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S A M P L ESPECIALIST MATHEMATICS
Written examination 2Day Date
Reading time: *.** to *.** (15 minutes) Writing time: *.** to *.** (2 hours)
Question 2Themaximaldomainandrangeofthefunctionwithrule f x x( ) = − +−3 4 1
21sin ( ) π arerespectively
A. [–�,2�]and 0, 12
B. 0, 12
and[–�,2�]
C. −
32π π, 3
2and −
12
, 0
D. 0, 12
and[0,3�]
E. −
12
, 0 and[–�,2�]
SECTION A – Multiple-choice questions
Instructions for Section AAnswerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrect forthequestion.Acorrectanswerscores1;anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.Unlessotherwiseindicated,thediagramsinthisbookarenot drawntoscale.Taketheacceleration due to gravitytohavemagnitudegms–2,whereg=9.8
SECTION A – continuedTURN OVER
Version3–July2016 3 SPECMATHEXAM2(SAMPLE)
Question 3Thefeaturesofthegraphofthefunctionwithrule f x
x xx x
( ) = − +− −
2
24 3
6include
A. asymptotesatx=1andx=–2B. asymptotesatx=3andx=–2C. anasymptoteatx=1andapointofdiscontinuityatx=3D. anasymptoteatx=–2andapointofdiscontinuityat x=3E. anasymptoteatx=3andapointofdiscontinuityatx=–2
Question 4Thealgebraicfraction
7 54 92 2
xx x
−− +( ) ( )
couldbeexpressedinpartialfractionformas
A. A
xB
x−( )+
+4 92 2
B. Ax
Bx
Cx−
+−
++4 3 3
C. A
xBx Cx−( )
+++4 92 2
D. Ax
Bx
Cx Dx−
+−( )
+++4 4 92 2
E. Ax
Bx
Cx−
+−( )
++4 4 92 2
Question 5OnanArganddiagram,asetofpointsthatliesonacircleofradius2centredattheoriginisA. { : }z C zz∈ = 2
B. { : }z C z∈ =2 4
C. { :Re( ) Im( ) }z C z z∈ + =2 2 4
D. { : }z C z z z z∈ +( ) − −( ) =2 2 16
E. { : Re( ) Im( ) }z C z z∈ ( ) + ( ) =2 2 16
Question 6ThepolynomialP(z)hasrealcoefficients.FouroftherootsoftheequationP(z)=0 are z =0,z =1–2i, z =1+2i and z =3i.TheminimumnumberofrootsthattheequationP(z)=0couldhaveisA. 4B. 5C. 6D. 7E. 8
Question 15A12kgmassmovesinastraightlineundertheactionofavariableforceF,sothatitsvelocityvms–1whenitisxmetresfromtheoriginisgivenby v x x= − +3 162 3 .TheforceFactingonthemassisgivenby
A. 12 3 32
2x x−
B. 12 3 162 3x x− +( )
C. 12 6 3 2x x−( )
D. 12 3 162 3x x− +
E. 12 3 3−( )x
SECTION A – continued
SPECMATHEXAM2(SAMPLE) 8 Version3–July2016
Question 16
Theacceleration,a ms–2,ofaparticlemovinginastraightlineisgivenby a vve
=log ( )
,wherevisthe
velocityoftheparticleinms–1attimetseconds.Theinitialvelocityoftheparticlewas5ms–1.Thevelocityoftheparticle,intermsoft, isgivenbyA. v = e2t
b. Ontheaxesbelow,sketchthegraphsoffand f –1,showingtheirpointsofintersection. 2marks
1
O 1 2 3 4
2
3
4
y
x
SECTION B
Instructions for Section BAnswerallquestionsinthespacesprovided.Unlessotherwisespecified,anexactanswerisrequiredtoaquestion.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.Taketheacceleration due to gravitytohavemagnitudegms–2,whereg=9.8
a. Verifybysubstitutionthatloge(N)=6–3e–0.4tsatisfiesthedifferentialequation
1 0 4 2 4 0NdNdt
Ne+ − =. log ( ) . 2marks
b. Findtheinitialnumberofmobilephonesownedinthecommunity.Giveyouranswercorrecttothenearestinteger. 1mark
c. Usingthismathematicalmodel,findthelimitingnumberofmobilephonesthatwouldeventuallybeownedinthecommunity.Giveyouranswercorrecttothenearestinteger. 2marks
e. i. Ifthemeanlifetimeofallcomputersisinfactμ =9.5hours,findPr * .X C> =( )µ 9 5 ,givingyouranswercorrecttothreedecimalplaces,whereC*isyouranswertopart d. 2marks
ii. Doestheresultinpart e.i.indicateatypeIortypeIIerror?Explainyouranswer. 1mark