Mathimatika g Gymnasiou Shmeiwseis Askiseis

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Μαθηματικα Γ γυμνασιου σημειωσεις και ασκησεις

Transcript of Mathimatika g Gymnasiou Shmeiwseis Askiseis

  • ( + )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)