Senior 3 Pre-Calculus Mathematics -...

Senior 3Pre-Calculus MathematicsCumulative Exercises

A Supplement toA Foundation forImplementation

Man itoba EducationEducation et Formation

fining profess onn 1.UManitoba

Cumulative Exercises Senior 3 Pre-Calculus Mathematics

CONTENTS

Acknowledgements iii

Introduction xi

Exercise New Topic Presented Outcome(s)

1 Quadratic Functions 1 A-1, A-2

2 Graphs of Quadratic Functions 1 4 A-l, A-2

3 Graphs of Quadratic Functions 2 6 A-2, A-3

4 Transformations of Quadratic Functions 1 9 A-3



7 Applications of Quadratic Functions 15 A-4

8 Trigonometric Equations 1 18 B-1

9 Trigonometric Equations 2 20 B-1

10 Trigonometric Equations and AmbiguousCase Problems 22 B-l, B-2

11 Ambiguous Case Problems 24 B-2

12 Review 1 26

13 Quadratic/ Trigonometric Equations 27 C-1, B-1

14 Quadratic Formula 29 C-1

15 Solving Quadratic Equations by Graphing 31 C-1

16 Nature of Roots 33 C-2

17 Nonlinear Equations 35 C-3, C-4

18 Radical Equations 37 C-5

Vii

Senior 3 Pre-Calculus Mathematics Cumulative Exercises


19 Rational/Absolute Value Equations 39

20 Review 2 41

21 Circles on a Coordinate Plane 43

22 Distance between Points and Lines 45

23 Verify and Prove Assertions in Plane Geometry 47

24 Systems of Linear Equations in Two Variables 49

25 Systems of Linear Equations in Three Variables 51

26 Systems of Nonlinear Equations 53

27 Graphing Linear Inequalities in Two Variables 55

28 Quadratic, Absolute Value , and Rational Inequalities 57

29 Review 3 59

30 Circle and Polygon Properties 1 61





35 Circle Properties 71

36 Polygon Properties 73

37 Wages (Hourly) 75

38 Wages (Commission and Net Income) 77

39 Property Tax 79

C-5

D-1

D-1

D-2

D-3

D-4

D-5

D-6

D-7

E-1, E-2, E-3

E-1, E-2, E-3

E-1, E-2, E-3

E-1, E-2, E-3

E-1, E-2, E-3

E-1, E-2, E-3

E-1, E-2, E-3

F-1

F-1

F-1

Viii



40 Unit Prices, Exchange Rates, and Reconciliation ofBank Statements 81 F-1, F-2

41 Budgeting 1 85 F-3

42 Budgeting 2 91 F-3

43 Exponential Growth 95 F-5

44 Interest 98 F-5

45 Inductive and Deductive Reasoning 100 G-1

46 R i 4 102ev ew

47 AND, OR, NOT, and Venn Diagrams 104 G-2

48 C l 107ounterexamp es G-3

49 Converses, Contrapositives, If. ..Then... 109 G-4

50 Direct and Indirect Reasoning 112 G-5

51 Operations and Compositions of Functions 116 H-1

52 Inverse Functions 118 H-2

53 Factor Theorem and Remainder Theorem 122 H-3

54 Graphs of Polynomial and Rational Functions 124 H-4

55 Review 5 127

56 Review 6 128

57 Review 7 130

58 Cumulative Review 131

ix

Senior 3 Pre-Calculus Mathematics

Notes

Cumulative Exercises

x


A-1, A-2

Exercise 1 : Quadratic Functions

Graph the following two sets of data on the same coordinate system. Join eachset of points with a smooth curve.

a.

b.

x -2 -1 0 1 2

y -1 0 -1 16

x -3 -2 0 2 3y 10 5 1 5 10

2. In graph la above, what would you expect the y-value to be when x = 3?

3. In graph lb above, what would you expect the x-value to be when y = 2?

4. A graph of a quadratic function is shown below. Each tick on the axis representsone unit.

a. What is the domain of the graph?b. What is the range of the graph?c. What are the coordinates of the vertex?d. What is the equation of the axis of symmetry?e. What are the zeroes of the function?f. What are the x-intercepts?g. What is the maximum value of this graph?h. What is the minimum value of this graph?

5. a. Graph the following functions on the same coordinate system.

i. y=x2 ii. y =x2 +3 iii. y=x2-2

b. State the similarities and differences of these graphs.

c. What are the coordinates of the vertex of each of these graphs?

d. If a similarly shaped graph had its vertex at (0, -4), what would be its equation?

6. Simplify each of the following expressions:

a. (-3x2)(4x') b. (_ 4xe)2 C.49x4

7x2

12c4d'ed. e. 7-2 f. -3"2

-9ed"

Continued

1


A-1, A-2

Exercise 1 : Quadratic Functions

7. Evaluate the following expressions if x = 2 and y = -3.

a. 5x - 3y + y2 b. 7x - 2y2 - (3x)°

8. Factor the following expressions:

a. x2+5x b. x2+5x+4

18x2c.

7y

c. 6x2 - 7x - 20

9. Rewrite the expression 5x + 3y = 4 to express x in terms of Y.

10. Find the area of the shaded region.

4 cm

2 cm

5 cm

11. You have been assigned the job of measuringthe height of the local water tower.Unfortunately, climbing makes you dizzy, soyou decide to do the whole job from groundlevel. From a point 47.3 metres from thebase of the tower, you find that you mustlook up at an angle of 53° to see the top ofthe tower. How high is the tower?

3 cm

4 cm

12. A local survey asked 100 subjects for their opinions on a zoning ordinance. Of the62 favorable responses, there were 40 males. Of the 38 unfavorable responses,there were 15 males. Find the probability of randomly selecting one of thesesubjects and getting a male response.

Continued

6 em.

2


y

A-1, A-2

Exercise 1: Quadratic Functions

13. Describe the domain and range using interval notation.

a.i

b. I

(1 1 1 1 , - - I I l x

S

C.

14. Which of the following describe a quadratic function?

a. f : x --+ 3x2 b. g: x -- 2x3 - 5

c. y=2x2-x+1

e,

5V


x5-T

.+1 I , 1 1 i.....1...^." 10-X 911

i ll.. i._. 1... 1 1 J( t1, x

-51

d. f(x)=3x-- 1X

3


A-1, A-2

Exercise 2 : Graphs of Quadratic Functions I

1. a. Graph the following functions on the same coordinate grid:

i. y = x2 ii. Y = 2x2 iii. Y =12 x2

b. What happens to the graph as the coefficient of x2 increases?

c. What are the coordinates of the vertex of each of these graphs?

2. Graph the following functions on the same coordinate grid:

a. y=x2 b. y=x2+3 c. y=x2-2

3. a. Graph the following functions on the same coordinate grid:

i. y=x2 ii. y=(x+3)2 iii. y=(x-2)2 iv y=(x+1)2

b. What are the coordinates of the vertices of these graphs?

c. Write the equation of a similarly shaped graph with its vertex at (8, 0).

4. Sketch the graph of the function y = 2 x2 +2.

5. Completely factor the following:

a. 2x2-'-S b. 60x2 --- 42x - 72

6. Find sin 0, cos 0, and tan 0 for the indicated angle in each of the triangles below.

a. K b. C.

7. Rewrite the expression 2y = 8 - x to express y in terms of x.

8. You lean a ladder 6.7 m long against a wall. It makes an angle of 63° with thelevel ground. How high up is the top of the ladder?

9. Bill obtained marks of 73%, 84%, and 79% on his first three math tests. Whatmark must he earn on his fourth math test so that his average for the four tests

will be 80%?

Continued

4


A-1, A-2

Exercise 2: Graphs of Quadratic Functions 1

10. If one man can jump a stream that is 3 metres wide, how wide a stream can 5men jump?

11. Jim and Kim each have money to buy ice-cream cones. Unfortunately, Jim is 240short of the price of a cone and Kim is 20 short. They decide to pool their moneyand buy a single cone, but they discover they still don't have enough money.What is the cost of an ice-cream cone?

12. A surveying crew is given the job of measuring the height of a mountain. From apoint on level ground, they measure an angle of elevation of 21°. They movecloser and find the angle of elevation is now 35°. How high is the mountain?(Hint: You may need to calculate some other numbers first.)

13. What is the equation of the line given by the following graph? (Give the equationin the general form.)

14. Solve for x (leave your answer as a reduced fraction). 2 x + 5 = 4 - 3 x

15. Simplify the following expressions by rationalizing the denominator.

b.4-_ 22-3,2

16. Given the points A(-4, 7) and B(8, 1), find the following:

a. slope of AB b. midpoint of AB c. length of AB


A-2, A-3

Exercise 3 : Graphs of Quadratic . Functions 2

. a. Graph the following functions on the same coordinate system:

i. Y = x2 ii. y = -x2 iii. Y = 2x2 iv. Y _ -2x2

b. What is the effect of the negative sign?

2. a. Graph the following functions.

i. y= (x+2)2+3 ii. y=(x+4 )2 -5 iii . y= (x-5)2+1

b. State the vertex of each graph.

c. State the equations of the axes of symmetry.

3. For each of the following parabolas, state

i. the direction of opening

ii. the coordinates of the vertex

iii. the equation of the axis of symmetry

iv. whether it is narrower or wider than y = x2

a. y = 2(x + 1)2 b. Y=- 1(x -1)2 +6 c. y = 2(x + 6)2 --10 d. y = 6(x -1)2 +8

4. Without making a table of values, sketch the graph of each of the followingfunctions . State the vertex and the equation of the axis of symmetry,

a. y=2(x+1)2 b. y=-1(x-1)2+6 c. y=2(x+6)2--10 d. y=6(x--

5. Factor each of the following expressions:

a. x2-x--6 b. x2-8x+15 c. 14x2+49x-105

Continued

6


A-2, A-3


6. Using trigonometric ratios, find the length of side x in each triangle below.(Round your answers to one decimal place.)

7. Express x in terms ofy in the expression 3x + y = 2.

C.

f.

8. You must order a new rope for the school's flagpole. You observe that the polecasts a shadow 11.6 m long on the ground. The angle of elevation of the sun is36°. What length of rope do you need to fit the height of the pole exactly?

9. What is the last digit of 53"?

10. Find the perimeter and area of the rectangle below.

3x

V x+1

x+6

Continued

7


A-2, A-3

Exercise 3 : Graphs of Quadratic Functions 2

x211. Simplify the expression

+3x + 2 x2 +x-6.

x2+4x+3 x2-4

12. Find the value of both a and L A inthe following triangle.

13. Simplify the expression (-3xy2)(--2x'y3)3.

14. Solve for x in the expression 3-J = 18.

15. Describe each inequality using interval notation.

a.

b.

C.

d.

e.

0

2

0-----

5

°- -----a^

-2 7

-3 10

0 6

a


A•3

Exercise 4 : Transformations of Quadratic Functions 1

1. a. Draw the graphs of the following quadratic functions:

y=x2-2x-3 y=x2+6x+5 iii. y= x2+6x+8

b. State the following for each of the graphs in Question la.

i. coordinates of the vertex

ii. equation of the axis of symmetry

iii, domain and range

iv x-intercepts

2. For each of the following quadratic equations, state

i. the coordinates of the vertex

ii. whether the graph opens upward or downward

a. y= 1(x -3)2 +5 b. y=- 2(x +4)2-7

c. y= 3(x+1)2+2 d. y=- 4(x-2)2+1

3. Completely factor each of the following expressions:

a. 2x2-20x+32 b. 4ax - 8bx

4. Using trigonometric ratios, find 0 in each of the triangles below. (Round your

answers to one decimal place.)

Continued

9


A-3

Exercise 4: Transformations of Quadratic Functions 1

5. Commercial airliners fly at an altitude of about 10 km. They start descendingtoward the airport when they are still quite far away, so that they will not haveto make a last minute dive at a steep angle.

a. If the pilot wants the plane's path to make an angle of 3° with the ground,how far from the airport must she start descending?

b. If she starts descending 300 kilometres from the airport, what angle will theplane's path make with the horizontal?

6. Simplify the following expressions:

2a.

_,2 v3b. 3x-4y+2x-(3x+7y)

7. An oil well is to be located on a hillside that slopes at 10° below horizontal. Thedesired rock formation has a dip of 27° to the hillside. The well is located 1200metres downhill from the nearest edge of the outcropping rock formation. Howdeep will the driller have to go out to reach the formation?

8. Find two numbers whose sum is 34 and whose difference between four times thelarger and twice the smaller is 37.

Continued

10

Curnuiative Exercises Senior 3 Pre-Calculus Mathematics

Ar-3

Exercise 4 : Transformations of Quadratic Functions I

9. Complete the chart.

y=(x-1)2+2 y-(x-.)'2.2 y=(x+1)2+2 y=(x+1)2 -2

Vertex

Equation ofaxis ofsymmetry

Domain

Range

Direction ofopening

Maximumor minimumy-values

10. Find the equation of the quadratic function, given that its vertex is (1, -2) and apoint on the graph is (-2, 16).

11. A labour study involves a sample of 12 mining companies, 18 constructioncompanies, 10 manufacturing companies, and 3 wholesale companies. If acompany is selected randomly from this sample group, find the probability ofgetting a construction company.

11


A-3


1. a. What value of k makes the expression a perfect square trinomial? Write eachexpression as the square of a binomial.

i. x2+8x+k ii. x2-8x+k iii. x2+20x+k

iv x2-2x+k v. x2-5x+k vi. x2+7x+k

b. List the steps that you used to find the value of k.

2. Find the vertex and describe the parabola for the following equations:

a. y=x2+6x-7 b. y= x2-4x-60 c. y=2x2+8x-10

d. y=3x2+24x+21 e. y=x2+5x+6 f. y=x2--3x-4

g. y=2x2+5x+2 h. y=-3x2+2x+1


a. 4x2 - 16y2 b. 25ab - 10ab2

4. Using trigonometric ratios, solve each of the triangles given below. (Round youranswers to one decimal place.)

a. A. b.

14

C.

5. Rearrange the equation y + 4x = 5 to express x in terms ofy.

6. Solve for x:

a. 23x- b. x v 12 = 27 + 2 108

7. A surveyor measures the three sides of a triangular field and gets 114, 165, and257 metres.

a. What is the measure of the largest angle of the triangle?

b. What is the area of the field?

8. Jon is three times as old as Cal, while Ron is 12 years older than Cal. Eight yearsfrom now, the sum of their ages will be 81. How old is each of the three people?

12


A-4


1. A farmer wishes to build a rectangular pen along one side of his barn. If he has80 metres of fencing, find the dimensions that will yield a maximum area.

2. Find two positive numbers whose sum is 13 if the sum of their squares is aminimum.

3. A projectile is shot straight up from a height of 6 m with an initial velocity of80 m/s. Its height in metres above the ground after t seconds is given by theequation h = 6 + 80t - 5t2. After how many seconds does the projectile reach itsmaximum height, and what is this height?

4. A survey found that 400 people will attend a theatre when the admission price is80 cents. The attendance decreases by 40 people for each 10 cents added to theprice. What price of admission will yield the greatest receipt?

5. Find two positive numbers whose sum is 13 and whose product is a maximum.

6. Completely factor the expression 72x2 + 106x - 126.

7. Using the law of cosines, solve each of the following triangles given below.(Round your answer to one decimal place.)

a. A b. P C.

8. a. A space probe is sent on a trajectory defined by the equation y = 9x2 - 30x + 25.

Meteors are expected to pass through the points (3, -16), (1, -4), and (-5, 400).Is there any chance of a collision? If so, at what point or points do thedangers lie?

b. Find the coordinates of the vertex of y = 9x2 - 30x + 25.

9. Express y in terms of x for the following equation: 3x = 2y - 1.

10. Solve for x in the equation: 5(2x + 1) + 3 = 3(x - 2).

11. Simplify the expression: (x - 4)2 - (3x + 2)(x - 4).

Continued

13


A-4


12. For equation y = 6x2 - 24x + 18,

a. find the coordinates of the vertex

b. find the equation of the axis of symmetry

c. find the coordinates of the x-intercepts

d. find the domain and range

e. sketch the graph

13. Find the area of the shaded region if the radius of the semi-circle is 4 cm.

14. Describe each solution to the inequality, using interval notation.

a. {xjx<-3orx?2}

b. {xI-10<x55{

c. {yly>-

d. {yay >5ory<-

e. {xj-5<x<--2or25x{

14


A-4

Exercise 7: Applications of Quadratic Functions

A rectangular field is to be enclosed by a fence and divided into three smallerplots by two fences parallel to one of the sides. Find the dimensions of thelargest such field if 1200 metres of fencing are available.

2. A ball is thrown vertically upward with an initial velocity of 20 m/s. It can beshown that the distanced in metres of the ball from the release point in time tseconds is given by d = -5t2 + 20t. Determine the maximum height attained bythe ball and the number of seconds required to attain this maximum height.

3. An orange grove now has 20 trees per hectare, and the average yield is 300oranges per tree. It is estimated that for each additional tree planted perhectare, the average yield per tree will be reduced by 10 oranges. How manytrees per hectare will produce the largest yield?

4. A dealer finds that he can sell 800 radio sets at $60.00 per set. However, forevery $2.00 drop in price he can sell 50 sets more. At what price per set shouldhe sell in order to get the largest cash return?

5. Factor the expression 14ab2 - 7ab.

6. Using the law of sines, solve each of the following triangles. (Round all answersto one decimal place.)

7. Express x in terms of y for the expression 2x + 3y - 1 = 0.

Continued

15


A-4


8. When surveyors measure land that slopes significantly, the distance that ismeasured will be longer than the horizontal distance drawn on the map. Supposethat the distance from the top edge of the Cibolo Creek bed to the edge of thewater is 37.8 m. The land slopes downward at 27° to the horizontal.

a. What is the horizontal distance from the top of the bank to the edge of thecreek?

b. How far is the surface of the creek below level of the surrounding land?

9. For the parabola given by the equation y = -3x2 + 24x + 27, find the

a. coordinates of the vertex

b. equation of the axis of symmetry

c. coordinates of the x-intercepts

d. domain and range

10. Find the number that is halfway between 0.8 and 11.

11. If 2` = 32, what is the value of x?

Continued

16


A-4

Exercise 7 : Applications of Quadratic Functions

12. Match each equation to its graph.

1. y-3x2

4. y+1=3(x+2)2

7. y=-3(x-2)2

2. y=- 11x 2 3.

5. y = 3 (x + 2)2 6.

8. y-1=-1(x-2)2

y = 3(x + 2)2

y+1=-3(x-2)2

C.a. b.

V

d. e.

g. h.

X-5 5

V

17


B-1

Exercise 8: Trigonometric Equations I

1. Determine the exact value of sin 8, cos 0, and tan 0 for the angle whose terminalarm passes through the given points:

a. P(5, 3) b. R(-3, 4) c. Q(8, -2) d. T(-3, -7)

2. Give the related angle for each angle given below:

a. 98° b. 120° c. 352° d. 263°

3. If sin 8 = 2 , state all possible angles for 0, where 0°5 0<_ 360°.

4. a. Complete the table of values for thefunction y = sin x.

b. Sketch the function on a cartesianplane . (Hint: The points are joinedby a smooth curve.)

c. Extend the graph to -360°.

x y

0°45°90°

135°180°225°270°315°360°

5. Repeat question 4, for the function y = cos x.

6. State in your own words a relationship between the graphs in questions 4 and 5.

7. An observer 2 km from the launching pad observes a vertically ascending missileat an angle of elevation of 21°. Five seconds later, the angle has increased to 35°.

a. How far did the missile travel du g the 5-second interval?

b. What was its average speed during this interval?

c. If it keeps going vertically at the same average speed, what will its angle ofelevation be 15 seconds after the first sighting?

8. Factor the following expressions:

a. 5x2-20 b. x4-81

Continued

18


B-1

Exercise 8 : Trigonometric Equations 1

9. Ms Brown's rent went up by 24%. By what percentage would it now have to go

back down if it were to return to its previous level?

10. Simplify the expression (2 v`3 -

11. Write the equation y = -2x2 + 8x + 5 in the form y = a(x -- h)2 + k.

12. Simplify the expression 3-2 + V.

13. For the function below,

y= Ix2 +4x+102

a. complete the square

b. sketch the graph

c. state the coordinates of the vertex

d. state the equation of the axis of symmetry

e. determine the maximum or minimum value

14. Solve for x. (Leave your answer as a reduced fraction.)

2x_ (2x-4)=2- 10(x-5)

15. A study of consumer smoking habits includes 200 married people (54 of whomsmoke), 100 divorced people (38 of whom smoke), and 50 adults who nevermarried (11 of whom smoke) (based on data from the U.S. Department of Healthand Human Services). If one subject is selected randomly from this sample, findthe probability of getting someone who

a, is divorced

b. smokes

19


B-i

Exercise 9: Trigonometric Equations 2

Determine the solution for each of the following trigonometric equations in theinterval 0°<- 0S 360°.

a. cos 0 = -3

c. tan 0 - 2 = 5

e. -3 sin 0 = 2

g.3cos0--2=0

i tan0106

k.4tanO-7=5tan0-6

2cos 0+1=-

2. Imagine you are the pilot of a commercial airliner. You find it necessary to detouraround a group of thundershowers. You turn at an angle of 21° to your originalpath, fly for a while, turn, and intercept your original path at an angle of 35°,70 km from where you left it.

a. How much farther did you have to go because of the detour?

b. What area is enclosed by the triangle?

N

b. sin O+1=0

d. 2 cos O= 2

ftan 0 5

2

h. 5tanO+4=0

7© km -

3. If417 +417 +417 +417 = 4X, find the value of x.

Continued

20


B-1


4. A function is given by f:x x2+6x-5, where 0<-x<6.

a. Draw the graph of the function.

b. State the domain and range.

c. State the coordinates of the vertex.

d. Give the equation of the axis of symmetry.

e. State the maximum or minimum value.

5. Simplify the expression: 2V-12 - 5V-2-7 + 348.

-1

6. Simplify the expression: 81 z +V8-- 32' + 32 .

7. If A has coordinates (7, 3) and B(5, 1), find

a. the midpoint of AB

b. the length of AB

c. the slope of AB

8. Solve for x. (Leave your answer as a reduced fraction.)

2(4x-1)-4(5-2x)=1-3(3x-1)

9. If the graph of y = 2(x - 2)2 - 4 is moved 2 units up and 3 units to the right, what

is the equation that represents this new position?

10. In which quadrant(s) is sin 0 < 0?

11. If the related angle 8 is 37°, what are the possible values of 8?

12, a. Graphy=cosx-2.

b. How does the graph of y = cos x compare to the graph of y = cos x -2?

c. How does the graph of y = cos x compare to the graph of y = cos x + k where k

is a constant?

21


B-1, B-2

Exercise 10: Trigonometric Equations and Ambiguous Case Problems

Find all solutions on the interval [0°, 3601 for each of the following trigonometricequations.

a. sin 0 = 0 b. tan O=-

c. 1 = -2cos 0

d. 3 tan 8 - 7 = 0

2. In A XYZ, y = 5, x = 4, and Z X = 27°. Find the possible values of

a. LY b. ZZ C. Z

3. In A ABC, a = 6, b = 5, and L A = 27°. Find the possible values of c.

4. In A DEF, d = 2, e = 5 , and L D = 27°. Find the possible values of f.

5. On a coordinate grid, points are located at A(0, 0) and C(12, 5), respectively Theline segment connecting A and C has a length of 13. If B is a point on the x-axis,and BC = 7, find two possible values for the length of AB.

6. Line segment AB has length 11 cm and is drawn at an angle of 44° to a horizontalline AE. A circle with its centre at B has a radius of 9 cm. The circle cuts AE atpoints C and D. Calculate the length of chord CD.

7. For the parabola with equation y = -5x2 - 20x + 60, find the




d. domain and range

Continued

22


B-1, B-2

Exercise 10 : Trigonometric Equations and Ambiguous Case Problems

8. Determine two numbers whose sum is 24 such that twice the square of thesmaller number plus the square of the larger number is a minimum.

9. Given points A(2, 4) and B(-3, -11), find the following:

a. slope of AB

b. equation of AB

10. How many fence posts are required to make a fence 240 m long if the posts areplaced 8 m apart?

11. Determine the value oft given points A, B, C, and D and the fact that AB isperpendicular to CD.

A(2, 3) B(6, 5) C(6, -1) D(5, t)

12. The length of one side of a square is increased by 10% and the other, side isdecreased by 10%. How does the area of the rectangle that is formed comparewith the area of the original square?

13. State all angles that have a related angle of 63°.

14. Graph the following: y = sin x + 3.


a.0 10

b.-8 -5 -6

a 0C.

-7 3 7

23


B-2

Exercise 11: Ambiguous Case Problems

1. Find the possible values of the indicated side-,

a. InAABC, LB=34°,a=4,andb3.Find c.

b. In A XYZ, L X = 13°, x = 12, and y = 15. Find z.

c. In A ABC, Z B = 34°, a = 4, and b = 5. Find c.

d. In A RST, L R = 130°, r = 20, and t = 16. Find s.

e. In ©MBT, L M = 170°, m = 19, and t = 11. Find b.

f InAABC, LB=34°,a =4,andb=2.Find c.

2. Find all the possible values of the indicated angle measure.

a. In A ABC, L A = 19°, a = 25, and c = 30. Find L C.

b. In A HDJ, L H = 28°, h = 50, and d = 20. Find L D.

c. InAXYZ,LX=58°,x=9.3,andz=7.5.Find ZZ.

d. In A BIG, L B = 39°, b = 900, and g = 1000. Find L I.

3. Examine the triangles in Questions la, lc, and if above (that is, A ABCs). Noticethat these triangles differ only in the length of b.

For each of these triangles, draw an accurate diagram according to the followinginstructions.

For each diagram, draw side a as the base so that it is 4 cm long. Then constructL B of measure 34° at one end of the base, c.

a. Use a compass to mark off the two possible triangles if b = 3 cm. Measure thetwo possible values of c. Your answer should be within + 0.1 can of thecalculated values found in Question la above.

b. Use a compass to mark off b = 5 cm, as in Question Ic. Measure the value of cand confirm that it agrees with the calculated value. Now extend segment ABbeyond angle B. Find the point on this segment where the 5-cm are cuts it.Show that the distance between this point and B equals the negative value ofc that is discarded in working through Question lc.

c. Use a compass to draw an arc of radius b = 2 cm, as in Question if. Showthat this misses the other side of angle B, and thus no such triangle exists.

Continued

24


B-2

Exercise 11: Ambiguous Case Problems

4. Graph the following equation: x = y2 + 2y - 1.

5. Factor the following expressions completely.

a. 15x2 - 7x - 36 b. (x - y)2 9(2x + y)2

6. Solve for x in the expression 5x2 -3x = 52z -4

7. Find the equation of a line that goes through the point P(-5,4) and is

perpendicular to the line y = 3 X+1.

S. Find all values of 0 such that 0° <- 0 :!^ 360° for the expression tan 0 = -1.

9. Tom can cut the lawn in 40 minutes, Dick can cut it in 30 minutes, and Harrytakes 60 minutes. How long will it take to cut the lawn if all three worktogether?

10. A car dealership can sell 20 cars per week at a profit of $2400 each. For every$300 the dealership increases its profit, it sells one less car per week. What isthe maximum profit the dealership can make? How many cars would thedealership then sell?

11. Graph the function y = -sin x over the interval [0°, 360°].

I

25


Exercise 12: Review 1

The height, h, in metres, after the launching of a rocket at any time, t, inseconds, is defined by the equation below. Find the maximum height reached bythe rocket and the time it takes to reach this height.

h=-3t2+9t+814

2. At a local beach, the lifeguard has 620 m of marker buoys to rope off a safeswimming area. Calculate the dimensions of the rectangular swimming area tocreate maximum swimming room if one side of the area is to be the beach.

3. If 65 apple trees were planted in an orchard, the average yield per tree would be1500 apples per year. For each additional tree planted in the orchard, the annualyield per tree drops by 20 apples. How many trees should be planted in order toproduce a maximum yield?

