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كتاب الانشطه - مصر- ترم اول - 2014

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منتدى الرياضيات المتقدمة
منتدى الرياضيات المتقدمة

كتاب الانشطه - مصر- ترم اول - 2014

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äÉ«°VÉjôdG äÉÑjQóàdG h ᣰûfC’G ÜÉàc ∫hC’G ≈°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 ﻳﺠﻮﺯ 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 ٨٤٩٧ / ٣١٠٢‬ ‫ﺍﻟﺮﻗﻢ ﺍﻟﺪﻭﻟﻰ 4 - 000 - 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 ﻓﻴﻤﺎﻳﻠﻰ:‬ ‫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 ‫ﺍﻟﺠ‪ ‬ﻭﺍﻟﻌﻼﻗﺎﺕ ﻭﺍﻟﺪﻭﺍﻝ‬ ‫1- 1‬ ‫ﺣﻞ ﻣﻌﺎدﻻت اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ ﻓﻰ ﻣﺘﻐﻴﺮ واﺣﺪ.‬ ‫1- 2‬ ‫ﻣﻘﺪﻣﺔ ﻋﻦ اﻷﻋﺪاد اﻟﻤﺮﻛﺒﺔ.‬ ‫1- 3‬ ‫ﺗﺤﺪﻳﺪ ﻧﻮع ﺟﺬرى اﻟﻤﻌﺎدﻟﺔ اﻟﺘﺮﺑﻴﻌﻴﺔ.‬ ‫1- 4‬ ‫اﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺟﺬري ﻣﻌﺎدﻟﺔ اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ وﻣﻌﺎﻣﻼت ﺣﺪودﻫﺎ.‬ ‫1- 5‬ ‫إﺷﺎرة اﻟﺪاﻟﺔ.‬ ‫21‬ ‫1- 6‬ ‫ﻣﺘﺒﺎﻳﻨﺎت اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ.‬ ‫41‬ ‫51‬ ‫71‬ ‫81‬ ‫................................................................................................................................................‬ ‫2‬ ‫...................................................................................................................................................................................................‬ ‫5‬ ‫.....................................................................................................................................................................‬ ‫7‬ ‫............................................................................................‬ ‫9‬ ‫............................................................................................................................................................................................................................................‬ ‫ﺗﻤﺎرﻳﻦ ﻋﺎﻣﺔ‬ ‫........................................................................................................................................................................................................‬ ‫.........................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر اﻟﻮﺣﺪة‬ ‫......................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر ﺗﺮاﻛﻤﻰ‬ ‫....................................................................................................................................................................................................................................‬ ‫‪IóMƒdG‬‬ ‫‪á«fÉãdG‬‬ ‫ﺍﻟﺘﺸﺎﺑﻪ‬ ‫2-1‬ ‫ﺗﺸﺎﺑﻪ اﻟﻤﻀﻠﻌﺎت‬ ‫02‬ ‫2-2‬ ‫ﺗﺸﺎﺑﻪ اﻟﻤﺜﻠﺜﺎت.‬ ‫22‬ ‫2-3‬ ‫اﻟﻌﻼﻗﺔ ﺑﻴﻦ ﻣﺴﺎﺣﺘﻰ ﺳﻄﺤﻰ ﻣﻀﻠﻌﻴﻦ ﻣﺘﺸﺎﺑﻬﻴﻦ.‬ ‫62‬ ‫2-4‬ ‫ﺗﻄﺒﻴﻘﺎت اﻟﺘﺸﺎﺑﻪ ﻓﻰ اﻟﺪاﺋﺮة‬ ‫82‬ ‫23‬ ‫43‬ ‫53‬ ‫.....................................................................................................................................................................................................................‬ ‫..........................................................................................................................................................................................................................‬ ‫ﺗﻤﺎرﻳﻦ ﻋﺎﻣﺔ‬ ‫...........................................................................................‬ ‫.............................................................................................................................................................................‬ ‫.........................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر اﻟﻮﺣﺪة‬ ‫......................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر ﺗﺮاﻛﻤﻰ‬ ‫....................................................................................................................................................................................................................................‬
‫‪IóMƒdG‬‬ ‫‪áãdÉãdG‬‬ ‫ﻧﻈﺮﻳﺎﺕ ﺍﻟﺘﻨﺎﺳﺐ ﻓﻰ ﺍﻟﻤﺜﻠﺚ‬ ‫3-1‬ ‫اﻟﻤﺴﺘﻘﻴﻤﺎت اﻟﻤﺘﻮازﻳﺔ واﻷﺟﺰاء اﻟﻤﺘﻨﺎﺳﺒﺔ‬ ‫3-2‬ ‫ﻣﻨﺼﻔﺎ اﻟﺰاوﻳﺔ ﻓﻰ اﻟﻤﺜﻠﺚ واﻷﺟﺰاء اﻟﻤﺘﻨﺎﺳﺒﺔ‬ ‫3-3‬ ‫ﺗﻄﺒﻴﻘﺎت اﻟﺘﻨﺎﺳﺐ ﻓﻰ اﻟﺪاﺋﺮة‬ ‫ﺗﻤﺎرﻳﻦ ﻋﺎﻣﺔ‬ ‫.................................................................................................................................................‬ ‫83‬ ‫.......................................................................................................................................‬ ‫14‬ ‫.......................................................................................................................................................................................‬ ‫.........................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر اﻟﻮﺣﺪة‬ ‫......................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر ﺗﺮاﻛﻤﻰ‬ ‫....................................................................................................................................................................................................................................‬ ‫‪IóMƒdG‬‬ ‫‪á©HGôdG‬‬ ‫34‬ ‫54‬ ‫64‬ ‫74‬ ‫ﺣﺴﺎﺏ ﺍﻟﻤﺜﻠﺜﺎﺕ‬ ‫4-1‬ ‫اﻟﺰاوﻳﺔ اﻟﻤﻮﺟﻬﺔ.‬ ‫4-2‬ ‫ﻃﺮق ﻗﻴﺎس اﻟﺰاوﻳﺔ.‬ ‫4-3‬ ‫اﻟﺪوال اﻟﻤﺜﻠﺜﻴﺔ‬ ‫4-4‬ ‫اﻟﻌﻼﻗﺎت ﺑﻴﻦ اﻟﺪوال اﻟﻤﺜﻠﺜﻴﺔ‬ ‫4-5‬ ‫اﻟﺘﻤﺜﻴﻞ اﻟﺒﻴﺎﻧﻰ ﻟﻠﺪوال اﻟﻤﺜﻠﺜﻴﺔ‬ ‫4-6‬ ‫إﻳﺠﺎد ﻗﻴﺎس زاوﻳﺔ ﺑﻤﻌﻠﻮﻣﻴﺔ داﻟﺔ ﻣﺜﻠﺜﻴﺔ‬ ‫..............................................................................................................................................................................................................................‬ ‫05‬ ‫.....................................................................................................................................................................................................................‬ ‫25‬ ‫.......................................................................................................................................................................................................................................‬ ‫55‬ ‫.............................................................................................................................................................................................‬ ‫75‬ ‫.......................................................................................................................................................................................‬ ‫06‬ ‫........................................................................................................................................................‬ ‫ﺗﻤﺎرﻳﻦ ﻋﺎﻣﺔ‬ ‫.........................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر اﻟﻮﺣﺪة‬ ‫......................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر ﺗﺮاﻛﻤﻰ‬ ‫اﺧﺘﺒﺎرات ﻋﺎﻣﺔ‬ ‫....................................................................................................................................................................................................................................‬ ‫.......................................................................................................................................................................................................................................................................‬ ‫إﺟﺎﺑﺎت ﺑﻌﺾ اﻟﺘﻤﺎرﻳﻦ‬ ‫.................................................................................................................................................................................................................................................‬ ‫16‬ ‫36‬ ‫46‬ ‫56‬ ‫66‬ ‫27‬
‫ﺍﻟﺠﺒﺮ‬ ‫‪IóMƒdG‬‬ ‫1‬ ‫ﺍﻟﺠﺒﺮ ﻭﺍﻟﻌﻼﻗﺎﺕ ﻭﺍﻟﺪﻭﺍﻝ‬ ‫‪Algebra, Relations and‬‬ ‫‪Functions‬‬ ‫دروس اﻟﻮﺣﺪة‬ ‫ﺍﻟﺪﺭﺱ )١ - ١(: ﺣﻞ ﻣﻌﺎﺩﻻﺕ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ ﻓﻰ ﻣﺘﻐﻴﺮ ﻭﺍﺣﺪ.‬ ‫ﺍﻟﺪﺭﺱ )١ - ٢(: ﻣﻘﺪﻣﺔ ﻋﻦ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻤﺮﻛﺒﺔ.‬ ‫ﺍﻟﺪﺭﺱ )١ - ٣(: ﺗﺤﺪﻳﺪ ﻧﻮﻉ ﺟﺬﺭ￯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ.‬ ‫ﺍﻟﺪﺭﺱ )١ - ٤(: ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺟﺬﺭ￯ ﻣﻌﺎﺩﻟﺔ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ ﻭﻣﻌﺎﻣﻼﺕ ﺣﺪﻭﺩﻫﺎ.‬ ‫ﺍﻟﺪﺭﺱ )١ - ٥(: ﺇﺷﺎﺭﺓ ﺍﻟﺪﺍﻟﺔ.‬ ‫ﺍﻟﺪﺭﺱ )١ - ٦(: ﻣﺘﺒﺎﻳﻨﺎﺕ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ.‬
‫ﺣﻞ ﻣﻌﺎدﻻت اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ ﻓﻰ ﻣﺘﻐﻴﺮ واﺣﺪ‬ ‫1-1‬ ‫‪Solving Quadratic Equations in One Variable‬‬ ‫‪k‬‬ ‫‪Oó©àe øe QÉ«àN’G :’hCG‬‬ ‫1 ﺍﻟﻤﻌﺎﺩﻟﺔ: )ﺱ – ١( )ﺱ + ٢( = ٠ ﻣﻦ ﺍﻟﺪﺭﺟﺔ:‬ ‫ب ﺍﻟﺜﺎﻧﻴﺔ‬ ‫أ ﺍﻷﻭﻟﻰ‬ ‫2 ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ = ﺱ ﻓﻰ ﺡ ﻫﻰ:‬ ‫ب }١{‬ ‫أ }٠{‬ ‫..................................................................................................................................‬ ‫ﺟ ﺍﻟﺜﺎﻟﺜﺔ‬ ‫د ﺍﻟﺮﺍﺑﻌﺔ‬ ‫.....................................................................................................................................‬ ‫ﺟ }- ١، ١{‬ ‫3 ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ٣ = ٠ ﻓﻰ ﺡ ﻫﻰ:‬ ‫ب }- ٣ {‬ ‫أ }-٣{‬ ‫د }0، ١{‬ ‫.................................................................................................................................‬ ‫4 ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٢ﺱ = -١ ﻓﻰ ﺡ ﻫﻰ:‬ ‫ب ‪z‬‬ ‫أ }-١{‬ ‫ﺟ } ٣ {‬ ‫د ‪z‬‬ ‫........................................................................................................................‬ ‫ﺟ }-١، ١{‬ ‫د }١{‬ ‫5 ﻳﻤﺜﻞ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ ﺍﻟﻤﻨﺤﻨﻰ ﺍﻟﺒﻴﺎﻧﻰ ﻟﺪﺍﻟﺔ ﺗﺮﺑﻴﻌﻴﺔ ﺩ.‬ ‫ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺩ)ﺱ( = ٠ ﻓﻰ ﺡ ﻫﻰ: ......................................‬ ‫ب }٤{‬ ‫أ }-٢{‬ ‫د }-٢، ٤{‬ ‫ﺟ ‪z‬‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫6 ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﻛﻞ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ ﻓﻰ ﺡ:‬ ‫ب ﺱ٢ + ٣ﺱ = ٠‬ ‫أ ﺱ٢ - ١ = ٠‬ ‫ﺟ )ﺱ – ٤(٢ = ٠‬ ‫............................................................‬ ‫............................................................‬ ‫...........................................................‬ ‫............................................................‬ ‫............................................................‬ ‫...........................................................‬ ‫............................................................‬ ‫............................................................‬ ‫...........................................................‬ ‫ﻫ ﺱ٢ + ٩ = ٠‬ ‫د ﺱ٢ - ٦ﺱ + ٩ = ٠‬ ‫و ﺱ )ﺱ+ ١( )ﺱ - ١( = ٠‬ ‫............................................................‬ ‫............................................................‬ ‫...........................................................‬ ‫............................................................‬ ‫............................................................‬ ‫...........................................................‬ ‫............................................................‬ ‫............................................................‬ ‫...........................................................‬ ‫¯‬ ‫−‬ ‫¯‬
‫¯‬ ‫7 ﻳﺒﻴﻦ ﻛﻞ ﺷﻜﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻵﺗﻴﺔ ﺍﻟﺮﺳﻢ ﺍﻟﺒﻴﺎﻧﻰ ﻟﺪﺍﻟﺔ ﻣﻦ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ.‬ ‫ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺍﻟﺤﻞ ﻟﻠﻤﻌﺎﺩﻟﺔ ﺩ )ﺱ( = ٠ ﻓﻰ ﻛﻞ ﺷﻜﻞ.‬ ‫ب‬ ‫أ‬ ‫ﺟ‬ ‫− −‬ ‫− − − − −‬ ‫−‬ ‫−‬ ‫− − −‬ ‫−‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫8 ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺍﻟﺤﻞ ﻟﻜﻞ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ ﻓﻰ ﺡ ﻭﺣﻘﻖ ﺍﻟﻨﺎﺗﺞ ﺑﻴﺎﻧﻴﺎ:‬ ‫ًّ‬ ‫ب ٢ﺱ٢ = ٣ – ٥ﺱ‬ ‫أ ﺱ٢ = ٣ﺱ + ٠٤‬ ‫............................................................‬ ‫............................................................‬ ‫ﺟ ٦ﺱ٢ = ٦ – ٥ﺱ‬ ‫د )ﺱ – ٣( = ٥‬ ‫٢‬ ‫............................................................‬ ‫............................................................‬ ‫و ١ ﺱ٢ - ٣ ﺱ = ١‬ ‫٥‬ ‫٢‬ ‫ﻫ ﺱ٢ + ٢ﺱ = ٢١‬ ‫............................................................‬ ‫............................................................‬ ‫9 ﺣﻞ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ ﻓﻰ ﺡ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻘﺎﻧﻮﻥ ﺍﻟﻌﺎﻡ ﻣﻘﺮﺑﺎ ﺍﻟﻨﺎﺗﺞ ﻟﺮﻗﻢ ﻋﺸﺮﻯ ﻭﺍﺣﺪ.‬ ‫ً‬ ‫ب ﺱ٢ – ٦ﺱ + ٧ = ٠‬ ‫أ ٣ﺱ٢ – ٥٦ = ٠‬ ‫............................................................‬ ‫............................................................‬ ‫ﺟ ﺱ٢ + ٦ﺱ + ٨ = ٠‬ ‫د ٢ﺱ٢+٣ﺱ–٤ = ٠‬ ‫............................................................‬ ‫............................................................‬ ‫و ٣ﺱ٢ – ٦ﺱ – ٤ = ٠‬ ‫ﻫ ٥ﺱ٢ – ٣ﺱ – ١ = ٠‬ ‫............................................................‬ ‫............................................................‬ ‫01 ﺃﻋﺪﺍﺩ: ﺇﺫﺍ ﻛﺎﻥ ﻣﺠﻤﻮﻉ ﺍﻷﻋﺪﺍﺩ ﺍﻟﺼﺤﻴﺤﺔ ﺍﻟﻤﺘﺘﺎﻟﻴﺔ )١ + ٢ + ٣ + ... + ﻥ(ﻳﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ ﺟـ = ﻥ )١ + ﻥ(‬ ‫٢‬ ‫ﻓﻜﻢ ﻋﺪﺩﺍ ﺻﺤﻴﺤﺎ ﻣﺘﺘﺎﻟﻴﺎ ﺑﺪﺀﺍ ﻣﻦ ﺍﻟﻌﺪﺩ ١ ﻳﻜﻮﻥ ﻣﺠﻤﻮﻋﻬﺎ ﻣﺴﺎﻭ ﻳﺎ:‬ ‫ً‬ ‫ً‬ ‫ً‬ ‫ً ً‬ ‫ب ١٧١‬ ‫أ ٨٧‬ ‫..............................................‬ ‫...............................................‬ ‫ﺟ ٣٥٢‬ ‫د ٥٦٤‬ ‫...............................................‬ ‫...............................................‬ ‫‪M‬‬ ‫−‬
‫11 ﻳﺒﻴﻦ ﻛﻞ ﺷﻜﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻵﺗﻴﺔ ﺍﻟﺮﺳﻢ ﺍﻟﺒﻴﺎﻧﻰ ﻟﺪﺍﻟﺔ ﻣﻦ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ ﻓﻰ ﻣﺘﻐﻴﺮ ﻭﺍﺣﺪ. ﺃﻭﺟﺪ ﻗﺎﻋﺪﺓ ﻛﻞ‬ ‫ﺩﺍﻟﺔ ﻣﻦ ﻫﺬه ﺍﻟﺪﻭﺍﻝ.‬ ‫ب‬ ‫أ‬ ‫ﺟ‬ ‫− − − −‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫− − − −‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫...............................................................‬ ‫...............................................................‬ ‫...............................................................‬ ‫21 ﺍﻛﺘﺸﻒ ﺍﻟﺨﻄﺄ: ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ )ﺱ – ٣(٢ = )ﺱ – ٣(.‬ ‫¯‬ ‫‪) a‬ﺱ – ٣(٢ = )ﺱ – ٣(‬ ‫–‬ ‫‪¯M‬‬ ‫ ‬ ‫‪F‬‬ ‫` ﺱ – ٣ = ١‬ ‫` ﺱ = ٤‬ ‫ﻣﺠﻤﻮﻋﺔ ﺍﻟﺤﻞ = }٤{‬ ‫!‬ ‫‪) a‬ﺱ – ٣(٢ = )ﺱ – ٣(‬ ‫` )ﺱ – ٣(٢ – )ﺱ – ٣( = ٠‬ ‫` )ﺱ – ٣(])ﺱ – ٣( – ١[ = ٠‬ ‫‪ :F‬ﺱ – ٣=٠ ﺃﻭ ﺱ – ٤=٠‬ ‫ﻣﺠﻤﻮﻋﺔ ﺍﻟﺤﻞ = }٣، ٤{‬ ‫ﺃﻱ ﺍﻟﺤﻠﻴﻦ ﺻﺤﻴﺢ? ﻟﻤﺎﺫﺍ?‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫31 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ: ﻗُﺬﻓﺖ ﻛﺮﺓ ﺭﺃﺳﻴﺎ ﺇﻟﻰ ﺃﻋﻠﻰ ﺑﺴﺮﻋﺔ ﻉ ﺗﺴﺎﻭﻯ ٤٫٩٢ ﻣﺘﺮ/ﺙ. ﺍﺣﺴﺐ ﺍﻟﻔﺘﺮﺓ ﺍﻟﺰﻣﻨﻴﺔ ﻥ ﺑﺎﻟﺜﺎﻧﻴﺔ ﺍﻟﺘﻰ‬ ‫ُ‬ ‫ًّ‬ ‫ﺗﺴﺘﻐﺮﻗﻬﺎ ﺍﻟﻜﺮﺓ ﺣﺘﻰ ﺗﺼﻞ ﺇﻟﻰ ﺍﺭﺗﻔﺎﻉ ﻑ ﻣﺘﺮﺍ، ﺣﻴﺚ ﻑ ﺗﺴﺎﻭﻯ ٢٫٩٣ ﻣﺘﺮﺍ ﻋﻠﻤﺎ ﺑﺄﻥ ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﻑ، ﻥ ﺗﻌﻄﻰ‬ ‫ُْ‬ ‫ً ً‬ ‫ً‬ ‫ﻛﺎﻵﺗﻰ ﻑ = ﻉ ﻥ – ٩٫٤ ﻥ٢.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
