WW HAESE & HARRIS PUBLICATIONS OO MMaatthheemmaattiiccss RR ffoorr tthhee iinntteerrnnaattiioonnaall ssttuuddeenntt KK MMaatthheemmaattiiccss HHLL ((CCoorree)) EE AAllssoo ssuuiittaabbllee ffoorr HHLL&& SSLLccoommbbiinneedd ccllaasssseess PPaauull UUrrbbaann DD JJoohhnn OOwweenn RRoobbeerrtt HHaaeessee SS SSaannddrraa HHaaeessee MMaarrkk BBrruuccee OO IInntteerrnnaattiioonnaall BBaaccccaallaauurreeaattee LL DDiipplloommaa PPrrooggrraammmmee UU TT Roger Dixon Valerie Frost II Robert Haese OO Michael Haese Sandra Haese NN SS Haese HarrisPublications & IBHL_WS MATHEMATICSFORTHEINTERNATIONALSTUDENT MathematicsHL(Core)–WORKEDSOLUTIONS InternationalBaccalaureateDiplomaProgramme RogerDixon B.Ed. ValerieFrost B.Sc.,Dip.Ed. RobertHaese B.Sc. MichaelHaese B.Sc.Hons.,Ph.D. SandraHaese B.Sc. Haese&HarrisPublications 3FrankCollopyCourt,AdelaideAirport, SA5950,AUSTRALIA Telephone: +618 83559444, Fax: +618 83559471 Email: [email protected] Web: www.haeseandharris.com.au NationalLibraryofAustraliaCardNumber&ISBN 1876543450 ©Haese&HarrisPublications2005 PublishedbyRaksarNomineesPtyLtd 3FrankCollopyCourt,AdelaideAirport, SA5950,AUSTRALIA FirstEdition 2005 CoverdesignbyPiotrPoturaj ComputersoftwarebyDavidPurton TypesetinAustraliabySusanHaese(RaksarNominees).TypesetinTimesRoman9/10\Qw_ Thetextbook,itsaccompanyingCDandthisbookoffullyworkedsolutionshavebeen developedindependentlyoftheInternationalBaccalaureateOrganization(IBO).These publicationsareinnowayconnectedwith,orendorsedby,theIBO. 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Disclaimer:Alltheinternetaddresses(URL’s)giveninthisbookwerevalidatthetimeofprinting. While the authors and publisher regret any inconvenience that changes of address may cause readers, no responsibility for any such changes can be accepted by either the authors or the publisher. IBSL_WS FOREWORD ThisbookgivesyoufullyworkedsolutionsforeveryquestionineachchapteroftheHaese&Harris Publications textbook Mathematics HL (Core) which is one of three textbooks in our series ‘Mathematics for the International Student’.The other two textbooks are Mathematics SL and MathematicalStudiesSL,andbooksoffullyworkedsolutionsareavailableforthosetextbooks also. Correctanswerscansometimesbeobtainedbydifferentmethods.Inthisbook,whereapplicable, eachworkedsolutionismodeledontheworkedexampleinthetextbook. Beawareofthelimitationsofcalculatorsandcomputermodellingpackages.Understandthatwhen