VCE General Maths Matrices 2021 Exam 2 Mini Test
VCAA General Maths Exam 2
This is the full VCE General Maths Exam with worked solutions. You can also try Mini-Tests, which are official VCAA exams split into short tests you can do anytime.
Number of marks: 12
Reading time: 3 minutes
Writing time: 18 minutes
Instructions
• Answer all questions in the spaces provided.
• Write your responses in English.
• In all questions where a numerical answer is required, you should only round your answer when instructed to do so.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
Matrices - 2021 - Exam 2 (Part 1)
Elena imports three brands of olive oil: Carmani (C), Linelli (L) and Ohana (O).
The number of 1 litre bottles of these oils sold in January 2021 is shown in matrix \(J\) below.
\[ J = \begin{bmatrix} 2800 \\ 1700 \\ 2400 \end{bmatrix} \begin{matrix} C \\ L \\ O \end{matrix} \]
a. What is the order of matrix \(J\)? 1 mark
b. Elena expected that in February 2021 the sales of all three brands of olive oil would increase by 5%.
She multiplied matrix \(J\) by a scalar value, \(k\), to determine the expected volume of sales for February.
Write down the value of the scalar \(k\). 1 mark
\(k = \)
The main computer system in Elena's office has broken down.
The five staff members, Alex (A), Brie (B), Chai (C), Dex (D) and Elena (E), are having problems sending information to each other.
Matrix \(M\) below shows the available communication links between the staff members.
In this matrix:
- the '1' in row \(A\), column \(B\) indicates that Alex can send information to Brie
- the '0' in row \(D\), column \(C\) indicates that Dex cannot send information to Chai.
a. Which two staff members can send information directly to each other? 1 mark
b. Elena needs to send documents to Chai.
What is the sequence of communication links that will successfully get the information from Elena to Chai? 1 mark
c. Matrix \(M^2\) below is the square of matrix \(M\) and shows the number of two-step communication links between each pair of staff members.
Only one pair of individuals has two different two-step communication links.
List each two-step communication link for this pair. 1 mark
A market research study of shoppers showed that the buying preferences for the three olive oils, Carmani (C), Linelli (L) and Ohana (O), change from month to month according to the transition matrix \(T\) below.
The initial state matrix \(S_0\) below shows the number of shoppers who bought each brand of olive oil in July 2021.
\[ S_0 = \begin{bmatrix} 3200 \\ 2000 \\ 2800 \end{bmatrix} \begin{matrix} C \\ L \\ O \end{matrix} \]
Let \(S_n\) represent the state matrix describing the number of shoppers buying each brand \(n\) months after July 2021.
a. How many of these 8000 shoppers bought a different brand of olive oil in August 2021 from the brand bought in July 2021? 1 mark
b. Using the rule \(S_{n+1} = T \times S_n\), complete the matrix \(S_1\) below. 1 mark
c. Consider the shoppers who were expected to buy Carmani olive oil in August 2021.
What percentage of these shoppers also bought Carmani olive oil in July 2021?
Round your answer to the nearest percentage. 1 mark
d. Write a calculation that shows that Ohana olive oil is the brand bought by 50% of these shoppers in the long run. 1 mark
e. Further research suggests more shoppers will buy olive oil in the coming months.
A rule to model this situation is \(R_{n+1} = T \times R_n + B\), where \(R_n\) represents the state matrix describing the number of shoppers \(n\) months after July 2021.
\(k\) represents the extra number of shoppers expected to buy Ohana olive oil each month.
If \(R_2 = \begin{bmatrix} 3333 \\ 2025 \\ 3642 \end{bmatrix}\), what is the value of \(k\)? 1 mark
Five staff members in Elena's office played a round-robin video game tournament, where each employee played each of the other employees once. In each game there was a winner and a loser.
A table of their one-step and two-step dominances was prepared to summarise the results.
| Staff member | One-step dominance | Two-step dominance |
|---|---|---|
| Ike (I) | 3 | 5 |
| Joelene (J) | 3 | 4 |
| Katie (K) | 1 | 1 |
| Leslie (L) | 1 | 2 |
| Mikki (M) | 2 | 4 |
Consider the results matrix shown below.
A '1' in this matrix shows that the player named in that row defeated the player named in that column.
A '0' in this matrix shows that the player named in that row lost to the player named in that column.
Use all of the information provided to complete the results matrix. 2 marks
End of Question and Answer Book
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