ΘΕΜΑΤΑ ΕΜΕ Γ ΛΥΚΕΙΟΥ 2014

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ΘΕΜΑΤΑ ΕΠΑΝΑΛΗΨΗΣ Γ ΛΥΚΕΙΟΥ ΕΜΕ 2014

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  • skas

    iarxeio

    pro.blogspot.gr

  • 1

    = +

    -

    - -

    + + + ... +=

    + + + ... +

    - + + =

    = = -1

    + ++ = -

    + = -

    :

    i 5 = i 41+1 = i , i 9 = i42+1 = i ,.., i 4-3 = i 4( -1)+1 = i

    z 2 = (x + xi)2 = x2 + 2x2i - x2 = 2x2i

    :

    z 4 + z 8 + z12 + ... + z 4

    i z 2 + i 5z 6 + i 9z10 + ... + i 4-3z 4-2 =

    z2 (z2 + z6 + z10 + ... + z4-2 )=

    i (z2 + z 6 + z 10 + ... + z4-2 ) =

    z 2 2x2i 2= = = 2x2 = Im(z )i i

  • 2

    :z3 = z z2 = (x + xi) 2x2i = -2x3 + 2x3i

    :

    z 3 - z2 + z + i = 0 (-2x 3 + 2x 3i) - 2x 2i + (x - xi) + i = 0

    - 2x3 + x = 0( x3 x ) ( x3 x2 x 1) i 0

    x3 x2 x 1 0 -2 + + 2 - 2 - + =

    2 - 2 - + =

    2x3-x = 0 2x3-x = 0 x (2x2 -1) = 0

    ( x3 x) x2 1 0 x2 1 0 x2 1 0

    2

    -

    - 2

    +

    =

    - 2

    +

    =

    2 - =

    2x2 -1= 0 2x2 =1 x2 = 1

    x = 2

    2 2:

    z1 = 2

    (1+ i)2

    z2 = - 2

    (1+ i)2

    :

    4 2z1 (z1) (1 i) (1 i 1) 4i 12

    = 2 = 1 + 2 =

    1 + 2 - 2 = 1 2 = - 2 4 4

    (1)

    z2 = - 2

    (1+ i) = -z12(2)

    :

    4 4 42 1 1

    (1)

    z = (-z) = z = -1

    :2 2 3 2 3 2

    1 1 2 2 1 1 2 2

    2 1 1 2

    (2)z + z+

    z + z=

    z + z + z + z=

    z z zz

    3 2 3 2

    = z1 +z1 + (-z1) + (-z1) =

    z1(- z1)3 2 3 2 2z1 z1 z1 z1 2z1

    2 21 1

    2z z

    + - += = = -

    - -

    :

    z12014 z1

    4503+2 (z14 )503

    z12

    (1)

    ( 1)503 z12 z1

    2= = = - = - =

    2

    (1 i) (1 i 1) i i 2 1 1

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

    2014 (2)

    2014 2014 (3)

    z2 = (-z1) =z1 = - i

    :2014 2014z1 + z2 = (-i)+ (-i) = -2i

  • 3

    - + =

    = + =

    =

    4w 2 -2w +1 = 0 D = (-2)2 - 4 4 1 = -12 < 0

    : w = 1

    i3

    4 4

    z1

    2

    = wz

    . z1 = 1 z1 0 , w 0 , (2)

    :

    4w + 1

    =2 4w 2 -2w + 1 =0 w =

    1

    i

    3

    z1 = 1

    i

    3

    w 4 4 z 2 4 4

    z1 =

    1 i

    3=

    1,

    z2 4 4 2

    (1 1)z1

    2

    z 2 2z 2= =

    , 0 , z1 z 2 , :

    2 2 2(O) = z1 = 1 = 1, (OB)2 = z2

    2 = 22 = 4

    :2 2 2 2 2 24z1 + z2 = 2z1z2 (z1- z2 ) = -3z1 |z1- z2| = 3 |z1| |z1- z2|= 3

    :2() = 3 () = 3

    :

    (OB)2 =(O)2 +(B)2 ,

    .

