STAGE 2 MATHEMATICAL APPLICATIONS
ASSESSMENT TYPE 1: SKILLS AND APPLICATIONS TASK 1
Purpose
To demonstrate your ability to:

accurately apply the mathematical concepts and relationships that you have learned in class to
solve a range of matrices questions set in different contexts

effectively and appropriately communicate relevant information within your solutions.
Description of assessment
This assessment allows you to show your skills in understanding and appropriate use of the
mathematical concepts and relationships in the following:
(a) Subtopic 4.1: Using Matrices to Organise Information – Costing and Stock Management
(b) Subtopic 4.2: Application of Matrices to Network Problems.
Assessment conditions
This is a supervised assessment. Provide complete working for all calculations. Use electronic
technology where appropriate.
Learning Requirements
Assessment Design Criteria
Capabilities
1.
Understand fundamental
mathematical concepts
and relationships.
Mathematical Knowledge and Skills and Their
Application
Communication
Identify, collect, and
organise mathematical
information relevant to
investigating and finding
solutions to
questions/problems taken
from social, scientific,
economic, or historical
contexts.

MKSA1 Knowledge of content and understanding of
mathematical concepts and relationships.

MKSA2 Use of mathematical algorithms and
techniques (implemented electronically where
appropriate) to find solutions to routine and complex
questions.
Recognise and apply the
mathematical techniques
needed when analysing
and finding a solution to a
question/problem in
context.
Mathematical Modelling and Problem-solving
Make informed use of
electronic technology to
provide numerical results
and graphical
representations.
2.
3.
4.
5.
Interpret results, draw
conclusions, and reflect on
the reasonableness of
these in the context of the
question/problem.
6.
Communicate
mathematical ideas and
reasoning using
appropriate language and
representations.
7.
Work both independently
and cooperatively in
planning, organising, and
carrying out mathematical
activities.
Page 1 of 10
The specific features are as follows:

Citizenship
Personal
Development
Work
Learning
MKSA3 Application of knowledge and skills to answer
questions in applied contexts.
The specific features are as follows:

MMP1 Application of mathematical models.

MMP2 Development of mathematical results for
problems set in applied contexts.

MMP3 Interpretation of the mathematical results in the
context of the problem.

MMP4 Understanding of the reasonableness and
possible limitations of the interpreted results, and
recognition of assumptions made.
Communication of Mathematical Information
The specific features are as follows:

CMI1 Communication of mathematical ideas and
reasoning to develop logical arguments.

CMI2 Use of appropriate mathematical notation,
representations, and terminology.
Stage 2 Mathematical Applications task for use in 2011
533565202 (revised January 2013)
© SACE Board of South Australia 2010
PERFORMANCE STANDARDS FOR STAGE 2 MATHEMATICAL APPLICATIONS
Mathematical Knowledge and
Skills and Their Application
Mathematical Modelling and Problemsolving
Communication of
Mathematical Information
A
Comprehensive knowledge of content
and understanding of concepts and
relationships.
Appropriate selection and use of
mathematical algorithms and techniques
(implemented electronically where
appropriate) to find efficient solutions to
complex questions.
Highly effective and accurate application
of knowledge and skills to answer
questions set in applied contexts.
Development and effective application of
mathematical models.
Complete, concise, and accurate solutions to
mathematical problems set in applied contexts.
Concise interpretation of the mathematical results in
the context of the problem.
In-depth understanding of the reasonableness and
possible limitations of the interpreted results, and
recognition of assumptions made.
Highly effective communication
of mathematical ideas and
reasoning to develop logical
arguments.
Proficient and accurate use of
appropriate notation,
representations, and
terminology.
B
Some depth of knowledge of content and
understanding of concepts and
relationships.
Use of mathematical algorithms and
techniques (implemented electronically
where appropriate) to find some correct
solutions to complex questions.
Accurate application of knowledge and
skills to answer questions set in applied
contexts.
Attempted development and appropriate application
of mathematical models.
Mostly accurate and complete solutions to
mathematical problems set in applied contexts.
Complete interpretation of the mathematical results in
the context of the problem.
Some depth of understanding of the reasonableness
and possible limitations of the interpreted results, and
recognition of assumptions made.
Effective communication of
mathematical ideas and
reasoning to develop mostly
logical arguments.
Mostly accurate use of
appropriate notation,
representations, and
terminology.
C
Generally competent knowledge of
content and understanding of concepts
and relationships.
Use of mathematical algorithms and
techniques (implemented electronically
where appropriate) to find mostly correct
solutions to routine questions.
Generally accurate application of
knowledge and skills to answer questions
set in applied contexts.
Appropriate application of mathematical models.
Some accurate and generally complete solutions to
mathematical problems set in applied contexts.
Generally appropriate interpretation of the
mathematical results in the context of the problem.
Some understanding of the reasonableness and
possible limitations of the interpreted results, and
some recognition of assumptions made.
Appropriate communication of
mathematical ideas and
reasoning to develop some
logical arguments.
Use of generally appropriate
notation, representations, and
terminology, with some
inaccuracies.
D
Basic knowledge of content and some
understanding of concepts and
relationships.
Some use of mathematical algorithms
and techniques (implemented
electronically where appropriate) to find
some correct solutions to routine
questions.
Sometimes accurate application of
knowledge and skills to answer questions
set in applied contexts.
Application of a mathematical model, with partial
effectiveness.
Partly accurate and generally incomplete solutions to
mathematical problems set in applied contexts.
Attempted interpretation of the mathematical results
in the context of the problem.
Some awareness of the reasonableness and possible
limitations of the interpreted results.
Some appropriate
communication of
mathematical ideas and
reasoning.
Some attempt to use
appropriate notation,
representations, and
terminology, with occasional
accuracy.
E
Limited knowledge of content.
Attempted use of mathematical
algorithms and techniques (implemented
electronically where appropriate) to find
limited correct solutions to routine
questions.
Attempted application of knowledge and
skills to answer questions set in applied
contexts, with limited effectiveness.
Attempted application of a basic mathematical model.
Limited accuracy in solutions to one or more
mathematical problems set in applied contexts.
Limited attempt at interpretation of the mathematical
results in the context of the problem.
Limited awareness of the reasonableness and
possible limitations of the results.
Attempted communication of
emerging mathematical ideas
and reasoning.
Limited attempt to use
appropriate notation,
representations, or
terminology, and with limited
accuracy.
Page 2 of 10
Stage 2 Mathematical Applications task for use in 2011
533565202 (revised January 2013)
© SACE Board of South Australia 2010
STAGE 2 MATHEMATICAL APPLICATIONS
SKILLS AND APPLICATIONS TASK 1 SOLUTIONS
MATRICES
1. (a) Draw and label a network with connectivity as shown in this matrix.
A B
A 0
B
C
D
E
0

