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Determine k so that the system
Kx + 15y = 12
2x + 5y = 1
Answer: k /=6
36. Determine k so that the system
8x + 2ky = 17
3x + 6y = 14
Answer: k /=8
37. Determine k so that the system
Kx + 4y = 10
3x + 6y = 15 has more the one solution
Answer: k = 2
38. Determine k so that the system
Kx – 2y = 11
5x + 2y = 5
Answer: k = 5;
39. Determine k so that the system
X + ky = 1
3x – 6y = 5
Answer: k = - 2;
40. Determine k so that the system
Kx + 15y = 12
2x + 5y = 1
Answer: k = 6;
41. Determine k so that the system
8x + 2ky = 17
3x + 6y = 14
Answer: k = 8;
42. Let A= 1 0 4 ,B= 0 0 3 Find AB(t)
0 2 1 2 1 0
Answer: AB = 12 2
3 0
43. Let A = (0 1 2),B = (3 4 2). Find AB(t)
Answer: AB(t) = 8
44. Let A = (0 1 2), B = (4 | 5| 3)Find AB(t)
Answer: AB(t) doesn’t exist
45. Let A = ( 2 0 | 0 1), f(x) = x2 – 2x + 3. Find f(A)
f(A) =( 3 0|0 2)
46. Let A = (3 0 | 0 3), f(x) = x3 – 9x + 1. Find f(A)
Answer: f(A) = (1 0 | 0 1)
47 Let A = (1 0 | 0 5), f(x) = 2x2 – 10 x + 10. Find f(A)
Answer: (2 0 | 0 10)
48. Given A = (1 3 | 0 4). Find a vector u = (x|y) such that Au = 2u
Answer: u = (0|0)
49. Given A = (2 0| 0 5). Find a vector u = (x|y) such that Au = 4u
Answer: u = (0|0)
50. Given A = (7 4| 0 3). Find a vector u = (x|y) such that Au = 6u
Answer: u = (0|0)
51. Write the vector v = (2,4) as a linear combination of the vectors e1 = (1,1), e2 = (0,1)
Answer: v = 2e1 + 2e2;
52. Write the vector v = (1, -2) as a linear combination of the vectors e1 = (1,1), e2 = (0,1)
Answer: v = e1 - 3e2;
53. Write the vector v = (3, -1) as a linear combination of the vectors e1 = (1,0), e2 = (1,1)
Answer: v = 4e1 - e2;
54. Write the vector v = (5, 2) as a linear combination of the vectors e1 = (1,0), e2 = (1,1)
Answer: v = 3e1 -+2e2;
55. Write M = (3 1 | 1 -1) as a linear combination of matrices
A = (1 1| 1 0), B = (0 0 | 1 1), C = (0 2| 0 -1)
Answer: M = 3A – 2B – C;
56. Write M = (4 0 | 0 -6) as a linear combination of matrices
A = (4 0 | 0 0), B = (0 0 | 0 1),
Answer: M = A – 6B;
57. Write M = (0 5 | 2 0) as a linear combination of matrices
A = (0 1 | 0 0), B = (0 0 | -1 0)
Answer: M = 5A – 2B;
58. Write M = (0 8 | 0 -3) as a linear combination of matrices
A = (0 2 | 0 0), B = (0 0 | 0 -1)
Answer: M = 4A + 3B;
59. Find the rank matrix A = (1 3 | 0 2 | 1 1 )
Answer: r(A) = 2;
60. Find the rank matrix A = (2 0 1 | 0 1 1)
Answer: r(A) = 2;
61. Find the rank matrix A = (2 4 0 | 0 1 5 | 0 1 2)
Answer: r(A) = 3;
62. Find the rank matrix A = (7 2 5 | 0 4 3 | 0 8 6)
Answer: r(A) = 2;
63. Find the rank matrix A = (1 3 0 | -2 -4 3 | 1 1 -3)
Answer: r(A) = 2;
64. Find the rank matrix A = (1 3 7 | -2 -4 0| 1 0 0)
Answer: r(A) = 3;
65. Find the rank matrix A = (2 1 6 2 | 0 3 5 0 | 0 0 1 2 | 0 0 1 1 )
Answer: r(A) = 4;
66. Find the rank matrix A = (2 1 6 2 | 1 3 5 0 | 7 2 0 0 | 3 0 0 0 )
