External Examination 2010
2010 MATHEMATICAL MATHEMATICAL STUDIES FOR OFFICE USE ONLY
Graphics calculator
SUPERVISOR CHECK
ATTAC TTACH H SACE REGISTRATION NUMBER LABEL
Brand
TO THIS BOX
Model Computer software
RE-MARKED
Thursday 4 November: 9 a.m.
Pages: 37 Questions: 16
Time: 3 hours Examination material: one 37-page question booklet one SACE registration number label Approved dictionaries, notes, calculators, and computer computer software may be used. Instructions to Students
1.
You will have 10 minutes to read the paper. paper. You 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 11, 35, and 36 if you need more space, making sure to label each answer clearly.
3.
The total total mark is approximately 145. The allocation allocation of of marks is shown below: Question
1
2
3
4
5
6
7
8
9
10
Marks
9
5
9
8
9
6
11
7
6
9
11 12 10 10
13
14
15
16
9
8
16 16
15 15
8
4.
Appropriate steps of logic and correct answers are required required for full marks.
5.
Show all working in this booklet. (You (You are are strongly advised consider incorrect should be crossed out with a single line.)
6.
Use only black or blue pens for all work other than graphs graphs and diagrams, for which which you may use a sharp dark pencil.
7.
State all answers correct to three significant significant figures, unless otherwise stated or as appropriate. appropriate.
8.
Diagrams, where given, are not necessarily drawn to scale.
9.
The list of mathematical formulae is on page 37. You 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 your SACE registration number number label to the box at the top of this page.
not
to use scribbling paper. Work that you
QUESTION 1
(a) Find
d y d x
if y =
x 5 − 2l n x .
(
)
There is no need to simplify your answer.
(3 marks)
(b) Find
∫ + 1
e
−2 x
dx.
x
(3 marks) (c) Find
∫
3 2 10 x +5 x e dx.
(3 marks)
2
QUESTION 2
⎡ 0 2 −1⎤ Consider the matrix A = ⎢1 1 −2⎥ , where k is some real number. ⎢ ⎥ ⎢⎣ 3 6 k ⎥⎦ (a) Find
A
.
(3 marks) (b) For what values of k does
A
−1
exist?
(2 marks)
3
PLEASE TURN OVER
QUESTION 3
Consider the function f ( x ) =
10
x 2
, where
x
> 0. The graph of y = f x ) is shown below:
y
ȅ 1
2
3
4
5
x
Let U 2 represent the overestimate of the area between the graph of y = f ( x ) and the x-axis from x = 1 to x = 5, calculated using two rectangles of equal width. (a) (i) On the graph above, draw two
unshaded
rectangles corresponding to
U 2 .
(1 mark) (ii) Find the value of U 2 . Show the calculations that support your answer.
(2 marks)
4
(b) (i) Find
∫
10 x
2
d x.
(1 mark) (ii) Using part (b)(i), calculate the exact value of the area between the graph of y = f ( x ) and the x-axis from x = 1 to x = 5.
(2 marks) (c) Calculate D2 , the difference between U 2 and the exact value of the area between the graph of y = f ( x ) and the x-axis from x = 1 to x = 5.
(1 mark) (d) Consider U 4 , the overestimate calculated using four rectangles of equal width for the area between the graph of y = f ( x ) and the x-axis from x = 1 to x = 5. (i) On the graph opposite, draw
unshaded rectangles
corresponding to
U 4 .
(1 mark) (ii)
U 4
gives a closer approximation than
U 2
to the exact value in part (b)(ii).
On the graph opposite, illustrate this statement by shading the difference between these approximations. (1 mark)
5
PLEASE TURN OVER
QUESTION 4
Consider the graph of y = f ( x ) shown below for 0 < x ≤ 8 : ,
y
2
ȅ 2
4
6
8
x
2
This graph has a non-stationary inflection point at ( 2, 0 ). It has no other inflection points and no stationary points. It has vertical asymptotes at x = 0 and x = 4. (a) For what values of x is the function increasing?
(2 marks) (b) For what what values of x is f ″ ( x ) < 0 ?
(2 marks)
6
(c) On the axes below, sketch a graph of y = f ′ ( x ) . y
ȅ 2
4
6
8
x
(4 marks)
7
PLEASE TURN OVER
QUESTION 5
A garment manufacturer imports a large batch of buttons. The proportion of these buttons that are defective is 0.037. (a) If a random sample of sixty buttons is taken, what is the probability that it will contain: (i) no defective buttons?
http://claybuttons.com s.com Source: http://claybutton
(1 mark) (ii) at most, three defective buttons?
