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Гродненский государственный университет им. Я. Купалы

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40.

x + 3x

+ 2x

− 2x

+ x

= −1

+ 5x

− 4x

x + 3x

= −1

x1 − 2x2 + x3 − x4 − x5 = 3

− 4x2

+ x3 + x4 − x5 = 3

42.

3x − 2x

+ 2x

− 3x

+ 5x

= 2

− x

2x + 2x

+ x

= 1

5x1 + x2 − x3 + 2x4 − x5 = −1

+ x3 − 3x4 = 4

2x1 − x2

44.

6x + 4x

+ 3x

+ 5x

+ 7 x

= 0

+ x

3x + 2x

+ 3x

+ 5x

= 0

9x1 + 6x2 + 5x3 + 7 x4 + 9x5 = 0

3x1 + 2x2 + 4x4 + 8x5 = 0

46.

x2 + 2x4

= 5

+ x4 = 4

+ 3x4

= 6

x + 2x

+ 3x

+ 14x

= 32

+ 5x2

+ 6x3 + 32x4 = 77

4x1

48.

3x + 2x

+ x

− 3x

= −2

+ x

= 7

x2 + 2x3 + 2x4 + 6x5 = 23

+ 4x2

+ 3x3 + 3x4 − x5 = 12

5x1

49.

2x1 − 3x2 + 8x3 − 4x4 = 1

x1 − x2 + 7 x3 − 2x4 = 2

4x1 + 2x2 + 19x3 + x4 = 86x1 − 5x2 + 11x3 − 3x4 = −3

51.

2x1 − 3x2 + 2x3 + 8x4 = 1

x1 + 3x2 + x3 + 4x4 = 2

4x1 + 3x2 + 4x3 − 4x4 = 0

x1 + 6x2 − x3 + 12x4 = 6

53.

2x1 − x2 + x3 − 4x4 = 0x1 + x2 − x3 + x4 = 2

4x1 + x2 + x3 − 2x4 = 4

5x1 + 2x2 − x4 = 6

55.

x1 + 2x2 + 3x3 + 4x4 = 0x1 + x2 − x3 − x4 = 0

3x1 − 2x2 + x3 − 5x4 = 0

x1 − 5x2 − x3 − 8x4 = 0

57.

x1 + x2 + 6x3 + 3x4 = −3

x2 + x3 + x4 = 3

5x2 − 7x3 + 4x4 = 5

2x1 + 3x2 + 5x3 + 2x4 = 1

50.

2x2

− x3 = −1

+ x2

= 1

2x1 + x2 − 2x3 + x4 = −5

− 2x2 + x3 − 2x4 + x5 = 6

+ x2

− 4x4 + 2x5 = 11

52.

− x

+ x

= 0

x1 + x2 + x3 = 0

2x1 + 2x2 + x5 = 0

− 2x

+ x

= 0

− x3 − x5 = 1

x1 + x2

54.

6x + 4x

+ 3x

+ 5x

+ 7 x

= 0

− x

3x + 2x

+ 3x

+ 5x

= 0

+ 6x2 + 5x3 − 3x4 + 2x5 = 0

3x1 + 2x2 + x3 + 4x4 + 8x5 = 0

56.

x1 + x2 − 2x3 − x4 = 0x1 + x2 + x3 + x4 = 1

x1 + x2 + 4x3 + 3x4 = 2x1 + x2 + 7 x3 + 5x4 = 3

1 − 3x2 + x3 − x4 = −1

58.x1 + x2 + x3 + x4 = 4

3x1 + 4x2 − 2x3 + 6x4 = 115x1 + 4x3 + 2x4 = 112x

59.

3x + x					+ 2x					− 2x						= 2
	1			2					3					4
x + x				+ x				− x				= 1
	1		2				3				4
2x1 − 4x2 − 3x3 + 6x4 = 7
		+ 5x2 + 6x3 − 6x4 = 6
7x1		+ 5x2 + 6x3 − 6x4 = 6
61.
x + x				+ x				− x				= 0
	1		2				3				4			= 0
x + 2x					− 3x					+ x				= 0
	1			2					3				4
x1 − x2 − x3 + x4 = 0
		+ x2			+ 4x3					+ 5x4 = 0
2x1		+ x2			+ 4x3					+ 5x4 = 0
63.
2x + x					+ 3x					+ x				= 0
	1			2					3				4
2x2 + 4x3 − 3x4 − 6 = 0
		+ 5x2 + 13x3 − x4 − 8 = 0
6x1		+ 5x2 + 13x3 − x4 − 8 = 0
		+ 3x2 + 7x3 − 2x4 + 5 = 0
2x1		+ 3x2 + 7x3 − 2x4 + 5 = 0
65.
2x + 8x						+ 2x					= 1
	1				2					3
x1 + x3				= 1

