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TSSM 2015 Page 1 of 29
FURTHER MATHEMATICS
Written Examination 2
(TSSM’s 2015 trial exam updated for the current study design)
Reading Time: 15 minutes
Writing Time: 1 Hour and 30 minutes
QUESTION AND ANSWER BOOK
Structure of book
Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners,
rulers, one approved graphics calculator or CAS (memory DOES NOT have to be cleared) and, if desired,
one scientific calculator, one bound reference (may be annotated). The reference may be typed or
handwritten (may be a textbook).
Students are not permitted to bring into the examination room: blank sheets of paper and/or white out
liquid/tape.
Materials Supplied
Question and answer book of 29 pages.
Working space provided throughout the book.
Instructions
Print your name in the space provided at the top of this page.
All written responses must be in English.
Core
Number of Number of questions Number of
questions to be answered marks
7 7 36
Module
Number of Number of modules Number of
modules to be answered marks
4 2 24
Total 60
Students are NOT permitted to bring mobile phones and/or any other electronic
communication devices into the examination room.
Figures
Words
‘2016 Examination Package’ -
Trial Examination 5 of 5
Letter
STUDENT NUMBER
THIS BOX IS FOR ILLUSTRATIVE PURPOSES ONLY
FURMATH EXAM 2
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This page is blank
FURMATH EXAM 2
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Instructions
This paper consists of a core and four modules. Students are to answer all questions in the core
and then select two modules and answer all questions within those modules.
You need not give numerical answers as decimals unless instructed to do so. Alternative forms
may involve e.g. π, surds, fractions.
Diagrams are not drawn to scale unless specified otherwise.
Page
Core............................................................................................................ 4
Module Page
Module 1: Matrices...................................................................................... 13
Module 2: Networks and Decision Mathematics......................................... 17
Module 3: Geometry and measurement……………................................ 21
Module 4: Graphs and relations................................................................... 25
FURMATH EXAM 2
TSSM 2015 Page 4 of 29
Core- Data Analysis
Question 1 (5 marks)
The back-to-back stem plot below shows the number of hours 14-year olds and 18-year olds
watch Television per week.
14-Year-Olds 18-Year-Olds
9 7 0 7 7 9 9
9 8 8 4 4 2 1 1 2 3 4 4 4 8 8 9 9
8 8 6 6 4 3 1 2 1 2 3 4 8
8 8 5 5 5 3 6
2 2 4 1
5 0
Key: 2|1 = 21 hours.
a. Find the range of time that 14 years old watch television.
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1 mark
b. Use mathematical evidence to identify any outliers in the 18 years old data. Describe the
shape of this set of data.
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2 marks
CORE - continued
FURMATH EXAM 2
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c. What percentage of 14 years old watch the television for at least 23 hours each week?
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1 mark
d. What is the average number of hours 14 years old watch television? Give your answer
correct to two decimal places.
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1 mark
Question 2 (7 marks)
The following scatterplot shows the percentage of population in farming jobs and the percentage
of population living in towns in 12 different countries.
The equation of the least squares regression line for the data in the above scatterplot is:
Percentage in towns 72 8384 0 4982 percentage in farming jobs 0 8968. . , r - .
CORE - continued
TURN OVER
FURMATH EXAM 2
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a. Write down the response variable.
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1 mark
b. Describe the relationship between percentage of population in farming jobs and percentage
of population living in towns.
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1 mark
c. Interpret the slope and the vertical intercept of the least squares regression line, in terms of
the variables.
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2 marks
d. What percentage of variation in the percentage of population living in towns can be
explained by the variation in the percentage of population in farming jobs?
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1 mark
e. Sketch the least-squares regression line on the scatterplot above.
1 mark
CORE - continued
FURMATH EXAM 2
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f. The percentage of population in farming jobs in one of the countries is 74%. Find the
residual value of percentage of population living in towns for this country. Give your answer
correct to two decimal places.
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1 mark
Question 3 (6 marks)
The following table shows the number of bacteria present in an experiment each day.
