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سلسلة ديماديما الباكالوريا علوم تجريبية

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  • W

    "!

    & %$#"! :

  • 2

  • 3

    @ @

    W&% W#"!

    4 &%-,+)*)#(J &%$# 5 2,-!1( J 0 /.

    6 " 8 5-43 01 5%/6

    21 /0509J !8,)+J 7,)+ 31 !:*" 41 #

    61 n#=!-

    22 + 42 #3?,

    62 #3@ 82 #(/#

    13 (5A 23 C#@B

    43 #( 63 5$,5

    83 FEED

  • 4

    '()*

    +(,-.-/01*' 2K4(05

    (,*'Wax b+( )0a

    + b

    a x

    %&a %&Ga ax b+

    +(,-.-/01*'Wax bx c+ +( )0a

    HBW( ) x ax bx c = + +

    ,,,, (+$(+$(+$(+$WWWW

    ( ) 0x x =

    %&%&%&%&( )x +,-+,-+,-+,-( )x

    0 < S =

    + x

    %&a ( )x

    =JK? L#$0@

    0 = { }2

    bS

    a

    =

    + ba

    x

    %&a

    %&a

    ( )x

    ( )

    2

    bx a x

    a

    = +

    = b - 4ac

    0 >

    { };1 2S x x= M$W

    1 2

    bx

    a

    =

    2 2

    bx

    a

    + =

    + 2x 1x

    x

    %&

    a

    G%&a

    %&

    a ( )x

    FOPQW1 2x x

  • 5

    6789 +:10;+( 2K4(05

    6789W

    =(#+L//$ab ( )2 2 22a b a ab b+ = + +

    ( )2 2 22a b a ab b = + ( )( )2 2a b a b a b = +

    ( )3 3 2 2 33 3a b a a b ab b+ = + + +

    ( )3 3 2 2 33 3a b a a b ab b = +

    ( )( )3 3 2 2a b a b a ab b = + + ( )( )3 3 2 2a b a b a ab b+ = + +

    (>>>

    ( )( )

    ( )

    xf x

    Q x

    = ( ) }0Q x >>>>( ){ / 0fD x x=

    ( )( )

    ( )

    xf x

    Q x

    = ( ) }0Q x ( )

    ( )/ 0f

    xD x

    Q x

    =

  • 6

    =)>?1( 2K4(05

    4(=( )* nn x x x x 7=18W

    0lim 0nx

    x

    = 0

    lim 0x

    x

    =

    >>>>

    1lim 0n

    x x=

    1lim 0n

    x x+=

    limx

    x+

    = +

    1lim 0

    x x+=

    O8S&O8S&O8S&O8S&nOTUX(#OTUX(#OTUX(#OTUX(#WWWW O8S&O8S&O8S&O8S&nOTU(!U(#OTU(!U(#OTU(!U(#OTU(!U(#WWWW lim n

    xx

    += +

    lim nx

    x

    = +

    0

    1lim nx x

    = +>>>>

    0

    1lim nx x

    = +>

    0

    1lim nx x

    = ^/

  • 7

    B1C=W

    ( ) ( )

    ( )( )

    00

    limlim 0 x x

    x x

    f x V x

    f xV x

    ==

    l

    l

    ( ) ( ) ( )

    ( )

    ( )

    ( )0 0

    0

    lim lim

    lim

    x x x x

    x x

    u x f x V x

    u x f x

    V x

    = ==

    l l

    l

    ( ) ( )

    ( )( )

    00

    limlim x x

    x x

    u x f x

    f xu x

    = += +

    ( ) ( )

    ( )( )

    00

    limlim x x

    x x

    u x V x

    f xV x

    = =

    ]Z."#>^/>>>l 0>>>l 0

  • 8

    41J2K4(05

    97K41JW :1W ( ) ( )

    00lim

    x xf f x f x

    =a40x

    E0D41JD41JW

    ( ) ( )0

    0limx x

    f f x f x>

    =^4L,a0x

    ( ) ( )0

    0limx x

    f f x f x 32D.4# 32>

    NHJ)W=f32^0g

  • 9

    3d3d3d3d( )f I 3d3d3d3dI

    f^0g#-^0g#-^0g#-^0g#-I f^0g4g-^0g4g-^0g4g-^0g4g-I [ ],a b ( ) ( )[ ];f a f b ( ) ( )[ ];f b f a

