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SENIOR 4PRE-CALCULUS MATHEMATICS
ANSWERS TO CUMULATIVE EXERCISESA Supplement to
A Foundation for Implementation
2000Manitoba Education and Training
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Manitoba Education and Training Cataloguing in PUblication Data510 Senior 4 pre-calculus mathematics. Answers to
cumulative exercises: a supplement to afoundation for implementation.ISBN 0-7711-2872-X1. Mathematics-Problems, exercises, etc.2. Calculus-Problems, exercises, etc. 3.Mathematics-Study and teaching (Secondary).4. Calculus-Study and teaching (Secondary).I. Manitoba. Dept. of Education and Training.
Copyright 2000, the Crown in Right ofManitoba as represented by the Ministerof Education and Training. Manitoba Education and Training, School ProgramsDivision, 1970Ness Avenue, Winnipeg, Manitoba, R3J OY9.Every effort has been made to acknowledge original sources and to comply withcopyright law. If cases are identified where this has not been done, please informManitoba Education and Training. Errors or omissions will be corrected in a futureedition. Sincere thanks to the authors and publishers who allowed their originalmaterial to be adapted or reproduced.
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
ACKNOWLEDGEMENTSManitoba Education and Training gratefully acknowledgesthe contributions ofthe followingindividuals in the development ofSenior 4 Pre-Calculus Mathematics Answers to CumulativeExercises: A Supplement toA Foundation for Implementation.Senior 2-4 Pre-Calculus Mathematics Revision CommitteeJohn BarsbyLynn BattenJoseph CombiadakisAbdou DaoudiDavid KlassenWilliam KorytowskiViviane LeonardJ ayesh ManiarDon NicholHilliard SawchukConnie ShaverKathy TetlockDon TrimBarb VuketsWayne WattAlan Wells
St. John's-RavenscourtDepartment ofMathematicsBureau de l'education francaiseNotre Dame CollegiateR.D. Parker CollegiateSisler HighCollege BeliveauOak Park HighWhitemouth SchoolRiversSilver Heights CollegiateSchool Programs DivisionDepartment ofMathematicsRiver East CollegiateSchool Programs DivisionWinnipeg
Independent SchoolsUniversity ofManitobaManitoba Education and TrainingMountain S.D. No 28Mystery Lake S.D. No. 2355Winnipeg S.D. No.1St. Boniface S.D. No.4Assiniboine South S.D. No.3Agassiz S.D. No. 13St. James-Assiniboia S.D. No.2Manitoba Education and TrainingUniversity ofManitobaRiver East S.D. No.9Manitoba Education and Training
Senior 4 Pre-Calculus Mathematics Cumulative Exercises CommitteeJohn BarsbyBarry BootheJudy BowesRuth DyckJohn EnnsLyn JannuskaDennis KasianDavid KlassenBill KorytowskiKelly LivesleyJoyce McCallumDon NicholChristine OttawaLarry RuppelGordon SmithKetri Wilkes
St. John's-RavenscourtFort Richmond CollegiateSt. John's-RavenscourtWestgate Mennonite CollegiateWestgate Mennonite CollegiateCrocus Plains RegionalRiver East CollegiateR.D. Parker CollegiateBalmoral HallUniversity ofWinnipegCollegiateMorden CollegiateWhitemouth SchoolSt. John's High SchoolRiver East CollegiateMacGregor CollegiateJohn Taylor Collegiate
Independent SchoolsFort Garry S.D. No.5Independent SchoolsIndependent SchoolsIndependent SchoolsBrandon S.D. No. 40River East S.D. No.9Mystery Lake S.D. No. 2355Independent SchoolsIndependent SchoolsWestern S.D. No. 47Agassiz S.D. No. 13Winnipeg S.D. No.1River East S.D. No.9Pine Creek S.D. No. 30St. James Assiniboia S.D. No.2
III
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
Senior 4 Pre-Calculus Mathematics Field Validation (Pilot) Project (1997-2000)Dave BarnesDarryl BohartBarry Boothe**Chris BoumfordJudy BowesPaul CarpenterPat ClarkKim DeamelDarren Deserranno*Ruth DyckJohn EnnsDean FavoniArlyn FilewichHarland GaringerJan GibsonEva GoldfeldChuck HatfieldJerry HildebrandErwin HildebrandtGreg J akubecLyn J annuskaBrian Johnston**Dennis KasianDoug KinneyDave KlassenPeter KulaDoug KutcherDon LepperKelly Livesley*Tanis MacdonaldBill MarshallJoyce McCallumPat McCreaRalph McKayPatty MillerLes MitchellMike Nantais
*Until June 1998
Neelin HighRiver East CollegiateFort Richmond CollegiateShaftesbury HighSt. John's-RavenscourtMorden CollegiateGordon Bell SchoolCarman CollegiateSwan Valley RegionalWestgate Mennonite CollegiateWestgate Mennonite CollegiateDakota CollegiateFort Richmond CollegiateHapnot CollegiateKelvin HighDaniel McIntyre CollegiateShaftesbury HighVincent Massey CollegiateRiver East CollegiateOak Park HighCrocus Plains RegionalWhitemouth SchoolRiver East CollegiateNeelin HighR.D. Parker CollegiateSpringfield CollegiateWhitemouth SchoolSanford CollegiateJohn Taylor CollegiateVincent Massey HighSwan Valley RegionalMorden CollegiatePortage CollegiateWindsor Park CollegiateR.D. Parker CollegiateFort Richmond CollegiateBoissevain School
**Until June 1999
Brandon S.D. No. 40River East S.D. No.9Fort Garry S.D. No.5Assiniboine South S.D. No.3Independent SchoolsWestern S.D. No. 47Winnipeg S.D. No.1Midland S.D. No. 25Swan Valley S.D. No. 35Independent SchoolsIndependent SchoolsSt. Vital S.D. No.6Fort Garry S.D. No.5Flin Flon S.D. No. 46Winnipeg S.D. No.1Winnipeg S.D. No.1Assiniboine South S.D. No.3Fort Garry S.D. No.5River East S.D. No.9Assiniboine South S.D. No.3Brandon S.D. No. 40Agassiz S.D. No. 13River East S.D. No.9Brandon S.D. No. 40Mystery Lake S.D. No. 2355Transcona-Springfield S.D. No. 12Agassiz S.D. No. 13Morris-Macdonald S.D. No. 19St. James-Assiniboia S.D. No.2Brandon S.D. No. 40Swan Valley S.D. No. 35Western S.D. No. 47Portage la Prairie S.D. No. 24St. Boniface S.D. No.4Mystery Lake S.D. No. 2355Fort Garry S.D. No.5Turtle Mountain S.D. No. 44
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Answers to Cumulative Exercl s Senior 4 Pre-Calculus Mathematics
Don Nichol*Joe Oczerklewicz**Rob PehuraNorm RosemanLarry RuppelJohn ScottMark SeftonGordon SmithEdith SteevesWanda StockfordJim StuparykHans TauchnitzEgon Ulrich**Ketri WilkesRick WilliamsonFrank Wilson
*Until June 1998
Whitemouth SchoolJohn Taylor CollegiateWinnipegosis CollegiateGrant Park HighRiver East CollegiateDakota CollegiateVincent Massey HighMacGregor CollegiateJoseph H. KerrMiami CollegiateKelvin HighOak Park HighSpringfield CollegiateJohn Taylor CollegiateArthur Meighen HighVincent Massey Collegiate
