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كتاب الانشطة مصر - ترم ثانى -2013-2014

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منتدى الرياضيات المتقدمة
منتدى الرياضيات المتقدمة

كتاب الانشطة مصر - ترم ثانى -2013-2014

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äÉ«°VÉjôdG äÉÑjQóàdG h ᣰûfC’G ÜÉàc ≈fÉãdG ≈°SGQódG π°üØdG iƒfÉãdG ∫hC’G ∞°üdG OGóYEGh ¿óªdG §«£îJh iQÉÑμdGh ¥ô£dG AÉ°ûfEG É¡æe IOó©àe ä’Éée ≈a á«∏ªY äÉ≤«Ñ£J äÉ«°VÉjô∏d ∫ƒ£dG ø«H Ö°SÉæJ ≥ah É¡d á©WÉ≤dG äɪ«≤à°ùªdG h äɪ«≤à°ùªdG iRGƒJ ≈∏Y óªà©J ≈àdG É¡£FGôN .º°SôdG ≈a ∫ƒ£dGh ≈≤«≤ëdG ¢ùjƒ°ùdG IÉæb ≈àØ°V ø«H §Hôj iòdG ΩÓ°ùdG iôHƒμd IQƒ°üdGh
‫‪OGóYEG‬‬ ‫‪ˆG ÜÉL OGDƒa ôªY /CG‬‬ ‫‪™Ñ°†dG ≥«aƒJ π«Ñf /O.CG ídÉ°U ìƒàØdG ƒHCG ±ÉØY /O.CG‬‬ ‫‪Qóæμ°SEG ¢SÉ«dEG º«aGÒ°S /CG‬‬ ‫‪π«FÉahQ ≈Ø°Uh ΩÉ°üY /O.Ω.CG‬‬ ‫‪á°ûÑc ¢ùfƒj ∫ɪc /CG‬‬ ‫ﺟﻤﻴﻊ ﺍﻟﺤﻘﻮﻕ ﻣﺤﻔﻮﻇﺔ ﻻ ﻳﺠﻮﺭ ﻧﺸﺮ ﺃ￯ ﺟﺰﺀ ﻣﻦ ﻫﺬﺍ ﺍﻟﻜﺘﺎﺏ ﺃﻭ ﺗﺼﻮﻳﺮﻩ ﺃﻭ ﺗﺨﺰﻳﻨﻪ ﺃﻭ ﺗﺴﺠﻴﻠﻪ‬ ‫ﺑﺄ￯ ﻭﺳﻴﻠﺔ ﺩﻭﻥ ﻣﻮﺍﻓﻘﺔ ﺧﻄﻴﺔ ﻣﻦ ﺍﻟﻨﺎﺷﺮ.‬ ‫ﺷﺮﻛﺔ ﺳﻘﺎرة ﻟﻠﻨﺸﺮ‬ ‫‪Ω .Ω .¢T‬‬ ‫ﺍﻟﻄﺒﻌــﺔ ﺍﻷﻭﻟﻰ ٣١٠٢/٤١٠٢‬ ‫ﺭﻗﻢ ﺍﻹﻳــﺪﺍﻉ ٠٥٩٧ / ٣١٠٢‬ ‫ﺍﻟﺮﻗﻢ ﺍﻟﺪﻭﻟﻰ 5 - 300 - 607 - 779 - 879‬
‫ﺑﻴﺎﻧﺎت اﻟﻄﺎﻟﺐ‬ ‫ﺍﻻﺳـــﻢ:‬ ‫.........................................................................................................................................................................‬ ‫ﺍﻟﻤﺪﺭﺳﺔ:‬ ‫ﺍﻟﻔﺼﻞ:‬ ‫......................................................................................................................................................................‬ ‫............................................................................................................................................................................‬
‫ﺍﻟﻤﻘﺪﻣﺔ‬ ‫بسم الل ّٰه الرحمن الرحيم‬ ‫ﻳﺴﻌﺪﻧﺎ وﻧﺤﻦ ﻧﻘﺪم ﻫﺬا اﻟﻜﺘﺎب أن ﻧﻮﺿﺢ اﻟﻔﻠﺴﻔﺔ اﻟﺘﻰ ﺗﻢ ﻓﻰ ﺿﻮﺋﻬﺎ ﺑﻨﺎء اﻟﻤﺎدة اﻟﺘﻌﻠﻴﻤﻴﺔ وﻧﻮﺟﺰﻫﺎ ﻓﻴﻤﺎ ﻳﻠﻰ:‬ ‫1‬ ‫اﻟﺘﺄﻛﻴﺪ ﻋﲆ أن اﻟﻐﺎﻳﺔ اﻷﺳﺎﺳﻴﺔ ﻣﻦ ﻫﺬه اﻟﻜﺘﺐ ﻫﻰ ﻣﺴﺎﻋﺪة املﺘﻌﻠﻢ ﻋﲆ ﺣﻞ املﺸﻜﻼت واﺗﺨﺎذ اﻟﻘﺮارات ﰱ ﺣﻴﺎﺗﻪ‬ ‫اﻟﻴﻮﻣﻴﺔ، واﻟﺘﻰ ﺗﺴﺎﻋﺪه ﻋﲆ املﺸﺎرﻛﻪ ﰱ املﺠﺘﻤﻊ.‬ ‫2‬ ‫اﻟﺘﺄﻛﻴﺪ ﻋﲆ ﻣﺒﺪأ اﺳﺘﻤﺮارﻳﺔ اﻟﺘﻌﻠﻢ ﻣﺪى اﻟﺤﻴﺎة ﻣﻦ ﺧﻼل اﻟﻌﻤﻞ ﻋﲆ إﻛﺴﺎب اﻟﻄﻼب ﻣﻨﻬﺠﻴﺔ اﻟﺘﻔﻜري اﻟﻌﻠﻤﻰ، وأن‬ ‫ﻳﻤﺎرﺳﻮا اﻟﺘﻌﻠﻢ املﻤﺘﺰج ﺑﺎملﺘﻌﺔ واﻟﺘﺸﻮﻳﻖ، وذﻟﻚ ﺑﺎﻻﻋﺘﻤﺎد ﻋﲆ ﺗﻨﻤﻴﺔ ﻣﻬﺎرات ﺣﻞ املﺸﻜﻼت وﺗﻨﻤﻴﺔ ﻣﻬﺎرات اﻻﺳﺘﻨﺘﺎج‬ ‫واﻟﺘﻌﻠﻴﻞ، واﺳﺘﺨﺪام أﺳﺎﻟﻴﺐ اﻟﺘﻌﻠﻢ اﻟﺬاﺗﻰ واﻟﺘﻌﻠﻢ اﻟﻨﺸﻂ واﻟﺘﻌﻠﻢ اﻟﺘﻌﺎوﻧﻰ ﺑﺮوح اﻟﻔﺮﻳﻖ، واملﻨﺎﻗﺸﺔ واﻟﺤﻮار، وﺗﻘﺒﻞ‬ ‫آراء اﻵﺧﺮﻳﻦ، واملﻮﺿﻮﻋﻴﺔ ﰱ إﺻﺪار اﻷﺣﻜﺎم، ﺑﺎﻹﺿﺎﻓﺔ إﱃ اﻟﺘﻌﺮﻳﻒ ﺑﺒﻌﺾ اﻷﻧﺸﻄﺔ واﻹﻧﺠﺎزات اﻟﻮﻃﻨﻴﺔ.‬ ‫3‬ ‫ﺗﻘﺪﻳﻢ رؤى ﺷﺎﻣﻠﺔ ﻣﺘﻤﺎﺳﻜﺔ ﻟﻠﻌﻼﻗﺔ ﺑني اﻟﻌﻠﻢ واﻟﺘﻜﻨﻮﻟﻮﺟﻴﺎ واملﺠﺘﻤﻊ)‪ (STS‬ﺗﻌﻜﺲ دور اﻟﺘﻘﺪﱡم اﻟﻌﻠﻤﻰ ﰱ ﺗﻨﻤﻴﺔ‬ ‫املﺠﺘﻤﻊ املﺤﲆ، ﺑﺎﻹﺿﺎﻓﺔ إﱃ اﻟﱰﻛﻴﺰ ﻋﲆ ﻣﻤﺎرﺳﺔ اﻟﻄﻼب اﻟﺘﴫﱡف اﻟﻮاﻋﻰ اﻟﻔﻌّﺎل ﺣِ ﻴﺎل اﺳﺘﺨﺪام اﻷدوات اﻟﺘﻜﻨﻮﻟﻮﺟﻴﺔ.‬ ‫4‬ ‫5‬ ‫6‬ ‫ﺗﻨﻤﻴﺔ اﺗﺠﺎﻫﺎت إﻳﺠﺎﺑﻴﺔ ﺗﺠﺎه اﻟﺮﻳﺎﺿﻴﺎت ودراﺳﺘﻬﺎ وﺗﻘﺪﻳﺮ ﻋﻠﻤﺎﺋﻬﺎ.‬ ‫ﺗﺰوﻳﺪ اﻟﻄﻼب ﺑﺜﻘﺎﻓﺔ ﺷﺎﻣﻠﺔ ﻟﺤﺴﻦ اﺳﺘﺨﺪام املﻮارد اﻟﺒﻴﺌﻴﺔ املﺘﺎﺣﺔ.‬ ‫اﻻﻋﺘﻤﺎد ﻋﲆ أﺳﺎﺳﻴﺎت املﻌﺮﻓﺔ وﺗﻨﻤﻴﺔ ﻃﺮاﺋﻖ اﻟﺘﻔﻜري، وﺗﻨﻤﻴﺔ املﻬﺎرات اﻟﻌﻠﻤﻴﺔ، واﻟﺒﻌﺪ ﻋﻦ اﻟﺘﻔﺎﺻﻴﻞ واﻟﺤﺸﻮ،‬ ‫واﻹﺑﺘﻌﺎد ﻋﻦ اﻟﺘﻌﻠﻴﻢ اﻟﺘﻠﻘﻴﻨﻰ؛ ﻟﻬﺬا ﻓﺎﻻﻫﺘﻤﺎم ﻳﻮﺟﻪ إﱃ إﺑﺮاز املﻔﺎﻫﻴﻢ واملﺒﺎدئ اﻟﻌﺎﻣﺔ وأﺳﺎﻟﻴﺐ اﻟﺒﺤﺚ وﺣﻞ املﺸﻜﻼت‬ ‫وﻃﺮاﺋﻖ اﻟﺘﻔﻜري اﻷﺳﺎﺳﻴﺔ اﻟﺘﻰ ﺗﻤﻴﺰ ﻣﺎدة اﻟﺮﻳﺎﺿﻴﺎت ﻋﻦ ﻏريﻫﺎ.‬ ‫‪:≈∏j Ée ÜÉàμdG Gòg ≈a ≈YhQ ≥Ñ°S Ée Aƒ°V ≈ah‬‬ ‫ﺗﻘﺪﻳﻢ ﺗﻤﺎرﻳﻦ ﺗﺒﺪأ ﻣﻦ اﻟﺴﻬﻞ إﱃ اﻟﺼﻌﺐ، وﺗﺸﻤﻞ ﻣﺴﺘﻮﻳﺎت ﺗﻔﻜري ﻣﺘﻨﻮﻋﺔ.‬ ‫ﺗﻨﺘﻬﻰ ﻛﻞ وﺣﺪة ﺑﺘﻤﺎرﻳﻦ ﻋﺎﻣﺔ ﻋﲆ اﻟﻮﺣﺪة واﺧﺘﺒﺎر ﻟﻠﻮﺣﺪة واﺧﺘﺒﺎر ﺗﺮاﻛﻤﻰ ﻳﺸﻤﻞ اﻟﻌﺪﻳﺪ ﻣﻦ اﻷﺳﺌﻠﺔ اﻟﺘﻰ ﺗﻨﻮﻋﺖ‬ ‫َ‬ ‫ﺑني اﻷﺳﺌﻠﺔ املﻮﺿﻮﻋﻴﺔ، واملﻘﺎﻟﻴﺔ وذات اﻹﺟﺎﺑﺎت اﻟﻘﺼرية، وﺗﺘﻨﺎول اﻟﻮﺣﺪات اﻟﺴﺎﺑﻖ دراﺳﺘﻬﺎ وﺷﻤﻞ اﻟﻜﺘﺎب اﺧﺘﺒﺎرات‬ ‫ﻧﻬﺎﻳﺔ ﻛﻞ ﻓﺼﻞ دراﳻ.‬ ‫ﻛﻤﺎ روﻋﻰ اﺳﺘﺨﺪام ﻟﻐﺔ ﻣﻨﺎﺳﺒﺔ ﰱ ﻛﺘﺎﺑﺔ املﺴﺎﺋﻞ اﻟﺮﻳﺎﺿﻴﺔ واﻟﺤﻴﺎﺗﻴﺔ ﻣﻌﺘﻤﺪًا ﻋﲆ ﻣﺎﺳﺒﻖ دراﺳﺘﻪ ﺑﺎﻟﺴﻨﻮات‬ ‫اﻟﺴﺎﺑﻘﺔ، وﰱ ﺿﻮء املﺤﺼﻮل اﻟﻠﻐﻮى ﻟﻄﻼب ﻫﺬا اﻟﺼﻒ.‬ ‫وأخير ًا ..نتمنى أن نكون قد وفقنا فى إنجاز هذا العمل لما فيه خير لأولادنا، ولمصرنا العزيزة.‬ ‫والل ّٰه من وراء القصد، وهو يهدى إلى سواء السبيل‬
‫‪äÉjƒàëªdG‬‬ ‫‪IóMƒdG‬‬ ‫‪≈dhC’G‬‬ ‫1- 1‬ ‫1- 2‬ ‫1- 3‬ ‫1- 4‬ ‫1- 5‬ ‫ﺍﻟﻤﺼﻔﻮﻓﺎﺕ‬ ‫ﺗﻨﻈﻴﻢ اﻟﺒﻴﺎﻧﺎت ﻓﻰ ﻣﺼﻔﻮﻓﺎت‬ ‫ﺟﻤﻊ وﻃﺮح اﻟﻤﺼﻔﻮﻓﺎت‬ ‫ﺿﺮب اﻟﻤﺼﻔﻮﻓﺎت‬ ‫2‬ ‫4‬ ‫5‬ ‫6‬ ‫8‬ ‫01‬ ‫21‬ ‫31‬ ‫..........................................................................................................................................................................................‬ ‫.........................................................................................................................................................................................................‬ ‫.............................................................................................................................................................................................................................‬ ‫اﻟﻤﺤﺪدات‬ ‫.........................................................................................................................................................................................................................................................‬ ‫اﻟﻤﻌﻜﻮس اﻟﻀﺮﺑﻰ ﻟﻠﻤﺼﻔﻮﻓﺔ‬ ‫...........................................................................................................................................................................................‬ ‫ﺗﻤﺎرﻳﻦ ﻋﺎﻣﺔ‬ ‫.........................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر اﻟﻮﺣﺪة‬ ‫......................................................................................................................................................................................................................................‬ ‫اﻻﺧﺘﺒﺎر اﻟﺘﺮاﻛﻤﻰ‬ ‫‪IóMƒdG‬‬ ‫‪á«fÉãdG‬‬ ‫............................................................................................................................................................................................................................‬ ‫ﺍﻟ‪‬ﻣﺠﺔ ﺍﻟﺨﻄﻴﺔ‬ ‫2-1‬ ‫اﻟﻤﺘﺒﺎﻳﻨﺎت اﻟﺨﻄﻴﺔ‬ ‫61‬ ‫2-2‬ ‫ﺣﻞ أﻧﻈﻤﺔ ﻣﻦ اﻟﻤﺘﺒﺎﻳﻨﺎت اﻟﺨﻄﻴﺔ ﺑﻴﺎﻧﻴٍّﺎ‬ ‫81‬ ‫2-3‬ ‫اﻟﺒﺮﻣﺠﺔ اﻟﺨﻄﻴﺔ واﻟﺤﻞ اﻷﻣﺜﻞ‬ ‫ﺗﻤﺎرﻳﻦ ﻋﺎﻣﺔ .........................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر اﻟﻮﺣﺪة ......................................................................................................................................................................................................................................‬ ‫اﻻﺧﺘﺒﺎر اﻟﺘﺮاﻛﻤﻰ ............................................................................................................................................................................................................................‬ ‫..........................................................................................................................................................................................................................‬ ‫.........................................................................................................................................................‬ ‫91‬ ‫12‬ ‫22‬ ‫32‬ ‫........................................................................................................................................................................................‬ ‫‪IóMƒdG‬‬ ‫‪áãdÉãdG‬‬ ‫ﺍﳴﺘﺠﻬﺎﺕ‬ ‫3-1‬ ‫اﻟﻜﻤﻴﺎت اﻟﻘﻴﺎﺳﻴﺔ واﻟﻜﻤﻴﺎت اﻟﻤﺘﺠﻬﺔ، واﻟﻘﻄﻌﺔ اﻟﻤﺴﺘﻘﻴﻤﺔ اﻟﻤﻮﺟﻬﺔ‬ ‫62‬ ‫3-2‬ ‫اﻟﻤﺘﺠﻬﺎت‬ ‫82‬ ‫3-3‬ ‫اﻟﻌﻤﻠﻴﺎت ﻋﻠﻰ اﻟﻤﺘﺠﻬﺎت‬ ‫03‬ ‫3-4‬ ‫ﺗﻄﺒﻴﻘﺎت اﻟﻤﺘﺠﻬﺎت‬ ‫ﺗﻤﺎرﻳﻦ ﻋﺎﻣﺔ .........................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر اﻟﻮﺣﺪة ......................................................................................................................................................................................................................................‬ ‫اﻻﺧﺘﺒﺎر اﻟﺘﺮاﻛﻤﻰ ............................................................................................................................................................................................................................‬ ‫..............................................................‬ ‫....................................................................................................................................................................................................................................................‬ ‫........................................................................................................................................................................................................‬ ‫23‬ ‫53‬ ‫63‬ ‫73‬ ‫.....................................................................................................................................................................................................................‬
‫‪IóMƒdG‬‬ ‫‪á©HGôdG‬‬ ‫ﺍﻟﺨﻂ ﺍﻟﻤﺴﺘﻘﻴﻢ‬ ‫4-1‬ ‫ﺗﻘﺴﻴﻢ ﻗﻄﻌﺔ ﻣﺴﺘﻘﻴﻤﺔ‬ ‫04‬ ‫4-2‬ ‫ﻣﻌﺎدﻟﺔ اﻟﺨﻂ اﻟﻤﺴﺘﻘﻴﻢ‬ ‫14‬ ‫4-3‬ ‫ﻗﻴﺎس اﻟﺰاوﻳﺔ ﺑﻴﻦ ﻣﺴﺘﻘﻴﻤﻴﻦ‬ ‫34‬ ‫4-4‬ ‫ﻃﻮل اﻟﻌﻤﻮد اﻟﻤﺮﺳﻮم ﻣﻦ ﻧﻘﻄﺔ إﻟﻰ ﺧﻂ ﻣﺴﺘﻘﻴﻢ‬ ‫54‬ ‫4-5‬ ‫اﻟﻤﻌﺎدﻟﺔ اﻟﻌﺎﻣﺔ ﻟﻠﻤﺴﺘﻘﻴﻢ اﻟﻤﺎر ﺑﻨﻘﻄﺔ ﺗﻘﺎﻃﻊ ﻣﺴﺘﻘﻴﻤﻴﻦ‬ ‫74‬ ‫84‬ ‫94‬ ‫05‬ ‫.........................................................................................................................................................................................................‬ ‫ﺗﻤﺎرﻳﻦ ﻋﺎﻣﺔ‬ ‫..........................................................................................................................................................................................................‬ ‫......................................................................................................................................................................................‬ ‫.......................................................................................................................‬ ‫.............................................................................................‬ ‫.........................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر اﻟﻮﺣﺪة‬ ‫......................................................................................................................................................................................................................................‬ ‫اﻻﺧﺘﺒﺎر اﻟﺘﺮاﻛﻤﻰ‬ ‫............................................................................................................................................................................................................................‬ ‫‪IóMƒdG‬‬ ‫‪á°ùeÉÿG‬‬ ‫ﺣﺴﺎﺏ ﺍﳴﺜﻠﺜﺎﺕ‬ ‫5-1‬ ‫اﻟﻤﺘﻄﺎﺑﻘﺎت اﻟﻤﺜﻠﺜﻴﺔ.‬ ‫25‬ ‫5-2‬ ‫ﺣﻞ اﻟﻤﻌﺎدﻻت اﻟﻤﺜﻠﺜﻴﺔ.‬ ‫35‬ ‫5-3‬ ‫ﺣﻞ اﻟﻤﺜﻠﺚ اﻟﻘﺎﺋﻢ اﻟﺰاوﻳﺔ.‬ ‫45‬ ‫5-4‬ ‫زواﻳﺎ اﻻرﺗﻔﺎع وزواﻳﺎ اﻻﻧﺨﻔﺎض‬ ‫65‬ ‫5-5‬ ‫اﻟﻘﻄﺎع اﻟﺪاﺋﺮى‬ ‫75‬ ‫5-6‬ ‫اﻟﻘﻄﻌﺔ اﻟﺪاﺋﺮﻳﺔ.‬ ‫85‬ ‫5-7‬ ‫اﻟﻤﺴﺎﺣﺎت.‬ ‫95‬ ‫16‬ ‫26‬ ‫36‬ ‫56‬ ‫17‬ ‫..................................................................................................................................................................................................................‬ ‫............................................................................................................................................................................................................‬ ‫....................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................‬ ‫.....................................................................................................................................................................................................................................‬ ‫................................................................................................................................................................................................................................‬ ‫................................................................................................................................................................................................................................................‬ ‫ﺗﻤﺎرﻳﻦ ﻋﺎﻣﺔ‬ ‫.........................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر اﻟﻮﺣﺪة‬ ‫......................................................................................................................................................................................................................................‬ ‫اﻻﺧﺘﺒﺎر اﻟﺘﺮاﻛﻤﻰ‬ ‫اﺧﺘﺒﺎرات ﻋﺎﻣﺔ‬ ‫............................................................................................................................................................................................................................‬ ‫.........................................................................................................................................................................................................................................................................‬ ‫إﺟﺎﺑﺎت ﺑﻌﺾ اﻟﺘﻤﺎرﻳﻦ‬ ‫..................................................................................................................................................................................................................................................‬
‫ﺍﻟﺠﺒﺮ‬ ‫‪IóMƒdG‬‬ ‫1‬ ‫ﺍﻟﻤﺼﻔﻮﻓﺎﺕ‬ ‫‪Matrices‬‬ ‫دروس اﻟﻮﺣﺪة‬ ‫ﺍﻟﺪﺭﺱ )١ - ١(: ﺗﻨﻈﻴﻢ ﺍﻟﺒﻴﺎﻧﺎﺕ ﻓﻰ ﻣﺼﻔﻮﻓﺎﺕ.‬ ‫ﺍﻟﺪﺭﺱ )١ - ٢(: ﺟﻤﻊ ﻭﻃﺮﺡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ.‬ ‫ﺍﻟﺪﺭﺱ )١ - ٣(: ﺿﺮﺏ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ .‬ ‫ﺍﻟﺪﺭﺱ )١ - ٤(: ﺍﻟﻤﺤﺪﺩﺍﺕ .‬ ‫ﺍﻟﺪﺭﺱ )١ - ٥(: ﺍﻟﻤﻌﻜﻮﺱ ﺍﻟﻀﺮﺑﻰ ﻟﻠﻤﺼﻔﻮﻓﺔ‬
