2015 HIGHER SCHOOL CERTIFICATE EXAMINATION

2600

Mathematics Extension 1

General Instructions

•Readingtime–5minutes

•Workingtime–2hours

•Writeusingblackpen

•Board-approvedcalculatorsmaybeused

•Atableofstandardintegralsisprovidedatthebackofthispaper

•InQuestions11–14,showrelevantmathematicalreasoningand/orcalculations

Total marks – 70

Section I Pages 2–5

10 marks

•AttemptQuestions1–10

•Allowabout15minutesforthissection

Section II Pages6–13

60 marks

•AttemptQuestions11–14

•Allowabout1hourand45minutesforthissection

– 2 –

Section I

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

Usethemultiple-choiceanswersheetforQuestions1–10.

1 Whatistheremainderwhen x3 − 6x isdividedby x+3?

(A) −9

(B) 9

(C) x2 − 2x

(D) x2 −3x+3

dN2 Giventhat N =100+ 80ekt, whichexpressionisequalto ?

dt

(A) k (100– N)

(B) k (180– N)

(C) k (N –100)

(D) k (N –180)

3 TwosecantsfromthepointPintersectacircleasshowninthediagram.

3

6

4NOT TOSCALE

x P

Whatisthevalueofx?

(A) 2

(B) 5

(C) 7

(D) 8

–3–

4 Arowingteamconsistsof8rowersandacoxswain.

Therowersareselectedfrom12studentsinYear10.

Thecoxswainisselectedfrom4studentsinYear9.

Inhowmanywayscouldtheteambeselected?

(A) 12C8 + 4C1

(B) 12P8 + 4P1

(C) 12C × 4C8 1

(D) 12P8 × 4P1

5 Whataretheasymptotesof y = 3x

x +1( ) x + 2( )?

(A) y = 0, x = −1, x = −2

(B) y = 0, x =1, x = 2

(C) y =3, x = −1, x = −2

(D) y =3, x =1, x = 2

6 Whatisthedomainofthefunction â (x) =sin−1 (2x)?

(A) −p ≤ x ≤ p

(B) −2 ≤ x ≤ 2

p p(C) − ≤ x ≤

4 4

1 1(D) − ≤ x ≤

2 2

–4–

7 Whatisthevalueofksuchthat1

4 − x2dx = π

30

k⌠

⌡⎮ ?

(A) 1

(B) 3

(C) 2

(D) 2 3

8 Whatisthevalueof limx→3

sin x − 3( )x − 3( ) x + 2( )

?

(A) 0

1(B)

5

(C) 5

(D) Undefined

9 Twoparticlesoscillatehorizontally.Thedisplacementofthefirstisgivenby x =3sin4t andthedisplacementofthesecondisgivenby x = asinnt. Inoneoscillation,thesecondparticlecoverstwicethedistanceofthefirstparticle,butinhalfthetime.

Whatarethevaluesofaandn?

(A) a =1.5, n = 2

(B) a =1.5, n = 8

(C) a =6, n = 2

(D) a =6, n = 8

– 5 –

10 Thegraphofthefunction y = cos 2t − π3

⎛⎝

⎞⎠ isshownbelow.

y

tPO

WhatarethecoordinatesofthepointP?

(A)5π12

, 0⎛⎝

⎞⎠

(B)2π3

, 0⎛⎝

⎞⎠

(C)11π12

, 0⎛⎝

⎞⎠

(D)7π6

, 0⎛⎝

⎞⎠

– 6 –

Section II

60 marksAttempt Questions 11–14Allow about 1 hour and 45 minutes for this section

AnswereachquestioninaSEPARATEwritingbooklet.Extrawritingbookletsareavailable.

In Questions 11–14, your responses should include relevant mathematical reasoning and/orcalculations.

Question 11 (15marks)UseaSEPARATEwritingbooklet.