    O=900

    (OA) = (OB)

    2,

    OBA 300

    = AOB 600

    =

  • 4

    + + - = +

    - =

    - = + -

    :

    2 2 ( )( )w1 w2 w1 w2 w1 w2 w1 w2 ( w1 w2 ) (w1 w2 )+ + - = + + + - - =

    = ( w1 + w2 )( w1 + w2 ) + ( w1 - w2 )( w1 - w2 ) =

    = w1w1 + w1w2 + w1w2 + w2w2 + w1w1 - w1w2 - w1w 2 + w 2w 2 =

    2 2= 2w1w1 + 2w2w2 = 2w1 + 2w2

    z = x + yi, x ,yR M(x, y)

    z -2i =1 z-( 0 +2i ) =1 ()=1, (z) z (0 ,2)z (0,2)

    1

    z

    (0,2) =1, z -2i =1

    z1, z 2 z -2i =1,:

    z1 - 2i =1 z2 - 2i =1

    w1 = z1 - 2i w 2 =z2 - 2i ()

    :

    (z1 - 2i) + ( z2 - 2i)2 + (z1 - 2i) - ( z2 - 2i)

    2 = 2 z1 - 2i

    2 + 2 z2 - 2i2

    2 2 2 2z1 + z2 - 4i + z1 - z2 =2 1 + 2 1

    2z1 + z2 - 4i +1= 4

    z1 + z2 - 4i = 3

  • 5

    ( - ) = ( - )

    = +

    =

    = -

    -

    (1):

    (3z -2i)2013 = (iz -6)2013 (3z-2i) 2013= ( iz -6) 2013

    ( 3z - 2i) = (i z - 6) ( 3z - 2i) 2 = (i z - 6) 2

    ( )( ) ( )( ) 3 z - 2 i 3 z + 2 i = iz -6 - iz -6

    9 z 2 + 6iz - 6i z + 4 = z 2 + 6iz - 6i z + 36

    8 z 2 = 32 z 2 = 4 z = 2

    :

    z = 2 z 2 = 4 z z = 4 4

    = z (3)z

    z = x + yi x ,yR ,:

    42014 (3)

    u = z - = (z- z)2014 =(2 yi)2014 = z

    = (2 y)2014 i2014 = (2 y)2014 i2 = - (2 y)2014 R

    u R

    :

    2014 2014 ( )2014 4028(3)2014

    u = z - 4 = (z- z) = z- z z + z = 2 z

    :

    u 24028

  • 6

    w1 = 1+ 2 z1 i w 2 = 1+ 2 z2 i, z1, z 2

    (1), z1 = z2 = 2 . z1 , z2 O(0, 0) r = 2 , z1 -z2 4

    :

    w1 - w 2 = 1+ 2z1i -1- 2z2i = 2i (z1 - z2 ) =

    = 2 i z1 - z2 = 2 z1 - z2 = 2 z1 - z2 2 4 = 8

    - + = + + + =

    + + +

    - + +

    + +

    :

    |z 2 - 2zi| +2|z| = 2+ |z + 2i|

    |z| |z - 2i| + 2|z| = 2+ |z - 2i|

    | z | | z - 2i | - | z - 2i | +2 | z | -2 = 0

    ( ) (|z - 2i| |z| -1 + 2 |z| -1) = 0

    (| z | -1)(| z - 2i | +2 ) = 0 |z| = 1 (1), | z - 2i | +2 0 z C: x2 + y2 =1 (0,0) r =1

    w :

    |w +3z | + | 2w +5z |= |w +3z | +2w + 5 z2

    |w +3z |+ w + 5 z (w +3z) - (w + 5 z) = 1 | z |= 1 1= 12 2 2 2 2

    1|w + 3z| + |2w + 5z|

    2

  • 7

    :(1)

    22| z -1 + z +1| | z +1 + z | +1 = 4.:

    | z -1 + z2 +1| | z -1+ z2 +1| = | z(z +1) | = | z | | z +1 | (2)

    | z |=1 (2) | z -1 + z2+1| | z +1| (3)

    | z -1 + z2+1| | z -1| (4)

    z = x + yi x,y[ -1,1] : | z | =1| z |2 =1 x 2 + y 2 =1 (5)

    (5)

    2 2 2 2| z +1| = (x +1) + y = x + y + 2x +1 = 1+ 2x +1= 2(1+x) = 2 1+x (6)

    (5)

    2 2 2 2| z -1| = (x -1) + y = x + y - 2x +1 = 1- 2x +1= 2(1-x) = 2 1-x (7)

    (3) (6) :

    | z -1 + z2+1| 2 1+x (8)

    (4) (7) :

    | z -1 + z2+1| 2 1-x (9)

    x[ 0 ,1] (8):| z -1 + z2+1| 2

    x[ -1, 0] (9):| z -1 + z2+1| 2

    :2 |z - 1 | + |z 2 + 1 | 4

    zC | z | =1 :

    |z 2 + uz + v|< 1 (10)

    (10) z =1 z = -1:|1+ u + v | | 2+2v|2>2|1+ v|| v+1|2|-1+ v|| v-1|| v+1|+ | v-1| | (v+1)- (v-1)| 2 >2

    .

