grade 12 cumulative exercises

SENIOR 4PRE-CALCULUS MATHEMATICS

CUMULATIVE EXERCISES

A Supplement toA Foundation for Implementation

2000Manitoba Education and Training

Manitoba Education and Training Cataloguing in Publication Data

510 Senior 4 pre-calculus mathematics. Cumulativeexercises: a supplement to a foundation forimplementation.

ISBN 0-7711-2870-3

1. 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 of Manitoba as represented by the Ministerof Education and Training. Manitoba Education and Training, School ProgramsDivision, 1970 Ness Avenue, Winnipeg, Manitoba, R3J OY9.

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Cumulative Exercises Senior 4 Pre-Calculus Mathematics

ACKNOWLEDGEMENTSManitoba Education and Training gratefully acknowledges the contributions of the followingindividuals in the development ofSenior 4 Pre-Calculus Mathematics Cumulative Exercises:A Supplement to A Foundation for Implementation (40S).

Senior 2-4 Pre-Calculus Mathematics Revision Committee

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Senior 4 Pre-Calculus Mathematics Cumulative Exercises

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v


Notes

vi


CONTENTS

Acknowledgements ui

Exercise New Topic Presented

1 Degree and Radian Measure 1

2 The Unit Circle 3

3 Special Angles and the Trigonometric Functions 5

4 Solving Trigonometric Equations on a Specified Interval

5 General Solution of Trigonometric Equations 9

6 Graphing Circular Functions 11

7 Translations 13

8 Horizontal and Vertical Stretches 16

9 Symmetry, Reflections, and Inverses 19

10 Graphing Reciprocals 22

11 Graphing Absolute Values 25

12 Practice with Transformations 27

13 Transformations with Trig Functions 30

14 Trigonometric Identities I 33

15 Trigonometric Identities II 35

16 Sum and Difference Identities I 37

17 Sum and Difference Identities II 39

18 DoubleAngle Identities 41

19 Exponential Functions 44

20 Solve Exponential Equations 46

Outcome(s)

A-I

A-2

A-3

7 A-4

A-5

A-6

B-1

B-2

B-3

B-4

B-5, B-6

B-6

B-7

C-l

C-l

C-2

C-2

C-2

D-l

D-2

vII


Outcome(s)

D-3, D-4

D-5

D-5

D-6

D-6

D-7, D-8

D-8

E-1

65 E-2

E-2

E-2

E-2

E-3

E-4

Exercise New Topic Presented

21 Logarithmic Functions 48

22 Logarithmic Theorems I 50

23 Logarithmic Theorems II 52

24 Exponential and Logarithmic Equations I 54

25 Exponential and Logarithmic Equations II 56

26 Natural Logarithms 58

27 Applications of the Exponential Function 60

28 Counting Principles 63

29 Permutation with Repetitions and Restrictions

30 Permutations 67

31 Circular Permutations 69

32 Permutations with Case Restrictions 71

33 Combinations 74

34 Binomial Theorem 76

35 Permutations, Combinations, and BinomialTheorem 78

36 Parabola 80

37 Circle and Ellipse 82

38 Hyperbola 84

39 Sample Spaces 87

E-2, E-3, E-4

F-1, F-2, F-3

F-1, F-2, F-3

F-1, F-2, F-3

G-1, G-2

40 Probability of Independent and Dependent Events 90 G-3

41 Combining Probabilities 92

42 Conditional Probability I 95

G-3

G-4

viii


Exercise

434445

46

47

48

49

50

New Topic Presented

Conditional Probability II 98

Probability Using Permutations and Combinations

Geometric Sequences 103

Geometric Series 106

Infinite Geometric Series 108

Review I 111

Review II 113

Review III 115

101

Outcome(s)

G-4

G-5

H-l

H-2

H-3

Ix


Notes

x


INTRODUCTION

These exercises provide experiences related to the student learning outcomes forSenior 4 Pre-Calculus Mathematics. These exercises have the following features:

• The exercises are cumulative. Each exercise starts with questions related tonew work, and then moves to questions based on previous topics. In the earlyexercises, the previous topics draw on material from Senior 2 Pre-CalculusMathematics and Senior 3 Pre-Calculus Mathematics.

• The cumulative nature of these exercises is designed to allow students tomaster concepts over a number of days, rather than all at once. In traditionallyarranged exercises, students may do 25 questions on a topic and then move on.In these exercises, students will do as many questions, but not all at once. Somemay feel they are moving on before they have mastered the previous topic.However, that mastery comes with time, and is reinforced frequently.

• This presentation of topics should also enhance retention since no topic everdisappears. It should also encourage students to see the connections among themany parts of Senior 4 Pre-Calculus Mathematics. Students find assessmentseasier with this cumulative approach, often requiring little formal review beforesummative assessment.

• As the course progresses, students frequently are asked to explain theirreasoning and to write simple proofs. Most exercises contain some problemsthat are aimed at a creative synthesis of ideas rather than simply a mastery ofskills.

• Schools on the semester system should spend about one day per exercise. Thisleaves a substantial number of days for assessment, group work, projects,mental mathematics, and other learning experiences. Schools not on thesemester system may require two class periods per exercise.

MaterialsScientific calculators are required throughout. Students should use spreadsheetsand graphing technology where appropriate.

Some parents may not approve of the use of playing cards and dice when teachinglessons in probability. Teachers may want to use number cards and number cubesinstead.

xl


Notes

xII


A-1

Exercise 1: Degree and Radian Measure

1. Convert the following to radian measure, expressing your answer in terms of n:a. 250 b. -1200 c. 4600 d. 3300

2. Convert the following values from radian measure to degree measure. Roundoff to one decimal place where necessary.

-7na.6

1mb.

12c. 2.634 d. -0.9825

3. Determine the supplement of 5n.12

4. Express the supplement of 1300 in radian measure. Leave your answer interms of n.

5. Find the complement of the third angle of an isosceles triangle where the two

equal angles each have a measure of 2n. Express in radians.7

6. If two parallel lines, intersected by a transversal, have interior alternateangles of 240 and n radians, find the value ofx.

x

7. Solve for x if the two angles referenced in the previous question were interiorangles on the same side of the transversal.

8. For what values of 0 in the interval 0 ~ 0 ~ 2n are the following conditions met?a. sin 0 < 0 and cos0 > 0 b. sin 0 > 0 and cos0 ~ 0c. tan 0 > 0 d. cos0 ~ 0

9. Determine the quadrant(s) in which the followingwould be true.a. sine < 0

b. cos0 = -J3 and tan 0 = --J23

10. In what quadrantts) is the point PCO) if we know thata. cos0 is negative?b. sin e < 0 and tan e > O?

Continued

1


A-1

Exercise 1: Degree and Radian Measure

11. Find the arc length of a bicyclewheel, given a radius of 24 inches and acentral angle of 48°.

12. An alien, power walking along the equator of his/her planet (gender is difficultto identify with aliens), travelled a distance of 12.3 balrogs. If the diameter ofthis planet is 16.4 balrogs, through what central angle did this alien travel?Express to the nearest tenth of a degree.

13. Simplify: 6.J12 + 2-J27.

14. Find an integral value ofx for which 3x-2 = 5x

-2•

1 1

15. Simplify: X3X6•

16. Solve the followingequations:

a. (x + 3)2 + 2 = 206 x-I

b.---=4x 2

17. A line has a slope of 2 and a y-intercept of 12. Find the x-intercept of this line.3

18. a. Completely factor x3 + 2X2 - 3x.b. Solve:x3 + 2X2 - 3x = o.

119. Find the points where the line x + y = 6 intersects the parabolay = - X2 + x.

2

20. Solve for x: ax + b = ex + d.

2


A-2

Exercise 2: The Unit Circle

1. Given that 0 ::;;e::;; 21l', determine the quadrant in which p(e) lies.

21l'a.

3Unb.6

c.-1l'

4Un

d.6

2. For the following, state a positive coterminal angle from 0°::;;e::;; 360°.a. 510° b. -390° c. 840° d. -210°

3. For the following, state a positive coterminal angle from 0 ::;;e::;; 21l'.

1911'_a.

61251l'

b.4

-131l'c.

6141l'

d.3

4. Determine the quadrant(s) in which the point P(8) will lie under the followingconditions:

c. sine> 0 and cose < 0

b. tan P « 0-3 4

d. sine =- and cose =-5 5

a. sin e is positive

5. Assuming that the point p(e) lies on the intersection of the unit circle and theline segment joining the origin to the point indicated, find the coordinates of P( 8).

a. (6, 8) b. (-9, 40) c. (-4, -12) d. (0, -3)

6. If (153'y ) is a point on the unit circle in quadrant IV, find the value ofy.

7. The point P( e) on the unit circle is not in Quadrant 1. If sin e = 187' find thevalue of cos e.

8. If cose =.J6 and sine = -M, find tan e. Rationalize the denominator and4 4

simplify.

9 I h . (.J5 2.J5J h .. I?. s t e pomt 5' -5- on t e unit CITC e.

10. If a wheel having a circumference of 30 metres rolls five metres, how manyradians has it turned? How many degrees has it turned?

Continued

3


A-2

Exercise 2: The Unit Circle

11. What is the area of a pastry used to cover a slice of mathberry pie having anouter crust arc of 7.6 em and a central angle of 30°?Express to the nearesttenth of a unit.

12. Write to explain in which quadrant the point P(e) lies if one knows the valueof (J in radians.

13. In Ll ABC, L B = 34°,BC = 4, and AC = 3. Find AB. How many answers arepossible?

14. If f(x) = ';x2 +7, find x so thatf(x) = 4.

15. If fix) = 2x + 3, find k so that fik + 2) = k + fik).

16. Let 0 be the origin and A the vertex ofy = X2 + 6x + 10. Find the length of OA.

The diagram shows the graph ofy = .§..7

17. x 6Vertical lines through C(6, 0) and D(2,0)

5meet this graph at B and A as shown.a. Find the area of trapezoid ABCD. 4

b. Find the perimeter of trapezoid ABCD. 3

2

1

1 3 4 5 7 8-1 D C

-2

18. Factor completely: X4 - 16.

19. Factor the following and solve for x: 12x2 - 25x + 12 = o.

20. Write the following equation in the form y = a(x - h)2 + k.(General form): y = 10x2 - 9x + 2. (Hint: Complete the square.)

4


A-3

Exercise 3: Special Angles and the Trigonometric Functions

1. Find the exact value for each of the following:

. na. Sln-6

b . 4n. Sln-3

c. cosn d. sec( -;)7ne. tan-4 (-5n)f. cot 4

2. Find the exact value for each of the following:a. cos450 b. sin 3000 c. tan 3300 d. csc 1350


a. (sin( -l:n ))-(cos 20n)-(tan l!n)b. (tan 2;Hcos(-!~))+(sin 3;)( tan 5:)

4. Show that the following statements are true:

a.

. n nSln- 1-cos-_------=3""-.. = 3

n . nl+cos- sm-3 3

~

-cosnb. sin n = 3

6 2

5. Given that tan e = is, find e where:

b. n ~ e ~ 3n2 2

3nc. - ~ e ~ 2n

2

6. Find the value(s) for e where 0 ~ e ~ 2n that satisfy the following:-J3

b. cose=-2

-1a. sine = -J2

c. tan e = -J3 and sin e < 0 -1d. cose = - and tan d < 02


25ncos--6

b . 9n. Sln-4

c. tan( -l:n ) (-35n)d. csc -6-

Continued

5


A-3

Exercise 3: Special Angles and the Trigonometric Functions

8. Solve for x and y: 4x - 3y = 103x + 2y =-1

9. The equal sides and equal angles of an isosceles triangle are 3.6 em and 54°respectively. Expressing your answer to the nearest tenth of a unit, finda. the perimeter. b. the area.

10. For what values of e in one positive revolution do both sin e and tan e havenegative values?

11. Express 1m radians to the nearest tenth of a degree.24

12. Express the size of the angles for a right isosceles triangle in radians as amultiple of n.

13. If the corrresponding angles for two lines intersected by a transversal

are 5n radians and 75°, determine if the lines are parallel.12

14. If sin e = ~ and cose < 0, find the valuers) for tan e.

15. If e = n + 2nn, where n E I, find the value for sine, coet), and tan e.2

16. Evaluate: tan 1~n - cos(-13m).

17. ABCDis a square and E is the midpoint of BC. Find, to the nearest degree,the measure of L AED.

18. Solve for x: .J2x + 3 -.Jx + 1= 1.

19. Find the perpendicular distance between the parallel lines 3x + 4y = 6and 3x + 4y = 1.

20. Sketch the graph of the followingrational function. State any intercepts andequations of asymptotes.

H(x) = 2x + 1x-4

6


A-4

Exercise 4: Solving Trigonometric Equations on a Specified Interval

1. Solve the following equations over the interval 0° ~ 8 ~ 360°.a. 2cos8= 2 b. 5tan8+ 4 = 0c. 4tan8 - 7 = 5tan 8 - 6 d. 2sin28 + sin 8 = 0

Solve the equations in questions 2-5 over the interval 0 ~ 8~ 2& Give theexact values.

2.2sin28-sin8=0 3. tan8+.J3 = 0

4. 2tan8+2.J3 = 0 5. 2cos8+.J3 = 0

Solve the equations in questions 6-8 over the interval ; s 8~ 321r.

6. 4cos28 = 1 7. 2 sin 8 + -J2 = 0

8. 2 cos" 8 - 5cos8 - 3 = 0

9 D . h I f h .. 2Jt' 7Jt' (-3Jt'). etermine t e exact va ue or t e expreSSIOn:SIn- .cos- .tan -- .3 6 4

10. Show that the following is true: 2cos2 Jt'-1 = cos" Jt'_ sin" Jt'.6 6 6

11 F· d 8 h . 8 -.J3 d Jt' 8 3Jt'. In were SIn = -- an - ~ ~ -.2 2 2

-312. If cos8 = - and tan 8 > 0, find the value for sin 8.4

13. Given P(27Jt')is a point on the unit circle, find the quadrant and the coordinatesof the point P using two different methods.

14. You have been given the following multiple choice question on a test: "Whichinterval satisfies the following condition: cos 8 > sin (J?"

tca. 0 ~ 8~ 2

Jt' Jt'b - < 8<-. 4 - 2Jt'c. 0 ~ 8~ -4

Jt'd. 0 ~ 8< -4

Jt'<8<Jt'e. 4 - - 2

Explain which answer is correct, which answers are incorrect, and why.

Continued

7


A-4

Exercise 4: Solving Trigonometric Equations on a Specified Interval

15. IfP(8) lies on the line segment joining the origin and the point (-6, -8) find cose.

16. Given the same information as in the previous question, find:

a. cos(e + ;) b. cos(e - 1[)

(-477r)17. Evaluate: sin -2- ·cos(-477r).

a. a > 0 and b2- 4ac > 0

b. a > 0 and b2 - 4ac < 0c. a < 0 and b2

- 4ac > 0d. a < 0 and b2 - 4ac < 0 1 2 3

18. Multiple Choice. The diagram on the rightshows the graph of y = ax2 + bx + c. Which ofthe following is true?

19. Find the domain and range of fix) = 3x2 + 6x - 5.

20. A line through (0, 2) intersects theparabola y = X2 at points A and B.If A is the point (-1, 1), find thecoordinates of B.

