There are 225 senior students studying Cg /3s-1bysw gnomputer Science and 169 senior students studying Mathematicgws3-1gn /byss at a university. According to an academic survey, 20 % of these students tell they will pursue a postgraduate degree, 8 % will start a business, 52 % will get employed, 15 % will start freelancing and the remaining students will become research assistants.

The column matrix

represents the number of senior students studying
Computer Science and Mathematics.

1.Write down a row matrix, R, to represent the percentages, in decimal form, of senior students who will choose one of the five routes after graduation.

2.Hence calculate the product A=SR. Give each element $a_{ij}$ of the matrix A correct to the nearest whole number.

3.In the context of this problem, explain what the element a$_{15}$ means.

The cost for textbooks per year for a computer science student is 1245 and for a mathematics student is 889.

4.Write down a matrix calculation that gives the total cost for textbooks paid by all the senior students studying Computer Science and Mathematics.

5.Hence calculate the total cost for all the textbooks. Give your answer correct to the nearest dollar.
Hence the total cost for the textbooks is   .

  The following transition diagram reflects the proportions of customers that Qat a;;:8-evv )cqtlrcg u d5o*qvar Airwacgvvvo la dru):qt;*85 ;qc-eys loses to its competitor airlines each year, and vice versa.


1.Construct a transition matrix T with elements in decimal form.

2.Interpret the meaning of the elements with values
2.1.
0.15
2.2.
0.75

Assume that the initial state of the market share is

3.Determine the market share of Qatar Airways after 5 years.
Hence the market share of Qatar Airways after 5 years will be   %.

  Domino's Pizza owns two pizzerias at Asia Mall and Metrocity shopping ce/b 2.zy)uhcbjy4ytg 2ntres. The number of Pacific Veggie, Pepperoni and Buffalo Chicken large pizzas sold during the last week at 2 / cyy4ty2ubzg).bjh the two pizzerias is shown in the table below.

The selling price of each type of pizza is shown in the table below.

1.Write down a matrix multiplication that finds the total amount of income from sales of the three types of pizzas that each pizzeria generated during the last week.

2.Hence find the total amount of income from sales of the three types of pizzas that each pizzeria generated during the last week. Give your answers correct to two decimal places.
Asia Mall:  
Metrocity:  .
​
​

​

  Data scientists, web duym5 1y jauf8n:l*l g(esigners and developers are paid according to an
industry
industry standard. The total annual salary spend for three tech startups paying to the industry standard are summarised in the table below.

Let x, y and z represent the salaries, in thousand dollars, for data scientists, web designers and web developers respectively.

1.Write down a system of three linear equations in terms of x, y and zthat represent the information in the table above.

2.Using matrices, solve the system of linear equations from part (a) to determine the salaries for the three roles.
Data Scientist:  
Web Designer:  
Web Developer:  

Data Quant is a tech startup that also pays to the industry standard and employs 10 data scientists, 4 web designers and 6 web developers.

3.Calculate the exact value of the total salary bill for Data Quant.The total salary bill for Data Quant is   .

  The Brown, Miller and Taylor families pay utility 5o2zmf vu)+2 gchn3cq bills for their houses each month. The table below shows the amount of electricity, water and gas consumed during January by each family, and the total cost o5u+ qv3 hmoz g2)f2nccf the utilities.


Let x, y and z represent the prices, in dollars, for 1 kWh of electricity, 1 $m^3$ of water and 1 $m^3$ of gas, respectively.

1.Write down a system of three linear equations in terms of x, y and z that represents the information in the table above.

2.Using matrices, find the price for each of the utility.
Electricity:    cents per kWh
Water:   dollars per $m^3$
Gas:   cents per $m^3$
​

The Smith family also pay utility bills each month. The table below shows the amount of electricity, water and gas consumed during January by the Smith family.

3.Calculate the total cost of the utilities for the Smiths.
The total cost of the utilities for the Smiths is ≈  .

  Taste of Home Magazine recommends using a combination onwka7f32w1bl f Cheddar, Brie and Swiss when putting together cheese boards for parties. The recommended total cheese board size for a party of 10 - 15 people is 1 kilogram. The table below shows the weight, in hundred of grams, of each kind of cheese required to make one kilogram of cheese combination, and lb af 3wnk712wthe cost of making each combination.


1.By setting up a system of linear equations and using matrices, find
the price per kilogram of each type of cheese.
Cheddar:
   per kg
Brie:
   per kg
Swiss:
   per kg
​
John prepares a cheese board with proportion of each cheese type, in hundred grams, as shown in the table below.

2.Calculate the amount of money John spent on cheese.
The amount John spent on cheese is   .

