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SPECIALIST MATHEMATICSWritten examination 1
Friday 10 November 2017 Reading time: 9.00 am to 9.15 am (15 minutes) Writing time: 9.15 am to 10.15 am (1 hour)
QUESTION AND ANSWER BOOK
Structure of bookNumber of questions
Number of questions to be answered
Number of marks
10 10 40
• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpenersandrulers.
• StudentsareNOTpermittedtobringintotheexaminationroom:anytechnology(calculatorsorsoftware),notesofanykind,blanksheetsofpaperand/orcorrectionfluid/tape.
Materials supplied• Questionandanswerbookof11pages• Formulasheet• Workingspaceisprovidedthroughoutthebook.
Instructions• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.• AllwrittenresponsesmustbeinEnglish.
At the end of the examination• Youmaykeeptheformulasheet.
Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2017
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2017
STUDENT NUMBER
Letter
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2017SPECMATHEXAM1 2
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3 2017SPECMATHEXAM1
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InstructionsAnswerallquestionsinthespacesprovided.Unlessotherwisespecified,anexactanswerisrequiredtoaquestion.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.Taketheacceleration due to gravitytohavemagnitudegms–2,whereg=9.8
Question 1 (3marks)Findtheequationofthetangenttothecurvegivenby3xy2+2y = xatthepoint(1,–1).
Question 2 (4marks)
Find 11 21
3
x xdx
+( )∫ , expressingyouranswerintheformlog ,eab
whereaandbarepositiveintegers.
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2017SPECMATHEXAM1 4
Question 3 (3marks)Letz3 + az2 + 6z + a=0,z ∈ C,whereaisarealconstant.
Giventhatz = 1 – iisasolutiontotheequation,findallothersolutions.
Question 4 (3marks)Thevolumeofsoftdrinkdispensedbyamachineintobottlesvariesnormallywithameanof298mLand astandarddeviationof3mL.Thesoftdrinkissoldinpacksoffourbottles.
Findtheapproximateprobabilitythatthemeanvolumeofsoftdrinkperbottleinarandomlyselected four-bottlepackislessthan295mL.Giveyouranswercorrecttothreedecimalplaces.
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Question 5 (4marks)Relativetoafixedorigin,thepointsB,CandDaredefinedrespectivelybythepositionvectors
b i j k c i j k and d i j,= − + = − + = −2 2 2, a whereaisarealconstant.
GiventhatthemagnitudeofangleBCDisπ3
,finda.
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2017SPECMATHEXAM1 6
Question 6 (3marks)
Let fx
x( ) .( )
=1
arcsin
Find f ′(x)andstatethelargestsetofvaluesofxforwhich f ′(x)isdefined.
Question 7 (4marks)
Thepositionvectorofaparticlemovingalongacurveattimetisgivenby
r i j( ) cos ( ) sin ( ) , .t t t t= + ≤ ≤3 3 04π
Findthelengthofthepaththattheparticletravelsalongthecurvefromt=0tot = π4
.
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CONTINUES OVER PAGE
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2017SPECMATHEXAM1 8
Question 8 –continued
Question 8 (4marks)
Aslopefieldrepresentingthedifferentialequationdydx
xy
=−+1 2
isshownbelow.
y
x1O 2–2 –1
2
–1
–2
1
a. Sketchthesolutioncurveofthedifferentialequationcorrespondingtothecondition y(–1)=1ontheslopefieldaboveand,hence,estimatethepositivevalueofxwheny=0. Giveyouranswercorrecttoonedecimalplace. 2marks
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b. Solvethedifferentialequation dydx
xy
=−+1 2
withtheconditiony(–1)=1.Expressyouranswer
intheformay3 + by + cx2 + d =0,wherea,b,canddareintegers. 2marks
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2017SPECMATHEXAM1 10
Question 9 (5marks)Aparticleofmass2kgwithinitialvelocity3 2
i j+ ms–1experiencesaconstantforcefor 10seconds.Theparticle’svelocityattheendofthe10-secondperiodis 43 18
i j− ms–1.
a. Findthemagnitudeoftheconstantforceinnewtons. 2marks
b. Findthedisplacementoftheparticlefromitsinitialpositionafter10seconds. 3marks
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END OF QUESTION AND ANSWER BOOK
Question 10 (7marks)
a. Showthat ddx
x xa
xa
x
a xarccos arccos ,
=
−
−2 2wherea>0. 1mark
b. Statethemaximaldomainandtherangeof f x x( ) arccos .=
2
2marks
c. Findthevolumeofthesolidofrevolutiongeneratedwhentheregionboundedbythegraphofy = f(x),andthelinesx=–2andy=0,isrotatedaboutthex-axis. 4marks
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SPECIALIST MATHEMATICS
Written examination 1
FORMULA SHEET
Instructions
This formula sheet is provided for your reference.A question and answer book is provided with this formula sheet.
Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.
Victorian Certificate of Education 2017
© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2017
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SPECMATH EXAM 2
Specialist Mathematics formulas
Mensuration
area of a trapezium 12 a b h+( )
curved surface area of a cylinder 2π rh
volume of a cylinder π r2h
volume of a cone 13π r2h
volume of a pyramid 13 Ah
volume of a sphere 43π r3
area of a triangle 12 bc Asin ( )
sine ruleaA
bB
cCsin ( ) sin ( ) sin ( )
= =
cosine rule c2 = a2 + b2 – 2ab cos (C )
Circular functions
cos2 (x) + sin2 (x) = 1
1 + tan2 (x) = sec2 (x) cot2 (x) + 1 = cosec2 (x)
sin (x + y) = sin (x) cos (y) + cos (x) sin (y) sin (x – y) = sin (x) cos (y) – cos (x) sin (y)
cos (x + y) = cos (x) cos (y) – sin (x) sin (y) cos (x – y) = cos (x) cos (y) + sin (x) sin (y)
tan ( ) tan ( ) tan ( )tan ( ) tan ( )
x y x yx y
+ =+
−1tan ( ) tan ( ) tan ( )
tan ( ) tan ( )x y x y
x y− =
−+1
cos (2x) = cos2 (x) – sin2 (x) = 2 cos2 (x) – 1 = 1 – 2 sin2 (x)
sin (2x) = 2 sin (x) cos (x) tan ( )tan ( )tan ( )
2 21 2
x xx
=−
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Circular functions – continued
Function sin–1 or arcsin cos–1 or arccos tan–1 or arctan
Domain [–1, 1] [–1, 1] R
Range −
π π2 2
, [0, �] −
π π2 2
,
Algebra (complex numbers)
z x iy r i r= + = +( ) =cos( ) sin ( ) ( )θ θ θcis
z x y r= + =2 2 –π < Arg(z) ≤ π
z1z2 = r1r2 cis (θ1 + θ2)zz
rr
1
2
1
21 2= −( )cis θ θ
zn = rn cis (nθ) (de Moivre’s theorem)
Probability and statistics
for random variables X and YE(aX + b) = aE(X) + bE(aX + bY ) = aE(X ) + bE(Y )var(aX + b) = a2var(X )
for independent random variables X and Y var(aX + bY ) = a2var(X ) + b2var(Y )
approximate confidence interval for μ x zsnx z s
n− +
,
distribution of sample mean Xmean E X( ) = µvariance var X
n( ) =σ 2
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SPECMATH EXAM 4
END OF FORMULA SHEET
Calculus
ddx
x nxn n( ) = −1 x dx n x c nn n=
++ ≠ −+∫ 1 1 1
1 ,
ddxe aeax ax( ) = e dx a e c
ax ax= +∫ 1
ddx
xxe
log ( )( ) = 1 1xdx x ce= +∫ log
ddx
ax a axsin ( ) cos( )( ) = sin ( ) cos( )ax dxa
ax c= − +∫ 1
ddx
ax a axcos( ) sin ( )( ) = − cos( ) sin ( )ax dxa
ax c= +∫ 1
ddx
ax a axtan ( ) sec ( )( ) = 2 sec ( ) tan ( )2 1ax dxa
ax c= +∫ddx
xx
sin−( ) =−
12
1
1( ) 1 0
2 21
a xdx xa c a−=
+ >
−∫ sin ,ddx
xx
cos−( ) = −−
12
1
1( ) −
−=
+ >
−∫ 1 02 2 1a x dxxa c acos ,
ddx
xx
tan−( ) =+
12
11
( ) aa x
dx xa c2 21
+=
+
−∫ tan( )
( )( ) ,ax b dx
a nax b c nn n+ =
++ + ≠ −+∫ 1 1 1
1
( ) logax b dxa
ax b ce+ = + +−∫ 1 1
product rule ddxuv u dv
dxv dudx
( ) = +
quotient rule ddx
uv
v dudx
u dvdx
v
=
−
2
chain rule dydx
dydududx
=
Euler’s method If dydx
f x= ( ), x0 = a and y0 = b, then xn + 1 = xn + h and yn + 1 = yn + h f (xn)
acceleration a d xdt
dvdt
v dvdx
ddx
v= = = =
2
221
2
arc length 12 2 2
1
2
1
2
+ ′( ) ′( ) + ′( )∫ ∫f x dx x t y t dtxx
t
t( ) ( ) ( )or
Vectors in two and three dimensions
r = i + j + kx y z
r = + + =x y z r2 2 2
� � � � �ir r i j k= = + +ddt
dxdt
dydt
dzdt
r r1 2. cos( )= = + +r r x x y y z z1 2 1 2 1 2 1 2θ
Mechanics
momentum
p v= m
equation of motion
R a= m
2017 Specialist Mathematics 1InstructionsFormula sheet