Embed Size (px)
Citation preview
External Examination 2013
FOR OFFICEUSE ONLY
SUPERVISORCHECK
RE-MARKED
ATTACH SACE REGISTRATION NUMBER LABELTO THIS BOX
Graphics calculator
Brand
Model
Computer software
Thursday 7 November: 1.30 p.m.
Time: 3 hours
Examination material: one 41-page question bookletone SACE registration number label
Approved dictionaries, notes, calculators, and computer software may be used.
Instructions to Students
1. You will have 10 minutes to read the paper. You must not write in your question booklet or use a calculator during this reading time but you may make notes on the scribbling paper provided.
2. Answer all parts of Questions 1 to 16 in the spaces provided in this question booklet. There is no need to fill all the space provided. You may write on pages 21, 33, 38, and 39 if you need more space, making sure to label each answer clearly.
3. The total mark is 145. The allocation of marks is shown below:
Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Marks 7 10 8 7 8 7 7 10 8 7 10 11 9 10 12 14
4. Appropriate steps of logic and correct answers are required for full marks.
5. Show all working in this booklet. (You are strongly advised not to use scribbling paper. Work that you consider incorrect should be crossed out with a single line.)
6. Use only black or blue pens for all work other than graphs and diagrams, for which you may use a sharp dark pencil.
7. State all answers correct to three significant figures, unless otherwise stated or as appropriate.
8. Diagrams, where given, are not necessarily drawn to scale.
9. The list of mathematical formulae is on page 41. You may remove the page from this booklet before the examination begins.
10. Complete the box on the top right-hand side of this page with information about the electronic technology you are using in this examination.
11. Attach your SACE registration number label to the box at the top of this page.
2013 MATHEMATICAL STUDIES
Pages: 41Questions: 16
22
3 PLEASE TURN OVER
QUESTION 1
(a) Find ddyx
for the following functions. There is no need to simplify your answers.
(i) y xx
= − +10 4 62 .
(2 marks)
(ii) y ex x= −2 4.
(2 marks)
(b) Find 3 2 5x x−( )∫ d .
(3 marks)
4
QUESTION 2
Consider the following matrices:
A m
Bn
nC
13 0
2 17 3
2 82 0 4
1 3−
⎡
⎣⎢
⎤
⎦⎥ =
−−
⎡
⎣⎢
⎤
⎦⎥ =
−
−
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
, , and ,
where m and n are real numbers.
Calculate:
(a) A B+ .
(1 mark)
(b) AB.
(2 marks)
(c) A−1.
(2 marks)
5 PLEASE TURN OVER
(d) (i) Find C in the form an b, where a and b are integers.
(3 marks)
(ii) For what value(s) of n will C−1 not exist?
(2 marks)
6
QUESTION 3
Heartworm disease — a parasitic condition that affects dogs — is spread by mosquitoes. The proportion of dogs affected by the disease varies from place to place. As part of a study, a random sample of 5400 dogs in the Adelaide area were tested for heartworm disease. The study found that 162 of those dogs had heartworm disease.
(a) Calculate p̂, the proportion of the sample of dogs that had heartworm disease.
(1 mark)
(b) Calculate a 95% confidence interval for p, the proportion of all dogs in the Adelaide area that have heartworm disease.
(2 marks)
A smaller, follow-up study is planned.
(c) What sample size would be necessary to obtain a confidence interval with a width of 0.032 or less for p, the proportion of all dogs in the Adelaide area that have heartworm disease?
(2 marks)
7 PLEASE TURN OVER
(d) As a cost-saving measure, a sample with n =150 is proposed.
(i) Calculate np̂.
(1 mark)
(ii) Comment on the validity of the confidence interval that would be obtained using this sample size.
(2 marks)
8
QUESTION 4
The graph of the relation x y y x2 2 4 26+ = is shown below:
O
y
x
(a) Show that the derivative of this relation is ddyx
x xyx y y
= −+
62
2
2 3 .
(3 marks)
9 PLEASE TURN OVER
(b) Find the equation of the normal to the relation at point 4 2, −( ).
(4 marks)
10
QUESTION 5
The general form of a cubic function is f x ax bx cx d( ) = + + +3 2 , where a, b, c, and d are real constants, with a ≠ 0. The derivative of this function is ′( ) = + +f x ax bx c3 22 .
