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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.
 

NETWORKS, FUR2 2014 VCAA 4
 

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

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

  1. What is the latest starting time of activity ?   (1 mark)

  2. What is the float time of activity ?   (1 mark)

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

Activity will take seven hours to complete.

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

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

Activity  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 .

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

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a.   

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

  
b.  


  

c.  


  

d.   


  


  

e.   

 


 

 

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. i. Write down the names of the towns at the start and at the end of Charlie’s path.   (1 mark)

    ii. What distance did he travel?   (1 mark)

Brianna will follow a Hamiltonian path from Bower to Attard.

  1. 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.

  1. Which other train line should be removed?

     

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

     
    NETWORKS, FUR2 2014 VCAA 32

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  1.  i. 
    ii.  
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a.i.   
  

a.ii.  


  

b.  

 

  
c.   

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.
 

     NETWORKS, FUR2 2014 VCAA 21
 

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.

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

     

     

    NETWORKS, FUR2 2014 VCAA 22
     

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.

  1. Explain why Andrew made this decision.   (1 mark)

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

NETWORKS, FUR2 2014 VCAA 23

  1.  i. Which task should be allocated to Andrew?   (1 mark)

  2. ii. How many hours in total are used to plan for the open day?   (1 mark)

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  1.  
    Networks, FUR2 2014 VCAA 2_2 answer

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a.    Networks, FUR2 2014 VCAA 2_2 answer

 

b.  

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


  

c.i.   
  

c.ii.   

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.
 

   Networks, FUR2 2015 VCAA 31
 

The directed network that shows these activities is shown below.
 

 Networks, FUR2 2015 VCAA 32
 

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

  1. What is the EST of activity ?   (1 mark)

  2. What is the latest starting time (LST) of activity ?   (1 mark)

  3. Given that the EST of activity is 22 minutes, what is the duration of activity ?   (1 mark)

  4. Write down, in order, the activities on the critical path.   (1 mark)

  5. Activities and can only be completed by either Cassie or Donna.

     

    One Monday, Donna is sick and both activities and 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)

  6. Every Friday, a special delivery to the five towns includes fresh seafood. This causes a slight change to activity , which then cannot start until activity has been completed.
      
    i.     Michael was the best player in 2014 and he considered purchasing cricket equipment that was valued at $750.
    On the directed graph below, show this change without duplicating any activity?   (1 mark)


    Networks, FUR2 2015 VCAA 32
     
  7. ii. 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)

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  6. i. 
    Networks, FUR2 2015 VCAA 3 Answer
    ii.
         

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a.   
 

b.  
 

c.   
 

d.  
 

e.   

 

f.i.    Networks, FUR2 2015 VCAA 3 Answer

 

f.ii.   

NETWORKS, FUR2 2015 VCAA 2

The factory supplies groceries to stores in five towns, , , , and , represented by vertices on the graph below.
 

Networks, FUR2 2015 VCAA 2
 

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 and 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 , after which the van will continue on to the other four towns before returning to the factory.

  1. i. The shortest possible circuit from the factory for this delivery run, starting from town , is not Hamiltonian.
     
    Complete the order in which these deliveries would follow this shortest possible circuit.   (1 mark)

     factory – – ___________________________ – factory

  2. ii. 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)

  3. 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 .   (1 mark)

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a.i.  

aii. 

b.   

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a.i.   
  

a.ii.   
  

b.  

NETWORKS, FUR2 2015 VCAA 1

A factory requires seven computer servers to communicate with each other through a connected network of cables.

The servers, , , , , , and , 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 and server ?   (1 mark)

  2. What is the cheapest cost, in dollars, of installing cables between server and server ?   (1 mark)

  3. 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)

     

  4. 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.
    i.     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
  5. ii. 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 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

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  4. i. 

 Networks, FUR2 2015 VCAA 12 Answer
d.ii.

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a.   
  

b.  


  

c.   

MARKER’S COMMENT: Many students had difficulty finding the minimum spanning tree, often incorrectly excluding or .
d.i.    Networks, FUR2 2015 VCAA 12 Answer

 

d.ii.   

