Aussie Maths & Science Teachers: Save your time with SmarterEd
Each day, members of a swim centre can choose to attend a morning session \((M)\), an afternoon session \((A)\) or no session \((N)\).
The transition diagram below shows the transition from day to day.
The transition diagram is incomplete.
Which one of the following transition matrices represents this transition diagram?
A new colony of endangered marsupials is established on a remote island.
For one week, the marsupials can feed from only one of three feeding stations: `A`, `B` or `C`.
On Monday, 50% of the marsupials were observed feeding at station `A` and 50% were observed feeding at station `B`. No marsupials were observed feeding `C`.
The marsupials are expected to change their feeding stations each day this week according to the transition matrix `T`.
`qquadqquadqquadqquad \ text(this day)`
`P = {:(qquad\ A quadquadqquad \ B quadquad \ C ),([(0.4,0.1, 0.2),(0.2,0.5,0.2),(0.4,0.4,0.6)]{:(A),(B),(C):} qquad text(next day)):}`
Let `S_n` represent the state matrix showing the percentage of marsupials observed feeding at each feeding station `n` days after Monday of this week.
The matrix recurrence rule `S_{n+1} = TS_n` is used to model this situation.
From Tuesday to Wednesday, the percentage of marsupials who are not expected to change their feeding location is
An offer to buy the Westmall shopping centre was made by a competitor.
One market research project suggested that if the Westmall shopping centre were sold, each of the three centres (Westmall, Grandmall and Eastmall) would continue to have regular shoppers but would attract and lose shoppers on a weekly basis.
Let `S_n` be the state matrix that shows the expected number of shoppers at each of the three centres `n` weeks after Westmall is sold.
A matrix recurrence relation that generates values of `S_n` is
`S_(n+1) = T xx S_n`
`{:(quad qquad qquad qquad qquad qquad qquad qquad text(this week)),(qquad qquad qquad qquad qquad qquad quad \ W qquad quad G qquad quad \ E),(text(where)\ T = [(quad 0.80, 0.09, 0.10),(quad 0.12, 0.79, 0.10),(quad 0.08, 0.12, 0.80)]{:(W),(G),(E):}\ text(next week,) qquad qquad S_0 = [(250\ 000), (230\ 000), (200\ 000)]{:(W),(G),(E):}):}`
Using values from the recurrence relation above, the graph below shows the expected number of shoppers at Westmall, Grandmall and Eastmall for each of the 10 weeks after Westmall is sold.
Give your answer correct to the nearest thousand. (1 mark)
Write down the week number in which this is expected to occur. (1 mark)
Round your answer to the nearest whole number. (1 mark)
Consider the matrix recurrence relation below.
`S_0 = [(30),(20),(40)], quad S_(n+1) = TS_n qquad text(where)\ T = [(j, 0.3, l),(0.2, m, 0.3),(0.4, 0.2, n)]`
Matrix `T` is a regular transition matrix.
Given the information above and that `S_1 = [(42),(28),(20)]`, which one of the following is true?
In a particular supermarket, the three top-selling magazines are Angel (A), Bella (B) and Crystal (C).
The transition diagram below shows the way shoppers at this supermarket change their magazine choice from week to week.
A transition matrix that provides the same information as the transition diagram is
| A. |
`{:(qquadqquadquad this\ week),(qquadquadAqquadquad\ Bqquadquad\ C),([(0.55,0.70,0.35),(0.70,0.60,0.40),(0.35,0.40,0.40)]{:(A),(B),(C):}qquad n\ext\ week):}`
|
B. |
`{:(qquadqquadquad this\ week),(qquadquadAqquadquad\ Bqquadquad\ C),([(0.55,0.60,0.25),(0.45,0.15,0.35),(0,0.25,0.40)]{:(A),(B),(C):} qquad n\ext\ week):}`
|
| C. |
`{:(qquadqquadquad this\ week),(qquadquadAqquadquad\ Bqquadquad\ C),([(0.55,0.25,0.35),(0.45,0.60,0.25),(0,0.15,0.40)]{:(A),(B),(C):} qquad n\ext\ week):}`
|
D. |
`{:(qquadqquadquad this\ week),(qquadquadAqquadquad\ Bqquadquad\ C),([(0.55,0.25,0.35),(0.45,0.60,0.25),(0.35,0.15,0.40)]{:(A),(B),(C):} qquad n\ext\ week):}`
|
| E. |
`{:(qquadqquadquad this\ week),(qquadquadAqquadquad\ Bqquadquad\ C),([(0.55,0.25,0),(0.45,0.60,0.25),(0,0.15,0.75)]{:(A),(B),(C):} qquad n\ext\ week):}`
|
A water park is open from 9 am until 5 pm.
