Intersecting
11. The angle between lines 31335 a=90
12. Find length of the median OC of triangle OAB if coordinates of vertices are O(0, 0), A(6, 0),
B(0, 10). OC=sqrt(34) AB=3;5;
13. Find equation of the line that passes through the origin and constitutes 1500 with the OX axis. y=-sqrt(3)x/3
14. Find length of the altitude (height) BD of triangle ABC if coordinates of vertices are A(-3, 0), B(2, 5), C(-3, 2). D=5
15. Find coordinates of the point of intersection of medians of the triangle ABC if coordinates of
vertices are A(-2, 0), B(0, 6), C(4, 0). M(2/3; 2)
16. Find eccentricity of the ellipse x2+4y2=256. E=sqrt(3)/2
17. Find eccentricity of the hyperbola x2 - 4y2=256. E=sqrt(5)/2
18. Find scalar product of two vectors a = (2,-1, 0) and b = (0, 5, 0). AB=-5
19. The angle between two vectors a = (-1,2, 0) and b = (1, - 4, 2) is cos(a)=-9/sqrt(105)
20. Vertices of triangle ABC are A(2, -1, -3), B(1, 3, 1), C(0, 0, 5). Find angle B in the triangle. cos(b)=-5/sqrt(858)
21. Find modulus of the vector product [ a, b ] where a = (1,3,2), b = (0,-1,0). A,B=sqrt(5)
22. Find coordinates of the vector product [ a, b ], where a = (1,2,5), b = (0,-2,0). A,B=10; -2
23. Find area of the triangle ABC if its vertices are A(1,-2,3), B(0,0,6), and C(6,2,0). sqrt(664)/2
24. Volume of the tetrahedron with vertices O(0,0,0), A(5,2,0) B(2,3,0), C(1,2,8) is 44/3
25. Find equation of the plane passing through the point A(1,0,0) and perpendicular to the vector
(2,1,1). P=2x+y+z-2=0
26. Find coordinates of the normal vector of the plane 5x-z=3. N(5, 0, -1)
27. Find equation of the plane through three points A(1,0,0), B(0,-2,0), C(0,0,1). -2x+y-2z+2=0
28. Find the distance from point O(0,0,0) to the plane x+2y+z=1. 1/sqrt(6)
29. Find the direction vector of the straight line in the space given by 43 P(0, 1, 1)
30. Find intersection point of the line and the plane 5x+y+z=5.55 x=1, y=0, z=0
31. Find slope of the straight line 1+5y=0. K=0
32. Find length of the line segment intercepting by straight line x+2y= -1 on OX-axis. L=1
33. The modulus of vector a = 2 i + j - k is sqrt(6)
34. Find the value of l so that vectors a = i + 2 j - k and b = 4 i - 2 j +l k will be orthogonal. l=0
35. Two vectors a = (1,l,2) and b = (-1,1,-5) are collinear if l equals l=-1
36. Absolute values of two vectors are | a |=1, | b |= 3 and their scalar product (a, b)= 2. Find themodulus of the vector product [ a, b ]. sqrt(5)
37. Two points A(1,2) and B(6,7) are given. Find coordinates of point C which divides the segment AB in the ratio 2:1. C(13/3, 16/3)
38. Let points A(1,1), B(3,4), C(3,-1) be the consecutive vertices of the parallelogram. Find
coordinates of the fourth vertex. D=5; 2
39. Find the angle between straight lines x-6=0 and x-4y+3=0. 1/17
40. Find equation of the straight line through point M(2,-3) and parallel to the straight line
3x+5y=0. L:3x+5y+9=0
41. Find equation of the line through the point of intersection of two lines 3x-y=0 and x+4y-2=0 and perpendicular to the line 2x+9y=0 -9x+2y+6/13=0
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42. Find equation of the plane which passes through the points M1(1,-3,1), M2(2,-1,2) and M3(4,-
2,6). y=-3, x=1
43. Straight line is given by 2x-y+5z-5=0 and x+3y-2z+8=0. Find canonical equation of the line. (x-1)/-13 = (y+3)/9 = z/7
44. Volume of parallelepiped constructed on vectors is equal to the
Absolute value of the triple product of the given vectors
45. Area of parallelogram constructed on vectors is equal to the
absolute value of vector product of the given vectors
46. If coordinates of two points are A(3,-3) and B(-4,1) find coordinates of vector
____ AB and its length. Sqrt(65)
47. If coordinates of two points are A(-1,2) and B(3,2) find coordinates of the unit vector 0
____ AB. AB=1; 0
48. If coordinates of two points are A(3,2) and B(-4,-10) find unit vector 0
____ AB. AB(-7/sqrt(193), -12/193)
49. If coordinates of three points are A(-2,1), B(4,-2), C(0,6) find coordinates of point D so that
____ ____ AB = DC. x=-6, y=9
50. If coordinates of two points are B(6,-2), C(0,8) find coordinates of point P lying on the line BC so that BP = PC. P=(3, 3)
