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تمارين حول الأعداد المركبة 2 .pdf



Nom original: تمارين حول الأعداد المركبة 2.pdf
Titre: تمارين لأعداد المركبة مصححة
Auteur: hani

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‫ ر ا اد ا آ‬

.

z0 =

z1
z2



z0

‫و‬

z2

01 ‫ا‬
. ‫ اص‬- ‫ و
ة د آ‬
= 1 + i ، z = 1 + i 3 ‫ ا اد ا
آ‬
1

. z :‫ال )( '" ـ‬+" ‫ ا‬,- %& #‫ و
ة ـ‬z " ‫ا‬
. z :‫ا " و
ة ـ‬
. ‫ ي‬2 ‫ ا‬345 ‫ ا‬6 z ‫اآ‬
. cos π ‫ و‬sin π 8 ' 9‫ا‬
2

1

0

0

12

.

z2 +2z

2



3
=0
4

12

(1
(2
(3
(4

02 ‫ا‬
. :9 ‫ ا‬345 ‫ ي و ا‬2 ‫ ا‬345 ‫ا‬

ا اد ا
آ ا
(د‬2 :;

. (‫ ي‬2 ‫ ا‬345 ‫ ا‬6 ‫ ا @ ل‬6A B ) ‫ ه=< ا
(د‬3 (1
. Arg ( z ) ≡ π [ 2π ] "#$ %& ‫ ا‬z ‫ ا "( و‬:C C@ ‫ ا‬3@ ‫ ا‬z :
" (2
2
2

2

1

. :9 ‫ ا‬345 ‫ ا‬6

‫اآ‬
03 ‫ا‬
. ‫=ران ا ) (ن د آ‬2 ‫ – ا‬:9 ‫ ا‬345 ‫ا‬
. Z = 3 + i ‫ ا د ا
آ‬
z2

+

z1

2 +i 2

. :9 ‫ ا‬345 ‫ ا‬6 Z ‫( اآ‬1
3+i
z =
‫ ا
(د‬3 (2
2

2 +i 2

04 ‫ا‬
. G (G ‫ ا‬H‫ ا ر‬I ‫ (د‬3
. z 3 − 2iz 2 + 2z − 4i = 0......... (∗) ‫
ا اد ا
آ ا
(د‬2 :;
. (∗) ‫
(د‬3 2i ‫ أن‬Iّ ) (1
. (∗) ‫ ل ا
(د‬8 ' 9‫ ا‬%& z 3 − 2iz 2 + 2z − 4i 3 (2
05 ‫ا‬
. :9 ‫ ا‬345 ‫ ا‬- sin α 3
5B (LBM ( G (G ‫ ا‬H‫ ا ر‬I ‫ (د‬3
. [0;π ] ‫(ل‬2
‫ ا‬6 ‫ إ‬:
' :C C ‫ د‬α
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(L L2 : ‫ ا‬z 3 − (1 − 2sin α ) z 2 + (1 − 2sin α ) z − 1 = 0.... (∗) ‫ ا
(د‬
.z
. (∗) ‫
(د‬3 1 ‫ أن‬Iّ ) (1
. (∗) ‫ ل ا
(د‬8 ' 9‫ ا‬%& z 3 − (1 − 2sin α ) z 2 + (1 − 2sin α ) z − 1 3 (2
.:9 ‫ ا‬345 ‫ ا‬6 BO‫( اآ ا اد ا
آ ا‬3
. z 3 = − sin α − i cos α ‫ ؛‬z 2 = − sin α + i cos α ‫ ؛‬z = 1
06 ‫ا‬
. ‫ ة‬LQ R (LB‫أ اد آ
ا‬
1

.

z = cos



+ i sin
7
7

‫ ا د ا
آ‬

. T = z 3 + z 5 + z 6 ‫ و‬S = z + z 2 + z 4 ST
. z k = z k −7 ، k U @V ‫ د‬3‫ آ‬3H‫ أ‬I ‫ أن‬Iّ ) (1
. T = S ‫ أن‬Iّ ) (2
. Im (S ) > 0 ‫ أن‬Iّ ) (3
. ST ‫ و‬S + T " ‫( ا‬4
2
‫ و‬S 8 ' 9‫ ا‬. x − ( S + T ) x + ST = 0 ‫ ا
(د‬3
07 ‫ا‬

