منتدى الرياضيات المتقدمة, profile picture
Uploaded byمنتدى الرياضيات المتقدمة
277 views

كتاب الانشطه - مصر- ترم اول - 2014

كتاب الانشطه - مصر- ترم اول - 2014

Embed presentation

Downloaded 14 times
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
‫‪ï‬‬ ‫4 ‪ C‬ﺏ ، ‪ E‬ﺟـ ﻭﺗﺮﺍﻥ ﻓﻰ ﺩﺍﺋﺮﺓ، ‪ C‬ﺏ ∩ ‪ E‬ﺟـ = }ﻫـ{ ﺣﻴﺚ ﻫـ ﺧﺎﺭﺝ ﺍﻟﺪﺍﺋﺮﺓ، ‪ C‬ﺏ = ٤ﺳﻢ، ‪ E‬ﺟـ = ٧ﺳﻢ،‬ ‫ﺏ ﻫـ = ٦ﺳﻢ. ﺃﺛﺒﺖ ﺃﻥ 9‪ E C‬ﻫـ + 9ﺟـ ﺏ ﻫـ، ﺛﻢ ﺃﻭﺟﺪ ﻃﻮﻝ ﺟـ ﻫـ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫5 ‪ C‬ﺏ ﺟـ، ‪ E‬ﻫـ ﻭ ﻣﺜﻠﺜﺎﻥ ﻣﺘﺸﺎﺑﻬﺎﻥ. ﺭﺳﻢ ‪ C‬ﺱ = ﺏ ﺟـ ﻟﻴﻘﻄﻌﻪ ﻓﻰ ﺱ، ﻭﺭﺳﻢ ‪ E‬ﺹ = ﻫـ ﻭ ﻟﻴﻘﻄﻌﻪ ﻓﻰ ﺹ.‬ ‫ﺃﺛﺒﺖ ﺃﻥ ﺏ ﺱ * ﺹ ﻭ = ﺟـ ﺱ * ﺹ ﻫـ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫6 ﻓﻰ ﺍﻟﻤﺜﻠﺚ ‪ C‬ﺏ ﺟـ، ‪ C‬ﺟـ < ‪ C‬ﺏ، ﻡ ∋ ‪ C‬ﺟـ ﺣﻴﺚ ‪ Cc)X‬ﺏ ﻡ( = ‪c) X‬ﺟـ( ﺃﺛﺒﺖ ﺃﻥ )‪ C‬ﺏ(٢ = ‪ C‬ﻡ * ‪ C‬ﺟـ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫7 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ‪ ،C‬ﺭﺳﻢ‬ ‫‪EC‬‬ ‫ﺏ‪E‬‬ ‫= ﺏ ﺟـ ﻟﻴﻘﻄﻌﻪ ﻓﻰ ‪ .E‬ﺇﺫﺍ ﻛﺎﻥ ‪ E‬ﺟـ = ١ ، ‪ ٢ ٦ = E C‬ﺳﻢ‬ ‫٢‬ ‫ﺃﻭﺟﺪ ﻃﻮﻝ ﻛﻞ ﻣﻦ ﺏ ‪ C ، E‬ﺏ ، ‪ C‬ﺟـ .‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫8‬ ‫:‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ‪،C‬‬ ‫‪C‬‬ ‫‪ = E C‬ﺏ ﺟـ ، ‪ E‬ﻫـ = ‪ C‬ﺏ ، ‪ E‬ﻭ = ‪ C‬ﺟـ‬ ‫ﺃﺛﺒﺖ ﺃﻥ:‬ ‫أ 9‪ E C‬ﻫـ + 9ﺟـ ‪ E‬ﻭ‬ ‫ب ﻣﺴﺎﺣﺔ ﺍﻟﻤﺴﺘﻄﻴﻞ ‪ C‬ﻫـ ‪ E‬ﻭ = ‪C‬ﻫـ * ﻫـ ﺏ * ‪ C‬ﻭ * ﻭ ﺟـ‬ ‫‪E‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
‫: ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻣﻨﻔﺮﺝ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ‪،C‬‬ ‫9‬ ‫‪ C‬ﺏ = ‪ C‬ﺟـ. ﺭﺳﻢ‬ ‫‪EC‬‬ ‫‪C‬‬ ‫= ‪ C‬ﺏ ﻭﻳﻘﻄﻊ ﺏ ﺟـ ﻓﻰ ‪.E‬‬ ‫ﺃﺛﺒﺖ ﺃﻥ: ٢)‪ C‬ﺏ(٢ = ﺏ ‪ * E‬ﺏ ﺟـ‬ ‫‪E‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫01 ﺗﻌﺒﺮ ﺍﻟﻤﺠﻤﻮﻋﺘﺎﻥ ‪ ،C‬ﺏ ﻋﻦ ﺃﻃﻮﺍﻝ ﺃﺿﻼﻉ ﻣﺜﻠﺜﺎﺕ ﻣﺨﺘﻠﻔﺔ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ.‬ ‫ﺍﻛﺘﺐ ﺃﻣﺎﻡ ﻛﻞ ﻣﺜﻠﺚ ﻣﻦ ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ C‬ﺭﻣﺰ ﺍﻟﻤﺜﻠﺚ ﺍﻟﺬﻯ ﻳﺸﺎﺑﻬﻪ ﻣﻦ ﺍﻟﻤﺠﻤﻮﻋﺔ ﺏ‬ ‫ﻣﺠﻤﻮﻋﺔ )ﺏ(‬ ‫ﻣﺠﻤﻮﻋﺔ )‪(C‬‬ ‫‪٥ ، ٤ ، ٢٫٥ C‬‬ ‫١ ٦ ، ٦ ، ٦‬ ‫ﺏ ٨ ، ٥٫٣١ ، ٤١‬ ‫٢ ٥ ، ٧ ، ١١‬ ‫ﺟـ ٥٢ ، ٥٣ ، ٥٥‬ ‫٣ ٥ ، ٨ ، ٠١‬ ‫‪١١ ، ١١ ، ١١ E‬‬ ‫٤ ٧ ، ٨ ، ٢١‬ ‫ﻫـ ٥٫٣ ، ٤ ، ٦‬ ‫٥ ٦١ ، ٧٢ ، ٨٢‬ ‫ﻭ ٨ ، ٦ ، ٠١‬ ‫ﺯ ٢٣ ، ٤٥ ، ٢٤‬ ‫11‬ ‫: ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻓﻴﻪ ‪ C‬ﺏ = ٦ﺳﻢ ، ﺏ ﺟـ = ٩ﺳﻢ ،‬ ‫‪E‬‬ ‫‪ C‬ﺟـ = ٥٫٧ﺳﻢ ، ‪ E‬ﻧﻘﻄﺔ ﺧﺎﺭﺟﺔ ﻋﻦ ﺍﻟﻤﺜﻠﺚ ‪ C‬ﺏ ﺟـ‬ ‫ﺣﻴﺚ ‪ E‬ﺏ = ٤ﺳﻢ ، ‪٥ = C E‬ﺳﻢ. ﺃﺛﺒﺖ ﺃﻥ:‬ ‫أ 9‪ C‬ﺏ ﺟـ + 9‪ E‬ﺏ ‪C‬‬ ‫ب ﺏ ‪ C‬ﻳﻨﺼﻒ ‪ E c‬ﺏ ﺟـ‬ ‫‪C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫ﺃﻛﻤﻞ:‬ ‫9 ‪ C‬ﺏ ﺟـ + 9‬ ‫ﻭﻣﻌﺎﻣﻞ ﺍﻟﺘﺸﺎﺑﻪ = .............................‬ ‫ﺳﻢ‬ ‫21‬ ‫‪C‬‬ ‫...............................‬ ‫ﺳﻢ ‪E‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
‫‪ï‬‬ ‫: ‪ C‬ﺏ ﺟـ + ﺱ ﺹ ﻉ، ﻫـ ﻣﻨﺘﺼﻒ ﺏ ﺟـ ،‬ ‫31‬ ‫‪C‬‬ ‫ﻡ ﻣﻨﺘﺼﻒ ﺹ ﻉ ، ﺟـ ‪ C = E‬ﺏ ، ﻉ ﻝ = ﺱ ﺹ ﺃﺛﺒﺖ ﺃﻥ:‬ ‫‪E‬‬ ‫أ 9‪ C‬ﻫـ ﺟـ + 9ﺱ ﻡ ﻉ‬ ‫ب ﺟـ ‪E‬‬ ‫‪ C‬ﻫـ‬ ‫ﻉﻝ = ﺱﻡ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫41 ‪ C‬ﺏ ﺟـ، ﺱ ﺹ ﻉ ﻣﺜﻠﺜﺎﻥ ﻣﺘﺸﺎﺑﻬﺎﻥ، ﺣﻴﺚ ‪ C‬ﺏ < ‪ C‬ﺟـ، ﺱ ﺹ < ﺱ ﻉ.‬ ‫ﻫـ، ﻝ ﻣﻨﺘﺼﻔﻰ ﺏ ﺟـ ، ﺹ ﻉ ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ، ﺭﺳﻢ ‪ C‬ﻭ = ﺏ ﺟـ ، ﺱ ﻡ = ﺹ ﻉ‬ ‫ﺃﺛﺒﺖ ﺃﻥ 9 ‪ C‬ﻫـ ﻭ + 9 ﺱ ﻝ ﻡ‬ ‫..................................................................................................................................................................................................................................‬ ‫51 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ، ‪ ∋ E‬ﺏ ﺟـ ﺣﻴﺚ )‪ = ٢(E C‬ﺏ ‪ E * E‬ﺟـ ، ﺏ ‪ = E C * C‬ﺏ ‪ C * E‬ﺟـ ﺃﺛﺒﺖ ﺃﻥ:‬ ‫ﺟ ‪ c) X‬ﺏ ‪ C‬ﺟـ( = ٠٩‪c‬‬ ‫أ 9 ‪ C‬ﺏ ‪9 + E‬ﺟـ ‪E C‬‬ ‫ب ‪ = E C‬ﺏ ﺟـ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫61 ﻳﺒﻴﻦ ﺍﻟﻤﺨﻄﻂ ﺍﻟﻤﻘﺎﺑﻞ ﻣﻮﻗﻊ ﻣﺤﻄﺔ ﺧﺪﻣﺔ ﻭﺗﻤﻮﻳﻦ ﺳﻴﺎﺭﺍﺕ ﻳﺮﺍﺩ‬ ‫ﺇﻗﺎﻣﺘﻬﺎ ﻋﻠﻰ ﺍﻟﻄﺮﻳﻖ ﺍﻟﺴﺮﻳﻊ ﻋﻨﺪ ﺗﻘﺎﻃﻊ ﻃﺮﻳﻖ ﺟﺎﻧﺒﻰ ﻳﺆﺩﻯ ﺇﻟﻰ‬ ‫ﺍﻟﻤﺪﻳﻨﺔ ﺟـ ﻭﻋﻤﻮﺩﻳﺎ ﻋﻠﻰ ﺍﻟﻄﺮﻳﻖ ﺍﻟﺴﺮﻳﻊ ﺑﻴﻦ ﺍﻟﻤﺪﻳﻨﺘﻴﻦ ‪ ،C‬ﺏ.‬ ‫ًّ‬ ‫أ ﻛﻢ ﻳﻨﺒﻐﻰ ﺃﻥ ﺗﺒﻌﺪ ﺍﻟﻤﺤﻄﺔ ﻋﻦ ﺍﻟﻤﺪﻳﻨﺔ ﺟـ?‬ ‫ب ﻣﺎ ﺍﻟﺒﻌﺪ ﺑﻴﻦ ﺍﻟﻤﺪﻳﻨﺘﻴﻦ ﺏ، ﺟـ?‬ ‫‪C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫ﻧﺸﺎط‬ ‫ﺍﺳﺘﺨﺪﺍﻡ ﺑﺮﻧﺎﻣﺞ ﺧﺮﺍﺋﻂ)‪ (Google Earth‬ﻟﺤﺴﺎﺏ ﺃﻗﺼﺮ ﺑﻌﺪ ﺑﻴﻦ ﻋﻮﺍﺻﻢ ﻣﺤﺎﻓﻈﺎﺕ ﺟﻤﻬﻮﺭﻳﺔ ﻣﺼﺮ ﺍﻟﻌﺮﺑﻴﺔ‬ ‫‪ïM‬‬ ‫−‬
‫اﻟﻌﻼﻗﺔ ﺑﻴﻦ ﻣﺴﺎﺣﺘﻰ ﺳﻄﺤﻰ ﻣﻀﻠﻌﻴﻦ ﻣﺘﺸﺎﺑﻬﻴﻦ‬ ‫2-3‬ ‫‪The Relation between the Area of two Similar Polygons‬‬ ‫1 ﺃﻛﻤﻞ:‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ 9‪ C‬ﺏ ﺟـ + 9ﺱ ﺹ ﻉ، ﻛﺎﻥ ‪ C‬ﺏ = ٣ ﺱ ﺹ ﻓﺈﻥ ‪9) W‬ﺱ ﺹ ﻉ( = ...............................‬ ‫ﻭ‬ ‫‪ C9) W‬ﺏ ﺟـ(‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ 9 ‪ C‬ﺏ ﺟـ + 9 ‪ E‬ﻫـ ﻭ، ‪ C 9) W‬ﺏ ﺟـ( = ٩ ‪ E 9)W‬ﻫـ ﻭ( ﻛﺎﻥ ‪ E‬ﻫـ = ٤ﺳﻢ ﻓﺈﻥ:‬ ‫ﻭ‬ ‫‪ C‬ﺏ = .............................. ﺳﻢ‬ ‫2 ﺍﺩﺭﺱ ﻛﻼ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ، ﺣﻴﺚ ﻙ ﺛﺎﺑﺖ ﺗﻨﺎﺳﺐ، ﺛﻢ ﺃﻛﻤﻞ:‬ ‫ًّ‬ ‫ب‬ ‫‪C‬‬ ‫أ‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫‪ C‬ﺏ ∩ ﺟـ ‪} = E‬ﻫـ{‬ ‫٢‬ ‫‪ C 9)W‬ﺟـ ﻫـ( = ٠٠٩ ﺳﻢ‬ ‫٢‬ ‫ﻓﺈﻥ: ‪ E 9)W‬ﻫـ ﺏ( = ............................... ﺳﻢ‬ ‫‪c)X‬ﺏ ‪ C‬ﺟـ( = ٠٩‪ = E C ،c‬ﺏ ﺟـ‬ ‫‪ E C 9)W‬ﺟـ( = ٠٨١ ﺳﻢ٢ ﻓﺈﻥ:‬ ‫٢‬ ‫‪ C 9)W‬ﺏ ﺟـ( = ............................... ﺳﻢ‬ ‫3 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ، ‪ C ∋E‬ﺏ ﺣﻴﺚ ‪ ٢ = E C‬ﺏ ‪ ،E‬ﻫـ ∋ ‪ C‬ﺟـ ﺣﻴﺚ ‪ E‬ﻫـ // ﺏ ﺟـ‬ ‫ﺇﺫﺍ ﻛﺎﻧﺖ ﻣﺴﺎﺣﺔ 9 ‪ E C‬ﻫـ = ٠٦ﺳﻢ٢. ﺃﻭﺟﺪ ﻣﺴﺎﺣﺔ ﺷﺒﻪ ﺍﻟﻤﻨﺤﺮﻑ ‪ E‬ﺏ ﺟـ ﻫـ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫4 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ﺏ، ﺭﺳﻤﺖ ﺍﻟﻤﺜﻠﺜﺎﺙ ﺍﻟﻤﺘﺴﺎﻭﻳﺔ ﺍﻷﺿﻼﻉ ‪ C‬ﺏ ﺱ، ﺏ ﺟـ ﺹ، ‪ C‬ﺟـ ﻉ‬ ‫ﺃﺛﺒﺖ ﺃﻥ: ‪ C9) W‬ﺏ ﺱ( + ‪9) W‬ﺏ ﺟـ ﺹ( = ‪ C9) W‬ﺟـ ﻉ(.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪C‬ﺏ‬ ‫5 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻓﻴﻪ ﺏ ﺟـ = ٤ ، ﺭﺳﻤﺖ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﻤﺎﺭﺓ ﺑﺮﺅﻭﺳﻪ. ﻣﻦ ﻧﻘﻄﺔ ﺏ ﺭﺳﻢ ﺍﻟﻤﻤﺎﺱ ﻟﻬﺬه ﺍﻟﺪﺋﺮﺓ ﻓﻘﻄﻊ‬ ‫٣‬ ‫‪ C 9) W‬ﺏ ﺟـ( ٧‬ ‫‪ C‬ﺟـ ﻓﻰ ﻫـ. ﺃﺛﺒﺖ ﺃﻥ: ‪ C 9) W‬ﺏ ﻫـ( = ٦١‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
‫‪M‬‬ ‫6 ‪ C‬ﺏ ﺟـ ‪ E‬ﻣﺘﻮﺍﺯﻯ ﺃﺿﻼﻉ ﺱ ∋ ‪ C‬ﺏ ، ﺱ ∌ ‪ C‬ﺏ ﺣﻴﺚ ﺏ ﺱ = ٢ ‪ C‬ﺏ، ﺹ ∋ ﺟـ ﺏ ، ﺹ ∌ ﺟـ ﺏ‬ ‫ﺣﻴﺚ ﺏ ﺹ = ٢ ﺏ ﺟـ ، ﺭﺳﻢ ﻣﺘﻮﺍﺯﻯ ﺍﻷﺿﻼﻉ ﺏ ﺱ ﻉ ﺹ ﺃﺛﺒﺖ ﺃﻥ: ‪ C) W‬ﺏ ﺟـ ‪١ = (E‬‬ ‫‪) W‬ﺱ ﺏ ﺹ ﻉ( ٤‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫7 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ﺏ، ﺏ ‪ C = E‬ﺟـ ﻳﻘﻄﻌﺔ ﻓﻰ ‪ ،E‬ﺭﺳﻢ ﻋﻠﻰ ‪ C‬ﺏ ، ﺏ ﺟـ ﺍﻟﻤﺮﺑﻌﺎﻥ‬ ‫ُ‬ ‫‪ C‬ﺱ ﺹ ﺏ، ﺏ ﻡ ﻥ ﺟـ ﺧﺎﺭﺝ ﺍﻟﻤﺜﻠﺚ ‪ C‬ﺏ ﺟـ.‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﻤﻀﻠﻊ ‪ C E‬ﺱ ﺹ ﺏ = ﺍﻟﻤﻀﻠﻊ ‪ E‬ﺏ ﻡ ﻥ ﺟـ‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ‪ C‬ﺏ = ٦ﺳﻢ، ‪ C‬ﺟـ = ٠١ﺳﻢ. ﺃﻭﺟﺪ ﺍﻟﻨﺴﺒﺔ ﺑﻴﻦ ﻣﺴﺎﺣﺘﻰ ﺳﻄﺤﻰ ﺍﻟﻤﻀﻠﻌﻴﻦ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫8 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ، ‪ C‬ﺏ ، ﺏ ﺟـ ، ‪ C‬ﺟـ ﺃﺿﻼﻉ ﻣﺘﻨﺎﻇﺮﺓ ﻟﺜﻼﺛﺔ ﻣﻀﻠﻌﺎﺕ ﻣﺘﺸﺎﺑﻬﺔ ﻣﺮﺳﻮﻣﺔ ﺧﺎﺭﺝ ﺍﻟﻤﺜﻠﺚ، ﻭﻫﻰ‬ ‫ﺍﻟﻤﻀﻠﻌﺎﺕ ﺑﻴﻦ ‪ ،N ،M‬ﻉ ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ.‬ ‫ﻓﺈﺫﺍ ﻛﺎﻧﺖ ﻣﺴﺎﺣﺔ ﺍﻟﻤﻀﻠﻊ ‪٤٠ = M‬ﺳﻢ٢، ﻭﻣﺴﺎﺣﺔ ﺍﻟﻤﻀﻠﻊ ‪ ٨٥= N‬ﺳﻢ٢، ﻭﻣﺴﺎﺣﺔ ﺍﻟﻤﻀﻠﻊ ﻉ = ٥٢١ﺳﻢ٢.‬ ‫ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﻤﺜﻠﺚ ‪ C‬ﺏ ﺟـ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫9 ‪ C‬ﺏ ﺟـ ‪ E‬ﻣﺮﺑﻊ ﻗﺴﻤﺖ ‪ C‬ﺏ ، ﺏ ﺟـ ، ﺟـ ‪ C E ، E‬ﺑﺎﻟﻨﻘﺎﻁ ﺱ، ﺹ، ﻉ، ﻝ ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ ﺑﻨﺴﺒﺔ ١ : ٣‬ ‫ﺃﺛﺒﺖ ﺃﻥ:‬ ‫‪ W‬ﺍﻟﻤﺮﺑﻊ ﺱ ﺹ ﻉ ﻝ‬ ‫=٥‬ ‫ب‬ ‫أ ﺍﻟﺸﻜﻞ ﺱ ﺹ ﻉ ﻝ ﻣﺮﺑﻊ‬ ‫٨‬ ‫‪ W‬ﺍﻟﻤﺮﺑﻊ ‪ C‬ﺏ ﺟـ ‪E‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫01 ﺻﺎﻟﺔ ﺃﻟﻌﺎﺏ ﻣﺴﺘﻄﻴﻠﺔ ﺍﻟﺸﻜﻞ ﺃﺑﻌﺎﺩﻫﺎ ٨ ﻣﺘﺮ، ٢١ ﻣﺘﺮ، ﺗﻢ ﺗﻐﻄﻴﺔ ﺃﺭﺿﻴﺘﻬﺎ ﺑﺎﻟﺨﺸﺐ، ﻓﻜﻠﻔﺖ ٠٠٢٣ ﺟﻨﻴﻪ.‬ ‫ﺍﺣﺴﺐ )ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﺘﺸﺎﺑﻪ( ﺗﻜﺎﻟﻴﻒ ﺗﻐﻄﻴﺔ ﺃﺭﺿﻴﺔ ﺻﺎﻟﺔ ﻣﺴﺘﻄﻴﻠﺔ ﺃﻛﺒﺮ ﺑﻨﻔﺲ ﻧﻮﻉ ﺍﻟﺨﺸﺐ ﻭﺑﻨﻔﺲ‬ ‫ﺍﻷﺳﻌﺎﺭ، ﺇﺫﺍ ﻛﺎﻥ ﺃﺑﻌﺎﺩﻫﺎ ٤١، ١٢ ﻣﻦ ﺍﻷﻣﺘﺎﺭ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
‫ﺗﻄﺒﻴﻘﺎت اﻟﺘﺸﺎﺑﻪ ﻓﻰ اﻟﺪاﺋﺮة‬ ‫2-4‬ ‫‪Applications of similarity in the circle‬‬ ‫1 ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻵﻟﺔ ﺍﻟﺤﺎﺳﺒﺔ ﺃﻭ ﺍﻟﺤﺴﺎﺏ ﺍﻟﻌﻘﻠﻰ، ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺱ ﺍﻟﻌﺪﺩﻳﺔ ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ.‬ ‫    )ﺍﻷﻃﻮﺍﻝ ﻣﻘﺪﺭﺓ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ(‬ ‫ﺟ‬ ‫ب‬ ‫أ‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫‪E‬‬ ‫................................................................................‬ ‫د‬ ‫‪E‬‬ ‫................................................................................‬ ‫................................................................................‬ ‫و‬ ‫ﻫ‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫‪C‬‬ ‫+‬ ‫‪E‬‬ ‫‪C‬‬ ‫................................................................................‬ ‫ز‬ ‫................................................................................‬ ‫ح‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫................................................................................‬ ‫ى‬ ‫ط‬ ‫‪E‬‬ ‫‪E‬‬ ‫................................................................................‬ ‫‪C‬‬ ‫................................................................................‬ ‫ك‬ ‫................................................................................‬ ‫ل‬ ‫‪E‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬ ‫................................................................................‬ ‫................................................................................‬
