2019 Specialist Mathematics Written examination 2 · SPECIALIST MATHEMATICS Written examination 2 Wednesday 5 June 2019 Reading time: 10.00 am to 10.15 am (15 minutes) Writing time:
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SPECIALIST MATHEMATICSWritten examination 2
Wednesday 5 June 2019 Reading time: 10.00 am to 10.15 am (15 minutes) Writing time: 10.15 am to 12.15 pm (2 hours)
Instructions for Section AAnswerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrectforthequestion.Acorrectanswerscores1;anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.Unlessotherwiseindicated,thediagramsinthisbookarenot drawntoscale.Taketheacceleration due to gravitytohavemagnitudegms–2,whereg=9.8
SECTION A – continuedTURN OVER
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Question 3The maximal domain and range of the function f (x) = a cos–1(bx) + c, where a, b and c are real constants with a > 0, b < 0 and c > 0, are respectivelyA. [0, π] and [–a, a]
B. [0, π] and [–a + c, a + c]
C. −
1 1b b
, and [c, aπ + c]
D. 1 1b b
, −
and [c, aπ + c]
E. 1 1b b
, −
and [–aπ + c, aπ + c]
Question 4Which one of the following statements is false for z1, z2 ∈ C?
A. z zz
z− = ≠12 0,
B. z z z z1 2 1 2+ > +
C. zz
z zz
z1
2
1 2
22 2 0= ≠,
D. z z z z1 2 1 2=
E. zz
zz
z1
2
1
22 0= ≠,
Question 5The circle defined by z a z i+ = +3 , where a ∈ R, has a centre and radius respectively given by
Inthecomplexplane,Listhelinewithequation z z i+ = − −2 1 3 .
a. Verifythatthepoint(0,0)liesonL. 1mark
b. ShowthatthecartesianformoftheequationofLis 3 .y x= − 2marks
c. ThelineLcanalsobeexpressedintheform z z z− = −1 1 ,wherez1 ∈ C.
Findz1incartesianform. 2marks
SECTION B
Instructions for Section BAnswerallquestionsinthespacesprovided.Unlessotherwisespecified,anexactanswerisrequiredtoaquestion.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.Taketheacceleration due to gravitytohavemagnitudegms–2,whereg=9.8
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SECTION B – continuedTURN OVER
d. Find, in cartesian form, the point(s) of intersection of L and the graph of z = 4. 2 marks
e. Sketch L and the graph of z = 4 on the Argand diagram below. 2 marks
5
4
3
2
1
–1
–2
–3
–4
–5
O–5 –4 –3 –2 –1 1 2 3 4 5
Im(z)
Re(z)
f. Find the area of the sector defined by the part of L where Re(z) ≥ 0, the graph of z = 4 where Re(z) ≥ 0, and the imaginary axis where Im(z) > 0. 1 mark
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SECTION B – Question 2–continued
Question 2 (10marks)
Considerthefunction f withrule f xx xx
( ) .=+ +−
2
21
1
a. i. Statetheequationsoftheasymptotesofthegraphof f. 2marks
ii. Statethecoordinatesofthestationarypointsandthepointofinflection.Giveyouranswerscorrecttotwodecimalplaces. 2marks
iii. Sketchthegraphof f fromx =–6tox =6(endpointcoordinatesarenotrequired)onthesetofaxesbelow,labellingtheturningpointsandthepointofinflectionwiththeircoordinatescorrecttotwodecimalplaces.Labeltheasymptoteswiththeirequations. 3marks
y
x–6 –4 –2 2 4 6
2
1
O
–1
–2
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SECTION B – continuedTURN OVER
Considerthefunction fk withrule f x x x kxk ( ) ,=+ +−
2
2 1wherek ∈ R.
b. Forwhatvaluesofkwill fk haveno stationary points? 2marks
c. Forwhatvalueof k willthegraphof fk haveapointofinflectionlocatedonthey-axis? 1mark
c. Findthedistance,inmetres,travelledbythepalletafter10seconds. 3marks
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SECTION B – continued
d. Whenthepalletreachesavelocityof3ms–1,waterispouredbackintothecontainerataconstantrateof2Lpersecond,whichinturnretardsthemotionofthepalletmovingdowntheplane.Let tbethetime,inseconds,afterthecontainerbeginstofill.
i. Writedown,intermsoft,anexpressionforthetotalmassofthehangingcontainerandthewateritcontainsaftertseconds.Giveyouranswerinkilograms. 1mark
ii. Showthattheaccelerationofthepalletdowntheplaneisgivenby g tt
( )5290
2−+
−ms for t∈[ )0 5, . 2marks
iii. Findthevelocityofthepalletwhent =4.Giveyouranswerinmetrespersecond,correcttoonedecimalplace. 2marks
b. WritedownsuitablenullandalternativehypothesesH0andH1respectivelytotestwhetherthemeantimetakenforthepainttodryislongerthanclaimed. 1mark
c. Writedownanexpressionforthepvalueofthestatisticaltestandevaluateitcorrecttothreedecimalplaces. 2marks
d. Usinga1%levelofsignificance,statewithareasonwhetherthecrashrepaircentreisjustifiedinbelievingthatthepaintcompany’sclaimofameantimetakenforitspainttodryof 3.55hoursistoolow. 1mark
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e. Atthe1%levelofsignificance,findthesetofsamplemeanvaluesthatwouldsupporttheconclusionthatthemeantimetakenforthepainttodryexceeded3.55hours.Giveyouranswerinhours,correcttothreedecimalplaces. 2marks
f. Ifthetrue meantimetakenforthepainttodryis3.83hours,findtheprobabilitythatthepaintcompany’sclaimisnotrejectedatthe1%levelofsignificance,assumingthestandarddeviationforthepainttodryisstill0.66hours.Giveyouranswercorrecttotwodecimalplaces. 1mark
END OF QUESTION AND ANSWER BOOK
SPECIALIST MATHEMATICS
Written examination 2
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.