Μιχαήλογλου Στέλιος, Τόλης Ευάγγελος - Τριγωνομετρία...
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math theory in trigonometry for b class in high school with solved probelms
Transcript of Μιχαήλογλου Στέλιος, Τόλης Ευάγγελος - Τριγωνομετρία...
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www.askisopolis.gr
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11
1.
1.1. ,
. - , .
(0 < < 90)
: AB
=
=
=
=
, 90 D Oxy.
x (- ) z , .
2 2(OM ) (x ) (y ) 0 = = + > y x y x , , , x y = = = = .
Ox - Oz (- ), . (x,y)
Oz 2 2(OM) x y 0= = + > , :
y x y x , , (x 0) , (y 0) x y
= = = = 360. Ox -
, 360 +D . , 360 +D . -
-
12
z , .
] :
( 360 ) + =D ( 360 ) + =D ( 360 ) + =D ( 360 ) + =D
, Oxy, .
1 (0,0) .
() 1= = y x y , x1 1
= = = = . y y -
x x -.
: 1 1 , 1 1
1 . -
y x
= = ,
x ZHy
= = . 1 2 -. - , , 1 2.
,
.
-
13
Oxy, -
, - .
-
(rad) , (O, R), R.
. 0 rad 180= . 3600 2 rad. S rad S = .
0 rad, 0 0 360 + 2 , + ] . :
(2 ) + = (2 ) + = (2 ) + = (2 ) + =
.
rad
0 0 0 1 0
30 6
12
32
3
3 3
45 4
22
2
2 1 1
60 3
32
12
3 33
90 2
1 0 0
180 0 1 0
270 32
1 0 0
360 2 0 1 0
-
14
______________________________________________________________
rad , :
180=
. rad - .
1. :
I. 7 rad6
II. 75 rad.
I. 180= 7
6= :
7 7 6
180=
6 o 6 7 180
110= = o 210=
II. o 180= = 75 :
75 5 12180= = 5
12=
______________________________________________________________ 360
3600, :360. . 360 = +D D D , Z 0 360
-
15
rad
rad
, :
. :
+ = = + = +
f : -
.
f : -
+ .
2. :
I. 2790 II. 253
III. 1776
I. 2790:360 7 270.
2790 7 360 270= +D D D . : 2790 270 1= = D D 0 02790 270= =
2790 270 0= =D D 2790 270 0= =D D II. 25:3 25 8 3 1= + .
25 8 3 1 1 8 83 3 3 3
+ = = + = +
8 2 :
25 3 3 3 2
= = 25 1 3 3 2
= = 25 3
3 3= = 25 3
3 3 3= =
-
16
III. 177:6 177 29 6 3= + . 177 3296 6
= + . 29 , 177
6 3 9 3
6 6 2+ = = .
: 177 3 3 16 6 2
= + = = 177 3 0
6 2= =
177 3 ,6 2
= 177 3 06 2
= = .
______________________________________________________________
. - . .. 2 - (, , ) .
3. 9 14x2 3< < , x x x x 0 > .
9 14 24 , 42 2 3 3= + = + .
x 2
23
, 2 ,
:
( )
x 0x 0 x 0
x x x x 0x 0 x 0x 0 x 0
+> < > >< > < >
.
2
32
x
.
.
02
12
3 4
x
y
y
++
+ +
-
17
4. x 2< < , : 2x x 2xx x x 0 + + < .
: 2x x 2xx x x 0 + + < ( )2x x 2x 1 x 0 + + < 2x(x 1) x 0 + <
x 2 2
2x 0 x(x 1) 0 x(x 1) x 0x 0
< < +
-
18
6. x : I. 2 x 3 5x 3+ < II. 2x 4 9 , = + + \ .
I. x\ : 2 2 x 3 5x 3 x 5x 6 0+ < + <
x = 2 5 6 0 + < (1) 2 ( 5) 4 1 6 25 24 1= = =
1,2
2 5 12 2 3
= = ==
(1) 2
-
19
7. rad : I. 210 II. 1845 III. 150 IV. 450.
(A.: I. 76
, II. 414
, III. 56
, IV. 52
)
8. :
I. 7 rad12
II. 11 rad6
III. rad8
IV. 40rad
(.: I. 105, II. 330, III. 22,5, IV. 7200
D)
9. 900 rad4
. rad .
(.: 3 x , y8 8
= = )
10. :
I. 1350 II. 27 rad III. 913
IV. 20064
(.: I. 1350 270 1,...= = D D , II. 27 0,...= = III. 91 3 ,...
3 3 2= = , IV. 2006 3 1,...
4 2= = )
11. ( )M 1, 3 , 2 , + ] .
(.: 3(2 )2
+ = , 1(2 ) ,...2
+ = , 3 )
12. 5 x 32< < , : x x x x > + .
13. 3 2
< < , : 1 3 4 0+ + + > . (: 1
-
20
15. : 2A 1 x= , 2B x 2 3= , 4 2 x 3= .
(. 0 A 1 , 6 B 2 , 3 5 )
16. 7 25
= + 13 25
= , ,] , - .
(: / 2)
17. x : 2x 4 4 3 , = + \ .
18. :
I. 360 150 , 360 210= = +D D D D II. 7 9 2 , 2 , ,4 4
= = + ] .
19. 24
= + 2 , , = + ] . , - .
(.: 1 2( ) 2 2= + )
-
21
:
1. 2 2 1+ = ( )
2. , 0 , 0.
= =
3. 1 = 4. :
22
2
1 = +
22
1 1
= +
_____________________________________________________________
1 : , :
() 2 2 1+ = ()
= , 1
= , .
2 : , :
() 2 21
1 = +
()
= = . , !!!
-
22
20. 3x5
= 3 x 22< < , x rad.
: 2
2 2 2 23 9 x x 1 x 1 x 15 25
+ = + = + =
2 29 16 4 x 1 x x25 25 5
= = = .
3 x 22< < , 4
x
-
23
22. x = 4x 3 x 22< < , x rad.
: 2
3x 2 x 2x 4x 24x x 41x xx x= 2
= <
-
24
24. 1213
= 2< < ,
rad.
(.: 513
= , 125
= , 512
= )
25. x = 2 3 x2
< < , x xKx x
+= + .
(.: 6 5K25
= )
26. 9 52< < 216 9= ,
1
+= + .
(.: 53
= )
27. 2 23 x x 0 = x 2< < ,
x rad.
(.: 1x2
= , 3x2
= , x 3= , 3x3
= )
28. 6x 8x 10 = x ,2
, x.
(.: 4x3
= )
-
25
_____________________________________________________________ : 1 : , -
. 2 : (-
). 3 : -
. 4 : -
.
, - .
. 1. :
2 2 x 1 x= , 2 2 x 1 x= 2. x, x, x x, :
x xx , xx x
= =
3. x x 1 = : 1xx
= 1xx
= 4. 1 1 ,
1 1 - ( )( + ) = 2 2 .
5. :
2 22 2
1 11 x , 1 x x x
= + = +
6. : 2 2 2 ( ) 2+ = + 3 3 3 ( ) 3( )+ = + +
, 2 2 x x 1+ = . 1 2 2 x x+ , :
2 2 21 2xx x x 2xx (x x) = + = .
-
26
29. \ :
. 3 4= +
2 4+= +
. 21
=
2= + .
. 4 : 2 2 1+ = 2 2 2 2
2 2 3 2 ( 3) ( 2)1 1 4 4 ( 4) ( 4) + + + = + = + + + +
2 2 2 2 2 2( 3) ( 2) ( 4) 6 9 4 4 8 16 + + = + + + + + = + + 2 10 3 0 = ,
:
10 112 10 16 7 10 4 72 2 2
= = = 5 2 7 5 2 7= + =
. , 1
1 0
2 0 2
+
1 = , : 2
22 21 1
1 2 2 2 = = + +
2 2 22 2 2 3 1 0= + + = , = 25 :
= 11 5 6
2 =3
=
-
27
30. :
. 2
2 2 1(x x) (x 1) 1x
+ =
. 3
2 x x xx 1 x
+ =+ .
. xxx
= , : 2 2 2 2 2(x x) (x 1) x 2xx x x 2x 1 + = + + + =
22 2
2x x x 2x x 2x 1x x
= + + + = ( 2x+2x=1)
2 2
2 2 x x2 2 2x x x
= + =
( x 2 2 x 1 x)= 2 2
21 x 1 x2 2 2xx x = + =
2 21 1 2 12 2 x 1 2x 1 2x 2x
x x x x + = + + =
2
22 1 11 1
x x x = + = .
. 2 21 x
1 x= + .
: 3
2 2 22
x 1 x xx x x x x xx x x1 x
+ = + = + = +
2 22 2 2x x x x 1 x x x
x x x x x x += + = = =
x xx
= = .
-
28
31. 0 < < 1 1 K , 1 1 + = = + .
: 2 = .
, : 2 2 2
2 21 (1 ) (1 ) (1 )1 (1 )(1 ) 1 + + + += = = = +
(/ ) 2 2
21 (1 ) (1 )1 (1 )(1 ) + = = =+ + .
:
2 2
2 2(1 ) (1 )
+ = =
1 1 + = 1 01 1
1 0+
0
-
29
x, 3 2 0+ = 32
= .
32
= 3 12
= + 1K2
=
33. x = , : 2 2
2( 1)( 2)2 x x 3xx .
1 + = +
x, , 2x :
22 2 2
2 2 x 3x2 x x 3xx x 2 x x
+ = + =
2 2 221 x(2 x 3x) ( x 3x 2)
1 x= + = + =+
22 21 ( 1)( 2)( 3 2)
1 1 = + =+ + .
-
30
34. :
. 21 1 xx
x 1 x = + .
22
1 3x 2x 1 3 x1 x x + =
.2x x x x
1 x x x = + V.
4 2 2 4
22 x x x x 1
3 x 1+ = .
35. = x B x,= : 2 2 2 2() () + = .
36. \ , 1x2 +=
3 2x 4=
(.: = 0 = -1)
37. 4 4 6 6 2 2f (x) x x x x x x= + .
(.: f(x) = 0)
38. : 2 2 x y 2x y 2+ + x, y\ .
39. :
I. 12
II. 2+ 0 2
< < .
40. : I. 2 21 2 ( ) 2( ) + + + II.