Answer: r(A) = 4;
67. Find a dimension of the solution space W of the homogeneous system
X – y + z = 0
X + y + 2z = 0
Answer: dim W = 1
68. Find a dimension of the solution space W of the homogeneous system
X – 5y + 2z = 0
3X - 15 y + 6z = 0
Answer: dim W = 2
69. Find a dimension of the solution space W of the homogeneous system
X +4y = 0
3x – 5y = 0
Answer: dim W = 0
70. Find a dimension of the solution space W of the homogeneous system
X + y + 2z + w = 0
3X + 3 y + 6z + 3w= 0
Answer: dim W = 3;
71. Find a dimension of the subspace W of R3 generated by vectors
U = (1,0,1), v = (1,1,1)
Answer: dim W = 2;
72. Find a dimension of the subspace W of R3 generated by vectors
U = (1,2,1), v = (0,1,4)
Answer: dim W = 2;
73. Find a dimension of the subspace W of R3 generated by vectors
U = (1,0,1), v = (5,0,5)
Answer: dim W = 1;
74. Find a dimension of the subspace W of R3 generated by vectors
U = (1,1,0), v = (1,1,1)
Answer: dim W = 3;
75. Find a dimension of the subspace W of R3 generated by vectors
U = (2,0,1), v = (1,0,1)
Answer: dim W = 2;
76. Let S, T be linear operators on R2 defined by S(x,y) = (x + y,0), T (x,) = (-y, x). Find (5S – 3T)(x,y)
Answer: (5S – 3T)(x,y) = (5x + 8y ,– 3x)
77. Let S, T be linear operators on R2 defined by S(x,y) = (x + y,0), T (x,) = (-y, x). Find (2S – T)(x,y)
Answer: (2S – T)(x,y) = (2x + 3y, – x)
78. Let S, T be linear operators on R2 defined by S(x,y) = (x - y,x), T (x,) = (-y, x+ y). Find (5S – 3T)(x,y)
Answer: (5S – 3T)(x,y) = (5x - 2y,2x – 3y)
79. Let S, T be linear operators on R2 defined by S(x,y) = (x - y,x), T (x,) = (-y, x+ y). Find (2S – T)(x,y)
Answer: (2S – T)(x,y) = (2x - y,x – y)
80. . Let S, T be linear operators on R2 defined by S(x,y) = (x + y,0), T (x,) = (-y, x). Find (S T)(x,y)
Answer: (ST)(x,y) = (x – y, 0)
81. Let S, T be linear operators on R2 defined by S(x,y) = (x + y,0), T (x,) = (-y, x). Find (TS)(x,y)
Answer: (TS)(x,y) = (0, x+y)
82. Let S, T be linear operators on R2 defined by S(x,y) = (y,x), T (x,y) = (x - y, x). Find ST
Answer: ST(x,y) = (x,x - y)
83. Which of the following mappings F : R2 -> R2 are linear?
Answer: F(x,y) = (2x,x - y)
84. Which of the following mappings F : R2 -> R2 are linear?
Answer: F(x,y) = (x – 3y, x + y)
85. Which of the following mappings F : R2 -> R2 are linear?
Answer: F(x,y) = (3x + 5y, 2y)
86. Find the matrix of the linear operator on R² defined by T(x,y) = (x + y, y) with respect to the usual basis.
Answer: (1 1 | 0 1)
87. Find the matrix of the linear operator on R² defined by T(x,y) = (y, x) with respect to the usual basis.
Answer: (0 1 | 1 0)
88. Find the matrix of the linear operator on R² defined by T(x,y) = (- x, y) with respect to the usual basis.
Answer: (- 1 0 | 0 1)
89. Find the matrix of the linear operator on R² defined by T(x,y) = (x, -y) with respect to the usual basis.
Answer: (1 0 | 0 -1)