0

0
1
1
0
1
0
1
C
0
0
0
0
1
D E
0
0
1
0
0
1
0
1

1
0
A
E
Requires
appropriate
representation
of a network to
demonstrate
communication
of mathematical
information.
B
D
C
(2 marks)
(b) One of the locations seems to be unusual. Which one is it and why?
If the nodes represented locations inside a video store what could this location represent?
B does not have any arcs leaving it  B could represent the checkout counter (or exit) in a
video store.
(2 marks)
Page 3 of 10
Stage 2 Mathematical Applications task for use in 2011
533565202 (revised January 2013)
© SACE Board of South Australia 2010
2. The table below shows some of the charges made for various services by telephone
companies.
Company
Local call
($)
Long distance
call ($)
Mobile call
($)
Text
message ($)
Message
bank ($)
W
0.29
1.00
0.58
0.15
0.20
X
0.27
0.95
0.69
0.12
0.20
Y
0.23
1.07
0.72
0.05
0.12
Z
0.26
0.87
0.83
0.10
0.15
(a) What service attracts the smallest charge?
Text message
(1 mark)
(b) A customer in one month made 45 local calls, 15 long distance calls, 106 mobile calls and
sent 345 text messages. Use matrix methods to find the cost for this customer at the
various companies. Which company is the cheapest for him?
0.29
0.27
Cost = C x N = 
 0.23

0.26
1.00
0.95
1.07
0.87
141 .28 
140 .94 

= 
119 .97 


147 .23 
0.58
0.69
0.72
0.83
0.15
0.12
0.05
0.10
 45 
0.20   
15
0.20   
 106 
0.12   
 345
0.15   
 0 
Cheapest company is Y charging $119.97.
(3 marks)
(c) His brother spends most of his working day on the road so makes use of the message
bank service. A typical month’s use would be 50 local calls, 120 long distance calls, 135
mobile calls, 200 text messages and 150 message bank uses. Use matrix methods to
find which company offers him the best deal?
Cost = C x B
 50 
120  272 .80 

 274 .65 

= C x 135   

 265 .10 
200  271 .95 

150  
Routine
questions (1and 2-step) that
require selection
and use of
mathematical
algorithms to
find solutions.
Together
Questions 1 and
2 give a first
impression of
knowledge of
content.
Company Y charging $265.10 is also cheapest for his brother.
(2 marks)
Page 4 of 10
Stage 2 Mathematical Applications task for use in 2011
533565202 (revised January 2013)
© SACE Board of South Australia 2010
3. Scott has the job of keeping the frozen pizza stocked in the freezer at the local supermarket.
The store keeps three brands of pizza, Apizza, Best and Choice in three different sizes Family,
Regular and Small. At the beginning of the day Scott knew that he had 11 Family, 9 Regular
and 6 Small pizzas of the Apizza brand, 8 Family, 12 Regular and 11 Small pizzas of the Best
brand and 14 Family, 10 Regular and 15 Small pizzas of the Choice brand.
(a) Represent this information as matrix B .
F
R
Moving from
information being
provided in an
unstructured format
to a structured
format (e.g. matrix
form) is considered a
complex process.
S
B = A 11 9 6 