2 2
2 21 1 2 2 + +++ +
III. 2 22 2 2 2
1 1 1 (1 ) 1 (1 )
+ + +
(.: I. 2, II. 1, III. 1)
-
31
41. 1,2 2 2(1 )x (1 )x (1 ) 0+ + + = , 1
: 1+2+12 = 1.
42. :
I. 22 21 1 4 (x x)
x x =
II. 4 4 2 2 2( x x)(x x) x x+ + = + III
4 2 2 2
4 2 2 2 x x x x 1 x x x x
+ + =+ +
43. x x + = , : I. xx III. 2 2 x x+ II. x x+ IV. 3 3 x x+
(.: I. 1
, II. 2+ , III. 2 2, IV. 3 3)
44. 3 2
< < , : 21 2 1
+ = + .
45. = , 2 2 2 2 2 3 ( ) = .
46. x = , : 2 2 3 x 2 x 5xx= + .
(.: 2(3 2)( 1)
1 = + )
47. : 5 2x x 5 + .
-
32
1
0 90.
:
2
+ 2
2
32
32
2
2
+ 32
32
+
-
33
_____________________________________________________________
++ = 180, + = 180 , :
( ) (180 ) + = = ( ) (180 ) + = =
, , , .
, : A B 90 902 2 2 2 2 2+ + = + =
: 90 2 2 2+ = =
90 2 2 2+ = = .
, . , -.
48. :
. ( ) 0+ + = . 2 2 12 2
++ = .
. 180 180 + + = + = ( ) (180 ) + = = (1)
(1) ( ) 0+ + = = .
. A B 90 902 2 2 2 2
++ + = = 90
2 2 2+ = = (2)
(2)
2 2 2 2 12 2 2 2
++ = + = .
-
34
1O
rad
, :.
,
+ . ,
-
, -
+ . :
2 , 3 , + 4 , 2
- ( ) , :
, , .
.
2
, :2 .
= 2 (), 2
, 2 2
2 2
= + = +
-
35
= 2+1 (), 3 2 ,
2
= 2 2
= + = (2 1) 2
+ + =
32 2 2 2
= + + = + .
2
3 2
- :
.
, 0 2
< < . .
() .
, , ,
2
.
49. :
. 3030 . 31 rad4
.
. 3030 : 360 3030 8 360 150= +D D D . , () - . - . : ( 3030 ) 3030 (8 360 150 ) = = + =D D D D
-
36
1150 (180 30 ) 302
= = = = D D D D
( 3030 ) 3030 150 = = =D D D 3(180 30 ) 30
2= = = D D D
( 3030 ) 3030 150 = = =D D D 3(180 30 ) ( 30 ) 30
3= = = =D D D ,
( 3030 ) 3030 150 (180 30 ) 30 3 = = = = =D D D D D D .
. 31:4 31 4 7 3= + , 31 374 4
= + . 7
, 314
3 74 4
+ = . :
31 7 2 2 4 4 4 4 2
= = = =
31 7 2 2 4 4 4 4 2
= = = =
31 7 2 14 4 4 4
= = = = ,
31 7 2 14 4 4 4
= = = = .
-
37
50. :
7 7( 5 ) ( 8 )2 2 1
5 ( 3 )2
+ + + = = .
. :
[ ] 5( 5 ) (5 ) (5 ) + = = = ( ) ( ) = = 7 7 3 2 2 2
= + = + + =
3 ( ) 2
= + = = ( 3 .)
7 3 3 2 2 2
+ = + + = + =
8( 8 ) (8 ) = + =
5 5 2 2 2 2 2
= = + = = .
3( 3 ) (3 ) ( ) = + = + =
: 2 2
2 2 ( ) 1 1
( )( ) + = = = = .
-
38
51. 0 2
< < , :
3 5 ( ) 2 1 2 2
+ + > .
:
3 2
+ = ( ) + = 5 2
2 2 = + =
2
= = :
2(1 ) > 2(1 )
> .
0 2
< < , >0, : 2 2 2 2(1 ) 2 2 > > 2 2 2 2 0+ >
2 2 2 2 0 + + > 2 21 2 0 ( 1) 0 + > > .
52. ,\ , 2 223 x 4 x 4( x) (9 x)
2 = + + x.
.
: 23 x 11 x2 2
= +
11 3 x x
2 = =
( x) x =
9(9 x) ( x) x = = .
-
39
: 2 2
2 2 2 2
x 4x 4x ( x)
x 4x 4x x ( 4 4 ) x
= + == + + = + +
x : 2 2 2 2 4 4 0 ( 2) 0 + + = + =
2
2
( 2) 0
0
2 0 2 0 0
= = ==
53. 1 4 4 3
+ + = , 4 4
= + .
4 4 2
+ + = , , 4 4+ .
:
1 1 4 4 3 4 2 4 3
+ + = + + =
1 4 4 3
+ + + =
:
2 2 1 2 4 4 4 4 9
+ + + + + + =
1 81 2 2 4 4 9 4 4 9
+ + + = + + =
4 4 4 4 9 2 4 4 9
+ + = + + =
4 4 4 4 9 9
+ = = .
-
40
54. : . A 0 20 40 60 80 100 120 140= + + + + + + + +D D D D D D D D 160 180+ +D D . 41 42 ...48 49= D D D D .
. :
(0 180 ) (20 160 ) (40 140 ) (60 120 )= + + + + + + + +D D D D D D D D
(80 100 )+ + =D D
[ ]1 ( 1) 20 (180 20 ) 40 (180 40 ) = + + + + + + D D D D D D ( )60 (180 60 ) (80 100 ) + + + + = D D D D D
0 (20 20 ) (40 40 ) (60 60 ) (80 80 ) 0= + + + + =D D D D D D D D .
. , :
41 42 ...45 ...48 49= =D D D D D
(41 49 ) (42 48 )...(44 46 ) 45= =D D D D D D D
41 (90 41 ) 42 (90 42 ) ... 44 (90 44 1 = = D D D D D D D D D
(41 41 )(42 42 )...(44 44 )= =D D D D D D 1 1...1 1 = .
-
41
55. , , , :
. ( ) 0+ + = . 12 2+ =
. 2 2 12 2+ + = . V. (2 ) 0+ + + = .
56. : I. 210 II. 1050 III. 315 IV. 3750.
(.: I. 0 0 01 1210 ,..., ( 1050 ) 330 ...,2 2
.I I= = =
III. 0 0 02 1315 ..., 3750 150 ...)2 2
IV.= = =
57. :
I. 7 rad6
II. 3 rad4
III. 2007 rad2
IV. 37 rad6
(.: I. 7 1 3 2 ..., ...,6 6 2 4 4 2
II. = + = = =
III. 2007 3 37 1 1002 1..., 6 ...2 2 6 6 2
IV. = + = = + = ).
58. :
(3 )(5 )(2006 )(2007 )A( 61 )(37 )(11 )(13 )
+ = + + +
5 7 9 11 2 2 2 2
21 19 17 15 2 2 2 2
+ + + + =
(.: = , = 1)
59. : ( )13 27( 9) 23
2 2 9( 11) 2
= +
-
42
60. 3
,
. (: = 6 + , = 0,1,...,5)
(.: 0, 3 3,2 2
)
61. 4
, ] . (.: 0, 1, 1, )
62. 3 55 10
+ + = , : 2 2 3
5 10 + + .
(.: 23)
63. 1 4 4 2
+ + = . :
I. 2 21 4 4
= + + II. 2 4 4
= +
(.: I. 1 = 1, II. 238
= ).
64. ,\ , 3 9 (11 x) x 2(2007 x) x2 2
= + + + + + + + .
(.: = 1, 12
= ).
65. : I. 0 1 2 ... 179 180 0+ + + + + =D D D D D II. 0 1 2 ... 359 0+ + + + =D D D D .
66. 4 x 5< < , : 7 x x 2(9 x) 22 2
+ > + .
-
43
67. 5 x x 212 12
+ + = , 2 25 x x
12 12 = + +
3 35 B x x12 12
= + + (.: = 2, = 2)
68. : f (x) x x 1= + 2 2(x) 2 x 2xx x 1= + . :
I. ( x) (x)+ = II. 3f ( x) f x2
+ = .
69. :
I. 3 3 ( ) 02 2 + + = II.
2
2
2
2
2 1 2
+ = + +
70. 2x 2 x 0 = 0 x2
< < , :
2007 33 x 2 x2 2
3(27 x) ( 3 x)
+ = + + .
(.: 3A5
= ).
71. 23 23 1 x x2 2 2
+ + = , 2 2 x x= + .
(.: 469
= ).
-
44
. () ():
1. 3 2
+ + = .
2. 17 5 6 6
= .
3. x 2< < , x x= .
4. 2 2(2 3x) (2x 1) 0 + + = .
5. 2 2(3 4x) (4 1) 0+ + = .
6. 2 2 5 112 12
+ = .
7. x x= .
8. ,2
+ = 2 = 2.
9. ( ) ( )+ = + .
10. , = 2, = 2.
11. 2< < , 2 > .
12. (1 2)(1 20) 0 + , \ .
-
45
. :
1. 32
= :
. 32
. 4 . 0 . 2 .
2. x 2< < ,
22 3 x x
x :
. 2x B. 0 . 4x E. 4x.
3. 10 30. :
. rad3
. 300 . 3 . 3 rad . 3 rad
4. x x 2+ = , x : . 0 . 90 . 180 . 270 . .
5. 2 2 3 x x2
+ : . 2 . 0 . 22x . 1. . 1 2x
6. = 5x, = 5x, 2 2x y+ = . 5 . 5 . 25 . 1 . 1
25
7. 20076
:
. 12
. 1 . 0 . 32
. 22
8. 2x x
1 x
+ (x 1) : . x2x . x+2x . x+2x . x2x E. x2x
9. 1 rad : . 60 . 360 57, 324 . 90 . 1
-
46
. - . .
. 616
. 288
. 276
. 484
. 373
1. 32
2. 3
3. 6
4. 2
5. 0
. .
. 27 2
+
. (2007 )
. 72 2
+
. 11 2
1.
2.
3.
4.
5.
6.
-
47
1.2.
f , >0 x A :
x T A, x T A+ f (x T) f (x T) f (x)+ = = .
f.
f (x) x= : \ : [1,1] = 2, (x 2) (x 2) x+ = = .
[0,2]. : :
: x2
= 12=
3x2
= 3 12= .
