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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
SECTION 1 – continued
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
SECTION 1 – continuedTURN OVER
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
2015MATHMETH(CAS)EXAM2 6
SECTION 1 – continued
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
× ×
7 2015MATHMETH(CAS)EXAM2
SECTION 1 – continuedTURN OVER
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
2015MATHMETH(CAS)EXAM2 8
SECTION 1 – continued
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
9 2015MATHMETH(CAS)EXAM2
SECTION 1 – continuedTURN OVER
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
2015MATHMETH(CAS)EXAM2 10
SECTION 1 – continued
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
11 2015MATHMETH(CAS)EXAM2
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
2015MATHMETH(CAS)EXAM2 12
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.
13 2015MATHMETH(CAS)EXAM2
SECTION 2 – Question 1–continuedTURN OVER
b. i. Findtheequationofthetangenttothegraphoffatthepoint P 1 45
, .
1mark
ii. FindthecoordinatesofpointsQandS. 2marks
c. FindthedistancePS andexpressitintheform bc,whereb andcarepositiveintegers. 2marks
2015 MATHMETH (CAS) EXAM 2 14
SECTION 2 – continued
y
x
4
S
QO
P 1 45
,
d. Find the area of the shaded region in the graph above. 3 marks
2015MATHMETH(CAS)EXAM2 16
SECTION 2 – Question 2–continued
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
17 2015MATHMETH(CAS)EXAM2
SECTION 2 – Question 2–continuedTURN OVER
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
2015 MATHMETH (CAS) EXAM 2 18
SECTION 2 – continued
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
19 2015MATHMETH(CAS)EXAM2
SECTION 2 – Question 3–continuedTURN OVER
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
2015 MATHMETH (CAS) EXAM 2 20
SECTION 2 – continued
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
21 2015MATHMETH(CAS)EXAM2
SECTION 2 – Question 4–continuedTURN OVER
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
2015MATHMETH(CAS)EXAM2 22
SECTION 2 – Question 4–continued
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
23 2015MATHMETH(CAS)EXAM2
SECTION 2 – continuedTURN OVER
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
2015MATHMETH(CAS)EXAM2 24
SECTION 2 – Question 5–continued
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
25 2015MATHMETH(CAS)EXAM2
SECTION 2 – Question 5–continuedTURN OVER
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
2015MATHMETH(CAS)EXAM2 26
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
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