Band 4 – Page 66

NETWORKS, FUR2 2008 VCAA 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.

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

Determine the length, in metres, of this minimal spanning tree. (1 mark)
How many different minimal spanning trees can be drawn for this network? (1 mark)

Show Answers Only

`text(or)`
`16\ text(metres)`
`2`

Show Worked Solution

`text(or)`

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

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

`= 16\ text(metres)`

c. `2`

NETWORKS, FUR2 2009 VCAA 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.

What is the earliest start time for activity `E`? (1 mark)
Write down the critical path for this project. (1 mark)
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.

Draw in activity `L` on the network diagram above. (1 mark)
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

`7`
`BDFGIK`
`H\ text(or)\ J\ text(can be delayed for)`
`text(a maximum of 3 weeks.)`
`text(25 weeks)`

Show Worked Solution

a. `7\ text(weeks)`

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

b. `BDFGIK`

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

`text(of 3 weeks.)`

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

`L\ text(now takes 7 weeks.)`

`:.\ text(Time for completion)`

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

`= 25\ text(weeks)`

NETWORKS, FUR2 2009 VCAA 3

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)
Steven wants to visit each landmark, but drive along each road only once. He will begin his journey at landmark `N`.
1. At which landmark must he finish his journey? (1 mark)
2. 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? (1 mark)
Cathy decides to visit each landmark only once.
1. Suppose she starts at `S`, then visits `R` and finishes at `T`.
  
  Write down the order Cathy will visit the landmarks. (1 mark)
2. Suppose Cathy starts at `S`, then visits `R` but does not finish at `T`.
  
  List three different ways that she can visit the landmarks. (1 mark)

Show Answers Only

`4`
1. `P\ text{(the other odd degree vertex)}`
2. `5`
1. `(SR)QPONU(T)`
2. `text(Other paths are)`
  `(SR)QPUTNO`
  `(SR)QPONTU`
  `(SR)TUNOPQ`
  `(SR)UTNOPQ`
  `text{(only 3 paths required)}`

Show Worked Solution

a. `4`

b.i. `P\ text{(the other odd degree vertex)}`

b.ii. `5 (N, T, R, P, U)`

c.i. `(SR)QPONU(T)`

c.ii. `text(Other paths are)`

`(SR)QPUTNO`

`(SR)QPONTU`

`(SR)TUNOPQ`

`(SR)UTNOPQ`

`text{(only 3 paths required)}`

NETWORKS, FUR2 2009 VCAA 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.

An adjacency matrix is constructed to represent the number of land borders between suburbs.

`{:({: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)]):}`

Explain why all values in the final row and final column 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.

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

`E\ text(has no land borders with other suburbs.)`
i. & ii.

Show Worked Solution

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

b.i. & ii.

NETWORKS, FUR2 2010 VCAA 4

In the final challenge, each of four teams has to complete a construction project that involves activities `A` to `I`.

Table 1 shows the earliest start time (EST), latest start time (LST) and duration, in minutes, for each activity.

The immediate predecessor is also shown. The earliest start time for activity `F` is missing.

What is the least number of activities that must be completed before activity `F` can commence? (1 mark)
What is the earliest start time for activity `F`? (1 mark)
Write down all the activities that must be completed before activity `G` can commence. (1 mark)
What is the float time, in minutes, for activity `G`? (1 mark)
What is the shortest time, in minutes, in which this construction project can be completed? (1 mark)
Write down the critical path for this network. (1 mark)

Show Answers Only

`2`
`9\ text(minutes)`
`A\ text(and)\ C`
`4\ text(minutes)`
`16\ text(minutes)`
`A-B-D-H`

Show Worked Solution

a. `2`

b.	`text(EST for)\ F`	`= 5 + 4`
		`= 9\ text(minutes)`

c. `A\ text(and)\ C`

d.	`text(Float time for)\ G`	`= 13 – 9`
		`= 4\ text(minutes)`

e. `text(Shortest construction time)`

`= 5 + 6 + 2 + 3`

`= 16\ text(minutes)`

f. `A-B-D-H`

NETWORKS, FUR2 2010 VCAA 3

The following network diagram shows the distances, in kilometres, along the roads that connect six intersections `A`, `B`, `C`, `D`, `E` and `F`.

If a cyclist started at intersection `B` and cycled along every road in this network once only, at which intersection would she finish? (1 mark)
The next challenge involves cycling along every road in this network at least once.

Teams have to start and finish at intersection `A`.

The blue team does this and cycles the shortest possible total distance.
1. Apart from intersection `A`, through which intersections does the blue team pass more than once? (1 mark)
2. How many kilometres does the blue team cycle? (1 mark)
The red team does not follow the rules and cycles along a bush path that connects two of the intersections.

This route allows the red team to ride along every road only once.

Which two intersections does the bush path connect? (1 mark)

Show Answers Only

`D`
1. `B, D\ text(and)\ F`
2. `32\ text(km)`
`B\ text(and)\ D`

Show Worked Solution

a. `D`

b.i. `text(Consider the path)`

`AFDEFBCDFBA`

`:.\ text(Passes through)\ B, D,\ text(and)\ F\ text(more)`

`text(than once.)`

b.ii. `text{Total distance (Blue)}`

`= 6 + 3 + 3 + 4 + 2 + 4 + 2 + 3 + 2 + 3`

`= 32\ text(km)`

c. `text(An Euler circuit can only exist when the)`

`text(degree of all vertices is even.)`

`:.\ text(The bush track joins)\ B\ text(and)\ D.`

NETWORKS, FUR2 2011 VCAA 4

Stormwater enters a network of pipes at either Dunlop North (Source 1) or Dunlop South (Source 2) and flows into the ocean at either Outlet 1 or Outlet 2.

On the network diagram below, the pipes are represented by straight lines with arrows that indicate the direction of the flow of water. Water cannot flow through a pipe in the opposite direction.

The numbers next to the arrows represent the maximum rate, in kilolitres per minute, at which stormwater can flow through each pipe.

Complete the following sentence for this network of pipes by writing either the number 1 or 2 in each box. (1 mark)

Determine the maximum rate, in kilolitres per minute, that water can flow from these pipes into the ocean at Outlet 1 and Outlet 2. (2 marks)

A length of pipe, show in bold on the network diagram below, has been damaged and will be replaced with a larger pipe.

The new pipe must enable the greatest possible rate of flow of stormwater into the ocean from Outlet 2.

What minimum rate of flow through the pipe, in kilolitres per minute, will achieve this? (1 mark)

Show Answers Only

`text(Storm water from Source 2 cannot reach Outlet 1)`
`text(Outlet 1: 700 kL/min)`
`text(Outlet 2: 700 kL/min)`
`text(300 kL per min)`

Show Worked Solution

a. `text(Storm water from Source 2 cannot reach Outlet 1)`

♦ Mean mark of all parts (combined) was 35%.

`text(The minimum cut includes the 200 kL/min pipe from Source 1.)`

`:.\ text(Maximum rates are)`

`text(Outlet 1: 700 kL/min)`

`text(Outlet 2: 700 kL/min)`

c. `text(The next smallest cut in the lower pipe system is 800.)`

`:.\ text(The minimum flow through the new pipe that will achieve)`

`text(this is 300 kL/min.)`

NETWORKS, FUR2 2011 VCAA 3

A section of the Farnham showgrounds has flooded due to a broken water pipe. The public will be stopped from entering the flooded area until repairs are made and the area has been cleaned up.

The table below shows the nine activities that need to be completed in order to repair the water pipe. Also shown are some of the durations, Earliest Start Times (EST) and the immediate predecessors for the activities.

What is the duration of activity `B`? (1 mark)
What is the Earliest Start Time (EST) for activity `D`? (1 mark)
Once the water has been turned off (Activity `B`), which of the activities `C` to `I` could be delayed without affecting the shortest time to complete all activities? (1 mark)

It is more complicated to replace the broken water pipe (Activity `E`) than expected. It will now take four hours to complete instead of two hours.

Determine the shortest time in which activities `A` to `I` can now be completed. (1 mark)

Turning on the water to the showgrounds (Activity `H`) will also take more time than originally expected. It will now take five hours to complete instead of one hour.

With the increased duration of Activity `H` and Activity `E`, determine the shortest time in which activities `A` to `I` can be completed. (1 mark)

Show Answers Only

`text(2 hours)`
`text(3 hours)`
`text(Activities)\ F and H`
`13\ text(hours)`
`14\ text(hours)`

Show Worked Solution

a.	`text(Duration of)\ B`	`= text(EST of)\ C`
		`= 2\ text(hours)`

♦ Mean mark of all parts (combined) was 42%.

b. `text(EST of)\ C = 3\ text(hours)`

c. `text(Activities)\ F and H`

MARKER’S COMMENT: Many students incorrectly included `G` in this answer (note that `G` is not on the critical path).

d. `text(Shortest time)\ (A\ text(to)\ I)`

`= 2 + 1 + 1 + 4 + 4 + 1`

`= 13\ text(hours)`

e. `text(New shortest time)`

`= 2 + 1 + 1 + 4 + 5 + 1`

`= 14\ text(hours)`

NETWORKS, FUR2 2011 VCAA 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.

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

On the diagram, show where these water pipes will be placed. (1 mark)
Calculate the total length, in metres, of water pipe that is required. ( 1 mark)

Show Answers Only

`510\ text(metres)`

Show Worked Solution

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

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

`+ 40 + 50 + 50`

`= 510\ text(metres)`

NETWORKS, FUR2 2011 VCAA 1

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)
How many different ways 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.

At which town will the inspection finish? (1 mark)

Another engineer decides to start and finish her road inspection at Dunlop.

If an assistant inspects two of the roads, this engineer can inspect the remaining six roads and visit each of the other five towns only once.

How many kilometres of road will the assistant need to inspect? (1 mark)

Show Answers Only

`200\ text(km)`
`6`
`text(Bredon)`
`240\ text(km)`

Show Worked Solution

a. `text{Farnham to Carrie (shortest)}`

`= 60 + 140`

`= 200\ text(km)`

b. `text(Different paths are)`

`FDC, FEDC, FEBC,`

`FEABC, FDEBC,`

`FDEABC`

`:. 6\ text(different ways)`

c. `text(A possible path is)\ DFEABCDEB\ text(and will finish)`

`text{at Bredon (the other odd-degree vertex).}`

d. `text(If the engineer’s path is)`

`DFEABCD,`

`text(Distance assistant inspects)`

`= 110 + 130`

`= 240\ text(km)`

Calculus, MET1 2006 VCAA 11

Part of the graph of the function `f: R-> R, f(x) = -x^2 + ax + 12` is shown below. If the shaded area is 45 square units, find the values of `a, m` and `n` where `m` and `n` are the `x`-axis intercepts of the graph of `y = f(x).` (5 marks)

Show Answers Only

`m = 6;\ \ \ n = – 2`

Show Worked Solution

`text(Shaded Area) = 45`

`45`	`= int_0^3 (– x^2 + ax + 12)\ dx`
`45`	`= [– 1/3x^3 + 1/2 ax^2 + 12x]_0^3`
`45`	`= – 9 + 9/2a + 36`
`9/2 a`	`= 18`
`:. a`	`= 36/9`
	`= 4`

`f(x) = – x^2 + 4x + 12`

`text(Factorise to find)\ x text(-intercepts:)`

`- (x^2 – 4x – 12)`	`=0`
`(x – 6) (x + 2)`	`=0`

`:. m = 6, and n = – 2`

NETWORKS, FUR2 2014 VCAA 4

To restore a vintage train, 13 activities need to be completed.

The network below shows these 13 activities and their completion times in hours.