: f ( x) ( x) x f (x) = = = , (0,0).
: x x (0,0), (,0), (2,0) x x (,0), ] , y y (0,0).
-
48
:
f (x) x= : \ : [1,1]. = 2, (x 2) x(x 2) x+ = = , -
[0,2].
: :
: x = 0 x = 2 0 = 2 = 1
x = = 1.
: : f ( x) ( x) x f (x) = = = , - y y .
: x x ,0
2
3 ,02
,02
+ ] y y (0,1).
-
49
:
f (x) x= : A x / x ,
2 = + \ ] , -
. : \ = , (x ) (x ) x+ = = , -
. , 2
+ ] ,
,2 2
. : -
:
: . : . : x
2= x
2= .
: f ( x) ( x) x f (x) = = = , (0,0).
: x x y y (0,0) x x (,0) ] .
x1
1
1
2
1
10
0
2
32
y
y
x
-
50
:
f (x) x= : { }A x / x ,= \ ] , . : \ . = , (x ) (x ) x+ = = ,
, (0,). : :
: .
: . : x = 0 x = . :
:
f ( x) ( x) x f (x) = = =
(0,0).
-
51
: y y , x x ,0
2
,02
+ ] . :
: f Cf. y f (x) c= , c>0,
xx +c ( ) c ( ).
y f (x)= Cf x x .
y f (x)= - Cf f (x) 0 , x x , Cf - f (x) 0< .
-
52
____________________________________________________
f >0 - x A x T A+ x T A
f (x T) f (x)+ = (1) , f (x T) f (x) = : (1) x xT, :
f (x T T) f (x T) f (x) f (x T) + = = . , x A :
f (x T) f (x T) f (x)+ = = . ,\ >0
x T , x T+ \ \ , x\ f (x T) f (x)+ = .
: , ,
. , -
. -
x . f, *]
f f.
1.1. f (x) 4x 36x= + = .
f (x) 4x 36x= + \ x , x + \ \ .
: [ ] [ ]f (x ) 4(x ) 3 6(x )+ = + + + =
(4x 4) 3(6x 6) 4x 36x f (x)= + + + = + = .
-
53
f (x ) f (x)+ = (1). (1) x x, :
f (x ) f (x ) f (x) f (x ) + = = (2) , (1), (2) f (x ) f (x ) f (x)+ = = .
1.2. f : \ \ x\ : f (x) f (x 1) f (x 2) ... f (x 2006) 0+ + + + + + + = .
f = 2007.
f \ , x 2007+ \ x 2007 \ , f (x 2007) f (x 2007) f (x)+ = = .
:
f (x) f (x 1) f (x 2) ... f (x 2006) 0+ + + + + + + = (1) x+2007 (1) x x+1, :
f (x 1) f (x 2) f (x 3) ... f (x 2007) 0+ + + + + + + + = (2) (1) (2) : f (x) f (x 1)+ + f (x 2)+ + ...+ f (x 2006)+ + f (x 1) + f (x 2) + f (x 3) + ... f ( 2007) + =
f (x) f (x 2007) 0 f (x 2007) f (x) + = + = (3) (3) x x2007, :
f (x 2007 2007) f (x 2007) f (x) f (x 2007) + = = . , :
f (x 2007) f (x 2007) f (x)+ = = .
-
54
1.3. f (x) 2x4
= + .
1.4. g(x) x x= + = 2.
1.5. 4 4h(x) x x= + T2
= .
1.6. : . f (x) 2x= . g(x) 3xx= . h(x) 2x 5x= + .
1.7. g : \ \ : g(x) g(x 1) g(x 3) g(x 4) 0+ + + + + + = , x\ .
g = 5.
1.8. f y f (x)= - h(x) f (x )= + , * , ] \ .
(: f , *] ).
-
55
____________________________________________________
f (x) x, g(x) (x), h(x) (x) c= = = + . -
, : 1. 2. 3. .
f = 2, g, h 2T
= .
, >0:
f(x) = x T = 2, ..: [ , ] g(x) = x(x) 2
= , ..: [ , ]
h(x) = x(x) + c 2
= , ..: [ c, c] + +
-
.
, , - 1 (x) 1 .
-
56
,>0 f g : f (x) x= : g(x) (x)= : : 2 :
2
: 30, , ,22 2
: 3 20, , ,
2 2
: 3,2 2
: ,
2
: 2
:
2
: 32
- : 32
xx : 0, , 2 xx : 0, , 2
1.9. 1f (x) x3
= , .
\ = 2. :
-
57
:
0,2
3 ,22
3,2 2
13
x2
= 13
3x2
= x x : (0,0), (,0) (2,0).
1.10. xf (x) 33
= .
f \ 2T 613
= = . -
: 3 , 31 1 2 2
3 3
= = = = 3 3 912 223
= = .
.
1.11. f (x) 3( 2x)= , 0 x 2 .
f 2 2 2 2 2
= = = = . O :
f (x) 3( 2x) 32x= = . :
x 0 4
2x
34
32x
0
0
1 0
0
1 0
03
2
+3
-
58
0 x 2, , - .
: :
3 5 7, , ,4 4 4 4
. :
3 5 70, , , , ,24 4 4 4
.
3 3x4
= 7x4
= .
3 x4
= 5x4
= .
1.12. xf (x) 4 32
= , .
f \ 2T 412
= = .
:
x x x1 1 3 3 3 7 4 3 12 2 2
1 f (x) 7 .
.
f: (,3) (0,) (3,4) 7 x = 3 1 x = .
-
59
____________________________________________________
f (x) x, g(x) (x), h(x) (x) c= = = + f = 2, g, h 2
= .
f(x) = x T = 2, ..: , g(x) = x(x) 2 = , ..: ,
h(x) = x(x) + c 2
= , ..: c, c + +
,>0, f g :
f (x) x= : g(x) (x)= : : 2 : 2
: (0,) : 0,
: (,2) : 2,
: 0,2 : 20,
: :
xx : 3,2 2
xx : 3,2 2
-
60
1.13. 1f (x) 2x2
= , .
f \ 2T 2
= = . , , :
3 3 3, ,2 2 2 4 2 2 2 2 4
= = = = =
:
x 0 4
2x
34
2x
1
2
12
1
12
0
012
0
0
1
12
f 0,2
.
,2
.
12
x = 0 x = , 12
x2
= .
-
61
1.14. : 7g(x) ( 3x) 3x2
= + + . I. g . II. g(x), 0 x 2 .
I. g \ . : ( 3x) 3x =
7 3 3x 3 3x 3x 3x 3x2 2 2 2
+ = + + = + + = + = .
g(x) : g(x) 3x 3x 23x= = . II. g(x) 23x= 2T
3= . :
[0,2] g 3 , 22 33
= .
:
2 4 50, , , , ,3 3 3 3
.
: 2 4 5, , , , ,23 3 3 3
.
2 x , x 3
= = 5x3
= , 2
2 4x 0, x , x3 3
= = = x = 2.
-
62
1.15. g(x) 23x 5= , .
g \ 2T3
= . :
1 3x 1 2 23x 2 7 23x 5 3 . .
g:
0,3
.
2,3 3
.
3 x = 0 2x3
= , 7 x3
= .
1.16. : I. f (x) x= II. g(x) x= III. h(x) x= .
I. x , x 0 x, x 0
f (x) f (x)( x), x 0 x, x 0
= = <
-
63
.
(x) x
x , x 0g(x) g(x) x
( x) , x 0 == =
-
64
1.17.
f (x) 2x 2x 1= + 0 x .
2x.
0 2x2
< < 0 x4
< < 2x 0> 3 2x 22< < 3 x
4< <
f (x) 2x 2x 1 22x 1= + = . 32x
2 2 3x
4 2 2x 0 f (x) 2x 2x 1 1= + = .
,
32x 1, x 0, ,4 4
f (x) 31 , x ,4 4
=
22x 1, 0,
4
3 ,4
3x ,4 4
y = 1.
f (x) 22x 1= 2T ,2
= = :
-
65
1.18. f (x) 23x= , .
(.: 0,6
2,2 3
,
,6 2
).
1.19. xg(x) 42
= , - .
(.: (0,2), (2,4)).
1.20. :
3h(x) 2x 2(2x )2
= +
. h(x) 32x= . . h(x), 0 x 2 . . h(x) 1= 0 x 2 .
(.: . 0,4
3 ,4
,
3,4 4
, . 4 ).
1.21. :
f (x) 2x= , g(x) 2x= , h(x) x= [ ]x 0,2 .
1.22. : . f (x) 20064x 5= + . g(x) 34x 1= +
. h(x) 2 x 58
= + V. (x) 4 35x= (.: . 2001 f (x) 2011 , . 2 g(x) 4
. 7 h(x) 3 , V.1 (x) 7 )
-
66
1.23. :
I. ( )f (x) 5 3x= II. xg(x) 2 2 = (.: . . : [5,5], 2 3T
3= , I. . 22, 2 , T 4 = ).
1.24. f (x) 3 42x= , .
1.25. xg(x) 2 13
= , -.
1.26. , , :
f (x) 2x= , g(x) 23x= , (x) 23x 1= + 0 x 2 .
1.27. 0 x 4 : 1f (x) 2x 12
= xg(x) 2 12
= + .
1.28. : I. f (x) x= II. g(x) x= II. h(x) x=
1.29.
f (x) 1 x x= + 0 x 2 .
-
67
____________________________________________________
f (x) (x), g(x) (x), 0= = > ) f (x) (x)= -
= ,
,2 2
.
> 0:
, ,2 2
- .
. x
2= x
2= .
x ,2
(x) ,+ x
,2
(x) . (0,0).
< 0: x x
,2 2
. :
-
68
) f (x) (x)= T
=
, 0,
.
> 0: 0,
,
.
.
x = 0 x
= . x 0, (x) ,+ x ,
(x) . (0,0).
< 0: x x , - 0,
. -
:
, - , x , .
-
69
1.30. , ( ) xf x 22
= - .
x x 2 2 2 + + , ] .
f { } x / x 2 , = + \ ] . T 21
2
= = , 2,
(,). . , x :
:
1.31. 1g(x) 2x3
= .
2x x2
.
A x / x , 2
= \ ] .
-
70
T2
= .
1f (x) 2x3
= . g(x) x x .
-
71
1.32. f (x) 23x= . (.:
,6 6
)
1.33. xg(x) 3
= .
