Ερωτήσεις επανάληψης Γ΄ Γυμνασίου Μαθηματικά

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Ε Ε Ε Ρ Ρ Ρ Τ Τ Τ Η Η Η Σ Σ Σ Ε Ε Ε Ι Ι Ι Σ Σ Σ Θ Θ Θ Ε Ε Ε Ρ Ρ Ρ Ι Ι Ι Α Α Α Σ Σ Σ Α Α Α Π Π Π Ο Ο Ο ΤΗΝ ΥΛΗ ΤΗΣ Γ ! ΤΑΞΗΣ Η φιλοσοφία είναι ένα παιχνίδι με στόχους και όχι κανόνες. Τα μαθηματικά είναι ένα παιχνίδι με κανόνες και όχι στόχους.

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Ερωτήσεις επανάληψης Γ΄ Γυμνασίου Μαθηματικά

Transcript of Ερωτήσεις επανάληψης Γ΄ Γυμνασίου Μαθηματικά

  • !

    .

    .

  • 2

    ((()))

    111 . 1. 1

    1. >1; > 1,

    .

    , = : 1 =

    0 = 1 0 =

    1

    0 = 1, 2, 3, ..

    2. ;

    , , :

    . = + .

    = . = ()

    .

    =

    .

    =

    . ( )

    =

    .

    3. ;

    - x , , .

    : = x x2 = x, > 0 0 = 0

    4. ;

    0 ( )2 = 2 =

    A 0 0, =

    A 0 0, =

  • 3

    5. A 0 0 , =

    = 2 = 2.

    :

    = ( )2 = ( )2 ( ) ( )2 2 = = .

    6. A 0 > 0 , =

    = 2 = 2, :

    =

    2

    =

    2

    ( )( )

    2

    2

    =

    =

    , .

    . 1. 2

    7. ;

    - .

    8. ;

    .

    9. ; , -

    -

    .

    10. ;

    .

    - ,

    .

  • 4

    11. ;

    . 12. ;

    .

    .

    13. ;

    .

    14. ;

    0. -

    .

    15. ;

    .

    16. ;

    . 17. ;

    .

    . 1. 3

    18. ;

    . 19. ;

    .

    20. ;

    0. -

    .

    . 1. 4

    21. : . ;

    . ;

  • 5

    :

    .

    .

    .

    .

    . 1. 5

    22. ;

    .

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

    ii. ( )2 = 2 2 + 2 iii. ( + )3 = 3 + 32 + 32 + 3

    iv. ( )3 = 3 32 + 32 3 v. ( )( + ) = 2 2 vi. 3 3 = ( )( 2 + + 2)

    vii. 3 + 3 = ( + )( 2 + 2)

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

    = 3 + 2 2 + 2 + 2 + 2 2 + 3 = 3 + 3 2 + 3 2 + 3 iv. ( )3 = ( ) 2( ) = ( 2 2 + 2 )( ) =

    = 3 22 + 2 2 + 2 2 3 = 3 3 2 + 3 2 3 v. ( )( + ) = 2 + 2 vi. ( )( 2 + + 2) = 3 + 2 + 2 2 2 3 = 3 3

    vii. ( + )( 2 + 2) = 3 2 + 2 + 2 2 + 3 = 3 + 3 . 1. 6

    24. ;

    - .

    25. ;

  • 6

    -

    , -

    -

    .

    + = ( + )

    , : + = ( + ) ( + ) =

    ( + )( ) ,

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

    .

    -

    :

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

    .

    :

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

    ( + )2 ( )2,

    :

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

    x2 + ( + )x +

    x2 + ( + )x + :

    x2 + ( + )x + = x2 + ( + )x + = (x + )(x + )

  • 7

    x2 + x + x + =

    x(x + ) + (x + ) =

    (x + )(x + )

    . 1. 7

    26. ; -

    (x) () (x) () (x)0 -

    (x) () (x) (), :

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

    (x) (x)

    (x) (x).

    . 1. 8

    27. (...) - (...)

    ;

    (...)

    ,

    .

    (...)

    , -

    .

    . 1. 9

    28. ; .

    29. ;

    .

    30. ; , , ,

    .

    . 1. 10

    31. ;

    .

    :

  • 8

    +

    = +

    = -

    +

    = +

    = -

    0

    =

    :

    =

    =

    0

    =

    :

    =

    =

    0

    222 . 2. 1

    32. 1 ;

    1 x + = 0 0.