.T

. :9 ‫ ا‬345 ‫ ا‬- ‫ ل‬L2
‫ ا‬3 )

tan α

3
5B (LBM ( G (G ‫ ا‬H‫ ا ر‬I ‫ (د‬3

.  − π ; π  ‫(ل‬2
‫ ا‬6 ‫ إ‬:
' :C C ‫ د‬
 2 2

(L L2 : ‫( ا‬1 + iz ) (1 − i tan α ) = (1 − iz ) (1 + i tan α ) .... ( ∗) ‫ ا
(د‬
.z
‫ أن‬8 ' 9‫ و ا‬1 + iz = 1 − iz ‫ن‬Y; (∗) ‫ ا
(د‬XC@ z ‫ أن إذا آ(ن‬Iّ ) (1
. z ∈R
3

3

.
.

 π π
β ∈ − ; 
 2 2

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S

‫ا ا و‬

1 + i tan α
= e i 2α
1 − i tan α

z = tan β

‫ أن‬Iّ ) (2

SZ ) (∗) ‫ ا
(د‬3 (3

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‫ ل‬.$
01 ‫ا‬
: z ‫ و
ة د ا
آ‬z (1
1

1

z 1 = 12 +
 z1 = 2


π
A rg ( z 1 ) ≡ [ 2π ]
3


#' ‫و‬

π
1
 2 = cos 3

 3 = sin π
 2
3

:

‫إذن‬

( 3)

2

1
3
z 1 = 2  + i

2 
2

z 2 ‫ا
آ‬

=2 •

#'

‫ و
ة د‬z
1

z 2 = 1 +1 = 2 •
2

#' ‫و‬








2
π
= cos
2
4
2
π
= sin
2
4

‫إذن‬

2

 2
 1
1 
2
z2 = 2
+i
=
2
+
i



 2
2 
2
 2


#'

z2 = 2


π
A rg ( z 2 ) ≡ [ 2π ]

4

:
‫و‬

z 0 ‫ا
آ‬

z0 =

‫ و
ة د‬z (2
1

z
z1
2
= 1 =
= 2
z2
z2
2

‫إذن‬

z0 =

z 
π π π
Arg  1  ≡ Arg ( z 1 ) − Arg ( z 2 ) ≡ − ≡ [ 2π ]
3 4 12
z2 
z0 = 2


π
A rg ( z 0 ) ≡ [ 2π ]

12

‫ ي‬2 ‫ ا‬345 ‫ ا‬6
z0 =

(

)

π
π 

z 0 = 2  cos + i sin 
12
12 


‫ا ا و‬

‫ه‬

π
12

z0

‫و‬

sin

π
12

‫إذن‬

)( ‫( آ‬3

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

cos

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z0

z1
z2

3 −1
2

‫ ' (ج‬9‫( ا‬4

:‫ ـ‬:G G
‫ ا‬345 ‫ا‬

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z0 =


π
cos =
 12

sin π =
 12

2

z

=

(

3 +1

=

2 2
3 −1
2 2

x2 +y2

)

2

www.youcefmaths.com ‫ ا ذ‬

3 +1
+i
2
6+ 2
4

6− 2
4

=

=x2

3 −1
2

‫ه‬

z0

:‫ ي ـ‬2 ‫ ا‬345 ‫ا‬


π
3 +1
 2 cos =

12
2

 2 sin π = 3 − 1

12
2

‫أي‬

. ‫ (ن‬C C ‫ دان‬y ‫ و‬x
+ y ‫ و‬z = ( x + iy ) = x
2

2

2

x 2 − y 2 + 2ixy + 2 ( x 2 + y 2 ) −

2

\;(4

02 ‫ا‬
، z = x + iy (1
− y + 2ixy ('
2

3
=0
4
3
3
2
 2
2
2
2
2
 x − y + 2x + 2 y −  + 2ixy = 0 \;(4 z + 2 z − = 0
4
4

3
3
2
 2
2
2
 3x + y −  + 2ixy = 0 \;(4 z + 2 z − = 0
4
4


x = 0
x = 0



3 ‫و‬
y =
y = −

2 
1 

x =
x
2 ‫و‬

 y = 0  y
. 1 et − 1 , i 3
2
2
2

‫أي‬

3
2

y =

1
2

‫أي‬

−i

3
2

=−

x =

=0

,

‫أو‬

3
2

1
2

‫أو‬

y =−

x =−

3
=0
4

8 ' "

