External Examination 2011
2011 MATHEMATICAL STUDIES FOR OFFICE USE ONLY
Graphics calculator
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ATTACH ATT ACH SACE REGISTRATION NUMBER LABEL TO THIS BOX
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Friday 4 November: 9 a.m.
Pages: 37 Questions: 17
Time: 3 hours Examination material: one 37-page question booklet one 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. 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 17 in the spaces provided in this question booklet. There is no need to fill all the space provided. You may write on pages 26 and 30 if you need more space, making sure to label each answer clearly.
3.
The total mark is approximately approximately 143. The The allocation of marks marks is shown below: Question
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Marks
8
6
7
7
4
8
7
9
12 12
9
6
8
11
7
13 13
10 10
11
4.
Appropriate steps steps of logic and correct correct answers answers are required for full marks.
5.
Show all working working in this booklet. (Y (You ou are are strongly 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 graphs and diagrams, diagrams, for which you may use a sharp dark pencil.
7.
State all answers correct to three significant figures, unless unless otherwise otherwise stated or as appropriate.
8.
Diagrams, where given, are not necessarily drawn to scale.
9.
The list of mathematical formulae 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 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.
QUESTION 1
(a) Find
∫
2 x
4
+ 5+
10
d x.
x
(3 marks)
(b) Find
d y d x
if y = 3e
− 0.1 x
+ 5 ln x.
(2 marks)
(c) Find
∫
x
9 − x dx. 2
(3 marks)
2
QUESTION 2
⎡ 2 (a) Find the determinant of the matrix ⎢ 1 ⎢ ⎢⎣ −1
k
k ⎤
8
2 .
3
⎥ ⎥ 1⎥ ⎦
(3 marks)
Consider the system of linear equations
+ k y + k z = 8 x + 8 y + 2 z = 0 − x + 3 y + z = − 4 2 x
where k is a real number.
(b) Express this system of linear equations in the form matrices.
AX
= B where A, X , and B are ,
(1 mark)
(c) Find all values of k for which the system of linear equations has a unique solution.
(2 marks)
3
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QUESTION 3 A toy rocket is launched vertically upwards. The height of the rocket s metres above the ground, at time t seconds after being launched, can be modelled by the function
( ) = − 4.9t 2 + 24t .
s t
The graph of y = s ( t ) is shown below: y 30
25
20
15
10
5
O
t 1
2
3
4
5
(a) Calculate the gradient of the line that joins the points on the graph of y = s ( t ) where t = 0 and t = 2. ,
(2 marks)
(b) What quantity associated with the motion of the rocket is measured by this gradient?
(2 marks)
4
(c) Find
s
′ ( 2).
(2 marks)
(d) What quantity associated with the motion of the rocket is measured by the value of s′ ( 2 )?
(1 mark)
5
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QUESTION 4 The committee of a large sporting association has proposed the major redevelopment of a stadium. For the redevelopment to be approved, all members of the association must vote on the proposal and at least three-quarters must vote yes. The committee wants to gain information about the likely outcome of the vote. A random sample of 150 members are asked whether they will vote yes or no. Of this sample, 119 plan to vote yes.
(a) What proportion of the sample of members is planning to vote yes?
(1 mark)
(b) Calculate a 95% confidence interval for p, the true proportion of members who are planning to vote yes.
(2 marks)
(c) Tick the appropriate box below to indicate your answer . answer . This confidence interval suggests that the redevelopment: will be approved by the vote. will not be approved by the vote. may or may not be approved by the vote. (1 mark)
(d) Justify the answer you indicated in part (c).
(1 mark)
6
(e) The committee plans to gain information from a larger sample of members. Determine the number of members who need to be sampled in order to obtain a 95% confidence interval with a width no greater than 0.065.
(2 marks)
7
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QUESTION 5 2 Find, from first principles, principles, f ′ ( x ) if f ( x ) = x
− 5 x.
(4 marks)
8
QUESTION 6 Consider the following 3 × 3 matrices: 0 ⎡ 1 ⎢ L = −1 6 ⎢ ⎢⎣ 2 −12
(a) (i)
Evaluate
0⎤
⎡2 ⎥ ⎢ 0 and U = 0 ⎥ ⎢ ⎢⎣ 0 1⎥ ⎦
1
a 0
−1⎤ ⎥ b . ⎥ 1⎥ ⎦
LU .