‫ﺗﻨﻈﻴﻢ اﻟﺒﻴﺎﻧﺎت ﻓﻰ ﻣﺼﻔﻮﻓﺎت‬ ‫1-1‬ ‫‪Organizing Data in Matrices‬‬ ‫1 ‪ ،D ، C‬ﺟـ ، ‪ E‬ﺃﺭﺑﻊ ﻣﺪﻥ ، ﻓﺈﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻤﺴﺎﻓﺔ ﺑﺎﻟﻜﻴﻠﻮ ﻣﺘﺮﺍﺕ ﺑﻴﻦ ﺃﻱ ﻣﺪﻳﻨﺘﻴﻦ ﻣﻮﺿﺤﺔ ﻓﻰ ﺍﻟﺠﺪﻭﻝ ﺍﻟﻤﻘﺎﺑﻞ.‬ ‫أ ﺍﻛﺘﺐ ﻣﺼﻔﻮﻓﺔ ﺗﻤﺜﻞ ﻫﺬه ﺍﻟﻤﻌﻠﻮﻣﺎﺕ.‬ ‫‪ C‬ﺏ ﺟـ‬ ‫ب ﺑﻔﺮﺽ ﺃﻥ ‪ M‬ﻫﻰ ﺍﻟﻤﺼﻔﻮﻓﺔ ﺍﻟﻤﻄﻠﻮﺑﺔ ﻓﻰ ) ﺃ ( ﺃﻭﺟﺪ ﻣﺎﻳﻠﻲ:‬ ‫‪٨٠ ٧٥ ٠ C‬‬ ‫١- ﺱ ، ﻣﺎﺫﺍ ﻳﻌﻨﻰ ﺫﻟﻚ? ..................................................................................................‬ ‫٢٣‬ ‫ﺏ ٥٧ ٠ ٦٥‬ ‫٢- ﺱ ، ﻣﺎﺫﺍ ﻳﻌﻨﻰ ﺫﻟﻚ? ...................................................................................................‬ ‫ﺟـ ٠٨ ٦٥ ٠‬ ‫٣٢‬ ‫٣- ﻣﺎ ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺱ ، ﺱ ? ........................................................................................‬ ‫٣٢‬ ‫٢٣‬ ‫ﺟ‬ ‫ﺍﻛﺘﺐ ﺟﻤﻴﻊ ﻋﻨﺎﺻﺮ ﺍﻟﺼﻒ ﺍﻟﺜﺎﻧﻲ ﻟﻠﻤﺼﻔﻮﻓﺔ ‪............................................................................................................ .M‬‬ ‫د ﺍﻛﺘﺐ ﺟﻤﻴﻊ ﻋﻨﺎﺻﺮ ﺍﻟﻌﻤﻮﺩ ﺍﻟﺜﺎﻧﻲ ﻟﻠﻤﺼﻔﻮﻓﺔ ‪ .M‬ﻣﺎﺫﺍ ﺗﺴﺘﻨﺘﺞ ﻣﻦ ﺍﻟﺒﻨﺪﻳﻦ )٤(، )٥(?‬ ‫.......................................................................................................................................................................................................................‬ ‫ﻫ ﺃﻭﺟﺪ ﺱ ﻙ ﻙ ﻋﻨﺪﻣﺎ ﻙ = ١، ٢، ٣ ﻣﺎﺫﺍ ﺗﻼﺣﻆ?‬ ‫.......................................................................................................................‬ ‫و ﺃﻛﻤﻞ ﻣﺎﻳﺄﺗﻰ:‬ ‫١- ‪ M‬ﻣﺼﻔﻮﻓﺔ ﻋﻠﻰ ﺍﻟﻨﻈﻢ .............................................................................................................................................................‬ ‫ﻟﺠﻤﻴﻊ ﻗﻴﻢ .........................................................................................................................................................‬ ‫٢- ﺱﻱ ﻫـ= ﺱ‬ ‫ﻫـ ﻱ‬ ‫2 ﻣﺎ ﻋﺪﺩ ﻋﻨﺎﺻﺮ ﻛﻞ ﻣﻦ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﺍﻵﺗﻴﺔ:‬ ‫أ ﻣﺼﻔﻮﻓﺔ ﻋﻠﻰ ﺍﻟﻨﻈﻢ ٢ * ٣ ................................................................................................................................................................‬ ‫ب‬ ‫ﻣﺼﻔﻮﻓﺔ ﻋﻠﻰ ﺍﻟﻨﻈﻢ ٢ * ٢ ................................................................................................................................................................‬ ‫ﺟ‬ ‫ﻣﺼﻔﻮﻓﺔ ﻋﻠﻰ ﺍﻟﻨﻈﻢ ٣ * ٢ ................................................................................................................................................................‬ ‫3 ﺃﻭﺟﺪ ﻗﻴﻢ ‪ ،C‬ﺏ، ﺟـ، ‪ E‬ﺇﺫﺍ ﻛﺎﻥ:‬ ‫٣‬ ‫٥ ‪٢ - C l = b‬‬‫أ ‪l‬‬ ‫‪٣-C‬‬ ‫ب ‪l‬‬ ‫٣‪٢ - E‬‬ ‫٥١‬ ‫٠‬ ‫٢ﺏ‬ ‫٢‪ + C‬ﺟـ‬ ‫ﺟـ‬ ‫‪l = b‬‬ ‫٣‪C‬‬ ‫٢ﺏ-‪E‬‬ ‫٢ﺏ + ١ ‪b‬‬ ‫٦١‬ ‫٠١‬ ‫٠١‬ ‫‪b‬‬ ‫...................................................................................................................‬ ‫...................................................................................................................‬ ‫4 ﺍﻟﺮﺑﻂ ﺑﺎﻟﺼﻨﺎﻋﺔ: ﻳﺒﻴﻦ ﺍﻟﺠﺪﻭﻝ ﺍﻟﻤﻘﺎﺑﻞ ﻋﺪﺩ ﺍﻟﻤﺼﺎﻧﻊ ﺍﻷﻫﻠﻴﺔ‬ ‫ﺍﻟﻌﺎﻣﻠﺔ ﻓﻰ ﻗﻄﺎﻋﻲ ﺻﻨﺎﻋﺔ ﺍﻷﻏﺬﻳﺔ ﻭﺍﻟﻤﺼﻨﻮﻋﺎﺕ ﺍﻟﺠﻠﺪﻳﺔ ﻓﻰ ﺛﻼﺙ‬ ‫ﻣﺪﻥ ﻣﺨﺘﻠﻔﺔ ﻣﻦ ﻣﺪﻥ ﺑﻌﺾ ﻣﺤﺎﻓﻈﺎﺕ ﺟﻤﻬﻮﺭﻳﺔ ﻣﺼﺮ ﺍﻟﻌﺮﺑﻴﺔ.‬ ‫أ ﻧﻈﻢ ﺍﻟﺒﻴﺎﻧﺎﺕ ﻓﻰ ﻣﺼﻔﻮﻓﺔ.‬ ‫.........................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬ ‫¯‬ ‫¯‬ ‫٤٤‬ ‫٨٢‬ ‫٧٣‬ ‫٨٦‬ ‫٢٥‬ ‫٤١‬
‫ب ﺍﺟﻤﻊ ﻋﻨﺎﺻﺮ ﻛﻞ ﻋﻤﻮﺩ، ﻣﺎ ﺗﻔﺴﻴﺮﻙ ﻟﻠﻨﺘﺎﺋﺞ ﺍﻟﺘﻰ ﺣﺼﻠﺖ ﻋﻠﻴﻬﺎ?‬ ‫......................................................................................................................................................................................................................‬ ‫ﺟ ﺍﺟﻤﻊ ﻋﻨﺎﺻﺮ ﻛﻞ ﺻﻒ. ﻫﻞ ﺍﻟﻨﺘﺎﺋﺞ ﺍﻟﺘﻰ ﺣﺼﻠﺖ ﻋﻠﻴﻬﺎ ﻳﻤﻜﻦ ﺃﻥ ﺗﺰﻭﺩﻧﺎ ﺑﺒﻴﺎﻧﺎﺕ ﺫﺍﺕ ﻣﻌﻨﻰ? ﻓﺴﺮ ﺇﺟﺎﺑﺘﻚ.‬ ‫..............................................................................................................................................................................................................................‬ ‫٤‬ ‫= ٤‬ ‫١‬‫5 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻣﻦ ‪ D ،C‬ﺇﺫﺍ ﻛﺎﻥ ‪l‬‬ ‫‪l b‬‬ ‫٣‬ ‫٢‪٣ ١ - C‬ﺏ + ١‬ ‫١ ‪b‬‬‫٧‬ ‫..................................................................................................................................................................................................................................‬ ‫6 ﺇﺫﺍ ﻛﺎﻥ ‪ ، b ٣- ٢ l = C‬ﺏ = ‪ b ١- E٢ l‬ﺣﻴﺚ ‪ = C‬ﺏ‬ ‫١ ٤‬‫٣ﻫـ ٤‬ ‫ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻣﻦ ‪ ، E‬ﻫـ.‬ ‫ﻣﺪ‬ ‫..................................................................................................................................................................................................................................‬ ‫7 ﺇﺫﺍ ﻛﺎﻧﺖ ‪p = C‬‬ ‫١‬ ‫٢‬ ‫٤‬ ‫٠‬ ‫١‬ ‫٥‬ ‫٢‬ ‫٣ ‪ ، f‬ﺏ = ‪p‬‬‫-١‬ ‫٣‬ ‫٢‬‫-٤‬ ‫٠‬ ‫٣‬ ‫-٥‬ ‫-٢‬ ‫٣ ‪f‬‬ ‫٥‬ ‫ﺃﻭﺟﺪ ‪ + C‬ﺏ  ،  ‪ - C‬ﺏ   ،  ‪٢+ C‬ﺏ  ،   ﺏ -٣‪C‬‬ ‫8 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ: ﺇﺫﺍ ﻛﺎﻧﺖ ‪C) = C‬ﺱ ﺹ( ﻟﻜﻞ ﺱ، ﺹ ∋} ١، ٢، ٣{ ﺍﻛﺘﺐ ﺍﻟﻤﺼﻔﻮﻓﺔ‪ C‬ﺇﺫﺍ ﻋﻠﻢ ﺃﻥ‪C‬ﺱ ﺹ = ﺹ - ﺱ،‬ ‫ﻣﺪ‬ ‫ﺛﻢ ﺃﻭﺟﺪ ‪C‬‬ ‫..................................................................................................................................................................................................‬ ‫ﻧﺸﺎط‬ ‫ﺃﻧﺸﺊ ﻣﺼﻔﻮﻓﺔ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺑﻴﺎﻧﺎﺕ ﺣﻴﺎﺗﻴﺔ ﺗﻜﻮﻥ ﻣﺠﺎﻣﻴﻊ ﻋﻨﺎﺻﺮ ﺃﻋﻤﺪﺗﻬﺎ ﺫﺍﺕ ﻣﻌﻨﻰ، ﻭﻣﺠﺎﻣﻴﻊ ﻋﻨﺎﺻﺮ ﺻﻔﻮﻓﻬﺎ‬ ‫ﻟﻴﺲ ﻟﻬﺎ ﻣﻌﻨﻰ. ﺃﺩﺧﻞ ﺑﻴﺎﻧﺎﺕ ﺍﻟﻤﺼﻔﻮﻓﺔ ﻋﻠﻰ ﺑﺮﻧﺎﻣﺞ ﺍﻟﺠﺪﺍﻭﻝ ﺍﻹﻟﻜﺘﺮﻭﻧﻴﺔ ﻭﺗﺤﻘﻖ ﻣﻦ ﺻﺤﺔ ﺍﻟﻤﺠﺎﻣﻴﻊ ﺍﻟﺘﻰ‬ ‫ﺣﺼﻠﺖ ﻋﻠﻴﻬﺎ، ﺛﻢ ﻓﺴﺮ ﻣﺎﺫﺍ ﺗﻌﻨﻰ ﻣﺠﺎﻣﻴﻊ ﺍﻷﻋﻤﺪﺓ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
‫ﺟﻤﻊ وﻃﺮح اﻟﻤﺼﻔﻮﻓﺎت‬ ‫1-2‬ ‫‪Adding and subtracting Matrices‬‬ ‫1 ﺇﺫﺍ ﻛﺎﻥ ‪ b ١- ٠ ٢- l = C‬ﻛﺎﻧﺖ ﻙ١ = ٢ ، ﻙ٢ = -١ ﻓﺄﻭﺟﺪ ﻛﻼ ﻣﻦ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﺍﻵﺗﻴﺔ: ﻙ١‪ ،C‬ﻙ٢‪C‬‬ ‫ﻭ‬ ‫ًّ‬ ‫٥‬ ‫٤‬ ‫٠‬ ‫..................................................................................................................................................................................................................................‬ ‫٨ ٤‬‫٧ ٠‬‫2 ﺇﺫﺍ ﻛﺎﻥ ‪l = C‬‬ ‫٥ ‪ ، b‬ﺏ = ‪ f ٧ ٠ p‬ﻓﺄﻭﺟﺪ ﻧﺎﺗﺞ ﺍﻟﻌﻤﻠﻴﺎﺕ ﺍﻵﺗﻴﺔ ﺇﻥ ﺃﻣﻜﻦ، ﻣﻊ ﺫﻛﺮ ﺍﻟﺴﺒﺐ ﻓﻰ‬‫٤ ٧ ٥‬ ‫٦ -٥‬‫ﺣﺎﻟﺔ ﺗﻌﺬﺭ ﺇﺟﺮﺍﺀ ﺍﻟﻌﻤﻠﻴﺔ‬ ‫ﻣﺪ‬ ‫ب ‪+C‬ﺏ‬ ‫أ ‪+C‬ﺏ‬ ‫ﺟ ‪ C‬ﻣﺪ + ﺏ‬ ‫-٤ -٢‬ ‫١‬ ‫-٢ -٤‬ ‫٤ -٣‬ ‫٠‬ ‫٥‬ ‫3 ﺇﺫﺍ ﻛﺎﻥ ‪ ، f ٢- ٠ p = N ، f ٦ ٣ p =M‬ﻉ = ‪ f ٢ ٣- p‬ﻓﺄﻭﺟﺪ ﺍﻟﻤﺼﻔﻮﻓﺔ ٣ ‪ + N - M‬ﻉ‬ ‫٠‬ ‫٤‬ ‫٦‬ ‫..................................................................................................................................................................................................................................‬ ‫4 ﺇﺫﺍ ﻛﺎﻥ: ‪p =C‬‬ ‫٢ -٦ ٢‬ ‫٤ ٨ -٦‬ ‫٢ -٤ ٨ ‪ ، f‬ﺏ = ‪٠ ١٠- ٤ p‬‬ ‫١ ٨ -٤‬‫٦ ٢١ ٠‬ ‫‪ f‬ﻓﺄﻭﺟﺪ ﺍﻟﻤﺼﻔﻮﻓﺔ ‪ M‬ﺑﺤﻴﺚ : ‪٣ - C٢ = M‬ﺏ‬ ‫..................................................................................................................................................................................................................................‬ ‫5 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ : ﺃﻭﺟﺪ ﻗﻴﻢ ﺃ، ﺏ، ﺟـ، ‪ E‬ﺍﻟﺘﻰ ﺗﺤﻘﻖ ﺍﻟﻤﻌﺎﺩﻟﺔ:‬ ‫٢ ﺃ ٣ = ٣ ﺃ ‪ ٤ - E‬ﺟـ‬ ‫‪l‬‬ ‫‪b‬‬ ‫‪l‬‬ ‫‪b‬‬ ‫‪l‬‬ ‫ﺟـ -٢‬ ‫٠‬ ‫٦ ﺏ‬ ‫٣‬ ‫ﺃ‬ ‫‪b‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫6 ﻣﺴﺄﻟﺔ ﻣﻔﺘﻮﺣﺔ : ﺍﺧﺘﺮ ﻣﻦ ﻋﻨﺪﻙ ﻣﺼﻔﻮﻓﺘﻴﻦ ‪ ،C‬ﺏ ﻟﻬﻤﺎ ﻧﻔﺲ ﺍﻟﻨﻈﻢ ، ﺛﻢ ﺃﺛﺒﺖ ﺃﻥ :‬ ‫ب )‪ + C‬ﺏ( ﻣﺪ = ‪C‬ﻣﺪ + ﺏ‬ ‫ﻣﺪ‬ ‫أ ‪ - C‬ﺏ = ‪-) + C‬ﺏ(‬ ‫ﺟ )‪ - C‬ﺏ( ﻣﺪ = ‪C‬ﻣﺪ - ﺏ‬ ‫ﻣﺪ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫ﻧﺸﺎط‬ ‫1 ﺍﻛﺘﺐ ﻣﺴﺄﻟﺔ ﺣﻴﺎﺗﻴﺔ ﻣﻦ ﻋﻨﺪﻙ ﻳﻤﻜﻦ ﺣﻠﻬﺎ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺟﻤﻊ ﺃﻭ ﻃﺮﺡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ.‬ ‫2 ﺍﺑﺤﺚ ﻓﻰ ﻣﻜﺘﺒﺘﻚ ﺍﻟﻤﺪﺭﺳﻴﺔ ﺃﻭ ﻋﻠﻰ ﺍﻟﺸﺒﻜﺔ ﺍﻟﺪﻭﻟﻴﺔ ﻟﻺﻧﺘﺮﻧﺖ ﺗﻄﺒﻴﻘﺎﺕ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﻓﻰ ﺍﻟﻌﻠﻮﻡ ﺍﻷﺧﺮﻯ.‬ ‫¯‬ ‫−‬ ‫¯‬
‫ﺿﺮب اﻟﻤﺼﻔﻮﻓﺎت‬ ‫1-3‬ ‫‪Matrix Multiplication‬‬ ‫1 ﺇﺫﺍ ﻛﺎﻥ ‪٣ = C‬‬ ‫‪l‬‬ ‫٦‬ ‫١ ‪ ، b‬ﺏ = ‪ b ٣ ١- l‬ﻓﺄﻭﺟﺪ ﻛﻼ ﻣﻤﺎﻳﺄﺗﻰ:‬ ‫ًّ‬ ‫٢‬ ‫٣‬ ‫٩‬ ‫أ ‪C‬ﺏ‬ ‫..........................................................................................................................................................................‬ ‫ب ﺏ‪C‬‬ ‫..........................................................................................................................................................................‬ ‫ﺟ )‪ + C‬ﺏ(‪C‬‬ ‫..........................................................................................................................................................................‬ ‫2 ﺇﺫﺍ ﻛﺎﻥ ٢‬ ‫‪l‬‬ ‫٤‬ ‫٨ ‪ b‬ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻣﻦ ﺱ، ﺹ:‬ ‫= ٧‬ ‫ﺱ ٧‬ ‫٣‬ ‫‪l b‬‬ ‫‪l b‬‬ ‫١١ ٨١‬ ‫٣ ﺹ‬ ‫٥‬ ‫..................................................................................................................................................................................................................................‬ ‫3 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ: ﺇﺫﺍ ﻛﺎﻥ ‪ ،C‬ﺏ ﻣﺼﻔﻮﻓﺘﻴﻦ، ﻛﺎﻧﺖ‬ ‫ﻭ‬ ‫ﺩﺍﺋﻤﺎ ﺃﻥ ‪= C‬‬ ‫ً‬ ‫ﺃﻭ ﺏ =‬ ‫ﻫﻰ ﺍﻟﻤﺼﻔﻮﻓﺔ ﺍﻟﺼﻔﺮﻳﺔ ، ‪ C‬ﺏ =‬ ‫ﺍﺗﺨﺬ ‪ ، b ٣ ٢- l = C‬ﺏ = ٣‬ ‫‪l‬‬ ‫٢ -٣‬ ‫٢‬ ‫، ﻓﻬﻞ ﻫﺬﺍ ﻳﻌﻨﻰ‬ ‫٦ ‪ b‬ﺛﻢ ﺍﻋﺮﺽ ﻟﺮﺃﻳﻚ ﺑﻌﺪ ﺫﻟﻚ.‬ ‫٤‬ ‫............................................................................................................................................................................................................................................‬ ‫............................................................................................................................................................................................................................................‬ ‫4 ﺍﻟﺮﺑﻂ ﺑﺎﻟﺴﻴﺎﺣﺔ: ﻳﺴﺘﻬﻠﻚ ﺃﺣﺪ ﺍﻟﻔﻨﺎﺩﻕ ﻓﻰ ﻣﺪﻳﻨﺔ ﺍﻟﻐﺮﺩﻗﺔ ﺍﻟﺴﻴﺎﺣﻴﺔ ﺍﻟﻜﻤﻴﺎﺕ ﺍﻵﺗﻴﺔ ﻣﻦ ﺍﻟﻠﺤﻮﻡ‬ ‫ﻭﺍﻟﺨﻀﺮﺍﻭﺍﺕ ﻭﺍﻟﻔﺎﻛﻬﺔ ﺑﺎﻟﻜﻴﻠﻮ ﺟﺮﺍﻡ، ﻓﻰ ﻭﺟﺒﺘﻰ ﺍﻟﻐﺪﺍﺀ ﻭﺍﻟﻌﺸﺎﺀ، ﻭﺫﻟﻚ ﺗﺒﻌﺎ ﻟﻠﺠﺪﻭﻝ ﺍﻟﺘﺎﻟﻰ:‬ ‫ً‬ ‫¯‬ ‫٠٠١‬ ‫٠٨‬ ‫٠٠٢‬ ‫٠٢١‬ ‫٠٥١‬ ‫٠٠١‬ ‫ﻓﺈﺫﺍ ﻛﺎﻥ ﻣﺘﻮﺳﻂ ﺳﻌﺮ ﺍﻟﻜﻴﻠﻮ ﺟﺮﺍﻡ ﻣﻦ ﺍﻟﻠﺤﻮﻡ ٥٦ ﺟﻨﻴﻬﺎ ﻭﻣﺘﻮﺳﻂ ﺳﻌﺮ ﺍﻟﻜﻴﻠﻮ ﺟﺮﺍﻡ ﻣﻦ ﺍﻟﺨﻀﺮﺍﻭﺍﺕ‬ ‫ً‬ ‫ﺃﺭﺑﻌﺔ ﺟﻨﻴﻬﺎﺕ ﻭﻣﺘﻮﺳﻂ ﺳﻌﺮ ﺍﻟﻜﻴﻠﻮ ﺟﺮﺍﻡ ﻣﻦ ﺍﻟﻔﺎﻛﻬﺔ ﻫﻮ ﺧﻤﺴﺔ ﺟﻨﻴﻬﺎﺕ ، ﻓﺄﻭﺟﺪ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺿﺮﺏ‬ ‫ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﺍﻟﺘﻜﺎﻟﻴﻒ ﺍﻟﻜﻠﻴﺔ ﻟﻠﻮﺟﺒﺘﻴﻦ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
‫اﻟﻤﺤﺪدات‬ ‫1-4‬ ‫‪Determinants‬‬ ‫1 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻣﻦ ﺍﻟﻤﺤﺪﺩﺍﺕ ﺍﻵﺗﻴﺔ:‬ ‫٧ ٥‬ ‫ب‬ ‫أ‬ ‫٣‬ ‫٢‬ ‫..............................................‬ ‫د‬ ‫ﺟ‬ ‫..............................................‬ ‫‪C‬‬ ‫‪+C‬ﺱ‬ ‫ﺏ+ﺹ ﺏ‬ ‫ﻫ‬ ‫..............................................‬ ‫ز‬ ‫١ ٢‬ ‫٣ -١‬ ‫ﺱ + ١ ﺱ٢ + ١‬ ‫ﺹ + ١ ﺹ٢ + ١‬ ‫..............................................‬ ‫٠ ٢٤ ٣‬ ‫٢ ٨١ ٧‬ ‫٠ ٨٢ ٣‬ ‫ح‬ ‫..............................................‬ ‫..............................................‬ ‫١ ٢‬ ‫١ ٤‬‫٠ ٧‬ ‫و‬ ‫..............................................‬ ‫٣ -٤ -٣‬ ‫٢ ٠ -١٣‬ ‫٥ ٠ ٢‬ ‫٦ -٣‬ ‫٩١ -٧‬ ‫٣‬ ‫٤‬ ‫٨‬ ‫..............................................‬ ‫٣١ ٣ ٣٢‬ ‫٠٣ ٧ ٥‬ ‫٠ ٠ ١‬ ‫ط‬ ‫..............................................‬ ‫2 ﺣﻞ ﻛﻞ ﻧﻈﺎﻡ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺨﻄﻴﺔ ﺍﻟﺘﺎﻟﻴﺔ ﺑﻄﺮﻳﻘﺔ ﻛﺮﺍﻣﺮ:‬ ‫ب ﺱ+ﺹ=٥‬ ‫أ ٢ﺱ-٣ﺹ=٥‬ ‫٢ ﺱ + ٥ ﺹ = ٦١‬ ‫٣ ﺱ + ٤ ﺹ = -١‬ ‫ﺟ ﺱ+٣ﺹ=٥‬ ‫٢ﺱ+٥ﺹ=٨‬ ‫د ٣ﺱ+٢ﺹ=٥‬ ‫٢ﺱ+ﺹ=٣‬ ‫ﻫ ٣ﺱ=١-٤ﺹ‬ ‫٥ ﺱ + ٢١ = ٧ ﺹ‬ ‫و ٢ﺱ=٣+٧ﺹ‬ ‫ﺹ=٥-ﺱ‬ ‫ز ٢ﺱ +ﺹ - ٢ﻉ =٠١‬ ‫٣ﺱ + ٢ﺹ + ٢ﻉ =١‬ ‫٥ﺱ + ٤ﺹ + ٣ﻉ = ٤‬ ‫ح ﺱ + ٢ﺹ - ٣ﻉ = ٦‬ ‫٢ﺱ-ﺹ-٤ﻉ=٢‬ ‫٤ﺱ + ٣ﺹ - ٢ ﻉ = ٤١‬ ‫ط ﺹ +٢ﺱ +٣ﻉ = ٦‬ ‫٢ﺱ - ﺹ +ﻉ = -٣‬ ‫ﺱ - ٢ﺹ + ٢ﻉ= -١١‬ ‫3 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻬﻨﺪﺳﺔ: ﺃﻭﺟﺪ ﻣﺴﺎﺣﺔ ﺳﻄﺢ ﺍﻟﻤﺜﻠﺚ ‪ C‬ﺏ ﺟـ ﺍﻟﺬﻯ ﻓﻴﻪ ‪ ،(٤ ،٢)C‬ﺏ )-٢، ٤( ﺟـ)٠، -٢(.‬ ‫................‬ ‫4 ﺃﻭﺟﺪ ﻣﺴﺎﺣﺔ ﺳﻄﺢ ﺍﻟﻤﺜﻠﺚ ﺱ ﺹ ﻉ ﺍﻟﺬﻯ ﻓﻴﻪ ﺱ )٣، ٣(، ﺹ )-٤، ٢(، ﻉ ) ١، -٤(.‬ ‫..................................................................................................................................................................................................................................‬ ‫5 ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻤﺤﺪﺩﺍﺕ ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﻨﻘﻂ )٣، ٥(، )٤، -١(، )٥، ٧( ﺗﻘﻊ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
‫ﻧﺸﺎط‬ ‫6 ﻹﻳﺠﺎﺩ ﻗﻴﻤﺔ ﺍﻟﻤﺤﺪﺩ‬ ‫٥ ٧ ٤‬ ‫٦ ٥٢ ١١‬ ‫٧ ٢٥ ١٢‬ ‫¯ ‪¯M‬‬ ‫ﻻﺣﻆ ﺃﻧﻨﺎ ﻛﺘﺒﻨﺎ ﺍﻷﻋﻤﺪﺓ ﺍﻟﺜﻼﺛﺔ ﻟﻠﻤﺤﺪﺩ ﻛﺮﺭﻧﺎ ﺍﻟﻌﻤﻮﺩﻳﻦ‬ ‫ﻭ‬ ‫ﺍﻷﻭﻟﻴﻦ.‬ ‫ﺍﺭﺳﻢ ﺧﻄﻮﻃًﺎ ﻗﻄﺮﻳﺔ ﻋﺒﺮ ﻛﻞ ﺛﻼﺛﺔ ﻋﻨﺎﺻﺮ ﻛﺎﻟﻤﺒﻴﻦ‬ ‫ﺑﺎﻷﺳﻬﻢ ﺍﻟﻤﻨﻘﻄﺔ ﻓﺘﻜﻮﻥ ﺍﻟﺤﺪﻭﺩ ﺍﻟﻨﺎﺗﺠﺔ ﻣﻦ ﻛﻞ ﺧﻂ‬ ‫ﻫﻰ ﺣﺪﻭﺩﺍ ﻓﻰ ﺍﻟﻤﻔﻜﻮﻙ ﻭﺍﻷﺳﻬﻢ ﺍﻟﻤﺘﺠﻬﺔ ﺇﻟﻰ ﺃﺳﻔﻞ‬ ‫ً‬ ‫ﺗﻜﻮﻥ ﺣﺪﻭﺩﻫﺎ ﺍﻟﻤﻨﺎﻇﺮﺓ ﻣﻮﺟﺒﺔ، ﺑﻴﻨﻤﺎ ﺗﻠﻚ ﺍﻟﻤﺘﺠﻬﺔ ﺇﻟﻰ‬ ‫ﺃﻋﻠﻰ ﺗﻜﻮﻥ ﺳﺎﻟﺒﺔ.‬ ‫ﺍﻟﻤﺤﺪﺩ = ٥ * ٥٢ *١٢ + ٧ * ١١ * ٧ + ٤ *٦*٢٥ - ٤ *٥٢ * ٧  - ٥ * ١١ * ٢٥ - ٧ * ٦ *١٢‬ ‫     = ٥٢٦٢ + ٩٣٥ + ٨٤٢١ - ٠٠٧ - ٠٦٨٢ - ٢٨٨‬ ‫      = -٠٣‬ ‫ﺣﺎول أن ﺗﺤﻞ‬ ‫ﺍﺳﺘﺨﺪﻡ ﺍﻟﻄﺮﻳﻘﺔ ﺍﻟﺴﺎﺑﻘﺔ ﻓﻰ ﻓﻚ ﻛﻞ ﻣﺤﺪﺩ ﻣﻤﺎ ﻳﻠﻰ ﻭﺇﻳﺠﺎﺩ ﻗﻴﻤﺘﻪ:‬ ‫أ 9=‬ ‫٣ ٥ ٧‬ ‫١١ ٩ ٣١‬ ‫٥١ ٧١ ٩١‬ ‫ب 9=‬ ‫٣١ ٣ ٣٢‬ ‫٠٣ ٧ ٣٥‬ ‫٩٣ ٩ ٠٧‬ ‫ﺟ 9=‬ ‫١ -٢ ١‬ ‫١ ٢ ٣‬ ‫٦ ٤ ٣‬ ‫د 9=‬ ‫٣ -٤ -٣‬ ‫٢ ٧ -١٣‬ ‫٥ -٩ ٢‬ ‫ﺗﺄﻛﺪ ﻣﻦ ﺻﺤﺔ ﺇﺟﺎﺑﺎﺗﻚ ﺑﺈﻳﺠﺎﺩ ﻗﻴﻤﺔ ﻛﻞ ﻣﺤﺪﺩ ﺑﺎﻟﻄﺮﻳﻘﺔ ﺍﻟﻤﻌﺘﺎﺩﺓ ﻭﻣﻘﺎﺭﻧﺔ ﻧﺘﺎﺋﺠﻚ .‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
‫اﻟﻤﻌﻜﻮس اﻟﻀﺮﺑﻰ ﻟﻠﻤﺼﻔﻮﻓﺔ‬ ‫1-5‬ ‫‪Multiplicative Inverse of a Matrix‬‬ ‫1 ﺑﻴﻦ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﺍﻟﺘﻰ ﻟﻬﺎ ﻣﻌﻜﻮﺳﺎﺕ ﺿﺮﺑﻴﺔ، ﻭﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﺍﻟﺘﻰ ﻟﻴﺲ ﻟﻬﺎ ﻣﻌﻜﻮﺳﺎﺕ ﺿﺮﺑﻴﺔ ﻓﻴﻤﺎ ﻳﻠﻰ، ﻭﺃﻭﺟﺪ‬ ‫ﺍﻟﻤﻌﻜﻮﺱ ﺇﻥ ﻭﺟﺪ.‬ ‫٢ ١‬ ‫أ ‪l‬‬ ‫-١ ١‬ ‫٢ ٦‬ ‫ب ‪l‬‬ ‫-١ ٣‬ ‫‪b‬‬ ‫..............................................‬ ‫٤‬ ‫ﻫ ‪l‬‬ ‫٣‬ ‫٢‬ ‫١‬ ‫١ ٠‬‫ﺟ ‪l‬‬ ‫٣ ٤‬ ‫‪b‬‬ ‫..............................................‬ ‫٢‬ ‫و ‪l‬‬ ‫٠‬ ‫‪b‬‬ ‫..............................................‬ ‫٠‬ ‫١‬ ‫٣‬ ‫ز ‪l‬‬ ‫٢‬ ‫‪b‬‬ ‫٦‬ ‫..............................................‬ ‫٢‬ ‫3 ﺇﺫﺍ ﻛﺎﻧﺖ ‪l = M‬‬ ‫٠‬ ‫..............................................‬ ‫٠‬ ‫٢‬ ‫‪b‬‬ ‫٤‬ ‫٢ ‪٢-C‬‬ ‫‪C‬‬ ‫‪ b‬ﻓﺄﺛﺒﺖ ﺃﻥ ‪l = ١- M‬‬ ‫٩‬ ‫٦‬ ‫‪b‬‬ ‫..............................................‬ ‫١‬ ‫٢‬ ‫٠‬ ‫١‬ ‫٢‬ ‫١‬ ‫..............................................‬ ‫٢‬ ‫ح ‪l‬‬ ‫٥‬ ‫..............................................‬ ‫2 ﻣﺎ ﻗﻴﻢ ‪ C‬ﺍﻟﺘﻰ ﺗﺠﻌﻞ ﻟﻜﻞ ﻣﻦ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ ﻣﻌﻜﻮﺳﺎ ﺿﺮﺑﻴﺎ‬ ‫ً‬ ‫ًّ‬ ‫‪٩ C‬‬ ‫‪١ C‬‬ ‫ﺟ ‪C l‬‬ ‫‪b‬‬ ‫‪b‬‬ ‫ب ‪l‬‬ ‫أ ‪l‬‬ ‫٣‬ ‫‪b‬‬ ‫..............................................‬ ‫..............................................‬ ‫٤‬ ‫٢‬ ‫د ‪l‬‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫‪b‬‬ ‫٣‬ ‫٦‬ ‫‪b‬‬ ‫..............................................‬ ‫د ‪b ٢- ١-C l‬‬ ‫١ ‪٢-C‬‬ ‫..............................................‬ ‫‪b‬‬ ‫..................................................................................................................................................................................................................................‬ ‫٢ ٠‬‫١‬ ‫‪l‬‬ ‫‪l=b‬‬ ‫4 ﺃﻭﺟﺪ ﺍﻟﻤﺼﻔﻮﻓﺔ‪ C‬ﺇﺫﺍ ﻛﺎﻥ: ‪١ ٣ C‬‬ ‫٠‬ ‫٠‬ ‫١‬ ‫‪b‬‬ ‫..................................................................................................................................................................................................................................‬ ‫١‬ ‫٢‬ ‫١ ‪b‬‬‫٣ ‪b‬‬‫‪l‬‬ ‫‪l‬‬ ‫5 ﺇﺫﺍ ﻛﺎﻧﺖ ‪١- = N ، ١ ٠ = M‬‬ ‫٣‬ ‫ﺃﺛﺒﺖ ﺃﻥ = )‪M١-N = ١- (NM‬‬ ‫-١‬ ‫..................................................................................................................................................................................................................................‬ ‫6 ﺣﻞ ﻛﻞ ﻧﻈﺎﻡ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺨﻄﻴﺔ ﺍﻟﺘﺎﻟﻴﺔ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ، ﺛﻢ ﺗﺤﻘﻖ ﻣﻦ ﺻﺤﺔ ﺍﻟﻨﺎﺗﺞ:‬ ‫ب ٢ﺱ-٧ﺹ=٣،ﺱ-٣ﺹ=٢‬ ‫أ ٤ ﺱ + ٣ ﺹ = ٦٢ ، ٥ ﺱ - ﺹ = ٤‬ ‫......................................................................................................................‬ ‫......................................................................................................................‬ ‫......................................................................................................................‬ ‫......................................................................................................................‬ ‫ﺟ ٢ ﺱ = ٣ + ٧ ﺹ، ﺹ = ٥ - ﺱ‬ ‫د ٢ ﺹ = ٥ - ٣ﺱ ، ٢ ﺱ = ٣ - ﺹ‬ ‫......................................................................................................................‬ ‫......................................................................................................................‬ ‫......................................................................................................................‬ ‫......................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