4. The difference between two numbers is 14, Find the two numbers so that theirproduct is a minimum.

5. Honest John's used car lot sells an average of 20 cars per week at an averageprice of $6400 each. Honest John would like to increase the average price by$300; however, he knows that his sales would fall by one car if he does. If thedealer's (Honest John's) cost per car is $4000, at what price should he sell thecars to maximize profits?

6. Solve for sin 0 = - on the interval 0°<_ 0 <_ 360°2

7. A terminal arm of an angle passes through the point (-3, 7). What is the value of

tan 0?

8. Determine the solution set for 6 cos 0 = 5 - cos 0, 0 e [180°, 360°}.

9. Graph y = --cos x + 1. State the domain and range in interval notation.

10. Ind ABC, L A = 41°, a = 23, and b = 28. Solve A ABC. (Express angles to the

nearest degree and lengths to one decimal place.)

11. What are the x-intercepts of y = 2 sin x on the interval [0°, 180°].

26


C-1, B-1

Exercise 13: QuadraticlTrigonometric Equations

1. Factor each of the following expressions completely:

a. 3x2+7x+2 b. x2-9

d. 2x2 - 16x + 32 e. sine 0 -

c. 25x2 - 100

f. tang 0 + 2 tan 0

2. State the roots of the following quadratic equations:

a. (x+3)(x-1)=0 b. (4x+7)(3x+1)=0

3. Solve these equations by factoring. Check your solutions.

a. x2-x-12=0 b. x2-9x+18=0 c. x2---x--20=0

d. 2x2+3x--2 =0 e. -x2-2x+3 =0

4. Rearrange each of the following equations and solve by factoring. Check yourroots.

a. 10x2=7x+12 b. 5x2+21x=54 c. 3x(x-2)-x(x+1)+5=0

d. x2+9x-21 =0 e. x2+9=02 2

5. Describe in words the steps necessary to solve the following quadratic equation:

17x + 15 = 4x2

6. Solve each of the following trigonometric equations on the interval 0° <- 0!5 360°.(Round your answers to one decimal place.)

a. cos0+1=2

c. (tan 0-2)(2 sin 0+ 1)=O

e. 2 sine 0+ 7 sin 0- 4= 0

g. sin20+2sin0+1=0

b. (2 sin 0 - 1)(sin 0 + 1) = 0

d. 4 cos' 0- 1 = 0

f. 3 sin 0 tan 0+ 2 tan 0= 0

Continued

27


C-1, B-1

Exercise 13: QuadraticfTrigonometric Equations

7. The children's slide at the park is 30 feet long and inclines 36° from thehorizontal. The ladder to the top is 18 feet long. How steep is the ladder, that is,what angle does it make with the horizontal? Assume the slide is straight andthat the bottom end of the slide is at the same level as the bottom end of theladder. (Hint: Draw a diagram.) Round off to one decimal place.

8. Simplify: 2;132 - 3--V'18 + 5v'50-

9, Find the equation of a line that passes through the point P(6, 2) and is parallel

to the line y = 3 x + 5. Leave your answer in standard form.

10. A square with sides of length 6 cm is circumscribed by a circle. Find the area of

the circle. (Leave the value r in your answer, that is, do not calculate the decimal

equivalent.)

11. One leg of a right triangle is 7 m longer than the other leg. The hypotenuse is17 m long. Find the length of each leg of the triangle.

12. Two (square) checkerboards together have an area of 169 square centimetres. Onehas sides that are 7 cm longer than the other. Find the length of the sides of each.

13. Find three consecutive odd integers such that the product of the second and the

third is 63.

14. If (16)(2X) = 6y -' and y = 8, then x = ?

15. The sides of the square are 4 units long. What is the area of the shaded region?

16. Sketch the graph ofy = x2 + 4x for x over the interval [2, 4j.

28


C-1

Exercise 14: Quadratic Formula

For each quadratic equation below, state the values of a, b, and c, where

ax2+bx+c = 0.

a. x2-2x-5=0 b. 3x2-2x+5=0 c. 5x2-3x=8

d. 2(x2-2x)-1=0 e. 5x2=9x f. 4-2x2=9x

g. -3 cost 0 + 2 cos 0 - 7 = 0 h. tan2 6 = 3

2. Solve these equations using the quadratic formula. Be sure to state the formulabefore substituting values into it.

a. x2+2x-15=0 b. 2w2-3w+1=0 c. 7w2-3w=0

d. 1 = 5x2 e. x2 - 0.1x - O.06 0 f. x2 - 7x - 1 = 0

g. sin2 0 + sin 0 - 1 = 0, 0 E [0°, 180°]

h. 18 sine 0 = 2 - 9 sin 0 , 0 c [90°, 360°]

3. Use the quadratic formula to find the roots of each equation below.

a. 3x2_6x-5=0 b. 2x2_4x-1=0

C. 9x2-8x-7=0 d. 2x2_x-3 = 0

4. Find the zeroes of the function f defined by

a. f:x-- 5x2-x-3 b. f(x) =2x2+6x-1

5. Find the roots of the quadratic equation 3x2 - 5x -- 1 = 0 to one decimal place.

6. a. Find the roots of the quadratic equation 6x2 + 5x -- 6 = 0

i. using the quadratic formula

ii. by factoring

b. Which method do you prefer and why?

Continued

29


C-1

Exercise 14 : Quadratic Formula

7. For the parabola defined by the equation y = 7x' + 70x + 63, find the




d. domain and range

8. A piece of wire 60 cm long is bent into the shape of a triangle. Find the angles ofthe triangle if two of the sides have lengths 24 cm and 20 cm.

9. Determine the solution set for each of the following trigonometric equations overthe interval 0°<_ 05 180°. (Round answers to two decimal places.)

a. cos' 0 - 1 = 0 b. (2 sin 0 -- 1)(tan 0 - 2) = 0 c. sin' 0 = sin 0

10. In an airport control tower, A, two planes at locations B and C, respectively, areregistered to be at the same altitude on a radar screen. The range finderdetermines one plane to bear N60°E at 100 km from the radar site while theother bears S50°E at 160 km from the radar site. How far apart are the planesfrom each other?

11. Mary is 5 times as old as Bill. Last year, she was 6 times as old as he was. Howold will each be in 2 years?

12. This pipe has an outside diameter of 14cm and is 60 cm long . If the pipe is 2 cmthick and is made of a material thatweighs 8 grams per cm3, how much doesthe pipe weigh?

13. The price of a radio is $50, and 40 are sold each day. For each $1.00 the price israised, the store sells one less radio. If each radio costs $18 to make, how muchshould the price be set at in order to maximize profit?

14. Graph the function y = cos x - 2 . State the domain and range in interval notation.

30


c-1

Exercise 15 : Solving Quadratic Equations by Graphing

1. a. Graph the quadratic function y = x2 _ 2x - 8.

b. Where does the graph intersect the x-axis?

c. What are the zeros of the function?d. What are the roots of the equation?e. Check each root.

2. Solve each of the following equations by graphing. Check each root.

a. x2+2x-8=0 b. x2+4x+3=0 c. x2+8x=-15

d.9-x2=0 e.2x2_12x+10=0

3. Find the zeroes of the quadratic function f (x) = x2 + 8x + 15 by

a. factoringb. using the quadratic formulac. graphing

4. Find the real number solution for

a. 3x2 - 48 = 0 b. 6x2 = 11x + 10

5. Solve these equations:

a. x4-5x2 + 4=0 b. x + 2)2-7x+2+12 =0x x

6. An observer 5.2 km from a launch padobserves a missile ascending.

a. At a particular time, the angle ofelevation is 31°. How high is themissile?

b. At this same time , how far is themissile from the observer?

c. What will the angle of elevation bewhen the missile reaches a heightof 30 km? 5.2 km ='

Continued

31


C-1

Exercise 15: Solving Quadratic Equations by Graphing

7. Solve each of the following trigonometric equations finding all solutions on theinterval 0° < 0< 180° to one decimal place.

a. cos O =2

b. 6 tang 0-19 tan 9 = -10 c. 13 2 sin 9

8. What is the value of m that would make each of the following equations a perfectsquare?

a. y=x2 + 2x+m b. y = x2-lOx+m

9. If 4j+ 4 is squared, what is the result?

10. For the triangle below, find the length of b and the measure of L A.

11. The area of a rectangle is 14 m2. If the length is doubled and the width is tripled,what is the area of the rectangle that is formed?

12. Two positive numbers differ by 4 and the sum of their squares is 136. Find thenumbers.

13. The length of a rectangle is 6 m more than its width. The area of the rectangle is

27 m2. Find the dimensions of the rectangle.

14. a. Men were once drafted into the U.S. army according to the random selectionof birthdays. If the 366 different possible birthdays are written on separateslips of paper and mixed in a bowl, find the probability of making oneselection and getting a birthday in May.

b. Using the same population of 366 different birthdays, find the probability ofmaking one selection that is the first day of a month.

15. In which quadrant(s) is cos 0 > 0?

32


C-2

Exercise 16 : Nature of Roots

If the discriminant of a quadratic equation has the given value, state thecharacteristics of the roots.

a. -15 b. 25 c. -9 d. 0

2. How many times would the graph of y = axe + bx + c (with a, b, and c as realnumbers) intersect the x-axis if the value of the discriminant is

a. negative b. zero c. positive

3. Determine the nature of the roots by calculating the discriminant for eachequation.

a. x2-8x+16=0 b. a2+2a+7=0

c. b2-16=0 d. 2x2+x=5

4. Determine the characteristics of the roots of the following equations:

a. 2 +4x+4=0 b. x21-x2-3=0 c. 2(x2-3)=4x

d. 6x2-x+2=0 e. 4x2-12x+9=0

5. Given 3x2 - mx + 3 = 0, for what values of m would the roots not be real?

6. Find value(s) of k so that each equation has real and equal roots.

a. kx2-6x+2=0 b. x2+(k-8)x+9=0

7. For what values of k will the equation 2x2 + 4x + (2 - k - k2) = 0 have exactlyone root?

Continued

33


C-2

Exercise 16: Nature of Roots

8. State the nature of the roots for each of the following parabolas:

9. Find all values of 0 in the following triangle.

C.

B

10. Find the sum and product of the roots of each equation.

a. 2x2---6x-7 =0 b. 0 =-3x2+2x---5

11. Find a quadratic equation whose roots are 7 and -3.

12. Find a quadratic equation whose roots are 2 + and 2 - Vd.

13. Find a quadratic equation whose roots have a sum of -5 and a product of 6.

14. Solve the following equations:

a. 9x2-36==0 b. 4p2 + 4p--3=0

15. In a collection of coins worth $9.13, there are twice as many dimes as quarters,four more nickels than dimes, and twice as many pennies as nickels. How manyof each kind of coin are in the collection?

16. A rectangular piece of cardboard is 5 cm longer than it is wide. A 3-cm by 3-cmsquare is cut out of each corner, and the four sides are folded up to form an open

box with a volume of 450 cm3. That were the length and width of the original

piece of cardboard?

34

Cum ulative Exercises Senior 3 Pre-Calculus Mathematics

C-3, C-4

Exercise 17: Nonlinear Equations

Solve the following equations:

a. x2 _2X = 24 b. x4 -

C. x2 =8 d. x4 -lOx2+9=0

2. Graph each function. (You may u se a graphing calculator.)

a. y= x2-2x---3 b. y=

C. y = x 8 d. y= x4- lOx2+9

3. For each function in Question 2, state its domain, range, and x-and y-intercepts.

4. Sketch the graph given by the equation y = - (x - 1)2 + 2.

5. Solve the equation x2 + 2x - 2 = 0 using the quadratic formula.

6. State the reason why each of the following statements is true.

a. L 1 = L 2 because ...

b. L3=L 4because ...

If A ABC = A DEF, then

AC = DF because ...

and L A = Z D because . .

1 2

Continued

35


C-3, C-4


7. Five centimetres are cut off (along one side of) a square sheet of paper and 8 cmare added to an adjacent side. The resulting rectangular sheet of paper has aperimeter of 98 cm.

a. What was the area of the original square?

b. What is the length of the diagonal of the resulting rectangular piece of paper?

8. Find the values of x, y, and 9 in the following diagram.

9. Find the solution for 6 tan20 - tan8 - 2 = 0 in the inverval 0° < 0:5 180°.

10. If x is any integer, then by which positive integers is x2 (x2 - 1) always divisible?

11. Convert the following sets to interval notation.

a. {x x>_7orx<- -2} b. y( y >-101 c. yl y <4}

1 12. Graph the functiony = 2 x. Calculate the related angle ify = 2 x is the terminalarm of 0.

36


C-5

Exercise 18: Radical Equations

1. Simplify each of the following expressions:

a. (* 2x 1)2

b, (5 + x )2 c. (2 + .J -

2. Find the real number solution for each of the following equations. Check yoursolutions.

a. x+2=-i2x+7 b. x= 2- 2x-5

c. ,/2x+3- x+1= 1 d. x2-3+1= 0

e. x= 3x -2+2 f. ^, x2+6x =2

g. 3x+2 -- 34-x -^-2 h. 1-x+V = x+l

3. Given the equation x2 + 2x - 2 = 0, calculate

a. the discriminant

b. the roots of the equation

4. Your cat is trapped on a tree branch 6.5 m above the ground. Your ladder is only6.7 m long. If you place the ladder's tip on the branch, what angle will the laddermake with the ground?

5. Given that A XOP - A XYZ in the diagram below, find the length of

a. XP

b. OY

Continued

37


C-5

Exercise 18 : Radical Equations

6. Simplify the following expression:

2y 2

E -y41

V

1

7. Without making a table of values, sketch the graph of each of the followingfunctions. State the range of each function.

a. y = -3(x + 3)2 b. Y = _ 2 (x - 1)2 +6 c. y = 6(x - 2)2

8. Define the quadratic equation (with integer coefficients) if the sum of its roots

is - 5 , and the product of its roots is - 3

9. The area of the trapezoid below is 60 cm2, and AD is parallel to BC. Find the

value of x.

2x

x

A x+6 ri B

10. Widgits are placed in boxes and the boxes are then packed in crates. The numberof widgits in each box is four less than the number of boxes in each crate. Findthe number of widgits in each box if a full crate contains 60 widgits.

38


C-5

Exercise 19: Rational/Absolute Value Equations

Solve each of the following rational equations . Check your solutions.

a. x=-2

x-3

C. x-4= 1x

e.2x + 1 + 3x+9 = 0

X-3 2x+3 2x2.3x-9

2. a. Solve the following absolute value equations.

i. 3xI =12 ii. 12x1-1=17

iv. I x2+4x_121=0 v.x

2

iii. 15x+21=-3

vi. ix-51=1 3x+71

b. Describe the steps involved in solving equation vi above.

3. Given the equation y = 4x2 - 48x + 128, find the following:




d. domain and range

4. Solve each of the following trigonometric equations on the interval00 <- 0 ^ 360°. (Round your answers to one decimal place.)

a. 3tanO-1=5 b. cos' 0-2cos0=2

Continued

2x-9 x 5+-b.

x-7 2 x-7

d. 3x` - 2 = 2x + 13x+1 3x+1

2+12 7x

x-3 x-3

39


C-5

Exercise 1 9: Rational/Absolute Value Equations

5. A cardboard box without a top is constructed by cutting squares out of thecorners of a rectangular piece of cardboard and then folding the flaps upward. If

each of the four corners has an area of 9 cm2, and if the length of the original

cardboard was 7 cm longer than its width, what were the original dimensions if

the volume is 684 cm'?

6. Find the measure of angle 0 in the triangle below.

7. Solve the equation 7x2 - 35 = 0 for x.

8. The Chin family drove to their lakeside cottage at 90 kmlh. They returned homeon the same highway at 60 km/h. If the round trip took 2 hours, how far does theChin family live from the lake?

9. Machinist X can do a job in 15 hours, Machinist Y can do the same job in 20hours, and Machinist Z can do the job in 12 hours . If all three work together onthe same job , how long will it take them to get the job done?

40



1. Sketch the function y = 2x2 - 8x - 10 and state the following:

a. vertex b. axis of symmetry c. min. or max. y-valued. domain e. range f. wide/narrow opening g. zeros

2. Given y = 3x2 + 12x - 8,

a. complete the square

b. find its zeroes of

3. Solve for x in each of the following triangles.

a. b.

x

C.

4. Find all the values of 0 on the interval {0° < 0< 3 60°}.

a. 2sin6+1= 0 b. 3sine6+10sin8-8=0

c. cost 9 + cos 0 = 0 d. 3 cos 9 tan 8 - tan 9 = 0

5. Find all possible values of the indicated angle measure. State whether onetriangle, two triangles, or no triangle is possible and why.

a. InOHDJ,LH=28°,h=50andd=20.Find LD.

b. In A BIG, L B = 39°, b = 900 and g = 1000. Find Z I and length i.

6. Solve for x. Leave the answer as a reduced fraction.

2 x - 3 (2x - 4) = 2 - 3 (x-5)3 5 10

Continued

41



7. a. Find the sum and product of the roots if the roots are 3 ± Vi-3.

b. Given these roots, what was the original quadratic equation?

8. Solve for x in the following:

a. 2x=3 5x+6-6 b.22x._-.22 + V -

3x = 5

c. 213x+11=6 d. 1 4xl=-3

e.x 3 x-2 1 9 2-x

2x-6 x2 --6x+9 3x-9 2x x2+6x _ 2x+12

9. To calculate the width of a river, a surveyor marks a base line AB that is 250 in ,along the river bank length. An object, C, is sighted on the other bank of theriver, making angles of 60° and 74° from A and B, respectively. Find the width ofthe river to the nearest metre.

10. From point T, a golfer aims a ball towards a hole at H that is 100 m away. Butthe ball is actually sliced in a direction 30° off course and lands at M, 60 m awayfrom T. If the next shot from point M is hit 50 m directly at the hole, will the ballgo in the hole? If not, how far away from the hole is the ball?

11. A rectangular field is to be enclosed by a fence and divided into three smallerplots by two fences parallel to one of the sides. Find the dimensions of thelargest field if 800 m of fencing is available.

42

Cumulative Exercises Senior 3 pre-Calculus Mathematics

D-1

Exercise 21 : Circles on a Coordinate Plane

1. Write equations for each of the following circles:

a. with centre (-2, 3) and radius 5

b. with centre (5, 0) and diameter 6

c. with centre (4, 3) and passing through (1, 2)

d. with diameter AB and given A(4, 3) and B(6, -1)

e. with centre (0, 0) and area 6ir

f. with centre (-1, 2) and circumference 10ir

2. The illustrated circle is centred at (3, 3). Find its equation.

3. The equation of the large circle is (x - 6)2 + y2 = 16. Find the equation of thesmall circle.

4. Find the centre and radius, and sketch the graph for the following circles.

a. x2+y2+4x-2y-4=0

b. x2 + y'2 + 6y - 12 = 0

c. x2+y2-1Ox-4y=0

Continued

43


D-1

Exercise 21 : Circles on a Coordinate Plane

5. Solve for x: 2x + 6 = -1.x-4 x+4

6. Solve forx: I x2 -261= 10.

7. Solve forx: x-2 =x-2.

8. The graph shows y = axe + bx + c. Which ofthe following is a true statement?

a. a>O,b2-4ac>0

b. a<0,b2--4ac>0

c. a > O, b2 - 4ac < 0

d. a<O,b2 -4ac<0

9. Sketch the graph of y = x2 -2x + 5. State the domain and the range.

10. Write a quadratic equation with roots 2 ± -.

11. Two sidewalks meet at right angles. At noon, Person A is 12 km north of theintersection, walking south at 2 km/h. Person B is 18 km east of the intersection,walking east at 4 km/h. At what time is the area of A AOB a maximum?

A

B

44


D-1

Exercise 22: Distances between Points and Lines

1. Calculate the distance between the following pairs of points:

a. (4, 6) and (6, 5) b. (-4, -2) and (2, 2)

2. Calculate the perpendicular distance from P(4, 6) to line 2x - y = 7.

3. Calculate the distance from the point P(-3, 2) to each of the following lines:

a. 3x-2y=8 b. 3x+2y=12

4. Find the midpoint between A(3, -4) and B(-15, 2).

5. A ship travels on a route represented by the line 2x - 2y + 7 = 0. A lighthouse issituated at point (5, -4). If the lighthouse can be seen anywhere within a radiusof 10 km, will the ship see the light?

6. Given A ABC with vertices at A(5, 4), B(7, -2), and C(-3, 4), find the

a. distance between the midpoints of sides AC and BC

b. length of the median from C

7. Solve the equation 3x2- 5x = 0 for x.

8. Two cars, starting from the intersection of two straight roads, travel along theroads at speeds of 55 km/hr and 65 kmlhr, respectively. If the angle ofintersection of the roads measures 72°, how far apart are the cars after 36

minutes?

9. Find the solution for each of the following trigonometric equations on theinterval 0° < 0< 360°. (Round your answers to one decimal place.)

a. cos 2 0 = 9 b. 2 cos 0 sin 0 + cos 0 = 0 . c. tan 2 0 = tan 0

10. Sketch the graph given by the equation y = (x + 2)2 - 3.

11. Solve the equation for x: ,I 14 - lOx + 3 = x

12. Solve the equation for x: x2 + (x + 2)2 = 452

Continued

45


D-1

Exercise 22 : Distances between Points and Lines

13. Triangles are proved congruent by the properties known as SSS, SAS, AAS, andASA. For each of the following pairs of triangles, state the reason why thetriangles can be said to be congruent.

C.

14. Write a quadratic equation having roots of -6 and 3.

15. A dog kennel owner has 108 in of chain link fence with which to enclose arectangular area and divide it into five pens of equal area as shown below.

a. What is the maximum area of each pen?

b. What are the dimensions of each pen?

c. If the enclosed rectangular area was to be divided instead into fourrectangular pens of equal area, will the layout of the pens affect themaximum area of each pen? What is this maximum area?

16. Find the centre and radius of x2 + y' + 12x - 6y + 20 = 0, and sketch the graph.

17. A circle has centre (-2, 4) and is tangent to the line x + y - 10 = 0. Find anequation for this circle.

46


D-2

Exercise 23: Verify and Prove Assertions in Plane Geometry

1. Three vertices of a rectangle ABCD are A(-9, 0), B(5, 4), and C(7, -3).

a. Find the coordinates of the fourth vertex of the rectangle.b. Find the perimeter of the rectangle.c. Find the area of the rectangle.

2. A triangle has vertices at A(-4, -2), B(2, -8), and C(4, 6). Is this a right triangle?Verify your answer.

3. Show that the quadrilateral with vertices A(-5, -2), B(1, -1), C(4, 4), andD(-2, 3) is a parallelogram.

4. Line 11 contains the points (x, 3) and (-2, 1). Line 11 is perpendicular to line 1,

which contains the points (5, -2) and (1, 4). Find the value of x. Explain yourrationale.

5. Line 1. contains the points (r, 3) and (-2, 1). Line 1. is parallel to line 14 which

contains the points (5, -2) and (1, 4). Find the value of r. Describe the proceduresused.

6. Solve each of the following trigonometric equations, finding all solutions on theinterval 0°< 0 S 360°.

3 sin 0a.

4

1b.3cos8-1=-2

7. Solve the following equation for x: 4200 + 4200 - 13 .x x + 100

8. Given that AC = EC and BC = DC,explain why it is that AB = ED.

4

9. Find the coordinates of the vertex of the quadratic function g(x) = -2x2 + x - 5with domain R.

10. Calculate the distance from the point (0, 4) to the line 2x = y + 3.

Continued

47


D-2

Exercise 23: Verify and Prove Assertions in Plane Geometry

11. Solve the following equation for x:

3(x-6)-3(2x- 1)=2-3(4-x)

12. Solve for y in the equation 5x - 2y = 4.

13. Solve this quadratic equation: 6y2 = -5y + 25.

14. Find the distance from the point A(3, 7) to the midpoint of the line segmentbetween point B(-2, 4) and point C(6, -2).

15. Describe the domain and range using interval notation.

4 1 1 1 *41 ..4

W

C. yI

d. ya

E IF ...Y t f 1 E Y 30 x

W T

16. Given the following graph, which function best describes it?

a. y=cosxb. y=sinx-1c. y=-sinxd. y =cosx - 1

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D-3

Exercise 24: Systems of Linear Equations in Two Variables

1. On the same coordinate grid, draw the graphs defined by the equations x + y = 8and x - y = 12. State the coordinates of the point of intersection.

2. For each system of equations below,

i. use the method of graphing to solve it

ii. check your solution

a. x+2y=10 b. y-2x=l c. 2x =y +22x--y=0 2y-4x=4 y=x-1

3. Solve the system given by the equations Zx + y = 5 and x - 3y = 6 using themethod of substitution.

4. For the following systems of equations, decide whether to substitute anexpression for x or for y, then solve.

a. 2x+3y=--4 b. 3y=x+11y-2x=4 x =y -5

5. For each system of equations below, decide which variable is more readilyeliminated, then solve the system.

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

6. Solve the following systems of equations using the addition-subtraction method.

a. 3x+2y=4 b. 2x+3y=48x-y=3 3x+2y=42

7. You now have a number of methods for solving systems of equations. Before yousolve the following systems, decide which method would be the most appropriate,

a. 2x+y=3 b. x-3y+7=0 c. 2a-3b-13=03x+2y=6 3x-2y=-7 3a-b-9=0

8. A field has the shape of a quadrilateral that is not a rectangle. Three sidesmeasure 50, 60, and 70 metres, and two angles measure 127° and 132°, asindicated in the diagram on the following page.

Continued

49


D-3

Exercise 24: Systems of Linear Equations in Two Variables

a. By dividing the quadrilateral into two triangles, find its area. You may haveto find some intermediate sides and angles first.

b. Find the length of the fourth side.

c. Find the measures of the other two angles.

9. Find the length of AB in the 10. If BD = CD, prove thatdiagram below. 4 ABC is isosceles.

A

11. Simplify and solve the following system of equations:

2(x -y ) - 3(x +y) = - 13 and 5 - 2(2x -y) = 3(x - 2y)

12. The paths of two ships are given by the following equations : Ship A, x + y = 8

and Ship B , x -y = 4. The paths of the two ships intersect at an island . What arethe coordinates of the location of the island?

13. Find the real number solution for 2x2 + 9x _ 18 = 0.

14. Write the equation y = 2x2 - 12x + 13 in the form y = a(x - h)2 + k using the

completing-the-square method.

15. State the value of sin 0, cos 0, and tan 0 for the angle below. (Round to thenearest hundredth.)

5o


D-4

Exercise 25 : Systems of Linear Equations in Three Variables

1. Solve the following systems of equations:

a. 3x-4y+5z=2 b. x+y+3z=12 c. 4x+3y-z=-74x+5y---3z =-5 2x+y+3z= 13 3x-2y+3z=-25x-3y+2z=-11 x-y+4z=11 x+y-z=-2

d. 2x+3y=13 e. 3x=6y-7 f. 2x-2y=63x-y=3 5x=-9y-18

4x+ly=-1

2. The total revenue, R, is a quadratic function of the price p of books sold,

represented by R = ap 2 + bp + c. Find the values of a, b, and c if the revenue is$6000 at a book price of $30, $6000 at a book price of $40, and $5000 at a bookprice of $50.

3. Solve the quadratic equation x2 - 6x + 4 = 0.

4. Find the value of the coefficients a, b, and c such that the three points (0, -5),

(1, -1), and (2, 5) he on the graph of the quadratic function y = axe + bx + c.

5. An observer 80 m above the surface of the water measures an angle ofdepression of 12° to a distant ship. How many metres is the ship from the base ofthe lighthouse?

6. Explain why each of the following is true:

a. L 1 = L 2 because ...

b. L1=L 3because...

c_ L2=L 4because...

4

Continued

51


D-4

Exercise 25: Systems of Linear Equations in Three Variables

7. The line y = x intersects the parabola y = 4x - x2 at the origin and at point Q.Let P be the vertex of the parabola.

a. Find the coordinates of Q.

b. Find the coordinates of P.

c. Find the length of OQ.

d. Find the distance from P to OQ.

e. Find the area of A OQP.