(3 marks) (ii) Find a and b if 1 ⎡ 2 ⎢ LU = −2 6 ⎢ ⎢⎣ 4 −12 12
−1⎤ −3⎥⎥ . 7⎥ ⎦
(3 marks)
(b) Find matrix P if LP = M
,
⎡ 1 1 −1⎤ ⎢ ⎥ where M = 1 2 1 . ⎢ ⎥ ⎢⎣ 2 −1 1⎥⎦
(2 marks) 9
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QUESTION 7 A portrait of René Descartes beside the ‘folium of Descartes’, which he first described in 1638, is shown on the Albanian stamp on the right. An example of the folium of of Descartes is the relation defined by the equation x
3
+ y 3 = 6 xy.
A graph graph of this relation relation is shown shown below:
Source:: http://curveba Source http://curvebank.calstatela.e nk.calstatela.edu du
O
(a) Find
d y d x
.
(4 marks)
10
(b) Find the equation of the tangent to the graph of this relation at point
( 3, 3) .
(3 marks)
11
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QUESTION 8 Various tests are used to measure literacy. The scores achieved by students in Literacy Test A can be modelled by a normal distribution with a mean of 62 and a standard deviation of 17.
(a) According to this model, the top 2% of scores achieved by students are greater than k . Find the value of k .
(2 marks)
Literacy Test B is developed. The scores achieved by students in this literacy test can be modelled by a different normal distribution.
(b) Literacy Test Test B is constructed so that the top 10% of scores achieved by students are greater than 120 and the bottom 10% of scores achieved by students are less than 60. (i)
Find the mean of the normal distributi distribution on that models the scores achieved by students in Literacy Test B.
(1 mark) (ii) Find the standard deviation of the normal distributi distribution on that models the scores achieved by students in Literacy Test B.
(3 marks)
12
(c) A student achieves a score of 86 in Literacy Test Test A and a score of 118 118 in Literacy Test B. In which test is the student’s performance better? Give a reason for your answer.
(3 marks)
13
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QUESTION 9
(a) Find the value(s) of x such that
x
3
− 6 x + 6 = 0.
(1 mark)
(b) Let f ( x ) =
x3 − 6 x + 6.
On the axes below, draw the graph of y = f ( x ) accurately marking axis intercepts and stationary points. ,
y 5
4
3
2
1
x
O –3
–2
–1
1
2
3
–1
(4 marks)
14
(c) Find the coordinates of the non-stationary inflection inflect ion point of the graph of y = f ( x ) to three significant figures.
,
(2 marks)
(d) Let
( )=
g x
x
3
− ax + a where a is a real number. ,
Determine all values of a for which
( )
g x
has no stationary points.
(5 marks)
15
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QUESTION 10
Consider the matrix
A =
⎡1 ⎢3 ⎣
3⎤
⎥.
9⎦
(a) Find: (i)
2
A .
(1 mark) (ii)
3
A .
(1 mark) (iii)
4
A .
(1 mark)
(b) On the basis of your results in part (a), complete the following conjecture conjectur e about A where n is a positive integer. A
n
n ,
= _____________________ × A (1 mark)
16
⎡1
Consider the matrix M = ⎢
⎢⎣ k
(c) (i)
Show that M
2
⎤ ⎥ where k is a real number. 2 k ⎥ ⎦ k
,
= (1 + k 2 ) M .
(2 marks) (ii) Hence derive an expression for M 3 in terms of M .
(2 marks)
(d) On the basis of your results in part (c), complete the following conjecture about where n is a positive integer. M
n
M
n ,
= ____________________ × M (1 mark)
17
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QUESTION 11 The photographs on the right show a type of timer. It consists of two roughly cylindrical glass chambers joined by a narrow neck. It contains a quantity of fine blue granules. In Figure 1 all the granules are in the lower chamber of the timer. When the timer is inverted, the granules trickle through the neck (Figure 2). The length of time measured by the timer corresponds to the time taken for all the granules to trickle through the neck. As a result of glass glass manufacturing manufacturing techniques, techniques, the ends ends of these timers are distinguishable from each other: one is flattened F in Figure 3) and the other is rounded ( R in Figure 3). In general, ( F the length of time measured by these timers varies, depending on which end is turned upwards .
