TASMANIAN QUALIFICATIONS AUTHORITY
PLACE LABEL HERE
Tasmanian Certificate of Education
MATHEMATICS – METHODS
Senior Secondary 5C
Subject Code: MME5C
External Assessment
2004
Part 1
Time: approximately 60 minutes
On the basis of your performance in this examination, the examiners will provide
a result on the following criterion taken from the syllabus statement:
Criterion 6
Demonstrate an understanding of polynomial, hyperbolic,
exponential and logarithmic functions.
Criterion 7
Demonstrate an understanding of circular functions.
Criterion 8
Use differential calculus in the study of functions.
Criterion 9
Use integral calculus in the study of functions.
Criterion 10
Demonstrate an understanding of binomial, hypergeometric and
normal probability distributions.
Pages:
Questions:
©
16
20
Copyright for part(s) of this examination may be held by individuals and/or organisations other
than the Tasmanian Qualifications Authority.
Mathematics – Methods (Part 1)
BLANK PAGE
Page 2
Mathematics – Methods (Part 1)
CANDIDATE INSTRUCTIONS
The 2004 Mathematics Methods 5C Formulae Sheet can be used throughout the examination.
No other printed material is allowed into the examination.
1.
ALL questions in this part should be attempted.
2.
Answers only may be given. Full marks will be awarded for a correct answer. In the case of
an incorrect answer, marks may be awarded where correct working towards a solution is
presented.
3.
Answers and any working must be written in the spaces provided on the examination paper.
4.
It is recommended that you spend approximately 60 minutes answering the questions in this
part.
5.
You are expected to provide a graphics calculator approved by the Tasmanian Qualifications
Authority.
Page 3
Mathematics – Methods (Part 1)
For
Marker
Use
Ony
SECTION A
This section assesses Criterion 6.
Question 1
If 2lnx − ln(x + 2) = 1+ lny find y in terms of x.
(2 marks)
................................................................................................................................................
€
................................................................................................................................................
................................................................................................................................................
................................................................................................................................................
Question 2
State the domain and range for the following function.
(2 marks)
y
(0, 4)
x = –2
y=3
x=2
x
................................................................................................................................................
................................................................................................................................................
................................................................................................................................................
Question 3
If f (x) =
3
+ 2, x ≠ 1 find the inverse function, f −1 (x) , given that it exists.
x −1
(2 marks)
................................................................................................................................................
€
€
................................................................................................................................................
................................................................................................................................................
Section A continues opposite.
Page 4
Mathematics – Methods (Part 1)
For
Marker
Use
Only
Section A (continued)
Question 4
The graph of the function f (x) = 2ln(x + 3) + 1 intersects the axes at the points ( a, 0) and
(0, b) .
(a)
Find the exact values of a and b.
(2 marks)
€
€
.............................................................................................................................................
€
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
(b)
Sketch f (x) . Label all relevant points and asymptotes.
€
Page 5
(2 marks)
Mathematics – Methods (Part 1)
For
Marker
Use
Only
SECTION B
This section assesses Criterion 7.
Question 5
The graph of the function f : [−4,4 ] → R, f (x) = Asin
values of A and C.
πx
+ C is shown below. Determine the
2
(2 marks)
y
€
3.0
2.0
1.0
-5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
5.0
x
-1.0
................................................................................................................................................
................................................................................................................................................
................................................................................................................................................
................................................................................................................................................
Question 6
The average temperature (T ˚C) at Sunnytown at time t [t hours after midnight] in January can
be modelled by the equation.
 πx

T = 12cos + 2 + 15
 12

What is the maximum and minimum temperature during the day at Sunnytown?
(2 marks)
................................................................................................................................................
€
................................................................................................................................................
................................................................................................................................................
................................................................................................................................................
Section B continues opposite.
Page 6
Mathematics – Methods (Part 1)
Section B (continued)
Question 7
How many solutions are there for the equation:
(2 marks)
2cos2x = 1, x ∈ [−π , π ]
................................................................................................................................................
................................................................................................................................................
€
................................................................................................................................................
................................................................................................................................................
