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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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  1.  
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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
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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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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.

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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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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.
     

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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.   

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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.  
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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.  


 

GRAPHS, FUR2 2012 VCAA 3

A company repairs phones and laptops.

Let   be the number of phones repaired each day

  be the number of laptops repaired each day.

It takes 35 minutes to repair a phone and 50 minutes to repair a laptop. 

The constraints on the company are as follows.

Constraint 1    

Constraint 2    

Constraint 3    

Constraint 4    

  1. Explain the meaning of Constraint 3 in terms of the time available to repair phones and laptops.  (1 mark)
  2. Constraint 4 describes the maximum number of phones that may be repaired relative to the number of laptops repaired.

     

    Use this constraint to complete the following sentence.

     

    For every ten phones repaired, at most _______ laptops may be repaired.  (1 mark)

The line  is drawn on the graph below.

GRAPHS, FUR2 2012 VCAA 3

  1. Draw the line    on the graph.  (1 mark)
  2. Within Constraints 1 to 4, what is the maximum number of laptops that can be repaired each day?  (1 mark)
  3. On a day in which exactly nine laptops are repaired, what is the maximum number of phones that can be repaired?  (1 mark)

The profit from repairing one phone is $60 and the profit from repairing one laptop is $100.

    1. Determine the number of phones and the number of laptops that should be repaired each day in order to maximise the total profit.  (2 marks)
    2. What is the maximum total profit per day that the company can obtain from repairing phones and laptops?  (1 mark) 
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a.  

 

b.  

 

 

 

c.    GRAPHS, FUR2 2012 VCAA 3 Answer

 

d.  

♦♦ Mean mark of parts (c) – (f) combined was 32%.

 

e.  

 

 

f.i.   

 

 

 

f.ii.  

GRAPHS, FUR2 2012 VCAA 1

The cost, , in dollars, of making phones, is shown by the line in the graph below.

GRAPHS, FUR2 2012 VCAA 1

    1. Calculate the gradient of the line, , drawn above.  (1 mark)
    2. Write an equation for the cost, , in dollars, of making phones.  (1 mark)

  2. The revenue, , in dollars, obtained from selling phones is given by  .

    1. Draw this line on the graph above.  (1 mark)
    2. How many phones would need to be sold to obtain $54 000 in revenue?  (1 mark)
  3. Determine the number of phones that would need to be made and sold to break even.  (1 mark) 
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a.i.  

 
 

 

a.ii.  

 

b.i.    GRAPHS, FUR2 2012 VCAA 1 Answer

 

b.ii.   
 
   

 

c.  

GEOMETRY, FUR2 2012 VCAA 4

has three triangular sections, as shown in the diagram below. 

Triangle is a right-angled triangle. 

Length is 10 m and length is 14 m. 

Angle = angle = angle = 30°
 

Geometry and trigonometry, FUR2 2012 VCAA 4
 

  1. Calculate the length, .

     

    Write your answer, in metres, correct to two decimal places.  (1 mark)

  2. Determine the area of triangle .

     

    Write your answer, in m², correct to one decimal place.  (1 mark)

  3. Triangles and are similar.

     

    The area of triangle is 35 m².

     

    Find the area of triangle , in m².  (2 marks)

  4. Determine angle .

     

    Write your answer, correct to the nearest degree.  (2 marks)

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  2. ²
  3. ²
Show Worked Solution

a.  

 
 

 

b.  

 
 
  ²

 

c.  

   
  ²

 

d.  

 

 

 

 

 

GEOMETRY, FUR2 2012 VCAA 3

A tree is growing near the block of land.

The base of the tree, , is at the same level as the corners, and , of the block of land.
 

Geometry and trigonometry, FUR2 2012 VCAA 3
 

  1. Show that, correct to two decimal places, distance is 41.81 metres.  (1 mark)
  2. From point , the angle of elevation to the top of the tree is 22°.

     

    Calculate the height of the tree.

     

    Write your answer, in metres, correct to one decimal place.  (1 mark) 

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a.   
   

 

 

 
 

 

b.  

