matematika 8 - udžbenik (krug)
DESCRIPTION
Matematika 8 - Udžbenik (Krug)TRANSCRIPT
-
I[ I ' ! I
.
(
-
8
8 .
^
-
: ,
8
: , , 244. 011-347 5577, -: [email protected]
:
: , , , , ,
:
:
:
: ,
13 978-86-7136-175-0
650-02-00299/2010-06 21. 07. 2010. , .
1 - ,
37.016 : 51(075.2)
, , 1954- 8 : 8.
/ . - 1. . - : , 2010 ( : 8 1). - 146 . ;24
3.000.
18 978-86-7136-175-0
158.3- 173075468
: 3000
: !,
-
VIII , 2009. . :
1. ,2. , ,3. ,4. ,5. ,6. ,7. ,8. ,9. ,
10. ,11. .
, - .
, 2010.
-
I
-
1. .......................................................................... 11.1. .......................................................................................... 21.2. .................................................................................... 71.3. ............................................ 10
2. , .......................................................................... 152.1. , .
................................................................................ 152.2. ; ......................................................... 202.3. , ,
................................................................................................................ 222.4. ..............................................! ....................................... 262.5. (, , )..................... 272.6. ......................................................................................................... 30
3. ............................................................................................ 33
3.1. ............................ 343.2. .............................................................................. 363.3. ....................... 393.4. ......................................................... ........................ 433.5. ........... .............................................................. 453.6.
..................................................................................................... 473.7. ........................... 50
4. .............................................................................................................. 554.1. : , , ............................................................... 554.2. ............................................................................................... 584.3. .......................................................................................... 604.4. ........................................................................................ 64
-
5. ....................................................................................................... 715.1. : , , ........................................................... 715.2. ............................................................................................ 745.3. .................................................................................... 755.4. .................................................................................... 80
6. .............................................................................. 876.1. = + ............................................ ................... 876.2. ; .......................................... . 896.3. ................................ 926.4. .................................. 93
7. 977.1.
............................................................................................................. 977.2.
...................................................................................................... 1007.3. .
................................................................. 103
8. . 1078.1. ........................................ 1078.2. .
................................................................... 1098.3.
; .............................................. 1118.4.
......................................................................................................... 119
9. ................................................................................................................. 1239.1. ....................................................... , ................. 1239.2. . ............................................................... 1259.3. .......................................................................................... 126
10. ................................................................................................................... 12910.1. ......................................................... 12910.2. . ................................................................... 13110.3. ............................................................................................ 133
11. ............................................................................................................... 13511.1. .................................................................................. 13511.2. () ................. 13611.3. ................................................................. 137
-
12. ................................................................................................. 1391. ...................................................................................... 1392. , .................................................................................... 1393. ........................................................... 1404. ............................................................................................................... .1415. ......................................................................................................... 1416. ........................................................................................ 1427. ........... ...................... 1438. .................................. 1439. ................................................................................................................. 14310. ............................................ .................................................................... 14411. ................................................................................................................. 144
................................................................................................................... 145
-
[/
-
1.
, , .
^^^ (. 1) , . = , /3 = (3\ 7 = 7 1 . ~ 1 11 . ,
~ ~ _
- . , , :
= \ , = - 1 = \.
1. 2 \ \. = | \ , || \ \ = , = 2 .
-
2 1.
/ . = /.^ / = / \ \ , \ ,
13
,
\
^ = 3 \ = 3 , \ = 3 -3 = 9 \ = 3 - 2 = 6 .
1.1.
- , , .
. .
3 , 7711 2 6 . - , .
^ 2 ~ \ 2 '
^ ^ 2 2 1 , - = . ,
\ 2 \ 2
2 + 2 2 1\2 \2 \2 \2 \2
_ \ + \2 = \\ 2 \ 2 \ 2
, 4, 5, 6 .
2 , 2 = 2 =
^ ' 8 =
-
1.1. 3
\ 4 5
= ^ .
I 2. 6 = 5 , = 8 \ = 4 . \ || \ \ .
5 8. . = , = ------, \ \ 4 \
4 8 32 5 \ 4 8 , . \ = = = 6,4.
5 5
6 7
3. 7 = 2 , \ = \ = . || \ \ , \ .
.
2 3 6 = 2 -3 , . = = 1 ,
6
\ = + \ = 1 + 3 = 4 .
-
. - . , 5 : 6 = : ( : = 6 : ). , , (. 8). \ \ = 5 ,
-
4 1.
= . \ . \ . : 6 = : .
. , , (. 9).
X1111X11
}
-X------
\\ : 2 [ 11 X X
9
, , .
I 4. ; . :) ; ) 2 : 3 .
. ) , \ , \ = \ 2 = 2 3 (. 10). $ , \ 2 \ , ^ , \ || 2 || 3 .
-
1.1. 5
\ 2 \ = \ 2 = 2 , \ .
\ _ \ 2 = 2 \ \ 2 2 2
\ , \ 2 2 3 , \, \ 2 2 .
) . 10 \, 2 , 3 , 4 5 (. 11) \ = \ 2 = 2 3 = 34 = 45 , \, 2 , 3 , 4 \ \ || 22 || 33 || 44 [| 5 . , , \ = \ 2 = 23 = 34 = 4 , 2 2 : 3, . 2 : 2 = 2 : 3 .
: , .
, , \ \ 1>
~\ ~ \ '
\ \ . - , , \ , , ' (. 12).
,
. _ , , 7 = 7 = ;, \ , .
'
\ \ \ ' .
'
-
6 1.
1. 13 .
) = , = 2 , = 12 .) = 30 , = , = 50 .
) = 8 , = 20 , = .
13 14
2. 14. :) = 4 , = , = 5 ;
) , 1 : = 3 : 4 = 2 ;
) , = 3 , 1
-
1.2. 7
5. .) .) .) : = 2 : 1 .) 5 : 2 .
6. \ \ 18 19 ?
1.2.
- .
^^ - , ^ .
.
5. ^ 20.
2 3
81) + 8 + 4
6 + 6 + 3
,
~ '.
|| \ \ = 11 = 1 1 .
-
1.
, , .
2 . = - .
. , .
^^ , , 1 1 1 .
6. 11,1 21 ?
. ~ ^ ^. , = =
12 9 ~ ~ ~ 3 ~ 3
_ = .
.11 /; I \ , , . = 3.
.
1 11 , .
_7. = 8, 6 = = 12, = 12 , = 9 = 18 ?
-
1.2. 9
. 1
8 2 612 3 ' 1>1 9
6 1 &1 1
.
23
112 2 18 ~ 3
, .
^^ , ^^ .
, . - .
1. = 6 = 4 , = 12 61 = 16 . .
2. 1 , 2 1 ,5 , 8 , 1 6 1 2 ?
3. 22?
22 23
4. 23 , = 19 , = 29 , = 32 , : = 5,8 = 3,8 . ,
-
10 1.
1.3.
. , , .
8 . 54, 36
^
180 - (90 + 36) = 180 - 126 = 54,
^ (. 24).
24 25
9.; = 4 = 9 . .
. (. 25)
= 180 - (90 + ) = 90 - = .
, ,
~ ~ 1 ~ '
= .
>2 = , 2 = 4-9,
2 = 36,
-
1.3. 11
- ., - - , (. 26). , ( ) ( - 9).
, 26. :
1) ~ :
_ . _ ~ ' ' ~ ^
2 =
2) ~ :
_ . _ ~ ~ ' ~
2 =
3) ~ :
. _ ~ ' ~
2 = -
[ 10.! 27.
. = 4 = 2 , 2 = ( 2)) -
16 8 = = = 8 . 2
-
12 1.
27 28
11. 6 28.
. = = 9 , = + = 3 + 9 = 12 . &2 = - ( 3))
6 = / = \/3 - 12 = /36 = 6 .
29
12. 6. = \ / 6.