4

8


A-5

Exercise 5: General Solution of Trigonometric Equations

Solve the following equations for 8where the domain is the set of all realnumbers .

1. . 8 1SIn ::-2

1Sin(~) = ~2. a. cos(38):: - b.

2

3. tan 8 = 0

4. sin28 = ~4

5. (1 + sin 8)(1- cose) = 0

6. 2sec8 + 4:: 0

7. 4csc8 + 6:: 14

8. (sin 8 - 1)(2see8 + 1) :: 0

Solve the equations in questions 9 and 10 in the interval - 21£"$ 8 $ 211'.

9.4sin28-3=0

10. 5tan 8 + 5 = 0

Solve the equations in questions 11and 12 in the interval 0 $ 8 $ 211'.

11. cos8 + cos28 + sin2= 0

12. 2 sin'' 8 + sin 8 - 1 = 0

13. Find the range of the function: fix) = X2 + 3x - 1.

14. As you stop your car at a traffic light, a pebble becomes wedged between thetire treads. The diameter of the wheel is 80 em. When the light changes, youtravel a distance of 130 m before discovering the pebble (you have excellenthearing!). How many revolutions has the pebble made?

Continued

9


A-5

Exercise 5: General Solution of Trigonometric Equations

15. A ball of diameter 50 em is sitting on a ring of diameter 48 cm. How far belowthe ring does the ball extend?

16. A regular five-sided polygon (i.e., a pentagon) is inscribed in a circle. Eachedge of the pentagon is four units long.

a. Find the radius of the circle to two decimal places.b. Find the area of the pentagon to two decimal places.

17. If $5000 is invested for five years at 6% compounded annually, what will bethe value of the investment at the end of five years?

218. Find the distance from the point (-1, 3) to the liney = 3 x+6.

19. Solve for x: ax + c = bix - c).

20. Solve for x, y, and z.x+y=7x-z=82x - y - 2z = 12

10


A-6

Exercise 6: Graphing Circular Functions

. . . -3n -n -n n n 3n1. Graph the function y = sm e by lettmg e = ... -- -- -- 0 - - - ...4' 2' 4' '4'2' 4'

2. Analyze the graph of the sine function as sketched in question 1 by identifyingthe following:a. amplitude b. period c. domain d. range e. zeroes

3. Graph the function fie) = cose using the same intervals for e as used for thesine function.

4. Analyze the graph of the cosine function as sketched in question 3 byidentifying the following:a. amplitude b. period c. domain d. range e. zeroes

5. How do the graphs of the sine function and cosine function compare?

6. Graph the function fix) = tanx.

7. By referring to the graph in question 6, analyze the tangent function byidentifying the following:a. amplitude b. period c. domain d. range e. zeroes

8. Locate where the tangent function is undefined and state the equations of theasymptotes.

9. Over the interval (-2n, 2n), state where the followingwould be true:a. sin e is increasing b. cose is decreasing

10. Solve the equation sin''e + sin e - 4 = 0 for 0 ~ e ~ 2n.

11. Solve graphically the equation sinx = -J3 in the interval 0 ~ x ~ 2n. Stateanswers as exact values. 2

12. Use technology to solve the followingequation in the interval 0 ~ e < 2n:

2sin"e = -sin e + 2. State answers to three decimal places.

13. If see e = -J7 and tan e < 0, find the exact value of csce.2

Continued

11


A-6

Exercise 6: Graphing Circular Functions

14. Through how many radians does the minute hand of a clock rotate between10:00 a.m. and 11:40 a.m.? Give your answer as an exact value.

15. If R = 21.7° and 8 lies in Q III, find two possible values of 8.

16. In the unit circle shown, the length of arc AB is 231!.Find the coordinates of B.

A

17. One plane is 20 km from the WinnipegAirport. A larger plane is 17 km from theairport. If the angle between the sightings is 110°,how far apart are the planes?

18. Find a linear function of the form /tx) = mx + b such that /t3) = 18 and /t2) = 24.

19. Find exact values for x and y.

20.2x

Letf(x) = --.x-2

a. Find /t3) and ttft3».b. Prove that ttftx» = x for all values ofx.

12


8-1

Exercise 7: Translations

1. Given fix) = X2, sketch the graph of:

a. fix) b. fix) + 3 c. fix - 2)

2. Given fix) = x3, sketch the graph of:

a. fix) b. fix) - 2 c. fix + 1)

3. Given fix) = _X2 + 2, sketch the graph of:

a. fix) b. fix) - 3 c. fix - 3)

4. Given the graph offix) below,sketch the graph of:

a. fix) - 2b. fix + 2)c. fix + 1) + 3

1

5. Sketch the graph ofy = 3 + sinx.

6. Sketch the graph ofy = sin(x - ~) +2.

7. State the range, period, and amplitude of the graphs on questions 5 and 6.

8. Given the sketch ofg(x) = fix + 2) + 1, sketch fix):

y

.(-3,2).

·(2, -2)

Continued

13


8-1


9. Given the graphs of fix) and g(x):

a. Express fix) as a function ofgix).b. Express g(x) as a function of fix).

10. Solve the followingequation over [0, 2n]:

-.J2sin3e=--

2

11. Find the approximate values of the equation 3sin''e + 5sin e - 2 = 0for 0 s e s 21r.

12. fix) = cosx + 1. Without drawing its graph, give the domain and range of fix).

E lai h e sin e . d d . h fi b 113. xp am ow tan -2 = 1 e IS emonstrate In t e gure e ow.+ cos

.• ---- -- --~-----~-----,- ..•...•.--t- ----- --- ~A(-l,O) CO, 0)

t

14

Continued


8-1


14. Write a trigonometric equation that has the given roots n, ~ over the domain[0, Jr].

15. Sketch the graph of f(x) = (x - 1)2(x + 5).

16. Write an equation for a polynomial function with x-intercepts of -2, 3, and 4,and a y-intercept of 8.

17. A circle with radius 3 units has its centre at (8, 12). Write equations for twovertical lines tangent to the circles.

18. The function f(x) is periodic with period 4. If f(0) = 3, f(1) = 5, f(2) = 4, andf(3) = 12, what is the value of f(75)?

19. /).ABC is isosceles as shown.a. What is the equation of the line through A and C?b. What is the equation of the line through Band C?

y.

~~------------------------~--------~x

20. Given f(x) = (x + 2)2 - 3, find, in standard form, the quadratic function whichwould be obtained by sliding the above graph two units left and four units up.

15


8-2

Exercise 8: Horizontal and Vertical Stretches

1. Given fix) =X2,sketch the graph of:a. fix) b. 2fix) c. fi2x)

2. Given fix) =x3, sketch the graph of:

a. fix) 1c. - f(x)

3

3. Given fix) = cos x, sketch the graph of the following and state the range,period, and amplitude.a. fix) b. 3fix) c. fi2x) d. fi3x) + 1

4. Given fix) =x2 + 2x, sketch the graph of:a. fix) b. fi2x) c. 3fix) - 1 d. fi3x) + 1

Given the graph of fix) below, describe each transformation in words and sketcheach graph:

-3 -2 3

5. 3f(x)

6. f(2x)

7. f(;)-2

8. Given the sketch of

g(x) = f(~ xJ sketch f(x).(5.00, 1.00)

Continued

16


8-2


9. Given the graphs of fix) and g(x):

a. express fix) as a function of g(x)b. express g(x) as a function of fix)

10. If the domain is the set of real numbers, what are the solutions for theequation 2 sin'' e + 3sin e + 1 = O?

n n11. If the roots of the trigonometric equation are 2"' 6" on [0, 2n], find the

equation.

12. Solve the following equation, given that the domain is the set of all realnumbers: sin" e + cos" e + 2 cose = o.

13. Draw the graph ofy = cos(x - ~) on the interval 0 ~ x ~ 2n .

14. Sketch fix) = X2 -: and give the domain and range of the function.x+

{X+3 if x '#-2

15. Sketch the graph off{x) = .4 if x =-2

16. Given that the formula for the area of a circular sector is A = ~2, where e is

the central angle, explain why the area of a circle is given by nr2.

17. Sketch the graphs ofy2 = x andy =.JX. How do these graphs differ?

Continued

17


8-2


18. If x - 3 is a factor of X2 + 7x + k, what is the value of k?

19. The area bounded by the line y = 4 x, the x-axis, and the vertical line x = k is5

19.6 units". Find the value ofk.

y.

20. a. If the radius of the circle is 12 and e = 1C radians, find the area ofsector AOB. 4

b. If the radius is rand e = 1C radians, express the area in terms ofr.3

c. Prove that the area of a sector is given by A = ..! fJr2, where r is the radiusand e is the central angle in radians. 2

~::..----~B

18


8-3

Exercise 9: Symmetry, Reflections, and Inverses

Given the graph of fix), sketch the graph of:

1. -fix) 2

2. flex)1

3. fi-x)

-1 1 2 3 4

-1

-2

4. State whether each of the followingis even, odd, or neither.a. fix) = 3x2 b. fix) = -4x2 + 3x c. fix) = cosxd. fix) = -sin x e. fix) = 13xI f. fix) = 7

5. a. For each of the equations below, indicate whether the graph is symmetricwith respect to the y-axis.

i. y = X2 ii. x = y2 iii. X2 + y2 = 1 iv. X2 + X4 = Y

b. How would you test whether the above can be symmetric with respect tothe x-axis?

6. a. Is the graph ofy = sinx symmetric with respect to either coordinate axis?b. Is the graph ofy = cosx symmetric with respect to either coordinate axis?

7. a. Write an equation for the line formed by reflecting y = 2x + 4 in the x-axis.b. Write an equation for the line formed by reflecting y = 2x + 4 in the y-axis.

8. Given f (x) = -r; + 2, state f -1(x), sketch both graphs on the same coordinate axes.

Continued

19


8-3


9. a Reflect in the y-axis.y

b. Reflect in the x-axis.y

1

10. a. The given graph is part of an EVEN function. Complete the graph.b. The given graph is part of an ODD function. Complete the graph.

y ,5 :,,4 :,3 :

- 2' - - - - - - - - - -:-- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -,

1 :,,~~~~r++-++~~~~~~~++-+~X

S 4 5 6 7 8 9 10 11 12 13 14 15-1

-2-3-4

11. Solve over the real numbers to two decimal places: sec ()= -2.

12. a. Stretch horizontallyby a factor of 2

b. Stretch verticallyby a factor of 3

c. Stretch horizontally1by a factor of -2

-2

..(2, -3)

(-4,~) :

-5

20

Continued


B-3


13. Sketch the graph ofy = 3 sin(x + n) in the interval 0 $ x $ 27t.

14. Solve this equation for ()with the domain being all real numbers.

cos2(}+ sin2(}+ 3sin(} = 3

15. Find the distance from the origin to the line x - 2y + 15 = o.

16. Find the values of ()in the equation 6sin'' ()+ 13sin ()= 5 for 0 $ ()$ 2n.

17. The graph represents the unit circle X2 + y2 = l.This is equivalent to y2 = 1- X2 or y = ±-J 1- X2 •

Sketch the graph of:

b. y = --Jl-x2

18. Sketch the graph ofy = -J x - 2. What geometric figure does this represent?

19. This parabola opens down and crosses the x-axisat x = 1 and x = 3. Write down the equations oftwo different parabolas with this property.

20. If the parabola in question 19 has its vertex at (2, 16), find its equation.

21


8-4

Exercise 10: Graphing Reciprocals

1. For each of the following functions, sketch the graph and specify the domain,range, and zeroes.

1a. f(x)=-

x-21

b. f(x) = --2x

2. Sketch the graph and specify the domain, range, and zeroes of:

1f(x) = x+3-4


1f(x) = X2 _ 4


-1f(x) = X2 +4


a. f(x) = sin x b. f(x) = _.1_sm x

. 16. GIven the graph off(x) below, sketch the graph of f(x)'

7. Solve the equation for 8 where 0° ~ 8 ~ 360°:sin 28 = -0.5794

8. If cos8 > 0 and tan 8 < 0, in what quadrant does 8 lie?

Continued

22


B-4


9. If sin 0 =.! and tan 0 < 0, find sec O.2

10. Multiple Choice: The statement which describes the vertical asymptotes ofy = tan x is:

2a. y = odd integral multiples of n.

2b. x = integral multiples of n.

2c. x = odd integral multiples of n .

2d. y = odd integral multiples of tt,

3

11. Change the following radian measures into degree measures:

a. 3n2

b. 57 c. -8.5 d. -22n

12. Solve this system of equations for a and f3 whereo s a s 2 nand 2 tan a - 4cosf3 = 4o ~ f3 ~ 2 n tana + 2cosf3=-1

1Sketch the graph of fix) = X2 _ 4

14. For what values of k will the system fix) = 2 1 4 and g(x) = k have:x -

13.

a. no solution b. exactly one solution c. more than one solution

15. If 0 < 0 < ~ and sinO = ~, prove that tanO = -Jb2~a2 .

16. Sketch the graph ofx = ~1- y2. (Hint: Recall that x" + y2 = 1 is the unit circle.)

Continued

23


8-4


17. Write an equation for a parabola with vertex (3,-6) crossing the x-axis at-1and 7.

18. Solve for 0 s 0 ~ 2n: sinO+ 2sinOcosO= 0

19. Given each graph, perform the indicated transformations to sketch a new graph.a. reflect in the y-axis b. reflect in the x-axis

c. stretch horizontally by a factor of .!2

1

d. stretch vertically by a factor of .!2

20. If LA = 45°, find an exact value for the length of Be.

c

24


B-5, B-6

Exercise 11: Graphing Absolute Values

For questions 1 to 7, sketch the graph and state the domain, range, andzeroes of the function.

1. y= 12x-ll 2. y = I-x -11

14. y = -Ix-21

2

6. y = -21 x + 31 + 31

7. v= I Ix-3

For questions 8 and 9, sketch the graph and state the domain, range, andperiod.

8. y = I sinx I9. y=-3Icosxl

10. Given fix) = X2, sketch the graph and state the zeroes of:

a. fix + 2) b. - 2fix) c. -2fix - 1) + 3

11. State the range, period, and amplitude ofy = 3sin(~ x )-2.12. Given fix) below, sketch the graph and state the domain and range of:

a. y = fix + 3)

b. y = fi-2x)

c. y = fi -x + 1)

d. y = fi2x - 2)

Continued

25


B-5, B-6

Exercise 11: Graphing Absolute Values

13. Solve the followingequation over the interval 0 $ 0 $ 2n:

4sin2 0 - 8cos 0 =-1

14. Write the equations of three different functions that have a range of [5, 00).

o 1- cosO15. Find a geometric interpretation for tan -2 = sin f .

• ~ ~ .....I-.l..---I -- ~A(-l, 0) C(l,O)

16. The equation of a parabola is: y = ax2 + c. If the points (2, 2) and (1, -3) lie onthe parabola, find the values of a and c.

17. The value of x + y is between 5 and 6. On a coordinate graph, shade in theregion in which the point (x, y) must be located.

