2007 - exam + solutions
Post on 07-Aug-2018
246 Views
Preview:
TRANSCRIPT
-
8/20/2019 2007 - Exam + Solutions
1/41
2007 MATHEMATICAL STUDIES
Thursday 8 November: 9 a.m.
Time: 3 hours
Examination material: one 37-page question booklet
one SACE registration number
Approved dictionaries, notes, calculators, and computer software may
Instructions to Students
1. You will have 10 minutes to read the paper. You must not write in your questio
during this reading time but you may make notes on the scribbling paper provid
2. Answer all parts of Questions 1 to 16 in the spaces provided in this question bo
all the space provided. You may write on pages 15, 35, and 36 if you need mor
each answer clearly.
3. The total mark is approximately 141. The allocation of marks is shown below:
Question 1 2 3 4 5 6 7 8 9 10 11 12 1
Marks 7 6 8 7 10 7 4 7 9 7 11 12 1
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 scrib
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 wh
ATTACH SACE REGISTRATION NUMBER LABEL
TO THIS BOX
SUPERVISOR
CHECK
FOR OFFICE
USE ONLY
RE-MARKED
-
8/20/2019 2007 - Exam + Solutions
2/41
-
8/20/2019 2007 - Exam + Solutions
3/41
QUESTION 1
(a) Findd
d
y
x for each of the following functions. There is no need to simpli
(i) y x x= +( )7 1034.
(ii) y e x x= −( )1 .
(b) Evaluate
k 2 0
1 2 3
2 1 0
− , where k is a real number.
-
8/20/2019 2007 - Exam + Solutions
4/41
QUESTION 2
The graph of y f = ( ), x where f ( ) x is the quadratic function f x ( ) = ax
below. Three regions of the area between the graph of y f = ( ) x and the
�
�
Region P has an area of 92 units2, Region Q has an area of 136 units
2, an
of 84 units2.
(a) (i) Write an equation involving a denite integral that relates to
(ii) Hence show that 28 9 3 138a c+ + =b .
-
8/20/2019 2007 - Exam + Solutions
5/41
When the areas of Region Q and Region R are used the following equations are
76 15 3148 21 3 126
a b ca b c
+ + =+ + =204
.
(b) Find the values of a, b, and c.
-
8/20/2019 2007 - Exam + Solutions
6/41
QUESTION 3
The wine in a percentage of all cork-sealed bottles is ‘corked’.
This means that the avour of the wine has been spoilt by a
chemical reaction in the cork.
(a) It is found that, under certain common conditions, the wine
in 4.1% of bottles is corked.
Of random selections of twelve bottles under such conditions,determine the proportion that will contain:
(i) exactly three bottles of corked wine.
(ii) no more than three bottles of corked wine.
(iii) no bottles of corked wine.
(iv) at least one bottle of corked wine.
-
8/20/2019 2007 - Exam + Solutions
7/41
Cork producers are using new methods to reduce the percentage of bottles o
(b) Find the value to which the percentage of bottles that contain corked wi
reduced so that no more than one-quarter of random selections of twelve
contain at least one bottle of corked wine. Give your answer to two sign
-
8/20/2019 2007 - Exam + Solutions
8/41
QUESTION 4
A graph of y e= 140 5. x is shown below:
�
An overestimate for the area between the graph above and the x-axis f
calculated using rectangles. The result, to two decimal places, is
2 1 85 2 5 02× + =. .× 13.74.
(a) On the graph above, draw the rectangles used to calculate this ove
(b) Calculate an overestimate for the area between the graph above an x = 2 to x = 6 using four rectangles, to two decimal places.
-
8/20/2019 2007 - Exam + Solutions
9/41
(c) The process of using more and more rectangles to improve this overesti
continued.
The overestimates approach a value A.
Determine the exact value of A.
-
8/20/2019 2007 - Exam + Solutions
10/41
QUESTION 5
Consider the following system of linear equations:
x y
x
x y k
− + =− + = −
+ + =
2 4 2
2 2
2 2 7
z
z
z
where k is a constant.
(a) Write this system of equations in augmented matrix form (i.e. det
form).
(b) Show, using clearly dened row operations, that this system of eq
reduced to
1 2 4
0 1 3
0 0 10
2
0
3
−−
+
k .
-
8/20/2019 2007 - Exam + Solutions
11/41
(c) For what value(s) of k does this system of equations have no solution?
(d) For all other values of k , solve this system of equations for x, y, and z,
-
8/20/2019 2007 - Exam + Solutions
12/41
QUESTION 6
Let f x x
( ) .= −1 1
The graph of y f x= ( ), and the chord through points A 1 0,( ) and B 3 23,( )
�
�
��
��
�
(a) Describe what the slope of chord AB represents.
-
8/20/2019 2007 - Exam + Solutions
13/41
(b) Find, from rst principles, ′ f ( )3 .
(c) Give a geometric interpretation for the value of ′ f ( )3 .
