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Networks, STD2 N3 2009 FUR2 4

A walkway is to be built across the lake.

Eleven activities must be completed for this building project.

The directed network below shows the activities and their completion times in weeks.
 

NETWORKS, FUR2 2009 VCAA 4
 

  1. What is the earliest start time for activity E?  (1 mark)

  2. Write down the critical path for this project.  (1 mark)

  3. The project supervisor correctly writes down the float time for each activity that can be delayed and makes a list of these times.

     

    Determine the longest float time, in weeks, on the supervisor’s list.  (1 mark)

A twelfth activity, L, with duration three weeks, is to be added without altering the critical path.

Activity L has an earliest start time of four weeks and a latest start time of five weeks.
 

NETWORKS, FUR2 2009 VCAA 4
 

  1. Draw in activity L on the network diagram above.  (1 mark)
  2. Activity L starts, but then takes four weeks longer than originally planned.

     

    Determine the total overall time, in weeks, for the completion of this building project.  (1 mark)

Show Answers Only
  1. `7`
  2. `BDFGIK`
  3. `H\ text(or)\ J\ text(can be delayed for)`
    `text(a maximum of 3 weeks.)`
  4.  
    NETWORKS, FUR2 2009 VCAA 4 Answer
  5. `text(25 weeks)`
Show Worked Solution

a.   `7\ text(weeks)`

♦ Mean mark of all parts (combined): 44%.

 

b.  `text(Scanning forwards and backwards)`
 

 
`text(Critical Path is)\ BDFGIK`

 

c.   `H\ text(or)\ J\ text(can be delayed for a maximum)`

`text(of 3 weeks.)`
 

d.    NETWORKS, FUR2 2009 VCAA 4 Answer

 

e.   `text(The new critical path is)\ BLEGIK.`

`=>\ text(Activity)\ L\ text(now takes 7 weeks.)`

`:.\ text(Time for completion)`

`= 4 + 7 + 1 + 5 + 2 + 6`

`= 25\ text(weeks)`

Networks, STD2 N2 SM-Bank 12

The following table shows the travelling time, in minutes, between towns which are directly connected by roads.

A dash indicates that towns are not directly connected.
 


 

  1. Draw a network diagram showing the information in this table.  (2 marks) 

  2. What is the shortest travelling time between A and E?  (1 mark)

Show Answers Only
  1.  
         
     
  2. `62\ text(minutes)`
Show Worked Solution
i.  

`text(Important to note that network diagrams)`

`text(do not need to be drawn to scale.)`

 

ii.   `text(One strategy – using Dijkstra’s algorithm:)`
    

 
`:.\ text(Shortest travelling time is the path)\ \ A – D – B – E`

`= 10 + 32 + 20`

`= 62\ text(minutes)`

Networks, STD2 N3 SM-Bank 49

The directed graph below shows the sequence of activities required to complete a project.

The time to complete each activity, in hours, is also shown.
 


 

  1. Find the earliest starting time, in hours, for activity `N`.   (2 marks)

To complete the project in minimum time, some activities cannot be delayed.

  1. Calculate the number of activities that cannot be delayed.  (1 mark)
Show Answers Only
  1. `12\ text(hours)`
  2. `4`
Show Worked Solution

i.   `text{Scanning forward:}`
 


 

`:.\ text(EST for)\ N` `= CGJ`
  `= 4 +3+5`
  `= 12`

 

ii.   `text(Critical path is:)\ CFHM`

`:. 4\ text(activities can’t be delayed.)`

Networks, STD2 N3 SM-Bank 17

The network diagram represents a system of roads connecting a shopping centre to the motorway.

Two routes from the shopping centre connect to A and one route connects D to F.

The number on the edge of each road indicates the number of vehicles that can travel on it per hour.
 


 

Draw additional road(s) on the diagram to maximise the capacity. Include the number of vehicles that can travel on each road.  (2 marks)

Show Answers Only

`text(Solutions could be:)`

Show Worked Solution

`text(Two possible solutions are:)`

`text(Note that the added roads above make the minimum cut/max)`

`text(flow increase to 170 vehicles per hour.)`

Networks, STD2 N3 SM-Bank 16 MC

The network diagram represents a system of roads connecting a shopping centre to the motorway.

Two routes from the shopping centre connect to A and one route connects to D to F.

The number on the edge of each road indicates the number of vehicles that can travel on it per hour.

At present, the capacity of the network from the shopping centre to the motorway is not maximised.

Which additional road(s) would increase the network capacity to its maximum?

A. A road from A to F with a capacity of 20 vehicles per hour
B. A road from B to E with a capacity of 30 vehicles per hour
C. A road from C to F with a capacity of 30 vehicles per hour and a road from E to F with a capacity of 60 vehicles per hour
D. A road from B to F with a capacity of 30 vehicles per hour and a road from D to F with a capacity of 30 vehicles per hour
Show Answers Only

`text(D)`

Show Worked Solution

`text(Consider option D:)`

`text(Adding these two roads increases the minimum cut/maximum)`

`text(flow to 170 vehicles per hour throughout the network.)`

`=>\ text(D)`

Networks, STD2 N3 SM-Bank 47

The directed graph below shows the sequence of activities required to complete a project.

All times are in hours.
 