‫ﻣﻘﺪﻣﺔ ﻋﻦ ا ﻋﺪاد اﻟﻤﺮﻛﺒﺔ‬ ‫1-2‬ ‫‪Complex Numbers‬‬ ‫1 ﺿﻊ ﻛﻼ ﻣﻤﺎ ﻳﺄﺗﻰ ﻓﻰ ﺃﺑﺴﻂ ﺻﻮﺭﺓ:‬ ‫ًّ‬ ‫ ٥٤‬‫٦٦‬ ‫ب ﺕ‬ ‫أ ﺕ‬ ‫......................................‬ ‫ﺟ ﺕ‬ ‫......................................‬ ‫٤ﻥ + ٢‬ ‫......................................‬ ‫د ﺕ‬ ‫٤ﻥ – ١‬ ‫......................................‬ ‫2 ﺑﺴﻂ ﻛﻼ ﻣﻤﺎ ﻳﺄﺗﻰ:‬ ‫ًّ‬ ‫أ‬ ‫-٨١ * -٢١‬ ‫ب ٣ ﺕ )- ٢ﺕ(‬ ‫..........................................‬ ‫ﺟ )- ٤ ﺕ( )- ٦ ﺕ(‬ ‫..........................................‬ ‫...........................................‬ ‫د )- ٢ ﺕ(٣ )- ٣ ﺕ(‬ ‫٢‬ ‫.........................................‬ ‫3 ﺃﻭﺟﺪ ﻧﺎﺗﺞ ﻛﻞ ﻣﻤﺎ ﻳﺄﺗﻰ ﻓﻰ ﺃﺑﺴﻂ ﺻﻮﺭﺓ:‬ ‫ٍّ‬ ‫أ )٣ + ٢ﺕ( + )٢ – ٥ ﺕ(   ب )٦٢ – ٤ﺕ( – )٩ – ٠٢ ﺕ(   ﺟ )٠٢ + ٥٢ ﺕ( – )٩ – ٠٢ ﺕ(‬ ‫.................................................................................       ..................................................................................      ..................................................................................‬ ‫4 ﺿﻊ ﻛﻼ ﻣﻤﺎ ﻳﺄﺗﻰ ﻋﻠﻰ ﺻﻮﺭﺓ ‪ + C‬ﺏ ﺕ‬ ‫ًّ‬ ‫ب )١ + ٢ﺕ٣( )٢ + ٣ ﺕ٥ + ٤ ﺕ٦(‬ ‫أ )٢ + ٣ ﺕ( – )١ – ٢ﺕ(‬ ‫5 ﺿﻊ ﻛﻼ ﻣﻤﺎ ﻳﺄﺗﻰ ﻋﻠﻰ ﺻﻮﺭﺓ ‪ + C‬ﺏ ﺕ‬ ‫ًّ‬ ‫٢‬ ‫ب ٤+ﺕ‬ ‫أ‬ ‫ﺕ‬ ‫ﺟ‬ ‫١+ﺕ‬ ‫......................................‬ ‫......................................‬ ‫......................................‬ ‫6 ﺣﻞ ﻛﻞ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ:‬ ‫ب ٤ ﺹ٢ + ٠٢ = ٠‬ ‫أ ٣ ﺱ٢ + ٢١ = ٠‬ ‫......................................‬ ‫٢ - ٣ﺕ‬ ‫٣+ﺕ‬ ‫ﺟ ٤ ﻉ٢ + ٢٧ = ٠‬ ‫......................................‬ ‫......................................‬ ‫د )٣ + ﺕ()٣ - ﺕ(‬ ‫٣-٤ﺕ‬ ‫......................................‬ ‫د ٣ ﺹ٢ + ٥١ = ٠‬ ‫٥‬ ‫......................................‬ ‫7 ﻛﻬﺮﺑﺎﺀ: ﺃﻭﺟﺪ ﺷﺪﺓ ﺍﻟﺘﻴﺎﺭ ﺍﻟﻜﻬﺮﺑﻰ ﺍﻟﻜﻠﻴﺔ ﺍﻟﻤﺎﺭ ﻓﻰ ﻣﻘﺎﻭﻣﺘﻴﻦ ﻣﺘﺼﻠﺘﻴﻦ ﻋﻠﻰ ﺍﻟﺘﻮﺍﺯﻯ ﻓﻰ ﺩﺍﺋﺮﺓ ﻛﻬﺮﺑﺎﺋﻴﺔ‬ ‫ﻣﻐﻠﻘﺔ ﺇﺫﺍ ﻛﺎﻧﺖ ﺷﺪﺓ ﺍﻟﺘﻴﺎﺭ ﻓﻰ ﺍﻟﻤﻘﺎﻭﻣﺔ ﺍﻷ ﻟﻰ ٤ – ٢ﺕ ﺃﻣﺒﻴﺮ، ﻭﻓﻰ ﺍﻟﻤﻘﺎﻭﻣﺔ ﺍﻟﺜﺎﻧﻴﺔ ٦ + ٣ﺕ ﺃﻣﺒﻴﺮ .................‬ ‫ﻭ‬ ‫٢+ﺕ‬ ‫8 ﺍﻛﺘﺸﻒ ﺍﻟﺨﻄﺄ: ﺃﻭﺟﺪ ﺃﺑﺴﻂ ﺻﻮﺭﺓ ﻟﻠﻤﻘﺪﺍﺭ: )٢ + ٣ﺕ(٢ )٢ – ٣ﺕ(‬ ‫¯‬ ‫)٢ + ٣ﺕ(٢)٢– ٣ﺕ( = )٤ + ٩ﺕ٢()٢ – ٣ﺕ(‬ ‫= )٤ – ٩()٢ – ٣ﺕ( = - ٥ )٢ – ٣ﺕ(‬ ‫= - ٠١ + ٥١ ﺕ‬ ‫)٢ + ٣ﺕ()٢ + ٣ﺕ()٢ – ٣ﺕ(‬ ‫= )٢ + ٣ﺕ( )٤ – ٩ﺕ٢(‬ ‫= )٢ + ٣ﺕ( )٤ + ٩( = ٣١)٢ + ٣ﺕ(‬ ‫= ٦٢ + ٩٣ ﺕ‬ ‫ﺃﻯ ﺍﻟﺤﻠﻴﻦ ﺻﺤﻴﺢ? ﻟﻤﺎﺫﺍ?‬ ‫............................................................................................................................................................................‬ ‫‪M‬‬ ‫−‬
‫ﻧﺸﺎط‬ ‫١-‬ ‫٢-‬ ‫٣-‬ ‫٤-‬ ‫٥-‬ ‫ﺑﺎﺳﺘﺨﺪﺍﻡ ﺃﺣﺪ ﺍﻟﺒﺮﺍﻣﺞ ﺍﻟﺮﺳﻮﻣﻴﺔ ﺍﺭﺳﻢ ﻣﻨﺤﻨﻰ ﺍﻟﺪﺍﻟﺔ ﺩ ﺣﻴﺚ ﺩ)ﺱ( = ﺱ٣ - ١ .‬ ‫ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ ﻳﻤﺜﻞ ﻣﻨﺤﻨﻰ ﺍﻟﺪﺍﻟﺔ، ﻫﻞ ﻳﻤﻜﻨﻚ ﺇﻳﺠﺎﺩ ﻣﺠﻤﻮﻋﺔ ﺣﻞ‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٣ -١ = ٠ ﻣﻦ ﺍﻟﺮﺳﻢ?‬ ‫ﻫﻞ ﺗﺘﻮﻗﻊ ﻭﺟﻮﺩ ﺟﺬﻭﺭ ﺃﺧﺮﻯ ﺑﺎﺳﺘﺜﻨﺎﺀ ﺍﻟﺠﺬﻭﺭ ﺍﻟﺘﻰ ﺣﺼﻠﺖ ﻋﻠﻴﻬﺎ ﻣﻦ‬ ‫ﺍﻟﺮﺳﻢ، ﻭﺫﻟﻚ ﻣﻦ ﺧﻼﻝ ﺩﺭﺍﺳﺘﻚ ﻟﻤﺠﻤﻮﻋﺎﺕ ﺍﻷﻋﺪﺍﺩ?‬ ‫ﻫﻞ ﻳﻤﻜﻨﻚ ﺣﻞ ﺱ٣ - ١ = ٠ ﺟﺒﺮ ﻳﺎ?‬ ‫ًّ‬ ‫ﺍﺳﺘﺨﺪﻡ ﻃﺮﻕ ﺍﻟﺘﺤﻠﻴﻞ ﺍﻟﺘﻰ ﺳﺒﻖ ﻟﻚ ﺩﺭﺍﺳﺘﻬﺎ ﻓﻰ ﺣﻞ ﻫﺬه ﺍﻟﻤﻌﺎﺩﻟﺔ.‬ ‫: ﺱ٣ - ١ = )ﺱ - ١()ﺱ٢ + ﺱ + ١( =٠‬ ‫¯‬ ‫¯‬ ‫٦- ﺗﻌﻠﻢ ﺃﻧﻪ ﻣﻦ ﺧﻮﺍﺹ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺇﺫﺍ ﻛﺎﻥ ‪ * C‬ﺏ * ﺟـ = ٠ ﻓﺈﻥ ‪ ، ٠ = C‬ﺏ = ٠، ﺟـ = ٠ ﻓﻬﻞ ﻳﻤﻜﻨﻚ ﺍﺳﺘﺨﺪﺍﻡ ﺫﻟﻚ‬ ‫ﻓﻰ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺴﺎﺑﻘﺔ?‬ ‫  ﺱ = ١  ﻭﻫﺬﺍ ﻳﻄﺎﺑﻖ ﺍﻟﺤﻞ ﺍﻟﺒﻴﺎﻧﻰ ﺃﻭ:‬ ‫ﺱ - ١ = ٠ ‬ ‫¯‬ ‫ﺱ٢ + ﺱ +١ = ٠ ﻫﻞ ﻳﻤﻜﻨﻚ ﺣﻞ ﻫﺬه ﺍﻟﻤﻌﺎﺩﻟﺔ ﺑﺎﻟﺘﺤﻠﻴﻞ?‬ ‫٧- ﺍﺳﺘﺨﺪﻡ ﻣﻔﻬﻮﻡ ﻣﻤﻴﺰ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﻟﺘﺤﺪﻳﺪ ﻧﻮﻉ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ﺱ + ١ = ٠ ﺣﻴﺚ ‪ ، ١ =C‬ﺏ = ١ ، ﺟـ = ١‬ ‫  ﺏ٢ - ٤ ‪ C‬ﺟـ > ٠‬ ‫ﺍﻟﻤﻤﻴﺰ )ﺏ٢- ٤ ‪C‬ﺟـ( = ١ - ٤ *١ *١ = -٣‬ ‫¯‬ ‫¯‬ ‫,‬ ‫٨- ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ﺱ + ١ = ٠ ﻓﻰ ﻣﺠﻤﻮﻋﺔ ﺍﻷﻋﺪﺍﺩ ﻛﺒﺔ .‬ ‫ﺍﻟﻤﺮ‬ ‫ﺱ = - ﺏ ! ﺏ ٢-٤‪C‬ﺟـ‬ ‫٢‪C‬‬ ‫ﻓﺘﻜﻮﻥ ﺱ = - ١ !‬ ‫٩- ﺍﻛﺘﺐ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٣ - ١ = ٠ ﻓﻰ ﻣﺠﻤﻮﻋﺔ ﺍﻷﻋﺪﺍﺩ ﻛﺒﺔ.‬ ‫ﺍﻟﻤﺮ‬ ‫ﻣﺠﻤﻮﻋﺔ ﺍﻟﺤﻞ ﻫﻰ}١، - ١ +‬ ‫٢*١‬ ‫-٣ ،‬ ‫١-‬‫٢*١‬ ‫-٣‬ ‫٢*١‬ ‫-٣ {‬ ‫٠١-ﻛﻢ ﻋﺪﺩ ﺍﻟﺠﺬﻭﺭ ﺍﻟﺤﻘﻴﻘﻴﺔ ﻛﻢ ﻋﺪﺩ ﺍﻟﺠﺬﻭﺭ ﻛﺒﺔ ?‬ ‫ﺍﻟﻤﺮ‬ ‫ﻭ‬ ‫١١- ﺃﻭﺟﺪ ﻣﺠﻤﻮﻉ ﺍﻟﺠﺬﻭﺭ ﺍﻟﺜﻼﺛﺔ ﻟﻠﻤﻌﺎﺩﻟﺔ - ﻣﺎﺫﺍ ﺗﻼﺣﻆ?‬ ‫٢١- ﺃﻭﺟﺪ ﺣﺎﺻﻞ ﺿﺮﺏ ﺍﻟﺠﺬﺭﻳﻦ ﺍﻟﺘﺨﻴﻠﻴﻴﻦ - ﻣﺎﺫﺍ ﺗﻼﺣﻆ?‬ ‫٣١- ﺃﻭﺟﺪ ﻣﺮﺑﻊ ﺃﺣﺪ ﺍﻟﺠﺬﺭﻳﻦ ﺍﻟﺘﺨﻴﻠﻴﻴﻦ ﻭﻗﺎﺭﻧﻪ ﻣﻊ ﺍﻟﺠﺬﺭ ﺍﻵﺧﺮ.‬ ‫٤١- ﻟﻤﺎﺫﺍ ﺃﻋﻄﻰ ﺍﻟﺤﻞ ﺍﻟﺒﻴﺎﻧﻰ ﺟﺬﺭﺍ ﻭﺍﺣﺪﺍ ﻓﻘﻂ، ﺑﻴﻨﻤﺎ ﺃﻋﻄﻰ ﺍﻟﺤﻞ ﺍﻟﺠﺒﺮﻯ ﺛﻼﺛﺔ ﺟﺬﻭﺭ ? ﻓﺴﺮ ﺫﻟﻚ.‬ ‫ً‬ ‫ً‬ ‫ِّ‬ ‫٥١- ﺍﺑﺤﺚ ﻓﻰ ﺍﻟﺸﺒﻜﺔ ﺍﻟﻌﻨﻜﺒﻮﺗﻴﺔ ﻋﻦ ﻛﻴﻔﻴﺔ ﺗﻤﺜﻴﻞ ﺟﺬﻭﺭ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﻜﻌﻴﺒﻴﺔ ﺑﻴﺎﻧﻴﺎ ﺑﻤﺎ ﻳﺘﻨﺎﺳﺐ ﻣﻊ ﻣﻌﻠﻮﻣﺎﺗﻚ.‬ ‫ًّ‬ ‫¯‬ ‫−‬ ‫¯‬
‫ﺗﺤﺪﻳﺪ ﻧﻮع ﺟﺬرى اﻟﻤﻌﺎدﻟﺔ اﻟﺘﺮﺑﻴﻌﻴﺔ‬ ‫1-3‬ ‫‪Determining The Type of Roots of a Quadratic Equation‬‬ ‫‪k‬‬ ‫‪:Oó©àe øe QÉ«àNG :’hCG‬‬ ‫1 ﻳﻜﻮﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٤ﺱ + ﻙ = ٠ ﻣﺘﺴﺎﻭﻳﻴﻦ ﺇﺫﺍ ﻛﺎﻧﺖ:‬ ‫ﺟ ﻙ=٨‬ ‫ب ﻙ=٤‬ ‫أ ﻙ=١‬ ‫............................................................................................‬ ‫2 ﻳﻜﻮﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٢ﺱ + ﻡ = ٠ ﺣﻘﻴﻘﻴﻴﻦ ﻣﺨﺘﻠﻔﻴﻦ ﺇﺫﺍ ﻛﺎﻧﺖ:‬ ‫ﺟ ﻡ<١‬ ‫ب ﻡ>١‬ ‫أ ﻡ=١‬ ‫د ﻙ = ٦١‬ ‫............................................................................‬ ‫3 ﻳﻜﻮﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﻝ ﺱ٢ – ٢١ﺱ + ٩ = ٠ ﻛﺒﻴﻦ ﺇﺫﺍ ﻛﺎﻧﺖ:‬ ‫ﻣﺮ‬ ‫ﺟ ﻝ=٤‬ ‫ب ﻝ>٤‬ ‫أ ﻝ<٤‬ ‫د ﻡ=٤‬ ‫...........................................................................................‬ ‫د ﻝ=١‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫4 ﺣﺪﺩ ﻋﺪﺩ ﺍﻟﺠﺬﻭﺭ ﻭﺃﻧﻮﺍﻋﻬﺎ ﻟﻜﻞ ﻣﻌﺎﺩﻟﺔ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻵﺗﻴﺔ:‬ ‫ب ٣ﺱ٢ + ٠١ﺱ - ٤ = ٠‬ ‫أ ﺱ٢ - ٢ﺱ + ٥ = ٠‬ ‫..................................................................................‬ ‫..................................................................................‬ ‫ﺟ ﺱ٢ – ٠١ﺱ + ٥٢ = ٠‬ ‫د ٦ﺱ٢ – ٩١ﺱ + ٥٣ = ٠‬ ‫..................................................................................‬ ‫..................................................................................‬ ‫و )ﺱ – ١( )ﺱ – ٧( = ٢ )ﺱ – ٣( )ﺱ – ٤(‬ ‫ﻫ )ﺱ – ١١( – ﺱ)ﺱ – ٦( = ٠‬ ‫..................................................................................‬ ‫..................................................................................‬ ‫5 ﺃﻭﺟﺪ ﺣﻞ ﻛﻞ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ ﻓﻰ ﻣﺠﻤﻮﻋﺔ ﺍﻷﻋﺪﺍﺩ ﻛﺒﺔ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻘﺎﻧﻮﻥ ﺍﻟﻌﺎﻡ.‬ ‫ﺍﻟﻤﺮ‬ ‫ٍّ‬ ‫ب ٢ﺱ٢ + ٦ﺱ + ٥ = ٠‬ ‫أ ﺱ٢ - ٤ﺱ + ٥ = ٠‬ ‫..................................................................................‬ ‫..................................................................................‬ ‫ﺟ ٣ﺱ٢ - ٧ﺱ + ٦ = ٠‬ ‫د ٤ﺱ٢ - ﺱ + ١ = ٠‬ ‫..................................................................................‬ ‫..................................................................................‬ ‫6 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻙ ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻟﺤﺎﻻﺕ ﺍﻵﺗﻴﺔ:‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ٤ﺱ + ﻙ = ٠ ﺣﻘﻴﻘﻴﻴﻦ ﻣﺨﺘﻠﻔﻴﻦ.‬ ‫.......................................................................................................................................................................................................................‬ ‫‪M‬‬ ‫−‬
‫١‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٣ﺱ + ٢ + ﻙ = ٠ ﻣﺘﺴﺎﻭﻳﻴﻦ.‬ ‫.......................................................................................................................................................................................................................‬ ‫ﺟ ﺇﺫﺍ ﻛﺎﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﻙ ﺱ٢ – ٨ﺱ + ٦١ = ٠ ﻛﺒﻴﻦ.‬ ‫ﻣﺮ‬ ‫..................................................................................................................................................................................................................................‬ ‫7 ﺇﺫﺍ ﻛﺎﻥ ﻝ، ﻡ ﻋﺪﺩﻳﻦ ﻧﺴﺒﻴﻴﻦ، ﻓﺄﺛﺒﺖ ﺃﻥ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ : ﻝ ﺱ٢ + )ﻝ – ﻡ( ﺱ – ﻡ = ٠ ﻋﺪﺩﺍﻥ ﻧﺴﺒﻴﺎﻥ.‬ ‫..................................................................................................................................................................................................................................‬ ‫8 ﻳﻘﺪﺭ ﻋﺪﺩ ﺳﻜﺎﻥ ﺟﻤﻬﻮﺭﻳﺔ ﻣﺼﺮ ﺍﻟﻌﺮﺑﻴﺔ ﻋﺎﻡ ٣١٠٢ ﺑﺎﻟﻌﻼﻗﺔ:‬ ‫ﻉ = ﻥ٢ + ٢٫١ ﻥ + ١٩ ﺣﻴﺚ )ﻉ( ﻋﺪﺩ ﺍﻟﺴﻜﺎﻥ ﺑﺎﻟﻤﻠﻴﻮﻥ، )ﻥ( ﻋﺪﺩ ﺍﻟﺴﻨﻮﺍﺕ‬ ‫.................................................................................................................‬ ‫أ ﻛﻢ ﻛﺎﻥ ﻋﺪﺩ ﺍﻟﺴﻜﺎﻥ ﻋﺎﻡ ٣١٠٢?‬ ‫.................................................................................................................‬ ‫ب ﻗﺪﺭ ﻋﺪﺩ ﺍﻟﺴﻜﺎﻥ ﻋﺎﻡ ٣٢٠٢.‬ ‫ﺟ‬ ‫ﻗﺪﺭ ﻋﺪﺩ ﺍﻟﺴﻨﻮﺍﺕ ﺍﻟﺘﻰ ﻳﺒﻠﻎ ﻋﺪﺩ ﺍﻟﺴﻜﺎﻥ ﻓﻴﻬﺎ ٤٣٣ ﻣﻠﻴ ﻧًﺎ. ...........................................................................................‬ ‫ﻮ‬ ‫ً‬ ‫د ﺍﻛﺘﺐ ﻣﻘﺎﻻ ﺗﻮﺿﺢ ﻓﻴﻪ ﺃﺳﺒﺎﺏ ﺍﻟﺰﻳﺎﺩﺓ ﺍﻟﻤﻄﺮﺩﺓ ﻓﻰ ﻋﺪﺩ ﺍﻟﺴﻜﺎﻥ ﻛﻴﻔﻴﺔ ﻋﻼﺟﻬﺎ.‬ ‫ﻭ‬ ‫9 ﺍﻛﺘﺸﻒ ﺍﻟﺨﻄﺄ: ﻣﺎ ﻋﺪﺩ ﺣﻠﻮﻝ ﺍﻟﻤﻌﺎﺩﻟﺔ ٢ﺱ٢ – ٦ ﺱ = ٥ ﻓﻰ ﺡ‬ ‫¯‬ ‫ﺏ٢– ٤‪C‬ﺟـ = )- ٦(٢ – ٤ * ٢ * ٥‬ ‫     = ٦٣ – ٠٤ = - ٤‬ ‫¯,‬ ‫ﺏ٢– ٤‪C‬ﺟـ = )- ٦(٢ – ٤ * ٢ )- ٥(‬ ‫     = ٦٣ +٠٤ = ٦٧‬ ‫¯,‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫01 ﺇﺫﺍ ﻛﺎﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ٢ )ﻙ - ١( ﺱ + )٢ﻙ + ١( =٠ ﻣﺘﺴﺎﻭﻳﻴﻦ، ﻓﺄﻭﺟﺪ ﻗﻴﻢ ﻙ ﺍﻟﺤﻘﻴﻘﻴﺔ، ﺛﻢ ﺃﻭﺟﺪ‬ ‫ﺍﻟﺠﺬﺭﻳﻴﻦ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫11 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ: ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ٦٣ ﺱ٢ – ٨٤ ﺱ + ٥٢ = ٠ ﻓﻰ ﻣﺠﻤﻮﻋﺔ ﺍﻷﻋﺪﺍﺩ ﻛﺒﺔ.‬ ‫ﺍﻟﻤﺮ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
‫اﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺟﺬرى ﻣﻌﺎدﻟﺔ اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ وﻣﻌﺎﻣﻼت ﺣﺪودﻫﺎ‬ ‫‪The relation between two roots of the second degree‬‬ ‫‪equation and the coefficients of its terms‬‬ ‫1-4‬ ‫‪k‬‬ ‫‪:≈JCÉjÉe πªcCG :’hCG‬‬ ‫1 ﺇﺫﺍ ﻛﺎﻥ ﺱ = ٣ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ﻡ ﺱ – ٧٢ = ٠ ﻓﺈﻥ ﻡ = .................................، ﺍﻟﺠﺬﺭ ﺍﻵﺧﺮ =‬ ‫................................‬ ‫2 ﺇﺫﺍ ﻛﺎﻥ ﺣﺎﺻﻞ ﺿﺮﺏ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ : ٢ ﺱ٢ + ٧ ﺱ + ٣ ﻙ = ٠ ﻳﺴﺎﻭﻯ ﻣﺠﻤﻮﻉ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ:‬ ‫٢‬ ‫ﺱ – )ﻙ + ٤( ﺱ = ٠ ﻓﺈﻥ ﻙ = ................................‬ ‫3 ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻟﺘﻰ ﻛﻞ ﻣﻦ ﺟﺬﺭﻳﻬﺎ ﻳﺰﻳﺪ ١ ﻋﻦ ﻛﻞ ﻣﻦ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٣ ﺱ + ٢ = ٠ ﻫﻰ‬ ‫...............................‬ ‫4 ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻟﺘﻰ ﻛﻞ ﻣﻦ ﺟﺬﺭﻳﻬﺎ ﻳﻨﻘﺺ ١ ﻋﻦ ﻛﻞ ﻣﻦ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٥ ﺱ + ٦ = ٠ ﻫﻰ‬ ‫...............................‬ ‫‪Oó©àe øe QÉ«àN’G :Ék«fÉK‬‬ ‫5 ﺇﺫﺍ ﻛﺎﻥ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ - ٣ ﺱ + ﺟـ = ٠ ﺿﻌﻒ ﺍﻵﺧﺮ ﻓﺈﻥ ﺟـ ﺗﺴﺎﻭﻯ‬ ‫د ٤‬ ‫ﺟ ٢‬ ‫ب -٢‬ ‫أ -٤‬ ‫.......................................................‬ ‫6 ﺇﺫﺍ ﻛﺎﻥ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ C‬ﺱ٢ – ٣ﺱ+ ٢ =٠ ﻣﻌﻜﻮﺳﺎ ﺿﺮﺑﻴﺎ ﻟﻶﺧﺮ، ﻓﺈﻥ ‪ C‬ﺗﺴﺎﻭﻯ‬ ‫ً‬ ‫ًّ‬ ‫ب ١‬ ‫أ ١‬ ‫د ٣‬ ‫ﺟ ٢‬ ‫٢‬ ‫٣‬ ‫...........................................‬ ‫7 ﺇﺫﺍ ﻛﺎﻥ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢– )ﺏ – ٣( ﺱ + ٥ = ٠ ﻣﻌﻜﻮﺳﺎ ﺟﻤﻌﻴﺎ ﻟﻶﺧﺮ، ﻓﺈﻥ ﺏ ﺗﺴﺎﻭﻯ‬ ‫ً‬ ‫ًّ‬ ‫د ٥‬ ‫ﺟ ٣‬ ‫ب -٣‬ ‫أ -٥‬ ‫........................‬ ‫‪k‬‬ ‫‪á«JB’G á∏Ä°SC’G øY ÖLCG :ÉãdÉK‬‬ ‫8 ﺃﻭﺟﺪ ﻣﺠﻤﻮﻉ ﻭﺣﺎﺻﻞ ﺿﺮﺏ ﺟﺬﺭﻯ ﻛﻞ ﻣﻌﺎﺩﻟﺔ ﻓﻴﻤﺎ ﻳﺄﺗﻰ:‬ ‫ب ٤ ﺱ٢ + ٤ ﺱ – ٥٣ = ٠‬ ‫أ ٣ ﺱ٢ + ٩١ ﺱ – ٤١ = ٠‬ ‫..................................................................................‬ ‫..................................................................................‬ ‫9 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ C‬ﺛﻢ ﺃﻭﺟﺪ ﺍﻟﺠﺬﺭ ﺍﻵﺧﺮ ﻟﻠﻤﻌﺎﺩﻟﺔ ﻓﻰ ﻛﻞ ﻣﻤﺎ ﻳﺄﺗﻰ:‬ ‫ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ  ﺱ٢ – ٢ ﺱ + ‪٠ = C‬‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ: ﺱ = - ١‬ ‫ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ  ‪ C‬ﺱ٢ – ٥ ﺱ + ‪٠ = C‬‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ: ﺱ = ٢‬ ‫........................................................‬ ‫........................................................‬ ‫01 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ ،C‬ﺏ ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ ﺇﺫﺍ ﻛﺎﻥ:‬ ‫......................................................................................................‬ ‫أ ٢، ٥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ‪ C‬ﺱ + ﺏ = ٠‬ ‫......................................................................................................‬ ‫ب -٣، ٧ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪C‬ﺱ٢ – ﺏ ﺱ - ١٢ = ٠‬ ‫ﺟ -١، ٣ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ C‬ﺱ٢ – ﺱ + ﺏ = ٠‬ ‫......................................................................................................‬ ‫٢‬ ‫٢‬ ‫د‬ ‫٣ ﺕ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ + ‪ C‬ﺱ + ﺏ = ٠ ..........................................................................................‬ ‫٣ ﺕ،-‬ ‫‪M‬‬ ‫−‬
‫11 ﺍﺑﺤﺚ ﻧﻮﻉ ﺍﻟﺠﺬﺭﻳﻦ ﻟﻜﻞ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ، ﺛﻢ ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﻛﻞ ﻣﻨﻬﺎ:‬ ‫ب ٢ﺱ٢ + ٣ﺱ + ٧ = ٠‬ ‫أ ﺱ٢ + ٢ﺱ – ٥٣ = ٠‬ ‫..................................................................................‬ ‫ﺟ ﺱ)ﺱ – ٤( + ٥ = ٠‬ ‫..................................................................................‬ ‫د ٣ﺱ)٣ﺱ – ٨( + ٦١ = ٠‬ ‫..................................................................................‬ ‫..................................................................................‬ ‫21 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺟـ ﺍﻟﺘﻰ ﺗﺠﻌﻞ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺟـ ﺱ٢ – ٢١ﺱ + ٩ = ٠ ﻣﺘﺴﺎﻭﻳﻴﻦ.‬ ‫..................................................................................................................................................................................................................................‬ ‫31 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ C‬ﺍﻟﺘﻰ ﺗﺠﻌﻞ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٣ﺱ + ٢ + ١ = ٠ ﻣﺘﺴﺎﻭﻳﻴﻦ.‬ ‫‪C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫41 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺟـ ﺍﻟﺘﻰ ﺗﺠﻌﻞ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ٣ ﺱ٢ – ٥ ﺱ + ﺟـ = ٠ ﻣﺘﺴﺎﻭﻳﻴﻦ، ﺛﻢ ﺃﻭﺟﺪ ﺍﻟﺠﺬﺭﻳﻦ.‬ ‫..................................................................................................................................................................................................................................‬ ‫51 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻙ ﺍﻟﺘﻰ ﺗﺠﻌﻞ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + )ﻙ - ١( ﺱ – ٣ = ٠ ﻫﻮ ﺍﻟﻤﻌﻜﻮﺱ ﺍﻟﺠﻤﻌﻰ ﻟﻠﺠﺬﺭ ﺍﻵﺧﺮ.‬ ‫..................................................................................................................................................................................................................................‬ ‫61 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻙ ﺍﻟﺘﻰ ﺗﺠﻌﻞ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ : ٤ ﻙ ﺱ٢ + ٧ ﺱ + ﻙ٢ + ٤ = ٠ ﻫﻮ ﺍﻟﻤﻌﻜﻮﺱ ﺍﻟﻀﺮﺑﻰ ﻟﻠﺠﺬﺭ ﺍﻵﺧﺮ.‬ ‫..................................................................................................................................................................................................................................‬ ‫71 ﻛﻮﻥ ﻣﻌﺎﺩﻟﺔ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ ﻛﺎﻵﺗﻰ :‬ ‫ب - ٥ ﺕ، ٥ ﺕ‬ ‫أ – ٢، ٤‬ ‫...................................................................................‬ ‫د ١ - ٣ﺕ ، ١ + ٣ﺕ‬ ‫...................................................................................‬ ‫ﺟ ٢،٣‬ ‫٣ ٢‬ ‫...................................................................................‬ ‫................................................................‬ ‫ﻫ ٣ - ٢ ٢ ﺕ ، ٣ + ٢ ٢ ﺕ‬ ‫...................................................................................‬ ‫81 ﺃﻭﺟﺪ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ ﺿﻌﻔﺎ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ٢ﺱ٢ – ٨ﺱ + ٥ = ٠‬ ‫..................................................................................................................................................................................................................................‬ ‫91 ﺃﻭﺟﺪ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻟﺘﻰ ﻛﻞ ﻣﻦ ﺟﺬﺭﻳﻬﺎ ﻳﺰﻳﺪ ﺑﻤﻘﺪﺍﺭ ١ ﻋﻦ ﻛﻞ ﻣﻦ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ : ﺱ٢ – ٧ﺱ – ٩ = ٠‬ ‫..................................................................................................................................................................................................................................‬ ‫02 ﺃﻭﺟﺪ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻟﺘﻰ ﻛﻞ ﻣﻦ ﺟﺬﺭﻳﻬﺎ ﻳﺴﺎﻭﻯ ﻣﺮﺑﻊ ﻧﻈﻴﺮه ﻣﻦ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ : ﺱ٢ + ٣ﺱ – ٥ = ٠‬ ‫..................................................................................................................................................................................................................................‬ ‫12 ﺇﺫﺍ ﻛﺎﻥ ﻝ، ﻡ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٧ ﺱ + ٣ = ٠ ﻓﺄﻭﺟﺪ ﻣﻌﺎﺩﻟﺔ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ:‬ ‫ﺟ ٢،٢‬ ‫د ﻝ + ﻡ، ﻝ ﻡ‬ ‫ب ﻝ + ٢، ﻡ + ٢‬ ‫أ ٢ ﻝ، ٢ ﻡ‬ ‫ﻝ ﻡ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