‫‪M‬‬ ‫¯‬ ‫‪ï‬‬ ‫2 ﻓﻰ ﺃﻯ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ ﺗﻘﻊ ﺍﻟﻨﻘﻂ ‪ ،C‬ﺏ، ﺟـ، ‪ E‬ﻋﻠﻰ ﺩﺍﺋﺮﺓ ﻭﺍﺣﺪﺓ? ﻓﺴﺮ ﺇﺟﺎﺑﺘﻚ.‬ ‫ِّ‬ ‫ٍّ‬ ‫)ﺍﻷﻃﻮﺍﻝ ﻣﻘﺪﺭﺓ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ(‬ ‫ﺟ‬ ‫ب‬ ‫‪C‬‬ ‫‪C‬‬ ‫أ‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫....................................................................................................................................................................................................................................................................................‬ ‫3 ﻓﻰ ﺃﻯ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ ‪ C‬ﺏ ﻣﻤﺎﺱ ﻟﻠﺪﺍﺋﺮﺓ ﺍﻟﻤﺎﺭﺓ ﺑﺎﻟﻨﻘﻂ ﺏ، ﺟـ، ‪.E‬‬ ‫ٍّ‬ ‫ﺟ‬ ‫‪ C‬ب‬ ‫‪C‬‬ ‫أ‬ ‫‪E‬‬ ‫٩٢‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫..............................................................................................................................................................................................................................................................................................‬ ‫ِ‬ ‫4 ﺩﺍﺋﺮﺗﺎﻥ ﻣﺘﻘﺎﻃﻌﺘﺎﻥ ﻓﻰ ‪ ،C‬ﺏ . ﺟـ ∋ ‪ C‬ﺏ ، ﺟـ ∌ ‪ C‬ﺏ ﺭﺳﻢ ﻣﻦ ﺟـ ﺍﻟﻘﻄﻌﺘﺎﻥ ﺟـ ﺱ ، ﺟـ ﺹ ﻣﻤﺎﺳﺘﺎﻥ‬ ‫ُ َ‬ ‫ﻟﻠﺪﺍﺋﺮﺗﻴﻦ ﻋﻨﺪ ﺱ، ﺹ. ﺃﺛﺒﺖ ﺃﻥ ﺟـ ﺱ = ﺟـ ﺹ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫.‬ ‫5‬ ‫: ﺍﻟﺪﺍﺋﺮﺗﺎﻥ ﻡ، ﻥ ﻣﺘﻤﺎﺳﺘﺎﻥ ﻋﻨﺪ ﻫـ‬ ‫‪ C‬ﺟـ ﻳﻤﺲ ﺍﻟﺪﺍﺋﺮﺓ ﻡ ﻋﻨﺪ ﺏ، ﻭﻳﻤﺲ ﺍﻟﺪﺍﺋﺮﺓ ﻥ ﻋﻨﺪ ﺟـ،‬ ‫‪ C‬ﻫـ ﻳﻘﻄﻊ ﺍﻟﺪﺍﺋﺮﺗﻴﻦ ﻋﻨﺪ ﻭ، ‪ E‬ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ‬ ‫ﺣﻴﺚ ‪ C‬ﻭ = ٤ﺳﻢ، ﻭ ﻫـ = ٥ﺳﻢ، ﻫـ ‪٧ = E‬ﺳﻢ.‬ ‫ﺃﺛﺒﺖ ﺃﻥ ﺏ ﻣﻨﺘﺼﻒ ‪ C‬ﺟـ‬ ‫‪E‬‬ ‫‪C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
‫6‬ ‫: ﻝ ∋ ﺱ ﺹ ﺣﻴﺚ ﺱ ﻝ = ٤ﺳﻢ،‬ ‫ﺹ ﻝ = ٨ﺳﻢ ، ﻡ ∋ ﺱ ﻉ ﺣﻴﺚ ﺱ ﻡ = ٦ﺳﻢ ، ﻉ ﻡ = ٢ﺳﻢ‬ ‫ﺃﺛﺒﺖ ﺃﻥ:‬ ‫أ 9ﺱ ﻝ ﻡ + 9ﺱ ﻉ ﺹ‬ ‫ب ﺍﻟﺸﻜﻞ ﻝ ﺹ ﻉ ﻡ ﺭﺑﺎﻋﻰ ﺩﺍﺋﺮﻯ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫٥‬ ‫7 ‪ C‬ﺏ ∩ ﺟـ ‪} = E‬ﻫـ{، ‪ C‬ﻫـ = ٢١ ﺏ ﻫـ، ‪ E‬ﻫـ = ٣ ﻫـ ﺟـ، ﺇﺫﺍ ﻛﺎﻥ ﺏ ﻫـ = ٦ﺳﻢ، ﺟـ ﻫـ = ٥ﺳﻢ.‬ ‫٥‬ ‫ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﻨﻘﻂ ‪ ،C‬ﺏ، ﺟـ، ‪ E‬ﺗﻘﻊ ﻋﻠﻰ ﺩﺍﺋﺮﺓ ﻭﺍﺣﺪﺓ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫8 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ، ‪ ∋ E‬ﺏ ﺟـ ﺣﻴﺚ ‪ E‬ﺏ = ٥ﺳﻢ، ‪ E‬ﺟـ = ٤ﺳﻢ. ﺇﺫﺍ ﻛﺎﻥ ‪ C‬ﺟـ = ٦ﺳﻢ. ﺃﺛﺒﺖ ﺃﻥ:‬ ‫أ ‪ C‬ﺟـ ﻣﻤﺎﺳﺔ ﻟﻠﺪﺍﺋﺮﺓ ﺍﻟﺘﻰ ﺗﻤﺮ ﺑﺎﻟﻨﻘﻂ ‪ ،C‬ﺏ، ‪.E‬‬ ‫ب 9‪ C‬ﺟـ ‪9 + E‬ﺏ ﺟـ ‪C‬‬ ‫ﺟ ‪ C9) W‬ﺏ ‪ C9) W : (E‬ﺏ ﺟـ( = ٥ : ٩‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫9 ﺩﺍﺋﺮﺗﺎﻥ ﻣﺘﺤﺪﺗﺎ ﻛﺰ ﻡ، ﻃﻮﻻ ﻧﺼﻔﻰ ﻗﻄﺮﻳﻬﻤﺎ ٢١ﺳﻢ، ٧ﺳﻢ، ﺭﺳﻢ ﺍﻟﻮﺗﺮ ‪ E C‬ﻓﻰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﻜﺒﺮﻯ ﻟﻴﻘﻄﻊ‬ ‫ﺍﻟﻤﺮ‬ ‫ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺼﻐﺮﻯ ﻓﻰ ﺏ ، ﺟـ ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ. ﺃﺛﺒﺖ ﺃﻥ: ‪ C‬ﺏ * ﺏ ‪٩٥ = E‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫01 ‪ C‬ﺏ ﺟـ ‪ E‬ﻣﺴﺘﻄﻴﻞ ﻓﻴﻪ ‪ C‬ﺏ = ٦ﺳﻢ، ﺏ ﺟـ = ٨ﺳﻢ. ﺭﺳﻢ ﺏ ﻫـ = ‪ C‬ﺟـ ﻓﻘﻄﻊ ‪ C‬ﺟـ ﻓﻰ ﻫـ، ‪ E C‬ﻓﻰ ﻭ.‬ ‫ب ﺃﻭﺟﺪ ﻃﻮﻝ ‪ C‬ﻭ .‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ )‪ C‬ﺏ(٢ = ‪ C‬ﻭ * ‪.E C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
‫‪M‬‬ ‫¯‬ ‫‪ï‬‬ ‫11 ﺍﻟﺮﺑﻂ ﻣﻊ ﺍﻟﺼﻨﺎﻋﺔ: ﻛُﺴﺮ ﺃﺣﺪ ﺗﺮﻭﺱ ﺁﻟﺔ ﻭﻻﺳﺘﺒﺪﺍﻟﻪ ﻣﻄﻠﻮﺏ‬ ‫ﻣﻌﺮﻓﺔ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮ ﺩﺍﺋﺮﺗﻪ. ﻳﺒﻴﻦ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ ﺟﺰﺀﺍ ﻣﻦ‬ ‫ً‬ ‫ﻫﺬﺍ ﺍﻟﺘﺮﺱ، ﻭﺍﻟﻤﻄﻠﻮﺏ ﺗﻌﻴﻴﻦ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮ ﺩﺍﺋ ﺗﻪ ................... .‬ ‫ﺮ‬ ‫‪C‬‬ ‫¯‬ ‫21 ﺍﻟﺮﺑﻂ ﻣﻊ ﺍﻟ ﻴﺌﺔ: ﻳﺒﻴﻦ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ ﻣﺨﻄﻄًّﺎ ﻟﺤﺪﻳﻘﺔ ﻋﻠﻰ‬ ‫ﺷﻜﻞ ﺩﺍﺋﺮﺓ ﺑﻬﺎ ﻃﺮﻳﻘﺎﻥ ﻳﺘﻘﺎﻃﻌﺎﻥ ﻋﻨﺪ ﻧﺎﻓﻮﺭﺓ ﺍﻟﻤﻴﺎه. ﺃﻭﺟﺪ ﺑﻌﺪ‬ ‫ُْ‬ ‫ﻧﺎﻓﻮﺭﺓ ﺍﻟﻤﻴﺎه ﻋﻨﺪ ﺍﻟﻤﺪﺧﻞ ﺟـ.‬ ‫‪E‬‬ ‫¯‬ ‫¯‬ ‫31 ﺍﻟﺮﺑﻂ ﻣﻊ ﺍﻟﻤﻨ ﻝ: ﺗﺴﺘﺨﺪﻡ ﻫﺪﻯ ﺷﺒﻜﺔ ﻟﺸﻰ ﺍﻟﻠﺤﻮﻡ ﻋﻠﻰ ﺷﻜﻞ‬ ‫ﺩﺍﺋﺮﺓ ﻣﻦ ﺍﻟﺴﻠﻚ، ﻃﻮﻝ ﻗﻄﺮﻫﺎ ٠٥ﺳﻢ، ﻳﺪﻋﻤﻬﺎ ﻣﻦ ﺍﻟﻮﺳﻂ ﺳﻠﻜﺎﻥ‬ ‫ﻣﺘﻮﺍﺯﻳﺎﻥ ﻭﻣﺘﺴﺎﻭﻳﺎﻥ ﻓﻰ ﺍﻟﻄﻮﻝ ﻛﻤﺎ ﻓﻰ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ، ﻭﺍﻟﺒﻌﺪ‬ ‫ﺑﻴﻨﻬﻤﺎ ٠١ﺳﻢ.‬ ‫ﺍﺣﺴﺐ ﻃﻮﻝ ﻛﻞ ﻣﻦ ﺳﻠﻜﻰ ﺍﻟﺪﻋﺎﻣﺔ. .........................................................‬ ‫‪C‬‬ ‫41 ﺍﻟﺮﺑﻂ ﻣﻊ ﺍﻻ ﺼﺎﻝ: ﺗﻨﻘﻞ ﺍﻷﻗﻤﺎﺭ ﺍﻟﺼﻨﺎﻋﻴﺔ ﺍﻟﺒﺮﺍﻣﺞ ﺍﻟﺘﻠﻴﻔﺰﻳﻮﻧﻴﺔ‬ ‫ﺇﻟﻰ ﻛﺎﻓﺔ ﻣﻨﺎﻃﻖ ﺍﻷﺭﺽ، ﻭﺗﺴﺘﺨﺪﻡ ﺃﻃﺒﺎﻕ ﺧﺎﺻﺔ ﻻﺳﺘﻘﺒﺎﻝ‬ ‫ﺇﺷﺎﺭﺍﺕ ﺍﻟﺒﺚ ﺍﻟﺘﻠﻴﻔﺰﻳﻮﻧﻰ، ﻭﻫﻰ ﺃﻃﺒﺎﻕ ﻣﻘﻌﺮﺓ ﻋﻠﻰ ﺷﻜﻞ ﺟﺰﺀ‬ ‫ﻣﻦ ﺳﻄﺢ ﻛﺮﺓ.‬ ‫ﻳﺒﻴﻦ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ ﻣﻘﻄﻌﺎ ﻓﻰ ﺃﺣﺪ ﻫﺬه ﺍﻷﻃﺒﺎﻕ، ﻃﻮﻝ ﻗﻄﺮه‬ ‫ً‬ ‫٠٨١ﺳﻢ، ﻭﺍﻟﻤﻄﻠﻮﺏ ﺣﺴﺎﺏ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮ ﻛﺮﺓ ﺗﻘﻌﺮه ﻡ ‪. C‬‬ ‫.......................................................................................................................................‬ ‫.......................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
‫ﺗﻤﺎﺭﻳﻦ ﻋﺎﻣﺔ‬ ‫1‬ ‫أ‬ ‫ب‬ ‫ﺟ‬ ‫د‬ ‫: ﺃﻯ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺘﺎﻟﻴﺔ ﻏﻴﺮ ﺻﺤﻴﺤﺔ:‬ ‫)‪ E = ٢(E C‬ﺏ * ‪ E‬ﺟـ‬ ‫)‪ C‬ﺏ(٢ = ﺏ ‪ * E‬ﺏ ﺟـ‬ ‫‪ C‬ﺟـ * ﺏ ﺟـ = ‪ C‬ﺏ * ‪E C‬‬ ‫‪ C‬ﺏ * ‪ C‬ﺟـ = ‪ * E C‬ﺏ ﺟـ‬ ‫‪C‬‬ ‫‪E‬‬ ‫..................................................................................................................................................................................................................................‬ ‫2‬ ‫: ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ‪ C ∋ E‬ﺏ ، ﻫـ ∋ ‪ C‬ﺟـ .‬ ‫ﺃﺛﺒﺖ ﺃﻥ 9‪ E C‬ﻫـ + 9‪ C‬ﺟـ ﺏ‬ ‫ﺛﻢ ﺃﻭﺟﺪ ﻃﻮﻝ ﻫـ ‪E‬‬ ‫‪C‬‬ ‫‪E‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫: ‪ C‬ﺏ ﻗﻄﺮ ﻓﻰ ﺍﻟﺪﺍﺋﺮﺓ ﻡ، ﻃﻮﻟﻪ ٢١ﺳﻢ‬ ‫3‬ ‫‪ C ∋ E‬ﺏ ﺣﻴﺚ ‪١٦ = E C‬ﺳﻢ، ﺟـ ﺗﻘﻊ ﻋﻠﻰ ﺍﻟﺪﺍﺋﺮﺓ‬ ‫ﺣﻴﺚ ﺟـ ‪٨ = E‬ﺳﻢ. ﺟـ ﻫـ = ‪ C‬ﺏ . ﺃﺛﺒﺖ ﺃﻥ:‬ ‫‪C‬‬ ‫‪E‬‬ ‫أ ﺟـ ‪ E‬ﻣﻤﺎﺳﺔ ﻟﻠﺪﺍﺋﺮﺓ ﻡ.‬ ‫ب 9‪ E‬ﺟـ ﺏ + 9‪ C E‬ﺟـ‬ ‫ﺟ ﺟـ ﻫـ = ٨٫٤ﺳﻢ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫4 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ﺏ. ﺏ ‪ C = E‬ﺟـ ، ‪ C‬ﺏ = ٥١ﺳﻢ، ‪٩ = E C‬ﺳﻢ. ﺭﺳﻢ ﻋﻠﻰ ‪ C‬ﺏ ، ﺏ ﺟـ ﻣﻦ‬ ‫ﺍﻟﺨﺎﺭﺝ ﺍﻟﻤﺮﺑﻌﺎﻥ ‪ C‬ﺏ ﺹ ﺱ، ﺏ ﺟـ ﻫـ ﻭ.‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﻤﻀﻠﻊ ‪ C E‬ﺱ ﺹ ﺏ + ﺍﻟﻤﻀﻠﻊ ‪ E‬ﺏ ﻭ ﻫـ ﺟـ.‬ ‫ب ﺃﻭﺟﺪ ‪) W‬ﺍﻟﻤﻀﻠﻊ ‪ C E‬ﺱ ﺹ ﺏ( : ‪) W‬ﺍﻟﻤﻀﻠﻊ ‪ E‬ﺏ ﻭ ﻫـ ﺟـ(‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
‫ﺗﻤﺎﺭﻳﻦ ﻋﺎﻣﺔ‬ ‫5‬ ‫: ﺍﻟﺪﺍﺋﺮﺗﺎﻥ ﻡ، ﻥ ﻣﺘﻘﺎﻃﻌﺘﺎﻥ ﻓﻰ ‪ ،C‬ﺏ‬ ‫‪ C‬ﺏ ∩ ﺟـ ‪ ∩ E‬ﻫـ ﻭ = }ﺱ{ ﺣﻴﺚ‬ ‫ﺱ ‪ E ٢ = E‬ﺟـ، ﻫـ ﻭ = ٠١ﺳﻢ، ﻭ ﺱ = ٦ ﺳﻢ‬ ‫‪E‬‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﺸﻜﻞ ﺟـ ‪ E‬ﻭ ﻫـ ﺭﺑﺎﻋﻰ ﺩﺍﺋﺮﻯ.‬ ‫ب ﺃﻭﺟﺪ ﻃﻮﻝ ﺟـ ‪E‬‬ ‫.................................................................................................................................‬ ‫‪C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫6‬ ‫: ﺩﺍﺋﺮﺗﺎﻥ ﻣﺘﺤﺪﺗﺎ ﻛﺰ،‬ ‫ﺍﻟﻤﺮ‬ ‫ﻭﺍﻷﻃﻮﺍﻝ ﺍﻟﻤﺒﻴﻨﺔ ﻟﻠﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ.‬ ‫ﺃﻭﺟﺪ ﻗﻴﻢ ﺱ، ﺹ ﺍﻟﻌﺪﺩﻳﺔ.‬ ‫............................................................................................................................................‬ ‫............................................................................................................................................‬ ‫¯‬ ‫7 ﺣﺪﻳﻘﺔ ﺣﻴﻮﺍﻥ: ﻓﻰ ﺭﺣﻠﺔ ﻣﺪﺭﺳﻴﺔ ﺇﻟﻰ ﺣﺪﻳﻘﺔ ﺍﻟﺤﻴﻮﺍﻥ ﺃﺭﺍﺩ‬ ‫ﺣﺴﺎﻡ ﺃﻥ ﻳﻌﺮﻑ ﺍﺭﺗﻔﺎﻉ ﺣﻴﻮﺍﻥ ﺍﻟﺰﺭﺍﻓﺔ. ﻭﺿﻊ ﺣﺴﺎﻡ ﻣﺮﺁﺓ‬ ‫ﻣﺴﺘﻮﻳﺔ ﻋﻠﻰ ﺍﻷﺭﺽ ﺗﺒﻌﺪ ﻋﻨﻪ ﻣﺘﺮﺍﻥ ﻭﻋﻦ ﺍﻟﺰﺭﺍﻓﺔ ٦ ﺃﻣﺘﺎﺭ،‬ ‫ﻓﺈﺫﺍ ﻛﺎﻥ ﺣﺴﺎﻡ ﻭﺍﻟﻤﺮﺁﺓ ﻭﺍﻟﺰﺭﺍﻓﺔ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ‬ ‫ﻭﺍﺭﺗﻔﺎﻉ ﺣﺴﺎﻡ ٥٫١ ﻣﺘﺮﺍ . ﻛﻢ ﻳﺒﻠﻎ ﺍﺭﺗﻔﺎﻉ ﺍﻟﺰﺭﺍﻓﺔ.‬ ‫ً‬ ‫¯‬ ‫¯‬ ‫.................................................................................................................................‬ ‫.................................................................................................................................‬ ‫8 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻔﻴ ﺎﺀ: ﺍﺣﺴﺐ ﻣﻌﺎﻣﻞ ﻣﻐﻴﺮ ﺍﻟﺒﻌﺪ، ﻭﺍﺣﺴﺐ ﻗﻴﻤﺔ ﺱ ﺍﻟﻌﺪﺩﻳﺔ ﻓﻰ ﻛﻞ ﺷﻜﻞ ﻣﻤﺎ ﻳﻠﻰ.‬ ‫ب‬ ‫أ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ‬ ‫1 ﺃﻛﻤﻞ ﻣﺎ ﻳﺄﺗﻲ:‬ ‫أ ﺍﻟﻤﻀﻠﻌﺎﻥ ﺍﻟﻤﺸﺎﺑﻬﺎﻥ ﻟﺜﺎﻟﺚ .............................................................................................................................................................‬ ‫ب‬ ‫ﺇﺫﺍ ﺗﻨﺎﺳﺒﺖ ﺃﻃﻮﺍﻝ ﺍﻷﺿﻼﻉ ﺍﻟﻤﺘﻨﺎﻇﺮﺓ ﻓﻰ ﻣﺜﻠﺜﻴﻦ ﻓﺈﻧﻬﻤﺎ ..................................................................................................‬ ‫ﺟ‬ ‫ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻨﺴﺒﺔ ﺑﻴﻦ ﻣﺤﻴﻄﻰ ﻣﻀﻠﻌﻴﻦ ﻣﺘﺸﺎﺑﻬﻴﻦ ٣ : ٥ ﻓﺈﻥ ﺍﻟﻨﺴﺒﺔ ﺑﻴﻦ ﻣﺴﺎﺣﺘﻴﻬﻤﺎ ...........................................‬ ‫د ﺇﺫﺍ ﺗﻘﺎﻃﻊ ﻭﺗﺮﺍﻥ ‪ C‬ﺏ ، ﺟـ ‪ E‬ﻟﺪﺍﺋﺮﺓ ﻓﻰ ﻧﻘﻄﺔ ﺱ ﻓﺈﻥ:‬ ‫................... * ................... = ................... *‬ ‫‪E‬‬ ‫...................‬ ‫ﻫ ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻤﺴﺘﻄﻴﻞ ‪ C‬ﺏ ﺟـ ‪ + E‬ﺍﻟﻤﺴﺘﻄﻴﻞ ﺱ ﺏ ﻉ ﺹ،‬ ‫‪١٥ = E C‬ﺳﻢ ، ﺟـ ‪٢٠ = E‬ﺳﻢ، ﺹ ﻉ = ٦١ﺳﻢ‬ ‫ﻓﺈﻥ: ﺱ ﻉ = ...........................................................‬ ‫‪C‬‬ ‫‪C‬‬ ‫2‬ ‫: ‪ C‬ﻫـ // ‪ E‬ﺟـ ، ‪ C‬ﺟـ = ﻫـ ‪} ∩ E‬ﺏ{،‬ ‫‪ C‬ﺏ = ٣ﺳﻢ، ﺏ ﺟـ = ٦ﺳﻢ، ﻫـ ‪١٢ = E‬ﺳﻢ‬ ‫ﻓﺄﻭﺟﺪ ﻃﻮﻝ ﻫـ ﺏ‬ ‫.......................................................................................................................................‬ ‫‪E‬‬ ‫.......................................................................................................................................‬ ‫‪E‬‬ ‫: ﺍﻟﻤﻀﻠﻊ ‪ C‬ﺏ ﺟـ ‪ + E‬ﺍﻟﻤﻀﻠﻊ ﺱ ﻫـ ﺟـ ﻭ‬ ‫3‬ ‫‪C‬‬ ‫ﺃﺛﺒﺖ ﺃﻥ ‪ C‬ﺏ // ﺱ ﻫـ‬ ‫١‬ ‫ﻭﺇﺫﺍ ﻛﺎﻧﺖ ﺱ ﻫـ = ٢ ‪C‬ﺏ، ﺟـ ﻭ = ٩ﺳﻢ ﻓﺄﻭﺟﺪ ﻃﻮﻝ ﻭ ‪E‬‬ ‫.......................................................................................................................................‬ ‫.......................................................................................................................................‬ ‫4 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻓﻴﻪ ﺱ ∋ ‪ C‬ﺏ ﺑﺤﻴﺚ ﻛﺎﻥ ‪ C‬ﺱ = ٨ﺳﻢ، ﺱ ﺏ = ٢١ﺳﻢ‬ ‫ﺹ ∋ ‪ C‬ﺟـ ، ﺑﺤﻴﺚ ﻛﺎﻥ ‪ C‬ﺹ = ٠١ﺳﻢ، ﺹ ﺟـ = ٦ﺳﻢ.‬ ‫ﺃﺛﺒﺖ ﺃﻥ:‬ ‫أ 9 ‪ C‬ﺏ ﺟـ + 9 ‪ C‬ﺹ ﺱ‬ ‫ب ﺍﻟﺸﻜﻞ ﺱ ﺏ ﺟـ ﺹ ﺭﺑﺎﻋﻰ ﺩﺍﺋﺮﻯ.‬ ‫‪C‬‬ ‫.......................................................................................................................................‬ ‫.......................................................................................................................................‬ ‫5 ‪ C‬ﺏ ، ﺟـ ‪ E‬ﻭﺗﺮﺍﻥ ﻓﻰ ﺩﺍﺋﺮﺓ ﻣﺘﻘﺎﻃﻌﺎﻥ، ﻓﻰ ﻫـ ﻓﺈﺫﺍ ﻛﺎﻥ ﻫـ ﻣﻨﺘﺼﻒ ‪ C‬ﺏ ، ﺟـ ﻫـ = ٤ﺳﻢ، ﻫـ ‪٩ = E‬ﺳﻢ‬ ‫..........................................................................................................................................................................‬ ‫ﻓﺄﻭﺟﺪ ﻃﻮﻝ ‪ C‬ﺏ .‬ ‫¯‬ ‫−‬ ‫¯‬
