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AQA D1 2006 January Q8

11 marks Moderate -0.8

8 Salvadore is visiting six famous places in Barcelona: La Pedrera $( L )$, Nou Camp $( N )$, Olympic Village $( O )$, Park Guell $( P )$, Ramblas $( R )$ and Sagrada Familia $( S )$. Owing to the traffic system the time taken to travel between two places may vary according to the direction of travel. The table shows the times, in minutes, that it will take to travel between the six places.

\backslashbox{From}{To}	La Pedrera ( $L$ )	Nou Camp (N)	Olympic Village ( $O$ )	Park Guell (P)	Ramblas (R)	Sagrada Familia ( $S$ )
La Pedrera $( L )$	-	35	30	30	37	35
Nou Camp $( N )$	25	-	20	21	25	40
Olympic Village ( $O$ )	15	40	-	25	30	29
Park Guell ( $P$ )	30	35	25	-	35	20
Ramblas ( $R$ )	20	30	17	25	-	25
Sagrada Familia ( $S$ )	25	35	29	20	30	-

Find the total travelling time for:
1. the route $L N O L$;
2. the route $L O N L$.
Give an example of a Hamiltonian cycle in the context of the above situation.
Salvadore intends to travel from one place to another until he has visited all of the places before returning to his starting place.
1. Show that, using the nearest neighbour algorithm starting from Sagrada Familia $( S )$, the total travelling time for Salvadore is 145 minutes.
2. Explain why your answer to part (c)(i) is an upper bound for the minimum travelling time for Salvadore.
3. Salvadore starts from Sagrada Familia ( $S$ ) and then visits Ramblas ( $R$ ). Given that he visits Nou Camp $( N )$ before Park Guell $( P )$, find an improved upper bound for the total travelling time for Salvadore.

AQA D1 2006 January Q9

8 marks Standard +0.3

AQA D1 2007 January Q1

10 marks Standard +0.3

1 The following network shows the lengths, in miles, of roads connecting nine villages. \includegraphics[max width=\textwidth, alt={}, center]{e47eb41e-0a4b-4865-a8ff-6c9978495ee0-02_856_1251_568_374}

Use Prim's algorithm, starting from $A$, to find a minimum spanning tree for the network.
Find the length of your minimum spanning tree.
Draw your minimum spanning tree.
State the number of other spanning trees that are of the same length as your answer in part (a).

AQA D1 2007 January Q2

6 marks Moderate -0.8

2 Five people $A , B , C , D$ and $E$ are to be matched to five tasks $R , S , T , U$ and $V$.
The table shows the tasks that each person is able to undertake.

Person	Tasks
$A$	$R , V$
$B$	$R , T$
$C$	$T , V$
$D$	$U , V$
$E$	$S , U$

Show this information on a bipartite graph.
Initially, $A$ is matched to task $V , B$ to task $R , C$ to task $T$, and $E$ to task $U$. Demonstrate, by using an alternating path from this initial matching, how each person can be matched to a task.

AQA D1 2007 January Q3

8 marks Easy -1.3

3 Mark is driving around the one-way system in Leicester. The following table shows the times, in minutes, for Mark to drive between four places: $A , B , C$ and $D$. Mark decides to start from $A$, drive to the other three places and then return to $A$. Mark wants to keep his driving time to a minimum.

\backslashbox{From}{To}	$\boldsymbol { A }$	$\boldsymbol { B }$	$\boldsymbol { C }$	$\boldsymbol { D }$
A	-	8	6	11
B	14	-	13	25
C	14	9	-	17
$\boldsymbol { D }$	26	10	18	-

Find the length of the tour $A B C D A$.
Find the length of the tour $A D C B A$.
Find the length of the tour using the nearest neighbour algorithm starting from $A$.
Write down which of your answers to parts (a), (b) and (c) gives the best upper bound for Mark's driving time.

