2010 Mathematical Studies Examination Paper

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External Examination 2010

2010 MATHEMATICAL STUDIES

Thursday 4 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 11, 35, and 36 if you need more space, making sure to label

each answer clearly.

3. The total mark is approximately 145. The allocation of marks is shown below:

Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Marks 9 5 9 8 9 6 11 7 6 9 10 8 9 8 16 15

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: 37

Questions: 16



2

QUESTION 1

(a) Findd

d

y

xif y x x= −( )5 2ln . There is no need to simplify your answer.

(3 marks)

(b) Find ∫ + −1 2

x

e x x

d .

(3 marks)

(c) Find ∫ + x e x

x2 103

5d .

(3 marks)



3 PLEASE TURN OVER

QUESTION 2

Consider the matrix A

k

=

−

−

⎡

⎣

⎢⎢

⎢

⎤

⎦

⎥⎥

⎥

0 2 1

1 1 2

3 6

, where k is some real number.

(a) Find A .

(3 marks)

(b) For what values of k does A−1 exist?

(2 marks)



4

QUESTION 3

Consider the function f x x

( ) =10

2, where x > 0. The graph of y f x= ) is shown below:

1 2 3 4 5

y

xȅ

Let U 2

represent the overestimate of the area between the graph of y f x= ( ) and the x-axis

from x =1 to x = 5, calculated using two rectangles of equal width.

(a) (i) On the graph above, draw two unshaded rectangles corresponding to U 2.

(1 mark)

(ii) Find the value of U 2. Show the calculations that support your answer.

(2 marks)



5 PLEASE TURN OVER

(b) (i) Find ∫ 10

2 x

xd .

(1 mark)

(ii) Using part (b)(i), calculate the exact value of the area between the graph of

y f x= ( ) and the x-axis from x =1 to x = 5.

(2 marks)

(c) Calculate D2 , the difference between U 2

and the exact value of the area between the

graph of y f x=( )

and the x-axis from x =1 to x = 5.

(1 mark)

(d) Consider U 4 , the overestimate calculated using four rectangles of equal width for the

area between the graph of y f x= ( ) and the x-axis from x =1 to x = 5.

(i) On the graph opposite, draw unshaded rectangles corresponding to U 4.

(1 mark)

(ii) U 4 gives a closer approximation than U 2

to the exact value in part (b)(ii).

On the graph opposite, illustrate this statement by shading the difference between

these approximations.

(1 mark)



6

QUESTION 4

Consider the graph of y f x= ( ) , shown below for 0 8< ≤ x :

2

2

2

4 6 8

x

y

ȅ

This graph has a non-stationary inflection point at 2 0,( ). It has no other inflection points and no

stationary points. It has vertical asymptotes at x = 0 and x = 4.

(a) For what values of x is the function increasing?

(2 marks)

(b) For what values of x is f x″ ( ) < 0?

(2 marks)



7 PLEASE TURN OVER

(c) On the axes below, sketch a graph of y f x= ′( ).

2 4 6 8

x

y

ȅ

(4 marks)



8

QUESTION 5

A garment manufacturer imports a large batch of buttons.

The proportion of these buttons that are defective is 0.037.

(a) If a random sample of sixty buttons is taken, what is

the probability that it will contain:

(i) no defective buttons? Source: http://claybuttons.com

(1 mark)

(ii) at most, three defective buttons?

(1 mark)

(b) The garment manufacturer will reject the batch of buttons if more than 5% of a random

sample of size n is defective (with sufficiently large n).

(i) What is the probability that this batch will be rejected if a random sample of sixty

buttons is taken?

(2 marks)

(ii) The garment manufacturer imports a second large batch of buttons. The proportion

of these buttons that are defective is 0.076.

(1) What is the probability that this batch will be rejected if a random sample of

sixty buttons is taken?

(2 marks)



9 PLEASE TURN OVER

(2) Show that there is a greater than 90% probability that this batch will be

rejected if a random sample of 220 buttons is taken.

(3 marks)



10

QUESTION 6

Consider the matrices P =−

−

⎡

⎣⎢

⎤

⎦⎥

4 0 2

0 3 1and Q

k

k = − −

−

⎡

⎣

⎢⎢

⎢

⎤

⎦

⎥⎥

⎥

1 0

2 0

0 3 1

.

Where possible, evaluate the following matrix expressions. If an expression cannot be

evaluated, describe the specific features of the matrices that prevent its evaluation.

(a) 3 P .

(1 mark)

(b) 3 P Q+ .

(1 mark)

(c) QP .

(1 mark)

(d) Q2.

(3 marks)



11 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 5(b)(i) continued’).





13 PLEASE TURN OVER

(d) Show that ′( ) =− − −

−( ) f x

x x

x

2 6 8

4

2

22

.

