رياضيات عامة 113 ريض
DESCRIPTION
ddTRANSCRIPT
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⎜⎜⎜
⎝
⎛
−−
−−−
=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−×+
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛×−=+−
2.46
2
012
8.04
2
601
52
0
613
4.02
1
305.0
2520
613
12)2 BA
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−
−
−
−=
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
−
−−−
−
−
−
=−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−
=−
2.01
5.0
5.10
25.0
24.0222
1
232
025.0
24.02
1
305.0
2)3 B
٢WFKE ٦WאאאKאW
)2(5.1)23)1 AA×
א ١١٣ אא
א אאא
אW 1Eא(1)אאאW
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛==×
1560
1839
33 AA
2Eא(1)אאאW
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛==×=
1560
1839
3)25.1()2(5.1 AAA
٢}٣W ٥Wאא
אאאא،אאאK ٧WאW
( ) ( ) .413
3.0021)2,
10413
3.0021)1⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−−=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛−
−= ba
אW 23023)103.0()40()1)2(())3(1()1 −=++−−=×+×+×−+−×=a
2EbאאK אאאאאK
٢}٤W ٦Wאkm×אnk ×F
אאאאEאnm×،אאאאאאאאאאK
א ١١٣ אא
א אאא
٨WאאW
.632312015
3021,
2321
11,
30
11
52
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−
−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−
= CBA
אW .)3,)2,)1 BCBAAB
אW
( ) ( )
( ) ( )⎟⎟⎠
⎞⎜⎜⎝
⎛−
=
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−
=315
41
221
31531
1315
221
01231
1012
)1 AB
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−−=
⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−⎟⎟⎠
⎞⎜⎜⎝
⎛−
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−⎟⎟⎠
⎞⎜⎜⎝
⎛−
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛
=654
638307
30
231
123
52
23
30
211
121
52
21
30
111
111
52
11
)2 BA
3Eאאא2אא3K
Wא(1)(2)אאאW BAAB ≠אK
٣WאK ٩Wא(1)אא٧W
( )3.0021
10413
−
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛−
=A
א ١١٣ אא
א אאא
אW
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−−
−−
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
××−××××−××××−×××−×−−×−×−
=
3020102.10843.00219.0063
3.0100102101103.040424143.010121113.03032313
A
W
( ) ( )3.0021
10413
10413
3.0021 −
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛−
≠
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛−
−
אאאאאأي n×1א1×nאK
١٠WאאW
( ) .4003
12,5.003,
21
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−==⎟⎟
⎠
⎞⎜⎜⎝
⎛−= CBA
אW ).()2,)()1 BCACAB
אW
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛ −−=
101256
4003
12
1065.003
)()1 CAB
2Eא1אאאW
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−==
101256
)()( CABBCA
٤W،אW
ACABCBABCACCBA
+=++=+
)()2)()1
אABCאאאאK
א ١١٣ אא
א אאא
אW ٧WAאnm×אtAאmn×،
Kאאאא ١١WאאאאאK אW
( ) .401032
,5.0
03
,21 ⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=−= ttt CBA
٣KאאW ٣}١אאW
٨WאAאK ١٢WאאW
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−=⎟⎟
⎠
⎞⎜⎜⎝
⎛ −=
510604
,316
035410
,102124301
,1132
,0111
ED
CBA
C33×وDالمصفوفات A22×وBالمصفوفات אE
٣}٢אW ٩Wאא،1א
אאKnIאאnn×،IאאK
א ١١٣ אא
א אאא
١٣W
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
1000010000100001
100010001
1001
432 III
٥WאאK ١٤WאW
.251430321
100010001
)2,100010001
251430321
)1⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−= BA
אW
.251430321
251430321
100010001
)2,251430321
100010001
251430321
)1⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−= BA
٤KאאW ١٠Wאא nn×אאא
אK(A)detK
Kאאא11×K
٤}١אא22×W ١١W22א×⎟⎟
⎠
⎞⎜⎜⎝
⎛=
dcba
AאאFאE
אאFאE،W
bcaddcba
(A) −==det
א ١١٣ אא
א אאא
١٥WאאאאW
0511
)2,3142
)1−
אW
5)5(0)15()01(0511
)2,246)14()32(3142
)1 =−−=−×−×=−
=−=×−×=
٤}٢אא33×W ١٢W33א×אAאאאא
FאEאאאאFאE،אאאאK
א⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
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333231
232221
131211
aaaaaaaaa
AW
122133112332132231322113312312332211
3231
2221
1211
333231
232221
131211
333231
232221
131211
det
aaaaaaaaaaaaaaaaaaaaaaaa
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−−−++=
==
١٦WאאאאW
.420315
201)2,
478362931
)1−
−−
אW
25524214321267224)324()137()968()729()833()461(
786231
478362931
478362931
)1