Determine the earliest starting time of activity `F`. (1 mark)

The minimum time in which all 13 activities can be completed is 21 hours.

What is the latest starting time of activity `L`? (1 mark)
What is the float time of activity `J`? (1 mark)

Just before they started restoring the train, the members of the club needed to add another activity, `X`, to the project.

Activity `X` will take seven hours to complete.

Activity `X` has no predecessors, but must be completed before activity `G` starts.

What is the latest starting time of activity `X` if it is not to increase the minimum completion time of the project? (1 mark)

Activity `A` can be crashed by up to four hours at an additional cost of $90 per our.

This may reduce the minimum completion time for the project, including activity `X`.

Determine the least cost of crashing activity `A` to give the greatest reduction in the minimum completion time of the project. (1 mark)

Show Answers Only

`text(7 hours)`
`text(18 hours)`
`text(2 hours)`
`text(4 hours)`
`$270`

Show Worked Solution

a. `5 + 2 = 7\ text(hours)`

♦ Mean mark for all parts (combined) was 42%.

b. `text(Latest starting time of)\ L`

`= text(Length of critical path – duration of)\ L`

`= 21 – 3`

`= 18\ text(hours)`

c. `text(Float time of)\ J`

`=\ text(LST – EST)`

`= 13 – 11`

`= 2\ text(hours)`

d. `X\ text(precedes)\ G`

`text(EST of)\ G = 11`

`:. text(LST of)\ X = 11`

`text(EST)\ text(of)\ X`

`= text(LST of)\ X – text(duration of)\ X`

`= 11 – 7`

`= 4\ text(hours)`

e. `text(Longer paths are)`

`A – C – G – K = 21\ text{hours (critical path)}`

`A – D – E – H – K = 20\ text(hours)`

`A – D – F – J – M = 19\ text(hours)`

`A – D – E – I – M = 18\ text(hours)`

`B – E – H – K = 18\ text(hours)`

`B – F – J – M = 17\ text(hours)`

`:.\ text(Reduce)\ \ A – C – G – K\ \ text(by 3 hours to get)`

`text{to 18 hours (equals}\ \ B – E – H – K)`

`:.\ text(Least cost)`	`= 3 xx 90`
	`= $270`

NETWORKS, FUR2 2014 VCAA 3

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 path through this network of train lines.

1. Write down the names of the towns at the start and at the end of Charlie’s path. (1 mark)
2. What distance did he travel? (1 mark)

Brianna will follow a Hamiltonian path from Bower to Attard.

What is the shortest distance that she can travel? (1 mark)

The train line between Derrin and Eden will be removed. If one other train line is removed from the network, Andrew would be able to follow an Eulerian circuit through the network of train lines.

Which other train line should be removed?

In the boxes below, write down the pair of towns that this train line connects. (1 mark)

Show Answers Only

1. `text(Bower, Eden)`
2. `910\ text(km)`
`270\ text(km)`
`text(Bower and Derrin)`

Show Worked Solution

a.i. `text(Bower, Eden or Eden, Bower)`

a.ii. `text(Distance)\ \ BDABCDEACE`

`= 160 + 130 + 80 + 70 + 60 + 40 + 100 + 150 + 120`

`= 910\ text(km)`

b. `text(Shortest Hamiltonian path is)\ BCDEA`

`text(Distance)`	`= 70 + 60 + 40 + 100`
	`= 270\ text(km)`

c. `text(Remove the line between Bower and Derrin.)`

NETWORKS, FUR2 2014 VCAA 2

Planning a train club open day involves four tasks.

Table 1 shows the number of hours that each club member would take to complete these tasks.

The Hungarian algorithm will be used to allocate the tasks to club members so that the total time taken to complete the tasks is minimised.

The first step of the Hungarian algorithm is to subtract the smallest element in each row of Table 1 from each of the elements in that row.

The result of this step is shown in Table 2 below.

Complete Table 2 by filling in the missing numbers for Andrew. (1 mark)

After completing Table 2, Andrew decided that an allocation of tasks to minimise the total time taken was not yet possible using the Hungarian algorithm.

Explain why Andrew made this decision. (1 mark)

Table 3 shows the final result of all steps of the Hungarian algorithm.

1. Which task should be allocated to Andrew? (1 mark)
2. How many hours in total are used to plan for the open day? (1 mark)

Show Answers Only

`text(The minimum number of lines to cover all zeros)`
`text(is less than four.)`
1. `text(Equipment)`
2. `36`

Show Worked Solution

b. `text(The minimum number of lines to cover all zeros)`

MARKER’S COMMENT: An answer of “there were not enough zeros” did not gain a mark.

`text(is less than four.)`

c.i. `text(Equipment)`

c.ii. `text(Total hours to plan)`

`= 8 +10 +10 + 8`

`= 36`

NETWORKS, FUR2 2015 VCAA 3

Nine activities are needed to prepare a daily delivery of groceries from the factory to the towns.

The duration, in minutes, earliest starting time (EST) and immediate predecessors for these activities are shown in the table below.

The directed network that shows these activities is shown below.

All nine of these activities can be completed in a minimum time of 26 minutes.

What is the EST of activity `D`? (1 mark)
What is the latest starting time (LST) of activity `D`? (1 mark)
Given that the EST of activity `I` is 22 minutes, what is the duration of activity `H`? (1 mark)
Write down, in order, the activities on the critical path. (1 mark)
Activities `C` and `D` can only be completed by either Cassie or Donna.

One Monday, Donna is sick and both activities `C` and `D` must be completed by Cassie. Cassie must complete one of these activities before starting the other.

What is the least effect of this on the usual minimum preparation time for the delivery of groceries from the factory to the five towns? (1 mark)
Every Friday, a special delivery to the five towns includes fresh seafood. This causes a slight change to activity `G`, which then cannot start until activity `F`
has been completed.
1. On the directed graph below, show this change without duplicating any activity. (1 mark)
2. What effect does the inclusion of seafood on Fridays have on the usual minimum preparation time for deliveries from the factory to the five towns? (1 mark)

Show Answers Only

`text(3 minutes)`
`text(4 minutes)`
`text(3 minutes)`
`A – C – F – H – I`
`text(The critical path is increased by)`
`text(7 minutes to 33 minutes.)`
2. `text(The critical path is increased by)`
  `text(2 minutes to 28 minutes.)`

Show Worked Solution

a. `text(3 minutes)`

b. `text(4 minutes)`

c. `text(3 minutes)`

d. `A – C – F – H – I`

e. `text(The critical path is increased by 7 minutes)`

`text(to 33 minutes.)`

f.i.

f.ii. `text(The critical path is increased by 2 minutes)`

`text(to 28 minutes.)`

NETWORKS, FUR2 2015 VCAA 2

The factory supplies groceries to stores in five towns, `Q`, `R`, `S`, `T` and `U`, represented by vertices on the graph below.

The edges of the graph represent roads that connect the towns and the factory.

The numbers on the edges indicate the distance, in kilometres, along the roads.

Vehicles may only travel along the road between towns `S` and `Q` in the direction of the arrow due to temporary roadworks.

Each day, a van must deliver groceries from the factory to the five towns.

The first delivery must be to town `T`, after which the van will continue on to the other four towns before returning to the factory.

1. The shortest possible circuit from the factory for this delivery run, starting with town `T`, is not Hamiltonian.
  
  Complete the order in which these deliveries would follow this shortest possible circuit. (1 mark)
  
  factory – `T` – ___________________________ – factory
2. With reference to the town names in your answer to part (a)(i), explain why this shortest circuit is not a Hamiltonian circuit. (1 mark)
Determine the length, in kilometres, of a delivery run that follows a Hamiltonian circuit from the factory to these stores if the first delivery is to town `T`. (1 mark)

Show Answers Only

1. `text(factory)\ −T − S −Q − R − S −U − text(factory)`
2. `text(The van passes through town)\ S\ text(twice.)`
`162\ text(km)`

Show Worked Solution

a.i. `text(factory)\ −T − S −Q − R − S −U − text(factory)`

a.ii. `text(The van passes through town)\ S\ text(twice.)`

b. `text(Hamiltonian circuit is)`

`text(factory)\ − T − S − R −Q −U − text(factory)`

`= 44 + 38 + 12 + 8 + 38 + 22`

`= 162\ text(km)`

NETWORKS, FUR2 2015 VCAA 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.

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.

What is the cost, in dollars, of installing the cable between server `L` and server `M`? (1 mark)
What is the cheapest cost, in dollars, of installing cables between server `K` and server `N`? (1 mark)
An inspector checks the cables by walking along the length of each cable in one continuous path.

To avoid walking along any of the cables more than once, at which vertex should the inspector start and where would the inspector finish? (1 mark)
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)
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)

Show Answers Only

`$300`
`$920`
`N\ text(and)\ P\ text{(or}\ P\ text(and)\ N)`
2. `$120`

Show Worked Solution

a. `$300`

b. `text(C)text(ost of)\ K\ text(to)\ N`

`= 440 + 480`

`= $920`

c. `N\ text(and)\ P\ text{(or}\ P\ text(and)\ N)`

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

d.i.

d.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`

MATRICES, FUR2 2006 VCAA 3

Market researchers claim that the ideal number of bookshops (`x`), sports shoe shops (`y`) and music stores (`z`) for a shopping centre can be determined by solving the equations

`2x + y + z = 12`

`x-y+z=1`

`2y-z=6`

Write the equations in matrix form using the following template. (1 mark)

`qquad[(qquadqquadqquad),(),()][(quad),(quad),(quad)] = [(quad),(quad),(quad)]`
Do the equations have a unique solution? Provide an explanation to justify your response. (1 mark)
Write down an inverse matrix that can be used to solve these equations. (1 mark)
Solve the equations and hence write down the estimated ideal number of bookshops, sports shoe shops and music stores for a shopping centre. (1 mark)

Show Answers Only

`[(2,1,1),(1,-1,1),(0,2,-1)][(x),(y),(z)] = [(12),(1),(6)]`
`text(Yes. See worked solutions.)`
`[(2,1,1),(1,-1,1),(0,2,-1)]^(-1) = [(-1,3,2),(1,-2,-1),(2,-4,-3)]`
`text(3 bookshops, 4 sports shoe shops, 2 music stores.)`

Show Worked Solution

a.	`[(2,1,1),(1,-1,1),(0,2,-1)][(x),(y),(z)] = [(12),(1),(6)]`

♦ Mean mark 35% for all parts (combined).

b.	`text(det)\ [(2,1,1),(1,-1,1),(0,2,-1)] = 1 != 0`

`:.\ text(A unique solution exists.)`

c. `text(By CAS,)`

`[(2,1,1),(1,-1,1),(0,2,-1)]^(-1) = [(-1,3,2),(1,-2,-1),(2,-4,-3)]`

d.	`[(x),(y),(z)]`	`= [(-1,3,2),(1,-2,-1),(2,-4,-3)][(12),(1),(6)]`
		`= [(3),(4),(2)]`

`:.\ text(Estimated ideal numbers are:)`

`text(3 bookshops)`

`text(4 shoe shops)`

`text(2 music stores)`

MATRICES, FUR2 2006 VCAA 2

A new shopping centre called Shopper Heaven (`S`) is about to open. It will compete for customers with Eastown (`E`) and Noxland (`N`).

Market research suggests that each shopping centre will have a regular customer base but attract and lose customers on a weekly basis as follows.

