Networks, SMB-014 MC
Networks, SMB-013 MC
Networks, SMB-012
A network diagram is drawn below.
- Starting at vertex `Z`, identify a trail that uses 6 edges and ends at vertex `V`. (1 mark)
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- Starting at vertex `V`, identify all six different paths that end at vertex `Y`. (2 marks)
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- Is the route identified as `YXUVXY` a circuit? Explain. (1 mark)
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Networks, SMB-011
A network diagram is drawn below.
- Starting at vertex `Q`, identify all five different paths that end at vertex `S`. (2 marks)
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- Starting at vertex `P`, identify four different cycles that exist in the network. (2 marks)
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- Starting at vertex `P`, what is the total number of cycles that exist in the network. (1 mark)
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Networks, SMB-010
- Starting at vertex `A`, identify all the different paths that finish at vertex `D`, using only three edges. (2 marks)
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- Starting at vertex `A`, identify a cycle route? (1 mark)
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- Explain the difference between a cycle and a circuit route. (1 mark)
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Networks, SMB-009
- Starting at vertex `A`, identify three different cycles in the above network. (2 marks)
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- Starting at vertex `A`, 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.
- Write down the names of the towns at the start and at the end of Charlie’s trail. (1 mark)
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- What distance did he travel? (2 marks)
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Networks, SMB-007
The city of Robville contains eight landmarks denoted as vertices `N` to `U` on the network diagram below. The edges on this network represent the roads that link the eight landmarks.
- Write down the degree of vertex `U`. (1 mark)
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- Steven wants to visit each landmark, but drive along each road only once. He will begin his journey at landmark `N`.
- At which landmark must he finish his journey? (1 mark)
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- 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 `A` to `J` on the graph below.
- Which one of the vertices on the graph has degree 4? (1 mark)
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For this graph, an Eulerian trail does not currently exist.
- 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: `P1`, `P2`, `P3`, `P4`, `P5` and `P6`.
The numbers on the edges represent the lengths, in metres, of the roads joining these locations.
- In this graph, what is the degree of the vertex at the entrance to the wildlife park? (1 mark)
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- What is the shortest distance, in metres, from the entrance to picnic area `P3`? (1 mark)
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A park ranger starts at the entrance and drives along every road in the park once.
- At which picnic area will the park ranger finish? (2 marks)
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- 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.
- In kilometres, what is the shortest distance between Farnham and Carrie? (1 mark)
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- How many different paths are there to travel from Farnham to Carrie without passing through any town more than once? (1 mark)
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An engineer plans to inspect all of the roads in this network.
He will start at Dunlop and inspect each road only once.
- At which town will the inspection finish? (2 marks)
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Networks, SMB-003
The following network diagram has a Eulerian trail.
Starting at vertex `D`, describe one Eulerian trail and at what vertex the trail finishes. (2 marks)
Networks, SMB-002
The following network diagram shows the distances, in kilometres, along the roads that connect six intersections `A`, `B`, `C`, `D`, `E` and `F`.
- A cyclist started at intersection `D` 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)
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- 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 `T`, `U`, `V`, `W`, `X`, `Y` and `Z` on the graph below.
The tracks between ramps `U` and `V` and between ramps `W` and `X` are rough, and cannot be used by skateboards.
- Describe a path that a skateboarder at ramp `V` could use to travel to ramp `T` that uses 4 edges only. (1 mark)
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- A skateboarder begins skating at ramp `W` and follows an Eulerian trail.
- 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.
- What is the minimum number of edges needed to connect the five houses? (1 mark)
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- 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-021
Networks, SMB-019
Networks, SMB-018
The network below can be represented as a planar graph.
Redraw the graph as a planar representation of the network, labelling each vertex. (2 marks)
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Networks, SMB-020 MC
Networks, SMB-017
The network below can be represented as a planar graph.
Redraw the graph as a planar representation of the network, labelling each vertex. (2 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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Networks, SMB-015
The network below can be represented as a planar graph.
Draw the planar graph representation of this network, labelling each vertex. (2 marks)
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Networks, SMB-014
Networks, SMB-013
Networks, SMB-012
A connected planar graph has 4 edges and 4 faces.
- Calculate the number of vertices for this graph. (2 marks)
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- Draw the planar graph. (2 marks)
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Networks, SMB-011 MC
A connected planar graph has 10 edges and 10 faces.
The number of vertices for this graph is
- `2`
- `5`
- `8`
- `12`
Networks, SMB-010 MC
A connected planar graph has seven vertices and nine edges.
The number of faces that this graph will have is
- 1
- 2
- 3
- 4
Networks, SMB-009 MC
A planar graph has five faces.
This graph could have
- six vertices and eight edges.
- eight vertices and five edges.
- eight vertices and six edges.
- five vertices and eight edges.
Networks, SMB-008
A planar graph has five vertices and six faces.
Calculate the number of edges in the graph. (2 marks)
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Networks, SMB-007 MC
A connected planar graph has 12 edges.
This graph could have
- 5 vertices and 6 faces.
- 5 vertices and 8 faces.
- 6 vertices and 8 faces.
- 6 vertices and 9 faces.
Networks, SMB-006
Networks, SMB-005
Networks, SMB-004
Networks, SMB-002 MC
In the graph above, the number of vertices of odd degree is
- `0`
- `1`
- `2`
- `3`
Networks, SMB-001 MC
The sum of the degrees of all the vertices in the graph above is
- `6`
- `9`
- `11`
- `12`
Special Properties, SMB-031
Special Properties, SMB-025
A quadrilateral is pictured below.
What is the value of `x`? (3 marks)
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Special Properties, SMB-026
A pentagon is pictured below, where one internal angle is a right angle.
What is the value of `x`? (3 marks)
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Special Properties, SMB-024
A quadrilateral is drawn below.
What is the value of `x`? (3 marks)
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Special Properties, SMB-030
Special Properties, SMB-029
A regular nonagon is pictured below.
What is the value of `x`? (2 marks)
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Special Properties, SMB-028
Special Properties, SMB-027
A regular pentagon is pictured below.
What is the value of `x`? (2 marks)
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Special Properties, SMB-024
A quadrilateral is drawn below.
What is the value of `x`? (3 marks)
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Special Properties, SMB-023
A pentagon is drawn below.
What is the value of `x`? (3 marks)
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Special Properties, SMB-022
A quadrilateral is drawn below.
What is the value of `x`? (2 marks)
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Special Properties, SMB-021
A five sided polygon is drawn below.
What is the value of `x`? (2 marks)
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Congruency, SMB-016
Special Properties, SMB-020 MC
`AB` is the diameter of a circle, centre `O`.
There are 3 triangles drawn in the lower semi-circle and the angles at the centre are all equal to `x^@`.
The three triangles are best described as:
- isosceles
- scalene
- right-angled
- equilateral
Special Properties, SMB-019
Special Properties, SMB-018
Special Properties, SMB-017
Special Properties, SMB-016
Special Properties, SMB-015
Special Properties, SMB-014
Special Properties, SMB-012
Special Properties, SMB-013 MC
Which statement is always true?
- Scalene triangles have two angles that are equal.
- All angles in a parallelogram are equal.
- The opposite sides of a trapezium are equal in length.
- The diagonals of a rhombus are perpendicular to each other.
Special Properties, SMB-011 MC
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