 

  1.  Find the number of activities that have exactly two immediate predecessors. (1 mark)

  2. Identify the critical path for this project. (3 marks)
  3. If Activity E is reduced by one hour, identify the two new critical paths. (1 mark)

Show Answers Only
  1. `2`
  2. `BEIL`
  3. `ADIL and CHKL`
Show Worked Solution

i.    `I\ text(and)\ J`

`:.\ text(2 activities have exactly two immediate predecessors.)`

 

ii.   `text(Scanning forwards:)`
 


 

`:.\ text(Initial critical path)\ BEIL`

 

iii.   `text(If)\ E\ text(reduced by 1 hour, critical path)\ BEIL`

`text(reduces to 19 hours.)`

`:.\ text(Other critical paths of 19 hours are:)`

`ADIL = 5 + 2 + 4 + 8 = 19`

`CHKL = 2 + 6 + 3 + 8 = 19`

Networks, STD2 N3 SM-Bank 48

The network shows the activities that are needed to complete a particular project.
 

networks-fur1-2009-vcaa-5-6-mc
 

  1. Find the total number of activities that need to be completed before activity L can begin.   (1 mark)

The duration of every activity is initially 5 hours. 

  1. If the completion times of both activity F and activity K are reduced to 3 hours each, calculate the effect on the completion time for the project.   (2 marks)

Show Answers Only
  1. `7`
  2. `text(Completion time is unchanged.)`
Show Worked Solution

i.   `A, B, C, D, E, H\ text(and)\ I\ text(must be completed before)\ L.`

`:.\ text(7 activities need to be completed.)`

 

ii.   `text{Scanning forward (all activities take 5 hours):}`
 


 

`text(Activity)\ F\ text(and)\ K\ text(are not on any critical path.)`

`:.\ text(Reducing either will not change the completion time)`

`text(for the project.)`

Networks, STD2 N3 2008 FUR1 8-9 MC

The network below shows the activities that are needed to finish a particular project and their completion times (in days).
 

networks-fur1-2008-vcaa-8-mc

 
Part 1

The earliest start time for Activity K, in days, is

A.     `7`

B.   `15`

C.   `16`

D.   `19`

 

Part 2

This project currently has one critical path.

A second critical path, in addition to the first, would be created by

A.   increasing the completion time of D by 7 days.

B.   increasing the completion time of G by 1 day.

C.   increasing the completion time of I by 2 days.

D.   decreasing the completion time of C by 1 day.

Show Answers Only

`text(Part 1:)\ C`

`text(Part 2:)\ A`

Show Worked Solution

`text(Part 1)`

`text(Scanning forwards:)`
 

 
`text(EST for Activity)\ K`

`=\ text(Duration)\ ACFI`

`= 2 + 5 + 6 + 3`

`= 16`

`=> C`

 

`text(Part 2)`

♦♦ Mean mark of Part 2 was 35%.

`text(Original critical path:)\ ACFHJL\ text{(22 days)}`
 

`text(Consider option)\ A,`

`text(New critical path:)\ ABDJL\ text{(22 days)}`

`=> A`

Networks, STD2 N2 SM-Bank 11

A network of roads between towns shows the travelling times in minutes between towns that are directly connected.
 

 
Complete the shaded cells in the following table so that it represents the information in this network.  (2 marks) 
 

Show Answers Only

Show Worked Solution


 
`text(Note the symmetry in this table across the diagonal.)`

Networks, STD2 N2 SM-Bank 9

 
 

In a separate diagram or on the diagram above, show the minimum spanning tree .  (2 marks)

Show Answers Only

Show Worked Solution

 

`text(One strategy – Using Kruskal’s algorithm:)`

`text(Edges 1 – 5: AB, BC, DG, DH and EI)`

`text(Edges 6 – 8: CD, EF and HI)`

`(text(note AC cannot be chosen → creates a cycle))`

`text(NB: There is more than one minimal spanning tree in this)`

`text{circuit (having a weight of 19).}`

Networks, STD2 N3 2015 FUR1 9 MC

The table below shows, in minutes, the duration, the earliest starting time (EST) and the latest starting time (LST) of eight activities needed to complete a project.
 

NETWORKS, FUR1 2015 VCAA 9 MC

 
Which one of the following directed graphs shows the sequence of these activities?

A.
B.
C.
D.
Show Answers Only

`B`

Show Worked Solution

`text(Consider option)\ B,`

`text(Scanning forwards:)`
  


 

`text(This network matches the table information.)`

`=> B`

Networks, STD2 N3 2011 FUR1 8 MC

The diagram shows the tasks that must be completed in a project.

Also shown are the completion times, in minutes, for each task.
 

 
The critical path for this project includes activities

A.   `B and I.`

B.   `C and H.`

C.   `D and E.`

D.   `F and K.`

Show Answers Only

`D`

Show Worked Solution

`text(Scanning forward:)`
 

`:.\ text(The critical path is)\ \ ACFIK.`

`=>  D`

Networks, FUR2 2013 VCE 3

The rangers at the wildlife park restrict access to the walking tracks through areas where the animals breed.

The edges on the directed network diagram below represent one-way tracks through the breeding areas. The direction of travel on each track is shown by an arrow. The numbers on the edges indicate the maximum number of people who are permitted to walk along each track each day.
 

NETWORKS, FUR2 2013 VCAA 31
 

  1. Starting at `A`, how many people, in total, are permitted to walk to `D` each day?  (1 mark)

One day, all the available walking tracks will be used by students on a school excursion.

The students will start at `A` and walk in four separate groups to `D`.

Students must remain in the same groups throughout the walk.

    1. Group 1 will have 17 students. This is the maximum group size that can walk together from `A` to `D`.

      Write down the path that group 1 will take.  (1 mark)

    2. Groups 2, 3 and 4 will each take different paths from `A` to `D`.

      Complete the six missing entries shaded in the table below.  (2 marks)

       


      NETWORKS, FUR2 2013 VCAA 32

Show Answers Only
  1. `37`
    1. `A  B  E  C  D`
    2.  `text{One possible solution is:}`
      Networks, FUR2 2013 VCAA 3_2 Answer1
Show Worked Solution
a.    `text(Maximum flow)` `=\ text(Minimum cut through)\ CD and ED`
    `= 24 + 13`
    `= 37`
♦ Mean mark of all parts (combined) was 41%.

