Algebra_final_preparation_questions
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1.Find slope of the line 7x-y+3=0.
2.Find equation of the plane that passes through point (1,-1,2) and is parallel to the plane 3x+y+z=6.
3.Find length of the vector a = (1, 2, -5).
4.Write equation of the line passing through the point (0,8) and parallel to the line y= - 3x-1.
5.Find slope of the line x-4y+2=0.
6.Find scalar product of vectors a and b if a 5 j 7k , b 2i 3 j k .
7.Find equation of the line in the (Oxy)-plane that passes through the point P(3,-2) and is perpendicular to the vector (-2i +4j)
8.Find the distance from point A(1, -1) to the line 3x+8y=5.
9.What is the equation of the line passing through the points A(0, -2) and B(-2, 0)?
10.What is the relative positions of the lines 5x-y=1 and x+11y= -1?
Intersecting Parallel Coincide
both have the same normal vector None of these
11.The angle between lines y 
3 x 1 and y 
33 x 5 is
12.Find length of the median OC of triangle OAB if coordinates of vertices are O(0, 0), A(6, 0), B(0, 10).
13.Find equation of the line that passes through the origin and constitutes 1500 with the OX axis.
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).
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).
16.Find eccentricity of the ellipse x2+4y2=256.
17.Find eccentricity of the hyperbola x2 - 4y2=256.
18.Find scalar product of two vectors a (2, 1, 0) and b (0, 5, 0) .
19.The angle between two vectors a ( 1, 2, 0) and b (1, 4, 2) is …
20.Vertices of triangle ABC are A(2, -1, -3), B(1, 3, 1), C(0, 0, 5). Find angle B in the triangle.
21.Find modulus of the vector product [ a,b ] where a (1,3,2) , b (0, 1,0) .
22.Find coordinates of the vector product [a,b ], where a (1,2,5) , b (0, 2,0) .
23.Find area of the triangle ABC if its vertices are A(1,-2,3), B(0,0,6), and C(6,2,0).
24.Volume of the tetrahedron with vertices O(0,0,0), A(5,2,0) B(2,3,0), C(1,2,8) is
25.Find equation of the plane passing through the point A(1,0,0) and perpendicular to the vector (2,1,1).
26.Find coordinates of the normal vector of the plane 5x-z=3.
27.Find equation of the plane through three points A(1,0,0), B(0,-2,0), C(0,0,1).
28.Find the distance from point O(0,0,0) to the plane x+2y+z=1.
29.Find the direction vector of the straight line in the space given by x 4 .
y 3
30.Find intersection point of the line x 1 y z and the plane 5x+y+z=5.
12 1
31.Find slope of the straight line 1+5y=0.
32.Find length of the line segment intercepting by straight line x+2y= -1 on OX-axis.
33.The modulus of vector a 2i j k is …
34. Find the value of so that vectors a i 2 j k and b 4i 2 j k will be orthogonal.
35.Two vectors a 1, ,2 and b 1,1, 5 are collinear if equals …
36.Absolute values of two vectors are | a | 1, | b | 3 and their scalar product a, b 2 . Find the modulus of the vector product [a, b ] .
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.
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.
39.Find the angle between straight lines x-6=0 and x-4y+3=0.
40.Find equation of the straight line through point M(2,-3) and parallel to the straight line 3x+5y=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
42. Find equation of the plane which passes through the points M1(1,-3,1), M2(2,-1,2) and M3(4,- 2,6).
43.Straight line is given by 2x-y+5z-5=0 and x+3y-2z+8=0. Find canonical equation of the line.
44.Volume of parallelepiped constructed on vectors is equal to the
scalar product of the given vectors
absolute value of vector product of the given vectors absolute value of the triple product of the given vectors sum of the given vectors
none of these
45. Area of parallelogram constructed on vectors is equal to the scalar product of the given vectors
absolute value of vector product of the given vectors absolute value of mixed product of the given vectors sum of the given vectors
none of these
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46. If coordinates of two points are A(3,-3) and B(-4,1) find coordinates of vector AB and its length.
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47. If coordinates of two points are A(-1,2) and B(3,2) find coordinates of the unit vector AB 0 .
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48.If coordinates of two points are A(3,2) and B(-4,-10) find unit vector AB 0 .
49.If coordinates of three points are A(-2,1), B(4,-2), C(0,6) find coordinates of point D so that
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AB DC .
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 .
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51. If coordinates of three points are A(-2,-1), B(3,-4), C(0,4) find length of the vector (2 AB BC ) .
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52. If coordinates of two points are A(3,4), B(-1,6) find modulus of the projection of AB to the
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vector CD (8, 1) .
53.Coordinates of vertices of triangle ABC are A(1,-6), B(5,-4), C(2,-3). Find length of median
AM .
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Find vector x which is collinear to the vector a (2, 1, 1) and satisfies the condition that |
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scalar product (x, a) 5 . |
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55. |
The angle between two vectors a and |
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56. |
Find the area of parallelogram constructed on the vectors a i |
6 j 2k and b 2i 3 j 4k . |
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57. |
Compute [k ,[ j, k ]], if i, j, k are standard basis vectors. |
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58.Compute [k , (i j)] , where i, j, k are standard basis vectors.
59.Compute [2 j, (i k )] , where i, j, k are standard basis vectors.
