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1080 Practice Tests Solutions
Chapter 1 Practice Test Solutions
1. m �3 � 2
1 � ��2� �13
2.
−12
−6
12
10
y � 3x � 7 or y � �3x � 7
y � 2 � �3x � 9
y � 2 � �3�x � 3�
Slope ��2 � ��5�
3 � 4�
�3
�1� �3
5. No, is not a function of For example, and both satisfy the equation.�0, �2��0, 2�x.y
3.
−9
−6
9
6
y � 3x � 2 or y � �3x � 2
y � 5 � �3x � 3
y � 5 � �3�x � 1� 4.
Slope of perpendicular line is
y �53 x � 7
y � 2 �53 �x � 3�
m �53.
y � �35 x �
75
5y � �3x � 7
3x � 5y � 7
6.
is not defined.
f �4� � �4 � 2��4 � 2�
�2
2� 1
f�2�
f�0� � �0 � 2��0 � 2�
�2
�2� �1 7. The domain of is all x � ±4.f�x� �
5
x2 � 16
8. The domain of consists of all satisfying
or t ≤ 4.4 � t ≥ 0
tg�t� � �4 � t 9.
is even.f�x� � 3 � x6
−6
−3
6
5
10.
is increasingon ��2, 2�.f
x
y
−1−2−4 1 2 3 4
−15
−20
5
10
15
20
f �x� � 12x � x3 11.
Relative minimum:
Relative maximum: ��0.577, 4.385��0.577, 3.615�
−6
−1
6
7
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Chapter 2 Practice Tests Solutions 1081
12. is a vertical shift of three units downward of y � x3.f�x� � x3 � 3 13. is a horizontal shift six units to the
right of y � �x.f�x� � �x � 6
14.
Domain: x ≥ 0
� g��x� � ��x �2� 2 � x � 2
�g � f ��x� � g� f�x�� 15.
The domain is all x � ±2.
� f
g ��x� �f�x�g�x�
�3x2
16 � x4
16.
−6
−4
6
4
f
g
y = x
�g � f��x� � g�3x � 1� ��3x � 1� � 1
3�
3x
3� x
� 3�x � 1
3 � � 1 � �x � 1� � 1 � x
� f � g��x� � f �x � 1
3 � 17.
−6
−4
6
4
y � �9 � x2
y2 � 9 � x2
x2 � 9 � y2
x � �9 � y2
y � �9 � x2, 0 ≤ x ≤ 3
18.
−6
−2
6
6
y � 0.882 � 0.912x
Chapter 2 Practice Test Solutions
1. intercepts:
intercept:
Vertex: �3, �4�
�0, 5�y-
x4 6 8
4
6
2−2
−2
−4
2
y�1, 0�, �5, 0�x- 2.
�b
2a�
90
2�.01�� 4500 units
a � 0.01, b � �90
3. Vertex:
Opening downward through
Standard form
� �2x2 � 4x � 5� �2�x2 � 2x � 1� � 7y � �2�x � 1�2 � 7
a � �2
5 � a � 7
5 � a�2 � 1�2 � 7
y � a�x � 1�2 � 7
�2, 5�
�1, 7�
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1082 Practice Tests Solutions
4. where is any real number.
y � ±�3x2 � 10x � 8�
ay � ±a�x � 2��3x � 4� 5. Leading coefficient:
Degree: 5
Moves down to the right and up to the left.
�3
6.
x � 0, x � ±1, x � ±2
0 � x�x � 1��x � 1��x � 2��x � 2�
0 � x�x2 � 1��x2 � 4�
0 � x�x4 � 5x2 � 4 �
0 � x5 � 5x3 � 4x 7.
f�x� � x3 � x2 � 6x
f�x� � x�x2 � x � 6�
f�x� � x�x � 3��x � 2�
8. Intercepts:
Moves up to the right.
Moves down to the left.
x4
−12
16
−16
21−3 −1
−8
y�0, 0�, �±2�3, 0�
x 0 1 2
y 16 11 0 �16�11
�1�2
9.
176
62x � 186
62x � 10
20x2 � 60x
20x2 � 2x
9x3 � 27x2
9x3 � 7x2
3x4 � 9x3
x � 3 ) 3x4 � 0x3 � 7x2 � 2x � 10
3x3 � 9x2 � 20x � 62 �176
x � 310.
5x � 13
�2x2 � 4x � 2
�2x2 � x � 11
x3 � 2x2 � x
x2 � 2x � 1 ) x3 � 0x2 � 0x � 11
x � 2 �5x � 13
x2 � 2x � 1
11.
3x5 � 13x4 � 12x � 1
x � 5� 3x4 � 2x3 � 10x2 � 50x � 262 �
1311
x � 5
�5
3
3
13
�15
�2
0
10
10
0
�50
�50
12
250
262
�1
�1310
�1311
12.
f ��6� � 15
�6
7
7
40
�42
�2
�12
12
0
150
15
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Chapter 2 Practice Tests Solutions 1083
13.
Possible rational roots:
is a zero.
Zeros: x � �2, x � �3, x � 5
0 � �x � 2��x � 3��x � 5�
0 � �x � 2��x2 � 2x � 15�
�2
�2
1
1
0
�2
�2
�19
4
�15
�30
30
0
±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30
0 � x3 � 19x � 30 14.
Possible rational roots:
is a zero.
Possible rational roots of
is a zero.
The zeros of are
Zeros:
x � �1
2�
�3
2i, x � �
1
2�
�3
2i
x � 3, x � �3,
x ��1 ± �3 i
2.x2 � x � 1
0 � �x � 3��x � 3��x2 � x � 1�
x � �3
�3
1
1
4
�3
1
4
�3
1
3
�3
0
x3 � 4x2 � 4x � 3: ±1, ±3
0 � �x � 3��x3 � 4x2 � 4x � 3�
x � 3
3
1
1
13
4
�812
4
�912
3
�9
9
0
±1, ±3, ±9
0 � x4 � x3 � 8x2 � 9x � 9
15.
Possible rational roots: ±1, ±3, ±5, ±15, ±12, ±3
2, ±52, ±15
2 , ±13, ±5
3, ±16, ±5
6
0 � 6x3 � 5x2 � 4x � 15
18.
� 1 � i
�2 � 2i1 � 1
2
1 � i�
21 � i
�1 � i1 � i
19.3 � i
2�
i � 14
�6 � 2i � i � 1
4�
54
�14
i
16.
Possible rational roots:
is a zero.