 
`:.\ text(A maximum of 37 people can walk)`

`text(to)\ D\ text(from)\ A.`

 

b.i.  `A  B  E  C  D`

 

b.ii.   `text(One solution using the second possible largest)`

  `text(group of 11 students and two groups from the)`

  `text(remaining 9 students is:)`
  

Networks, FUR2 2013 VCAA 3_2 Answer1

Networks, STD2 N3 SM-Bank 35

The following directed graph shows the flow of water, in litres per minute, in a system of pipes connecting the source to the sink.
 

 
Calculate the maximum flow, in litres per minute, from the source to the sink.  (2 marks)

Show Answers Only

`18`

Show Worked Solution
`text(Maximum Flow)` `=\ text(Capacity of Minimum Cut)`
  `= 2 + 10 + 6`
  `= 18\ \ text(litres/minute)`

Networks, STD2 N3 SM-Bank 34

The arrows on the diagram below show the direction of the flow of waste through a series of pipelines from a factory to a waste dump.

The numbers along the edges show the number of megalitres of waste per week that can flow through each section of pipeline.
 

NETWORKS, FUR1 2015 VCAA 4 MC

 
The minimum cut is shown as a dotted line.

Calculate the capacity of this cut, in megalitres of waste per week.  (2 marks)

Show Answers Only

`26`

Show Worked Solution

`text(Flows from the waste dump side of the minimum cut to)`

`text(the factory side are ignored.)`
 

`:.\ text{Minimum Cut}`

`= 5 + 2 + 12 + 7`

`= 26\ \ text(ML/week)`

Networks, STD2 N3 2012 FUR1 6 MC

networks-fur1-2012-vcaa-6-mc

 
In the directed network diagram above, all vertices are reachable from every other vertex.

All vertices would still be reachable from every other vertex if we remove the edge in the direction from

A.  `Q` to `U`

B.  `R` to `S`

C.  `S` to `T`

D.  `T` to `R`

Show Answers Only

`A`

Show Worked Solution

`text(Consider option B:)`

`text(If R to S is removed, vertex S cannot be reached from other)`

`text(vertices. All vertices only remain reachable from all other vertices)`

`text(if Q to U is removed.)`

`rArr A`

Networks, STD2 N3 2006 FUR1 6 MC

networks-fur1-2006-vcaa-6-mc

 
In the directed graph above the weight of each edge is non-zero.

The capacity of the cut shown is

A.   a + b + c + d + e

B.   a + c + d + e

C.   a + b + c + e

D.   a + b + c – d + e

Show Answers Only

`C`

Show Worked Solution

`text(Flows in the opposite direction are not counted when)`

`text(calculating the capacity of a cut.)`
 

`:.\ text(Capacity of the cut) = a + b + c + e`

`rArr C`

Networks, STD2 N3 2013 FUR1 9 MC

Alana, Ben, Ebony, Daniel and Caleb are friends. Each friend has a different age.

The arrows in the graph below show the relative ages of some, but not all, of the friends. For example, the arrow in the graph from Alana to Caleb shows that Alana is older than Caleb.
 

 
Using the information in the graph, it can be deduced that the second-oldest person in this group of friends is

A.   Alana

B.   Ben

C.   Caleb

D.   Ebony

Show Answers Only

`B`

Show Worked Solution

`text(Completing the graph, we can deduce that Alana)`

`text(must be older than Daniel, etc…)`
 

vcaa-networks-fur1-2013-9i

 
`:.\ text(Oldest to youngest is:)`

`text(Alana, Ben, Daniel, Caleb, Ebony.)`

`=>  B`

Networks, STD2 N2 SM-Bank 8

 
Highlight the minimal spanning tree of this network on the diagram above, or in a separate diagram.  (2 marks)

Show Answers Only

Show Worked Solution

`text(One Strategy – Using Prim’s algorithm:)`

`text(6 vertices → 5 edges on spanning tree)`

`text(Starting at vertex A)`

`text(Edge 1: AC)`

`text(Edge 2: AD)`

`text{Edge 3: ED  (reject CD which creates a cycle)}`

`text(Edge 4: EF)`

`text(Edge 5: BF)`
 

Networks, STD2 N2 SM-Bank 7 MC

 
How many spanning trees are possible for this network?

A. `3`
B. `8`
C. `14`
D. `30`
Show Answers Only

`text(C)`

Show Worked Solution

`text(S)text(ince there are 6 vertices, each spanning tree will have)`

`text{5 edges (with no cycles).}`

`text(Consider if A and B are not connected, possible spanning)`

`text(trees are:)`

 
`text(3 other spanning trees exist when BF is not connected, and likewise)`

`text(when EF and ED are not connected.)`

`text(Finally, 2 other spanning trees exist when AD and AC are removed, and)`

`text(when AD and CD are removed.)`

`:.\ text(Total spanning trees)` `= 4 xx 3 + 2`
  `= 14`

Networks, STD2 N2 SM-Bank 6

 
Describe the shortest path between A and J in the network above and its weight.  (2 marks)

Show Answers Only

`text(Shortest path is:)`

`text(A – B – E – F – J)`

`text(Weight of shortest path is 15.)`

Show Worked Solution

`text(One Strategy: Using Dijkstra’s algorithm)`
 


 

 `text(Shortest path is:)`

`text(A – B – E – F – J)`

`:.\ text(Weight of shortest path)`

`= 4 + 5 + 4 + 2`

`= 15`

Networks, STD2 N2 SM-Bank 5

A network of roads is pictured below, with the distances of each road represented, in kilometres, on each edge.
 

  
A driver wants to travel from A to H in the shortest distance possible.

