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Национальный минерально-сырьевой университет «Горный»

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attachments_22-12-2011_21-07-09 / kontrol 1

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.				3 2			1
								= ( 3)( 1)1+1		2		1	+

				4	2 1
				4	2 1						3	1
	2	1		0	3		1
				0	3		1
+ 4( 1)2+1			+ 0( 1)3+1			2	1		= ( 3) 1 1		+ 4( 1)( 1) + 0 = 3 + 4 = 1.
						2	1
	3	1				2 1

! ". # $ %

! & '. ( ) ! ",

, * .

2. #

2x + 2 y z = 3,4x + 5z = 19,

2x + y + z = 7.

. .% ! – !,

$&& 0 ! , ) , !, ! $

*0.

	2	2	1				1+2		4 5						3+2	2 1



D =	4 0		5		=		2( 1)		2 1		+ 0 +1( 1)					4 5	=
	2	1	1
	2	1	1
= 2( 1)( 6) + 0 +1( 1) 14 = 12 + 0 14 = 2.
2 D 0, , ! % !
& ':								Dy
		x =		D	x	,	y =	Dy		,	z =	D	z	,
		x =				,	y =	D		,	z =			,
				D				D				D

D – ! , Dx , Dy , Dz - !,

! "6 % ! % *0 $&& 0 !

"6 *0 * . .%

Dx , Dy , Dz .

Dx =	3	2	1	= 2; Dy =	2	3	1	= 4; Dz =	2	2	3	= 6.
	3	2	1		2	3	1		2	2	3
	19 0		5		4	19 5			4	0	19
	7	1	1		2	7	1		2	1	7

2 *,

183

x =	2	= 1;	y =	4	= 2;	z =	6	= 3.
	2			2			2

! , ! % x = 1, y = 2, z = 3

2 1+ 2 2 3 = 3,
	4 1+ 5 3 = 19,

	2 1+ 2 + 3 = 7.

. % * ), , .

2 ! ! ) !

! ( . 1.3 . 46-60).

3. #

2x + 2 y z = 3,

4x + 5z = 19,

2x + y + z = 7

! 6 " * 0.

. . "6 0:

2 2 1 A = 4 0 5 2 1 1

- 0 $&& 0 ! (0 ); x

X = y - 0-* 0 ; z

B = 19 - 0-* 0 * . $ % )

* ! &: A X = B . # $ . : % $

) * % 0 A 1, * " 0 ( ! ! ), 0 A 1 6): A 1 A X = A 1 B.

184

1 0 0

2 ! ! " * 0 A 1 A = E , E = 0 1 0 -

0 0 1

% 0, ! :

E X = A 1 B X = A 1 B ( EX = X .).

2 , 0- 0 X %				!	& X = A 1 B.
<* % 0 A 1 6,			! 0
%: D(A) 0, % % ! &:
A 1 =	1	A11	A21	A31

		A	A	A	,

	D( A)	12	22	32
		A	A	A
		13	23	33

ik – * ! % $ ik 0 A(i, k = 1, 2,3). <! 0

2 2 1

D( A) = 4	0				5 = 2			(! 2).
2	1				1
2 *, D(A)									0, , A 1															6.					.%
* ! % $ 0 :
A = ( 1)2				0 5		= 5;				A = ( 1)3						4 5					= 6;			A = ( 1)4				4 0				= 4;
A = ( 1)2				0 5		= 5;				A = ( 1)3						4 5					= 6;			A = ( 1)4				4 0				= 4;
11				1	1					12						2	1							13				2		1
				1	1											2	1											2		1
A = ( 1)3			2 1				= 3;		A = ( 1)4						2 1							= 4;		A = ( 1)5					2 2				= 2;
A = ( 1)3			2 1				= 3;		A = ( 1)4						2 1							= 4;		A = ( 1)5					2 2				= 2;
21		1			1					22					2		1							23					2	1
		1			1										2		1												2	1
A = ( 1)4			2		1		= 10;	A			= ( 1)5			2			1			= 14;				A		= ( 1)6	2			2	= 8.
A = ( 1)4			2		1		= 10;	A			= ( 1)5			2			1			= 14;				A		= ( 1)6	2			2	= 8.
31			0		5			32						4			5							33			4			0
			0		5									4			5										4			0
>			* ,											! )												*				0
*					! %					$													! " %
. < "										5 3 10									2,5 1,5 5
							A 1 =	1		5 3 10							=		2,5 1,5 5
								1		6		4 14						3 2 7							.
								2		6		4 14						3 2 7							.
								2		4		2	8							2			1	4

, ! * % 0. : % $ * %,

A 1 A = E. 0 A 1 A ! ! ) %

0, ) $ ik ! % C = A B

185

! $ i- 0 "6

$ k- 0 ..

