TSSM 2013 Page 1 of 27
FURTHER MATHEMATICS Units 3 & 4 – Written examination 1
(TSSM’s 2013 trial exam updated for the current study design)
Reading time: 15 minutes
Writing time: 1 hour and 30 minutes
MULTIPLE-CHOICE QUESTION BOOK
Structure of book
Section Number of
questions
Number of questions
to be answered
Number of
modules
Number of modules
to be answered
Number of
marks
A
24
24
24
B 32 16 4 2 16
Total 40
Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners,
rulers, one bound reference, one approved graphic calculator or approved CAS calculator or CAS software
and if desired, one scientific calculator
Students are NOT permitted to bring into the examination room: blank sheets of paper and/or white out
liquid/tape.
Materials supplied
Question book of 27 pages.
Instructions
Print your name in the space provided on the top of this page.
All written responses must be in English.
Students are NOT permitted to bring mobile phones and/or any other unauthorised
electronic communication devices into the examination room.
Figures
Words
‘2016 Examination Package’ -
Trial Examination 3 of 5
Letter
STUDENT NUMBER
THIS BOX IS FOR ILLUSTRATIVE PURPOSES ONLY
FURMATH EXAM 1
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This page is blank
FURMATH EXAM 1
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SECTION A
Instructions for Section A
Answer all questions.
Choose the response that is correct for the question.
A correct answer scores 1, an incorrect answer scores 0.
Marks will not be deducted for incorrect answers.
No marks will be given if more than one answer is completed for any question.
Core: Data analysis
Question 1
The mean weight of nine blocks of chocolate is 170g. The total weight of these nine chocolates is
closest to:
A. 0.2 kg
B. 1.3 kg
C. 1.5 kg
D. 1.6 kg
E. 1.8 kg
Question 2
The following information was recorded for a class about test results in Biology and Chemistry.
The test results were normally distributed.
subject mean standard deviation
Biology 54 15
Chemistry 78 5
If Tara obtained a mark of 81 in the Biology test, the mark that she needs to obtain on the
Chemistry test to achieve the same standard score for both the subjects is:
A. 81
B. 82
C. 83
D. 87
E. 9
SECTION A - continued
TURN OVER
FURMATH EXAM 1
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Question 3
The table below shows the percentage of students in two-age groups who regularly eat food
cooked at one or more of three locations.
Location Age group
15-19 years 20-24 years
home 95% 95%
cafes 85% 18%
restaurants 38% 74%
For the students surveyed, which one of the following statements, supports the argument that the
location of food cooked is associated with the age group of the students?
A. 95% of students aged 15-19 years ate home cooked food, but only 38% of 15-19 year olds ate
at restaurants.
B. 85% of students aged 15-19 years ate food at a café.
C. 95% of students aged 15-19 years ate home cooked food, but only18% of 20-24 year olds ate
at restaurants
D. The percentage of students who ate food at a café decreased from 85% for those aged 15-19
year olds to 18% for those aged 20-24 years.
E. The percentage of students who ate home cooked food was 95% for both the age groups.
Question 4
The number of siblings in each of the 20 households is recorded in the frequency table below.
Number of siblings Frequency
0 6
1 9
2 3
3 1
4 0
5 1
For this sample of households, the percentage of households with at least one sibling is:
A.
B.
C.
D.
E.
SECTION A - continued
FURMATH EXAM 1
TSSM 2013 Page 5 of 27
The following information relates to Questions 5 and 6
The following data shows the change in blood pressure of a rabbit over time.
0.5, 4, 10, 26, 37, 32, 1.25, 0.75, 4, 9, 25, 37
A log scale is used to represent the above data in a histogram.
Question 5
The data represented on a log scale is:
A. positively skewed
B. negatively skewed
C. symmetric
D. bimodal
E. approximately symmetric
Question 6
The value of 0.6 on a log scale is closest to:
A. 4.5
B. 9
C. 32
D. 26
E. 4
Question 7
For a set of bivariate data, involving the variables x and y, the following information was
obtained.
