Precalculus
Final Exam Review Questions
3 Days
DAY 1
1.) Find the values of θ for which
this equation is true: sin θ = 0
a. 180°k
b. 90° + 180°k
c. 0° + 360°k
d. 90° + 270°k
2.) Find the values of θ for which
this equation is true: cot θ = 0
a. 180°k
b. 90° + 180°k

c. 0° + 360°k
d. 90° + 270°k
cos 
cot  
sin 
3.) Which cosine equation has an
amplitude of 2, period of 180°, and
the phase shift of 0°.
a. y = 4cos 2θ
b. y = 2cos 4θ
c. y = 4cos 4θ

d. y = 2cos 2θ
360
 180
k
k 2
4.) Which sine equation has an
amplitude of 3, period of 360°, and
the phase shift of 90°.
a. y = cos (3θ – 180°)
360
 360
k
b. y = 3cos (θ – 30°)
k 1
c. y = 3cos (θ – 90°)
c
 90
1
d. y = 3cos (θ – 360°)
c  90
5.) Which graph represents this
equation y  2sin( x  45)
a.
b.

c.
d.
6.) Which graph represents this


equation y  1 cos
 180
2
a.
2
b.

c.
d.

7.) Find the values of x if
0° ≤ x ≤ 360°, and satisfy this
equation: x = arcsin ½
a. 30°, 150°
1
sin x 
2
b. 0°, 90°
x  ????
c. 30°, 210°
d. 0°, 30°,90°
8.) Find the values of x if
0° ≤ x ≤ 360°, and satisfy this
equation: x  arctan 3
3
a. 30°, 150°

b. 0°, 90°
c. 30°, 210°
d. 0°, 30°, 90°

3
tan x 
3
x  ????
9.) Evaluate sec (cos-1 ½). Assume
all angles are in Quadrant I (for cos,
sin, tan)
a. 3/2
b. 2
c. 1/2
d. 4/5
1
cos x 
2
x  60
sec(60)  2
10.) Evaluate cos (cot-1 4/3). Assume
all angles are in Quadrant I (for cos,
sin, tan)
a. 3/2
b. 2
c. 1/2
d. 4/5
5
3
4


2 
2 
tanarcsin
 cotarccos

2 
2 


11.) Evaluate
.
Assume all angles are in Quadrant I
(for cos, sin,
tan)

a. 3/2
b. 2
c.
3
3

d. 0
2
arcsin
 45
2
2
arccos
 45
2
tan( 45)  cot(45)
1 1
0

3
3 
1
tanarcsin
 cos

2
2 

12.) Evaluate
.
Assume all angles are in Quadrant I
(for cos, sin,
 tan)
a. 3/2
b. 2
c.
3
3
d. 0

3
arcsin
 60
2
3
1
cos
 30
2
tan(60  30)
 tan 60
3

3
13.) State the domain of y = Cos-1 x
a. All real numbers
b. -1 ≤ x ≤ 1
c. 0° < x < 180°
d. -90° < x < 90°
14.) State the domain of y = sin-1 x
a. All real numbers
b. -1 ≤ x ≤ 1
c. 0° < x < 180°
d. -90° < x < 90°
15.) State the domain of
y = Cos-1 x + 1
a. All real numbers
b. -1 ≤ x ≤ 1
c. 0° < x < 180°
d. -90° < x < 90°
16.) Determine a counterexample for
the following statement:
Arccos (x) = Arccos (-x) for -1 ≤ x ≤ 1
a. x = 2
b. x = -1
c. x = 0
d. x = 1
17.) Determine a counterexample for
the following statement:
Sin-1 (x) = -Sin-1 (-x) for -1 ≤ x ≤ 1
a. x = 2
IT’s a TRUE STATEMENT
b. x = -1
c. x = 0
d. x = 1
There is no counter example
18.) Find the inverse of the function:
y = 2x + 7
a.
x 2
y
7
b.
y  2x  7
c.
x 7
y
2
d.
x 2
y
7