. = + 1? , = = 6 (. 29). 5 - , & 8
, , = . , ( ), 1) 2 = , 2 = 6, . = / 6.
- &.
- :
2 + 62 = 2,
& , . 2) 3) ,
.
-
1.3. 13
(. 30)2 =
2 + &2 = '5 + -
62 = ,
:
1 = ( + ) = = 2.
31
1. (. 31) 10 , = 6 . , , .
2. :25 144
= 9 = . :
) ; ) ; ) .
3. - 2 3 . .
4. = / .
-
14 1.
-
2. ,
2 . 1 . , .
- , . : , , , , 2 ,. . . , : , , , . . . . , , . . . , ,!, 2 , . . . , , 1 '
6 2 , ( ) .
3 ( ) .
( ); 6 . , . ^ .
1
2
-
16 2. ,
"1|[|11111!11111111||||!!|| ||| |!||||1|||!||/
, .
:
.
- . .
.
1. , , >, . ?
. ( , ) , . 4. . , 4-3. , ( , ) ,
, (3, 7 , . . . , (. 5).
4 5
-
2.1. , 17
5 , , N . , N 6 , , N , : .
, N , . : .
6 . , - , . ., , - .
6
, (. 7 . 8). . , , .
7 8
, .
.
, .
. .
-
18 2. ,
.
, , - (. 9) . , , , ( ) ,
.
2 , , (. 10), . .
10 11
, , - , (. 11). , , .
6 (. 12). 6 . . ( - 6). ,
12
.
( (. 13) ( ) (
-
192.1. ,
^ 14
14. ( ) ( ).
.
1. \ \ \ (. 15). , \ , , ? , \ , , \, \ 1
15 16
2. , \ ^ (. 16). ?
3. , , , , .) ?) , , ?
4. , , . ? .
-
20 2. ,
5. ?(I) .
(II) .(III) .
2.2 . ;
6 . , , . , (. 17).
17
18
6 , . .
, . , , \ (. 18). - .
4. ' (. 19) , - (. 20).
19 20
-
2.2. ; 21
5 . ^ 21 :
) \) ;) .
21
5
. ) , , .) 9, , ^ .) : 1: , ^ .
1. . , ?
2. 22 . :) \) .
-
22 2. ,
2.3 . , ,
. .1) (. 23).
|| .
2) ( = { } ) 24. .
3) (. 25). 2.1 ( ) . .
, . , , .
, .
(. 26).
23 24
25 26
-
2.3. , 23
& , .
6. / (. 27). ^ - ,
. , \ || ]_, .! .
7
\
27
, - (. 28).
( 1 ) .
- . (. 29).
, , 6 , , .
.
-
24 2. ,
7. ( ) 30. 6 , .
8. 31. - 6 , ( , , , , ,. . . ), .
30
31 32
(. 32). \ _1_ \ 4- , \ , . \ .
(. 33).
-
2.3. , 25
' . ' .
10. (. 34). : ' = ' = ' .
1. . :) ; ) ?
2. , , ?
) 6 , || 6.
) 6 , 1 .
) , .
3. ^ (. 35) :
) ^ -,
) ^ .
/
1
/ _____ 7
4. 5 , 7 (. 36). .
-
26 2. ,
2.4.
(. 37).
/3 , . .
/ 3 :
1) /3 . /3 . /3 = (. 37).
2) (3 (. 38). . \ \ 0 .
/3 37 , 2.1 , , = /3. || /3. , .
. , .
11. , ,
37 38
-
2.5. 27
. . , , , .
1. , ?) , (3 7 || /3 /3 [| 7 , || 7 .) ,
.) ,
.) (3 ,
/3 .
2. (3 (. 39). :
) , /3;
) , || /3;) ,
(3. 39
2.5. (, , )
, (. 40) . ' . ' .
40 41
-
28 2. ,
(. 41) , ' ' . . ' ' ' .
, | (. 42), (. 43). ' , .
, , , , , . , , (. 44).
44
12. (. 45). :
) \ , \ , \ , ;) , , , ,
,;) .. ) , , , .) , , \ ;
.) .
-1/ 7
1
/1_____ 7
%
45
-
2.5. 29
13. ' (. 46), ' , ' . 8 , 12 = 5 , '' .
. ' || ' . (. 47) ' ' , = ' ' , = ' - ' = 12 8 = 4 .
= / 2 - 2 = /52 - 42 = /9 = ,
'' = 3 .
1. ' ' :) ; ) 7
2. . : ' = 8 = , '' = 12 , .3.
(. 48). =4 , :
) ' = 60;) 1 ' = 45;) 1 ' = 30.
48
-
30 2. ,
2.6 .
, . . - ( 49).
7
0
/
- . - , , , .
50.
50
- , , , . , , .
-
2.6. 31
14. 51 . - . : , , , , .
12 : , , , , , , , , 1 1 , , \ 11 8 : , , . , , 1 , 1 >1 .
51
1. . .
2. ?
3. , 52?
52
-
*
32 2 .
I
-
3.
. - . , ,
5, ^ - 7, 4 ^ 6\/3 ,
, , ( ) . ,
1 + 2, + , + &, 2 , \/2 + 5,
9
2 - 1 ,X / - 3,
2 22 + 5
. -
, . :
- 6 6 + ( + ) = ( + ) +
6 = 6 ( ) = ( 6) - ( 1) + ) = - + -
( ) ( ) ( ) ( ) ( )
-
34 3.
, & , , () . , ,
(I2 + 1) = + 1 ,
+ 1 ( + 3
2 + 2 + 2.
(2 + 1) + 1 + 3 + 1 , ---------1--------- + 2 () , .
+ 6,
6 .
3.1 .
. , (, ). , ,
14 - 13 = 3, 5 + 2 = , /9 = 3, 1 2 - - - 4 = 6 + 8: 2,
, ,
2 = 1, 3 - 2 - 5 = 5, 0 : 5 = 5.
, , , . . , , , . . . .
, ,
+ 2 = 4
-
3.1. 35
2 - . , .
1 .
2 1^ + 3 = - , 2 ~ 2 = ( - ) ( + ),
2 + 1 - 2 - = 0, 7 = 5 , 4 ^ = 1 1
. , , .
, . , ,
+ 6 = +
-
36 3.
1. , ?) 5 -2 3 - 62 : 12 = 37; 6 ) 4 - 5 -3 + 6 = 3.
2. 4 :5 9 - 4 2( + 1) + 4
+ ^ = 2 6) " 3 + 5-1 = ^ ?
3. {2, 1,0,1, 2}
2 + 3 + . _ ^ ?
3.2.
3.1 . .
. , , .
4. 2 5 = 11 7 + 6 = 62 . 8 . = + 2 = 1 .
- , ( ) = () / ( ) = () () ( ) .
5.
3( + 3) 2( 1) 11 = 5.
. :
+ 9 2 + 2 - 11 = 5, ! 11 - 11= 5,
= 5.
-
3.2. 37
, 1\ = 3( + 3) 2( 1) 11 12 = 12 1 \ . . 5, ,
3(5 + 3) - 2(5 1) 11 = 3 8 - 2 4 11 = 24 - 8 11 = 5.
, - , , .
6.
2 + 7 = 14 + .
. 7
2 + 7 7 = 1 4 + 7, 1 = 7.
7 .
1. ( ) = () ( ) ( ) , ( ) + () = () + ( ) .
, 6, () , - . , 1 .
1 '. , . .
7.
11 = 26 + 2.
. 11 , 2 , .
2 = 26 + 11.
-
38 3.
( 2 + 11 - 1 .)
= 37. 37.
8.
. 6. -
+ 2 7 = 6 - 4, + 14 = 24, = 24 14,
= 10, 10
= .
.
10
2. ( ) = () , - ( ) = ( ) .
1 3 , , - 3 = 2 12 = 3
4. .
1. :) 5 + 3 4 + 3; ) 2 + 2 + 4 ( + 2)2;) : + 3 2 + 6 ?