‫¯‬ ‫7 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻬﻨﺪﺳﺔ: ﺍﻟﺨﻂ ﺍﻟﻤﺴﺘﻘﻴﻢ ﺍﻟﺬﻯ ﻣﻌﺎﺩﻟﺘﻪ ﺹ + ‪ C‬ﺱ = ﺣـ ﻳﻤﺮ ﺑﺎﻟﻨﻘﻄﺘﻴﻦ )١، ٥(، )٣، ١(، ﺍﺳﺘﺨﺪﻡ‬ ‫ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﻹﻳﺠﺎﺩ ﻗﻴﻤﺔ ﺍﻟﺜﺎﺑﺘﻴﻦ ‪ ،C‬ﺣـ .‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫8 ﺍﻟﺮﺑﻂ ﺑﺎﻟﺤﻴﺎﺓ: ﻳﺸﺘﺮﻯ ﺳﺎﺋﻖ ﺩﺭﺍﺟﺔ ﺑﺨﺎﺭﻳﺔ ٤٢ ﻟﺘﺮﺍ ﻣﻦ ﺍﻟﺒﻨﺰﻳﻦ ﻭ ٥ ﻟﺘﺮﺍﺕ ﻣﻦ ﺍﻟﺰﻳﺖ ﺑﻤﺒﻠﻎ ٦٥ ﺟﻨﻴﻬﺎ‬ ‫ً‬ ‫ً‬ ‫ﻟﺘﻤﻮﻳﻦ ﺩﺭﺍﺟﺘﻪ، ﺑﻴﻨﻤﺎ ﻳﺸﺘﺮﻯ ﺳﺎﺋﻖ ﺩﺭﺍﺟﺔ ﺑﺨﺎﺭﻳﺔ ﺃﺧﺮﻯ ٨١ ﻟﺘﺮﺍ ﻣﻦ ﺍﻟﺒﻨﺰﻳﻦ، ٠١ ﻟﺘﺮﺍﺕ ﻣﻦ ﺍﻟﺰﻳﺖ ﺑﻤﺒﻠﻎ‬ ‫ً‬ ‫٧٦ ﺟﻨﻴﻬﺎ ﻟﺘﻤﻮﻳﻦ ﺩﺭﺍﺟﺘﻪ، ﺍﺳﺘﺨﺪﻡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﻓﻰ ﺇﻳﺠﺎﺩ ﺛﻤﻦ ﻛﻞ ﻣﻦ ﻟﺘﺮ ﺍﻟﺒﻨﺰﻳﻦ ﻭﻟﺘﺮ ﺍﻟﺰﻳﺖ، ﺇﺫﺍ ﻋﻠﻤﺖ‬ ‫ً‬ ‫ﺃﻧﻬﻤﺎ ﻳﺴﺘﺨﺪﻣﺎﻥ ﻧﻔﺲ ﺍﻟﻨﻮﻋﻴﺔ ﻣﻦ ﺍﻟﺒﻨﺰﻳﻦ ﻭﺍﻟﺰﻳﺖ.‬ ‫.......................................................................................................................................................................................................................‬ ‫.......................................................................................................................................................................................................................‬ ‫9 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻬﻨﺪﺳﺔ: ﻳﻤﺮ ﺍﻟﻤﻨﺤﻨﻰ ﺹ = ‪C‬ﺱ٢ + ﺏ ﺱ ﺑﺎﻟﻨﻘﻄﺘﻴﻦ )٢، ٠( ، )٤، ٨(، ﺍﺳﺘﺨﺪﻡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﻹﻳﺠﺎﺩ‬ ‫ﺍﻟﺜﺎﺑﺘﻴﻦ ‪ ،C‬ﺏ .‬ ‫.......................................................................................................................................................................................................................‬ ‫.......................................................................................................................................................................................................................‬ ‫01 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ: ﻧﺼﻒ ﺍﻟﻔﺮﻕ ﺑﻴﻦ ﻋﺪﺩﻳﻦ ﻫﻮ ٢ ﻭﻣﺠﻤﻮﻉ ﺍﻟﻌﺪﺩ ﺍﻷﻛﺒﺮ ﻭﺿﻌﻒ ﺍﻟﻌﺪﺩ ﺍﻷﺻﻐﺮ ﻫﻮ ٣١. ﺑﺎﺳﺘﺨﺪﺍﻡ‬ ‫ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﺃﻭﺟﺪ ﺍﻟﻌﺪﺩﻳﻦ.‬ ‫.......................................................................................................................................................................................................................‬ ‫.......................................................................................................................................................................................................................‬ ‫.......................................................................................................................................................................................................................‬ ‫ﻧﺸﺎط‬ ‫ﺍﻛﺘﺐ ﻣﺴﺄﻟﺔ ﻣﻦ ﻋﻨﺪﻙ ﻳﺤﺘﺎﺝ ﺣﻠﻬﺎ ﺇﻟﻰ ﺗﻜﻮﻳﻦ ﻧﻈﺎﻡ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺨﻄﻴﺔ ، ﺛﻢ ﺣﻠﻬﺎ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ.‬ ‫.............................................................................................................................................................................................................................................‬ ‫.............................................................................................................................................................................................................................................‬ ‫.............................................................................................................................................................................................................................................‬ ‫.............................................................................................................................................................................................................................................‬ ‫.............................................................................................................................................................................................................................................‬ ‫.............................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
‫ﲤﺎﺭﻳﻦ ﻋﺎﻣﺔ‬ ‫٥ ٢‬ ‫١ ‪b‬‬‫1 ﺇﺫﺍ ﻛﺎﻧﺖ ‪l = C‬‬ ‫٤ ٠ ١‬ ‫أ ﻣﺎ ﻧﻈﻢ ﺍﻟﻤﺼﻔﻮﻓﺔ ‪......................................................................................................................................................................... ?C‬‬ ‫ب ﺍﻛﺘﺐ ﻋﻨﺎﺻﺮ ﺍﻟﺼﻒ ﺍﻷﻭﻝ ﻟﻠﻤﺼﻔﻮﻓﺔ ‪.C‬‬ ‫ﺟ‬ ‫ﺍﻛﺘﺐ ﻋﻨﺎﺻﺮ ﺍﻟﻌﻤﻮﺩ ﺍﻟﺜﺎﻟﺚ ﻓﻰ ﺍﻟﻤﺼﻔﻮﻓﺔ ‪....................................................................................................................... .C‬‬ ‫د ﺍﻛﺘﺐ ﺍﻟﻌﻨﺎﺻﺮ: ‪..................................................................................................................................................... C ، C ، C ، C‬‬ ‫١١ ٢٢ ١٣ ٢١‬ ‫.................................................................................................................................‬ ‫2 ﻋﺒﺮ ﻋﻦ ﺇﺣﺪﺍﺛﻴﺎﺕ ﺍﻟﻨﻘﻂ ﺍﻟﺘﺎﻟﻴﺔ ﺑﻤﺼﻔﻮﻓﺔ ﻣﻦ ﺍﻟﻨﻈﻢ ١ * ٢‬ ‫‪ ، (٣ ،٢) C‬ﺏ )-١، ٠(، ﺟـ)-٢، -٣(‬ ‫..................................................................................................................................................................................................................................‬ ‫3 ﻣﺎ ﻋﺪﺩ ﻋﻨﺎﺻﺮ ﻛﻼ ﻣﺼﻔﻮﻓﺔ ﻣﻤﺎﻳﻠﻰ?‬ ‫ًّ‬ ‫أ ﻣﺼﻔﻮﻓﺔ ﻣﻦ ﺍﻟﻨﻈﻢ ٣ * ٢‬ ‫.................................................................................................................‬ ‫ب ﻣﺼﻔﻮﻓﺔ ﻣﺮﺑﻌﺔ ﻣﻦ ﺍﻟﻨﻈﻢ ٢ * ٢.‬ ‫.................................................................................................................‬ ‫4 ﺣﻞ ﻛﻞ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ:‬ ‫٩‬ ‫‪ l‬ﺱ +٥‬ ‫‪l=b‬‬ ‫أ‬ ‫٤‬ ‫ﺱ-ﺹ‬ ‫‪b‬‬ ‫ب ‪٣ - C‬ﺏ ﺏ‬ ‫٠‬ ‫‪l‬‬ ‫٢ ‪l=b‬‬ ‫٤‬ ‫ﺏ +‪C‬‬ ‫.................................................................................................................‬ ‫١‬ ‫٢‬ ‫‪b‬‬ ‫.................................................................................................................‬ ‫5 ﺍﻟﺮﺑﻂ ﺑﺎﻟﺼﻨﺎﻋﺔ: ﻳﻨﺘﺞ ﻣﺼﻨﻊ ﻟﺸﺎﺷﺎﺕ ﺍﻟﺘﻠﻴﻔﺰﻳﻮﻥ ﺛﻼﺛﺔ ﺃﻧﻮﺍﻉ ٢٣ ﺑﻮﺻﺔ ، ٢٤ ﺑﻮﺻﺔ ، ٨٤ ﺑﻮﺻﺔ، ﻭﻟﻠﻤﺼﻨﻊ‬ ‫ﻓﺮﻋﺎﻥ ﺃ، ﺏ، ﻛﺎﻥ ﻋﺪﺩ ﺍﻟﺸﺎﺷﺎﺕ ﺍﻟﺘﻰ ﺃﻧﺘﺠﻬﺎ ﻛﻞ ﻓﺮﻉ ﻣﻦ ﻛﻞ ﻧﻮﻉ ﺧﻼﻝ ﺷﻬﺮﻯ ﻳﻨﺎﻳﺮ ﻭﻓﺒﺮﺍﻳﺮ ﻋﺎﻡ ٣١٠٢‬ ‫ﻭ‬ ‫ﻛﻤﺎ ﻳﻮﺿﺢ ﺫﻟﻚ ﻓﻰ ﺍﻟﺠﺪﻭﻟﻴﻦ ﺍﻟﺘﺎﻟﻴﻴﻦ ، ﻋﺒﺮ ﻋﻦ ﺇﻧﺘﺎﺝ ﺍﻟﺸﻬﺮﻳﻦ ﻣﻌﺎ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ.‬ ‫ً‬ ‫٢٣ﺑﻮﺻﺔ ٢٤ ﺑﻮﺻﺔ ٨٤ ﺑﻮﺻﺔ‬ ‫ﺃ‬ ‫٠٠٦‬ ‫٠٠٧‬ ‫٠٥٨‬ ‫‪b‬‬ ‫‪l‬‬ ‫ﺏ‬ ‫٠٥٥‬ ‫٠٠٦‬ ‫٠٥٧‬ ‫ﺷﻬﺮ ﻳﻨﺎﻳﺮ ٣١٠٢‬ ‫٢٣ﺑﻮﺻﺔ ٢٤ ﺑﻮﺻﺔ ٨٤ ﺑﻮﺻﺔ‬ ‫ﺃ‬ ‫٠٥٥‬ ‫٠٥٦‬ ‫٠٠٨‬ ‫‪b‬‬ ‫‪l‬‬ ‫ﺏ‬ ‫٠٠٦‬ ‫٠٠٧‬ ‫٠٤٨‬ ‫ﺷﻬﺮ ﻳﻨﺎﻳﺮ ٣١٠٢‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫٢ -١‬ ‫١ -٢‬ ‫6 ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ: ‪. b ٥ ٠ l = b ٤- ٣ l + C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
‫ﲤﺎﺭﻳﻦ ﻋﺎﻣﺔ‬ ‫7 ﺃﻭﺟﺪ ﺃ، ﺏ، ﺟـ، ﺩ ﺣﻴﺚ‬ ‫٢ ١‬‫‪b‬‬ ‫أ ‪ l b ٢- ٣- l‬ﺃ ﺏ ‪l b‬‬ ‫١ ٥ + ﺟـ ﺩ = ٤ -٣‬ ‫ﺃ‬ ‫١‬ ‫ب‬ ‫‪ p -f ٠ p‬ﺏ ‪p = f‬‬ ‫٠‬ ‫١‬‫١‬ ‫.................................................................................................................‬ ‫‪f‬‬ ‫.................................................................................................................‬ ‫ﺟـ‬ ‫١‬‫١ ٢ ٣‬ ‫١ ٢ ٣‬ ‫8 ﺇﺫﺍ ﻛﺎﻥ: ‪= N ، f ٢- ٤ ٦ p = M‬‬ ‫‪١- ٣- ٢- p‬‬ ‫٠ ٢ ١‬ ‫٠ -١ -٦‬ ‫أ ٢‪I٢-N٣+M‬‬ ‫.................................................................................................................‬ ‫ب ‪( I ٥ - N) - M‬‬ ‫٣‬ ‫9 ﺃﻭﺟﺪ ﺱ، ﺹ ﺣﻴﺚ: ‪l‬‬ ‫١‬ ‫‪ f‬ﻓﺄﻭﺟﺪ :‬ ‫.................................................................................................................‬ ‫٠‬ ‫ﺱ‬ ‫٢‬ ‫‪l= b‬‬ ‫‪lb‬‬ ‫١‬ ‫ﺹ‬ ‫٤‬ ‫‪b‬‬ ‫01 ﺑﻴﻦ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﻟﻜﻞ ﻣﻦ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ ﻣﻌﻜﻮﺱ ﺿﺮﺑﻰ ﺛﻢ ﺃﻭﺟﺪه‬ ‫١‬ ‫أ ‪l‬‬ ‫٣‬ ‫٢‬ ‫٤‬ ‫‪b‬‬ ‫..........................................‬ ‫٠‬ ‫ب ‪l‬‬ ‫٣‬ ‫٢‬ ‫٠‬ ‫ﺃ‬ ‫ﺏ ‪b‬‬‫ﺟ ‪l‬‬ ‫ﺃ ﺏ‬ ‫‪b‬‬ ‫..........................................‬ ‫..........................................‬ ‫11 ﺣﻞ ﻛﻞ ﻧﻈﺎﻡ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺨﻄﻴﺔ ﺍﻟﺘﺎﻟﻴﺔ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ‬ ‫ب ٢ ﺱ - ٣ ﺹ -٣ = ٠‬ ‫أ ﺱ+ﺹ=٣‬ ‫٥ ﺱ + ٤ ﺹ - ٩١ = ٠‬ ‫ﺱ-ﺹ=٥‬ ‫......................................................‬ ‫......................................................‬ ‫٣‬ ‫د ‪l‬‬ ‫٢‬ ‫٩‬ ‫٦‬ ‫‪b‬‬ ‫..........................................‬ ‫ﺟ ﺹ = ١١ - ٥ﺱ‬ ‫ﺱ=٣-٥ﺹ‬ ‫......................................................‬ ‫١ -٢‬ ‫٤ ٣‬ ‫21 ﺇﺫﺍ ﻛﺎﻥ )‪C‬ﺏ(-١= ١ ‪ b ٣ ١- l‬ﻛﺎﻥ ‪ b ٢ ١ l = C‬ﻓﺄﻭﺟﺪ ﺏ - ١.‬ ‫ﻭ‬ ‫٥‬ ‫..................................................................................................................................................................................................................................‬ ‫31 ﺣﻞ ﻧﻈﺎﻡ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ ﺑﺎﺳﺘﺨﺪﺍﻡ ﻃﺮﻳﻘﺔ ﻛﺮﺍﻣﺮ: ٦١ ﺱ + ٠٢ ﺹ = ٩٦ ، ٢١ ﺱ + ٤ ﺹ =٧٢.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ‬ ‫1 ﺇﺫﺍ ﻛﺎﻥ ‪ ، b ١ ٣ l = C‬ﺏ = ‪ ، b ٢ ١- l‬ﺝ = ‪ b ١- ٢ l‬ﺃﻭﺟﺪ:‬ ‫-٢ ٥‬ ‫٦ -٣‬ ‫-٣ ٤‬ ‫ب ﻛﻼ ﻣﻦ ﺍﻟﻤﺼﻔﻮﻓﺘﻴﻦ: ‪ C‬ﺏ ، ﺏ ‪C‬‬ ‫ًّ‬ ‫أ ‪٢+C‬ﺏ-٣ﺝ‬ ‫ﺟ ﺗﺤﻘﻖ ﺃﻥ: ‪) C‬ﺏ + ﺝ( = ‪ C‬ﺏ + ‪ C‬ﺝ‬ ‫ﻣﺪ ﻣﺪ‬ ‫د ﺗﺤﻘﻖ ﺍﻥ )‪C‬ﺏ(ﻣﺪ = ﺏ ‪C‬‬ ‫2 ﺗﺒﻴﻊ ﻣﻜﺘﺒﺔ ٣ ﻣﺠﻤﻮﻋﺎﺕ ﻣﻦ ﺍﻷﺳﻄﻮﺍﻧﺎﺕ ﺍﻟﻤﺪﻣﺠﺔ ‪،CD's‬‬ ‫ﻭﻳﺒﻴﻦ ﺍﻟﺠﺪﻭﻝ ﺍﻟﺘﺎﻟﻰ ﺗﻜﻠﻔﺔ ﻛﻞ ﻣﺠﻤﻮﻋﺔ ﻭﺳﻌﺮ ﺑﻴﻌﻬﺎ،‬ ‫‪ï‬‬ ‫‪M‬‬ ‫٠٠١‬ ‫٠٨‬ ‫ﻓﺈﺫﺍ ﺑﺎﻋﺖ ﺍﻟﻤﻜﺘﺒﺔ ٠٤ ﻣﺠﻤﻮﻋﺔ ﻣﻦ ﺍﻷﺳﻄﻮﺍﻧﺎﺕ ﺍﻟﻤﺪﻣﺠﺔ‬ ‫٥٨‬ ‫٥٦‬ ‫ﻟﻤﻮﺿﻮﻋﺎﺕ ﻋﻠﻤﻴﺔ، ٤٦ ﻣﺠﻤﻮﻋﺔ ﻣﻦ ﺍﻷﺳﻄﻮﺍﻧﺎﺕ ﺍﻟﻤﺪﻣﺠﺔ‬ ‫٠١١‬ ‫٠٩‬ ‫ﻟﻤﻮﺿﻮﻋﺎﺕ ﺛﻘﺎﻓﻴﺔ، ٥٤ ﻣﺠﻤﻮﻋﺔ ﻣﻦ ﺃﺳﻄﻮﺍﻧﺎﺕ ﻣﺪﻣﺠﺔ‬ ‫ﻟﻘﺼﺺ ﻋﺎﻟﻤﻴﺔ.‬ ‫أ ﻧﻈﻢ ﺍﻟﺒﻴﺎﻧﺎﺕ ﻓﻰ ﻣﺼﻔﻮﻓﺘﻴﻦ، ﺛﻢ ﺍﺳﺘﺨﺪﻡ ﺿﺮﺏ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﻹﻳﺠﺎﺩ ﺍﻟﺘﻜﻠﻔﺔ ﺍﻟﻜﻠﻴﺔ ﻟﻸﺳﻄﻮﺍﻧﺎﺕ ﺍﻟﻤﺪﻣﺠﺔ.‬ ‫ب ﺍﺳﺘﺨﺪﻡ ﺿﺮﺏ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﻹﻳﺠﺎﺩ ﺍﻟﻤﺒﻠﻎ ﺍﻟﻜﻠﻰ ﺍﻟﺬﻯ ﺳﺘﺤﺼﻞ ﻋﻠﻴﻪ ﺍﻟﻤﻜﺘﺒﺔ ﻣﻦ ﺑﻴﻊ ﺍﻟﻤﺠﻤﻮﻋﺎﺕ‬ ‫¯‬ ‫ﺍﻟﻤﺸﺎﺭ ﺇﻟﻴﻬﺎ ﻣﻦ ﺍﻷﺳﻄﻮﺍﻧﺎﺕ ﺍﻟﻤﺪﻣﺠﺔ.‬ ‫ﺟ ﺍﺳﺘﺨﺪﻡ ﺍﻟﻌﻤﻠﻴﺎﺕ ﻋﻠﻰ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﻹﻳﺠﺎﺩ ﺭﺑﺢ ﺍﻟﻤﻜﺘﺒﺔ ﻣﻦ ﺑﻴﻊ ﻣﺠﻤﻮﻋﺎﺕ ﺍﻷﺳﻄﻮﺍﻧﺎﺕ ﺍﻟﻤﺸﺎﺭ ﺇﻟﻴﻬﺎ.‬ ‫3 ﺍﺳﺘﺨﺪﻡ ﻃﺮﻳﻘﺔ ﻛﺮﺍﻣﺮ ﻟﺤﻞ ﻛﻞ ﻧﻈﺎﻡ ﻣﻤﺎﻳﻠﻰ:‬ ‫ب ٣ﺱ-ﺹ+٥=٠‬ ‫أ ﺱ+ﺹ=٤‬ ‫ﺱ + ٢ﺹ +١ = ٠‬ ‫٢ﺱ - ﺹ = ١‬ ‫ﺟ ٢ ﺱ + ﺹ - ﻉ   = ٣‬ ‫ ٣ﺱ  + ٢ ﺹ + ﻉ = ٤‬‫٤ ﺱ + ٢ ﺹ - ﻉ   = ٨‬ ‫4 ﺑﻴﻦ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﻟﻜﻞ ﻣﺼﻔﻮﻓﺔ ﻣﻤﺎﻳﻠﻰ ﻣﻌﻜﻮﺱ ﺿﺮﺑﻰ ﺃﻡ ﻻ، ﻭﻓﻲ ﺣﺎﻟﺔ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﻟﻬﺎ ﻣﻌﻜﻮﺱ ﺿﺮﺑﻲ ﺃﻭﺟﺪه:‬ ‫٢‬ ‫١ ‪b‬‬‫أ ‪l‬‬ ‫٣ ٢‬ ‫٣‬ ‫ب ‪l‬‬ ‫٤‬ ‫٥‬ ‫٦‬ ‫‪b‬‬ ‫5 ﺣﻞ ﻛﻞ ﻧﻈﺎﻡ ﻣﻤﺎﻳﻠﻰ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ:‬ ‫ب ٣ﺱ-ﺹ=٠‬ ‫أ ﺱ + ٢ﺹ = ٨‬ ‫٥ ﺱ + ٢ ﺹ = ٢٢‬ ‫٢ ﺱ - ﺹ = -٩‬ ‫١‬ ‫ﺟ ‪l‬‬ ‫٢‬ ‫٢‬ ‫١‬ ‫‪b‬‬ ‫١ ١‬ ‫د ‪l‬‬ ‫-١ -١‬ ‫‪b‬‬ ‫ﺟ ٤ ﺱ = - ٦ﺹ‬ ‫٨ﺱ-٧=٢ﺹ‬ ‫6 أ ﺣﻞ ﻧﻈﺎﻡ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ ﺑﺎﺳﺘﺨﺪﺍﻡ ﻃﺮﻳﻘﺔ ﻛﺮﺍﻣﺮ.‬ ‫٣ﺹ = ١ + ﻉ - ﺱ ، ٢ ﺱ = -٢ﺹ + ﻉ ، ٣ ﺱ + ﺹ + ٢ ﻉ = -١‬ ‫ب ﻣﻊ ﺧﺎﻟﺪ ٥٢ ﻗﻄﻌﺔ ﻧﻘﺪﻳﺔ ﻣﻦ ﻓﺌﺔ ﺃﺭﺑﺎﻉ ﻭﺍﻧﺼﺎﻑ ﺍﻟﺠﻨﻴﻪ، ﻛﺎﻥ ﻗﻴﻤﺔ ﻣﺎ ﻣﻌﻪ ٥٫٨ ﻣﻦ ﺍﻟﺠﻨﻴﻪ، ﺍﺳﺘﺨﺪﻡ‬ ‫ﻭ‬ ‫ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﻓﻰ ﺇﻳﺠﺎﺩ ﻋﺪﺩ ﺍﻷﺭﺑﺎﻉ ﻭﺍﻷﻧﺼﺎﻑ ﺍﻟﺘﻰ ﻣﻌﻪ.‬ ‫¯‬ ‫−‬ ‫¯‬
‫ﺍﺧﺘﺒﺎﺭ ﺗﺮﺍﻛﻤﻰ‬ ‫‪:≈JCÉj Ée πªcCG :’hCG‬‬ ‫٢‬ ‫1 ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ، b ٢- l = C‬ﺏ = )٢ ٥( ﻓﺈﻥ )ﺏ‪(C‬ﻣﺪ =‬ ‫..........................................................................................................................‬ ‫2 ﺇﺫﺍ ﻛﺎﻥ ‪  I = b ١- ١ l b ١ ٤ l‬ﻓﺈﻥ ﺱ =‬ ‫ﺱ‬ ‫.........................................................................................................................‬ ‫-٣‬ ‫٣ ١ ‬ ‫‪٢- ١ l‬‬ ‫‪ b‬ﻓﺈﻥ ‪= ٢C‬‬ ‫3 ﺇﺫﺍ ﻛﺎﻥ ‪= C‬‬ ‫٣ ٢‬ ‫..............................................................................................................................................................‬ ‫‪:Oó©àe øe QÉ«àN’G á∏Ä°SCG :Ék«fÉK‬‬ ‫٣‬ ‫ﻣﺪ ﻣﺪ‬ ‫‪...................................................................................... = C‬‬ ‫4 ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ،C‬ﺏ ﻣﺼﻔﻮﻓﺘﻴﻦ ﺣﻴﺚ ‪C‬ﺏ = ‪ b ١- ٤ l‬ﻓﺈﻥ ﺏ‬ ‫٥‬ ‫٥ ٤‬ ‫٥‬ ‫٣ ٤‬ ‫٣‬ ‫‪b‬‬ ‫١ ‪b‬‬‫‪b‬‬ ‫١ ‪b‬‬‫د ‪l‬‬ ‫ﺟ ‪l‬‬ ‫ب ‪l‬‬ ‫أ ‪l‬‬ ‫١ ٣‬‫١ ٥‬‫٤ ٣‬ ‫٤ ٥‬ ‫5 ﺇﺫﺍ ﻛﺎﻥ:‬ ‫أ -٣‬ ‫٢-ﺱ ٢‬ ‫٣ ﺱ + ٢ = ١ ﻓﺈﻥ ﺱ ﺗﺴﺎﻭﻯ‬‫ب ٣‬ ‫..................................................................................................................................‬ ‫ﺟ !٣‬ ‫د !٤‬ ‫‪k‬‬ ‫‪≈JCÉj Ée øY ÖLCG :ÉãdÉK‬‬ ‫6 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻣﻦ ﺍﻟﻤﺤﺪﺩﺍﺕ ﺍﻵﺗﻴﺔ:‬ ‫أ‬ ‫٣ ٤‬‫٢ -٥‬ ‫ب‬ ‫٢ -٢‬ ‫٥ ٤‬ ‫ﺟ‬ ‫٢ -٢ ٣‬ ‫٤ ١ -١‬ ‫١ ٢ -١‬ ‫٣‬ ‫٥‬ ‫٢‬ ‫د‬ ‫١ -٤‬ ‫٠ -١‬ ‫٢ ٠‬ ‫7 ﺣﻞ ﻛﻞ ﻣﻦ ﺃﻧﻈﻤﺔ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ ﺑﻄﺮﻳﻘﺔ ﻛﺮﺍﻣﺮ‬ ‫ﺟ ﺱ - ٢ ﺹ - ﻉ =٠ د ٣ ﺱ = ٢ ﺹ + ٣ + ﻉ‬ ‫ب ٣ﺱ + ﺹ = ٥‬ ‫أ ٢ ﺱ - ٤ = -ﺹ‬ ‫ﺱ + ﺹ - ٢ﻉ = -١‬ ‫٢ﺱ + ٣ ﺹ = ٨‬ ‫ﺹ=٣ﺱ-٦‬ ‫٢ﺱ-ﺹ+٤=ﻉ‬ ‫ﺱ + ٤ﺹ +٧ﻉ = ٦‬‫ﺹ + ﻉ = -ﺱ + ٣‬ ‫8 ﺍﺳﺘﺨﺪﻡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ‪ ،C‬ﺏ، ﺝ ﻟﺘﺤﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﻛﻞ ﻣﻤﺎﻳﺄﺗﻰ ﺻﺤﻴﺤﺎ ﺃﻡ ﻻ.‬ ‫ً‬ ‫‪، ٣ ٢ =C‬ﺏ= ٠ ٥ ،ﺝ= ٤ ١‬ ‫‪b‬‬ ‫‪l‬‬ ‫‪b‬‬ ‫‪l‬‬ ‫‪b‬‬ ‫‪l‬‬ ‫٦ -١‬ ‫-١ -٣‬ ‫٢‬ ‫٣‬ ‫ب ‪)C‬ﺏ + ﺝ( = ‪C‬ﺏ + ‪C‬ﺝ.‬ ‫أ ‪)C‬ﺏ ﺝ( = )‪ C‬ﺏ( ﺝ.‬ ‫9 ﺣﻞ ﻧﻈﺎﻡ ﺍﻟﻤﻌﺎﺩﻟﺘﻴﻦ ﺍﻵﻧﻴﺘﻴﻦ ﺍﻟﺘﺎﻟﻴﺘﻴﻦ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ‬ ‫٥ﺹ = ١ - ٢ﺱ   ،   ٣ﺱ = ٢ - ٧ﺹ‬ ‫01 ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ، b ١ ٢ l = C‬ﺏ = ‪ b ٣ ١ l‬ﻓﺄﻭﺟﺪ ﻛﻼ ﻣﻦ:‬ ‫ًّ‬ ‫٠ -١‬ ‫-١ ٢‬ ‫ب ﺏ‪C‬‬ ‫أ ‪C‬ﺏ‬ ‫٢‬ ‫د ﺏ‬ ‫ﻫ ‪C‬ﻣﺪ ﺏ‬ ‫‪ïM‬‬ ‫−‬ ‫ﺟ ‪C‬‬ ‫ﻣﺪ‬ ‫و ‪C‬ﺏ‬ ‫٢‬