There are three activities, the pool `(P)`, the slide `(S)` and the water jets `(W)`, at the water park.
Children have been found to change their activity at the water park each half hour, as shown in the transition matrix, `T`, below.
`{:(qquad qquad qquad quadtext(this half year)),(qquad qquad qquad quad P qquad qquad S qquad quad W), (T = [(0.80,0.20,0.40),(0.05,0.60,0.10),(0.15,0.20,0.50)] {:(P),(S),(W):} text( next half year)):}`
A group of children has come to the water park for the whole day.
The percentage of these children who are expected to be at the slide `(S)` at closing time is closest to
Consider the matrix recurrence relation below.
`S_0 = [(40),(15),(20)], \ S_(n + 1) = TS_n` where `T = [(0.3,0.2,V),(0.2,0.2,W),(X,Y,Z)]`
Matrix `T` is a regular transition matrix.
Given the above and that `S_1 = [(29),(13),(33)]`, which of the following expressions is not true?
Each week, the 300 students at a primary school choose art (A), music (M) or sport (S) as an afternoon activity.
The transition matrix below shows how the students’ choices change from week to week.
`{:(qquadqquadqquadqquadtext(this week)),((qquadqquadqquadA,quadM,quadS)),(T=[(0.5,0.4,0.1),(0.3,0.4,0.4),(0.2,0.2,0.5)]{:(A),(M),(S):}quad{:text(next week):}):}`
Based on the information above, it can be concluded that, in the long term
Families in a country town were asked about their annual holidays.
Every year, these families choose between staying at home (H), travelling (T) and camping (C).
The transition diagram below shows the way families in the town change their holiday preferences from year to year.
A transition matrix that provides the same information as the transition diagram is
| A. | `{:(qquadqquadqquad\ text(from)),((qquadH,qquadT,qquadC)),([(0.30,0.75,0.65),(0.75,0.50,0.20),(0.65,0.20,0.60)]{:(H),(T),(C):}quad{:text(to):}):}` | B. | `{:(qquadqquadqquad\ text(from)),((qquadH,qquadT,qquadC)),([(0.30,0.30,0.40),(0.45,0.50,0),(0.25,0.20,0.60)]{:(H),(T),(C):}quad{:text(to):}):}` |
| C. | `{:(qquadqquadqquad\ text(from)),((qquadH,qquadT,qquadC)),([(0.30,0.30,0.40),(0.45,0.50,0.20),(0.25,0.20,0.60)]{:(H),(T),(C):}quad{:text(to):}):}` | D. | `{:(qquadqquadqquad\ text(from)),((qquadH,qquadT,qquadC)),([(0.30,0.30,0.40),(0.45,0.50,0.20),(0.25,0.20,0.40)]{:(H),(T),(C):}quad{:text(to):}):}` |
| E. | `{:(qquadqquadqquad\ text(from)),((qquadH,qquadT,qquadC)),([(0.30,0.45,0.25),(0.30,0.50,0.20),(0.40,0,0.60)]{:(H),(T),(C):}quad{:text(to):}):}` |
A new shopping centre called Shopper Heaven (`S`) is about to open. It will compete for customers with Eastown (`E`) and Noxland (`N`).