51. If coordinates of three points are A(-2,-1), B(3,-4), C(0,4) find length of the vector (2)
___ ____ AB - BC. sqrt(365)
52. If coordinates of two points are A(3,4), B(-1,6) find modulus of the projection of
___ AB to the vector (8, 1)___ CD = -.
53. Coordinates of vertices of triangle ABC are A(1,-6), B(5,-4), C(2,-3). Find length of median
AM. sqrt(50)/2
54. Find vector x which is collinear to the vector a = (2, -1,1) and satisfies the condition that
scalar product (x, a) = 5. x=5/3, y=-5/6, z=5/6
55. The angle between two vectors a and b is 1200 and | a |= 6, | b |= 4. Compute
2 (a + b). 28
56. Find the area of parallelogram constructed on the vectors a = i + 6 j - 2 k and b = 2 i - 3 j - 4 k. 5sqrt(45)
57. Compute [ k,[ j, k ]], if i, j, k are standard basis vectors. [j]
58. Compute [ k, (i + j)], where i, j, k are standard basis vectors. j-i
59. Compute [2 j, (i - k)], where i, j, k are standard basis vectors. -2k-2i
60. Find the area of parallelogram constructed on the vectors a = 2 i - 4 j + k and b = 2 i + 3 j - k. sqrt(213)
61. Find the volume of parallelepiped constructed on the vectors a = i + j + 3 k, b = 6 i - j + k,
c = 3 i - 4 j + k -63
62. Find an equation of the straight line through the point A(-1,4) and parallel to the straight line
x-5y=5. x-5y+21=0
63. Find an equation of the straight line through the point A(1,-2) and perpendicular to the straight line x-3y=1. 3x+y-1=0
64. Find an equation of the straight line through the point A(-2,5) and perpendicular to the straight line x+5y=4. -5x+y+15=0
65. Two vectors are a (4;-1) and b (2;4). Find coordinates of the vector a -3 b. (-2, -13)
66. Find the point of intersection of the straight line 2 x + 5 y - 6 = 0 and OX-axis. x=3
67. Find semi-axes of the ellipse 4 x2 + 25 y2 = 400 a=4, b=10
68. Find slope of the line that is perpendicular to the straight line 9 + 23 = 0. 2/9
69. Canonical equation of the straight line is 1425.Find coordinates of the direction vector of the line. P=(4, -5)
70. Indicate the equation of the straight line passing through the origin.
71. Find the distance from point M(-1, 5) to the straight line 2x-3y+10=0. 7/sqrt(13)
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72. Find the angle between straight lines y=5x+1 and y=0.5x-1. -9/7
73. Find eccentricity of the ellipse225291 4/5
74. Find eccentricity of the hyperbola 292161 5/3
75. Find directrix of the ellipse 7521002361 + - 12.5
76. Find directrix of the hyperbola 225291 +-25/34
77. Find vector product of the given two vectors a = (3,0,8)and b = (-2,1,0). -8i-16j+3k
78. Find equation of the plane which is parallel to the plane (OXY).
Cz+D=0
79. What is the relative positions of the planes x+y-z+1=0, 5x+5y-5z+6=0?
Parallel
80. Find the distance from point M(1,2,3) to the plane 2x+5y-1=0. 11/sqrt(129)
81. Find the angle between the planes 3x-y+9z-4=0 and 5x+3y-5z+2=0
82. Find coordinates of the direction vector of the straight line:
83. Find parametric form of the equation
84. What is the relative positions of the plane 2x-3y+z-1=0 and the straight line
Straight line is perpendicular to the plane
Straight line is parallel to the plane
Straight line lies in the plane
Straight line intersects the plane
None of these
85. Find equation of the straight line passing through the point M(1,5,-1) and parallel to the line
86. Represent the straight line in canonical form.
87. Find angle between the straight lines
and
88. Find angle between the straight line
and the plane 4x+2y+z-5=0.