The equation of the least squares regression line is closest to:
A.
B.
C.
D.
E.
SECTION A - continued
TURN OVER
FURMATH EXAM 1
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Question 8
The number of bikes parked in a 30-space bike shed is counted each day for a month. Which one
of the following would be an appropriate representation to display the data recorded?
A. a dot plot
B. a bar chart
C. a stem plot
D. a histogram
E. a frequency table
Question 9
For the time series given in the table below, the three-moving mean centred at x = 5 is:
x 1 2 3 4 5 6 7 8 9 10
y 4 5 4 4 8 6 9 10 9 12
A. 4
B. 5
C. 6
D. 7
E. 8
Question 10
For the histogram shown below, which of the following statements is most likely to be true?
A. The mode would exceed the mean.
B. The mean would exceed the mode.
C. The mean would equal the mode.
D. The lower quartile would equal the median.
E. The interquartile range would exceed the range.
SECTION A - continued
FURMATH EXAM 1
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Question 11
The graph below shows a plot of the residuals against the x-values for a particular data set (x, y).
The residuals were determined after a least squares regression line was fitted to the data.
Which of the following is true?
A. An x2 transformation should be carried out on the original data set.
B. A log transformation should be carried out on the original data set.
C. The actual y-value for the first x-value falls above the least squares regression line.
D. The actual y-values for the last two x-values are the same distance from the least squares
regression line.
E. There are equal numbers of points above and below the least squares regression line.
Question 12
The quarterly sales for a cheese producing company and seasonal indices are as follows.
quarter (2000) Q1 Q2 Q3 Q4
sales ($1000s) 3600 3200 3100 3800
seasonal index 1.04 0.8 1.2
The deseasonalised figure in $1000s for the second quarter is closest to
A. 3072
B. 3200
C. 3333
D. 3462
E. 3875
SECTION A - continued
TURN OVER
FURMATH EXAM 1
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Question 13
The product moment correlation, r, for the relationship between police expenditure and the crime
rate is – 0.54. This means that
A. only 54% of crime is solved
B. increased police expenditure has a 54% chance of decreasing the crime rate.
C. there is no relationship between police expenditure and the crime rate.
D. there is a strong negative relationship between police expenditure and the crime rate.
E. police expenditure and the crime rate are moderately negatively related.
Question 14
The coefficient of determination between two variables is 0.8674. The least squares regression
line describing the association between these two variables is given by
The value of product moment correlation coefficient between the two variables is closest to
A. 0.9313
B. – 0.4317
C. 0.4317
D. – 0.9313
E. 12.56
Question 15
The distribution of weights of items produced by a manufacturing process can be approximated
by a normal distribution with a mean of 90 grams and a standard deviation of 1 gram. The
percentage of the items will either weigh less than 88 grams or more than 93 grams is
A. 2.5%
B. 2.65%
C. 5.0%
D. 0.15%
E. 16.0%
SECTION A - continued
FURMATH EXAM 1
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Question 16
The following data shows the relationship between number of cigarettes smoked and life
expectancy.
Number of cigarettes Life expectancy
5 80
23 78
25 60
48 53
17 85
8 84
4 73
26 79
11 81
19 75
14 68
35 72
29 58
4 92
23 65
The least squares regression line is used to predict life expectancy from number of cigarettes
smoked. The slope of this line indicates
A. life expectancy increases as the number of cigarettes smoked increases
B. life expectancy decreases by 85.7 as the number of cigarettes smoked increases
C. life expectancy decreases by 0.63 year with every increase in number of cigarettes
D. increase in number of cigarettes causes life expectancy to decrease by 0.63
E. increase in life expectancy tends to decrease the number of cigarettes by 0.63
Core: Recursion and financial modelling
Question 17
A businessman aims to increase profits by 10% each year. The profit, , in dollars, after n years,
would be modelled by the rule:
A.
B.
C.
D.
E.