Day Number Bacteria
1 35500
2 21100
3 19700
4 16600
5 14200
6 10600
7 10400
8 6000
9 5600
10 3800
11 3600
12 3200
13 2100
14 1900
15 1500
CORE - continued
TURN OVER
FURMATH EXAM 2
TSSM 2015 Page 8 of 29
The scatterplot of the above data is also shown below.
a. What kind of transformation should the student make to improve linearity?
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1 mark
b. Perform the transformation and write down the least squares regression line, with
coefficients correct to four decimal places.
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2 marks
c. Using the least squares regression line of the transformed data, find the number of bacteria
on the fifth day.
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2 marks
CORE - continued
FURMATH EXAM 2
TSSM 2015 Page 9 of 29
d. Explain why using the regression line to predict the number of bacteria on the 20th
day is not
reliable.
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1 mark
Question 4 (6 marks)
The following table shows the actual demand of a product (in thousands) over a period of 2
years.
Year Spring Summer Autumn Winter
2014 205 140 375 575
2015 475 275 685 965
a. Describe the trend plot of the above data.
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1 mark
The seasonal index for Spring is 0.74 and seasonal index for Summer is 0.45.
b. Find the seasonal index for Autumn.
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2 marks
c. What does seasonal index for Spring indicate about the demand of the product?
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1 mark
CORE - continued
TURN OVER
FURMATH EXAM 2
TSSM 2015 Page 10 of 29
The least squares regression line of the deseasonalised demand is given by the equation
d. Assuming that the trend continues in this manner, in which season will the de-seasonalised
demand of the product become 769 000?
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2 marks
Total 24 marks
Core – Recursion and financial modelling
Question 5 (4 marks)
Fairy Amusement Park has been increasing their ticket prices each year by 10%. The ticket price
in 2010 was $45.
a. How much did the price of the ticket increase by in 2011?
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1 mark
b. What was the price of the ticket in 2013?
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1 mark
c. Write down a recurrence relation that calculates the ticket price, after n years in terms of
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1 mark
CORE - continued
FURMATH EXAM 2
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d. In which year will the ticket price become more than $100?
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1 mark
Question 6 (5 marks)
Suzie inherits $75 000 from her parents and invests in an account for six years. The compound
interest is calculated on a quarterly basis. The value of the investment after six years is $116 395.
a. How much interest did Suzie earn in six years?
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1 mark
b. What was the annual rate of interest for Suzie’s investment? Give your answer correct to one
decimal place.
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1 mark
At the end of 6 years, Suzie re invested $116 395 into another account at the rate of 7.8% per
annum compounded monthly. At the end of each month she also added $480 to her account.
c. How much did Suzie have in her new account after one month, to the nearest dollar?
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1 mark
CORE - continued
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FURMATH EXAM 2
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d. How much did Suzie have in her new account after 12 months? Give your answer to the
nearest dollar.
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2 marks
Question 7 (3 marks)
Suzie decides to take out a loan to buy a house. She takes out a loan of $200 000 at 6.95% per
annum compounded monthly and makes monthly repayments of $1350.
a. How much does she owe after three years?
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1 mark
b. How many years will it take for her to pay off the loan?
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2 marks
Total 12 marks
END OF CORE
FURMATH EXAM 2
TSSM 2015 Page 13 of 29
Module 1: Matrices
Question 1 (6 marks)
The local bakery shop sells three types of pasties- beef, chicken and vegetable. The number of
each type of pasties sold on four different days is given in the table below.
The matrix that shows how many of each type the shop sold on each of the four days is
represented below.
3046
6478
157913
N
a. Write down the order of matrix N.
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1 mark
b. How many vegetable pasties were sold on Thursday?
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1 mark
c. What does the sum of all the elements of the second column of matrix N represent?
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1 mark
MODULE 1 – continued
TURN OVER
FURMATH EXAM 2
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The local shop sells each of the pasties according to the following cost matrix.