89. Find angle between the given two planes: x-5y+5=0 and 2x-y+5z-16=0.
90. Find distance from point (2,5) to the line 4x+8y-5=0.
91. Find angle between the straight lines y = 2 x + 4 and y = -3 x 1.
92. Find equation of the straight line that passes through the origin and is parallel to the straight
line y=4x+6.
93. Find equation of the plane through the point (-2,8,3) and parallel to the plane x-6y+5z-1=0.
94. Find equations of asymptotes of the hyperbola 2 x 2 - 3 y 2 = 16.
95. If C(-3,-14) is the midpoint of the line segment AB and coordinates of A(-5,-7), find coordinates
of point B.
96. Let A (4,6), B (-4,0), C (-1,-6) be vertices of the triangle. Find equation of side BC.
97. Let A (4,6), B (-4,0), C (-1,-4) be vertices of the triangle. Find equation of the altitude from
vertex A.
98. Let A (4,6), B (-4,0), C (-1,-4) be vertices of the triangle. Find equation of the median
dropped from vertex B.
99. Find equation of the straight line through the point A(-1,-3) if the angle between it and the Xaxis
is 600.
100. Find equation of the straight line through the point A(-1,-5) if the angle between it and the Xaxis
is 300.
101. Find equation of the straight line through the point A(-1,-5) if the angle between it and the Xaxis
is 1800.
102. Find distance from the origin to the straight line: 9 x -15 y +10 = 0.
103. Find distance from the origin to the straight line x+y=0.
104. Find equation of the plane that passed through the point (2,-5,3) and is parallel to the
coordinate plane XOZ.
105. Find equation of the plane through point A(1,2-9) and is parallel to the XOY-plane.
106. Find distance between two parallel planes: 11 x - 2 y -10 z + 30 = 0, 11 x - 2 y -10 z - 45 = 0.
107. Find the volume of the tetrahedron if its vertices are A (0,0,2), B (3,0,5), C (1,1,0), D (4,1,6).
108. Coordinates of vertices of triangle ABC are A(1,6), B(-5,2), C(2,-3). Find angle at vertex B.
109. Find angle between the line
and the plane 4x-2y-2z+7=0.
110. Find canonical equation of the straight line: x - y -3 z - 2 = 0, x - 2 y + z + 4 = 0.
111. Find equation of the line through point M(6,-2) and parallel to the OY-axis.
112. Find canonical equation of the line passing through point M(1,0,-2) and parallel to the vector
s = 2 i - 5 j.
113. Find the values of αand β such that vector a = (3,-1,a) is perpendicular to the vector
b = (2,b,1) if b = 5.
