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20-Oct-2015

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Μαθηματικα Γ γυμνασιου σημειωσεις και ασκησεις

Transcript

• ( + )2 = 2 + 2 + 2

x2 + x + = 0 , 0

x = 2 4

2

• 1

.

-

-

.. ...

• 1: 3

1.

1.1. ;

: :

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

+ 0 = .1 =

+ () = 0 1 = 1 , 0

.( + ) = . + . .0 = 0

1.2. .

: : :

2 : = ...

= 1 1 = . 0, 0 = 1 = 1

.

( ) = ( ) , , 0

- :

1. . = +

2.

=

3. . = (.)

4.

= ( ) 5. () =

1.3. ;.

: , , , . = x , x2 = ( )2 = . 0 = 0.

1.4. = , , 0.

: : 1 : ( )2 = ( ) ( )2 2 = . 2 : ( )2 = .

• 4 1:

( )2 = ( )2 = .

1.5.

=

0 > 0. : :

1 : 2

= ( )( )

2

2

=

2 : 2

=

2

= 2

=

.

1.5.1. > 0 > 0 + +

+ = + 0. ___________________________________________________________________________________________ 1.6.

= 2003 6 10 2(4 3)3( ) 3( )x x y

x z y z +

+ + 2(x +13 ) 2y ,

x + y = 2003. [. =-2003]

1.7.

= ( ) 115 172412 18 3236 [. =1]

1.8. :

(55 + 55 + 55 + 55 + 55)2.(22 . 22 . 22)2 10.

[. 1012] 1.9. :

(33)3 : 33 + 33 . 33 + 333

3

3. [. 37]

1.10. 3 42 2x yx y+ = 5 -

= 2 22x y

xy+ .

[. =3] 1.11. x = 0,03 y = 310 , -

= 37

3 5 211( ) :

xx yy

. [. =1000]

1.12. x = 2 y = 101 , = 51.(x1 y1 2x1y)

[. - 4825 ]

1.13. 31 21 13 7 4+ + + + = 6. 1.14. 24 6 9 16 = 12. 1.15. : ) 2 3 24 ) 1 12 453 5 4

[. )12 )3] 1.16. :

) 20 615 2

)

[. )2 )] 1.17. :

16 + 2 81 82

+ 2 2 8 7.

[. 14=.7] 1.18. , , - :

2 249 2 4 + + . [. 7]

• 1: 5

1.19. :

= 4 122+ = 8 202

[. =2+ 3 , =4- 5 ]

1.20. x, y , - : 5x 3xy 2y 3x y 3 3 3x y .

1.21. - 12 cm 3 cm.

[. =3cm2] 1.22. :

i) 6 11 6 11 + ii) 19 3 19 3+

[. i)5 ii)4] 1.23. = 3 + 2 = 12 8 , - : + , , ..

[. +=3 3 - 2 , -=3 2 - 3 , =2] 1.24.

i) = 3 3 273+ ii) = 8 2 5 2

8 8+ +

[. i)=6 ii)= 12 ]

1.25. :

3 53 5+ +

3 53 5+

[. 7] 1.26. :

x 3 3 = 5 (1x) [. x=1]

1.27. :

15 1

x =

15 1

x++

[. x= 5 ] 1.28.

5 7 3 13 : 31 3

+ + 6. 1.29. :

5 3 32 1560 :3 5 15 + = 1.

1.30. 27 cm 12 cm 2 3 cm.

5 cm; [. 3 ]

3.1. ( )212

+ ( )21

2 2+ = 1.

3.2. , , .

32

9 2 50

8

3.3. 10 10

4 118 48 4

++

[. 16]

• 6 1:

2.

-

2.1. ;

: , .

2.1.1. , : 5x , 4 1 , 3 2 + 9 , 2,64 + 3 2.2. ;

: , .

2.2.1. : = 2 32 5 8

3(2)2 5(2) 8 = 3.4 + 10 8 = 12 + 10 8 = 14. 2.3. ;

: .

2.3.1. : 3 , , 35 x

2 , 2 34 , 4,35629.

2.4. ;

: .

2.4.1. : 32 25 , 925 , 2,725 .

2.5. ;

: - .

2.5.1. : 52 , 2 , 22 :

52 + 2 + (22) = (5 + 1 2)2 = 42. 2.6. ;

: .

2.6.1. : 3x2 1 , 532 + 3,222 , 62 5 3 + 43 25.

• 4: 7

2.7. ; : .

2.7.1. : 22 , 34

54 , 5x :

(22).( 34 54).(x) = 2( 34 )(1)

2.5.43.x) = 32 479x.

2.8. ;

: , , .

2.8.1. : (1) 1252 : (33) = 1252 . 31

3 = 5 2

312

3 =

25

312

3

= 42.

(2) (10x2y2) : (5xy4) = (10x2y2) . 41

5xy = 2 2

410

5x y xy

=

22

4105

yx x y

= 2

22xy

.

2.8.2. :

. ___________________________________________________________________________________________ 2.9.

i) 3 32 ii) 1,4x4y5 iii) 22x

y

iv) 4x6-3 v) x 1x vi) (1+ 5 )

2.10. x+y = 2 -: x(x3) + y(y3) + 2xy

[. 10] 2.11. ) 3x2y 7x2y 5x3y5 + 7x2y + + = 5x2y ) 0,632 + 32 35

32 + + 423 =

232 2.12.

(5+4)43 ;

2.13.

2x2 , 12 xy2 , x2y , yx2 , 2x2y2 ,

2

3x , 12 x

2

. 2.14. :

i) 2x2y 3x2y x2y + 21 x2y

ii) 13 xy3 xy3 23 y

3x + 35

6xy

iii) 43 324 2324 1,25342 + 1 45

324

2.15. 3x23y+2 , 2x3y4. , - .

[. =3, =1] 2.16. , : 2+1x2 + 53x+2

2.17. , : 32x2y+7 + 22x+1y12

2.18. :

i) 2xy2 . (3x2y4) ii) ( )323 . ( )234 y iii) 2 325 x y

. ( )yx 35815 iv) 2xy2 . (4x3y) . (x4y6) v) xy . (22) . 435 . (

21 32)

2.19. :

i) ( )3212 . (235) . (34)2 ii) (2x3y2)2 . ( )32 412 x y

• 8 1:

iii) (323)3 . (42)2 . ( )25 613 v) ( ) ( ) ( )3 2 44 2 5 4 3 6 23 2 x y xy ax y

2.20. :

i) 12x8 : (6x5) ii) (2497) : (653) iii) (1823) : (93) iv) 10xy4z3 : (15x3y4z5)

2.21. :

i) [2x3y4 . (6x2y)] : (4x5y7) ii) [(423x) . 5x3) . (

21 6x)] : (14115x4)

iii) (32 3x5)3 : ( 9

4 4x4)2

iv) ( ) ( ) ( )2 3 22 3 4 21 2 3: :3 3 4

• 3: - & 9

3.

-

3.1. ;

: , .

3.1.1. : 52 3 +6 + 2 + 9 2 + 3 2 + = 42 2 + 14 + 3. ___________________________________________________________________________________________ 3.2.

i) 4x2 3x3 + 5x 2x2 + 7 x 2 + 3x2 + 4x3 ii) 5x4 3x2 + 2x 7x4 3x3 1 2x + 4 5x2 iii) x2 + 2x + 3x2 x + 5 4x iv) 23x 62x2 + 53x + 3 24 + x4

3.3. = x23x+1 , = x2+2x+5 = 2x2x3 i) + ii) + + iii)

3.4. :

i) 3x2 [(5x3x) + 4x2 (2x2+6)] + (2x25x) ii) 53 (22+3) + [ (3224) + (2+6)]

3.5. :

i) (2x3) [2x (x22)] {x2[3x+4(x21)]} ii)3x2{x2[x(1x3)]+2x}{2x3 + [x+(x23)3x2]1}4

• 10 1:

4.

4.1. ;

: , - .

4.1.1. : (22).(3 12

32 + 4) = 632 232 + 82.

4.2. ;

: , .

4.2.1. : ( 2).(2 3 4) = 3 32 4 22 + 6 + 8

= 3 52 + 2 + 8 4.2.2. : ,

. ___________________________________________________________________________________________ 4.3. :

(x22y)3y + (xy+y2)(x) + (x+y)(2xy) (x+3)2y2. x = 2 y = 1.

4.4.

2x(2+1) (x+1)(2) x(2+2) + , x.

4.5. . ) (2x+3)(x2+x1) (x21)(x+2) 2x3 , x = 2. ) (x2y2xy2)(2xy) 2x3(x+y) (xy)(2y3), x = 1, y = 2.