3 118. Solve for x: 2 +- = 1.x +x x

419. Sketch the graph ofy = 3--.

x

20. Prove that the graphs of x + y = 2 and y = X2 - 2x + 3 do not intersect.

26


8-6

Exercise 12: Practice with Transformations

For questions 1 through 6, assume that the graph of (is given, anddescribe how the graphs of the following functions are obtained.

l. a. y = 4fix) b. y = -fix)

1 b. y = -3fix)2. a. y=-f(x)4

1 b. y = fi2x)3. a. y=--f(x)5

4. a. y = 2fix) + 1 b. y = -fix) + 6

5. a. y = 4fix + 1) b. y = 3fix - 2)

1 16. a. y = - f(x)-5 b. y=-f(x)+4

2 3

-Jx-27. Sketch the graph of: f( x) = 2X2

4

if x ~ 3if 1::;; x < 3

if x < 0x

8. Sketch the graph ofy = x + Ix I.9. Sketch the graph, and state the range, period, and amplitude of:

a. fix) = sin x b. fix) = 2 sin x c. fix) = sin 2x

10. Sketch the graph, and state the range and period of:

a. fix) = Icosx I b. fix) = cos" X

11. Sketch the graph of:fix) = l:~::::~~~~~l)(x_n)3 on [n, 00)

Continued

27


8-6


12. Find the coordinates of the centre of the circle that is tangent to the y-axis, tothe line y - 7 = 0, and to the line 2x - y - 2 = o.

Given the graph of fix) below, sketch the graph and state the domain andrange of:

14. y = f(~x)

13. y = 1 f(x) 1-3

5 U divisi . 2x + 3 . th .L'. Q R1. se IVISlonto write y = In e rorm y = +--.x-I x-I

. 2x+316. Use the result from question 15 to sketch the graph ofy =--x-I

17. Multiple Choice. A circle and a square have equal areas. If the perimeter ofthe square is P, then the circumference of the circle is:

a. nP Pb.-Jipc.

2P

d. -J1i e. 2n..JP

Continued

28


8-6


18. Multiple Choice. The graphs ofy = sin x and 51[Y = 2x intersect at k differentpoints. The value of k is:a. 1 b. 3 c. 4 d. 5 e. 7

19. -J132 -122 = V125. Solve for n.

20. Given the circle with centre 0 and a diameter of 20 decimetres, find the areaof the shaded region.

29


B-7

Exercise 13: Transformations with Trig Functions

1. This graph represents y = fix). Use the sinefunction to find an equation for fix).

2. Repeat question 1, using a cosine function to find an equation for fix).

3. Use the sine function to find anequation for this graph.

4. Use the cosine function to find an equation for the graph in question 3.

5. Write an equation for this graph.

-3.14 3.14 6.28 9.42

-1

-2

6. The depth of water in a harbour is given by the equation d(t) = -4.5 cos(O.16m)+ 13.7, where d(t) is the depth, in metres, and t is the time, in hours, after lowtide.a. Sketch the graph of d(t).b. What is the period of the tide, from one high tide to the next?c. A bulk carrier needs at least 14.5 metres of water to dock safely. For how

many hours per cycle can the bulk carrier dock safely?

Continued

30


8-7


7. The average daily maximum temperature in Vancouver follows a sinusoidalpattern with a highest value of 23.60 C on July 26, and a lowest value of 4.20 Con January 26. Express the maximum temperature as a function of cosine.

8. Use the results from question 7 to find the expected maximum temperature forMay 26.

9. Use the results from question 7 to find how many days will have an expectedmaximum of at least 21.00 C.

10. Determine the measure of the largest angle in a triangle with sides 8, 9, and 11.

11. Change the following degree measures into radian measures:a. 1630 b. 1890 c. 2160 d. 3520

{1- X2 for - 3~ x < 0

12. Sketch the graph off(x) = r:-v x + 1 for 0 ~ x < 2

13. Solve for r: mx = J:..(~+-.!:.).m r p

14. On the left below is a list of functions. On the right are descriptions of how thegraphs of these functions can be formed from the graph of f(x). Match eachfunction with its description.a. y = -f(x) 1. Stretch horizontally by a factor of 3.b. y = f(-x) 2. Stretch vertically by a factor of 3.c. y = f(3x) 3. Compress horizontally by a factor of 3.d. y = 3f(x) 4. Reflect in the x-axis.

e. y = f( % x ) 5. Reflect in the y-axis.

15. Sketch the graph off(x) = ~. You might wish to use division to express itx+1

in the formf(x) = Q+~.x+1

16. Graph: y = 3 - 1x + 21.Continued

31


8-7


17. a. A triangle is uniquely determined if S.A.S. data(two sides and the included angle) are given.Prove that the area of a triangle are given by:

A = ~ ab sinO

if a and b are two sides and 0 is the anglebetween them.

b. Use the formula from 17a to find the area of a triangle with sides oflength 15 and 20 and an included angle of 45°.

18. The horizontal line y = k meets the parabola y = X2 + 8x at one point only. Whatis the value of k?

Fx +2Y -1919. Solve for x and y: r: --i x - 2Y = 3

20. a. How many different parabolascross the x-axis at (0, 0) and (2, O)?

b. Write equations for two suchparabolas.

-1 3

32


C-1

Exercise 14: Trigonometric Identities I

1. Express each of the followingentirely in terms of sin 8 .

a. csc8 b. cos28 c. cot28

2. Express each of the followingentirely in terms of cos 8 .

a. sec8 b. sin28 c. tan28

3. Express each of the followingin terms of sin 8 or cos 8 or both.

a. sin8csc8 b. sin28+ \sec 8

c. l-csc28

Prove that each of the following is an identity.

4. cosx secx = 1 5. cscx sinx = 1 6. tan 8 cot8 = 1

7. cot8 sin 8 = cos8 8. tan8cos8= sin 8

9. Solve the equation: 4sin8+2.J3 = 0 for the interval 0 $ 8 $ 2n-.

10. Solve the equation: 3cos8 + 2 = 0 for the interval 0°$ 8 $ 360°.Give answers toone decimal place.

11. Graph both sides of the followingequation: _1_ - cosx = tan 8 sin 8cosx

a. What do you notice about the graphs?b. Verify algebraically that the above is an identity.

12. Identify the quadrant ofP(8) for each of the followingvalues of ():

8= 1mc.3

a. 8 = 600° b. 8 = -400° d. 8 =-9

-313. If sin8 = - and cos8 > 0, find the exact value of tan 8.4

-8n-14. If 8 = -, find the coordinates ofP(8).

3

Continued

33


C-1

Exercise 14: Trigonometric Identities I

15. Find the exact value(s) of ()in the interval zr ~ () ~ 2rc:

2cos"()= .,j2 cos()

16. Solve for x with domain over the reals: cosx = -..J3 .2

17. Solve for (),0 ~ ()~ 2rc:

tan 2(} =-1

18. Use the graph of fix) to evaluate fTf(4)].

y

(-3,0) 1 (2,0)y =f(x)

19. Iff(x) = _3_, find an expression for ["1(x).x+1

20. The line y = mx, m > 0, touches the parabola y = X2 + 9 y.at only one point. Find the value of m.

.-----~-------.x

34


C-1

Exercise 15: Trigonometric Identities II

Prove the following identities.

1. (1 + sin x )( 1- sin x) = 12sec x

2. cos" X - sin" x = 1 - 2 sin" x

3. cos28 = (1 - sin 8)(1 + sin 8)

4. (1 - sec 8)(1 + sec8) = - tan28

5. 2see" x = 1 + 11- sin x 1+ sin x

6. see" x - csc" X = tan" x - cot" X

7. cos" X - sin" x = 1 - 2 sin" x

9. Solve the equation for 8 where 0°~ 8 ~ 360°:

sec8 -1 1 II---= +sec e2

10. Solve the equation over the interval 0 ~ 8 ~ 21r . State your answer to fourdecimal places.3 - 6 tan 8 = tan 8 + 6

11. Find the value of sin 8 if cos8 = ~~ and tan 8> O.

12. Solve graphically the following equation for 8 over the set of real numbers.(Give answers to four decimal places.)

6 sin2 8 + 10sin 8 - 4 = 0

Continued

35


c-t

Exercise 15: Trigonometric Identities II

13. Solve algebraically the following equation for ()over the set of real numbers.

1. sin ()- -J3 = 04 8

14. Prove the following identity using two different methods.

sec2t - 1 . 2t2t = SInsec

15. Express each of the following in radian measure.a. 2250 b. 2160 c. 1250 d. 1050

16. Express each of the following in degree measure.

27ra. -3

57rb.6

47rc.3

37rd.4

17. If fix) = 2x + 3, find the y-intercept of the graph of y = Mx)].

18. Multiple Choice. If a < 0 and b2 - 4ac > 0, which of the following graphscould represent y = ax? + bx + c?

v c. d.a. b.

19. Factor x3 - 2x2+ 3x - 6.

20. Iff(x) = x+3 ,findf-l(5).x-2

36


C-2

Exercise 16: Sum and Difference Identities I

1. Find the exact value of sin( 5lt )by letting a = 1t and ~ = 1t in the sum formula12 6 4

for sinfrr+B].

2. Find the exact value of cos(~) by setting a = 1(; and f3 = 1(; in the relevantformula. 12 3 4

3. Using the known relationships for sin(a + f3) and cos(a + f3), derive the formulafor tan(a + f3) and express in terms of tangent only.

4. Using a = 1(; and f3 = 1(; , find:3 4

. 71(;a. SID-

1271(;

b. cos-12

71(;c. tan-

12

5. Determine a formula for the following, expressing in terms of tangent only:a. cot(a + f3) b. tan(a - f3)

6. Using the formula derived in question 3, find tan 71(; . Let a = 1(; and f3 = 1(; •12 3 4

Compare this result to your solution for question 4c.

7. Find the exact value of cos 105°by using 45° and 60°.

8. Given that sin a = 2., and cos f3 = ~ and neither pea) nor P(f3) are in25 41

quadrant I, find:

a. sin(a+ f3) b. cos(a+ f3)d. in what quadrant is the point pea + f3)?

c. sec(a + f3)

9. Graph the following equation: Sin(t + 3;) = -cost

a. What do you notice about the graphs?b. Verify algebraically that the above equation is an identity.

sin218° + cos" 18°10. Find the exact value of 2

1-cos 210°

Continued

37


C-2

Exercise 16: Sum and Difference Identities I

11. If tan e = ~ and sin e ~ 0, find see e.

12. Solve the equation for e where 0 ~ e ~ 2n: 4 cos"e + cose - 3 = O.

Prove the identities in questions 13 to 15.

13. sIn x 1-cosx=

l+cosx sm xsIn x + cosx = 114.cscx secx

15. 1----. - = tan x sec xcscx +srn x

16. Find the distance from line 2x - 3y +7 = 0 to the point on the unit circle at n.3

17. Sketch the graph of y = X2(X - 2). x+1 x18. Solve for x: -- +-- = -1.x-I x+l

19. The graph represents a function fix).

a. If the equation fix) = k has exactly fourroots, what can be said about k?

b. If the equation fix) = k has exactly tworoots, what can be said about k?

y.

20. A cone has a height of 8 em and a radius of 6 cm. When it is cut along the lineAB and flattened, it forms the shape shown in figure 2.

a. How long is AB?b. What is the length of the major arc BC in

figure 2?c. How many degrees are in L CAB?d. What was the surface area of the cone?

A

ebBBFigure 1 Figure 2

38


C-2

Exercise 17: Sum and Difference Identities II

1. Express COs(;+e) as a function of e only.

2. Expand and simplify tan( e - ~) .

3. Write sec(: +e) as a function of e only.

4. Prove the following identities:

a. sin( t - ~) = - cost b. cos(t + 3;) = sint

5. Give the exact value for the following:

. 51T IT 5IT. ITa. Sin-COS--COS-Sln-16 16 16 16

b. cos33° cos27°- sin 33°sin 27°

6. Given sin a = -4 and cos~ = -5 where a and ~ are in the third quadrant,5 13

a. Find the coordinates ofP(a - 13).b. In which quadrant does the terminal side of (a - 13) lie?

7. Show the followingis true: sin( e - ~) +cos(e - ; ) = -J3sin e.

8. Show that the following statement is false:

tan( e + :) - tan( e - 3:) = 1

9. Prove the identity: sin(a + 13) + sin(a - 13) = 2sin acos 13·

10. Show that the followingrelationship is always true:

cos(x + y) cos(x - y) = cos" X - sin" y

Continued

39


C-2

Exercise 17: Sum and Difference Identities II

11. Prove that the following statement is true for all values of a and f3:

t( f3) cot a cot f3 - 1co a + = -----'---cota +cotf3

12. If cos a = ~ and tan f3 = 3 with neither a nor f3 in Quadrant I, find tan( a - f3).13 4

13. Create the right-hand side of the following equation: sin t = 1 1 to makeit an identity usinga. sum or difference of cosinesb. sum or difference of sinesc. Prove the identity in a. algebraically

14. Solve the following equations over the real numbers, expressing your answerto two decimal places:

2sin28- 5sin8- 3 = 0

15. If 8 = -924°, find the related angle for 8.

16. Find the approximate values for 8 if 0 ~ 8 ~ 21t'and 3cos2 8 = cos8 (to twodecimal places).

17. If x3 - y3 = (x - y)(x2 + xy + y2), find the factors of sin"x - cos"x.

18. If x3 + y3 = (x + y)(x2 - xy + y2), find the factors of tan" x + cot"x.

19. Prove the following identity: see"x(l - sin"x) = 1.

40 ,


C-2

Exercise 18: Double Angle Identities

1. a. Starting with the identity for sinCa+ {3),prove that sin 20 = 2sin 0 cosO.

b. Starting with the identity for cos(a + {3), prove that cos20 = cos20 - sin'' O.

2. Use question lb. to prove the following:

a. cos20 = 2cos2 0 - 1 b. cos20 = 1 - 2sin20

3. Derive a relationship for tan 20.

4. Evaluate tan (120°)using the above double angle formula and setting 0 = 60°.

5. Evaluate cos 1t to two decimal places, given that cos~ = 0.95. Use a double5 10

angle formula to find this result.

6. Prove this identity: csc2x - cot2x = tanx.

7. If sin 0 = : and cos0 = ~, find:

a. sin20 b. cos20

8. If 0 is the angle indicated in the diagram, find an exact value for sin20.

9. Graphically solve the following equation for 0 in [0, 21l].

2cos20 -.J3 = 0

State answers to three decimal places.

Continued

41


C-2


.. . csc4x-cot4x10. Prove this identity: 2 2 + cot" x = csc"x.

csc x+cot x

11. An even function fix) has the property that fi-x) = fix). Using the subtractionformula for cos e, prove that cos e is an even function. Start with cos(-e) =cos(O- e).

12. An odd function fix) has the property that fi-x) = -f(x). Using the subtractionformula for sin e, prove that sin e is an odd function. Start with sin(-e) =sin(O- e).

13. If sin e = ~2 , and e lies in Quadrant IV, find the exact value of cot e.

. 2n 2nSIn-+cos-14. Find the exact value of: -. ~~'-n-----'2"'!=~~

SIn-'COS-3 3

• 3 3

15 P the f 11 . 'd tit SIn x + cos x -_1- sinx cosx,. rove e 10 OWIng1 en 1 y: .sm x + cosx

16. Solve the following equation for e where the domain is the set of real numbers.

tan e =-J3

17. Solve for ewhere the domain is the set of real numbers. Round the answersto four decimal places.

csc" e sin" e + 3sin e + 3 = 5

18. Express each as a function of x only:

a. sin(~ -x) b. cos(; -x)

Continued

42


C-2


19. a. Use the graph of thefunction y = fix) to draw thegraph ofy == (1 (x).

b. Determine if y = fix) is a one-to-one function.

y

1y = fix)

20. Given y = 3(x - 4)2 + 2,a. State the name of the function.b. State the coordinates of the vertex.c. Describe how it has been shifted from the original equation y = X2.

