2. Find the equation of the plane through (1,-1,3) parallel to the plane 3x+y+z=7.

1. Find slope of the line 7x-2y+3=0.

k =

2. Find the equation of the plane through (1,-1,3) parallel to the plane 3x+y+z=7.

3x + y + z - 5=0

3. Find length of the vector a = (1, 2, 3).

4. Write equation of the line passing through the point (0,6) and parallel to the line y= -2x-1.

2x + y – 6 = 0

5. Find slope of the line 5x-4y+2=0.

k =

6. Find scalar product of vectors and if a = 5 j + 3 k, b = 2 i - 3 j + k.

-12

7. Find the equation of the line in the (Oxy)-plane that passes through the point P(1,-2) and is perpendicular to the vector (-2 i +3 j)

2x-3y -8 = 0;

8. Find slope of the line y+2x=1.

k = -2

9. Find the distance from point A(1, 1) to the line 3x+4y=5.

10. What is the equation of the line passing through the points A(0, -1) and B(-1, 0)?

x+ y +1 =0

11. What is the relative positions of the lines 2x-y=1 and x+y= -1?

Пересек (tga=3)

12. The angle between lines y=x and y = is

= arcos ( ) Kazi: =arctg ) тута по формулам позырить надо, возможно оба правильны))

13. Find length of the median OC of triangle OAB if coordinates of vertices are O(0, 0), A(8, 0), B(0, 6).

5

14. Find equation of the line that passes through the origin and constitutes 1200 with the OX axis.

y = - x

15. Find length of the altitude (height) BD of triangle ABC if coordinates of vertices are A(-3, 0),

B(2, 5), C(3, 2).

16. Find coordinates of the point of intersection of medians of the triangle ABC if coordinates of vertices are A(-2, 0), B(0, 4), C(2, 0).

(1, )

17. Indicate the line that is defined by the equation + - 4x+6y-3=0.

ELLIPSE Kazi: =16 Circle Этот вопрос стоит рассмотреть

18. Find eccentricity of the ellipse +4 =16.

19. The distance between the foci of an ellipse is equal to 8, and minor semi-axis equals 3. Find the length of major semi-axis a.

5

20. Find eccentricity of the hyperbola - 4 =16.

21. Find radius of the circle + - 4x = 0.

22. Indicate equation of the parabola passing through the points (0, 0), (-1, 2), and symmetrical about axis OX.

= - 4x

23. Indicate equation of the parabola passing through the points (0, 0), (2, -4), and symmetrical about axis OY.

= - y

24. Equation of the directrix of the parabola =8(x+2) is

x = -6

25. Find scalar product of two vectors a = (1,-1, 0) and b = (0, 1, 0).

-1

26. The angle between two vectors a = (-1,1, 0) and b = (1, - 2, 2) is

Kazi: перепроверено! Мой ответ верен!

27. Vertices of triangle ABC are A(2, -1, 3), B(1, 1, 1), C(0, 0, 5). Find angle B in the triangle.

28. Vertices of triangle ABC are A(2, -1, 3), B(1, 1, 1), C(0, 0, 5). How can this triangle be classified?

РАВНОБЕДРЕННЫЙ Isosceles Kazi: Прямоугольный +равнобедренныйàcos(AB;AC) = 0; обратить внимание на вторую половину ответа!!!

29. Find modulus of the vector product [ a, b ] where a = (1,2,1), b = (0,-1,0).

30. Find coordinates of the vector product [ a, b ] where a = (1,2,1), b = (0,-1,0).

(1, 0 -1)

31. Find area of the triangle ABC if its vertices are A(1,-2,8), B(0,0,4), and C(6,2,0).

7

32. Volume of the tetrahedron with vertices O(0,0,0), A(5,2,0) B(2,5,0), C(1,2,4) is

14

33. Find equation of the plane passing through the point A(1,0,0) and perpendicular to the vector

(1,1,1).

x + y + z – 1 = 0

34. Find coordinates of the normal vector of the plane 2x-z=1.

(2, 0, -1)