 

MATRICES, FUR2 2006 VCAA 3

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

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

     

     

     

     

  2. Do the equations have a unique solution? Provide an explanation to justify your response.   (1 mark)

  3. Write down an inverse matrix that can be used to solve these equations.   (1 mark)

  4. 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)

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  1.  
  2.  
  3.  
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a.   
♦ Mean mark 35% for all parts (combined).

 

b.   

 

 

c.   

 

d. 

MATRICES, FUR2 2006 VCAA 2

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

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

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

     

     

     

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.

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

     

     

     

  2. Use and 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)

  3. 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)
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  1.  
  2.  
  3.  
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a.    

 

b.    

 

c.  

 

 

d.  

 

 

MATRICES, FUR2 2006 VCAA 1

A manufacturer sells three products, , and , through outlets at two shopping centres, Eastown () and Noxland (). 

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

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

The matrix , shown below, gives the selling price, in dollars, of products , , .

  1.   i. Evaluate the matrix , where .   (1 mark)

  2.  ii. What information does the elements of matrix provide?   (1 mark)

  3. Explain why the matrix is not defined.   (1 mark)

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a.  
 

b.i.   
   
   

 
b.ii.  

   
 

c.   

MATRICES, FUR2 2007 VCAA 2

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

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

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

  1. 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
 


 

  1. What proportion of eggs turn into juveniles each week?   (1 mark)

    1. Evaluate the matrix product     (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.
    5. When does this first occur?   (1 mark)

    6. Write down the exact steady-state matrix for this population.  (1 mark)

  3. If the study is repeated with unsterilised adult insects, eggs will be laid and potentially grow into adults.
  4. Assuming 30% of adults lay eggs each week, the population matrix after one week, , is now given by
  6. where     and  
     

    1. Determine  (1 mark)

    2. This pattern continues. The population matrix after weeks, , is given by
    4. Determine the number of live eggs in this insect population after two weeks.  (1 mark)

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  3.  
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a.  
 

b.   
 

c.i.   
   
   

 
c.ii.   
 

c.iii.   
   

 

 

c.iv. 

 

 

c.v.  

 

d.i.  
   

 

♦♦ Mean mark for part (d) was 30%.
d.ii.  

 

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.
 

MATRICES, FUR2 2007 VCAA 1
 

  1. Write down a 2 x 3 matrix that displays the fat, carbohydrate and protein content (in columns) of bread and margarine.   (1 mark)

  2. and are two matrices defined as follows.
     
             

    1. Evaluate the matrix product  .   (1 mark)

    2. Determine the order of matrix product  .   (1 mark)

Matrix 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 displays the energy content per serving of the four foods: bread, margarine, peanut butter and honey.

    1. Explain the information that the matrix product provides.   (1 mark)

  1. The number of serves of bread (), margarine (), peanut butter () and honey () 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
      
         
      
    Solve the matrix equation to find the values , , and .   (2 marks)

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  1.  


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a.   

 

b.i.   
   

 

b.ii.   

 

b.iii.  
 

 

c.   
   

 

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

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

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

  1. Calculate the number of students who are expected to attend the extra singing rehearsals in week 3.   (1 mark)

  2. 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)

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a.  

 

 

 

b.   

 

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 .
 


 

  1. 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 shown below.
 


 

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

MATRICES, FUR2 2009 VCAA 3

    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)

  2. Evaluate the product  , where is the state matrix at the end of the first week.   (1 mark)

  3. 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)

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a.  


 

b.i.   
 

b.ii.   
 

b.iii.   


 

c.   
   

 

d.     
     

 

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.

MATRICES, FUR2 2009 VCAA 21

For this function

    • student tickets cost 
    • teacher tickets cost 
    • parent tickets cost  .
  1. Use the information in Table 1 to complete the following matrix equation by inserting the missing values in the shaded boxes.   (1 mark)

     
         MATRICES, FUR2 2009 VCAA 22
     

  2. Use the matrix equation to find the cost of a teacher ticket to the school function.   (2 marks)

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Show Worked Solution

a.  
 

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

 

MATRICES, FUR2 2010 VCAA 4

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

The matrix 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.
 


 

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

where is the matrix     

    1. Determine , 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.

and so on such that

  1. Determine the attendance matrix (with the elements written correct to the nearest whole number) for game 10.   (1 mark)

  2. 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.

  1. 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)

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  2.  

  3.  