‫ﺍﺧﺘﺒﺎﺭ ﺗﺮﺍﻛﻤﻰ‬ ‫11 ﺃﺟﺮ ﺍﻟﻌﻤﻠﻴﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ ﺇﻥ ﺃﻣﻜﻦ ﻣﻊ ﺫﻛﺮ ﺍﻟﺴﺒﺐ ﻓﻰ ﺣﺎﻟﺔ ﺗﻌﺬﺭ ﺇﺟﺮﺍﺀ ﺍﻟﻌﻤﻠﻴﺔ:‬ ‫ﺱ -ﺹ‬‫ﺱ ﺹ‬ ‫‪l‬‬ ‫أ ‪ l‬ﻉ ﻝ ‪- + b‬ﻉ -ﻝ‬ ‫١‬ ‫٢ -٣‬ ‫ﺟ‬ ‫‪l‬‬ ‫‪٢ b ٥ ٤ l‬‬ ‫٣‬ ‫٠‬ ‫٠‬ ‫١‬ ‫٤ -٦‬‫٤ ٦‬ ‫ب‬ ‫‪l‬‬ ‫‪٩ ٣ - b ٢- ٣- l‬‬ ‫‪b‬‬ ‫‪b‬‬ ‫د‬ ‫-١ ٢‬ ‫١‬ ‫‪l‬‬ ‫١‬ ‫21 ﺇﺫﺍ ﻛﺎﻧﺖ ‪ b ١ ١- l = N ، b ٣ ٠ l = M‬ﺃﺛﺒﺖ ﺃﻥ:‬ ‫٠‬ ‫٣‬ ‫‪C‬‬ ‫١ -٢ ٣‬ ‫٠ ١ ٤‬ ‫-٣ ٢ ١‬ ‫‪b‬‬ ‫‪b‬‬ ‫١ -٢ ٣‬ ‫‪l‬‬ ‫٤ ٥ ٦‬ ‫‪b‬‬ ‫‪MN!NM‬‬ ‫٠‬ ‫31 ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ٠ l = N ، b ٤ ٠ l = M‬ﺏ ‪ b‬ﺣﻴﺚ ‪ C‬ﺏ ! ٠‬ ‫ﺍﺛﺒﺖ ﺃﻧﻪ ﻟﻜﻞ ﻣﻦ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ‪ N M ، N ، M‬ﻣﻌﻜﻮﺱ ﺿﺮﺑﻰ ﻭﺃﻭﺟﺪه.‬ ‫41 ﺍﺷﺘﺮﻯ ﻛﺮﻳﻢ ﻣﻦ ﺇﺣﺪﻯ ﺍﻟﻤﻜﺘﺒﺎﺕ ٠١ ﺃﺳﻄﻮﺍﻧﺎﺕ ﻣﺪﻣﺠﺔ)‪٦ ،(CD‬ﺃﻗﻼﻡ ﺟﺎﻑ، ٤ ﺃﻗﻼﻡ ﺭﺻﺎﺹ، ﺍﺷﺘﺮﻯ ﺯﻣﻴﻠﻪ‬ ‫ﺳﺎﻣﻰ ٨ ﺃﺳﻄﻮﺍﻧﺎﺕ ﻣﺪﻣﺠﺔ، ٥ ﺃﻗﻼﻡ ﺟﺎﻑ، ٣ ﺃﻗﻼﻡ ﺭﺻﺎﺹ ﻣﻦ ﻧﻔﺲ ﺍﻷﻧﻮﺍﻉ ﺍﻟﺘﻰ ﺍﺷﺘﺮﺍﻫﺎ ﻛﺮﻳﻢ، ﻓﻴﻤﺎ ﻛﺎﻥ‬ ‫ﺳﻌﺮ ﺍﻟﺒﻴﻊ ﻫﻮ ﺟﻨﻴﻬﻴﻦ ﻟﻸﺳﻄﻮﺍﻧﺔ ﺍﻟﻤﺪﻣﺠﺔ، ٠٥٫١ ﻣﻦ ﺍﻟﺠﻨﻴﻪ ﻟﻠﻘﻠﻢ ﺍﻟﺠﺎﻑ، ٥٧٫٠ ﻣﻦ ﺍﻟﺠﻨﻴﻪ ﻟﻠﻘﻠﻢ ﺍﻟﺮﺻﺎﺹ.‬ ‫أ ﺍﻛﺘﺐ ﻣﺼﻔﻮﻓﺔ ﺗﻤﺜﻞ ﻣﺸﺘﺮﻳﺎﺕ ﻛﺮﻳﻢ ﻭﺳﺎﻣﻰ، ﺛﻢ ﺍﻛﺘﺐ ﻣﺼﻔﻮﻓﺔ ﺗﻮﺿﺢ ﺃﺳﻌﺎﺭ ﻛﻞ ﺳﻠﻌﺔ ﺗﻢ ﺷﺮﺍﺅﻫﺎ.‬ ‫ب ﺍﻛﺘﺐ ﻣﺼﻔﻮﻓﺔ ﺗﻮﺿﺢ ﺍﻟﻤﺒﺎﻟﻎ ﺍﻟﺘﻰ ﺩﻓﻌﻬﺎ ﻛﻞ ﻣﻦ ﻛﺮﻳﻢ ﻭﺳﺎﻣﻰ ﺛﻤﻨﺎ ﻟﻤﺸﺘﺮﻭﺍﺗﻬﻢ.‬ ‫ً‬ ‫ﺟ ﻣﺎ ﺍﻟﻤﺒﻠﻎ ﺍﻟﺬﻱ ﺩﻓﻌﻪ ﻛﻞ ﻣﻦ ﻛﺮﻳﻢ ﻭﺳﺎﻣﻰ ﺛﻤﻨﺎ ﻟﻤﺸﺘﺮﻭﺍﺗﻬﻢ?‬ ‫ً‬ ‫51 ﺇﺫﺍ ﻛﺎﻥ ﺇﻧﺘﺎﺝ ﺛﻼﺛﺔ ﺃﻗﺴﺎﻡ ﻣﺨﺘﻠﻔﺔ ﻣﻦ ﻣﺼﻨﻊ ﻹﻧﺘﺎﺝ ﻟﻌﺐ‬ ‫ﺍﻷﻃﻔﺎﻝ ﻓﻰ ٣ ﺷﻬﻮﺭ ﻣﺘﺘﺎﻟﻴﺔ ﻣﺪﻭﻧًﺎ ﺑﺎﻟﺠﺪﻭﻝ ﺍﻟﻤﻘﺎﺑﻞ، ﻛﻤﺎ‬ ‫ﻳﻮﺿﺢ ﺍﻟﺠﺪﻭﻝ ﺳﻌﺮ ﺑﻴﻊ ﻛﻞ ﻟﻌﺒﺔ ، ﻣﺎ ﺍﻟﻤﺼﻔﻮﻓﺔ ﺍﻟﺘﻰ ﺗﻤﺜﻞ‬ ‫ﺩﺧﻞ ﺍﻟﻤﺼﻨﻊ ﻣﻦ ﻣﻨﺘﺠﺎﺕ ﺍﻷﻗﺴﺎﻡ ﺍﻟﺜﻼﺛﺔ ﻓﻰ ﺍﻟﺸﻬﻮﺭ‬ ‫ﺍﻟﺜﻼﺛﺔ ﺑﻔﺮﺽ ﺃﻧﻪ ﻗﺪ ﺗﻢ ﺑﻴﻊ ﺇﻧﺘﺎﺝ ﺍﻟﻤﺼﻨﻊ ﺑﺎﻟﻜﺎﻣﻞ?‬ ‫ﻣﺎ ﻣﻘﺪﺍﺭ ﺍﻟﺪﺧﻞ ﺍﻟﻜﻠﻲ ﻟﻠﻤﺼﻨﻊ?‬ ‫‪M‬‬ ‫¯‬ ‫¯‬ ‫٠٦٣‬ ‫٠٠٤‬ ‫٠٨٣ ٨٢ ﺟﻨﻴﻪ‬ ‫٠٨٤‬ ‫٠٠٥‬ ‫٠٥٤ ٧٣ ﺟﻨﻴﻪ‬ ‫٠٧٥‬ ‫٠٠٦‬ ‫٠٥٥ ٢٣ ﺟﻨﻴﻪ‬ ‫:‬ ‫ﺭﻗﻢ ﺍﻟﺴﺆﺍﻝ‬ ‫١‬ ‫٢‬ ‫٣‬ ‫٤‬ ‫٥‬ ‫٦‬ ‫٧‬ ‫٨‬ ‫٩ ٠١ ١١ ٢١ ٣١ ٤١ ٥١‬ ‫١- ٣‬ ‫١- ٣‬ ‫١- ٥‬ ‫١- ٥‬ ‫١- ٣‬ ‫١- ٣‬ ‫١- ٤‬ ‫١- ٣‬ ‫١- ٤‬ ‫١- ٤‬ ‫١- ٢‬ ‫١- ٢‬ ‫١- ٣‬ ‫١- ٣‬ ‫١- ٣‬ ‫ﺭﻗﻢ ﺍﻟﺪﺭﺱ‬ ‫¯‬ ‫¯‬ ‫−‬ ‫¯‬ ‫¯‬
‫ﺍﻟﺠﺒﺮ‬ ‫‪IóMƒdG‬‬ ‫-‬ ‫2‬ ‫ﺍﻟﺒﺮﻣﺠﺔ ﺍﻟﺨﻄﻴﺔ‬ ‫‪Linear Programing‬‬ ‫دروس اﻟﻮﺣﺪة‬ ‫ﺍﻟﺪﺭﺱ )٢ - ١(: ﺍﻟﻤﺘﺒﺎﻳﻨﺔ ﺍﻟﺨﻄﻴﺔ.‬ ‫ﺍﻟﺪﺭﺱ )٢ - ٢(: ﺣﻞ ﺃﻧﻈﻤﺔ ﻣﻦ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ ﺍﻟﺨﻄﻴﺔ ﺑﻴﺎﻧﻴﺎ.‬ ‫ﺍﻟﺪﺭﺱ )٢ - ٣(: ﺍﻟﺒﺮﻣﺠﺔ ﺍﻟﺨﻄﻴﺔ ﻭﺍﻟﺤﻞ ﺍﻷﻣﺜﻞ.‬ ‫‪ïM‬‬ ‫−‬
‫اﻟﻤﺘﺒﺎﻳﻨﺎت اﻟﺨﻄﻴﺔ‬ ‫2-1‬ ‫‪Linear Inequalities‬‬ ‫1 ﺻﻞ ﻛﻞ ﻣﺘﺒﺎﻳﻨﺔ ﺑﺎﻟﺮﺳﻢ ﺍﻟﺒﻴﺎﻧﻲ ﺍﻟﺬﻱ ﻳﻤﺜﻞ ﻣﺠﻤﻮﻋﺔ ﺣﻠﻬﺎ )ﺍﺧﺘﺒﺮ ﺍﻟﻨﻘﻄﺔ )٠، ٠( ﻓﻰ ﻛﻞ ﻣﺘﺒﺎﻳﻨﺔ(.‬ ‫ﺟ‬ ‫ب‬ ‫د‬ ‫أ‬ ‫٢- ﺹ > ﺱ + ١‬ ‫١- ﺹ ‪ H‬ﺱ + ١‬ ‫٣- ﺹ < ﺱ + ١‬ ‫٤- ﺹ ‪ G‬ﺱ + ١‬ ‫2 ﺍﺧﺘﺒﺮ ﺃﻳﺎ ﻣﻦ ﺍﻟﻨﻘﻂ ﻫﻮ ﺣﻞ ﻟﻠﻤﺘﺒﺎﻳﻨﺔ:‬ ‫ًّ‬ ‫] )٠، ١( ، )٣، ٩( ، )–١، ٠( [ .........................................................................................‬ ‫أ ﺹ ‪ ٢ G‬ﺱ +٣‬ ‫] )٠، ١( ، )٣، ٩( ، )–١، ٠( [.........................................................................................‬ ‫ب ﺹ>٢ﺱ+٣‬ ‫3 ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﻛﻞ ﻣﻦ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ:‬ ‫ب ﺹ<٢ﺱ–٣‬ ‫أ ﺹ‪H‬ﺱ+٢‬ ‫ﺟ ﺱ+٣ﺹ‪٦H‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫4 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻤﺴﺘﻬﻠﻚ: ﺍﻓﺘﺮﺽ ﺃﻧﻚ ﺗﺮﻳﺪ ﺷﺮﺍﺀ ﻭﺭﻕ ﺯﻳﻨﺔ؛ ﻟﺘﺰﻳﻦ ﻓﺼﻠﻚ ﺍﻟﺪﺭﺍﺳﻲ ﻟﻌﻤﻞ ﺣﻔﻠﺔ ﻷﻭﺍﺋﻞ ﺍﻟﻄﻠﺒﺔ،‬ ‫ﻓﺈﺫﺍ ﻛﺎﻥ ﺛﻤﻦ ﺍﻟﻠﻔﺔ ﻣﻦ ﻭﺭﻕ ﺍﻟﺰﻳﻨﺔ ﺫﻫﺒﻲ ﺍﻟﻠﻮﻥ ﻫﻮ ٥ ﺟﻨﻴﻬﺎﺕ، ﻭﺛﻤﻦ ﺍﻟﻠﻔﺔ ﻣﻦ ﻭﺭﻕ ﺍﻟﺰﻳﻨﺔ ﺍﻷﺯﺭﻕ ﺍﻟﻠﻮﻥ‬ ‫ﻫﻮ ٣ ﺟﻨﻴﻬﺎﺕ، ﻭﺃﻧﻚ ﺗﺮﻳﺪ ﺻﺮﻑ ٨٤ ﺟﻨﻴﻬﺎ ﻋﻠﻰ ﺍﻷﻛﺜﺮ؛ ﻟﺸﺮﺍﺀ ﻭﺭﻕ ﺍﻟﺰﻳﻨﺔ، ﻓﻜﻢ ﻟﻔﺔ ﻣﻦ ﻛﻞ ﻧﻮﻉ ﻳﻤﻜﻨﻚ‬ ‫ً‬ ‫..........................................................................................................................................................................‬ ‫ﺷﺮﺍﺅﻫﺎ? ﻓﺴﺮ ﺇﺟﺎﺑﺘﻚ.‬ ‫..................................................................................................................................................................................................................................‬ ‫ﻧﺸﺎط )١(‬ ‫ﺻﻒ ﻟﺰﻣﻴﻠﻚ ﺍﻟﺬﻯ ﻛﺎﻥ ﻏﺎﺋﺒﺎ ﺃﺛﻨﺎﺀ ﺷﺮﺡ ﻫﺬﺍ ﺍﻟﺪﺭﺱ ﻟﻤﺮﺿﻪ، ﻛﻴﻒ ﻳﻤﻜﻦ ﺗﻤﺜﻴﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺔ ﺱ - ﺹ ‪ ٢ G‬ﺑﻴﺎﻧﻴﺎ‬ ‫ًّ‬ ‫ً‬ ‫ﻭﺍﻟﺘﺤﻘﻖ ﻣﻦ ﺻﺤﺔ ﺍﻹﺟﺎﺑﺔ.‬ ‫¯‬ ‫−‬ ‫¯‬
‫‪M‬‬ ‫اﻟﺮﺑﻂ ﺑﺎﻟﺘﻜﻨﻮﻟﻮﺟﻴﺎ‬ ‫ﻧﺸﺎط )٢(‬ ‫:‬ ‫ﻳﻤﻜﻨﻚ ﺍﺳﺘﺨﺪﺍﻡ ﺧﺎﺻﻴﺔ ‪ Draw‬ﻓﻰ ﺍﻵﻟﺔ ﺍﻟﺤﺎﺳﺒﺔ ﺍﻟﺒﻴﺎﻧﻴﺔ ‪ graphing colculator‬ﻟﺮﺳﻢ‬ ‫ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ، ﺣﻴﺚ ﻳﻌﺘﻤﺪ ﺗﺮﺗﻴﺐ ﺍﺩﺧﺎﻝ ﺍﻟﺒﻴﺎﻧﺎﺕ ﻋﻠﻰ ﻣﺎ ﺇﺫﺍ ﻛﻨﺖ ﺳﺘﻈﻠﻞ ﻓﻮﻕ ﺍﻟﺨﻂ‬ ‫ﺍﻟﺤﺪﻯ ﺃﻡ ﺃﺳﻔﻠﻪ؛ ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺘﻈﻠﻴﻞ ﺃﺳﻔﻞ ﺍﻟﺨﻂ ﺍﻟﺤﺪﻯ 1‪ y‬ﺍﺳﺘﺨﺪﻡ ﺍﻟﺘﻈﻠﻴﻞ )1‪،(ymin, y‬‬ ‫ﻭﻋﻨﺪﻣﺎ ﺗﻈﻠﻞ ﻓﻮﻕ 1‪ y‬ﺍﺳﺘﺨﺪﻡ ﺍﻟﺘﻈﻠﻴﻞ )‪ (y1, ymax‬ﻭﻟﺴﺖ ﻓﻰ ﺣﺎﺟﺔ ﻻ ﺳﺘﺨﺪﺍﻡ ﺍﻷﻗﻮﺍﺱ‬ ‫ﺍﻟﻤﻐﻠﻘﺔ ﻗﺒﻞ ﺍﻟﻀﻐﻂ ﻋﻠﻰ ﻣﻔﺘﺎﺡ ‪ENTER‬‬ ‫ﻻﺣﻆ‬ ‫¯‬ ‫‪¯ y‬‬ ‫,‪¯ x‬‬ ‫¯‬ ‫¯‬ ‫+‬ ‫>‬ ‫3 + ‪y < 2x‬‬ ‫ﻣـﺜـﺎل‬ ‫5 ﻣﺜﻞ ﻛﻞ ﻣﺘﺒﺎﻳﻨﺔ ﺑﻴﺎﻧﻴﺎ:‬ ‫ًّ‬ ‫1 – ‪B y = 0.5x‬‬ ‫3 + ‪A y < 2x‬‬ ‫ﺍﻟﺘﻈﻠﻴﻞ ﻓﻮﻕ ﺍﻟﺨﻂ ﺍﻟﺤﺪﻯ 1‪ y‬ﻟﻮﺟﻮﺩ ﻋﻼﻣﺔ ﺃﻛﺒﺮ ﻣﻦ‬ ‫1‬ ‫–‬ ‫‪X,T,i‬‬ ‫,‬ ‫5‬ ‫1‬ ‫.‬ ‫1‬ ‫5‬ ‫‪Draw‬‬ ‫‪2nd‬‬ ‫‪Y-VARS‬‬ ‫7‬ ‫‪ENTRER‬‬ ‫0‬ ‫=‪Y‬‬ ‫ﺍﻟﺘﻈﻠﻴﻞ ﺃﺳﻔﻞ 1‪ y‬ﻟﻮﺟﻮﺩ ﻋﻼﻣﺔ ﺃﻗﻞ ﻣﻦ‬ ‫‪F‬‬ ‫$‬ ‫#‬ ‫$‬ ‫3‬ ‫#‬ ‫‪2nd‬‬ ‫1‬ ‫+‬ ‫7‬ ‫,‬ ‫‪$ VARS‬‬ ‫)‪(y1, ymax‬‬ ‫) ‪,y‬‬ ‫1 ‪min‬‬ ‫‪# (y‬‬ ‫‪ENTRER‬‬ ‫‪X,T,i‬‬ ‫1‬ ‫4‬ ‫1‬ ‫2‬ ‫‪Draw‬‬ ‫1‬ ‫‪Y-VARS‬‬ ‫=‪Y‬‬ ‫‪2nd‬‬ ‫‪VARS‬‬ ‫‪2nd‬‬ ‫× ﻳﻤﻜﻨﻚ ﺍﻟﺘﺤﻜﻢ ﻓﻰ ﺩﺭﺟﺔ ﺍﻟﺘﻈﻠﻴﻞ ﺑﺈﺩﺧﺎﻝ ﻋﺪﺩ ﺻﺤﻴﺢ ﻣﻦ )‪ 1(Dark‬ﺇﻟﻰ )‪ ،8(Light‬ﺃﺿﻒ ﻓﺎﺻﻠﺔ )‪(Comma‬‬ ‫ﻭﺍﻟﻌﺪﺩ ﺍﻟﺼﺤﻴﺢ ﻗﺒﻞ ﺍﻟﻀﻐﻂ ﻋﻠﻰ ﻣﻔﺘﺎﺡ ‪. ENTRER‬‬ ‫× ﻻ ﺗﻀﻊ ﺍﻵﻟﺔ ﺍﻟﺤﺎﺳﺐ ﺍﻟﺒﻴﺎﻧﻴﺔ ﺗﻤﻴﺰﺍ ﺑﻴﻦ ﺍﻟﺨﻂ ﺍﻟﺤﺪﻯ ﺍﻟﻤﺘﺼﻞ ﻭﺍﻟﻤﻘﻄﻊ؛ ﻟﺬﺍ ﻳﺠﺐ ﻋﻠﻴﻚ ﺃﻥ ﺗﺤﺪﺩ ﺇﺫﺍ ﻛﺎﻥ‬ ‫ً‬ ‫ﺍﻟﺨﻂ ﺍﻟﺤﺪﻯ ﻣﺘﺼﻼ ﺃﻡ ﻣﺘﻘﻄﻌﺎ ﻋﻨﺪ ﺭﺳﻤﻚ ﻟﻠﻤﺘﺒﺎﻳﻨﺔ ﻓﻰ ﻛﺮﺍﺳﺘﻚ.‬ ‫ً‬ ‫ً‬ ‫ﺣﺎول أن ﺗﺤﻞ‬ ‫1 ﺍﺳﺘﺨﺪﻡ ﺍﻵﻟﺔ ﺍﻟﺤﺎﺳﺒﺔ ﺍﻟﺒﻴﺎﻧﻴﺔ؛ ﻟﺮﺳﻢ ﻛﻞ ﻣﺘﺒﺎﻳﻨﺔ ﻣﻤﺎ ﻳﻠﻲ:‬ ‫3 + ‪C y H -x‬‬ ‫1 + ‪B y > 2x‬‬ ‫‪A y<x‬‬ ‫21 ‪F 2x + 3y G‬‬ ‫4‪E x-yH‬‬ ‫5‪D yG‬‬ ‫‪ïM‬‬ ‫−‬
‫ًّ‬ ‫ﺣﻞ أﻧﻈﻤﺔ ﻣﻦ اﻟﻤﺘﺒﺎﻳﻨﺎت اﻟﺨﻄﻴﺔ ﺑﻴﺎﻧﻴﺎ‬ ‫2-2‬ ‫‪Solving systems of Linear Inequalities Graphcally‬‬ ‫1 ﺃﻯ ﻧﻈﺎﻡ ﻣﻤﺎ ﻳﺄﺗﻰ ﻟﻪ ﻣﻨﻄﻘﺔ ﺍﻟﺤﻞ ﺍﻟﻤﻮﺿﺤﺔ ﻓﻰ‬ ‫ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ:‬ ‫أ ﺱ+ﺹ‪٣H‬‬ ‫ب ﺱ+ﺹ<٣‬ ‫ﺹ< ﺱ-٣‬ ‫ﺹ‪H‬ﺱ-٣‬ ‫ﺟ ﺱ+ﺹ‪٣G‬‬ ‫ﺹ>ﺱ-٣‬ ‫د ﺱ+ﺹ>٣‬ ‫ﺹ‪G‬ﺱ-٣‬ ‫2 ﺣﻞ ﻛﻞ ﻧﻈﺎﻡ ﻣﻦ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ ﺍﻟﺨﻄﻴﺔ ﺑﻴﺎﻧﻴﺎ:‬ ‫ًّ‬ ‫ب ﺹ-ﺱ<٠‬ ‫أ ﺱ‪٤H‬‬ ‫٢ ﺱ + ٢ ﺹ ‪١٢ H‬‬ ‫ﺹ>ﺱ+٢‬ ‫ﺹ>٦+٢ﺱ‬ ‫ﺱ + ٢ ﺹ ‪٢- G‬‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫ﺟ ﺱ+٤ﺹ<٤‬ ‫٤ﺱ+ﺹ‪٢G‬‬ ‫ﺱ-ﺹ>١‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫3 ﺃﻋﻄﻰ ﺍﻷﺳﺘﺎﺫ ﻛﺮﻳﻢ ﻟﺘﻼﻣﻴﺬه ﺯﻣﻨﺎ ﻗﺪﺭه ٠٦ ﺩﻗﻴﻘﺔ ﻹﺟﺎﺑﺔ ﺍﺧﺘﺒﺎﺭ ﻓﻰ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ، ﻳﺠﺐ ﺃﻥ ﻳﺠﻴﺐ ﺍﻟﺘﻼﻣﻴﺬ‬ ‫ً‬ ‫ﻋﻦ ٤ ﺃﺳﺌﻠﺔ ﻋﻠﻰ ﺍﻷﻗﻞ ﻣﻦ ﺍﻟﻘﺴﻢ )ﺃ(، ٣ ﺃﺳﺌﻠﺔ ﻋﻠﻰ ﺍﻷﻗﻞ ﻣﻦ ﺍﻟﻘﺴﻢ )ﺏ(، ﺑﺤﻴﺚ ﻻﺗﻘﻞ ﻋﺪﺩ ﺍﻷﺳﺌﻠﺔ ﺍﻟﻤﺠﺎﺑﺔ‬ ‫ﻣﻦ ﺍﻟﻘﺴﻤﻴﻦ ﻣﻌﺎ ﻋﻦ ٠١ ﺃﺳﺌﻠﺔ. ﻓﺈﺫﺍ ﺍﺳﺘﻐﺮﻗﺖ ﻫﻨﺎﺀ ٤ ﺩﻗﺎﺋﻖ ﻹﺟﺎﺑﺔ ﻛﻞ ﺳﺆﺍﻝ ﻓﻰ ﺍﻟﻘﺴﻢ )ﺃ(، ٥ ﺩﻗﺎﺋﻖ ﻹﺟﺎﺑﺔ‬ ‫ً‬ ‫ً‬ ‫ﻛﻞ ﺳﺆﺍﻝ ﻓﻰ ﺍﻟﻘﺴﻢ )ﺏ(. ﻛﻢ ﺳﺆﺍﻻ ﻓﻰ ﻛﻞ ﻗﺴﻢ ﺣﺎ ﻟﺖ ﻫﻨﺎﺀ ﺍﻹﺟﺎﺑﺔ ﻋﻨﻪ?.................................................................‬ ‫ﻭ‬ ‫4 ﺍﻟﺘﻔﻜﻴﺮ ﺍﻟﻨﺎﻗﺪ:‬ ‫أ ﺍﻛﺘﺐ ﻧﻈﺎﻣﺎ ﻣﻦ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ ﺍﻟﺨﻄﻴﺔ، ﻭﺍﻟﺘﻲ ﻳﻜﻮﻥ ﺣﻠﻬﺎ ﻫﻮ ﺧﻂ ﻣﺴﺘﻘﻴﻢ.‬ ‫ً‬ ‫ب ﺑﺪﻭﻥ ﺗﻤﺜﻴﻞ ﺑﻴﺎﻧﻲ، ﻓﺴﺮ ﻟﻤﺎﺫﺍ ﻧﻘﻄﺔ ﺗﻘﺎﻃﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﻴﻦ ﺍﻟﺤﺪﻳﻴﻦ ﻓﻰ ﺍﻟﻨﻈﺎﻡ: ٢ ﺱ + ﺹ < ٢، ﺱ - ﺹ ‪٣ G‬‬ ‫ﻟﻴﺴﺖ ﺣﻼ ﻟﻬﺬﺍ ﺍﻟﻨﻈﺎﻡ. ............................................................................................................................................................................‬ ‫ًّ‬ ‫........................................................‬ ‫ﻧﺸﺎط‬ ‫5 ﺍﻛﺘﺐ ﻣﺴﺄﻟﺔ ﻣﻦ ﻋﻨﺪﻙ ﻳﺤﺘﺎﺝ ﺣﻠﻬﺎ ﺇﻟﻰ ﻛﺘﺎﺑﺔ ﻧﻈﺎﻡ ﻣﻦ ﻣﺘﺒﺎﻳﻨﺘﻴﻦ ﺧﻄﻴﺘﻴﻦ ﻓﻰ ﻣﺠﻬﻮﻟﻴﻦ ﺛﻢ ﺣﻞ ﻫﺬﺍ ﺍﻟﻨﻈﺎﻡ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
‫اﻟﺒﺮﻣﺠﺔ اﻟﺨﻄﻴﺔ واﻟﺤﻞ ا ﻣﺜﻞ‬ ‫2-3‬ ‫‪Linear programing and optimization‬‬ ‫1 ﺍﺧﺘﺮ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ ﻣﻦ ﺑﻴﻦ ﺍﻹﺟﺎﺑﺎﺕ ﺍﻟﻤﻌﻄﺎﺓ:‬ ‫أ ﺍﻟﻨﻘﻄﺔ ﺍﻟﺘﻰ ﺗﻨﺘﻤﻲ ﺇﻟﻰ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ: ﺱ < ٢ ، ﺹ < ١ ، ﺱ + ﺹ ‪ ٣ G‬ﻫﻰ: ............‬ ‫‪`(٣ ،١) ، (٢ ،٣) ، (٢ ،١) ، (١ ،٣) j‬‬ ‫ب‬ ‫ﺍﻟﻨﻘﻄﺔ ﺍﻟﺘﻰ ﺗﻜﻮﻥ ﻋﻨﺪﻫﺎ ﻟﻠﺪﺍﻟﺔ ‪٤٠ = S‬ﺱ + ٠٢ﺹ ﻗﻴﻤﺔ ﻋﻈﻤﻰ ﻫﻰ: .................................................................‬ ‫‪`(٠ ،٢٥) ، (١٠ ،١٥) ، (٤- ،٠) ، (٠ ،٠) j‬‬ ‫ﺟ‬ ‫ﺍﻟﻨﻘﻄﺔ ﺍﻟﺘﻰ ﺗﻜﻮﻥ ﻋﻨﺪﻫﺎ ﻟﻠﺪﺍﻟﺔ ﻡ = ٥٣ﺱ + ٠١ﺹ ﻗﻴﻤﺔ ﺻﻐﺮﻯ ﻫﻰ: ...................................................................‬ ‫‪`(١٠ ،٢٠) ، (٤٠ ،٠) ، (١٠ ،٠) ، (٠ ،٠) j‬‬ ‫2 ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﺮﺳﻢ ﺍﻟﺒﻴﺎﻧﻲ ﺍﻟﻤﻘﺎﺑﻞ، ﺃﻭﺟﺪ ﻗﻴﻤﺘﻲ ﺱ، ﺹ ﺍﻟﺘﻰ ﺗﺠﻌﻞ ﻗﻴﻤﺔ‬ ‫ﺩﺍﻟﺔ ﺍﻟﻬﺪﻑ ‪٣ = S‬ﺱ + ٢ﺹ ﻗﻴﻤﺔ ﺻﻐﺮﻯ، ﺛﻢ ﺃﻭﺟﺪ ﻫﺬه ﺍﻟﻘﻴﻤﺔ.‬ ‫‪C‬‬ ‫....................................................................................................................................................‬ ‫....................................................................................................................................................‬ ‫‪E‬‬ ‫3 ﻣﺜﻞ ﻛﻼ ﻣﻦ ﺍﻷﻧﻈﻤﺔ ﺍﻟﺘﺎﻟﻴﺔ ﺑﻴﺎﻧﻴﺎ، ﺛﻢ ﺃﻭﺟﺪ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻌﻈﻤﻰ ﺃﻭ ﺍﻟﻘﻴﻤﺔ‬ ‫ًّ‬ ‫ًّ‬ ‫ﺍﻟﺼﻐﺮﻯ ﻟﺪﺍﻟﺔ ﺍﻟﻬﺪﻑ ﺗﺒﻌﺎ ﻟﻤﺎ ﻫﻮ ﻣﻌﻄﻰ.‬ ‫ً‬ ‫ب ٢ﺱ + ﺹ ‪٦ H‬‬ ‫أ ﺱ+ﺹ‪٥H‬‬ ‫ﺹ‪١G‬‬ ‫ﺱ‪٢G‬‬ ‫ﻗﻴﻤﺔ ﺻﻐﺮﻯ ﻟﺪﺍﻟﺔ ﺍﻟﻬﺪﻑ ‪٢ = S‬ﺱ + ٣ﺹ‬ ‫−‬ ‫ﺱ‪١G‬‬ ‫ﺹ‪٢G‬‬ ‫ﻗﻴﻤﺔ ﻋﻈﻤﻰ ﻟﺪﺍﻟﺔ ﺍﻟﻬﺪﻑ ‪٢ = S‬ﺱ + ٣ﺹ‬ ‫4 ﺍﻟﺮﺑﻂ ﺑﺎﻟﺼﻨﺎﻋﺔ: ﺍﻓﺘﺮﺽ ﺃﻧﻚ ﺗﺼﻨﻊ ﻭﺗﺒﻴﻊ ﻣﺮﻃﺒﺎ ﻟﻠﺠﻠﺪ، ﻭﺇﺫﺍ ﻛﺎﻥ ﺗﺼﻨﻴﻊ ﻋﺒﻮﺓ ﺍﻟﻤﺮﻃﺐ ﺍﻟﻌﺎﺩﻯ ﻳﺴﺘﻠﺰﻡ‬ ‫ُ‬ ‫ً‬ ‫٣‬ ‫٢ﺳﻢ٣ ﻣﻦ ﺍﻟﺰﻳﺖ، ١ﺳﻢ٣ ﻣﻦ ﺯﺑﺪﺓ ﺍﻟﻜﺎﻛﺎﻭ، ﻛﺎﻥ ﺗﺼﻨﻴﻊ ﻋﺒﻮﺓ ﺍﻟﻤﺮﻃﺐ ﻣﻦ ﺍﻟﻨﻮﻉ ﺍﻟﻤﻤﺘﺎﺯ ﻳﺴﺘﻠﺰﻡ ١ﺳﻢ‬ ‫ﻭ‬ ‫ﻣﻦ ﺍﻟﺰﻳﺖ، ٢ﺳﻢ٣ ﻣﻦ ﺯﺑﺪﺓ ﺍﻟﻜﺎﻛﺎﻭ، ﺳﻮﻑ ﻳﻜﻮﻥ ﺭﺑﺤﻚ ﻫﻮ ٠١ ﺟﻨﻴﻬﺎﺕ ﻟﻜﻞ ﻋﺒﻮﺓ ﻣﻦ ﺍﻟﻨﻮﻉ ﺍﻟﻌﺎﺩﻯ، ٨‬ ‫ﺟﻨﻴﻬﺎﺕ ﻟﻜﻞ ﻋﺒﻮﺓ ﻣﻦ ﺍﻟﻨﻮﻉ ﺍﻟﻤﻤﺘﺎﺯ. ﻓﺈﺫﺍ ﻛﺎﻥ ﻟﺪﻳﻚ ٤٢ ﺳﻢ٣ ﻣﻦ ﺍﻟﺰﻳﺖ، ٨١ ﺳﻢ٣ ﻣﻦ ﺯﺑﺪﺓ ﺍﻟﻜﺎﻛﺎﻭ، ﻓﻤﺎ‬ ‫ﻋﺪﺩ ﺍﻟﻌﺒﻮﺍﺕ ﺍﻟﺘﻰ ﻳﻤﻜﻨﻚ ﺗﺼﻨﻴﻌﻬﺎ ﻣﻦ ﻛﻞ ﻧﻮﻉ؛ ﺣﺘﻰ ﺗﺤﺼﻞ ﻋﻠﻰ ﺃﻛﺒﺮ ﺭﺑﺢ ﻣﻤﻜﻦ، ﻭﻣﺎ ﻫﺬﺍ ﺍﻟﺮﺑﺢ?‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
‫5 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻤﻬﻦ: ﻟﺪﻯ ﺃﺣﺪ ﺍﻟﺨﻴﺎﻃﻴﻦ ٠١ ﺃﻣﺘﺎﺭ ﻣﻦ ﻗﻤﺎﺵ ﺍﻟﻜﺘﺎﻥ، ٦ ﺃﻣﺘﺎﺭ ﻣﻦ ﻗﻤﺎﺵ ﻗﻄﻨﻰ، ﻭﻳﺮﻳﺪ ﺍﻟﺨﻴﺎﻁ‬ ‫ﺗﻔﺼﻴﻞ ﻧﻮﻋﻴﻦ ﻣﻦ ﺍﻟﻤﻼﺑﺲ ﻣﻦ ﺍﻟﻤﻮﺍﺩ ﺍﻟﻤﺘﻮﺍﻓﺮﺓ ﻟﺪﻳﻪ، ﺍﻟﻨﻮﻉ ﺍﻷﻭﻝ ﻣﻦ ﺍﻟﻤﻼﺑﺲ ﻳﺤﺘﺎﺝ ﺇﻟﻰ ﻣﺘﺮ ﻭﺍﺣﺪ ﻣﻦ‬ ‫ﺍﻟﻜﺘﺎﻥ، ﻭﻣﺘﺮ ﻭﺍﺣﺪ ﻣﻦ ﺍﻟﻘﻄﻦ، ﻭﻳﺤﻘﻖ ﺭﺑﺤﺎ ﻗﺪﺭه ٣ ﺟﻨﻴﻬﺎﺕ، ﺑﻴﻨﻤﺎ ﻳﺤﺘﺎﺝ ﺍﻟﻨﻮﻉ ﺍﻟﺜﺎﻧﻰ ﺇﻟﻰ ٢ ﻣﺘﺮ ﻣﻦ‬ ‫ً‬ ‫ﺍﻟﻜﺘﺎﻥ ﻭﻣﺘﺮ ﻭﺍﺣﺪ ﻣﻦ ﺍﻟﻘﻄﻦ، ﻭﻳﺤﻘﻖ ﺭﺑﺤﺎ ﻗﺪﺭﺓ ٤ ﺟﻨﻴﻬﺎﺕ. ﻣﺎ ﺍﻟﻜﻤﻴﺔ ﺍﻟﺘﻰ ﻳﺠﺐ ﻋﻠﻴﻪ ﺗﻔﺼﻴﻠﻬﺎ ﻣﻦ ﻛﻞ‬ ‫ً‬ ‫ﻧﻮﻉ ﺣﺘﻰ ﻳﺤﻘﻖ ﺍﻟﺨﻴﺎﻁ ﺃﻛﺒﺮ ﺭﺑﺢ ﻣﻤﻜﻦ? .....................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫6 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻤﻮﺳﻴﻘﻰ: ﻳﻨﺘﺞ ﺃﺣﺪ ﻣﺼﺎﻧﻊ ﺍﻵﻻﺕ ﺍﻟﻤﻮﺳﻴﻘﻴﺔ ﻧﻮﻋﻴﻦ ﻣﻦ ﺁﻻﺕ ﺍﻟﻨﻔﺦ، ﻳﺤﺘﺎﺝ ﺗﺼﻨﻴﻊ ﺍﻟﻨﻮﻉ ﺍﻷﻭﻝ‬ ‫ﺇﻟﻰ ٥٢ ﻭﺣﺪﺓ ﻣﻦ ﺍﻟﻨﺤﺎﺱ، ٤ ﻭﺣﺪﺍﺕ ﻣﻦ ﺍﻟﻨﻴﻜﻞ، ﻭﻳﺤﺘﺎﺝ ﺗﺼﻨﻴﻊ ﺍﻟﻨﻮﻉ ﺍﻟﺜﺎﻧﻲ ٥١ ﻭﺣﺪﺓ ﻣﻦ ﺍﻟﻨﺤﺎﺱ، ٨‬ ‫ﻭﺣﺪﺍﺕ ﻣﻦ ﺍﻟﻨﻴﻜﻞ، ﻓﺈﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻜﻤﻴﺔ ﺍﻟﻤﺘﺎﺣﺔ ﻓﻰ ﺍﻟﻤﺼﻨﻊ ﻓﻰ ﺃﺣﺪ ﺍﻷﻳﺎﻡ ٥٩ ﻭﺣﺪﺓ ﻣﻦ ﺍﻟﻨﺤﺎﺱ، ٢٣ ﻭﺣﺪﺓ‬ ‫ﻭ‬ ‫ﻣﻦ ﺍﻟﻨﻴﻜﻞ، ﻛﺎﻥ ﺭﺑﺢ ﺍﻟﻤﺼﻨﻊ ﻓﻰ ﺍﻵﻟﺔ ﻣﻦ ﺍﻟﻨﻮﻉ ﺍﻷﻭﻝ ﻫﻮ ٠٦ ﺟﻨﻴﻬﺎ ﻭﺭﺑﺤﻪ ﻓﻰ ﺍﻵﻟﺔ ﻣﻦ ﺍﻟﻨﻮﻉ ﺍﻟﺜﺎﻧﻲ ٨٤‬ ‫ً‬ ‫ﺟﻨﻴﻬﺎ، ﻓﻤﺎ ﻋﺪﺩ ﺍﻵﻻﺕ ﺍﻟﺘﻰ ﻳﺠﺐ ﺃﻥ ﻳﻨﺘﺠﻬﺎ ﺍﻟﻤﺼﻨﻊ ﻣﻦ ﻛﻞ ﻧﻮﻉ ﺣﺘﻰ ﻳﺤﻘﻖ ﺃﻛﺒﺮ ﺭﺑﺢ ﻣﻤﻜﻦ?‬ ‫ً‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫7 ﺍﻟﺮﺑﻂ ﺑﺎﻟﺴﻴﺎﺣﺔ: ﺃﻗﺎﻣﺖ ﺇﺣﺪﻯ ﻛﺎﺕ ﺍﻟﺴﻴﺎﺣﺔ ﺟﺴﺮﺍ ﺟﻮ ﻳﺎ ﻟﻨﻘﻞ ﺍﻟﺴﺎﺋﺤﻴﻦ. ﺫﻟﻚ ﻟﻨﻘﻞ ٠٠٦١ ﺳﺎﺋﺢ، ٠٩‬ ‫ﺷﺮ‬ ‫ًّ‬ ‫ً‬ ‫ﻃﻨﺎ ﻣﻦ ﺍﻷﻣﺘﻌﺔ ﺑﺄﻗﻞ ﺗﻜﻠﻔﺔ، ﻛﺎﻥ ﺍﻟﻤﺘﺎﺡ ﻧﻮﻋﻴﻦ ﻣﻦ ﺍﻟﻄﺎﺋﺮﺍﺕ ‪ ،C‬ﺏ ﻛﺎﻥ ﻋﺪﺩ ﺍﻟﻄﺎﺋﺮﺍﺕ ﺍﻟﻤﺘﺎﺣﺔ ﻣﻦ ﺍﻟﻨﻮﻉ‬ ‫ﻭ‬ ‫ﻭ‬ ‫ًّ‬ ‫‪ ١٢ ،C‬ﻃﺎﺋﺮﺓ، ﻭﻋﺪﺩ ﺍﻟﻄﺎﺋﺮﺍﺕ ﺍﻟﻤﺘﺎﺣﺔ ﻣﻦ ﺍﻟﻨﻮﻉ ﺏ ٩ ﻃﺎﺋﺮﺍﺕ، ﻛﺎﻧﺖ ﺍﻟﺤﻤﻮﻟﺔ ﻛﺎﻣﻠﺔ ﻟﻠﻄﺎﺋﺮﺓ ﻣﻦ ﺍﻟﻨﻮﻉ ‪C‬‬ ‫ﻭ‬ ‫٠٠٢ ﺷﺨﺺ، ٦ ﺃﻃﻨﺎﻥ ﻣﻦ ﺍﻷﻣﺘﻌﺔ، ﻭﺍﻟﺤﻤﻮﻟﺔ ﺍﻟﻜﺎﻣﻠﺔ ﻟﻠﻄﺎﺋﺮﺓ ﻣﻦ ﺍﻟﻨﻮﻉ ﺏ ٠٠١ ﺷﺨﺺ، ٥١ ﻃﻨﺎ ﻣﻦ ﺍﻷﻣﺘﻌﺔ،‬ ‫ًّ‬ ‫ﻭ‬ ‫ﻛﺎﻥ ﺇﻳﺠﺎﺭ ﺍﻟﻄﺎﺋﺮﺓ ﻣﻦ ﺍﻟﻨﻮﻉ ‪ C‬ﻫﻮ ٠٠٠ ٠٢٣ ﺟﻨﻴﻪ، ﻭﻣﻦ ﺍﻟﻨﻮﻉ ﺏ ﻫﻮ ٠٠٠ ٠٥١ ﺟﻨﻴﻪ، ﻓﻜﻢ ﻃﺎﺋﺮﺓ ﻣﻦ ﻛﻞ‬ ‫ﻛﺔ ﺍﺳﺘﺌﺠﺎﺭﻫﺎ?‬ ‫ﻧﻮﻉ ﻳﻤﻜﻦ ﻟﻠﺸﺮ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫أﻧﺸﻄﺔ‬ ‫ً‬ ‫8 ﺍﺭﺟﻊ ﺇﻟﻰ ﻣﻜﺘﺒﺘﻚ ﺍﻟﻤﺪﺭﺳﻴﺔ ﺃﻭ ﺍﻟﺸﺒﻜﺔ ﺍﻟﺪﻭﻟﻴﺔ ﻟﻠﻤﻌﻠﻮﻣﺎﺕ )ﺍﻹﻧﺘﺮﻧﺖ( ﻭﺍﻛﺘﺐ ﻣﺜﺎﻻ ﻳﻮﺿﺢ ﺍﺳﺘﺨﺪﺍﻣﺎﺕ‬ ‫ﺍﻟﺒﺮﻣﺠﺔ ﺍﻟﺨﻄﻴﺔ ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻟﻤﺠﺎﻻﺕ ﺍﻟﺘﺎﻟﻴﺔ:‬ ‫ﺟ ﺇﺩﺍﺭﺓ ﺍﻟﻮﻗﺖ‬ ‫ب ﺍﻟﺼﻨﺎﻋﺔ‬ ‫أ ﺍﻻﻗﺘﺼﺎﺩ‬ ‫د ﺑﺤﻮﺙ ﺍﻟﻌﻤﻠﻴﺎﺕ‬ ‫..................................................................................................................................................................................................................................‬ ‫9 ﺍﻛﺘﺐ ﻣﺴﺄﻟﺔ ﻣﻦ ﻋﻨﺪﻙ ﻳﺘﻄﻠﺐ ﺣﻠﻬﺎ ﻛﺘﺎﺑﺔ ﺃﺭﺑﻊ ﻣﺘﺒﺎﻳﻨﺎﺕ ﺧﻄﻴﺔ، ﺛﻢ ﻣﺜﻞ ﻣﻨﻄﻘﺔ ﺍﻟﺤﻞ ﺑﻴﺎﻧﻴﺎ. ﺍﻛﺘﺐ ﺩﺍﻟﺔ‬ ‫ًّ‬ ‫ﺍﻟﻬﺪﻑ ﻟﻤﺴﺄﻟﺘﻚ، ﻭﺣﺪﺩ ﻣﺘﻰ ﻳﻜﻮﻥ ﻟﻬﺎ ﻗﻴﻤﺔ ﻋﻈﻤﻰ، ﺃﻭ ﻗﻴﻤﺔ ﺻﻐﺮﻯ، ﺛﻢ ﺃﻭﺟﺪ ﻫﺎﺗﻴﻦ ﺍﻟﻘﻴﻤﺘﻴﻦ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
‫ﺗﻤﺎﺭﻳﻦ ﻋﺎﻣﺔ‬ ‫1 ﺣﺪﺩ ﺻﺤﺔ ﺃﻭ ﺧﻄﺄ ﻛﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺘﺎﻟﻴﺔ، ﻭﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﻄﺄ، ﻭﺿﺢ ﺳﺒﺐ ﺍﻟﺨﻄﺄ ﻭﺇﺫﺍ ﻛﺎﻧﺖ ﺻﻮﺍﺑﺎ‬ ‫ً‬ ‫ﺃﻋﻂ ﺃﻣﺜﻠﺔ ﺗﻮﺿﺤﻬﺎ.‬ ‫( ........................................‬ ‫)‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ ‪ ،C‬ﺏ ∋ ﺡ ﻛﺎﻥ ‪ < C‬ﺏ ، ‪ ،C‬ﺏ ﻣﻮﺟﺒﺎﻥ ﻣﻌﺎ ﻓﺈﻥ ‪ < ٢C‬ﺏ٢.‬ ‫ﻭ‬ ‫ً‬ ‫( ........................................‬ ‫)‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ‪ ،C‬ﺏ ∋ ﺡ ﻛﺎﻥ ‪ < C‬ﺏ ، ‪ ،C‬ﺏ ﺳﺎﻟﺒﺎﻥ ﻣﻌﺎ ﻓﺈﻥ ‪ < ٢C‬ﺏ٢.‬ ‫ﻭ‬ ‫ً‬ ‫١‬ ‫١‬ ‫( ........................................‬ ‫ﺟ ﺇﺫﺍ ﻛﺎﻥ ‪ ،C‬ﺏ ∋ ﺡ ﻛﺎﻥ ‪ < C‬ﺏ ، ‪ ،C‬ﺏ ﻣﺘﺤﺪﺍﻥ ﻓﻰ ﺍﻹﺷﺎﺭﺓ ﻓﺈﻥ > . )‬ ‫ﻭ‬ ‫ﺏ‬ ‫‪C‬‬ ‫١‬ ‫١‬ ‫د ﺇﺫﺍ ﻛﺎﻥ ‪ ،C‬ﺏ ∋ ﺡ ﻛﺎﻥ ‪ < C‬ﺏ ، ‪ ،C‬ﺏ ﻣﺨﺘﻠﻔﺘﺎﻥ ﻓﻰ ﺍﻹﺷﺎﺭﺓ ﻓﺈﻥ > ﺏ . )‬ ‫ﻭ‬ ‫‪C‬‬ ‫(‬ ‫........................................‬ ‫2 ﺣﻞ ﻛﻼ ﻣﻦ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ ﺍﻵﺗﻴﺔ، ﻭﻣﺜﻞ ﺍﻟﺤﻞ ﻋﻠﻰ ﺧﻂ ﺍﻷﻋﺪﺍﺩ:‬ ‫ًّ‬ ‫أ ٢ﺱ + ٤ > ٥ﺱ - ٥ ..........................................................................................................................................................................‬ ‫.......................................................................................................................................................................................................................‬ ‫ب ٧ > ٥ﺱ + ٢ ‪١٢ H‬‬ ‫..........................................................................................................................................................................‬ ‫.......................................................................................................................................................................................................................‬ ‫3 ﻣﺜﻞ ﺑﻴﺎﻧﻴﺎ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﻛﻞ ﻣﻦ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ:‬ ‫ًّ‬ ‫ب‬ ‫ﺹ ‪- G‬ﺱ - ٤‬ ‫أ ﺹ ‪٢ H‬ﺱ + ١‬ ‫4 ﺣﻞ ﻛﻞ ﻧﻈﺎﻡ ﻣﻦ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ ﺑﻴﺎﻧﻴﺎ:‬ ‫ًّ‬ ‫أ ﺹ>ﺱ+٣‬ ‫ب ﺱ+ﺹ<٢‬ ‫٢ﺱ - ﺹ > ١‬ ‫ﺹ<٦+ﺱ‬ ‫ﺟ ﺹ > -٢ﺱ + ٣‬ ‫ﺟ ٢ﺱ + ٣ﺹ ‪١٢ H‬‬ ‫ﺱ‪٢G‬‬ ‫ﺹ‪١G‬‬ ‫5 ﺃﻭﺟﺪ ﻣﻦ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ ﻗﻴﻤﺘﻰ ﺱ، ﺹ ﺍﻟﺘﻰ ﺗﺠﻌﻞ ﻗﻴﻤﺔ‬ ‫ﺍﻟﺪﺍﻟﺔ ‪١ = S‬‬ ‫ﺱ + ﺹ ﻗﻴﻤﺔ ﺻﻐﺮﻯ. ...................................................................‬ ‫٢‬ ‫.................................................................................................................................................‬ ‫.................................................................................................................................................‬ ‫6 ﺍﻟﺮﺑﻂ ﺑﺎﻟﺼﻨﺎﻋﺔ ﻭﺭﺷﺔ ﺻﻐﻴﺮﺓ ﻟﻌﻤﻞ ﺍﻷﻭﺍﻧﻲ ﺍﻟﻤﻌﺪﻧﻴﺔ، ﺗﺼﻨﻊ ﻧﻮﻋﻴﻦ ﻣﻦ ﺍﻷﻭﺍﻧﻰ ‪ ،C‬ﺏ، ﻭﺗﺤﺘﺎﺝ ﺍﻵﻧﻴﺔ ‪ C‬ﺇﻟﻰ‬ ‫٠١ ﺩﻗﺎﺋﻖ ﻋﻤﻞ ﻣﻦ ﺍﻟﻤﺎﻛﻴﻨﺔ ﺍﻷﻭﻟﻰ، ٢١ ﺩﻗﻴﻘﺔ ﻋﻤﻞ ﻣﻦ ﺍﻟﻤﺎﻛﻴﻨﺔ ﺍﻟﺜﺎﻧﻴﺔ، ﺑﻴﻨﻤﺎ ﺗﺤﺘﺎﺝ ﺍﻵﻧﻴﺔ ﺏ ﺇﻟﻰ ٥١ ﺩﻗﻴﻘﺔ‬ ‫ﻋﻤﻞ ﻣﻦ ﺍﻟﻤﺎﻛﻴﻨﺔ ﺍﻷﻭﻟﻰ، ٠١ ﺩﻗﺎﺋﻖ ﻣﻦ ﺍﻟﺜﺎﻧﻴﺔ، ﻓﺈﺫﺍ ﻛﺎﻥ ﺭﺑﺢ ﺍﻵﻧﻴﺔ )ﺃ( ٤ ﺟﻨﻴﻬﺎﺕ ﻭﺭﺑﺢ ﺍﻵﻧﻴﺔ )ﺏ( ﻫﻮ‬ ‫٥ ﺟﻨﻴﻬﺎﺕ، ﻓﻤﺎ ﻋﺪﺩ ﺍﻷﻭﺍﻧﻲ ﻣﻦ ﻛﻞ ﻧﻮﻉ ﺣﺘﻰ ﻳﻜﻮﻥ ﺍﻟﺮﺑﺢ ﺃﻛﺒﺮ ﻣﺎ ﻳﻤﻜﻦ، ﻋﻠﻤﺎ ﺑﺄﻥ ﻛﻼ ﻣﻦ ﺍﻟﻤﺎﻛﻴﻨﺘﻴﻦ‬ ‫ًّ‬ ‫ً‬ ‫ﻻﺗﻌﻤﻞ ﺃﻛﺜﺮ ﻣﻦ ﺛﻤﺎﻧﻲ ﺳﺎﻋﺎﺕ?‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
كتاب الانشطة مصر - ترم ثانى -2013-2014
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كتاب الانشطة مصر - ترم ثانى -2013-2014

  • 1.