(3 marks)

(e) Hence show that f x( ) has no stationary points.

(2 marks)



14

QUESTION 8

A company decided to gather information about the health of its employees. The company

arranged for a nurse to complete a study of the systolic blood pressure of a random sample

of its employees.

Blood pressure is measured in millimetres of mercury (mmHg). It is given that the systolic

blood pressure of the company’s employees has a standard deviation of V = 28 mmHg.

(a) The nurse measured the systolic blood pressure of the sample of employees and

calculated the following 95% confidence interval for their mean systolic blood pressure:

118 17 140 13. . .≤ ≤P

(i) What was the sample mean?

(1 mark)

(ii) How many employees were in the sample?

(2 marks)

(b) The company wanted to reduce the width of the confidence interval, and so the nurse

took a second random sample. The mean systolic blood pressure of the second sample,

of eighty employees, was found to be 132.4 mmHg.

Calculate a 95% confidence interval based on the second sample.

(2 marks)



15 PLEASE TURN OVER

(c) High blood pressure is indicated by systolic blood pressure that is higher than

140 mmHg.

On the basis of the confidence interval calculated in part (b), the company made the

claim that, on average, its employees did not suffer from high blood pressure.

Was this reasonable? Explain your answer.

(2 marks)



16

QUESTION 9

The graph below shows functions f x( ) and g x( ), which intersect when x = 0, x = 2, and x = 3:

A

B

C

D

1 2 3 4

y

xȅ

Region A has an area of 22 units2 and Regions A and B have a combined area of 56 units2.

(a) Find:

(i) f x x( )∫ d .

0

2

(1 mark)

(ii) f x g x x( ) − ( )∫ d .0

2

(1 mark)

g x( )

f x( )



17 PLEASE TURN OVER

It is known that f x x( ) = −∫ d 18

2

3

and g x x( ) = −∫ d 13

2

3

.

(b) Find the area of Region C and the area of Region D.

(2 marks)

(c) Find f x g x x( ) − ( )∫ d0

3

.

(2 marks)



18

QUESTION 10

(a) The graph of y x= 32, between x = −1 and x =1, fits within the rectangle ABCD, as

shown in the diagram below:

A

D

B

x

y

C ȅ

(i) Determine the area of the rectangle ABCD.

(1 mark)

(ii) Determine the area of the shaded region between y x= 3

2

and the x-axis, as shownin the diagram above.

(2 marks)



19 PLEASE TURN OVER

(b) The graph of y kx= 2 where k > 0, between x a= − and x a= , fits within the rectangle

ABCD, as shown in the diagram below:

A

Da a

B

x

y

C ȅ

(i) Determine the area of the rectangle ABCD.

(2 marks)

(ii) Determine the area of the shaded region between y kx= 2 and the x-axis, as shown

in the diagram above.

(3 marks)

(iii) What is the relationship between the area of the rectangle and the area of the

shaded region between y kx= 2 and the x-axis?

(1 mark)



20

QUESTION 11

The amount of money that individual employees spend

in a workplace cafeteria each lunchtime is recorded.

It is found that the amount spent per employee ( X )

can be modelled by a normal distribution with meanP = $ .5 85 and standard deviation V = $ . .1 33

The distribution of X is graphed below:

dollars

(a) On the horizontal axis of the graph of the normal density curve above, write numbersto illustrate the distribution of X .

(1 mark)

Suppose that each lunchtime sixty-five employees spend an amount of money in the cafeteria,

and that these amounts constitute a random sample of amounts spent per employee.

Let X 65

be the average of the amounts spent per employee during a randomly chosen lunchtime.

(b) (i) Write down the mean and the standard deviation of the distribution of X 65.

(2 marks)

(ii) Sketch the distribution of X 65

on the graph above.

(2 marks)

Source: www.klauscherarchitects.com



21 PLEASE TURN OVER

(iii) Determine the proportion of lunchtimes for which the average amount spent per

employee will be between $5.50 and $6.20.

(1 mark)

(c) (i) Determine the probability that, for a randomly chosen lunchtime, the average

amount spent per employee will be $5.21 or less.

(1 mark)

(ii) Using your answer to part (c)(i), explain whether or not the cafeteria manager

should consider $5.21 to be an unusually low value.

(1 mark)

One explanation for an unusually low value of X 65

is fraud by cafeteria staff.

The manager wants to detect possible cases of fraud and so decides on a dollar value below

which all values of X 65

will be considered unusually low.

(d) Calculate the value below which 1% of the average amounts spent per employee will

fall.

(2 marks)



22

QUESTION 12

An example of a number puzzle called a ‘tri-sum’

is shown on the right.