  Three Internet Servijjmbkg 5 sga)whhqcig++(+er*0 5.gi ce Providers (ISPs) are available in a small town. During the year, ISP A is expected to retain 85 % of its customers; 10 % will be lost to ISP B and 5 % to ISP C. ISP B is expected to retain 80 % of its customers; 10 % will be lost to each of the other two ISPs. ISP C is expected to retain 75gwj gis+e+ikahc.mggq5)hj r 5 0+ (b* % of its customers; 15 % will be lost to ISP A and 10 % to ISP B.

1.Write down a transition matrix that describes the exchange of market shares between the three ISPs during the year.

The current market share held by ISP A is 0.2, by ISP B is 0.3 and by ISP C is 0.5.

2.Find the market share held by each ISP after one year.
ISP A:   .
ISP B:   .
ISP C:   .
​
3.Find the market share held by each ISP after five years if the same trend of market share exchanges between the three ISPs continues.
Hence the market share held by each ISP after five years are
ISP A:  .
ISP B:  .
ISP C:  .

  When a ball is thrown from the top of a tall building, its heidx iyjr0hlhe4,.6k0kq ;qr1 vght above the ground afte;d 0lxj.hyk0 re r64i1qqhvk,r t seconds is given by s(t)=at$^2$+bt+c , where a,b,c∈R and s(t) is measured in metres. After 1 second, the ball is 84.3 m above the ground; after 2 seconds, 93.9 m; after 8 seconds, 42.3 m.

1.1Write down a system of three linear equations in terms of a, b and c.

1.2.Hence find the values of a, b and c.
Hence we find a=−  ,b=  ,c=  .

2.Find the height of the building.
s(t)=  m.
3.Find the time it takes for the ball to hit the ground.
t is(≈)    seconds.

  When a coin is thrown from the top of a skyscraper, itskg-ri 0we1 xx+fj9l ) dwtqu5* height above the ground after t seconds is given by s(tt)kd wg e+5j-uxl0 9qw*frx1i)=at$^2$+bt+c , where a,b,c∈R and s(t) is measured in metres. After 1 second, the coin is 179.3 m above the ground; after 2 seconds, 188.2 m; after 6 seconds, 159.8 m.

1.1.Write down a system of three linear equations in terms of a, b and c.

1.2.Hence find the values of a, b and c.
Hence we find a=−  ,b=  ,c=  

2.Find the height of the skyscraper.
s(t)=  m.
3.Find the time it takes for the coin to hit the ground.
t is(≈)    seconds.

  Two competing radio stations, station A and station B, each hap ;mjo,es u:7je-g8y lve 50 % of the listener market at some point in time. Over each one-year pe:8gm j,op;us7j e -elyriod, station A manages to take away 15 % of station B's share, and station B manages to take away 10 % of station A's share.

1.Write down a transition matrix that describes the exchange of market shares between the two stations over each one-year period.

2.Find the market share held by each station after one year.
Station A:  
Station B:  

3.Write down the market shares of stations A and B over a five-year period.

4.Find the market share held by each station in the long term if the same trend of market share exchanges between the two stations continues indefinitely.
Station A:  
Station B:  

  Austin allocates a portion of his employmexs87n 6f0bbqt2g yr ,tnt salary each month to investing and in2gy0btn tsq,8 6fx7brvests this money into two stock funds: A and B. He adjusts his investment portfolio each month according to the following transition diagram.


1.Construct a transition matrix T with elements in decimal form.

2.Interpret the meaning of the elements with values
2.1.
0.1
2.2.
0.7

The initial state of his investment portfolio is 100% in stock fund B.

3.1.Find the investment proportion in stock fund A after 3 months.
The investment proportion in stock A after 3 months will be   %
3.2.Determine the long term steady state proportion of his investment between the two stock funds.
Stock fund A:  
Stock fund B:  

  The vegetables sold at supermarkets in a town are sup;c5chhr3a7d) eso d/ qplied by three major retail suppliers: A, B and C. According to an analysis report, supplier ) do5 ;acce3hsdq7/hr A retains 80 % of their customers each year and lose 15 % to supplier B and 5 % to supplier C. Meanwhile, supplier B retains 70 % of their customers each year and lose 20 % to supplier A and 10 % to supplier C. Supplier C retains 75 % of their customers each year and lose 10 % to supplier A and 15 % to supplier B.

The report also shows that suppliers A, B and C currently hold a market share of 50 %, 25 % and 25 %, respectively.