Zeros: x � 1, x �23, x � 5
0 � �x � 1��3x � 2��x � 5�
0 � �x � 1��3x2 � 17x � 10�
x � 1
1
3
3
�20
3
�17
27
�17
10
�10
10
0
±1, ±2, ±5, ±10, ±13, ±2
3, ±53, ±10
3
0 � 3x3 � 20x2 � 27x � 10
0 � x3 �203 x2 � 9x �
103 17.
Possible rational roots:
is a zero.
is a zero.
� �x � 1��x � 2��x � �5i��x � �5i� f �x� � �x � 1��x � 2��x2 � 5�x � �2
�2 1
1
2 �2
0
5 0
5
10�10
0
x � 1
1 1
1
11
2
3 2
5
5 5
10
�1010
0
±1, ±2, ±5, ±10
f �x� � x4 � x3 � 3x2 � 5x � 10
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1084 Practice Tests Solutions
22. z �kx2
�y
23.
Vertical asymptote:
Horizontal asymptote:
intercept: �1, 0�x-
y �1
2
–3 –2 –1 1 2 3
–2
–1
1
3
4
x
yx � 0
f �x� �x � 1
2x24.
Vertical asymptote:
Slant asymptote:
intercepts: �±2
�3, 0�x-
–3 –2 2 3
–3
1
2
3
x
yy � 3x
x � 0
f �x� �3x2 � 4
x
21. 3i
1
1
4
3i
4 � 3i
9
12i � 9
12i
36
�36
0
20.
� x5 � 14x4 � 83x3 � 256x2 � 406x � 260
� �x � 2��x2 � 6x � 10��x2 � 6x � 13�
� �x � 2���x � 3�2 � 1���x � 3�2 � 4�
f �x� � �x � 2��x � �3 � i���x � �3 � i���x � �3 � 2i���x � �3 � 2i��
25. is a horizontal asymptote since the degree ofthe numerator equals the degree of the denominator.There are no vertical asymptotes.
y � 8 26. is a vertical asymptote.
so is a slant asymptote.y � 4x � 2
4x2 � 2x � 7
x � 1� 4x � 2 �
9
x � 1
x � 1
27.
Vertical asymptote:
Horizontal asymptote:
intercept: �0, �1
5�y-
y � 0
x � 5
2 6 8 10 12
−6
−4
−2
2
4
6
x
yf�x� �x � 5
�x � 5�2�
1
x � 5
Chapter 3 Practice Test Solutions
1.
� �3�8 �5� 25 � 32
x � 853
x35 � 8 2.
x � �3
x � 1 � �4
3x�1 � 3�4
3x�1 �181
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Chapter 3 Practice Tests Solutions 1085
3.
−3 −2 −1 1 2 3−1
1
2
3
4
5
x
y
f�x� � 2�x � � 12�x
0 1 2
4 2 1 14
12f �x�
�1�2x
4.
−3 −2 −1 1 2 3−1
2
3
4
5
x
y
g�x� � ex � 1
0 1 2
1.14 1.37 2 3.72 8.39g�x�
�1�2x
5.
(a)
(b)
(c) A � 5000e�0.09��3� $6549.82
A � 5000�1 �0.09
4 �4�3�
$6530.25
A � 5000�1 �0.09
12 �12�3�
$6543.23
A � P�1 �r
n�nt
6.
log7 1
49� �2
7�2 �1
49
7.
x � �2
x � 4 � �6
2x�4 � 2�6
2x�4 �164
x � 4 � log2 164 8.
� �0.1464
� 14�3�0.3562� � 2�0.8271��
� 14�3 logb 2 � 2 logb 5�
� 14�logb 2
3 � logb 52�
� 14�logb 8 � logb 25�
logb 4� 8
25 �14 logb
825
9. 5 ln x �1
2 ln y � 6 ln z � ln x5 � ln �y � ln z6 � ln�x5z6
�y�
10. log9 28 �log 28
log 9 1.5166 11.
N � 100.6646 4.62
log10 N � 0.6646
12.
1 2 3 4
−2
−1
1
2
x
y 13. Domain:
x < �3 or x > 3
�x � 3��x � 3� > 0
x2 � 9 > 0
14.
1 3 4 5 6
−3
−2
−1
1
2
3
x
y
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1086 Practice Tests Solutions
15. since ln xln y
� logy x.ln x
ln y� ln�x � y� 16.
x � log5 41 �ln 41
ln 5 2.3074
5x � 41
17.
x � �1 or x � 2
0 � �x � 1��x � 2�
0 � x2 � x � 2
x � x2 � �2
5x�x2� 5�2
5x�x2�
125
x � x2 � log5 125 18.
(extraneous solution) x � �1
x � 4
�x � 1��x � 4� � 0
x2 � 3x � 4 � 0
x2 � 3x � 4
x�x � 3� � 22
log2�x�x � 3�� � 2
log2 x � log2�x � 3� � 2
19.
or
ex �2.4779ex 2.4779
xx ln 0.08392xx ln 11.9161
ex 0.08392ex 11.9161
ex �12 ± �144 � 4
2
e2x � 12ex � 1 � 0
e2x � 1 � 12ex
ex�ex � e�x� � 12ex
ex � e�x
3� 4 20.
t 5.3319 yr or 5 yr 4 mo
ln 2
0.13� t
ln 2 � 0.13t
2 � e0.13t
12,000 � 6000e0.13t
A � Pert
21. There are two points of intersection:
,
�1.731, 1.647�−4
−4
8
4�0.0169, �2.983�
22. y � 1.0597x1.9792
Chapter 4 Practice Test Solutions
1. 350� � 350�� �
180�� �35�
182.
5�
9�
5�
9�
180�
�� 100�
3.
135.2367�
135� 14� 12 � �135 �1460 �
123600�� 4.
�22� 34� 8
� ��22� 34� � 0.14�60� �
� �22� 34.14�
�22.569� � ��22� � 0.569�60�� �
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Chapter 3 Practice Tests Solutions 1087
5.
tan �y
x� ±
�5
2
x � 2, r � 3, y � ±�9 � 4 � ±�5
cos �2
36.
65� or 13�
36
� arcsin 0.9063
sin � 0.9063
7.
ta 20ºx �35
tan 20� 96.1617
tan 20� �35
x 8. is in Quadrant III.