4.6. : ) 3x(x21) 4x2(x+2) 3x + 4(x21) ) 5x2(x32x2+4) +(12x)(4x3) x(x1) 2x ) 2(2+2) 3 ()(3) 42 ) 3[x2 (x+4) 3] 2x2[x + (x2)] 5

) 2{ [2(+4)] + 2} 3(22) 4.7. , : ) * .(427+8) = 283 492 + 56 ) * .(3x2+8x7) = 36x5 + * * ) 523( * 92 + * ) = 2057 * + 49

4.8. : ) (3x+1)(2x3) ) (3x22x+4)(2x5) ) (x+2)(x2)(x+5) ) (x+1)(x+2)(x+3)

4.9. : ) (2)(1)(23) ) (3)(4+1)(2) ) (x1)(2x5)(x2+5x1) ) (+3)(2)(221)(42)

4.10. :

• 4: 11

i) (x3)(1x) + (x2x+2)(x2+x2) ii) (+1)(2) (22+3)(3+22)

4.11. = 2+1 , = +2 = 1 - : . , . , . . + . ..

4.12. = 4x1 , = x2 = x2+x1 - : K. , . , . K. .(. .).

4.13. :

(x2y)(4x3y) ( ) ( ) ( )211 372 5 2 23 3x y x y y xy 4.14. x , 2x 2

x .

x. .

[. 34 ]

• 12 1:

5.

5.1. ;

: .

5.1.1. :

(i) ( 2) = 2 2 (ii) (x + 2)(x + 5) = x2 + 7x +10 (iii) 2 + 2 = ( + )2 2

5.2. ( + )2 = 2 + 2 + 2 .

: : ( + )2 = ( + )( + ) = 2 + + + 2 = 2 + 2 + 2 .

5.2.1. : (i) ( + 3)2 = 2 + 2.3 + 32 = 2 + 6 + 9.

(ii) (5x + 4y)2 = (5x)2 + 2.5x.4y + (4y)2 = 25x2 + 40xy + 16y2. (iii) (23 + 14

2)2 = (23)2 + 2.23. 14 2 + ( 14

2)2 = 46 + 32 + 116 4.

5.3. ( )2 = 2 2 + 2 .

: : ( )2 = ( )( ) = 2 + 2 = 2 2 + 2 .

5.3.1. : (i) ( 11)2 = 2 2.11 + 112 = 2 22 + 121.

(ii) (6x2 + 7)2 = (7 6x2)2 = (7)2 2.7.6x2 + (6x2)2 = 492 84x2 + 36x4.

(iii) 24

342 =

24342

+ = 24

342 + =

24

2 + 2

. 42 .43 + (43)2

= 8

4 + 443 + 1626.

• 5: 13

5.4. ( + )3 = 3 + 32 + 32 + 3 .

: : ( + )3 = ( + )( + )2 = ( + )(2 + 2 + 2) = 3 + 22 + 2 + 2 + 22 + 3 = 3 + 32 + 32 + 3 .

5.5. ( )3 = 3 32 + 32 3 .

: : ( )3 = ( )( )2 = ( )(2 2 + 2) = 3 22 + 2 2 + 22 3 = 3 32 + 32 3 .

5.6. ( + )( ) = 2 2 .

: : ( + )( ) = 2 + 2 = 2 2 .

5.6.1. : (i) (9x + 2y)(9x 2y) = (9x)2 (2y)2 = 812x2 42y2.

(ii) (35 + 32)(35 + 32) = (32 35)( 32 + 35) = (32)2 (35)2 = 64 910.

5.7. (i) 3 3 = ( )(2 + + 2) . (ii) 3 + 3 = ( + )(2 + 2) .

: (i) 2 : ( )(2 + + 2) = 3 + 2 + 2 2 2 3 = 3 3 .

(ii) 2 : ( + )(2 + 2) = 3 2 + 2 + 2 2 + 3 = 3 + 3 .

___________________________________________________________________________________________ 5.8. : ) (x + 5)2 ) (2x 3)2 ) (3 7)2 ) (3xy + 2)2 ) (x2 4y)2 ) (23 2)2

5.9. : ) (

23 x2 + 4xy)2 ) (7

23 2)2

) (3 12 )2 ) ( 2

34a

+ 23)2

) ( 56 x2y4 2xy )

2 ) ( 12 23 2

6

)2

5.10. :

) (2 + 24)2 ) (2x3 + 2

4y )2

) ( 23 2x3 34

5)2 ) ( 14 x2y3 2

116x y

)2

5.11. ( + )2 + 2( + ) . 5.12. - - : ) x2 + 2x + ) 4x2 + 4xy + ) 92 6 + ) x2 + 4y2 +

• 14 1:

) 252 + 42 + ) 812 + 492 ) 40xy + 16x2 + ) 60 + 92 + ) 1232 + 44 +

5.13. - - : ) 2 + ) 2 + 14 +

) x2 + 65 x + ) 496 + 8 +

) 94 x2 6xy + ) 254 a

2 5 +

5.14. (+)2 = 2 + 2 + 2, - , . 5.15. - = (2+3)2 (23)2 24, . 5.16.i) 2 + 2 2.

ii) : 2 + 2 + 2 + + .

5.17. : ) (4x+3)(4x3) ) (+2)(2) ) (523)(5+23) ) (2x2y43a3x)(2x2y4+3a3x)

5.18. : ) (7a+4)(47) ) ( 25 x

2y3 2)(2 + 52 x2y3)

) (4x3 92y)(4x3 92y) ) ( 2 x3 a4)(a4 2 x3)

5.19. : ) (x+y1)(x+y+1) ) (2x+y3)(2x+y+3) ) (2+3)(2+3) ) (x2+x 2 +1)(x2x 2 +1) ) (x+y)(+x++y)

5.20. = 2 1 , = 2 + 1 , :

i) . ii) + iii) 2 2( )

+

+ [. i)1 ii)2 2 iii)6]

5.21. : ) (x + 4)3 ) (2x 3)3 ) (2x 3y)3 ) (3xy + 2)3 ) (x2 + 2y)3 ) (2a3 x2)3 5.22. : ) (+3)2 (2)2 52 ) (x)2 ()(+3x) (x)2 ) (x3+2y2)2 (y2+2x3)2 + (x32y2)(x3+2y2)

) (2x+3)2 + (2x3)2 + (2x+3)(2x3) 3(x5)2 5.23. : ) (x+3)2 +(x3)2+(x2)2 (x+2)2 + (x+3)(x3)

(x+2)(x2) ) (2x+5)2 (x5)2 + (3x1)2 (2x+1)2 (2x3)(2x+3) ) (2x+1)2 + (2x+1)(2x1) (x+3)(x3)

(x3)(x3) ) (x2+1)2 + (2x23)2 (3x2+4)2 + (x22)2 +

(x23)(x2+3) 5.24. :

(3x+2)(3x2)(x1) (2x1)3 x(x2)(x5) + 5 = 10(x1)2.

5.25. = 2 4 15 = 6 10 :

i) 2 2. ii) -.

5.26. = 7 2 6 7 2 6 + = 3

4 14 + 3

4 14+ . -.

[. =30, =30] 5.27. - , , 1. .

5.28. = 10 + 5 2 : x2 20x + 50 = 0.

5.29. x2 6xy + 11y2 8y + 8 = 0 ) : (x 3y)2 + 2(y 2)2 = 0. ) (x,y) .

[. )(6,2)] 5.30. : ) 3 3 = ( )3 + 3( ). ) 3 + 3 = ( + )3 3( + ).

5.31. : ) (x1)3 (3x+2)3 x(x2)(x+2) ) (x+y)3 y(xy)(x+y) x(xy) ) (x+2)3 3x(x1)2 + (x1)(x+1)(x2)

5.32. :

i) (x+1)3 + 3(x+1)2 (x1) + 3(x+1)(x1)2 + (x1)3 ii) (xy)3 3(xy)2 (x+y) + 3(xy)(x+y)2 (x+y)3 iii) (+)3 ()3 3(+)2 () + 3(+)()2

• 5: 15

iv) 35

2 5yx +

352 5

yx + + 325 5

2 5 2 5y yx x + +

325 5

2 5 2 5y yx x +

5.33. : ) (+)2 ()2 = 4 ) (2+2)2 (2)2 = (22)2

) ( )22+ ( )22 = ) (x+3y)2 (x2y)2 5y2 = 10xy ) (2+4)(x2+1) (x+2)2 = (2x)2 ) (2+2)(x2+y2) = (x+y)2 + (yx)2

5.34. :

) 2( 1)

2 +

2( 1)2

= 3

5.35. > , 2 + 2, 2 2 2 .