For a specific function f x( ) the graph of y f x= ( ) has tangents:
• y x= +14 14 at point −( )1 0,
• y x= − +112
17 at point 2 6,( ).
The graph of y f x= ( ), as well as these tangents and their points of contact with the curve (points of tangency), is shown on the axes below:
O x
y
( 1, 0)
(2, 6)
(a) (i) What is the slope of the tangent to the curve at point 2 6,( )?
(1 mark)
(ii) Use your answer to part (a)(i) to show that 24 8 2 11a b c+ + = − .
(1 mark)
11 PLEASE TURN OVER
The equation 8 4 2 6a b c d+ + + = can also be obtained by using point 2 6,( ).
(b) Using the information about point −( )1 0, on page 10, obtain two other equations that involve a, b, c, and d.
(2 marks)
(c) By solving the four equations from parts (a) and (b), find the equation of f x( ).
(2 marks)
(d) The equation of a seventh-order polynomial
g x ax bx cx dx ex fx gx h( ) = + + + + + + +7 6 5 4 3 2
can be found, based on the equations of a number of its tangents and their points of tangency.
What is the least number of tangents needed to find this equation? Give a reason for your answer.
(2 marks)
12
QUESTION 6
When a radial-arm saw (as shown on the right) is used, its cutting edge (as indicated by the red dot) moves forwards and then backwards along a straight line.
During a particular cutting procedure, the velocity of the cutting edge of the saw, in metres per second, can be modelled by the function
v t t t t( ) = − +0 05 0 38 0 6243 2. . . ,
where t represents the time in seconds from the start of the cutting procedure and 0 5 2≤ ≤t . .
(a) On the axes below, draw a graph of y v t= ( ).
0.4
2
0.2
0.2
0.4
1 543tO
y
(1 mark)
(b) For what values of t is the cutting edge of the saw at rest?
(2 marks)
cutting edge
RADIAL ARM SAW
13 PLEASE TURN OVER
(c) Calculate v t t( )∫ d0
5 2..
(1 mark)
(d) Interpret your answer to part (c) in the context of the motion of the cutting edge of the saw.
(1 mark)
(e) Find the total distance travelled by the cutting edge of the saw during the cutting procedure.
(2 marks)
14
QUESTION 7
Consider the following system of linear equations:
xxx
yky
zzkz
2 34 1
52
−+
+ =− =+ = .
(a) (i) Show, using clearly defined row operations, that this system of linear equations can be reduced to
100
010
43
2 4
111− −
−+
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥k k.
(4 marks)
15 PLEASE TURN OVER
(ii) Hence find the value of k for which this system of linear equations has no solution.
(1 mark)
(b) If the system of linear equations has a unique solution with z =1, what is the value of k?
(2 marks)
16
QUESTION 8
The graph of y f x= ( ), for f x x( ) = +( )2 1ln , is shown below:
O x
y
1
1 2 3
2
3
An overestimate for the area between the graph of y f x= ( ), the x-axis, and the line x = 2 is to be calculated using four rectangles of equal width.
(a) (i) On the graph above, draw the rectangles used to obtain this overestimate.(1 mark)
(ii) Calculate this overestimate, giving your answer to three significant figures.
(2 marks)
17 PLEASE TURN OVER
(b) Write down an expression for the exact area between the graph of y f x= ( ), the x-axis, and the line x = 2.
(1 mark)
(c) Show by differentiation that
2 1 2 1 2 1 2ln ln lnx x x x x x c+( ) = +( ) + +( ) − +∫ d ,
where c is a real constant.
(3 marks)
(d) Hence find the exact area between the graph of y f x= ( ), the x-axis, and the line x = 2. Give your answer in the form p q rln + , where p, q, and r are integers.
(3 marks)
18
QUESTION 9
Consider the graph y f x= ( ) that is shown below. Its stationary points are points B and D, with x -coordinates x b x d= =and respectively. Its inflection points are points A, C, and E, with x-coordinates x a x c x e= = =, , and respectively. Its x-intercept is at point C. It has an asymptote with equation y = 0.
a
A
B
C
D
E
b cxO
y
ed
(a) For what value(s) of x is f x( ) increasing?
(2 marks)
(b) For what value(s) of x is f x( ) decreasing most rapidly?
