Tajribi Math Sx (110)
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اﻻﻣﺘﺤــﺎن اﻟﺘﺠﺮﻳﺒـﻲ ﻟﻠﺴﻨﺔ اﻟﺜﺎﻧﻴﺔ ﻣﻦ ﺳﻠﻚ اﻟﺒﺎآﺎﻟﻮرﻳــﺎ 2006 -2005 اﻟﺜﺎﻧﻮﻳﺔ :اﻟﺤﺴﻦ اﻟﺜﺎﻧﻲ اﻟﺘﺄهﻴﻠﻴﺔ
اﻟﺸﻌﺒﺔ :اﻟﻌﻠـــــﻮم ﺗﺠﺮﻳـﺒﻴـــــــﺔ
اﻟﻤــﺎدة :اﻟﺮﻳﺎﺿﻴـــــــــﺎت اﻟﻤﺴﺘﻮى :اﻟﺜــﺎﻧﻴــﺔ ﺛﺎﻧـﻮي
ﺗﻤﺮﻳﻦ 1 .I
1 ⎞⎛ 1 g ( x ) = L n ⎜1 + ⎟ − ⎝ x ⎠ x +1
ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ اﻟﻌﺪدﻳﺔ gاﻟﻤﻌﺮﻓﺔ ﺑﻤﺎ ﻳﻠﻲ :
.1ﺑﻴﻦ أن Dg = ]−∞, −1[ ∪ ]0, +∞[ : و ) lim g ( x
.2أﺣﺴﺐ lim g ( x ) :
∞x→+
.3ﺑﻴﻦ أن lim g ( x ) = +∞ :
∞x→−
x →−1−
.4أﻋﻂ ﺟﺪول ﺗﻐﻴﺮات اﻟﺪاﻟﺔ gواﺳﺘﻨﺘﺞ أن 0 : .II
و ) lim g ( x
x → 0+
): g ( x
x≠0
ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ اﻟﻌﺪدﻳﺔ fاﻟﻤﻌﺮﻓﺔ ﺑﻤﺎ ﻳﻠﻲ :
;
.1ﺑﻴﻦ أن D f = ]−∞, −1[ ∪ [ 0, +∞[ : .2أﺣﺴﺐ lim f ( x ) :
و ) lim f ( x
x →−1−
∞x →+
1 ) ﺿﻊ x
) ( ∀x ∈ D g
⎧ ⎞⎛ 1 ⎟ ⎪ f ( x ) = xL n ⎜ 1 + ⎠⎝ x ⎨ ⎪ f ( 0) = 0 ⎩
= (X
.3أدرس اﺗﺼﺎل fﻋﻠﻰ ﻳﻤﻴﻦ . 0 .4أدرس ﻗﺎﺑﻠﻴﺔ اﺷﺘﻘﺎق fﻋﻠﻰ ﻳﻤﻴﻦ 0ﺛﻢ أﻋﻂ ﺗﺄوﻳﻼ هﻨﺪﺳﻴﺎ. .5أدرس ﺗﻐﻴﺮات . f .6
أرﺳﻢ )
( C fﻣﻨﺤﻨﻰ اﻟﺪاﻟﺔ fﻓﻲ ﻣﻌﻠﻢ ﻣﺘﻌﺎﻣﺪ ﻣﻤﻨﻈﻢ ) . ( O, i , j n
v ⎞⎛ 1 .IIIﻧﻀﻊ ﻟﻜﻞ nﻣﻦ * un = ⎜ 1 + ⎟ :و ) vn = Ln ( unو . wn = n n ⎠⎝ n .1ﺗﺤﻘﻖ أن vn = f ( n ) :وأن ) ( vnﻣﺘﺘﺎﻟﻴﺔ ﺗﺰاﻳﺪﻳﺔ ﻗﻄﻌﺎ و 0 ≺ vn ≺ 1ﻟﻜﻞ nﻣﻦ .2أﺣﺴﺐ lim vn :
∞n →+
.3أﺣﺴﺐ :
*
.
