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Networks, STD2 N2 2024 HSC 18

The diagram shows a network with weighted edges.
 

  1. Draw a minimum spanning tree for this network and determine its weight.   (2 marks)
     


  1. Is it possible to find another spanning tree with the same weight? Give a reason for your answer.   (1 mark)

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a.
         

 
b.   

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a.
         

b.   

♦ Mean mark (b) 45%.

Networks, STD2 N2 2023 HSC 19

A network of running tracks connects the points , as shown. The number on each edge represents the time, in minutes, that a typical runner should take to run along each track.


  1. Which path could a typical runner take to run from point to point in the shortest time?  (2 marks)

  2. A spanning tree of the network above is shown.
  3.  
  5. Is it a minimum spanning tree? Give a reason for your answer.  (2 marks)

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a.   

b.   

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a.   
 

 
   
   

 
b.   


 


 

♦ Mean mark (b) 47%.

Networks, STD2 N2 2022 HSC 20

The table below shows the distances, in kilometres, between a number of towns.
 

  1. Using the vertices given, draw a weighted network diagram to represent the information shown in the table.  (2 marks)
     

     
  2. A tourist wishes to visit each town.
  3. Draw the minimum spanning tree which will allow for this AND determine its length.  (3 marks)
     

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  2.   
     
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a. 

 

b.   

Networks, STD2 N2 2021 HSC 23

The network diagram shows the travel times in minutes along roads connecting a number of different towns.
 


 

  1. Draw a minimum spanning tree for this network and determine its length.  (3 marks)

  2. How long does it take to travel from Queentown to Underwood using the fastest route?  (1 mark)

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a.  

 

 
 
b.  
   

Networks, STD2 N2 2020 HSC 18

The diagram represents a network with weighted edges.
 


 

  1. Draw a minimum spanning tree for this network and determine its length.   (3 marks)

  2. The network is revised by adding another vertex, . Edges and have weights of 12 and 10 respectively, as shown.
     

   
 

What is the length of the minimum spanning tree for this revised network?   (1 mark)

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a.     








 


 

 

 

b.     

 

Networks, STD2 N2 2019 HSC 30

The network diagram shows the tracks connecting 8 picnic sites in a nature park. The vertices to represents the picnic sites. The weights on the edges represent the distance along the tracks between the picnic sites, in kilometres.
 


 

  1. Each picnic site needs to provide drinking water. The main water source is at site .

     

    Draw a minimum spanning tree and calculate the minimum length of water pipes required to supply water to all the sites if the water pipes can only be laid along the tracks.  (2 marks)

  2. One day, the track between and is closed. State the vertices that identify the shortest path from to that avoids the closed track.  (1 mark)

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a.   


 

 

b.   

Networks, STD2 N2 SM-Bank 2

A school is designing a computer network between five key areas within the school.

The cost of connecting the rooms is shown in the diagram below.
  


 

  1. Complete the spanning tree below that creates the school's network at a minimum cost.  (1 mark)
      


     
     

  2. What is the minimum cost of the network?  (1 mark)

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i.   


  

 

ii.   
   

Networks, STD2 N2 SM-Bank 14

Water will be pumped from a dam to eight locations on a farm.

The pump and the eight locations (including the house) are shown as vertices in the network diagram below.

The numbers on the edges joining the vertices give the shortest distances, in metres, between locations.
 

NETWORKS, FUR2 2012 VCAA 1

 

  1. How many vertices on the network diagram have an odd degree?  (1 mark)

     

    The total length of all edges in the network is 1180 metres.

     

    The total length of pipe that supplies water from the pump to the eight locations on the farm is a minimum.

     

    This minimum length of pipe is laid along some of the edges in the network.

  2. On the diagram below, draw the minimum length of pipe that is needed to supply water to all locations on the farm.  (2 marks)
     
     
    NETWORKS, FUR2 2012 VCAA 1

  3. What is the mathematical term that is used to describe this minimum length of pipe?  (1 mark)

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  2.  
    NETWORKS, FUR2 2012 VCAA 1 Answer

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i.  

MARKER’S COMMENT: Many students, surprisingly, had problems with part (i).
 

ii. 


 

NETWORKS, FUR2 2012 VCAA 1 Answer


iii.   

