2012 Specialist Mathematics Written examination 2 · 2012 SPECMATH EXAM 2 10 SECTION 1 –...
Transcript of 2012 Specialist Mathematics Written examination 2 · 2012 SPECMATH EXAM 2 10 SECTION 1 –...
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SPECIALIST MATHEMATICSWritten examination 2
Monday 12 November 2012 Reading time: 3.00 pm to 3.15 pm (15 minutes) Writing time: 3.15 pm to 5.15 pm (2 hours)
QUESTION AND ANSWER BOOK
Structure of bookSection Number of
questionsNumber of questions
to be answeredNumber of
marks
12
225
225
2258
Total 80
• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpeners,rulers,aprotractor,set-squares,aidsforcurvesketching,oneboundreference,oneapprovedCAScalculatororCASsoftwareand,ifdesired,onescientificcalculator.CalculatormemoryDOESNOTneedtobecleared.
• StudentsareNOTpermittedtobringintotheexaminationroom:blanksheetsofpaperand/orwhiteoutliquid/tape.
Materials supplied• Questionandanswerbookof25pageswithadetachablesheetofmiscellaneousformulasinthe
centrefold.• Answersheetformultiple-choicequestions.
Instructions• Detachtheformulasheetfromthecentreofthisbookduringreadingtime.• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Checkthatyournameandstudent numberasprintedonyouranswersheetformultiple-choice
questionsarecorrect,andsignyournameinthespaceprovidedtoverifythis.
• AllwrittenresponsesmustbeinEnglish.
At the end of the examination• Placetheanswersheetformultiple-choicequestionsinsidethefrontcoverofthisbook.
Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2012
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2012
2012SPECMATHEXAM2 2
SECTION 1–continued
Question 1Thegraphwithequation y
x x=
− −1
2 62hasasymptotesgivenby
A. x = − 32, x =2andy = 1
B. x = − 32and x =2only
C. x =32, x =–2andy = 0
D. x = − 32, x =2andy = 0
E. x =32and x =–2only
Question 2Arectangleisdrawnsothatitssideslieonthelineswithequationsx =–2,x=4,y =–1andy =7.Anellipseisdrawninsidetherectanglesothatitjusttoucheseachsideoftherectangle.Theequationoftheellipsecouldbe
A. x y2 2
9 161+ =
B. ( ) ( )x y+
++
=1
93
161
2 2
C. ( ) ( )x y−
+−
=1
93
161
2 2
D. ( ) ( )x y+
++
=1 1
2 2
363
64
E. ( ) ( )x y−
+−
=1
363
641
2 2
SECTION 1
Instructions for Section 1Answerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrect forthequestion.Acorrectanswerscores1,anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.Taketheacceleration due to gravitytohavemagnitudegm/s2,whereg=9.8.
3 2012 SPECMATH EXAM 2
SECTION 1 – continuedTURN OVER
Question 3The graph of y = f (x) is shown below.
y
x
2
1
–1
O
–2
–2 –1 1 2
All of the axes below have the same scale as the axes in the diagram above.
The graph of yf x
=1( )
is best represented by
A.
y
xO
B.
y
xO
C.
y
xO
D.
xO
y
E.
y
xO
2012SPECMATHEXAM2 4
SECTION 1–continued
Question 4Thedomainandrangeofthefunctionwithrule f x x( ) ( )= − +arccos 2 1
2π arerespectively
A. [–2,0]and[0,π ]
B. [–2,0]and π π2
32
,
C. [0,1]and[0,π ]
D. [0,1]and π π2
32
,
E. [0,π ]and[0,1]
Question 5
If z = 2 45
cis _ π andw = z9,then
A. w =16 2 365
cis π
B. w = −16 25
cis π
C. w =16 2 45
cis π
D. w = −9 25
cis π
E. w = 9 2 45
cis π
Question 6Foranycomplexnumberz,thelocationonanArganddiagramofthecomplexnumberu i z= 3 canbefoundby
A. rotatingzthrough 32π inananticlockwisedirectionabouttheorigin
B. reflectingzaboutthex-axisandthenreflectingaboutthey-axis
C. reflectingzaboutthey-axisandthenrotatinganticlockwisethrough π2 abouttheorigin
D. reflectingzaboutthex-axisandthenrotatinganticlockwisethroughπ2 abouttheorigin
E. rotatingzthrough 32π inaclockwisedirectionabouttheorigin
5 2012SPECMATHEXAM2
SECTION 1–continuedTURN OVER
Question 7Thesetofpointsinthecomplexplanedefinedby| z + 2i | = | z |correspondstoA. thepointgivenbyz = –iB. thelineIm(z) = –1C. thelineIm(z) = –iD. thelineRe(z) = –1E. thecirclewithcentre–2iandradius1
Question 8Ifz = a + bi,wherebothaandbarenon-zerorealnumbersandz ∈ C,whichofthefollowingdoesnot representarealnumber?A. z z+
B. | z |C. z z
D. z2 – 2abi
E. z z z z−( ) +( )
Question 9Euler’sformulaisusedtofindy2,where
dydx
x= cos( ),x0=0,y0=1andh=0.1
Thevalueofy2 correcttofourdecimalplacesisA. 1.1000andthisisanunderestimateofy (0.2)B. 1.1995andthisisanoverestimateofy (0.2)C. 1.1995andthisisanunderestimateofy (0.2)D. 1.2975andthisisanunderestimateofy (0.2)E. 1.2975andthisisanoverestimateofy (0.2)
2012 SPECMATH EXAM 2 6
SECTION 1 – continued
Question 10The diagram that best represents the direction field of the differential equation dy
dxxy= is
A.