‫ﺍﺧﺘﺒﺎﺭ ﺗﺮﺍﻛﻤﻰ‬ ‫‪Oó©àe øe QÉ«àN’G á∏Ä°SCG‬‬ ‫1 ﺇﺫﺍ ﻛﺎﻥ ٢ﺱ +١١ = ٣ ﻓﺈﻥ ١١ - ﺱ ﺗﺴﺎﻭﻯ:‬ ‫٢‬ ‫ﺱ+‬ ‫ب‬ ‫ﺻﻔﺮﺍ‬ ‫أ -٠١‬ ‫ً‬ ‫ﺟ ٥‬ ‫2 ﻣﺴﺘﻌﻴﻨﺎ ﺑﻤﻌﻄﻴﺎﺕ ﺍﻟﺸﻜﻞ، ﻓﺈﻥ ﺱ ﺗﺴﺎﻭﻯ:‬ ‫ً‬ ‫ب‬ ‫٨١‬ ‫أ ٢٣‬ ‫د ١٥‬ ‫ﺟ ٧٢‬ ‫د ٠١‬ ‫+‬ ‫‪c‬‬ ‫− ‪c‬‬ ‫‪c‬‬ ‫−‬ ‫3 ﻣﺴﺘﻌﻴﻨﺎ ﺑﻤﻌﻄﻴﺎﺕ ﺍﻟﺸﻜﻞ، ﻓﺈﻥ ﺱ ﺗﺴﺎﻭﻯ:‬ ‫ً‬ ‫ب ١١‬ ‫أ ٥‬ ‫د ٤١‬ ‫ﺟ ٢١‬ ‫‪E‬‬ ‫4‬ ‫أ ٥ ﺳﻢ‬ ‫ﺟ ٨ ﺳﻢ‬ ‫: ‪ C‬ﺏ = ٢١ﺳﻢ، ﺟـ ﻫـ = ٤ ﺳﻢ، ﻓﺈﻥ ﻫـ ‪ E‬ﺗﺴﺎﻭﻯ:‬ ‫ب ٦ ﺳﻢ‬ ‫د ٩ ﺳﻢ‬ ‫‪C‬‬ ‫5 ﻣﺴﺘﻄﻴﻼﻥ ﻣﺘﺸﺎﺑﻬﺎﻥ ﺑﻌﺪﺍ ﺍﻷﻭﻝ ٠١ ﺳﻢ، ٨ ﺳﻢ، ﻭﻣﺤﻴﻂ ﺍﻟﺜﺎﻧﻰ ٨٠١ ﺳﻢ ﻓﺈﻥ ﻃﻮﻝ ﺍﻟﻤﺴﺘﻄﻴﻞ ﺍﻟﺜﺎﻧﻰ ﻳﺴﺎﻭﻯ:‬ ‫د ٦٣ ﺳﻢ‬ ‫ﺟ ٠٣ ﺳﻢ‬ ‫ب ٤٢ ﺳﻢ‬ ‫أ ٨١ ﺳﻢ‬ ‫‪C‬‬ ‫‪:Iô«°ü≤dG äÉHÉLE’G äGP á∏Ä°SC’G‬‬ ‫6‬ ‫: ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻣﻦ ﺱ، ﺹ‬ ‫ﺍﻷﻃﻮﺍﻝ ﻣﻘﺪﺭﺓ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ.‬ ‫+‬ ‫‪E‬‬ ‫−‬ ‫7 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻓﻴﻪ ‪ C‬ﺏ = ‪ C‬ﺟـ، ‪ ∋ E‬ﺏ ﺟـ . ﺭﺳﻢ ‪ E‬ﻫـ = ‪ C‬ﺏ ، ‪ E‬ﻭ = ‪ C‬ﺟـ .‬ ‫ﺏ ﻫـ ‪ E‬ﻫـ‬ ‫ﺃﺛﺒﺖ ﺃﻥ: ﺟـ ﻭ =‬ ‫‪E‬ﻭ‬ ‫‪ïM‬‬ ‫−‬
‫ﺍﺧﺘﺒﺎﺭ ﺗﺮﺍﻛﻤﻰ‬ ‫8‬ ‫: ‪ C‬ﺏ = ‪ C‬ﺟـ ، ‪ = E C‬ﺏ ﺟـ‬ ‫‪ c) X‬ﺏ( = ٠٣‪ C ، c‬ﺟـ = ٦ ﺳﻢ‬ ‫‪C‬‬ ‫ﺃﻭﺟﺪ ﻃﻮﻝ ﻛﻞ ﻣﻦ: ‪ C‬ﺏ ، ﺏ ‪E C ، E‬‬ ‫‪c‬‬ ‫‪E‬‬ ‫‪:á∏jƒ£dG äÉHÉLE’G äGP øjQɪàdG‬‬ ‫‪ E‬ﻫـ‬ ‫‪ C‬ﻫـ‬ ‫9 ‪ C‬ﺏ ﺟـ ‪ E‬ﺷﺒﻪ ﻣﻨﺤﺮﻑ ﺗﻘﺎﻃﻊ ﻗﻄﺮﺍه ﻓﻰ ﻫـ، ﺇﺫﺍ ﻛﺎﻥ ‪ // E C‬ﺏ ﺟـ ﺃﺛﺒﺖ ﺃﻥ: ﻫـ ﺟـ = ﻫـ ﺏ‬ ‫: ‪ C‬ﺏ ﺟـ ‪ E‬ﻣﺴﺘﻄﻴﻞ، ﻡ ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٦ ﺳﻢ‬ ‫01‬ ‫‪E‬‬ ‫ﻭﺗﻤﺲ ‪ C‬ﺏ ﻋﻨﺪ ﻫـ، ﺟـ ‪ E‬ﻋﻨﺪ ﻭ.‬ ‫ﺭﺳﻢ ﻡ ﺹ // ‪ C‬ﺏ ﻭﻳﻘﻄﻊ ﺍﻟﺪﺍﺋﺮﺓ ﻓﻰ ﺱ، ‪ E C‬ﻓﻰ ﺹ.‬ ‫ﺇﺫﺍ ﻛﺎﻥ: ﺱ ﺹ = ٢ﺳﻢ، ‪ C 9) W‬ﻫـ ﻡ( = ١‬ ‫‪ C 9) W‬ﺏ ﺟـ( ٤‬ ‫ﺃﻭﺟﺪ ﻃﻮﻝ ﺏ ﻫـ ، ﺏ ﺟـ‬ ‫‪C‬‬ ‫‪?á«aÉ°VCG IóYÉ°ùªd êÉàëJ πg‬‬ ‫ﺇﻥ ﻟﻢ ﺗﺴﺘﻄﻊ ﺍﻹﺟﺎﺑﺔ ﻋﻦ ﺃﻯ ﻣﻦ ﺍﻷﺳﺌﻠﺔ ﺍﻟﺴﺎﺑﻘﺔ ﻓﺎﺭﺟﻊ ﻟﻠﺠﺪﻭﻝ ﺍﻟﺘﺎﻟﻰ:‬ ‫١‬ ‫¯‬ ‫٢‬ ‫−‬ ‫٣‬ ‫¯‬ ‫٤‬ ‫٥‬ ‫٦‬ ‫٧‬ ‫٨‬ ‫٩‬ ‫٠١‬
‫-‬ ‫‪IóMƒdG‬‬ ‫3‬ ‫ﻧﻈﺮﻳﺎﺕ ﺍﻟﺘﻨﺎﺳﺐ ﻓﻰ ﺍﻟﻤﺜﻠﺚ‬ ‫‪The Triangle Proportionality Theorems‬‬ ‫ﻣﻌﺒﺪ ﺣﺘﺸﺒﺴﻮت )اﻷﻗﺼﺮ(‬ ‫دروس اﻟﻮﺣﺪة‬ ‫ﺍﻟﺪﺭﺱ )٣ - ١(: ﺍﻟﻤﺴﺘﻘﻴﻤﺎﺕ ﺍﻟﻤﺘﻮﺍﺯﻳﺔ ﻭﺍﻷﺟﺰﺍﺀ ﺍﻟﻤﺘﻨﺎﺳﺒﺔ.‬ ‫ﺍﻟﺪﺭﺱ )٣ - ٢(: ﻣﻨﺼﻔﺎ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ﺍﻟﻤﺜﻠﺚ ﻭﺍﻷﺟﺰﺍﺀ ﺍﻟﻤﺘﻨﺎﺳﺒﺔ.‬ ‫ﺍﻟﺪﺭﺱ )٣ - ٣(: ﺗﻄﺒﻴﻘﺎﺕ ﺍﻟﺘﻨﺎﺳﺐ ﻓﻰ ﺍﻟﺪﺍﺋﺮﺓ.‬ ‫‪ïM‬‬ ‫−‬
‫اﻟﻤﺴﺘﻘﻴﻤﺎت اﻟﻤﺘﻮازﻳﺔ وا ﺟﺰاء اﻟﻤﺘﻨﺎﺳﺒﺔ‬ ‫3-1‬ ‫‪Parallel lines and proportional parts‬‬ ‫1‬ ‫‪C‬‬ ‫‪ E‬ﻫـ // ﺏ ﺟـ ﺃﻛﻤﻞ:‬ ‫ﺏ‬ ‫ﺟـ ﻫـ‬ ‫‪C‬‬ ‫‪EC‬‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ ‪ E‬ﺏ = ٥ ﻓﺈﻥ : ﺏ ‪ ،  ............. = E‬‬ ‫= .............‬ ‫٣‬ ‫ﻫـ ‪C‬‬ ‫.............‬ ‫.............‬ ‫ﺟـ ﻫـ‬ ‫ﺏ‪E‬‬ ‫ﻫـ‬ ‫= .............  ، ‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ‪ ٤ = C‬ﻓﺈﻥ :‬ ‫= .............‬ ‫‪ C‬ﺟـ‬ ‫2‬ ‫.............‬ ‫ﻫـ ‪C‬‬ ‫٧‬ ‫.............‬ ‫‪E‬‬ ‫‪C‬ﺏ‬ ‫‪ E‬ﻫـ // ﺏ ﺟـ . ﺣﺪﺩ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺼﺤﻴﺤﺔ ﻣﻦ ﻣﺎ ﻳﻠﻲ:‬ ‫‪C‬ﺏ‬ ‫أ ‪E‬ﺏ‬ ‫ﺟ ‪C‬ﺏ‬ ‫ﺏ‪E‬‬ ‫ﻫ ‪ C‬ﺟـ‬ ‫‪EC‬‬ ‫ﺏ‪E‬‬ ‫ب ‪EC‬‬ ‫‪ C‬ﻫـ = ﻫـ ﺟـ‬ ‫‪C‬ﺏ‬ ‫‪ C‬ﺟـ‬ ‫د ﺏ‪= E‬‬ ‫ﺟـ ﻫـ‬ ‫و ﺟـ ﻫـ ‪ C‬ﺟـ‬ ‫ﺏ‪C = E‬ﺏ‬ ‫‪ C‬ﻫـ‬ ‫= ﻫـ ﺟـ‬ ‫‪ C‬ﺟـ‬ ‫= ‪ C‬ﻫـ‬ ‫‪C‬ﺏ‬ ‫= ‪ C‬ﻫـ‬ ‫3‬ ‫‪E‬‬ ‫‪ E‬ﻫـ // ﺏ ﺟـ . ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺱ ﺍﻟﻌﺪﺩﻳﺔ )ﺍﻷﻃﻮﺍﻝ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ(.‬ ‫ﺟ‬ ‫ب‬ ‫‪C‬‬ ‫أ‬ ‫‪E‬‬ ‫‪E‬‬ ‫+‬ ‫‪E‬‬ ‫د‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫ﻫ‬ ‫‪C‬‬ ‫و‬ ‫+‬ ‫+‬ ‫‪E‬‬ ‫+‬ ‫4‬ ‫: ‪ C‬ﺏ // ‪ E‬ﻫـ ، ‪ C‬ﻫـ ∩ ﺏ ‪} = E‬ﺟـ{‬ ‫‪ C‬ﺟـ = ٦ﺳﻢ، ﺏ ﺟـ = ٤ﺳﻢ، ﺟـ ‪٣ = E‬ﺳﻢ‬ ‫ﺃﻭﺟﺪ ﻃﻮﻝ ﺟـ ﻫـ‬ ‫¯‬ ‫−‬ ‫¯‬ ‫‪C‬‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪E‬‬
‫5 ﺱ ﺹ ∩ ﻉ ﻝ = }ﻡ{، ﺣﻴﺚ ﺱ ﻉ // ﻝ ﺹ ، ﻓﺈﺫﺍ ﻛﺎﻥ ﺱ ﻡ = ٩ﺳﻢ، ﺹ ﻡ = ٥١ﺳﻢ، ﻉ ﻝ = ٦٣ ﺳﻢ.‬ ‫ﺃﻭﺟﺪ ﻃﻮﻝ ﻉ ﻡ .‬ ‫6 ﻟﻜﻞ ﻣﻤﺎ ﻳﺄﺗﻰ: ﺍﺳﺘﺨﺪﻡ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ ﻭﺍﻟﺒﻴﺎﻧﺎﺕ ﺍﻟﻤﻌﻄﺎﺓ ﻹﻳﺠﺎﺩ ﻗﻴﻤﺔ ﺱ:‬ ‫أ ‪ ، ٤ = E C‬ﺏ ‪ ،  ٨ = E‬ﺟـ ﻫـ = ٦ ، ‪ C‬ﻫـ = ﺱ.‬ ‫ب ‪ C‬ﻫـ = ﺱ ، ﻫـ ﺟـ = ٥ ، ‪ = E C‬ﺱ - ٢ ، ‪ E‬ﺏ = ٣.‬ ‫ﺟ ‪ C‬ﺏ = ١٢ ، ﺏ ﻭ = ٨ ، ﻭ ﺟـ = ٦ ، ‪ = E C‬ﺱ.‬ ‫‪C‬‬ ‫‪E‬‬ ‫د ‪ = E C‬ﺱ  ، ﺏ ﻭ = ﺱ + ٥ ، ٢‪ E‬ﺏ = ٣ﻭ ﺟـ = ٢١.‬ ‫7 ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ، ﺣﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﺱ ﺹ // ﺏ ﺟـ‬ ‫ب‬ ‫أ‬ ‫ﺟ‬ ‫‪C‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫8 ﺱ ﺹ ﻉ ﻣﺜﻠﺚ ﻓﻴﻪ ﺱ ﺹ = ٤١ﺳﻢ، ﺱ ﻉ = ١٢ﺳﻢ، ﻝ ∋ ﺱ ﺹ ﺑﺤﻴﺚ ﺱ ﻝ = ٦٫٥ﺳﻢ،‬ ‫ﻡ ∋ ﺱ ﻉ ﺣﻴﺚ ﺱ ﻡ = ٤٫٨ﺳﻢ. ﺃﺛﺒﺖ ﺃﻥ ﻝ ﻡ // ﺹ ﻉ‬ ‫9 ﻓﻰ ﺍﻟﻤﺜﻠﺚ ‪ C‬ﺏ ﺟـ، ‪ C ∋ E‬ﺏ ، ﻫـ ∋ ‪ C‬ﺟـ ، ٥‪ C‬ﻫـ = ٤ ﻫـ ﺟـ.‬ ‫ﺇﺫﺍ ﻛﺎﻥ ‪ ١٠ = E C‬ﺳﻢ، ‪ E‬ﺏ = ٨ﺳﻢ. ﺣﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ‪ E‬ﻫـ // ﺏ ﺟـ . ﻓﺴﺮ ﺇﺟﺎﺑﺘﻚ.‬ ‫01 ‪ C‬ﺏ ﺟـ ‪ E‬ﺷﻜﻞ ﺭﺑﺎﻋﻰ ﺗﻘﺎﻃﻊ ﻗﻄﺮﺍه ﻓﻰ ﻫـ. ﻓﺈﺫﺍ ﻛﺎﻥ ‪ C‬ﻫـ = ٦ﺳﻢ، ﺏ ﻫـ = ٣١ﺳﻢ، ﻫـ ﻭ = ٠١ﺳﻢ،‬ ‫ﻫـ ‪٧٫٨ = E‬ﺳﻢ. ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﺸﻜﻞ ‪ C‬ﺏ ﺟـ ‪ E‬ﺷﺒﻪ ﻣﻨﺤﺮﻑ.‬ ‫11 ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﻘﻄﻌﺔ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺍﻟﻤﺮﺳﻮﻣﺔ ﺑﻴﻦ ﻣﻨﺘﺼﻔﻰ ﺿﻠﻌﻴﻦ ﻓﻰ ﻣﺜﻠﺚ ﻳﻮﺍﺯﻯ ﺿﻠﻌﻪ ﺍﻟﺜﺎﻟﺚ، ﻭﻃﻮﻟﻬﺎ ﻳﺴﺎﻭﻯ‬ ‫ﻧﺼﻒ ﻃﻮﻝ ﻫﺬﺍ ﺍﻟﻀﻠﻊ.‬ ‫21 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ، ‪ C ∋ E‬ﺏ ﺣﻴﺚ ٣‪ E ٢ = E C‬ﺏ، ﻫـ ∋ ‪ C‬ﺟـ ﺣﻴﺚ ٥ ﺟـ ﻫـ = ٣ ‪ C‬ﺟـ، ﺭﺳﻢ ‪ C‬ﺱ ﻳﻘﻄﻊ ﺏ ﺟـ‬ ‫ﻓﻰ ﺱ. ﺇﺫﺍ ﻛﺎﻥ ‪ C‬ﻭ = ٨ﺳﻢ، ‪ C‬ﺱ = ٠٢ﺳﻢ، ﺣﻴﺚ ﻭ ∋ ‪ C‬ﺱ . ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﻨﻘﻂ ‪ ،E‬ﻭ، ﻫـ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ.‬ ‫ﻫـ‬ ‫‪E‬‬ ‫31 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ، ‪ ∋ E‬ﺏ ﺟـ ، ﺑﺤﻴﺚ ﺏﺟـ = ٣ ، ﻫـ ∋ ‪ ، E C‬ﺑﺤﻴﺚ ‪ ، ٣ = E C‬ﺭﺳﻢ ﺟـ ﻫـ ﻓﻘﻄﻊ ‪ C‬ﺏ ﻓﻰ ﺱ،‬ ‫٧‬ ‫٤‬ ‫‪E‬‬ ‫‪C‬‬ ‫ﺭﺳﻢ ‪ E‬ﺹ // ﺟـ ﺱ ﻓﻘﻄﻊ ‪ C‬ﺏ ﻓﻰ ﺹ. ﺃﺛﺒﺖ ﺃﻥ ‪ C‬ﺱ = ﺏ ﺹ.‬ ‫41 ‪ C‬ﺏ ﺟـ ‪ E‬ﻣﺴﺘﻄﻴﻞ ﺗﻘﺎﻃﻊ ﻗﻄﺮﺍه ﻓﻰ ﻡ. ﻫـ ﻣﻨﺘﺼﻒ ‪ C‬ﻡ ، ﻭ ﻣﻨﺘﺼﻒ ﻡ ﺟـ . ﺭﺳﻢ ‪ E‬ﻫـ ﻳﻘﻄﻊ ‪ C‬ﺏ ﻓﻰ ﺱ،‬ ‫ﻭﺭﺳﻢ ‪ E‬ﻭ ﻳﻘﻄﻊ ﺏ ﺟـ ﻓﻰ ﺹ. ﺃﺛﺒﺖ ﺃﻥ: ﺱ ﺹ // ‪ C‬ﺟـ .‬ ‫‪ïM‬‬ ‫−‬
‫51 ﺍﻛﺘﺐ ﻣﺎ ﺗﺴﺎﻭﻳﻪ ﻛﻞ ﻣﻦ ﺍﻟﻨﺴﺐ ﺍﻟﺘﺎﻟﻴﺔ ﻣﺴﺘﺨﺪﻣﺎ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ:‬ ‫ً‬ ‫أ‬ ‫ﺟ‬ ‫ﻫ‬ ‫ز‬ ‫‪ C‬ﺏ = ‪ E‬ﻫـ‬ ‫ﺏ ﺟـ‬ ‫ﻡ‪E‬‬ ‫ﻡ‪C‬‬ ‫= ................‬ ‫‪C‬ﺏ‬ ‫................‬ ‫ﻡﺏ‬ ‫=‬ ‫‪ C‬ﺏ ‪ E‬ﻫـ‬ ‫ﺏ ﺟـ ﻫـ ﻭ‬ ‫ﻡ ﺏ = ................‬ ‫................‬ ‫ب ‪ C‬ﺟـ‬ ‫ﺏ ﺟـ = ﻫـ ﻭ‬ ‫................‬ ‫‪C‬‬ ‫................‬ ‫ﺟـ‬ ‫د ‪= C‬‬ ‫‪ C‬ﺏ ‪ E‬ﻫـ‬ ‫و ﻡ ﺟـ ﻡ ﻭ‬ ‫= ................‬ ‫‪E‬‬ ‫‪ C‬ﺟـ‬ ‫ح ‪ E‬ﻭ = ‪ C‬ﺟـ‬ ‫................‬ ‫ﻡﻭ‬ ‫61 ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ، ﺍﺣﺴﺐ ﻗﻴﻢ ﺱ، ﺹ ﺍﻟﻌﺪﺩﻳﺔ )ﺍﻷﻃﻮﺍﻝ ﻣﻘﺪﺭﺓ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ(‬ ‫ﺟ‬ ‫ب‬ ‫أ‬ ‫+‬ ‫+‬ ‫−‬ ‫−‬ ‫+‬ ‫+‬ ‫:‬ ‫71‬ ‫‪E‬‬ ‫‪ C‬ﺏ ∩ ﺟـ ‪} = E‬ﻡ{، ﻫـ ∋ ﻡ ﺏ ،‬ ‫‪C‬‬ ‫ﻭ ∋ ﻡ ‪ C ، E‬ﺟـ // ﻭ ﻫـ // ‪ E‬ﺏ‬ ‫ﺃﻭﺟﺪ:‬ ‫أ ﻃﻮﻝ ﻡ ﻭ‬ ‫ب ﻃﻮﻝ ‪ C‬ﻡ‬ ‫ﻭ‬ ‫81 ‪ C‬ﺏ ∩ ﺟـ ‪} = E‬ﻫـ{ ، ﺱ ∋ ‪ C‬ﺏ ، ﺹ ∋ ﺟـ ‪ ، E‬ﻛﺎﻥ ﺱ ﺹ // ﺏ ‪ C // E‬ﺟـ‬ ‫ﺃﺛﺒﺖ ﺃﻥ: ‪ C‬ﺱ * ﻫـ ‪ = E‬ﺟـ ﺹ * ﻫـ ﺏ‬ ‫91 ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ، ﺍﺣﺴﺐ ﻗﻴﻢ ﺱ، ﺹ ﺍﻟﻌﺪﺩﻳﺔ:‬ ‫ب‬ ‫‪C‬‬ ‫أ‬ ‫−‬ ‫‪E‬‬ ‫+‬ ‫+‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫02 ‪ C‬ﺏ ﺟـ ‪ E‬ﺷﻜﻞ ﺭﺑﺎﻋﻰ ﻓﻴﻪ ‪ C‬ﺏ // ﺟـ ‪ ، E‬ﺗﻘﺎﻃﻊ ﻗﻄﺮﺍه ﻓﻰ ﻡ، ﻧﺼﻒ ﺏ ﺟـ ﻓﻰ ﻫـ،‬ ‫ﻭﺭﺳﻢ ﻫـ ﻭ // ﺏ ‪ ، C‬ﻭﻳﻘﻄﻊ ﺏ ‪ E‬ﻓﻰ ﺱ ، ‪ C‬ﺟـ ﻓﻰ ﺹ ، ‪ E C‬ﻓﻰ ﻭ.‬ ‫ﺃﺛﺒﺖ ﺃﻥ:‬ ‫أ ﻫـ ﺹ = ١ ‪ C‬ﺏ.‬ ‫٢‬ ‫¯‬ ‫−‬ ‫+‬ ‫+‬ ‫ﺟ‬ ‫+‬ ‫ب ‪C‬ﺹ ﺏﺱ‬ ‫=‬ ‫ﺟـ ﻡ ‪ E‬ﻡ‬ ‫−‬ ‫¯‬ ‫−‬
‫ﻣﻨﺼﻔﺎ اﻟﺰواﻳﺔ ﻓﻰ اﻟﻤﺜﻠﺚ وا ﺟﺰاء اﻟﻤﺘﻨﺎﺳﺒﺔ‬ ‫3-2‬ ‫‪Angle Bisectors and Proportional Parts‬‬ ‫: ‪ E C‬ﻳﻨﺼﻒ ‪ . Cc‬ﺃﻛﻤﻞ:‬ ‫1‬ ‫‪E‬‬ ‫أ ﺏﺟـ =‬ ‫‪E‬‬ ‫ب ‪ C‬ﺟـ =‬ ‫.....................................‬ ‫‪C‬ﺏ‬ ‫ﺟ ﺏ‪E‬‬ ‫= ........................................‬ ‫ﺏ‪C‬‬ ‫‪C‬‬ ‫........................................‬ ‫د ‪ C‬ﺏ * ﺟـ ‪= E‬‬ ‫‪E‬‬ ‫........................‬ ‫2 ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ، ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺱ )ﺍﻷﻃﻮﺍﻝ ﻣﻘﺪﺭﺓ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ(‬ ‫ب‬ ‫‪C‬‬ ‫أ ‪C‬‬ ‫+‬ ‫−‬ ‫‪E‬‬ ‫‪E‬‬ ‫......................................................................................................‬ ‫‪C‬‬ ‫ﺟ‬ ‫......................................................................................................‬ ‫د‬ ‫+‬ ‫‪C‬‬ ‫+‬ ‫+‬ ‫‪E‬‬ ‫‪E‬‬ ‫......................................................................................................‬ ‫......................................................................................................‬ ‫3 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻣﺤﻴﻄﻪ ٧٢ﺳﻢ، ﺭﺳﻢ ﺏ ‪ E‬ﻳﻨﺼﻒ ‪ c‬ﺏ ﻭﻳﻘﻄﻊ ‪ C‬ﺟـ ﻓﻰ ‪.E‬‬ ‫ﺇﺫﺍ ﻛﺎﻥ ‪٤ = E C‬ﺳﻢ، ﺟـ ‪٥ = E‬ﺳﻢ، ﺃﻭﺟﺪ ﻃﻮﻝ ﻛﻞ ﻣﻦ ‪ C‬ﺏ ، ﺏ ﺟـ ، ‪E C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫4 ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺱ، ﺛﻢ ﺃﻭﺟﺪ ﻣﺤﻴﻂ 9‪ C‬ﺏ ﺟـ.‬ ‫ﺟ‬ ‫ب‬ ‫‪C‬‬ ‫‪C‬‬ ‫أ‬ ‫+‬ ‫‪E‬‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪E‬‬ ‫5 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻓﻴﻪ ‪ C‬ﺏ = ٨ﺳﻢ، ‪ C‬ﺟـ = ٤ﺳﻢ، ﺏ ﺟـ = ٦ﺳﻢ، ﺭﺳﻢ ‪ E C‬ﻳﻨﺼﻒ ‪ C‬ﻭﻳﻘﻄﻊ ﺏ ﺟـ ﻓﻰ ‪ ،E‬ﻭﺭﺳﻢ‬ ‫‪ C‬ﻫـ ﻳﻨﺼﻒ ‪ Cc‬ﺍﻟﺨﺎﺭﺟﺔ ﻭﻳﻘﻄﻊ ﺏ ﺟـ ﻓﻰ ﻫـ ﺃﻭﺟﺪ ﻃﻮﻝ ﻛﻞ ﻣﻦ ‪ E‬ﻫـ ، ‪ C ، E C‬ﻫـ .‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
