2015 HIGHER SCHOOL CERTIFICATE EXAMINATION

Mathematics

General Instructions

•Readingtime–5minutes

•Workingtime–3hours

•Writeusingblackpen

•Board-approvedcalculatorsmaybeused

•Atableofstandardintegralsisprovidedatthebackofthispaper

•InQuestions11–16,showrelevantmathematicalreasoningand/orcalculations

2590

Total marks – 100

Section I Pages 2–6

10 marks

•AttemptQuestions1–10

•Allowabout15minutesforthissection

Section II Pages7–17

90 marks

•AttemptQuestions11–16

•Allowabout2hoursand45minutesforthissection

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Section I

10 marksAttempt Questions 1–10Allow about 15 minutes for this section

Usethemultiple-choiceanswersheetforQuestions1–10.

1 Whatis0.005 233 59writteninscientificnotation,correctto4significantfigures?

(A) 5.2336×10–2

(B) 5.234×10–2

(C) 5.2336×10–3

(D) 5.234×10–3

2 Whatistheslopeofthelinewithequation 2x −4y +3=0?

(A) − 2

1(B) −

2

1(C)

2

(D) 2

3 Thefirstthreetermsofanarithmeticseriesare3,7and11.

Whatisthe15thtermofthisseries?

(A) 59

(B) 63

(C) 465

(D) 495

–3–

54 TheprobabilitythatMel’ssoccerteamwinsthisweekendis .

72

TheprobabilitythatMel’srugbyleagueteamwinsthisweekendis .3

Whatistheprobabilitythatneitherteamwinsthisweekend?

2(A)

21

10(B)

21

13(C)

21

19(D)

21

5 Usingthetrapezoidalrulewith4subintervals,whichexpressiongivestheapproximateareaunderthecurve y = xe x between x = 1 and x = 3?

1(A) (e1 + 6e1.5 + 4e2 +10e2.5 + 3e3

4)

1(B) (e1 + 3e1.5 + 4e2 + 5e2.5 + 3e3

4)

1(C) (e1 + 6e1.5 + 4e2 +10e2.5 + 3e3

2)

1(D) (e1 + 3e1.5 + 4e2 + 5e2.5 + 3e3

2)

–4–

6 Whatisthevalueofthederivativeof y =2sin3x–3tanx at x =0?

(A) –1

(B) 0

(C) 3

(D) –9

7 The diagram shows the parabola y =4x − x2 meeting the line y = 2x at (0,0) and(2,4).

(4, 0)

(2, 4)

y = 2x

y = 4x – x2

(0, 0)

y

x

Whichexpressiongivestheareaoftheshadedregionboundedbytheparabolaandtheline?

⌠ 2

(A) x2⎮ − 2x dx⌡0

⌠ 2

(B) 2x − x2⎮ dx⌡0

⌠ 4

(C) x2⎮ − 2x dx⌡0

⌠ 4

(D) 2x − x2⎮ dx⌡0

– 5 –

8 Thediagramshowsthegraphof y = e x (1+ x).

−1 O x

1

y

−2

Howmanysolutionsaretheretotheequation e x (1+ x) =1− x 2?

(A) 0

(B) 1

(C) 2

(D) 3

9 Aparticleismovingalongthex-axis.Thegraphshowsitsvelocityvmetrespersecondattimetseconds.

O

8

4 t

v

When t =0 thedisplacementxisequalto2metres.

Whatisthemaximumvalueofthedisplacementx ?

(A) 8m

(B) 14m

(C) 16m

(D) 18m

– 6 –

210 Thediagramshowstheareaunderthecurve y = from x =1 to x = d.x

1 d

y

xO

Whatvalueofdmakestheshadedareaequalto2?

(A) e

(B) e +1

(C) 2e

(D) e2

– 7 –

Section II

90 marksAttempt Questions 11–16Allow about 2 hours and 45 minutes for this section

Answereachquestionintheappropriatewritingbooklet.Extrawritingbookletsareavailable.

In Questions 11–16, your responses should include relevant mathematical reasoning and/orcalculations.

Question 11 (15marks)UsetheQuestion11WritingBooklet.

(a) Simplify 4x − (8− 6x). 1

(b) Factorisefully 3x2–27. 2

8(c) Express witharationaldenominator. 2

2 + 7

1 1 1(d) Findthelimitingsumofthegeometricseries1− + − +! . 2

4 16 64

5(e) Differentiate (ex + x) . 2

(f) Differentiate y = (x + 4)lnx . 2

π⌠ 4

(g) Evaluate⎮ cos2x dx. 2⌡0

⌠ x(h) Find⎮ dx. 2

⌡ x2 − 3

–8–

Question 12 (15marks)UsetheQuestion12WritingBooklet.