Figure 1
The manufacturer of the timers wanted to know whether or not the difference in the length of time measured is the result of a systematic bias associated with the flattened end or the rounded end. To investigate this, twelve timers were randomly selected. For each timer, the time measured when the flattened end was turned upwards ( F seconds) and the time measured when the rounded end was turned upwards ( R seconds) were recorded. The difference D = F − R was calculated. The results are shown in the table below.
F
Figure 2
R Figure 3
Timer
Time in seconds when flattened end turned F ) upwards ( F
Time in seconds when rounded end turned R) upwards ( R
1
247
238
9
2
219
271
−52
3
251
310
−59
4
252
223
29
5
240
249
−9
6
228
206
22
7
232
253
−21
8
241
218
23
9
224
215
9
10
245
269
−24
11
223
287
−64
12
231
284
−53
18
D
=F−R
(a) Find
X D
for the data in the table on the page opposite.
(1 mark)
A plot plot of the D values is shown below:
–65 –60 –55 –50 –45 –40 –35 –30 –25 –20 –15 –10
–5
0
5
10
15
20
25
30
(b) Outline the reason why testing the null hypothesis H 0 : P D
=0
corresponds to the manufacturer’s question of whether or not the difference in the length of time measured is the result of a systematic bias associated with the flattened end or the rounded end.
(1 mark)
(c) Using a two-tailed Z -test -test at the 0.05 level of significance, draw a conclusion about the existence or otherwise of a systematic bias associated with the flattened end or the rounded end of the timers. Assume that the
V D
= 34.6 seconds.
(4 marks) 19
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QUESTION 12 The graph of y =
( )
g x
is shown below for x > 0. It has x-intercepts at x = a x = b and x = c. ,
,
y
M
O
a
b
x
c N
Let f ( x ) be any function such that f ′ ( x ) = g ( x ) . (a) The region bounded by the graph of y = (marked M ) has an area of 2 units 2.
( ) and
g x
the x-axis between
x
= a and x = b
Show that f ( b ) = f ( a ) + 2.
(2 marks)
20
(b) The region bounded by the graph of y = g (marked N ) also has an area of 2 units 2.
x
and the x-axis between x = b and
=
Show that f ( c ) = f ( a ) .
(3 marks)
(c) Using the results from parts (a) and (b), sketch on the axes below a graph of y = f ( x ) for x > 0 given that f ( a ) = −1. ,
y 4
3
2
1
O
a
b
c
x
–1
–2
–3
–4
(3 marks)
21
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QUESTION 13 The weight of individual Vegie Sausages varies. Let W represent the weight of a randomly selected Vegie Sausage. W can be modelled by a normal distribution with mean P = 55 grams and standard deviation V = 7 grams.
(a) What is the probability that a randomly selected Vegie Sausage will have a weight of less than 50 grams?
(1 mark) Let W 6 represent the average weight of six randomly selected Vegie Sausages. (b) (i)
State the distributi distribution on of W 6 .
(2 marks) (ii) Calculate
(
P W 6
≤ 50 ) to verify that P (W 6 ≤ 50 ) = 0.04 to two decimal places. ,
(1 mark)
Vegie Sausages are sold in packets labelled as containing six sausages with a net weight of 300 grams. Part of the label is shown on the right.
Source:: Packaging for S anitarium Vegie Sausages Source
Suppose that six Vegie Sausages are randomly selected during the packaging process. If the combined weight of the six Vegie Sausages is less than or equal to 300 grams, a seventh sausage is added to the packet, as shown on the left.
22
(
)
(c) Outline the reason why P W 6 ≤ 50 is the probability that a seventh sausage will be added to a randomly selected packet of Vegie Sausages.
(1 mark)
The production of 900 packets of Vegie Sausages is planned. It is likely that some of the 900 packets will need to have a seventh sausage added. It is likely therefore that more than 5400 sausages will be required. The production manager needs to know how many sausages will be required so that the 900 packets can be produced. Let Y represent the number of packets that will need to have a seventh sausage added. The distribution of Y can be modelled by a binomial distribution with
(d) (i)
n
= 900 and p = 0.04.