B 8 12 11


C 14 10 15
(2 marks)
Matrix E shows the stock at the end of the day.
F R
A 6 8
S
5
E=
 4 6 5
B 

C 7 4 9
(b) Calculate
N  B  E and explain what information N contains.
F
N=B–E=
R
S
A 5
1 1
 4 6 6
B 



7
6
6
C 
Matrix N shows how many of each brand and size pizzas are sold on the day.
(2 marks)
(c) The pizzas are priced so that all brands make the same profit for each of the three pizzas.
The Family pizza gives a profit of $2.35, the Regulars earn $1.70 and the Small pizzas
give a profit of $1.05. Write the information as a column matrix and use it to help find the
total profits made on the pizza for the day.
2.35
P  1.70 
1.05 
Total profit = 1 1 1  N  P
5 1 1 2.35
 1 1 1 4 6 6  1.70   73.35
7 6 6 1.05 
 total profit = $73.35.
(3 marks)
(d) Show how you can use the matrices above to determine:
(i) what size pizza seems to be the most popular
Adding up the columns of matrix N there were 16 Family, 13 Regular and 13 Small
pizzas sold  Family pizza is most popular.
(ii) what brand has the largest market share on the day.
Adding up the rows of matrix N Choice brand has the largest market share with 19
pizzas sold.
(2 marks)
Page 5 of 10
Stage 2 Mathematical Applications task for use in 2011
533565202 (revised January 2013)
© SACE Board of South Australia 2010
4. The diagram represents the lines of communication between 5 sections in a manufacturing
facility.
(a) Draw up a connectivity matrix R to describe the network below.
0
0

R= 0

0
01
1
0
1
0
1
0
1
0
1
0
1
0
0
0
0
0
0
1

1
0
B
A
E
C
D
(2 marks)
(b) Calculate
0
0

R 2  1

1
0
2
R and explain what information it contains.
0
1
1
2
1
2
0
1
0
1
0
0
0
0
1
1
1
0

1
0
R 2 contains the number of 2-stage communication lines between the sections, i.e.
communication between two sections via another, e.g. A talks to B then talks to C.
(2 marks)
(c) What is the minimum number of stages it takes for B to communicate with A? Explain.
BC  E  A
 3 stages are needed for B to communicate with A.
(1 mark)
(d) Calculate the matrix Q  R  R 2  R 3 .
1
1

Q  1

2
1
Page 6 of 10
4
2
4
4
3
2
2
2
3
3
1
0
1
1
1
3
1 
2

2
2
Provides an
opportunity to
interpret
mathematical
results in the
context of the
question. There
are similar
opportunities in
each question
that provide
evidence of
application of
knowledge and
skills to answer
questions in
applied contexts.
Stage 2 Mathematical Applications task for use in 2011
533565202 (revised January 2013)
© SACE Board of South Australia 2010
(i) What is the meaning of Q4, 2  4 ?
Q4,2  4 means that there are 4 ways that D can communicate with B in three or
less stages.
(ii) One division still does not communicate with another in either 1, 2 or 3 stage
connections. Which is it?
B to D as Q2, 4  0
(3 marks)
(e) What single step alteration or addition do you think would improve the system? Why?
(Answers will vary). Give B direct communication with D. This means that all sections can
communicate with each other in 3 or less stages.
(2 marks)
(f) Explain the limitations of using the matrix model in this situation.
The matrix model indicates how many but not what the communication links are between
the sections so some communication links between sections pass back and forth, e.g.
Q2,2  2 as B  C  E  B and B  C  B.
Need to be careful if you also considered 4 stage links, e.g. B  C  B  C would
describe a link from B to C but it has already been counted as a 2 stage link.
Our matrix model doesn’t give the number of unique links. The model assumes that direct
and indirect links are equally important as no higher weighting has been given to direct
links.
Part f) is one of
several
opportunities to
discuss the
limitations of the
interpreted
results. Each
response must
relate to the
context of the
question.
(2 marks)
Page 7 of 10
Stage 2 Mathematical Applications task for use in 2011
533565202 (revised January 2013)
© SACE Board of South Australia 2010
5. An enterprising farmer decides to set up a small factory to supply the local vineyards with her
innovative machines for use in grape production. She makes a small cultivator for ploughing
between rows of vines, a mower for keeping down the grass and a spray unit for spraying the
vines for downy mildew.
She developed the following matrix M, which shows the number of units of different materials
needed to make each design.
M = Cultivator
Mower
Sprayer
Steel
Paint
Wheels Labour
7
5