Question 8
Consider the function of the graph below.
x=π4
x = 3π
4
x = 3π 5π , 0
4
4
(
(π 4 , 0)
€
€
€
(0,−2)
€
x = 7π 4
€
€
€
(a)
)
What is the period of the function y?
(2 marks)
.......................................................................................................................................
.......................................................................................................................................
.......................................................................................................................................
.......................................................................................................................................
(b)
Write down a possible equation for the function y.
(2 marks)
.......................................................................................................................................
.......................................................................................................................................
.......................................................................................................................................
.......................................................................................................................................
Page 7
For
Marker
Use
Only
Mathematics – Methods (Part 1)
For
Marker
Use
Only
SECTION C
This section assesses Criterion 8.
Question 9
Find the derivative of
(a)
f (x) = 4 x 3 + 2x 2 − 9.
(1 mark)
.............................................................................................................................................
.............................................................................................................................................
€
.............................................................................................................................................
(b)
g(x) = ln(3x 4 + 5) .
(1 mark)
.............................................................................................................................................
€
.............................................................................................................................................
.............................................................................................................................................
Question 10
Let f : R → R be a function so that f ′(−1) = 0 and f ′(x) > 0 when x < −1 and f ′(x) < 0
when x > −1. State what occurs at x = −1 on the graph of f.
(2 marks)
.......................................................................................................................................................
€
€
€
€
€
€ .......................................................................................................................................................
€
.......................................................................................................................................................
.......................................................................................................................................................
Section C continues opposite.
Page 8
Mathematics – Methods (Part 1)
For
Marker
Use
Only
Section C (continued)
Question 11
Given the graph of the function f (x) below. Sketch a graph to indicate the derivative
of f (x) .
(2 marks)
f(x)
€
9.0
€
8.0
7.0
6.0
5.0
4.0
3.0
2.0
1.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
x
-1.0
Question 12
(x + 1)
If f (x) = 2
find and interpret f ′(3) .
x −4
€
(4 marks)
................................................................................................................................................
€
................................................................................................................................................
................................................................................................................................................
................................................................................................................................................
Page 9
Mathematics – Methods (Part 1)
For
Marker
Use
Only
SECTION D
This section assesses Criterion 9.
Question 13
If f ′(x) = (2x + 1) 4 , find f (x) given f (0) = 1.
(2 marks)
.......................................................................................................................................................
€
.......................................................................................................................................................
€
€
.......................................................................................................................................................
.......................................................................................................................................................
Question 14
Find the indefinite integral:
∫ [sin 3x − 6e 3x ]dx .
(2 marks)
.......................................................................................................................................................
.......................................................................................................................................................
€
.......................................................................................................................................................
.......................................................................................................................................................
Question 15
5
If
∫
1
5
f (x)dx = 3, find
∫ [2 f (x) + 1]dx .
(2 marks)
1
.......................................................................................................................................................
€
.......................................................................................................................................................
€
.......................................................................................................................................................
.......................................................................................................................................................
Section D continues opposite.
Page 10
Mathematics – Methods (Part 1)
For
Marker
Use
Only
Section D (continued)
Question 16
Find the area enclosed between the curves f (x) = (x −1) 2 −1 and g(x) = 9 − (x − 3) 2 .
(4 marks)
.......................................................................................................................................................
€
€
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
Page 11
Mathematics – Methods (Part 1)
For
Marker
Use
Only
SECTION E
This section assesses Criterion 10.
Question 17
At a particular cinema, 70% of all tickets sold are Adult with the remaining sold as
Concession tickets. A random sample of 15 moviegoers was taken. What is the probability
that 10 of them had Adult tickets?
(2 marks)
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
Question 18
In question 17, Adult tickets are sold for $13 and Concession tickets for $9. Calculate the
mean cost of a ticket for the cinema.
(2 marks)
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
Section E continues opposite.
Page 12
Mathematics – Methods (Part 1)
Section E (continued)
Question 19
In the fast food outlet ‘Fast Fry’, the number of customers served in any 30 minute period is
normally distributed with a mean of 250 people and a standard deviation of 35 people. Find
the probability that in a random half hour fewer than 200 customers will be served. (2 marks)
................................................................................................................................................
