ﺑﺴﻢ ﺍﷲ ﺍﻟﺮﺣﻤﻦ ﺍﻟﺮﺣﻴﻢ ﻭﺍﻟﺼﻼﺓ ﻭﺍﻟﺴﻼﻡ ﻋﻠﻰ ﺃﺷﺮﻑ ﺍﻟﻤﺨﻠﻮﻗﻴﻦ ﻣﺤﻤﺪ ﺳﻴﺪ ﺍﻟﻤﺮﺳﻠﻴﻦ ﻭﻋﻠﻰ ﺁﻟﻪ ﻭﺻﺤﺒﻪ ﺃﺟﻤﻌﻴﻦ ﺃﻣﺎ ﺑﻌﺪ ٬ ﻳﺴﺮﻧﻲ ﺃﻥ ﺃﻗﺪﻡ ﻟﻜﻢ ﻫﺬﺍ ﺍﻟﻌﻤﻞ ﺍﻟﻤﺘﻮﺍﺿﻊ ﻭﻫﻮ ﻋﺒﺎﺭﺓ ﻋﻠﻰ ﻣﻠﺨﺼﺎﺕ ﻣﻊ ﺗﻘﻨﻴﺎﺕ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ ﻟﻤﺴﺘﻮﻯ ﺍ ﻟﺠﺬﻉ ﺍﻟﻤﺸﺘﺮﻙ ﻋﻠ ﻤ ﻲ ﻣﺠﻤﻌﺔ ﻓﻲ ﻛﺘﺎﺏ ﻭﺍﺣﺪ ﻭﻫﻲ ﻟﻸﺳﺘﺎﺫ ﺣﻤﻴﺪ ﺑﻮﻋﻴﻮﻥ sefroumaths.site.voila.fr
ﺗﺠﻤﻴﻊ ﻭﺗﺮﺗﻴﺐ ALMOHANNAD
ــــ دئ اــــ ﺏـــــــــ ت
(2 . 0 ". 6 -! !3! . 0 , ! ". 0 . a ". c $) b ". c a ". b $ !%& . 6 -! !3! 8 ". 1 -! . 9 ". . 1 , ! " # 1 -
(* (* (* (* (* (*
25,11,9,5,4,3,2 () &* + (3 " (a $ # α r 7..... 7 α 3 7 α 2 7 α1 7 α 0 $ -
{0,1, 2,3, 4,5, 6, 7,8,9}
%-! -! -& α r α r −1 ...α 0 6 - ............ 7 α1 ! "# 7 α 0 ! "#
- (b : - a = α r α r −1 ...α 0 -! α 0 ∈ {0, 2, 4, 6,8} $ !%& 2 6.-! . a (* 3 / α 0 + α1 + α 2 + + α r $ !%& 3 6.-! . a (* 4 / α 0α1 $ !%& 4 6.-! . a (*
α 0 ∈ {0,5} $ !%& 5 6.-! . a (* 9 / α 0 + α1 + α 2 + + α r $ !%& 9 6.-! . a (* $
a !%& 3 6.-! . (* 11 / (α 0 + α 2 + α 4 + ......) − (α1 + α 3 + α 5 + .....)
{
}
α1α 0 ∈ 00, 25,50, 75 $ !%& 25 6.-! . a (*
. $"% & * (4 . $ 1 $ $ b a $ - ! " " 1 " # , b a $ - 3! 4 -! " .-! . a ∧ b PGCD (a , b ) - . 5 4
. / )' 0 - (5 . a ≥ b IN * $ b a $ : 6 - 6 # ; # PGCD (a, b ) $ ".-! < ". "= b a 6. $ " # !%, # -! . " 1 # *> , PGCD (a, b ) : ? -! @%, A * $ a r1
b q1
r1
r2
q2
q3
r1
r2
...
...
...
...
