# User manual | Senior 3 Pre-Calculus Mathematics

```Senior 3
Pre-Calculus Mathematics
Cumulat ive E xerci s e s
A Supplement to
A Foundation for
Implementation
Manitoba
Education
fining
Education
et Formation
profess onn 1.U
Manitoba
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
CONTENTS
Acknowledgements
Introduction
iii
xi
Outcome(s)
New Topic Presented
Exercise
1
2
4
A-l, A-2
3
6
A-2, A-3
4
9
A-3
5
12
A-3
6
13
A-4
7
8
Trigonometric Equations 1
18
B-1
9
Trigonometric Equations 2
20
B-1
10
1
A-1, A-2
A-4
15
Trigonometric Equations and Ambiguous
Case Problems
B-l, B-2
22
B-2
24
11
Ambiguous Case Problems
12
Review 1
13
14
15
16
Nature of Roots
17
Nonlinear Equations
18
26
C-1, B-1
27
C-1
29
31
C-1
C-2
33
35
37
C-3, C-4
C-5
Vii
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
Viii
Outcome(s)
New Topic Presented
Exercise
C-5
39
19
Rational/Absolute Value Equations
20
Review 2
21
Circles on a Coordinate Plane
22
Distance between Points and Lines
23
Verify and Prove Assertions in Plane Geometry
24
Systems of Linear Equations in Two Variables
25
Systems of Linear Equations in Three Variables
26
Systems of Nonlinear Equations
27
Graphing Linear Inequalities in Two Variables
28
Quadratic, Absolute Value , and Rational Inequalities
29
Review 3
30
Circle and Polygon Properties 1
61
E-1, E-2, E-3
31
Circle and Polygon Properties 2
63
E-1, E-2, E-3
32
Circle and Polygon Properties 3
65
E-1, E-2, E-3
33
Circle and Polygon Properties 4
67
E-1, E-2, E-3
34
Circle and Polygon Properties 5
69
E-1, E-2, E-3
35
Circle Properties
36
Polygon Properties
37
Wages ( Hourly)
38
Wages ( Commission and Net Income)
39
Property Tax
41
D-1
43
D-1
45
D-2
47
49
D-3
51
D-4
D-5
53
55
D-6
57
D-7
59
E-1, E-2, E-3
71
73
E-1, E-2, E-3
75
79
F-1
77
F-1
F-1
Cumulative Exercises
Senior 3 Pre- Calculus Mathematics
Exercise
40
New Topic Presented
Outcome(s)
Unit Prices, Exchange Rates, and Reconciliation of
Bank Statements
81
F-1, F-2
41
Budgeting 1
85
F-3
42
Budgeting 2
91
F-3
43
Exponential Growth
44
Interest
45
Inductive and Deductive Reasoning
46
Rev i ew 4
47
AND, OR, NOT, and Venn Diagrams
48
C ounterexamp l es
49
Converses, Contrapositives, If. ..Then...
50
Direct and Indirect Reasoning
51
Operations and Compositions of Functions
52
Inverse Functions
53
Factor Theorem and Remainder Theorem
54
Graphs of Polynomial and Rational Functions
55
Review 5
127
56
Review 6
128
57
Review 7
130
58
Cumulative Review
95
F-5
98
F-5
100
G-1
102
104
G-2
107
G-3
G-4
109
112
G-5
116
118
H-1
H-2
122
H-3
124
H-4
131
ix
Cumulative Exercises
Senior 3 Pre - Calculus Mathematics
N otes
x
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
A-1, A-2
Graph the following two sets of data on the same coordinate system. Join each
set of points with a smooth curve.
a.
x
-2
y
b.
x
y
-3
10
-1
-1
0
0
1
-1
2
-2
5
0
1
2
5
3
10
16
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 represents
one 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
d.
12c4d'e
-9ed"
e. 7 -2
f. -3 "2
Continued
1
Cumulative Exercises
Senior 3 Pre- Calculus Mathematics
A-1, A-2
7. Evaluate the following expressions if x = 2 and y = -3.
c.
b. 7x - 2y2 - (3x)°
a. 5x - 3y + y2
18x2
7y
8. Factor the following expressions:
c. 6x2 - 7x - 20
b. x2+5x+4
a. x2+5x
9. Rewrite the expression 5x + 3y = 4 to express x in terms of Y.
10. Find the area of the shaded region.
4 cm
6 em.
2 cm
3 cm
5 cm
4 cm
11. You have been assigned the job of measuring
the height of the local water tower.
Unfortunately, climbing makes you dizzy, so
you decide to do the whole job from ground
level. From a point 47.3 metres from the
base of the tower, you find that you must
look up at an angle of 53° to see the top of
the tower. How high is the tower?
12. A local survey asked 100 subjects for their opinions on a zoning ordinance. Of the
62 favorable responses, there were 40 males. Of the 38 unfavorable responses,
there were 15 males. Find the probability of randomly selecting one of these
subjects and getting a male response.
Continued
2
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
A-1, A-2
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
d. f(x)=3x-- 1
X
e,
y
x
5-T
5V
.+1 I , 1 1 i.....1...^." 10-X
1
91
i
ll.. i._.
1...
1
1
J( t1, x
-51
3
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
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
1
iii. Y = 2 x2
ii. Y = 2x2
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
c. y=x2-2
b. y=x2+3
3. a. Graph the following functions on the same coordinate grid:
i. y=x2
ii. y=(x+3)2
iv y=(x+1)2
iii. y=(x-2)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 the
level 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. What
mark must he earn on his fourth math test so that his average for the four tests
will be 80%?
Continued
4
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
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 5
men jump?
11. Jim and Kim each have money to buy ice-cream cones. Unfortunately, Jim is 240
short of the price of a cone and Kim is 20 short. They decide to pool their money
and 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 a
point on level ground, they measure an angle of elevation of 21°. They move
closer 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 equation
in 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-_ 2
2-3,2
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
Cumulative Exercises
Senior 3 Pre -Calculus Mathematics
A-2, A-3
Exercise 3 : Graphs of Quadratic . Functions 2
. a. Graph the following functions on the same coordinate system:
iii. Y = 2x2
ii. y = -x2
i. Y = x2
iv. Y _ -2x2
b. What i s 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 following
functions. 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
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
A-2, A-3
Exercise 3: Graphs of Quadratic Functions 2
6. Using trigonometric ratios, find the length of side x in each triangle below.
C.
f.
7. Express x in terms ofy in the expression 3x + y = 2.
8. You must order a new rope for the school's flagpole. You observe that the pole
casts a shadow 11.6 m long on the ground. The angle of elevation of the sun is
36°. 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
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
A-2, A-3
Exercise 3: Graphs of Quadratic Functions 2
.
11. Simpli fy the expression
x2
+3x + 2 x2 +x-6
x2+4x+3
x2-4
12. Find the value of both a and L A in
the 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.
0
2
b.
-----^ 0
5
°- -----a
-2
7
-3
10
0
6
C.
d.
e.
a
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
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
Continued
9
Cumulative Exercises
Senior 3 Pre- Calculus Mathematics
A-3
Exercise 4: Transformations of Quadratic Functions 1
5. Commercial airliners fly at an altitude of about 10 km. They start descending
toward the airport when they are still quite far away, so that they will not have
to 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 the
plane's path make with the horizontal?
6. Simplify the following expressions:
a.
2
_,2
v3
b. 3x-4y+2x-(3x+7y)
7. An oil well is to be located on a hillside that slopes at 10° below horizontal. The
desired rock formation has a dip of 27° to the hillside. The well is located 1200
metres downhill from the nearest edge of the outcropping rock formation. How
deep 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 the
larger and twice the smaller is 37.
Continued
10
Senior 3 Pre-Calculus Mathematics
Curnuiative Exercises
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 of
axis of
symmetry
Domain
Range
Direction of
opening
Maximum
or min i mum
y-values
10. Find the equation of the quadratic function, given that its vertex is (1, -2) and a
point on the graph is (-2, 16).
11. A labour study involves a sample of 12 mining companies, 18 construction
companies, 10 manufacturing companies, and 3 wholesale companies. If a
company is selected randomly from this sample group, find the probability of
getting a construction company.
11
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
A-3
Exercise 5 : Transformations of Quadratic Functions 2
1. a. What value of k makes the expression a perfect square trinomial? Write each
expression 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:
y= x2-4x-60
a. y=x2+6x-7
b.
d. y=3x2+24x+21
e. y=x2+5x+6
g. y=2x2+5x+2
h. y=-3x2+2x+1
c. y=2x2+8x-10
f. y=x2--3x-4
3. Factor each of the following expressions:
b. 25ab - 10ab2
a. 4x2 - 16y2
4. Using trigonometric ratios, solve each of the triangles given below. (Round your
a.
C.
b.
A.
14
5. Rearrange the equation y + 4x = 5 to express x in terms ofy.
6. Solve for x:
a. 2 3x-
b. x v 12 =
27 + 2 108
7. A surveyor measures the three sides of a triangular field and gets 114, 165, and
257 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 years
from now, the sum of their ages will be 81. How old is each of the three people?
12
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
A-4
Exercise 6: Transformations of Quadratic Functions 3
1. A farmer wishes to build a rectangular pen along one side of his barn. If he has
80 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 a
minimum.
3. A projectile is shot straight up from a height of 6 m with an initial velocity of
80 m/s. Its height in metres above the ground after t seconds is given by the
equation h = 6 + 80t - 5t2. After how many seconds does the projectile reach its
maximum height, and what is this height?
4. A survey found that 400 people will attend a theatre when the admission price is
80 cents. The attendance decreases by 40 people for each 10 cents added to the
price. 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.
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 the
dangers 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
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
A-4
Exerci se 6: Transformations of Quadratic Functions 3
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
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
A-4
Exercise 7: Applications of Quadratic Functions
A rectangular field is to be enclosed by a fence and divided into three smaller
plots by two fences parallel to one of the sides. Find the dimensions of the
largest 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 be
shown that the distanced in metres of the ball from the release point in time t
seconds is given by d = -5t2 + 20t. Determine the maximum height attained by
the 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 300
oranges per tree. It is estimated that for each additional tree planted per
hectare, the average yield per tree will be reduced by 10 oranges. How many
trees per hectare will produce the largest yield?
4. A dealer finds that he can sell 800 radio sets at \$60.00 per set. However, for
every \$2.00 drop in price he can sell 50 sets more. At what price per set should
he 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 answers
to one decimal place.)
7. Express x in terms of y for the expression 2x + 3y - 1 = 0.
Continued
15
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
A-4
Exercise 7: Applications of Quadratic Functions
8. When surveyors measure land that slopes significantly, the distance that is
measured will be longer than the horizontal distance drawn on the map. Suppose
that the distance from the top edge of the Cibolo Creek bed to the edge of the
water 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 the
creek?
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
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
A-4
Exercise 7: Applications of Quadratic Functions
12. Match each equation to its graph.
1. y-3x2
1x 2
2. y=- 1
3. y = 3(x + 2)2
4. y+1=3(x+2)2
5. y
6. y+1=-3(x-2)2
7. y=-3(x-2)2
8. y-1=-1(x-2)2
a.
= 3 (x + 2)2
b.
C.
V
d.
e.
g.
h.
X
5
-5
V
17
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
B-1
Exercise 8: Trigonometric Equations I
1. Determine the exact value of sin 8, cos 0, and tan 0 for the angle whose terminal
arm passes through the given points:
a. P(5, 3)
c. Q(8, -2)
b. R(-3, 4)
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 the
function y = sin x.
b. Sketch the function on a cartesian
plane . (Hint: The points are joined
by a smooth curve.)
x
y
0°
45°
90°
135°
180°
225°
270°
315°
360°
c. Extend the graph to -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 missile
at 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 of
elevation be 15 seconds after the first sighting?
8. Factor the following expressions:
a. 5x2-20
b. x4-81
Continued
18
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
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
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
2
+4x+10
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
2x_ (2x-4)=2- 10(x-5)
15. A study of consumer smoking habits includes 200 married people (54 of whom
smoke), 100 divorced people (38 of whom smoke), and 50 adults who never
married (11 of whom smoke) (based on data from the U.S. Department of Health
and Human Services). If one subject is selected randomly from this sample, find
the probability of getting someone who
a, is divorced
b. smokes
19
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
B-i
Exercise 9: Trigonometric Equations 2
Determine the solution for each of the following trigonometric equations in the
interval 0°<- 0S 360°.
a. cos 0 = c.
b. sin O+1=0
3
d. 2 cos O= 2
tan 0 - 2 = 5
tan 0 5
e. -3 sin 0 = 2
f
g.3cos0--2=0
h. 5tanO+4=0
i
tan010
6
2
2cos 0+1=-
k.4tanO-7=5tan0-6
2. Imagine you are the pilot of a commercial airliner. You find it necessary to detour
around a group of thundershowers. You turn at an angle of 21° to your original
path, 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
3. If417
+417
+417
+417
= 4X, find the value of x.
Continued
20
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
B-1
Exercise 9: Trigonometric Equations 2
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.
7 + 348.
12 - 5V-25. Simplify the expression: 2V-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
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
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
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 trigonometric
equations.
a. sin 0 = 0
b. tan O=-
c.
d. 3 tan 8 - 7 = 0
1 = -2
cos 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 The
line 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 horizontal
line AE. A circle with its centre at B has a radius of 9 cm. The circle cuts AE at
points C and D. Calculate the length of chord CD.
7. For the parabola with equation y = -5x2 - 20x + 60, find the
a. coordinates of the vertex
b. equation of the axis of symmetry
c. coordinates of the x-intercepts
d. domain and range
Continued
22
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
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 the
smaller 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 are
placed 8 m apart?
11. Determine the value oft given points A, B, C, and D and the fact that AB is
perpendicular to CD.
B(6, 5)
A(2, 3)
C(6, -1)
D(5, t)
12. The length of one side of a square is increased by 10% and the other, side is
decreased by 10%. How does the area of the rectangle that is formed compare
with 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.
15. Describe each solution to the inequality, using interval notation.
a.
0
10
b.
-8
C.
-7
-5
-6
a
0
3
7
23
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
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). Notice
that these triangles differ only in the length of b.
For each of these triangles, draw an accurate diagram according to the following
instructions.
For each diagram, draw side a as the base so that it is 4 cm long. Then construct
L 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 the
two possible values of c. Your answer should be within + 0.1 can of the
calculated values found in Question la above.
b. Use a compass to mark off b = 5 cm, as in Question Ic. Measure the value of c
and confirm that it agrees with the calculated value. Now extend segment AB
beyond 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 of
c 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. Show
that this misses the other side of angle B, and thus no such triangle exists.
Continued
24
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
B-2
Exercise 11: Ambiguous Case Problems
4. Graph the following equation: x = y2 + 2y - 1.
5. Factor the following expressions completely.
b. (x - y)2
a. 15x2 - 7x - 36
6. Solve for x in the expression 5x2
-3x
9(2x + y)2
= 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 Harry
takes 60 minutes. How long will it take to cut the lawn if all three work
together?
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 is
the maximum profit the dealership can make? How many cars would the
dealership then sell?
11. Graph the function y = -sin x over the interval [0°, 360°].
I
25
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
Exercise 12: Review 1
The height, h, in metres, after the launching of a rocket at any time, t, in
seconds, is defined by the equation below. Find the maximum height reached by
the rocket and the time it takes to reach this height.
h=-3t2+9t+81
4
2. At a local beach, the lifeguard has 620 m of marker buoys to rope off a safe
swimming area. Calculate the dimensions of the rectangular swimming area to
create 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 be
1500 apples per year. For each additional tree planted in the orchard, the annual
yield per tree drops by 20 apples. How many trees should be planted in order to
produce a maximum yield?
4. The difference between two numbers is 14, Find the two numbers so that their
product is a minimum.
5. Honest John's used car lot sells an average of 20 cars per week at an average
price 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 the
dealer's (Honest John's) cost per car is \$4000, at what price should he sell the
cars 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
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
C-1, B-1
1. Factor each of the following expressions completely:
a. 3x2+7x+2
b. x2-9
c. 25x2 - 100
d. 2x2 - 16x + 32
e. sine 0 -
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
d. 2x2+3x--2 =0
e. -x2-2x+3 =0
c. x2---x--20=0
4. Rearrange each of the following equations and solve by factoring. Check your
roots.
a. 10x2=7x+12
b. 5x2+21x=54
d. x2+9x-21 =0
2
2
e. x2+9=0
c. 3x(x-2)-x(x+1)+5=0
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°.
a. cos0+1=2
b. (2 sin 0 - 1)(sin 0 + 1) = 0
c. (tan 0-2)(2 sin 0+ 1)=O
d. 4 cos' 0- 1 = 0
e. 2 sine 0+ 7 sin 0- 4= 0
f. 3 sin 0 tan 0+ 2 tan 0= 0
g. sin20+2sin0+1=0
Continued
27
Senior 3 Pre- Calculus Mathematics
Cumulative Exercises
C-1, B-1
7. The children's slide at the park is 30 feet long and inclines 36° from the
horizontal. 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 and
that the bottom end of the slide is at the same level as the bottom end of the
ladder. (Hint: Draw a diagram.) Round off to one decimal place.