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60. |
Find the area of parallelogram constructed on the vectors a 2i |
4 j k and b 2i 3 j k . |
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61. |
Find the volume of parallelepiped constructed on the vectors a i |
j 3k , b 6i j k , |
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c3i 4 j k
62.Find an equation of the straight line through the point A(-1,4) and parallel to the straight line x-5y=5.
63.Find an equation of the straight line through the point A(1,-2) and perpendicular to the straight line x-3y=1.
64.Find an equation of the straight line through the point A(-2,5) and perpendicular to the straight line x+5y=4.
65.Two vectors are a (4; 1) and b (2;4) . Find coordinates of the vector a 3b .
66.Find the point of intersection of the straight line 2x 5y 6 0 and OX-axis.
67.Find semi-axes of the ellipse 4x2 + 25y2 = 400
68.Find slope of the line that is perpendicular to the straight line 9х + 2у – 3 = 0.
69. Canonical equation of the straight line is |
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of the line. |
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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.
72.Find the angle between straight lines y=5x+1 and y=0.5x-1.
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Find eccentricity of the ellipse |
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74. |
Find eccentricity of the hyperbola |
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75. |
Find directrix of the ellipse |
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76. |
Find directrix of the hyperbola |
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77. |
Find vector product of the given two vectors a (3,0,8) and |
b ( 2,1,0). |
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78. |
Find equation of the plane which is parallel to the plane (OXY). |
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Ax+By=0 |
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C=0
Ax+D=0
Cz+D=0
Ax+Cz=0
79. What is the relative positions of the planes x+y-z+1=0 , 5x+5y-5z+6=0? intersect
coincide skewed parallel
none of these
80. Find the distance from point M(1,2,3) to the plane 2x+5y-1=0.
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: 2x y z 0 |
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83. |
Find parametric form of the equation |
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84. What is the relative positions of the plane 2x-3y+z-1=0 and the straight line
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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 |
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x 3 4t |
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86. |
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Represent the straight line y 1 2t in canonical form. |
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z 5 t |
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87. |
Find angle between the straight lines |
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88. |
Find angle between the straight line |
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and the plane 4x+2y+z-5=0. |
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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 = 2x + 4 and y = -3x – 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 2x2 - 3y2 = 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. |
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99.Find equation of the straight line through the point A(-1,-3) if the angle between it and the X- axis is 600.
100.Find equation of the straight line through the point A(-1,-5) if the angle between it and the X- axis is 300.
101.Find equation of the straight line through the point A(-1,-5) if the angle between it and the X- axis is 1800.
102.Find distance from the origin to the straight line: 9x 15y 10 0 .
103.Find distance from the origin to the straight line –x+y=0. |
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104.Find equation of the plane that passed through the point |
(2, 5,3) and is parallel to the |
coordinate plane XOZ. |
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105.Find equation of the plane through point A(1,2-9) and is parallel to the XOY-plane. |
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106.Find distance between two parallel planes: 11x 2y 10z 30 0, |
11x 2y 10z 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.
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109. Find angle between the line |
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y 3 t and the plane 4x-2y-2z+7=0. |
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z 4 2t |
110.Find canonical equation of the straight line: x y 3z 2 0, x 2y 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 2i 5 j .
113. Find the values of α and β such that vector a (3, 1, ) is perpendicular to the vector
b(2, ,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
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
117.Find AB, if A=
118.Find AB, if
119.Find AB, if
120. Find AB, if |
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121.Find AB, if
122.Find AB, if
123.Find AB, if
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124. |
Solve the system 2x 3y 1 . |
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5x 9 y 3 |
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2x y 3z 5 |
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125. |
Solve the system 3x 2 y 2z 5 . |
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5x 3y z 16 |
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2x 3y 2z 5
126. Solve the system .4x y 4z 1
127.Determine k so that the system
128.Determine k so that the system
129.Determine k so that the system
kx 4 y 10 |
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3x 8y |
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kx 2 y 11 |
has the unique solution. |
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5x y 5 |
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x ky 1 |
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3x 8y |
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130.Let
131.Let
132.Let
1 0 4
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f (x) x2 3x 3. Find |
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f (x) x3 8x 1. Find |
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133. Given matrix |
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Au 2u . |
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134. Given |
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135.Write the vector (2,5) as a linear combination of the vectors e1 (1,3), e2 (0,1) .
136.Write the vector v (1, 6) as a linear combination of the vectors e1 (1,1), e2 (0,1) .
137.Write the vector v ( 5,3) as a linear combination of the vectors e1 (1,0), e2 (1,1) .
1
138. Find rank of matrix A 01
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 .
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147. Evaluate the determinant of the matrix |
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148. |
Evaluate the determinant of the matrix . |
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149. |
Let |
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150. |
Let |
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151. |
Let |
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A 1 . |
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152. |
Let |
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A 1 . |
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153. |
Find detA , if A= |
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154. |
Find detA , if |
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155. |
Find , if |
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156. Find rank of the matrix |
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157.Find rank of the matrix .
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158. |
Given matrix A= |
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Find A A 1 . |
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159. |
Find A 1 A if |
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160. |
Find the matrix 6A if |
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161. |
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Calculate |
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162. If you interchange any two rows in a determinant then … the determinant will change in the sign
the value of the determinant will not be changed the value of the determinant will be equal to 0 the value of the determinant will be equal to 1
the value of the determinant will be multiplied by 2
163. |
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164. |
Find the matrix |
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165. |
Calculate |
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166. |
Find A 1 |
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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 |
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1 |
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AB . |
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