MATHEMATICAL METHODS (CAS) · 2019-03-14 · MATHEMATICAL METHODS (CAS) Written examination 2 Thursday 5 November 2015 Reading time: 3.00 pm to 3.15 pm (15 minutes) Writing time:

MATHEMATICAL METHODS (CAS)Written examination 2

Thursday 5 November 2015 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

1 22 22 222 5 5 58

Total 80

• Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners, rulers, a protractor, set squares, aids for curve sketching, one bound reference, one approved CAS calculator (memory DOES NOT need to be cleared) and, if desired, one scientifi c calculator. For approved computer-based CAS, their full functionality may be used.

• Students are NOT permitted to bring into the examination room: blank sheets of paper and/or correction fl uid/tape.

Materials supplied• Question and answer book of 26 pages with a detachable sheet of miscellaneous formulas in the

centrefold.• Answer sheet for multiple-choice questions.

Instructions• Detach the formula sheet from the centre of this book during reading time. • Write your student number in the space provided above on this page.• Check that your name and student number as printed on your answer sheet for multiple-choice

questions are correct, and sign your name in the space provided to verify this.

• All written responses must be in English.

At the end of the examination• Place the answer sheet for multiple-choice questions inside the front cover of this book.

Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2015

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certifi cate of Education2015

STUDENT NUMBER

Letter

2015MATHMETH(CAS)EXAM2 2

SECTION 1 – continued

Question 1Letf:R → R, f(x)=2sin(3x)–3.Theperiodandrangeofthisfunctionarerespectively

A. period = 23�andrange=[–5,–1]

B. period = 23�andrange=[–2,2]

C. period = �3andrange=[–1,5]

D. period = 3πandrange=[–1,5]

E. period = 3πandrange=[–2,2]

Question 2Theinversefunctionof f R f x

x: ( , ) , ( )− ∞ → =

+2 1

2is

A. f–1:R+ → R f xx

− = −12

1 2( )

B. f–1:R\{0} → R f xx

− = −12

1 2( )

C. f–1:R+ → R f xx

− = +12

1 2( )

D. f–1:(–2,∞)→ R f–1(x)=x2 + 2

E. f–1:(2,∞)→ R f xx

− =−

121

2( )

SECTION 1

Instructions for Section 1Answerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrect forthequestion.Acorrectanswerscores1,anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.

3 2015 MATHMETH (CAS) EXAM 2

SECTION 1 – continuedTURN OVER

Question 3

y

xb

cd

O

The rule for a function with the graph above could be A. y = –2(x + b)(x – c)2(x – d)

B. y = 2(x + b)(x – c)2(x – d)

C. y = –2(x – b)(x – c)2(x – d)

D. y = 2(x – b)(x – c)(x – d)

E. y = –2(x – b)(x + c)2(x + d)

Question 4Consider the tangent to the graph of y = x2 at the point (2, 4).Which of the following points lies on this tangent?A. (1, –4)B. (3, 8)C. (–2, 6)D. (1, 8)E. (4, –4)

2015 MATHMETH (CAS) EXAM 2 4


Question 5Part of the graph of y = f (x) is shown below.

y

xO

The corresponding part of the graph of the inverse function y = f –1(x) is best represented by

y

x

y

x

y

x

y

x

y

x

A. B.

C. D.

E.

O O

O

O

O

5 2015MATHMETH(CAS)EXAM2


Question 6ForthepolynomialP(x)=x3–ax2–4x+4,P(3)=10,thevalueofaisA. –3B. –1C. 1D. 3E. 10

Question 7Therangeofthefunctionf:(–1,2]→ R,f(x)=–x2 + 2x –3isA. RB. (–6,–3]C. (–6,–2]D. [–6,–3]E. [–6,–2]

Question 8Thegraphofafunctionf:[–2,p]→ Risshownbelow.

y

x

(p, p)

(–2, 0)y = f (x)

O

Theaveragevalueoffovertheinterval[–2,p]iszero.

Theareaoftheshadedregionis 258.

Ifthegraphisastraightline,for0≤x≤p,thenthevalueofpisA. 2

B. 5

C. 54

D. 52

E. 254



Question 9Thegraphoftheprobabilitydensityfunctionofacontinuousrandomvariable,X,isshownbelow.

x

y

16

2 aO

Ifa>2,thenE(X)isequaltoA. 8B. 5C. 4D. 3E. 2

Question 10Thebinomialrandomvariable,X,hasE(X)=2andVar X( ) = 4

3.

