2019 MM1-ANS - VCAA past exam answer PDF

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SUPERVISOR TO ATTACH PROCESSING LABEL HERE

Victorian Certificate of Education 2019

Letter STUDENT NUMBER

MATHEMATICAL METHODS Written examination 1 Wednesday 6 November 2019 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 book Number of questions

Number of questions to be answered

Number of marks

9

9

40

• Studentsaretowriteinblueorblackpen. • Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers, sharpenersandrulers. • StudentsareNOTpermittedtobringintotheexaminationroom:anytechnology(calculatorsor software),notesofanykind,blanksheetsofpaperand/orcorrectionfluid/tape. Materials supplied • Questionandanswerbookof13pages • Formulasheet • Workingspaceisprovidedthroughoutthebook. Instructions • Writeyourstudent numberinthespaceprovidedaboveonthispage. • Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale. • AllwrittenresponsesmustbeinEnglish. At the end of the examination • Youmaykeeptheformulasheet. Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room. ©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2019

2

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2019MATHMETHEXAM1

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3

2019MATHMETHEXAM1

Instructions Answerallquestionsinthespacesprovided. Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegiven,unlessotherwisespecified. Inquestionswheremorethanonemarkisavailable,appropriateworkingmustbeshown. Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.

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Question 1 (4marks) 1 1 . Let f :  , ∞  → R, f ( x) = x 3 3 −1   a.

i. Findf′(x).

1mark



ii. Findanantiderivativeoff (x).

1mark

b.

Letg : R \ {− 1} → R , g ( x ) =



Evaluateg′(1).

sin(π x ) . x+1 2marks

TURN OVER

2019MATHMETHEXAM1

4

Question 2 (4marks) 1 1 → R , f (x) = . a. Let f : R \ 3 3x − 1

{}

Findtheruleoff –1.

b.

Statethedomainoff –1.

c.

LetgbethefunctionobtainedbyapplyingthetransformationTtothefunction f,where

2marks

1mark

 x   x  c  T     =   +     y   y  d  

andc,d∈R.



Findthevaluesofcanddgiventhatg=f–1.

1mark

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5

2019MATHMETHEXAM1

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Question 3 (3marks) Theonlypossibleoutcomeswhenacoinistossedareaheadoratail.Whenanunbiasedcoinistossed,the probabilityoftossingaheadisthesameastheprobabilityoftossingatail. Johasthreecoinsinherpocket;twoareunbiasedandoneisbiased.Whenthebiasedcoinistossed,the 1 probabilityoftossingaheadis . 3 Jorandomlyselectsacoinfromherpocketandtossesit. a.

Findtheprobabilitythatshetossesahead.

b.

Findtheprobabilitythatsheselectedanunbiasedcoin,giventhatshetossedahead.

2marks

1mark

TURN OVER

6

Question 4 (4marks) x x a. Solve1− cos  = cos  for x ∈ [−2 π , π ]. 2  2  

b.

2marks

x Thefunction f : [ −2π, π ] → R, f ( x) = cos  isshownontheaxesbelow. 2 y 2

1 2�

−�

O

x �

1 2



Let g : [ −2π ,π ] → R , g (x ) = 1 − f ( x ).



Sketchthegraphofgontheaxesabove.Labelallpointsofintersectionofthegraphsof fandg,and theendpointsofg,withtheircoordinates.

2marks

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2019MATHMETHEXAM1

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7

2019MATHMETHEXAM1

Question 5 (5marks)

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Let f : R \ {1} → R, f ( x) =

2 + 1. ( x − 1) 2

a.

i. Evaluatef (–1).

1mark



ii. Sketchthegraphoffontheaxesbelow,labellingallasymptoteswiththeirequations.

2marks

y 6 5 4 3 2 1 5

4

3

2

1

O 1

x 1

2

3

4

5

2 3 4

b.

Findtheareaboundedbythegraphoff,thex-axis,thelinex=–1andthelinex=0.

2marks

TURN OVER

2019MATHMETHEXAM1

8

Question 6 (3marks) Fredownsacompanythatproducesthousandsofpegseachday.Herandomlyselects41pegsthatare producedononedayandfindseightfaultypegs. Whatistheproportionoffaultypegsinthissample?

b.

Pegsarepackedeachdayinboxes.Eachboxholds12pegs.LetP betherandomvariablethat representstheproportionoffaultypegsinabox. 1 Theactualproportionoffaultypegsproducedbythecompanyeachdayis . 6 1 Find Pr  Pˆ   .Expressyouranswerintheforma(b)n,whereaandbarepositiverationalnumbers 6  andnisapositiveinteger.

