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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