External Examination 2008
2008 MATHEMATICAL STUDIES FOR OFFICE USE ONLY
Graphics calculator
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ATTACH SACE REGISTRATION NUMBER LABEL
Brand
TO THIS BOX
Model Computer software
RE-MARKED
Thursday 6 November: 9 a.m.
Pages: 41 Questions: 17
Time: 3 hours Examination material: one 41-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 29, 33, 39, and 40 if you need more space, making sure to label each answer clearly.
3.
The total total mark is approximately 146. The allocation allocation of of marks is shown below: Question
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Marks
7
4
8
4
4
8
7
8
6
7
11
12 12
9
11
14 14
9
17
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 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 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 for each of the following functions. There is no need to simplify your answers. d x 7
(i) y 2 3 x5 .
(2 marks) (ii) y
e 4 x x 2 9
.
(3 marks)
(b) Find the exact values of x for which 1n x 5 1n x 0 .
(2 marks)
2
QUESTION 2
Let P
3
1 0
0.5 3 1
and Q
2 2 1
3
0 1
.
(a) Find 2 P – Q .
(2 marks) (b) (i) Write down a matrix X , such that PX can be calculated.
(1 mark) (ii) What is the order of the resulting matrix PX ?
(1 mark)
3
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QUESTION 3
Let A
6 4 3
n
, B
1
0 1
0
2
1
3
5
, and C 1 2 . 2
5
(a) (i) For what value(s) of n does A1 not exist?
(3 marks) (ii) Find A1 in terms of n.
(2 marks) (b) (i) Find BC .
(1 mark)
4
(ii) Does C B1 ? Give a reason for your answer.
(2 marks)
5
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QUESTION 4
Find
x 10 x 2 dx .
(4 marks)
6
QUESTION 5
Consider the function f ( x x). The sign diagrams for f ( x) and f ( x) are shown below:
f ( x)
2
f ( x)
x
x
4
1
4
Points A (−2, 7), B (1, 3), and C (4, 0) lie on the graph of y f ( x). Sketch a graph of y f ( x) . y
8 A
6
4 B
2
C
Ο
6
4
2
2
4
x
6
2
4
6
8 (4 marks) 7
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QUESTION 6
Let f ( x) x x x, where x 0 . (a) Find f ( x).
(2 marks) Let Ax , f ( x) be any point on the graph of y f ( x) and let B be the fixed point 3, f (3). The graph of y f ( x) and a chord AB are shown below: y 3
B 2
1
x
Ο
A
1
2
3
(b) On the graph above, draw the chord AB when A is placed at 2, f (2).
8
(1 mark)
(c) Complete the third column of of the following table, giving your answers correct to two decimal places.
x-coordinate of A A
x-coordinate of B B
2.5
3
2.9
3
2.99
3
slope of the chord AB
(4 marks) As A approaches B, the slope of the chord AB approaches a limiting value k . (d) Determine the exact value of k .
(1 mark)
9
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QUESTION 7
An upper triangular matrix is a square matrix containing all zeros below its leading diagonal. The following are examples of 3 3 upper triangular matrices: 2 3 1
3
0 4
5 4
20 20
A 0
1
5
B 0
2
1
C 0
2 15
0
0
4
0
0
5
0
0 10
.
(a) Find: (i) A .
(1 mark) (ii) B .
(1 mark) (iii) C .
(1 mark) (b) On the basis of your answers to part (a), make a conjecture about about the value of the determinant of all 3 3 upper triangular matrices.
(2 marks)
10
(c) Prove the conjecture you made in part (b).
(2 marks)
11
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QUESTION 8
A wholesaler who purchases large batches of roof tiles from a manufacturer has a policy that the proportion of first-grade tiles in each batch purchased should be at least 0.8. The wholesaler randomly samples fifty tiles from each batch and accepts a batch only if the sample contains forty or more first-grade tiles.
This photograph cannot be reproduced here for copyright reasons.