AQA D1 2007 January Q4

8 marks Moderate -0.8

A student is using a bubble sort to rearrange seven numbers into ascending order.
Her correct solution is as follows:
Initial list 18 17 13 26 10 14 24
After 1st pass 17 13 18 10 14 24 26
After 2nd pass 13 17 10 14 18 24 26
After 3rd pass 13 10 14 17 18 24 26
After 4th pass 10 13 14 17 18 24 26
After 5th pass 10 13 14 17 18 24 26
Write down the number of comparisons and swaps on each of the five passes.
Find the maximum number of comparisons and the maximum number of swaps that might be needed in a bubble sort to rearrange seven numbers into ascending order.

AQA D1 2007 January Q5

8 marks Easy -1.2

5 A student is using the following algorithm with different values of $A$ and $B$.

Line 10	Input $A , B$
Line 20	Let $C = 0$ and let $D = 0$
Line 30	Let $C = C + A$
Line 40	Let $D = D + B$
Line 50	If $C = D$ then go to Line 110
Line 60	If $C > D$ then go to Line 90
Line 70	Let $C = C + A$
Line 80	Go to Line 50
Line 90	Let $D = D + B$
Line 100	Go to Line 50
Line 110	Print $C$
Line 120	End

1. Trace the algorithm in the case where $A = 2$ and $B = 3$.
2. Trace the algorithm in the case where $A = 6$ and $B = 8$.
State the purpose of the algorithm.
Write down the final value of $C$ in the case where $A = 200$ and $B = 300$.

AQA D1 2007 January Q6

13 marks Moderate -0.8

6 [Figure 1, printed on the insert, is provided for use in this question.]
Dino is to have a rectangular swimming pool at his villa.
He wants its width to be at least 2 metres and its length to be at least 5 metres.
He wants its length to be at least twice its width.
He wants its length to be no more than three times its width.
Each metre of the width of the pool costs $\pounds 1000$ and each metre of the length of the pool costs $\pounds 500$. He has $\pounds 9000$ available. Let the width of the pool be $x$ metres and the length of the pool be $y$ metres.

Show that one of the constraints leads to the inequality $$2 x + y \leqslant 18$$
Find four further inequalities.
On Figure 1, draw a suitable diagram to show the feasible region.
Use your diagram to find the maximum width of the pool. State the corresponding length of the pool.

AQA D1 2007 January Q7

14 marks Standard +0.3

7 [Figure 2, printed on the insert, is provided for use in this question.]
The network shows the times, in seconds, taken by Craig to walk along walkways connecting ten hotels in Las Vegas. \includegraphics[max width=\textwidth, alt={}, center]{e47eb41e-0a4b-4865-a8ff-6c9978495ee0-07_1435_1267_525_351} The total of all the times in the diagram is 2280 seconds.

1. Craig is staying at the Circus ( $C$ ) and has to visit the Oriental ( $O$ ). Use Dijkstra's algorithm on Figure 2 to find the minimum time to walk from $C$ to $O$.
2. Write down the corresponding route.
1. Find, by inspection, the shortest time to walk from $A$ to $M$.
2. Craig intends to walk along all the walkways. Find the minimum time for Craig to walk along every walkway and return to his starting point.

AQA D1 2007 January Q8

8 marks Standard +0.3

The diagram shows a graph $\mathbf { G }$ with 9 vertices and 9 edges. \includegraphics[max width=\textwidth, alt={}, center]{e47eb41e-0a4b-4865-a8ff-6c9978495ee0-08_188_204_411_708} \includegraphics[max width=\textwidth, alt={}, center]{e47eb41e-0a4b-4865-a8ff-6c9978495ee0-08_184_204_415_1105} \includegraphics[max width=\textwidth, alt={}, center]{e47eb41e-0a4b-4865-a8ff-6c9978495ee0-08_183_204_612_909}
1. State the minimum number of edges that need to be added to $\mathbf { G }$ to make a connected graph. Draw an example of such a graph.
2. State the minimum number of edges that need to be added to $\mathbf { G }$ to make the graph Hamiltonian. Draw an example of such a graph.
3. State the minimum number of edges that need to be added to $\mathbf { G }$ to make the graph Eulerian. Draw an example of such a graph.
A complete graph has $n$ vertices and is Eulerian.
1. State the condition that $n$ must satisfy.
2. In addition, the number of edges in a Hamiltonian cycle for the graph is the same as the number of edges in an Eulerian trail. State the value of $n$.

