Name: __________________________ Date: _____________
1. Match the parametric equations to the curve they represent and the corresponding
rectangular equation. Adjust the domain of the resulting rectangular equation if
necessary.
2. Analyze the curves below and determine the way(s) the plane curves differ from each
other. Make use of sketchs of the graphs to aid in your analysis.
3. Which set of parametric equations represents the following line or conic?
Use x = h + a sec θ and y = k + b tan θ .
Hyperbola:
vertices
foci
( –1, 7 ) , ( 3, 7 ) ,
( –7, 7 ) , ( 9, 7 )
4. Which set of parametric equations represents the graph of the following rectangular
equation using t = 5 − x ?
y = x2 + 3
5. Use a graphing utility to graph the curve represented by the parametric equations.
Page 1
6. Match the graph to a set of parametric equations.
7. A projectile is launched from ground level at an angle of θ with the horizontal. The
initial velocity is v0 feet per second and the path of the projectile is modeled by the
parametric equations
x = (v0 cos θ )t and y = (v0 sin θ )t − 16t 2 .
Use a graphing utility to graph the paths of a projectile launched from ground level with
the values given for θ and v0 . Use the graph to approximate the maximum height and
range of the projectile to the nearest foot.
θ = 55° ,
v0 = 152 feet per second
Page 2
8. Which group of polar coordinates contains only representations of the point shown in
the graph?
9. Convert the point from polar coordinates to rectangular coordinates. Round answer to
three decimal places if necessary.
π⎞
⎛
⎜ –3, ⎟
3⎠
⎝
10. Which answer is a set of polar coordinates for the following rectangular coordinates?
Answers are rounded to three decimal places, if necessary.
( 7,
– 2)
11. Which answer is a rectangular form of the given polar equation?
r=
2
3 + sin θ
Page 3
12. Identify the type of polar graph.
13. Test for symmetry with respect to θ = π/2, the polar axis, and the pole.
r = 9 cos 7θ
14. Test for symmetry with respect to θ = π/2, the polar axis, and the pole.
r=
8
4 + sin θ
15. Find any zeros of r on the interval 0 ≤ θ < 2π .
r = 2 – 2 cos θ
16. Find the maximum value of r on the interval 0 ≤ θ < 2π .
r 2 = 9sin 2θ
Page 4
17. Sketch the graph of the polar equation using symmetry, zeros, maximum r-values and
any other additional points.
Use either grid below for your graph, whichever is more convenient.
18. Use a graphing utility to graph the polar equation. Describe the resulting graph.
r = 3cos ( 7θ − 4 )
Page 5
19. Find an interval for θ for which the graph is traced only once.
Page 6
20. Use a graphing utility to graph the polar equation and show that the indicated line is an
asymptote.
Name of Graph
Polar Equation
Asymptote
Conchoid
Use either grid below for your graph, whichever is more convenient.
Page 7
21. Write the polar equation of the conic for e = 1.5 . Identify the conic.
r=
5e
1 + e cos θ
22. Match the graph to its polar equation.
Page 8
23. Identify the conic and sketch its graph.
Use either grid below for your graph, whichever is more convenient.
Page 9
24. Use a graphing utility to graph the rotated conic.
Use either grid below for your graph, whichever is more convenient.
Page 10
25. Find a polar equation of the conic with the given characteristics and with one focus at
the pole.
Conic
Ellipse
Eccentricity
1
e=
3
Directrix
y=2
26. Find a polar equation of the conic with the given characteristics and with one focus at
the pole.
Conic
Hyperbola
Vertices
3π ⎞
⎛ 12 π ⎞ ⎛
⎜ , ⎟ , ⎜ –12,
⎟
2 ⎠
⎝ 7 2⎠ ⎝
27. Find a polar equation of the conic with the given characteristics and with the focus at
the pole.
Conic
Parabola
Vertex
⎛ π⎞
⎜ 5, ⎟
2⎠
⎝
28. If a new comet were discovered with an eccentricity of e ≈ 0.537 and whose length of
the major axis of orbit is approximately 3.08 astronomical units, how close would the
comet come to the sun (perihelion distance)? The perihelion distance is defined by the
equation r = a (1 − e) . Round answer to three decimal places, if necessary.