8. Solve the following equation for x: x +1 +1= 30

9. Find the coordinates of the point(s) where the graph of y = x2 + 8x + 15 crossesthe x-axis.

10. Find the product (3J - 2J)(5 - 3 ).

11. Solve for x: x2 -1= 5.

12. Determine the characteristics of the roots of the equation 3 x2 + 2 x -1 = 0.

13. Solve the equation 2x2 + 7x = 0 for x.

14. The two small circles have equations (x - 4)2 + y2 = 9 and (x T- 10)2 + y2 = 9. Findthe equation of the large circle.

52


D-5

Exercise 26 : Systems of Nonlinear Equations

Find the solution to the following system of equations: y = x2

a. graphicallyy=8 x2

b. algebraically

2. Graphically, find the solution for the system of equations y = 3x + 2 and y = 2x2.

3. Find the point of intersection of the circle x2 + y2 = 18 and the line y = x.

4. Solve the systems:

a. x2 + y2 =25 b. X2 + 3y2 = 30x2 -y2

ry= 1 `2x2 +y2 255

5. For the parabola defined by the equation y = x2 + 6x, find the




d. domain and range

6. Solve each of the following trigonometric equations on the interval 0° < 0 < 180°.

a. 2 cos 8 + = 0 b. (cos 0 - 2)(2 sin 0 - -)(cos 0 - 1) = 0

7. Determine the nature of the roots of the equation x2 + 5 = 3x.

8. Two scuba divers are swimming 6 m below the surface of the water. When theyare 20 m apart they see a shark directly below them. If the angle of depressionfrom the first diver to the shark is 47° and the angle of depression from thesecond diver to the shark is 40°, how far is each diver from the shark?

9. A jet flew from Halifax to Vancouver, a distance of 4200 km. On the return tripthe speed was increased by 100 km/h. If the total trip took 13 hours, what wasthe speed of the plane on the first leg of the trip (from Halifax to Vancouver)?

Continued

53


D-5

Exercise 26 : Systems of Nonlinear Equations

10. Prove that the quadrilateral defined by the lines l1: 4y = 3x - 6, 12: 4x + 3y = 33l3: 4y = 3x + 19, and l4: 4x + 3y - 8 = 0 is a square.

11. Solve the equation x2 - 6x + 4 = 0.

12. The sum of twice a number and three times its square is 261. Find the number.

1) [ 2 )

13. Solve: rx^ xi = 4.\ J l J

14. Solve: ( x2 + 4x -12

15. State the value of k that makes the trinomial x2 + 18x + k a perfect square.

16. Describe the domain and range using interval notation,

a. b.I

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D-6

Exercise 27: Graphing Linear Inequalities in Two Variables

1. Graph these lines on the same coordinate system: l1: 5x - 6y = 30, l2: 5x + 2y = 8,

and 13: y = -3.

2. Graph the inequality y > x + 2.

3. Sketch the region defined by y < x + 2 and 5x - 2y < 10.

4. Sketch the region defined by 5x - 6y >- 6, 3x + y < 4, and y >_ _3.

5. a. Draw the region defined by x ? 0, y > 0, 3x + y < 4, and y - 2x > -1.

b. Name the coordinates of the vertices in part a.

6. Solve the equation 4x + 5 - 2x-- 6 = 3 for x.

7. Calculate the distance from the origin to the line 3x + 4y = 6.

8. A ball is dropped from the top of a building that is 70 m high. The height of the

ball at time t (in seconds) is given by h = 70 - 4.9t2.

a. What is the height of the ball after 3 seconds have elapsed?

b. When will the ball strike the ground?

9. For what values(s) of k will the equation 2x2 +4X + (2 --- k - k2) = 0 have exactly

one root?

10. Solve the system of equations:

4a+3b- c=-7

3a-2b+3c=-10

a+ b- c=-2

11. The sum of two numbers is 181. Three times the larger plus twice the smallerequals 459. Find the numbers.

Continued

55


D-6

Exercise 27 : Graphing Linear Inequalities in Two Variables

12. a. Using a protractor, measure the interior angles of each of the polygons below.Find the sum of the interior angles for each polygon.

b. Write a general statement about polygons that describes the relationshipbetween the number of sides and the sum of the interior angles.

13. Solve the following system graphically:

y= 2x-4andy= 2x2-4

14. Expand and simplify this expression : (-v"2 + 3V3- )2.

15. Solve the following system .: y =x2 -x-2

y=x-3

56


D-7

Exercise 28: Quadratic, Absolute Value , and Rational Inequalities

a. Sketch the graph of Ax) = x' + 2x - 3.

b. Using the graph above, state the solution of

i. x2+2x-3 _ 0 ii. x2+2x-3<0

c. Explain how you arrived at your answer in part b.

2. Find the solution for the following. (Leave answer in set notation and intervalnotation.)

a. x2+3x-450

b. 2x2+3x-5>0

3. Solve the following inequalities. (Leave answer in interval notation.)

a. x2-x-20<0

b. x2 + 3x > 18

4. A block bordering Market Street is a right triangle, as indicated in the figurebelow. You walk around the block, which takes 125 paces on Market Street and102 paces on Pine Street. (Round all answers to one decimal place.)

a. At what angle do Pine and Market Streets intersect?

b. How many paces must you take on Front Street to complete the trip?

Market St.

5. State the roots of the equation (x - 6)(x + 7) = 0.

6. Solve the inequality (x+

< 0. Check your solutions.x + 1)

Continued

57


D-7

Exercise 28: Quadratic, Absolute Value , and Rational Inequalities

7. Solve the inequality 2 X+1 > 0. Check your solutions.X -x-2

8. Solve the inequality x2 4 0.

9. a. What is the first step in solving the inequality 2x + 5 < x + 17x+1 x-1

b. Solve the inequality. Check your solutions.

10. Solve the inequality I x I > 3 and sketch its solution on a number line.

11. Solve 12x + 3 1 < 5 and sketch its solution on a number line.

12 Solve the following inequalities:

a. 1 5x -31 2 b. 12x 1 +1<5 c. 13-4x 1 >9 d. 14x+ 8 1>1 - 4 1


a. 5x2 - 20 b. 5x2- 5y2

14. Both of these circles have centres labelled O. Using a protractor, measure L ABCand L AOC for each circle. Determine the ratio of the measure of L AOC to themeasure of L ABC for each circle.

15. A plane took 4 hours to fly 1920 km when it had a tail wind. Flying back againstthe same wind, and travelling at the same air speed, the plane took 1 hourlonger. Find the speed of the wind and the plane's air speed.

16. Graph the function y = sin (x + 90°).

58


Exercise 29 : Review 3

1. Sketch the function y = --3x2 - 12x + 7 and state the following:

a. vertexb. axis of symmetryc. min. or max. y-valued. direction of openinge. domainf. rangeg. wide/narrow openingh. zeros

2. Find the values of 0 between 0° and 360° given

a. sin 0=4-3/2

b.3cosO - 1=-2cosO

c. 6 tang 0 -- 11 tan 0 = -3

d. cos' 0 = cos 0

3. Solve each system of equations:

a. y=3x - 9 b. 4x=2-2y

2y=x2-10 -3y+2x=13

4. Given the points A(3, -2) and B(-5, 4), find the slope, midpoint , and distance

between them.

5. Find the solution by graphing:

a. y=x2+2x-4

.Y=-x-1

b. y < xz - 2 (Shade in the area and state the solution/zeros.)

Continued

59



6. The roots of a quadratic equation are 4± NZ.

a. Find the sum and product of the roots.

b. Given these roots, what was the original quadratic equation?

7. Find the value of k in the quadratic equation 0 = 6x2 --- 2kx + 3 if there is only

one solution, (Hint: Find the discriminant.)

8. Ambiguous triangles:

a. In A DEF, d = 2, e = 5, and Z D = 27°. Find all possible values of side f.

b. In A BYE, b = 8, e = 6, and Z E = 15°. Find all possible values of side y.

9. A dealer finds that he can sell 800 radios at $60.00 per set. However, for every$2.00 drop in price he can sell 50 more radios. At what price per radio should hesell to get the maximum cash return?

10. The sum of three numbers is 18. The third number is five times the sum of thefirst two numbers. The sum of the third number, three times the first number,and twice the second number equals 17. Write the three equations and solve forthe numbers.

11. y+2 < 5

12. -31x +4>10

13. (x + 3)(x - 4) > 0

(x2.25)

14.2x2 -5x-3 <0

3x2 +5x- 2

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E-1, E-2, E-3

Exercise 30 : Circle and Polygon Properties 1

a. In circle centre C, D is the midpoint of any chord AB.What can you conclude about AC and BC?

b. What can you conclude about A ADC and A BDC?c. Why is DC perpendicular to AB?d. Will the centre of a circle always lie on the

perpendicular bisector of a chord? Explain.

2. This circle has a centre at 0, and OB I AC, with OB = 4 and BC = 3. Find thelength of

a. ABb. ACc. the radius of the circled. the diameter of the circle

3. Solve the following equation: 15x2 + 14x = 8.

4. This circle has a centre at 0, AB = 12, OE I CD, and CD = 8. Find the length of

a. ODb. CEc. OE

5. This circle has a centre at 0, OP I AB,AC = 16, and AP = OP. Find the length of

a. OPb. OCc. ABd. AP

Continued

61


E-1, E-2, E-3

Exercise 30: Circle and Polygon Properties 1

6. Solve the following equation:1

+ 2 = 3.x-1 x+1

7. You have found one-third of a rim of an antique wagon wheel, and you wish toconstruct a replica of the antique. How would you find the radius of the wheel?

8. This circle has its centre at O.

a. Find the measure of L BOD.b. Find the measure of L COD.c. What is the relationship between the

measures of L BAC and L BOC?

9. Solve this linear system using any appropriate method: 3a - 2b = -10 andb+15=3a.

10. Solve the equation. 10x2 --- 9x = -2.

11. Sketch the following equation, and find its vertex, axis of symmetry, x-intercepts,domain, and range.

1x2-2x+13

12. Solve each of the following trigonometric equations on the interval 0° < B < 180°.(Round answers to one decimal place.)

a. 3 tang 0+ 7 tan 8+ 2= 0 b. cos3 U cos 8= 0 c. 4 sine B- 1= 0

13. Calculate the discriminant of each of the following and determine the nature ofthe roots:

a. x2-25=0 b. 0=3x2+5x+6 c. 2x2+5x+2=0

14. A circle with radius 5 crosses the x-axis at (4, 0) and (10, 0).

a. Find the coordinates of the centre.b. Find an equation for the circle.

62


E-1, E-2, E-3


Construct a circle with its centre at O. Draw a diameter AB. Plot a point Canywhere on the circle. Construct line segments AC and BC.

a. What is the measure of the angle at C?b. If AC = 5 and BC = 12, then find the length of AB.c. What is the length of OC?

2. This circle has its centre at 0, OB = 5, and BC = 6.Find the length of

a. ABb. AC

3. This circle has its centre at 0, AC = 5, and OC = 6.5.

a. Find the length of AB,b. Find the length of BC.c. Find the area of 0 ABC.d. Find the area of the circle.

4. This circle has its centre at 0, L I = 44°, and Z 2 = 98°.

a. Find the measures of Z 3 and L 4.b. What is the relationship between the measures of

Z GFH and L GOH?

5. This circle has its centre at O.

a. Find the measure of Z BOD in terms of x.b. Find the measure of L COD in terms of Y.c. What is the measure of Z BAC?d. What is the measure of Z BOG?

Continued

63


E-1, E-2, E-3


6. Solve this equation for x: x = I2x + 1.

7. Solve the linear system defined by the equations 2x + 5y - 8 and 3x - y = 12.

8. Spokes OD, OF, and OE of lengths 12, 6, and 10 radiate from a common point O.Angles DOF and FOE are each 20°. Find the area of d DEF.

9. Solve:

a. x9 + 3x- 18 0 b. x2 - x - 20 0

10. Find the minimum value of the function g(x) = 4x' + x + 3.

11. Solve: + X-1 -X

x + 2) _ x +2

12. This circle has its centre at 0,L BAO = 20°, and .Z CAO = 15°.

Find the measure of L BOD.

13. If A ABC has vertices A(-3, 4), B(4, 1), and C(-4, -5), find the length of thealtitude from vertex A to side BC.

14. Solve: 2x - 1 > 0.x+3 x+3

15. Sketch the region described by (x - 2)z + y2 < 4.

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E-1, E-2, E-3


1. This circle has its centre at 0, and L A = 27°.

a. What is the measure of L BOD and why?

b. What is the measure of Z E and why?

E

2. Using a protractor, measure Z 1 and L 2 for each of the following circles. Whatdid you discover?

3. The centre of this circle is at O.

a. What is the measure of L C? Why?

b. What is the measure of L D?

4. Given that AB is the diameter of this circle,prove that the area of the circle is given by

Aa2 + b2

4

5. Solve the linear system defined by the equations 3x + 21 = 5y and 2y + 3 = 3x.

6. A triangle is given by the vertices P(-5, 4), Q ( 1, 8), and R(-1, -2). Does the

perpendicular from P to QR bisect QR? Explain your answer.Continued


E-1, E-2, E-3


7. Solve each of the following trigonometric equations, finding all solutions on theinterval 0°< 6 < 180°. Round solutions to two decimal places where necessary.

a. tan 8 = 0 b. 5 tan 0 = -1

8. Solve this equation for x: x = N 2x - 3 + 3.

9. Solve this system of equations: x + y = 4y + z =-82x-z=15

c. sin 6 = 0.6493

10. A cube which measures 5 cm on each side is painted blue. The cube is cut into1 cm' cubes. Determine the number of 1 cm' cubes with:

a. three blue faces b. two blue faces c. one blue face d. no blue faces

11. Solve the following quadratic equations:

a. 2x2+9x+6=0 b. 3x2-6x-4=0

12. Tom Anderson's company plans on selling tickets to the U2 Concert. Thecompany is trying to decide on a price for the ticket. The company is consideringa price of $60/ticket if 1000 or less people were to purchase tickets. For every 100tickets sold over 1000, the ticket would be discounted by $3.00. What ticket pricewill yield a maximum profit?

13. ABCD is a square with AB = BC = 4. E is themidpoint of AD. Coordinate axes are drawn asshown.

a. Find equations for lines DB and EC.

b. Find the coordinates of F.

c. Find the area of 0 DCF.

14. For each of the following, describe its domain and range using interval notation.

a. Y b. Y

•

0

1/066


E-1, E-2, E-3


1. a. Construct a circle of radius approximately 4 cm.b. Construct a diameter of the circle and label the end points A and B.c. Draw a point C anywhere on the circumference of the circle.d. Complete A ACB.e. Determine the measure of L ACB.

2. If AB is a diameter of this circle, and 0 is the

centre,

a. what is the measure of L AOB? Explain why.b. what is the measure of Z D? Explain why.

If AC is tangent at B, the measure ofarc BD is 108°, and L CBE = 32°, findthe measure of each of the followingangles: L 1, Z 2, L 3, Z 4, and Z 5.

4. If AC is tangent at B, L ABD = 48°, and L CBE = 60°,

find the measures of L 2, L 3, and L 4.

5.

Given that AC is tangent at B, and thatthe measure of the are FEB is 248°, findthe measures of angles 1, 2, and 3.

Continued

67


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6. Given that AB is tangent at B, AD = BD,and L 1 = 70°, find the measures of L 2,L3,L4, and L5.

7. Determine the value of y given that OP is parallel to RS and given that thecoordinates of the quadrilateral OPRS are 0(4, 5), P(6, 4), R(2, 7), and S(-3, y).

8. Find the solution of the inequality -x2 + 16 ? 0.

9. Solve the linear system given by m - 3n = 11 and 2m = 5n + 19.

10. Solve this system of equations: x2 + y2 = 16 and x = - 4.

11. Find the zeroes of the function f(x) = 4 + 5x - W.

12. Given BC is tangent at B, and BEbisects L ABD (L 1 = L 2), verifythat BC =- CE.

13. The side of one square is 5 m longer than the side of another square. The area ofthe square with the longer sides is 153 m2 greater than the area of the othersquare. Find the length of the sides of each square.

14. Solve each of the following trigonometric equations on the interval 0° 5 9 < 360°.

a. (2 cos 0 - 1)(3 tan 0 - 5) = 0 b. 2 sine 0 - 3 sin 8 = -- 1

c. 6tang6-8tan6-8=0

15. Find an equation for a circle with centre (6, 9) and with the y-axis as a tangentto the circle.

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1.

Given that this circle has its centre at 0, AB istangent at A, and L 1 = 49°, find L 2.

2.

Given that this circle has its centre at 0, CD istangent at C, and OC = DC, find G 3.

3. Given that this circle has its centre at 0,radius = 5, distance OA is 20, and the linethrough A is tangent at B, determine thedistance AB.

4. Describe how you can construct a tangent to a circle at a given point on the circleif you are given the centre of the circle.

5. Given that this circle has its centre at 0, AD istangent at C, and Z ODC = 40°, verify L 1 = 50°.

6. Given AB and AC are tangents at B and C,respectively, and L 1 = 40°, find L 2.

Continued

69


E-1, E-2, E-3


7. Given that AD and AC are tangents at Band C, and L A = 30°, find Z 1.

8. Given that this circle has its centre at 0,tangent segments PQ and PR at Q and R,respectively, radius = 8, and QP = 15, findthe distances OP, OR, and RP.

9. Given that AB is tangent at A, chordDA 1 AB, and C is a point on the circle,find the measures of Z 1 and L 2.

10. Two tangent segments to a circle, from a point in the exterior, form an angle of60°. If the diameter of the circle is 10, how long are the tangent segments?

11. Find the x-intercepts of y = 3x2 _ 9x.

12. Solve the linear system given by 6 = 6x --- 11y and 4x - 5y = -2.

13. Solve and check: 2y + 5 _ y - 2 = 3.

14. Solve each of the following trigonometric functions on the interval 0° _< 0:5 180'.(Round answers to one decimal place.)

a. 2 tan O- 1 = 0 b. 3 sin 8- 1= 1

15. Factor the expression: a2 - b2 - 2bc - c2

16. Find the area of a triangle if the vertices of the triangle have coordinates (1, 3),(6, 0), and (0, -5).

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E-1, E-2, E-3

Exercise 35 : Circle Properties

1. Given: Inscribed quadrilateral ABCDLA= 68°LD=2LB

Find the measure ofa. L Cb. L Bc. L D

2. Given: circle centre 0LDCO=30°LABO=20°L BOC = 100°

Find the measure ofa. L ABCb. L A

c. L D

3. Given : circle centre 0A OQR is equilateralL ORS = 35°L PQO = 28°

Find the measure ofa. L ORQb. L PQRc. L Sd. L P

4. Given: EB is tangent at BAB is the diameterL1_L2

Verify: EB = BD

Continued

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E-1, E-2, E-3

Exercise 35 : Circle Properties

5. Given: circle center 0QP and PR are tangent segments

Verify: PQ = PR

6. The perimeter of a right triangle is 36 cm. If the hypotenuse is 15 cm find thelength of the other two sides.

x2 _ 2y = 01system of equations for x and y:7. Solve thisx+2y=6

8. To approximate the distance between two points A and B on opposite sides of aswamp , a surveyor selects a point C and measures it to be 140 m from A and260 m from B. Then she measures the angle ACB, which turns out to be 49°.What is the calculated distance from A to B?

9. Solve each of the following using an appropriate method. State the method usedin solving:

a. 3x2-10x+3=0 c.2s2=8s-7

b. 10a2--21a+9=0 d.5x216=0

10. Find the value(s) ofp for which the expression x2 + (p + 3)x + 2p + 3 is a perfect

square.

11. a. Calculate the distance between the points (-5, -4) and (8, -1).b. Determine the slope of the line between these two points.

12. Solve x2 - 4x = 21 by the method of your choice. Why did you select that method?

y<x

13. Find the area of the region determined by the graph of the solutions of x < 6

y>0

14. Individuals questioned in surveys are often chosen by computer programs thatrandomly select telephone numbers. Assume that a computer generatesrandomly the last digit of a telephone number. Find the probability that the lastdigit is an 8.

15. Graph the function y = cos(x - 90°) on 0°, 360°].

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Exercise 36: Polygon Properties

1. A polygon has 10 sides. What is the sum of the interior angles?



4. A polygon has n sides. What is the sum of the interior angles?

5. The sum of the interior angles of polygon is 1080°. How many sides does thepolygon have?

6. The sum of the interior angles of a polygon is 4500°. How many sides does thepolygon have?

7. The sum of the interior angles of a polygon is S. In terms of S, how many sidesdoes the polygon have?

8. Given AB is tangent to a circle at X, andCD contains the centre of the circle, drawthe circle.

9. Show that points A(1, 6), B(-3, -14), and C(2, 11) are collinear.

10. Solve: 12-3x I<- 1.

11. Solve this system of equations: 7 - 2y 1 = 0

x-2y209 3

12. Solve each of the following trigonometric equations on the interval 0° <- O<_ 180°.(Round answers to two decimal places.)

° 0a. - -==0 b. 2+3 sin 8=4 c. 2 tan 0-2 = 5 tan 02

13. Determine the roots of 10x2 -9x = -2.

Continued

73


E-1, E-2, E-3

Exercise 36 : Polygon Properties

14. Given: AC is a tangent to circle G at B.BE is a diameter.BDIBF

Fft=78°

Find the measure of

a. L EBF

b. B

c. L FBC

d. ED

e. .D

f. L DBE

g. L ABD

15. Solve: 56t2 + 14 = 65t.

16. Joe invested part of his inheritance of $335 000 at 7% per annum and theremainder at 10% per annum. After 1 year, the total interest from theseinvestments was $24 500. How much did Joe invest at each rate?

17. Solve: 1+XX -I

1--x x+1

18. The equation of the large circle is(x -- 8)2 + y2 = 64. If the small circles

have !equal radii, find the equationsthat represent them.

74


F-i

Exercise 37: Wages (Hourly)

Compute the weekly gross earnings for these employees. Time and a half is paidfor all hours over 40 in a given week.

a.

b.

C.

Employee Name Hourly Rate Mon. Tues. Wed. Thurs. Fri.

J. MacAdoo $10.20 8 9 10 11 7.5

S. Kashin $12.15 7 7 12 10 8

P. Dyck $20.00 8 8 9 12 9

2. Compute the weekly gross earnings for these employees. Time and a half is paidfor all hours over eight in a given day.

a.

b.

C.

Employee Name Hourly Rate Mon. Tues. Wed. Thurs. Fri.

J. Rees $ 8.20 9 8.5 7 10 7.5

S. L'Heureux $12.25 8 9.5 10 9 8.5

P Bennett $18.00 10 8 9 11 9.5

3. One particular week an assembly line worker worked 54 hours (40 hours wereregular hours, 6 hours were at time and a half, and the remainder was at doubletime). Find the worker's gross pay if the regular rate of pay was $14.00 per hour.

4. A waiter earns $5 .75 an hour for a 40-hour work week. He makes time and ahalf for overtime. One week he worked 45 hours and made $185 in tips . Find hisgross pay for the week.

5. Solve each equation

a. 3x2-5x+2=0 b. 4x2-11x--45=0

6. The tallest free-standing structure in the world is the 553-m CN Tower inToronto. Suppose that at a certain time of day it casts a shadow 1100 in long onthe ground. What is the angle of elevation of the sun at that time of day?

7. Determine the coordinates of the midpoint between the points A(12, 7) and B(6, --3).

Continued

75


F-I

Exercise 37: Wages (Hourly)

8. Given : L M = 75°

Mx =90°-G-ft = 70°

Find the measure of

a. L 1 e. L 5

b. Z2 f.

c. L 3 g. M

d. /4

9. Solve the linear system:$x-3y= 6

Cx+12y=

10. Solve :1

x = 2x

11. Solvefor each of the following trigonometric equations on the interval0° < 0 <_ 180°. (Round answers to two decimal places.)

a. 3 sine 0 - sin 0 = 0 b. (2 cos 0 - 1)(3 tan0 + 2) = 0 c. tan 2 0 - 9 = 0

12. A jeep is travelling on a road running due east. An enemy gun is spotted 800 maway in a direction 24° north of east. The gun has a range of 500 m.

a. How much further east can the jeep safely travel?

b. What length of road is within the range of the gun?

13. Find the distance between the point P(1, 3) and the line y = 3 x+2.

14. Solve for x: 2x+5I=11.

15. Solve for x:.t-x-

+2x+7 = 8.

16. Find the intersection of the line y = 4x - 11 and the parabola y = x' - 3x + 1.

-244

76


F-1

Exercise 38 : Wages (Commission and Net Income)

1. If a commission of 12% were paid on all sales, what commissions could be earnedon sales of

a. $740.50? b. $1345.99? $654.38?

2. A salesperson receives 8% commission on the first $1000 of sales, and 15% on allsales in excess of $1000 dollars. If sales for the past week amounted to $5000,what was the salesperson's total commission?

3. A salesclerk working in the appliance section of a department store receives aregular salary of $250 a week , plus 5% commission on sales in excess of $900.Last week's sales amounted to $3150 . What were the salesperson's total earningsfor the week?

4. Fred's monthly salary is $700. In addition, he receives 5% commission on thefirst $12 000 of his sales, and 7% commission rate on all sales over $12 000. Lastmonth Fred sold $24 000 worth of products. What was his gross pay?

5. Wendy works at an electronics store, earning $7.10 an hour, plus 6% commissionon sales up to $1000, 9% on sales from $1000 to $2000 and 12% on sales over$2000. How much did she earn if she sold $2600 worth of stereos and worked for40 hours?

6. The rate of Canada Pension Plan (CPP) contributions is 2.6% of taxable income.The rate of Employment Insurance (EI) is 3.05% of taxable income. Income tax iscalculated on taxable incomes as follows:

Earnings Tax Rate

$0-$550 17%

$551-$1138 26%

$1139 - 29%n

a. George earns $10.40 per hour for a 40-hour week. He pays union dues of$7.50 per week. What is his net pay?

b. May earns $10.60 per hour for a 43-hour week. What is her net pay?

Bill earns $5.60 per hour for a 20-hour week. What is his net pay?

Continued

77


F-1

Exercise 38 : Wages (Commission and Net Income)

7. Employee A can complete an assignment in 10 hours and employee B cancomplete this assignment in 8 hours. If employee B begins 3 hours afteremployee A has started, find the total time needed for the two employees to dothe complete job together.

8. Solve the linear system:6x= 12-3y

1x=-5

9. Two runners start from the same point at 12:00 noon, one of them heads north at6 km/hr and the other heads 68° east of north at 8 km/hr. What is the distancebetween them at 3:00 p.m. that afternoon?

10. Solve: Vx+2+ x-1= 4x+1.

11. Given A ABC with A(5, 4), B(7, -2), C(-3, 4).

b. Find the length of the line between the midpoints of AC and BC.

b. Find the length of the median from C.

12. An aquarium has a base of 60 cm by 40 cm. If 36 000 cm3 of water are pouredinto the aquarium, what is the depth of the water?

13. Solve for x: 5x2 + 10x - 3 = 0.

14. Solve for x: 22 - 3 -- 7X -4 2x-4 _ 2x+4

15.. Find the vertex, x-intercepts, domain, and range of y = 3x2 - 8x + 4.

16. Solve for x: x+2 < 8.x-5

78


F-1

Exercise 39: Property Tax

The Gysels own a home valued at $90 000. The rate of assessment is 45%. Themill rate was 62 mills and there was a local improvement tax of $180.00 forsidewalk reconstruction. What was the total tax bill for the family?

2. At the time of purchase, the Walchuck's home was assessed at $80 000. A taxassessor reassessed the house at $90 000. Assuming a mill rate of 55 mills, findthe amount of general tax increase resulting from the reassessment.

3. A ratepayer has a house valued at $85 000. The rate of assessment is 45%. Thelot has a 15 m frontage. Local improvements are charged as follows: sewer$3.87/r and sidewalks $2.50/m. Find the ratepayer's tax bill before school taxes ifthe municipal mill rate is 70 mills.

4. The rate of property tax in mills for a municipality can be found using theformula:

Mill Rate =Total Tax to Be Raised

x 1000Total Assessed Value of Property

Calculate the mill rate to the nearest whole mill for each of the years listed in aRural Municipality.