19.) Write a cosine equation with a phase shift 0 to
represent a simple harmonic motion with initial
position = -7, amplitude = 7, and period = 4
a.
y  7cos 4t


b. y  7cos t
2
c.


d.
y  7cos 4t

y  7cos t
2
20.) Write a sine equation with a phase shift 0 to
represent a simple harmonic motion with initial
position = 0, amplitude = 22, and period = 12
a.
y  sin

6
t
b. y  22sin12t


c. y  22sin  t
6
d. y  22sin12t
21.) Solve for 0° ≤ θ ≤ 90°: If cot θ = 2, find tan θ
a. 1/2
b. 2/3
c. 1
d. 0
22.) Solve for 0° ≤ θ ≤ 90°: If tan θ = 1, find cot θ
a. 1/2
b. 2/3
c. 1
d. 0
23.) Solve for 0° ≤ θ ≤ 90°: If sin θ = 40/41,
find tan θ
a. 1/2
b. 0
c. 9/40
d. 40/9
24.) SIMPLIFY
a.
2
tan x



b.
c.
1
cot x
2

d. csc x
tan x csc x
sec x
25.) SIMPLIFY
a. 1 sin A



b. 1sin A
c.
cos A
2

d. cos A
cos 2 A
1 sin A
26.) Find a numerical value of one trig function.
2tan x  cot x
a.

2
cot x  
2
b.
2
cos x  
2
c.
3
cos x  
2
d.
2
tan x  
2


27.) Find a numerical value of one trig function.
sin x  2cos x
a.
csc x  2




b.
tan x  2
c.
cos x  2
d.
2
tan x  
2
28.) Use the sum or difference identity to find the
exact value of cos 255°
cos255  cos(225  30)
a.
6 2
4
b.
2 6
4

c.

d.
2 6
4
6 2
4
 cos225cos 30  sin 225sin 30
 2  3   2 1 
 
  
 
 2  2   2 2 
 6  2


4
4
29.) Use the sum or difference identity to find the
exact value of sin 195°
sin195  sin(150  45)
a.
6 2
4
b.
2 6
4

c.

d.
2 6
4
6 2
4
 sin150cos 45  cos150sin 45
1  2   3  2 
    
 
2  2   2  2 
2  6


4
4
30.) Use the sum or difference identity to find the
exact value of tan (-105°)
tan( 105)  tan( 45 150)
tan 45  tan150
1 tan 45tan150
1  3 3

 3 

1 1
3


3  3 


3

 3  3   3 

 
 

3  3   3  3  3 


3



a.
6 2
4
b.
2 3



c.
d.
2 3
6 2
4
3  3 3  3  9  3 3  3 3  3

 

3  3 3  3  9  3 3  3 3  3
12  6 3
6
2 3

31.) If tan x = 4/3 and cot y = 5/12, find sin (x – y)
5
4
x
x
a.
416
425
13
3
y
5
b.
56
33
c.
87
425


d.
16
65
sin( x  y)
4  5  3 12 
      
5 13  5 13 
20 36


65 65
12
32.) If sin x = 8/17 and tan y = 7/24, find cos (x – y)
17
x
x
a.
416
425
8
25
15
y
24
b.
56
33
c.
87
425


d.
16
65
cos(x  y)
15 24   8  7 
      
17 25  17 25 
360 56


425 425
7
33.) If tan θ = 5/12 and θ is in Quadrant III, find the
exact value of cos 2θ
12
a.
120
169
b.
56
33

c.

d.
119
169
16
65
x
5
θ
13
cos2  cos2   sin 2 
12   5 
     
13  13 
144 25


169 169
2
2
34.) If tan θ = 5/12 and θ is in Quadrant III, find the exac
value of sin 2θ
a.
120
169
b.
150
140
c.
119
169
d.
16
65


