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Math C30
Final Exam
You have 2.5 hours to complete this exam.
Name __________________________________
Teacher ________________________________
Period ________
Date _______________
Page 1 of 15
PART I: Multiple Choice
Enter the CAPITAL letter of the response that corresponds to the correct
answer on the answer sheet provided. (30 marks)
1)
2)
3)
4)
5)
6)
The locus of points equidistant from a point and a line is a(an) ****.
A.
circle
B.
hyperbola
C.
ellipse
D.
parabola
The set of all points in a plane equidistant from a given point is a(an) ****.
A.
parabola
B.
circle
C.
ellipse
D.
hyperbola
The figure represented by x2 + 3y2 + 6x - 12y = 14 is a(an) ****.
A.
parabola
B.
circle
C.
ellipse
D.
hyperbola
The equation for the circle with centre (-3,0) and radius of 3 is ****.
A.
x2 + y2 + 3x = 9
B.
x2 + y2 + 6x = 9
C.
x2 + y2 + 3x = 0
D.
x2 + y2 + 6x = 0
The graph of the parabola 4y = -2y2 - x - 3 will be a curve that opens
A.
to the left
B.
to the right
C.
upward
D.
downward
The equation of a parabola with focus (-4, -3), and directrix x  2 is
A. y 
1
( x  3) 2  1
12
B. y  
1
( x  3) 2  1
12
C. x 
1
( y  3) 2  1
12
D. x  
1
( y  3) 2  1
12
Page 2 of 15
7)
8)
9)
An example of a quadrantal angle is ****.
A.
670°
B.
640°
C.
630°
D.
610°
If (6, 8) is a point on the terminal arm of angle  , drawn in standard position, then sin 
is equal to
A. 0.8000
B. 0.6000
C. 8.0000
D. 6.0000
cos 45 is equal to
A.
2
2
B. 1
C. –1
10)
11)
D.
2
If θ is an angle in standard position where sin θ < 0 and tan θ > 0, then the terminal arm
of θ lies in quadrant ****.
A.
I
B.
II
C.
III
D.
IV
If sin  
12
and  is an acute angle, then cos is equal to
13
A.
5
12
B.
5
13
C.
13
12
D.
12
35
12)
A.

to the left
3
C. 3 to the left
13)
14)
15)
16)
17)
Page 3 of 15
The phase shift of y  3 cos(3x   ) is
B.

to the right
3
D. 3 to the right
The reference angle associated with 186° is **** degrees.
A.
6
B.
74
C.
106
D.
254
If the angle θ is in standard position, and 90° < θ < 180°; which of the following
trigonometric functions have negative values?
A.
sin θ and cos θ
B.
sin θ and tan θ
C.
tan θ and cos θ
D.
cos θ
Two ways of describing the heading/bearing for the line of travel in the diagram are ***.
A)
210° or 330°
B)
W30°S or 210°
C)
S60°W or 210°
D)
S60°W or 240°
30°
The period of the function y = 4 cos ¼ x is ****.
A.
8π
B.
4π
C.
2
3
D.

3
If csc θ = 5/3 and tan θ < 0, then sec θ = ****.
A.
5/4
B.
4/5
C.
-3/5
D.
-5/4
Page 4 of 15
18)
19)
20)
21)
22)
Which of the following is an equation of an asymptote for y = tan x?
A.
x=0
B.
y=0
C.
x=

2
D.
y=

2
The graph shown BEST represents the equation of *****.
A)
y  sin 2 x
B)
y  cos 2 x
C)
1
y  sin x
2
D)
1
y  cos x
2
Triangle ABC, with A = 30° and b = 10 cm would have two solutions if the value of side
a, in cm, was equal to ****.
A.
4
B.
5
C.
8
D.
11
The area of a triangle ABC with b = 4 cm, c = 5 cm and A = 63° is **** cm2.
A.
17.82
B.
9.08
C.
8.91
D.
4.54
If points P (1,-4) and Q (9,-4) are endpoints of a diameter of a circle, then the center of
the circle is at****.
A.
(-1,0)
B.
(5,-4)
C.
(5,0)
D.
(-5,-4)
Page 5 of 15
( x  2)
( y  1)

