1

() i2 = 1i2 = 1i2 = 1

1.1 + i, , R :) i(1 2i) + 2(i 1) ) (i2 2i+ 3)(i2 + i 1)) (i2 + i)(1+ 2i) ) (i2 i)(1 i)(2+ i2)(2 i2)) i(2 i)+2i(1 i)3i(i22) ) (i1)(3i2+2i+5)) (3i2 7i+ 5)(4i2 2i+ 7) + (1 i2)(2+ 3i 5i2)) ix(i x 2) i(2x 3) + i(x2 3), x R

1.2 + i, , R :

)5

1 2i) i

1i

)1+ 2i2 i

)3 7i1+ i

+7 3i1 i

)3+ i2i

)1+ i

2

2 i)

3i2 + i2i 1

) i + i

)1 i

3

3+ i)

(2+ 3i)(1 i)1 2i

)1 2i3+ i

1+ 2i1+ 3i

)(2 5i)(1+ i) (2+ 3i)

3+ 4i)

1+ 2i2 i

+2i

1.3 + i, , R :) (i 3)2 + (2+ i)2 ) (3+ 4i)2 + (4 3i)2 + i2) (1

2i)2 ) (3+ 4i)2(1 i) + (3 4i)2(1+ i)

) (1 i)(1 2i)(1+ i)(1+ 2i) + i2 ) (1 2i)3) (1+ i)3 (1 i)3 ) (2+ 3i)3 + (2 3i)3) (2 i)5(2+ i)6

1.4 pi :) (i 1)i1 2(1 i)2 ) (1 2i)2 + (1+ 2i)2) (6 4i)(1 i)1 + (17+ 19i)(1 5i)1

1.5 + i, , R :

)(2+ i)2

3+ i)

(1 i1+ i

)2)

(1 5i3 2i

)2+

(4 2i1 3i

)2)

(1 i)3 + (1+ i)3

(1 i)3 (1+ i)3)

(i 32

)3

(i+ 32

)3)

(3+ 3ii 1

)2)

22 +22i4 22

22i4

1.6 x R, pi :

(x 1 i)(x 1+ i)(x+ 1+ i)(x+ 1 i) = x4 + 4

1.7 z = 3+4i w = 23i, + i, , R, :

z+ w, zw,26w

25wz

1.8 z = 2 3i, + i,

, R, w = z+ 2iz 2i

1.9 :

z =(1 i)2

1+ i w =

z2 + 1z

pi w

1.10 z =1

1 i+

11+ i

+i2.

pi 4z2 8z+ 5 = 0

1.11 z = 9+ xi, x > 0. x z2 z3

1.12 a, b, c C :a

7+ 4i=

b1+ 2i

=c

1+ 3i

pi a

a 6b+ 2c= 1+ 2i

1.13 pi :

)1+ i1

1 i+

1 i1

1+ i+

1+ i1 i1

+1 i1+ i1

= 4

)1i

(1i 1

)

(1i i

) (1i+ 1

)= 1+ 3i

)(1+ i)n

(1 i)n2= 2in1, n N

) (1+ i)4n = (1 i)4n, n N

1.14 z C, : Re(z) > 0 Re ( 1z

)> 0

1.15 , > 0 n N (+i)n = (i)n) pi n 3) n = 3, pi

=

33

1.16 :

z =1

1+ + i, pi , 2pi+ pi, Z

pi (Re(z))2+(Im(z))2 =1

2(1+ )

1.17 z =12+

32

i w = 12+

32

i. pi- pi:

1

A =1z3+

1w3

+3z2w

+3zw2

1.18 z1 = 1 + i z2 = 1 i pi

E(z) =z2 + z+ 1z3 + z2 + 1

pi E(z1) E(z2)

1.19 (1, 1), pi - :

) z =1+ + i

1

1+ i1

1 + i

1+

1 i1+

) w =1 i

1+

1 + i1+

1+ i

1

1+ + i1

1.20 z = 12+ i

32

w =12+ i

32.

pi :) 1+ z+ z2 = 0 z3 = 1) 1+ z+ z2 + z3 + z4 + z5 + z6 = 1) w2n = zn, n N) w300 = 1 w333 = 1) z2016 +

1z2016

= z2013 +1

z2013= 2

1.21 z =1+ i

3

2

pi:

E = (+ z+ z2)(+ z2 + z), , , R

) pi z3 = 1) pi pi E

1.22 a, b, c pi c = (a+ bi)3 107i

()

1.23 z = (2x + y + 5) + 2i w = (4x 2y + 4) + (3x 4y)i. z = w, pi pi x, y

1.24 :

z = 6i (3 4i)x 3yi (3i 2)x+ (4 2yi)

x, y R, pi z = 0

1.25 pi x, y pi :) x 2+ (y 3)i = (1+ 3i)(y+ i)) (x+ y) + (x2 y2)i = 5+ 5i) x(1+ xi) + 2(1+ xi) = 2(3y 1)i y(1+ yi)) x(x+ 2y+ i) + 1+ yi = 6x+ 3i) (x2 + y2i)i+ x2 = 9(1+ i) + y

) (x+ yi)2 =12+ 5i

i

)x

2+ i+

y1 2i

=6 2i4 3i

) (x+ i)2 +4i= (i x)2 + y+ 1

)x yi1+ i

+1i=

x2+

11 i

) (x+ yi)2 =55+ 10i1 2i

)x

1+ 2i+

y3+ 2i

=5+ 6i1+ 8i

1.26 f (z) = z3 + 2z2 + z+ 4 i. , R, pi f (1+ i) = 0

() z R Im(z) = 0z R Im(z) = 0z R Im(z) = 0 z I Re(z) = 0z I Re(z) = 0z I Re(z) = 01.27 pi pi

pi:

) (1 i)24 ) (1+ i3)3 )

(1

1 i

11+ i

)2) (3+ i)6

1.28 z =1+ i1 i

, pi R. pi :

) w = z+1z

pi

) u = z1z

1.29 :

z =(1+ 3i)(2 3i) (4+ i)

2x+ i, x R

) pi - z) x, pi :

) z R ) z I1.30 pi pi pipi,

x R pi :) z R ) z I

2

) z = 6+2x2i3(x+6i) ) z = (2 x+3xi)(i x)) z = (2x+ i)(1+ i)2 + (x2 + 4)i+ x

) z =x+ 2x+ i

) z =1 i

3

x+ (x+ 1)i) z =

x+ ix xi

) z =x+ ix

3+ i

1.31 z = x+ yi, x, y R, pi :

(z 1)(iz 1) R x = y x y = 11.32 , R :

z =+ i i

w =+ i i

pi :) z+ w I ) zw R ) z = w || = ||

() z 0 Re(z) 0z 0 Re(z) 0z 0 Re(z) 0 Im(z) = 0Im(z) = 0Im(z) = 01.33 z C, z2 0, pi z R1.34 z , z , i, -

pi z3 iz+ i

< 0

1.35 z, pi - pi :

) z2 + z 0 ) z2 4z+ 3 < 0 ) z+ 1z< 0

() in = iin = iin = i

1.36 + i, , R pi :

)1+ ii5

+i2 ii7

+i4 ii9

+i 1i3

)(1i3+ i

) (1i7 1

)+

(1i5 i

) (1i9 1

)1.37 pi pi pi:

) i31 + i46 + i265 ) i2012 + i2013 + i2014 + i2015

) 5i2 + 7i78 4i31 + 7i 3i44 + 15

)1i+

2i2+

3i3+

4i4+

5i5+

6i6+

7i7

)14i3

1

5i13+

12i23

+1

3i37+

16i49

) i10 +1i15

+ i20 +1i25

+ i30 +1i35

1.38 + i, , R pi :) (1+ i)20 ) (1+ i)21 ) (2+ 2i)20 + (2 2i)20

1.39 , , , 4 pipi, pi i+++ = 1

1.40 pi pi pi:) i57 + i58 + i59 + + i213 ) i i2 i3 i2015) i+ i2 + i3 + + i4, N

1.41 + i, , R :

z =(1 9i5 4i

)28+

(2+ 3i3+ 2i

)171.42 :

z =(13+ 7i7 13i

)36+

(21 4i4+ 21i

)53 pi [z Re(z)]2015 = i

1.43 z =12

12i, pi z127

1.44 n Z, pi (3+ i)6n = (3 i)6n

1.45 n Z, pi :) i3n+31 = in5 ) i3n13 = i3n ) i3n2 = i32n

1.46 n N, pi :) 2in + (1)n = n 1 ) 2in 3(1)n = n 3) 8in + 5(1)n = n+ 5 ) 2i2n+4 + 5(1)n = 4n 1

1.47 pi , . pi- : (+ i)2014 + ( i)2014 = 0

1.48 f (n) = inz, z C n N. pi f (3) + f (8) + f (13) + f (18) = 0

1.49 2in = 3n 20, n N, pi :) n = 6 ) (3+ 5i)n + (5 3i)n = 0

1.50 in+2 i = 0, n N, pi- (1+ i)n+1 pi

1.51 f (n) = in(1 i), n N. pi :) f (4n) + f (4n+ 1) + f (4n+ 2) + f (4n+ 3) = 0) f (1) + f (2) + f (3) + + f (101) = 1+ i

1.52 f (z) = i(z2 + 1) + z, z C. pi:) f (i) f (i)) pi ( f (i))n + ( f (i))n, n N

3

1.53 z =3+ i w = 1

3i.

) pi :

)zw= i )

z wz+ w

=zw, R

) z22 + w22 = 0) n N, pi zn + wn = 0

1.54 n N, pi :) in + in+1 + in+2 + in+3 =

1in+

1in+1

+1

in+2+

1in+3

) i2n+i2n+1+i2n+2+i2n+3 =1i2n

1

i2n+1+

1i2n+2

1

i2n+3

1.55 n N, pi pi-:) i5n+7 ) i3n+1 + i2n + i5n+3 ) (1+ in)(1+ i2n)

) in+ i3n+2+ i2n+1+ in+3 ) in+1+ i2n+3+ i3n+ in+2

1.56 z =2(1 i)2

. pi:) z8 = 1 ) 1+ z+ z2 + z3 + + z79 = 0

1.57 z C z2 iz 1 = 0. pi pi:

A = z63 z42 + z33 + z120

1.58 pi 3+ 1

22

+

3 122

i72 = 1

1.59 n N, pi :

)(1+ i1 i

)2n+

(1 i1+ i

)2n= 2(1)n

)(4 2i1+ 2i

)n+

(6+ 2i1 3i

)n=

{0, n pi2n+1in, n

)( x+ i1 xi

)2n+

( i x1+ xi

)2n= 2(1)n, x, y R

)(xi yx+ yi

)4n

(x+ yixi y

)4n+1= 1+ i, x, y R

1.60 n N, :) (2 3i)n + (3+ 2i)n = 0 )

(2 i1+ 2i

)3n+1= 1

)(2 3i3+ 2i

)2n+

(4+ 5i5 4i

)2n= 2n

27n+13

1.61 f (z) = z3n, n N) n = 10, pi pi- f (1+ i

3) f (

3 i)

) n pi f (3+ 4i) + f (4 3i) = 0

1.62 pi 99=0

(1)(100 )i+1

() z = w, zn = wn z = w, zn = wn z = w, zn = wn

1.63 z,w z2 + w2 = 0. pi- z2014 + w2014 = 0

1.64 z,w pi 2z = (1+ i)w. pi z4 + w4 = 0.

1.65 z C, z2 + z+ 10 = 0. pi (2z+ 1)10 < 0

()

1.66 a, b, c a, b , 0, pi- :

b+ ca R c+ a

b R a

b< R

pi a+ b+ c = 0

1.67 z,w C z + w = 1 z3 + w3 = 2. pi z100 + w100 = 1

1.68 z C z4 + z2 + 1 = 0 , , N, pi z6 + z6+2 + z6+4 = 0

1.69 a, b, c :

a+ b+ c = a2 + b2 + c2 = 0

pi :) ab+ bc+ ca = 0 ) a4 + b4 + c4 = 0) a4b4 + b4c4 + c4a4 = 0

1.70 z C z2 + z + 1 = 0. , , C, pi :) (+ )(z+ z2)(z2 + z) = 3 + 3) (z2 + z+ )3 = (z2 + z+ )3

() pi

1.71 f : C C, pi 2 f (z) + f (iz) = z2, z C. pi f (z) = z2, z C

1.72 f : C C, pi - f (z) + f (iz) + f (iz) = z, z C. pi f (z) = z, z C

1.73 f : C C, pi f (z) + f (z) = z, z C, 3 = 1. pi f (z) =

z + 1

4

2

()

2.74 z =1+ i

3

2. pi

z2 = z

2.75 z1 = 2 i z2 = 3+ 2i. w = z1 + z1 z2 u =

z1z2

2.76 x + yi, x, y R :

) w = (1+ i)3 (1 i)3 ) z =(2+ i)2 + (2 i)2

1+ i

2.77 z C, pi :(1+ zz

)

(z+

1z

)= 1 z

2.78 x, y R, z,w , :) z = x2 + (y+ 3)i w = xy 2(x+ i)) z = 3+ (3x+ y)i z = 2y x 5i) z = (x2 + i)i w = x(5i x) + 4i

2.79 z, x+ yi, x, y R, :

) z =(1+ 2i)2(3 i)

i(1+ i)5) z =

1 i1+ i

(1+ i)20

(1 i)10

2.80 z =(3+ i)2

1+ i.

pi -

w =1+ zz

2.81 z C I, pi Re( zz+ z

)=

12

2.82 z,w C, pi (zw zw)2 0

2.83 f (z) =2z+ iz 2i

, z , 2i. f (9 5i) = w, :

) pi w =32

32i

) pi 23 w

20142.84 z,w Re(z) Im(w) , 0.