(1 mark)
19 PLEASE TURN OVER
(c) On the axes below, draw a graph of y f x= ′( ).
a b c d exO
y
(3 marks)
(d) Draw a sign diagram of the second derivative function ′′( )f x .
(2 marks)
20
QUESTION 10
An airline knows from historical data that only 92% of purchasers of tickets for commuter flights attempt to board their ticketed flight. On the basis of this knowledge, the airline oversells flights: tickets for a commuter flight that holds 230 travellers are sold to 235 purchasers.
(a) What is the probability that exactly 230 of the 235 purchasers will attempt to board this flight?
(2 marks)
(b) What is the probability that more than 230 of the 235 purchasers will attempt to board this flight?
(2 marks)
(c) The airline is investigating the likely effect of different levels of overselling.
What is the greatest number of tickets that the airline can sell for a commuter flight so that there is a less than 2% probability that more than 230 travellers will attempt to board the flight?
(3 marks)
21 PLEASE TURN OVER
You may write on this page if you need more space to finish your answers. Make sure to label each answer carefully (e.g. ‘Question 8(a)(ii) continued’).
22
QUESTION 11
The use of emotive language is believed to increase pain tolerance, as measured by the length of time that a person can tolerate pain.
To test this belief, a study involving ten volunteers was undertaken. The pain tolerance of the volunteers was measured twice, by timing how long in seconds they could keep their left hand in ice water. On one of the two occasions the volunteers were instructed to use emotive language while keeping their hand in the ice water; on the other occasion they were instructed to use only the word ‘table’. The two measurements were taken a number of hours apart, and the order in which they were taken was chosen at random.
The following pain tolerance times (in seconds) were recorded:
Volunteer With emotive language (L)
Without emotive language (N)
Difference D L N= −
1 58 61 3
2 66 57 9
3 62 51 11
4 58 55 3
5 66 56 10
6 61 48 13
7 55 59 4
8 61 61 0
9 59 54 5
10 64 61 3
(a) Find the value of the sample mean of the differences XD.
(1 mark)
To test the hypotheses HH
D
DA
0 00
::
=≠
,
a two-tailed Z-test, at the 0.05 level of significance, is to be applied, where D is the mean of the random variable D.
(b) Interpret the null hypothesis in the context of this study.
(2 marks)
23 PLEASE TURN OVER
(c) Assume that D can be modelled by a normal distribution with a standard deviation of = 6.
Determine whether or not, on the basis of this study, the null hypothesis should be rejected.
(3 marks)
(d) What can you conclude from your answer to part (c)?
(1 mark)
(e) Calculate a 95% confidence interval for the mean of D.
(2 marks)
(f ) How does this confidence interval relate to the result of the Z-test?
(1 mark)
24
QUESTION 12
Consider the function f x xx
x( ) =+( )
≠ −21
12 , where .
(a) On the axes below, sketch the graph of y f x= ( ). Clearly show the axis intercept(s) and the vertical asymptote(s).
2
1
1
2
3
4
1 2 3 4 5 6xO
y
(3 marks)
(b) Find the x-coordinate and the y-coordinate of the stationary point of f x xx
( ) =+( )2
1 2 .
(1 mark)
Now consider the function g x xx
x( ) =+( )
≠ −31
12 , where .
(c) Find the x-coordinate and the y-coordinate of the stationary point of g x xx
( ) =+( )3
1 2 .
(1 mark)
25 PLEASE TURN OVER
(d) Complete the following table.
Function2
1 2x
x +( )3
1 2x
x +( )4
1 2x
x +( )5
1 2x
x +( )x-coordinate of stationary point
y-coordinate of stationary point
(1 mark)
(e) Make a conjecture about the coordinates of the stationary point of the function
h x axx
( ) =+( )1 2 ,
x ≠ −1where and a is a real number, a ≠ 0.
(1 mark)
(f ) Prove or disprove the conjecture you made in part (e).
(4 marks)
26
QUESTION 13
Telephone calls to an enquiries line are placed in a queue and are answered in order by a single server. For this enquiries line, it is known that call time (T ), the time taken to complete a call, varies according to a distribution with a mean of =1 2. minutes and a standard deviation of = 0 8. minutes.
Let Tn represent the average call time for a random sample of n calls.