	A 1 A =					2,5	1,5						5								2 2					1		=
						2,5	1,5						5								2 2					1
						3 2 7															4 0 5
						2		1				4									2 1				1
	5 + 6 10				5 5 2,5 + 7,5 5																					1 0 0
	5 + 6 10				5 5 2,5 + 7,5 5																					1 0 0
=	6 8 + 14 6 + 7									3 10 + 7													=			0 1 0					= E.
	4 4 + 8 4 + 4									2 5 + 4																0 0 1
>, * %					0 A 1													. ? !
" 0 X:								5														7,5 + 28,5 35
X = A 1 B =				2,5	1,5									3																			1
				2,5	1,5									3																			1
				3 2 7										19				=				9 38 + 49										=	2	.
				2 1 4										7									6 19 + 28										3
! :															2 + 4 3
A X =			2 2 1						1		=													4 + 15			=		3	= B.
			2 2 1						1																				3
		4 0 5							2																				19
		2			1 1				3						2 + 2 + 3														7
@ 0 X . 2 *,
						X =				x						=	1
										x							1
										y									2	,
										z							3

, :

x = 1,

y = 2,z = 3.

, . , ,

2 ! ! ) !

! ( . 2 . 61-76).
	r	r		r	r
4. ? AB	= 5m + 11n		AC = 2m + 6n !

ABC. ? ! 6 ABC , ! 6 "

186

BC, m n 1 2 ,

, * m n , 135°.

. 1) ? ! 6 S ABC. 6 , ! , ! % ! %,

=1 ×

S 2 AB AC .

AC .

: %

! ! ! %:

r r

AB×

AC = ( 5m

+11n)×

(2m + 6n) = 10m

× m + 22n

m 30m ×

n + 66n ×

= 0; ! ! )

m ×

m = 0,

n ×

! % ! ! ),

30m × n=

30n× m. < "

AB×

AC =

22n ×

m +

30n ×

m = 52n ×

? !

2 1 sin1350 = 52

AB× AC

= 52

= 52.

>, S =

= 26.

2) ? ABC, BC . >

! ) % ,

AB+ BC = AC ,

= AC

AB = (2m + 6n) ( 5m +11n) =

2m +

6n +

5m 11n

= 7m

5n.

? ! ! &:

BC BC.

% ! BC *%. ?

r r

BC BC = (7m 5n) (7m

5n) = 49m

m 35n m

35m n

+ 25n

r r

> , m m

n n

m = m n,

r n.

187

+ 25 (

BC BC = 49

= 49 12

70 1 2 cos1350

2 70m n + 25

2) =

= 49 + 70 + 50 = 169.

49 70

+ 25 2

BC =

= 13.

2 *,

169

3) ? h ABC, ! 6 " .

& ! 6 S =

h BC, h =

6 S :

S = 26, BC = 13. >,

h =

2 26

= 4.

! "

2 ! ! ) !

! ( . 3.3. . 89-97).

5. > !, ! %6

M1 (1, 2,3),	M 2 (2, 1,0) ! % ! %	x 1	=	y + 2	=	z + 2
		3
! " Oxy .			2		1

.	? M3 ( x, y, z)	- ! %

! % ! " Oxy . : % $

! % ! ! , * !

Oxy z = 0, ! % ! % :
x = 1+ 3t,
	+ 2t,
y = 2
	+ t,
z = 2

z = 0.

G					!	t = 2,
x = 7, y = 2, z = 0 .		2	*,	M3 (7, 2, 0) -		! %
! % ! " Oxy .
> !, ! %6 M1, M2 , M3 .
(	M (x, y, z)		- 6 %		!,

188

M1M = (x 1, y + 2, z 3), M1M 2 = (2 1, 1 ( 2), 0 3)= (1, 1, 3)

M1M 3 = (6,4, 3) - !, ,

! ":		M1M 3 (M1M 2 × M1M 3 )= 0
&
	x 1	y + 2	z 3	= 0.
	x 1	y + 2	z 3
	1	1	3
	6	4	3

# $ ! ! $ ! , !

9(x 1) 15(y + 2) 2(z 3) = 0

9x 15y 2z 33 = 0.

!, ! %6

A(2; 1;1)

x y + 2z +1 = 0,

! % "

3x y z +1 = 0.