−=−−−++=××−××−××−××+××+××=
=
א ١١٣ אא
א אאא
180)6(02004)054()132()210()252()030()411(
201501
420315
201
420315
201)2
−=−−−−−+−=××−−×−×−−××−−××+×−×+××−=
−
−
−−
−=
−−
−
١٧WאאאאW
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−=⎟⎟
⎠
⎞⎜⎜⎝
⎛ −=
510604
,316
035410
,102124301
,1132
,0111
ED
CBA
אW
107150722000131
65
0
316035410
316035410
)det(
1000)12(00202
0
241
102124301
102124301
)det(
5321132
)det(,1)1(00111
)det(
−=−−−−+=−−
−=−
−=
=−−−−++−=−−−
−−
=−
−−
=
−=−−=−
==−−=−
=
D
C
BA
)det(EEK ٦WK
١٨WאאאKאאאאW ).det()4),det()3),det()2),det()1 DCCDBAAB
א ١١٣ אא
א אאא
אW אאאאKא
אW
107010107)det()det()det()4107010710)det()det()det()3
515)det()det()det()2551)det()det()det()1
−=×−=×=−=−×=×=
−=×−=×=−=−×=×=
CDDCDCCDABBABAAB
5KאW ١٣WAאא1−A J–
،אW IAAAA == −− 11
אא22×
٧WאAאא 22×W
٨Wאabcdא0det ≠−= bcadAאW
dcbaacbd
bcadacbd
dcba ⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−
=−
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−
=⎟⎟⎠
⎞⎜⎜⎝
⎛−1
אא22×אK ١٩Wאא22× אW
.3162
,4321
)2,0213
)1 ⎟⎟⎠
⎞⎜⎜⎝
⎛ −=⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛ −= CBA
אW 1EאAאW
א ١١٣ אא
א אאא
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=⎟⎟⎠
⎞⎜⎜⎝
⎛−
=−
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−−
=⎟⎟⎠
⎞⎜⎜⎝
⎛ − −
5.115.00
23210
021332
)1(0
0213 1
2EאBאW
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=
−
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−
=⎟⎟⎠
⎞⎜⎜⎝
⎛−
−
=⎟⎟⎠
⎞⎜⎜⎝
⎛−
5.05.112
21324
43211324
4321 1
3EאCאK
١WאאW
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−−−
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−−−
=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−
−−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
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−=
324116
010,
423015
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2513
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011DCBA
אא )3(2)5),(3)4,23)3,)2,32)1 BICDADABCDA ttttt −++−++−
٢WאאאW
.321352103
501241
432)4,
71
01
3423
14
71
32
01
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,23
3412
241532121
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14
214213
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⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−−
−
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−
−
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−
−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−
٣WאאאאW
א ١١٣ אא
א אאא
.100410321
)6,522113021
)5,021212113
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,6370
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52)2,
5321
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−−−
−−
−−
٤WאאW
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=612
126M
אW ).det()3).det()2).det()1 32 MMM
٥WאW
,21105
)2,64
21)1 ⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−= BA
٦WאאW
.66
95,
2115
,0132
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−
=⎟⎟⎠
⎞⎜⎜⎝
⎛−= CBA
אPQRW .)3,)2,2)1 2 CRAIBQAQBAP ==++=
٧WאאאW
AAB⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛= −
001010100
1
٨WאאW
.421
035,
231012
,0132
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−= CBA
אא CBACBBAAB ttt +− −− 11 )4,)3,)2,)1
אא
א
٣
א ١١٣ אא
א א
- ٣٢ -
אאWא
אאWאאK
אאWאאאW o אאאא؛ o ؛אאאא o ؛אאא o אאאK
אאאWאא٨٠٪K אאWאK
א ١١٣ אא
א א
- ٣٣ -
אאWא
אאאאאאאאKאאאאא?אא?אאאאאא
Fא٨٢٥EאאאאKאאאאאאאK
אאאK
١Kא ١WאאFKEא
K ١W712א =+x3=xلxK
3=xx3א71)3(2 =+אK
א،אאאאK
٢W3=x2=x0652 =+− xxK
אאאאאxאאאאWثابتx =K
אאאאW
• אאא،אאאאאאW11532 −=++ xx1137 −=+xK
• אאW273 =−x93 =xK • אאאאאW
1065
=x12=x
א ١١٣ אא
א א
- ٣٤ -
٢KאאאאאW ٢Wאאאאאא ) אא (אW
0=+ baxabא0≠a
٣WאאW).1(5)15)(2()3,0643)2,952)1 +=++=−=+ xxxxxx
אW 1Eא5אא2W
44259552 =⇔=⇔−=−+ xxx
2E 6א34א
43W
8)6(34
43
346
436066
4306
43
=⇔⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛⎟⎠⎞
⎜⎝⎛⇔=⇔+=+−⇔=− xxxxx
3Eאאאא2,5,5 2 xxאא6W
31
31
6226026
5211552115)1(5)15)(2( 22
−=⇔−=−=⇔−=⇔=+⇔
=+⇔+=++⇔+=++
xxxx
xxxxxxxxxx
WאאאאK
אאאאאאאאKאאאאאאאא
אאאאK
٤WאאW
.5
55
1)2,3524
3)1
−=
−+
−−
=− xx
xx
xx
x
אW 1E3≠xאאאKא)3( −xאאאאK