13
35.) Use a half-angle identity to find the value of sin 12
a.  2  3
2
b.
2 3

2
c.
2 3
2
d.
2 3
2



36.) Use a half-angle identity to find the value of cos
a.  2  3
2
b.
2 3

2
c.
2 3
2
d.
2 3
2



19
12
37.) Solve for 0° ≤ x ≤ 180°: 2sin 2 x  sin x  0
a. 0°, 90°
b. 30°, 150°
c. 0°, 180°
d. 120°

38.) Solve for 0° ≤ x ≤ 180°: 2cos2 x  sin x 1
a. 0°, 90°
b. 30°, 150°
c. 0°, 180°
d. 120°

DAY 2
39.) Solve for 0° ≤ x ≤ 180°:
a. 0°, 90°
b. 30°, 150°
c. 0°, 180°
d. 120°

2sin 2 x  cos 2 x  0
40.) Write the equation 5x – y + 3 = 0 in normal form.
a.
5
1
3
x
y
0
26
26
26
b.
1
5
3
x
y
0
26
26
26
c.
5
1
3
x
y
0
26
26
26
d.
5
1
3
x
y
0
26
26
26



41.) Write the equation 5x + y = 7 in normal form.
a.
5
1
3
x
y
0
26
26
26
b.
5
1
7
x
y
0
26
26
26
c.
5
1
3
x
y
0
26
26
26
d.
5
1
3
x
y
0
26
26
26



42.) Write the standard form of the equation of a line for
which the length of the normal is 3 and the
normal makes a 60° angle with the positive x-axis.
a. x  y  5 2  0



b. x  3y  6  0
c.
3x  y  4  0
d.
x  3y  64  0
43.) Write the standard form of the equation of a line for
which the length of the normal is 2 and the
normal makes a 150° angle with the positive x-axis.
a.


b. x  3y  6  0
c. x  y  5 2  0
d.

3x  y  4  0
x  3y  64  0
44.) Write the standard form of the equation of a line for
which the length of the normal is 32 and the
normal makes a 120° angle with the positive x-axis.
a.


b. x  3y  6  0
c. x  y  5 2  0
d.

3x  y  4  0
x  3y  64  0
45.) Find the distance in units between P(-3, 5) and
12x + 5y – 3 = 0

a.
0.9
b.
14
13
c.
4.2


d.
1.9
46.) Find the distance in units between P(-5, 0) and
x – 3y + 11 = 0

a.
0.9
b.
14
13
c.
4.2


d.
1.9
v
47.) v has a magnitude of 1.5 cm and a amplitude of
135°. Find the magnitude of its vertical and horizontal
components.
a.


c.
x  2.1 cm
b.
y  3.6 cm

x  2.3 cm
y  1.4 cm

x  1.8 cm
y  1.8 cm
x  1.1 cm
d. y  1.1 cm
v
v has a magnitude of 4.3 cm and a amplitude of
48.)
330°. Find the magnitude of its vertical and horizontal
components.

a.

c.
x  2.1 cm
b.
y  3.6 cm

x  2.3 cm
y  1.4 cm

x  1.8 cm
y  1.8 cm
x  1.1 cm
d. y  1.1 cm
v
49.) v has a magnitude
of 4.2 m.
v
the magnitude of w ?
a.

8.4 m




b.
12.6 m
c.
4.3 m
d.
2.6 m

If
v
v
w  3v , what is
v
50.) v has a magnitude
v of 4.2 m.
is the magnitude of w ?
a.

8.4 m




b.
12.6 m
c.
4.3 m
d.
2.6 m

v
v
If w  2v , what
51.) Find the ordered pair that represents the vector
from A(-2, 5) to B(1, 3). ?
a.
(3, -2)
b. (7, 5)
c. (5, -5)
d. (2, 6)
52.) Find the ordered pair that represents the vector
from A(-9, 2) to B(-4, -3). ?
a.
(3, -2)
b. (7, 5)
c. (5, -5)
d. (2, 6)
53.) If AB is a vector from A(12, -4) to B(19, 1), find
the magnitude of AB




a. 2 10

b. 5 2
c.
13
d.
74
54.) If AB is a vector from A(-9, 2) to B(-4, -3), find
the magnitude of AB




a. 2 10

b. 5 2
c.
13
d.
74
55.) Write the CD as the sum of unit vectors for points
C(-1, 2) and D(3, 5).
a.