YearAssessed Value

of Real PropertyBudget

Requirement

a. 1994 $780 000 000 $69 000 000

b. 1995 $852 000 000 $82 000 000

c. 1996 $945 000 000 $95 000 000

5. Solve: 2- 3 =1.x x+1

6. Find the real number solution for ,,13x = x.

7. Find the vertex, axis of symmetry, x.intercepts, domain, and range of

y=--3x2--x+2.

8. Draw the region defined by 3x _.. y < 4 and x - 2y > 2.

Continued

79


F-1


9. Solve and check: x + 1 < 1.x-2


1x-y =8

11. Find all solutions for the following trigonometric equations on the interval90°<_ 0 :!^ 2700,

a. cos 0 = 1 b. 3 sin 0 = - 22

12. In the figure, QR and QS aretangent segments to the circle withcentre P. QP intersects the circle atM. Prove that M is equidistant fromthe tangent segments.

13. Solve for x: 3x2 + l Ox -- 7 = 0.

c. 2 tan 0 =5 d. tan 2 0_9=0

14. Find the distance between P(-2, 1) and the line 2x - 3y + 5 = 0. (Express youranswer in simplest radical form.)

15. Describe each solution to the inequality, using interval notation,

6 12

b.-5 2 8

C.

80


F-1, F-2

Exercise 40: Unit Prices , Exchange Rates , and Reconciliationof Bank Statements

1. A 355-ml can of soft drink costs 85o and a 1000-ml bottle costs $1.89. Find thecost per millilitre of each type of purchase.

2. If a 5.2-kg box of soap costs $12.49 and an 8.7-kg box of soap costs $17.85, whichis a better buy? Defend your answer.

3. Find the unit cost of each of the following:

a. 780 g of type A costs $14.65b. 390 g of type B costs $12.49c. 1580 g of type C costs $25.95

4. The value of a Canadian dollar in terms of the American dollar is 72¢.

a. If you exchange $250 Canadian for American dollars, how much would youreceive?

b. If you decide to buy an article in Grand Forks with a price tag of $28, howmuch is it worth in Canadian dollars?

c. A hotel in North Dakota advertises daily rates of $38. How much is that inCanadian dollars?

d. Can you find a simple method of converting American prices into approximateCanadian prices without using a calculator? Describe your procedure and givean example.

5. You are planning a trip to the United States and estimate you will need $200.00U.S. The listed cost is $1.00 Canadian equals 73¢ U.S. How much will it cost youin Canadian funds?

6. Complete a cheque book record for the following:

The balance on Sept. 8 is $998.43. The following cheques were issued:

Sept. 9, Cheque 234 to Kate's Department Store for $48.00; Sept. 13, Cheque 244to Gas Depot for $43.87; Sept. 20, Cheque 245 to Hydro for $66.98; Sept. 25, adeposit was made for $200.00; Sept. 30, Cheque 246 to Dales Rental Agency for$475.00.

7. Complete the table below for finding the cost of credit for using a departmentstore charge account for the period shown. Monthly credit charges are 1.4% ofthe balance due.

MONTH

PREVIOUS

BALANCE

PAYMENT

MADE

PURCHASES

CHARGED

BALANCEDUE

CREDIT

CHARGE

NEW

BALANCE

February $586.00 $100.00 $ 93.00March 200.00 121.75April $275.00 13.17May $200.00 $ 87.13

Continued

81


F-1, F-2

Exercise 40 : Unit Prices , Exchange Rates , and Reconciliationof Bank Statements

8. Complete a reconciliation statement form for this account.

ACCU CREDITBALANCE FORWARD

DATE

20 1 08 350 00

DESCRIPTION DEBITS CREDITS DAY Mo. BALANCE

Deposit 452 51 21 08 802 51Cheque 191 102 90 25 08 699 61Cheque 192 141 12Cheque 193 24 88 27 08 558 49Cheque 194 56 70 476 91Deposit 215 00 691 91Deposit 280 00 30 08 971 91Cheque 195 125 45Service Charge 8175 311 08 837 71

CHEQUE CHEQUES ISSUED TO OR CHEQUE DEPOSIT DEDUCT/ADD BALANCE I''WD

DATE No. DESCRIPTION OF DEPOSIT AMOUNT AMOUNT CHEQUES/DEPS 350 00

Aug. 452 51 CHQ - /DEP + 452 51

21Deposit

BALANCE 802 51

E102 90 CHQ - /DEP + 102 90

25 191 ssoBALANCE 699 61

141 12 CHQ - /DEP + 141 1225 192 Wires

BALANCE 558 49

T l h24 88 CHQ - /DEP + 24 88

27 193 onee epBALANCE 533 61

d56 70 CHQ - /DEP+ 56 70

27 194 roHyBALANCE 476 91

D i215 00 CHQ - /DEP + 215 00

27 epos tBALANCE 691 91

iD280 00 CHQ - /DEP + 280 00

30 teposALANCE 971 91

Sept. ' kSh125 45 CHQ - /DEP + 125 45

1195 sPete ac

BALANCE 846 46

I 211 11 CHQ - /DEP + 211 113 196 nsurance

BALANCE 635 35

iD 2400 00 CHQ - /DEP + 2000 006 epos t

BALANCE 2635 35

S854 00 CHQ - /DEP + 854 00

7 197 earsBALANCE 1781 35

198 G57 10 CHQ - /DEP + 57 10

7 asBALANCE 1724 2

hiL ' Cl146 58 CHQ - /DEP + 146 58

8 199 ngynn s otBALANCE 1577 7

Continued

82


F-I, F-2

Exercise 40: Unit Prices, Exchange Rates, and Reconciliationof Bank Statements

STATEMENT OF RECONCILIATION

Bank Reconciliation

Balance from statement:

Add:

Total additions:

Subtotal:

Subtract:

Total additions:

Subtotal:

This should agree with the balance shown in your record bookafter service charge is deducted:

9. A rectangular room , 5 m by 11 m, has an open -beam ceiling . The two parts of theceiling make angles of 65° and 32° with the horizontal. Find the total area of theceiling.

10. Solve: 14--2xI_8.

11. Use the discriminant to determine the nature of the roots of 2x2 _X + 4 = 0.Continued

83


F-1, F-2

Exercise 40 : Unit Prices , Exchange Rates , and Reconciliationof Bank Statements

12. Given the diagram with circle centre G , tangent line AC, $^ = 120°. Find themeasure of

a. L CBF b. L FBE c. L EBC d. 13-DP

13. Solve the following equation (to one decimal place ): w2 + 1.4w - 7.35 = 0.

14. a. A circle has a radius of 12. What is the length of a chord that is determinedby an are of 90°?

b. A chord has a measure of 15. If one are determined by the chord has ameasure of 90°, what is the radius of the circle?

x+y+z=8

15. Solve the system of equations: 2x - 3y + z = 23

i x-y+3z = 1

16. Solve for x: 82' = 32 +5

17. The equation of the circle is x2 + y2 = 64.

Find the area of the square.

84


F-3

Exercise 41: Budgeting 1

Dean Charles earns a net weekly salary of $645.25. The family receives amonthly child tax benefit cheque that amounts to $42.50 per child . There arefour children in the family. The family's expenses are as listed below.

a. monthly mortgage payment ............................. $ 625.00

b. monthly car payment .................................... 213.50

c. average monthly telephone bill ............................. 17.40

d. average monthly hydro bill ................................ 120.00

e. yearly car insurance premium ............................. 822.00

£ monthly life insurance premium ............................ 18.00

g. property taxes for the year ............................... 1925.00

h, yearly home insurance premium ........................... 275.00

i, food (average per month) ................................. 425.00

j. clothing expenses for the year ............................. 725.00

k. average car maintenance for the year ....................... 340.00

1. gasoline per month ....................................... 80.00

in. entertainment per year ................................... 750.00

n. gift spending per year ........................... .... 630.00

o, newspapers and periodicals (per year) ...................... 210.00

p. water bill - paid quarterly ................................ 115.00

Using the information provided, prepare an estimated monthly budget for theCharles family on the blank budget form on the following page.

Continued

85


5. Personal Financesa. Personal Loan $b. Investments $c. RRSP ' $d. Life Insurance $e. Charities $f. Credit Card Payments $g. Service Charges $h. Savings " $i. Other Personal Finances $Total Personal Finances #5 $

F-3

Exercise 41 : Budgeting 1

1. Incomea. Regular Monthly Income _..,

b. Spouse's Regular Monthly Income

c. Additional Income _,,..

d. Other Income _Total Monthly Income #1

2. Housing Expensesa. Mortgage or Rent $b. Property Tax $

c. Home/Property Insurance $

d. Repairs/Maintenance $

e. Other Housing Expenses $Total Housing Expenses #2 $

3. Utilitiesa. Hydro

b. Gasc. Phoned. Watere. OtherTotal Utilities

4. Transportationa. Public Transportb. Car Loanc. Car Fueld. Car Maintenancee. Car Insurancef. Other TransportationTotal Transportation

6. Personal Expensesa. Groceries _____b. Clothing $c. Entertainment $d. Gifts $e. Vacations $f. Other Personal Expenses $Total Personal Expenses #6 .$.

#3 $. 7. Other Expensesa.b.


$ C. .$ Total Other Expenses #7 $___$

Total Monthly Expenses #8 $

$#4 $

Income minus Expenses (#1 - #8) #9 $

Comments:

* Note 1: Financial analysts advise that RRSP contributions should start early.

Note 2 : Financial analysts advise that a reserve fund of two or three months of income should besaved for emergencies . Generally, it could take several years to build up a reserve fund.Reserve Fund Calculation . Calculate two months of income and divide by the number of months it willtake you to achieve it.

Continued

86


F-3


2_ Charlie and Bonny Wood are both employed. Bonny receives a weekly salary of$391.82 after deductions. Charlie receives a weekly salary of $381.42. The familyreceives a monthly child tax benefit cheque that amounts to $107.72. Thefamily's expenses are as listed below.

a. monthly first mortgage payment .......................... $531.50

b. monthly second mortgage payment ......................... 201.65

c. monthly car payment .................................... 237.75

c. average monthly telephone bill ............................. 20.20

d, average monthly hydro bill ................................ 200.00

e. yearly car insurance premium ............................. 770.00

f. monthly life insurance premium ............................ 22.00

g, home is assessed for property tax purposesat $80 000; the mill rate is 22.35 mills ..........................

h. annual home insurance based on a home value ...................of $60 000 at a cost of $0.42 per $100 . ........................ .

i. food (average per month) ................................. 740.00

j. clothing expenses for the year ............................ 1200.00

k. average car maintenance for the year ....................... 460.00

1. gasoline per month ...................................... 140.00

m. entertainment per month ................................. 180.00

n. newspapers and periodicals (per year) ....................... 102.00

o. average monthly credit card payment ....................... 200.00

p. water bill - paid quarterly ................................ 135.00

Using the information provided, prepare an estimated monthly budget for theWood family on the blank budget form on the following page.

Continued

87


F-3


1. Income

a. Regular Monthly Income $

b. Spouse's Regular Monthly Income $

c. Additional Income $

d. Other Income $Total Monthly Income #1 $

2. Housing Expensesa. Mortgage or Rentb. Property Tax $

c. Home/Property Insurance $

d. Repairs/Maintenance $e. Other Housing Expenses $Total Housing Expenses #2 $

3. Utilitiesa. Hydro $b. Gas $c. Phoned. Watere. OtherTotal Utilities #3 $

4. Transportationa. Public Transport $b. Car Loan $c. Car Fuel Sd. Car Maintenance $e. Car Insurance $

I. Other Transportation $Total Transportation #4 $

5. Personal Financesa. Personal Loanb. Investmentsc. RRSP "d, Life Insurancee. Charitiest. Credit Card Payments $g. Service Chargesh. Savings **i. Other Personal Finances $Total Personal Finances #5 $

6. Personal Expensesa. Groceries $b. Clothingc. Entertainment Sd. Gifts Se. Vacations $f. Other Personal Expenses $_Total Personal Expenses #6 $

7. Other Expensesa. $

b. $

$C.

Total Other Expenses #7 $


Income minus Expenses (#1 - #8) #9$

Comments:

* Note 1: Financial analysts advise that RRSP contributions should start early.

Note 2: Financial analysts advise that a reserve fund of two or three months of income should besaved for emergencies . Generally, it could take several years to build up a reserve fund.Reserve Fund Calculation : Calculate two months of income and divide by the number of months it willtake you to achieve it.

Continued

88


F-3


3. A sheet of paper has a perimeter of 40 cm. Its area is 99 cm2. Find its

dimensions.

4. If AB = 12, AC = 13, and AD = 15, find the area of A ACD.


6. Solve each of the following trigonometric equations over the interval 0°<_ 0 < 180°.Round answers to two decimal places.

a.2tan 8=-23

b.3sin0--1=1 c. 3 tang + 2 tan 0 = 2

7. Given: Diameter ACBC tangent at CCD bisects L ACB.Points A, D, and B lieon a straight line

Prove: DC = DB.

8. A PQR has vertices at P(-2, 1), Q(1, 5), and R(5, 2).

a. Is A PQR isosceles?

b. What is the length of the longest median?

9. Find the vertex, x-intercepts , domain , and range of y = -3x2 + 4x + 3.

Continued

89


F-3


10. Find the length(s) of BC.

11. How many squares are there in this figure?

12. Find the region on a graph where y ? 2x+ 1 or y < 2 x+3.

13. SecurCard has an annual fee of $20 and a finance charge of 19.8% per year onthe unpaid balance. In May, SecurCard charged Mable the annual fee and afinance charge on her unpaid balance of $324.00. Find the total of Mable'smonthly statement.

90


F-3


Erica and Tom Elsimatesky are both employed. Erica receives a weekly salary of$301.60 after deductions. Tom nets $310.50 per week. The family receives amonthly child tax benefit cheque that amounts to $26.93 per child for each oftheir two children. The family's expenses are as listed below.

Expenses for the family include:

a. monthly mortgage payment ................................................................$725.00

b. monthly car payment ............................................................................186.40

c, average monthly telephone bill ..............................................................18.60

d. average monthly hydro bill ...................................................................225.00

e. yearly car insurance premium ..............................................................720.00

f, semi-annual life insurance premium ...................................................120.00

g. home is assessed for property tax purposes

at $30 000; the mill rate is 61 mills ...........................................................

h. annual home insurance based on a home value of$50 000 at a cost of $0.62 per $100 ............................................................

i. monthly boat payment ..........................................................................130.00

j. food (average per month) .....................................................................525.00

k. clothing expenses for the year ..............................................................650.00

1. average car maintenance for the year ..................................................560.00

m. gasoline per month ................................................................................100.00

n. entertainment per year .........................................................................600.00

o. yearly vacation .......................................................................................940.00

............................. 144.00p. newspapers and periodicals (per year) ....................

q, average monthly credit card payment .................................................200.00

r. gift spending per year ...........................................................................850.00

s. baby-sitting (average per month) .........................................................200.00

Using the information provided, prepare an estimated monthly budget for theElsimatesky family on the blank budget form on the following page.

Continued

91


F-3


1. incomea. Regular Monthly Income $

b. Spouse's Regular Monthly Income $c. Additional Income $

d, Other Income $Total Monthly income #1 $

2. Housing Expensesa. Mortgage or Rent $b. Property Tax $c. Home/Property Insurance $d. Repairs/Maintenance $

e. Other Housing Expenses $Total Housing Expenses #2 $.

3. Utilitiesa. Hydro $b. Gas $c. Phone $d. Water $e. Other $Total Utilities #3 $

4. Transportationa. Public Transport $b. Car Loan $

c. Car Fuel $d. Car Maintenance $

e. Car Insurance $

t. Other Transportation $Total Transportation #4 $ ,

5. Personal Financesa. Personal Loanb. Investmentsc. RRSPd. Life Insurancee. Charitiesf. Credit Card Paymentsg. Service Chargesh. Savings "i. Other Personal FinancesTotal Personal Finances #5 $

6. Personal Expensesa. Groceriesb. Clothingc. Entertainmentd. Giftse. Vacationst. Other Personal ExpensesTotal Personal Expenses #6 $.

7. Other Expensesa. $b. $C.

Total Other Expenses #7 $


Income minus Expenses (#1 - #8) #9 $

Comments:

" Note 1: Financial analysts advise that RRSP contributions should start early.

" Note 2: Financial analysts advise that a reserve fund of two or three months of income should besaved for emergencies . Generally, it could take several years to build up a reserve fund.Reserve Fund Calculation : Calculate two months of income and divide by the number of months it willtake you to achieve it.

Continued

92


F-3


2. A submarine at the surface of the ocean makes an emergency dive. Its pathmakes an angle of 21° with the surface.

a. If it goes for 300 meters along its downward path, how deep will it go? Whathorizontal distance is it from its starting point?

b. How many metres must it go along its downward path to reach a depth of1000 meters?

3. Solve: x2 - 2y = 0

3x+2y.-10

4. Solve: 2x2 + 5x 8 = 0.

5. Show that no parabola (y = axe + bx + c) can pass through the set of points (1, 2),(4, 8) and (1, -4).

6. Given : DE is tangent at CAB\\DE

Prove: A ABC is isosceles

7. A store sells 60 tape recorders a day at $80.00 each . (They cost $54.00 to make.)For every increase in cost of $1.00, the number sold decreases by 1. What is thelargest possible profit?

8. a. Find the equation of the line through (1, 7) and parallel to the line

y=4x+5.

b. Find the equation of the line through (.1, 7) and perpendicular to the line

y=4x+5.

9. What is the distance between the parallel lines 5x + 2y -- 7 = 0 and5x+2y+8=0?

Continued

93


F-3


10. Mrs. Murray wants to sell a particular bolt costing $0.25 each together withanother type of bolt costing $0.40 each. She plans on charging $3.10 for themixture. The number of $0.25 bolts is two more than the number of $0.40 bolts.How many of each type of bolt is she planning to include in the package?

11. Solve: 2x+5 < x+1x+1 x-1


a0 s-----

.-30 70

b E.

9

--6 4

0.C .

-8 0 6

13. Graph the function y = -cos 0 + 2 on 0 E [0°, 360°].

94


F-5

Exercise 43: Exponential Growth

. The growth of the value of a RRSP is as shown in the table.

Time (years) Value ($)0 50001 54002 58323 62994 68025 73476 7934

a. Esti ate the time needed to reach $ 10 000.

b. Estimate the value of the RRSP after 10 years.

2. Sally invests $4000 in a bond that pays 6% interest, compounded annually. Makea table showing the value of the investment over the 5 years. Plot the data andestimate the value of the investment after 9 years.

3. If you put $100 in the bank for 8 years, how much will it be worth at the end ofthat time at

a. 3.2% interest, compounded annually?

b. 5.4% interest, compounded annually?

4. River City's present population of 1000 is expected to grow exponentially overthe next 10 years at 4% per year. What is the expected population at the end ofthat time?

5. Find the vertex, the axis of symmetry, x-intercepts, domain , and range ofy=2xZ-11x+5.

6. A Canadian dollar is worth 72iZ U.S. A stereo sells for $750 in Minneapolis. Whatis its value in Canadian funds?

Continued

95


F-5

Exercise 43 : Exponential Growth

4x-3y+6z=--9

7. Solve the system of equations: 2x + 4y - 3z = -10

3x+2y-4z=--11

8. Find the intersection of+y2 = 25

+y=13

9. A diameter and a chord of a circle have the same endpoint A. If the diameter is40 cm and the chord is 24 cm, how far is the chord from the center of the circle?

10. Solve and check: 4-4=3x+2 x3

11. Determine the solution for each of the following trigonometric equations. (Roundanswers to two decimal places.)

a. (4 cos' 0 -1)(3 sin 0 + 1) = 0, 0° < 0:5 180°

b. tang 0 - tan 0 = 2, 0°:!^ 0:5 360°

c. cos 0 sin 0 --- cos 0 = 0, -180'<_ 0 5 180°

12. Find a quadratic equation for which the sum of the roots is 3 and the product4

is3

13. Using analytic geometry, prove that the diagonals of a parallelogram bisect eachother.

Continued

96


F-5


14. Given: AB = 70

E-50, =80ED = DCLCAB=35°

a. Find the measure of each numbered angle (L 1...L 9).

b. Find the measure of each of the following arcs: ED, DC, BC, AE, AB,

97


F-5

Exercise 44 : Interest

1. a. If $6000 was invested for 3 years at 7% simple interest, to what amount willit grow?

b. If $3000 was invested for 6 years at 4% simple interest, how much interest isgenerated?

c. If $10 000 was invested for 6 months at 9% simple interest, how muchinterest is generated?

d. Determine how long it will take a $1500 deposit to earn $630 interest at 6%simple interest.

e. How much will a $2500 deposit be worth if it is invested for 5 years at 6 3/4%simple interest?

f. What interest rate will generate $665 interest after 8 years on a $1750deposit?

g. What principal will generate $324 interest at 3% simple interest after 9years?

2. If $6000 was invested for 3 years at 6%, what will be the value of the investmentafter 3 years

a. using simple interest?

b. assuming it is compounded annually?

3. A man invests $12 000 for 5 years compounded annually. If the rate of interest is9%, how much interest will be earned during the 5 years?

4. Ms. Jones invested $8000 for 1 year. At the end of the year, her investment had avalue of $8800. What rate of interest did she receive?

5. Determine the effective rate on a loan of $1000 at 10% per year compoundedsemi-annually.

6. Determine the effective rate on a loan of $2000 at 12% per year compoundedquarterly.

7. Mr. Smith invested in a 1-year term deposit paying interest at the rate of 4% perannum. How much did he invest if he earned $750 interest during the year?

Continued

98


F-S


8. A bank offers an interest rate of 6% per year, compounded annually. A secondbank offers an interest rate of 6% per year, compounded quarterly. If $5000 weredeposited for 12 years, in each bank, how much more income would be gained inthe second bank than in the first?

9. Find the roots of a = X24 2

10. Find the vertex, axis of symmetry, x-intercepts, domain and range ofy=-6x2+7x+5.

11. The midpoint of EF is (5, 1) and one endpoint is given by E (-1, 0). Find thecoordinates of F.

12. Solve: x3- 2x2 - 15x > 0.

13. Solve : ,10y + 16 = 3y.

14. Given: circle with centre Fchord ACAB is tangent at A.BC is tangent at C.ZABE=15°

a. Find the measures of all thenumbered angles (Z 1... G 5).

b. Find the measures of €1 and AE.

15. A sample consists of 200 business calculators (eight of which are defective) and150 scientific calculators (nine of which are defective). If one calculator isselected randomly from this sample, find the probability that it is defective.

99


G-1

Exercise 45: Inductive and Deductive Reasoning

1. Which of the following examples of reasoning are inductive and which aredeductive?

a. Every time we have a club meeting, I have a test in school the next day.

b. Susan's father brought her early to school each day. She noticed that MsTaylor, her math teacher, arrived at 7:30 each day for several weeks. Susansaid, "Ms Taylor always arrives at 7:30."

c. All students in senior high must enroll in physical education. John is astudent, so he concludes that he will take physical education.

d. The sun has risen each morning from time immemorial. We can be certain itwill rise tomorrow.

e. Anyone who likes to play football likes to play basketball. Sheeva likes toplay football. We conclude that she likes to play basketball.

f. Triangle ABC is an equilateraltriangle . We can conlude thatAB = AC.

A

B` IC

g. Joe counted the number of cars of different colours that passed his house in15 minutes. More than half the cars were white. He decided that white is themost popular colour for cars.

2. Which of the above conclusions are valid?

3. Triangle PQR has vertices at P(1, 4), Q(-5, 2), and R(-1, -4). Show that the linejoining the midpoints of any two sides is parallel to the third side.

4. Find the zeroes ofx - 6 -2x-3= 0.

3x+4 x+2

5. Solve the following system: {x2 + y2 = 4

x-2y=4

6. Two points, A and B, are on ground level and in line with the base, C, of a tower.The angles of elevation of the top of the tower at A and B are 21° and 35°,respectively. How tall is the tower if A and B are 300 feet apart?

Continued

1 00


G-1


7. Given: AB is tangent at BAC is tangent at CD is the midpoint of tangent BC

Verify: L1=Z2


a. {x x € Real Numbers} b. y l y < 0} c. xI -4<x<2

9. Write the equation of the line that has a slope of 3 and an x-intercept of 5.

10. Find the vertex, x-intercepts, axis of symmetry, domain, and range for thefollowing quadratic function.

y=-3x2+8x-2

11. Solve for 0 where 0° < 0 < 360°. Express your answer(s) to the nearest tenth.

10Cos' 6+11 cos0+1=0

12. Harland's GIC pays him 6% simple interest.

a. How much interest will he earn on $4000 deposited in this account after 1year?

b. What will be his balance in this account at the end of the year?

13. Use the discriminant to find the number of solutions for each of the followingquadratic equations.

a. x2-2x+2=0

b. x2-6x=17

14. A credit card company charges a daily finance charge of 0.0722% on all cashadvances. How much would your finance charge be if you borrowed $200.00 for60 days using the cash advance?

101



1. Given the quadratic function y = axe + b, under what conditions for a and b will

its graph pass through the

a. origin?

b. point (-1, 1)?

2. Create a quadratic function with vertex (2, 3) and having a minimum value.

3. The vertex of a parabola is (-1, -1). A point on the parabola is (4, 7).

a. Determine the quadratic equation that defines this parabola.

b. What are the x-intercepts of this parabola?

4. Solve: 2 tan 0 - 3 = 5 tan 0 - 1 on the interval [0°, 360 °] .

5. A frisbee is thrown straight up into the air from a position 2 m above groundlevel. The height h in metres after a given time t in seconds is given by the

equation h = 2 + 6t - 2t2.

a. What is the maximum height the frisbee will reach?

b. If it is caught 2 m above the ground, how long will it have been in the air?

c. Approximately how much longer would it have taken to hit the ground?

6. Solve 2 sin 2 0 + 7 sin 0- 4 = 0 on 0, 360°] .

7. Solve:30 - 5 -

X 2 -9 _ x-3

Continued

102



8. Given a circle with its centre at 0, diameter AB, and CD = 50°, find the followingangle measures.

L1=

L2=

L3

L4

L5

9. If $6500 is put into an account that earns 6% per annum, compounded quarterlyfor 20 years, how much interest would you make over the 20 years?

10. Solve the equation 3x +7 - x--5 = 4.

11. Solve the system of linear inequalities graphically.

-2x-3y<6 and 3-y+x>0

12. Given that AB = 12 cm and 0 is thecircle's centre, find

a. OD =

b. AB

c. ACB =

103


G-2

Exercise 47: AND, OR , NOT, and Venn Diagrams

Draw two overlapping circles. Label one "A", and the other "B".

a. Shade the region that is in A n B.

b. Mark with xs the region that is in A u B.c. Place os in the region that is not in B.

2. Draw three overlapping circles and label them A,

B, and C.

a. Mark xs in the region enclosed in A n B.

b. Mark os in the region enclosed in C v B.

c. Mark *s in the region that is not in A.

3. Everyone in a class of 30 students wears at least one of braces or glasses. If 18wear braces and 3 wear braces and glasses, how many wear only glasses?

4. Each member of a sports club plays at least one of the following sports: soccer,baseball, or tennis. Find the number of members the club has if the clubsecretary reported the following facts at the last meeting:

• 163 members play tennis • 36 members play tennis and baseball

• 13 members play tennis and soccer • 6 members play all three sports• 11 members play soccer and baseball • 208 members play baseball or tennis

• 98 play soccer or baseball

5. In a class of 20 boys, 10 boys play hockey, 14 boys play football, and 6 of themplay both hockey and football. How many boys do not play either of these games?

6. In a class of 28 students, 16 students received a B in mathematics, 14 studentsreceived a B in English, and 11 students received a B in both mathematics andEnglish. How many students did not receive a B in either of these subjects?

7. Solve and check : '3x + 1= ,/5x + 1.

8. Solve each of the following trigonometric equations over the interval 0°<_ 0< 360°.

a. cos 0 = - 7 b. 3 sin 0 - 2 = -1 c. tan (0 + 41°) =

Continued

104


G-2

Exercise 47: AND , OR, NOT, and Venn Diagrams

9. C is the centre of the circle shown, and F is a point on the circle such thatquadrilateral BCDF is a 2-cm by 3-cm rectangle. Find the area in squarecentimetres of the shaded region.