(a) The distributions of T T5 15and are represented by the two histograms below:
Histogram A Histogram B
0 1 2 3 4average call time (minutes)
0 1 2 3 4average call time (minutes)
Which histogram (A or B) represents the distribution of T15? Give a reason for your answer.
(1 mark)
(b) According to the central limit theorem, the distribution of Tn will be approximately normal if n is sufficiently large.
Is n = 25 sufficiently large in this case? Give a reason for your answer.
(2 marks)
27 PLEASE TURN OVER
(c) (i) Write down the mean and standard deviation of T25.
(2 marks)
(ii) Hence calculate P T25 1 32≤( ). .
(1 mark)
(d) The enquiries line has thirty calls in the queue.
Calculate the probability that the thirty calls will be completed in 33 minutes or less.
(3 marks)
28
QUESTION 14
A factory uses water during a 24-hour day. Each day the factory uses water in the same way. For each day, the factory’s rate of water use, t hours after midnight, can be modelled by the function
R t t t t( ) = − +( )0 001 48 5766 5 4. litres per hour, where 0 24≤ ≤t .
The graph of y R t= ( ) is shown below (centre). The rate of water use for the days before and after is shown on either side, to illustrate the repeated nature of the rate of water use:
6
2000
1000
3000
4000
5000
12 18 24
y
O t
(a) Determine R 10( ).
(1 mark)
(b) Determine the amount of water used during one day. Give your answer to four significant figures.
(2 marks)
29 PLEASE TURN OVER
(c) To meet its water needs, the factory is supplied with water at a constant rate of 1820 litres per hour. When the factory is using less than 1820 litres per hour, the unused water is stored in a holding tank (as shown on the right) for times when the factory is using more than 1820 litres per hour.
(i) Show that this rate of supply is sufficient to meet the daily water needs of the factory.
(1 mark)Source: © Epantha/Dreamstime.com
(ii) On the graph on page 28, represent the supply of water to the factory.(1 mark)
(d) (i) Solve R t( ) =1820.
(1 mark)
(ii) Hence determine the interval(s) of time during each day when the amount of water stored in the holding tank is increasing.
(1 mark)
(e) During each day the amount of water in the holding tank varies between 0 and k litres.
Find the value of k.
(3 marks)
30
QUESTION 15
The life cycle of beetles consists of three stages, as shown in the diagram on the right:• adult beetles lay eggs• the eggs hatch into larvae• the larvae mature into adult beetles.
A population of beetles is monitored for 26 weeks.
The number of eggs in the population, in hundreds, after x weeks can be modelled by the function
f x e xx( ) = ≥− −( )5 00 5 3 2. , where .
The graph of y f x= ( ) is shown below. It has two inflection points, marked A and B.
adults
eggslarvaeSources: Eggs © Inventori/Dreamstime.com.Larvae © Vitaserendipity/Dreamstime.com.
Adults © Aboikis69/Dreamstime.com.
O
y
x5
5
10 15 20 25
A B
(a) According to this model, after how many weeks is the number of eggs greatest?
(1 mark)
(b) Find the x-coordinates of the inflection points A and B.
(2 marks)
31 PLEASE TURN OVER
The number of larvae in the same beetle population, in hundreds, after x weeks is to be modelled by the function g x( ). It is decided that a condition upon g x( ) will be that it has an inflection point at point C with the same x-coordinate as point B, as shown below:
O
y
x5
5
10 15 20 25
A
C
B
(c) What does this inflection point condition upon g x( ) mean about the number of eggs and the number of larvae in the beetle population, according to these models?
(1 mark)
To model the number of larvae in a way that satisfies the inflection point condition, a function of the form g x x e kx( ) = −0 05 4. , where k is a positive real number, is to be used.
(d) Show that ′′( ) = − +( )−g x x e k x kxkx0 05 8 122 2 2. .
(4 marks)
32
(e) Hence find the value of k that satisfies the inflection point condition.
(4 marks)
33 PLEASE TURN OVER
You may write on this page if you need more space to finish your answers. Make sure to label each answer carefully (e.g. ‘Question 8(a)(ii) continued’).
34
QUESTION 16
A player plays a simple board game, using the board shown below:
1 2 3 4 5 6 7 8
The player is represented by a counter that is placed on square 1 at the start of a game.
A game consists of turns. In each turn the player rolls a die that returns an outcome of 1 or 2 with equal probability (three sides of the die indicate 1, and three sides indicate 2).
