mathematics general 2€¦ · higher school certificate examination mathematics general 2. 1060...

2015 HIGHER SCHOOL CERTIFICATE EXAMINATION

Mathematics General 2

1060

General Instructions

•Readingtime–5minutes

•Workingtime–2 12hours

•Writeusingblackpen

•Board-approvedcalculatorsmaybeused

•Aformulaeanddatasheetisprovidedatthebackofthispaper

•InQuestions26−30,showrelevantmathematicalreasoningand/orcalculations

Total marks – 100

Section I Pages2–12

25 marks

•AttemptQuestions1–25

•Allowabout35minutesforthissection

Section II Pages13–36

75 marks

•AttemptQuestions26–30

•Allowabout1hourand55minutesforthissection

–2–

Section I

25 marksAttempt Questions 1–25Allow about 35 minutes for this section

Usethemultiple-choiceanswersheetforQuestions1–25.

1 Whatis1 560 200kmwritteninscientificnotationcorrecttotwosignificantfigures?

(A) 1.56×104km

(B) 1.6×105km

(C) 1.56×106km

(D) 1.6×106km

2 Whichofthefollowingis 4x+3y−x−5y initssimplestform?

(A) 3x−2y

(B) 3x+8y

(C) 5x−2y

(D) 5x+8y

3 Gayle’sgrosspayeachweekis$952.25.

Thefollowingdeductionsaretakenfromhergrosspayeachweek:

• tax$180.93

• superannuation$85.70

• unionmembership$21.40

• healthfund$38.15.

WhatisGayle’snetpayeachweek?

(A) $326.18

(B) $626.07

(C) $771.32

(D) $952.25

–3–

4 On a school report, a student’s record of completing homework is graded using thefollowingcodes.

C = consistentlyU = usuallyS = sometimesR = rarelyN = never

Whattypeofdataisthis?

(A) Categorical,ordinal

(B) Categorical,nominal

(C) Quantitative,continuous

(D) Quantitative,discrete

5 Whichofthefollowinggraphsbestrepresentstheequation y=x3+1?

y

x

y

x

y

x

(A) (B)

(C) (D)

1

y

x

1

1

1

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6 The times, in minutes, that a large group of students spend on exercise per day arepresentedinthebox-and-whiskerplot.

40302010 50 60 70 80 90 100 110 120

Whatpercentageofthesestudentsspendbetween40minutesand60minutesperdayonexercise?

(A) 17%

(B) 20%

(C) 25%

(D) 50%

7 ThediagramshowsaradialsurveyofafieldABCD.

B051°

O

C160°

A 300°

D 224°

North

NOT TOSCALE90 m80

m

130 m110 m

IntriangleAOB,whatisthesizeof AOB?

(A) 51°

(B) 111°

(C) 125°

(D) 249°

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8 TheLouvrePyramidinParishasasquarebasewithsidelength35mandaperpendicularheightof22m.

22 m

35 m

Whatisthevolumeofthispyramid,tothenearestm3?

(A) 257m3

(B) 1027m3

(C) 8983m3

(D) 26 950m3

9 Fromthetopofacliff67metresabovesealevel,theangleofdepressionofabuoyis42°.

67 m

NOT TOSCALE

buoy

Howfaristhebuoyfromthebaseofthecliff,tothenearestmetre?

(A) 60m

(B) 74m

(C) 90m

(D) 100m

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10 Apieceofmachinery,initiallyworth$56 000,depreciatesat8%perannum.

Whichgraphbestshowsthesalvagevalueofthispieceofmachineryovertime?

00 10 20

Time in yearsV

alue

($’

000)

(B)

30

10

20

30

40

50

60

70

00 10 20

Time in years

Val

ue (

$’00

0)

(C)

30

10

20

30

40

50

60

70

00 10 20

Time in years

Val

ue (

$’00

0)

(A)

30

10

20

30

40

50

60

70

00 10 20

Time in years

Val

ue (

$’00

0)

(D)

30

10

20

30

40

50

60

70

11 Whichofthefollowingis3x0+5xinitssimplestform?

(A) 6x

(B) 8x

(C) 1+5x

(D) 3+5x

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12 Thelengthofafishwasmeasuredtobe49cm,correcttothenearestcm.