Pr(X=1)isequalto

A. 13

6

B. 23

6

C. 13

23

2

×

D. 6 13

23

5

× ×

E. 6 23

13

5

× ×



Question 11

Thetransformationthatmapsthegraphof y x= +8 13 ontothegraphof y x= +3 1 isa

A. dilationbyafactorof2fromthey-axis.

B. dilationbyafactorof2fromthex-axis.

C. dilationbyafactorof 12fromthex-axis.

D. dilationbyafactorof8fromthey-axis.

E. dilationbyafactorof12 fromthey-axis.

Question 12Aboxcontainsfiveredballsandthreeblueballs.Johnselectsthreeballsfromthebox,withoutreplacingthem.TheprobabilitythatatleastoneoftheballsthatJohnselectedisredis

A. 57

B. 5

14

C. 728

D. 1556

E. 5556



Question 13Thefunctionf isaprobabilitydensityfunctionwithrule

f xae xae x

x

( ) =≤ ≤< ≤

otherwise

0 11 2

0

ThevalueofaisA. 1

B. e

C. 1e

D. 12e

E. 1

2 1e −

Question 14ConsiderthefollowingdiscreteprobabilitydistributionfortherandomvariableX.

x 1 2 3 4 5

Pr(X=x) p 2p 3p 4p 5p

ThemeanofthisdistributionisA. 2

B. 3

C. 72

D. 113

E. 4



Question 15

If g x dx( ) =∫ 200

5and 2 90

0

5g x ax dx( ) ,+( ) =∫ thenthevalueofais

A. 0B. 4C. 2D. –3E. 1

Question 16Letf(x)=axmandg(x)=bxn,wherea,b,mandnarepositiveintegers.Thedomainoff=domainofg=R.Iff ′ (x)isanantiderivativeofg(x),thenwhichoneofthefollowingmustbetrue?

A. mnisaninteger

B. nmisaninteger

C. abisaninteger

D. ba isaninteger

E. n–m=2

Question 17Agraphwithrulef(x)=x3–3x2 + c,wherecisarealnumber,hasthreedistinctx-intercepts.ThesetofallpossiblevaluesofcisA. RB. R+

C. {0,4}D. (0,4)E. (–∞,4)

Question 18Forwhichoneofthefollowingfunctionsistheequation f x y f x y f x f y( ) ( ) ( ) ( )+ − − = 4 truefor all x R y R∈ ∈and ?A. f(x)=x2

B. f(x)=|2x|

C. f(x)=ex

D. f(x)=x3

E. f(x)=x



Question 19

If f x tx

( ) = +( )∫ 2

04 dt,thenf ′(–2)isequalto

A. 2

B. − 2

C. 2 2D. −2 2E. 4 2

Question 20Iff(x–1)=x2–2x +3,thenf(x)isequaltoA. x2–2B. x2 + 2C. x2–2x + 2D. x2–2x+4E. x2–4x+6

Question 21Thegraphsofy=mx + candy=ax2willhavenopointsofintersectionforallvaluesofm,canda suchthatA. a >0andc > 0

B. a >0andc < 0

C. a >0andc > −ma

2

4

D. a <0andc > −ma

2

4

E. m >0andc > 0


END OF SECTION 1TURN OVER

Question 22Thegraphsofthefunctionswithrulesf(x)andg(x)areshownbelow.

y

x

y = g(x)y

x

y = f (x)

OO

Whichoneofthefollowingbestrepresentsthegraphofthefunctionwithruleg(–f (x))?

y

x

y

x

A. B.

C. D.y

x

y

x

E.y

x

O

O

O

O

O


SECTION 2 – Question 1–continued

Question 1 (9marks)

Let f R R f x x x: ( ) .→ = −( ) −( ), 15

2 52 ThepointP 1 45

,

isonthegraphoff,asshownbelow.

ThetangentatPcutsthey-axisatSandthex-axisatQ.

y

x

4

S

QO

P 1 45

,

a. Writedownthederivativef ′(x)off(x). 1mark

SECTION 2

Instructions for Section 2Answerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegivenunlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.


SECTION 2 – Question 1–continuedTURN OVER

b. i. Findtheequationofthetangenttothegraphoffatthepoint P 1 45

, .

1mark

ii. FindthecoordinatesofpointsQandS. 2marks

c. FindthedistancePS andexpressitintheform bc,whereb andcarepositiveintegers. 2marks



y

x

4

S

QO

P 1 45

,

d. Find the area of the shaded region in the graph above. 3 marks

15 2015 MATHMETH (CAS) EXAM 2


CONTINUES OVER PAGE



Question 2 (14marks)Acityislocatedonariverthatrunsthroughagorge.Thegorgeis80macross,40mhighononesideand30mhighontheotherside.Abridgeistobebuiltthatcrossestheriverandthegorge.Adiagramforthedesignofthebridgeisshownbelow.

y

X (–40, 40) E

F Y (40, 30)

A–40

B40

x

60Q N

MP

θ O

Themainframeofthebridgehastheshapeofaparabola.Theparabolicframeismodelledby

y x= −60 380

2 andisconnectedtoconcretepadsatB(40,0)andA(–40,0).

