Questions — AQA D1 (170 questions)

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AQA D1 2016 June Q6
7 marks Moderate -0.5
6 A connected graph is semi-Eulerian if exactly two of its vertices are of odd degree.
  1. A graph is drawn with 4 vertices and 7 edges. What is the sum of the degrees of the vertices?
  2. Draw a simple semi-Eulerian graph with exactly 5 vertices and 5 edges, in which exactly one of the vertices has degree 4 .
  3. Draw a simple semi-Eulerian graph with exactly 5 vertices that is also a tree.
  4. A simple graph has 6 vertices. The graph has two vertices of degree 5 . Explain why the graph can have no vertex of degree 1.
AQA D1 2016 June Q7
17 marks Moderate -0.5
7 A company operates a steam railway between six stations. The minimum cost (in euros) of travelling between pairs of stations is shown in the table below.
  1. On Figure 1 below, use Prim's algorithm, starting from \(P\), to find a minimum spanning tree for the graph connecting \(P , Q , R , S , T\) and \(U\). State clearly the order in which you select the vertices and draw your minimum spanning tree. \section*{Question 7 continues on page 20} \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1}
    \(\boldsymbol { P }\)\(\boldsymbol { Q }\)\(\boldsymbol { R }\)\(\boldsymbol { S }\)\(\boldsymbol { T }\)\(\boldsymbol { U }\)
    \(\boldsymbol { P }\)-14711612
    \(\boldsymbol { Q }\)14-810910
    \(\boldsymbol { R }\)78-121315
    \(\boldsymbol { S }\)111012-511
    \(\boldsymbol { T }\)69135-10
    \(\boldsymbol { U }\)1210151110-
    \end{table}
  2. Another station, \(V\), is opened. The minimum costs (in euros) of travelling to and from \(V\) to each of the other stations are added to the table in part (a), as shown in Figure 2(i) below. Further copies of this table are shown in Figure 2(ii). Arjen is on holiday and he plans to visit each station. He intends to board a train at \(V\) and visit all the other stations, once only, before returning to \(V\).
    1. By first removing \(V\), obtain a lower bound for the minimum travelling cost of Arjen's tour. (You may use Figure 2(i) for your working.)
    2. Use the nearest neighbour algorithm twice, starting each time from \(V\), to find two different upper bounds for the minimum cost of Arjen's tour. State, with a reason, which of your two answers gives the better upper bound. (You may use Figure 2(ii) for your working.)
    3. Hence find an optimal tour of the seven stations. Explain how you know that it is optimal. Answer space for question 7(b) \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Figure 2(ii)}
      \(\boldsymbol { P }\)\(Q\)\(\boldsymbol { R }\)\(\boldsymbol { S }\)\(T\)\(\boldsymbol { U }\)\(V\)
      \(\boldsymbol { P }\)-1471161215
      \(Q\)14-81091018
      \(\boldsymbol { R }\)78-12131514
      \(\boldsymbol { S }\)111012-51114
      \(T\)69135-1017
      \(\boldsymbol { U }\)1210151110-12
      \(V\)151814141712-
      \end{table}
AQA D1 2016 June Q8
13 marks Easy -1.2
8 Nerys runs a cake shop. In November and December she sells Christmas hampers. She makes up the hampers herself, in two sizes: Luxury and Special. Each day, Nerys prepares \(x\) Luxury hampers and \(y\) Special hampers.
It takes Nerys 10 minutes to prepare a Luxury hamper and 15 minutes to prepare a Special hamper. She has 6 hours available, each day, for preparing hampers. From past experience, Nerys knows that each day:
  • she will need to prepare at least 5 hampers of each size
  • she will prepare at most a total of 32 hampers
  • she will prepare at least twice as many Luxury hampers as Special hampers.
Each Luxury hamper that Nerys prepares makes her a profit of \(\pounds 15\); each Special hamper makes a profit of \(\pounds 20\). Nerys wishes to maximise her daily profit, \(\pounds P\).
