Bac Blanc Sujet2 Se2008

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‫اﻟﺠﻤﻬﻮرﻳﺔ اﻟﺠﺰاﺋﺮﻳﺔ اﻟﺪﻳﻤﻘﺮاﻃﻴﺔ اﻟﺸﻌﺒﻴﺔ‬ ‫ﻣﺘﻘﻦ ﻋﻠﻲ ﺟﻌﻔﺮ ‪ -‬ﺗﺎﺟﻨﺎﻧﺖ‬

‫ﻣﺪﻳﺮﻳﺔ اﻟﺘﺮﺑﻴﺔ ﻟﻮﻻﻳﺔ ﻣﻴﻠﺔ‬

‫اﻟﺒﻜﺎﻟﻮرﻳﺎ اﻟﺘﺠﺮﻳﺒﻴﺔ ﻓﻲ ﻣﺎدة اﻟﺮﻳﺎﺿﻴﺎت)))))))))))))))))))))))))))))†‪2008@ðbß@ñŠë‬‬ ‫اﻟﺸﻌﺒﺔ‪ 3 :‬ﻋﻠﻮم ﺗﺠﺮﻳﺒﻴﺔ‬ ‫اﻟﺘﻤﺮﻳﻦ اﻷول‪) :‬‬

‫اﻟﻤﺪة‪3 :‬ﺳﺎﻋﺎت‬

‫ﻨﻘﺎﻁ(‬

‫ﻨﻌﺘﺒﺭ ﻓﻲ ﻤﺠﻤﻭﻋﺔ ﺍﻷﻋﺩﺍﺩ ﺍﻝﻤﺭﻜﺒﺔ ‪ ℂ‬ﺍﻝﻤﻌﺎﺩﻝﺔ )‪:(E‬‬

‫‪z + 4(1 − i ) z ² − 2(1 + 6i ) z − 8 − 4i = 0‬‬ ‫‪3‬‬

‫‪ -1‬ﺒﺭ ﻫﻥ ﺃﻥ ﺍﻝﻤﻌﺎﺩﻝﺔ )‪ (E‬ﺘﻘﺒل ﺤﻼ ﺘﺨﻴﻠﻴﺎ ﺼﺭﻓﺎ ‪ z1‬ﻴﻁﻠﺏ ﺘﻌﻴﻴﻨﻪ‪.‬‬ ‫‪ -2‬ﺤل ﻓﻲ ‪ ℂ‬ﺍﻝﻤﻌﺎﺩﻝﺔ )‪ z2 ) , (E‬ﺍﻝﺤل ﺍﻝﺫﻱ ﻝﻪ ﺃﻜﺒﺭ ﺠﺯﺀ ﺘﺨﻴﻠﻲ‪ z3 ،‬ﺍﻝﺤل ﺍﻵﺨﺭ(‬ ‫‬ ‫‪ C ، B , A -3‬ﺜﻼﺙ ﻨﻘﺎﻁ ﻤﻥ ﺍﻝﻤﺴﺘﻭﻱ ﺍﻝﻤﻨﺴﻭﺏ ﺇﻝﻰ ﻤﻌﻠﻡ ﻤﺘﻌﺎﻤﺩ ﻭ ﻤﺘﺠﺎﻨﺱ ) ‪ (o, i , j‬ﻝﻭﺍﺤﻘﻬﺎ ﻋﻠﻰ ﺍﻝﺘﺭﺘﻴﺏ ‪، z1‬‬

