Questions — AQA D1 (170 questions)

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AQA D1 2012 June Q4
8 marks Easy -1.2
4 The edges on the network below represent some major roads in a city. The number on each edge is the minimum time taken, in minutes, to drive along that road.
    1. Use Dijkstra's algorithm on the network to find the shortest possible driving time from \(A\) to \(J\).
    2. Write down the corresponding route.
  2. A new ring road is to be constructed connecting \(A\) to \(J\) directly. Find the maximum length of this new road from \(A\) to \(J\) if the time taken to drive along it, travelling at an average speed of \(90 \mathrm {~km} / \mathrm { h }\), is to be no more than the time found in part (a)(i). \section*{(a)(i)} \includegraphics[max width=\textwidth, alt={}, center]{1258a6d3-558a-46dc-a916-d71f71b175ff-08_912_1276_1053_429}
AQA D1 2012 June Q5
8 marks Standard +0.3
5 The network below shows some streets in a town. The number on each edge shows the length of that street, in metres. Leaflets are to be distributed by a restaurant owner, Tony, from his restaurant located at vertex \(B\). Tony must start from his restaurant, walk along all the streets at least once, before returning to his restaurant. \includegraphics[max width=\textwidth, alt={}, center]{1258a6d3-558a-46dc-a916-d71f71b175ff-10_643_1353_625_340} The total length of the streets is 2430 metres.
  1. Find the length of an optimal Chinese postman route for Tony.
  2. Colin also wishes to distribute some leaflets. He starts from his house at \(H\), walks along all the streets at least once, before finishing at the restaurant at \(B\). Colin wishes to walk the minimum distance. Find the length of an optimal route for Colin.
  3. David also walks along all the streets at least once. He can start at any vertex and finish at any vertex. David also wishes to walk the minimum distance.
    1. Find the length of an optimal route for David.
    2. State the vertices from which David could start in order to achieve this optimal route.
AQA D1 2012 June Q6
7 marks Moderate -0.8
6 The complete graph \(K _ { n } ( n > 1 )\) has every one of its \(n\) vertices connected to each of the other vertices by a single edge.
  1. Draw the complete graph \(K _ { 4 }\).
    1. Find the total number of edges for the graph \(K _ { 8 }\).
    2. Give a reason why \(K _ { 8 }\) is not Eulerian.
  3. For the graph \(K _ { n }\), state in terms of \(n\) :
    1. the total number of edges;
    2. the number of edges in a minimum spanning tree;
    3. the condition for \(K _ { n }\) to be Eulerian;
    4. the condition for the number of edges of a Hamiltonian cycle to be equal to the number of edges of an Eulerian cycle.
AQA D1 2012 June Q7
11 marks Moderate -0.8
7 Rupta, a sales representative, has to visit six shops, \(A , B , C , D , E\) and \(F\). Rupta starts at shop \(A\) and travels to each of the other shops once, before returning to shop \(A\). Rupta wishes to keep her travelling time to a minimum. The table shows the travelling times, in minutes, between the shops.
AQA D1 2012 June Q8
8 marks Easy -1.2
8 The following algorithm finds an estimate of the value of the number represented by the symbol e:
Line 10Let \(A = 1 , B = 1 , C = 1\)
Line 20Let \(D = A\)
Line 30Let \(C = C \times B\)
Line 40Let \(D = D + ( 1 / C )\)
Line 50If \(B = 4\) then go to Line 80
Line 60Let \(B = B + 1\)
Line 70Go to Line 30
Line 80Print 'An estimate of e is', \(D\)
Line 90End
  1. Trace the algorithm.
  2. A student miscopied Line 70 . His line was
    Line 70 Go to Line 10
    Explain what would happen if his algorithm were traced.
AQA D1 2012 June Q9
14 marks Moderate -0.3
9 Ollyin is buying new pillows for his hotel. He buys three types of pillow: soft, medium and firm. He must buy at least 100 soft pillows and at least 200 medium pillows.
He must buy at least 400 pillows in total.
