AQA Further AS Paper 2 Discrete (Further AS Paper 2 Discrete) 2021 June

1. A project consists of three activities \(A\), \(B\) and \(C\) An activity network for the project is shown in the diagram below. \includegraphics{figure_1} Find the value of \(x\) Circle your answer. [1 mark] 5 \quad 7 \quad 8 \quad 12
2. Find the value of \(y\) Circle your answer. [1 mark] 5 \quad 7 \quad 8 \quad 15

The set \(S\) is given by \(S = \{0, 2, 4, 6\}\)

1. Construct a Cayley table, using the grid below, for \(S\) under the binary operation addition modulo 8 [3 marks] \includegraphics{figure_2}
2. State the identity element for \(S\) under the binary operation addition modulo 8 [1 mark]

The diagram shows a network of pipes. Each pipe is labelled with its upper capacity in \(\mathrm{m}^3 \mathrm{s}^{-1}\) \includegraphics{figure_3}

1. Find the value of Cut \(X\) [1 mark]
2. Find the value of Cut \(Y\) [1 mark]
3. Add a supersink \(T\) to the network. [2 marks]

The binary operation \(*\) is defined as $$a * b = ab + 1 \quad \text{where } a, b \in \mathbb{R}$$

1. Prove that \(*\) is commutative on \(\mathbb{R}\) [2 marks]
2. Prove that \(*\) is not associative on \(\mathbb{R}\) [3 marks]

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]

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]

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]

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]