‫‪. z3 ، z2‬‬ ‫ﺃ( ﺃﺜﺒﺕ ﻭﺠﻭﺩ ﺘﺸﺎﺒﻪ ﻤﺒﺎﺸﺭ ﻴﺤﻭل ﺍﻝﻨﻘﻁﺔ ‪ A‬ﺇﻝﻰ ﺍﻝﻨﻘﻁﺔ ‪ B‬ﻭ ﺍﻝﻨﻘﻁﺔ ‪ B‬ﺇﻝﻰ ‪ C‬ﺜﻡ ﻋﻴﻥ ﻋﻨﺎﺼﺭﻩ ﺍﻝﻤﻤﻴﺯﺓ‪.‬‬ ‫ﺏ( ﻋﻴﻥ ﺍﻝﻌﺩﺩﻴﻥ ﺍﻝﺤﻘﻴﻘﻴﻴﻥ ‪ β ، α‬ﺤﺘﻰ ﻴﻜﻭﻥ ﻤﺭﺠﺢ ﺍﻝﻨﻘﺎﻁ ‪ C ، B ، A‬ﺍﻝﻤﺭﻓﻘﺔ ﻋﻠﻰ ﺍﻝﺘﺭﺘﻴﺏ ﺒﺎﻝﻤﻌﺎﻤﻼﺕ ‪β ، α ، 1‬‬ ‫ﻫﻭ ﺍﻝﻨﻘﻁﺔ ‪ H‬ﺫﺍﺕ ﺍﻝﻼﺤﻘﺔ )‪(-2+2i‬‬ ‫اﻟﺘﻤﺮﻳﻦ اﻟﺜﺎﻧﻲ‪) :‬‬

‫ﻨﻘﺎﻁ(‬

‫  ‬ ‫ﻓﻲ ﺍﻝﻔﻀﺎﺀ ﺍﻝﻤﻨﺴﻭﺏ ﺇﻝﻰ ﻤﻌﻠﻡ ﻤﺘﻌﺎﻤﺩ ﻭ ﻤﺘﺠﺎﻨﺱ ) ‪ ( o, i , j , k‬ﻨﻌﺘﺒﺭ ﺴﻁﺢ ﺍﻝﻜﺭﺓ )‪ (S‬ﺍﻝﺘﻲ ﻤﻌﺎﺩﻝﺘﻬﺎ ‪:‬‬

‫‪ x ² + y ² + z ² − 2 x − 4 y − 6 z + 8 = 0‬ﻭ ﺍﻝﻤﺴﺘﻭﻱ )‪ (P‬ﺍﻝﺫﻱ ﻤﻌﺎﺩﻝﺘﻪ ﻫﻲ‪x − y + 2 z + 1 = 0 :‬‬ ‫‪ (1‬ﺒﻴﻥ ﺃﻥ ﻤﺭﻜﺯ ﺍﻝﻜﺭﺓ )‪ (S‬ﻫﻭ ﺍﻝﻨﻘﻁﺔ )‪ Ω(1, 2, 3‬ﻭ ﺃﻥ ﻨﺼﻑ ﻗﻁﺭﻫﺎ ﻴﺴﺎﻭﻱ ‪. 6‬‬ ‫‪ (2‬ﺘﺤﻘﻕ ﻤﻥ ﺃﻥ ﺍﻝﻤﺴﺘﻭﻱ )‪ (P‬ﻤﻤﺎﺱ ﻝﻠﻜﺭﺓ )‪.(S‬‬ ‫‪ (3‬ﺃ‪ .‬ﻋﻴﻥ ﺘﻤﺜﻴﻼ ﻭﺴﻴﻁﻴﺎ ﻝﻠﻤﺴﺘﻘﻴﻡ ) ∆ ( ﺍﻝﻤﺎﺭ ﻤﻥ ‪ Ω‬ﻭ ﺍﻝﻌﻤﻭﺩﻱ ﻋﻠﻰ )‪.(P‬‬ ‫ﺏ‪ .‬ﻋﻴﻥ ﺇﺤﺩﺍﺜﻴﺎﺕ ﺍﻝﻨﻘﻁﺔ ‪ ω‬ﻨﻘﻁﺔ ﺘﻤﺎﺱ )‪ (P‬ﻭ )‪.(S‬‬ ‫اﻟﺘﻤﺮﻳﻦ اﻟﺜﺎﻟﺚ‪) :‬‬