Whatisthepercentageerrorinthismeasurement,correcttoonesignificantfigure?

(A) ±0.01%

(B) ±0.5%

(C) ±1%

(D) ±2%

13 Whatistheequationofthelinel?

2

O

y

x1

(A) y=−2x+2

(B) y=2x+2

(C) y=− x2

+2

(D) y=x2

+2

14 Stockholmislocatedat59°N18°EandDarwinislocatedat13°S131°E.

What is the time difference between Stockholm and Darwin? (Ignore time zones anddaylightsaving.)

(A) 184minutes

(B) 288minutes

(C) 452minutes

(D) 596minutes

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15 Acameracosts$449,including12%GST.

WhatisthepriceofthecamerawithoutGST,correcttothenearestdollar?

(A) $395

(B) $401

(C) $437

(D) $503

16 Theprobabilityofwinningagameis710

.

Whichexpressionrepresentstheprobabilityofwinningtwoconsecutivegames?

(A) ×10 9

7 6(B) ×

10 10

7 7(C) ×

10 9

7 7(D) ×

10 10

7 6

17 Whatamountmustbeinvestednowat4%perannum,compoundedquarterly,sothatinfiveyearsitwillhavegrownto$60 000?

(A) $8919

(B) $11 156

(C) $49 173

(D) $49 316

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18 AStudentRepresentativeCouncil(SRC)consistsoffivemembers.Threeofthemembersarebeingselectedtoattendaconference.

Inhowmanywayscanthethreemembersbeselected?

(A) 10

(B) 20

(C) 30

(D) 60

19 The table shows the life expectancy (expected remainingyearsof life) for females atselectedagesinthegivenperiodsoftime.

Life expectancy for females

Period of time

Remaining years of life

atage0 atage25 atage45 atage65

1965–1967 74.2 51.2 32.3 15.7

1975–1977 76.6 53.1 34.0 17.1

1985–1987 79.2 55.4 36.1 18.6

1995–1997 81.3 57.1 37.7 19.8

In1975,a45-year-oldfemaleusedtheinformationinthetabletocalculatetheagetowhichshewasexpectedtolive.Twentyyearslatersherecalculatedtheagetowhichshewasexpectedtolive.

Whatisthedifferencebetweenthetwoagesshecalculated?

(A) 2.7years

(B) 3.1years

(C) 3.7years

(D) 5.8years

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20 Amachineproducescylindricalpipes.Themeanofthediametersofthepipesis8cmandthestandarddeviationis0.04cm.

Assuminganormaldistribution,whatpercentageofcylindricalpipesproducedwillhaveadiameterlessthan7.96cm?

(A) 16%

(B) 32%

(C) 34%

(D) 68%

21 Fourcards,labelled2,3,5and7,areplacedonatabletoformafour-digitnumber.

Howmanydifferentnumbersgreaterthan3000canbeformed?

(A) 6

(B) 12

(C) 18

(D) 24

22 Theareaofthetriangleshownis250cm2.

44°

28 cm

30 cmNOT TOSCALE

x cm

Whatisthevalueofx,correcttothenearestwholenumber?

(A) 11

(B) 18

(C) 22

(D) 24

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23 Thenumberof‘standarddrinks’invariousglassesofwineisshown.

Number of standard drinks

White Wine Red Wine

small glass large glass small glass large glass

0.9 1.4 1.0 1.5

Awomanweighing62kgdrinksthreesmallglassesofwhitewineandtwolargeglassesofredwinebetween8pmand1am.

What would be her blood alcohol content (BAC) estimate at 1am, correct to threedecimalplaces?

(A) 0.030

(B) 0.037

(C) 0.046

(D) 0.057

24 Considertheequation2x3

−4=5x2

+1.

Whichofthefollowingwouldbeacorrectstepinsolvingthisequation?

(A) −3=3 2

2x 5x(B) = +5

3 2

15x(C) 2x−4= +3

2

4x(D) −8=5x+2

6

2x 5x

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25 An insurance company offers customers the following discounts on the basic annualpremiumforcarinsurance.