    zo C |zo | =1,2|zo + uzo + v|1

  • 8

    -

    - - =

    - + = + -

    - - + +

    = ( - ) - - - - -

    :

    112

    3

    2

    13i

    22

    1

    2

    3i1

    2

    3i1 25

    2522252525

    ==

    -+

    =-=

    -=

    -

    ( )25

    z 3 3i 2 1 i 3

    z 3 3i 22

    - - - = - + =

    z ( 3 , 3) = 2 z z) x Re( - k , x = 3

    ( , ) = 2:

    Re(z)- x x - Re(z) x +

    3- 2 Re(z) 3+ 2 1 Re(z) 5

    Re(z) 1 z =1 + 3i , 5 z = 5 + 3i

    :

    z (3 + 3i) = 2z=x+yi

    - (x - 3)2 + (y - 3)2 = 4

    (x - 3)2 4 x -3 2 -2 x - 3 2 1 x 5

    Re(z) 1 z =1+3i , 5 z =5 +3i

  • 9

    :

    2 w -3+3i = iz +3-3i 2 (w -3-3i) = i(z -3-3i) )

    2w -3-3i = i z -3-3i 2w -3-3i =2 w -3-3i =1 w - (3+3i) =1 w ( 3 , 3) r1 =1

    z w () ():

    z - w = ( z - (3+3i)) - ( w - (3+3i)) = u - v

    z (3 3i)u = - + v = w - (3+ 3i) u =2 v =1:

    2 1 3vuvuwz - = - + = + =

    2 1 1vuvuwz - = - - = - =

    3w1 z -

    .

    :z + w = (z - (3+ 3i)) + ( w - (3+ 3i)) + 6(1 + i) = u + v +6(1+ i)

    :

    3262612i)6(1vui)6(1vuwz + = + + + + + + = + + = +

    ( ) ( )z + w = u + v +6(1 + i) 6(1 + i) -u + v 6(1 + i) - u +v = 6 2 -3 = 6 2 -3

    6 2 3wz36 2 - + +

  • 10

    :

    ) ( ) 6 2 3(MNw)(zwz + = - - = B = +

    ) 6 2 3()(MNw)(zwz + = - - = AG = - ,

    u = z - (3+ 3i), :23i)(3u = z - + = (1)

    u = 2 u 2 = 4 u u = 4 u = 4 (2)

    u z1 , z2 z,:

    ( 1 2 )1 2

    1 1 k = z - z - =

    z - 3 - 3i z - 3 - 3i

    (( 1 ) ( 2 ))1 2

    1 1 = z -3-3i - z -3-3i - = z -3-3i z -3-3i

    ( ) ( )( 2 )

    1 2 1 21 2 1 2

    1 1 1 4 4 = u - u - = u - u - = u u 4 u u

    = 1 ( u1 -u2 )( u1 - u2 ) =

    1 ( u1 - u2 )( u1 - u2 ) = 1 u1 - u2 24 4 4 k :

    k = 1u1 - u2

    2 0 4

    k1

    u1 u22 1 ( u1 u2 )

    2 (1)1(2 2)2 4

    4 4 4= - + = + =

    0 k 4

    + = +

    =

    + > <

    - + + + =

  • 11

    :

    iz +8 = 2 2iz +1 iz +8 2 = 4 2iz +1 2

    (iz + 8)(iz + 8) = 4 (2 iz + 1)(2 iz + 1)

    (iz + 8)(-iz + 8) = 4(2iz +1)(-2iz +1)

    z2 + 8iz- 8iz + 64 =16 z 2 + 8iz- 8iz + 4

    15z2 = 60 z 2 = 4 z = 2

    z (0,0) = 2

    :z1 = 2 z2 = 2

    :

    1 2 1 2 1 2 1 2z + x z > 3 z + x z2 > 3 (z + x z ) (z + x z ) > 3

    1 2 1 2 1 1 2 1 2 2(z + x z ) ( z + xz ) > 3 z2 + x z z + x z z + x 2 z

    2 > 3

    1 2 1 2 1 2 1 24x2 + (z z + z z ) x +1 > 0 4x2 + (z z + z z ) x +1 > 0

    1 24x2 + 2Re(zz ) x +1 > 0 , xR

    4x2 + 2Re(z1z2 ) x +1 :

    0 ( 2 -1)( z 2 -4) > 0 (3)

    ( -1,1) 2 -1< 0 (3) :

    z 2 - 4 < 0 z 2< 4 z < 2 ,, z > 2

  • 12

    z 2 ,.

    z = 2 , z,

    (1)C : x 2 + y 2 = 4 , () .