    [ [,a b ( ) ( ); limx b

    f a f x

    ( ) ( )lim ;

    x bf x f a

    ] ],a b ( ) ( )lim ;x a

    f x f b+

    ( ) ( ); lim

    x af b f x

    +

    ] [,a b ( ) ( )lim ; limx bx a

    f x f x+

    ( ) ( )lim ; lim

    x b x af x f x +

    [ [,a + ( ) ( ); limx

    f a f x+

    ( ) ( )lim ;

    xf x f a

    +

    ] [,a + ( ) ( )lim ; limxx a

    f x f x+ +

    ( ) ( )lim ; lim

    x x af x f x

    ++

    ] ],a ( ) ( )lim ;x

    f x f a

    ( ) ( ); lim

    xf a f x

    ] [,a ( ) ( )lim ; limx x a

    f x f x

    ( ) ( )lim ; lim

    xx af x f x

    ( ) ( )lim ; limx x

    f x f x +

    ( ) ( )lim ; lim

    x xf x f x

    +

    9O76PW c8S&f32^4[ ],a bD//$(#+jTU=(#L 47( )f a( )f b

    D//$(#+g^#d=3[ ],a bMfW ( )f =

    W

    c8S&f32^4[ ],a b( ) ( ) 0f a f b

  • 10

    S7J 2K4(05

    +(KS7J8TW

    3/&(Of(#a6/%* g0x"c8S&W ( ) ( )0

    0 0

    limx x

    f x f x

    x x

    "

    #ep(#^,-"]Z.fa0x! j!W( )0'f x

    +D%#U0#4+J+D%#0#&W(W

    =fa6/%* g(0x #^nq,(fC4UPr0/a0xD.W ( )( ) ( )0 0 0'y f x x x f x= + #u^U!D,8W ( ) ( )( ) ( )0 0 0'u x f x x x f x= +

    #^n@,s#^,-fC4UPr0/a0x#t!/-D.fu0x

    E0DS7J8TJE0DS7J8TW

    3/&(OfaL,^6/%* g0x"c8S&( ) ( )

    0

    0

    0

    limx x

    f x f x

    x x

    >

    "

    #ep(#^,-"]Z.f^L,a0x! j!W ( )0'f xd

    3/&(Of^6/%* g a0x"c8S&W ( ) ( )

    0

    0

    0

    limx x

    f x f x

    x x

    0 <

    limn

    n

    += + lim 0

    nn

    +=

    (Z#=( )nqa)W q W

    1q > 1q = 1 1q < < 1q

    lim nn

    q+

    = + lim 1nn

    q+

    = lim 0nn

    q+

    = ( )nq

    "CG

    ^7b+W

  • 20

    4(Nc 2K4(05

    4;D(Nc4(W

    :1W

    =f32^U!(#(I

    OP3/F#>P(D.f3d^I

    OOh!pe/yS&W

    F/%* g3d^6I ( ) ( )'x I F x f x =

    NHW

    3dZ.^>P(+P(f3d^IOTUW

    #>3#Hzf^U!IDWW ( ) ( )x F x k k+

    =f+PI

    =0x=!4I0y=!4

    $>P(#-#F#f3d^I

    D#PL(fg3d^IOTUD^W

    F G+#>P(f g+^3dI

    kF#>P(kf^3dI

  • 21

    +J4(

  • 22

    Q^,2K4(05

    9TD+/1W :1W

    =f32^4(IF#>P(f3d^I ab3d==!4I

    #+-f=ak&bD//(#.W

    ( ) ( )[ ] ( ) ( )ba

    bf x dx F x F b F a

    a= =

    NHW 9fW

    ( ) 0a

    f x dxa

    = ( ) ( )a b

    f x dx f x dxb a

    =

    ( ) ( ) ( )b b

    k kf x dx k f x dxa a

    = ( ) ( )[ ] ( ) ( )b b b

    f x g x dx f x dx g x dxa a a

    + = + 4T.W

    ( ) ( ) ( )b c b

    f x dx f x dx f x dxa a c

    = + B1C/W

    O8S&W [ ] ( ), 0x a b f x

    OTUW( ) 0b f x dxa

    O8S&W[ ] ( ) ( ),x a b f x g x

    OTUW( ) ( )b bf x dx g x dxa a

    9#07W

    =f32^4([ ],a b

    3d^#0@,/D.D//(#W ( )1 b f x dxab a

    F\ c8#W =uv32^6/%*L gL(IL#Mfuvd^L43I

    ab3d==!4I

    ( ) ( ) ( ) ( )[ ] ( ) ( )bab bu x v x dx u x v x u x v x dx

    a a =

    \))