(z + w) I (z2 + w2) R, pi z+ w = 0

2.85 z,w C z+ w R zw R, pi z = w

2.86 z , 0 w =zzzz.

pi : ) w I ) 2 Im(w) 22.87 a, b, c C, pi :

) a Im(bc) + b Im(ca) + c Im(ab) = 0) Im(ab) = Im(bc) = Im(ca) , 0, a+ b+ c = 0

() f (z, z) = 0f (z, z) = 0f (z, z) = 0

2.88 z pi pi pi pi :) Im(z+ 1) + iRe(2 z) = 2 i) Re(2z+ 1) + iIm(3 z) = 5 2i) [Re(z) + 3i][Im(z) i] = 3 4i z) 3(1+ i) z 2z = [Re(z)+ 5i]i+ [Im(z)+ 2i](1 i)

2.89 z pi pi pi pi :) z(1+ i) = 1+ i z ) z+ z+ i(z+ i) + 2 = 0) 2z+ z = 3+ i ) (3+ 2i)z (1 2i)z = 4 4i) z+ i(z z) + 3 = 0 ) zz+ 2i(z+ z) = 4(5+ 2i)) z i(z+ z) 1 = 0 ) i(z+ z) + i(z z) = 6i 4) (i 1)(2z i) = z 1 ) zz+ 5(z z) = 61+ 50i) (1+ i)2z+ 5 = z+ 4i

2.90 pi :) z2 + z = 0 ) z2 + 4z+ 4 = 0) z2 + 2z2 + z z+ 9 = 0 ) zz+ z z z2 = 18) z2 + 2z+ 1 = 0 ) z2 2z+ 1 2i = 0) iz2 2z i = 0

2.91 1+ z+ z+ z2 + z2 + z3 = 0

() z R z = zz R z = zz R z = z z I z = zz I z = zz I z = z2.92 a, b, c C, pi pi -

pi:

) z = ab+ ab ) z =a+ ba

+a+ ba

) z = (ab+ c)(ab+ c) ) z = iab+ ba

ab ba

) z = (a b)15 + (a b)15

2.93 z C, pi pi pi:

) w =z iz+ i

+z+ iz i

) w =z2 + z2 z z

zz+ 1) w = z2 zz+ z2 ) w = (z i)2 + (z+ i)2

5

) w =z+ 1z

+z+ 1z

) w =zz(z+ z)(z z)

2zz

) w =z i zz i z

2.94 pi pi -:) (3i 1)9 + (3i+ 1)9 ) (2 i

3)5 (2+ i

3)5

2.95 z,w C, pi pi - :) u = zw zw ) u = (z w)5 (z w)5

) u =z+ w1 zw

z+ w1 zw

) u =(z+ i)2 (z i)2

(z+ i)3 + (z i)3

) u =z2 z2

1+ zz2.96 z :

f (z) = (1 i)z+ (1+ i)z+ 1

pi f (z) = f (z) z R2.97 , , , pi

w = ( )( ) ( )( )

2.98 z , 1 w =z

z+ 1.

pi : w R z R2.99 z , i w =

z+ iz i

. pi- : w R z I

2.100 n N, pi pi pi:) (1+ i)n + (1 i)n ) (3+ 4i)4n + (4+ 3i)4n

)(+ i)n

i+( i)n

+ i, , R

)(1 i)n

(2+ i)(3 i)+

(1+ i)n

(2 i)(3+ i)+ (5 2i)n(5+ 2i)n

)(19+ 7i9 i

)n+

(20+ 5i7+ 6i

)n2.101 z w z+w , 0.

pi u = i 1 zwz+ w

2.102 n N z = (1+i)n+(1i)n+in pi, pi n

2.103 f (z) = z7 2z8 + 1, z C. pi f (z) + f (z) R

2.104 R Im(+ 2z 2z

)= 0, z ,

2,

pi z R

2.105 z C, pi :(z+

1z

) R z = 1

z z R

2.106 z , i, R. pi- :

z+ iiz+

I z I2.107 z, z , 0

w =1 zz2

. pi :

w R z R z z = 2Re(z)2.108 z,w :

z7 = 4+ 3i w7 = 3+ 4i

u =zw, pi : ) u < R ) u < I

2.109 z = x+yi w = y+xi, x, y R. z5 R, pi w5 I

2.110 z C - :

w =zz+

zz

pi :) w R 2 w 2 ) w = 2, z R) w = 2, z I

2.111 a, b c. pi- :) z = a(b+ c) + b(c+ a) + c(a+ b) pi) w = a(b c) + b(c a) + c(a b)

2.112 pi x, (1+ xi)7 R. pi (x+ i)7 I

2.113 z,w C, (z + w) R (z w) I. pi :) zw 0) n N, : ) zn + wn R ) zn wn I

2.114 z C (z i)(z + i) = 1. pi w =

(z 2i)3

(z i)3 i3 pi

2.115 z,w C u = (zw)(zw). pi- : u R z R (z w) I

6

2.116 f (z) =(1+ z)n

1+ zn, z C

n N) pi f

(1z

)= f (z), z , 0

) zz = 1, pi f (z) R) n, pi f (i) pi f (i) R

2.117 a, b, c pi :

a+ b = c, b+ c = a c+ a = b

pi : ) a+ b+ c = 0 ) a, b, c I2.118 a, b, c pi

:

a = b+ c, b = c+ a c = a+ b

pi : ) a + b + c = 0 ) a, b, c R) a = b = c = 0

2.119 , , R , w C, pi w+ w+ = 0. pi w R

2.120 pi , n (+ i)n + (+ i)n R. pi n = 4, N

2.121 z =2+ 2

3i

1+ i. zn R,

n N, pi n pipi 12.

2.122 z,w z , w :

a =1 zwz w

, b =1+ zwz w

i c =z+ wz w

pi :) a2 + b2 + c2 = 1) a, b, c R, z 1

w

3

() pi

3.123 pi :) (1+ i)z = 2 ) z (1 i)(1+ i) = i(z 2)) 2iz (1+ i)z = 2 ) (3+ i)z+ (2+ i)2 = 7+ 2i

) (2+3i)z2 = (1+2i)+2z+ i ) (1 i)2z = 4i2) (i z)(1+ 2i) + (1 iz)(3+ 4i) = 14+ 6i) (1 i)z+ z = 2(1+ i)2 + 7 ) iz+ z = i2 + i3) 3iz (1 i)(z+ 1) = (5+ i)2 + 23i2

3.124 pi :

)z+ 31 i

z 11+ i

= 2i ) 2ziz+ 3i1 2i

= 9+ 2i

) z+iz

1 i= (1+ i)3 )

z 1i+ 1

=z+ 1i 1

i

3.125 z1 = 3 i z2 = 2+ i, :

zz1+ z2 = (z1 z2)2

3.126 f (z) = (1+ i)z+3 i, z C.) :

) f (z) = 0 ) f (z) = z ) f (z) = f (1 i)) x, y R, f (x 2yi) = f (2 4i)

3.127 (1 i)30z = (1+ i)33 216i

() pi

3.128 z,w pi -pi :) z+ w = 4+ i 2z w = 1+ 5i) z+ iw = i 3z+ 2w = 3+ 5i) (1+ 2i)z iw = 7 (3 i)z w = 2 10i) w iz = 3(1+ i) z 2iw = 3(1 i)) (1+ i)z+ (1 i)w = 2+ i 3z 2iw = 4 i) (1 i)z+ iw = i(4+3i) iz(2+ i)w = 23i) 3z+ (1+ i)w = 2+ 3i (2+ i)z+ (1+ i)w = 1

3.129 z1, z2, z3 pi- : z1 = z2+iz3, z2 = z3+iz1 z3 = z1+iz2

()

3.130 pi :) z2+1 = 0 ) z22z+4 = 0 ) z26z+25 = 0) z2 + 4z+ 8 = 0 ) z2 = 4z 13

3.131 pi :

)(z+ 7iz 3i

)2+ 9 = 0 )

(z iz+ i

)2 3

(z iz+ i

)+ 2 = 0

3.132 pi :) z2 = 2i ) z2 = 5 12i ) z2 2iz 1+ 2i = 0) z2 (5+ i)z+ 12+ 5i = 0 ) z2 iz+ 6 = 0) z2 5z+ 7 i = 0 ) z(z i) = 2(iz+ 1)) (z2 4z+ 5) + i(z+ 1) = 0) (1+ i)z2 2

2iz (1 i) = 0

7

3.133 , R, z2 + z+ = 0 z1 = 2 i.

3.134 f (z) =z 1+ i

z2 2z+ 2, z C

) pi z -

) f (z) =z 1+ i

2

3.135 z1, z2 z2 2z+ 10 = 0. pi :

)1z1+

1z2=

15

) z21 z2+z1z22 = 20 ) z

21+z

22 = 16

3.136 w1 = 4 + 7i w2 = 2i2020 + 3i2019.

x2 (4+ )x+ 2 = 0, , R

w1w2, :

) , )

z+ z i

= i(1+ i)

3.137 z1, z2 z +3z= 2.

pi :

(z1 z222

)2016+

(z1 + z22

)2017+

(z1z23

)2018= 3

3.138 pi :

)(z 1z i

)2+ 4 = 0 )

(z 3i2+ i

)2+

(z+ 3i2 i

)2= 0

3.139 z1, z2 z2+4z+8 = 0, pi :

z1 + z2 + 4iz1z2 + 8i

I

3.140 z1, z2 z2 + 2z+ 2 = 0.

w =(z1 + z2)4 + 2i(z1z2)2 2i

+ i, , R.

3.141 z1z= i3

) pipi ) z1 Re(z1) < 0, pi z31 = 1 pi pi- z20151 + z

20151

3.142 , z2+ 5z+ 13 = 0 M(0, 1), pi (MA) (MB) = 13

3.143 z1, z2 z2+z+ = 0, , , R , 0. 2 = 2++2, pi z21 (1+ z2) + z

22(1+ z1) = 1

3.144 z2 + 2z+ 2 = 0) z1, z2) pi z41 = z

42 = 4

) n N, zn1 + zn2 = 03.145 z2 + z+ = 0, , C.

z1, z2, pi :) , R) z2 z = 0 pi

3.146 z24 z+4 = 0, 0 pi) z1, z2 pipi ) pi pi:

S 1 =1z1+

1z2

S 2 = z41 + z42

) , S 2 = 0

() pi

3.147 pi z2016 + z + i = 0, , > 0, pi

3.148 , R pi z3 + z + = 0 1 + i. ;

3.149 pi P(z) = z3 + 7iz2 z 7i, z C) pi P(1) = 0) pi Q(z) P(z) = (z 1)Q(z) z C) P(z) = 0

3.150 pi :) z31 = 0 ) z3+z22 = 0 ) z33z2+2 = 0) z3 + 3z2 + 3z+ 2 = 0 ) z3 3z2 + z+ 2 = 0) z4 = 1 ) z4 + 2z3 + 2z2 + 2z+ 1 = 0) z6 + 2z5 + 2z4 + 2z3 + z2 + (z+ 1)2 = 0

3.151 R, pi :) z3 2(1+ )z2 + (1+ 4)z 2 = 0

) z2 +1z2= 2

3.152 z1, z2, z3

8

z3 z2 + z 1 = 0

pi z20161 + z20162 + z

20163 = 3

3.153 pi :

)(z+ iz i

)3+

(z+ iz i

)2+

z+ iz i

+ 1 = 0

)(z+ 1i

)3 7

(z+ 1i

)2+ 16

z+ 1i

10 = 0

3.154 pi :) (z2 + 1)2 + (z2 z 1)2 = 0) (z2 4z+ 5)2 + (z+ 1)2 = 0) (3z 1)4 + (z 2)4 = 0

3.155 z1, z2 - z3 = 1+

3, pi

z1z2< R

3.156 pi:

P(z) = z3 (5+ 2i)z2 + (11+ 5i)z 12i

) pi ) P(z)

3.157 pi - z4+z2(z1)2+(z1)4 = 0

3.158 n N n , pi - (z + 1)n + (z 1)n = 0 pi

3.159 z z4 + z2 + 1 = 0 (1)) pi : ) z6 = 1 ) z2018 = z2) (1)

3.160 z0

nxn + n1xn1 + + 1x+ 0 = 0

0, 1, ..., n R n , 0, pi z0 pi

3.161 zn = , n N n 2, , pi R

3.162 z C z5 = 1, pi :

)z

1+ z2+

z2

1+ z4+

z3

1+ z+

z4

1+ z3= 2

)z

1 z2+

z2

1 z4+

z3

1 z+

z4

1 z3= 0, z , 1

)z4

z 1+

z3

z2 1+

z2

z3 1+

z4

z 1= 1, z , 1

3.163 z5 + z3 + z + 3 = 0. z1, z2, z3, z4, z5 , -pi :

(1+ z21 )(1+ z22)(1+ z

23)(1+ z

24)(1+ z

25) = 10

()

3.164 z,w pi pi pipi:) z2 = w w2 = z ) z3+w5 = 0 z2 w4 = 1

3.165 z,w pi- z2 = w w2 = z.

) pi : ) z3 = 1 ) w = 1z

) z,w

3.166 pi pi pipi, :) z16 + 2z15 + 1 = 0 (z+ 1)2 + z2 + z = 0) z16 + 2z14 + 1 = 0 (z2 + 1)2 + z3 + z = 0) z3 + 4z = 3z2 + 2 z10 32z+ 32 = 0

4 pi

()

4.167 pi z pi pi pipi:) Re(z) = 2 ) Im(z) = 3 ) Im(z) = 3 Re(z) > 3 ) Im(z) = 1 2 Re(z) 2

4.168 pi z, :) z+ z = 2 ) z z = 2i ) 1+ Im(z) = 2Re(z)) Im(z2) = 0 ) z2 = z2 ) z2 + 4z = z2 + 4z) (2+ 5i)z+ (2 5i)z = 2 ) Re[(2 i)z] = 4) Re(z+ 2) = Im(z) ) Re(2+ iz) = Im(zi) + 2Re(z)) Re(z+ 5+ 7i) = Im(z 2+ 6i) ) z3 = z3

) (1+ i3)z = 2z

4.169 R, pi z, pi pi- pipi:) z = 5+ + (12+ 3)i ) z = 4 (1 )i) z = 2+ (1 2)i ) z = 2 + 1 (2 2)i

) z = (4+ 3i) + (13i

14

)) z =

2 i1+ 2i

4.170 z = 1++ i w = 1 + i, R. , z w, pi .

9

4.171 z = (1 + t) 2ti, t R w = iz + 2. pi z w

4.172 z = (1 + i) + 1 i, R. :) pi z

) , R z = + i1+ i

4.173 pi z pi w ,:) w = z(1 2i) ) w =

iz+ 1

) w =2

z+ 1+ i

4.174 pi z pi w pi,:) w = iz+2z ) w = (zi)(z1) ) w = z2+z+1) w =

zz+ i

) w =z+ 1z 1

) w =z+ i

z 1+ 2i

) w =z+ 2zz2

) w =z+ 1z+ 3

) w =2 2iz+ 1

4.175 z w = (z i)(z+zi). w y = x, pi z

4.176 z w = zz z2. pi z, Re(w) + Im(w) = 0

4.177 z C w = (1z)(1 iz). M(z) pi Re(w) + Im(z2) = 0

4.178

z =x+ yi1 i

+2 4i1+ i

, x, y R

pi M(x, y), z y = x.

4.179 z = 2x(y+1)i w =z 2z+ 2i

.

w R, pi M(x, y) - pi .

4.180 z = + 1 + (2 1)i, pi R) pi pi - z () : y = 2x3) () pi- pi A(0, 2) ()

) z pi pi-

4.181 z 2x + 3y + 7 = 0, pi w, pi (2+ i)w = z 1+ 2i

4.182 z pipi () : y = x + 1, pi w = (2 + i)z + z + i pi pi .

4.183 z , pi w, pi pi pipi:

) w =z 1z+ i

) w =z+ i1+ iz

4.184 :

z2 + (2 2)z+ (52 2+ 1) = 0, R

:) z1, z2 pipi ) pi z1, z2

()

4.185 pi z, :) zz = z+ z ) 2zz+ 3(z+ z) + 2i(z z) + 2 = 0) zz+ i(z z) = 3(2 z z) ) zz+ i(z z) = 3) zz 3z 5z = 2iIm(z) ) zz+ z+ z = i(z z)) (z )(z+ ) = (z+ )( z), > 0) (z (1+ i))(z (1 i)) = 9

4.186 R, pi - z, pi pipi, pi -:) z = (1 ) + (2+ )i) z = 3+ 2+ (4+ 2)i) z = 1+ 2i+ 5(+ i)

4.187 z,w z , 1, w =z+ 1z 1

.

