tcsh trial 09 final.doc

8/10/2019 TCSH TRIAL 09 final.doc

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Friday 18 September: 7 a.m.

Time: 3 hours 10 minutes

Examination material: one 35-page question booklet

Approved dictionaries, notes, calculators and computer software may be used

Instructions to Candidates

1. You will have 10 minutes to rea the paper. You must not write in !our question booklet or use a

"al"ulator uring this reaing time but !ou ma! make notes on the s"ribbling paper provie.

#. $nswer allparts o% &uestions 1 to 15 in the spa"es provie in this question booklet. There is no

nee to %ill all the spa"es provie. You ma! write on pages #' #3 an 3( i% !ou nee more spa"e'

make sure to label ea"h answer "learl!.

3. The total mark is approximatel! 1() The allo"ation o% marks is shown below:

&uestion 1 # 3 ( 5 * ) + , 10 11 1# 13 1( 15

arks , , 5 1# 1# ) ) 11 11 , 5 10 , 15 1*

(. $ppropriate steps o% logi" an "orre"t answers are require %or %ull marks.

5. how all working in this booklet. /You are strongl! avise not to use s"ribbling paper. ork that

!ou "onsier in"orre"t shoul be "rosse out with a single line.

*. 2se onl! bla"k or blue pens %or all work other than graphs an iagrams' %or whi"h !ou ma! use

a sharp ark pen"il.

). tate all answers "orre"t to three signi%i"ant %igures' unless otherwise state or as appropriate.

+. iagrams' where given' are not ne"essaril! rawn to s"ale.

,. The list o% mathemati"al %ormulae is on page 35. You ma! remove this page %rom the booklet

be%ore the examination begins.

10. rite !our name' stuent number an group in the spa"e provie at the top o% this page.

11. 4omplete the box on the top right-han sie o% this page with in%ormation about the ele"troni"

te"hnolog! !ou are using in this examination.

TRIAL EA!I"ATI#" $%%& !AT'E!ATICAL ST()IES

S#(T' A(STRALIA" !ATRIC(LATI#"

ame : 66666666666666666666666666666666666

tuent o: 66666666666666666666666666666666666

7roup : 66666666666666666666666666666666666

7raphi"s 4al"ulator

8ran 666666666666666

oel 66666666666666

4omputer o%tware

9ages: 35

&uestions: 15


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You may write in this section if you need additional space. Clearly number each question and part

attempted in this section (for example, Question 1(c(i.

#


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*(ESTI#" 1

There is no nee to simpli%! answers in parts /a' /b an /".

/a in dydx

i%3

1ln #xy

x+

=

+

.

/3 marks

/b indy

dxi% ()/ xexy += .

/3 marks

/" in dxx

x ++ (

1/* # .

/3 marks

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*(ESTI#" $

;et

=

11

#0

11

x

x

x

A an

=

100

010

001

I .

/a in A? #I.

/# marks

/b how that et/A? #I @x3? #x#?x.

/( marks

(


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/" Aen"e' or otherwiese' %in the values o%xsu"h that /A? #I+B 1 oes not exist.

/3 marks

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*(ESTI#" ,

The %ollowing iagram shows part o% the graph o% a %un"tiony@ f C/x:

/a =n the same set o% axes as the graph o%y@ f C/x' raw the "orresponing part o% the graph o%

! @f/x.

/# marks

*

x @b

y@ f C/x


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The linex@ b' where bis a positive "onstant' interse"ts the graph o%y@ f C/x. upposef C/x is

s!mmetri"al b! the origin through a rotation o% 1+0D an Adxxfb

=0 / ' whereA 0.

/b in' in terms o% $'

i. 0

/b

dxxf

/1 markii.

b

bdxxf /

/1 mark

iii. the area boune between the graph o% y@ f C/x an thex-axis %romx@ Bbtox@ b.

/1 mark

*(ESTI#" -

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The Y.=.7 a"tor! has ma"hines that ispense i%%erent %avore

!ogurt rink into bottles. The %a"tor! %ills re"!"le bottles with its

!ogurt rink. $ ranom sample o% 1#0 bottles was "he"ke to see i%

the! were "lean enough to use.The supervisor reporte that 5F o% the 1#0 bottles were reGe"te

be"ause the! were not "lean enough.

/a in a ,5F "on%ien"e interval %or the proportion o% all bottles that were reGe"te be"ause

the! were not "lean enough. Hnterpret !our answer.

/# marks

The manager oubts the "laim o% the supervisor. he took a sample o% 150 bottles an 13 were %oun

not "lean enough.