35. Find equation of the plane through three points A(1,0,0), B(0,-1,0), C(0,0,2).

2x – 2y + z – 2 = 0

36. Find the distance from point O(0,0,0) to the plane x+y+z=1.

d = -

37. Find equation of the plane passing through the axis OY and point A(4,0,3).

z – 3 = 0

38. Find the direction vector of the straight line in the space given by (0, 0, 1)

39. Find intersection point of the line = = and the plane x+y+z=5.

(2,2, 1)

40. Which of the following surfaces is defined by the equation + = 2pz

ЭЛЛИПТИЧЕСКИЙ ПАРАБОЛОИД

41. Find scalar product if coordinates of points are A(-3, -1), B(0, -4), C(2, 3), D(4, 1).

42. Find slope of the straight line 2x-1=0.

k = -

43. Find slope of the straight line 1+y=0.

k = 0

44. Find length of the line segment intercepting by straight line x+y= -1 on OX-axis.

45. Coordinates of two points are A(2,2) and B(4,8). Find ordinate of the midpoint of segment AB.

(3, 5)

46. The modulus of vector a = 2 i + j - 2 k is

47. Find the value of l so that vectors a = i + 2 j - 2 k and b = 4 i -1 j +l k will be perpendicular.

1

48. Two vectors a = (1,l,2) and b = (- 2,2,-4) are collinear if l equals

5 Kazi: -1 я лично уверен в своем ответе!!!

49. Absolute values of two vectors are | a |=1, | b |= 2 and their scalar product (a, b)=1. Find the modulus of the vector product [ a, b ].

50. Two points A(1,2) and B(3,4) are given. Find coordinates of point C which divides the segment AB in the ratio 2:1.

( , )

51. Let points A(1,1), B(2,2), C(3,-1) be the consecutive vertices of the parallelogram. Find coordinates of the fourth vertex.

D (2, - 2) Kazi: (0,4) ßЗуб даю я здесь решил правильно!Не *бет ответ (0,4)!!!

52. Find the angle between straight lines x-5=0 and x-y+3=0.

53. Find equation of the straight line through point M(2,-1) and parallel to the straight line 2x+3y=0.

2x + 3y – 1 = 0

54. 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+7y=0

91x – 26y – 2 = 0

55. Vertices of triangle ABC are A(-2, 3, 1), B(-2, -1, 4), C(-2, -4, 0). Find angle C in the triangle.

56. Find equation of the plane which passes through the points M1(1,3,-1), M2(2,1,-2) and M3(4,2,- 6).

3x + 2y – z – 10 = 0 Kazi: 9x+2y-5z-10=0 походу Азим не правильно решил

57. Straight line is given by 2x-y+3z-5=0 and 4x+3y-2z+8=0. Find canonical equation of the line.

= = Если чо, НЕ паникуй ^_^ <<<<By Alish)))

58. Volume of parallelepiped constructed on vectors is equal to the

Kazi: V= or V= я от себя добавил так что не помешает/)))

59. If coordinates of two points are A(1,1) and B(3,-2) find coordinates of vector AB and its length.

AB = (2, -3)

60. If coordinates of two points are A(3,2) and B(-4,1) find coordinates of vector AB and its length.

(-7, -1) 5

61. If coordinates of two points are A(-2,2) and B(3,-2) find unit vector AB 0.

e = = cos i+ cos j Kazi: cos ; cos = ; answer:

62. If coordinates of two points are A(3,2) and B(-4,1) find unit vector AB 0.

Cos = Kazi:

63. If coordinates of three points are A(-2,1), B(3,-2), C(0,4) find coordinates of point D so that AB = DC.

(-5, 7)

64. If coordinates of three points are A(3,2), B(-4,1), C(2,-1) find coordinates of point D so that AB = DC.

Аналог Kazi: (9, 0)

65. If coordinates of two points are A(-2,1), B(3,-2) and number l = 5 find coordinates of point K so that AK: KB =l.

K(- 3/2; 1) Kazi: ( ) чё т здесь не то….