  1. Show that \(x\) and \(y\) must satisfy the inequality \(2 x + 3 y \leqslant 72\).
  2. In addition to \(x \geqslant 5\) and \(y \geqslant 5\), write down two more inequalities that model the constraints above.
  3. On the grid opposite draw a suitable diagram to enable this problem to be solved graphically. Indicate a feasible region and the direction of an objective line.
    1. Use your diagram to find the number of each type of hamper that Nerys should prepare each day to achieve a maximum profit.
    2. Calculate this profit.
      \includegraphics[max width=\textwidth, alt={}]{fb95068f-f76d-492a-b385-bce17b26ae30-27_2490_1719_217_150}
      \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}
AQA D1 2011 January Q1
7 marks Easy -1.2
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{figure_1}
  1. Represent this information in an adjacency matrix. [2]
  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. [5]
AQA D1 2011 January Q2
6 marks Easy -1.8
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 list: 10, 7, 4, 22, 13, 16, 19, 5 After 1st pass: 7, 4, 5, 10, 22, 13, 16, 19 After 2nd pass: 4, 5, 7, 10, 13, 16, 19, 22 After 3rd pass: 4, 5, 7, 10, 13, 16, 19, 22
  1. State the pivots used for the 2nd pass. [2]
  2. Write down the number of comparisons on each of the three passes. [3]
  3. Explain whether the student has completed the algorithm. [1]
AQA D1 2011 January Q3
9 marks Moderate -0.8
The following network shows the lengths, in miles, of roads connecting nine villages, \(A\), \(B\), ..., \(I\). \includegraphics{figure_3}
    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. [4]
    2. State the length of your minimum spanning tree. [1]
    3. Draw your minimum spanning tree. [2]
  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]
AQA D1 2011 January Q4
10 marks Moderate -0.3
The network below shows some paths on an estate. The number on each edge represents the time taken, in minutes, to walk along a path.
    1. Use Dijkstra's algorithm on the network to find the minimum walking time from \(A\) to \(J\). [6]
    2. Write down the corresponding route. [1]
  2. A new subway is constructed connecting \(C\) to \(G\) directly. The time taken to walk along this subway is \(x\) minutes. The minimum time taken to walk from \(A\) to \(G\) is now reduced, but the minimum time taken to walk from \(A\) to \(J\) is not reduced. Find the range of possible values for \(x\). [3]
\includegraphics{figure_4}
AQA D1 2011 January Q5
8 marks Moderate -0.3
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{figure_5} 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 \(A\). [1]
  2. Find the length of an optimal Chinese postman route around the estate, starting and finishing at \(A\). [5]
  3. For an optimal Chinese postman route, state:
    1. the number of times that the vertex \(F\) would occur; [1]
    2. the number of times that the vertex \(H\) would occur. [1]
AQA D1 2011 January Q6
5 marks Easy -1.2
  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 \(K_5\). [1]
    2. State the number of edges in a minimum spanning tree for the graph \(K_5\). [1]
    3. State the number of edges in a Hamiltonian cycle for the graph \(K_5\). [1]
  2. A simple graph \(G\) has six vertices and nine edges, and \(G\) is Eulerian. Draw a sketch to show a possible graph \(G\). [2]
AQA D1 2011 January Q7
10 marks Moderate -0.8