    z = 0 , (1) 16 = 0 ,. z 0 :

    z2 0

    z4 2z3 8z 16 0

    z2 28 16 2 16 4zz z2

    0 zz2

    zz

    0 (4)+ + + = + + + = + + 2a + =

    z + 4

    = u (5), :z

    4 2 2 4 16 2 2 16 2z u z z 2 u z 2 u 8 ,z z z z +

    2

    = + 2 + = + = -

    (4) :

    (u2- 8) + 2au =0 u2+ 2au -8 =0 (6):

    D = 42 +32 = 4(2 +8) >0 (6) :

    2 8 2u = -

    2a

    2

    +

    = - a +82

    (7)

    (5) :

    ( )(7)z2 - uz + 4 = 0 z2 + a 2 +8 z + 4 = 0 (8):

    ( ) ( )2D = a 2 +8 -16 = 2 +2 +8 2a 2 +8 -16 = 2 2 -4 a 2 +8 -1

  • 13

    y = 0 , z = x + 0 i = xR () z =2z= 2

    z = 2 :

    24 + 2 23 + 8 2 + 16 = 0 = -1,, (-1,1)

    z = - 2 :

    (-2)4 + 2 (-2)3 + 8 (-2) + 16 = 0 =1,, (-1,1)

    y 0

    x = 0 , z = 0+ yi = yi I () z =2z= 2i

    z = 2i :

    (2i)4 + 2 (2i)3 + 8 2i + 16 = 0

    24 i4 + 2 23i3 +8 2i + 16 = 0

    16-16i + 16i + 16 = 0 32 = 0 ,.

    z = - 2i :

    (-2i)4 + 2 (-2i)3 + 8 (-2i) + 16 = 0

    24 i4 + 2 (-2)3 i3 -8 2i + 16 = 0

    16 +16i -16i +16 = 0 32 = 0 ,. x 0 (1) x + yi xy 0

    [ ]= =

    - =

    =

    = -

    - [ ]

    z = x + yi, x , yR M(x , y).

    :

    z -2i = z x +yi -2i = x +yi x +(y-2)i = x +yi

    x2 + (y - 2)2 = x2 + y 2 x2 + (y - 2)2 = x2 + y 2

    x2 + y2 -4y+ 4 = x2 + y2 -4y+ 4 = 0 4y = 4 y =1

  • 14

    z y =1 , z = x + i , x R:

    w z2 1

    (x i)2 1

    (x i)2 x -i

    z x i (x i)(x i)= - = + - = + - =

    + + -

    2 x - i 2 x 1= x + 2xi -1- 2 =

    x -1-

    2

    +

    2x +

    2

    i

    x +1 x +1 x +1 :

    32w R 2x + 1

    = 0 2x + 2x +1 = 0 (1)x +1

    (1) x R, x=Re(z)

    g(x ) =2x 3 + 2x +1 , xR

    g g(x)=6x2 + 2,xR

    x R g(x) > 0 , g R

    :

    g R

    ( )x x xlimg(x) = lim 2x3 + 2x +1 = lim 2x3 = - - - -

    ( )x x xlimg(x) = lim 2x3 + 2x +1 = lim 2x3 = + + + +

    g g(R) = R, R,

    g( ) = 0 , g.

    z 2 = x2 +1 , -1 f (x2+1) 1

    f [1,2 f(1) < f(2) , f] [1 , 2] [ ]Re(z) 0,1 ,:

    2 2f

    0 x 1 0 x 1 1 x +1 2

    f (1) f (x2 +1) f (2) -1 f (x2 +1) 1

    - 2 = 2 + 2 -

    = = + = - +

  • 15

    2 + 2 = =

    = - -

    - = - =

    = +

    :2 ( ) ( )z-w = z-w z-w = ( z-w)( z -w) = zz - zw - zw + ww == z 2+ w 2-(zw+ zw) = z 2+ w 2-( zw+ zw) = z 2+ w 2-2Re(zw) (1)

    :

    z2 w

    2 2Re zw) z

    2 w

    2 2Re zw) 0

    (1)

    + = ( + - ( =

    z - w 2 = 0 z - w = 0 z - w = 0 z = w

    :

    1

    1

    f (2) = 3 f (2) = 3z =w 3+ f (2)i =f -(2)+3i

    f(3) 2f -(2) 3

    == f ,

    . f ,:

    f

    2 < 3f (2) < f (3) 3 < 2. f .