(1 mark) (b) The garment manufacturer will reject reject the batch of buttons if more than 5% of a random sample of size n is defective (with sufficiently large n). (i) What is the probability probability that this batch will be rejected if a random sample of sixty buttons is taken?
(2 marks) (ii) The garment manufacturer imports a second large batch of buttons. The proportion of these buttons that are defective is 0.076. (1) What is the probability that this batch will be rejected if a random random sample of sixty buttons is taken?
(2 marks)
8
(2) Show that there is a greater than 90% probability that this batch batch will be rejected if a random sample of 220 buttons is taken.
(3 marks)
9
PLEASE TURN OVER
QUESTION 6
⎡ 1 0 k ⎤ ⎡ 4 0 −2 ⎤ Consider the matrices P = ⎢ and Q = ⎢ −2 − k 0 ⎥ . ⎥ ⎢ ⎥ ⎣ 0 3 −1⎦ ⎢⎣ 0 3 −1 ⎥⎦ Where possible, evaluate the following matrix expressions. If an expression cannot be evaluated, describe the specific features of the matrices that prevent its evaluation. (a)
3 P .
(1 mark) (b)
3 P + Q.
(1 mark) (c) QP .
(1 mark) (d) Q 2 .
(3 marks) 10
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 5(b)(i) continued’).
11
PLEASE TURN OVER
QUESTION 7
Consider the function f ( x ) =
2x + 3
x 2 − 4
, where
≠ ± 2.
x
(a) Find the x-intercept and the y-intercept of the graph of y = f ( x ) .
(2 marks) (b) Find the coordinates of the inflection point of the graph graph of y = f ( x ) . Give these coordinates correct to two decimal places.
(2 marks) An incomplete graph of y = f ( x ) is shown below: y
ȅ
x
1
2
3
4
5
(c) Clearly showing the information you have found in parts (a) and (b), complete complete the graph above by sketching y = f ( x ) for −2 < x < 2 . (2 marks) 12
(d) Show that f ′ ( x ) =
−2 x 2 − 6 x − 8
( x
2
− 4)
2
.
(3 marks) (e) Hence show that f ( x ) has no stationary points.
(2 marks)
13
PLEASE TURN OVER
QUESTION 8
A company decided to gather information about the health of its employees. The company arranged for a nurse to complete a study of the systolic blood pressure of a random sample of its employees. Blood pressure is measured in millimetres of mercury (mmHg). It is given that the systolic blood pressure of the company’s company’s employees has a standard deviation of V = 28 mmHg. (a) The nurse measured the systolic blood blood pressure of the sample of employees and calculated the following 95% confidence interval for their mean systolic blood pressure: 118.17
≤
P
≤ 140.13 .
(i) What was the sample mean?
(1 mark) (ii) How many employees were in the sample?
(2 marks) (b) The company wanted to reduce the width of the confidence confidence interval, and so the nurse took a second random sample. The mean systolic blood pressure of the second sample, of eighty employees, was found to be 132.4 mmHg. Calculate a 95% confidence interval based on the second sample.
(2 marks)
14
(c) High blood pressure is indicated by systolic blood pressure that is higher than 140 mmHg. On the basis of the confidence interval calculated in part (b), the company made the claim that, on average, its employees did not suffer from high blood pressure. Was this reasonable? Explain your answer.
(2 marks)
15
PLEASE TURN OVER
QUESTION 9
The graph below shows functions f ( x ) and x
= 3:
( ),
g x
which intersect when
y
x
= 0,
x
= 2, and
f ( x )
( )
g x
x
ȅ
A
1
2
3
4
C
D
B
Region A has an area of 22 units 2 and Regions A and B have a combined area of 56 units 2. (a) Find: 2
(i)
∫ f ( x ) dx. 0
(1 mark) 2
(ii)
∫ f ( x ) − g ( x ) dx. 0
(1 mark)
16
3
It is known that
3
∫ f ( x ) dx = −18 and ∫ g ( x ) dx = −13. 2
2
(b) Find the area of Region C and the area of Region D.
(2 marks) 3
(c) Find
∫ f ( x ) − g ( x ) dx. 0
(2 marks)
17
PLEASE TURN OVER
QUESTION 10
(a) The graph of y = 3 x 2 , between shown in the diagram below:
x
= −1 and
x
= 1, fits within the rectangle ABCD, as
y A
B
D
ȅ
C
x
(i) Determine the area of the rectangle ABCD.