/b 4onu"t a h!pothesis testing at 5F signi%i"ant level to Gusti%! whether the proportion o%

bottles that are not "lean enough is i%%erent %rom 5F.

i. tate the null an alternative h!pothesis.

/# marks

ii. tate the null istribution o% the test statisti"s.

/1 mark

iii. 4al"ulate the test statisti"s an etermine whether the null h!pothesis shoul be reGe"te.

/3 marksiv. ake a "on"lusion on the proportion o% bottles that are not "lean enough.

+


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/1 mark

v. The manager "laims that the proportion o% all bottles that are not "lean enough is more than

5F. Iusti%! the "laim using the results %rom h!pothesis testing.

/# marks

/" uggest one wa! to alter the result o% !our h!pothesis testing in part /b' using the same

sample. o "al"ulation neee.

/1 mark

*(ESTI#"

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$ parti"le travels along a straight line %or ( se"ons. ;et ( )tv be the velo"it! o% the parti"le t se"ons a%ter the motion starts. The velo"it! is a quarati" an is measure in meters per se"on.

$ graph o% ( )tvy= is shown below.

/a i. orm a s!stem o% equations using the above in%ormation' an show that the velo"it! %un"tion

( )tv is given b! ( ) 53# += tttv .

/( marks

ii. in the time when the parti"le is at rest.

10

( )5'3C( )3'1!

( ),'("

A

y

t


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/1 mark

/b =n the graph above' ini"ate all times %or whi"h the a""eleration o% the parti"le is negative.

/1 mark

/" i. 2sing two re"tangles' %in an unerestimate %or (

#1/ dttv .

/# marks

ii. 2sing two re"tangles' %in an overestimate %or (

#1/ dttv .

/# marks

iii. Explain the meaning o% (

#1/ dttv in terms o% the motion o% the parti"le.

/# marks

*(ESTI#" /

7iven the equation 5## +=+ xyxy .

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/a how that1#

# #

+

=xy

yx

dx

dy.

/3 marks

/b in the equation o% the normal to 5## +=+ xyxy at one o% the points where #=y .

1#


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/( marks

*(ESTI#" 7

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The histogram below shows the istribution o% test marks /# obtaine b! a large group o% stuents in

the emester Examination #00,. The test marks has mean (5 an stanar eviation 1+.). The %ull

mark o% the test is *0 an the minimum passing mark is 35.

$ tea"her took groups o% %ort! stuents an their mean test marks / (0# are as shown:

/a 8ase on the histograms' state' giving reasons' whether sample siJe o% %ort! stuents is

su%%i"ientl! large.

/# marks

The tea"her then took groups o% 100 stuents to stu! the istribution o% their mean test marks. The

mean test marks is approximatel! normal./b i. etermine the mean an stanar eviation o% 100# .

1(

Test ark'#

Test ark' (0#


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/# marks

ii. in$r/ 100# 50.

/1 mark

/" s. 9errington has a "lass o% 100 stuents. he is ver! prou o% the a"hievement o% her

stuents' an announ"e that her "lass average mark is at least (0 marks in the emester

Examination #00,. etermine how true is her "laim.

/# marks

*(ESTI#" 8

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/a 7iven ( )#

3#1

x

xxf

+= .

i. ket"h the graph o% ( )xf on the axes below %or B 5 KxK 5. 4learl! label the a!smptote anthe inter"epts.

/3 marks

ii. inf C/x an etermine the turning point o% ( )xf .

/3 marks

/b or ( ) #(31 xxx% += ' %in the values o%x when%/x @ 0 an% C/x @ 0.

1*


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/# marks

/" or ( )#

5(1

x

xxh

+= ' %in the values o%x when h/x @ 0 an h C/x @ 0.

/1 mark

/ or ( )#

1/1

x

xmmxp

++= ' one o% itsx&inter"epts is 1.

2sing !our results %rom parts /a to /"' make a "onGe"ture on the other x & inter"ept o% p/x

an the value o%xwhen p C/x @ 0 .

/# marks

*(ESTI#" &

9;E$E T2E


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Aartamas 7ol% 4lub is hosting a pro%essional gol% tournament. The gol% balls

use in this tournament must meet a set o% stanars. =ne o% these stanars is

the istan"e travelle. hen a ball is hit b! a me"hani"al evi"e with a 10-

egree angle o% laun"h' a ba"kspin o% (# revolutions per se"on' an a ball

velo"it! o% )# metres per se"on' the istan"e the ball travels ma! not ex"ee#**.3 metres. anu%a"turers want to evelop balls that will travel as "lose to

the #**.3 metres as possible without ex"eeing that istan"e.

