VCE General Maths Networks and Decision Mathematics 2022 Exam 2 Mini Test
VCAA General Maths Exam 2
This is the full VCE General Maths Exam with worked solutions. You can also try Mini-Tests, which are official VCAA exams split into short tests you can do anytime.
Number of marks: 12
Reading time: 3 minutes
Writing time: 18 minutes
Instructions
• Answer all questions in the spaces provided.
• Write your responses in English.
• In all questions where a numerical answer is required, you should only round your answer when instructed to do so.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
Networks and decision mathematics - 2022 - Exam 2
Joe owns a holiday park with seven cabins.
The diagram below shows a network of roads in the holiday park.
Joe's office is labelled as a vertex on the network. The other vertices – \(A, B, C, D, E, F\) and \(G\) – represent the cabins.
In the morning, Joe leaves his office and visits each cabin once only before returning to the office.
a. Write down a route that Joe could follow. 1 mark
b. Later in the day, Joe will leave his office and travel along each road once only to check the road conditions.
i. Which vertex will Joe finish on? 1 mark
ii. What is the mathematical term for this route? 1 mark
Joe plans to add a new cabin to the holiday park.
This project requires 10 activities to be completed.
The network below shows these 10 activities and their completion times in weeks.
a. How many of these activities have two immediate predecessors? 1 mark
b. What is the minimum completion time, in weeks, for this project? 1 mark
c. It is possible to reduce the completion time of two activities.
One activity can have its completion time decreased by two weeks and another activity can have its completion time decreased by one week.
These two changes result in the minimum completion time being reduced by three weeks.
Complete the table below, showing the two activities that could have their completion times reduced and the reduction in individual activity completion time that would achieve the three-week reduction. 1 mark
| Activity | Reduction in completion time (1 week or 2 weeks) |
|---|---|
d. The holiday park has had some changes to its roads to accommodate the new cabin.
The adjacency matrix below shows road connections between the office and each cabin. The new cabin, \(H\), is included in the matrix.
\[ \begin{array}{c@{}c} & \begin{array}{@{}ccccccccc@{}} \text{office} & A & B & C & D & E & F & G & H \end{array} \\ \begin{matrix} \text{office} \\ A \\ B \\ C \\ D \\ E \\ F \\ G \\ H \end{matrix} & \left[ \begin{array}{@{}ccccccccc@{}} 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 \end{array} \right] \end{array} \]
On the diagram below, add the new cabin, \(H\), and any additional roads to the network. 2 marks
A series of pipelines is installed to allow for the flow of stormwater from the holiday park.
The capacity of these pipes, in litres per minute, is shown in the directed network below.
When considering the possible flow of stormwater through this network, many different cuts can be made.
a. Determine the capacity of Cut 1, shown above. 1 mark
b. What is the maximum flow of stormwater, in litres per minute, from the source to the sink? 1 mark
Joe would like to increase the maximum flow through this network.
The maximum flow through this network may be increased either by reversing the direction of flow through one pipe or by increasing the capacity of one pipe.
c. The direction of flow is reversed through one pipe.
Complete the following sentence by filling in the boxes provided. 1 mark
The pipe that should have its flow reversed to cause the largest increase in flow from source to sink is the pipe that runs from vertex to vertex .
d. The capacity of one pipe is increased.
Complete the following sentences by filling in the boxes provided. 1 mark
The pipe that should have its capacity increased to cause the largest increase in flow from source to sink is the pipe that runs from vertex to vertex . Its new capacity, in litres per minute, should be at least .
End of Question and Answer Book
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