5.36. > 1 , , 2 12

2 12

+ . 5.37. : ) (+)2 + (+)2 + (+)2 2 2 2 = (++)2 ) (x2+y2+xy)2 = x2y2 + (x+y)2(x2+y2) ) (++)2 + (+)2 + (+)2 + ()2 =

4(2+2+2) 5.38. :

(3x2+2)2 (2x2+3)2 2(2x21)3 + 27x2 = 5(x41) (x22)(4x2+1)2

5.39. : ) 9992 + 1998 + 1 ) 10012 + 1 2002 ) 1001 . 999 = (1000+1)( ) ) 993 + 3.992 + 3.99 + 1 ) 1013 3.1012 + 303 1

5.40. :

i) (2 + )2 = + + 92 ii) (3 + )2 = + + 4x4 iii) ( 3y3)2 = 25x4 + iv) ( 3)2 = 242 + v) ( + )2 = 42 + 94 + vi) ( )2 = 9x2 9x +

5.41. :

i) ( 53 + )( 82) =

ii) ( )( + ) = 1218x2 49 2y6

iii) ( )( 21 x3 + ) = 324

5.42. :

i) ( + )3 = 643 + + + 1253 ii) ( + )3 = 83 + 122 + + iii) ( 3)3 = 1000x3 +

5.43. :

i) (+1)3 = (3)2 + (31)2 ii) (+)3 = (3)2 + (3)2 iii) (x3+y3)2 (x2+y2)3+3x2y2(x+y)2 = (2xy)3

5.44. x = 22 , y = 2 = 2+2 , - x2 + y2 = 2.

5.45. = (x+1)2 , = (x1)2 = +8 - : + = (x+3)2

5.46. x + y = 3 xy = 1 , - : i) x2 + y2 ii) x3 + y3

[.i)11 ii)36] 5.47. x + y = xy = , :

i) x2 + y2 = 2 2 ii) x3 + y3 = 3 3 5.48. = 5 + 2 = 5 2 , : i) = 2 3 + 2

ii) = 2 2 +

5.49. 99.

[.49 , 50] 5.50. 1

= 7 , -

: i) 2 + 2

1

ii) 3 31

. [.i)51 ii)364]

5.51. x > 0 (x + 1x )

2 = 9 ,

x3 + 31x . [.18]

5.52. :

i) 2 + 2 2 ii) ( + )2 4 iii)

2 12

x + x

• 16 1:

iv) + 1

2 , > 0 ;

5.53. .

A

x

yz

-

, : x2 + y2 = z2.

5.54. :

A

B

x

x

y

y

, : x2 + y2 = 2.

5.55. - : ( + 2)2 = 2 + 4 +4.

5.56. x, y, z x, = 3x + 2y + 2z , = 2x + y + 2z , = 2x + 2y + z .

5.57. . , (+1)(+2)(+3) + 1 = (2+3+1)2.

5.58. x.y = 6 x2y + xy2 + x +y = 35 x2 + y2.

[. 13] 5.59. :

= 2 10 + 272 8 + 8. ) - .

) , - .

[. )=10, =2] 5.60. x, y, z, w x2 + 10y2 + 10z2 + 9w2 = 6(xy + yz + zx). ) - 0.

) x w.

[. x=27w] 5.61. , - : 2 + 2 + 2 2 4 6 + 14 = 0.

[. =1, =2, =3] 5.62. :

x

y . = x , = y , = 2x + y. () - 25 ()

. : i) xy

ii) ()()

[. i)2 ii) 225 ]

• 6: 17

6.

6.1. ;

: , .

____________________________________________________________________________________________

6.2. : ) 2 4 ) 6x3 + 3x2 ) 12x2y + 6xy2 3xy ) 152 25 + 302

6.3. : ) (x1) x +1 ) (2) + 6 3 ) (2x+3y) 7(2x+3y) (3y+2x)

6.4. : ) ()2 + (+)2() 2(2+2) ) (52)(4x3y) + (3y4x)

6.5. :

3+3 3+2 + 3 19.

6.6. - ) x + 2ay + 3x + 6y ) 5y 5 + y ) 4y 2y + 2 ) 15x + 6x 20y 8y ) xy 2x + 3y2 6y ) x3 + x2 + x + 1 ) 5 + 4 + 3 + 2 + + 1

6.7. : ) + 1 ) x2 + xy x y ) 33 62 + 5 10 ) 3 2 2 + 3 ) 8xy3 24y2 7xy + 21

) 2x2 + x 2xy y ) x y x + y ) x3 5x2 + 3x 15

6.8. : ) x2 + (5y)x 5y ) () () ) x(x3) + y(x) (x3) ) xy(2+2) + (x2+y2) ) (22) + (22)

6.9. : ) (+)(+) ) x 2y x + 2y + x 2y ) x x + y + y x y ) 2 2 + + 1 ) x2y + xy + xy + y + x2 2

6.10. x , y

. 2

211

xy

++ =

xy

x, y .

6.11. : ) 812 492 ) 3xy3 27x3y ) 45x2y4 802 ) 202 15

) 2 2

9x y 14

) 1 4x2 ) 1 x8 ) 166 498

• 18 1:

6.12. : ) 10012 1 )

2 21540 4631077

) 2 262,31 37,69

24,62 ) 2 2225,12 72,12297,24

) 2 2

2 22004 18341924 1914

6.13. : ) 2 2 + + 1 ) 81x4 16y4 ) x8 1 ) 27x 3x3 ) 20x 5x3

6.14. : ) 3 52 4 + 20 ) 2x2y 18y + 3x2 27 ) 2x2 42 x2 + 4 ) 5 1 + 4 ) x2 + x2 y2 y2 ) x3 3x(y + z)2 ) 92x2 42 92x2 + 42

6.15. - i) (5x6)2 9 ii) (3)2 162 iii) (32)2 (32)2 iv) (x5y)2 9(x+2y)2 v) 9(2)2 16(+2)2 vi) x3 x(y z)2

6.16. x ,

(x2+3x+1)2 1 - .

6.17. : ) (4x23x18)2 (4x2+3x)2 ) 5(4x2) (x2)2 ) (53x)(x+4) + (3x5)(2x3) +9x2 25

6.18.

3 - .

6.19. = 3 2 + 1.

i) . ii) , .

[. ii)=2] &

6.20. i) 3 125 ii) 83 + 27 iii) 273 + 643 iv) 2163 1 v) x4 64x

6.21.

i) 64x3y6 + 27 ii) 6

3

18

iii) 6 + 643

6.22. : ) 1 2x + x2 ) 2 4y + 4y2 ) 9x2 + 48xy + 64y2 ) 916 x

2 35 xy + 4

25 y2

) x2 x + 14

) 3xy +

2

9y +

2

4x

6.23. - ) (+2) + 1 ) 9(x+y)2 6y(x+y) + y2 ) x + 1 + 2 x , x > 0 ) y2 2 3 xy + 3x2

6.24. - ) 2x2 12x + 18 ) x2 4x3 + 4x4 ) x3 + 2x2 + x + xy + y

6.25. ) x2 10x + 24 ) 2 + 12 + 27 ) y2 + 3y 10 ) 2 + 2 15 ) 3x2 6x 24 ) 2x2 2x 40 ) 14 5x x2 ) x2 + 2x + 15 ) x2 17x 84 ) x2 + 6x + 5

6.26. = x2 5x + 6 , x > 3.

6.27. ) x2 + 9x + 142 ) 2 4 + 32 ) 2 + 4 52

6.28. : ) x2 2 + 6 9

• 6: 19

) x2 4y2 + 4y 1 ) 9 92 2 + 6 ) 92 12 + 4 42 ) 1 + 2 +2 2 2 2 ) 9y2 4x2 + 4x 1 ) 4x2 36x2 + 81 y2 ) 16y2 9x2 + 12x 4

6.29. :

i) 2 + 2 + 2( 8) ii) 22 x2y2 + 2 2 2 2xy iii) y2 4y + 32 2x x2

6.30. :

i) 4 + 2 + 1 ii) 4 + 22 + 4 iii) 4 1122 + 4 iv) x4 + 3x2 + 4 v) y4 2y2 + 1 vi) 254 + 3122 + 164

6.31. :

i) 4 + 44 + 322 ii) x4 + 9y2 + 5x2y2 iii) 4 132 + 36

6.32. : ) x3 + 12x2 + 48x + 64 ) 83 122 + 6 1 ) 3x + 62x + 12x + 8x

6.33. ) (x+y)3 3(x+y)2.y + 3(x+y).y2 y3 ) (+)3 + ()3 + 3(+)2.() + 3(+).()2

6.34. : ) (2x1)(x+3) + 1 2x ) (+)(2) + (2)2 (242) ) (x29)2 (x+3)2 ) (2x25x+2)2 4x2 + 4x 1 ) 53 95