2.
2( 1) 2 1 + 2( + 1) - ( - 2) = 7.
3. :) 2 = 16 = 24; ) 1 = 4 1 + 3 = 5;) + 5 = ; + 1 + 2 = 3( + 1) ?
-
3.3. 39
3.3.
. 4, 6, 7 8 , , , , , . 1 , 1 ' 2, 3.2, = , . .
9.
. 12 ( 2). .
4( - 4) + 6( + 1) - 12 = 12( - 3).
:
4 16 + : + 6 12 = 12; 36,10. - 22 = 12 - 36.
22 , 12 ( 1 ') , :
4 ( 2( + 1) 3 + 4
1 = 3.
10 12 = 36 + 22, 2 = -14.
~ (. 2) ( 2)
7 .
10.
1 15 2 3
-
40 3.
. (2,3 ,5 ,6) = 30 30 ( 2) .
6 15 = 10 + 5.
, ( 1') :
6 - 10 = 5 + 15.
( )
- 4 = 20
4 ( 2)
20
~ 4 5 .
. = 5.
- - - , () ( 2) ( 1'). , .
( + 6)2 = ? + 2 + 62 ( 6)2 = 2 26 62 2 62 = ( 6)( + 6) ( 6)2 = 2 62
2 62(!=
( ) ( ) ( ) ( )
( )
11.
6 2 + ( + 1)( 1) :1
7 + 26 12
.
6 - 2 + 2 - I6
1 -7 + 2
5 = 1 -
127 + 2
6 12
-
3.3. 41
12,
2 (5 - ) = 1 2 - (7+ 2).
10 - 2 = 12 - 7 - 2.
10 7 ,
-2 + 7 = 12 - 2 - 10,5 = 0,
0 = 5 = 0.
0.
, , . , - 12 13.
12.
3 + 6 = 3( + 1) + 3.
.
3 + 6 = 3 + 3 + 3,3 3 = 6 6,
0 = 0 .
, - .
; 13.
( + I )2 + ( + 2)2 + ( + )2 + ( + 4)2 = (2 + 5)2.
.
2 + 2 + 1 + 2 + 4 + 4 + 2 + 6 + 9 + 2 + 8 + 16 = 42 + 20 + 25,
42 + 20 + 30 = 42 + 20 + 25,
42 + 20 - 42 - 20 = 25 - 30,0 = -5 .
-
42 3.
0 = 5. .
3.1
+ = 0. (*) 9-13, - :
1) 0, = 6, . = . (*);
2) = 6 = 0 , 0 = 0 (*) ;
3) = 0 6 / 0 , 0 = 6 (*) . .
, - .
14. \ + 3| 3 = 2 4.
. + 3, + 3 ^ 0, . ^ 3,
+ 3 = \( ( + 3), + 3 < 0, . < 3,
. , , ^ 3.
+ 3 3 = 2 4, . 2 = 4,
= 4, = 4.
4 3 . < 3.
( + 3) 3 = 2 4, 3 3 = 2 4,
2 = 4 + 3 + 3, = 2,
2
3' =
-
3.4. 43
,
3. , - 4.
1. :
) - = 2; ) 2 7 = 15; ) 4 + 3 = 2 + 9;
) 5(2 + 3) 3(4 + 3) = + 5;) 3( + 3) + 2 = 4 + 9;) (15 + 2 (5 + 11)) = 2 8 .
2. : - 1 6 - 5
) * 0 , 5 - ;
4 2( + 1) 5( 3) 11 + 43) ~ ~ + -----1--------1 --- ------8 + 2 7
3. :
) || + ^ = 3;
) | + 3| = 2 (2 - ) ;) 2 | + 1 | + 3 = 0.
3.4.
, - . , :
3 > 2; - 5 < 0; ~ 3,62 < 3,623; 2 < 2,8 7
:1 12 ^ 4
-5 ^ -7 ; 2 > 2 .
, , (< , < > ) . :
5 + 7 < 11, + ^ < 5 + 2, 6 - 1 ^ 2 - 2 + 2, 7 > 110
-
44 3.
. , ,
+ & < 0, + 6 ^ 0, + & ^ 0, + 6 > 0,
6 .
15.
12 + 5 > 0, 3 x ^ 7 5 , - | < 1 +6 5 3
,
:2 + 4 + 4 > 0, 2^/+ < 7 , :3 + : ^ 0
.
16. 5: 11 > 3, . 5 , 5 -5 11 > 3, 1, 5 1 11 > 3 .
, , - . .
1. , ?
) 5 + 3 -2 > 1 5 ; ) | + | + | ^ 1;
) - 5 - 2 ^ < 11 + 2 -
2.
3.
5 ^ 2( 1) + 6 :
) 8 ; ) - 2 ; ) 1,5;
12 + 4 < 7 1 ?
) 2^ ; ) /2 ?
. 3{2; 1; 0,3}
-
3.5. 45
3.5.
3.2 . , , , . - .
17.
( + 1)(2 - + 1) - ( - 1)(2 + + 1) + 2 < 5 .
.
3 + 2 - 2 + + 1 - (3 - 2 + 2 - + 1) + - 2 < 5,
3 + 1 3 + 1 + 2 < 5, < 5.
< 5, (,5).
, -
. , .
, () () , - () < () () < / ( ) , 17.
, , , 1 , 1' 2 3.2. ,
- , , .
-
46 3.
18. 18+2; ^ 3
2 ^ 3 18.
18 , . ^ 2 1 . [21,+).
, 2 3.2, , .
19. 3 < 5 - , , 2 , 6 < 10 . , , - 1, 3 < - 5 . ,
- , .
, , .
, < > , ^ > .
20. :
1. 2 + 5 < 6 6 + 15 18( 3 );
2. 14 + 12 > 2 7 + 6 > ( | ) ;
3. 5 < 0 5 > 02 2
( 1 );
4' 3 + 2:< - 4 5( 6 ).
1.
( + 2)2 - ( - 2)2 + 3 x ^ 5 .
-
3.6. 47
2. ?) 1 + 1 ) 4 ;) 3 < 7 < 10;) 2 + 6 < 5 4 + 12 > 10;) ; - 1 < 7 + 3 - + 1 < 7 3;
) -5 + - ^ 1 5 ^ > + 1;
) 18 + 6 > 0 > 3 .
3.6 .
, . - , , . , .
21.
14 > 11 + 24.
. 11 .
14 11 > 24, . > 24.
3
> 8.
(
8 , . (8, + ), 1 .
8
1
-
48 3.
22.
7 - 11(:+ 2) < 22: + 5.
. " ,
7 ; 22 ^ 2 + 5.
2 22 . , :
7 1\ 2 < 5 + 22, .6 ^ 27.
6 (. | ) . ^
27 > - ~ , .
^ 4,5.
[4,5; +) 2.
-------------- , - ... 1............-......-4 ,5
2
23.
3
. - 2 3, . 6, :
- 6 - - -6 > 4 - 6 , .
2 ^ 24, ^ 24.
1
^ 24.
-
3.6. 49
(, 24] (. 3).
- 2 4
3
24.
2(3 ) 3(2 + ) < 5 + 1.
.
6 2 6 < 5 + 1,2 + 5 < 1,
0 < 1.
. .
25.;
3( + 4) 5 < 2 2.
.
+ 12 5 < 2 2, 5 + 2 < 2 12,
0 < - 10 .
. - .
- :
1) :
(,), (,], (,+) [,+),
;2) ; 3) , . .
-
50 3.
1. :) 2,4 < 0,6; ) 16 ^ 9 6 ;) 1 5 > 3 + 2; ) 13 2 5(2 + ) + 3
:) 2 1 ; ) 12 - 4
?
:) 3( 1) + 6 ^ 1 + 3( + 2); ) 4 > 12(3 - 1) + 4(1
:, 1 2 ,
) - < 5; ) (- ~ 3') ^ 5;. 2 , 1 + 3
) ----------------- - > 1' 2 3
3.7.