(a) Findthesolutionsof 2sinq =1 for 0≤ q ≤ 2p. 2

(b) ThediagramshowstherhombusOABC.

ThediagonalfromthepointA(7,11)tothepointCliesonthelinel1.

Theotherdiagonal,fromtheoriginOtothepointB,liesonthelinel2which

hasequation y = − x

3.

O

C

BD

A(7,11)

x

y1

2

NOT TO SCALEy = −

x

3

(i) Showthattheequationofthelinel1is y =3x –10. 2

(ii) Thelinesl1andl2intersectatthepointD.

FindthecoordinatesofD.

2

Question 12 continues on page 9

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Question12(continued)

x2 + 3(c) Find ƒ ′(x) ,where ƒ (x) = .

x −12

(d) For what values of k does the quadratic equation x2–8x + k=0 have realroots?

2

x2 ⎛ 1⎞(e) Thediagramshowstheparabola y = withfocus S 0, .Atangenttothe2 ⎝ 2⎠

parabolaisdrawnat P 1,12

⎛ ⎞⎝ ⎠ .

O

y

x

y =x2

2

P 1,12

S 0,12

NOT TOSCALE

(i) FindtheequationofthetangentatthepointP. 2

(ii) Whatistheequationofthedirectrixoftheparabola? 1

(iii) ThetangentanddirectrixintersectatQ.

ShowthatQliesonthey-axis. 1

(iv) ShowthatrPQSisisosceles. 1

End of Question 12

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Question 13 (15marks)UsetheQuestion13WritingBooklet.

(a) ThediagramshowsrABCwithsides AB =6cm, BC =4cmand AC =8cm.

6 cm

4 cm8 cm

C

BA

NOT TO SCALE

7 (i) Showthat cosA = . 1

8

(ii) ByfindingtheexactvalueofsinA,determinetheexactvalueofthearea 2ofrABC.

(b) (i) Findthedomainandrangeforthefunction ƒ (x) = 9 − x2 . 2

(ii) Onanumberplane,shadetheregionwherethepoints(x,y)satisfyboth 2

oftheinequalities y ≤ 9 − x2 and y ≥ x.

(c) Considerthecurve y = x3 − x2 − x + 3.

(i) Findthestationarypointsanddeterminetheirnature. 4

⎛ 1 70 ⎞ (ii) Giventhatthepoint P , liesonthecurve,provethatthereisa⎝ 3 27⎠pointofinflexionatP.

2

(iii) Sketch the curve, labelling the stationary points, point of inflexionand y-intercept.

2

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Question 14 (15marks)UsetheQuestion14WritingBooklet.

(a) In a theme park ride, a chair is released from a height of 110metres andfalls vertically. Magnetic brakes are applied when the velocity of the chairreaches−37 metrespersecond.

110 m

Magnetic brakes NOT TOSCALE

Release point

Theheightof thechairat time t seconds isxmetres.Theaccelerationof thechairisgivenby x!! = −10 .Atthereleasepoint,t = 0,x =110and x! = 0 .

(i) Usingcalculus,showthat x = −5t2 +110. 2

(ii) Howfarhasthechairfallenwhenthemagneticbrakesareapplied? 2

Question 14 continues on page 12

–12–

Question14(continued)

(b) Weatherrecordsforatownsuggestthat:

5• ifaparticulardayiswet(W),theprobabilityofthenextdaybeingdryis

61

• ifaparticulardayisdry(D),theprobabilityofthenextdaybeingdryis .2

In a specific week Thursday is dry. The tree diagram shows the possibleoutcomesforthenextthreedays:Friday,SaturdayandSunday.

W

Friday Saturday Sunday

D

W

D

W

D

W

D

W

D

W

D

W

D

12

12

12

16

56

16

56

12

12

16

56

12

12

12

2 (i) ShowthattheprobabilityofSaturdaybeingdryis . 1

3

(ii) WhatistheprobabilityofbothSaturdayandSundaybeingwet? 2

(iii) WhatistheprobabilityofatleastoneofSaturdayandSundaybeingdry? 1

Question 14 continues on page 13

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Question14(continued)

(c) Sam borrows $100 000 to be repaid at a reducible interest rate of 0.6% permonth.Let $An be the amount owing at the endof nmonths and$M be themonthlyrepayment.

(i) Showthat A2 = 100 000(1.006)2 − M (1+1.006) . 1

⎛ (1.006)n −1⎞ (ii) Showthat A = 100 000(1.006)n − Mn ⎜⎝ 0.006 ⎟ . 2

⎠

(iii) Sammakesmonthlyrepaymentsof$780.

Showthataftermaking120monthlyrepayments theamountowing is$68 500tothenearest$100.