Market research suggests that each shopping centre will have a regular customer base but attract and lose customers on a weekly basis as follows.
80% of Shopper Heaven customers will return to Shopper Heaven next week
12% of Shopper Heaven customers will shop at Eastown next week
8% of Shopper Heaven customers will shop at Noxland next week
76% of Eastown customers will return to Eastown next week
9% of Eastown customers will shop at Shopper Heaven next week
15% of Eastown customers will shop at Noxland next week
85% of Noxland customers will return to Noxland next week
10% of Noxland customers will shop at Shopper Heaven next week
5% of Noxland customers will shop at Eastown next week
`qquad{:(qquadqquadqquadtext(this week)),((qquadqquadqquadS,E,N)),(T = [(qquadqquadqquadqquad),(),()]{:(S),(E),(N):}{:qquadtext(next week):}):}`
During the week that Shopper Heaven opened, it had 300 000 customers.
In the same week, Eastown had 120 000 customers and Noxland had 180 000 customers.
`qquadK_0 = [(qquadqquadqquad),(),()]{:(S),(E),(N):}`
`qquadK = [(194\ 983),(150\ 513),(254\ 504)]`
In 2009, the school entered a Rock Eisteddfod competition.
When rehearsals commenced in February, all students were asked whether they thought the school would make the state finals. The students’ responses, ‘yes’, ‘no’ or ‘undecided’ are shown in the initial state matrix `S_0`.
`S_0 = [(160),(120),(220)]{:(text(yes)),(text(no)),(text(undecided)):}`
Each week some students are expected to change their responses. The changes in their responses from one week to the next are modelled by the transition matrix `T` shown below.
`{:(qquadqquadqquadtext( response this week)),(qquadqquadquadtext( yes no undecided)),(T = [(0.85quad,0.35quad,0.60),(0.10quad,0.40quad,0.30),(0.05quad,0.25quad,0.10)]{:(text(yes)),(text(no)),(text(undecided)):}qquad{:(text(response)),(text(next week)):}):}`
The following diagram can also be used to display the information represented in the transition matrix `T`.
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.
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):}):}`
There are 30 children in a Year 6 class. Each week every child participates in one of three activities: cycling (C), orienteering (O) or swimming (S).
The activities that the children select each week change according to the transition matrix below.
`{:({:qquadqquadqquadqquadtext(this week):}),(qquadqquadqquad\ Cqquad\ OqquadquadS),(T = [(0.5,0.3, 0.3), (0.1,0.6,0.2), (0.4,0.1,0.5)]{:(C), (O), (S):}qquadtext(next week)):}`
From the transition matrix it can be concluded that
A. in the first week of the program, ten children do cycling, ten children do orienteering and ten children do swimming.
B. at least 50% of the children do not change their activities from the first week to the second week.
C. in the long term, all of the children will choose the same activity.
D. orienteering is the most popular activity in the first week.
E. 50% of the children will do swimming each week.
A coffee shop sells three types of coffee, Brazilian (B), Italian (I) and Kenyan (K). The regular customers buy one cup of coffee each per day and choose the type of coffee they buy according to the following transition matrix, `T`.
`{:({:qquadqquadqquadtext(choose today):}),(qquadqquadqquad\ BquadqquadIquadqquadK),(T = [(0.8,0.1,0.1), (0,0.8,0.1), (0.2,0.1,0.8)]{:(B), (I), (K):} qquadtext(choose tomorrow)):}`
On a particular day, 84 customers bought Brazilian coffee, 96 bought Italian coffee and 81 bought Kenyan coffee.
If these same customers continue to buy one cup of coffee each per day, the number of these customers who are expected to buy each of the three types of coffee in the long term is
A transition matrix, `T`, and a state matrix, `S_2`, are defined as follows.
`T=[(0.5,0,0.5),(0.5,0.5,0),(0,0.5,0.5)]\ \ \ \ \ S_2=[(300),(200),(100)]`
If `S_2 = TS_1`, the state matrix `S_1` is