90. Find the matrix of the linear operator on R² defined by T(x,y) = (x – y , y) with respect to the usual basis.
Answer: (1 -1 | 0 1)
91. Let T be the linear operator on R² defined by T(x,y) = (x + y,y). Find dim(KerT)
Answer: dim( KerT) = 0;
92. Let T be the linear operator on R² defined by T(x,y) = (0,y). Find dim(KerT)
Answer: dim( KerT) = 1;
93. Let T be the linear operator onR² defined by T(x,y) = (x,0). Find dim(KerT)
Answer: dim( KerT) = 1;
94. . Let T be the linear operator on R² defined by T(x,y) = (x + y,y). Find dim(Im T)
Answer: dim( Im T) = 2;
95. Let T be the linear operator on R² defined by T(x,y) = (0, y). Find dim(Im T)
Answer: dim( Im T) = 1;
96. Let T be the linear operator on R² defined by T(x,y) = (x, 0). Find dim(Im T)
Answer: dim( Im T) = 1;
97. For the matrix (1 2 1 | 0 3 1 | 0 0 0 ), find the cofactor of the entry a33
Answer: 3;
98. For the matrix (1 4 2 | 0 3 1 | 0 5 0 ),find the cofactor of the entry a33
Answer: - 2;
99. For the matrix (1 2 1 | 0 3 1 | 0 0 0 ),find the cofactor of the entry a21
Answer: 0;
100. Evaluate the determinant of the matrix
A = (1 1 1 1 | 0 2 4 0 | 0 0 1 0 | 0 0 0 1 )
Answer: det A = 2;
101. Evaluate the determinant of the matrix
A = (1 1 1 1 | 3 2 4 0 | 6 8 1 0 | 0 0 0 0 )
Answer: det A = 0;
102. Evaluate the determinant of the matrix
A = (1 1 1 1 | 5 2 4 7 | 6 8 1 0 | 3 3 3 3 )
Answer: det A = 0;
103. Evaluate the determinant of the matrix
A = (1 5 2 1 | 0 2 4 0 | 0 0 3 7 | 0 0 0 1 )
Answer: det A = 6;
104. Evaluate the determinant of the matrix
A = (1 6 1 0 | 3 2 4 0 | 6 8 1 0 | 5 3 0 0 )
Answer: det A = 0
105. Let A = (1 2| -1 0 ). Find A -1(îáðàòíàÿ)
Answer: A-1 = (0 -1 | ½ ½ )
106. Let A = (3 0| 0 2 ).Find A -1(îáðàòíàÿ)
Answer: A-1 = (1/3 0 | 0 ½ )
107. Let A = (0 -1| -1 0 ). Find A -1(îáðàòíàÿ)
Answer: A-1 = (0 -1 | -1 ½ )
108. Let A = (4 0| 0 3 ). Find A -1(îáðàòíàÿ)
Answer: A-1 = (1/4 0| 0 1/3 )
109. Let A = (4 1| 8 2 ). Find A -1(îáðàòíàÿ)
Answer: A-1 = (1 3 | 0 1 )
110. Let A = (3 9| 5 15 ). Find A -1(îáðàòíàÿ)
Answer: Doesn’t exist
111. Let T be the linear operator on R² defined by T(x,y) = (x + y,y).Find det T.
Answer: det T = 1;
112. Let T be the linear operator on R²defined by T(x,y) = (y,x).Find det T.
Answer: det T = - 1;
113. Let T be the linear operator on R² defined by T(x,y) = (- x, y).Find det T.
Answer: det T = - 1;
114. Let T be the linear operator on R² defined by T(x,y) = (x, - y). Find det T.
Answer: det T = - 1;