  4.  

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a.i.   
   
   

 

a.ii.   


 

b.   
   
   
   

 

c.   

 


 

d.  


 

MATRICES, FUR2 2010 VCAA 3

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

The equations are   
  1. These equations can be written equivalently in matrix form.
  2. Complete the missing information below.   (1 mark)

 

 

This matrix equation can be solved in the following way.

 

 

  1. Determine the value of shown in the matrix equation above.   (1 mark)

  2. How many rebounds is Oscar predicted to have in his next game?   (1 mark)

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a.   

 

b.   
   

 

 

c.   
   

 

 

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 (), 150 players are doing moderate training () and 60 players are doing light training (). The state matrix, , shows the number of players who are undertaking each type of training in the first week
 


 

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.
 

MATRICES, FUR2 2010 VCAA 2
 

  1. 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 .
 


 

  1. Determine how many players will be doing heavy training in week two.   (1 mark)

  2. Determine how many fewer players will be doing moderate training in week three than in week one.   (1 mark)

  3. 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) 
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  1.  

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a.  

 

b.   
   
   

 

 

c.  

 
 

 

 

d.  

 

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.

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

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.
  1. Write a row matrix, , that shows the number of each type of successful shot for goal that Oscar had in that game.   (1 mark)

  2. Matrix is found by multiplying matrix with matrix so that  
  3. Evaluate matrix .   (1 mark)

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

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a.  
 

b.   
   
   

 
c.   

NETWORKS, FUR2 2012 VCAA 3

Four tasks, , , and , 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.
 

Networks, FUR2 2012 VCAA 3_1
 

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.
 

Networks, FUR2 2012 VCAA 3_2
 

  1. Complete step 1 for task 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.
 

Networks, FUR2 2012 VCAA 3_3
 

  1. 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)

  2. 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)

     

     
        Networks, FUR2 2012 VCAA 3_4

          Networks, FUR2 2012 VCAA 3_4
     

  3. Write the name of the task that each person should do for the optimal allocation of tasks.   (2 marks)

     

Networks, FUR2 2012 VCAA 3_5

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  2.  

  3.  
    Networks, FUR2 2012 VCAA 3_1 Answer
  4.  
    Networks, FUR2 2012 VCAA 3_2 Answer
Show Worked Solution

a.  

 

b.  

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

 

c.    Networks, FUR2 2012 VCAA 3_1 Answer

 

d.    Networks, FUR2 2012 VCAA 3_2 Answer

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.

 

NETWORKS, FUR2 2012 VCAA 2
  

  1. Determine the earliest starting time, in days, for activity .   (1 mark)

  2. A dummy activity starts at the end of activity .

     

    Explain why this dummy activity is used on the network diagram.   (1 mark)

  3. Determine the earliest starting time, in days, for activity .   (1 mark)

  4. In order, list the activities on the critical path.   (1 mark)

  5. Determine the latest starting time, in days, for activity .   (1 mark)

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Show Worked Solution
a.   
   
♦ Mean mark of all parts (combined) 47%.

 

b.  

 

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

 

d.  

 

e.  

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

 

 

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.
 

NETWORKS, FUR2 2012 VCAA 1
 

    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.
    4. A journey starts and finishes at the house and travels along every edge in the network.
    5. 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)

       

     NETWORKS, FUR2 2012 VCAA 1

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

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  1. i.  
    ii. 
    iii.
  2. i.
    NETWORKS, FUR2 2012 VCAA 1 Answer
    ii.

Show Worked Solution

a.i.   

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

  

a.ii.  
 

a.iii.   

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

   

   

   

   

 

 

b.i.    NETWORKS, FUR2 2012 VCAA 1 Answer

 

b.ii.   

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 displays the number of juvenile female ducks () and the number of adult female ducks () that are introduced to the wetlands at the start of the breeding program.

  1. 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 () and the number of adult female ducks () in the colony at the end of Year 1 of the breeding program is determined using the matrix equation

In this equation, is the breeding matrix

  1. Determine   (1 mark)

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

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

 

MATRICES, FUR2 2011 VCAA 3

    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.
    2. Plot the corresponding points on the graph.   (2 marks)

    3. Use matrices to determine the expected total number of female ducks in the colony in the long term.
    4. Write your answer correct to the nearest whole number.   (1 mark)

The breeding matrix 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

with a different breeding matrix

and the initial state matrix

models this situation.

