Algevra b Lykeioy Biblio Mathiti

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Transcript of Algevra b Lykeioy Biblio Mathiti

  • :

    . .

    : 1991 : 1992, 1993, 1994, 1995, 1996, 1997, 1998, 2012 .

  • . , . , . . , .

  • . (2010) , - . , . , . , . . . : , (2010), - . , . , . , . . . - (2010), - (2010).

    :

    1 , -, . , .

    2 , , , - .

    3 - . , -, . - .

    4 . 5 , .

  • 1: 1.1 .......................................................................9 1.2 ..............................................................24

    2: 2.1 -- ............................30 2.2 - ...........................40

    3: 3.1 ..................................................49 3.2 ...........................................60 3.3 1 ......................................................65 3.4 ..................................................73 3.5 ............................................83 3.6 ...........................89 3.7 2 .......................................97 3.8 ...................103 3.9 f(x)=x+x ..................................................108 3.10 .......................................................................114

    4: - 4.1 ..................................................................................128 4.2 ................................................................132 4.3 .....................................140 4.4 .......150

    5: 5.1 ....................................................................160 5.2 ...................................................................................173 5.3 ..............................................................181

    .........................................................193 - ...................................195

  • 1

    1.1

    x + y =

    x + y = , 0 0, , . : :

    0, , :

    x y y x

    y x ,

    + = = +

    = +

    -

    =

    y'y

    .

    :

    9 0, (. ),

    x+y=, =0

    0 x

    y

    x+y=, 0

    x

    y

    0

  • 10 1:

    9 = 0 , y =

    x'x y'y

    (. ).

    = 0 ( 0 ),

    - y'y

    x'x

    .

    :

    9 x 2y 2 =

    1

    2 =

    y'y 1 .

    9 y=2 x'x - y'y 2 .

    9 x=2 y'y - x'x 2.

    .

    1y x 1

    2=

    x+y=, =0

    x

    y

    0

    y= x1y

    0 2-1

    x

    12

    y=2

    x

    y

    2

    0

    x=2

    x

    y

    0 2

    x+y=, =0

    x

    y

    0

    y= x1y

    0 2-1

    x

    12

    y=2

    x

    y

    2

    0

    x=2

    x

    y

    0 2

    x+y=, =0

    x

    y

    0

    y= x1y

    0 2-1

    x

    12

    y=2

    x

    y

    2

    0

    x=2

    x

    y

    0 2

    x+y=, =0

    x

    y

    0

    y= x1y

    0 2-1

    x

    12

    y=2

    x

    y

    2

    0

    x=2

    x

    y

    0 2

    x x .

    = =

  • 1.1 11

    , (4,-1) x 2y 6, = 4 2( 1) 4 2 6. = + = , , (16,5), ( 10, 8)

    1k, k 3 , k2

    .

    2 x 2 x+y= x+y= , - , , 2 x 2

    x y

    'x ' y '

    + = + =

    . . . . : .

    () (') , .. (),

    ( ) ' ( ') + , () - 0, (') '.

    ( ) ' ( ') + () (').

    - 1o :

    9 0, : = = 9 = = , + = +.

    , , :

    x 2y 6 (1)

    3x 4y 8 (2)

    = + =

  • 12 1:

    , ( )

    , .. (1) x.

    x

    3(2y 6) 4y 8 6y 18 4y 8

    10y 10

    y 1

    + + = + + = = =

    y :

    x 2( 1) 6 4= + =

    (4, 1).

    ,

    .

    - , .

    , x = 4 y 1,= :1 : 4 2(1) = 6

    2 : 3 4 4( 1) 12 4 8 + = =

    ( ) , :

    x 2y 6

    3x 4y 8

    = + + =

    x 2y 6

    y 1

    = + =

  • 1.1 13

    , - , : . :

    x 2( 1) 6 x 2 6 x 4 = + = = . (4, 1) ( - ).

    2 x 2

    . , - .

    ,

    , , . 2 x 2 . , 2 x 2 , , , .

    3x 6y 18

    3x 4y 8

    + = + =

    x 2y 6 ( 3)

    3x 4y 8 1

    = + =

    3x 6y 3x 4y 18 8 10y 10 y 1 + + + = + = =

    x 2y 6

    3x 4y 8

    = + =

    x y

    'x ' y '

    + = + =

    A(4, 1)

    3x+4y=8 x2y=6y

    O x

  • 14 1:

    2 x 2 -.