(.: 3 3,2 2
)
1.34. xh(x) 22
= . (.: (0,2))
1.35. 1(x) 3x2
= .
(.: 0,3
)
-
72
____________________________________________________
.
( ) ( )f x f x ,+ + f - , :
x + , x + f x x + + . :
x 0 22
x
32
x 0 22
x
32
x 0 22
x
32
x 0 22
x
32
:
'' ( ) ( ) f1 2 l 2f x f x x x< < ''
'' ( ) ( ) f1 2 l 2f x f x x x< > ''
1.36. 1 20 x x ,< < < 1 2 x x .3 3
1 2 1 2 1 20 x x 0 x x x x 0< < < > > > > > >
1 2 x x 03 3 3
> > > 2 1 x x 03 3 3 < < < .
1 x3 , 2 x 0,
3 3 , 1 , -
f(x) = x . , :
2 1 2 1 x x x x 3 3 3 3 < >
-
73
1.37. :
. 2x x 012 6
+ + > x 0,6
. 2 x 2x6 3
+ < 2x ,2 3
. 50 x 0 2x 2x6 3 12 12 12
+ (1)
0 x x x6 6 6 6 3
+ + (2)
(1), (2) 2x ,x12 6
+ + 1 ( )f x x= :
x 2x x 0 2x x12 6 12 6
+ + > + > +
2x x12 6
+ > + 2x x x6 12 12
> > .
x 0,6
:
, x , .12 6
. 2 2 2 5x x x2 3 2 6 6 3 6 3 6 6 + + + + (3)
2 4 2 2 4 2 2x 2x 2x 2x2 3 3 3 3 3 3 3 3 (4)
(3), (4) x6
+ , 22x3
(0, ) (1, 2 ), g(x) = x . :
2 x 2x6 3
+ < 2 2 5x 2x 2x x x6 3 6 3 6
+ > + > <
-
74
2x , ,2 3 :
, 2x ,2 3 .
-
75
1.38. 1 20 x x < < < : I. 1x
4 2x
4 II. 1 x
3 2 x
3
(.: I. 1 2x x ,4 4> II. 1 2 x x
3 3 > )
1.39. x,y 3, ,4 4 :
I. N x, 4
, 3 .4
II. x, y.
(A: I. 3 x ,4 4 II. x y)
1.40. :
I. x x , x ,6 4 2
<
II. 2x x , x 0,10 3 4
+ > +
III. x x , x ,8 4 4 2
<
(.: I. , II. 7x 0, ,30
III. )
-
76
____________________________________________________ .
, , ( ) .
( ) ( )f x x c= + ( ) ( )g x x c= + - 2T
= p c, p c, , ( ) ( )h x x=
T
= .
(x0,y0), ( )0 0f x y= -.
1.41. - :
II.
III. IV.
.
-
77
.
( ) ( ) max 1 1f x x f 2 2= = = 2 2 2 2
= = =
, ( ) 1f x 2x2
= ( ) ( )1f x 2x2
=
.
( ) ( ) maxg x x 0 g 3 3= > = = 2 2 1= = = . , ( )g x 3x.=
. x x ( ) ( ) x x 0= > max 2 2= =
2 2 2 3
= = 3= . , ( ) x 23x= . V. ( ) ( )h x x= > 0, > 0.
1T 2 2 , 2
= = = 1h(x) x2
= .
,1 ,2
h 1 1 12 4
= = = .
, ( ) xh x .2
=
1.42. ( ) 2xf x x3
= + \ 0, .> \ f 3 y y 1, f.
maxf ,= + 3+ = (1) (0,1), :
( )f 0 1= 0 1 1. + = = = 1 (1) : 2= . ( ) 2xf x 2 1.
3= +
-
78
1.43.
( ) ( )f x x = + ( ) ( ) ( )g x 1 2 x , 0 = + + > . ,\ , f,g .
, > 0, + > 0, + 2 > 0, + 1 > 0. , max maxf g 1 1= = + = + (1). :
f g2 2 1 1 T 2 2 2
= = = + = + + + + + = 2
(1) = 3.
1.44. ( ) tU t 220 10 ,2
= + U Volt t sec. I. . II. . III. 0 t 4 .
I. U U(t) = c + (t),
[ ] [ ] c, c, 10 220,10 220 210,230 + + = + + = , 210V 230V 230 210 = 20V.
II. 2 4.12
= =
III. :
-
79
1.45. 2006
( ) t t 1,2 0,1 ,3
= + (t) t .
I. .
II. (t).
III. - ;
V. 0 t 12. V. ;
I. t t1 1 0,1 0,1 0,13 3
( )t1,2 0,1 1,2 0,1 1,2 0,1 1,1 t 1,33
+ +
, 2006 1,3 1,1 = 0,2 .
II. H 2 6T 6 3
= = = .
III. 5 , :
( ) 5 5 1,2 0,1 1,2 0,1 2 1,2 0,13 3 3
11,2 0,1 1,2 0,05 1,25 .2
= + = + = + = = + = + =
IV. :
V. 3 6 ( - ) 9 12 ( - ).
-
80
1.46. ( ) [ ]f x x,x 0, .=
I. ,.
II. x f -
(.: I. = 4, = 2,
II. x4
= , 3x4
= )
1.47. ( ) ( ) [ ]g x x ,x 0,4 .= +
I. ,,.
II. x f .
(.: I. = 1, = 2, 1 ,2
= II. x = 0 x = 4, x = 2)
1.48. ( ) x 2x , 0.= + > ,,\ 4 5A ,
6 2 .
(.: = 3, = 1)
1.49. ( ) ( )f x x .= ,\ f 2
A , 5 3 .6
(.: = 5, = 2)
1.50. , ,\ ( )f x 2x= + A , 1
4
B , 4 .8
(.: = 1, = 3)
x12
0
y
2
2
x23
2
0
3
3
y
1
4
-
81
1.51. ( ) ( ) ( )f x 3 2 2 1 x= + + [ ]g(x) ( 1) ( 2)x= + + , 0> . , ,\ f - g g f.
(.: = 3, 92
= )
1.52. 2m. ,
( ) th t 2 ,3
= + t sec. I. . II. . III. 0 t 6.
(.: I. 2m II. T = 6sec)
1.53. To
( ) tf t 20 43
= + , t 0 t 24. I. .
II. ;
III. 30m ( ), 8m 12 .
IV. 1 .. 5 .. ;
(.: I. T = 6 , II. 24m, 16m, III. IV. 7.., 11.., 13.., 17.., 19.., 23..)
1.54.
( ) tA t 4 2 ,2
= 0 t 4 11.. 3 ..
I. .
II. .
III. ;
-
82
IV. S A(t) 4 , - ;
(.: I. 6, 2, III. 1.. 6 , IV. 12 .. 2...).
1.55. - ( )
( ) tE t 6 3 ,6
= t ( t = 1 , t = 2 - )
I. .
II. .
III. 0 t 12 . IV. .
(.: I. 75 , II. 90 , IV. )
-
83
. () ():
1. H ( )f x x= 2. H ( )f x x= . 3. H ( )f x x= . 4. H ( )f x x= 2. 5. H ( )f x 25x 8= + [10, 4] 6. .
6 8>
7. ( ) 3f x x2
=
( ) ( )g x x= o x .\
8. ( ) xf xx
= [1,1].
9. H ( )g x 2x 4x= . 10. H ( ) ( )f x 70x= y 69= y 69.=
-
84
. :
1. f . f : . ( )f x 22x= B. ( )f x 22x= . ( )f x 2x= . ( ) xf x 2
2=
. ( ) xf x 22
=
2. - ( )f x x= ( )g x x= [ ]0,2 . ( ) ( ) [ ]f x g x , x 0,2 ,= : . 3x x
4 4= = . 3x x
2 2= =
. 5x x4 4
= = . 7x x4 4
= =
. x x 2= =
3. H :
. . 6 . 3
. 23
. 2
4. ; . ( )f x x x= + . ( )f x x x= . ( ) x 2x= . ( ) 2g x x= E. ( ) 2 x x=
-
85
. -:
. ( ) xf x 22
=
. ( )f x x 3= . ( )f x 22x 5= + . ( )f x x=
1.
,2 2
2. ( )0,
3.
0,2
4. ( ),3
5. ,2 2
6. ( ),3
. :
. ( ) xf x 33
=
. ( )f x 3x=
. ( ) xf x 24
=
. ( ) xf x 2
=
1. 6
2. 2
3. 8
4. 8
5. 23
6. 3
-
86
-
87
1.3. ,
. .
. :
x =
1> , 1< 1> , , x\ - 1 x 1 .
1 1 1 , = , x = . -
, - :
x 2 x ,
x 2
= += = + ]
, , x 0x = 0 =
( )( )x 2 x 2 2 1 , = = + = + ] ` 2, 2 + 1 ( ). - : x ,= ]
x = 1 x 2
= 3x 2 x 2 , 2 2
= + = + ]
: x 2 ,2
= + ]
-
88
x = 1 3x 2
= 3 3 x 2 x 2 2 , 2 2 2
= + = + = ]
32
2
. , -
x 22
= ,] .
x =
x = : > 1, < 1 > 1,
x\ 1 x 1 . 1, 1 1 , ,
= , x = . , - :
x 2
x ,x 2
= += = ]
x 0= x 2
=
( ) x 2 x 2 2 2 1 2 2 2 2
= + = = + = + 2, 2 1 -
, : x 2
= + , ]
x = 1 x 0 x 2 0 = = x 2= , ] x = 1 x =
( )x 2 x 2 ,= + = ] , - ,
x 2 = + , ]
-
89
x = , x =
\ , , - , x = x = .
, :
x
x ,x
= = + = ]
x = 0 x 0 x 0x
= = x = , ]
x = 0 x 0 x 0x
= = x 2
= + , ]
x = x = 2+ x = 2 + , ] x = 0 x = , ] x = 1
x = 2 + , 2
]
x = 1 x = 2 , 2
]
x = x 2 x 2 , = + = ]
x = 0 x , 2
= + ]
x = 1 x = 2, ] x = 1 x = 2 + , ] x = x = + , ] x = 0 x = , ] x = x = + , ] x = 0
x , 2
= + ]
-
90
- , .
: , . (x), x
2 + ( ) x 0 .
A (x), x ( ) x 0 .
.