Theroadacrossthegorgeismodelledbyacubicpolynomialfunction.

a. Findtheangle,θ,betweenthetangenttotheparabolicframeandthehorizontalatthepointA(–40,0)tothenearestdegree. 2marks



TheroadfromX to YacrossthegorgehasgradientzeroatX(–40,40)andatY(40,30),andhas

equation y x x= − +3

25600316

35.

b. Findthemaximumdownwardsslopeoftheroad.Giveyouranswerintheform−mn wheremandnarepositiveintegers. 2marks

Twoverticalsupportingcolumns,MNandPQ,connecttheroadwiththeparabolicframe.Thesupportingcolumn,MN,isatthepointwheretheverticaldistancebetweentheroadandtheparabolicframeisamaximum.

c. Findthecoordinates(u,v)ofthepointM,statingyouranswerscorrecttotwodecimalplaces. 3marks

Thesecondsupportingcolumn,PQ,hasitslowestpointatP(–u,w).

d. Find,correcttotwodecimalplaces,thevalueofwandthelengthsofthesupportingcolumnsMNandPQ. 3marks



For the opening of the bridge, a banner is erected on the bridge, as shown by the shaded region in the diagram below.

y

60

E

F

x40–40 O

e. Find the x-coordinates, correct to two decimal places, of E and F, the points at whichthe road meets the parabolic frame of the bridge. 3 marks

f. Find the area of the banner (shaded region), giving your answer to the nearest square metre. 1 mark



Question 3(11marks)Maniisafruitgrower.Afterhisorangeshavebeenpicked,theyaresortedbyamachine,accordingtosize.Orangesclassifiedasmediumaresoldtofruitshopsandtheremainderaremadeintoorangejuice.Thedistributionofthediameter,incentimetres,ofmediumorangesismodelledbyacontinuousrandomvariable,X,withprobabilitydensityfunction

f x x x x( ) = −( ) −( ) ≤ ≤

34

6 8 6 8

0

2

otherwise

a. i. Findtheprobabilitythatarandomlyselectedmediumorangehasadiameter greaterthan7cm. 2marks

ii. Manirandomlyselectsthreemediumoranges.

Findtheprobabilitythatexactlyoneoftheorangeshasadiametergreaterthan7cm.

Expresstheanswerintheformab,whereaandbarepositiveintegers. 2marks

b. Findthemeandiameterofmediumoranges,incentimetres. 1mark



For oranges classifi ed as large, the quantity of juice obtained from each orange is a normally distributed random variable with a mean of 74 mL and a standard deviation of 9 mL.

c. What is the probability, correct to three decimal places, that a randomly selected large orange produces less than 85 mL of juice, given that it produces more than 74 mL of juice? 2 marks

Mani also grows lemons, which are sold to a food factory. When a truckload of lemons arrives at the food factory, the manager randomly selects and weighs four lemons from the load. If one or more of these lemons is underweight, the load is rejected. Otherwise it is accepted.It is known that 3% of Mani’s lemons are underweight.

d. i. Find the probability that a particular load of lemons will be rejected. Express the answer correct to four decimal places. 2 marks

ii. Suppose that instead of selecting only four lemons, n lemons are selected at random from a particular load.

Find the smallest integer value of n such that the probability of at least one lemon being underweight exceeds 0.5 2 marks



Question 4 (9marks)Anelectronicscompanyisdesigninganewlogo,basedinitiallyonthegraphsofthefunctions

f x x g x x x( ) sin( ) ( ) sin( ), .= = ≤ ≤2 12

2 0 2and for π

Thesegraphsareshowninthediagrambelow,inwhichthemeasurementsinthexandydirectionsareinmetres.

y

x32��

2� �2O

2

–2

Thelogoistobepaintedontoalargesign,withtheareaenclosedbythegraphsofthetwofunctions(shadedinthediagram)tobepaintedred.

a. Thetotalareaoftheshadedregions,insquaremetres,canbecalculatedas a x dxsin( ) .0

π

∫ Whatisthevalueofa? 1mark



Theelectronicscompanyconsiderschangingthecircularfunctionsusedinthedesignofthelogo.

Itsnextattemptusesthegraphsofthefunctionsf(x)=2sin(x)and h x x( ) sin( ),= 13

3 for0≤x≤2π.

b. Ontheaxesbelow,thegraphofy=f(x)hasbeendrawn.