Re(w) = Im(w), pi z

4.188 z , 2 w =z iz 2

. pi :) w R, z pi pi A(2, 0)) w I, z pi pi A(2, 0)

10

4.189 pi z, :

)z

iz 2 R ) z

2

z+ i I ) z

2

z 2 R

)(z

1z

) I ) z i

z 1 I ) 1 z

1 iz R

)z

z+ 4 I )

(z+

4z

) R ) z+ 2i

z i I

)z+ 1z 2i

I ) z2 z+ 1z2 + 1

R ) z 1 iiz+ 1

R

)i(z2 + 1)z+ 1

I ) z(z )1 z

R, (0, 1)

4.190 pi z, :

) Re(z2 1) = 2Im2(z) ) Re(z) + Im(1iz

)= 0

) Re(z+ 1z 1

)= 2 ) Re

(z+ 3z 1

)= Im

(z+ 3z 1

)) Im

(1

z+ 1

)=

16

) Im(z

1z

)= 2Re(i z)

) Re(z+

2

z

)= (+ 1)Re(z), > 0

) Im(

z z

)= (1 )Im(z), R

) Im(

1 z

)=

12, R

4.191 z1 = 3+ i z2 = 1+ 3i. pi z, pi pi pipi:

)z z1z z2

I ) (z z1)(z z1) = z2z2

4.192 R, pi pi pi-pi, pi - z pi :

) z =+ 1+ i

) z = 2i+ 2i

) z =+ 2i1+ i

) z =1+

1+ 2+ i

4.193 z = x 2+ yi w = x+ yi. pi M(x, y) pi

zw

.

4.194 z,w w =z+ 1z 1

. z x + y = 1, pi w.

4.195 , z,w - pi zw = 2i. () : x+y+ 1 = 0, pi- pi .

4.196 z x2 + y2 = 4, pi -

w = z+4iz

pi .

4.197 z,w C, w , 0, pi :

2Im(w) + wRe(z) + iw[1+ Im(z)] = 0

pi z - (0,0) 1.

4.198 z =1

2+ + i, R.

pi z - pi

() pi

4.199 pi z, :) 4(z+ z) = (z z)2 ) [Im(z)]2 = 2(z+ z)) (z z)2+ 4(z+ z) = 0 ) 2zz = z2+ z2+ 8(z+ z)

4.200 R, pi - z, pi pipi:) z = (+ i)2 + 1 ) z = ( 1)2 + 2(1 )i

4.201 z pi- y = x2, > 0, pi w = iz

4.202 pi - z = + i2, R pi.

() -pi

4.203 pi z, :) z2 + z2 = 14zz 48 ) 10zz 3(z2 + z2) = 16) z2 + z2 = zz+ 1 ) Re(z2 + z) = 1+ Re(z)) z2 z2 = 8i ) 2(z+ z)2 + 3(z z)2 = 24) (z2 1) I ) z2 = 1+ 2iRe(z) Im(z)) z2 + z2 + 2(1 i)zz = 2i, R

4.204 z w = 2z z. Re(w2) = 1, pi - z

11

4.205 R, pi z pi pi pipi:) z = 3 + 4i) z = (3+ 4) + (4+ )i

4.206 :

z = x+ yi w = 2x+ 3+ (3y 5)i, x, y R

pi w (3,-5) 6, - pi z

4.207 pi z, pi :

)1

z2 + 9 I ) i z+ 1

z i R ) z 3z+ 2

z 2 I

)i

z2 + 1 R ) 1

2(z+z)2(z2+z2)+(zz)+16 I

4.208 > 0 (0, 1). z , pi w pi 2w = (1+ )z+ (1 )z

4.209 pi z pi pi pi-pi:) z = + i

1

, (pi

2,pi

2

)) z = + i

1

, (0, pi)

4.210 z = x + yi, x, y R

w = x 11(y+ 2)i u = x 1+ (y 2)i

R w = i u, pi z pi-.

4.211 z = x + yi, x, y R

w = ( i)z+ 3Re(z) u =2(1+ 2)1 i

R w = u, pi z .

4.212 z,w pi 3w (Im(w)) i = z+ 2i. z 4x2 + 9y2 = 36, pi w .

4.213 z,w w = z+22

z,

> 0. z O(0, 0) , pi- w pi

4.214 w = +i, , R, pi :

x2

25+

y2

9= 1

z = x+ yi, x, y R, w :

3z+ 2Re(z) = 5+ 6i+ w

pi pi - z K(1, 2) = 1.

4.215 0 < < pi pi pi :) z2 10 z 92 + 25 = 0) 2 z2 2 z+ 5 42 = 0) (2)z2 (2)z+ 2 2 = 0) (2)z2 2 z+ 5 42 = 0

4.216 z , 0 y = x, pi

w = z+1z

5

()

(i) - |zw| = |z||w||zw| = |z||w||zw| = |z||w|

5.217 :

) z = (2 3i) + (6 3i) ) z = (12i)(3+

3i)

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

3+ 2i(1 i)2 2 i

) z =43 4i

1+3i

) z =(1 2i)2 + 2+ 3i(3 i)2 7+ 5i

) z = (3 4i)(1 xix i

)5, x R

5.218 z = 2 i w =3 i.

zw zw

12

5.219 z,w C Re(z) Im(w) = Im(z) Re(w), pi |zw| = |Re(zw)|

5.220 z C, pi :

|Im(z)| =

|z|2 Re(z2)

2

5.221 z = (1+ xi)(x+ i)+ (1 xi)2, x R.) z R, pi x = 1) x pi |z| = 8

x2 + 1

5.222 z C, |z| = 1. w =

z2(z i)1+ zi

5.223 z,w C w = z1+ |z|

, pi

z =w

1 |w|

5.224 z = 1 i w =z 2z4

, pi

4|w| = |z|

5.225 z, pi :) z = |z| 1+ (|z| 2)i ) z = |z| 1+ |1 2i|i) z+ |z| = 2+ 8i

5.226 z,w pi |z| = |w| = 1, zw R z

w I

5.227 z C w = z+ |z|2(Re(z) + |z|)

. pi-

w2 = z

5.228 z = x + yi, x, y R, pi 2 (x+ iy2)3 I

5.229 z,w C, pi :

|z|2 |w|2 = (Re(zw))2 + (Im(zw))2

5.230 z1, z2 C w = z1 + z2+

, , > 0.

pi |w z1|+ |w z2| = |z1 z2|

5.231 zn =(1+ i

3)3

(3 i)n

, n N. n pi :) |zn| = 641 ) zn I

5.232 z C {1}, |z| = 1. pi pi R, z = 1+ i

1 i

5.233 z,w, u C, :

zw+ wu+ uz = 1 |(z+ w)(w+ u)(u+ z)| = 10

pi |(1+ z2)(1+ w2)(1+ u2)|

5.234 n N, :

zn =(1+

i1

) (1+

i2

)

(1+

in

) pi |zn+1 zn| = 1

(ii)

5.235 z2+2z+ = 0, > > 0.) z1, z2 pipi ) pi |z1| = |z2| = 1

5.236 z2 (2)z + 1 = 0, pi

(0,pi

4

). z1, z2

pi

|z1 + z2| =|z1|+ |z2|

3

z1, z2.

5.237 [2, 2] z,w z2 zw+ w2 = 0. pi |z| = |w|

(iii)

5.238 :) |z+ i2| = 2iz ) |z2| = z ) |z 1| = z i) z+2z = |z+2| ) |z|2 = z+1 i ) |z|+z = 2+ i) |z|2 2z+ 2i = 0 ) |z i| = 2z ) |z 2| = z) z+ |z+ i| = 8+ 3i ) 3z |z+ 1+ 3i| = 1+ 3i) z2 2iz+ |z| = 6 ) |z|2 2(z z) = 2(5 6i)

5.239 :) 4z2 + 8|z|2 = 8 ) z2 + z+ |z+ 1| = 0) |z| z = 1+ 2i ) |z|2 3z+ 2z = 21+ 15i) z+ |z 9 6i| = 17+ 2i ) 2z |z| = 1 8i) z2 + 2|z| 8z+ 15 = 0 ) z2 + z+ |z+ 1| = 0) z2 + |z| = 0 ) 3(z z) + 2|z|2 = 26+ 12i

5.240 z pi - :

|z 1 2i|+ |z 2|i = |z+ 1 3i|i+ 22

5.241 R, pi :) |z|2 z 1+ i = 0 ) z2 3|z|+ 2 = 0) 4|z 1|2 4iz+ 1+ 4i = 0

13

5.242 z |z| = 1 - w = (z+ z)(1 i)n + i(z z)(1+ i)n, n Z pi

5.243 z, pi pi pipi:) Re(iz) = 0 |z+ i| = 2) z = i z |z+ 1| = |z|) |z|+ |i z| = 2 Re(z2) = 1) |z+ 2z| =

10 |z|2 + Re(z2) = 2Im(z)

) |z| = 1 |z2 + z2| = 1) |z| = 1 |z2 + z2 + z2z+ zz2 4| = 6) (i|z|3i)(z2 z) = 3i+ z3 z2+ z i|z| piz2012 + z1000 + 1 = 0

5.244 pi :

) z2 = 4z+ |z|2 +16|z|3

)zz+

zz=

z|z|+

|z|z

) z2 + 2+ i = (1+ i)|z| ) |z2 + 4|2 + 6|z+ 2|2 = 30) 8z3 + 12z|z|2 + 6z|z|2 + z3 8 = 0) 2014(z i)2(z+ i) 2013(z+ i)2(z i) = 1

5.245 z,w C, pi -: (

z+1z

) (|w|+

1|w|

)= 4 |w| < 1

pi |z| = 1

(iv) |z|2 = zz|z|2 = zz|z|2 = zz

5.246 z C, pi :)

z|z|+

|z|z R ) z

|z|

|z|z I

5.247 z1, z2, z3 C |z1| = |z2| = |z3| = 1 z1 + z2 + z3 = 1, pi

1z1+

1z2+

1z3= 1

5.248 z. pi :) |zi| = |z+i| z R ) |1iz| = |1+iz| z R

5.249 , R , , pi :

|z i| = |z i| Im(z) = + 2

5.250 z C, |z| = 2, pi pi- |iz+ 2|2 + |2i z|2

5.251 z C. pi :

z2 2iz |z|2 = 0 Im(z) = 1

5.252 z C, pi :) |z+ i z|2 = 2|z|2 + 2Im(z2)) |z 3i|2 |z+ i|2 = 8 Re(8i z)

5.253 z,w C, pi :

|zw| Re(zw) =|z w|2 (|z| |w|)2

2

5.254 z,w w , 0. pi:) |z+ w|2 = |z|2 + |w|2,

zw I

) |z+ w| = |z w|, zw I

) |z+ w| = |z|+ |w|, zw 0

) |z+ iw| = |z|+ |w|, ( zw

)2 0

5.255 z,w C |z| = 3, |w| = 4 |z w| = 5, pi 16z2 + 9w2 = 0

5.256 z C |z| = 1, pi :

|1+ z2| = |2 |1 z|2|

5.257 z,w) pi :

|z+ w|2 + |z w|2 = 2(|z|2 + |w|2)

) pi pi pi:

) |z+w| =3 |z| = |w| = 1, |zw| = 1

) a = |z|+|w|+i(|z||w|) b = |z+w|+i|zw|,

|a| = |b|

) |z 2i| = 1 |z| = 2, |z+ 2i| =15

v) |z w| = |z| = |w|, |z+ w| =3|z|

v) |z+ w| =2|z| |z w| = 2|w|

v) |z 2| = |w 2| = 2 |z w| = 4,

|z+ w| = 4

v) R w , 0, :

4 z2 + w

2 + |z 2w|2 = 2|w|2 ( zw2 + 4)

v) a = 3

|z3 + 3z2i i 3z|2

b = 3

|z3 3z2i i3 3z|2, a+b = 2(|z|2+1)

5.258 z,w C , R, pi :

|z+ w|2 + |z w|2 = (2 + 2)(|z|2 + |w|2)

14

5.259 pi pi pi:) z C, (1+ z)(1 z) I |z| = 1) z , i,

z iz+ i

I |z| = 1) z C R, (1+ z)(1 z)

z z R |z| = 1

) z , 1, z iz 1

I |2z 1 i| = 25.260 z,w C, pi :

) z , w, i(z+ w)z w

R |z| = |w|) z , w,

z+ wz w

I |z| = |w|) w , 1,

z zw1 w

R z R |w| = 15.261 z,w C, |z| = |w| = 1, pi :

)(z w)2

zw 0 ) u+ zwu (z+ w)

z w I, u C

5.262 z C, pi :) |z+ 8| = 2|z+ 2|, |z| = 4) |z 2| = |2z 1|, |z| = 1) |z 8i| = 2|2z i|, |z| = 2) |2z i| = |iz+ 2|, |z| = 1) |z 1| = |z |, R {1, 1}, |z| = 1) |z+ 9+ 3i| = |3z+ 3+ i|, |z| =

10

5.263 z C, pi :) |z 10| = 3|z 2|, |z 1| = 3) |z+ 3| = 2|z 3|, |z 5| = 4) |z i| = 2|z+ i|, |3z+ 5i| = 4) |z+ 1 2i| = |2z+ 2+ i|, |z+ 1| = 1

5.264 z,w C, pi z|w| w|z|zw+ |zw|

I

5.265 z,w C. pi :) |z+ w|2 = |z|2 + |w|2 + 2Re(zw)) |z+ w|2 |z w|2 = 4Re(zw)) |zw 1|2 |z w|2 = (|z|2 1)(|w|2 1)) (|z|+ |w|)2 |z+ w|2 = 2(|zw| Re(zw))

5.266 z,w C, z , w, pi :

z+ wz w

=|z|2 |w|2

|z w|2+ i

zw zw|z w|2

5.267 z C |z22iz1|+|z2+2iz1| = 2|z2+1|. pi : ) |z i| = |z+ i| ) z R

5.268 z,w |w| , 1

u = ( + i)z w1 zw

, R

pi |u| = 1 |z| = 15.269 z1, z2, z3,

pi pi :

z1 + z3 , 0 |z1 + z2 + z3| = |z2 + z3| = |z1|

z1

z2 + z3

5.270 z,w C, |z| = |w| = > 0 pi :) (z+ w)

(1z+

1w

) R ) z+ w

2 + zw R

)(z+ wz w

)2 R, z , w ) z

3 + w3

(z w)3 I, z , w

5.271 z,w C z + w , 0, zw < R |z| =|w| = . R, z+ w+ zw

R

5.272 z,w, u C :

|z+ u w|2 + |z u+ w|2 = 2|z|2

pi u = w

5.273 z w, w , 0. pi- :

) Re( zw

)=

Re(zw)|w|2

) Im( zw

)=

Im(zw)|w|2

5.274 z, pi :) |z 1| = |z i| = |z 2| ) |z| = |z|2 = |1 z|) |z 1| = |z+ i| |z 2| = 2|z+ 3|) |z+ i| = |z+ 5 4i| |z 6i| = |z|

) |z| =2|z|= |z 2| ) |z| = 1

z 1+ i2 = 2

) |z| =1+ i

3

iz

= |2 z|5.275 z,w C, pi pi pi-

:) |z+ w| = |z w| zw I) |1 zw| = |z w| |z| = 1 |w| = 1) |z 2|2 + |z+ 2|2 = 10 |z| = 1) |z+ w|2 = 2zw |z w|2 = 2zw) |z|2 + |w|2 = |z w|2 |z+ w| = |z w|) |z+ w| = |z|+ |w| |z w| = ||z| |w||)

zw+

wz I |z+ w| = |z w|, R

5.276 z,w, u C, pi :

|z w|2 + |z u|2 = 2z u+ w2

2 + |u w|2215

5.277 z,w C, |z| = |w| = 1. pi: z+ zw

2 + z zw2 = 4

5.278 z,w C, pi : |w||z| z+ |z||w|w = |z+ w|

5.279 z,w. pi :) |1+ zw|2 + |z w|2 = (1+ |z|2)(1+ |w|2)

) |z w| =(1+ |z|2)(1+ |w|2), zw = 1

5.280 z,w z , w u =

z+ wz w

. pi :) zw R, u R ) |z| = |w|, u I

5.281 z,w C z , iw w , 0. |z2+w2| = |z2 2izww2|, pi

zw R

5.282 z C, pi :

|z+ |z||+ |z |z|| = 2|z| z R5.283 z C, |z+ i|+ |z i| = 2(|z|+ 1).

pi z+ 1z

= 25.284 z pi -

|z 3|+ |z+ 3| = 10 |z| = 4

5.285 z C, |z 1| = 3 |z| = 1. pi : ) z2 + z+ 1 = 0 ) z2013 = 1

5.286 z C, |z+ i|+ |z i| = |z+ 1|+ |z 1|, pi |Re(z)| = |Im(z)|

5.287 z C z < R w = 1+ z+ z2

1 z+ z2 pi, pi |z| = 1.