...
rn
0
(I IN = {0,1, 2,3, 4,5.......}
IN * = {1, 2,3, 4,5.......}
– (II a (1 . k ∈ IN a = 2 k a (2 . k ∈ IN a = 2 k − 1 a = 2 k + 1
(3 . ! "# $ !%& $ (a . ! "# $ !%& $ (b . a + b $) $ b a $ !%& (* (c . a + b $) $ b a $ !%& (* . a + b $) b $ a $ !%& (* . ab $) $ b a $ !%& (* (d . ab $) $ b a $ !%& (* . ab $) b $ a $ !%& (* *+! , $) $ $ b a $ !%& (e .
(III . $ $ b a $ - ! " (1 a $ !%& b - / 0 a -! $& . . k ∈ IN a = b k
(2 . / 0 0 (* . 0 , ! / 0 - 0 (* $) c / 0 b b / 0 a $ !%& (* . b - / 0 a
# $"% ! (3 . $ 1 $ $ b a $ - ! " / 0 2 , b a $ - 23! 4 -! / 0-! PPCM (a, b ) - . 5 4 " 1 . a ∨b
(4
PPCM (a, b ) = a $) b - / 0 a -! $ !%& (* PPCM (a, a ) = a (*
& ' (IV . $ $ b a $ - ! " (1 ". b -! $& 7 b 6.- # a -! $& . a b / 0 a $ !%& a . b / a . k ∈ IN a = b k
(V - a - ! ! " (1 . a 1 . $ #
(2 . 8 - a -! , B. - (a p ≤ a B. -! p 6 -3! !3! 8 . - 1 a $) a ". !3! @%, $ !%& . - a $) a ". C !3! @%, 8 ; !%& , 100 $ 23! 6 -3! !3! (b 47 ,43 ,41 ,37 ,31 ,29 ,23 ,19 ,17 ,13 ,11 ,7 ,5 ,3 ,2 .97 ,89 ,83 ,79 ,73 ,71 ,67 ,61 ,59 ,53 7 , p ≠ 2 - (c . - E - 1 -! (d 2
0 2 3 (4 $ ." (3 < 6. a ≥ 2 : a = p1α1 . p 2α 2 . p 3α3 ...... p rα r
. 6 - ! p r 7..... 7 p 3 7 p 2 7 p1 . 6 1 6 ! α r 7..... 7 α 3 7 α 2 7 α1 . 6 - ! F! -& a -! 4 9 6 -! @%, 54 = 2 × 3
3
54 27 9 3 1
$%& $%&
2 5
2 3 3 3
- :54 -! 49-
. + " (4 F! , b a $ - 23! 4 -! / 0-! (a b a 9 $ 6 -! 1 6 -! 6 -3! ! -! . E -& 6 ! -! F! , b a $ - 3! 4 -! " .-! (b . E 2 -& 6 b a 9 $ 6 -! 6 -3! 76 ∨ 632 و76 ∧ 632
76 2 38 2 19 19 1
76 = 22.19
632 316 158 79 1
2 5 $ -
2 2 2 79
632 = 23.79
76 ∨ 632 = 23.19.79 = 12008
:-
76 ∧ 632 = 22 = 4
$%&
a ≥ 2 $ - (c ! F! -& a -! 4 9 a = p1 . p 2 . p 3α3 ...... p rα r . 6 - (1 + α1 )(1 + α 2 ) (1 + α r ) , a -! "!# α1
α2
ا ب ا :
&'( IK وIJ + 2 (a .(... + α IJ + β IK = 0 + IJ = α IK 9) 3; &*2 IK وIJ : 2 ! .9 AC وAB &8-. < = IK = 6AB − 3AC IJ = 2AB − AC 9 * . IK = 3IJ 3IJ = 6 AB − 3DC = IK + MA = 3MB > 2 & M 989 ( ABC ) (b
: 8 @* & 8 M 73 3? MA = 3 ( MA + AB ) MA = 3MB
MA − 3MA = 3 AB
3 2
. AM = AB 2 AM = 3 AB −2MA = 3 AB
(A