66. If coordinates of two points are A(3,2), B(-4,1) and number l = find coordinates of point K lying on line AB so that AK: KB =l.

k= (- 5/3; 4/3) Kazi: ( ) И ЗДЕСЬ ТОЖЕ

67. If coordinates of two points are B(3,-2), C(0,4) find coordinates of point P lying on line BC so that BP = PC.

ß---- Чё За х*йня??^^ Kazi: P(3/2; 1) мой ответ адекватнее ^^

68. If coordinates of two points are B(-4,1), C(2,-1) find coordinates of point P lying on line BC so that BP = PC.

P (-1; 0)

69. If coordinates of three points are A(-2,1), B(3,-2), C(0,4) find length of the vector (2) AB - BC.

Kazi:

70. If coordinates of three points are A(3,2), B(-4,1), C(2,-1) find length of the vector (2) AB - BC.

Kazi:

71. If coordinates of three points are A(1,-2), B(0,-1), C(3,-4) find length of the vector (2) AB - BC.

Kazi: 4

72. If coordinates of two points are A(3,4), B(-1,6) find projection of AB on the CD = (7, -1).

Kazi: 3

73. Vectors OA = a and OB = b in which | a |= 2, | b |= 4 and the angle between these vectors is

. In the triangle AOB find the angle between median OMand side OA.

74. Coordinates of vertices of triangle ABC are A(1,6), B(-5,4), C(2,-3). Find length of median AM.

75. Vertices of triangle ABC are A(1,-2), B(-4, 5), C(5,8). Find the angle between median BM and side BA of the triangle.

Cos = 49/

76. Find vector x which is collinear to the vector a = (2,1, -1) and satisfies the condition that scalar product (x, a) = 3.

Kazi: (3, )

77. The angle between two vectors a and b is 1200 and | a |= 3, | b |= 4. Compute 2 (a + b).

78. Find the area of parallelogram constructed on the vectors a = i + 5 j - 2 k and b = 2 i - 2 j - 4 k.

12

79. Coordinates of three points are A(2,-1,2), B(1,2,-1) and C(3,2,1). Find coordinates of the vector

product

(-6;4;6) or (-12, 8, 12)

80. Coordinates of three points are A(1,-1,2), B(5,-6,2) and C(1,3,-1). Find area of triangle ABC.

S=25/2

81. Compute [ k,[ j, k ]], if i, j, k are standard basis vectors.

j

82. Compute [ k, (i + j)], where i, j, k are standard basis vectors.

j - i

83. Compute [2 j, (i - k)], where i, j, k are standard basis vectors.

-2k - 2i

84. Find the area of parallelogram constructed on the vectors a = 3 i - 4 j + k and b = 2 i + 2 j - k.

85. Find the volume of parallelepiped constructed on the vectors a = i + 2 j + 3 k, b = 4 i - j + k,

c = 3 i - 4 j + k

86. Find the volume of tetrahedron constructed on the vectors a = i + j + k, b = 2 i - j - k,

c = -3 i + 3 j + k

87. Find an equation of the straight line through the point A(-1,2) and parallel to the straight line

x-3y=5.

( )

88. Find an equation of the straight line through the point A(4,-1) and parallel to the straight line

3x+2y=7.

( )

89. Find an equation of the straight line through the point A(-1,2) and perpendicular to the straight

line 2x-y=1

.

90. Find an equation of the straight line through the point A(4,-1) and perpendicular to the straight

line x-2y=2

( )

91. Find an equation of the straight line through the point A(2,-3) and perpendicular to the straight

line 4x-5y= -3.

( )

92. Find an equation of the straight line through the point A(1,2) and perpendicular to the straight

line 5x-y= -1.

( )

93. Find an equation of the straight line through the point A(-2,3) and perpendicular to the straight

line x+5y=4.

( )

94. Find an equation of the straight line through the point A(-3,4) and perpendicular to the straight

line x+y=5.

( )

95. If coordinates of two points are A (4;-3)and B (-4;5) find coordinates of vector

____

AB.