29. Determine the order of the matrix.
[9 –6 3]
30. Determine the order of the matrix.
⎡ 4 –9 –3⎤
⎢ –1 –7 –2 ⎥
⎣
⎦
31. Write the augmented matrix for the system of linear equations.
⎧x – 3 y – 4z = 8
⎪
–9 y + 3z = –6
⎨
⎪x
– 2 z = –1
⎩
Page 11
32. Write the system of linear equations represented by the augmented matrix. (Use
variables x, y, z, and w.)
⎡ –5 0 0 4 –7 ⎤
⎢ –9 5 0 0 6 ⎥
⎢
⎥
⎢ 0 –3 7 2 9 ⎥
⎢
⎥
⎣ 0 0 –1 –9 –7 ⎦
33. Fill in the blank using elementary row operations to form a row-equivalent matrix.
⎡ –1 –1 1 ⎤
⎢ 5 7 –2 ⎥
⎣
⎦
⎡ –1 –1 1⎤
⎢
⎥
3⎦
⎣0
34. Identify the elementary row operation being performed to obtain the new rowequivalent matrix.
Original Matrix
New Row-Equivalent Matrix
⎡ –3 2 9 ⎤
⎡ –18 37 49 ⎤
⎢ 3 –7 –8⎥
⎢ 3 –7 –8⎥
⎣
⎦
⎣
⎦
35. Perform the indicated row operations on the matrix. Show the final result.
⎡ 1 –1 4 ⎤
⎢ 2 –1 4 ⎥
⎢
⎥
⎢⎣ –7 4 –16 ⎥⎦
Add –2 times R1 to R2.
Add 7 times R1 to R3.
36. Determine whether the matrix is in row-echelon form. If it is, determine if it is also in
reduced row-echelon form.
⎡1 –1 9 6 ⎤
⎢0 1 0 9 ⎥
⎢
⎥
⎢⎣0 0 1 –9 ⎥⎦
Page 12
37. Write the matrix in reduced row-echelon form.
⎡ 8 –2 7 –42 ⎤
⎢ 5 –9 7 –65⎥
⎢
⎥
⎢⎣ –8 –2 7 22 ⎥⎦
38. Use the matrix capabilities of a graphing utility to write the matrix in reduced rowechelon form.
⎡ 2 6 –22 ⎤
⎢1 1 1 ⎥
⎢
⎥
⎢⎣1 6 –29 ⎥⎦
39. Write the system of linear equations represented by the augmented matrix. Then use
back-substitution to solve. (Use variables x, y, and z.)
⎡1 –5 5 27 ⎤
⎢0 1 –2 –9 ⎥
⎢
⎥
⎢⎣0 0 1 4 ⎥⎦
40. An augmented matrix that represents a system of linear equations (in variables x, y, and
z) has been reduced using Gauss-Jordan elimination. Write the solution represented by
the augmented matrix.
⎡1 0 0 –5⎤
⎢0 1 0 –4 ⎥
⎢
⎥
⎣⎢0 0 1 9 ⎦⎥
41. Use matrices to solve the system of equations (if possible). Use Gaussian elimination
with back-substitution or Gauss-Jordan elimination.
⎧ 8 x + 7 y – 3z = 47
⎪
⎨ –3x – 7 y + 9 z = –15
⎪ 3 x – 9 y – 9 z = –33
⎩
42. Use matrices to solve the system of equations (if possible). Use Gaussian elimination
with back-substitution or Gauss-Jordan elimination.
⎧ x – 4 y + 2 z = –2
⎨
⎩ –8 x + 31 y + z = 8
Page 13
43. Use the matrix capabilities of a graphing utility to reduce the augmented matrix
corresponding to the system of equations, and solve the system.
⎧ x + 4 y + 8 z = –53
⎪ – x – 6 y + 2 z = 25
⎪
⎨
⎪ –2 x – 8 y – 16 z = 106
⎪⎩ – x + 6 y + z = –43
44. Use the matrix capabilities of a graphing utility to reduce the augmented matrix
corresponding to the system of equations, and solve the system.
⎧ x – 3 y + 3 z = –5
⎪4 x – 8 y – 6 z = 80
⎪
⎨
⎪2 x – 6 y + 6 z = –5
⎪⎩4 x – y + 5 z = 0
45. Determine whether the two systems of linear equations yield the same solutions. If so,
find the solutions using matrices.