3
2

‫إذن‬

1
2

‫ إذن‬12x

:‫أر) ل ه‬

y2−

3
=0
4
−3 = 0

2

z2 +2z

2



z2 +2z

2

:

%1‫ أ‬

x =0

:



3H‫ أ‬I

y =0

‫ ا
(د‬3 CB

3
=0
4

(2
(9 : ] ‫_ء< ا‬H ‫ ;( و‬V : ]B

−i

H : ] ‫_ء< ا‬H ‫ ;( و‬V : ]B
.

z2 =i

3
2

‫و‬

z1 = −

1
2

i

‫ن‬Y;

3
2

‫ ن‬

3
2

‫ ن‬

1
− ∈ R+
2


3
π
Arg  −i
 ≡ − [ 2π ] •
2 
2

 3 π
Arg  i
 ≡ [ 2π ]
 2  2

‫و‬

1
− ∈ R−
2

: :9 ‫ ا‬345 ‫ ا‬6
z1 + z

.

z1 + z

2

=e

i


3

‫إذن‬


Arg ( z 1 + z 2 ) ≡
[ 2π ]
3

‫و )
(أن‬

z1 + z
2

2

)( ‫• آ‬

1
3
= − +i
2
2
2

‫و‬

z1 + z

2

2
 1  3
=  −  + 
 =1
 2   2 

03 ‫ا‬
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π
 3
i
1
π
π

3 + i = 2 
+ i  = 2  cos + i sin  = 2e 6 ‫ إذن‬3 + i =
2
2
6
2


π
 2
i
2  π
π
2 + i 2 = 2 
+i
 = 2  cos + i sin  = 2e 4 ‫ إذن‬2 + i 2 =
2  
4
4
 2

.

2e

Z =

2e

6

π

=e

r 2e i 2α = e

−i

‫أي‬

12

r 2e i 2α = e

r = 1


π
α = − 24 + k π

−i

π

‫إذن‬

.e

i

23π
24

‫و‬

e

i

23π
24

−i

π
24

−i

‫و‬

π

24

#'

i

π
6

−i

π
4

=e

−i

π
12

z = r e iα

3+i

z2 =

k ∈Z

2

2

(1

=2

#'

2 +i 2

ST (2

‫ا
(د‬

r 2 = 1


π
2α = − + 2k π

12

8 ' "

π

: k = 0 3H‫ أ‬I
24
#' α = − π + π = 23π : k = 1 3H‫ أ‬I
24
24
(
‫ ه‬I z 2 = 3 + i ‫ ا
(د‬3 CB
2 +i 2

z = 1 ×e

z = 1 ×e

\;(4B

12

+ 12 = 2

4

.
π

2

( 2) + ( 2)

π

i

i

( 3)

α =−

04 ‫ا‬
( 2i ) − 2i ( 2i ) + 2 ( 2i ) − 4i = −8i + 8i + 4i − 4i = 0 (1
. (∗) ‫
(د‬3 2i ‫ ا
(د )∗( إذن‬XC@B 2i
:‫
(د )∗( إذن‬3 2i (2
z − 2iz + 2z − 4i = ( 2i ) − 2i ( 2i ) + 2 ( 2i ) − 4i \;(4B ( ∗)
z − ( 2i ) − 2iz + 2i ( 2i ) + 2z − 2 ( 2i ) = 0 \;(4B
. z − ( 2i )  − 2i  z − ( 2i )  + 2 [ z − 2i ] = 0 \;(4B
A − B = ( A − B ) ( A + B + AB ) : ‫=آ أن‬B
:‫إذن‬
( z − 2i )  z + ( 2i ) + 2iz  − 2i ( z − 2i )( z + 2i ) + 2 [ z − 2i ] = 0 \;(4B ( ∗)
( z − 2i )  z − 4 − 4iz  − 2i ( z − 2i )( z + 2i ) + 4 [ z − 2i ] = 0 \;(4B
( z − 2i )  z + 2  = 0 \;(4B
3