    1 2x, x R f

    x1 f (x2)- f (x1)< - f (x2):

    f (+)x1< x2 2x1< 2x2 2x1 - f (x1) < 2x2 - f (x2) - f (x1) < - f (x2 ) - f (x1) < - f (x2 )

    f ( 2x1 - f (x1)) > f ( 2x2 - f (x2 )):

    f ( 2x1 - f(x1)) > f ( 2x2 - f(x2)) (+) f (2x1 f (x1)) x1 f (2x2 f (x2)) x2 g(x1)>g(x2)x1 x2

    - - > - - - > -

    g .

    xR :

    ( ) ( )f -1(x) =2x-f(x) f 2x-f(x) =x f 2x-f(x) - x = 0 g(x) = 0

    f R,

    [1,4 , 1 (1 ,4) ,]

  • 16

    1 1 1

    f (4) f (1) f (4) f (1)f ( ) f 3f f (4) f (1)

    4 1 3=

    - ( ) =

    - ( ) = -

    -(2)

    xR :

    g(x) = f ( 2x- f (x)) - x

    2x-f(x) R,,

    f (2x- f(x)) R, g R,

    .

    g

    [ 2 ,3] , 2 (2,3) ,

    2 2g g(3) g(2)

    g g(3) g(2)3 2

    ( ) = -

    ( ) = --

    (3)

    x=2 g(2) = f ( 2 2- f (2)) - 2 = f (4-3)- 2 = f (1)- 2 , f (1) -g(2) =2 (4)

    x=3 g(3) = f ( 2 3- f (3)) -3 = f (6- 2) -3 = f (4) -3 , f (4) - g(3) =3 (5)

    (2) (3) :

    ( ) ( )3f (1) - g( 2) = f (4)-f (1) - g(3)-g(2)

    ( ) ( )(4),(5)

    3f (1) - g ( 2) = f (4)-g(3) - f (1)-g(2)

    3f (1) - g ( 2) = 3 - 2 3f (1) = g ( 2) +1

    - + =

    + =

    p q -

    pq = > *

    [ - ]>

  • 17

    :

    D = ( -6 sunq)2 - 4( 5sun2q + 4) = 36 sun2q - 20 sun2q -16 = -16 (1 - sun2q) = -16hm2q:

    1,2z 6sunq i4hmq

    3 2 i2

    = = sunq hmq

    z1,2 = x +yi ,x,yR , :

    x =x 3 3

    yy 22

    sunq = sunq

    = hmq = hmq

    (1)

    :2 2(1)

    2 2 1x y

    19 4

    sun q + hm q = + =

    z1,z2 x2 y2

    + =19 4

    ,

    qR

    :

    z1 - z2 = 4ihmq = 4hmq , 0 4p

    q

    :

    0p

    0 2

    0 4 2 24 2

    q hmq hmq

    z1 - z2 2 2, z1 - z2 2 2

    2

    pq = z1 =2i, z1

    n = 2 n in

    n = 4k , k N * z1n = 2 n R

    n = 4k +1, k N* z1n = 2 n iR

    n =4k + 2 , kN* z1n = -2 n R

    n =4k +3 , kN* z1n = -2 n iR

    n = 2m, m N*

    :2 2 2 ( 2 ) y 0x y

    1 4x2 9y2 36 y2 36- 4x y2

    4 9- x > y2

    9 x29 4 9 9 3

    + = + = = = = -

  • 18

    M ( x, y) C ,:

    ( ) ( ) ( ) ( ) 22

    d M,A = x - 2 2 + y -3 2 = x - 2 2 + 2

    9 - x -3 3

    ( ) 22

    f(x)= x-2 2 +2

    9-x -33 [ ]-1,2

    ,

    x1, x2 [ -1,2 ,:]m = f(x1) M = f(x2 ) , m f ( x ) M

    C ,.

    - = ( ) + - - - =

    =

    =

    - + =

    - - =

    = ( ) -1

    2 2 2x +yi -1 =x +1 (x -1) +y =x +1 (x -1) +y = x +1

    x2 - 2x +1+ y2 = x2 + 2x +1y2 = 4x, x 0

    z y2 = 4x

    w = +i ,R (2) :2 2 2 2 ( 2 2 ) + i - 2 - + i - 2i = 8 ( - 2) + - + ( - 2) = 8

    2 - 4 + 4+2 -2 -2 + 4 - 4 = 8 -4 + 4 -8 = 0 - + 2 = 0

    w x-y+2=0

  • 19

    u = x + yi x, yR (3) :2 u0 2 2(x yi) 2x 2y

    2 2 2 2 2 2w = +i = +i = -

    = = -

    u x + yi x + y x + y x + y

    - + 2 = 0 ,:

    2 22 2 2 2

    2x 2y+ +2 =0 2x +2y +2x+2y =0

    x + y x + y2 2

    x 2 + y2 +

    x +

    y =

    0

    x + 1 +

    y + 1 =

    1 2 2 2

    x = y = 0 , u = 0 ,.

    u K - 1,-

    1 2 2 2

    =2

    ,

    O(0,0)

    ( x1, y1 ) ,:(yy1 =2 x + x1 ) 2x - y1y+2x1 =0

    1

    = 2

    y

    ():

    11

    =1 2

    =1 y = 2y

    x1 =1

    ( )M1,2 ,(),().

