbd-06_1072_1027_548_502}

Find the length of an optimal Chinese postman route around the nine villages, starting and finishing at $A$.
For an optimal Chinese postman route corresponding to your answer in part (a), state:
1. the number of times village $E$ would be visited;
2. the number of times village $I$ would be visited.

AQA D1 2013 January Q4

11 marks Easy -1.2

4 The following network shows the lengths, in miles, of roads connecting nine villages, $A , B , \ldots , I$. A programme of resurfacing some roads is undertaken to ensure that each village can access all other villages along a resurfaced road, while keeping the amount of road to be resurfaced to a minimum. \includegraphics[max width=\textwidth, alt={}, center]{d666b2d9-cb14-4d29-a842-8c87f1b25dbd-08_1008_1043_589_497}

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. State the length of your minimum spanning tree.
3. Draw your minimum spanning tree.
Given that Prim's algorithm is used with different start vertices, state the final edge to be added to the minimum spanning tree if:
1. the start vertex is $E$;
2. the start vertex is $G$.
Given that Kruskal's algorithm is used to find the minimum spanning tree, state which edge would be:
1. the first to be included in the tree;
2. the last to be included in the tree.

AQA D1 2013 January Q5

10 marks Moderate -0.8

5 The feasible region of a linear programming problem is defined by $$\begin{aligned} x + y & \leqslant 60 \\ 2 x + y & \leqslant 80 \\ y & \geqslant 20 \\ x & \geqslant 15 \\ y & \geqslant x \end{aligned}$$

On the grid opposite, draw a suitable diagram to represent these inequalities and indicate the feasible region.
In each of the following cases, use your diagram to find the maximum value of $P$ on the feasible region. In each case, state the corresponding values of $x$ and $y$.
1. $P = x + 4 y$
2. $P = 4 x + y$

AQA D1 2013 January Q6

10 marks Easy -1.2

6 The network opposite shows some roads connecting towns. The number on each edge represents the length, in miles, of the road connecting a pair of towns.

1. Use Dijkstra's algorithm on the network to find the minimum distance from $A$ to $J$.
2. Write down the corresponding route.
The road $A J$ is a dual carriageway. Ken drives at 60 miles per hour on this road and drives at 50 miles per hour on all other roads. Find the minimum time to travel from $A$ to $J$.
\includegraphics[max width=\textwidth, alt={}]{d666b2d9-cb14-4d29-a842-8c87f1b25dbd-15_2487_1714_221_152}

AQA D1 2013 January Q7

8 marks Moderate -0.8

A simple connected graph $X$ has eight vertices.
1. State the minimum number of edges of the graph.
2. Find the maximum number of edges of the graph.
A simple connected graph $Y$ has $n$ vertices.
1. State the minimum number of edges of the graph.
2. Find the maximum number of edges of the graph.
A simple graph $Z$ has six vertices and each of the vertices has the same degree $d$.
1. State the possible values of $d$.
2. If $Z$ is connected, state the possible values of $d$.
3. If $Z$ is Eulerian, state the possible values of $d$.

	Task 1	Task 2	Task 3	Task 4	Task 5	Task 6
$\boldsymbol { A }$	0	0	1	0	1	1
B	0	1	0	1	0	0
$\boldsymbol { C }$	0	1	0	0	0	1
$\boldsymbol { D }$	0	0	0	1	0	0
$E$	1	0	1	0	1	0
$\boldsymbol { F }$	0	0	0	1	1	0

Show this information on a bipartite graph.
Initially, $A$ is matched to task 3, $B$ to task 4, $C$ to task 2 and $E$ to task 5. From this initial matching, use the maximum matching algorithm to obtain a complete matching. List your complete matching.

AQA D1 2008 June Q2

7 marks Easy -1.2

Use a quick sort to rearrange the following letters into alphabetical order. You must indicate the pivot that you use at each pass.
P
B
M
N
J
K
R
D
(5 marks)
1. Find the maximum number of swaps needed to rearrange a list of 8 numbers into ascending order when using a bubble sort.
  (1 mark)
2. A list of 8 numbers was rearranged into ascending order using a bubble sort. The maximum number of swaps was needed. What can be deduced about the original list of numbers?
  (1 mark)

AQA D1 2008 June Q3

10 marks Easy -1.8

1. State the number of edges in a minimum spanning tree of a network with 11 vertices.
2. State the number of edges in a minimum spanning tree of a network with $n$ vertices.
The following network has 11 vertices, $A , B , \ldots , K$. The number on each edge represents the distance, in miles, between a pair of vertices. \includegraphics[max width=\textwidth, alt={}, center]{4c5c963b-0183-4dc7-9054-b2c7a3eb8c1b-03_1468_1239_721_404}
1. Use Prim's algorithm, starting from $A$, to find a minimum spanning tree for the network.
2. Find the length of your minimum spanning tree.
3. Draw your minimum spanning tree.

\cline { 2 - 7 } \multicolumn{1}{c\|}{}	\(\mathbf { 1 }\)	\(\mathbf { 2 }\)	\(\mathbf { 3 }\)	\(\mathbf { 4 }\)	\(\mathbf { 5 }\)	\(\mathbf { 6 }\)
\(\boldsymbol { A }\)	1	1	0	0	1	1
\(\boldsymbol { B }\)	0	1	0	0	1	1
\(\boldsymbol { C }\)	1	0	0	0	1	1
\(\boldsymbol { D }\)	0	1	1	1	0	1
\(\boldsymbol { E }\)	0	0	0	0	1	1
\(\boldsymbol { F }\)	0	0	0	0	0	1

	1	2	3	4	5
A	0	0	0	0	1
\(\boldsymbol { B }\)	0	0	0	1	1
\(\boldsymbol { C }\)	0	1	0	1	1
\(\boldsymbol { D }\)	0	0	0	1	1
\(\boldsymbol { E }\)	1	1	1	0	1

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AQA D1 2005 January Q4

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