‫22 ﻣﺴﺎﺣﺎﺕ: ﻗﻄﻌﺔ ﺃﺭﺽ ﻋﻠﻰ ﺷﻜﻞ ﻣﺴﺘﻄﻴﻞ ﺑﻌﺪﺍه ٦، ٩ ﻣﻦ ﺍﻷﻣﺘﺎﺭ، ﻳﺮﺍﺩ ﻣﻀﺎﻋﻔﺔ ﻣﺴﺎﺣﺔ ﻫﺬه ﺍﻟﻘﻄﻌﺔ ﻭﺫﻟﻚ‬ ‫ﺑﺰﻳﺎﺩﺓ ﻃﻮﻝ ﻛﻞ ﺑﻌﺪ ﻣﻦ ﺃﺑﻌﺎﺩﻫﺎ ﺑﻨﻔﺲ ﺍﻟﻤﻘﺪﺍﺭ.ﺃﻭﺟﺪ ﺍﻟﻤﻘﺪﺍﺭ ﺍﻟﻤﻀﺎﻑ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫32 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ: ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﻗﻴﻢ ﺟـ ﻓﻰ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ٧ ﺱ٢ + ٤١ ﺱ + ﺟـ = ٠ ﺑﺤﻴﺚ ﻳﻜﻮﻥ ﻟﻠﻤﻌﺎﺩﻟﺔ:‬ ‫أ ﺟﺬﺭﺍﻥ ﺣﻘﻴﻘﻴﺎﻥ ﻣﺨﺘﻠﻔﺎﻥ.‬ ‫ب ﺟﺬﺭﺍﻥ ﺣﻘﻴﻘﻴﺎﻥ ﻣﺘﺴﺎﻭﻳﺎﻥ.‬ ‫ﺟ ﺟﺬﺭﺍﻥ ﻛﺒﺎﻥ.‬ ‫ﻣﺮ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫42 ﺍﻛﺘﺸﻒ ﺍﻟﺨﻄﺄ: ﺇﺫﺍ ﻛﺎﻥ ﻝ + ١، ﻡ + ١ ﻫﻤﺎ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ٥ﺱ + ٣ = ٠ ﻓﺄﻭﺟﺪ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻟﺘﻰ‬ ‫ﺟﺬﺭﺍﻫﺎ ﻝ، ﻡ.‬ ‫¯‬ ‫‪) a‬ﻝ + ١( + )ﻡ+١( = - ٥‬ ‫` ﻝ + ﻡ = - ٧،‬ ‫`ﻝ+ﻡ+٢=-٥‬ ‫‪) a‬ﻝ + ١()ﻡ + ١( = ٣ ` ﻝ ﻡ + )ﻝ + ﻡ( + ١ = ٣‬ ‫`ﻝﻡ=٩‬ ‫`ﻝﻡ–٧+١=٣‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ ﻫﻰ: ﺱ٢ + ٧ﺱ + ٩ = ٠‬ ‫¯‬ ‫‪ a‬ﻝ + ﻡ = - ٥، ﻝ ﻡ = ٣‬ ‫` )ﻝ +١ ( + )ﻡ + ١(   = ﻝ+ ﻡ + ٢‬ ‫              = - ٥ + ٢ = -٣،‬ ‫‪) a‬ﻝ+١()ﻡ + ١( = ﻝ ﻡ + )ﻝ + ﻡ( + ١‬ ‫              = ٣ – ٣ + ١ = ١‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ ﻫﻰ: ﺱ٢ + ٣ﺱ + ١ = ٠‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫52 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ: ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻔﺮﻕ ﺑﻴﻦ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ﻙ ﺱ + ٢ﻙ = ٠ ﻳﺴﺎﻭﻯ ﺿﻌﻒ ﺣﺎﺻﻞ ﺿﺮﺏ ﺟﺬﺭﻯ‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ٣ ﺱ + ﻙ = ٠ ﻓﺄﻭﺟﺪ ﻙ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪M‬‬ ‫−‬
‫إﺷﺎرة اﻟﺪاﻟﺔ‬ ‫1-5‬ ‫‪Sign of a Function‬‬ ‫‪k‬‬ ‫‪:≈JCÉj Ée πªcCG :’hCG‬‬ ‫1 ﺍﻟﺪﺍﻟﺔ ﺩ، ﺣﻴﺚ ﺩ)ﺱ( = - ٥ ﺇﺷﺎﺭﺍﺗﻬﺎ .................................................... ﻓﻰ ﺍﻟﻔﺘﺮﺓ‬ ‫....................................................‬ ‫2 ﺍﻟﺪﺍﻟﺔ ﺩ، ﺣﻴﺚ ﺩ)ﺱ( = ﺱ٢ + ١ ﺇﺷﺎﺭﺍﺗﻬﺎ .................................................... ﻓﻰ ﺍﻟﻔﺘﺮﺓ‬ ‫3 ﺍﻟﺪﺍﻟﺔ ﺩ، ﺣﻴﺚ ﺩ)ﺱ( = ﺱ٢ – ٦ ﺱ + ٩ ﻣﻮﺟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ‬ ‫4 ﺍﻟﺪﺍﻟﺔ ﺩ، ﺣﻴﺚ ﺩ)ﺱ( = ﺱ – ٢ ﻣﻮﺟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ‬ ‫5 ﺍﻟﺪﺍﻟﺔ ﺩ، ﺣﻴﺚ ﺩ)ﺱ( = ٣ – ﺱ ﺳﺎﻟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ‬ ‫....................................................‬ ‫....................................................‬ ‫....................................................‬ ‫....................................................‬ ‫6 ﺍﻟﺪﺍﻟﺔ ﺩ، ﺣﻴﺚ ﺩ)ﺱ( = - )ﺱ – ١( )ﺱ +٢( ﻣﻮﺟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ‬ ‫7 ﺍﻟﺪﺍﻟﺔ ﺩ، ﺣﻴﺚ ﺩ)ﺱ( = ﺱ٢ + ٤ ﺱ – ٥ ﺳﺎﻟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ‬ ‫....................................................‬ ‫....................................................‬ ‫8 ﺍﻟﺸﻜﻞ ﺍﻟﻤﺮﺳﻮﻡ ﻳﻤﺜﻞ ﺩﺍﻟﺔ ﻣﻦ ﺍﻟﺪﺭﺟﺔ ﺍﻷﻭﻟﻰ ﻓﻰ ﺱ:‬ ‫أ ﺩ)ﺱ( ﻣﻮﺟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ ....................................................‬ ‫ب‬ ‫ﺩ)ﺱ( ﺳﺎﻟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ ....................................................‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫9 ﺍﻟﺸﻜﻞ ﺍﻟﻤﺮﺳﻮﻡ ﻳﻤﺜﻞ ﺩﺍﻟﺔ ﻣﻦ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ ﻓﻰ ﺱ:‬ ‫أ ﺩ)ﺱ( = ٠ ﻋﻨﺪﻣﺎ ﺱ ∋ ....................................................‬ ‫ب‬ ‫ﺩ)ﺱ( < ٠ ﻋﻨﺪﻣﺎ ﺱ ∋ ....................................................‬ ‫ﺟ‬ ‫ﺩ)ﺱ( > ٠ ﻋﻨﺪﻣﺎ ﺱ ∋ ....................................................‬ ‫¯‬ ‫−‬ ‫¯‬ ‫− − −‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬
‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫01 ﻓﻰ ﺍﻟﺘﻤﺎﺭﻳﻦ ﻣﻦ أ ﺇﻟﻰ ن ﻋﻴﻦ ﺇﺷﺎﺭﺓ ﻛﻞ ﻣﻦ ﺍﻟﺪﻭﺍﻝ ﺍﻵﺗﻴﺔ:‬ ‫ب ﺩ)ﺱ( = ٢ﺱ‬ ‫.......................................‬ ‫أ ﺩ)ﺱ( = ٢‬ ‫د ﺩ)ﺱ( =٢ﺱ+٤‬ ‫.......................................‬ ‫ﺟ ﺩ)ﺱ( = - ٣ﺱ‬ ‫.......................................‬ ‫و ﺩ)ﺱ( = ﺱ‬ ‫ح ﺩ)ﺱ( = ﺱ٢ – ٤‬ ‫.......................................‬ ‫.......................................‬ ‫ﻫ ﺩ)ﺱ( =٣ – ٢ﺱ‬ ‫٢‬ ‫ز ﺩ)ﺱ( = ٢ﺱ‬ ‫ط ﺩ)ﺱ( = ١ – ﺱ‬ ‫.......................................‬ ‫ى ﺩ)ﺱ( = )ﺱ – ٢( )ﺱ + ٣(‬ ‫.......................................‬ ‫......................................‬ ‫ل ﺩ)ﺱ( = ﺱ٢– ﺱ – ٢‬ ‫.......................................‬ ‫.......................................‬ ‫ن ﺩ)ﺱ( = - ٤ ﺱ٢ + ٠١ ﺱ – ٥٢‬ ‫.......................................‬ ‫٢‬ ‫.......................................‬ ‫.......................................‬ ‫٢‬ ‫ك ﺩ)ﺱ( = )٢ ﺱ – ٣(‬ ‫م ﺩ)ﺱ( = ﺱ٢– ٨ ﺱ + ٦١‬ ‫٢‬ ‫.......................................‬ ‫11 ﺍﺭﺳﻢ ﻣﻨﺤﻨﻰ ﺍﻟﺪﺍﻟﺔ ﺩ)ﺱ( = ﺱ٢ – ٩ ﻓﻰ ﺍﻟﻔﺘﺮﺓ ] - ٣، ٤ [، ﻭﻣﻦ ﺍﻟﺮﺳﻢ ﻋﻴﻦ ﺇﺷﺎﺭﺓ ﺩ)ﺱ(.‬ ‫21 ﺍﺭﺳﻢ ﻣﻨﺤﻨﻰ ﺍﻟﺪﺍﻟﺔ ﺩ)ﺱ( = – ﺱ٢ + ٢ ﺱ + ٤ ﻓﻰ ﺍﻟﻔﺘﺮﺓ ]- ٣، ٥[، ﻭﻣﻦ ﺍﻟﺮﺳﻢ ﻋﻴﻦ ﺇﺷﺎﺭﺓ ﺩ)ﺱ(.‬ ‫31 ﺍﻛﺘﺸﻒ ﺍﻟﺨﻄﺄ: ﺇﺫﺍ ﻛﺎﻧﺖ ﺩ)ﺱ( = ﺱ + ١، ﺭ)ﺱ( = ١ – ﺱ٢ ﻓﻌﻴﻦ ﺍﻟﻔﺘﺮﺓ ﺍﻟﺘﻰ ﺗﻜﻮﻥ ﻓﻴﻬﺎ ﺍﻟﺪﺍﻟﺘﺎﻥ‬ ‫ﻣﻮﺟﺒﺘﻴﻦ ﻣﻌﺎ.‬ ‫ً‬ ‫¯‬ ‫¯‬ ‫ﺗﺠﻌﻞ ﺩ)ﺱ( = ٠‬ ‫ﺱ=-١‬ ‫ﺩ)ﺱ( ﻣﻮﺟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ [- ١، ∞]،‬ ‫ﺗﺠﻌﻞ ﺭ)ﺱ( = ٠‬ ‫ﺱ=!١‬ ‫ﺭ)ﺱ( ﻣﻮﺟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ [- ١، ١]‬ ‫ﻟﺬﻟﻚ ﻓﺈﻥ ﺍﻟﺪﺍﻟﺘﻴﻦ ﺗﻜﻮﻧﺎﻥ ﻣﻮﺟﺒﺘﻴﻦ ﻣﻌﺎ ﻓﻰ ﺍﻟﻔﺘﺮﺓ‬ ‫ً‬ ‫[- ١، ∞] ∪ [- ١، ١] = [- ١، ∞]‬ ‫ﺗﺠﻌﻞ ﺩ)ﺱ( = ٠‬ ‫ﺱ=-١‬ ‫ﺩ)ﺱ( ﻣﻮﺟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ [- ١، ∞]،‬ ‫ﺗﺠﻌﻞ ﺭ)ﺱ( = ٠‬ ‫ﺱ=!١‬ ‫ﺭ)ﺱ( ﻣﻮﺟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ [- ١، ١]‬ ‫ﻟﺬﻟﻚ ﻓﺈﻥ ﺍﻟﺪﺍﻟﺘﻴﻦ ﺗﻜﻮﻧﺎﻥ ﻣﻮﺟﺒﺘﻴﻦ ﻣﻌﺎ ﻓﻰ ﺍﻟﻔﺘﺮﺓ‬ ‫ً‬ ‫[- ١، ∞] ∩ [- ١، ١] = [- ١، ١]‬ ‫ﺃﻯ ﺍﻹﺟﺎﺑﺘﻴﻦ ﻳﻜﻮﻥ ﺻﺤﻴﺤﺎ? ﻣﺜﻞ ﻛﻼ ﻣﻦ ﺍﻟﺪﺍﻟﺘﻴﻦ ﺑﻴﺎﻧﻴﺎ ﻭﺗﺄﻛﺪ ﻣﻦ ﺻﺤﺔ ﺍﻹﺟﺎﺑﺔ.‬ ‫ً ِّ ًّ‬ ‫ًّ‬ ‫..................................................................................................................................................................................................................................‬ ‫41 ﻣﻨﺎﺟﻢ ﺍﻟﺬﻫﺐ: ﻓﻰ ﺍﻟﻔﺘﺮﺓ ﻣﻦ ﻋﺎﻡ ٠٩٩١ ﺇﻟﻰ ٠١٠٢ ﻛﺎﻥ ﺇﻧﺘﺎﺝ ﺃﺣﺪ ﻣﻨﺎﺟﻢ ﺍﻟﺬﻫﺐ ﻣﻘﺪﺭﺍ ﺑﺎﻷﻟﻒ ﺃﻭﻗﻴﺔ‬ ‫ً‬ ‫ﻳﺘﺤﺪﺩ ﺑﺎﻟﺪﺍﻟﺔ ﺩ : ﺩ)ﻥ( = ٢١ ﻥ٢ - ٦٩ ﻥ + ٠٨٤ ﺣﻴﺚ ﻥ ﻋﺪﺩ ﺍﻟﺴﻨﻮﺍﺕ، ﺩ)ﻥ( ﺍﻧﺘﺎﺝ ﺍﻟﺬﻫﺐ‬ ‫: ﺍﺑﺤﺚ ﺇﺷﺎﺭﺓ ﺩﺍﻟﺔ ﺍﻹﻧﺘﺎﺝ ﺩ. ...........................................................................................................................................................‬ ‫: ﺧﻼﻝ ﺍﻷﻋﻮﺍﻡ ﻣﻦ ٠٩٩١ﺇﻟﻰ ٠١٠٢ ﻓﻰ ﺃﻯ ﺍﻷﻋﻮﺍﻡ ﻛﺎﻥ ﺇﻧﺘﺎﺝ ﺍﻟﺬﻫﺐ ﻳﺘﻨﺎﻗﺺ? .................................................‬ ‫: ﺧﻼﻝ ﺍﻷﻋﻮﺍﻡ ﻣﻦ ٠٩٩١ﺇﻟﻰ ٠١٠٢ ﻓﻰ ﺃﻯ ﺍﻷﻋﻮﺍﻡ ﻛﺎﻥ ﺇﻧﺘﺎﺝ ﺍﻟﺬﻫﺐ ﻳﺘﺰﺍﻳﺪ? ....................................................‬ ‫‪M‬‬ ‫−‬
‫ﻣﺘﺒﺎﻳﻨﺎت اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ‬ ‫1-6‬ ‫‪Quadratic Inequalities‬‬ ‫¯‬ ‫1 ﺱ٢ ‪٩ H‬‬ ‫:‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫2 ﺱ٢ - ١ ‪٠ H‬‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫3 ٢ﺱ – ﺱ٢ > ٠‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫4 ﺱ٢ + ٥ ‪١ H‬‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫5 )ﺱ - ٢( )ﺱ - ٥( > ٠‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫6 ﺱ )ﺱ + ٢( - ٣ ‪٠ H‬‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫7 )ﺱ - ٢(٢ ‪٥ - H‬‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫8 ٥ – ٢ﺱ ‪ H‬ﺱ‬ ‫٢‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫9 ﺱ٢ ‪ ٦ G‬ﺱ – ٩‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫01 ٣ ﺱ٢ ‪ ١١ H‬ﺱ + ٤‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫11 ﺱ٢ - ٤ ﺱ + ٤ ‪٠ G‬‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫21 ٧ + ﺱ٢ - ٤ ﺱ > ٠‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
‫ﺗﻤﺎﺭﻳﻦ ﻋﺎﻣﺔ‬ ‫‪k‬‬ ‫‪:IÉ£©ªdG äÉHÉLE’G ø«H øe áë«ë°üdG áHÉLE’G ôàNG :’hCG‬‬ ‫1 ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٦ ﺱ + ٩ = ٠ ﻓﻰ ﺡ ﻫﻰ :‬ ‫ﺟ }-٣، ٣{‬ ‫ب }٣{‬ ‫أ }-٣{‬ ‫..............................................................................................................‬ ‫2 ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ٤ = ٠ ﻫﻰ :‬ ‫ب }٢{‬ ‫أ }-٢{‬ ‫3 ﺃﺑﺴﻂ ﺻﻮﺭﺓ ﻟﻠﻤﻘﺪﺍﺭ )١ – ﺕ(٤ ﻫﻮ :‬ ‫ب ٤‬ ‫أ -٤‬ ‫د ‪z‬‬ ‫............................................................................................................................................‬ ‫ﺟ }-٢، ٢{‬ ‫د }-٢ﺕ، ٢ﺕ{‬ ‫...................................................................................................................................................‬ ‫ﺟ -٤ ﺕ‬ ‫4 ﺇﺫﺍ ﻛﺎﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٤ﺱ + ﻙ = ٠ ﺣﻘﻴﻘﻴﻴﻦ ﻭﻣﺨﺘﻠﻔﻴﻦ ﻓﺈﻥ:‬ ‫ﺟ ﻙ=٤‬ ‫ب ﻙ>٤‬ ‫أ ﻙ<٤‬ ‫د ٤ﺕ‬ ‫..................................................................................‬ ‫5 ﺇﺫﺍ ﻛﺎﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٢١ﺱ + ﻡ = ٠ ﻣﺘﺴﺎﻭﻳﻴﻦ ﻓﺈﻥ ﻡ ﺗﺴﺎﻭﻯ:‬ ‫ﺟ ٦‬ ‫ب -٦‬ ‫أ -٦٣‬ ‫د ﻙ‪٤G‬‬ ‫..............................................................................‬ ‫د ٦٣‬ ‫6 ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ ٢ – ٣ﺕ ، ٢ + ٣ﺕ ﻫﻰ :‬ ‫أ ﺱ٢ + ٤ﺱ + ٣١ = ٠ ب ﺱ٢ – ٤ﺱ + ٣١ = ٠ ﺟ ﺱ٢ + ٤ﺱ – ٣١ = ٠ د ﺱ٢ – ٤ﺱ – ٣١ = ٠‬ ‫...........................................................................................................‬ ‫7 ﺇﺫﺍ ﻛﺎﻧﺖ ﺩ : ]- ٢ ، ٤[ # ‪ I‬ﺣﻴﺚ ﺩ)ﺱ( = ٢ – ﺱ ﻓﺈﻥ ﺇﺷﺎﺭﺓ ﺍﻟﺪﺍﻟﺔ ﺩ ﺳﺎﻟﺒﺔ ﻓﻰ:‬ ‫د [٢ ، ٤[‬ ‫ﺟ ]٢ ، ٤[‬ ‫ب ]- ٢ ، ٢[‬ ‫أ ]-٢ ، ٢]‬ ‫8 ﺇﺫﺍ ﻛﺎﻥ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ) ﻡ + ٢( ﺱ + ٣ = ٠ ﻣﻌﻜﻮﺳﺎ ﺟﻤﻌﻴﺎ ﻟﻠﺠﺬﺭ ﺍﻵﺧﺮ ﻓﺈﻥ ﻡ ﺗﺴﺎﻭﻯ:‬ ‫ً‬ ‫ًّ‬ ‫د‬ ‫٣‬ ‫ﺟ ٢‬ ‫ب -٢‬ ‫أ -٣‬ ‫9 ﺇﺫﺍ ﻛﺎﻥ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ٢ ﺱ٢ + ٧ ﺱ + ﻙ = ٠ ﻫﻮ ﺍﻟﻤﻌﻜﻮﺱ ﺍﻟﻀﺮﺑﻰ ﻟﻠﺠﺬﺭ ﺍﻵﺧﺮ ﻓﺈﻥ ﻙ ﺗﺴﺎﻭﻯ:‬ ‫د ٧‬ ‫ﺟ ٢‬ ‫ب -٢‬ ‫أ -٧‬ ‫01 ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺔ ﺱ٢ + ﺱ – ٢ > ٠ ﻫﻰ :‬ ‫ب ]- ٢ ، ١[‬ ‫أ [- ٢ ، ١]‬ ‫ﺟ ﺡ – ]-٢ ، ١[‬ ‫د ﺡ – [-٢ ، ١]‬ ‫‪O á«©«HôJ ádGód ≈fÉ«ÑdG π«ãªàdG πHÉ≤ªdG πμ°ûdG πãªj :Ék«fÉK‬‬ ‫11 ﺃﻛﻤﻞ ﻣﺎﻳﺄﺗﻰ :‬ ‫أ ﻣﺪﻯ ﺍﻟﺪﺍﻟﺔ ﺩ ﻫﻮ .............................................................................................‬ ‫ب‬ ‫ﺍﻟﻘﻴﻤﺔ ﺍﻟﻌﻈﻤﻰ ﻟﻠﺪﺍﻟﺔ ﺩ = ............................................................................‬ ‫ﺟ‬ ‫ﻧﻮﻉ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺩ )ﺱ( = ٠ .............................................................‬ ‫د ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺩ)ﺱ( = ٠ ﻫﻰ ..................................................‬ ‫ﻫ ﺩ)ﺱ( < ٠ ﻋﻨﺪﻣﺎ ﺱ ∋ ...............................................................................‬ ‫و‬ ‫ﺩ)ﺱ( > ٠ ﻋﻨﺪﻣﺎ ﺱ ∋ ...............................................................................‬ ‫ز‬ ‫ﺩ)ﺱ( = ٠ ﻋﻨﺪﻣﺎ ﺱ = ..................................................................................‬ ‫‪M‬‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫− −‬
‫ﺗﻤﺎﺭﻳﻦ ﻋﺎﻣﺔ‬ ‫21 ﺍﻛﺘﺐ ﻗﺎﻋﺪﺓ ﺍﻟﺪﺍﻟﺔ ﺍﻟﺘﻰ ﺗﻤﺮ ﺑﺎﻟﻨﻘﺎﻁ )- ٣، ٠( ، )٢، ٠( ، )٢، ١(‬ ‫..................................................................................................................................................................................................................................‬ ‫31 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ :‬ ‫أ ﺍﻛﺘﺐ ﻧﻘﺎﻁ ﺗﻘﺎﻃﻊ ﻣﻨﺤﻨﻰ ﺍﻟﺪﻭﺍﻝ ﺍﻟﺘﻰ ﻗﺎﻋﺪﺗﻬﺎ ﺹ = ﺱ٢ ، ﺹ = ﺱ‬ ‫.......................................................................................................................................................................................................................‬ ‫ب ﺍﻛﺘﺐ ﻧﻘﺎﻁ ﺗﻘﺎﻃﻊ ﻣﻨﺤﻨﻰ ﺍﻟﺪﻭﺍﻝ ﺍﻟﺘﻰ ﻗﺎﻋﺪﺗﻬﺎ ﺹ = - ﺱ٢، ﺹ = - ﺱ ﻣﺎﺫﺍ ﺗﻼﺣﻆ ? ﻓﺴﺮ ﺇﺟﺎﺑﺘﻚ.‬ ‫.......................................................................................................................................................................................................................‬ ‫‪k‬‬ ‫‪á«JB’G á∏Ä°SC’G øY ÖLCG :ÉãdÉK‬‬ ‫41 ﺑﻴﻦ ﻧﻮﻉ ﺟﺬﺭﻯ ﻛﻞ ﻣﻌﺎﺩﻟﺔ ﻣﻤﺎ ﻳﺄﺗﻰ، ﺛﻢ ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﻛﻞ ﻣﻌﺎﺩﻟﺔ.‬ ‫ب )ﺱ – ١(٢ = ٤‬ ‫أ ﺱ٢ – ٢ﺱ = ٠‬ ‫........................................................‬ ‫د ﺱ٢ + ٣ﺱ – ٨٢ = ٠‬ ‫........................................................‬ ‫ﺟ ﺱ٢ – ٦ ﺱ+ ٩ = ٠‬ ‫........................................................‬ ‫........................................................‬ ‫ﻫ ٦ﺱ )ﺱ – ١( = ٦ – ﺱ‬ ‫........................................................‬ ‫51 ﺣﻞ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻘﺎﻧﻮﻥ ﺍﻟﻌﺎﻡ ﻣﻘﺮﺑﺎ ﺍﻟﻨﺎﺗﺞ ﻷﻗﺮﺏ ﺭﻗﻤﻴﻦ ﻋﺸﺮﻳﻴﻦ.‬ ‫ً‬ ‫ب ﺱ٢ – ٣)ﺱ -٢( = ٥‬ ‫أ ﺱ٢ + ٤ﺱ + ٢ = ٠‬ ‫........................................................‬ ‫........................................................‬ ‫61 ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ ﻓﻰ ﻣﺠﻤﻮﻋﺔ ﺍﻷﻋﺪﺍﺩ ﻛﺒﺔ .‬ ‫ﺍﻟﻤﺮ‬ ‫ب ﺱ٢ + ٢ﺱ + ٢ = ٠‬ ‫أ ﺱ٢ + ٩ = ٠‬ ‫........................................................‬ ‫ﺟ ﺱ٢ + ٤ﺱ + ٥ = ٠‬ ‫........................................................‬ ‫71 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ ،C‬ﺏ ﻓﻰ ﻛﻞ ﻣﻤﺎ ﻳﺄﺗﻰ :‬ ‫أ )٧ – ٣ﺕ( – )٢ + ﺕ( = ‪ + C‬ﺏ ﺕ‬ ‫ﺟ ٢ ٠١ﺕ = ‪ + C‬ﺏ ﺕ‬ ‫+‬ ‫........................................................‬ ‫ب )٢ – ٥ﺕ()٣ + ﺕ( = ‪ + C‬ﺏ ﺕ‬ ‫‬‫د ٦١ -٤ﺕ = ‪ + C‬ﺏ ﺕ‬ ‫ﺕ‬ ‫81 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻡ ﻓﻰ ﻛﻞ ﻣﻤﺎ ﻳﺄﺗﻰ :‬ ‫٢‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ٢ﺱ + ﻡ ﺱ + ٨١ = ٠ ﻣﺘﺴﺎﻭﻳﻴﻦ ..............................................................................................‬ ‫٢‬ ‫ب‬ ‫ﺇﺫﺍ ﻛﺎﻥ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ + ٣ ﺱ + ﻙ = ٠ ﺿﻌﻒ ﺍﻟﺠﺬﺭ ﺍﻵﺧﺮ ..............................................................‬ ‫91 ﺍﺑﺤﺚ ﺇﺷﺎﺭﺓ ﺍﻟﺪﺍﻟﺔ ﺩ ﻓﻰ ﻛﻞ ﻣﻤﺎ ﻳﺄﺗﻰ :‬ ‫أ ﺩ)ﺱ( = ﺱ٢ – ٢ ﺱ – ٨‬ ‫........................................................‬ ‫02 ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺍﻟﺤﻞ ﻟﻜﻞ ﻣﻦ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ ﺍﻵﺗﻴﺔ :‬ ‫أ ﺱ٢ – ﺱ – ٢١ < ٠‬ ‫........................................................‬ ‫¯‬ ‫−‬ ‫¯‬ ‫ب ﺩ)ﺱ( = ٤ – ٣ﺱ – ﺱ‬ ‫........................................................‬ ‫ب ﺱ٢ – ٧ﺱ + ٠١ ‪٠ H‬‬ ‫........................................................‬ ‫٢‬
‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ‬ ‫‪k‬‬ ‫‪: Oó©àe øe QÉ«àNC’G :’hCG‬‬ ‫1 ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٤ﺱ = -٤ ﻓﻰ ﺡ ﻫﻰ:‬ ‫ب }٢{‬ ‫أ }-٢{‬ ‫2 ﺣﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺔ ﺱ٢ + ٩ < ٦ﺱ ﻓﻰ ﺡ ﻫﻰ:‬ ‫ب ﺡ – }٣{‬ ‫أ ﺡ‬ ‫.......................................................................................................................‬ ‫ﺟ }-٢، ٢{‬ ‫د ‪z‬‬ ‫............................................................................................................................................‬ ‫ﺟ [- ٣، ٣]‬ ‫3 ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ٢ﺱ٢ – ٥ﺱ + ٣ = ٠‬ ‫أ ﺣﻘﻴﻘﻴﺎﻥ ﻣﺘﺴﺎﻭﻳﺎﻥ ب ﺣﻘﻴﻘﻴﺎﻥ ﻣﺨﺘﻠﻔﺎﻥ‬ ‫د ﺡ – ]-٣، ٣[‬ ‫......................................................................................................................................................‬ ‫ﺟ ﻛﺒﺎﻥ‬ ‫ﻣﺮ‬ ‫4 ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ )١ + ﺕ(، )١ – ﺕ( ﻫﻰ :‬ ‫أ ﺱ٢ – ٢ﺱ + ٢ = ٠ ب ﺱ٢ + ٢ﺱ – ٢ = ٠ ﺟ ﺱ٢ + ٢ﺱ + ٢ = ٠‬ ‫د ﻛﺒﺎﻥ ﻭ ﻣﺘﺮﺍﻓﻘﺎﻥ‬ ‫ﻣﺮ‬ ‫.........................................................................................................‬ ‫د ﺱ٢ – ٢ﺱ – ٢ = ٠‬ ‫‪á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫5 ﺇﺫﺍ ﻛﺎﻥ )‪(٣ + C‬ﺱ٢ + )٢ – ‪ (C‬ﺱ + ٤ = ٠ ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ ‪ C‬ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻟﺤﺎﻻﺕ ﺍﻵﺗﻴﺔ:‬ ‫أ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﻣﻌﻜﻮﺱ ﺟﻤﻌﻰ ﻟﻠﺠﺬﺭ ﺍﻵﺧﺮ.‬ ‫.......................................................................................................................................................................................................................‬ ‫ب ﻣﺠﻤﻮﻉ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﻳﺴﺎﻭﻯ ٦.‬ ‫.......................................................................................................................................................................................................................‬ ‫6‬ ‫٢ ٢‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ ﻝ ، ﻡ ﻫﻤﺎ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٦ﺱ + ٤ = ٠ ﻓﺄﻭﺟﺪ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ ﻝ، ﻡ.‬ ‫.......................................................................................................................................................................................................................‬ ‫ب ﺍﺑﺤﺚ ﺇﺷﺎﺭﺓ ﺍﻟﺪﺍﻟﺔ ﺩ، ﺣﻴﺚ ﺩ)ﺱ( = ٨ – ٢ﺱ – ﺱ‬ ‫٢‬ ‫.......................................................................................................................................................................................................................‬ ‫7‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ٣ = ٥ﺱ ﺣﻘﻴﻘﻴﺎﻥ ﻣﺨﺘﻠﻔﺎﻥ، ﺛﻢ ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﻓﻰ ﺡ‬ ‫ﻣﻘﺮﺑﺎ ﺍﻟﻨﺎﺗﺞ ﻷﻗﺮﺏ ﺛﻼﺛﺔ ﺃﺭﻗﺎﻡ ﻋﺸﺮﻳﺔ.‬ ‫ً‬ ‫.........................................................................................................................................................................................................................‬ ‫ب ﺃﻭﺟﺪ ﺣﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺔ : ﺱ٢ – ٥ﺱ – ٤١ ‪٠ H‬‬ ‫.......................................................................................................................................................................................................................‬ ‫8 ﺗﻄ ﻴﻘﺎﺕ ﻓﻴ ﺎﺋﻴﺔ: ﺃُﻃْﻠﻖ ﺻﺎﺭﻭﺥ ﺭﺃﺳﻴﺎ ﺇﻟﻰ ﺃﻋﻠﻰ ﺑﺴﺮﻋﺔ ٨٩ ﻣﺘﺮﺍ/ﺛﺎﻧﻴﺔ، ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺍﻟﻤﺴﺎﻓﺔ‬ ‫ًّ‬ ‫ً‬ ‫٢‬ ‫ﺍﻟﻤﻘﻄﻮﻋﺔ ﻑ ﺑﺎﻟﻤﺘﺮ ﻭﺍﻟﺰﻣﻦ ﻥ ﺑﺎﻟﺜﺎﻧﻴﺔ ﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ : ﻑ = ٨٩ ﻥ – ٩٫٤ ﻥ ﻓﺄﻭﺟﺪ :‬ ‫أ ﺍﻟﻤﺴﺎﻓﺔ ﺍﻟﺘﻰ ﻳﻘﻄﻌﻬﺎ ﺍﻟﺼﺎﺭﻭﺥ ﻓﻰ ﺛﺎﻧﻴﺘﻴﻦ. ............................................................................................................................‬ ‫ب ﺍﻟﺰﻣﻦ ﺍﻟﺬﻯ ﻳﺴﺘﻐﺮﻗﻪ ﺍﻟﺼﺎﺭﻭﺥ ﺣﺘﻰ ﻳﻘﻄﻊ ﻣﺴﺎﻓﺔ ٤٫٠٧٤ ﻣﺘﺮﺍ. ﺑﻤﺎ ﺗﻔﺴﺮ ﻭﺟﻮﺩ ﺇﺟﺎﺑﺘﻴﻦ?‬ ‫ً‬ ‫‪M‬‬ ‫−‬
‫ﺍﺧﺘﺒﺎﺭ ﺗﺮﺍﻛﻤﻲ‬ ‫1 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻙ ﺍﻟﺘﻰ ﺗﺠﻌﻞ ﻟﻠﻤﻌﺎﺩﻟﺔ ٣ﺱ٢ + ٤ﺱ + ﻙ = ٠ ﺟﺬﺭﻳﻦ :‬ ‫أ ﺣﻘﻴﻘﻴﻴﻦ ﻣﺘﺴﺎﻭﻳﻴﻦ ......................................‬ ‫ب‬ ‫ﺣﻘﻴﻘﻴﻴﻦ ﻣﺨﺘﻠﻔﻴﻦ ......................................‬ ‫ﻛﺒﻴﻦ ......................................‬ ‫ﺟ ﻣﺮ‬ ‫2 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻙ ﺍﻟﺘﻰ ﺗﺠﻌﻞ:‬ ‫٢‬ ‫أ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ – ﻙ ﺱ + ﻙ + ٢ = ٠ ﺿﻌﻒ ﺍﻟﺠﺬﺭ ﺍﻵﺧﺮ. .......................................................................‬ ‫٢‬ ‫ب‬ ‫ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ – ﻙ ﺱ + ٨ = ٠ ﻳﺰﻳﺪ ﻋﻦ ﺍﻟﺠﺬﺭ ﺍﻵﺧﺮ ﺑﻤﻘﺪﺍﺭ ٢. ......................................................‬ ‫ﺟ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ﻙ ﺱ + ٣ = ٠ ﻳﺰﻳﺪ ﻋﻦ ﺍﻟﻤﻌﻜﻮﺱ ﺍﻟﻀﺮﺑﻰ ﻟﻠﺠﺬﺭ ﺍﻵﺧﺮ ﺑﻤﻘﺪﺍﺭ ١.‬ ‫3 ﺇﺫﺍ ﻛﺎﻥ ﻝ، ﻡ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٣ﺱ + ٢ = ٠ ﻓﺄﻭﺟﺪ ﻣﻌﺎﺩﻟﺔ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ :‬ ‫ﺟ ١ ١‬ ‫د ﻝ + ﻡ، ﻝ ﻡ‬ ‫ب ﻝ + ١، ﻡ + ١‬ ‫أ ٣ ﻝ، ٣ ﻡ‬ ‫ﻝ، ﻡ‬ ‫..................................................................................................................................................................................................................................‬ ‫١ ١‬ ‫4 ﺇﺫﺍ ﻛﺎﻥ ﻝ ، ﻡ ﻫﻤﺎ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ٦ﺱ٢ – ٥ ﺱ +١ = ٠ ﻓﻜﻮﻥ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ ﻝ، ﻡ.‬ ‫..................................................................................................................................................................................................................................‬ ‫5 ﺍﺭﺳﻢ ﻣﻨﺤﻨﻰ ﺍﻟﺪﺍﻟﺔﺩ، ﺣﻴﺚ ﺩ)ﺱ( = ﺱ٢– ٤ ﻓﻰ ﺍﻟﻔﺘﺮﺓ ]- ٣،٣[ ﻭﻣﻦ ﺍﻟﺮﺳﻢ ﻋﻴﻦ ﺇﺷﺎﺭﺓ ﺩ ﻓﻰ ﻫﺬه ﺍﻟﻔﺘﺮﺓ.‬ ‫6 ﺍﺭﺳﻢ ﻣﻨﺤﻨﻰ ﺍﻟﺪﺍﻟﺔ ﺩ، ﺣﻴﺚ ﺩ)ﺱ( = ٦ – ٥ﺱ – ٤ﺱ٢ ﻓﻰ ﺍﻟﻔﺘﺮﺓ ]-٣،٢[ ﻭﻣﻦ ﺍﻟﺮﺳﻢ ﻋﻴﻦ ﺇﺷﺎﺭﺓ ﺩ ﻓﻰ ﻫﺬه ﺍﻟﻔﺘﺮﺓ.‬ ‫7 ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺍﻟﺤﻞ ﻟﻠﻤﺘﺒﺎﻳﻨﺎﺕ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻵﺗﻴﺔ:‬ ‫ﺟ )ﺱ - ٢(٢ ‪٩ - G‬‬ ‫ب ﺱ٢ - ٦ ﺱ < - ٥‬ ‫أ ﺱ٢ + ٤ ﺱ + ٤ > ٠‬ ‫.................................................................‬ ‫د ٣ – ٢ﺱ ‪ G‬ﺱ‬ ‫..................................................................‬ ‫.................................................................‬ ‫و ٢ﺱ٢ - ٧ﺱ ‪١٥ H‬‬ ‫ﻫ ﺱ٢ ‪١٠ H‬ﺱ – ٥٢‬ ‫٢‬ ‫.................................................................‬ ‫..................................................................‬ ‫.................................................................‬ ‫8 ﺃﻋﻤﺎﻝ ﺗﺠﺎ ﺔ: ﺇﺫﺍ ﻛﺎﻥ ﻋﺪﺩ ﺍﻟﻮﺣﺪﺍﺕ ﺍﻟﻤﻨﺘﺠﺔ ﻭﺍﻟﻤﺒﺎﻋﺔ ﻣﻦ ﺳﻠﻌﺔ ﻣﻌﻴﻨﺔ ﻓﻰ ﺍﻷﺳﺒﻮﻉ ﻫﻰ ﺱ ﻣﻠﻴﻮﻥ ﻭﺣﺪﺓ‬ ‫ﻛﺎﻥ ﺳﻌﺮ ﺑﻴﻊ ﺍﻟﻮﺣﺪﺓ ﻫﻮ ﻉ ﺣﻴﺚ ﻉ = ٢ – ﺱ، ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺘﻜﺎﻟﻴﻒ ﺍﻟﻜﻠﻴﺔ ﺍﻟﻼﺯﻣﺔ ﻹﻧﺘﺎﺝ ﺱ ﻣﻠﻴﻮﻥ ﻭﺣﺪﺓ‬ ‫ﻭ‬ ‫ﻓﻰ ﺍﻷﺳﺒﻮﻉ ﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ ﺕ = )٣٫٠ + ٥٫٠ﺱ( ﻣﻠﻴﻮﻥ ﻭﺣﺪﺓ ﻓﺄﻭﺟﺪ :‬ ‫أ ﺩﺍﻟﺔ ﺍﻹﻳﺮﺍﺩ ﺍﻟﻜﻠﻰ )ﻯ( ..................................................................‬ ‫ب‬ ‫ﺩﺍﻟﺔ ﺍﻟﺮﺑﺢ )ﺭ( ..................................................................‬ ‫ﺟ‬ ‫ﺃﻭﺟﺪ ﺱ ﻋﻨﺪ ﻣﺴﺘﻮﻯ ﺭﺑﺢ ٢٫٠ ﻣﻠﻴﻮﻥ ﺟﻨﻴﻪ. ........................................................................................................................‬ ‫9 ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ٣ + ١ = C‬ﺕ  ،  ﺏ = - ١ – ﺕ، ﺟـ = - ٢ - ٣ + ﺕ ﻓﺄﺛﺒﺖ ﺃﻥ: ﺟـ - ﺏ = )‪ – C‬ﺏ(ﺕ‬ ‫:‬ ‫‪M‬‬ ‫ﺭﻗﻢ ﺍﻟﺴﺆﺍﻝ‬ ‫١‬ ‫ﺃ، ﺏ‬ ‫١-٣‬ ‫ﺭﻗﻢ ﺍﻟﺪﺭﺱ‬ ‫¯‬ ‫٢‬ ‫ﺟـ‬ ‫١- ٢‬ ‫−‬ ‫¯‬ ‫٣‬ ‫٤‬ ‫٥‬ ‫٦‬ ‫٧‬ ‫٨‬ ‫٩‬ ‫١- ٤‬ ‫١-٤‬ ‫١-٤‬ ‫١-٥‬ ‫١- ٥‬ ‫١-٦‬ ‫١-١‬ ‫١-٢‬
‫-‬ ‫‪IóMƒdG‬‬ ‫2‬ ‫ﺍﻟﺘﺸﺎﺑﻪ‬ ‫‪Similarity‬‬ ‫دروس اﻟﻮﺣﺪة‬ ‫ﺍﻟﺪﺭﺱ )٢ - ١(: ﺗﺸﺎﺑﻪ ﺍﻟﻤﻀﻠﻌﺎﺕ.‬ ‫ﺍﻟﺪﺭﺱ )٢ - ٢(: ﺗﺸﺎﺑﻪ ﺍﻟﻤﺜﻠﺜﺎﺕ.‬ ‫ﺍﻟﺪﺭﺱ )٢ - ٣(: ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﻣﺴﺎﺣﺘﻰ ﺳﻄﺤﻰ ﻣﻀﻠﻌﻴﻦ ﻣﺘﺸﺎﺑﻬﻴﻦ.‬ ‫ﺍﻟﺪﺭﺱ )٢ - ٤(: ﺗﻄﺒﻴﻘﺎﺕ ﺍﻟﺘﺸﺎﺑﻪ ﻓﻰ ﺍﻟﺪﺍﺋﺮﺓ.‬ ‫‪ïM‬‬ ‫−‬
‫ﺗﺸﺎﺑﻪ اﻟﻤﻀﻠﻌﺎت‬ ‫2-1‬ ‫‪Similarity of Polygons‬‬ ‫1 ﺑﻴﻦ ﺃﻳﺎ ﻣﻦ ﺃﺯﻭﺍﺝ ﺍﻟﻤﻀﻠﻌﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ ﺗﻜﻮﻥ ﻣﺘﺸﺎﺑﻬﺔ، ﻭﺍﻛﺘﺐ ﺍﻟﻤﻀﻠﻌﺎﺕ ﺍﻟﻤﺘﺸﺎﺑﻬﺔ ﺑﺘﺮﺗﻴﺐ‬ ‫ًّ‬ ‫ﺍﻟﺮﺅﻭﺱ ﺍﻟﻤﺘﻨﺎﻇﺮﺓ، ﻭﺣﺪﺩ ﻣﻌﺎﻣﻞ ﺍﻟﺘﺸﺎﺑﻪ )ﺍﻷﻃﻮﺍﻝ ﻣﻘﺪﺭﺓ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ(.‬ ‫‪C‬‬ ‫ب‬ ‫أ‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪E c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫...........................................................................................‬ ‫...........................................................................................‬ ‫...........................................................................................‬ ‫.................................................................................‬ ‫ﺟ‬ ‫د‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪E‬‬ ‫...........................................................................................‬ ‫...........................................................................................‬ ‫...........................................................................................‬ ‫...........................................................................................‬ ‫2 ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻤﻀﻠﻊ ‪ C‬ﺏ ﺟـ ‪ + E‬ﺍﻟﻤﻀﻠﻊ ﺱ ﺹ ﻉ ﻝ، ﺃﻛﻤﻞ:‬ ‫أ ‪C‬ﺏ‬ ‫ﺏ ﺟـ = ﺹ ﻉ‬ ‫ﺟ ﺏ ﺟـ + ﺹ ﻉ‬ ‫ﺹﻉ =‬ ‫................‬ ‫ب ‪C‬ﺏ*ﻉﻝ=ﺱﺹ*‬ ‫................ + ﻝ ﺱ‬ ‫ﻝﺱ‬ ‫ﻣﺤﻴﻂ ﺍﻟﻤﻀﻠﻊ‬ ‫د‬ ‫ﻣﺤﻴﻂ ﺍﻟﻤﻀﻠﻊ.........................‬ ‫.........................‬ ‫.........................‬ ‫ﺱﺹ‬ ‫=‬ ‫‪C‬ﺏ‬ ‫3 ﺍﻟﻤﻀﻠﻊ ‪ C‬ﺏ ﺟـ ‪ + E‬ﺍﻟﻤﻀﻠﻊ ﺱ ﺹ ﻉ ﻝ. ﻓﺈﺫﺍ ﻛﺎﻥ: ‪ C‬ﺏ = ٢٣ﺳﻢ، ﺏ ﺟـ = ٠٤ﺳﻢ، ﺱ ﺹ = ٣ﻡ - ١،‬ ‫ﺹ ﻉ = ٣ﻡ +١. ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻡ ﺍﻟﻌﺪﺩﻳﺔ. ................................................................................................................................................‬ ‫4 ﻣﺴﺘﻄﻴﻞ ﺑﻌﺪﺍه ٠١ﺳﻢ، ٦ﺳﻢ. ﺃﻭﺟﺪ ﻣﺤﻴﻂ ﻭﻣﺴﺎﺣﺔ ﻣﺴﺘﻄﻴﻞ ﺁﺧﺮ ﻣﺸﺎﺑﻪ ﻟﻪ ﺇﺫﺍ ﻛﺎﻥ:‬ ‫ب ﻣﻌﺎﻣﻞ ﺍﻟﺘﺸﺎﺑﻪ ٤٫٠‬ ‫أ ﻣﻌﺎﻣﻞ ﺍﻟﺘﺸﺎﺑﻪ ٣‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
‫‪ï‬‬ ‫5 ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ ﺍﻟﻤﻀﻠﻊ ﻡ١ + ﺍﻟﻤﻀﻠﻊ ﻡ٢ + ﺍﻟﻤﻀﻠﻊ ﻡ٣.‬ ‫ﺃﻭﺟﺪ ﻣﻌﺎﻣﻞ ﺗﺸﺎﺑﻪ ﻛﻞ ﻣﻦ ﺍﻟﻤﻀﻠﻊ ﻡ١، ﺍﻟﻤﻀﻠﻊ ﻡ٢ ﻟﻠﻤﻀﻠﻊ ﻡ٣.‬ ‫ب‬ ‫أ‬ ‫...........................................................................................‬ ‫...........................................................................................‬ ‫...........................................................................................‬ ‫...........................................................................................‬ ‫6 ﺍﻟﻤﻀﻠﻌﺎﺕ ﺍﻟﺜﻼﺛﺔ ﺍﻟﺘﺎﻟﻴﺔ ﻣﺘﺸﺎﺑﻬﺔ. ﺃﻭﺟﺪ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻌﺪﺩﻳﺔ ﻟﻠﺮﻣﺰ ﺍﻟﻤﺴﺘﺨﺪﻡ ﻓﻰ ﺍﻟﻘﻴﺎﺱ.‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫..................................................................................................................................................................................................................................‬ ‫7 ﻋﻠﺒﺔ ﻋﻠﻰ ﺷﻜﻞ ﻣﺴﺘﻄﻴﻞ ﺫﻫﺒﻰ ﻃﻮﻟﻪ ٢٫٦١ﺳﻢ. ﺍﺣﺴﺐ ﻋﺮﺽ ﺍﻟﻌﻠﺒﺔ ﻷﻗﺮﺏ ﺳﻨﺘﻴﻤﺘﺮ.‬ ‫..................................................................................................................................................................................................................................‬ ‫8 ﻣﺴﺘﻄﻴﻼﻥ ﻣﺘﺸﺎﺑﻬﺎﻥ ﺑﻌﺪﺍ ﺍﻷﻭﻝ ٨ﺳﻢ، ٢١ﺳﻢ، ﻭﻣﺤﻴﻂ ﺍﻟﺜﺎﻧﻰ ٠٠٢ﺳﻢ. ﺃﻭﺟﺪ ﻃﻮﻝ ﺍﻟﻤﺴﺘﻄﻴﻞ ﺍﻟﺜﺎﻧﻰ ﻭﻣﺴﺎﺣﺘﻪ.‬ ‫ُ‬ ‫..................................................................................................................................................................................................................................‬ ‫ﻧﺸﺎط‬ ‫9 ﻫﻨﺪﺳﺔ ﻣﻌﻤﺎ ﺔ: ﻳﻮﺿﺢ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ ﻣﺨﻄﻄًﺎ‬ ‫ﻹﺣﺪﻯ ﺍﻟﻮﺣﺪﺍﺕ ﺍﻟﺴﻜﻨﻴﺔ ﺑﻤﻘﻴﺎﺱ ﺭﺳﻢ ١ : ٠٥١ ﺃﻭﺟﺪ:‬ ‫......................................................‬ ‫أ ﺃﺑﻌﺎﺩ ﺣﺠﺮﺓ ﺍﻻﺳﺘﻘﺒﺎﻝ.‬ ‫.................................................................‬ ‫ب ﺃﺑﻌﺎﺩ ﺣﺠﺮﺓ ﺍﻟﻨﻮﻡ.‬ ‫......................................................‬ ‫ﺟ ﻣﺴﺎﺣﺔ ﺣﺠﺮﺓ ﺍﻟﻤﻌﻴﺸﺔ.‬ ‫د ﻣﺴﺎﺣﺔ ﺍﻟﻮﺣﺪﺓ ﺍﻟﺴﻜﻨﻴﺔ. ......................................................‬ ‫‪ïM‬‬ ‫−‬ ‫¯‬ ‫¯‬ ‫¯‬ ‫‪M‬‬
‫ﺗﺸﺎﺑﻪ اﻟﻤﺜﻠﺜﺎت‬ ‫2-2‬ ‫‪Similarity Of Triangles‬‬ ‫1 ﺍﺫﻛﺮ ﺃﻯ ﺍﻟﺤﺎﻻﺕ ﻳﻜﻮﻥ ﻓﻴﻬﺎ ﺍﻟﻤﺜﻠﺜﺎﻥ ﻣﺘﺸﺎﺑﻬﻴﻦ، ﻭﻓﻰ ﺣﺎﻟﺔ ﺍﻟﺘﺸﺎﺑﻪ ﺍﺫﻛﺮ ﺳﺒﺐ ﺍﻟﺘﺸﺎﺑﻪ.‬ ‫ﺟ‬ ‫ب‬ ‫‪C‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫أ‬ ‫‪c‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫‪c‬‬ ‫................................................................‬ ‫د‬ ‫................................................................‬ ‫................................................................‬ ‫‪E‬‬ ‫و‬ ‫ﻫ‬ ‫‪C‬‬ ‫................................................................‬ ‫................................................................‬ ‫2 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺮﻣﺰ ﺍﻟﻤﺴﺘﺨﺪﻡ ﻓﻰ ﺍﻟﻘﻴﺎﺱ:‬ ‫‪C‬‬ ‫ب‬ ‫أ‬ ‫‪C‬‬ ‫................................................................‬ ‫ﺟ‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫................................................................‬ ‫................................................................‬ ‫‪C‬‬ ‫: ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ‪ = E C‬ﺏ ﺟـ‬ ‫3‬ ‫: ﺃﻛﻤﻞ: 9‪ C‬ﺏ ﺟـ + 9 ........................... + 9‬ ‫...........................‬ ‫: ﺇﺫﺍ ﻛﺎﻥ ﺱ ، ﺹ، ﻉ، ﻝ،ﻡ، ﻥ ﻫﻰ ﺃﻃﻮﺍﻝ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ‬ ‫ﻭﺍﻟﻤﻌﻴﻨﺔ ﺑﺎﻟﺸﻜﻞ: ﻓﺄﻛﻤﻞ ﺍﻟﺘﻨﺎﺳﺒﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ:‬ ‫ﺱ‬ ‫ﺟ ﻡ‬ ‫ﻝ‬ ‫ب ﺱ‬ ‫ﻡ‬ ‫ﺱ‬ ‫أ‬ ‫ﻝ = ...............‬ ‫ﻉ = ...............‬ ‫ﻉ = ...............‬ ‫...............‬ ‫ﺱ‬ ‫ﻫ ............... =‬ ‫ﺱ‬ ‫¯‬ ‫و‬ ‫−‬ ‫¯‬ ‫...............‬ ‫ﺹ =‬ ‫ﺹ‬ ‫...............‬ ‫................................................................‬ ‫...............‬ ‫ﻝ‬ ‫ز ﺱ =‬ ‫ﻉ‬ ‫‪E‬‬ ‫...............‬ ‫د ﻝ‬ ‫............... = ﻝ‬ ‫...............‬ ‫ﻝ‬ ‫ح ﺱ =‬ ‫ﺹ‬
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
كتاب الانشطه - مصر- ترم اول - 2014
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كتاب الانشطه - مصر- ترم اول - 2014

  • 1.