**Until June 1999Manitoba Education and Training StaffSchool Programs DivisionCarole Bilyk Consultant
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
Notes
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Answers to Cumulative Exercises SenIor 4 Pre-Calculus Mathematics
CONTENTSAcknowledgements ~HExercise New Topic Presented
1 Degree and Radian Measure 12 The Unit Circle 23 SpecialAngles and the TrigonometricFunctions 34 SolvingTrigonometricEquations on a SpecifiedInterval5 General Solution ofTrigonometricEquations 66 Graphing Circular Functions 77 Translations 98 Horizontal and Vertical Stretches 119 Symmetry, Reflections,and Inverses 1510 Graphing Reciprocals 1811 Graphing AbsoluteValues 2112 Practice with Transformations 2413 Transformations with Trig Functions 2714 Trigonometric Identities I 2915 Trigonometric Identities II 3116 Sum and Difference Identities I 3217 Sum and Difference Identities II 3418 DoubleAngle Identities 3519 Exponential Functions 3720 SolveExponential Equations 40
Outcome(s)A-IA-2A-3
5 A-4A-5A-6B-1B-2B-3B-4
B-5,B-6B-6B-7C-lC-lC-2C-2C-2D-lD-2
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Senior4 Pre-CalculusMathematics Answersto Cumulativeexercises
Exercise NewTopicPresented21 Logarithmic Functions 4222 Logarithmic Theorems I 4523 Logarithmic Theorems II 4724 Exponential and Logarithmic Equations I 4925 Exponential and Logarithmic Equations II 5126 Natural Logarithms 5227 Applications of the Exponential Function 5428 Counting Principles 5629 Permutation with Repetitions and Restrictions30 Permutations 5831 Circular Permutations 6032 Permutations with Case Restrictions 6133 Combinations 6234 Binomial Theorem 6335 Permutations, Combinations, and BinomialTheorem 6436 Parabola 6537 Circle and Ellipse 6738 Hyperbola 6939 Sample Spaces 71
Outcome(s)D-3, D-4D-5D-5D-6D-6
D-7, D-8D-8E-1
57 E-2E-2E-2E-2E-3E-4
E-2, E-3, E-4F-1, F-2, F-3F-1, F-2, F-3F-1, F-2, F-3G-1, G-2G-30 Probability of Independent and Dependent Events 73
41 Combining Probabilities 7542 Conditional Probability I 77
G-3G-4
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
Exercise NewTopicPresented43 Conditional Probability II 7944 Probability Using Permutations and Combinations 8145 Geometric Sequences 8346 GeometricSeries 8547 Infinite Geometric Series 8748 ReviewI 8949 ReviewII 9050 ReviewIII 92
Outcome(s)G-4G-5H-lH-2H-3
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Notes
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
Exercise 1: Degree and Radian MeasureI. 5n b. -2na. - -36 32. a. -210 b. 165
3. 7n 4. 5n- -12 188. a. (3; , 2n) b. [; , n)9. a. III, IV b. IV10. a. II, III b. IIIII. 32n-512. 85.913. 18.J314. x=2
115. X2
23nc . 9c. 150.9n5. 6. 7.514
16. a. x = -3 3-v '2 -7.J97b. x=--- 217. x = -1818. a. x(x + 3)(x- 1)19. (-6, 12) and (2, 4)
20. d-bx=--a+c
b. x = 0, x = 1,x = -3
A-1
1md. 6d. -56.3
157. 13
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Senior 4 Pre-calculus Mathematics Answers to Cumulative Exercises
A-2Exercise 2: The Unit Circle1. a. II2. a. 1500
3. a. 7n-64. a. I, II
5. a. ( : ' ! )
b. IV c. IV
5nb. 41mc . 6
d. Id. 1500
d. 2n-3d. IV
b. 3300 c. 1200
b. II, IV c. II
(-9 40)b --. 41' 41 c. ( - - J lO - 3 - J lO )10' 10 d. (0,-1)
-126. y=-13-157. cos8=- 17 8. tan 8 = -.Ji53
9. yes
10. n radians; 6003
11. 55.2 em"12. Students could explain the following:
division of unit circle values of each quadrantal angles how to determine if greater than one revolution approximate and/or exact values
13. AB = 5.315 or AB = 1.317 (two possible answers)
14. x = 3 15. k = 4 16.0A=M17. a. 8 b. 2.[5+818. (x - 2)(x + 2)(x2 + 4)
4 319. (3x-4)(4x-3) =0; x=-, x=-3 4 (9 ) 2 120. y = 10 x - 20 - 40
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Answers to Cumulative exercises Senior 4 Pre-Calculus Mathematics
A-3Exercise 3: Special Angles and the Trigonometric Functions1. 1 b . - -13 c. -1 d. 2 e. -1 f. -1. - --2 22 . - J 2 b . - -13 - -13 d. - J 2. - -- c. --2 2 33. -13 b . 3.J6 +2 - 13a. -3 64. a. Both sides reduce to - 13 . b . Both sides reduce to .!.3 2
6 b .71t {},does not existc.6
b . 1t I 1 1 t 41t d. 21t6' - c. - -6 3 3b . - J 2 - -13 d. 2c. --2 3
tt5. a.
51t 71t6. a. 4' 47. a. -1 328. (x, y) = (1, -2)9. a. 11.4 em b. 6.2 em"
11. 82.5
12 1t 1t 1t di. -, -, - ra lans24413. Yes, lines are parallel.
Continued3
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
A-3Exercise 3: Special Angles and the Trigonometric Functions
- - - / 214.4
15. sin ( J = 1, cos ( J = 0, tan ( J is undefined116. 2
17. 53
18. x = 3 or x = - 119. 1
20. H(x) = 2x + 1x-4x=4+1
11- - - -1-- - - - - ...= 21
x-intercept (- ~, 0)
y-intercept (0, - ~)
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Answers to Cumulative exercises Senior 4 Pre-Calculus MathematicsA-4
Exercise 4: Solving Trigonometric Equations on a Specified Interval
2. n 5n 3. 2n 5n 4. 2n 5n 5. 5n 7n0, -, -, n,2n 3- 3- 6-6 6 3 3 66. 2n 4n 7. 5n 8. 2n 4n3- - 3-3 4 39. 3--4
10. Both sides reduce to ~.2
II. 4n-312. . -~Slne= --4
13. P(27n)= (-1, 0) which is between Quadrants II and III. See full solutions fordifferent methods.
14. The answer is d. See full solutions for explanations.-315. cose=-5
4a . 517.-1
3b. 56.
18. B19. Domain: R Range:y ~-820. (2, 4)
5
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative exercises
A-5Exercise 5: General Solution of Trigonometric Equations
1. 8 = n + 2kn, 5n + 2kn, where k is an integer6 6n 2kn 5n 2kn ..2. a. 8 =-+--,-+--, where k ISan Integer9 3 9 3
b. ( J = 2n + 4kn, 4n +Akn; where k is an integer3 33. 8= kn; where k is an integer
4. 8= n + 2kn, 5n + 2kn, 7n + 2kn, 1m + 2kn, where k is an integer6 6 6 6
5. 8 = 3n + 2kn, 2kn, where k is an integer2
6. ( J = 2n + 2kn, 4n + 2kn, where k is an integer3 3
7. 8= n +2kn, 5n +2kn, where k is an integer6 68. 8= n+ 2kn, where k is an integer2
n 2n 4n 5n9. 8=-- - -3' 3' 3' 3 3n 7n -at -5n10. 8= 4' 4' 4'411. 8 = n 12 ( J _ n 5n 3n. -"6'6'2
{ -13}13. Range: y ~4 14. 51.73 revolutions 15. 18 em16. a. 3.40 b. 27.53 u2
17. $6691.13 7 7m18. Ji3 or ~ -bc-c19. a-b 20. x = 3, y = 4, Z = -56
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
A-6Exercise 6: Graphing Circular Functions1.