114. What is the relative position of two straight lines: x-y-1=0 and 8x-8y-8=0.
Have one common point
Coincide
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Have no common point
Perpendicular each other
None of these
115. What is the relative position of two straight lines: 4x-y-1=0 and x+y-2=0.
116. Find determinant 1234025900372480 76
117. Find AB, if A=458151 B=152334 AB=20 67 14 14
118. Find AB, if A=3512 B=1617 AB=21718
119. Find AB, if A=110315 B=073410 AB=311217
120. Find AB, if A=246 B=731 AB=14 6 2 28 12 4 42 18 6
121. Find AB, if
122. Find AB, if
123. Find AB, if
124. Solve the system
125. Solve the system
126. Solve the system
127. Determine k so that the system 4103815 has the unique solution. K=3/2
128. Determine k so that the system 21155 has the unique solution. K=-10
129. Determine k so that the system 1385has the unique solution. K=-8/3
130. Let A=104031 B=003250 AB=122315
131. Let A=2001 f(x)=233 1001
135. Write the vector J = (2,5) as a linear combination of the vectors (1,3), (0,1) 1 2 e = e =. x=2e1-e2
136. Write the vector v = (1,-6) as a linear combination of the vectors (1,1), (0,1) 1 2 e = e =.
137. Write the vector v = (-5,3) as a linear combination of the vectors (1,0), (1,1) 1 2 e = e =.
138. Find rank of matrix A=150211. R=2
139. Find rank of matrix.
140. Find rank of matrix.
141. Find the rank of matrix.
142. Find the rank of matrix.
143. Find the rank of matrix.
144. Find rank of matrix.
145. Evaluate the determinant of the matrix 1000/0200/0420/1111
146. Evaluate the determinant of the matrix 0000/0186/0423/1111
147. Evaluate the determinant of the matrix 4444/0186/7425/1111
148. Evaluate the determinant of the matrix.
149. Let A=10/13 Find A-1.
150. Let A=02/60 Find A-1
151. Let A=07/40 Find A-1
152. Let A=5/15/39 Find A-1
153. Find detA, if A=4 3/2 2.
154. Find detA, if A=1 3 2/4 1 3/2 5 2.
155. Find, if A=3 2/ 1 4.
156. Find rank of the matrix 1 0 0 0 5/0 0 0 0 0/2 0 0 0 11.
157. Find rank of the matrix.
158. Given matrix A=-4 0 1/2 -1 3/3 2 2 Find A × A-1.
159. Find A × A-1 if A=-4 0 1/2 -1 3/3 2 2
160. Find the matrix 6A if A 3 2/2 4
161. Calculate (2 1/-3 5)*(3/1)
162. If you interchange any two rows in a determinant then
the determinant will change in the sign
163. Let 2 -7/8 -9 Find the transposed matrix.
164. Find the matrix 5A if A 1 3/-3 7
165. Calculate (2 -3/-1 2)*(-1/-2)
166. Find -1 A if A=3 4/1 2
167. Find value of y so that the vectors a = (-4; 10) and b = (2; y)will be linearly dependent.
168. What is the value of determinant of identity matrix?
169. Let A=1 2/-1 3 B=0 -1/4 5 Find AB.
170. Let A=1 9/-1 2 B=0 -1/4 5 Find A + B.
171. Let A=6 1/5 1Find -1 A.
172. Solve the equation x2+(2x 5/3 1)=0
173. The rank of a matrix is equal to...
174. Find the product of the (2 -1/1 4)*(1 0/3 4)
175. Find A2 if A=4 3/-5 -4
176. Find the scalar product of vectors 1 2 3 a = - e + 2 e + 3 e and 1 2 b = 7 e - e.
177. Let m v, v,..., v 1 2 be vectors of a linear space R. The vectors m v, v,..., v 1 2 are linearly dependent
if
there exist real numbers m a, a,..., a 1 2, not all of them 0, such that
they are orthogonal.
178. A basis for n -dimensional linear space R is formed by...
a set of n linearly independent vectors from R
179. Let (1;1), (1; 0) 1 2 e = e = - be a basis of a linear space. Find the coordinates of the vector
x = (5;8) in this basis.
180. Let (1;2), (3; 5) 1 2 e = - e = - be a basis of a linear space. Find the coordinates of the vector
x = (4;- 7) in this basis.
181. Let A be a matrix of size (6x3). Then rank of the matrix A can be the following:
182. Let A=2 3 -1/0 1 2 Find the determinant of A.
183. The linear span of four vectors x, y, z, u is
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the set of all linear combinations of these vectors
the set consisting only of these vectors
the sum of these vectors
the set of all scalar products of these vectors
the set of vectors that are orthogonal to each of these vectors
184. Determine the dimension of the subspace formed by the solutions of the system
185. Let {, } {, } 1 2 1 2 e e and e ¢ e ¢ be old and new bases respectively in a 2-dimensional linear space,
and let 1 1 2 e ¢ = 2 e - 5 e, 2 1 2 e ¢ = e + 5 e. Then the transition matrix from the old basis to new is
186. Let vector 1 2 x = 2 e - 3 e be given. Find resolution of this vector in the new basis 1 2 e ¢, e ¢ if
1 1 2 e ¢ = e - e,2 1 2 e ¢ = -3 e + 4 e.
187. Let {, } {, } 1 2 1 2 e e and e ¢ e ¢ be old and new bases respectively in a 3-dimensional linear space,
and let 1 1 2 3 e ¢ = 5 e + 2 e - e, 2 1 2 3 e ¢ = -3 e + 2 e + 4 e, 3 1 2 3 e ¢ = e - 6 e + 2 e. Then the transition matrix
from the old basis to new basis is...
188. Find the length of the vector 1 2 3 x = e 3 - e 2 + 6 e.
189. Let A be a matrix of size 2x7. Then rank of the matrix A can be the following:
190. Find determinant: A=sinx cosx/-cosx sinx
191. What object will you get in the result of addition of two vectors?
Line segment
Number
vector
matrix
none of these
193. What is the dimension of the vector space of square matrices of size 2?
194. What is the rank of matrix 0 0/0 0
195. What operations are defined for elements of the Vector Space?
Addition and multiplication by a number
Four arithmetic operations
Only division
None of these
only addition
196. From expression n n v = l v +l v +K+l v 1 2 2 3 3 we can conclude the following