  • 2. äÉ«°VÉjôdG äÉÑjQóàdG h ᣰûfC’G ÜÉàc ∫hC’G ≈°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 ﻳﺠﻮﺯ 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 ٨٤٩٧ / ٣١٠٢‬ ‫ﺍﻟﺮﻗﻢ ﺍﻟﺪﻭﻟﻰ 4 - 000 - 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 ﻓﻴﻤﺎﻳﻠﻰ:‬ ‫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 ‫ﺍﻟﺠ‪ ‬ﻭﺍﻟﻌﻼﻗﺎﺕ ﻭﺍﻟﺪﻭﺍﻝ‬ ‫1- 1‬ ‫ﺣﻞ ﻣﻌﺎدﻻت اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ ﻓﻰ ﻣﺘﻐﻴﺮ واﺣﺪ.‬ ‫1- 2‬ ‫ﻣﻘﺪﻣﺔ ﻋﻦ اﻷﻋﺪاد اﻟﻤﺮﻛﺒﺔ.‬ ‫1- 3‬ ‫ﺗﺤﺪﻳﺪ ﻧﻮع ﺟﺬرى اﻟﻤﻌﺎدﻟﺔ اﻟﺘﺮﺑﻴﻌﻴﺔ.‬ ‫1- 4‬ ‫اﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺟﺬري ﻣﻌﺎدﻟﺔ اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ وﻣﻌﺎﻣﻼت ﺣﺪودﻫﺎ.‬ ‫1- 5‬ ‫إﺷﺎرة اﻟﺪاﻟﺔ.‬ ‫21‬ ‫1- 6‬ ‫ﻣﺘﺒﺎﻳﻨﺎت اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ.‬ ‫41‬ ‫51‬ ‫71‬ ‫81‬ ‫................................................................................................................................................‬ ‫2‬ ‫...................................................................................................................................................................................................‬ ‫5‬ ‫.....................................................................................................................................................................‬ ‫7‬ ‫............................................................................................‬ ‫9‬ ‫............................................................................................................................................................................................................................................‬ ‫ﺗﻤﺎرﻳﻦ ﻋﺎﻣﺔ‬ ‫........................................................................................................................................................................................................‬ ‫.........................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر اﻟﻮﺣﺪة‬ ‫......................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر ﺗﺮاﻛﻤﻰ‬ ‫....................................................................................................................................................................................................................................‬ ‫‪IóMƒdG‬‬ ‫‪á«fÉãdG‬‬ ‫ﺍﻟﺘﺸﺎﺑﻪ‬ ‫2-1‬ ‫ﺗﺸﺎﺑﻪ اﻟﻤﻀﻠﻌﺎت‬ ‫02‬ ‫2-2‬ ‫ﺗﺸﺎﺑﻪ اﻟﻤﺜﻠﺜﺎت.‬ ‫22‬ ‫2-3‬ ‫اﻟﻌﻼﻗﺔ ﺑﻴﻦ ﻣﺴﺎﺣﺘﻰ ﺳﻄﺤﻰ ﻣﻀﻠﻌﻴﻦ ﻣﺘﺸﺎﺑﻬﻴﻦ.‬ ‫62‬ ‫2-4‬ ‫ﺗﻄﺒﻴﻘﺎت اﻟﺘﺸﺎﺑﻪ ﻓﻰ اﻟﺪاﺋﺮة‬ ‫82‬ ‫23‬ ‫43‬ ‫53‬ ‫.....................................................................................................................................................................................................................‬ ‫..........................................................................................................................................................................................................................‬ ‫ﺗﻤﺎرﻳﻦ ﻋﺎﻣﺔ‬ ‫...........................................................................................‬ ‫.............................................................................................................................................................................‬ ‫.........................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر اﻟﻮﺣﺪة‬ ‫......................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر ﺗﺮاﻛﻤﻰ‬ ‫....................................................................................................................................................................................................................................‬
  • 7. ‫‪IóMƒdG‬‬ ‫‪áãdÉãdG‬‬ ‫ﻧﻈﺮﻳﺎﺕ ﺍﻟﺘﻨﺎﺳﺐ ﻓﻰ ﺍﻟﻤﺜﻠﺚ‬ ‫3-1‬ ‫اﻟﻤﺴﺘﻘﻴﻤﺎت اﻟﻤﺘﻮازﻳﺔ واﻷﺟﺰاء اﻟﻤﺘﻨﺎﺳﺒﺔ‬ ‫3-2‬ ‫ﻣﻨﺼﻔﺎ اﻟﺰاوﻳﺔ ﻓﻰ اﻟﻤﺜﻠﺚ واﻷﺟﺰاء اﻟﻤﺘﻨﺎﺳﺒﺔ‬ ‫3-3‬ ‫ﺗﻄﺒﻴﻘﺎت اﻟﺘﻨﺎﺳﺐ ﻓﻰ اﻟﺪاﺋﺮة‬ ‫ﺗﻤﺎرﻳﻦ ﻋﺎﻣﺔ‬ ‫.................................................................................................................................................‬ ‫83‬ ‫.......................................................................................................................................‬ ‫14‬ ‫.......................................................................................................................................................................................‬ ‫.........................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر اﻟﻮﺣﺪة‬ ‫......................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر ﺗﺮاﻛﻤﻰ‬ ‫....................................................................................................................................................................................................................................‬ ‫‪IóMƒdG‬‬ ‫‪á©HGôdG‬‬ ‫34‬ ‫54‬ ‫64‬ ‫74‬ ‫ﺣﺴﺎﺏ ﺍﻟﻤﺜﻠﺜﺎﺕ‬ ‫4-1‬ ‫اﻟﺰاوﻳﺔ اﻟﻤﻮﺟﻬﺔ.‬ ‫4-2‬ ‫ﻃﺮق ﻗﻴﺎس اﻟﺰاوﻳﺔ.‬ ‫4-3‬ ‫اﻟﺪوال اﻟﻤﺜﻠﺜﻴﺔ‬ ‫4-4‬ ‫اﻟﻌﻼﻗﺎت ﺑﻴﻦ اﻟﺪوال اﻟﻤﺜﻠﺜﻴﺔ‬ ‫4-5‬ ‫اﻟﺘﻤﺜﻴﻞ اﻟﺒﻴﺎﻧﻰ ﻟﻠﺪوال اﻟﻤﺜﻠﺜﻴﺔ‬ ‫4-6‬ ‫إﻳﺠﺎد ﻗﻴﺎس زاوﻳﺔ ﺑﻤﻌﻠﻮﻣﻴﺔ داﻟﺔ ﻣﺜﻠﺜﻴﺔ‬ ‫..............................................................................................................................................................................................................................‬ ‫05‬ ‫.....................................................................................................................................................................................................................‬ ‫25‬ ‫.......................................................................................................................................................................................................................................‬ ‫55‬ ‫.............................................................................................................................................................................................‬ ‫75‬ ‫.......................................................................................................................................................................................‬ ‫06‬ ‫........................................................................................................................................................‬ ‫ﺗﻤﺎرﻳﻦ ﻋﺎﻣﺔ‬ ‫.........................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر اﻟﻮﺣﺪة‬ ‫......................................................................................................................................................................................................................................‬ ‫اﺧﺘﺒﺎر ﺗﺮاﻛﻤﻰ‬ ‫اﺧﺘﺒﺎرات ﻋﺎﻣﺔ‬ ‫....................................................................................................................................................................................................................................‬ ‫.......................................................................................................................................................................................................................................................................‬ ‫إﺟﺎﺑﺎت ﺑﻌﺾ اﻟﺘﻤﺎرﻳﻦ‬ ‫.................................................................................................................................................................................................................................................‬ ‫16‬ ‫36‬ ‫46‬ ‫56‬ ‫66‬ ‫27‬
  • 8. ‫ﺍﻟﺠﺒﺮ‬ ‫‪IóMƒdG‬‬ ‫1‬ ‫ﺍﻟﺠﺒﺮ ﻭﺍﻟﻌﻼﻗﺎﺕ ﻭﺍﻟﺪﻭﺍﻝ‬ ‫‪Algebra, Relations and‬‬ ‫‪Functions‬‬ ‫دروس اﻟﻮﺣﺪة‬ ‫ﺍﻟﺪﺭﺱ )١ - ١(: ﺣﻞ ﻣﻌﺎﺩﻻﺕ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ ﻓﻰ ﻣﺘﻐﻴﺮ ﻭﺍﺣﺪ.‬ ‫ﺍﻟﺪﺭﺱ )١ - ٢(: ﻣﻘﺪﻣﺔ ﻋﻦ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻤﺮﻛﺒﺔ.‬ ‫ﺍﻟﺪﺭﺱ )١ - ٣(: ﺗﺤﺪﻳﺪ ﻧﻮﻉ ﺟﺬﺭ￯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ.‬ ‫ﺍﻟﺪﺭﺱ )١ - ٤(: ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺟﺬﺭ￯ ﻣﻌﺎﺩﻟﺔ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ ﻭﻣﻌﺎﻣﻼﺕ ﺣﺪﻭﺩﻫﺎ.‬ ‫ﺍﻟﺪﺭﺱ )١ - ٥(: ﺇﺷﺎﺭﺓ ﺍﻟﺪﺍﻟﺔ.‬ ‫ﺍﻟﺪﺭﺱ )١ - ٦(: ﻣﺘﺒﺎﻳﻨﺎﺕ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ.‬
  • 9. ‫ﺣﻞ ﻣﻌﺎدﻻت اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ ﻓﻰ ﻣﺘﻐﻴﺮ واﺣﺪ‬ ‫1-1‬ ‫‪Solving Quadratic Equations in One Variable‬‬ ‫‪k‬‬ ‫‪Oó©àe øe QÉ«àN’G :’hCG‬‬ ‫1 ﺍﻟﻤﻌﺎﺩﻟﺔ: )ﺱ – ١( )ﺱ + ٢( = ٠ ﻣﻦ ﺍﻟﺪﺭﺟﺔ:‬ ‫ب ﺍﻟﺜﺎﻧﻴﺔ‬ ‫أ ﺍﻷﻭﻟﻰ‬ ‫2 ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ = ﺱ ﻓﻰ ﺡ ﻫﻰ:‬ ‫ب }١{‬ ‫أ }٠{‬ ‫..................................................................................................................................‬ ‫ﺟ ﺍﻟﺜﺎﻟﺜﺔ‬ ‫د ﺍﻟﺮﺍﺑﻌﺔ‬ ‫.....................................................................................................................................‬ ‫ﺟ }- ١، ١{‬ ‫3 ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ٣ = ٠ ﻓﻰ ﺡ ﻫﻰ:‬ ‫ب }- ٣ {‬ ‫أ }-٣{‬ ‫د }0، ١{‬ ‫.................................................................................................................................‬ ‫4 ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٢ﺱ = -١ ﻓﻰ ﺡ ﻫﻰ:‬ ‫ب ‪z‬‬ ‫أ }-١{‬ ‫ﺟ } ٣ {‬ ‫د ‪z‬‬ ‫........................................................................................................................‬ ‫ﺟ }-١، ١{‬ ‫د }١{‬ ‫5 ﻳﻤﺜﻞ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ ﺍﻟﻤﻨﺤﻨﻰ ﺍﻟﺒﻴﺎﻧﻰ ﻟﺪﺍﻟﺔ ﺗﺮﺑﻴﻌﻴﺔ ﺩ.‬ ‫ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺩ)ﺱ( = ٠ ﻓﻰ ﺡ ﻫﻰ: ......................................‬ ‫ب }٤{‬ ‫أ }-٢{‬ ‫د }-٢، ٤{‬ ‫ﺟ ‪z‬‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫6 ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﻛﻞ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ ﻓﻰ ﺡ:‬ ‫ب ﺱ٢ + ٣ﺱ = ٠‬ ‫أ ﺱ٢ - ١ = ٠‬ ‫ﺟ )ﺱ – ٤(٢ = ٠‬ ‫............................................................‬ ‫............................................................‬ ‫...........................................................‬ ‫............................................................‬ ‫............................................................‬ ‫...........................................................‬ ‫............................................................‬ ‫............................................................‬ ‫...........................................................‬ ‫ﻫ ﺱ٢ + ٩ = ٠‬ ‫د ﺱ٢ - ٦ﺱ + ٩ = ٠‬ ‫و ﺱ )ﺱ+ ١( )ﺱ - ١( = ٠‬ ‫............................................................‬ ‫............................................................‬ ‫...........................................................‬ ‫............................................................‬ ‫............................................................‬ ‫...........................................................‬ ‫............................................................‬ ‫............................................................‬ ‫...........................................................‬ ‫¯‬ ‫−‬ ‫¯‬
  • 10. ‫¯‬ ‫7 ﻳﺒﻴﻦ ﻛﻞ ﺷﻜﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻵﺗﻴﺔ ﺍﻟﺮﺳﻢ ﺍﻟﺒﻴﺎﻧﻰ ﻟﺪﺍﻟﺔ ﻣﻦ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ.‬ ‫ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺍﻟﺤﻞ ﻟﻠﻤﻌﺎﺩﻟﺔ ﺩ )ﺱ( = ٠ ﻓﻰ ﻛﻞ ﺷﻜﻞ.‬ ‫ب‬ ‫أ‬ ‫ﺟ‬ ‫− −‬ ‫− − − − −‬ ‫−‬ ‫−‬ ‫− − −‬ ‫−‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫8 ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺍﻟﺤﻞ ﻟﻜﻞ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ ﻓﻰ ﺡ ﻭﺣﻘﻖ ﺍﻟﻨﺎﺗﺞ ﺑﻴﺎﻧﻴﺎ:‬ ‫ًّ‬ ‫ب ٢ﺱ٢ = ٣ – ٥ﺱ‬ ‫أ ﺱ٢ = ٣ﺱ + ٠٤‬ ‫............................................................‬ ‫............................................................‬ ‫ﺟ ٦ﺱ٢ = ٦ – ٥ﺱ‬ ‫د )ﺱ – ٣( = ٥‬ ‫٢‬ ‫............................................................‬ ‫............................................................‬ ‫و ١ ﺱ٢ - ٣ ﺱ = ١‬ ‫٥‬ ‫٢‬ ‫ﻫ ﺱ٢ + ٢ﺱ = ٢١‬ ‫............................................................‬ ‫............................................................‬ ‫9 ﺣﻞ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ ﻓﻰ ﺡ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻘﺎﻧﻮﻥ ﺍﻟﻌﺎﻡ ﻣﻘﺮﺑﺎ ﺍﻟﻨﺎﺗﺞ ﻟﺮﻗﻢ ﻋﺸﺮﻯ ﻭﺍﺣﺪ.‬ ‫ً‬ ‫ب ﺱ٢ – ٦ﺱ + ٧ = ٠‬ ‫أ ٣ﺱ٢ – ٥٦ = ٠‬ ‫............................................................‬ ‫............................................................‬ ‫ﺟ ﺱ٢ + ٦ﺱ + ٨ = ٠‬ ‫د ٢ﺱ٢+٣ﺱ–٤ = ٠‬ ‫............................................................‬ ‫............................................................‬ ‫و ٣ﺱ٢ – ٦ﺱ – ٤ = ٠‬ ‫ﻫ ٥ﺱ٢ – ٣ﺱ – ١ = ٠‬ ‫............................................................‬ ‫............................................................‬ ‫01 ﺃﻋﺪﺍﺩ: ﺇﺫﺍ ﻛﺎﻥ ﻣﺠﻤﻮﻉ ﺍﻷﻋﺪﺍﺩ ﺍﻟﺼﺤﻴﺤﺔ ﺍﻟﻤﺘﺘﺎﻟﻴﺔ )١ + ٢ + ٣ + ... + ﻥ(ﻳﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ ﺟـ = ﻥ )١ + ﻥ(‬ ‫٢‬ ‫ﻓﻜﻢ ﻋﺪﺩﺍ ﺻﺤﻴﺤﺎ ﻣﺘﺘﺎﻟﻴﺎ ﺑﺪﺀﺍ ﻣﻦ ﺍﻟﻌﺪﺩ ١ ﻳﻜﻮﻥ ﻣﺠﻤﻮﻋﻬﺎ ﻣﺴﺎﻭ ﻳﺎ:‬ ‫ً‬ ‫ً‬ ‫ً‬ ‫ً ً‬ ‫ب ١٧١‬ ‫أ ٨٧‬ ‫..............................................‬ ‫...............................................‬ ‫ﺟ ٣٥٢‬ ‫د ٥٦٤‬ ‫...............................................‬ ‫...............................................‬ ‫‪M‬‬ ‫−‬
  • 11. ‫11 ﻳﺒﻴﻦ ﻛﻞ ﺷﻜﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻵﺗﻴﺔ ﺍﻟﺮﺳﻢ ﺍﻟﺒﻴﺎﻧﻰ ﻟﺪﺍﻟﺔ ﻣﻦ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ ﻓﻰ ﻣﺘﻐﻴﺮ ﻭﺍﺣﺪ. ﺃﻭﺟﺪ ﻗﺎﻋﺪﺓ ﻛﻞ‬ ‫ﺩﺍﻟﺔ ﻣﻦ ﻫﺬه ﺍﻟﺪﻭﺍﻝ.‬ ‫ب‬ ‫أ‬ ‫ﺟ‬ ‫− − − −‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫− − − −‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫...............................................................‬ ‫...............................................................‬ ‫...............................................................‬ ‫21 ﺍﻛﺘﺸﻒ ﺍﻟﺨﻄﺄ: ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ )ﺱ – ٣(٢ = )ﺱ – ٣(.‬ ‫¯‬ ‫‪) a‬ﺱ – ٣(٢ = )ﺱ – ٣(‬ ‫–‬ ‫‪¯M‬‬ ‫ ‬ ‫‪F‬‬ ‫` ﺱ – ٣ = ١‬ ‫` ﺱ = ٤‬ ‫ﻣﺠﻤﻮﻋﺔ ﺍﻟﺤﻞ = }٤{‬ ‫!‬ ‫‪) a‬ﺱ – ٣(٢ = )ﺱ – ٣(‬ ‫` )ﺱ – ٣(٢ – )ﺱ – ٣( = ٠‬ ‫` )ﺱ – ٣(])ﺱ – ٣( – ١[ = ٠‬ ‫‪ :F‬ﺱ – ٣=٠ ﺃﻭ ﺱ – ٤=٠‬ ‫ﻣﺠﻤﻮﻋﺔ ﺍﻟﺤﻞ = }٣، ٤{‬ ‫ﺃﻱ ﺍﻟﺤﻠﻴﻦ ﺻﺤﻴﺢ? ﻟﻤﺎﺫﺍ?‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫31 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ: ﻗُﺬﻓﺖ ﻛﺮﺓ ﺭﺃﺳﻴﺎ ﺇﻟﻰ ﺃﻋﻠﻰ ﺑﺴﺮﻋﺔ ﻉ ﺗﺴﺎﻭﻯ ٤٫٩٢ ﻣﺘﺮ/ﺙ. ﺍﺣﺴﺐ ﺍﻟﻔﺘﺮﺓ ﺍﻟﺰﻣﻨﻴﺔ ﻥ ﺑﺎﻟﺜﺎﻧﻴﺔ ﺍﻟﺘﻰ‬ ‫ُ‬ ‫ًّ‬ ‫ﺗﺴﺘﻐﺮﻗﻬﺎ ﺍﻟﻜﺮﺓ ﺣﺘﻰ ﺗﺼﻞ ﺇﻟﻰ ﺍﺭﺗﻔﺎﻉ ﻑ ﻣﺘﺮﺍ، ﺣﻴﺚ ﻑ ﺗﺴﺎﻭﻯ ٢٫٩٣ ﻣﺘﺮﺍ ﻋﻠﻤﺎ ﺑﺄﻥ ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﻑ، ﻥ ﺗﻌﻄﻰ‬ ‫ُْ‬ ‫ً ً‬ ‫ً‬ ‫ﻛﺎﻵﺗﻰ ﻑ = ﻉ ﻥ – ٩٫٤ ﻥ٢.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