Type of discount Discount

Multi-policydiscount(Ownerhasmorethan2insurancepolicieswiththecompany)

15%

No-claimbonus(Ownerhashadatleast5yearswithoutaninsuranceclaim)

20%

CombinedCTPandcomprehensiveinsurance(Ownerhasbothinsuranceswiththecompany)

$50

Ifacustomer iseligible formore thanonediscount, subsequentdiscountsareappliedto the already discounted premium. The combined compulsory third party (CTP) andcomprehensiveinsurancediscountisalwaysappliedlast.

Jamie has three insurance policies, including combined CTP and comprehensiveinsurance,withthiscompany.Hehasusedthiscompanyfor8yearsandhehasnevermadeaclaim.

Thebasicannualpremiumforhiscarinsuranceis$870.

HowmuchwillJamieneedtopayafterthediscountsareapplied?

(A) $482.44

(B) $515.50

(C) $541.60

(D) $557.60

©2015BoardofStudies,TeachingandEducationalStandardsNSW

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Centre Number

Student Number

26903103711061 15235



Section II

75 marksAttempt Questions 26–30Allow about 1 hour and 55 minutes for this section

Answer the questions in the spacesprovided. These spaces provideguidance for the expected length ofresponse.

Your responses should includerelevant mathematical reasoningand/orcalculations.

Extra writing space is provided onpages35and36.Ifyouusethisspace,clearly indicate which question youareanswering.

Write your Centre Number andStudent Number at the top of thispage.

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Question 26 (15marks)

(a) A farmer used the ‘capture-recapture’ technique to estimate the number ofchickens he had on his farm. He captured, tagged and released 18 of thechickens.Later, he caught26chickens at randomand found that4hadbeentagged.

2

Whatistheestimateforthetotalnumberofchickensonthisfarm?

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(b) Clark’sformulaisusedtodeterminethedosageofmedicineforchildren.

Dosage =weightinkg × adultdosage

70

2

Theadultdailydosageofamedicinecontains3150mgofaparticulardrug.

A child who weighs 35kg is to be given tablets each containing 525mg ofthisdrug.

Howmanytabletsshouldthischildbegivendaily?

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Question26(continued)

(c) Twocitieslieonthesamemeridianoflongitude.Oneis40°northoftheother.

Whatisthedistancebetweenthetwocities,correcttothenearestkilometre?

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(d) Afamilycurrentlypays$320forsomegroceries.

Assumingaconstantannualinflationrateof2.9%,calculatehowmuchwouldbepaidforthesamegroceriesin5years’time.

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(e) The table shows the relative frequency of selecting each of the differentcolouredjellybeansfrompacketscontaininggreen,yellow,black,redandwhitejellybeans.

Colour Relative frequency

Green 0.32

Yellow 0.13

Black 0.14

Red

White 0.24

(i) Whatistherelativefrequencyofselectingaredjellybean?

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(ii) Based on this table of relative frequencies, what is the probability ofNOTselectingablackjellybean?

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(f) Approximately71%ofEarth’s surface is coveredbywater.AssumeEarth isasphere.

2

Calculatethenumberofsquarekilometrescoveredbywater.

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Question 26 (continued)

(g) Pat’s mobile phone plan is shown. 3

$49 per month includes:

• $550 calls

• 500MB data

• unlimited SMS and MMS messages

Pay any amount in excess of

$550

Excess data charged at

$0.0293 per MB

Last month Pat: • made calls to the value of $561 • sent 152 SMS messages • sent 37 MMS messages • used 1.7 GB of data.

What was the total of Pat’s phone bill for last month?

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(a) Ataparticulartimeduringtheday,atowerofheight19.2metrescastsashadow.Atthesametime,apersonwhois1.65metrestallcastsashadow5metreslong.

2

1.65 m

NOT TOSCALE

19.2 m

5 mShadow of tower

Whatisthelengthoftheshadowcastbythetoweratthattime?

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(b) Apatientrequires2400mLoffluidtobedeliveredataconstantratebymeansofadripover12hours.EachmLoffluidisequivalentto15drops.