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. $$\begin{array}{c|ccccc} & A & B & C & D & E \\ \hline A & - & 3 & 11 & 15 & 5 \\ B & 3 & - & 18 & 12 & 4 \\ C & 11 & 18 & - & 5 & 16 \\ D & 15 & 12 & 5 & - & 10 \\ E & 5 & 4 & 16 & 10 & - \end{array}$$
  1. Find the travelling time for the tour \(BACDEB\). [1]
  2. Use the nearest neighbour algorithm, starting at \(B\), to find another upper bound for the travelling time for Fred's tour. [3]
  3. By deleting \(C\), find a lower bound for the travelling time for Fred's tour. [4]
  4. Sketch a network showing the edges that give you the lower bound in part (c) and comment on its significance. [2]
AQA D1 2011 January Q8
7 marks Easy -1.2
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 INT(3.8) = 3. Line 10: Let \(X = 0\) Line 20: Input \(A\), \(B\) Line 30: If INT\((A/2) = A/2\) then go to Line 50 Line 40: Let \(X = X + B\) Line 50: If \(A = 1\) then go to Line 90 Line 60: Let \(A =\) INT\((A/2)\) Line 70: Let \(B = 2 \times B\) Line 80: Go to Line 30 Line 90: Print \(X\) Line 100: End
  1. Trace the algorithm in the case where the input values are \(A = 20\) and \(B = 8\). [4]
  2. State the purpose of the algorithm. [1]
  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. [2]
AQA D1 2011 January Q9
13 marks Moderate -0.3
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. [4]
  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 \(x + 3y \leqslant 100\) \(3x + 5y \leqslant 200\) \(x + 5y \geqslant 80\) [2]
    2. On the grid opposite, draw a suitable diagram to represent Herman's situation, indicating the feasible region. [4]
    3. Use your diagram to find the maximum total number of hampers that Herman can pack on that day. [2]
    4. Find the number of each type of hamper that Herman packs that corresponds to your answer to part (b)(iii). [1]
AQA D1 2010 June Q1
4 marks Easy -1.2
  1. Draw a bipartite graph representing the following adjacency matrix. [2 marks] $$\begin{array}{c|ccccc} & 1 & 2 & 3 & 4 & 5 \\ \hline A & 1 & 0 & 0 & 0 & 1 \\ B & 0 & 1 & 1 & 1 & 0 \\ C & 0 & 1 & 1 & 1 & 0 \\ D & 1 & 0 & 0 & 0 & 1 \\ E & 1 & 0 & 0 & 0 & 1 \\ \end{array}$$
  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]
AQA D1 2010 June Q2
9 marks Easy -1.8
    1. Use a bubble sort to rearrange the following numbers into ascending order, showing the list of numbers after each pass. 6 \quad 2 \quad 3 \quad 5 \quad 4 [3 marks]
    2. Write down the number of comparisons on the first pass. [1 mark]
    1. Use a shuttle sort to rearrange the following numbers into ascending order, showing the list of numbers after each pass. 6 \quad 2 \quad 3 \quad 5 \quad 4 [4 marks]
    2. Write down the number of comparisons on the first pass. [1 mark]
AQA D1 2010 June Q3
11 marks Easy -1.2
The network shows 10 towns. The times, in minutes, to travel between pairs of towns are indicated on the edges. \includegraphics{figure_3}
  1. Use Kruskal's algorithm, showing the order in which you select the edges, to find a minimum spanning tree for the 10 towns. [6 marks]
  2. State the length of your minimum spanning tree. [1 mark]
  3. Draw your minimum spanning tree. [3 marks]
  4. If Prim's algorithm, starting at \(B\), had been used to find the minimum spanning tree, state which edge would have been the final edge to complete the minimum spanning tree. [1 mark]
AQA D1 2010 June Q4
14 marks Moderate -0.3
The network below shows 13 towns. The times, in minutes, to travel between pairs of towns are indicated on the edges. The total of all the times is 384 minutes.
