Matrices

[Maximum mark: 5]

There are $225$ senior students studying Computer Science and $169$ senior students studying Mathematics at a university. According to an academic survey, $20$ % of these $\text{students}$ 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 $\text{research}$ assistants.

The column matrix

represents the number of senior students studying

Computer Science and Mathematics.

1. Write down a row matrix, $\bm{R}$, to represent the percentages, in decimal form, of senior students who will choose one of the five routes after graduation. [1]
   

2. Hence calculate the product $\bm{A} = \bm{SR}$. Give each element $a_{ij}$ of the matrix $\bm{A}$ correct to the nearest whole number. [1]
   

3. In the context of this problem, explain what the element $a_{15}$ means. [1]
   

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. [1]
   

5. Hence calculate the total cost for all the textbooks. Give your answer correct to the nearest dollar. [1]
   

[Maximum mark: 4]

Domino's Pizza owns two pizzerias at Asia Mall and Metrocity shopping centres. The number of Pacific Veggie, Pepperoni and Buffalo 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]
   

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. [2]
   

[Maximum mark: 6]

The Brown, Miller and Taylor families 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, 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]
   

2. Using matrices, find the price for each of the utility. [2]
   

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. [2]
   

[Maximum mark: 6]

When a ball is thrown from the top of a tall building, its height above the ground after $t$ seconds is given by
$s(t) = at^2 + bt + c\,,$ where $a,b,c \in \mathbb{R}$ 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. 1. Write down a system of three linear equations in terms of $a$, $b$ and $c$.

   2. Use matrices to find the values of $a$, $b$ and $c$. [3]
      

2. Find the height of the building. [1]
   

3. Find the time it takes for the ball to hit the ground. [2]