8. Simplify: 2;132 - 3--V'18 + 5v'509, 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 is
17 m long. Find the length of each leg of the triangle.
12. Two (square) checkerboards together have an area of 169 square centimetres. One
has 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 of y = x2 + 4x for x over the interval [2, 4j.
28
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
C-1
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
h. tan2 6 = 3
g. -3 cost 0 + 2 cos 0 - 7 = 0
2. Solve these equations using the quadratic formula. Be sure to state the formula
before substituting values into it.
a. x2+2x-15=0
b. 2w2-3w+1=0
d. 1 = 5x2
e. x2 - 0.1x - O.06
c. 7w2-3w=0
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
ii. by factoring
b. Which method do you prefer and why?
Continued
29
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
C- 1
7. For the parabola defined by the equation y = 7x' + 70x + 63, find the
a. coordinates of the vertex
b. equation of the axis of symmetry
c. coordinates of the x-intercepts
d. domain and range
8. A piece of wire 60 cm long is bent into the shape of a triangle. Find the angles of
the 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 over
the 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, are
registered to be at the same altitude on a radar screen. The range finder
determines one plane to bear N60°E at 100 km from the radar site while the
other bears S50°E at 160 km from the radar site. How far apart are the planes
from each other?
11. Mary is 5 times as old as Bill. Last year, she was 6 times as old as he was. How
old will each be in 2 years?
12.
This pipe has an outside diameter of 14
cm and is 60 cm long. If the pipe is 2 cm
thick and is made of a material that
weighs 8 grams per cm3, how much does
the pipe weigh?
13. The price of a radio is \$50, and 40 are sold each day. For each \$1.00 the price is
raised, the store sells one less radio. If each radio costs \$18 to make, how much
should 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
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
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
d.9-x2=0
e.2x2_12x+10=0
c. x2+8x=-15
3. Find the zeroes of the quadratic function f (x) = x2 + 8x + 15 by
a. factoring
c. 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 =0
x
x
6. An observer 5.2 km from a launch pad
observes a missile ascending.
a. At a particular time, the angle of
elevation is 31°. How high is the
missile?
b. At this same time , how far is the
missile from the observer?
c.
What will the angle of elevation be
when the missile reaches a height
of 30 km?
5.2 km
='
Continued
31
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
C-1
Exercise 15: Solving Quadratic Equations by Graphing
7. Solve each of the following trigonometric equations finding all solutions on the
interval 0° < 0< 180° to one decimal place.
a. cos O =
2
3
b. 6 tang 0-19 tan 9 = -10
c.
1
2 sin 9
8. What is the value of m that would make each of the following equations a perfect
square?
a. y=x2 + 2x+m
b. y = x2-lOx+m
9. Ifj+
4 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 the
numbers.
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 selection
of birthdays. If the 366 different possible birthdays are written on separate
slips of paper and mixed in a bowl, find the probability of making one
selection and getting a birthday in May.
b. Using the same population of 366 different birthdays, find the probability of
making one selection that is the first day of a month.
15. In which quadrant(s) is cos 0 > 0?
32
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
C-2
Exercise 16 : Nature of Roots
If the discriminant of a quadratic equation has the given value, state the
characteristics 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 real
numbers) 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 each
equation.
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:
2 +4x+4=0
b. x21-x2-3=0
d. 6x2-x+2=0
e. 4x2-12x+9=0
a.
c.
2(x2-3)=4x
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 exactly
one root?
Continued
33
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
C-2
Exercise 16: Nature of Roots
8. State the nature of the roots for each of the following parabolas:
C.
9. Find all values of 0 in the following triangle.
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 many
of 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-cm
square 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. x 2 _2X = 24
C.
x2
=8
b. x4 -
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
d.
=
x
8
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 ...
1
2
b. L3=L 4because ...
If A ABC = A DEF, then
AC = DF because ...
and L A = Z D because . .
Continued
35
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
C-3, C-4
Exercise 17: Nonlinear Equations
7. Five centimetres are cut off (along one side of) a square sheet of paper and 8 cm
are added to an adjacent side. The resulting rectangular sheet of paper has a
perimeter of 98 cm.
a. What was the area of the original square?
b. What i s 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 >-1 01
c.
yl y <4}
y = 2 x. Calculate the related angle if y = 2
12.Graphtefuncio x is the terminal
arm of 0.
36
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
C-5
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 your
solutions.
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.
x -^-2
3x+2 -- 34-
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 only
6.7 m long. If you place the ladder's tip on the branch, what angle will the ladder
make with the ground?
5. Given that A XOP - A XYZ in the diagram below, find the length of
a. XP
b. OY
Continued
37
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
C-5
6. Simplify the following expression:
2y 2
E
1
1
-y4
V
7. Without making a table of values, sketch the graph of each of the following
functions. 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 number
of widgits in each box is four less than the number of boxes in each crate. Find
the number of widgits in each box if a full crate contains 60 widgits.
38
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
C-5
Exercise 19: Rational/Absolute Value Equations
Solve each of the following rational equations . Check your solutions.
a. x=
2x-9 + x
5
-b.
x-7 2 x-7
-2
x-3
d.
C.
x-4= 1
x
e.
2x + 1 + 3x+9 = 0
X-3 2x+3 2x2.3x-9
3x` - 2 = 2x + 1
3x+1
3x+1
2+12
7x
x-3
x-3
2. a. Solve the following absolute value equations.
i.
3xI =12
ii. 12x1-1=17
v.
iv. I x2+4x_121=0
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:
a. coordinates of the vertex
b. equation of the axis of symmetry
c. coordinates of the x-intercepts
d. domain and range
4. Solve each of the following trigonometric equations on the interval
00 <- 0 ^ 360°. (Round your answers to one decimal place.)
a. 3tanO-1=5
b. cos' 0-2cos0=2
Continued
39
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
C-5
Exercise 1 9: Rational/Absolute Value Equations
5. A cardboard box without a top is constructed by cutting squares out of the
corners 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 home
on the same highway at 60 km/h. If the round trip took 2 hours, how far does the
Chin family live from the lake?
9. Machinist X can do a job in 15 hours, Machinist Y can do the same job in 20
hours, and Machinist Z can do the job in 12 hours . If all three work together on
the same job , how long will it take them to get the job done?
40
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
Exercise 20: Review 2
1. Sketch the function y = 2x2 - 8x - 10 and state the following:
a. vertex
b. axis of symmetry
c. min. or max. y-value
d. 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.
C.
x
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 one
triangle, 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
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
Exercise 20: Review 2
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:
b.
a. 2x=3 5x+6-6
c.
213x+11=6
e.
3
x
2x-6 x2 --6x+9
22x._-.22 + V 3x = 5
d. 1 4xl=-3
x-2
3x-9
1
2x
2-x
9
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 the
river, making angles of 60° and 74° from A and B, respectively. Find the width of
the river to the nearest metre.
10. From point T, a golfer aims a ball towards a hole at H that is 100 m away. But
the ball is actually sliced in a direction 30° off course and lands at M, 60 m away
from T. If the next shot from point M is hit 50 m directly at the hole, will the ball
go 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 smaller
plots by two fences parallel to one of the sides. Find the dimensions of the
largest 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 the
small 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
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
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 of
the 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 the
intersection, 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
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
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 is
situated at point (5, -4). If the lighthouse can be seen anywhere within a radius
of 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 the
roads at speeds of 55 km/hr and 65 kmlhr, respectively. If the angle of
intersection 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 the
interval 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
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
D-1
Exercise 22 : Distances between Points and Lines
13. Triangles are proved congruent by the properties known as SSS, SAS, AAS, and
ASA. For each of the following pairs of triangles, state the reason why the
triangles 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 a
rectangular 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 four
rectangular pens of equal area, will the layout of the pens affect the
maximum 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 an
equation for this circle.
46
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
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?
3. Show that the quadrilateral with vertices A(-5, -2), B(1, -1), C(4, 4), and
D(-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 your
rationale.
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 procedures
used.
6. Solve each of the following trigonometric equations, finding all solutions on the
interval 0°< 0 S 360°.
a.
3 sin 0
4
1
b.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 - 5
with domain R.
10. Calculate the distance from the point (0, 4) to the line 2x = y + 3.
Continued
47
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
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 segment
between 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.
y
d.
y
I
a
E I
F
...Y
f
t
1
E
T
W
16. Given the following graph, which function best describes it?
a. y=cosx
b. y=sinx-1
c. y=-sinx
d. y =cosx - 1
48
Y
30 x
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
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 = 8
and 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
a. x+2y=10
2x--y=0
b. y-2x=l
2y-4x=4
c. 2x =y +2
y=x-1
3. Solve the system given by the equations Zx + y = 5 and x - 3y = 6 using the
method of substitution.
4. For the following systems of equations, decide whether to substitute an
expression for x or for y, then solve.
a. 2x+3y=--4
y-2x=4
b. 3y=x+11
x =y -5
5. For each system of equations below, decide which variable is more readily
eliminated, then solve the system.
a. x+y=4
x-2y=1
b. 3x-2y=4
x-2y=4
6. Solve the following systems of equations using the addition-subtraction method.
a. 3x+2y=4
x-y=3
b. 2x+3y=48
3x+2y=42
7. You now have a number of methods for solving systems of equations. Before you
solve the following systems, decide which method would be the most appropriate,
a. 2x+y=3
3x+2y=6
b. x-3y+7=0
3x-2y=-7
c. 2a-3b-13=0
3a-b-9=0
8. A field has the shape of a quadrilateral that is not a rectangle. Three sides
measure 50, 60, and 70 metres, and two angles measure 127° and 132°, as
indicated in the diagram on the following page.
Continued
49
Cumulative Exercises
Senior 3 Pre -Calculus Mathematics
D-3
Exercise 24: Systems of Linear Equations in Two Variables
a. By dividing the quadrilateral into two triangles, find its area. You may have
to find some intermediate sides and angles first.
b. Find the length of the fourth side.
c. Find the measures of the other two angles.
10. If BD = CD, prove that
4 ABC is isosceles.
9. Find the length of AB in the
diagram below.
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 are
the 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 the
nearest hundredth.)
5o
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
D-4
Exercise 25 : Systems of Linear Equations in Three Variables
1. Solve the following systems of equations:
a. 3x-4y+5z=2
4x+5y---3z =-5
5x-3y+2z=-11
b. x+y+3z=12
2x+y+3z= 13
x-y+4z=11
c. 4x+3y-z=-7
3x-2y+3z=-2
x+y-z=-2
d. 2x+3y=13
3x-y=3
e. 3x=6y-7
5x=-9y-18
f. 2x-2y=6
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 book
price 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 of
depression of 12° to a distant ship. How many metres is the ship from the base of
the lighthouse?
6. Explain why each of the following is true:
a. L 1 = L 2 because ...
4
b. L1=L 3because...
c_ L2=L 4because...
Continued
51
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
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 crosses
the x-axis.
10. Find the product (3J - 2J)(5
11. Solve for x:
- 3 ).
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. Find
the equation of the large circle.
52
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
D-5
Exercise 26 : Systems of Nonlinear Equations
Find the solution to the following system of equations: y = x2
y=8
x2
a. graphically
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:
b. X2 + 3y2 = 30
a. x2 + y2 =25
ry
x2 -y2 = 1
`2x2 +y2
255
5. For the parabola defined by the equation y = x2 + 6x, find the
a. coordinates of the vertex
b. equation of the axis of symmetry
c. coordinates of the x-intercepts
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 they
are 20 m apart they see a shark directly below them. If the angle of depression
from the first diver to the shark is 47° and the angle of depression from the
second 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 trip
the speed was increased by 100 km/h. If the total trip took 13 hours, what was
the speed of the plane on the first leg of the trip (from Halifax to Vancouver)?
Continued
53
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
D-5
Exercise 26 : Systems of Nonlinear Equations
10. Prove that the quadrilateral defined by the lines l1: 4y = 3x - 6, 12: 4x + 3y = 33
l3: 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) [
13. Solve: rx^
\
2)
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
54
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
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 smaller
equals 459. Find the numbers.
Continued
55
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
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 relationship
between 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 :
15. Solve the following system .:
56
(-v"2 + 3 V3- )2.
x2 -x-2
y=
y=x-3
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
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
2. Find the solution for the following. (Leave answer in set notation and interval
notation.)
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 figure
below. You walk around the block, which takes 125 paces on Market Street and
102 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
x + 1)
Continued
57
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
D-7
Exercise 28: Quadratic, Absolute Value , and Rational Inequalities
7. Solve the inequality 2 X+1
X -x-2
8. Solve the inequality x2 4
0.
9. a. What is the first step in solving the inequality 2x + 5 < x + 17
x-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
13. Factor each of the following expressions:
a. 5x2 - 20
b. 5x2- 5y2
14. Both of these circles have centres labelled O. Using a protractor, measure L ABC
and L AOC for each circle. Determine the ratio of the measure of L AOC to the
measure of L ABC for each circle.
15. A plane took 4 hours to fly 1920 km when it had a tail wind. Flying back against
the same wind, and travelling at the same air speed, the plane took 1 hour
longer. Find the speed of the wind and the plane's air speed.
16. Graph the function y = sin (x + 90°).
58
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
Exercise 29 : Review 3
1. Sketch the function y = --3x2 - 12x + 7 and state the following:
a.
b.
c.
d.
e.
vertex
axis of symmetry
min. or max. y-value
direction of opening
domain
f. range
g. wide/narrow opening
h. zeros
2. Find the values of 0 between 0° and 360° given
3/2
a. sin 0=4b.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
2y=x2-10
b. 4x=2-2y
-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
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
Exercise 29: Review 3
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 he
sell to get the maximum cash return?
10. The sum of three numbers is 18. The third number is five times the sum of the
first 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 for
the numbers.
11.
y+2 < 5
12. -31x +4>10
13. (x + 3)(x - 4) > 0
(x2.25)
14.
60
2x2 -5x-3 <0
3x2 +5x- 2
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
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?
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 the
length of
a. AB
b. AC
c. the radius of the circle
d. 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. OD
b. CE
c. OE
5. This circle has a centre at 0, OP I AB,
AC = 16, and AP = OP. Find the length of
a.
b.
c.
d.
OP
OC
AB
AP
Continued
61
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
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 to
construct 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 and
b+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+1
3
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 of
the 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
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
E-1, E-2, E-3
Exercise 31 : Circle and Polygon Properties 2
Construct a circle with its centre at O. Draw a diameter AB. Plot a point C
anywhere 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. AB
b. 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
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
E-1, E-2, E-3
Exercise 31 : Circle and Polygon Properties 2
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 the
altitude 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.
64
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
E-1, E-2, E-3
Exercise 32: Circle and Polygon Properties 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. What
did 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
A
a2
+
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
Continued
Cumulative Exercises
Senior 3 Pre- Calculus Mathematics
E-1, E-2, E-3
Exercise 32: Circle and Polygon Properties 3
7. Solve each of the following trigonometric equations, finding all solutions on the
interval 0°< 6 < 180°. Round solutions to two decimal places where necessary.
b. 5 tan 0 = -1
a. tan 8 = 0
c. sin 6 = 0.6493
8. Solve this equation for x: x = N 2x - 3 + 3.
9. Solve this system of equations: x + y = 4
y + z =-8
2x-z=15
10. A cube which measures 5 cm on each side is painted blue. The cube is cut into
1 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. The
company is trying to decide on a price for the ticket. The company is considering
a price of \$60/ticket if 1000 or less people were to purchase tickets. For every 100
tickets sold over 1000, the ticket would be discounted by \$3.00. What ticket price
will yield a maximum profit?
13. ABCD is a square with AB = BC = 4. E is the
midpoint of AD. Coordinate axes are drawn as
shown.
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.
b.
Y
Y
•
0
66
1/0
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
E-1, E-2, E-3
Exercise 33: Circle and Polygon Properties 4
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 of
arc BD is 108°, and L CBE = 32°, find
the measure of each of the following
angles: 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 that
the measure of the are FEB is 248°, find
the measures of angles 1, 2, and 3.
Continued
67
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
E-1, E-2, E-3
Exercise 33 : Circle and Polygon Properties 4
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 the
coordinates 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 BE
bisects L ABD (L 1 = L 2), verify
that BC =- CE.
13. The side of one square is 5 m longer than the side of another square. The area of
the square with the longer sides is 153 m2 greater than the area of the other
square. 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 tangent
to the circle.
68
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
E-1, E-2, E-3
Exercise 34: Circle and Polygon Properties 5
1.
Given that this circle has its centre at 0, AB is
tangent at A, and L 1 = 49°, find L 2.
2.
Given that this circle has its centre at 0, CD is
tangent 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 line
through A is tangent at B, determine the
distance AB.
4. Describe how you can construct a tangent to a circle at a given point on the circle
if you are given the centre of the circle.
5. Given that this circle has its centre at 0, AD is
tangent 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
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
E-1, E-2, E-3
Exercise 34: Circle and Polygon Properties 5
7. Given that AD and AC are tangents at B
and 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, find
the distances OP, OR, and RP.