80% of Shopper Heaven customers will return to Shopper Heaven next week
12% of Shopper Heaven customers will shop at Eastown next week
8% of Shopper Heaven customers will shop at Noxland next week

76% of Eastown customers will return to Eastown next week
9% of Eastown customers will shop at Shopper Heaven next week
15% of Eastown customers will shop at Noxland next week

85% of Noxland customers will return to Noxland next week
10% of Noxland customers will shop at Shopper Heaven next week
5% of Noxland customers will shop at Eastown next week

Enter this information into transition matrix `T` as indicated below (express percentages as proportions, for example write 76% as 0.76). (2 marks)

`qquad{:(qquadqquadqquadtext(this week)),((qquadqquadqquadS,E,N)),(T = [(qquadqquadqquadqquad),(),()]{:(S),(E),(N):}{:qquadtext(next week):}):}`

During the week that Shopper Heaven opened, it had 300 000 customers.

In the same week, Eastown had 120 000 customers and Noxland had 180 000 customers.

Write this information in the form of a column matrix, `K_0`, as indicated below. (1 mark)

`qquadK_0 = [(qquadqquadqquad),(),()]{:(S),(E),(N):}`
Use `T` and `K_0` to write and evaluate a matrix product that determines the number of customers expected at each of the shopping centres during the following week. (2 marks)
Show by calculating at least two appropriate state matrices that, in the long term, the number of customers expected at each centre each week is given by the matrix (2 marks)

`qquadK = [(194\ 983),(150\ 513),(254\ 504)]`

Show Answers Only

`{:((qquadqquadqquad\ S,qquadE,qquadN)),(T = [(0.8,0.09,0.10),(0.12,0.76,0.05),(0.08,0.15,0.85)]{:(S),(E),(N):}):}`
`K_0 = [(300\ 000),(120\ 000),(180\ 000)]{:(S),(E),(N):}`
`TK_0 = [(268\ 800),(136\ 200),(195\ 000)]`
`text(See Worked Solutions)`

Show Worked Solution

a.	`{:((qquadqquadqquad\ S,qquadE,qquadN)),(T = [(0.8,0.09,0.10),(0.12,0.76,0.05),(0.08,0.15,0.85)]{:(S),(E),(N):}):}`

b.	`K_0 = [(300\ 000),(120\ 000),(180\ 000)]{:(S),(E),(N):}`

c. `text(Customers expected at each centre the next week,)`

`TK_0`	`= [(0.80,0.09,0.10),(0.12,0.76,0.05),(0.08,0.15,0.85)][(300\ 000),(120\ 000),(180\ 000)]`
	`= [(268\ 800),(136\ 200),(195\ 000)]`

d. `text(Consider)\ \ T^nK_0\ \ text(when)\ n\ text(large),`

`text(say)\ n=50, 51`

`T^50K_0`	`= [(0.8,0.09,0.10),(0.12,0.76,0.05),(0.08,0.15,0.85)]^50[(300\ 000),(120\ 000),(180\ 000)]`
	`= [(194\ 983),(150\ 513),(254\ 504)]`

`T^51K_0`	`= [(0.8,0.09,0.10),(0.12,0.76,0.05),(0.08,0.15,0.85)]^51[(300\ 000),(120\ 000),(180\ 000)]`
	`= [(194\ 983),(150\ 513),(254\ 504)]`
	` = T^50K_0`

MATRICES, FUR2 2006 VCAA 1

A manufacturer sells three products, `A`, `B` and `C`, through outlets at two shopping centres, Eastown (`E`) and Noxland (`N`).

The number of units of each product sold per month through each shop is given by the matrix `Q`, where

`{:((qquadqquadqquad\ A,qquadquadB,qquad\ C)),(Q=[(2500,3400,1890),(1765,4588,2456)]{:(E),(N):}):}`

Write down the order of matrix `Q`. (1 mark)

The matrix `P`, shown below, gives the selling price, in dollars, of products `A`, `B`, `C`.

`P = [(14.50),(21.60),(19.20)]{:(A),(B),(C):}`

1. Evaluate the matrix `M`, where `M = QP`. (1 mark)
2. What information does the elements of matrix `M` provide? (1 mark)
Explain why the matrix `PQ` is not defined. (1 mark)

Show Answers Only

`2 xx 3`
1. `M = QP = [(135\ 320.5),(171\ 848.5)]`
2. `text(The total of selling products)\ A, B,and C`
  `text(at each of Eastown and Noxland.)`
`PQ\ text(is not defined because the number of)`
`text(columns in)\ P !=\ text(the number of rows in)\ Q.`

Show Worked Solution

a. `2 xx 3`

b.i.	`M`	`= QP`
		`= [(2500,3400,1890),(1765,4588,2456)][(14.50),(21.60),(19.20)]`
		`= [(135\ 320.5),(171\ 848.5)]`

b.ii. `text(The total revenue from selling products)\ A, B,`

`text(and)\ C\ text(at each of Eastown and Noxland.)`

c. `PQ\ text(is not defined because the number of)`

`text(columns in)\ P !=\ text(the number of rows in)\ Q.`

MATRICES, FUR2 2007 VCAA 2

To study the life-and-death cycle of an insect population, a number of insect eggs (`E`), juvenile insects (`J`) and adult insects (`A`) are placed in a closed environment.

The initial state of this population can be described by the column matrix

`S_0 = [(400),(200),(100),(0)]{:(E),(J),(A),(D):}`

A row has been included in the state matrix to allow for insects and eggs that die (`D`).

What is the total number of insects in the population (including eggs) at the beginning of the study? (1 mark)

In this population

- eggs may die, or they may live and grow into juveniles
- juveniles may die, or they may live and grow into adults
- adults will live a period of time but they will eventually die.

In this population, the adult insects have been sterilised so that no new eggs are produced. In these circumstances, the life-and-death cycle of the insects can be modelled by the transition matrix

`{:(qquadqquadqquadqquadquadtext(this week)),((qquadqquadqquadE,quad\ J,quadA,\ D)),(T = [(0.4,0,0,0),(0.5,0.4,0,0),(0,0.5,0.8,0),(0.1,0.1,0.2,1)]{:(E),(J),(A),(D):}):}`

What proportion of eggs turn into juveniles each week? (1 mark)
1. Evaluate the matrix product `S_1 = TS_0` (1 mark)
2. Write down the number of live juveniles in the population after one week. (1 mark)
3. Determine the number of live juveniles in the population after four weeks. Write your answer correct to the nearest whole number. (1 mark)
4. After a number of weeks there will be no live eggs (less than one) left in the population.
  When does this first occur? (1 mark)
5. Write down the exact steady-state matrix for this population. (1 mark)
If the study is repeated with unsterilised adult insects, eggs will be laid and potentially grow into adults.

Assuming 30% of adults lay eggs each week, the population matrix after one week, `S_1`, is now given by

`qquad S_1 = TS_0 + BS_0`

where `B = [(0,0,0.3,0),(0,0,0,0),(0,0,0,0),(0,0,0,0)]` and `S_0 = [(400),(200),(100),(0)]{:(E),(J),(A),(D):}`
1. Determine `S_1` (1 mark)
  
  This pattern continues. The population matrix after `n` weeks, `S_n`, is given by
  
  `qquad qquad qquad S_n = TS_(n - 1) + BS_(n - 1)`
2. Determine the number of live eggs in this insect population after two weeks. (1 mark)

Show Answers Only

`700`
`50text(%)`

1. `[(160),(280),(180),(80)]{:(E),(J),(A),(D):}`
2. `280`
3. `56`
4. `text(7th week)`
5. `[(0),(0),(0),(700)]`
1. `[(190),(280),(180),(80)]`
2. `130`

Show Worked Solution

a. `400 + 200 + 100 + 0 = 700`

b. `50text(%)`

c.i.	`S_1`	` = TS_0`
		`= [(0.4,0,0,0),(0.5,0.4,0,0),(0,0.5,0.8,0),(0.1,0.1,0.2,1)][(400),(200),(100),(0)]`
		`= [(160),(280),(180),(80)]{:(E),(J),(A),(D):}`

c.ii. `280`

c.iii.	`S_4`	` = T^4S_0`
		`= [(10.24),(56.32),(312.96),(320.48)]{:(E),(J),(A),(D):}\ \ \ text{(by graphics calculator)}`

`:. 56\ text(juveniles still alive after 4 weeks.)`

c.iv.	`text(Each week, only 40% of eggs remain.)`
	`text(Find)\ \ n\ \ text(such that)`

`400 xx 0.4^n`	`< 1`
`0.4^n`	`<1/400`
`n`	`> 6.5`

`:.\ text(After 7 weeks, no live eggs remain.)`

c.v. `text(Consider)\ \ n\ \ text{large (say}\ \ n = 100 text{)},`

`[(0.4, 0, 0, 0), (0.5, 0.4, 0, 0), (0, 0.5, 0.8, 0), (0.1, 0.1, 0.2, 1)]^100 [(400), (200), (100), (0)] ~~ [(0), (0), (0), (700)]`

d.i.	`S_1`	`= TS_0 + BS_0`
		`= [(160),(280),(180),(80)] + [(0,0,0.3,0),(0,0,0,0),(0,0,0,0),(0,0,0,0)][(400),(200),(100),(0)]`
		`= [(190),(280),(180),(80)]`

♦♦ Mean mark for part (d) was 30%.

d.ii.	`S_2`	`= TS_1 + BS_1`
		`= [(130), (207), (284), (163)]`

`:.\ text(There are 130 live egss after 2 weeks.)`

MATRICES, FUR2 2007 VCAA 1

The table below displays the energy content and amounts of fat, carbohydrate and protein contained in a serve of four foods: bread, margarine, peanut butter and honey.

Write down a 2 x 3 matrix that displays the fat, carbohydrate and protein content (in columns) of bread and margarine. (1 mark)
`A` and `B` are two matrices defined as follows.

`A = [(2,2,1,1)]` `B = [(531),(41),(534),(212)]`
1. Evaluate the matrix product `AB`. (1 mark)
2. Determine the order of matrix product `BA`. (1 mark)
  
  Matrix `A` displays the number of servings of the four foods: bread, margarine, peanut butter and honey, needed to make a peanut butter and honey sandwich.
  
  Matrix `B` displays the energy content per serving of the four foods: bread, margarine, peanut butter and honey.
3. Explain the information that the matrix product `AB` provides. (1 mark)
The number of serves of bread (`b`), margarine (`m`), peanut butter (`p`) and honey (`h`) that contain, in total, 53 grams of fat, 101.5 grams of carbohydrate, 28.5 grams of protein and 3568 kilojoules of energy can be determined by solving the matrix equation

`[(1.2,6.7,10.7,0),(20.1,0.4,3.5,12.5),(4.2,0.6,4.6,0.1),(531,41,534,212)][(b),(m),(p),(h)] = [(53),(101.5),(28.5),(3568)]`

Solve the matrix equation to find the values `b`, `m`, `p` and `h`. (2 marks)

Show Answers Only

`[(1.2,20.1,4.2),(6.7,0.4,0.6)]`
1. `[1890]`
2. `underset (4 xx 1) B xx underset (1 xx 4) A = underset (4 xx 4) (BA)`
3. `BA\ text(provides the total energy)`
  `text(content of the servings of these)`
  `text(four foods in one sandwich.)`
`b = 4, m = 4, p = 2, h = 1`

Show Worked Solution

a.	`[(1.2,20.1,4.2),(6.7,0.4,0.6)]`

b.i.	`AB`	`= [(2, 2, 1, 1)] [(531), (41), (534), (212)]`
		`= [1890]`

b.ii. `underset (4 xx 1) B xx underset (1 xx 4) A = underset (4 xx 4) (BA)`

b.iii.	`BA\ text(provides the total energy content of the)`
	`text(servings of these four foods in one sandwich.)`

c.	`[(b),(m),(p),(h)]`	`= [(1.2,6.7,10.7,0),(20.1,0.4,3.5,12.5),(4.2,0.6,4.6,0.1),(531,41,534,212)]^(-1)[(53),(101.5),(28.5),(3568)]`
		`= [(4),(4),(2),(1)]\ \ \ text{(by graphics calculator)}`

`:. b = 4, m = 4, p = 2\ text(and)\ h = 1.`

MATRICES, FUR2 2009 VCAA 4

A series of extra rehearsals commenced in April. Each week participants could choose extra dancing rehearsals or extra singing rehearsals.

