2007 mathematical studies examination paper

2007 MATHEMATICAL STUDIES

Thursday 8 November: 9 a.m.

Time: 3 hours

Examination material: one 37-page question bookletone 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. 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 15, 35, and 36 if you need more space, making sure to label 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 13 14 15 16Marks 7 6 8 7 10 7 4 7 9 7 11 12 10 13 10 13

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 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 and diagrams, for 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 37. 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 SACE registration number label to the box at the top of this page.

Pages: 37Questions: 16

ATTACH SACE REGISTRATION NUMBER LABEL TO THIS BOX

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QUESTION 1

(a) Find ddyx

for each of the following functions. There is no need to simplify your answers.

(i) y x x= +( )7 103 4.

(2 marks)

(ii) y e xx= −( )1 .

(2 marks)

(b) Evaluate k 2 01 2 32 1 0

− , where k is a real number.

(3 marks)

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QUESTION 2

The graph of y f= ( ),x where f ( )x is the quadratic function f x bx c( ) ,= + +ax2 is shown below. Three regions of the area between the graph of y f= ( )x and the x-axis are also shown.

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Region P has an area of 92 units2, Region Q has an area of 136 units2, and Region R has an area of 8� units2.

(a) (i) Write an equation involving a definite integral that relates to the areaofRegionP.

(2 marks)

(ii) Hence show that 28 9 3 138a c+ + =b .

(3 marks)

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When the areas of Region Q and Region R are used the following equations are obtained:

76 15 3148 21 3 126

a b ca b c

+ + =+ + =

204.

(b) Find the values of a, b, and c.

(1 mark)

6

QUESTION 3

The wine in a percentage of all cork-sealed bottles is ‘corked’. This means that the flavour 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.

(2 marks)

(ii) no more than three bottles of corked wine.

(1 mark)

(iii) no bottles of corked wine.

(1 mark)

(iv) at least one bottle of corked wine.

(2 marks)

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Cork producers are using new methods to reduce the percentage of bottles of corked wine.

(b) Find the value to which the percentage of bottles that contain corked wine must be reduced so that no more than one-quarter of random selections of twelve bottles contain at least one bottle of corked wine. Give your answer to two significant figures.

(2 marks)

8

QUESTION 4

A graph of y e= 14

0 5. x is shown below:

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An overestimate for the area between the graph above and the x-axis from x = 2 to x = 6 is 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 overestimate.(2 marks)

(b) Calculate an overestimate for the area between the graph above and the x-axis from x = 2 to x = 6 using four rectangles, to two decimal places.

(2 marks)

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(c) The process of using more and more rectangles to improve this overestimate is continued.

The overestimates approach a value A.

Determine the exact value of A.

(3 marks)

10

QUESTION 5

Consider the following system of linear equations:

x yxx y k

− + =− + = −

+ + =

2 4 22 2

2 2 7

z

z

z

where k is a constant.

(a) Write this system of equations in augmented matrix form (i.e. detached coefficient form).

(1 mark)

(b) Show, using clearly defined row operations, that this system of equations can be reduced to

1 2 40 1 30 0 10

203

−−

+

k.

(4 marks)

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(c) For what value(s) of k does this system of equations have no solution?

(1 mark)

(d) For all other values of k, solve this system of equations for x, y, and z, in terms of k.

(4 marks)

12

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, ,( ) are shown below:

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(a) Describe what the slope of chord AB represents.

(1 mark)

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

(5 marks)

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

(1 mark)

14

QUESTION 7

The graph of the derivative y f x= ′( ) for x ≥ 0 is shown below:

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(a) On the graph above, mark and label point A where f '' ( ) .x = 0 (1 mark)

(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.

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(3 marks)

y f x= ′( )

y f x= ( )

15 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 3(a)(ii) continued’).

16

QUESTION 8

Australian government legislation states that products containing more than 10 grams per kilogram of genetically modified material must be identified as genetically modified.

It has been discovered that a canola crop contains genetically modified canola. The grower needs to know whether or not the crop must be identified as genetically modified.

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 fifty sections and x, the amount of genetically modified canola per kilogram in each section, is measured.

The mean amount of genetically modified canola in the fifty sections is x = 9.21 grams per kilogram.

(a) (i) Calculate a 95% confidence interval for the amount of genetically modified canola in the crop. Assume that the population has a standard deviation of σ = 0 74. grams per kilogram.

(2 marks)

(ii) The grower claims that the crop will not need to be identified as genetically modified.

Do you think this claim is justified?

(2 marks)

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(b) How many sections of the crop need to be sampled to obtain a 95% confidence interval with a width of no more than 0.1 grams per kilogram?

(3 marks)

18

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 graph of y f x= ( ), as shown below:

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(a) Complete the following table. Let A represent the area of triangle PQR.

k A f x xk

( )d4∫

4 0 0

5

7 3 32

2 3

8

(4 marks)

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(b) It is conjectured that f x x mn A

k

( ) .d4∫ = ×

(i) Write down integer values for m and n.

(1 mark)

(ii) Prove this conjecture for k ≥ 4.

(4 marks)

20

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. Of the 520 parents surveyed, 384 state they are too tired after work to do some of the things that they would like to do with their children.

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 sufficient evidence that the proportion of all Australian parents who are too tired after work to do some of the things that they would like to do with their children is different from three-quarters.

(a) State the null hypothesis.

(1 mark)

(b) State the alternative hypothesis.

(1 mark)

(c) State the null distribution of the test statistic.

(1 mark)

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

(3 marks)

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

(1 mark)

22

QUESTION 11

Let the function f x xx

( ) ln= 2 , x > 0.

(a) On the axes below, draw the graph of y f x= ( ), clearly showing any axis intercept(s).