Describe the possible paths she can take, and the total distance she must travel.  (2 marks)

Show Answers Only

`text(Shortest distance paths:)`

`text(A – E – H   or   A – F – G – H)`

`text(Shortest distance is 11 km.)`

Show Worked Solution

`text(One Strategy: Dijkstra’s algorithm)`
  


 
`text(Shortest distance paths:)`

`text(A – E – H   or   A – F – G – H)`

`text(Shortest distance is 11 km.)`

Networks, STD2 N2 SM-Bank 4


 

Complete the minimal spanning tree of the network above on the diagram below.  (2 marks)
 

Show Answers Only

Show Worked Solution

`text(One Strategy: Using Prim’s algorithm)`

`text(6 vertices → need 5 edges.)`

`text{Starting at vertex A (any vertex can be chosen)}`

`text(1st Edge: AB)`

`text{2nd Edge: AC (BF also possible)}`

`text(3rd Edge: CD)`

`text(4th Edge: BF)`

`text(5th Edge: AS)`
 

Networks, STD2 N2 SM-Bank 3

The diagram below is a connected network.
 

 
Complete the diagram below to show the minimal spanning tree of this network.  (2 marks)
 

Show Answers Only

`text(or)`

Show Worked Solution

`text(One Strategy – Kruskul’s algorithm)`

`text(There are 6 vertices, so we need 5 edges.)`

`text{Edge 1: BC (least weight) – could have chosen ED}`

`text{Edge 2: DE (next least weight)}`

`text{Edge 3: AF (could have been CD)}`

`text(Edge 4: CD)`

`text{Edge 5: CF or EF (reject CE as it creates a cycle)}`

`text(or)`

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)

Show Answers Only
  1.  
         
     
  2. `$1500`
Show Worked Solution

i.   `text(One Strategy: Using Prim’s Algorithm)`

`text(Starting vertex – Staff Room)`

`text(1st edge: Staff Room – Library)`

`text(2nd edge: Library – School Office)`

`text(3rd edge: Staff Room – IT Staff)`

`text(4th edge: Library – Computer Room)`
  

 

ii.    `text(Minimum Cost)` `= 300 + 300 + 400 + 500`
    `= $1500`

Networks, STD2 N2 SM-Bank 15

Four children, James, Dante, Tahlia and Chanel each live in a different town. 

The following is a map of the roads that link the four towns, `A`, `B`, `C` and `D`.
 

NETWORKS, FUR2 2008 VCAA 21

 
James’ father has begun to draw a network diagram that represents all the routes between the four towns on the map. This is shown below.


NETWORKS, FUR2 2008 VCAA 22

In this network, vertices represent towns and edges represent routes between towns that do not pass through any other town.

One more edge needs to be added to complete this network. Draw in this edge clearly on the diagram above.  (1 mark)

Show Answers Only

networks-fur2-2008-vcaa-2-answer

Show Worked Solution

`text(Two routes exist that join town A directly to town B.)`
 

networks-fur2-2008-vcaa-2-answer

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)

Show Answers Only
  1. `2`
  2. `1250\ text(m)`
     
    NETWORKS, FUR2 2012 VCAA 1 Answer

  3. `text(Minimum spanning tree)`
Show Worked Solution

i.   `2\ text{(the house and the top right vertex)}`

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

ii.  `text(One Strategy – Using Prim’s algorithm:)`

`text(Starting at the house)`

`text(1st edge: 50)`

`text{2nd edge: 40 (either)}`

`text(3rd edge: 40)`

`text(4th edge: 60  etc…)`
 

NETWORKS, FUR2 2012 VCAA 1 Answer


iii.   `text(Minimum spanning tree)`

Networks, FUR2 2016 VCE 1

A map of the roads connecting five suburbs of a city, Alooma (`A`), Beachton (`B`), Campville (`C`), Dovenest (`D`) and Easyside (`E`), is shown below.
 


 

  1. Starting at Beachton, which two suburbs can be driven to using only one road?  (1 mark)

A graph that represents the map of the roads is shown below.
 

 
One of the edges that connects to vertex `E` is missing from the graph.

    1. Add the missing edge to the graph above.  (1 mark)

       

      (Answer on the graph above.)

    2. Explain what the loop at `D` represents in terms of a driver who is departing from Dovenest.  (1 mark)
Show Answers Only
  1. `text(Alooma and Easyside.)`

    2. `text(The loop represents that a driver can take a route out of)`
      `text(Dovenest and return home without going through another)`
      `text(suburb or turning back.)`
Show Worked Solution

a.   `text(Alooma and Easyside.)`

 

b.i.   

`text(Draw a third edge between Easyside and Dovenest.)`

 

b.ii. `text(The loop represents that a driver can take a)`

♦♦ Mean mark 30%.

`text(route out of Dovenest and return home without)`

`text(going through another suburb or turning back.)`

Networks, STD2 N2 2009 FUR2 1

The city of Robville is divided into five suburbs labelled as `A` to `E` on the map below. 

A lake which is situated in the city is shaded on the map.

 

NETWORKS, FUR2 2009 VCAA 11

A table is constructed to represent the number of land borders between suburbs.

If there is no land border between two suburbs, the table records a '0'. If there is a single land border between two suburbs, the table records a '1', and if there are two separate land borders between the same two suburbs, the table records a '2'.
 

`{:({:qquadqquadAquadBquadCquadDquadE:}),({:(A),(B),(C),(D),(E):}[(0,1,1,1,0),(1,0,1,2,0),(1,1,0,0,0),(1,2,0,0,0),(0,0,0,0,0)]):}`

 

  1. Explain why all values in the final row and final column of the table are zero.  (1 mark)

In the network diagram below, vertices represent suburbs and edges represent land borders between suburbs. 