A matrix equation used to determine the number of students expected to attend these extra rehearsals is given by

`L_(n + 1) = [(0.85,0.25),(0.15,0.75)] xx L_n - [(5),(7)]`

where `L_n` is the column matrix that lists the number of students attending in week `n`.

The attendance matrix for the first week of extra rehearsals is given by

`L_1 = [(95),(97)]{:(text(dancing)),(text(singing)):}`

Calculate the number of students who are expected to attend the extra singing rehearsals in week 3. (1 mark)
Of the students who attended extra rehearsals in week 3, how many are not expected to return for any extra rehearsals in week 4? (1 mark)

Show Answers Only

`text(68 students)`
`12`

Show Worked Solution

a. `text(Using matrix equation,)`

`L_2`	`= T xx L_1 – [(5),(7)]`
	`= [(0.85,0.25),(0.15,0.75)][(95),(97)] – [(5),(7)]`
	`= [(100),(80)]`

`L_3`	`= [(0.85,0.25),(0.15,0.75)][(100),(80)] – [(5),(7)]`
	`= [(100),(68)]`

`:.\ text(68 students are expected to attend singing)`

`text(in week 3.)`

b.	`L_4`	`= [(0.85,0.25),(0.15,0.75)][(100),(68)] – [(5),(7)]`
		`= [(97),(59)]`

`:.\ text(Students expected not to return)`

`= (100 + 68) – (97 + 59)`

`= 12`

MATRICES, FUR2 2009 VCAA 3

In 2009, the school entered a Rock Eisteddfod competition.

When rehearsals commenced in February, all students were asked whether they thought the school would make the state finals. The students’ responses, ‘yes’, ‘no’ or ‘undecided’ are shown in the initial state matrix `S_0`.

`S_0 = [(160),(120),(220)]{:(text(yes)),(text(no)),(text(undecided)):}`

How many students attend this school? (1 mark)

Each week some students are expected to change their responses. The changes in their responses from one week to the next are modelled by the transition matrix `T` shown below.

`{:(qquadqquadqquadtext( response this week)),(qquadqquadquadtext( yes no undecided)),(T = [(0.85quad,0.35quad,0.60),(0.10quad,0.40quad,0.30),(0.05quad,0.25quad,0.10)]{:(text(yes)),(text(no)),(text(undecided)):}qquad{:(text(response)),(text(next week)):}):}`

The following diagram can also be used to display the information represented in the transition matrix `T`.

1. Complete the diagram above by writing the missing percentage in the shaded box. (1 mark)
2. Of the students who respond ‘yes’ one week, what percentage are expected to respond ‘undecided’ the next week when asked whether they think the school will make the state finals? (1 mark)
3. In total, how many students are not expected to have changed their response at the end of the first week? (2 marks)
Evaluate the product `S_1 = TS_0`, where `S_1` is the state matrix at the end of the first week. (1 mark)
How many students are expected to respond ‘yes’ at the end of the third week when asked whether they think the school will make the state finals? (1 mark)

Show Answers Only

`500`
1. `text(25%)`
2. `text(5%)`
3. `206`
`S_1 = [(310),(130),(60)]`
`361`

Show Worked Solution

a. `text(Total students attending)`

`= 160 + 120 + 220`

`= 500`

b.i. `text(25%)`

b.ii. `text(5%)`

b.iii. `text(Students not expected to change)`

`= 0.85 xx 160 + 0.4 xx 120 + 0.1 xx 220`

`= 206`

c.	`S_1`	`=TS_0`
		`= [(0.85,0.35,0.60),(0.10,0.40,0.30),(0.05,0.25,0.10)][(160),(120),(220)]`
		`= [(310),(130),(60)]`

d.	`S_3`	`= T^3 S_0`
		`= [(0.85,0.35,0.60),(0.10,0.40,0.30),(0.05,0.25,0.10)]^3[(160),(120),(220)]`
		`= [(361),(91.1),(47.9)]`

`:. 361\ text(students expected to respond “yes” at end of week 3.)`

MATRICES, FUR2 2009 VCAA 2

Tickets for the function are sold at the school office, the function hall and online.

Different prices are charged for students, teachers and parents.

Table 1 shows the number of tickets sold at each place and the total value of sales.

For this function

- student tickets cost `$x`
- teacher tickets cost `$y`
- parent tickets cost `$z`.

Use the information in Table 1 to complete the following matrix equation by inserting the missing values in the shaded boxes. (1 mark)
Use the matrix equation to find the cost of a teacher ticket to the school function. (2 marks)

Show Answers Only

`text(35 and 2)`
`$32`

Show Worked Solution

a. `text(35 and 2)`

MARKER’S COMMENT: Simply writing the column matrix in part (b) did not earn full marks. Students must extract the required data.

b.	`[(x),(y),(z)]`	`= [(283,28,5),(35,4,2),(84,3,7)]^(-1)[(8712),(1143),(2609)]`
		`= [(27),(32),(35)]`

`:.\ text(C)text(ost of a teacher ticket = $32)`

MATRICES, FUR2 2010 VCAA 4

The Dinosaurs (`D`) and the Scorpions (`S`) are two basketball teams that play in different leagues in the same city.

The matrix `A_1` is the attendance matrix for the first game. This matrix shows the number of people who attended the first Dinosaur game and the number of people who attended the first Scorpion game.

`A_1 = [(2000),(1000)]{:(D),(S):}`

The number of people expected to attend the second game for each team can be determined using the matrix equation

`A_2 = GA_1`

where `G` is the matrix `{:(qquadqquadqquadtext(this game)),((qquadqquadqquadD,qquad\ S)),(G = [(1.2,-0.3),(0.2,0.7)]{:(D),(S):}qquad{:text(next game):}):}`

1. Determine `A_2`, the attendance matrix for the second game. (1 mark)
2. Every person who attends either the second Dinosaur game or the second Scorpion game will be given a free cap. How many caps, in total, are expected to be given away? (1 mark)

Assume that the attendance matrices for successive games can be determined as follows.

`A_3 = GA_2`

`A_4 = GA_3`

and so on such that `A_(n + 1) = GA_n`

Determine the attendance matrix (with the elements written correct to the nearest whole number) for game 10. (1 mark)
Describe the way in which the number of people attending the Dinosaurs’ games is expected to change over the next 80 or so games. (1 mark)

The attendance at the first Dinosaur game was 2000 people and the attendance at the first Scorpion game was 1000 people.

Suppose, instead, that 2000 people attend the first Dinosaur game, and 1800 people attend the first Scorpion game.

Describe the way in which the number of people attending the Dinosaurs’ games is expected to change over the next 80 or so games. (1 mark)

Show Answers Only

1. `A_2 = [(2100),(1100)]`
2. `3200\ text(people)`
`A_10 = [(2613),(1613)]`
`text(Attendance at the Dinosaur’s games increases gradually)`

`text(to 3000, at which level it remains steady.)`
`text(Attendence at the Dinosaur’s games decreases)`

`text(gradually to 600, where it remains steady.)`

Show Worked Solution

a.i.	`A_2`	`= GA_1`
		`= [(1.2,-0.3),(0.2,0.7)][(2000),(1000)]`
		`= [(2100),(1100)]`

a.ii. `text(Total attending second games)`

`= 2100 + 1100`

`= 3200\ text(people)`

b.	`A_10`	`= GA_9`
		`= G^9A_1`
		`= [(1.2,-0.3),(0.2,0.7)]^9[(2000),(1000)]`
		`= [(2613),(1613)]`

c.	`A_80`	`= G^79A_1`
		`= [(3000),(2000)]`

`A_81`	`= G^80A_1`
	`= [(3000),(2000)]`

`:.\ text(Attendance at the Dinosaur’s games increases)`

`text(gradually to 3000, at which level it remains)`

`text(steady.)`

d. `text(Using the new initial attendences,)`

`A_80 = [(1.2,-0.3),(0.2,0.7)]^79[(2000),(1800)] = [(600),(400)]`

`A_81 = [(1.2,-0.3),(0.2,0.7)]^80[(2000),(1800)] = [(600),(400)]`

`:.\ text(Attendence at the Dinosaur’s games decreases)`

`text(gradually to 600, where it remains steady.)`

MATRICES, FUR2 2010 VCAA 3

The basketball coach has written three linear equations which can be used to predict the number of points, `p`, rebounds, `r`, and assists, `a`, that Oscar will have in his next game.

The equations are `p + r + a`	`= 33`
`2p - r + 3a`	`= 40`
`p + 2r + a`	`= 43`

These equations can be written equivalently in matrix form.

Complete the missing information below. (1 mark)

`[(qquadqquad),(qquadqquad),(qquadqquad)][(p),(r),(a)] = [(33),(40),(43)]`

This matrix equation can be solved in the following way.

`[(p),(r),(a)] = [(7,-1,-4),(-1,0,1),(x,1,3)][(33),(40),(43)]`

Determine the value of `x` shown in the matrix equation above. (1 mark)
How many rebounds is Oscar predicted to have in his next game? (1 mark)

Show Answers Only

`[(1,1,1),(2,-1,3),(1,2,1)][(p),(r),(a)] = [(33),(40),(43)]`
`-5`
`10`

Show Worked Solution

a.	`[(1,1,1),(2,-1,3),(1,2,1)][(p),(r),(a)] = [(33),(40),(43)]`

b.	`[(7,-1,-4),(-1,0,1),(x,1,3)]`	`=[(1,1,1),(2,-1,3),(1,2,1)]^(-1)`
		`= [(7,-1,-4),(-1,0,1),(-5,1,3)]`

`:.x = -5`

c.	`[(p),(r),(a)]`	`= [(7,-1,-4),(-1,0,1),(-5,1,3)][(33),(40),(43)]`
		`= [(19),(10),(4)]`

`:.\ text(Oscar is predicted to have 10 rebounds in the next game.)`

MATRICES, FUR2 2010 VCAA 2

The 300 players in Oscar’s league are involved in a training program. In week one, 90 players are doing heavy training (`H`), 150 players are doing moderate training (`M`) and 60 players are doing light training (`L`). The state matrix, `S_1`, shows the number of players who are undertaking each type of training in the first week

`S_1 = [(90),(150),(60)]{:(H),(M),(L):}`

The percentage of players that remain in the same training program, or change their training program from week to week, is shown in the transition diagram below.

What information does the 20% in the diagram above provide? (1 mark)

The information in the transition diagram above can also be written as the transition matrix `T`.

