ΜΑΘΗΜΑΤΙΚΑ ΓΕΝΙΚΗΣ ΠΑΙΔΕΙΑΣ Γ ΛΥΚΕΙΟΥ ,MATHEMATICA.GR

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Transcript of ΜΑΘΗΜΑΤΙΚΑ ΓΕΝΙΚΗΣ ΠΑΙΔΕΙΑΣ Γ ΛΥΚΕΙΟΥ ,MATHEMATICA.GR

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    2010 11

    24

    2011

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    www.mathematica.gr

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    ii),

    1 2 2009 1 2 20091 2 2009 x 1 x 1 ... x 1 2009 x x ...xy y ... yy 1 x 1 2 12009 2009 2009

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    ii. fi% xi.

    iii. 2 6 ;

    iv. 2 5 ;

    v. 1 5 ;

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    ix.

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    5

    :

    iii. 2 6 20%+30%=50%

    iv. 2 5 20%+15%=35%

    v. 1 5 5%+20%+15%=40%

    vi. 5 45%.

    : x

    5 60 x 6 5

    2x 90 x 45%x 30 5 4

    vii. 5 100%-45%=55%

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    .

    14.75

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    7

    : f(x) = x - 2 , :

    x -3, :

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    x 3 x 3

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    fC (-6, -2) : 22 : y f( 6) f ( 6)(x 6 : y 3x 16)

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    ) 5 ,

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    12

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    P M P A P A P A140 140 140

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    , N A A A

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    3 21 1 3 5f (x) x x P(A B) x ,x R

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    ii. : 2B P A B P B P A B 5 (3)

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    1 1 3

    P A B P A P A B P A B5 5 5

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    (3) (4) 1 2

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    :

    ={ , 2, 4, 5 , 10}

    ={ x 2ln(x x 1) 0 }

    i.

    ii. ={3,4} ={2,3,4}

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    iii. P(A B) P(A B)

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    x =2 x =3 x = 4. , B 2,3,4

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    3 4 7

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    . (3+4):2=3,5

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    f x=10. : f (x) f (10) , x 5 . :

    f( ) f(10) , 5 10P(A ) P(A ) , 5 . ,

    =10 P(A ) . : 0 10 .

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    2 2 2

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    13,5%+14%=27,5%, : (27,5*800)/100=220

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    1) .

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    ( )

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    1

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    ( )

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    1. x

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    .

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    19 ( )

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    25

    ( ):

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    2s 21 16 5 s 5 .

    : 0 21 . 0 0 0

    21 (xgastone)

    . .

    .

    :

    16

    16 + 2x

    16 ln(ex)

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    : |

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    x 1/e e15

    .

    ) ln ex

    2x

    )

    ln exf x

    2x ln ex

    ) x

    ) .

    ( )

    ) ix : iy : , :

    i i i ix y

    16 x y 16 (1)

    i ix

    16 2x x 16 2x (2)

    i ix

    16 2x x 16 2x (2)

    (2), (3) (1) . ln ex2x

    ) ,

    ln ex

    ln ex2xP Aln ex 2x ln ex

    12x

    )

    15

    ln ex 1f (x) , x e

    2x ln ex e

    ,

    2

    2 1 ln exf x ...

    2x ln ex

    x = 1 f

    ) , lne 1

    P A2 lne 3

    1 2

    P B P A 1 P A 1 2P A3 3

    22 (panathas13)

    , ' 80% 60%

    . ' ,

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    3/5.

    ) .

    ) ' ,

    .

    ) 5, ' .

    ( )

    .

    :

    * 0,8 0,4 ,

    * 0,6 0,2 .

    ) (-) 3/5, :

    0, 4a 0,8 3... a

    a 5

    ) 0,6, :

    0,6 0,60,3 30%

    a 2

    .

    ) : 0,2=5, = 25, 50 .

    23 ( )

    , , 17, . 2

    16,

    )

    ) , .

    )( ) , 1 2 1w 3 4 2w , 1 2w ,w 0 , :

    i) O 15 16

    ii) ,

    : 2 1w 3w

    iii) ,

    .

    (sorfan)

    ) ( )

    17 17 32 152

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    28

    ) R 2 2 2

    , : ,15,17, 2 .

    15 ( ), 3 4

    2 17 15 ( + 2 17,

    ), = 15, : 15, 15, 17, 17

    )( )

    1 1 2 2 1 2.

    1 1 2 2 1 2

    15 w 15 w 17 w 17 w 15 w 17 wx

    w w w w w w

    i) : .15 x 17

    , 1 2. 21 2

    15 w 17 w15 x 15 0 w

    w w

    1 2. 11 2

    15 w 17 wx 17 17 0 w

    w w

    .15 x 17

    ii) , .15 15 17 17

    x 164

    ,

    . .x x 0,5 16 0,5 16,5

    21 2

    . 1 2 1 2 2 1 2 1

    1 2

    15 w 17 wx 16,5 16,5 15 w 17 w 16,5 w 16,5 w 0,5 w 1,5 w w 3w

    w w

    iii) ,

    1 2w w

    11 2 1 1. .

    1 2 1 1 1

    15 17 w15 w 17 w 15 w 17 wx 16 xw w w w 2w

    : i iii , ,

    .min maxx x x , .x : . .1 2 vw w ... w x x , 1 2 vw ,w ,...,w

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    24 ( ..)

    ={1,2,3,,10}

    P 1 P 2 P 3 P 4 2P 5 , 1

    P 9 P 10 P 12

    P 5 P 6 P 7 3P 8 .

    2A x / x 7x 10 0 , A B 3,4,5 , = {1, 8}

    2

    3 2

    2x 1

    4 2 x 3 / lim 9 20 11

    2x 5x 3

    .

    )

    ) , , .

    ) A , B,A B

    ( , Rania )

    ) P(8)=m P(5)=P(6)=P(7)=3m P(1)=P(2)=P(3)=P(4)=6m P(9)=P(10)=3m

    ,10

    i 1

    1P(i) 1 m 15m 24m 1 m

    40

    :1

    P(8)40 ,

    3P(5) P(6) P(7) P(9) P(10) 40

    6P(1) P(2) P(3) P(4)

    40

    ) A x / (x 2)(x 7) 0 A 2,3,4,5,6,7

    27

    P(A) P(2) P(3) P(4) P(5) P(6) P(7)40

    B A B A

    B B (A A ) (B A ) (B A) 1,3, 4,5,8

    22P(B) P(1) P(3) P(4) P(5) P(8)

    40

    2

    2x 1

    4x 4lim 1

    (2x 3)(x 1)(4 2 x 3)

    orner 1 vieta 1,2,6

    ) :

    ={2, 3, 4, 5, 6, 7} ' ={1, 8, 9}

    ={1, 3, 4, 5, 8} ' = {2, 6, 7, 9, 10}

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    ={1, 2, 6} ' = {3, 4, 5, 7, 8, 9, 10}

    :

    A 2,3,4,5,6,7,8,9,10

    B 3,4,5,8

    A B 9 .

    6 3 1 34P(A ) P(2) P(3) P(4) P(5) P(6) P(7) P(8) P(9) P(10) 3 5

    40 40 40 40

    6 3 1 16

    P( B) P(3) P(4) P(5) P(8) 2 40 40 40 40

    3P(A B ) P(9)

    40