STAGE 2 GENERAL MATHEMATICS
ASSESSMENT TYPE 1: SKILLS AND APPLICATIONS TASKS
TOPIC 1: MODELLING WITH LINEAR RELATIONSHIPS
Purpose
To demonstrate your ability to:

understand mathematical concepts and relationships from within Topic 1: Modelling with Linear
Relationships

select and apply mathematical techniques and algorithms to find solutions to problems

interpret results, draw conclusions, and consider the reasonableness of solutions in context

communicate mathematically and present mathematical information.
This assessment allows you to show your skills in understanding and appropriate use of the mathematical concepts,
process and strategies in the following:
(a) Subtopic 1.1: Simultaneous Linear Equations
(b) Subtopic 1.2: Linear Programming
Assessment Conditions
This is a supervised assessment of 50 minutes working time.
CALCULATOR and NOTES can be used for the entire test.
Provide complete working for all calculations.
Assessment Design Criteria
Concepts and Techniques
CT 1
Knowledge and understanding of concepts and relationships.
CT 2
Selection and application of mathematical techniques and algorithms to find solutions to problems in a variety
of contexts.
CT 4
Use of electronic technology to find solutions to mathematical problems.
Reasoning and Communication
RC 1
Interpretation of mathematical results.
RC 3
Use of appropriate notations, representations and terminology.
Page 1 of 8
Performance Standards for Stage 2 General Mathematics
Concepts and Techniques
A Comprehensive knowledge and understanding of concepts and
relationships.
Highly effective selection and application of mathematical techniques and
algorithms to find efficient and accurate solutions to routine and complex
problems in a variety of contexts.
Successful development and application of mathematical models to find
concise and accurate solutions.
Appropriate and effective use of electronic technology to find accurate
solutions to routine and complex problems.
B Some depth of knowledge and understanding of concepts and relationships.
Mostly effective selection and application of mathematical techniques and
algorithms to find mostly accurate solutions to routine and some complex
problems in a variety of contexts.
Reasoning and Communication
Comprehensive interpretation of mathematical results in the context of the
problem.
Drawing logical conclusions from mathematical results, with a comprehensive
understanding of their reasonableness and limitations.
Proficient and accurate use of appropriate mathematical notation,
representations, and terminology.
Highly effective communication of mathematical ideas and reasoning to
develop logical and concise arguments.
Formation and testing of appropriate predictions, using sound mathematical
evidence.
Mostly appropriate interpretation of mathematical results in the context of the
problem.
Drawing mostly logical conclusions from mathematical results, with some depth
of understanding of their reasonableness and limitations.
Attempted development and successful application of mathematical models
to find mostly accurate solutions.
Mostly accurate use of appropriate mathematical notation, representations,
and terminology.
Mostly appropriate and effective use of electronic technology to find mostly
accurate solutions to routine and some complex problems.
Mostly effective communication of mathematical ideas and reasoning to
develop mostly logical arguments.
Formation and testing of mostly appropriate predictions, using some
mathematical evidence.
C Generally competent knowledge and understanding of concepts and
relationships.
Generally effective selection and application of mathematical techniques
and algorithms to find mostly accurate solutions to routine problems in
different contexts.
Generally appropriate interpretation of mathematical results in the context of
the problem.
Drawing some logical conclusions from mathematical results, with some
understanding of their reasonableness and limitations.
Application of mathematical models to find generally accurate solutions.
Generally appropriate use of mathematical notation, representations, and
terminology, with reasonable accuracy.
Generally appropriate and effective use of electronic technology to find
mostly accurate solutions to routine problems.
Generally effective communication of mathematical ideas and reasoning to
develop some logical arguments.
Formation of an appropriate prediction and some attempt to test it using
mathematical evidence.
D Basic knowledge and some understanding of concepts and relationships.
Some interpretation of mathematical results.
Some selection and application of mathematical techniques and algorithms
to find some accurate solutions to routine problems in context.
Drawing some conclusions from mathematical results, with some awareness of
their reasonableness.
Some application of mathematical models to find some accurate or partially
accurate solutions.
Some appropriate use of mathematical notation, representations, and
terminology, with some accuracy.
Some appropriate use of electronic technology to find some accurate
solutions to routine problems.
Some communication of mathematical ideas, with attempted reasoning and/or
arguments.