AQA D1 2008 January Q1

5 marks Easy -1.2

1 Five people, $A , B , C , D$ and $E$, are to be matched to five tasks, $J , K , L , M$ and $N$. The table shows the tasks that each person is able to undertake.

Person	Task
$A$	$J , N$
$B$	$J , L$
$C$	$L , N$
$D$	$M , N$
$E$	$K , M$

Show this information on a bipartite graph.
Initially, $A$ is matched to task $N , B$ to task $J , C$ to task $L$, and $E$ to task $M$. Complete the alternating path $D - M \ldots$, from this initial matching, to demonstrate how each person can be matched to a task.
(3 marks)

AQA D1 2008 January Q2

9 marks Easy -1.2

2 [Figure 1, printed on the insert, is provided for use in this question.]
The feasible region of a linear programming problem is represented by $$\begin{aligned} x + y & \leqslant 30 \\ 2 x + y & \leqslant 40 \\ y & \geqslant 5 \\ x & \geqslant 4 \\ y & \geqslant \frac { 1 } { 2 } x \end{aligned}$$

On Figure 1, draw a suitable diagram to represent these inequalities and indicate the feasible region.
Use your diagram to find the maximum value of $F$, on the feasible region, in the case where:
1. $F = 3 x + y$;
2. $F = x + 3 y$.

AQA D1 2008 January Q3

10 marks Moderate -0.5

3 The diagram shows 10 bus stops, $A , B , C , \ldots , J$, in Geneva. The number on each edge represents the distance, in kilometres, between adjacent bus stops. \includegraphics[max width=\textwidth, alt={}, center]{92175666-ef7a-4dca-9cdb-ebde1b40b2c9-03_595_1362_422_331} The city council is to connect these bus stops to a computer system which will display waiting times for buses at each of the 10 stops. Cabling is to be laid between some of the bus stops.

Use Kruskal's algorithm, showing the order in which you select the edges, to find a minimum spanning tree for the 10 bus stops.
State the minimum length of cabling needed.
Draw your minimum spanning tree.
If Prim's algorithm, starting from $A$, had been used to find the minimum spanning tree, state which edge would have been the final edge to complete the minimum spanning tree.

AQA D1 2008 January Q4

12 marks Standard +0.3

4 [Figure 2, printed on the insert, is provided for use in this question.]
The network shows 11 towns. The times, in minutes, to travel between pairs of towns are indicated on the edges. \includegraphics[max width=\textwidth, alt={}, center]{92175666-ef7a-4dca-9cdb-ebde1b40b2c9-04_1762_1056_516_484} The total of all of the times is 308 minutes.

1. Use Dijkstra's algorithm on Figure 2 to find the minimum time to travel from $A$ to $K$.
2. State the corresponding route.
Find the length of an optimum Chinese postman route around the network, starting and finishing at $A$. (The minimum time to travel from $D$ to $H$ is 40 minutes.)

AQA D1 2008 January Q5

10 marks Easy -1.8

5 [Figure 3, printed on the insert, is provided for use in this question.]

James is solving a travelling salesperson problem.
1. He finds the following upper bounds: $43,40,43,41,55,43,43$. Write down the best upper bound.
2. James finds the following lower bounds: 33, 40, 33, 38, 33, 38, 38 . Write down the best lower bound.
Karen is solving a different travelling salesperson problem and finds an upper bound of 55 and a lower bound of 45 . Write down an interpretation of these results.
The diagram below shows roads connecting 4 towns, $A , B , C$ and $D$. The numbers on the edges represent the lengths of the roads, in kilometres, between adjacent towns. \includegraphics[max width=\textwidth, alt={}, center]{92175666-ef7a-4dca-9cdb-ebde1b40b2c9-05_451_1034_1160_504} Xiong lives at town $A$ and is to visit each of the other three towns before returning to town $A$. She wishes to find a route that will minimise her travelling distance.
1. Complete Figure 3, on the insert, to show the shortest distances, in kilometres, between all pairs of towns.
2. Use the nearest neighbour algorithm on Figure 3 to find an upper bound for the minimum length of a tour of this network that starts and finishes at $A$.
3. Hence find the actual route that Xiong would take in order to achieve a tour of the same length as that found in part (c)(ii).