⎧ x – 5 y – z = 18
⎪
y + z = –9
⎨
⎪
z = –5
⎩
⎧ x + 4 y – 3z = 1
⎪
y + 6 z = –34
⎨
⎪
z = –5
⎩
46. The currents in an electrical network are given by the solutions of the system
⎧ I1 + I 2 − I3 = 0
⎪
+ 8I 3 = 44
⎨ I1
⎪
2 I 2 + I3 = 7
⎩
where I1, I2, and I3 are measured in amperes. Solve the system of equations using
matrices.
Page 14
47. Use a system of equations to find the specified equation that passes through the points.
Solve the system using matrices.
Parabola: y = ax2 + bx + c
48. Find x and y.
⎡ –4 x ⎤ ⎡ –4 8 ⎤
⎢ y –8⎥ = ⎢ –3 –8⎥
⎣
⎦ ⎣
⎦
49. Find x and y.
⎡x + 4
⎢ 9
⎢
⎢⎣ –1
8 y ⎤ ⎡ –2 x + 13 –7 16 ⎤
3 ⎥⎥ = ⎢⎢ 9
18 3 ⎥⎥
y + 9 6 ⎥⎦ ⎢⎣ –1
11 6 ⎥⎦
–7
6x
50. If possible, find A – B.
⎡ 3 –4 ⎤
⎡8 3⎤
A=⎢
,B = ⎢
⎥
⎥
⎣ 4 –3⎦
⎣ –8 –5⎦
51. If possible, find 4A + 2B.
⎡0 –3 –9 ⎤
⎡ –8 –5 –9 ⎤
A=⎢
,B = ⎢
⎥
⎥
⎣1 –4 6 ⎦
⎣ –3 0 –6 ⎦
52. Evaluate the expression.
⎡3 7 ⎤ ⎡ –3 9 ⎤ ⎡ –5 7 ⎤
⎢3 –5⎥ + ⎢ –7 –2 ⎥ + ⎢ –2 6 ⎥
⎣
⎦ ⎣
⎦ ⎣
⎦
Page 15
53. Evaluate the expression.
9
[1 8 2] + [ 4 –5 –3]