9. Given that AB is tangent at A, chord
DA 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 of
60°. 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).
70
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
E-1, E-2, E-3
Exercise 35 : Circle Properties
LA= 68°
LD=2LB
Find the measure of
a. L C
b. L B
c. L D
2. Given: circle centre 0
LDCO=30°
LABO=20°
L BOC = 100°
Find the measure of
a. L ABC
b. L A
c. L D
3. Given: circle centre 0
A OQR is equilateral
L ORS = 35°
L PQO = 28°
Find the measure of
a. L ORQ
b. L PQR
c. L S
d. L P
4. Given: EB is tangent at B
AB is the diameter
L1_L2
Verify: EB = BD
Continued
71
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
E-1, E-2, E-3
Exercise 35 : Circle Properties
5. Given: circle center 0
QP 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 the
length of the other two sides.
1 x2 _ 2y = 0
7. Solve this system of equations for x and y:
x+2y=6
8. To approximate the distance between two points A and B on opposite sides of a
swamp, a surveyor selects a point C and measures it to be 140 m from A and
260 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 used
in solving:
a. 3x2-10x+3=0
c.2s2=8s-7
b. 10a2--21a+9=0
d.5x216=0
10. Find the value(s) of p 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 that
randomly select telephone numbers. Assume that a computer generates
randomly the last digit of a telephone number. Find the probability that the last
digit is an 8.
15. Graph the function y = cos(x - 90°) on 0°, 360°].
72
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
E-1, E-2, E-3
Exercise 36: Polygon Properties
1. A polygon has 10 sides. What is the sum of the interior angles?
2. A polygon has 14 sides. What is the sum of the interior angles?
3. A polygon has 102 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 the
polygon have?
6. The sum of the interior angles of a polygon is 4500°. How many sides does the
polygon have?
7. The sum of the interior angles of a polygon is S. In terms of S, how many sides
does the polygon have?
8. Given AB is tangent to a circle at X, and
CD contains the centre of the circle, draw
the 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-2y20
9
3
12. Solve each of the following trigonometric equations on the interval 0° <- O<_ 180°.
(Round answers to two decimal places.)
a.
°- 0 -==0
2
b. 2+3 sin 8=4
c. 2 tan 0-2 = 5 tan 0
13. Determine the roots of 10x2 -9x = -2.
Continued
73
Senior 3 Pre -Calculus Mathematics
Cumulative Exercises
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 the
remainder at 10% per annum. After 1 year, the total interest from these
investments was \$24 500. How much did Joe invest at each rate?
17. Solve: 1+X
1--x
X -I
x+1
18. The equation of the large circle is
(x -- 8)2 + y2 = 64. If the small circles
have !equal radii, find the equations
that represent them.
74
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
F-i
Exercise 37: Wages (Hourly)
Compute the weekly gross earnings for these employees. Time and a half is paid
for all hours over 40 in a given week.
Employee Name
Hourly Rate
a.
\$10.20
b.
S. Kashin
\$12.15
C.
P. Dyck
\$20.00
Mon.
8
Tues.
Wed.
Thurs.
Fri.
9
10
7.5
7
8
7
8
12
9
11
10
8
12
9
2. Compute the weekly gross earnings for these employees. Time and a half is paid
for all hours over eight in a given day.
Employee Name
Hourly Rate
b.
J. Rees
S. L'Heureux
\$ 8.20
\$12.25
C.
P Bennett
\$18.00
a.
Mon.
9
8
10
Tues.
Wed.
8.5
9.5
8
7
Thurs.
10
Fri.
7.5
10
9
9
11
8.5
9.5
3. One particular week an assembly line worker worked 54 hours (40 hours were
regular hours, 6 hours were at time and a half, and the remainder was at double
time). 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 a
half for overtime. One week he worked 45 hours and made \$185 in tips . Find his
gross 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 in
Toronto. Suppose that at a certain time of day it casts a shadow 1100 in long on
the 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
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
F-I
Exercise 37: Wages (Hourly)
8. Given: L M = 75°
Mx =9 0°
-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:
10. Solve:
1
x=
\$x-3y= 6
Cx+12y= -244
2x
11. Solvefor each of the following trigonometric equations on the interval
0° < 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 m
away 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.
76
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
F-1
Exercise 38 : Wages (Commission and Net Income)
1. If a commission of 12% were paid on all sales, what commissions could be earned
on sales of
a. \$740.50?
\$654.38?
b. \$1345.99?
2. A salesperson receives 8% commission on the first \$1000 of sales, and 15% on all
sales 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 a
regular 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 earnings
for the week?
4. Fred's monthly salary is \$700. In addition, he receives 5% commission on the
first \$12 000 of his sales, and 7% commission rate on all sales over \$12 000. Last
month 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% commission
on 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 for
40 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 is
calculated on taxable incomes as follows:
Earnings
Tax Rate
\$0-\$550
\$551-\$1138
17%
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
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
F-1
Exercise 38: Wages (Commission and Net Income)
7. Employee A can complete an assignment in 10 hours and employee B can
complete this assignment in 8 hours. If employee B begins 3 hours after
employee A has started, find the total time needed for the two employees to do
the complete job together.
6x= 12-3y
8. Solve the linear system:
1
x=-5
9. Two runners start from the same point at 12:00 noon, one of them heads north at
6 km/hr and the other heads 68° east of north at 8 km/hr. What is the distance
between 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 poured
into the aquarium, what is the depth of the water?
13. Solve for x: 5x2 + 10x - 3 = 0.
14. Solve for x: 22 - 3 -- 7
X -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
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
F-1
Exercise 39: Property Tax
The Gysels own a home valued at \$90 000. The rate of assessment is 45%. The
mill rate was 62 mills and there was a local improvement tax of \$180.00 for
sidewalk 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 tax
assessor reassessed the house at \$90 000. Assuming a mill rate of 55 mills, find
the 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%. The
lot 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 if
the municipal mill rate is 70 mills.
4. The rate of property tax in mills for a municipality can be found using the
formula:
Mill Rate =
Total Tax to Be Raised
Total Assessed Value of Property
x 1000
Calculate the mill rate to the nearest whole mill for each of the years listed in a
Rural Municipality.
Year
a. 1994
b. 1995
c. 1996
Budget
Requirement
\$69 000 000
\$82 000 000
\$95 000 000
Assessed Value
of Real Property
\$780 000 000
\$852 000 000
\$945 000 000
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
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
F-1
Exercise 39: Property Tax
9. Solve and check: x + 1 < 1.
x-2
1x-y =8
10. Solve the system of equations:
11. Find all solutions for the following trigonometric equations on the interval
90°<_ 0 :!^ 2700,
a. cos 0 = 1
2
b. 3 sin 0 = - 2
c.
2 tan 0
=5
d. tan 2 0_9=0
12. In the figure, QR and QS are
tangent segments to the circle with
centre P. QP intersects the circle at
M. Prove that M is equidistant from
the tangent segments.
13. Solve for x: 3x2 + l Ox -- 7 = 0.
14. Find the distance between P(-2, 1) and the line 2x - 3y + 5 = 0. (Express your
15. Describe each solution to the inequality, using interval notation,
6
12
-5
2
b.
C.
80
8
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
F-1, F-2
Exercise 40: Unit Prices , Exchange Rates , and Reconciliation
of Bank Statements
1. A 355-ml can of soft drink costs 85o and a 1000-ml bottle costs \$1.89. Find the
cost 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, which
3. Find the unit cost of each of the following:
a. 780 g of type A costs \$14.65
b. 390 g of type B costs \$12.49
c. 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 you
b. If you decide to buy an article in Grand Forks with a price tag of \$28, how
much is it worth in Canadian dollars?
c. A hotel in North Dakota advertises daily rates of \$38. How much is that in
d. Can you find a simple method of converting American prices into approximate
an example.
5. You are planning a trip to the United States and estimate you will need \$200.00
U.S. The listed cost is \$1.00 Canadian equals 73¢ U.S. How much will it cost you
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 244
to Gas Depot for \$43.87; Sept. 20, Cheque 245 to Hydro for \$66.98; Sept. 25, a
deposit 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 department
store charge account for the period shown. Monthly credit charges are 1.4% of
the balance due.
MONTH
February
March
April
May
PREVIOUS
BALANCE
PAYMENT
PURCHASES
CHARGED
\$586.00
\$100.00
200.00
\$275.00
\$200.00
\$ 93.00
121.75
13.17
\$ 87.13
BALANCE
DUE
CREDIT
CHARGE
NEW
BALANCE
Continued
81
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
F-1, F-2
Exercise 40: Unit Prices , Exchange Rates, and Reconciliation
of Bank Statements
8. Complete a reconciliation statement form for this account.
ACCU CREDIT
BALANCE
DESCRIPTION
102
141
24
56
215 00
280 00
125 45
81 75
CHEQUES ISSUED TO OR
CHEQUE
No.
DESCRIPTION OF DEPOSIT
AMOUNT
191
Esso
192
Wires
27
193
Te lep h one
194
Hydro
Sept.
1
195
Pete 's Shac k
3
196
Insurance
7
197
Sears
7
198
G as
8
199
Lynn 's Cl ot hi ng
BALANCE
802 51
699 61
27 08
30 08
558
476
691
971
311 08
837 71
DEPOSIT
AMOUNT
49
91
91
91
BALANCE I''WD
CHEQUES/DEPS
350 00
CHQ - /DEP +
CHQ - /DEP +
452 51
802 51
102 90
BALANCE
699 61
CHQ - /DEP +
24 88
CHQ - /DEP +
141 12
558 49
24 88
BALANCE
533 61
56 70
CHQ - /DEP+
56 70
476 91
215 00
691 91
141 12
BALANCE
BALANCE
215 00
CHQ - /DEP +
BALANCE
280 00
CHQ - /DEP +
ALANCE
CHQ - /DEP +
125 45
BALANCE
211 11
CHQ - /DEP +
BALANCE
2400 00
Depos i t
6
Mo.
21 08
25 08
102 90
Deposit
30
DAY
350 00
BALANCE
Deposit
27
DATE
20 1 08
452 51
Deposit
25
27
90
12
88
70
CHEQUE
Aug.
21
25
CREDITS
452 51
Deposit
Cheque 191
Cheque 192
Cheque 193
Cheque 194
Deposit
Deposit
Cheque 195
Service Charge
DATE
DEBITS
F ORWARD
CHQ - /DEP +
BALANCE
854 00
CHQ - /DEP +
BALANCE
57 10
CHQ - /DEP +
BALANCE
146 58
CHQ - /DEP +
BALANCE
280 00
971 91
125 45
846 46
211 11
635 35
2000 00
2635 35
854 00
1781 35
57 10
1724 2
146 58
1577 7
Continued
82
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
F-I, F-2
Exercise 40: Unit Prices, Exchange Rates, and Reconciliation
of Bank Statements
STATEMENT OF RECONCILIATION
Bank Reconciliation
Balance from statement:
Subtotal:
Subtract:
Subtotal:
This should agree with the balance shown in your record book
after service charge is deducted:
9. A rectangular room , 5 m by 11 m, has an open -beam ceiling . The two parts of the
ceiling make angles of 65° and 32° with the horizontal. Find the total area of the
ceiling.
10. Solve: 14--2xI_8.
11.
Use the discriminant to determine the nature of the roots of 2x2 _X + 4 = 0.
Continued
83
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
F-1, F-2
Exercise 40 : Unit Prices, Exchange Rates , and Reconciliation
of Bank Statements
12. Given the diagram with circle centre G, tangent line AC, \$^ = 120°. Find the
measure 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 determined
by an are of 90°?
b. A chord has a measure of 15. If one are determined by the chord has a
measure 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
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
F-3
Exercise 41: Budgeting 1
Dean Charles earns a net weekly salary of \$645.25. The family receives a
monthly child tax benefit cheque that amounts to \$42.50 per child . There are
four 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 the
Charles family on the blank budget form on the following page.
Continued
85
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
F-3
Exercise 41 : Budgeting 1
1. Income
_..,
a. Regular Monthly Income
b. Spouse's Regular Monthly Income
_,,..
_
d. Other Income
#1
Total Monthly Income
2. Housing Expenses
a. Mortgage or Rent
b. Property Tax
c. Home/Property Insurance
d. Repairs/Maintenance
e. Other Housing Expenses
Total Housing Expenses
\$
\$
\$
\$
\$
#2 \$
3. Utilities
a. Hydro
b. Gas
c. Phone
d. Water
e. Other
Total Utilities
#3 \$.
4. Transportation
a. Public Transport
b. Car Loan
c. Car Fuel
d. Car Maintenance
e. Car Insurance
f. Other Transportation
Total Transportation
\$
\$
\$
\$
#4 \$
5. Personal Finances
a. 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 \$
6. Personal Expenses
a. Groceries
b. Clothing
c. Entertainment
e. Vacations
f. Other Personal Expenses
Total Personal Expenses
_____
\$
\$
\$
\$
\$
#6 .\$.
7. Other Expenses
a.
b.
C.
Total Other Expenses
#7 \$___
Total Monthly Expenses
#8 \$
.
Income minus Expenses (#1 - #8) #9 \$
* 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 be
saved 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 will
take you to achieve it.
Continued
86
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
F-3
Exercise 41 : Budgeting 1
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 family
receives a monthly child tax benefit cheque that amounts to \$107.72. The
family'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 purposes
at \$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 the
Wood family on the blank budget form on the following page.
Continued
87
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
F-3
Exercise 41 : Budgeting 1
1. Income
\$
a. Regular Monthly Income
b. Spouse's Regular Monthly Income \$
\$
\$
d. Other Income
#1 \$
Total Monthly Income
2. Housing Expenses
a. Mortgage or Rent
b. Property Tax
c. Home/Property Insurance
d. Repairs/Maintenance
e. Other Housing Expenses
Total Housing Expenses
3. Utilities
a. Hydro
b. Gas
c. Phone
d. Water
e. Other
Total Utilities
4. Transportation
a. Public Transport
b. Car Loan
c. Car Fuel
d. Car Maintenance
e. Car Insurance
I. Other Transportation
Total Transportation
\$
\$
\$
\$
#2 \$
\$
\$
#3 \$
\$
\$
S
\$
\$
\$
#4 \$
5. Personal Finances
a. Personal Loan
b. Investments
c. RRSP "
d, Life Insurance
e. Charities
t. Credit Card Payments
g. Service Charges
h. Savings **
i. Other Personal Finances
Total Personal Finances
\$
\$
#5 \$
6. Personal Expenses
a. Groceries
b. Clothing
c. Entertainment
e. Vacations
f. Other Personal Expenses
Total Personal Expenses
\$
S
S
\$
\$_
#6 \$
7. Other Expenses
a.
b.
C.
Total Other Expenses
\$
\$
\$
#7 \$
Total Monthly Expenses
#8 \$
Income minus Expenses (#1 - #8) #9\$
* 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 be
saved 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 will
take you to achieve it.
Continued
88
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
F-3
Exercise 41 : Budgeting 1
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.
5. Solve the system of equations:
6. Solve each of the following trigonometric equations over the interval 0°<_ 0 < 180°.
Round answers to two decimal places.
a.2tan 8=-2
3
b.3sin0--1=1
c. 3 tang + 2 tan 0 = 2
7. Given: Diameter AC
BC tangent at C
CD bisects L ACB.
Points A, D, and B lie
on 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
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
F-3
Exercise 41: Budgeting 1
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 on
the unpaid balance. In May, SecurCard charged Mable the annual fee and a
finance charge on her unpaid balance of \$324.00. Find the total of Mable's
monthly statement.
90
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
F-3
Exercise 42: Budgeting 2
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 a
monthly child tax benefit cheque that amounts to \$26.93 per child for each of
their 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,
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
p. newspapers and periodicals (per year) .................... ............................. 144.00
q, average monthly credit card payment .................................................200.00
r.
s. baby-sitting (average per month) .........................................................200.00
Using the information provided, prepare an estimated monthly budget for the
Elsimatesky family on the blank budget form on the following page.
Continued
91
Cumulative Exercises
Senior 3 Pre -Calculus Mathematics
F-3
Exercise 42: Budgeting 2
1. income
a. Regular Monthly Income
b. Spouse's Regular Monthly Income
d, Other Income
#1
Total Monthly income
\$
\$
\$
\$
\$
2. Housing Expenses
a. Mortgage or Rent
b. Property Tax
c. Home/Property Insurance
d. Repairs/Maintenance
e. Other Housing Expenses
Total Housing Expenses
\$
\$
#2 \$.
3. Utilities
a. Hydro
b. Gas
c. Phone
d. Water
e. Other
Total Utilities
\$
\$
\$
\$
\$
#3 \$
4. Transportation
a. Public Transport
b. Car Loan
c. Car Fuel
d. Car Maintenance
e. Car Insurance
t. Other Transportation
Total Transportation
\$
\$
\$
\$
\$
\$
\$
\$
\$
#4 \$ ,
5. Personal Finances
a. 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 \$
6. Personal Expenses
a. Groceries
b. Clothing
c. Entertainment
e. Vacations
t. Other Personal Expenses
Total Personal Expenses
#6 \$.