5.288 z w =1 iz+ z2

1+ iz+ z2.

pi : |w| = 1 |z| = 1 z R5.289 f (z) = z+

1z, pi z C.

pi :

) f (z) I z I ) f ( z1z2

) I |z1+z2| = |z1z2|

5.290 a, b, c C |a| = |b| = |c| = 1. pi- :

) u =a+ bc

+b+ ca

+c+ ab

pi-

) v =a bc

+b ca

+c ab

-) pipi a + b + c = 0, pi

Re(ab+

bc+

ca

)=

32

5.291 z1, z2, z3 C, pi :

|z1| = |z2| = |z3| = 1 1+ z1z2z3z1 + z2 + z3

R

pi z1z2z3 = 1 z1z2 + z2z3 + z3z1 =z1 + z2 + z3

5.292 z,w R, zw = 1. pi :z w2 + i+ z w2 i

= |z|+ |w|5.293 z,w, u C, z2 = wu. pi

:

|w+ u+ 2z|+ |w+ u 2z| = 2|w|+ 2|u|

5.294 z,w, u C, w2 = z2 u2. pi- :

|z w|+ |z+ w| = |z u|+ |z+ u|

5.295 z1, z2, ..., zn :

|z1| = |z2| = = |zn| = > 0

pi

w = (z1 + z2)(z2 + z3) (zn1 + zn)(zn + z1)

pi

5.296 z1, z2, z3, ..., zn, pi . pi (

1+z2z1

) (1+

z3z2

)

(1+

znzn1

) (1+

z1zn

) R

5.297 z,w C, |z| = |w| |z + w| = |z| + |w|. pi z = w.

5.298 z,w pi :) z+ w+ 2zw = 1 |z| = |w| = 1) z+ w+ 1 = zw |z| = |w| = 1

16

5.299 z,w, u C, pi :) |z w|2 + |w u|2 + |u z|2 + |z+ w+ u|2 == 3(|z|2 + |w|2 + |u|2)) |z+w|2+|w+u|2+|u+z|2 = |z|2+|w|2+|u|2+|z+w+u|2) |z+w+u|2+|z+w+u|2+|zw+u|2+|z+wu|2 == 4(|z|2 + |w|2 + |u|2)

5.300 z,w C, |z2 w2| = 2|zw|. pi- :

||z+ w| |z w|| = 22||z| |w||

5.301 z,w C, |z| = |w| = |z+w|) pi z3 w3 = 0) pi

zw

5.302 1, 2, ..., n > 0 z C :1

z+ 1+

2z+ 2

+ + nz+ n

= 0

pi z R5.303 z,w pi

: |z| = |w| = 1 |z+ w+ 2| = |zw 1|

5.304 , z,w |z| = |w| AOB = 450, pi z2 + w2 =

2zw

(v) |z| = |z||z| = |z||z| = |z|

5.305 z C |i z| + |z + i| = 2, pi |z i| = 1

5.306 z C, pi :) |z2 + i| = |i z2| ) |z+ |z|2| = |z2 + z|

5.307 z1, z2, z3 C |z1| = |z2| = |z3| = 1 z1 + z2 + z3 = 4i, pi

w =1z1+

1z2+

1z3

5.308 z,w C, |z| = |w| = 1 R, pi- |zw+ | = |zw+ 1|

5.309 z,w C, |z| = |w| = 1, pi |z+ w+ zw 1| = |z+ w zw+ 1|

5.310 |z1| = |z2| = |z3| = 2 |z1 + z2 + z3| = 4, pi |z1z2 + z2z3 + z3z1| = 8.

5.311 z1, z2, z3 C, |z1| = |z2| = |z3| = 1, pi |z1 + z2 + z3| = |z1z2 + z2z3 + z3z1|

5.312 z1, z2, z3 C, :

z1 + z2 + z3 =32

|z1 1| = |z2 1| = |z3 1| = 3

pi |z1z2+z2z3+z3z1| = 3|z1+z2+z33|

(vi) z = w,z = w,z = w, |z| = |w||z| = |w||z| = |w|

5.313 z C, 8 + z2 = (3z2 6)i, z

5.314 z C z4 + (z 1)4 = 0, pi Re(z) =

12

5.315 z C, (z 2)n = (z + 2)n, n N, pi z I

5.316 n N, pi pi pipi:) (1+ i)n = 16 ) 8i(1+ i)n = (

3 i)n

) (3 i)n = 8i ) (2i)n + (2i)n = 2048

) (1+ 2i)n = 3+ 4i ) + in = 0, , > 0) (+ i)2n = (+ i)n+1, pi , R) n(2+ 3i)n + (3 2i)n + (3+ 2i)n = 0

5.317 n N, (1+ i)n = 2n

5.318 z C {i} n N (z 3iiz+ 1

)n=

1 i3

2,

pi Im(z) = 2

5.319 z,w C zw

wz= i, pi

|z| = |w|

5.320 z,w C (zw)2+w2 = 0. pi:

) Re(zw) =|w|2

2) |w| =

2|z|

5.321 z,w C z20+w20 = 0. pi:) |z| = |w| ) w , 0,

( zw

)30 I

5.322 pi pi , pi :) (1+ iz)7 = (z+ i)5

) (1+ iz)n =1+ i

3

3+ i, n N

) (1+ iz)2+1 = (i+ z)2+1, , N , ,

5.323 z C, 5(z + i)n = (4 + 3i)(1 + iz)n, n N, pi z R

5.324 z1, z2 z2+z+ = 0, , , R > 2. pi :) z1, z2 < R) (w z1)n = (w z2)n, n N n > 1, w R

17

5.325 z = x + yi, x, y R, pi (z 1)22 = (z i)22. pi :) x = y ) z22 I

5.326 z C (1+z)2n+(1z)2n = (1z2)n, pin N n 2. pi |1 z| = |1+ z|

5.327 z , i pi (z+ i)3 + 2i(z i)2 = 0. pi :

) |z+ i| = 2 )(z+ i)2 + 4

z+ i R

5.328 pi , , 0 z :(z i

z+ i

)n=+ i i

, pi n N

pi : ) |z i| = |z+ i| ) z R5.329 pi :

) z3 = |z| ) z3 |z| 6 = 0 ) 2z3 = |z|+ 1) z3 = z ) z3 = z|z| ) 2z6 = |z|+ 1

5.330 z n pi :

zn = 1 (z+ 1)n = 1

pi :) |z| = |z+ 1| = 1 ) z2 + z+ 1 = 0

5.331 z C, z14 = 27z11, pi z25 R5.332 z C, [(1 2i)z]7 = (z + 2)7, pi

: )5|z| = |z+ 2| ) |2z 1| =

5

5.333 z C, z100 + iz99 iz + 1 = 0. pi- |z| = 1

5.334 z,w C, z2+zw+w2 = 0. pi |z+ w| = |z| = |w|

5.335 z C, 1+ 2z+ 22z2+ + 2nzn = 0, n N. pi (1+ 2z)(1 2z) I

(vii)

5.336 z,w pi :

z+20iw

= 5+ i w+12iz= 4+ 10i

|zw|

5.337 z1, z2 C |z1| = |z2|. x, y pi xy = z1 xy = z2

()

(i) pi

5.338 :

z = 8i

(3+ 4i)4 w =

+ 17i(4 3i)4

, R

pi - pipi

(ii)

5.339 , , z1 = 1 + 3i, z2 = 3 3i z3 = 1 9i . pi .

5.340 z , 0 w = t(3 1)z,

t R. t pi , =

5.341 pi - z25(1+ i)z+ 13i = 0

5.342 z,w C, 2zw+

wz= 2.

pi :) z,w

) |z| =22

|w|

5.343 a, b, c :

c = za+ (1 z)b, a , b z = 12+ i

32

pi a, b, c - .

5.344 R -:

z1 = + i(1+ ) z2 = (1+ ) + i

, z1, z2 pi .

5.345 z C z < R z < I, pi z, z,

z2

z -

18

5.346 a, b, c - pi a + b + c = 0 |b| = |c|. pi a, b, c -

5.347 ,, , , -. ,, - , pi Re[(+ 2)( )] = 0

5.348 ,, , , -. ,, , pi pi C R || = 1 (1 )+ = 0

(iii)

5.349 pi pi pipi, pi z1, z2, z3 - :) z1 = 3+ 4i, z2 = 5+ 5i z3 = 2+ 6i) z1 =

4 2i1 i

, z2 =6+ 2i1 3i

z3 =2 6i2 i

5.350 z,w z , 0 pi w = z(1 + i), > 0. pi z,w - , pi .

5.351 , z,w , 0 z , w. pi pi pi :) AOB = 900 ) |z w| = |z+ w|) |z|2 + |w|2 = |z+ w|2 ) zw+ zw = 0

5.352 pi pi pipi, pi z,w, u, pi , - :) 2u = iz+ (2 i)w ) (1 2i)u = w 2iz

5.353 z,w C, pi -, pi pipi, :) z,w u = iz(1 i)w, z , w) z i, 1+ iz (1+ i)(z i), z , i

5.354 , , , , -, pi :

||2=

||

2=

||

22

, A = 900, pi :

Re()Re()

=12

5.355 z1, z2 z21 + z

22 = 0. pi :

) |z1| = |z2| ) |z1 + z2| = |z1 z2|) , z1, z2

5.356 z1, z2, z3 , :

|z1 z2|2 + |z1 z3|2 = |z2 z3|2

pi z2 + z3 = 0

5.357 , , z1, z1, z3 , pi - . pi :

A = 900 |z2 + z3 z1|2 = |z2|2 + |z3|2 |z1|2(iv) pi

5.358 pi pi- , pi -:) z2 2i z = 0, z , 0 ) z3 + 3z2 + 3z+ 9 = 0) z3 = 27

5.359 z,w C 2z = (1+ i3)w. w x2 + y2 = 4, pi- :) z ) pi

5.360 z,w C, pi z + w, z w z + i

3w

pi .

5.361 z, pi pi pipi, pi :) i, z, iz ) 1, z, z2

5.362 z,w C z2 + w2 = zw, pi pi z,w pi

5.363 :

19

(2x)z2 (6x)z+ 42x+ 9 = 0, x (0,pi

2

)) z1, z2 pipi -.) , z1, z2 , -:) pi x, -pi ,) pi x pi;

5.364 :

z2 (2+1)z+ 22 = 0, [0, 2pi)

) z1, z2 pipi ) , z1, z2 , pi pi pi.

5.365 , , z1, z2, z3 pi :

z1 z2z3 z1

= 12 i

32

pi pi

5.366 a, b, c C, |a| = |b| = |c| a, b, c pi , pi ab, bc, ca pi pi .

5.367 3 = 1, , 1, pi z,w z 2w z , w pi

5.368 a, b, c pi :

|a| = |b| = |c| = Re(ab) = Re(bc) = 2

2

, , a, b, c, pi :) pi)

) Re(ca) = 2

2

(v)

5.369 z,w |z| = 1 |w| = 2. z,w - 600, :) pi pi z w) pi : ) |z+ w| ) |2z+ w|

5.370 , z,w , pi :

|z w|2 +3|z| |w| = |z|2 + |w|2

pi (OAB) =|zw|4

5.371 z,w C zw+

wz=3, pi-

O(0, 0), A(z) B(w).

5.372 , , z1, z2, z3. ,

= A =z2 z3z1 z2

2 , pi 5(z22 + z

23) = 4z1(z3 + z2 z1) + 6z2z3

(vi) pi

5.373 pi pi pipi, pi z1, z2, z3, z4 pi:) z1 = 4+ i, z2 = 6+ 5i, z3 = 4+ 11i z4 = 2+ 7i) z1 = 1+ 2i, z2 = 5+ 8i, z3 = 7+ 12i z4 = 3+ 6i

5.374 pi

z2 z+ 1 = 0 z2 +3z+ 1 = 0

pi.

5.375 , a , 2 + 2 = 0. pi - , , +

5.376 z,w z , w. pi z,w, z + (z w)i w+ (z w)i

5.377 , , C pi - . , , , , , pi pi- pi, Im[( )] = 0 | | = ||

20

() pi

(i)

5.378 pi z, pi :) |z+ 2| = |z+ 1 i| ) |z 1 2i| = |z 3+ 4i|) |z+ i| = |z 3+ i| ) |iz 3+ 2i| = |iz+ 2 i|) |z+ z|2 = 2|z|2 ) |2iz 1+ i| = |2iz+ 1+ i|

5.379 z C n N, z2n = |z|n(z + 1)n. pi z 2x+ 1 = 0

5.380 f (z) =z 3iz+ i

, z , i) f (z) = 1, pi z) f (z) < 1, pi Im(z) > 1

5.381 f (z) = iz+ 1, pi z C) f (z) = z) | f (z)| = |z|, pi z

5.382 z,w. pi z, pi pi- pipi:) |w 1| = 1 zw = i) |w| = 1 zw = z+ 1+ wi

5.383 z,w pi :

2|z| = z+ z

wiw 2

R

) pi - z,w) z = w

5.384 z,w C, |z+w| = |zw|. pi u = zw.