    :

    9 2x 3y 3

    4x 9y 1

    + = =

    2y x 1

    34 1

    y x9 9

    = + =

    ,

    , .

    ,

    (1, 0,3) .

    ,

    - 1(1, )

    3.

    9 x 2y 3

    2x 4y 5

    + = + =

    1 3y x

    2 21 5

    y x2 4

    = + =

    , ,

    .

    9

    y 1 2x

    4x 2y 2

    + = =

    y 2x 1y 2x 1

    = =

    , .

    (k,2k 1), k .

    y

    x0

    21

    1

    1

    y

    x0

    1

    2

    y

    x0

    2

    1

    ,

  • 1.1 15

    , 2 x 2 :9 9 9 .

    2 x 2

    - 2 x 2 .

    0 0. - :

    1 2 -

    1

    =

    2

    '

    '

    =

    '

    '

    , ' ' 0 , 1 2

    ,

    ' ' ' 'x x x

    ' ' ' '

    ( ' ' )x ' '

    ' 'x

    ' '

    + = + =

    =

    =

    :

    ' ' ' ' ' 'y

    ' ' ( ' ' )

    ( ' ' )

    ( ' ' )

    + + = + =

    =

    x y

    'x ' y '

    + = + =

    1

    2

    y x ( )

    ' 'y x ( )

    ' '

    = + = +

  • 16 1:

    ' 'y

    ' '

    =

    , ,

    '

    '

    =

    , - =0 , 1 2

    , . . =0 ' =0.

    :

    ' ' 0

    ' ' 0 = .

    ' ' ,

    :

    .

    D, x , : .

    , D, - y , :

    .

    ' ' ' '(x, y) ,

    ' ' ' '

    =

    x y

    'x ' y '

    + = + =

    ' ' ' '(x, y) ,

    ' ' ' '

    =

    D' '

    =

    D ' '' '

    = =

    xD ' '' '

    = =

    yD ' '' '

    = =

  • 1.1 17

    - , , :

    :

    9

    ,

    .

    : .

    , (x, y) (4, 1)= .

    9

    - . 2,

    2x 3y 40. =

    .

    x 2y 6

    3x 4y 8

    = + =

    1 2D 1 4 3( 2) 4 6 10 0

    3 4

    = = = + =

    x

    6 2D 24 16 40

    8 4

    = = + =

    yD 10y 1D 10

    = = =

    y

    1 6D 8 18 10

    3 8= = =

    xD 40x 4D 10

    = = =

    2x 402x 3y 40 y

    3

    = =

    D0, , (x,y)

    D=0, .

    xDxD

    =

    x y

    'x ' y '

    + = + =

    yDyD

    =

    2 3D 2( 6) 4( 3) 12 12 0

    4 6

    = = = + =

    2x 3y 40

    4x 6y 80

    = =

    2x 3y 40,

    2x 3y 40

    = =

  • 18 1:

    9

    .

    .

    1

    - , . , . , . , .

    D, Dx, D

    y. :

    2k 40k, , k3

    3x y 11

    9x 3y 6

    + = + =

    3 1D 9 9 0

    9 3= = =

    3x y 11

    3x y 2

    + = + =

    2

    x y 1

    x 2y

    = =

    22

    x

    2 2 2 2y 2

    1D 2 ( 2)

    2

    1 1D 2( 1) 2

    2

    1D ( 1) (1 1) (2 )

    = = + =

    = = + =

    = = = + =

  • 1.1 19

    , D = 0 . :

    :9 D0, 0 2, (x, y), :

    .

    , 0 2 ,

    .

    9 D = 0 , = 0 = 2 , . : = 0 ,

    . = 2 ,

    . , ,

    3 x 3 x+y+z=0 , - , , , . . 2x+3y+z=6 - (2, 1,5) , 22+3(1)+5=6.

    D 0 ( 2) 0= =

    xD 2 ( 2) 1xD ( 2) ( 2)

    = = = =

    2 2

    yD (2 ) ( 2)yD ( 2) ( 2)

    = = = =

    0x y 1 y 1

    0x 2y 0 y 0

    = = = =

    2 =0 =

    2x y 1 2x y 1

    4x 2y 2 2x y 1

    = = = =

    .( )k,2k 1 , k

    1 ,

    2x y 1 y 2x 1 = =

  • 20 1:

    :

    1 1 1 1x y z + + = , 2 2 2 2x y z + + = 3 3 3 3x y z + + = , - , , 3 x 3

    - 2 x 2 .