  1. 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.

  1. Assuming the same breeding matrix, , 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)

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  3.  
    1.  
      MATRICES, FUR2 2011 VCAA 3 Answer

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a.  


 

b.   
   

 

c.i.    MATRICES, FUR2 2011 VCAA 3 Answer


 

c.ii.  

 


 

d.  

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


 

e.   
   

 

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 (), birds (), lizards () and frogs () in the wetland that has been sprayed with insecticide are displayed in the matrix below.
 


 

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 below.
 


 

  1. Evaluate the matrix product  .   (1 mark)

  2. Use the information in matrix to determine the number of birds that have been killed by the insecticide.   (1 mark)

  3. Evaluate the matrix product  , where .   (1 mark)

  4. In the context of the problem, what information does matrix contain?   (1 mark)

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

 
b.  
 

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

d.  

 

MATRICES, FUR2 2011 VCAA 1

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

MATRICES, FUR2 2011 VCAA 11

  1. Referring to insects, birds or lizards
  2.  i. what does the 1 in column , row , of matrix indicate?   (1 mark)

  3. ii. what does the row of zeros in matrix indicate?   (1 mark)

The diagram below shows the feeding paths for insects (), birds (), lizards () and frogs ().

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

Matrix has not been completed.
 

MATRICES, FUR2 2011 VCAA 12

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

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a.i.   

a.ii.   

b.
     

Show Worked Solution

a.i.   
 

a.ii.   

   
 

b.   

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 () may die () or they may live and eventually become baby trout ()
    • baby trout () may die () or they may live and eventually become adult trout ()
    • adult trout () may die () 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 , where
 


 

  1. Use the information in the transition matrix 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)

       

          Matrices, FUR2 2013 VCAA 2_a

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


 

Let  represent the state matrix describing the trout population after years.

  1. Using the rule  , determine each of the following.

     

    1.    (1 mark)

    2. the number of adult trout predicted to be in the population after four years   (1 mark)

  2. The transition matrix 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)

  3. 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    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.

  1. To take breeding into account, assume that 50% of the adult trout lay 500 eggs each year.
  2. The matrix describing the population after one year, , is now given by the new rule
  4. where      
    1. Use this new rule to determine .   (1 mark)

  5. This pattern continues so that the matrix describing the population after years, , is given by the rule
  6.      
     

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

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a.i.   

a.ii. 

b.i.   

b.ii.

c.i.   

c.ii.  

d.   

e.i.   

e.ii.   

Show Worked Solution

a.i.   


 

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

Matrices-FUR2-2013-VCAA-2_a Answer
b.i.   
   

 

b.ii.   
   

 

 

c.i.   


 


 

c.ii.   


 


 

d.   

 

 

e.i.   
   
   
   

 

e.ii.   
   

       
   

MATRICES, FUR2 2013 VCAA 1

Five trout-breeding ponds, , , , and , are connected by pipes, as shown in the diagram below.
 

Matrices, FUR2 2013 VCAA 1 

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

In matrix

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

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

  1. Find the sum of the elements in row 3 of matrix .   (1 mark)

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

The pipes connecting pond to pond and pond to pond are removed.

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

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

        
             Matrices, FUR2 2013 VCAA 1_c

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  3.  
Show Worked Solution

a.  
 

b.  


 

c.   

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.
 

     VCAA MATRICES FUR2 2008 7i

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.   

B.   

C. 

D. 

E. 

 

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. 

B. 

C. 

D. 

E. 

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♦♦♦ Mean mark 25%.

 
 

 

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. 

B. 

C. 

D. 

E. 

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MATRICES, FUR1 2009 VCAA 5 MC

, , , and are five intersections joined by roads as shown in the diagram below.

Some of these roads are one-way only.
 

MATRICES, FUR1 2009 VCAA 5 MC

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

In this matrix

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


 

Cars can travel in both directions between intersections

A.   and

B.   and

C.   and

D.   and

E.   and

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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.
 

     MATRICES, FUR1 2009 VCAA 3 MC 1
 

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

A.  

B.  

C.  

D.  

E.  

 

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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
 


 

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.

