Questions Further AS Paper 2 Discrete (64 questions)

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AQA Further AS Paper 2 Discrete 2021 June Q5
7 marks Moderate -0.8
An adjacency matrix for the simple graph \(G\) is shown below. \includegraphics{figure_5}
  1. Using the adjacency matrix, explain why \(G\) is not a complete graph. [2 marks]
  2. State, with a reason, whether \(G\) is Eulerian, semi-Eulerian or neither. [2 marks]
  3. Draw a graph that is the complement of \(G\) [3 marks]
AQA Further AS Paper 2 Discrete 2021 June Q6
6 marks Standard +0.3
Vaya and Wynne are playing a zero-sum game. The game is represented by the pay-off matrix for Vaya. \includegraphics{figure_6}
  1. Find the play-safe strategies for Vaya and Wynne. Fully justify your answer. [4 marks]
  2. Vaya and Wynne decide not to play their play-safe strategies. Deduce the best possible outcome for Wynne. [2 marks]
AQA Further AS Paper 2 Discrete 2021 June Q7
7 marks Challenging +1.2
A jeweller is making pendants. Each pendant is made by bending a single, continuous strand of wire. Each pendant has the same design as shown below. \includegraphics{figure_7} The lengths on the diagram are in millimetres. The sum of these lengths is 240 mm As the jeweller does not cut the wire, some sections require a double length of wire.
  1. The jeweller makes a pendant by starting and finishing at \(B\) Find the minimum length of the strand of wire that the jeweller needs to make the pendant. Fully justify your answer. [4 marks]
  2. The jeweller makes another pendant of the same design. Find the minimum possible length for the strand of wire that the jeweller would need. [2 marks]
  3. By considering the differences between the pendants in part (a) and part (b), state one reason why the jeweller may prefer the pendant in part (a). [1 mark]
AQA Further AS Paper 2 Discrete 2021 June Q8
5 marks Standard +0.8
A linear programming problem is set up to maximise \(P = ax + y\) where \(a\) is a constant. \(P\) is maximised subject to three linear constraints which form the triangular feasible region shown in the diagram below. \includegraphics{figure_8} The vertices of the triangle are \((1, 6)\), \((5, 11)\) and \((13, 9)\) \(P\) is maximised at \((5, 11)\) Find the range of possible values for \(P\) [5 marks]
AQA Further AS Paper 2 Discrete 2024 June Q1
1 marks Easy -1.8
A connected planar graph has \(v\) vertices, \(e\) edges and \(f\) faces. Which one of the formulae below is correct? Circle your answer. [1 mark] \(v + e + f = 2\) \quad\quad \(v - e + f = 2\) \quad\quad \(v - e - f = 2\) \quad\quad \(v + e - f = 2\)
AQA Further AS Paper 2 Discrete 2024 June Q2
1 marks Easy -1.8
Find an expression for the number of edges in the complete bipartite graph, \(K_{m,n}\) Circle your answer. [1 mark] \(\frac{m}{n}\) \quad\quad \(m - n\) \quad\quad \(m + n\) \quad\quad \(mn\)
AQA Further AS Paper 2 Discrete 2024 June Q3
1 marks Easy -1.8
Which one of the graphs shown below is semi-Eulerian? Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_3}
AQA Further AS Paper 2 Discrete 2024 June Q4
4 marks Moderate -0.8
The set \(S\) is defined as \(S = \{1, 2, 3, 4\}\)
  1. Complete the Cayley Table shown below for \(S\) under the binary operation multiplication modulo 5 [2 marks]
    \(\times_5\)1234
    1
    2
    3
    4
  2. State the identity element for \(S\) under multiplication modulo 5 [1 mark]
  3. State the self-inverse elements of \(S\) under multiplication modulo 5 [1 mark]
AQA Further AS Paper 2 Discrete 2024 June Q5