$ manu%a"turer that wishes to suppl! gol% balls to the Aartamas 7ol% 4lub has etermine that the

istan"es travelle %or the balls it prou"es are normall! istribute with a stanar eviation o% #.5*

metres. This manu%a"turer has a new pro"ess that allows it to set the mean istan"e the ball will travel.

/a H% the manu%a"turer wants to be ,,F "ertain that a ranoml! sele"te ball will not ex"ee the

maximum istan"e o% #**.3 metres' what is the largest mean that "an be use in the

manu%a"turing pro"essL

/3 marks

/b i. H% the manu%a"turer sets the mean istan"e travelle to be equal to #*3 metres' %in

the probabilit! that a ball that is ranoml! sele"te %or testing will travel too %ar.

/1 mark

ii. $ssume the mean istan"e travelle is #*3 metres an that %ive balls are inepenentl! teste.

in the probabilit! that less than two o% the %ive balls will ex"ee the maximum istan"e o%

#**.3 metres.

/# marks

1+
http://www.google.com.my/imgres?imgurl=http://3.bp.blogspot.com/_GSL_0b5iric/SgmYJS4ympI/AAAAAAAAADI/D3TforW9TCE/s400/golf.jpg&imgrefurl=http://otrsportsman.blogspot.com/&h=320&w=400&sz=29&tbnid=kjb1klKhbjFaNM:&tbnh=99&tbnw=124&prev=/images%3Fq%3Dgolf,%2Bimage&hl=en&usg=__hjCESJUu_Ct0Ti_OHrYI2p8VxtE=&ei=ygORSvCAMMGOkAWH-uG9Cg&sa=X&oi=image_result&resnum=4&ct=image


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The se"retar! o% the Aartamas 7ol% 4lub visits the manu%a"turer an is assure that the mean istan"e

travelle b! their gol% balls is #*3 metres. Ae makes a ranom sele"tion o% *( gol% balls.

/" H% the sample mean is #*#.+ metres' etermine a ,5F "on%ien"e interval %or the mean

istan"e travelle b! the gol% balls.

/1 mark

/ i. Aow large a sample shoul the manu%a"turer use to be ,5F sure that the estimation o% mean

is within 1 metre o% #*3 metresL

/3 marks

ii. ue to tight buget an time "onstrain' the manu%a"turer sample onl! hal% the minimum

siJe. hat is the e%%e"t to the with o% a ,5F "on%ien"e interval %or the mean istan"e

travelle b! the gol% ballsL

/1 mark

9;E$E T2E


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*(ESTI#" 1%

9art o% the graph o% f/x @ # B eBx is as shown below.

The line segment$Qis rawn %rom the point$/#'f/# to the point Q/''f/'.

/a i. in the graient o%$Qin terms o% '.

/# marks

ii. 8! "onsiering the graient o%$Q' write an expression %or the graient o% the tangent to the

graph o%y@f /x at the point$.

/1 mark

#0

$Q

'


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The graphs o%%/x @ B ln /# Bx anf /x are as shown below.

/b in the axes inter"epts o%%/x in exact%orm.

/# marks

/" i. rite own a e%inite integral %or the area o% the region en"lose betweeny@f /x an

y@%/x %or 0 KxK 1.

/# marks

ii. in this area.

/1 mark

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*(ESTI#" 1$

Hn an amusement park there is a small train "alle 4ooper whi"h oes a "ir"uit o% the park. The train

must "omplete six "ir"uits between ,.00 am an 1#.00 noon. The management pre%ers 4ooper to

"omplete a "ir"uit in less than #5 minutes. Ht is known that the probabilit! 4ooper to "omplete a"ir"uit in less than #5 minutes is 0.+.

/a i. in the probabilit! that o% the six "ir"uits "omplete' at least ( o% them take less than #5

minutes ea"h.

/# marks

ii. in the probabilit! that exa"tl! 3 or ( out o% the six "ir"uits "omplete ea"h taking less than

#5 minutes.

/# marks

or s"heuling reasons the management wants to know the time' bminutes' %or whi"h the probabilit!

o% exa"tl! 3 or ( out o% the six "ir"uits "omplete ea"h taking less than bminutes' is maximise.

;et$r/M b @p' an let Qbe the probabilit! that exa"tl! 3 or ( "ir"uits ea"h take less than b

minutes.

Ht is given that Q/p @ 5/(p3Bp( /1 Bp#.

/b tate the values o%psu"h that the probabilit! %un"tion' Q/p is e%ine.

/1 mark

#(


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/" The erivative o% Q/p is Q C/p @ 30p#/1 Bp/p#B (p? #.

i. 2sing "al"ulus' etermine the value o%p su"h that Q/p is maximise.

