Questions D1 (899 questions)

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AQA D1 2005 January Q1
1 A student is using the algorithm below.
LINE 10INPUT \(A , B\)
LINE 20LET \(C = A - B\)
LINE 30LET \(D = A + B\)
LINE 40LET \(E = ( D * D ) - ( C * C )\)
LINE 50LET \(F = E / 4\)
LINE 60PRINT \(F\)
LINE 70END
Trace the algorithm in the case where \(A = 5\) and \(B = 3\).
AQA D1 2005 January Q2
2
  1. Use a bubble sort algorithm to rearrange the following numbers into ascending order, showing the new arrangement after each pass. $$\begin{array} { l l l l l l l l } 19 & 3 & 7 & 20 & 2 & 6 & 5 & 15 \end{array}$$
  2. Write down the number of comparisons and the number of swaps during the first pass.
    (2 marks)
AQA D1 2005 January Q3
3 A local council is responsible for gritting roads. The diagram shows the length, in miles, of the roads that have to be gritted.
\includegraphics[max width=\textwidth, alt={}, center]{76bccb26-f2ec-4798-bb6b-89c922f9651a-03_671_686_488_669} Total length \(= 87\) miles The gritter is based at \(A\), and must travel along all the roads, at least once, before returning to \(A\).
  1. Explain why it is not possible to start from \(A\) and, by travelling along each road only once, return to \(A\).
  2. Find an optimal 'Chinese postman' route around the network, starting and finishing at \(A\). State the length of your route.
AQA D1 2005 January Q4
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.
  1. Show this information on a bipartite graph.
  2. 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
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}
  1. Use Prim's algorithm, starting from \(A\), to find the minimum spanning tree for the network.
  2. State the length of your minimum spanning tree.
  3. Draw your minimum spanning tree.
  4. 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
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}
  1. Use Dijkstra's algorithm on Figure 1 to find the minimum time to travel from \(A\) to \(J\).
  2. 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
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.
\(\boldsymbol { A }\)\(\boldsymbol { B }\)\(\boldsymbol { C }\)\(\boldsymbol { D }\)\(\boldsymbol { E }\)\(\boldsymbol { F }\)
\(\boldsymbol { A }\)-869127
\(\boldsymbol { B }\)8-1014138
\(\boldsymbol { C }\)610-71610
\(\boldsymbol { D }\)9147-155
\(\boldsymbol { E }\)12131615-11
\(\boldsymbol { F }\)7810511-
    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 \(A\).
  2. By deleting \(A\), find a lower bound for the length of a minimum tour.
  3. 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
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, \(\pounds P\).
  1. Show that one of the constraints leads to the inequality $$2 x + y \leqslant 600$$
  2. Formulate this situation as a linear programming problem.
  3. On Figure 2, draw a suitable diagram to enable the problem to be solved graphically, indicating the feasible region and an objective line.
  4. Use your diagram to find the maximum daily profit.
  5. 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 2011 January Q1
1 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]{d1e453d6-0abb-4a7e-87c1-28275074dd08-02_1040_620_644_703}
  1. Represent this information in an adjacency matrix.
  2. Initially, \(B\) is assigned to task 5, \(D\) to task 2, \(E\) to task 4 and \(F\) to task 3. Demonstrate, by using an algorithm from this initial matching, how each person can be allocated to a task.
AQA D1 2011 January Q2
2 A student is using a quicksort algorithm to rearrange a set of numbers into ascending order. She uses the first number in each list (or sublist) as the pivot. Her correct solution for the first three passes is as follows.
Initial list1074221316195
After 1st pass7451022131619
After 2nd pass4571013161922
After 3rd pass4571013161922
  1. State the pivots used for the 2nd pass.
  2. Write down the number of comparisons on each of the three passes.
  3. Explain whether the student has completed the algorithm.
AQA D1 2011 January Q3
3 The following network shows the lengths, in miles, of roads connecting nine villages, \(A , B , \ldots , I\).
\includegraphics[max width=\textwidth, alt={}, center]{d1e453d6-0abb-4a7e-87c1-28275074dd08-06_832_858_388_591}
    1. Use Prim's algorithm starting from \(E\), 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.
  2. On a particular day, village \(B\) is cut off, so its connecting roads cannot be used. Find the length of a minimum spanning tree for the remaining eight villages.
