chapter 1 practice test solutions - king philip regional ...€¦ · chapter 2 practice tests...

26
1080 Practice Tests Solutions Chapter 1 Practice Test Solutions 1. m 3 2 1 2 1 3 2. 12 6 12 10 y 3x 7 or y 3x 7 y 2 3x 9 y 2 3x 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 3x 1 4. Slope of perpendicular line is y 5 3 x 7 y 2 5 3 x 3 m 5 3 . y 3 5 x 7 5 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 x 2 16 8. The domain of consists of all satisfying or t 4. 4 t 0 t gt 4 t 9. is even. f x 3 x 6 6 3 6 5 10. is increasing on 2, 2. f x y 1 2 4 1 2 3 4 15 20 5 10 15 20 f x 12x x 3 11. Relative minimum: Relative maximum: 0.577, 4.385 0.577, 3.615 6 1 6 7 © Houghton Mifflin Company. All rights reserved.

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Page 1: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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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Page 2: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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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Page 3: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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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Page 4: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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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Page 5: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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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Page 6: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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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Page 7: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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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Page 8: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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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Page 9: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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°

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©

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Page 10: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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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Page 11: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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

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Page 12: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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

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Page 13: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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

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Page 14: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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

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Page 15: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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.

Page 16: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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�

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Page 17: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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�

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Page 18: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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

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Page 19: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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

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Page 20: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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.

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Page 21: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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

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Page 22: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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�

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Page 23: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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

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�

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Page 24: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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�

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Page 25: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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

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Page 26: Chapter 1 Practice Test Solutions - King Philip Regional ...€¦ · Chapter 2 Practice Tests Solutions 1081 12. is a vertical shift of three units downward of y x3. f x x3 3 13

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

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