S-exp-i Math Devoir S1 1
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ﺛﺎﻧﻮﻳﺔ اﻟﻤﺨﺘﺎر اﻟﺴﻮﺳﻲ اﻟﻤﻌﺎزﻳﺰ ﻧﻴﺎﺑﺔ اﻟﺨﻤﻴﺴﺎت
اﻷ ﺳﺘﺎذ :ﻋﻠﻲ اﻟﺸﺮﻳﻒ ﻗﺴﻢ :اﻷوﻟﻰ ﺑﺎآﺎﻟﻮرﻳﺎ ع.ت 11 – 12 – 2004
ﻓﺮض ﻡﺤﺮوس رﻗﻢ 2د 1 ﻡﺎدة اﻟﺮﻳﺎﺿﻴﺎت ﻡﺪة اﻹﻧﺠﺎز -ﺳﺎﻋﺔ -
" اﻟﺘﻤﺮﻳﻦ رﻗﻢ 8 ) : 1ن (
u n +1 = 2 + u n ﻧﻌﺘﺒﺮ اﻟﻤﺘﺘﺎﻟﻴﺔ اﻟﻌﺪدﻳﺔ (u n )n∈INاﻟﻤﻌﺮﻓﺔ ﺏﻤﺎ ﻳﻠﻲ , n ∈ IN : u 0 = 0 ( 1أ – ﺏﻴﻦ أ ن:
∀n ∈ IN : 0 ≤ u n ≤ 2
) 2ن(
ب – ﺏﻴﻦ أ ن ) (u nﺕﺰاﻳﺪﻳﺔ
) 1ن(
(2ﻟﺘﻜﻦ ) (v nاﻟﻤﺘﺘﺎﻟﻴﺔ اﻟﻌﺪدﻳﺔ اﻟﻤﻌﺮﻓﺔ ﺏﻤﺎ ﻳﻠﻲ :
∀n ∈ IN : v n = 2 − u n
1 0 ≤ v n +1 ≤ .v n 2
أ – ﺏﻴﻦ أ ﻧﻪ ﻟﻜﻞ nﻡﻦ : IN
)1.5ن(
n
1 ب – ﺁ ﺳﺘﻨﺘﺞ أن ﻟﻜﻞ nﻡﻦ 0 ≤ v n ≤ .v 0 : IN 2
) 1.5ن(
ج – ﺁ ﺳﺘﻨﺘﺞ ﻧﻬﺎﻳﺔ اﻟﻤﺘﺘﺎﻟﻴﺔ ) (v nﺛﻢ ﻧﻬﺎﻳﺔ اﻟﻤﺘﺎﻟﻴﺔ ) . (u n
) 2ن(
" اﻟﺘﻤﺮﻳﻦ رﻗﻢ 12 ) : 2ن(
)
(
اﻟﻤﺴﺘﻮى Pﻡﻨﺴﻮب إﻟﻰ م.م.م . O; i; j ﻧﻌﺘﺒﺮ ﻓﻲ Cاﻟﻤﻌﺎدﻟﺔ :
(E θ ) : z 2 − (1 + i.sin 2θ)z + 1 i sin 2θ = 0 2
π π ) ( θ ∈ − , 2 2
(1ﺣﻞ ﻓﻲ Cاﻟﻤﻌﺎدﻟﺔ ) (E θو أﻋﻂ اﻟﺤﻞ اﻟﻤﺰدوج.
) 4ن(
(2ﻟﺘﻜﻦ Mو ' ' Mﺻﻮرﺕﻲ اﻟﺤﻠﻴﻦ ' zو ' ' zو Iﻡﻨﺘﺼﻒ ]' '. [M' M − π π أ -ﻡﺎ هﻲ ﻡﺠﻤﻮﻋﺔ اﻟﻨﻘﻂ Iﻋﻨﺪﻡﺎ ﻳﺘﻐﻴﺮ θﻓﻲ , 2 2
.
) 2ن(
ب -ﺏﺮهﻦ ﻋﻠﻰ أن ﻡﺠﻤﻮﻋﺔ اﻟﻨﻘﻄﺘﻴﻦ ' Mو ' ' Mهﻲ داﺋﺮة ﻳﺠﺐ ﺕﺤﺪﻳﺪهﺎ.