AQA D1 2008 January Q6

10 marks Easy -1.8

6 A student is solving cubic equations that have three different positive integer solutions.
The algorithm that the student is using is as follows:
Line 10 Input $A , B , C , D$ Line $20 \quad$ Let $K = 1$ Line $30 \quad$ Let $N = 0$ Line $40 \quad$ Let $X = K$ Line 50 Let $Y = A X ^ { 3 } + B X ^ { 2 } + C X + D$ Line 60 If $Y \neq 0$ then go to Line 100
Line $70 \quad$ Print $X$, "is a solution"
Line $80 \quad$ Let $N = N + 1$ Line 90 If $N = 3$ then go to Line 120
Line $100 \quad$ Let $K = K + 1$ Line 110 Go to Line 40
Line 120 End

Trace the algorithm in the case where the input values are:
1. $A = 1 , B = - 6 , C = 11$ and $D = - 6$;
2. $A = 1 , B = - 10 , C = 29$ and $D = - 20$.
Explain where and why this algorithm will fail if $A = 0$.

AQA D1 2008 January Q7

11 marks Easy -1.2

7 The numbers 17, 3, 16 and 4 are to be sorted into ascending order.
The following four methods are to be compared: bubble sort, shuttle sort, Shell sort and quick sort (with the first number used as the pivot). A student uses each of the four methods and produces the correct solutions below. Each solution shows the order of the numbers after each pass.

\multirow[t]{4}{*}{Solution 1}	17	3	16	4
	3	17	16	4
	3	16	17	4
	3	4	16	17
\multirow[t]{3}{*}{Solution 2}	17	3	16	4
	16	3	17	4
	3	4	16	17
\multirow[t]{4}{*}{Solution 3}	17	3	16	4
	3	16	4	17
	3	16	4	17
	3	4	16	17
\multirow[t]{4}{*}{Solution 4}	17	3	16	4
	3	16	4	17
	3	4	16	17
	3	4	16	17

Write down which of the four solutions is the bubble sort, the shuttle sort, the Shell sort and the quick sort.
For each of the four solutions, write down the number of comparisons and swaps (exchanges) on the first pass.

AQA D1 2008 January Q8

8 marks Standard +0.3

8 Each day, a factory makes three types of hinge: basic, standard and luxury. The hinges produced need three different components: type $A$, type $B$ and type $C$. Basic hinges need 2 components of type $A , 3$ components of type $B$ and 1 component of type $C$. Standard hinges need 4 components of type $A , 2$ components of type $B$ and 3 components of type $C$. Luxury hinges need 3 components of type $A$, 4 components of type $B$ and 5 components of type $C$. Each day, there are 360 components of type $A$ available, 270 of type $B$ and 450 of type $C$. Each day, the factory must use at least 720 components in total.
Each day, the factory must use at least $40 \%$ of the total components as type $A$.
Each day, the factory makes $x$ basic hinges, $y$ standard hinges and $z$ luxury hinges.
In addition to $x \geqslant 0 , y \geqslant 0 , z \geqslant 0$, find five inequalities, each involving $x , y$ and $z$, which must be satisfied. Simplify each inequality where possible.

AQA D1 2009 January Q1

10 marks Moderate -0.8

1 The following network shows the lengths, in miles, of roads connecting 11 villages, $A , B , \ldots , K$. \includegraphics[max width=\textwidth, alt={}, center]{6360ed01-76da-4265-8bc8-53ffe391704e-2_915_1303_591_365}

Starting from $G$ and showing your working at each stage, use Prim's algorithm to find a minimum spanning tree for the network.
State the length of your minimum spanning tree.
Draw your minimum spanning tree.

AQA D1 2009 January Q2

7 marks Moderate -0.8

2 Six people, $A , B , C , D , E$ and $F$, are to be allocated to six tasks, 1, 2, 3, 4, 5 and 6. The following bipartite graph shows the tasks that each of the people is able to undertake. \includegraphics[max width=\textwidth, alt={}, center]{6360ed01-76da-4265-8bc8-53ffe391704e-3_401_517_429_751} \includegraphics[max width=\textwidth, alt={}, center]{6360ed01-76da-4265-8bc8-53ffe391704e-3_408_520_943_751}

Represent this information in an adjacency matrix.
Initially, $B$ is assigned to task $1 , C$ to task $2 , D$ to task 4, and $E$ to task 5 . Demonstrate, by using an algorithm from this initial matching, how each person can be allocated to a task.