  • 12. ‫ﻣﻘﺪﻣﺔ ﻋﻦ ا ﻋﺪاد اﻟﻤﺮﻛﺒﺔ‬ ‫1-2‬ ‫‪Complex Numbers‬‬ ‫1 ﺿﻊ ﻛﻼ ﻣﻤﺎ ﻳﺄﺗﻰ ﻓﻰ ﺃﺑﺴﻂ ﺻﻮﺭﺓ:‬ ‫ًّ‬ ‫ ٥٤‬‫٦٦‬ ‫ب ﺕ‬ ‫أ ﺕ‬ ‫......................................‬ ‫ﺟ ﺕ‬ ‫......................................‬ ‫٤ﻥ + ٢‬ ‫......................................‬ ‫د ﺕ‬ ‫٤ﻥ – ١‬ ‫......................................‬ ‫2 ﺑﺴﻂ ﻛﻼ ﻣﻤﺎ ﻳﺄﺗﻰ:‬ ‫ًّ‬ ‫أ‬ ‫-٨١ * -٢١‬ ‫ب ٣ ﺕ )- ٢ﺕ(‬ ‫..........................................‬ ‫ﺟ )- ٤ ﺕ( )- ٦ ﺕ(‬ ‫..........................................‬ ‫...........................................‬ ‫د )- ٢ ﺕ(٣ )- ٣ ﺕ(‬ ‫٢‬ ‫.........................................‬ ‫3 ﺃﻭﺟﺪ ﻧﺎﺗﺞ ﻛﻞ ﻣﻤﺎ ﻳﺄﺗﻰ ﻓﻰ ﺃﺑﺴﻂ ﺻﻮﺭﺓ:‬ ‫ٍّ‬ ‫أ )٣ + ٢ﺕ( + )٢ – ٥ ﺕ(   ب )٦٢ – ٤ﺕ( – )٩ – ٠٢ ﺕ(   ﺟ )٠٢ + ٥٢ ﺕ( – )٩ – ٠٢ ﺕ(‬ ‫.................................................................................       ..................................................................................      ..................................................................................‬ ‫4 ﺿﻊ ﻛﻼ ﻣﻤﺎ ﻳﺄﺗﻰ ﻋﻠﻰ ﺻﻮﺭﺓ ‪ + C‬ﺏ ﺕ‬ ‫ًّ‬ ‫ب )١ + ٢ﺕ٣( )٢ + ٣ ﺕ٥ + ٤ ﺕ٦(‬ ‫أ )٢ + ٣ ﺕ( – )١ – ٢ﺕ(‬ ‫5 ﺿﻊ ﻛﻼ ﻣﻤﺎ ﻳﺄﺗﻰ ﻋﻠﻰ ﺻﻮﺭﺓ ‪ + C‬ﺏ ﺕ‬ ‫ًّ‬ ‫٢‬ ‫ب ٤+ﺕ‬ ‫أ‬ ‫ﺕ‬ ‫ﺟ‬ ‫١+ﺕ‬ ‫......................................‬ ‫......................................‬ ‫......................................‬ ‫6 ﺣﻞ ﻛﻞ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ:‬ ‫ب ٤ ﺹ٢ + ٠٢ = ٠‬ ‫أ ٣ ﺱ٢ + ٢١ = ٠‬ ‫......................................‬ ‫٢ - ٣ﺕ‬ ‫٣+ﺕ‬ ‫ﺟ ٤ ﻉ٢ + ٢٧ = ٠‬ ‫......................................‬ ‫......................................‬ ‫د )٣ + ﺕ()٣ - ﺕ(‬ ‫٣-٤ﺕ‬ ‫......................................‬ ‫د ٣ ﺹ٢ + ٥١ = ٠‬ ‫٥‬ ‫......................................‬ ‫7 ﻛﻬﺮﺑﺎﺀ: ﺃﻭﺟﺪ ﺷﺪﺓ ﺍﻟﺘﻴﺎﺭ ﺍﻟﻜﻬﺮﺑﻰ ﺍﻟﻜﻠﻴﺔ ﺍﻟﻤﺎﺭ ﻓﻰ ﻣﻘﺎﻭﻣﺘﻴﻦ ﻣﺘﺼﻠﺘﻴﻦ ﻋﻠﻰ ﺍﻟﺘﻮﺍﺯﻯ ﻓﻰ ﺩﺍﺋﺮﺓ ﻛﻬﺮﺑﺎﺋﻴﺔ‬ ‫ﻣﻐﻠﻘﺔ ﺇﺫﺍ ﻛﺎﻧﺖ ﺷﺪﺓ ﺍﻟﺘﻴﺎﺭ ﻓﻰ ﺍﻟﻤﻘﺎﻭﻣﺔ ﺍﻷ ﻟﻰ ٤ – ٢ﺕ ﺃﻣﺒﻴﺮ، ﻭﻓﻰ ﺍﻟﻤﻘﺎﻭﻣﺔ ﺍﻟﺜﺎﻧﻴﺔ ٦ + ٣ﺕ ﺃﻣﺒﻴﺮ .................‬ ‫ﻭ‬ ‫٢+ﺕ‬ ‫8 ﺍﻛﺘﺸﻒ ﺍﻟﺨﻄﺄ: ﺃﻭﺟﺪ ﺃﺑﺴﻂ ﺻﻮﺭﺓ ﻟﻠﻤﻘﺪﺍﺭ: )٢ + ٣ﺕ(٢ )٢ – ٣ﺕ(‬ ‫¯‬ ‫)٢ + ٣ﺕ(٢)٢– ٣ﺕ( = )٤ + ٩ﺕ٢()٢ – ٣ﺕ(‬ ‫= )٤ – ٩()٢ – ٣ﺕ( = - ٥ )٢ – ٣ﺕ(‬ ‫= - ٠١ + ٥١ ﺕ‬ ‫)٢ + ٣ﺕ()٢ + ٣ﺕ()٢ – ٣ﺕ(‬ ‫= )٢ + ٣ﺕ( )٤ – ٩ﺕ٢(‬ ‫= )٢ + ٣ﺕ( )٤ + ٩( = ٣١)٢ + ٣ﺕ(‬ ‫= ٦٢ + ٩٣ ﺕ‬ ‫ﺃﻯ ﺍﻟﺤﻠﻴﻦ ﺻﺤﻴﺢ? ﻟﻤﺎﺫﺍ?‬ ‫............................................................................................................................................................................‬ ‫‪M‬‬ ‫−‬
  • 13. ‫ﻧﺸﺎط‬ ‫١-‬ ‫٢-‬ ‫٣-‬ ‫٤-‬ ‫٥-‬ ‫ﺑﺎﺳﺘﺨﺪﺍﻡ ﺃﺣﺪ ﺍﻟﺒﺮﺍﻣﺞ ﺍﻟﺮﺳﻮﻣﻴﺔ ﺍﺭﺳﻢ ﻣﻨﺤﻨﻰ ﺍﻟﺪﺍﻟﺔ ﺩ ﺣﻴﺚ ﺩ)ﺱ( = ﺱ٣ - ١ .‬ ‫ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ ﻳﻤﺜﻞ ﻣﻨﺤﻨﻰ ﺍﻟﺪﺍﻟﺔ، ﻫﻞ ﻳﻤﻜﻨﻚ ﺇﻳﺠﺎﺩ ﻣﺠﻤﻮﻋﺔ ﺣﻞ‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٣ -١ = ٠ ﻣﻦ ﺍﻟﺮﺳﻢ?‬ ‫ﻫﻞ ﺗﺘﻮﻗﻊ ﻭﺟﻮﺩ ﺟﺬﻭﺭ ﺃﺧﺮﻯ ﺑﺎﺳﺘﺜﻨﺎﺀ ﺍﻟﺠﺬﻭﺭ ﺍﻟﺘﻰ ﺣﺼﻠﺖ ﻋﻠﻴﻬﺎ ﻣﻦ‬ ‫ﺍﻟﺮﺳﻢ، ﻭﺫﻟﻚ ﻣﻦ ﺧﻼﻝ ﺩﺭﺍﺳﺘﻚ ﻟﻤﺠﻤﻮﻋﺎﺕ ﺍﻷﻋﺪﺍﺩ?‬ ‫ﻫﻞ ﻳﻤﻜﻨﻚ ﺣﻞ ﺱ٣ - ١ = ٠ ﺟﺒﺮ ﻳﺎ?‬ ‫ًّ‬ ‫ﺍﺳﺘﺨﺪﻡ ﻃﺮﻕ ﺍﻟﺘﺤﻠﻴﻞ ﺍﻟﺘﻰ ﺳﺒﻖ ﻟﻚ ﺩﺭﺍﺳﺘﻬﺎ ﻓﻰ ﺣﻞ ﻫﺬه ﺍﻟﻤﻌﺎﺩﻟﺔ.‬ ‫: ﺱ٣ - ١ = )ﺱ - ١()ﺱ٢ + ﺱ + ١( =٠‬ ‫¯‬ ‫¯‬ ‫٦- ﺗﻌﻠﻢ ﺃﻧﻪ ﻣﻦ ﺧﻮﺍﺹ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺇﺫﺍ ﻛﺎﻥ ‪ * C‬ﺏ * ﺟـ = ٠ ﻓﺈﻥ ‪ ، ٠ = C‬ﺏ = ٠، ﺟـ = ٠ ﻓﻬﻞ ﻳﻤﻜﻨﻚ ﺍﺳﺘﺨﺪﺍﻡ ﺫﻟﻚ‬ ‫ﻓﻰ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺴﺎﺑﻘﺔ?‬ ‫  ﺱ = ١  ﻭﻫﺬﺍ ﻳﻄﺎﺑﻖ ﺍﻟﺤﻞ ﺍﻟﺒﻴﺎﻧﻰ ﺃﻭ:‬ ‫ﺱ - ١ = ٠ ‬ ‫¯‬ ‫ﺱ٢ + ﺱ +١ = ٠ ﻫﻞ ﻳﻤﻜﻨﻚ ﺣﻞ ﻫﺬه ﺍﻟﻤﻌﺎﺩﻟﺔ ﺑﺎﻟﺘﺤﻠﻴﻞ?‬ ‫٧- ﺍﺳﺘﺨﺪﻡ ﻣﻔﻬﻮﻡ ﻣﻤﻴﺰ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﻟﺘﺤﺪﻳﺪ ﻧﻮﻉ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ﺱ + ١ = ٠ ﺣﻴﺚ ‪ ، ١ =C‬ﺏ = ١ ، ﺟـ = ١‬ ‫  ﺏ٢ - ٤ ‪ C‬ﺟـ > ٠‬ ‫ﺍﻟﻤﻤﻴﺰ )ﺏ٢- ٤ ‪C‬ﺟـ( = ١ - ٤ *١ *١ = -٣‬ ‫¯‬ ‫¯‬ ‫,‬ ‫٨- ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ﺱ + ١ = ٠ ﻓﻰ ﻣﺠﻤﻮﻋﺔ ﺍﻷﻋﺪﺍﺩ ﻛﺒﺔ .‬ ‫ﺍﻟﻤﺮ‬ ‫ﺱ = - ﺏ ! ﺏ ٢-٤‪C‬ﺟـ‬ ‫٢‪C‬‬ ‫ﻓﺘﻜﻮﻥ ﺱ = - ١ !‬ ‫٩- ﺍﻛﺘﺐ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٣ - ١ = ٠ ﻓﻰ ﻣﺠﻤﻮﻋﺔ ﺍﻷﻋﺪﺍﺩ ﻛﺒﺔ.‬ ‫ﺍﻟﻤﺮ‬ ‫ﻣﺠﻤﻮﻋﺔ ﺍﻟﺤﻞ ﻫﻰ}١، - ١ +‬ ‫٢*١‬ ‫-٣ ،‬ ‫١-‬‫٢*١‬ ‫-٣‬ ‫٢*١‬ ‫-٣ {‬ ‫٠١-ﻛﻢ ﻋﺪﺩ ﺍﻟﺠﺬﻭﺭ ﺍﻟﺤﻘﻴﻘﻴﺔ ﻛﻢ ﻋﺪﺩ ﺍﻟﺠﺬﻭﺭ ﻛﺒﺔ ?‬ ‫ﺍﻟﻤﺮ‬ ‫ﻭ‬ ‫١١- ﺃﻭﺟﺪ ﻣﺠﻤﻮﻉ ﺍﻟﺠﺬﻭﺭ ﺍﻟﺜﻼﺛﺔ ﻟﻠﻤﻌﺎﺩﻟﺔ - ﻣﺎﺫﺍ ﺗﻼﺣﻆ?‬ ‫٢١- ﺃﻭﺟﺪ ﺣﺎﺻﻞ ﺿﺮﺏ ﺍﻟﺠﺬﺭﻳﻦ ﺍﻟﺘﺨﻴﻠﻴﻴﻦ - ﻣﺎﺫﺍ ﺗﻼﺣﻆ?‬ ‫٣١- ﺃﻭﺟﺪ ﻣﺮﺑﻊ ﺃﺣﺪ ﺍﻟﺠﺬﺭﻳﻦ ﺍﻟﺘﺨﻴﻠﻴﻴﻦ ﻭﻗﺎﺭﻧﻪ ﻣﻊ ﺍﻟﺠﺬﺭ ﺍﻵﺧﺮ.‬ ‫٤١- ﻟﻤﺎﺫﺍ ﺃﻋﻄﻰ ﺍﻟﺤﻞ ﺍﻟﺒﻴﺎﻧﻰ ﺟﺬﺭﺍ ﻭﺍﺣﺪﺍ ﻓﻘﻂ، ﺑﻴﻨﻤﺎ ﺃﻋﻄﻰ ﺍﻟﺤﻞ ﺍﻟﺠﺒﺮﻯ ﺛﻼﺛﺔ ﺟﺬﻭﺭ ? ﻓﺴﺮ ﺫﻟﻚ.‬ ‫ً‬ ‫ً‬ ‫ِّ‬ ‫٥١- ﺍﺑﺤﺚ ﻓﻰ ﺍﻟﺸﺒﻜﺔ ﺍﻟﻌﻨﻜﺒﻮﺗﻴﺔ ﻋﻦ ﻛﻴﻔﻴﺔ ﺗﻤﺜﻴﻞ ﺟﺬﻭﺭ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﻜﻌﻴﺒﻴﺔ ﺑﻴﺎﻧﻴﺎ ﺑﻤﺎ ﻳﺘﻨﺎﺳﺐ ﻣﻊ ﻣﻌﻠﻮﻣﺎﺗﻚ.‬ ‫ًّ‬ ‫¯‬ ‫−‬ ‫¯‬
  • 14. ‫ﺗﺤﺪﻳﺪ ﻧﻮع ﺟﺬرى اﻟﻤﻌﺎدﻟﺔ اﻟﺘﺮﺑﻴﻌﻴﺔ‬ ‫1-3‬ ‫‪Determining The Type of Roots of a Quadratic Equation‬‬ ‫‪k‬‬ ‫‪:Oó©àe øe QÉ«àNG :’hCG‬‬ ‫1 ﻳﻜﻮﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٤ﺱ + ﻙ = ٠ ﻣﺘﺴﺎﻭﻳﻴﻦ ﺇﺫﺍ ﻛﺎﻧﺖ:‬ ‫ﺟ ﻙ=٨‬ ‫ب ﻙ=٤‬ ‫أ ﻙ=١‬ ‫............................................................................................‬ ‫2 ﻳﻜﻮﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٢ﺱ + ﻡ = ٠ ﺣﻘﻴﻘﻴﻴﻦ ﻣﺨﺘﻠﻔﻴﻦ ﺇﺫﺍ ﻛﺎﻧﺖ:‬ ‫ﺟ ﻡ<١‬ ‫ب ﻡ>١‬ ‫أ ﻡ=١‬ ‫د ﻙ = ٦١‬ ‫............................................................................‬ ‫3 ﻳﻜﻮﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﻝ ﺱ٢ – ٢١ﺱ + ٩ = ٠ ﻛﺒﻴﻦ ﺇﺫﺍ ﻛﺎﻧﺖ:‬ ‫ﻣﺮ‬ ‫ﺟ ﻝ=٤‬ ‫ب ﻝ>٤‬ ‫أ ﻝ<٤‬ ‫د ﻡ=٤‬ ‫...........................................................................................‬ ‫د ﻝ=١‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫4 ﺣﺪﺩ ﻋﺪﺩ ﺍﻟﺠﺬﻭﺭ ﻭﺃﻧﻮﺍﻋﻬﺎ ﻟﻜﻞ ﻣﻌﺎﺩﻟﺔ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻵﺗﻴﺔ:‬ ‫ب ٣ﺱ٢ + ٠١ﺱ - ٤ = ٠‬ ‫أ ﺱ٢ - ٢ﺱ + ٥ = ٠‬ ‫..................................................................................‬ ‫..................................................................................‬ ‫ﺟ ﺱ٢ – ٠١ﺱ + ٥٢ = ٠‬ ‫د ٦ﺱ٢ – ٩١ﺱ + ٥٣ = ٠‬ ‫..................................................................................‬ ‫..................................................................................‬ ‫و )ﺱ – ١( )ﺱ – ٧( = ٢ )ﺱ – ٣( )ﺱ – ٤(‬ ‫ﻫ )ﺱ – ١١( – ﺱ)ﺱ – ٦( = ٠‬ ‫..................................................................................‬ ‫..................................................................................‬ ‫5 ﺃﻭﺟﺪ ﺣﻞ ﻛﻞ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ ﻓﻰ ﻣﺠﻤﻮﻋﺔ ﺍﻷﻋﺪﺍﺩ ﻛﺒﺔ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻘﺎﻧﻮﻥ ﺍﻟﻌﺎﻡ.‬ ‫ﺍﻟﻤﺮ‬ ‫ٍّ‬ ‫ب ٢ﺱ٢ + ٦ﺱ + ٥ = ٠‬ ‫أ ﺱ٢ - ٤ﺱ + ٥ = ٠‬ ‫..................................................................................‬ ‫..................................................................................‬ ‫ﺟ ٣ﺱ٢ - ٧ﺱ + ٦ = ٠‬ ‫د ٤ﺱ٢ - ﺱ + ١ = ٠‬ ‫..................................................................................‬ ‫..................................................................................‬ ‫6 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻙ ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻟﺤﺎﻻﺕ ﺍﻵﺗﻴﺔ:‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ٤ﺱ + ﻙ = ٠ ﺣﻘﻴﻘﻴﻴﻦ ﻣﺨﺘﻠﻔﻴﻦ.‬ ‫.......................................................................................................................................................................................................................‬ ‫‪M‬‬ ‫−‬
  • 15. ‫١‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٣ﺱ + ٢ + ﻙ = ٠ ﻣﺘﺴﺎﻭﻳﻴﻦ.‬ ‫.......................................................................................................................................................................................................................‬ ‫ﺟ ﺇﺫﺍ ﻛﺎﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﻙ ﺱ٢ – ٨ﺱ + ٦١ = ٠ ﻛﺒﻴﻦ.‬ ‫ﻣﺮ‬ ‫..................................................................................................................................................................................................................................‬ ‫7 ﺇﺫﺍ ﻛﺎﻥ ﻝ، ﻡ ﻋﺪﺩﻳﻦ ﻧﺴﺒﻴﻴﻦ، ﻓﺄﺛﺒﺖ ﺃﻥ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ : ﻝ ﺱ٢ + )ﻝ – ﻡ( ﺱ – ﻡ = ٠ ﻋﺪﺩﺍﻥ ﻧﺴﺒﻴﺎﻥ.‬ ‫..................................................................................................................................................................................................................................‬ ‫8 ﻳﻘﺪﺭ ﻋﺪﺩ ﺳﻜﺎﻥ ﺟﻤﻬﻮﺭﻳﺔ ﻣﺼﺮ ﺍﻟﻌﺮﺑﻴﺔ ﻋﺎﻡ ٣١٠٢ ﺑﺎﻟﻌﻼﻗﺔ:‬ ‫ﻉ = ﻥ٢ + ٢٫١ ﻥ + ١٩ ﺣﻴﺚ )ﻉ( ﻋﺪﺩ ﺍﻟﺴﻜﺎﻥ ﺑﺎﻟﻤﻠﻴﻮﻥ، )ﻥ( ﻋﺪﺩ ﺍﻟﺴﻨﻮﺍﺕ‬ ‫.................................................................................................................‬ ‫أ ﻛﻢ ﻛﺎﻥ ﻋﺪﺩ ﺍﻟﺴﻜﺎﻥ ﻋﺎﻡ ٣١٠٢?‬ ‫.................................................................................................................‬ ‫ب ﻗﺪﺭ ﻋﺪﺩ ﺍﻟﺴﻜﺎﻥ ﻋﺎﻡ ٣٢٠٢.‬ ‫ﺟ‬ ‫ﻗﺪﺭ ﻋﺪﺩ ﺍﻟﺴﻨﻮﺍﺕ ﺍﻟﺘﻰ ﻳﺒﻠﻎ ﻋﺪﺩ ﺍﻟﺴﻜﺎﻥ ﻓﻴﻬﺎ ٤٣٣ ﻣﻠﻴ ﻧًﺎ. ...........................................................................................‬ ‫ﻮ‬ ‫ً‬ ‫د ﺍﻛﺘﺐ ﻣﻘﺎﻻ ﺗﻮﺿﺢ ﻓﻴﻪ ﺃﺳﺒﺎﺏ ﺍﻟﺰﻳﺎﺩﺓ ﺍﻟﻤﻄﺮﺩﺓ ﻓﻰ ﻋﺪﺩ ﺍﻟﺴﻜﺎﻥ ﻛﻴﻔﻴﺔ ﻋﻼﺟﻬﺎ.‬ ‫ﻭ‬ ‫9 ﺍﻛﺘﺸﻒ ﺍﻟﺨﻄﺄ: ﻣﺎ ﻋﺪﺩ ﺣﻠﻮﻝ ﺍﻟﻤﻌﺎﺩﻟﺔ ٢ﺱ٢ – ٦ ﺱ = ٥ ﻓﻰ ﺡ‬ ‫¯‬ ‫ﺏ٢– ٤‪C‬ﺟـ = )- ٦(٢ – ٤ * ٢ * ٥‬ ‫     = ٦٣ – ٠٤ = - ٤‬ ‫¯,‬ ‫ﺏ٢– ٤‪C‬ﺟـ = )- ٦(٢ – ٤ * ٢ )- ٥(‬ ‫     = ٦٣ +٠٤ = ٦٧‬ ‫¯,‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫01 ﺇﺫﺍ ﻛﺎﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ٢ )ﻙ - ١( ﺱ + )٢ﻙ + ١( =٠ ﻣﺘﺴﺎﻭﻳﻴﻦ، ﻓﺄﻭﺟﺪ ﻗﻴﻢ ﻙ ﺍﻟﺤﻘﻴﻘﻴﺔ، ﺛﻢ ﺃﻭﺟﺪ‬ ‫ﺍﻟﺠﺬﺭﻳﻴﻦ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫11 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ: ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ٦٣ ﺱ٢ – ٨٤ ﺱ + ٥٢ = ٠ ﻓﻰ ﻣﺠﻤﻮﻋﺔ ﺍﻷﻋﺪﺍﺩ ﻛﺒﺔ.‬ ‫ﺍﻟﻤﺮ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
  • 16. ‫اﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺟﺬرى ﻣﻌﺎدﻟﺔ اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ وﻣﻌﺎﻣﻼت ﺣﺪودﻫﺎ‬ ‫‪The relation between two roots of the second degree‬‬ ‫‪equation and the coefficients of its terms‬‬ ‫1-4‬ ‫‪k‬‬ ‫‪:≈JCÉjÉe πªcCG :’hCG‬‬ ‫1 ﺇﺫﺍ ﻛﺎﻥ ﺱ = ٣ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ﻡ ﺱ – ٧٢ = ٠ ﻓﺈﻥ ﻡ = .................................، ﺍﻟﺠﺬﺭ ﺍﻵﺧﺮ =‬ ‫................................‬ ‫2 ﺇﺫﺍ ﻛﺎﻥ ﺣﺎﺻﻞ ﺿﺮﺏ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ : ٢ ﺱ٢ + ٧ ﺱ + ٣ ﻙ = ٠ ﻳﺴﺎﻭﻯ ﻣﺠﻤﻮﻉ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ:‬ ‫٢‬ ‫ﺱ – )ﻙ + ٤( ﺱ = ٠ ﻓﺈﻥ ﻙ = ................................‬ ‫3 ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻟﺘﻰ ﻛﻞ ﻣﻦ ﺟﺬﺭﻳﻬﺎ ﻳﺰﻳﺪ ١ ﻋﻦ ﻛﻞ ﻣﻦ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٣ ﺱ + ٢ = ٠ ﻫﻰ‬ ‫...............................‬ ‫4 ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻟﺘﻰ ﻛﻞ ﻣﻦ ﺟﺬﺭﻳﻬﺎ ﻳﻨﻘﺺ ١ ﻋﻦ ﻛﻞ ﻣﻦ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٥ ﺱ + ٦ = ٠ ﻫﻰ‬ ‫...............................‬ ‫‪Oó©àe øe QÉ«àN’G :Ék«fÉK‬‬ ‫5 ﺇﺫﺍ ﻛﺎﻥ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ - ٣ ﺱ + ﺟـ = ٠ ﺿﻌﻒ ﺍﻵﺧﺮ ﻓﺈﻥ ﺟـ ﺗﺴﺎﻭﻯ‬ ‫د ٤‬ ‫ﺟ ٢‬ ‫ب -٢‬ ‫أ -٤‬ ‫.......................................................‬ ‫6 ﺇﺫﺍ ﻛﺎﻥ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ C‬ﺱ٢ – ٣ﺱ+ ٢ =٠ ﻣﻌﻜﻮﺳﺎ ﺿﺮﺑﻴﺎ ﻟﻶﺧﺮ، ﻓﺈﻥ ‪ C‬ﺗﺴﺎﻭﻯ‬ ‫ً‬ ‫ًّ‬ ‫ب ١‬ ‫أ ١‬ ‫د ٣‬ ‫ﺟ ٢‬ ‫٢‬ ‫٣‬ ‫...........................................‬ ‫7 ﺇﺫﺍ ﻛﺎﻥ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢– )ﺏ – ٣( ﺱ + ٥ = ٠ ﻣﻌﻜﻮﺳﺎ ﺟﻤﻌﻴﺎ ﻟﻶﺧﺮ، ﻓﺈﻥ ﺏ ﺗﺴﺎﻭﻯ‬ ‫ً‬ ‫ًّ‬ ‫د ٥‬ ‫ﺟ ٣‬ ‫ب -٣‬ ‫أ -٥‬ ‫........................‬ ‫‪k‬‬ ‫‪á«JB’G á∏Ä°SC’G øY ÖLCG :ÉãdÉK‬‬ ‫8 ﺃﻭﺟﺪ ﻣﺠﻤﻮﻉ ﻭﺣﺎﺻﻞ ﺿﺮﺏ ﺟﺬﺭﻯ ﻛﻞ ﻣﻌﺎﺩﻟﺔ ﻓﻴﻤﺎ ﻳﺄﺗﻰ:‬ ‫ب ٤ ﺱ٢ + ٤ ﺱ – ٥٣ = ٠‬ ‫أ ٣ ﺱ٢ + ٩١ ﺱ – ٤١ = ٠‬ ‫..................................................................................‬ ‫..................................................................................‬ ‫9 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ C‬ﺛﻢ ﺃﻭﺟﺪ ﺍﻟﺠﺬﺭ ﺍﻵﺧﺮ ﻟﻠﻤﻌﺎﺩﻟﺔ ﻓﻰ ﻛﻞ ﻣﻤﺎ ﻳﺄﺗﻰ:‬ ‫ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ  ﺱ٢ – ٢ ﺱ + ‪٠ = C‬‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ: ﺱ = - ١‬ ‫ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ  ‪ C‬ﺱ٢ – ٥ ﺱ + ‪٠ = C‬‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ: ﺱ = ٢‬ ‫........................................................‬ ‫........................................................‬ ‫01 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ ،C‬ﺏ ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ ﺇﺫﺍ ﻛﺎﻥ:‬ ‫......................................................................................................‬ ‫أ ٢، ٥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ‪ C‬ﺱ + ﺏ = ٠‬ ‫......................................................................................................‬ ‫ب -٣، ٧ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪C‬ﺱ٢ – ﺏ ﺱ - ١٢ = ٠‬ ‫ﺟ -١، ٣ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ C‬ﺱ٢ – ﺱ + ﺏ = ٠‬ ‫......................................................................................................‬ ‫٢‬ ‫٢‬ ‫د‬ ‫٣ ﺕ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ + ‪ C‬ﺱ + ﺏ = ٠ ..........................................................................................‬ ‫٣ ﺕ،-‬ ‫‪M‬‬ ‫−‬
  • 17. ‫11 ﺍﺑﺤﺚ ﻧﻮﻉ ﺍﻟﺠﺬﺭﻳﻦ ﻟﻜﻞ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ، ﺛﻢ ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﻛﻞ ﻣﻨﻬﺎ:‬ ‫ب ٢ﺱ٢ + ٣ﺱ + ٧ = ٠‬ ‫أ ﺱ٢ + ٢ﺱ – ٥٣ = ٠‬ ‫..................................................................................‬ ‫ﺟ ﺱ)ﺱ – ٤( + ٥ = ٠‬ ‫..................................................................................‬ ‫د ٣ﺱ)٣ﺱ – ٨( + ٦١ = ٠‬ ‫..................................................................................‬ ‫..................................................................................‬ ‫21 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺟـ ﺍﻟﺘﻰ ﺗﺠﻌﻞ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺟـ ﺱ٢ – ٢١ﺱ + ٩ = ٠ ﻣﺘﺴﺎﻭﻳﻴﻦ.‬ ‫..................................................................................................................................................................................................................................‬ ‫31 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ C‬ﺍﻟﺘﻰ ﺗﺠﻌﻞ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٣ﺱ + ٢ + ١ = ٠ ﻣﺘﺴﺎﻭﻳﻴﻦ.‬ ‫‪C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫41 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺟـ ﺍﻟﺘﻰ ﺗﺠﻌﻞ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ٣ ﺱ٢ – ٥ ﺱ + ﺟـ = ٠ ﻣﺘﺴﺎﻭﻳﻴﻦ، ﺛﻢ ﺃﻭﺟﺪ ﺍﻟﺠﺬﺭﻳﻦ.‬ ‫..................................................................................................................................................................................................................................‬ ‫51 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻙ ﺍﻟﺘﻰ ﺗﺠﻌﻞ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + )ﻙ - ١( ﺱ – ٣ = ٠ ﻫﻮ ﺍﻟﻤﻌﻜﻮﺱ ﺍﻟﺠﻤﻌﻰ ﻟﻠﺠﺬﺭ ﺍﻵﺧﺮ.‬ ‫..................................................................................................................................................................................................................................‬ ‫61 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻙ ﺍﻟﺘﻰ ﺗﺠﻌﻞ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ : ٤ ﻙ ﺱ٢ + ٧ ﺱ + ﻙ٢ + ٤ = ٠ ﻫﻮ ﺍﻟﻤﻌﻜﻮﺱ ﺍﻟﻀﺮﺑﻰ ﻟﻠﺠﺬﺭ ﺍﻵﺧﺮ.‬ ‫..................................................................................................................................................................................................................................‬ ‫71 ﻛﻮﻥ ﻣﻌﺎﺩﻟﺔ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ ﻛﺎﻵﺗﻰ :‬ ‫ب - ٥ ﺕ، ٥ ﺕ‬ ‫أ – ٢، ٤‬ ‫...................................................................................‬ ‫د ١ - ٣ﺕ ، ١ + ٣ﺕ‬ ‫...................................................................................‬ ‫ﺟ ٢،٣‬ ‫٣ ٢‬ ‫...................................................................................‬ ‫................................................................‬ ‫ﻫ ٣ - ٢ ٢ ﺕ ، ٣ + ٢ ٢ ﺕ‬ ‫...................................................................................‬ ‫81 ﺃﻭﺟﺪ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ ﺿﻌﻔﺎ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ٢ﺱ٢ – ٨ﺱ + ٥ = ٠‬ ‫..................................................................................................................................................................................................................................‬ ‫91 ﺃﻭﺟﺪ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻟﺘﻰ ﻛﻞ ﻣﻦ ﺟﺬﺭﻳﻬﺎ ﻳﺰﻳﺪ ﺑﻤﻘﺪﺍﺭ ١ ﻋﻦ ﻛﻞ ﻣﻦ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ : ﺱ٢ – ٧ﺱ – ٩ = ٠‬ ‫..................................................................................................................................................................................................................................‬ ‫02 ﺃﻭﺟﺪ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻟﺘﻰ ﻛﻞ ﻣﻦ ﺟﺬﺭﻳﻬﺎ ﻳﺴﺎﻭﻯ ﻣﺮﺑﻊ ﻧﻈﻴﺮه ﻣﻦ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ : ﺱ٢ + ٣ﺱ – ٥ = ٠‬ ‫..................................................................................................................................................................................................................................‬ ‫12 ﺇﺫﺍ ﻛﺎﻥ ﻝ، ﻡ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٧ ﺱ + ٣ = ٠ ﻓﺄﻭﺟﺪ ﻣﻌﺎﺩﻟﺔ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ:‬ ‫ﺟ ٢،٢‬ ‫د ﻝ + ﻡ، ﻝ ﻡ‬ ‫ب ﻝ + ٢، ﻡ + ٢‬ ‫أ ٢ ﻝ، ٢ ﻡ‬ ‫ﻝ ﻡ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