SECTION A - continued
TURN OVER
FURMATH EXAM 1
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Question 18
A tank contains 18 000 litres of water. The water flows out at a constant rate of 120 litres per
minute. The amount of water left in this tank after three hours is:
A. 0 L
B. 3600 L
C. 7200 L
D. 10 800 L
E. 14 400 L
Question 19
Rachelle decides to improve her fitness level by cycling each day. The time in minutes Mn that
Rachelle cycles on the nth day is modelled by the recurrence relation where
. The number of minutes Rachelle cycles on day 4 is closest to:
A. 23
B. 25
C. 26
D. 27
E. 68
Question 20
A sequence is defined by the recurrence relation . The rule that
defines the nth term of this sequence is:
A.
B.
C.
D.
E.
Question 21
Shelly invests $45000 at 4% per annum for 5 years compounded annually. The total amount of
interest earned is closest to:
A. $1800
B. $2100
C. $9000
D. $9750
E. $54750
SECTION A - continued
FURMATH EXAM 1
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Question 22
Jane borrows $80 000 for 10 years at 5.6% per annum, compounded monthly, with monthly
repayments of $555. Which one of the following is true?
A. The loan will be fully paid out in 10 years.
B. Weekly repayments of $132 compounded weekly would reduce the period of the loan.
C. The amount of interest paid each month during the loan increases.
D. The balance of the loan will be $40 000 at the end of five years.
E. The loan will be fully paid out in 12 years.
Question 23
An investment of $1200 for one year returns $140.38 interest. The simple interest rate per annum
for this investment is closest to:
A. 0.88%
B. 3.21%
C. 4.38%
D. 11.70%
E. 15.99%
Question 24
Shia buys a new television. The advertised price was $2700. He chooses to pay a deposit of $500
and monthly repayments of $115 over 2 years.
The total cost of the television under this arrangement is closest to:
A.
B.
C.
D.
E.
END OF SECTION A
TURN OVER
FURMATH EXAM 1
TSSM 2013 Page 12 of 27
SECTION B
Instructions for Section B
Select two modules and answer all questions within the modules.
Choose the response that is correct or best answers the question and mark this response on the
multiple-choice response sheet.
A correct answer scores 1, an incorrect answer scores 0.
No mark will be given if more than one answer is completed for any question.
Marks will not be deducted for incorrect answers.
Module Page
Module 1: Matrices...................................................................................... 13
Module 2: Networks and Decision Mathematics......................................... 17
Module 3: Geometry and measurement……………................................ 21
Module 4: Graphs and relations................................................................... 24
SECTION B - continued
FURMATH EXAM 1
TSSM 2013 Page 13 of 27
Module 1: Matrices
Question 1
If
00
46
31
A and the matrix product
53
41
14
XA then order of matrix X is:
A. 22 B. 32
C. 13
D. 23
E. 33
Question 2
If
14
28A and
26
13B then the matrix
26
1236AB
The element “6” in the matrix AB is correctly obtained by:
A. 6134
B. 6134
C. 4286
D. 6134
E. 6134
SECTION B – continued
TURN OVER
FURMATH EXAM 1
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Question 3
Consider the following system of three simultaneous equations
63
234
123
zx
zyx
zyx
The solution of the simultaneous equations above is given by:
A.
6
2
1
536
404
112
8
1
z
y
x
B.
6
2
1
536
404
112
z
y
x
C.
6
2
1
103
341
123
z
y
x
D.
6
2
1
103
341
123
8
1
z
y
x
E.
6
2
1
113
341
123
z
y
x
Question 4
Which one of the following is not true for the matrix equation CAB , where
4
6,
42
31Cand
x
yBA
A. 2|| A
B. The product AB is defined.
C. The product AC is defined.
D. A is a singular matrix.
E. y = -6 and x = -4
SECTION B - continued
FURMATH EXAM 1
TSSM 2013 Page 15 of 27
Question 5
Let
66
10,
12
55,
54
32CBA
The matrix X such that CABX can be found by:
A. CABX 1
B. CBAX 1
C. CBAX 1)(
D. 1)( ABCX
E. CABX 1)(
The following information relates to Questions 6 and 7
A railway knows that 250 goods wagons will be needed to carry goods from point A to point B.
