2008 Mathematical Studies Examination Paper

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2008 MATHEMATICAL STUDIES

Thursday 6 November: 9 a.m.

Time: 3 hours

Examination material: one 41-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 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 mark is approximately 146. The allocation 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 9 11 14 9 17

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 41. 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.

External Examination 2008

FOR OFFICE

USE ONLY

SUPERVISOR

CHECK

RE-MARKED

ATTACH SACE REGISTRATION NUMBER LABEL

TO THIS BOX

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Model

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Pages: 41

Questions: 17



2

QUESTION 1

(a) Findd

d

y

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

(i) y x 2 3 57.

(2 marks)

(ii) y e

x

x

4

2 9.

(3 marks)

(b) Find the exact values of x for which .1 5 1 0n n x x

(2 marks)



3 PLEASE TURN OVER

QUESTION 2

Let P Q

3

0 5

1

3

0

1

2

1

2

0

3

1..and

(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)



4

QUESTION 3

Let An

B C

6 4

3

1

0

0

2

1

1

3 5

1 2

2 5

, , and .

(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)



5 PLEASE TURN OVER

(ii) Does C B 1 ? Give a reason for your answer.

(2 marks)



6

QUESTION 4

Find x x x10 2 d .

(4 marks)



7 PLEASE TURN OVER

QUESTION 5

Consider the function f ( x). The sign diagrams for f x( ) and f x( ) are shown below:

f x( ) x

2 4

f x( ) x

41

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 ( ).

A

B

C x

y

46 2 42 6

8

6

4

2

2

4

6

8

Ο

(4 marks)



8

QUESTION 6

Let f x x x x( ) , where x 0.

(a) Find f x( ).

(2 marks)

Let A x f x, ( ) be any point on the graph of y f x ( ) and let B be the fixed point 3 3, ( ) f .

The graph of y f x ( ) and a chord AB are shown below:

A

B

1

2

3

y

x

1 2 3

Ο

(b) On the graph above, draw the chord AB when A is placed at 2 2, ( ) . f (1 mark)



9 PLEASE TURN OVER

(c) Complete the third column of the following table, giving your answers correct

to two decimal places.

x-coordinate of A x-coordinate of B slope of the chord AB

2.5 3

2.9 3

2.99 3

(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)



10

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:

A B C

2

0

0

3

1

0

1

5

4

3

0

0

0

2

0

4

1

5

5

0

0

4

2

0

220

15

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 the value of thedeterminant of all 3 3 upper triangular matrices.

(2 marks)



11 PLEASE TURN OVER

(c) Prove the conjecture you made in part (b).

(2 marks)



12

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.

(a) Consider a batch of tiles in which the proportion of first-grade tiles is 0.75.

Suppose thatfi

fty 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 of first-grade tiles is 0.8.

Determine the probability that the batch will be accepted.

(2 marks)

Source: The Tile Man Inc. website,www.thetileman.com

This photograph cannot be reproduced

here for copyright reasons.



13 PLEASE TURN OVER

(c) Consider a 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 calculated for more values 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 shown

below:

p

0.5

0.6

0.7

0.8

0.9

1.0

0.81 0.85 0.89 0.93

acceptance

probability

A( p)

(i) Which one of the following functions provides the best model for A p( )?

Tick the appropriate box.

A p p p( ) . . . 1 09 2 05 1 952

A p p p( ) . . . 0 526 0 796 0 7022

A p p p( ) . . . 32 55 59 45 26 142 (1 mark)

(ii) The manufacturer of the 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)



14

QUESTION 9

Consider the function f x x( ) ,1

where x 0.

(a) A graph of y f x ( ) is shown below:

A

B

1 2 4 m

y

xΟ

Find the value of m if the shaded regions marked A and B are equal in area.

(2 marks)



15 PLEASE TURN OVER

(b) Consider the general case where the shaded regions marked A and B are equal in area,

as shown below:

A

B x

a b c d

y

Ο

Derive a relationship for ba

in terms of c and d .

(4 marks)



16

QUESTION 10

(a) Consider the matrices A X

a

b

cd

0 1 0 0

0 0 1 0

0 0 0 11 0 0 0

and .

(i) Calculate AX .

(1 mark)

(ii) Describe, in words, the effect of pre-multiplying any 4 1 matrix by matrix A.

(1 mark)

(iii) Hence explain, in words, why A X X 4 .

(1 mark)

(iv) Hence deduce a positive value of n for which A An 1.

(2 marks)



17 PLEASE TURN OVER

(b) Matrix B is a square matrix such that B

a

b

c

d

e

c

d

e

a

b

.

Write down matrix B.

(2 marks)



18

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.

(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)



19 PLEASE TURN OVER

(b) Using the 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 to claim that p 2

3exactly.

(1 mark)

(iii) Using your answer to part (b), suggest a reasonable claim for the possible value

of p.

(1 mark)



20

QUESTION 12

The graph of x xy y2 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.

Q

P

x

y

5

5

5 5Ο

(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 dd y x

y x y x

2

2.

(3 marks)



21 PLEASE TURN OVER

(d) For the case where k 2, use the result in part (c) to find the equation of

the tangent to the curve at point:

(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)



22

QUESTION 13

The graphs of y e x and y x1n

for x 0 are shown.

The line segment AB with equation

y x e 2 2 meets these graphs

at P e,2 2 and Q e ,2 2

.

(a) State the exact coordinates of

points A and B, the axis intercepts

of the line segment AB.

y

x

A

P

Q

B

2

2

4

6

8

10

4 6 8 10Ο

(2 marks)

(b) Calculate the exact value of the area of the blue region between y x e 2 2 and

y e x , from x 0 and x 2.