 1 , are at
5
6
2
23)
24)
25)
The foci of the conic whose equation is
A. (2  11,1)
B. (2  11,1)
C. (2,1  11)
D. (2,1)
The vertices of the ellipse given by 4 x 2  25 y 2  16 x  150 y  141  0 are
A. (2,-2) and (2,8)
B. (0,-2) and (0,8)
C. (-3,3) and (7,3)
D. (-3, 0) and (7,0)
1
The amplitude and period of the graph of y  2 cos x are:
3
A. amplitude = 2; period is 6
C. amplitude =
26)
Express
1
; period is 2
3
B. amplitude = 2; period is
1
3
D. amplitude = 2; period is 3
sin 2
in simplest form
2 sin 
1  cos 2 
A.
2 sin 
27)
2
B.
sin 
2
C. cos
The solution to the system of equations
x2  4  y
3x 2  y 2  1
D. 1
represents the intersection points of
A. an ellipse and a line
B. a line and a parabola
C. a parabola and a hyperbola
D. a hyperbola and a circle
28)
Page 6 of 15
The maximum number of intersection points between a circle and an ellipse is
A. 1
29)
32)
34)
3
2
B.
3
4
C.
1
4
D.
3
2
cot 
B)
sin 
C)
cos
D)
sec
If you decide something by looking for a pattern and then making a prediction, you are
using
A)
inductive reasoning
B)
deductive reasoning
C)
mathematical deduction
D)
interpretive reasoning
How many counterexamples are required to disprove a theory?
A)
33)
D. 4
tan csc = *****.
A)
31)
C. 3
The numerical value of cos 2 30 is
A.
30)
B. 2
0
B)
1
C)
2
D)
3
Which of the following could represent an odd number?
A)
2 x2  6 x  4
B)
2 x2  x  4
C)
2x2  6x  1
D)
2 x2  6 x  2
Sam has discovered that the induction step of his proof is not valid. What does he know?
A)
His hypothesis must be incorrect.
B)
His basis step is incorrect.
C)
His deductive step is wrong.
D)
All of the above.
Page 7 of 15
PART II: Show all work in the space provided.
Chapter 1: Circular Functions
2
1)
a)
Express
in degrees. ( /1)
5
b)
Express 225° in radians. Leave your answer in terms of π. ( /1)
2)
Convert 11.5235 to degrees, minutes, and seconds. ( /1)
3)
Evaluate the following using exact values. ( /5)
a) cos 60°
b) sin 300°
c) tan 135°
d) csc π
Page 8 of 15
2
e) sec
3
4)
Find θ for the following if 0< θ  360°. Round to the nearest degree ( /8)
a) sin  
b) cos  
1
2
2
2
c) sin  cos  sin   0
d) 2sin 2   3sin   1  0
5)
A Ferris Wheel with a diameter of 12 m makes one revolution every 30 seconds. What is
the linear speed of a section of the rim wheel? (give answer in m/s correct to two decimal
places) ( /2)
Page 9 of 15
Chapter 2: Graphs of Circular Functions
1)
For the following, list the amplitude, period, phase shift and vertical shift. Then graph the
function. ( /5)
1
y  3cos x  2
2
Amplitude = _____________
Period = _____________
Phase shift = _____________
Vertical shift = _____________
2)
Write an equation for the sine function with the following properties. ( /2)
amplitude =
period = 2π
Phase shift =
1
2

6
Chapter 3: Triangles
1)
A vertical pole 3 m tall casts a shadow 1.8 m long. What is the angle of elevation of the
sun? ( /3)
Page 10 of 15
2)
In ABC, a = 2 m, b = 3 m, C = 30°. Find the area of ABC.
3)
Find the indicated part of ABC. Give all answers to one decimal place. ( /4)
a)
a = 6 cm, b = 10 cm, C = 60. Find c.
b)
b = 4 m, c = 2 m, B = 48. Find C.
( /2)
Chapter 4: Identities
1)
Express the following as a single trig ratio of a single angle, then evaluate, using exact
values. ( /4)
a)
sin 70° cos 20° + cos 70° sin 20°
b)
cos2 15 - sin2 15 
Page 11 of 15
2)
a) If α is a first quadrant angle and sin α = 3/5, find cos α. ( /2)
b) If β is a second quadrant angle and sin β = 5/13, find cos β. ( /2)
c) Use parts a) and b) to find sin (α + β). ( /2)
3)
Prove the following identities: ( /9)
a)
sin x cos x cot x  1  sin 2 x
b)
tan x(sin x cot x  cos x)  2sin x
Page 12 of 15
c)
sin y  sin y cos y  sin y
3
2
Chapter 5: Conics
1)
Find the centre and radius of the circle given by the equation x2 + y2 - 8x + 6y - 24 = 0.
( /3)
Centre =
Radius =
2)
Find each of the following for the parabola x  2 
1
2
 y  3 and graph it with
20
everything labeled. ( /7)
focal length (p) = ____________
vertex = ___________________
focus = ____________________
directrix = __________________
axis of symmetry = ___________
3)
Page 13 of 15
For the following system of equations, identify each conic, state the number of possible
answers, then solve the system. ( /3)
x 2  y 2  25
x2  y  5  0
4)
Conic: _______________
Conic: _______________
 x  3
Graph the ellipse
16
2
 y  2

25
Number of solutions: ____________
Solution set:___________________
2
 1 and find: ( /5)
center = ___________________
foci = ___________________
vertices (major axis) = ___________________
Page 14 of 15
Chapter 6: Proof
1.
Use mathematical induction to prove: ( /3)
1  4  7  . . .  (3n  2) 
2.
n(3n  1)
2
Use integer properties to show that the product of any 2 consecutive numbers is even.
( /2)
Page 15 of 15
Name: ________________
Teacher: _____________
Math C30 Multiple Choice Answer Sheet
1.
A
B
C
D
18.
A
B
C
D
2.
A
B
C
D
19.
A
B
C
D
3.
A
B
C
D
20.
A
B
C
D
4.
A
B
C
D
21.
A
B
C
D
5.
A
B
C
D
22.
A
B
C
D
6.
A
B
C
D
23.
A
B
C
D
7.
A
B
C
D
24.
A
B
C
D
8.
A
B
C
D
25.
A
B
C
D
9.
A
B
C
D
26.
A
B
C
D
10.
A
B
C
D
27.
A
B
C
D
11.
A
B
C
D
28.
A
B
C
D
12.
A
B
C
D
29.
A
B
C
D
13.
A
B
C
D
30.
A
B
C
D
14.
A
B
C
D
31.
A
B
C
D
15.
A
B
C
D
32.
A
B
C
D
16.
A
B
C
D
33.
A
B
C
D
17.
A
B
C
D
34.
A
B
C
D