♦♦ Exact data unavailable although this part was highlighted as “very poorly answered”.

 
GEOMETRY, FUR2 2012 VCAA 3 Answer

 
 

GEOMETRY, FUR2 2012 VCAA 1

A rectangular block of land has width 50 metres and length 85 metres.

  1. Calculate the area of this block of land.

     

    Write your answer in m².  (1 mark)

In order to build a house, the builders dig a hole in the block of land.

The hole has the shape of a right-triangular prism, .

The width = 20 m, length = 25 m and height = 4 m are shown in the diagram below.

GEOMETRY, FUR2 2012 VCAA 11

  1. Calculate the volume of the right-triangular prism, .

     

    Write your answer in m³.  (1 mark)

Once the right-triangular prism shape has been dug, a fence will be placed along the two sloping edges,  and , and along the edges and .

Geometry and trigonometry, FUR2 2012 VCAA 1_2

  1. Calculate the total length of fencing that will be required. 

     

    Write your answer, in metres, correct to one decimal place.  (1 mark)

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  1. ²
  2. ³
Show Worked Solution
a.   
    ²

 

b.   
   
    ²

 

  ³

 

c.  

GEOMETRY, FUR2 2012 VCAA 1

 
 

 

MARKER’S COMMENT: A poorly done question with many students failing to include the two 20 metre sections of fencing.

CORE*, FUR2 2013 VCAA 3

Hugo paid $7500 for a second bike under a hire-purchase agreement.

A flat interest rate of 8% per annum was charged.

He will fully repay the principal and the interest in 24 equal monthly instalments.

  1. Determine the monthly instalment that Hugo will pay.

     

    Write your answer in dollars, correct to the nearest cent.   (2 marks)

  2. Find the effective rate of interest per annum charged on this hire-purchase agreement.

     

    Write your answer as a percentage, correct to two decimal places.   (1 mark)

  3. Explain why the effective interest rate per annum is higher than the flat interest rate per annum.   (1 mark)

  4. The value of his second bike, purchased for $7500, will be depreciated each year using the reducing balance method of depreciation.

     

    One year after it was purchased, this bike was valued at $6375.

     

    Determine the value of the bike five years after it was purchased. 

     

    Write your answer, correct to the nearest dollar.   (2 marks)

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Show Worked Solution
a.   
   
   
 
 

 

♦ Mean mark of all parts combined was 37%.
b.   
   
   

  
c.   


  

d.   
   
   

CORE*, FUR2 2013 VCAA 2

Hugo won $5000 in a road race and invested this sum at an interest rate of 4.8% per annum compounding monthly.

  1. What is the value of Hugo’s investment after 12 months?
  2. Write your answer in dollars, correct to the nearest cent.   (1 mark)

    1. Suppose instead that at the end of each month Hugo added $200 to his initial investment of $5000.
    2. Find the value of this investment immediately after the 12th monthly payment of $200 is made.
    3. Write your answer in dollars, correct to the nearest cent.   (1 mark)
    4. Assume Hugo follows the investment that is described in part b.i.
    5. Determine the total interest he would earn over the 12-month period.
    6. Write your answer in dollars, correct to the nearest cent.   (1 mark)

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a.  

♦ Mean mark for all parts combined was 39%.


 

 

b.i.   

 

 

b.ii.  

CORE*, FUR2 2013 VCAA 1

Hugo is a professional bike rider.

The value of his bike will be depreciated over time using the flat rate method of depreciation.

The graph below shows his bike’s initial purchase price and its value at the end of each year for a period of three years.
 

  1. What was the initial purchase price of the bike?   (1 mark)
    1. Show that the bike depreciates in value by $1500 each year.   (1 mark)
    2. Assume that the bike’s value continues to depreciate by $1500 each year.
    3. Determine its value five years after it was purchased.   (1 mark)

The unit cost method of depreciation can also be used to depreciate the value of the bike.

In a two-year period, the total depreciation calculated at $0.25 per kilometre travelled will equal the depreciation calculated using the flat rate method of depreciation as described above.