) y2 + 4x 16x2 y 6.35. :

i) 2 + 2 + 2 ii) x2 + (2+1)x + 2 + iii) 2 + 9(22) + 6 iv) 2 + 2 2 2 2 + 2 v) 2 + (+1) +(+1) + 2

6.36.

i) 2 + 2 + 2 x2 + 4x 4

ii) 9x2 4y2 4y 1 iii) x2 y2 2 +2x 2y + 1 iv) x4 + 5x2y2 + 9y4 v) 9(x + y)2 6y(x + y) + y2 vi) (3 1) 2(2 1) ( 1)2 vii) ( 1)3 (2 4) + (4 2)

6.37. :

i) (x25x+6)(x1) (x3)(x21) ii) 4( + )2 (2 2 2 + 2)2 iii) (52 +2 3)2 (2 2 3)2 iv) (3x 6)(x2 1) (5x 10)(x 1)2 v) (2 9)2 ( + 3)2

6.38. :

(2 + 3 + 1)2 1. [.(+1)(+2)(+3)]

6.39. ) 4 4 (+)()3 ) (2 + 1)2 42 ) 6 1 ) (2 + 2 2)2 422

6.40. ) 42 4 +1 2(21) + 183 92 ) 5 + 24 + 3 2 2 1

6.41. :

i) 1 x3 ii) (2+x+x2)2 1 iii) (2+x+x2)2 x2 iv) x3 + 2x2 3

6.42.

i) 3 + 3 2 2 ii) 2 2 + 2 2 + 2 2 iii) 2 + 2 + 2 + 2 + 2 + 2 + 2 iv) 32 23 + 32 23 + 32 23

6.43. i) - , >

= 2 2(2 1) (2 1)

4 + +

. ii) , , > 1 - 1 .

[. ii)=1, =0] 6.44. (105 + 25)2 (105 25)2 = 10 , , .

[. =7]

• 20 1:

6.45. : = 2 4 6 8 ... 20023 6 9 12 ... 3003

+ + + + ++ + + + + .

[. = 23 ]

6.46.

= 3 + 3 2 2. 6.47. :

= (2 + x + x2) x3 (: 1.)

[. =(1+x+x2)(4+x2)] 6.48. ) :

x4 + 4y4. ) x, y

y 2, x4 + 4y4 .

6.49.

S = 1002 992 + 982 972 + + 22 12 . ) 1002 992 = 100 + 99 - S = 100 + 99 + 98 + 97 + + 2 + 1.

) S Gauss .

[. )S=5050]

• 7: 21

7.

____________________________________________________________________________________________ 7.1. :

i) (x3 x2 21x + 45) : (x + 5) ii) (18x3 + 9x2 50x 25) : (3x 5) iii) (2x3 3x2 17x 12) : (2x + 3) iv) (153 142 + 7 + 6) : (5 + 2)

7.2. :

i) (x3 + 4x2 18x + 2) : (x2 + 1) ii) (3 + 42 11 30) : (2 6) iii) (2y4 + 8y3 16y + 8) : (2y2 4) iv) (2x5 + 4x6 + x2 6 + 5x 3x3) : (x 3 + x2 + 2x3)

7.3. :

i) [(3x+5)2 + (2x+3)2 3x(2x+4) (x+1)2] : (3x 2) ii) [(x29)2 (x+5)(x3)2] : (x2 + x 12) iii)

7.4. , - x2 x + 1 x4 x2 + 2x 1.

7.5. , - x + 3 x3 5x2 + 7x + 95.

7.6. (x) = 2x2 5x + 3, -: [(x) + (x2) (x1)] : (x 3)

7.7. (x) = x3 2x2 + 3x 1, -: [(x+1) + (x1) (x)] : (x 2)

• 22 1:

8.

... & ...

____________________________________________________________________________________________ 8.1. ... ... -: ) 1232 , 1523 , 643 ) 82x3 , 43x5 , 12 3x3 ) 32( )2 , 62( )2( + )

8.2. ... ... -: ) 4(x2 y2) , 6(x + y)2 , 3(x y)2 ) 2 2 , ( )2 , 3 3 ) 3 62 + 12 8 , 2 4 , 2 2 ) 2 3 + 2 , 2 + 3 4 , 3 , 2 2 + 1

• 9: 23

9.

9.1. ;

: , -.

: 1x , 2

3

+ , 2

453

x yxy y

.

____________________________________________________________________________________________ 9.2. o

i) 233

22

ii) 2

3 23

6 18

iii) 2

24

3 6a

iv)

2 2

242

a

9.3. o

i) 3 34 4

ii)

2

24

12 3x xyxy y

iii) 3

2x xx x

+ iv)

2

29

4 3x

x x

+ 9.4. o

i) 2

21

(1 )x

x

ii) 2 2

2 24 4

4

+

iii) a x x x x

+ + iv)

3 3

2 2

+

+ 9.5. o

i) 2 2

2 29 4

9 12 4x y

x xy y

+ ii) 2 2

27 714 49

xy y xx x + +

iii) 2 2

2 249 16

49 56 16x y

x xy y

+ iv) 3 3 2 2

3 33 3 6

6 6

+

9.6.

i) 4

2819

ii)

2

26 94 3

x xx x

+ +

iii) 2 2( 1) ( 1)

1 ++ + + iv)

2 2 2

2( 4) ( 2)

4 3x x

x x +

+

9.7. = 2

24 4 482 8 6x xx x+ + .

i) x ; ii) ; iii) x ;

[. i)x 1 3 iii)x=-4] 9.8. - ;

2

214 49

49

+

[. ]

9.9. = 2 2

2 2(3 2 ) (2 3 )(5 4 ) (4 5 )

, . > 12 .

9.10. o

i) 3 2

22 2

3 2x x x

x x+ + + ii)

2

3 4( ) +

iii) 4 4

3 2 2 32 2 2 2 x x

a a

+ + + iv) 2 2

2 2

a

9.11. :

5 4 4 5

4 3 2 2 3x x y xy y

x yx y x y x + +

9.12. = 2 2

2( 2) 2 4

9 4x x x

x +

i) x; ii) .

• 24 1:

iii) = 1. [. iii)x=-2]

9.13. = 2

27 6

36x x

x + ,

= 2

27 6

1x x

x+ +

. i) x -;

ii) . iii) : (+)2 ()2 = 4.

9.14. :

7 4 3 6

5 2 3 4 42x x y xy

x x y x y xy +

+ = x2 + xy + y2

• 10: 25

10.

____________________________________________________________________________________________

10.1. :

i) 2 23 2 2 3 2

45 9 4x x xx x+ ii)

2

2 21 1:x x

++

10.2. :

i) 2

4 2123

6 5 15

ii)

2

216 3 12: 3 98 16

+ +

iii) 2 2

2 23 6 12 3:

1 2 2 2 4

+ + [. i) 325 ii)

-3-4 iii)

21- ]

10.3. :

A = 2 2 4 2 3

3 3 2 2:x y y xy y

x y x xy y +

+ + . [. A=1]

10.4. :

23

3 3 2 2 248 2: :x yx x

x y x xy y xy y + + . [. 1]

10.5. = 3 2

24

5 6x xx x

+

i) x ; ii) .

10.6. :

2 2

22 2 2 2:5 4 16 25 40

+ + + + + =

(4 5)( 2 )2

+ +

.

10.7. :

i) 35

1595+ ii)

4

4+ 1

[. i)-31 ii)

41 ]

10.8. : i) 1 1x+ +

21x 2

41x

ii) 2x y + 2 24 y

y x

iii) 1 2 + 1

2+ 24

4 iv) 3

1+ + 3

1 2

69

v) xx

+

xx

2( )( )( )

x x

[. i)1x

3+ ii) yx

2+ iii) 2

2+ iv) 3

2+ v)2]

10.9. x , y, z , : 1( )( )x y x z +

1( )( )y z y x +

1( )( )z x z y = 0.

10.10. :

i) x yx y+ 2 2

4xyx y

ii) 2 2

2

+ +

2 2

2 22

+ +

iii) 2

2

+ 2 2 + + 22a +

iv) 2

xy + 2

2( ) xx xy+

2

2( ) yxy y+

v) 1 2

32

+ 2

44 a

[. i)yxyx

+ ii)-

iii)-

iv)1 v)-2

2+ ]

10.11. :

i) 223+ 33

2a + 2

5 36 6a+

ii) 2 21

+ 21

+ 21

2 2

iii)

2

2 +

+2 2 2

44

a

• 26 1:

iv) 23

12 + xx + 2

12 + xx + 4

12 x

v) 2

122

xx

x 6

122 +

+xx

x + 44

422 +

+xx

x

10.12. :

i) 2296

+ +3

3 +

3

ii) 2xx + 33 +x

x 22

2

xxx

iii)

3

+1 + 32

42

[. i) +3 3

ii) x

3(x 2) iii)

32)2(

]

10.13. = 1 2

69

xx +

33x .

i) x ; ii) .