) 2ن(
ج -ﺏﺮهﻦ أﻧﻪ إذا آﺎن " M ' ≠ Mﻓﺈن اﻟﻤﺴﺘﻘﻴﻢ )" (M' Mﻟﻪ اﺕﺠﺎﻩ ﻏﻴﺮ ﻡﺮﺕﺒﻂ ﺏﻘﻴﻢ . θ
) 2ن(
د θ -ﻡﻌﻠﻮم ﺁ ﺳﺘﻨﺘﺞ ﻡﻤﺎ ﺳﺒﻖ ﻃﺮﻳﻘﺔ ﺏﺴﻴﻄﺔ ﻹﻧﺸﺎء Iو ' Mو ". M
) 2ن(
http://arabmaths.ift.fr
اﻷ ﺳﺘﺎذ :ﻋﻠﻲ اﻟﺸﺮﻳﻒ ﻗﺴﻢ :اﻷوﻟﻰ ﺑﺎآﺎﻟﻮرﻳﺎ ع.ت 11 – 12 – 2004
ﻓﺮض ﻡﺤﺮوس رﻗﻢ 2د 1 ﻡﺎدة اﻟﺮﻳﺎﺿﻴﺎت ﻡﺪة اﻹﻧﺠﺎز -ﺳﺎﻋﺔ -
" اﻟﺘﻤﺮﻳﻦ رﻗﻢ 8 ) : 1ن (
u n +1 = 2 + u n ﻧﻌﺘﺒﺮ اﻟﻤﺘﺘﺎﻟﻴﺔ اﻟﻌﺪدﻳﺔ (u n )n∈INاﻟﻤﻌﺮﻓﺔ ﺏﻤﺎ ﻳﻠﻲ , n ∈ IN : u 0 = 0 ( 1أ – ﺏﻴﻦ أ ن:
∀n ∈ IN : 0 ≤ u n ≤ 2
) 2ن(
ب – ﺏﻴﻦ أ ن ) (u nﺕﺰاﻳﺪﻳﺔ
) 1ن(
(2ﻟﺘﻜﻦ ) (v nاﻟﻤﺘﺘﺎﻟﻴﺔ اﻟﻌﺪدﻳﺔ اﻟﻤﻌﺮﻓﺔ ﺏﻤﺎ ﻳﻠﻲ :
∀n ∈ IN : v n = 2 − u n
1 0 ≤ v n +1 ≤ .v n 2
أ – ﺏﻴﻦ أ ﻧﻪ ﻟﻜﻞ nﻡﻦ : IN
)1.5ن(
n
1 ب – ﺁ ﺳﺘﻨﺘﺞ أن ﻟﻜﻞ nﻡﻦ 0 ≤ v n ≤ .v 0 : IN 2
) 1.5ن(
ج – ﺁ ﺳﺘﻨﺘﺞ ﻧﻬﺎﻳﺔ اﻟﻤﺘﺘﺎﻟﻴﺔ ) (v nﺛﻢ ﻧﻬﺎﻳﺔ اﻟﻤﺘﺎﻟﻴﺔ ) . (u n
) 2ن(
" اﻟﺘﻤﺮﻳﻦ رﻗﻢ 12 ) : 2ن(
)
(
اﻟﻤﺴﺘﻮى Pﻡﻨﺴﻮب إﻟﻰ م.م.م . O; i; j ﻧﻌﺘﺒﺮ ﻓﻲ Cاﻟﻤﻌﺎدﻟﺔ :
(E θ ) : z 2 − (1 + i.sin 2θ)z + 1 i sin 2θ = 0 2
π π ) ( θ ∈ − , 2 2
(1ﺣﻞ ﻓﻲ Cاﻟﻤﻌﺎدﻟﺔ ) (E θو أﻋﻂ اﻟﺤﻞ اﻟﻤﺰدوج.
) 4ن(
(2ﻟﺘﻜﻦ Mو ' ' Mﺻﻮرﺕﻲ اﻟﺤﻠﻴﻦ ' zو ' ' zو Iﻡﻨﺘﺼﻒ ]' '. [M' M − π π أ -ﻡﺎ هﻲ ﻡﺠﻤﻮﻋﺔ اﻟﻨﻘﻂ Iﻋﻨﺪﻡﺎ ﻳﺘﻐﻴﺮ θﻓﻲ , 2 2
.
) 2ن(
ب -ﺏﺮهﻦ ﻋﻠﻰ أن ﻡﺠﻤﻮﻋﺔ اﻟﻨﻘﻄﺘﻴﻦ ' Mو ' ' Mهﻲ داﺋﺮة ﻳﺠﺐ ﺕﺤﺪﻳﺪهﺎ.
) 2ن(
ج -ﺏﺮهﻦ أﻧﻪ إذا آﺎن " M ' ≠ Mﻓﺈن اﻟﻤﺴﺘﻘﻴﻢ )" (M' Mﻟﻪ اﺕﺠﺎﻩ ﻏﻴﺮ ﻡﺮﺕﺒﻂ ﺏﻘﻴﻢ . θ
) 2ن(
د θ -ﻡﻌﻠﻮم ﺁ ﺳﺘﻨﺘﺞ ﻡﻤﺎ ﺳﺒﻖ ﻃﺮﻳﻘﺔ ﺏﺴﻴﻄﺔ ﻹﻧﺸﺎء Iو ' Mو ". M
) 2ن(
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