VCE General Maths Matrices 2024 Exam 1 Mini Test
VCAA General Maths Exam 1
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: 8
Reading time: 3 minutes
Writing time: 18 minutes
Instructions
• Answer all questions in pencil on your Multiple-Choice Answer Sheet.
• Choose the response that is correct for the question.
• A correct answer scores 1; an incorrect answer scores 0.
• Marks will not be deducted for incorrect answers.
• No marks will be given if more than one answer is completed for any question.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
Matrices - 2024
Matrix \(J\) is a \(2 \times 3\) matrix.
Matrix \(K\) is a \(3 \times 1\) matrix.
Matrix \(L\) is added to the product \(JK\).
The order of matrix \(L\) is
- A. \(1 \times 3\)
- B. \(2 \times 1\)
- C. \(2 \times 3\)
- D. \(3 \times 2\)
A market stall sells three types of candles.
The cost of each type of candle is shown in matrix \(C\) below.
\( C = \begin{bmatrix} 25 & 32 & 43 \end{bmatrix} \)
Towards the end of the day, the cost of each item is discounted by 15%.
Which one of the following expressions can be used to determine each discounted price?
- A. \(0.15C\)
- B. \(0.85C\)
- C. \(8.5C\)
- D. \(15C\)
Consider the following matrix, where \(h \neq 0\).
\( \begin{bmatrix} 4 & g \\ 8 & h \end{bmatrix} \)
The inverse of this matrix does not exist when \(g\) is equal to
- A. \(-2h\)
- B. \( \frac{h}{2} \)
- C. \(h\)
- D. \(2h\)
A primary school is hosting a sports day.
Students represent one of four teams: blue (\(B\)), green (\(G\)), red (\(R\)) or yellow (\(Y\)).
Students compete in one of three sports: football (\(F\)), netball (\(N\)) or tennis (\(T\)).
Matrix \(W\) shows the number of students competing in each sport and the team they represent.
\[ \begin{array}{c@{}c@{}c} & \begin{array}{@{}cccc@{}} B & G & R & Y \end{array} & \\ W= & \begin{bmatrix} 85 & 60 & 64 & 71 \\ 62 & 74 & 80 & 64 \\ 63 & 76 & 66 & 75 \end{bmatrix} & \begin{matrix} F \\ N \\ T \end{matrix} \end{array} \]
Matrix \(W\) is multiplied by the matrix \( \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} \) to produce matrix \(X\).
Element \(x_{31}\) indicates that
- A. 210 students represent the blue team.
- B. 210 students compete in netball.
- C. 280 students compete in tennis.
- D. 280 students compete in football.
A tennis team consists of five players: Quinn, Rosie, Siobhan, Trinh and Ursula.
When the team competes, players compete in the order of first, then second, then third, then fourth.
The fifth player has a bye (does not compete).
On week 1 of the competition, the players competed in the following order.
| First | Second | Third | Fourth | Bye |
|---|---|---|---|---|
| Quinn | Rosie | Siobhan | Trinh | Ursula |
This information can be represented by matrix \(G_1\), shown below.
\( G_1 = \begin{bmatrix} Q & R & S & T & U \end{bmatrix} \)
Let \(G_n\) be the order of play in week \(n\).
The playing order changes each week and can be determined by the rule \(G_{n+1} = G_n \times P\)
where \( P = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{bmatrix} \)
Which player has a bye in week 4?
- A. Quinn
- B. Rosie
- C. Siobhan
- D. Trinh
Data has been collected on the female population of a species of mammal located on a remote island.
The female population has been divided into three age groups, with the initial population (at the time of data collection), the birth rate, and the survival rate of each age group shown in the table below.
| 0–2 | 2–4 | 4–6 | |
|---|---|---|---|
| Initial population | 2100 | 6400 | 4260 |
| Birth rate | 0 | 1.8 | 1.2 |
| Survival rate | 0.7 | 0.6 | 0 |
The Leslie matrix (\(L\)) that may be used to model this particular population is
- A. \( L = \begin{bmatrix} 0 & 1.8 & 0 \\ 0.7 & 0 & 1.2 \\ 0 & 0.6 & 0 \end{bmatrix} \)
- B. \( L = \begin{bmatrix} 0 & 1.8 & 1.2 \\ 0.7 & 0 & 0 \\ 0 & 0.6 & 0 \end{bmatrix} \)
- C. \( L = \begin{bmatrix} 0 & 1.8 & 1.2 \\ 0.7 & 0.6 & 0 \\ 0 & 0 & 0 \end{bmatrix} \)
- D. \( L = \begin{bmatrix} 2100 & 6400 & 4260 \\ 0 & 1.8 & 1.2 \\ 0.7 & 0.6 & 0 \end{bmatrix} \)
The matrix below shows the results of a round-robin chess tournament between five players: \(H, I, J, K\) and \(L\). In each game, there is a winner and a loser.
Two games still need to be played.
A '1' in the matrix shows that the player named in that row defeated the player named in that column.
For example, the 1 in row 4 shows that player \(K\) defeated player \(L\).
A '...' in the matrix shows that the player named in that row has not yet competed against the player in that column.
At the end of the tournament, players will be ranked by calculating the sum of their one-step and two-step dominances.
The player with the highest sum will be ranked first. The player with the second-highest sum will be ranked second, and so on.
Which one of the following is not a potential outcome after the final two games have been played?
- A. Player \(I\) will be ranked first.
- B. Player \(I\) will be ranked fifth.
- C. Player \(J\) will be ranked first.
- D. Player \(J\) will be ranked fifth.
A large sporting event is held over a period of four consecutive days: Thursday, Friday, Saturday and Sunday.
People can watch the event at four different sites throughout the city: Botanical Gardens (\(G\)), City Square (\(C\)), Riverbank (\(R\)) or Main Beach (\(M\)).
Let \(S_n\) be the state matrix that shows the number of people at each location \(n\) days after Thursday.
The expected number of people at each location can be determined by the matrix recurrence rule
\( S_{n+1} = TS_n + A \)
where \( T = \begin{bmatrix} 0.4 & 0.2 & 0.4 & 0 \\ 0.4 & 0.1 & 0.3 & 0.3 \\ 0.1 & 0.4 & 0.1 & 0.2 \\ 0.1 & 0.3 & 0.2 & 0.5 \end{bmatrix} \) and \( A = \begin{bmatrix} 300 \\ 200 \\ 100 \\ 300 \end{bmatrix} \)
Given the state matrix \( S_3 = \begin{bmatrix} 5620 \\ 6386 \\ 4892 \\ 6902 \end{bmatrix} \)
the number of people watching the event at the Botanical Gardens (\(G\)) from Thursday to Sunday has
- A. decreased by 162
- B. decreased by 212
- C. increased by 124
- D. increased by 696
End of Multiple-Choice Question Book
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