1.Find the market share held by each supplier after three years.
Supplier A:  
Supplier B:  
Supplier C:  
2.Determine the steady state market share held by each supplier if the same trend remains unchanged.
Supplier A:  
Supplier B:  
Supplier C:  

  Laura creates a list of her favorite songs that includes three genre /ubr*3s*rovn c:1x tkyhq/m+s: Jazz, Slow Rock and Country. After her current song ends she randomly selects the next song and the probabilities of genre of the next song are outline s:h 1*3 /r/ *qubct+xkyrmvnod in the following table.

Laura starts her day with a Slow Rock song and is now listening to her fourth song.

1.Determine the genre of music she is currently most likely listening to.
the genre Laura is most likely listening to is  ----待选择----JazzSlow RockCountry .
2.Determine which genre of music she listens to most over the long term.
Therefore, over the long term, Laura listens to  ----待选择----JazzSlow RockCountry  most often.

  A discrete dynamical system is described by the following t ;k fe pn3z/j+m jyqnux70p 17.fvdso1ransition matrix, T, j 7pzqum;snp+f1 1nj.f7y30odv x /e k
$ T = \begin{vmatrix} 0.3 & 0.8 \\ 0.7 & 0.2 \end{vmatrix} $

The state of the system is defined by the proportions of population with a particular characteristic.

1.Use the characteristic polynomial of T to find its eigenvalues.

2.Find the corresponding eigenvectors of T.
Hence we get $ X_1 = \begin{vmatrix} -x \\ y \end{vmatrix} $
Hence we get $ X_2 = \begin{vmatrix} a \\ b \end{vmatrix} $
x=  ,y=  ; a=  ,b=  .
3.Hence find the steady state matrix s of the system.
$ S = \begin{vmatrix} x \\ y \end{vmatrix} $
x=  ,y=  .

  A biologist conducts an experiment to study the pollination preference of bummb)+.jp7zb*6 opet y s*m ug,yblebees' on different epm, mbsjg6 )z+*byo p7y tu*.floral types. In a flight cage,
240 bumblebees are free to choose between two species of floral: A. majus striatum or A. majus pseudomajus. The changes of pollination behaviors of these bumblebees after every minute are reflected in the following table.

Initially, 150 bumblebees choose A. majus striatum and 90 bumblebees choose A. majus pseudomajus.

1.Write down the initial state $s_0$ and the transition matrix T.
$ s_0 = \begin{vmatrix} a \\ b \end{vmatrix} $; a=  ,b=  .

2.Determine $T_{s_0}$ and interpret the result.
$T_{s_0}$=$\begin{vmatrix} a \\ b \end{vmatrix} $; a=  ,b=  .

3.Find the eigenvalues and corresponding eigenvectors of T.
$ X_1 = \begin{vmatrix} -a \\ b \end{vmatrix} $; a=  ,b=  .
$ X_2 = \begin{vmatrix} x \\ y \end{vmatrix} $; x=  ,y=  .
4.1.Write an expression for the number of bumblebees choosing to pollinate on A. majus pseudomajus after n minutes, n∈N.
the number of bumblebees choosing to pollinate on A. majus pseudomajus (the second row) after n minutes is
$A(n)=-x(y)^n+z; x=  ,y=  ,z=  .
4.2.Hence find the number of bumblebees choose to pollinate on A. majus pseudomajus in the long term.
The number is   .

  An information techno2 kcdo;sj:sd dlxq9)ov00x ailc:* h4logy (IT) company offers paid travelling vacation to its 160 employees every year. The employees can choose between travelling domestically or internationally. It is observed that 50 % of the employees who choose to travel domestically one year, choose internationally the next year. Conversely, 30 % of those who choose to travel internationally one year change to travel domestically the following year. For this year, 80 employees chose travelling domeslv: c*c idl)4a:9k0 dsxj0o 2 sho;dqxtically and 80 employees chose travelling internationally.

1.Write down the initial state $s_0$ and the transition matrix T.
$ T = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} $; $a_1$=  ,$b_1$=  ; $a_2$=  ,$b_2$=  .
$ s_0 = \begin{vmatrix} x_1 \\ y_1 \end{vmatrix} $ ; $x_1$=  ,$y_1$=  .

2.Determine $Ts_0$ and interpret the result.
$ Ts_0 = \begin{vmatrix} x_1 \\ y_1 \end{vmatrix} $ ; $x_1$=  ,$y_1$=  .
3.Find the eigenvalues and corresponding eigenvectors of T.
$ X_1 = \begin{vmatrix} -x_1 \\ y_1 \end{vmatrix} $ ; $x_1$=  ,$y_1$=  .
$ X_2 = \begin{vmatrix} x_2 \\ y_2 \end{vmatrix} $ ; $x_2$=  ,$y_2$=  .

