exasmjuin04.pdf

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ab bc ac x x n n 2ln b c z z z
    
3
/
1
 
ﺔﺤـــــﻔﺼﻟا 
4
تﺎﻋﺎﺳ  
 ةﺪﻣ
زﺎﺠﻧلإا
 
ﻟا نﺎـــﺤﺘﻣلاا
 ﺪﺣﻮﻤﻟا ﻲﻨﻃﻮ
 ةدﺎﻬﺷ ﻞﻴﻨﻟ
ﺎﻳرﻮـــــــــﻟﺎﻜﺒﻟا 
ﻟا
 ةروﺪ
ﺔﻳدﺎﻌﻟا
 :
ﻮﻴﻧﻮﻳ
 
2004
 
ﺔﻴﺑﺮﻐﻤﻟا ﺔﻜﻠﻤﻤﻟا 
 ﺔﻴﻨﻃﻮﻟا ﺔﻴﺑﺮﺘﻟا ةرازو
بﺎﺒﺸﻟاو 
 
10 
 
ﻞـــﻣﺎـﻌﻤﻟا 
ةدﺎﻤﻟا     
 :
تﺎﻴﺿﺎﻳﺮﻟا 
ﺔﺒﻌﺸﻟا     
 :
ﺔﻴﺿﺎﻳر مﻮﻠﻋ
)
أ
 (
و
)ب
(
ﺔﺠﻣﺮﺒﻠﻟ ﺔﻠﺑﺎﻗ ﺮﻴﻏ ﺔﺒﺳﺎﺣ لﺎﻤﻌﺘﺳﺎﺑ ﺢﻤﺴﻳ 
 
ﻦﻳﺮﻤﺘﻟا
1
)  
3
ﻂﻘﻧ
(
  
1
-
 
 ﻦﻜﻴﻟ
n
 ﺎﻴﻌﻴﺒﻃ ﺎﺤﻴﺤﺻ ادﺪﻋ 
  
أ 
(
  
 نﺎآ اذا ﻪﻧ أ ﻦﻴﺑ
n
 نﺎﻓ ﺎﻳدﺮﻓ 
[ ]
2
1
8
n ≡
  
ب
 
(
 
 نﺎآ اذا ﻪﻧ أ ﻦﻴﺑ
n
 نﺎﻓ ﺎﻴﺟوز 
[ ]
2
0
8
n ≡
 وأ 
[ ]
2
4
8
n ≡
 
2
-
 
 ﻦﻜﻴﻟ
a
 و 
b
 و 
c
ﺔﻳدﺮﻓ ﺔﻴﻌﻴﺒﻃ ﺔﺤﻴﺤﺻ اداﺪﻋأ 
.
  
أ 
(
  
 نأ ﻦﻴﺑ
2
2
2
a
b
c
+
+
 لاﻣﺎآ ﺎﻌﺑﺮﻣ ﺲﻴﻟ 
)
ﻲﻌﻴﺒﻃ ﺢﺒﺤﺻ دﺪﻌﻠﻟ ﻊﺑﺮﻣ ﺲﻴﻟ يأ
(
  
ب
 
(
 
 نأ ﻦﻴﺑ
(
)
[ ]
2
6
8
ab bc ac
+
+
≡
 
)
 ﺔﻈﺣلاﻣ ﻦﻜﻤﻳ
(
)2
2
2
2
2
2
a b c
a
b
c
ab abc
ac
+ +
=
+
+
+
+
+
(
  
ج               
 (
 نأ ﺞﺘﻨﺘﺳا
(
)
2 ab bc ac
+
+
لاﻣﺎآ ﺎﻌﺑﺮﻣ ﺲﻴﻟ 
.
  
د               
 (
 نأ ﻦﻴﺑ
ab bc ac
+
+
لاﻣﺎآ ﺎﻌﺑﺮﻣ ﺲﻴﻟ 
.
  
 
  
 
ﻦﻳﺮﻤﺘﻟا
2
)  
3
ﻧ
ﻂﻘ
(
  
 ﻦﻜﺘﻟ
E
 ﻞﻜﺷ ﻰﻠﻋ ﺐﺘﻜﺗ ﻲﺘﻟا تﺎﻓﻮﻔﺼﻤﻟا ﺔﻋﻮﻤﺠﻣ 
1
1
3
1
0
a
a
a
a
M
a




−








=






  
 و
F
 ﻞﻜﺷ ﻰﻠﻋ ﺐﺘﻜﺗ ﻲﺘﻟا تﺎﻓﻮﻔﺼﻤﻟا ﺔﻋﻮﻤﺠﻣ 
1
1
3
3
a
a
a
a
N
a
a




−




=






−
−


  
 ﺚﻴﺣ    
a∈\
.
  
1
-
 
أ
 (
 نأ ﻦﻴﺑ
(
)
*2
;
a
b
ab
a b
M M M
∀
∈
×
=
\
  
ب
 (
 ﻦﻜﻴﻟ
ϕ
 ﻦﻣ فﺮﻌﻤﻟا ﻖﻴﺒﻄﺘﻟا 
*\
 ﻮﺤﻧ  
E
 ﺚﻴﺤﺑ 
( )
a
a
M
ϕ
=
  
 نأ ﻦﻴﺑ    
ϕ
ﺎﺸﺗ 
 ﻦﻣ ﻞآ
(
)
*;×
\
 ﻮﺤﻧ 
(
)
;E ×
  
ﻴﻨﺒﻟا ﺞﺘﻨﺘﺳا    
 ـﻟ ﺔﻳﺮﺒﺠﻟا ﺔ
(
)
;E ×
  
2
-
 
أ
 (
 نأ ﻦﻴﺑ
(
)
*2
;
a
b
 b
a
a b  
N  
N  M
∀
∈
×
=
\
  
ب
 (
 ﻊﻀﻧ
G E F
= ∪
 .
 نأ ﻦﻴﺑ
(
)
;G ×
ةﺮﻣز 
  
ج
 (
 ﻞه
(
)
;G ×
؟ﺔﻴﻟدﺎﺒﺗ ةﺮﻣز 
  
 
ﻦﻳﺮﻤﺘﻟا
3
) 
3.50
ن 
(
  
   
1
-
 ﻲﻓ ﻞﺣ 
^
 ﺔﻟدﺎﻌﻤﻟا    
2
1 1
z
z
+ + =
  
   
2
-
 يﺪﻘﻋ دﺪﻋ ﻞﻜﻟ 
z
 ﺚﻴﺣ 
( )
( )
cos
sin
i
z
e
i
θ
θ
θ
=
=
+
 ﻊﻣ 
π θ π
− ≤ ≤
 و 
2
3
π
θ ≠
 و 
2
3
π
θ ≠ −
  
 ﻊﻀﻧ      
2
1
'
1
z
z
z
=
+ +
  
أ    
 (
 نأ ﻖﻘﺤﺗ
(
)
2
1
1
z
z
z
z
z
+ +
=
+ +
 
 
 
 
ب    
 (
 ةﺪﻤﻋو رﺎﺒﻌﻣ ﺐﺴﺟأ
'
z
 ﺔﻟلاﺪﺑ 
θ
  
ج     
 (
 ﻊﻀﻧ
'
z
x
iy
= +
 ﺚﻴﺣ 
(
)
2
;x y ∈\
  
 نأ ﻦﻴﺑ         
(
)2
2
2
1 2
x
y
x
+
= −
  
    
د  
 (
 ﺔﻄﻘﻨﻟا نأ ﺞﺘﻨﺘﺳا
M
 ﻖﺤﻠﻟا تاذ 
'
z
ﻪﻴﺑرﺎﻘﻣ و ﻪﻴﺳأر و ﻩﺰآﺮﻣ ﺪﻳﺪﺤﺗ ﻢﺘﻳ لﻮﻟﺬه ﻰﻟا ﻲﻤﺘﻨﺗ 
.
  
  
ﻦﻳﺮﻤﺘﻟا
4
) 
10
ﻂﻘﻧ
(
  
        
(I
ﺮﺒﺘﻌﻧ 
f
ﺔﻓﺮﻌﻤﻟا ﺔﻟاﺪﻟا 
ﻰﻠﻋ 
*\
 
ـﺑ 
 :
( )
x
e
f x
x
−
=
  
1
-
 
 تﺎﻳﺎﻬﻧ ﺐﺴﺣأ
f
 ﺎﻬﻔﻳﺮﻌﺗ عﻮﻤﺠﻣ تاﺪﺤﻣ ﺪﻨﻋ 
  
2
-
 
تاﺮﻴﻐﺗ سردأ
f
 
 
3
-
 
 ﻦﻜﻴﻟ
(
)C
 ﺔﻟاﺪﻠﻟ ﻞﺜﻤﻤﻟا ﻰﻨﺤﻨﻤﻟا 
f
ﻨﻤﻣ ﺪﻣﺎﻌﺘﻣ ﻢﻠﻌﻣ ﻲﻓ 
ﻢﻈ
 
أ
 (
 ﻰﻨﺤﻨﻤﻠﻟ ﺔﻴﺋﺎﻬﻧلاﻟا عوﺮﻔﻟا سردأ
(
)C
  
 ﺊﺸﻧأ ج
(
)C
  
        
(II
 ﻦﻜﺘﻟ 
(
)
n
u
ـﺑ ﺔﻓﺮﻌﻤﻟا ﺔﻳدﺪﻌﻟا ﺔﻴﻟﺎﺘﺘﻤﻟا 
 :
(
)
(
)
2
1
0
;
1
n
u
n
n
n
n
n
u
u
f u
u e
u
−
+
∀ ∈
=
=
=
`
  
1
-
 
 نأ ﻦﻴﺑ
(
)
1
x
x
e
x
∀ ∈
≥ +
\
  
2
-
 
 نأ ﺞﺘﻨﺘﺳا
( )
2
0
1
x
x
x f x
x
∀
≤
+
;
 
3
-
 
أ
 (
 نأ ﻦﻴﺑ ﻊﺟﺮﺘﻟﺎﺑ نﺎهﺮﺒﻟا لﺎﻤﻌﺘﺳﺎﺑ
1
0
1
n
n
u
n
∀ ∈
≤
+
`
≺
 
ب
 (
 نأ ﻦﻴﺑ
(
)
n
u
 ﺎﻬﺘﻳﺎﻬﻧ دﺪﺣو ﺔﺑرﺎﻘﺘﻣ 
  
            
4
-
 ﻞآ ﻞﺟأ ﻦﻣ ﻊﻀﻧ 
n
 ﻦﻣ 
*`
     :
1
0
n
n
k
k
v
u
−
=
= ∑
  
أ               
-
 ﻪﻧأ ﻦﻴﺑ 
 :
*
1
ln
n
n
n
v
u


∀ ∈
=




`
  
ب              
-
 ﺔﻳﺎﻬﻧ دﺪﺣ 
(
)
n
v
  
        
(III
ﻟاﺪﻟا ﺮﺒﺘﻌﻧ  
ﺔ
 
F
ﻓﺮﻌﻤﻟا  
ﺔ
 ﻰﻠﻋ 
[
[
0;+∞
 
ﻲﻠﻳ ﺎﻤﺑ
:
  
                                                
( )
( )
2
2
4
0
x
x
F x
f t dt
x
= ∫
;
 و 
( )
( )
0
2ln 2
F
=
  
1
-
 
أ
 (
 ﻖﻘﺤﺗ
 نأ
(
)
( )
2
2
4 1
0
2ln 2
x
x
x
dt
t
∀
=
∫
;
 
    
ب
(
 ﺔﺠﻴﺘﻧ لﺎﻤﻌﺘﺳﺎﺑ 
(1
II
−
 ،
  نأ ﻦﻴﺑ
(
)
0
1 0
t
t
t
e−
∀
− ≤
− ≤
;
  
2
-
 
أ
 (
 نأ ﻦﻴﺑ
( )
( )
2
0
3
2ln 2
0
x
x
F x
∀
−
−
≤
;
≺
 
   
ب
 (
نأ ﺞﺘﻨﺘﺳا
F
 ﺔﻠﺼﺘﻣ 
و
 ﻰﻠﻋ قﺎﻘﺘﺷلاﻟ ﺔﻠﺑﺎﻗ 
 ﻲﻓ ﻦﻴﻤﻴﻟا
0
  
3
-
 
أ
 (
 نأ ﻦﻴﺑ
( )
1
t
t
f
t
e−
∀ ≥
≺
  
ب    
 (
 ﺞﺘﻨﺘﺳا
( )
lim
x
F x
→+∞
  
4
-
أ 
 (
 نأ ﻦﻴﺑ
أ
F
 
قﺎﻘﺘﺷلاﻟ ﺔﻠﺑﺎﻗ
 ﻰﻠﻋ 
]
[
0;+∞
 ﺐﺴﺣأ و 
( )
'F x
  
ب    
 (
 تاﺮﻴﻐﺗ لوﺪﺟ ﻂﻋأ
F
  
ج    
 (
ﺊﺸﻧأ
 ﻰﻨﺤﻨﻤﻟا 
 
F
C
ﻢﻈﻨﻤﻣ ﺪﻣﺎﻌﺘﻣ ﻢﻠﻌﻣ ﻲﻓ 
.
  
5
-
 
 ﻦﻜﺘﻟ
G
 ﻰﻠﻋ ﺔﻓﺮﻌﻤﻟا ﺔﻟاﺪﻟا 
]
[
0;+∞
ـﺑ 
:
    
( )
( )
4
ln
x
t
x
G x
e
t dt
−
= ∫
  
 
 
 
 
 
 
 
 
ﺔﺤﻔﺻ
3
/
2
 
  
أ 
(
  
 نأ ﻦﻴﺑ
( )
(
)
(
)
( )
4
0
ln 4
ln
x
x
x
G x
F
x
e
x
e
x
−
−
∀
=
−
+
;
  
ب
 
(
 
 ﺐﺴﺣأ
(
)
( )
4
0
lim
ln
x
x
x
e
e
x
+
−
−
→
−
 
ت
 
(
 
ﻨﺘﺳا
ﺘ
 ﺞ
( )
0
lim
x
G x
+
→
 
 
 
 
 
 
 
 
ﺔﺤﻔﺻ
3
/
3