- , , , . :
1) ( ) .
2) .
3) - .
4) .
26. 7 3, 10 , 9. ?
. . .
-
3.7. 51
.
7-: + 3 10 9.
.
7 + 3 = 10 9.
. :
7 10 = 9 3,- = - 12 ,
-1 2
= 4.
4.
27. 16 2 440 . 250 , 120. , ?
. . . . 16 . .
250 + 120 (16 ) = 2440.
.
250 + 1920 - 120 = 2440,130 = 520,
520 = ---- ,
130 = 4.
, 4 16 4 = 12 .
28. 12 , 4 . .
. . . . 4.
-
52 3.
.
( - 4 )2 + 122 = 2.
.
2 - 8 + 16 + 144 = 2,
8 + 160 = 0,
8 = 160,
-1 6 0- 8
= 20.
20 .
29. 10 . 10 / , . ?
. . V . .
5 = V I . > 10, ( + 10) 8 .
. -
10 = 8( + 10).
. :
10 = 8 + 80, 10 8 = 80,
2 = 80,
V = 40.
4 0 / .
1. 8 2. 5 , 31. ?
-
3.7. 53
2. 2 800 , 2 300. . ?
3. 4 , 2 . .
4. 24 ( ). 5 , ?
-
3. I 54
-
4.
4 .1 . : , ,
. ( 1) , .
1
- .
-
56 4.
, ( 2).
2
. (. 2 2), (. 26 2), , (. 2), , . . . . ( ) , . .
, - . .
, (. 3).
) )
3
. , , , .
-
4.1. : , , 57
. 2, 2 2 , 26 2 ; . , . 36 . , .
.
1. I 4 1 1 1 . ^^^, 1 1 , 1 1 ^. : , , , ^^, 1 1 ^ ^ , : , 1 . .
4 5
, , , (. 5). , .
. - - , , - ,
/ / 111 \ 1 / \ /
11111
11111
11
111
/ ----111
-
^ \ / \ / \ /
6
- .
-
58 4.
1 .
V111
\
/ 7
1
7 ^ . - ? , ? , ? ?
2. , : ) ; ) ?
7
3. 48 . ?
4. = , 6 = 4 = 12 .
4 .2 .
2. | ( ) (. 8) , (. 9). .
(. 10) (. 106).
-
4.2. 59
( ) 11, .
/-------V \ \
\____ __ /
11
1 . . 12 ?
12
2. :) ;) ;) .
-
60 4.
4 .3 .
, = 62, , 6 , = 2 + 21> + 2 (. 13).
13
. , . () , . ,
= 2 +
4 .3 .1 .
- . , = 2 , (. 14).
14
-
4.3. 61
= 2 + = 22 + = 2 ( + 2 ).
-
= 2{ + 2).
3.
-
62 4.
^
= /3, 2 2
= 2 + = 2 * + 3 = + 3.4 2
= / + .
5. (. ) = = 4 .
.,2
= 2 + = 2 + 2 = 8 + 48 = 8(\/3 + 6) 2.
4 .3 .3 .
16
, . (. 16)
2 2
-
4.3. 63
- , (. 17).
/ \
,
= 2 + 2 32\/3
2+ 6 = 32\/3 + 6 = 3(\/3 + 2 ).
= 3(\/3 + 2).
6. = 3 \ /3 . = 10 . .
. - = \/3 (. 18). 3\/3 = / = .
= 3(\/3 + 2) = 3 6(6\/3 + 2-10)
= 36(3\/3 + 10) 2.
18
-
64 4.
1. 16 2, 5 . .
2. - = 2 = 8 , = 12 .
3. = 4 , = 5 . .
4. - = 2 = 8 ?
5. - = 10 6 = 8 = , = 9 .
4 .4 .
7. = 4 , 6 = 2 = . , . , . 1 (. 19).
19
6 = 4 2 = 8 1 . 8 . = 3, - &- = 4- 2- 3 = 24 . 24 3 .
, , : 3 , 4 3 , 3 , :
1 3 = 1000 3 = 1 000 000 3,
1 3 = 1000 3 = 1 000 000 3,
1 3 = 1 000 3.
-
4.4. 65
7 , V = 6 . V = , .
, ,
V = 6 .
V = - II,
( = -), ( = ) .
\ \ \ 1 1 11 (. 20). \ \ - , \ \ \ . , \ \ \ , 0 \ 0 \ \ \ \ \ , \ \ \ .
20 21
V*, , , V \ \ \ . ,
= 2 V, - = 2 V,
V = . , = - , V = . V = .
-
66 4.
. . , 21 . ,
V = !+ 2 + + 4 = + 2 + + 4 = {\ + 2 + + 4) = .
,
V = .
1. : ) = 1 2 ; ) = 2-\/2 .2. = 6 = 5 ,
= 190 2 . ?3. , V - , .
14 2 130 !2 13 5
V 520 13 25/23
4 .4 .1 .
V = . .
8. .
-
4.4. 67
. ,
V = = 2.
9. = 13 ?2 = 12. = 5 .
. = = = 5 = 13 6 5 = 9 0 3.&
. . = 21 , = 1 1 , = = 13 . = 10 . .
. = (. 22)
6
. = -------- = 5 . - :
22
= /2 - 2 = / ^ - 52 = /169 - 25 = / = 12.
( ) 1) 21 11
= = -----------12 = 192 2,
V = 192 10 = 1 920 3
4 .4 .2 .
- , (. 23).
V = =2\ / 3
II,
2/ 3 .
23
-
68 4.
=2\/3
.
. 11. = 9 ,
-
4.4. 69
. = 6 ,
6 = 72,72 72
= - = = 4 , 6 -3 18
3 3
V = - 2\/3 = - 16/3 3 = 72 /3.
1. & = /2 , 15 . .
2. : ) , ) - = 2 = 5 .
3. ^ = ^ = 8 . > = 10 . .
4. = 4 , . .
4 .4 .3 .
V - , .
= .
.
20 , 15 1 . = 2,6 / 3?
. , V = 20-15-1 = 300 3 . = V = 300 2,6 = 780 .
14. ( ) 7 2 ? ( = 0,85 / 3 .)
13.
-
70 4.
. V = - 2%/3 = 4\/3 7 = 42\/3 ~ 72,75 3 =
72 750 000 3 , = V ~ 72 750 000 0,85 = 61837 500 ~ 61,841.
15. 25 12 6,5 3,51 . .
-. V = 25-12-6,5 = 1950 3 ,
=
3510V 1950
= 1 ,8/ 3.
16. - = = 12 = 364,34 . ?
. V =2
46,77 3 . = 4
364,34V 46,77
= -\/3 12 = 27\/3 ~ 4
7,8 / 3.
1. = 20 . 0 ,7 / 3 .
2. 69, 21 . 84 ? = 0,84 / 3 .
3. = 5 . 80? ( 8 ,9 / 3 .)
-
5.
5.1 . : , ,
1 2 . . . 8 (. 1). ^ ^, 8 2 3 , , 8 . - 1 2 . . . , 5 .
- .
5
2.
2540. , . . ? ? .
-
72 5.
, (. ), (. 36), (. ), (. ), .
3
. , .
, .
, . , (. 4).
] 5 . 51, , , , 3 0 . : , , , : , , 8 0 . 8 0 .
. ( ) (. 6).
-
5.1. : , , 73
, . , - - , - - , (. 7).
7
, (. 8).
1. 9 . ? , ? , ?
-
74 5.
2. = 5 , = 8 . (. 10).
3. - 12 . ?
4.2 .
, . 11 - , , 12 .
11 12
( ). 13 .
13
-
5.3. 75
1. :
) = 5 = 4 ;
) = 4 = 3 .
2. , .
3. ? ?
5.3.
. , () - , . ,
= +
5 .3 .1 .
- . , = 2 ,
, = 4- = 2. ,
= + = 2 + 2 .