115. Let T be the linear operator on R² defined by T(x,y) = (x - y,y). Find det T.
Answer: det T = 1;
116. Find all eigenvalues of the matrix (4 1 | 0 3)
Answer: λ1 = 4, λ2 = 3;
117. Find all eigenvalues of the matrix (5 1 | 0 1)
Answer: λ1 = 5, λ2 = 1;
118. Find all eigenvalues of the matrix (7 1 | 0 6)
Answer: λ1 = 7, λ2 = 6;
119. Find all eigenvalues of the matrix (0 1 | - 5 0)
Answer: doesn’t exist;
120. Find all eigenvalues of the matrix (9 1 | 0 4)
Answer: λ1 = 9, λ2 = 4;
121. Find the slope and the segment cut out on the y-axis of the straight line 2x – y + 3 = 0
Answer: 2; 3
122. Find the slope and the segment cut out on the y-axis of the straight line x + y - 5 = 0
Answer: - 1; 5
123. Find the slope and the segment cut out on the y-axis of the straight line x – 2y + 2 = 0
Answer: 1/2; 1
124. Find the equation of the straight line that passes through the origin and is parallel to the straight line y = 4x + 3;
Answer: y = 4x
125. Find the equation of the straight line that passes through the origin and is parallel to the straight line y = 1/2x + 1;
Answer: y = 1/2x
126. Find the equation of the straight line that passes through the point (1, -2) and is parallel to the straight line y = 4x + 3;
Answer: y = 4x – 6
127. Find the equation of the straight line that passes through the point ( 1, 0) and is parallel to the straight line y = 2x - 1;
Answer: y = 2x – 2
128. Find the equation of the straight line that passes through the point (-1, 1) and is parallel to the straight line y = 1/2x + 1;
Answer: y = -2x – 1
129. Find the equation of the straight line that passes through the point (0, 2) and is parallel to the straight line y = -3x;
Answer: y = 1/3x + 2
130. Find the equation of the straight line that passes through the point (2, 1) and is parallel to the straight line y = x + 5;
Answer: y = - x + 3
131. Find the equation of the straight line that passes through the point (6, 1) and is parallel to the straight line y = 1/3x + 2;
Answer: y = - 3x + 19;
132. Find the equation of the straight line that passes through points (- 1, 1) and (0, 1)
Answer: y = 1;
133. Find the equation of the straight line that passes through points (2, 3) and (0, 0)
Answer: 3x – 2y = 0;
134. Find the equation of the straight line that passes through points (- 1, 4) and (-1, 3)
Answer: x = - 1;
135. Find the equation of the straight line that passes through points (1, 1) and (3, 3)
Answer: y = x;
136. Find the equation of the straight line if it passes through the point (2, 2) and the angle between it and the x-axis is equal to pi/4.
Answer: y = x;
137. Find the equation of the straight line if it passes through the point (sqrt3, 2) and the angle between it and the x-axis is equal to pi/6.
Answer: y = 1/sqrt3x + 1;
138. Find the equation of the straight line if it passes through the point (- 2, 2) and the angle between it and the x-axis is equal to pi/2.
Answer: x = -2;
139. Find the equation of the straight line if it passes through the point (-3, 4) and the angle between it and the x-axis is equal to 0.
Answer: y = 4;
140. Find the angle between straight lines 3x – y = 0, 2x + y + 5 = 0.
Answer: pi/4
141. Find the angle between straight lines y = x + 1 = 0, y + x + 2 = 0.
Answer: pi/2
142. Find the angle between straight lines 4x – 5y + 7 = 0, 8x – 10y + 5 = 0.
Answer: 0;
143. Find the angle between straight lines y = 2x, y = - 1/2x + 5.
Answer: pi/2
144. Find the angle between straight lines (x – 1)/3 = (y + 3)/2 and (x + 2)/2 = (y – 3)/3.
Answer: arccos 12/13
145. Find the angle between straight lines x/4 = (y - 3)/3 and (x + 1)/1 = (y + 3)/2.
Answer: arccos 2/sqrt5
146. Find the angle between straight lines x/5 = (y - 3)/6 and (x + 7)/6 = (y - 3)/-5.
Answer: pi/2
147. Find the distance between the straight line 4x – 15y – 11 = 0 and the point P(4, -2)
Answer: 35/sqrt241
148. Find the distance between the straight line 12x + 5y - 7 = 0 and the point P(2, 7)
Answer: 4;
149. Find the distance between the straight line 9x – 12y + 2 = 0 and the point P(-3, 5)
Answer:17/3
150. Find the distance between the straight line 4x – 3y – 7 = 0 and the point P(0, 1)
Answer: 2;
151. Put the equation 16x² + 25y² = 400 in standard form and find foci.
Answer: x²/25 + y²/16 = 1; F1(-3, 0), F2(3, 0)
152. Put the equation 7x² + 16y² = 112 in standard form and fined foci.
Answer: x²/16 + y²/7 = 1; F1(-3, 0), F2(3, 0)
153. Put the equation 3x² + 4y² = 24 in standard form and fined foci.
Answer: x²/8 + y²/6 = 1; F1(-sqrt2, 0), F2(sqrt2, 0)
154. Find the standard-form equation of the ellipse if it’s center is the origin and foci:
(+-sqrt2, 0), major axis vertices: (+-2,0)
Answer: x²/4 + y²/2 = 1;
155. Find the standard-form equation of the ellipse if it’s center is the origin and foci:
(+-2, 0), eccentricity:0,25.