/( marks

ii. Aen"e' or otherwise' %in the maximum value o% Q.

/1 mark

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*(ESTI#" 1,

To stu! the li%e-an-eath "!"le o% an inse"t population' a number o% inse"t eggs /)' Guvenile inse"ts

/* an ault inse"ts /A are pla"e in a "lose invironment.

The initial state o% this population "an be es"ribe b! the "olumn matrix

"

A

*

)

=

0

100

#00

(00

0S

$ row has been in"lue in the state matrix to allow %or inse"ts an eggs that ie /".

/a etermine the total number o% inse"ts in the population /in"luing eggs at the beginning o%

the stu!.

/1 mark

Hn this population

Eggs ma! ie' or the! ma! live an grow into Guveniles

Iuveniles ma! ie' or the! ma! live an grow into aults

$ults will live a perio o% time but the! will eventuall! ie.

Hn this stu!' the aults inse"ts have been sterilise so that no new eggs are prou"e. Hn these

"ir"umstan"es' the li%e-an-eath "!"le o% the inse"ts "an be moelle b! the transtion matrix

next week

1#.01.01.0

0+.05.00

00(.05.0

000(.0

weekthis

"

A

*

)

"A*)

=T

/b i. Evaluate the matrix prou"t S1@ TS0

/1 mark

ii. rite own the number o% live Guveniles in the population a%ter one week.

#*


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/1 mark

/" i. etermine the number o% live Guveniles in the population a%ter %our weeks. rite !our

answer "orre"t to the nearest whole number.

/# marks

ii. $%ter a number o% weeks there will be no live eggs /less than one le%t in the population.

hen oes this %irst o""urL

/# marks

H% the stu! is repeate with unsterilise ault inse"ts' eggs will be lai an potentiall! grow into

aulsts. $ssuming 30F o% aults la! eggs ea"h week' the population matrix a%ter one week' S1' is now

given b!

S1@ TS0?0S0 where

"

A

*

)

=

0

100

#00

(00

0S

/ etermine the matrix 0.

/# marks

*(ESTI#" 1-

/a 4onsier the %ollowing s!stem o% equations using elementar! row operations' where 'is a real

s"alar.

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x?yB+@ 1

x? #y? '+@ 3

#x? 'y? #+@ (

2sing elementar! row operations' show that the s!stem "an be reu"e to

+

*#

#

1

*00

110

111

# '''

'

/3 marks

/b Aen"e etermine the values o% '%or whi"h the s!stem has

i. a unique solution.

#+


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/# marks

ii. no solution.

/1 mark

iii. more than one solution.

/1 mark

/" in the solution to the s!stem o% equations when '@ 1.

/# marks

/ or the matri"es A@

#

13

a' 0@

b

#an C@

5

,'

i. %in the inverse o% Ain terms o% a.

9;E$E T2E


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/# marks

ii. state the restri"tion on asu"h that A0 @Chas a unique solution.

/1 mark

iii. %in the values o% aan b i% A0@ C.

/3 marks

*(ESTI#" 1

Ele"troni"


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use prin"iple helps make motorist more aware o% the true "ost o% riving. This wa!' roa usage "an be

optimise.

The lan authorit! o% a metropolitan is stu!ing the number o% vehi"les passing through the pri"ing

points uring the peak perio %rom ).00 am. The stu! team wishes to "hoose between twosuggestions %or the rate' in vehi"les per hour' at whi"h motorists pass through the pri"ing points.

The two suggestions are:

oel I :te

y(51

(000+

= an oel II: ttey ,.010000 =

wherey is the rate o% vehi"les per hour an t the number o% hours a%ter ).00 am.

/a h! moel IIis "onsiere as the most appropriate %or this situationL

/# marks

/b ket"h the graph o% the moel II%or the time %rom ).00 am until 1# noon.

/# marks

8ase on the moel II'

/" at approximatel! what time is the rate o% vehi"les passing through the pri"ing points greatestL

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/# marks

/ The graph o%y"hanges shape at

(5.300)',

#0. 7iven an interpretation o% this point in

this "ontext.

/# marks

/e Ht is thought that the "urrent number o% pri"ing points "oul be reu"e to "ater %or a

maximum o% *0 vehicles per minute. 8etween what times woul this be insu%%i"ient to"ope with the numbers o% vehi"les wanting to pass throughL

/3 marks

/% i. rite own an expression %or the number o% vehi"les whi"h passe through the pri"ing points

between ).00 am an 10.00 am.

3#


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/# marks

ii. Evaluate an interpret this expression.

/3 marks



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