Soft pillows cost \(\pounds 4\) each. Medium pillows cost \(\pounds 3\) each. Firm pillows cost \(\pounds 4\) each.
He wishes to spend no more than \(\pounds 1800\) on new pillows.
At least \(40 \%\) of the new pillows must be medium pillows.
Ollyin buys \(x\) soft pillows, \(y\) medium pillows and \(z\) firm pillows.
  1. In addition to \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\), find five inequalities in \(x , y\) and \(z\) that model the above constraints.
  2. Ollyin decides to buy twice as many soft pillows as firm pillows.
    1. Show that three of your answers in part (a) become $$\begin{aligned} 3 x + 2 y & \geqslant 800 \\ 2 x + y & \leqslant 600 \\ y & \geqslant x \end{aligned}$$
    2. On the grid opposite, draw a suitable diagram to represent Ollyin's situation, indicating the feasible region.
    3. Use your diagram to find the maximum total number of pillows that Ollyin can buy.
    4. Find the number of each type of pillow that Ollyin can buy that corresponds to your answer to part (b)(iii).
      \includegraphics[max width=\textwidth, alt={}]{1258a6d3-558a-46dc-a916-d71f71b175ff-20_2256_1707_221_153}
AQA D1 2013 June Q1
7 marks Moderate -0.8
1 Six people, Andy, Bob, Colin, Dev, Eric and Faisal, are to be allocated to six tasks, \(1,2,3,4,5\) and 6 . The following table shows the tasks that each person is able to undertake.
PersonTask
Andy1,3
Bob1,4
Colin2,3
Dev\(4,5,6\)
Eric\(2,5,6\)
Faisal1,3
  1. Represent this information on a bipartite graph.
  2. Initially, Bob is allocated to task 1, Colin to task 3, Dev to task 5 and Eric to task 2. Demonstrate, by using an alternating path algorithm from this initial matching, how each person can be allocated to a different task.
AQA D1 2013 June Q2
5 marks Easy -1.8
2
  1. Use the quicksort algorithm to rearrange the following numbers into ascending order, showing the new arrangement after each pass. You must indicate the pivot(s) being used on each pass. $$2 , \quad 12 , \quad 17 , \quad 18 , \quad 5 , \quad 13$$
  2. For the first pass, write down the number of comparisons.
AQA D1 2013 June Q3
9 marks Moderate -0.5
3 The following network shows the lengths, in miles, of roads connecting ten villages, \(A , B , C , \ldots , J\). \includegraphics[max width=\textwidth, alt={}, center]{77c4efd4-a905-48f3-a6f7-36b0e47dbc6d-06_899_1458_397_285}
    1. Use Kruskal's algorithm, showing the order in which you select the edges, to find a minimum spanning tree for the network.
    2. Find the length of your minimum spanning tree.
    3. Draw your minimum spanning tree.
  2. Prim's algorithm from different starting points produces the same minimum spanning tree. State the final edge that would be added to complete the minimum spanning tree if the starting point were:
    1. \(A\);
    2. \(F\).
AQA D1 2013 June Q4
12 marks Moderate -0.8
4 Sarah is a mobile hairdresser based at \(A\). Her day's appointments are at five places: \(B , C , D , E\) and \(F\). She can arrange the appointments in any order. She intends to travel from one place to the next until she has visited all of the places, starting and finishing at \(A\). The following table shows the times, in minutes, that it takes to travel between the six places.
\cline { 2 - 7 } \multicolumn{1}{c|}{}\(\boldsymbol { A }\)\(\boldsymbol { B }\)\(\boldsymbol { C }\)\(\boldsymbol { D }\)\(\boldsymbol { E }\)\(\boldsymbol { F }\)
\(\boldsymbol { A }\)-1511142712
\(\boldsymbol { B }\)15-13192415
\(\boldsymbol { C }\)1113-101912
\(\boldsymbol { D }\)141910-2615
\(\boldsymbol { E }\)27241926-27
\(\boldsymbol { F }\)1215121527-
  1. Sarah decides to visit the places in the order \(A B C D E F A\). Find the travelling time of this tour.
  2. Explain why this answer can be considered as being an upper bound for the minimum travelling time of Sarah's tour.
  3. Use the nearest neighbour algorithm, starting from \(A\), to find another upper bound for the minimum travelling time of Sarah's tour.
