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NETWORKS, FUR1 2021 VCAA 8 MC

A network of roads connecting towns in an alpine region is shown below.

The distances between neighbouring towns, represented by vertices, are given in kilometres.
 

The region receives a large snowfall, leaving all roads between the towns closed to traffic.

To ensure each town is accessible by car from every other town, some roads will be cleared.

The minimal total length of road, in kilometres, that needs to be cleared is

  1. 361 if  `x` = 50 and  `y` = 55
  2. 361 if  `x` = 50 and  `y` = 60
  3. 366 if  `x` = 55 and  `y` = 55
  4. 366 if  `x` = 55 and  `y` = 60
  5. 371 if  `x` = 55 and  `y` = 65
Show Answers Only

`B`

Show Worked Solution

`text{A partial minimal spanning tree can be drawn:}`
 

`text{Consider each option:}`

♦ Mean mark 48%.

`A:\ text{If} \ x=50 \ text{(include),} \ y = 55 \ text{(include)}`

   `-> \ text{Total length} = 251 + 50 + 55 != 361 \ text{(incorrect)}`

`B:\ text{If} \ x=50 \ text{(include),} \ y = 60 \ text{(include)}`

   `-> \ text{Total length} = 251 + 50 + 55 = 356 \ text{(correct)}`

`text{Similarly, options} \ C, D, E \ text{can be shown to be incorrect.}`

`=> B`

NETWORKS, FUR1 2020 VCAA 5 MC

The network below shows the distances, in metres, between camp sites at a camping ground that has electricity.

The vertices `A` to `I` represent the camp sites.
 


 

The minimum length of cable required to connect all the camp sites is 53 m.

The value of `x`, in metres, is at least

  1.  5
  2.  6
  3.  8
  4.  9
  5. 11
Show Answers Only

`D`

Show Worked Solution

`text(One strategy – Using Prim’s Algorith)`

`text(Starting at)\ A`

`text(1st edge) : AH = 6`

`text(2nd edge) : HG = 5`

`text(then …)\ AB = 7,\ GI = 9,\ IE = 6,\ EF = 5`

`DE = 8,\ CD = 7`
 

`text {Total length = 53 m (not including}\ x text{)}`

`text(If)\ \ x < 9, x\ text(would replace)\ GI\ text(and minimum`

`text(length would be less than 53m.)`

`=>  D`

NETWORKS, FUR1 2016 VCAA 4 MC

The minimum spanning tree for the network below includes the edge with weight labelled `k`.
 

 
The total weight of all edges for the minimum spanning tree is 33.

The value of `k` is

  1. `1`
  2. `2`
  3. `3`
  4. `4`
  5. `5`
Show Answers Only

`E`

Show Worked Solution

`text(Minimum spanning tree)`

`text(Total weight)` `= k + 5 + 5 + 3 + 2 + 4 + 2 + 1 + 6`
`33` `= k + 28`
`:. k` `= 5`

`=> E`

NETWORKS, FUR1 2007 VCAA 7 MC

The minimal spanning tree for the network below includes two edges with weightings `x` and `y.`
 

 
The length of the minimal spanning tree is 19.

The values of `x` and `y` could be

A.   `x = 1 and y = 7`

B.   `x = 2 and y = 5`

C.   `x = 3 and y = 5`

D.   `x = 4 and y = 5`

E.   `x = 5 and y = 6`

Show Answers Only

`C`

Show Worked Solution

`text(Minimal spanning tree is:)`
 

vcaa-networks-fur1-2007-7i

 
`:.\ text(Minimal spanning tree)`

`19` `= y + 3 + x + 2 + 1 + 5`
`8` `= x + y`

 
`=>  C`