5.385 z,w w , 1 pi :

|2zw 2z+ w 1| = |2zw 2z+ 2w 2|

pi z

(ii)

5.386 pi z, pi :) |2z 1| = 4 ) |z+ 1+ 2i| = 3 ) |z i| = 2) |(1 i)z+ 2i| = 2 ) |3 2iz+ 4z| = 7

) 4z2+ i

+ 5 = |z5|

5.387 pi z pi :) |z 3 4i| 5 ) 1 |z 3 4i| 2) 2 < |2z 2+ 6i| < 4 ) |1 z|2 1 |z|2

5.388 pi z, pi :) |4z i| = 2|z+ i| ) |z 1|2 + |z+ 1|2 = 10) |z+ 1| = 2|z 2| ) |z 1 i| = 2|z 1+ i|) 2|z| = |z 3| ) |z 2i|2 + |z 3+ i|2 = 13) 2|z 3 2i|2 + 2|z+ 1 3i|2 = 21

5.389 z, pi :

|z 4| 52

|z 3i| 52

5.390 z,w C, pi w, pi pi pipi:) |z i| = 2 w = 4z+ i) |z| =

5 w = (2 i)z+ 3i

) |z| = 4 2iw = z 1 i) |z 1| = 2 (w+ 2i)z = 2w) |z 3+ 4i| = 5 zw = 1) |z 2i| = 3 w = 4z i

) |z| = 1 w =z 1z

5.391 z pi |z2+ i| = 3. pi :) z ) z+ 3 2i ) iz+ 1

5.392 z, pi |z3+ i| = 2. pi w, pi pipi:) w = z+ 1 ) w = z 3+ 5i ) w = 2z 4+ 3i

5.393 z (0,0) 1, pi

w =z iiz+

, R

5.394 z,w C |z| = 1 z2 = 2+ izw) pi z iw) pi w

5.395 z,w zw iw = z (2 + 3i). z (0,1) (2,3), pi w -

21

5.396 pi z pi (z 2i) = 4, N 4 .

5.397 z , 0, pi -

Re(1z

)=

14

) pi z |z 2| = 2) Im(z) = 1, pi pi- z

5.398 z1, z2, z3 - , > 0 z1+z2+z3 , 0. pi

z =z1z2 + z2z3 + z3z1

z1 + z2 + z3

.

5.399 z pi - 2|z| = |z 3|) pi z) pi

w =z 3z+ 1

pi .

5.400 z, pi |z+ 3| = 2|z 3|. pi:

) z ) w =z 9z 5

5.401 z , i w =z+ 2iz i

.

pi z, pi- pipi:) |w| = 2 ) w I Im(w) > 0

5.402 z0 , 0 - z pi |z z0| = |z|. pi w pi zw = 1

5.403 z pi pi , pi w, pi zw = 1,

5.404 z, O(0, 0) 4, pi w = z2 + 3i pi pi .

5.405 z pi :

|z i| = 1 Im(z) 1

) pi z) pi

w =12

(z i

1z i

) y y

5.406 pi z, pi |z2 + 1| = 2|z|.

5.407 pi z , 0, pi :z+ 1z + 2

+ z+ 1z 2 = 4

(iii) pi

5.408 pi z, pi- pipi:) |z+ 2| = |Re(z) 2| ) |4z i| = 1+ 4Im(z)

) |z|2 (Im(z) +

12

)2+

54= 0

) (|z| 2i)(|z|+ 2i) = (Re(z) + 2)2

5.409 p R z C, pi :) |2z p| = |z + z + p|, z pi y2 = 2px) |2z pi| = |z z + pi|, z pi x2 = 2py

5.410 z,w C w = z2, R |z 1| = 1,

pi w

5.411 z pi :

(2z+ z+ i)10 = 3 i2

50 (z i)10 pi :) |2z+ z+ i| = |z i|) z pi pi

5.412 z C, |z|2 (Im(z) + 1)2 + 1 = 0. :) pi z) z pi

8

22

(iv) -pi

5.413 pi z, pi- pipi:) |z 1|+ |z+ 1| = 4 ) |z 3i|+ |z+ 3i| = 10) ||z 4i| |z+ 4i|| = 6 ) |z 2| = |z+ 2|+ 1

) |z10i||z+10i| = 12 )1

|z i|+

1|z+ i|

=4

|z2 + 1|

) |z |+ |z+ | = 4, > 0 )z+ 1z

= |z|5.414 z pi

|z+ 6|+ |z 6| = 20

) pi z) pi pi |9z z|

5.415 > > 0, pi z pi :

|(+ )z ( )z| = 2

()

(i)

5.416 z , pi pi :) |z i| < |z+ i| Im(z) > 0) |z| < |1 z| 2Re(z) < 1) |iz+ 1| > |z| 2Im(z) < 1) |6z i| |2+ 3iz| |z| 1

3

5.417 z C {1}, pi :

Re(

2z+ 1

)> 1 |z| < 1

5.418 z C Re(z) > 1, pi :z 12 < 12

5.419 z,w C, pi pi -:) |z w| > |z+ w| Re(z) Re(w) < 0) |z w| |z w| Im(z) Im(w) 0

5.420 z C, |z| 1, pi :

|2z z|2 + |2z+ z|2 10

5.421 z,w , pi (z2 3z+ 5)(w2 3w+ 5) 0

5.422 z,w C |z| < |w| |z w| > |1 zw|, pi |z| < 1 |w| > 1

5.423 a, b, z,w C |a| 1, |b| = 1 w = b(a z)az 1

.

pi |w| 1 |z| 15.424 z C, pi pi -

:

)z 2iz

< |z| Im(z2) < 1)

z+ 1z < |z| Re(z2) < 12

5.425 a, b, c C |a| 1, |b| 1 |c| 1, pi |ab|+ |bc|+ |ca| 3|abc|

5.426 z,w C, pi :) |z|2 + |w|2 zw+ zw) 2|zw| zw+ zw 2|zw|) 2(|z|2 + |w|2) (z+ w)2) (1+ |z|2)(1+ |w|2) (|z|+ |w|)2) 2|z+ w|2 6|z|2 + 3|w|2

5.427 z,w C, |z| 1 |w| 1, pi: ) |z|+ |w| 1+ |zw| ) |z w| |1 zw|

5.428 z,w, x, y C, pi :

(|z|2 + |w|2)(|x|2 + |y|2) |zx+ wy|2

5.429 z C |z| = 2 w = z2 z 4. pi :

) |w| = 21+ 4(Im(z))2 ) |w| 217

) (z2 z 4)(16z2

4z 4

) 68

5.430 z,w C |z| = |w| = 2 x R |z+ xw| > 1, pi |Re(zw)| < 2

3

5.431 z,w C pi - zw R 8z|w| w = z2|w|w. pi 0 < |w| 16

5.432 z,w C, pi pi pi-:) |z| 1, |w| 1 |z w| = 3, |z+ w| 1) |z| 5, |w| 5 |z + w| = 6, |z w| 8

5.433 z,w C, |z| = |w| = 1, pi :

|z+ w|+ |z w| 22

5.434 z C, Re(z) 0, pi

23

2|1+ z| 1+ |z|

5.435 z,w C, Re(z) 0 Re(w) 0, pi- |z|2 + |w|2 + zw+ z w 0

5.436 z,w C |w| = 1, pi :

|z w| 1 12(zw+ zw)

5.437 z,w C, |z + w| = 1. pi

Re(zw) 14

5.438 z,w C ww+1 = |zwiw|. pi |z i| 2

5.439 z C, pi |z + 1| 12

|z2 + 1| 1

5.440 , R ( + ) , 0 1+

1< 0,

pi :

|z+ w|2

+ |z|

2

+

|w|2

5.441 z,w C, |w| = 1, pi

|z w| 12(zw+ zw) 1

5.442 z = x + yi, x, y R. pi :

||x| |y|| |z| |x|+ |y| 2|z|

5.443 (,1) (1,+) z C.) pi :

|z+ i| > |z+ i| |z| < 1) z4 + iz3 + z+ i = 0, pi |z| = 1

5.444 z |z| = 1. pi |z2 + z 1| 5

5.445 a, b, c pi |a| = |b| = |c| = 1 a + b + c = 3. z C [Im(z)]2 1, pi :

|z a|2 + |z b|2 + |z c|2 3

5.446 :

A = {z C : z = + i, > 0, |z| < 1}

pi z A pi x A z =

1 x1+ x

5.447 z1, z2, ..., zn C {i}, n N pi :z1 iz1 + i

+ z2 iz2 + i+ + zn izn + i

< 1 pi :

)(z1 + z2 + + zn) i(z1 + z2 + + zn) + i

< 1) pi pi pi z1, z2, ..., zn pi pi

5.448 , > 0, + = z,w C, pi- |z|2 + |w|2 |z+ w|2

5.449 z,w C > 0, pi :

|z+ w|2 (1+ )|z|2 +(1+

1

)|w|2

5.450 z C, pi : 1z iz+ 1z zi

12 iz zi

5.451 x, y R, x2 + y2 = 1. z,w C,

pi : zx2 + wy

2 |z+ w|25.452 z,w C, |z| = |w| z + w > 0, pi-

zw > 0

(ii)

5.453 z,w C, pi :

||z| |w|| |z w| |z|+ |w|

5.454 z C. pi :) |z| = 3 w = 6+ 8i, 7 |z+ w| 13) |z| = 4 w = 3 4i, 1 |z w| 9) |z 1| = 3 5 |2z 3| 7) |z| = 3 w = 8+6i, 24 |2z+3w| 36) |z 3i| = 3, 2 |z 4| 8) |z 6i| 4, 6 |z+ 8| 14) |z 2+ 3i| = 2, 3 |z 5+ 7i| 7

5.455 a = 3+ 5i, b = 7+ 5i z C, pi |z a|+ |z b| 4

24

5.456 z,w C |z| = 1 |w| = 2, pi |2z+ (3+ 4i)w| 12

5.457 z,w C, pi :) |z| = |w| = 1, |zw+ zw| 2) |z| = 1, |zw |z|| 1+ |w|

5.458 z C |z| 3 1 R, pi |2z + z2| 2

5.459 z C, pi pi pi:) |z| 0. pi 310 m 1

25.477 a, b, c a + b + c = 0

|a| = |b| = |c| = 1. z C, pi :

3|z| |z a|+ |z b|+ |z c| 3(|z|+ 1)

5.478 z,w C u = |z|w + |w|z pi |u| = |z|2 + |w|2. pi |z| = |w|

5.479 z,w C, |z| = 1 |w| = 2. pi- :

(z+ w)(1z+

4w

) 9

5.480 a, b, c |a| = 1, |b| = 2 |c| = 3. pi :

) (a+ b+ c) (1a+

4b+

9c

) 36

) |a b c|+ 1a + 4b + 9c

25.481 ) z C, pi :

|z 2i|+ |z+ 3i| |z|+ |z+ 5i|

) w C |w| = 1, pi :

25

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

5.482 z C z+ 2z

= 3, pi :|z|+

2|z| 17

5.483 z C, |z| = 1, pi :) |3z 2| = |2z 3| ) |3z 2|+ |5z2 2z 1| 1

5.484 z C |z| = 1, pi :

|2013z 1|+ |2014 z+ 2013z2| 2014

5.485 z ,12

w , 12, pi :

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

12+

|z||z+ w|

5.486 z1, z2, z3 C |z1| = |z2| = |z3| = 1, pi : 1z1 + 1z2 + 1z3

35.487 z1, z2, ..., zn

(0,0) 1, pi : 1z1 + 1z2 + + 1zn = |z1 + z2 + + zn| n

5.488 z C R, pi zz+ |z|+

zz |z| 1

5.489 z C |z| = > 2, pi : 1z2 + z+ 1 12 1

5.490 z C :

|z 1|+ |z|+ |z+ 2|+ |z+ 3| = 8

pi |z+ 1| 25.491 , , C :

K = {z C/|z |+ |z | ||}

z1, z2 K [0, 1], pi z = z1 + (1 )z2

5.492 z C n N :

|z 1|+ |z+ 1|+ |z2|+ |z+2|+ + |zn|+ |z+n| =

= 2n2 + 4n

pi |z| n+ 25.493 z,w C, pi :

|z+ w| z|z| + w|w|

2(|z|+ |w|)5.494 , R, + = 1 z,w C, |z| =

|w| = 1, pi :

|z+ w| |z+ w|2

5.495 z,w C, |z| = |w| x 1, pi:

(x+ 1)|z+ w| 2|xz+ w|

5.496 z,w C, |z + w| = |zw|, pi |z| 2 |w| 2

5.497 z C, |z| = 2, pi :

|z4 5z2 + 6| 2

5.498 z1, z2, z3, z4 C, pi :

|z1 z2| |z3 z4|+ |z2 z3| |z1 z4| |z1 z3| |z2 z4|

5.499 z,w C, pi :) |z|+ |w| |z|+ |w|+ ||z| |w|| |z+ w|+ |z w|)

|z|+ |w|+

||z| |w||

|z+ w|+

|z w|

5.500 , , , z C, pi :

|z |+ |z |+ |z | | |+ | |+ | |2

5.501 z,w u = |z|w + |w|z, |u| = |z|2 + |w|2. pi |z| = |w|

5.502 z C, pi :) |1+ z|+ |1 z| 2(1+ |z|)) |1+ z| |z| |1+ z|2) |z2 + 2z+ 2|+ |z 1|+ |z2 + z| 3) |z2 + z|+ |z2 + 1|+ 3|z+ 1| 2) |z+ 1|+ |z+ 2| |z|+ |z+ 3|) |z+ 2|+ |z+ 3| |z|+ |z+ 1|+ 4) |z+ 3|+ |z+ 4| |z+ 1| |z 2| 8) 2|z+ 5| |z|+ |z+ 1|+ 9) |z|+ |z+ 1| |z 1|+ |z+ 2|) |z+ 2|+ |z 1| |z+ 1|+ |z|+ 2

26

5.503 z C, |z2 + 1| 1 |z + 1| 1. pi :) |z2 + 2z+ 1| 1 ) |z| 1

5.504 z,w C x (0,pi

2

), pi :

|z|x

+|w|x

|z+ w|

5.505 z,w C = pi3+ i

pi

3, pi

|z w| |z|+ |z w|

5.506 z C, z4 4z+ 6 = 0, pi |z| > 1.

5.507 z,w C [0, 1], pi :

|z+ w| |z+ (1 )w|+ |(1 )z+ w| |z|+ |w|

5.508 z C, |z| = 1, pi :) |1+ z|+ |1+ z2|+ |1+ z3| 2) |1+ 3z2014|+ |1 z2014|+ |1+ 2z2014|+ |1+ z2014| 5) 2|1+ z|+ |1+ z2|+ |1+ z3|+ |1+ z4|+ |1+ z5| 4

5.509 z,w C, |z| = |w| = 1, pi :

|z+ 1|+ |w+ 1|+ |zw+ 1| 2

5.510 z C, |z2+ 1| = 2|z+ 1|. pi |z| 3

5.511 z C, pi :

) z2 + 1z2

2, z+ 1z 2

) z3 + 1z3

2, z+ 1z 2

5.512 z,w C, z3 + z2w = z w. |w| 2, pi |z| 2.