The diagram has been started but is not finished.

 

Networks, FUR2 2009 VCAA 1_1

  1. The network diagram is missing one edge and one vertex. 

     

    On the diagram

    1. draw the missing edge  (1 mark)
    2. draw and label the missing vertex.  (1 mark)

Show Answers Only
  1. `E\ text(has no land borders with other suburbs.)`
  2. i. & ii.
    Networks, FUR2 2009 VCAA 1 Answer
Show Worked Solution

a.   `E\ text(has no land borders with other suburbs.)`

 

b.i. & ii.    Networks, FUR2 2009 VCAA 1 Answer

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) 

Show Answers Only
  1. `4`
  2.  
    networks-fur2-2007-vcaa-1-answer
Show Worked Solution

a.   `text(Minimum number of edges = 4)`

 

b.   `text(One of many possibilities,)`

networks-fur2-2007-vcaa-1-answer

Networks, FUR1 2015 VCE 6 MC

The map below shows all road connections between five towns, `U`, `V`, `W`, `X` and `Y`.
 

NETWORKS, FUR1 2015 VCAA 6 MC1

 
A graph, shown below, was constructed to represent this map.


NETWORKS, FUR1 2015 VCAA 6 MC2

 
A mistake has been made in constructing this graph.

This mistake can be corrected by

A.   drawing another edge between `V` and `W`.

B.   drawing a loop at `W`.

C.   removing the loop at `V`.

D.   removing one edge between `X` and `V`.

Show Answers Only

`A`

Show Worked Solution

`text(There are two routes directly between V and W and therefore)`

`text(a second edge should connect them.)`

`=> A`

Networks, STD2 N2 2014 FUR1 3 MC

The diagram below shows the network of roads that Stephanie can use to travel between home and school.

The numbers on the roads show the time, in minutes, that it takes her to ride a bicycle along each road.

Using this network of roads, the shortest time that it will take Stephanie to ride her bicycle from home to school is 

A.  `12\ text(minutes)`

B.  `13\ text(minutes)`

C.  `14\ text(minutes)`

D.  `15\ text(minutes)`

Show Answers Only

`C`

Show Worked Solution

`text(Using Djikstra’s algorithm:)`

`text(Shortest time riding)`

`=3+2+3+4+2`

`=14\ text(minutes)`

 
`=> C`

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)

Show Answers Only

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 2007 FUR1 4 MC

The following network shows the distances, in kilometres, along a series of roads that connect town `A` to town `B.`
 

 
The shortest distance, in kilometres, to travel from town `A` to town `B` is

A.     `9`

B.   `10`

C.   `11`

D.   `12`

Show Answers Only

`B`

Show Worked Solution

`text(Using Djikstra’s algorithm:)`

 
`text(Shortest path)`

`= 4 + 1 + 3 + 2`

`= 10`
 

`=>  B`

Networks, FUR2 2012 VCE 1

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
 

a.i.  Determine the shortest distance between the house and the pump.  (1 mark)

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

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.

b.i.  On the diagram below, draw the minimum length of pipe that is needed to supply water to all locations on the farm.  (1 mark)

 

   
  NETWORKS, FUR2 2012 VCAA 1
 

b.ii. What is the mathematical term that is used to describe this minimum length of pipe in part b.i.?  (1 mark)

Show Answers Only

a.i.  `160\ text(m)`

a.ii. `2`

b.i.  `1250\ text(m)`

NETWORKS, FUR2 2012 VCAA 1 Answer

b.ii. `text(Minimal spanning tree)`

Show Worked Solution

a.i.   `text(Shortest distance)`

`=70 + 90`

`= 160\ text(m)`

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

 

a.ii.   `2\ text{(the house and the top right vertex)}`

 

b.i.    NETWORKS, FUR2 2012 VCAA 1 Answer

 

b.ii.   `text(Minimal spanning tree)`

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)


 

Show Answers Only

a.   `text{Minimum spanning tree}`

b.    
       

Show Worked Solution

a.   `text{Minimum spanning tree.}`

 

b.   `text(Using Kruskal’s Algorithm)`

`text(Edge 1: 50)`

`text(Edges 2–3: 100)`

`text{Edge 4: 150 (ignore the 150 edge that creates a circuit)}`

`text{Edges 5–6: 200  (ignore the 200 edge that creates a circuit)} `

`text(Edge 7: 300)`

`text(Edge 8: 400)`
 
        

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, `J`, `K`, `L`, `M`, `N`, `O` and `P`, 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 `L` and server `M`?  (1 mark)

  2. What is the cheapest cost, in dollars, of installing cables between server `K` and server `N`?  (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 `P` 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

Show Answers Only
  1. `$300`
  2. `$920`
  3. `N\ text(and)\ P\ text{(or}\ P\ text(and)\ N)`
    1.  
      Networks, FUR2 2015 VCAA 12 Answer
    2. `$120`
Show Worked Solution

a.   `$300`

 

b.   `text(Minimum cost of)\ K\ text(to)\ N`

`= 440 + 480`

`= $920`
 

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

c.i.  `text(Using Prim’s Algorithm:)`

`text(Starting at Vertex)\ L`

`text{1st Edge: L → M (300)}`

`text{2nd Edge: L → K (360)}`

`text{3rd Edge: K → J (250)}`

`text{4th Edge: J → P (200)  etc…}`
 

Networks, FUR2 2015 VCAA 12 Answer


c.ii.   `text(Disconnect)\ J – P\ text(and)\ O – P`

`text(Savings) = 200 + 400 = $600`

`text(Add in)\ M – N`

`text(C)text(ost) = $480`

`:.\ text(Net savings)` `= 600 – 480`
  `= $120`

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) 
Show Answers Only
  1.  
    NETWORKS, FUR2 2011 VCAA 2 Answer
  2. `510\ text(metres)`
Show Worked Solution

a. `text(Using Prim’s Algorithm)`

`text(Starting at bottom right vertex)`

`text{1st Edge: 50}`

`text{2nd Edge: 40`

`text(3rd Edge: 50)`

`text(4th Edge: 40   etc…)`
 

NETWORKS, FUR2 2011 VCAA 2 Answer


b.   `text(Total length of water pipe)`

`= 50+40+50+40+50+60+40+60+60+60`

`= 510\ text(metres)`

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)

Show Answers Only
  1.  