= 2 + 2 .
-
76 5.
2. = 8 = 3 .
. - 8 0 (. 14). - , - .
( (2 \ 22 /
-
= ^ 2 + 0 2 = 2 + ( | ) 2 = 7 9 + 1 6 = /25 = 5 ,
= + = 2 + 2 = 64 + 2 8 5 = 144 2 .
- .
3. - 1) , . = 32 6 = 10 , = 12 , .
. 8 0 (. 15). \ 2 :
\ = \ 2 + ( ^ ) = 1 2 2 + ( | ) 2 = /144 + 256 = 7400 = 20 ,
2 = ^ 2 + ( ~ = 122 + ( ) 2 = 7144 + 25 = 7169 = 13 .
15
-
5.3. 77
= 6 = 32 10 = 320 2, ,
= 2 + 2 ~ = 2 + = 32 13 + 10 20 = 416 + 200 = 616 2., = + 320 + 616 = 936 2 .
1. :) = 10 = 8 ;) = = 5 ;) = = 4 .
2. - = 8 2 , = 32 2.
3. - = 2 .
5.3 .2 .
- , ( , , ).
16 - . - . ,
= + =2/3 + 3 -----.
4 2
=2/3 ----- 1- 3 -----.
4 2
-
78 5.
4. = 13 = 12 .
. 16 8 . :
2 = 2 + 2,
132 = 122 + 2,
169 = 144 + 2,
2 = 25, = 5 .
17 ,
(. 17) : = 2 : 1 = - \ / 3 ,
= = - - \ / 3 = /. , /3 = 5, 3 3 2 6 6
6 -5 30 / 30\/3 -* = + = + = 11/5'
2/ 3 300\/3 10/3 13 ~ ~ ~ = ^---+3 - = 4~ ~2------ = 7 5 ^ + 1 9 5 / 3 = 2 7 0 /3 2.
5. - = 2 \ / , = 10 . .
.
(. 17) = = ^ ^/ = - / , \/3 = 2 \/3 , =
_ 2/ 3 ~ 4 + '
36\/3 6 - 1 0 - , , > - + 3 - = 9\/3 + 90 = 9 ( /3 + 10) 2.
1. . - .
-
5.3. 79
2. 16 , 10 . .
3. = 1 0 \/ = 12 .
5 .3 .3 .
, (. 18).
3 2 = 6 ^ V:,
-
, = 6 = 3. ,
= + = ~ 2/ 3 + 3.
18
-
= ^ 2\ / 3 + 3.
6. = 2 = 3 \ /3 .
24^3
3 3. = - 2\/3 + = 22\/3 + 3 2 3\/3 = 6\/3 + 18\/3 =
2
7. = , = .
. 18 -
, = \/3 = - \ / 3 = . , 8 (. 18): 2 = 2 + 2 = 3 + (3\/3)2 = 9 + 9 3 = 36,
-
80 5.
= /36 = 6
1. . - , .
2. 8 11 . : ) ; ) .
3. = , . .
, , , (. 19).
, , . . , , . , , . , , :
5.4.
19
-
5.4. 81
= \
.
5.4 .1 .
. , = 2
V = - = - 2.3 3
8. = 4\/2 5 = 5 .
. 8 20. - , .
-
82 5.
1 - 233 149 . .
2. 12 , 10 .
3. 9 1 2 , 12,5 . .
5 .4 .2 .
-? \ / 3 . . ,
= - . ,
1
4
1 2\ / 3 = 3 ' 3 4
2\
12
10. = 8 = 10 (. 21).
/. = 4
5.3.2. , 2 = 2 2 = 10282 = 100 64 = 36, \ .3 6-6 36/3 -
= . = 6 = -= = - -^ ,
21
-
5.4. 83
11. = 15 , = 9 . .
. 8 0 (. 22) 2 = 2 - 2 = 152 - 92 = 225 - 81 = 144, = /1 4 4 = 12 .
2 3 . \*^> , = - - = ,
3-12
= 12
\/3" 7 / - /
= 1 2 \/ . ,
1 , = 1 (12 . 9 = 3 2 4 ^ 3 . 3 3 4 3 4
1. 2 , 12 2 . ?
2. (21 + 4\/3) 2 , - 21 2 . : ) ; ) ;) ; ) .
3. = 2 . : ) ; ) .
5 .4 .3 .
= 6 = - 2\/3 . ,
V = - = - - 2\ / 3 = ^ 2\/ .3 3 2 2
-
84 5.
12. = 4 5 = 5 .
. 2 (. 23), = ^ 5 2 - 42 = ^25 - 16 = /9 = 3
. - 1 42' - 3 = 24:3.
13. = 2 \ / , = /2 . .
$
2
. 2>, /
= 3 2\/3 = -------= = 5 . 3 2\ / 3
24 , =
^ = 2/ ^ = .
8 .
2 = 2 - 2 = 52 - 2 = 25 - 9 = 16,
= 4 . ,
V = - = - - 2\/3 = \ { 2 ) 2 4 = 24\/ 3.3 3 2 2
-
5.4. 85
1. 12 .
2. = 2 . . .
3. 4 4 \/ 3 = 4 . : ) ; ) ; ) .
-
5. 86
-
6.
6 .1 . = + 1
, . = ( , 0).
I 1. = = .
.
1~ 2
X 1 0 - 1 2 - 2 4 = 3 3 0 - 3 6 - 6 12
= ~ \ 12 01
___2___ - 1 1 - 2
- , . (. 1) 16)).
-
6.
- .
= + 5,
6 . , .
2. . - 4 / . ?
. 4 , - 4 + 3 . , = 4 + 3 , , .
X 0 1 2 3
= 4 + 3 3 7 11 15
| 3.: 120 . 25 . ?
. . = 120 25. {0,1,2, 3,4}. .
X 0 1 2 3 4
= 120 - 25 120 95 70 45 20
1. 250 100 . , ?
2. 8 0 / . ( )
-
6.2. ; 89
, 85 ?
3. = 2 + 5. - .
X 0 1 2 - 1 - 3
6.2. ;
= . > 0 , < 0 , 1 6.1. .
4. - = 2 + 1 .
. = 2 . (. 2). - = 2 + 1 = 2. - .
X - 2 - 1 0 1 2 3
= 2 - 4 - 2 0 2 4 6
= 2 + 1 - 3 - 1 1 3 5 7
= 2 + 1 = 2 . , = 2 + 1 = 2 (. 2).
-
90 6.
, = + 6 = .
= + 1> .
3
, - . , = 2 + 1 , , = 3, = 2 ( -3 ) + 1 = - 5 I = 1 , = 2-1 + 1 = 3 , (3, 5) (1,3) , . = 2 + 1 (. 3).
... ......1 . . 1 1 5.! - + 2 > ~
= 1 .2
. .
X 0 2
= ~ \ ~ 1 - 1 - 2
X 0 2
= - \ 0 - 1X 0 2
= ~ \ + 2 2 1
4
- - , - - - (. 4). - - . ( - .().
. - .
-
6.2. ; 91
6 . '
) = 3 1 ; ) = | + 2 ; ) = 5.
. ) = 0, . - 1 = 0. = 1, = \ . = 1 | .
_ 2) \ + 2 0 - \ = - 2 , . = , = 8 . ,
= \ + 2 8 .) 5 = 0, = 0, = 5 0. 5 6.
5
(-) .
1. :) = 6 ; ) = 2 - 3 ; ) = - - 1 ] ) = 2 + 5;) = ; ) = \ - 1; ) = - | + 1; ) = \ + 2.
.
2. = \ + 1, = \ - 2 = \ .
3. :) = 2 - 4; ) = - + 3.
, .
-
92 6.
6.3.
. - = + 6, , + + = 0, , 6 6 ^ 0 .
7. 2 + 5 = 0 . .
. 2 + 5 = 0
3 = 2 + 5, .2 5
= - + -
8. 2 + 4 = 0 .