‫6 ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ: ﺃﺛﺒﺖ ﺃﻥ ﺱ ﺹ // ﺏ ﺟـ‬ ‫‪E‬‬ ‫أ‬ ‫ب‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫......................................................................................................‬ ‫......................................................................................................‬ ‫......................................................................................................‬ ‫......................................................................................................‬ ‫7 ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ، ﺃﺛﺒﺖ ﺃﻥ ﺏ ﻫـ ﻳﻨﺼﻒ ‪ Cc‬ﺏ ﺟـ.‬ ‫ب‬ ‫‪C‬‬ ‫أ‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫......................................................................................................‬ ‫......................................................................................................‬ ‫......................................................................................................‬ ‫......................................................................................................‬ ‫‪C‬‬ ‫8‬ ‫: ﻫـ ‪ // E‬ﺱ ﺹ // ﺏ ﺟـ ،‬ ‫‪ * E C‬ﺏ ﺱ = ‪ C‬ﺟـ * ﻫـ ﺱ.‬ ‫ﺃﺛﺒﺖ ﺃﻥ ‪ C‬ﺹ ﻳﻨﺼﻒ ‪c‬ﺟـ ‪.E C‬‬ ‫‪E‬‬ ‫.................................................................................................................................‬ ‫.................................................................................................................................‬ ‫9 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ‪ ∋ E‬ﺏ ﺟـ ، ‪ ∌ E‬ﺏ ﺟـ ﺣﻴﺚ ﺟـ ‪ C = E‬ﺏ. ﺭﺳﻢ ﺟـ ﻫـ // ‪ C E‬ﻭﻳﻘﻄﻊ ‪ C‬ﺏ ﻓﻰ ﻫـ، ﻭﺭﺳﻢ‬ ‫ﻫـ ﻭ // ﺏ ﺟـ ﻭﻳﻘﻄﻊ ‪ C‬ﺟـ ﻓﻰ ﻭ ﺃﺛﺒﺖ ﺃﻥ ﺏ ﻭ ﻳﻨﺼﻒ ‪ Cc‬ﺏ ﺟـ‬ ‫..................................................................................................................................................................................................................................‬ ‫01‬ ‫: ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻓﻴﻪ ‪ C‬ﺏ = ٦ﺳﻢ، ‪ C‬ﺟـ = ٩ﺳﻢ،‬ ‫ﺏ ﺟـ = ٠١ﺳﻢ. ‪ ∋ E‬ﺏ ﺟـ ﺑﺤﻴﺚ ﺏ ‪٤ = E‬ﺳﻢ .‬ ‫ﺭﺳﻢ ﺏ ﻫـ = ‪ E C‬ﻭﻳﻘﻄﻊ ‪ C ، E C‬ﺏ ﻓﻰ ﻫـ، ﻭ ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ.‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ ‪ E C‬ﻳﻨﺼﻒ ‪.Cc‬‬ ‫ب ﺃﻭﺟﺪ ‪ C9) W‬ﺏ ﻭ( : ‪9) W‬ﺟـ ﺏ ﻭ(‬ ‫‪C‬‬ ‫‪E‬‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
‫ﺗﻄﺒﻴﻘﺎت اﻟﺘﻨﺎﺳﺐ ﻓﻰ اﻟﺪاﺋﺮة‬ ‫3-3‬ ‫‪Applications of Proportionality in the Circle‬‬ ‫1 ﺣﺪﺩ ﻣﻮﻗﻊ ﻛﻞ ﻣﻦ ﺍﻟﻨﻘﻂ ﺍﻟﺘﺎﻟﻴﺔ ﺑﺎﻟﻨﺴﺒﺔ ﺇﻟﻰ ﺍﻟﺪﺍﺋﺮﺓ ﻡ، ﻭﺍﻟﺘﻰ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٠١ﺳﻢ، ﺛﻢ ﺍﺣﺴﺐ ﺑﻌﺪ ﻛﻞ‬ ‫ُ َ‬ ‫ﻧﻘﻄﺔ ﻋﻦ ﻛﺰ ﺍﻟﺪﺍﺋﺮﺓ.‬ ‫ﻣﺮ‬ ‫ﺟ ‪X‬ﻡ)ﺟـ( = ﺻﻔﺮ‬ ‫ب ‪X‬ﻡ)ﺏ( = ٦٩‬ ‫أ ‪X‬ﻡ) ‪٣٦ - = ( C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫2 ﺃﻭﺟﺪ ﻗﻮﺓ ﺍﻟﻨﻘﻄﺔ ﺍﻟﻤﻌﻄﺎﺓ ﺑﺎﻟﻨﺴﺒﺔ ﺇﻟﻰ ﺍﻟﺪﺍﺋﺮﺓ ﻡ، ﻭﺍﻟﺘﻰ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ‪:H‬‬ ‫.................................................................................................................‬ ‫أ ﺍﻟﻨﻘﻄﺔ ‪ C‬ﺣﻴﺚ ‪ C‬ﻡ = ٢١ﺳﻢ ، ‪ ٩ = H‬ﺳﻢ‬ ‫.................................................................................................................‬ ‫ب ﺍﻟﻨﻘﻄﺔ ﺏ ﺣﻴﺚ ﺏ ﻡ = ٨ ﺳﻢ ، ‪ ١٥ = H‬ﺳﻢ‬ ‫.................................................................................................................‬ ‫ﺟ ﺍﻟﻨﻘﻄﺔ ﺟـ ﺣﻴﺚ ﺟـ ﻡ = ٧ ﺳﻢ ، ‪ ٧ = H‬ﺳﻢ‬ ‫.................................................................................................................‬ ‫د ﺍﻟﻨﻘﻄﺔ ‪ E‬ﺣﻴﺚ ‪ E‬ﻡ = ٧١ ﺳﻢ، ‪ ٤ = H‬ﺳﻢ‬ ‫3 ﺇﺫﺍﻛﺎﻥ ﺑﻌﺪ ﻧﻘﻄﺔ ﻋﻦ ﻛﺰ ﺩﺍﺋﺮﺓ ﻳﺴﺎﻭﻯ ٥٢ﺳﻢ ﻭﻗﻮﺓ ﻫﺬه ﺍﻟﻨﻘﻄﺔ ﺑﺎﻟﻨﺴﺒﺔ ﺇﻟﻰ ﺍﻟﺪﺍﺋﺮﺓ ﻳﺴﺎﻭﻯ ٠٠٤.‬ ‫ﻣﺮ‬ ‫ﺃﻭﺟﺪ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮ ﻫﺬه ﺍﻟﺪﺍﺋﺮﺓ.  ..............................................................................................................................................‬ ‫ﻣﺮ‬ ‫4 ﺍﻟﺪﺍﺋﺮﺓ ﻡ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٠٢ﺳﻢ. ‪ C‬ﻧﻘﻄﺔ ﺗﺒﻌﺪ ﻋﻦ ﻛﺰ ﺍﻟﺪﺍﺋﺮﺓ ﻣﺴﺎﻓﺔ ٦١ﺳﻢ، ﺭﺳﻢ ﺍﻟﻮﺗﺮ ﺏ ﺟـ‬ ‫ﺣﻴﺚ ‪ ∋ C‬ﺏ ﺟـ ، ‪ C‬ﺏ = ٢ ‪ C‬ﺟـ. ﺇﺣﺴﺐ ﻃﻮﻝ ﺍﻟﻮﺗﺮ ﺏ ﺟـ .‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫5‬ ‫: ﺍﻟﺪﺍﺋﺮﺗﺎﻥ ﻡ، ﻥ ﻣﺘﻘﺎﻃﻌﺘﺎﻥ ﻓﻰ ‪ ،C‬ﺏ‬ ‫ﺣﻴﺚ ‪ C‬ﺏ ∩ ﺟـ ‪ ∩ E‬ﻫـ ﻭ = }ﺱ{، ﺱ ‪ E ٢ = E‬ﺟـ ، ﻫـ ﻭ = ٠١ﺳﻢ،‬ ‫‪X‬ﻥ )ﺱ( = ٤٤١.‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ ‪ C‬ﺏ ﻣﺤﻮﺭ ﺃﺳﺎﺳﻰ ﻟﻠﺪﺍﺋﺮﺗﻴﻦ ﻡ، ﻥ.‬ ‫ب ﺃﻭﺟﺪ ﻃﻮﻝ ﻛﻞ ﻣﻦ ﺱ ﺟـ ، ﺱ ﻭ‬ ‫ﺟ ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﺸﻜﻞ ﺟـ ‪ E‬ﻭ ﻫـ ﺭﺑﺎﻋﻰ ﺩﺍﺋﺮﻯ.‬ ‫‪E‬‬ ‫‪C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
‫6 ﻣﺴﺘﻌﻴﻨﺎ ﺑﻤﻌﻄﻴﺎﺕ ﺍﻟﺸﻜﻞ، ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺮﻣﺰ ﺍﻟﻤﺴﺘﺨﺪﻡ ﻓﻰ ﺍﻟﻘﻴﺎﺱ.‬ ‫ً‬ ‫ب‬ ‫‪E‬‬ ‫‪C‬‬ ‫أ‬ ‫‪c‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫ﺟ‬ ‫‪c‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫................................................................‬ ‫د‬ ‫................................................................‬ ‫‪E‬‬ ‫ﻫ‬ ‫و‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫‪c‬‬ ‫ز‬ ‫................................................................‬ ‫ح‬ ‫‪E‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫ط‬ ‫‪E‬‬ ‫− ‪c‬‬ ‫‪C‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫+‬ ‫‪C‬‬ ‫................................................................‬ ‫− ‪c‬‬ ‫‪c‬‬ ‫................................................................‬ ‫+‬ ‫‪E‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪E c‬‬ ‫................................................................‬ ‫+‬ ‫‪c‬‬ ‫................................................................‬ ‫‪c‬‬ ‫................................................................‬ ‫‪c‬‬ ‫7‬ ‫: ‪c)X‬ﺏ ‪ C‬ﺟـ( = ٣٣‪c)X ،c‬ﺏ ‪ E‬ﺟـ( = ٠٧‪،c‬‬ ‫‪ C )X‬ﺏ ( = ٤٩‪ )X ، c‬ﺟـ ﺹ ( = ٠٠١‪ c‬ﺃﻭﺟﺪ ﻗﻴﺎﺱ ﻛﻞ ﻣﻦ:‬ ‫أ ﺱﺹ‬ ‫ب ‪C‬ﺱ‬ ‫ﺟ ‪c‬ﺏ ﻫـ ﺟـ‬ ‫‪c‬‬ ‫‪E‬‬ ‫‪c‬‬ ‫‪C‬‬ ‫‪c‬‬ ‫..................................................................................................................................................................................................................................‬ ‫8 ﺍﻟﺮﺑﻂ ﻣﻊ ﺍﻟﺼﻨﺎﻋﺔ: ﻣﻨﺸﺎﺭ ﺩﺍﺋﺮﻯ ﻟﻘﻄﻊ ﺍﻟﺨﺸﺐ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮ‬ ‫ﺩﺍﺋﺮﺗﻪ ٠١ﺳﻢ. ﻳﺪﻭﺭ ﺩﺍﺧﻞ ﺣﺎﻓﻈﺔ ﺣﻤﺎﻳﺔ، ﻓﺈﺫﺍ ﻛﺎﻥ ‪c)X‬ﺏ ‪= (E C‬‬ ‫٥٤‪ )X ،c‬ﺏ ‪ c١٥٥ = ( E‬ﺃﻭﺟﺪ ﻃﻮﻝ ﻗﻮﺱ ﻗﺮﺹ ﺍﻟﻤﻨﺸﺎﺭ ﺧﺎﺭﺝ ﺣﺎﻓﻈﺔ‬ ‫ﺍﻟﺤﻤﺎﻳﺔ.‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪E‬‬ ‫‪C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫‪C‬‬ ‫9 ﺍ ﺼﺎﻻﺕ: ﺗﺘﺒﻊ ﺍﻹﺷﺎﺭﺍﺕ ﺍﻟﺘﻰ ﺗﺼﺪﺭ ﻋﻦ ﺑﺮﺝ ﺍﻻﺗﺼﺎﻻﺕ ﻓﻰ ﻣﺴﺎﺭﻫﺎ‬ ‫ﺷﻌﺎﻋﺎ، ﻧﻘﻄﺔ ﺑﺪﺍﻳﺘﻪ ﻋﻠﻰ ﻗﻤﺔ ﺍﻟﺒﺮﺝ، ﻭﻳﻜﻮﻥ ﻣﻤﺎﺳﺎ ﻟﺴﻄﺢ ﺍﻷﺭﺽ،‬ ‫ً‬ ‫ً‬ ‫ﻛﻤﺎ ﻓﻰ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ. ﺣﺪﺩ ﻗﻴﺎﺱ ﺍﻟﻘﻮﺱ ﺍﻟﻤﺤﺼﻮﺭ ﺑﺎﻟﻤﻤﺎﺳﻴﻦ‬ ‫ﺑﻔﺮﺽ ﺃﻥ ﺍﻟﺒﺮﺝ ﻳﻘﻊ ﻋﻠﻰ ﻣﺴﺘﻮﻯ ﺳﻄﺢ ﺍﻟﺒﺤﺮ، ‪c)X‬ﺟـ ‪ C‬ﺏ( = ٠٨‪c‬‬ ‫¯‬ ‫−‬ ‫¯‬
‫ﺗﻤﺎﺭﻳﻦ ﻋﺎﻣﺔ‬ ‫1 ﺃﻛﻤﻞ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺘﺎﻟﻴﺔ:‬ ‫أ‬ ‫ب‬ ‫ﺟ‬ ‫د‬ ‫ﻫ‬ ‫ﺍﻟﻤﻨﺼﻔﺎﻥ ﺍﻟﺪﺍﺧﻠﻰ ﻭﺍﻟﺨﺎﺭﺟﻰ ﻟﺰﺍﻭﻳﺔ ﻭﺍﺣﺪﺓ‬ ‫ﻣﻨﺼﻔﺎﺕ ﺯﻭﺍﻳﺎ ﺍﻟﻤﺜﻠﺚ ﺗﺘﻘﺎﻃﻊ ﻓﻰ ..............................................................................................................................................‬ ‫ﺇﺫﺍ ﺭﺳﻢ ﻣﺴﺘﻘﻴﻢ ﻳﻮﺍﺯﻯ ﺃﺣﺪ ﺃﺿﻼﻉ ﻣﺜﻠﺚ، ﻭﻳﻘﻄﻊ ﺍﻟﻀﻠﻌﻴﻦ ﺍﻵﺧﺮﻳﻦ ﻓﺈﻧﻪ .........................................................‬ ‫ﺍﻟﻤﻨﺼﻒ ﺍﻟﺨﺎﺭﺟﻰ ﻟﺰﺍﻭﻳﺔ ﺭﺃﺱ ﺍﻟﻤﺜﻠﺚ ﺍﻟﻤﺘﺴﺎﻭﻯ ﺍﻟﺴﺎﻗﻴﻦ .................................... ﻗﺎﻋﺪﺓ ﺍﻟﻤﺜﻠﺚ.‬ ‫ﺇﺫﺍ ﻛﺎﻧﺖ ﻗﻮﺓ ﺍﻟﻨﻘﻄﺔ ‪ C‬ﺑﺎﻟﻨﺴﺒﺔ ﻟﻠﺪﺍﺋﺮﺓ ﻡ ﻛﻤﻴﺔ ﺳﺎﻟﺒﺔ، ﻓﺈﻥ ﻧﻘﻄﺔ ‪ C‬ﺗﻘﻊ .....................................................................‬ ‫......................................................................................................................‬ ‫2 ﻣﺴﺘﻌﻴﻨﺎ ﺑﻤﻌﻄﻴﺎﺕ ﺍﻟﺸﻜﻞ، ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺮﻣﺰ ﺍﻟﻤﺴﺘﺨﺪﻡ ﻓﻰ ﺍﻟﻘﻴﺎﺱ.‬ ‫ً‬ ‫ب‬ ‫+‬ ‫أ‬ ‫+‬ ‫‪E‬‬ ‫‪c‬‬ ‫−‬ ‫‪c‬‬ ‫ﺟ‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪c‬‬ ‫‪C‬‬ ‫−‬ ‫3 ﺩﺍﺋﺮﺗﺎﻥ ﻡ، ﻥ ﻣﺘﻘﺎﻃﻌﺘﺎﻥ ﻓﻰ ‪ ،C‬ﺏ.‬ ‫ﻫـ ‪ E‬ﻣﻤﺎﺱ ﻣﺸﺘﺮﻙ ﻟﻠﺪﺍﺋﺮﺗﻴﻦ ﻡ، ﻥ ﻋﻨﺪ ‪ ،E‬ﻫـ ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ،‬ ‫ﺏ ‪ E ∩ C‬ﻫـ = }ﺟـ{‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ: ﺏ ﺟـ ﻣﺤﻮﺭ ﺃﺳﺎﺳﻰ ﻟﻠﺪﺍﺋﺮﺗﻴﻦ.‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ‪ C‬ﺏ = ٩ﺳﻢ، ‪) X‬ﺟـ( = ٦٣، ﺃﻭﺟﺪ ﻃﻮﻝ ﺟـ ‪ ، C‬ﺟـ ‪E‬‬ ‫‪E‬‬ ‫‪C‬‬ ‫ﻥ‬ ‫4‬ ‫ﺃﺣﺪ ﺍﻟﺤﻮﺍﺟﺰ ﺍﻟﻤﺮﻭﺭﻳﺔ ‪ C‬ﺏ ﺟـ ‪ E‬ﻋﻠﻰ ﺷﻜﻞ‬ ‫ﻣﺴﺘﻄﻴﻞ ﻭﻣﻜﻮﻥ ﻣﻦ ﻣﺘﻮﺍﺯﻳﺔ ﻭﻣﺘﻄﺎﺑﻘﺔ، ﻭﻋﻠﻰ ﺃﺑﻌﺎﺩ ﻣﺘﺴﺎﻭﻳﺔ،‬ ‫ﻭﻣﺜﺒﺖ ﺑﻪ ﺩﻋﺎﻣﺘﺎﻥ ‪ C‬ﺟـ ، ﺏ ‪ ، E‬ﺗﻘﻄﻌﺎﻥ ﺃﺣﺪ ﺍﻟﻘﻀﺒﺎﻥ ﺍﻟﺮﺃﺳﻴﺔ ﻓﻰ‬ ‫ﻭ، ﻫـ ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ ﻓﺈﺫﺍ ﻛﺎﻥ ‪ C‬ﺏ = ٠٢١ﺳﻢ ﺃﻭﺟﺪ ﻃﻮﻝ ﻫـ ﻭ .‬ ‫5 ﻫﻨﺪﺳﺔ ﻣﻌﻤﺎ ﺔ: ﻣﻦ ﻧﻘﻄﺔ ‪ C‬ﻭﺍﻟﺘﻲ ﺗﺒﻌﺪ ٦٫١ ﻣﺘﺮﺍ ﻋﻦ ﻗﺎﻋﺪﺓ ﻗﻨﻄﺮﺓ‬ ‫ً‬ ‫ﺗﻌﻠﻮ ﺑﺎﺏ ﻣﻨﺰﻝ، ﻭﺟﺪ ﺃﻥ ﻗﻮﺓ ﺍﻟﻨﻘﻄﺔ ‪ C‬ﺑﺎﻟﻨﺴﺒﺔ ﻟﺪﺍﺋﺮﺓ ﻗﻮﺱ ﺍﻟﻘﻨﻄﺮﺓ‬ ‫ﻳﺴﺎﻭﻯ ٤٫٦ ﻣﺘﺮ ﻣﺮﺑﻊ.‬ ‫أ ﺃﻭﺟﺪ ﻃﻮﻝ ﻗﺎﻋﺪﺓ ﺍﻟﻘﻨﻄﺮﺓ )ﺏ ﺟـ(.‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ﺍﺭﺗﻔﺎﻉ ﺍﻟﻘﻨﻄﺮﺓ ﻳﺴﺎﻭﻯ ٠٨ﺳﻢ، ﻓﺄﻭﺟﺪ ﻗﻮﺓ ﺍﻟﻨﻘﻄﺔ ‪E‬‬ ‫ﺑﺎﻟﻨﺴﺒﺔ ﻟﺪﺍﺋﺮﺓ ﺍﻟﻘﻨﻄﺮﺓ ﻭﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ.‬ ‫‪ïM‬‬ ‫−‬ ‫‪E‬‬ ‫‪C‬‬ ‫¯‬ ‫‪E‬‬ ‫‪C‬‬
‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ‬ ‫1 ﻣﺴﺘﺨﺪﻣﺎ ﻣﻌﻄﻴﺎﺕ ﺍﻟﺸﻜﻞ، ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺮﻣﺰ ﺍﻟﻤﺴﺘﺨﺪﻡ ﻓﻰ ﺍﻟﻘﻴﺎﺱ.‬ ‫ً‬ ‫ب‬ ‫‪E‬‬ ‫‪C‬‬ ‫أ‬ ‫‪C‬‬ ‫ﺟ‬ ‫−‬ ‫‪E‬‬ ‫‪E‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪C‬‬ ‫+‬ ‫‪c‬‬ ‫: ‪ C c‬ﺟـ ﺏ ﻗﺎﺋﻤﺔ، ﺏ ﺟـ // ‪ E‬ﻫـ‬ ‫2‬ ‫ﺟـ ‪ // E‬ﻫـ ﻭ . ﺃﺛﺒﺖ ﺃﻥ:‬ ‫٢‬ ‫‪ C‬ﻭ * ‪ C‬ﺏ = )‪ C‬ﻫـ(٢ + )ﻫـ ‪(E‬‬ ‫‪E‬‬ ‫‪C‬‬ ‫3 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ، ﻥ ﻧﻘﻄﺔ ﺩﺍﺧﻞ ﺍﻟﻤﺜﻠﺚ. ﻧﺼﻔﺖ ﺍﻟﺰﻭﺍﻳﺎ ‪ C‬ﻥ ﺏ، ﺏ ﻥ ﺟـ ، ﺟـ ﻥ ‪C‬‬ ‫ﺑﻤﻨﺼﻔﺎﺕ ﻻﻗﺖ ‪ C‬ﺏ ، ﺏ ﺟـ ، ﺟـ ‪ C‬ﻓﻰ ‪ ،E‬ﻫـ، ﻭ ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ.‬ ‫‪ E C‬ﺏ ﻫـ ﺟـ ﻭ‬ ‫*‬ ‫*‬ ‫ﺃﺛﺒﺖ ﺃﻥ:‬ ‫‪ E‬ﺏ ﻫـ ﺟـ ﻭ ‪C‬‬ ‫=١‬ ‫4 ‪ C‬ﻧﻘﻄﺔ ﺧﺎﺭﺝ ﺍﻟﺪﺍﺋﺮﺓ ﻡ، ‪ C‬ﺏ ﻣﻤﺎﺱ ﻟﻠﺪﺍﺋﺮﺓ ﻋﻨﺪ ﺏ.‬ ‫ﺭﺳﻢ ‪ C‬ﺟـ ، ‪ C‬ﻫـ ﻳﻘﻄﻌﺎﻥ ﺍﻟﺪﺍﺋﺮﺓ ﻓﻰ ﺟـ، ‪ ،E‬ﻫـ، ﻭ ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ،‬ ‫‪ C‬ﺟـ = ٤ﺳﻢ، ﻫـ ﻭ = ٩ﺳﻢ.‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ ‪X‬ﻡ) ‪ ٣٦ = ( C‬ﺃﻭﺟﺪ ﻃﻮﻝ ﻛﻞ ﻣﻦ ‪ C‬ﺏ ، ‪ C‬ﻫـ ، ﺟـ ‪E‬‬ ‫ب ﺇﺫﺍ ﻛﺎﻧﺖ ﺱ ∋ ﺟـ ‪ E‬ﺣﻴﺚ ﺟـ ﺱ = ٢ﺳﻢ ﺃﻭﺟﺪ ‪X‬ﻡ)ﺱ(، ‪X‬ﻡ )‪.(E‬‬ ‫‪E‬‬ ‫‪C‬‬ ‫5 ‪ E C‬ﻣﺘﻮﺳﻂ ﻓﻰ 9‪ C‬ﺏ ﺟـ، ﺟـ ﺱ ﻳﻨﺼﻒ ‪ E C c‬ﺏ ﻭﻳﻘﻄﻊ ‪ C‬ﺏ ﻓﻰ ﺱ، ‪ E‬ﺹ ﻳﻨﺼﻒ ‪ E Cc‬ﺟـ ﻭﻳﻘﻄﻊ‬ ‫‪ C‬ﺟـ ﻓﻰ ﺹ.‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ: ﺱ ﺹ // ﺏ ﺟـ‬ ‫ب ﺇﺫﺍ ﺭﺳﻢ ‪ E‬ﻉ = ﺱ ﺹ ﻭ ﻳﻘﻄﻌﻪ ﻓﻰ ﻉ، ﻛﺎﻥ ﺱ ﻉ = ٩ﺳﻢ، ﻉ ﺹ = ٦١ﺳﻢ‬ ‫ﻭ‬ ‫ﺃﻭﺟﺪ ﻃﻮﻝ ﻛﻞ ﻣﻦ : ‪ E‬ﺱ ، ‪ E‬ﺹ .‬ ‫¯‬ ‫−‬ ‫¯‬