Reference angle:6�
5� � �
�
5 or 36�
�6�
5,
9. csc 3.92 �1
sin 3.92 �1.4242 10. lies in Quadrant III.
so
sec ��37
�1 �6.0828.
y � �6, x � �1, r � �36 � 1 � �37,
tan � 6 �6
1,
11. Period:
Amplitude: 3
3ππ 5π 7πx
3
2
1
−1
−3
y
4� 12. Period:
Amplitude: 2
2ππ 3π 4πx
2
−1
−2
y
2� 13. Period:
ππx
2
2
1
y
�
2
14. Period:
2ππ 3πx
2
1
−1
−2
y
2� 15.
−4
−20
4
20
��
16.
−4
−40
4
40
��
17.
��
2
sin � 1
� arcsin 1 18.
�1.249 or �71.565º
tan � �3
� arctan��3�
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1088 Practice Tests Solutions
19.
sin ��19�35
0.736835
4
θ
x = 35 − 16
= 19
sin�arccos 4
�35� 20.
cos ��16 � x2
4
16 − x2
x4
θ
cos�arcsin x
4�
21. Given
b � 12 cos 40� 9.193
cos 40� �b
12
a � 12 sin 40� 7.713
sin 40� �a
12
B � 90� � 40� � 50�
A � 40�, c � 12 22. Given
b �21.3
tan 83.16� 2.555
tan 83.16� �21.3
b
c �21.3
sin 83.16� 21.453
sin 83.16� �21.3
c
A � 90� � 6.84� � 83.16�
B � 6.84�, a � 21.3
23. Given
B � 90� � 29.055� � 60.945�
A � arctan 5
9 29.055º
tan A �5
9
10.296
c � �25 � 81 � �106
a � 5, b � 9 24.
67°
x 20
x � 20 sin 67� 18.41 feet
sin 67� �x
2025.
250 ft5°
5°
x
0.541 mi
2857.513 feet
x �250
tan 5�
tan 5� �250
x
Chapter 5 Practice Test Solutions
1. is in Quadrant III.
cot x �11
4tan x �
4
11
sec x � ��137
11cos x � �
11
�137� �
11�137
137
csc x � ��137
4sin x � �
4
�137� �
4�137
137
y � �4, x � �11, r � �16 � 121 � �137
tan x �4
11, sec x < 0 ⇒ x 2.
�sec2 x � csc2 x
csc2 x � sec2 x� 1
�sec2 x � csc2 x
csc2 x �1
cos2 x
�sec2 x � csc2 x
csc2 x �1
sin2 x�
sin2 x
cos2 x
sec2 x � csc2 x
csc2 x�1 � tan2 x��
sec2 x � csc2 x
csc2 x � �csc2 x� tan2 x©
Hou
ghto
n M
ifflin
Com
pany
. All
right
s re
serv
ed.
Chapter 5 Practice Tests Solutions 1089
3.
� ln�tan2 � � 2 ln�tan �
� ln�sin cos
cos sin � � ln�sin2
cos2 � ln�tan � � ln�cot � � ln
�tan � �cot �
4. is true since
cos��
2� x� � sin x �
1
csc x.
cos��
2� x� �
1
csc x
5.
� sin2 x�1� � sin2 x
sin4 x � �sin2 x� cos2 x � sin2 x�sin2 x � cos2 x� 6. �csc x � 1��csc x � 1� � csc2 x � 1 � cot2 x
10. (a)
��2
4��3 � 1�
��3
2��2
2�
1
2��2
2
� sin 60� cos 45� � cos 60� sin 45�
sin 105� � sin�60� � 45�� (b)
�2�3 � 4
�2� 2 � �3
��3 � 1
1 � �3�
1 � �3
1 � �3�
2�3 � 1 � 3
1 � 3
tan 15� � tan�60� � 45�� �tan 60� � tan 45�
1 � tan 60� tan 45�
11. �sin 42�� cos 38� � �cos 42�� sin 38� � sin�42� � 38�� � sin 4�
12. tan� ��
4� �tan � tan��4�
1 � �tan � tan��4��
tan � 1
1 � tan �1��
1 � tan
1 � tan
13.
� �x��x� � ��1 � x2 ���1 � x2 � � x2 � �1 � x2� � 2x2 � 1
sin�arcsin x � arccos x� � sin�arcsin x� cos�arccos x� � cos�arcsin x� sin�arccos x�
14. (a)
(b) tan�300�� � tan�2�150��� �2 tan 150�
1 � tan2 150��
�2�33
1 � �13�� ��3
cos�120�� � cos�2�60��� � 2 cos2 60� � 1 � 2�1
2�2
� 1 � �1
2
7.cos2 x
1 � sin x�
1 � sin x
1 � sin x�
cos2 x�1 � sin x�1 � sin2 x
�cos2 x�1 � sin x�
cos2 x� 1 � sin x
8.
�2 � 2 cos
sin �1 � cos ��
2
sin � 2 csc
�1 � 2 cos � cos2 � sin2
sin �1 � cos �
1 � cos
sin �
sin
1 � cos �
�1 � cos �2 � sin2
sin �1 � cos �
9. tan4 x � 2 tan2 x � 1 � �tan2 x � 1�2 � �sec2 x�2 � sec4 x
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1090 Practice Tests Solutions
15. (a)
(b) tan �
12� tan
�6
2�
sin��6�1 � cos��6�
�12
1 � �32�
1
2 � �3� 2 � �3
sin 22.5� � sin 45�
2��1 � cos 45�
2��1 � �22
2��2 � �2
2
16. lies in Quadrant II
�� 2
10�
1
�5�
�5
5
cos
2��1 � cos
2��1 � �35�
2
⇒ cos � �3
5.sin �
4
5, 17.
�1
8�1 � cos 4x�
�1
8�2 � �1 � cos 4x��
�1
4�1 �1 � cos 4x
2 �
�1
4�1 � cos2 2x�
�sin2 x� cos2 x �1 � cos 2x
2�
1 � cos 2x
2
18. 6�sin 5� cos 2 � 6 12�sin�5 � 2� � sin�5 � 2��� � 3�sin 7 � sin 3�
19. sin�x � �� � sin�x � �� � 2�sin ��x � �� � �x � ���
2 � cos ��x � �� � �x � ���
2� 2 sin x cos � � �2 sin x
20.sin 9x � sin 5x
cos 9x � cos 5x�
2 sin 7x cos 2x
�2 sin 7x sin 2x� �
cos 2x
sin 2x� �cot 2x
21.