    ( x, y) ,:

    ( )

    y22 2

    y 2x-y+2 4

    - + y -4y+8 y -4y+8d M, = = = =

    1+1 2 4 2 4 2

    2f(y) = y -4y+8, yR. f R yR:

    f (y) = 2y - 4:

    f (y) =0 2y - 4 =0 y =2 f (y) >0 2y - 4 >0 y >2

    f (y) f .

    x - 2 +

    f (y) - 0 +f (y)

    .

    f y =2 . y=2 x =1, ( )M 1,2().

  • 20

    :

    y 0

    x > 0 (5) :

    x2 x+ 5 - 2 = 0 x x + 5 x - 2 = 0

    x x

    g(x) =x x +5 x -2, x >0

    g ( 0,+ ) g (x) = 3 x + 5 , x >02 2 x

    :

    g (x) >0 x ( 0,+)

    g ( ) = 0,+:

    x 0 ( )

    x 0 ( )limg x = lim x x +5 x - 2 = -2

    + +

    x ( )

    x ( )limg x = lim x x +5 x - 2 = +

    + +

    g ( )g() = -2,+ 0g() g , g(x) = 0

    ( )A = 0,+ z z = w Im ( z)

    , + =

    = --

  • 21

    0 , = 0, 2

  • 22

    e -x > 0 x > 0 , (x) > 0 (0, + ) .

    (0) = -1

    lim (x) lim x3 x2 x e -x x x

    = + + - = ++ +

    limx3

    x2

    x = lim x3 = +

    x x + +

    + +lim e -x =0

    x +

    ([0, + )) = [-1, + ). 0 (0) 0 , xo (0 , ) +

    , , j(xo ) = 0

    xo (0 , ) + , f (xo ) g(xo ) = a (1)

    h h = (0 , xo)v (xo , +) . h

    (1)

    h(x)=g(x)f (xo ) - = g(x)f (xo ) -g(xo )f (xo ) =g (x) -g(xo )f (xo )(x - xo ) f (xo )(x - xo ) x - xo

    g(x)(x xo ) g(x) g(xo ) 1 g(x) g(xo ) h(x) = -

    -

    2

    + =

    g(x) -

    -(x - xo ) x - xo

    x - xo

    (2)

    x > xo, [ xo ,x] g

    ... 1 (xo ,x) , g(1) = g(x) - g(xo )

    x - xo (3)

    (2) (3) :

    h(x) = x

    1x

    ( g(x) -g(1))- o g

    g(x) = ex +xex =(x+1)ex

    g(x) = ex + (x +1)ex =(x+2)ex >0 x > xo > 0

    g , x > x1 g(x) > g(1)

    o1h(x) = x

    1x

    (g(x) -g( )) >0-

    x > xo > 0

    h (xo , + )

    0 < x < xo , [x ,x ]o g

    ... 2 (x, xo) , g(2) = g(x)-g(xo )

    x - xo (4)

  • 23

    (2) (4) :

    h(x) = x

    1x

    ( g(x) -g(2))- o g

    g(x) = ex +xex =(x+1)ex

    g(x) = ex + (x +1)ex =(x+2)ex >0 x (0 , xo)

    g , x< x2 g(x)0-

    x (0 , xo)

    h (0 ,xo )

    + =

    =

    + +

    + + = = =

    +

    :

    zw + 2i = z w

    zw + 2i 2 = z 2 w 2

    (zw + 2i) ( zw + 2i) = z 2 w 2 (zw + 2i) ( zw + 2i) = z 2 w 2 (zw +2i) ( zw - 2i) = z 2 w 2

    z2 w 2 + 2i zw - 2i zw + 4 = z 2 w 2

    2i (zw - zw) = -4 2i ( zw - zw ) = -4 2i 2i Im(zw) = -4 -4 Im(zw) = -4 Im(zw) =1

  • 24

    :

    zw + 2i = z w zw + 2i = z w

    zw=uzw - 2i = zw u - 2i = u u - (0 +2i) = u - (0 +0i)

    u A

    (0, 0) A(0,2), y =1, Im(u) =1Im(zw) =1

    :

    Im(zw) =1

    :

    zw = x + i xR (1)

    (1) z 0 w = x + i

    xR,:z

    2x + i x +1x + iw = w = w =

    z z z(2)