1

(iv) Immediately aftermaking the120th repayment,Sammakes aone-offpayment,reducingtheamountowingto$48 500.Theinterestrateandmonthlyrepaymentremainunchanged.

After how many more months will the amount owing be completelyrepaid?

3

End of Question 14

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Question 15 (15marks)UsetheQuestion15WritingBooklet.

(a) The amount of caffeine, C, in the human body decreases according to theequation

dC =−0.14C,

dt

whereCismeasuredinmgandtisthetimeinhours.

− dC (i) Show that C = Ae 0.14t is a solution to =−0.14C, where A is

dtaconstant.1

When t =0, thereare130mgofcaffeineinLee’sbody.

(ii) FindthevalueofA. 1

(iii) WhatistheamountofcaffeineinLee’sbodyafter7hours? 1

(iv) WhatisthetimetakenfortheamountofcaffeineinLee’sbodytohalve? 2

Question 15 continues on page 15

–15–

Question15(continued)

(b) ThediagramshowsrABCwhichhas a right angle atC.ThepointD is themidpointofthesideAC.ThepointEischosenonABsuchthatAE = ED.ThelinesegmentEDisproducedtomeetthelineBCatF.

NOT TOSCALE

C

A

D

E

BF

Copyortracethediagramintoyourwritingbooklet.

(i) ProvethatrACB issimilartorDCF. 2

(ii) ExplainwhyrEFBisisosceles. 1

(iii) ShowthatEB = 3AE. 2

(c) Waterisflowinginandoutofarockpool.Thevolumeofwaterinthepoolattimet hoursisV litres.Therateofchangeofthevolumeisgivenby

dV = 80sin(0.5t) .dt

Attime t = 0,thevolumeofwaterinthepoolis1200 litresandisincreasing.

(i) Afterwhattimedoesthevolumeofwaterfirststarttodecrease? 2

(ii) Findthevolumeofwaterinthepoolwhen t = 3. 2

(iii) Whatisthegreatestvolumeofwaterinthepool? 1

End of Question 15

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Question 16 (15marks)UsetheQuestion16WritingBooklet.

(a) The diagram shows the curve with equation y = x2 – 7x + 10. The curveintersectsthex-axisatpointsAandB.ThepointConthecurvehasthesamey-coordinateasthey-interceptofthecurve.

O A B

C

y

x

(i) Findthex-coordinatesofpointsAandB. 1

(ii) WritedownthecoordinatesofC. 1

⌠ 2

(iii) Evaluate x2⎮ ( − 7x +10)dx . 1⌡0

(iv) Hence,orotherwise,findtheareaoftheshadedregion. 2

Question 16 continues on page 17

–17–

Question16(continued)

(b) A bowl is formed by rotating the curve y =8loge (x –1) about the y-axisfor 0≤ y ≤6.

3

2

NOT TOSCALE

O

6

y

x

Findthevolumeofthebowl.Giveyouranswercorrectto1decimalplace.

(c) Thediagramshowsacylinderofradiusxandheighty inscribedinaconeofradiusRandheightH,whereRandHareconstants.

y

R

Hx

1 Thevolumeofaconeofradiusrandheighthis π r2h .

3

Thevolumeofacylinderofradiusrandheighthispr2h.

(i) Showthatthevolume,V,ofthecylindercanbewrittenasH

V = π x2 (R − x) .R

3

(ii) Byconsidering the inscribedcylinderofmaximumvolume,showthat4

thevolumeofanyinscribedcylinderdoesnotexceed ofthevolume9

ofthecone.

4

End of paper

BLANK PAGE

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BLANK PAGE

–19–

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STANDARD INTEGRALS

©2015BoardofStudies,TeachingandEducationalStandardsNSW

⌠ 1xn dx = xn+1

⎮ , n ≠ −1; x ≠ 0, if n < 0⌡ n +1

⌠ 1⎮ dx = ln x , x > 0⌡ x

⌠eax 1

⎮ dx = eax , a ≠ 0⌡ a

⌠ 1⎮ cosax dx = sinax , a ≠ 0⌡ a

⌠ 1⎮ sinax dx = − cosax , a ≠ 0⌡ a

⌠sec2 1

⎮ ax dx = tanax , a ≠ 0⌡ a

⌠ 1⎮ secax tanax dx = secax , a ≠ 0⌡ a

⌠ 1 1= tan−⎮ dx 1 x

, a ≠ 0⌡ 2 + a aa x2

⌠ 1 x⎮ dx = sin−1 , a > 0, − a < x < a⌡ a2 a− x2

⌠ 1⎮ dx = ln x + x2 − a2 , x > a > 0⌡ x2 − a2

( )

⌠ 1⎮ dx = ln(x + x2 + a2

⌡ x2 + a2)

NOTE : ln x = log xe , x > 0