    `text(or)`
  2. `16\ text(metres)`
  3. `2`
Show Worked Solution

a.  `text(Using Kruskal’s Algorithm)`

`text{Edges 1-3: 2}`

`text{Edges 4-5: 3  (2 edges with weight 3 create a circuit and are ignored)`

`text(Edge 6: 4)` 
 

`text(or)`

 

b.   `text(Length of minimal spanning tree)`

`= 2+2+2+3+3+4`

`= 16\ text(metres)`
 

c.   `2`

Networks, STD2 N2 2007 FUR1 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`

Show Answers Only

`C`

Show Worked Solution

`text{Using Kruskal’s Algorithm (as a guide),}`

`text(Minimum spanning tree must be:)`
 


 

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

 
`=>  C`

Networks, STD2 N2 2006 FUR1 4 MC

networks-fur1-2006-vcaa-4-mc-1

 
The minimal spanning tree for the network above will include the edge that has a weight of

A.     `3`

B.     `6`

C.     `8`

D.     `9`

Show Answers Only

`D`

Show Worked Solution

`text(Using Kruskal’s Algorithm:)`

`text{Edge 1: 1  (minimum weight)}`

`text(Edge 2: 2)`

`text{Edge 3: 4  (edge weight 3 creates a circuit and is ignored)}`

`text(Edge 4: 5   etc…)`
 

`:.\ text(Minimum spanning tree is:)`

 
`rArr D`

Networks, FUR1 2009 VCE 1 MC

Consider the following graph.
 

networks-fur1-2009-vcaa-1-mc

 
The smallest number of edges that need to be added to make this a connected graph is

A.   `1`

B.   `2`

C.   `3`

D.   `4`

Show Answers Only

`C`

Show Worked Solution

`text(A connected graph is one where any two vertices can be)`

`text(connected by a path.)`

`=>  C`

Networks, STD2 N2 SM-Bank 05 MC

A simple connected graph has 3 edges has 4 vertices.

This graph must be

A.   a cycle.

B.   a tree.

C.   a graph that contains a loop.

D.   a graph that contains a circuit.

Show Answers Only

`=> B`

Show Worked Solution

`text(Consider the graph below,)`

 

networks-fur1-2008-vcaa-4-mc-answer

`=> B`

Calculus, MET1 SM-Bank 35

 

The diagram shows two parallel brick walls  `KJ`  and  `MN`  joined by a fence from  `J`  to  `M`.  The wall  `KJ`  is  `s`  metres long and  `/_KJM=alpha`.  The fence  `JM`  is  `l`  metres long.

A new fence is to be built from  `K`  to a point  `P`  somewhere on  `MN`.  The new fence  `KP`  will cross the original fence  `JM`  at  `O`.

Let  `OJ=x`  metres, where  `0<x<l`.

  1. Show that the total area,  `A`  square metres, enclosed by  `DeltaOKJ`  and  `DeltaOMP`  is given by
  2.    `A=s(x-l+l^2/(2x))sin alpha`.  (3 marks)
  3. Find the value of  `x`  that makes  `A`  as small as possible. Justify the fact that this value of  `x`  gives the minimum value for  `A`.  (3 marks)
  4. Hence, find the length of  `MP`  when  `A`  is as small as possible.    (1 mark)
Show Answers Only
  1. `text{Proof  (See Worked Solutions)}`
  2. `l/sqrt2`
  3. `(sqrt2-1)s\ \ text(metres)`
Show Worked Solution
i.

`A=text(Area)\  Delta OJK+text(Area)\ Delta OMP`

♦♦♦ Although mean mark data for question parts is not available before 2009, students found this question extremely challenging.

`text(Using sine rule)`

`text(Area)\ Delta OJK=1/2\ x s sin alpha` 

`text(Area)\ DeltaOMP =>text(Need to find)\ \ MP`

`/_OKJ` `=/_MPO\ \ text{(alternate angles,}\ MP\ text(||)\ KJtext{)}`
`/_PMO` `=/_OJK=alpha\ \ text{(alternate angles,}\ MP\ text(||)\ KJtext{)}`

`:.\ DeltaOJK\ text(|||)\ Delta OMP\ \ text{(equiangular)}`

`=>x/s` `=(l-x)/(MP)\ ` ` text{(corresponding sides of similar triangles)}`
`MP` `=(l-x)/x *s`  
`text(Area)\ Delta\ OMP` `=1/2 (l-x)* MP * sin alpha`
  `=1/2 (l-x)*((l-x))/x* s  sin alpha`
`:. A`  `=1/2 x*s sin alpha+1/2 (l-x)*((l-x))/x* s sin alpha`
  `=s sin alpha(1/2 x+1/2 (l-x)*((l-x))/x)`
  `=s sin alpha(1/2 x+(l-x)^2/(2x))`
  `=s sin alpha(1/2 x+l^2/(2x)-l+1/2 x)`
  `=s(x-l+l^2/(2x))sin alpha\ \ \ \ text(… as required)`