    :(2 )

    z2 + w 2 + Re( zw) 3

    x 2 12z 2 x 3

    z+

    + +

    z4 + x2 +1+ x z 2 3 z 2

    ( )2y z

    z4 x 3 z

    2 x2 1 0

    =

    + - + +

    y2 + ( x - 3 ) y + x2 +1 0 (3)

    y2 + ( x - 3) y + x2 +1 y :

    ( )2 = x - 3 - 4(x2 +1) = x2 - 2 3x + 3- 4x2 - 4 == -3x2 - 2 3x-1= -(3x2 +2 3x+1) = -( 3x+1)2 0

    (3) x,yR, z 2 + w 2 + Re(zw) 3

    < 0,() z 2 + w 2 + Re(zw) > 3,,= 0

  • 25

    :

    0 ( 3x 1)2 0 3x 1 0 x 1

    x 3

    33= - + = + = = - = - (4)

    :

    ( )z 2 + w 2 + Re(zw) = 3 y2 + x - 3 y + x2 +1= 0= 0 ,:

    2(4) y z 2

    3

    y-x + 3

    y 3+ 3

    y 2 3 =

    z 2 3

    2 2 3 3= = = = (5)

    (2) :

    (2),(5)21

    w2 x +1

    w2 3

    +1w

    2 2w

    2 2 3

    32 3 3z2

    3

    = = = = (6)

    (5) (6) : z2 = w 2 =

    2 3

    3

    :

    z2w

    2 z

    2w

    2

    z2w

    2 Re(zw) z

    2 w

    2 Re(zw)

    = =+ +

    zw2

    (1) x2 +1 x2 +1= zw

    2 + Re(zw)= x2 +1+ x

    = x2 + x +1

    x2 1

    f (x) = 2 +

    ,x + x +1

    xR

    xR:

    2x(x2 x 1) (x2 1)(2x 1) x2 12 2 2 2f (x) (x x 1) (x x 1)

    + + - + + -= =

    + + + +

    :

    f (x) = 0 x2 -1= 0 x2 =1 x= 1

    f (x) >0 x2-1>0 x2 >1 x >1 x < -1 x >1

    :

    ( 1)2 1 2

    f ( 1) 2 2( 1) ( 1) 1 1

    - +- = = =

    - + - +

    12 1 2

    2f (1) 1 1 1 3

    += =

    + +

    x x x

    x2 1 x2limf (x) = lim 2

    + = lim 2 =1 x + x +1 x

  • 26

    f (x) f

    .

    f x1 = -1 f(-1) =2 x2 =1

    f (1) = 2

    , x R 2

    3 f(x) 2

    3

    z

    2w

    2

    z2w

    2 +Re(zw)

    2

    3 2

    + - + - + =

    + =

    = + -

    + + = + +

    - - + - + - +

    +

    - =

    - =

    z = x + yi x ,yR :

    z + 2 - 4i 2 + z - 2 + 4i 2 = 58

    x - -1 1 +

    f (x) + 0 - 0 +

    f(x)

    .

    2 2

    3

    .

    1

    1

  • 27

    x + yi + 2 - 4i 2 + x + yi - 2 + 4i 2 = 58

    x + 2 + (y - 4)i 2 + x - 2 + (y + 4)i 2 = 58

    (x + 2)2 + y - 4)2 + (x - 2)2 + (y + 4)2 = 58

    x2 + 4x + 4+ y2 -8y +16+ x2 - 4x + 4+ y2 +8y +16 = 58

    2x2 + 2y2 =18 x2 + y2 = 9 C: x2 + y2 = 32

    z O(0,0) = 3 x =0 :

    02 + y2 = 32 y2 = 32 y = 3 y = -3 yy A(0,3) B(0, 3)-

    z O(0,0) = 3 ,:

    z = 3 z 2 = 9 z z = 9 z = 9

    z = 9

    z z(1)

    :9 (1) 9

    w =z + = + z =wz z

    w .:

    9 (1)w = z + = z + z = 2Re(z) (2)

    z

    z = 3 ,(2)

    -3 Re(z) 3 -6 2Re(z) 6 - 6 w 6

    z1 ,z2 z3 :

    1 1 11

    9zz = 9 z = ,

    z 2 2 2 2

    9z z = 9 z =

    z 3 3 3 3

    9z z = 9 z =

    z

    :