    #ok& =( ), ,o i j

    $#$ .u A(#+0$D.0/ oL"|i

    j

    1. .u A i j=

  • 23

    =f32^4([ ],a b ^ nL 4$fC+>U7

    ,.(=ZL,/W x a=x b=

    D.W ( ) . .b f x dx u Aa

    =fg32^L4L([ ],a b LnL 4$fCgC7

    >U,.(=ZL,/+Wx a=x b=D.W

    D.W ( ) ( ) . .b f x g x dx u Aa

    NHJ)W o@o@o@o@ DnA-DnA-DnA-DnA- $*$*$*$* $$$$D|

  • 24

    0g4(2K4(05

    0g(> :1W

    (#>#D.vK >

    ] [0;

    ln y

    x y

    x y x e

    +

    = =

    ] [ ] [

    ( )

    ( )

    0; 0;

    ln ln ln

    ln ln

    1ln ln

    ln ln ln

    r

    x y

    xy x y

    x r x

    xx

    xx y

    y

    + +

    = +

    =

    = =

    ( )r

    O8S&nX(# OTUW( )* ln lnnx x n x = ;:0W

    ####fD,8U!D,8U!D,8U!D,8U!WWWW #1!-,2#1!-,2#1!-,2#1!-,2fD.D.D.D.WWWW ( ) ( )[ ]lnf x u x= ( ) }0u x > { /f uD x x D=

    ( ) ( )( )2lnf x u x=

    ( ) ( )lnf x u x= ( ) }0u x { /f uD x x D=

    A=W

    ( )lim lnx

    x+

    = + lnlim 0nx

    x

    x+=

    ( )0

    lim lnx

    x

    = >

    ( )0

    lim ln 0nx

    x x

    =>

    1

    lnlim 1

    1x

    x

    x=

    ( )0

    ln 1lim 1x

    x

    x

    +=

    ( )n *

    41JW

    #lnx x3d^4] [0;+

    =uU!(^32I c8S&u32^40g

  • 25

    S7JW

    #lnx x6/%* g^] [0;+

    #:

    ] [ ( )1

    0; lnx xx

    + =

    =uU!(^32I c8S&u32^6/%* g0g

  • 26

    c4(2K4(05

    0g(> :1W

    #@K > > > > > > >

    x y x y x ye e e + =

    ( )r ( )rx rxe e= 1 xx ee

    =

    xx y

    y

    ee

    e

    =

    :0;W

    ####fD,8U!D,8U!D,8U!D,8U!WWWW #1!-,2#1!-,2#1!-,2#1!-,2fD.D.D.D.WWWW ( ) xf x e= fD =

    ( )( )u x

    f x e= { }/f uD x x D= A=W

    lim xx

    e+

    = +

    lim 0xx

    e

    =

    limx

    nx

    e

    x+

    = +

    ( )lim 0n xx

    x e

    =

    0

    1lim 1

    x

    x

    e

    x

    =

    ( )n *

    41JW

    #xx e^4

    =u(U!^32I c8S&u^4d3I#OTU( )u xx e3d^4I

  • 27

    S7JW

    #xx e g 6/%*^#W( )x xx e e =

    =uU!(^32I c8S&u^6/%* g d3IOTUW#( )u xx e3d^6/%* gI

    #W ( )( ) ( ) ( )'u x u xx I ue x e = (Q#/0lnW

    (Uhcaa)W}{1a +

    :1W

    ##aog lq@@#^,-a ! C!Wexpa

    +HBx=( )exp xa x a= NHW

    lnx x ax a e = ( )xaog a x=l

    ] [ ( )0; og xax a a + = l ( ) 2; x yx y a a x y = =

    ] [0;x y + ( )aog y l

    xa y x= =

    ( ) 2;x y x y x ya a a + =

    ( )r ( )rx rxa a= 1 xx aa

    = x

    x yy

    aa

    a

    =

    =1&W

    1a > 0 1a< < x ya a x y> > x ya a x y< <

    lim xx

    a+

    = +

    lim 0xx

    a

    =

    lim 0xx

    a+

    =

    lim xx

    a

    = +

    0

    1lim ln

    x

    x

    aa

    x

    =

    7X#W ( ) ( )lnx xa a a =

  • 28

    (7+(c 2K4(05

    D.#/(#,2W } 1i = ( ){ / ; z a ib a b= = + i(7+(P@8W

    =z a ib= +$#/(#MW( ); a b a ib+v#/(#Y; ^,-z (#a(#D//;^,z! j!W( )Re z (#b(#D;^,z! j!W ( )Im z