 

ii.   `text(Find)\ x\ text(such that)\ A\ text(is a minimum)`

MARKER’S COMMENT: Students who could not complete part (i) are reminded that they can still proceed to part (ii) and attempt to differentiate the result given.
Note that `l` and `alpha` are constants when differentiating. 
`A` `=s(x-l+l^2/(2x))sin alpha`
`(dA)/(dx)` `=s(1-l^2/(2x^2))sin alpha`

`text(MAX/MIN when)\ (dA)/(dx)=0`

`s(1-l^2/(2x^2))sin alpha` `=0`
`l^2/(2x^2)` `=1`
`2x^2` `=l^2`
`x^2` `=l^2/2`
`x` `=l/sqrt2,\ \ \ x>0`

 

`(d^2A)/(dx^2)=s((l^2)/(2x^3))sin alpha`

`text(S)text(ince)\ \ 0<alpha<90°\ \ =>\ sin alpha>0,\ \ l>0\ \ text(and)\ \  x>0`

`(d^2A)/(dx^2)>0\ \ \ =>text(MIN at)\ \ x=l/sqrt2`

 

iii.   `text(S)text(ince)\ \ MP=((l-x))/x s\ \ text(and MIN when)\ \ x=l/sqrt2`

`MP` `=((l-l/sqrt2)/(l/sqrt2))s xx sqrt2/sqrt2`
  `=((sqrt2 l-l))/l s`
  `=(sqrt2-1)s\ \ text(metres)`

 

`:.\ MP=(sqrt2-1)s\ \ text(metres when)\ A\ text(is a MIN.)`

Calculus, MET1 SM-Bank 34

2005 8a

A cylinder of radius  `x`  and height  `2h`  is to be inscribed in a sphere of radius  `R`  centred at  `O`  as shown.

  1. Show that the volume of the cylinder is given by
  2.    `V = 2pih(R^2 − h^2).` (1 mark)
  3. Hence, or otherwise, show that the cylinder has a maximum volume when
  4.    `h = R/sqrt3.`  (3 marks)
Show Answers Only
  1. `text{Proof (See Worked Solutions).}`
  2. `text{Proof (See Worked Solutions).}`
Show Worked Solution
i.    `V` `= pir^2h\ \ \ \ text{(general form)}`
    `= pix^2 xx 2h`

 

`text(Using Pythagoras:)`

`R^2` `= h^2 + x^2`
`x^2` `= R^2 − h^2`
`:.V` `= 2pih(R^2 − h^2)\ \ …text(as required.)` 

 

ii. `V` `= 2pih(R^2 − h^2)`
    `= 2piR^2h − 2pih^3`
  `(dV)/(dh)` `= 2piR^2 − 3 xx 2pih^2`
    `= 2piR^2 − 6pih^2`
  `(d^2V)/(dh^2)` `= −12pih`

 
`text(Max or min when)\ (dV)/(dh) = 0`

`2piR^2 − 6pih^2` `= 0`
`6pih^2` `= 2piR^2`
`h^2` `= R^2/3`
`h` `= R/sqrt3,\ \ h > 0`

 
`text(When)\ h = R/sqrt3`

`(d^2V)/(dh^2) = −12pi xx R/sqrt3 < 0`

`:.\ text(Volume is a maximum when)\ h = R/sqrt3\ text(units.)`

Calculus, MET1 SM-Bank 30

A function is given by  `f(x) = 3x^4 + 4x^3 - 12x^2`.

  1. Find the coordinates of the stationary points of  `f(x)`  and determine their nature.   (3 marks)
  2. Hence, sketch the graph  `y = f(x)`   showing the stationary points.   (2 marks)
  3. For what values of  `x`  is the function increasing?   (1 mark)
  4. For what values of  `k`  will  `f(x) = 3x^4 + 4x^3 - 12x^2 + k = 0`  have no solution?   (1 mark)
Show Answers Only
  1. `text(MAX at)\ (0,0)`
  2. `text(MIN at)\ text{(1,–5)}`
  3. `text(MIN at)\ text{(–2,–32)}`
  4. 2UA HSC 2012 14ai
     
  5. `f(x)\ text(is increasing for)\ -2 < x < 0\ text(and)\ x > 1`
  6. `text(No solution when)\ k > 32`
Show Worked Solution
i. `f(x)` `= 3x^4 + 4x^3 -12x^2`
  `f prime (x)` `= 12x^3 + 12x^2 – 24x`
  `f″(x)` `= 36x^2 + 24x – 24`

 

`text(Stationary points when)\ f prime (x) = 0`

`=> 12x^3 + 12x^2 – 24x` `=0`
`12x(x^2 + x – 2)` `=0`
`12x (x+2) (x -1)` `=0`

 

`:.\ text(Stationary points at)\ x=0,\ 1\ text(or)\ –2`

`text(When)\ x=0,\ \ \ \ f(0)=0`
`f″(0)` `= -24 < 0`
`:.\ text{MAX at  (0,0)}`

 

`text(When)\ x=1`

`f(1)` `= 3+4\ – 12 = -5`
`f″(1)` `= 36 + 24\ – 24 = 36 > 0`
`:.\ text{MIN at  (1,–5)}`

 

`text(When)\ x=–2`

`f(–2)` `=3(–2)^4 + 4(–2)^3 – 12(–2)^2`
  `= 48 -32\ – 48`
  `= -32`
`f″(–2)` `= 36(–2)^2 + 24(–2) -24`
  `=144 – 48 – 24 = 72 > 0`
`:.\ text{MIN at  (–2,–32)}`

 

ii.  2UA HSC 2012 14ai

 

♦ Mean mark (HSC) 42%
MARKER’S COMMENT: Be careful to use the correct inequality signs, and not carelessly include `>=` or `<=` by mistake.

 

iii. `f(x)\ text(is increasing for)`
  `-2 < x < 0\ text(and)\ x > 1`

 

iv.   `text(Find)\ k\ text(such that)`

♦♦♦ Mean mark (HSC) 12%.