x
y
–4 –3 –2 –1 O 1 2 3 4
–3–2–1
1234
–4
B.
x
y
–4 –3 –2 –1 O 1 2 3 4
–4
–3
–2
–1
1
2
3
4
C.
y
–4 –3 –2 –1 O 1 2 3 4
–4
–3
–2
–1
1
2
3
4
x
D.
x–4 –3 –2 –1 O 1 2 3 4
–4
–3
–2
–1
1
2
3
4
y
E.
x–4 –3 –2 –1 O 1 2 3 4
–4–3–2–1
1234
y
7 2012SPECMATHEXAM2
SECTION 1–continuedTURN OVER
Question 11
If d ydx
x x2
22= − and dy
dx= 0 at x=0,thenthegraphofywillhave
A. alocalminimumat x =12
B. alocalmaximumatx =0andalocalminimumatx = 1
C. stationarypointsofinflectionatx =0andx=1,andalocalminimumatx = 32
D. astationarypointofinflectionatx =0,nootherpointsofinflectionandalocalminimumatx = 32
E. astationarypointofinflectionatx =0,anon-stationarypointofinflectionatx =1andalocalminimum
at x = 32
Question 12Thevolumeofthesolidofrevolutionformedbyrotatingthegraphofy = 9 1 2− −( )x aboutthex-axisisgivenby
A. 4 3 2π ( )
B. π 9 1 2
3
3− −( )
−∫ ( )x dx
C. π 9 1 2
2
4− −( )
−∫ ( )x dx
D. π 9 1 2 2
2
4− −( )
−∫ ( )x dx
E. π 9 1 2
4
2
− −( )−∫ ( )x dx
2012SPECMATHEXAM2 8
SECTION 1–continued
Question 13
Usingasuitablesubstitution, sin ( )cos ( )3 4
0
3x x dx
π
∫ canbeexpressedintermsofuas
A. u u du6 4
0
3−( )∫
π
B. u u du6 4
1
12
−( )∫
C. u u du6 4
12
1
−( )∫
D. u u du6 4
0
32
−( )∫
E. u u du4 6
0
32
−( )∫
Question 14Aparticleisactedonbytwoforces,oneof6newtonsactingduesouth,theotherof4newtonsactinginthedirectionN60°W.Themagnitudeoftheresultantforce,innewtons,actingontheparticleisA. 10
B. 2 7
C. 2 19
D. 52 24 3−
E. 52 24 3+
9 2012 SPECMATH EXAM 2
SECTION 1 – continuedTURN OVER
Question 15The vectors a i j k~ ~ ~ ~= + −2 3m and b i j k~ ~ ~ ~= − +m2 are perpendicular for
A. m = −23
and m = 1
B. m = −32
and m = 1
C. m =23
and m = –1
D. m =32
and m = –1
E. m = 3 and m = –1
Question 16The distance between the points P (–2, 4, 3) and Q (1, –2, 1) isA. 7
B. 21
C. 31
D. 11
E. 49
Question 17If u~ ~ ~ ~= − +2 2i j k and v~ ~ ~ ~= − +3 6 2i j k, the vector resolute of v~ in the direction of u~ is
A. 2049
3 6 2( )~ ~i j k~
− +
B. 203
2 2( ~ ~ ~i j k )− +
C. 207
3 6 2( )~ ~ ~i j k− +
D. 209
2 2( )~ ~ ~i j k− +
E. 19
2 2( )~ ~ ~− + −i j k
2012SPECMATHEXAM2 10
SECTION 1–continued
Question 18Thevelocity–timegraphforthefirst2secondsofthemotionofaparticlethatismovinginastraightlinewithrespecttoafixedpointisshownbelow.