AQA D1 2009 January Q3

14 marks Standard +0.3

3 [Figure 1, printed on the insert, is provided for use in this question.]
The diagram shows roads connecting some places of interest in Berlin. The numbers represent the times taken, in minutes, to walk along the roads. \includegraphics[max width=\textwidth, alt={}, center]{6360ed01-76da-4265-8bc8-53ffe391704e-4_1427_1404_502_319} The total of all walking times is 167 minutes.

Mia is staying at $D$ and is to visit $H$.
1. Use Dijkstra's algorithm on Figure 1 to find the minimum time to walk from $D$ to $H$.
2. Write down the corresponding route.
Each day, Leon has to deliver leaflets along all of the roads. He must start and finish at $A$.
1. Use your answer to part (a) to write down the shortest walking time from $D$ to $A$.
2. Find the walking time of an optimum Chinese Postman route for Leon. (6 marks)

AQA D1 2009 January Q4

18 marks Moderate -0.8

4 [Figure 2, printed on the insert, is provided for use in this question.]
Each year, farmer Giles buys some goats, pigs and sheep.
He must buy at least 110 animals.
He must buy at least as many pigs as goats.
The total of the number of pigs and the number of sheep that he buys must not be greater than 150 .
Each goat costs $\pounds 16$, each pig costs $\pounds 8$ and each sheep costs $\pounds 24$.
He has $\pounds 3120$ to spend on the animals.
At the end of the year, Giles sells all of the animals. He makes a profit of $\pounds 70$ on each goat, $\pounds 30$ on each pig and $\pounds 50$ on each sheep. Giles wishes to maximize his total profit, $\pounds P$. Each year, Giles buys $x$ goats, $y$ pigs and $z$ sheep.

Formulate Giles's situation as a linear programming problem.
One year, Giles buys 30 sheep.
1. Show that the constraints for Giles's situation for this year can be modelled by $$y \geqslant x , \quad 2 x + y \leqslant 300 , \quad x + y \geqslant 80 , \quad y \leqslant 120$$ (2 marks)
2. On Figure 2, draw a suitable diagram to enable the problem to be solved graphically, indicating the feasible region and the direction of the objective line.
  (8 marks)
3. Find Giles's maximum profit for this year and the number of each animal that he must buy to obtain this maximum profit.
  (3 marks)

AQA D1 2009 January Q5

6 marks Moderate -0.8

5 A student is using the algorithm below to find an approximate value of $\sqrt { 2 }$.
Line 10 Let $A = 1 , B = 3 , C = 0$ Line $20 \quad$ Let $D = 1 , E = 2 , F = 0$ Line 30 Let $G = B / E$ Line $40 \quad$ Let $H = G ^ { 2 }$ Line 50 If $( H - 2 ) ^ { 2 } < 0.0001$ then go to Line 130
Line 60 Let $C = 2 B + A$ Line 70 Let $A = B$ Line 80 Let $B = C$ Line 90 Let $F = 2 E + D$ Line 100 Let $D = E$ Line 110 Let $E = F$ Line 120 Go to Line 30
Line 130 Print ' $\sqrt { 2 }$ is approximately', $B / E$ Line 140 Stop
Trace the algorithm.

AQA D1 2009 January Q6

7 marks Challenging +1.2

6 A connected graph $G$ has five vertices and has eight edges with lengths $8,10,10,11,13,17$, 17 and 18.

Find the minimum length of a minimum spanning tree for $G$.
Find the maximum length of a minimum spanning tree for $G$.
Draw a sketch to show a possible graph $G$ when the length of the minimum spanning tree is 53 .

AQA D1 2009 January Q7

13 marks Challenging +1.2

7 Liam is taking part in a treasure hunt. There are five clues to be solved and they are at the points $A , B , C , D$ and $E$. The table shows the distances between pairs of points. All of the distances are functions of $x$, where $\boldsymbol { x }$ is an integer. Liam must travel to all five points, starting and finishing at $A$.