`{:(qquadqquadqquadquad\ text(this week)),((qquadqquadqquadH,quadM,\ L)),(T = [(0.5,0.1,0.1),(0.2,0.6,0.5),(0.3,0.3,0.4)]{:(H),(M),(L):}qquad{:text(next week):}):}`

Determine how many players will be doing heavy training in week two. (1 mark)
Determine how many fewer players will be doing moderate training in week three than in week one. (1 mark)
Show that, after seven weeks, the number of players (correct to the nearest whole number) who are involved in each type of training will not change. (1 mark)

Show Answers Only

`text(It means that 20% of the players doing heavy training one)`

`text(week will switch to moderate training the next.)`
`text(66 players)`
`text(6 fewer players)`
`text(See Worked Solutions)`

Show Worked Solution

a. `text(It means that 20% of the players doing heavy)`

`text(training one week will switch to moderate)`

`text(training the next.)`

b.	`S_2`	`= TS_1`
		`= [(0.5,0.1,0.1),(0.2,0.6,0.5),(0.3,0.3,0.4)][(90),(150),(60)]`
		`= [(66),(138),(96)]`

`:. 66\ text(players will be in hard training)`

`text(in week 2.)`

c. `text(150 in moderate training in week 1.)`

`text(In week 3,)`

`S_3`	`= T^2S_1`
	`= [(0.5,0.1,0.1),(0.2,0.6,0.5),(0.3,0.3,0.4)]^2[(90),(150),(60)]`
	`= [(56.4),(144),(99.6)]`

`:.\ text(The reduction in players training moderately)`

`= 150 – 144`

`= 6`

d. `text(Need to show steady numbers for consecutive)`

`text(weeks 8 and week 9,)`

`S_8 = T^7S_1 = [(50),(150),(100)]`

`S_9 = T^8S_1 = [(50),(150),(100)]`

`:. S_8 = S_9`

`text{(i.e. player numbers don’t change after week 7.)}`

MATRICES, FUR2 2010 VCAA 1

In a game of basketball, a successful shot for goal scores one point, two points, or three points, depending on the position from which the shot is thrown.

`G` is a column matrix that lists the number of points scored for each type of successful shot.

`G = [(1),(2),(3)]`

In one game, Oscar was successful with

- 4 one-point shots for goal
- 8 two-point shots for goal
- 2 three-point shots for goal.

Write a row matrix, `N`, that shows the number of each type of successful shot for goal that Oscar had in that game. (1 mark)
Matrix `P` is found by multiplying matrix `N` with matrix `G` so that `P = N xx G`

Evaluate matrix `P`. (1 mark)
In this context, what does the information in matrix `P` provide? (1 mark)

Show Answers Only

`N = [(4, 8, 2)]`
`P = [26]`
`text(The total points scored by Oscar in the game.)`

Show Worked Solution

a. `N = [(4, 8, 2)]`

b.	`P`	`= NG`
		`= [(4, 8, 2)][(1),(2),(3)]`
		`= [26]`

c. `text(The total points scored by Oscar in the game.)`

NETWORKS, FUR2 2012 VCAA 3

Four tasks, `W`, `X`, `Y` and `Z`, must be completed.

Four workers, Julia, Ken, Lana and Max, will each do one task.

Table 1 shows the time, in minutes, that each person would take to complete each of the four tasks.

The tasks will be allocated so that the total time of completing the four tasks is a minimum.

The Hungarian method will be used to ﬁnd the optimal allocation of tasks. Step 1 of the Hungarian method is to subtract the minimum entry in each row from each element in the row.

Complete step 1 for task `X` by writing down the number missing from the shaded cell in Table 2. (1 mark)

The second step of the Hungarian method ensures that all columns have at least one zero.

The numbers that result from this step are shown in Table 3 below.

Following the Hungarian method, the smallest number of lines that can be drawn to cover the zeros is shown dashed in Table 3.

These dashed lines indicate that an optimal allocation cannot be made yet.

Give a reason why. (1 mark)
Complete the steps of the Hungarian method to produce a table tasks can be made.

Two blank tables have been provided for working if needed. (1 mark)
Write the name of the task that each person should do for the optimal allocation of tasks. (2 marks)

Show Answers Only

`17`
`text(Allocating four tasks to four people requires)`

`text(four lines and there are only three.)`

Show Worked Solution

a. `17`

b. `text(Allocating four tasks to four people)`

♦♦ Exact data unavailable although examiners highlighted part (b) as “poorly answered”.

`text(requires four lines and there are)`

`text(only three.)`

NETWORKS, FUR2 2012 VCAA 2

Thirteen activities must be completed before the produce grown on a farm can be harvested.

The directed network below shows these activities and their completion times in days.

Determine the earliest starting time, in days, for activity `E`. (1 mark)
A dummy activity starts at the end of activity `B`.

Explain why this dummy activity is used on the network diagram. (1 mark)
Determine the earliest starting time, in days, for activity `H`. (1 mark)
In order, list the activities on the critical path. (1 mark)
Determine the latest starting time, in days, for activity `J`. (1 mark)

Show Answers Only

`12\ text(days)`
`F\ text(has)\ B\ text(as a predecessor while)\ G\ text(and)\ H`
`text(have)\ B\ text(and)\ C\ text(as predecessors.)`
`text(S)text(ince there cannot be 2 activities called)\ B,`
`text{a dummy activity is drawn as an extension of}`
`B\ text(to show that it is also a predecessor of)\ G\ text(and)`
`H\ text{(with zero time).}`
`15\ text(days)`
`A-B-H-I-L-M`
`25\ text(days)`

Show Worked Solution

a.	`text(EST of)\ E`	`= 10 + 2`
		`= 12\ text(days)`

♦ Mean mark of all parts (combined) 47%.

b. `F\ text(has)\ B\ text(as a predecessor while)\ G\ text(and)\ H`

`text(have)\ B\ text(and)\ C\ text(as predecessors.)`

`text(S)text(ince there cannot be 2 activities called)\ B,`

`text{a dummy activity is drawn as an extension of}`

`B\ text(to show that it is also a predecessor of)\ G\ text(and)`

`H\ text{(with zero time).}`

♦♦ Exact data unavailable but “few students” were able to correctly deal with the dummy activity in this question.

c.	`text(EST of)\ H`	`= 10 + 5`
		`= 15\ text(days)`

d. `text(The critical path is)`

`A-B-H-I-L-M`

e. `text(The shortest time to complete all the activities)`

MARKER’S COMMENT: A correct calculation based on an incorrect critical path in part (d) gained a consequential mark here. Show your working!

`= 10 + 5 + 4 + 3 + 4 + 2`

`= 28\ text(days)`

`:.\ text(LST of)\ J`	`= 28 − 3`
	`= 25\ text(days)`

NETWORKS, FUR2 2012 VCAA 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.

1. Determine the shortest distance between the house and the pump. (1 mark)
2. How many vertices on the network diagram have an odd degree? (1 mark)
3. The total length of all edges in the network is 1180 metres.
  
  A journey starts and finishes at the house and travels along every edge in the network.
  
  Determine the shortest distance travelled. (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.

1. On the diagram below, draw the minimum length of pipe that is needed to supply water to all locations on the farm. (1 mark)
2. What is the mathematical term that is used to describe this minimum length of pipe in part i.? (1 mark)

Show Answers Only

1. `160\ text(m)`
2. `2`
3. `1250\ text(m)`
2. `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)}`

a.iii. `text{An Eulerian path is possible if it starts at}`

♦♦ “Very poorly answered”.
MARKER’S COMMENT: An Euler circuit is optimal but not possible here because of the two odd degree vertices.

`text{the house (odd vertex) and ends at the top}`

`text{right vertex (the other odd vertex). However,}`

`text{70 metres must be added to return to the}`

`text{house.}`

`:.\ text(Total distance)`	`= 1180 + 70`
	`= 1250\ text(m)`

b.i.

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

MATRICES, FUR2 2011 VCAA 3

A breeding program is started in the wetlands. It is aimed at establishing a colony of native ducks.

The matrix `W_0` displays the number of juvenile female ducks (`J`) and the number of adult female ducks (`A`) that are introduced to the wetlands at the start of the breeding program.

`W_0 = [(32),(64)]{:(J),(A):}`

In total, how many female ducks are introduced to the wetlands at the start of the breeding program? (1 mark)

The number of juvenile female ducks (`J`) and the number of adult female ducks (`A`) in the colony at the end of Year 1 of the breeding program is determined using the matrix equation

`W_1 = BW_0`

In this equation, `B` is the breeding matrix

`{:((qquadqquadqquad\ J,qquadA)),(B = [(0,2),(0.25,0.5)]{:(J),(A):}):}`

Determine `W_1` (1 mark)

The number of juvenile female ducks (`J`) and the number of adult female ducks (`A`) in the colony at the end of Year `n` of the breeding program is determined using the matrix equation

`W_n = BW_(n - 1)`

The graph below is incomplete because the points for the end of Year 3 of the breeding program are missing.

1. Use the matrices to calculate the number of juvenile and the number of adult female ducks expected in the colony at the end of Year 3 of the breeding program.
  
  Plot the corresponding points on the graph. (2 marks)
2. Use matrices to determine the expected total number of female ducks in the colony in the long term.
  
  Write your answer correct to the nearest whole number. (1 mark)

The breeding matrix `B` assumes that, on average, each adult female duck lays and hatches two female eggs for each year of the breeding program.

If each adult female duck lays and hatches only one female egg each year, it is expected that the duck colony in the wetland will not be self-sustaining and will, in the long run, die out.

The matrix equation

`W_n = PW_(n - 1)`

with a different breeding matrix

`{:((qquadqquadqquad\ J,qquadA)),(P = [(0,1),(0.25,0.5)]{:(J),(A):}):}`

and the initial state matrix

`W_0 = [(32),(64)]{:(J),(A):}`

models this situation.

During which year of the breeding program will the number of female ducks in the colony halve? (1 mark)

Changing the number of juvenile and adult female ducks at the start of the breeding program will also change the expected size of the colony.

Assuming the same breeding matrix, `P`, determine the number of juvenile ducks and the number of adult ducks that should be introduced into the program at the beginning so that, at the end of Year 2, there are 100 juvenile female ducks and 50 adult female ducks. (2 marks)

Show Answers Only

`96`
`W_1 = [(128),(40)]`
2. `144`
`text(year 5)`
`400\ text(juvenile ducks and 0 adult)`
`text(ducks should be introduced.)`

Show Worked Solution

a. `text(Total female ducks introduced)`

`= 32 + 64`

`= 96`

b.	`W_1`	`= BW_0`
		`= [(0,2),(0.25,0.5)][(32),(64)]`
		`= [(128),(40)]`

c.i.

`W_n = BW_(n – 1)`

`W_2 = [(0,2),(0.25,0.5)][(128),(40)] = [(80),(52)]`

`W_3 = [(0,2),(0.25,0.5)][(80),(52)] = [(104),(46)]`

`:.\ text{Plot (3, 104) and (3, 46)}`

c.ii. `text(Consider)\ n = text(50 and 51,)`

`W_50 = [(0,2),(0.25,0.5)]^49[(32),(64)] = [(96),(48)]`

`W_51 = [(0,2),(0.25,0.5)]^50[(32),(64)] = [(96),(48)]`

`:.\ text(Total female ducks in the long term)`

`= 96 + 48`

`= 144`

d. `text(Initial female ducks = 96)`

♦♦♦ Mean mark of parts (d)-(e) was 20%.

`W_n = PW_(n – 1)`

`W_4 = [(0,1),(0.25,0.5)]^3[(32),(64)] = [(28),(33)]`

`:. 51\ text(ducks at end of year 4.)`

`W_5 = [(0,1),(0.25,0.5)]^4[(32),(64)] = [(23),(18.5)]`

`:. 41.5\ text(ducks at end of year 5.)`

`:.\ text{Numbers halve (drop below 48) in year 5.}`

e.	`text(Let )`	`a = text(initial juvenile ducks)`
		`b = text(initial adult ducks)`

`text(Find)\ a\ text(and)\ b\ text(such that,)`

`[(0,1),(0.25,0.5)]^2[(a),(b)]= [(100),(50)]`

`[(0.25,0.5),(0.125,0.5)][(a),(b)] = [(100),(50)]`

`:. [(a),(b)]`	`= [(0.25,0.5),(0.125,0.5)]^(-1)[(100),(50)]`
	`= [(400),(0)]`

`:. 400\ text(juvenile ducks and 0 adult ducks)`

`text(should be introduced.)`

MATRICES, FUR2 2011 VCAA 2

To reduce the number of insects in a wetland, the wetland is sprayed with an insecticide.