Source: The Tile Man Inc. website, www.thetileman.com
(a) Consider a batch of tiles in which the proportion of first-grade tiles is 0.75. Suppose that
fifty
tiles are randomly sampled from the batch.
Let X be the number of first-grade tiles in the sample. (i) Find P ( X 39).
(2 marks) (ii) Hence
find
the probability that the batch will be accepted.
(1 mark)
(b) Consider a batch of tiles in which the proportion proportion of first-grade tiles is 0.8. Determine the probability that the batch will be accepted.
(2 marks)
12
(c) Consider a batch batch of tiles in which the proportion of first-grade tiles is p. The acceptance probability A( p), which depends on p, is A( p) P( X 40). The value of A( p) has been been calc calcula ulated ted for for more more value valuess of p, with the results shown below: p
0.82
0.85
0.88
0.91
0.93
A( p)
0.719
0.880
0.968
0.996
0.999
A graph of these values and the graph of an algebraic model for A( p) are sh shown below: 1.0 0.9
acceptance probability A( p)
0.8 0.7 0.6 0.5
p 0.81
0.85
0.89
0.93
(i) Which one of the following functions provides the best model for A( p )? Tick the appropriate box. A( p ) 1.09 p 2 2.05 p 1.95 A( p ) 0.526 p 2 0.796 p 0.702 A( p ) 32 .55 p 2 59.45 p 26.14
(1 mark)
(ii) The manufacturer of the roof roof tiles wants the wholesaler to accept the batches with a probability of 0.99. Using your choice of model for A( p), determine the minimum value of p that will achieve this. Give your answer correct to three decimal places.
(2 marks)
13
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QUESTION 9 1
Consider the function f ( x) x , where x 0 . (a) A graph of y f ( x) is shown below: y
A B
Ο
1
2
4
x m
Find the value of m if the shaded regions marked A and B are equal in area.
(2 marks)
14
(b) Consider the general case case where the shaded regions regions marked A and B are equal in area, as shown below: y
A B x
Ο
a
Derive a relationship for
b
c
d
b in terms of c and d . a
(4 marks)
15
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QUESTION 10
0 1 0 0 (a) Consider the matrices A
0 0 1 0 0 0 0 1
a
and X
1 0 0 0
b c
.
d
(i) Calculate AX .
(1 mark) (ii) Descri Describe, be, in words, words, the effe effect ct of pre-m pre-multip ultiplying lying any 4 1 matrix by matrix A.
(1 mark) (iii) Hence explain, in words, why A4 X X .
(1 mark) (iv) Hence deduce a positive value of n for which An A1.
(2 marks)
16
(b) Matrix B is a square matrix such that B
a
c
b
d
c e . d
a
e
b
Write down matrix B.
(2 marks)
17
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QUESTION 11
A newspaper article stated that ‘two out of three South Australians do not know when daylight saving ends’. A survey of 2467 South Australians is conducted. Of those surveyed, 1608 do not know when daylight saving ends. (a) A two-tailed Z-test, at the 0.05 level of significance, is to be applied to the survey data, to determine whether or not there is suf ficient evidence that the proportion of all South Australians who do not know when daylight saving ends is different from two out of three. (i) State the null hypothesis.
(1 mark) (ii) State the alternative hypothesis. hypothesis.
(1 mark) (iii) State the null distribution of the test statistic.
(1 mark) (iv) Determine whether or not the null hypothesis should be rejected.
(3 marks)
18
(b) Using the survey survey data, calculate a 95% confidence interval for p, the proportion of all South Australians who do not know when daylight saving ends.
(2 marks) (c) (i) Do you consider, from your answers to parts (a) and (b), that the survey data provide evidence that the newspaper’s statement is wrong?
(1 mark) (ii) Explain why it is not reasonable reasonable to claim that p 2 exactly. 3
(1 mark) (iii) Using your answer to part (b), suggest a reasonable claim for the possible value of p.