  • 2. äÉ«°VÉjôdG äÉÑjQóàdG h ᣰûfC’G ÜÉàc ≈fÉãdG ≈°SGQódG π°üØdG iƒfÉãdG ∫hC’G ∞°üdG OGóYEGh ¿óªdG §«£îJh iQÉÑμdGh ¥ô£dG AÉ°ûfEG É¡æe IOó©àe ä’Éée ≈a á«∏ªY äÉ≤«Ñ£J äÉ«°VÉjô∏d ∫ƒ£dG ø«H Ö°SÉæJ ≥ah É¡d á©WÉ≤dG äɪ«≤à°ùªdG h äɪ«≤à°ùªdG iRGƒJ ≈∏Y óªà©J ≈àdG É¡£FGôN .º°SôdG ≈a ∫ƒ£dGh ≈≤«≤ëdG ¢ùjƒ°ùdG IÉæb ≈àØ°V ø«H §Hôj iòdG ΩÓ°ùdG iôHƒμd IQƒ°üdGh
  • 3. ‫‪OGóYEG‬‬ ‫‪ˆG ÜÉL OGDƒa ôªY /CG‬‬ ‫‪™Ñ°†dG ≥«aƒJ π«Ñf /O.CG ídÉ°U ìƒàØdG ƒHCG ±ÉØY /O.CG‬‬ ‫‪Qóæμ°SEG ¢SÉ«dEG º«aGÒ°S /CG‬‬ ‫‪π«FÉahQ ≈Ø°Uh ΩÉ°üY /O.Ω.CG‬‬ ‫‪á°ûÑc ¢ùfƒj ∫ɪc /CG‬‬ ‫ﺟﻤﻴﻊ ﺍﻟﺤﻘﻮﻕ ﻣﺤﻔﻮﻇﺔ ﻻ ﻳﺠﻮﺭ ﻧﺸﺮ ﺃ￯ ﺟﺰﺀ ﻣﻦ ﻫﺬﺍ ﺍﻟﻜﺘﺎﺏ ﺃﻭ ﺗﺼﻮﻳﺮﻩ ﺃﻭ ﺗﺨﺰﻳﻨﻪ ﺃﻭ ﺗﺴﺠﻴﻠﻪ‬ ‫ﺑﺄ￯ ﻭﺳﻴﻠﺔ ﺩﻭﻥ ﻣﻮﺍﻓﻘﺔ ﺧﻄﻴﺔ ﻣﻦ ﺍﻟﻨﺎﺷﺮ.‬ ‫ﺷﺮﻛﺔ ﺳﻘﺎرة ﻟﻠﻨﺸﺮ‬ ‫‪Ω .Ω .¢T‬‬ ‫ﺍﻟﻄﺒﻌــﺔ ﺍﻷﻭﻟﻰ ٣١٠٢/٤١٠٢‬ ‫ﺭﻗﻢ ﺍﻹﻳــﺪﺍﻉ ٠٥٩٧ / ٣١٠٢‬ ‫ﺍﻟﺮﻗﻢ ﺍﻟﺪﻭﻟﻰ 5 - 300 - 607 - 779 - 879‬
  • 5. ‫ﺍﻟﻤﻘﺪﻣﺔ‬ ‫بسم الل ّٰه الرحمن الرحيم‬ ‫ﻳﺴﻌﺪﻧﺎ وﻧﺤﻦ ﻧﻘﺪم ﻫﺬا اﻟﻜﺘﺎب أن ﻧﻮﺿﺢ اﻟﻔﻠﺴﻔﺔ اﻟﺘﻰ ﺗﻢ ﻓﻰ ﺿﻮﺋﻬﺎ ﺑﻨﺎء اﻟﻤﺎدة اﻟﺘﻌﻠﻴﻤﻴﺔ وﻧﻮﺟﺰﻫﺎ ﻓﻴﻤﺎ ﻳﻠﻰ:‬ ‫1‬ ‫اﻟﺘﺄﻛﻴﺪ ﻋﲆ أن اﻟﻐﺎﻳﺔ اﻷﺳﺎﺳﻴﺔ ﻣﻦ ﻫﺬه اﻟﻜﺘﺐ ﻫﻰ ﻣﺴﺎﻋﺪة املﺘﻌﻠﻢ ﻋﲆ ﺣﻞ املﺸﻜﻼت واﺗﺨﺎذ اﻟﻘﺮارات ﰱ ﺣﻴﺎﺗﻪ‬ ‫اﻟﻴﻮﻣﻴﺔ، واﻟﺘﻰ ﺗﺴﺎﻋﺪه ﻋﲆ املﺸﺎرﻛﻪ ﰱ املﺠﺘﻤﻊ.‬ ‫2‬ ‫اﻟﺘﺄﻛﻴﺪ ﻋﲆ ﻣﺒﺪأ اﺳﺘﻤﺮارﻳﺔ اﻟﺘﻌﻠﻢ ﻣﺪى اﻟﺤﻴﺎة ﻣﻦ ﺧﻼل اﻟﻌﻤﻞ ﻋﲆ إﻛﺴﺎب اﻟﻄﻼب ﻣﻨﻬﺠﻴﺔ اﻟﺘﻔﻜري اﻟﻌﻠﻤﻰ، وأن‬ ‫ﻳﻤﺎرﺳﻮا اﻟﺘﻌﻠﻢ املﻤﺘﺰج ﺑﺎملﺘﻌﺔ واﻟﺘﺸﻮﻳﻖ، وذﻟﻚ ﺑﺎﻻﻋﺘﻤﺎد ﻋﲆ ﺗﻨﻤﻴﺔ ﻣﻬﺎرات ﺣﻞ املﺸﻜﻼت وﺗﻨﻤﻴﺔ ﻣﻬﺎرات اﻻﺳﺘﻨﺘﺎج‬ ‫واﻟﺘﻌﻠﻴﻞ، واﺳﺘﺨﺪام أﺳﺎﻟﻴﺐ اﻟﺘﻌﻠﻢ اﻟﺬاﺗﻰ واﻟﺘﻌﻠﻢ اﻟﻨﺸﻂ واﻟﺘﻌﻠﻢ اﻟﺘﻌﺎوﻧﻰ ﺑﺮوح اﻟﻔﺮﻳﻖ، واملﻨﺎﻗﺸﺔ واﻟﺤﻮار، وﺗﻘﺒﻞ‬ ‫آراء اﻵﺧﺮﻳﻦ، واملﻮﺿﻮﻋﻴﺔ ﰱ إﺻﺪار اﻷﺣﻜﺎم، ﺑﺎﻹﺿﺎﻓﺔ إﱃ اﻟﺘﻌﺮﻳﻒ ﺑﺒﻌﺾ اﻷﻧﺸﻄﺔ واﻹﻧﺠﺎزات اﻟﻮﻃﻨﻴﺔ.‬ ‫3‬ ‫ﺗﻘﺪﻳﻢ رؤى ﺷﺎﻣﻠﺔ ﻣﺘﻤﺎﺳﻜﺔ ﻟﻠﻌﻼﻗﺔ ﺑني اﻟﻌﻠﻢ واﻟﺘﻜﻨﻮﻟﻮﺟﻴﺎ واملﺠﺘﻤﻊ)‪ (STS‬ﺗﻌﻜﺲ دور اﻟﺘﻘﺪﱡم اﻟﻌﻠﻤﻰ ﰱ ﺗﻨﻤﻴﺔ‬ ‫املﺠﺘﻤﻊ املﺤﲆ، ﺑﺎﻹﺿﺎﻓﺔ إﱃ اﻟﱰﻛﻴﺰ ﻋﲆ ﻣﻤﺎرﺳﺔ اﻟﻄﻼب اﻟﺘﴫﱡف اﻟﻮاﻋﻰ اﻟﻔﻌّﺎل ﺣِ ﻴﺎل اﺳﺘﺨﺪام اﻷدوات اﻟﺘﻜﻨﻮﻟﻮﺟﻴﺔ.‬ ‫4‬ ‫5‬ ‫6‬ ‫ﺗﻨﻤﻴﺔ اﺗﺠﺎﻫﺎت إﻳﺠﺎﺑﻴﺔ ﺗﺠﺎه اﻟﺮﻳﺎﺿﻴﺎت ودراﺳﺘﻬﺎ وﺗﻘﺪﻳﺮ ﻋﻠﻤﺎﺋﻬﺎ.‬ ‫ﺗﺰوﻳﺪ اﻟﻄﻼب ﺑﺜﻘﺎﻓﺔ ﺷﺎﻣﻠﺔ ﻟﺤﺴﻦ اﺳﺘﺨﺪام املﻮارد اﻟﺒﻴﺌﻴﺔ املﺘﺎﺣﺔ.‬ ‫اﻻﻋﺘﻤﺎد ﻋﲆ أﺳﺎﺳﻴﺎت املﻌﺮﻓﺔ وﺗﻨﻤﻴﺔ ﻃﺮاﺋﻖ اﻟﺘﻔﻜري، وﺗﻨﻤﻴﺔ املﻬﺎرات اﻟﻌﻠﻤﻴﺔ، واﻟﺒﻌﺪ ﻋﻦ اﻟﺘﻔﺎﺻﻴﻞ واﻟﺤﺸﻮ،‬ ‫واﻹﺑﺘﻌﺎد ﻋﻦ اﻟﺘﻌﻠﻴﻢ اﻟﺘﻠﻘﻴﻨﻰ؛ ﻟﻬﺬا ﻓﺎﻻﻫﺘﻤﺎم ﻳﻮﺟﻪ إﱃ إﺑﺮاز املﻔﺎﻫﻴﻢ واملﺒﺎدئ اﻟﻌﺎﻣﺔ وأﺳﺎﻟﻴﺐ اﻟﺒﺤﺚ وﺣﻞ املﺸﻜﻼت‬ ‫وﻃﺮاﺋﻖ اﻟﺘﻔﻜري اﻷﺳﺎﺳﻴﺔ اﻟﺘﻰ ﺗﻤﻴﺰ ﻣﺎدة اﻟﺮﻳﺎﺿﻴﺎت ﻋﻦ ﻏريﻫﺎ.‬ ‫‪:≈∏j Ée ÜÉàμdG Gòg ≈a ≈YhQ ≥Ñ°S Ée Aƒ°V ≈ah‬‬ ‫ﺗﻘﺪﻳﻢ ﺗﻤﺎرﻳﻦ ﺗﺒﺪأ ﻣﻦ اﻟﺴﻬﻞ إﱃ اﻟﺼﻌﺐ، وﺗﺸﻤﻞ ﻣﺴﺘﻮﻳﺎت ﺗﻔﻜري ﻣﺘﻨﻮﻋﺔ.‬ ‫ﺗﻨﺘﻬﻰ ﻛﻞ وﺣﺪة ﺑﺘﻤﺎرﻳﻦ ﻋﺎﻣﺔ ﻋﲆ اﻟﻮﺣﺪة واﺧﺘﺒﺎر ﻟﻠﻮﺣﺪة واﺧﺘﺒﺎر ﺗﺮاﻛﻤﻰ ﻳﺸﻤﻞ اﻟﻌﺪﻳﺪ ﻣﻦ اﻷﺳﺌﻠﺔ اﻟﺘﻰ ﺗﻨﻮﻋﺖ‬ ‫َ‬ ‫ﺑني اﻷﺳﺌﻠﺔ املﻮﺿﻮﻋﻴﺔ، واملﻘﺎﻟﻴﺔ وذات اﻹﺟﺎﺑﺎت اﻟﻘﺼرية، وﺗﺘﻨﺎول اﻟﻮﺣﺪات اﻟﺴﺎﺑﻖ دراﺳﺘﻬﺎ وﺷﻤﻞ اﻟﻜﺘﺎب اﺧﺘﺒﺎرات‬ ‫ﻧﻬﺎﻳﺔ ﻛﻞ ﻓﺼﻞ دراﳻ.‬ ‫ﻛﻤﺎ روﻋﻰ اﺳﺘﺨﺪام ﻟﻐﺔ ﻣﻨﺎﺳﺒﺔ ﰱ ﻛﺘﺎﺑﺔ املﺴﺎﺋﻞ اﻟﺮﻳﺎﺿﻴﺔ واﻟﺤﻴﺎﺗﻴﺔ ﻣﻌﺘﻤﺪًا ﻋﲆ ﻣﺎﺳﺒﻖ دراﺳﺘﻪ ﺑﺎﻟﺴﻨﻮات‬ ‫اﻟﺴﺎﺑﻘﺔ، وﰱ ﺿﻮء املﺤﺼﻮل اﻟﻠﻐﻮى ﻟﻄﻼب ﻫﺬا اﻟﺼﻒ.‬ ‫وأخير ًا ..نتمنى أن نكون قد وفقنا فى إنجاز هذا العمل لما فيه خير لأولادنا، ولمصرنا العزيزة.‬ ‫والل ّٰه من وراء القصد، وهو يهدى إلى سواء السبيل‬
  • 6. ‫‪äÉjƒàëªdG‬‬ ‫‪IóMƒdG‬‬ ‫‪≈dhC’G‬‬ ‫1- 1‬ ‫1- 2‬ ‫1- 3‬ ‫1- 4‬ ‫1- 5‬ ‫ﺍﻟﻤﺼﻔﻮﻓﺎﺕ‬ ‫ﺗﻨﻈﻴﻢ اﻟﺒﻴﺎﻧﺎت ﻓﻰ ﻣﺼﻔﻮﻓﺎت‬ ‫ﺟﻤﻊ وﻃﺮح اﻟﻤﺼﻔﻮﻓﺎت‬ ‫ﺿﺮب اﻟﻤﺼﻔﻮﻓﺎت‬ ‫2‬ ‫4‬ ‫5‬ ‫6‬ ‫8‬ ‫01‬ ‫21‬ ‫31‬ ‫..........................................................................................................................................................................................‬ ‫.........................................................................................................................................................................................................‬ ‫.............................................................................................................................................................................................................................‬ ‫اﻟﻤﺤﺪدات‬ ‫.........................................................................................................................................................................................................................................................‬ ‫اﻟﻤﻌﻜﻮس اﻟﻀﺮﺑﻰ ﻟﻠﻤﺼﻔﻮﻓﺔ‬ ‫...........................................................................................................................................................................................‬ ‫ﺗﻤﺎرﻳﻦ ﻋﺎﻣﺔ‬ ‫.........................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر اﻟﻮﺣﺪة‬ ‫......................................................................................................................................................................................................................................‬ ‫اﻻﺧﺘﺒﺎر اﻟﺘﺮاﻛﻤﻰ‬ ‫‪IóMƒdG‬‬ ‫‪á«fÉãdG‬‬ ‫............................................................................................................................................................................................................................‬ ‫ﺍﻟ‪‬ﻣﺠﺔ ﺍﻟﺨﻄﻴﺔ‬ ‫2-1‬ ‫اﻟﻤﺘﺒﺎﻳﻨﺎت اﻟﺨﻄﻴﺔ‬ ‫61‬ ‫2-2‬ ‫ﺣﻞ أﻧﻈﻤﺔ ﻣﻦ اﻟﻤﺘﺒﺎﻳﻨﺎت اﻟﺨﻄﻴﺔ ﺑﻴﺎﻧﻴٍّﺎ‬ ‫81‬ ‫2-3‬ ‫اﻟﺒﺮﻣﺠﺔ اﻟﺨﻄﻴﺔ واﻟﺤﻞ اﻷﻣﺜﻞ‬ ‫ﺗﻤﺎرﻳﻦ ﻋﺎﻣﺔ .........................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر اﻟﻮﺣﺪة ......................................................................................................................................................................................................................................‬ ‫اﻻﺧﺘﺒﺎر اﻟﺘﺮاﻛﻤﻰ ............................................................................................................................................................................................................................‬ ‫..........................................................................................................................................................................................................................‬ ‫.........................................................................................................................................................‬ ‫91‬ ‫12‬ ‫22‬ ‫32‬ ‫........................................................................................................................................................................................‬ ‫‪IóMƒdG‬‬ ‫‪áãdÉãdG‬‬ ‫ﺍﳴﺘﺠﻬﺎﺕ‬ ‫3-1‬ ‫اﻟﻜﻤﻴﺎت اﻟﻘﻴﺎﺳﻴﺔ واﻟﻜﻤﻴﺎت اﻟﻤﺘﺠﻬﺔ، واﻟﻘﻄﻌﺔ اﻟﻤﺴﺘﻘﻴﻤﺔ اﻟﻤﻮﺟﻬﺔ‬ ‫62‬ ‫3-2‬ ‫اﻟﻤﺘﺠﻬﺎت‬ ‫82‬ ‫3-3‬ ‫اﻟﻌﻤﻠﻴﺎت ﻋﻠﻰ اﻟﻤﺘﺠﻬﺎت‬ ‫03‬ ‫3-4‬ ‫ﺗﻄﺒﻴﻘﺎت اﻟﻤﺘﺠﻬﺎت‬ ‫ﺗﻤﺎرﻳﻦ ﻋﺎﻣﺔ .........................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر اﻟﻮﺣﺪة ......................................................................................................................................................................................................................................‬ ‫اﻻﺧﺘﺒﺎر اﻟﺘﺮاﻛﻤﻰ ............................................................................................................................................................................................................................‬ ‫..............................................................‬ ‫....................................................................................................................................................................................................................................................‬ ‫........................................................................................................................................................................................................‬ ‫23‬ ‫53‬ ‫63‬ ‫73‬ ‫.....................................................................................................................................................................................................................‬
  • 7. ‫‪IóMƒdG‬‬ ‫‪á©HGôdG‬‬ ‫ﺍﻟﺨﻂ ﺍﻟﻤﺴﺘﻘﻴﻢ‬ ‫4-1‬ ‫ﺗﻘﺴﻴﻢ ﻗﻄﻌﺔ ﻣﺴﺘﻘﻴﻤﺔ‬ ‫04‬ ‫4-2‬ ‫ﻣﻌﺎدﻟﺔ اﻟﺨﻂ اﻟﻤﺴﺘﻘﻴﻢ‬ ‫14‬ ‫4-3‬ ‫ﻗﻴﺎس اﻟﺰاوﻳﺔ ﺑﻴﻦ ﻣﺴﺘﻘﻴﻤﻴﻦ‬ ‫34‬ ‫4-4‬ ‫ﻃﻮل اﻟﻌﻤﻮد اﻟﻤﺮﺳﻮم ﻣﻦ ﻧﻘﻄﺔ إﻟﻰ ﺧﻂ ﻣﺴﺘﻘﻴﻢ‬ ‫54‬ ‫4-5‬ ‫اﻟﻤﻌﺎدﻟﺔ اﻟﻌﺎﻣﺔ ﻟﻠﻤﺴﺘﻘﻴﻢ اﻟﻤﺎر ﺑﻨﻘﻄﺔ ﺗﻘﺎﻃﻊ ﻣﺴﺘﻘﻴﻤﻴﻦ‬ ‫74‬ ‫84‬ ‫94‬ ‫05‬ ‫.........................................................................................................................................................................................................‬ ‫ﺗﻤﺎرﻳﻦ ﻋﺎﻣﺔ‬ ‫..........................................................................................................................................................................................................‬ ‫......................................................................................................................................................................................‬ ‫.......................................................................................................................‬ ‫.............................................................................................‬ ‫.........................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر اﻟﻮﺣﺪة‬ ‫......................................................................................................................................................................................................................................‬ ‫اﻻﺧﺘﺒﺎر اﻟﺘﺮاﻛﻤﻰ‬ ‫............................................................................................................................................................................................................................‬ ‫‪IóMƒdG‬‬ ‫‪á°ùeÉÿG‬‬ ‫ﺣﺴﺎﺏ ﺍﳴﺜﻠﺜﺎﺕ‬ ‫5-1‬ ‫اﻟﻤﺘﻄﺎﺑﻘﺎت اﻟﻤﺜﻠﺜﻴﺔ.‬ ‫25‬ ‫5-2‬ ‫ﺣﻞ اﻟﻤﻌﺎدﻻت اﻟﻤﺜﻠﺜﻴﺔ.‬ ‫35‬ ‫5-3‬ ‫ﺣﻞ اﻟﻤﺜﻠﺚ اﻟﻘﺎﺋﻢ اﻟﺰاوﻳﺔ.‬ ‫45‬ ‫5-4‬ ‫زواﻳﺎ اﻻرﺗﻔﺎع وزواﻳﺎ اﻻﻧﺨﻔﺎض‬ ‫65‬ ‫5-5‬ ‫اﻟﻘﻄﺎع اﻟﺪاﺋﺮى‬ ‫75‬ ‫5-6‬ ‫اﻟﻘﻄﻌﺔ اﻟﺪاﺋﺮﻳﺔ.‬ ‫85‬ ‫5-7‬ ‫اﻟﻤﺴﺎﺣﺎت.‬ ‫95‬ ‫16‬ ‫26‬ ‫36‬ ‫56‬ ‫17‬ ‫..................................................................................................................................................................................................................‬ ‫............................................................................................................................................................................................................‬ ‫....................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................‬ ‫.....................................................................................................................................................................................................................................‬ ‫................................................................................................................................................................................................................................‬ ‫................................................................................................................................................................................................................................................‬ ‫ﺗﻤﺎرﻳﻦ ﻋﺎﻣﺔ‬ ‫.........................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر اﻟﻮﺣﺪة‬ ‫......................................................................................................................................................................................................................................‬ ‫اﻻﺧﺘﺒﺎر اﻟﺘﺮاﻛﻤﻰ‬ ‫اﺧﺘﺒﺎرات ﻋﺎﻣﺔ‬ ‫............................................................................................................................................................................................................................‬ ‫.........................................................................................................................................................................................................................................................................‬ ‫إﺟﺎﺑﺎت ﺑﻌﺾ اﻟﺘﻤﺎرﻳﻦ‬ ‫..................................................................................................................................................................................................................................................‬
  • 8. ‫ﺍﻟﺠﺒﺮ‬ ‫‪IóMƒdG‬‬ ‫1‬ ‫ﺍﻟﻤﺼﻔﻮﻓﺎﺕ‬ ‫‪Matrices‬‬ ‫دروس اﻟﻮﺣﺪة‬ ‫ﺍﻟﺪﺭﺱ )١ - ١(: ﺗﻨﻈﻴﻢ ﺍﻟﺒﻴﺎﻧﺎﺕ ﻓﻰ ﻣﺼﻔﻮﻓﺎﺕ.‬ ‫ﺍﻟﺪﺭﺱ )١ - ٢(: ﺟﻤﻊ ﻭﻃﺮﺡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ.‬ ‫ﺍﻟﺪﺭﺱ )١ - ٣(: ﺿﺮﺏ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ .‬ ‫ﺍﻟﺪﺭﺱ )١ - ٤(: ﺍﻟﻤﺤﺪﺩﺍﺕ .‬ ‫ﺍﻟﺪﺭﺱ )١ - ٥(: ﺍﻟﻤﻌﻜﻮﺱ ﺍﻟﻀﺮﺑﻰ ﻟﻠﻤﺼﻔﻮﻓﺔ‬
  • 9. ‫ﺗﻨﻈﻴﻢ اﻟﺒﻴﺎﻧﺎت ﻓﻰ ﻣﺼﻔﻮﻓﺎت‬ ‫1-1‬ ‫‪Organizing Data in Matrices‬‬ ‫1 ‪ ،D ، C‬ﺟـ ، ‪ E‬ﺃﺭﺑﻊ ﻣﺪﻥ ، ﻓﺈﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻤﺴﺎﻓﺔ ﺑﺎﻟﻜﻴﻠﻮ ﻣﺘﺮﺍﺕ ﺑﻴﻦ ﺃﻱ ﻣﺪﻳﻨﺘﻴﻦ ﻣﻮﺿﺤﺔ ﻓﻰ ﺍﻟﺠﺪﻭﻝ ﺍﻟﻤﻘﺎﺑﻞ.‬ ‫أ ﺍﻛﺘﺐ ﻣﺼﻔﻮﻓﺔ ﺗﻤﺜﻞ ﻫﺬه ﺍﻟﻤﻌﻠﻮﻣﺎﺕ.‬ ‫‪ C‬ﺏ ﺟـ‬ ‫ب ﺑﻔﺮﺽ ﺃﻥ ‪ M‬ﻫﻰ ﺍﻟﻤﺼﻔﻮﻓﺔ ﺍﻟﻤﻄﻠﻮﺑﺔ ﻓﻰ ) ﺃ ( ﺃﻭﺟﺪ ﻣﺎﻳﻠﻲ:‬ ‫‪٨٠ ٧٥ ٠ C‬‬ ‫١- ﺱ ، ﻣﺎﺫﺍ ﻳﻌﻨﻰ ﺫﻟﻚ? ..................................................................................................‬ ‫٢٣‬ ‫ﺏ ٥٧ ٠ ٦٥‬ ‫٢- ﺱ ، ﻣﺎﺫﺍ ﻳﻌﻨﻰ ﺫﻟﻚ? ...................................................................................................‬ ‫ﺟـ ٠٨ ٦٥ ٠‬ ‫٣٢‬ ‫٣- ﻣﺎ ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺱ ، ﺱ ? ........................................................................................‬ ‫٣٢‬ ‫٢٣‬ ‫ﺟ‬ ‫ﺍﻛﺘﺐ ﺟﻤﻴﻊ ﻋﻨﺎﺻﺮ ﺍﻟﺼﻒ ﺍﻟﺜﺎﻧﻲ ﻟﻠﻤﺼﻔﻮﻓﺔ ‪............................................................................................................ .M‬‬ ‫د ﺍﻛﺘﺐ ﺟﻤﻴﻊ ﻋﻨﺎﺻﺮ ﺍﻟﻌﻤﻮﺩ ﺍﻟﺜﺎﻧﻲ ﻟﻠﻤﺼﻔﻮﻓﺔ ‪ .M‬ﻣﺎﺫﺍ ﺗﺴﺘﻨﺘﺞ ﻣﻦ ﺍﻟﺒﻨﺪﻳﻦ )٤(، )٥(?‬ ‫.......................................................................................................................................................................................................................‬ ‫ﻫ ﺃﻭﺟﺪ ﺱ ﻙ ﻙ ﻋﻨﺪﻣﺎ ﻙ = ١، ٢، ٣ ﻣﺎﺫﺍ ﺗﻼﺣﻆ?‬ ‫.......................................................................................................................‬ ‫و ﺃﻛﻤﻞ ﻣﺎﻳﺄﺗﻰ:‬ ‫١- ‪ M‬ﻣﺼﻔﻮﻓﺔ ﻋﻠﻰ ﺍﻟﻨﻈﻢ .............................................................................................................................................................‬ ‫ﻟﺠﻤﻴﻊ ﻗﻴﻢ .........................................................................................................................................................‬ ‫٢- ﺱﻱ ﻫـ= ﺱ‬ ‫ﻫـ ﻱ‬ ‫2 ﻣﺎ ﻋﺪﺩ ﻋﻨﺎﺻﺮ ﻛﻞ ﻣﻦ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﺍﻵﺗﻴﺔ:‬ ‫أ ﻣﺼﻔﻮﻓﺔ ﻋﻠﻰ ﺍﻟﻨﻈﻢ ٢ * ٣ ................................................................................................................................................................‬ ‫ب‬ ‫ﻣﺼﻔﻮﻓﺔ ﻋﻠﻰ ﺍﻟﻨﻈﻢ ٢ * ٢ ................................................................................................................................................................‬ ‫ﺟ‬ ‫ﻣﺼﻔﻮﻓﺔ ﻋﻠﻰ ﺍﻟﻨﻈﻢ ٣ * ٢ ................................................................................................................................................................‬ ‫3 ﺃﻭﺟﺪ ﻗﻴﻢ ‪ ،C‬ﺏ، ﺟـ، ‪ E‬ﺇﺫﺍ ﻛﺎﻥ:‬ ‫٣‬ ‫٥ ‪٢ - C l = b‬‬‫أ ‪l‬‬ ‫‪٣-C‬‬ ‫ب ‪l‬‬ ‫٣‪٢ - E‬‬ ‫٥١‬ ‫٠‬ ‫٢ﺏ‬ ‫٢‪ + C‬ﺟـ‬ ‫ﺟـ‬ ‫‪l = b‬‬ ‫٣‪C‬‬ ‫٢ﺏ-‪E‬‬ ‫٢ﺏ + ١ ‪b‬‬ ‫٦١‬ ‫٠١‬ ‫٠١‬ ‫‪b‬‬ ‫...................................................................................................................‬ ‫...................................................................................................................‬ ‫4 ﺍﻟﺮﺑﻂ ﺑﺎﻟﺼﻨﺎﻋﺔ: ﻳﺒﻴﻦ ﺍﻟﺠﺪﻭﻝ ﺍﻟﻤﻘﺎﺑﻞ ﻋﺪﺩ ﺍﻟﻤﺼﺎﻧﻊ ﺍﻷﻫﻠﻴﺔ‬ ‫ﺍﻟﻌﺎﻣﻠﺔ ﻓﻰ ﻗﻄﺎﻋﻲ ﺻﻨﺎﻋﺔ ﺍﻷﻏﺬﻳﺔ ﻭﺍﻟﻤﺼﻨﻮﻋﺎﺕ ﺍﻟﺠﻠﺪﻳﺔ ﻓﻰ ﺛﻼﺙ‬ ‫ﻣﺪﻥ ﻣﺨﺘﻠﻔﺔ ﻣﻦ ﻣﺪﻥ ﺑﻌﺾ ﻣﺤﺎﻓﻈﺎﺕ ﺟﻤﻬﻮﺭﻳﺔ ﻣﺼﺮ ﺍﻟﻌﺮﺑﻴﺔ.‬ ‫أ ﻧﻈﻢ ﺍﻟﺒﻴﺎﻧﺎﺕ ﻓﻰ ﻣﺼﻔﻮﻓﺔ.‬ ‫.........................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬ ‫¯‬ ‫¯‬ ‫٤٤‬ ‫٨٢‬ ‫٧٣‬ ‫٨٦‬ ‫٢٥‬ ‫٤١‬