ﺛﻢ اﺳﻨﺘﺞ . lim un ∞n →+
S n = w1 + w2 + w3 + ................. + wnﺑﺪﻻﻟﺔ nواﺣﺴﺐ . lim S n ∞n →+
ﺗﻤﺮﻳﻦ 2 ﻓﻲ اﻟﻔﻀﺎء اﻟﻤﻨﺴﻮب إﻟﻰ ﻣﻌﻠﻢ ﻣﺘﻌﺎﻣﺪ ﻣﻤﻨﻈﻢ ﻣﺒﺎﺷﺮ O, i , j , k
)
واﻟﻤﺴﺘﻮى ذو اﻟﻤﻌﺎدﻟﺔ :
(
ﻧﻌﺘﺒﺮ اﻟﻨﻘﻄﺔ )A (1,1,1
( P) : x − 2 y + 2z + 8 = 0
.1ﺗﺤﻘﻖ أن A ∉ ( P ) : .2ﺣﺪد إﺣﺪاﺛﻴﺎت اﻟﻨﻘﻄﺔ Hاﻟﻤﺴﻘﻂ اﻟﻌﻤﻮدي ﻟــ Aﻋﻠﻰ ) . ( P .3ﻟﺘﻜﻦ ) ( Sاﻟﻔﻠﻜﺔ ذات اﻟﻤﺮآﺰ Aوﺗﻘﻄﻊ اﻟﻤﺴﺘﻮى ) ( Pوﻓﻖ اﻟﺪاﺋﺮة ذات اﻟﻤﺮآﺰ Hواﻟﺸﻌﺎع r = 4
.aﺣﺪد Rﺷﻌﺎع اﻟﻔﻠﻜﺔ ) . ( S .bﺣﺪد ﻣﻌﺎدﻟﺔ دﻳﻜﺎرﺗﻴﺔ ﻟﻠﻔﻠﻜﺔ ) . ( S
اﻻﻣﺘﺤــﺎن اﻟﺘﺠﺮﻳﺒـﻲ ﻟﻠﺴﻨﺔ اﻟﺜﺎﻧﻴﺔ ﻣﻦ ﺳﻠﻚ اﻟﺒﺎآﺎﻟﻮرﻳــﺎ 2006 -2005 اﻟﺜﺎﻧﻮﻳﺔ :اﻟﺤﺴﻦ اﻟﺜﺎﻧﻲ اﻟﺘﺄهﻴﻠﻴﺔ
اﻟﺸﻌﺒﺔ :اﻟﻌﻠـــــﻮم ﺗﺠﺮﻳـﺒﻴـــــــﺔ
اﻟﻤــﺎدة :اﻟﺮﻳﺎﺿﻴـــــــــﺎت اﻟﻤﺴﺘﻮى :اﻟﺜــﺎﻧﻴــﺔ ﺛﺎﻧـﻮي
ﺗﻤﺮﻳﻦ 3 .1أﺣﺴﺐ اﻟﺘﻜﺎﻣﻼت اﻟﺘﺎﻟﻴﺔ : π
dx
x2
) (1 − 2 x
3 2
.2ﺑﻴﻦ أن : .3أﺣﺴﺐ اﻟﺘﻜﺎﻣﻞ :
2
∫= I
و
dx
1
x2 − 2 x 3x 2 − x3
2
∫= J
2
cos x ∫=K dx cos 2 x − 2 0
و
1
e −2 x ex −x = − 1 + e 1 + e− x 1 + ex Ln 2 e −2 x ∫ =L dx −x + e 1 0
.4ﺑﺎﺳﺘﻌﻤﺎل اﻟﻤﻜﺎﻣﻠﺔ ﺑﺎﻷﺟﺰاء أﺣﺴﺐ اﻟﺘﻜﺎﻣﻞ :
Ln (1 + e − x ) dx
−x
Ln 2
∫e
=R
0
ﺗﻤﺮﻳﻦ 4 ﻧﻌﺘﺒﺮ اﻟﺤﺪودﻳﺔ :
P ( z ) = z + ( 3 − 2i ) z + (1 − 4i ) z − 1 − 2i 2
3
.1أﺣﺴﺐ )P ( −1
) ∈(z
.2أوﺟﺪ اﻟﻌﺪدﻳﻦ اﻟﻌﻘﺪﻳﻴﻦ aو bﺑﺤﻴﺚ P ( z ) = ( z + 1) ( z 2 + az + b ) :
.3ﺣﻞ ﻓﻲ
اﻟﻤﻌﺎدﻟﺔ P ( z ) = 0 :
.4ﻧﻀﻊ A ( −1) :و ) B ( iو ) . C ( −2 + i .aأﺛﺒﺖ أن ABCﻣﺜﻠﺜﺎ ﻗﺎﺋﻢ اﻟﺰاوﻳﺔ وﻣﺘﺴﺎوي اﻟﺴﺎﻗﻴﻦ.