`3x^4 + 4x^3 – 12x^2 + k = 0\ \ text(has no solution)`

`k\ text(is the vertical shift of)\ \ y = 3x^4 + 4x^3 – 12x^2`

`=>\ text(No solution if it does not cross the)\ x text(-axis.)`

`:.\ text(No solution when)\ \ k > 32`

Calculus, MET1 2014 ADV 13a

  1. Differentiate  `3 + sin 2x`.    (1 mark)
  2. Hence, or otherwise, find  `int (cos2x)/(3 + sin 2x)\ dx`.    (2 marks)
Show Answers Only
  1. `2 cos 2x`
  2. `ln (3 + sin 2x)^(1/2) + C`
Show Worked Solution
a. `y` `= 3 + sin 2x`
  `dy/dx` `= 2 cos 2x`

 

b. `int (cos 2x)/(3 + sin 2x)\ dx`
  `= 1/2 int (2 cos 2x)/(3 + sin 2x)\ dx`
  `= 1/2  ln (3 + sin 2x) + C\ \ \ \ \ text{(from part (i))}`

Calculus, MET1 2010 ADV 2e

Given that `int_0^6 ( x + k ) \ dx = 30`, and `k` is a constant, find the value of `k`.   (2 marks)

Show Answers Only

`k = 2`

Show Worked Solution
`int_0^6 ( x + k ) \ dx` `= 30`
`int_0^6 ( x + k ) \ dx` `= [ 1/2\ x^2 + kx ]_0^6`
`30` `= [(1/2xx 6^2 + 6 xx k ) – 0 ]`
`30` `= 18 + 6k`
`6k` `= 12`
`:.  k` `= 2`

Calculus, MET1 2009 ADV 2biii

Evaluate  `int_1^4 x^2 + sqrtx\ dx`.   (3 marks)

 

 

Show Answers Only

`77/3`

 

Show Worked Solution

`int_1^4 x^2 + sqrtx\ \ dx`

`= int_1^4 (x^2 + x^(1/2))\ dx`

`= [1/3 x^3 + 1/(3/2) x^(3/2)]_1^4`

`= [(x^3)/3 + 2/3x^(3/2)]_1^4`

`= [((4^3)/3 + 2/3 xx 4^(3/2))\ – (1/3 + 2/3)]`

`= [(64/3 + 16/3) – 3/3]`

`= [80/3 – 3/3]`

`= 77/3`

Calculus, MET1 2011 ADV 4d

  1. Differentiate  `y=sqrt(9-x^2)`  with respect to  `x`.   (2 marks)
  2. Hence, or otherwise, find  `int (6x)/sqrt(9-x^2)\ dx`.    (2 marks)
Show Answers Only
  1. `- x/sqrt(9-x^2)`
  2. `-6 sqrt(9-x^2) + c`
Show Worked Solution
IMPORTANT: Some students might find calculations easier by rewriting the equation as `y=(9-x^2)^(1/2)`.
a.    `y` `= sqrt(9-x^2)`
    `= (9-x^2)^(1/2)`

 
`text{Using the function of a function rule (or “chain” rule)}`

`dy/dx` `=1/2 xx (9-x^2)^(-1/2) xx d/dx (9-x^2)`
  `= 1/2 xx (9-x^2)^(-1/2) xx -2x`
  `=-x/sqrt(9-x^2)`

 

b.    `int (6x)/sqrt(9-x^2)\ dx` `= -6 int (-x)/sqrt(9-x^2)\ dx`
    `= -6 sqrt(9-x^2) + c`

Calculus, MET1 2011 VCAA 1b

If  `g(x) = x^2 sin (2x)`,  find  `g prime (pi/6).`  (2 marks)

Show Answers Only

`(sqrt 3 pi)/6 + pi^2/36`

Show Worked Solution

`g(x) = x^2 sin (2x)`

MARKER’S COMMENT: Incorrect determination of exact trig values lost many students marks here.

`text(Using Product Rule:)`

`(fh)′` `= f′ h + fh′`
`g prime (x)` `= 2 x sin (2x) + 2x^2 cos (2x)`
   
`:. g prime (pi/6)` `= 2 (pi/6) sin (pi/3) + 2 (pi/6)^2 cos (pi/3)`
  `= pi/3 xx sqrt 3/2 + pi^2/18 xx 1/2`
  `= (sqrt 3 pi)/6 + pi^2/36`

Calculus, MET1 2015 VCAA 1b

Let  `f(x) = (log_e(x))/(x^2)`.

  1. Find  `f′(x)`.  (2 marks)
  2. Evaluate  `f′(1)`.  (1 mark)
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  1. `(1 – 2log_e(x))/(x^3)`
  2. `1`
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i.   `text(Using Quotient Rule:)`

`(h/g)′` `= (h′g – hg′)/(g^2)`
`f′(x)` `= ((1/x)x^2 – log_e(x)*2x)/(x^4)`
  `= (1 – 2log_e(x))/(x^3)`

 

ii.    `f′(1)` `= (1 – 2log_e(1))/(1^3)`
    `= 1`

Calculus, MET1 2012 VCAA 1b

If  `f(x) = x/(sin(x))`,  find  `f prime (pi/2).`  (2 marks)

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`1`

Show Worked Solution

`text(Using Quotient Rule:)`

`(h/g)′` `= (h′ g – h g′)/g^2`
`f prime (x)` `= (1 xx sin (x) – x cos (x))/(sin x)^2`
`:. f prime (pi/2)` `= (sin (pi/2) – pi/2 xx cos (pi/2))/(sin(pi/2))^2`
  `= (1 – 0)/1^2`
  `= 1`