7. Other Expenses
a.
b.
C.
Total Other Expenses
#7 \$
Total Monthly Expenses
#8 \$
Income minus Expenses (#1 - #8)
#9 \$
\$
\$
" 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 be
saved 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 will
take you to achieve it.
Continued
92
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
F-3
Exercise 42: Budgeting 2
2. A submarine at the surface of the ocean makes an emergency dive. Its path
makes an angle of 21° with the surface.
a. If it goes for 300 meters along its downward path, how deep will it go? What
horizontal distance is it from its starting point?
b. How many metres must it go along i ts downward path to reach a depth of
1000 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 C
AB\\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 the
largest 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 and
5x+2y+8=0?
Continued
93
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
F-3
Exercise 42: Budgeting 2
10. Mrs. Murray wants to sell a particular bolt costing \$0.25 each together with
another type of bolt costing \$0.40 each. She plans on charging \$3.10 for the
mixture. 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+1
x-1
x+1
12. Describe each solution to the inequality, using interval notation.
0
a.
b.
s-----
-30
70
--6
4
E
0
.9
C.
-8
0
6
13. Graph the function y = -cos 0 + 2 on 0 E [0°, 360°].
94
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
F-5
Exercise 43: Exponential Growth
. The growth of the value of a RRSP is as shown in the table.
Time (years)
0
1
2
3
4
5
6
Value (\$)
5000
5400
5832
6299
6802
7347
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. Make
a table showing the value of the investment over the 5 years. Plot the data and
estimate 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 of
that 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 over
the next 10 years at 4% per year. What is the expected population at the end of
that time?
5. Find the vertex, the axis of symmetry, x-intercepts, domain , and range of
y=2xZ-11x+5.
6. A Canadian dollar is worth 72iZ U.S. A stereo sells for \$750 in Minneapolis. What
is its value in Canadian funds?
Continued
95
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
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 is
40 cm and the chord is 24 cm, how far is the chord from the center of the circle?
4-4=3
x3
10. Solve and check: x+2
11. Determine the solution for each of the following trigonometric equations. (Round
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 product
4
is
3
13. Using analytic geometry, prove that the diagonals of a parallelogram bisect each
other.
Continued
96
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
F-5
Exercise 43 : Exponential Growth
14. Given: AB = 70
E-50, =80
ED = DC
LCAB=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
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
F-5
Exercise 44 : Interest
1. a. If \$6000 was invested for 3 years at 7% simple interest, to what amount will
it grow?
b. If \$3000 was invested for 6 years at 4% simple interest, how much interest is
generated?
c. If \$10 000 was invested for 6 months at 9% simple interest, how much
interest 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 \$1750
deposit?
g. What principal will generate \$324 interest at 3% simple interest after 9
years?
2. If \$6000 was invested for 3 years at 6%, what will be the value of the investment
after 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 is
9%, 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 a
value of \$8800. What rate of interest did she receive?
5. Determine the effective rate on a loan of \$1000 at 10% per year compounded
semi-annually.
6. Determine the effective rate on a loan of \$2000 at 12% per year compounded
quarterly.
7. Mr. Smith invested in a 1-year term deposit paying interest at the rate of 4% per
annum. How much did he invest if he earned \$750 interest during the year?
Continued
98
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
F-S
Exercise 44 : Interest
8. A bank offers an interest rate of 6% per year, compounded annually. A second
bank offers an interest rate of 6% per year, compounded quarterly. If \$5000 were
deposited for 12 years, in each bank, how much more income would be gained in
the 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 of
y=-6x2+7x+5.
11. The midpoint of EF is (5, 1) and one endpoint is given by E (-1, 0). Find the
coordinates of F.
12. Solve: x3- 2x2 - 15x > 0.
13. Solve : ,10y + 16 = 3y.
14. Given: circle with centre F
chord AC
AB is tangent at A.
BC is tangent at C.
ZABE=15°
a. Find the measures of all the
numbered 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) and
150 scientific calculators (nine of which are defective). If one calculator is
selected randomly from this sample, find the probability that it is defective.
99
Cumulative Exercises
Senior 3 Pre- Calculus Mathematics
G-1
Exercise 45: Inductive and Deductive Reasoning
1. Which of the following examples of reasoning are inductive and which are
deductive?
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 Ms
Taylor, her math teacher, arrived at 7:30 each day for several weeks. Susan
said, "Ms Taylor always arrives at 7:30."
c. All students in senior high must enroll in physical education. John is a
student, so he concludes that he will take physical education.
d. The sun has risen each morning from time immemorial. We can be certain it
will rise tomorrow.
e. Anyone who likes to play football likes to play basketball. Sheeva likes to
play football. We conclude that she likes to play basketball.
f.
Triangle ABC is an equilateral
triangle . We can conlude that
AB = AC.
A
B`
IC
g. Joe counted the number of cars of different colours that passed his house in
15 minutes. More than half the cars were white. He decided that white is the
most 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 line
joining the midpoints of any two sides is parallel to the third side.
4. Find the zeroes of
x - 6 -2x-3= 0.
x+2
3x+4
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
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
G-1
Exercise 45: Inductive and Deductive Reasoning
7. Given: AB is tangent at B
AC is tangent at C
D is the midpoint of tangent BC
Verify: L1=Z2
8. Describe each solution to the inequality, using interval notation.
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 the
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 1
year?
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 following
a. x2-2x+2=0
b. x2-6x=17
14. A credit card company charges a daily finance charge of 0.0722% on all cash
advances. How much would your finance charge be if you borrowed \$200.00 for
60 days using the cash advance?
101
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
Exercise 46 : Review 4
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 ground
level. 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
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
Exercise 46: Review 4
8. Given a circle with its centre at 0, diameter AB, and CD = 50°, find the following
angle measures.
L1=
L2=
L3
L4
L5
9. If \$6500 is put into an account that earns 6% per annum, compounded quarterly
for 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 the
circle's centre, find
a. OD =
b. AB
c. ACB =
103
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
G-2
Exercise 4 7: 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 18
wear 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 club
secretary reported the following facts at the last meeting:
• 36 members play tennis and baseball
• 163 members play tennis
• 6 members play all three sports
• 13 members play tennis and soccer
• 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 them
play 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 students
received a B in English, and 11 students received a B in both mathematics and
English. 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
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
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 that
quadrilateral BCDF is a 2-cm by 3-cm rectangle. Find the area in square
10. Dennis Murray earns \$18 500 per year after deductions. His wife earns a takehome pay of \$248.56 a week. The Murrays have one child for whom Mrs. Murray
receives a child tax benefit that amounts to \$323.16 per year.
The Murrays have recently bought a home for which they are making monthly
payments of \$425.00, and monthly payments of \$165.00 on a 3-year bank loan
that helped to finance the purchase. Taxes are \$730.00 per year, gas heat is
estimated at \$740.00 a year, electricity and water bills are approximately \$83.50
for a 2-month period, the telephone bill averages approximately \$10.50 per
month, and home insurance is \$242.00 per year.
The Murrays are also making payments of \$180.00 per month on a one-year loan
they made to finance the purchase of furniture. Insurance on the car costs
\$288.00 per year, and gasoline averages \$105 per month. Additional expenses
include: food, \$420.00 per month; clothing, \$795.00 per year; entertainment,
\$330 per year; holiday gift purchases, \$165.00 per year; newspapers, books and
magazines, \$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
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
G-2
Exercise 47: AND, OR, NOT, and Venn Diagrams
12. For each parabola, state
a. whether the parabola opens upward or downward
b. the coordinates of the vertex
c. 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 F
EB and BD are tangents at A and C, respectively
* 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 15o-
G
6
1
8
2
14. Write the equation of a line that is perpendicular to the x-axis and passes
through the point (4, -5).
15. Graph the region that satisfies the inequalities x2 + y2 > 9 and x2 + y2 -< 16.
106
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
G-3
Exercise 48: Counterexamples
1. Ravi concluded that whenever he added two prime numbers, the sum was
always 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 for
x < 0 and conjectured that y = xx is undefined for x < 0. Find an example that
would support her conjecture. Find a counterexample to show that her conjecture
is false.
2
49 can be reduced to x + 7, the functions
4. Frank claims that since f (x) = x
x-7
x2 -49
g(x) = x + 7 are the same. Find a value of x that is a
f (x) =
x-7 and
counterexample.
5. Notice that x2 + x + 41 produces the prime number 43 if x = 1, the prime number
47 if x = 2, and the prime number 53 if x = 3. One might assume it always
produces primes for positive integral values of x. Find a counterexample to prove
this is wrong.
6. Everyone in a class of 25 students must take either Latin or French. There are
18 students taking French, three of whom take Latin as well. How many
students are taking Latin?
7. In a gathering of 18 men it was discovered that eight of them could speak
French and eleven could speak English. There were four who could speak both
these 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 the
basketball team, 26 on the baseball team, 29 are on the football team, and 8 are
on all three teams. Furthermore, 14 play basketball and baseball, 15 play
baseball and football, and 12 play football and basketball. How many members
are there altogether?
9. A Canadian dollar is worth 72¢ U.S. A pair of shoes cost \$75.00 in Fargo. What is
10. Solve and check: 2x2 - 1= 4x+6
x-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
Senior 3 Pre- Calculus Mathematics
Cumulative Exercises
G-3
Exercise 48: Counterexamples
12. The account in which Jane deposits her money pays 4.25% simple interest
annually.
a. How much interest will she earn on a deposit of \$5000 left in her account for
6 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 with
two circles and a square. The shaded part is
gold. 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
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
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 angles
are congruent.
b. If two sides of a triangle are congruent, then the angles opposite these sides
are 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 were
established:
• 13 women spoke English
• 12 women spoke Spanish
• 4 women spoke English and Spanish
• I woman spoke English, French,
• 16 women spoke French
• 7 women spoke English and French
• 5 women spoke French and Spanish
and Spanish
How many could not speak any of these languages?
Continued
109
Cumulative Exercises
Senior 3 Pre -Calculus Mathematics
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.
9. Sketch and solve the following system:
J y2 _ 3x
2x-y=3
10. Given: BF is a tangent at D
L EDB = 58°
AR = 110°
CD = 80°
Find the measure of all
indicated angles (L 1... L 6).
11. The total surface area of the rectangular solid
shown is 36 m2. Find the value of x.
x+2
x
Continued
110
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
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 AB
b. midpoint of BC
c. length of the median from C
d. 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
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
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. Both
Al 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 are
base angles of an isosceles triangle. What can be concluded about the sizes of Z 1
and 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. You
go to the library and she is not there. What do you then know? What type of
proof (indirect or direct) did you use?
5. Write each of the following in "If...then..." form.
a.
b.
c.
d.
e.
All multiples of 6 are multiples of 3.
All people born in 1810 are now dead.
When it is sunny, my family always goes on a picnic.
Vertically opposite angles are congruent.
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. The
width 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 of
the floor if 44 m2 of the floor are not covered by the rug.
Continued
112
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
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
7
x
12. Find the roots of 2x2 + 5x - 3 = 0.
13. Prove the following statement:
If the measure of the angle determined by two tangent segments to a circle from
a point in the exterior is 60°, then the tangent segments form an equilateral
triangle 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 of
radius 6 cm. What is the area between the circle
and 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
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
G-5
Exercise 50: Direct and Indirect Reasoning
ACLU CREDIT
B ALANCE FORWARD
DESCRIPTION
520 15
Deposit
Cheque 346
Cheque 347
Deposit
Cheque 348
Cheque 350
Deposit
Cheque 349
Cheque 351
Cheque 353
Service Charge
DATE
425 00
57 66
80 89
42 38
103 56
420 15
144
125
36
14
01
03
08
10
13
14
15
19
23
28
28
BALANCE
11
11
11
11
11
11
11
11
11
11
11
647 33
222 33
164 67
245 56
203 18
99 62
519 77
375 43
250 43
214 28
199 53
CHEQUES ISSUED TO OR
CHEQUE
DEPOSIT
DESCRIPTION OF DEPOSIT
AMOUNT
AMOUNT
CHEQUESIDEPS
127 18 1
CHQ - IDEP +
520 15
647 33
425 00
222 33
57 66
164 67
42 38
122 2
520 15
D epos it
346
R en t
6
347
Super Foo ds
10
348
Utilities
10
D epos it
12
Depos it
349
350
20
352
Sporting Goods
25
353
Wa l mart
354
ALA.NCE
BALANCE
Dep osit
28
Service Charge
CHQ - IDEP +
42 38
BALANCE
80 89
CHQ - IDEP +
144 34
CHQ - /DEP +
103 56
CHQ - /DEP +
BALANCE
375 43
36 15
CHQ - IDEP +
AI.ANCE
CHQ - /DEP +
BALANCE
CHQ - IDEP +
54 76
CxQ - /DEP +
125 00
250 43
17 86
232 57
36 15
196 42
54 76
141 66
45 00
186 66
420 15
CxQ - IDEP +
ALANCE
BALANCE
125 00
17 86
BALANCE
BALANCE
45 00
14 75
1
BALANCE FWD
80 89
203 18
420 15
623 3
144 34
478 99
103 56
BALANCE
Groc eries
30
CHQ - /DEP +
57 66
Thi essen 's Dep t . Store
r ippe d up
AI ANCE
CHQ - IDEP +
425 00
Car Repa i r
351
30
DAY Mo.
No.
1
15
34
00
15
75
127 18
CHEQUE
Nov.
1
12
CREDITS
DEBITS
D ATE
01 11
=
CHQ - IDEP+
BALANCE
CxQ - /DEP +
BALANCE
14 75
171 191
Continued
114
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
G-5
Exercise 50 : Direct and Indirect Reasoning
STATEMENT OF RECONCILIATION
Bank Reconciliation
Balance from statement:
Subtotal:
Subtract:
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. She
decides to take a rest and lands on the top of the minute hand of the wall clock
at 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.
f3)
-
a. f(1) =
b. f(2) =
c.
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.
d. f(3) - g(-1)
f (f (x))
e. g(O) f NO
3. Given functions f and g such that ft) = x - 1 and g(x) = 2x2, determine
b. g(f (3 ))
a. f(g(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 f and g such that f (x) =
a. f (g(x))
6. Solve: x +
b. g(&))
and g(x) = x -1, determine
c
g(5)
f(9)
x
-2 = 4.
-
7. Three mutually exclusive circles have radii
4, 5, and 6, respectively. (See diagram.)
a. Find the angles of the triangle whose
vertices are the circles' centres.
b. Find the area of the white region
between the circles.
Continued
116
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
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 three
times 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 the
vertices 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 the
lawn." 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 tape
on a spool with radius 1 cm. The tape is
0.02 cm thick and 1.5 cm wide. The
combined radius of the spool and the
tape is 3 cm. Approximate the length of
the tape on the spool in metres.
17. Describe each solution to the inequality, using interval notation.
a.
yI y >- -3
x#5, xe R}
c.
{yl5 >_y>
117
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
H-2
Exercise 52: Inverse Functions
1. For each of the following functions, specify the inverse function.
a. Multiplying by 5
{(x, y) y = 3x + 2
b. {(4, 5), (6, 6), (7, 8)}
}
d. {(x, y) f y = 4 - x}
2. For each of the following functions, f, define its inverse, f ;.
a. f (x) =
x
3
b. f(x)=x2 +1 andx?0
c. f(x)=
3
x-2
3. Given Ax) = 3x + 7, determine
a. f'(1)
b. f-1(8)
c. f -1(3a + 7)
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-one
correspondence.
5. Explain why f (x) = 2x + 1 and g(x) = X
1 are inverses of each other.
6. Given :L ABC = L FDE
BC = DE
AC//EF
Verify: a. A ABC = A FDE
b. AB//DF
Continued
118
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
H-2
Exercise 52: Inverse Functions
7. Tony Hill earns \$335.75 net per week. His wife, Natalie, earns \$337.75 net per
week. The family receives a child tax benefit of \$36.75 per month. The family's
expenses are as follows:
Expenses for the family include:
monthly mortgage payment ..........................
.. \$715.40
monthly car payment .................................... 206.10
average monthly telephone bill ............................. 23.00
other monthly utilities ................................... 305.20
yearly car insurance premium ............................. 610.00
home is assessed for property tax purposes
at \$40 000; the mill rate is 60 mills ............................
g. home insurance (yearly premium) .......................... 249.40
h. monthly boat payment ................................... 130.00
i. student loan repayment per month ......................... 100.00
j. food (average per month) ................................. 560.00
k, clothing expenses for the year ............................. 830.00
1. average car maintenance per month ......................... 35.00
m. gasoline per month ...................................... 120.00
n. entertainment per year .................................. 2600.00
o. yearly vacation ........................................ 2000.00
p. newspapers and periodicals (per year) ....................... 250.00
q. average monthly credit card payment ....................... 200.00
r. holiday gift purchases per year ............................ 500.00
s. baby-sitting (average per year) ............................. 400.00
a.
b,
c.
d.
e.
f.
a. Prepare an estimated monthly budget for the Hill family using the blank
budget form on the following page.
b. As the financial planner for the Hill family, you notice that the Hills have no
why this logic is faulty.
c. The Hills are a little concerned about their present financial position. Mr. Hill
suggests they reduce their reserve fund payment to balance their budget.