10. Dennis Murray earns $18 500 per year after deductions. His wife earns a take-home pay of $248.56 a week. The Murrays have one child for whom Mrs. Murrayreceives a child tax benefit that amounts to $323.16 per year.

The Murrays have recently bought a home for which they are making monthlypayments of $425.00, and monthly payments of $165.00 on a 3-year bank loanthat helped to finance the purchase. Taxes are $730.00 per year, gas heat isestimated at $740.00 a year, electricity and water bills are approximately $83.50for a 2-month period, the telephone bill averages approximately $10.50 permonth, and home insurance is $242.00 per year.

The Murrays are also making payments of $180.00 per month on a one-year loanthey made to finance the purchase of furniture. Insurance on the car costs$288.00 per year, and gasoline averages $105 per month. Additional expensesinclude: food, $420.00 per month; clothing, $795.00 per year; entertainment,$330 per year; holiday gift purchases, $165.00 per year; newspapers, books andmagazines, $125 per year; car maintenance, $255.00 per year; and vacation,$850.00 per year. Prepare a monthly budget for the Murray family for the month

of April.

11. How wide of a uniform white border should be left on a sheet of paper measuring

7 cm by 11 cm if 45 cm' is required for the printed matter?

Continued

105


G-2

Exercise 47: AND, OR, NOT, and Venn Diagrams

12. For each parabola, state

a. whether the parabola opens upward or downwardb. the coordinates of the vertexc. the equation of the axis of symmetry

i. y+3=x2 ii. y-3=--(x+2)2 W. y+1 = (x_5)2

13. Given: circle with centre FEB and BD are tangents at A and C, respectivelyLEAD=40°

* AFC 150()

a. Find the measure of all numbered angles (Z 1 ...L 8).

b. Find the measures of GA, -G--G, and GAS.

40°3

5 4

F

G

15o-

6

1 8 2

14. Write the equation of a line that is perpendicular to the x-axis and passesthrough the point (4, -5).

15. Graph the region that satisfies the inequalities x2 + y2 > 9 and x2 + y2 -< 16.

106


G-3

Exercise 48: Counterexamples

1. Ravi concluded that whenever he added two prime numbers, the sum wasalways even. Find a counterexample to prove his conjecture wrong.

2. Give a counterexample to prove that the conjecture 1 < 1 is false.X

3. Mary used a graphing calculator to graph y = xx. She found the screen blank forx < 0 and conjectured that y = xx is undefined for x < 0. Find an example thatwould support her conjecture. Find a counterexample to show that her conjectureis false.

2

4. Frank claims that since f (x) = x 49 can be reduced to x + 7, the functions

x2 -49x-7

f (x) =x-7 and

g(x) = x + 7 are the same. Find a value of x that is a

counterexample.

5. Notice that x2 + x + 41 produces the prime number 43 if x = 1, the prime number47 if x = 2, and the prime number 53 if x = 3. One might assume it alwaysproduces primes for positive integral values of x. Find a counterexample to provethis is wrong.

6. Everyone in a class of 25 students must take either Latin or French. There are18 students taking French, three of whom take Latin as well. How manystudents are taking Latin?

7. In a gathering of 18 men it was discovered that eight of them could speakFrench and eleven could speak English. There were four who could speak boththese languages. How many could not speak either of these languages?

8. Of the members of three athletic teams in a certain school, 21 are on thebasketball team, 26 on the baseball team, 29 are on the football team, and 8 areon all three teams. Furthermore, 14 play basketball and baseball, 15 playbaseball and football, and 12 play football and basketball. How many membersare there altogether?

9. A Canadian dollar is worth 72¢ U.S. A pair of shoes cost $75.00 in Fargo. What isits value in Canadian funds?

10. Solve and check: 2x2 - 1= 4x+6x-3 x-3

11. Find the vertex, axis of symmetry, x-intercepts, and range of the following

quadratic function: y = 10x2 + 13x - 3.

Continued

107


G-3


12. The account in which Jane deposits her money pays 4.25% simple interestannually.

a. How much interest will she earn on a deposit of $5000 left in her account for6 months?

b. What will be her balance in the account at the end of this time?

13. Given : EC is tangent at D

Verify: L 5 =_ L ADC

14. The face of Brian's watch is decorated withtwo circles and a square. The shaded part isgold. One side of the square measures 20 mm,

a. What is the radius of the smaller circle?b. What is the area of the gold?c. Calculate the radius of the larger circle.

15. Find the distance between the lines 3x - 5y + 7 = 0 and 6x - l0y - 2 = 0.

16. Given the information in the triangle below, solve the triangle.

A

17. Graph the function y = 2 sin 0, 0 E [-180°, 270°].

108


G-4

Exercise 49: Converse , Contrapositives, if...Then...

1. For each of the following statements, write the converse of the statement.Determine the truth of the statement and its converse.

a. If you can operate a car, you can fly a plane.b. If a child is less than 6 years old, the child believes in the tooth fairy.c. If you are a successful basketball player in college, you are taller than

average.d. If you studied home economics in school, you are a good cook.e. If it is raining, visibility is poor.f. If a girl goes to a party, she wears high-heel shoes.g. If a person likes pizza, he will like spaghetti.h. If two sides of a triangle are congruent, the angles opposite those sides are

congruent.i. If two angles are right angles, they are congruent.

2. For each of the following statements, write the contrapositive of the statement.

a. If two angles of a triangle are congruent, then the sides opposite these anglesare congruent.

b. If two sides of a triangle are congruent, then the angles opposite these sidesare congruent.

c. If two angles are supplements of congruent angles, then they are congruent.d. If two angles are complements of congruent angles, then they are congruent.e. If a triangle is equilateral, then it is equiangular.f. If a triangle is equiangular, then it is equilateral.g. If a point is on the perpendicular bisector of the segment, then it is

equidistant from the end points of the segment.h. If M is the midpoint of AB, then d(A, M) = d(B, M).i. If P is between A and B, then d(A, P) + d(P, B) = d(A, B).

3. Thirty-four women attended an international conference. These facts wereestablished:

• 13 women spoke English • 16 women spoke French• 12 women spoke Spanish • 7 women spoke English and French• 4 women spoke English and Spanish • 5 women spoke French and Spanish• I woman spoke English, French,

and Spanish

How many could not speak any of these languages?

Continued

109


G-4

Exercise 49: Converse , Contrapositives , If...Then...

4. Find the roots of x2 - 6x + I = 0 to the nearest tenth.

5. Solve 1 3x -6 > 3.

6. Solve I 7x-3 1=3-7x.

7. Determine the nature of the roots of 12x2 - x - 6 = 0.

8. In the figure, AC is 10 m longer than CB. Determine the length of CD.

J y2 _ 3x9. Sketch and solve the following system:

10. Given: BF is a tangent at DL EDB = 58°

AR = 110°

CD = 80°

Find the measure of allindicated angles (L 1... L 6).

2x-y=3

11. The total surface area of the rectangular solid

shown is 36 m2. Find the value of x.

x

x+2

Continued

110


G-4

Exercise 49 : Converse , Contra positives , If...Then...

12. Given A ABC with vertices at A(5, 4), B(-3, 6), C(1, -4), find the

a. slope of ABb. midpoint of BCc. length of the median from Cd. length of AC

13. Solve for x: x + 1 - x + 3 < 0.x+2 x+4

14. Graph the region represented by these inequalities:y> x +1

x2+y2 <9

111


G-5

Exercise 50 : Direct and Indirect Reasoning

1. Why are indirect proofs referred to as "the process of elimination"?

2. In a murder investigation, there are only three suspects: Al, Ben, and Tom. BothAl and Tom have alibis. What can be concluded? What type of proof (indirect or

direct) was used?

3. In the diagram, L 1 and L 2 are vertically opposite angles, and L 2 and L 3 arebase angles of an isosceles triangle. What can be concluded about the sizes of Z 1and Z 3? What type of proof (indirect or direct) did you use?

4. Your best friend says she'll meet you either at the library or the laboratory. Yougo to the library and she is not there. What do you then know? What type ofproof (indirect or direct) did you use?

5. Write each of the following in "If...then..." form.

a. All multiples of 6 are multiples of 3.b. All people born in 1810 are now dead.

c. When it is sunny, my family always goes on a picnic.

d. Vertically opposite angles are congruent.

e. Base angles of an isosceles triangle are congruent.

f. All even numbers larger than 2 are the sum of two primes.

6. Write the converse of each of the statements in Question 5. Which are true?

7. Write the contrapositive of each of the statements in Question 5. Which are true?

8. The length of a rectangular floor is 4 m less than three times its width. Thewidth of a rectangular area rug on the floor is 2 m less than the floor's width.The length of the rug is 2 m greater than twice its own width. Find the area ofthe floor if 44 m2 of the floor are not covered by the rug.

Continued

112


G-5

Exercise 50: Direct and Indirect Reasoning

9. Solve: 1 +x=3.x

10. Solve: jj`2x2 = 'J-15x - 25.

11. In the adjacent squares shown, the vertices A, B, and C lie in a straight line.Find the value of x.

C_-

4

12. Find the roots of 2x2 + 5x - 3 = 0.

13. Prove the following statement:

7 x

If the measure of the angle determined by two tangent segments to a circle froma point in the exterior is 60°, then the tangent segments form an equilateraltriangle with the chord joining the points of tangency.

14. Without solving the equation, determine the nature of the roots:

a. 3x2--7x+5=0 b. 2x2-13x+15=0

15. A regular hexagon is inscribed in a circle ofradius 6 cm. What is the area between the circleand the regular hexagon?

16. On the following pages, you are given a bank statement, a chequebook record,and a statement of reconciliation. Complete the statement of reconciliation.

Continued

113


G-5


ACLU CREDITBALANCE FORWARD

DATE

01 11 127 18

DESCRIPTION DEBITS CREDITS DAY Mo. BALANCE

Deposit 520 15 01 11 647 33Cheque 346 425 00 03 11 222 33Cheque 347 57 66 08 11 164 67Deposit 80 89 10 11 245 56Cheque 348 42 38 13 11 203 18Cheque 350 103 56 14 11 99 62Deposit 420 15 15 11 519 77Cheque 349 144 34 19 11 375 43Cheque 351 125 00 23 11 250 43Cheque 353 36 15 28 11 214 28Service Charge 14 75 28 11 199 53

CHEQUE CHEQUES ISSUED TO OR CHEQUE DEPOSIT DEDUCT/ADD BALANCE FWD

DATE No. DESCRIPTION OF DEPOSIT AMOUNT AMOUNT CHEQUESIDEPS 127 18 1

Nov. itD520 15 CHQ - IDEP + 520 15

1 epos AI ANCE 647 33

1 346 R t425 00 CHQ - IDEP + 425 00

enALA.NCE 222 33

6 347 F dS57 66 CHQ - /DEP + 57 66

uper oo sBALANCE 164 67

10 348 Utiliti 42 38 CHQ - IDEP + 42 38es

BALANCE 122 2

10 itD80 89 CHQ - IDEP + 80 89epos

BALANCE 203 18

12 itD420 15 CxQ - IDEP + 420 15

eposALANCE 623 3

12 349 iC R144 34 CHQ - /DEP + 144 34ar repa

BALANCE 478 99

15 350 Thi ' t StD103 56 CHQ - /DEP + 103 56

essen . ores epBALANCE 375 43

351 i d u125 00 CHQ - IDEP + 125 00pr ppe

AI.ANCE 250 43

20 352 tin GoodS17 86 CHQ - /DEP + 17 86

por g sBALANCE 232 57

25 353 W l t36 15 CHQ - IDEP + 36 15

a marBALANCE 196 42

30 354 Groc ri54 76 CxQ - /DEP + 54 76e es

BALANCE 141 66

30 De osit45 00 CHQ - IDEP+ 45 00

pBALANCE 186 66

28 Service Char e1

14 75

=

CxQ - /DEP + 14 75gBALANCE 171 191

Continued

114


G-5


STATEMENT OF RECONCILIATION

Bank Reconciliation

Balance from statement:

Add:

Total additions:

Subtotal:

Subtract:

Total additions:

Subtotal:

This should agree with the balance shown in your record book:

17. Harriet the fly is having a busy day, bothering math students all day. Shedecides to take a rest and lands on the top of the minute hand of the wall clockat exactly 3 o'clock.

a. Sketch a graph of Harriet's height in relation to the centre of the clock vs.time for 1 hour. The minute hand is 12 cm long.

b. Which trigonometric function best represents this curve?

115

Senior 3 Pre-Calculus Mathematics . Cumulative Exercises

H-1

Exercise 51: Operations and Compositions Functions

Given the functions f and g such that f= {(1, 8), (2, 9), (3, 9)} and

g = 1(8, 12), (9, 14)}, fill in the blanks.

a. f(1) = b . f(2) = c. f3) -

d. g(8) = e. g(9) = f. g(f(1)) _

g• g(i2 )) - h. g(f(3)) = i. f(1) + g(9) _

2. Suppose the functions f and g are defined as follows: Ax) = 2x + 1 and g(x) = 3x2 -find each of the following:

a. g(f(x)) b. f(g(x)) c. f (f (x)) d. f(3) - g(-1) e. g(O) f NO

3. Given functions fand g such that ft) = x - 1 and g(x) = 2x2, determine

a. f(g(3 )) b. g(f (3 )) c. f(3 + g(3))

4. Given functions f and g such that f (x) = x2 + 1 and g(x) = 2x - 3,

a. define the function composed of g with f.

b. define the function composed of f with g.

5. Given functions fand g such that f (x) = and g(x) = x -1, determine

a. f (g(x)) b. g(&)) c g(5)f(9)

6. Solve: x + x--2 = 4.

7. Three mutually exclusive circles have radii4, 5, and 6, respectively. (See diagram.)

a. Find the angles of the triangle whosevertices are the circles' centres.

b. Find the area of the white regionbetween the circles.

Continued

116


H-1

Exercise 51 : Operations and Compositions Functions

8. Solve the following algebraically.x2 + 2y2 = 18

xy = 4

9. Find the distance from the line 2x + 5y = 2 to the point (3, -1).

10. For what value of k will the sum of the roots of the following equation be 8?

x2-(k2-2k)x+3=0

11. The sum of the ages of Flavio and Inga is 36 years. The difference between threetimes Flavio's age and twice Inga's age is 28 years. How old is each person?

12. A quadrilateral PQRS has vertices at P(5, -6), Q(3, 0), R(-1, 2), and S(-5, -4).Verify that the midpoints of each of the sides of this quadrilateral form thevertices of a parallelogram.

13. Verify that 1 + NI 1- c is a root of x 2 - 2 x + c = 0.

14. Bill's parents said, "You may borrow the car if you clean your room or mow thelawn." Bill mows the lawn. May he borrow the car?

15. Given that the zeroes of a function are 1, 3, and -5, find the polynomial function.

16. A manufacturer sells clear plastic tapeon a spool with radius 1 cm. The tape is0.02 cm thick and 1.5 cm wide. Thecombined radius of the spool and thetape is 3 cm. Approximate the length ofthe tape on the spool in metres.


a. yI y >- -3 x#5, xe R} c. {yl5 >_y>

117


H-2

Exercise 52: Inverse Functions

1. For each of the following functions, specify the inverse function.

a. Multiplying by 5 b. {(4, 5), (6, 6), (7, 8)}

{(x, y) y = 3x + 2} d. {(x, y) f y = 4 - x}

2. For each of the following functions, f, define its inverse, f ;.

a. f (x) =x

3b. f(x)=x2 +1 andx?0 c. f(x)=

3. Given Ax) = 3x + 7, determine

a. f'(1) b. f-1(8) c. f -1(3a + 7)

3x-2

4. a. Sketch the graph of a quadratic function or define g(x) with vertex at (1, 2)and a = 2.

b. Sketch g"1(x).

c. Why is g- (x) not a function? Explain with reference to one-to-onecorrespondence.

5. Explain why f (x) = 2x + 1 and g(x) = X 1 are inverses of each other.

6. Given :L ABC = L FDEBC = DEAC//EF

Verify: a. A ABC = A FDE

b. AB//DF

Continued

118


H-2


7. Tony Hill earns $335.75 net per week. His wife, Natalie, earns $337.75 net perweek. The family receives a child tax benefit of $36.75 per month. The family'sexpenses are as follows:

Expenses for the family include:

a. monthly mortgage payment .......................... .. $715.40b, monthly car payment .................................... 206.10c. average monthly telephone bill ............................. 23.00d. other monthly utilities ................................... 305.20e. yearly car insurance premium ............................. 610.00f. home is assessed for property tax purposes

at $40 000; the mill rate is 60 mills ............................g. home insurance (yearly premium) .......................... 249.40h. monthly boat payment ................................... 130.00i. student loan repayment per month ......................... 100.00j. food (average per month) ................................. 560.00k, clothing expenses for the year ............................. 830.001. average car maintenance per month ......................... 35.00m. gasoline per month ...................................... 120.00n. entertainment per year .................................. 2600.00o. yearly vacation ........................................ 2000.00p. newspapers and periodicals (per year) ....................... 250.00q. average monthly credit card payment ....................... 200.00r. holiday gift purchases per year ............................ 500.00s. baby-sitting (average per year) ............................. 400.00

a. Prepare an estimated monthly budget for the Hill family using the blankbudget form on the following page.

b. As the financial planner for the Hill family, you notice that the Hills have nolife insurance. Upon questioning Mr. Hill about this, he states, "I am not tooworried about this yet; I am still a young healthy man." Explain to Mr. Hillwhy this logic is faulty.

c. The Hills are a little concerned about their present financial position. Mr. Hillsuggests they reduce their reserve fund payment to balance their budget.Suggest to the Hill family other areas in which you feel the family couldreduce spending and still balance their budget.

Continued

119


H-2


Incomea. Regular Monthly Income $

b. Spouse's Regular Monthly Income $

c. Additional Income

d. Other IncomeTotal Monthly Income #1

2. Housing Expensesa. Mortgage or Rentb. Property Taxc. Home/Property Insurance $d. Repairs/Maintenance

e. Other Housing ExpensesTotal Housing Expenses #2 $

3. Utilitiesa. Hydrob. Gasc. Phone

d. Watere. OtherTotal Utilities #31

4. Transportationa. Public Transportb. Car Loanc. Car Fueld. Car Maintenancee. Car Insurancef. Other TransportationTotal Transportation

5. Personal Financesa. Personal Loanb. Investmentsc. RRSPd. Life Insurancee. Charitiesf. Credit Card Payments $g. Service Chargesh. Savings "i. Other Personal FinancesTotal Personal Finances #5 $

6. Personal Expensesa. Groceriesb. Clothing $c. Entertainmentd. Giftse, Vacationsf. Other Personal ExpensesTotal Personal Expenses #6 .

7. Other Expensesa. $b. $

C.Total Other Expenses #7 $


Income minus Expenses (#1 - #8) #9 $#4

Comments:

Note 1: Financial analysts advise that RRSP contributions should start early.

Note 2 : Financial analysts advise that a reserve fund of two or three months of income should besaved for emergencies . Generally, it could take several years to build up a reserve fund.Reserve Fund Calculation : Calculate two months of income and divide by the number of months it willtake you to achieve it.

Continued

120


H-2

Exercise 52 : Inverse Functions

8. Compute the discriminant and tell whether the equation has none, one, or twosolutions.

a. x2-7x+12=0 b. 3x2=5x-3

9. Solve this system of equations: 2x+y = -6Ix-5y = 8

10. On a number line, indicate the region corresponding to each of the following:

a. (x<2)or (x<5) b. (x < 2 ) and (x < 5)

c. (x<2)and (x>5) d. (x:2) or (x>5)

e. (x<5)andnot (x>2) f. (x<4andx<-1)and(x>-5)

11. Solve: x+4+ x-1=5.

12. Two identical boxes are filled with equal numbers of marbles. The marbles arecoloured green or yellow. The ratio of green to yellow marbles is 7:2 in Box 1 and8:1 in Box 2. If there are 90 yellow marbles in total, how many green marblesare in Box 2?

13. Assuming that the half-life of a radioactive substance is 1690 years, whatfraction of an initial amount of the substance will remain after

a. 3380 years? b. 5070 years?

121


H-3

Exercise 53: Factor Theorem and Remainder Theorem

Given the polynomial fix) = x3 + 2x2 - 5x - 6, use the factor theorem to determinewhether

a. (x + 1) is a factor of f(x)

b. (x - 3) is a factor of &)

2. Verify whether or not (x + 1) is a factor of g(x) = x4 - 9x3 + 18x2 -- 3.

3. Factor Ax) =x3- 2x2+3x-6.

4. Divide x4+ 6x' - 9x + 2 by x - 1.

5. Find the remainder for each of the following divisions:

a. (a3+3a2 - 9a - 12) +(a + 4)

b. (4m3 + 7m2 - 3m - 20) -:- (4rn - 5)

6. Find each remainder:

a. (x3 + 5x2 - 7x + 1) -1- (x + 2)(x --1)

b. (2x3+x2 -4x-2)_(x2 +4x+3)

7. Find the inverse function of f(x) = 2x + 5.

8. Two ships are meeting at a landmark. The path of the first ship is 2x + 3y = 48and the path of the second ship is 3x +2 y = 42. Where do the ships meet?

9. At a ski resort, a hill slants 20° from the horizontal. The chair lift running up thehill is supported by a 50-m vertical pole. A support cable runs from the top of thepole to an anchor located 88 m directly downhill from the base of the pole. Howlong is the cable?

10. A Canadian dollar is worth 720 U.S. A set of golf clubs in North Dakota ismarked at $420.00 US. What is its cost in Canadian funds?

Continued

i

122


H-3


11. A person can row a boat 9 km downstream in 2 hours. Rowing back upstream, ittakes 3 hours to return to the starting point. Find the speed with which the boatrowed through the water and the speed of the current, assuming that both ofthese are constant.

current current

12. Solve the following equation: 2x2 + 5x + 1 = 0.

13. Solve: 2x--1 = x2.

14. Determine the inverse of the function defined by 4x - 2y = 8. Sketch both thefunction and its inverse on the same coordinate system. What do you notice?

15. Solve and check:x2

-9 ? 0.x2 -x-2

16. If P is the centre of a circle with radius 10 cm, and chord AB is 6 cm from thecentre, how long is chord AB?

17. How many zeroes are in the product of the first 500 natural numbers?

123


H-4

Exercise 54: Graphs of Polynomial and Rational Functions

1. Which of the following graphs could be graphs of polynomial functions and whichcould be graphs of rational functions?

a. 1 b.20

d.

2. a. Find the x-and y-intercepts of the function fix) = x(x - 1)(x + 1).

b. Sketch the graph of f x).

3. What is the domain and the range of the function f(x) = (x + 4)(x2+x --- 2)? Sketch

the graph.

4. Compare the graphs of y = 1 and y = 1 .x+2 (x+2)2

5. a. Graph y = x2 - 1. What are the zeroes of this function?

b. Sketch the graph of y =x21 What do you notice about the zeroes of

y = x2 - 1 and the asymptotes of y = 2?

X

Continued

124


H-4

Exercise 54 : Graphs of Polynomial and Rational Functions

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

b. For fl x) = 2x3 - 3x2 - 3x + 2, find the x-intercepts.

c. Sketch the graph of fix).

7. Find all solutions for each of the following trigonometric equations on theinterval 0°<_ 0 S 360°. (Round to one decimal place.)

a. 2 sin 0 = --v'2 b . cos 0 + 1 c. Cos 8 =cost 0 -1

8. Solve the system algebraically:

9. Solve: x2 -- 4x - 2 = 0.

10. George has a lottery ticket, number 7. He wins if his number is less than 10 andless than 5. Does he win?

11. Write equations for lines that are at a distance of 3 units from the linex - Sy+ 10=0.

12. Two friends were comparing the different pay scales paid by the two companiesfor which they work. Each company pays at a rate of time and a half forovertime.

Company A: Paid employees overtime after 40 hours in a week.

Company B: Paid employees overtime after 8 hours in a day.

Suppose they worked the following hours during the week. Compare the total paybetween Company A and Company B if the employees earned $16.00 per hour.

Monday Tuesay Wednesday Thursday Friday

11 7 11 12 11

Continued

125


H-4


13. Given : P is the centre of the circleCE is a diameterL1=35°

fAB = 100°

Find the measure of all numbered angles (L 2 ... L 5).

14. Calculate the roots of the quadratic equation to the nearest tenth.

2-x = 3(4-x)

x 2+x

15. The numbers 64 and 729 both have an unusual property. Each of these numbersis both a perfect square and a perfect cube.

a. Find two other numbers that have this property.

b. How might you generate numbers that have this property?

iP

I

126



What is the remainder when you divide the polynomial (x3 - 3x2+ 6x + 5)

by(x-2)?

2. Use the remainder theorem to find the remainder when x5 - 4x3 + 2x + 3 isdivided by

a. x-1 b. x+2

3. Find the remainder when (4x3 - 6x + 5) is divided by (2x - 1).

4. Factor the expression 2x3 + 3x2 - 32x + 15.

5. Find the values of a and b if the remainder is 2x + 3 when x5 + 4x' + ax + b

is divided by x2 - 1.

6. The polynomial P(x) = 4x3 + bx2 + ex + 11 has a remainder of -7 when divided by(x + 2), and a remainder of 14 when divided by (x -1). Find the values of b and c.

7. Given: 0 is the centre of the circleQR = RPLBST=60°ST is tangent at TAP is tangent at PAB is tangent at Q

Find the measure of all numberedangles.

8. Solve A ABC if you are given that L A = 36°, a = 9.4, and b = 13.1.

127

Senior 3 Pre-Calcul us Mathematics



1. Find the remainders when

a. x3+3x2-4x +2is dividedbyx-1

b. x3-2x2+ 5x+ 8 is dividedbyx-2

c. x5 + x - 9 is divided by x + 1

d. x3+3x2+3x+1 is divided by x + 2

e. 4x3 - 5x + 4 is divided by 2x - 1

f. 4x3+6x2+3x+2 is divided by 2x+3

2. Find the values of a in the expressions below when the following conditions aresatisfied.

a. x3 + axe + 3x - 5 has remainder -- 3 when divided by x - 2

b. x3 + x2 + ax + 8 is divisible by x - 1

c. x3 + x2 - tax + a2 has remainder 8 when divided by x - 2

d. x4-3x2+2x+ a is divisible by x + 1

e. x3 - 3x2 + ax + 5 has remainder 17 when divided by x - 3

f. x5 + 4x4 - 6x2 + ax + 2 has remainder 6 when divided by x + 2

3. Show that 2x3 + x2 - 13x + 6 is divisible by x - 2, and, hence, find the other

factors of the expression.

4. Show that 12x3 + 16x2 - 5x - 3 is divisible by 2x - 1, and find the factors of theexpression.

5. Factor:

a. x3-2x2-5x+6 b. x3.4x2+x+6 c. 2x3+x2-Sx-4

d. 2x3+5x2+x-2 e. 2x3+11x2+17x+6 f. 2x3--x2+2x-1

Continued

128



6. Find the values of a and b if ax4 + bx3 - 8x2+ 6 has remainder 2x + 1 when

divided by x2 - 1.

7. The expression px4 + qx3 + 3x2 - 2x + 3 has remainder x + 1 when divided by

x2 - 3x + 2. Find the values of p and q .

8. The expression axe + bx + c is divisible by x - 1, has remainder 2 when divided

by x + 1, and has remainder 8 when divided by x - 2. Find the values of a, b,

and c.

9. Both x - 1 and x + 1 are factors of the expression x3 + ax2 + bx + c, and theexpression leaves a remainder of 12 when divided by x - 2. Find the values of a,b, and c.

10. Susan must wash the dishes and polish her shoes if she wants to go out. Shewashes the dishes. Can she go out?

129

Senior 3 Pre-Caicuius Mathematics


Let j(x) = x2 and k(x) = x3. Does j(k(x)) = k(j(x)) for all x?


