exam 2 - section 2 - 2012

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PERTH MODERN SCHOOL

Trial WACE Examination, 2012

Quetion!An"er #oo$let

MATHEMAT%CSSPEC%AL%ST &C!&D

Se'tion T"o(Cal'ulator)aume*

Student Number: In figures

In words ______________________________________

Your name ______________________________________

Teacher ______________________________________

Time allo"e* +or ti e'tionReading time before commencing work: ten minutesWorking time for this section: one hundred minutes

Material re-uire*!re'ommen*e* +or ti e'tionTo be provided by the supervisor

This Question/Answer ook!et"ormu!a Sheet #retained from Section $ne%

To be provided by the candidateStandard items: &ens' &enci!s' &enci! shar&ener' eraser' correction f!uid/ta&e' ru!er' high!ighters

S&ecia! items: drawing instruments' tem&!ates' notes on two unfo!ded sheets of A( &a&er'and u& to three ca!cu!ators satisf)ing the conditions set b) the *urricu!um*ounci! for this e+amination,

%m.ortant note to 'an*i*ate

No other items ma) be used in this section of the e+amination, It is /our res&onsibi!it) to ensurethat )ou do not ha-e an) unauthorised notes or other items of a non.&ersona! nature in thee+amination room, If )ou ha-e an) unauthorised materia! with )ou' hand it to the su&er-isore+ore reading an) further,

SOLT%ONS

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CALCLATOR)ASSMED & MATHEMAT%CS SPEC%AL%ST &C!&D

Stru'ture o+ ti .a.er

SectionNumber of uestionsa-ai!ab!e

Number of uestions tobe answered

Working time#minutes%

0arksa-ai!ab!e

1ercentageof e+am

Section $ne:*a!cu!ator.free 2 2 34 34 55

Section Two:*a!cu!ator.assumed

65 65 644 644 72

Total 634 644

%ntru'tion to 'an*i*ate

6, The ru!es for the conduct of Western Austra!ian e+terna! e+aminations are detai!ed in theYear 12 Information Handbook 2012 , Sitting this e+amination im&!ies that )ou agree toabide b) these ru!es,

8, Write )our answers in the s&aces &ro-ided in this Question/Answer ook!et, S&are &agesare inc!uded at the end of this book!et, The) can be used for &!anning )our res&onsesand/or as additiona! s&ace if reuired to continue an answer,• 1!anning: If )ou use the s&are &ages for &!anning' indicate this c!ear!) at the to& of the

&age,• *ontinuing an answer: If )ou need to use the s&ace to continue an answer' indicate in

the origina! answer s&ace where the answer is continued' i,e, gi-e the &age number,"i!! in the number of the uestion#s% that )ou are continuing to answer at the to& of the&age,

5, So" all /our "or$in 'learl/, Your working shou!d be in sufficient detai! to a!!ow )ouranswers to be checked readi!) and for marks to be awarded for reasoning, Incorrectanswers gi-en without su&&orting reasoning cannot be a!!ocated an) marks, "or an)uestion or &art uestion worth more than two marks' -a!id working or 9ustification isreuired to recei-e fu!! marks, If )ou re&eat an answer to an) uestion' ensure that )oucance! the answer )ou do not wish to ha-e marked,

(, It is recommended that )ou *o not ue .en'il' e+ce&t in diagrams,

See next .ae

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MATHEMAT%CS SPEC%AL%ST &C!&D 3 CALCLATOR)ASSMED

Se'tion T"o( Cal'ulator)aume* 4100 Mar$5

This section has tirteen 41&5 uestions, Answer all uestions, Write )our answers in the s&aces&ro-ided,

Working time for this section is 644 minutes,

Quetion 6 47 mar$5

In two residentia! suburbs' A and ' from 6;( to 6' the median house &rice' M do!!ars'

increased at a rate gi-en b)dM

kM dt

= ' where t is the time' in )ears and k is a constant s&ecific

to each suburb,

"or suburb A' the median &rice at the start of 6;( was

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CALCLATOR)ASSMED 8 MATHEMAT%CS SPEC%AL%ST &C!&D

Quetion 9 48 mar$5

The fo!!owing >es!ie matri+' L ' a&&!ies to a &o&u!ation of beet!es in which the fema!e beet!es inthe &o&u!ation !i-e for a ma+imum of 5 )ears and on!) &ro&agate in their third )ear of !ife,

6883

4 4 3

4 44 4

L

=

#a% What is the &robabi!it) that a newborn fema!e beet!e wi!! sur-i-e to the 5rd )ear of its !ife=#6 mark%

#b% Initia!!) there are 344 fema!es in each age grou&, ?ow man) fema!es wi!! there bea!together after 8 )ears= #8 marks%

#c% *omment on the !ong.term &o&u!ation of fema!e beet!es &redicted b) this mode!, #8 marks%

See next .ae

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Quetion 10 4: mar$5

#a% A triang!e with -ertices at ( , )6 6 A ' ( , )5 6 B and ( , )5 (C is ref!ected in the x .a+is and then

rotated 4° antic!ockwise about the origin,

#i% "ind the matri+ T that wi!! combine these two transformations in the order gi-en,#5 marks%

#ii% "ind the coordinates of C after transformation b) T , #6 mark%

#b% Another transformation matri+ is gi-en b).

. .