v v
2i  7 j

v
v
b. 2i  7 j
v v
c. 2i  5 j
d.
v v
2i  5 j
v v
v
56.) Find an ordered pair to represent u  v  2w in
v
v
each equation if v = (1, -3, -8) and w = (3, 9, -1)
a. (2,12,7)




b. (5,39,29)
c. (5,21,6)
d. (1,33,38)


v
v
v
57.) Find an ordered pair to represent u  4v  3w in
v
v
each equation if v = (1, -3, -8) and w = (3, 9, -1)
a. (2,12,7)




b. (5,39,29)
c. (5,21,6)
d. (1,33,38)


58.) Find the ordered triple that represents the vector
from A(8, 1, 1) to B(4, 0, 1). ?
a.
(3, -2, 10)
b. (7, 5, -3)
c. (5, -5, 2)
d. (-4, -1, 0)
59.) Find the inner product of: (3, 5)  (4, -2)



a.
0
b.
9
c.
7
d.
2
60.) Find the inner product of: (4, 2)  (-3, 6)



a.
0
b.
9
c.
7
d.
2
61.) Find the inner product of: (7, -2, 4)  (3, 8, 1)



a.
0
b.
9
c.
7
d.
2
62.) Find the cross product: (7, 2, 1) x (2, 5, 3)
a. (9,6,0)

b. (4,12,16)
c. (1,19,31)


d. (8,19,2)
63.) Find the cross product: (-1, 0, 4) x (5, 2, -1)
a. (9,6,0)

b. (4,12,16)
c. (1,19,31)


d. (8,19,2)
64.) Name the polar curve of: r = 5 + 2cos θ
a. cardiod

b. spiral
c.
lim acon
d.
rose


65.) Name the polar curve of: r = 3 + 3sin θ
a. cardiod

b. spiral
c.
lim acon
d.
rose


66.) Name the polar curve of: r = 2θ
a. cardiod

b. spiral
c.
lim acon
d.
rose


 3 
67.) Convert 0,  into polar coordinates.
 2 

1 
,  
a. 
4 

  
1, 
b. 
 6 
c.

3  
 , 
2 2 
d. 2, 0
 3 1 
68.) Convert 
,  into polar coordinates.
 2 2 

1 
,  
a. 
4 

  
1, 
b. 
 6 
c.

3  
 , 
2 2 
d. 2, 0
69.) Convert
coordinates.
 13,  0.59 into rectangular
a. 1,1




b. 2,  5
c. 3,  2.01
d. 1.72, 3.01
70.) Convert
 2, 45 into rectangular coordinates.
a. 1,1




b. 2,  5
c. 3,  2.01
d. 1.72, 3.01
71.) Change this polar equation
rectangular equation.
a. x  y  144 
2



2
b. y  4
c.
x  2
d. y  x
  45 into a
72.) Change this polar equation
rectangular equation.
a. x  y  144 
2



2
b. y  4
c.
x  2
d. y  x
rsin   4 into a
2
2
x

y
7
73.) Change this rectangular equation
into a polar equation.
a. r   7



b. r  5csc
c. r 10sec
d.
5
r  sin 
2

74.) Change this rectangular equation
polar equation.
a. r   7



b. r  5csc
c. r 10sec
d.
5
r  sin 
2

x  10 into a
75.) Identify the conic section:
x 2  y 2  6y  8x  24  0
a. parabola




b. circle
c. hyperbola
d. ellipse
76.) Identify the conic section:
x 2  6x  4 y  9  0
a. parabola




b. circle
c. hyperbola
d. ellipse
77.) Identify the conic section:
27x 2  9y 2  6y 108x  82  0
a. parabola