16x2 − 10x4 = 21
x + 10x						+ x				+ 10x						= 2
	1				2				3						5
x − 4x				2	+ 6x				3	+ 4x				4		− 2x	5	= −1
	1			2					3					4			5

60.
x	+ x		= 3
1		4
x1 + x3			= 0
2x + 3x					+ x		= 0
	1			3		5
2x3 + 3x4 + x6 = 7
x	+ 2x			+ 2x			= 0
1			3			5
	+ 2x3			+ 2x6 = −7
x2	+ 2x3			+ 2x6 = −7
62.
6x + 4x					+ 3x			+ 5x			+ 7 x			= 0
	1			2	− x		3			4			5
3x + 2x					− x		+ 3x			+ 5x			= 0
	1			2		3			4			5

9x1 + 6x2 + x3 + 7 x4 + 9x5 = 03x1 + 2x2 + x3 + 4x4 + 8x5 = 0

64.

3x + x					+ 3x							+ 5x					+ 4x				= 0
		1	2		− x					3						4				5
x		− 2x			− x					+ 3x						+ x			= 0
	1		2					3					4					5			5.
x1 − x2 + x3 − 3x4 + 2x5 = 0
		− 2x2			+ x3 + 4x4 + 6x5 = 0
x1		− 2x2			+ x3 + 4x4 + 6x5 = 0
66.
3x + 4x							− x					− x					= 0
		1				2					3			4			= 1
2x + x						+ x					+ 4x						= 1
		1		2					3						4
x1 + 5x2 − x3 + x4 = 2

5x1 − 3x2 + 6x3 + 3x4 = 3

67.

2x1 + 6x2 − 3x3 + 4x4 = 03x1 + 4x2 + 5x3 + 7 x4 = 14x1 + 2x2 + 13x3 + 10x4 = 05x1 + 21x3 + 13x4 = 3

68.

2x2 + 4x3 − 3x4 − 6 = 0

2x1 + x2 + 3x3 + x4 = 0

6x1 + 5x2 + 13x3 − x4 − 8 = 02x1 + 3x2 + 7x3 − 2x4 + 5 = 0

70.

x1 + x2 − 3x4 − x5 = 0x1 − x2 + 2x3 − x4 = 0

4x1 − 2x2 + 6x3 + 3x4 − 4x5 = 02x1 + 4x2 − 2x3 + 4x4 − 7x5 = 0

72.

x1 + 7 x2 − 5x3 − 5x4 + 5x5 = 0x1 − 2x2 + x3 − x4 − x5 = 0

2x1 + x2 − x3 − x4 + x5 = 03x1 − x2 − 2x3 + x4 − x5 = 0

74.

3x1 + 3x2 − 3x3 − 3x4 + 4x5 = 2

2x1 + x2 − x3 − x4 + x5 = 1x1 − x2 + x3 + x4 − 2x5 = 0

4x1 − 5x2 − 5x3 − 5x4 + 7x5 = 3

76.

2x + 3x				+ x		+ x		= 1
	1		2		3	− x	4	= 1
x	+ 2x		+ 3x			− x		= 1
1		2			3		4	5.
3x1 + 2x2				+ x3 − x4				+ x5 = 1

2x1 + 2x2 + 2x3 − x4 − 2x5 = 1

69.

4x + x			− 3x				+ 6x							= 0
	1	2		− x		3	+ 5x					4	= 0
2x + 3x				− x			+ 5x						= 0
	1		2			3						4
3x1 − x2 + 2x3 − 7 x4 = 0
	− 2x2		+ 4x3 − 7 x4 = 0
x1	− 2x2		+ 4x3 − 7 x4 = 0
71.
4x + 11x					− 13x					+ 16x						= 0
	1			2					3						4
3x + 4x				− 5x				+ 7 x						= 0
	1		2				3						4
2x1 − 3x2 + 3x3 − 2x4 = 0

7x1 − 2x2 + x3 + 3x4 = 0
73.
− x1 + 2x2 + x3 + x4 + 3x5 = 0
	− x2 + 2x3 + 3x4 + 4x5 = 0
x1	− x2 + 2x3 + 3x4 + 4x5 = 0
			− x3 + 2x4 = 0
2x1 + x2			− x3 + 2x4 = 0
x + 5x			− 8x				− 5x						− 12x					= 0
1		2				3					4						5
3x − 7 x			2	+ 8x			3	+ 8x					4	+ 13x				5	= 0
	1		2				3						4					5

.75.