AB=(-8,8)

96. Two vectors are a (4;-1) and b (2;5). Find coordinates of the vector a -3 b.

(-2, -16)

97. Find the value of x so that the vectors a (x;2) and b (1;-3) are collinear.

(x=- )

98. Find the value of x so that vectors a (x;2) and b (1;-3)are orthogonal.

(x=6)

99. Find the point of intersection of the straight line 2 x +3 y -6 = 0 and OX-axis.

(3,0)

100. Write the equation of the straight line through the points A(-1,1) and B(0,2 ).------------------------------à ( )

101. If two vectors are a (3;-2) and b (1;1) find coordinates of the vector 2 a - b .------------------------------à(5; -5)

102. Find the point of intersection of the straight line 3 x - 4 y +12 = 0 and OY-axis. ------------------------à (0;3)

103. Find slope of the line y= -5x+3 .-------------------------------------------------------------------------------------à(k=-5)

104. Find equation of the line through two points A(1,-1) and B(2,0) .-------------------------------------------à( )

105. Find semi-axis of the ellipse 16 x2 + 25 y2 = 400----------------------------------------------------------à (b= )

106. Find slope of the line that is perpendicular to the straight line 5х + 2у – 3 = 0. -------------à (k= ) Kazi: k= -2/5 разница в знаках

107. Determine the form of the curve 9 x2 – 25 y2 = 225 and its parameters. ------------------------------à (hyperbola )

108. Canonical equation of the straight line is. Find coordinates of the direction vector of the line.----à n(2,-3)

109. Indicate the equation of the straight line passing through the origin.------------------------à??? Подобные отметки говорят что это ни кто не решал!

110. Indicate the normalized factor for the straight line -3x+4y+5=0 .---------------------------------à( )

111. Find the distance from point M(1,-2) to the straight line 4x-3y+10=0 .-------------------------à(4)

112. Find the angle between straight lines y=x+1 and y=0.5x-1 .------------------------------------------------à(α= )

113. Indicate the position of the straight line 2x+3y=0.-------------------------------------------à???

114. Find coordinates of the foci of ellipse ------------------------------------à(F1(12, 0) F2(-12, 0))

115. Find coordinates of the foci of hyperbola -----------------------------à(F1(13, 0) F2(-13,0))

116. Find parameter of the parabola y2=4x. ----------------------------------------------à(P=2)

117. Find eccentricity of the ellipse ------------------------------------------à (E= )

118. Find eccentricity of the hyperbola ---------------------------------à (E= )

119. Find directrix of the ellipse -------------------------------------------à (D= )

120. Find directrix of the hyperbola -------------------------------------- à( )

121. Find directrix of the parabola y2 =8x. -----------------------------------------à (d=-2)

122. Find asymptotes of the hyperbola ------------------------------à ( )

123. If the directrix of parabola x= -3 find the equation of the parabola.-----à (y=12x)

124. Find tangent line to the parabola y 2 x 2 = at point O(0,0).------------------à??????

Нашёл по формулам с инета: Уравнение касательной в точке так что нулевой ответ

125. Find vector product of the given two vectors a =(2,0,3) and b =(-2,1,0).------------------------------à (3,6,-2)

126. Find the equation of the median of triangle ABC dropped from vertex B if coordinates of

vertices are A(2,4), B(-2,0), C(4,2). ---------------------------------------------------------------------------------à (5y-3x-6)

127. Find the area of the triangle formed by the straight line 2x+3y-6=0 and the first quadrant. ---------------à????

128. What is the eccentricity of a circle?---------------------------à (E=1) Nurbek KADR! ^^ Kazi’s Answer: Для окружности полагают e = 0. (wikipedia)

129. Find equation of the plane which is parallel to the plane (OXY).-------------------------------------à (среди вариантов ответа видно будет)

130. What is the relative positions of the planes x+y-z+1=0, 2x+2y-2z+1=0?----------------------------------------à?????????????

Kazi: Parallel <<< я всё таки добился этого ответа

131. Normalize the equation of the plane 3х+4у- 11 z -5=0. --------------------------------à???????????????????????????

Kazi: Cos a= 3/ cos b= 4/ cos c = -11/ p=-5/

Answer: 3/ x + 4/ y - 11/ z - 5/ = 0

132. Find the distance from point M(1,2,3) to the plane 3x+4y-1=0.-----------------------------------------à (d=2)

Kazi: (чисто вбивал Кази)

133. Coordinates of two points are A(-5,0,0), B(-1,2,-3) and equation of the plane π: х-2y+4z-1=0.------------à A и B не сидят на трубе ^^

What are the positions of points A and B with respect to the plane?