Determine the probability that none of the packets will need to have a seventh sausage added.
(1 mark) (ii) Determine the probability that that no more than than thirty-five of the packets packets will need to have a seventh sausage added.
(2 marks)
(e) How many Vegie Vegie Sausages will be required so that there is a greater than 99% chance that all 900 packets can be produced?
(3 marks)
23
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QUESTION 14
Consider the functions f ( x ) = 8 e and x
The graphs of y = f ( x ) and y =
( ) = 4e 2
g x
x
for x ≥ 0.
( ) and the line x = k where k > ln 2 The graphs intersect at point ( ln 2,16 ) . g x
,
,
are shown below.
y
R2
y = f ( f ( x x)) x = k R1
y = g ( g ( x) x)
x
O
Let R1 represent the area of the region bounded by the graphs of these two functions from x = 0 to x = ln 2. Let R represent the area of the region bounded by the graphs of these two 2 functions from x = ln 2 to x = k .
(a) Explain why
∫ f ( x )− g ( x ) k
0
dx
= R1 − R2 .
(2 marks)
24
(b) Find the exact value of k , where k > 0 such that ,
R1
= R2 .
(5 marks)
25
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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 11(c) continued’).
26
QUESTION 15 Consider the metal rod shown in the diagram below. One end is kept at a fixed temperature of 0°C and the other end is kept at a fixed temperature of 100°C. Heat is conducted along the rod and, assuming there is no temperature loss, each point along the rod reaches its own constant temperature after a certain time. 0ºC
A
100ºC
B
C
D
E
Consider five equally spaced points on this rod, labelled A, B, C , D, and E . Points A and E , at the ends of the rod, have fixed temperatures of 0°C and 100°C. The constant temperatures at points B, C , and D can be found by using the fact that the constant temperature at each point is equal to the average of the temperatures of its two neighbouring points. Let x, y, and z represent the constant temperatures at points B, C , and D respectively.
(a) Show that the following equation can be determined by considering the constant temperature at point D:
− y + 2 z = 100.
(1 mark)
(b) Determine two additional equations, one that represents the constant temperature at point B and one that represents the constant temperature at point C .
(2 marks)
27
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(c) Find the constant temperatures at points B, C , and D.
(1 mark) Suppose instead that one end of the metal rod is kept at a fixed temperature of p°C, where 0 < p < 100, and the other is still kept at a fixed temperature of 100°C: pºC
A
100ºC
B
C
D
E
(d) Show that the system of equations for temperatures x, y, and z can be represented by the augmented matrix
⎡ 2 −1 ⎢ 1 −2 ⎢ ⎢⎣ 0 −1
0 1 2
p ⎤
⎥ ⎥ 100 ⎥ ⎦
0 .
(2 marks)
28
(e) Use clearly defined row operations to solve this system of equations for temperatures x, y, and z.
(5 marks)
(f ) Hence find fixed temperature p if the constant temperature at point B is 55°C.
(2 marks)
29
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 11(c) continued’).
30
QUESTION 16 Let C ( x ) =
ax x
3
+b
,
where a and b are positive numbers and
x
≥ 0.
(a) Find C ′ ( x ) .
(2 marks) (b) Graphs of y = C ( x ) all have a shape similar to that of the graph shown below, and all have a single maximum value. y
O
x
Show that this maximum value occurs when x =
3
b
.
2
(2 marks)
31
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When coal is burnt at a power station, sulphur dioxide is released into the air. The ground-level concentration of sulphur dioxide varies, depending on the distance from the power station. The photograph below shows coal-burning Power Station P , which is approached along line PQ :
P
Q
Source:: Adapted from www.dreamstime. Source www.dreamstime.com com
The ground-level concentration of sulphur dioxide, C ( x ) in grams per square metre, at a distance of x x kilometres from Power Station P , along line PQ, can be modelled by the function C ( x) =
ax x
3
+b
.
It is known that the ground-level concentration of sulphur dioxide reaches a maximum level of 15 grams per square metre at a distance of 1.2 kilometres from Power Station P , along line PQ.