[. i)x -3 3 ii)= xx+3

]

10.14.

= 2

26

9x x

x +

2 1xx+ 2 2 33

x xx x+ +

) x ; ) , .

[. )x 0,-3 3 )= 2(x+1)x

]

10.15. :

i) 2 2

2 21 1 x yx y x y

ii) 2 21 1:

( ) ( ) + + + + +

iii) 2 2

2 212 + + + + + +

iv) 2 24 2 44 2: 1y y xyx x

+ + +

10.16. :

2

2 24

+

+ .

10.17. :

( )2 2 2 31 1 2 4:6 4 2 4xx x x x x x x+ + + [. 3x+3 ]

10.18. :

i) 1 11x

x

+

ii) 2

11

11x

x

10.19. :

i)

2 2

2

1 1

+

ii) 21

1 1

+ ++ +

[. i)- ii)-] 10.20. :

i) 1

1

+ + +

ii) 2

+ + + +

[. i) ii) 1

]

10.21. :

i) a

a

a

a

+

++1

1

11

ii) 31

+ +

+

++1

3 +

a++1

3

[. i)12

12

ii)6]

10.22. :

2 2 2 2

3 31 1:

+ +

[. ] 10.23. =

x+11 , = x1

1 ,

= xx

+

10.24. : = 2 + , x =

+ ,

y = + z =

+ , :

y = 2xzx z+ . 10.25. ) :

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

11+ ) (1

1+ 1

2+ ) ) :

11 2 3 +

12 3 4 +

13 4 5 + +

11999 2000 2001

= 14 1

4000 + 1

4002

• 10: 27

10.26. :

2 244 42 2

:4 42 2

+ + + + + + + +

. [. 5]

10.27. x , y, , :

2

( )( )x

x y x + 2

( )( )y

y y x + 2

( )( )

x y = 1.

• 29

x + = 0

-

• 1: x + = 0 31

1.

x + = 0

1.1. ;

; : x + y = . .

___________________________________________________________________________________________ 1.1. : ) (3 x).21 = x ) 101.(x 3) = 5.102 .

[. )1 )3,5]

1.2. , - -, ; 6

, + 30, = ,, .

[. 5] 1.3. 15.

[. 7,8] 1.4. :

3310 + 2 1

3x+ 2( 3)5

x+ = 3 26x+ 2 110

x [. x=60]

1.5. : ) x 3 3 = 3 (1x) ) 2 1

5 1x =

2 15 1x++

[. )1 ) 52 ]

1.6. 21

xx+

11

xx+ = 2

21

x .

) - ;

) x = 3.

[. )-1,1 )=5] 1.7. :

2x

12

12

x+ +

13 313

x+ + 152 = 0 .

[. x=-1]

• 32 2: -

2.

2

2.1. 2 .

: x2 + x + = 0 0.

x2 + x + = 0 4.x2 + 4.x + 4. = 0 [ 4] 42x2 + 4x = 4 [ 2 ] (2x)2 + 2.2x. = 4 (2x)2 + 2.2x. + 2 = 2 4 [ 1 2 ] (2x + )2 = 2 4 2 4 () : (2x + )2 = : > 0 :

2x + = 2x = x = 2

= 0 :

2x + = 0 2x =

x = 2

< 0 ().

2.1.1. :

x2 + x + = 0 0 > 0 x = 2

= 0 x = 2

< 0 ().

• 2: 2 33

___________________________________________________________________________________________ 2.2. :

i) x2 x = 0 ii) 2x2 7x = 0 iii) 4x2 + 36x = 0 iv) 3x2 = 5x

[. i)0,1 ii)0, 72 iii)0,-9 iv)0,53 ]

2.3. :

i) x2 1 = 0 ii) 3x2 27 = 0 iii) 6x2 + 12 = 0 iv) x2 12 = 0 v) 5x2 10 = 0 vi) 8x2

225 = 0

[. i)-1,1 ii)-3, 3 iii) iv)-2 3 ,2 3 v)- 2 , 2 vi)- 54 ,

54 ]

2.4. :

i) 300x2 600.25 = 0 ii) 200x2 = 2000x [. i)-8,8 ii)0,10]

2.5. :

i) x(x2 4) = 0 ii) 4x(x2 + 6) = 0 iii) (x 5)(3x2 147) = 0 iv) (2x + 1)(8 x2) = 0

[. i)-2,0,2 ii)0 iii)-7,5,7 iv)-2 2 ,2 2 ,- 12 ]

2.6. :

i) x2 9x +14 = 0 ii) x2 12x +20 = 0 iii) 12x2 x 6 = 0 iv) 3x2 5x 2 = 0

[. i)2,7 ii)2, 10 iii) 34 ,-23 iv)2,-

13 ]

2.7. :

i) 1000x2 + 4000 = 4000x ii) 400x2 + 300x = 1000 iii) 3.210.x2 5.210.x + 212 = 0

[. i)2 ii)-2, 54 iii)]

2.8. :

i) (x + 2)2 = x + 2 ii) (x + 2)2 + x2 4 = 0 iii) (x 1)2 + (x + 2)2 = 29 iv) x2 + (x+1)2 = (x + 2)2 v) (2x + 1)2 + (3x + 1)2 = (5x + 2)2

[. i)-1,-2 ii)0,-2 iii)3,-4 iv)3,-1 v)- 13 ,-12 ]

2.9. :

25(x + 2)2 9(x 2)2 = 0. [. -8,- 12 ]

2.10. ) :

= (2x 3)2 8x2 7(1 x x2) ) = 0.

[. )A=3x2-5x+2 )1, 23 ]

2.11. :

i) (x + 1)(x2 + 2x 8) = 0 ii) (2x 1)(x2 + 3)(x2 + x 2) = 0 iii) x3 2x2 9x + 18 = 0 iv) 2x3 + x2 3x = 0

[. i)-4, -1, 2, -2 ii)-2, 1, 12 iii)-32 , 0, 1]

2.12. :

(x25x+4)2 + (2x26x+4)2 = 0. [. 1]

2.13. : ) x2 + ( 2 1)x 2 = 0 ) x2 + 3 3 x 12 = 0 ) 4x3 4 3 x2 x + 3 = 0

[. )1, 2 ) 3 ,-4 3 ) 3 , 12 ,-12 ]

2.14. ) : x(x 5 ) = 5. ) x1 , x2 -, : x1.x2.

[. )-5] 2.15. = (x29)2 (x+5)(x3)2 = (x1)2 4 i) , . ii) x

;

iii)

.

iv) x 6 = x 3.

v) x 2

9x = 3B. [. i)A=(x-3)2(x+1)(x+4) B=(x+1)(x-3) ii)x -1,x 3

iii)(x-3)(x+4) iv)x=2 v)x=-5/2]

2.16. = 2 4x x

x ,

= 2( 3)( 9)

3x x

x

+ = 3 24 4x x x

x +

i) x ;

ii) , . iii) 2 = .

[. i) x 0, x -3 ii)A=x-4 , B=(x-3)2, =(x-2)2 iii)x=-1, x=3]

2.17. = 2x + 4x2 ,

= 1 4x2 K = AB , = 2

3 21 2

4 8 4x x

x x x+ ++ +

i) x ;

ii) . iii) = .

• 34 2: -

[. i) x 12 , x 0 -1 iii)x= 14 ]

2.18. :

= x3 x2 4x + 4 = (x3)2 4. i)

x -.

ii)

= 12.

[. i)x 1 5, =(x-2)(x+2)x-5 ii)x=-16, x=4]

2.19. :

(x + 1)3 3(x 1)2 2(x 1)(x + 1) = (x 1)3 + 8 [. -7 , 1]

2.20. x.

(x )2 + (x + )2 = (x )(x + ) + 4x . [. ,3]

2.21. :

(x2 4)(x 1) = (x2 1)(x 2) [. 1, 2]

2.22. :

22

+ 3

= 2.

i) . ii) , :

2x x = 14 . [. i)=2 ii)-1,2]

2.23. :

2 2( 3 1)5 x x+ + = 5. [. -3,-2,-1,0]

2.24. i) :

= x3 3x + 2 = x5 x3 x + 1. ii) : 1x+ = . [. i)=(x-1)2(x+2), B=(x-1)2(x+1)(x2+x+1) ii)x=1]

• 3: 2 35

3.

2

___________________________________________________________________________________________ 3.1. x + 3 x 3. 40 cm2. i) - .

ii) x. iii) .