2. Given Ax) = 2x - 6 and g(x) = 2 x + 3, determine each of the following:

a. g(f(7)) b. g(f(-3)) c. f (g(8)) d. f(g( -43 ))

e. g(f(1000)) f. f (g(428)) g. g(f (a)) h. f(g(a))

3. Let s(x) = xI + 1 and t(x) = x - 3. Does t(s(x)) = s(t(x)) for all x?

4. Suppose that s(x) = 2 - x and t(x) = -x - 2.

a. Define the function composed of t with s.

b. Define the function composed of s with t.

c. Does s(t(x)) = t(s(x)) for all x?

5. Given f (x) = x - 2 and g(x) = 2x, determine each of the following if it exists:

a. g(f(6))

e. A g(9))

b. g(f(5))

f. f(g(5))

c. g(f(1))

g. f(g(-1))

d. g(f(-2))

h. f(g(-3))

6. Given that Ax) = 3x + 4 and g(x) = x2 - 1, determine each of the following:

a. g(f(2 )) b. /(g(2)) c . g(t1)) d . g(f(---2))

e. g(f Ca )) £ fg(a)) g . Ma)) h . g(g(a))

7. Susan has a lottery ticket, number 8. She wins if her number is less than 10 orless than 5. Does she win?

8. A language teacher has a box containing 20 books. Some of the books are new.Five of the books are in English. Ten of the books have red covers . Three of theEnglish books have red covers . Two of the English books are new. Four of thebooks with red covers are new . One of the new English books has a red cover.There are three books which are not new, are not in English , and do not have redcovers . How many new books are there?

130


Exercise 58: Cumulative Review

For each of the following questions (1, 2, and 3), sketch its graph and state thefollowing:

a. axis of symmetry b. vertexc. whether max or min d. max/min valuee. y-intercept f. x-intercept (roots, zeroes)g.i.

domaindirection of opening

h. range

1. ftx)=3x2+4 2. y= -2(x--2)2-5 3. y=2x2-4x-7

4. Determine the type of graph and sketch the graph of the following equations:

a. 3x+2y=4 b. f1x)=-(x-2)2+3

c. y=x2+5x+6 d. (x - 2)4 + (y + 1)2 = 12

5. The graph of the quadratic function is fix) = (x + 2)2 - 3 is moved one unit to theright and four units down. State the equation of the resulting graph.

6. For what value ofp is the equation y = x2 + 7x + p a perfect square?

7. Computer programs are sold for $20.00 each if 300 people buy them. For every$5.00 increase in price, 30 fewer people buy them. Using algebra, find thenumber of programs sold for a maximum profit. Also find the price of theprogram and the maximum revenue.

8. a. In what quadrants is sin 0 positive? Negative?

b. In what quadrants is cos 6 positive? Negative?

c. In what quadrants is tan 0 positive? Negative?

9. Find the following values:

a. cos 42° b. sin 45° c. tan 100°

`2d i 3, s n e, cos f. tan 6.53 2

Continued

131


Exercise 58 : Cumulative Review

10. a. If sin 0 = 0.63777, find 8.

b. If cos 8 = 0.01991, find 8.

c. If sin 0 = 2 , find 0.

11. a. Solve the following equations for 0°:!^ 0 < 180°.

i. 2sin6-1=0 ii. Cos' 6-1=0

b. Solve the following equations for 0° <_ 0< 360°.

i. 2 tang 8 - tang - 1 = 0 ii. 2 cost 0 + cos 0=0

12, a. In A ABC, L B = 150°, a = 100, and c = 300. Find side b.

b. In A ABC, a = 30, b = 20, and c = 40. Find the smallest angle.

13. Two planes leave an airport at the same time. One flies due east at 600 km/h,the other flies northwest at 400 km/h. How far apart are they after 2 hours?

14. a. In A ABC, a = 16, L A = 35°, and L B = 65°. Find L C and side b.

b. InAABC, a=2,c=3 .2,and LC=125°. Find LBandLA.

15. A 6-m loading ramp whose angle with the horizontal is 25° is to be replaced witha newer, longer ramp whose angle of inclination is 10°. How long is the newer,longer ramp?

16. Explain how you would recognize the ambiguous case when solving a triangle.

17. In A ABC, b = 16, c = 25 and L B = 30°. Find all the possible measures of L C,L A, and side a.

18. InAABC , a=7,c=6,andLC=31. 8°. Find side b ,LA,andLB.

Continued

132



19. The perimeter of the isoscelestriangle ABC is 54 cm, and AC = BC.If AD = 5 cm, and D, E, and F arepoints of tangency, find length BC.

20. a. Find the sum of the measures of the interior angles for the following figure.

b. Find the sum of the interior angles of a 70-sided polygon.

c. If the interior angles of a polygon add to 7020°, how many sides does it have?

21. Find the area of this circle.4 is the centre.AB=3OC=3'2

Continued

133




22. If L 0 = 1500 and 0 is the centre, 23. If 0 is the centre and B is a point of

find the measure of Z B. tangency, find L 1, L 2, and are BEF.

24. A circle has a centre at 0, FG is a tangent,AB//CD, are AC = 20°, L DCF = 60°,arc EF = 30°, and are AB = 70°. Find themeasure of

a. L BAD h. arc CADb. L EOF i. are EDFc. L DCE j. are CEd. L OFG k. arc CFDe. L DFG L arc EFDf. L CDE m. are FCFg. are BD

25. For the diagram to the right, find themeasure of

a. L OCB e. are BDb. L BDC f. are BCc.d.

L BADL DBO

g. are BCD

Continued

134



26. State the number of *s in each of the following:


a. A and Bb. Aor Bc. Ad. Only Be. Not in Af. Only A

27. Thirty-five students were surveyed. Of those, 19 indicated they are takingchemistry, 8 are taking chemistry and biology, while 7 are taking biology andphysics. Nine are taking chemistry and physics. Five students are takingchemistry, biology, and physics. Twenty-nine students are taking chemistry orbiology. If 28 students are taking biology or physics, find the number of studentstaking only physics. (Include a complete Venn diagram as part of your solution.)

28. Fill in the following blanks using either the word inductive or the worddeductive.

a. Using reasoning, we take an accepted general rule andapply it to a specific case or instance.

b. In reasoning, we use specific cases or instances toformulate a general rule.

29. Statement: If a triangle is equilateral, then it i s also isosceles.

a. Is the above statement true or false?

b. State the converse of the statement and indicate whether it is true or false.

c. State the contrapositive of the original statement and indicate whether it istrue or false.

30. Statement: Every relation is a function.

Use a counterexample to show that the above statement is false.

Continued

135



31. Given: AB # AC and L I = L 3.

Using an indirect proof, show thatL2#L4.

32. Calculate the (shortest) distance from a point located at (2, 5) to the linedescribed by 3x -- y = 4.

33. A wallet contains a total of 20 coins, consisting of only nickels and quarters. Thetotal value of the coins is $2.40. How many nickels and how many quarters arein the wallet?

34. Solve the following system of equations algebraically.

y=x2-1

x+2y-4= 0

35. Solve the following system of equations for x, y, and z:

x + 7y - 2z = -1- 4x 3y+z=83x--5y+6z=7

36. Sketch the following inequalities and determine the solution of the systemgraphically.

y <-(x-2)2+ 1

2x - 3y <6

37. Solve for x:

a. Ix-31<1

38. Solve for x:

a. x2-2x-3>0

c. x2-3x-10>0

b. 3x+2 ?8

b. x2+3x-4<0

d. x2-x -12<0

Continued

136



39. If f(x) = x2 - 3x, find each of the following:

a. A-3) b. f(5) c. f(0)

d. f(1/2) e. A--112) f f(2x)

g. f(x - 3) h. f(3 - x) i. )T 1/X)

40. Given the functions ft) = 2x - 3 and g(x) = 3x + 2, find

a. f(1)+g(1) b. g(2) - f(2) c. g(-2)f (-2)

d. f(g(x)) e. g(l0))

41. Given h(x) = 3x + 7, find (inverse)

a. h-1(x)

42. Sketch the following:

a. f(x) = x(x - 1)(x + 3)

C. f (X) = (x - 1)2(x +4)2

e. f (x) =

f. f(f(x))

b. h-1(2)

b. / Tx) = x(x + 2)2

d. f (x) = 2x+4X - 1

3f.

f(x) _ 2+1 2

1g . f(x)=

2-4x-5x

i. f(x)=x3+4x2+x-6

k. f(x)=x3+5x2+2x-8

X -4

h. f(x)=x'-4x

j. f(x)=x3-7x-6

43. Find the remainder when x + 1 is divided into x3 + 4x2 - 5x + 1.

44. Find the other factors for f(x) = x3 + x2 - 17x + 15 if one of the factors is x - 3.

Continued

137




45. Find the value of k such that x + 4 is a factor of ftx) = x' + 5x2 + kx - 8.

46. Solve f o r x: j + 7 = 10.

47. Solve for x: Vix _-1 = 3x + 2.

48. Solve for x:x+4 = V7x + 1

2 4

49. Solve for x: x + 2 = 2x +7.

50. Solve for x: 2x + 3 - x + 1

51. Solve for m: 2m + 3

52. Solve for t: I 3t - 4 1 = 2.

53. Solve for x: I 2x + 1 1

4 x54. Solve for x:

55. Solve for x: 5 + 2 = 6x+3 x x+1

56. Solve for x:2x + 1 = 3x + 9 _ 0.

x-3 2x+3 2x2-3x-9

57. Solve for x:+x x2 + 2x x+2 '

58. Find the inverse of the function f (x) =X

3x+1

Continued

138



59. Find the value of x.

60. A beacon from the top of a 18.6-m lighthouse illuminates a boat in the water. Ifthe beam of light makes a 19.7° angle with the boat, how far is the boat from thelighthouse?

1 4(-2)+5(-l)-6 I61. Given:16-+25

a. What does this formula represent?

b. State the equation of the line.

62. Find y of the point (4, y) that is true for the function, fix) = x2 - 8x.

63. Complete the square: y = -2x2 + 8x - 5.

64. For the equation y = -2x2 + 8x - 5, state the value of the discriminant and thenature of the roots. State the sum and the product of the roots.

65. A diameter of a circle has the endpoints (2, 4) and (-6, 2).

a. Find the length of the diameter.

b. Find the coordinates of the centre.

c. Find the slope of the diameter.

d. Find the equation of this circle.

e. Find the circumference of this circle.

66. For the equation y = 2x2 + 3x - 2, one of the factors is x + 2. Find the other factor.

Continued

139




67. Find the measures of L 1, L 2, and the sum of L 3 and L 4, given thatL1=5x+4and/2=9x+8.

68_ Find the equation of the following graphs:

c.

69. Fredrick earns $10.25 per hour with time and a half for time worked over 40hours. He worked the following hours: Tues., 8.5; Wed., 9.75; Thurs., 8; Fri., 0;Sat., 0; Sun., 10; and Mon., 12. He pays 25% of his gross salary toward incometax. He also has the following deductions: CPP, $8.35; UIC, $9.20; Blue Cross,$11.22; and Union Dues, $5.70. Calculate his gross pay and his net pay.

70. Find the area of the shaded region in the figure below if the area of the square is

20 cm2.

Continued

140

Cumulative Exercises Senior 3 pre-Calculus Mathematics


71. a. If you are given four vertices of a quadrilateral, how would you prove that itis a rhombus?

b. If you are given four vertices of a quadrilateral, how would you prove that itis a rectangle?

c. If you are given four vertices of a quadrilateral, how would you prove that itis a parallelogram?

72. Factor completely: y = x3 - 2x2 - 5x + 6.

73. Given the equation y = axe + bx + c, b = 0 and the points (2, -3) and (-1, 3) that

go through its graph, find the values of a, b, and c, and state in an equationform.

74. Find the simple interest earned if $5000 is invested at 10.5% per year for

a. 6 months b. 18 months c. 14 days d. 1 year

75. Complete the following chart for a the first five payments on a loan of $5000 at8% per annum, and payments are $300 per month:

MonthlyPayment Principal

PaymentMade

Interest8% per year

AmountOwing

$5000.00

1 $5000.00 $300.00

2 $300.00

3 $300.00

4 $300.00

5 $300.00

76. Graph the following functions:

a. y=cosx b. y=-sinx+4 c. y= cos(x- 45°) d, y=-3sinx

141

SENIOR 3PRE-CALCULUS MATHEMATICS

ANSWERS TO CUMULATIVE EXERCISES

A Supplement toA Foundation for Implementation

1999

Manitoba Education and Training

Manitoba Education and Training Cataloguing in Publication Data

510 Senior 3: pre-calculus mathematics. Answers tocumulative exercises : a supplement to afoundation for implementation

(Renewing education : new directions)

ISBN 0-7711-2222-5

1. Mathematics-Problems, exercises, etc.2. Calculus-Problems, exercises, etc. 3.Mathematics-Study and teaching (Secondary).4. Calculus-Study and teaching (Secondary).1. Manitoba. Dept, of Education and Training.II. Series

Copyright © 1999, the Crown in Right of Manitoba as represented by the Minister ofEducation and Training. Manitoba Education and `braining, School ProgramsDivision, 1970 Ness Avenue, Winnipeg, Manitoba, R3J OY9.

Every effort has been made to acknowledge original sources and to comply withcopyright law. If cases are identified where this has not been done, please informManitoba Education and Training. Errors or omissions will be corrected in a futureedition. Sincere thanks to the authors and publishers who allowed their originalmaterial to be adapted or reproduced.

Answers to Cumulative Exercises Senior 3 Pre-Calculus Mathematics

A-1, A-2

Exercise 1: Quadratic Functions

1. a. y = x2 b. y = x2 + 1

-5 5

3. +1

b. [-4, oo) c. (4, -4) d. x=4 e, 2, 6

g. Does not have a max. h. -4

b. Similarities: all are parabolas, open up,same basic shape, same domainDifferences: vertices, x- and y-intercepts,range

c. i. (0, 0)

ii. (0, 3)

iii. (0, -2)

d. y x2-4

6. a. -12x5 b. 16xi2 c. 7x2

7. a. 28 b. -5 C.24

7

d.--4c3d2e

e.1

f. 13 49 9

8. a. x(x + 5) b. (x + 4)(x + 1) c. (3x + 4)(2x - 5)

119. X= 4

- 3y10. 57 cm2 11. 62.8 m 12.5 20

13. a. D: (--cx, oo), R: [1, oo) b. D: (-, 3], R: (_oo, o)

c. D: [-1, 51, R: [0, 3] d. D: (-3, -1] u [1, 3), R: [-1] v [1]

14. A, G, F

I

Senior 3 Pre-Calculus Mathematics Answers to Cumulative Exercises

A-1, A-2


Note: For each of the questions 1, 2, and 3, the graphs should be on the samegrid.

b. The parabolas get narrower.

c. All three are (0, 0).

FIr-i-++ +i i i5

2. a.b.

C.

b. Vertices: i. (0, 0); ii. (-3, 0); iii. (2, 0); iv. (-1, 0)

c. y = k(x - 8)2 where k is any value except 0

4. ^ 10f

Continued

2


A-1, A-2

Exercise 2 : Graphs of Quadratic Functions 1

5. a. 2(x - 2)(x + 2) b. 6(5x + 4)(2x - 3)

46. a. sin0 = 4, cos0 = 3, tanO =

12 5 12b. sin 0 = 13 , cos e = 13 , tan 0 = 5

c. sin 0 = 2,[2-9 5V-

29 tan 0 229 , cos0 =-29- 1

= 5

7. y=8-x

2 ory=x

8. 6.0 m

9. 84%

10. It is dependent upon how far each man can jump.

11. 250

12. h = 430.8

13. 11x + 6y - 66 = 0 or 11x + 6y = 66

14. x = 1217

15.2

3

16. a. -12

b.-1-5 -4

7

b. (2, 4) c. 6 -J5

3


A-2, A-3


5 5

2. a. i.

b. i. Vertex: (-2, 3)ii. Vertex: (-4, -5)iii. Vertex: (5, 1)

ii.

b. The negative sign inverts theparabola (reflects it in the x-axis).

c, i. Axis of Symmetry: x = -2ii. Axis of Symmetry: x = -4iii. Axis of Symmetry: x = 5

3. a. Opens up; Vertex: (-1, 0); Axis of Symmetry: x = -1, narrower

b. Opens down; Vertex: (1, 6); Axis of Symmetry: x = 1, wider

c. Opens up; Vertex: (-6, -10); Axis of Symmetry: x = -6, narrower

b. Opens up; Vertex: (1, 8); Axis of Symmetry: x = 1, narrower

4. a. Vertex: (-1, 0);Axis of Symmetry: x = -

b. Vertex: (1, 6);Axis of Symmetry: x = 1

Continued

4


A-2, A-3


c. Vertex: (-6, -10);Axis of Symmetry: x = -6

5. a. (x+2)(x-3) b. (x-5)(x-3)

d. Vertex: (1, 8);Axis of Symmetry: x

c. 7(2x - 3)(x + 5)

6. a. 5.0 b. 13.2 c. 9.4 d. 23.9 e. 6.1 f. 3.5

7. 2-Y3

8. 8.4 m

9. 3

10. Perimeter = 26 units; Area = 36 units2

11. 1

12. G A = 102.2°, a = 13.4 units

13. 24 x22y 11

14. 36

15. a. (2, 51 b. (--, -2) u (7, °°)

d. [0, 61 e. (-6, -31 u [2, 8)

c. (--°°, -31 u [10,


A-3


b. i.

11.

ii.

Vertex: (1, -4); Vertex: (-3, -4);Axis of Sym: x = 1; Axis of Sym: x = -3;Domain: (x x e 9tt); Domain: (x I X E 9t};Range: (y Iy r=- 9t, y ? ---4}; Range: (y I Y E 91, y > -4);x-intercepts: 3, -1 x--intercepts: -5, -1

in.

Vertex: (-3, -1);Axis of Sym: x = -3;Domain: (x I x e 9t );Range : (y I y e 91, y ? -1);x-intercepts: -4, -2

2. a. Vertex: (3, 5); opens up b. Vertex: (-4, -7); opens down

c. Vertex: (-1, 2); opens up d. Vertex: (2, 1); opens down

3. a. 2(x -- 2)(x - 8) b . 4x(a - 2b)

4. a. 8 = 36 .9° b. 8 = 18.2° c. 6 = 62.9°

5. a. 190.8 km b. q = 1.9°

6. a. -2(I +) b. 2x-11y

7. 682.1 m

8. 16.5 and 17.5

d. 0 = 56.9°

Continued

6


A-3


9. y= (x-1) +2 y=-(x_1)2-2 y=(x+l)2+2 y(x+l)2-2

Vertex (1, 2) (1, -2) (-1, 2) (-1, -2)

Equation ofAxis of x=1 x=1 x_-1 x_-1Symmetry

Domain (._,,,o, W) (-001 CO) (--00, 00)

Range [2, (-,-21 [2, -) [-2, oo)

Direction ofOpening

up down up up

Maximumor Minimum

nun max min min

y values2y- y=-2 y= 2 y=-2

10. y=2(x-1)2---2

7


A-3


a. i. 16 ii . 16 iii . 100 iv. 1 v.25

vi.49

4 42 2

5 (x+4)2 (x-4)2 (x+10)2 (x-1)2 Ix- 2

(

X+ 2^

b. To find the value of k, (i) divide b by a

(ii) divide the quotient by 2

(iii) square " b :- 2"a

2. Axis of x- y- Width (as compared toVertex Symmetry intercept(s) intercept(s) Opening y = x2)

(-3, -16) x = -3 1,-7 -7 up same

(2, -64) x = 2 -6, 10 -60 up same

(-2, -18) x = -2 -5, 1 -10 up narrower

(-4,-27) x = -4 -7,-1 21 up narrower

5

2,

4 )x = - -2, -3 6 up same

252 , x = 2 -1, 4 -4 up same

N 8x=- 4 -2, - 2 2 up narrower

13, 3x = 3 1, -- 1 down narrower

a.

b.

C.

d.

e.

f.

g.

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

4. a. AC = 26.4, AB = 22.4, L C = 58°

b. PR = 11.4,ZR=52.1°, LP=37.9°

c. YZ = 3.0, XZ = 5.8, LZ=59°

5.5-y

4

6. a. x 2

7. a. 133.4°

8. Jon 27, Cal 9, Ron 21

b.x=15

2

b. 6836.2 m2

8


A-4


Length = 40 m, Width = 20 m

2.13

'

133. 8 sec 326 m

T 2,

6. 2(9x - 7)(4x + 9)

4. 90c 5. 132'

7. a. LA= 108.2°,ZB=49.5°, LC=22.3°b. PQ 72.5,LP=15.9°,LQ=14.1°c. AC 12.5,LA=43.9°, LC=76.1°

8. a. When x = 3, y = 16, b.53 l

, 0)whenx=1,y=4,when x = -5, y = 400.Collision is at (-5, 400).

10. x = -2

11. -2x2+2x+24

12. a. (2, -6 ) b. x=2 c . (3, 0), (1, 0) d. Domain: (x I x € 9i};Range : (y i y > --6, y E 9;}

e.

5

I-5 t

13. 32 - 8,r= 6.9cm2

14. a. (--, -3) u [2, o) b. (-10, 51d. (5, oo) u (-oo, -7) e. (-5, -2] u [2, o)

c. (-8, 00)

9


A-4


1. 150mx300m

2.2see, 20m

3. 25 trees, 6250 oranges

4. $46

5. 7ab(2b - 1)

6. a. BC 20.9,LC=55.5°, AB = 17.4

b. LR= 18.2°, LP= 131.8°, RQ= 11.9

1- 3y7. x

2

8. a. 33.7 m b. 17.1 m

9. Vertex: (4, 75);Axis of Symmetry: x = 4;Coordinates of x-intercepts: (9, 0) and (-1, 0)Domain : {x I X E 9Z};Range : {y i y < 75, y r= 9Z}

10. 1

12, lc, 2e, 3h, 4b, 5a, 6g, 7f, 8d

10


B-i


sin 0 cos 0

3,/34 5134a.

34 34

b. 45

3

17 4,(-17C. -

17 17

d. -7V-58

58

3V-5-858

2. a. 82° b. 600

3. 210°, 330-

4. a.

b.

x y - sin x

0° 0.00000450 0.70710900 1,00000

135° 0.70710180° 0.00000225° -0.7071270° -1315° -0.70713600 0.00000

-360 -180

tan 0

3

5

C. 8° d. 83°

2

90 180 270 /360

-2+

Continued

11


B-I


5. a.

b.

x y = c©s x

0° 1.0000045° 0.70710900 0.00000

135° -0.70711800 -1225° -0.70712700 0.000003150 0.70710360° 1.00000

6. The graphs have the same shape, the same domain and range, however, thegraph of cos x is shifted left/right by 90°.

7. a. 0.632 km b. 0.126 km/sec

8. a. 5(x - 2)(x + 2) b. (x - 3)(x + 3)(x2 + 9)

9. 19.4%

10. 17 - 4 15

11. Y = -2(x -- 2 )2 + 13

12.

c. 53°

Continued

12


B-#


13. a. y= (x+4)2+2 b.

i i--fr`--5

c. Vertex: (-4, 2)

d. Axis of Symmetry: x = - 4

e. Minimum Value: 2

14.-33

7

15. a. 100 = 0.29 b. 103 = 0.29350 350

13



1. a. 131.8°, 228.2° b. 270°

e. 221.8°, 318.2° f. 84.3°, 264.3°

Answers to Cumulative Exercises

B-1

c. 81.9°, 261.9° d. 0°, 360-

g. 48.2°, 311.8° h. 141.3°, 321.3-

i. 80.5°, 260.5° j. 104.5°, 255.5° k. 135°, 315-

2. a. 8.7 km b. 605.5 km2

3. x = 18

4. a. b. Domain: [0, 61; Range: [-5, 4]

c. Vertex: (3, 4)

d. Axis of Symmetry: x = 3

e. Maximum Value: 4Maximum Value: - 5

5. 43

6.7

2

7. a. (6, 2)

9. y= 2(x-5 )2 -2

12. a.

-360

b. 2J c. 1

10. Quadrants III and IV

2-F

-180

83

104

11. 37',143-,217-,323-

180

b. The graph ofy = cos x - 2 is 2 units lower.

c. The graph ofy = cos x + k moves k units above y = cos x, k > 0.

It is k units below if k < 0.

360

14


B-1, B-2

Exercise 10: Trigonometric Equations and Ambiguous Case Problems

1. a. 180°, 360°, 0 b. 135°, 315° c. 120°, 240° d. 66.8°, 246.8-

2. a. 34.6° or 145.4° b. 118.4° or 7.6° c. 7.8 or 1.2

3. 10.0

4. Triangle is impossible.

5. 16.9, 7.1

6. 9.5 cm

7. Vertex: (-2, 80);Axis of Symmetry: x = -2;x-intercepts: -6, 2;Domain: {x I x ` } or (-oo, co);Range: ( , 801 or (y Iy E 9Z, y 5 80)

8, 8 and 16

9. Slope = 3,y=3x-2.

10. 31

11. t = 1

12. 99% of the original area

13. 63°, 117°, 243°, 297° for angles from [0°, 360°]

14.

-µ360 -180 1 180 360

15. a. (-, 0) u [10, oo) b. (-o, -8) u [-5,---6I c. [-7, 3) a (7, co)

15


b. (-5x - 4y)(7x + 2y)

B-2

Exercise 11 : Ambiguous Case Problems

1. a. 1.3 or 5.3 b. 3.1 or 26.1 c. 7.8

d. 5.5 e. 8.0 f. No such triangleis possible

2. a. 23.0° or 157.0° b. 10.8°

d. 96.6° or 5.4°

3. (Measure triangles drawn to ensure correct measures.)

4.

5. a. (5x - 9)(3x + 4)

6. 4 or 1

8. 135°, 315°

10. $58 800, 14 cars

U.

-2+


c. 43.1

7. y=-3x+lor3x+5y-5=0

40 minutes

16



1. Maximum Height = 27 m, Time = 1.5 seconds

2. Length = 310 in, Width = 155 m

3. 70 trees

4. ±7

5. $8200

6. 60',120-

7.

8. 315.58°

9. Domain: (-, oo); Range: [0, 21

2

180 360

-2 t

10. GB=53°, C=86°,AB=35 orLB=127°,ZC=12°,AB=7.3

11. 0°, 180°

17


C-1, B-1

Exercise 13: Quadratic/Trigonometric Equations

1. a. (3x + 1)(x + 2) b. (x + 3)(x - 3)

d. 2(x - 4)2

2. a. x = -3, 1 b. x=- 4 )

e. (sin 0 + 1)(sin 0 - 1)

c. 25(x + 2)(x - 2)

f. tan 8(tan 0 + 2)

3. a. x = 4, -3 b. x == -6, -3 c. x = 5, --4 d. x - , - 2 e. x = 1, -3

34. a. x5 b. x = 5 , -6 c. x = 2 , 1 d. x = , -5 e. No solution

5. Rearrange into the form 0 = 4x2 - 17x - 15, factor, set each factor equal to zero,

and solve.

6. a. 0°, 360° b. 30°, 150°, 270° c. 63.4°, 210°, 243.4°, 330°

d. 60°, 1200, 2400, 300° e. 30°, 1500 f. 0°, 180°, 221.8°, 318.2°, 360°

g. 270°

7. 78.4°

10. 18n cm2

13. 5, 7, 9 or -7, ---9, -11

8. 24V 9. 2x-3y=6

11. 8 m, 15 m 12. 5 cm, 12 cm

14. -4 15. 9 units2

18


C-1

Exercise 14: Quadratic Formula

1. a. a=l,b=-2,c=--5 b. a=3,b=-2,c=5 c. a=5,b=-3,c=-8d. a=2,b=--4,c=-1 e. a=5,b=-9,c=0 £ a=2,b=9,c=-4g. a=-3,b=2,c=-7 h. a=1,b=0,c=-3

2. a. 3, -5 b. 2 , 1 c. 0, 7 d. 5

e. --0.2, 0.37± 5

-2g. 38.2°, 141.8° h. 170.4°, 221 .8°, 318.2°

3±2 b2± 4± 79

d.3

3. a.3

b.2

c.-9 . -1, 2

4.1±Vi

a.10

5. 1.8,-0.2

b.--3± 11

2

7. Vertex: (-5, -112); Axis of Symmetry: x = -5; x-intercepts: -1, - 9;

Domain: {x x e 9t ; Range: {y I y € 91, y ? -1121

8. 55.8°, 82.8°, 41.4-

9. a. 0°, 180° b. 30°, 150°, 63.43-

10. 157.0 km 11. Bill 7, May 27

14.