The player moves the counter the number of squares shown on the die. If the counter reaches a square where an arrow starts, the player then moves the counter to the square where the arrow ends. The player’s game ends when the player’s counter reaches the last square.
It can be seen from the board shown above that, if a counter is on square 1 at the start of a turn, there is a 0.5 probability that the player will roll:• 1, resulting in a move to square 2• 2, resulting in a move to square 3 and then to square 5, because of the arrow leading
from square 3 to square 5.
(a) (i) If a counter is on square 2 at the start of a turn, which two squares could it be on at the end of the turn?
(1 mark)
(ii) State why a counter cannot be on square 3 at the start of a turn.
(1 mark)
35 PLEASE TURN OVER
Let andA B= [ ] =1 0 0 0 0 0 0 0
0 0 5 0 0 0 5 0 0 00 5 0 0 0 0 5 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0
. .. .
00 00 0 0 0 0 5 0 0 0 50 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 1
. .
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
.
(b) How does row 2 of matrix B relate to your answer to part (a)(i)?
(1 mark)
(c) (i) Calculate AB.
(1 mark)
It is known that AB p q2 14
0 0 0 0 0= ⎡⎣⎢
⎤⎦⎥.
(ii) Calculate AB2 and write down the values of p and q.
(1 mark)
(iii) What does the value of p represent?
(2 marks)
36
(d) What is the probability that a player’s game will end within eight turns?
(2 marks)
A different board (shown below) is designed for the simple board game described on page 34:
1 2 3 4 5 6
(e) Write down matrices C and D so that CD n calculates the probabilities associated with the position of a counter after n turns.
(3 marks)
37 PLEASE TURN OVER
(f ) Use matrices C and D to show that it is likely to take more turns for a player’s game to end with this board than with the board shown on page 34.
(2 marks)
38
You may write on this page if you need more space to finish your answers. Make sure to label each answer carefully (e.g. ‘Question 8(a)(ii) continued’).
39 PLEASE TURN OVER
You may write on this page if you need more space to finish your answers. Make sure to label each answer carefully (e.g. ‘Question 8(a)(ii) continued’).
© SACE Board of South Australia 2013
40
41
You may remove this page from the booklet by tearing along the perforations so that you can refer to it while you write your answers.
LIST OF MATHEMATICAL FORMULAE FOR USE INSTAGE 2 MATHEMATICAL STUDIES
Standardised Normal Distribution
A measurement scale X is transformed into a standard scale Z, using the formula
Z X
where is the population mean and is the standard deviation for the population distribution.
Con dence Interval — MeanA 95% con dence interval for the mean of a normal population with standard deviation , based on a simple random sample of size n with sample mean x , is
xn
xn
1 96 1 96. . .
For suitably large samples, an approximate 95% con dence interval can be obtained by using the sample standard deviation s in place of .
Sample Size — Mean
The sample size n required to obtain a 95% con dence interval of width w for the mean of a normal population with standard deviation is
nw
2 21.96 .
Con dence Interval — Population ProportionAn approximate 95% con dence interval for the population proportion p, based on a large simple random sample of size n with sample proportion
p Xn, is
pp pn
p pp pn
1 961 1
. .1.96
Sample Size — Proportion
The sample size n required to obtain an approximate 95% con dence interval of approximate width w for a proportion is
nw
p p2 1 96 12. .
(p is a given preliminary value for the proportion.)
Binomial Probability
P X k C p pkn k n k1
where p is the probability of a success in one trial and the possible values of X are k n0 1, , . . . and
C nn k k
n n n kkk
n 1 1. . ..
Binomial Mean and Standard DeviationThe mean and standard deviation of a binomial
count X and a proportion of successes p Xn
are
X np p p
X np p1 pp pn1
where p is the probability of a success in one trial.
Matrices and Determinants
If then andAa bc d
A A ad bc=⎡
⎣⎢⎢
⎤
⎦⎥⎥ = = −det
AA
d bc a
1 1 .
Derivatives
f x y f x yxdd
xn
e kx
ln x xelog
nxn 1
ke kx
1x
Properties of Derivativesdd
dd
xf x g x f x g x f x g x
xf xg x
f x g x f x g x
g x
xf g x f g x g x
2
dd
Quadratic Equations
If thenax bx c x b b aca
2 0 42
2
+ + = =− −