� 12�2�cos u� sin v� � �cos u� sin v
12�sin�u � v� � sin�u � v�� �
12 �sin u� cos v � �cos u� sin v � ��sin u� cos v � �cos u� sin v��
22.
or
x �7�
6 or
11�
6 x �
�
6 or
5�
6
sin x � �12
sin x �12
sin x � ±1
2
sin2 x �1
4
4 sin2 x � 1 23.
or
�2�
3 or
5�
3 �
�
4 or
5�
4
tan � ��3 tan � 1
�tan � 1��tan � �3� � 0
tan2 � ��3 � 1� tan � �3 � 0
24.
or
x ��
6 or
5�
6 x �
�
2 or
3�
2
sin x �12
cos x � 0
cos x�2 sin x � 1� � 0
2�sin x� cos x � cos x � 0
sin 2x � cos x
©H
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ton
Miff
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ny. A
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rese
rved
.
Chapter 6 Practice Tests Solutions 1091
25.
or
x � 0.6524 or 3.7940 x 1.3821 or 4.5237
tan x � 3 � �5 tan x � 3 � �5
tan x �6 ± �20
2� 3 ± �5
tan x ����6� ± ���6�2 � 4�1��4�
2�1�
tan2 x � 6 tan x � 4 � 0
Chapter 6 Practice Test Solutions
1.
c � sin 128�� 100
sin 12�� 379.012
a � sin 40�� 100
sin 12�� 309.164
C � 180� � �40� � 12�� � 128� 2.
b � sin 22.819�� 20
sin 150�� 15.513
B 180� � �150� � 7.181�� � 22.819�
A 7.181�
sin A � 5�sin 150�
20 � � 0.125
3.
6.894 square units
� 12�3��6� sin 130�
Area �12ab sin C 4.
Since and is acute, the triangle has nosolution.
Aa < h
a � 10
h � b sin A � 35 sin 22.5� 13.394
5.
C 180� � �62.627� � 73.847�� � 43.526�
B 73.847�
cos B ��49�2 � �38�2 � �53�2
2�49��38� 0.2782
A 62.627�
cos A ��53�2 � �38�2 � �49�2
2�53��38� 0.4598 6.
B 180� � �12.85� � 29�� � 138.15�
A 12.85�
cos A ��300�2 � �218�2 � �100�2
2�300��218� 0.97495
c 218
47,522.8176
c2 � �100�2 � �300�2 � 2�100��300� cos 29�
7.
� 11.273 square units
� �8.2�8.2 � 4.1��8.2 � 6.8��8.2 � 5.5�
Area � �s�s � a��s � b��s � c�
s �a � b � c
2�
4.1 � 6.8 � 5.5
2� 8.2 8.
12°°4070
x
168
x 190.442 miles
11,977.6266
x2 � �40�2 � �70�2 � 2�40��70� cos 168�
9. w � 4�3i � j� � 7��i � 2j� � 19i � 10j 10.
�5�34
34i �
3�34
34j
v�v�
�5i � 3j
�25 � 9�
5
�34i �
3
�34j
©H
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Miff
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ny. A
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hts
rese
rved
.
1092 Practice Tests Solutions
11.
96.116�
cos ��3
�61�13
�u� � �61, �v� � �13
u � v � 6�2� � 5��3� � �3
u � 6i � 5j, v � 2i � 3j 12.
x30°
v
y
4�i cos 30� � j sin 30�� � 4��3
2i �
1
2j�� �4�3, 2�
13. projvu � �u � v�v�2 �v �
�10
20��2, 4� � �1, �2�
14.
Since is in Quadrant IV,
z � 5�2�cos 315� � i sin 315��.
� 315�
z
tan ��5
5� �1
r � �25 � 25 � �50 � 5�2 15.
� �3�2 � 3�2i
z � 6���2
2� i
�2
2 �cos 225� � �
�2
2 sin 225� � �
�2
2
16.
� 28�cos 30� � i sin 30�� � 14�3 � 14i
�7�cos 23� � i sin 23����4�cos 7� � i sin 7��� � 7�4��cos�23� � 7�� � i sin�23� � 7���
17.9�cos
5�
4� i sin
5�
4 �3�cos � � i sin ��
�9
3�cos�5�
4� �� � i sin�5�
4� ��� � 3�cos
�
4� i sin
�
4� �3�2
2�
3�2
2i
18.
� 4096�cos 360� � i sin 360�� � 4096
�2 � 2i�8 � �2�2�cos 45� � i sin 45���8 � �2�2 �8�cos�8��45�� � i sin �8��45���
19.
The cube roots of are:
For
For
For k � 2, 3�8�cos �3 � 4�
3� i sin
�3 � 4�
3 � � 2�cos 13�
9� i sin
13�
9 �.
k � 1, 3�8�cos �3 � 2�
3� i sin
�3 � 2�
3 � � 2�cos 7�
9� i sin
7�
9 �.
k � 0, 3�8�cos �3
3� i sin
�3
3 � � 2�cos �
9� i sin
�
9�.
z
z � 8�cos �
3� i sin
�
3�, n � 3
©H
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ton
Miff
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ny. A
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hts
rese
rved
.
Chapter 7 Practice Tests Solutions 1093
20.
For
For
For
For k � 3, cos 3�2 � 6�
4� i sin
3�2 � 6�
4� cos
15�
8� i sin
15�
8.
k � 2, cos 3�2 � 4�
4� i sin
3�2 � 4�
4� cos
11�
8� i sin
11�
8.
k � 1, cos 3�2 � 2�
4� i sin
3�2 � 2�
4� cos
7�
8� i sin
7�
8.
k � 0, cos 3�2
4� i sin
3�2
4� cos
3�
8� i sin
3�
8.
x4 � �i � 1�cos 3�
2� i sin
3�
2 �
Chapter 7 Practice Test Solutions
1.
y � �3
x � 4
4x � 16
x � �3x � 15� � 1
� x3x
� y �
� y �
115
⇒ y � 3x � 15
2.
x � �1
y �23
�3y � 2�2 � 0
9y2 � 12y � 4 � 0
9y2 � 18y � 9 � 6y � 5
�3y � 3�2 � 6y � 5
� x � 3y �
x2 � 6y �
�35
⇒ x � 3y � 3
3.
z � 6 � x � y � 3
x � 18 � 4y � �2
y � 5
�21y � �105
6�18 � 4y� � 3y � 3
x � 18 � 4y
� x2x5x
�
�
�
yy
2y
�
�
�
z3zz
�
�
�
60
�3
⇒ z � 6 � x � y2x � y � 3�6 � x � y� � 05x � 2y � �6 � x � y� � �3
⇒⇒
�x6x
�
�
4y3y
�
�
�183
4.
y � 70 y � 40
x � 40 or x � 70
0 � �x � 40��x � 70�
0 � x2 � 110x � 2800
x�110 � x� � 2800
�x � y �
xy �
1102800
⇒ y � 110 � x 5.