134. Find the angle between the planes 3x-y+7z-4=0 and 5x+3y-5z+2=0----------- à (cos(π1,π2)=-23/59)

135. Find an equation of the plane passing through the point M0(2,0,3) and parallel to the plane-----------------à x+y-2=0

(XOY).

136. Find coordinates of the direction vector of the straight line:

S(-3,-3,3)

137. Find equation of the straight line through the given two points: M(5,3,7) and N(5,0,0).

= =

138. Find parametric form of the equation

X=1 y=2t+2 z=-5t-3

139. What is the relative positions of the given two straight lines:

=

Answers: Skewed// Intersecting// parallel//

coincides// none of these

140. What is the condition for the parallelism of the straight line and the

plane Ах+Ву+Cz+D=0? Не сссы! Тут ответишь сразу, если знаешь геометрию ^^

141. What is the relative positions of the plane x-3y+4z-1=0 and the straight line

o Straight line is perpendicular to the plane

o Straight line is parallel to the plane

o Straight line lies in the plane

o Straight line intersects the plane

o None of these

142. Find equation of the straight line passing through the point M(1,5,-1) and parallel to the line

S(-1,3,-1);

143. Represent the straight line in canonical form.

= =

144. Find angle between the straight lines and

145. Find angle between the straight line and the plane 4x+2y+2z-5=0.

arccos(

146. Point A given in polar coordinates: A (6, ). Find rectangular coordinates of point A.

147. In the triangle ABC find equation of the altitude (height) AH if: A(1,-8), B(7,2), C(-1,2). XD Писец, Триндец, Апокалипсис!

Список получившихся ответов:

1)

2) 10

148. Find angle between the given two planes: x-3y+5=0 and 2x-y+5z-16=0.

Arccos( ) Not my answer: 30 /6

149. Find angle between the given two planes: x+3y+z-1=0 and x+z-1=0

Arccos(

150. Find angle between the given two planes: 4x-5y+3z-1=0 and x-4y-z+9=0

Arccos(

151. Find distance between points A (4, -5) and B (7, -1).

Ans: 5 4 3 2 7

152. Find distance from point (2,5) to the line 6x+8y-5=0.

D=4,7

153. Find equation of the line through points (-1,2) and (2,1).

o x + 3y - 7 = 0

o x + 3y - 5 = 0

o x - 3y - 5 = 0

o -x + 3y + 5 = 0

o None of these

154. Find angle between the straight lines y = 2 x + 4 and y = 3 x – 1.

Arccos( )

155. Find cosine of the angle between vectors____ AB and___ AC if A (0,3,-6), B (9,3,6), C (12,3,3).

Arccos(

156. Find cosine of the angle between vectors AB and AC if A (3,3,-1), B (5,1,-2), C (4,1,-3)

arccos

157. Find the equation of the straight line that passes through the origin and is parallel to the straight line y=2x+6.

Y=2x

158. Put the equation 3 +4 =24 in canonical form and find distance between foci.

=1 c=

159. Find equation of the plane through the point (-2,7,3) and parallel to the plane x-4y+5z-1=0.

x-4y+5z+6=0

160. Find equations of asymptotes of the hyperbola 2 x 2 - 3 y 2 = 6.

k=

161. Parabola y2=2px passes through the point A(2,4). Find parameter of the parabola.

p=4

162. Find foci of the curve

C=

163. If C(-9,-12) is the midpoint of the line segment AB and coordinates of A(-5,-7), find

coordinates of point B.

B(-13, -17)

164. Find the length of the minor axis of the ellipse with center (0,0), foci (0,±3) and vertex (0,5).

b=

165. Find equation of a circle which center C(2,-3) and radius is R=6.

166. Let A (4,6), B (-4,0), C (-1,-4) be vertices of the triangle. Find the equation of side AB.

6x-8y+24=0

167. Let A (4,6), B (-4,0), C (-1,-4) be vertices of the triangle. Find the equation of side BC.

3y+4y+16=0

168. Let A (4,6), B (-4,0), C (-1,-4) be vertices of the triangle. Find the equation of the height

from vertex A.