  1. Determine the number of kilometres the bike travels in the two-year period.   (1 mark)

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  2. i. 
    ii.

Show Worked Solution

a.  
  

b.i.   

 

  
b.ii.   

 

  
c.   

 

  

GRAPHS, FUR2 2013 VCAA 4

The school group may hire two types of camp sites: powered sites and unpowered sites.

Let   be the number of powered camp sites hired

  be the number of unpowered camp sites hired.

Inequality 1 and inequality 2 give some restrictions on and .

inequality 1   

inequality 2   

There are 48 students to accommodate in total.

A powered camp site can accommodate up to six students and an unpowered camp site can accommodate up to four students.

Inequality 3 gives the restrictions on and based on the maximum number of students who can be accommodated at each type of camp site.

inequality 3  

  1. Write down the values of and in inequality 3.  (1 mark)

     


          Graphs and relations, FUR2 2013 VCAA 4

School groups must hire at least two unpowered camp sites for every powered camp site they hire.

  1. Write this restriction in terms of and as inequality 4.  (1 mark)

     

The graph below shows the three lines that represent the boundaries of inequalities 1, 3 and 4.

GRAPHS, FUR2 2013 VCAA 4

  1. On the graph above, show the points that satisfy inequalities 1, 2, 3 and 4.  (1 mark)
  2. Determine the minimum number of camp sites that the school would need to hire.  (1 mark)
  3. The cost of each powered camp site is $60 per day and the cost of each unpowered camp site is $30 per day.

     

    1. Find the minimum cost per day, in total, of accommodating 48 students.  (1 mark)

School regulations require boys and girls to be accommodated separately.

The girls must all use one type of camp site and the boys must all use the other type of camp site.

  1. Determine the minimum cost per day, in total, of accommodating the 48 students if there is an equal number of boys and girls.  (1 mark)
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  3.  
    GRAPHS, FUR2 2013 VCAA 4 Answer
Show Worked Solution

a.   

 

b.  

 

c.    GRAPHS, FUR2 2013 VCAA 4 Answer

 

d.  

♦♦ Mean mark of parts (c)-(e) combined was 22%.

 

 

e.i.  

 

 

e.ii.   

 

GRAPHS, FUR2 2013 VCAA 3

A rock-climbing activity will be offered to students at the camp on one afternoon.

Each student who participates will pay $24.

The organisers have to pay the rock-climbing instructor $260 for the afternoon. They also have to pay an insurance cost of $6 per student.

Let be the total number of students who participate in rock climbing.

  1. Write an expression for the profit that the organisers will make in terms of .  (1 mark)
  2. The organisers want to make a profit of at least $500.

     

    Determine the minimum number of students who will need to participate in rock climbing.  (1 mark)

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Show Worked Solution

a.  

 

 

b.  

GEOMETRY, FUR2 2014 VCAA 3

The chicken coop contains a circular water dish.

Water flows into the dish from a water container.

The water container is in the shape of a cylinder with a hemispherical top.

The water container and the dish are shown in the diagrams below.
 

GEOMETRY, FUR2 2014 VCAA 3
 

The cylindrical part of the water container has a diameter of 10 cm and a height of 15 cm.

The hemisphere has a radius of 5 cm.

  1. What is the surface area of the hemispherical top of the water container?

     

    Write your answer, correct to the nearest square centimetre.  (1 mark)

  2. What is the maximum volume of water that the water container can hold?

     

    Write your answer, correct to the nearest cubic centimetre.  (2 marks)

The eating space of the chicken coop also has a feed container.

 

The feed container is similar in shape to the water container.

 

The volume of the water container is three-quarters of the volume of the feed container.

 

The surface area of the water container is 628 cm².

  1. What is the surface area of the feed container?

     

    Write your answer, correct to the nearest square centimetre.  (2 marks) 

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  1. ²
  2. ³
  3. ²
Show Worked Solution
a.   
   
    ²²

 

b.   
   
    ³
 
   
    ³

 

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

³³

 

c.   
 
 

 

²

GEOMETRY, FUR2 2013 VCAA 3

A grassed region in the athletics ground is shown shaded in the diagram below.