א ١١٣ אא
א א
- ٣٥ -
4624
66246
552455243524)3(
3)3(
=⇔=⇔=⇔
+−=+⇔−=⇔−−
−=−
−
xxx
xxxxxxx
xxx
xx
אא3אאאK
2E5≠xאאאKאא)5( −xאאאK
510255552552555
5)5(5
)5(1)5(5
5)5(5
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אW ١ER=fDK ٢ER=fRאאKK ٣EK ٤Eאא0=bא),0( bא0≠bK
١٦WאאאWRR: →fxxf 2)( −=K אW
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
א ١١٣ אאא
א אא
١٧WאאאWRR: →f5.12)( +−= xxfK אW
(4אאאWאWRR: →fcxbxaxfy ++== 2)(0≠ab
cאאאK אW
١ER=fDK ٢ER≠fRאאKK ٣Eא0=bK ٤Eאאא0== cbK
١٨WאאאWRR: →f5.12)( 2 +−= xxfK אW
-2 -1 1 2
-3
-2
-1
1
2
-2 -1 1 2 3
-3
-2
-1
1
2
3
א ١١٣ אאא
א אא
(4אאאWאK
אאWRR: →f112)(
−+
=xxxf
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112)(
−+
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K٧WאאאxאאW xsinא،אאאא xcosאאאאאK xtanאאאאאאא
xcotאאאאאאא
-10 -5 5 10
-10
-7.5
-5
-2.5
2.5
5
7.5
10
א ١١٣ אאא
א אא
(1אאWWsinאWRR:sin →xy sin=KאK
אW ١ER=fDK ٢E]1,1[−=fRאאW1sin1 ≤≤− x ٣Exx sin)sin( −=−K ٤Exx sin)2sin( =+ πאπ2K ٥EאאW
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cossintan ==KK
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7,2
5,2
3,2
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5,{R LLππππππ
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5
7.5
10
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10
א ١١٣ אאא
א אא
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1)(,RR:)4,1)(,RR:)3
,)(,NN:)2,)(,NN:)133
33
−=→+=→
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xxffxxff
xxffxxff
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1)(,RR:)22sin)(,RR:)1
xxffxxxffx
xffxxff
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xxffxxxffx
xffxxff
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)(,RR:)42cos3sin)(,RR:)3
1)(,RR:)22sin)(,RR:)1
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.3cos2tan)4,41cos2sin)3,
41cos)2,2sin)1 xxxxxx +−
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אאאאאאאא
א
א
אא
٥
א ١١٣ אא
א אאאא
אאWאאאא
אאWאאאאאאאאKK
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אאאWאא٨٠٪K אאWK
א ١١٣ אא
א אאאא
אאאא
אאאאאאאאאFאאאאEאא،NapierJohnא
K١٦١٤KאאאאK
א،אאאאאאאKא
١KאW ١WxnxnW
n xxxxn ×××= L W 10 =x
nمرفوع للقوة xأو nأس xويقرأ xللعدد nيسمى القوة nxالرمز • .يسمى األس nيسمى األساس و العدد xالعدد . nxفي الرمز •١WאW
342 )3()3)2()23)1 −− אW
27333)3()3
162222)2()2
9333)1
3
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nn
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א ١١٣ אא
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271
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161
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א אאאא
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אx n xnx1
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====
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−−−−−−−−
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−
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א،אxK
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א אאאא
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10
20
30
40
50
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xx xfyxfyxfy ====== − π אW
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5918118159
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4 هو 4 16− ناتج القيمة d) 3 اليمكن حسابها c) 2− b) 0 a)
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هو 42ناتج القيمة 8a) 42× b) 2222 ××× c) 16 d)
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3) Gwyn Davies and Gordon Hick, Mathematics for scientific and technical students,
Addison Wesley Longman, Harlow, England, 1998. 4) Anders Hald, A History of Probability and Statistics and Their Applications before
1750, John Wiley and Sons, New York, 1989.
5) Alexander Schrijver, Theory of Linear and Integer Programming, John Wiley & Sons,
Chichester, England, 1986.
6) Seymour Lipschutz and Marc Lipson, Discrete Mathematics, McGraw-Hill, New York,
1997. 7) Peter Tebbutt, Basic Mathematics, John Wiley & Sons, Chichester, England, 1998.
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