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Networks, SMB-012

A network diagram is drawn below.
 

  1. Starting at vertex , identify a trail that uses 6 edges and ends at vertex .   (1 mark)

  2. Starting at vertex , identify all six different paths that end at vertex .   (2 marks)

  3. Is the route identified as a circuit? Explain.  (1 mark)

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Networks, SMB-011

A network diagram is drawn below.
 

  1. Starting at vertex , identify all five different paths that end at vertex .   (2 marks)

  2. Starting at vertex , identify four different cycles that exist in the network.   (2 marks)

  3. Starting at vertex , what is the total number of cycles that exist in the network.  (1 mark)

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Networks, SMB-010

  1. Starting at vertex , identify all the different paths that finish at vertex , using only three edges.   (2 marks)

  2. Starting at vertex , identify a cycle route?   (1 mark)

  3. Explain the difference between a cycle and a circuit route.   (1 mark)

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Networks, SMB-009

  1. Starting at vertex , identify three different cycles in the above network.   (2 marks)

  2. Starting at vertex , how many different cycles exist in this network?   (1 mark)

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Networks, SMB-008

The diagram below shows a network of train lines between five towns: Attard, Bower, Clement, Derrin and Eden.

The numbers indicate the distances, in kilometres, that are travelled by train between connected towns.
 

 

Charlie followed an Eulerian trail through this network of train lines.

  1. Write down the names of the towns at the start and at the end of Charlie’s trail.  (1 mark)
  2. What distance did he travel?  (2 marks)
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Networks, SMB-007

The city of Robville contains eight landmarks denoted as vertices to on the network diagram below. The edges on this network represent the roads that link the eight landmarks.
 

  

  1. Write down the degree of vertex .  (1 mark)

  2. Steven wants to visit each landmark, but drive along each road only once. He will begin his journey at landmark .
  3. At which landmark must he finish his journey?  (1 mark)

  4. Regardless of which route Steven decides to take, how many of the landmarks (including those at the start and finish) will he see on exactly two occasions?  (2 marks)

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Networks, SMB-006

In one area of the town of Zenith, a postal worker delivers mail to 10 houses labelled as vertices to on the graph below.
 

  1. Which one of the vertices on the graph has degree 4?  (1 mark)

For this graph, an Eulerian trail does not currently exist.

  1. For an Eulerian trail to exist, what is the minimum number of extra edges that the graph would require, giving reasons.  (2 marks)

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Networks, SMB-005

The vertices in the network diagram below show the entrance to a wildlife park and six picnic areas in the park: , , , , and .

The numbers on the edges represent the lengths, in metres, of the roads joining these locations.

 

  1. In this graph, what is the degree of the vertex at the entrance to the wildlife park?  (1 mark)

  2. What is the shortest distance, in metres, from the entrance to picnic area ?  (1 mark)

A park ranger starts at the entrance and drives along every road in the park once.

  1. At which picnic area will the park ranger finish?  (2 marks)

  2. What mathematical term is used to describe the route the park ranger takes?  (1 mark)

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Networks, SMB-004

Aden, Bredon, Carrie, Dunlop, Enwin and Farnham are six towns.

The network shows the road connections and distances between these towns in kilometres.
 

  1. In kilometres, what is the shortest distance between Farnham and Carrie?  (1 mark)

  2. How many different paths are there to travel from Farnham to Carrie without passing through any town more than once?  (1 mark)

An engineer plans to inspect all of the roads in this network.

He will start at Dunlop and inspect each road only once.

  1. At which town will the inspection finish?  (2 marks)

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Networks, SMB-002

The following network diagram shows the distances, in kilometres, along the roads that connect six intersections , , , , and .
 

  1. A cyclist started at intersection and cycled along every road in this network once only. What route would the cyclist take and at which intersection would she finish?  (3 marks)

  2. What is another name for this type of trail?   (1 mark)

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Networks, SMB-001

The network diagram below describes a skateboard park with seven ramps.

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

 

The tracks between ramps and and between ramps and are rough, and cannot be used by skateboards.

  1. Describe a path that a skateboarder at ramp could use to travel to ramp that uses 4 edges only.   (1 mark)

  2. A skateboarder begins skating at ramp and follows an Eulerian trail.
  3. What trail does the skateboarder take?  (2 marks)

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Networks, SMB-023

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.
 

  
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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Networks, SMB-022

The following graph with five vertices is a complete graph.
 

How many edges must be removed so that the graph will have the minimum number of edges to remain connected. Explain your answer.   (3 marks)

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Networks, SMB-016

The network below can be represented as a planar graph.
 

Complete the partial graph drawn below, adding the missing edges so that it is a planar representation of the above network.   (3 marks)
  

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Circle Geometry, SMB-019

The diagram shows a large semicircle with diameter    and two smaller semicircles with diameters    and  , respectively, where    is a point on the diameter  . The point    is the centre of the semicircle with diameter  .

The line perpendicular to    through    meets the largest semicircle at the point  . The points    and    are the intersections of the lines    and    with the smaller semicircles. The point    is the intersection of the lines    and  .
 

Explain why is a rectangle.   (3 marks)

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Circle Geometry, SMB-017

In the diagram, the points , , and are on the circumference of a circle, whose centre lies on . The chord intersects the diameter at . The tangent at passes through the point .

It is given that    and  .

 

 

  1. What is the size of  ?   (1 mark)

  2. What is the size of  ?   (2 marks)

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Networks, STD2 N2 2015 FUR1 5 MC

The graph below represents a friendship network. The vertices represent the four people in the friendship network: Kwan (K), Louise (L), Milly (M) and Narelle (N).

An edge represents the presence of a friendship between a pair of these people. For example, the edge connecting K and L shows that Kwan and Louise are friends.

Which one of the following graphs does not contain the same information.
 
 

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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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NETWORKS, FUR2 2007 VCAA 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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