3x − 2x

− x

+ x

− 2x

= 0

− 2x

+ x

− x

+ x

= 0

2x1 + x2 − x3 + 2x4 − 3x5 = 0

− 5x2

+ x3 − 2x4 + 2x5 = 0

2x1

77.

+ 2x

− x

= 4

+ x2 + x3 = 6

+ x2

+ 3x3 = 15

4x1

− x

+ x

= 3

+ 5x

− x

= 14

78.

x1 − 2x2 + x3 − x4 − x5 = 3

x1 + 3x2 + 5x3 − 4x4 = −1

x1 + 3x2 + 2x3 − 2x4 + x5 = −1

x1 − 4x2 + x3 + x4 − x5 = 3

80.

5x1 + x2 − x3 + 2x4 − x5 = −1

2x1 + 2x2 − x3 + x4 = 1

3x1 − 2x2 + 2x3 − 3x4 + 5x5 = 2

2x1 − x2 + x3 − 3x4 = 4

82.

9x1 + 6x2 + 5x3 + 7 x4 + 9x5 = 03x1 + 2x2 + x3 + 3x4 + 5x5 = 06x1 + 4x2 + 3x3 + 5x4 + 7 x5 = 03x1 + 2x2 + 4x4 + 8x5 = 0

84.

x3 + 3x4

= 6

+ x4

= 4

+ 2x4

= 5

x + 2x

+ 3x

+ 14x

= 32

+ 6x3 + 32x4 = 77

4x1 + 5x2

86.

x2 + 2x3 + 2x4 + 6x5 = 23

+ x

= 7

3x1 + 2x2 + x3 + x4 − 3x5 = −2

+ 3x3 + 3x4 − x5 = 12

5x1 + 4x2

79.

4x + 2x

+ 19x

+ x

= 8

− x

+ 7 x

− 2x

= 2

2x1 − 3x2 + 8x3 − 4x4 = 1

+ 11x3 − 3x4 = −3

6x1 − 5x2

1 + 3x2 + 4x3 − 4x4 = 0

81.x1 + 3x2 + x3 + 4x4 = 2

2x1 − 3x2 + 2x3 + 8x4 = 14x

x1 + 6x2 − x3 + 12x4 = 6

83.

4x1 + x2 + x3 − 2x4 = 4x1 + x2 − x3 + x4 = 2

2x1 − x2 + x3 − 4x4 = 0

5x1 + 2x2 − x4 = 6

85.

3x1 − 2x2 + x3 − 5x4 = 0

x1 + x2 − x3 − x4 = 0x1 + 2x2 + 3x3 + 4x4 = 0x1 − 5x2 − x3 − 8x4 = 0

87.

5x2 − 7x3 + 4x4 = 5

x2 + x3 + x4 = 3

x1 + x2 + 6x3 + 3x4 = −3

2x1 + 3x2 + 5x3 + 2x4 = 1

88.

2x + x

− 2x

+ x

= −5

+ x2

= 1

2x2 − x3

= −1

− 2x2 + x3 − 2x4 + x5 = 6

+ x2

− 4x4 + 2x5 = 11

90.

2x + 2x

+ x

= 0

+ x2 + x3 = 0

− x2

+ x4 = 0

2x − 2x

+ x

= 0

+ x

− x

= 1

92.

x1 + 6x2 + 5x3 − 3x4 + 2x5 = 0

3x1 + 2x2 − x3 + 3x4 + 5x5 = 06x1 + 4x2 + 3x3 + 5x4 + 7 x5 = 03x1 + 2x2 + x3 + 4x4 + 8x5 = 0

94.

x1 + x2 + 4x3 + 3x4 = 2x1 + x2 + x3 + x4 = 1

x1 + x2 − 2x3 − x4 = 0x1 + x2 + 7 x3 + 5x4 = 3

3x + 4x

− 2x

+ 6x

= 11

+ x

= 4

96.

2x1 − 3x2 + x3 − x4 = −1

+ 4x3

+ 2x4

= 11

5x1

2x − 4x

− 3x

+ 6x

= 7

= 1

+ x

− x

89.

3x1 + x2 + 2x3 − 2x4 = 2

+ 5x2

+ 6x3

− 6x4 = 6

7x1

91.

16x2 −10x4 = 21

+ x3

= 1

+ 2x3

2x1 + 8x2

= 1

+ 10x2

+ x3 +10x5

= 2

x − 4x + 6x + 4x − 2x = −1

93.