‫ﺍﺧﺘﺒﺎﺭ ﺗﺮﺍﻛﻤﻲ‬ ‫‪Oó©àe øe QÉ«àN’G á∏Ä°SCG‬‬ ‫ﺱ‬ ‫1 ﺇﺫﺍ ﻛﺎﻥ ٦ = ٩ ﻓﺈﻥ ﺱ ﺗﺴﺎﻭﻯ:‬ ‫٢‬ ‫ب ٦١‬ ‫أ ٢١‬ ‫ﺟ ٧٢‬ ‫2 ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ﺱ - ٠٢ = ﺻﻔﺮ ﻫﻤﺎ:‬ ‫ب ٤، -٥‬ ‫أ ٢، -٠١‬ ‫ﺟ ٥، -٤‬ ‫د ١٨‬ ‫د -٤، ٥‬ ‫‪C‬‬ ‫3 ﺇﺫﺍ ﻛﺎﻥ ‪ E‬ﻫـ // ﺏ ﺟـ ﻓﺈﻥ ‪ C‬ﺟـ ﻳﺴﺎﻭﻯ:‬ ‫ب ٤ﺳﻢ‬ ‫أ ٣ﺳﻢ‬ ‫د ٠١ﺳﻢ‬ ‫ﺟ ٦ﺳﻢ‬ ‫‪E‬‬ ‫4 ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻤﺴﺘﻘﻴﻤﺎﺕ ﻝ١، ﻝ٢، ﻝ٣ ﻣﺘﻮﺍﺯﻳﺔ، ﻳﻘﻄﻌﻬﺎ ﺍﻟﻤﺴﺘﻘﻴﻤﺎﻥ‬ ‫ﻡ، ﻡ/ ﻭﺍﻷﻃﻮﺍﻝ ﻣﻘﺪﺭﺓ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ ﻓﺈﻥ ﺱ ﺗﺴﺎﻭﻯ:‬ ‫ب ٣‬ ‫أ ٥‬ ‫د ٢‬ ‫ﺟ ٧‬ ‫−‬ ‫+‬ ‫5‬ ‫‪ E C‬ﻳﻨﺼﻒ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺨﺎﺭﺟﺔ‬ ‫ﻋﻨﺪ ‪ C‬ﻓﺈﻥ ﻃﻮﻝ ﺟـ ‪ E‬ﻳﺴﺎﻭﻯ:‬ ‫ب ٠١ﺳﻢ‬ ‫أ ٥ﺳﻢ‬ ‫د ٨١ﺳﻢ‬ ‫ﺟ ٢١ﺳﻢ‬ ‫‪C‬‬ ‫‪E‬‬ ‫6 ﺍﻟﺪﺍﺋﺮﺓ ﻡ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٥ﺳﻢ، ‪ E C‬ﻣﻤﺎﺱ ﻟﻠﺪﺍﺋﺮﺓ ﻋﻨﺪ ‪،E‬‬ ‫‪١٢ = E C‬ﺳﻢ ﻓﺈﻥ ﻃﻮﻝ ‪ C‬ﺟـ ﻳﺴﺎﻭﻯ:‬ ‫ب ٢١ﺳﻢ‬ ‫أ ٧ﺳﻢ‬ ‫د ٨١ﺳﻢ‬ ‫ﺟ ٥١ﺳﻢ‬ ‫7 ﺇﺫﺍ ﻛﺎﻧﺖ ﻣﺴﺎﺣﺔ ﺳﻄﺢ 9‪ E C‬ﻫـ = ٦١ﺳﻢ‬ ‫ﻓﺈﻥ ﻣﺴﺎﺣﺔ ﺳﻄﺢ ﺍﻟﻤﺜﻠﺚ ‪ C‬ﺏ ﺟـ = .......................... ﺳﻢ٢.‬ ‫أ ٦١‬ ‫ب ٢٣‬ ‫ﺟ ٤٦‬ ‫د ٨٢١‬ ‫‪E‬‬ ‫‪C‬‬ ‫٢‬ ‫‪ïM‬‬ ‫−‬ ‫‪C‬‬ ‫‪E‬‬
‫ﺍﺧﺘﺒﺎﺭ ﺗﺮﺍﻛﻤﻰ‬ ‫‪:Iô«°ü≤dG äÉHÉLE’G äGP á∏Ä°SC’G‬‬ ‫8‬ ‫:‬ ‫‪ C‬ﺏ // ﺟـ ‪ ، E‬ﺏ ﻫـ = ٢ﺳﻢ، ﺟـ ﻫـ = ٣ﺳﻢ،‬ ‫‪١٠ = E C‬ﺳﻢ. ﺃﻭﺟﺪ ﻃﻮﻝ ﻫـ ‪E‬‬ ‫‪C‬‬ ‫‪E‬‬ ‫: ﺏ ﻫـ ﻳﻨﺼﻒ ‪c‬ﺏ،‬ ‫9‬ ‫ﻭﻳﻘﻄﻊ ‪ C‬ﺟـ ﻓﻰ ﻫـ. ‪ C‬ﺏ = ٦ﺳﻢ، ﺟـ ‪٥ = E‬ﺳﻢ، ‪٧٫٥ = C E‬ﺳﻢ‬ ‫ﺏ ﺟـ = ٤ﺳﻢ . ﺃﺛﺒﺖ ﺃﻥ ‪ E‬ﻫـ ﻳﻨﺼﻒ ‪ E C c‬ﺟـ.‬ ‫‪E‬‬ ‫‪C‬‬ ‫:‬ ‫01‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪ C‬ﺏ ، ﺟـ ‪ E‬ﻭﺗﺮﺍﻥ ﻓﻰ ﺍﻟﺪﺍﺋﺮﺓ، ‪ C‬ﺏ ∩ ﺟـ ‪} = E‬ﻫـ{‬ ‫ﺃﺛﺒﺖ ﺃﻥ 9‪ C‬ﻫـ ﺟـ + 9‪ E‬ﻫـ ﺏ‬ ‫‪á∏jƒ£dG äÉHÉLE’G äGP øjQɪàdG‬‬ ‫11‬ ‫: ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻓﻴﻪ ‪ C‬ﺏ = ٢ ﺏ ﺟـ = ٢١ﺳﻢ،‬ ‫‪ C‬ﺟـ = ٩ﺳﻢ، ‪ C ∋ E‬ﺏ ﺣﻴﺚ ‪٣ = E C‬ﺳﻢ،‬ ‫ﻫـ ∋ ‪ C‬ﺟـ ﺣﻴﺚ ‪ C‬ﻫـ = ٤ﺳﻢ.‬ ‫ﺃﺛﺒﺖ ﺃﻥ 9 ‪ C‬ﺏ ﺟـ + 9 ‪ C‬ﻫـ ‪E‬‬ ‫ﺛﻢ ﺃﻭﺟﺪ ﻃﻮﻝ ﻫـ ‪. E‬‬ ‫‪C‬‬ ‫‪E‬‬ ‫21 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ، ‪ ∋ E‬ﺏ ﺟـ ، ‪ ∌ E‬ﺏ ﺟـ ، ﺭﺳﻢ ‪ E‬ﻭ ﻓﻘﻄﻊ ‪ C‬ﺟـ ، ‪ C‬ﺏ ﻓﻰ ﻫـ، ﻭ ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ‬ ‫ﻭﺏ‬ ‫ﻓﺈﺫﺍ ﻛﺎﻥ ﺍﻟﺸﻜﻞ ﺏ ﺟـ ﻫـ ﻭ ﺭﺑﺎﻋﻴﺎ ﺩﺍﺋﺮ ﻳﺎ ﺃﺛﺒﺖ ﺃﻥ ﺏ ‪ = E‬ﺟـ ﻫـ .‬ ‫ًّ‬ ‫ًّ‬ ‫‪ E‬ﻫـ‬ ‫‪?á«aÉ°VEG IóYÉ°ùªd êÉàëJ πg‬‬ ‫ﺃﻥ ﻟﻢ ﺗﺴﺘﻄﻊ ﺇﺟﺎﺑﺔ ﺃﻯ ﻣﻦ ﺍﻷﺳﺌﻠﺔ ﺍﻟﺴﺎﺑﻘﺔ ﻓﺎﺭﺟﻊ ﻟﻠﺠﺪﻭﻝ ﺍﻟﺘﺎﻟﻲ:‬ ‫١‬ ‫−‬ ‫¯‬ ‫٢‬ ‫٣‬ ‫٤‬ ‫٥‬ ‫٦‬ ‫٧‬ ‫٨‬ ‫٩‬ ‫٠١‬ ‫١١‬ ‫٢١‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫¯‬
‫‪IóMƒdG‬‬ ‫4‬ ‫ﺣﺴﺎﺏ ﺍﻟﻤﺜﻠﺜﺎﺕ‬ ‫‪Trigonometry‬‬ ‫دروس اﻟﻮﺣﺪة‬ ‫ﺍﻟﺪﺭﺱ )٤ - ١(: ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﻮﺟﻬﺔ.‬ ‫ﺍﻟﺪﺭﺱ )٤ - ٢(: ﻃﺮﻕ ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ.‬ ‫ﺍﻟﺪﺭﺱ )٤ - ٣(: ﺍﻟﺪﻭﺍﻝ ﺍﻟﻤﺜﻠﺜﻴﺔ.‬ ‫ﺍﻟﺪﺭﺱ )٤ - ٤(: ﺍﻟﻌﻼﻗﺎﺕ ﺑﻴﻦ ﺍﻟﺪﻭﺍﻝ ﺍﻟﻤﺜﻠﺜﻴﺔ.‬ ‫ﺍﻟﺪﺭﺱ )٤ - ٥(: ﺍﻟﺘﻤﺜﻴﻞ ﺍﻟﺒﻴﺎﻧﻰ ﻟﻠﺪﻭﺍﻝ ﺍﻟﻤﺜﻠﺜﻴﺔ.‬ ‫ﺍﻟﺪﺭﺱ )٤ - ٦(: ﺇﻳﺠﺎﺩ ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ﺑﻤﻌﻠﻮﻣﻴﺔ ﺩﺍﻟﺔ ﻣﺜﻠﺜﻴﺔ.‬
‫اﻟﺰاوﻳﺔ اﻟﻤﻮﺟﻬﺔ‬ ‫4-1‬ ‫‪Directed Angle‬‬ ‫:‬ ‫1‬ ‫أ ﺗﻜﻮﻥ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﻮﺟﻬﺔ ﻓﻰ ﻭﺿﻊ ﻗﻴﺎﺳﻰ ﺇﺫﺍ ﻛﺎﻥ‬ ‫ب‬ ‫ﻳﻘﺎﻝ ﻟﻠﺰﺍﻭﻳﺔ ﺍﻟﻤﻮﺟﻬﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﺃﻧﻬﺎ ﻣﺘﻜﺎﻓﺌﺔ ﺇﺫﺍ ﻛﺎﻥ .............................................................................‬ ‫.................................................................................................‬ ‫ﺟ ﺗﻜﻮﻥ ﺍﻟﺰﺍﻭﻳﺔ ﻣﻮﺟﺒﺔ ﺇﺫﺍ ﻛﺎﻥ ﺩﻭﺭﺍﻥ ﺍﻟﺰﺍﻭﻳﺔ.................................. ﻭﺗﻜﻮﻥ ﺳﺎﻟﺒﺔ ﺇﺫﺍ ﻛﺎﻥ ﺩﻭﺭﺍﻥ ﺍﻟﺰﺍﻭﻳﺔ‬ ‫د ﺇﺫﺍ ﻭﻗﻊ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻟﻠﺰﺍﻭﻳﺔ ﺍﻟﻤﻮﺟﻬﺔ ﻋﻠﻰ ﺃﺣﺪ ﻣﺤﺎﻭﺭ ﺍﻹﺣﺪﺍﺛﻴﺎﺕ ﺗﺴﻤﻰ ..................................‬ ‫ﻫ ﺇﺫﺍ ﻛﺎﻥ )‪ (i‬ﺯﺍﻭﻳﺔ ﻣﻮﺟﻬﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ، ﻥ∋ ‪ N‬ﻓﺈﻥ )‪ + i‬ﻥ * ٠٦٣‪ (c‬ﺗﺴﻤﻰ ﺑﺎﻟﺰﻭﺍﻳﺎ‬ ‫و‬ ‫ﺃﺻﻐﺮ ﻗﻴﺎﺱ ﻣﻮﺟﺐ ﻟﻠﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ ٠٣٥‪ c‬ﻫﻮ ..................................‬ ‫ز ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ ٠٣٩‪ c‬ﺗﻘﻊ ﻓﻰ ﺍﻟﺮﺑﻊ‬ ‫2 ﺃﻱ ﻣﻦ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﻤﻮﺟﻬﺔ ﺍﻵﺗﻴﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻲ ‬ ‫..................................‬ ‫......................................................................................................................‬ ‫ب‬ ‫ﺟ‬ ‫د‬ ‫3 ﺃﻭﺟﺪ ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﻮﺟﻬﺔ ‪ i‬ﺍﻟﻤﺸﺎﺭ ﺇﻟﻴﻬﺎ ﻓﻰ ﻛﻞ ﺷﻜﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ:‬ ‫ﺟ‬ ‫ب‬ ‫د‬ ‫أ‬ ‫‪i‬‬ ‫‪i‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫..................................‬ ‫..................................‬ ‫ح ﺃﺻﻐﺮ ﻗﻴﺎﺱ ﻣﻮﺟﺐ ﻟﻠﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ –٠٩٦‪ c‬ﻫﻮ‬ ‫أ‬ ‫...............................‬ ‫‪c‬‬ ‫‪i‬‬ ‫..............................................................‬ ‫..............................................................‬ ‫..............................................................‬ ‫.............................................................‬ ‫4 ﻋﻴﻦ ﺍﻟﺮﺑﻊ ﺍﻟﺬﻯ ﺗﻘﻊ ﻓﻴﻪ ﻛﻞ ﻣﻦ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﺘﻰ ﻗﻴﺎﺳﺎﺗﻬﺎ ﻛﺎﻵﺗﻰ:‬ ‫أ ٤٢‪       c‬ب ٥١٢‪       c‬ﺟ - ٠٤‪       c‬د -٠٢٢‪       c‬ﻫ ٠٤٦‪c‬‬ ‫........................................  ........................................    ........................................    ........................................     ........................................‬ ‫¯‬ ‫‪i‬‬ ‫−‬ ‫¯‬
‫5 ﺿﻊ ﻛﻼ ﻣﻦ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻵﺗﻴﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ، ﻣﻮﺿﺤﺎ ﺫﻟﻚ ﺑﺎﻟﺮﺳﻢ:‬ ‫ًّ‬ ‫ً‬ ‫أ ٢٣‪       c‬ب ٠٤١‪       c‬ﺟ - ٠٨‪       c‬د -٠١١‪       c‬ﻫ -٥١٣‪c‬‬ ‫6 ﻋﻴﻦ ﺃﺣﺪ ﺍﻟﻘﻴﺎﺳﺎﺕ ﺍﻟﺴﺎﻟﺒﺔ ﻟﻜﻞ ﺯﺍﻭﻳﺔ ﻣﻦ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻵﺗﻴﺔ:‬ ‫ﺟ ٠٩‪c‬‬ ‫ب ٦٣١‪c‬‬ ‫أ ٣٨‪c‬‬ ‫........................................‬ ‫د ٤٦٢‪c‬‬ ‫........................................‬ ‫........................................‬ ‫و ٠٧٠١‪c‬‬ ‫ﻫ ٤٦٩‪c‬‬ ‫........................................‬ ‫........................................‬ ‫........................................‬ ‫7 ﻋﻴﻦ ﺃﺻﻐﺮ ﻗﻴﺎﺱ ﻣﻮﺟﺐ ﻟﻜﻞ ﺯﺍﻭﻳﺔ ﻣﻦ ﺍﻟﺰﺍﻭﻳﺎ ﺍﻵﺗﻴﺔ:‬ ‫ﺟ -٥١٣‪c‬‬ ‫ب -٧١٢‪c‬‬ ‫أ -٣٨١‪c‬‬ ‫8 ﺃﻯ ﻣﻦ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﻤﻮﺟﻬﺔ ﻓﻰ ﺍﻷﺯﻭﺍﺝ ﺍﻟﻤﺮﺗﺒﺔ ﺍﻵﺗﻴﺔ ﻓﻰ ﺍﻟﺸﻜﻞ‬ ‫ﺍﻟﻤﻘﺎﺑﻞ ﻓﻰ ﻭﺿﻊ ﻗﻴﺎﺳﻰ? ﻟﻤﺎﺫﺍ?‬ ‫أ ) ﻭ‪ ، C‬ﻭ ‪( E‬‬ ‫ب ) ﻭ ﺯ ، ﻭ ﺟـ (‬ ‫ﺟ ) ‪C‬ﺏ ، ‪ C‬ﺟـ (‬ ‫‪E‬‬ ‫د ) ﻭ ﻫـ ، ﻭ ‪( E‬‬ ‫ﻫ ) ﻭ‪ ، E‬ﻭ ﺯ (‬ ‫د -٠٧٥‪c‬‬ ‫و ) ﻭﺏ ، ﻭﺯ (‬ ‫‪C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫9 ﻳﺪﻭﺭ ﺃﺣﺪ ﻻﻋﺒﻰ ﺍﻟﺠﻤﺒﺎﺯ ﻋﻠﻰ ﺟﻬﺎﺯ ﺍﻷﻟﻌﺎﺏ ﺑﺰﺍﻭﻳﺔ ﻗﻴﺎﺳﻬﺎ ٠٠٢‪ c‬ﺍﺭﺳﻢ ﻫﺬه ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ‬ ‫ﻛﺎﻥ ﻣﻊ‬ ‫01 ﺍﻛﺘﺸﻒ ﺍﻟﺨﻄﺄ: ﺍﻛﺘﺐ ﻗﻴﺎﺱ ﺃﺻﻐﺮ ﺯﺍﻭﻳﺔ ﺑﻘﻴﺎﺱ ﻣﻮﺟﺐ ﻭﺯﺍﻭﻳﺔ ﺃﺧﺮﻯ ﺑﻘﻴﺎﺱ ﺳﺎﻟﺐ ﺗﺸﺘﺮ‬ ‫ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻟﻠﺰﺍﻭﻳﺔ )-٥٣١‪(c‬‬ ‫¯‬ ‫ﺃﺻﻐﺮ ﺯﺍﻭﻳﺔ ﺑﻘﻴﺎﺱ ﻣﻮﺟﺐ = -٥٣١‪ c٤٥ = c١٨٠+ c‬ﺃﺻﻐﺮ ﺯﺍﻭﻳﺔ ﺑﻘﻴﺎﺱ ﻣﻮﺟﺐ = -٥٣١‪c٢٢٥ = c٣٦٠+ c‬‬ ‫ﺃﺻﻐﺮ ﺯﺍﻭﻳﺔ ﺑﻘﻴﺎﺱ ﺳﺎﻟﺐ = -٥٣١‪ c٣١٥- = c١٨٠ - c‬ﺃﺻﻐﺮ ﺯﺍﻭﻳﺔ ﺑﻘﻴﺎﺱ ﺳﺎﻟﺐ = -٥٣١‪c٤٩٥- = c٣٦٠ - c‬‬ ‫ﺃﻯ ﺍﻹﺟﺎﺑﺘﻴﻦ ﺻﺤﻴﺢ ? ﻓﺴﺮ ﺇﺟﺎﺑﺘﻚ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
‫ﻃﺮق ﻗﻴﺎس اﻟﺰاوﻳﺔ‬ ‫4-2‬ ‫‪Methods of measuring the angle‬‬ ‫‪k‬‬ ‫‪:Oó©àe øe QÉ«àNG :’hCG‬‬ ‫1 ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ ٠٦‪ c‬ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﺗﻜﺎﻓﺊ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ:‬ ‫ب ٠٤٢‪c‬‬ ‫أ ٠٢١‪c‬‬ ‫ﺟ ٠٠٣‪c‬‬ ‫١٣‬ ‫2 ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ ٦‪ r‬ﺗﻘﻊ ﻓﻰ ﺍﻟﺮﺑﻊ:‬ ‫٩‬‫3 ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ ٤‪ r‬ﺗﻘﻊ ﻓﻰ ﺍﻟﺮﺑﻊ:‬ ‫ﺟ ﺍﻟﺜﺎﻟﺚ‬ ‫د ﺍﻟﺮﺍﺑﻊ‬ ‫.............................................................................................................................................‬ ‫ب ﺍﻟﺜﺎﻧﻰ‬ ‫أ ﺍﻷﻭﻝ‬ ‫د ٠٢٤‪c‬‬ ‫....................................................................................................................................‬ ‫ب ﺍﻟﺜﺎﻧﻰ‬ ‫أ ﺍﻷﻭﻝ‬ ‫........................................................................‬ ‫ﺟ ﺍﻟﺜﺎﻟﺚ‬ ‫د ﺍﻟﺮﺍﺑﻊ‬ ‫4 ﺇﺫﺍ ﻛﺎﻥ ﻣﺠﻤﻮﻉ ﻗﻴﺎﺳﺎﺕ ﺯﻭﺍﻳﺎ ﺃﻯ ﻣﻀﻠﻊ ﻣﻨﺘﻈﻢ ﺗﺴﺎﻭﻯ ٠٨١ ْ)ﻥ – ٢( ﺣﻴﺚ ﻥ ﻋﺪﺩ ﺍﻷﺿﻼﻉ، ﻓﺈﻥ ﻗﻴﺎﺱ‬ ‫ﺯﺍﻭﻳﺔ ﺍﻟﻤﺨﻤﺲ ﺍﻟﻤﻨﺘﻈﻢ ﺑﺎﻟﻘﻴﺎﺱ ﺍﻟﺪﺍﺋﺮﻯ ﺗﺴﺎﻭﻯ: .....................................................................................................................‬ ‫أ‬ ‫ب ٧‪r‬‬ ‫٢‬ ‫‪r‬‬ ‫٣‬ ‫ﺟ‬ ‫5 ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ ٧‪ r‬ﻗﻴﺎﺳﻬﺎ ﺍﻟﺴﺘﻴﻨﻰ ﻳﺴﺎﻭﻯ:‬ ‫٣‬ ‫ب ٠١٢‪c‬‬ ‫أ ٥٠١‪c‬‬ ‫٣‪r‬‬ ‫٥‬ ‫........................................................................................................................‬ ‫ﺟ ٠٢٤‪c‬‬ ‫6 ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻘﻴﺎﺱ ﺍﻟﺴﺘﻴﻨﻰ ﻟﺰﺍﻭﻳﺔ ﻫﻮ ٨٤ َ ٤٦ ْ ﻓﺈﻥ ﻗﻴﺎﺳﻬﺎ ﺍﻟﺪﺍﺋﺮﻯ ﻳﺴﺎﻭﻯ:‬ ‫أ ٨١٫٠‬ ‫ب ٦٣٫٠‬ ‫‪E‬‬ ‫‪E‬‬ ‫د‬ ‫٢‪r‬‬ ‫٣‬ ‫ﺟ ٨١٫٠ ‪r‬‬ ‫د ٠٤٨‪c‬‬ ‫.......................................................................‬ ‫د ٦٣٫٠ ‪r‬‬ ‫7 ﻃﻮﻝ ﺍﻟﻘﻮﺱ ﻓﻰ ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻗﻄﺮﻫﺎ ٤٢ ﺳﻢ ﻭﻳﻘﺎﺑﻞ ﺯﺍﻭﻳﺔ ﻛﺰﻳﺔ ﻗﻴﺎﺳﻬﺎ ٠٣‪ c‬ﻳﺴﺎﻭﻯ:‬ ‫ﻣﺮ‬ ‫د ٥‪ r‬ﺳﻢ‬ ‫ﺟ ٤‪ r‬ﺳﻢ‬ ‫ب ٣‪ r‬ﺳﻢ‬ ‫أ ٢‪ r‬ﺳﻢ‬ ‫............................................‬ ‫8 ﺍﻟﻘﻮﺱ ﺍﻟﺬﻯ ﻃﻮﻟﻪ ٥‪r‬ﺳﻢ ﻓﻰ ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٥١ﺳﻢ ﻳﻘﺎﺑﻞ ﺯﺍﻭﻳﺔ ﻛﺰﻳﺔ ﻗﻴﺎﺳﻬﺎ ﻳﺴﺎﻭﻯ:‬ ‫ﻣﺮ‬ ‫ب ٠٦‪c‬‬ ‫أ ٠٣‪c‬‬ ‫ﺟ ٠٩‪c‬‬ ‫.................‬ ‫د ٠٨١‪c‬‬ ‫9 ﺇﺫﺍ ﻛﺎﻥ ﻗﻴﺎﺱ ﺇﺣﺪﻯ ﺯﺍﻭﻳﺎ ﻣﺜﻠﺚ ٥٧‪ c‬ﻭﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ﺃﺧﺮﻯ ﻓﻴﻪ ‪ r‬ﻓﺈﻥ ﺍﻟﻘﻴﺎﺱ ﺍﻟﺪﺍﺋﺮﻯ ﻟﻠﺰﺍﻭﻳﺔ ﺍﻟﺜﺎﻟﺜﺔ‬ ‫٤‬ ‫ﻳﺴﺎﻭﻯ: .................................................................................................................................................................................................................‬ ‫د ٥‪r‬‬ ‫ﺟ ‪r‬‬ ‫ب ‪r‬‬ ‫أ ‪r‬‬ ‫٣‬ ‫٤‬ ‫٦‬ ‫٢١‬ ‫¯‬ ‫−‬ ‫¯‬