[. i)4x ii)x=7 iii)28cm] 3.2. - 2x m , 3 (x+1) m 4 m. .

[. 6+2 3 m] 3.3. - 5cm 12cm. - .

[. 3cm, 4cm] 3.4. , 2cm - 10cm.

[. 6cm , 8cm] 3.5. - 1cm , - 2cm - . , .

[. 3cm 4cm 5cm , 12cm , 6cm2] 3.6. - 10m , - .

[. 6m, 8m]

3.7. = 900 :

= 2 1+ = 2 1

+ > 1.

. [. 2+2]

3.8.

2cm . 60cm2, .

[. 34cm , 13cm] 3.9. - . .

[. 3, 4, 5] 3.10. 10 . ;

[. 15] 3.11. - 100. .

[. 6,8] 3.12. = 2x7

cm , = x5 cm 27 cm2. .

[. 9cm, 3cm] 3.13. // = =

900. = = x2 cm , = x+10 cm - 112 cm2 , .

[. ==8cm, B=20cm]

• 36 2: -

4.

4.1. ;.

: , .

___________________________________________________________________________________________ 4.2. : ) 4 2x+ +

12x = 2 4

xx

) 22

2x x+ = 21

5 6x x+ +

) 131x+ 1

1 x = 25 3

1x

x

[. )3/2 ) )] 4.3. -: 810x = 10

xx

108

x = 10 xx

[. ] 4.4. : ) 2

24x 2

12x x = 2

42x

x x+

) 5 3x+ 1

2x+ = 22 1

5 6x

x x+

+ +

) 2x x = 22 42

xx x

+

) 23xx+ +

23

xx+ =

2

22( 6)

9x

x+

[. )3 ) ) ) -3,3] 4.5. = 2

22

xx x+ , =

32 4x = 2

142( 4)

xx .

) x .

) = . [. ) x 0 -2. x 2, -

x 2 ) 0,-2,2] 4.6. :

) 2 2x

x+ + 2

2x

x+ = 2

) 1xx + 2

xx 2

12x x = 0

) 2 3xx 25 3xx

= 2 32 8x xx+ + 2

) 1 2x 2

1x+ = 23

2x x [. )2 )1, - 12 )1, -0,8 )]

4.7. : ) 4 1x+ 2

4 31

xx

= 2

) 1 21+x

12 x = 2

24

xx

) 123 2x 8

3 2x+ = 22 334 9

xx

[. )- 22 , 22 )0 )2] 4.8. 2

22x x+

32 4x+

, x = 1; ;

[. 76 , -127 ]

4.9.

= 2

22

414

xx x

x

+ +

, = 11xx+

3

3x

x x .

) x .

) 2x (x+1). = 0.

• 4: 37

[. ) x 0, 2 -2. x 0, 1 -1 )4]

4.10. :

x2 2 1

1+ x + 2 + 2 = 0

1. [. -1, 23 ]

4.11. : ) 12

xx

21

xx = 2

13 2x x +

) 13xx + 1 =

2

23 45 6

x xx x

+ +

) 13xx+

23

xx =

2

239

xx

+ [. ) )1,4 )1,-6]

4.12. : ) 2

5 45 6

xx x

+ + 2

10 13 27

xx

+ =

53 9x

) 24

2x x + 5

2 2xx++ =

23 6

xx

) 23xx++

11

xx+ = 2

42 3x x+

) 2

23

4x

x 212

xx x

+ =

22

xx+ + 2

[. )1034 )3 ) )1,

52 ]

4.13. :

) 2 1x

x+ + 1x

x + 1 = 2

23 5

1xx

) 22 1

3x

x x++ + 2

49

xx =

43 x+

23 x

) 2

27

1x

x +

31x+ =

21x

) 2x x + 4 2x = 2

82x x

[. )-4 ) )2 )-1] 4.14. :

i) 23 1

1xx

2

2x x =

2 3 2x xx

+ ii)

[. i)-3 ii)]

• 38 2: -

5.

-

5.1. ;.

: , : > 0 >< 0 < = 0 =

5.2. ,

. > + > + . : : ( + ) ( + ) = + = > 0 , > . + > + .

5.3. ,

. > > 0 . > . . :

: . . = .( ) > 0 , > 0 > 0. . > . .

5.4. , -

. >< 0 . > . . :

: . . = .( ) < 0 , < 0 > 0. . < . .

5.5. ,

. > > + > + . : : ( + ) ( + ) = + = ( ) + ( ) > 0 , > 0 > 0. + > + .

5.6. > > > . ( )

: > > 0 (1). , > > 0 (2). (1) (2) , : + > 0 > 0 > .

5.7. >< .

:

• 5: - 39

: > (1). < (1). < .

5.8. > , , 1

< 1

.

:

, > 0 1

> 0.

: > 1

> 1

1

> 1

.

___________________________________________________________________________________________ 5.9. x > y ,

34 x 5z 34 y 5z.

5.10. > ,

8,3 6 8,3 6. 5.11. > 0 : + > . 5.12. x < z 0 < y < :

x 1y < z 1

.

5.13. x > 3

= 2( 3)x = 2x. [A.

• 41

• 1: 43

1.

1.1. y = x + ;

: y = x + y = x.

1.2. x + = ;

: x + = .

___________________________________________________________________________________________ 1.1. , (52,2+1) y = x+6.

[. =1] 1.2. 1 , 2

y = (2+4)x + 5 y = 4x 3 . , .

[. =2] 1.3. y = 32x + 1

y = (+4)x 52 ; [. =-1 =4/3]

1.4. 1 : y = (34)x + 2 6. 1 : ) (2,2) ) xx ) ) 2 : y = 8x + 7 ) xx 3 ) yy 4 )

3 : y = (3 1) 2 1

2 x +

[. )=2 )= 43 )=3 )=4 )=1811 )=5 )=

73 ]

1.5. 4x + 3y = 12.

i) . ii) .

iii) .

[.ii)A(3,0) B(0,4) iii)=12,E=6] 1.6. 2 2

x + (1)y = 12 (3,2). : i) . ii) .

iii) o . [.i)=-2 ii)A(6,0) B(0,-4) iii)E=12]

1.7. 1 , 2

y = 22x + 3 + 1 y = (53)x + 9 , , . 1 (1,6). i) , .. ii) - .

[.i)==1]

• 44 3:

2.

___________________________________________________________________________________________ 2.1. :

) 4

1x y

y+ = = )

25

xx y

= + = ) 2 6y x

x y= + =

[. )(x,y)=(3,1) )(x,y)=(2,3) )(x,y)=(2,2)] 2.2. :

) 2 3

y xy x

= = ) 2

2 5y xx y= + + =

[. )(x,y)=(3,3) )(x,y)=(2,4)] 2.3. , .

i) 2 4

3x yy x+ = = + ii)

3 22 0

x yx y

= + =

iii) 3 2 54 6 7

x yy x = = + iv)

1 2 322 8 12 0

x y

x y

= =

2.4. 2( ) 4 05( ) 10 0

x yx y+ = + = -

; .

• 3: 45

3.

___________________________________________________________________________________________ 2.1. .

i)3 1

3 2 11x y

x y= + = ii)

2 431

x yx y = + =

[. i)(x,y)=(-5,-2) ii) (x,y)=(22,9)] 2.2. :

i)2 2

7 13 0x y

x y = + = ii)

5 452 3 25

x yy x+ = =

[. i)(x,y)=(8,3) ii) (x,y)=(5,20)]

2.3. : 2 1

3 5( 1)x yx y = = + .

[. (x,y)=(5,2)] 2.4. :

i)2[ (4 )] 20 0

4 0x y

x y + = + = ii)

2(2 3 ) 3(2 3 ) 104 3 4(6 2 ) 3

x y x yx y y x+ = + = +

[. i)x=-8, y=2 ii)x= 52 ,y=1] 2.5. :

i)

1 12 2

3 2 192 3 2

x y

x y

+ = = ii)

0,3 1,5 0,3 00,3 0,2

x yy x

+ = =

[. i)(x,y)=(5,-3) ii)(x,y)=(4,1)] 2.6. :

i)4 16 4

7

x y x

x y

= + = ii)

1 4 122 3 0

x y

x y

+ = + =

[. i)x=2, y=5 ii)x=-2,y= 12 ] 2.7. :

i)2 3 6

3 4

x yyx

= = ii)

3 2 1 3 22 32 2 1

5 3

x y x y

x y x y

+ = + + + =

[. i)(-3,-4) ii)(-1,-2)] 2.8. :

i)

2 2 3322 12 3

x y

yx

+ + = ++ = ii)

2 23 ( 1)3( 2 ) 2

x y yx y

+ + + = + =

[. i)(2,1) ii)] 2.9. :

i)

2 13 6 2

72

x y x y

x y

+ = + = ii)

2 4 03 2 2

3 5 5

x yx y y x x

+ + = + + = +

[. i)(x,y)=(4,5) ii)(x,y)=(-2,0)]

2.10. (2 1) (4 1) 3( 1) ( 2) 2 x y x y + + = + + =

x, y. , , - (1,1).

