Vectors and matrices — Question 11

QCAA 2020, Paper 2 · 5 marks

Q11 · 2020 · Technology-activeSimple familiar5 marks

QUESTION 11 (5 marks)

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

To rank the teams at the end of the competition, the organisers constructed a dominance matrix,

N

, that is partially completed.

By allocating 1 to represent ‘defeated’ and 0 to represent either ‘was defeated by’ or ‘no result’, complete matrix

N

[1 mark]

Use the model

N+N^2

to rank the teams in individual places from first to fifth.

[2 marks]

Identify a limitation of the ranking model used in 11b).

[1 mark]

Describe a mathematical refinement of the ranking model identified in 11c).

[1 mark]

WORKED SOLUTION

5 marks

ANSWER

N=\begin{bmatrix}0&0&0&1&0\\1&0&1&0&1\\1&0&0&0&1\\0&1&1&0&1\\1&0&0&0&0\end{bmatrix}

; model values D 9, B 7, A 4, C 4, E 2.

Worked solution

N=\begin{bmatrix}0&0&0&1&0\\1&0&1&0&1\\1&0&0&0&1\\0&1&1&0&1\\1&0&0&0&0\end{bmatrix}.

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}.

Row sums are A 4, B 7, C 4, D 9, E 2, so D is first, B second, A and C are tied, and E is fifth.

c) The model does not provide individual positions from first to fifth.

d) The model could be improved by including weightings in the calculations, e.g.

N+\frac12N^2

correctly completes matrix

N

[1 mark]

calculates

N+N^2

[1 mark]

ranks the teams to show that teams A and C are tied

[1 mark]

identifies a limitation of the organiser’s model based on the model

N+N^2

[1 mark]

correctly describes a suitable mathematical refinement

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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