(1 mark)
19
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QUESTION 12
The graph of x 2 xy y 2 12 is shown below. The line x k , where k is a constant, intersects the graph of this relation at points P and Q. The case where k 2 is shown. y
5
P
x
Ο
5
5
Q 5
(a) On the graph above, show the case where k 3.
(1 mark)
(b) Calculate the coordinates of P and Q for the case where k 1.
(2 marks) (c) Show that
d y d x
y 2 x
2 y x
.
(3 marks) 20
(d) For the case where where k 2, use the result in part (c) to the tangent to the curve at point:
find
the equation of
(i) P (2, 4) .
(2 marks) (ii) Q (2, 2).
(2 marks) (iii) The tangents in parts (d)(i) and (d)(ii) intersect at point T . Show that PTQ is an isosceles triangle.
(2 marks)
21
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QUESTION 13
The graphs of y e x and y 1n x
y 10
for x 0 are shown. The line segment AB with equation
A
8
P
2
y x e 2 meets these graphs
at P 2, e2 and Q e2 , 2 .
6
(a) State the exact coordinates coordinates of
4
points A and B, the axis intercepts
Q
2
of the line segment AB.
B
Ο
2
4
6
8
10
x
(2 marks) (b) Calculate the exact value of the area of the blue region between y x e2 2 and y e x , from x 0 and x 2.
(4 marks)
22
(c) Give a reason why the area of the blue region is equal to the area of of the green region enclosed by the graph of y 1n x, the the lin linee seg segme ment nt AB, and the x-axis.
(1 mark) (d) Hence calculate the area area of the yellow yellow region.
(2 marks)
23
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QUESTION 14
A pain relief drug is sold in tablet form. When a person swallows a tablet, some of the drug is absorbed into the bloodstream and is then broken down by the body. Let C (t ) be the concentration (milligrams per litre) of the drug in the bloodstream of an average person t minutes after a tablet is swallowed, where: C (t ) 0.075t 2 e0.05t for t 0.
(a) On the axes below below,, draw a graph graph of y C (t ) , showing intercepts, turning-points, and shape. C
20
10
t
Ο
20
40
60
80
100
120
(3 marks) The area between the graph of y C (t ) and the time axis is known as AUC (area under curve). AUC is used as a measure of drug exposure, that is, a measure of how much and for how long’ a drug stays in the body body.. ‛
(b) Determine the AUC value that the tablet produces produces in an average person person within 60 minutes of being swallowed.
(2 marks)
24
The same pain relief drug can be administered by intravenous drip. Let D(t ) be the concentration (milligrams per litre) of the drug in the bloodstream of an average person t minutes after an intravenous drip is inserted, where: D(t )
14 1
e0.05t for t 0. Source: DCC Healthcare website, www.dcc.ie
(c) On the axes in part (a), draw a graph of y D(t ), showing shape and intersection points. (d) A doctor needs to know the time for which an intravenous drip should be inserted so so that it will produce the same AUC value that the tablet produces in an average person within 60 minutes of being swallowed. Let k be the time required. (i) Write down an equation that involves a de finite integral and has k as its solution.
(2 marks) (ii) Find the value of k to the nearest minute.
(2 marks)
25
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QUESTION 15
(a) (i)
Z is
distributed normally with mean in the diagram below:
0
and standard deviation
a
Given that
as shown
Z
a
P( a Z a) 0.9, find
1,
the value of a.
(2 marks) (ii) Suppose that deviation .
Y is
distributed normally with unknown mean
Given that
,
and standard
find .