[. =-1/2, =9/2)] 2.11. :

i) 2( 2 ) 3( 3 ) 832( 2 ) ( 2 ) 49

x y x yy x x y+ = =

ii) 3 2 7

2(3 2 ) 4(2 3 ) 42x y

x y y x+ = + + + =

[. i)x=5, y=-6 ii)] 2.12. :

i) 32 14 8 2

5 94 1 14 6 2

yx x

yx x

+ = + = + ii)

[. i)x=2,y=3 ii)] 2.13. :

i)

3 2 12 5 3612 3

x y x y yx

x y xx y

+ + + = + + + + = ii)

3 2 2 42 3 3

2 3 45

x y x

x yx y

+ = + + =

[. i)(6,4) ii)(2,3)]

2.14. 22 5 3

2 5 1x y

x y = = .

.

• 46 3:

[. =2 =-2]

2.15. 23 3 3x yx y

= = . . -.

[. =1]

2.16. : 2 3 5

22 3

x yyx

+ = + =

[. x= 2 ,y= 3 ] 2.17. 5x 4y = 7. - ) , . ) , . ) , .

2.18. :

i) 2 29 5

3 2x yx y = + = ii)

1 1 7127

x yx y

+ = + =

[. i)x=3,y=- 13 ii)(3,4), (4,3)] 2.19. :

i) 2 21

1y x

x xy y = + = ii) 2 2

2 57

x yx y xy

+ = + + = [. i)(-1,0), (2,3) ii)(2,1), (3,-1)]

2.20. :

i) 23 2 7

2 3x y

x xy y+ = + = ii) 2

3 55 9x yx y+ = =

[. i)(1,2), ( 85 , 1110 ) ii)(2,1), (53,-16)] 2.21. i) ,

y = 2x + 3 (0,1).

ii)

: 2

2 312

y x

y xy

= + + =

[. i)=3 ii)(x,y)=(1,3), (- 116 ,- 83 )] 2.22. :

i)

1 1 32 1 23 2 42 1 2

x y

x y

=+ + =+ ii)

4 12 1

1 1 02 2

x y x y

x y x y

+ = + + = +

[. i)(- 14 ,1) ii)(5,-15)] 2.23. :

= x + 4 , = 4x y = y + 2. .

[. 7] 2.24. - , 10 cm 1 cm .

[. 3cm 4cm] 2.25. , 5 3 2 - 8. ;

[. 17] 2.26. y = x + (2,5) (4,1).

[. y=-3x+11] 2.27. , ,

x2 + x + = 0 x = 3 x = 7. [. =4, =-21]

2.28. f (x) = x2 + x + .

f (1) = 2 2 f (x) = 0 , .

[. =-1, =2, ] 2.29. y = x + (,1) (1,3).

[. y=-x-2] 2.30. 9 -, . . 9 , - . ;

[. 33 12 ] 2.31. = 3x+6 , = x2 , = 5y+4 = y x . .

[. 20] 2.32. = x2 , = 5 y2 , = x + y = 1 . - .

[. ==1 ==4] 2.33. x2 + x + = 0

16 3 + 1

xx =

2 11

x+ , , .

[. =1,=- 154 =-3,=-74 ]

2.34. 9. 27 - . ;

[. 36]

• 47

y = x2

y = x2 + x + , 0

• 1: y = x2 49

1.

y = x2

___________________________________________________________________________________________ 1.1. f (x) = 20x2. :

) 2 2( ) ( )f f

++ = 20.

) ( ) ( )( ) ( )f f

f f + +

+ = 2. 1.2. y = x2 y = x + 6.

[. (-2,4) , (3,9)] 1.3. y = (26)x2 ;

[. >3] 1.4. y = 3 57

x2 xx;

[. > 35 ]

• 50 4:

2.

y = x2 + x +

____________________________________________________________________________________________ 2.1. f (x) = x2 3x + 1. , , x 11.

[. x=-2, x=5] 2.2. f (x) = 3x2 (+1)x 2. ) , (1,5);

) 1 x 4.

[. )=5] 2.3. f (x) = x2 4x + 5.

i) , - (1,8).

ii) -.

iii) - .

[. i)=-1] 2.4. y = x2 + 2x + .

i) , (1,5).

ii) . [. i)=2, =0]

2.5. y = x + 3

y = x2 2x + 3. i) -.

ii) .

[. (0,3) (3,6)] 2.6. y = 2x2 + x +

y = x . (1,2) : i) , . ii) .

[. i)=1, =-1 ii)B(-1,0)] 2.7. , - . 10cm.

E

B

x x .

[. 5cm] 2.8. - 12 cm. i) 2x cm, x.

ii) - x.

iii) - 0 x 6.

iv) . [. i)=12-2x ii)E(x)=-2x2+12x iv)x=3, ==6cm]

2.9. f (x) = x2 9x + 12. x : i) f (x) = 6 ii) f (x) = x iii) f (x) = f (x) iv) f (x) = f (x+1)

[. i)x=3, x=6 ii)x=2, x=6 iii)x=0 iv)x=4] 2.10. - y = x2 + x + , A(1,3).

• 51

-

• 1: 53

1.

1.1. ;

. : . .

, , , -. : < + < + < + .

1.2. . . : : 1) -

, .

= = =

.

2)

, .

= = =

.

3) , -

.

= = =

.

• 54 1:

1.3. .

: : 1)

, .

2) , .

___________________________________________________________________________________________ 1.4. , . - .

1.5. , , , . - - .

1.6. . :

+ + > 2 + + .

1.7. .

1.8. :

) < 2 + .

) + + < + + . 1.9. . , -:

+ + > 2 + + .

1.10. - .

1.11. = . - =

31 =

31 .

, - .

1.12. = .

, = . , -.

1.13. = . , = . = .

1.14. = . - . .

1.15. , - , , = = . ) . ) . ) -.

1.16. . , = = . = .

1.17. . , . - : ) = l . ) .

1.18. lx y . x y , , - , = , = . : ) = ) = ) lx y .

• 1: 55

1.19. . . - , . - = .

1.20. . . , .

1.21. . - .

1.22. lOx y . - x y . : ) . ) lAMB . ) .

1.23. .

1.24. . = .

• 56 1:

2.

2.1. , . :

//. // , , . .

___________________________________________________________________________________________ 2.2. , , . .

2.3. . , . .

2.4. , . , - = 12 =

12 .

= 14 .

2.5. , . . - .

2.6. , , , , , . . .

2.7. , ( < ) . , . ) .

) = = 2 .

) = = 2 .

) = 2+ .

) = 2 .

2.8. . , . - .

• 2: 57

2.9. - , , , -. .

2.10. = 900, = 12 cm = 2x+2 cm. - 2x1 cm, - .

• 58 1:

3.

3.1. ; . : , .

A

B

1 , 2 , 3 4. :

= =

.

___________________________________________________________________________________________ 3.2. - //. = 2x+1, = x, = x2+2 = 4x2+x. x.

[. x= 17 ]

3.3. = 90 = 8cm , = 6cm. // = 4cm. .

[. =3,2cm]

3.4. = .

A

B

E

: i) //

ii)

=

3.5. 1 2. - = 2cm, = x cm , = x+1 cm AE = 3cm, .

[. 2cm] 3.6. - = = .

B

2 +1x

x

x2+2

4 + x x2

• 3: 59

E

. = 2,5 m = 8 m .

[. =10m]

• 60 1:

5.

5.1. ;

: , .

5.2. .

: 1) , ,

. 2) , , -

. ___________________________________________________________________________________________ 5.3. 5cm , 3cm 15cm , 9cm. .

5.4. - = 7,2dm , = 40cm , = = 650 , = 72cm , = 4dm. .

5.5. - = 8cm , = 2cm , = = 700 , = 10cm , = 4cm. .

5.6. 14 . = 4cm , = 5cm

. 5.7. = 12 cm = 18 cm. - = 8 cm . = 10 cm, , .

[. =12cm, =6cm, =15cm] 5.8. = l ,

1 = 2 , = 4 cm = 9 cm.

1

2

4 cm

9 cm .

[. 6cm] 5.9. 6 cm - = 2 cm. , .

[. =18cm] 5.10. -: = , = , = 4cm , = x+6 cm , = x+2 cm , = 4x+8 cm. i) . ii) x. iii) = 13 .

5.11. , .