5.513 a, b, c C, pi :

|(1 a)(1 b)(1 c)| 1 |a| |b| |c|

5.514 z1, z2, ..., zn C, pi :

|1+ z1|+ |z1+ 2z2|+ + |(n 1)zn1+ nzn| 1 n|zn|

() -

(i)

5.515 pi pi pipi, |z| z pi :) Re(z) + 1 = Im(z) ) |z 5+ 3i| = |z 1 i|) |iz+ 3| = |z 4| ) |z+ 4i|2 |z+ 3|2 = 17) z = (+ 2) + (2+ 1)i, R

5.516 z C Re(z) = Im(z) + 3, pi |z|2 8

5.517 () pi pi z1, z2 pi z1+ z2 = 12i z1 z2 = 34i. w pi () .

5.518 z pi :

2|z+ 1| = |(1+ i3)z 3

3+ 3i|

:) pi z) |z|

5.519 z C, |z 4|+ |z 3i| = 5.) pi z) |z|

(ii) -

5.520 z 3x + 4y 60 = 0. |z 3 4i|

5.521 z = (2)+(1+2)i, R. :) pi z) z pi ) |z 1 i|

5.522 z, pi - |z+ 2 3i| = |z+ 1 i|. :) pi z) z, pi pi -pi pi A(1, 1)

5.523 z C |2iz 2 6i| = 2|z 5+ 3i|) pi z) |z|) pi |z 2+ 3i| 25

27

5.524 z, pi |z 1| = |z i| |z 2i|+ |z+ 3 4i|

(iii)

5.525 pi - z w |z w|, pi pipipi:) z(43i)+z(4+3i) = 2 4(w+w)3i(ww) =22) |z 2| = |z+ 2 3i| |w+ i| = |w 4+ 4i|

5.526 z |z2|+ Im(z2) = 1) pi z (1) (2)) z1, z2 (1) (2), pi |z1 z2|

2

5.527 z,w C, w = 3z i z+4. z y = x 2, :) pi w) pi |z w| 52

5.528 z1 = + ( 1)i z2 = ( 3) i, R.) pi z1 z2 pi .) |z1 z2|) = 2, pi pi

K = 1+(z1z2

)20145.529 z,w pi

:

|z+ 2| = |iz+ 2| w = (1+ i) +2+ ii, R

) pi - z w

) pi |z w| 22

(iv)

5.530 pi pi pipi, - z, pi :) |z 2| = 1 ) 2|z|2 = (1 i)z+ (1+ i)z) |z 4 3i| = 2 ) |z|2 + 2i(z z) + 3 = 0

5.531 z1, z2 C, |z|2 = 2Re(z), pi |z1 z2| 2

5.532 z,w C, |w| = 1 z5 = 22w5i. pi :) z ) 32 |z| 72

5.533 z pi - :

|z 1 3i|2 + |z 3 3i|2 = 4

) pi z) pi 4 |z z| 8

5.534 z,w pi |w| = 2 z = 2w + 2i. |z w|

5.535 z pi - :

Re[(z 1)(z+ 1)] = z+ z

) pi z) z1, z2 pi pipi - z, pi |z1 z2| 2

2

5.536 z, :

(z 3 4i)(z 3+ 4i) = 25

:) pi z) z -

5.537 z pi - :

|z 1+ 2i| =22

:) pi w = 2z+ 1+ i) w pi

5.538 z C pi |z i| = 2|z 1|) pi z) pi

5 2 |z| 5+ 2

5.539 z pi :

|(3+ 4i)z+ 25|+ |z+ 3+ 4i| = 6

28

) pi z) pi z pi ) z1, z2 pi pi , |z1 z2|

5.540 z1, z2 |z1 z2| = 2 w pi :

(w z1)(w z1) + (w z2)(w z2) = 4

) pi w) pi |wz2|

5.541 :

z = (3+2) + (3+

2)i, pi [0, 2pi)

) pi pi - z) pi pipi z, pi pi

(v) -

5.542 z C, pi , pi pipipi:) A = |z 4 3i|, |z| = 2) A = |z 1 i|, |z+ 2+ 3i| = 3) A = |z 2+ 5i|, |z+ 1+ i| = 2) A = |z 6+ 2i|, |z+ 2i| 1

5.543 z C. pi :) |z 2 i| 5, 8 |z 14 6i| 18) |z 1 i| < 5, |z 10 13i| > 10) |z 1+ 2i| < 3, 2 < |z+ 2 2i| < 8

5.544 z C |z| = 1, pi |z2 + 2z|

5.545 z pi - |z i|2 + |z + i|2 = 4. pi 4 |z 3+ 4i| 6

5.546 z = x + (y 1)i, x, y R pi |z 1 i| 1 = 0) pi w = x+ yi) pi

2 1 |w 2 i| 2+ 1

5.547 z pi (2z 3)n = (z 3i)n, n N.) pi z pi -) z1, z2 pi pi , pi |z1 z2| 2

5

) w = 2+5i, z

pi |z w|

5.548 z C, |(3 + 4i)z + 4 3i| = 10. :) pi z) pi u = |iz2+3i|) z pi pi u

5.549 z C |z + 1 i| = 1. pi:

|z 1 2i|2 + |z 5+ 4i|2

(vi) -

5.550 z,w C, pi- |z w| pi pipipi:) |z 1| = |z i| |w 2| = 1) (1+ i)z+ (1 i)z+ 4 = 0 (w 2)(w 2) = 1) |z 1 i| =

2 zw = 6

5.551 z,w pi :

|z+ 1| 1 w = 1 t2

+ i1+ t

2, t R

- |z w|

5.552 z,w C, zw = 2w + 1. z (1,0) = 1, :) pi w) |z w|

5.553 z,w pi - :

|(1+ i)z 1| =2 |w+ 3i| = |w 1+ 2i|

:) pi z,w) |z w|

29

5.554 z,w pi :

zz+ 2|z| = 3 w = ( 3)i, R

) pi z w

) pi |z w| 32 22

) pi |2z w| 32 42

5.555 z,w pi - :

|z 2 4i| = 2 |w 2+ 4i| = |w 6|

pi:) |z w| ) |z w 1+ 3i|

(vii)

5.556 pi pi pipi, |z w|) |z+ 2| = 1 |w 4i| = 2) |2z 4i| = 2 |w 3+ 3i| = 2) |z+ 1 2i| = 2 |w 6 2i| = 3) |z 1 i| = 1 w = 2z+ 3

5.557 z , pi :

) w =z 2i2iz+ 1

) |z w| 2

5.558 z,w pi - :

|z 3+ 2i| 3 |(1+ i)w+ 2| 22

pi |z w| 105.559 z,w C :

1+ izz+ i

R |iw+ 1| = |2w+ i|

:) pi z,w) pi z,w

5.560 z,w pi :

|z 1 4i| 22 |iw+ 11 2i| 32

) |z w|) pi z = w

5.561 z C, |z 1| 1 |z 2| = 1, pi 1 |z| 3

5.562 z,w C |z| = 2 |w 3 + 4i| = 5. pi |2z+ w|

(viii) pi-

5.563 z = 2+ 2i w = t 1+ 2ti, , t R. :) pi z,w) |z w|

5.564 z,w pi :

|4z 1| = 1+ Re(z) Re(w) + Im(2w) = 6

) pi - z w) pi |z w| 5

5.565 z,w pi :

z2 + z2 + 4Re(z) = 2|z|2 |w+ 1|2 |w 2i|2 = 15

) pi - z w) pi |z w| 5

(ix) pi-

5.566 z,w pi :

|z+ 1| = 1 Re(z) w 2 =(1+ 3i6+ 2i

)7) pi z) pi |z w| 1

30

(x)

5.567 z C |z 4|+ |z+ 4| = 10) pi (C) z) z1, z2 (C) pi , pi 6 |z1z2| 10

5.568 z pi - :

1|z 3i|

+1

|z+ 3i|=

10z2 + 9

) pi z) z1, z2 pi pipi - z |z1 z2| = 6, pi |z1+ z2| 8

5.569 z,w pi |z3|+ |z+3| = 10 |w| = 4. pi|z w|

5.570 z C, |iz 3| + |iz + 3| = 10. pi :) z , pi , ) 4 |z| 5 ) 16 |z2 9| 25

(xi) pi

5.571 z C |z+ 5| |z 5| = 8, pi |z| 4

5.572 z pi :

|z+ 2i| |z 2i| = 23

) pi z) pi |z 1| 6

6

()

6.573 pi pi pipi, pi z,w :) zw = zw ) |z| = |w| = |Re(zw)| = 1

6.574 , z,w -. pi pi pi:)

zw R OM//ON ) z

w I OMON

) |z+ w| = |z|+ |w| OM ON6.575 pi pi pipi,

pi - z, pi pi , :) 1+ i, z, z+ i ) z, z+ 1, z2 ) 1, iz, 1 z2) 1, z, 1+ z2

6.576 A1, A2, A3, A4 - z1, z2, z3, z4 . pi :

A1A2//

A3A4 z2 z1z4 z3 R

6.577 , C f (z) = z+ , z C. z1, z2, z3 , pi - f (z1), f (z2), f (z3) -

6.578 z C, pi - z,z

1z

.

6.579 z,w C zw = 1, pi z,w

6.580 a, b, c pi - pi ab+ bc+ ca = 1. pi :

)b ac a

R) a, b, c

6.581 z C R a, b R a , b a + b + 1 > 0. pi z2, az+b, bz+a |z 1| =

a+ b+ 1

6.582 z C :

w =(z|z|+ 1

)2n u =

(z|z| 1

)2n, n N

pi w, u

6.583 , z,w. pi :

(z+ w)2

zw R , , |OA| = |OB|

31

6.584 z , 0 w =

z, > 0.

, z,w , pi :)OM2 ON = OM

) , ,

6.585 a, b, c C |a| = |b| = |c| = 1 |a+b+c| = 3, pi a = b = c

()

6.586 z = 2 + 3i w = 1 + 4i , . () : 3x + y 5 = 0, MAB = 900

6.587 z,w , .) pi

OA OB = Re(zw)

) pi x x pi pi - z 1 z i) pi z, z1 z i

6.588 , z2 + 2z + = 0, R. w = 1 i, pi |

MA| |MB| = 4

6.589 z , z 1 2z + i .

AMAB

6.590 z w = 1 + i. , z+w zw , pi z,

OA OB = 2

6.591 z , 0 , z z2 1 . pi z,

OAOB

6.592 A1, A2, A3, A4 - z1, z2, z3, z4 . pi :

A1A2A3A4 z2 z1z4 z3 I

6.593 [0, pi] - z2 2 z+ 1 = 0 (1)) z1, z2 (1)) M1,M2 z1, z2 -, ,

OM1OM2

) pi , .

6.594 , , , || = ||. z = ( + )n w = ( )n n N, pi :) n , , ) n pi,

() pi

6.595 , , , z1, z2, z3, z4 . pi pi z1 z2 =z4 z3

6.596 , , z, z+ w,w

. z+ iwz iw

-, pi pi .

7 pipi

()

7.597 z , 1 w =1 zz 1

) z = 2+ i, |w|) pi |w| = 1

) pi u =w 1z 1

pi) w = 1, pi z

7.598 pipi, OA -

z , 0 OB -

w = z, =1+ i

3

2) pi 3 = 1) pi -pi) pi z3 + w3 = 0 z2 + w2 = zw) z , pi w

7.599 z , 1 w =z+ iz+ 1

) |w| = 1, pi Re(z) = Im(z)) w I, pi - z) |w| = 2, pi z - , pi

32

7.600 z pi :

|z 1| = 10 z+ 13

= 12 w 3wz + 6 = 6z w, :) w) pi u -pi :(

2u32

) (u

34

)= 6ww

7.601 z |z 3 6i| = 2|z|) pi |z+ 1+ 2i| = 2

5

) z1, z2 pi pipi - z, pi |z1 z2| 4

5

) z1, z2 pi pipi - z, |z1 z2| = 4

5, pi :

|z1 + z2|n = 2n 5 n2 , pi n N

7.602 z,w, u pi- :

w = z+ i z u = z+ iz

) pi :

) |w|2 = 2|z|2 + 2Im(z2)

) |w+ u| 4|z|) |w u| 22|z|

) |w| =2, pi

z

7.603 f (z) =1z, z C

) pi ( f (i))2015

) z 1, pi f (z)

) pi | f (z) f (z)| 2|z|

) | f (z) z| = 3|z|, pi z

7.604 z pi |z i|2 + |z+ i|2 = 4 w = (4+ 4i)z) , , i,i z -, pi AMB = 900) pi z) pi w) pi |z w| = 5

7.605 z = + i, [0, 2pi) :

w = 3z+ 10+4iz

) pi - z w) pi 5 Re(w) 15 5 Im(w) 5) |z w|

7.606 w pi - 3(w+ w) + 2i75(w w) = 26i90) pi w) w0 pi - ) z C |z+ i10|2+ |zw0|2 = 6, |z+ 1 5i|

7.607 z,w pi - :

|w| = 1 z =1+ 2ww 2

) pi z

K(43, 0

)

=53

) z1, z2 pi pipi - z, z1 , z2, pi

u =3z1 + 3z2 + 83z1 3z2

) pi 1z

) pi |12z2 + 7z 12| = 25|z|

7.608 R z , i w =

z+ iiz+

. pi :) w , z ) |w| = 1, z R) w R, pi z

7.609 z1, z2 - w = |z1z2|2 + (z1z2 + z1z2)i pi y = x) pi z1z2 K(1, 0) = 1) z = 2+ 2i Im(z1z2) > 0, :

33

) z1z2 z1z2) pi (z1z2)20 + (z1z2)20 = 211

) z , z1z2 z1z2, pi (MAB) = 1

7.610 z,w pi :

w =(4 3i3+ 4i

)13+ (2 i)2 + 1+ 2i

|z+ 3i|+ |i z+ 3| = 2|1+ i|

) pi Re(w) = 4 Im(w) = 3) pi z - K(0,3) = 1) pi pi z pi ) |z w|

7.611 f (z) = z+ i z, z C) z , 0, pi

f (z)z 2

) pi f (i), f (1 + i) -

) f (z) , 0, pi f (z)f (z)

2014 = 1) f (z) + f (z) = 4, :

) pi z

) |z|

7.612 f : C C pi:

2 f (z) + f (1 z) = 3z2 7z+ 14, z C

) pi f (z) = z2 5z + 6, z C) f (z) = 5z+ 9+ 4i) f (z) = 54z pi- ) f (z) > 0) pi f (z) f (z) 0, z C

7.613 f (z) =2iz+ 9 12i

z 2i, z , 2i

w1 = z 2i w2 = f (z) 2i.) pi w1w2 = 5 12i

) w1 = w2, w1,w2) :

) f (z) = 2 2i ) f (z2) =8 7i6+ 5i

) f (z) R, pi z

7.614 f (z) =iz 1z+ 1

, z C {1}) pi ( f (i))12 = 64) z pi :

f (z) =12

12i

) | f (z)| = 1 | f (w)| = 2, - |z w|

7.615 f (z) = |z+ 2i|, z C) pi :

) f (z) = |i z+ 2| ) f (z) = f (z)) f (1+i+z) = f (2+3i+z), pi z pi ) pi pi :

) f (z) = f (z) z R) f 2(z) + f 2(z) = 10 |z| = 1

) :

z+ f (z 2i) = f 2(1 i) + i f (i)

7.616 f (z) =(z i)(z+ i)

z z,

z C Im(z) , 0) pi f (z) I) pi f

(1z

)= f (z)