    ( ) . .

    .

    33. x + = 0 ;

    0, x + = x =

    = 0, 0 x + = 0 0x = (),

    = 0, = 0, x + = 0 0x = 0 ( ).

    . 2. 2

    34. 2 , ;

    x2 + x + = 0 , , 0.

    .

    x .

  • 9

    35.

    x2 + x + = 0 , , 0.

    - x2 + x + = 0 , ,

    0 : x2 + x + = 0

    42x2 + 4x + 4 = 0 [ 4]

    42x2 + 4x = 4 [ ]

    42x2 + 4x + 2 = 2 4 [ 2]

    (2x)2 + 22x + 2 = 2 4 [ (2 + )2] (2 + )2 = 2 4

    2 4 -

    (2 + )2 = 2 4 (2 + )2 = (i)

    0 (i) : (2 + )2 = ( ) 2 2x + = 2x =

    x =

    2

    < 0 ( I ) x 2 + x + = 0 , ,

    0 x = 2

    0

    36. : . ;

    . ;

    . ;

    2 + + = 0 , , , 0 = 2 4:

    . x =

    2 , > 0

    . x =

    2

    , = 0

    . , < 0

  • 10

    37. x2 + x + x2 + x + = 0

    0 1, 2;

    1, 2 x2 + x + = 0 0 x2 + x + :

    x2 + x + = (x 1)( x2) . 2. 4

    38. ; , .

    ,

    .

    . 2. 5

    39. ( ) ;

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

    > 0, . < 0, . = 0, . 40. ;

    > ( < ) , -, ( ) .

    < < ( > > ) ( )

    < > ( > < ) ( )

    x - y, x < < y.

    x , - x .

    41. ;

    , - . > , + > + .

  • 11

    , .

    > > , + > + .

    , .

    > > 0, > >

    .

    - , .

    > < 0, < <

    .

    .

    , , , > > > 333 . 3. 1

    42. ;

    x + y= . x + y= (x, y)

    .

    43. x + y= 0 0 ;

    x + y = 0 0 :

    , x + y = .

    x + y = - .

    44. ;

    . y = k k 0 . y = 0

    . x = k k 0 . x = 0

  • 12

    . y = k k 0 xx yy (0, k )

    . y = 0 xx.

    . x = k k 0 yy xx (k, 0)

    . x = 0 yy.

    45. x + y= 0 0 xx yy;

    xx 0, , x + y =

    xx, y = 0 x x + 0 = x = x =

    . (

    , 0)

    yy 0, B, x + y =

    yy, x = 0 y 0 +y = y = y =

    . B(0,

    )

    . 3. 2

    46. ; . x y;

    . x y;

    . x y;

    .

    , , , , 0. x + y = x + y =

    . x y

    (x0, y0) .

    . x

    y (x0, y0)

    .

    47. x y , , ;

    x

    y

    :

    .

  • 13

    , .

    () .

    444 . 4. 1

    48. y = x2 > 0;

    y = x2 > 0 - .

    y = x2 > 0 - (0, 0) xx ,

    x y 0.

    y = x2 > 0 y = 0, x = 0, x y, y =

    x2 > 0 yy. , . . 4. 1

    49. y = x2 < 0;

    y = x2 < 0 - .

    y = x2 < 0 - (0, 0) xx ,

    x y 0.

    y = x2 > 0 y = 0, x = 0, x y, y =

    x2 < 0 yy. , . . 4. 2

    50. ;

    y = x2 + x + 0.

  • 14

    51. y = x2 + x + 0;

    = x2 + x + 0 :

    ( 2

    ,

    4) = 2 4

    - x =

    2

    > 0, y = x2 + x + y = 4

    x =

    2

    < 0, y = x2 + x + y = 4

    x =

    2

    555 52. ; , -

    .

    53. ;

    Venn.

    54. ; , .

    55. ; ,

    .

    56. ; . -

    .

    57. , ; ,

    .

    58. , ; ,

    .

    59. ;

  • 15

    .

    60. ; , -

    .

    61. ; -

    .

    62. ; .

    ,

    .

    63. ;

    .

    -

    .

    64. ;

    = 0.

    65. ;

    .

    66. P() - ;

    - o P( ) =

    =

    ()

    ()

    : P( ) = ()

    ()= 1 P() = ( )

    ()= 0

    () 1. 67. ;

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

  • 16

    ((( ---)))

    111 . 1. 1

    68. ; A -

    .