3.10 4.80 3.50Beef Chicken Vegetable
C
d. Explain why the product CN is defined.
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1 mark
e. Find the product matrix CN and explain what that represents.
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2 marks
MODULE 1 - continued
FURMATH EXAM 2
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Question 2 (6 marks)
There are three different bakery shops in the town – A, B and C. Regular customers change their
preference of buying bakery products from one shop to another each week. A transition diagram
below shows the percentage of customers who are expected to change their preferred shop from
one week to another.
a. Write down a matrix T that represents the transition from one week to another.
2 marks
b. Of all the customers who shop at B, what percentage of customers prefer to shop at A the
following week?
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1 mark
MODULE 1 – continued
TURN OVER
B
C
A
FURMATH EXAM 2
TSSM 2015 Page 16 of 29
In a particular week, the number of customers who shop at each of the three shops is given by the
following matrix.
225
195
240
1S
c. How many customers change their preference to shop C in the following week?
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1 mark
d. Calculate 1
2
3 STS and explain what this matrix represents.
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2 marks
Total 12 marks
END OF MODULE 1
FURMATH EXAM 2
TSSM 2015 Page 17 of 29
Module 2: Networks and decision mathematics
Question 1 (5 marks)
A goods train company has four trains located at four different sheds. The trains need to carry
goods to four different towns A, B, C and D. The distances in kilometres between the sheds and
the towns are given below.
Train/ Town A B C D
1 90 75 75 80
2 35 85 55 65
3 125 95 90 105
4 45 110 95 115
The Hungarian algorithm will be used to decide which train will travel to which town so that the
total time taken to is minimised.
The table obtained after applying the first step of the Hungarian algorithm (subtracting the
smallest element in each row of above table from each of the elements in that row) is given
below.
Train/ Town A B C D
1 15 0 b 5
2 0 50 20 30
3 a 5 0 15
4 0 65 50 70
a. Write down the values of a and b.
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2 marks
b. Explain why an allocation of trains to towns cannot be made from the second table.
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1 mark
MODULE 2 - continued
TURN OVER
FURMATH EXAM 2
TSSM 2015 Page 18 of 29
The final table obtained by applying the Hungarian Algorithm is given below.
Train/ Town A B C D
1 40 0 5 0
2 0 25 0 0
3 55 0 0 5
4 0 40 30 40
Using this table there are 2 different allocations of trains to towns which give the minimum
possible distances that the trains need to travel.
c. What are the two allocations and what is the minimum distance travelled by the trains?
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2 marks
MODULE 2 - continued
FURMATH EXAM 2
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Question 2 (3 marks)
Below is a map of the offices of a telecommunication company at various locations and the
approximate distances (in kilometres) between the offices.
Each office needs a link to the head office at C but that link doesn’t need to be a direct link.
a. Draw the minimal spanning tree for the network above.
2 marks
MODULE 2 - continued
TURN OVER
23.4
14.5
17.8
21.6
29.8
24.9
40.1
12.7
16.8
15.3
10.7
20.6
17.0
15.5
K
J
I
H
G
F
E
D C
B
A
FURMATH EXAM 2
TSSM 2015 Page 20 of 29
b. What is the minimum distance in order to connect the offices?
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1 mark
Question 3 (4 marks)
The following network shows the various activities A – H required to complete a project. The
numbers in the brackets represent the time, in hours, required to complete the activity.
The precedence table of the above network is also given below.
Activity Time (in hours) Depends on
A 3 -
B 5 -
C 2 A
D 3 A
E 3
F 5
G 1
H 2 F, G
a. Complete the table above.
2 marks
b. Write down the critical path of the entire project.
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2 marks
Total 12 marks
END OF MODULE 2
FURMATH EXAM 2
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Module 3: Geometry and measurement
Question 1 (2 marks)
A table has a trapezoidal top as shown in the diagram below. The non-parallel sides of the table
top are of equal length. The thickness of the table top is 1.5 cm.
a. Find the area of the table top.
