| Mia | | | X | Y | Z | | | \multirow{3}{*}{Li} | X | 5 | - 6 | 0 | | \cline { 2 - 5 } | Y | - 2 | 3 | 4 | | \cline { 2 - 5 } | Z | - 1 | 4 | 8 | | \cline { 2 - 5 } |
| Mia | | X | Y | Z | | | \multirow{2}{*}{Li} | X | 4 | | | | \cline { 2 - 5 } | Y | 11 | | 5 | | \cline { 2 - 5 } | Z | 10 | 5 | 1 | | \cline { 2 - 5 } |
The game can be converted into a zero-sum game, this means that the total number of points won by Li and Mia is the same for each combination of strategies.- Complete the table in the Printed Answer Booklet to show the points won by Mia.
- Convert the game into a zero-sum game, giving the pay-offs for Li .
Use dominance to reduce the pay-off matrix for the game to a \(2 \times 2\) table. You do not need to explain the dominance.
Mia knows that Li will choose his play-safe strategy.Determine which strategy Mia should choose to maximise her points.
5 A linear programming problem is formulated as below.
Maximise \(\quad \mathrm { P } = 4 \mathrm { x } - \mathrm { y }\)
subject to \(2 x + 3 y \geqslant 12\)
\(x + y \leqslant 10\)
\(5 x + 2 y \leqslant 30\)
\(x \geqslant 0 , y \geqslant 0\)- Identify the feasible region by representing the constraints graphically and shading the regions where the inequalities are not satisfied.
- Hence determine the maximum value of the objective.
The constraint \(x + y \leqslant 10\) is changed to \(x + y \leqslant k\), the other constraints are unchanged.
Determine, algebraically, the value of \(k\) for which the maximum value of \(P\) is 3 .
Do not draw on the graph from part (a) and do not use the spare grid.Determine, algebraically, the other value of \(k\) for which there is a (non-optimal) vertex of the feasible region at which \(P = 3\).
Do not draw on the graph from part (a) and do not use the spare grid.
6 Sarah is having some work done on her garden.
The table below shows the activities involved, their durations and their immediate predecessors. These durations and immediate predecessors are known to be correct.
| Activity | Immediate predecessors | Duration (hours) | | A Clear site | - | 4 | | B Mark out new design | A | 1 | | C Buy materials, turf, plants and trees | - | 3 | | D Lay paths | B, C | 1 | | E Build patio | B, C | 2 | | F Plant trees | D | 1 | | G Lay turf | D, E | 1 | | H Finish planting | F, G | 1 |
- Use a suitable model to determine the following.
- The minimum time in which the work can be completed
The activities with zero float
[0pt]
(ii) State one practical issue that could affect the minimum completion time in part (a)(i). [1]
Sarah needs the work to be completed as quickly as possible. There will be at least one activity happening at all times, but it may not always be possible to do all the activities that are needed at the same time.Determine the earliest and latest times at which building the patio (activity E) could start.
There needs to be a 2-hour break after laying the paths (activity D). During this time other activities that do not depend on activity D can still take place.Describe how you would adapt your model to incorporate the 2-hour break.
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