(2 marks)
A large drill core is taken from the site of a potential zinc mine. The zinc content of the drill core is not distributed evenly but occurs in small concentrated sections. To estimate the zinc content of this drill core, a small amount of the drill core is sampled. This photograph cannot be reproduced here for copyright reasons. To obtain a representative sample, the entire drill core is crushed into small fragments and thoroughly mixed before a sample is taken. Source: Dynasty
Metals & Mining Inc. website, www www.dynastymining.com .dynastymining.com
26
(b) Suppose that the drill core has been crushed to form a large large non-normal population of fragments of approximately equal mass. Let X be the zinc content of one such fragment (in grams per kilogram), with mean X 11.7 and standard deviation X 10.9 . Let X 5 be the average zinc content of a sample of five randomly chosen fragments. (i) Two histograms, histograms, A and B, are shown shown below. below. One of the histograms corresponds to the distribution of X and the other to the distribution of X 5. Histogram A
0
10
20
Histogram B
30
40
0
percentage zinc (by weight)
20
40
60
80
10 0
percentage zinc (by weight)
State, giving reasons, which of the two histograms corresponds to the distribution of X .
(2 marks) (ii) Find the mean X and the standard deviation X . 5
5
(2 marks) (iii) Use the normal distribution to
find
an approximate value for
5
28.
(1 mark)
27
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(iv) Is the answer to part (b)(iii) an underestimate or an overestimate of the exact value of
5
28?
Explain your answer with reference to the histograms in part (b)(i).
(2 marks) (c) Suppose that the zinc content content of a second drill core is to be estimated. Let Y n be the average zinc content of a sample of n randomly chosen fragments from the second drill core. Assume that the distribution of Y n is approximately normal and that Y has a standard deviation of Y 10.9. A geologist wants to claim, with 90% probability, that Y n is within kilogram of the population mean Y .
2.5 grams per
(i) Find the minimum number of fragments, n, that should be sampled for this claim to be made.
(2 marks) (ii) Using the value of n that you obtained in part (c)(i) and the histograms in part (b)(i), explain whether or not the assumption of approximate normality is reasonable.
(1 mark)
28
29
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QUESTION 16
A publisher plans to launch a new monthly home improvement magazine. The information in the table below is known about similar home improvement magazines. Let x represent the selling price of a copy of a home improvement magazine in dollars. Let n represent the number of these magazines sold per month.
x
4.50
5.50
6.20
6.80
7.50
7.95
n
116 000
97 000
75 0 0 0
66 000
5 4 00 0
46 000
This information is represented in the graph below: 210000 180000 150000
n
120000 90000 60000 30000 0 0
1
2
3
4
5
6
7
8
9
10 10
x
An algebraic model for the relationship between n and x is n 20 600 x 208 000 .
(a) Monthly revenue is the amount of money earned from magazines sold per month. Determine the monthly revenue if the new magazine has a selling price of $5.00.
(2 marks)
30
(b) Let R be the monthly revenue if the new magazine has a selling price of x. (i) Write down an expression for R.
(1 mark) (ii) Draw a sketch of R R versus x for 0 x 8 , marking an appropriate scale on the vertical axis below. R
x
Ο
2
4
6
8
(2 marks) (iii) Determine the selling price that will return the maximum monthly revenue.
(1 mark) (c) The cost of printing copies of the new magazine will depend on the number number printed. If all copies printed are sold, this cost can be de fined in terms of n. If n copies of the magazine are printed, let c represent the cost of printing each of 000017 173 3n these copies, where c 2.88e0.0000 . Write down an expression for T , the total cost of printing n magazines.
(1 mark)
31
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(d) Let P be the profit, where P R T . Determine the selling price of the magazine that will maximise P .
(2 marks)
32
33
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QUESTION 17
(a) (i) Represent the following following system system of equations in augmented augmented matrix form.
x y z 1 0.007 x 0.006 y 0.009 z 0.007 0.5 x 0.6 y k z 0.66.
(1 mark) (ii) Show, using clearly de defined row operations, that this system can be reduced to 1
1
0
1
0 0
1
2 10k 3
1 0
.
1.6
(5 marks)
34
(iii) Hence show that this system has the following solution:
x
10k 7.8 10k 3
, y
3.2 10k 3
,
z
1.6 10k 3
.
(3 marks)
Question 17 continues on page 36.