4y-3y+2=0

169. Let A (4,6), B (-4,0), C (-1,-4) be vertices of the triangle. Find equation of the median

dropped from vertex C.

y-7y-3=0

170. Find equation of the straight line through the point A(-1,-3) if the angle between it and the

Y-axis is 450.

x-y-2=0 or x+y-2=0

171. Find equation of the straight line through the point A(-1,-3) if the angle between it and the эти три примера прошу рассмотреть на всякий)))

Y-axis is 300.

x- y-2=0 or x+ y-2=0

172. Find equation of the straight line through the point A(-1,-3) if the angle between it and the

Y-axis is 1800.

y = -3

173. Two points A (-3,1), B (3,-7) are given. Find coordinates of point M on Y-axis such that

straight lines AM and BM are mutually perpendicular.

Y=22

174. Find distance from the origin to the straight line: 9 x -12 y +10 = 0.

D=2/3

175. Find angle between straight lines 2 x + y -5 = 0 and 6 x - 2 y + 7 = 0.

45

176. Find distance from the origin to the straight line x-y=0.

D=0

177. Determine a so that the lines 3 ax -8 y +13 = 0 and (a +1) x - 2 ay - 21 = 0 are parallel.

a1=2; a2= -2/3

178. Find the equation of the straight line through the point P (-5,2) and perpendicular to the

straight line 4 x - y + 3 = 0.

x+4y+18=0

179. Find equation of the plane that passed through the point (2,-5,3) and is parallel to the

coordinate plane XOZ.

x+z-5=0

180. Find equation of the plane through the three points M1(1, 2, -1); M2(-1, 0, 4); M3(-2, -1, 1).

11x-11y+12z+23=0 11x+11y-33=0

181. Find equation of the plane through point A(1,2-4) and is parallel to the XOY-plane.

X+y-3=0

182. Find distance between two parallel planes: 5 x + 3 y - 4 z + 15 = 0; 15 x + 9 y - 12 z - 5 = 0.

D=-

183. Find distance between two parallel planes: 11 x - 2 y -10 z +15 = 0, 11 x - 2 y -10 z - 45 = 0.

D=-4

184. Find the volume of the tetrahedron if its vertices are A (0,0,2), B (3,0,5), C (1,1,0), D (4,1,2).

1/2

185. Coordinates of vertices of triangle ABC are A(1,6), B(-5,4), C(2,-3). Find angle at vertex B.

Arccos(

186. Find angle between the line and the plane 4x-2y-2z+7=0.

60

187. Find the angle between the planes 3y-z=0 and 2y-z=0.

Arccos( )

188. Find canonical equation of the line x - y - z - 2 = 0, x - 2 y + z + 4 = 0.

= = Если чё не ссы ^_^

189. Find equation of the line through point M(5,-2) and parallel to the OY-axis.

X=5

190. In the triangle ABC find length of the altitude (height) AH if: A(1,-8), B(7,2), C(-5,2)

191. Find equation of the plane through the point (2,-1,5) and parallel to the plane x-3y+5z-1=0.

x-3y+5z-24=0

192. Find equation of the plane passing through the point (2,5,3) and is parallel to the coordinate

plane XOY.

X+Y-7=0

193. Find point of intersection of straight line and plane x + 2 y + 3 z -14 = 0.

M(1,2,3)

194. Find area of the parallelogram formed by vectors a and b: a = p + 2 q, b = 3 p - q; Тупо не знаем как решать XD

p =1, q = 2, (p ^ q) =p / 6.

195. Find area of the parallelogram formed by vectors a and b: a = 3 p + q, b = p - 2 q;

p = 4, q =1, (p ^ q) =p / 4.

196. Find area of the parallelogram formed by vectors a and b: a =10 p + q, b = 3 p - 2 q;

p = 4, q =1, (p ^ q) =p / 6.

-46 ß вроде как))))

197. Find canonical equation of the line passing through point M(1,0,-2) and parallel to the vector

s = 2 i - 3 j.

2x-3y-1=0 and Not my answer: x=1 хеее тупанули))) каноническое уравнение:

198. Find angle between straight lines And

45

199. Find the values of α and β such that vector a = (3,-1,a) is perpendicular to the vector

b = (2,b,1) if b = 3.

a=-4 a=-8

200. Find distance between two points A(5,2) and B(3,-3).

D=

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