(c) Using your result from part (b), find the value of b in the function that models the ground-level concentration of sulphur dioxide in relation to Power Station P .
(2 marks)
32
(d) Find the corresponding value of a.
(2 marks)
(e) A person approaches Power Station P , along line PQ. At what distance from Power Station P is the person first exposed to a ground-level concentration of sulphur dioxide of 0.4 grams per square metre, according to this model?
(2 marks)
33
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QUESTION 17 k where k > 0. x The graph of y = f ( x ) for k = 1 is shown below. The tangent and the normal to the graph, at the Let f ( x ) = k + point where
x
,
= 1 are also shown. ,
y
5
O –5
5
x
–5
(a) Estimate the x-coordinate of the point where the normal to the graph of y = f ( x ) at the point where x = 1 intersects the graph again. ,
,
(1 mark) For f ( x ) = k +
k with k = 2 the graph of y = f ( x ) is shown below. The tangent and the normal x ,
,
to the graph, at the point where
x
= 1 are also shown. ,
y
5
O
– 5
x
5
– 5
34
(b) Estimate the x-coordinate of the point where the normal to the graph of y = f ( x ) at the point where x = 1 intersects the graph again. ,
,
(1 mark)
(c) For f ( x ) = k + where
x = 1,
k with x ,
k = 3
,
has equation
the normal to the graph of y = f ( x ) at the point ,
x − 3 y = −17.
Find the x-coordinate of the point where this normal intersects the graph again.
(2 marks)
(d) Using your results from parts (a), (b), and (c), complete the following table.
k
x-coordinate
y = f ( x )
,
of the point where the normal to the graph of at the point where x = 1 intersects the graph again ,
1
2
3
On the basis of these results, the following incomplete conjecture is made: For f ( x ) = k +
k the normal to the graph of y = f ( x ) at the point where x ,
,
x = 1,
intersects the graph again at a single point, and the x-coordinate of that single point is
.
x =
Complete this conjecture by filling in the box above. (1 mark)
35
PLEASE TURN OVER
(e) Prove or disprove the conjecture made in part (d).
(6 marks)
© SACE Board of South Australia 2011
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 ST AGE 2 MATHEMATICAL MATHEMATICAL STUDIES ST UDIES
Standardised Normal Distribution
Binomial Probability nk
P X k Ck n p k 1 p
A measurement measurement scale X is transformed into a standard scale Z , using the formula
Z
where p is the probability of a success in one trial and the possible values of 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
k
count X and a proportion of successes p
simple random sample of size n with sample mean x , is
b N b x 1.96
n n 1 . . . n k 1
T
n
.
X n
the sample standard deviation s in place of T .
p p
N X
np
N
T X
np 1 p
T
p
p 1 p n
Matrices and Determinants If A =
Sample Size — Mean
⎡ a ⎢ ⎢ c ⎣
b ⎤
⎥ then det A = A = ad − bc and
d ⎥⎦
The sample size n required to obtain a
A1
95% con¿dence interval of width w for the mean of a normal population with standard deviation T is
¥ 2s1.96T ´µ2 n ¦¦ µµ . ¦§ ¶ w
1
d b
A c
a
.
Derivatives
f a x
f x y
Con¿dence Interval — Population Proportion An approximate 95% con¿dence interval for the
x
n
nx
e
kx
ke kx 1 x
ln x log e x
population proportion p, based on a large simple
dy d x
n1
random sample of size n with sample proportion
X n
, is
p 1.96
Properties of Derivatives
p 1 p n
b p b p 1.96
p 1 p n
are
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
p
.
The mean and standard deviation of a binomial
A 95% con¿dence interval for the mean N of a normal population with standard deviation T , based on a
T
n k k
Binomial Mean and Standard Deviation
Con¿dence Interval — Mean
x 1.96
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 f x g a x d ® ®¬ ®» 2 d x ® g x ®- g x ®®¼
Sample Size — Proportion The sample size n required to obtain an approximate
d
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
( p is a given preliminary value for the proportion.)
37
2
+ bx + c = 0 then x =
−b
2
b − 4ac 2a
MATHEMATICAL STUDIES 2011 ACKNOWLEDGMENT
The photograph for Question 16 on page 32 is © Dreamstime/Fesus.
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.
