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8/16/2019 Exam 2 - Section 2 - Soln 2012
1/21
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,
8/16/2019 Exam 2 - Section 2 - Soln 2012
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8/16/2019 Exam 2 - Section 2 - Soln 2012
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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
8/16/2019 Exam 2 - Section 2 - Soln 2012
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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
8/16/2019 Exam 2 - Section 2 - Soln 2012
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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
6 8 6
8 3 3× =
8
344 6444
344 6834
344 644
L
=
Tota! @ 8534 beet!es
The tota! number of fema!e beet!es starts at 6344 initia!!)'then near!) doub!es after 6 )ear to 834' fa!!s back to 8534after 8 )ears and then returns to 6344 after 5 )ears' with344 in each age grou& as at the start of the c)c!e,
This c)c!e then re&eats end!ess!)' according to the mode!,
8/16/2019 Exam 2 - Section 2 - Soln 2012
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MATHEMAT%CS SPEC%AL%ST &C!&D 7 CALCLATOR)ASSMED
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
− = − −
,
Betermine the area of triang!e ABC after it has been transformed b) T andthen b) R , #5 marks%
See next .ae
4 6 6 4 4 6
6 4 4 6 6 4T
− = × = −
'( , )
4 6 5 (
6 4 ( 5
( 5C
=
$rigina! area of triang!e A* @ 5 s units,
Beterminant of T is .6' so no change in area,
Beterminant of R is 4,57' so fina! area . .4 57 5 6 4;= × = s units,
8/16/2019 Exam 2 - Section 2 - Soln 2012
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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% C+&ress ( ) f x without the use of abso!ute -a!ue bars, #5 marks%
or f(x) = (x + 2)2
+ (3 – 2x)2
#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
( ) ( )
( ) ( ) ( ) .
( ) ( ) .
( ) .
.
8 5 8 8
8 5 8 8 6 3
8 5 8 6 3
6 5 8
3 8 6 3
5 6 6 3
x x x
f x x x x
x x x
x x
f x x x
x x
− + + − < −= + + − − ≤ ≤ + − − >
− < −= − − ≤ ≤ − >
3 3 4
3 5 6 5
So!ution:
4 5
x x x
x x x
x
+ = − ⇒ =+ = − ⇒ =
≤ ≤
8/16/2019 Exam 2 - Section 2 - Soln 2012
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MATHEMAT%CS SPEC%AL%ST &C!&D 6 CALCLATOR)ASSMED
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 thetangent to ( ) f x when ; x = ,
x3 64
y
64
84
See next .ae
( ) ( )
.
( )
'( )
'( )
( )
( )
8
4 83
8
88
88
; ; 88
4 (8 8
8
; 8
8
;8
Tangent:
8 ;8
88
Area:
88
; ; (
( ;
x
f e
e f x
e f
e y e x
e y x e
e f x dx x e dx
e e
e
=
=
=
− = −
= −
− −
= − −
= −
∫ ∫
units8
8/16/2019 Exam 2 - Section 2 - Soln 2012
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CALCLATOR)ASSMED 9 MATHEMAT%CS SPEC%AL%ST &C!&D
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 horiDonta! ground ha!f a second after the ba!! is dro&&ed=
68
(
(,:t8
+
See next .ae
"inddx
dt when .4 3t = ,
Esing simi!ar triang!es'
.
.
( )
.
.
.
8
8
5
68 (
(
(;
(
8 (;
(
4 3
27;4637 2
(
t
x
xt
dx
dt t
t
dx
dt
=
=
−=
== − ≈ −
>ence s&eed of shadow is 637,2 m/s,
8/16/2019 Exam 2 - Section 2 - Soln 2012
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MATHEMAT%CS SPEC%AL%ST &C!&D 10 CALCLATOR)ASSMED
Quetion 13 46 mar$5
#a% Ese &roof b) contradiction to &ro-e that 8 is irrationa!, #( marks%
#b% Ese 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
et andOP RQ OR PQ= = = =a ,
Then andOQ RP = + = −a a ,
So dot &roduct of diagona!s is
( ) ( )
8 8
4 since
OQ RP • = + • −
= −
= =
a a
a
a
>ence the diagona!s are &er&endicu!ar,
Assume that 8 is rationa! and can be written as the sim&!ified fractiona
b
'
where a and b are integers with no common factors,
.