 

1mark

2marks

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a.



9

2019MATHMETHEXAM1

Question 7 (4marks) Thegraphoftherelationy = 1− x 2 isshownontheaxesbelow.Pisapointonthegraphofthisrelation, Aisthepoint(–1,0)andBisthepoint(x,0). y

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y  1  x2

A (–1, 0)

P (x, y)

B (x, 0)

a.

FindanexpressionforthelengthPBintermsofxonly.

b.

FindthemaximumareaofthetriangleABP.

x

1mark

3marks

TURN OVER

2019MATHMETHEXAM1

10

Question 8 (4marks) Thefunction f :R→R, f (x)isapolynomialfunctionofdegree4.Partofthegraphof fisshownbelow. Thegraphof ftouchesthex-axisattheorigin. y  1  , 1   2 

a.

O

x (1, 0)

Findtheruleoff.

1mark

Letgbeafunctionwiththesameruleas f. Let h : D → R, h ( x ) = log e ( g (x )) − log e ( x 3 + x 2 ) ,whereDisthemaximaldomainofh. b.

StateD.

1mark

Question 8 –continued

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(–1, 0)

 1  , 1   2 



11

Statetherangeofh.

2marks

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c.

2019MATHMETHEXAM1

TURN OVER

2019MATHMETHEXAM1

12

Question 9 (9marks)

a.

Statetheruleofg ( f ( x)) .

1mark

b.

Findthevaluesofxforwhichthederivativeof g ( f ( x) ) isnegative.

2marks

c.

Statetheruleof f ( g ( x )) .

d.

Solve f ( g ( x )) = 0.

1mark

2marks

Question 9–continued

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Considerthefunctions f : R → R, f ( x) = 3 + 2 x − x2 and g : R → R, g( x) = e x.

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

13

e.

Findthecoordinatesofthestationarypointofthegraphof f ( g( x)) .

f.

Statethenumberofsolutionstog ( f (x )) + f ( g ( x)) =0 .

END OF QUESTION AND ANSWER BOOK

2019MATHMETHEXAM1

2marks

1mark

Victorian Certificate of Education 2019

MATHEMATICAL METHODS Written examination 1

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.

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2019

MATHMETH EXAM

2

ateatica etod forua Mensuration area of a trapezium

1 ( a + b) h 2

volume of a pyramid

1 Ah 3

curved surface area of a cylinder

2  

volume of a sphere

4 3 πr 3

volume of a cylinder

2

area of a triangle

1 bc sin ( A ) 2

volume of a cone

1 2 πr h 3

Calculus d n n− 1 x = nx dx

∫ x dx = n + 1 x

1

n +1

d n−1 (ax + b )n = an ( ax + b ) dx

∫ ( ax + b)

dx =

1 (ax + b )n + 1 + c , n ≠ − 1 a( n + 1)

d ax e = ae ax dx

∫

1 d ( loge (x )) = x dx

∫ x dx = log (x ) + c , x > 0

d (sin (ax) ) = a cos (ax ) dx

∫sin (ax)dx = − a cos(ax) + c

d ( cos (ax )) = − a sin (ax) dx

∫ cos (ax)dx = a sin (ax ) + c

( )

(

n

)

( )

e axdx =

n

+ c, n ≠ −1

1 ax +c ae

1

e

1

1

d a 2 = a sec ( ax ) (tan (ax) ) = 2 dx cos ( ax)

product rule

d (uv )= udv + v du dx dx dx

chain rule

dy dy du  dx du dx

quotient rule

du dv −u v d  u dx dx  = dx  v  v2

3

MATHMETH EXAM

Probability Pr(A  B) = Pr(A) + Pr(B) – Pr(A  B)

Pr(A) = 1 – Pr(A) Pr(A|B) =

Pr ( A∩ B ) Pr(B )

  E(X)

mean

var(X) =  2 = E((X –  )2) = E(X 2) – 2

variance

Probability distribution discrete

Pr(X  x)  p(x)

continuous

Pr(a < X < b ) =

Mean

Variance

   x p(x)

∫

b

f (x )dx

a

µ=

∫

 2   (x –  )2 p(x)

∞

−∞

x f ( x) dx

σ2=

∫

∞

( x − µ ) 2 f ( x) dx

−∞

Sample proportions P 

X n

standard deviation

mean sd( ˆP) 

p(1  p) n

approximate confidence interval

E(P ) = p   ˆp  z  

END OF FORMULA SHEET

pˆ 1  pˆ n

, ˆp  z

pˆ 1  pˆ  n

   ...