grade 12 answers

8/4/2019 Grade 12 Answers

1/100

SENIOR 4PRE-CALCULUS MATHEMATICS

ANSWERS TO CUMULATIVE EXERCISESA Supplement to

A Foundation for Implementation

2000Manitoba Education and Training


2/100

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.


3/100

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


4/100

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

Iv


5/100

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

Lee-Ila Bothe

Diane Cooley

Grant Moore

Katharine Tetlock

Wayne Watt

Murielle White

Coordinator

Project Manager

Publications Editor

Consultant

Project Leader

Desktop Publisher

Agassis S.D. No. 13St. James-Assiniboia S.D. No.2Duck Mountain S.D. No. 34Winnipeg S.D. No. 1River East S.D. No.9St. Vital S.D. No.6Brandon S.D. No. 40Pine Creek S.D. No. 30Swan Lake S.D. No. 2309Midland S.D. No. 25Winnipeg S.D. No. 1Assiniboine South S.D. No.3Transcona-Springfield S.D. No. 12St. James-Assiniboia S.D. No.2Portage la Prairie S.D. No. 24Fort Garry S.D. No.5

Curriculum UnitProgram Development BranchProduction Support UnitProgram Development BranchCurriculum UnitProgram Development BranchProduction Support UnitProgram Development BranchLearning Technologies UnitProgram Development BranchCurriculum UnitProgram Development BranchProduction Support UnitProgram Development Branch

v


6/100


Notes

vi


7/100

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

vII


8/100

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

viII


9/100


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

Ix


10/100


Notes

x


11/100


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

1


12/100

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

2


13/100

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


14/100


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, - ~)

4


15/100

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


16/100

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


17/100


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


18/100


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.

8


19/100


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


20/100

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


21/100


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


22/100

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


23/100


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


24/100

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

14


25/100


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


26/100


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


27/100


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


28/100


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


29/100


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


30/100


Exercise 10: Graphing Reciprocals114 a --


31/100


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


32/100

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


33/100

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


34/100


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


35/100


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


36/100


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


37/100


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


38/100

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.

28


39/100


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


40/100


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


41/100

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.


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

31


42/100

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


43/100

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


44/100


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.

34


45/100


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

35


46/100


Exercise 18: Double Angle Identities2 .J3 -614. 3


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 ,


47/100


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


48/100


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


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


49/100


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

39


50/100


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


Continued40,


51/100


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.

41


52/100


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


53/100


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


54/100


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

44


55/100


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


56/100


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

46


57/100


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


58/100


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


59/100


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


60/100


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


50


61/100

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)

51


62/100

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


63/100

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

53


64/100

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

54


65/100

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

55


66/100


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


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.

56


67/100


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

57


68/100


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

58


69/100


E-2Exercise 30: Permutations17.

~.28 -3.14-1

-2

18 AB= 21l'. 3

19.

44

-2

20. k = 4

59


70/100


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

60


71/100


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


72/100


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


73/100

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

63


74/100

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

64


75/100


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


76/100

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

66


77/100

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

67


78/100


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


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

68 .


79/100

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\ /\ /\/ \/ \/ \/ ,/--------',/ ,,,

69


80/100

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 ,


81/100

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


82/100


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

72


83/100


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


84/100


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

74


85/100

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


86/100


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"

76


87/100


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


88/100


89/100


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


90/100


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

80


91/100


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


92/100


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

82


93/100


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


94/100

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

84


95/100


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


96/100

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


16. x = 917. X2 + y2 - 8x - 8y + 7 = 018. 117a. 256x16 - 4 X ll + x6 b. 1120x4

19. 109824020. x = 2

86


97/100

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


98/100


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

. . . . . . . . .


99/100


100/100

grade 12 answers

Documents