R 9

R

K

{ }0,1,2,3,...= .

To { }..., 3, 2, 1, 0, 1, 2, 3,...= .

*

0 . { }* 1,2,3,...= .

Q

,

*

-

.

.. 4 1 1

2, 0,25, 0,333... 0, 3 ...2 4 3= = =

, .

. .. 2 1,41, 3 1,73, 3,14 ...

R

.

, .

R .1

R

, = : 0= =

:

i. =

ii. = =

iii. = = =

, :

0 ( 0 )

: i.

ii. =

iii.

: 0< < , .

.

, :

( ) = + 1

: , 0

= =

A - 110

H -

-

,

.

-

-

,-

,

.

, -

, -

.

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

( ) ( ) = =

-

.

, -

:

... , *

= + + +

... , *

=

1= : 1 =

1 = .

1. + +

2. , 0 >

3. , 0

iii v :

.

! .

! .

5. i. 0 0 =

ii. 0 0 0= = =

iii. 0 0 0

6. i.

= ,

= ,, , 0

ii.

= ,

:

= = ,, , 0

7. i. 0> 0 0> + >

ii. 0< 0 0< +

0>

ii. , 0 <

0

ii. , , : 1 1

< * :

i. = =

ii. < > +

) x 1 y xy 1 x y< < + < +

A6. ,,

0> > 0> .

:

+ > x x>

. 0 x 1< < > x x , = + , .

:

( ) x x x x x 0 x x 1 0+ > > >

x 0 x 0> >

x 1 x 1 x 1 0> > >

. ().

13

.2

2 . -

:

. . .

= , 2

1 1 : = =

0 , 0 1=

1

=

: = = .

, .

.. ( )2 2

3 3 = 3 3 .

o :

i)

+

= ii) : =

iii) ( )

= iv) ( ) =

v)

=

vi)

=

,

:

.

:

0 0

0

0 0. .. 19600 0=

0 .

.. ( )4

2 , ( )4

3

, 4

7

, .

14 A - 2

A7. : x, y

x 3, y 2> > : x 1 y 12 5 20 = .

A8. () :

( ) ( ) ( ) ( ) 1 2 3

1 1 1 1 0+ + +

+ + + = .

A9.

:

( ) ( )

( )

1 22 6

1 1

2,5 5 2 0,4

8 0,1

=

A10. ,y ,

( ) ( )x y3 5

2 1 12 2

= + .

A11.

:

i) ( )100

x 2

2 1

=

iii) ( )4x 6

1 1

=

ii) ( )x 1

5

1 1

+

=

iv) ( )2x 10

1 1+

=

A12. A * x, y, -

:

x y y xx y

y x

1

+ + +

=

A13. 0, x, y x x=

x y= :

y

1y

=

A14. 2 1 0, 0+ + = -

:

i) 3

1=

ii) 2007 2007 2+ =

2

-

( -

).

i) ( )

22 1 2

3 2 4

ii)

3 4

2 2

2:

5

i) ( )

22 1 2 4 2 4

7

3 2 4 3 2 4

= =

ii) 3 4 3 2 3 2

2 2 2 4 2 4

2 5 5 5:

2 5 2 2

= = =

1A ( )

43 2

=

( )3

4 5

2 3 2

=

3 2 :

( ) ( )3

4 123 3 2 3 2 36 24 12

= = =

2 212 3 15 10 20

2

2 3 2 13 26

= = =

36 24 12 26

3 2 16 24 38

20

:

= =

15

.3

. (

).

.

( )2 2 2 2 + = + +

( )2 2 2 2 = +

( )( ) 2 2 + =

( )( ) ( )2x x x x + + = + + +

( )3 3 2 2 3 3 3 + = + + +

( )3 3 2 2 3 3 3 = +

( )( )3 3 2 2 + = + +

( )( )3 3 2 2 = + + :

( )( ) 1 2 2 1 ... = + + + +

:

Euller

( ) ( ) ( ) ( )2 2 23 3 3 1 3

2 + + = + + + +

: 0+ + = 3 3 3 3= = + + =

Lagrange

( )( ) ( ) ( )2 22 2 2 2 + + + =

.

16 A - 3

A15. N :

i) ( ) ( )2 2

4x 3 3x 4+

ii) ( ) ( )3 3

3 2 3 2+

iii) ( )( ) ( )2

2 4x 2 4x 4x 1+

iv) ( )( ) 2x 1 x 1 x 5x + +

3

2

1 0 + + =

3

+ + ++ + + = + +

3

3

1 1 1 3

+ + ++ + + =

= + + + =

= + + + = + +

i) N :

( ) ( ) ( )

( ) ( ) ( )

3 3 3

3

+ + =

=

ii) :

( ) ( ) ( )3 3 3

x 2 3x 4 6 4x 0 + + =

i) E ( ) ( ) ( ) 0 + + =

Euller :

( ) ( ) ( )

( )( )( )

3 3 3

3

+ + =

=

ii) O

( ) ( ) ( )x 2 3x 4 6 4x 0 + + = :

( ) ( ) ( )

( )( )( )

3 3 3x 2 3x 4 6 4x

3 x 2 3x 4 6 4x

+ + =

=

:

( )( )( )3 x 2 3x 4 6 4x 0 =

x 2 0 = 3x 4 0 = 6 4x 0 =

x 2= 4

x3

= 3

x2

=

( )2 2 2 2

+ + = + +

0 1 1 1

0 + + =

( ) ( )2 2 2 2 2 + + = + + + + + ,

, :

0+ + =

0 :

1 1 10

+ + =

A16. N :

i) ( ) ( )2 2

x y x y 4xy+ =

ii) ( ) ( ) ( )2 2 22 2 5 2 3 4+ + + = +

iii) ( )( ) ( )( )3 4 4 2 + + = +

iv) ( ) ( ) ( )2 2

2 2 2 2 2 2 4 2+ + = +

17

.4

-

. ,

.