2. a. I b. 21 r c. {8 E 9t] d. {-I:5: y :5:I} e. kn; where k E I3. f(8)
-2
4. a. I b. 21 r c. te E 9t} d. {-I:5: f ( e) :5:1} 1 re. 2 + k1r, where k E I5. Identical amplitude, period, domain, and range. The curves would coincide ifthe cosine curve was shifted ; units to the right.
6.
Continued7
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
A-6Exercise 6: Graphing Circular Functions7. a. undefinedb. TCc. all real numbers excepts odd integral multiples of ;d. all real numberse. kT C where k E I
TC8. x = (2k + 1) " 2 ' where k E I
9. a. (-2TC -3TC)u (-TC TC)u (3TC 2TC), 2 2'2 2' b. (- 2TC, - TC ) U0, TC)
10. No solution.11. Insert on same axes y = sin x. Indicate coordinates ofpoints of intersection.- J 3y= -2
An . TC d 2TCswer IS x = - an -3 3
12. Insert the graph ofy = 2sin"0 + sin 0 - 2 and indicate zeroes.0= 0.896, 2.246
-.J2i13. cscO=-- 310TC14. 3 16. B ( . ! - - J 3 J2' 2
17. 30.36 km 18. fix) = -6x + 36 19. y = 9 , x = 9-J320. a. fi3) = 6,M3)) = 3 b. See full solutions for proof.
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
Exercise 7: Translations1. a.
2. a.
3. a.
4. a.
2
b .
2
2
2
. c.
8-1
c .
-2
-2 2
2
-2
b.
-2
c.
2
1
-2
b . c .
6
-2
b.
2
-2
1
Continued9
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises8-1
Exercise 7: Translations5 . 5 6. 5
4
2 7 r 2 7 r7 r -1-2 -2
7. Question 5:Range: {2s y s 4}Period: 2nAmplitude: 1
8.
Question 6:Range: {1::;y s3}Period: 2nAmplitude: 1
9. a. fix) = g(x) + 4b. g(x) = fix) - 4
5n Tt: 13n 15n 21n 23n10. ()= 12' 12' 12' 12' --r2' 12 11. ()= 2.8018, 0.339812. Domain: {x E 9\} Range: {Os y s 2}13. See full solution.14. Two possible answers are: (cos()+ l)(sin ()- 1) = 0
cos"()+ cos()= 0Other answers are possible.
115. y. 16. fix) = 6 " (x + 2)2(X - 3)(x - 4)1or fix) = 3"(x + 2)(x - 3)(x - 4)
x 17. x = 5 and x = 1118. 1219. a. y = 2x
b. y = - 2x + 1620. X2+ 8x + 1710
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
8-2Exercise 8: Horizontal and Vertical Stretches1. a.
2. a.
3. a.
-5 5
b.
b .
6.28
-5
-5
-5
c .
-5
-5 5
5
3
2
-2-3
Range: {-I s y s I}Period: 2nAmplitude: 1
c .
-5 5
32
-2
-3
Range: {-I s y s I}Period: nAmplitude: 1
-5
c .
5-5
b.
Range: {-3 ~y s 3}Period: 2nAmplitude: 3
d.
-1
-2
-3
Range: {O~y s 2}P . d 2neno: 3Amplitude: 1
Continued11
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Senior 4 Pre-Calculu8 Mathematics Answers to Cumulative Exercises
8-2Exercise 8: Horizontal and Vertical Stretches4. a. b. c.
-3 -2 -3
d.
-1 2
-1
5.Vertical stretch by a factor of 3.
6.3 Horizontal stretch by a factor of ~ .
-2 -1 2-1
-2
-3
7. Horizontal stretch by a factor of 3.
Continued12
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
8-2Exercise 8: Horizontal and Vertical Stretches8. . ..(-.1.5,.2)
9. a. fix) =g(2x) b. g(x) = f ( ~ x)10. 76n+ 2kn, l!n + 2kn, 32n+ 2kn, where k is an integer11. 2cos" 0 - .J 3 cos0 = 0
2n 4n12. 0 =3+ 2kn,3+ 2kn, n + 2kn, where k E I13.
14. I YIIIIIIIIIII/IiI
Domain: {x 1 x;/:- 3}Range: {y Iy $ ; - 11.65 and y ~ -O.34}Note: Values for range are approximate.
Continued13
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Senior 4 Pre-Calculu8 Mathematic. Answer. to Cumulative ExerclB-2
Exercise 8: Horizontal and Vertical Stretches15. (-2,4)
16. See full solutions. (Hint: What is the measure of the central angle?)17. 5 5 y 2 = x is a parabola, but
y =.J; is a semi-parabola.-5 5 10 15
y = - V x-5 -5
18. k =-3019. k = 7
20. a. 181 r 1 2b. 6 nr . !orc. 2
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
8-3Exercise 9: Symmetry, Reflections, and Inverses1. 2 . 2 3. 2
-1
-2 2-2
-1
4. a. even b. neither c. even d. odd e. even f. even5. a. Equations i., iii., and iv. are symmetric with respect to the y-axis.b. Replace y with - :y . If there is no change in x, then the graph is symmetricalwith the x-axis.
6. a. y = sin x is symmetrical to neither axis.b. y = cosx is symmetrical to the y-axis.7. a. y = -2x - 4 b. y = -2x + 48.
9. a . . b.. 1 1
Continued15
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
8-3Exercise 9: Symmetry, Reflections, and Inverses10. a. x =-3 x=3
~6
543 y=24 5 6
b . x=-3 x=3
y==2
11. () = 2.09 + 2k1t, 4.19 + 2k1t , where k is an integer.12. a. b .
-5 5-2 -1 1 2 4 5
-3
-5
c.
16Continued
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
B-3Exercise 9: Symmetry, Reflections, and Inverses13.
7t
14. e = 0.7298 + 2krc, 2.4118 + 2krc , where k is an integer.15. 3.J516. e = 0.34 or 2.8017. a.
18.
-1
b .
This is a semi-parabola.
2
19. Any parabola in the form y = a(x - l)(x - 3).20. y = -16(x - l)(x - 3)
17
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
8-4Exercise 10: Graphing Reciprocals1. a. 3
2
~ 3 4 5IIII
Domain: Reals, x :t 2Range: Reals, y:t 0Zeroes: None
2.-1
27 ~ -5 -4 -2 -1IIIIIIII- - - - - - - - - - r - - - - - ~IIIIIII
-2
-5
-7
3. 32
3 4
4'~
b.-- 3 -2 -1 3 4 5
-1
------~ - - - - - - - - - - - -
Domain: Reals, x :t 0Range: Reals, y:t-21Zeroes: 2 "
Domain: Reals, x:t-3Range: Reals, y:t-4
-11Zeroes: 4
Domain: {x E 9t I x :t 2}Range: ( - 0 0 , ~1)u(O, 0 0 )Zeroes: None
Domain: {x E 9t}Range: [~1 , 0)Zeroes: None
18Continued
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
B-4Exercise 10: Graphing Reciprocals5. a.
Domain: {x E ~}Range: {-I sy $ ; I}Zeroes: n 1C , n E I
6. \ ....1~
~. . . ,8. IV9. -2.J3
310. C11. a. 2700 b. 3265.86012. a = 0.4636, 3.605, f J = 2.419, 3.86413. J! .li !L "
1 ( 0 , 1 - 1 )1 11 11 1+ +
-3.14 .:.14 6.28Domain: {x E ~ IX::l; n 1C , n E I}Range: {y s -1 or y ~ 1}Zeroes: None
c. -487.0r d. -39600
Continued19
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises8-4
Exercise 10: Graphing Reciprocals114 a --
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
B-5, B-6Exercise 11: Graphing Absolute Values1.
Domain: RealsRange. y z 0
1Zeroes: 2 "
4.3
-3 -2 -1 1 2 3 4 5 6-1-2
Domain: RealsRange. y z 0Zeroes: 2
7 .