................................................................................................................................................
................................................................................................................................................
................................................................................................................................................
................................................................................................................................................
................................................................................................................................................
................................................................................................................................................
................................................................................................................................................
Question 20
A U-Beaut chocolate bar is labelled as having 50g of pure milk chocolate. There is variation
in this with a sample producing a mean of 51g and a standard deviation of 2g. A normal
distribution curve is shown below with weight of chocolate bars W along the horizontal axis.
W
(a)
Shade the region of the normal distribution indicating the proportion of chocolate bars
that weigh more then 50g.
(2 marks)
(b)
Calculate this proportion.
(2 marks)
.......................................................................................................................................
.......................................................................................................................................
.......................................................................................................................................
.......................................................................................................................................
Page 13
For
Marker
Use
Only
Mathematics – Methods (Part 1)
BLANK PAGE
Page 14
Mathematics – Methods (Part 1)
BLANK PAGE
Page 15
Mathematics – Methods (Part 1)
NORMAL DISTRIBUTION TABLE
Z score =
score – mean
standard deviation
Page 16
TASMANIAN QUALIFICATIONS AUTHORITY
PLACE LABEL HERE
Tasmanian Certificate of Education
MATHEMATICS – METHODS
Senior Secondary 5C
Subject Code: MME5C
External Assessment
2004
Part 2
Time: approximately 24 minutes
On the basis of your performance in this examination, the examiners will provide
a result on the following criterion taken from the syllabus statement:
Criterion 10
Pages:
Questions:
©
Demonstrate an understanding of binomial, hypergeometric and
normal probability distributions.
7
3
Copyright for part(s) of this examination may be held by individuals and/or organisations other
than the Tasmanian Qualifications Authority.
Mathematics – Methods (Part 2)
BLANK PAGE
Page 2
Mathematics – Methods (Part 2)
CANDIDATE INSTRUCTIONS
The 2004 Mathematics Methods 5C Formulae Sheet can be used throughout the examination.
No other printed material is allowed into the examination.
1.
ALL questions in this part should be attempted.
2.
You are advised to show relevant working towards each answer.
3.
Answers and any working must be written in the spaces provided on the examination paper.
4.
It is recommended that you spend approximately 24 minutes answering the questions in this
part.
5.
You are expected to provide a graphics calculator approved by the Tasmanian Qualifications
Authority.
Page 3
Mathematics – Methods (Part 2)
Question 21
A university study investigated the increase in heart rates (measured in beats per minute) of
people undertaking a particular exercise. The increases in heart rate were normally distributed
with a mean of 40 and a standard deviation of 9. Classifications were then made as follows:
Increase
< 22
22 ≤ x < 31
31 ≤ x < 49
x ≥ 50
(a)
Classification
Very fit
Fit
Average fitness
Unfit
Determine the proportion of people classified as very fit.
(2 marks)
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
(b)
Alex likes to believe she is in the top 10% of fitness. What heart rate increase would she
need for her to be in the top 10% of fitness?
(3 marks)
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
(c)
€
Another university study has determined that the proportion of people classified as unfit
(x ≥ 50) is 15% and the proportion of people classified as very fit (x < 22) is 10%. Find
the mean and standard deviation of the increases in heart rate for this university study.
(4 marks)
.............................................................................................................................................
€
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
Page 4
For
Marker
Use
Only
Mathematics – Methods (Part 2)
Question 22
A class of 28 students is made up of 12 male and 16 female students. A committee of 7
students is to be selected at random to plan an end of year function. What is the probability
that the committee has four or more male students?
(6 marks)
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
.......................................................................................................................................................
Page 5
For
Marker
Use
Only
Mathematics – Methods (Part 2)
Question 23
It is known that 5% of Australian teenagers participate in the aviation industry. A random
sample of 60 teenagers is taken and asked whether they participate in the aviation industry.
What is the probability that:
(a)
Exactly one teenager participates in the aviation industry.
(2 marks)
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
(b)
No more than one teenager participates in the aviation industry.
(2 marks)
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
(c)
At least one teenager participates in the aviation industry.