( )
8 8
8 8 8 8
8 8
8 8 is an e-en integer,
If is e-en then it can be written as 8
>ence 8 8 ( 8
>ence 8 is an e-en integer,
ut if and are both e-en integers then the) ha-e 8 as a
aa b a
b
a a n
n b n b
b n b
a b
= ⇒ = ⇒
=
= ⇒ =
= ⇒
common factor'
which contradicts the assum&tion that the) ha-e no common factors,
Thus the assum&tion is incorrect' and 8 must be irrationa!,
as $1QR is a rhombus
8/16/2019 Exam 2 - Section 2 - Soln 2012
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CALCLATOR)ASSMED 11 MATHEMAT%CS SPEC%AL%ST &C!&D
Quetion 18 412 mar$5
A*BC"F> 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 C"F> in the form c• =r n , #5 marks%
#b% "ind a -ector euation for the !ine &assing through A and C, #8 marks%
See next .ae
A
*B
C "
F>
65 6 68
82 8 83
7 5
68 65
83 82 ;;3
7
68
83 ;;3
AE
c OE
= = − = − −
= • = • =
− −
• = −
n
n
r
uuur
uuur
( )8 5 68 83
OA AE λ
λ
= += + + + + −
r
i ; $ i ; $
8/16/2019 Exam 2 - Section 2 - Soln 2012
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MATHEMAT%CS SPEC%AL%ST &C!&D 12 CALCLATOR)ASSMED
#c% The &oint 1 !ies on the !ine through AC so that the siDe of HPD∠ is 4°, "ind the shortest&ossib!e distance from A to 1,
#2 marks%
See next .ae
65 ; 3
82 7 55
7 7 4
3 68 (
55 83 56
4 5
6 68 2 68 ;
8 83 ; 83 7
5 7
OH OE EH
HP
DP
HP DP
λ
λ
λ
λ λ
λ λ
λ λ
− = + = + = −
− = − = − − + + − +
= + − = −
− − −
•
uuur uuur uuur
uuur
uuur
uuur uur
( ) ( ) ( ) ( ) ( ) ( )
. , .
.
.
.
.
8
4
68 ( 68 ; 83 56 83 7 5 7 4
;34 ;34 657 4
4 8 4 ;
68 8 (
4 8 83 3
6 ;
5( 3 ;5 units
AP
AP
λ λ λ λ λ λ
λ λ
λ λ
=
− + + − − + − + − − =
− + == =
= × =
= ≈
u
uuur
uuur
6 ? 68
8 ? 83
5 G :
G
3
55
4
@
68 G (
83 G 7
G : ? 5
..
8/16/2019 Exam 2 - Section 2 - Soln 2012
13/21
CALCLATOR)ASSMED 1& MATHEMAT%CS SPEC%AL%ST &C!&D
Quetion 17 49 mar$5
et sinw cos θ θ = + and sin ! cos φ φ = + ,
#a% Ese Cu!erHs formu!a to e+&ress the &roduct w! in e+&onentia! form, #6 mark%
#b% Ese w and ! 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%
See next .ae
( ) w! e e eθ φ θ φ += =
( ) ( )
( ) ( ) ( )
( )
( ) cos sin cos sin
cos sin cos cos sin sin cos sin sin cosCuating imaginar) &arts gi-es
sin cos sin sin cos
as reuired,
e
θ φ θ θ φ φ
θ φ θ φ θ φ θ φ θ φ θ φ
θ φ θ φ θ φ
+ = + +
+ + + = − + +
+ = +
sin cos
cos cos sin sin cos
cos sin
sin
8
8
5
6
( ( 8
5( ( (
5( (
(
d
d
c
π π
π π π θ θ θ θ
π π θ θ θ
π θ
= =
+ + ÷ ÷
= + + ÷ ÷
= + + ÷
∫
∫
8/16/2019 Exam 2 - Section 2 - Soln 2012
14/21
MATHEMAT%CS SPEC%AL%ST &C!&D 13 CALCLATOR)ASSMED
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%
See next .ae
cos68 8(5
a aπ
= ⇒ =
cos
( ) sin
( ) cos
8
8
8(5
8(5 5
8(5 5
S>05
x
x
x
x x
π
π π
π π
π
= ÷
= − ÷
= − ÷ ÷
= − ⇒ ÷
&
&&
&&
0a+imum s&eed when &assing through origin' so
sin
( ) sin
65
8(5 5
; s&eed ; cm/s
x
x
π
π π
π π
= ÷
= − ÷
= − ⇒ =
&
&
1eriod is5
87 seconds
π
π = ' so in 74 seconds wi!!
mo-e through e+act!) 64 c)c!es,
Am&!itude is 8( cm' so in one com&!ete c)c!emo-es 8( ( 7× = cm,
>ence wi!! tra-e! 64 7 74× = cm,
8/16/2019 Exam 2 - Section 2 - Soln 2012
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CALCLATOR)ASSMED 18 MATHEMAT%CS SPEC%AL%ST &C!&D
Quetion 16 49 mar$5
A com&!e+ ineua!it) is gi-en b) 5 5 5 5 ! + − ≤ ,
#a% Sketch the region in the com&!e+ &!ane defined b) this ineua!it), #5 marks%
Re(!).7 .5 5 7
"#(!)