-1 1 2 3 4 5 6 7 8-1
Domain: x * 3Range:y > 0Zeroes: None
3
2 .2
2
3 .
-3 -2-1
Domain: RealsRangery z 0Zeroes: -1
5.
1 2
-5 -4 -3 -2 -1 1 2 3 4 5
Domain: RealsRange. y e 0Zeroes: 3
6. 4
-2
Domain: RealsRange.y e 0Zeroes: -2
8.
Domain: RealsRange: 0 ~y s 1Period: tt
-3
Domain: RealsRange:y s3
-3 -9Zeroes: - -2 ' 2
9.
Domain: RealsRange: -3 ~y ~ 0Period: n
Continued21
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises8-5,8-6
Exercise 11: Graphing Absolute Values10. a. b. c.
2
-3 -2 -1-1
Zeroes:-2 Zeroes: 011. Range: {-5 s y s I}
Period:4nAmplitude: 3
12. a. b.
Domain: -6 ~x ~ 1Range: -2 ~y s 2
c .
Domain: -4 ~x s 3Range: -1 ~y ~ 3
n 5n13. e = 1.0472, 5.2360 or e = 3'3
Domain: -2 ~x s ~Range: -2 ~y ~ 2
d.
-1 3
-1Domain: " " 2 ~x s 3Range: -2 ~y ~ 2
Zeroes: 1..J1:5
22Continued
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Answers to Cumulative Exercises Senior 4 Pre-Calculus MathematicsB-5, B-6
Exercise 11: Graphing Absolute Values14. a. fix) = X2 + 5
(Others possible)b. fix) = I x - 3 1 + 5 c. fix) = ~5x-3 +5
15. See solution.
16. 5 -14a=- c=--3' 317.
(x, y) lies on the strip between the linesx + y = 5 and x + y = 6
-2 -1-1
18. 219.
~2
~ -4 -2 2 4 6 8 10 12-2-4~-8
20. See full solutions for proof.
23
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises8-6
Exercise 12: Practice with Transformationsl. a. vertical stretch
b . reflect in x-axis2 . a. vertical shrink
b . vertical stretch and reflect in x-axis3. a. vertical shrink and reflect in x-axis
b . horizontal compression4. a. vertical stretch, shift one unit up
b . reflect in x-axis, shift six units up
5. a. shift one unit left, stretch verticallyb . shift two units right, stretch vertically6. a. vertical shrink, shift five units down
b . vertical shrink, shift four units up7. j . . . . . . ...... . . . . . . . .......
. ~1"
8. 43
2
-1 2 3
Continued24
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
8-6Exercise 12: Practice with Transformations9. a.
-2
Range: {-I SY S 1}P e r i o d : 21l"A m p l i t u d e : 1
10. a.2
-1
Range: {OSy S I}P e r i o d : 1l "
1 1 .
b.
Range: {-2 SY S 2}P e r i o d : 21l"A m p l i t u d e : 2b.
-1
Range: {OSY S I}P e r i o d : 1l "
+-~~--+---4---~--~---+--~x
12. (1.72, 5.28)
c.
Range: {-I SY S I}P e r i o d : 1 l "A m p l i t u d e : 1c .
-a14 3.14 6.28
Range: {2SY S 3}P e r i o d : 1 l "
Continued25
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
8-6Exercise 12: Practice with Transformations13. 1
Domain: {-3 s x s 4}Range: {-3 s y s -l}
515. y = 2+--x-I16. x=l: ~
246 8
17. C18. E19. n = 320. 34.91 dm"
14.
{-9 ~x s 12}{-2 ~y s 2}
26
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
B-7Exercise 13: Transformations with Trig Functions
4. Y = 3cos2x
5. Y = 2+2 S i n ( ~ X ) or y = 2cos [~ ( x - ; ) ] + 26. a.
20
5
5 10 15 20
7. Y= 13.9-9.7COS(:~(t-26))8. 18.59. 87 days10. 80.41
b. 12.5 hoursc. 5.54 hours
Continued27
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative ExercisesB-7
Exercise 13: Transformations with Trig Functions11. a. 2.8449 b. 3.2987 c. 3.7699 d. 6.143612.
14. a. 4 b. 5 c. 3 d. 2 e. 115. = 1~: 16. 432
-5 --4 -3 -21~
=2t 123456~4-
-2-3
17. a. See full solutions for proof.b . 75 -- /2
18. k = -16
19. y = 3, x = 12120. a. Infinitely many
b. y = x(x - 2),y = 5x(x - 2) and infinitely many others.
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
Exercise 14: Trigonometric Identities I1. 1a. sine
2. 1a. cose
1-sin" ec. sin'' e1-cos" ec. cos" eI- Ic. sin" e
a. are identicalb. See full solutions.
c. IV d. III
C-1
Continued
3. a. 1 b . 1
4. See full solutions for proof.5. See full solutions for proof.6. See full solutions for proof.7. See full solutions for proof.8. See full solutions for proof.
9. e = 47t' 57t'3 ' 310. e = 131.8, 228.211.
12. a. III b. IV-3-fi13. -- 7
29
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
C-1Exercise 14: Trigonometric Identities I
14. ( - 1 - . J 3 J2' 215. 0= 3n 7n2 ' 4
16. 0 = 5: + 2kn, 7: + 2kn, where k is an integer.
17. 0 = 3n 7n 1m 15n8 ' 8' 8 ' 8
18. 019. r-1(x) = (3 - x)x20. m = 6
30
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Answers to Cumulative Exercises .Senior 4 Pre-Calculus MathematicsC-1
Exercise 15: Trigonometric Identities II1. See full solutions for proof.3. See full solutions for proof.5. See full solutions for proof.7. See full solutions for proof.9. e =109.5, 250.510. e =2.7367, 5.878311. . e 5SIn =-1312.
2. See full solutions for proof.
4. See full solutions for proof.6. See full solutions for proof.8. See full solutions for proof.
e = 0.3398 + 2kn, 2.8018 + 2kn,where k is an integer
13. e = n + 2kn, 2n + 2kn, where k is an integer3 314. See full solutions for proof.15. a. 3.92699 b. 3.7699 c. 2.1817 d. 1.823616. a. 120 b. 150 c. 240 d. 13517. y = 918. C
20. r-l(5) = 134
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative ExercisesC-2
Exercise 16: Sum and Difference Identities I
. J 2 + - J 62. 4
t ( f 3 ) tan a +tan f 33. an a+ =----..!...-1- tanatanf3. J 2 + - J 6
4. a. 4. J 2 - - J 6
b. 4
t( f 3 ) 1- tan a tan f 35. a. co a+ =------=--tana+tanf36. -2 - -J3
. J 2 - - J 67. 4
8. a. 1023102564b . 1025
9.
a. identicalb. See full solutions for proof.
10. 4
c. -2 - -J3
b. tan(a _ f 3 ) = tan a - tan f 31+tan a tan f 3
c . 102564 d. 1
. ( 3 1 t ' )=sm t+2y = - cost
3 2Continued
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Answers to Cumulative Exercises Senior 4 Pre-Calculus MathematicsC-2
Exercise 16: Sum and Difference Identities I- - J8511. sec8=-- 7
12. 8 = 0.7227, 3.1416, 5.560513. See full solutions for proof.14. See full solutions for proof.15. See full solutions for proof.16. d = 1.498
-1 3
17.
18. -1x=O orx=- 319. a. O
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative ExercisesC-2
Exercise 17: Sum and Difference Identities II
(n a ) cos8 - - J 3 sin 81. cos - + u = ------3 2
2. tan(8 _ n) = - J 3 tan 8 -16 - J 3 + tan 8
3. sec(n + 8) = J24 cos8 - sin 8 4. See full solutions for proof.5. a. J2 b. 1- -2 26. a. (63 -16) b. Quadrant IV65' 657. See full solutions for proof. 8. See full solutions for proof.9. See full solutions for proof. 10. See full solutions for proof.11. See full solutions for proof.
