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Algevra b lukeiou

### 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 .