) z pi:

f(1z

)=

z z2

+ i

pi:

) z

) w = 2z2 + 1

7.617 z,w pi w+ 2+ i = (1+ i)|z|) z = w, z) pi z : ) w R ) w I) pi

2|z| |w|+ 5

7.618 z,w pi :

34

w =

5z

|z| w2 = 3+ i, > 0

) pi = 4) pi 5z (3+ 4i)z = 0) pi z) |z 3+ 4i|

7.619 z,w pi :

|z+ 2i| = |w+ 2i| = 2 |z w| = 4

pi :) z w ) |z+ w| = 4 ) |w| = 1, |z+ 3w| = 2

3

7.620 z,w pi :

z1+ i

+z

1 i= 2 Re(w2) + (Im(w))2 = Re(w)

) pi z = 2i w = 1+ i) pi (w z)30 + w30 = 0) n pi :

3( z2

)n= 4n 27

) pi u = w+ ( 1)z, R

7.621 z , 0 w = 1z

, z,w .) pi , , -) pi w+ 1 =

z 1z

) |z 1| = 1, pi |w+ 1| = |w|) |w|

7.622 z1, z2 z+1z= 0 -

w pi :

|wz1|+ |wz2| = |z1 z2|

) z1, z2 pi- z2n1 = z

2n2 , n N

) pi w) w1,w2 pi pipi - w, pi :

w1 w21+ w1w2

I

) pi 4 |w 3+ 4i| 67.623 z,w pi

:

z+ w = 2 z 2w = 1 i

) z w) pi z30 + w30 = 0) pi u pi :

|u z|2 + |u w|2 = 4

) :

v =(u 1

1u 1

)4 pi : ) v R ) v 16

7.624 z,w pi :

|z+ 2| = |iz+ 2| w = (1+ i) +2+ ii, R

) pi - z w

) pi |z w| 22

) u |4u+1+3i| =10

) pi u) v1, v2 pi

pi w u

) pi |v1|2 + |v2|2 =52

7.625 f (z) =12(z+ i z), z C

) f (z) = z, pi z) pi f ( f (z)) = f (z)

) pi f (z) z1 i

=12(Im(z) Re(z))

7.626 f (z) = iz (1+ i)z, z C) pi f (1i) f (2+i)) f (z) = f (1+ i)) pi z -pi f (z) = z) pi :

) f(1z

)=

f (z)zz

) f (z) I, z I) f (z) = f (z), z R

35

7.627 z pi :

4(Re(z))2 z2 z2 = 2

) pi |z| = 1) z , i u C, pi

w =u+ izu z i

z i

) pi m pi m(z 1) = 3(i z)

7.628 z pi :

Re( zz 1

)= Im

( zz 1

)) pi z) z0 pi -

) pi pi(z02

)20167.629 z = 2+

3 i

3,

R w < R pi- |w| = w+ w, R) pi z (2,0) =

3

) pi w) pi z = w ) |z z|

7.630 z pi :

(2z 3i)5 (z 3i)5 = 0

) pi :

) |z i| = 1 ) z , i, (z i

1z i

)5 I

) z1, z2 pi pipi - z |z1 z2| =

2, pi

|z1 + z2| 14

7.631 f (z) =z+ 2iz 2

, z , 2

) ( f (2 2i))2016) f (z) , :

) pi z , pi

) |z|

) pi

w = z 1+ i+2

z 1+ i

pi, 22 w 22

7.632 z pi |z 3i| = |z+ 1 2i|) pi z) z1, z2 Im(z1) > 0, pi pi

10 pi

2i) w =

1z1+iz22, pi

w,w2,w3

7.633 z1 = 1 2i z2 = 3+ 4i) pi -

z2z1

) , R pi x2 + x+ = 0

z2z1

) z pi |z z1| =

) pi z

) pi iz2z1

pipi pi

) w =iz 4 27i

z z2, pi :

w+ 5iw 5i

I

7.634 pi , 3 = 25

z,w w = +

4i

|iz 3i 4| = 5.) pi - z w) z pi -) |z w|) pi 1 |z 3| 9

36

7.635 z2 2z+ 2 + 3 = 0, R) pi pi- ) , z1, z2 (OMN) =

3, :

) z1, z2) pi z31 = z

32 = 8

) pi z20131 + z20132 = 2

2014

7.636 w pi :

(w+ w)(3 2i) + (w w)(5 i) = 8+ 6i

) w

) z1 =(w2

)50

z pi |z z1| = |z |z1||

) Re(z1) Im(z1)

) pi Re(z) = Im(z)

) z , 0, pi pi-

A =zz+

zz

7.637 z2 2z+ 22 = 0, > 0, pi z1, z2 Im(z1) > 0) z1, z2 pi z1 = iz2) n N, pi z4n+21 + z4n+22 = 0) z41 = 4, ) pi z1, z2

7.638 f (z) =3Re(z) + 4Im(z)

z,

z C) | f (z)| = 3, pi z) Re(z) = Im(z), R, pi :

) | f (z)| =|3+ 4|2 + 1

) | f (z)| 5

7.639 z = i + i

, R) pi z = i) n N, pi:

(1 z)n = (1+ z)n

) pi 1+ z+ z2 + + z99 = 07.640 z2 + 6z + 10 = 0, pi

z1, z2, Im(z1) < 0.) z1, z2 pipi ) w C, |w z1| = |w+ z26i|,:

) pi w

) w pi

) pi (w 2)2014 I7.641 2z2 + 2z+ 1 = 0

) pipi ) z1 Im(z1) > 0, pi- z21 z

31

) pi - z1, z21 , z

31 pi

pi .

7.642 z2 + z = 0, , R z1 z2 = 1+

2i2013

.

) pi < 0) , ) pi Im(z20151 + z

20152 ) = 0

Re(z20151 z20152 ) = 0

) pi|w z1|+ |w z2|, w C) (1 + iu)n =

z1z2, n N, pi

u < R

7.643 z =1

2+ i, R. pi-

:) z+ z = 4zz) pi

z (C) K(14, 0

)

=14

) z w =

2 4i

(C))

12+ i

,1

2 4i

12+ 2010i

7.644 z1 = + i z2 = i, R. pi :

37

) z1 = iz2 |z1| = |z2|) w =

z1z2, :

) 1+ w101 + w102 + w103 = 0) (1+ w)100 = (1 w)100

) z1, z2 ) 4 |z1 3 4i| 6) z3 + w2 = 0 pi ) z3 pi pi

, z903 = 1 z3 +1z3= 1

7.645 z,w pi |z2 i| = 1 |w+ 1+ i| = 2. :) pi - z w) |z+ 2w|) |z 1+ 2i|

7.646 z,w |z w| = |z| |w| = 1

) pi Re(zw) =12

) > 0 pi w = (1 2i)z

) =12, pi :

) |z| =22

) z,w

7.647 z,w C, w , 0 |z + w| = |z w|. pi :)

zw

) z , w, izz w

+ wz+ w 1

) w = 1+ i, z y = x

7.648 z,w C, |z| = |w| = 2.) pi z =

2z

) pi |z+w+zw2| = |z+wzw+2|) pi :

z+ w+ zw 2 = 0 z+ w zw+ 2 = 0) z+ w = 2, z,w

7.649 f (z) =z i|z|z+ i|z|

) z C pi f (z)) [ f (1)]2014) pi f (z) R z I

7.650 f (z) =z(z 2i)|z 2|

, z C{2}. :) f (1 i)) pi z, :

) f (z) R ) | f (z)| = |z|

7.651 z2+z+ = 0, , R, pi z1 =

5+ 33i

1 23i

) pi - z1) pi z31 = 8) , ) , (2,0), pi pi

7.652 f (z) =iz 2+ 4i

z i, z , i

u = z i w = f (z) i.) pi uw = 3+ 4i) z2 = uw) pi z, f (z) R

7.653 z,w :

|z 12 6i| = |z 4| 2|w+ i|2 = |w i|2 + 17

) pi z pi w) |z w|

7.654 z,w -pi :

zw+

wz= 1

pi :) z3 + w3 = 0 ) |z| = |w|) z,w , pi

)( zw

)20+

(wz

)20= 1

7.655 z = x + yi, x, y R :

f (z) =(2+

32i)z

52zi

38

) Re( f (z)) Im( f (z))) pi f (z)) pi | f (z)| = |x 2y|

5

) | f (z)| =5, pi

z

7.656 z,w C, pi -:

|zi 12| = |3z 4i| w =z 8z 2

) z w

) pi w2 + 4

w R

) pi |w 3+ 4i|

7.657 z , 1 -

w =2+ i z1 z

.

) z = 2, pi w2012 R) pi

w 2w+ i

= z

) z -, pi w) w pi, - pi z

7.658 f (z) =5 12i 2iz

z 2i, z , 2i

) f (z) = z f (z) = 5+2i) f (z) R, pi z

7.659 f (n) = 1+ i+ i2+ + in, n N) pi | f (n)| 2) n f (n) = 0

7.660 f (z) = (z i)(z+ 1), z C) pi

f (i)f (1)

2015) pi f (zi) = f (z), z C) pi z, Re( f (z)) = Re(z+ zi)

7.661 f (z) = |z + i| z pi f (z) = 2 f (z)) pi z) pi |3z 5i| = 4) z pi - ) pi w = z 3

7.662 z , 2i f (z) =z3 + 8iz 2i

) pi f (z) = z2 + 2iz 4) f (1+ i)) :

) f (z) = |z|2 + 2iz 4 ) f (z) = 2iz+ z 6) |z| = 1, pi - f (z) = 7

7.663 z C f (z) = 2z 1z2

) f(1z

)= 2

) pi f (z) R z R Re(z) = |z|2) pi z, pi f (z) R) pi - pi pi pi -.

7.664 f (n) = inz, n N z C) pi :

f (4n) + f (4n+ 1) + f (4n+ 2) + f (4n+ 3) = 0

) f (8) f (3) (1,1) = 4, pi z) z = 1+ i

3, pi pi

0, z, f (4n + 3) , pi pi

7.665 f (z) =z+ 2iz 2i

, z C {2i}) pi | f (z)| = 1, z , 2i) f (z) = f (z), pi z I) z0 = 2i3 2 f (i), :

) pi w,

|w+ z0| =

i f (z0)z01+ i3

) w pi

7.666 f (z) = z2 + 2z+ 3, z C) x, y R, f (x 2yi) = 2) f (z) = 0) , R, f (z) = z+ , 1 2i) z1 f (z) = 2z + 2, Im(z1) < 0, pi pi-

A = z531 +1z741

39

7.667 z2 2t z+ 5 4t = 0, t [0, pi]) z1, z2 pipi ) pi z1, z2) pi|z1 z2| |z1 + z2|

7.668 f (z) =2z i zz 1

, z , 1. pi :) f (1 i) = 3+ 3i ) ( f (1 i))2016 R) f (1 i), f (1 + i)

7.669 z C {1, 0} f (z) =

z 1z+ 1

) pi ( f (i))90

) pi f (z) + f(1z

)= 0

) f (z) = f (z), pi z R) pi z, pi | f (z)| = | f (2i)|

7.670 z1, z2,w :

|z1| = |z2| = 2, z21 + z22 , 0 w =

2z1z2z21 + z

22

) pi : ) w R ) z1z2 + z2z1

2) w = 2, pi pi z1, z2 pi

7.671 |2w+ 1| 4w = 1+ 8i) w pipi ) pi z, pi :

zz+ 4Re(zw) + 2 = i56 i72

) z1, z2 -pi pi , - |z1 z2|

7.672 3z2+ z+ = 0, , R, z1 = 1+ i z2. pi :) = 6 = 6) z21 + z

22 = 0 z

20161 + z

20162 = 2

1009

7.673 z , 0, pi z3 + |z| = 0 (1)) pi z) (1)

) z1, z2 pi (1), pi pi A =z20131 + z

20132

7.674 z,w |z| = 1 w = z2+ iz) |w| Im(z)) pi u = w iz) z |w|

7.675 z, pi |z|2 + z(1+ i) + z(1 i) + 1 = 0 (1)) pi z) pi

2 1 |z| 2+ 1

) z1, z2 pi pi (1), pi|z1 z2|

7.676 z = x + yi, x, y R z , 2i. w =

z 2z+ 2i

, :

) pi Re(w) =x2 + y2 2x+ 2yx2 + (y+ 2)2

Im(w) =2x+ 2y+ 4x2 + (y+ 2)2

) w pi, :

) pi z

) |z|

) z ) w , :

) pi z

) |z|

) z1, z2 -

7.677 z,w w =z 2iz 2

) z = 3 i, pi w100) w , - pi z) w Re(z) , Im(z), |z|) Re(z) = Im(z), w

7.678 z pi :

|z 2 3i|2 + |i z 3 6i|2 = 16

) pi z) |(4 3i)z 25|)

40

z) z0 pi pi - w1,w2 :

w2 + 2(|z0| 5)w+ 9 = 0

) pi w1,w2 < R

) w1,w2

) pi w1 3iw1 + 3i

I

7.679 z C, z , 1 :

f (z) =2+ i z1 z

) f (2)) pi w = ( f (2))2014 pi

) pi f (z) 2f (z) + i

= |z|) |z| = 1, pi M - f (z), pi .

7.680 z , i

f (z) =iz 1z i

) pi f ( f (z)) = z) f (z) = f (z), pi z R) | f (z)| = 2, pi z) | f (z1)| = | f (z2)| = 2, pi

|z1 z2| 837.681 z = (2+ 1)+ (3 2)i,

R) pi z) z pi ) pi z pi ) pi 3 |z 4+ i| 7

7.682 z pi |z+ 8| = 2|2z+ 1| (1)) pi |z| = 2) - 2z+ 1) z1, z2 pi pi (1), z1 , z2, pi

w =z1 + z

2

(z1 + z2) pi

7.683 pi , > > 0 z2 + 2z+ = 0, pi z1 z2.) z1 z2) z1 z2) pi w =

z1 + z2z1 z2

) u (u z1) = (u z2), N, pi u R

7.684 z,w pi :

|z i| = |2iz 1| 1+ 1+ iw

= 1) pi (C1) z) pi (C2) w) , (C1) (C2), pi |

OM|2 + |

ON|2 = 4

7.685 z pi |z 2i| = |z|) pi z) pi pipi z, z1, z2 pi 2) pi z41 + z

42 = 16

7.686 f (z) =z 3iz+ 1

, z , 1) z0 f (z) = 2 2i, pi pi A = (z20160 + i)

4032

) pi z, :

| f (z)| =1 3i2

300) pi pi | f (z) 1| |z+ 1|) z (-1,0)

10, -

pi f (z)

7.687 f (z) =z3 iz+ i

, z , i

) pi f (z) = z2 iz 1) f (z) = (2 i)z 21 3+ 4i, :

) , R) n N, n < 100,

pi (+ i)n = (5+ 3i)n

7.688 z,w pi :

41

|2w+ 1| 4w = 1+ 8i z = w( i), R

) w) pi z) |z|) z1, z2 pipi pi, z1 , z2 |z1| = |z2|, pi- z1 + z2

7.689 z1, z2 C z1 + z2 = 1 z31 + z32 = 2) z1, z2) pi - 0, z1 z2) pi z1001 + z

1002 = 1

7.690 z pi :

Re(2z+

8z

)= 4Re(z)

) pi z) Re(z) , 0, pi

w = 2z+8z

pi 8 w 8

7.691 z2 + z + = 0 (1) R, pi z1 z2) w =

z1 + z2 + i1+ z1z2i

, :

) pi |w| = 1

) pi w2012) (1) + i,

7.692 z , 0 w = z +

z,

R. z1 = 1 i3

w = , (1) R, :) , (1)) pi (1) z3 + 8 = 0) n pi (iz1)3n = i(z1)3n) w R, pi z

7.693 z,w pi :

z+ |z+ 2 i| = 6 3i zw = 6 8i

) pi z = 1 3i) pi |iz w|) pi z2014 + w2014 = 0) u =

w+ zw z

, R. pi u2n R, n N

7.694 f (z) =z3 iz+ i

, z , i.