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MATRICES, FUR1 2010 VCAA 2 MC

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

The matrix


 

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

The matrix
 


 

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

The matrix product, , gives

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

MATRICES, FUR1 2011 VCAA 5-6 MC 1
 

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.
 

MATRICES, FUR1 2011 VCAA 5-6 MC 2

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.    

B.   

C.   

D.  

E.  

 

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.

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♦ Mean mark 48%.

 

 

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.
 


 

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.

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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.
 

     MATRICES, FUR1 2012 VCAA 7 MC1 
 

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
 

MATRICES, FUR1 2012 VCAA 7 MC ab

MATRICES, FUR1 2012 VCAA 7 MC cd

MATRICES, FUR1 2012 VCAA 7 MC e

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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.
 

MATRICES, FUR1 2012 VCAA 6 MC1
  

Let   
 

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

A.   

B.   

C.   

D.   

E.   

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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, .
 


 

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
 

MATRICES, FUR1 2013 VCAA 3 MC abc

MATRICES, FUR1 2013 VCAA 3 MC de

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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.     B.    
       
C.     D.       
       
E.        

 

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MATRICES, FUR1 2006 VCAA 5 MC

A company makes Regular (), Queen () and King () 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 , where
 


 

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  by forming a matrix product with the matrix, , where
 


 

This new price matrix is

MATRICES, FUR1 2006 VCAA 5 MC ab 1

MATRICES, FUR1 2006 VCAA 5 MC cd 1

MATRICES, FUR1 2006 VCAA 5 MC e

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MATRICES, FUR1 2007 VCAA 6 MC

A colony of fruit bats feeds nightly at three different locations, and

Initially, the number of bats from the colony feeding at each of the locations was as follows.

MATRICES, FUR1 2007 VCAA 6 MC

The bats change feeding locations according to the following transition matrix

 

If this pattern of feeding continues, the number of bats feeding at location in the long term will be closest to

A.   

B.   

C.   

D.   

E.   

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MATRICES, FUR1 2011 VCAA 7 MC

Each night, a large group of mountain goats sleep at one of two locations,  or .

On the first night, equal numbers of goats are observed to be sleeping at each location.

From night to night, goats change their sleeping locations according to a transition matrix .

It is expected that, in the long term, more goats will sleep at location than location .

Assuming the total number of goats remains constant, a transition matrix  that would predict this outcome is

MATRICES, FUR1 2011 VCAA 7 MC ab

MATRICES, FUR1 2011 VCAA 7 MC cd

MATRICES, FUR1 2011 VCAA 7 MC e

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MATRICES, FUR1 2014 VCAA 8 MC

Wendy will have lunch with one of her friends each day of this week.

Her friends are Angela (A), Betty (B), Craig (C), Daniel (D) and Edgar (E).

On Monday, Wendy will have lunch with Craig.

Wendy will use the transition matrix below to choose a friend to have lunch with for the next four days of the week.

 

The order in which Wendy has lunch with her friends for the next four days is

A.   Angela, Betty, Craig, Daniel

B.   Daniel, Betty, Angela, Craig

C.   Daniel, Betty, Angela, Edgar

D.   Edgar, Angela, Daniel, Betty

E.   Edgar, Daniel, Betty, Angela 

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MATRICES, FUR1 2014 VCAA 5 MC

Students from Year 7 and Year 8 in a school sold trees to raise funds for a school trip.

The number of large, medium and small trees that were sold by each year group is shown in the table below.

The large trees were sold for $32 each, the medium trees were sold for $26 each and the small trees were sold for $18 each.

A matrix product that can be used to calculate the amount, in dollars, raised by each year group by selling trees is

VCAA MATRICES FUR1 2014 5aii

VCAA MATRICES FUR1 2014 5aiii

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MATRICES, FUR2 2015 VCAA 3

A new model for the number of students in the school after each assessment takes into account the number of students who are expected to leave the school after each assessment.

After each assessment, students are classified as beginner (), intermediate (), advanced () or left the school ().

Let matrix be the transition matrix for this new model.

Matrix , shown below, contains the percentages of students who are expected to change their ability level or leave the school after each assessment.
  