43


0-1

Exercise 19: Exponential Functions

1. a. Sketch the graph of the function fix) = 3X•

b. State the domain and range of the function.c. As the x values increase, what happens to the y-values?

2. a. Sketch the graphs of the functions f( x) = 2x and f( x) = ( ..!.r .b. Describe the similarities between the two graphs. 2c. How are the two graphs different?

3. a. Sketch the graph of the function fix) = 3(2X).

b. State the domain and range of the function.c. Find the x and y intercepts of the function.

4. Sketch the graphs offix) = 5\ g(x) = 5x-2

, and h(x) = 5X- 2 on the same set of

axes.

5. From the graph of fix) = 2\ estimate the value of:a. 21.3 b. 25.1

6. From the graph of fix) = 3\ estimate the missing value of k, given the points:

a. (k,6) b. (k, 10)

7. a. Sketch the graphs of fix) = 4\ g(x) = 4-\ and h(x) = -4X.b. How is each of these graphs different?

8. a. Sketch the function fix) = 2x - 3.b. Find and sketch the inverse of fix) = 2x - 3.

9 P he identi 1- cos"(J . 2 (J 2 (J. rove tel entity: 2 = sm cos .1+ tan (J

10. Solve the followingequation for (J where 0 :::;;(J:::;; 2n:

2tan2 (J + sec (J = 1

11. Graph y = 2sin4x for 0 :::;;x:::;; 27t.

12. Graph y = - co{ x - ;) for 0 :::;;(J :::;; 27r.

Continued

44


0-1

Exercise 19: Exponential Functions

13. Sketch the graph ofy =--Jx.

14. Sketch the graphs of:

a. y = 3+--Jx b. y = -Jx+2

15. Sketch the graph ofy = 4 - --Jx .

16. This graph represents y = fix).Sketch the graph ofy = 1fix) I.

17. Given sin(t) = -3 , find the coordinates of P(2t) if tt ~ t ~ 3n .8 2

18. Solve for ()if sin ()= cos()on 0 ~ ()~ 2n.

(-4,3) .19. This graph represents y = t(x).

Sketch the graph ofy = to x I).• 1

(4,0)

(0, ~2) •

20. A sinusoidal curve passes through (:' 0), (541t,0), and (941t,0) with a

maximum value at (341t,3). Find two equations for this curve: one in terms of

sine and the second in terms of cosine.

45


0-2

Exercise 20: Solve Exponential Equations

For questions 1 through 8, solve the given equation for x.

l. 2x= 32

2. 23x-5 = 16

3. 54>:-7= 125

4.3x2 +4x = -.!...

27

5. _1_=813x-1

6. 2(52X-9

) = 250

7.38x = -.!.

81

8. 323x-2 = 16

9. Determine which two of the three given functions are equivalent:

a. y = 2-x b. y = _2x c. y =(~J10. Use the graph of the function jlx) = 4X to sketchg(x) = 4x-3 and h(x) = 4X_ 3.

11. Sketch the graph of the function fix) = 2X- 3. State the x- and y-intercepts.

12. Sketch the graph of the function fix) = 31-x• State the x- andy-intercepts,

asymptotes, domain, and range.

.. . sec2x13. Prove this identity: 2 = csc"x.sec x-I

Continued

46


0-2

Exercise 20: Solve Exponential Equations

14. Iff(x) = X2 - 9, graphy = f(~r

15. Determine whether tan ()is an even or an odd function usingtan(-(} ) = tan(O - (}).

16. If sin( ()+ 2; ) + cos(()+ 5; ) = Asin ()+ Bcos(), find numerical values for A and B.

17. Sketch the graph ofy = .! sin 2x for - tt ~ ()~ n,2

18. Sketch the graph ofy = 2cos(2x- 1l') for 0 ~ ()~ 21l'.

19. This is the graph ofy = fix). Sketch the graph ofy = 2f(x) - 2.

1

20. Using the graph in question 19, sketch the graph ofy = If(x) I+ 1.

47


0-3,0-4

Exercise 21: Logarithmic Functions

1. Express each of the following in logarithmic form:

a. 34 = 81 b. 16 = 24 d. 2-3 = -.!8

2. Express each of the following in exponential form:

c. loglOO.Ol = -2

3. Evaluate each of the following:a. log416 b. logg3 c. log0J28

4. Graph fix) = 3X and its inverse, (-lex).

5. Sketch fix) = log2(x-1) and state the domain, range, intercepts, andasymptotes.

6. Sketch fix) = log5(x) + 3 and state the domain, range, intercepts, andasymptotes.

7. Sketch fix) = logi3 - x) and state the domain, range, intercepts, andasymptotes.

8. Sketch fix) = logix - 1) + 1 and state the domain, range, intercepts, andasymptotes.

For questions 9 through 12, solve the given equations for x.

9. 2x2 = 16

10. 82x+ 1 = 64

11. _1_=644x-2

12. (~J 27=--

125

Continued

48


0-3,0-4

Exercise 21: Logarithmic Functions

13. Sketch the graph of fix) = 3-X• State any asymptotes, x and y-intercepts, the

domain, and range.

14. Sketch the graph offix) = 2 - 3%-1. State the domain, range, asymptotes, andthe y-intercept.

15. Sketch the graph ofy = -3 sin(28 + 1T:) in the interval 0 S 8 S 21T:.

16. Sketch the graph ofy = Ix - 41.

17. Sketch the graph ofy = Ix - 41-2.

18. Sketch the graph ofy = Ilx - 41-21.

19. Solve the equation sin 8 + 2sin 8cos8 = 0 where 0 S 8 S 21T:.

20. If cos()= -0.491 where 0 S 8 S 41T:, find all values of 8.

49


0-5

Exercise 22: Logarithm Theorems I

1. Write as a single logarithm.

a. log; 2 + log, 32 1b. 4 log x - slog( X2 + 1)+ 2log(x-I)

2. Write as a single logarithm.

1a. - slog108 b. 3 logx - 2log y - 4 log t +1.logb

2

3. Use the Laws of Logarithms to rewrite each of the following:

X3y4b. log-6-

Z

5. Sketchy = logx" -logx.

6. Express log, V36 in terms of log, 6.

7. Given logs2 = 0.33333 and logs3 = 0.52832, use properties of logarithms to findlogs12 and logs36.

8a3b4

8. Write in terms of log a, log b, and log c: log ~

9. Express each of the following in logarithmic form:

a. 64 = 1296 b. 5-3 = _1_125

10. Express each of the following in exponential form:

b. log{6~) =-3a. logs 64 = 2

11. Graph f( x) = log1(x). State the domain, range, intercepts, and asymptotes.2

Continued

50


0-5

Exercise 22: Logarithm Theorems I

12. Evaluate each of the following:

a. log2(6~) b. log, 81= 2

13. Solve for x: 39x = ~.27

14. Solve for x: (:5 r = ~

15. For the graphs ofy = 3x-1 and y = !(3X), find the domains, ranges, x-intercepts,

and y-intercepts.

. . see"x-tan2 x .16. Prove the identity: 2 = sin" x.l+cot x

17. Solve the equation tan'' ()+ sec2()= 3 for 0 ~ ()~ 21l'.

18. a. Sketch the graph ofy = -sin rex.b. What is the period of this function?

19. a. Sketch the graph ofy = 3cos(2x + 1l').

b. What is the period?c. What is the horizontal shift?

20. 1a. Sketch the graph ofy = -.

xb. 1Sketch the graph ofy = 3+ --.

x-3

51


0-5

Exercise 23: Logarithmic Theorems II

1. Express each of the following as a single logarithm:a. log25 + log27 + log26 b. log54 + log56 - log, 3

2. Express each of the following as a single logarithm:1 1

a. 2 log37 - [log, 14 + log, 35) b. -log2 4 + -log2 272 3

3. You are given log, 2 = 0.3010; log, 3 = 0.4771; log, 7 = 0.8451. Find the valuesof each:

a. log, 6 b. log, 1414

C. logb-3

d. log, V96

log x4. Prove: log x = ab log, b

log a5. True or false? -- = log a -log blog b

6. Evaluate logs7 correct to six decimal places.

7. Expand as a sum and difference of logarithmic expressions:

8. Express the following as a single logarithm:4 log, x - 2 log, Y + 3 log, t - 4 log, k

9. Solve for x: 252%+1 = 125.

10. Solve for x: 22%+2= 1~'

11. Express in logarithmic form: 3-2 = !.12. Express in exponential form: log, 32 = 5.

Continued

52


0-5

Exercise 23: Logarithmic Theorems II

13. Solve for ()in the followingequation if the domain is all real numbers.

cos"()+ sin'' ()+ 2sin ()= 4

14. Sketch the graph ofy = 4COS(1lX + n).

15. a. Sketch the graph ofy = Ix I.b. Sketch the graph ofy = Ix 1- 2.c. Sketch the graph ofy = 2 -Ix I.

16 Sketch the graph ofy = 1 + sin x .

17. Find the value of ()in degrees.

8 ()

3

4

18. Iffix) = 2x + 3, findf-l(x).

19. Multiple Choice: Which of the followingequations represents a relation inwhich y is a function of x?

a. X2 + y2 = 1 b. X2 + 3y = x c. x + 3y2 = 2y d. x2y2 = 16

20. Sketch the graph of (x - I? = 9.

53


0-6

Exercise 24: Exponential and Logarithmic Equations I

1. Solve for x and check:

a. log, x = 2 b. log x 25 = 2-4c. log, 16=-3

2. Solve for x and check:

a. log~G)'= x

3. Which of the following equations have the same solutionis)?4

a. log x 16 = 3 b. log, X = 2 c. log162= x2

d. log32X =-5

4. Evaluate each of the following:

5. Solve each of the following for x and check your solutions.1

a. log x ~ = 4 b. log, X -log3 4 = log312 c. 5x-2 = 1-

6. Solve for x and check: log5(x2- 4x) = 1.

7. Solve for x and check: log313 - 2x I =2.

8. Solve for x and check: log(x + 21) + log x = 2.

9. Which pairs of the following functions are equivalent?a. fix) = _22-x b. g(x) = 4x-2 c. hex) = 22x-4

10. Solve the equation for x: 46x = 1:....64

11. Find the values of cot 0 if cosO = ~ and sinO < O.5

12. Graph y = log, (-x) and state the domain, range, intercepts, and asymptotes.

Continued

54


0-6

Exercise 24: Exponential and Logarithmic Equations I

13. Prove this identity: _l_+tanx = secxcscx.tan x

14. Sketch the graph ofy = 2sin(3x+ 3;).

15. Sketch the graph ofy = IX2 - 11.

16. Sketch the graph ofy = 4 + Ix - 21.

17. Solve the equation (x + 2)2= (x - 1)2+ X2.

18. Find an equation for the followingfunction.

19. Find the inverse of the functionf(x) = _3_.x-2

20. Show that log(sin600)= l:(log 3-log4).2

55


0-6

Exercise 25: Exponential and Logarithmic Equations II

For questions 1 through 5, solve for x and check your solutions.

5. log1 X + log1(5x - 28) = -2- -7 7

For questions 6 through 8, solve for x to two decimal places.

6. 4x = 15

7. 52.>:-3 = 8

8. 63x = 22x-3

9. Solve for x: 3x = 2· 52-x•

10. Verify: log.! = -log5.5

For questions 11through 14, solve the equations for x.

11. logx 10M = 32

12. log127 = x9

13. log., 0.0001 = x

14. log, 6 + log, 9 = log, X

Continued

56


0-6

Exercise 25: Exponential and Logarithmic Equations II

15. Solve for x: 103%-1 = 110000

16. If 0 < e < 1t and sine = -J8 , find the exact value ofP(2e) on the unit circle.2 9

17. Sketch y = x3•

18. Sketch y = 2 + (x - 3)3.

19. Convert 5.5 radians to degrees.

20. Several trig values of 8, measured in radians, on the interval [0, 2n] have the

value -../3 . One such value is sin 4n. Find all other possible answers.2 3

""-- 57


0-7,0-8

Exercise 26: Natural Logarithms

1. Find the value of:1

b. eO.67 c. In(9.43) d. In(0.0036)

2. Sketch fix) = 2X and g(x) = 3X on the same axes. Where would the sketch ofhex) = e" fit?

3. Sketch fix) = ex,g(x) = ex-3, and hex) = eX_ 3. State the domain, range,

intercepts, and asymptotes.

4. Sketch fix) = e". Sketch the inverse of fix) = e". State the domain, range,intercepts, and asymptotes.

5. Use the laws oflogarithms to write In(f(x))as an expression of naturallogarithms.

6. Solve for x.a. e-O.Olx = 27 b. eln(l-x) = 2x

d. e2x-1 = 5

7. A radioactive substance is decaying according to y = A(e-O.2t ) where y is theamount of material remaining after t years.a. If the initial amount A = 80 grams, then how much is left after three years?b. The half-life of a substance is the time taken for half of it to decompose.

Find the half-life of this substance in which A = 80 grams.

8. Suppose that $1000 is invested at 10%interest, compounded continuously.How long will it take for this investment to double?

9. Solve for n: 53n+ 1 = 625.

9

10. Express in logarithm form: 22 = 16-J2.

Continued

58


0-7,0-8

Exercise 26: Natural Logarithms

11 P he identit l-cos2(J 2(J. rove tel enti y: 2 = cos .

see (J-l

12. Solve for x. Check: log, 8-12 = x.

13. Solve for x. Check: log5n25n2 = x.

14. Solve for x. Check: log5(3x + 1) + log5(x - 3) = 3.

15. Solve for x: log(x3- 1)-log( X2 + x + 1)= 1.

16. Expand using the laws of logarithms.

I 4(x - 5)og------'--~x3(x + 6)

17. Sketch the graph ofy = 3 + 2sin2(J.

18 S I fi 5" 62"-". 0 ve or x: -1 - = .7-"

19. Find values ofA, B, and C such that a maximum ofy = A sin(x + B) + C occursat (0, 0).

20. If sin( (J + :) + cos( (J - 7;) = P sin (J +Q cos (J, find numerical values for P and Q.

59


0-8

Exercise 27: Applications of the Exponential Function

1. Convert each expression into the logarithm of a single expression.

a. ~lnx-2In(x-1)-~ln(x2+1) b. In(x3-1)-ln(x2+x+1)

2. Solve for k. Leave in terms of natural logarithms.

a. 5000 = 50e2k b. A = Ae4k

3

3. A $5000 investment earns interest at the annual rate of 8.4% compoundedmonthly.a. What is the investment worth after one year?b. What is the investment worth after 10 years?c. How much interest was earned in 10 years?d. What sum of money must be invested now so that $20 000 is available in

five years if the rate is 8.4% compounded monthly?

4. The population of gophers in a field can be modelled by the equation:

P = 100(1.1)n

where n is measured in years. Plot the graph for a 10-year period. How manygophers will there be after 20 years? How long will it take for the gopherpopulation to double?

5. Suppose that $2000 is invested at 8%, compounded continuously. How longwill it take for the investment to triple?

6. A radioactive substance is decaying according to the formula:

y = Aekx

where x is time in years. The initial amount is A = 10 grams and eight gramsremain after five years.a. Find k. (Leave your answer in natural logarithms.)b. Estimate the amount remaining after 10 years.c. Find the half-life to the nearest tenth of a year.