6312. tan(a - f3 ) =-1613. Possible solutions:
a. sint = cos(t- ;) or cos(t+ 3 ; )b. sin t = sin(t + 2n) or sin(t - 4n)c. See full solutions for possible proof.
14. 8 = 3.66 + 2kn , 5.76 + 2kn , where k is an integer.
16. 8= 1.23, 1.57, 4.71, 5.0517. (sin x - cosx)(sin2 x + sin x cosx + cos"x)18. (tan x + cotxntan" x + cot"X - 1)19. See full solutions for proof.
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
C-2Exercise 18: Double Angle Identities1. See full solutions for proof.
2. See full solutions for proof.tan 28 = 2tan83. I-tan28
4. - -135. 0.816. See full solutions for proof.
7 . 28 24. a. SIn = 25 -7b. cos28 = -25
8. sin28 = 12M49
9. y = cos28y=~
For alternate method, see solutions.8 = 0.262, 2.880, 3.403, 6.021
10. See full solutions for proof.11. See full solutions for proof.12. See full solutions for proof.
- J 513. cot8=--2Continued
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative ExercisesC-2
Exercise 18: Double Angle Identities2 .J3 -614. 3
15. See full solutions for proof.
16. ( J = ; + k1C , where k is an integer.17. ( J = 0.3398 + 2k1C, 2.8018 + 2k1C, where k is an integer18. sin( ~ - x ) = cos x b. cos(~ - x ) = sin x
19. a. b. y = f l(X) is not a function, therefore,y = fix) is not one-to-one.
20. a. quadraticb. (4, 2)c. four units to the right and two units up
36 ,
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
0-1Exercise 19: Exponential Functions1. a. b. D: {x Ix E 9\}
R : {y Iy > O }c. The y-values increase.
-2 -1 2
f(x) = ( ; J- __ t-++-+ _b. Both pass through (0,1). Both havethe same domain and range. Bothhave the same horizontal asymptotes.
2. a.
-1
c. f ix) = 2%s an increasing function;f ix) = (~) % is a decreasing function.
3. a. b. D: {x I x E 9\}R : {y Iy > O }
c. x-intercepts: noney-intercept: 3
-3 -2 -1
4.
5. a. 2.5 b. 34.3
Continued37
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
0-1Exercise 19: Exponential Functions
14. a. 5 b . 24
3-2 -1 2 3
2 -1
6. a. 1.6 b. 2.17. a.
8. a.
4
9. See full solutions for proof.
10. f) = 0, 2.3005, 3.9827, 21r11.
-1 234
b. f ix) = 4"increasing functiony-intercept is 1R : {yIy > O }
g(x) = 4 -"decreasing functiony-intercept is 1R : {y Iy > O }
hex) =-4"decreasing functiony-intercept is -1R : {y Iy < O }
12. 13. 2
234-1
38
Continued
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
Exercise 19: Exponential Functions15. 5
0-1
16.
-- 3 -2 -1 2 3
4
-1 2 3 4
17. ( 2 3 3 ~ J32' 3218. o = ~ 5n4' 419.
1
20. Twopossible answers are:
f ( x) = 3sin( x - :)f ( x) = 3 cos(x _ 3 : )Other answers possible.
-1
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
0-2Exercise 20: Solve Exponential Equations1. x=5 2. x=3 3 x = 2.54. x = -3,-1 5. x =-3 6. x=67 . x = -0.5 8. 14 9. a. and c.x = 1510.
11. 2x-intercept: noney-intercept: !-2 -1 -1
12. x-intercept: noney-intercept: 3Domain: {x 1 x E 9t}Range: fy Iy > O}Horizontal Asymptote: y = 0
-1 234
-1
13. See full solutions for proof.
Continued40,
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
0-2Exercise 20: Solve Exponential Equations14. 6
42
-Q -4 -2 2 4 6-2
-4
15. odd16. A = -1 and B = 017.
18. 2-1
20.9.
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
0-3,0-4Exercise 21: Logarithmic Functionsl. a. log381= 4 b. log216= 4 c. logl( ~ ) = 2 d . log{~) =-3- 1642 . a. 2 4 = 16 b. 43 = 64 c. 10-2= 0.01 d . 5-1 = 1 .53. 2 b. 1 c. 6 d . 1. -24.
-2
5.
-1
-1
-2
6.
-2
5
Domain: (1, 00)Range: {y E 9t}y-intercept: nonex-intercept: 2Asymptote: x = 1
3 Domain: (0, 00)Range: {y E 9t}y-intercept: nonex-intercept: 1~5Asymptote: x = 0
2
1
31 1 2-1
42Continued
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
0-3,0-4Exercise 21: Logarithmic Functions7. 3
2Domain: (-00,3)Range: {y E 9t}y-intercept: 2
-2 -1 3 x-intercept: 0.7925-1 Asymptote: x = 3-2
8. 3 Domain: (1, 00)2
Range: {y E 9t}y-intercept: nonex-intercept: ~-2 -1
-1 Asymptote: x = 1-2
9. x = 2 10. 1 11. x =-1 12. x=3= -213. 3 Domain: {x 1 x E 9t}
Range: {y Iy > O }y-intercept: 1x-intercept: noneAsymptote: y = 0
-2 -1 2 3 4
14. 3
-2 -1 3
Domain: {x 1 x E 9t}Range: {y Iy < 2}y-intercept: ~Asymptote: y = 2
2
Continued43
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
0-3,0-4Exercise 21: Logarithmic Functions15. 16. 3
2 3 4 5 6 7
2
-1
-1
17. 3
-1
-1
-2
18. 3
2 3 41-1
2n 4n19. e = 0, n, 2n, "3'320. e = 2.08, 4.20, 8.37, 10.48
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
0-5Exercise 22: Logarithmic Theorems I
1. a. 3
(X3- J b )b. log y2t4
13. a. log3x + " 2 log3Y b. 3logx + 4logy - 6log z
c. log,6 + log,X4. See full solutions for proof.5.
-1
y=logx
41
-2
26. " 5 log36 7. 1.195, 1.723 8. log8 + 3 loga + 4log b - 6log c - log 5
9. a. log, 1296 = 4 b. log5(_I_) =-312510. a. 82 = 6411.
Domain: (0,00)Range: {y E 9t}y-intercept: nonex-intercept: 1Asymptote: x = 0
Continued45
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
0-5Exercise 22: Logarithmic Theorems I12. a. x =-6 b. x = 913. -1x= -3 14.
-1x=- 2
15. The graphs are identical since !3X) = 3-1(3X) = 3x-1Domain: {y E 9t}Range: (0,00)x-intercept: noney-intercept: !16. See full solutions for proof. n 3n 5n 'lt:17. 8=0, 4' 4' 4' 4
18. a. 19. a.
b. Period = 2 b. Period = xc. Horizontal Shift = ~n
20. a. b.
-2 -1\-1t2
5
4 3 . + . , ~ , .
2
-1 4 5 6
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
0-5Exercise 23: Logarithmic Theorems II1. a. log2210 b. log58
b. log26
3. a. 0.7781 b. 1.1461 c. 0.669 d. 0.66074. See full solutions for proof.5. False6. 0.935785
17. log, 3 + log, b + 2"loga(c+ 1)-Iog, 4 - 2loga d
8. ( x't3 Jog, y2k49. 1x= -4
10. x =-3
11. log3(!) =-2
12. 25= 3213. { }14.
Continued47
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
Exercise 23: Logarithmic Theorems II15. a. b. 22
2
0-5
c .