‫ﻨﻘﺎﻁ(‬

‫ﻜﻴﺱ ﺒﻪ ‪ 9‬ﻜﺭﺍﺕ ﻤﻨﻬﺎ ‪ 2‬ﺴﻭﺩﺍﺀ ‪ 3‬ﺒﻴﻀﺎﺀ ‪ 4‬ﺤﻤﺭﺍﺀ ‪.‬‬ ‫ﻨﺴﺤﺏ ﻜﺭﺘﻴﻥ ﻤﻥ ﺍﻝﻜﻴﺱ ﻓﻲ ﺁﻥ ﻭﺍﺤﺩ ﻭﻨﻔﺭﺽ ﺃﻨﻪ ﻋﻨﺩ ﺴﺤﺏ ﻜﺭﺓ ﺴﻭﺩﺍﺀ ﻨﺴﺠل ‪ 3‬ﻨﻘﺎﻁ‪ ,‬ﻭ ﻋﻨﺩ ﺴﺤﺏ ﻜﺭﺓ ﺒﻴﻀﺎﺀ ﻨﺴﺠل‬ ‫ﻨﻘﻁﺔ ﻭﺍﺤﺩﺓ ‪ ,‬ﻭ ﻋﻨﺩ ﺴﺤﺏ ﻜﺭﺓ ﺤﻤﺭﺍﺀ ﻨﻔﻘﺩ ﻨﻘﻁﺘﻴﻥ ‪.‬‬ ‫ﻨﻌﺘﺒﺭ ﺍﻝﻤﺘﻐﻴﺭ ﺍﻝﻌﺸﻭﺍﺌﻲ ‪ X‬ﺍﻝﺫﻱ ﻴﺭﻓﻕ ﺒﻜل ﺴﺤﺏ ﻤﺠﻤﻭﻉ ﺍﻝﻨﻘﺎﻁ ﺍﻝﻤﺴﺠﻠﺔ‪.‬‬ ‫‪ -1‬ﻋﻴﻥ ﻗﻴﻡ ﺍﻝﻤﺘﻐﻴﺭ ﺍﻝﻌﺸﻭﺍﺌﻲ ‪. X‬‬ ‫‪ -2‬ﻤﺎ ﻫﻭ ﻗﺎﻨﻭﻥ ﺍﻻﺤﺘﻤﺎل ﻝﻠﻤﺘﻐﻴﺭ ﺍﻝﻌﺸﻭﺍﺌﻲ ‪ X‬؟‬ ‫‪ -3‬ﺍﺤﺴﺏ ﺍﻷﻤل ﺍﻝﺭﻴﺎﻀﻲ ﻝﻠﻤﺘﻐﻴﺭ ﺍﻝﻌﺸﻭﺍﺌﻲ ‪.X‬‬ ‫اﻟﺘﻤﺮﻳﻦ اﻟﺮاﺑﻊ‪) :‬‬

‫ﻨﻘﺎﻁ(‬

‫‪ g (I‬ﺩﺍﻝﺔ ﻋﺩﺩﻴﺔ ﻤﻌﺭﻓﺔ ﻋﻠﻰ [∞‪ ]0, +‬ﺒﺎﻝﺸﻜل ‪g ( x ) = x − 1 + 2 ln x :‬‬ ‫‪ .1‬ﺃﺩﺭﺱ ﺘﻐﻴﺭﺍﺕ ﺍﻝﺩﺍﻝﺔ ‪ ، g‬ﻭ ﺃﺤﺴﺏ )‪.g(1‬‬ ‫‪ .2‬ﺍﺴﺘﻨﺘﺞ ﺇﺸﺎﺭﺓ )‪ g(x‬ﻋﻠﻰ [∞‪. ]0, +‬‬