Suggest to the Hill family other areas in which you feel the family could
reduce spending and still balance their budget.
Continued
119
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
H-2
Exercise 52: Inverse Functions
Income
\$
a. Regular Monthly Income
b. Spouse's Regular Monthly Income \$
d. Other Income
#1
Total Monthly Income
2. Housing Expenses
a. Mortgage or Rent
b. Property Tax
c. Home/Property Insurance
d. Repairs/Maintenance
e. Other Housing Expenses
Total Housing Expenses
3. Utilities
a. Hydro
b. Gas
c. Phone
d. Water
e. Other
Total Utilities
\$
#2 \$
#31
5. Personal Finances
a. 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
6. Personal Expenses
a. Groceries
b. Clothing
c. Entertainment
e, Vacations
f. Other Personal Expenses
Total Personal Expenses
#5 \$
\$
#6 .
7. Other Expenses
a.
4. Transportation
a. Public Transport
b. Car Loan
c. Car Fuel
d. Car Maintenance
e. Car Insurance
f. Other Transportation
Total Transportation
\$
b.
\$
\$
C.
Total Other Expenses
#7 \$
Total Monthly Expenses
#8 \$
Income minus Expenses (#1 - #8) #9 \$
#4
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 be
saved 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 will
take you to achieve it.
Continued
120
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
H-2
Exercise 52 : Inverse Functions
8. Compute the discriminant and tell whether the equation has none, one, or two
solutions.
a. x2-7x+12=0
b. 3x2=5x-3
9. Solve this system of equations: 2x+y = -6
Ix-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 are
coloured green or yellow. The ratio of green to yellow marbles is 7:2 in Box 1 and
8:1 in Box 2. If there are 90 yellow marbles in total, how many green marbles
are in Box 2?
13. Assuming that the half-life of a radioactive substance is 1690 years, what
fraction of an initial amount of the substance will remain after
a. 3380 years?
b. 5070 years?
121
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
H-3
Exercise 53: Factor Theorem and Remainder Theorem
Given the polynomial fix) = x3 + 2x2 - 5x - 6, use the factor theorem to determine
whether
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 = 48
and 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 the
hill is supported by a 50-m vertical pole. A support cable runs from the top of the
pole to an anchor located 88 m directly downhill from the base of the pole. How
long is the cable?
10. A Canadian dollar is worth 720 U.S. A set of golf clubs in North Dakota is
marked at \$420.00 US. What is its cost in Canadian funds?
i
Continued
122
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
H-3
Exercise 53: Factor Theorem and Remainder Theorem
11. A person can row a boat 9 km downstream in 2 hours. Rowing back upstream, it
takes 3 hours to return to the starting point. Find the speed with which the boat
rowed through the water and the speed of the current, assuming that both of
these 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 the
function and its inverse on the same coordinate system. What do you notice?
x2
15. Solve and check:
-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 the
centre, how long is chord AB?
17. How many zeroes are in the product of the first 500 natural numbers?
123
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
H-4
Exercise 54: Graphs of Polynomial and Rational Functions
1. Which of the following graphs could be graphs of polynomial functions and which
could be graphs of rational functions?
a.
b.
1
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.
.
and y =
1
4. Compare the graphs of 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
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
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 the
interval 0°<_ 0 S 360°. (Round to one decimal place.)
c. Cos 8 =cost 0 -1
b . cos 0 + 1
2
a. 2 sin 0 = --v'
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 and
less than 5. Does he win?
11. Write equations for lines that are at a distance of 3 units from the line
x - Sy+ 10=0.
12. Two friends were comparing the different pay scales paid by the two companies
for which they work. Each company pays at a rate of time and a half for
overtime.
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 pay
between 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
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
H-4
Exercise 54: Graphs of Polynomial and Rational Functions
13. Given : P is the centre of the circle
CE is a diameter
L1=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 numbers
is both a perfect square and a perfect cube.
a. Find two other numbers that have this property.
iP
b. How might you generate numbers that have this property?
I
126
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
Exercise 55: Review 5
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 is
divided 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 circle
QR = RP
LBST=60°
ST is tangent at T
AP is tangent at P
AB is tangent at Q
Find the measure of all numbered
angles.
8. Solve A ABC if you are given that L A = 36°, a = 9.4, and b = 13.1.
127
Cumulative Exercises
Senior 3 Pre-Calcul us Mathematics
Exercise 56: Review 6
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 are
satisfied.
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 the
expression.
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
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
Exercise 56: Review 6
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 the
expression 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. She
washes the dishes. Can she go out?
129
Cumulative Exercises
Senior 3 Pre-Caicuius Mathematics
Exercise 57: Review 7
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:
(
g( 43
- ))
a. g(f(7))
b. g(f(-3))
c. f (g(8))
d. f
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:
g(f(5))
a. g(f(6))
b.
e. A g(9))
f. f(g(5))
c. g(f(1))
d. g(f(-2))
g. f(g(-1))
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 or
less 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 the
English books have red covers . Two of the English books are new. Four of the
books 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 red
covers . How many new books are there?
130
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
Exercise 58: Cumulative Review
For each of the following questions (1, 2, and 3), sketch its graph and state the
following:
a. axis of symmetry
c. whether max or min
b. vertex
d. max/min value
f. x-intercept (roots, zeroes)
e. y-intercept
g. domain
i. direction of opening
1. ftx)=3x2+4
h. range
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 the
right and four units down. State the equation of the resulting graph.
6. For what value of p 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 the
number of programs sold for a maximum profit. Also find the price of the
program 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°
d , sin
`2
3
b. sin 45°
e, cos
3
2
c. tan 100°
f. tan 6.5
Continued
131
Cumulative Exercises
Senior 3 Pre- Calculus Mathematics
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 with
a 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
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
Exercise 58: Cumulative Review
19. The perimeter of the isosceles
triangle ABC is 54 cm, and AC = BC.
If AD = 5 cm, and D, E, and F are
points 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=3
OC=3'2
Continued
133
Cumulative Exercises
Senior 3 Pre- Calculus Mathematics
Exercise 58: Cumulative Review
22. If L 0 = 1500 and 0 is the centre,
find the measure of Z B.
23. If 0 is the centre and B is a point of
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 the
measure of
a.
b.
c.
d.
e.
f.
g.
L EOF
L DCE
L OFG
L DFG
L CDE
are BD
i. are EDF
j. are CE
k. arc CFD
L arc EFD
m. are FCF
25. For the diagram to the right, find the
measure of
a. L OCB
e. are BD
b. L BDC
d. L DBO
f. are BC
g. are BCD
Continued
134
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
Exercise 58: Cumulative Review
26. State the number of *s in each of the following:
a. A and B
b. Aor B
c. A
d. Only B
e. Not in A
f. Only A
27. Thirty-five students were surveyed. Of those, 19 indicated they are taking
chemistry, 8 are taking chemistry and biology, while 7 are taking biology and
physics. Nine are taking chemistry and physics. Five students are taking
chemistry, biology, and physics. Twenty-nine students are taking chemistry or
biology. If 28 students are taking biology or physics, find the number of students
taking 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 word
deductive.
reasoning, we take an accepted general rule and
a. Using
apply it to a specific case or instance.
reasoning, we use specific cases or instances to
b. In
formulate 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 is
true or false.
30. Statement: Every relation is a function.
Use a counterexample to show that the above statement is false.
Continued
135
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
Exercise 58: Cumulative Review
31. Given: AB # AC and L I = L 3.
Using an indirect proof, show that
L2#L4.
32. Calculate the (shortest) distance from a point located at (2, 5) to the line
described by 3x -- y = 4.
33. A wallet contains a total of 20 coins, consisting of only nickels and quarters. The
total value of the coins is \$2.40. How many nickels and how many quarters are
in 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=8
3x--5y+6z=7
36. Sketch the following inequalities and determine the solution of the system
graphically.
y <-(x-2)2+ 1
2x - 3y <6
37. Solve for x:
a. Ix-31<1
b.
3x+2 ?8
38. Solve for x:
a. x2-2x-3>0
b. x2+3x-4<0
c. x2-3x-10>0
d. x2-x -12<0
Continued
136
Senior 3 Pre-Calculus Mathematics
Cumulative Exercises
Exercise 58 : Cumulative Review
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)
d.
f(g(x))
g(-2)
f (-2)
b. g(2) - f(2)
c.
e. g(l0))
f. f(f(x))
41. Given h(x) = 3x + 7, find (inverse)
b. h-1(2)
a. h-1(x)
42. Sketch the following:
a. f(x) = x(x - 1)(x + 3)
b. / Tx) = x(x + 2)2
C. f (X) = (x - 1)2(x +4)2
d. f (x) = 2x+4
X-1
e. f (x) =
g. f(x)=
i.
3
+1
1
x 2-4x-5
f(x)=x3+4x2+x-6
f.
2
f(x) _
2
X -4
h. f(x)=x'-4x
j. f(x)=x3-7x-6
k. f(x)=x3+5x2+2x-8
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
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
Exercise 58: Cumulative Review
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
48. Solve for x:
x+4 = V7x + 1
4
2
49. Solve for x: x + 2 =
50. Solve for x:
3x + 2.
2x +7.
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
54. Solve for x:
4
1
x
55. Solve for x:
5 +2=6
x+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
Cumulative Exercises
Senior 3 Pre-Calculus Mathematics
Exercise 58: Cumulative Review
59. Find the value of x.
60. A beacon from the top of a 18.6-m lighthouse illuminates a boat in the water. If
the beam of light makes a 19.7° angle with the boat, how far is the boat from the
lighthouse?
61. Given:
1 4(-2)+5(-l)-6
16
+25
I
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 the
nature 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
Cumulative Exercises
Senior 3 Pre- Calculus Mathematics
Exercise 58: Cumulative Review
67. Find the measures of L 1, L 2, and the sum of L 3 and L 4, given that
L1=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 40
hours. 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 income
tax. 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
Senior 3 pre-Calculus Mathematics
Cumulative Exercises
Exercise 58: Cumulative Review
71. a. If you are given four vertices of a quadrilateral, how would you prove that it
is a rhombus?
b. If you are given four vertices of a quadrilateral, how would you prove that it
is a rectangle?
c. If you are given four vertices of a quadrilateral, how would you prove that it
is 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 equation
form.
74. Find the simple interest earned if \$5000 is invested at 10.5% per year for
a. 6 months
c. 14 days
b. 18 months
d. 1 year
75. Complete the following chart for a the first five payments on a loan of \$5000 at
8% per annum, and payments are \$300 per month:
Monthly
Payment
Principal
Payment
Interest
8% per year
Amount
Owing
\$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 3
PRE-CALCULUS MATHEMATICS
ANS WERS TO C UMULA TIVE EXERCISES
A Supplement to
A Foundation for Implementation
1999
Manitoba Education and Training
Manitoba Education and Training Cataloguing in Publication Data
510
Senior 3: pre-calculus mathematics. Answers to
cumulative exercises : a supplement to a
foundation 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 of
Education and Training. Manitoba Education and `braining, School Programs
Division, 1970 Ness Avenue, Winnipeg, Manitoba, R3J OY9.
Every effort has been made to acknowledge original sources and to comply with
copyright law. If cases are identified where this has not been done, please inform
Manitoba Education and Training. Errors or omissions will be corrected in a future
edition. Sincere thanks to the authors and publishers who allowed their original
material to be adapted or reproduced.
Senior 3 Pre-Calculus Mathematics
A-1, A-2
b. y = x2 + 1
1. a. y = x2
-5
5
3. +1
b. [-4, oo)
c. (4, -4)
d. x=4
g. Does not have a max.
e, 2, 6
h. -4
b. Similarities: all are parabolas, open up,
same basic shape, same domain
Differences: vertices, x- and y-intercepts,
range
c. i.
ii.
(0, 0)
(0, 3)
iii. (0, -2)
d.
6. a. -12x5
b. 16xi2
c. 7x2
y
x2-4
d.
--4c3d2e
3
e.
1
49
f.
24
7
7. a. 28
b. -5
C.
8. a. x(x + 5)
b. (x + 4)(x + 1)
c. (3x + 4)(2x - 5)
9. X= 4
- 3y
5
10. 57 cm2
13. a. D: (--cx, oo), R: [1, oo)
c. D: [-1, 51, R: [0, 3]
11. 62.8 m
1
9
11
12. 20
b. D: (-, 3], R: (_oo, o)
14. A, G, F
d. D: (-3, -1] u [1, 3), R: [-1] v [1]
I
Senior 3 Pre-Calculus Mathematics
A-1, A-2
Exercise 2: Graphs of Quadratic Functions 1
Note: For each of the questions 1, 2, and 3, the graphs should be on the same
grid.
b. The parabolas get narrower.
c. All three are (0, 0).
-i-++
FIr +i i i
5
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
Senior 3 Pre-Calculus Mathematics
A-1, A-2
Exercise 2 : Graphs of Quadratic Functions 1
b. 6(5x + 4)(2x - 3)
5. a. 2(x - 2)(x + 2)
6. a. sin0 = 4, cos0 = 3, tanO =
4
12
5
12
b. sin 0 = 13 , cos e = 13 , tan 0 = 5
5V,[29
29 tan 0= 2
5
c. sin 0 = 2 29 , cos0 =
- 29
-1
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 = 12
17
15.
2
3
16. a. -1
2
b.
-1-5-4
7
b. (2, 4)
c. 6-J5
3
Senior 3 Pre-Calculus Mathematics
A-2, A-3
Exercise 3: Graphs of Quadratic Functions 2
parabola (reflects it in the x-axis).
5
5
2. a. i.
b. i. Vertex: (-2, 3)
ii. Vertex: (-4, -5)
iii. Vertex: (5, 1)
3. a.
b.
c.
b.
ii.
c, i. Axis of Symmetry: x = -2
ii. Axis of Symmetry: x = -4
iii. Axis of Symmetry: x = 5
Opens up; Vertex: (-1, 0); Axis of Symmetry: x = -1, narrower
Opens down; Vertex: (1, 6); Axis of Symmetry: x = 1, wider
Opens up; Vertex: (-6, -10); Axis of Symmetry: x = -6, narrower
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
Senior 3 Pre-Calculus Mathematics
A-2, A-3
Exercise 3: Graphs of Quadratic Functions 2
c. Vertex: (-6, -10);
Axis of Symmetry: x = -6
5. a. (x+2)(x-3)
6. a. 5.0
d. Vertex: (1, 8);
Axis of Symmetry: x
b. (x-5)(x-3)
b. 13.2
c. 9.4
c. 7(2x - 3)(x + 5)
d. 23.9
e. 6.1
7. 2-Y
3
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 x 22y 11
14. 36
15. a. (2, 51
d. [0, 61
b. (--, -2) u (7, °°)
e. (-6, -31 u [2, 8)
c. (--°°, -31 u [10,
f. 3.5
Senior 3 Pre-Calculus Mathematics
A-3
Exercise 4: Transformations of Quadratic Functions 1
11.
b. i.
Vertex: (1, -4);
Axis of Sym: x = 1;
Domain: (x x e 9tt);
Range: (y Iy r=- 9t, y ? ---4};
x-intercepts: 3, -1
ii.
in.
Vertex: (-3, -4);
Axis of Sym: x = -3;
Domain: (x I X E 9t};
Range: (y I Y E 91, y > -4);
x--intercepts: -5, -1
Vertex: (-3, -1);
Axis of Sym: x = -3;
2. a. Vertex: ( 3, 5); opens up
c. Vertex: (-1, 2); opens up
3. a. 2(x -- 2)(x - 8)
b. 4x(a - 2b)
4. a. 8 = 36 .9°
b. 8 = 18.2°
5. a. 190.8 km
b. q = 1.9°
6. a. -2(I +)
b. 2x-11y
Domain: (x I x e 9t );
Range : (y I y e 91, y ? -1);
x-intercepts: -4, -2
b. Vertex: (-4, -7); opens down
d. Vertex: (2, 1); opens down
c. 6 = 62.9°
d. 0 = 56.9°
7. 682.1 m
8. 16.5 and 17.5
Continued
6
Senior 3 Pre-Calculus Mathematics
A-3
Exercise 4: Transformations of Quadratic Functions 1
9.
y= (x-1)
Vertex
+2 y=-(x_1)2-2 y=(x+l)2+2 y(x+l)2-2
(1, 2)
(1, -2)
(-1, 2)
(-1, -2)
x=1
x=1
x_-1
x_-1
(-001 CO)
(--00, 00)
(-,-21
[2, -)
[-2, oo)
up
down
up
up
nun
max
y=-2
min
y= 2
min
Equation of
Axis of
Symmetry
(._,,,o,
Domain
Range
Direction of
Opening
Maximum
or Minimum
y values
[2,
y- 2
W)
y=-2
10. y=2(x-1)2---2
7
Senior 3 Pre-Calculus Mathematics
A-3
Exercise 5 : Transformations of Quadratic Functions 2
ii . 16
a. i. 16
iv. 1
iii . 100
v.