  4. By deleting \(A\), find a lower bound for the minimum travelling time of Sarah's tour.
  5. Sarah thinks that she can reduce her travelling time to 75 minutes. Explain why she is wrong.
AQA D1 2013 June Q5
17 marks Standard +0.3
5 The network on the page opposite shows the times, in minutes, taken by police cars to drive along roads connecting 12 places, \(A , B , \ldots , L\). On a particular day, there are three police cars in the area at \(A , E\) and \(J\). There is an emergency at \(G\) and all three police cars drive to \(G\).
    1. Use Dijkstra's algorithm on the network, starting from \(\boldsymbol { G }\), to find the minimum driving time for each of the police cars to arrive at \(G\).
    2. For each of the police cars, write down the route corresponding to the minimum driving time in your answer to part (a)(i).
  2. Each day, a police car has to drive along each road at least once, starting and finishing at \(A\). For an optimal Chinese postman route:
    1. find the driving time for the police car;
    2. state the number of times that the vertex \(B\) would appear.
      \includegraphics[max width=\textwidth, alt={}]{77c4efd4-a905-48f3-a6f7-36b0e47dbc6d-13_2486_1709_221_153}
AQA D1 2013 June Q6
9 marks Easy -1.2
6 A student is tracing the following algorithm. The function INT gives the integer part of any number, eg \(\operatorname { INT } ( 2.3 ) = 2\) and \(\operatorname { INT } ( 6.7 ) = 6\). Line 10 Input \(A , B\) Line \(20 \quad\) Let \(C = \operatorname { INT } ( A \div B )\) Line 30 Let \(D = B \times C\) Line \(40 \quad\) Let \(E = A - D\) Line 50 If \(E = 0\) then go to Line 90
Line 60 Let \(A = B\) Line \(70 \quad\) Let \(B = E\) Line 80 Go to Line 20
Line 90 Print \(B\) Line 100 Stop
  1. Trace the algorithm when the input values are:
    1. \(A = 36\) and \(B = 16\);
    2. \(A = 11\) and \(B = 7\).
  2. State the purpose of the algorithm.
AQA D1 2013 June Q7
16 marks Moderate -0.3
7 Paul is a florist. Every day, he makes three types of floral bouquet: gold, silver and bronze. Each gold bouquet has 6 roses, 6 carnations and 6 dahlias.
Each silver bouquet has 4 roses, 6 carnations and 4 dahlias.
Each bronze bouquet has 3 roses, 4 carnations and 4 dahlias.
Each day, Paul must use at least 420 roses and at least 480 carnations, but he can use at most 720 dahlias. Each day, Paul makes \(x\) gold bouquets, \(y\) silver bouquets and \(z\) bronze bouquets.
  1. In addition to \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\), find three inequalities in \(x , y\) and \(z\) that model the above constraints.
  2. On a particular day, Paul makes the same number of silver bouquets as bronze bouquets.
    1. Show that \(x\) and \(y\) must satisfy the following inequalities. $$\begin{aligned} & 6 x + 7 y \geqslant 420 \\ & 3 x + 5 y \geqslant 240 \\ & 3 x + 4 y \leqslant 360 \end{aligned}$$
    2. Paul makes a profit of \(\pounds 4\) on each gold bouquet sold, a profit of \(\pounds 2.50\) on each silver bouquet sold and a profit of \(\pounds 2.50\) on each bronze bouquet sold. Each day, Paul sells all the bouquets he makes. Paul wishes to maximise his daily profit, \(\pounds P\). Draw a suitable diagram, on the grid opposite, to enable this problem to be solved graphically, indicating the feasible region and the direction of the objective line.
      (6 marks)
    3. Use your diagram to find Paul's maximum daily profit and the number of each type of bouquet he must make to achieve this maximum.