v
t1 2
20
–20
–40
Theparticle’svelocityvismeasuredincm/s.Initiallytheparticleisx0cmfromthefixedpoint.Thedistancetravelledbytheparticleinthefirst2secondsofitsmotionisgivenby
A. vdt0
2
∫
B. vdt x0
2
0∫ +
C. vdt vdt1
2
0
1
∫ ∫−
D. | |vdt0
2
∫
E. vdt vdt x1
2
0
1
0∫ ∫− +
Question 19Abodyismovinginastraightlineand,aftertseconds,itisxmetresfromtheoriginandtravellingatvms–1.Giventhatv = x,andthatt =3wherex =–1,theequationforx intermsoft isA. x = et–3
B. x = – e3–t
C. x t= −2 5
D. x t= − −2 5
E. x = – et–3
11 2012SPECMATHEXAM2
SECTION 1–continuedTURN OVER
Question 20Particlesofmass3kgandmkgareattachedtotheendsofalightinextensiblestringthatpassesoverasmoothpulley,asshown.
3 kg
m kg
Iftheaccelerationofthe3kgmassis4.9m/s2upwards,thenA. m=4.5B. m=6.0C. m=9.0D. m=13.5E. m=18.0
Question 21Aparticleofmass3kgisactedonbyavariableforce,sothatitsvelocityvm/swhentheparticleisxmfromtheoriginisgivenbyv = x2.Theforceactingontheparticlewhenx =2,innewtons,isA. 4B. 12C. 16D. 36E. 48
2012SPECMATHEXAM2 12
END OF SECTION 1
Question 22A12kgmassissuspendedinequilibriumfromahorizontalceilingbytwoidenticallightstrings.Eachstringmakesanangleof60°withtheceiling,asshown.
60° 60°
12 kg
Themagnitude,innewtons,ofthetensionineachstringisequaltoA. 6gB. 12 gC. 24 gD. 4 3 gE. 8 3 g
13 2012SPECMATHEXAM2
SECTION 2 – Question 1–continuedTURN OVER
Question 1Acurveisdefinedbytheparametricequations
x=cosec(θ) + 1 y=2cot(θ)
a. Showthatthecurvecanbeexpressedinthecartesianform
( )x y− − =1
412
2
2marks
SECTION 2
Instructions for Section 2Answerallquestionsinthespacesprovided.Unlessotherwisespecifiedanexactanswerisrequiredtoaquestion.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.Taketheacceleration due to gravitytohavemagnitudegm/s2,whereg=9.8.
2012SPECMATHEXAM2 14
SECTION 2 – Question 1–continued
b. Sketchthecurvedefinedbytheparametricequations
x =cosec(θ) + 1 y =2cot(θ)
labellinganyasymptoteswiththeirequations.
5
4
3
2
1
O
–1
–2
–3
–4
–5
–5 –4 –3 –2 –1 1 2 3 4 5
y
x
4marks
15 2012SPECMATHEXAM2
SECTION 2–continuedTURN OVER
c. Theregionboundedbythecurveandthelinex =3isrotatedaboutthex-axis. Findthevolumeofthesolidformed.
2marks
d. Findthegradientofthecurvewhereθ π=
76.
3marks
2012SPECMATHEXAM2 16
SECTION 2 – Question 2–continued
Question 2a. Giventhatcos ,π
123 22
=
+ showthatsin .π12
2 32
=
−
2marks
b. i. Expressz i13 22
2 32
=+
+− inpolarform.
ii. Writedownz14inpolarform.
1+1=2marks
c. OntheArganddiagrambelow,shadetheregiondefinedby
z z z: ) )Arg( Arg(z) Arg(1 14≤ ≤{ }∩{z:1≤| z |≤2}, z C∈ .
Im(z)
Re(z)O–1–2–3 321
3
2
1
–1
–2
–3
2marks
17 2012SPECMATHEXAM2
SECTION 2–continuedTURN OVER
d. Findtheareaoftheshadedregioninpart c.
2marks
e. i. Findthevalue(s)ofnsuchthatRe(z1n) =0,where z i1
3 22
2 32
=+
+− .
ii. Findz1nforthevalue(s)ofnfoundinpart i.
3+1=4marks
2012SPECMATHEXAM2 18
SECTION 2 – Question 3–continued
Question 3Acaracceleratesfromrest.ItsspeedafterTsecondsisVms–1,where
V T= −17
61tan ,π T≥0
a. Writedownthelimitingspeed ofthecarasT → ∞.
1mark
b. Calculate,correcttothenearest0.1ms–2,theaccelerationofthecarwhenT=10.