25
4
(
2
2
(x-1)2
49
vi.
Ix 5(x+4)2- 10
-2
4
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
Vertex
y-
x-
Width (as compared to
Symmetry intercept(s) intercept(s) Opening
x2)
(-3, -16)
x = -3
1,-7
-7
up
same
b.
(2, -64)
x=2
-6, 10
-60
up
same
C.
(-2, -18)
x = -2
-5, 1
-10
up
narrower
d.
(-4,-27)
x = -4
-7,-1
21
up
narrower
e.
,
-2, -3
6
up
same
x=2
-1, 4
-4
up
same
x=- 4
-2, - 2
2
up
narrower
x=3
1, --
1
down
narrower
g.
2 4 )
2,
25
x=-
5
N 8
3, 3
1
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-y
4
6. a. x
2
7. a. 133.4°
8. Jon 27, Cal 9, Ron 21
8
=
a.
f.
5.
y
b.
x=15
2
b. 6836.2 m2
Senior 3 Pre-Calculus Mathematics
A-4
Exercise 6 : Transformations of Quadratic Functions 3
Length = 40 m, Width = 20 m
2.
13 13
T' 2
3. 8 sec , 326 m
5. 13
2'
4. 90c
6. 2(9x - 7)(4x + 9)
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,
whenx=1,y=4,
when x = -5, y = 400.
Collision is at (-5, 400).
b.
5
, 0)
3 l
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)
d. (5, oo) u (-oo, -7)
b. (-10, 51
e. (-5, -2] u [2, o)
c. (-8, 00)
9
Senior 3 Pre-Calculus Mathematics
A-4
Exercise 7: Applications of Quadratic Functions
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
7. x
1- 3y
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
Senior 3 Pre-Calculus Mathematics
B-i
Exercise 8 : Trigonometric Equations 1
a.
cos
3,/34
34
5134
34
17
-
3
5
4,(17
17
d. -
tan 0
3
b. 4
5
C.
0
sin 0
17
3V-58
58
7V58
58
2. a. 82°
b.
600
C. 8°
d. 83°
3. 210 °, 3304. a.
x
y - sin x
0°
450
900
135°
180°
225°
270°
315°
3600
0.00000
0.70710
1,00000
0.70710
0.00000
-0.7071
-1
-0.7071
0.00000
b.
2
-360
90
-180
180
270
/360
-2+
Continued
11
Senior 3 Pre-Calculus Mathematics
B -I
Exercise 8: Trigonometric Equations 1
5. a.
x
0°
1.00000
45°
900
135 °
1800
225°
2700
3150
360°
0.70710
0.00000
-0.7071
-1
-0.7071
0.00000
0.70710
1.00000
b.
6. The graphs have the same shape, the same domain and range, however, the
graph 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)
c. 53°
9. 19.4%
10. 17 - 4 15
11. Y = -2(x -- 2 )2 + 13
12.
Continued
12
Senior 3 Pre-Calculus Mathematics
B-#
Exercise 8 : Trigonometric Equations 1
b.
13. a. y= (x+4)2+2
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
350
b. 103 = 0.29
350
13
Senior 3 Pre-Calculus Mathematics
B-1
Exercise 9 : Trigonometric Equations 2
1. a. 131.8°, 228.2°
e. 221.8°, 318.2°
i.
80.5°, 260.5°
2. a. 8.7 km
b. 270°
c. 81.9°, 261.9°
d. 0°, 360-
f. 84.3°, 264.3°
g. 48.2°, 311.8°
h. 141.3°, 321.3-
k. 135°, 315-
104.5°, 255.5°
j.
b. 605.5 km2
3. x = 18
b. Domain: [0, 61; Range: [-5, 4]
4. a.
c. Vertex: (3, 4)
d. Axis of Symmetry: x = 3
e. Maximum Value: 4
Maximum Value: - 5
5.
6.
7. a. (6, 2)
9.
y=
2(x-5 )2 -2
b. 2J
c.
43
7
2
1
12. a.
83
104
11. 37',143-,217-,323-
2-F
-360
-180
180
b. The graph of y = cos x - 2 is 2 units lower.
c. The graph of y = cos x + k moves k units above y = cos x, k > 0.
It is k units below if k < 0.
14
360
Senior 3 Pre-Calculus Mathematics
B-1, B-2
Exercise 10: Trigonometric Equations and Ambiguous Case Problems
1. a. 180°, 360°, 0
b. 135°, 315°
c. 120°, 240°
2. a. 34.6° or 145.4°
b. 118.4° or 7.6°
c. 7.8 or 1.2
d. 66.8°, 246.8-
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
15. a. (-, 0) u [10, oo)
1
b. (-o, -8) u [-5,---6I
180
360
c. [-7, 3) a (7, co)
15
Senior 3 Pre-Calculus Mathematics
B-2
Exercise 11 : Ambiguous Case Problems
1. a. 1.3 or 5.3
d. 5.5
2. a. 23.0° or 157.0°
b. 3.1 or 26.1
c. 7.8
e. 8.0
f. No such triangle
is possible
b. 10.8°
c. 43.1
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+
16
b. (-5x - 4y)(7x + 2y)
7. y=-3x+lor3x+5y-5=0
40 minutes
Senior 3 Pre-Calculus Mathematics
Exercise 12: Review 1
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',1207.
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
Senior 3 Pre-Calculus Mathematics
C-1, B-1
1. a. (3x + 1)(x + 2)
d. 2(x - 4)2
b. (x + 3)(x - 3)
c. 25(x + 2)(x - 2)
e. (sin 0 + 1)(sin 0 - 1)
f. tan 8(tan 0 + 2)
2. a. x = -3, 1
b. x=- 4 )
3. a. x = 4, -3
b. x == -6, -3
c. x = 5, --4
d. x - , - 2 e. x = 1, -3
b. x = 5 , -6
c. x = 2 , 1
d. x =
4. a. x5
3
, -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°
d. 60°, 120 0, 240 0, 300°
b. 30°, 150°, 270°
c. 63.4°, 210°, 243.4°, 330°
e. 30 °, 1500
f. 0°, 180°, 221.8°, 318.2°, 360°
g. 270°
7.
10.
78.4°
18n cm2
13. 5, 7, 9 or -7, ---9, -11
18
8. 24V
9. 2x-3y=6
11. 8 m, 15 m
12. 5 cm, 12 cm
14. -4
15. 9 units2
Senior 3 Pre-Calculus Mathematics
C-1
1. a. a=l,b=-2,c=--5
d. a=2,b=--4,c=-1
g. a=-3,b=2,c=-7
2. a. 3, -5
b. 2 , 1
d.
g. 38.2°, 141.8°
h. 170.4°, 221 .8°, 318.2°
7± 5
-2
3±2
3
b
b.
2±
2
b.
--3± 11
2
1±Vi
4. a.
10
c. a=5,b=-3,c=-8
£ a=2,b=9,c=-4
c. 0, 7
e. --0.2, 0.3
3. a.
5.
b. a=3,b=-2,c=5
e. a=5,b=-9,c=0
h. a=1,b=0,c=-3
c.
5
4± 79
9-
3
d.
. -1, 2
1.8,-0.2
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°, 41.49. a. 0°, 180°
10. 157.0 km
b. 30°, 150°, 63.4311. Bill 7, May 27
c. 0°, 180°, 90°
12. 36.2 kg
13. \$54
14.
Domain : (-, to); Range: [-1.5, 0.51
19
Senior 3 Pre-Calculus Mathematics
C-1
Exercise 15 : Solving Quadratic Equations by Graphing
1. a.
2. a. x=(-4,2)
b. x=-3,-1
c. x=-5,-3
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±
d. x=3,-3
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
14. a.
31
366
20
b.
12
366
e. x=1,5
Senior 3 Pre-Calculus Mathematics
C-2
Exercise 16: Nature of Roots
1. a. No real roots
b. Two real roots
c. No real roots
2. a. zero
b. once
c. twice
3. a. Discriminant = 0; One real root
c. Discriminant = 64; Two real roots
4. a. Discriminant = 8; Two real roots
c. Discriminant = 64; Two rational roots
d. One real root
b. Discriminant = -24; No real roots
d. Discriminant = 41; Two real roots
b. Discriminant = -55; No real roots
d. Discriminant = -47; No real roots
e. Discriminant = 0; One real root
5. Real numbers in (-6, 6)
b. 2, 14
6. a. 9
7. 0,-1
c. One real root
b. No real roots
8. a. Two real roots
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
Senior 3 Pre-Calculus Mathematics
C-3, C-4
Exercise 1 7: Nonlinear Equations
b . ±1
a. 6,-4
d. ±1., ± 3
c. ±2V 2
2. Complete the square to find the vertex; solve the quadratic equation to find the
x-intercepts.
-5+
-2+
20+
10+
-10
10
--20
3,
Domain
Range
x-intercept(s)
y-intercept(s)
a.
Real Numbers
Real Numbers > -4
-1, 3
-3
b.
Real Numbers
Real Numbers ? -1
-1, 1
-1
C.
Real Numbers
Real Numbers
2
-8
d.
Real Numbers
y ? -15
-3,-1,1,3
9
Continued
22
Senior 3 Pre-Calculus Mathematics
C-3, C-4
Exercise 17: Nonlinear Equations
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°, 33.710. 1, 2, 3, 4, 6, 12
11, a. (--mob, -21 or [7, «^}
b. (-10, o)
c. (-°°, 41
12. 26.6°
23
Senior 3 Pre-Calculus Mathematics
C-5
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=4
5
. a. 2x-1
3. a. 12
b. -1±5
4. 76.00
5. a. 14
b. 22
6. 0
C.
5+
5
R:
(yIyE
Sit , y
8. 5x2+4x-3=0
9. x=4
10. 6
24
< 0)
R: {y1yc R ,y<6}
R: (ylyE 9,y?31
Senior 3 Pre-Calculus Mathematics
C-5
Exercise 19: Rational/Absolute Value Equations
1. a. 1, 2
b. -4
d. 3
e. --1
2. a. i.
4, -4
iv. 2, -6
ii. 9,-9
v.
3,3
vi. -6,
b. First, set the expressions equal; then, set the first equal to the negative of the
second.
3. a. (6, -16)
b. x=6
4. a. 63 . 4°, 243.4°
c. (4, 0), (8, 0)
d. Domain: (x I x E 9R);
Range: (x j y E 9i, y ? -16)
b. 137. 1°, 222.9°
5. W=18cm , L=25cm
6. 18.7°
7. ±,f58. 72 km
9. 5 hrs
25
Senior 3 Are-Calculus Mathematics
Exercise 20 : Review 2
1. a.
b.
c.
d.
e.
Vertex: (2, -18)
Axis of Symmetry: x = 2
Minimum of -18
Domain: (_o , e)
Range: [-18, -)
f. Narrower than y = x2
g. Zeroes: 5, -1
6 ± 2 15
3
2. a. y = 3(x + 2)2 - 20
b.
3, a. Impossible
b. 16.9
4. a. 210 0, 330°
b. 41.8°, 138.2-
c. 90°, 180°, 270°
5. a. L D = 10.8°
6.
d. 0°, 70.5°, 180°, 289.5°, 360b. G I. 96.6° or 5.4°, i = 1420.6 or 134.5
-33
7. a. Sum: 6; Product: -4
8. a. x = 6, -
b. x--6x-4=0
4
d. Impossible
9. 289.3 m
10. No, 6.6 m
11. 100mby200m
26
c. 22.4°
f. a=4,-3
Senior 3 Pre-Calculus Mathematics
D-1
Exercise 21 Circles on a Coordinate Plane
1. a. (x+2)2+(y-3)2=25
d. (x-5)2+(y-1)2=5
b. (x-5)2+y2=9
c. (x-4)2+(y-3)2=10
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
20
5. x=-- , 2
9. Domain: {x xe91
b. Centre (0, -3), r =
6. x = ±6, ±4
; Range : {y I yE.9 , y? 4
7. x=2,3
c. Centre (5, 2), r = 29
8. D
or [4, -)
10. x2-4x+1 =0
11. 12:45 pm
27
Senior 3 Pre-Calculus Mathematics
D-1
Exercise 22: Distance between Points and Lines
a. V 5
b. 2
33
2.
5
3. a.
21, 13
13
b.
17 ,[13
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°
10.
b. 90°, 210°, 270°, 3300
c. 0°, 60°, 180°, 240°, 360°
-5+
12. 14, -16
11. No solution
13. a. SAS
b. SSS
c. AAS, ASA, SAS
14. x2+3x-18=0
15. a. 48.6 m2
b. 5.4 m x 9 m
16. Centre (-6, 3), Radius = 5
17. (x+2)2+(y--4)2=32
28
c. 81 m2
d. ASA
Senior 3 Pre-Calculus Mathematics
D-2
Exercise 23 : Verify and Prove Assertions in Plane Geometry
1. a. (-7, -7)
c. 106
b. 6,153
2. Yes; slope AB = -1; slope AC = 1.
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°
7. 600,
b. 109.5°, 250.5°
700
13
8. Vertically opposite angles are congruent and SAS.
9. Vertex:
(1 _39)
4'
8
9
S
11.
5
13.
12. Y= 5x2 4
5
- f
14. -07
15. a. Domain: [-2, 2); Range: [-1, 2)
c. Domain:
Range: [-1, 1)
b. Domain: [1, oo); Range: (---co, oo)
d. Domain: [-2, 2) v (2, 31; Range: [11
16. c
29
Senior 3 Pre-Calculus Mathematics
D-3
Exercise 24: Systems of Linear Equations in Two Variables
Intersection: (10, -2)
2. a.
Intersection: (2, 4)
Intersection: (1, 0)
No intersection
(No solution)
3. (3, -1)
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)
c. (2, -3)
8. a. 4573.2 m2
b. 132.8
c. 52.9° and 48.1°
9. 20
11. (3, 2)
10 . A ABC - A ACD by SAS, and AB - AC
12. (6,2)
13. 3 , - 6
15. cos 0 = 0.63, sin 0 = -0.78, tan 0 = -1.25
30
14. y= 2(x - 3)2 - 5
Senior 3 Pre-Calculus Mathematics
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
d. (2, 3)
c. X = -
11
,y=-3,z=9
f. (4, -6)
3
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 parallel
lines 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. 21V15 - 77
11. ± 26
12. Discriminant is positive; Two real roots
13. 0,- 7
14. (x--w7)2+y2=36
31
Senior 3 Pre-Calculus Mathematics
D-5
Exercise 26: Systems of Nonlinear Equations
1. (2, 4), (-2, 4)
-1
2,
) , (2, 8)
3. (3, 3), (-3, -3)
b. (±3, ± ^i7 )
3. (±4, ±3)
b. Axis of Symmetry: x = -6
5. a. Vertex: (-6, -18)
c. x-intercepts: 0, -12
d. Domain: (x I x E 9Z}; Range: ly I y e 91, y ? -18}
b. 0°, 60°, 120-
6. a. 150°
7. Discriminant = -11; No real roots
8. 14.65mand12.87 m
9. 600 kmAhr
10 . slop e 1 1 = 3 ; slo pe l = --- 4 ; slo p e 13 =- 3 ; slope 1, =
4
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 -29
3
13. 4
14. 2, -6
15. 81
16. a. Domain:
32
Range: (-, 2]
b. Domain: [-1, 2), Range: [-2, 1)
Senior 3 Pre-Calculus Mathematics
Exercise 27: Graphing Linear Inequalities in Two Variables
2.
5.
Continued
33
Senior 3 Pre-Calculus Mathematics
D-6
Exercise 27: Graphing Linear Inequalities in Two Variables
6. 5,11
7.
6
5
b. 3.8 sec
8. a. 25.9 m
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= 120 0, 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°.
13.
14. 29 + 6,i6
34
t
D
t
t
/
Graphical solutions are (0, -4) and (1, -2).
15. x=1,y=-2or(1,-2)
Senior 3 Pre-Calculus Mathematics
D-7
Exercise 28 : Quadratic, Absolute Value, and Rational Inequalities
1. a.
k
II
b. i.
f
(x I x < - 3 or x ? 1,x E R} or
(-, -3)
u
[ 1, 00)
3<xC1,xe 9R) or f-3, 11
2. a. Ix j-4 <x<1,x (=- SR) or (-4, 11
x< - orx>1, xe91 or
5
L) 1, 00
3. a. (-4, 5)
b,
4. a. 35.30
b. 72.3 paces
)
(--, --61 u [3,
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
-5
-4
-3
i
-2
---1
€--0
i
1
2
----- ;- ---+
3
4
5
6
{xIx<-3orx>3,xE 9N}
Continued
35
Senior 3 Pre-Calculus Mathematics
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
5
b. {x1 -2<x<-2, xE9Z
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.