b. circle
c. hyperbola
d. ellipse
78.) Identify the conic section:
x 2  4 y 2 10x 16y  5  0
a. parabola




b. circle
c. hyperbola
d. ellipse
79.) What is the correct vertex of this conic section :
x 2  2x 1  8y 16
a. (0, 4)




b. (2, 7)
c. (1,  3)
d. (1, 2)
80.) What is the correct vertex of this conic section :
y 2  6y  9  16 16x
a. (0, 4)




b. (2, 7)
c. (1,  3)
d. (1, 2)
81.) Which conic section has a directrix of y = 0
a.
1
2
y  x 1  2
8
b.
y  x 1  2
2


c.
d.
1 2
x y
2
1
2
y  x 1  5
5
82.) Which conic section has a directrix of x = 1/2
a.
1
2
y  x 1  2
8
b. y  x 1  2
2


c.
d.
1 2
x y
2
1
2
y  x 1  5
5

83.) Which conic section has focal point 4,  2  3



a.
x 2  2y 2  x  4 y 10 0
b.
4 x 2  y 2  32x  4 y  64  0
c.
4 x 2  9y 2  8x  36y  4  0
d.
x 2  5y 2  x  8y  50  0

DAY 3

84.) Which conic section has focal point
a. x 2  2y 2  x  4 y 10  0
1 

5, 2
Center is (1, 2)
Center is (½, 
-1)
b. 4 x 2  y 2  32x  4 y  64  0
Center is (4, -2)
c.
4 x 2  9y 2  8x  36y  4  0
Center is (1, 2)
d.
x 2  5y 2  x  8y  50  0
Center is (½, -4/5)
Just match
up the centers
85.) What is the eccentricity of this conic section:
4 x 2  9y 2  54 y  45  0
a.



5
3
9
81
4(x 2 )  9(y 2  6y  __)
 45  __

4(x 2 )  9(y  3) 2  36
b. 1
36
36
x
(y  3)

1
9
4
2
c.
36
4.5 
2
a3
d.
3
b2

c 5
Eccentricity = c/a
86.) What is the eccentricity of this conic section:
(y 1) 2  3(x  4)
a.
5

3
b.
1
c.
4.5
d.
3



87.) Which standard form equation of an hyperpola ha
slant asymptotes: y   3 x
2

2
a.
x2 y

1
1
9
b.
(x  6) 2 (y  3) 2

1
36
9

c.

d.
(x  2) 2 y 2

1
64
81
x2 y2

1
36 81
88.) Which standard form equation of an hyperpola ha
slant asymptotes: y   1 x
2

2
a.
x2 y

1
1
9
b.
(x  6) 2 (y  3) 2

1
36
9

c.

d.
(x  2) 2 y 2

1
64
81
x2 y2

1
36 81
1
3
89.) Express using radicals: 2 2 a 2b

a. x 6 y 3z 3
2
4 3
b. x y z

c.
2a3b5


d.
3
22 a2b2
5
2
90.) Express using radicals:
a. x 6 y 3z 3
2
4 3
b. x y z

c.
2a3b5


d.
3
22 a2b2

x y 
6
3
1
2
z
3
2
91.) Express using rational exponents:
1024a
1
a. 27 5 x 2 y



b. 12x
3
c. 32a
3
d.
c
7

y
2
3
5
3
92.) Write this in logarithmic form:
2  32
5

a.
1
log 5
 3
125
b.
1
log 6
 3
216
c. log 3 27  3


d. log 2 32  5
93.) Write this in logarithmic form:
1
6 
216
3
a.
1
log 5
 3
125
b. log 6


1
216
 3
c. log 3 27  3


d. log 2 32  5
94.) Evaluate each expression:
6



a.
3
b.
3
c.

5
d.
256
log6 5
95.) Evaluate each expression:
log 8 8



a.
3
b.
3
c.

5
d.
256
256
96.) Solve:
log 4 (3x)  log 4 27
a.
3
 b.
9
c.
5
d.
256



97.) Solve:



a.