3

2

3

2

3

3

3

2

2

3

2

3

2

3

2

2

2

2

2

2

2

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‫أو‬
= ( i 2 ) ‫أو‬

\;(4B
\;(4B
z
z = 2i
z = i 2 ‫ أو‬z = −i 2 ‫ أو‬z = 2i \;(4B
. i 2 ‫ و‬−i 2 ، 2i :‫& ل ه‬M& (∗) ‫ ا
(د‬3 CB
05 ‫ا‬
− (1 − 2sin α ) × 1 + (1 − 2sin α ) × 1 − 1 = 1 − 1 + 2sin α + 1 − 2sin α − 1 = 0 (1
. (∗) :‫ ـ‬3 1 ‫ ا
(د )∗( إذن‬XC@ 1
(2
: (' z ‫ د آ‬3‫ آ‬3H‫ أ‬I ‫ إذن‬، (∗) :‫ ـ‬3 1 •
z 2 = −2

2

13

z = 2i

2

2

z 3 − (1 − 2sin α ) z 2 + (1 − 2sin α ) z − 1 = ( z − 1) ( az 2 + bz + c )

= az 3 + (b − a ) z 2 + (c − b ) z − c
a = 1

b = 2sin α
c = 1


.

a = 1
b − a = −1 + 2 sin α


c − b = 1 − 2sin α
c = 1

‫أي‬

‫ أن‬8 ' "

z 3 − (1 − 2sin α ) z 2 + (1 − 2sin α ) z − 1 = ( z − 1) ( z 2 + 2sin α z + 1)

#' ‫و‬
( z − 1) ( z + 2sin α z + 1) = 0 \;(4B ( ∗) ‫• ا
(د‬
z + 2sin α z + 1 = 0 ‫ أو‬z = 1 \;(4B
∆ = ( 2sin α ) − 4 ‫
_ه( ه‬: z + 2sin α z + 1 = 0 ‫ ا
(د‬3@
∆ = 4 ( sin α − 1) = −4 cos α = ( 2cos α ) ‫أي‬
(
‫ ه‬I 3 CB ‫ه=< ا
(د‬
2

2

2

2

2

2

2

−2sin α + 2i cos α
= − sin α + i cos α
2
−2sin α − 2i cos α
z '' =
= − sin α − i cosα ‫و‬
2

z '=

.

− sin α − i cos α

‫؛‬

‫ ؛‬1 : ‫& ل‬M& (∗) ‫ ا
(د‬3 CB
:9 ‫ ا‬345 ‫ ا‬6 z ‫ و‬z ، z )( ‫( آ‬3

− sin α + i cos α

3

2

1

z 1 = 1 = e io •
z 2 = − sin α + i cos α = i ( cos α + i sin α ) = i e



i

π

=e e
2



=e

z 3 = − sin α − i cos α = z 2 = e

 π
i α + 
2


 π
−i α + 
2





06 ‫ا‬
(1
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 2k π 
 2k π 
 2k π

 2k π

z k = cos 
− 2π  + i sin 
− 2π 
 + i sin 
 = cos 
 7 
 7 
 7

 7

 2 ( k − 7)π 
 2 ( k − 7)π 
k −7
z k = cos 
 + i sin 
=z
7
7





(2
‫ أن‬6
(3

T =z3 +z5 +z6 =z3 +z5 +z6

.T

‫إذن‬

=z3 +z5 +z6 =z4 +z2 +z =S

‫ن‬78

z k = z 7 −k

z k = z k −7



π
+ sin
− sin
7
7
7

2π  π 
>0 •
sin
∈ 0;  ‫ ن‬sin
7
7  2
‫ _ا ة‬sin ‫ ن ا ّا‬sin 4π > sin π •
7
7
Im ( S ) = sin

 π
0; 2 

6

.

1
z −1
. S + T = z z = z z = −1
1− z
1− z
1−

Im ( S ) > 0

: ‫إذن‬
(4


S +T = z + z 2 + z 4 + z 3 + z 5 + z 6
1− z 6
2
3
4
5
6
S +T = z + z + z + z + z + z = z
1− z

‫إذن‬

z6 =

‫ن‬Y;

1
z

z 6 ×z =1

‫أي‬

z 7 =1

‫و )
(أن‬

ST = ( z + z 2 + z 4 )( z 3 + z 5 + z 6 )



ST = z 4 + z 6 + z 7 + z 5 + z 7 + z 8 + z 7 + z 9 + z 10
z 8 = z 7 × z = z
 9
7
2
2
‫ن‬Y; z 7 = 1 ‫و )
(أن‬
z = z × z = z
z 10 = z 7 × z 3 = z 3