-
91
_____________________________________________________________ ( ) ( )f x f x = , f ,
. ,
. :
( )( ) ( )( ) ( ) ( )( ) ( ) x 2 x
x x , x 2 x = += = +
]
( )( ) ( )( ) ( ) ( )( ) ( ) x 2 x
x x , x 2 x = += =
]
( )( ) ( )( )( )( ) ( )( ) ( ) ( )
x x x x ,
x x
= = + = ]
x.
1.56. :
. 24x 1 0 = . 2x 13
+ =
. : 1 24x 1 = 0 24x 1 4x 4x 2 6
= = =
4x = 2+ 4x 2 , 6 6
= + ]
2 5 5x 4x 2 , x x , 4 24 6 2 24 2 24
= + = + = + = + ] ]
. : 2x+ 13
= :
2 2x 2 2x 2 2x 2 x , 3 3 3 3
+ = + = + = + = + ]
-
92
1.57. :
I. 32x 3 0 = II. 2x x4 3
+ =
I. . 2x 0 .
:
3 32x 3 2x 2x 2x x3 3 3 2 6
= = = = + = + .
2x 2 02 6 3 3
= + = + = ( 1 3 ). , :
x , 2 6
= + ]
II. , 2x+ 04
x 03
:
7 2x x 2x x 2x x x 4 3 4 3 3 4 12
+ = + = + = =
:
7 11 11 2x+ 2 2 04 12 4 12 12
= + = =
7 11 11 x 03 12 3 12 12
= = =
, 7x , 12
= ] .
-
93
1.58. :
I. 23x 1 0 = II. 2 x 1 04
=
III. 3 2x 3 06
=
(.: I. 2 x3 9
= + , II. 5x 212
= + , 13x 212
= + , III. x2 6
= + )
1.59. :
I. 2x x 03 6
+ = II. 2x 2x4 4
+ =
(.: I. 2 7x 2 , x2 3 18
= + = + , II. x2
= )
1.60. :
I. 3x x3 6
+ = + II. 2x x 06 3
+ =
III. x x 04 6
+ = IV. 3 2x x5 5
+ = +
(A.: I. x2 12
= , II. x 12
= + , III. x 24
= , IV. 2x 25
= + 2 x3 15
= + )
-
94
______________________________________________________________ () -
(), -. , :
( )( )( )
x x
x x x
x x
= = = = = =
:: x = ,:
( )x =
1.61. :
I. 23x 1 0+ = II. 2x 1 0+ =
I. 123x 1 0 23x 1 3x2
+ = = = 23x 3x 3x 3 3 3
= = = 2 23x 2 3x 23 3
= + = 2 2 2 2x x , 3 9 3 9
= + = ] II. 2x, 2x 0 . :
2x 1 0 2x 1 2x 2x 2x 4 4 4
+ = = = = = x , .2 8
= ] :
2x 2 02 8 4 4
= = =
x , 2 8
= ]
-
95
1.62. :
I. x x 03 4
+ + + = II. 2x x 05
+ =
I. x x 0 x x3 4 3 4
+ + + = + = + x x3 4
+ = x 2 x3 4
+ = x 2 x3 4
+ = + + + 72x 212
= 110x 212
= + 7x 24
= 1124
= ] ,
7x , 24
= ] .
II. : 2x 0 x 05
. : 2x x 0 2x x 2x x 2x x5 5 5 5
+ = = = + = + x , 5
= ] :
2 2 22x 2 2 05 5 5 5
= = = =
2 2 x 5 5 5 5 5 5 5
= = + = + = =
2 0.5
=
, x , .5
= ]
-
96
1.63. :
I. 24x 1 0+ = II. 2 4x 1 03
+ =
(A.: I. x2 24
= 7x2 24
= + , II. 13x2 48
= + 5x2 48
= )
1.64. :
I. 3 x 3 03
+ = II. 3x 2x 04
+ + =
(.: I. 2x 3
= + , II. 2 x5 20
= 2 3x5 20
= + )
1.65. :
I. 3 2x 3 03
+ = II. x x 04 6
+ + =
III. 3x x 03 4
+ + + =
(.: I. x2 12
= + , II. 13x 24
= , III. 7x2 48
= 11x 24
= )
1.66. : ( )
( )3 x x
122 x 2 2
+ = + .
(.: x , )4
= ]
-
97
_____________________________________________________________
, -, -. , ( , , ) -, (x) B(x) (x) = 0, :
( ) ( ) ( )( )( )( )
A x 0
A x B x x ... 0 x 0
x 0...
= = = =
.
1.67. : I. 2 x x 0+ = II. 22x x x=
I. x x 0 . : ( )2 x x 0 x x 1 0 x 0+ = + = = (1) x = 1 (2)
(1) : x 0 x , ()= = ] x 1 0= = . (2) :
( ) x 1 x x x , 4 4 4
= = = = ]
2x 04 4 2
= = = . II. :
( ) ( )( )2 22x x x 0 x 2 x 1 0 x 2x 1 2x 1 0 = = + = ( ) ( ) ( )x 0 1 2x 1 0 2 2x 1 0 3= = + =
( )1 : x 0 x ,= = ]
-
98
(2) :
1 22x 1 0 2x 1 x x22
= = = = x 4
=
x 2 x 2 , 4 4
= + = ] .
A (3) : 1 22x 1 0 x x22
+ = = =
x x 4 4
= = 3x 4
=
3 3x 2 x 2 , 4 4
= + = ] .
1.68. :
I. 2 2 2x x 03
+ = II. 1 x x 4x 1 x+ + =+
I. : 2 2 2x x 03
+ =
2x x 2x x 03 3
+ + + =
( ) ( ) 2x x 0 1 2x x 0 23 3
+ = + + =
(1) :
2x x 2x 2 x 2x 2 x , 3 3 3
= + = + + = ]
2 x 2 x , 3 3 9
= + = ]
(2) : 2x x 2x x3 3
= + = +
2 2 22x x 2x x 2x 2 x 2x 2 x3 3 3 3
= = = + = +
2 2 2x x 2 , 3 9 3
= + = ] .
-
99
II. . x 0 x 1 x 0 x 1 x 2 . + + :
( ) ( ) ( )2 2 2 21 x x 4x 1 x 1 2x x x 4x 1 x+ + = + + + + = + ( ) ( )1 2x 1 4x 1 x 2 2x 4x 1 x 0 + + = + + + =
2(1 x) 4x(1 x) 0 (1 x)(2 4x) 0 + + = + = 2 4x 0 1 x 0 = + = ( ) , 1 2 4x 0 x x
2 6 = = =
5 5x 2 x 2 x 2 x 2 , 6 6 6 6
= + = + = + = + ]
, .
1.69. : ( )( )2 2 25 x x x x 3 x 0 + + = .
2 2 x 1 x= .
:
( ) ( ) ( )25x x 1 x x 1 3 1 x 0 + + = ( )( ) ( )( )25 x x 1 x 1 3 1 x 1 x 0 + + + = ( )( ) ( )( )25 x x 1 x 1 3 x 1 x 1 0 + + = ( )( )( )2x 1 x 1 5 x 3 0 + = x 1 0 x 1 x 2, = = = ]
x 1 0 x 1 x 2 , + = = = + ]
2 2 3 35 x 3 0 x x5 5
= = = .
-
100
30 15
< < 0,2
3 ,5
= :
3x x 5
= = x 2
, x 2
= + = ] -
( )3x 5
= = = x 2
, x 2
= + = +]
, ,
-
101
1.70. :
I. 6xx 3 2 3x 3x =
II. 2 x x 0+ =
III. 22 x x 0+ =
IV. 22x x x 0 = (.: I. 2 x 2 , x ,
3 3= = + ] , II. 3x x 2
2= = + ,
III. 2x x 22 3
= + = , IV. 3 5x x 2 x 2 x 2 , 2 4 4 4
= + = = + = + ] )
1.71. : x 1 x 2x 2 x 1
+=+ .
(.: 7x 2 x 2 , 6 6
= = + ] )
1.72. :
I. 2 x 2x 0 =
II. 3 x x=
III. 1 xx x x 0+ =
IV. 2xx 2x 2x 2 0 + = (.: I. x = , II. x x
4= = + , III. x 2 x 2
2= = + ,
IV. x 2 x 2 , 4 2
= = + ] ),
1.73. : xx 21 x
+ = + .
(.: 2x 2 , 3
= ] )
-
102
1.74. : 3x 2x= .
(.: x x 2 , 6
= = ] )
1.75. : ( )( )2 2 24 x x x x 3 x 0+ + = (.: 2 4x x 2 x 2 x 2 ,
2 3 3 3= + = = + = + ] )
-
103
_____________________________________________________________ ( )( ) ( )( ) ( )( ) ( )( ) x x , x x= =
, - , - .
:
( ) ( ) ( ) ( ) x x x x2
= = ( ) ( )
( ) ( )
x 2 x2
, x 2 x2
= + = + +
]
- x = .
( ) ( ) ( ) ( ) ( ) ( ) x x x x x x ,2 2
= = = + ] - . (), - .
, .., ( )x x = = ( ) x x 2 2
= = +
1.76. :
I. 2x x 07 3
+ + = II. 5x 3x 0+ =
I. 2x x 2x x7 3 7 2 3
+ = + + = +
2x x 2x x7 2 3 7 6
+ = + =
2x 2 x 2x 2 x7 6 7 6
+ = + + = + +
-
104
3x 2 x 2 6 7 6 7
= + = +
2 29x x 2 , 3 126 42
= + = + ]
II. :
5x 3x 0+ =
( )3x 5x 3x 5x= =
( )3x 5x2
=
3x 5x 5x 3x2 2
= + + =
5x 2 3x 5x 2 3x2 2
+ = + + =
2x 2 8x 22 2
= =
x x , 4 4 16
= = ]
1.77. : x2x 1= .
, .
x x 0 , x 2
+ (1) ] 2x 2x 0 ,
( )2x , 22
+ ]
x 2x 1 = , x 0 x. :
x 2x 1 12xx x x = =
:
, - , - , - .
-
105
x x 1 = 1xx
=
2x x 2x x 2x x 3x x , 2 2 2 3 6
= = = + = + = + ] (1) :
4 2 12 6 4 12 4 3 13 6 2+ + + + + +
(2) :
2 6 12 4 2 6 3 4 6+ 3 6 2 3 3 2 4
+ + + + + + +
4 2 1 2 1 4 4
+ + + +
2 14++ ] .