(4 marks)



23 PLEASE TURN OVER

(c) Give a reason why the area of the blue region is equal to the area of the green region

enclosed by the graph of y x1n , the line segment AB, and the x-axis.

(1 mark)

(d) Hence calculate the area of the yellow region.

(2 marks)



24

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 t e t ( ) . . 0 075 2 0 05 for t 0.

(a) On the axes below, draw a graph of y C t ( ), showing intercepts, turning-points, and

shape.

20 40 60 80 100 120t

C

20

10

Ο

(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.

(b) Determine the AUC value that the tablet produces in an average person within

60 minutes of being swallowed.

(2 marks)



25 PLEASE TURN OVER

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 e t ( ) .14 1 0 05 for t 0.

(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 sothat 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 definite integral and has k as its solution.

(2 marks)

(ii) Find the value of k to the nearest minute.

(2 marks)

Source: DCC Healthcare website,

www.dcc.ie



26

QUESTION 15

(a) (i) Z is distributed normally with mean 0 and standard deviation 1, as shown

in the diagram below:

a a Z

Given that P a Z a( ) . 0 9, find the value of a.

(2 marks)

(ii) Suppose that Y is distributed normally with unknown mean and standard

deviation .

Given that , 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.

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.dynastymining.com

This photograph cannot be reproduced

here for copyright reasons.



27 PLEASE TURN OVER

(b) Suppose that the drill core has been crushed to form a 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, A and B, are shown below. One of the histograms corresponds to

the distribution of X and the other to the distribution of X 5.

0 10 20 30 40 0 20 40 60 80 100

Histogram A Histogram B

percentage zinc (by weight) 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 5and the standard deviation X 5

.

(2 marks)

(iii) Use the normal distribution to find an approximate value for .5 28

(1 mark)



28

(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 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 2 5. grams per

kilogram of the population mean Y .

(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)



29 PLEASE TURN OVER



30

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 75 000 66 000 54 000 46 000

This information is represented in the graph below:

210000

180000

150000

120000

90000

60000

30000

0 1 2 3 4 5 6 7 8 9 10

x

n

0

An algebraic model for the relationship between n and x is

n x 20600 208000.

(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)



31 PLEASE TURN OVER

(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 versus x for 0 8 x , marking an appropriate scale on the

vertical axis below.

R

x2 4 6 8

Ο

(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 printed.

If all copies printed are sold, this cost can be defined in terms of n.

If n copies of the magazine are printed, let c represent the cost of printing each of

these copies, where c e n 2 88 0 0000173. . .

Write down an expression for T , the total cost of printing n magazines.

(1 mark)



32

(d) Let P be the profit, where P R T .

Determine the selling price of the magazine that will maximise P .

(2 marks)



33 PLEASE TURN OVER



34

QUESTION 17

(a) (i) Represent the following system of equations in augmented matrix form.

x y

x y

x y k

z

z

z

1

0 007 0 006 0 009 0 007

0 5 0 6 0 66

. . . .

. . . .

(1 mark)

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

1 1

0 1

0 0

1 1

0

1 6

2

10 3k .

.

(5 marks)



35 PLEASE TURN OVER

(iii) Hence show that this system has the following solution:

xk

k y

k k

10 7 8

10 3

3 2

10 3

1 6

10 3

.,

.,

.. z

(3 marks)

Question 17 continues on page 36.



36

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 .

(b) The coal-mining company aims to produce blended coal containing 0.7% sulphur and

60% carbon.

The proportion of coal required from each mine can be determined by solving the

following system of equations:

x y

x y

x y

z

z

z

1

0 007 0 006 0 009 0 007

0 5 0 6 0 7 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)

Source: EduPic Graphical

Resource website, www.edupic.net



37 PLEASE TURN OVER

(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 following system of equations:

x y

x y

x y

z

z

z

1

0 007 0 006 0 009 0 007

0 5 0 6 0 7 0 66

. . . .

. . . . .

(1 mark)

(ii) Explain what your answer to part (d)(i) means about producing blended coal

containing 0.7% sulphur and 66% carbon.

(1 mark)

Question 17 continues on page 38.



38

(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)



39 PLEASE TURN OVER



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

A measurement scale X is transformed into a

standard scale Z , using the formula

Z X

where is the population mean and is the

standard deviation for the population distribution.

Condence Interval — Mean

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

xn

xn

1 96 1 96. . .

For suitably large samples, an approximate

95% condence interval can be obtained by using

the sample standard deviation s in place of .

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

nw

2

21.96

.

Condence Interval — Population Proportion

An approximate 95% condence interval for the

population proportion p, based on a large simple

random sample of size n with sample proportion

p X

n, is

p p p

n p p

p p

n

1 96

1 1. .1.96

Sample Size — Proportion

The sample size n required to obtain an approximate

95% condence interval of approximate width w for

a proportion is

2

Binomial Probability

P X k C p pk n k n k

1

where p is the probability of a success in one trial

and the possible values of X are k n 0 1, , .. . and

C

n

n k k

n n n k

k k n

1 1. . ..

Binomial Mean and Standard Deviation

The mean and standard deviation of a binomial

count X and a proportion of successes p X

n

are

X np p p

X np p 1 p p p

n

1

where p is the probability of a success in one trial.

Matrices and Determinants

If then and Aa b

c d A A ad bc det

A A

d bc a

1 1 .

Derivatives

f x y f x y

x

d

d

xn

e kx

ln x xe log

nxn1

ke kx

1

x

Properties of Derivatives

d

d

d

d

x f x g x f x g x f x g x

x

f x

g x

f x g x f x g x

g x

x f g x f g x g x

2

d

d

2008 Mathematical Studies Examination Paper

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