Networks, STD2 N2 2007 FUR2 1

A new housing estate is being developed.

There are five houses under construction in one location.

These houses are numbered as points 1 to 5 below.
  

NETWORKS, FUR2 2007 VCAA 1

 
The builders require the five houses to be connected by electrical cables to enable the workers to have a supply of power on each site.

  1. What is the minimum number of edges needed to connect the five houses?  (1 mark)

  2. On the diagram above, draw a connected graph with this number of edges.  (1 mark) 

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  2.  
    networks-fur2-2007-vcaa-1-answer
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a.   

 

b.   

networks-fur2-2007-vcaa-1-answer

Networks, STD2 N2 2017 FUR2 3a

While on holiday, four friends visit a theme park where there are nine rides.

On the graph below, the positions of the rides are indicated by the vertices.

The numbers on the edges represent the distances, in metres, between rides.
 

Electrical cables are required to power the rides.

These cables will form a connected graph.

The shortest total length of cable will be used.

 

  1. Give a mathematical term to describe a graph that represents these cables.   (1 mark)

  2. Draw in the graph that represents these cables on the diagram below.   (1 mark)


 

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a.   

b.    
       

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a.   

 

b.   


 
        

Networks, STD2 N2 2015 FUR2 1

A factory requires seven computer servers to communicate with each other through a connected network of cables.

The servers, , , , , , and , are shown as vertices on the graph below.
 

Networks, FUR2 2015 VCAA 11

 
The edges on the graph represent the cables that could connect adjacent computer servers.

The numbers on the edges show the cost, in dollars, of installing each cable.

  1. What is the cost, in dollars, of installing the cable between server and server ?  (1 mark)

  2. What is the cheapest cost, in dollars, of installing cables between server and server ?  (1 mark)

  3. The computer servers will be able to communicate with all the other servers as long as each server is connected by cable to at least one other server.

    1. The cheapest installation that will join the seven computer servers by cable in a connected network follows a minimum spanning tree.

       

      Draw the minimum spanning tree on the plan below.  (1 mark) 
       

      Networks, FUR2 2015 VCAA 12
       

    2. The factory’s manager has decided that only six connected computer servers will be needed, rather than seven.

       

      How much would be saved in installation costs if the factory removed computer server  from its minimum spanning tree network?

       

      A copy of the graph above is provided below to assist with your working.  (1 mark)

      Networks, FUR2 2015 VCAA 12

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    1.  
      Networks, FUR2 2015 VCAA 12 Answer
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a.   

 

b.  


 

MARKER’S COMMENT: Many students had difficulty finding the minimum spanning tree, often incorrectly excluding or .

c.i.  


 

Networks, FUR2 2015 VCAA 12 Answer


c.ii.   

 

Networks, STD2 N2 2011 FUR2 2

At the Farnham showgrounds, eleven locations require access to water. These locations are represented by vertices on the network diagram shown below. The dashed lines on the network diagram represent possible water pipe connections between adjacent locations. The numbers on the dashed lines show the minimum length of pipe required to connect these locations in metres.
 

NETWORKS, FUR2 2011 VCAA 2 

 
All locations are to be connected using the smallest total length of water pipe possible.

  1. On the diagram, show where these water pipes will be placed.  (1 mark)
  2. Calculate the total length, in metres, of water pipe that is required.  ( 1 mark) 
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  1.  
    NETWORKS, FUR2 2011 VCAA 2 Answer
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a.


 

NETWORKS, FUR2 2011 VCAA 2 Answer


b.  

Networks, STD2 N2 2008 FUR2 1

James, Dante, Tahlia and Chanel are four children playing a game.

In this children’s game, seven posts are placed in the ground.

The network below shows distances, in metres, between the seven posts.

The aim of the game is to connect the posts with ribbon using the shortest length of ribbon.

This will be a minimal spanning tree.

 

NETWORKS, FUR2 2008 VCAA 11

  1. Draw in a minimal spanning tree for this network on the diagram below.  (1 mark)


NETWORKS, FUR2 2008 VCAA 12

  1. Determine the length, in metres, of this minimal spanning tree.  (1 mark)

  2. How many different minimal spanning trees can be drawn for this network?  (1 mark)

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b.   


 

c.