‫‪¯I‬‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫01 ﺃﻭﺟﺪ ﺑﺪﻻﻟﺔ ‪ r‬ﺍﻟﻘﻴﺎﺱ ﺍﻟﺪﺍﺋﺮﻯ ﻟﻠﺰﻭﺍﻳﺎ ﺍﻟﺘﻰ ﻗﻴﺎﺳﺎﺗﻬﺎ ﻛﺎﻵﺗﻰ:‬ ‫.........................................‬ ‫ب  ٠٤٢‪c‬‬ ‫.........................................‬ ‫أ ٥٢٢‪c‬‬ ‫ﺟ‬ ‫.........................................‬ ‫د  ٠٠٣‪c‬‬ ‫٥٣١‪......................................... c‬‬‫.........................................‬ ‫و  ٠٨٧‪c‬‬ ‫.........................................‬ ‫ﻫ ٠٩٣‪c‬‬ ‫11 ﺃﻭﺟﺪ ﺑﺎﻟﺮﺍﺩﻳﺎﻥ ﺍﻟﻘﻴﺎﺱ ﺍﻟﺪﺍﺋﺮﻯ ﻟﻠﺰﻭﺍﻳﺎ ﺍﻟﺘﻰ ﻗﻴﺎﺳﺎﺗﻬﺎ ﻛﺎﻵﺗﻰ، ﻣﻘﺮﺑﺎ ﺍﻟﻨﺎﺗﺞ ﻟﺜﻼﺛﺔ ﺃﺭﻗﺎﻡ ﻋﺸﺮﻳﺔ:‬ ‫ً‬ ‫ﺟ‬ ‫٨٤ ً ٠٥ َ ٠٦١‪c‬‬ ‫ب ٨١ َ ٥٢‪c‬‬ ‫أ ٦٫٦٥‪c‬‬ ‫.................................................‬ ‫.................................................‬ ‫21 ﺃﻭﺟﺪ ﺍﻟﻘﻴﺎﺱ ﺍﻟﺴﺘﻴﻨﻰ ﻟﻠﺰﻭﺍﻳﺎ ﺍﻟﺘﻰ ﻗﻴﺎﺳﺎﺗﻬﺎ ﻛﺎﻵﺗﻰ، ﻣﻘﺮﺑﺎ ﺍﻟﻨﺎﺗﺞ ﻷﻗﺮﺏ ﺛﺎﻧﻴﺔ:‬ ‫ً‬ ‫‪E‬‬ ‫‪E‬‬ ‫ب ٧٢٫٢‬ ‫أ ٩٤٫٠‬ ‫.................................................‬ ‫.................................................‬ ‫.................................................‬ ‫١ ‪E‬‬ ‫ﺟ -٢٣‬ ‫.................................................‬ ‫31 ﺇﺫﺍ ﻛﺎﻧﺖ ‪ i‬ﺯﺍﻭﻳﺔ ﻛﺰﻳﺔ ﻓﻰ ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ‪ H‬ﻭﺗﺤﺼﺮ ﻗﻮﺳﺎ ﻃﻮﻟﻪ ﻝ :‬ ‫ﻣﺮ‬ ‫ً‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ ‪ ٢٠ = H‬ﺳﻢ، ‪ c٧٨ َ ١٥ ً ٢٠ = i‬ﺃﻭﺟﺪ ﻝ. .......................................................... )ﻷﻗﺮﺏ ﺟﺰﺀ ﻣﻦ ﻋﺸﺮﺓ(‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ﻝ = ٣٫٧٢ ﺳﻢ، ‪ c٧٨ َ ٠ ً ٢٤ = i‬ﺃﻭﺟﺪ ‪) ......................................................... .H‬ﻷﻗﺮﺏ ﺟﺰﺀ ﻣﻦ ﻋﺸﺮﺓ(‬ ‫41 ﺯﺍﻭﻳﺔ ﻛﺰﻳﺔ ﻗﻴﺎﺳﻬﺎ ٠٥١‪ c‬ﻭﺗﺤﺼﺮ ﻗﻮﺳﺎ ﻃﻮﻟﻪ ١١ ﺳﻢ، ﺍﺣﺴﺐ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮ ﺩﺍﺋﺮﺗﻬﺎ )ﻷﻗﺮﺏ ﺟﺰﺀ ﻣﻦ ﻋﺸﺮﺓ(‬ ‫ﻣﺮ‬ ‫ً‬ ‫..................................................................................................................................................................................................................................‬ ‫51 ﺃﻭﺟﺪ ﺍﻟﻘﻴﺎﺱ ﺍﻟﺪﺍﺋﺮﻯ ﻭﺍﻟﻘﻴﺎﺱ ﺍﻟﺴﺘﻴﻨﻰ ﻟﻠﺰﺍﻭﻳﺔ ﻛﺰﻳﺔ ﺍﻟﺘﻰ ﺗﻘﺎﺑﻞ ﻗﻮﺳﺎ ﻃﻮﻟﻪ ٧٫٨ ﺳﻢ ﻓﻰ ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻧﺼﻒ‬ ‫ﺍﻟﻤﺮ‬ ‫ً‬ ‫ﻗﻄﺮﻫﺎ ٤ ﺳﻢ. ................................................................................................................................................................................................................‬ ‫61 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻬﻨﺪﺳﺔ: ﻣﺜﻠﺚ ﻗﻴﺎﺱ ﺇﺣﺪﻯ ﺯﻭﺍﻳﺎه ٠٦‪ c‬ﻭﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ﺃﺧﺮﻯ ﻣﻨﻪ ﻳﺴﺎﻭﻯ ‪ r‬ﺃﻭﺟﺪ ﺍﻟﻘﻴﺎﺱ‬ ‫٤‬ ‫ﺍﻟﺪﺍﺋﺮﻯ ﻭﺍﻟﻘﻴﺎﺱ ﺍﻟﺴﺘﻴﻨﻰ ﻟﺰﺍﻭﻳﺘﻪ ﺍﻟﺜﺎﻟﺜﺔ. .........................................................................................................................................‬ ‫71 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻬﻨﺪﺳﺔ: ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٤ ﺳﻢ، ﺭﺳﻤﺖ ‪ Cc‬ﺏ ﺟـ ﺍﻟﻤﺤﻴﻄﻴﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ ٠٣‪ c‬ﺃﻭﺟﺪ‬ ‫ﻃﻮﻝ ﺍﻟﻘﻮﺱ ﺍﻷﺻﻐﺮ ‪ C‬ﺟـ .......................................................................................................................................................................‬ ‫81 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻬﻨﺪﺳﻴﺔ: ﻓﻰ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ ﺇﺫﺍ ﻛﺎﻥ ﻣﺴﺎﺣﺔ ﺍﻟﻤﺜﻠﺚ ﻡ ‪ C‬ﺏ‬ ‫ﺍﻟﻘﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ﻡ = ٢٣ ﺳﻢ٢ ﻓﺄﻭﺟﺪ ﻣﺤﻴﻂ ﺍﻟﺸﻜﻞ ﻣﻘﺮﺑﺎ ﺍﻟﻨﺎﺗﺞ ﻷﻗﺮﺏ‬ ‫ً‬ ‫ﺭﻗﻤﻴﻦ ﻋﺸﺮﻳﻴﻦ ............................................................................................................................‬ ‫‪C‬‬ ‫‪ïM‬‬ ‫−‬
‫91 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻬﻨﺪﺳﺔ: ‪ C‬ﺏ ﻗﻄﺮ ﻓﻰ ﺩﺍﺋﺮﺓ ﻃﻮﻟﻪ ٤٢ ﺳﻢ ، ﺭﺳﻢ ﺍﻟﻮﺗﺮ ‪ C‬ﺟـ ﺑﺤﻴﺚ ﻛﺎﻥ ﻕ)‪c‬ﺏ ‪C‬ﺟـ( = ٠٥‪c‬‬ ‫ﺃﻭﺟﺪ ﻃﻮﻝ ﺍﻟﻘﻮﺱ ﺍﻷﺻﻐﺮ ‪ C‬ﺟـ ﻣﻘﺮ ﺑﺎ ﺍﻟﻨﺎﺗﺞ ﻷﻗﺮﺏ ﺭﻗﻤﻴﻴﻦ ﻋﺸﺮﻳﻴﻦ. ..........................................................................‬ ‫ً‬ ‫02 ﻣﺴﺎﻓﺎﺕ: ﻛﻢ ﺍﻟﻤﺴﺎﻓﺔ ﺍﻟﺘﻰ ﺗﻘﻄﻌﻬﺎ ﻧﻘﻄﺔ ﻋﻠﻰ ﻃﺮﻑ ﻋﻘﺮﺏ ﺍﻟﺪﻗﺎﺋﻖ ﺧﻼﻝ ٠١ ﺩﻗﺎﺋﻖ ﺇﺫﺍ ﻛﺎﻥ ﻃﻮﻝ ﻫﺬﺍ‬ ‫ﺍﻟﻌﻘﺮﺏ ٦ ﺳﻢ?‬ ‫..................................................................................................................................................................................................................................‬ ‫12 ﻓﻠﻚ: ﻗﻤﺮ ﺻﻨﺎﻋﻰ ﻳﺪﻭﺭ ﺣﻮﻝ ﺍﻷﺭﺽ ﻓﻰ ﻣﺴﺎﺭ ﺩﺍﺋﺮﻯ ﺩﻭﺭﺓ ﻛﺎﻣﻠﺔ ﻛﻞ ٦ ﺳﺎﻋﺎﺕ، ﻓﺈﺫﺍ ﻛﺎﻥ ﻃﻮﻝ ﻧﺼﻒ‬ ‫ﻗﻄﺮ ﻣﺴﺎﺭه ﻋﻦ ﻛﺰ ﺍﻷﺭﺽ ٠٠٠٩ ﻛﻢ، ﻓﺄﻭﺟﺪ ﺳﺮﻋﺘﻪ ﺑﺎﻟﻜﻴﻠﻮﻣﺘﺮ ﻓﻰ ﺍﻟﺴﺎﻋﺔ.‬ ‫ﻣﺮ‬ ‫..................................................................................................................................................................................................................................‬ ‫22 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻬﻨﺪﺳﺔ: ﻓﻰ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ:‬ ‫‪ C‬ﺏ ، ‪ C‬ﺟـ ﻣﻤﺎﺳﺎﻥ ﻟﻠﺪﺍﺋﺮﺓ ﻡ، ‪ c) X‬ﺟـ‪ C‬ﺏ ( = ٠٦‪ C ،c‬ﺏ = ٢١ ﺳﻢ.‬ ‫ﺃﻭﺟﺪ ﻷﻗﺮﺏ ﻋﺪﺩ ﺻﺤﻴﺢ ﻃﻮﻝ ﺍﻟﻘﻮﺱ ﺍﻷﻛﺒﺮ ﺏ ﺟـ .‬ ‫‪C‬‬ ‫‪c‬‬ ‫.....................................................................................................................................................‬ ‫32 ﺍﻟﺮﺑﻂ ﺑﺎﻟﺰﻣﻦ: ﺗﺴﺘﺨﺪﻡ ﺍﻟﻤﺰﻭﻟﺔ ﺍﻟﺸﻤﺴﻴﺔ ﻟﺘﺤﺪﻳﺪ ﺍﻟﻮﻗﺖ ﺃﺛﻨﺎﺀ ﺍﻟﻨﻬﺎﺭ ﻣﻦ‬ ‫ﺧﻼﻝ ﻃﻮﻝ ﺍﻟﻈﻞ ﺍﻟﺬﻯ ﻳﺴﻘﻂ ﻋﻠﻰ ﺳﻄﺢ ﻣﺪﺭﺝ ﻹﻇﻬﺎﺭ ﺍﻟﺴﺎﻋﺔ ﻭﺃﺟﺰﺍﺋﻬﺎ،‬ ‫ﻓﺈﺫﺍ ﻛﺎﻥ ﺍﻟﻈﻞ ﻳﺪﻭﺭ ﻋﻠﻰ ﺍﻟﻘﺮﺹ ﺑﻤﻌﺪﻝ ٥١‪ c‬ﻟﻜﻞ ﺳﺎﻋﺔ.‬ ‫أ ﺃﻭﺟﺪ ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺑﺎﻟﺮﺍﺩﻳﺎﻥ ﺍﻟﺘﻰ ﻳﺪﻭﺭ ﺍﻟﻈﻞ ﻋﻨﻬﺎ ﺑﻌﺪ ﻣﺮﻭﺭ ٤ ﺳﺎﻋﺎﺕ.‬ ‫......................................................................................................................................................................‬ ‫ب ﺑﻌﺪ ﻛﻢ ﺳﺎﻋﺔ ﻳﺪﻭﺭ ﺍﻟﻈﻞ ﺑﺰﺍﻭﻳﺔ ﻗﻴﺎﺳﻬﺎ ٢‪ r‬ﺭﺍﺩﻳﺎﻥ?‬ ‫٣‬ ‫....................................‬ ‫ﺟ ﻣﺰﻭﻟﺔ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٤٢ ﺳﻢ، ﺃﻭﺟﺪ ﺑﺪﻻﻟﺔ ‪ r‬ﻃﻮﻝ ﺍﻟﻘﻮﺱ ﺍﻟﺬﻯ ﻳﺼﻨﻌﻪ ﺩﻭﺭﺍﻥ ﺍﻟﻈﻞ ﻋﻠﻰ ﺣﺎﻓﺔ‬ ‫ﺍﻟﻘﺮﺹ ﺑﻌﺪ ﻣﺮﻭﺭ ٠١ ﺳﺎﻋﺎﺕ. ......................................................................................................................................................‬ ‫42 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ: ﻣﺴﺘﻘﻴﻢ ﻳﺼﻨﻊ ﺯﺍﻭﻳﺔ ﻗﻴﺎﺳﻬﺎ ‪ r‬ﺭﺍﺩﻳﺎﻥ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﻟﺪﺍﺋﺮﺓ ﺍﻟﻮﺣﺪﺓ ﻣﻊ ﺍﻻﺗﺠﺎه ﺍﻟﻤﻮﺟﺐ‬ ‫٣‬ ‫ﻟﻤﺤﻮﺭ ﺍﻟﺴﻴﻨﺎﺕ. ﺃﻭﺟﺪ ﻣﻌﺎﺩﻟﺔ ﻫﺬﺍ ﺍﻟﻤﺴﺘﻘﻴﻢ. .................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
‫اﻟﺪوال اﻟﻤﺜﻠﺜﻴﺔ‬ ‫4-3‬ ‫‪Trigonometric Functions‬‬ ‫‪k‬‬ ‫‪:Oó©àe øe QÉ«àN’G :’hCG‬‬ ‫٣‬ ‫1 ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺔ ‪ i‬ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﻟﺪﺍﺋﺮﺓ ﺍﻟﻮﺣﺪﺓ ﻳﻤﺮ ﺿﻠﻌﻬﺎ ﺍﻟﻨﻬﺎﺋﻰ ﺑﺎﻟﻨﻘﻄﺔ ) ١ ، ٢ (‬ ‫٢‬ ‫ﻓﺈﻥ ﺟﺎ ‪ i‬ﺗﺴﺎﻭﻯ: ...........................................................................................................................................................................................‬ ‫ﺟ ٣‬ ‫أ ١‬ ‫د ٢‬ ‫ب ١‬ ‫٢‬ ‫٢‬ ‫٣‬ ‫٣‬ ‫2 ﺇﺫﺍ ﻛﺎﻧﺖ ﺟﺎ ‪ ١ = i‬ﺣﻴﺚ ‪ i‬ﺯﺍﻭﻳﺔﺣﺎﺩﺓ ﻓﺈﻥ ‪ (ic)X‬ﺗﺴﺎﻭﻯ‬ ‫٢‬ ‫ﺟ ٠٦‪c‬‬ ‫ب ٥٤‪c‬‬ ‫أ ٠٣‪c‬‬ ‫3 ﺇﺫﺍ ﻛﺎﻧﺖ ﺟﺎ ‪ ،١ - = i‬ﺟﺘﺎ ‪ ٠ = i‬ﻓﺈﻥ ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ‪ i‬ﺗﺴﺎﻭﻯ‬ ‫ﺟ ٣‪r‬‬ ‫أ ‪r‬‬ ‫ب ‪r‬‬ ‫...........................................................................................‬ ‫د ٠٩‪c‬‬ ‫..............................................................................................‬ ‫د ٢‪r‬‬ ‫٢‬ ‫٢‬ ‫4 ﺇﺫﺍ ﻛﺎﻧﺖ ﻗﺘﺎ ‪ ٢ = i‬ﺣﻴﺚ ‪ i‬ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ ﻓﺈﻥ ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ‪ i‬ﺗﺴﺎﻭﻯ‬ ‫ﺟ ٥٤‪c‬‬ ‫ب ٠٣‪c‬‬ ‫أ ٥١‪c‬‬ ‫٣‬ ‫5 ﺇﺫﺍ ﻛﺎﻧﺖ ﺟﺘﺎ ‪ ، ١ = i‬ﺟﺎ ‪ ٢ - = i‬ﻓﺈﻥ ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ‪ i‬ﺗﺴﺎﻭﻯ‬ ‫٢‬ ‫ﺟ ٥‪r‬‬ ‫ب ٥‪r‬‬ ‫٢‪r‬‬ ‫أ‬ ‫٦‬ ‫٣‬ ‫....................................................................‬ ‫.........................................................................................‬ ‫٣‬ ‫6 ﺇﺫﺍ ﻛﺎﻧﺖ ﻇﺎ ‪ ١ = i‬ﺣﻴﺚ ‪ i‬ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ ﻣﻮﺟﺒﺔ ﻓﺈﻥ ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ‪ i‬ﺗﺴﺎﻭﻯ‬ ‫ﺟ ٥٤‪c‬‬ ‫ب ٠٣‪c‬‬ ‫أ ٠١‪c‬‬ ‫7 ﻇﺎ ٥٤‪ + c‬ﻇﺘﺎ ٥٤‪ - c‬ﻗﺎ ٠٦‪ c‬ﺗﺴﺎﻭﻯ‬ ‫ب ١‬ ‫أ ﺻﻔﺮﺍ‬ ‫ً‬ ‫٢‬ ‫د‬ ‫١١‪r‬‬ ‫٦‬ ‫..................................................................‬ ‫د ٠٦‪c‬‬ ‫........................................................................................................................................................‬ ‫ﺟ‬ ‫٣‬ ‫د ١‬ ‫٢‬ ‫٣‬ ‫8 ﺇﺫﺍ ﻛﺎﻧﺖ ﺟﺘﺎ ‪ ٢ = i‬ﺣﻴﺚ ‪ i‬ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ ﻓﺈﻥ ﺟﺎ ‪ i‬ﺗﺴﺎﻭﻯ‬ ‫أ ١‬ ‫ﺟ ٢‬ ‫ب ١‬ ‫٢‬ ‫د ٠٦‪c‬‬ ‫٣‬ ‫.............................................................................................‬ ‫٣‬ ‫د‬ ‫٣‬ ‫٢‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫9 ﺃﻭﺟﺪ ﺟﻤﻴﻊ ﺍﻟﺪﻭﺍﻝ ﺍﻟﻤﺜﻠﺜﻴﺔ ﻟﻠﺰﺍﻭﻳﺔ ‪ i‬ﺍﻟﻤﺮﺳﻮﻣﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ، ﻭﺍﻟﺘﻰ ﻳﻤﺮ ﺿﻠﻌﻬﺎ ﺍﻟﻨﻬﺎﺋﻰ ﺑﺎﻟﻨﻘﺎﻁ ﺍﻵﺗﻴﺔ.‬ ‫٥‬ ‫أ )٢، ٣ (‬ ‫٣‬ ‫٣‬ ‫ﺟ )- ٢ ، ١ (‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫ب ) ٢ ،- ٢ (‬ ‫د )- ٣ ، - ٤ (‬ ‫٥ ٥‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