v وu 1 ( ) u .!" v AB = − BA 2 (.$% &'() AB + BC = AC 3 . A = B ) AB = 0 4 u +v v v u + v ** $+ 5 u $-. v u , ./0+ 1 u /0+ 1 ( ABCD ) (23 6 5* 4 C D :& 3% I AB = DC (a A B AD = BC (b AC = AB + AD (c .7- [ BD] [ AC ] 3 (d B I A [ AB ] & 7- I 7 1 AI = AB 2
IA = − IB (*
(*
IA + IB = 0
AI = IB (* 1 BI = BA 2
(*
(*
: 1 AI = AB 2
$ [ AB ] 7- I (a
+ 2 + [ AB ] 7- I + 2 (b IA + IB = 0 BC [ ] 7- I 989 ( ABC ) 1 . AI = AB + AC C 2
A
(
I
B
B
*
.989 ( ABC ) 9 [ AC ] 7- J [ AB ] 7- I
A I
)
8
J C
1 IJ = BC 2
*
v وu (a 10 . v وu (b . u = α v + v = α u AC وAB 4 & C وBوA (c AC = α AB + AB = α AC CDوAB 4 ( CD ) ( AB ) (d .
( BC ) ( D) . 7 7 ( ABC ) (3 N ( AC ) M ( AB) A AM AN MN : ! = = AB AC BC M N AB AC BC = = AM AN MN B C MA NA MN = ≠ " MB NC BC . 86 I 3* 8 9 ( ABCD) (4
B
A I D
IA IB AB = = : ! IC ID CD IC ID AB = = IA IB CD AI AB C BI = ≠ " BD AC CD
. (I O ( L) ( D) ( P) M (D ) M ( L) M ' . ( D) M M M ' M ' (L ) . ( D) ( L)
. !" # $ % #& ' ( L) M (a . O ' ( D) M (b ) ' ( D) ( L) ( (c . ( P) * ( P) p . p( M ) = M ' - M ' M ' + ,$ ! ( p . ( D) ⊥ ( L) ,$ (d . ( L)
. ! (II
: " # "! (5 + 3 ( L3 ) ( L2 ) ( L1 ) (a # 6 ( D ') ( D) A A ' C B A (L1 ) B B ' . C' B' A' (L 2 ) C C ' (L 3 ) ( L1 ) //( L2 ) ( L1 ) //( L2 ) //( L3 ) . AB A ' B ' ,$ = AC A ' C ' ( AC ) N ( AB ) M . 7 7 ( ABC ) (b
( MN ) //( BC ) .
AM AN = ,$ AB AC
: " < +": 4;3;2;1 +": (1 . 0 < = 9 . > !# (5) ": . D C B A + ,$ (2 AB = kCD ? AB = kCD
:
/0 1 * ( (1 {( A,α ), ( B, β )} /0 G ,$ p(G ) = G ' p( B) = B ' p( A) = A '
{( A ',α ), ( B ', β )} [ A ' B ']
/0 G' .
:
3" 1 * ( (2 /0 I' . [ AB ] 3" I ,$ p( B) = B ' p( A) = A ' 4*
:
#0 1 * ( (3 A ' B ' = kC ' D ' . AB = kCD ,$ D C B A " ' D ' C' B' A' .
(III + 5 ( L4 ) ( L3 ) ( L2 ) ( L1 ) (1 C B A # 6 ( D ') ( D) : ! . D ' C' B' A' D A A ' CA C ' A ' AB A ' B ' B B ' ...... BD = B ' D ' CD = C ' D ' C C ' D D ' ( L3 ) ( L2 ) ( L1 ) (2 ( D ') ( D) + 3 A A ' C B A # 6 (L1 ) B ' . C' B' A' (L 2 ) B : ! C ' (L 3 ) C CB C ' B ' AB A ' B ' ...... = = AB A ' B ' AC A ' C '
IR ا- ا ب . '",.ور ا01( ا4
ا*يb ,ﺝ. د ا$% ا. هa د$%& '*ر ا )ﺏ+ ا. a ∈ IR .
( a )n = a n
a = b
a
و
b
. x = − a أوx =
ab =
(∗
x2 = x a
a ≥ 0 (∗ ab = a b
(∗
a a = b b
(∗
.
. IR ا ب
"ی
a = b ,4 و. b 2 = a : 123ی . ــــــت%ﺥ + . IR b وa (a
a 2 = ( a ) 2 = a (∗
.
+
x ∈ IR (b
b : نab > 0 ( إذا آ نc
a یـــــــx 2 = a a ∈ IR + (d
. '2ـــــــــﺱ4( ا5
. IR ( ا ا ب1 . IR d وc وb وa a + c = b + c یـــــــa = b (a (c ≠ 0) a.c = b .c یـــــــa = b (b a + c = b + d a = b ـــــن و ( إذا آ ن وc a.c = b .d c = d . b = 0 أوa = 0 یـــــــa.b = 0 (d . b ≠ 0 وa ≠ 0 یـــــــa.b ≠ 0 (e a c (a ≠ 0 وb ≠ 0) a.d = b .c = یـــــــ (g b d a c ac a c ad + bc . = و (h + = b d bd b d bd a a a 1 b b = a ⋅ d (i = وb = و a a c b c c bc b d
IR ( ا ى2
: إذا آ ن82 إذا وd وc ' ' ن6 74 b وa دی$%ل إن ا.2 (a
a b = c d a a a : ن1 = 2 = = n : ( إذا آ نb b1 b 2 bn a1 a2 a k a + k 2a2 + + k n an = = = n = 1 1 b1 b 2 b n k 1b1 + k 2b 2 + + k n b n
. 5 6ئ ا81( ا6 x 22د ﺡ$: ; آ: ( "یa k ≤ x < k + 1 : 7% یk + 1 أوE (x ) = k ,4 وx د$%& >3=@ئ ا+ اA < یk <7د ا$%ا [x ] = k
وk % ﺏ44 < دی$: ر ﺏ.=3
.
: '(*ﺡ x ; B )ة$ﺝ.< ا*ي ی7د ا$% ا. هx د$%& >3=@ئ ا+∗( ا . IR x ; E ( x ) ≤ x < E ( x ) + 1 (∗
.
IR
( اـــــــــــــــII ـــــــت%( ﺥ1
a − b ≥ 0 یـــــــa ≥ b (∗ (a a − b ≤ 0 یـــــــa ≤ b (∗ a − b > 0 یـــــــa > b (∗ (b a − b < 0 یـــــــa < b (∗ . a = b أوa < b 7% یa ≤ b (∗ (c . >3) ﺹD % واa ≤ b نa < b ∗( إذا آ ن a + c ≥ b + c یـــــــa ≥ b (∗ (d a + c > b + c یـــــــa > b (∗ a ≤ b . a ≤ c ن ( ∗( إذا آ ن وe b ≤ c
(a ≠ 0) a = 1 (∗ a = 1 (∗ * (n ∈ IN − {1} ) a n = a . a. a..... a (∗ 1
0
( "یa
n fois