  • 10. ‫ب ﺍﺟﻤﻊ ﻋﻨﺎﺻﺮ ﻛﻞ ﻋﻤﻮﺩ، ﻣﺎ ﺗﻔﺴﻴﺮﻙ ﻟﻠﻨﺘﺎﺋﺞ ﺍﻟﺘﻰ ﺣﺼﻠﺖ ﻋﻠﻴﻬﺎ?‬ ‫......................................................................................................................................................................................................................‬ ‫ﺟ ﺍﺟﻤﻊ ﻋﻨﺎﺻﺮ ﻛﻞ ﺻﻒ. ﻫﻞ ﺍﻟﻨﺘﺎﺋﺞ ﺍﻟﺘﻰ ﺣﺼﻠﺖ ﻋﻠﻴﻬﺎ ﻳﻤﻜﻦ ﺃﻥ ﺗﺰﻭﺩﻧﺎ ﺑﺒﻴﺎﻧﺎﺕ ﺫﺍﺕ ﻣﻌﻨﻰ? ﻓﺴﺮ ﺇﺟﺎﺑﺘﻚ.‬ ‫..............................................................................................................................................................................................................................‬ ‫٤‬ ‫= ٤‬ ‫١‬‫5 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻣﻦ ‪ D ،C‬ﺇﺫﺍ ﻛﺎﻥ ‪l‬‬ ‫‪l b‬‬ ‫٣‬ ‫٢‪٣ ١ - C‬ﺏ + ١‬ ‫١ ‪b‬‬‫٧‬ ‫..................................................................................................................................................................................................................................‬ ‫6 ﺇﺫﺍ ﻛﺎﻥ ‪ ، b ٣- ٢ l = C‬ﺏ = ‪ b ١- E٢ l‬ﺣﻴﺚ ‪ = C‬ﺏ‬ ‫١ ٤‬‫٣ﻫـ ٤‬ ‫ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻣﻦ ‪ ، E‬ﻫـ.‬ ‫ﻣﺪ‬ ‫..................................................................................................................................................................................................................................‬ ‫7 ﺇﺫﺍ ﻛﺎﻧﺖ ‪p = C‬‬ ‫١‬ ‫٢‬ ‫٤‬ ‫٠‬ ‫١‬ ‫٥‬ ‫٢‬ ‫٣ ‪ ، f‬ﺏ = ‪p‬‬‫-١‬ ‫٣‬ ‫٢‬‫-٤‬ ‫٠‬ ‫٣‬ ‫-٥‬ ‫-٢‬ ‫٣ ‪f‬‬ ‫٥‬ ‫ﺃﻭﺟﺪ ‪ + C‬ﺏ  ،  ‪ - C‬ﺏ   ،  ‪٢+ C‬ﺏ  ،   ﺏ -٣‪C‬‬ ‫8 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ: ﺇﺫﺍ ﻛﺎﻧﺖ ‪C) = C‬ﺱ ﺹ( ﻟﻜﻞ ﺱ، ﺹ ∋} ١، ٢، ٣{ ﺍﻛﺘﺐ ﺍﻟﻤﺼﻔﻮﻓﺔ‪ C‬ﺇﺫﺍ ﻋﻠﻢ ﺃﻥ‪C‬ﺱ ﺹ = ﺹ - ﺱ،‬ ‫ﻣﺪ‬ ‫ﺛﻢ ﺃﻭﺟﺪ ‪C‬‬ ‫..................................................................................................................................................................................................‬ ‫ﻧﺸﺎط‬ ‫ﺃﻧﺸﺊ ﻣﺼﻔﻮﻓﺔ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺑﻴﺎﻧﺎﺕ ﺣﻴﺎﺗﻴﺔ ﺗﻜﻮﻥ ﻣﺠﺎﻣﻴﻊ ﻋﻨﺎﺻﺮ ﺃﻋﻤﺪﺗﻬﺎ ﺫﺍﺕ ﻣﻌﻨﻰ، ﻭﻣﺠﺎﻣﻴﻊ ﻋﻨﺎﺻﺮ ﺻﻔﻮﻓﻬﺎ‬ ‫ﻟﻴﺲ ﻟﻬﺎ ﻣﻌﻨﻰ. ﺃﺩﺧﻞ ﺑﻴﺎﻧﺎﺕ ﺍﻟﻤﺼﻔﻮﻓﺔ ﻋﻠﻰ ﺑﺮﻧﺎﻣﺞ ﺍﻟﺠﺪﺍﻭﻝ ﺍﻹﻟﻜﺘﺮﻭﻧﻴﺔ ﻭﺗﺤﻘﻖ ﻣﻦ ﺻﺤﺔ ﺍﻟﻤﺠﺎﻣﻴﻊ ﺍﻟﺘﻰ‬ ‫ﺣﺼﻠﺖ ﻋﻠﻴﻬﺎ، ﺛﻢ ﻓﺴﺮ ﻣﺎﺫﺍ ﺗﻌﻨﻰ ﻣﺠﺎﻣﻴﻊ ﺍﻷﻋﻤﺪﺓ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
  • 11. ‫ﺟﻤﻊ وﻃﺮح اﻟﻤﺼﻔﻮﻓﺎت‬ ‫1-2‬ ‫‪Adding and subtracting Matrices‬‬ ‫1 ﺇﺫﺍ ﻛﺎﻥ ‪ b ١- ٠ ٢- l = C‬ﻛﺎﻧﺖ ﻙ١ = ٢ ، ﻙ٢ = -١ ﻓﺄﻭﺟﺪ ﻛﻼ ﻣﻦ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﺍﻵﺗﻴﺔ: ﻙ١‪ ،C‬ﻙ٢‪C‬‬ ‫ﻭ‬ ‫ًّ‬ ‫٥‬ ‫٤‬ ‫٠‬ ‫..................................................................................................................................................................................................................................‬ ‫٨ ٤‬‫٧ ٠‬‫2 ﺇﺫﺍ ﻛﺎﻥ ‪l = C‬‬ ‫٥ ‪ ، b‬ﺏ = ‪ f ٧ ٠ p‬ﻓﺄﻭﺟﺪ ﻧﺎﺗﺞ ﺍﻟﻌﻤﻠﻴﺎﺕ ﺍﻵﺗﻴﺔ ﺇﻥ ﺃﻣﻜﻦ، ﻣﻊ ﺫﻛﺮ ﺍﻟﺴﺒﺐ ﻓﻰ‬‫٤ ٧ ٥‬ ‫٦ -٥‬‫ﺣﺎﻟﺔ ﺗﻌﺬﺭ ﺇﺟﺮﺍﺀ ﺍﻟﻌﻤﻠﻴﺔ‬ ‫ﻣﺪ‬ ‫ب ‪+C‬ﺏ‬ ‫أ ‪+C‬ﺏ‬ ‫ﺟ ‪ C‬ﻣﺪ + ﺏ‬ ‫-٤ -٢‬ ‫١‬ ‫-٢ -٤‬ ‫٤ -٣‬ ‫٠‬ ‫٥‬ ‫3 ﺇﺫﺍ ﻛﺎﻥ ‪ ، f ٢- ٠ p = N ، f ٦ ٣ p =M‬ﻉ = ‪ f ٢ ٣- p‬ﻓﺄﻭﺟﺪ ﺍﻟﻤﺼﻔﻮﻓﺔ ٣ ‪ + N - M‬ﻉ‬ ‫٠‬ ‫٤‬ ‫٦‬ ‫..................................................................................................................................................................................................................................‬ ‫4 ﺇﺫﺍ ﻛﺎﻥ: ‪p =C‬‬ ‫٢ -٦ ٢‬ ‫٤ ٨ -٦‬ ‫٢ -٤ ٨ ‪ ، f‬ﺏ = ‪٠ ١٠- ٤ p‬‬ ‫١ ٨ -٤‬‫٦ ٢١ ٠‬ ‫‪ f‬ﻓﺄﻭﺟﺪ ﺍﻟﻤﺼﻔﻮﻓﺔ ‪ M‬ﺑﺤﻴﺚ : ‪٣ - C٢ = M‬ﺏ‬ ‫..................................................................................................................................................................................................................................‬ ‫5 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ : ﺃﻭﺟﺪ ﻗﻴﻢ ﺃ، ﺏ، ﺟـ، ‪ E‬ﺍﻟﺘﻰ ﺗﺤﻘﻖ ﺍﻟﻤﻌﺎﺩﻟﺔ:‬ ‫٢ ﺃ ٣ = ٣ ﺃ ‪ ٤ - E‬ﺟـ‬ ‫‪l‬‬ ‫‪b‬‬ ‫‪l‬‬ ‫‪b‬‬ ‫‪l‬‬ ‫ﺟـ -٢‬ ‫٠‬ ‫٦ ﺏ‬ ‫٣‬ ‫ﺃ‬ ‫‪b‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫6 ﻣﺴﺄﻟﺔ ﻣﻔﺘﻮﺣﺔ : ﺍﺧﺘﺮ ﻣﻦ ﻋﻨﺪﻙ ﻣﺼﻔﻮﻓﺘﻴﻦ ‪ ،C‬ﺏ ﻟﻬﻤﺎ ﻧﻔﺲ ﺍﻟﻨﻈﻢ ، ﺛﻢ ﺃﺛﺒﺖ ﺃﻥ :‬ ‫ب )‪ + C‬ﺏ( ﻣﺪ = ‪C‬ﻣﺪ + ﺏ‬ ‫ﻣﺪ‬ ‫أ ‪ - C‬ﺏ = ‪-) + C‬ﺏ(‬ ‫ﺟ )‪ - C‬ﺏ( ﻣﺪ = ‪C‬ﻣﺪ - ﺏ‬ ‫ﻣﺪ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫ﻧﺸﺎط‬ ‫1 ﺍﻛﺘﺐ ﻣﺴﺄﻟﺔ ﺣﻴﺎﺗﻴﺔ ﻣﻦ ﻋﻨﺪﻙ ﻳﻤﻜﻦ ﺣﻠﻬﺎ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺟﻤﻊ ﺃﻭ ﻃﺮﺡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ.‬ ‫2 ﺍﺑﺤﺚ ﻓﻰ ﻣﻜﺘﺒﺘﻚ ﺍﻟﻤﺪﺭﺳﻴﺔ ﺃﻭ ﻋﻠﻰ ﺍﻟﺸﺒﻜﺔ ﺍﻟﺪﻭﻟﻴﺔ ﻟﻺﻧﺘﺮﻧﺖ ﺗﻄﺒﻴﻘﺎﺕ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﻓﻰ ﺍﻟﻌﻠﻮﻡ ﺍﻷﺧﺮﻯ.‬ ‫¯‬ ‫−‬ ‫¯‬
  • 12. ‫ﺿﺮب اﻟﻤﺼﻔﻮﻓﺎت‬ ‫1-3‬ ‫‪Matrix Multiplication‬‬ ‫1 ﺇﺫﺍ ﻛﺎﻥ ‪٣ = C‬‬ ‫‪l‬‬ ‫٦‬ ‫١ ‪ ، b‬ﺏ = ‪ b ٣ ١- l‬ﻓﺄﻭﺟﺪ ﻛﻼ ﻣﻤﺎﻳﺄﺗﻰ:‬ ‫ًّ‬ ‫٢‬ ‫٣‬ ‫٩‬ ‫أ ‪C‬ﺏ‬ ‫..........................................................................................................................................................................‬ ‫ب ﺏ‪C‬‬ ‫..........................................................................................................................................................................‬ ‫ﺟ )‪ + C‬ﺏ(‪C‬‬ ‫..........................................................................................................................................................................‬ ‫2 ﺇﺫﺍ ﻛﺎﻥ ٢‬ ‫‪l‬‬ ‫٤‬ ‫٨ ‪ b‬ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻣﻦ ﺱ، ﺹ:‬ ‫= ٧‬ ‫ﺱ ٧‬ ‫٣‬ ‫‪l b‬‬ ‫‪l b‬‬ ‫١١ ٨١‬ ‫٣ ﺹ‬ ‫٥‬ ‫..................................................................................................................................................................................................................................‬ ‫3 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ: ﺇﺫﺍ ﻛﺎﻥ ‪ ،C‬ﺏ ﻣﺼﻔﻮﻓﺘﻴﻦ، ﻛﺎﻧﺖ‬ ‫ﻭ‬ ‫ﺩﺍﺋﻤﺎ ﺃﻥ ‪= C‬‬ ‫ً‬ ‫ﺃﻭ ﺏ =‬ ‫ﻫﻰ ﺍﻟﻤﺼﻔﻮﻓﺔ ﺍﻟﺼﻔﺮﻳﺔ ، ‪ C‬ﺏ =‬ ‫ﺍﺗﺨﺬ ‪ ، b ٣ ٢- l = C‬ﺏ = ٣‬ ‫‪l‬‬ ‫٢ -٣‬ ‫٢‬ ‫، ﻓﻬﻞ ﻫﺬﺍ ﻳﻌﻨﻰ‬ ‫٦ ‪ b‬ﺛﻢ ﺍﻋﺮﺽ ﻟﺮﺃﻳﻚ ﺑﻌﺪ ﺫﻟﻚ.‬ ‫٤‬ ‫............................................................................................................................................................................................................................................‬ ‫............................................................................................................................................................................................................................................‬ ‫4 ﺍﻟﺮﺑﻂ ﺑﺎﻟﺴﻴﺎﺣﺔ: ﻳﺴﺘﻬﻠﻚ ﺃﺣﺪ ﺍﻟﻔﻨﺎﺩﻕ ﻓﻰ ﻣﺪﻳﻨﺔ ﺍﻟﻐﺮﺩﻗﺔ ﺍﻟﺴﻴﺎﺣﻴﺔ ﺍﻟﻜﻤﻴﺎﺕ ﺍﻵﺗﻴﺔ ﻣﻦ ﺍﻟﻠﺤﻮﻡ‬ ‫ﻭﺍﻟﺨﻀﺮﺍﻭﺍﺕ ﻭﺍﻟﻔﺎﻛﻬﺔ ﺑﺎﻟﻜﻴﻠﻮ ﺟﺮﺍﻡ، ﻓﻰ ﻭﺟﺒﺘﻰ ﺍﻟﻐﺪﺍﺀ ﻭﺍﻟﻌﺸﺎﺀ، ﻭﺫﻟﻚ ﺗﺒﻌﺎ ﻟﻠﺠﺪﻭﻝ ﺍﻟﺘﺎﻟﻰ:‬ ‫ً‬ ‫¯‬ ‫٠٠١‬ ‫٠٨‬ ‫٠٠٢‬ ‫٠٢١‬ ‫٠٥١‬ ‫٠٠١‬ ‫ﻓﺈﺫﺍ ﻛﺎﻥ ﻣﺘﻮﺳﻂ ﺳﻌﺮ ﺍﻟﻜﻴﻠﻮ ﺟﺮﺍﻡ ﻣﻦ ﺍﻟﻠﺤﻮﻡ ٥٦ ﺟﻨﻴﻬﺎ ﻭﻣﺘﻮﺳﻂ ﺳﻌﺮ ﺍﻟﻜﻴﻠﻮ ﺟﺮﺍﻡ ﻣﻦ ﺍﻟﺨﻀﺮﺍﻭﺍﺕ‬ ‫ً‬ ‫ﺃﺭﺑﻌﺔ ﺟﻨﻴﻬﺎﺕ ﻭﻣﺘﻮﺳﻂ ﺳﻌﺮ ﺍﻟﻜﻴﻠﻮ ﺟﺮﺍﻡ ﻣﻦ ﺍﻟﻔﺎﻛﻬﺔ ﻫﻮ ﺧﻤﺴﺔ ﺟﻨﻴﻬﺎﺕ ، ﻓﺄﻭﺟﺪ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺿﺮﺏ‬ ‫ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﺍﻟﺘﻜﺎﻟﻴﻒ ﺍﻟﻜﻠﻴﺔ ﻟﻠﻮﺟﺒﺘﻴﻦ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
  • 13. ‫اﻟﻤﺤﺪدات‬ ‫1-4‬ ‫‪Determinants‬‬ ‫1 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻣﻦ ﺍﻟﻤﺤﺪﺩﺍﺕ ﺍﻵﺗﻴﺔ:‬ ‫٧ ٥‬ ‫ب‬ ‫أ‬ ‫٣‬ ‫٢‬ ‫..............................................‬ ‫د‬ ‫ﺟ‬ ‫..............................................‬ ‫‪C‬‬ ‫‪+C‬ﺱ‬ ‫ﺏ+ﺹ ﺏ‬ ‫ﻫ‬ ‫..............................................‬ ‫ز‬ ‫١ ٢‬ ‫٣ -١‬ ‫ﺱ + ١ ﺱ٢ + ١‬ ‫ﺹ + ١ ﺹ٢ + ١‬ ‫..............................................‬ ‫٠ ٢٤ ٣‬ ‫٢ ٨١ ٧‬ ‫٠ ٨٢ ٣‬ ‫ح‬ ‫..............................................‬ ‫..............................................‬ ‫١ ٢‬ ‫١ ٤‬‫٠ ٧‬ ‫و‬ ‫..............................................‬ ‫٣ -٤ -٣‬ ‫٢ ٠ -١٣‬ ‫٥ ٠ ٢‬ ‫٦ -٣‬ ‫٩١ -٧‬ ‫٣‬ ‫٤‬ ‫٨‬ ‫..............................................‬ ‫٣١ ٣ ٣٢‬ ‫٠٣ ٧ ٥‬ ‫٠ ٠ ١‬ ‫ط‬ ‫..............................................‬ ‫2 ﺣﻞ ﻛﻞ ﻧﻈﺎﻡ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺨﻄﻴﺔ ﺍﻟﺘﺎﻟﻴﺔ ﺑﻄﺮﻳﻘﺔ ﻛﺮﺍﻣﺮ:‬ ‫ب ﺱ+ﺹ=٥‬ ‫أ ٢ﺱ-٣ﺹ=٥‬ ‫٢ ﺱ + ٥ ﺹ = ٦١‬ ‫٣ ﺱ + ٤ ﺹ = -١‬ ‫ﺟ ﺱ+٣ﺹ=٥‬ ‫٢ﺱ+٥ﺹ=٨‬ ‫د ٣ﺱ+٢ﺹ=٥‬ ‫٢ﺱ+ﺹ=٣‬ ‫ﻫ ٣ﺱ=١-٤ﺹ‬ ‫٥ ﺱ + ٢١ = ٧ ﺹ‬ ‫و ٢ﺱ=٣+٧ﺹ‬ ‫ﺹ=٥-ﺱ‬ ‫ز ٢ﺱ +ﺹ - ٢ﻉ =٠١‬ ‫٣ﺱ + ٢ﺹ + ٢ﻉ =١‬ ‫٥ﺱ + ٤ﺹ + ٣ﻉ = ٤‬ ‫ح ﺱ + ٢ﺹ - ٣ﻉ = ٦‬ ‫٢ﺱ-ﺹ-٤ﻉ=٢‬ ‫٤ﺱ + ٣ﺹ - ٢ ﻉ = ٤١‬ ‫ط ﺹ +٢ﺱ +٣ﻉ = ٦‬ ‫٢ﺱ - ﺹ +ﻉ = -٣‬ ‫ﺱ - ٢ﺹ + ٢ﻉ= -١١‬ ‫3 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻬﻨﺪﺳﺔ: ﺃﻭﺟﺪ ﻣﺴﺎﺣﺔ ﺳﻄﺢ ﺍﻟﻤﺜﻠﺚ ‪ C‬ﺏ ﺟـ ﺍﻟﺬﻯ ﻓﻴﻪ ‪ ،(٤ ،٢)C‬ﺏ )-٢، ٤( ﺟـ)٠، -٢(.‬ ‫................‬ ‫4 ﺃﻭﺟﺪ ﻣﺴﺎﺣﺔ ﺳﻄﺢ ﺍﻟﻤﺜﻠﺚ ﺱ ﺹ ﻉ ﺍﻟﺬﻯ ﻓﻴﻪ ﺱ )٣، ٣(، ﺹ )-٤، ٢(، ﻉ ) ١، -٤(.‬ ‫..................................................................................................................................................................................................................................‬ ‫5 ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻤﺤﺪﺩﺍﺕ ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﻨﻘﻂ )٣، ٥(، )٤، -١(، )٥، ٧( ﺗﻘﻊ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
  • 14. ‫ﻧﺸﺎط‬ ‫6 ﻹﻳﺠﺎﺩ ﻗﻴﻤﺔ ﺍﻟﻤﺤﺪﺩ‬ ‫٥ ٧ ٤‬ ‫٦ ٥٢ ١١‬ ‫٧ ٢٥ ١٢‬ ‫¯ ‪¯M‬‬ ‫ﻻﺣﻆ ﺃﻧﻨﺎ ﻛﺘﺒﻨﺎ ﺍﻷﻋﻤﺪﺓ ﺍﻟﺜﻼﺛﺔ ﻟﻠﻤﺤﺪﺩ ﻛﺮﺭﻧﺎ ﺍﻟﻌﻤﻮﺩﻳﻦ‬ ‫ﻭ‬ ‫ﺍﻷﻭﻟﻴﻦ.‬ ‫ﺍﺭﺳﻢ ﺧﻄﻮﻃًﺎ ﻗﻄﺮﻳﺔ ﻋﺒﺮ ﻛﻞ ﺛﻼﺛﺔ ﻋﻨﺎﺻﺮ ﻛﺎﻟﻤﺒﻴﻦ‬ ‫ﺑﺎﻷﺳﻬﻢ ﺍﻟﻤﻨﻘﻄﺔ ﻓﺘﻜﻮﻥ ﺍﻟﺤﺪﻭﺩ ﺍﻟﻨﺎﺗﺠﺔ ﻣﻦ ﻛﻞ ﺧﻂ‬ ‫ﻫﻰ ﺣﺪﻭﺩﺍ ﻓﻰ ﺍﻟﻤﻔﻜﻮﻙ ﻭﺍﻷﺳﻬﻢ ﺍﻟﻤﺘﺠﻬﺔ ﺇﻟﻰ ﺃﺳﻔﻞ‬ ‫ً‬ ‫ﺗﻜﻮﻥ ﺣﺪﻭﺩﻫﺎ ﺍﻟﻤﻨﺎﻇﺮﺓ ﻣﻮﺟﺒﺔ، ﺑﻴﻨﻤﺎ ﺗﻠﻚ ﺍﻟﻤﺘﺠﻬﺔ ﺇﻟﻰ‬ ‫ﺃﻋﻠﻰ ﺗﻜﻮﻥ ﺳﺎﻟﺒﺔ.‬ ‫ﺍﻟﻤﺤﺪﺩ = ٥ * ٥٢ *١٢ + ٧ * ١١ * ٧ + ٤ *٦*٢٥ - ٤ *٥٢ * ٧  - ٥ * ١١ * ٢٥ - ٧ * ٦ *١٢‬ ‫     = ٥٢٦٢ + ٩٣٥ + ٨٤٢١ - ٠٠٧ - ٠٦٨٢ - ٢٨٨‬ ‫      = -٠٣‬ ‫ﺣﺎول أن ﺗﺤﻞ‬ ‫ﺍﺳﺘﺨﺪﻡ ﺍﻟﻄﺮﻳﻘﺔ ﺍﻟﺴﺎﺑﻘﺔ ﻓﻰ ﻓﻚ ﻛﻞ ﻣﺤﺪﺩ ﻣﻤﺎ ﻳﻠﻰ ﻭﺇﻳﺠﺎﺩ ﻗﻴﻤﺘﻪ:‬ ‫أ 9=‬ ‫٣ ٥ ٧‬ ‫١١ ٩ ٣١‬ ‫٥١ ٧١ ٩١‬ ‫ب 9=‬ ‫٣١ ٣ ٣٢‬ ‫٠٣ ٧ ٣٥‬ ‫٩٣ ٩ ٠٧‬ ‫ﺟ 9=‬ ‫١ -٢ ١‬ ‫١ ٢ ٣‬ ‫٦ ٤ ٣‬ ‫د 9=‬ ‫٣ -٤ -٣‬ ‫٢ ٧ -١٣‬ ‫٥ -٩ ٢‬ ‫ﺗﺄﻛﺪ ﻣﻦ ﺻﺤﺔ ﺇﺟﺎﺑﺎﺗﻚ ﺑﺈﻳﺠﺎﺩ ﻗﻴﻤﺔ ﻛﻞ ﻣﺤﺪﺩ ﺑﺎﻟﻄﺮﻳﻘﺔ ﺍﻟﻤﻌﺘﺎﺩﺓ ﻭﻣﻘﺎﺭﻧﺔ ﻧﺘﺎﺋﺠﻚ .‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
  • 15. ‫اﻟﻤﻌﻜﻮس اﻟﻀﺮﺑﻰ ﻟﻠﻤﺼﻔﻮﻓﺔ‬ ‫1-5‬ ‫‪Multiplicative Inverse of a Matrix‬‬ ‫1 ﺑﻴﻦ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﺍﻟﺘﻰ ﻟﻬﺎ ﻣﻌﻜﻮﺳﺎﺕ ﺿﺮﺑﻴﺔ، ﻭﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﺍﻟﺘﻰ ﻟﻴﺲ ﻟﻬﺎ ﻣﻌﻜﻮﺳﺎﺕ ﺿﺮﺑﻴﺔ ﻓﻴﻤﺎ ﻳﻠﻰ، ﻭﺃﻭﺟﺪ‬ ‫ﺍﻟﻤﻌﻜﻮﺱ ﺇﻥ ﻭﺟﺪ.‬ ‫٢ ١‬ ‫أ ‪l‬‬ ‫-١ ١‬ ‫٢ ٦‬ ‫ب ‪l‬‬ ‫-١ ٣‬ ‫‪b‬‬ ‫..............................................‬ ‫٤‬ ‫ﻫ ‪l‬‬ ‫٣‬ ‫٢‬ ‫١‬ ‫١ ٠‬‫ﺟ ‪l‬‬ ‫٣ ٤‬ ‫‪b‬‬ ‫..............................................‬ ‫٢‬ ‫و ‪l‬‬ ‫٠‬ ‫‪b‬‬ ‫..............................................‬ ‫٠‬ ‫١‬ ‫٣‬ ‫ز ‪l‬‬ ‫٢‬ ‫‪b‬‬ ‫٦‬ ‫..............................................‬ ‫٢‬ ‫3 ﺇﺫﺍ ﻛﺎﻧﺖ ‪l = M‬‬ ‫٠‬ ‫..............................................‬ ‫٠‬ ‫٢‬ ‫‪b‬‬ ‫٤‬ ‫٢ ‪٢-C‬‬ ‫‪C‬‬ ‫‪ b‬ﻓﺄﺛﺒﺖ ﺃﻥ ‪l = ١- M‬‬ ‫٩‬ ‫٦‬ ‫‪b‬‬ ‫..............................................‬ ‫١‬ ‫٢‬ ‫٠‬ ‫١‬ ‫٢‬ ‫١‬ ‫..............................................‬ ‫٢‬ ‫ح ‪l‬‬ ‫٥‬ ‫..............................................‬ ‫2 ﻣﺎ ﻗﻴﻢ ‪ C‬ﺍﻟﺘﻰ ﺗﺠﻌﻞ ﻟﻜﻞ ﻣﻦ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ ﻣﻌﻜﻮﺳﺎ ﺿﺮﺑﻴﺎ‬ ‫ً‬ ‫ًّ‬ ‫‪٩ C‬‬ ‫‪١ C‬‬ ‫ﺟ ‪C l‬‬ ‫‪b‬‬ ‫‪b‬‬ ‫ب ‪l‬‬ ‫أ ‪l‬‬ ‫٣‬ ‫‪b‬‬ ‫..............................................‬ ‫..............................................‬ ‫٤‬ ‫٢‬ ‫د ‪l‬‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫‪b‬‬ ‫٣‬ ‫٦‬ ‫‪b‬‬ ‫..............................................‬ ‫د ‪b ٢- ١-C l‬‬ ‫١ ‪٢-C‬‬ ‫..............................................‬ ‫‪b‬‬ ‫..................................................................................................................................................................................................................................‬ ‫٢ ٠‬‫١‬ ‫‪l‬‬ ‫‪l=b‬‬ ‫4 ﺃﻭﺟﺪ ﺍﻟﻤﺼﻔﻮﻓﺔ‪ C‬ﺇﺫﺍ ﻛﺎﻥ: ‪١ ٣ C‬‬ ‫٠‬ ‫٠‬ ‫١‬ ‫‪b‬‬ ‫..................................................................................................................................................................................................................................‬ ‫١‬ ‫٢‬ ‫١ ‪b‬‬‫٣ ‪b‬‬‫‪l‬‬ ‫‪l‬‬ ‫5 ﺇﺫﺍ ﻛﺎﻧﺖ ‪١- = N ، ١ ٠ = M‬‬ ‫٣‬ ‫ﺃﺛﺒﺖ ﺃﻥ = )‪M١-N = ١- (NM‬‬ ‫-١‬ ‫..................................................................................................................................................................................................................................‬ ‫6 ﺣﻞ ﻛﻞ ﻧﻈﺎﻡ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺨﻄﻴﺔ ﺍﻟﺘﺎﻟﻴﺔ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ، ﺛﻢ ﺗﺤﻘﻖ ﻣﻦ ﺻﺤﺔ ﺍﻟﻨﺎﺗﺞ:‬ ‫ب ٢ﺱ-٧ﺹ=٣،ﺱ-٣ﺹ=٢‬ ‫أ ٤ ﺱ + ٣ ﺹ = ٦٢ ، ٥ ﺱ - ﺹ = ٤‬ ‫......................................................................................................................‬ ‫......................................................................................................................‬ ‫......................................................................................................................‬ ‫......................................................................................................................‬ ‫ﺟ ٢ ﺱ = ٣ + ٧ ﺹ، ﺹ = ٥ - ﺱ‬ ‫د ٢ ﺹ = ٥ - ٣ﺱ ، ٢ ﺱ = ٣ - ﺹ‬ ‫......................................................................................................................‬ ‫......................................................................................................................‬ ‫......................................................................................................................‬ ‫......................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