• 5: 61

i) , , .

ii) - .

iii) l = l . 5.12. = 900, - = 8 cm = 10 cm. . i) .

ii) , - .

[. i)=2,4cm , =3,2cm] 5.13. = 6 x , = x(x+3) = x2 + 5x. ) x 2; ) x - 15 cm.

) x , = 2.

[. ) )x=1 )10cm, 8cm, 12cm]

• 62 1:

6.

6.1. ;

: .

___________________________________________________________________________________________ 6.2. 490cm2.

= 25 . -

, - . . i) , .

ii) .

iii) - .

[. i)40cm2 ii)250cm2 iii)200cm2]

• 63

• 1: 0 00 180 65

1.

1.1. , ;

: , .

, =

=

.

, .

, =

=

. , .

, =

=

.

1.1.1. : , . 1.2. :

(900 ) = (900 ) = . : : =

=

, = .

, -, + = 900 = 900 . (900 ) = . (900 ) = . , 900 : (900 ) = . (900 ) = .

1.2.1. : 540 = (900 360) = 360.

730 = (900 170) = 170.

B

B

• 66 7:

1.3. . :

O xx

y

M(x,y)

x

y

(x, y) , lOMx = . = , :

= y

, = x

, = yx (x 0). 1.4. , 1 1.

: x y > x > y . 1 , 1 0. , : 1 1 1 1.

1.5. .

:

. . 1

2 3 4 x + + x + + x + +

____________________________________________________________________________________________ 1.6. :

= 250 350 + 550 650 = 2x + 3(900x) 5x

= 2 0 2 0

100 0 0 45 4 30 20072006 45 260

+

[. =0, =0, =-1] 1.7. (3,0) , (2,2) , (6,8) , (0,5). lx , lx , lx lx .

1.8. : i) = 3x 5 ii) = 4 2x iii) = 7x + 6x iv) = 2x 3x

1.9. - y = 22x + 1.

[. 1, 3]

1.10. - y = 42x 1.

[. -1, 3]

• 1: 0 00 180 67

2.

1800 :

(1800) = (1800) = (1800) =

: (i) 1300 = (1800500) = 500 0,766 (ii) 1150 = (1800650) = 650 0,423 (iii) 980 = (1800820) = 820 7,115. ____________________________________________________________________________________________ 2.1. 350 = 0,819 : ) 550 ) 1450

[. )0,819 )-0,819] 2.2. :

= 0 0 0 0

0 0 0(90 ) (90 ) (180 ) 45 (180 ) (180 ) 150

[. 2] 2.3. , , , : ) (+) = ) (+) = ) (+) = B

2.4. x, : ) 2x 3 = 0 , 00 < x < 1800 ) 2x = 1 , 900 < x < 1800 ) 3x + 3 = 0 , 900 < x < 1800 ) x = 1 , 900 < x < 1800

2.5. x, : ) 2x = 0,25 , 00 x 1800 ) 42x = 3 , 900 x 1800 ) 2x 2x + 1 = 0 , 00 x 1800 ) 22x + x 1 = 0 , 00 x 1800 ) (2x 2 )(x1) = 0 , 00 x 1800

2.6. 900 < < 1800 2(1 ) + 4 + 3 = 3( 3 2) + 6 .

[. = 32 ,=-12 ,=- 3 , =120

0]

• 68 7:

3.

3.1. : =

.

:

O xx

y

M(x,y)

x

y

lOMx = , = y

= x

= yx

= y x =

yx = .

, =

.

, 0, 900 2700. 3.2. : 2 + 2 = 1.

:

O xx

y

M(x,y)

A

: x2 + y2 = 2

2

2x

+ 2

2y

= 2

2

2x

+

2y

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

____________________________________________________________________________________________ 3.3. i) : = 13 13 7 4 + + = 41 29 19 9 ii) 2 =

,

: = 2

22

.

[. i)=3,=6 ii)= 34 ]

3.4. x = 35 0

0 < x < 900, :

i) T x x. ii) T :

A = 2

23 5

x x x

x+ .

• 3: 69

3.5. x = 513 90

0 x 1800, -: i) T x x. ii) T :

A = 25 3

2 3x x x

x+ +

+ . 3.6. x 00 < x < 900, - :

= 2

2 2 2

4 3x x x

x x

. [. = 111 ]

3.7. = 3 = 4 0 < < 14 ,

: = 252 + (5 1)3.

[. =1] 3.8. x2 + .x + 2 = 0 . - , 4 -.

[. =- 2 55 , =55 , =-2]

3.9. = 2 2

2 2 +

, > > 0 900 < x < 1800, : ) )

[. )=- +2 22

)= 2 2 -2 ]

3.10. : 2 (1 + 2) = 1. 3.11. : ( 3) = 3. 3.12. i) :

2 1 4 3 23 2

2 3( 1)

x y x y x

y x

+ + = + =

ii) (x,y) : 4(x.+2006y.)2 + (2006y.+2x.)2 = 4.

[. i)(x,y)=(1,0)] 3.13. : 1 + 2 = 2

1

.

3.14. :

( + )2 ( )2 = 4.. 3.15. :

(3 + 4)2 + (4 3)2 = 25.

3.16. :

+ = 1 + .

3.17. : x.x.x + 2

11+ x

= 1.

3.18. : (1 2

1

)(1 21

) = 1.

3.19. :

+ =

1 1

+ .

3.20. : 2

2 1 1

+ =

2 2.

3.21. = 3. , = 3. , = 3 : 2 + 2 + 2 = 9.

3.22. : 11 +

11 + = 2

2

.

3.23. : 1+

+ 1

= 1

.

3.24. : 1

1

= 1

.

3.25. : 1 +

1

= 2

.

3.26. : 2 2 = 2 . 2. 3.27. : 4 2 = 4 2. 3.28. : 4x 4x = 22x 1. 3.29. :

i) 4 4

2

1

ii) 2

4 42 1

[. i)2 ii)1]

3.30. : 1 2

1+

= .

3.31. :

( +

)( 1

)( 1

) = 1.

3.32. :

1 xx +

2

xx x = x + x.

3.33. 00 < x < 900 x = 2x 1 x.

[. x= 45 , x=35 , x=

43 ]

• 70 7:

3.34. i) = 1213 90

0 1800 - .

ii) : ( ) ( ) 1

( ) 1 x y

x y = = .

[. i)= 513 , =-512 ii)(

52 ,

124 )]

3.35. x, 3x + 4x = 5.

[. = 35 , =45 , =

34 ]

• 6: & 71

4.

&

4.1. .

:

, , , : =

=

.

. =

= . (1)

=

= . (2)

(1) (2)

. = . . = . =

.

=

.

, =

=

.

4.2. .

:

, , , : 2 = 2 + 2 2. 2 = 2 + 2 2. 2 = 2 + 2 2..

. 2 = 2 + 2 (1) =

= . (2)

=

= ..

= = . (3) (1) , (2) (3) 2 = (.)2 + ( .)2 = 2.2 + 2 2. + 2.2 = 2.(2+2) + 2 2. = 2 + 2 2..

2 = 2 + 2 2. 2 = 2 + 2 2..

• 72 7:

4.2.1. : : 2 = 2 + 2 2. 2. = 2 + 2 2

= 2 2 2

2

+ .

= 2 2 2

2

+ =

2 2 2

2

+ .

___________________________________________________________________________________________ 4.3. . l = x l = y :

xy =

=

.

4.4. > 900 = 450 , = 10 cm = 5 2 cm. .

[. =300, =1050]

4.5. = 2 , = 3

= 600. i) , - .

ii) ; iii) ,

= 10cm. iv) .

[. i) =900, =300 iii)=5cm, =5 3 cm iv)= 25 32 cm

2]

4.6. :

2 = 2 + 2. = 900.

4.7. = 3 cm , = 1 cm = 300 , : i) . ii) .

[. i)=1 ii)1200] 4.8. = 2cm , = 2 cm = 450. -.

4.9. = 31 , = x + 1 , = x + 2 = 600. .

[. =5, =6]

4.10. = 2 , -.

4.11. :

+ = . 4.12. :

= 2 2 . 4.13. : = 3, = 5 = 7. = 1200.

[. 1200] 4.14. , , : = 2 2 + + , .

[. 1200] 4.15. = 4 , = 4 = 1200. i) . ii) B. iii) .

[. i)=7 ii)= 3 314 iii)=15 34 ]

• , 1 , , Pascal (1623-1662) , 1653. .

1 = 20 1+1 = 21 1+2+1 = 22 1+3+3+1 = 23 1+4+6+4+1 = 24

Fibonacci. , 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 1 = 1 , 2 = 1 n+2 = n+1 + n.

, , . 1+9 = 10 1+5+15 = 21 1+6+21+56 = 84

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