(a) Find sin2x dx⌠

⌡⎮ . 2

(b) Calculatethesizeoftheacuteanglebetweenthelines y = 2x+5 and y=4–3x. 2

(c) Solvetheinequality4

x + 3≥ 1. 3

(d) Express 5cosx −12sinx intheform Acos(x + a), where 0≤ a ≤ p

2. 2

(e) Usethesubstitution u = 2x–1 toevaluatex

(2x −1)2dx

1

2⌠

⌡⎮ . 3

(f) Considerthepolynomials P (x) = x3 − kx2 + 5x +12 and A(x) = x −3.

(i) GiventhatP (x)isdivisiblebyA(x),showthat k = 6. 1

(ii) FindallthezerosofP (x)whenk = 6. 2

– 7 –

Question 12 (15marks)UseaSEPARATEwritingbooklet.

(a) Inthediagram,thepointsA,B,CandDareonthecircumferenceofacircle,whosecentreOliesonBD.ThechordACintersectsthediameterBDatY.ThetangentatDpassesthroughthepointX.

Itisgiventhat CYB =100° and DCY =30°.

100°

NOT TOSCALE

OA

B

C D

YX

30°

Copyortracethediagramintoyourwritingbooklet.

(i) Whatisthesizeof ACB ? 1

(ii) Whatisthesizeof ADX ? 1

(iii) Find,givingreasons,thesizeof CAB. 2

Question 12 continues on page 8

– 8 –

Question12(continued)

(b) ThepointsP (2ap,ap2 )andQ (2aq,aq2 )lieontheparabola x2 =4ay.

TheequationofthechordPQisgivenby ( p + q)x – 2y – 2apq =0.(DoNOTprovethis.)

(i) ShowthatifPQisafocalchordthen pq = –1. 1

(ii) If PQ is a focal chord and P has coordinates (8a,16a), what are thecoordinatesofQintermsofa?

2

(c) Apersonwalks2000metresduenorthalonga roadfrompointA topointB.ThepointAisdueeastofamountainOM,whereMisthetopofthemountain.ThepointOisdirectlybelowpointMandisonthesamehorizontalplaneastheroad.TheheightofthemountainabovepointOishmetres.

FrompointA,theangleofelevationtothetopofthemountainis15°.

FrompointB,theangleofelevationtothetopofthemountainis13°.

15°

13°

NOT TOSCALEO

A

B

M

h m

N20

00 m

(i) Showthat OA = hcot15°.

1

(ii) Hence,findthevalueofh. 2

Question 12 continues on page 9

– 9 –

Question12(continued)

(d) Akitchenbenchisintheshapeofasegmentofacircle.Thesegmentisboundedby an arcof length200cmand a chordof length160cm.The radiusof thecircleisrcmandthechordsubtendsanangleqatthecentreO ofthecircle.

r cm r cm

200 cm

NOT TOSCALE

q

160 cm

O

(i) Showthat 1602 = 2r 2 (1−cosq). 1

(ii) Hence,orotherwise,showthat 8q 2 +25cosq − 25 =0. 2

(iii) Taking q1 = p as a first approximation to the value of q, use oneapplicationofNewton’smethodtofindasecondapproximationtothevalueofq.Giveyouranswercorrecttotwodecimalplaces.

2

End of Question 12

–10–

Question 13 (15marks)UseaSEPARATEwritingbooklet.

(a) A particle is moving along the x-axis in simple harmonic motion. Thedisplacementoftheparticleisxmetresanditsvelocityisvms–1.Theparabolabelowshowsv2asafunctionofx.

v2

11

O 3 7 x

(i) Forwhatvalue(s)ofxistheparticleatrest? 1

(ii) Whatisthemaximumspeedoftheparticle? 1

(iii) Thevelocityvoftheparticleisgivenbytheequation 3

v2 = n2 a2 − x − c( )2( ) wherea,candnarepositiveconstants.

Whatarethevaluesofa,candn?

Question 13 continues on page 11

–11–

Question13(continued)

(b) Considerthebinomialexpansion

2x + 13x

⎛⎝

⎞⎠

18

= a0x18 + a1x16 + a2x14 +!

wherea0,a1,a2,...areconstants.