) .)
X 0 2 - 2 - 4
) .
9. X , 20 . - , .
-
6.4. 93
. 2 + 2 = 20, + = 10, = 10 .
1 . 2 2> + 6 = 0. ) - . ) .
2. 2 5 = 0 , .
3. , . 30 .
6.4 .
. , , - .
10. = + 6 4 (1 ,3 ). 6, .
. 4 (1 ,3) .
, . 3 = - + 6.1 5 5
6 = 3 = . -^ 0 = + .
- - 4 (1 ,3 ) , , ( 1,2).
X 1 -1
3 2
- 7.
7
-
94 6.
= + , - . , ( 4 5). = > 0 - , < 0 . = + > 0 - , < 0 .
11. 8 9 > 0 , < 0 .
> 0 ( . 8) = + - . < 0
_ 2 ( . 9) = + 6 . .
--------------------- --- ___%. = 0, = + 6 -_ = 6
-------------------- ------------ -. 10 .
10
{ 3) + 5 : ) ; ) .
. ) - 3 > 0, . > 3. ) 3 < 0 , . < 3 .
12.
-
6.4. 95
6.2 . .
13. = 2 1.) . )
, . ) .
. ) 2 1 = 0, . 2 = 1, | .
) 2 1 > 0 2: > 1, . > | , 2 1 < 0 2 < 1, . < ^ > | , < | .
) - ,
X 0 1/2
- 1 0
11. - ~ . .
> ^ -, , < | -, < \ .
21. = - + 6 (0 ,1). &
. . ?
2. :3
) = 2 - 5 ; ) = - 1; ) = 5; ) + 2 - 2 = 0,
- , .
-
23. = - - 2 .
0 :) = 0 ; ) > 0 ; ) < 0 .
4. :) = (2 + 1) + 5 ;) = ( - | + 3) + - 2 .
5. { 1,3) = + 5.
5 6.
-
7.
7.1.
, , . , .
1. 500 . , 100 . , . .
500
400
300
200
100
0
1
-
98 7.
1 - . = 500 - 100:, {0 ,1 ,2 ,3 ,4 ,5} .
2. = 4, = ? . .
12
1 1,5 2,1 3
= 4 2 4 6 8,4 12
= 214
1 2,25 4,41 9
, (. 2 . 3).
2,25--1 .....
12
4I I--- -11,52,1 3
3
-
3. 4 10 18 .
) ?) ?) 14 17 ?) , ?
-
7.1. 99
. ) 12 13 .' ) 18 .
) 0,5.) 10 12 , 13 16
17 18 .
1 . . - , .
X 10 20 35 57 72
2. 2 - . V .
1 1,5 2 2,4 3,2
V
.
3. V = 3 . -
-
100 7.
1 1,5 2 2,5V
.
7.2 .
- . . . , - . - .
4.; - .
5 3
4 7
3 11
2 4
1 1
, . .
5
-
7.2. 101
5. 90 , 40 , 23 , 11 - , . . (. 6).
6. 7 - . (), (), ( ), (15) .
1. .
12
7
5
1
0
8 .
-
102 7.
. . . . .
8
.
2. 240 . 86, - 55, - 28, . .
----- 1-------- 1--------1-20 40 60 80
9
10 0 120
3. 9 .
) , ?)
?
-
7.3. 103
7.3. ^ .
. , , , .
, . - , . , , , , .
7. | , .
34 35 36 37 38 39 40 41 42
19 183 296 375 411 398 255 56 9
, .
, . . , .
[ 8. :
5, 5, 2, 3, 4, 5.
()
5 + 5 + 2 + 3 + 4 + 5 __ 24 __ 6 6
,
^ 1}
-
104 7.
( )
+ 2 + 3 + +
.
9. 4 7.2 . , . .
. 26 . , - , - , - - ,
. . ' .
10. ! 12 . :
, 16, .
, - () . , , 8 :
, 4.
. : . :
3 -5 + 7- 4 + 11- 3 + 4 - 2 + 1 -1 _ 14 + 28 + 33 + 8 + 1 26 26
2, 3, 4, 5, 5, 5.
- = 4,5. ,
-
7.3. 105
. :2, 3, 3, 5, 6, 7, 7, 7, 9, 10.
()
2 + 3 + 3 + 5 + 6 + 7 + 7 + 7 + 9 + 10 __ 5910 ~ 10 - '
5,9 .
( - 6 7),
6 + 7 2
,5.
1. :) 10, 80, 30, 50, 60; ) 10, 70, 20, 40, 70, 90.
2. :
3 6
5 8
4 10
4 13
, ?
3. 80 - . , 84?
-
:106
-
8.
8.1 .
3.1 - , . , ,
+ 5 = 0,
6 . , 2 + 3 = 0, - 7 = 0, /2 . .----1------ = 0. 4 5 + 6 = 0, . , , .
.
+ + = 0,
, 6 .
1.
12 + 5 = 0,1
- - = 4, 2 6
,
+ 4 2 + 5 _ 7/ + 5
2 + 2 = 9, ( + )2 = 2 + + 2, / + 2 =
, .
-
108 8.
2. 2 + / 8 = 0.
. 1 , 2 ,
2 -1 + 3- 2 8 = 0,2 + 6 - 8 = 0 ,
0 = 0 ., . (,) = (1,2) . , , . , (2,1) ,
2 -2 + 3 -1 8 = 4 + 3 - 8 = - 1 ^ 0 .
(,) (0 , ) .
3. + 7 = 0
(7,0), (5,2), (8 ,-1 ) ,
. = 7 - , (, 7 - ) , , . , .
/ 22\\ ' )
1
-
18.2. 109
. 6.3, ++ = 0 . , = 7 - 1. , , = 7 .
1. , , 6 . .
2. = 2. (3 ,-1 ) , (6,0), ( -2 ,0 ) , ( -1 ,3 ) , (0,6), (0 ,-2 ) , (5,5) ?
3. 2 = 1. .
8.2 . .
,
+ = 10,2 = 1 .
. - . (1,3) . ,
+ 2 = 7 + 2 = 11
(, ) + 2 7 11.
,
. \ + 0\ + 1 = 1), 2 + 62 + 2 = 0,
-
110 8.
. (,) . , , , .
(.,), 0 0 , .
.
, , :
. - 2 , , 3.
, (. ), . 3.2 3.3. , ( 1 3.2), ( 2 3.2), , ( 1 3.3).
- , .
1. (1, 1) :
2 = 5, + = 11
6 4 = 10, 12; + = 33
2 + 3 = 1 + = 0,15
) 2 1 ^ - ^
-
8.3. 111
2. /3 (1 , 2)
2 + + 3 = ,5 = (3.
3.
( - 3)( + 3) - ( + I )2 = 2 - 2,
(2 + 1)(2 - 1) + 5 42 = ( + I )2 ( I )2 - 4.
.
8.3. .
, ,
+ &1 = 1, + 62 = 2,
1, 61, !,2, 62, 2 .
.
8 .3 .1 .
4.
2 + = 16, 2 = 1 .
2 , = 1 + 2 .
2 + 3 16, = 1 + 2 .
-
112 8.
1 + 2
2(1 + 2) + = 16, = 2 + 1 .
- .
2 + + = 16, = 2 + 1 ,
,
7 = 14, = 2, = 2 + 1 = 2 + 1 .
= 2, = 2 - 2 + 1 = 5,
(5, 2).
. . . , - , .
5.
2 + = 7, 4 = 5.
. = 7 2. 7 2 - .
= 7 - 2 , - 4(7 - 2) = 5.
: = 7 - 2, _ = 7 - 2,
28 + 8 = 5, ' 11 = 5 + 28,
= 7 - 2 , 11 = 33,
,
= 7 - 2 , = 3,
-
8.3. 113
, = 7 2 - 3 = 1 , 3.
(3,1).
6.
+ 2 3
= 4,
~2 -
434 '
. 3, 4:
+ 2 = 12,
4 ~ (2 - ) = 3.