A tri-sum is made up of three overlapping arms.

Each arm contains two unknowns and one

numerical value.

To solve a tri-sum, it is necessary to find the

unknowns x, y, and z such that, for each arm, the

sum of the unknowns is equal to the numerical

value.

(a) To solve the tri-sum shown above, one equation that must be satisfied is x y+ =143.

(i) Write down the two other equations that must be satisfied.

(1 mark)

(ii) Hence find the values of x, y, and z .

(2 marks)

(b) All tri-sums can be represented by the diagram below:

y z

x

a

c

b

y z

x

117

166

143



23 PLEASE TURN OVER

The system of equations associated with this representation of all tri-sums can be

written as

1 0 1

1 1 0

0 1 1

a

b

c

⎡

⎣

⎢⎢

⎢

⎤

⎦

⎥⎥

⎥

.

Solve this system of equations for x, y, and z in terms of a, b, and c.

(5 marks)



24

QUESTION 13

The rate of extraction of the world’s crude oil can be modelled by Hubbert’s Curve.

This model, in gigabarrels per year, is

R t

e

e

t

t ( ) =+( )

− +

− +

111

1

0 0625 9

0 0625 92

.

.,

where t represents time in years since 1865.

The graph of Hubbert’s Curve is shown below:

ȅ

10

50 100 150 200

20

30

40

50

R

t

(a) (i) Using R t ( ) , write down an expression that will determine the amount of theworld’s crude oil extracted in the decade from 2000 to 2010.

(1 mark)

(ii) Evaluate the expression you wrote down in part (a)(i).

(1 mark)



25 PLEASE TURN OVER

(b) ‘Peak oil’ is defined as the time when the maximum rate of extraction of the world’s

crude oil is reached.

Find the year in which peak oil is reached, as predicted by Hubbert’s Curve.

(2 marks)

(c) Let A t ( ) represent the total amount of the world’s crude oil extracted by time t :

A t

et

( ) =+ − +

1776

10 0 62 5 9.

.

(i) Show that ′( ) = ( ) A t R t .

(3 marks)

(ii) State the total amount of crude oil available for extraction as predicted by A t ( ).

(1 mark)

(iii) What percentage of this total amount was extracted in the decade from

2000 to 2010?

(1 mark)



26

QUESTION 14

An American scientist is studying the mating habits of

Drosophila melanogaster (a species of fruit fly).

She wants to know whether or not this species has a

preference for mating partner, based on the fly’s place of

origin.

In the study the scientist captures thirty male and thirty female

Drosophila from Alabama and thirty male and thirty female

Drosophila from Grand Bahama Island.

The flies are all released together in a closed environment and carefully observed.

Of the 246 matings that are observed:

140 are between a male and a female from the same place of origin•

106 are between a male and a female from different places of origin.•

Using these data, a two-tailed Z -test, at the 0.05 level of significance, is applied to determine

whether or not there is sufficient evidence that Drosophila have a preference for mating partner,

based on place of origin.

The null hypothesis and the alternative hypothesis are

H p

H p A

00 5

0 5

: .

: . .

=

≠

(a) Interpret the null hypothesis in terms of Drosophila’s preference for mating partner.

(1 mark)

(b) Calculate the proportion of matings between Drosophila from the same place of origin,as observed in this closed environment.

(1 mark)

Source: http://tolweb.org

This photograph cannot be

reproduced here for copyright

54275159_1a756047cf_o.jpg

reasons. It can be found at

http://tolweb.org/tree/ToLimages/



27 PLEASE TURN OVER

(c) The Z -test produces a P -value of 0.0302 (correct to three significant figures).

(i) Explain what this P -value represents.

(2 marks)

(ii) State whether or not the null hypothesis should be rejected. Give a reason for your answer.

(2 marks)

(d) On the basis of parts (b) and (c), what can you determine about Drosophila’s preference

for mating partner in this closed environment?

(2 marks)



28

QUESTION 15

A vehicle hire company is planning to offer 1-day hire of removal vans.

On each day, in each town in the area, some vans will be hired out for

local moves and some vans will be hired out for moves between towns.

The company assumes that, on each day:

in Town A, half of the vans will be hired out locally and•

half will be hired out for moves to Town B

in Town B, one-third of the vans will be hired out locally,•

one-third will be hired out for moves to Town A, and

one-third will be hired out for moves to Town C

in Town C, half of the vans will be hired out locally and half will be•

hired out for moves to Town B.

The company also assumes that all vans will be hired each day, and that all

vans will be returned in time for the start of the next day.

The company plans to start on Day 0 with ten vans in Town A,

thirty vans in Town B, and twenty vans in Town C.

(a) Using the company’s assumptions, write down the calculation that determines the

number of vans that the company will have in Town B at the start of Day 1.