-2 -1-1
16.
17. 141.34
18. fl(X) = x ; 319. B20.
48
-1
6
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
0-6Exercise 24: Exponential and Logarithmic Equations I1. a. x = 252. a. x=43. b. and d.4. a. x = 105. a. x = 256. x = -1,57. x = -3,68 . x=49. b.and c .10. -1x= -211. -3cote = 412. 2
-J -2
b. x = 5 1c. X= 8 "b. {x E~Ix> 0, x:l: I} c. x = 0
b. x = 729 c. x = 1b. x = 48 c. x = 2
Domain: (--00, 0)Range: {y E ~}y-intercept nonex-intercept -1Asymptote: x = 0
2
13. See full solutions for proof.14.
Continued49
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
0-6Exercise 24: Exponential and Logarithmic Equations I15.
-2 -1 2
-1
16.
-5 5 10
17. x = 32.J3
18. y = 3sin(2x -TC) or y = 3CO{2x+ ;)319. y = 2 + -x
20. See full solutions for proof.
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Answe rs to C umula tiv e E xe rc is es Senio r 4 P re -C alc ulu s Ma th ema tic s0-6
Exercise 25: Exponential and Logarithmic Equations II1. x = 5 2. x = 5
4. x = 2 5. x = 77. x = 2.15
3. x = 36. x = 1.95
8. x = -0.52 9. x = 1.44510. See full solutions for proof.11. x = 1012. -3x= -213. x =-414. x = 5415. x =-1
16. (65 4 . J146)81' 8117. 18.
2
19. 315.13
6
4
2
-2
-4
20. sin( 4;), sin( 5;), cos(5;), cos(7;), tan(2.428), tan(5.570)
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises0-7,0-8
Exercise 26: Natural Logarithms1. a. 221.41
5. 1a. 2 [31nx + In(x + 1)]
6. x = -329.58369 b. 1a. X= -37. a. 43.90 grams after 3 years.
2.
3 .
4 .
b. 0.512 c. 2.244 d. -5.627
-2 2 4
The graph of h(x) is between fix) and g(x)since 2 < e < 3.
6
Property fix) I!(x) h(x)Domain {x I x E 5 R } {x I x E 5 R } {x I x E 5 R }Range {y I y > O} {y I y > O} {y I y > -3}x-intercept none none In3 = = 1.1y-intercept 1 = = 0.05 -2Asymptote(s) y=O y=O y =-3
-2
-2
Property fix) jl(X)Domain {x I x E 5 R } {x I x> O}Range {Yly>O} {y lYE 5 R }x-intercept none 1y-intercept 1 noneAsymptote(s) y=O x=O
1b. In(x - 1) + 21n(x + 3) - "21n(x2+ 2)c. x = 8 d. x = 1.30472
b. x = 3.466 years
52
Continued
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Answers to Cumulative exercises Senior 4 Pre-Calculus Mathematics
0-7,0-8Exercise 26: Natural Logarithms8. t = 6.93 years9. n=l
910. log, 16-./2= -211. See full solutions for proof.
12. 7x= -213. x=214. x=815. x = 1116. log 4 + log(x - 5) - 3 log x - log(x + 6)1 7 . 6
3:14 6.28-1-2
18. 1.0341 r19. B = 2'A = 2, and C =-2
20. P = - ./ 2 - 1 and Q = - ./ 2 - - J 32 2
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative exercises0-8
Exercise 27: Applications of the Exponential Function
1. a. In[ ~ ]( x _ 1 ) 2 V x2 +1 b. In(x-1)
2 k = In 100. a. 2 -In 3b. k = 43. a. A = $5436.55 after 1 yearb. A = $11 547.99 after 10 yearsc. Interest earned after 10 years is $ 6547.99d. P = $13 160.18 is the initial investment.
4.After 20 years, there will be 673gophers. The gophers' population willdouble after 7.3 years.
60
303 6
5. t = 13.73 years
6. k= In (0.8)a. 57. t = 39.83 years8. H = 2.0 X 10-7
9. a.
b. y = 6.4 grams c. x = 15.5 years
b. No x-intercept. The y-intercept is 1.
2 4
-2Continued
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics0-8
Exercise 27: Applications of the Exponential Function10. See full solutions for proof.11. x = 312. x = 313. a. x = 0.321 b. x = 0.32114. 15.
2
2 6
-2
17. e = 0.2527, 2.889, 3.871, 5.554418. 5
19. D320. x = 0, 5, - 2
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
E-1Exercise 28: Counting Principles1. a. 72. a. k+33. n=54. a. 11, 111, V5. 4 ways7. 240 meals9. a. 6 ways10. 240 ways11. c .
b. 31 c. 504 d. 210b. 7(r + 2)(r + 1)r2
b. Answers will vary6. 1320 ways8. 60 ways
b. 6 ways c. 12 ways
12. See full solutions for proof.
13. 2x= -314. x = 4.5115. See full solutions for proof.16. e = 1.8235, 4.459717. T = 17.3 years18. See full solutions for proof.
19. a. k = 0.12945 b. P = 78.19 kPa20.
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
E-2Exercise 29: Permutations with Repetitions and Restrictions1. a. 3360
2. 693003. 6 ways4. a. 60 ways5. 17 576 000 plates6. a. 1257. 488. 729. a. 38410. 34 other routes
111. log 17 = --49 2
b. 1663200
b. 24 ways c. 24 ways d. 48 ways
b. 15
b. 1152
12. See full solutions for proof.113. 2
15. (0, 2) and (-2, 0)16. 8117. 0.918. e = 1.1988, 2.1404, 4.3403, 5.281919. 2 x4 - 5x3 + 5x - 220. a. 24 644 bacteria b. 4.39 hours
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
E-2Exercise 30: Permutations1. a. 202. a. n(n- 1)3. n=74. a. 720 ways5 . 30240 ways6. 288 ways7. 336 ways8. a. 129. a. 1680 "words"10. See full solutions.
b. 210b. n-2
b. 72 ways
b. 2b. 210 "words" c. 180 "words"
11. [0, 1]
12. 5-1= !13. See full solutions for proof.14. See full solutions for proof.15. fix): R: {y I y > OJ;y-intercept is 1
g(x): R: {y I y < OJ;y-intercept is -116. x = 4.11
Continued
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
E-2Exercise 30: Permutations17.
~.28 -3.14-1
-2
18 AB= 21l'. 3
19.
44
-2
20. k = 4
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
E-2Exercise 31: Circular Permutations1. a. 5040 ways b. 1440ways2. 2880 ways3. The history classroom has the greater number of seating arrangements.4. 3 bracelets7. a. 7208. 523 numbers9. a. 42010. 1440011. a. In a row12. See full solutions for proof.13. 2
14.3 -1
x= - and x = -2 215. 2
5. 21600 6. 420b. 220 c. 420
b. 160 c. 14
b. There are 21 600 more ways to sit in a row.