7. When the population growth of a city was first studied, the population was22 000. Five years later, it was 24 000. If the population grows exponentially,how long will it take for the population to double?

Continued

60


0-8


8. The pH of a substance is defined by:

pH = -log[ H+] where [H+] is the hydrogen ion concentration in moles per litre.If the pH of a substance is 6.62, find its ion concentration.

9. a. Sketch the graph of the function: y = 3-X

b. State the x- and y-intercepts.

. . . sinx cosx10. Prove the Identity: --+-- = 1.cscx secx

111. Solve and check: logx 9 = -2.

12. Solve and check: log3 ( 27 ) = x."5 125

13. If 5(23X) = 471

- X, solve for x:

a. using logarithms to base 10.b. using logarithms to base e.

14. Sketch the graph ofy = 2X- 2.

15. On the same coordinate system, sketch the graphs of: y = x; y = x3; andy = Vx.

Youmay want to use a graphing calculator or make a table of values. What doyou observe?

17. Find all values of 8, 0 ~ 8 ~ 27rsuch that 5sin8 - 12cos28 + 10 = O.

1 118. Ifa+b = 8, and ab = 10, find the value of -+-.

a b

Continued

61


0-8


19. Multiple Choice. Which of the following is not equal to tanx?a. tan(x + n) b. tanix - n)

20. Solve for x: -15x =- 7x2- 2x3•

62


E-1

Exercise 28: Counting Principles

Simplify the following. (Do not use a calculator.)

1. 7! b. (31)! 9! d. 10!a. - c. - --6! (30)! 6! 6!4!

2. a. (k + 3)! b. 7!(r + 2)!r(k + 2)! 6!(r - I)!

3. Solve n! = 20(n - 2)!

4. a. Which of the followingexpressions is equal to n? (There can be severalanswers.)

n! n! (n + I)! n1.(n + I)!

11.(n - I)!

111.n! n+l

IV.(ri + I)! v. n2(n - I)!

n! n!

b. Write two other expressions involving factorials which are equal to n.

5. A nickel and a dime are tossed on a table. In how many ways can they land?

6. If there are 12 runners entered in a race, in how many ways can first, second,and third place be awarded?

7. Pizza Barn offers three choices of salad, 20 kinds of pizza, and four differentdesserts. How many different three-course meals can one order?

8. A freshman student must take a modern language, a natural science, a socialscience, and English. If there are four different modern languages, five naturalsciences, three social sciences, but each student must take the same Englishcourse, how many different ways can she select her course of study?

9. Suppose that the executive of the Manitoba Association of MathematicsTeachers consists of three women and two men. In how many ways can apresident and a secretary be chosen if:a. the president is to be female and the secretary male?b. the president is to be male and the secretary female?c. the president and secretary are to be of the opposite sex?

Continued

63


E-1

Exercise 28: Counting Principles

10. There are five main roads between the cities A and B, and four between BandC. In how many ways can a person drive from A to C and return, goingthrough B on both trips, without driving on the same road twice?

11. Multiple Choice. If fix) = x-3, then flex) is equal to1

C. X 3

12. Prove the identity: sin" x - cos"X = sin'' x - cos''x.

13. Solve and check: log, V4 = x.

14. Solve: 2(3l = 5x-l.

15. Show that log(_2_)::/; log 2 .1.08 log 1.08

16. Solve the equation for ()where 0 :::;;():::;;27r.

4cos2()-7cos()- 2 = 0

17. A certain radioactive substance decays according to S = Soe-0.04t where So isthe initial amount of the substance and S is the amount of the substance leftafter t years. If there were 50 grams of the radioactive substance to begin with,how long will it take for half of it to decay?

18. Show that when 8 = 80e-<>·04' is solved for t, the result is t = -(25) In( :0).

19. The atmospheric pressure P at height h kilometres is given by P = Poe-kh.

The pressure at sea level is Po= 101.3 kPa.a. IfP = 89 kPa when h = 1, find k.b. Calculate the pressure at a height of 2 km.

20. Sketch fix) = -IX2 - 11.

64


E-2

Exercise 29: Permutation with Repetitions and Restrictions

1. How many distinguishable permutations are there of the letters in thefollowingwords?a. AARDVARK b. BOOKKEEPERS

2. How many distinct ways can three red flags, two blue flags, two green flags,and four yellow flags be arranged in a row?

3. How many different arrangements can be made using the letters in the wordTEETH, if the "word"must start with H?

4. Using the letters of the word FACTOR(without repetition), how manyfour-letter code words can be formed:a. starting with R?b. with vowels in the two middle positions?c. with only consonants?d. with vowels and consonants alternating?

5. Manitoba license plate numbers consist of three letters followedby threedigits. How many different plates could be issued?

6. Consider the digits 1, 3, 5, 7, 9. If repetitions are allowed, finda. how many three-digit numbers can be formed.b. how many three-digit numbers can be formed if the numbers must be less

than 600 and divisible by five.

7. How many ways can Amanda, Basil, Charles, Dennis, and Edna sit in a row ifBasil and Edna insist on sitting together?

8. How many ways can the people in the previous question be seated if Charlesand Edith insist on not sitting together?

9. How many ways can four married couples sit in a row on a park bench ifa. every husband and wife must sit together?b. the men and women must alternate?

Continued

65


E-2

Exercise 29: Permutation with Repetitions and Restrictions

10. North B,----,---,----.--

East West

The diagram shows a road map. Theindicated route from A to B can bedescribed as WNNWWNW,meaning: "Goone block west, two blocks north, twoblocks west, one block north, and one blockwest." How many other routes are therefrom A to B if you must stay on the roads?A South

1

11. Express in logarithmic form: (:9r2 = 7.

12. Prove this identity: sec4x - tantx = 1 + 2tan2x.

13. Solve and check: logloo10 = x.

14. Use the laws oflogarithms to write in expanded form: log,( ";,~2J.15. The line x - y + 2 = 0 intersects the circle X2 + y2 - 4 = 0 in two points. Find the

coordinates of these points.

17. Solve. Express your answer to one decimal place. 53x = 63

18. Solve the equation tan'' (J - tan (J - 4 = 0 for 0 $ (J$ 21r.

19. Find the 4th degree polynomial whose roots are 1, -1, ~, and 2.

20. The number of bacteria in a certain culture t hours from now grows accordingto the formula: A = 800(3Ya. What will be the bacteria count 3.12 hours from now?b. When will the bacteria count reach 100 ODD?

66


E-2

Exercise 30: Permutations

1. Evaluate:

a. 5P2

2. Write an algebraic expression for:

b. n~nPz

3. Solve for n: 2n + nP2 = 56

4. Three brothers and three sisters are lining up to be photographed. How manyarrangements are there:a. altogether?b. with brothers and sisters in alternating positions?

5. Five students walk into a French classroom with 10 desks. How manydifferent seating arrangements are possible?

6. Winnipeg Stadium has four entrances and nine exits. In how many ways cantwo people enter together but leave by different exits?

7. How many ways can the offices of chairperson, secretary, and treasurer befilled by members of a committee of eight people?

8. How many five-digit numbers can be found from 1,2,3,4,5 if:a. the odd digits occupy the odd places?b. the odd digits occupy the odd places in ascending order?

9. a. How many four-letter "words" are possible using the letters inTHURSDAY?

b. How many end in the letter Y?c. How many do not start with an R and end with a Y?

10. Explain the meaning of sP3. Why does 3PS not make sense?

11. State the range of fix) = cos" x.

12. Express in exponential form: log5(~) = -1.

Continued

67


E-2

Exercise 30: Permutations

4 114. If log, x = n2 and log, b = -, show that n = -. (b > O,b"# 1).n 4

15. Find the range and y-interceptrs) for the graphs off(x) = e-X andg(x) = =e":

16. Solve for x: e2x-5 = 25.

17. Sketch the graph ofy = 2 - 2cos(~ x ).

18.

The line y = 1intersects the graph ofy = sin x2

at points A and B. Find the length of AB.

19. Sketch the graph ofy = 1X2 - 41.

20. The horizontal line y = k intersects the graph ofy = 1X2 - 41 in exactly threepoints. What is the value of k?

68


E-2

Exercise 31: Circular Permutations

1. a. How many ways can eight people be seated around a circular table?b. How many ways can they be seated if Bob and Raj insist on sitting next to

each other?

2. Five men and five women sit around a circular table, men and womenalternating. How many ways can this be done?

3. In both a science classroom and a history classroom there are 12 desks. In thescience class students are seated in a circle, and in the history class studentsare seated in a row. Which classroom has the greater number of seatingarrangements?

4. How many ways can four beads of different colours be arranged to form abracelet?

5. How many ways can three good friends be seated together around a circulartable with 10 chairs if Brad refuses to sit beside them and five other peopleare to be seated?

6. How many four-digit numbers larger than 5600 can be made using the digits0,1,2,5,6,8,9?

7. Using the numbers 1, 2, 3, 5, 6, 8, 0 (no repetitions):a. how many four-digit numbers are possible?b. how many are divisible by five?c. how many are even?

8. How many numbers less than 700 have no repetition of digits?

9. Using the digits 2, 2, 2, 3, 3, 4, 5:a. how many seven-digit numbers can be found?b. how many are greater than 3 400 OOO?c. how many are greater than 3 400 000 and divisible by five?

10. How many ways can five men and three women be arranged in a row if thereis a man at each end of the row?

Continued

69


E-2

Exercise 31: Circular Permutations

11. A class has 10 students. Three of them belong to the Scouts and always sittogether. Three others belong to Girl Guides and always sit together. Thedesks in period one form a row of 10. The desks in period 2 form a circle of 10.

a. In which class do they have the greater number of seating arrangements?b. If the number of circular arrangements is x, and the number of row

arrangements is y, what is x - y ?

12. Prove the identity: [sin" x+cos" xr = 1.

13. Simplify: 2 log, 3 + ~ log, 16.2

32 -

14. Solve for x: eX = eXe4•

15. Sketch the graph off(x) = ex+1• Then find the inverse off(x) graphically

and algebraically.

(n+2)! 5716. Solve for n: ( ) =8! n-2! 16

17. Solve for n: (n + I)! - 30 = O.(n -I)!

. 118. Solve for x if 5sm

X = - and x E 9\.5

19. Multiple Choice. Which of the following is not equal to sin x?

a. sin(x + 27r) b. sin(-x) c. cos(; -x) d. sin(x-67r)

20. Multiple Choice. Which of the following is not equal to cos x?

a. cos(x+ 27r) b. cos(-x) c. sin(~ -x) d. cos(x+7r)

70


E-2

Exercise 32: Permutations with Case Restrictions

1. There are 11 chairs in a row. In how many ways can five people be seated ifthey sit in consecutive chairs?

2. A book collector has five different books by Dickens, three different plays byShakespeare, and three different novels by Danielle Steele. She also has ashort story by Margaret Laurence. How many ways can they be arranged on ashelf if the books by each author are to be kept together?

3. a. How many ways can eight people be seated around a circular table ifGeorge and Monica insist on sitting together?

b. If, in addition, Nicky and Brent refuse to be seated together, how manyways can this be done?

4. How many ways can four boys and two girls sit in a movie theatre row (whichcontains six seats), ifa. one boy must be seated on each end?b. all the boys insist on sitting together?

5. a. How many five-letter "words" are possible using the letters in WINTER?b. How many contain "I" as the second letter?c. How many do not start with an E?

6. If all the letters in the word BARRIERare rearranged,a. find the number of permutations.b. find the number of arrangements beginning with the letter R.c. find the number of arrangements beginning with exactly one R.

7. Eight boys are to be arranged in a row. Twoparticularly unruly boys are notpermitted to sit together nor are they allowed to sit at either end of the row.How many ways can this be done?

8. How many four-digit numbers greater than 5364 are possible using the digits1, 2, 3, 5, 7, 8?

9. A school bus can seat 14 people on each side of the aisle. The boys always siton the left; the girls always sit on the right. When the bus arrives at a stop, itcontains nine boys and seven girls. How many ways can four new boys andfive new girls be seated?

Continued

71


E-2


10. Six boys and six girls are walking counter-clockwise and single-file in a ring.a. How many ways can they be arranged?b. How many ways can they be arranged if the girls are all together?c. How many ways can they be arranged with the boys and girls alternating?

11. Evaluate (sin :6)( cos~~) - (cos :6)( sin ~~). Express your answer as an

exact value.

12. Solve:

13. In which quadrant is P(4)?

14. Solve for x:

9x =...!.

a.27

15. Select the pair of functions that are equivalent:a. fix) = (0.5y-4 b. fix) = - 24-x c. fix) = 2-(x - 4)

16. Solve and check: log, 2t + log, 3t = log, 24, (b > 0, b ::j:. 1).

17. Find the exact solutions of the equation:2sin'' ()+ cos()= 1 in the interval 0 $ ()$ 2Jr.

18. At 100oe,hot water cools in a room with temperature 200e according toT = 20 + 80e-O·03t where T is temperature t minutes after coolingbegins. Howlong will it take for the water to cool to 400e in the room?

Continued

72


E-2


19. Write a sine and cosine equation for the sinusoidal graph given.

y8

TC-1 "6-2-3-4

20. The electricity supplied to your house is called "alternating current" becausethe current varies sinusoidally with time. The frequency of the sinusoid is 60cycles per second. Suppose that at time t = 0 seconds the current is at itsmaximum, i = 5 amperes.

a. Write an equation expressing current in terms of time.b. What is the current when t = 0.01?

73


E-3

Exercise 33: Combinations

1. Evaluate:

a. 4C2

2. a. Evaluate sC2 and sC6•

b. Evaluate 5C2 and 5C3.

c. What do you notice about the answers in a. and b.?d. Explain why nCr = nCn-r.

3. Solve the equations:

a. nC2 = nCS

4. Lotto 6-49 is a lottery in which one selects six numbers from 1 to 49. Howmany ways can this be done?

5. On a collegebaseball squad, there are three catchers, five pitchers, seveninfielders, and seven outfielders. How many different baseball nines can beformed?

6. a. A poker hand consists of five cards. How many different poker hands arepossible?

b. How many of them contain all red cards?c. How many of them have exactly two hearts and two clubs?

7. How many different sums of money are possible using at least three coins froma collection consisting of one penny, one nickel, one dime, one quarter, and oneloonie?

8. An investment club has a membership of four women and six men. A researchcommittee of three is to be formed. In how many ways can this be done ifa. there are to be two women and one man on the committee?b. there is to be at least one woman on the committee?c. all three are to be the same sex?

9. Suppose that a bag contains four black and seven white balls. (Assume theballs are distinguishable; for example, they may be numbered.) In how manyways can you draw from the bag a group of three balls consisting ofa. one black and two white balls?b. balls ofjust one colour?c. at least one black ball?

Continued

74


E-3

Exercise 33: Combinations

10. A lady gives a dinner party for six of her nine friends.a. How many ways can she choose her six guests?b. How many ways can she do this if Dorothy and Lori cannot attend

together?

12. Ten children are playing "Farmer in the Dell." One child stands in the centrewhile the others form a ring around him. How many arrangements arepossible?