‫01 ﺇﺫﺍ ﻛﺎﻥ ‪ i‬ﻫﻮ ﻗﻴﺎﺱ ﺯﺍﻭﻳﻪ ﻣﻮﺟﻬﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ، ﻭﺍﻟﺘﻰ ﻳﻤﺮ ﺿﻠﻌﻬﺎ ﺍﻟﻨﻬﺎﺋﻰ ﺑﺪﺍﺋﺮﺓ ﺍﻟﻮﺣﺪﺓ ﻓﺄﻭﺟﺪ‬ ‫ﺟﻤﻴﻊ ﺍﻟﺪﻭﺍﻝ ﺍﻟﻤﺜﻠﺜﻴﺔ ﻟﻠﺰﺍﻭﻳﺔ ‪ i‬ﻓﻰ ﺍﻟﺤﺎﻻﺕ ﺍﻵﺗﻴﺔ:‬ ‫ﺣﻴﺚ ‪٠ < C‬‬ ‫أ )٣ ‪(C٤ - ،C‬‬ ‫ب ) ٣ ‪(C٢- ،C‬‬ ‫٣‪r‬‬ ‫ﺣﻴﺚ ٢ > ‪r٢ > i‬‬ ‫٢‬ ‫.................................................................................................................‬ ‫.................................................................................................................‬ ‫11 ﺍﻛﺘﺐ ﺇﺷﺎﺭﺍﺕ ﺍﻟﺪﻭﺍﻝ ﺍﻟﻤﺜﻠﺜﻴﺔ ﺍﻵﺗﻴﺔ:‬ ‫ب ﻇﺎ ٥٦٣‪c‬‬ ‫أ ﺟﺎ ٠٤٢‪c‬‬ ‫....................................‬ ‫ﺟ ﻗﺘﺎ ٠١٤‪c‬‬ ‫....................................‬ ‫٩‪r‬‬ ‫....................................‬ ‫ﻫ ﻗﺎ - ٩‪r‬‬ ‫٤‬ ‫د‬ ‫ﻇﺘﺎ ٤‬ ‫....................................‬ ‫و ﻇﺎ‬ ‫....................................‬ ‫-٠٢‪r‬‬ ‫٩‬ ‫...................................‬ ‫:‬ ‫21‬ ‫‪r‬‬ ‫٣‪r‬‬ ‫‪r‬‬ ‫أ ﺟﺘﺎ ٢ * ﺟﺘﺎ ٠ + ﺟﺎ ٢ * ﺟﺎ ٢‬ ‫ب ﻇﺎ٢ ٠٣‪ ٢ + c‬ﺟﺎ٢ ٥٤‪ + c‬ﺟﺘﺎ٢ ٠٩‪c‬‬ ‫......................................................................................................‬ ‫......................................................................................................‬ ‫31 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻔﻴ ﺎﺀ: ﻋﻨﺪ ﺳﻘﻮﻁ ﺃﺷﻌﺔ ﺍﻟﻀﻮﺀ ﻋﻠﻰ ﺳﻄﺢ‬ ‫ﺷﺒﻪ ﺷﻔﺎﻑ، ﻓﺈﻧﻬﺎ ﺗﻨﻌﻜﺲ ﺑﻨﻔﺲ ﺯﺍﻭﻳﺔ ﺍﻟﺴﻘﻮﻁ ﻭﻟﻜﻦ‬ ‫ﺍﻟﺒﻌﺾ ﻣﻨﻬﺎ ﻳﻨﻜﺴﺮ ﻋﻨﺪ ﻣﺮﻭﺭه ﺧﻼﻝ ﻫﺬﺍ ﺍﻟﺴﻄﺢ. ﻛﻤﺎ‬ ‫ﻓﻰ ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ:‬ ‫ﺇﺫﺍ ﻛﺎﻥ ﺟﺎ ‪ = ١i‬ﻙ ﺟﺎ‪،٢i‬ﻛﺎﻧﺖ ﻙ = ٣ ، ‪c٦٠ = ١i‬‬ ‫ﻓﺄﻭﺟﺪ ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ‪................................................................................ . i‬‬ ‫٢‬ ‫‪U‬‬ ‫‪i‬‬ ‫‪F‬‬ ‫‪i‬‬ ‫‪i‬‬ ‫41 ﺍﻛﺘﺸﻒ ﺍﻟﺨﻄﺄ: ﻃﻠﺐ ﺍﻟﻤﻌﻠﻢ ﻣﻦ ﻃﻼﺏ ﺍﻟﻔﺼﻞ ﺇﻳﺠﺎﺩ ﻧﺎﺗﺞ ٢ ﺟﺎ ٥٤‪.c‬‬ ‫¯‬ ‫٢ ﺟﺎ ٥٤‪ = c‬ﺟﺎ ٢ * ٥٤‪c‬‬ ‫  = ﺟﺎ ٠٩‪١ = c‬‬ ‫ﺃﻯ ﺍﻹﺟﺎﺑﺘﻴﻦ ﺻﺤﻴﺢ?ﻭﻟﻤﺎﺫﺍ?‬ ‫٢ ﺟﺎ ٥٤ = ٢ * ١ = ٢ *‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫         = ٢‬ ‫......................................................................................................................................................................‬ ‫51 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ: ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺔ ‪ i‬ﻣﺮﺳﻮﻣﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ، ﺣﻴﺚ ﻇﺘﺎ ‪ ،١ - = i‬ﻗﺘﺎ ‪ . ٢ = i‬ﻫﻞ ﻣﻦ‬ ‫٣‪r‬‬ ‫? ﻓﺴﺮ ﺇﺟﺎﺑﺘﻚ. .......................................................................................................................‬ ‫ﺍﻟﻤﻤﻜﻦ ﺃﻥ ﻳﻜﻮﻥ ﻕ)‪٤ = (ic‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
‫اﻟﻌﻼﻗﺎت ﺑﻴﻦ اﻟﺪوال اﻟﻤﺜﻠﺜﻴﺔ‬ ‫4-4‬ ‫‪Relations between trigonometric functions‬‬ ‫‪k‬‬ ‫‪:≈JCÉjÉe πªcCG :’hCG‬‬ ‫2 ﻇﺎ ) ٠٨١‪= (i - c‬‬ ‫...............................‬ ‫1 ﺟﺘﺎ )٠٨١ ‪= (i +c‬‬ ‫4 ﺟﺎ )٠٦٣‪= ( i + c‬‬ ‫...............................‬ ‫5 ﺟﺎ )٠٩ ‪= (i +c‬‬ ‫...............................‬ ‫6 ﻇﺘﺎ ) ٠٩‪= (i - c‬‬ ‫...............................‬ ‫7 ﻗﺎ ) ٠٧٢‪= (i+ c‬‬ ‫...............................‬ ‫8 ﺟﺘﺎ )٠٧٢‪= (i - c‬‬ ‫3 ﻗﺘﺎ )٠٦٣‪= (i - c‬‬ ‫...............................‬ ‫...............................‬ ‫...............................‬ ‫‪v‬‬ ‫‪IOÉM ájhGR ¢SÉ«≤H ≈JCÉj ɪe Óc πªcCG :Ék«fÉK‬‬ ‫9 ﺟﺎ ٥٢‪ = c‬ﺟﺘﺎ ...............................‪c‬‬ ‫01 ﺟﺘﺎ ٧٦‪ = c‬ﺟﺎ ...............................‪c‬‬ ‫11 ﻇﺎ ٢٤‪ = c‬ﻇﺘﺎ ...............................‪c‬‬ ‫21 ﻗﺘﺎ ٣١‪ = c‬ﻗﺎ ...............................‪c‬‬ ‫31 ﺇﺫﺍ ﻛﺎﻥ ﻇﺘﺎ٢‪ = i‬ﻃﺎ‪ i‬ﺣﻴﺚ ٠‪ c٩٠ >i>c‬ﻓﺈﻥ ‪=(i c) X‬‬ ‫...............................‬ ‫41 ﺇﺫﺍ ﻛﺎﻥ ﺟﺎ ٥‪ = i‬ﺟﺘﺎ٤‪ i‬ﺣﻴﺚ ‪ i‬ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ ﻣﻮﺟﺒﺔ ﻓﺈﻥ ‪c............................... = i‬‬ ‫51 ﺇﺫﺍ ﻛﺎﻥ ﻗﺎ ‪ = i‬ﻗﺎ)٠٩‪ (i - c‬ﻓﺈﻥ ﻇﺘﺎ ‪= i‬‬ ‫...............................‬ ‫61 ﺇﺫﺍ ﻛﺎﻥ ﻇﺎ ٢‪ = i‬ﻇﺘﺎ٣‪ i‬ﺣﻴﺚ ‪ ] r ، ٠[∋ i‬ﻓﺈﻥ ‪= =(i c) X‬‬ ‫٢‬ ‫71 ﺇﺫﺍ ﻛﺎﻥ ﺟﺘﺎ ‪ = i‬ﺟﺎ٢‪ i‬ﺣﻴﺚ ‪ i‬ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ ﻣﻮﺟﺒﺔ ﻓﺈﻥ ﺟﺎ٣‪= i‬‬ ‫...............................‬ ‫‪E‬‬ ‫...............................‬ ‫‪k‬‬ ‫‪:Oó©àe øe QÉ«àN’G :ÉãdÉK‬‬ ‫81 ﺇﺫﺍ ﻛﺎﻧﺖ ﻇﺎ )٠٨١‪ ١ = (i + c‬ﺣﻴﺚ ‪ i‬ﻗﻴﺎﺱ ﺃﺻﻐﺮ ﺯﺍﻭﻳﺔ ﻣﻮﺟﺒﺔ ﻓﺈﻥ ﻗﻴﺎﺱ ‪ i‬ﻳﺴﺎﻭﻯ‬ ‫د ٥٣١‪c‬‬ ‫ﺟ ٠٦‪c‬‬ ‫ب ٠٣‪c‬‬ ‫أ ٥٤‪c‬‬ ‫91 ﺇﺫﺍ ﻛﺎﻥ ﺟﺘﺎ ٢‪ = i‬ﺟﺎ‪ i‬ﺣﻴﺚ ‪ ] r ،٠[∋ i‬ﻓﺈﻥ ﺟﺘﺎ ٢‪ i‬ﺗﺴﺎﻭﻯ‬ ‫٢‬ ‫ﺟ ٣‬ ‫ب ١‬ ‫أ ١‬ ‫٢‬ ‫٢‬ ‫............................................................................................‬ ‫٢‬ ‫02 ﺇﺫﺍ ﻛﺎﻥ ﺟﺎ ‪ = a‬ﺟﺘﺎ ‪ ،b‬ﺣﻴﺚ ‪ b ،a‬ﺯﺍﻭﻳﺘﺎﻥ ﺣﺎﺩﺗﺎﻥ ﻓﺈﻥ ﻇﺎ)‪ (b + a‬ﺗﺴﺎﻭﻯ‬ ‫أ ١‬ ‫ب ١‬ ‫ﺟ ٣‬ ‫٣‬ ‫د ١‬ ‫.............................................................‬ ‫د ﻏﻴﺮ ﻣﻌﺮﻭﻑ‬ ‫12 ﺇﺫﺍ ﻛﺎﻥ ﺟﺎ ٢‪ = i‬ﺟﺘﺎ ٤‪ i‬ﺣﻴﺚ ‪ i‬ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ ﻣﻮﺟﺒﺔ ﻓﺈﻥ ﻇﺎ)٠٩‪ (i٣ - c‬ﺗﺴﺎﻭﻯ‬ ‫ب ١‬ ‫د‬ ‫ﺟ ١‬ ‫أ -١‬ ‫......................................................‬ ‫٣‬ ‫‪ïM‬‬ ‫−‬ ‫٣‬
‫22 ﺇﺫﺍ ﻛﺎﻥ ﺟﺘﺎ)٠٩‪ ١ = (i + c‬ﺣﻴﺚ ‪ i‬ﻗﻴﺎﺱ ﺃﺻﻐﺮ ﺯﺍﻭﻳﺔ ﻣﻮﺟﺒﺔ ﻓﺈﻥ ﻗﻴﺎﺱ ‪ i‬ﻳﺴﺎﻭﻯ‬ ‫٢‬ ‫ﺟ‬ ‫ب‬ ‫د‬ ‫٠٣٣‪c‬‬ ‫٠٤٢‪c‬‬ ‫٠١٢‪c‬‬ ‫أ ٠٥١‪c‬‬ ‫..................................................‬ ‫‪á«JB’G á∏Ä°SC’G øY ÖLCG :Ék©HGQ‬‬ ‫32 ﺃﻭﺟﺪ ﺇﺣﺪﻯ ﻗﻴﻢ ‪ i‬ﺣﻴﺚ٠‪c٩٠ > i H‬ﺍﻟﺘﻰ ﺗﺤﻘﻖ ﻛﻼ ﻣﻦ ﺍﻵﺗﻰ:‬ ‫ًّ‬ ‫.................................................................................................................‬ ‫أ ﺟﺎ)٣‪ = (c١٥ + i‬ﺟﺘﺎ)٢‪(c٥ - i‬‬ ‫ب ﻗﺎ)‪ =       (c٢٥ + i‬ﻗﺘﺎ)‪(c١٥ + i‬‬ ‫.................................................................................................................‬ ‫ﺟ ﻇﺎ)‪ =     (c٢٠ + i‬ﻇﺘﺎ )٣‪(c٣٠ + i‬‬ ‫.................................................................................................................‬ ‫د ﺟﺘﺎ ‪ =       c٢٠ + i‬ﺟﺎ ‪c٤٠ + i‬‬ ‫.................................................................................................................‬ ‫٢‬ ‫٢‬ ‫42 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻣﻤﺎ ﻳﺄﺗﻰ:‬ ‫ب ﻗﺘﺎ ٥٢٢‬ ‫أ ﺟﺎ ٠٥١‪c‬‬ ‫.....................................‬ ‫ﻫ ﻗﺘﺎ‬ ‫.....................................‬ ‫١١‪r‬‬ ‫و ﺟﺎ‬ ‫٦‬ ‫.....................................‬ ‫٧‪r‬‬ ‫٤‬ ‫.....................................‬ ‫ﺟ ﻗﺎ٠٠٣‪c‬‬ ‫.....................................‬ ‫ز ﻇﺘﺎ‬ ‫-٢‪r‬‬ ‫٣‬ ‫.....................................‬ ‫د ﻇﺎ ٠٨٧‪c‬‬ ‫.....................................‬ ‫ح ﺟﺘﺎ‬ ‫-٧‪r‬‬ ‫٤‬ ‫.....................................‬ ‫52 ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻟﻠﺰﺍﻭﻳﺔ ‪ i‬ﺍﻟﻤﺮﺳﻮﻣﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻲ ﻳﻘﻄﻊ ﺩﺍﺋﺮﺓ ﺍﻟﻮﺣﺪﺓ ﻓﻰ ﺍﻟﻨﻘﻄﺔ ﺏ )- ٣ ، ٤ (‬ ‫٥ ٥‬ ‫ﻓﺄﻭﺟﺪ:‬ ‫ب ﺟﺘﺎ ) ‪(i - r‬‬ ‫أ ﺟﺎ)٠٨١‪(i + c‬‬ ‫٢‬ ‫.................................................‬ ‫.................................................‬ ‫د ﻗﺘﺎ ) ٣‪(i - r‬‬ ‫٢‬ ‫ﺟ ﻇﺎ )٠٦٣‪(i -c‬‬ ‫.................................................‬ ‫.................................................‬ ‫62 ﺍﻛﺘﺸﻒ ﺍﻟﺨﻄﺄ: ﺟﻤﻴﻊ ﺍﻹﺟﺎﺑﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ ﺻﺤﻴﺤﺔ ﻣﺎﻋﺪﺍ ﺇﺟﺎﺑﺔ ﻭﺍﺣﺪﺓ ﻓﻘﻂ ﺧﻄﺄ، ﻓﻤﺎ ﻫﻰ:‬ ‫١- ﺟﺘﺎ ‪ i‬ﺗﺴﺎﻭﻯ‬ ‫.................................................................................................................................................................................................‬ ‫ب ﺟﺎ ) ٠٧٢‪(i - c‬‬ ‫أ ﺟﺎ )‪(c٢٧٠ - i‬‬ ‫٢- ﺟﺎ ‪ i‬ﺗﺴﺎﻭﻯ‬ ‫أ ﺟﺘﺎ ) ‪( i - r‬‬ ‫٢‬ ‫ﺟ ﺟﺘﺎ )٠٦٣‪(i - c‬‬ ‫د ﺟﺘﺎ ) ٠٦٣ ‪(i +c‬‬ ‫...................................................................................................................................................................................................‬ ‫٣- ﻇﺎ ‪ i‬ﺗﺴﺎﻭﻯ‬ ‫ب ﺟﺎ ) ‪(i - r‬‬ ‫٣‪r‬‬ ‫ﺟ ﺟﺘﺎ ) ٢ + ‪(i‬‬ ‫‪r‬‬ ‫د ﺟﺎ ) ٢ + ‪(i‬‬ ‫.....................................................................................................................................................................................................‬ ‫ب ﻇﺘﺎ ) ٠٧٢‪(i - c‬‬ ‫أ ﻇﺘﺎ ) ٠٩‪(i-c‬‬ ‫¯‬ ‫−‬ ‫¯‬ ‫ﺟ ﻇﺎ )٠٧٢‪(i - c‬‬ ‫د ﻇﺎ ) ٠٨١ ‪(i +c‬‬
‫72 ﺍﻟﺮﺑﻂ ﺑﺎﻟﺘﻜﻨﻮﻟﻮﺟﻴﺎ: ﻋﻨﺪ ﺍﺳﺘﺨﺪﺍﻡ ﻛﺮﻳﻢ ﺣﺎﺳﻮﺑﻪ ﺍﻟﻤﺤﻤﻮﻝ‬ ‫ﻛﺎﻧﺖ ﺯﺍﻭﻳﺔ ﻣﻴﻠﻪ ﻣﻊ ﺍﻷﻓﻘﻰ ٢٣١‪ c‬ﻛﻤﺎ ﻫﻮ ﻣﻮﺿﺢ ﺑﺎﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ.‬ ‫أ ﺍﺭﺳﻢ ﺍﻟﺸﻜﻞ ﺍﻟﺴﺎﺑﻖ ﻓﻰ ﺍﻟﻤﺴﺘﻮﻯ ﺍﻹﺣﺪﺍﺛﻰ، ﺑﺤﻴﺚ ﺗﻜﻮﻥ‬ ‫ﺍﻟﺰﺍﻭﻳﺔ ٢٣١‪ c‬ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﺛﻢ ﺃﻭﺟﺪ ﺯﺍﻭﻳﺘﻬﺎ ﺍﻟﻤﻨﺘﺴﺒﺔ.‬ ‫............................................................................................................................................‬ ‫‪c‬‬ ‫ب ﺍﻛﺘﺐ ﺩﺍﻟﺔ ﻣﺜﻠﺜﻴﺔ ﻳﻤﻜﻦ ﺍﺳﺘﺨﺪﺍﻣﻬﺎ ﻓﻰ ﺇﻳﺠﺎﺩ ﻗﻴﻢ ‪ ،C‬ﺛﻢ ﺃﻭﺟﺪ‬ ‫ﻗﻴﻤﺔ ‪ C‬ﻷﻗﺮﺏ ﺳﻨﺘﻴﻤﺘﺮ.‬ ‫‪C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫ﺃﻟﻌﺎﺏ: ﺗﻨﺘﺸﺮ ﻟﻌﺒﺔ ﺍﻟﻌﺠﻠﺔ ﺍﻟﺪﻭﺍﺭﺓ ﻓﻰ ﻣﺪﻳﻨﺔ ﺍﻟﻤﻼﻫﻰ، ﻭﻫﻰ‬ ‫ﻋﺒﺎﺭﺓ ﻋﻦ ﻋﺪﺩ ﻣﻦ ﺍﻟﺼﻨﺎﺩﻳﻖ ﺗﺪﻭﺭ ﻓﻰ ﻗﻮﺱ ﺩﺍﺋﺮﻯ ﻳﺒﻠﻎ‬ ‫ﻛﺔ‬ ‫ﻧﺼﻒ ﻗﻄﺮه ٢١ ﻣﺘﺮﺍ، ﻓﺈﺫﺍ ﻛﺎﻥ ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﺸﺘﺮ‬ ‫ً‬ ‫ﻣﻊ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ٥‪r‬‬ ‫٤ .‬ ‫أ ﺍﺭﺳﻢ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ ٥‪ r‬ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ.‬ ‫٤‬ ‫‪C‬‬ ‫...........................................................................................................‬ ‫ب ﺍﻛﺘﺐ ﺩﺍﻟﺔ ﻣﺜﻠﺜﻴﺔ ﻳﻤﻜﻦ ﺍﺳﺘﺨﺪﺍﻣﻬﺎ ﻓﻰ ﺇﻳﺠﺎﺩ ﻗﻴﻤﺔ‬ ‫‪ C‬ﺛﻢ ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ C‬ﺑﺎﻟﻤﺘﺮ ﻷﻗﺮﺏ ﺭﻗﻤﻴﻦ ﻋﺸﺮﻳﻴﻦ.‬ ‫...........................................................................................................‬ ‫82 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ:‬ ‫أ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺔ ‪ i‬ﻣﺮﺳﻮﻣﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ، ﺣﻴﺚ ﻇﺘﺎ ‪ ، ١- = i‬ﻗﺘﺎ ‪ . ٢ = i‬ﻓﻬﻞ ﻳﻤﻜﻦ ﺃﻥ‬ ‫٣‪r‬‬ ‫ﻳﻜﻮﻥ ‪ ? ٤ = (ic) X‬ﻓﺴﺮ ﺇﺟﺎﺑﺘﻚ?‬ ‫.......................................................................................................................................................................................................................‬ ‫٣‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ﺟﺘﺎ ) ٣‪ ، ٢ = (i - r‬ﺟﺎ ) ‪ ١ = (i + r‬ﻓﺄﻭﺟﺪ ﺃﺻﻐﺮ ﻗﻴﺎﺱ ﻣﻮﺟﺐ ﻟﻠﺰﺍﻭﻳﺔ ‪.i‬‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫.......................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