4 7 4

6 8 4 7 R

− = − −

,

@etermine the area of triang!e ABC after it has been transformed b) T andthen b) R , #5 marks%

See next .ae

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CALCLATOR)ASSMED : MATHEMAT%CS SPEC%AL%ST &C!&D

Quetion 11 4: mar$5

A function is defined as ( ) 8 5 8 f x x x= + + − ,

#a% +&ress ( ) f x without the use of abso!ute -a!ue bars, #5 marks%

#b% Sketch the gra&h of ( ) f x #8 marks%

x.64 .3 3 64

y

3

64

#c% So!-e ( ) 3 f x x≤ + , #8 marks%

See next .ae

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Quetion 12 47 mar$5

"ind the e+act area bounded b) the x .a+is' the y .a+is' the function .( ) 4 838 x f x e= and the

tangent to ( ) f x when ; x = ,

x3 64

y

64

84

See next .ae

units8

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Quetion 1& 47 mar$5

A !ight is &ositioned at the to& of a -ertica! &ost 68 m high, A sma!! ba!! is dro&&ed from the sameheight as the !ight but at a &oint ( m awa),

If the distance tra-e!!ed b) the ba!! t seconds after re!ease is gi-en b) . 8( t ' how fast is the

shadow of the ba!! mo-ing a!ong the horiBonta! ground ha!f a second after the ba!! is dro&&ed=

68

(

(,:t8

+

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Quetion 13 46 mar$5

#a% Cse &roof b) contradiction to &ro-e that 8 is irrationa!, #( marks%

#b% Cse a -ector method to &ro-e that the diagona!s of the rhombus $1QR are &er&endicu!ar,

#( marks%

$

1 Q

R

See next .ae

as $1QR is a rhombus

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Quetion 18 412 mar$5

A*@"D? is a rectangu!ar &rism with

8 5OA = + +i ; $

( 8 2OB = + +i ; $

2 ; OD = − + +i ; $

65 82 7OE = + −i ; $

#a% "ind a -ector euation for the &!ane "D? in the form c• =r n , #5 marks%

#b% "ind a -ector euation for the !ine &assing through A and , #8 marks%

See next .ae

A

*@

"

D?

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#c% The &oint 1 !ies on the !ine through A so that the siBe of HPD∠ is 4°, "ind the shortest&ossib!e distance from A to 1,

#2 marks%

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..

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CALCLATOR)ASSMED 1& MATHEMAT%CS SPEC%AL%ST &C!&D

Quetion 17 49 mar$5

>et sinw cos iθ θ = + and sin z cos iφ φ = + ,

#a% Cse u!erEs formu!a to e+&ress the &roduct wz in e+&onentia! form, #6 mark%

#b% Cse w and z to show that sin( ) sin cos cos sinθ φ θ φ θ φ + = + , #( marks%

#c% ?ence show thatcos sin

cos

855 sin

( (8 8d c

π θ θ π θ θ θ

+ + = + + ÷ ÷ ÷

∫ , #( marks%

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Quetion 1: 46 mar$5

A &artic!e mo-es a!ong the x .a+is' with dis&!acement x cm from the origin' after t seconds'

gi-en b) cos5

t x a

π = ÷

' where a is a &ositi-e constant, After 6 second' the &artic!e is 68 cm from

the origin,

#a% "ind the -a!ue of a , #6 mark%

#b% Show that the motion of the &artic!e is sim&!e harmonic, #8 marks%

#c% "ind the s&eed of the &artic!e as it &asses through the origin, #8 marks%

#d% "ind the distance tra-e!!ed b) the &artic!e during the first minute of its motion, #5 marks%

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Quetion 16 49 mar$5

A com&!e+ ineua!it) is gi-en b) 5 5 5 5 z i+ − ≤ ,

#a% Sketch the region in the com&!e+ &!ane defined b) this ineua!it), #5 marks%

Re(z).7 .5 5 7

Im(z)

.:

.7

.5

5

75 5

#b% "ind the minimum and ma+imum -a!ues of z , #5 marks%

#c% "ind the minimum and ma+imum -a!ues of arg z , #5 marks%

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Quetion 19 410 mar$5

The -e!ocit) of a bod) mo-ing in a straight !ine is gi-en b) 5 (dx

xdt

= + ' where x is the

dis&!acement' in metres' from a fi+ed reference &oint at time t seconds, When 6t = ' 8 x = ,

#a% "ind an e+&ression for x in terms of t , #3 marks%

#b% What is the e+act -e!ocit) of the bod) when

#i% 5 x = = #6 mark%

#ii% 5t = = #8 marks%

#c% What is the acce!eration of the bod) when 6t = = #8 marks%

See next .ae

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CALCLATOR)ASSMED 1: MATHEMAT%CS SPEC%AL%ST &C!&D

Quetion 20 4: mar$5

>et ( )3 5 2

3 5 63

n n n P n = + + ,

#a% -a!uate ( )6 P and ( )( P , #6 mark%

#b% 1ro-e b) induction that ( ) P n is a!wa)s an integer' when n is a &ositi-e integer, #7 marks%

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A**itional "or$in .a'e

Question number: _________

En* o+ -uetion

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A**itional "or$in .a'e

Question number: _________

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2012 Tem.late

This e+amination &a&er ma) be free!) co&ied' or communicated on an intranet' for non.commercia! &ur&oses withineducationa! institutes that ha-e &urchased the &a&er from WA +amination 1a&ers &ro-ided that WA +amination

1a&ers is acknow!edged as the co&)right owner, Teachers within &urchasing schoo!s ma) change the &a&er &ro-idedthat WA +amination 1a&erEs mora! rights are not infringed,

*o&)ing or communication for an) other &ur&oses can on!) be done within the terms of the *o&)right Act or with &rior written &ermission of WA +amination &a&ers,

Published by WA Examination PapersPO ox !!" #laremont WA $%10

exam 2 - section 2 - 2012

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