Dimensions: 60� � 25�
y � 60 or y � 25
x � 25 or x � 60
0 � �x � 25��x � 60�
0 � x2 � 85x � 1500
x�85 � x� � 1500
�2x � 2y �
xy �
170
1500
⇒ y �170 � 2x
� 85 � x
2
©H
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ton
Miff
lin C
ompa
ny. A
ll rig
hts
rese
rved
.
1094 Practice Tests Solutions
6.
y �x � 23
3� �2
x � 17
�2xx
�
�
15y3y
�
�
423
⇒⇒
2x5x7x
�
�
15y15y
�
�
�
4115119
7.
�2319
�38
19�
15
19
y � 2 � x
x �45
57�
15
19
57x � 45
� x38x
�
�
y19y
� 2� 7
⇒⇒ 19x � 19y �
38x � 19y �
387
8.
y � �2��0.112 � 0.4x� � 0.20
x �0.0129
0.43� 0.03
�0.4x0.3x
�
�
0.5y0.7y
�
�
0.112�0.131
⇒⇒
0.28x0.15x0.43x
�
�
0.35y0.35y
�
�
�
0.0784�0.0655
0.0129
y2 ��0.131 � 0.3x�
0.7−1.5 1.5
−1
1y1 � 2�0.112 � 0.4x�
9. Let amount in 11% fund and amount in 13% fund.
y � $10,500
x � $6500
�0.02x � �130
0.11x � 0.13�17,000 � x� � 2080
0.11x � 0.13y � 2080
x � y � 17,000 ⇒ y � 17,000 � x
y �x �
�
10. Using a graphing utility, you obtain Analytically,
y � ax � b �1114 x �
17
b �14�1 � 2 �33
42�� � �17
a �3342 �
1114
4b �42a � �33
2b � 22a � 17 ⇒ �4b � 44a � �34
4b � 22a � 1 ⇒ �4b � 42a � �31
n � 4, �4
i�1
xi � 2, �4
i�1
yi � 1, �4
i�1
xi2 � 22, �
4
i�1
xi yi � 17
�1, 1�, ��1, �2�, ��2, �1�.�4, 3�,y � 0.7857x � 0.1429.
11.
Answer: y � �5x � 3z � 0
3Eq.2 � Eq.3�x � y
�3y�5y
�
� z �
�
�21525
�2Eq.1 � Eq.2�x � y�3y
4y
�
� z �
� 3z �
�215
�20
Equation 1Equation 2Equation 3
x �
2x �
yy
4y
�
� z �
� 3z �
�211
�20©
Hou
ghto
n M
ifflin
Com
pany
. All
right
s re
serv
ed.
Chapter 7 Practice Tests Solutions 1095
12.
Answer: z �14
y � �34
x �12
Eq.2 � Eq.3�4x � y
�3y� 5z �
� 7z �
16z �
444
Eq.1 � 2Eq.2�Eq.2 � Eq.3
�4x � y�3y
3y
� 5z �
� 7z �
� 9z �
440
Equation 1Equation 2Equation 3
4x �
2x �
2x �
yy
4y
� 5z �
� z �
� 8z �
400
13.
Let then x �3 � 3a
5 and y �
8 � 7a
5.z � a,
x �9 � 9z
15�
3 � 3z
5
15x � 2y � 19z � 9
12x � 2y � 10z � 4
13x � 2y � 10z � 5
y �8 � 7z
5
5y � 7z � 8
6x � 2y � 5z � 2 ⇒ �6x � y � 5z � �2
3x � 2y � 5z � 5 ⇒ �6x � 4y � 2z � 10�
14.
At
At
At
Thus, y � 2x2 � 3x � 1.
b � 3
a � 2
�2 � �a � b�2, 13�: 13 � a�2�2 � b�2� � 1 ⇒ 14 � 4a � 2b ⇒ �7 � �2a � b
�1, 4�: 4 � a�1�2 � b�1� � 1 ⇒ 5 � a � b ⇒ 5 � a � b
�0, �1�: �1 � a�0�2 � b�0� � c ⇒ c � �1
y � ax2 � bx � c passes through �0, �1�, �1, 4�, and �2, 13�.
15. passes through
Thus,
s �12�6�t2 � 16t � 25 � 3t2 � 16t � 25.
v0 � �16
s0 � 25
a � 6
At �3, 4�: 4 �92 a � 3v0 � s 0 ⇒
At �2, 5�: 5 � 2a � 2v0 � s0 ⇒
At �1, 12�: 12 �12a � v0 � s0 ⇒
�1, 12�, �2, 5�, and �3, 4�.s �12at2 � v0t � s0
2Eq.1 � Eq.2
3Eq.2 � 2Eq.3�
12a
�a
�3a
� v0 � s0
� s0
� s0
� 12
� 19
� 7
�Eq.2 � Eq.3�
12a
�a
�2a
� v0 � s0
� s0
�
�
�
12
19
�12
16.
2R2 � R1→ �1
0
0
1
�2
�3�
�3R1 � R2→ �1
0
�2
1
4
�3�
�1
3
�2
�5
4
9�
©H
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ton
Miff
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ny. A
ll rig
hts
rese
rved
.
1096 Practice Tests Solutions
17.
Answer: x � �4, y � 3
�6R2 � R1→ �1
0
0
1
..
.
..
.�4
3�
�
113R2→
�1
0
6
1
..
.
..
.14
3�
�2R1 � R2→ �
1
0
6
�13
..
.
..
.14
�39�
�R2 � R1→ �1
2
6
�1
..
.
..
.14
�11�
�3
2
5
�1
..
.
..
.3
�11�
2x � 5y � �11
3x � 5y � �13 18.
Answer: x � 6, y � �5
R2 � R1 →
�R2→
�R2 � R3→ �1
0
0
0
1
0
..
.
..
.
..
.
6
�5
0�
�3R1 � R2→
�2R1 � R3→ �1
0
0
1
�1
1
..
.
..
.
..
.
1
5
�5�
R3
R1
�1
3
2
1
2
3
..
.
..
