Mathematics HL Core Worked Solutions

792 Pages·2007·6.62 MB·English

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: [phone], Fax: [phone]
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.
Thisbookiscopyright.ExceptaspermittedbytheCopyrightAct(anyfairdealingforthepurposes
ofprivatestudy,research,criticismorreview),nopartofthispublicationmaybereproduced,stored
in a retrieval system, or transmitted in any form or by any means, electronic, mechanical,
photocopying,recordingorotherwise,withoutthepriorpermissionofthepublisher.Enquiriestobe
madetoHaese&HarrisPublications.
Copyingforeducationalpurposes:Wherecopiesofpartorthewholeofthebookaremadeunder
Part VB of the CopyrightAct, the law requires that the educational institution or the body that
administers it has given a remuneration notice to Copyright Agency Limited (CAL). For
information,contacttheCopyrightAgencyLimited.
Acknowledgements:Whileeveryattempthasbeenmadetotraceandacknowledgecopyright,the
authors and publishers apologise for any accidental infringement where copyright has proved
untraceable. Theywouldbepleasedtocometoasuitableagreementwiththerightfulowner.
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