  • 16. ‫¯‬ ‫7 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻬﻨﺪﺳﺔ: ﺍﻟﺨﻂ ﺍﻟﻤﺴﺘﻘﻴﻢ ﺍﻟﺬﻯ ﻣﻌﺎﺩﻟﺘﻪ ﺹ + ‪ C‬ﺱ = ﺣـ ﻳﻤﺮ ﺑﺎﻟﻨﻘﻄﺘﻴﻦ )١، ٥(، )٣، ١(، ﺍﺳﺘﺨﺪﻡ‬ ‫ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﻹﻳﺠﺎﺩ ﻗﻴﻤﺔ ﺍﻟﺜﺎﺑﺘﻴﻦ ‪ ،C‬ﺣـ .‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫8 ﺍﻟﺮﺑﻂ ﺑﺎﻟﺤﻴﺎﺓ: ﻳﺸﺘﺮﻯ ﺳﺎﺋﻖ ﺩﺭﺍﺟﺔ ﺑﺨﺎﺭﻳﺔ ٤٢ ﻟﺘﺮﺍ ﻣﻦ ﺍﻟﺒﻨﺰﻳﻦ ﻭ ٥ ﻟﺘﺮﺍﺕ ﻣﻦ ﺍﻟﺰﻳﺖ ﺑﻤﺒﻠﻎ ٦٥ ﺟﻨﻴﻬﺎ‬ ‫ً‬ ‫ً‬ ‫ﻟﺘﻤﻮﻳﻦ ﺩﺭﺍﺟﺘﻪ، ﺑﻴﻨﻤﺎ ﻳﺸﺘﺮﻯ ﺳﺎﺋﻖ ﺩﺭﺍﺟﺔ ﺑﺨﺎﺭﻳﺔ ﺃﺧﺮﻯ ٨١ ﻟﺘﺮﺍ ﻣﻦ ﺍﻟﺒﻨﺰﻳﻦ، ٠١ ﻟﺘﺮﺍﺕ ﻣﻦ ﺍﻟﺰﻳﺖ ﺑﻤﺒﻠﻎ‬ ‫ً‬ ‫٧٦ ﺟﻨﻴﻬﺎ ﻟﺘﻤﻮﻳﻦ ﺩﺭﺍﺟﺘﻪ، ﺍﺳﺘﺨﺪﻡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﻓﻰ ﺇﻳﺠﺎﺩ ﺛﻤﻦ ﻛﻞ ﻣﻦ ﻟﺘﺮ ﺍﻟﺒﻨﺰﻳﻦ ﻭﻟﺘﺮ ﺍﻟﺰﻳﺖ، ﺇﺫﺍ ﻋﻠﻤﺖ‬ ‫ً‬ ‫ﺃﻧﻬﻤﺎ ﻳﺴﺘﺨﺪﻣﺎﻥ ﻧﻔﺲ ﺍﻟﻨﻮﻋﻴﺔ ﻣﻦ ﺍﻟﺒﻨﺰﻳﻦ ﻭﺍﻟﺰﻳﺖ.‬ ‫.......................................................................................................................................................................................................................‬ ‫.......................................................................................................................................................................................................................‬ ‫9 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻬﻨﺪﺳﺔ: ﻳﻤﺮ ﺍﻟﻤﻨﺤﻨﻰ ﺹ = ‪C‬ﺱ٢ + ﺏ ﺱ ﺑﺎﻟﻨﻘﻄﺘﻴﻦ )٢، ٠( ، )٤، ٨(، ﺍﺳﺘﺨﺪﻡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﻹﻳﺠﺎﺩ‬ ‫ﺍﻟﺜﺎﺑﺘﻴﻦ ‪ ،C‬ﺏ .‬ ‫.......................................................................................................................................................................................................................‬ ‫.......................................................................................................................................................................................................................‬ ‫01 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ: ﻧﺼﻒ ﺍﻟﻔﺮﻕ ﺑﻴﻦ ﻋﺪﺩﻳﻦ ﻫﻮ ٢ ﻭﻣﺠﻤﻮﻉ ﺍﻟﻌﺪﺩ ﺍﻷﻛﺒﺮ ﻭﺿﻌﻒ ﺍﻟﻌﺪﺩ ﺍﻷﺻﻐﺮ ﻫﻮ ٣١. ﺑﺎﺳﺘﺨﺪﺍﻡ‬ ‫ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﺃﻭﺟﺪ ﺍﻟﻌﺪﺩﻳﻦ.‬ ‫.......................................................................................................................................................................................................................‬ ‫.......................................................................................................................................................................................................................‬ ‫.......................................................................................................................................................................................................................‬ ‫ﻧﺸﺎط‬ ‫ﺍﻛﺘﺐ ﻣﺴﺄﻟﺔ ﻣﻦ ﻋﻨﺪﻙ ﻳﺤﺘﺎﺝ ﺣﻠﻬﺎ ﺇﻟﻰ ﺗﻜﻮﻳﻦ ﻧﻈﺎﻡ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺨﻄﻴﺔ ، ﺛﻢ ﺣﻠﻬﺎ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ.‬ ‫.............................................................................................................................................................................................................................................‬ ‫.............................................................................................................................................................................................................................................‬ ‫.............................................................................................................................................................................................................................................‬ ‫.............................................................................................................................................................................................................................................‬ ‫.............................................................................................................................................................................................................................................‬ ‫.............................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
  • 17. ‫ﲤﺎﺭﻳﻦ ﻋﺎﻣﺔ‬ ‫٥ ٢‬ ‫١ ‪b‬‬‫1 ﺇﺫﺍ ﻛﺎﻧﺖ ‪l = C‬‬ ‫٤ ٠ ١‬ ‫أ ﻣﺎ ﻧﻈﻢ ﺍﻟﻤﺼﻔﻮﻓﺔ ‪......................................................................................................................................................................... ?C‬‬ ‫ب ﺍﻛﺘﺐ ﻋﻨﺎﺻﺮ ﺍﻟﺼﻒ ﺍﻷﻭﻝ ﻟﻠﻤﺼﻔﻮﻓﺔ ‪.C‬‬ ‫ﺟ‬ ‫ﺍﻛﺘﺐ ﻋﻨﺎﺻﺮ ﺍﻟﻌﻤﻮﺩ ﺍﻟﺜﺎﻟﺚ ﻓﻰ ﺍﻟﻤﺼﻔﻮﻓﺔ ‪....................................................................................................................... .C‬‬ ‫د ﺍﻛﺘﺐ ﺍﻟﻌﻨﺎﺻﺮ: ‪..................................................................................................................................................... C ، C ، C ، C‬‬ ‫١١ ٢٢ ١٣ ٢١‬ ‫.................................................................................................................................‬ ‫2 ﻋﺒﺮ ﻋﻦ ﺇﺣﺪﺍﺛﻴﺎﺕ ﺍﻟﻨﻘﻂ ﺍﻟﺘﺎﻟﻴﺔ ﺑﻤﺼﻔﻮﻓﺔ ﻣﻦ ﺍﻟﻨﻈﻢ ١ * ٢‬ ‫‪ ، (٣ ،٢) C‬ﺏ )-١، ٠(، ﺟـ)-٢، -٣(‬ ‫..................................................................................................................................................................................................................................‬ ‫3 ﻣﺎ ﻋﺪﺩ ﻋﻨﺎﺻﺮ ﻛﻼ ﻣﺼﻔﻮﻓﺔ ﻣﻤﺎﻳﻠﻰ?‬ ‫ًّ‬ ‫أ ﻣﺼﻔﻮﻓﺔ ﻣﻦ ﺍﻟﻨﻈﻢ ٣ * ٢‬ ‫.................................................................................................................‬ ‫ب ﻣﺼﻔﻮﻓﺔ ﻣﺮﺑﻌﺔ ﻣﻦ ﺍﻟﻨﻈﻢ ٢ * ٢.‬ ‫.................................................................................................................‬ ‫4 ﺣﻞ ﻛﻞ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ:‬ ‫٩‬ ‫‪ l‬ﺱ +٥‬ ‫‪l=b‬‬ ‫أ‬ ‫٤‬ ‫ﺱ-ﺹ‬ ‫‪b‬‬ ‫ب ‪٣ - C‬ﺏ ﺏ‬ ‫٠‬ ‫‪l‬‬ ‫٢ ‪l=b‬‬ ‫٤‬ ‫ﺏ +‪C‬‬ ‫.................................................................................................................‬ ‫١‬ ‫٢‬ ‫‪b‬‬ ‫.................................................................................................................‬ ‫5 ﺍﻟﺮﺑﻂ ﺑﺎﻟﺼﻨﺎﻋﺔ: ﻳﻨﺘﺞ ﻣﺼﻨﻊ ﻟﺸﺎﺷﺎﺕ ﺍﻟﺘﻠﻴﻔﺰﻳﻮﻥ ﺛﻼﺛﺔ ﺃﻧﻮﺍﻉ ٢٣ ﺑﻮﺻﺔ ، ٢٤ ﺑﻮﺻﺔ ، ٨٤ ﺑﻮﺻﺔ، ﻭﻟﻠﻤﺼﻨﻊ‬ ‫ﻓﺮﻋﺎﻥ ﺃ، ﺏ، ﻛﺎﻥ ﻋﺪﺩ ﺍﻟﺸﺎﺷﺎﺕ ﺍﻟﺘﻰ ﺃﻧﺘﺠﻬﺎ ﻛﻞ ﻓﺮﻉ ﻣﻦ ﻛﻞ ﻧﻮﻉ ﺧﻼﻝ ﺷﻬﺮﻯ ﻳﻨﺎﻳﺮ ﻭﻓﺒﺮﺍﻳﺮ ﻋﺎﻡ ٣١٠٢‬ ‫ﻭ‬ ‫ﻛﻤﺎ ﻳﻮﺿﺢ ﺫﻟﻚ ﻓﻰ ﺍﻟﺠﺪﻭﻟﻴﻦ ﺍﻟﺘﺎﻟﻴﻴﻦ ، ﻋﺒﺮ ﻋﻦ ﺇﻧﺘﺎﺝ ﺍﻟﺸﻬﺮﻳﻦ ﻣﻌﺎ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ.‬ ‫ً‬ ‫٢٣ﺑﻮﺻﺔ ٢٤ ﺑﻮﺻﺔ ٨٤ ﺑﻮﺻﺔ‬ ‫ﺃ‬ ‫٠٠٦‬ ‫٠٠٧‬ ‫٠٥٨‬ ‫‪b‬‬ ‫‪l‬‬ ‫ﺏ‬ ‫٠٥٥‬ ‫٠٠٦‬ ‫٠٥٧‬ ‫ﺷﻬﺮ ﻳﻨﺎﻳﺮ ٣١٠٢‬ ‫٢٣ﺑﻮﺻﺔ ٢٤ ﺑﻮﺻﺔ ٨٤ ﺑﻮﺻﺔ‬ ‫ﺃ‬ ‫٠٥٥‬ ‫٠٥٦‬ ‫٠٠٨‬ ‫‪b‬‬ ‫‪l‬‬ ‫ﺏ‬ ‫٠٠٦‬ ‫٠٠٧‬ ‫٠٤٨‬ ‫ﺷﻬﺮ ﻳﻨﺎﻳﺮ ٣١٠٢‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫٢ -١‬ ‫١ -٢‬ ‫6 ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ: ‪. b ٥ ٠ l = b ٤- ٣ l + C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
  • 18. ‫ﲤﺎﺭﻳﻦ ﻋﺎﻣﺔ‬ ‫7 ﺃﻭﺟﺪ ﺃ، ﺏ، ﺟـ، ﺩ ﺣﻴﺚ‬ ‫٢ ١‬‫‪b‬‬ ‫أ ‪ l b ٢- ٣- l‬ﺃ ﺏ ‪l b‬‬ ‫١ ٥ + ﺟـ ﺩ = ٤ -٣‬ ‫ﺃ‬ ‫١‬ ‫ب‬ ‫‪ p -f ٠ p‬ﺏ ‪p = f‬‬ ‫٠‬ ‫١‬‫١‬ ‫.................................................................................................................‬ ‫‪f‬‬ ‫.................................................................................................................‬ ‫ﺟـ‬ ‫١‬‫١ ٢ ٣‬ ‫١ ٢ ٣‬ ‫8 ﺇﺫﺍ ﻛﺎﻥ: ‪= N ، f ٢- ٤ ٦ p = M‬‬ ‫‪١- ٣- ٢- p‬‬ ‫٠ ٢ ١‬ ‫٠ -١ -٦‬ ‫أ ٢‪I٢-N٣+M‬‬ ‫.................................................................................................................‬ ‫ب ‪( I ٥ - N) - M‬‬ ‫٣‬ ‫9 ﺃﻭﺟﺪ ﺱ، ﺹ ﺣﻴﺚ: ‪l‬‬ ‫١‬ ‫‪ f‬ﻓﺄﻭﺟﺪ :‬ ‫.................................................................................................................‬ ‫٠‬ ‫ﺱ‬ ‫٢‬ ‫‪l= b‬‬ ‫‪lb‬‬ ‫١‬ ‫ﺹ‬ ‫٤‬ ‫‪b‬‬ ‫01 ﺑﻴﻦ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﻟﻜﻞ ﻣﻦ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ ﻣﻌﻜﻮﺱ ﺿﺮﺑﻰ ﺛﻢ ﺃﻭﺟﺪه‬ ‫١‬ ‫أ ‪l‬‬ ‫٣‬ ‫٢‬ ‫٤‬ ‫‪b‬‬ ‫..........................................‬ ‫٠‬ ‫ب ‪l‬‬ ‫٣‬ ‫٢‬ ‫٠‬ ‫ﺃ‬ ‫ﺏ ‪b‬‬‫ﺟ ‪l‬‬ ‫ﺃ ﺏ‬ ‫‪b‬‬ ‫..........................................‬ ‫..........................................‬ ‫11 ﺣﻞ ﻛﻞ ﻧﻈﺎﻡ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺨﻄﻴﺔ ﺍﻟﺘﺎﻟﻴﺔ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ‬ ‫ب ٢ ﺱ - ٣ ﺹ -٣ = ٠‬ ‫أ ﺱ+ﺹ=٣‬ ‫٥ ﺱ + ٤ ﺹ - ٩١ = ٠‬ ‫ﺱ-ﺹ=٥‬ ‫......................................................‬ ‫......................................................‬ ‫٣‬ ‫د ‪l‬‬ ‫٢‬ ‫٩‬ ‫٦‬ ‫‪b‬‬ ‫..........................................‬ ‫ﺟ ﺹ = ١١ - ٥ﺱ‬ ‫ﺱ=٣-٥ﺹ‬ ‫......................................................‬ ‫١ -٢‬ ‫٤ ٣‬ ‫21 ﺇﺫﺍ ﻛﺎﻥ )‪C‬ﺏ(-١= ١ ‪ b ٣ ١- l‬ﻛﺎﻥ ‪ b ٢ ١ l = C‬ﻓﺄﻭﺟﺪ ﺏ - ١.‬ ‫ﻭ‬ ‫٥‬ ‫..................................................................................................................................................................................................................................‬ ‫31 ﺣﻞ ﻧﻈﺎﻡ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ ﺑﺎﺳﺘﺨﺪﺍﻡ ﻃﺮﻳﻘﺔ ﻛﺮﺍﻣﺮ: ٦١ ﺱ + ٠٢ ﺹ = ٩٦ ، ٢١ ﺱ + ٤ ﺹ =٧٢.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
  • 19. ‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ‬ ‫1 ﺇﺫﺍ ﻛﺎﻥ ‪ ، b ١ ٣ l = C‬ﺏ = ‪ ، b ٢ ١- l‬ﺝ = ‪ b ١- ٢ l‬ﺃﻭﺟﺪ:‬ ‫-٢ ٥‬ ‫٦ -٣‬ ‫-٣ ٤‬ ‫ب ﻛﻼ ﻣﻦ ﺍﻟﻤﺼﻔﻮﻓﺘﻴﻦ: ‪ C‬ﺏ ، ﺏ ‪C‬‬ ‫ًّ‬ ‫أ ‪٢+C‬ﺏ-٣ﺝ‬ ‫ﺟ ﺗﺤﻘﻖ ﺃﻥ: ‪) C‬ﺏ + ﺝ( = ‪ C‬ﺏ + ‪ C‬ﺝ‬ ‫ﻣﺪ ﻣﺪ‬ ‫د ﺗﺤﻘﻖ ﺍﻥ )‪C‬ﺏ(ﻣﺪ = ﺏ ‪C‬‬ ‫2 ﺗﺒﻴﻊ ﻣﻜﺘﺒﺔ ٣ ﻣﺠﻤﻮﻋﺎﺕ ﻣﻦ ﺍﻷﺳﻄﻮﺍﻧﺎﺕ ﺍﻟﻤﺪﻣﺠﺔ ‪،CD's‬‬ ‫ﻭﻳﺒﻴﻦ ﺍﻟﺠﺪﻭﻝ ﺍﻟﺘﺎﻟﻰ ﺗﻜﻠﻔﺔ ﻛﻞ ﻣﺠﻤﻮﻋﺔ ﻭﺳﻌﺮ ﺑﻴﻌﻬﺎ،‬ ‫‪ï‬‬ ‫‪M‬‬ ‫٠٠١‬ ‫٠٨‬ ‫ﻓﺈﺫﺍ ﺑﺎﻋﺖ ﺍﻟﻤﻜﺘﺒﺔ ٠٤ ﻣﺠﻤﻮﻋﺔ ﻣﻦ ﺍﻷﺳﻄﻮﺍﻧﺎﺕ ﺍﻟﻤﺪﻣﺠﺔ‬ ‫٥٨‬ ‫٥٦‬ ‫ﻟﻤﻮﺿﻮﻋﺎﺕ ﻋﻠﻤﻴﺔ، ٤٦ ﻣﺠﻤﻮﻋﺔ ﻣﻦ ﺍﻷﺳﻄﻮﺍﻧﺎﺕ ﺍﻟﻤﺪﻣﺠﺔ‬ ‫٠١١‬ ‫٠٩‬ ‫ﻟﻤﻮﺿﻮﻋﺎﺕ ﺛﻘﺎﻓﻴﺔ، ٥٤ ﻣﺠﻤﻮﻋﺔ ﻣﻦ ﺃﺳﻄﻮﺍﻧﺎﺕ ﻣﺪﻣﺠﺔ‬ ‫ﻟﻘﺼﺺ ﻋﺎﻟﻤﻴﺔ.‬ ‫أ ﻧﻈﻢ ﺍﻟﺒﻴﺎﻧﺎﺕ ﻓﻰ ﻣﺼﻔﻮﻓﺘﻴﻦ، ﺛﻢ ﺍﺳﺘﺨﺪﻡ ﺿﺮﺏ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﻹﻳﺠﺎﺩ ﺍﻟﺘﻜﻠﻔﺔ ﺍﻟﻜﻠﻴﺔ ﻟﻸﺳﻄﻮﺍﻧﺎﺕ ﺍﻟﻤﺪﻣﺠﺔ.‬ ‫ب ﺍﺳﺘﺨﺪﻡ ﺿﺮﺏ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﻹﻳﺠﺎﺩ ﺍﻟﻤﺒﻠﻎ ﺍﻟﻜﻠﻰ ﺍﻟﺬﻯ ﺳﺘﺤﺼﻞ ﻋﻠﻴﻪ ﺍﻟﻤﻜﺘﺒﺔ ﻣﻦ ﺑﻴﻊ ﺍﻟﻤﺠﻤﻮﻋﺎﺕ‬ ‫¯‬ ‫ﺍﻟﻤﺸﺎﺭ ﺇﻟﻴﻬﺎ ﻣﻦ ﺍﻷﺳﻄﻮﺍﻧﺎﺕ ﺍﻟﻤﺪﻣﺠﺔ.‬ ‫ﺟ ﺍﺳﺘﺨﺪﻡ ﺍﻟﻌﻤﻠﻴﺎﺕ ﻋﻠﻰ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﻹﻳﺠﺎﺩ ﺭﺑﺢ ﺍﻟﻤﻜﺘﺒﺔ ﻣﻦ ﺑﻴﻊ ﻣﺠﻤﻮﻋﺎﺕ ﺍﻷﺳﻄﻮﺍﻧﺎﺕ ﺍﻟﻤﺸﺎﺭ ﺇﻟﻴﻬﺎ.‬ ‫3 ﺍﺳﺘﺨﺪﻡ ﻃﺮﻳﻘﺔ ﻛﺮﺍﻣﺮ ﻟﺤﻞ ﻛﻞ ﻧﻈﺎﻡ ﻣﻤﺎﻳﻠﻰ:‬ ‫ب ٣ﺱ-ﺹ+٥=٠‬ ‫أ ﺱ+ﺹ=٤‬ ‫ﺱ + ٢ﺹ +١ = ٠‬ ‫٢ﺱ - ﺹ = ١‬ ‫ﺟ ٢ ﺱ + ﺹ - ﻉ   = ٣‬ ‫ ٣ﺱ  + ٢ ﺹ + ﻉ = ٤‬‫٤ ﺱ + ٢ ﺹ - ﻉ   = ٨‬ ‫4 ﺑﻴﻦ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﻟﻜﻞ ﻣﺼﻔﻮﻓﺔ ﻣﻤﺎﻳﻠﻰ ﻣﻌﻜﻮﺱ ﺿﺮﺑﻰ ﺃﻡ ﻻ، ﻭﻓﻲ ﺣﺎﻟﺔ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﻟﻬﺎ ﻣﻌﻜﻮﺱ ﺿﺮﺑﻲ ﺃﻭﺟﺪه:‬ ‫٢‬ ‫١ ‪b‬‬‫أ ‪l‬‬ ‫٣ ٢‬ ‫٣‬ ‫ب ‪l‬‬ ‫٤‬ ‫٥‬ ‫٦‬ ‫‪b‬‬ ‫5 ﺣﻞ ﻛﻞ ﻧﻈﺎﻡ ﻣﻤﺎﻳﻠﻰ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ:‬ ‫ب ٣ﺱ-ﺹ=٠‬ ‫أ ﺱ + ٢ﺹ = ٨‬ ‫٥ ﺱ + ٢ ﺹ = ٢٢‬ ‫٢ ﺱ - ﺹ = -٩‬ ‫١‬ ‫ﺟ ‪l‬‬ ‫٢‬ ‫٢‬ ‫١‬ ‫‪b‬‬ ‫١ ١‬ ‫د ‪l‬‬ ‫-١ -١‬ ‫‪b‬‬ ‫ﺟ ٤ ﺱ = - ٦ﺹ‬ ‫٨ﺱ-٧=٢ﺹ‬ ‫6 أ ﺣﻞ ﻧﻈﺎﻡ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ ﺑﺎﺳﺘﺨﺪﺍﻡ ﻃﺮﻳﻘﺔ ﻛﺮﺍﻣﺮ.‬ ‫٣ﺹ = ١ + ﻉ - ﺱ ، ٢ ﺱ = -٢ﺹ + ﻉ ، ٣ ﺱ + ﺹ + ٢ ﻉ = -١‬ ‫ب ﻣﻊ ﺧﺎﻟﺪ ٥٢ ﻗﻄﻌﺔ ﻧﻘﺪﻳﺔ ﻣﻦ ﻓﺌﺔ ﺃﺭﺑﺎﻉ ﻭﺍﻧﺼﺎﻑ ﺍﻟﺠﻨﻴﻪ، ﻛﺎﻥ ﻗﻴﻤﺔ ﻣﺎ ﻣﻌﻪ ٥٫٨ ﻣﻦ ﺍﻟﺠﻨﻴﻪ، ﺍﺳﺘﺨﺪﻡ‬ ‫ﻭ‬ ‫ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ﻓﻰ ﺇﻳﺠﺎﺩ ﻋﺪﺩ ﺍﻷﺭﺑﺎﻉ ﻭﺍﻷﻧﺼﺎﻑ ﺍﻟﺘﻰ ﻣﻌﻪ.‬ ‫¯‬ ‫−‬ ‫¯‬
  • 20. ‫ﺍﺧﺘﺒﺎﺭ ﺗﺮﺍﻛﻤﻰ‬ ‫‪:≈JCÉj Ée πªcCG :’hCG‬‬ ‫٢‬ ‫1 ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ، b ٢- l = C‬ﺏ = )٢ ٥( ﻓﺈﻥ )ﺏ‪(C‬ﻣﺪ =‬ ‫..........................................................................................................................‬ ‫2 ﺇﺫﺍ ﻛﺎﻥ ‪  I = b ١- ١ l b ١ ٤ l‬ﻓﺈﻥ ﺱ =‬ ‫ﺱ‬ ‫.........................................................................................................................‬ ‫-٣‬ ‫٣ ١ ‬ ‫‪٢- ١ l‬‬ ‫‪ b‬ﻓﺈﻥ ‪= ٢C‬‬ ‫3 ﺇﺫﺍ ﻛﺎﻥ ‪= C‬‬ ‫٣ ٢‬ ‫..............................................................................................................................................................‬ ‫‪:Oó©àe øe QÉ«àN’G á∏Ä°SCG :Ék«fÉK‬‬ ‫٣‬ ‫ﻣﺪ ﻣﺪ‬ ‫‪...................................................................................... = C‬‬ ‫4 ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ،C‬ﺏ ﻣﺼﻔﻮﻓﺘﻴﻦ ﺣﻴﺚ ‪C‬ﺏ = ‪ b ١- ٤ l‬ﻓﺈﻥ ﺏ‬ ‫٥‬ ‫٥ ٤‬ ‫٥‬ ‫٣ ٤‬ ‫٣‬ ‫‪b‬‬ ‫١ ‪b‬‬‫‪b‬‬ ‫١ ‪b‬‬‫د ‪l‬‬ ‫ﺟ ‪l‬‬ ‫ب ‪l‬‬ ‫أ ‪l‬‬ ‫١ ٣‬‫١ ٥‬‫٤ ٣‬ ‫٤ ٥‬ ‫5 ﺇﺫﺍ ﻛﺎﻥ:‬ ‫أ -٣‬ ‫٢-ﺱ ٢‬ ‫٣ ﺱ + ٢ = ١ ﻓﺈﻥ ﺱ ﺗﺴﺎﻭﻯ‬‫ب ٣‬ ‫..................................................................................................................................‬ ‫ﺟ !٣‬ ‫د !٤‬ ‫‪k‬‬ ‫‪≈JCÉj Ée øY ÖLCG :ÉãdÉK‬‬ ‫6 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻣﻦ ﺍﻟﻤﺤﺪﺩﺍﺕ ﺍﻵﺗﻴﺔ:‬ ‫أ‬ ‫٣ ٤‬‫٢ -٥‬ ‫ب‬ ‫٢ -٢‬ ‫٥ ٤‬ ‫ﺟ‬ ‫٢ -٢ ٣‬ ‫٤ ١ -١‬ ‫١ ٢ -١‬ ‫٣‬ ‫٥‬ ‫٢‬ ‫د‬ ‫١ -٤‬ ‫٠ -١‬ ‫٢ ٠‬ ‫7 ﺣﻞ ﻛﻞ ﻣﻦ ﺃﻧﻈﻤﺔ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ ﺑﻄﺮﻳﻘﺔ ﻛﺮﺍﻣﺮ‬ ‫ﺟ ﺱ - ٢ ﺹ - ﻉ =٠ د ٣ ﺱ = ٢ ﺹ + ٣ + ﻉ‬ ‫ب ٣ﺱ + ﺹ = ٥‬ ‫أ ٢ ﺱ - ٤ = -ﺹ‬ ‫ﺱ + ﺹ - ٢ﻉ = -١‬ ‫٢ﺱ + ٣ ﺹ = ٨‬ ‫ﺹ=٣ﺱ-٦‬ ‫٢ﺱ-ﺹ+٤=ﻉ‬ ‫ﺱ + ٤ﺹ +٧ﻉ = ٦‬‫ﺹ + ﻉ = -ﺱ + ٣‬ ‫8 ﺍﺳﺘﺨﺪﻡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ‪ ،C‬ﺏ، ﺝ ﻟﺘﺤﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﻛﻞ ﻣﻤﺎﻳﺄﺗﻰ ﺻﺤﻴﺤﺎ ﺃﻡ ﻻ.‬ ‫ً‬ ‫‪، ٣ ٢ =C‬ﺏ= ٠ ٥ ،ﺝ= ٤ ١‬ ‫‪b‬‬ ‫‪l‬‬ ‫‪b‬‬ ‫‪l‬‬ ‫‪b‬‬ ‫‪l‬‬ ‫٦ -١‬ ‫-١ -٣‬ ‫٢‬ ‫٣‬ ‫ب ‪)C‬ﺏ + ﺝ( = ‪C‬ﺏ + ‪C‬ﺝ.‬ ‫أ ‪)C‬ﺏ ﺝ( = )‪ C‬ﺏ( ﺝ.‬ ‫9 ﺣﻞ ﻧﻈﺎﻡ ﺍﻟﻤﻌﺎﺩﻟﺘﻴﻦ ﺍﻵﻧﻴﺘﻴﻦ ﺍﻟﺘﺎﻟﻴﺘﻴﻦ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ‬ ‫٥ﺹ = ١ - ٢ﺱ   ،   ٣ﺱ = ٢ - ٧ﺹ‬ ‫01 ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ، b ١ ٢ l = C‬ﺏ = ‪ b ٣ ١ l‬ﻓﺄﻭﺟﺪ ﻛﻼ ﻣﻦ:‬ ‫ًّ‬ ‫٠ -١‬ ‫-١ ٢‬ ‫ب ﺏ‪C‬‬ ‫أ ‪C‬ﺏ‬ ‫٢‬ ‫د ﺏ‬ ‫ﻫ ‪C‬ﻣﺪ ﺏ‬ ‫‪ïM‬‬ ‫−‬ ‫ﺟ ‪C‬‬ ‫ﻣﺪ‬ ‫و ‪C‬ﺏ‬ ‫٢‬