(i) Findanexpressionfora2. 2

(ii) Findanexpressionforthetermindependentofx. 2

(c) Provebymathematicalinductionthatforallintegers n ≥ 1, 3

12!

+ 23!

+ 34!

+! + n

n +1( )!= 1 − 1

n +1( )!.

(d) Let â (x) =cos−1 (x) +cos−1 (−x), where −1≤ x ≤1.

(i) Byconsideringthederivativeofâ (x),provethatâ (x)isconstant. 2

(ii) Hencededucethat cos−1 (−x) = p − cos−1 (x). 1

End of Question 13

–12–

Question 14 (15marks)UseaSEPARATEwritingbooklet.

(a) AprojectileisfiredfromtheoriginOwithinitialvelocityVms−1atanangleq tothehorizontal.Theequationsofmotionaregivenby

x = Vtcosq, y = Vtsinq − 12

gt 2. (DoNOTprovethis.)

y

O

V

qx

(i) Showthatthehorizontalrangeoftheprojectileis V 2sin2q

g. 2

Aparticularprojectileisfiredsothat q = p

3.

(ii) Find the angle that this projectile makes with the horizontal when 2

t = 2V

3 g.

(iii) Statewhetherthisprojectileistravellingupwardsordownwardswhen 1

t = 2V

3 g.Justifyyouranswer.

Question 14 continues on page 13

–13–

Question14(continued)

(b) Aparticleismovinghorizontally.InitiallytheparticleisattheoriginOmovingwithvelocity1ms−1.

Theaccelerationoftheparticleisgivenby !!x = x −1,wherexisitsdisplacementattimet.

(i) Showthatthevelocityoftheparticleisgivenby !x = 1− x . 3

(ii) Findanexpressionforxasafunctionoft. 2

(iii) Findthelimitingpositionoftheparticle. 1

(c) TwoplayersAandBplayaseriesofgamesagainsteachothertogetaprize.Inanygame,eitheroftheplayersisequallylikelytowin.

Tobeginwith,thefirstplayerwhowinsatotalof5gamesgetstheprize.

(i) Explain why the probability of player A getting the prize in exactly

7gamesis 64

⎛⎝

⎞⎠

12

⎛⎝

⎞⎠

7

.

1

(ii) WriteanexpressionfortheprobabilityofplayerAgettingtheprizeinatmost7games.

1

(iii) Suppose now that the prize is given to the first player to win a totalof (n +1)games,wherenisapositiveinteger.

ByconsideringtheprobabilitythatAgetstheprize,provethat

2

n

n⎛⎝⎜

⎞⎠⎟

2n +n +1

n⎛⎝⎜

⎞⎠⎟

2n−1 +n + 2

n⎛⎝⎜

⎞⎠⎟

2n−2 +! +2n

n⎛⎝⎜

⎞⎠⎟= 22n .

End of paper

BLANK PAGE

–14–

BLANK PAGE

–15–

–16–

STANDARD INTEGRALS

©2015BoardofStudies,TeachingandEducationalStandardsNSW

xn dx = 1n +1

xn+1 , n ≠ −1; x ≠ 0, if n < 0⌠

⌡⎮

1x

dx = ln x , x > 0⌠

⌡⎮

eax dx = 1a

eax , a ≠ 0⌠

⌡⎮

cosax dx = 1a

sinax , a ≠ 0⌠

⌡⎮

sinax dx = − 1a

cosax , a ≠ 0⌠

⌡⎮

sec2 ax dx = 1a

tanax , a ≠ 0⌠

⌡⎮

secax tanax dx = 1a

secax , a ≠ 0⌠

⌡⎮

1

a2 + x2dx = 1

atan−1 x

a, a ≠ 0

⌠

⌡⎮

1

a2 − x2dx = sin−1 x

a, a > 0, − a < x < a

⌠

⌡⎮

1

x2 − a2dx = ln x + x2 − a2( ), x > a > 0

⌠

⌡⎮

1

x2 + a2dx = ln x + x2 + a2( )⌠

⌡⎮

NOTE : ln x = loge x , x > 0