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SPECIALIST MATHEMATICSWritten examination 1
Friday 8 November 2019 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• Questionandanswerbookof12pages• 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.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2019
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2019
STUDENT NUMBER
Letter
2019SPECMATHEXAM1 2
THIS PAGE IS BLANK
3 2019SPECMATHEXAM1
TURN OVER
InstructionsAnswerallquestionsinthespacesprovided.Unlessotherwisespecified,anexactanswerisrequiredtoaquestion.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.Taketheacceleration due to gravitytohavemagnitudegms–2,whereg=9.8
Question 1 (4marks)
Solvethedifferentialequationdydx
yee
x
x=+
21
2
2 giventhaty(0)=π.
Question 2 (3marks)
Findallvaluesofxforwhich| | x x− = +4
27.
2019SPECMATHEXAM1 4
Question 3 (3marks)Amachineproduceschocolateintheformofacontinuouscylinderofradius0.5cm.Smallercylindricalpiecesarecutparalleltoitsend,asshowninthediagrambelow.Thelengthsofthepiecesvarywithameanof3cmandastandarddeviationof0.1cm.
a. Findtheexpectedvolumeofapieceofchocolateincm3. 1mark
b. Findthevarianceofthevolumeofapieceofchocolateincm6. 1mark
c. Findtheexpectedsurfaceareaofapieceofchocolateincm2. 1mark
5 2019SPECMATHEXAM1
TURN OVER
Question 4 (3marks)ThepositionvectorsoftwoparticlesAandBattimetsecondsaftertheyhavestartedmovingaregivenby
r i j and r iA Bt t a t t t t t( ) ( ) arccos~ ~ ~= −( ) + +
= −( ) + 2 31
3 2
j
~respectively,whereaisarealconstant
and0≤t≤2.
Findthevalueofaiftheparticlescollideaftertheyhavestartedmoving.
2019SPECMATHEXAM1 6
Question 5 (6marks)Thegraphof f (x)=cos2(x)+cos(x)+1overthedomain0≤x≤2πisshownbelow.
2 23
252 3
−2O
y
x
4
3
2
1
–1
a. i. Find f ′(x). 1mark
ii. Hence,findthecoordinatesoftheturningpointsofthegraphintheinterval(0,2π). 2marks
b. Sketchthegraphofyf x
=1( )
onthesetofaxesabove.Clearlylabeltheturningpointsand
endpointsofthisgraphwiththeircoordinates. 3marks
7 2019SPECMATHEXAM1
TURN OVER
Question 6 (3marks)Findthevalueofdforwhichthevectors
a i j k b i j k= − + = − + −2 3 4 2 4 8, and
c i j k= − + +6 2 d are linearly dependent.
2019SPECMATHEXAM1 8
Question 7 (5marks)
a. Showthat3 3 2 36
− = −
i cis π . 1mark
b. Find 3 33
−( )i ,expressingyouranswerintheformx+iy,wherex,y ∈ R. 2marks
c. Findtheintegervaluesofnforwhich 3 3−( )i nisreal. 1mark
d. Findtheintegervaluesofnforwhich 3 3−( ) =i ain
,whereaisarealnumber. 1mark
9 2019SPECMATHEXAM1
Question 8 (4marks)
Findthevolumeofthesolidofrevolutionformedwhenthegraphofy = 1 21 2++
xxisrotatedaboutthex-axis
overtheinterval[0,1].
TURN OVER
2019SPECMATHEXAM1 10
Question 9–continued
Question 9 (4marks)a. Alightinextensiblestringisconnectedateachendtoahorizontalceiling.Amassof
mkilogramshangsinequilibriumfromasmoothringonthestring,asshowninthediagrambelow.Thestringmakesanangleαwiththeceiling.
α
m
α
Expressthetension,Tnewtons,inthestringintermsofm,gandα. 1mark
11 2019SPECMATHEXAM1
TURN OVER
b. Adifferentlightinextensiblestringisconnectedateachendtoahorizontalceiling.Amassof mkilogramshangsfromasmoothringonthestring.AhorizontalforceofFnewtonsisappliedtothering.Thetensioninthestringhasaconstantmagnitudeandthesystemisinequilibrium.Atoneendthestringmakesanangleβ withtheceilingandattheotherendthestringmakesanangle2β withtheceiling,asshowninthediagrambelow.
m
2
F
ShowthatF = mg 1−
cos( )sin ( )
ββ
. 3marks
2019SPECMATHEXAM1 12
Question 10 (5marks)
Finddydxatthepoint
π π6 3
,
forthecurvedefinedbytherelationsin( ) cos( )x y xy2 2 3 2
+ =π
.
Giveyouranswerintheformππ−
( )a b
a b + ,wherea,b ∈ Z+.
END OF QUESTION AND ANSWER BOOK
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 2019
© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2019
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 2x x
x=
−
3 SPECMATH EXAM
TURN OVER
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 z snx z s
n− +
,
distribution of sample mean Xmean E X( ) = µvariance var X
n( ) = σ2
SPECMATH EXAM 4
END OF FORMULA SHEET
Calculus
ddx
x nxn n( ) = −1 x dxn
x c nn n=+
+ ≠ −+∫ 11
11 ,
ddxe aeax ax( ) = e dx
ae cax ax= +∫ 1
ddx
xxelog ( )( ) = 1 1
xdx 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 x
a c a−
=
+ >−∫ sin ,
ddx
xx
cos−( ) = −
−
12
1
1( ) −
−=
+ >−∫ 1 0
2 21
a xdx x
a c acos ,
ddx
xx
tan−( ) =+
12
11
( ) aa x
dx xa c2 2
1
+=
+
−∫ tan
( )( )
( ) ,ax b dxa n
ax b c nn n+ =+
+ + ≠ −+∫ 11
11
( ) 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 1 2 2 2
1
2
1
2
+ ′( ) ′( ) + ′( )∫ ∫f x dx x t y t dtx
x
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