    ,

    - .

    . 69. , ; :

    , , , , ,

    .

    A

    A

    :

    , , , , , . 70. , , , .

    -

    .

    , ,

    .

    .

    >90

  • 17

    , ,

    .

    .

    -

    , ,

    .

    71. ; ,

    (

    ) .

    A

    =

    :

    l

    =

    =

    =

    A= = =

    72. ; ( )

    (. . .)

    ,

    .

    A

    =

    :

    = A = = =

    + +

    ( . . . )

    -

    .

    A

    =

    :

    =

    A = =

    =

    +

    +

  • 18

    (. . .) ,

    .

    A

    =

    :

    =

    =

    =

    =

    +

    +

    73. ;

    ( )

    ,

    .

    :

    l l o=A = A 90 = =

    =

    ++

    , -

    -

    .

    :

    l l o= A =A 90 = =

    =

    ++

    , -

    .

    :

    1

    l l

    o=A =A 90 =

    =

    =

    ++

  • 19

    , -

    .

    :

    l l

    o= A =A 90 =

    =

    =

    ++

    , -

    .

    :

    l l

    o= A =A 90 =

    =

    =

    ++

    74. ;

    .

    .

    75. ;

    . . 76.

    , .

    . -

    . = .

    A

    N M

    // . , -

    ,

    . = .

    B

  • 20

    . 1. 2

    77. ;

    , -

    , = .

    - .

    78. , , ;

    ,

    =

    =

    , , , .

    , , , -

    .

    79. ; , , ,

    . :

    -

    .

    = =

    -

    .

    = = =

    -

    .

    + +

    = = =

    . 1. 3 80. -

    .

    ,

    .

    1 // 2 // 3

    =

    =

    1

    2

    3

  • 21

    - , .

    //

    =

    :

    ,

    .

    =

    //

    . 1. 4

    81. ;

    = . 82. ;

    .

    .

    : , > 1

    , 0 < < 1

    , = 1.

    . 1. 4

    83. ; , -

    .

    -

    ( ) .

    ,

    = , = , = , l = l , = l

    = = = = =

    .

    .

    A

    B

    A

    B

    A

    B

  • 22

    84. ; .

    . , 1. . . 85. ;

    - (-

    ) .

    A

    , = +

    + , = , =

    = =

    () -

    .

    86. ; ( )

    , .

    A

    = , = , ,, +

    +

    = = =

    . 1. 5

    87. ;

    .

    111 . 2. 1

    88. ;

    (0 / / 180 ) x,

    .

    A ( x, y ) nxOM = = = 2 2x + y - :

    B

    A

    B

  • 23

    y =

    yM(x, y)

    x =

    x y =

    x

    x

    y

    1 +1. 1 / / 1 1 / / 1

    . A (x, y) 1 , > 0, >0, >0 A (x, y) 2 , > 0,

  • 24

    90. ;

    180 - : (180 ) = (180 ) = (180 ) = 91. :

    . 2 +2 = 1 . =

    .

    2 2 + = 2 2y x+

    =

    2 2

    2 2y x+

    = y

    M(x, y)A(y)2 2

    2y + x

    = 2 2

    2

    y + x

    =

    2 2

    2 +

    x x B(x)

    2 2

    2 +

    = 2

    2

    =

    2

    2

    =1

    y

    .

    yM(x, y)

    y y y= = = =

    x x x

    A(y)

    x x B(x)y

    92. .

    : = =

    ( )

    ( = 90) :

    =

    (1) =

    ( = 90) :

    =

    (2) =

    ( 1 ), ( 2 ) :

  • 25

    = =

    (3)

    =

    (4)

    (3), (4)

    = =

    93. .

    2 = 2 + 2 2 2 = 2 + 2 2 2 = 2 + 2 2

    ( ) 2 = 2 + 2 2.

    ( = 90) :

    =

    (1) =

    :

    2 + 2 = 2 (2)

    ( = 90 ) :

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

    2 + 2 2 + 2 2 + 2 2 (2

    =) (1

    =)

    2 + 2 2 :

    2 = 2 + 2 2 2 = 2 + 2 2

    x , , . = ( ( A ( 0 ( 0, = A ( 0 ( 0, = , .

    . . = . 2 + 2 ( 2