  1. An incomplete transition diagram for matrix is shown below.
  2. Complete the transition diagram by adding the missing information.   (2 marks)


     

    Matrices, FUR2 2015 VCAA 3

The number of students at each level, immediately before the first assessment of the year, is shown in matrix below.

Matrix , repeated below, contains the percentages of students who are expected to change their ability level or leave the school after each assessment.

  1. What percentage of students is expected to leave the school after the first assessment?   (1 mark)

  2. How many advanced-level students are expected to be in the school after two assessments.
  3. Write your answer correct to the nearest whole number.   (1 mark)

  4. After how many assessments is the number of students in the school, correct to the nearest whole number, first expected to drop below 50?   (1 mark)

Another model for the number of students in the school after each assessment takes into account the number of students who are expected to join the school after each assessment.

Let be the state matrix that contains the number of students in the school immediately after assessments.

Let be the matrix that contains the number of students who join the school after each assessment. 

Matrix is shown below.

The expected number of students in the school after assessments can be determined using the matrix equation

where


 

  1. Consider the intermediate-level students expected to be in the school after three assessments.
  2. How many are expected to become advanced-level students after the next assessment?
  3. Write your answer correct to the nearest whole number.   (2 marks)

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  1.  
Show Worked Solution
a.   

 

b.  

 

♦♦ Mean mark of parts (b)-(e) was 32%.


 

c.  

 

 

d.  


 

MARKER’S COMMENT: Understand why the common error is incorrect!
e.  
   
 
   

 

MATRICES, FUR2 2012 VCAA 3

When a new industrial site was established at the beginning of 2011, there were 350 staff at the site. 

The staff comprised 100 apprentices (), 200 operators () and 50 professionals (). 

At the beginning of each year, staff can choose to stay in the same job, move to a different job at the site or leave the site ().

The number of staff in each category at the beginning of 2011 is given in the matrix

The transition diagram below shows the way in which staff are expected to change their jobs at the site each year.

Matrices, FUR2 2012 VCAA 3

  1. How many staff at the site are expected to be working in their same jobs after one year?   (1 mark)

The information in the transition diagram has been used to write the transition matrix .

  1. Explain the meaning of the entry in the fourth row and fourth column of transition matrix .   (1 mark)

If staff at the site continue to change their jobs in this way, the matrix  will contain the number of apprentices (), operators (), professionals () and staff who leave the site () at the beginning of the th year.

  1. Use the rule    to find

     

    1.    (1 mark)

    2. the expected number of operators at the site at the beginning of 2013   (1 mark)

    3. the beginning of which year the number of operators at the site first drops below 30   (1 mark)

    4. the total number of staff at the site in the longer term.   (1 mark)

Suppose the manager decides to bring 30 new apprentices, 20 new operators and 10 new professionals to the site at the beginning of each year.

The matrix will then be given by

  where     and  

  1. Find the expected number of operators at the site at the beginning of 2013.   (2 marks) 

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a. 

♦ Mean mark for parts (a)-(d) (combined) was 37%.


 

b.   

MARKER’S COMMENT: Most students couldn’t explain the meaning of the 1.0 figure in this context.

 

c.i.  
   
   

 

c.ii.  
   

 

 

c.iii.  
   
 
   

 

MARKER’S COMMENT: “In the 10th year” recieved no marks – make sure you answer with specific years when actual years are used.

 
c.iv.  


 

d.  
   
     
 
   
MARKER’S COMMENT: A common error was  . Know why this is not correct!

 

MATRICES, FUR2 2008 VCAA 2

The following transition matrix, , is used to help predict class attendance of History students at the university on a lecture-by-lecture basis.

  

is the attendance matrix for the first History lecture.

indicates that 540 History students attended the first lecture and 36 History students did not attend the first lecture.

  1. Use and to

     

    1. determine the attendance matrix for the second lecture.   (1 mark)

    2. predict the number of History students attending the fifth lecture.   (1 mark)

  2. Write down a matrix equation for in terms of , and .   (1 mark)

The History lecture can be transferred to a smaller lecture theatre when the number of students predicted to attend falls below 400.

  1. For which lecture can this first be done?   (1 mark)

  2. In the long term, how many History students are predicted to attend lectures?   (1 mark)

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a.i.  
   
   

 

  ii.  
   
   

 

 
b.  
 

c.   
 


 

d.