Onthesameaxes,drawthegraphofy=h(x). 2marks

2

− 13

13

O

2

1

–1

–2

x

y

� �

c. Stateasequenceoftwotransformationsthatmapsthegraphofy=f(x)tothegraphof y=h(x). 2marks



Theelectronicscompanynowconsidersusingthegraphsofthefunctionsk(x)=msin(x)and

q xn

nx( ) sin( ),= 1 wheremandnarepositiveintegerswithm≥2and0≤x≤2π.

d. i. Findtheareaenclosedbythegraphsofy=k(x)andy=q(x)intermsofmandn if n is even.

Giveyouranswerintheformam bn

+ 2 ,whereaandbareintegers.2marks

ii. Findtheareaenclosedbythegraphsofy=k(x)andy=q(x)intermsofmandn if n is odd.

Giveyouranswerintheformam bn

+ 2 ,whereaandbareintegers.2marks



Question 5(15marks)

a. Let S t e et t

( ) ,= +−

2 8323 where0≤t≤5.

i. FindS(0)andS(5). 1mark

ii. TheminimumvalueofSoccurswhent=loge(c).

StatethevalueofcandtheminimumvalueofS. 2marks

iii. Ontheaxesbelow,sketchthegraphofSagainsttfor0≤t≤5.Labeltheendpointsandtheminimumpointwiththeircoordinates. 2marks

S

t

10

8

6

4

2

O 1 2 3 4 5



iv. FindthevalueoftheaveragerateofchangeofthefunctionSovertheinterval [0,loge(c)]. 2marks

LetV :[0,5]→ R,V t de d et t

( ) ,= + −( )−

32310 whered isarealnumberand d ∈ ( , ).0 10

b. Iftheminimumvalueofthefunctionoccurswhent=loge (9),findthevalueofd. 2marks

c. i. Findthesetofpossiblevaluesofdsuchthattheminimumvalueofthefunctionoccurswhent=0. 2marks

ii. Findthesetofpossiblevaluesofdsuchthattheminimumvalueofthefunctionoccurswhent=5. 2marks


END OF QUESTION AND ANSWER BOOK

d. IfthefunctionVhasalocalminimum(a,m),where0≤a≤5,itcanbeshownthat

m k d d= −( )2

1023

13 .

Findthevalueofk. 2marks

MATHEMATICAL METHODS (CAS)

Written examinations 1 and 2

FORMULA SHEET

Instructions

Detach this formula sheet during reading time.

This formula sheet is provided for your reference.

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2015

MATHMETH (CAS) 2

THIS PAGE IS BLANK

3 MATHMETH (CAS)

END OF FORMULA SHEET

Mathematical Methods (CAS)Formulas

Mensuration

area of a trapezium: 12a b h+( ) volume of a pyramid:

13Ah

curved surface area of a cylinder: 2π rh volume of a sphere: 43

3π r

volume of a cylinder: πr 2h area of a triangle: 12bc Asin

volume of a cone: 13

2π r h

Calculusddx

x nxn n( ) = −1

x dx

nx c nn n=

++ ≠ −+∫ 1

111 ,

ddxe aeax ax( ) =

e dx a e cax ax= +∫ 1

ddx

x xelog ( )( ) = 1 1x dx x ce= +∫ log

ddx

ax a axsin( ) cos( )( ) = sin( ) cos( )ax dx a ax c= − +∫ 1

ddx

ax a axcos( )( ) −= sin( ) cos( ) sin( )ax dx a ax c= +∫ 1

ddx

ax aax

a axtan( )( )

( ) ==cos

sec ( )22

product rule: ddxuv u dv

dxv dudx

( ) = + quotient rule: ddx

uv

v dudx

u dvdx

v

=

−

2

chain rule: dydx

dydududx

= approximation: f x h f x h f x+( ) ≈ ( ) + ′( )

ProbabilityPr(A) = 1 – Pr(A′) Pr(A ∪ B) = Pr(A) + Pr(B) – Pr(A ∩ B)

Pr(A|B) = Pr

PrA BB∩( )( ) transition matrices: Sn = Tn×S0

mean: µ = E(X) variance: var(X) = σ 2 = E((X – µ)2) = E(X 2) – µ2

Probability distribution Mean Variance

discrete Pr(X = x) = p(x) µ = ∑ x p(x) σ 2 = ∑ (x – µ)2 p(x)

continuous Pr( ) ( )a X b f x dxa

b< < = ∫ µ =

−∞

∞

∫ x f x dx( ) σ µ2 2= −−∞

∞

∫ ( ) ( )x f x dx

Documents

MATHEMATICAL METHODS (CAS) · 2019-03-14 · MATHEMATICAL METHODS (CAS) Written examination 2 Thursday 5 November 2015 Reading time: 3.00 pm to 3.15 pm (15 minutes) Writing time: