1 2 1 3 2 2 2 1 2 2 1 1 2 1 1 12 n u 4 v n 3 v 2 v v v u u 1 vu =− 29 n v n u = 2

ﺔﻴﺒﻌﺸﻟﺍ ﺔﻴﻁﺍﺭﻘﻤﻴﺩﻟﺍ ﺔﻴﺭﺌﺍﺯﺠﻟﺍ ﺔﻴﺭﻭﻬﻤﺠﻟﺍ

ﺩﻌﺒ ﻥﻋ ﻥﻴﻭﻜﺘﻟﺍﻭ ﻡﻴﻠﻌﺘﻠﻟ ﻲﻨﻁﻭﻟﺍ ﻥﺍﻭﻴﺩﻟﺍ ﺔﻴﻨﻁﻭﻟﺍ ﺔﻴﺒﺭﺘﻟﺍ ﺓﺭﺍﺯﻭ ﻯﻭﺘﺴﻤﻟﺍ ﻥﺎﺤﺘﻤﺍ

– ﺎﻤ ﺓﺭﻭﺩ ﻱ

2011

ﺔﺒﻌﺸﻟﺍﻭ ﻯﻭﺘﺴﻤﻟﺍ :

ﺔﻴﺒﻴﺭﺠﺘ ﻡﻭﻠﻋ ﻱﻭﻨﺎﺜ 3 ﺓﺩﺎﻤﻟﺍ

: ﺕﺎﻴﻀﺎﻴﺭ ﺕﻴﻗﻭﺘﻟﺍ

: ﺎﺴ 8 - ﺎﺴ 10 C/4

لﻭﻷﺍ ﻥﻴﺭﻤﺘﻟﺍ :

) ﻘﻨ 04 ﺎ ﻁ (

ﻟ

( )

un ﻥﻜﺘ ﺔﻓﺭﻌﻤ ﺔﻴﻟﺎﺘﺘﻤ ﺎﻤﻜ

ﻲﻠﻴ :

0 2

u = ﻭ ﻥﻤ لﺠﺃ لﻜ ﺩﺩﻋ ﻲﻌﻴﺒﻁ :n

1

1 9

2 4

n n

u ₊ = u + .

ﻭ

( )

vn ﻥﻜﺘﻟ ﺔﻓﺭﻌﻤﻟﺍ ﺔﻴﻟﺎﺘﺘﻤﻟﺍ ﻥﻤ

لﺠﺃ لﻜ ﺩﺩﻋ ﻲﻌﻴﺒﻁ n

ـﺒ :

2 9

n n

v = u − .

(1 ﺏﺴﺤﺃ u1

، u2

ﻡﺜ v0

، v1

، v2

.

(2 ﻥﻫﺭﺒ ﻥﺃ ﻟﺎﺘﺘﻤﻟﺍ

( )

^v_n ﻴﺔ ﺏﻠﻁﻴ ﺔﻴﺴﺩﻨﻫ ﻌﺘ

ﻴﻴ ﻥ ﺎﻬﺴﺎﺴﺃ .

(3 ﺩﺠ ﺓﺭﺎﺒﻋ ﺩﺤﻟﺍ ﻡﺎﻌﻟﺍ vn

ﺔﻟﻻﺩﺒ . n

(4 ﺞﺘﻨﺘﺴﺍ ﺓﺭﺎﺒﻋ ﺩﺤﻟﺍ ﻡﺎﻌﻟﺍ un

ﺔﻟﻻﺩﺒ .n

ﻲﻨﺎﺜﻟﺍ ﻥﻴﺭﻤﺘﻟﺍ :

) ﻘﻨ 04 ﺎ ﻁ (

ﻰﻠﻋ ﺱﻴﻜ ﻱﻭﺘﺤﻴ ﺎﻬﻨﻤ ﺓﺭﻜ 12

: ﻡﺎﻗﺭﻷﺍ لﻤﺤﺘ ﺀﺎﻀﻴﺒ ﺕﺍﺭﻜ ﺙﻼﺜ , 1

, 1 ﻡﺎﻗﺭﻷﺍ لﻤﺤﺘ ﺀﺍﺭﻤﺤ ﺕﺍﺭﻜ ﻊﺒﺭﺃﻭ 2

, 1 , 1 , 2 ﻡﺎﻗﺭﻷﺍ لﻤﺤﺘ ﺀﺍﺭﻀﺨ ﺕﺍﺭﻜ ﺱﻤﺨﻭ 2 , 1

, 2 , 2 , 2 .3

ﺱﻴﻜﻟﺍ ﻥﻤ ﻥﻴﺘﺭﻜ ﺩﺤﺍﻭ ﻥﺁ ﻲﻓﻭ ﺎﻴﺌﺍﻭﺸﻋ ﺏﺤﺴﻨ ﻭ

ﻥﻴﺘﺜﺩﺎﺤﻟﺍ ﺭﺒﺘﻌﻨ :

" A ﻥﻭﻠﻟﺍ ﺱﻔﻨ ﻥﻤ ﻥﻴﺘﺭﻜ ﺏﺤﺴ "

ﻭ " B لﻗﻷﺍ ﻰﻠﻋ ﺀﺍﺭﻀﺨ ﺓﺭﻜ ﺏﺤﺴ "

(1 ﺙﺩﺍﻭﺤﻟﺍ ﻥﻤ ﺔﺜﺩﺎﺤ لﻜ لﺎﻤﺘﺤﺍ ﺏﺴﺤﺃ :

، A ، B A∩B .

 

(2 ﻥﺎﺘﺜﺩﺎﺤﻟﺍ لﻫ ﻭ A

؟ ﻥﺎﺘﻠﻘﺘﺴﻤ B .

ﺙﻟﺎﺜﻟﺍ ﻥﻴﺭﻤﺘﻟﺍ ) :

ﻁﺎﻘﻨ 05 (

لﺍﺅﺴ لﻜﻟ ﺔﺒﺎﺠﺇ

ﺔﺤﻴﺤﺼ ﺓﺩﺤﺍﻭ .

ﺭﺘﺨﺍ ﺒﺎﺠﻹﺍ ﺔ ﺭﻴﺭﺒﺘﻟﺍ ﻊﻤ ﺔﺤﻴﺤﺼﻟﺍ

( 1 ﺏﻜﺭﻤﻟﺍ ﺩﺩﻌﻟﺍ ﻥﻜﻴﻟ ﺙﻴﺤ Z

: i Z

Z+ =6−2 .

ـﻟ ﻱﺭﺒﺠﻟﺍ لﻜﺸﻟﺍ Z

ﻭﻫ :

ﺃ

( i 3 2 8− ﺏ

( i 3 2 8− − ﺟ (ـ 8 2

3 i

− + ﺩ

( i 3 2 8+

(2 ﺏﻜﺭﻤﻟﺍ ﻱﻭﺘﺴﻤﻟﺍ ﻲﻓ .

ﻁﻘﻨﻟﺍ ﺔﻋﻭﻤﺠﻤ ﺔﻘﺤﻼﻟﺍ ﺕﺍﺫ M

iy x z = + ﻕﻘﺤﺘ ﻲﺘﻟﺍﻭ

i z z−1 = +

ﻲﻫ :

ﺃ (

−1

= x y ﺏ ( x y=− ﺟ (ـ +1

−

= x y

ﺩ ( x y=

(3 ﻥﻜﻴﻟ ﻲﻌﻴﺒﻁ ﺩﺩﻋ n

. ﺩﺩﻌﻟﺍ

(

^{1+i 3}

)

ⁿ

ﻥﺎﻜ ﺍﺫﺍ ﻁﻘﻓ ﻭ ﺍﺫﺍ ﺎﻴﻘﻴﻘﺤ ﻥﻭﻜﻴ لﻜﺸﻟﺍ ﻰﻠﻋ ﺏﺘﻜﻴ n

:

Z k∈

ﺃ (

1 3k+

ﺏ

(

2 3k+

ﺟ

(ـ

k

3

ﺩ (

k

6

(4 ﺔﻟﺩﺎﻌﻤﻟﺍ ﻥﻜﺘﻟ )

....(

3

6 E

z z z

−

= − ﻊﻤ

C Z∈

ﺔﻟﺩﺎﻌﻤﻟﺍ ﻩﺫﻫ لﻭﻠﺤ ﺩﺤﺃ )

ﻭﻫ (E :

ﺃ

(

2 2−i

ﺏ (

i

2

ﺟ (ـ

−i

1

ﺩ (

−i

−1

1 / 2

ﻊﺒﺍﺭﻟﺍ ﻥﻴﺭﻤﺘﻟﺍ ) :

ﻁﺎﻘﻨ 07 (

ﺔﻟﺍﺩﻟﺍ ﺭﺒﺘﻌﻨ ﻰﻠﻋ ﺔﻓﺭﻌﻤﻟﺍ f

ﻲﻠﻴ ﺎﻤﻜ R :

ex

x x

f = − +

1 ) 1

. (

ﻲﻤﺴﻨ

( )

^C

ﺔﻟﺍﺩﻟﺍ ﻰﻨﺤﻨﻤ ﺱﻨﺎﺠﺘﻤ ﻭ ﺩﻤﺎﻌﺘﻤ ﻡﻠﻌﻤ ﻲﻓ f

) لﻭﻁﻟﺍ ﺓﺩﺤ ﻭ . ( 3cm

(1 ﺏﺴﺤﺍ )

( lim f x

x→+∞

ﻭ ) ( lim f x

x→−∞

.

( 2 ﻥﺃ ﻥﻴﺒ : 0 ) ( ' x لﻜ لﺠﺍ ﻥﻤ f ﻥﻤ x

R .

( 3 ﻰﻨﺤﻨﻤﻟﺍ ﻥﺃ ﻥﻴﺒ

( )

^C

ﻥﻴﺒﺭﺎﻘﻤ ﻥﻴﻤﻴﻘﺘﺴﻤ لﺒﻘﻴ

( )

^∆

( )

^∆' ﻭ ﺏﻴﺘﺭﺘﻟﺍ ﻰﻠﻋ ﺎﻤﻬﺘﻟﺩﺎﻌﻤ x

y= ﻭ

−1

= x .y

( 4 ﻰﻨﺤﻨﻤﻠﻟ ﻲﺒﺴﻨﻟﺍ ﻊﻀﻭﻟﺍ ﺱﺭﺩﺍ

( )

C

ﻥﻴﻤﻴﻘﺘﺴﻤﻟﺍ ﻭ

( )

∆

( )

∆' ﻭ .

( 5 ﺔﻟﺩﺎﻌﻤﻟﺍ ﻥﺃ ﻥﻴﺒ 0

) (x = ﺍﺩﻴﺤﻭ ﻼﺤ لﺒﻘﺘ f

ﺙﻴﺤ α 2 0≺α ≺ 1

ﻥﺃ ﻕﻘﺤﺘ ﻡﺜ α :

α 1

1= + .e

( 6 ﺊﺸﻨﺃ

( )

^C

( )

^∆ ﻭ

( )

^∆' ﻭ ) . ﺫﺨﺄﻨ

4 .

≈0

.(α

2

/ 2