ST = z 4 + z 6 + 1 + z 5 + 1 + z + 1 + z 2 + z 3
ST = 3 + S = 3 − 1 = 2

#' ‫و‬

ST = 3 + z + z 2 + z 3 + z 4 + z 5 + z 6

‫إذن‬
‫أي‬
(5

‫ ا
(د‬3 •
− 4ST : ‫ا

_ ه‬

x 2 − ( S + T ) x + ST = 0
∆ = (S + T

)

2

− 4ST = S 2 + T 2 + 2ST

∆ = S 2 + T 2 − 2ST = ( S − T

.

medyoucef@gmail

‫ا ا و‬

x 2 =T

‫و‬

)

2

: (
‫ن ه‬M@ ‫ا‬
T ‫ و‬S ‫; ب‬$ •

x1 = S

www.youcefmaths.com ‫ ا ذ‬

medyoucef@gmail

.

(

∆= i 7

‫و‬

x '=

‫ا ا و‬

www.youcefmaths.com ‫ ا ذ‬

\;(4B x − (S + T ) x + ST = 0 ‫ا
(د‬

_ه( ه‬: x + x + 2 = 0 ‫ ا
(د‬3@

x 2 + x + 2= 0

) ‫ = ∆ أي‬−7
2

−1 − i 7
2

2

2

(
‫ن ه‬M@ ‫ا‬
.


(G ‫ال ا‬+" ‫ ا‬I 8 ' " ، {S ;T } =  −1 − i
2


x '' =

7 −1 + i 7 
;

2


−1 + i 7
2

‫ '( إذن‬


−1 + i 7
S =

2

T = −1 − i 7

2

‫أن‬

07 ‫ا‬
(1
(1 + iz ) (1 − i tan α ) = (1 − iz ) (1 + i tan α ) (' •
(1 + iz ) × 1 − i tan α = (1 − iz ) × 1 + i tan α ‫إذن‬
3

3

3

.

1 + iz = 1 − iz

‫أي‬

1 + iz

1 + i ( x + iy ) = 1 − i ( x + iy )
y =0

‫( إذن‬1 − y )

3

U bB
2

= 1 − iz

3

‫ن‬Y;

3

1 + iz = 1 − iz

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


 1 − i tan α = 1 + i tan α

3
3
 (1 + iz ) = 1 + iz

 (1 − iz )3 = 1 − iz 3


‫و )
(أن‬

aM ‫ ا‬، z = x+i y ST •
‫أي‬
‫( أي‬1 − y ) + ix = (1 + y ) − ix
. z ∈ R ‫ أي‬z = x #' ‫و‬

sin α
1 + i tan α
cos α + i sin α
cos α + i sin α
e iα
cos
α
=
=
=
= −i α = e i 2α
sin α cos α − i sin α cos ( −α ) + i cos ( −α ) e
1 − i tan α
1− i
cos α
1+ i

 1 + iz  1 + i tan α

 =
 1 − iz  1 − i tan α
3

medyoucef@gmail

‫ا ا و‬

(2

\;(4B (∗) ‫(ا
(د‬3

www.youcefmaths.com ‫ ا ذ‬

medyoucef@gmail

e i 6 β = e i 2α

‫( أي‬e β )
i2

3

‫ا ا و‬

= e i 2α

www.youcefmaths.com ‫ ا ذ‬

4B (∗) ‫ن‬Y;

α kπ

β = +
3
3

k ∈ Z

1 + i tan α
i 2α
1 − i tan α = e

1 + iz = 1 + i tan β = e i 2 β
1 − iz 1 − i tan β

‫أي‬

6 β = 2α + 2k π

k ∈ Z

β=

 α + 2π 
z 3 = tan 

 3 

،

α +π 
z 2 = tan 

 3 

medyoucef@gmail

،

‫ا ا و‬

π

3

:

‫ أن‬8 ' " ‫و‬
k =0

3H‫ أ‬I

α +π

: k = 1 3H‫ أ‬I
3
α 2π α + 2π
β= +
=
: k = 2 3H‫ أ‬I
3
3
3
α 
z 1 = tan   : :‫ ل ا
(د )∗ ( ه‬
3
β=

.

α

α

‫و )
(أن‬

3

+

3

=

www.youcefmaths.com ‫ ا ذ‬


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