1 (∗ an ـــــــت%( ﺥb * . Z n وm وIR b وa (a 1 a − n = n (∗ a m . a n = a m + n (∗ a n n (ab ) = a .b n (∗ (a m ) n = a mn (∗ n a an a ( ) n = n (∗ = a n − m (∗ m b b a 2 2 a = b نa = b ( إذا آ نb 2 2 . a = b ا رة نb وa وa = b ( إذا آ نc a −n =
. a = −b أوa = b یـــــــa = b (d ی "! أن أنa = b : أن '*ﺡ(ـــــ 2
ا رةb وa وa = b 2
2
2
' ـــــت ه,- (3 (a + b ) 2 = a 2 + 2ab + b 2 (a (a − b ) 2 = a 2 − 2ab + b 2 (b a 2 − b 2 = (a − b )(a + b ) (c (a + b )3 = a 3 + 3a 2b + 3ab 2 + b 3 (d (a − b )3 = a 3 − 3a 2b + 3ab 2 − b 3 (e a 3 − b 3 = (a − b )(a 2 + ab + b 2 ) (f a 3 + b 3 = (a + b )(a 2 − ab + b 2 ) (g
ت9ــــــــ1.( ا3
[a, b ] = {x ∈ IR / a ≤ x ≤ b } [a, b [ = {x ∈ IR / a ≤ x < b } ]a, b ] = {x ∈ IR / a < x ≤ b } ]a, b [ = {x ∈ IR / a < x < b } [a, +∞[ = {x ∈ IR / x ≥ a} ]a, +∞[ = {x ∈ IR / x > a} ]−∞, a ] = {x ∈ IR / x ≤ a} ]−∞, a[ = {x ∈ IR / x < a}
(a (a (a
a ≤ b a + c < b + d ن ∗( إذا آ ن و c < d
(a (a
a ≤ b ac ≤ bc ن ( ∗( إذا آ ن وg c ≥ 0
(a (a (a
ـــــــــــ:;( ا4
وa ≤ x < b وa < x < b : وﺕ ت4 اF وﺕ4 ; آ: "ی . b − a K4%6 x د$%& )اPQ ﺕA < ﺕa ≤ x ≤ b وa < x ≤ b
. 'ــــــــ, .' ا. ( ا5 م.2 ، r FB$ ﺏx د$%& 8)ی4 ﺏF)ﺏ2 F B x 0 ( إذا أرد أن أنi (a
0 ≤ x − x 0 ≤ r $+76 وx − x 0 )PQ4ﺏ م.2 ، r FB$ ﺏx د$%& ﺏ )اطF)ﺏ2 F B x 0 ( إذا أرد أن أنii .
a ≤ b ∗( إذا آ ن و b < c a ≤ b . >3) ﺹD % واa + c ≤ b + d ن ( ∗( إذا آ ن وf c ≤ d . a < c ن
− r ≤ x − x 0 ≤ 0 $+76 وx − x 0 )PQ4ﺏ
م.2 ، r FB$ ﺏx د$%& F)ﺏ2 F B x 0 ( إذا أرد أن أنiii
− r ≤ x − x 0 ≤ r $+76 وx − x 0 )PQ4ﺏ . x − x 0 ≤ r 7%ی
a ≤ b ∗( إذا آ ن و c ≤ 0 0 ≤ a ≤ b . >3) ﺹD % واac ≤ bd ن ( ∗( إذا آ ن وf 0 ≤ c ≤ d 0 ≤ a ≤ b ac < bd ن ∗( إذا آ ن و 0 < c < d 1 1 ≥ یـــــــa ≤ b (∗ . b > 0 وa > 0 (i a b 1 1 ≥ یـــــــa ≤ b (∗ . b < 0 وa < 0 (j a b a 2 ≤ b 2 یـــــــa ≤ b (∗ b ≥ 0 وa ≥ 0 (k . ac ≥ bc ن