-
8/20/2019 2007 - Exam + Solutions
14/41
QUESTION 7
The graph of the derivative y f x= ′( ) for x ≥ 0 is shown below:
�
(a) On the graph above, mark and label point A where f ''( ) . x = 0
(b) The graph of y f x= ( ) passes through point B.
On the axes below, sketch the graph of y f x= ( ) for x ≥ 0.
�
y f x= ′( )
y f x= ( )
-
8/20/2019 2007 - Exam + Solutions
15/41
You may write on this page if you need more space to nish your answers. M
label each answer carefully (e.g. ‘Question 3(a)(ii) continued’).
-
8/20/2019 2007 - Exam + Solutions
16/41
QUESTION 8
Australian government legislation states that productscontaining more than 10 grams per kilogram of
genetically modied material must be identied as
genetically modied.
It has been discovered that a canola crop contains
genetically modied canola. The grower needs to know
whether or not the crop must be identied as genetically
modied.The crop is divided into sections. These sections, taken
collectively, can be considered to be a population.
Tests are undertaken on a randomly chosen sample of fty sections and x
genetically modied canola per kilogram in each section, is measured.
The mean amount of genetically modied canola in the fty sections is x
kilogram.
(a) (i) Calculate a 95% condence interval for the amount of genetic
in the crop. Assume that the population has a standard deviati
per kilogram.
(ii) The grower claims that the crop will not need to be identied
modied.
Do you think this claim is justied?
-
8/20/2019 2007 - Exam + Solutions
17/41
(b) How many sections of the crop need to be sampled to obtain a 95% con
with a width of no more than 0.1 grams per kilogram?
-
8/20/2019 2007 - Exam + Solutions
18/41
QUESTION 9
Let f x x( ) = − 4 .
Points P k , ,0( ) Q k f k , ( ) ,( ) and R 4 0, ,( ) where k ≥ 4, are marked on the gshown below:
�
(a) Complete the following table. Let A represent the area of triangle
k A f
k
4∫
4 0
5
73 3
22
-
8/20/2019 2007 - Exam + Solutions
19/41
(b) It is conjectured that f x xm
n A
k
( ) .d 4∫ = ×
(i) Write down integer values for m and n.
(ii) Prove this conjecture for k ≥ 4.
-
8/20/2019 2007 - Exam + Solutions
20/41
QUESTION 10
A newspaper article makes the following claim:
Parents too tiredThree-quarters of Australian parents
say they are too tired after work to
do some of the things that they would
like to do with their children.
Source: Adapted from Advertiser , Adelaide,
12 February 2007
A survey is undertaken to investigate the impact of work on family life. O
surveyed, 384 state they are too tired after work to do some of the things
to do with their children.
A two-tailed Z -test, at the 0.05 level of signicance, is to be applied to thdetermine whether or not there is sufcient evidence that the proportion o
parents who are too tired after work to do some of the things that they wo
their children is different from three-quarters.
(a) State the null hypothesis.
(b) State the alternative hypothesis.
-
8/20/2019 2007 - Exam + Solutions
21/41
(d) Determine whether or not the null hypothesis should be rejected.
(e) What can you conclude from your answer to part (d)?
-
8/20/2019 2007 - Exam + Solutions
22/41
QUESTION 11
Let the function f x x
x( ) ln=
2, x > 0.
(a) On the axes below, draw the graph of y f x= ( ), clearly showing a
�
� �
(b) Calculate the area enclosed by the graph of y f x= ( ), the x-axis, a
(c) (i) Show thatd
d x
x
x
x
x
12
+
= −
ln ln.
-
8/20/2019 2007 - Exam + Solutions
23/41
(ii) Hence, or otherwise, nd the exact value ofln
. x
x x
e
2
1
d ∫ Simplify you
-
8/20/2019 2007 - Exam + Solutions
24/41
QUESTION 12
Let D =b
a
0
0
.
(a) (i) Find D2.
(ii) Find D3.
(iii) Hence write down Dn.
Let P =
1
2
1
2
1
0
and A = a b a
b
−
0.
1
-
8/20/2019 2007 - Exam + Solutions
25/41
(ii) Show that PDP A− =1 .
(c) Simplify the right-hand side of the following equation, giving your answ
P, P−1
, and D.
times
A PDP PDP PDP PDP
n
n = ( )( )( ) ( )− − − −1 1 1 1
(d) Find An , using your results from part (a) and part (c).
-
8/20/2019 2007 - Exam + Solutions
26/41
QUESTION 13
(a) For the curve 2 02 2
x xy y− + = , where x and y are real numbers, s
d
d
y
x
y x
xy=
−−
24
1 2.
The graph of the curve 2 02 2 x xy y− + = is shown below:
�
�
��
��
-
8/20/2019 2007 - Exam + Solutions
27/41
(ii) Show that, at one of the points you found in part (b)(i), the curve h
tangent.
(c) Show that, for all a < 0, there are two points on the curve 2 2 2 x xy y− + y a= .