4 marks Moderate -0.3
A network of roads connects the villages \(A\), \(B\), \(C\), \(D\), \(E\), \(F\) and \(G\) The weight on each arc in the network represents the distance, in miles, between adjacent villages. The network is shown in the diagram below. \includegraphics{figure_5}
  1. Draw, in the space below, the spanning tree of minimum total length for this road network. [3 marks]
  2. Find the total length of the spanning tree drawn in part (a). [1 mark]
AQA Further AS Paper 2 Discrete 2024 June Q6
4 marks Easy -1.2
A Young Enterprise Company decides to sell two types of cakes at a breakfast club. The two types of cakes are blueberry and chocolate. From its initial market research, the company knows that it will: • sell at most 200 cakes in total • sell at least twice as many blueberry cakes as they will chocolate cakes • make 20p profit on each blueberry cake they sell • make 15p profit on each chocolate cake they sell. The company's objective is to maximise its profit. Formulate the Young Enterprise Company's situation as a linear programming problem. [4 marks]
AQA Further AS Paper 2 Discrete 2024 June Q7
5 marks Standard +0.3
The binary operation \(\nabla\) is defined as \(a \nabla b = a + b + ab\) where \(a, b \in \mathbb{R}\)
  1. Determine if \(\nabla\) is commutative on \(\mathbb{R}\) Fully justify your answer. [2 marks]
  2. Prove that \(\nabla\) is associative on \(\mathbb{R}\) [3 marks]
AQA Further AS Paper 2 Discrete 2024 June Q8
7 marks Standard +0.3
The diagram below shows a network of pipes. \includegraphics{figure_8} The uncircled numbers on each arc represent the capacity of each pipe in m³ s⁻¹ The circled numbers on each arc represent an initial feasible flow, in m³ s⁻¹, through the network. The initial flow through pipe \(SD\) is \(x\) m³ s⁻¹ The initial flow through pipe \(DC\) is \(y\) m³ s⁻¹ The initial flow through pipe \(CB\) is \(z\) m³ s⁻¹
  1. By considering the flows at the source and the sink, explain why \(x = 7\) [3 marks]
    1. State the value of \(y\) [1 mark]
    2. State the value of \(z\) [1 mark]
  3. Prove that the maximum flow through the network is at most 27 m³ s⁻¹ [2 marks]
AQA Further AS Paper 2 Discrete 2024 June Q9
6 marks Standard +0.3
Robert, a project manager, and his team of builders are working on a small building project. Robert has divided the project into ten activities labelled \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(G\), \(H\), \(I\) and \(J\) as shown in the precedence table below:
ActivityImmediate Predecessor(s)Duration (Days)
\(A\)None1
\(B\)None1
\(C\)\(A\)10
\(D\)\(A\)2
\(E\)\(B, D\)5
\(F\)\(E\)6
\(G\)\(E\)1
\(H\)\(F\)1
\(I\)\(F\)2
\(J\)\(C, G, H, I\)4
  1. On the opposite page, construct an activity network for the project and fill in the earliest start time and latest finish time for each activity. [4 marks]
  2. Robert claims that the project can be completed in 20 days. Comment on the validity of Robert's claim. [2 marks]
AQA Further AS Paper 2 Discrete 2024 June Q10
7 marks Challenging +1.2
Bilal and Mayon play a zero-sum game. The game is represented by the following pay-off matrix for Bilal, where \(x\) is an integer.
Mayon
\(\mathbf{M_1}\)\(\mathbf{M_2}\)\(\mathbf{M_3}\)
\(\mathbf{B_1}\)\(-2\)\(-1\)\(1\)
Bilal \quad \(\mathbf{B_2}\)\(4\)\(-3\)\(1\)
\(\mathbf{B_3}\)\(-1\)\(x\)\(0\)
The game has a stable solution.
  1. Show that there is only one possible value for \(x\) Fully justify your answer. [6 marks]
  2. State the value of the game for Bilal. [1 mark]