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(3 marks)

(b) Calculate the area enclosed by the graph of y f x= ( ), the x-axis, and the line x e= .

(1 mark)

(c) (i) Show that ddx

xx

xx

12

+

= −ln ln .

(3 marks)

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(ii) Hence, or otherwise, find the exact value of ln .xx

xe

21

d∫ Simplify your answer.

(4 marks)

24

QUESTION 12

Let D = b

a0

0

.

(a) (i) Find D2.

(1 mark)

(ii) Find D3.

(1 mark)

(iii) Hence write down Dn.

(1 mark)

Let P =

12

12

1

0

and A = a b a

b−

0.

(b) (i) Find P−1, giving the elements in exact form.

(2 marks)

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(ii) Show that PDP A− =1 .

(2 marks)

(c) Simplify the right-hand side of the following equation, giving your answer in terms of P, P−1, and D.

times

A PDP PDP PDP PDP

n

n = ( )( )( ) ( )− − − −1 1 1 1

(2 marks)

(d) Find An , using your results from part (a) and part (c).

(3 marks)

26

QUESTION 13

(a) For the curve 2 02 2x xy y− + = , where x and y are real numbers, show that

ddyx

y xxy

= −−

2 41 2

.

(3 marks)

The graph of the curve 2 02 2x xy y− + = is shown below:

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(b) (i) Find the x-coordinates of the points on the curve where y = −1.

(2 marks)

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(ii) Show that, at one of the points you found in part (b)(i), the curve has a vertical tangent.

(2 marks)

(c) Show that, for all a < 0, there are two points on the curve 2 02 2x xy y− + = where y a= .

(3 marks)

28

QUESTION 14

Premium Instant Coffee is produced in sachets. The net weight of the coffee in a randomly chosen sachet can be modelled by W, a normally distributed random variable with a mean of µ = 5 6. grams and a standard deviation of σ = 0 2. grams.

The distribution of W is graphed below:

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(a) On the horizontal axis of the graph of the normal density curve above, write numbers to illustrate the distribution of W. (1 mark)

These coffee sachets are sold in packs of twenty. Let W 20 be the average of the net weights of the sachets in a randomly chosen pack.

(b) (i) Write down the mean and the standard deviation of the distribution of W 20.

(2 marks)

(ii) On the axes of the graph above, sketch the distribution of W 20.(2 marks)

(c) On the graph above, illustrate the fact that P W P W≤( ) > ≤( )5 5 5 520. . . (1 mark)

29 PLEASE TURN OVER

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 contain less than its labelled weight.

(2 marks)

The coffee sachets are also sold in bulk in catering boxes. These boxes contain 240 sachets.

(e) (i) If 0.1% of catering boxes contain less than k grams, find k to the nearest whole gram.

(3 marks)

(ii) Would it be appropriate to label the catering boxes as containing 1.3 kilograms net?

Give a reason for your answer.

(2 marks)

30

QUESTION 15

Let f x xe x( ) = −32 for x ≥ 0.

(a) Find ′f x( ).

(3 marks)

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

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31 PLEASE TURN OVER

(b) (i) Find the exact values of the slope and the y-intercept of the tangent to the graph of y f x= ( ) at the point where x =1.

(3 marks)

(ii) Draw the tangent on the graph opposite. (1 mark)

(c) Find, correct to three decimal places, the x-coordinate of the point of inflection of the graph of y f x= ( ).

(2 marks)

(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 the one passing through the point of inflection of the graph of y f x= ( ).

Explain why this claim seems reasonable.

(1 mark)

32

QUESTION 16

The life cycle of salmon can be divided into three main stages: eggs, smolts (immature fish), and adult salmon.

It takes 2 years for eggs to hatch and mature into smolts and then 2 more years for smolts to mature into adult salmon.

Fisheries scientists have studied the population of a type of salmon living in a particular lake and have developed the following simple model for the female component of this population of salmon:

1.6% of eggs hatch and survive for the 2 years needed to mature into smolts.4.1% of the smolts survive for the 2 years needed to mature into adult salmon.Adult salmon spawn 1500 female eggs and then die.

The scientists estimate that in 2006 the female component of this population of salmon was made up of 384 000 eggs, 7100 smolts, and 257 adult salmon.

(a) Using the model above for this population of salmon, calculate the number of female eggs, smolts, and adult salmon that there will be in 2008.

(2 marks)

Let L =

0 0 15000 016 0 00 0 041 0.

. and X =

3840007100

257.

(b) (i) Evaluate the matrix product LX.

(1 mark)

•••

33 PLEASE TURN OVER

(ii) Using L and X, complete the following table.

Year 2006 2008 2010 2012

Number of female adult salmon 257

(2 marks)

(c) Describe the way that the number of female adult salmon changes in the years after 2006.

(1 mark)

A type of trout also lives in this lake and has a life cycle similar to that of salmon. Fisheries scientists observe that a proportion of female adult trout survive their first spawning of eggs and return 2 years later to have a second spawning. The scientists have developed the following 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 smolts.3.5% of the smolts survive for the 2 years needed to mature into adult trout.At their first 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 then die.

The scientists estimate that in 2006 the female component of this population of trout was made up of 75 600 eggs, 12 100 smolts, 417 first-spawning adult trout, and 63 second-spawning adult trout.

(d) Write down the matrices L and X such that the matrix multiplication LX will calculate the number of female eggs, smolts, first-spawning adults, and second-spawning adults in this population of trout in 2008.

(4 marks)

•••••

34

(e) Using the matrices defined in part (d), find the number of female adult trout that there will be in 2012, according to this model.

(3 marks)

35 PLEASE TURN OVER


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2007 mathematical studies examination paper

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