1mark
c. Calculate,correcttothenearestsecond,thetimeittakesforthecartoacceleratefromrestto25ms–1.
2marks
19 2012SPECMATHEXAM2
SECTION 2 – Question 3–continuedTURN OVER
Afteracceleratingto25ms–1,thecarstaysatthisspeedfor120secondsandthenbeginstodeceleratewhilebraking.Thespeedofthecartsecondsafterthebrakesarefirstappliedisvms–1where
dvdt
t= −− 1100
145 2( ),
untilthecarcomestorest.d. i. Findvintermsoft.
ii. Findthetime,inseconds,takenforthecartocometorestwhilebraking.
2+2=4marks
2012SPECMATHEXAM2 20
SECTION 2–continued
e. i. Writedowntheexpressionsforthedistancetravelledbythecarduringeachofthethreestagesofitsmotion.
ii. Findthetotaldistancetravelledfromwhenthecarstartstoacceleratetowhenitcomestorest. Giveyouranswerinmetrescorrecttothenearestmetre.
2+1=3marks
21 2012 SPECMATH EXAM 2
SECTION 2 – Question 4 – continuedTURN OVER
Question 4The position vector of the International Space Station (S), when visible above the horizon from a radar tracking location (O) on the surface of Earth, is modelled by
r i j~ ~ ~( ) sin ( . . ) cos ( . . )t t t= −( ) + −( ) −( )6800 1 3 0 1 6800 1 3 0 1 6400π π ,,
for t∈[ ]0 0 154, . ,
where i~ is a unit vector relative to O as shown and j~ is a unit vector vertically up from point O. Time t is
measured in hours and displacement components are measured in kilometres.
i∼
j∼
P
S hy
xO
r(t)∼
a. Find the height, h km, of the space station above the surface of Earth when it is at point P, directly above point O.
Give your answer correct to the nearest km.
1 mark
b. Find the acceleration of the space station, and show that its acceleration is perpendicular to its velocity.
3 marks
2012 SPECMATH EXAM 2 22
SECTION 2 – continued
c. Find the speed of the space station in km/h. Give your answer correct to the nearest integer.
2 marks
d. Find the equation of the path followed by the space station in cartesian form.
2 marks
e. Find the times when the space station is at a distance of 1000 km from the radar tracking location O. Give your answers in hours, correct to two decimal places.
3 marks
23 2012SPECMATHEXAM2
SECTION 2 – Question 5–continuedTURN OVER
Question 5Atherfavouritefunpark,Katherine’sfirstactivityistoslidedowna10mlongstraightslide.Shestarts fromrestatthetopandacceleratesataconstantrate,untilshereachestheendoftheslidewithavelocityof 6ms–1.a. Howlong,inseconds,doesittakeKatherinetotraveldowntheslide?
1mark
Whenatthetopoftheslide,whichis6mabovetheground,Katherinethrowsachocolateverticallyupwards.Thechocolatetravelsupandthendescendspastthetopoftheslidetolandonthegroundbelow.Assumethatthechocolateissubjectonlytogravitationalaccelerationandthatairresistanceisnegligible.b. Iftheinitialspeedofthechocolateis10m/s,howlong,correcttothenearesttenthofasecond,doesit
takethechocolatetoreachtheground?
2marks
c. AssumethatittakesKatherinefoursecondstorunfromtheendoftheslidetowherethechocolatelands.
Atwhatvelocitywouldthechocolateneedtobepropelledupwards,ifKatherineweretoimmediatelyslidedowntheslideandruntoreachthechocolatejustasithitstheground?
Giveyouranswerinms–1,correcttoonedecimalplace.
2marks
2012SPECMATHEXAM2 24
SECTION 2 – Question 5–continued
Katherine’snextactivityistorideaminispeedboat.Tostopatthecorrectboatdock,sheneedstostoptheengineandallowtheboattobeslowedbyairandwaterresistance.Attimetsecondsaftertheenginehasbeenstopped,theaccelerationoftheboat,ams–2,isrelatedtoitsvelocity,vms–1,by
a v= − −1
10196 2 .
Katherinestopstheenginewhenthespeedboatistravellingat7m/s.d. i. Findanexpressionforvintermsoft.
ii. Findthetimeittakesthespeedboattocometorest. Giveyouranswerinsecondsintermsof π.
25 2012SPECMATHEXAM2
iii. Findthedistanceittakesthespeedboattocometorest,fromwhentheengineisstopped. Giveyouranswerinmetres,correcttoonedecimalplace.
3+2+3=8marks
END OF QUESTION AND ANSWER BOOK
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 2012
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