, x3 6
= + 3 1= + .
-
106
1.78. :
I. 3x 2x4
= II. 2x 3x=
III. x x3
= + IV. 2x x4
= +
(.: I. 2 3 x x 25 20 4
= + = + II. 2 x x 25 10 2
= + = +
III. x2 12
= + IV. x 3 2, 3 12
= + + ] ] )
1.79. : 2 2 4x 2x 0 = . (.: x x ,
3 12 4= = ] )
1.80. : 2x1 03x
+ = .
(.: 2 x 2 x , 2 5 10
= = ] )
1.81. : 2 2 x 3x2
+ = , x4
= .
(.: x 4 2, 4
= + ] ]
x 2 1, 2
= + ] ] )
-
107
_____________________________________________________________
-
, .
2 2 x 1 x= 2 2 x 1 x= - .
1.82. : I. ( )22 x 2 4 x 2 2 0+ = II. 2 22 x 5x x =
I. x = , : ( )22 2 4 2 2 0+ = ( )( )( ) ( )
2
2 2
2 22
2 4 16 2
2 8 2 4 16 2
2 8 2 4 2 4
= + == + + == + + = +
( ) ( ) 11,2
2
2 1 12 4 2 4 24
2
= += =
2 3 3x x x x x 2 ,
2 4 4 4 4 = = = = = ]
II. x , 2 2 x 1 x= - x. :
( )2 2 2 2 2 22 x 5x x 2 x 5x 1 x 2 x 5x 1 2 x = = = + 20 2 x 5x 3= +
:
- - .
-
108
x = , 22 5 3 0+ = .
1,2
3, 5 7 25 24 49, 14
2
= + = = =
, 1x2
= 5x x 2 x 2 , 6 6 6
= = + = + ] .
1.83. :
I. 3 2 x 2 x x+ = II. 3 21 x 3 x 42 x + =
I. 2 2 x 1 x= , : ( )3 2 3 2 x 2 1 x x x 2 2 x x 0+ = + =
x = , : 3 2 2 2 0+ =
( ) ( ) ( )( ) ( )( )( )2 2 2 2 0 2 1 0 2 1 1 0 = = + = = 2 = 1 = 1
: x 2= x 1 x 2, = = ] x 1 x 2 , = = + ]
II. 221 1 x
x= + x x
2 =
: 3 2 x 1 x 3x 4+ + = x 0 x = , :
( ) ( )3 2 3 2 2 1 3 4 3 3 0 1 3 1 0+ + = + = + + = ( )( ) ( )( )( )2 1 3 0 1 3 3 0 1 3 3
+ = + + = = = =
: x 1 x x x , 4 4 4
= = = = ]
x 3 x , 6 6
= = = + ]
x 3 x , 6 6 6
= = = = ]
, x 0 .
-
109
1.84. : 2 27 x 3 4 x x 2 x = +
. 1 x 1 ,
x 3 0 < 2 x 0 > . :
( ) ( )2 2 x 1 x
2 27 x 3 4 x x 2 x= + = +
2 27 x 3 4 x 1 x 2 x+ = + 2 24 x x 4 1 x 2 x 0 + + + + =
2 23 x 2x 1 0 3 x 2x 1 0 + + = = 2 x = ,: 1 1, :
23 2 1 0, = = 4 + 12 = 16, 1,21
2 4 163
= =
, x 1 x 2 , 2
= = + ] 1x .3
= 0,2
:13
=
( )x x = = x 2 x 2 ,= = + + ] .
-
110
1.85. :
I. 22 x 3x 1 0 + = II. 23 x 2 3x 3 =
III. ( )2 22 x 1 2 x 02 + + = (.: I. 5x 2 x 2 x 2
2 6 6= + = + = + ,
II. x x 6 3
= + = , III. x 2 x 2 , 4 3
= = ]
1.86. :
I. 22 x 1 5x+ = II. 26 x x 5 0+ = (.: I. 5x 2 x 2 ,
6 6= + = + ] ,
II. 1x 2 x 2 x 2 , 0, 3 2 3
= = + = + = ] )
1.87. : x 3x 2 3+ = (.: x ,
3= + ] )
1.88. : 21 1 3x
x+ =
(.: x x 0, 24 2
= + = + = )
1.89. :
I. 4x 1 7 x 3x 1 x 115 10 5 3
+ + + =
II. 3 2 x 2 x 8x 0 =
III. 4 24 x 5 x 4 0+ = (.: I. x 2 = + , II. x = ,
III. 5 7x 2 x 2 x 2 x 2 , 2 6 6 6
= = + = + = + ] )
-
111
_____________________________________________________________
. , , -:
) .
) 1, 2 2 x x+ , .
) , - . .
) x 0 , x 2x 3x x 2x 3x .
) , :
( )22 2 2+ = , ( ) ( )33 3 3 = ( )2 2 0 0 0 0+ = + = + = = =
1.90. : 8x4x 1= .
8x 0 4x 0 . :
18x4x 1 8x 14x
= = 8x 4x 8x 4x 4x x ,4
= = + = = ] .
4x 4 04
= = = . , .
-
112
1.91. : 2 22 x 3xx 3 x 1 + = .
x = 0, 2 2 12 x 1 x2
= = , 2 2 x x 1+ = 1 0 1
2 + = , x 0 , 2x
:
22
1 1 x x
2 22
2 2 2 2 22 x 3xx 3 x 1 12 x 3x 3 x x x x x
= +
+ = + =
2 2 22 x 3x 3 1 x x 3x 2 0. + = + + =
x = , 2 1,2 23 1 3 2 0, 1, 12 + = = = =
x = 2 (1) x = 1 (2)
(1) 0,2
2= , x = x , = + ]
(2) x 1 x 4
= = x , 4
= + ]
1.92. : ( )2 1 x x+ = .
( )2 1 x 0+ x 0 . :
( )2 2 x 1 x
22 1 x x= + = ( ) 22 1 x 1 x+ = ( ) ( )( )2 1 x 1 x 1 x+ = +
( ) ( )( )2 1 x 1 x 1 x 0+ + = ( ) ( )1 x 2 1 x 0+ = ( )( )1 x 1 x 0+ + =
1 x 0 + = x 1 x 2 , 2
= = ] ,
2 02
= .
-
113
1.93. : 22xx
1 x= + .
21 x 0+ x 0 . 2 x ,
2
22
x x1 x
= + . :
( )( )2
2 2 22
2 2 2 22
1 x 0 22
2
(2x) x 4 x x(1 x) 1 x 1 x
4 x x1 x
+
= =+ + +
= +
( )2 2 2 2 4 2 x 1+ x 4 x x x 4 x 0= + = 4 2 2 2 x 3 x 0 x( x 3) 0 = = 2 x(x 3)(x 3) 0 x 0 x , + = = = ]
x 3 x , 3 3
= = = + ]
x 3 x , 3 3
= = = ] .
1.94. :
I. 2006 2004 2x x 0+ = II. 4 4 1 x x2
+ =
I. 2006 2x 0 2004 x 0 , 2006 2x 0= 2004 x 0= ,
2x 0 2x x2
= = = , ] x 0 x 2
= = + , ] .
x , 2
= + ]
II. ( ) ( )2 22 2 1 x x 2+ = ( 2 2 + = = ( )2 2+ )
2x, - .
-
114
( )22 2 2 2 2 21 1 x x 2 x x 1 2 x x2 2+ = = 2 2 12 x x 2= ( )2 2 2 2 2 4 4 21 1 1 1 x x x 1 x x x 0 x x 04 4 4 4= = = + =
22 21 1 2 x 0 x x
2 2 2 = = =
:
2 3x x 2 x 2 , 2 4 4 4
= = = + = + ]
2 5x x 2 x 2 , 2 4 4 4 4
= = = = = + ]
1.95. : 5 4 x x 1+ = .
x 1 x 1 , 5 2 x x 4 2 x x . 5 2
4 2
O x x
x x
( )( )
24 2 22
22
x 2x 1 0 x 1 0x 1 0 x 1
x 1 0
+ = =
x = 1 0 = 1 .
x = 1 2 13
= , . , x = 1.
-
116
1.98. :
I. x x 0 =
II. x 3x 0+ =
III. x 3x= (.: I. x , x , x ,
4 3 3= + = = ] )
1.99. : ( )2 1 x x = . (.: x 2 , = ] )
1.100. : 6 5 x x 1+ = . (.: x 2 x ,
2= = + ] )
1.101. : 2000 x 2x 0+ = . (.: x x ,
2 4= = + ] )
1.102. : 3x6x 1= . (.: )
1.103. : x 12
= .
(.: x 2 , 3
= ] )
1.104. : 2
4 x x 12
= + . (.: x = 0)
1.105. : ( ) x 0= . (.: x ,
2= + ] )
1.106. : ( )2 x 1 3 xx 3x 0 + + = . (.: x x ,
4 3= + = + ] )
-
117
_____________________________________________________________
- [, ], : x = f(), ] , x , ( ) f , ] . - ,\ .
, , .
.
1.107. : x x 03
+ = (0,2).
x x 0 x x x x3 3 2 3
+ = = + = +
x x x x x 2 x2 3 6 6
= = = + x 2 x,6
= + + ]
5x (1) 0x 212 6
= + = +
, (1) 0 x 2< < , :
0 2 212 12 12
< + < < < 23 1 23 12 12 12 12
< < < <
: ( )1 0 x
12= =
( )1 13 1 x 12 12
= = + =
-
118
1.108. [ ]x 0,2 ( ) g x 3 5 x3
= + .
( ) x c+ c+ , g 5 3 8+ + = . , [ ]x 0,2 ( ) g x 8 3 5 x 8 5 x 5
3 3 = + = =
( ) x 1 x 2 x 2 13 3 3
= = = + , ] .
5 1 50 x 2 0 2 2 2 23 3 3 3 3
+ 1 56 6
,
5 16 6
, = 0, = 0 (1) x3
= .
1.109. [0,] : 2x 1 x+ = .
. : 2x 1 0+ x 0 . :
( )2 2 22x 1 x 2x 1 1 x x 2x 0 x x 2 0+ = + = + = + = x 0= x 2= ()
: x 0 x = = , ] . [ ]x 0, , 0 0 1 = 0 = 1 x = 0 x = . - .
x = 0 20 1 0 1 0 0 0 1 0, x=0 + x = , , 0 1 0 . , [0,] x = 0.