(1 mark)

Let L =

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

1

2

1

3

1

2

1

3

1

2

1

3

1

2

0

0

and X =

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

10

30

20

.

(b) (i) Evaluate LX.

(1 mark)

B

A

C



29 PLEASE TURN OVER

(ii) Interpret the values obtained from your calculation in part (b)(i).

(2 marks)

(iii) Hence calculate how many vans will be in each of the three towns on Day 2.

(2 marks)

(c) Describe what the company can expect to happen to the number of vans in the

three towns over a long period of time.

(2 marks)

The company changes its plans and decides to start on Day 0 with twenty vans in Town A,

twenty vans in Town B, and twenty vans in Town C.

(d) (i) Write down the new matrix X .

(1 mark)

(ii) Comment on what the company can now expect to happen to the number of vans

in the three towns over a long period of time.

(1 mark)



30

(e) The company changes its assumptions about the hire of vans in Town B.

In relation to the new assumptions, let L

a

b

c

=

⎡

⎣

⎢⎢⎢⎢

⎢

⎤

⎦

⎥⎥⎥⎥

⎥

1

2

1

2

1

2

1

2

0

0

, where 0 1≤ ≤a b c, , .

(i) Explain why a b c+ + must equal 1.

(1 mark)

(ii) If X =

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

18

24

18

, solve LX X = for a, b, and c.

(3 marks)

(iii) How should the company interpret the values of a, b, and c in part (e)(ii)?

(2 marks)



31 PLEASE TURN OVER

Question 16 starts on page 32.



32

QUESTION 16

The circle C , with centre 0 0,( ) and radius 1 unit, has equation x y2 2

1+ = , where − ≤ ≤1 1 x ,

as shown in the diagram below:

1 2 3ȅ x

y

(a) (i) Findd

d

y

x.

(2 marks)



33 PLEASE TURN OVER

(ii) Find the equation of the tangent to the circle C at the point where x = 0 3. and

y < 0. Give your answer correct to three significant figures.

(3 marks)

(iii) On the Cartesian plane on the page opposite, draw the graph of the tangent to the

circle C at the point where x = 0 3. and y < 0.

(1 mark)

Let f x x( ) = 2.

(b) On the Cartesian plane on the page opposite, draw the graph of y f x= ( ).(1 mark)

(c) Show that the equation of the tangent to the graph of y f x= ( ) at the point where

x k = is given by y kx k = −22, where k is any real number.

(3 marks)



34

The tangent to the circle C at the point where x t = and y < 0, with − < <1 1t , has equation

yt

t

x

t

=−

−−1

1

12 2

.

(d) Hence or otherwise, find the values of t for which the tangent to the circle C is also

tangential to the graph of y f x= ( ).

(5 marks)



35 PLEASE TURN OVER





36



© SACE Board of South Australia 2010



37

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

N

T

where N is the population mean and T is the

standard deviation for the population distribution.

Con¿dence Interval — Mean

A 95% con¿dence interval for the mean N of a

normal population with standard deviation T , based

on a simple random sample of size n with sample

mean x , is

xn

xn

b1 96 1 96. . .T

NT

b

For suitably large samples, an approximate

95% con¿dence interval can be obtained by using

the sample standard deviation s in place of T .

Sample Size — Mean

The sample size n required to obtain a

95% con¿dence interval of width w for the

mean of a normal population with standard

deviation T is

nw

s2

21.96T ¥

§¦¦¦

´

¶µµµ .

Con¿dence 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

p X n

, is

p p p

n p p

p p

n

b b

1 96

1 1. .1.96

Sample Size — Proportion

The sample size n required to obtain an approximate

95% con¿dence interval of approximate width w for

a proportion is

n

w

p ps¥

§

¦¦¦

´

¶

µµµ

2 1 961

2.

.

( p

is a given preliminary value for the proportion.)

Binomial Probability

P X k C p pk

n k n k

1where p is the probability of a success in one trial

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

ÕC

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

nare

N X np N p p

T X np p 1 T 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 b

c a

1 1.

Derivatives

f x y a 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

\ ^ a a

«¬®®

-®®

º»®®

¼®®

a a

a a

f x g x f x g x

g x

x f g x f g x g x

2

d

d

Quadratic Equations

b bIf thenax bx c x

ac

a

2 04

2

2



MATHEMATICAL STUDIES 2010

ACKNOWLEDGMENT

The photograph for Question 5 on page 8 is Copyright Creative Impressions in Clay.

The SACE Board of South Australia has made every effort to trace copyright holders. If however,any material has been incorrectly acknowledged, we apologise and invite the copyright holder tocontact us.

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2010 Mathematical Studies Examination Paper