AQA Further AS Paper 2 Discrete (Further AS Paper 2 Discrete) 2024 June

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 \(v + e - f = 2\)

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 \(mn\)

Which one of the graphs shown below is semi-Eulerian? Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_3}

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]

   \(\times_5\)	1	2	3	4
   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]

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]

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]

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]

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]
2. 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]

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:

Activity	Immediate Predecessor(s)	Duration (Days)
\(A\)	None	1
\(B\)	None	1
\(C\)	\(A\)	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]

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]