VCE General Maths Networks and Decision Mathematics 2023 Exam 1 Mini Test

Question 1 [2023 Exam 1 Q33]

Consider the following graph.

How many of the following five statements are true?

• The graph is a tree.
• The graph is connected.
• The graph contains a path.
• The graph contains a cycle.
• The sum of the degrees of the vertices is eight.

• A. 1
• B. 2
• C. 3
• D. 4
• E. 5

Question 2 [2023 Exam 1 Q34]

A bipartite graph is typically used to display which one of the following?

• A. the allocation of tasks on a construction site
• B. the path used to visit five different construction sites
• C. the total distance travelled between two construction sites
• D. the critical path of activities to be completed in a construction project
• E. the minimum length of cable required to connect six construction sites

Question 3 [2023 Exam 1 Q35]

Consider the weighted graph shown below.

The weight of the minimum spanning tree is

• A. 30
• B. 32
• C. 40
• D. 42
• E. 52

Question 4 [2023 Exam 1 Q36]

Four employees, Anthea, Bob, Cho and Dario, are each assigned a different duty by their manager. The time taken for each employee to complete duties 1, 2, 3 and 4, in minutes, is shown in the table below.

	Duty 1	Duty 2	Duty 3	Duty 4
Anthea	8	7	7	8
Bob	10	8	10	9
Cho	8	9	7	10
Dario	7	7	8	9

The manager allocates the duties so as to minimise the total time taken to complete the four duties. The minimum total time taken to complete the four duties, in minutes, is

• A. 29
• B. 30
• C. 31
• D. 32
• E. 33

Question 5 [2023 Exam 1 Q37]

The adjacency matrix below represents a planar graph with five vertices.

\[ \begin{array}{@{}c@{\,}c} & \begin{array}{ccccc} J & K & L & M & N \end{array} \\ \begin{matrix} J \\ K \\ L \\ M \\ N \end{matrix} & \left[ \begin{array}{ccccc} 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 2 & 1 & 1 \\ 0 & 2 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 & 0 \end{array} \right] \end{array} \]

The number of faces on the planar graph is

• A. 5
• B. 7
• C. 9
• D. 15
• E. 17

Question 6 [2023 Exam 1 Q38]

A particular building project has ten activities that must be completed. These activities and their immediate predecessor(s) are shown in the table below.

Activity	Immediate predecessor(s)
A	–
B	–
C	A
D	A
E	B
F	D, E
G	C, F
H	F
I	D, E
J	H, I

A directed graph that could represent this project is

Question 7 [2023 Exam 1 Q39]

Use the following information to answer Questions 39 and 40.

The network below shows the one-way paths between the entrance, \(A\), and the exit, \(H\), of a children's maze. The vertices represent the intersections of the one-way paths. The number on each edge is the maximum number of children who are allowed to travel along that path per minute.

Cuts on this network are used to consider the possible flow of children through the maze. The capacity of the minimum cut would be

• A. 20
• B. 23
• C. 24
• D. 29
• E. 30

Question 8 [2023 Exam 1 Q40]

One path in the maze is to be changed. Which one of these five changes would lead to the largest increase in flow from entrance to exit?

• A. increasing the capacity of flow along the edge \(CE\) to 12
• B. increasing the capacity of flow along the edge \(FH\) to 14
• C. increasing the capacity of flow along the edge \(GH\) to 16
• D. reversing the direction of flow along the edge \(CF\)
• E. reversing the direction of flow along the edge \(GF\)