(1 mark) (ii) Determine the area of the shaded region between in the diagram above.
y
= 3x 2 and the x-axis, as shown
(2 marks)
18
(b) The graph of y = k x 2 where k > 0, between ABCD, as shown in the diagram below:
x
= − a and
x
= a fits within the rectangle ,
y A
D
a
B
C x a
ȅ
(i) Determine the area of the rectangle ABCD.
(2 marks) (ii) Determine the area of the shaded region between y = k x 2 and the x-axis, as shown in the diagram above.
(3 marks) (iii) What is the relationship between the area of the rectangle and the area of the shaded region between y = k x 2 and the x-axis?
(1 mark)
19
PLEASE TURN OVER
QUESTION 11
The amount of money that individual employees spend in a workplace cafeteria each lunchtime is recorded. It is found that the amount spent per employee ( X ) can be modelled by a normal distribution with mean P = $5.85 and standard deviation V = $1.33. The distribution of X X is graphed below: www.klauscherarchitects.com scherarchitects.com Source: www.klau
dollars
(a) On the horizontal axis of the graph of the normal normal density curve above, write numbers to illustrate the distribution of X . (1 mark) Suppose that each lunchtime sixty-five employees spend an amount of money in the cafeteria, and that these amounts constitute a random sample of amounts spent per employee. Let X 65 be the average of the amounts spent per employee during a randomly chosen lunchtime. (b) (i) Write down the mean and the standard deviation of the distribution of X 65 .
(2 marks) (ii) Sketch the distribution of X 65 on the graph above. (2 marks)
20
(iii) Determine the proportion of lunchtimes for which the average amount spent per employee will be between $5.50 and $6.20.
(1 mark) (c) (i) Determine the probability that, for for a randomly chosen lunchtime, lunchtime, the average amount spent per employee will be $5.21 or less.
(1 mark) (ii) Using your answer to part (c)(i), (c)(i), explain whether or not the cafeteria manager should consider $5.21 to be an unusually low value.
(1 mark) One explanation for an unusually low value of X 65 is fraud by cafeteria staff. The manager wants to detect possible cases of fraud and so decides on a dollar value below which all values of X 65 will be considered unusually low low.. (d) Calculate the value below which 1% of the average amounts spent per employee will fall.
(2 marks)
21
PLEASE TURN OVER
QUESTION 12
An example of a number puzzle called a ‘tri-sum’ is shown on the right.
117
A tri-sum is made up of three overlapping arms. Each arm contains two unknowns and one numerical value.
x
To solve a tri-sum, it is necessary to find the unknowns x, y, and z such that, for each arm, the sum of the unknowns is equal to the numerical value.
y
z
166
143
(a) To solve the tri-sum shown above, above, one equation that must be be satisfied is
x + y
= 143.
(i) Write down the two other equations that must be satisfied.
(1 mark) (ii) Hence find the values of x, y, and z .
(2 marks) (b) All tri-sums can be represented by the diagram below:
a x
y
z
b
22
c
The system of equations associated with this representation of all tri-sums can be written as
⎡1 ⎢1 ⎢ ⎢⎣ 0
0
1
a⎤
1
0
b .
1
1
⎥ ⎥ c⎥ ⎦
Solve this system of equations for x, y, and z in terms of a, b, and c.
(5 marks)
23
PLEASE TURN OVER
QUESTION 13
The rate of extraction of the world’s crude oil can be modelled by Hubbert’s Curve. This model, in gigabarrels per year, is
( )=
R t
111e
(1 + e
−0.0625 0625t +9
2 −0.0625 0625t +9
)
,
where t represents time in years since 1865. The graph of Hubbert’s Curve is shown below: R
50
40
30
20
10
ȅ
t
50
100
150
200
(a) (i) Using R ( t ) write down an expression that will determine the amount of the world’s crude oil extracted in the decade from 2000 to 2010. ,
(1 mark) (ii) Evaluate the expression you wrote down in part (a)(i).
(1 mark)
24
(b) ‘Peak oil’ is defined as the time when the maximum rate of extraction of the world’s world’s crude oil is reached. Find the year in which peak oil is reached, as predicted by Hubbert’s Curve.