16. n = 19 17. n = 5
18. x = 3; + 2kn, where k is an integer.19. B 20. D
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
E-2Exercise 32: Permutations with Case RestrictionsI. 840 ways
2. 1036803. a. 14404. a. 288 ways5. a. 720 ways6. a. 840 ways7. 14400 ways8. 1509. 302400 ways10. a . 39 916 800 ways
II. .J 2-212. x=313. Quadrant III
14. 3a. x=-- 215. a. and c.16. t=217. 2n 4n8=0'3'3,2n18. 46.21 minutes
b. 960b. 144 waysb. 120 ways c. 600 waysb. 360 ways c. 240 ways
b. 518 400 ways c. 86 400 ways
1b. x = " 3
20. a. i = 5 cos (120m) b. i = -4.045 amperes61
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
E-3Exe rc is e 33: Combin ations1. a. 6 b. 20 d. 212. a. Both equal 28.b. Both equal 10.c. The answers, in each pair, are the same.d. The number ofways of choosing r objects from n is the same as the number
ofways of leaving behind the n - r remaining objects.3. a. n = 10 b. x = 44. 139838165. 18,3756. a. 2598960 hands b. 65 780 hands c. 158 184 hands7. 168. a. 36 ways b. 100ways9. 84 ways b. 39ways10. a. 84 ways b. 49ways11. n = 1512. 403,20013. See full solution for proof.14. 199584015. x=2
c. 24 waysc. 130 ways
16. 517. a. 151351200 b. 1081080018. 132
19. a. 4..[59 b .1-9
b . -7..[5--272220. a. 2762
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Answers to Cumulative Exercises Senior 4 Pre-Calculus MathematicsE 4
Exercise 34: Binomial Theorem
3. a. 512x9 + 2304x8 + 4608x74. -76545a55. 210a4bB
9. 9Cr(3af-r(6~~ J10. 372736x2O11. 450450 ways12. 12600 ways13. a. 80ways14. a. 13300
356. 4y
b. 1260waysb. 1596000
15. a. 0 ways or impossibleb. 32
16. See full solutions for proof.17. a. 666 863.3118. a. 253 games
19 ()= ~ 51t. 6' 620. a. $8635.70d. $ 8901.38
b. 1.83 daysb. 506 games
b. $8832.16e. $8902.16
b. 2048x22 - 11264x21 + 28160x20
7. 924
c. 2520ways
c. $8878.56
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative ExercisesE-2, E-3, E-4
Exercise 35: Permutations, Combinations, and Binomial Theorem1. 3276 committees2. a. 79 833 600 ways b. 3,628,800 ways c. 13 305 600 ways
4. 20C9 (x )11( -2y )98. a . 709. a . 2810. 132011. 12612. n=713.
5. 840x2 4956. 16 7. -42240x14
b. 56b. 49 c. 1
Domain: {x Ix E ~}Range: {y Iy > O }Horizontal Asymptote: y = 0x-intercept: noney-intercept: 1The graphs of fix) and g(x) are the same.
-1
-1
14. x = 6 15 I (9.3)(8.6). og% 19.1 16. 24017. 21018. 3619. a. 330 b. 46220. x = 0.739
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
F-1, F-2, F-3Exercise 36: Parabola1. a.
2. a.
b.
i=4x
c. d .
b. y .
2(y - 3) = 4(x + 5)
2(y +2) = -4(x -2)(2,-2)
Continued65
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative ExercisesF-1, F-2, F-3
Exercise 36: Parabola3. a. y
2(x -2) = -8(y + 1)
Vertex (2, -1)4.
-1 2(y +1) = 20x
Vertex (0, -1)56. a = 4 " 7. (3, - 2)
110. y + 3 = -7(x - 2)212. 231
11. 144013. 672
14. x =-7 15. 52= 25
16. x = 3 17 231. 1618. See full solutions for proof.
20 e Tt: 2k Ll.n 2k k . t. =-+ 'ft, --+ 'ft, IS an In eger.6 6
b . y.
-32(x +3) = 4y
Vertex (-3, 0)5. y .
Vertex (2, 0)
8. y = 2 9. x =-2
19. 7.79 years
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Answe rs to C umula tiv e E xe rc is es Sen io r 4 P re -C alc ulu s Ma th ema tic s
F-1, F-2, F-3Exercise 37: Circle and Ellipse
1. (x - 3? + (y + 1)2 = 4
3. (x - 2)2 + (y + 2)2 = 524. a. (x + 1)2 + (y - 5)2 = 1
C(-l, 5),r:1x
5. a. b.
6. a. (Centre (2.0) b .
-2
Centre (-3, 2)Continued
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
F-1, F-2, F-3Exercise 37: Circle and Ellipse7. a.
-2b . 10
-2
-s Centre (-1, 2)-8 (Centre (2, -5)-8
X2 y28. a. Center: (0, 0); Equation: 36 + " 9 = 1
9. See full solutions for proof.
10. (x - 3)2 (y _ 2)2 =149 2511. See full solutions for proof.13. See full solutions for proof.15. 360 numbers17. 1.7227119. B = 219.820. a. b.
. (X-2)2 (y_3)2b. Center: (2, 3); Equation: 36 + 4 = 1
12. n = 614. x = 0,-116. (x - 5)2+ (y - 5)2= 2518. 0.835
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Answers to Cumulative Exercises Senior 4 Pre-Calculus MathematicsF-1, F-2, F-3
Exercise 38: Hyperbola1. a.
2. a.
Centre (0, 0)... -2
-2
3. a. ,, 11,,, . .,, 12r',- ---" 10,,' , I .,,I , ,I ;L _,,,,,,
-1 -10,-8 -< I - - ' -a ".,, - , ,, ,, ,, - - ' ,4.
X2 y211 6 3 6
5. (y _ 1)2 (X - 2)2 125 49
b.
Continued
b. 2ll
8 12 18
Centre (~, -7)
-20
b. //\ /'---------(1\ /
1 \ / 11 \ / 11 \ / 11 \ / 1\ /\ /\/ \/ \/ \/ ,/--------',/ ,,,
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative ExercisesF-1, F-2, F-3
Exercise 38: Hyperbola6. a. ellipse b. hyperbolae. parabola f. ellipse
7 . a. ellipse b. hyperbola8. a. x
x2_y~2x2_y~1
x2_y~O.1
10. k = 1.611. 2012. x = 813. 104 = 1000014. See full solutions for proof.15. 290304016. See full solutions for proof.17. x = 15
18. a. 1.571, 3.142
2lx620. 2
c. parabola d. hyperbola
c. circle d. parabolab. y.
7 rb. 2' 7 r
70 ,
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Answers to Cumulative Exercises Senior 4 Pre-Calculus MathematicsG-1, G-2
Exercise 39: Sample Spaces1. a. H,Tb. 1, 2, 3, 4, 5, 6c. (H, H), (H, T), (T, H), (T, T)d. (1, 1), (1, 2), (1, 3), (1, 4) ,(1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4) ,(2, 5), (2, 6),(3, 1), (3, 2), (3, 3), (3, 4) ,(3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4) ,(4, 5), (4, 6),(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4) ,(6, 5), (6, 6)e. (H,H,H), (H,H,T), (H,T,H), (T,H,H), (T,T,T), (T,T,H), (T,H,T), (H,T,T)
2. a. (5, 7), (5, 8), (5, 9), (6, 7), (6, 8), (6, 9), (7, 7), (7, 8), (7, 9), (8, 7), (8, 8), (8, 9)b. (3)(4)= 12c. 2 points1d. 12
3. a. Independent b. Dependentc. Independent d. Independent
4. 1-P45. a. 5236. a. 10
b. 13 1- c . -52 52b. 7 1 (Certainty)c .10b. 10-36
67. a. 36
168. 529. See full solutions for proof.
Continued71
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
G-1, G-2Exercise 39: Sample Spaces10. x = 211. x = 7.83812. 2.613
b. 120 c. 6th term d. 11 terms14. (y - 2)2 = 12(x - 3)
8
16
-8
-16
15. a. parabola with vertical axis of symmetryb. parabola with horizontal axis of symmetryc. hyperbolad. circlee. oblique linef. ellipseg. horizontal lineh. vertical line
16. a. 19.43 days b. 10.9 g17. a. 5.85 and 3.57 b. 2.08 and 5.2218. 1 330 560ways19. 8 = 0.8411+ 2kn; 2.3005 + 2kn; 3.9827 + 2kn; 5.4421 + 2ktr, where k is an
integer.20. $2439.35
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
G-3Exercise 40: Probability of Independent and Dependent Events1. 1a. -1692. 1-33. 1a. -44. 1a. -45. 5-96. 7-8
1b. 221
b. 5 13- c . -24 24b. 1-16
17. 6
59. 93110. 66
11. (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4),(4, 1), (4, 2), (4, 3), (4, 4)
12. Answers will vary.
13 !. 2314. 13
15. 3.14 units"
Continued73
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
G-3Exercise 40: Probability of Independent and Dependent Events16. x = 0, 617. {(x) /?(x)
Domain {x I x e ~} {x I x e ~}Range {y I y > O } {y I y > O }Horizontal y=O y=OAsymptote Increasing Increasing
x-intercept none noney-intercept 1 418. See full solutions for proof.