Attempted formation of a prediction with limited attempt to test it using
mathematical evidence.
E Limited knowledge or understanding of concepts and relationships.
Limited interpretation of mathematical results.
Attempted selection and limited application of mathematical techniques or
algorithms, with limited accuracy in solving routine problems.
Limited understanding of the meaning of mathematical results, their
reasonableness or limitations.
Attempted application of mathematical models, with limited accuracy.
Limited use of appropriate mathematical notation, representations, or
terminology, with limited accuracy.
Attempted use of electronic technology, with limited accuracy in solving
routine problems.
Attempted communication of mathematical ideas, with limited reasoning.
Limited attempt to form or test a prediction.
Page 2 of 8
QUESTION 1
(7 marks)
Using algebraic methods (substitution or elimination), solve the following sets of simultaneous equations.
a) 𝑦 = 3𝑥 − 5
2𝑥 − 𝑦 = 6
(3 marks)
b) 2𝑥 + 3𝑦 = 8
−5𝑥 + 4𝑦 = −20
(4 marks)
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QUESTION 2
(5 marks)
When visiting the school canteen, James purchases 5 sour straps and 6 chocolate eclairs, this costs him a
total of $2.55. Regina visits the same canteen and purchases 10 sour straps and 4 chocolate eclairs, this
costs her a total of $2.70.
a) If 𝑥 represents the cost of one sour strap and 𝑦 represents the cost of one chocolate eclair, write
two equations connecting 𝑥 and 𝑦.
(2 marks)
b) Solve the simultaneous equations from part a).
(2 marks)
c) Interpret your solutions to part b).
(1 mark)
QUESTION 3
(3 marks)
Solve the following simultaneous linear equations by graphing them below:
𝑦 = 2𝑥 + 4
1
𝑦 =− 𝑥−1
2
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QUESTION 4
(5 marks)
a) Using the graph above, shade the feasible region that is defined by the following constraints:
𝑥 + 2𝑦 ≥ 4
d 𝑥+𝑦 ≤8
𝑦≥0
𝑥≥2
(1 mark)
b) State the coordinates of the vertices of the feasible region.
(2 marks)
c) Find the coordinates of the point that maximises the objective function 𝑃 = 4𝑥 + 5𝑦 over the
feasible region and state the maximum value achieved.
(2 marks)
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QUESTION 5
(5 marks)
A company produces two kinds of dining tables, a 6-seater dining table and an 8-seater dining table.
It takes 2 hours to manufacture the parts for the 6-seater, 1 hour to assemble it, 1 hour to sand it and
0.5 hours to stain it.
It takes 3 hours to manufacture the parts for the 8-seater, 1.5 hours to assemble it, 1.5 hours to sand it,
and 0.5 hours to stain it
Per month, the company has available: 160 hours for manufacturing parts, 100 hours for assembling the
tables, 120 hours for sanding the tables and 60 hours for staining the tables.
Let x be the number of 6-seater dining tables and let y be the number of 8-seater dining tables.
List the 6 constraints that define this situation.
QUESTION 6
(16 marks)
The Nuts Galore Company has developed two new products called ‘Chocolate Delight’ and ‘Peanut
Extravagance’.
The products contain different mixtures of chocolate-coated sultanas, toffee-coated peanuts and salted
cashews. The respective mixtures are given in the table below.
Chocolate-coated
sultanas (grams)
Toffee-coated
peanuts (grams)
Salted cashews
(grams)
Chocolate Delight
400
200
300
Peanut Extravagance
200
400
300
Nuts Galore receives 6 kilograms (6000 grams) of chocolate-coated sultanas, 6 kilograms of toffeecoated peanuts and 5.4 kilograms of salted cashews per day.
Let x represent the number of packets of Chocolate Delight filled per day.
Let y represent the number of packets of Peanut Extravagance filled per day.
a) Complete the following constraints:
Toffee-coated peanuts: 200𝑥 + _______ ≤ 6000
Salted cashews: _______ + 300𝑦 ≤ 5400
(2 marks)
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b) Interpret the meaning of the following constraint: 400𝑥 + 200𝑦 ≤ 6000.
Number of packets of Peanut Extravaganza filled per day
(2 marks)
y
c) On the following set of axes, shade the feasible region defined by the constraints identified in
parts a) and b), and by 𝑥 ≥ 0, 𝑦 ≥ 0.
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
x
Number of packets of Chocolate Delight filled per day
(4 marks)
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d) Nuts Galore makes a profit of $4 for every packet of Chocolate Delight it sells and $5 for every
packet of Peanut Extravagance it sells. Write an objective function for the daily profit P.
(1 mark)
e) Determine the optimal solution for maximising profit, and interpret this optimal solution in
context.
(2 marks)
f)
i) Considering the graph of the feasible region, which of the three ingredients is wasted in this
optimal solution?
g)
(1 mark)
ii) Calculate the amount of wasted ingredient in this optimal solution.
h)
(1 mark)
i)
The manager of Nuts Galore decides to increase the price of Chocolate Delight and now
makes $5 profit per packet sold.
Write the new objective function for this scenario.
ii)
(1 mark)
ii)
Using calculations, determine the impact this new objective function has had on the
scenario. What is the new optimal solution?
(2 marks)
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