How many QCAA Specialist Mathematics questions cover Further matrices?

AusGrader has 49 QCAA Specialist Mathematics questions on Further matrices, all with instant AI grading and detailed marking feedback.

QCAA (QCE) Specialist Mathematics — Further matrices Questions & Answers | AusGrader

Q2

2022

QCAA

Paper 2

1 mark

Q2

1 mark

The win/draw/loss results after a netball competition involving five teams is represented in matrix M.

Losing teams

$\begin{matrix} & P & Q & R & S & T \\ P & 0 & 1 & 2 & 0 & 2 \\ Q & 1 & 0 & 0 & 1 & 1 \\ R & 0 & 2 & 0 & 0 & 0 \\ S & 2 & 1 & 2 & 0 & 2 \\ T & 0 & 1 & 2 & 0 & 0 \end{matrix}$

Key: Team P drew with Team Q, defeated Team R and Team T, and lost to Team S

The model $\mathbf{M} + \mathbf{M}^2 + \mathbf{M}^3$ is used to rank the teams. The final positions from first to fifth are

A

S, Q, P, R, T

B

S, Q, P, T, R

C

S, P, Q, T, R

D

S, P, Q, R, T

Q9

2022

QCAA

Paper 2

1 mark

Q9

1 mark

Consider the matrix equation.

$\mathbf{X} \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$

Matrix $\mathbf{X}$ is

A

$\begin{bmatrix} 0 & 1 & 1 \\ 1 & -1 & 2 \\ -1 & 0 & 2 \end{bmatrix}$

B

$\begin{bmatrix} 0 & 1 & -1 \\ 1 & -1 & 0 \\ 1 & 2 & 2 \end{bmatrix}$

C

$\begin{bmatrix} 2 & 2 & 1 \\ 4 & 3 & 3 \\ 5 & 5 & 5 \end{bmatrix}$

D

$\begin{bmatrix} 2 & 4 & 5 \\ 2 & 3 & 5 \\ 1 & 3 & 5 \end{bmatrix}$

Q12

2022

QCAA

Paper 2

5 marks

Q12

A scientist collects data for a species of tree frog in a protected area. Details for the female tree frog population are shown in the table.

Age (years)	0–1	1–2	2–3	3–4
Population in Year 1	150	101	84	62
Birth (breeding) rate	0.4	0.7	0.5	0.1
Survival rate	0.6	0.3	0.2	0

The scientist uses a Leslie matrix model to make predictions about the female tree frog population.

Q12a

1 mark

State the initial population matrix.

Initial population matrix $= N_1 = \begin{bmatrix} 150 \\ 101 \\ 84 \\ 62 \end{bmatrix}$

Marking Criteria

Descriptor	Marks
Correctly states the initial population matrix	1

Q12b

1 mark

Determine the Leslie matrix.

Leslie matrix $= L = \begin{bmatrix} 0.4 & 0.7 & 0.5 & 0.1 \\ 0.6 & 0 & 0 & 0 \\ 0 & 0.3 & 0 & 0 \\ 0 & 0 & 0.2 & 0 \end{bmatrix}$

Marking Criteria

Descriptor	Marks
Correctly determines the Leslie matrix	1

Q12c

3 marks

A species is considered to be endangered if the female population in a restricted area is predicted to fall to less than 125 in the next 20 years.
Determine whether this species of tree frog is considered to be endangered.

Consider the population in Year 20
Using matrix facility of GDC
$N_{20} = L^{19}N_1$
$\approx \begin{bmatrix} 62.7 \\ 39.7 \\ 12.6 \\ 2.7 \end{bmatrix}$

Female population in Year 20 $\approx 119$

The female population is less than 125 within the 20-year period so the species is considered to be endangered.