(2 marks) (c) Let
()
A t
represent the total amount of the world’s crude oil extracted by time t :
( )=
A t
(i) Show that
1776 1+ e
62 5t +9 −0.0 62
.
A′ t
( ) = R ( t ) .
(3 marks) (ii) State the total amount of crude oil available for extraction as predicted predicted by
( ).
A t
(1 mark) (iii) What percentage of this total amount was extracted in the decade from 2000 to 2010?
(1 mark)
25
PLEASE TURN OVER
QUESTION 14
An American scientist is studying the mating habits of Drosophila melanogaster (a species of fruit fly). She wants to know whether or not this species has a preference for mating partner, partner, based on the fly’s fly’s place of origin. In the study the scientist captures thirty male and thirty female Drosophila from Alabama and thirty male and thirty female Drosophila from Grand Bahama Island.
This photograph cannot be reproduced here for copyright reasons. It can be found at http://tolweb.org/tree/ToLimages/ 54275159_1a756047cf_o.jpg
Source: http://tolweb.org
The flies are all released together in a closed environment and carefully observed. Of the 246 matings that are observed: • 140 are between a male and a female from the same place of origin • 106 are between a male and a female from different places of origin. Using these data, a two-tailed Z -test, -test, at the 0.05 level of significance, is applied to determine whether or not there is sufficient evidence that Drosophila have a preference for mating partner, based on place of origin. The null hypothesis and the alternative hypothesis are H 0 : p = 0.5 H A : p ≠ 0.5.
(a) Interpret the null hypothesis in terms of Drosophila’s preference for mating partner.
(1 mark) (b) Calculate the proportion of matings between Drosophila from the same place of origin, as observed in this closed environment.
(1 mark)
26
(c) The Z -test -test produces a P -value -value of 0.0302 (correct to three significant figures). (i) Explain what this P -value -value represents.
(2 marks) (ii) State whether or not the null hypothesis hypothesis should be rejected. Give a reason for your answer.
(2 marks) (d) On the basis of parts (b) (b) and (c), what can you determine about Drosophila’s ’s preference preference for mating partner in this closed environment?
(2 marks)
27
PLEASE TURN OVER
QUESTION 15
A vehicle hire company is planning to offer 1-day hire of removal vans.
A
On each day, in each town in the area, some vans will be hired out for local moves and some vans will be hired out for moves between towns. The company assumes that, on each day: • in Town A, half of the vans will be hired out locally and half will be hired out for moves to Town B • in Town B, one-third of the vans will be hired out locally, one-third will be hired out for moves to Town A, and one-third will be hired out for moves to Town C • in Town C, half of the vans will be hired out locally and half will be hired out for moves to Town B.
B
The company also assumes that all vans will be hired each day, and that all vans will be returned in time for the start of the next day. The company plans to start on Day 0 with ten vans in Town A, thirty vans in Town B, and twenty vans in Town C.
C
(a) Using the company’s company’s assumptions, write down the calculation that determines the number of vans that the company will have in Town B at the start of Day 1.
(1 mark)
⎡1 ⎢2 ⎢1 Let L = ⎢ 2 ⎢ ⎢⎣0
1 3
0
1
1
3
2
1
1
3
2
⎤ ⎥ ⎥ and ⎥ ⎥ ⎥⎦
⎡ 10 ⎤ ⎢ ⎥ X = 30 . ⎢ ⎥ ⎢⎣ 20 ⎥⎦
(b) (i) Evaluate LX.
(1 mark)
28
(ii) Interpret the values obtained from from your calculation in part (b)(i).
(2 marks) (iii) Hence calculate how many vans will be in each of the three towns on Day 2.
(2 marks) (c) Describe what the company can expect expect to happen to the number number of vans in the three towns over a long period of time.
(2 marks) The company changes its plans and decides to start on Day 0 with twenty vans in Town A, twenty vans in Town B, and twenty vans in Town C. (d) (i) Write down the new matrix X .
(1 mark) (ii) Comment on what the company can now expect to happen to the number of vans in the three towns over a long period of time.
(1 mark)
29
PLEASE TURN OVER
(e) The company changes its assumptions about about the hire of vans in To Town wn B.
⎡1 ⎢2 ⎢1 In relation to the new assumptions, let L = ⎢2 ⎢ ⎢⎣ 0
(i) Explain why
a+b+c
a b c
0⎤
⎥ 1⎥ , where 0 ≤ a, b, c ≤ 1. 2⎥ 1⎥ 2⎥ ⎦
must equal 1.