The number of insects (`I`), birds (`B`), lizards (`L`) and frogs (`F`) in the wetland that has been sprayed with insecticide are displayed in the matrix `N` below.

`{:((qquadqquadqquadqquadI,qquadquad B,qquadL,\ qquadF)),(N = [(100\ 000, 400,1000,800)]):}`

Unfortunately, the insecticide, that is used to kill the insects can also kill birds, lizards and frogs. The proportion of insects, birds, lizards and frogs that have been killed by the insecticide are displayed in the matrix `D` below.

`{:(qquadqquadqquadquadquadtext(alive before spraying)),((qquadqquadqquadqquadI,qquad\ B,qquad\ L,qquad\ F)),(D = [(0.995,0,0,0),(0,0.05,0,0),(0,0,0.025,0),(0,0,0,0.30)]{:(I),(B),(L),(F):}{:qquadtext(dead after spraying):}):}`

Evaluate the matrix product `K = ND`. (1 mark)
Use the information in matrix `K` to determine the number of birds that have been killed by the insecticide. (1 mark)
Evaluate the matrix product `M = KF`, where `F = [(0),(1),(1),(1)]`. (1 mark)
In the context of the problem, what information does matrix `M` contain? (1 mark)

Show Answers Only

`K = [(99\ 500,20,25,240)]`
`20`
`M = [285]`
`M\ text(contains the combined number)`
`text(of birds, lizards and frogs that died.)`

Show Worked Solution

a.	`K`	`= ND`
		`= [(99\ 500,20,25,240)]`

MARKER’S COMMENT: In part (a), separating matrix elements by commas or dots is not correct notation.

b. `text(Birds dead after spraying) = 20`

c.	`M`	`= KF`
		`= [(99\ 500,20,25,240)][(0),(1),(1),(1)]`
		`= [0 + 20 + 25 + 240]`
		`= [285]`

MARKER’S COMMENT: The ability of many students to interpret the result of a matrix product was poor.

d. `text(Matrix)\ M\ text(contains the combined number)`

`text(of birds, lizards and frogs that died.)`

MATRICES, FUR2 2011 VCAA 1

The diagram below shows the feeding paths for insects (`I`), birds (`B`) and lizards (`L`). The matrix `E` has been constructed to represent the information in this diagram. In matrix `E`, a 1 is read as "eat" and a 0 is read as "do not eat".

Referring to insects, birds or lizards
1. what does the 1 in column `B`, row `L`, of matrix `E` indicate? (1 mark)
2. what does the row of zeros in matrix `E` indicate? (1 mark)

The diagram below shows the feeding paths for insects (`I`), birds (`B`), lizards (`L`) and frogs (`F`).

The matrix `Z` has been set up to represent the information in this diagram.

Matrix `Z` has not been completed.

Complete the matrix `Z` above by writing in the seven missing elements. (1 mark)

Show Answers Only

`text(Birds eat lizards)`
`text(Insects, birds or lizards do not eat birds)`
`{:((qquadqquadquadI,B,L,F)),(Z = [(0,1,1,1),(0,0,0,0),(0,1,0,0),(0,1,1,0)]{:(I),(B),(L),(F):}):}`

Show Worked Solution

a.i. `text(The 1 represents that birds eat lizards.)`

a.ii. `text(It indicates that insects, birds or lizards DO NOT)`

`text(eat birds.)`

b.	`{:((qquadqquadquadI,B,L,F)),(Z = [(0,1,1,1),(0,0,0,0),(0,1,0,0),(0,1,1,0)]{:(I),(B),(L),(F):}):}`

MATRICES, FUR2 2013 VCAA 2

10 000 trout eggs, 1000 baby trout and 800 adult trout are placed in a pond to establish a trout population.

In establishing this population

- eggs (`E`) may die (`D`) or they may live and eventually become baby trout (`B`)
- baby trout (`B`) may die (`D`) or they may live and eventually become adult trout (`A`)
- adult trout (`A`) may die (`D`) or they may live for a period of time but will eventually die.

From year to year, this situation can be represented by the transition matrix `T`, where

`{:(qquadqquadqquadqquadqquadtext(this year)),((qquadqquadqquadE,quad\ B,quad\ A,\ D)),(T = [(0,0,0,0),(0.4,0,0,0),(0,0.25,0.5,0),(0.6,0.75,0.5,1)]):}{:(),(),(E),(B),(A),(D):}{:(),(),(qquadtext(next year)):}`

Use the information in the transition matrix `T` to
1. determine the number of eggs in this population that die in the first year (1 mark)
2. complete the transition diagram below, showing the relevant percentages. (2 marks)

The initial state matrix for this trout population, `S_0`, can be written as

`S_0 = [(10\ 000),(1000),(800),(0)]{:(E),(B),(A),(D):}`

Let `S_n` represent the state matrix describing the trout population after `n` years.

Using the rule `S_n = T S_(n – 1)`, determine each of the following.
1. `S_1` (1 mark)
2. the number of adult trout predicted to be in the population after four years (1 mark)
The transition matrix `T` predicts that, in the long term, all of the eggs, baby trout and adult trout will die.
1. How many years will it take for all of the adult trout to die (that is, when the number of adult trout in the population is first predicted to be less than one)? (1 mark)
2. What is the largest number of adult trout that is predicted to be in the pond in any one year? (1 mark)
Determine the number of eggs, baby trout and adult trout that, if added to or removed from the pond at the end of each year, will ensure that the number of eggs, baby trout and adult trout in the population remains constant from year to year. (2 marks)

The rule `S_n = T S_(n – 1)` that was used to describe the development of the trout in this pond does not take into account new eggs added to the population when the adult trout begin to breed.

To take breeding into account, assume that 50% of the adult trout lay 500 eggs each year.

The matrix describing the population after one year, `S_1`, is now given by the new rule

`S_1 = T S_0 + 500\ M\ S_0`

where `T=[(0,0,0,0),(0.40,0,0,0),(0,0.25,0.50,0),(0.60,0.75,0.50,1.0)], M=[(0,0,0.50,0),(0,0,0,0),(0,0,0,0),(0,0,0,0)]\ text(and)\ S_0=[(10\ 000),(1000),(800),(0)]`
1. Use this new rule to determine `S_1`. (1 mark)

This pattern continues so that the matrix describing the population after `n` years, `S_n`, is given by the rule

`S_n = T\ S_(n – 1) + 500\ M\ S_(n – 1)`

Use this rule to determine the number of eggs in the population after two years (2 marks)

Show Answers Only

1. `6000`
2. `text(See Worked Solutions)`
1. `text(See Worked Solutions)`
2. `text(See Worked Solutions)`
1. `text(13 years)`
2. `text(Largest number of adult trout is 1325.)`
`text(Add 10 000 eggs, remove 3000 baby trout and add 150)`

`text(150 adult trout to keep the population constant.)`
1. `text(See Worked Solutions)`
2. `text(See Worked Solutions)`

Show Worked Solution

a.i. `text(60% of eggs die in 1st year,)`

`:.\ text(Eggs that die in year 1)`

`= 0.60 xx 10\ 000`

`= 6000`

MARKER’S COMMENT: A 100% cycle drawn at `D` was a common omission. Do not draw loops and edges of 0%!

a.ii.

b.i.	`S_1`	`= TS_0`
		`= [(0,0,0,0),(0.4,0,0,0),(0,0.25,0.5,0),(0.6,0.75,0.5,1)][(10\ 000),(1000),(800),(0)]`
		`= [(0),(4000),(650),(7150)]`

b.ii.	`S_4`	`= T^4S_0`
		`= [(0,0,0,0),(0.4,0,0,0),(0,0.25,0.5,0),(0.6,0.75,0.5,1)]^4[(10\ 000),(1000),(800),(0)]`
		`= [(0),(0),(331.25),(11\ 468.75)]`

`:. 331\ text(trout is the predicted population)`

`text(after 4 years.)`

c.i.

`S_12 = T^12S_0 = [(0),(0),(1.29),(11\ 791)]`

`S_13 = T^13S_0 = [(0),(0),(0.65),(11\ 799)]`

`:.\ text{It will take 13 years (when the trout}`

`text{population drops below 1).}`

c.ii.

`S_1 = TS_0 = [(0),(4000),(650),(7150)]`

`text(After 1 year, 650 adult trout.)`

`text(Similarly,)`

`S_2 = T^2S_0 = [(0),(0),(1325),(10\ 475)]`

`S_3 = T^3S_0 = [(0),(0),(662.5),(11\ 137.5)]`

`S_4 = T^4S_0 = [(0),(0),(331),(11\ 469)]`

`:.\ text(Largest number of adult trout = 1325.)`

d.	`S_0 – S_1 = [(10\ 000),(1000),(800),(0)] – [(0),(4000),(650),(7150)] = [(10\ 000),(−3000),(150),(−7150)]`

`:.\ text(Add 10 000 eggs, remove 3000 baby trout)`

`text(and add 150 adult trout to keep the)`

`text(population constant.)`

e.i.	`S_1`	`= TS_0 + 500MS_0`
		`= [(0),(4000),(650),(7150)] + 500 xx [(0,0,0.5,0),(0,0,0,0),(0,0,0,0),(0,0,0,0)][(10\ 000),(1000),(800),(0)]`
		`= [(0),(4000),(650),(7150)] + 500[(400),(0),(0),(0)]`
		`= [(200\ 000),(4000),(650),(7150)]`

e.ii.

`S_2`

`= TS_1 + 500MS_1`

`= [(0,0,0,0),(0.4,0,0,0),(0,0.25,0.5,0),(0.6,0.75,0.5,1)][(200\ 000),(4000),(650),(7150)]`

`+ 500 xx [(0,0,0.5,0),(0,0,0,0),(0,0,0,0),(0,0,0,0)][(200\ 000),(4000),(650),(7150)]`

`= [(162\ 500),(80\ 000),(1325),(130\ 475)]`

MATRICES, FUR2 2013 VCAA 1

Five trout-breeding ponds, `P`, `Q`, `R`, `X` and `V`, are connected by pipes, as shown in the diagram below.

The matrix `W` is used to represent the information in this diagram.

`{:({:\ qquadqquadqquadPquadQquad\ Rquad\ Xquad\ V:}),(W = [(0,1,1,1,0), (1,0,0,1,0),(1,0,0,1,0),(1,1,1,0,1),(0,0,0,1,0)]):}{:(),(P),(Q),(R),(X),(V):}`

In matrix `W`

• the 1 in column 1, row 2, for example, indicates that a pipe directly connects pond `P` and pond `Q`

• the 0 in column 1, row 5, for example, indicates that pond `P` and pond `V` are not directly connected by a pipe.

Find the sum of the elements in row 3 of matrix `W`. (1 mark)
In terms of the breeding ponds described, what does the sum of the elements in row 3 of matrix `W` represent? (1 mark)

The pipes connecting pond `P` to pond `R` and pond `P` to pond `X` are removed.

Matrix `N` will be used to show this situation. However, it has missing elements.