.
..
.
1
8
�3�
�2
3
1
3
2
1
..
.
..
.
..
.
�3
8
1�
2x � 2y � �1
3x � 2y � �8
2x � 3y � �3
19.
Answer: x � 1, y � �2, z � �2
�3R3 � R1 →
6R3 � R2→
�14R3→
�1
0
0
0
1
0
0
0
1
..
.
..
.
..
.
1
�2
�2�
�R2 � R3→ �
1
0
0
0
1
0
3
�6
�4
..
.
..
.
..
.
�5
10
8�
�2R1 � R2→�3R1 � R3→
�1
0
0
0
1
1
3
�6
�10
..
.
..
.
..
.
�5
10
18�
�1
2
3
0
1
1
3
0
�1
..
.
..
.
..
.
�5
0
3�
3x � y � 5z � 3
2x � y � 3z � �0
2x � y � 3z � �5
�
�20. �1
2
4
0
5
�3� �1
0
�1
6
�7
2� � � �4
5
�12
6�
21.
� � �3
�27
13
�1�
3A � 5B � 3� 9
�4
1
8� � 5�6
3
�2
5� 22.
� � �4
�21
0
2�
� � 9
28
0
1� � �21
49
0
7� � �8
0
0
8�
� �3
7
0
1� �3
7
0
1� � �21
49
0
7� � �8
0
0
8�
f�A� � �3
7
0
1�2
� 7�3
7
0
1� � 8�1
0
0
1�
©H
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rese
rved
.
Chapter 7 Practice Tests Solutions 1097
23. False
� A2 � 3AB � BA � 3B2
�A � B��A � 3B� � A�A � 3B� � B�A � 3B�
24.
A�1 � ��5
3
2
�1�
2R2 � R1→
�R2→ �1
0
0
1
..
.
..
.�5
3
2
�1�
�3R1 � R2→
�1
0
2
�1
..
.
..
.1
�3
0
1�
�1
3
2
5
..
.
..
.1
0
0
1�
25.
A�1 � � 1
�3
3
�1
�1
2
12
1
�32�
12R3 � R1→R3 � R2→
�32R3→
�1
0
0
0
1
0
0
0
1
..
.
..
.
..
.
1
�3
3
�1
�1
2
12
1
�32�
�13R2 � R1→
13R2→
�4R2 � R3→
�1
0
0
0
1
0
13
23
�23
..
.
..
.
..
.
2
�1
�2
�13
13
�43
0
0
1�
�3R1 � R2→�6R1 � R3→
�1
0
0
1
3
4
1
2
2
..
.
..
.
..
.
1
�3
�6
0
1
0
0
0
1�
�1
3
6
1
6
10
1
5
8
..
.
..
.
..
.
1
0
0
0
1
0
0
0
1� 26. (a)
(b)
x � �19, y � 11
X � A�1B � ��5
3
2
�1��3
�2� � ��19
11� 3x � 5y � �2
x � 2y � �3
x � �18, y � 11
X � A�1B � ��5
3
2
�1��4
1� � ��18
11�
�2R2 � R1→
�R2→ �1
0
0
1
..
.
..
.�5
3
2
�1�
�3R1 � R2→�1
0
2
�1
..
.
..
.1
�3
0
1�
�1
3
2
5
..
.
..
.1
0
0
1� 3x � 5y � 1
x � 2y � 4
27. �63 �1
4� � 24 � ��3� � 27 28.
� 74
�156 3
9
2
�1
0
�5� � 1��45� � ��3���25� � ��1���44�
29. �1032 4
1
5
0
2
�2
�1
6
3
0
1
1� � �7 30. ��60000 4
5
0
0
0
3
1
2
0
0
0
4
7
9
0
6
8
3
2
1�� � 6�5��2��9��1� � 540
31.
� 15.5 square units
Area �1
2�053 7
0
9
1
1
1� �1
2 �31� 32.
or 3 x � y � 5 � 0
� x
2
�1
y
7
4
1
1
1� � 3x � 3y � 15 � 0©H
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hts
rese
rved
.
1098 Practice Tests Solutions
33. x �
� 4
11
�7
5� �62 �7
5� �97
4434. z �
�301 0
1
�1
1
3
2� �301 0
1
�1
1
4
0� �14
1135.
�12,769.747
77,515.530 0.1647
y �
�721.4
45.9
33.77
19.85� �721.4
45.9
�29.1
105.6�
Chapter 8 Practice Test Solutions
1.
Terms:1
3,
1
6,
1
20,
1
90,
1
504
a5 �2�5�7!
�10
5040�
1
504
a4 �2�4�6!
�8
720�
1
90
a3 �2�3�5!
�6
120�
1
20
a2 �2�2�4!
�4
24�
1
6
a1 �2�1�3!
�2
6�
1
3
an �2n
�n � 2�!
2. an �n � 3
3n3. �
6
i�1
�2i � 1� � 1 � 3 � 5 � 7 � 9 � 11 � 36
4.
Terms: 23, 21, 19, 17, 15
a5 � a4 � d � 15
a4 � a3 � d � 17
a3 � a2 � d � 19
a2 � a1 � d � 21
a1 � 23, d � �2 5.
a50 � 12 � �50 � 1�3 � 159
5an � a1 � �n � 1�d5a1 � 12, d � 3, n � 50
6.
S200 �200
2�1 � 200� � 20,100
Sn �n
2�a1 � an�
a200 � 200
a1 � 1 7.
Terms: 7, 14, 28, 56, 112
a5 � a1r4 � 112
a4 � a1r3 � 56
a3 � a1r2 � 28
a2 � a1r � 14
a1 � 7, r � 2
©H
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Miff
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ny. A
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.
Chapter 8 Practice Tests Solutions 1099
10. For Assume that Now for
Thus, for all integers n ≥ 1.1 � 2 � 3 � 4 � . . . � n �n�n � 1�
2
��k � 1��k � 2�
2.
�k�k � 1�
2�
2�k � 1�2
1 � 2 � 3 � 4 � . . . � k � �k � 1� �k�k � 1�
2� k � 1
n � k � 1,1 � 2 � 3 � 4 � . . . � k �k�k � 1�
2.n � 1, 1 �
1�1 � 1�2
.
8.
Sn �a1�1 � rn�
1 � r�
6�1 � �2�3�10�1 � �2�3�
17.6879
�9
n�0
6�2
3�n
, a1 � 6, r �2
3, n � 10 9.