-
119
1.110. :
I. 2 3x 1 0 ,4 2
=
II. 3 2x 3 0 0,3 2
+ + =
III. 2 2x 1 0,5 2
=
(A.: I. 23 31x ,x36 36
= = , II. x4
= , III. 11x60
= )
1.111. 3x x3 4
= + ,3 2
. (.: )
1.112. 2x 1 x = [,3] . (.: 5x
2= )
1.113. 2 2 2x x 0+ = ,2 2
. (.: x = 0)
1.114. : 23 x 3x 1 x + = [0,3]. (.: x = 0 x = 2)
1.115. ( ]1 x x 0,2 = . (.: x x 2
2= = )
1.116. 20044x + 20062x = 0 [ ]0,2 . (.: 3 5 7x , x ,x ,x
4 4 4 4= = = = )
-
120
______________________________________________________________
( ) ( )f x , f x , f (x)\ f(x), f(x), ( ) ( )f x f x
2 +
f(x) , g(x) 0g(x)
f (x), f (x) 0
1.117. ( ) f x x 2x4
= .
3x x , 4 2 4
+ + ] 2x x , 2
]
, 3 A x / x x ,4 2
= + \ ] .
1.118. ( ) 2x 3xg xx 1 2x 1
= + .
( )x x 1 , \ ] ( )x x 2 ,
2 + \ ]
( )( )
x 1 3x 1 0 x 1
x 1 4
( )12x 1 0 x 52
(3): ( ) x x 6 4 4
+ ]
(4): x x (7), 4 4 4
= ]
(5): x x 2 x 2 , 3 3 3
+ ]
-
121
(1), (2) : x , 2
]
(6), (7) : x , 2 4
+ ] :
A x / x ,x ,x 2 x 2 ,,2 2 4 3 3
= + + \ ]
-
122
1.119. ( ) 3f x 2xx
= + .
(.: x / x , 2
= + \ ] )
1.120. ( ) g x x x3 4
= +
(.: 5 x / x , x , ,6 4
= + \ ] )
1.121. ( ) 222 3x 1h x 2 x 1= .
I. .
II. ( )h x 0= . (.: I. 3 5 x / x 2 , x 2+ , x 2
4 4 4 = + + \ ] ,
II. 2 2 x x3 12 3 4
= = )
1.222. :
( ) 1 1f x x 1 x 1x x
= + + .
I. .
II. (0, ) f(x) = 2.
(.: I. { } x / x ,= \ ] , II. x4
= )
-
123
_____________________________________________________________
- . :
x f(x) = 0. ( )f x 0> (
f(x) .
f,g ( ) ( )f x g x= .
1.123. - :
( ) tf t 15 2 , t , 0 t 63
= + <
I. -.
II. 160.000 ;
I. : t t1 1 2 2 23 3
( )t15 2 15 2 15 2 13 f t 173
+ +
f 17. , :
( )f t 17= t t t t15 2 17 2 2 1 2 t 63 3 3 3
+ = = = = =
0 t 6 0 6 6 0 1< < <
-
124
, = 1 t = 6, 6 . II. , 160.000 = (16
. . ). :
( ) t t t 1 t f t 16 15 2 16 2 1 3 3 3 2 3 3
= + = = = =
t 2 t 6 13 3
t t 6 123 3
= + = +
= =
0 t 6 , t = 1 t = 5.
1.124. 2003
( ) tf t 8 3 ,4
= + t 0 t 36 . I. ; II. ; III.
; IV. 500 2003
2004. ;
I. H - . -. :
t t t1 1 3 3 3 8 3 8 3 8 34 4 4
+ + ( )5 f t 11 . , 11 5 6 = .
II. ( ) t t t t f t 11 8 3 11 3 3 1 2 t 8 2.4 4 4 4 2
= + = = = = + = +
1 170 8 2 36 2 8 34 0,1,2,3,4
4 4
+ =] = 0 t = 2. , 2 2003.
-
125
III. ( )f 0 8 30=8= + . :
( ) t t tf t 8 8 3 8 0 4 4 4
= + = = =
t 4. 0 4 36 0 9.=
:
0 t 0 . 2003 5 t 20 . 2004 1 t 4 . 2003 6 t 24 . 2004 2 t 8 . 2003 7 t 28 . 2005 3 t 12 . 2003 8 t 32 . 2005 4 t 16 . 2004 9 t 36 . 2005
= = = == = = == = = == = = == = = =
IV. ( ) ( )6f 6 8 3 8 3 1 54
= + = + = ( ) 24f 24 8 34
= + = 8 3 0 8= + = . 3 , 500 3 1500 = .
-
126
E
1.125. . h cm t sec :
( ) th t 20 10 0 t 244
= +
I. 1t 2sec,= 2t 5sec,= 3t 13sec.=
II. .
III. 15;
(.: I. ( ) ( ) ( ) ( )h 2 30cm,h 5 h 13 20 5 2 cm= = = , II. 10 cm t = 6 t = 14 t = 22 sec,
30 cm t = 2 t = 10 t = 18 sec,
III. 14 22 38 46 62 70t , , , , , sec3 3 3 3 3 3
= )
1.126. C -
( )1 tT t 4 48= , t 0 .
I. .
II. 24.
III.
( )2 t t 4 4 .4= 4 -.
(.: I. T = 16 , II. 8C 8 .. , 0 C 4 ..,
III. 9.20 .. 2.40 ..)
-
127
1.127.
2005 ( ) t t 10 36
= , - t 0 t 12 (t = 1 I, t = 2 , )
I. 2005 ;
II. ;
III. 8,5;
IV. 1000 2005 . ;
(A.: I. , II. ,
III. , IV. 3000 )
-
128
A. () ():
1. 1x2
= ,2
23
.
2. 2 2 x 5 x 2+ = 1x2
= . 3. x 1 x 1 = . 4. 2 2 x 2x 0+ = .
5. 2 2
2 x x
x 2x 3= + x\ .
6. x x= x\ . 7. ( ) x 0 x , = = ] .
B. : 1. 2 x 1= :
. x = . x 22
= + . x 22
= . 4
. 22
2. x 5= :
. 5 5< . 5 5 2
+ . \
3. ( ) 1f x2x 1
= \ :
. 26
. 2 . 23
. . 12
4. 2 x 5x 6 0 + = ; . . 0 . 1 . 2 . 5
5. 23x 4 5= + : . 2 . 2= . 2< . 2 .
-
129
. :
. 1 32x 3
=
. 2x 1x
=
. x 2x5
= . 22 x x 1 0 =
1. 7x3 30
= +
2. 2 x 2, x3 3
= = +
3. 7x 30
= +
4. x 2 , x 23 3
= = +
5. 4x 2 , x 23
= + =
6. 4x 2 , x 23 3
= + =
. :
. ( ) 23f x x 1= . ( ) xg x
1+x=
. ( ) h x x4
=
. ( ) ( )2 x x
x 2x-1=
1. 3x 4
+
2. 2x , x 2 , x 22 3 3
+ + +
3. x 22
4. 5x , x 2 , x 22 6 6
+ + +
5. x , x2 2 4
+ +
6. x4
-
130
-
131
1.4
o 5 715 , 75 , 105 ,...,12 12 12
- :
( )( )( )( )
+ = + = + = = +
, ,, 2
+ ,] :
( ) ( ) , 1 1
+ + = = +
, ,, , , ] :
( ) ( ) 1 1 ,
++ = =+
_____________________________________________________________
( ) , - . , , , . - .
( -), .
-
132
1.128 :
. x y x y4 4 4 4
=
.
5 2x 2x6 6B
5 1 2x 2x6 6
= +
I. ( ) + = -
x, y,4 4 :
x y x y4 4 4 4
= = x y x y4
A4 2
= + = ( ) ( ) x y x y2
= + = +
II. ( ) 1
= + :
5 2x 2x6 6
5 1 x 2x6 6
= +
5 2x 2x6 6
=
( ) 5 2x 2x 4x 6 6
= + = = ( ) ( ) 4x 4x 4x = = .
1.129. :
22 99 .5 5 18 18
= +
2 2 2 5 2 5 9 9 9 18 9 18 5 5 5 5
18 18 18 18
= =
=
2 5 9 18 2 0.5 5 5 5 18 18 18 18
+ = =
-
133
1.130 :
I. 75 II. 7 rad12
III. 49 rad12
I. o o o75 45 30 ,= + :
( ) 2 3 2 1 6 275 45 30 45 30 45 30 2 2 2 2 4+= + = + = + = ( ) 2 3 2 1 6 275 45 30 45 30 45 30 2 2 2 2 4= + = = =
( )( )( )
2
6 26 275 6 2 8 2 12475 2 3
475 6 2 6 2 6 2 6 24
+ ++ += = = = = = + +
( )( ) 1 1 2 3 2 375 2 3175 2 3 2 3 2 3 = = = = = + +
II. 7 ,12 3 4
= + :
7 3 2 2 1 6 2 12 3 4 3 4 4 3 2 2 2 2 4
+ = + = + = + =
7 1 2 3 2 2 6 12 3 4 3 4 3 4 2 2 2 2 4
= + = = =
( )( )( )
2 3 17 3 1 3 1 2 33 4 2 3 12 3 4 1 31 3 1 1 3 1 31 3 4
+ ++ + + = + = = = = = +
( )7 1 1 1 2 3 3 2712 2 3 2 312
= = = = = +
III. 49 4 ,12 12
= + :
49 2 3 1 2 6 2 4 12 12 12 4 6 4 6 6 4 2 2 2 2 4
= + = = = = =
-
134
49 2 3 2 1 6 2 12 12 4 6 4 6 4 6 2 2 2 2 4
+ = = = + = + =
( )( )( )
2 6 2 6 249 8 2 1212 4 2 312 12 46 2 6 2 6 212 4
= = = = = = + +
( )( )49 1 1 2 3 2 312 12 2 3 2 3 2 312
+= = = = = + +
1.131 : 3 5 x x x x4 4 4 4
= +
. -. :
3 3 3 x x x x4 4 4 4
= = + = +
5 x x x4 4 4
= + =
x x x x4 2 4 2 4 4
= = + = +
:
x x x x4 4 4 4
= + + =
x x x x 2x4 4 4 4
= + = + + =
-
135
1.132. x y 3 x y 1,+ = = (x + y).