Geometry and Trig, FUR2 2013 VCAA 3_1

The perimeter of the grassed region comprises two parallel lines, and , each 100 m in length, and two semi-circles, and .

In total, the perimeter of the grassed region is 400 m.

  1. The diameter of the semi-circle is 63.66 m, correct to two decimal places.

     

    Show how this value could be obtained.  (1 mark)

  2. Determine the area of the grassed region, correct to the nearest square metre.  (1 mark)

A running track, shown shaded in the diagram below, surrounds the grassed region. This running track is 8 m wide at all points.

GEOMETRY, FUR2 2013 VCAA 3

  1. The running track is to be resurfaced with special rubber material that is 0.1 m deep.

     

    Find the volume of rubber material that is needed to resurface the running track.

     

    Write your answer, correct to the nearest cubic metre.  (2 marks)

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  2. ²²
  3. ³³
Show Worked Solution

a.   

 
 

 

b.   
   
   
    ²²

 

c.  

²

 

♦ Mean mark of part (c) 34%.

³³

GEOMETRY, FUR2 2013 VCAA 2

A concrete staircase leading up to the grandstand has 10 steps.

The staircase is 1.6 m high and 3.0 m deep.

Its cross-section comprises identical rectangles.

One of these rectangles is shaded in the diagram below.

Geometry and Trig, FUR2 2013 VCAA 2_1

  1. Find the area of the shaded rectangle in square metres.  (1 mark)

The concrete staircase is 2.5 m wide.

GEOMETRY, FUR2 2013 VCAA 2

  1. Find the volume of the solid concrete staircase in cubic metres.  (2 marks)
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  1. ²
  2. ³
Show Worked Solution

a.  

MARKER’S COMMENT: Keep units in metres as many students made mistakes converting cm² to m².

  ²

 

b.  

  ²
  ³

GEOMETRY, FUR2 2013 VCAA 1

A spectator, , in the grandstand of an athletics ground is 13 m vertically above point .

Competitor , on the athletics ground, is at a horizontal distance of 40 m from .
  

Geometry and Trig, FUR2 2013 VCAA 1_1
 

  1. Find the distance, , correct to the nearest metre.  (1 mark)

Competitor  is 40 m from and competitor is 52 m from .

The angle is 113°.
 

Geometry and Trig, FUR2 2013 VCAA 1_2
 

    1. Calculate the distance, , correct to the nearest metre.  (1 mark)
    2. Find the area of triangle , correct to the nearest square metre.  (1 mark)
  2. Determine the angle of elevation of spectator from competitor , correct to the nearest degree.

    Note that , and are on the same horizontal level.  (1 mark)
     

Geometry and Trig, FUR2 2013 VCAA 1_3

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    2. ²²
Show Worked Solution

a.  

 
 

 

b.i.   

 
 

 

b.ii.  

 
 
  ²²

 

c.    Geometry-FUR2-2013-VCAA-1 Answer
 
 

CORE*, FUR2 2014 VCAA 3

The cricket club had invested $45 550 in an account for four years.

After four years of compounding interest, the value of the investment was $60 000.

  1. How much interest was earned during the four years of this investment?   (1 mark)

Interest on the account had been calculated and paid quarterly.

  1. What was the annual rate of interest for this investment?
  2. Write your answer, correct to one decimal place.   (1 mark)

The $60 000 was re-invested in another account for 12 months.

The new account paid interest at the rate of 7.2% per annum, compounding monthly.

At the end of each month, the cricket club added an additional $885 to the investment.

    1. The equation below can be used to determine the account balance at the end of the first month, immediately after the $885 was added.
    2. Complete the equation by filling in the boxes.   (1 mark)


      BUSINESS, FUR2 2014 VCAA 3
    3. What was the account balance at the end of 12 months?
    4. Write your answer, correct to the nearest dollar.   (1 mark)

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Show Worked Solution

a.  

 

♦ Mean mark of all parts (combined) was 40%.

b.  

 

 

c.i.   

 

c.ii.