Σημειώσεις Τριγωνομετρίας Β΄ Λυκείου

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Άλγεβρα Β΄ Λσκείοσ Τριγωμομετρία Στέλιος Μιταήλογλοσ Εσάγγελος Τόλης www.askisopolis.gr
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Transcript of Σημειώσεις Τριγωνομετρίας Β΄ Λυκείου

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  • 11

    1.

    1.1. ,

    . - , .

    (0 < < 90)

    : AB

    =

    =

    =

    =

    , 90

    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 + . , 360 + . -

  • 12

    z , .

    :

    ( 360 ) + = ( 360 ) + = ( 360 ) + = ( 360 ) + =

    , 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 = + , Z 0 360 < .

    .

  • 15

    rad

    rad

    , :

    . :

    + = = + = +

    : -

    .

    : -

    + .

    2. :

    I. 2790 II. 253

    III. 1776

    I. 2790:360 7 270.

    2790 7 360 270= + .

    : 2790 270 1= = 0 02790 270= =

    2790 270 0= = 2790 270 0= =

    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 33 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 06 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

    22x 0 x(x 1) 0 x(x 1) x 0

    x 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

    )

    9. 900 rad4

    . rad .

    (.: 3 x , y8 8

    = = )

    10. :

    I. 1350 II. 27 rad III. 913

    IV. 20064

    (.: I. 1350 270 1,...= = , 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= = + 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. : 2

    22

    1

    =+

    2 21

    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 0

    1 11 0+

    0

  • 29

    x, 3 2 0+ = 32

    = .

    32

    = 3 12

    = +

    1K2

    =

    33. x = , :

    2 22

    ( 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 = +

    . 221 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= + . , () - . - . : ( 3030 ) 3030 (8 360 150 ) = = + =

  • 36

    1150 (180 30 ) 302

    = = = =

    ( 3030 ) 3030 150 = = = 3(180 30 ) 30

    2= = =

    ( 3030 ) 3030 150 = = = 3(180 30 ) ( 30 ) 30

    3= = = = ,

    ( 3030 ) 3030 150 (180 30 ) 30 3 = = = = = .

    . 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= + + + + + + + + 160 180+ + . 41 42 ...48 49= .

    . :

    (0 180 ) (20 160 ) (40 140 ) (60 120 )= + + + + + + + +

    (80 100 )+ + =

    [ ]1 ( 1) 20 (180 20 ) 40 (180 40 ) = + + + + + +

    ( )60 (180 60 ) (80 100 ) + + + + =

    0 (20 20 ) (40 40 ) (60 60 ) (80 80 ) 0= + + + + = .

    . , :

    41 42 ...45 ...48 49= =

    (41 49 ) (42 48 )...(44 46 ) 45= =

    41 (90 41 ) 42 (90 42 ) ... 44 (90 44 1 = =

    (41 41 )(42 42 )...(44 44 )= = 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+ + + + + = II. 0 1 2 ... 359 0+ + + + = .

    66. 4 x 5< < , :

    7 x x 2(9 x) 22 2

    + > +

    .

  • 43

    67. 5 x x 212 12

    + + =

    ,

    2 25 x x12 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 . 125

    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 ,02

    3 ,02

    ,02

    +

    y y (0,1).

  • 49

    :

    f (x) x=

    : A x / x ,2

    = +

    , -

    . : = , (x ) (x ) x+ = = , -

    . , 2

    + ,

    ,2 2

    .

    : - :

    : . : .

    : x2

    = x2

    = .

    : 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 ,02

    ,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= 2T3

    = . :

    [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= + = .

    32x2 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 3T3

    = , 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

    -

    .

    .

    x2

    = x2

    = .

    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 212

    = = , 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

    :

    '' ( ) ( )f

    1 2 l 2f x f x x x

    < < ''

    '' ( ) ( )f

    1 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. 1x4

    2x4

    II. 1 x3 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

    .

    ( ) ( ) max1 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 , 14

    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 22

    =

    . ( ) xf x 22

    =

    2. - ( )f x x= ( )g x x= [ ]0,2 . ( ) ( ) [ ]f x g x , x 0,2 ,= :

    . 3x x4 4

    = = . 3x x2 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 12

    2 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 22

    = = + ,

    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 22

    = = + ,

    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 4

    2 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 12

    + = , 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 1,223 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= = = + = 2

    2 21 1 2 x 0 x x2 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 53 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] .

    (.: 5x2

    = )

    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 22

    = = )

    1.116. 20044x + 20062x = 0 [ ]0,2 .

    (.: 3 5 7x , x ,x ,x4 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. ( )2

    22 3x 1h x2 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 -

    ( )1tT t 4 48

    = , t

    0 .

    I. .

    II. 24.

    III.

    ( )2t 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 x1+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 y

    2 = + = +

    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 3

    175 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 2

    2 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 .=

    (.: 12

    = )

    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 y

    2+

    = )

    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 =

    . ,

    : 2

    2 22 2

    1 , 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 , 513

    =

  • 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 8

    17

    = = =

    :

    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 1010

    = 17

    = , :

    . 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 + + + + + + =

    ( ) ( )( )

    ( ) ( )( )

    22

    2 2

    + + += + + + =

    + +

    ( ) ( ) ( )2

    21

    1 2 2

    =+

    = + + + + + ==========

    ( )( ) ( )22

    1 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. 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 2

    2 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 2

    2 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 y

    A x, y2y

    + =

    D y / y ,2

    =

    I. 2 2.=

    II. y.

    III. ( ) A 2x A 2x .8

    =

    (.: . A 2x,= . x , 4 32

    = + )

    1.164.