Domain : (-, to); Range: [-1.5, 0.51

c. 0°, 180°, 90°

12. 36.2 kg 13. $54

19


C-1

Exercise 15 : Solving Quadratic Equations by Graphing

1. a.

2. a. x=(-4,2) b. x=-3,-1 c. x=-5,-3 d. x=3,-3 e. x=1,5

3. a. x = -5, -3 b. x = -5, -3 c. x = -5, -3

4. a. t4 b. - 2 . 5

5. a. t2 , t1 b. 1,2,2±

6. a. 3.1 km b. 6.1 km c. 80.2°

7. a. 48 . 2° b. 33. 7°, 68.2° c. 9.6°, 170.4°

8. a. 1 b. 25

9. 6

10, b=6 .5,LA=89.4°

11. 84 m2

12. 6 and 10

13. W=3m,L=9m

31 1214. a.

366 366

15. Quadrants I and IV

b.

20


C-2

Exercise 16: Nature of Roots

1. a. No real roots b. Two real roots c. No real roots d. One real root

2. a. zero b. once c. twice

3. a. Discriminant = 0; One real root b. Discriminant = -24; No real roots

c. Discriminant = 64; Two real roots d. Discriminant = 41; Two real roots

4. a. Discriminant = 8; Two real roots b. Discriminant = -55; No real roots

c. Discriminant = 64; Two rational roots d. Discriminant = -47; No real roots

e. Discriminant = 0; One real root

5. Real numbers in (-6, 6)

6. a. 9 b. 2, 14

7. 0,-1

8. a. Two real roots b. No real roots c. One real root

9. 21.9-,158.1-

10. a. Sum = 3, Product = - 2 7 b. Sum = 2 , Product = 5

11. x2-4x-21=0

12. x2-4x+1=0

13. x2+5x+6=0

14. a. ± 2 b.

15. 15 quarters, 30 dimes, 34 nickels, 68 pennies

16. W=16cm,L=21 cm

21


C-3, C-4

Exercise 1 7: Nonlinear Equations

a. 6,-4 b . ±1 c. ±2V 2 d. ±1., ± 3

2. Complete the square to find the vertex; solve the quadratic equation to find thex-intercepts.

-5+

20+

10+

-10

-2+

--20

3,

a.

b.

C.

d.

10

Domain Range x-intercept(s) y-intercept(s)

Real Numbers Real Numbers > -4 -1, 3 -3

Real Numbers Real Numbers ? -1 -1, 1 -1

Real Numbers Real Numbers 2 -8

Real Numbers y ? -15 -3,-1,1,3 9

Continued

22


C-3, C-4


4.

5. -1± J3

6. a. Vertically opposite angles are congruent.

b. Base angles of an isosceles triangle are congruent.

c. Corresponding parts of congruent triangles are congruent.

7. a. 529 cm2 b. 35.8

8. x=12 . 7,6=22 . 8°,andy=15.8

9. 153 .4°, 33.7-

10. 1, 2, 3, 4, 6, 12

11, a. (--mob, -21 or [7, «^} b. (-10, o) c. (-°°, 41

12. 26.6°

23


C-5

Exercise 18: Radical Equations

. a. 2x-1 b . 25 + 10' + x c. x - 1+ 4 5

2. a. x = 1 b. No solution c. 3, -1 d. No solution

e. x=6 £ x=2,x=-8 g. x=2 h. x=0,x=45

3. a. 12 b. -1±5

4. 76.00

5. a. 14 b. 22

6. 0

C.

5+

5

R: (yIyE Sit , y < 0)

8. 5x2+4x-3=0

9. x=4

10. 6

R: {y1yc R ,y<6} R: (ylyE 9,y?31

24


C-5

Exercise 19 : Rational/Absolute Value Equations

1. a. 1, 2 b. -4

d. 3 e. --1

2. a. i. 4, -4 ii. 9,-9

iv. 2, -6 v.3 , 3

vi. -6,

b. First, set the expressions equal; then, set the first equal to the negative of thesecond.

3. a. (6, -16) b. x=6 c. (4, 0), (8, 0) d. Domain: (x I x E 9R);Range: (x j y E 9i, y ? -16)

4. a. 63 . 4°, 243.4°

5. W=18cm , L=25cm

6. 18.7°

7. ±,f5-

8. 72 km

9. 5 hrs

b. 137. 1°, 222.9°

25

Senior 3 Are-Calculus Mathematics Answers to Cumulative Exercises


1. a. Vertex: (2, -18)b. Axis of Symmetry: x = 2c. Minimum of -18d. Domain: (_o , e)e. Range: [-18, -)f. Narrower than y = x2g. Zeroes: 5, -1

2. a. y = 3(x + 2)2 - 20 b.6 ± 2 15

3

3, a. Impossible b. 16.9 c. 22.4°

4. a. 2100, 330° b. 41.8°, 138.2-

c. 90°, 180°, 270° d. 0°, 70.5°, 180°, 289.5°, 360-

5. a. L D = 10.8° b. G I. 96.6° or 5.4°, i = 1420.6 or 134.5

-336.

7. a. Sum: 6; Product: -4 b. x--6x-4=0

8. a. x = 6, -4

d. Impossible

9. 289.3 m

10. No, 6.6 m

11. 100mby200m

f. a=4,-3

26


D-1

Exercise 21 Circles on a Coordinate Plane

1. a. (x+2)2+(y-3)2=25 b. (x-5)2+y2=9 c. (x-4)2+(y-3)2=10

d. (x-5)2+(y-1)2=5 e. x2+y2=6 f. (x+1)2+(y-2)2=25

2. (x - 3)2 + (y - 3)2 = 9

3. (x - 1)2 + y2 = 1

4. a. Centre (--2, 1), r = 3 b. Centre (0, -3), r = c. Centre (5, 2), r = 29

20, 25. x=--

9. Domain: {x xe91

10. x2-4x+1 =0

11. 12:45 pm

6. x = ±6, ±4 7. x=2,3

; Range : {y I yE.9, y? 4 or [4, -)

8. D

27


D-1

Exercise 22: Distance between Points and Lines

21, 13 17,[1-3a. V 5 b. 2 33 2. 5 3. a.

13b.

13

4. (--6,-1)

5. Yes, distance reduces to 8.8 km.

6. a. 10 b. 3 10

7. 0, 3

8. 42.6 km

9. a. 70.5°, 109.5°, 250.5°, 289.5°

b. 90°, 210°, 270°, 3300

c. 0°, 60°, 180°, 240°, 360°

10.

-5+

11. No solution 12. 14, -16

13. a. SAS b. SSS c. AAS, ASA, SAS d. ASA

14. x2+3x-18=0

15. a. 48.6 m2 b. 5.4 m x 9 m c. 81 m2

16. Centre (-6, 3), Radius = 5

17. (x+2)2+(y--4)2=32

28


D-2

Exercise 23 : Verify and Prove Assertions in Plane Geometry

1. a. (-7, -7) b. 6,153

2. Yes; slope AB = -1; slope AC = 1.

c. 106

3. Slope AB = slope CD = 6 , and slope BC = slope DA = . Opposite sides are parallel.

4. x = 1. The product of the slopes of the lines is -

5. r=- 10 The slopes are equal.

6. a. 206.4°, 333.6° b. 109.5°, 250.5°

7007. 600,

13

8. Vertically opposite angles are congruent and SAS.

9. Vertex:(1 _39)

4' 8

11.S

912. Y= 5x2 4

5 513.

- f14. -07

15. a. Domain: [-2, 2); Range: [-1, 2) b. Domain: [1, oo); Range: (---co, oo)

c. Domain: Range: [-1, 1) d. Domain: [-2, 2) v (2, 31; Range: [11

16. c

29


D-3

Exercise 24 : Systems of Linear Equations in Two Variables

Intersection: (10, -2)

2. a.

Intersection: (2, 4)

3. (3, -1)

No intersection(No solution)

4. a. (-2, 0) b. (-2, 3)

5. a. (3, 1) b. (0, -2)

6. a. (2, -1) b. (6, 12)

7. a. (0, 3) b. (-1, 2)

8. a. 4573.2 m2 b. 132.8

Intersection: (1, 0)

c. (2, -3)

c. 52.9° and 48.1°

9. 20 10 . A ABC - A ACD by SAS, and AB - AC

11. (3, 2) 12. (6,2) 13. 3 , - 6 14. y= 2(x - 3)2 - 5

15. cos 0 = 0.63, sin 0 = -0.78, tan 0 = -1.25

30


D-4

Exercise 25 : Systems of Linear Equations in Three Variables

1. a. x=-2 , y=3,z=4 b . x=1,y=2 , z=3 c. X = -11

,y=-3,z=9

d. (2, 3)3

f. (4, -6)

2. a. a=-5 , b=350,c=0

3. 3±,F5

4. a. a=1,b=3,c=-5

5. 376 m

6. a. Corresponding angles of parallel lines are congruent.

b. Vertically opposite angles are congruent, and corresponding angles of parallellines are congruent.

c. Interior alternate angles of parallel lines are congruent.

7. a. (3, 3) b. (2, 4) c. 3 d. e. 3 units2

8. 15, -30

9. (-3, 0), (-5, 0)

10. 21V-15 - 77

11. ± 26

12. Discriminant is positive; Two real roots

13. 0,- 7

14. (x--w7)2+y2=36

31


D-5

Exercise 26: Systems of Nonlinear Equations

1. (2, 4), (-2, 4)

-1, (2, 8)2, )

3. (3, 3), (-3, -3)

3. (±4, ±3) b. (±3, ± ^i7 )

5. a. Vertex: (-6, -18) b. Axis of Symmetry: x = -6

c. x-intercepts: 0, -12 d. Domain: (x I x E 9Z}; Range: ly I y e 91, y ? -18}

6. a. 150° b. 0°, 60°, 120-

7. Discriminant = -11; No real roots

8. 14.65mand12.87 m

9. 600 kmAhr

= 3 =- 3 ; slope 1e 1 =e l = --- 4 ; slosloe 110 slo ,3p1 ; p. p4 3 4 3

Since two pairs of slopes are negative reciprocals, the figure is a rectangle.

11 and 12 intersect at A(6, 3); 12 and 1a intersect at B(3, 7); 13 and 14 intersect at

C(-1, 4). Since adjacent sides AB and BC both have length 5, the rectangle is a

square.

11. 3 ± J

12. 9or -293

13. 4

14. 2, -6

15. 81

16. a. Domain: Range: (-, 2] b. Domain: [-1, 2), Range: [-2, 1)

32


D-6

Exercise 27 : Graphing Linear Inequalities in Two Variables

6. 5,11

7.65

8. a. 25.9 m b. 3.8 sec

9. 0'-1

10. a=-3,b=2,c=1

11. 97 and 84

12. a. i. L A = 64°, L B = 36°, Z C = 80°. Total = 180°.

ii. L A= 1200, LB=109°, GC=52°, LD =79°. Total = 360°.

iii. L A = 106°, L B = 122°, L C = 91 °, L D = 114°, L E = 107°. Total = 540°.

iv. LA= 125°,LB=120°, LC= 1.46°,GD=111°, LE= 109°, LF= 109°.Total = 720°.

b. If n = number of sides, the number of degrees = (n - 2) • 180°.

t t / Graphical solutions are (0, -4) and (1, -2).13. t D

14. 29 + 6,i6 15. x=1,y=-2or(1,-2)

34


D-7

Exercise 28 : Quadratic, Absolute Value, and Rational Inequalities

1. a. k II f b. i. (x I x < - 3 or x ? 1,x E R} or

2. a. Ix j-4 <x<1,x (=- SR) or (-4, 11

5L) 1, 00)

3. a. (-4, 5) b, (--, --61 u [3,

4. a. 35.30 b. 72.3 paces

5. 6,-7

6. (-3, -1)

7. (2,-]

8. (-2, 2)

9. a. Move the term from the right side to the left side.

b. [-3, -1) a (1, 21

10.;--^-i i €--- i ----- ;- ---+-5 -4 -3 -2 ---1 0 1 2 3 4 5 6

{xIx<-3orx>3,xE 9N}

Continued

(-, -3) u [1, 00)

3<xC1,xe 9R) or f-3, 11

x< - orx>1, xe91 or

35


D-7

Exercise 28: Quadratic , Absolute Value , and Rational Inequalities

11. 3 1 E 1 1 1 7 1 1 f 1 1

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

{xI --4<x<1,xE 9t}

6<x<2, xE9Z b. {x1 -2<x<-2, xE9Z5

x<--3 orx>3, xE9 F d. {x x<--3 orx>-1, xE9t}

13. a. 5(x-2)(x+2) b. 5(x-y)(x+y)

14. a. L ABC = 50°, L AOC = 100° b. L ABC = 34° , L AOC = 68°

Ratio = 2

15. Wind: 48 km/h.Plane: 432 km/h

16.

Ratio 2

36



1. a. Vertex (-2, 19)

b. Axis of Symmetry: x = -2

c. Maximum at 19

d. Opens down

e. Domain: (-oo, -)

f. Range: (-co, 19]

g. Narrower than y = x2

h. Zeroes:


2. a. 240°, 300° b. 78.5°, 281.5°

c. 18.4°, 56.3°, 198.4°, 236.3° d. 00, 90°, 270-

3. a. ((4, 3), (2, -3)1 b. (2, -3)

4. Slope: , Distance: 10, Midpoint (-1, 1)

5. a. f(-2,-4),(1.5,1.25)j

--5 5

b. (-°°, --) u (Vi, °)

Continued

37



6. a. Sum is 8, product is 11. b. x2 - 8x + 11 = 0

7. k=±3J2

8. a. Impossible b. y = 2 or 13.3

9. $46/radio

10. (-4, 7, 15)

1L [-7, 3

12. No solution

13. (-o, -5) u [-3, 41 v (5,

14.

38


E-1, E-2, E-3


a. AC = BC, equal radii

b. A ADC _ A BDC, SSS or SAS

c. Since L 1 and L 2 are corresponding angles of congruent triangles , L 1 = L 2.

If L 1 and Z 2 are both congruent and supplementary, they are both 90°,

therefore DC I A.B.

d. Every point on the perpendicular bisector of a chord is equidistant from theendpoints of the chord.

2. a. 3

5. a. 4-J 2 b. 8

d. 10

c. 2v5

8J d. 4-v2 6. 3±V336

7. Draw 2 chords (not parallel.) Construct the perpendicular bisectors of each.Their intersection is the centre. Be careful!

8. a. 70° b. 60° c. The inscribed angle is one-half of the central angle.

9.

11. Vertex: (-3,4)Axis of Symmetry: x = -3

x-intercepts: - 3 ± 2i

Domain : x I x E 9tj

Range : yI yE9I, y54

13. a. 100; Two real roots

2

b. -47; No real roots

12. a. 116.6°, 161.6°

b. 0°, 90°, 180°c. 30°, 150°

c. 9; Two real roots

(rational) (rational)

14. Centre: (7, 4), Equation: (x - 7)2 + (y - 4)2 = 25

or Centre: (7, -4), Equation: (x -- 7)2 + (y + 4)2 = 25

b.6 c.5

4. a. 6 b. 4

39


E-1, E-2, E-3


a. 90°

2. a. 10

b. 13 c. 6.5

b. 8

133 b 12 169n or 4230d 25n. a. .

. .c.4

4. a. L 3 = 22°, L 4 = 49° b. Inscribed angle equals one-half the central angle.

5. a. LBOD = 2x b. LCOD = 2y c. x+y d. 2x+2y

6. 1 + /

7. (4,0) 8. 16

9. a. ---6<x<3or (-6, 3) b. x<-4,x>5or(--o,-4)u(5,c)

10. 4716

11. X - 12. 40°

1333 14 (-oo 3)u 2 00 xe caor ^xI x<--3 orx>. , , ,

15. Centre (2, 0), Radius = 2

40


E-1, E-2, E-3


1. a. 54°, central angle = twice inscribed angle.

b. 27°, inscribed angles subtended by the same are are equal.

2_ a. Both are 46°. b. Both are 40°.

3. a. 59°; inscribed angle is one-half the central angle

4. L C is a right angle because it is subtended by a diameter.

b. 59°

2 2 2 2

Pythagoras: AB = a2 + b2 so radius = a and area = a 4+b

19, 8)5. I

6. Midpoint of QR = (0, 3), slope MP = - 5 , slope QR = 5.

Slopes of MP and RQ are negative reciprocals, so lines are perpendicular and thealtitude from P bisects QR.

7. a. 0°, 180°

8. 6

9. (3,1,-9)

10. a. 8

b. 168.69° c. 40.49°, 139.51°

b. 36 c. 54

11. a.-9± 33 b 3± J21

4 3

12. $45.00

13. a. Eq. of DB: y = x; Eq. of EC: y=-x+2

14. a. D: [2], R: (-2, 2] b. D: (--oo, «o), R: ( oo, -1]

d. 27

c. Area = 83

41


E-1, E-2, E-3


1. 90°

2. a. 180°

b. 90°; the inscribed angle is equal to one-half the central angle

3. L1=94°,L2=32°,L3=54°,L4= 54°,L5=22°

4. L2=60°,L3=72°,L4=48°

5. L1=56°,L2=82°,L3=42°

6. L2=35°,L3=75°,L4=35°,L5=35°

19

8. {x -4<x<4, xe9}

9. (2, -3)

10. (-4, 0)

1 411. The zeroes are:

2

12. Since BC is tangent , L 3 = L 9 by the tangent-chord angle theorem . GivenL1=L2,then L2+L3=L1+L9. Aswell , L7=L1+ L9 becauseL7isanexternal angle for A ABE. Therefore , L 7 = L 2 + L 3 and thus BC = CE sincethey are sides opposite congruent angles in B BCE.

13. 12. 8ma.nd17.8m

14. a. 60°, 59°, 239°, 300°

b. 30°, 90°, 150°

c. 63.4°, 146.3°, 243.4°, 326.3°

15. (x-6)2+(y®9)2=36

42


E-1, E-2, E-3


1. 41°

2. 45°

3. 5-J 15

4. Join the point to the centre. Construct a line perpendicular to this radius at theend point.

5. Since AD is a tangent and OC is a radius , L OCD = 90°. It is given thatL ODC = 40°. Since the sum of three angles in A OCD totals 180°, that leaves 50°forL1.

6. L2=100°

7. 105°

8. OP=17, OR=.8,RP=15

9. L 1 = 20°, L 2 = 70°

10 8.7

11. O and 3

1812.

26,

7 7

2,38

14. a. 26.6°

15. (a-b-c)(a+b+c)

16. 21.5 units'

b. 41.8°, 138.2°

43


E-1, E•2, E-3

Exercise 35: Circle Properties

1. LC= 112°,LB= 60°,LD=120°

2. L ABC = 60°, Z A = 110°, L D = 120°

3. LORQ=60°, LPQR=88°, LS=92°, LP=85°

4. L EBA is a right angle since EB is a tangent and AB is a diameter. L ACB is aright angle since it is an inscribed angle subtended by the diameter AB.Therefore , Z EBA = L ACB. It is given that Z 1 = L 2. Therefore , Z CDA = L AEBsince they are the third angles in two triangles with two pairs of congruentangles . Since Z CDA = L EDB (vertically opposite angles are congruent),L EDB = L AEB by the transitive property . This makes EB = BD.

5. A OQP = A ORP by hypotenuse-leg (OQ = OR since they are radii, L Q and L Rare right angles since the radius to a tangent is perpendicular to the tangent andOP is a common hypotenuse). Therefore, PQ = PR.

6. 9 cm 12 cm 7. (2 2) and 3 - f or x = 2 2 and x = -3

8. 198.6 m

10. -1, 3

11. a. ,I 718

12. -3,7

15.

b.3 3 4±-J2

5 ' 2C.

2

b. 313

d. ±45

13. 18 sq. units 14.1

10

44


E-1, E-2, E-3

Exercise 36 : Polygon Properties

1. 14400

2. 21600

3. 18 000°

4. 180 (n-2)

5. 8 sides

6. 27 sides

7. S +2180

8. Construct a perpendicular to AB at X. Use the intersection with CD as the centre.

9. Slope AB = 5, slope of BC = 5. Therefore, A, B, and C are collinear.

10. 3, 1

11. x=-7,y=-3

12. a. 78.5° b. 41 .8°, 138.2-

14. a. 39° b. 102° c. 51°

15.8 , 7

16. $300 000 at 7%, $35 000 at 10%

17. No solution

18. (x --4)2 + y2 = 16, (x -12)2 + y2 = 16

c. 146.3°

d. 102° e. 78° f. 51° g. 390

45


F-i

Exercise 37 : Wages (Hourly)

$492.15

2. a. $358.75

b. $558.90

b. $581.88

c. $980.00

c. $922.50

3. $910

4. $458.13

5. a. b. 5, - 4

6. 26.7°

7. (9, 2)

8. a. 60° b. 3 50 c. 105° d. 40° e. 45° f. 80° g. 120°

9. x=0,y=-2

410 .

11. a. 0°, 180°, 19.47°, 160.53° b. 60°, 146.31° c. 71.57°, 108.43°

12. 351 m, 759 m

13.1

5

14. 3,-8

15. 9

16. (3, 1) and (4, 5)

46


F-1

Exercise 38: Wages (Commission and Net Income)

1. a. $88.86 b. $161.52 c. $78.53

2. $680

3. $362.50

4. $2140

5. $506

6. a. $315.97 b. $352.56 c. $86.63

7. 55 hrs9

8. x= 7, y=-3

b. 3J10

13.-5±2 10

5

14.6

5

15.. Vertex: 4, - 4); x-intercepts : 3 , 2; Domain: Real Numbers ; Range:

16. (-oo, 5) v (6, -

47


F-1


1. $2691 2. $550 3. $2773.50

4. a. 88 b. 96 c. 101

5. -1-t 3

6. 0, 1

1 25, 12 , x-intercepts: 27. Vertex: - 6

1Axis of Symmetry: x =

8.

11. a. No solution b. 221.8°

d. Out of range in Quad I and IV1108.4°, 251.6°

c. 251.6°

12. Join PR and PS . A PRQ = d PSQ by hypotenuse -leg (legs PR and PS are

congruent radii , angles at R and S are right angles since the radius and tangentare perpendicular, and QP is a common hypotenuse ). This bisects the angle at Q.Construct perpendiculars from M to QR and M to QS . These new triangles arecongruent by AAS, making M the same distance from each tangent.

13._5± i46

3

15. a. [6, 12] b. (-5, 2) V [8]

14.2_v13

13

x E Jl ; R: (__I 12 )I

9. x<2

10. (4,-6)

c. (-oo, -12] v (5, 20)

48


F-1, F-2

Exercise 40 : Unit Prices, Exchange Rates, and Reconciliationof Bank Statements

1. 0.2390/mI, 0.189V/ml

2. The second box costs $2.05/kg compared to $2.40/kg for the first. The second isbetter.

3. a. 1.878¢/g b. 3.203g/g c. 1.6420/g

4. a. $180 US b. $38.89

d. Add one-third the cost plus a bit.

5. $273.97 Cdn.

6.

c. $52.78

CHEQUE CHEQUES ISSUED TO OR CHEQUE DEPOSIT DEDUCT/ADD BALANCE FWDDATE No. DESCRIPTION OF DEPOSIT AMouNT AMOUNT CHEQUESIDEPS 998 43

Sept.234 The Bay

48 00 CHQ - /DEP + 48 009 BALANCE 950 43

13 244 Esso 43 87 CHQ - /DEp + 43 87BALANCE 906 56

20 245 Hydro 66 , 98 CHQ - /DEP + 66 98BALANCE 839 58

25 Deposit200 00 CHQ - /DEP + 200 00

BALANCE 1039 58

30 246 Dales Rental Agency `175 00 CHQ /DEP + 475 00BALANCE 5641 58

7.MONTH

PreviousBalance

PAYMENT

MADE

PURCHASES

CHARGED

BALANCEDUE

CREDIT

CHARGE

NEW

BALANCEFebruary $586.00 $100.00 $ 93.00 $579.00 $8.11 $587.11March 587.11 200 .0 $121.75 $508.86 7.12 515.98April $515.98 $275.00 13.17 254.15 3.56 257.71May $257.71 $200.00 $ 87.13 $144.84 2.03 $146.87

Continued

49


211.11

854.00

57.10

146.58

$ 1268.79

F-1, F-2

Exercise 40 : Unit Prices, Exchange Rates , and Reconciliationof Bank Statements

8.

Bank Reconciliation

Balance from statement : $ 837.71

Add deposits: 2000.00

Total additions: $2000.00

Subtotal:

Subtract withdrawals:

Total subtractions:

Subtotal:

This should equal balance shown in your record book:

9. 79.7 m2

10. 6, -2

11. Discriminant = -31; No real roots

12. a. 60° b. 30°

13. 2.1, -3.5

14. a. 12-2

15. x=7,y=-2,z=3

b. 15x'22

c. 90°


$2000.00

$2837.71

$ 1268.79

$ 1568.92

$1568.92

d. 240° e. 180°

16. 25 17 . 256 units2

50


F-3


1.

1. Incomea. Regular Monthly Income $ 2796.08b. Spouse's Regular Monthly Income Sc. Additional Income $ 170.00d. Other Income $Total Monthly Income #1 $ 2966.08

2. Housing Expensesa. Mortgage or Rent $ 625.00b. Property Tax $___ 160.42c. Home/Property Insurance $ 22.92

d. Repairs/Maintenance $e. Other Housing Expenses $Total Housing Expenses #2 $ 808.34

3. Utilitiesa. Hydro $ 120.00b. Gas $c. Phone $ 17,40d. Water $ 3&33e. Other $Total Utilities #3 .1 .73

4. Transportationa. Public Transportb. Car Loanc. Car Fueld. Car Maintenancee. Car Insurancef. Other Transportation

$ 213.50$ .80.00

$ 2d'a$......_.....U 0

Total Transportation #4 $ x,33

5. Personal Financesa. Personal Loanb. Investments $c. RASPd. Life insurancee. Charitiesf. Credit Card Paymentsg. Service Chargesh. Savingsi. Other Personal FinancesTotal Personal Finances #5 $ 18.00

6. Personal Expensesa. Groceries 42500b. Clothing $ 60.42c. Entertainment $ 62,50d. Gifts $ 52.50e. Vacations $f. Other Personal Expenses $Total Personal Expenses #6 $ 600.42

7. Other Expensesa. Newspaper/Per. $ _ 17.50b. BabysittingC. $Total Other Expenses #7 $ 17.50

Total Monthly Expenses #8 $ 2010,32

Income minus Expenses (#1 - #8) #9 $ 955.76

Comments: Savings of $955.76/month

Continued

51


F-3


2.

1. Incomea. Regular Monthly Income $ 169739

b. Spouse's Regular Monthly Income $ 1652.82c. Additional Income $ 107.72d. Other Income $Total Monthly Income #1 $ 3458.43

2. Housing Expensesa. Mortgage or Rent $ 733.15b. Property Tax $ 149.00c. Home/Property Insurance $21.00d. Repairs/Maintenance $e. Other Housing Expenses $Total Housing Expenses #2 $ 903.15

3. Utilitiesa. Hydro $ 200.00

b. Gas $c. Phone $ 20.20d. Water $ 45.00e. Other $Total Utilities #3 $ 265.20

4. Transportationa. Public Transport $b. Car Loan $ 237.75c. Car Fuel $ 140.QQd. Car Maintenance 3$.33e. Car Insurance $ 64.17f. Other Transportation $Total Transportation #4 $ 480.25

3. W=9cm,L=llcm

4. 24 units2

5. (3.2, 7.8)

6. a. 161.57°

5. Personal Financesa. Personal Loan $b, Investments $

c. RRSP $d. Life Insurance $ 22.00e. Charities $f. Credit Card Payments $ 200.00g. Service Charges $h. Savings $i. Other Personal Finances $Total Personal Finances #5 $ 222.00

6. Personal Expensesa. Groceries $ 740.00b. Clothing $ 100.00c. Entertainment $ 180.00d. Gifts $e. Vacations $ Tf. Other Personal Expenses $Total Personal Expenses #6 $ 1020.00

7. Other Expensesa. Newspaper $ 8.50

b.