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1 mark
b. Find the perimeter of the table top correct to two decimal places.
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1 mark
MODULE 3 – continued
TURN OVER
FURMATH EXAM 2
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Question 2 (2 marks)
The angle that the longest side of the table top makes with the non-parallel side is 83°.
a. Find the distance between the opposite corners of the table top. Give your answer correct to
three decimal places.
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1 mark
These table tops are packed in air-tight boxes.
b. If the volume of one box is 3.3 m3, how many table tops are in one box?
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1 mark
MODULE 3 - continued
83°
FURMATH EXAM 2
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Question 3 (6 marks)
The table tops are manufactured in Town A and shipped to two different towns B and P. One
ship starts from A and sails for 32 km at a bearing of 38° to reach town B. Another ship sails for
25 km to reach Town P. The bearing of A from P is 124° as shown on the diagram below.
a. Find the bearing of P from A.
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1 mark
b. How far North of A is the second ship when it reaches P? Give your answer to the nearest
metre.
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1 mark
MODULE 3 – continued
TURN OVER
FURMATH EXAM 2
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c. Find the distance in km, correct to one decimal place, between the Towns B and P.
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1 mark
d. If a ship travels from B to P, what bearing would it take? Give your answer to the nearest
degree.
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3 marks
Question 4 (2 marks)
There are two similar hemi-spherical bowls on the table top. The ratio of their volumes is 3 : 2.
a. Find the ratio of their surface areas.
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1 mark
b. If the surface area of the larger bowl is 24cm2, find the radius of the smaller bowl correct to
one decimal place.
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1 mark
Total 12 marks
END OF MODULE 3
FURMATH EXAM 2
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Module 4: Graphs and relations
Question 1 (4 marks)
A firm manufactures wood screws and metal screws. All the screws have to pass through a
threading machine and a slotting machine.
A box of wood screws requires 3 minutes on the slotting machine and 2 minutes on the threading
machine.
A box of metal screws requires 2 minutes on the slotting machine and 8 minutes on the threading
machine.
In a week, each machine is available for 24 hours
Let the number of wood screw boxes be x and the number of metal screw boxes by y.
One of the constraints is drawn below on the axes.
a. Write down the constraint on the threading machine.
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1 mark
b. Sketch all the remaining constraints on the graph above and label all the extreme points
shading the feasible region.
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2 marks
MODULE 4 – continued
TURN OVER
FURMATH EXAM 2
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There is a profit of $10 per box on wood screws and $17 per box on metal screws.
c. Find the maximum profit.
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1 mark
Question 2 (4 marks)
Another firm manufactures metal screws in boxes. The cost to produce b boxes of metal screws
follows the following rule
where b is the number of boxes produced.
a. Explain what the number 20 means in this context.
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1 mark
Each box is sold for $28.
b. Write down the profit function in terms of b.
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1 mark
MODULE 4 - continued
FURMATH EXAM 2
TSSM 2015 Page 27 of 29
The cost function is drawn on the axes below.
c. Draw the revenue function on the graph above and find the break-even point.
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2 marks
MODULE 4 – continued
TURN OVER
FURMATH EXAM 2
TSSM 2015 Page 28 of 29
Question 3 (4 marks)
A wholesale t-shirt manufacturer charges the following prices for t-shirt orders.
$20 per shirt for shirt orders up to 20 shirts.
$15 per shirt for shirt between 21 and 40 shirts.
$10 per shirt for shirt orders between 41 and 80 shirts.
$5 per shirt for shirt orders over 80 shirts.
a. Sketch the step function that describes the cost of each shirt against the number of shirts
ordered.
2 marks
b. How much would an order of 75 shirts cost?
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1 mark
MODULE 4 - continued
FURMATH EXAM 2
TSSM 2015 Page 29 of 29
The cost function of the above graph can be written as
{
}
c. Write down the values of a and b.
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1 mark
Total 12 marks
END OF QUESTION AND ANSWER BOOK