4.1.Write an expression for the number of employees who choose travelling internationally after n years, n∈N.
an expression for the number of employees who choose travelling internationally (the second row) after n years is

I(n)=-a(b)$^n$+100 ; a=  ;b=  ;c=  .

4.2.Hence find the long term steady state number of employees to choose to travel internationally.

  Two grocery stores, store A and store-g*.g ep5sb kd B, serve in a small city. Each year, store A keeps 30 % of its customers while 70 % of them switch to sto*.b 5gd-kgsepre B. Store B keeps 60% of its customers while 40 % of them switch to store A.

1.Write down a transition matrix T representing the proportions of the customers moving between the two stores.
$ T = \begin{vmatrix} a & b \\ c & d \end{vmatrix} $ ; a=  ,b=  ,c=  ,d=  .

At the end of 2019, store A had 8360 customers while store B had 6820 customers.

2.Find the distribution of the customers between the two stores after two years.
Store A:  
Store B:  
​
3.1.Show that the eigenvalues of T are $λ_1$ =1 and $λ_2$ =−0.1.

3.2.Find a corresponding eigenvector for each eigenvalue from part (c) (i).
$ X_1 = \begin{vmatrix} a \\ b \end{vmatrix} $ ; a=  ,b=  .
$ X_2 = \begin{vmatrix} c \\ -d \end{vmatrix} $ ; c=  ,d=  .
3.2.Hence express T in the form T=PDP$^{−1}$.

4.show that


where n∈Z$^+$.

5.Hence find an expression for the number of customers buying groceries from store A after n years, where n∈Z$^+$

6.Verify your formula by finding the number of customers buying groceries from store A after two years and comparing with the value found in part (b).

7.Write down the long-term number of customers buying groceries from store A.
The number is   .

  A city has two major security guard companies-h5; t c+gp (zmz(g:uci4hpvm, company A and company B. Each year, 15 % of customers using company A move to company B and 5 % of thum4:+m z(gthzpivgc c; p 5(-he customers using company B move to company A. All additional losses and gains of customers by the companies can be ignored.

1.Write down a transition matrix T representing the movements between the two companies in a particular year.
$ T = \begin{vmatrix} a & b \\ c & d \end{vmatrix} $ ; a=  ,b=  ,c=  ,d=  .
2.1.Find the eigenvalues and corresponding eigenvectors of T.
$ X_1 = \begin{vmatrix} a \\ b \end{vmatrix} $ , a=  ,b=  ;
$ X_2 = \begin{vmatrix} c \\ -d \end{vmatrix} $ , c=  ,d=  .
2.2.Hence write down matrices P and D such that T=PDP$^{-1}$.

Initially company A and company B both have 3600 customers.

3.Find an expression for the number of customers company A has after n years, where n∈Z.
Hence the number of customers company A has after n years is
C(n)=a+b(c0.8$^n$) ; a  , b=   , c=  .

4.Hence write down the number of customers that company A can expect to have in the long term.
The number is   .

  Zoologists have been collec6pn*nd9 /* ggxzh:uhbting data about the migration habits of a particular species of mammals in two regions; region X and region Y. Each year 30% of the mammals move from r uz g *g6pbnnd/h9x:*hegion X to region Y and 15 % of the mammals move from region Y to region X. Assume that there are no mammal movements to or from any other neighboring regions.

1.Write down a transition matrix T representing the movements between the two regions in a particular year.
$ T = \begin{vmatrix} a & b \\ c & d \end{vmatrix} $ ; a=  , b=  , c=  , d=  .
2.1.Find the eigenvalues of T.
Hence the eigenvalues of T are λ$_1$=   and λ$_2$=  .
​
2.2.Find a corresponding eigenvector for each eigenvalue of T.
$ X_1 = \begin{vmatrix} a \\ b \end{vmatrix} $ ; a=  , b=  .
$ X_2 = \begin{vmatrix} A \\ -B \end{vmatrix} $ ; A=  , B=  .

2.3.Hence write down matrices P and D such that T=PDP$^{-1}$.
$ P = \begin{vmatrix} a & b \\ c & -d \end{vmatrix} $ ; a=  , b=  , c=  , d=  .
$ D = \begin{vmatrix} A & B \\ C & D \end{vmatrix} $ ; A=  , B=  , C=  , D=  .

Initially region X had 12600 and region Y had 16200 of these mammals.

3.Find an expression for the number of mammals living in region Y aftern years, where n∈Z$^+$ .

4.Hence write down the long-term number of mammals living in region Y.
The number is   .