  3. On another day, Paul again makes the same number of silver bouquets as bronze bouquets, but he makes a profit of \(\pounds 2\) on each gold bouquet sold, a profit of \(\pounds 6\) on each silver bouquet sold and a profit of \(\pounds 6\) on each bronze bouquet sold. Find Paul's maximum daily profit, and the number of each type of bouquet he must make to achieve this maximum.
    (3 marks) Turn over -
AQA D1 Q3
Easy -1.8
3
    1. State the number of edges in a minimum spanning tree of a network with 10 vertices.
    2. State the number of edges in a minimum spanning tree of a network with \(n\) vertices.
  2. The following network has 10 vertices: \(A , B , \ldots , J\). The numbers on each edge represent the distances, in miles, between pairs of vertices. \includegraphics[max width=\textwidth, alt={}, center]{194d16e0-8e05-45c0-8948-99808440ed2a-004_1294_1118_785_445}
    1. Use Kruskal's algorithm to find the minimum spanning tree for the network.
    2. State the length of your spanning tree.
    3. Draw your spanning tree.
AQA D1 Q4
Moderate -0.3
4 The diagram shows the feasible region of a linear programming problem. \includegraphics[max width=\textwidth, alt={}, center]{194d16e0-8e05-45c0-8948-99808440ed2a-005_1349_1395_408_294}
  1. On the feasible region, find:
    1. the maximum value of \(2 x + 3 y\);
    2. the maximum value of \(3 x + 2 y\);
    3. the minimum value of \(- 2 x + y\).
  2. Find the 5 inequalities that define the feasible region.
AQA D1 Q5
Moderate -0.8
5 [Figure 1, printed on the insert, is provided for use in this question.]
The network shows the times, in minutes, to travel between 10 towns. \includegraphics[max width=\textwidth, alt={}, center]{194d16e0-8e05-45c0-8948-99808440ed2a-006_412_1561_568_233}
  1. Use Dijkstra's algorithm on Figure 1 to find the minimum time to travel from \(A\) to \(J\).
    (6 marks)
  2. State the corresponding route.
    (1 mark)
AQA D1 Q7
Moderate -0.8
7 Stella is visiting Tijuana on a day trip. The diagram shows the lengths, in metres, of the roads near the bus station. \includegraphics[max width=\textwidth, alt={}, center]{194d16e0-8e05-45c0-8948-99808440ed2a-007_1539_1162_495_424} Stella leaves the bus station at \(A\). She decides to walk along all of the roads at least once before returning to \(A\).
  1. Explain why it is not possible to start from \(A\), travel along each road only once and return to \(A\).
  2. Find the length of an optimal 'Chinese postman' route around the network, starting and finishing at \(A\).
  3. At each of the 9 places \(B , C , \ldots , J\), there is a statue. Find the number of times that Stella will pass a statue if she follows her optimal route.
AQA D1 2006 January Q1
7 marks Moderate -0.8
1
  1. Draw a bipartite graph representing the following adjacency matrix.
    (2 marks)
    \(\boldsymbol { U }\)\(V\)\(\boldsymbol { W }\)\(\boldsymbol { X }\)\(\boldsymbol { Y }\)\(\boldsymbol { Z }\)
    \(\boldsymbol { A }\)101010
    \(\boldsymbol { B }\)010100
    \(\boldsymbol { C }\)010001
    \(\boldsymbol { D }\)000100
    \(\boldsymbol { E }\)001011
    \(\boldsymbol { F }\)000110
  2. Given that initially \(A\) is matched to \(W , B\) is matched to \(X , C\) is matched to \(V\), and \(E\) is matched to \(Y\), use the alternating path algorithm, from this initial matching, to find a complete matching. List your complete matching.