‫‪ (II‬ﻝﺘﻜﻥ ‪ f‬ﺍﻝﺩﺍﻝﺔ ﺍﻝﻌﺩﺩﻴﺔ ﺍﻝﻤﻌﺭﻓﺔ ﻋﻠﻰ [∞‪ ]0, +‬ﺒﺎﻝﺸﻜل ‪f ( x ) = x − 2 + (ln x )² − ln x :‬‬

‫)‬

‫ ‬

‫(‬

‫)‪ (CC‬ﺍﻝﻤﻨﺤﻨﻰ ﺍﻝﺒﻴﺎﻨﻲ ﺍﻝﻤﻤﺜل ﻝﻠﺩﺍﻝﺔ ‪ f‬ﻓﻲ ﻤﺴﺘﻭﻱ ﻤﻨﺴﻭﺏ ﺇﻝﻰ ﻤﻌﻠﻡ ﻤﺘﻌﺎﻤﺩ ﻭ ﻤﺘﺠﺎﻨﺱ ‪. o, i , j‬‬

‫) ‪g( x‬‬ ‫‪ (1‬ﺒﻴﻥ ﺃﻨﻪ ﻤﻥ ﺃﺠل ﻜل ﻋﺩﺩ ﺤﻘﻴﻘﻲ ﻤﻭﺠﺏ ﺘﻤﺎﻤﺎ ‪ x‬ﻝﺩﻴﻨﺎ‬ ‫‪x‬‬ ‫‪ (2‬ﺍﺩﺭﺱ ﺘﻐﻴﺭﺍﺕ ﺍﻝﺩﺍﻝﺔ ‪ ، f‬ﻋﻴﻥ ﻨﻬﺎﻴﺘﻲ ﺍﻝﺩﺍﻝﺔ ‪ f‬ﻋﻨﺩ ‪+∞, 0‬‬

‫= ) ‪f '( x‬‬

‫‪1 ‬‬ ‫‪ ‬‬

‫‪ 9‬‬ ‫‪‬‬ ‫‪‬‬

‫‪ (3‬ﺃﺜﺒﺕ ﺃﻥ )‪ (CC‬ﻴﻘﻁﻊ ﻤﺤﻭﺭ ﺍﻝﻔﻭﺍﺼل ﻓﻲ ﻨﻘﻁﺘﻴﻥ ﻓﺎﺼﻠﺘﺎﻫﻤﺎ ‪x1 , x0‬ﺤﻴﺙ ‪ x0 ∈  ,1 :‬ﻭ ‪. x1 ∈  2, ‬‬ ‫‪4‬‬ ‫‪e‬‬ ‫‪ (4‬ﺃﺩﺭﺱ ﻭﻀﻌﻴﺔ ﺍﻝﻤﻨﺤﻨﻲ ﺒﺎﻝﻨﺴﺒﺔ ﺇﻝﻰ ﺍﻝﻤﺴﺘﻘﻴﻡ )∆( ﺍﻝﺫﻱ ﻤﻌﺎﺩﻝﺘﻪ‪y = x :‬‬ ‫‪ (5‬ﺒﺭﻫﻥ ﺃﻨﻪ ﺘﻭﺠﺩ ﻨﻘﻁﺔ ‪ A‬ﻴﻜﻭﻥ ﻋﻨﺩﻫﺎ ﺍﻝﻤﻤﺎﺱ )‪ (D‬ﻝـ)‪ (CC‬ﻤﻭﺍﺯﻴﺎ ﻝـ )∆( ﻭ ﺍﻜﺘﺏ ﻤﻌﺎﺩﻝﺔ ﻝﻪ‪.‬‬ ‫‪ (6‬ﺃﻨﺸﺊ) ‪ (CC‬و )‪.(D‬‬ ‫ﺍﻷﺴﺘﺎﺫ‪ :‬ﻳﻮﺳﻔﻲ ك‬

‫ا ‪2/2‬‬

‫إ ‬

‫‬

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