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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

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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.

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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 .

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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.

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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.

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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

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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.

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(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.

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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 .

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(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,

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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.

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(b) Find, from  rst principles,  ′ f  ( )3 .

(c) Give a geometric interpretation for the value of ′ f  ( )3 .

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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= ( )

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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’).

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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?

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(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?

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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

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(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.

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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.

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(d) Determine whether or not the null hypothesis should be rejected.

(e) What can you conclude from your answer to part (d)?

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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.

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(ii) Hence, or otherwise, nd the exact value ofln

. x

 x x

e

2

1

d ∫   Simplify you

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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

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(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).

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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:

 

�

�

��

��

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(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= .

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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

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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

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QUESTION 15

Let  f x xe  x( ) =   −32

  for  x ≥ 0.

(a) Find ′ f x( ).

The graph of  y f x= ( ) is shown below:

�

�

�

 

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(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

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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 .

••

•

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(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

•

•

•

•

•

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(e) Using the matrices dened in part (d), nd the number of female

will be in 2012, according to this model.

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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’).

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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’).

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