Mathimatika g Gymnasiou Shmeiwseis Askiseis

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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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