36
Ratio
2
Senior 3 Pre-Calculus Mathematics
Exercise 29: Review 3
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:
b. 78.5°, 281.5°
2. a. 240 °, 300°
d. 00, 90°, 270-
c. 18.4°, 56.3°, 198.4°, 236.3°
b. (2, -3)
3. a. ((4, 3), (2, -3)1
4. Slope:
, Distance: 10, Midpoint (-1, 1)
5. a. f(-2,-4),(1.5,1.25)j
--5
b. (-°°, --) u (Vi, °)
5
Continued
37
Senior 3 Pre-Calculus Mathematics
Exercise 29 : Review 3
6. a. Sum is 8, product is 11.
b. x2 - 8x + 11 = 0
7. k=±3J2
8. a. Impossible
10. (-4, 7, 15)
1L [-7, 3
12. No solution
13. (-o, -5) u [-3, 41 v (5,
14.
38
b. y = 2 or 13.3
Senior 3 Pre-Calculus Mathematics
E-1, E-2, E-3
Exercise 30: Circle and Polygon Properties 1
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 the
endpoints of the chord.
b.6
c.5
d. 10
4. a. 6
b. 4
c. 2v5
2. a. 3
b. 8
5. a. 4-J 2
8J
d. 4-v2
6. 3±V33
6
7. Draw 2 chords (not parallel.) Construct the perpendicular bisectors of each.
Their intersection is the centre. Be careful!
b. 60°
8. a. 70°
9.
c. The inscribed angle is one-half of the central angle.
2
12. a. 116.6°, 161.6°
11. Vertex: (-3,4)
Axis of Symmetry: x = -3
x-intercepts: - 3 ± 2i
Domain: x I x E 9tj
Range:
b. 0 90°,
°,
18 0°
c. 30°, 150°
yI yE9I, y54
13. a. 100; Two real roots
b. -47; No real roots
(rational)
c. 9; Two real roots
(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
39
Senior 3 Pre-Calculus Mathematics
E-1, E-2, E-3
Exercise 31 : Circle and Polygon Properties 2
a. 90°
2. a. 10
3 . a. 13
b. 13
c. 6.5
b. 8
b . 12
d . 169n or 42 25n
.
4
c. 30
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 + /
8. 16
7. (4,0)
9. a. ---6<x<3or (-6, 3)
b. x<-4,x>5or(--o,-4)u(5,c)
10. 47
16
11. X -
13
33
12. 40°
14 . (-oo ,
15. Centre (2, 0), Radius = 2
40
3)u 2 , 00 or ^xI x<--3 orx>
, xe ca
Senior 3 Pre-Calculus Mathematics
E-1, E-2, E-3
Exercise 32: Circle and Polygon Properties 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
b. 59°
4. L C is a right angle because it is subtended by a diameter.
2
Pythagoras: AB =
5. I
19
a2 + b2 so radius =
a
2
2
and area = a 4+
b
2
, 8)
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 the
altitude from P bisects QR.
7. a. 0°, 180°
b. 168.69°
c. 40.49°, 139.51°
b. 36
c. 54
8. 6
9. (3,1,-9)
10. a. 8
11. a.
-9± 33
4
b
d. 27
3± J21
3
12. \$45.00
13. a. Eq. of DB: y = x; Eq. of EC: y=-x+2
14. a. D: [2], R: (-2, 2]
c. Area = 8
3
b. D: (--oo, «o), R: ( oo, -1]
41
Senior 3 Pre-Calculus Mathematics
E-1, E-2, E-3
Exercise 33: Circle and Polygon Properties 4
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)
11. The zeroes are:
1 4
2
12. Since BC is tangent , L 3 = L 9 by the tangent-chord angle theorem . Given
L1=L2,then L2+L3=L1+L9. Aswell , L7=L1+ L9 becauseL7isan
external angle for A ABE. Therefore , L 7 = L 2 + L 3 and thus BC = CE since
they 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
Senior 3 Pre-Calculus Mathematics
E-1, E-2, E-3
Exercise 34 : Circle and Polygon Properties 5
1. 41°
2. 45°
3. 5-J 15
4. Join the point to the centre. Construct a line perpendicular to this radius at the
end point.
5. Since AD is a tangent and OC is a radius , L OCD = 90°. It is given that
L 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
26,
12.
7
18
7
2,38
14. a. 26.6°
b. 41.8°, 138.2°
15. (a-b-c)(a+b+c)
16. 21.5 units'
43
Senior 3 Pre-Calculus Mathematics
E-1, E•2, E-3
Exerci se 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 i s a diameter. L ACB is a
right 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 AEB
since they are the third angles in two triangles with two pairs of congruent
angles . 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 R
are right angles since the radius to a tangent is perpendicular to the tangent and
OP 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
b.
33
5'2
C.
4±-J2
2
d.
±4
5
10. -1, 3
11. a. ,I 18
7
12. -3,7
15.
44
b. 3
13
13. 18 sq. units
14.
1
10
Senior 3 Pre-Calculus Mathematics
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 +2
180
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°
14. a. 39°
15.
b. 41 .8°, 138.2-
b. 102°
c. 51°
d. 102°
c. 146.3°
e. 78°
f. 51°
g. 390
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
45
Senior 3 Pre-Calculus Mathematics
F-i
Exercise 37 : Wages (Hourly)
\$492.15
b. \$558.90
c. \$980.00
2. a. \$358.75
b. \$581.88
c. \$922.50
3. \$910
4. \$458.13
b. 5, - 4
5. a.
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
10 . 4
11. a. 0°, 180°, 19.47°, 160.53°
12. 351 m, 759 m
13.
1
5
14. 3,-8
15. 9
16. (3, 1) and (4, 5)
46
b. 60°, 146.31°
c. 71.57°, 108.43°
Senior 3 Pre-Calculus Mathematics
F-1
Exercise 38: Wages (Commission and Net Income)
1. a. \$88.86
b. \$161.52
c. \$78.53
b. \$352.56
c. \$86.63
2. \$680
3. \$362.50
4. \$2140
5. \$506
6. a. \$315.97
7. 55 hrs
9
8. x= 7, y=-3
b. 3J10
13.
-5±2 10
5
14.
6
5
15.. Vertex:
4, - 4); x-i ntercepts: 3 , 2; Domain: Real Numbers ; Range:
16. (-oo, 5) v (6, -
47
Senior 3 Pre-Calculus Mathematics
F-1
Exercise 39: Property Tax
3. \$2773.50
2. \$550
1. \$2691
c. 101
b. 96
4. a. 88
5. -1-t 3
6. 0, 1
1
25
7. Vertex: - 6 , 12 , x-intercepts: 2
Axis of S ym metry: x =
x E Jl ; R: (__I 12 )I
1
9. x<2
8.
10. (4,-6)
11. a. No solution
c. 251.6°
b. 221.8°
d. Out of range in Quad I and IV1108.4°, 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 tangent
are 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 are
congruent by AAS, making M the same distance from each tangent.
13.
_5± i46
3
15. a. [6, 12]
48
14.
b. (-5, 2) V [8]
2_v13
13
c. (-oo, -12] v (5, 20)
Senior 3 Pre-Calculus Mathematics
F-1, F-2
Exercise 40: Unit Prices, Exchange Rates, and Reconciliation
of 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 is
better.
3. a. 1.878¢/g
b. 3.203g/g
c. 1.6420/g
4. a. \$180 US
b. \$38.89
c. \$52.78
d. Add one-third the cost plus a bit.
5. \$273.97 Cdn.
6.
CHEQUE
DATE
No.
Sept.
9
234
13
244
CHEQUES ISSUED TO OR
DESCRIPTION OF DEPOSIT
CHEQUE
DEPOSIT
AMouNT
AMOUNT
48 00
The Bay
CHEQUESIDEPS
CHQ - /DEP +
BALANCE
Esso
43 87
CHQ - /DEp +
Hydro
66 , 98
CHQ - /DEP +
BALANCE
20
245
BALANCE
25
200 00
Deposit
CHQ - /DEP +
BALANCE
30
246
Dales Rental Agency
`175 00
CHQ
/DEP
+
BALANCE
7.
MONTH
February
March
April
May
Previous
Balance
\$586.00
587.11
\$515.98
\$257.71
PAYMENT
BALANCE FWD
998 43
48 00
950 43
43 87
906 56
66 98
839 58
200 00
1039 58
475 00
5641 58
PURCHASES
CHARGED
BALANCE
DUE
CREDIT
CHARGE
NEW
BALANCE
\$100.00
200 .0
\$275.00
\$200.00
\$ 93.00
\$121.75
13.17
\$ 87.13
\$579.00
\$508.86
254.15
\$144.84
\$8.11
7.12
3.56
2.03
\$587.11
515.98
257.71
\$146.87
Continued
49
Senior 3 Pre-Calculus Mathematics
F-1, F-2
Exercise 40 : Unit Prices, Exchange Rates , and Reconciliation
of Bank Statements
8.
Bank Reconciliation
\$ 837.71
Balance from statement :
2000.00
\$2000.00
\$2000.00
\$ 2837.71
Subtotal:
211.11
854.00
Subtract withdrawals:
57.10
146.58
\$ 1268.79
\$ 1268.79
Total subtractions:
Subtotal:
\$ 1568.92
This should equal balance shown in your record book:
\$1568.92
9. 79.7 m2
10. 6, -2
11. Discriminant = -31; No real roots
12. a. 60°
c. 90°
b. 30°
d. 240°
13. 2.1, -3.5
14. a. 12-2
15. x=7,y=-2,z=3
50
b. 15x'2
2
16. 25
17 . 256 units2
e. 180°
Senior 3 Pre-Calculus Mathematics
F-3
Exercise 41: Budgeting 1
1.
1. Income
a. Regular Monthly Income
b. Spouse's Regular Monthly Income
d. Other Income
Total Monthly Income
#1
2. Housing Expenses
a. Mortgage or Rent
b. Property Tax
c. Home/Property Insurance
d. Repairs/Maintenance
e. Other Housing Expenses
Total Housing Expenses
170.00
\$
\$ 2966.08
\$
\$
b. Gas
4. Transportation
a. Public Transport
b. Car Loan
c. Car Fuel
d. Car Maintenance
e. Car Insurance
f. Other Transportation
Total Transportation
S
\$
\$
625.00
\$___ 160.42
22.92
\$
\$
\$
#2 \$
808.34
3. Utilities
a. Hydro
c. Phone
d. Water
e. Other
Total Utilities
\$ 2796.08
120.00
\$
17,40
\$
\$
3&33
#3
.1
.73
\$
213.50
\$
.80.00
\$ 2d'a
\$......_.....U 0
5. Personal Finances
a. Personal Loan
b. Investments
c. RASP
d. Life insurance
e. Charities
f. Credit Card Payments
g. Service Charges
h. Savings
i. Other Personal Finances
Total Personal Finances
\$
#5 \$
18.00
6. Personal Expenses
a. Groceries
b. Clothing
c. Entertainment
e. Vacations
f. Other Personal Expenses
Total Personal Expenses
\$
\$
\$
\$
\$
#6 \$
42500
60.42
62,50
52.50
600.42
7. Other Expenses
a. Newspaper/Per.
b. Babysitting
C.
Total Other Expenses
\$
#7 \$
Total Monthly Expenses
#8 \$ 2010,32
\$_
17.50
17.50
Income minus Expenses (#1 - #8) #9 \$ 955.76
#4 \$
x,33
Continued
51
Senior 3 Pre-Calculus Mathematics
F-3
Exercise 41 : Budgeting 1
2.
1. Income
a. Regular Monthly Income
b. Spouse's Regular Monthly Income
d. Other Income
#1
Total Monthly Income
2. Housing Expenses
a. Mortgage or Rent
b. Property Tax
c. Home/Property Insurance
d. Repairs/Maintenance
e. Other Housing Expenses
Total Housing Expenses
3. Utilities
a. Hydro
b. Gas
c. Phone
d. Water
e. Other
Total Utilities
\$ 169739
\$ 1652.82
\$
107.72
\$
\$ 3458.43
733.15
\$
149.00
\$
\$21.00
\$
\$
#2 \$ 903.15
#3
\$
\$
\$
\$
\$
200.00
\$
265.20
20.20
45.00
5. Personal Finances
a. 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 \$
6. Personal Expenses
a. Groceries
b. Clothing
c. Entertainment
e. Vacations
f. Other Personal Expenses
Total Personal Expenses
740.00
\$
\$
100.00
180.00
\$
\$
\$
T
\$
#6 \$ 1020.00
c. Car Fuel
d. Car Maintenance
e. Car Insurance
f. Other Transportation
Total Transportation
200.00
222.00
7. Other Expenses
a. Newspaper
4. Transportation
a. Public Transport
b. Car Loan
22.00
\$
8.50
\$
#7 \$
8.50
b.
\$
\$
237.75
C.
Total Other Expenses
\$ 140.QQ
\$
\$
#4 \$
3\$.33
64.17
Total Monthly Expenses
#8 \$ 2899.10
Income minus Expenses (#1 - #8) #9 \$ 559.33
480.25
3. W=9cm,L=llcm
4. 24 units2
5. (3.2, 7.8)
6. a. 161.57°
b. 41.81°, 138.19°
c. 28.8°, 129.3°
Continued
52
Senior 3 Pre-Calculus Mathematics
F-3
Exercise 41: Budgeting 1
7. Line segments AC and BC are perpendicular since BC is a tangent and AC is a
diameter. L ACB is bisected , making each angle 45°. Angle ADC is a right angle
since it is an inscribed angle subtended by a diameter . Therefore, L B is 45° since
the three angles in A BCD total 180°. Since L DCB and Z B are both 45°, the
sides 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
Senior 3 Pre-Calculus Mathematics
F-3
Exercise 42: Budgeting 2
1. Income
\$ 1306.93
a. Regular Monthly income
b. Spouse's Regular Monthly Income \$ 1345.50
\$
53,86
d. Other Income
#1 \$ 2706,29
Total Monthly Income
2. Housing Expenses
a. Mortgage or Rent
b. Property Tax
c. Home/Property Insurance
d. Repairs/Maintenance
e. Other Housing Expenses
Total Housing Expenses
\$
#2 \$
3. Utilities
a. Hydro
b. Gas
c. Phone
d. Water
e. Other
Total Utilities
\$ 225. 00
\$
18.60
\$
\$
\$
#3 \$
243.60
4. Transportation
a, Public Transport
b. Car Loan
c. Car Fuel
d. Car Maintenance
e. Car Insurance
f. Other Transportation
Total Transportation
725.00
\$
\$5:,\$0
25.83
\$
903.33
\$
\$
5. Personal Finances
a. Personal Loan
b. Investments
c. RRSP
d. Life Insurance
e. Charities
I. Credit Card Payments
g. Service Charges
h. Savings
i. Other Personal Finances
Total Personal Finances
6. Personal Expenses
a. Groceries
b. Clothing
c. Entertainment
e. Vacations
f, Other Personal Expenses
Total Personal Expenses
\$
\$
\$
\$
20.00
\$
\$
.00
\$
\$
\$
#5 \$
220.00
\$
\$
\$
\$
\$
#6 \$
778.33
\$
\$
12.00
200.00
7. Other Expenses
a. Newspaper
b. Babysitting
\$
C.
Total Other Expenses
\$
186.40
100.00
\$
\$
\$
#4 \$
M&
60.00
130.00
523.07
Total Monthly Expenses
525.00
54.17
50.00
70.83
78.33
#7 \$
212.00
#8 \$ 2850.33
Income minus Expenses (#1 - #8) #9
174 4)
Comments: Need to reduce expenses by \$174.04
2, a. 107.5 m, 280.1 m
b. 2790.4 m
3. (2, 2) and (-5, 5 ,J
4
-5±-0'89
4
Continued
54
Senior 3 Pre-Calculus Mathematics
F-3
Exercise 42: Budgeting 2
5. Since points (1, 2) and (1, -4) are on the same vertical line, they cannot belong to
6. Since DE is tangent at C, L 3 = Z 5 by the tangent-chord angle theorem. Since
AB 1/ DE, Z 4 -= L 3 because interior alternate angles of parallel lines are
congruent . 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
4
4
9
b,
y_--4x+25
3
3
15,129
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
360
55
Senior 3 Pre-Calculus Mathematics
F-5
Exercise 43 : Exponential Growth
b. About \$11 000 (exactly \$10 794)
1. a. 9 years
2,
Time (years)
Value (\$)
0
1
2
3
4
5
4000.00
4240.00
4494.40
4764.06
5049.91
5352.90
After 9 years = \$6800.
b. \$ 152.31
3, a. \$ 128.66
4. 1480
4 , -
5. Vertex:
D: {xl
xE
81);
Axis of Symmetry: x = 11 ; x-intercepts: 1
} , R: (yI y
81,
8
y
6. \$1041.67
7. x = -3, y = -1, z = 0
9. 16
10. - 4 , 2
8. (4, -3), (-4, -3), (3, 4), (-3, 4)
c. ±90°
b. 63.43°, 135°, 243.43°, 315°
11. a. 60°, 120°
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 has
length = p = AD, and is horizontal , C(t + p, q). (See
diagram .) Find the coordinates of the midpoints of
both AC and BD.