yourcalculatorgivesananswerthatisdifferentfromtheansweryoufindinthebook,youhavenot necessarilymadeamistake,butthebookmaynotbewrongeither. WehavealistoferrataforMathematicsHL(Core)onourwebsite.Pleasecontactusifyouhave anyadditionstothislist. RLD VF RCH PMH SHH e-mail: [email protected] web: www.haeseandharris.com.au IBSL_WS TABLEOFCONTENTS BACKGROUNDKNOWLEDGE 5 Chapter 1 FUNCTIONS 33 Chapter 2 SEQUENCES&SERIES 48 Chapter 3 EXPONENTS 73 Chapter 4 LOGARITHMS 89 Chapter 5 NATURALLOGARITHMS 105 Chapter 6 GRAPHINGANDTRANSFORMINGFUNCTIONS 114 Chapter 7 QUADRATICEQUATIONSANDFUNCTIONS 128 Chapter 8 COMPLEXNUMBERSANDPOLYNOMIALS 171 Chapter 9 COUNTINGANDBINOMIALTHEOREM 218 Chapter 10 MATHEMATICALINDUCTION 232 BACKGROUNDKNOWLEDGE- TRIGONOMETRYWITHRIGHTANGLEDTRIANGLES 249 Chapter 11 THEUNITCIRCLEANDRADIANMEASURE 264 Chapter 12 NONRIGHTANGLEDTRIANGLETRIGONOMETRY 278 Chapter 13 PERIODICPHENOMENA 286 Chapter 14 MATRICES 323 Chapter 15 VECTORSIN2AND3DIMENSIONS 377 Chapter 16 COMPLEXNUMBERS 420 Chapter 17 LINESANDPLANESINSPACE 450 Chapter 18 DESCRIPTIVESTATISTICS 496 Chapter 19 PROBABILITY 517 Chapter 20 INTRODUCTIONTOCALCULUS 539 Chapter 21 DIFFERENTIALCALCULUS 546 Chapter 22 APPLICATIONSOFDIFFERENTIALCALCULUS 583 Chapter 23 DERIVATIVESOFEXPONENTIALANDLOGARITHMICFUNCTIONS 624 Chapter 24 DERIVATIVESOFCIRCULARFUNCTIONSANDRELATEDRATES 646 Chapter 25 INTEGRATION 672 Chapter 26 INTEGRATION(AREASANDOTHERAPPLICATIONS) 693 Chapter 27 CIRCULARFUNCTIONINTEGRATION 719 Chapter 28 VOLUMESOFREVOLUTION 731 Chapter 29 FURTHERINTEGRATIONANDDIFFERENTIALEQUATIONS 739 Chapter 30 STATISTICALDISTRIBUTIONS 761 IBSL_WS Background knowledge EXERCISEA p p p p p p p 1 a 3£ 5 b ( 3)2 c 2 2£ 2 d 3 2£2 2 p p p p p p p = 3£5 = 3£ 3 =2( 2£ 2) =(3£2)( 2£ 2) p = 15 =3 =2£2 =6£2 =4 =12 p p p p p 12 12 18 e 3 7£2 7 f p g p h p p p 2 6 3 =(3£2)( 7£ 7) r r r =6£7 = 12 = 12 = 18 2 6 3 =42 p p p = 6 = 2 = 6 p p p p p p p p 2 a 2 2+3 2 b 2 2¡3 2 c 5 5¡3 5 d 5 5+3 5 p p p p =(2+3) 2 =(2¡3) 2 =(5¡3) 5 =(5+3) 5 p p p p =5 2 =¡ 2 =2 5 =8 5 p p p p p p p p p e 3 5¡5 5 f 7 3+2 3 g 9 6¡12 6 h 2+ 2+ 2 p p p p =(3¡5) 5 =(7+2) 3 =(9¡12) 6 =3£ 2 p p p p =¡2 5 =9 3 =¡3 6 =3 2 p p p p 3 a 8 b 12 c 20 d 32 p p p p = 4£2 = 4£3 = 4£5 = 16£2 p p p p p p p p = 4£ 2 = 4£ 3 = 4£ 5 = 16£ 2 p p p p =2 2 =2 3 =2 5 =4 2 p p p p e 27 f 45 g 48 h 54 p p p p = 9£3 = 9£5 = 16£3 = 9£6 p p p p p p p p = 9£ 3 = 9£ 5 = 16£ 3 = 9£ 6 p p p