    (2 marks)
    \includegraphics[max width=\textwidth, alt={}, center]{d1e453d6-0abb-4a7e-87c1-28275074dd08-08_2486_1724_221_143}
AQA D1 2011 January Q5
5 Norris delivers newspapers to houses on an estate. The network shows the streets on the estate. The number on each edge shows the length of the street, in metres. Norris starts from the newsagents located at vertex \(A\), and he must walk along all the streets at least once before returning to the newsagents.
\includegraphics[max width=\textwidth, alt={}, center]{d1e453d6-0abb-4a7e-87c1-28275074dd08-10_1045_931_552_557} The total length of the streets is 1215 metres.
  1. Give a reason why it is not possible to start at \(A\), walk along each street once only, and return to \(A\).
  2. Find the length of an optimal Chinese postman route around the estate, starting and finishing at \(A\).
  3. For an optimal Chinese postman route, state:
    1. the number of times that the vertex \(F\) would occur;
    2. the number of times that the vertex \(H\) would occur.
      \includegraphics[max width=\textwidth, alt={}]{d1e453d6-0abb-4a7e-87c1-28275074dd08-11_2486_1714_221_153}
AQA D1 2011 January Q6
6
  1. The complete graph \(K _ { n }\) has every one of its \(n\) vertices connected to each of the other vertices by a single edge.
    1. Find the total number of edges in the graph \(K _ { 5 }\).
    2. State the number of edges in a minimum spanning tree for the graph \(K _ { 5 }\).
    3. State the number of edges in a Hamiltonian cycle for the graph \(K _ { 5 }\).
  2. A simple graph \(G\) has six vertices and nine edges, and \(G\) is Eulerian. Draw a sketch to show a possible graph \(G\).
    \includegraphics[max width=\textwidth, alt={}]{d1e453d6-0abb-4a7e-87c1-28275074dd08-12_1844_1714_863_153}
AQA D1 2011 January Q7
7 Fred delivers bread to five shops, \(A , B , C , D\) and \(E\). Fred starts his deliveries at shop \(B\), and travels to each of the other shops once before returning to shop \(B\). Fred wishes to keep his travelling time to a minimum. The table shows the travelling times, in minutes, between the shops.
\cline { 2 - 6 } \multicolumn{1}{c|}{}\(\boldsymbol { A }\)\(\boldsymbol { B }\)\(\boldsymbol { C }\)\(\boldsymbol { D }\)\(\boldsymbol { E }\)
\(\boldsymbol { A }\)-311155
\(\boldsymbol { B }\)3-18124
\(\boldsymbol { C }\)1118-516
\(\boldsymbol { D }\)15125-10
\(\boldsymbol { E }\)541610-
  1. Find the travelling time for the tour \(B A C D E B\).
  2. Use the nearest neighbour algorithm, starting at \(B\), to find another upper bound for the travelling time for Fred's tour.
  3. By deleting \(C\), find a lower bound for the travelling time for Fred's tour.
  4. Sketch a network showing the edges that give you the lower bound in part (c) and comment on its significance.
AQA D1 2011 January Q8
8 A student is tracing the following algorithm with positive integer values of \(A\) and \(B\).
The function INT gives the integer part of a number, eg INT(2.3) \(= 2\) and \(\operatorname { INT } ( 3.8 ) = 3\).
Line 10Let \(X = 0\)
Line 20Input \(A , B\)
Line 30If \(\operatorname { INT } ( A / 2 ) = A / 2\) then go to Line 50
Line 40Let \(X = X + B\)
Line 50If \(A = 1\) then go to Line 90
Line 60Let \(A = \operatorname { INT } ( A / 2 )\)
Line 70Let \(B = 2 \times B\)
Line 80Go to Line 30
Line 90Print \(X\)
Line 100End
  1. Trace the algorithm in the case where the input values are \(A = 20\) and \(B = 8\).
  2. State the purpose of the algorithm.
  3. Another student changed Line 50 to Line 50 If \(A = 1\) then go to Line 80
    Explain what would happen if this algorithm were traced.
