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Networks, STD2 N2 2021 HSC 2 MC

Consider the network diagram.
 

What is the sum of the degrees of all the vertices in this network?

  1.  5
  2.  8
  3.  14
  4.  16
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`D`

Show Worked Solution

`text(Working from)\ A\ text(to)\ E:`

COMMENT: This simple question caused problems for many with mean mark just 58%.
`text{Sum of degrees}` `= 4 + 3 + 4 + 2 + 3`
  `= 16`

 
`=> D`

Networks, STD2 N2 2020 HSC 9 MC

Team `A` and Team `B` have entered a chess competition.

Team `A` and `B` have three members each. Each member of Team `A` must play each member of Team `B` once.

Which of the following network diagrams could represent the chess games to be played?
 

 

 

  
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`B`

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♦ Mean mark 38%.

`text(Vertices = players)`

`text(Edges = games between 2 players)`

`text(S)text(ince each player plays once against the three players)`

`text(in the other team, each vertex must be degree 3.)`

`=> \ B`

Networks, STD2 N2 2009 FUR1 8 MC

An undirected connected graph has five vertices.

Three of these vertices are of even degree and two of these vertices are of odd degree.

One extra edge is added. It joins two of the existing vertices.

In the resulting graph, it is not possible to have five vertices that are

A.   all of even degree.

B.   all of equal degree.

C.   one of even degree and four of odd degree.

D.   four of even degree and one of odd degree. 

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`D`

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`text(Consider an example of the graph)`

♦♦♦ Mean mark 25%.

`text{described (below):}`
 

matrices-fur1-2009-vcaa-8-mc-answer
 

`A\ text(is possible – join)\ V\ text(and)\ Z`

`B\ text(is possible – join)\ V\ text(and)\ Z`

`C\ text(is possible – join)\ W\ text(and)\ Y`

`D\ text(is NOT possible)`

`=>  D`

Networks, STD2 N2 SM-Bank 37

The map of Australia shows the six states, the Northern Territory and the Australian Capital Territory (ACT).
  

In the network diagram below, each of the vertices `A` to `H` represents one of the states or territories shown on the map of Australia. The edges represent a border shared between two states or between a state and a territory.
 

  1. In the network diagram, what is the order of the vertex that represents the Australian Capital Territory (ACT)?  (1 mark)

  2. In the network diagram, Queensland is represented by which letter? Explain why.  (2 marks)

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i.    `1`

ii.   `text{NSW is Vertex B (it is connected to the ACT – Vertex D)}`

`=> C\ text{is Victoria as it has degree 2}`

`:.\ text(Queensland is vertex)\ A\ text(as it is connected to)\ B\ text(and has degree 3.)`

Show Worked Solution

i.     `text {ACT has 1 border (with NSW)}`

`:.\ text(Degree of ACT’s vertex = 1)`
 

ii.   `text{NSW is Vertex B (it is connected to the ACT – Vertex D)}`

`=> C\ text{is Victoria as it has degree 2}`

`:.\ text(Queensland is vertex)\ A\ text(as it is connected to)\ B\ text(and has degree 3.)`

Networks, STD2 N2 2017 FUR1 2 MC

Two graphs, labelled Graph 1 and Graph 2, are shown below.
 

 
The sum of the degrees of the vertices of Graph 1 is

  1. two less than the sum of the degrees of the vertices of Graph 2.
  2. one less than the sum of the degrees of the vertices of Graph 2.
  3. equal to the sum of the degrees of the vertices of Graph 2.
  4. two more than the sum of the degrees of the vertices of Graph 2.
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`C`

Show Worked Solution

`text(Graph 1)`

`∑\ text(degrees)\ = 3 + 3 + 3 + 3 = 12`

`text(Graph 2)`

`∑\ text(degrees)\ = 2 + 2 + 2 + 2 + 2 + 2 = 12`

`=> C`