2x1 + 3x3 + x5 = 0

x1 + x3 = 0

x1 + x4 = 3

2x3 + 3x4 + x6 = 7

x1 + 2x3 + 2x5 = 0

x2 + 2x3 + 2x6 = −7

95.

9x + 6x + x + 7x + 9x = 0
	1	2	3	4	5
3x + 2x − x + 3x + 5x = 0
	1	2	3	4	5
6x1 + 4x2 + 3x3 + 5x4 + 7 x5 = 0
		+ 2x2 + x3		+ 4x4 + 8x5 = 0
3x1		+ 2x2 + x3		+ 4x4 + 8x5 = 0
97.
x − x + x − 3x + 2x = 0
	1	2	3	4	5
x − 2x − x + 3x + x = 0
	1	2	3	4	5
3x1 + x2 + 3x3 + 5x4 + 4x5 = 0

	− 2x2 + x3 + 4x4 + 6x5 = 0
x1	− 2x2 + x3 + 4x4 + 6x5 = 0
99.
x + 5x − x + x = 2
1		2	3	4
2x + x + x + 4x = 1
	1	2	3	4
3x1 + 4x2 − x3 − x4 = 0
		− 3x2 + 6x3		+ 3x4 = 3
5x1		− 3x2 + 6x3		+ 3x4 = 3

4x + 2x			+ 13x + 10x = 0		100.
	1	2	3	4	6x1 + 5x2 + 13x3 − x4 − 8 = 0
3x1		+ 4x2 + 5x3 + 7 x4		= 1
98.
98.		+ 6x2 − 3x3 + 4x4		= 0	2x1 + x2 + 3x3 + x4 = 0
2x1					2x1 + x2 + 3x3 + x4 = 0
2x1					2x + 4x − 3x − 6 = 0
		+ 21x3 +13x4 = 3			2x + 4x − 3x − 6 = 0
5x1		+ 21x3 +13x4 = 3				2	3	4
							+ 3x2 + 7 x3 − 2x4 + 5 = 0
					2x1		+ 3x2 + 7 x3 − 2x4 + 5 = 0
Задача 6.		Даны координаты вершин треугольника						ABC . Найти:

а) длину стороны АВ б) уравнение высоты СД и ее длину

в) уравнение медианы АМ г) точку пересечения высоты СД и медианы АМ

д) угол С в ABC . Сделать чертеж.

Вариант	А	В	С
1	(7,1)	(9,2)	(0,3)
2	(2,-4)	(8,1)	(-1,3)
3	(-1,1)	(2,5)	(6,-2)
4	(-8,9)	(1,-3)	(6,5)
5	(1,3)	(10,-9)	(15,1)
6	(-2,4)	(8,-8)	(13,2)
7	(-2,11)	(7,-1)	(10,5)
8	(-9,12)	(1,0)	(5,10)
9	(-4,8)	(5,-3)	(10,6)
10	(3,1)	(-3,-1)	(11,4)
11	(-2,0)	(2,-3)	(7,7)
12	(2,1)	(5,-4)	(9,6)
13	(1,1)	(8,1)	(6,8)
14	(-4,-1)	(2,-5)	(5,3)
15	(-1,-1)	(3,-4)	(7,6)
16	(-5,61)	(0,-3)	(8,4)
17	(-4,8)	(1,-2)	(6,9)
18	(2,0)	(5,-3)	(10,5)

19	(-4,7)	(0,-1)	(7,8)
20	(-7,6)	(2,-6)	(7,4)
21	(-5,7)	(4,-5)	(9,5)
22	(-3,5)	(6,7)	(11,3)
23	(-6,10)	(3,-2)	(8,8)
24	(-3,8)	(4,-3)	(9,6)
25	(-2,6)	(0,0)	(11,9)
26	(-5,6)	(0,-5)	(3,10)
27	(-1,0)	(5,-4)	(9,7)
28	(-4,7)	(4,-2)	(8,9)
29	(-6,0)	(5,-7)	(6,8)
30	(-4,0)	(6,-3)	(7,6)
31	(0,3)	(9,2)	(7,1)
32	(-1,3)	(8,1)	(2,-4)
33	(6,-2)	(2,5)	(-1,1)
34	(6,5)	(1,-3)	(-8,9)
35	(15,1)	(10,-9)	(1,3)
36	(13,2)	(8,-8)	(-2,4)
37	(10,5)	(7,-1)	(-2,11)
38	(5,10)	(1,0)	(-9,12)
39	(10,6)	(5,-3)	(-4,8)
40	(11,4)	(-3,-1)	(3,1)
41	(7,7)	(2,-3)	(-2,0)
42	(9,6)	(5,-4)	(2,1)
43	(6,8)	(8,1)	(1,1)
44	(5,3)	(2,-5)	(-4,-1)
45	(7,6)	(3,-4)	(-1,-1)
46	(8,4)	(0,-3)	(-5,61)
47	(6,9)	(1,-2)	(-4,8)
48	(10,5)	(5,-3)	(2,0)
49	(7,8)	(0,-1)	(-4,7)
50	(7,4)	(2,-6)	(-7,6)
51	(9,5)	(4,-5)	(-5,7)
52	(11,3)	(6,7)	(-3,5)
53	(8,8)	(3,-2)	(-6,10)

