There are 225 senior students studying Computer Science and 169 seny;v- ru3+ltpi /:gd4u5gli zkior students studying Mathematics at a university. According to an academic survey,l;-z4+t3:rp gigul/d v uyki5 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 custo(gd+5oq2k zw vmers that Qatar Airways loses to its competitor aio v+zg 5qkd(w2rlines 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 Metrocitykeny)/ido6 8/-ndtvx u7c 5sx shopping centres. The number of Pacific Veggie, Pepperoni and Buffxv5uk6/ctxe / nn)7d d8ys o-ialo Chicken large pizzas sold during the last week at 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 designers and developers are::iu1 u+ymk vm,ec; bl 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;75 9jdgs jz:phoeml9 pay utility bills for their houses each month. The table below shows the amount of electricity, water and gas consumed during January by each family,:;ph9ed l 57zjsjm go9 and the total cost of 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 of C )fbi1 cvo2*gdheddar, 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 the cost of maki1g2fdcvo) b*ing 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 Service Providers (ISPs) are available in a small town. Dkqo(qi v (a 0seg1u6ll0 as(0e+6lbqduring 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 7(ikeaq0e0lauds( q vl1s+o 0l gb(q6 65 % 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, syl 61(xpih 1eits height above the ground after t seconds ish ylx161si (pe 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, its height aboveembz * 25ov-tq-s f:t)gelst; the ground after t seconds ist:t b-s qmg2e 5vt;o-*efzl) s given by s(t)=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 have 50 % of the lisb,ut /xsrw9 f(5)fb(ku1ewtm tener market at some point in time9 (k r/wxu b5bt ),w(fm1etfus. Over each one-year period, 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 empl0mf4q-pkh7aym x tezp4uhs(i 2yt+/) oyment salary each month to investing and inveh i47h (4pay0q)/ emk2uy zsft-xmpt+sts 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 supplied by three0:i- 4tfsnx d q8h3nguv()2x9 qklepn major retail suppliers: A, B and C. According to an analysis report, supplier A retains 80 % of their customers each year and lose 15 % to supplier B and 5 % to supplier C. Meanwhile, supplierni: xl-f(x32s pkuqdhnqge48v9 t)n0 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,76 zcfkhf+ ri genres: Jazz, Slow Rock and Country. After her current song ends she randomly self+h i fkc6z,r7ects the next song and the probabilities of genre of the next song are outlined 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 vm2buwtcal8 + 6a-2cs described by the following transition matrix, T, +u6awmvl 8t2c bs-2 ac
$ 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 bp 5o 5;f :jmdukre4d(jumblebees'5pj4 uer;o(k: dd fjm5 on different 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 technology (crv/z.l 5ojsrp wg*l3*im9)iIT) 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. Convemllijr v* *.3g /oic)s9w5 zrprsely, 30 % of those who choose to travel internationally one year change to travel domestically the following year. For this year, 80 employees chose travelling domestically 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 B, serve in a smal- ijx6+dj 49hu2idv j4n rgja0l city. Each year, store A keeps 30 jr iudx+4-n2d9vjjajihg6 04 % of its customers while 70 % of them switch to store 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, company A and company B. jrbtyz dl1 l96sm/v(9Each year, 15 % of customers using company A move to company B and 5 % of the customers using company B 96ld1mz y (lv/bs9jrt 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 collectorb7vm k;+yvtqh455 bing 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 region X to region Y and 15 % of the mammals move from region Y to region X. Assume that there are no mammal mov4vbt v7hk;byq mo55r+ements 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   .