MATHEMATICAL METHODS (CAS) Methods (CAS) Formulas Mensuration area of a trapezium: 1 2 ()ab+ h...

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Transcript of MATHEMATICAL METHODS (CAS) Methods (CAS) Formulas Mensuration area of a trapezium: 1 2 ()ab+ h...

  • MATHEMATICAL METHODS (CAS)Written examination 1

    Wednesday 4 November 2015 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,sharpeners,rulers.

    StudentsareNOTpermittedtobringintotheexaminationroom:notesofanykind,blanksheetsofpaper,correctionfluid/tapeoracalculatorofanytype.

    Materials supplied Questionandanswerbookof16pages,withadetachablesheetofmiscellaneousformulasinthe

    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.

    VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2015

    SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2015

    STUDENT NUMBER

    Letter

  • 2015MATHMETH(CAS)EXAM1 2

    THIS PAGE IS BLANK

  • 3 2015MATHMETH(CAS)EXAM1

    TURN OVER

    Question 1 (4marks)a. Lety=(5x+1)7.

    Find dydx. 1mark

    b. Let f x xxe( ) log ( ) .= 2

    i. Findf(x). 2marks

    ii. Evaluatef(1). 1mark

    InstructionsAnswerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegivenunlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmustbeshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.

  • 2015MATHMETH(CAS)EXAM1 4

    Question 2 (3marks)

    Let = f xx

    ( ) ,1 3 wherex0.

    Giventhatf (e)=2,find f (x).

  • 5 2015MATHMETH(CAS)EXAM1

    TURN OVER

    Question 3 (2marks)

    Evaluate 11

    4

    xdx

    .

  • 2015MATHMETH(CAS)EXAM1 6

    Question 4continued

    Question 4 (6marks)Considerthefunction f R f x x x: , , ( ) .[ ] = + ( )3 2 12 3 4

    3 2

    a. Findthecoordinatesofthestationarypointsofthefunction. 2marks

  • 7 2015MATHMETH(CAS)EXAM1

    TURN OVER

    Theruleforfcanalsobeexpressedas f x x x( ) .= ( ) +( )12

    1 2 2

    b. Ontheaxesbelow,sketchthegraphoff, clearlyindicatingaxisinterceptsandturningpoints. Labeltheendpointswiththeircoordinates. 2marks

    y

    x8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8

    87654321

    1

    O

    2345678

    c. Findtheaveragevalueoffovertheinterval0x2. 2marks

  • 2015 MATHMETH (CAS) EXAM 1 8

    Question 5 (3 marks)On any given day, the depth of water in a river is modelled by the function

    h t t t( ) sin ,= +

    14 8 12

    0 24

    where h is the depth of water, in metres, and t is the time, in hours, after 6 am.

    a. Find the minimum depth of the water in the river. 1 mark

    b. Find the values of t for which h(t) = 10. 2 marks

  • 9 2015MATHMETH(CAS)EXAM1

    TURN OVER

    Question 6 (3marks)LettherandomvariableXbenormallydistributedwithmean2.5andstandarddeviation0.3LetZbethestandardnormalrandomvariable,suchthatZ~N(0,1).

    a. FindbsuchthatPr(X>3.1)=Pr(Z