Marking Criteria

Descriptor	Marks
Calculates a matrix representing the female population within a 20-year period	1
Calculates female population for a year within a 20-year period	1
Makes a suitable decision whether the species is considered endangered	1

Q12

2024

QCAA

Paper 2

4 marks

Q12

A system of linear equations is given by

$x - 2y - 2z = -6$
$-3x - y + z = 2$
$2x + 3y - 5z = 10$

Q12a

1 mark

Express the system of equations as a matrix equation of the form $AX = B$ , where $A$ is a $3 \times 3$ matrix and both $X$ and $B$ are $3 \times 1$ column vectors.

$\begin{bmatrix} 1 & -2 & -2 \\ -3 & -1 & 1 \\ 2 & 3 & -5 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -6 \\ 2 \\ 10 \end{bmatrix}$

Marking Criteria

Descriptor	Marks
correctly expresses the equations of the form $AX = B$	1

Q12b

1 mark

Use matrix algebra to express $X$ in terms of $A$ and $B$ .

Descriptor	Marks
correctly expresses X in terms of A and B	1

Q12c

1 mark

Use your result from Question 12b) to determine the solution of the system of equations.

Using GDC

$\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -2 \\ 3 \\ -1 \end{bmatrix}$

Marking Criteria

Descriptor	Marks
determines solution to equations	1

Q12d

1 mark

Verify your result from Question 12c) using one of the given linear equations.

Descriptor	Marks
verifies result by substituting result from 12c) into one of the given linear equations	1

Q11

2020

QCAA

Paper 2

5 marks

Q11

Teams A, B, C, D and E participated in a competition with the following results:

A defeated D.
B defeated A, C and E.
C defeated A and E.
D defeated B, C and E.
E defeated A.

To rank the teams at the end of the competition, the organisers constructed a dominance matrix, N, that is partially completed.

Q11a

1 mark

By allocating 1 to represent 'defeated' and 0 to represent either 'was defeated by' or 'no result', complete matrix N.

	A	B	C	D	E
A
B
C
D
E

Rows: Winning teams, Columns: Losing teams

\begin{matrix} & \text{A} & \text{B} & \text{C} & \text{D} & \text{E} \\ \text{A} & 0 & 0 & 0 & 1 & 0 \\ \text{B} & 1 & 0 & 1 & 0 & 1 \\ \text{C} & 1 & 0 & 0 & 0 & 1 \\ \text{D} & 0 & 1 & 1 & 0 & 1 \\ \text{E} & 1 & 0 & 0 & 0 & 0 \end{matrix}

Marking Criteria

Descriptor	Marks
correctly completes matrix N	1

Q11b

2 marks

The organisers need to rank the teams into individual places from first to fifth place.
They decide to use the ranking model $\mathbf{N} + \mathbf{N}^2$ to achieve this.

Use the model $\mathbf{N} + \mathbf{N}^2$ to rank the teams.

$N + N^2 = \begin{bmatrix} 0 & 1 & 1 & 1 & 1 \\ 3 & 0 & 1 & 1 & 2 \\ 2 & 0 & 0 & 1 & 1 \\ 3 & 1 & 2 & 0 & 3 \\ 1 & 0 & 0 & 1 & 0 \end{bmatrix}$

Ranking teams using the model $N + N^2$

Team	Model value	Rank position
A	4	3
B	7	2
C	4	3
D	9	1
E	2	5

Marking Criteria

Descriptor	Marks
calculates N + N²	1
ranks the teams to show that teams A and C are tied	1

Q11c

1 mark

Use the result from 11b) to identify a limitation of the organisers' ranking model.

The limitation of the ranking model is that it does not provide individual positions from first to fifth.

Marking Criteria

Descriptor	Marks
identifies a limitation of the organiser's model based on the model N + N²	1

Q11d

1 mark

State a mathematical refinement the organisers could consider to overcome the limitation of the ranking model identified in 11c).

The ranking model could be improved by including weightings in the calculations (e.g. $N + \frac{1}{2}N^2$ )

Marking Criteria

Descriptor	Marks
correctly describes a suitable mathematical refinement	1

QCAA Specialist Mathematics Further matrices

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