) pi f (z) = z2 iz 1) , R pi f (z) = ( i)z+ i2018 3+ 4i) w =

iz+ 1z+ i

pi, :

) pi |z|

) pi | f (z)| 3

) pi f (iz)f (i z)

= 17.695 f (z) =

z+ 7iz 1

, z , 1) pi f (3 3i) ( f (3 3i))2011) w |w 13+ 3i| = 11+ i) Re( f (z)) = Im( f (z)), :

) pi z

) pi- |zw0|, pi w0 ()

7.696 z2 z + = 0, , R, pi z1 = 2+ i z2.) , ) pi z20141 + z

20142 R

) , ,

z1, z2, z3 =z1z2+

17+ i5

, pi ) |w z1| = |w z1|, pi w R) pi w, pi |w z2|+ |w z2| = 10

7.697 z(t) =1

1+ 2ti, t R.

pi :) z(t) + z(t) = 2z(t) z(t)) z(t) (C), pi ) t , 0, z(t)

z(

14t

) pipi

(C)

42

7.698 z() =

1+ i, R

) pi z() (C)

K(0,

12

) =

12

) z() pi - ) pi z()

z(1

) pi-

pi (C)) pi z(1), z(1) z(2012)

7.699 z1, z2 C z1 , z2 z1 , z2.

w =z1 + z2z1 z2

) pi : |w| = 1 z1z2 I

) |z1| = |z2|, pi w I (z1 z2z1 + z2

)2014< 0

7.700 f (z) = (z + i)(z + i), piz C) z1 = 1+ i, z2 = f (z1) z3 = f ( f (z1))) , , z1, z2, z3 -, pi -) pi z pi-pi: ) f (z) R ) f (z) I

7.701 z C, Re(1z

)=

14. pi :

) Im(z) = 1, Re(z)) pi z pipi) |z|

) z1, z2 C, Re(1z1

)= Re

(1z2

)=

14,

|z1 z2|

7.702 z , i w =z+ iz i

. w I,:) pi z - = 1

) z1, z2, z3 pi pipi - z. pi :

) |z1 + 2z2 + 3z3| = |3z1z2 + z2z3 + 2z3z1|

) |z1 z2|

2= |z3 z1| = 1, |z2 z3| =

3

7.703 z2 2z+ 2 + 3 = 0, R) pi pi ) , (OMN) =

3, :

) z1, z2) pi z31 = z

32 = 8

) pi z20131 + z20132 = 2

2014

7.704 ) pi (z+ 3+ i)(z 4+ 2i)) z2 (1 3i)z = 14 2i) z1, z2 pipi , Re(z1) > 0 , , z1, z2 z3 = 3+ i , :

) pi -

) pi M(z), pi :

MA2 + 2MB2 = 2M2 + 30

7.705 z2 z + 9 = 0, R pi z1, z2 < R.) pi :

) 6 < < 6 ) (z171 + z172 ) R

) z1, z2) ,

z1z2+

z2z1= 2

) = 0 Im(z1) > 0, pi w pi :

|w z1| = 4+ |w z2|

7.706 z , 0

w =1z

.) pi z ) pi z4 < 0

) z2 1z

= 2, :) pi |z| = 1

) pi Im((z+ i)5

z5 + i

)= 0

) u =3z

4zi

43

7.707 z1, z2, z3 :

z3 + (1 )z2 + (1 )z = 0, R

z1, z2, z3 , pi :

) = 12

) n N, (zn1 + zn2 + zn3) R

7.708 z,w pi :

|w| = 2 z = w+4iw

) pi z) z1, z2, z3 pi pipi - z, z1 + z2 + z3 , 0, pi :

)z18=

1z1,z28=

1z2

z38=

1z3

) (z1 + z2)(1z1+

1z2

) 4

)z1z2 + z2z3 + z3z1z1 + z2 + z3

= 22v)

z1 z2z3

+z2 z3

z1+

z3 z1z2

I

()

7.709 z,w pi - :

z 2i+ 4|z 2i|z+ 2i

=5

z+ 2i |iw 2 2i| = |i w|

) pi z w) pi z) |z w| |z w|) u v pi pi w z pi |u|2 + |v|2 = 10

7.710 w1,w2 |w1| = |w2| =2

Re(w1w2) > 0 z1, z2 z2 |w1 w2|z+ 1 = 0. pi :) z1, z2 < R) |z1| = |z2| = 1) (z1 z2)2 = |w1 w2|2 4

7.711 z,w pi :

|z 40| = |z 30i| |w 3 4i| = 1

) pi |z| , |w|) |z+ w|) |z| |w|) pi u = 6+ 8i z

7.712 z,w pi :

|z 1| = |z i| w =i 1z i

) pi - z w) z pi |w2 2i| = 2) n |z| 1+ in = (1463n2)i, pi zn3 R

7.713 z,w -pi :

2014|z| z+ 2014|w| w = 2014|z+w| (z+ w)

pi :) |z| = |w| = |z+ w|

zw R

) ,, - z,w, z+w , , ,

ABO

7.714 z,w z , 0 pi w2 2|z|w+ 2|z|2 = 0. pi:) w < R ) |w|2 = 2|z|2 ) w2 + w2 = 0) w2n + w2n = 0, n pi

7.715 f (z) =z iz

, z C) f (z) R, :

) pi z I) zn = 1, n N, pi z = i

) f (z) f (z) = 5, :) pi z

) | f (z) 1|) z2 + 4 f 2(z) = 0

7.716 z1, z2 z4=2z+ 1

) z1 z2

) pi (z1z2

)2012+

(z2z1

)2014= 0

) w C |w z1|2 + |w z2|2 = 16, :

44

) pi w pi |w 2| = 2

) w1,w2 pi pipi - w |w1 w2| = 4, pi |w1 + w2| = 4

) w1,w2,w3 pi pipi w, pi :w1 + w2 + w3 64

= 1w1 2 + 1w2 2 + 1w3 2

7.717 z (-2,1) = 1 w = 4z+ 8) pi iz1+2i) pi w) pi 0 |w| 8) pi w w =

|w|2

4 i

) |3w216|

7.718 z C {i} w = z2 2iz 1z2 + 1

. w , :) pi w =

z iz+ i

) pi |z| = 1) z1, z2 pi pipi - z, pi

v =(z1 + z2)7

z71 + z72

pi) w R :

wi+ w11 = + i

7.719 z C (z 4 + 2i)(z 4 6i) I w pi :

w 2z = 4 i2018 (1+ i)32

(8i)5

) pi w = 2z+ 4+ 2i) pi - z w) |z w|) , z w u , pi 1 |u 8+ 5i| 3

7.720 z pi -:

|z 2|2 + z(2+ 3i) + z(2 3i) + 1 = 0

) pi z) pi 1 |z| 5) w =

8z 3i

z

) pi w

) pi (z w)2 0) pi z = w Im(z) = 3iv) , z,w z , w,

pi x x

v) u , , pi :

|u z| |u w|+ |u 3i|2 = 4

7.721 u1, u2 pi- u1 + u2 = 3 + 9i y = x+7 y = 5 . - z1, z2 w z51 = u1, z

52 = u2 w =

u1u2

) u1 u2) pi w pi - ) pi w2 < R) w2 z2 + z + = 0, pi = 1

7.722 ) z,w ,. pi , zw R) pi z pi -

z i,1

z 2

) z0 pi- , - n zn0 = 5n 31) u C |u| = 1, z (), |u 2z|

7.723 z1, z2 z +1z= 1

w pi - Re(w2) = ww 8Re(w)

45

) z1, z2 pi z31 = z

32 = 1

) pi w pi y2 = 4x) pi z1, z2 pipi pi pi) u

2|u+ z1z2|+ z111 + z112 = 0

pi 2|w u| 17.724 z pi -

= 1

w =2z+ iiz 2

) pi : ) |w| = 1 ) |z+ w| 1) w < R, R w+

w R

) z1, z2 pi pipi - z |z1z2| =

3, pi |z1+z2| = 1

) pi |w+ 4|+ |3iw 1| 57.725 z,w, u pi-

:

w =z 2z 2i

, z , 2i |2u 3 3i| =2

) w = + i, , R, pi :

w2 R = 0 = 0) w2 R, pi z) pi u) pi z = u

7.726 z pi - :

|z+ 2i| = 1+ 2Im(z)

) pi z) pi |z+ 2i| |z 2i| = 2) pi |z2 + 4| = 2 + |z|2 - pi w = z2 + 2) z1, z2 pi pipi - z |z1z2| =

3, pi |z1+z2| 1

7.727 z pi zn0 = 3+4i z

n+10 = 2+11i, n N

z pi

w =zn

z z0 + 2

) z0 n) pi z) u = w z i

) z < I, pi u) u1, u2, u3 pi pipi

u, pi :

u1 + u2 + 3u3 , 0 3+ 4iu1 + u2 + 3u3

17.728 f (z) = z2+ z+ 3,

z C , R , 0, pi

i2. pi :

w =f (z)

2iz 1

) pi = 4 = 0) pi w = 2iz 1) |z 2| = 1, :

) pi z) pi 2 |z w| 8

7.729 z,w, u pi- :

2z+ |z+ 1+ 2i| = 1+ 2i w+z10

32u = 0

pi :) z = 1+ i) w, u pi pi ) |w u| =

2|w|

) w, u

7.730 z pi - :

(3z 2)5 (2z 3)5 = 0

pi :) |z| = 1

) w =(z i)9

z9 i pi

) |3z 1| = |z 3|) |3z 1|+ |2z2 z+ 6| 1) u = z

4z

pi 3 |u| 5

46

7.731 z,w pi :

z 2i|z 2i|

=

3

z+ 2i w =

5( i)3 4i

, R

) pi z) pi

u = z 2i+3

z 2i

x x) pi w |w|) pi 5|z w| 11 53

7.732 z3+z2+ 2 = 0, pi z0 = 1 + i w pi :

2Re(w) , 1 |w2 + w+ |w|2| = |i w|

) pi = 1 = 1) pi w R) u =

w 7w+ 7

, pi :

) |u| = 1 ) v =u2 + iu 1u2 iu 1

, v R

) |u2 + u 1| 57.733 z,w pi

:

z =2 i

, R w = z 1

) pi z) pi 2|z|2 3|z| 2 0) w1,w2 pi pipi w :

u =w1w2

+w2w1

pi : ) u R ) u = 2Re(w1w2)) z1, z2 pi pipi - z |z1 z2| = 2, |z1 + z2|

7.734 z |z i| = 1+ Im(z) :

f (w) =w2 + 1w+ i

, w C {i} | f (w)| = 1

) pi - z w) z0 pi :

|z0 i| = 1+ Im(z0) |iz0 + 1| = 1

) u pi (1+ i)u+ (1 i)u = 4

) pi u

) pi |z u| 22

7.735 z2 z + 2 = 0, R, pi z1, z2 < R) z1, z2 ) pi :

z1z2+

z2z1= 2

) = 0, :

) pi w pi :

|w z1| |w z2| = 2, Im(z1) < 0

) pi |w 1| 62

) pi w2

7.736 z pi :

(2z i)7 (iz+ 2)7 = 0

pi :) |z| = 1) w = 2z1 (-1,0) = 2

) u =(z

1z

)15, u I |Im(u)| 215

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

7.737 z,w pi :

|z| = 1, |w| = 3 |z+ w| = 4

47

) pi |z w| = 2

) pi (z+ w+ 1) (1z+

9w+ 1

) 25

) pi u pi 2Re(u2) = |z w|) pi u2 x = 1

7.738 z,w pi :

|iz+ 2| = |z+ 4| |4iw+ 1| 1 = 4Im(w)

) pi - z w) |z34i| z pi pipi

) pi |z w| 25

5

7.739 a, b, c pi pipi - |a| = |b| = |c| = > 0.) pi

w =(1

ac

) (1

ba

) (1

cb

) ) pi |ab+ bc+ ca| = |a+ b+ c|

) z = b+(1)c, R. pi :

) b, c, z

) |za| 12

|ab| |ac|

7.740 z,w C R z2 + w2 = zw.) pi |z| = |w| zw R) , z,w , pi :

) 2 < < 2

)

) AOB = 450, =2

7.741 z,w pi - , 0 < , 1, pi :

|z 2i| = |z i| z(w 2i) = 22 + wi

) pi pi - z (C1) ) pi w (C2) pi (C1), pi ) pi 2|z+ w+ 2i| = |w+ 4z zwi|

) pi 2z w2z+ w

I) , pi (C1) w pi (C2), pi -

13 MA

MB 3

7.742 z pi 2z4+z = z4+2z.) pi : ) z4 + z4 = z+ z ) z4 = z) 2z4 + z = z4 + 2z) z1, z2, z3 pi- , pi pi z1, z2, z3

7.743 z pi - :

32 + i22013 z6

= 64) pi |z| = 2) z1, z2 pi pi , pi :

)z1 + z2i4 z1z2i

I ) (z1 + z2)(1z1+

1z2

) 4

) w C, 2w = z 6 + 8i, pi:

) w , pi

) Re(w) Im(w) < 0) u C, z2 = 3+ 2iuz, pi :

) u , E(

3, 0) E (

3, 0)

)14 |u| 7

47.744 z pi

2z i z = 3) z

) pi (z+ izz 1

)2016= 1

) w pi

|w+ z| =

52 +

112

i z

48

) pi w

) pi 5 |w| 35

) pi pi - w1,w2 pi pipi pi

7.745 z |z 2i| = |2zi + 1|. pi :) z pi

) z5 + 1z5 1

) 1 Im(z) 1)

z 1z + 2i+ z 1z 2i

= 4) w C, zw , 1,

z+ w1+ zw = 1

7.746 w |z|2 + 2Im(wz) + 4 = 0, pi pi z, z0) z0 |w|) pi z20140 + w

2014 = 0) pi u pi |u + 2i| |u 2i| = |z0| u pi

7.747 z pi :

|z+ |z||2 + |z |z||2 + 2z2 = 0

) pi z I) z pi pipi :

1+ z+ z2 + + z9 = 1 i

) pi w pi :

w wi =iz z

) w pi -, pi x2 + |w| 2x, x R

7.748 a, b, c pi :

|a| = |b| = |c| = 1 a+ b+ c = 1

) z = (a+ b)(b+ c)(c+ a), pi :

) z R ) (iz+ 2013i3)2 0) pi pi:

A =1a+

1b+