Complete matrix `N` below by filling in the missing elements in row 1 and column 1. (1 mark)

Show Answers Only

`2`
`text(The sum means that 2 other)`
`text(ponds connect directly to pond)\ R.`
`{:({:qquadqquadqquadPquadQquadRquadXquadV:}),(N = [(0,1,0,0,0),(1,0,0,1,0),(0,0,0,1,0),(0,1,1,0,1),(0,0,0,1,0)]):}{:(),(P),(Q),(R),(X),(V):}`

Show Worked Solution

a. `1+ 0 + 0 +1+ 0 = 2`

b. `text(The sum means that 2 other ponds)`

`text(connect directly to pond)\ R.`

c.	`{:({:qquadqquadqquadPquadQquad\ Rquad\ Xquad\ V:}),(N = [(0,1,0,0,0),(1,0,0,1,0),(0,0,0,1,0),(0,1,1,0,1),(0,0,0,1,0)]):}{:(),(P),(Q),(R),(X),(V):}`

MATRICES, FUR1 2008 VCAA 7-9 MC

A large population of mutton birds migrates each year to a remote island to nest and breed. There are four nesting sites on the island, A, B, C and D.

Researchers suggest that the following transition matrix can be used to predict the number of mutton birds nesting at each of the four sites in subsequent years. An equivalent transition diagram is also given.

`{:(qquad qquad qquad qquad {:text(this year):}), (qquad qquad quad quad \ {:(A,\ \ B,\ \ C,\ D):}), (T = [(0.4, 0, 0.2, 0),(0.35, 1, 0.15, 0), (0.15, 0, 0.55, 0), (0.1, 0, 0.1, 1)] {:(A), (B), (C), (D):} quad {:text(next year):}):}`

Part 1

Two thousand eight hundred mutton birds nest at site C in 2008.

Of these 2800 mutton birds, the number that nest at site A in 2009 is predicted to be

A. `560`

B. `980`

C. `1680`

D. `2800`

E. `3360`

Part 2

This transition matrix predicts that, in the long term, the mutton birds will

A. nest only at site A.

B. nest only at site B.

C. nest only at site A and C.

D. nest only at site B and D.

E. continue to nest at all four sites.

Part 3

Six thousand mutton birds nest at site B in 2008.

Assume that an equal number of mutton birds nested at each of the four sites in 2007. The same transition matrix applies.

The total number of mutton birds that nested on the island in 2007 was

A. `6000`

B. `8000`

C. `12\ 000`

D. `16\ 000`

E. `24\ 000`

Show Answers Only

`text(Part 1:) A`

`text(Part 2:) D`

`text(Part 3:) D`

Show Worked Solution

`text(Part 1:)`

`text(20% of birds at site)\ A\ text(in 2008 are predicted)`

`text(to move to site)\ C.`

`:.\ text(Number of birds)`

`= 20 text(%) xx 2800`

`= 560`

`=> A`

`text(Part 2:)`

`text(Consider)\ n\ text{large (say}\ n = 50 text{)},`

`T^50 = [(0, 0, 0, 0), (0.75, 1, 0.66, 0), (0, 0, 0, 0), (0.25, 0, 0.33, 0)]`

`=> D`

`text(Part 3:)`

`text(Let)\ \ x = text(mutton birds at each site in 2007)`

`text(In 2008, 6000 birds nest at)\ B.`

♦♦♦ Mean mark 25%.

`text(Using the diagram,)`

`x + 0.35x + 0.15x`	`= 6000`
`1.5x`	`= 6000`
`x`	`= 4000`

`:.\ text(Total number nested in 2007)`

`= 4 xx 4000`

`= 16\ 000\ text(birds)`

`=> D`

MATRICES, FUR1 2008 VCAA 6 MC

The solution of the matrix equation `[(0, – 3, 2), (1, 1, 1), (– 2, 0, 3)] [(x), (y), (z)] = [(11), (5), (8)]` is

A.	`[(1), (24), (2)]`	B.	`[(2), (– 1), (4)]`

C.	`[(2), (1), (3)]`	D.	`[(– 11), (4/3), (8)]`

E.	`[(11), (5), (8)]`

Show Answers Only

`B`

Show Worked Solution

`[(0, – 3, 2), (1, 1, 1), (– 2, 0, 3)] [(x), (y), (z)]= [(11), (5), (8)]`

`:. [(x), (y), (z)]`	`= [(0, – 3, 2), (1, 1, 1), (– 2, 0, 3)]^-1 [(11), (5), (8)]`
	`= [(2), (– 1), (4)]`

`=> B`

MATRICES, FUR1 2008 VCAA 4 MC

Matrix `A` is a `1 xx 3` matrix.

Matrix `B` is a `3 xx 1` matrix.

Which one of the following matrix expressions involving `A` and `B` is defined?

A. `A + 1/3 B`

B. `2B xx 3A`

C. `A^2 B`

D. `B^-1`

E. `B - A`

Show Answers Only

`B`

Show Worked Solution

`text(Consider)\ B,`

`underset (3 xx 1) (2B) xx underset (1 xx 3) (3A)`

`text(S) text(ince the number of columns in)`

`text(matrix)\ 2B\ text(equals the number of)`

`text(rows in matrix)\ 3A,\ text(their product)`

`text(is defined.)`

`=> B`

MATRICES, FUR1 2008 VCAA 3 MC

The cost prices of three different electrical items in a store are $230, $290 and $310 respectively.

The selling price of each of these three electrical items is 1.3 times the cost price plus a commission of $20 for the salesman.

A matrix that lists the selling price of each of these three electrical items is determined by evaluating

A. `1.3 xx [(230), (290), (310)] + [20]`

B. `1.3 xx [(230), (290), (310)] + 1.3 xx 20`

C. `1.3 xx [(230), (290), (310)] + [(20), (20), (20)]`

D. `1.3 xx [(230), (290), (310)] + 1.3 xx [(20), (20), (20)]`

E. `1.3 xx [(230 + 20), (290 + 20), (310 + 20)]`

Show Answers Only

`C`

Show Worked Solution

`=> C`

MATRICES, FUR1 2008 VCAA 2 MC

Apples cost $3.50 per kg, bananas cost $4.20 per kg and carrots cost $1.89 per kg.

Ashley buys 3 kg of apples, 2 kg of bananas and 1 kg of carrots.

A matrix product to calculate the total cost of these items is

A. `[(3), (2), (1)] [(3.50), (4.20), (1.89)]`

B. `[(3, 2, 1)] [(3.50, 4.20, 1.89)]`

C. `[(3.50 xx 2, 4.20 xx 3, 1.89 xx 1)]`

D. `[(3), (2), (1)] [(3.50, 4.20, 1.89)]`

E. `[(3.50, 4.20, 1.89)] [(3), (2), (1)]`

Show Answers Only

`E`

Show Worked Solution

`text(The “total” cost must be a 1 × 1 matrix,)`

`[(3.50, 4.20, 1.89)] [(3), (2), (1)]`

`= [3.50 xx 3 + 4.20 xx 2 + 1.89 xx 1]`

`= [20.79]`

`=> E`

MATRICES, FUR1 2009 VCAA 9 MC

`T =[[0.8,0.3],[0.2,0.7]]` is a transition matrix.

`S_3 = [[1150],[850]]` is a state matrix.

If `S_3 = TS_2`, then `S_2` equals

A. `[[1000],[1000]]`

B. `[[1090],[940]]`

C. `[[1100],[900]]`

D. `[[1150],[850]]`

E. `[[1175],[825]]`

Show Answers Only

`C`

Show Worked Solution

`S_3`	`= TS_2`
`:. S_2`	`= T^(−1)S_3`
	`= [(0.8,0.3),(0.2,0.7)]^(−1)[(1150),(850)]`
	`= [(1.4,−0.6),(−0.4,1.6)][(1150),(850)]`
	`= [(1100),(900)]`

`=> C`

MATRICES, FUR1 2009 VCAA 5 MC

`A`, `B`, `C`, `D` and `E` are five intersections joined by roads as shown in the diagram below.

Some of these roads are one-way only.

The matrix below indicates the direction that cars can travel along each of these roads.

In this matrix

- 1 in column `A` and row `B` indicates that cars can travel directly from `A` to `B`
- 0 in column `B` and row `A` indicates that cars cannot travel directly from `B` to `A` (either it is a one-way road or no road exists).

`{:(text(from intersection)),({:quad\ Aquad\ Bquad\ Cquad\ Dquad\ E:}),([(0,0,0,0,0),(1,0,0,0,0),(0,1,0,1,1),(1,0,0,0,0),(0,0,1,1,0)]):}{:(),(),(A),(B),(Cqquadtext(to intersection)),(D),(E):}`

Cars can travel in both directions between intersections

A. `A` and `D`

B. `B` and `C`

C. `C` and `D`

D. `D` and `E`

E. `C` and `E`

Show Answers Only

`E`

Show Worked Solution

`e_(CE) = e_(EC) = 1`

`:.\ text(Cars can travel both ways)`

`text(between)\ E\ text(and)\ C.`

`=> E`

MATRICES, FUR1 2009 VCAA 3 MC

The number of people attending the morning, afternoon and evening sessions at a cinema is given in the table
below. The admission charges (in dollars) for each session are also shown in the table.

A column matrix that can be used to list the number of people attending each of the three sessions is

A. `[25,56,124]`

B. `[[25],[56],[124]]`

C. `[12,15,20]`

D. `[[12],[15],[20]]`

E. `[[25,56,124],[12,15,20]]`

Show Answers Only

`B`

Show Worked Solution

`text(S)text(ince the answer must be a “column” matrix)`

`text(of the attendees.)`

`=> B`

MATRICES, FUR1 2010 VCAA 8 MC

`m` and `n` are positive whole numbers.

Matrix `P` is of order `m xx n.`

Matrix `Q` is of order `n xx m.`

The matrix products `PQ` and `QP` are both defined

A. for no values of `m` and `n`

B. when `m` is equal to `n` only

C. when `m` is greater than `n` only

D. when `m` is less than `n` only

E. for all values of `m` and `n.`

Show Answers Only

`E`

Show Worked Solution

`underset (m xx n) P xx underset (n xx m) Q = underset (m xx m)(PQ)`

`:. PQ\ text(is defined for all values)`

`text(of)\ m and n.`

`text(Similarly,)`

`underset (n xx m)Q xx underset (m xx n) P = underset (n xx n) (QP)`

`=> E`

MATRICES, FUR1 2010 VCAA 7 MC

A new colony of several hundred birds is established on a remote island. The birds can feed at two locations, A and B. The birds are expected to change feeding locations each day according to the transition matrix

`{:(qquad qquad qquad {:text(this day):}), (qquad qquad qquad {: A\ \ \ \ \ B:}), (T = [(0.4, 0.3),(0.6, 0.7)] {:(A), (B):} {:quad text(next day):}):}`

In the beginning, approximately equal numbers of birds feed at each site each day.

Which of the following statements is not true

A. 70% of the birds that feed at B on a given day will feed at B the next day

B. 60% of the birds that feed at A on a given day will feed at B the next day.

C. In the long term, more birds will feed at B than at A.

D. The number of birds that change feeding locations each day will decrease over time to zero

E. In the long term, some birds will always be found feeding at each location.

Show Answers Only

`D`

Show Worked Solution

`A, B, and C\ \ text(are all true.)`

`text(Consider)\ \ D and E,`

`text(If)\ \ n\ \ text{is large (say}\ n = 100 text{),}`

`[(0.4, 0.3), (0.6, 0.7)]^100 = [(0.333, 0.333), (0.666, 0.666)]`

`:.\ text(Birds will always be feeding at both locations and)`

`text(some will be changing locations each day.)`

`=> D`

MATRICES, FUR1 2010 VCAA 5 MC

A system of three simultaneous linear equations is written in matrix form as follows.