. *6 % ! %

x x

y y

z z

ãäå s = (m; n; p) - ! %"6

! %, a

M 0 (x0 , y0 , z0 ) - , ) 6 % $ ! %. 2 ! % %

)

! %,

! x y + 2z +1 = 0

! 3x y z +1 = 0 , ! %"6		s
! ! %	$ ! N1 = (1; 1; 2)
N 2 = (3; 1; 1) , !$ ) *

		r	r
		r	r
r		i	j
r	=	1	1
s	=	1	1
		3	1

2 = ir

1	2	r	1	2		r	1	1	=	r	r	r

		j			+ k					3i	+ 7 j + 2 k .
1	1		3	1			3	1
'					M 0 (x0 , y0 , z0 )
			,					"6			! % "
x0 y0 + 2z0				+1 = 0		.	.* %
		z0 +1 = 0
3x0 y0

#. 4.1.1		!,		!,	! )
			x	y + 1 = 0,

	z0 = 0,	!	0	0
			3x0 y0 + 1 = 0,
x0 = 0, y0 = 1. L, M 0 (0;1;0),		r
		s = (3;7; 2) %

189

! % "

y 1

2 !

! %6 ! % "

! "

M (x, y, z) , M 0 M = (x; y 1; z),

= (3;7; 2)

( . 4.1.1),

M0 A = (2; 2;1), s

M0M (M0 A×

s )= 0.

uuuuur

y 1

M0 M

(M0 A×

s )=

2 1

= x

2 1

( y 1)

2 1

+ z

2 2

= 11x y + 1 + 20 z.

2. . ! 11x + y 20z 1 = 0.

! ! " %

2 ! ! ) !

! ( . 3.4, 3.5 . 97-114).

7. ? ! %

x2 + y 2 6x +1 = 0 x + y2 = 5. M	y	M1
. > ).		M1
. > ).

. <! . M


x2 + y2 6x +1 = 0;		x2 6x + 9 + y2 +1 = 9;							x
( x 3)2 + y2 = 8	! %		) O				1		5

								3
0 (3; 0) R = 2					.			3
0 (3; 0) R = 2				2	.
M " x + y2 = 5			y2 = (x 5)			M	2
	! *,		%			M	2	#. 4.1.2
	! *,		%
Ox ,		!
Ox ,		!

, % (5,0). ' ! %

% %" % % :

				2	+ y	2	6x + 1 = 0,
		x			+ y		6x + 1 = 0,

		x + y2 = 5.
%%	y2 = 5 x				%				!,		!
x2 7x + 6 = 0 ,		x = 1,			x = 6 .2 !			x	= 1,	y 2 = 4,	y = ±2.
		1			2			1
x = 6		y2	= 5 x				%	.		2	*,
2

190

) ! * ! " %

M1 (1;2) M 2 (1; 2) ( . 4.1.2).

. ! % "

! ! 6 !,

" ! *6 !

! % ( " ) ).

8. 2 !

x2 + y2 (z 2)2 0,x2 + y2 z,

z 2.

<! !, "6, * $ .

. M x2 + y2 (z 2)2 = 0 !

" Oz , 6 Oz 2 ( . 4.1.3). < *

! . <*N , ) 6

z	z
	2

y		y		y
O	x	O	x	O
x		#.4.1.4		#. 4.1.5
#. 4.1.3
Oz , %	x2 + y2 (z 2)2 0 .		*,

x2 + y2 = z ( . 4.1.4), * ! ,

% x2 + y2 z . 2

A(0; 0; 2) %" $ , , ,

!, ) 6 ! *.

? 0,	z 2	! !,	) )
!	z = 2.	x2 + y2 = z	x2 + y2 (z 2)2 = 0

! " % ! z = 1 ! ) x2 + y2 = 1.

<*N %% $ , !,

, . 4.1.5.

191

9. > ,

	x	2	+ y	2	16,

2 + 2 2

x y z 12.

M !, "6 . <!, !

% ! % ! " % $ !.

z		z	z
z			z
				2
O	4 y	O 2 3	y	O	4	y
				2	3
x	x		x	2
x
#. 4.1.6		#. 4.1.7	#. 4.1.8
#. 4.1.6		#. 4.1.7
.	M	x2 + y2 = 16	0			" Oz ,

! %"6 % % % ) 4 0

( . 4.1.6). M x2 + y2 z2 = 12 !

! * ( 6 % - Oz ), " " ( ! "

z = 0 ) 12 =23 ( . 4.1.7). <, % ! % ! * ) ) , ! %"6 % 0. <!, ! % ! " % !. : %

$ " : x2 + y2 !

! *. 16 z2 = 12 z2 = 4, z = ±2 .

? . 4.1.8 * ) , ) 0,

! ! *, ! " % !

) % 0 Oz , R = 4,

! ) ! % z = 2 z = 2 .

4.1.2. * + ,

-" .2

* -

* ! !

>! "6 :

192

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