0.8088
b.
0.4815
c.
2.2843
d. 10.0795
3  85
5x
98.) Solve:



a.

0.8088
b.
0.4815
c.
2.2843
d. 10.0795
1.8
x 5
 19.8
99.) Solve:
a.
0.8088
b.
0.4815
c.
2.2843




d. 10.0795
x  log 3 12.3
100.) Solve:



a.

17.63
b.
42.92
c.
19.52
d.
48.52
ln 4.5  ln e
0.031t
101.) Which sequence below is arithmetic. :
a. 5, 10, 20, …
b. 9, 3, 1, ..
c. 1.5, 3, 4.5, …
d. 2, 4, 8, …
102.) Which sequence below is geometric. :
a. 4, 8, 12, …
b. 9, 3, 1, ..
c. 1.5, 3, 4.5, …
d. -5, -3, -1, …
103.) Find 16th term in the sequence:
1.5, 2, 2.5, …
a. 9
b. -5
c. -25
d. 10
104.) Find 19th term in the sequence:
11, 9, 7, …
a. 9
b. -5
c. -25
d. 10
105.) Find 9th term in the sequence:
2, 2, 2 2,...




a.
12 2
b.
15 2
c.
16 2
d. 10
106.) What is the sum of the first 11 terms of the
arithmetic sequence: :
-3 – 1 + 1 + …
a.
56

b.
77

c.
74

d. 59
107.) What is the sum of the first 9 terms of the
geometric sequence: :
0.5 + 1 + 2 + …
a.
255.5
b.
235

c.
265.5

d. 270

108.) Evaluate the limit of
1  2n
lim
n
5n
a.
0

3

b.

c.
1
d.
2
5

109.) Evaluate the limit of
(n  2)(2n 1)
lim
n
n2
a.
2

3

b.

c.
1
d.
2
5

110.) Find the sum of this infinite geometric series:
2 1 1 1
    ...
3 3 6 12
0
a.




3
b.
c.
4
3
d.
2
5
111.) Find the sum of this infinite geometric series:
2 4 8
   ...
7 7 7
a.
0



c.
d.

3
b.
4
3
Does not exist
112.) Evaluate the limit of
x2  x  6
lim
n3
x 3
a.
2

3

b.

c.
1
d.
5

113.) Evaluate the limit of
x2
lim 4
x2 x  4
a.
2

0

b.

c.
1
d.
5

114.) Evaluate the limit of
1 for
f g(x) as x approaches
f(x) = 2x + 1 and g(x) = x – 3
a.



3
0
b.
c.
1
d.
5

115.) Evaluate the limit of
1 for
f g(x) as x approaches
f(x) = 3x – 4 and g(x) = 2x + 5
a.



b.
3
0
c.
17
d.
5

116.) Find the derivative of:
f (x)  (2x  3)(x  5)
a.


b.
2x(2x 2 1)
(x 2 1) 4
c.
4x  7


18x 2  26x  5
d.
4x
4 x 2 1
117.) Find the derivative of:
f (x)  x (x 1)
2

a.
18x 2  26x  5
b.
2x(2x 2 1)
(x 2 1) 4
c.
4x  7


d.

4x
4 x 2 1
2
3
118.) Find the derivative of:
f (x)  4 x 1
2

a.
18x 2  26x  5
b.
2x(2x 2 1)
(x 2 1) 4
c.
4x  7


d.

4x
4 x 2 1
119.) Find the integral of:
 2x 12 dx

a.
18x 2  26x  C
b.
1 5
x  5x  C
5
c.
x 2 12x  C



d.
2 1 x  C
2
120.) Find the integral of:


a.
18x 2  26x  C
b.
1 5
x  5x  C
5
c.
x 2 12x  C



d.
2 1 x  C
2
x 4  5 dx
121.) Find the integral of:

a.
18x 2  26x  C
b.
1 5
x  5x  C
5
c.
x 2 12x  C



d.
2 1 x  C
2

2x
1 x
2
dx