`[(1, – 2, 0), (1, 0, 3), (0, 2, – 1)] [(x), (y), (z)] = [(4), (11), (– 5)]`

One of the three linear equations is

A. `x - 2y + z = 4`

B. `x + y + 3z = 11`

C. `2x - y = – 5`

D. `x + 3z = 11`

E. `3y - z = – 5`

Show Answers Only

`D`

Show Worked Solution

`text(Expanding the matrix,)`

`x – 2y`	`= 4`
`x + 3z`	`= 11`
`2y – z`	`= – 5`

`=> D`

MATRICES, FUR1 2010 VCAA 4 MC

Which matrix expression results in a matrix that contains the sum of the numbers 2, 5, 4, 1 and 8?

A. `[(1),(1),(1),(1),(1)] xx [(2, 5, 4, 1, 8)]`

B. `[(2, 5, 4, 1, 8)] xx [(1),(1),(1),(1),(1)]`

C. `[(1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)] xx [(2, 0, 0, 0, 0), (0, 5, 0, 0, 0), (0, 0, 4, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 8)]`

D. `[(1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)] xx [(2),(5),(4),(1),(8)]`

E. `[(1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)] + [(2, 0, 0, 0, 0), (0, 5, 0, 0, 0), (0, 0, 4, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 8)]`

Show Answers Only

`B`

Show Worked Solution

`text(The result needs to be a 1 × 1 matrix.)`

`=> B`

MATRICES, FUR1 2010 VCAA 2 MC

Peter bought only apples and bananas from his local fruit shop.

The matrix

`{:(qquad qquad quad {:\ text(A B):}), (N = [(3, 4)]):}`

lists the number of apples (A) and bananas (B) that Peter bought

The matrix

`C = [(0.37), (0.43)] {:(A), (B):}`

lists the cost (in dollars) of one apple and one banana respectively

The matrix product, `NC`, gives

the total amount spent by Peter on the fruit that he bought.
the total number of pieces of fruit that Peter bought.
the individual amounts that Peter spent on apples and bananas respectively.
the total number of pieces of fruit that Peter bought and the total amount that he spent.
the individual number of apples and bananas that Peter bought and the individual amounts that Peter spent on these apples and bananas respectively.

Show Answers Only

`A`

Show Worked Solution

`NC\ \ text(is a 1 × 1 matrix that gives the total amount)`

`text(spent on both fruits.)`

`=> A`

MATRICES, FUR1 2011 VCAA 5-6 MC

Two politicians, Rob and Anna, are the only candidates for a forthcoming election. At the beginning of the election campaign, people were asked for whom they planned to vote. The numbers were as follows.

During the election campaign, it is expected that people may change the candidate that they plan to vote for each week according to the following transition diagram.

Part 1

The total number of people who are expected to change the candidate that they plan to vote for one week after the election campaign begins is

A. `828`

B. `1423`

C. `2251`

D. `4269`

E. `6891`

Part 2

The election campaign will run for ten weeks.

If people continue to follow this pattern of changing the candidate they plan to vote for, the expected winner after ten weeks will be

A. Rob by about 50 votes.

B. Rob by about 100 votes.

C. Rob by fewer than 10 votes.

D. Anna by about 100 votes.

E. Anna by about 200 votes.

Show Answers Only

`text(Part 1:)\ C`

`text(Part 2:)\ E`

Show Worked Solution

`text(Part 1)`

`text(Students expected to change)`

`= 25text(%) xx 5692 + 24text(%) xx 3450`

`= 2251`

`=> C`

`text(Part 2)`

♦ Mean mark 48%.

`text(After 1 week,)`

`[(0.75,0.24),(0.25,0.76)][(5692),(3450)] = [(5097),(4045)]`

`text(After 10 weeks,)`

`[(0.75,0.24),(0.25,0.76)]^10[(5692),(3450)] = [(4479),(4663)]`

`:.\ text(Anna is ahead by)`

`4663 – 4479 = 184\ text(votes)`

`=> E`

MATRICES, FUR1 2012 VCAA 8 MC

There are 30 children in a Year 6 class. Each week every child participates in one of three activities: cycling (C), orienteering (O) or swimming (S).

The activities that the children select each week change according to the transition matrix below.

`{:({:qquadqquadqquadqquadtext(this week):}),(qquadqquadqquad\ Cqquad\ OqquadquadS),(T = [(0.5,0.3, 0.3), (0.1,0.6,0.2), (0.4,0.1,0.5)]{:(C), (O), (S):}qquadtext(next week)):}`

From the transition matrix it can be concluded that

A. in the first week of the program, ten children do cycling, ten children do orienteering and ten children do swimming.

B. at least 50% of the children do not change their activities from the first week to the second week.

C. in the long term, all of the children will choose the same activity.

D. orienteering is the most popular activity in the first week.

E. 50% of the children will do swimming each week.

Show Answers Only

`B`

Show Worked Solution

`text(50% stay in cycling, 60% stay in orienteering)`

`text(and 50% stay in swimming.)`

`rArr B`

MATRICES, FUR1 2012 VCAA 7 MC

A store has three outlets, A, B and C. These outlets sell dresses, jackets and skirts made by the fashion house Ocki.

The table below lists the number of Ocki dresses, jackets and skirts that are currently held at each outlet.

A matrix that shows the total number of Ocki dresses (D), jackets (J) and skirts (S) in each size held at the three outlets is given by

Show Answers Only

`B`

Show Worked Solution

`text(By consolidating the 3 outlets for the)`

`text(same items in the same size.)`

`rArr B`

MATRICES, FUR1 2012 VCAA 6 MC

The table below shows the number of classes and the number of students in each class at each year level in a secondary school.

Let `F= [1 quad 1 quad 1 quad 1], \ G= [(1),(1),(1),(1)],\ M= [7 quad 5 quad 6 quad 4],\ N= [(7),(5),(6),(4)],\ P= [(22,0,0,0), (0,20,0,0), (0,0,18,0), (0,0,0,24)]`

A matrix product that displays the total number of students in Years 9 – 12 at this school is

A. `M xx P xx F`

B. `P xx G xx M`

C. `F xx P xx N`

D. `P xx N xx F`

E. `F xx N xx P`

Show Answers Only

`C`

Show Worked Solution

`text(Total student matrix will produce)`

`text(a 1 × 1 matrix.)`

`text(Consider)\ C,`

`F`	`xx`	`P`	`xx`	`N`	`=`	`FPN`
`1 xx 4`		`4 xx 4`		`4 xx 1`		`1 xx 1`

`A, B\ text(and)\ E\ text(produce undefined matrices)`

`text(and)\ D\ text(produces a 4 × 4.)`

`rArr C`

MATRICES, FUR1 2012 VCAA 2 MC

If `A = [(8,1), (4,2)]` and `B= [(3,12),(6,0)],` then the matrix `AB = [(30,96), (24,48)].`

The element 24 in the matrix `AB` is correctly obtained by calculating

A. `4 × 6 + 2 × 0`

B. `4 × 3 + 2 × 6`

C. `3 × 4 + 12 × 1`

D. `4 × 2 + 8 × 2`

E. `8 × 3 + 1 × 0`

Show Answers Only

`B`

Show Worked Solution

`[(8,1),(4,2)][(3,12),(6,0)] = [(30,96),(24,48)]`

`e_21\ text(in)\ AB\ text(is calculated.)`

`4 xx 3 + 2 xx 6 = 24`

`rArr B`

MATRICES, FUR1 2013 VCAA 3 MC

A coffee shop sells three types of coffee, Brazilian (B), Italian (I) and Kenyan (K). The regular customers buy one cup of coffee each per day and choose the type of coffee they buy according to the following transition matrix, `T`.

`{:({:qquadqquadqquadtext(choose today):}),(qquadqquadqquad\ BquadqquadIquadqquadK),(T = [(0.8,0.1,0.1), (0,0.8,0.1), (0.2,0.1,0.8)]{:(B), (I), (K):} qquadtext(choose tomorrow)):}`

On a particular day, 84 customers bought Brazilian coffee, 96 bought Italian coffee and 81 bought Kenyan coffee.

If these same customers continue to buy one cup of coffee each per day, the number of these customers who are expected to buy each of the three types of coffee in the long term is

Show Answers Only

`B`

Show Worked Solution

`text(Consider)\ n\ text(large)\ (n= 50),`

`[(B),(I),(K)]`	`= [(0.8,0.1,0.1),(0,0.8,0.1),(0.2,0.1,0.8)]^50[(84),(96),(81)]`
	`= [(87),(58),(116)]`

`rArr B`

MATRICES, FUR1 2013 VCAA 4 MC

`2.8x + 0.7y`	`= 10`
`1.4x + ky`	`= 6`

The set of simultaneous linear equations above does not have a solution if `k` equals

A. `– 0.35`

B. `– 0.250`

C. `0`

D. `0.25`

E. `0.35`

Show Answers Only

`E`

Show Worked Solution

`[(2.8,0.7),(1.4,k)][(x),(y)] = [(10),(6)]`

`text(det) [(2.8,0.7),(1.4,k)] = 2.8k – 0.7 xx 1.4`

`text(No solution if det) = 0,`

`0`	`= 2.8k – 0.98`
`k`	`= (0.98)/2.8`
	`= 0.35`

`rArr E`

MATRICES, FUR1 2014 VCAA 7 MC

A transition matrix, `T`, and a state matrix, `S_2`, are defined as follows.

`T=[(0.5,0,0.5),(0.5,0.5,0),(0,0.5,0.5)]\ \ \ \ \ S_2=[(300),(200),(100)]`

If `S_2 = TS_1`, the state matrix `S_1` is

Show Answers Only

`D`

Show Worked Solution

`TS_1`	`= S_2`
`:. S_1`	`= T^-1 S_2`
	`= [(0.5, 0, 0.5), (0.5, 0.5, 0), (0, 0.5, 0.5)]^-1 [(300), (200), (100)]`
	`= [(400), (0), (200)]`

`=>D`

MATRICES, FUR1 2006 VCAA 8 MC

Australians go on holidays either within Australia or overseas.

Market research shows that

95% of those who had their last holiday in Australia said that their next holiday would be in Australia
20% of those who had their last holiday overseas said that their next holiday would also be overseas.

A transition matrix that could be used to describe this situation is

A.	`[(0.95),(0.20)]`	B.	`[(0.95),(0.05)] + [(0.20),(0.80)]`

C.	`[(0.95,0.95),(0.20,0.20)]`	D.	`[(0.95,0.20),(0.05,0.80)]`

E.	`[(0.95,0.80),(0.05,0.20)]`

Show Answers Only

`E`

Show Worked Solution

`{:(\ text(last holiday)),(qquad\ AqquadquadO),([(0.95,0.8),(0.05,0.2)]{:(A),(O):}qquadtext(next holiday)):}`

`rArr E`

MATRICES, FUR1 2006 VCAA 5 MC

A company makes Regular (`R`), Queen (`Q`) and King (`K`) size beds. Each bed comes in either the Classic style or the more expensive Deluxe style.

The price of each style of bed, in dollars, is listed in a price matrix `P`, where

`{:(qquadqquadqquad\ RquadqquadQquadqquadK),(P = [(145, 210, 350), (185, 270, 410)]{: (text (Classic)), (text (Deluxe)) :}):}`

The company wants to increase the price of all beds.

A new price matrix, listing the increased prices of the beds, can be generated from `P` by forming a matrix product with the matrix, `M`, where

`M = [(1.2,0), (0, 1.35)]`

This new price matrix is

Show Answers Only

`D`

Show Worked Solution

`MP`	`= [(1.2,0),(0,1.35)][(145,210,350),(185,270,410)]`
	`= [(174,252,420),(249.75,364.50,553.50)]`

`rArr D`