Howmanydropsperminuteneedtobedelivered?

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(c) Ariana’s parents have given her an interest‑free loan of $4800 to buy a car. She will pay them back by paying $x immediately and $y every month until she has repaid the loan in full.

After 18 months Ariana has paid back $1510, and after 36 months she has paid back $2770.

This information can be represented by the following equations.

x + 18y = 1510

x + 36y = 2770

(i) Solve these equations simultaneously to find the values of x and y.

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(ii) How many months will it take Ariana to repay the loan in full?

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(d) Inasmallbusiness,thesevenemployeesearnthefollowingwagesperweek:

$300, $490, $520, $590, $660, $680, $970.

(i) Isthewageof$970anoutlierforthissetofdata?Justifyyouranswerwithcalculations.

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(ii) Eachemployeereceivesa$20payincrease.

Whateffectwillthishaveonthestandarddeviation?

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(e) A 42 megabyte (MB) file is to be downloaded at a rate of 500 kilobits per second (kbps), where 1 kilobit = 1000 bits.

How long would it take to download this file? Give your answer in minutes and seconds, correct to the nearest second.

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(a) Thediagramshowsanannulus.

1

10 cm NOT TOSCALE

3 cm

Calculatetheareaoftheannulus.

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(b) The results of two tests are normally distributed. The mean and standard deviation for each test are displayed in the table.

2

Mathematics English

x 70 75

s 6.5 8

Kristoff scored 74 in Mathematics and 80 in English. He claims that he has performed better in English.

Is Kristoff correct? Justify your answer using appropriate calculations.

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(c) Threeequallyspacedcross-sectionalareasofavaseareshown. 3

30 cm A2 = 180 cm2

A3 = 35 cm2

A1 = 45 cm2

UseSimpson’sruletofindtheapproximatecapacityofthevaseinlitres.

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5(d) The formulaC= (F−32) isused toconvert temperaturesbetweendegrees

9Fahrenheit(F)anddegreesCelsius(C ).

2

Convert3°CtotheequivalenttemperatureinFahrenheit.

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(e) The shoe size and height of ten students were recorded.

Shoe size 6 7 7 8 8.5 9.5 10 11 12 12

Height 155 150 165 175 170 170 190 185 200 195

(i) Complete the scatter plot AND draw a line of fit by eye.

2

6 7 8 9Shoe size

Hei

ght (

cm)

10 11 12 13

130

140

150

160

170

180

190

200

210

(ii) Use the line of fit to estimate the height difference between a student who wears a size 7.5 shoe and one who wears a size 9 shoe.

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(iii) A student calculated the correlation coefficient to be 1 for this set of data. Explain why this cannot be correct.

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(f) Acharity seeks to raisemoneyby telephoningpeople at random froma callcentreandaskingthemtodonate.

Overtheyears,thischarityhasfoundthattheamountofmoneyraised($A)isrelatedtothenumberoftelephonecallsmade(n).Agraphofthisrelationshipisshown.

1000

0

2000

3000

4000

A

2000 400 600

Number of calls

Am

ount

rai

sed

($)

800 1000n

Itcoststhecharity$2100perweektorunthecallcentre.Italsocostsanaverageof50centspertelephonecall.

(i) Writeanequationtorepresentthetotalcost,C,ofrunningthecallcentreforaweekinwhichnphonecallsaremade.

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(ii) Bygraphingthisequationontheaxesabove,determinethenumberofphonecallsthecharityneedstomakeinordertobreakeven.

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7551310372

Question 29 (15 marks)

(a) On 20 August, tickets were purchased for $425 using a credit card. No other purchases were made using this card in August. Simple interest was charged at a rate of 18.4% per annum. There was no interest‑free period. The period for which interest was charged included the date of purchase and the date of payment.

What amount was paid when the account was paid in full on 31 August?

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(b) Jamal borrowed $350 000 to be repaid over 30 years, with monthly repayments of $1880. However, after 10 years he made a lump sum payment of $80 000. The monthly repayment remained unchanged. The graph shows the balances owing over the period of the loan.