13. Prove the identity: 1+ tan2

8 = see" 8-l.1+cot28

14. How many arrangements can be made of the letters in BABBLING BABY?

15. Solve and check: log2(x - 1) + log2(x + 2) = 2.

16. Without using the change ofbase formula, evaluate log, 25+ log, \1512.

17. a. How many permutations can be made from the letters in CINCINNATIOHIO?

b. How many begin with a T?

18. How many four-digit numbers greater than 5687 can be found using only thedigits 0, 2, 4, 5, 6, 8 (no repetition is allowed).

19. If 0°< 8 < 45° and sin 8 = 2 , find exact values for:3

a. sin28 b. cos28

20. If 0°< 8 < 45° and sin8 = 2, find exact values for:3

a. sin38 b. cos38

75


E-4

Exercise 34: Binomial Theorem

1. Write the complete expansion for:a. (a + b)3 b. (a + b)4

2. Write the complete expansion for (2x _ y)4.

3. Write and simplify the first three terms for each of the following:a. (2x + 1)9 b. (2x2 - X)l1

4. Find the fourth term of (3a2_ :)

7

•

5. Find the 7th term of (a + b)10.

6. Find the 5th term of ( ~ - : J' .

(1)12

7. Find the middle term of 2x - 2x

8. Find the 6th term of (2y + X)l1.

9. Find the (r + 1)8t term of (3a _ 6!2 )9

10. Find the term containing x20 in (2x _X4)14.

11. A group of 15 treasure hunters comes to a clearing in the forest from whichthere are three exits. It is decided that seven will go left, four go by the middleroute, and the rest go to the right. How many ways can this be accomplished?

12. How many different bracelets consisting of six beads can be made from 10differently coloured beads?

13. A tennis club has 10 boys and eight girls as members.a. How many matches are possible with a boy against a girl?b. How many matches are possible with two boys against two girls?c. How many matches are possible with a boy and girl against another boy

and girl?

Continued

76


E-4

Exercise 34: Binomial Theorem

14. The English alphabet consists of five vowels and 21 consonants.a. In how many ways can a selection of letters be made consisting of two

different vowels and three different consonants?b. How many "words" of five letters can be made from two different vowels

and three different consonants?c. How many "words" of five different letters can be made if each "word"

contains at least three consonants? (Do not simplify).

15. a. How many ways can six people be seated around a circular table if twoparticular people insist on sitting opposite each other, two insist on beingtogether, and the remaining two insist on being apart?

b. How many ways can this be done if there were eight chairs around thetable?

16 P he identi tan x. rove tel entity: ---secx-1

secx+ 1tan x

17. A bacterial culture is growing according to the formulay = 10000eo.6x where x isthe time in days.

a. Find the number of bacteria after one week.b. How long will it take for the culture to triple in size?

18. There are 23 teams in Division A of German soccer.a. If each team plays each other team once, how many games will be played?b. If each team plays each other twice (not necessarily in consecutive games),

how many games will be played?

19. Find the exact solutions of the equation: 2 cos8 = 3tan 8, 0 < 8 < 21!.

20. If $4000 is invested today, how much will it be worth after 10 years at aninterest rate of 8% if interest is compounded:a. annually b. quarterly c. monthly d. daily e. continuously

77


E-2, E-3, E-4

Exercise 35: Permutations, Combinations, and Binomial Theorem

1. In how many ways can a dog team of five be chosen from 10 huskies and eightretrievers, so that the majority of the dog team would be composed ofretrievers?

2. A baseball team is to be formed from a squad of 12 people. Two teams made upof the same nine people are different if at least some of the people are assigneddifferent positions. In how many ways can a team be formed ifa. there are no restrictions?b. only two of the people can pitch and these two cannot play any other

position?c. only two of the people can pitch but they can also play any other position?

3. Write the complete expansion for (2x - ~J4. What is the 10th term of (x - 2y)20? Do not simplify.

5. Find the term containingx' in ( ; - :' r6. Find the term with no xs in ( 2X4 - 2!2 )12

7. Find the term containing X14 in (2x - X2)11.

8. A student council consists of a president and eight other members. A yearbookcommittee of five is to be selected from this group.a. How many ways can this be done if the president must be on the

committee?b. How many ways can this be done if the president is not on the committee?

9. A hockey team has nine forwards and three are needed to form a forward line.a. How many possible forward lines involve their top goal scorer?b. If Mario and Serge are two of the forwards, how many lines include at least

one of them?c. How many lines involve their worst three players?

Continued

78


E-2, E-3, E-4

Exercise 35: Permutations, Combinations, and Binomial Theorem

10. How many different five-letter arrangements can be made from the letters inJANUARY?

11. How many rectangles are formed when seven vertical lines are intersected byfour horizontal lines?

12. Solve for n in the equation: (n + 2)C4= 6(nC2).

13. Sketch the graph off(x) = (~J.Determine the domain, range, asymptotes,

and all intercepts. Compare this graph tog(x) = 5-X•

14. Solve and check: log2a(4a2)3 = x.

15. Express as a product or quotient oflogarithms: log, 9.3 + log, 8.6 = log; 19.1.

16. A man and his wife invite four couples to dinner. After the host and hostess sitat the table ends, the guests sit four to a side of the table. If men and womenalternate around the table and no man sits next to his wife, how many seatingarrangements are possible?

17. How many different signals consisting of seven flags can be made using threewhite, two red, and two blue flags?

18. Without considering special cases, how many straight lines are determined bynine points?

19. If a committee of five is to be selected from 12 persons,a. find the number of ways this can be done if a particular person must be on

the committee.b. find the number of ways this can be done if a particular person must not be

on the committee.

20. Use your calculator to solve the equation x - cos x = 0 for 0 :::;;x:::;; 21t'. State theanswer accurate to three decimal places.

79


F-1, F-2, F-3

Exercise 36: Parabola

1. Sketch the graphs of the following parabolas:

a. 4x - y2 = 0 b. 2x2 + Y = 0 c. 4x + y2 = 0 d. 2X2 - Y = 0

2. Sketch the graphs of the following parabolas. State the coordinates of thevertex.

a. (y - 3)2 = 4(x + 5) b. (y + 2)2 = -4(x - 2)

3. Sketch the graph of the following parabolas. State the coordinate of the vertex.

a. (x - 2)2 = -8(y + 1) b. (x + 3)2 = 4y

4. Consider the parabola y2 - 20x + 2y + 1 = O. Find the coordinates of the vertexand sketch the graph.

5. Consider the parabola X2 - 4x + 8y + 4 = O.Find the coordinates of the vertexand sketch the graph.

6. Find the value of a if the the parabola y = ax2 passes through (2,5).

7. The parabola y2 - X + 4y + k = 0 passes through the point (12, 1). Find thevertex of the parabola.

8. Find an equation for the axis of symmetry for the parabola (y - 2)2 = 8(x + 3).

9. Find an equation for the axis of symmetry for the parabola X2 + 4x + 2 - y = O.

10. Find an equation for the vertical parabola with vertex (2, -3) passing throughthe point (9, -10).

11. How many arrangements can be made from the letters in the word DRAUGHTif the vowels must not be separated?

12. From six huskies and seven retrievers, how many ways can one form a team offive dogs if there must be four or five retrievers?

13. How many different bracelets consisting of five beads can be made from eightdifferently coloured beads?

114. Solve for x: --1 = 64.2%+

Continued

80


F-1, F-2, F-3

Exercise 36: Parabola

15. Express in exponential form: log, 25 = 2.

16. Solve and check: log3x = 3 -log3(x + 6).

17. Find and simplify the term containing no xs in the expansion of (x + 2~) 12

18 P this id tit cosO sinO sinO-1. rove IS 1 en 1 y: ----- = ---cscO tan 0 see0

19. How long will it take to double your money if it is invested at 9% andcompounded quarterly?

20. Solve for 0 over the real numbers: cot20 + 2sin 0 = csc20 - 2.

81


125F-1, F-2, F-3

Exercise 37: Circle and Ellipse

1. Find the equation of a circle with centre (3, -1) and radius two units andsketch.

2. Find the equation of a circle with centre at the origin and passing through thepoint (4, -5).

3. Given two points A(6, -8) and B(-2, 4), find the equation of the circle whosediameter is AB.

4. Find the centre and radius of each of the following circles and sketch:

a. X2 + y2 + 2x - 10y + 25 = 0 b. 4x2 + 4y2 + 4x - 12y + 1 = 0

5. Sketch the graphs of the following ellipses.

a. 3x2 + y2 = 12 b. X2 + 4y2 = 16

6. Sketch the graphs and state the coordinates of the centre for the followingellipses.

a (x - 2)2 + ~ = 1. 4 9 b. (x + 3)2 + (y - 2)2 = 1

25 16

7. Sketch the graphs and state the coordinates of the centre for the followingellipses.

a. 4x2 + 9y2 - 16x + 90y + 205 = 0b. 49x2 + 16y2 + 98x - 64y - 671 = 0

8. Find the centre of the ellipse with AB and CD as major and minor axes,respectively. Then write the equation of the ellipse.a. A(6, 0); B(-6, 0); C(O, 3); D(O,-3)b. A(-4, 3); B(8, 3); C(2, 1); D(2, 5)

(x - h)2 (y - k )29. Show that the equation for an ellipse, 2 + b2 = 1, reduces to the

aequation of a circle if a = b.

10. Find the equation of the ellipse whose vertices are (-4, 2) and (10, 2) andwhose minor axis has length 10.

n(n - 3)11. Prove that the number of diagonals in an n-sided polygon is 2

Continued

82


F-1, F-2, F-3

Exercise 37: Circle and Ellipse

12. Solve for n: 2nC3 = 44.nC2 3

13. Prove this identity: see" x csc" x = see" x + csc" x.

214. Solve for x: b" +x = 1.

15. How many numbers greater than a million can be formed with the digits0, 2, 2, 3, 3, 3, 4? (Only the indicated repetitions are allowed.)

16. A circle has an area of 251l'units" and is tangent to both the x-axis and y-axis.Find one possible equation for this circle.

17. Using the change of base formula, evaluate log516.

18. Given log, 2 = 0.63 and log, 5 = 1.465, use the laws oflogarithms to evaluate5log, -.2

19. Solve the equation for ()where 0° :s; (}:s; 360°:

()see 2" = -2.9413

20. • Y. Given the graph of fix):

x) a. Graphy = 3 - 2fix)

b. Graph y = fi-x).

83


F-1, F-2, F-3

Exercise 38: Hyperbola

1. Sketch the graphs of the following hyperbolas.

a. y2 _ X2 = 1 b. 2x2 _ y2 = 8

2. Sketch the graphs and state the coordinates of the centre for the followinghyperbolas.

v2 X2a. .t...- - - - 1 = 09 16

b. -(x + 6)2 + (y + 7)2 = 19 25

3. Sketch the graphs and state the coordinates of the centre for the followinghyperbolas.

a. X2 - y2 + 4x + 16y - 69 = 0b. 25x2 - 16y2 - 200x - 96y - 144 = 0

4. A hyperbola has vertices at (4,0) and (-4,0), and one of its asymptotes has

slope ~. Find its equation.

5. Find the equation of the hyperbola whose centre is at (2, 1), one vertex is at(2, -4), and the equation of one asymptote is 5x - 7y = 3.

6. Identify each of the following equations as representing a parabola, ellipse, orhyperbola. State whether the graph is stretched or opens along the x-axis ory-aXIS.

X2 y2a. -+-=1

2 8c. y2 = 12xe. X2 + 4y = 0

b. 2x2 - y2 + 32 = 0

d. X2 - 2y2 = 4f. 3x2 + 2y2 = 18

7. Identify each of the following equations as representing:

i. Circle ii. Parabola iii. Ellipse

a. X2 + 4y2 - 2x + 32y =-61b. 4X2 - y2 + 90x + 8y + 200 = 0c. 2x2 + 2y2 + 64x - 3y = 400d. x = y2 + 4y + 2

IV. Hyperbola

Continued

84


F-1, F-2, F-3


8. a. On the same coordinate system, sketch the graphs of X2 - y2 = 2, X2 - y2 = 1,and X2 - y2 = O.l.

b. Sketch the graph of X2 - y2 = o. Describe this graph in words.

9.

10.

11. Simplify:

12. Solve for x: In(x + 5) + In 5 = In 65.

A rectangle is circumscribed around2 2

the ellipse :6 + ~ = 1, and a circle is

circumscribed around the rectangle.Find the equation of the circle.

P is the point (10, 0). A and B are onthe parabola y2 = kx. If the area ofL1 AOB is 40, find the value of k.

13. Express in exponential form: loglo10000 = 4.

14. Prove the identity: _1_ + _1_ = 1cos" () sin" () sin" ()- sin" ()

15. In how many ways can 10 examination papers be stacked so that the paperswith the highest and lowest scores never come together?

Continued

85


F-1, F-2, F-3


116. Show that log e = -1-.n 10

17. Solve and check: 10g5(x + 1) - 10g58 = 10g5(x - 3) - 10g56.

18. Solve the equation cos x + 1 = sin x on [0, 2lr], using technology.a. State your answer(s) to three decimals.b. Examine your answers for part a. What do you suppose the exact answers

are? Confirm your guess by checking.

19. Solve the following equation where 0°~ 8 ~ 360°:

3tan28-7sec8 =-5

20. Find the term containing x' in (2x' - 2~r

86


G-1, G-2

Exercise 39: Sample Spaces

1. Sketch the sample space for each of the followingevents.a. Tossing a coin.b. Rolling a 6-sided die.c. Tossing two coins simultaneously.d. Rolling two 6-sided dice.e. Tossing three coins simultaneously.

2. A bus is scheduled to arrive at a train station at any time between 07:05 and07:08 inclusive. A train is scheduled to arrive between 07:07 and 07:09inclusive. The arrival of a bus at 07:06 and a train at 07:09 can be representedby the point (6, 9). Times are expressed in whole minutes.a. Sketch the sample space for the situation described above.b. How many points are there in the sample space?c. How many points have the bus and train arriving at the same time?d. What is the probability of the bus arriving after the train?

3. Classify the followingevents as independent or dependent.a. Tossing a coin and rolling a die.b. Cutting a deck of cards, removing a card, cutting the deck again, and

drawing another card.c. Rolling a die two consecutive times.d. Selecting a card from a deck of cards, replacing the card, and then selecting

another card from the deck.

4. If the probability that an event will occur is P, what is the probability that itwill not occur?

5. One card is drawn from a standard deck of 52 cards. What is the probability thata. it will be a jack?b. it will be a club?c. it will be a jack of clubs?

6. A box contains three red balls and seven blue balls.a. If one ball is drawn, what is the probability that it will be red?b. If one ball is drawn, what is the probability that it will be blue?c. What is the sum of these probabilities? Why?

7. In a single cast with two dice, what is the probability that the sum will bea. less than five? b. five or less?

Continued

87


G-1, G-2


8. If one card is drawn from a deck of 52 cards, what is the probability that itwill be a jack or higher? (Acesare high cards.)

9. Prove the identity: 1 + 1 = 2csc2x.l+cosx 1-cosx

10. Solve and check: 10g3X2 + 10g3x3 = 10g316x.

12. Given that 10gb 3 = 0.613 and log, 4 = 0.774, find log, 108.

13. In the expansion of (p + q)lO:

a. write the first three terms and the last three terms.b. what is the coefficientof the term containing p7?c. which term contains q5?d. how many terms are there in the complete expansion?