.bاﺳﺘﻨﺘﺞ ﻗﻴﺎﺳﺎ ﻟﻠﺰاوﻳﺔ )
(
. BA, BC
) ﺿﻊ .( t = sin x
اﻟﺸﻌﺒﺔ :اﻟﻌﻠـــــﻮم ﺗﺠﺮﻳـﺒﻴـــــــﺔ
اﻟﻤــﺎدة :اﻟﺮﻳﺎﺿﻴـــــــــﺎت اﻟﻤﺴﺘﻮى :اﻟﺜــﺎﻧﻴــﺔ ﺛﺎﻧـﻮي
ﺗﻤﺮﻳﻦ 1 .I
1 ⎞⎛ 1 g ( x ) = L n ⎜1 + ⎟ − ⎝ x ⎠ x +1
ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ اﻟﻌﺪدﻳﺔ gاﻟﻤﻌﺮﻓﺔ ﺑﻤﺎ ﻳﻠﻲ :
.1ﺑﻴﻦ أن Dg = ]−∞, −1[ ∪ ]0, +∞[ : و ) lim g ( x
.2أﺣﺴﺐ lim g ( x ) :
∞x→+
.3ﺑﻴﻦ أن lim g ( x ) = +∞ :
∞x→−
x →−1−
.4أﻋﻂ ﺟﺪول ﺗﻐﻴﺮات اﻟﺪاﻟﺔ gواﺳﺘﻨﺘﺞ أن 0 : .II
و ) lim g ( x
x → 0+
): g ( x
x≠0
ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ اﻟﻌﺪدﻳﺔ fاﻟﻤﻌﺮﻓﺔ ﺑﻤﺎ ﻳﻠﻲ :
;
.1ﺑﻴﻦ أن D f = ]−∞, −1[ ∪ [ 0, +∞[ : .2أﺣﺴﺐ lim f ( x ) :
و ) lim f ( x
x →−1−
∞x →+
1 ) ﺿﻊ x
) ( ∀x ∈ D g
⎧ ⎞⎛ 1 ⎟ ⎪ f ( x ) = xL n ⎜ 1 + ⎠⎝ x ⎨ ⎪ f ( 0) = 0 ⎩
= (X
.3أدرس اﺗﺼﺎل fﻋﻠﻰ ﻳﻤﻴﻦ . 0 .4أدرس ﻗﺎﺑﻠﻴﺔ اﺷﺘﻘﺎق fﻋﻠﻰ ﻳﻤﻴﻦ 0ﺛﻢ أﻋﻂ ﺗﺄوﻳﻼ هﻨﺪﺳﻴﺎ. .5أدرس ﺗﻐﻴﺮات . f .6
أرﺳﻢ )
( C fﻣﻨﺤﻨﻰ اﻟﺪاﻟﺔ fﻓﻲ ﻣﻌﻠﻢ ﻣﺘﻌﺎﻣﺪ ﻣﻤﻨﻈﻢ ) . ( O, i , j n
v ⎞⎛ 1 .IIIﻧﻀﻊ ﻟﻜﻞ nﻣﻦ * un = ⎜ 1 + ⎟ :و ) vn = Ln ( unو . wn = n n ⎠⎝ n .1ﺗﺤﻘﻖ أن vn = f ( n ) :وأن ) ( vnﻣﺘﺘﺎﻟﻴﺔ ﺗﺰاﻳﺪﻳﺔ ﻗﻄﻌﺎ و 0 ≺ vn ≺ 1ﻟﻜﻞ nﻣﻦ .2أﺣﺴﺐ lim vn :
∞n →+
.3أﺣﺴﺐ :
*
.