At the end of each week, it finds that 10% of the wagons that started at A, ended at point B, and
8% of the wagons that started at point B ended at point A.
Question 6
A transition matrix that could represent this situation is:
A.
2.01.0
8.09.0
B. 0.9 0.08
0.1 0.92
C.
92.09.0
08.01.0
D.
2.09.0
8.01.0
E.
08.09.0
92.01.0
SECTION B – continued
TURN OVER
FURMATH EXAM 1
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Question 7
If equal number of wagons started at both points, the number of wagons located at point B at the
end of two weeks is:
A. 120
B. 121
C. 129
D. 130
E. 131
Question 8
If RTSP , matrix P is of order 33 and
1009
841
323
T then the order of R is:
A. 23
B. 32
C. 33
D. 22
E. cannot be found
END OF MODULE 1
FURMATH EXAM 1
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Module 2: Networks and decision mathematics
Question 1
The sum of the degrees of all the vertices in the network below is:
A. 6
B. 7
C. 8
D. 9
E. 10
Question 2
Which one of the following statements is true regarding the network shown below?
A. ABCDEFG is a Hamiltonian circuit.
B. Only one Hamiltonian path exists.
C. CBAGFEDC is an Eulerian circuit.
D. At least two Eulerian paths exist.
E. There are no circuits.
SECTION B – continued
TURN OVER
G F
E
D
C B
A
FURMATH EXAM 1
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Question 3
A connected planar graph has 12 edges. This graph could have:
A. 5 vertices and 6 faces.
B. 5 vertices and 8 faces.
C. 6 vertices and 8 faces.
D. 6 vertices and 9 faces.
E. 7 vertices and 9 faces.
The following information relates to Questions 4 and 5
Consider the following table.
Activity Time Immediate predecessor
A 12 -
B 11 A
C 13 -
D 24 A, C
E 11 D, B
F 21 E
Question 4
The activity/ies that come before activity E is/are:
A. D and B.
B. A, D and B.
C. A, C and D.
D. A, B, C and D.
E. F.
Question 5
The earliest completion time for the network is:
A. 55
B. 56
C. 68
D. 69
E. 70
SECTION B - continued
FURMATH EXAM 1
TSSM 2013 Page 19 of 27
Question 6
Consider the network flow diagram below representing the flow of water between a reservoir (A)
and a town (H). The other nodes (B…G) represent pumping stations. One of the four dotted lines
shown on the diagram is the minimum cut for this network.
The maximum flow of water from A to H is:
A. 100
B. 110
C. 120
D. 130
E. 140
Question 7
Which of the following statements is true?
A. A Hamiltonian circuit is a tree.
B. A tree can contain multiple edges or loops.
C. A Hamiltonian path is not a tree.
D. A tree can visit the same vertex more than once.
E. A tree can have one edge.
SECTION B - continued
TURN OVER
FURMATH EXAM 1
TSSM 2013 Page 20 of 27
Question 8
What additional edge should be added to the planar graph in the figure below to make an Euler
circuit possible?
A. CA
B. FA
C. FD
D. EB
E. none of the above
END OF MODULE 2
F
A B
E
C E
D
FURMATH EXAM 1
TSSM 2013 Page 21 of 27
Module 3: Geometry and measurement
Question 1
Which one of these can be used to find the value of in the diagram on the right?
A. ( )
B. ( ) ( )
C. ( )
D. ( ) ( )
E. ( )
35°
Question 2
If the value of ( )
then the area of the triangle is:
A. 27 square units
B. 55 square units
C. 88 square units
D. 110 square units
E. 176 square units
Question 3 The combined total surface area of a hemisphere of radius 28mm mounted on top of a right
circular cone, with slant height 40mm is closest to:
A. 8444.6 mm2
B. 9221 mm2
C. 14146.5 mm2
D. 50271.1 mm2
E. 16609.5 mm2
SECTION B - continued
TURN OVER
θ
16
11
6
θ B
C A 9
FURMATH EXAM 1
TSSM 2013 Page 22 of 27
The following information relates to Questions 4 and 5
A girl scout walks on a bearing of for 10km. Another scout starts from the same initial
position and walks due west until she is due north of the 1st scout.