a≤ b
x د$%) اPQ4م ﺏ.2 x د$%& F)ﺏ2 F B د$3 ( إذا أرد أنb : & أن یT474< 7 و ه: a ≤ x ≤ b r = b − a FB$ ﺏx د$%& 8)ی4 ﺏF)ﺏ2 اF 2 ه اa (i r = b − a FB$ ﺏx د$%& ﺏ )اطF)ﺏ2 اF 2 ه اb (ii b −a a +b r= FB$ ﺏx د$%& F)ﺏ2 اF 2ه ا (iii 2 2 '( *ﺡ(ــــــc : F 4)ات اPQ4ى ا$ إﺡ7ی$ V )ة إذا آx د$%& F)ﺏ2 F B $ی$3ی ﺕ r FB$ ﺏx د$%& 8)ی4 ﺏF)ﺏ2 F B x 0 ن.46 و0 ≤ x − x 0 ≤ r (i r FB$ ﺏx د$%& ﺏ )اطF)ﺏ2 F B x 0 ن.46 و− r ≤ x − x 0 ≤ 0 (ii
یـــــــa ≤ b (∗
$+76 و
x − x 0 ≤ r أو− r ≤ x − x 0 ≤ r (iii
r FB$ ﺏx د$%& F)ﺏ2 F B x 0 ن.46و
. ( ا ی ا"=يd . IR x n
E (10 x ) )يW%د ا$%( اi 10n −n . 10 FB$ ﺏ n E (10 x ) د$%& ﺏ )اطF)ﺏ2 اF)یW% اF 2 اA <ی + 1 )يW%د ا$%( اi 10n −n . 10 FB$ ﺏx
x د$%& 8)ی4 ﺏF)ﺏ2 اF)یW% اF 2 اA <ی
a 2 ≥ b 2 یـــــــa ≤ b (∗ b ≤ 0 وa ≤ 0 (l a 2 ≤ b 2 یـــــــa ≤ b IR b وa (m b = 0 وa = 0 نa + b = 0 ا رة وb وa ( إذا آ ن ـn '(*ﺡ رن2 ، F%*ور ا )ﺏ+ اA&: ی ن.43 یb وa دی$%إذا آ ن ا 2 2 G ﺙb وa إ رة1234 وb وa رن2 ی "! أنb وa . (l ( وk 4 ﺹI ; ا%4<
. ' -.' ا.( ا ــــــــــ2 K @) د ا*ي$% ه اx د$%& F2&J اF 2 ا. IR x :
"ی
x ; x ≥0 x = : &)ف ﺏ ی% واx ب −x ; x ≤ 0 . K< هx د$%& F2&J اF 2 ن اx ≥ 0 ∗( إذا آ ن: 7%ی . K& ﺏ2 هx د$%& F2&J اF 2 ن اx ≤ 0 ∗( إذا آ ن ـــــــــــت%ﺥ x ≥ 0 (∗ −x = x (∗ (a x x n = (∗ x n = x (∗ xy = x y (∗ y y . .
x = −r أوx = r یـــــــx = r (∗ (b
x = − y أوx = y یـــــــx = y (∗ − r ≤ x ≤ r یـــــــx ≤ r (∗ (c x ≤ − r أوx ≥ r یـــــــx ≥ r (∗
III u
/ A :3/ 1 u 1 # A # ) # (D ) u ! # M # " i A . ( D ) D ( A, u ) 1 u AM .
u
AM
: !"
