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STUDENT NUMBER Letter
SPECIALIST MATHEMATICSWritten examination 1
Friday 8 November 2013 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
9 9 40
• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpeners andrulers.
• Studentsarenotpermittedtobringintotheexaminationroom:notesofanykind,acalculatorofanytype,blanksheetsofpaperand/orwhiteoutliquid/tape.
Materials supplied• Questionandanswerbookof11pageswithadetachablesheetofmiscellaneousformulasinthe
centrefold.• Workingspaceisprovidedthroughoutthebook.
Instructions• Detachtheformulasheetfromthecentreofthisbookduringreadingtime.• Writeyourstudent numberinthespaceprovidedaboveonthispage.
• AllwrittenresponsesmustbeinEnglish.
Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2013
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2013
2013SPECMATHEXAM1 2
THIS PAGE IS BLANK
3 2013SPECMATHEXAM1
TURN OVER
InstructionsAnswerallquestionsinthespacesprovided.Unlessotherwisespecified,anexactanswerisrequiredtoaquestion.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.Taketheacceleration due to gravitytohavemagnitudegm/s2,whereg=9.8.
Question 1 (3marks)Abodyofmass10kgisheldinplaceonasmoothplaneinclinedat30°tothehorizontalbyatensionforce,Tnewtons,actingparalleltotheplane.a. Onthediagrambelow,showallotherforcesactingonthebodyandlabelthem. 1mark
T
30°
b. FindthevalueofT. 2marks
2013SPECMATHEXAM1 4
Question 2 (4marks)
Evaluatex
x xdx
−− +∫
55 62
0
1
.
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Question 3 (4marks)ThecoordinatesofthreepointsareA(–1,2,4),B(1,0,5)andC(3,5,2).
a. FindAB. 1mark
b. ThepointsA,BandCaretheverticesofatriangle. Provethatthetrianglehasarightangleat A. 2marks
c. Findthelengthofthehypotenuseofthetriangle. 1mark
2013SPECMATHEXAM1 6
Question 4 (6marks)a. Statethemaximaldomainandtherangeofy=arccos(1–2x). 2marks
b. Sketchthegraphofy=arccos(1–2x)overitsmaximaldomain.Labeltheendpointswiththeircoordinates. 2marks
y
xO
c. Findthegradientofthetangenttothegraphofy=arccos(1–2x)at x = 14. 2marks
7 2013SPECMATHEXAM1
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Question 5 (5marks)Acontainerofwaterisheatedtoboilingpoint(100°C)andthenplacedinaroomthathasaconstanttemperatureof20°C.Afterfiveminutesthetemperatureofthewateris80°C.
a. UseNewton’slawofcooling dTdt
k T= − −( )20 ,whereT °Cisthetemperatureofthewater
attime tminutesafterthewaterisplacedintheroom,toshowthate k− =5 34. 2marks
b. Findthetemperatureofthewater10minutesafteritisplacedintheroom. 3marks
2013SPECMATHEXAM1 8
Question 6 (4marks)Findthevalueofc,wherec R,suchthatthecurvedefinedby
y ex
cx
2132
+−
=−( )
hasagradientof2wherex =1.
9 2013SPECMATHEXAM1
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Question 7 (6marks)Thepositionvectorr( )~ t ofaparticlemovingrelativetoanoriginOattimetsecondsisgivenby
r i j~( ) sec( ) tan( ) , ,~ ~t t t t= + ∈
4 2 0
2π
wherethecomponentsaremeasuredinmetres.
a. Showthatthecartesianequationofthepathoftheparticleis x y2 2
16 41− = . 2marks
b. Sketchthepathoftheparticleontheaxesbelow,labellinganyasymptoteswiththeirequations. 2marks
x
y
O
c. Findthespeedoftheparticle,inms–1,when t = π4. 2marks
2013SPECMATHEXAM1 10
Question 8 (4marks)Findallsolutionsofz4–2z2+4=0,z Cincartesianform.
11 2013SPECMATHEXAM1
END OF QUESTION AND ANSWER BOOK
Question 9 (4marks)
Theshadedregionbelowisenclosedbythegraphofy =sin(x)andthelinesy =3xandx = π3 .
Thisregionisrotatedaboutthex-axis.
1
2
3
x =y = 3x
y = sin(x)
O
y
x
π3
π3
Findthevolumeoftheresultingsolidofrevolution.
SPECIALIST MATHEMATICS
Written examinations 1 and 2
FORMULA SHEET
Directions to students
Detach this formula sheet during reading time.
This formula sheet is provided for your reference.
© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2013
SPECMATH 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: 43 π r
3
area of a triangle: 12 bc Asin
sine rule: aA
bB
cCsin sin sin
= =
cosine rule: c2 = a2 + b2 – 2ab cos C
Coordinate geometry
ellipse: x ha
y kb
−( )+
−( )=
2
2
2
2 1 hyperbola: x ha
y kb
−( )−
−( )=
2
2
2
2 1
Circular (trigonometric) functionscos2(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
+ =+
−1 tan( ) tan( ) tan( )tan( ) tan( )
x y x yx 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=
−
function sin–1 cos–1 tan–1
domain [–1, 1] [–1, 1] R
range −
π π2 2
, [0, �] −
π π2 2
,
3 SPECMATH
Algebra (complex numbers)z = x + yi = r(cos θ + i sin θ) = r cis θ
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)
Calculusddx
x nxn n( ) = −1
∫ =
++ ≠ −+x dx
nx c nn n1
111 ,
ddxe aeax ax( ) =
∫ = +e dx
ae cax ax1
ddx
xxelog ( )( )= 1
∫ = +1xdx x celog
ddx
ax a axsin( ) cos( )( )=
∫ = − +sin( ) cos( )ax dxa
ax c1
ddx
ax a axcos( ) sin( )( )= −
∫ = +cos( ) sin( )ax dxa
ax c1
ddx
ax a axtan( ) sec ( )( )= 2
∫ = +sec ( ) tan( )2 1ax dx
aax c
ddx
xx
sin−( ) =−
12
1
1( )
∫
−=
+ >−1 0
2 21
a xdx x
a c asin ,
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
1tan
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
constant (uniform) acceleration: v = u + at s = ut +12
at2 v2 = u2 + 2as s = 12
(u + v)t
TURN OVER
SPECMATH 4
END OF FORMULA SHEET
Vectors in two and three dimensions
r i j k~ ~ ~ ~= + +x y z
| r~ | = x y z r2 2 2+ + = r
~ 1. r~ 2 = r1r2 cos θ = x1x2 + y1y2 + z1z2
Mechanics
momentum: p v~ ~= m
equation of motion: R a~ ~= m
friction: F ≤ µN