  1. Use Dijkstra's algorithm on the network below, starting from \(M\), to find the minimum time to travel from \(M\) to each of the other towns. [7 marks]
    1. Find the travelling time of an optimum Chinese postman route around the network, starting and finishing at \(M\). [6 marks]
    2. State the number of times that the vertex \(F\) would appear in a corresponding route. [1 mark]
\includegraphics{figure_4}
AQA D1 2010 June Q5
10 marks Moderate -0.8
Phil, a squash coach, wishes to buy some equipment for his club. In a town centre, there are six shops, \(G\), \(I\), \(N\), \(R\), \(S\) and \(T\), that sell the equipment. The time, in seconds, to walk between each pair of shops is shown in the table. Phil intends to check prices by visiting each of the six shops before returning to his starting point. $$\begin{array}{c|cccccc} & G & I & N & R & S & T \\ \hline G & - & 81 & 82 & 86 & 72 & 76 \\ I & 81 & - & 80 & 82 & 68 & 73 \\ N & 82 & 80 & - & 84 & 70 & 74 \\ R & 86 & 82 & 84 & - & 74 & 70 \\ S & 72 & 68 & 70 & 74 & - & 64 \\ T & 76 & 73 & 74 & 70 & 64 & - \\ \end{array}$$
  1. Use the nearest neighbour algorithm starting from \(S\) to find an upper bound for Phil's minimum walking time. [4 marks]
  2. Write down a tour starting from \(N\) which has a total walking time equal to your answer to part (a). [1 mark]
  3. By deleting \(S\), find a lower bound for Phil's minimum walking time. [5 marks]
AQA D1 2010 June Q6
17 marks Standard +0.3
Phil is to buy some squash balls for his club. There are three different types of ball that he can buy: slow, medium and fast. He must buy at least 190 slow balls, at least 50 medium balls and at least 50 fast balls. He must buy at least 300 balls in total. Each slow ball costs £2.50, each medium ball costs £2.00 and each fast ball costs £2.00. He must spend no more than £1000 in total. At least 60% of the balls that he buys must be slow balls. Phil buys \(x\) slow balls, \(y\) medium balls and \(z\) fast balls.
  1. Find six inequalities that model Phil's situation. [4 marks]
  2. Phil decides to buy the same number of medium balls as fast balls.
    1. Show that the inequalities found in part (a) simplify to give $$x \geq 190, \quad y \geq 50, \quad x + 2y \geq 300, \quad 5x + 8y \leq 2000, \quad y \leq \frac{1}{3}x$$ [2 marks]
    2. Phil sells all the balls that he buys to members of the club. He sells each slow ball for £3.00, each medium ball for £2.25 and each fast ball for £2.25. He wishes to maximise his profit. On Figure 1 on page 14, draw a diagram to enable this problem to be solved graphically, indicating the feasible region and the direction of an objective line. [7 marks]
    3. Find Phil's maximum possible profit and state the number of each type of ball that he must buy to obtain this maximum profit. [4 marks]
AQA D1 2010 June Q7
6 marks Easy -1.2
A student is testing a numerical method for finding an approximation for \(\pi\). The algorithm that the student is using is as follows. Line 10 \quad Input \(A\), \(B\), \(C\), \(D\), \(E\) Line 20 \quad Let \(A = A + 2\) Line 30 \quad Let \(B = -B\) Line 40 \quad Let \(C = \frac{B}{A}\) Line 50 \quad Let \(D = D + C\) Line 60 \quad Let \(E = (D - 3.14)^2\) Line 70 \quad If \(E < 0.05\) then go to Line 90 Line 80 \quad Go to Line 20 Line 90 \quad Print '\(\pi\) is approximately', \(D\) Line 100 \quad End Trace the algorithm in the case where the input values are $$A = 1, \quad B = 4, \quad C = 0, \quad D = 4, \quad E = 0$$ [6 marks]
AQA D1 2010 June Q8
4 marks Moderate -0.8
A simple connected graph has six vertices.
  1. One vertex has degree \(x\). State the greatest and least possible values of \(x\). [2 marks]
  2. The six vertices have degrees $$x - 2, \quad x - 2, \quad x, \quad 2x - 4, \quad 2x - 4, \quad 4x - 12$$ Find the value of \(x\), justifying your answer. [2 marks]