	$\boldsymbol { A }$	$\boldsymbol { B }$	$\boldsymbol { C }$	$\boldsymbol { D }$	$\boldsymbol { E }$
A	-	$x + 6$	$2 x - 4$	$3 x - 7$	$4 x - 14$
$\boldsymbol { B }$	$x + 6$	-	$3 x - 7$	$3 x - 9$	$x + 9$
$\boldsymbol { C }$	$2 x - 4$	$3 x - 7$	-	$2 x - 1$	$x + 8$
$\boldsymbol { D }$	$3 x - 7$	$3 x - 9$	$2 x - 1$	-	$2 x - 2$
E	$4 x - 14$	$x + 9$	$x + 8$	$2 x - 2$	-

The nearest point to $A$ is $C$.
1. By considering $A C$ and $A B$, show that $x < 10$.
2. Find two other inequalities in $x$.
The nearest neighbour algorithm, starting from $A$, gives a unique minimum tour $A C D E B A$.
1. By considering the fact that Liam's tour visits $D$ immediately after $C$, find two further inequalities in $x$.
2. Find the value of the integer $x$.
3. Hence find the total distance travelled by Liam if he uses this tour.

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Person	Tasks
\(A\)	\(R , V\)
\(B\)	\(R , T\)
\(C\)	\(T , V\)
\(D\)	\(U , V\)
\(E\)	\(S , U\)

Person	Task
\(A\)	\(J , N\)
\(B\)	\(J , L\)
\(C\)	\(L , N\)
\(D\)	\(M , N\)
\(E\)	\(K , M\)

	\(\boldsymbol { A }\)	\(\boldsymbol { B }\)	\(\boldsymbol { C }\)	\(\boldsymbol { D }\)	\(\boldsymbol { E }\)
A	-	\(x + 6\)	\(2 x - 4\)	\(3 x - 7\)	\(4 x - 14\)
\(\boldsymbol { B }\)	\(x + 6\)	-	\(3 x - 7\)	\(3 x - 9\)	\(x + 9\)
\(\boldsymbol { C }\)	\(2 x - 4\)	\(3 x - 7\)	-	\(2 x - 1\)	\(x + 8\)
\(\boldsymbol { D }\)	\(3 x - 7\)	\(3 x - 9\)	\(2 x - 1\)	-	\(2 x - 2\)
E	\(4 x - 14\)	\(x + 9\)	\(x + 8\)	\(2 x - 2\)	-

\backslashbox{From}{To}	La Pedrera ( \(L\) )	Nou Camp (N)	Olympic Village ( \(O\) )	Park Guell (P)	Ramblas (R)	Sagrada Familia ( \(S\) )
La Pedrera \(( L )\)	-	35	30	30	37	35
Nou Camp \(( N )\)	25	-	20	21	25	40
Olympic Village ( \(O\) )	15	40	-	25	30	29
Park Guell ( \(P\) )	30	35	25	-	35	20
Ramblas ( \(R\) )	20	30	17	25	-	25
Sagrada Familia ( \(S\) )	25	35	29	20	30	-

Initial list	18	17	13	26	10	14	24
After 1st pass	17	13	18	10	14	24	26
After 2nd pass	13	17	10	14	18	24	26
After 3rd pass	13	10	14	17	18	24	26
After 4th pass	10	13	14	17	18	24	26
After 5th pass	10	13	14	17	18	24	26

Line 10	Input \(A , B\)
Line 20	Let \(C = 0\) and let \(D = 0\)
Line 30	Let \(C = C + A\)
Line 40	Let \(D = D + B\)
Line 50	If \(C = D\) then go to Line 110
Line 60	If \(C > D\) then go to Line 90
Line 70	Let \(C = C + A\)
Line 80	Go to Line 50
Line 90	Let \(D = D + B\)
Line 100	Go to Line 50
Line 110	Print \(C\)
Line 120	End

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AQA D1 2007 January Q6

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AQA D1 2008 January Q1

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AQA D1 2009 January Q1

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AQA D1 2009 January Q7