    jNHj)W O8S&W ( )Im 0z =OTUzD//$(#. O8S&W ( )Re 0z =( )Im 0z OTUzU!>(#^,

    E(7+(i1W =zz L#/=(#

    ( ) ( )Im Imz z =( ) ( )Re Rez z z z = = #/0i(7+(QW oJ#ok& v#/=( )1 2, ,o e e

    i(7+(bkW =z a ib= +M$#/(#W( ); a b

    (#eU!zv#/(#.Wz a ib= ( )M z( )M zD//n,

  • 29

    ( )

    ( )

    *

    ' 0

    n nz z n

    z z

    z zz

    z z

    =

    =

    =

    1 1

    z z z z

    z z

    z z

    =

    =

    =

    /XQ#(>gi(7+(c8W

    =zj->#K?#/(#M v#/(##,z."@g#$PW( ),1 OMe

    ! j!Wargz tW[ ]arg 2z pi=

    =z#K?#/(# HBr z=[ ]arg 2z pi=

    #DEE+pv#/(z.W ( ) [ ]cos sin ,z r i r = + =

    v#/(#@ zD.Wiz re =

    NHJ)W D//$(#EE a#K?

    0a > 0a AYNW

    ( ) ( ) ( )cos sin cos sinnn

    i n n i n

    + = +

    ( )1cos2

    i ie e = +

    ( )1sin2

    i ie ei

    =

    +#/)z z a =

    a)( )a W ((((WWWW 22223$,3$,3$,3$,((((WWWW

    0a > { };S a a= 0a = { }0S = z z a = 0a < { };S i a i a=

  • 30

    +#/)W 0z az bz c + + =

    a)Wab c77)+(A( )0a

    ((((WWWW (3$,2(3$,2(3$,2(3$,2WWWW

    0 > ;2 2

    b bS

    a a

    + =

    0 = { }2b

    Sa

    =

    ( )

    2

    2

    0

    4

    z az bz c

    b ac

    + + =

    =

    0 < ;2 2

    b i b iS

    a a

    + =

    (7+(c%9(6O6&W

    D@#C"D@#C"D@#C"D@#C" #/g*#/g*#/g*#/g* UAB B AAB z z=

    I0/14[ ];A B 2

    A BI

    z zz

    +=

    qg( );AB AC ( ) [ ]; arg 2c AB A

    z zAB AC

    z zpi

    ABC,// C AB A

    z z

    z z

    ABCD#/ D A B CB A D C

    z z z z

    z z z z

    PD A D C

    B A B C

    z z z z

    z z z z

    #/g*#/g*#/g*#/g* D@#C"D@#C"D@#C"D@#C"

    Az z r = ( )0r >

    AM r= M.8!r!#k&D,-A"%r

    A Bz z z z = AM BM= M@k&D,-[ ]AB

    ;2

    C A

    B A

    z zr

    z z

    pi =

    ABCaogMEA

    [ ]1;C AB A

    z z

    z z

    =

    ABCvMEaLgA

    1;2

    C A

    B A

    z z

    z z

    pi =

    ABCaogLgvMEA

    1;3

    C A

    B A

    z z

    z z

    pi =

    ABC:*AvME

  • 31

    "&J+#

    2K

    4(05

    A(A(A(A(WWWW A(,+A(,+A(,+A(,+WWWW

    'y ay b= + ( )0a

    ( ) axb

    y x ea

    =

    ( )