- (x + y), (x+y) . .
( ) ( )( )
( )22 2 22 2 2 2
x y 3 x y 2xy 3
x y 1 x y 2xy 1
++ = + + = = + =
( )2 2 2 2 x x y y 2 xy xy 4+ + + + = ( ) ( ) ( )1 1 2 x y 4 2 x y 2 x y 1+ + + = + = + =
-
136
1.133. :
I. 4 13 4 13 7 14 7 14
II. ( ) ( ) 67 22 67 22 + III.
13 13 14 143 3 7 7
+
(.: . 0, . 2 ,2
. 0)
1.134. : 50 70 40 20 .= (.: 1
2= )
1.135. : 5 2 A x x x x .3 3 6 3
= +
(.: 12
= = )
1.136. x y 3 y x 1,+ = + = ( ) x y .+ (.: ( ) x y 1+ = )
1.137. x y x y , :+ = + = I. (x y) II. 2 2 4+
(.: ( ) 2 2 2 x y2
+ = )
1.138. : I. 195 II. 1005
(.: . 6 2 6 2195 , 195 , 195 2 3, 195 2 34 4 += = = = +
. 6 2 6 21005 , 1005 , 1005 2 3, 1005 3 24 4+ = = = = )
-
137
1.139. :
I. 11 rad12
II. 29 rad12
(.: . 11 6 2 11 6 2 11 11 , , 3 2, 2 312 4 12 4 12 12
+= = = =
. 29 6 2 29 6 2 29 29 , , 2 3, 2 312 4 12 4 12 12
+ = = = + = )
-
138
______________________________________________________________ , -
, , :
2 2 1 = 2 2 1 = .
, :
22 2
2 21 ,
1 1 = =+ +
.
, ( ) .
1.140. 3 0 5 2
= < < 12 ,13 2
= < < - ( ) + ( ) .
. - :
22 2 2 2 23 9 16 4 1 1 1
5 25 25 5 + = + = = = =
1 , 45
=
2
2 2 2 2 212 144 25 1 , 1 1 13 169 169
+ = + = = = 5=
13 . 2 , 5
13=
-
139
:
( ) 3 12 4 5 16 ,5 13 5 13 65 + = + = + =
( ) 4 12 3 5 33 5 13 5 13 65 = + = + =
1.141. 3 15 5 , 2 , 2 2 17 12< < < < = = , ( ) +
( ) .
: 2
2 2 2 2 215 225 64 1 1 1 17 289 289 + = + = = =
1 , 8 ,17
= 15
1517 8 817
= = =
:
22 2
1 1 1 1 144 25 169 1691 5 11 144 14412
= = = = =+ ++
4 , 1213
= .
,
= = 5 12 5 12 13 13
= =
:
( ) 8 12 15 5 21 17 13 17 13 221
+ = = =
( )15 5
1408 12 15 51 17118 12
+ = = = + +
-
140
1.142. 0 x , y 2 2
< < < < 2x2
= , 3y5
= : . ( ) x y+ II. ( ) x y
(.: ( ) ( )2I. x y II. x y 710
+ = = )
1.143. 3 15 12 , 2 , 2 2 17 13< < < < = = :
. ( ) + II. ( ) (.: ( ) ( )21 171I. II.
221 140+ = = )
1.144. 12 30 x , y x ,y2 2 13 4
< < < < = = , : . ( ) ( ) x y x y= + II. ( ) ( )B x y x y= +
(.: 2079 896I. , II. 4225 4225
= = )
1.145. 1 4 3
+ = = ,
(.: 12
= )
1.146. , 0,4
3 10
10= 1
7= , :
. 13
= II. 324
= III. 2 4
+ =
1.147. = 20, = 16 = 5 ( )
( ) + (.: ( ) ( )56 63 ,
65 16 = + = )
20
16 5
-
141
_____________________________________________________________
, : 1 : , -
, .
2 : , - .
3 : - . -
, , , .
1.148. x,y\ :
( ) ( )2 2 2 x y 2xy x y x y+ + + = +
: ( )2 2 x y 2xy xy xy+ + =
2 2 2 2 x y 2xyxy 2 x y+ + = 2 2 2 2 2 2 x y 2xyxy x y x y+ + =
( ) ( )2 2 2 2 x 1 y y 1 x 2xyxy + + = 2 2 2 2 x y y x 2xyxy+ + =
( ) ( )2 2xy yx x y+ = +
-
142
1.149. ( ) ( )
+ + = + .
+ + += =+ ( ) ( )
+ + += =+
( )( )
+ += = ++
1.150. ( ) ( ) ( )2 2f x x 2 x x x= + + + -.
x -. :
( ) ( ) ( )2 2f x x 2 x x x= + + + = ( ) ( )2 2x x 2x x x x= + + + =
2 2 2 2 2 2 2 x x 2xx 2xx 2 x x= + + + = 2 2 2 2 2 x x x= + =
( )2 2 2 2 x x 1 = + = ( )2 2 2 2 2 2 2 2 x x x x 1= + = + = ( ) 2f x =
, f .
1.151. : 2 2
2 2 6 3 931 6 3
=
( )( )2 2 = + , : ( )( )( )( )
2 2
2 2
6 3 6 3 6 31 63 1 631 6 3
+ = = +
6 3 6 31 63 1 63
+ = = + ( ) ( ) 6 3 6 3 93+ =
-
143
1.152. : 1 1 3.4 4
=
1 = 1
=
: 1 1 1 11 14 4 4 4
= =
1 1 4 1 4 4 4 44
= = + =
( )4 1 4 3 4
+= = =
1.153. , 2x x 0,+ + = : ( ) ( ) ( ) ( )2 2 + + + + + + =
Vieta , 2x x 0,+ + = : 1
+ = = , 1
= =
( )1 1
++ = =
( ) ( ) ( )2 2 ( ) + + + + + + ( )2 + ( ) + .
:
( ) ( ) ( ) ( )2 2 + + + + + + =
( ) ( )( )( ) ( )
( )2
22 2
+ + += + + + = + +
( ) ( ) ( )2
21
1 2 2
= + = + + + + + ==========
( ) ( ) ( )2
21
1 = + + + + = + +
-
144
( )( )
( )( )
( ) ( )( )
2 22 22 2
2 2 2 22
2
1 1 1 1 1 1 1 1 1
1
+ = + = + +
( )( )
( )( )( )
2222 2 2
2 22 2
1 1
1 1
+ + + = = = + +
1.154. 3AH AB.4
=
. 2 .
4 3 = +
II. l 45 ,= 2x 7 x 2 0 = .
I. , ,
= =
( ) 3 3 1 4 4
= = =
= ( ) 1
= = +
2
3 1 4 4 3 31 1 4 4
= =
+ +
2 2
1 4 4 3 4 3
4
= =+ +
II. 45 ,= 1,= 21 1 ,
74 3 1= =+
2x 7x 2 0 = :
2 21x 7 x 2 1 0 x x 2 07
= = x 2 x 1= =
-
145
1.155. : . ( ) ( ) 2 2 + = II. ( ) ( ) 2 22 2
+ =
III. 2( ) ( ) ( )
+ = ++ +
IV. 2 2 22 2 3 x x x3 3 2
+ + + = V. ( ) ( ) ( ) ( ) 2 1 2 1 2 1 2 1 2 + + + =
1.156. :
I. 1 x 1 x x x 2x x 4 x x 4
+ + + + = +
II. 1 x 2x x1 x2x 3x+ =
III. 1 1 8x5x 3x 5x 3x
=+ +
1.157. : I. 2 2 2 2 4 4 53 = II.
2 2
2 2 4 5 3
1 4 =
III. 2 2
2 2 7 5 1 12 2 5 7
=
IV. ( ) ( ) 2 22 2 x y x y x y x y+ =
1.158. , 4, 5, 9, :
54 49 59 1 =
1.159. , ( ) ( )2x 6 2 x 1 6 2 0 + + = , ( ) + ( + ).
(.: ( ) 1 4
+ = + = + )
-
146
1.160. , , ,2 2
22x 2 3x 1 0 + = , -
:
( ) ( ) ( ) ( )2 2 2 2 3 3 = + + + + + + (.: = 3)
1.161. : ( ) ( ) ( )2 2g x x 2x x x= + + + .
(.: ( ) 2g x = )
1.162. : 2 2 2 3 3
= + + + .
(.: 32
= )
1.163. ( ) ( ) ( )2 2 x y x yA x, y2y
+ = D y / y ,2
= \ ] I. 2 2.= II. y.
III. ( ) A 2x A 2x .8
=
(.: . A 2x,= . x , 4 32
= + ] )
1.164. 2 .3
=
I. 2 .
3 2 = +
II. l oB 60 ,= : 2 x 2 x 0.
=
(.: . 1x 1 x3
= = )
-
147
1.165. : 8 5 3 853. =
1.166. 1 2 ... 0+ + + = 1 2 ... 0+ + + = , : ( ) ( ) ( )1 2 ... 0 + + + =
1.167. :
2x, x 4x.= = = :
I. 819
= II. >
1.168. : 3, 4, x= = = lK I. 1.+ = II. x.
(.: . 3x2
= )
2x
x
4x
x
4
3
-
148
______________________________________________________________
-. + + = A B 2 2 2 2+ + = . , , -
, - . -:
( ) ( )( ) ( )( ) ( )( ) ( )
+ = =+ = = + = = + = =
2 2 2 2
2 2 2 2
2 2 2 2
2 2 2 2
+ = = + = = + = = + = =
. ,
, , , , .
1.169. + + = , .2 2 2 2 2 2+ + =
,2 2 2 2 2 2 2 2
+ + = + + = + = :
1 2 2 2 2 2 2 2 2 2
+ = = +
-
149
1 1 2 2 2 2 2 2 2 2 2 2 2
= = +
+
2 2 2 2 2 2
+ + =
1.170. ,+ + = : 2 2 2 2 2+ + =
,+ + = + =
( ) ( ) + = = + =
, 2 2 2 2 2 2 + + =
( )( ) ( )2 2 2 2 21 1 1 2 + + =2 2 2 2 2 2 21 1 2 + + + = 2 2 22 2 = + +
1.171. 4
+ = , : ( )( )1 1 2, + =
,4 4
+ = =
4
= ( ) 1 4 14 1 1
4
+ + = + = =
(1) :