C. $Total Other Expenses #7 $ 8.50

Total Monthly Expenses #8 $ 2899.10



b. 41.81°, 138.19° c. 28.8°, 129.3°

Continued

52


F-3


7. Line segments AC and BC are perpendicular since BC is a tangent and AC is adiameter. L ACB is bisected , making each angle 45°. Angle ADC is a right anglesince it is an inscribed angle subtended by a diameter . Therefore , L B is 45° sincethe three angles in A BCD total 180°. Since L DCB and Z B are both 45°, thesides BD and ED opposite them are congruent.

8. a. Yes, since PQ = QR = 5 b.5

2

9. Vertex 3 2, 3 3 i; x-intercepts : 2 t 3 13 ; D: Real Numbers; R:

13. $349.35 or $349.45

53


$ C. $

$ 186.40 Total Other Expenses #7 $ 212.00

$ 100.00

$ M& Total Monthly Expenses #8 $ 2850.33$ 60.00$ 130.00 Income minus Expenses (#1 - #8) #9 174 4)

#4 $ 523.07

F-3


1. Incomea. Regular Monthly income $ 1306.93b. Spouse's Regular Monthly Income $ 1345.50c, Additional Income $ 53,86d. Other IncomeTotal Monthly Income #1 $ 2706,29

2. Housing Expensesa. Mortgage or Rent $ 725.00b. Property Tax $5:,$0c. Home/Property Insurance $ 25.83d. Repairs/Maintenancee. Other Housing Expenses $Total Housing Expenses #2 $ 903.33

3. Utilitiesa. Hydro $ 225.00b. Gas $

c. Phone $ 18.60d. Water $e. Other $Total Utilities #3 $ 243.60

4. Transportationa, Public Transport

b. Car Loanc. Car Fueld. Car Maintenancee. Car Insurancef. Other TransportationTotal Transportation

2, a. 107.5 m, 280.1 m

3. (2, 2) and (-5, 5 ,J

4-5±-0'89

4


5 . Personal Financesa. Personal Loan $b. Investments $c. RRSP $d. Life Insurance $ 20.00e. Charities $I. Credit Card Payments $ .00g. Service Charges $h. Savings $i. Other Personal Finances $Total Personal Finances #5 $ 220.00

6. Personal Expensesa. Groceries $ 525.00b. Clothing $ 54.17

c. Entertainment $ 50.00d. Gifts 70.83e. Vacations $ 78.33f, Other Personal Expenses $Total Personal Expenses #6 $ 778.33

7. Other Expensesa. Newspaper $ 12.00b. Babysitting $ 200.00

Comments : Need to reduce expenses by $174.04

b. 2790.4 m

Continued

54


F-3


5. Since points (1, 2) and (1, -4) are on the same vertical line, they cannot belong tothe quadratic function.

6. Since DE is tangent at C, L 3 = Z 5 by the tangent-chord angle theorem. SinceAB 1/ DE, Z 4 -= L 3 because interior alternate angles of parallel lines arecongruent . The transitive property makes L 4 -= L 5. This makes AC = BC, and

the triangle is isosceles.

7. $1849 when they sell 43 tape recorders

8. a. y= 3x+25 b,y_--4x+25

4 4 3 3

915,12-9

29

10. 6 of the 250 bolts, 4 of the 400 bolts

11. (-3, -1) u (1, 2)

12. a. (-OQ, -30) a [70, oo) b. [-6, 4) c. (_ o, -8) u [0, 61

13.

2

18© 360

55


F-5


1. a. 9 years

2,

b. About $11 000 (exactly $10 794)

Time (years) Value ($)

0 4000.001 4240.002 4494.403 4764.064 5049.915 5352.90

After 9 years = $6800.

3, a. $ 128.66 b. $ 152.31

4. 1480

5. Vertex : 4 , -81);

Axis of Symmetry: x = 11 ; x-intercepts: 1

D: {xl xE } , R: (yI y81,

y8

6. $1041.67 7. x = -3, y = -1, z = 0 8. (4, -3), (-4, -3), (3, 4), (-3, 4)

9. 16 10. - 4 , 2

11. a. 60°, 120° b. 63.43°, 135°, 243.43°, 315° c. ±90°

12. 3x2--2x-4=0

13. Set up the coordinate system so that corner A is (0, 0)and corner D is (p , 0). Since side BC is parallel to AD,theY-coordinates of B and C will both be the same,say q. Let B have coordinates (t, q). Since BC haslength = p = AD, and is horizontal , C(t + p, q). (Seediagram .) Find the coordinates of the midpoints ofboth AC and BD.

Both are (t + p2 2

14. a. L1=70°, L2 =40°, L3=35°,L4 =70°, L5 =20°, L6= 140°, L7 =20°,L8=35°,L9=75°

b. D=40°,b40SG=70°,A^= 140°, AB=70°

y

x

56


F-5


1. a. $7260 b. $720 c. $450 d. 7 years

e. $3343.75 f. 4.75% g. $1200

2. a. $7080.00 b. $7146.10

3. $6463.49

4. 10%

5. 10.25%

6. 12.55%

7. $18 750

8. $156.41

9.2± 22

3

10. Vertex: l 7 , 249 ); Axis of Symmetry: x = 7 ; x-intercepts: 5 ,12

D: {x I x E 91 1 ; or R: fy I y< 169 or 169 124 24

11. F(11, 2)

12. (-3, 0 ) v (5, o)

13. 2

14. a. L1=15°,L2=75°,L3=15°,L4=75°,L5=90°

b. are CE = 75°, are AE = 75°

15.17350

57


G-1


1. a. Inductive b. Inductive c. Deductive d. Inductive

e. Deductive f. Deductive g. Inductive

2. C,E,F

3. Find the midpoints of any two sides. Find the slope between them. Calculate theslope of the third side. Compare.

4. x=0,-

5. x=0,y=-2andx =1.6,y=-1.2

6. 254.9 feet

7. AB and AC are tangents from A. Therefore, AB _= AC since tangents from thesame point are congruent . BD = CD since D is the midpoint. A ABD = A ACDby SSS. Then L 1 = L 2 by corresponding parts of congruent triangles arecongruent.

8. a. (--oo, co) b. (---oo, 0) c. [-4, 21

109. y= x-3

3

11. 95.7°, 180°, 264.3-

12. a. $240 b. $4240

13. a. Discriminant = -4; No real roots or two imaginary roots

b. Discriminant = 104; Two real roots or two real unequal roots

14. $8.66

58



1. a. b=0,aE 91 b. Ifa=1-borb=1-a

2. y=a(x- 2)2+3,a>0

25(x+1)2-1 b.

x=4+45-,x`23. a. y= 8

4. 146.3°, 326.3-

5. a. 6.5 m

6. 300, 150°

7. x=2

b. 3 sec c. 0.3 sec

8. Z I=55°,/2=90°,L3=30°,L4=60°,L5=70°

9. $14 889.31

10. x=14orx=6

11.

12. a. 5.29 units b. 97.2° c. 262.8°

59


G-2

Exercise 47: AND, OR, NOT, and Venn Diagrams

0oxoxo

oxoxoxoxoxoxoxoxoxoxx0x0x

oxa

3. 12 4. 218 5. 2

8. a. 115.4°, 244.6° b. 19.5°, 160.5°

9. 4.2 cm2

10. See budget form on next page.

11. 1 cm

12. i. a. up b. (0, -3) c. X 0ii, a. down b. (-2, 3) c. x = -2iii. a. up b. (5, -1) c. X=5

C

6. 9 7. 0,3

c. 4°, 184°

13. L1=10°,L2=30°,L3=75°,L4=15°,L5=50°,L6=15°,L7=90°,L8=75°,are GA = 80°, are GC = 130°, are GAC = 230°

14. x = 4

15.

Continued

60


G-2

Exercise 47: AND, OR, NOT and Venn Diagrams

10.

1. Incomea. Regular Monthly Income $ 1541.67b. Spouse's Regular Monthly Income $ 1977.09c. Additional Income $ 26.93d. Other Income $Total Monthly Income #1 $ 2645.69

2. Housing Expensesa. Mortgage or Rent $ 425.00b. Property Tax $ 0.0c. Home/Property Insurance $ MIZd. Repairs/Maintenance $e. Other Housing Expenses $Total Housing Expenses #2 $ 506.00

3. Utilitiesa. Hydro $ 41.75b. Gas $ 61.66c. Phone $ 10.50d. Water $ 41.75e. Other $Total Utilities #3 $ 155&Q

4. Transportationa. Public Transport $b. Car Loan $c. Car Fuel $ 105.00d. Car Maintenance $ 21.25e. Car Insurance $ 24,00f. Other Transportation $Total Transportation #4 $ 150.25

5. Personal Financesa. Personal Loan $ 345.00b. Investmentsc. RRSP $d. Life Insurance $e. Charities $f. Credit Card Payments $g. Service Charges $_h. Savings $i. Other Personal Finances $Total Personal Finances #5 $ 345.00

6. Personal Expensesa. Groceries 420.00b. Clothing 66.25c. Entertainment $ 27.50d. Gifts $ 13.75e. Vacations $ 70.63f. Other Personal Expenses $Total Personal Expenses #6 $ 598.33

7. Other Expensesa. Newspaper $_ 10.42b. $C. $.,.Total Other Expenses #7 $ 10.42




61


G-3


- 0.5)-°.5,

5. x=41(1763=43x41) 6. 10

8. 43 9. $104.17

r

10.

111. Vertex: (-0.65, - 7.225); Axis of Symmetry: x = -0.65; x-intercepts: 1 ,

Range: {y I y e 91, y >_ -7.2251

12. a. $106.25 b. $5106.25

13. Consider A CBD and A CDA . Since EC is a tangent, Z 3 = Z 7 because of the

tangent-chord theorem . Obviously, L 4 = L 4 by the reflexive property. So twoangles in A CBD (L 3 and L 4) are congruent to two angles in A CDA (Z 7 and

L 4) and then their third angles , namely L 5 and L ADC are congruent.

14. a. 10 mm b. 85.8 mm2

15. 4 3417

c. 10f mm

16, LC= 16.3° or 163 .7°,LA=148. 7° or 1.3°, BC= 46 .2 or 2.0

270

62


G-4

Exercise 49: Converse, Contrapositives, It...Then...

a. If you can fly a plane, then you can operate a car.Both are false.

b. If a child believes in the tooth fairy, then the child is less than 6 years old.Probably both are false.

c. If you are taller than average, then you are a successful basketball player incollege.Both are false.

d. If you are a good cook, then you studied home economics in school.Probably both are false.

e. If visibility is poor, then it is raining.Statement is true, converse is false.

f. If a girl wears high-heel shoes, then she goes to a party.Both are false.

g. If a person likes spaghetti, then the person likes pizza.Both are false.

h. If the angles opposite two sides of a triangle are congruent, then the twosides of the triangle are congruent.Both are true.

i. If two angles are congruent, then the angles are right angles.Statement is true, but its converse is false.

2. a. If the sides opposite two angles of a triangle are not congruent, then the twoangles are not congruent,

b. If the angles opposite two sides of a triangle are not congruent, then the twosides are not congruent.

c. If two angles are not congruent, then the two angles are not supplements ofcongruent angles.

d. If two angles are not congruent, then the two angles are not complements ofcongruent angles.

e. If a triangle is not equiangular, then the triangle is not equilateral.f. If a triangle is not equilateral, then the triangle is not equiangular.g. If a point is not equidistant from the endpoints of a segment, then it is not

on the perpendicular bisector of the segment.h. If d(A, M) # d(B, M) then M is not the midpoint of A.B.i. If d(A, P) + d(P, B) # d(A, B) then P is not between A and B.

3. 8 4. 3 ± 2-`2 5, x>3,x<1

6. x < 3 7 . Two real roots 8. 14.27

Continued

63


G-4

Exercise 49 : Converse, Contrapositives , If...Then...

9. (3, 3) and

10. L1=85°,L2=95°,L3=55°,L4=113°,L5=82°,L6=122°

11. - 1.4

12. a.- 4

b. (-1, 1) c. 9 d. 4i

13. (--, -4) v (-2,

64


G-5

Exercise 50: Direct and Indirect Reasoning

A list of all the possibilities is made, and then, one by one, possibilities areshown to be impossible and are eliminated until only one possibility remains.

2. Ben is the murderer. Indirect proof

3. Z 1 = Z 3. Direct proof

4. She is at the laboratory. Indirect proof

5. a. If a number is a multiple of 6, then it is a multiple of 3.b. If a person was born in 1810, then that person is now dead.c. If it is sunny, then my family always goes on a picnic.d. If two angles are vertically opposite, then the angles are congruent.e. If two angles are base angles of an isosceles triangle, then the angles are

congruent.f. If a number is even and larger than 2, then it is the sum of two primes.

6, a. If a number is a multiple of 3, then it is a multiple of 6. Falseb. If a person is now dead, then that person was born in 1810. Falsec. If my family always goes on a picnic, then it is sunny. Falsed. If two angles are congruent, then they are vertically opposite. Falsee. If two angles are congruent, then they are base angles of

an isosceles triangle. Falsef. If a number is the sum of two primes, then it is even and

larger than 2. False

7. a. If a number is a not multiple of 3, then it is not a multiple of 6. Trueb. If a person is not dead , then that person was not born in 1810. Truec. If my family does not go on a picnic, then it is not sunny. Trued. If two angles are not congruent, then they are not vertically

opposite. Truee. If two angles are not congruent, then they are not base angles

of an isosceles triangle. Truef. If a number is not the sum of two primes, then it is neither even

nor larger than 2. True

8. 84 m2 9.3± 5

210. - -5

11. x = 12.25 12. 3

Continued

65

Senior 3 Pro-calculus Mathematics Answers to Cumulative Exercises

G-5

Exercise 5d : Direct and Indirect Reasoning

13. Since PA and PB are tangents from P , PA = PB. The angle at P is 60°, and theangles at A and B total 120°, because the three angles in the triangle total 180°.L A = L B since PA = PB . Therefore , all three angles are 60° and all 3 sides are

congruent , making the triangle equilateral.

14. a. Discriminant = -11; No real roots b. Discriminant = 49; Two real roots

15. 36n - 540 cm2 or = 19.6 cm2

16.

Bank Reconciliation

Balance from statement: $ 199.53

Add:

- deposits not cleared in statement $ 45.00

Total additions: $ 45,00

Subtotal (add): $ 244.53

Subtract:

- withdrawals/cheques 17.86

not cleared in statement .54.76

Total subtractions: $ 72.62 72.62

Subtotal (minus): $ 171.91

This should agree with the balance shown in your record book: $ 171.91

b. y=12cosx

66


H-1

Exercise 51: Operations and Compositions Functions

1. a. 8 b. 9 c. 9

d. 12 e. 14 f 12

g. 14 h. 14 i. 22

2. a. 12x2 + 12x + 1 b. 6x2 -3 c. 4x + 3

e. -4

b. 8

b. 4x2 - 12x + 10

5. a. b.x -1 C.4

3

c. 20

6. 3

7. a. 70 . 5°, 59.0°, 50.5° b. 3.85 units2

8. x= , y=2 , x= -, y=-2 , x= 4, y = L x =-4, y=-1

6

29

10. -2, 4 11. Flavio is 20 , and Inga is 16.

12. The midpoint of PQ is (4, - 3), of QR is (1, 1 ), of RS is (-3, --1), and of SP is (0, - 5)

The slopes of consecutive midpoints are. - 4 , 1,''

- 4 , 1 . Because opposite sides3 2

have the same slope, opposite sides are parallel.3 2

13. Use the quadratic formula: x = 2± (-2)2 _ 4(1)(c)2(1)

15. ftx)= x3+x2- 17x+ 15

17. a. [-3, oo)

14. Yes

16. 12.57 m

b. (-- , 5) U (5, c) c. (3, 51

67


H-2


1. a. Divide by 5. b. {(5, 4), (6, 6), (8, 7)

c. {(x ) I y = x 3 2 d. (x=y) f y = 4- x}

2. a. 1(x) = 3x b. f 1 + x -1 x > 1 c. 2x+3

x

3. a. -2 b.1

3c. a

y

2+

-2+

-2 i 2

c. For any value of x > 2, there are two possible y-values.

5. ftg(x)) = x as does g(flx)).

2 4 -

6. a. AC 11 EF makes L ACE = L FEC since they are interior alternate angles. Thismakes L ACB = L FED since they are complements of these congruentangles. These angles, plus the congruent angles and sides in the givens makeA ABC =- A FDE by ASA.

b. AB 11 DF since the interior alternate angles, Z ABC -= L FDE are congruent.

Continued

68


H-2

Exercise 52 : Inverse Functions

7.

1. Income 5. Personal Finances

a. Regular Monthly Income $ 1454.92 a. Personal Loan

b. Spouse's Regular Monthly Income $ 1463.58 b. Investments

c. Additional Income $ 36.75 c. RRSPd. Other Income $ d. Life Insurance

Total Monthly Income #1 $ 2955.25 e. Charities

2. Housing Expensesa. Mortgage or Rent $ 71 5.440b. Property Tax x:00c. Home/Property Insurance $ 20.78d. Repairs/Maintenance $e. Other Housing Expenses $Total Housing Expenses #2 $ 936.18

3. Utilitiesa. Hydrab. Gasc. Phone $ 23.00

d. Water $e. Other $ 305.20Total Utilities #3 $ 328,20

4. Transportationa. Public Transport $

b. Car Loan $ 206.10c. Car Fuel $ 120.00d. Car Maintenance $ 15.aoe. Car Insurance $ 50.83f. Other Transportation $ 130.00Total Transportation #4 $ 541.93

$ 100.00$$$$

f. Credit Card Payments $ 200,00

g. Service Charges $

h. Savings $i. Other Personal Finances $

Total Personal Finances #5 $ 300.00

6. Personal Expensesa. Groceriesb. Clothing 69.16c. Entertainment $ 216,67d. Gifts $ 41.67e. Vacations $ 166.67f. Other Personal Expenses $Total Personal Expenses #6 $ 1.0 ,4.11

7. Other Expensesa. Newspaper $ 20.83b. Babysitting $ 33,33C.Total Other Expenses #7 $ 54.16


Income minus Expenses (#1 - #8) #9 $ (259.39)

Comments : The Hills need to be concerned abouttheir financial position.

8. a. Discriminant = 1; Two real roots b. Discriminant = -11; No real roots

9. (-2, -2)

Continued

69


H-2


10. a. -- -- - 1 } '-5 -4 -3 -2 -1 0 1 2 3 4 5 6

-2 -1 0 1 2 3 4 5 6Empty Set

-5 -4 -3 -2 -1 0 1 2 3 4 5 6CY __d. _^ 1 1 1 ^_ T T F f 4

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

e. F- F j -^ ---- - ---------i--F--3 4 5 6-5 --4 -3 -2 -1

l0

1 2'

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

11. 5

12. 240

13. a. 1 14 8

70


H-3


. f(-1) = 0, (x + 1) is a factor. b. ft 3) = 24, (x - 3) is not a factor.

2. a. g(-1) = 25, (x + 1) is not a factor.

3. (x2 + 3)(x -- 2)

4. x3+7x2+7x-2

5. a. 8 b. -5

6. a. -9x + 9 b. 18x + 19

X-51 (x) _7. f

2

8. x=6,y=12

9. 115.1 m

10. $583.33

11. 3.75 km/h, 0.75 km/h

12.-5 ± 17

8

14. f-'(x)=x+4

13. 1, -1±/

The lines intersect on the line y = x.

15. (--oo, -3] u (-1, 2) a [3, -] 16. 16 17. 124

71

Senior 3 Pro-Calculus Mathemattcs

c. Rational

H-4

Exercise 54 : Graphs of Polynomial and Rational Functions

1. a. Polynomial b. Rational

2. a. x-intercepts : 0, 1, -1; y-intercept: 0

3. Domain : {x x E 39t1; Range: ly I y e 3t}

4. a.

5. a. k 4

-2+

Zeroes are ±1

b.

b.

-5

b.


ti 1o+

II

d. Polynomial

11y 5

The zeroes ofy = x2 -1 and the

asymptotes of y = x21 are both x - ±1.

Continued

72


H-4


6. a. (x + 1 )(x - 2)(2x - 1)

C.

7. a. 225°, 315° b. 120°, 240-

8. (2, 3), (-2, 3), (2, -3), (-2, -3)

9. 2 ± 46-

10. No

11. x-5y+10- 3 26 =0 , x-5y+10+3 26 =0

12. A: $928 B: $936

13. L2=55°,L3=40°,L4=35°,L5=70°

14. 3 ±

15. Various answers are possible.

b. -1, 2, 2

c. 128.2°, 231.8°

73



1, 13

2. a. 2

3.

4. (x - 3)(2x - 1)(x + 5)

5. a= 1,b

6. b=2,c=-3

7. L1=30°,L2=30°,L3=120°,L4=90°,L5=60°,L6=30°,L7=90°

8. LB=55°orLB= 125°

LC=89°orLC= 19°

c = 16.0orc=5.2

74



1. a. 2

d. -1

2. a. -3

d. 4

3. (x - 2)(2x - 1)(x + 3)

4. (2x - 1)(3x + 1)(2x + 3)

b. 18

e. 2

b. -10 c. 2

e. 4 f. 2

5. a. (x - 1)(x - 3)(x + 2) b. (x + 1)(x - 3)(x - 2)

c. (x - 2)(2x + 1)(x + 2) d. (x + 2)(2x - 1)(x + 1)

e. (x + 2)(2x + 1)(x + 3) f. (2x -- 1)(x2 + 1)

6. a=3,b=2

7. p=1,`2...-3

8. a=3,b=--1,c=-1

9. a=2,b=-1,c=-2

10. No

75




1. Yes, both = x6.

32. a. 7 b. -3 c. 8 d. e. 1000

4

£ 428 g. a h. a

3. t(s(x )) = x2 - 2, s(t(x)) = x2 - 6x + 10

4. a. t(s(x)) = x - 4

b. s(t(x)) = x + 4

c. No

5. a. 4 b. 2v3 c. Does not exist.

d. Does not exist. e. 4 2

g. Does not exist. h. Does not exist.

b. 13

e. 9a2 + 24a + 15

h. a4 - 2a2

c. 48

£ 3a2+1

76



2+

_2

b. (0, 4)

f. none

2.

2

c. minimum d. 4

g. 1

y = 2(x-1)2 - 9


e. 4

X E 'R} h. { y) y E 9t, y > 4} i. up

a. x=2b. (2, 5)c. maximumd. 5e. 13f. none

g. {xx€9t}

h. }yIy€9t, y:5 -5}i. down

a. x=1b. (1, 9)c. minimumd. 9e. -7

f 2±3J

2g. {x Ix E1}h. f y (y € 9t, y >_ -9}i. up

Continued



4. a. b. Y

(2,3)

x

linear

3x + 2y = 4

y=x2+5x+6

parabola

5. y=(x+1)2-7

6.494


y=-(x-2)2+3

parabola

(x - 2)2 + (y + 1)2 = 12

circle

7. 210 programs sold at $35.00 for a maximum profit of $7350.

8. Function I II III IVa. sine + + -- -b. cosine + - - +c. tangent + - + -

Continued

78




9. a. 0.74314 b. 0.70710 c . -5.67128

d. 0.00822 e . 0.99966 f. 0.11393

10. a. 39.63° or 140 . 37° b. 88 .86° or 271 . 14° c. undefined

11. a. i. 30°, 150° ii. 0°, 180-

b. i. 153 . 4°, 333.4°, 45°, 225° ii. 90°, 270°, 120°, 240°

12. a. 389 . 82 b. 28.96°

13. 1854.1 km

14. a. 80°, 25.28 b . 30.8°, 24.2°

15. 14.6 m

16. Given angle , side , side, where the given side closer to the given angle is largerthan the opposite side.

17. LC =51.4°or 128 .6° a=31 .64or 11.68L A = 98. 6° or 21.4°

18. LA=37. 9°, 142 . 1°;LB=110 . 3°, 6.1°;b=10.7, 1.2

19. 22

20. a. 720°

21. 106.81 or 34 rr

b. 12 240° c. 41

22. 105°

23. L 1 = 20°, L 2 = 70°, are BEF = 220°

24. a. 10° b. 30° c. 75° d. 90°

e. 60° f 50° g. 20° h. 110°

i. 330° j. 100° k. 250° 1. 150°

m. 360°

Continued

79



25. a. 40°

e. 140°

26. a. 2 b. 9

27, 6 Physics

b. 50°

£ 100°

c. 110° d. 20°

g. 220°

c. 5 d. 4 e. 4 £ 3

Biology

28. a. Deductive b. Inductive

29, a. True

b. If a triangle is isosceles, then it is also equilateral. (False)

c. If a triangle is not isosceles, then it is not equilateral. (True)

30. Two possible counter examples are {(4 , 2), (5, 1), (5, -1)} and y = ±V.

31. Start by assuming that Z 2 is equal to Z 4. Then L 1 and L 3. Then Z ABC isequal to L ACB because of angle addition. AB = AC because sides opposite equalangles are equal. This however contradicts the given AB = AC and thereforeL2# L4.

32. 3100 33. 13 nickels, 7 quarters 34. , 4 , (-2, 3)^(2

35. (-2, 1, 3) 36.

Continued

80



37_ a. IxI xe9t, 2<x<4


xe91 , x?2, x<

38. a. {x I x E 9Z, x > 3, x < .1} or (-o, - 1) u (3,

b. Ix xE9, --4<_x< 1 or[-4, 1]

c. {x x E%, x?5, x<-2} or(-°, -2]v[5, oo]

d. IX x€9Z,-4<x<_31or(---4,3)

39. a. 18 b. 10 c. 0

d. - 5 e.f. 4x2 - 6x

4

1 3g. x' - 9x + 18 h. x2- 3x i.

2- -

x X

440. a. 4 b. 7 c. -

7

d. 6x + 1 e. -7

41. a. h-'(x) = x - 7

f. 4x-9

b. h-'(2)3

10

3

Continued

81



42. a.

-F--10

10

D -4--10 -10

43. 9

44. (x -1)(x + 5)

45. k = 2

46. 3

47. No real solution

0

-10

A

10

10

Continued

b.


82


Exercise

48. 5

49. 1 (not -3)

50. -1, 3

51. -

52. 2, 3

53. No answer

54. -4, 3

55. 2,3

56. -1

57. -1

58. f-'(x) _

59. 7.69

x

Cumulative Review

1-3x

Senior 3 pre-Calculus Mathematics

60. 51.95 m

61. a. Shortest distance (perpendicular) from a point to a line

b. 4x+5y-6=9

62. y = -16

63. y=-2(x-2)2+3

64. Discriminant = 24; Two real roots; Sum of roots = 4; Product of roots =

Continued

83



65. a. 2 7 b. (-2, 3) C. =4

d. (x+2)2+ (y-3)2= 17 e. 25.91

66. 2x-1

67. x=12 ,61=64°,L2=116°,L3+L4=180°

68. a. y 5 -

b. y<2x-6

c. (x+2)2+(y-1)2<9

69. Gross pay = $536.84; Deductions = $167.26; Net pay = $369.58

70. Area of circle = 31.37 cm2; Area of shaded region = 11.37 cm2

71. a. The lengths of four sides are equal.

b. Lengths of opposite sides are equal and slopes of adjacent sides are negativereciprocals.

c. Opposite sides are equal in length and slope.

72. (x-1)(x+2)(x--3)

73. y=2x2+5

74. a. $262.50 b. $787.50 c. $20.14 d. $525.00

Continued

84


Exercise 58 : Cumu lative Review

75.


MonthlyPayment Principal

PaymentMade

Interest8%'%% per year

AmountOwing

$5000.001 $5000.00 $300.00 $33.33 $4733.332 $4733.33 $300.00 $31.56 $4464.893 $4464.89 $300.00 $29.77 $4194.654 $4194.65 $300.00 $27.96 $3922.625 $3922.62 $300.00 $26.15 $3648.77

76. a.

C.

-2+

d.

b.

85

Senior 3 Pre-Calculus Mathematics -...

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