7
54. Use the matrix capabilities of a graphing utility to evaluate the expression.
⎛ ⎡3.98 9.76 ⎤ ⎡ –5.2 –2.93⎤ ⎞
–5 ⎜ ⎢
⎥−⎢
⎥⎟
⎝ ⎣ 5.5 –4.54 ⎦ ⎣ –4.4 –4.74 ⎦ ⎠
55. Use the matrix capabilities of a graphing utility to evaluate the expression.
⎡ –1⎤
⎡5⎤
⎡ –7 ⎤
2 ⎢⎢ –5⎥⎥ + 9 ⎢⎢ 4⎥⎥ – 2 ⎢⎢ –3⎥⎥
⎢⎣ –3⎥⎦
⎢⎣ 4⎥⎦
⎢⎣ 7 ⎥⎦
56. Solve for X in the equation given.
–22 ⎤
⎡ –7 –2 ⎤
⎡ 8
–5 X = A − B, A = ⎢
and B = ⎢
⎥
⎥
⎣ –6 1 ⎦
⎣ –16 11 ⎦
57. Solve for X in the equation given.
⎡ –2 –6 –9 ⎤
⎡ –5 –3 –2 ⎤
12 A + 16 B = 4 X , A = ⎢
and B = ⎢
⎥
⎥
⎣ –3 8 8 ⎦
⎣ 8 –2 7 ⎦
58. If possible, find AB.
⎡ 2 5⎤
⎡ –2 ⎤
A = ⎢⎢ 7 6 ⎥⎥ , B = ⎢ ⎥
⎣3⎦
⎢⎣ –5 0 ⎥⎦
59. If possible, find AB.
⎡ 7 –8⎤
⎡ –8⎤
A = ⎢ ⎥ , B = ⎢⎢ 3 –2 ⎥⎥
⎣4⎦
⎢⎣ –7 –1⎥⎦
Page 16
60. Use the matrix capabilities of a graphing utility to find AB, if possible.
⎡ 4 1 –7 ⎤
⎡ 7 –7 –6⎤
⎢
⎥
A = ⎢7 8 –3⎥ , B = ⎢⎢ 3 –6 –6⎥⎥
⎢⎣7 3 –6 ⎥⎦
⎢⎣ –1 –5 6 ⎥⎦
61. Use the matrix capabilities of a graphing utility to find AB, if possible.
⎡ –8 6 ⎤
⎢ 4 2⎥
⎥ , B = ⎡ –9 6 1 –6 ⎤
A=⎢
⎢ 8 –4 –2 –1⎥
⎢ –1 9 ⎥
⎣
⎦
⎢
⎥
⎣ –1 9 ⎦
62. Of the products AB, BA, A2, and B2, which ones are possible for the given matrices?
⎡ –4 ⎤
A = ⎢⎢ 3 ⎥⎥ , B = [5 6 1]
⎢⎣ –1⎥⎦
63. Of the products AB, BA, A2, and B2, which ones are possible for the given matrices?
⎡ –5 –5⎤
⎡ –7 –7 ⎤
A=⎢
,B = ⎢
⎥
⎥
⎣ –7 –9 ⎦
⎣ –4 –5 ⎦
64. Find A2. (Note: A2 = AA.)
⎡ –5 –2 ⎤
A=⎢
⎥
⎣2 4⎦
65. Find A2. (Note: A2 = AA.)
⎡3⎤
A=⎢ ⎥
⎣2⎦
66. Evaluate the expression.
⎡ 0 5 ⎤ ⎡ –3 5 ⎤ ⎡ 3 3 ⎤
⎢ 4 –5⎥ ⎢ –5 4 ⎥ ⎢ 0 –5⎥
⎣
⎦⎣
⎦⎣
⎦
Page 17
67. Evaluate the expression.
⎡6⎤
⎢ –3⎥ 5 6 + 9 2
] [ ])
⎢ ⎥ ([
⎢⎣ –7 ⎥⎦
68. Write the system of linear equations as a matrix equation AX = B, and use Gauss-Jordan
elimination on the augmented matrix [ A# B ] to solve for the matrix X.
⎧ x + 5 y = –44
⎨
⎩5 x – 4 y = 12
69. Write the system of linear equations as a matrix equation AX = B, and use Gauss-Jordan
elimination on the augmented matrix [ A# B ] to solve for the matrix X.
⎧ x1 + 8 x2 + 9 x3 = 12
⎪
⎨9 x1 – 7 x2 + 4 x3 = –60
⎪2 x – 2 x – 6 x = 18
2
3
⎩ 1
70. A lawn-and-garden store sells three types of fertilizer in 50-pound bags: Feed & Weed,
Winterizer Blend, and 17-17-17. The number of bags that were sold last weekend is
represented by A.
Feed & Weed
Winterizer
17-17-17
19
24 ⎤ Friday
⎡14
A = ⎢⎢11
21
30 ⎥⎥ Saturday
⎢⎣12
21
20 ⎥⎦ Sunday
The selling price per bag and the profit per bag for the three types of fertilizer are
represented by B (values are in dollars).
Selling Price
Profit
1.50 ⎤ Feed & Weed
⎡11.25
⎢
B = ⎢12.50
1.70 ⎥⎥ Winterizer
⎢⎣ 8.25
0.60 ⎥⎦ 17 − 17 − 17
Use matrix multiplication to find the total revenue and the total profit for the weekend.
⎡ 5 21⎤
71. Given matrix A = ⎢
. Find A−1 the inverse matrix.
⎥
⎣ –10 7 ⎦
Page 18
⎡ –2 6 2 ⎤
72. Given that matrix A = ⎢⎢ 4 10 0 ⎥⎥ . Find the inverse matrix.
⎢⎣ 6 2 –4 ⎥⎦
8 ⎤
⎡ 4
73. Find the inverse of the matrix ⎢
⎥.
⎣ –12 –20 ⎦
⎡3 3 3 ⎤
74. Find the inverse of the matrix ⎢⎢9 15 12 ⎥⎥ .
⎢⎣9 18 15⎥⎦
75. Use the matrix capabilities of a graphing utility to find the inverse of the matrix
⎡ −4 −5 3⎤
1 ⎢
−4 −8 3⎥⎥ (if it exists).