2
~ + = - 6, + 2 = 3.
- ( 3 - 2 ) + = - 6, = 3 2,
= 3 - 2 :
- 3 + 2/ + / = -6 ,
= 3 - 2 ,
- = -3 , = 3 - 2 ,
= 1, = 3 2 (1) = 5. (5, 1).
8 .3 .2 .
( ) , , .
13 = 654 6 = 8
, ,
4 6 = 8, 9 + 6 = 57
-
114 8.
. - , . .
.
4 6 = 8,9 + 6 = 57
(6 6 ), , . .
7. | -
+ = 5, + 3 = 11 .
. 1 1. 4 = 16, , , :
4 = 16, + = 5.
= 4, + = 5,
= 4,
+ 4 = 5, = 4, = 1.
-
(1,4).
, , .
8 . -
5 4 = 19, + = 2.
-
1158.3.
. 4,
5 4 = 19,4 + 4 = 8.
(4 4). , ,
5 - 4 = 19, 5 4 = 19,9 = 27, ' -=3,
5 3 - 4 = 19, - 4 = 19 - 15, = 3] . = 1 , = 3 .
(3, 1).
I 9.]
14 - 3 = 25, + 2 = 8.
. . () , . 2, 3 . , , ; .
( 28 - 6 = 50,\ 9 + 6 = 24
37 = 74,\ 9 + 6 = 24
( = 2,\ 9-2 + 6 = 24
( = 2 ,
\ = 1 -
(2,1).
-
116 8.
10.
2 + 5 = 7, + 4 = 7.
. . - 3 , 2.
- 21,14
-7 ,14
1 ,14
8 .3 .3 .
11.
+ 3 = 11 ,5 = 7.
6, , , - . . (. 2).
- . , ( ),
X 1 3
= 5 7 - 2 8
X 0 511
= 311~3~
2
.- 15// = \ 6 + 8 =
( - 7 =\ 6 + 8 =
I = \ 6 + 8 =
(1,1).
-
1178.3.
, . . (2,3). , , :
2 + 3 - 3 = 11,5 2 - 3 = 7.
, , . .
- . , 11 , . 12 13 , , .
12. + 2 = 5,
2 + 4 = 10.
. (. 3).
X 0 5
5/2 0
-
118 8.
5 . = -
(, - ~ ) * 11 '
13.|
- 2 = 3, + 6 = 6.
. 4. .
. - 3 - 2 = - 2 .
-
8.4. 119
, :1) , - -
.2) ,
.3) , - .
1 . : + 2 7,
5 = 10; 3 3
)4 + = 22,
5 = 6;
- 1 =2 2 - 2 + + 1 _ 2
1 2
3 2
2. : 5 4 = 1,
' 5 + 7 = 2;
( + 5)( - 3) = + 2, ( - 1)( + 3) = + 2.
2 = 1.) ,
5* + = - 1 ;
3. : + 5 = 14, . + 2 = 9,
) - = 6 ; ^ X + \ = 3;
- 2 = 1 , 4 8 = 4.
8.4 .
3.7 , . :
1) ( ) .
-
120 8.
2) .
3) - .
4) .
14. 132 , . .?
. 1) , .
2) + 5 .3) :
+ = 132 = 5 .
4)
+ = 132, = 5 .
. . ,
( + / = 132,1 = 5
5 + = 132,I =
6 = 132,\ = 5
22,\ = 5 22 = 110.
110, 22 .
15. 10 , , 20 , . ?
-
8.4. 121
. , .
- 1 0 = + 10 ,\ + 20 = ( - 20) 3
= 20,\ - = -8 0
, - 1 , .
( - = 20,\ + = 80
( = 20,\ 2 = 100
( - = 20,
\ = 50
= 70, = 50.
, 70, 50 .
16. , 2 0 / , , 12 /. ?
. V , V I . V + = 20, V - = 12. :
( V + V I = 20, \ V V I = 12
( V + V I = 20, \ 2 = 32
V = 16, V I = 4.
1 6 / .
-
122 8.
1. 47, 23. ?
2. 50. , 2 5. ?
3. - , , .- , . ?
4. 26 , 6 , 2 . .
-
9.
9.1.
(. 1). . , , , , , .
1 2
\ 2 - 2 , 0 \ 0 2 2 (. 2). \ 2 , 2 .
\ &2 , 12 , , . , , , (. 3).
\ /2 (, ). 12 .
-
124 9.
\ 2 (. 3) , \ 2 . 2 ,- . ( ). . .
(. 4) . , , . , .
3
4 5
. . , . - (. 5).
-
9.2. . 125
3. 25 2. .
9.2. .
, ( 4.2 5.2). - , .
7
. (. 7), - . ' ' ' ( ' = 2), . 2 . , .
, . , ,
= 2 + .
= 2 = 2 ,
= 2 2 + 2
1. = 2 , = 5 .
-
126 9.
.
= 2 27 + 2
= 2 227 + 2 2 57 = 87 + 207
= 28712.
2. = 4 6 = 2 . .
. = = 4 = 6 = 2 (. 8). ,
8
= 2 27 + 2
= 2 42 + 2 4 2
= 327 + 17
= 487 2.
1. = 1 = .
2. :) = 5 , = 1 2 ; ) = 2, = 5 .
3. 9 2 . .
4. = , = 6072 . .
9 .3 .
.
-
9.3. 127
, .
, . = 2,
. - - , , (. 9). , . .
3. 1 6 22 ? ( = 3,14.)
. 2 = 16 , = 8 ,
V = 2 = 82 7 22 = 14083 4421,12 3 4,42^.
4.1 10. = 7 ,8 / 3. ( = 3,14.)
.
V = ,
V = 2
9
10
= = 7,8 / 3 320 3 7 837,44 7,8 ^.
-
128 9.
1. 367 3, . .
2. 40 50,241? . ? ( 7 = 3,14.)
3. - 6 4 .
4. 1 = - = 2,5 . - . ( 2 ,7 / 3 ; 7 = 3,14.)
-
10.
10.1.
1 8 0 (. 1). 8 , X .
, 8 . 8 0 . , , , . . 8 0 .
7 1 2 3
(. 2). 8 . 5 . . , , .
1. = 12 , = 9 (. 3). .
-
130 10.
. 8 0 3 5 , ,
52 = 122 + 92 = 144 + 81 = 225,
5 = \/225 = 15.
, , , (. 3). ,
, ,
52 = 2 + 2.
- (. 4). , - . , - .
- . , - . (25 (. 5).
-
10.2. . 131
10.2. .
(. 6).
2 ~ 360
~ 180 '
6
= I ~2~'
5 .
7
5 , (. 7). 5 , , . 2. . , .
, , , ,
= + .
5 27
, , 5 27 =
2= 57.
-
132 10.
, = 2,
= 2 7 + 57
2. = 4 , 5 . .
. 2 = 2 2 = 2 5 16 = 9, = /9 = 3 . ,
= 2 7 + 8 = 327 + 5 37 = 97 + 157 = 247 2.
3. 12 120 . .
. (. 8). ,
. 87
12 120 180
27 ,
= 27,
8 = = 4 .
2
,
= 2 + 8 = 427 + 12 4 7 = 167 + 487 = 64-7 2.
1. :) = 12 , 5 = 1 3 ; ) = 8 , = ;) = 15 , 5 = 25 .
2. 127, 8 . :) ; ) .
3. 9072, 2572 . .
4. 9 - 120.
-
10.3. 133
10.3.
- . , - . , .
- \
, . = 27,
V = ^ 2#3
.
- , 7. , (. 9).
4. , 10 . ? ( = 3,14.)
. V = - 2 = 257 30 = 3 3
2507 3 785 3 . 10
-
134 10.
,
5. 207, . .
. 2 = 20 = 10 . = \3 = 10/3,
V = - * 1 2 = - 100 10/3 = 1 0 0 0 ^ 3 4.3 3 3
1. 5 , 13 . .
