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MATRICES, FUR1 2019 VCAA 8 MC

An airline parks all of its planes at Sydney airport or Melbourne airport overnight.

The transition diagram below shows the change in the location of the planes from night to night.
 


 

There are always `m` planes parked at Melbourne airport.

There are always `s` planes parked at Sydney airport.

Of the planes parked at Melbourne airport on Tuesday night, 12 had been parked at Sydney airport on Monday night.

How many planes does the airline have?

  1. 25
  2. 37
  3. 62
  4. 65
  5. 85
Show Answers Only

`E`

Show Worked Solution

`text(48% of planes parked in Sydney on Monday are parked)`

`text(in Melbourne on Tuesday.)`

`0.48 s` `= 12`
`s` `= 25`

 
`text(S) text(ince there are always)\ m\ text(planes parked in Melbourne)`

`=>\ text(12 from Melbourne must transfer to Sydney.)`

`0.2 m` `= 12`
`m` `= 60`

 

`:.\ text(Total planes)` `= 25 + 60`
  `= 85`

 
`=>  E`

MATRICES, FUR2 2017 VCAA 2

Junior students at a school must choose one elective activity in each of the four terms in 2018.

Students can choose from the areas of performance (`P`), sport (`S`) and technology (`T`).

The transition diagram below shows the way in which junior students are expected to change their choice of elective activity from term to term.

 

  1. Of the junior students who choose performance (`P`) in one term, what percentage are expected to choose sport (`S`) the next term?  (1 mark)

Matrix `J_1` lists the number of junior students who will be in each elective activity in Term 1.
 

`J_1 = [(300),(240),(210)]{:(P),(S),(T):}`
 

  1. 306 junior students are expected to choose sport (`S`) in Term 2.
     
    Complete the calculation below to show this.  (1 mark)
     


     

  2. In Term 4, how many junior students in total are expected to participate in performance (`P`) or sport (`S`) or technology (`T`)?  (1 mark)
Show Answers Only
  1. `text(40%)`
  2.  `300 xx 0.4 + 240 xx 0.6 + 210 xx 0.2 = 306`
  3. `750`
Show Worked Solution

a.   `text(40%)`
 

b.  `300 xx 0.4 + 240 xx 0.6 + 210 xx 0.2 = 306`

♦♦ Mean mark part (c) 30%.

MARKER’S COMMENT: No matrix calculations were required here.


c.   `text(Each term, every student will do)\ P\ text(or)\ S\ text(or)\ T.`

`:. text(Total students (Term 4))` `=\ text(Total students (Term 1))`
  `= 300 + 240 + 210`
  `= 750`

MATRICES, FUR2 2010 VCAA 2

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

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

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

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 `T`.
 

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

  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) 
Show Answers Only
  1. `text(It means that 20% of the players doing heavy training one)`

     

    `text(week will switch to moderate training the next.)`

  2. `text(66 players)`
  3. `text(6 fewer players)`
  4. `text(See Worked Solutions)`
Show Worked Solution

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

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

`text(training the next.)`

 

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

 

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

`text(in week 2.)`

 

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

`text(In week 3,)`

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

 

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

`= 150 – 144`

`= 6`

 

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

`text(weeks 8 and week 9,)`

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

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

 

`:. S_8 = S_9`

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

MATRICES, 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.     `828`

B.   `1423`

C.   `2251`

D.   `4269`

E.   `6891`

 

Part 2

The election campaign will run for ten weeks.

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

A.   Rob by about 50 votes.

B.   Rob by about 100 votes.

C.   Rob by fewer than 10 votes.

D.   Anna by about 100 votes.

E.   Anna by about 200 votes.

Show Answers Only

`text(Part 1:)\ C`

`text(Part 2:)\ E`

Show Worked Solution

`text(Part 1)`

`text(Students expected to change)`

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

`= 2251`

`=> C`

 

`text(Part 2)`

♦ Mean mark 48%.

`text(After 1 week,)`

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

 

`text(After 10 weeks,)`

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

 

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

`4663 – 4479 = 184\ text(votes)`

`=> E`