    \includegraphics[max width=\textwidth, alt={}]{d1e453d6-0abb-4a7e-87c1-28275074dd08-17_2486_1714_221_153}
AQA D1 2011 January Q9
9 Herman is packing some hampers. Each day, he packs three types of hamper: basic, standard and luxury. Each basic hamper has 6 tins, 9 packets and 6 bottles.
Each standard hamper has 9 tins, 6 packets and 12 bottles.
Each luxury hamper has 9 tins, 9 packets and 18 bottles.
Each day, Herman has 600 tins and 600 packets available, and he must use at least 480 bottles. Each day, Herman packs \(x\) basic hampers, \(y\) standard hampers and \(z\) luxury hampers.
  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, simplifying each inequality.
  2. On a particular day, Herman packs the same number of standard hampers as luxury hampers.
    1. Show that your answers in part (a) become $$\begin{aligned} x + 3 y & \leqslant 100
      3 x + 5 y & \leqslant 200
      x + 5 y & \geqslant 80 \end{aligned}$$
    2. On the grid opposite, draw a suitable diagram to represent Herman's situation, indicating the feasible region.
    3. Use your diagram to find the maximum total number of hampers that Herman can pack on that day.
    4. Find the number of each type of hamper that Herman packs that corresponds to your answer to part (b)(iii).
      (1 mark)
AQA D1 2012 January Q1
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
2
  1. Draw a bipartite graph representing the following adjacency matrix.
    \cline { 2 - 7 } \multicolumn{1}{c|}{}\(\mathbf { 1 }\)\(\mathbf { 2 }\)\(\mathbf { 3 }\)\(\mathbf { 4 }\)\(\mathbf { 5 }\)\(\mathbf { 6 }\)
    \(\boldsymbol { A }\)110011
    \(\boldsymbol { B }\)010011
    \(\boldsymbol { C }\)100011
    \(\boldsymbol { D }\)011101
    \(\boldsymbol { E }\)000011
    \(\boldsymbol { F }\)000001
  2. 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
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}
  1. Use Kruskal's algorithm to find a minimum spanning tree for the network. (5 marks)
  2. State the length of your minimum spanning tree.
  3. There are two minimum spanning trees for this network. Draw both of these minimum spanning trees.
AQA D1 2012 January Q4
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.
  1. Find the length of an optimal Chinese postman route for the policeman.
  2. 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
5 The feasible region of a linear programming problem is determined by the following: $$\begin{aligned} y & \geqslant 20
x + y & \geqslant 25
5 x + 2 y & \leqslant 100
y & \leqslant 4 x
y & \geqslant 2 x \end{aligned}$$
  1. On Figure 1 opposite, draw a suitable diagram to represent the inequalities and indicate the feasible region.
  2. 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
6 The network below shows the lengths of roads, in miles, connecting 10 towns, \(A , B , \ldots , J\).
  1. Use Dijkstra's algorithm on the network to find the shortest distance from \(A\) to \(J\). Show all your working at each vertex.
  2. Write down the corresponding route.
  3. 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
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.
  1. 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.
  4. 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-2641627
    \(\boldsymbol { B }\)2-831526
    \(\boldsymbol { C }\)68-102232
    \(\boldsymbol { D }\)4310-1223
    \(\boldsymbol { F }\)16152212-20
    G2726322320-
    \end{table}
    \includegraphics[max width=\textwidth, alt={}]{5a414265-6273-41c5-ad5f-f6316bd774d0-17_2486_1714_221_153}
AQA D1 2012 January Q8
8 Four distinct positive integers are \(( 3 x - 5 ) , ( 2 x + 3 ) , ( x + 1 )\) and \(( 4 x - 13 )\).
  1. Explain why \(x \geqslant 4\).
  2. 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 \(x\).
  3. 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
1
  1. Draw a bipartite graph to represent the following adjacency matrix.
    12345
    A00001
    \(\boldsymbol { B }\)00011
    \(\boldsymbol { C }\)01011
    \(\boldsymbol { D }\)00011
    \(\boldsymbol { E }\)11101
  2. 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)