Answer: x²/64 + y²/60 = 1;
156. Find the standard-form equation of the ellipse if it’s center is the origin and foci:
(0, +-4), major axis vertices: (0, +-5)
Answer: x²/27 + y²/36 = 1;
157. Find the standard-form equation of the ellipse if it’s center is the origin and major axis vertices: (0, +-70), eccentricity: 0,1.
Answer: x²/441 + y²/490 = 1;
158. Find the standard-form equation of the ellipse if it’s center is the origin and major axis vertices: (+-10, 0), eccentricity: 0,5.
Answer: x²/100 + y²/75 = 1;
159. Find the standard-form equation of the ellipse if it’s center is the origin and the focus:
(sqrt5, 0), corresponding directrix: x = 9/sqrt5.
Answer: x²/9 + y²/4 = 1;
160. Find the standard-form equation of the ellipse if it’s center is the origin and the focus:
(-4, 0), corresponding directrix: x = -16.
Answer: x²/64 + y²/48 = 1;
161. The equation of the ellipse is 25x² + 169y² = 4225. Find foci.
Answer: F1(-12,0), F2(12,0);
162. The equation of the ellipse is 25x² + 169y² = 4225. Find foci.
Answer: F1(-12,0), F2(12,0);
163. The equation of the ellipse is 25x² + 169y² = 4225. Find the eccentricity.
Answer: e = 12/13;
164. The equation of the ellipse is 7x² + 16y² = 112. Find the eccentricity.
Answer: e = 3/4;
165. Put the equation 16x² - 25y² = 400 in standard form and find foci.
Answer: x²/25 - y²/16 = 1; F1(-sqrt41, 0), F2(sqrt41, 0)
166. Put the equation 7x² - 16y² = 112 in standard form and find foci.
Answer: x²/16 - y²/7 = 1; F1(-sqrt23, 0), F2(sqrt23, 0)
167. Put the equation 3x² - 4y² = 24 in standard form and fined foci.
Answer: x²/8 - y²/6 = 1; F1(-sqrt14, 0), F2(sqrt14, 0)
168. Find the standard-form equation of the hyperbola if its center is the origin and its foci:
(+-sqrt2,0), asymptotes: y = +- x;
Answer: x² - y² = 1
169. Find the standard-form equation of the hyperbola if its center is the origin and its foci:
(+-2,0), asymptotes: y = +-1/sqrt3 x;
Answer: x²/3 - y² = 1
170. Find the standard-form equation of the hyperbola if its center is the origin and its foci:
(+-3,0), eccentricity: 3
Answer: x² - y²/8 = 1;
171. Find the standard-form equation of the hyperbola if its center is the origin and its foci:
(+-5,0), eccentricity: 1,25
Answer: x²/16 - y²/9 = 1;
172. Find the standard-form equation of the hyperbola if its center is the origin and its major axis vertices: (+-9,0), eccentricity: 3
Answer: x²/9 - y²/72= 1;
173. Find the standard-form equation of the hyperbola if its center is the origin and its major axis vertices: (+-9,0), eccentricity: 2
Answer: x²/16 - y²/48 = 1;
174. Find the standard-form equation of the hyperbola if its center is the origin and its major axis vertices: (+-4,0), asymptotes: y = +-1/2x
Answer: x²/16 - y²/4 = 1;
175. Find the standard-form equation of the parabola if the parabola is symmetric about x-axis and passed through the origin and the point (1, -4).
Answer: y² = 16x
176. Find the standard-form equation of the parabola if the parabola is symmetric about x-axis, the focus is the point (0, 2) and the vertex is the origin.
Answer: x² = 8y
177. Find the standard-form equation of the parabola if the parabola is symmetric about x-axis and passed through the origin and the point (4, -2).
Answer: x² = -8y
178. Find points of intersection y² = 18x and the straight line 6x + y – 6 = 0.
Answer: (2, -6), (1/2, 3);
179. Find the standard-form equation of the parabola if the parabola is symmetric about x-axis and passed through the origin and the point (1, -2).
Answer: y² = 4x
180. Find the standard-form equation of the parabola if the parabola is symmetric about x-axis and passed through the origin and the point (3, 1).
Answer: y² = 1/3x
Date: 2015-12-11; view: 1084
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