S �a1
1 � r�
1
1 � 0.03�
1
0.97�
100
97 1.0309
�
n�0
�0.03�n, a1 � 1, r � 0.03
11. For Assume that Then
Thus, for all integers n ≥ 4.n! > 2n
� 2k�1.
�k � 1�! � �k � 1��k!� > �k � 1�2k > 2 � 2k
k! > 2k.n � 4, 4! > 24. 12. 13C4 �13!
�13 � 4�!4!� 715
13.
� x5 � 15x4 � 90x3 � 270x2 � 405x � 243
�x � 3�5 � x5 � 5x4�3� � 10x3�3�2 � 10x2�3�3 � 5x�3�4 � �3�5
14. 12C5x7��2�5 � �25,344x7 15. 30P4 �
30!
�30 � 4�!� 657,720
16. ways6! � 720 17. 12P3 � 1320
18.
� 636 �
16
P�2� � P�3� � P�4� �136 �
236 �
336 19. P�K, B10� �
452 � 2
51 �2
663
20. Let
P�A� � � 1 � P�A� 0.1395
P�A� � � 9971000�50 0.8605
A � probability of no faulty units.
©H
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ton
Miff
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ny. A
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rved
.
1100 Practice Tests Solutions
Chapter 9 Practice Test Solutions
1.
Vertex:
Focus:
Directrix: y � �3
�3, �1�
�3, �2�
�x � 3�2 � 4�1��y � 2� ⇒ p � 1
�x � 3�2 � 4y � 8
x2 � 6x � 9 � 4y � 1 � 9
x2 � 6x � 4y � 1 � 0 2. Vertex:
Focus:
Vertical axis; opens downward with
x2 � 4x � 4y � 24 � 0
x2 � 4x � 4 � �4y � 20
�x � 2�2 � 4��1��y � 5�
�x � h�2 � 4p�y � k�
p � �1
�2, �6�
�2, �5�
3.
Horizontal major axis
Center:
Foci:
Vertices:
Eccentricity: e ��3
2
�3, �4�, ��1, �4�
�1 ± �3, �4�(1, �4)
a � 2, b � 1, c � �3
�x � 1�2
4�
�y � 4�2
1� 1
�x � 1�2 � 4�y � 4�2 � 4
�x2 � 2x � 1� � 4�y2 � 8y � 16� � �61 � 1 � 64
x2 � 4y2 � 2x � 32y � 61 � 0 4. Vertices:
Eccentricity:
Center:
Vertical major axis
x2
27�
y2
36� 1
b2 � �6�2 � �3�2 � 27
a � 6, e �c
a�
c
6�
1
2 ⇒ c � 3
�0, 0�
e �1
2
�0, ±6�
5.
Center:
Vertical transverse axis
Vertices:
Foci:
Asymptotes: y � 4 ±1
4�x � 3�
��3, 4 ± �17���3, 5�, ��3, 3�
��3, 4�
a � 1, b � 4, c � �17
�y � 4�2
1�
�x � 3�2
16� 1
16�y � 4�2 � �x � 3�2 � 16
16�y2 � 8y � 16� � �x2 � 6x � 9� � �231 � 256 � 9
16y2 � x2 � 6x � 128y � 231 � 0
6. Vertices:
Foci:
Center:
Horizontal transverse axis
�0, 2�
�±5, 2�
�±3, 2�
x2
9�
�y � 2�2
16� 1
�x � 0�2
9�
�y � 2�2
16� 1
a � 3, c � 5, b � 4
©H
ough
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Miff
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ny. A
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rved
.
Chapter 9 Practice Tests Solutions 1101
7.
Ellipse centered at the origin
�x� �2
53�
�y� �2
52� 1
3�x� �2
5�
2�y� �2
5� 1
6�x� �2 � 4�y� �2 � 10 � 0
5�x� �2
2�
10x�y�
2�
5�y� �2
2� �x� �2 � �y� �2 �
5�x� �2
2�
10x�y�
2�
5�y� �2
2� 10 � 0
5�x� � y�
�2 �2
� 2�x� � y�
�2 ��x� � y�
�2 � � 5�x� � y�
�2 �2
� 10 � 0
y � x� sin �
4� y� cos
�
4�
x� � y�
�2
x � x� cos �
4� y� sin
�
4�
x� � y�
�2
2 ��
2 ⇒ �
�
4
cot 2 �5 � 5
2� 0
A � 5, B � 2, C � 5
x
y ′ x ′
−2 2
2
−2
y5x2 � 2xy � 5y2 � 10 � 0
8. (a)
Ellipse
B2 � 4AC � ��2�2 � 4�6��1� � �20 < 0
A � 6, B � �2, C � 1
6x2 � 2xy � y2 � 0 (b)
Parabola
B2 � 4AC � �4�2 � 4�1��4� � 0
A � 1, B � 4, C � 4
x2 � 4xy � 4y2 � x � y � 17 � 0
9.
�x � 3�2
4�
�y � 1�2
25� 1
�x � 3
�2 �2
� �y � 1
5 �2
� 1
x � 3
�2� sin ,
y � 1
5� cos
x � 3 � 2 sin , y � 1 � 5 cos 10.
y � �e2t �2� �x�2 � x2, x > 0, y > 0
x > 0, y > 0
x � e2t, y � e4t
11. Polar:
Rectangular: ��1, 1�
y � �2 sin 3�
4� �2� 1
�2� � 1
x � �2 cos 3�
4� �2��
1
�2� � �1
��2, 3�
4 � 12. Rectangular:
Polar: ��2, 5�
6 � or �2, 11�
6 �
�5�
6 or �
11�
6
tan ��1
�3� �
�3
3
r � ±���3 �2 � ��1�2 � ±2
��3, �1�
©H
ough
ton
Miff
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ny. A
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rese
rved
.
1102 Practice Tests Solutions
13. Rectangular:
r �12
4 cos � 3 sin
r�4 cos � 3 sin � � 12
Polar: 4r cos � 3r sin � 12
4x � 3y � 12 14.
Rectangular:
x2 � y2 � 5x � 0
x2 � y2 � 5x
r2 � 5r cos
Polar: r � 5 cos
15.
Cardioid
Symmetry: Polar axis
Maximum value of
Zero of r: r � 0 when � 0.
r � 2 when � �.�r�:
3π2
0π
1
1
1
(2, )π
π2r � 1 � cos
0
r 0 1 2 1
3�
2�
�
2
16.