(1 mark)
⎡ 18 ⎤ ⎢ ⎥ (ii) If X = 24 , solve LX = X for a, b, and c. ⎢ ⎥ ⎢⎣ 18 ⎥⎦
(3 marks) (iii) How should the company interpret the values of a, b, and c in part (e)(ii)?
(2 marks)
30
Question 16 starts on page 32.
31
PLEASE TURN OVER
QUESTION 16
The circle C , with centre ( 0, 0 ) and radius 1 unit, has equation as shown in the diagram below:
x
2
+ y 2 = 1, where −1 ≤ x ≤ 1,
y
ȅ
1
2
3
x
(a) (i) Find
d y
.
d x
(2 marks)
32
(ii) Find the equation of the tangent tangent to the circle C at the point where y < 0. Give your answer correct to three significant figures.
x
= 0.3 and
(3 marks) (iii) On the Cartesian plane on the page opposite, draw the graph of the tangent to the circle C at the point where x = 0.3 and y < 0. (1 mark) Let f ( x ) = x 2 . (b) On the Cartesian plane on the the page opposite, draw the graph graph of y = f ( x ) . (1 mark) (c) Show that the equation equation of the tangent to the graph of y = f ( x ) at the point where 2 x = k is given by y = 2 k x − k , where k is any real number.
(3 marks)
33
PLEASE TURN OVER
The tangent to the circle C at the point where x = t and y < 0, with −1 < t < 1, has equation y
=
t 1 − t
2
x
−
1
.
1 − t
2
(d) Hence or otherwise, find the values of t for which the tangent to the circle C is also tangential to the graph of y = f ( x ) .
(5 marks)
34
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 5(b)(i) continued’).
35
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 carefully (e.g. ‘Question 5(b)(i) continued’).
© SACE Board of South Australia 2010
36
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 IN STAGE 2 MATHEMATICAL STUDIES
Standardised Normal Distribution
Binomial Probability nk
A measurement scale X scale X is is transformed into a standard scale Z scale Z , using the formula Z
P X k Ck n p k 1 p
where p where p is the probability of a success in one trial and the possible values of X are X are k 0, 1, .. . . . n and
X N T
C k n
where N is the population mean and T is the standard deviation for the population distribution.
n
n k k
n n 1 . . . n k 1 k
.
Con¿dence Interval — Mean
Binomial Mean and Standard Deviation
A 95% con ¿dence interval for the mean N of a normal population with standard deviation T , based on a simple random sample of size n with sample mean x mean x,, is
The mean and standard deviation of a binomial X count X count X and and a proportion of successes p are n
x 1.96
T
n
b N b x 1.96
T
n
.
p p
N X
np
N
T X
np 1 p
T
p
n where p where p is the probability of a success in one trial.
For suitably large samples, an approximate 95% con¿dence interval can be obtained by using the sample standard deviation s deviation s in place of T .
Matrices and Determinants
If A
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 T is
A 1
¥ 2s1.96T ´µ2 n ¦¦ µµ . ¦§ ¶ w
a
b
c
d
1
then det A A ad bc and d b
A c
a
.
Derivatives
f x y
f a x
x n
n x n1
Con¿dence Interval — Population Proportion
e kx
An approximate 95% con ¿dence interval for the population proportion p proportion p,, based on a large simple random sample of size n with sample proportion
ln x ln x log e x
ke kx 1 x
p
X n
, is
p 1.96
p 1 p
d y d x
Properties of Derivatives
p 1 p n
b p b p 1.96
p 1 p n
d .
d x
\ f x g x^ f a x g x f x g a x
®® f x º®® f a x g x g x f x g a x d « ¬ » 2 d x ® g x g x ®- g x ®®¼
Sample Size — Proportion
d
The sample size n required to obtain an approximate 95% con¿dence interval of approximate width w for a proportion is
d x
¥ 2s1.96 ´µ2 n ¦¦ p 1 p . ¦§ w µµ¶
f g x f a g x g a x
Quadratic Equations
If ax 2 bx c 0 then x
( p is a given preliminary value for the proportion.)
37
b
b 2 4ac 2a
MATHEMATICAL STUDIES 2010 ACKNOWLEDGMENT
The photograph for Question 5 on page 8 is Copyright Creative Impressions in Clay.
The SACE Board of South Australia has made every effort to trace copyright holders. If however, any material has been incorrectly acknowledged, we apologise and invite the copyright holder to contact us.