( )( )2

x 1 x 1 x 1 +

.. ( )2x 4 2 x 2 = . -

2.

1.

. 3 2xy 3xy 2x 6xy + . 2 22x y 6xy 8xy +

:

. x.

:

( )3 2 3 2xy 3xy 2x 6xy x y 3y 2 6y + = + . 2xy.

:

( )2 22x y 6xy 8xy 2xy x 3 4y + = +

2.

. xy 3x 2y 6 + . 3 23x x y 6x 2y +

. .

( ) ( ) ( )( )xy 3x 2y 6 x y 3 2 y 3 y 3 x 2 + = + = +

. ( ) ( )3 2 3 23x x y 6x 2y 3x x y 6x 2y + = + =

( ) ( ) ( )( )2 2x 3x y 2 3x y x 2 3x y= + = +

3. ( )

( )

2 2 2

2 2 2

2

2

+ = + +

= +

3.

i) 2 225x 20xy 4y + ii) 2 29x 24xy 16y +

iii) 2 216x 40xy 25y+ +

-

1 :

(

)

2 :

(

)

3 :

18 A - 4

( ) ( ) ( )2 2 22 2

25x 20xy 4y 5x 2 5x 2y 2y 5x 2y + = + =

ii) ( ) ( ) ( )2 2 22 2

9x 24xy 16y 3x 2 3x 4y 4y 3x 4y + = + =

iii) ( ) ( ) ( )2 2 22 2

16x 40xy 25y 4x 2 4x 5y 5y 4x 5y+ + = + + = +

3. ( ) ( )2 2 = +

3.

i) 2 225x 4y ii) 4 816x y

i ) ( ) ( ) ( )( )2 22 2

25x 4y 5x 2y 5x 2y 5x 2y = = +

ii) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )( )

2 24 8 2 4 2 4 2 4

222 4 2 2 4 2 2

16x y 4x y 4x y 4x y

4x y 2x y 4x y 2x y 2x y

= = + =

= + = + +

3. - ( ) ( )

( ) ( )

3 3 2 2

3 3 2 2

+ = + +

= + +

3.

i) 3x 64 ii) 38x 27+

i ) ( )( ) ( )( )3 3 3 2 2 2x 64 x 4 x 4 x 4x 4 x 4 x 4x 16 = = + + = + +

ii) ( ) ( ) ( ) ( )( )33 3 2 2 2

8x 27 2x 3 2x 3 2x 2x 3 3 2x 3 4x 6x 9 + = + = + + = + +

3. 2 ( ) ( ) ( )2x x x x + + + = + +

, ,

2 , 2x x , 0+ + .

: 2x x + + ,

: ( ) ( )( )2x x x x + + + = + + , , -

.

19

3.

i) 2x 5x 6 + ii) 2x 4x 3+ + iii) 2x x 2

i ) 2, 3 ( ) ( )2 3 5 + = ( ) ( )2 3 6 = .

( )( )2x 5x 6 x 2 x 3 + =

ii) O 1,3 1 3 4+ = 1 3 3 = .

: ( )( )2x 4x 3 x 1 x 3+ + = + +

iii) O 2,1 ( )2 1 1 + = ( )2 1 2 = .

: ( )( )2x x 2 x 2 x 1 = +

4.

i) 2

2x 18 ii) 2 2x 2xy y 9 + +

iii) 2 2x y x 2xy y + + iv) 9 7 5 3x x x x +

i) ( ) ( ) ( )( )2 2 2 22x 18 2 x 9 2 x 3 2 x 3 x 3 = = = +

ii) ( ) ( ) ( )

( ) ( ) ( ) ( )

2 22 2 2 2 2x 2yx y 9 9 x 2xy y 9 x y 3 x y

3 x y 3 x y 3 x y 3 x y

+ + = + = = =

= + = + +

iii) ( ) ( ) ( ) ( ) ( )( )22 2

x y x 2xy y x y x y x y 1 x y x y 1 x y + + = + = + = +

iv) ( ) ( ) ( )

( )( ) ( )( )( )

( ) ( ) ( )

9 7 5 3 3 6 4 2 3 4 2 2

3 2 4 3 2 2 2

2 23 2

x x x x x x x x 1 x x x 1 x 1

x x 1 x 1 x x 1 x 1 x 1

x x 1 x 1 x 1

+ = + = =

= = + =

= + +

4 ( )

i) 2 2x 2xy 3y+ ii) 4 2

x 5x 9+ +

i ) 23y 2 2y 4y

( ) ( ) ( )( )

( )( )

2 22 2 2 2 2x 2xy 3y x 2xy y 4y x y 2y x y 2y x y 2y

x 3y x y

+ = + + = + = + + + =

= +

ii) 2x

4 :

20 A - 4

( )

( )( ) ( )( )

24 2 4 2 2 2 4 2 2 2 2

2 2 2 2

x 5x 9 x 5x 9 x x x 6x 9 x x 3 x

x 3 x x 3 x x x 3 x x 3

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

= + + + = + + +

A17. :

i) ( )4x x 2y x 2y +

ii) 2 2 2 25 x y x 5 y+

iii) 2 236x 49y

iv) 4 416x y

v) 225x 20x 4 +

vi) 3