14. Complete the square and sketch the graph of:y2 - 12x - 4y + 40 = O.

15. Match each of the equations on the left with one of the geometric figures onthe right.a. y = X2 + 3x - 2 circleb. x + y2 = 5 ellipsec. 4x2- 9y2 = 36 hyperbolad. X2 + y2 + 6x + 8y = 4 parabola with horizontal axis of symmetrye. y = 3x + 6 parabola with vertical axis of symmetryf. 6x2 + 5y2 = 30 vertical lineg. y = 4 horizontal lineh. x = 3 oblique line

16. If 20 g of a radioactive material decomposesexponentially to 14 gin 10 days:a. find the half-life of the material.b. find the amount left after 17 days.

17. Find all values of e, 0 ~ e ~ 2n such that:a. sine = -0.419 b. tan e = -1.79

Continued

88


G-1, G-2


18. How many ways can 11 people be seated around two circular tables, one withsix chairs and the other with five chairs?

19. Solve the following equation if the domain is all real numbers: csc2

~ - 1 = 4csc e 9

20. If $1200 is invested at 6% compounded semi-annually for 12 years, how muchwould it be worth?

89


G-3

Exercise 40: Probability of Independent and Dependent Events

1. Twocards are drawn from a well-shufiled ordinary deck of 52 cards. Find theprobability that they are both aces if the first card isa. replaced.b. not replaced.

2. A fair die is tossed twice. Find the probability of getting a 4, 5, or 6 on the firsttoss and a 1, 2, 3, or 4 on the second toss.

3. One bag contains four white marbles and two black marbles; another containsthree white marbles and five black marbles. If one marble is drawn from eachbag, find the probability thata. both are whiteb. both are blackc. one is white and one is black.

4. One card is drawn from a deck of 52 cards. It is then replaced and a secondcard is drawn from the pack. What is the chancea. that both cards will be red?b. that both cards will be hearts?

5. There are ten tickets in a hat, numbered from 1 to 10. If two tickets are drawnwithout replacement, what is the probability that the sum of the numbers ofthem will be odd?

6. If a coin is tossed three times, what is the probability that it does not fall"tails" all three times?

7. A business woman wrote three letters and addressed three correspondingenvelopes. Her secretary put the letters in the envelopes without checking theaddresses. What is the probability that each letter was placed in its correctenvelope?

8. Write the first four terms of(2x3 -3y2f.

9. If a box contains two red, three white, and four blue marbles, what is theprobability that a marble drawn will be red or white?

10. A box contains five yellow candies and seven black candies. Only two candiesare drawn from the box. What is the probability that the two candies will bothbe yellow or both be black?

Continued

90


G-3

Exercise 40: Probability of Independent and Dependent Events

11. Sketch a sample space for rolling two 4-sided dice.

12. Give an example of two dependent events.

13. What is the probability of getting an odd number on a single throw of a 6-sideddie?

14. What is the probability of getting a face card on a single cut of a deck of cards?

15. Find (to two decimal places) the area of the circle whose equation is:X2 + y2 _ 8x - 4y + 19 = O.

16. Solve for x: 42x= 2x(x-2).

17. Compare the graphs of fix) = 2X and g(x) = 4(2X).

19. Find the middle term and the next term in (2 x _ --;)103 2x

20. Simplify: eln3-ln2

•

91


G-3

Exercise 41: Combining Probabilities

1. If the probability of winning a game is J:.., what is the probability of notwinning the game? 31

2. A shootout consists of Teams A and B taking alternate shots on goal. The firstteam to score wins. Team A has a probability of 0.3 of scoring with anyoneshot. Team B has a probability of 0.4 of scoring with anyone shot.a. If Team A shoots first, what is the probability of Team B winning on its first

shot?b. If Team A shoots first, what is the probability of Team A winning on its

third shot?

3. What is the probability of drawing a face card on a single cut of a deck ofcards?

4. What is the probability of getting a sum greater than nine on a single throw oftwo dice?

5. A bag contains four red candies and seven black ones. Twocandies are drawnfrom the bag. What is the probability that they will both be red if the firstcandy is eaten before the second one is drawn?

6. What is the probability of getting a king or a red card on a single cut of astandard deck of 52 cards?

7. What is the probability of rolling a five with a die and tossing a head with acoin?

8. What is the probability of two children in a two-child family both being girls?

9. What is the probability of drawing a red card and then a black card out of adeck of 52 playing cards if the first card is not replaced before the second cardis drawn?

10. A track coach buys three stopwatches.a. If one out of every 200 stopwatches is defective, what is the probability that

all three of the new stop watches are defective?b. What is the probability that all three work properly?

11. Sketch the graph of 4x2 - 9y2 + 32x + 18y + 91 = O.

Continued

92


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12. A bag contains five red balls and seven black balls. A ball is drawn from thebag, replaced, and then a second ball is drawn. Are these events independentor dependent?

13. Determine the probability p for each of the following events:a. an odd number appears in a single toss of a fair die.b. an ace, ten of diamonds, or two of spades appears in drawing a single card

from a well-shuffled, ordinary deck of 52 cards.c. a tail appears in the next toss of a coin if, out of 100 tosses, 56 were heads.

14. Solve the equation for 0 ~ 8 ~ 21r. Give exact values for 8.

tan28 = sec8 + 1

15. Match each of the geometric figures below on the left with one of the equationson the right.

a. circle 3x2 + 2y2 = 6b. parabola 4x - 2y + 5 = 0c. hyperbola y=x2-2x+3d. ellipse X2 _ y2 = 1e. semi-parabola x=3f. horizontal line X2 + y2 = 4g. vertical line y =Ixlh. oblique line y-2=0l. a V-shaped figure y =.JX

16. Solve: log, x = O.

7 P he id tit cosx sinx sinx-11. rovet e i en 1 y: -----=---cscx tanx secx

Continued

93


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18. Find the area of the shaded region.

19. Find the term containing -;. in (2X7 _ ~JI0X 3 2x

20. Write 500 as a power of 11.

94


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Exercise 42: Conditional Probability I

1. A student randomly selects one of the followingthree boxes with the numberof coloured marbles indicated:

a 3 red2 blue

b. 3 red3 blue

c. 2 red3 blue

She then selects a marble at random from that box. What is the probabilitythat the marble will be red?

2. In room 1 there are 12 boys and eight girls. In room 2 there are seven boysand nine girls. If I select a student at random from one of the rooms, what isthe probability that the student is a girl?

3. There are two boxes. Box 1 contains two red marbles and one green marble.Box 2 contains one red marble and one blue marble. A box is selected atrandom and then a marble is selected from that box. What is the probabilitythat the marble is red?...••

4. Urn I contains five red, three white, and two blue marbles. Urn II containsthree red and seven blue marbles. We throw a die to determine which urn toselect. If the die shows a "I" or "2",we use Urn I, otherwise Urn II. A marble isdrawn at random from the chosen urn. Find PfR), pew), PCB).

5. There are three urns, I, II, III. Urn I contains three chips, numbered 1, 2, 3.Urn II contains two chips, numbered 1, 2. Urn III contains two chips,numbered 2, 4. An urn is chosen at random and a chip is drawn at random.What is the probability that the chip drawn is numbered 2?

6. Referring to exercise 5:a. find P(chip has an even number).b. find P(chip has a number less than three).

7. Seeing two gum machines, a boy doesn't know which to use. He flips a coin to

decide. It happens that machine A gives three pieces with probability! ' and one

piece with probability: . Machine B gives one or two pieces equally

often. Find the probabilities that the boy receives one, two, or three pieces.

Continued

95


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8. Kate and Jane playa simple game. Kate has two disks, each one red on oneside and green on the other. Jane has one such disk. At a given signal, Janeand Kate each put a disk on the table. If they show the same colour, Katetakes them both; if the colours are different, Jane takes both of them. Theyplay until one player has no more disks or until they have compared disksthree times.a. Make a tree to show the progress of the game.

b. We know that the probability that Jane wins any particular play is .!.. Findthe probability that Jane wins the game. 2

c. Find the probability that Kate wins.d. Find the probability that neither wins.

9. How many five-card hands can be dealt from a deck of 52 so thata. four of the cards have the same face value?b. three of the cards have the same face value and the other two have

different face values?

10. Solve for x: 7%+ 1 = 343.

11. A bag contains five red marbles, six blue marbles, and seven yellow marbles.a. Sketch a sample space for drawing two marbles from the bag.b. Does it make a difference to the sample space if the marble is replaced or

not after each trial?c. If each marble is not replaced, are the events dependent or independent?d. What is the probability that the first marble out of the bag is red or blue?

12. If the probability that the Blackhawks will win the Stanley Cup is ~ and that

the Maple Leafs will win the Cup is 4 , what is the probability thateither the9

Hawks or the Leafs will win the Cup?

13. If the probability of Man of War winning a race is ..!. and the probability of

Citation winning the same race is ~, what is the p~obability that either7

Man of War or Citation will win this race?

14. What is the probability that neither of the horses mentioned will win the racein question 13?

Continued

96


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15. Complete the following:a. If A and B are mutually exclusive events, then P(AAND B) = _

b. If A represents the complement ofA, then P(A)+ P(A) = _c. For two independent events A and B, peAand then B) = _d. For two dependent events A and B, peAand then B) = _

16. Sketch the graph of: 25x2 - 49y2 - 150x - 196y - 1196 = O.

17. Prove this identity: see e - tan e sin e = cose.18. Solve and check: log, 25 = -2.

19. If log2(cosx) = log3(~) and 0 ::;;e ::;;2Jr, find the value(s) ofx.

20. What is the sum of the angles in a 10-sided polygon?

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Exercise 43: Conditional Probability II

1. Twin brothers, Ed and Jim, deliver the evening newspaper six nights a week.Ed delivers on two nights, chosen at random, and Jim on the other nights. Thejride by a house on their bicycles and throw the newspaper onto the porch. Theprobability that Ed hits the door is 3 , and the probability that Jim hits the

5door is ~. One night, while Mr. Jones is watching TV before dinner, he hears

10a paper crash against the door. He sighs to Mrs. Jones, It must be Ed's nightwith the papers. What is the probability that he is right?

2. A factory has four machines producing axe handles. Machine I produces 30% ofthe output; machine II produces 25%; machine III produces 20%; and machineIV produces the rest. Defective handles produced by each machine are 5%, 4%,3%, and 2%, respectively. A handle chosen at random from the total output ofthe factory is examined and found to be defective. What is the probability thatit was made by machine I?

3. In a two-year college, 60% of the students are freshmen, and 40% aresophomores. Of the freshmen, 70% are boys. Of the sophomores, 80% are boys.A student is chosen at random. Find the probability that the student isa. a girl.b. a freshmen, given that a girl was chosen.

4. It is known that 10% of a population has a certain disease. A blood test for thedisease gives a correct diagnosis 95% of the time. The test is equally reliablefor persons with or without the disease. What is the probability that a personwhose blood test shows the disease actually has the disease?

5. Seeing two gum machines, a boy doesn't know which to use. He flips a coin todecide.

It happens that machine A gives three pieces with probability.! and one piece5

with probability 4 . Machine B gives one or two pieces equally often. The boy5

uses one machine without noticing which one. He receives one piece of gum.What is the probability that he used machine A?

6. There are three urns, I, II, III. Urn I contains three chips, numbered 1, 2, 3.Urn II contains two chips, numbered 1, 2. Urn III contains two chips,numbered 2, 4. An urn is chosen at random and a chip is drawn. What is theprobability that Urn II was selected, given that the chip drawn is numbered "2"?

Continued

98


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7. The probability that Jane will be on time for her math class on Day 1 is .!..4

However, if she is on time one day, she is less concerned about punctuality the

next day and the probability of punctuality is .!.. If she is late one day she makes2

a greater effort the next day and the probability of punctuality is ~. If Jane is4

on time on Monday, find the probability that she is on time on Wednesday.

8. Solve for x: eln(4x - 1) = 7.

9. Graph: Y=3CO{X- ~}

10. Solve for x: a= - 4) = 243.

11. The probability that Gallant Fox will win the first race is 2 and that Nashua5

will win the second race is .!.. What is the probability that both horses will wintheir respective races? 3

12. What is the probability that both horses mentioned in question 11will not wintheir respective races?

13. What is the probability of drawing any ace or the queen of hearts from a deckof cards?

14. Here is a game you might play. An urn contains seven red and three greenballs. You are to select a ball, note its colour, and replace it. Your opponent isthen to select a ball. R, is the event that you select red; R, is the event thatyour opponent selects red; Gy is the event that you select green, etc. Youwin ifthe ball your opponent selects has the same colour as the ball you selected.What is your probability of winning? Make a tree diagram of possibleoutcomes.

15. Graph fix) = log, (X2) and state the domain, range, intercepts, and asymptotes.

16. Solve and check: 1 - log(x - 4) = log(x + 5).

Continued

99


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2 2

17. The area of the ellipse :2 + ~ = 1 is given by ttab, What is the area of the

ellipse 25x2 + 9y2 - 225 = O?

18. This is the graph of fix). Make a graph ofg(x) = 2fix + 1) - 2.

19. Prove the identity: sin(a - fJ)· cos fJ +cos(a - fJ)· sin fJ = sin a.

20. If.!< logloX < 2, what are the possible values for x?2

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Exercise 44: Probability Using Permutations and Combinations

1. What is the probability of holding all four aces in a five-card hand dealt from astandard 52-card deck?

2. Three people form a lineup at the grocery store. What is the probability thatthey line up in descending order of age?

3. A committee of five people is to be selected from 10 males and eight females.What is the probability that there are exactly three males on the committee?

4. A family of five children is known to have at least two girls. What is theprobability of this family having exactly four girls?

5. Five books, each of a different colour, including one red and one green book,are placed on a shelf. What is the probability of the red book being at one endand the green book at the other?

6. Nine horses are entered in a race. What is the probability of choosing the first,second, and third place finishes in order?

7. The school yearbook is to be produced by a committee of two boys and threegirls, chosen by lot from five boys and six girls. One of the boys is theboyfriend of one of the girls. What is the probability that both will be chosen tobe on the committee?

8. What is the probability of getting exactly four face cards when drawing fivecards from a standard deck of cards?

9. If the letters in TORONTO are rearranged, what is the probability that the Tsare together?

10. If all the letters in BANANAare rearranged, what is the probability that theNs are not together?

11. A ball is drawn at random from a box containing six red balls, four white balls,and five blue balls. Determine the probability that it isa. red b. white c. blued. not red e. red or white

Continued

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Exercise 44: Probability Using Permutations and Combinations

12. The probability that Tony will purchase a house in Winnipeg is 2. The9

probability that if he moves to Winnipeg he will marry Angelina is JL. What is20

the probability that Tony will settle in Winnipeg and marry Angelina?

13. What is the probability of rolling a three or a five with one roll of a die?

14. John is writing a math test. He estimates that the probability of getting the

answer to the next question correct if the previous one was correct is 4. However,5

if the previous answer was wrong, the probability of getting the next answer

correct is only 2. If the probability of getting the answer to the first question5

correct is ~, find4

a. the probability that the answer to the second question is correct.b. the probability that the answer to the third question is correct.

15 P his identi tan" x + 1 2. rove t IS 1 entity: = sec x - tan x.tan z + 1

( 5)2 (+ 1)216. Sketch the graph of: x ;6 + y 64 = 1.