    A(A(A(A(WWWW ,"(,"(,"(,"(WWWW +

  • 32

    Mm&(Z 2K4(05

    oJ#ok& B=Z.6@a!%

  • 33

    U(( )S.0gP#$P[ ]AB@5 .##y=mD~U W ( ) . 0M S AM BM =

    $*W( )S.8!14[ ]AB"%2

    AB

    knR71( ),S R`( ) : 0ax by cz d + + + =

    =H,/8!,v(^( ) HBW( )( );d H d= =

    ( )P( )S 5Oh/

    ( )PqJ( )S 0/aH

    ( )PH0/( )S !(eU( )C

    .8!WH

    "%W2 2r R d=

    knR71( ),S RO7( )W =H8!,v(,/o/^( ) HBW( )( );d H d= =

    o/( )( )S

    5Oh/ o/( )qJ( )S

    0/aH o/( )H0/( )S

    LL0/a

  • 34

    +(2K4(05

    0;QMW

    :1W

    ",2DE,d!>(#.E! j!WCardE NH)W 0Card =

    NHW

    ABO"O,2 ( ) ( )Card A B CardA CardB Card A B = +

    0;O0W

    :1W

    =A",2=E o,A ,|,

  • 35

    (81Cj8W

    =np==!4*( )p n

  • 36

    J0)J2K4(05

    %9

    i04i04i04i04D,$5D,$5D,$5D,$5WWWW ]]]]WWWW p !} +

  • 37

    QRX40)JJE-().7W :1W =ABMfp !|G L0

  • 38

    Q#^,)>?1(2K4(05

    #*M(+JO74( W

    #BE8T.W

    -1 cos 1

    -1 sin 1

    cos sin 1

    x

    x

    x x

    + =

    sintan

    cos1

    1 tan cos

    xx

    x

    xx

    =

    + =

    ( )

    ( )

    ( )

    cos 2 cos

    sin 2 sin

    tan tan

    x k x

    x k x

    x k x

    pi

    pi

    pi

    + =

    + =

    + =

    AJ+W

    - 2x a kpi= + Pcos cos 2x a x a kpi= = + ( )- 2x a kpi pi= + P sin sin 2x a x a kpi= = +

    ( ) tan tanx a x a k kpi= = +

    2

    pi

    3

    pi

    4

    pi

    6

    pi 0 x

    1 3

    2

    2

    2

    1

    2 0 sinx

    0 1

    2

    2

    2

    3

    2 1 cosx

    3 1 3

    3 0 tanx

    2x

    pi+

    2x

    pi +xpi xpi x

    cosx cosx -sinx sinx -sinx sin

    sinx sinx - cosx - cosx cosx cos

  • 39

    !0;/_eNW ( )

    ( )

    ( )

    cos cos cos - sin sin

    sin sin cos cos sin

    tan tantan

    1 - tan tan

    a b a b a b

    a b a b a b

    a ba b

    a b

    + =

    + = +

    ++ =

    ( )

    ( )

    ( )

    cos - cos cos sin sin

    sin - sin cos - cos sin

    tan - tantan -

    1 tan tan

    a b a b a b

    a b a b a b

    a ba b

    a b

    = +

    =

    =+

    LMW

    cos 2 cos - sin

    2 cos - 1

    1 - 2 sin

    sin 2 2 sin cos

    2 tantan 2

    1 - tan

    a a a

    a

    a

    a a a

    aa

    a

    =

    =

    =

    =

    =

    1 cos 2cos

    21 - cos 2

    sin 2

    aa

    aa

    +=

    =

    HA W tan2

    at =

    2sin

    1

    1 - cos

    1 2

    tan1 -

    ta

    t

    ta

    t

    ta

    t

    =+

    =+

    =

    !0;r'F( /_WF( r'!0;/_W

    ( ) ( )[ ]

    ( ) ( )[ ]

    ( ) ( )[ ]

    ( ) ( )[ ]

    1cos cos cos cos -

    21

    sin sin cos cos2

    1sin cos sin sin

    21

    cos sin sin - sin2

    a b a b a b

    a b a b a b

    a b a b a b

    a b a b a b

    = + +

    = +

    = +

    = +

    cos cos 2cos cos2 2

    cos cos 2 sin sin2 2

    sin sin 2 sin cos2 2

    sin sin 2cos sin2 2

    p q p qp q

    p q p qp q

    p q p qp q

    p q p qp q

    + + = + =

    + + = + =

    /_Wcos sina x b x+( ) ( ), 0,0a b

    ( )

    cos sin cos sin

    cos

    a ba x b x a b x x

    a b a b

    a b x

    + = + + + +

    = +

    M$e/D//$(#W

    sin

    b

    a b =

    +cos

    a

    a b =

    +