⎢
12
⎢⎣ 1 2 0 ⎥⎦
76. Use the matrix capabilities of a graphing utility to find the inverse of the matrix
⎡ 5 1 15 1 ⎤
⎢ −3 10 −3 20 ⎥
⎢
⎥ (if it exists).
⎢ 5 1 15 1 ⎥
⎢
⎥
⎣ −3 10 −3 20 ⎦
⎡ –1 –5⎤
77. Find the inverse of the matrix ⎢
⎥ (if it exists).
⎣ –3 5 ⎦
⎡ –32 –18⎤
78. Find the inverse of the matrix ⎢
⎥ (if it exists).
⎣ –16 –9 ⎦
Page 19
79. Solve the system of linear equations
⎧ –5 x – 2 y = 3
⎨
⎩ –10 x + 5 y = –2
⎡ 1
⎢– 9
using the inverse matrix ⎢
⎢– 2
⎢⎣ 9
–
2⎤
45 ⎥
⎥.
1 ⎥
9 ⎥⎦
80. Solve the system of linear equations
⎧7 x + 7 y + 7 z = 2
⎪
⎨21x + 35 y + 28 z = –6
⎪21x + 42 y + 35 z = 4
⎩
⎡ 1 1 −1⎤
1⎢
using the inverse matrix ⎢ −3 2 −1⎥⎥ .
7
⎢⎣ 3 −3 2 ⎥⎦
81. Solve the system of linear equations
⎧8 x1 – 16 x2 – 8 x3 – 16 x4
⎪24 x – 40 x – 16 x – 24 x
⎪ 1
2
3
4
⎨
⎪16 x1 – 40 x2 – 16 x3 – 40 x4
⎪⎩ –8 x1 + 32 x2 + 32 x3 + 88 x4
=
=
=
=
0
15
–10
0
⎡ −24 7 1 −2 ⎤
⎢
⎥
1 ⎢ −10 3 0 −1⎥
using the inverse matrix
.
8 ⎢ −29 7 3 −2 ⎥
⎢
⎥
⎣ 12 −3 −1 1 ⎦
⎧4 x + y = 2
82. Solve the system of linear equations ⎨
using an inverse matrix.
⎩8 x – 5 y = –6
⎧ –4 x + 12 y + 4 z = 1
⎪
= 2 using an inverse matrix.
83. Solve the system of linear equations ⎨8 x + 20 y
⎪12 x + 4 y – 8 z = –1
⎩
Page 20
84. The owner of the "Crazy 'Bout Nuts" shop wants to create his own blend of mixed nuts.
To do so, he mixes peanuts ($4 per pound), pecans ($5 per pound), and cashews ($9 per
pound) to obtain 140 pounds of mixed nuts costing $6 per pound. If he wants the
amount of peanuts to be twice that of the pecans, how many pounds of each type of nut
should he use? Use the matrix capabilities of a graphing utility to solve the resulting
system of linear equations. Round answers to nearest hundredth of a pound.
⎧ x + y + z = 140
⎪
⎨4 x + 5 y + 9 z = 840
⎪x − 2 y
= 0
⎩
85. A chemist has three acid solutions of varying strengths. The first contains 15% acid, the
second 24%, and the third 46%. He wishes to use all three solutions and obtain 52
gallons of a 37% acid solution. If the chemist wants to use twice as much 24% solution
as the 15% solution, how many gallons of each solution must he use? Round answers to
nearest gallon.
Page 21
86. Consider the circuit shown in the figure below. The currents I1 , I 2 , and I 3 ( in amperes )
+ 6 I 3 = E1
⎧3I1
⎪
2 I 2 + 6 I 3 = E2 , where
are the solutions to the system of linear equations ⎨
⎪I + I − I = 0
2
3
⎩1
E1 and E2 are voltages of 6 and 15 volts, respectively. Use the inverse of the coefficient
matrix of this system to find the unknown currents. Round answers to nearest tenth.
[Hint: If a current value turns out to be negative, it simply means that the current flow is
in the opposite direction from that indicated in the figure below.]
I1
I2
3Ω
E1
6Ω
2Ω
E2
I3
⎡ 3 0⎤
87. Find the determinant of the matrix ⎢
⎥.