2. 5 9 . .
3. . 2,5 , 4 , , 2,2 . /3 .
4. - 9 2 . .
-
11.
11.1.
, (. 1).
. . .
. .
, , - ( ). .
, . .
-
136 11.
, , - ( ). 2, .
- (. 3) . . , , , , , .
11.2. ( )
. , (1 .
(I < (. 4) ' . 0 0 ' = . .
, (. 5), . . , , , , .
-
11.3. 137
. .
, , (. 7).
1. 7 . ?
2. = 5 . ,
-
138 11.
. 1. = 642 .
.
. 64 = 42, 2 = = 16, = / =4 _ 2567
4 . = - 4 7 - .
2 127. .
. 2 = 12
= = . 2
, :
= 42 = 4 62 = 4 = 1442,
V = - 37 = - 637 = 2883.3 3
7
3. = 19 / 3 4 . = 3,14.)
? (
.
V 3 ' = 0,033 3 19 / 3 0,63 .
0,033 3 ,
1. 8 . .
2. 6373 . ?
3. 2 . 2 , : ) , ) ?
4. . - ? ( = 10,5 / 3 .)
-
12.
1.
1.1
1. ) = 4 ; ) = 1 0 ;) = 4 .
2. ) 7,5 ; ) 8 ; ) 9 .3. = 20, = 6 .4. . 18 - ; . 19 - .
1.2
1. = 12 , 0 \ = 48 .2. .3. .4. = 5, > = 6 ,4 .
1.3
1 . = 8 ; = 3,6 ; =6,4 ; /1 = 4 ,8 .
2. ) = ~ ; ) = 12 , 6 =
5 ; ) 0 = 30 , = 30 2 .
3 . = \ / .
2. ,
2.1
1 . \ .2 . 3 .3. ) 10; ) 4.4. .5. (II) (III).
2.2
1. - . - .
2. ) 8 , , , 8 ; ) 5>, 5 .
2.3
1 . ) ; ) .2. ) ; ), ) .3. ) , , \ \ , ;
) >, 1) \ , \ \ , , , , \ .
-
140
4. . 3.3
2.4
1 . ), ) ; ), ) .
2.5
1 . ) || ; ) || .
2. = 13 .3. ) 2 ; ) 2^ - , ) 2\ / . 1 . ) ; ), ) .
1 . ) 14; ) 1 1 ; ) 3; ) - 1 ; ) 0 ; ) - 1 .
2. ) 0; ) 7.
3. ) ; ) - ; ) 5, 13 '
3 .4
2.6
2. 4.3. , -
.
3.
2. ),),),) ; ) - .
33. - 2 , - - .
3.5
1 . ) , ) .
), ) ).
3.1 3.6
1 . ) ; ) .2. ) ; ) .3. 0.
1 . ) < 1 ; ) >
) < .
3.22. ) > ^ ; ) < 3.
1 . ) ; ) ; ) .3. ) ; ) < ^
612. 6. 4. ) < 35; ) <
3. ) ; ) ; ) . ) > 9.
-
141
3 .7 4 .4
1 . 11 .2. 125,
625 .3. 8 .4. 2700 .
4.
4.1
1 . 12 . , , >, , , 1!, 1 1 , >14.1 , ^ , \ , 1 >1 . ^^^!- - 1 , >1 , 1 .
2. ) 10, 5, 10; ) 12, 6, 12.3. 16 2 .4. 13 .
1. ) 1728 3 ; ) 16 /23 .2. 150 3 .3.
14 2 130 4 2 5\/2 2
13 4 5
V 182 3 520 3 2 5 \ /2 3
4 .4 .1 4 .4 .2
1. 135 3 .2. ) 5 / 3 ; ) 3 0 \/3 3 .3. 144 3 .4. 8 3 .
4 .4 .3
1. 2800 .2. 4800 .3. 138 .
4.2 5.
1 . .5.1
4.3
1. = 112 2 .2. = 288 2.3. = (60 + 8\/3) 2 .4. 9(16 + 3\/3) 2.5. = 4 4 4 2 .
1 . 12 , : , , , , , : 8 } 8 , , 5 ,8 , 8 . , 8 , 8 , , , 57+4.
2. / = 3 .
-
142
3. /2 .
5.2
3. .
2. ) 4 ; ) - ;
) ~ / 3 .
3. ) )
5.3 .1 5 .4 .3
1. ) 260 2 ; ) 84 2 ; ) 96 2 .
2. = /.3. (4 + 4/3) 2 .
5.3 .2
1. 9 \ / 2.2. 16(4\/3 + 9) 2 .3. 2 7 0 \/ 2 .
5 .3 .3
1 . 21\ / .2. ) 13 ; ) 24(13 + 4\/3) 2 .
3. ^ ( 1 + ^ ) 2.
5.4 .1
1. 2 696 354 3 .2. 384 3 .3. 360 3.
1 . - /4 3 . 3
1. 4 3 2 \/3 3 .2. 3 .
^ 133. ) ; ) ;
) 6(4\/3 + 13) 2.
6.
6.1
1. = 250 + 100.2. = 85 + 80.3.
0 1 2 - 1 - 3
5 3 1 7 11
6.2
1 . , ; ; .3. : ) 2; 6 )3 .
6.3
21 . ) = - + 2 ; ) :0 = - 3
2. = 2 5.3. + 2 = 30.
5.4 .2
-
143
6.4
1 . 6 = 1 ; .3. ) 3; ) > 3; ) < 3.
4. ) > ^ ; ) > 6 .
5. = + 2 .
7.
7.1
1. = 90 .2 .
1 1,5 2 2,4 3,2V 6 9 12 14,4 19,8
1 1,5 2 2,5
V 1 3,375 8 15,625
7.2
3. ) - , . ) .
7.3
8.
8.1
1 . + = 6 .2. (3 ,-1 ) , (6,0), (0 ,-2 ) -
.
8.2
1. ) ; ) .2. = 3, = - 11.3. ( | , 5).
8.3
1. ) ( -1 ,5 ) ; ) (4,2); ) (1,1).
2. ) (1 , 1); ) (0, - 1); )
8.4
1. 35 12.2. 35 15.3. .4. 8,25 4,75 .
9.
9.1
1. ) 46 50; ) 50 55.2. 9,375 9.3 . 1 0 0 .
1. ) ; ) .2. .3. = 5 , = 2 ,5 .
-
144
9.2 2. 757 3 .3. 1 ,6 1 .
2. ) 17072 ; ) 100072 . 4. 97 3 .3. 13,57 2 .4. = 7 . 11.
11.29.3
1 . V = 547 3 .1 . .
2. 40 .2. = 4 .
3. = 807 2 , V = 967 3 . 3. .4. = 2331,45 .
10. 11.3
9 , 256 31 . = 647 " , V = 7 3 .
1 10.1 2. 510100 000 2 .224
3. ) 487 2 ; ) - 7 31 . .2. . 4. V" 113 040 3 ,
3. = 12 . ~ 11187.
4. = , 5 = , = 3 \ /3 .
10.2
1. ) 30072 ; ) 14472;) 9007 2 .
2. ) 5 = 10 ; ) = 97 2.3. = 12 .4. = 9 2 , = 6\/2
10.3
1. 10073 .
-
72 72 56 123 136 123 129 72 56 129 71 12 30 56 36 45 () 34 110 92 ( ) 72 30 ( ) 56 124 129 92 34 1 101 129 135 35 44 88 33
104 112 114 20 125 131 74 58 34 90 103 136 124 129 56 72 27 123 129 56 123 129 71 124 26 71 30 123 23 22 () 57 73 57 73
-
146
136 56 25 () 94 35 110 110 1 104 9 100
-
30 100 135 2 30 56 103 () 104 135 3
|
1. 2. , 3. E ...4. 5. 6. 7. ...8. ...9. 10. 11. 12.