  • 18. ‫22 ﻣﺴﺎﺣﺎﺕ: ﻗﻄﻌﺔ ﺃﺭﺽ ﻋﻠﻰ ﺷﻜﻞ ﻣﺴﺘﻄﻴﻞ ﺑﻌﺪﺍه ٦، ٩ ﻣﻦ ﺍﻷﻣﺘﺎﺭ، ﻳﺮﺍﺩ ﻣﻀﺎﻋﻔﺔ ﻣﺴﺎﺣﺔ ﻫﺬه ﺍﻟﻘﻄﻌﺔ ﻭﺫﻟﻚ‬ ‫ﺑﺰﻳﺎﺩﺓ ﻃﻮﻝ ﻛﻞ ﺑﻌﺪ ﻣﻦ ﺃﺑﻌﺎﺩﻫﺎ ﺑﻨﻔﺲ ﺍﻟﻤﻘﺪﺍﺭ.ﺃﻭﺟﺪ ﺍﻟﻤﻘﺪﺍﺭ ﺍﻟﻤﻀﺎﻑ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫32 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ: ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﻗﻴﻢ ﺟـ ﻓﻰ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ٧ ﺱ٢ + ٤١ ﺱ + ﺟـ = ٠ ﺑﺤﻴﺚ ﻳﻜﻮﻥ ﻟﻠﻤﻌﺎﺩﻟﺔ:‬ ‫أ ﺟﺬﺭﺍﻥ ﺣﻘﻴﻘﻴﺎﻥ ﻣﺨﺘﻠﻔﺎﻥ.‬ ‫ب ﺟﺬﺭﺍﻥ ﺣﻘﻴﻘﻴﺎﻥ ﻣﺘﺴﺎﻭﻳﺎﻥ.‬ ‫ﺟ ﺟﺬﺭﺍﻥ ﻛﺒﺎﻥ.‬ ‫ﻣﺮ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫42 ﺍﻛﺘﺸﻒ ﺍﻟﺨﻄﺄ: ﺇﺫﺍ ﻛﺎﻥ ﻝ + ١، ﻡ + ١ ﻫﻤﺎ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ٥ﺱ + ٣ = ٠ ﻓﺄﻭﺟﺪ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻟﺘﻰ‬ ‫ﺟﺬﺭﺍﻫﺎ ﻝ، ﻡ.‬ ‫¯‬ ‫‪) a‬ﻝ + ١( + )ﻡ+١( = - ٥‬ ‫` ﻝ + ﻡ = - ٧،‬ ‫`ﻝ+ﻡ+٢=-٥‬ ‫‪) a‬ﻝ + ١()ﻡ + ١( = ٣ ` ﻝ ﻡ + )ﻝ + ﻡ( + ١ = ٣‬ ‫`ﻝﻡ=٩‬ ‫`ﻝﻡ–٧+١=٣‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ ﻫﻰ: ﺱ٢ + ٧ﺱ + ٩ = ٠‬ ‫¯‬ ‫‪ a‬ﻝ + ﻡ = - ٥، ﻝ ﻡ = ٣‬ ‫` )ﻝ +١ ( + )ﻡ + ١(   = ﻝ+ ﻡ + ٢‬ ‫              = - ٥ + ٢ = -٣،‬ ‫‪) a‬ﻝ+١()ﻡ + ١( = ﻝ ﻡ + )ﻝ + ﻡ( + ١‬ ‫              = ٣ – ٣ + ١ = ١‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ ﻫﻰ: ﺱ٢ + ٣ﺱ + ١ = ٠‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫52 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ: ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻔﺮﻕ ﺑﻴﻦ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ﻙ ﺱ + ٢ﻙ = ٠ ﻳﺴﺎﻭﻯ ﺿﻌﻒ ﺣﺎﺻﻞ ﺿﺮﺏ ﺟﺬﺭﻯ‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ٣ ﺱ + ﻙ = ٠ ﻓﺄﻭﺟﺪ ﻙ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪M‬‬ ‫−‬
  • 19. ‫إﺷﺎرة اﻟﺪاﻟﺔ‬ ‫1-5‬ ‫‪Sign of a Function‬‬ ‫‪k‬‬ ‫‪:≈JCÉj Ée πªcCG :’hCG‬‬ ‫1 ﺍﻟﺪﺍﻟﺔ ﺩ، ﺣﻴﺚ ﺩ)ﺱ( = - ٥ ﺇﺷﺎﺭﺍﺗﻬﺎ .................................................... ﻓﻰ ﺍﻟﻔﺘﺮﺓ‬ ‫....................................................‬ ‫2 ﺍﻟﺪﺍﻟﺔ ﺩ، ﺣﻴﺚ ﺩ)ﺱ( = ﺱ٢ + ١ ﺇﺷﺎﺭﺍﺗﻬﺎ .................................................... ﻓﻰ ﺍﻟﻔﺘﺮﺓ‬ ‫3 ﺍﻟﺪﺍﻟﺔ ﺩ، ﺣﻴﺚ ﺩ)ﺱ( = ﺱ٢ – ٦ ﺱ + ٩ ﻣﻮﺟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ‬ ‫4 ﺍﻟﺪﺍﻟﺔ ﺩ، ﺣﻴﺚ ﺩ)ﺱ( = ﺱ – ٢ ﻣﻮﺟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ‬ ‫5 ﺍﻟﺪﺍﻟﺔ ﺩ، ﺣﻴﺚ ﺩ)ﺱ( = ٣ – ﺱ ﺳﺎﻟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ‬ ‫....................................................‬ ‫....................................................‬ ‫....................................................‬ ‫....................................................‬ ‫6 ﺍﻟﺪﺍﻟﺔ ﺩ، ﺣﻴﺚ ﺩ)ﺱ( = - )ﺱ – ١( )ﺱ +٢( ﻣﻮﺟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ‬ ‫7 ﺍﻟﺪﺍﻟﺔ ﺩ، ﺣﻴﺚ ﺩ)ﺱ( = ﺱ٢ + ٤ ﺱ – ٥ ﺳﺎﻟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ‬ ‫....................................................‬ ‫....................................................‬ ‫8 ﺍﻟﺸﻜﻞ ﺍﻟﻤﺮﺳﻮﻡ ﻳﻤﺜﻞ ﺩﺍﻟﺔ ﻣﻦ ﺍﻟﺪﺭﺟﺔ ﺍﻷﻭﻟﻰ ﻓﻰ ﺱ:‬ ‫أ ﺩ)ﺱ( ﻣﻮﺟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ ....................................................‬ ‫ب‬ ‫ﺩ)ﺱ( ﺳﺎﻟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ ....................................................‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫9 ﺍﻟﺸﻜﻞ ﺍﻟﻤﺮﺳﻮﻡ ﻳﻤﺜﻞ ﺩﺍﻟﺔ ﻣﻦ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ ﻓﻰ ﺱ:‬ ‫أ ﺩ)ﺱ( = ٠ ﻋﻨﺪﻣﺎ ﺱ ∋ ....................................................‬ ‫ب‬ ‫ﺩ)ﺱ( < ٠ ﻋﻨﺪﻣﺎ ﺱ ∋ ....................................................‬ ‫ﺟ‬ ‫ﺩ)ﺱ( > ٠ ﻋﻨﺪﻣﺎ ﺱ ∋ ....................................................‬ ‫¯‬ ‫−‬ ‫¯‬ ‫− − −‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬
  • 20. ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫01 ﻓﻰ ﺍﻟﺘﻤﺎﺭﻳﻦ ﻣﻦ أ ﺇﻟﻰ ن ﻋﻴﻦ ﺇﺷﺎﺭﺓ ﻛﻞ ﻣﻦ ﺍﻟﺪﻭﺍﻝ ﺍﻵﺗﻴﺔ:‬ ‫ب ﺩ)ﺱ( = ٢ﺱ‬ ‫.......................................‬ ‫أ ﺩ)ﺱ( = ٢‬ ‫د ﺩ)ﺱ( =٢ﺱ+٤‬ ‫.......................................‬ ‫ﺟ ﺩ)ﺱ( = - ٣ﺱ‬ ‫.......................................‬ ‫و ﺩ)ﺱ( = ﺱ‬ ‫ح ﺩ)ﺱ( = ﺱ٢ – ٤‬ ‫.......................................‬ ‫.......................................‬ ‫ﻫ ﺩ)ﺱ( =٣ – ٢ﺱ‬ ‫٢‬ ‫ز ﺩ)ﺱ( = ٢ﺱ‬ ‫ط ﺩ)ﺱ( = ١ – ﺱ‬ ‫.......................................‬ ‫ى ﺩ)ﺱ( = )ﺱ – ٢( )ﺱ + ٣(‬ ‫.......................................‬ ‫......................................‬ ‫ل ﺩ)ﺱ( = ﺱ٢– ﺱ – ٢‬ ‫.......................................‬ ‫.......................................‬ ‫ن ﺩ)ﺱ( = - ٤ ﺱ٢ + ٠١ ﺱ – ٥٢‬ ‫.......................................‬ ‫٢‬ ‫.......................................‬ ‫.......................................‬ ‫٢‬ ‫ك ﺩ)ﺱ( = )٢ ﺱ – ٣(‬ ‫م ﺩ)ﺱ( = ﺱ٢– ٨ ﺱ + ٦١‬ ‫٢‬ ‫.......................................‬ ‫11 ﺍﺭﺳﻢ ﻣﻨﺤﻨﻰ ﺍﻟﺪﺍﻟﺔ ﺩ)ﺱ( = ﺱ٢ – ٩ ﻓﻰ ﺍﻟﻔﺘﺮﺓ ] - ٣، ٤ [، ﻭﻣﻦ ﺍﻟﺮﺳﻢ ﻋﻴﻦ ﺇﺷﺎﺭﺓ ﺩ)ﺱ(.‬ ‫21 ﺍﺭﺳﻢ ﻣﻨﺤﻨﻰ ﺍﻟﺪﺍﻟﺔ ﺩ)ﺱ( = – ﺱ٢ + ٢ ﺱ + ٤ ﻓﻰ ﺍﻟﻔﺘﺮﺓ ]- ٣، ٥[، ﻭﻣﻦ ﺍﻟﺮﺳﻢ ﻋﻴﻦ ﺇﺷﺎﺭﺓ ﺩ)ﺱ(.‬ ‫31 ﺍﻛﺘﺸﻒ ﺍﻟﺨﻄﺄ: ﺇﺫﺍ ﻛﺎﻧﺖ ﺩ)ﺱ( = ﺱ + ١، ﺭ)ﺱ( = ١ – ﺱ٢ ﻓﻌﻴﻦ ﺍﻟﻔﺘﺮﺓ ﺍﻟﺘﻰ ﺗﻜﻮﻥ ﻓﻴﻬﺎ ﺍﻟﺪﺍﻟﺘﺎﻥ‬ ‫ﻣﻮﺟﺒﺘﻴﻦ ﻣﻌﺎ.‬ ‫ً‬ ‫¯‬ ‫¯‬ ‫ﺗﺠﻌﻞ ﺩ)ﺱ( = ٠‬ ‫ﺱ=-١‬ ‫ﺩ)ﺱ( ﻣﻮﺟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ [- ١، ∞]،‬ ‫ﺗﺠﻌﻞ ﺭ)ﺱ( = ٠‬ ‫ﺱ=!١‬ ‫ﺭ)ﺱ( ﻣﻮﺟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ [- ١، ١]‬ ‫ﻟﺬﻟﻚ ﻓﺈﻥ ﺍﻟﺪﺍﻟﺘﻴﻦ ﺗﻜﻮﻧﺎﻥ ﻣﻮﺟﺒﺘﻴﻦ ﻣﻌﺎ ﻓﻰ ﺍﻟﻔﺘﺮﺓ‬ ‫ً‬ ‫[- ١، ∞] ∪ [- ١، ١] = [- ١، ∞]‬ ‫ﺗﺠﻌﻞ ﺩ)ﺱ( = ٠‬ ‫ﺱ=-١‬ ‫ﺩ)ﺱ( ﻣﻮﺟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ [- ١، ∞]،‬ ‫ﺗﺠﻌﻞ ﺭ)ﺱ( = ٠‬ ‫ﺱ=!١‬ ‫ﺭ)ﺱ( ﻣﻮﺟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ [- ١، ١]‬ ‫ﻟﺬﻟﻚ ﻓﺈﻥ ﺍﻟﺪﺍﻟﺘﻴﻦ ﺗﻜﻮﻧﺎﻥ ﻣﻮﺟﺒﺘﻴﻦ ﻣﻌﺎ ﻓﻰ ﺍﻟﻔﺘﺮﺓ‬ ‫ً‬ ‫[- ١، ∞] ∩ [- ١، ١] = [- ١، ١]‬ ‫ﺃﻯ ﺍﻹﺟﺎﺑﺘﻴﻦ ﻳﻜﻮﻥ ﺻﺤﻴﺤﺎ? ﻣﺜﻞ ﻛﻼ ﻣﻦ ﺍﻟﺪﺍﻟﺘﻴﻦ ﺑﻴﺎﻧﻴﺎ ﻭﺗﺄﻛﺪ ﻣﻦ ﺻﺤﺔ ﺍﻹﺟﺎﺑﺔ.‬ ‫ً ِّ ًّ‬ ‫ًّ‬ ‫..................................................................................................................................................................................................................................‬ ‫41 ﻣﻨﺎﺟﻢ ﺍﻟﺬﻫﺐ: ﻓﻰ ﺍﻟﻔﺘﺮﺓ ﻣﻦ ﻋﺎﻡ ٠٩٩١ ﺇﻟﻰ ٠١٠٢ ﻛﺎﻥ ﺇﻧﺘﺎﺝ ﺃﺣﺪ ﻣﻨﺎﺟﻢ ﺍﻟﺬﻫﺐ ﻣﻘﺪﺭﺍ ﺑﺎﻷﻟﻒ ﺃﻭﻗﻴﺔ‬ ‫ً‬ ‫ﻳﺘﺤﺪﺩ ﺑﺎﻟﺪﺍﻟﺔ ﺩ : ﺩ)ﻥ( = ٢١ ﻥ٢ - ٦٩ ﻥ + ٠٨٤ ﺣﻴﺚ ﻥ ﻋﺪﺩ ﺍﻟﺴﻨﻮﺍﺕ، ﺩ)ﻥ( ﺍﻧﺘﺎﺝ ﺍﻟﺬﻫﺐ‬ ‫: ﺍﺑﺤﺚ ﺇﺷﺎﺭﺓ ﺩﺍﻟﺔ ﺍﻹﻧﺘﺎﺝ ﺩ. ...........................................................................................................................................................‬ ‫: ﺧﻼﻝ ﺍﻷﻋﻮﺍﻡ ﻣﻦ ٠٩٩١ﺇﻟﻰ ٠١٠٢ ﻓﻰ ﺃﻯ ﺍﻷﻋﻮﺍﻡ ﻛﺎﻥ ﺇﻧﺘﺎﺝ ﺍﻟﺬﻫﺐ ﻳﺘﻨﺎﻗﺺ? .................................................‬ ‫: ﺧﻼﻝ ﺍﻷﻋﻮﺍﻡ ﻣﻦ ٠٩٩١ﺇﻟﻰ ٠١٠٢ ﻓﻰ ﺃﻯ ﺍﻷﻋﻮﺍﻡ ﻛﺎﻥ ﺇﻧﺘﺎﺝ ﺍﻟﺬﻫﺐ ﻳﺘﺰﺍﻳﺪ? ....................................................‬ ‫‪M‬‬ ‫−‬
  • 21. ‫ﻣﺘﺒﺎﻳﻨﺎت اﻟﺪرﺟﺔ اﻟﺜﺎﻧﻴﺔ‬ ‫1-6‬ ‫‪Quadratic Inequalities‬‬ ‫¯‬ ‫1 ﺱ٢ ‪٩ H‬‬ ‫:‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫2 ﺱ٢ - ١ ‪٠ H‬‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫3 ٢ﺱ – ﺱ٢ > ٠‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫4 ﺱ٢ + ٥ ‪١ H‬‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫5 )ﺱ - ٢( )ﺱ - ٥( > ٠‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫6 ﺱ )ﺱ + ٢( - ٣ ‪٠ H‬‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫7 )ﺱ - ٢(٢ ‪٥ - H‬‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫8 ٥ – ٢ﺱ ‪ H‬ﺱ‬ ‫٢‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫9 ﺱ٢ ‪ ٦ G‬ﺱ – ٩‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫01 ٣ ﺱ٢ ‪ ١١ H‬ﺱ + ٤‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫11 ﺱ٢ - ٤ ﺱ + ٤ ‪٠ G‬‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫21 ٧ + ﺱ٢ - ٤ ﺱ > ٠‬ ‫..........................................................................................................................................................................‬ ‫..........................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
  • 22. ‫ﺗﻤﺎﺭﻳﻦ ﻋﺎﻣﺔ‬ ‫‪k‬‬ ‫‪:IÉ£©ªdG äÉHÉLE’G ø«H øe áë«ë°üdG áHÉLE’G ôàNG :’hCG‬‬ ‫1 ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٦ ﺱ + ٩ = ٠ ﻓﻰ ﺡ ﻫﻰ :‬ ‫ﺟ }-٣، ٣{‬ ‫ب }٣{‬ ‫أ }-٣{‬ ‫..............................................................................................................‬ ‫2 ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ٤ = ٠ ﻫﻰ :‬ ‫ب }٢{‬ ‫أ }-٢{‬ ‫3 ﺃﺑﺴﻂ ﺻﻮﺭﺓ ﻟﻠﻤﻘﺪﺍﺭ )١ – ﺕ(٤ ﻫﻮ :‬ ‫ب ٤‬ ‫أ -٤‬ ‫د ‪z‬‬ ‫............................................................................................................................................‬ ‫ﺟ }-٢، ٢{‬ ‫د }-٢ﺕ، ٢ﺕ{‬ ‫...................................................................................................................................................‬ ‫ﺟ -٤ ﺕ‬ ‫4 ﺇﺫﺍ ﻛﺎﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٤ﺱ + ﻙ = ٠ ﺣﻘﻴﻘﻴﻴﻦ ﻭﻣﺨﺘﻠﻔﻴﻦ ﻓﺈﻥ:‬ ‫ﺟ ﻙ=٤‬ ‫ب ﻙ>٤‬ ‫أ ﻙ<٤‬ ‫د ٤ﺕ‬ ‫..................................................................................‬ ‫5 ﺇﺫﺍ ﻛﺎﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٢١ﺱ + ﻡ = ٠ ﻣﺘﺴﺎﻭﻳﻴﻦ ﻓﺈﻥ ﻡ ﺗﺴﺎﻭﻯ:‬ ‫ﺟ ٦‬ ‫ب -٦‬ ‫أ -٦٣‬ ‫د ﻙ‪٤G‬‬ ‫..............................................................................‬ ‫د ٦٣‬ ‫6 ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ ٢ – ٣ﺕ ، ٢ + ٣ﺕ ﻫﻰ :‬ ‫أ ﺱ٢ + ٤ﺱ + ٣١ = ٠ ب ﺱ٢ – ٤ﺱ + ٣١ = ٠ ﺟ ﺱ٢ + ٤ﺱ – ٣١ = ٠ د ﺱ٢ – ٤ﺱ – ٣١ = ٠‬ ‫...........................................................................................................‬ ‫7 ﺇﺫﺍ ﻛﺎﻧﺖ ﺩ : ]- ٢ ، ٤[ # ‪ I‬ﺣﻴﺚ ﺩ)ﺱ( = ٢ – ﺱ ﻓﺈﻥ ﺇﺷﺎﺭﺓ ﺍﻟﺪﺍﻟﺔ ﺩ ﺳﺎﻟﺒﺔ ﻓﻰ:‬ ‫د [٢ ، ٤[‬ ‫ﺟ ]٢ ، ٤[‬ ‫ب ]- ٢ ، ٢[‬ ‫أ ]-٢ ، ٢]‬ ‫8 ﺇﺫﺍ ﻛﺎﻥ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ) ﻡ + ٢( ﺱ + ٣ = ٠ ﻣﻌﻜﻮﺳﺎ ﺟﻤﻌﻴﺎ ﻟﻠﺠﺬﺭ ﺍﻵﺧﺮ ﻓﺈﻥ ﻡ ﺗﺴﺎﻭﻯ:‬ ‫ً‬ ‫ًّ‬ ‫د‬ ‫٣‬ ‫ﺟ ٢‬ ‫ب -٢‬ ‫أ -٣‬ ‫9 ﺇﺫﺍ ﻛﺎﻥ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ٢ ﺱ٢ + ٧ ﺱ + ﻙ = ٠ ﻫﻮ ﺍﻟﻤﻌﻜﻮﺱ ﺍﻟﻀﺮﺑﻰ ﻟﻠﺠﺬﺭ ﺍﻵﺧﺮ ﻓﺈﻥ ﻙ ﺗﺴﺎﻭﻯ:‬ ‫د ٧‬ ‫ﺟ ٢‬ ‫ب -٢‬ ‫أ -٧‬ ‫01 ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺔ ﺱ٢ + ﺱ – ٢ > ٠ ﻫﻰ :‬ ‫ب ]- ٢ ، ١[‬ ‫أ [- ٢ ، ١]‬ ‫ﺟ ﺡ – ]-٢ ، ١[‬ ‫د ﺡ – [-٢ ، ١]‬ ‫‪O á«©«HôJ ádGód ≈fÉ«ÑdG π«ãªàdG πHÉ≤ªdG πμ°ûdG πãªj :Ék«fÉK‬‬ ‫11 ﺃﻛﻤﻞ ﻣﺎﻳﺄﺗﻰ :‬ ‫أ ﻣﺪﻯ ﺍﻟﺪﺍﻟﺔ ﺩ ﻫﻮ .............................................................................................‬ ‫ب‬ ‫ﺍﻟﻘﻴﻤﺔ ﺍﻟﻌﻈﻤﻰ ﻟﻠﺪﺍﻟﺔ ﺩ = ............................................................................‬ ‫ﺟ‬ ‫ﻧﻮﻉ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺩ )ﺱ( = ٠ .............................................................‬ ‫د ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺩ)ﺱ( = ٠ ﻫﻰ ..................................................‬ ‫ﻫ ﺩ)ﺱ( < ٠ ﻋﻨﺪﻣﺎ ﺱ ∋ ...............................................................................‬ ‫و‬ ‫ﺩ)ﺱ( > ٠ ﻋﻨﺪﻣﺎ ﺱ ∋ ...............................................................................‬ ‫ز‬ ‫ﺩ)ﺱ( = ٠ ﻋﻨﺪﻣﺎ ﺱ = ..................................................................................‬ ‫‪M‬‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫− −‬
  • 23. ‫ﺗﻤﺎﺭﻳﻦ ﻋﺎﻣﺔ‬ ‫21 ﺍﻛﺘﺐ ﻗﺎﻋﺪﺓ ﺍﻟﺪﺍﻟﺔ ﺍﻟﺘﻰ ﺗﻤﺮ ﺑﺎﻟﻨﻘﺎﻁ )- ٣، ٠( ، )٢، ٠( ، )٢، ١(‬ ‫..................................................................................................................................................................................................................................‬ ‫31 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ :‬ ‫أ ﺍﻛﺘﺐ ﻧﻘﺎﻁ ﺗﻘﺎﻃﻊ ﻣﻨﺤﻨﻰ ﺍﻟﺪﻭﺍﻝ ﺍﻟﺘﻰ ﻗﺎﻋﺪﺗﻬﺎ ﺹ = ﺱ٢ ، ﺹ = ﺱ‬ ‫.......................................................................................................................................................................................................................‬ ‫ب ﺍﻛﺘﺐ ﻧﻘﺎﻁ ﺗﻘﺎﻃﻊ ﻣﻨﺤﻨﻰ ﺍﻟﺪﻭﺍﻝ ﺍﻟﺘﻰ ﻗﺎﻋﺪﺗﻬﺎ ﺹ = - ﺱ٢، ﺹ = - ﺱ ﻣﺎﺫﺍ ﺗﻼﺣﻆ ? ﻓﺴﺮ ﺇﺟﺎﺑﺘﻚ.‬ ‫.......................................................................................................................................................................................................................‬ ‫‪k‬‬ ‫‪á«JB’G á∏Ä°SC’G øY ÖLCG :ÉãdÉK‬‬ ‫41 ﺑﻴﻦ ﻧﻮﻉ ﺟﺬﺭﻯ ﻛﻞ ﻣﻌﺎﺩﻟﺔ ﻣﻤﺎ ﻳﺄﺗﻰ، ﺛﻢ ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﻛﻞ ﻣﻌﺎﺩﻟﺔ.‬ ‫ب )ﺱ – ١(٢ = ٤‬ ‫أ ﺱ٢ – ٢ﺱ = ٠‬ ‫........................................................‬ ‫د ﺱ٢ + ٣ﺱ – ٨٢ = ٠‬ ‫........................................................‬ ‫ﺟ ﺱ٢ – ٦ ﺱ+ ٩ = ٠‬ ‫........................................................‬ ‫........................................................‬ ‫ﻫ ٦ﺱ )ﺱ – ١( = ٦ – ﺱ‬ ‫........................................................‬ ‫51 ﺣﻞ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻘﺎﻧﻮﻥ ﺍﻟﻌﺎﻡ ﻣﻘﺮﺑﺎ ﺍﻟﻨﺎﺗﺞ ﻷﻗﺮﺏ ﺭﻗﻤﻴﻦ ﻋﺸﺮﻳﻴﻦ.‬ ‫ً‬ ‫ب ﺱ٢ – ٣)ﺱ -٢( = ٥‬ ‫أ ﺱ٢ + ٤ﺱ + ٢ = ٠‬ ‫........................................................‬ ‫........................................................‬ ‫61 ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻵﺗﻴﺔ ﻓﻰ ﻣﺠﻤﻮﻋﺔ ﺍﻷﻋﺪﺍﺩ ﻛﺒﺔ .‬ ‫ﺍﻟﻤﺮ‬ ‫ب ﺱ٢ + ٢ﺱ + ٢ = ٠‬ ‫أ ﺱ٢ + ٩ = ٠‬ ‫........................................................‬ ‫ﺟ ﺱ٢ + ٤ﺱ + ٥ = ٠‬ ‫........................................................‬ ‫71 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ ،C‬ﺏ ﻓﻰ ﻛﻞ ﻣﻤﺎ ﻳﺄﺗﻰ :‬ ‫أ )٧ – ٣ﺕ( – )٢ + ﺕ( = ‪ + C‬ﺏ ﺕ‬ ‫ﺟ ٢ ٠١ﺕ = ‪ + C‬ﺏ ﺕ‬ ‫+‬ ‫........................................................‬ ‫ب )٢ – ٥ﺕ()٣ + ﺕ( = ‪ + C‬ﺏ ﺕ‬ ‫‬‫د ٦١ -٤ﺕ = ‪ + C‬ﺏ ﺕ‬ ‫ﺕ‬ ‫81 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻡ ﻓﻰ ﻛﻞ ﻣﻤﺎ ﻳﺄﺗﻰ :‬ ‫٢‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ٢ﺱ + ﻡ ﺱ + ٨١ = ٠ ﻣﺘﺴﺎﻭﻳﻴﻦ ..............................................................................................‬ ‫٢‬ ‫ب‬ ‫ﺇﺫﺍ ﻛﺎﻥ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ + ٣ ﺱ + ﻙ = ٠ ﺿﻌﻒ ﺍﻟﺠﺬﺭ ﺍﻵﺧﺮ ..............................................................‬ ‫91 ﺍﺑﺤﺚ ﺇﺷﺎﺭﺓ ﺍﻟﺪﺍﻟﺔ ﺩ ﻓﻰ ﻛﻞ ﻣﻤﺎ ﻳﺄﺗﻰ :‬ ‫أ ﺩ)ﺱ( = ﺱ٢ – ٢ ﺱ – ٨‬ ‫........................................................‬ ‫02 ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺍﻟﺤﻞ ﻟﻜﻞ ﻣﻦ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ ﺍﻵﺗﻴﺔ :‬ ‫أ ﺱ٢ – ﺱ – ٢١ < ٠‬ ‫........................................................‬ ‫¯‬ ‫−‬ ‫¯‬ ‫ب ﺩ)ﺱ( = ٤ – ٣ﺱ – ﺱ‬ ‫........................................................‬ ‫ب ﺱ٢ – ٧ﺱ + ٠١ ‪٠ H‬‬ ‫........................................................‬ ‫٢‬