⎣ –9 6 ⎦
2 ⎤
⎡3
⎢5
5 ⎥
88. Find the determinant of the matrix ⎢
⎥.
⎢ –3 – 2 ⎥
3 ⎦⎥
⎣⎢
89. Use the matrix capabilities of a graphing utility to find the determinant of the matrix
⎡ 0.6 –1.8 1.8⎤
⎢ 2.4 1.2
0 ⎥⎥ .
⎢
⎢⎣ –1.2 –4.2 3 ⎥⎦
Page 22
90. Use the matrix capabilities of a graphing utility to find the determinant of the matrix
⎡5 7.5 2.5⎤
⎢0 12.5 –5 ⎥ .
⎢
⎥
⎢⎣0 0
–5 ⎥⎦
⎡ –3 2 –8⎤
91. Find the minor M 13 and its cofactor C13 of the matrix ⎢⎢ 3 –2 6 ⎥⎥ .
⎢⎣ –1 3 –6 ⎥⎦
⎡ –9 6 –24 ⎤
92. Find the determinant of ⎢⎢ 9 –6 18 ⎥⎥ by the method of expansion by cofactors.
⎢⎣ –3 9 –18⎥⎦
⎡ 6 –2 0 ⎤
93. Find the determinant of ⎢⎢ –4 4 0 ⎥⎥ .
⎢⎣ 2 16 2 ⎥⎦
3
3⎤
⎡0
⎢
94. Find the determinant of ⎢ 6 –6 0 ⎥⎥ .
⎢⎣ –3 –24 –9 ⎥⎦
95. Use the matrix capabilities of a graphing utility to find the determinant of the matrix
⎡2 0 0 0 0⎤
⎢ 0 –4 0 0 0 ⎥
⎢
⎥
⎢0 0 6 0 0 ⎥ .
⎢
⎥
⎢ 0 0 0 –2 0 ⎥
⎢⎣ 0 0 0 0 4 ⎥⎦
Page 23
96. Use the matrix capabilities of a graphing utility to find the determinant of the matrix
–1 0 2
–3⎤
⎡1
⎢0 0 –1 –4 0 ⎥
⎢
⎥
⎢0 2 3 5
–1⎥ .
⎢
⎥
⎢ –1 0 0 –7 0 ⎥
⎢⎣0 0 0 –1 0 ⎥⎦
⎡0 –1 –2 ⎤
97. Given A = ⎢⎢3 2 –1⎥⎥ , find A .
⎢⎣0 –4 –1⎥⎦
⎡0 –1 –2 ⎤
⎡ –3 2 0 ⎤
⎢
⎥
98. Given A = ⎢3 2 –1⎥ and B = ⎢⎢ –1 1 –2 ⎥⎥ , find BA .
⎢⎣0 –4 –1⎥⎦
⎢⎣ –3 –1 –1⎥⎦
99. Evaluate the determinant of the matrix below to determine which of the following
makes the equation true.
w –7 x
=
8y
z
100. Solve for x given the following equation involving a determinant.
x+2 5
=0
−1 x + 8
101. Evaluate the determinant
8 x x ln x
in which the entries are functions.
8 7 + ln x
102. Use Cramer's Rule to solve the following system of linear equations:
⎧10 x – 15 y + 10 z = 5
⎪
⎨ –15 x + 10 y + 5 z = 4
⎪20 x + 5 y – 15 z = 8
⎩
Page 24
103. Use a determinant to find the area of the triangle shown below.
104. Determine a positive value for y such that a triangle with vertices (–6,8), (1,1) , and
(0, y ) has an area of 14 square units.
105. You inherited a triangular piece of property after your Uncle Izzy passed away. You
want to know the size of land, so you "step it off" to estimate the square footage. From
the southernmost vertex A, you travel north 240 feet then west 280 feet (for vertex C),
and from the southernmost vertex A, you travel 400 feet north then 90 feet west (for
vertex B). Use a graphing utility to approximate the number of square feet of land that
you have inherited.
B
C
A
Page 25
106. A triangular region of farmland has become overrun with deer and will be open to the
public for hunting to reduce the population. In order to know how many hunters to
allow on the land at one time, you need to know the area of the region in square miles.