  • 21. ‫ﺍﺧﺘﺒﺎﺭ ﺗﺮﺍﻛﻤﻰ‬ ‫11 ﺃﺟﺮ ﺍﻟﻌﻤﻠﻴﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ ﺇﻥ ﺃﻣﻜﻦ ﻣﻊ ﺫﻛﺮ ﺍﻟﺴﺒﺐ ﻓﻰ ﺣﺎﻟﺔ ﺗﻌﺬﺭ ﺇﺟﺮﺍﺀ ﺍﻟﻌﻤﻠﻴﺔ:‬ ‫ﺱ -ﺹ‬‫ﺱ ﺹ‬ ‫‪l‬‬ ‫أ ‪ l‬ﻉ ﻝ ‪- + b‬ﻉ -ﻝ‬ ‫١‬ ‫٢ -٣‬ ‫ﺟ‬ ‫‪l‬‬ ‫‪٢ b ٥ ٤ l‬‬ ‫٣‬ ‫٠‬ ‫٠‬ ‫١‬ ‫٤ -٦‬‫٤ ٦‬ ‫ب‬ ‫‪l‬‬ ‫‪٩ ٣ - b ٢- ٣- l‬‬ ‫‪b‬‬ ‫‪b‬‬ ‫د‬ ‫-١ ٢‬ ‫١‬ ‫‪l‬‬ ‫١‬ ‫21 ﺇﺫﺍ ﻛﺎﻧﺖ ‪ b ١ ١- l = N ، b ٣ ٠ l = M‬ﺃﺛﺒﺖ ﺃﻥ:‬ ‫٠‬ ‫٣‬ ‫‪C‬‬ ‫١ -٢ ٣‬ ‫٠ ١ ٤‬ ‫-٣ ٢ ١‬ ‫‪b‬‬ ‫‪b‬‬ ‫١ -٢ ٣‬ ‫‪l‬‬ ‫٤ ٥ ٦‬ ‫‪b‬‬ ‫‪MN!NM‬‬ ‫٠‬ ‫31 ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ٠ l = N ، b ٤ ٠ l = M‬ﺏ ‪ b‬ﺣﻴﺚ ‪ C‬ﺏ ! ٠‬ ‫ﺍﺛﺒﺖ ﺃﻧﻪ ﻟﻜﻞ ﻣﻦ ﺍﻟﻤﺼﻔﻮﻓﺎﺕ ‪ N M ، N ، M‬ﻣﻌﻜﻮﺱ ﺿﺮﺑﻰ ﻭﺃﻭﺟﺪه.‬ ‫41 ﺍﺷﺘﺮﻯ ﻛﺮﻳﻢ ﻣﻦ ﺇﺣﺪﻯ ﺍﻟﻤﻜﺘﺒﺎﺕ ٠١ ﺃﺳﻄﻮﺍﻧﺎﺕ ﻣﺪﻣﺠﺔ)‪٦ ،(CD‬ﺃﻗﻼﻡ ﺟﺎﻑ، ٤ ﺃﻗﻼﻡ ﺭﺻﺎﺹ، ﺍﺷﺘﺮﻯ ﺯﻣﻴﻠﻪ‬ ‫ﺳﺎﻣﻰ ٨ ﺃﺳﻄﻮﺍﻧﺎﺕ ﻣﺪﻣﺠﺔ، ٥ ﺃﻗﻼﻡ ﺟﺎﻑ، ٣ ﺃﻗﻼﻡ ﺭﺻﺎﺹ ﻣﻦ ﻧﻔﺲ ﺍﻷﻧﻮﺍﻉ ﺍﻟﺘﻰ ﺍﺷﺘﺮﺍﻫﺎ ﻛﺮﻳﻢ، ﻓﻴﻤﺎ ﻛﺎﻥ‬ ‫ﺳﻌﺮ ﺍﻟﺒﻴﻊ ﻫﻮ ﺟﻨﻴﻬﻴﻦ ﻟﻸﺳﻄﻮﺍﻧﺔ ﺍﻟﻤﺪﻣﺠﺔ، ٠٥٫١ ﻣﻦ ﺍﻟﺠﻨﻴﻪ ﻟﻠﻘﻠﻢ ﺍﻟﺠﺎﻑ، ٥٧٫٠ ﻣﻦ ﺍﻟﺠﻨﻴﻪ ﻟﻠﻘﻠﻢ ﺍﻟﺮﺻﺎﺹ.‬ ‫أ ﺍﻛﺘﺐ ﻣﺼﻔﻮﻓﺔ ﺗﻤﺜﻞ ﻣﺸﺘﺮﻳﺎﺕ ﻛﺮﻳﻢ ﻭﺳﺎﻣﻰ، ﺛﻢ ﺍﻛﺘﺐ ﻣﺼﻔﻮﻓﺔ ﺗﻮﺿﺢ ﺃﺳﻌﺎﺭ ﻛﻞ ﺳﻠﻌﺔ ﺗﻢ ﺷﺮﺍﺅﻫﺎ.‬ ‫ب ﺍﻛﺘﺐ ﻣﺼﻔﻮﻓﺔ ﺗﻮﺿﺢ ﺍﻟﻤﺒﺎﻟﻎ ﺍﻟﺘﻰ ﺩﻓﻌﻬﺎ ﻛﻞ ﻣﻦ ﻛﺮﻳﻢ ﻭﺳﺎﻣﻰ ﺛﻤﻨﺎ ﻟﻤﺸﺘﺮﻭﺍﺗﻬﻢ.‬ ‫ً‬ ‫ﺟ ﻣﺎ ﺍﻟﻤﺒﻠﻎ ﺍﻟﺬﻱ ﺩﻓﻌﻪ ﻛﻞ ﻣﻦ ﻛﺮﻳﻢ ﻭﺳﺎﻣﻰ ﺛﻤﻨﺎ ﻟﻤﺸﺘﺮﻭﺍﺗﻬﻢ?‬ ‫ً‬ ‫51 ﺇﺫﺍ ﻛﺎﻥ ﺇﻧﺘﺎﺝ ﺛﻼﺛﺔ ﺃﻗﺴﺎﻡ ﻣﺨﺘﻠﻔﺔ ﻣﻦ ﻣﺼﻨﻊ ﻹﻧﺘﺎﺝ ﻟﻌﺐ‬ ‫ﺍﻷﻃﻔﺎﻝ ﻓﻰ ٣ ﺷﻬﻮﺭ ﻣﺘﺘﺎﻟﻴﺔ ﻣﺪﻭﻧًﺎ ﺑﺎﻟﺠﺪﻭﻝ ﺍﻟﻤﻘﺎﺑﻞ، ﻛﻤﺎ‬ ‫ﻳﻮﺿﺢ ﺍﻟﺠﺪﻭﻝ ﺳﻌﺮ ﺑﻴﻊ ﻛﻞ ﻟﻌﺒﺔ ، ﻣﺎ ﺍﻟﻤﺼﻔﻮﻓﺔ ﺍﻟﺘﻰ ﺗﻤﺜﻞ‬ ‫ﺩﺧﻞ ﺍﻟﻤﺼﻨﻊ ﻣﻦ ﻣﻨﺘﺠﺎﺕ ﺍﻷﻗﺴﺎﻡ ﺍﻟﺜﻼﺛﺔ ﻓﻰ ﺍﻟﺸﻬﻮﺭ‬ ‫ﺍﻟﺜﻼﺛﺔ ﺑﻔﺮﺽ ﺃﻧﻪ ﻗﺪ ﺗﻢ ﺑﻴﻊ ﺇﻧﺘﺎﺝ ﺍﻟﻤﺼﻨﻊ ﺑﺎﻟﻜﺎﻣﻞ?‬ ‫ﻣﺎ ﻣﻘﺪﺍﺭ ﺍﻟﺪﺧﻞ ﺍﻟﻜﻠﻲ ﻟﻠﻤﺼﻨﻊ?‬ ‫‪M‬‬ ‫¯‬ ‫¯‬ ‫٠٦٣‬ ‫٠٠٤‬ ‫٠٨٣ ٨٢ ﺟﻨﻴﻪ‬ ‫٠٨٤‬ ‫٠٠٥‬ ‫٠٥٤ ٧٣ ﺟﻨﻴﻪ‬ ‫٠٧٥‬ ‫٠٠٦‬ ‫٠٥٥ ٢٣ ﺟﻨﻴﻪ‬ ‫:‬ ‫ﺭﻗﻢ ﺍﻟﺴﺆﺍﻝ‬ ‫١‬ ‫٢‬ ‫٣‬ ‫٤‬ ‫٥‬ ‫٦‬ ‫٧‬ ‫٨‬ ‫٩ ٠١ ١١ ٢١ ٣١ ٤١ ٥١‬ ‫١- ٣‬ ‫١- ٣‬ ‫١- ٥‬ ‫١- ٥‬ ‫١- ٣‬ ‫١- ٣‬ ‫١- ٤‬ ‫١- ٣‬ ‫١- ٤‬ ‫١- ٤‬ ‫١- ٢‬ ‫١- ٢‬ ‫١- ٣‬ ‫١- ٣‬ ‫١- ٣‬ ‫ﺭﻗﻢ ﺍﻟﺪﺭﺱ‬ ‫¯‬ ‫¯‬ ‫−‬ ‫¯‬ ‫¯‬
  • 22. ‫ﺍﻟﺠﺒﺮ‬ ‫‪IóMƒdG‬‬ ‫-‬ ‫2‬ ‫ﺍﻟﺒﺮﻣﺠﺔ ﺍﻟﺨﻄﻴﺔ‬ ‫‪Linear Programing‬‬ ‫دروس اﻟﻮﺣﺪة‬ ‫ﺍﻟﺪﺭﺱ )٢ - ١(: ﺍﻟﻤﺘﺒﺎﻳﻨﺔ ﺍﻟﺨﻄﻴﺔ.‬ ‫ﺍﻟﺪﺭﺱ )٢ - ٢(: ﺣﻞ ﺃﻧﻈﻤﺔ ﻣﻦ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ ﺍﻟﺨﻄﻴﺔ ﺑﻴﺎﻧﻴﺎ.‬ ‫ﺍﻟﺪﺭﺱ )٢ - ٣(: ﺍﻟﺒﺮﻣﺠﺔ ﺍﻟﺨﻄﻴﺔ ﻭﺍﻟﺤﻞ ﺍﻷﻣﺜﻞ.‬ ‫‪ïM‬‬ ‫−‬
  • 23. ‫اﻟﻤﺘﺒﺎﻳﻨﺎت اﻟﺨﻄﻴﺔ‬ ‫2-1‬ ‫‪Linear Inequalities‬‬ ‫1 ﺻﻞ ﻛﻞ ﻣﺘﺒﺎﻳﻨﺔ ﺑﺎﻟﺮﺳﻢ ﺍﻟﺒﻴﺎﻧﻲ ﺍﻟﺬﻱ ﻳﻤﺜﻞ ﻣﺠﻤﻮﻋﺔ ﺣﻠﻬﺎ )ﺍﺧﺘﺒﺮ ﺍﻟﻨﻘﻄﺔ )٠، ٠( ﻓﻰ ﻛﻞ ﻣﺘﺒﺎﻳﻨﺔ(.‬ ‫ﺟ‬ ‫ب‬ ‫د‬ ‫أ‬ ‫٢- ﺹ > ﺱ + ١‬ ‫١- ﺹ ‪ H‬ﺱ + ١‬ ‫٣- ﺹ < ﺱ + ١‬ ‫٤- ﺹ ‪ G‬ﺱ + ١‬ ‫2 ﺍﺧﺘﺒﺮ ﺃﻳﺎ ﻣﻦ ﺍﻟﻨﻘﻂ ﻫﻮ ﺣﻞ ﻟﻠﻤﺘﺒﺎﻳﻨﺔ:‬ ‫ًّ‬ ‫] )٠، ١( ، )٣، ٩( ، )–١، ٠( [ .........................................................................................‬ ‫أ ﺹ ‪ ٢ G‬ﺱ +٣‬ ‫] )٠، ١( ، )٣، ٩( ، )–١، ٠( [.........................................................................................‬ ‫ب ﺹ>٢ﺱ+٣‬ ‫3 ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﻛﻞ ﻣﻦ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ:‬ ‫ب ﺹ<٢ﺱ–٣‬ ‫أ ﺹ‪H‬ﺱ+٢‬ ‫ﺟ ﺱ+٣ﺹ‪٦H‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫4 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻤﺴﺘﻬﻠﻚ: ﺍﻓﺘﺮﺽ ﺃﻧﻚ ﺗﺮﻳﺪ ﺷﺮﺍﺀ ﻭﺭﻕ ﺯﻳﻨﺔ؛ ﻟﺘﺰﻳﻦ ﻓﺼﻠﻚ ﺍﻟﺪﺭﺍﺳﻲ ﻟﻌﻤﻞ ﺣﻔﻠﺔ ﻷﻭﺍﺋﻞ ﺍﻟﻄﻠﺒﺔ،‬ ‫ﻓﺈﺫﺍ ﻛﺎﻥ ﺛﻤﻦ ﺍﻟﻠﻔﺔ ﻣﻦ ﻭﺭﻕ ﺍﻟﺰﻳﻨﺔ ﺫﻫﺒﻲ ﺍﻟﻠﻮﻥ ﻫﻮ ٥ ﺟﻨﻴﻬﺎﺕ، ﻭﺛﻤﻦ ﺍﻟﻠﻔﺔ ﻣﻦ ﻭﺭﻕ ﺍﻟﺰﻳﻨﺔ ﺍﻷﺯﺭﻕ ﺍﻟﻠﻮﻥ‬ ‫ﻫﻮ ٣ ﺟﻨﻴﻬﺎﺕ، ﻭﺃﻧﻚ ﺗﺮﻳﺪ ﺻﺮﻑ ٨٤ ﺟﻨﻴﻬﺎ ﻋﻠﻰ ﺍﻷﻛﺜﺮ؛ ﻟﺸﺮﺍﺀ ﻭﺭﻕ ﺍﻟﺰﻳﻨﺔ، ﻓﻜﻢ ﻟﻔﺔ ﻣﻦ ﻛﻞ ﻧﻮﻉ ﻳﻤﻜﻨﻚ‬ ‫ً‬ ‫..........................................................................................................................................................................‬ ‫ﺷﺮﺍﺅﻫﺎ? ﻓﺴﺮ ﺇﺟﺎﺑﺘﻚ.‬ ‫..................................................................................................................................................................................................................................‬ ‫ﻧﺸﺎط )١(‬ ‫ﺻﻒ ﻟﺰﻣﻴﻠﻚ ﺍﻟﺬﻯ ﻛﺎﻥ ﻏﺎﺋﺒﺎ ﺃﺛﻨﺎﺀ ﺷﺮﺡ ﻫﺬﺍ ﺍﻟﺪﺭﺱ ﻟﻤﺮﺿﻪ، ﻛﻴﻒ ﻳﻤﻜﻦ ﺗﻤﺜﻴﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺔ ﺱ - ﺹ ‪ ٢ G‬ﺑﻴﺎﻧﻴﺎ‬ ‫ًّ‬ ‫ً‬ ‫ﻭﺍﻟﺘﺤﻘﻖ ﻣﻦ ﺻﺤﺔ ﺍﻹﺟﺎﺑﺔ.‬ ‫¯‬ ‫−‬ ‫¯‬
  • 24. ‫‪M‬‬ ‫اﻟﺮﺑﻂ ﺑﺎﻟﺘﻜﻨﻮﻟﻮﺟﻴﺎ‬ ‫ﻧﺸﺎط )٢(‬ ‫:‬ ‫ﻳﻤﻜﻨﻚ ﺍﺳﺘﺨﺪﺍﻡ ﺧﺎﺻﻴﺔ ‪ Draw‬ﻓﻰ ﺍﻵﻟﺔ ﺍﻟﺤﺎﺳﺒﺔ ﺍﻟﺒﻴﺎﻧﻴﺔ ‪ graphing colculator‬ﻟﺮﺳﻢ‬ ‫ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ، ﺣﻴﺚ ﻳﻌﺘﻤﺪ ﺗﺮﺗﻴﺐ ﺍﺩﺧﺎﻝ ﺍﻟﺒﻴﺎﻧﺎﺕ ﻋﻠﻰ ﻣﺎ ﺇﺫﺍ ﻛﻨﺖ ﺳﺘﻈﻠﻞ ﻓﻮﻕ ﺍﻟﺨﻂ‬ ‫ﺍﻟﺤﺪﻯ ﺃﻡ ﺃﺳﻔﻠﻪ؛ ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺘﻈﻠﻴﻞ ﺃﺳﻔﻞ ﺍﻟﺨﻂ ﺍﻟﺤﺪﻯ 1‪ y‬ﺍﺳﺘﺨﺪﻡ ﺍﻟﺘﻈﻠﻴﻞ )1‪،(ymin, y‬‬ ‫ﻭﻋﻨﺪﻣﺎ ﺗﻈﻠﻞ ﻓﻮﻕ 1‪ y‬ﺍﺳﺘﺨﺪﻡ ﺍﻟﺘﻈﻠﻴﻞ )‪ (y1, ymax‬ﻭﻟﺴﺖ ﻓﻰ ﺣﺎﺟﺔ ﻻ ﺳﺘﺨﺪﺍﻡ ﺍﻷﻗﻮﺍﺱ‬ ‫ﺍﻟﻤﻐﻠﻘﺔ ﻗﺒﻞ ﺍﻟﻀﻐﻂ ﻋﻠﻰ ﻣﻔﺘﺎﺡ ‪ENTER‬‬ ‫ﻻﺣﻆ‬ ‫¯‬ ‫‪¯ y‬‬ ‫,‪¯ x‬‬ ‫¯‬ ‫¯‬ ‫+‬ ‫>‬ ‫3 + ‪y < 2x‬‬ ‫ﻣـﺜـﺎل‬ ‫5 ﻣﺜﻞ ﻛﻞ ﻣﺘﺒﺎﻳﻨﺔ ﺑﻴﺎﻧﻴﺎ:‬ ‫ًّ‬ ‫1 – ‪B y = 0.5x‬‬ ‫3 + ‪A y < 2x‬‬ ‫ﺍﻟﺘﻈﻠﻴﻞ ﻓﻮﻕ ﺍﻟﺨﻂ ﺍﻟﺤﺪﻯ 1‪ y‬ﻟﻮﺟﻮﺩ ﻋﻼﻣﺔ ﺃﻛﺒﺮ ﻣﻦ‬ ‫1‬ ‫–‬ ‫‪X,T,i‬‬ ‫,‬ ‫5‬ ‫1‬ ‫.‬ ‫1‬ ‫5‬ ‫‪Draw‬‬ ‫‪2nd‬‬ ‫‪Y-VARS‬‬ ‫7‬ ‫‪ENTRER‬‬ ‫0‬ ‫=‪Y‬‬ ‫ﺍﻟﺘﻈﻠﻴﻞ ﺃﺳﻔﻞ 1‪ y‬ﻟﻮﺟﻮﺩ ﻋﻼﻣﺔ ﺃﻗﻞ ﻣﻦ‬ ‫‪F‬‬ ‫$‬ ‫#‬ ‫$‬ ‫3‬ ‫#‬ ‫‪2nd‬‬ ‫1‬ ‫+‬ ‫7‬ ‫,‬ ‫‪$ VARS‬‬ ‫)‪(y1, ymax‬‬ ‫) ‪,y‬‬ ‫1 ‪min‬‬ ‫‪# (y‬‬ ‫‪ENTRER‬‬ ‫‪X,T,i‬‬ ‫1‬ ‫4‬ ‫1‬ ‫2‬ ‫‪Draw‬‬ ‫1‬ ‫‪Y-VARS‬‬ ‫=‪Y‬‬ ‫‪2nd‬‬ ‫‪VARS‬‬ ‫‪2nd‬‬ ‫× ﻳﻤﻜﻨﻚ ﺍﻟﺘﺤﻜﻢ ﻓﻰ ﺩﺭﺟﺔ ﺍﻟﺘﻈﻠﻴﻞ ﺑﺈﺩﺧﺎﻝ ﻋﺪﺩ ﺻﺤﻴﺢ ﻣﻦ )‪ 1(Dark‬ﺇﻟﻰ )‪ ،8(Light‬ﺃﺿﻒ ﻓﺎﺻﻠﺔ )‪(Comma‬‬ ‫ﻭﺍﻟﻌﺪﺩ ﺍﻟﺼﺤﻴﺢ ﻗﺒﻞ ﺍﻟﻀﻐﻂ ﻋﻠﻰ ﻣﻔﺘﺎﺡ ‪. ENTRER‬‬ ‫× ﻻ ﺗﻀﻊ ﺍﻵﻟﺔ ﺍﻟﺤﺎﺳﺐ ﺍﻟﺒﻴﺎﻧﻴﺔ ﺗﻤﻴﺰﺍ ﺑﻴﻦ ﺍﻟﺨﻂ ﺍﻟﺤﺪﻯ ﺍﻟﻤﺘﺼﻞ ﻭﺍﻟﻤﻘﻄﻊ؛ ﻟﺬﺍ ﻳﺠﺐ ﻋﻠﻴﻚ ﺃﻥ ﺗﺤﺪﺩ ﺇﺫﺍ ﻛﺎﻥ‬ ‫ً‬ ‫ﺍﻟﺨﻂ ﺍﻟﺤﺪﻯ ﻣﺘﺼﻼ ﺃﻡ ﻣﺘﻘﻄﻌﺎ ﻋﻨﺪ ﺭﺳﻤﻚ ﻟﻠﻤﺘﺒﺎﻳﻨﺔ ﻓﻰ ﻛﺮﺍﺳﺘﻚ.‬ ‫ً‬ ‫ً‬ ‫ﺣﺎول أن ﺗﺤﻞ‬ ‫1 ﺍﺳﺘﺨﺪﻡ ﺍﻵﻟﺔ ﺍﻟﺤﺎﺳﺒﺔ ﺍﻟﺒﻴﺎﻧﻴﺔ؛ ﻟﺮﺳﻢ ﻛﻞ ﻣﺘﺒﺎﻳﻨﺔ ﻣﻤﺎ ﻳﻠﻲ:‬ ‫3 + ‪C y H -x‬‬ ‫1 + ‪B y > 2x‬‬ ‫‪A y<x‬‬ ‫21 ‪F 2x + 3y G‬‬ ‫4‪E x-yH‬‬ ‫5‪D yG‬‬ ‫‪ïM‬‬ ‫−‬
  • 25. ‫ًّ‬ ‫ﺣﻞ أﻧﻈﻤﺔ ﻣﻦ اﻟﻤﺘﺒﺎﻳﻨﺎت اﻟﺨﻄﻴﺔ ﺑﻴﺎﻧﻴﺎ‬ ‫2-2‬ ‫‪Solving systems of Linear Inequalities Graphcally‬‬ ‫1 ﺃﻯ ﻧﻈﺎﻡ ﻣﻤﺎ ﻳﺄﺗﻰ ﻟﻪ ﻣﻨﻄﻘﺔ ﺍﻟﺤﻞ ﺍﻟﻤﻮﺿﺤﺔ ﻓﻰ‬ ‫ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ:‬ ‫أ ﺱ+ﺹ‪٣H‬‬ ‫ب ﺱ+ﺹ<٣‬ ‫ﺹ< ﺱ-٣‬ ‫ﺹ‪H‬ﺱ-٣‬ ‫ﺟ ﺱ+ﺹ‪٣G‬‬ ‫ﺹ>ﺱ-٣‬ ‫د ﺱ+ﺹ>٣‬ ‫ﺹ‪G‬ﺱ-٣‬ ‫2 ﺣﻞ ﻛﻞ ﻧﻈﺎﻡ ﻣﻦ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ ﺍﻟﺨﻄﻴﺔ ﺑﻴﺎﻧﻴﺎ:‬ ‫ًّ‬ ‫ب ﺹ-ﺱ<٠‬ ‫أ ﺱ‪٤H‬‬ ‫٢ ﺱ + ٢ ﺹ ‪١٢ H‬‬ ‫ﺹ>ﺱ+٢‬ ‫ﺹ>٦+٢ﺱ‬ ‫ﺱ + ٢ ﺹ ‪٢- G‬‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫ﺟ ﺱ+٤ﺹ<٤‬ ‫٤ﺱ+ﺹ‪٢G‬‬ ‫ﺱ-ﺹ>١‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫3 ﺃﻋﻄﻰ ﺍﻷﺳﺘﺎﺫ ﻛﺮﻳﻢ ﻟﺘﻼﻣﻴﺬه ﺯﻣﻨﺎ ﻗﺪﺭه ٠٦ ﺩﻗﻴﻘﺔ ﻹﺟﺎﺑﺔ ﺍﺧﺘﺒﺎﺭ ﻓﻰ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ، ﻳﺠﺐ ﺃﻥ ﻳﺠﻴﺐ ﺍﻟﺘﻼﻣﻴﺬ‬ ‫ً‬ ‫ﻋﻦ ٤ ﺃﺳﺌﻠﺔ ﻋﻠﻰ ﺍﻷﻗﻞ ﻣﻦ ﺍﻟﻘﺴﻢ )ﺃ(، ٣ ﺃﺳﺌﻠﺔ ﻋﻠﻰ ﺍﻷﻗﻞ ﻣﻦ ﺍﻟﻘﺴﻢ )ﺏ(، ﺑﺤﻴﺚ ﻻﺗﻘﻞ ﻋﺪﺩ ﺍﻷﺳﺌﻠﺔ ﺍﻟﻤﺠﺎﺑﺔ‬ ‫ﻣﻦ ﺍﻟﻘﺴﻤﻴﻦ ﻣﻌﺎ ﻋﻦ ٠١ ﺃﺳﺌﻠﺔ. ﻓﺈﺫﺍ ﺍﺳﺘﻐﺮﻗﺖ ﻫﻨﺎﺀ ٤ ﺩﻗﺎﺋﻖ ﻹﺟﺎﺑﺔ ﻛﻞ ﺳﺆﺍﻝ ﻓﻰ ﺍﻟﻘﺴﻢ )ﺃ(، ٥ ﺩﻗﺎﺋﻖ ﻹﺟﺎﺑﺔ‬ ‫ً‬ ‫ً‬ ‫ﻛﻞ ﺳﺆﺍﻝ ﻓﻰ ﺍﻟﻘﺴﻢ )ﺏ(. ﻛﻢ ﺳﺆﺍﻻ ﻓﻰ ﻛﻞ ﻗﺴﻢ ﺣﺎ ﻟﺖ ﻫﻨﺎﺀ ﺍﻹﺟﺎﺑﺔ ﻋﻨﻪ?.................................................................‬ ‫ﻭ‬ ‫4 ﺍﻟﺘﻔﻜﻴﺮ ﺍﻟﻨﺎﻗﺪ:‬ ‫أ ﺍﻛﺘﺐ ﻧﻈﺎﻣﺎ ﻣﻦ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ ﺍﻟﺨﻄﻴﺔ، ﻭﺍﻟﺘﻲ ﻳﻜﻮﻥ ﺣﻠﻬﺎ ﻫﻮ ﺧﻂ ﻣﺴﺘﻘﻴﻢ.‬ ‫ً‬ ‫ب ﺑﺪﻭﻥ ﺗﻤﺜﻴﻞ ﺑﻴﺎﻧﻲ، ﻓﺴﺮ ﻟﻤﺎﺫﺍ ﻧﻘﻄﺔ ﺗﻘﺎﻃﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﻴﻦ ﺍﻟﺤﺪﻳﻴﻦ ﻓﻰ ﺍﻟﻨﻈﺎﻡ: ٢ ﺱ + ﺹ < ٢، ﺱ - ﺹ ‪٣ G‬‬ ‫ﻟﻴﺴﺖ ﺣﻼ ﻟﻬﺬﺍ ﺍﻟﻨﻈﺎﻡ. ............................................................................................................................................................................‬ ‫ًّ‬ ‫........................................................‬ ‫ﻧﺸﺎط‬ ‫5 ﺍﻛﺘﺐ ﻣﺴﺄﻟﺔ ﻣﻦ ﻋﻨﺪﻙ ﻳﺤﺘﺎﺝ ﺣﻠﻬﺎ ﺇﻟﻰ ﻛﺘﺎﺑﺔ ﻧﻈﺎﻡ ﻣﻦ ﻣﺘﺒﺎﻳﻨﺘﻴﻦ ﺧﻄﻴﺘﻴﻦ ﻓﻰ ﻣﺠﻬﻮﻟﻴﻦ ﺛﻢ ﺣﻞ ﻫﺬﺍ ﺍﻟﻨﻈﺎﻡ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
  • 26. ‫اﻟﺒﺮﻣﺠﺔ اﻟﺨﻄﻴﺔ واﻟﺤﻞ ا ﻣﺜﻞ‬ ‫2-3‬ ‫‪Linear programing and optimization‬‬ ‫1 ﺍﺧﺘﺮ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ ﻣﻦ ﺑﻴﻦ ﺍﻹﺟﺎﺑﺎﺕ ﺍﻟﻤﻌﻄﺎﺓ:‬ ‫أ ﺍﻟﻨﻘﻄﺔ ﺍﻟﺘﻰ ﺗﻨﺘﻤﻲ ﺇﻟﻰ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ: ﺱ < ٢ ، ﺹ < ١ ، ﺱ + ﺹ ‪ ٣ G‬ﻫﻰ: ............‬ ‫‪`(٣ ،١) ، (٢ ،٣) ، (٢ ،١) ، (١ ،٣) j‬‬ ‫ب‬ ‫ﺍﻟﻨﻘﻄﺔ ﺍﻟﺘﻰ ﺗﻜﻮﻥ ﻋﻨﺪﻫﺎ ﻟﻠﺪﺍﻟﺔ ‪٤٠ = S‬ﺱ + ٠٢ﺹ ﻗﻴﻤﺔ ﻋﻈﻤﻰ ﻫﻰ: .................................................................‬ ‫‪`(٠ ،٢٥) ، (١٠ ،١٥) ، (٤- ،٠) ، (٠ ،٠) j‬‬ ‫ﺟ‬ ‫ﺍﻟﻨﻘﻄﺔ ﺍﻟﺘﻰ ﺗﻜﻮﻥ ﻋﻨﺪﻫﺎ ﻟﻠﺪﺍﻟﺔ ﻡ = ٥٣ﺱ + ٠١ﺹ ﻗﻴﻤﺔ ﺻﻐﺮﻯ ﻫﻰ: ...................................................................‬ ‫‪`(١٠ ،٢٠) ، (٤٠ ،٠) ، (١٠ ،٠) ، (٠ ،٠) j‬‬ ‫2 ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﺮﺳﻢ ﺍﻟﺒﻴﺎﻧﻲ ﺍﻟﻤﻘﺎﺑﻞ، ﺃﻭﺟﺪ ﻗﻴﻤﺘﻲ ﺱ، ﺹ ﺍﻟﺘﻰ ﺗﺠﻌﻞ ﻗﻴﻤﺔ‬ ‫ﺩﺍﻟﺔ ﺍﻟﻬﺪﻑ ‪٣ = S‬ﺱ + ٢ﺹ ﻗﻴﻤﺔ ﺻﻐﺮﻯ، ﺛﻢ ﺃﻭﺟﺪ ﻫﺬه ﺍﻟﻘﻴﻤﺔ.‬ ‫‪C‬‬ ‫....................................................................................................................................................‬ ‫....................................................................................................................................................‬ ‫‪E‬‬ ‫3 ﻣﺜﻞ ﻛﻼ ﻣﻦ ﺍﻷﻧﻈﻤﺔ ﺍﻟﺘﺎﻟﻴﺔ ﺑﻴﺎﻧﻴﺎ، ﺛﻢ ﺃﻭﺟﺪ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻌﻈﻤﻰ ﺃﻭ ﺍﻟﻘﻴﻤﺔ‬ ‫ًّ‬ ‫ًّ‬ ‫ﺍﻟﺼﻐﺮﻯ ﻟﺪﺍﻟﺔ ﺍﻟﻬﺪﻑ ﺗﺒﻌﺎ ﻟﻤﺎ ﻫﻮ ﻣﻌﻄﻰ.‬ ‫ً‬ ‫ب ٢ﺱ + ﺹ ‪٦ H‬‬ ‫أ ﺱ+ﺹ‪٥H‬‬ ‫ﺹ‪١G‬‬ ‫ﺱ‪٢G‬‬ ‫ﻗﻴﻤﺔ ﺻﻐﺮﻯ ﻟﺪﺍﻟﺔ ﺍﻟﻬﺪﻑ ‪٢ = S‬ﺱ + ٣ﺹ‬ ‫−‬ ‫ﺱ‪١G‬‬ ‫ﺹ‪٢G‬‬ ‫ﻗﻴﻤﺔ ﻋﻈﻤﻰ ﻟﺪﺍﻟﺔ ﺍﻟﻬﺪﻑ ‪٢ = S‬ﺱ + ٣ﺹ‬ ‫4 ﺍﻟﺮﺑﻂ ﺑﺎﻟﺼﻨﺎﻋﺔ: ﺍﻓﺘﺮﺽ ﺃﻧﻚ ﺗﺼﻨﻊ ﻭﺗﺒﻴﻊ ﻣﺮﻃﺒﺎ ﻟﻠﺠﻠﺪ، ﻭﺇﺫﺍ ﻛﺎﻥ ﺗﺼﻨﻴﻊ ﻋﺒﻮﺓ ﺍﻟﻤﺮﻃﺐ ﺍﻟﻌﺎﺩﻯ ﻳﺴﺘﻠﺰﻡ‬ ‫ُ‬ ‫ً‬ ‫٣‬ ‫٢ﺳﻢ٣ ﻣﻦ ﺍﻟﺰﻳﺖ، ١ﺳﻢ٣ ﻣﻦ ﺯﺑﺪﺓ ﺍﻟﻜﺎﻛﺎﻭ، ﻛﺎﻥ ﺗﺼﻨﻴﻊ ﻋﺒﻮﺓ ﺍﻟﻤﺮﻃﺐ ﻣﻦ ﺍﻟﻨﻮﻉ ﺍﻟﻤﻤﺘﺎﺯ ﻳﺴﺘﻠﺰﻡ ١ﺳﻢ‬ ‫ﻭ‬ ‫ﻣﻦ ﺍﻟﺰﻳﺖ، ٢ﺳﻢ٣ ﻣﻦ ﺯﺑﺪﺓ ﺍﻟﻜﺎﻛﺎﻭ، ﺳﻮﻑ ﻳﻜﻮﻥ ﺭﺑﺤﻚ ﻫﻮ ٠١ ﺟﻨﻴﻬﺎﺕ ﻟﻜﻞ ﻋﺒﻮﺓ ﻣﻦ ﺍﻟﻨﻮﻉ ﺍﻟﻌﺎﺩﻯ، ٨‬ ‫ﺟﻨﻴﻬﺎﺕ ﻟﻜﻞ ﻋﺒﻮﺓ ﻣﻦ ﺍﻟﻨﻮﻉ ﺍﻟﻤﻤﺘﺎﺯ. ﻓﺈﺫﺍ ﻛﺎﻥ ﻟﺪﻳﻚ ٤٢ ﺳﻢ٣ ﻣﻦ ﺍﻟﺰﻳﺖ، ٨١ ﺳﻢ٣ ﻣﻦ ﺯﺑﺪﺓ ﺍﻟﻜﺎﻛﺎﻭ، ﻓﻤﺎ‬ ‫ﻋﺪﺩ ﺍﻟﻌﺒﻮﺍﺕ ﺍﻟﺘﻰ ﻳﻤﻜﻨﻚ ﺗﺼﻨﻴﻌﻬﺎ ﻣﻦ ﻛﻞ ﻧﻮﻉ؛ ﺣﺘﻰ ﺗﺤﺼﻞ ﻋﻠﻰ ﺃﻛﺒﺮ ﺭﺑﺢ ﻣﻤﻜﻦ، ﻭﻣﺎ ﻫﺬﺍ ﺍﻟﺮﺑﺢ?‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
  • 27. ‫5 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻤﻬﻦ: ﻟﺪﻯ ﺃﺣﺪ ﺍﻟﺨﻴﺎﻃﻴﻦ ٠١ ﺃﻣﺘﺎﺭ ﻣﻦ ﻗﻤﺎﺵ ﺍﻟﻜﺘﺎﻥ، ٦ ﺃﻣﺘﺎﺭ ﻣﻦ ﻗﻤﺎﺵ ﻗﻄﻨﻰ، ﻭﻳﺮﻳﺪ ﺍﻟﺨﻴﺎﻁ‬ ‫ﺗﻔﺼﻴﻞ ﻧﻮﻋﻴﻦ ﻣﻦ ﺍﻟﻤﻼﺑﺲ ﻣﻦ ﺍﻟﻤﻮﺍﺩ ﺍﻟﻤﺘﻮﺍﻓﺮﺓ ﻟﺪﻳﻪ، ﺍﻟﻨﻮﻉ ﺍﻷﻭﻝ ﻣﻦ ﺍﻟﻤﻼﺑﺲ ﻳﺤﺘﺎﺝ ﺇﻟﻰ ﻣﺘﺮ ﻭﺍﺣﺪ ﻣﻦ‬ ‫ﺍﻟﻜﺘﺎﻥ، ﻭﻣﺘﺮ ﻭﺍﺣﺪ ﻣﻦ ﺍﻟﻘﻄﻦ، ﻭﻳﺤﻘﻖ ﺭﺑﺤﺎ ﻗﺪﺭه ٣ ﺟﻨﻴﻬﺎﺕ، ﺑﻴﻨﻤﺎ ﻳﺤﺘﺎﺝ ﺍﻟﻨﻮﻉ ﺍﻟﺜﺎﻧﻰ ﺇﻟﻰ ٢ ﻣﺘﺮ ﻣﻦ‬ ‫ً‬ ‫ﺍﻟﻜﺘﺎﻥ ﻭﻣﺘﺮ ﻭﺍﺣﺪ ﻣﻦ ﺍﻟﻘﻄﻦ، ﻭﻳﺤﻘﻖ ﺭﺑﺤﺎ ﻗﺪﺭﺓ ٤ ﺟﻨﻴﻬﺎﺕ. ﻣﺎ ﺍﻟﻜﻤﻴﺔ ﺍﻟﺘﻰ ﻳﺠﺐ ﻋﻠﻴﻪ ﺗﻔﺼﻴﻠﻬﺎ ﻣﻦ ﻛﻞ‬ ‫ً‬ ‫ﻧﻮﻉ ﺣﺘﻰ ﻳﺤﻘﻖ ﺍﻟﺨﻴﺎﻁ ﺃﻛﺒﺮ ﺭﺑﺢ ﻣﻤﻜﻦ? .....................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫6 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻤﻮﺳﻴﻘﻰ: ﻳﻨﺘﺞ ﺃﺣﺪ ﻣﺼﺎﻧﻊ ﺍﻵﻻﺕ ﺍﻟﻤﻮﺳﻴﻘﻴﺔ ﻧﻮﻋﻴﻦ ﻣﻦ ﺁﻻﺕ ﺍﻟﻨﻔﺦ، ﻳﺤﺘﺎﺝ ﺗﺼﻨﻴﻊ ﺍﻟﻨﻮﻉ ﺍﻷﻭﻝ‬ ‫ﺇﻟﻰ ٥٢ ﻭﺣﺪﺓ ﻣﻦ ﺍﻟﻨﺤﺎﺱ، ٤ ﻭﺣﺪﺍﺕ ﻣﻦ ﺍﻟﻨﻴﻜﻞ، ﻭﻳﺤﺘﺎﺝ ﺗﺼﻨﻴﻊ ﺍﻟﻨﻮﻉ ﺍﻟﺜﺎﻧﻲ ٥١ ﻭﺣﺪﺓ ﻣﻦ ﺍﻟﻨﺤﺎﺱ، ٨‬ ‫ﻭﺣﺪﺍﺕ ﻣﻦ ﺍﻟﻨﻴﻜﻞ، ﻓﺈﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻜﻤﻴﺔ ﺍﻟﻤﺘﺎﺣﺔ ﻓﻰ ﺍﻟﻤﺼﻨﻊ ﻓﻰ ﺃﺣﺪ ﺍﻷﻳﺎﻡ ٥٩ ﻭﺣﺪﺓ ﻣﻦ ﺍﻟﻨﺤﺎﺱ، ٢٣ ﻭﺣﺪﺓ‬ ‫ﻭ‬ ‫ﻣﻦ ﺍﻟﻨﻴﻜﻞ، ﻛﺎﻥ ﺭﺑﺢ ﺍﻟﻤﺼﻨﻊ ﻓﻰ ﺍﻵﻟﺔ ﻣﻦ ﺍﻟﻨﻮﻉ ﺍﻷﻭﻝ ﻫﻮ ٠٦ ﺟﻨﻴﻬﺎ ﻭﺭﺑﺤﻪ ﻓﻰ ﺍﻵﻟﺔ ﻣﻦ ﺍﻟﻨﻮﻉ ﺍﻟﺜﺎﻧﻲ ٨٤‬ ‫ً‬ ‫ﺟﻨﻴﻬﺎ، ﻓﻤﺎ ﻋﺪﺩ ﺍﻵﻻﺕ ﺍﻟﺘﻰ ﻳﺠﺐ ﺃﻥ ﻳﻨﺘﺠﻬﺎ ﺍﻟﻤﺼﻨﻊ ﻣﻦ ﻛﻞ ﻧﻮﻉ ﺣﺘﻰ ﻳﺤﻘﻖ ﺃﻛﺒﺮ ﺭﺑﺢ ﻣﻤﻜﻦ?‬ ‫ً‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫7 ﺍﻟﺮﺑﻂ ﺑﺎﻟﺴﻴﺎﺣﺔ: ﺃﻗﺎﻣﺖ ﺇﺣﺪﻯ ﻛﺎﺕ ﺍﻟﺴﻴﺎﺣﺔ ﺟﺴﺮﺍ ﺟﻮ ﻳﺎ ﻟﻨﻘﻞ ﺍﻟﺴﺎﺋﺤﻴﻦ. ﺫﻟﻚ ﻟﻨﻘﻞ ٠٠٦١ ﺳﺎﺋﺢ، ٠٩‬ ‫ﺷﺮ‬ ‫ًّ‬ ‫ً‬ ‫ﻃﻨﺎ ﻣﻦ ﺍﻷﻣﺘﻌﺔ ﺑﺄﻗﻞ ﺗﻜﻠﻔﺔ، ﻛﺎﻥ ﺍﻟﻤﺘﺎﺡ ﻧﻮﻋﻴﻦ ﻣﻦ ﺍﻟﻄﺎﺋﺮﺍﺕ ‪ ،C‬ﺏ ﻛﺎﻥ ﻋﺪﺩ ﺍﻟﻄﺎﺋﺮﺍﺕ ﺍﻟﻤﺘﺎﺣﺔ ﻣﻦ ﺍﻟﻨﻮﻉ‬ ‫ﻭ‬ ‫ﻭ‬ ‫ًّ‬ ‫‪ ١٢ ،C‬ﻃﺎﺋﺮﺓ، ﻭﻋﺪﺩ ﺍﻟﻄﺎﺋﺮﺍﺕ ﺍﻟﻤﺘﺎﺣﺔ ﻣﻦ ﺍﻟﻨﻮﻉ ﺏ ٩ ﻃﺎﺋﺮﺍﺕ، ﻛﺎﻧﺖ ﺍﻟﺤﻤﻮﻟﺔ ﻛﺎﻣﻠﺔ ﻟﻠﻄﺎﺋﺮﺓ ﻣﻦ ﺍﻟﻨﻮﻉ ‪C‬‬ ‫ﻭ‬ ‫٠٠٢ ﺷﺨﺺ، ٦ ﺃﻃﻨﺎﻥ ﻣﻦ ﺍﻷﻣﺘﻌﺔ، ﻭﺍﻟﺤﻤﻮﻟﺔ ﺍﻟﻜﺎﻣﻠﺔ ﻟﻠﻄﺎﺋﺮﺓ ﻣﻦ ﺍﻟﻨﻮﻉ ﺏ ٠٠١ ﺷﺨﺺ، ٥١ ﻃﻨﺎ ﻣﻦ ﺍﻷﻣﺘﻌﺔ،‬ ‫ًّ‬ ‫ﻭ‬ ‫ﻛﺎﻥ ﺇﻳﺠﺎﺭ ﺍﻟﻄﺎﺋﺮﺓ ﻣﻦ ﺍﻟﻨﻮﻉ ‪ C‬ﻫﻮ ٠٠٠ ٠٢٣ ﺟﻨﻴﻪ، ﻭﻣﻦ ﺍﻟﻨﻮﻉ ﺏ ﻫﻮ ٠٠٠ ٠٥١ ﺟﻨﻴﻪ، ﻓﻜﻢ ﻃﺎﺋﺮﺓ ﻣﻦ ﻛﻞ‬ ‫ﻛﺔ ﺍﺳﺘﺌﺠﺎﺭﻫﺎ?‬ ‫ﻧﻮﻉ ﻳﻤﻜﻦ ﻟﻠﺸﺮ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫أﻧﺸﻄﺔ‬ ‫ً‬ ‫8 ﺍﺭﺟﻊ ﺇﻟﻰ ﻣﻜﺘﺒﺘﻚ ﺍﻟﻤﺪﺭﺳﻴﺔ ﺃﻭ ﺍﻟﺸﺒﻜﺔ ﺍﻟﺪﻭﻟﻴﺔ ﻟﻠﻤﻌﻠﻮﻣﺎﺕ )ﺍﻹﻧﺘﺮﻧﺖ( ﻭﺍﻛﺘﺐ ﻣﺜﺎﻻ ﻳﻮﺿﺢ ﺍﺳﺘﺨﺪﺍﻣﺎﺕ‬ ‫ﺍﻟﺒﺮﻣﺠﺔ ﺍﻟﺨﻄﻴﺔ ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻟﻤﺠﺎﻻﺕ ﺍﻟﺘﺎﻟﻴﺔ:‬ ‫ﺟ ﺇﺩﺍﺭﺓ ﺍﻟﻮﻗﺖ‬ ‫ب ﺍﻟﺼﻨﺎﻋﺔ‬ ‫أ ﺍﻻﻗﺘﺼﺎﺩ‬ ‫د ﺑﺤﻮﺙ ﺍﻟﻌﻤﻠﻴﺎﺕ‬ ‫..................................................................................................................................................................................................................................‬ ‫9 ﺍﻛﺘﺐ ﻣﺴﺄﻟﺔ ﻣﻦ ﻋﻨﺪﻙ ﻳﺘﻄﻠﺐ ﺣﻠﻬﺎ ﻛﺘﺎﺑﺔ ﺃﺭﺑﻊ ﻣﺘﺒﺎﻳﻨﺎﺕ ﺧﻄﻴﺔ، ﺛﻢ ﻣﺜﻞ ﻣﻨﻄﻘﺔ ﺍﻟﺤﻞ ﺑﻴﺎﻧﻴﺎ. ﺍﻛﺘﺐ ﺩﺍﻟﺔ‬ ‫ًّ‬ ‫ﺍﻟﻬﺪﻑ ﻟﻤﺴﺄﻟﺘﻚ، ﻭﺣﺪﺩ ﻣﺘﻰ ﻳﻜﻮﻥ ﻟﻬﺎ ﻗﻴﻤﺔ ﻋﻈﻤﻰ، ﺃﻭ ﻗﻴﻤﺔ ﺻﻐﺮﻯ، ﺛﻢ ﺃﻭﺟﺪ ﻫﺎﺗﻴﻦ ﺍﻟﻘﻴﻤﺘﻴﻦ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
  • 28. ‫ﺗﻤﺎﺭﻳﻦ ﻋﺎﻣﺔ‬ ‫1 ﺣﺪﺩ ﺻﺤﺔ ﺃﻭ ﺧﻄﺄ ﻛﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺘﺎﻟﻴﺔ، ﻭﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﻄﺄ، ﻭﺿﺢ ﺳﺒﺐ ﺍﻟﺨﻄﺄ ﻭﺇﺫﺍ ﻛﺎﻧﺖ ﺻﻮﺍﺑﺎ‬ ‫ً‬ ‫ﺃﻋﻂ ﺃﻣﺜﻠﺔ ﺗﻮﺿﺤﻬﺎ.‬ ‫( ........................................‬ ‫)‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ ‪ ،C‬ﺏ ∋ ﺡ ﻛﺎﻥ ‪ < C‬ﺏ ، ‪ ،C‬ﺏ ﻣﻮﺟﺒﺎﻥ ﻣﻌﺎ ﻓﺈﻥ ‪ < ٢C‬ﺏ٢.‬ ‫ﻭ‬ ‫ً‬ ‫( ........................................‬ ‫)‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ‪ ،C‬ﺏ ∋ ﺡ ﻛﺎﻥ ‪ < C‬ﺏ ، ‪ ،C‬ﺏ ﺳﺎﻟﺒﺎﻥ ﻣﻌﺎ ﻓﺈﻥ ‪ < ٢C‬ﺏ٢.‬ ‫ﻭ‬ ‫ً‬ ‫١‬ ‫١‬ ‫( ........................................‬ ‫ﺟ ﺇﺫﺍ ﻛﺎﻥ ‪ ،C‬ﺏ ∋ ﺡ ﻛﺎﻥ ‪ < C‬ﺏ ، ‪ ،C‬ﺏ ﻣﺘﺤﺪﺍﻥ ﻓﻰ ﺍﻹﺷﺎﺭﺓ ﻓﺈﻥ > . )‬ ‫ﻭ‬ ‫ﺏ‬ ‫‪C‬‬ ‫١‬ ‫١‬ ‫د ﺇﺫﺍ ﻛﺎﻥ ‪ ،C‬ﺏ ∋ ﺡ ﻛﺎﻥ ‪ < C‬ﺏ ، ‪ ،C‬ﺏ ﻣﺨﺘﻠﻔﺘﺎﻥ ﻓﻰ ﺍﻹﺷﺎﺭﺓ ﻓﺈﻥ > ﺏ . )‬ ‫ﻭ‬ ‫‪C‬‬ ‫(‬ ‫........................................‬ ‫2 ﺣﻞ ﻛﻼ ﻣﻦ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ ﺍﻵﺗﻴﺔ، ﻭﻣﺜﻞ ﺍﻟﺤﻞ ﻋﻠﻰ ﺧﻂ ﺍﻷﻋﺪﺍﺩ:‬ ‫ًّ‬ ‫أ ٢ﺱ + ٤ > ٥ﺱ - ٥ ..........................................................................................................................................................................‬ ‫.......................................................................................................................................................................................................................‬ ‫ب ٧ > ٥ﺱ + ٢ ‪١٢ H‬‬ ‫..........................................................................................................................................................................‬ ‫.......................................................................................................................................................................................................................‬ ‫3 ﻣﺜﻞ ﺑﻴﺎﻧﻴﺎ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﻛﻞ ﻣﻦ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ:‬ ‫ًّ‬ ‫ب‬ ‫ﺹ ‪- G‬ﺱ - ٤‬ ‫أ ﺹ ‪٢ H‬ﺱ + ١‬ ‫4 ﺣﻞ ﻛﻞ ﻧﻈﺎﻡ ﻣﻦ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ ﺑﻴﺎﻧﻴﺎ:‬ ‫ًّ‬ ‫أ ﺹ>ﺱ+٣‬ ‫ب ﺱ+ﺹ<٢‬ ‫٢ﺱ - ﺹ > ١‬ ‫ﺹ<٦+ﺱ‬ ‫ﺟ ﺹ > -٢ﺱ + ٣‬ ‫ﺟ ٢ﺱ + ٣ﺹ ‪١٢ H‬‬ ‫ﺱ‪٢G‬‬ ‫ﺹ‪١G‬‬ ‫5 ﺃﻭﺟﺪ ﻣﻦ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ ﻗﻴﻤﺘﻰ ﺱ، ﺹ ﺍﻟﺘﻰ ﺗﺠﻌﻞ ﻗﻴﻤﺔ‬ ‫ﺍﻟﺪﺍﻟﺔ ‪١ = S‬‬ ‫ﺱ + ﺹ ﻗﻴﻤﺔ ﺻﻐﺮﻯ. ...................................................................‬ ‫٢‬ ‫.................................................................................................................................................‬ ‫.................................................................................................................................................‬ ‫6 ﺍﻟﺮﺑﻂ ﺑﺎﻟﺼﻨﺎﻋﺔ ﻭﺭﺷﺔ ﺻﻐﻴﺮﺓ ﻟﻌﻤﻞ ﺍﻷﻭﺍﻧﻲ ﺍﻟﻤﻌﺪﻧﻴﺔ، ﺗﺼﻨﻊ ﻧﻮﻋﻴﻦ ﻣﻦ ﺍﻷﻭﺍﻧﻰ ‪ ،C‬ﺏ، ﻭﺗﺤﺘﺎﺝ ﺍﻵﻧﻴﺔ ‪ C‬ﺇﻟﻰ‬ ‫٠١ ﺩﻗﺎﺋﻖ ﻋﻤﻞ ﻣﻦ ﺍﻟﻤﺎﻛﻴﻨﺔ ﺍﻷﻭﻟﻰ، ٢١ ﺩﻗﻴﻘﺔ ﻋﻤﻞ ﻣﻦ ﺍﻟﻤﺎﻛﻴﻨﺔ ﺍﻟﺜﺎﻧﻴﺔ، ﺑﻴﻨﻤﺎ ﺗﺤﺘﺎﺝ ﺍﻵﻧﻴﺔ ﺏ ﺇﻟﻰ ٥١ ﺩﻗﻴﻘﺔ‬ ‫ﻋﻤﻞ ﻣﻦ ﺍﻟﻤﺎﻛﻴﻨﺔ ﺍﻷﻭﻟﻰ، ٠١ ﺩﻗﺎﺋﻖ ﻣﻦ ﺍﻟﺜﺎﻧﻴﺔ، ﻓﺈﺫﺍ ﻛﺎﻥ ﺭﺑﺢ ﺍﻵﻧﻴﺔ )ﺃ( ٤ ﺟﻨﻴﻬﺎﺕ ﻭﺭﺑﺢ ﺍﻵﻧﻴﺔ )ﺏ( ﻫﻮ‬ ‫٥ ﺟﻨﻴﻬﺎﺕ، ﻓﻤﺎ ﻋﺪﺩ ﺍﻷﻭﺍﻧﻲ ﻣﻦ ﻛﻞ ﻧﻮﻉ ﺣﺘﻰ ﻳﻜﻮﻥ ﺍﻟﺮﺑﺢ ﺃﻛﺒﺮ ﻣﺎ ﻳﻤﻜﻦ، ﻋﻠﻤﺎ ﺑﺄﻥ ﻛﻼ ﻣﻦ ﺍﻟﻤﺎﻛﻴﻨﺘﻴﻦ‬ ‫ًّ‬ ‫ً‬ ‫ﻻﺗﻌﻤﻞ ﺃﻛﺜﺮ ﻣﻦ ﺛﻤﺎﻧﻲ ﺳﺎﻋﺎﺕ?‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