4
2
4
3
3
3
2
15 
18 

14 

She also made a cost matrix C of the unit cost in dollars.
38 
9
C 
85 
 
42 
Her initial orders look very promising. These original orders are shown in the matrix O.
O  11 19 23
Use matrix methods to find:
(a) The quantities needed to fill the initial orders.
7 2 4 15
Q  O  M  11 19 23 5 3 3 18  264 148 147 829 
4 3 2 14 
(2 marks)
(b) The cost of each machine.
Each machine cost:
38 
7 2 4 15    1254 
9
E  M  C  5 3 3 18      1228 
85 
4 3 2 14     937 
42 
Question 5
provides an
opportunity to
demonstrate
highly effective
and accurate
application of
matrix
knowledge and
skills to find
solutions to
questions set in
applied
contexts.
Parts a) and b)
do not provide
the formulae
and therefore
require use of
mathematical
algorithms and
techniques
(implemented
electronically) to
find solutions to
complex
questions.
Cost of production of: cultivator $1254, mower $1228, sprayer $937.
(2 marks)
Page 8 of 10
Stage 2 Mathematical Applications task for use in 2011
533565202 (revised January 2013)
© SACE Board of South Australia 2010
(c) The total cost of materials needed to fill the initial orders.
Total Cost,
T=OxE
1254 
 11 19 23 1228   58677 
 937 
Total Cost = $58677
(2 marks)
(d) The retail price of each machine if she aims for a 60% profit on each item.
Price
P = 1.6 x E
1254  2006 .40 
 1.6  1228   1964 .80 
 937  1499 .20 
Retail prices: cultivator, $2006.40; mower $1964.80; sprayer $1499.20.
(2 marks)
(e) Calculate the total income from the initial orders.
Total Income, I = O x P
2006 .40 
 11 19 23 1964 .80   93883 .20
1499 .20 
Total Income = $93883.20
(2 marks)
(f) Use your answers to calculate the total profit made on the initial orders.
Total Profit = Total Income – Total Cost
=I–T
= 93883.20 – 58677
Total Profit = $35206.20
(2 marks)
(g) How reasonable is your answer. Discuss the limitations of using matrix models.
Reasonableness: Model doesn’t take into account all costs, e.g. overhead costs, transport,
wastage, so that the actual profit may be different.
It is a useful/efficient method when all the machines have the same raw materials. It lets us
organise the data efficiently but care needs to be taken to multiply matrices in the correct
order to get meaningful results. In some parts of the question it would have been just as
easy to calculate results without the use of matrix methods.
(2 marks)
Page 9 of 10
Stage 2 Mathematical Applications task for use in 2011
533565202 (revised January 2013)
© SACE Board of South Australia 2010
6
Four children are observed in a play situation and an impression of any dominance relations
is made. This has been represented on the network diagram.
(a) Represent this information in the form of a matrix D.
0
1
D
1

1
A
1 0 1
0 1 1 
0 0 1

0 0 0
B
D
C
(2 marks)
(b) Calculate D 2 .
2
2
D2  
1

0
0 1 1
1 0 2
1 0 1

1 0 1
(2 marks)
2
(c) Describe what is meant by D2, 4 and verify it from the diagram.
D22, 4  2 indicates that B has 2 second order dominances over D as B  A  D and
BC D
(2 marks)
2
(d) Use the model S  D  12 D to decide the ranking of leadership among the children.
1

2
S 1
1 2

 1
1
1
2
1
2
1
2
1
2
1 12 

1 2
0 1 12 

0 12 
4
5 1 
V   12 
3 2 
 
2
Ranking from most dominant to least dominant is B, A, C, D.
(3 marks)
(e) Why is
1
2
1
2
D 2 used in the supremacy matrix S.
D 2 is used to show that first order dominance is twice as significant as second order
Parts a), b) and
d) provide the
mathematical
results that are
used to inform
the development
of logical
arguments
about
dominance
relationships
through highly
effective
communication
of mathematical
ideas and
reasoning.
dominance when ranking for leadership.
(1 mark)
(f) Briefly discuss any limitations of this mathematical model.
The model uses an impression of dominance which may not actually exist in reality.
Second stage dominance includes being given power over itself from A  B  A and
A  C  A . If another child entered the group the group dynamics may change. Also
group dynamics likely to change with classroom environment.
(2 marks)
Page 10 of 10
Stage 2 Mathematical Applications task for use in 2011
533565202 (revised January 2013)
© SACE Board of South Australia 2010