In order to estimate the area of the region, you travel from the southernmost vertex C
north 30 miles then west 20 miles (for vertex B), and from the southernmost vertex C
you travel 56 miles north then 9 miles east (for vertex A). Use a graphing utility to
approximate the number of square miles of land.
107. Use a determinant to find a point collinear with the points (2, –7) and (8, –3) .
108. Use a determinant to find the point that is not collinear with the points
(1,9), and (–1, 7).
109. Use a determinant to find y such that (6, –15), (12, y ) , and (15, – 6) are collinear.
110. Use a determinant to find an equation of the line passing through the points
(–5, –21) and (–1, –5) .
Page 26
Answer Key
1.
2. Answers may vary.
Curve (a)
3.
Curve (b)
x = 1 + 2sec θ
y = 7 + 2 15 tan θ
4.
x = 5−t
y = (5 − t ) + 3
2
Page 27
5.
6.
7. maximum height ≈ 242 feet range ≈ 678 feet
8.
9.
10.
( –1.5, –2.598)
( 7.28, –0.278)
11. 9 x 2 + 8 y 2 + 4 y − 4 = 0
12.
13. symmetric with respect to the polar axis
14. symmetric with respect to θ = π/2
π 7π
15. zeros:
,
4 4
16. maximum value of r :
r = 3 where θ =
Page 28
π
5π
4 4
,
17.
18. rose curve with 7 petals, rotated 4 radians
19.
20.
21. hyperbola:
r=
7.5
1 + 1.5 cos θ
22.
Page 29
23.
24.
2
3 + sin θ
12
r=
3 + 4sin θ
10
r=
1 + sin θ
approximately 0.713 astronomical units
1×3
2×3
25. r =
26.
27.
28.
29.
30.
31.
⎡1 –3 –4 8 ⎤
⎢0 –9 3 –6 ⎥
⎢
⎥
⎢⎣1 0 –2 –1⎥⎦
Page 30
32.
+ 4w
⎧ –5 x
⎪ –9 x + 5 y
⎪
⎨
–3 y + 7 z + 2w
⎪
⎪⎩
– z – 9w
=
=
=
=
–7
6
9
–7
33.
⎡ –1 –1 1⎤
⎢ 0 2 3⎥
⎣
⎦
34. Add –5 times R2 to R1.
⎡1 –1 4 ⎤
⎢0 1 –4 ⎥
35.
⎢
⎥
⎢⎣0 –3 12 ⎥⎦
36. row-echelon form
37.
⎡1 0 0 –4 ⎤
⎢0 1 0 5 ⎥
⎢
⎥
⎢⎣0 0 1 0 ⎥⎦
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
⎡1 0 7 ⎤
⎢0 1 –6 ⎥
⎢
⎥
⎣⎢0 0 0 ⎦⎥
x = 2, y = –1, z = 4
x = –5, y = –4, z = 9
x = 4, y = 3, z = 2
(66a + 30, 17a + 8, a) where a is any real number
x = 3, y = –6, z = –4
no solution
The systems yield different solutions.
I1 = 4, I2 = 1, I3 = 5
48. x = 8, y = –3
49. x = 3, y = 2
⎡ –5 –7 ⎤
50. ⎢
⎥
⎣12 2 ⎦
⎡ –16 –22 –54 ⎤
51. ⎢
⎥
⎣ –2 –16 12 ⎦
⎡ –5 23⎤
52. ⎢
⎥
⎣ –6 –1⎦
Page 31
3⎤
⎡ 37 37
53. ⎢
– ⎥
7
7⎦
⎣7
⎡ –45.9 –63.45⎤
54. ⎢
–1 ⎥⎦
⎣ –49.5
55.
⎡57 ⎤
⎢32 ⎥
⎢ ⎥
⎢⎣16 ⎥⎦
⎡ 3 –4 ⎤
56. ⎢
⎥
⎣ –2 2 ⎦
⎡ –26 –30 –35⎤
57. ⎢
⎥
⎣ 23 16 52 ⎦
⎡11⎤
58. ⎢⎢ 4 ⎥⎥
⎢⎣10 ⎥⎦
59. not possible
–72 ⎤
⎡38 1
⎢
60. ⎢76 –82 –108⎥⎥
⎢⎣64 –37 –96 ⎥⎦
61.
62.
63.
64.
65.
66.