AQA D1 2006 January Q2
5 marks Easy -1.2
2 Use the quicksort algorithm to rearrange the following numbers into ascending order. Indicate clearly the pivots that you use. $$\begin{array} { l l l l l l l l } 18 & 23 & 12 & 7 & 26 & 19 & 16 & 24 \end{array}$$
AQA D1 2006 January Q3
15 marks Easy -2.0
3
    1. State the number of edges in a minimum spanning tree of a network with 10 vertices.
    2. State the number of edges in a minimum spanning tree of a network with \(n\) vertices.
  2. The following network has 10 vertices: \(A , B , \ldots , J\). The numbers on each edge represent the distances, in miles, between pairs of vertices. \includegraphics[max width=\textwidth, alt={}, center]{4a186c87-5f84-4ec3-8cc3-a0ed8721b040-03_1294_1118_785_445}
    1. Use Kruskal's algorithm to find the minimum spanning tree for the network.
    2. State the length of your spanning tree.
    3. Draw your spanning tree.
AQA D1 2006 January Q4
8 marks Moderate -0.8
4 The diagram shows the feasible region of a linear programming problem. \includegraphics[max width=\textwidth, alt={}, center]{4a186c87-5f84-4ec3-8cc3-a0ed8721b040-04_1349_1395_408_294}
  1. On the feasible region, find:
    1. the maximum value of \(2 x + 3 y\);
    2. the maximum value of \(3 x + 2 y\);
    3. the minimum value of \(- 2 x + y\).
  2. Find the 5 inequalities that define the feasible region.
AQA D1 2006 January Q5
7 marks Moderate -0.8
5 [Figure 1, printed on the insert, is provided for use in this question.]
The network shows the times, in minutes, to travel between 10 towns. \includegraphics[max width=\textwidth, alt={}, center]{4a186c87-5f84-4ec3-8cc3-a0ed8721b040-05_412_1561_568_233}
  1. Use Dijkstra's algorithm on Figure 1 to find the minimum time to travel from \(A\) to \(J\).
    (6 marks)
  2. State the corresponding route.
    (1 mark)
AQA D1 2006 January Q6
7 marks Easy -1.8
6 Two algorithms are shown. \section*{Algorithm 1}
Line 10Input \(P\)
Line 20Input \(R\)
Line 30Input \(T\)
Line 40Let \(I = ( P * R * T ) / 100\)
Line 50Let \(A = P + I\)
Line 60Let \(M = A / ( 12 * T )\)
Line 70Print \(M\)
Line 80Stop
\section*{Algorithm 2}
Line 10Input \(P\)
Line 20Input \(R\)
Line 30Input \(T\)
Line 40Let \(A = P\)
Line 50\(K = 0\)
Line 60Let \(K = K + 1\)
Line 70Let \(I = ( A * R ) / 100\)
Line 80Let \(A = A + I\)
Line 90If \(K < T\) then goto Line 60
Line 100Let \(M = A / ( 12 * T )\)
Line 110Print \(M\)
Line 120Stop
In the case where the input values are \(P = 400 , R = 5\) and \(T = 3\) :
  1. trace Algorithm 1;
  2. trace Algorithm 2.
AQA D1 2006 January Q7
13 marks Moderate -0.5
7 Stella is visiting Tijuana on a day trip. The diagram shows the lengths, in metres, of the roads near the bus station. \includegraphics[max width=\textwidth, alt={}, center]{4a186c87-5f84-4ec3-8cc3-a0ed8721b040-06_1539_1162_495_424} Stella leaves the bus station at \(A\). She decides to walk along all of the roads at least once before returning to \(A\).
  1. Explain why it is not possible to start from \(A\), travel along each road only once and return to \(A\).
  2. Find the length of an optimal 'Chinese postman' route around the network, starting and finishing at \(A\).
  3. At each of the 9 places \(B , C , \ldots , J\), there is a statue. Find the number of times that Stella will pass a statue if she follows her optimal route.
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 \(( L )\)-3530303735
Nou Camp \(( N )\)25-20212540
Olympic Village ( \(O\) )1540-253029
Park Guell ( \(P\) )303525-3520
Ramblas ( \(R\) )20301725-25
Sagrada Familia ( \(S\) )2535292030-
  1. Find the total travelling time for:
    1. the route \(L N O L\);
    2. the route \(L O N L\).
  2. Give an example of a Hamiltonian cycle in the context of the above situation.
  3. 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.