/ M ∈ D ( A, u ) (a
( ∆ ) # .) ( D ) (b . (D)
. AB 1 A ( AB ) ) # (c (D ) i B A . % &' 2
:($ u ( a, b )
(D) :
1 # A ( x , y ) # ) # ( D ) 0
x = x0 + at y = y0 + bt
( t ∈ IR )
0
- ( D ) ) ! 2 # $
I ( i, j )
1 u . B = ( i, j ) 2 # $#% ( x, y ) # u = xi + y j !" i
.j
. u x u ( x, y ) y
B
u
&
.& u # '$#% #(% : !" #(% . j i * u # ) B = ( i, j ) . u ( x, y ) ( x, y ) - u $#% +, u = xi + y j . B = ( i, j ) 3
j ( 0,1)
i (1, 0 ) (a v ( x′, y′ ) u ( x, y ) # / (b α u (α x,α y ) u − v ( x − x′, y − y′ ) u + v ( x + x′, y + y′ ) v ( x′, y′ ) u ( x, y ) # / (c /# . B & v u # (* : ! 3/ # det ( u, v ) 1 2(# x x′ = xy′ − yx′ det u , v = y y′ , #(% v u # (* det u , v ≠ 0 . j i (1 α = β = 0 +, α i + β j = 0 #(% (* . β = β ′ α = α ′ +, α i + β j = α ′i + β ′ j #(% (*
( )
# # " - ( D ) / 2 # # $ # #(. t ∈ IR 6 (1 + 3t , 2 − 4t ) !" $##
'$## !" 8 IR t 9" ! / . (D) . M ( 4, −2 ) ∈ ( D ) (% y = −2 x = 4 t = 1 "' . ) $ 3 1 # A ( x , y ) # ) # ( D ) (a : ! : ( D ) / !" 8! u ( a, b ) det ( AM , u ) = 0
/ M ( x, y ) ∈ ( D ) 0
0
x − x0
a
y − y0
b
=0
b ( x − x0 ) − a ( y − y0 )
/
/ : Ax + By + C = 0 !" / !" 8 ) ( D ) / - ( A, B ) ≠ ( 0, 0 ) . ( D ) : Ax + By + C = 0 ( D ) : a x + b y + c = 0 " # / (b . u ( −b, a ) 1 ) ( D ) # ( D ) #(% (* (c (D ) c i (1, 0 ) # +, 8,;# j . y = c !" 1/ 1 O i # ( D ) #(% (* (D ) j ( 0,1) # +, #;# j . x = c !" 1/ 1 O i c 1 # o ( 0, 0 ) # ) # - 8,;# (* . y = 0 1/ i (1, 0 )
#(%
( ) : !"
# +, 65$ C B A ' #(% (2 . ! . AC AB j
i
o
$ II 6 ( o, i, j ) 6!$ !/ 1
. ( ) )!/ # / 2 OM # 7 # M # $#% ( x, y ) # OM = xi + y j !" R = o, i, j
x M y
( o, i, j ) )!/ !
M ( x, y )
R
)!/ !
M
'$#% #(% : !" #(% . j i * OM ) . M ( x, y ) +, . R = ( o, i, j ) )!/ # / 3 B ( x , y ) A ( x , y ) # / . AB ( x − x , y − y ) (* # '$#% +, [ AB ] 38 I #(% (* M
OM = xi + y j
B
B
A
A
B
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ا ــت ا ــ دـــــ (Iا ـــــــــــــ آــــــ (Aﺕ!"# Ωو kد م M ' . ا آ ا ي آ Ωو "! kه $ا!# ا ي ﺏـ ) h (Ω, kوا ي یﺏ( M آ* ( P ) Mﺏ ' Mﺏ + Ω " . ΩM ' = k ΩM (Bا ( ا'' &ة ﺕ$ن ا ن ' Mو ' $0 Nرﺕ ا Mو 12 Nا$ا ﺏ آ hإذا و ( 6إذا و 3د k ≠ 1ﺏ + . M ' N ' = k MN (Cﺥ ( ــــ ت hﺕ آ آ Ωو "! k . h (M ) = M ' (1ﺕ ΩM ' = k ΩM 76
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! ا! f #ا ي T0ا 22ه : y ' = 3y − 4 x ' = x أ *3ﺕ ی + ! f !Wا ( ا> ة ﺏ * ا?X y ' = y x = −1 3x + 2 = x إذن fﺕ !* 0ة و ة ی ی y =2 3 y − 4 = y ه ). Ω( −1, 2 M (x , y ) YZ LVو )' M (x ', yﺏ h (M ) = M ' + x ' = 3x + 2 . و ی ) ΩM '( x + 1, y '− 2ی ی إذن y ' = 3y − 4 ) ΩM '(3x + 2 + 1,3 y − 4 − 2ی )ΩM '(3x + 3,3 y − 6
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