2

50 10 15 20 25 30

50 0000

100 000150 000200 000250 000300 000350 000

Years

Loa

n ba

lanc

e ($

)

Over the period of the loan, how much less did Jamal pay by making the lump sum payment?

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(c) Theimageshowsarectangularfarmshedwithaflatroof.

Thewidthoftheshedindicatedbythedottedlinewasmeasuredusinganonlinerulertool,andfoundtobeapproximately12metres.

(i) Bymeasurementandcalculation,showthat theareaof theroofof theshedisapproximately216m2.

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(ii) Alltherainthatfallsontothisroofisdivertedintoacylindricalwatertankwhichhasadiameterof3.6m.Duringastorm,5mmofrainfallsontotheroof.

Calculatetheincreaseinthedepthofwaterinthetankduetotherainthatfallsontotheroofduringthestorm.

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(d) Data from 200 recent house sales are grouped into class intervals and a cumulative frequency histogram is drawn.

370

50

100

150

200

380 390 400 410

House price ($’000)

Cum

ulat

ive

freq

uenc

y

(i) Use the graph to estimate the median house price.

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(ii) By completing the table, calculate the mean house price. 3

Class Centre ($’000)

Frequency

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(e) Adiverspringsupwardsfromadivingboard,thenplungesintothewater.Thediver’sheightabovethewaterasitvarieswithtimeismodelledbyaquadraticfunction.Graphingsoftwareisusedtoproducethegraphofthisfunction.

2

Explainhowthegraphcouldbeusedtodeterminehowhighabovetheheightofthedivingboardthediverwaswhenhereachedthemaximumheight.

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Question 30 (15 marks)

(a) The energy consumption of a computer in standby mode is 21 watts. The cost of electricity is 31 cents per kWh.

A school computer room has 20 computers.

How much will the school save by switching off all 20 computers during 11 weeks of school holidays?

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(b) On a tray there are 12 hard‑centred chocolates (H ) and 8 soft‑centred chocolates (S ). Two chocolates are selected at random. A partially completed probability tree is shown.

3

1220

H

S

H

S

H

S

What is the probability of selecting one of each type of chocolate?

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(c) Thetablegivesthepresentvalueinterestfactorsforanannuityof$1perperiod,forvariousinterestrates(r)andnumbersofperiods(N).

Table of present value interest factors

r

N

Interestrateperperiod(asadecimal)

0.0075 0.0080 0.0085 0.0090 0.0095

70 54.30462 53.43960 52.59397 51.76724 50.95891

71 54.89293 54.00754 53.14226 52.29657 51.46995

72 55.47685 54.57097 53.68593 52.82118 51.97618

73 56.05643 55.12993 54.22502 53.34111 52.47764

74 56.63169 55.68446 54.75957 53.85641 52.97438

(i) Oscarplans to invest$200eachmonthfor74months.His investmentwillearninterestattherateof0.0080(asadecimal)permonth.

Use the information in the table to calculate the present value of thisannuity.

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(ii) Lucyisusingthesametabletocalculatetheloanrepaymentforhercarloan.Herloanis$21 500andwillberepaidinequalmonthlyrepaymentsover6years.Theinterestrateonherloanis10.8%perannum.

Calculatetheamountofeachmonthlyrepayment,correcttothenearestdollar.

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(d) Claireisdrivingonamotorwayataspeedof110kilometresperhourandhastobrakesuddenly.Shehasareactiontimeof2secondsandabrakingdistanceof59.2metres.

Calculateherstoppingdistance.

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(e) FrompointS,whichis1.8mabovetheground,apulleyatP isusedtoliftaflat object F. The lengths SP and PF are 5.4m and 2.1m respectively. TheanglePSCis108°.

h

F

P

S108°

5.4 m

C

2.1 m

1.8 m

E

NOT TOSCALE

(i) ShowthatthelengthPCis6.197m,correctto3decimalplaces.

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(ii) Calculateh,theheightoftheobjectabovetheground.

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4

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Section II extra writing space

If you use this space, clearly indicate which question you are answering.

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0968310377OfficeUseOnly–DoNOTwriteanything,ormakeanymarksbelowthisline.

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