-
8/20/2019 2007 - Exam + Solutions
28/41
QUESTION 14
Premium Instant Coffee is produced in sachets. The net weight of the chosen sachet can be modelled by W , a normally distributed random v
of µ = 5 6. grams and a standard deviation of σ = 0 2. grams.
The distribution of W is graphed below:
(a) On the horizontal axis of the graph of the normal density curve a
illustrate the distribution of W .
These coffee sachets are sold in packs of twenty. Let W 20 be the average o
the sachets in a randomly chosen pack.
(b) (i) Write down the mean and the standard deviation of the distrib
-
8/20/2019 2007 - Exam + Solutions
29/41
The packs of twenty coffee sachets are labelled as containing 110 grams net.
(d) Find the probability that a randomly chosen pack of twenty sachets will conlabelled weight.
The coffee sachets are also sold in bulk in catering boxes. These boxes contain 2
(e) (i) If 0.1% of catering boxes contain less than k grams, nd k to the n
gram.
(ii) Would it be appropriate to label the catering boxes as containing 1.
Give a reason for your answer
-
8/20/2019 2007 - Exam + Solutions
30/41
QUESTION 15
Let f x xe x( ) = −32
for x ≥ 0.
(a) Find ′ f x( ).
The graph of y f x= ( ) is shown below:
�
�
�
-
8/20/2019 2007 - Exam + Solutions
31/41
(b) (i) Find the exact values of the slope and the y-intercept of the tangent
y f x= ( ) at the point where x =1.
(ii) Draw the tangent on the graph opposite.
(c) Find, correct to three decimal places, the x-coordinate of the point of in
graph of y f x= ( )
.
(d) Consider all tangents to the graph of y f x= ( ).
It has been claimed that the tangent with the greatest y-intercept will be
-
8/20/2019 2007 - Exam + Solutions
32/41
QUESTION 16
The life cycle of salmon can be divided into three main stages: eggs, ssh), and adult salmon.
It takes 2 years for eggs to hatch and mature into smolts and then 2 more
mature into adult salmon.
Fisheries scientists have studied the population of a type of salmon living
and have developed the following simple model for the female componen
salmon:
1.6% of eggs hatch and survive for the 2 years needed to mature int4.1% of the smolts survive for the 2 years needed to mature into adu
Adult salmon spawn 1500 female eggs and then die.
The scientists estimate that in 2006 the female component of this populati
made up of 384 000 eggs, 7100 smolts, and 257 adult salmon.
(a) Using the model above for this population of salmon, calculate theggs, smolts, and adult salmon that there will be in 2008.
Let L =
0 0 1500
0 016 0 0
0 0 041 0
.
.
and X =
384000
7100
257
.
(b) (i) Evaluate the matrix product LX .
••
•
-
8/20/2019 2007 - Exam + Solutions
33/41
(ii) Using L and X , complete the following table.
Year 2006 2008 2010 201
Number of female
adult salmon257
(c) Describe the way that the number of female adult salmon changes in thafter 2006.
A type of trout also lives in this lake and has a life cycle similar to that of salmo
scientists observe that a proportion of female adult trout survive their rst spawn
and return 2 years later to have a second spawning. The scientists have develope
simple model for the female component of this population of trout:
1.3% of eggs hatch and survive for the 2 years needed to mature into smo
3.5% of the smolts survive for the 2 years needed to mature into adult trou
At their rst spawning the adult trout produce 2500 female eggs.
15% of adult trout survive for 2 more years to have a second spawning.
At their second spawning the adult trout produce 1500 female eggs and the
The scientists estimate that in 2006 the female component of this population of
up of 75 600 eggs, 12 100 smolts, 417 rst-spawning adult trout, and 63 second-
trout.
(d) Write down the matrices L and X such that the matrix multiplication LX
the number of female eggs, smolts, rst-spawning adults, and second-spa
thi l ti f t t i 2008
•
•
•
•
•
-
8/20/2019 2007 - Exam + Solutions
34/41
(e) Using the matrices dened in part (d), nd the number of female
will be in 2012, according to this model.
-
8/20/2019 2007 - Exam + Solutions
35/41
You may write on this page if you need more space to nish your answers. M
label each answer carefully (e.g. ‘Question 3(a)(ii) continued’).
-
8/20/2019 2007 - Exam + Solutions
36/41
You may write on this page if you need more space to nish your ans
label each answer carefully (e.g. ‘Question 3(a)(ii) continued’).
-
8/20/2019 2007 - Exam + Solutions
37/41
�
�
.
� �
X Z
�
, x
.96.196.1n
xn
x �
�
�
�
.96.12
2
�
wn
�
�
,ˆn
X p �
�
k X P )(
!)!(
!
k k n
nC
nk
�
np X ��
(1 ) X np p �
a b
Ac d
A
.11
�
� ac
bd A
A
y x f �)(
n x
kxe
x x elogln �
-
8/20/2019 2007 - Exam + Solutions
38/41
-
8/20/2019 2007 - Exam + Solutions
39/41
-
8/20/2019 2007 - Exam + Solutions
40/41
-
8/20/2019 2007 - Exam + Solutions
41/41
top related