⎡120 –72 –20
⎢ –20 16
0
⎢
⎢ 81 –42 –19
⎢
⎣ 81 –42 –19
AB and BA only
AB, BA, A2, and B2
⎡ 21 2 ⎤
⎢ –2 12 ⎥
⎣
⎦
not possible
⎡ –75 –175⎤
⎢ 39
39 ⎥⎦
⎣
42 ⎤
–26 ⎥⎥
–3 ⎥
⎥
–3 ⎦
⎡ 84 48 ⎤
67. ⎢⎢ –42 –24 ⎥⎥
⎢⎣ –98 –56 ⎥⎦
⎡ –4 ⎤
68. X = ⎢ ⎥
⎣ –8⎦
Page 32
69.
70.
71.
72.
⎡1⎤
X = ⎢⎢ 7 ⎥⎥
⎢⎣ –5⎥⎦
total revenue $1789.25
total profit $203.60
3⎤
⎡1
− ⎥
⎢
35
35
A−1 = ⎢
⎥
1 ⎥
⎢2
⎢⎣ 49 49 ⎥⎦
⎡ −10 7 −5 ⎤
1 ⎢
−1
A = ⎢ 4 −1 2 ⎥⎥
18
⎢⎣ −13 10 −11⎥⎦
1 ⎡ −5 −2 ⎤
4 ⎢⎣ 3 1 ⎥⎦
⎡ 1 1 −1⎤
1⎢
74. ⎢ −3 2 −1⎥⎥
3
⎢⎣ 3 −3 2 ⎥⎦
⎡ –8 8 12 ⎤
75. ⎢⎢ 4 –4 0 ⎥⎥
⎢⎣ 0 4 16 ⎥⎦
73.
76. does not exist
1 ⎡5 5 ⎤
77. – ⎢
20 ⎣3 –1⎥⎦
78. does not exist
⎡ 11 ⎤
–
⎡ x ⎤ ⎢ 45 ⎥
79. ⎢ ⎥ = ⎢
⎥
⎣ y⎦ ⎢ – 8 ⎥
⎢⎣ 9 ⎥⎦
⎡ 8⎤
⎢– ⎥
⎡ x⎤ ⎢ 7 ⎥
22
80. ⎢⎢ y ⎥⎥ = ⎢ – ⎥
⎢ 7⎥
⎢⎣ z ⎥⎦ ⎢
⎥
⎢ 32 ⎥
⎣⎢ 7 ⎦⎥
Page 33
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
⎡ 95 ⎤
⎢ 8 ⎥
⎡ x1 ⎤ ⎢ 45 ⎥
⎥
⎢x ⎥ ⎢
⎢
⎢ 2⎥ = 8 ⎥
⎢ x3 ⎥ ⎢ 75 ⎥
⎥
⎢ ⎥ ⎢
⎢
⎥
8
x
⎣ 4⎦
⎢ 35 ⎥
⎢– ⎥
⎣ 8⎦
⎡1⎤
⎡x⎤ ⎢ 7 ⎥
⎢ y ⎥ = ⎢10 ⎥
⎣ ⎦ ⎢ ⎥
⎢⎣ 7 ⎥⎦
⎡1⎤
⎡ x⎤ ⎢ 4⎥
⎢ ⎥ ⎢ ⎥
⎢ y⎥ = ⎢ 0 ⎥
⎢⎣ z ⎥⎦ ⎢ 1 ⎥
⎢ ⎥
⎣2⎦
peanuts: 60.00 lb; pecans: 30.00 lb; cashews: 50.00 lb
15%: 6 gallons; 24%: 12 gallons; 46%: 33 gallons
I1 = –1.2 amps; I 2 = 2.8 amps; I 3 = 1.6 amps
18
4
5
–0.432
–312.5
3 –2
M 13 =
=7
–1 3
C13 = 7
–378
32
–324
384
–7
21
399
8y
z
99. −
w –7 x
100. x = –7, – 3
101. 56x
92.
93.
94.
95.
96.
97.
98.
Page 34
102. x =
17
29
151
, y=
, z=
15
21
105
103.
104.
105.
106.
107.
108.
109.
110.
6
45,200 square feet
695 square miles
(–1, –9)
(4,11)
y = –9
–4 x + y = –1
Page 35