  • 24. ‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ‬ ‫‪k‬‬ ‫‪: Oó©àe øe QÉ«àNC’G :’hCG‬‬ ‫1 ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٤ﺱ = -٤ ﻓﻰ ﺡ ﻫﻰ:‬ ‫ب }٢{‬ ‫أ }-٢{‬ ‫2 ﺣﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺔ ﺱ٢ + ٩ < ٦ﺱ ﻓﻰ ﺡ ﻫﻰ:‬ ‫ب ﺡ – }٣{‬ ‫أ ﺡ‬ ‫.......................................................................................................................‬ ‫ﺟ }-٢، ٢{‬ ‫د ‪z‬‬ ‫............................................................................................................................................‬ ‫ﺟ [- ٣، ٣]‬ ‫3 ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ٢ﺱ٢ – ٥ﺱ + ٣ = ٠‬ ‫أ ﺣﻘﻴﻘﻴﺎﻥ ﻣﺘﺴﺎﻭﻳﺎﻥ ب ﺣﻘﻴﻘﻴﺎﻥ ﻣﺨﺘﻠﻔﺎﻥ‬ ‫د ﺡ – ]-٣، ٣[‬ ‫......................................................................................................................................................‬ ‫ﺟ ﻛﺒﺎﻥ‬ ‫ﻣﺮ‬ ‫4 ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ )١ + ﺕ(، )١ – ﺕ( ﻫﻰ :‬ ‫أ ﺱ٢ – ٢ﺱ + ٢ = ٠ ب ﺱ٢ + ٢ﺱ – ٢ = ٠ ﺟ ﺱ٢ + ٢ﺱ + ٢ = ٠‬ ‫د ﻛﺒﺎﻥ ﻭ ﻣﺘﺮﺍﻓﻘﺎﻥ‬ ‫ﻣﺮ‬ ‫.........................................................................................................‬ ‫د ﺱ٢ – ٢ﺱ – ٢ = ٠‬ ‫‪á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫5 ﺇﺫﺍ ﻛﺎﻥ )‪(٣ + C‬ﺱ٢ + )٢ – ‪ (C‬ﺱ + ٤ = ٠ ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ ‪ C‬ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻟﺤﺎﻻﺕ ﺍﻵﺗﻴﺔ:‬ ‫أ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﻣﻌﻜﻮﺱ ﺟﻤﻌﻰ ﻟﻠﺠﺬﺭ ﺍﻵﺧﺮ.‬ ‫.......................................................................................................................................................................................................................‬ ‫ب ﻣﺠﻤﻮﻉ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﻳﺴﺎﻭﻯ ٦.‬ ‫.......................................................................................................................................................................................................................‬ ‫6‬ ‫٢ ٢‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ ﻝ ، ﻡ ﻫﻤﺎ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٦ﺱ + ٤ = ٠ ﻓﺄﻭﺟﺪ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ ﻝ، ﻡ.‬ ‫.......................................................................................................................................................................................................................‬ ‫ب ﺍﺑﺤﺚ ﺇﺷﺎﺭﺓ ﺍﻟﺪﺍﻟﺔ ﺩ، ﺣﻴﺚ ﺩ)ﺱ( = ٨ – ٢ﺱ – ﺱ‬ ‫٢‬ ‫.......................................................................................................................................................................................................................‬ ‫7‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ٣ = ٥ﺱ ﺣﻘﻴﻘﻴﺎﻥ ﻣﺨﺘﻠﻔﺎﻥ، ﺛﻢ ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﻓﻰ ﺡ‬ ‫ﻣﻘﺮﺑﺎ ﺍﻟﻨﺎﺗﺞ ﻷﻗﺮﺏ ﺛﻼﺛﺔ ﺃﺭﻗﺎﻡ ﻋﺸﺮﻳﺔ.‬ ‫ً‬ ‫.........................................................................................................................................................................................................................‬ ‫ب ﺃﻭﺟﺪ ﺣﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺔ : ﺱ٢ – ٥ﺱ – ٤١ ‪٠ H‬‬ ‫.......................................................................................................................................................................................................................‬ ‫8 ﺗﻄ ﻴﻘﺎﺕ ﻓﻴ ﺎﺋﻴﺔ: ﺃُﻃْﻠﻖ ﺻﺎﺭﻭﺥ ﺭﺃﺳﻴﺎ ﺇﻟﻰ ﺃﻋﻠﻰ ﺑﺴﺮﻋﺔ ٨٩ ﻣﺘﺮﺍ/ﺛﺎﻧﻴﺔ، ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺍﻟﻤﺴﺎﻓﺔ‬ ‫ًّ‬ ‫ً‬ ‫٢‬ ‫ﺍﻟﻤﻘﻄﻮﻋﺔ ﻑ ﺑﺎﻟﻤﺘﺮ ﻭﺍﻟﺰﻣﻦ ﻥ ﺑﺎﻟﺜﺎﻧﻴﺔ ﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ : ﻑ = ٨٩ ﻥ – ٩٫٤ ﻥ ﻓﺄﻭﺟﺪ :‬ ‫أ ﺍﻟﻤﺴﺎﻓﺔ ﺍﻟﺘﻰ ﻳﻘﻄﻌﻬﺎ ﺍﻟﺼﺎﺭﻭﺥ ﻓﻰ ﺛﺎﻧﻴﺘﻴﻦ. ............................................................................................................................‬ ‫ب ﺍﻟﺰﻣﻦ ﺍﻟﺬﻯ ﻳﺴﺘﻐﺮﻗﻪ ﺍﻟﺼﺎﺭﻭﺥ ﺣﺘﻰ ﻳﻘﻄﻊ ﻣﺴﺎﻓﺔ ٤٫٠٧٤ ﻣﺘﺮﺍ. ﺑﻤﺎ ﺗﻔﺴﺮ ﻭﺟﻮﺩ ﺇﺟﺎﺑﺘﻴﻦ?‬ ‫ً‬ ‫‪M‬‬ ‫−‬
  • 25. ‫ﺍﺧﺘﺒﺎﺭ ﺗﺮﺍﻛﻤﻲ‬ ‫1 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻙ ﺍﻟﺘﻰ ﺗﺠﻌﻞ ﻟﻠﻤﻌﺎﺩﻟﺔ ٣ﺱ٢ + ٤ﺱ + ﻙ = ٠ ﺟﺬﺭﻳﻦ :‬ ‫أ ﺣﻘﻴﻘﻴﻴﻦ ﻣﺘﺴﺎﻭﻳﻴﻦ ......................................‬ ‫ب‬ ‫ﺣﻘﻴﻘﻴﻴﻦ ﻣﺨﺘﻠﻔﻴﻦ ......................................‬ ‫ﻛﺒﻴﻦ ......................................‬ ‫ﺟ ﻣﺮ‬ ‫2 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻙ ﺍﻟﺘﻰ ﺗﺠﻌﻞ:‬ ‫٢‬ ‫أ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ – ﻙ ﺱ + ﻙ + ٢ = ٠ ﺿﻌﻒ ﺍﻟﺠﺬﺭ ﺍﻵﺧﺮ. .......................................................................‬ ‫٢‬ ‫ب‬ ‫ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ – ﻙ ﺱ + ٨ = ٠ ﻳﺰﻳﺪ ﻋﻦ ﺍﻟﺠﺬﺭ ﺍﻵﺧﺮ ﺑﻤﻘﺪﺍﺭ ٢. ......................................................‬ ‫ﺟ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ﻙ ﺱ + ٣ = ٠ ﻳﺰﻳﺪ ﻋﻦ ﺍﻟﻤﻌﻜﻮﺱ ﺍﻟﻀﺮﺑﻰ ﻟﻠﺠﺬﺭ ﺍﻵﺧﺮ ﺑﻤﻘﺪﺍﺭ ١.‬ ‫3 ﺇﺫﺍ ﻛﺎﻥ ﻝ، ﻡ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٣ﺱ + ٢ = ٠ ﻓﺄﻭﺟﺪ ﻣﻌﺎﺩﻟﺔ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ :‬ ‫ﺟ ١ ١‬ ‫د ﻝ + ﻡ، ﻝ ﻡ‬ ‫ب ﻝ + ١، ﻡ + ١‬ ‫أ ٣ ﻝ، ٣ ﻡ‬ ‫ﻝ، ﻡ‬ ‫..................................................................................................................................................................................................................................‬ ‫١ ١‬ ‫4 ﺇﺫﺍ ﻛﺎﻥ ﻝ ، ﻡ ﻫﻤﺎ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ٦ﺱ٢ – ٥ ﺱ +١ = ٠ ﻓﻜﻮﻥ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ ﻝ، ﻡ.‬ ‫..................................................................................................................................................................................................................................‬ ‫5 ﺍﺭﺳﻢ ﻣﻨﺤﻨﻰ ﺍﻟﺪﺍﻟﺔﺩ، ﺣﻴﺚ ﺩ)ﺱ( = ﺱ٢– ٤ ﻓﻰ ﺍﻟﻔﺘﺮﺓ ]- ٣،٣[ ﻭﻣﻦ ﺍﻟﺮﺳﻢ ﻋﻴﻦ ﺇﺷﺎﺭﺓ ﺩ ﻓﻰ ﻫﺬه ﺍﻟﻔﺘﺮﺓ.‬ ‫6 ﺍﺭﺳﻢ ﻣﻨﺤﻨﻰ ﺍﻟﺪﺍﻟﺔ ﺩ، ﺣﻴﺚ ﺩ)ﺱ( = ٦ – ٥ﺱ – ٤ﺱ٢ ﻓﻰ ﺍﻟﻔﺘﺮﺓ ]-٣،٢[ ﻭﻣﻦ ﺍﻟﺮﺳﻢ ﻋﻴﻦ ﺇﺷﺎﺭﺓ ﺩ ﻓﻰ ﻫﺬه ﺍﻟﻔﺘﺮﺓ.‬ ‫7 ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺍﻟﺤﻞ ﻟﻠﻤﺘﺒﺎﻳﻨﺎﺕ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻵﺗﻴﺔ:‬ ‫ﺟ )ﺱ - ٢(٢ ‪٩ - G‬‬ ‫ب ﺱ٢ - ٦ ﺱ < - ٥‬ ‫أ ﺱ٢ + ٤ ﺱ + ٤ > ٠‬ ‫.................................................................‬ ‫د ٣ – ٢ﺱ ‪ G‬ﺱ‬ ‫..................................................................‬ ‫.................................................................‬ ‫و ٢ﺱ٢ - ٧ﺱ ‪١٥ H‬‬ ‫ﻫ ﺱ٢ ‪١٠ H‬ﺱ – ٥٢‬ ‫٢‬ ‫.................................................................‬ ‫..................................................................‬ ‫.................................................................‬ ‫8 ﺃﻋﻤﺎﻝ ﺗﺠﺎ ﺔ: ﺇﺫﺍ ﻛﺎﻥ ﻋﺪﺩ ﺍﻟﻮﺣﺪﺍﺕ ﺍﻟﻤﻨﺘﺠﺔ ﻭﺍﻟﻤﺒﺎﻋﺔ ﻣﻦ ﺳﻠﻌﺔ ﻣﻌﻴﻨﺔ ﻓﻰ ﺍﻷﺳﺒﻮﻉ ﻫﻰ ﺱ ﻣﻠﻴﻮﻥ ﻭﺣﺪﺓ‬ ‫ﻛﺎﻥ ﺳﻌﺮ ﺑﻴﻊ ﺍﻟﻮﺣﺪﺓ ﻫﻮ ﻉ ﺣﻴﺚ ﻉ = ٢ – ﺱ، ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺘﻜﺎﻟﻴﻒ ﺍﻟﻜﻠﻴﺔ ﺍﻟﻼﺯﻣﺔ ﻹﻧﺘﺎﺝ ﺱ ﻣﻠﻴﻮﻥ ﻭﺣﺪﺓ‬ ‫ﻭ‬ ‫ﻓﻰ ﺍﻷﺳﺒﻮﻉ ﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ ﺕ = )٣٫٠ + ٥٫٠ﺱ( ﻣﻠﻴﻮﻥ ﻭﺣﺪﺓ ﻓﺄﻭﺟﺪ :‬ ‫أ ﺩﺍﻟﺔ ﺍﻹﻳﺮﺍﺩ ﺍﻟﻜﻠﻰ )ﻯ( ..................................................................‬ ‫ب‬ ‫ﺩﺍﻟﺔ ﺍﻟﺮﺑﺢ )ﺭ( ..................................................................‬ ‫ﺟ‬ ‫ﺃﻭﺟﺪ ﺱ ﻋﻨﺪ ﻣﺴﺘﻮﻯ ﺭﺑﺢ ٢٫٠ ﻣﻠﻴﻮﻥ ﺟﻨﻴﻪ. ........................................................................................................................‬ ‫9 ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ٣ + ١ = C‬ﺕ  ،  ﺏ = - ١ – ﺕ، ﺟـ = - ٢ - ٣ + ﺕ ﻓﺄﺛﺒﺖ ﺃﻥ: ﺟـ - ﺏ = )‪ – C‬ﺏ(ﺕ‬ ‫:‬ ‫‪M‬‬ ‫ﺭﻗﻢ ﺍﻟﺴﺆﺍﻝ‬ ‫١‬ ‫ﺃ، ﺏ‬ ‫١-٣‬ ‫ﺭﻗﻢ ﺍﻟﺪﺭﺱ‬ ‫¯‬ ‫٢‬ ‫ﺟـ‬ ‫١- ٢‬ ‫−‬ ‫¯‬ ‫٣‬ ‫٤‬ ‫٥‬ ‫٦‬ ‫٧‬ ‫٨‬ ‫٩‬ ‫١- ٤‬ ‫١-٤‬ ‫١-٤‬ ‫١-٥‬ ‫١- ٥‬ ‫١-٦‬ ‫١-١‬ ‫١-٢‬
  • 26. ‫-‬ ‫‪IóMƒdG‬‬ ‫2‬ ‫ﺍﻟﺘﺸﺎﺑﻪ‬ ‫‪Similarity‬‬ ‫دروس اﻟﻮﺣﺪة‬ ‫ﺍﻟﺪﺭﺱ )٢ - ١(: ﺗﺸﺎﺑﻪ ﺍﻟﻤﻀﻠﻌﺎﺕ.‬ ‫ﺍﻟﺪﺭﺱ )٢ - ٢(: ﺗﺸﺎﺑﻪ ﺍﻟﻤﺜﻠﺜﺎﺕ.‬ ‫ﺍﻟﺪﺭﺱ )٢ - ٣(: ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﻣﺴﺎﺣﺘﻰ ﺳﻄﺤﻰ ﻣﻀﻠﻌﻴﻦ ﻣﺘﺸﺎﺑﻬﻴﻦ.‬ ‫ﺍﻟﺪﺭﺱ )٢ - ٤(: ﺗﻄﺒﻴﻘﺎﺕ ﺍﻟﺘﺸﺎﺑﻪ ﻓﻰ ﺍﻟﺪﺍﺋﺮﺓ.‬ ‫‪ïM‬‬ ‫−‬
  • 27. ‫ﺗﺸﺎﺑﻪ اﻟﻤﻀﻠﻌﺎت‬ ‫2-1‬ ‫‪Similarity of Polygons‬‬ ‫1 ﺑﻴﻦ ﺃﻳﺎ ﻣﻦ ﺃﺯﻭﺍﺝ ﺍﻟﻤﻀﻠﻌﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ ﺗﻜﻮﻥ ﻣﺘﺸﺎﺑﻬﺔ، ﻭﺍﻛﺘﺐ ﺍﻟﻤﻀﻠﻌﺎﺕ ﺍﻟﻤﺘﺸﺎﺑﻬﺔ ﺑﺘﺮﺗﻴﺐ‬ ‫ًّ‬ ‫ﺍﻟﺮﺅﻭﺱ ﺍﻟﻤﺘﻨﺎﻇﺮﺓ، ﻭﺣﺪﺩ ﻣﻌﺎﻣﻞ ﺍﻟﺘﺸﺎﺑﻪ )ﺍﻷﻃﻮﺍﻝ ﻣﻘﺪﺭﺓ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ(.‬ ‫‪C‬‬ ‫ب‬ ‫أ‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪E c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫...........................................................................................‬ ‫...........................................................................................‬ ‫...........................................................................................‬ ‫.................................................................................‬ ‫ﺟ‬ ‫د‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪E‬‬ ‫...........................................................................................‬ ‫...........................................................................................‬ ‫...........................................................................................‬ ‫...........................................................................................‬ ‫2 ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻤﻀﻠﻊ ‪ C‬ﺏ ﺟـ ‪ + E‬ﺍﻟﻤﻀﻠﻊ ﺱ ﺹ ﻉ ﻝ، ﺃﻛﻤﻞ:‬ ‫أ ‪C‬ﺏ‬ ‫ﺏ ﺟـ = ﺹ ﻉ‬ ‫ﺟ ﺏ ﺟـ + ﺹ ﻉ‬ ‫ﺹﻉ =‬ ‫................‬ ‫ب ‪C‬ﺏ*ﻉﻝ=ﺱﺹ*‬ ‫................ + ﻝ ﺱ‬ ‫ﻝﺱ‬ ‫ﻣﺤﻴﻂ ﺍﻟﻤﻀﻠﻊ‬ ‫د‬ ‫ﻣﺤﻴﻂ ﺍﻟﻤﻀﻠﻊ.........................‬ ‫.........................‬ ‫.........................‬ ‫ﺱﺹ‬ ‫=‬ ‫‪C‬ﺏ‬ ‫3 ﺍﻟﻤﻀﻠﻊ ‪ C‬ﺏ ﺟـ ‪ + E‬ﺍﻟﻤﻀﻠﻊ ﺱ ﺹ ﻉ ﻝ. ﻓﺈﺫﺍ ﻛﺎﻥ: ‪ C‬ﺏ = ٢٣ﺳﻢ، ﺏ ﺟـ = ٠٤ﺳﻢ، ﺱ ﺹ = ٣ﻡ - ١،‬ ‫ﺹ ﻉ = ٣ﻡ +١. ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻡ ﺍﻟﻌﺪﺩﻳﺔ. ................................................................................................................................................‬ ‫4 ﻣﺴﺘﻄﻴﻞ ﺑﻌﺪﺍه ٠١ﺳﻢ، ٦ﺳﻢ. ﺃﻭﺟﺪ ﻣﺤﻴﻂ ﻭﻣﺴﺎﺣﺔ ﻣﺴﺘﻄﻴﻞ ﺁﺧﺮ ﻣﺸﺎﺑﻪ ﻟﻪ ﺇﺫﺍ ﻛﺎﻥ:‬ ‫ب ﻣﻌﺎﻣﻞ ﺍﻟﺘﺸﺎﺑﻪ ٤٫٠‬ ‫أ ﻣﻌﺎﻣﻞ ﺍﻟﺘﺸﺎﺑﻪ ٣‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
  • 28. ‫‪ï‬‬ ‫5 ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ ﺍﻟﻤﻀﻠﻊ ﻡ١ + ﺍﻟﻤﻀﻠﻊ ﻡ٢ + ﺍﻟﻤﻀﻠﻊ ﻡ٣.‬ ‫ﺃﻭﺟﺪ ﻣﻌﺎﻣﻞ ﺗﺸﺎﺑﻪ ﻛﻞ ﻣﻦ ﺍﻟﻤﻀﻠﻊ ﻡ١، ﺍﻟﻤﻀﻠﻊ ﻡ٢ ﻟﻠﻤﻀﻠﻊ ﻡ٣.‬ ‫ب‬ ‫أ‬ ‫...........................................................................................‬ ‫...........................................................................................‬ ‫...........................................................................................‬ ‫...........................................................................................‬ ‫6 ﺍﻟﻤﻀﻠﻌﺎﺕ ﺍﻟﺜﻼﺛﺔ ﺍﻟﺘﺎﻟﻴﺔ ﻣﺘﺸﺎﺑﻬﺔ. ﺃﻭﺟﺪ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻌﺪﺩﻳﺔ ﻟﻠﺮﻣﺰ ﺍﻟﻤﺴﺘﺨﺪﻡ ﻓﻰ ﺍﻟﻘﻴﺎﺱ.‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫..................................................................................................................................................................................................................................‬ ‫7 ﻋﻠﺒﺔ ﻋﻠﻰ ﺷﻜﻞ ﻣﺴﺘﻄﻴﻞ ﺫﻫﺒﻰ ﻃﻮﻟﻪ ٢٫٦١ﺳﻢ. ﺍﺣﺴﺐ ﻋﺮﺽ ﺍﻟﻌﻠﺒﺔ ﻷﻗﺮﺏ ﺳﻨﺘﻴﻤﺘﺮ.‬ ‫..................................................................................................................................................................................................................................‬ ‫8 ﻣﺴﺘﻄﻴﻼﻥ ﻣﺘﺸﺎﺑﻬﺎﻥ ﺑﻌﺪﺍ ﺍﻷﻭﻝ ٨ﺳﻢ، ٢١ﺳﻢ، ﻭﻣﺤﻴﻂ ﺍﻟﺜﺎﻧﻰ ٠٠٢ﺳﻢ. ﺃﻭﺟﺪ ﻃﻮﻝ ﺍﻟﻤﺴﺘﻄﻴﻞ ﺍﻟﺜﺎﻧﻰ ﻭﻣﺴﺎﺣﺘﻪ.‬ ‫ُ‬ ‫..................................................................................................................................................................................................................................‬ ‫ﻧﺸﺎط‬ ‫9 ﻫﻨﺪﺳﺔ ﻣﻌﻤﺎ ﺔ: ﻳﻮﺿﺢ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ ﻣﺨﻄﻄًﺎ‬ ‫ﻹﺣﺪﻯ ﺍﻟﻮﺣﺪﺍﺕ ﺍﻟﺴﻜﻨﻴﺔ ﺑﻤﻘﻴﺎﺱ ﺭﺳﻢ ١ : ٠٥١ ﺃﻭﺟﺪ:‬ ‫......................................................‬ ‫أ ﺃﺑﻌﺎﺩ ﺣﺠﺮﺓ ﺍﻻﺳﺘﻘﺒﺎﻝ.‬ ‫.................................................................‬ ‫ب ﺃﺑﻌﺎﺩ ﺣﺠﺮﺓ ﺍﻟﻨﻮﻡ.‬ ‫......................................................‬ ‫ﺟ ﻣﺴﺎﺣﺔ ﺣﺠﺮﺓ ﺍﻟﻤﻌﻴﺸﺔ.‬ ‫د ﻣﺴﺎﺣﺔ ﺍﻟﻮﺣﺪﺓ ﺍﻟﺴﻜﻨﻴﺔ. ......................................................‬ ‫‪ïM‬‬ ‫−‬ ‫¯‬ ‫¯‬ ‫¯‬ ‫‪M‬‬
  • 29. ‫ﺗﺸﺎﺑﻪ اﻟﻤﺜﻠﺜﺎت‬ ‫2-2‬ ‫‪Similarity Of Triangles‬‬ ‫1 ﺍﺫﻛﺮ ﺃﻯ ﺍﻟﺤﺎﻻﺕ ﻳﻜﻮﻥ ﻓﻴﻬﺎ ﺍﻟﻤﺜﻠﺜﺎﻥ ﻣﺘﺸﺎﺑﻬﻴﻦ، ﻭﻓﻰ ﺣﺎﻟﺔ ﺍﻟﺘﺸﺎﺑﻪ ﺍﺫﻛﺮ ﺳﺒﺐ ﺍﻟﺘﺸﺎﺑﻪ.‬ ‫ﺟ‬ ‫ب‬ ‫‪C‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫أ‬ ‫‪c‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫‪c‬‬ ‫................................................................‬ ‫د‬ ‫................................................................‬ ‫................................................................‬ ‫‪E‬‬ ‫و‬ ‫ﻫ‬ ‫‪C‬‬ ‫................................................................‬ ‫................................................................‬ ‫2 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺮﻣﺰ ﺍﻟﻤﺴﺘﺨﺪﻡ ﻓﻰ ﺍﻟﻘﻴﺎﺱ:‬ ‫‪C‬‬ ‫ب‬ ‫أ‬ ‫‪C‬‬ ‫................................................................‬ ‫ﺟ‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫................................................................‬ ‫................................................................‬ ‫‪C‬‬ ‫: ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ‪ = E C‬ﺏ ﺟـ‬ ‫3‬ ‫: ﺃﻛﻤﻞ: 9‪ C‬ﺏ ﺟـ + 9 ........................... + 9‬ ‫...........................‬ ‫: ﺇﺫﺍ ﻛﺎﻥ ﺱ ، ﺹ، ﻉ، ﻝ،ﻡ، ﻥ ﻫﻰ ﺃﻃﻮﺍﻝ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ‬ ‫ﻭﺍﻟﻤﻌﻴﻨﺔ ﺑﺎﻟﺸﻜﻞ: ﻓﺄﻛﻤﻞ ﺍﻟﺘﻨﺎﺳﺒﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ:‬ ‫ﺱ‬ ‫ﺟ ﻡ‬ ‫ﻝ‬ ‫ب ﺱ‬ ‫ﻡ‬ ‫ﺱ‬ ‫أ‬ ‫ﻝ = ...............‬ ‫ﻉ = ...............‬ ‫ﻉ = ...............‬ ‫...............‬ ‫ﺱ‬ ‫ﻫ ............... =‬ ‫ﺱ‬ ‫¯‬ ‫و‬ ‫−‬ ‫¯‬ ‫...............‬ ‫ﺹ =‬ ‫ﺹ‬ ‫...............‬ ‫................................................................‬ ‫...............‬ ‫ﻝ‬ ‫ز ﺱ =‬ ‫ﻉ‬ ‫‪E‬‬ ‫...............‬ ‫د ﻝ‬ ‫............... = ﻝ‬ ‫...............‬ ‫ﻝ‬ ‫ح ﺱ =‬ ‫ﺹ‬