ﺛﻢ اﺳﻨﺘﺞ . lim un ∞n →+
S n = w1 + w2 + w3 + ................. + wnﺑﺪﻻﻟﺔ nواﺣﺴﺐ . lim S n ∞n →+
ﺗﻤﺮﻳﻦ 2 ﻓﻲ اﻟﻔﻀﺎء اﻟﻤﻨﺴﻮب إﻟﻰ ﻣﻌﻠﻢ ﻣﺘﻌﺎﻣﺪ ﻣﻤﻨﻈﻢ ﻣﺒﺎﺷﺮ O, i , j , k
)
واﻟﻤﺴﺘﻮى ذو اﻟﻤﻌﺎدﻟﺔ :
(
ﻧﻌﺘﺒﺮ اﻟﻨﻘﻄﺔ )A (1,1,1
( P) : x − 2 y + 2z + 8 = 0
.1ﺗﺤﻘﻖ أن A ∉ ( P ) : .2ﺣﺪد إﺣﺪاﺛﻴﺎت اﻟﻨﻘﻄﺔ Hاﻟﻤﺴﻘﻂ اﻟﻌﻤﻮدي ﻟــ Aﻋﻠﻰ ) . ( P .3ﻟﺘﻜﻦ ) ( Sاﻟﻔﻠﻜﺔ ذات اﻟﻤﺮآﺰ Aوﺗﻘﻄﻊ اﻟﻤﺴﺘﻮى ) ( Pوﻓﻖ اﻟﺪاﺋﺮة ذات اﻟﻤﺮآﺰ Hواﻟﺸﻌﺎع r = 4
.aﺣﺪد Rﺷﻌﺎع اﻟﻔﻠﻜﺔ ) . ( S .bﺣﺪد ﻣﻌﺎدﻟﺔ دﻳﻜﺎرﺗﻴﺔ ﻟﻠﻔﻠﻜﺔ ) . ( S
اﻻﻣﺘﺤــﺎن اﻟﺘﺠﺮﻳﺒـﻲ ﻟﻠﺴﻨﺔ اﻟﺜﺎﻧﻴﺔ ﻣﻦ ﺳﻠﻚ اﻟﺒﺎآﺎﻟﻮرﻳــﺎ 2006 -2005 اﻟﺜﺎﻧﻮﻳﺔ :اﻟﺤﺴﻦ اﻟﺜﺎﻧﻲ اﻟﺘﺄهﻴﻠﻴﺔ
اﻟﺸﻌﺒﺔ :اﻟﻌﻠـــــﻮم ﺗﺠﺮﻳـﺒﻴـــــــﺔ
اﻟﻤــﺎدة :اﻟﺮﻳﺎﺿﻴـــــــــﺎت اﻟﻤﺴﺘﻮى :اﻟﺜــﺎﻧﻴــﺔ ﺛﺎﻧـﻮي
ﺗﻤﺮﻳﻦ 3 .1أﺣﺴﺐ اﻟﺘﻜﺎﻣﻼت اﻟﺘﺎﻟﻴﺔ : π
dx
x2
) (1 − 2 x
3 2
.2ﺑﻴﻦ أن : .3أﺣﺴﺐ اﻟﺘﻜﺎﻣﻞ :
2
∫= I
و
dx
1
x2 − 2 x 3x 2 − x3
2
∫= J
2
cos x ∫=K dx cos 2 x − 2 0
و
1
e −2 x ex −x = − 1 + e 1 + e− x 1 + ex Ln 2 e −2 x ∫ =L dx −x + e 1 0
.4ﺑﺎﺳﺘﻌﻤﺎل اﻟﻤﻜﺎﻣﻠﺔ ﺑﺎﻷﺟﺰاء أﺣﺴﺐ اﻟﺘﻜﺎﻣﻞ :
Ln (1 + e − x ) dx
−x
Ln 2
∫e
=R
0
ﺗﻤﺮﻳﻦ 4 ﻧﻌﺘﺒﺮ اﻟﺤﺪودﻳﺔ :
P ( z ) = z + ( 3 − 2i ) z + (1 − 4i ) z − 1 − 2i 2
3
.1أﺣﺴﺐ )P ( −1
) ∈(z
.2أوﺟﺪ اﻟﻌﺪدﻳﻦ اﻟﻌﻘﺪﻳﻴﻦ aو bﺑﺤﻴﺚ P ( z ) = ( z + 1) ( z 2 + az + b ) :
.3ﺣﻞ ﻓﻲ
اﻟﻤﻌﺎدﻟﺔ P ( z ) = 0 :
.4ﻧﻀﻊ A ( −1) :و ) B ( iو ) . C ( −2 + i .aأﺛﺒﺖ أن ABCﻣﺜﻠﺜﺎ ﻗﺎﺋﻢ اﻟﺰاوﻳﺔ وﻣﺘﺴﺎوي اﻟﺴﺎﻗﻴﻦ.
.bاﺳﺘﻨﺘﺞ ﻗﻴﺎﺳﺎ ﻟﻠﺰاوﻳﺔ )
(
. BA, BC
) ﺿﻊ .( t = sin x
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