.:
.7
.5
5
75 5
#b% "ind the minimum and ma+imum -a!ues of ! , #5 marks%
#c% "ind the minimum and ma+imum -a!ues of arg ! , #5 marks%
See next .ae
Radius of circ!e is 5,
Bistance from #4' 4% to circ!e centre is ( )8 85 5 5 7+ = ,
0in ! is 7 5 5− =
0a+ ! is 7 5 :+ =
0inimum -a!ue is8
π ,
0a+imum -a!ue is3
88 7 7
π π π + × = ,
a
5
7sin a @ 5/7
a @π
7
8/16/2019 Exam 2 - Section 2 - Soln 2012
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MATHEMAT%CS SPEC%AL%ST &C!&D 17 CALCLATOR)ASSMED
Quetion 19 410 mar$5
The -e!ocit) of a bod) mo-ing in a straight !ine is gi-en b) 5 (dx
xdt
= + ' where x is the
dis&!acement' in metres' from a fi+ed reference &oint at time t seconds, When 6t = ' 8 x = ,
#a% "ind an e+&ression for x in terms of t , #3 marks%
#b% What is the e+act -e!ocit) of the bod) when
#i% 5 x = = #6 mark%
#ii% 5t = = #8 marks%
#c% What is the acce!eration of the bod) when 6t = = #8 marks%
See next .ae
ln( )
,
(
(
( (
( (
6
5 (
65 (
(
5 (
6 8 66
5 ( 66
66 5(
t
t
t
dx dt x
x t c
x ae
t x a e
x e e
e x
−
−
−
=+
+ = +
+ =
= = ⇒ =
+ =
−=
∫ ∫
( )5 ( 5 63 m/s$ = + =
( )( 5 ( ;
;;
66 5 66
( (
665 ( 5 66 m/s
(
e e x
e$ e
− −= =
= + × = +
( ( ))
8
5 (
(
( 5 ( 8
(( m/s
$ x
d$ dx
dt dt
= +
=
= × +
=
66e
; G 5
(
66e;
8/16/2019 Exam 2 - Section 2 - Soln 2012
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CALCLATOR)ASSMED 1: MATHEMAT%CS SPEC%AL%ST &C!&D
Quetion 20 4: mar$5
et ( )3 5 2
3 5 63
n n n P n = + + ,
#a% C-a!uate ( )6 P and ( )( P , #6 mark%
#b% 1ro-e b) induction that ( ) P n is a!wa)s an integer' when n is a &ositi-e integer, #7 marks%
See next .ae
( )
( )
6 6
( 88;
P
P
=
=
( ) ( )
( ) ,
( ) ( ) ( )( )
(
3 5
3 5
6 6 is an integer when 6
2 Assume that is an integer where is a &ositi-e integer' so that3 5 63
k?6 is the ne+t consecuti-e integer after k,
6 6 2 66
3 5 63
6
P P n n
k k k P k k "
k k k P k
P k
= ⇒ =
+ + =
+ + ++ = + +
+ )
( )
( )
( )
( ) ( )
3 ( 5 8 5 8
3 5( 8
3 5( 5 8
( 5 8
3 64 64 3 6 5 5 6 2 2
3 5 63
2 526 5 63 5 63
26 8 5 8 6
3 5 63
6 8 5 8 6
>ence' if is an integer' then 6 is a!so an integer' as bo
k k k k k k k k k
k k k P k k k
k k k P k k k k k
P k " k k k k
P k P k
+ + + + + + + + += + +
+ = + + + + +
+ = + + + + + + +
+ = + + + + +
+
( ) ( )
th and are integers,
Since 6 is an integer' then 8 must be an integer' and so on for a!! &ositi-e integer ,
" k
P P n
8/16/2019 Exam 2 - Section 2 - Soln 2012
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MATHEMAT%CS SPEC%AL%ST &C!&D 16 CALCLATOR)ASSMED
A**itional "or$in .a'e
Question number: _________
En* o+ -uetion
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CALCLATOR)ASSMED 19 MATHEMAT%CS SPEC%AL%ST &C!&D
A**itional "or$in .a'e
Question number: _________
8/16/2019 Exam 2 - Section 2 - Soln 2012
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MATHEMAT%CS SPEC%AL%ST &C!&D 20 CALCLATOR)ASSMED
A**itional "or$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 C+amination 1a&ers &ro-ided that WA C+amination
1a&ers is acknow!edged as the co&)right owner, Teachers within &urchasing schoo!s ma) change the &a&er &ro-idedthat WA C+amination 1a&erHs 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 C+amination &a&ers,
Published by WA Examination PapersPO ox !!" #laremont WA $%10