54	(9,6)	(4,-3)	(-3,8)
55	(11,9)	(0,0)	(-2,6)
56	(3,10)	(0,-5)	(-5,6)
57	(9,7)	(5,-4)	(-1,0)
58	(8,9)	(4,-2)	(-4,7)
59	(6,8)	(5,-7)	(-6,0)
60	(7,6)	(6,-3)	(-4,0)
61	(9,2)	(7,1)	(0,3)
62	(8,1)	(2,-4)	(-1,3)
63	(2,5)	(-1,1)	(6,-2)
64	(1,-3)	(-8,9)	(6,5)
65	(10,-9)	(1,3)	(15,1)
66	(8,-8)	(-2,4)	(13,2)
67	(7,-1)	(-2,11)	(10,5)
68	(1,0)	(-9,12)	(5,10)
69	(5,-3)	(-4,8)	(10,6)
70	(-3,-1)	(3,1)	(11,4)
71	(2,-3)	(-2,0)	(7,7)
72	(5,-4)	(2,1)	(9,6)
73	(8,1)	(1,1)	(6,8)
74	(2,-5)	(-4,-1)	(5,3)
75	(3,-4)	(-1,-1)	(7,6)
76	(0,-3)	(-5,61)	(8,4)
78	(1,-2)	(-4,8)	(6,9)
79	(5,-3)	(2,0)	(10,5)
80	(0,-1)	(-4,7)	(7,8)
81	(2,-6)	(-7,6)	(7,4)
82	(4,-5)	(-5,7)	(9,5)
83	(6,7)	(-3,5)	(11,3)
84	(3,-2)	(-6,10)	(8,8)
85	(4,-3)	(-3,8)	(9,6)
86	(0,0)	(-2,6)	(11,9)
87	(0,-5)	(-5,6)	(3,10)
88	(5,-4)	(-1,0)	(9,7)
89	(4,-2)	(-4,7)	(8,9)

90	(5,-7)	(-6,0)	(6,8)
91	(6,-3)	(-4,0)	(7,6)
92	(0,3)	(7,1)	(9,2)
93	(-1,3)	(2,-4)	(8,1)
94	(6,-2)	(-1,1)	(2,5)
95	(6,5)	(-8,9)	(1,-3)
96	(15,1)	(1,3)	(10,-9)
97	(13,2)	(-2,4)	(8,-8)
98	(10,5)	(-2,11)	(7,-1)
99	(5,10)	(-9,12)	(1,0)
100	(10,6)	(-4,8)	(5,-3)

Задача 7. Даны четыре точки А, В, С и S. Найти: уравнения а) плоскости АВС б) прямой АВ

в) прямой SN, перпендикулярной к плоскости АВС г) косинус угла между плоскостями АВС и ВСS д) объем пирамиды АВСS

е) уравнение прямой SD, параллельной прямой АВ ж) площадь грани АВС

Вариант	А	В	С	S
1	(1,2,-1)	(-1,0,4)	(-2,-1,1)	(1,0,1)
2	(1,-3,4)	(0,-2,1)	(1,1,-1)	(5,6,7)
3	(1,2,-0.5)	(2,1,3)	(0,-1,-1)	(1,6,8)
4	(1,2,3)	(4,-1,-2)	(4,0,3)	(4,3,6)
5	(1,-1,-3)	(0,6,1)	(2,2,-2)	(0,-3,6)
6	(1,1,4)	(2,-1,0)	(3,2,1)	(-1,4,3)
7	(2,1,-3)	(1,1,0)	(-1,2,7)	(1,8,2)
8	(2,3,-10)	(1,-1,-9)	(0,-1,-4)	(5,8,1)
9	(2,1,3)	(0,0,2)	(1,1,1)	(6,9,3)

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