Rose curve with four petals
Symmetry: Polar axis, and pole
Maximum value of
Zeros of r: r � 0 when � 0, �
2, �,
3�
2.
�r�: �r� � 5 when ��
4,
3�
4,
5�
4,
7�
4.
��
2,
3π2
0π4
4−5,
5,
5,
−5,
)
)
)
)
)
)
)
)
74π
54π
π4
34π
π2r � 5 sin 2
17.
so the graph is an ellipse.
3π2
0π1
1
π2
e �1
6< 1,
r �12
1 � �16� cos
r �3
6 � cos
0
r1
2
3
7
1
2
3
5
3�
2�
�
2
18. Parabola
Vertex:
Focus:
r �12
1 � sin
12 � p
6 �p
2
6 �p
1 � sin��2�
r �p
1 � sin
r �ep
1 � e sin
e � 1
�0, 0�
�6, �
2�
©H
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Miff
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ny. A
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.
Chapter 10 Practice Tests Solutions 1103
Chapter 10 Practice Test Solutions
1. Let
Side
Side
Side
26 � 21 � 5
BC 2 � AB2 � AC2
���1�2 � ��2 � 2�2 � ��1 � 4�2 � �1 � 16 � 9 � �26BC:
�02 � 22 � 12 � �5AC:
�12 � 22 � 42 � �21AB:
C � �0, �2, �1�.B � �1, 2, �4�,A � �0, 0, 0�,
2.
x2 � �y � 4�2 � �z � 1�2 � 25
�x � 0�2 � �y � 4�2 � �z � 1�2 � 52 3.
Center:
Radius: 4
��1, 0, 2�
�x � 1�2 � y2 � �z � 2�2 � 16
�x2 � 2x � 1� � y2 � �z2 � 4z � 4� � 1 � 4 � 11
4.
� ��11, �9, 17�
� �1, 0, �1� � �12, 9, �18�
u � 3v � �1, 0, �1� � 3�4, 3, �6� 5.
��12 v�� � �12 � 22 � ��3�2 � �14
12 v �12 �2, 4, �6� � �1, 2, �3�
6.
� 2 � 1 � 6 � 9
u � v � �2, 1, �3� � �1, 1, �2� 7. Because u and v are parallel.
v � ��3, �3, 3� � �3�1, 1, �1� � �3u,
8.
v � u � ��u � v� � ��2, �5, �1�
u � v � � i�1
1
j0
�1
k23� � �2, 5, 1� 9.
Volume � �u � �v � w�� � ��2� � 2
� �4 � 1 � 1 � �2
� 1��4� � 1��1� � 1�1�
u � �v � w� � �101 1�1
0
114�
10.
x � 2 � 2t, y � �3, z � 4 � t
v � ��2 � 0�, �3 � ��3�, 4 � 3� � �2, 0, 1� 11.
x � y � 1 � 0
x � 1 � y � 2 � 0
1�x � 1� � 1�y � 2� � 0�z � 3� � 0
12.
Plane:
x � 2y � z � 0
1�x � 0� � 2�y � 0� � �z � 0� � 0
n � AB\
� AC\
� � i11
j12
k13� � �1, �2, 1�
AB\
� �1, 1, 1�, AC\
� �1, 2, 3� 13.
n1 � n2 � 3 � 4 � 1 � 0 ⇒ Orthogonal planes
n1 � �1, 1, �1�, n2 � �3, �4, �1�
14. on plane,
D ��PQ
\
� n��n�
��1 � 2 � 5��1 � 4 � 1
�2�6
��63
PQ\
� �1, 1, �5�n � �1, 2, 1�, Q � �1, 1, 1�, P � �0, 0, 6�
©H
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ton
Miff
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ny. A
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rved
.
1104 Practice Tests Solutions
Chapter 11 Practice Test Solutions
1.
limx→3
x � 3x2 � 9
0.1667
2.9 2.99 3 3.01 3.1
0.1695 0.1669 ? 0.1664 0.1639f �x�
x 2.
−5 5
1
−0.5
limx→0
�x � 4 � 2
x
14
3. limx→2
ex�2 � e2�2 � e0 � 1 4.
� limx→1
�x2 � x � 1� � 3
limx→1
x3 � 1x � 1
� limx→1
�x � 1��x2 � x � 1�
x � 1
5.
−5 5
4
−2
limx→0
sin 5x
2x 2.5 6. The limit does not exist. If
then for and forx < �2.
f �x� � �1x > �2,f �x� � 1
f �x� ��x � 2�x � 2
,
7.
m � limh→0
1
�4 � h � 2�
1�4 � 2
�14
�1
�4 � h � 2, h � 0
�h
h��4 � h � 2�
��4 � h� � 4
h��4 � h � 2�
��4 � h � 2
h��4 � h � 2�4 � h � 2
��4 � h � 2
h
msec �f �4 � h� � f �4�
h8.
� limh→0
3hh
� limh→0
3 � 3
� limh→0
3x � 3h � 1 � 3x � 1
h
� limh→0
�3�x � h� � 1� � �3x � 1�
h
f ��x� � limh→0
f �x � h� � f �x�
h
9. (a)
(b)
(c) limx→
�x�1 � x
� �1
limx→�
x2
x2 � 3� 1
limx→
3x4 � 0 10.
limn→
an � limn→
1 � n2
2n2 � 1� �
12
a4 �1 � 16
33� �
1533
a1 � 0, a2 �1 � 48 � 1
� �13
, a3 �1 � 918 � 1
� �819
,
11. �25
i�1
i2 � �25
i�1
i �25�26��51�
6�
25�26�2
�25�26�
6�51 � 3� �
25�26��54�6
� 5850
©H
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Miff
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ny. A
ll rig
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rese
rved
.
Chapter 11 Practice Tests Solutions 1105
12.
limn→
S�n� �1
3
�n
i�1
i2
n3�
1
n3 �
n
i�1
i2 � 1
n3�n�n � 1��2n � 1�6 � �
2n2 � 3n � 1
6n2� S�n�
13. Width of rectangles:
Height:
A � limn→
An � 1 �1
3�
2
3
� 1 �1
n2 n�n � 1��2n � 1�
6An �
n
i�1
�1 �i2
n2�1
n� �
n
i�1
1
n� �
n
i�1
i2
n3
� 1 � � i
n�2
f �a ��b � a�i
n � � f � i
n�
b � a
n�
1
n
©H
ough
ton
Miff
lin C
ompa
ny. A
ll rig
hts
rese
rved
.
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