Both are (t + p
2
y
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°
56
x
Senior 3 Pre-Calculus Mathematics
F-5
Exercise 44 : Interest
1. a. \$7260
e. \$3343.75
b. \$720
c. \$450
f. 4.75%
g. \$1200
d. 7 years
b. \$7146.10
2. a. \$7080.00
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: f y
I
y< 169 or
24
169
24
1
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.
17
350
57
Senior 3 Pre-Calculus Mathematics
G-1
Exercise 45: Inductive and Deductive Reasoning
1. a. Inductive
b. Inductive
c. Deductive
e. Deductive
f. Deductive
g. Inductive
d. Inductive
2. C,E,F
3. Find the midpoints of any two sides. Find the slope between them. Calculate the
slope 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 the
same point are congruent . BD = CD since D is the midpoint. A ABD = A ACD
by SSS. Then L 1 = L 2 by corresponding parts of congruent triangles are
congruent.
8. a. (--oo, co)
9. y=3 x-
b. (---oo, 0)
c. [-4, 21
10
3
11. 95.7 °, 180°, 264.312. 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
Senior 3 Pre-Calculus Mathematics
Exercise 46 : Review 4
1. a. b=0,aE 91
b. Ifa=1-borb=1-a
2. y=a(x- 2)2+3,a>0
3. a. y= 8 (x+1)2-1
25
b.
x=4+45-,x`2
4. 146.3°, 326.35. a. 6.5 m
b. 3 sec
c. 0.3 sec
6. 300, 150°
7. x=2
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
Senior 3 Pre-Calculus Mathematics
G-2
Exercise 47: AND, OR, NOT, and Venn Diagrams
0
oxoxo
oxoxo
xoxox
oxoxo
xoxox
x0x0x
oxa
C
3. 12
5. 2
4. 218
8. a. 115.4°, 244.6°
6. 9
7. 0,3
c. 4°, 184°
b. 19.5°, 160.5°
9. 4.2 cm2
10. See budget form on next page.
11. 1 cm
12. i. a. up
ii, a. down
iii. a. up
b. (0, -3)
b. (-2, 3)
b. (5, -1)
c. X 0
c. x = -2
c. X=5
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
Senior 3 Pre-Calculus Mathematics
G-2
Exercise 47: AND, OR, NOT and Venn Diagrams
10.
1. Income
a. Regular Monthly Income
b. Spouse's Regular Monthly Income
d. Other Income
#1
Total Monthly Income
\$ 1541.67
\$ 1977.09
26.93
\$
\$
\$ 2645.69
2. Housing Expenses
a. Mortgage or Rent
b. Property Tax
c. Home/Property Insurance
d. Repairs/Maintenance
e. Other Housing Expenses
Total Housing Expenses
\$
\$
\$
\$
\$
#2 \$
3. Utilities
a. Hydro
b. Gas
c. Phone
d. Water
e. Other
Total Utilities
\$
\$
\$
\$
\$
#3 \$
4. Transportation
a. Public Transport
b. Car Loan
c. Car Fuel
d. Car Maintenance
e. Car Insurance
f. Other Transportation
Total Transportation
425.00
0.0
MIZ
506.00
41.75
61.66
10.50
41.75
155&Q
\$
\$
\$
105.00
\$
21.25
24,00
\$
\$
#4 \$ 150.25
5. Personal Finances
a. 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
6. Personal Expenses
a. Groceries
b. Clothing
c. Entertainment
e. Vacations
f. Other Personal Expenses
Total Personal Expenses
\$
345.00
\$
\$
\$
\$
\$_
\$
\$
#5 \$
345.00
\$
\$
\$
\$
#6 \$
420.00
66.25
27.50
13.75
70.63
598.33
7. Other Expenses
a. Newspaper
b.
C.
Total Other Expenses
\$.,.
#7 \$
Total Monthly Expenses
#8 \$ 1765.66
\$_
\$
10.42
10.42
Income minus Expenses (#1 - #8) #9 \$ 880.03
61
Senior 3 Pre-Calculus Mathematics
G-3
Exercise 48: Counterexamples
- 0.5)-°.5,
r
6. 10
5. x=41(1763=43x41)
10.
9. \$104.17
8. 43
11. Vertex: (-0.65, - 7.225); Axis of Symmetry: x = -0.65; x-intercepts: 11,
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 two
angles 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
c.
15. 4 34
17
16, LC= 16.3° or 163 .7°,LA=148. 7° or 1.3°, BC= 46 .2 or 2.0
270
62
10f mm
Senior 3 Pre-Calculus Mathematics
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 in
college.
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 two
sides 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 two
angles are not congruent,
b. If the angles opposite two sides of a triangle are not congruent, then the two
sides are not congruent.
c. If two angles are not congruent, then the two angles are not supplements of
congruent angles.
d. If two angles are not congruent, then the two angles are not complements of
congruent 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
7 . Two real roots
8. 14.2
Continued
63
Senior 3 Pre- Calculus Mathematics
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
13. (--, -4) v (-2,
64
b. (-1, 1)
c. 9
d. 4i
Senior 3 Pre-Calculus Mathematics
G-5
Exercise 50: Direct and Indirect Reasoning
A list of all the possibilities is made, and then, one by one, possibilities are
shown 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.
b.
c.
d.
e.
If a number is a multiple of 6, then it is a multiple of 3.
If a person was born in 1810, then that person is now dead.
If it is sunny, then my family always goes on a picnic.
If two angles are vertically opposite, then the angles are congruent.
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.
b.
c.
d.
e.
If a number is a multiple of 3, then it is a multiple of 6.
If a person is now dead, then that person was born in 1810.
If my family always goes on a picnic, then it is sunny.
If two angles are congruent, then they are vertically opposite.
If two angles are congruent, then they are base angles of
an isosceles triangle.
f. If a number is the sum of two primes, then it is even and
7. a.
b.
c.
d.
If a number is a not multiple of 3, then it is not a multiple of 6.
If a person is not dead , then that person was not born in 1810.
If my family does not go on a picnic, then it is not sunny.
If two angles are not congruent, then they are not vertically
opposite.
e. If two angles are not congruent, then they are not base angles
of an isosceles triangle.
f. If a number is not the sum of two primes, then it is neither even
nor larger than 2.
11. x = 12.25
False
False
larger than 2.
8. 84 m2
False
False
False
False
9.
3± 5
2
12.
3
10. -
True
True
True
True
True
True
-5
Continued
65
Senior 3 Pro-calculus Mathematics
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 the
angles 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
\$ 199.53
Balance from statement:
- deposits not cleared in statement
\$
45.00
\$
45,00
\$ 244.53
Subtract:
17.86
- withdrawals/cheques
not cleared in statement
Total subtractions:
.54.76
\$
72.62
Subtotal (minus):
This should agree with the balance shown in your record book:
72.62
\$ 171.91
\$ 171.91
b. y=12cosx
66
Senior 3 Pre-Calculus Mathematics
H-1
Exercise 51: Operations and Compositions Functions
b. 9
c. 9
d. 12
e. 14
f 12
g. 14
h. 14
i. 22
b. 6x2 -3
c. 4x + 3
1. a. 8
2. a. 12x2 + 12x + 1
e. -4
c. 20
b. 8
b. 4x2 - 12x + 10
5. a.
b.x -1
C.
4
3
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 sides
3' 2' 3 2
have the same slope, opposite sides are parallel.
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
Senior 3 Pre-Calculus Mathematics
H-2
Exercise 52: Inverse Functions
b. {(5, 4), (6, 6), (8, 7)
1. a. Divide by 5.
c. {(x
2. a.
I
) y=x32
d.
b. f 1
1(x) = 3x
b.
3. a. -2
( x =y) f y = 4- x}
2x+3
x
c.
+ x -1 x > 1
1
3
c. a
y
2+
2
4
-
-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)).
6. a. AC 11 EF makes L ACE = L FEC since they are interior alternate angles. This
makes L ACB = L FED since they are complements of these congruent
angles. These angles, plus the congruent angles and sides in the givens make
A ABC =- A FDE by ASA.
b. AB 11 DF since the interior alternate angles, Z ABC -= L FDE are congruent.
Continued
68
Senior 3 Pre-Calculus Mathematics
H-2
Exercise 52 : Inverse Functions
7.
1. Income
a. Regular Monthly Income
b. Spouse's Regular Monthly Income
d. Other Income
#1
Total Monthly Income
2. Housing Expenses
a. Mortgage or Rent
b. Property Tax
c. Home/Property Insurance
d. Repairs/Maintenance
e. Other Housing Expenses
Total Housing Expenses
3. Utilities
a. Hydra
b. Gas
c. Phone
d. Water
e. Other
Total Utilities
4. Transportation
a. Public Transport
b. Car Loan
c. Car Fuel
d. Car Maintenance
e. Car Insurance
f. Other Transportation
Total Transportation
\$ 1454.92
\$ 1463.58
\$
36.75
\$
\$ 2955.25
\$
\$
\$
\$
#2 \$
\$
\$
\$
#3 \$
\$
\$
\$
\$
\$
\$
#4 \$
71 5.440
x:00
20.78
936.18
23.00
305.20
328,20
206.10
120.00
15.ao
50.83
130.00
541.93
5. Personal Finances
a. 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
6. Personal Expenses
a. Groceries
b. Clothing
c. Entertainment
e. Vacations
f. Other Personal Expenses
Total Personal Expenses
\$
\$
\$
\$
\$
\$
\$
\$
\$
#5 \$
100.00
200,00
300.00
69.16
\$
216,67
41.67
\$
\$
166.67
\$
#6 \$ 1.0 ,4.11
7. Other Expenses
a. Newspaper
b. Babysitting
C.
Total Other Expenses
\$
\$
20.83
33,33
#7 \$
54.16
Total Monthly Expenses
#8 \$ 3214.64
Income minus Expenses (#1 - #8)
#9 \$ (259.39)
their financial position.
8. a. Discriminant = 1; Two real roots
b. Discriminant = -11; No real roots
9. (-2, -2)
Continued
69
Senior 3 Pre-Calculus Mathematics
H-2
Exercise 52: Inverse Functions
1 0. a.
-- --5
-5
d.
-4
-3
-3
-2
-1
-2
-1
-2
-1
_^
1
-4
-3
-2
--4
-5
-4
-3
-2
-3
-2
12. 240
70
1
0
1
Empty Set
0
1
2
3
4
5
6
2
3
4
5
6
2
3
4
5
1
1
-1
0
^_
1
T T
2
3
6
__
F
f
4
4
5
6
F- F j -^ ---- - ---------i--F--5
11. 5
13. a. 1
4
0
CY
-5
e.
'
- 1 }
-4
1
8
-1
-1
l0
0
1
2'
3
4
5
6
1
2
3
4
5
6
Senior 3 Pre-Calculus Mathematics
H-3
Exercise 53: Factor Theorem and Remainder Theorem
. 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
7. f 1 (x) _
X-5
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
13. 1, -1±/
14. f-'(x)=x+4
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
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
c. Rational
d. Polynomial
b.
3. Domain: {x x E 39t1; Range: ly I y e 3t}
ti 1o+
b.
4. a.
I
I
11
-5
5. a.
k
4
y
5
b.
-2+
Zeroes are ±1
The zeroes of y = x2 -1 and the
asymptotes of y = x21 are both x - ±1.
Continued
72
Senior 3 Pre-Calculus Mathematics
H-4
Exercise 54: Graphs of Polynomial and Rational Functions
b. -1, 2, 2
6. a. (x + 1 )(x - 2)(2x - 1)
C.
7. a. 225°, 315°
b. 120°, 240-
c. 128.2°, 231.8°
8. (2, 3), (-2, 3), (2, -3), (-2, -3)
9. 2 ± 4610. 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 ±
73
Senior 3 Pre-Calculus Mathematics
Exercise 55: Review 5
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
Senior 3 Pre-Calculus Mathematics
Exercise 56: Review 6
1. a. 2
b. 18
d. -1
2. a. -3
d. 4
e. 2
b. -10
c. 2
e. 4
f. 2
3. (x - 2)(2x - 1)(x + 3)
4. (2x - 1)(3x + 1)(2x + 3)
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
Senior 3 Pre-Calculus Mathematics
Exercise 57: Review 7
1. Yes, both = x6.
2. a. 7
£ 428
b. -3
c. 8
g. a
h. a
d.
3
4
e. 1000
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
d. Does not exist.
e. 4
g. Does not exist.
h. Does not exist.
2
b. 13
c. 48
e. 9a2 + 24a + 15
£ 3a2+1
h. a4 - 2a2
76
c. Does not exist.
Senior 3 Pre-Calculus Mathematics
Exercise 58: Cumulative Review
2+
_2
2
b. (0, 4)
c. minimum
f. none
g. 1
2.
d. 4
h.
X E 'R}
{ y) y E 9t,
e. 4
y > 4}
i. up
a. x=2
b. (2, 5)
c. maximum
d. 5
e. 13
f. none
g. {xx€9t}
y = 2(x-1)2 - 9
h.
}yIy€9t, y:5 -5 }
i.
down
a. x=1
b. (1, 9)
c. minimum
d. 9
e. -7
f
2±3J
2
g. {x Ix
E1 }
h. f y (y € 9t, y >_ -9}
i.
up
Continued
Senior 3 Pre-Calculus Mathematics
Exercise 58: Cumulative Review
b.
4. a.
Y
(2,3)
x
3x + 2y = 4
y=-(x-2)2+3
linear
parabola
(x - 2)2 + (y + 1)2 = 12
y=x2+5x+6
circle
parabola
5. y=(x+1)2-7
6.
49
4
7. 210 programs sold at \$35.00 for a maximum profit of \$7350.
8. Function
a. sine
b. cosine
c. tangent
I
+
+
II
+
-
+
-
III
-+
IV
+
Continued
78
Senior 3 Pre-Calculus Mathematics
Exercise 58 : Cumulative Review
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°
ii. 0°, 180-
11. a. i. 30°, 150°
b. i. 153 . 4°, 333.4°, 45°, 225°
12. a. 389. 82
c. undefined
ii. 90°, 270°, 120°, 240°
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 larger
than the opposite side.
17. LC =51.4°or 128 .6°
a=31 .64or 11.68
L 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°
b. 12 240°
c. 41
21. 106.81 or 34 rr
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°
j. 100°
k. 250°
1. 150°
i.
330°
m. 360°
Continued
79
Senior 3 Pre-Calculus Mathematics
Exercise 58: Cumulative Review
25. a. 40°
e. 140°
c. 110°
£ 100°
g. 220°
d. 4
c. 5
b. 9
26. a. 2
27, 6
b. 50°
d. 20°
e. 4
£3
Physics
Biology
b. Inductive
28. a. Deductive
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 is
equal to L ACB because of angle addition. AB = AC because sides opposite equal
angles are equal. This however contradicts the given AB = AC and therefore
L2# L4.
32. 3100
33. 13 nickels, 7 quarters
35. (-2, 1, 3)
36.
34.
^(2
, 4 , (-2, 3)
Continued
80
Senior 3 Pre-Calculus Mathematics
Exercise 58: Cumulative Review
xe91 , x?2, x<
37_ a. IxI xe9t, 2<x<4
38. a. {x I x E 9Z, x > 3, x <
b. Ix xE9, --4<_x<
1
10
3
.1} or (-o, - 1) u (3,
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)
b. 10
c. 0
d. - 5
e.
f. 4x2 - 6x
g. x' - 9x + 18
h. x2- 3x
i.
39. a. 18
4
1
x2
40. a. 4
d. 6x + 1
41. a. h-'(x) = x - 7
3
-X
4
b. 7
c. 7
e. -7
f. 4x-9
b. h-'(2)
3
Continued
81
Senior 3 Pre-Calculus Mathematics
Exercise 58: Cumulative Review
A
b.
42. a.
0
10
-F-10
D
10
-4--10
10
-10
10
43. 9
44. (x -1)(x + 5)
45. k = 2
46. 3
47. No real solution
Continued
82
Ex e rci se
Senior 3 pre-Calculus Mathematics
Cumulative Review
48. 5
49. 1 (not -3)
50. -1, 3
51. 52. 2,
3
54. -4, 3
55. 2,3
56. -1
57. -1
58. f-'(x) _
x
1-3x
59. 7.69
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
Senior 3 Pre-Calculus Mathematics
Exercise 58: Cumulative Review
65. a. 2
b. (-2, 3)
7
d. (x+2)2+ (y-3)2= 17
C. =
4
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 negative
reciprocals.
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
Senior 3 Pre-Calculus Mathematics
Exercise 58 : Cumu lative Review
75.
Monthly
Payment
Principal
Payment
Interest
8%'%% per year
Amount
Owing
\$5000.00
1
\$5000.00
\$300.00
\$33.33
\$4733.33
2
\$4733.33
\$300.00
\$31.56
\$4464.89
3
\$4464.89
\$300.00
\$29.77
\$4194.65
4
\$4194.65
\$300.00
\$27.96
\$3922.62
5
\$3922.62
\$300.00
\$26.15
\$3648.77
76. a.
b.
C.
d.
-2+
85
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