-252 94519. Middle term: ~; next term: 2x8320. 2
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Answers to Cumulative exercises Senior 4 Pre-Calculus MathematicsG-3
Exercise 41: Combining Probabilities1. 30-312. a . 0.28
3. 3-135. 6-55
7. 1-12139. 51
110. a. 8000000
11 (y_1)2 _ (x+ 4)2 = 1. 4 9
12. independent
13. 1a. 2
b . 0.05292
4. 1-6
6. 7-138. 1-
4
7880599b. 8000000
, /, /, /, /, /", /r-,-- -- /fI', // II ' . . / II /, I'-r-+-~~-r-ty~~-~,~T-;--r-+-+-'x/ __ -4 __~
//////
3b. 26 1c. " 2
Continued75
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
G-3Exercise 41: Combining Probabilities14. rr 5rr
8="3' 3rr15. a . X2 + y 2 = 4
d. 3x2 + 2y 2 = 6g. x=3
16. x=l
17. See full solutions for proof.19. 1215--2x720. 112.5917
b. y = X2 - 2x + 3e. y =-rx f. y - 2=0h. 4x - 2y + 5 = 0 i. y = I x l
18. 98rr units? or 307.88 units"
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
Exercise 42: Conditional Probability I11 . 2772 . 16073. 12
4 P(R) = !!..P(W) = .L.P(B) = ~. 30' 10' 1545. 9
11 136. P(Even) = 18; P(< 3) = 1813 1 17. P(l) = 20; P(2) = 4; P(3) = 10
G-4
8. a. First Play
12different
Second Play Third Play
yJ (Jane wins)J~
KK (Kate wins)
J (neither wins)
1b. 49. a. 624
10. x = 2
5c . 8b. 54912
1d. 8
Continuedn
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
G-4Exercise 43: Conditional Probability II1. 3-42. 0.423 . a . 0.264. 0.68
5. 8-1 36. 3-87. 5-88. x=29.
b. 0.69
y = 3C o s ( x - ~)y.54
10. x = 5,-1211. 15
Continued79
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
G-4Exercise 43: Conditional Probability II12. 2-
513. 5-5214. 29-5015.
16. x = 517. 15n18.
Domain: {x I x - : : j : O}Range: {y I y E 9\}x-intercept: 1,-1y-intercept: noneAsymptote: x = 0
19. See full solutions for proof.20. .JiO < x < 100
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
G-5Exercise 44: Probability Using Permutations and Combinations1. 1.8 X 10-5 2. 1-63. 0.39 4. 5-265. 1 6. 1- -10 5047. 1 8. 0.0085
9. 2 10.2- -7 3
11. 2 b. 4 1a . - - c. -5 15 3d. 3 2- e. -5 3
12. 1-1013. 1-314. 7 b. 17a . - -10 2515. See full solutions for proof.
Continued81
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
G-SExercise 44: Probability Using Permutations and Combinations
y.16. (5,7)
(-1, -1)
(5,-9)
17. ()= 3.6470 + 2kn, 6.0305 + 2kn, where k is an integer.18. r = 9.13%19. 116 396 28020. 2.523658
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
H-1Exercise 45: Geometric Sequencesl. a. arithmetic, d = 2 b. geometric, r = 2 c. arithmetic, d = -3d. t . 1 neither f. geometric, r = . J 2eome rIC,r = 2 " e.g. geometric, r = -5
2. 3, 9, 27, 81: geometric3. a. 2,4, 8,... b. 1,4,16, ... c. 6, 1 8 , 54, ...
c. fix) = (_2)%-1d. 8,4,2, ...
4. a. fix) = 2.1:+1 b. f ix) = 2(W)
( 1 ) % ( 1 ) % - 1d. f(x) = 20 2 or f(x) = 10 25. a. 3846. a. $10 600, $11 236, $11 910.16
b. $10000(l.06)nc. 12 years
7. See full solutions for proof.8. 17
9. 1 Ck =-610. 5-1811. 17csc()=--812. a. circle b. hyperbola
d. line e. ellipse
13. 15 b. 15a. - -28 32
c. parabolaf. semi-parabola
Continued83
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative ExercisesH-1
Exercise 45: Geometric Sequences14. a. 0y+e
hastra~5 -ve0.001
b. 0.02093 c. 0.00095 d. 0.04539
0.999 0~02 +venot trait0.98 -ve
15. See full solutions for proof.16. x = 3
7r 5 7r17. ()= 3' 318. y = ~+1 - 419. True. (See full solutions for proof.)20. a. 12600 b. 4200
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Answers to Cumulative Exercises Senior 4 Pre-Calculus Mathematics
H-2Exercise 46: Geometric Series1. a. 2 + 4 + 6 + 8 + 10 = 30 b. 1 + 4 + 9 + 16 = 30
c. -2 + 0 + 2 + 4 + 6 + 8 = 18 d. 2 + 4 + 8 + 16 = 301 1 15e. 2 + 1 + - + - = -2 4 4
2. Both series are the same: 6 + 12 + 24 + 488b. I(4+2k)
k= l
Note: other expressions are possible.4. a. and c. are arithmetic; d. and e. are geometric.5. a. 384 b. 7656. a. 1000, 1050, 1102.5 b. 33065.957. a. 1023 b. 255.75 c. 906.64
d. 36 X2(X20 - 1)e.X2 -1
8. 995.819. 14.77
10. 4 b. 1 1a. - - c. -13 9 1611. a. 650 b. 140612. a., b., and c. are independent.
13. 3-2
Continued85
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative ExercisesH-2
Exercise 46: Geometric Series14.
Domain: (3,00)Range: 9\x-intercept: 4y-intercept: noneAsymptote: x = 3
5
15 20
15. See full solutions for proof.
16. x = 917. X2 + y2 - 8x - 8y + 7 = 018. 117a. 256x16 - 4 X ll + x6 b. 1120x4
19. 109824020. x = 2
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Answers to Cumulative Exercises Senior 4 Pre-Calculus MathematicsH-3
Exercise 47: Infinite Geometric SeriesI. 1 b. 7.875 7.9922 d . 8. - c.82. 12 b. 4 18 d . 2a. c. -33. a. 16 b. 3
4. 16-35. S~= 2~~ or 16(-12+2)
6. 1 0 - J 3 -J3S = -J3 or 20 3 + 30ee 2- 3
7. 27 b. 14metres. 32 metres8. a. 2184 b. 4334.059. 10 levels
10. See full solutions for proof.II. 4-1312. 1-12013.
-1
x-intercept: 0.415y-intercept: -!Domain: {x Ix E 9\}Range: {y Iy>-l}
-4 -3 -2 -1
-2
Continued87
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Senior 4 Pre-Calculus Mathematics Answers to Cumulative Exercises
Exercise 47: Infinite Geometric Series1 114. logsl "9 = - "2
15. See full solutions for proof.16. 3.19 x 10-1117. x = 1.5
H-3
19 (x + 3)2_ y 2 = 1. 9
20. x = 6
. . . . . . . . .
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