(2 marks)
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
(d)
At least ten teenagers participate in the aviation industry
(2 marks)
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
Page 6
For
Marker
Use
Only
Mathematics – Methods (Part 2)
BLANK PAGE
Page 7
TASMANIAN QUALIFICATIONS AUTHORITY
PLACE LABEL HERE
Tasmanian Certificate of Education
MATHEMATICS – METHODS
Senior Secondary 5C
Subject Code: MME5C
External Assessment
2004
Part 3
Time: approximately 24 minutes
On the basis of your performance in this examination, the examiners will provide
a result on the following criterion taken from the syllabus statement:
Criterion 6
Demonstrate an understanding of polynomial, hyperbolic,
exponential and logarithmic functions.
Criterion 8
Use differential calculus in the study of functions.
Pages:
Questions:
©
7
2
Copyright for part(s) of this examination may be held by individuals and/or organisations other
than the Tasmanian Qualifications Authority.
Mathematics – Methods (Part 3)
BLANK PAGE
Page 2
Mathematics – Methods (Part 3)
CANDIDATE INSTRUCTIONS
The 2004 Mathematics Methods 5C Formulae Sheet can be used throughout the examination.
No other printed material is allowed into the examination.
1.
ALL questions in this part should be attempted.
2.
You are advised to show relevant working towards each answer.
3.
Answers and any working must be written in the spaces provided on the examination paper.
4.
It is recommended that you spend approximately 24 minutes answering the questions in this
part.
5.
You are expected to provide a graphics calculator approved by the Tasmanian Qualifications
Authority.
Page 3
Mathematics – Methods (Part 3)
Question 24
The weight of a baby was recorded at various time in the first twelve months and was as
follows:
Age (months)
Weight (kg)
(a)
0
3.3
1
4.1
2
4.9
3
5.5
6
7.2
12
9.5
Initially it was assumed a function f (x) = ax + b would be the best model for the data
recorded. Using the first and last piece of data [(0, 3.3) and (12, 9.5)] show that
a = 6.61 and b = 10.89.
(2 marks)
.............................................................................................................................................
€
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
(b)
Sketch the graph of f (x) = 6.61x + 10.89 , showing all intercepts.
(2 marks)
€
Question 24 continues opposite.
Page 4
For
Marker
Use
Only
Mathematics – Methods (Part 3)
Question 24 (continued)
(c)
State the domain and range of f (x) . Is it appropriate for the data?
(3 marks)
.............................................................................................................................................
.............................................................................................................................................
€
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
(d)
A logarithmic model was then established using regression analysis and found to be
g(x) = 4.92 + 1.84lnx . Find the weight of the baby at 12 months using this model.
(2 marks)
.............................................................................................................................................
€
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
(e)
Using both models, calculate the age at which a weight of 8 kg would be expected.
(2 marks)
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
(f)
Determine the rate of change in weight at 6 months for each of the two models.
(4 marks)
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
Page 5
For
Marker
Use
Only
Mathematics – Methods (Part 3)
Question 25
A virus is introduced to control a rabbit population of 10 000. The number of rabbits N(t)
contracting the virus by time t has been modelled by the equation:
N(t) =
10000
2 + 9998e−0.4t
,t ≥ 0.
€
(t is the number of days after the virus reached the rabbit population.)
(a)
Sketch the graph €of the function N(t) on the axes below by completing the following
table or otherwise
(2 marks)
t
N
0
€
10
20
20
30
30
40
50
10 000
5 000
t (days)
10
40
50
Question 25 continues opposite.
Page 6
For
Marker
Use
Only
Mathematics – Methods (Part 3)
Question 25 (continued)
(b)
Show that N ′(t) =
39992000e−0.4t
(2 + 9998e−0.4t ) 2
.
(4 marks)
.............................................................................................................................................
.............................................................................................................................................
€
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
(c)
Find N ′(t) for.
(i)
(2 marks)
t = 0............................................................................................................................
....................................................................................................................................
€
€ (ii)
t = 20..........................................................................................................................
....................................................................................................................................
(d)€ What can be said about the rate at which rabbits are contracting the virus from the 40 th
day onward?
(2 marks)
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
Page 7
For
Marker
Use
Only