    1 2 3 1 2 3 1 2 31 2 3

    9 9 9z +z +z = z +z +z = z +z +z = + + =

    z z z

    z2z3 z1z3 z1z2 z2z3 z1z3 z1z2

    1 2 3 1 2 3 1 2 3

    = 9

    1+

    1+

    1=

    9

    + + =

    9

    + + =

    z z z zz z z z z

    = 9 z2z3 +z1z3 +z1z2 =

    z2z3 +z1z3 +z1z23 3 3 3

    :

    z2z3 + z1z3 + z1z2z1 + z2 + z3 = 3:

    z2z3 +z1z3 +z1z2 = 3 z1 + z2 + z3

  • 28

    :

    vsunq -ihmq = z(w - z) + z(w - z) + 2z 2 - w 2 + 3

    vsunq - ihmq = z w - z z + z w - z z + 2z z - w 2 + 3

    vsunq- ihmq = z w + z w - w 2 +3

    vsunq- ihmq = z w + z w - w w +3

    vsunq - i hmq =w (z + z -w) +3

    vsunq - i hmq = 2x ( 2x -2x) + 3

    vsunq - ihmq = 3 vsunq = 3 + ihmq (3)

    R +

    , Z sunq 2

    0 ,

    (3) sunq :

    v = 3

    + ihmq

    v =

    3+ iejq

    sunq sunq sunq

    v = a + bi , R, :

    3 1

    3

    a= a =

    sunq sunq ejq = b

    :

    2 , R 21

    =1+ ej qsun q

    + , Z

    2

    :

    2 2a 1 2 a 2 1

    9 9= + b -b =

    v x2

    C : - y2 =19

    M(a,b) x2

    C : - y2 =19

    A(0,3) M(a,b) :

    d(A,M) = d = (a - 0)2 + (b -3)2 = a2 + (b -3)2 = a2 + b2 - 6b +9 (4)

    M(a,b)C :2 2a 2 1

    a 1 2 2 9 9 29 9

    -b = = + b a = + b

    a2 (4) :

    d = 9 + 9b2 + b2 - 6b + 9 = 10b2 - 6b +18

    d(b) = 10b2 - 6b +18 , R

  • 29

    R :

    (10 2 6 18) 20 6 10 32 2 2

    d( )2 10 6 18 2 10 6 18 10 6 18

    b = b - b + = b - = b -

    b - b + b - b + b - b +

    :

    d (b) =0 10b-3=0 10b = 3 b = 310

    d (b) >0 10b-3>0 10b > 3 b > 310

    d (b) d .

    d = 3

    10

    = 3

    10

    :

    2 2 2a -

    3 =1 a

    =1+ 9

    a2 =

    9

    109

    a

    =

    3 1099 10 9 100 100 10

    A(0,3) :

    1

    3 109 3 ,

    10 10M

    M2

    -

    3 109, 3

    10 10

    = + -

    ( + ) - = =

    ( - ) = =

    =

    - +

    - 3

    10+

    d (b) - 0 +d(b)

  • 30

    >

    =

    ( )

    :

    (1+ 2i)w -5 = 3 5 (1+ 2i)w - 5 = 3 5 1+ 2i

    1+ 2i w - 5(1-

    2i)

    = 3 5 w - (1- 2i) = 35

    w

    (1, - 2) R = 3

    :z=x+yi

    f(z) =f(z) z +1+i = z +1-i z -z = 2i 2yi = 2i y = 1z y =1

    y =1,-2 -1

    d(K,e) = = 3 = R.1

    w z z = w

    w = x + yi

    (1, - 2) R =3 , (x -1)2 +(y +2)2 = 9 y = 0 :

    (x -1)2 = 5 x = 5 +1 x = - 5 +1 w ,:

    w = 5 +1+ 0i = 5 +1 w = - 5 +1+ 0i = - 5 +1

    w- (5 - 6i) w (A 5,-6) , 5 - 6i w- (5 - 6i) :

    2 2w 5 6imax

    (K ) R (5 1) + ( 6 2) 3- + = A + = - - + + =

    = 16 +16 + 3 = 32 + 3 = 4 2 + 3

    q z xx ,:

    y y=1 1

    x xefq = efq = , x >0

  • 31

    x t :1

    efq(t) =x(t)

    t :

    2 2

    1

    q (t) =

    -

    x (t)

    sun q(t) x (t)(1)

    :

    22

    1= 1+ e j q(t)

    sun q(t)(2)

    (1) (2) :

    ( 1 2 (t)) (t) x2(t)x (t)

    + ej q q = - (3)

    to M (3,1),

    t = to (3) :

    ( 1 2 (to )) (to ) x2(to )x (to )

    + ej q q = - (4)

    :x(to ) = 3 , x (to ) = 1

    2

    o 2o

    1 101+ (t ) 1 1 1ef q = +

    x (t )=

    +

    9=

    9 (4) :

    o o

    10 q(t ) = -

    1 q(t ) = -

    1rad/s

    9 9 10 M yy z xx .