Question 4
How far did the 2nd
scout walk?
A. 3.7 km
B. 4.2 km
C. 5.8 km
D. 6.7 km
E. 7.4 km
Question 5
The distance between the two scouts is closest to:
A. 7.4314 km
B. 6.6913 km
C. 4.4832 km
D. 4.9791 km
E. 11.1061 km
Question 6
The arc length of a circle with radius 4.2 cm is 4π cm. The angle that this arc subtends at the
centre is closest to:
A.
B.
C.
D.
E.
SECTION B - continued
FURMATH EXAM 1
TSSM 2013 Page 23 of 27
Question 7
If the height of a rectangle with diagonal equal to 76 cm is x cm, then a similar rectangle with
diagonal equal to 38 cm has a height given by:
A.
cm
B.
cm
C.
cm
D.
cm
E.
cm
Question 8
The length of the side of an equilateral triangle with area 0.4 cm2 is closest to:
A.
B.
C.
D.
E.
END OF MODULE 3
TURN OVER
FURMATH EXAM 1
TSSM 2013 Page 24 of 27
Module 4: Graphs and relations
Question 1
At a café, one sandwich costs $2.40 and one drink costs $3.00. A customer purchased five
sandwiches and a number of drinks at a total cost of $24.00. The number of drinks purchased
was:
A. 4
B. 5
C. 6
D. 9
E. 10
Question 2
The shaded area in the graph below represents the feasible region for a linear programming
problem.
The minimum value of the objective function for this feasible region is:
A.
B.
C.
D.
E.
SECTION B - continued
FURMATH EXAM 1
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Question 3
The graph of y against
is shown below:
The rule connecting x and y is:
A.
B.
C.
D.
E.
Question 4
The confectionery store makes both dark and white chocolates. It makes at least twice as many
dark chocolates as white chocolates. If x represent the number of dark chocolates and y represent
the number of white chocolates, an inequality representing the situation is:
A. 2 xy
B. xy 2
C. xy 2
D. yx 2
E. yx 2
SECTION B - continued
TURN OVER
(2, 5)
1/x
y
FURMATH EXAM 1
TSSM 2013 Page 26 of 27
Question 5
Which of the following rules could be used to describe the graph above?
A.
64,12
40,532
x
xxy
B.
64,12
40,532
xx
xxy
C.
64,12
40,532
xx
xxy
D.
64,12
40,5.232
xx
xxy
E.
64,12
40,5.232
x
xxy
Question 6
A linear inequation has an x-intercept of 5 and a y-intercept of 6. Also, the point (0, 0) is not
included in the solution region. The linear inequation is:
A. 3065 xy
B. 3065 xy
C. 3056 xy
D. 3056 xy
E. 30 xy
SECTION B - continued
FURMATH EXAM 1
TSSM 2013 Page 27 of 27
Question 7
The required region shown in the following graph is best described by:
A. 0,0,032,842 yxxyyx
B. 0,0,023,024 yxxyyx
C. 0,0,032,842 yxxyyx
D. 0,0,032,842 yxxyyx
E. 0,0,032,842 yxxyyx
Question 8
A manufacturer produces 2 softwares, Playpal (x) and Playshow (y), and requires two stages in
the production of the software, programming and manufacturing.
The programming stage requires 4 workers per hour to build a Playpal, but 6 workers per hour to
build Playshow. There are up to 72 workers per hour available. The manufacturing stage requires
8 workers per hour for Playpal but 5 workers per hour for Playshow with up to 88 workers per
hour available.
The constraint regarding the programming of each software is:
A.
B.
C.
D.
E.
END OF MULTIPLE-CHOICE QUESTION BOOK