p =3 3 =3 5 =4 3 =3 6 p p p p i 50 j 80 k 96 l 108 p p p p = 25£2 = 16£5 = 16£6 = 36£3 p p p p p p p p = 25£ 2 = 16£ 5 = 16£ 6 = 36£ 3 p p p p =5 2 =4 5 =4 6 =6 3 p p p p p p 4 a 4 3¡ 12 b 3 2+ 50 c 3 6+ 24 p p p p p p =4 3¡ 4£3 =3 2+ 25£2 =3 6+ 4£6 p p p p p p =4 3¡2£ 3 =3 2+5£ 2 =3 6+2£ 6 p p p p p p =4 3¡2 3 =3 2+5 2 =3 6+2 6 p p p =2 3 =8 2 =5 6 p p p p p p p d 2 27+2 12 e 75¡ 12 f 2+ 8¡ 32 p p p p p p p =2 9£3+2 4£3 = 25£3¡ 4£3 = 2+ 4£2¡ 16£2 p p p p p p p =6 3+4 3 =5 3¡2 3 = 2+2 2¡4 2 p p p =10 3 =3 3 =¡ 2 IBHL_WS 6 MathematicsHL–BACKGROUNDKNOWLEDGE 5 a p1 b p6 c p7 d p10 e p10 2 3 2 5 2 p p p p p = p1 £ p2 = p6 £ p3 = p7 £ p2 = p10 £ p5 = p10 £ p2 2 2 3 3 2 2 5 5 2 2 p p p p p = 2 = 6 3 = 7 2 = 10 5 = 10 2 2 3 2 5 2 p p p =2 3 =2 5 =5 2 p f p18 g p12 h p5 i p14 j 2p3 6 3 7 7 2 p p p p p p = p18 £ p6 = p12 £ p3 = p5 £ p7 = p14 £ p7 = 2p3 £ p2 6 6 3 3 7 7 7 7 2 2 p p p p p = 18 6 = 12 3 = 5 7 = 14 7 = 2 6 6 3 7 7 2 p p p p =3 6 =4 3 =2 7 = 6 EXERCISEB 1 a 259 b 259000 c 2:59 =2:59£102 =2:59000£105 =2:59£1 =2:59£102 =2:59£105 =2:59£100 d 0:259 e 0:000259 f 40:7 =02:59¥10 =00002:59¥104 =4:07£10 =2:59£10¡1 =2:59£10¡4 =4:07£101 g 4070 h 0:0407 i 407000 =4:070£103 =004:07¥102 =4:07000£105 =4:07£103 =4:07£10¡2 =4:07£105 j 407000000 k 0:0000407 =4:07000000£108 =000004:07¥105 =4:07£108 =4:07£10¡5 2 a 149500000000m b 0:0003mm c 0:001mm =1:49500000000£1011 =0003:£10¡4 =001:£10¡3 =1:495£1011 m =3£10¡4 mm =1£10¡3 mm d 15 milliondegrees e 300000times =15000000oC =3£100000 =1:5000000£107 oC =3£105 times =1:5£107 oC 3 a 4£103 b 5£102 c 2:1£103 =4£1000 =5£100 =2:100£103 =4000 =500 =2100 d 7:8£104 e 3:8£105 f 8:6£101 =7:8000£104 =3:80000£105 =8:6£10 =78000 =380000 =86 g 4:33£107 h 6£107 =4:3300000£107 =6£10000000 =60000000 =43300000 IBHL_WS MathematicsHL–BACKGROUNDKNOWLEDGE 7 4 a 4£10¡3 b 5£10¡2 c 2:1£10¡3 d 7:8£10¡4 =004:¥103 =05:¥102 =002:1¥103 =0007:8¥104 =0:004 =0:05 =0:0021 =0:00078 e 3:8£10¡5 f 8:6£10¡1 g 4:33£10¡7 h 6£10¡7 =00003:8¥105 =8:6¥101 =0000004:33¥107 =0000006:¥107 =0:000038 =0:86 =0:000000433 =0:0000006 5 a 9£10¡7 m b 6:130£109 people c 1£105 lightyears =0000009:¥107 =6:130000000£109 =1£100000 =0:0000009m =6130000000people =100000 lightyears d 1£10¡5 mm =00001:¥105 =0:00001 mm 6 a (3:42£105)£(4:8£104) b (6:42£10¡2)2 =(3:42£4:8)£(105£104) =(6:42)2£(10¡2)2 =16:416£109 =41:2164£10¡4 =1:6416£1010 =4:12164£10¡3 =1:64£1010 (2d.p.) =4:12£10¡3 (2d.p.) c 3:16£10¡10 d (9:8£10¡4)¥(7:2£10¡6) 6£107 3:16 10¡10 = 9:8£10¡4 = £ 7:2£10¡6 6 107 =0:52¹6£10¡17 = 9:8 £ 10¡4 =5:2¹6£10¡18 7:2 10¡6 =5:27£10¡18 (2d.p.) =1:36¹1£102 =1:36£102 (2d.p.) 1 e f (1:2£103)3 3:8£105 =(1:2)3£(103)3 1 = £10¡5 =1:728£109 3:8 =2:63£10¡6 (2d.p.) =1:73£109 (2d.p.) 7 a 1day=24 hours b 1week = 7days i.e., missiletravels 5400£24 = 7£24 hours =129600 = 168hours =1:296£105 i.e., missiletravels 5400£168 +1:30£105 km =907200 =9:072£105 c 2years = 2£365:25 days +9:07£105 km = 730:5days = 730:5£24 hours = 17532hours i.e., missiletravels 5400£17532 =94672800 =9:46728£107 +9:47£107 km IBHL_WS 8 MathematicsHL–BACKGROUNDKNOWLEDGE 8 a distance=speed£ time b distance=speed£time time=1minute=60 seconds time=1day = 24 hours so, lighttravels (3£108)£60 = 24£60£60seconds =180£108 = 86400seconds =1:80£1010 m = 8:64£104 seconds i.e., lighttravels (3£108)£(8:64£104) =3£8:64£1012 c distance=speed£ time =25:92£1012 time= 1year = 365:25days +2:59£1013 m = 365:25£8:64£104 sec ffrombg = 3155:76£104 + 3:16£107 sec i.e., lighttravels (3£108)£(3:156£107) =3£3:156£1015 =9:468£1015 +9:47£1015 m EXERCISEC 1 a fx: x>5g reads ‘thesetofallxsuchthatxisgreaterthan5’ b fx: x63g reads ‘thesetofallxsuchthatxislessthanorequalto3’ c fy: 0<y<6g reads ‘thesetofally suchthaty liesbetween0and6’ d fx: 26 x 6 4g reads ‘the set of all x such that x is greater than or equal to 2, but less thanorequalto4’ e ft: 1<t<5g reads ‘thesetofalltsuchthattliesbetween1and5’ f fn: n<2 or n>6g reads ‘theset ofalln suchthatn islessthan2orgreaterthanor equalto6’ 2 a fx: x>2g b fx: 1<x65g c fx: x6¡2 or x>3g d fx: x2Z, ¡16x63g e fx: x2Z, 06x65g f fx: x<0g 3 a b 2 3 4 5 6 7 8 9 10 (cid:2)(cid:3) (cid:2)(cid:4) (cid:2)(cid:5) (cid:6) (cid:5) (cid:4) (cid:3) c d (cid:2)(cid:7) (cid:4) (cid:2)(cid:3) (cid:2)(cid:4) (cid:2)(cid:5) (cid:6) (cid:5) (cid:4) e (cid:8) EXERCISED 1 a 3x+7x¡10 b 3x+7x¡x c 2x+3x+5y =10x¡10 =9x =5x+5y d 8¡6x¡2x e 7ab+5ba f 3x2+7x3 =8¡8x =7ab+5ab =3x2+7x3 =12ab i.e., cannotbesimplified 2 a 3(2x+5)+4(5+4x) b 6¡2(3x¡5) =6x+15+20+16x =6¡6x+10 =22x+35 =16¡6x IBHL_WS MathematicsHL–BACKGROUNDKNOWLEDGE 9 c 5(2a¡3b)¡6(a¡2b) d 3x(x2¡7x+3)¡(1¡2x¡5x2) =10a¡15b¡6a+12b =3x3¡21x2+9x¡1+2x+5x2 =4a¡3b =3x3¡16x2+11x¡1 3a2b3 p 3 a 2x(3x)2 b c 16x4 d (2a2)3£3a4 9ab4 p p =2x£9x2 = 16£ x4 =23£(a2)3£3a4 3£a£a£b£b£b p =18x3 = 3£3£a£b£b£b£b =4£ (x2)2 =8£a6£3a4 a =4x2 =24a10 = 3b EXERCISEE x 1 a 2x+5 = 25 b 3x¡7 > 11 c 5x+16 = 20 d ¡7 = 10 3 ) 2x = 20 ) 3x > 18 ) 5x = 4 x ) x = 10 ) x > 6 ) x = 4 ) 3 = 17 5 ) x = 51 3x¡2 e 6x+11 < 4x¡9 f = 8 g 1¡2x > 19 h 1x+1 = 2x¡2 5 2 3 )) 2xx << ¡¡2100 ) 3x¡2 = 40 ))¡22xx 6> ¡1818 ) 36x¡ 46x = ¡3 ) 3x = 42 ) ¡1x = ¡3 ) x 6 ¡9 6 ) x = 14 ) x = 18 2 3x i ¡ = 1(2x¡1) Multiplyingeachtermby ) 8¡9x = 6(2x¡1) 3 4 2 theLCDof12gives ) 8¡9x = 12x¡6 ) 14 = 21x i.e., x= 2 3 2 a x+2y = 9 ..... (1) b 2x+5y = 28 ..... (1) x¡y = 3 ..... (2) x¡2y = 2 ..... (2) Multiplying(2)by2gives Multiplying(2)by¡2gives x+2y = 9 2x+5y = 28 2x¡2y = 6 ¡2x+4y = ¡4 ) 3x = 15 faddingg ) 9y = 24 faddingg ) x = 5 ) y = 24 = 8 9 3 Substituting x=5 into(2)gives Substituting y= 8 into(2)gives 5¡y = 3 3 ) y = 2 x¡2(83) = 2 ) x¡ 136 = 2 andso x= 232 ) x=5 and y=2 ) x= 232 and y= 83 c 7x+2y = ¡4 ..... (1) d 5x¡4y = 27 ..... (1) 3x+4y = 14 ..... (2) 3x+2y = 9 ..... (2) Multiplying(1)by¡2gives Multiplying(2)by2gives ¡14x¡4y = 8 5x¡4y = 27 3x+4y = 14 6x+4y = 18 ) ¡11x = 22 faddingg ) 11x = 45 faddingg ) x = ¡2 ) x = 45 11 Substituting x=¡2 into(2)gives Substituting x= 45 into(1)gives 3(¡2)+4y = 14 11 ) ¡6+4y = 14 5(4151)¡4y = 27 ) 21215 ¡27=4y ) 4y = 20 and ) y=5 ) 4y = ¡72 and ) y=¡18 11 11 ) x=¡2 and y=5 ) x= 45 and y=¡18 11 11 IBHL_WS 10 MathematicsHL–BACKGROUNDKNOWLEDGE x y e x+2y = 5 ..... (1) f + = 5 ..... (1) 2 3 2x+4y = 1 ..... (2) x y + = 1 ..... (2) Multiplying(1)by¡2gives 3 4 ¡2x¡4y = ¡10 Multiplying(1)by18 and(2)by ¡24 gives 2x+4y = 1 9x+6y = 90 ..... (3) ) 0 = ¡9 faddingg ¡8x¡6y = ¡24 ) x = 66 faddingg whichisabsurd ) thereare nosolutions Substituting x=66 into(3)gives 9£66+6y = 90 ) 6y = 90¡594=¡504 ) y = ¡84 ) x=66 and y=¡84 EXERCISEF 1 a 5¡(¡11) b j5j¡j¡11j c j5¡(¡11)j =5+11 =5¡11 =j5+11j =16 =¡6 =j16j =16 ¯ ¯ ¯ ¯ d (¡2)2+11(¡2) e j¡6j¡j¡8j f j¡6¡(¡8)j =j4¡22j =6¡8 =j¡6+8j =j¡18j =¡2 =j2j =18 =2 2 a jaj = j¡2j b jbj = j3j c jajjbj = j¡2jj3j = 2 = 3 = 2£3 = 6 d jabj = j¡2£3j e ja¡bj = j¡2¡3j f jaj¡jbj = j¡2j¡j3j = j¡6j = j¡5j = 2¡3 = 6 = 5 = ¡1 g ja+bj = j¡2+3j h jaj+jbj = j¡2j+j3j i jaj2 = j¡2j2 = j1j = 2+3 = 22 = 1 = 5 = 4 ¯ ¯ ¯ ¯ ¯c¯ ¯¡4¯ jcj j¡4j 4 j a2 = (¡2)2=4 k ¯ ¯ = ¯ ¯=j2j=2 l = = =2 a ¡2 jaj j¡2j 2 3 a jxj = 3 b jxj = ¡5 c jxj = 0 ) x = §3 but jxj > 0 forallx ) x = 0 (propertyofmodulus) ) nosolution d jx¡1j = 3 e j3¡xj = 4 f jx+5j = ¡1 ) x¡1 = §3 ) 3¡x = §4 but jx+5j > 0 forallx ) x = 1§3 ) ¡x = ¡3§4 (propertyofmodulus) ) x = ¡2 or 4 ) x = 3¨4 ) nosolution ) x = ¡1 or 7 IBHL_WS