17. Solve for (J over the real number set for: tan" (J + 4 sin (J = see" (J - 2.

18. Alaine wishes to invest $8000 for five years so that she will have a finalamount of $12 500. If the money is compounded semi-annually, what interestrate is required for this investment?

19. There are 20 students in a senior class who need to register for a science class.Unfortunately, there are only eight spaces left in biology, six in physics, andsix in chemistry. How many ways can the classes be filled?

20. Use the change of base formula to evaluate the logarithm to six decimalplaces: log, 92

, 102.


H-l

Exercise 45: Geometric Sequences

1. For each sequence below, indicate whether it is geometric, arithmetic, orneither. If it is geometric, give the value of r (the common ratio) and if it isarithmetic, give the value of d (the common difference).

a. 4, 6, 8, 10, ... b. 3, 6, 12, 24, ...

d. 80, 40, 20, 10, e. 1, 4, 9, 16, 25, ...g. 1, -5, 25, -125, .

c. 18, 15, 12, 9, .

f. 2, -fB, 4, -/32, .

2. Consider the exponential function fix) = 3x• Find fi1), fi2), fi3), fi4). What type

of sequence is this?

3. Write the first three terms of the geometric sequences generated by each of thefollowing exponential functions:

a. fix) = 2X b. g(x) = 4x-1 c. hex) = 2·3x d. F(x) = 16(~J

4. Write exponential functions which generate each of the following geometricsequences:

a. 4, 8, 16, 32, ...

c. 1, -2, 4, -8, 16, ...

b. 6, 18, 54, 162, ...5 5

d. 10,5, 2' 4' ...

5. a. Find the 8th term of the sequence 3, 6, 12, 24, ...b. Find the nth term of the above sequence.

6. A sum of $10 000 is invested at 6% compounded annually.

a. What is the value of the investment at the end of years 1, 2, 3?b. What is the value of the investment at the end of year n?c. How many years will it take for the investment to double?

7. If a, b, c form a geometric sequence, show that log a, log b, and log c form anarithmetic sequence.

8. What is the value of 61og6 17?

9. If 0 < k < it, and log(cos k) = log 3 - log 4 , find k.2

10. If two dice are thrown together, what is the probability that their sum will begreater than eight?

Continued

103


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11. Find the value of csc8if tan 8 > 0, and cos8 = _15.17

12. Match each of the equations on the left with one of the geometric figures onthe right.

a. X2 + y2 - 2x + 6y = 0 ellipseX2 y2

b. 4 - 9= 1 circle

c. 2x(x + 3) = y parabola

d. ~ + L = 1 semi-parabola2 3

e. 3x2 + 2y2 - 5 = 0 hyperbola

f. y=.,,)x-2 line

13. A bucket contains five white balls and three black ones. Twoballs are drawn.a. If the first ball is not replaced before the second one is drawn, what is the

probability that one will be white and one will be black?b. If the first ball is replaced before the second one is drawn, what is the

probability that one will be white and one will be black?

14. An unusual genetic trait occurs in 0.1% of the population. The reliability ofatest to discover the trait is: if the person has the trait then the test is positive95% of the time, but the test is also positive 2% of the time for those who donot have the trait.a. Construct a tree diagram to represent the population and the testing

reliability for the genetic trait.b. If a person is selected at random, what is the probability that the person

will test positive?c. If a person is selected at random, what is the probability that the person

will have the trait and test positive?d. If a person is selected at random, what is the probability that the person

who tests positive has the trait?

.. sin38+csc38 .15. Prove the identity: = sin'' 8+cot"8.sinfl+ csc8

16. Solve and verify: log7(2x+ 2) - log7(x- 1) = log7(x+ 1).Continued

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17. Solve the equation for ()where 0 ~ ()~ 2n:

4cos()+ 1_ 2cos()- 1 = 13 2

18. Given fix) as per diagram:

-5

Find an equation which represents the function fix).

19. True or false? (Show your proofl)a

logs 2" = logs a-2log5 bb

20. a. How many bracelets can be made by putting six different-coloured beads ona ring if 10 different colours are available?

b. How many are there if the red and green beads must be used?

105


H-2

Exercise 46: Geometric Series

1. For each of the following,write the series in expanded form and find its sum.5

a. L2kk=l

6

c. L(2k-4)k=l

4

d. L2kk=l

5 42. Write L 12(2)k-3 and L3(2k) in expanded form. What do you observe?

k=2 k=l

3. Write the followingseries using sigma notation.a. 3 + 9 + 27 + 81 b. 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20c. 3 + 6 + 12 + 24 + 48 d. -1 + 4 - 9 + 16 - 25 + 36

4. Examine the series in question 1 above. Which of these series are arithmetic?Which are geometric?

5. Consider the sequence 3, 6, 12, 24, ...

a. Find the 8th term.b. Find the sum of eight terms.

6. Consider the geometric sequence with t1 = 1000 and r = 1.05.a. Write the first three terms.b. Find the sum of the first 20 terms, expressing your answer to two decimal

places.

7. Find the sum of the first 10 terms for each of the following series. If thenumerical answer is not an integer, state it accurate to two decimal places.a. 1 + 2 + 4 + ... b. 128 + 64 + 32 + ...

8c. 8 + 12 + 18 + 27 + ... d. 24 + 8 + - + ...3

178. Evaluate L8(1.2)k

k=3

6 159. Evaluate L-k

k=12

10. Find the probability of:a. getting a card lower than a six on a single cut of a deck of cards.b. getting a sum of five on a single throw of two dice.c. getting four heads on four successive tosses of a coin.

Continued

106


H-2

Exercise 46: Geometric Series

11. In earlier mathematics courses, you learned that the sum of an arithmetic

series is: S = n (t1 + tn). Use this to find:2

20

a. I(3k+ 1)k=l

40

b. I(2k-6)k=4

12. Which of the followingpairs of events would you classify as independent?a. Rolling a die and cutting a deck of cards.b. A worker is well trained. A worker meets the production quota.c. Twoconsecutive tosses of a coin.d. Drawing two balls from a bag of seven red balls and three blue balls if the

first ball is not replaced before the second draw.

13. Solve for x: 3(52x- 1) = 75.

14. Graph fix) = -log(x - 3). State the domain, range, intercepts, and asymptotes.

5 P he id . sin x cosx . 2 21. rove tel entity: --+-- = sm x+cos x.cscx secx

16. Solve and check: 31og24 = x.

17. A circle has its centre in Quadrant I and cuts the x-axis at (1,0) and (7,0). Ifthe centre lies four units above the x-axis, find the equation of the circle.

18. a. Find the first three terms of (2~2_4x3)

8

•

b. Find the first term that does not have an x in the denominator.

19. How many five-card hands can be formed from a deck of 52 cards with onlyone pair the same and the other three cards having different face values?

20. Solve and check: log(2x + 1) -log(x + 3) = o.

107


H-3

Exercise 47: Infinite Geometric Series

1. Consider the series 4 +2 + 1+ .!.+ ...2

a. Find the sixth term.b. Find the sum of six terms.c. Find, to four decimal places, the sum of 10 terms.d. Find the sum to infinity.

2. Find the sum for each of the infinite geometric series:

8 8a. 8+-+-+ ...3 9

8c. 6+4+ 3 + ...

3 3b. 6-3+---+ ...2 4

111d. 1--+---+ ...248

3. Find:

~ 16a. L-k

k=12b. f 1~

k=23

4. A 4-by-4 square is divided into four congruentsquares. The bottom left square is shaded. Thetop right square is divided into four congruentsquares and the bottom left is shaded. Thisprocess is repeated indefinitely. What is thetotal area shaded?

5. A square measures 4 by 4. The midpoint ofits sides are joined to form a new square.This process is continued indefinitely. Whatis the sum of the perimeter of all thesquares?

1111111111111

108

Continued


H-3


c6. ~ ABC is a right triangle with L. A = 30°,L. C = 60°,

and BC = 10.A flea at B travels along a lineperpendicular to AC. When it reaches AC, it turnsand travels back to AB, taking the perpendicularroute. It travels back and forth between AB and ACin this manner until it reaches A. How far does ittravel?

7. A ball is dropped from a height of two metres. Each time it strikes the ground,it rises three-quarters of the distance it has fallen.a. How far does it rise after it strikes the ground the third time?b. How far does it travel before it comes to rest?

8. Find:10

b. L 15(1.6tk=2

9. An organization sets up a telephonetree. Each employee phones exactlytwo other employees. How manylevels (see diagram) are needed toreach all 1000 employees?

LEVEL 1

LEVEL 2

LEVEL 3

10. If a, b, c form an arithmetic sequence, prove that 2a, 2b

, 2C is a geometricsequence.

11. On a single cut of a deck of cards, what is the probability of getting a heart ora queen?

12. Five men check their coats at a wedding reception but lose their tickets. If thecoats are handed out in a random way, what is the probability that each getshis own coat?

13. Sketch the graph of f(x) = 3(2.>:-2)- 1. State the intercepts, the domain, andrange.

Continued

109


H-3


_~ 114. Express in logarithm form: 81 2 = -.

9

15. Prove that sin20+ sin40+ sin60+ ...= tan20.

16. A multiple choice test has 20 questions with five choices for each question. If astudent guesses every answer, what is the probability of getting 18 out of 20on the test?

17. Solve and verify. Express your answer to one decimal place.

18. Solve the equation for 0 where 0° $ 0 $ 360°: -J3 - tan 40 = O.

19. Complete the square and sketch the graph of: X2 - 9y2 + 6x = O.

20. Solve for x: log7(x + 1) + log7(x - 5) = 1.

110


Exercise 48: Review I

1. A committee of five shareholders is to be chosen by lot from seven men andfive women. What is the probability that the committee will consist of threemen and two women?

2. If a coin is tossed twice, what is the probability that it will land on "tails" bothtimes? If two coins are tossed simultaneously, what is the probability they willboth show "tails"?

3. For a given e, cot e > 0 and sine = - 172 ' find sec e.

4. Find the probability of getting a face card or a heart on a single cut of a deckof cards.

tan x5. Prove the identity: +1 + secx

1+ secx=tanx

2sin x

6. Solve and check: log, Vi6 = x.

1 27. Solve and check: In x = "2ln 4 + SIn 8.

8. Graph y = 2X and find the domain, range, zeroes, and y-interceptts).

9. Write as a single logarithm: ~ (log, A + 3 log, B) - 2 (logs C + log, D).

10. Given that log, 2 = p, log, 3 = q, and log, 5 = r, find log, ~~ in terms of p, q,and r.

11.

y = fix)

Multiple Choice. The diagram at the left represents y = fix). The diagram atthe right represents

a. v = Ifix) I b. y=filxl) c. y=-fix) d. y=fi-x)Continued

111


Exercise 48: Review I

12. The sum of an infinite geometric series is 30. If the common ratio is !,findthe first term.

13. Find the centre of the ellipse 2X2 + 3y2 - 6x + 18y - 12 = O.

14. a. How many arrangements can be made using all the letters in the wordCHEESE?

b. If one of these arrangements is chosen at random, what is the probabilitythat it starts and ends with an E?

15. 4

-4

Write an equation for this curve usinga. the cosine function. b. the sine function.

5 25 12516. Consider the geometric series: -1 + 4 - 16 + 64 - ...

Sue said that the sum to infinity was four. Do you agree? Explain.

317. Express 5" revolutions in

a. degrees. b. radians.

3 rr 5 tt18. Ifsina= 5",0 < a < 2' and sinf3= 13,0 < f3 < 2' find cos(a + f3).

19. Youare dealt a seven-card hand. What is the probability that it consists of twopairs and three of a kind? For example, one hand could be two kings, two 3sand three 7s. Do not simplify your answer.

20. Find an exact value for (tan rr+cos rr)sin 7rr.363

112


Exercise 49: Review II

1. a. Give a sample space which describes the result of tossing a coin four times.b. Use the sample space to find the probability of getting at least three tails

when a coin is tossed four times.

2. Prove the identity: tanx + 1 = sinx + C?sx.1- tanx cosx - smx

3. Four cards are dealt from a standard deck of 52 cards. Find the probabilitythat the hand contains exactly two face cards.

4. A square has vertices (-4, 1), (2, 1), (2, 7), (-4, 7). Find the equation of thecircle which is inscribed in this square.

5. Find the solution(s) and check.

log5(x2 + 2x + 5) - log5(x - 5) = 2

6. Find the exact values of ()for the equation where 0 ~ ()~ 21t.

tan'' ()= tan ()

7. Solve for ()over the real numbers: tan" ()- see()- 1 = o.

8. Write an equation representing this semicircle:

y.

9. Use the laws of logarithms to write as a natural logarithm: {(x) = x-J X2 + 1

10. Find the values of ()for the equation whose domain is all real numbers:

2sin'' ()- sin ()= 3

11. If two dice are thrown together, what is the probability that they will give asum of eight?

Continued

113


Exercise 49: Review II

12. Solve for x: (%T = 8.

13. a. A die is thrown twice. What is the probability of throwing a five first andthen a four?

b. A die is thrown twice. What is the probability that one throw will give afive and the other a four?

14. Evaluate: log1 32.2

15. Prove this identity: tan a + cot a = sec a csc a.

16. Solve and check: 31og3x = 4.

17. Rewrite the expression in a form with no logarithms of products, quotients, orpowers:

18. a. On the same set of axes, draw the graphs of the equations X2 + y2 = 9 and2X2 + y2 = 13.

b. Verifyyour answer by solving the system algebraically.

19. a. How many ways can 11children sit in a row if three good friends must betogether?

b. How many ways could this be done in a circle?

20. Sketch the graph of:a. y = log2x

(-~ --fil

21. If P(8) = -3-' -3-)' find the exact value of sin(8 + n).

114


Exercise 50: Review III

• 10 241. FlndI~.

k=32

2. Solve -J2x+ 1--Jx+4 = 1.

3. Sketch the graph of Ix I+ Iy 1=4.

34. If 00 < 8 < 1800, and cos8 = - -, find exact values for:

4a. sin28b. cos28

5. The line x + y = 8 intersects the circle (x - 1)2+ y2 = 25 at points A and B. Findthe coordinates of these points.

6. If C is the centre of the circle in question 5, find the measure of L ACB to thenearest tenth of a degree.

7. For a period of its life, a tree grows according to the formula D = Doekt where Dis the diameter in centimetres of the tree t years after the beginning of theperiod. After two years, the diameter of the tree is 15.62 cm. After five years,the diameter is 21.724 cm. Find the value of Do and k.

8. Find log, 200. (Express your answer with four decimal places of accuracy.)

9. How many digits are there in the number 45362?

10. Solve for 0 :::;;8:::;;2n: sin 8 + 2sin 8 cos8 = O.

11. Write and simplify the first three terms in the binomial expansion Of(2x _ ~ y)7.

12. Three boys and three girls are sitting on a bench. How many arrangementsare possible if the sexes must alternate?

Continued

115


Exercise 50: Review III

14. Consider a geometric sequence where t3 = 16 and t5 = 10.

a. Find r.ee

b. Find L/k. Give your answer accurate to two decimal places.k=l

16. Suppose a, b, and c form a geometric sequence. If abc = 8, find b. Is it possibleto find a and c?

18. How many four-letter "words" can be formed using four of the letters from theword CANADIAN?

19. If logM = x and log.M = y, prove that logabM=~.x+y

2cos2820. Prove that = cot 8 - tan 8.sin 28

116

grade 12 cumulative exercises

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