35
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QUESTION 17 continued
The sulphur and carbon content of coal varies, depending on where the coal is mined. A coal-mining company plans to blend coal from different mines in order to produce coal with specific sulphur and carbon content. The coal to be blended comes from mines X , Y , and Z . The coal from mine X contains 0.7% sulphur and 50% carbon. The coal from mine Y contains 0.6% sulphur and 60% carbon. The coal from mine Z contains 0.9% sulphur and 70% carbon. Let x represent the proportion of coal to be obtained from mine X . Let y represent the proportion of coal to be obtained from mine Y . Let z represent the proportion of coal to be obtained from mine Z . Source: EduPic Graphical Resource website, www.edupic.net
(b) The coal-mining company aims to produce blended coal coal containing 0.7% sulphur sulphur and 60% carbon. The proportion of coal required from each mine can be determined by solving the following system of equations:
x y z 1 0.007 x 0.006 y 0.009 z 0.007 0.5 x 0.6 y 0.7 z 0.6. (i) Give an interpretation of the equation x y z 1 .
(1 mark) (ii) Determine the proportion of coal required from each mine.
(2 marks)
36
(c) Write down the system of equations that could be solved to determine the proportion of coal required from mines X , Y , and Z in order to produce blended coal containing 0.67% sulphur and 59% carbon.
(1 mark)
(d) The coal-mining company is asked to produce blended coal containing 0.7% sulphur and 66% carbon. (i) Solve the the following following system of equations:
x y z 1 0.007 x 0.006 y 0.009 z 0.007 0.5 x 0.6 y 0.7 z 0.66.
(1 mark) (ii) Explain what your answer to part (d)(i) means about producing producing blended coal containing 0.7% sulphur and 66% carbon.
(1 mark)
Question 17 continues on page 38.
37
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(iii) The coal-mining company believes that it can produce blended coal containing 0.7% sulphur and 66% carbon from mines X , Y , and Z by washing the coal from mine Z to increase its proportion of carbon without changing its sulphur content. Let k represent the proportion of carbon in the coal from mine Z after the coal-washing process. Using the results from part (a)(iii) on page 35, determine the minimum value of k that must be achieved by the washing process in order to produce blended coal containing 0.7% sulphur and 66% carbon from mines X , Y , and Z .
(2 marks)
38
39
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© SACE Board of South Australia 2008
40
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 is transformed into a standard scale Z , using the formula
Z
P X k Ck n p k 1 p
where p is the probability of a success in one trial and the possible values of X . . . n and X are k 0, 1, ..
X
C k n
where is the population mean and is the standard deviation for the population distribution.
n n 1 . . . n k 1 n . k n k k
Condence Interval — Mean
Binomial Mean and Standard Deviation
A 95% condence 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
The mean and standard deviation of a binomial X count X and a proportion of successes p are n p p X np
x 1.96
n
x 1.96
n
.
X
p 1 p
p
Matrices and Determinants
If A
Sample Size — Mean
The sample size n required to obtain a 95% condence interval of width w for the mean of a normal population with standard deviation is
1 A
21.96 2 n . w
a
b
c
d
1
then det A A ad bc and
d b
A c
a
.
Derivatives
f x y
Condence Interval — Population Proportion
An approximate 95% con dence interval for the population proportion p, based on a large simple random sample of size n with sample proportion
f x
d y d x
n1
x
n
nx
e
kx
ke kx 1 x
ln x log e x
X , is n
p 1.96
n where p is the probability of a success in one trial.
For suitably large samples, an approximate 95% condence interval can be obtained by using the sample standard deviation s in place of .
p
np 1 p
Properties of Derivatives
p 1 p n
p p 1.96
p 1 p n
d .
d x
f x g x f x g x f x g x
f x f x g x g x f x g x d 2 d x g x g x g x
Sample Size — Proportion
d
The sample size n required to obtain an approximate 95% condence interval of approximate width w for a proportion is
d x
21.96 2 1 . n w p p
f g x f g x g x
Quadratic Equations 2
If ax bx c 0 then x
( p is a given preliminary value for the proportion.) 41
b
2
b 4ac 2a