‫اﻟﺘﻤﺜﻴﻞ اﻟﺒﻴﺎﻧﻰ ﻟﻠﺪوال اﻟﻤﺜﻠﺜﻴﺔ‬ ‫4-5‬ ‫‪Graphing trigonometric functions‬‬ ‫‪k‬‬ ‫‪:≈JCÉjÉe πªcCG :’hCG‬‬ ‫1 ﻣﺪﻯ ﺍﻟﺪﺍﻟﺔ ﺩ ﺣﻴﺚ ﺩ)‪ = (i‬ﺟﺎ‪ i‬ﻫﻮ‬ ‫................................‬ ‫2 ﻣﺪﻯ ﺍﻟﺪﺍﻟﺔ ﺩ ﺣﻴﺚ ﺩ)‪ ٢ = (i‬ﺟﺎ‪ i‬ﻫﻮ‬ ‫................................‬ ‫3 ﺍﻟﻘﻴﻤﺔ ﺍﻟﻌﻈﻤﻰ ﻟﻠﺪﺍﻟﺔ ﻉ ﺣﻴﺚ ﻉ)‪٤ = (i‬ﺟﺎ‪ i‬ﻫﻰ‬ ‫................................‬ ‫4 ﺍﻟﻘﻴﻤﺔ ﺍﻟﺼﻐﺮﻯ ﻟﻠﺪﺍﻟﺔ ﻫـ ﺣﻴﺚ ﻫـ)‪٣ = (i‬ﺟﺘﺎ‪ i‬ﻫﻰ‬ ‫................................‬ ‫‪.É¡d ôXÉæªdG πμ°ûdG QGƒéH á«ã∏ãe ádGO πc IóYÉb ÖàcG :Ék«fÉK‬‬ ‫‪r‬‬ ‫‪r‬‬ ‫‪r‬‬ ‫‪r‬‬ ‫−‬ ‫−‪r− r‬‬ ‫− ‪r −r‬‬ ‫ﺷﻜﻞ )١( ﺍﻟﻘﺎﻋﺪﺓ ﻫﻰ:‬ ‫‪r‬‬ ‫‪r‬‬ ‫‪r‬‬ ‫‪r‬‬ ‫−‬ ‫−‪r− r‬‬ ‫− ‪r − r‬‬ ‫ﺷﻜﻞ )٢( ﺍﻟﻘﺎﻋﺪﺓ ﻫﻲ:‬ ‫........................................................................................................‬ ‫........................................................................................................‬ ‫‪k‬‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :ÉãdÉK‬‬ ‫5 ﺃﻭﺟﺪ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻌﻈﻤﻰ ﻭﺍﻟﻘﻴﻤﺔ ﺍﻟﺼﻐﺮﻯ، ﺛﻢ ﺍﺣﺴﺐ ﺍﻟﻤﺪﻯ ﻟﻜﻞ ﺩﺍﻟﺔ ﻣﻦ ﺍﻟﺪﻭﺍﻝ ﺍﻵﺗﻴﺔ :‬ ‫أ ﺹ = ﺟﺎ‪i‬‬ ‫.......................................................................................................................................................................................................................‬ ‫ب ﺹ = ٣ ﺟﺘﺎ‪i‬‬ ‫.......................................................................................................................................................................................................................‬ ‫ﺟ ﺹ = ٣ ﺟﺎ‪i‬‬ ‫٢‬ ‫.......................................................................................................................................................................................................................‬ ‫6 ﻣﺜﻞ ﻛﻞ ﻣﻦ ﺍﻟﺪﻭﺍﻝ ﺹ = ٤ ﺟﺘﺎ‪ ، i‬ﺹ = ٣ ﺟﺎ‪ i‬ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻵﻟﺔ ﺍﻟﺤﺎﺳﺒﺔ ﺍﻟﺮﺳﻮﻣﻴﺔ ﺃﻭ ﺑﺄﺣﺪ ﺑﺮﺍﻣﺞ ﺍﻟﺤﺎﺳﻮﺏ‬ ‫ﺍﻟﺮﺳﻮﻣﻴﺔ ﻭﻣﻦ ﺍﻟﺮﺳﻢ ﺃﻭﺟﺪ :‬ ‫ب ﺍﻟﻘﻴﻢ ﺍﻟﻌﻈﻤﻰ ﻭﺍﻟﻘﻴﻢ ﺍﻟﺼﻐﺮﻯ ﻟﻠﺪﺍﻟﺔ.‬ ‫أ ﻣﺪﻯ ﺍﻟﺪﺍﻟﺔ.‬ ‫.‬ ‫.................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬ ‫.................................................................................................‬
‫إﻳﺠﺎد ﻗﻴﺎس زاوﻳﺔ ﺑﻤﻌﻠﻮﻣﻴﺔ داﻟﺔ ﻣﺜﻠﺜﻴﺔ‬ ‫‪Finding the measure of an angle given the‬‬ ‫4-6‬ ‫‪value of one of its functions‬‬ ‫‪k‬‬ ‫‪:Oó©àe øe QÉ«àN’G :’hCG‬‬ ‫1 ﺇﺫﺍ ﻛﺎﻥ ﺟﺎ ‪ ٠٫٤٣٢٥ = i‬ﺣﻴﺚ ‪ i‬ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ ﻣﻮﺟﺒﺔ ﻓﺈﻥ ‪ (i) c X‬ﺗﺴﺎﻭﻯ‬ ‫ﺟ ٨٨٣٫٢٣‪c‬‬ ‫ب ٧٤٣٫٤٦‪c‬‬ ‫أ ٦٢٦٫٥٢‪c‬‬ ‫2 ﺇﺫﺍ ﻛﺎﻥ ﻇﺎ ‪ ١٫٨ = i‬ﻛﺎﻧﺖ ٠٩‪ c٣٦٠ H iHc‬ﻓﺈﻥ ‪ (i) c X‬ﺗﺴﺎﻭﻯ‬ ‫ﻭ‬ ‫ﺟ ٥٤٩٫٠٤٢‪c‬‬ ‫ب ٥٥٠٫٩١١‪c‬‬ ‫أ ٥٤٩٫٠٦‪c‬‬ ‫.............................................................‬ ‫د ٦١٣٫٦٤‪c‬‬ ‫............................................................................‬ ‫د ٥٥٠٫٩٩٢‪c‬‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫1 ﺇﺫﺍ ﻗﻄﻊ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻟﻠﺰﺍﻭﻳﺔ ‪ i‬ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﺩﺍﺋﺮﺓ ﺍﻟﻮﺣﺪﺓ ﻓﻰ ﺍﻟﻨﻘﻄﺔ ﺏ، ﻓﺄﻭﺟﺪ ﻛﻼ ﻣﻦ‬ ‫ًّ‬ ‫ﺟﺘﺎ ‪ ،i‬ﺟﺎ ‪ i‬ﻓﻰ ﺍﻟﺤﺎﻻﺕ ﺍﻵﺗﻴﺔ:‬ ‫٣‬ ‫أ ﺏ )١، ٢ (‬ ‫٢‬ ‫ب ﺏ) ١ ، - ١ (‬ ‫٢‬ ‫٢‬ ‫٦ ٨‬ ‫ﺟ ﺏ )- ٠١ ، ٠١ (‬ ‫..................................................................‬ ‫..................................................................‬ ‫..................................................................‬ ‫..................................................................‬ ‫..................................................................‬ ‫................................................................‬ ‫2 ﺇﺫﺍ ﻗﻄﻊ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻟﻠﺰﺍﻭﻳﺔ ‪ i‬ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﺩﺍﺋﺮﺓ ﺍﻟﻮﺣﺪﺓ ﻓﻰ ﺍﻟﻨﻘﻄﺔ ﺏ، ﻓﺄﻭﺟﺪ ﻛﻼ ﻣﻦ‬ ‫ًّ‬ ‫ﻗﺎ‪ ،i‬ﻗﺘﺎ‪ i‬ﻓﻰ ﺍﻟﺤﺎﻻﺕ ﺍﻵﺗﻴﺔ:‬ ‫٢‬ ‫٢‬ ‫ﺟ ﺏ )- ٥ ، - ٢١‬ ‫ب ﺏ)- ١ ، - ٢‬ ‫٣١ ٣١ (‬ ‫(‬ ‫أ ﺏ) ٢ ،- ٢ (‬ ‫٥‬ ‫٥‬ ‫..................................................................‬ ‫..................................................................‬ ‫..................................................................‬ ‫..................................................................‬ ‫..................................................................‬ ‫................................................................‬ ‫3 ﺇﺫﺍ ﻗﻄﻊ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻟﻠﺰﺍﻭﻳﺔ ‪ i‬ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﺩﺍﺋﺮﺓ ﺍﻟﻮﺣﺪﺓ ﻓﻰ ﺍﻟﻨﻘﻄﺔ ﺏ، ﻓﺄﻭﺟﺪ ﻛﻼ ﻣﻦ‬ ‫ًّ‬ ‫ﻇﺎ ‪ ،i‬ﻇﺘﺎ ‪ i‬ﻓﻰ ﺍﻟﺤﺎﻻﺕ ﺍﻵﺗﻴﺔ:‬ ‫ﺟ ﺏ )- ٤ ، - ٣ (‬ ‫ب ﺏ) ٣ ،- ٥ (‬ ‫أ ﺏ) ١ ،- ٣ (‬ ‫٥‬ ‫٥‬ ‫٠١‬ ‫٠١‬ ‫٤٣‬ ‫٤٣‬ ‫..................................................................‬ ‫..................................................................‬ ‫..................................................................‬ ‫..................................................................‬ ‫..................................................................‬ ‫................................................................‬ ‫4 ﺇﺫﺍ ﻗﻄﻊ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻟﻠﺰﺍﻭﻳﺔ ‪ i‬ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﺩﺍﺋﺮﺓ ﺍﻟﻮﺣﺪﺓ ﻓﻰ ﺍﻟﻨﻘﻄﺔ ﺏ‬ ‫ﻓﺄﻭﺟﺪ: ‪ (ic)X‬ﺣﻴﺚ ٠ ْ > ‪ ْ ٣٦٠ > i‬ﻋﻨﺪﻣﺎ:‬ ‫٣‬ ‫أ ﺏ ) ٢ ، ١(‬ ‫٢‬ ‫ب ﺏ)- ١ ، ١ (‬ ‫٢‬ ‫..................................................................‬ ‫٢‬ ‫..................................................................‬ ‫‪ïM‬‬ ‫−‬ ‫٦‬ ‫ﺟ ﺏ ) ٠١ ، -٨ (‬ ‫٠١‬ ‫..................................................................‬
‫5 ﺃﻭﺟﺪ ﺑﺎﻟﻘﻴﺎﺱ ﺍﻟﺴﺘﻴﻨﻰ ﺃﺻﻐﺮ ﺯﺍﻭﻳﺔ ﻣﻮﺟﺒﺔ ﺗﺤﻘﻖ ﻛﻼ ﻣﻦ:‬ ‫ًّ‬ ‫ب ﺟﺘﺎ-١ ٦٣٤٫٠‬ ‫أ ﺟﺎ-١ ٦٫٠‬ ‫..................................................................‬ ‫د ﻗﺎ-١ )- ٤٦٣٢٫٢(‬ ‫ﺟ ﻇﺎ-١ ٢٥٥٤٫١‬ ‫..................................................................‬ ‫ﻫ ﻇﺘﺎ-١ ٨١٢٦٫٣‬ ‫..................................................................‬ ‫..................................................................‬ ‫و ﻗﺘﺎ-١ )-٤٠٠٦٫١(‬ ‫..................................................................‬ ‫6 ﺇﺫﺍ ﻛﺎﻧﺖ ٠‪ c٣٦٠ H iHc‬ﻓﺄﻭﺟﺪ ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ‪ i‬ﻟﻜﻞ ﻣﻤﺎ ﻳﺄﺗﻰ:‬ ‫ب ﺟﺘﺎ-١ )- ٢٤٦٫٠(‬ ‫أ ﺟﺎ-١ )٦٥٣٢٫٠(‬ ‫..................................................................‬ ‫..................................................................‬ ‫ﺟ ﻇﺎ-١ )- ٦٥٤١٫٢(‬ ‫..................................................................‬ ‫..................................................................‬ ‫7 ﺇﺫﺍ ﻛﺎﻥ ﺟﺎ ‪ ١ = i‬ﻛﺎﻧﺖ ٠٩‪c١٨٠ H iHc‬‬ ‫٣ﻭ‬ ‫أ ﺍﺣﺴﺐ ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ‪ i‬ﻷﻗﺮﺏ ﺛﺎﻧﻴﺔ‬ ‫.......................................................................................................................................................................................................................‬ ‫ب ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻣﻦ: ﺟﺘﺎ‪ ، i‬ﻇﺎ‪ ، i‬ﻗﺎ‪. i‬‬ ‫ٍّ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫8 ﺳﻼﻟﻢ: ﺳﻠﻢ ﻃﻮﻟﻪ ٥ ﺃﻣﺘﺎﺭ ﻳﺴﺘﻨﺪ ﻋﻠﻰ ﺟﺪﺍﺭ ﻓﺈﺫﺍ ﻛﺎﻥ ﺍﺭﺗﻔﺎﻉ ﺍﻟﺴﻠﻢ ﻋﻦ‬ ‫ﺳﻄﺢ ﺍﻷﺭﺽ ﻳﺴﺎﻭﻯ ٣ ﺃﻣﺘﺎﺭ ﻓﺄﻭﺟﺪ ﺑﺎﻟﺮﺍﺩﻳﺎﻥ ﺯﺍﻭﻳﺔ ﻣﻴﻞ ﺍﻟﺴﻠﻢ ﻋﻠﻰ ﺍﻷﻓﻘﻰ.‬ ‫‪ci‬‬ ‫.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫9 ﺃﻭﺟﺪ ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ‪ i‬ﺑﺎﻟﻘﻴﺎﺱ ﺍﻟﺴﺘﻴﻨﻰ ﻓﻰ ﻛﻞ ﺷﻜﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻵﺗﻴﺔ:‬ ‫ب‬ ‫أ‬ ‫ﺟ‬ ‫‪i‬‬ ‫‪i‬‬ ‫............................................................‬ ‫¯‬ ‫−‬ ‫............................................................‬ ‫¯‬ ‫‪i‬‬ ‫............................................................‬
‫ﲤﺎﺭﻳﻦ ﻋﺎﻣﺔ‬ ‫‪:ø«jô°ûY ø«ªbQ ÜôbC’ èJÉædG ÉkHô≤e á«JB’G á∏Ä°SC’G øY ÖLCG‬‬ ‫1 ﺣﻮﻝ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻵﺗﻴﺔ ﻣﻦ ﺩﺭﺟﺎﺕ ﺇﻟﻰ ﺭﺍﺩﻳﺎﻥ:‬ ‫ِّ‬ ‫أ ٠٢١‪c‬‬ ‫ب ٨٫٤٦‪c‬‬ ‫   ...............................‬ ‫               ...............................‬ ‫ﺟ ٦٣ َ ٠٢٢‪c‬‬ ‫................................‬ ‫2 ﺣﻮﻝ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻵﺗﻴﺔ ﻣﻦ ﺭﺍﺩﻳﺎﻥ ﺇﻟﻰ ﺩﺭﺟﺎﺕ:‬ ‫أ ٥‪r‬‬ ‫٣‬ ‫ب - ٣‪r‬‬ ‫٢‬ ‫       ...............................‬ ‫      ...............................‬ ‫ﺟ ٢١٫١‬ ‫‪E‬‬ ‫      ...............................‬ ‫3 ‪ i‬ﺯﺍﻭﻳﺔ ﻛﺰﻳﺔ ﻓﻰ ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ‪ H‬ﻭﺗﺤﺼﺮ ﻗﻮﺳﺎ ﻃﻮﻟﻪ ﻝ:‬ ‫ﻣﺮ‬ ‫ً‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ ‪ ٨ = H‬ﺳﻢ،  ‪ E١٫٢ = i‬ﺃﻭﺟﺪ ﻝ.‬ ‫    ‬ ‫................................................................................................‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ﻝ = ٦٢ ﺳﻢ، ‪ ١٨ = H‬ﺳﻢ ﺃﻭﺟﺪ ‪ i‬ﺑﺎﻟﺪﺭﺟﺎﺕ.‬ ‫...............................................................................................‬ ‫4 ﺑﺪﻭﻥ ﺍﺳﺘﺨﺪﺍﻡ ﺍﻵﻟﺔ ﺍﻟﺤﺎﺳﺒﺔ ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻣﻤﺎ ﻳﺄﺗﻰ:‬ ‫٣١‬ ‫أ ﻇﺎ ٠٢١‪    c‬ب ﺟﺎ ) ٦‪    ( r‬ﺟ ﺟﺘﺎ ٠٣٣‪    c‬د ﻇﺘﺎ )- ٠٠٣‪    (c‬ﻫ ﻗﺘﺎ )- ‪( r‬‬ ‫٣‬ ‫..............................        ..............................            ..............................            ..............................               ..............................‬ ‫5 ﺃﻭﺟﺪ ﺟﻤﻴﻊ ﺍﻟﺪﻭﺍﻝ ﺍﻟﻤﺜﻠﺜﻴﺔ ﻟﻠﺰﺍﻭﻳﺔ ‪ i‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻣﺮﺳﻮﻣﺎ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﻭﻳﻤﺮ ﺑﻜﻞ ﻧﻘﻄﺔ‬ ‫ً‬ ‫ﻣﻦ ﺍﻟﻨﻘﺎﻁ ﺍﻵﺗﻴﺔ:‬ ‫ﺟ )- ٣‬ ‫د )- ٥ ، ٢(‬ ‫٢ ، - ٢(‬ ‫ب - )٥، - ٢١(‬ ‫أ )٤، ٣(‬ ‫.............................................‬ ‫6‬ ‫.............................................‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ:‬ ‫: ﺟﺎ ٠٦ = ٢ ﺟﺎ ٠٣‪ c‬ﺟﺘﺎ ٠٣‪c‬‬ ‫.............................................‬ ‫.............................................‬ ‫: ﺟﺘﺎ ٠٠٣‪ ٢ = c‬ﺟﺎ٢ ٠٦‪١- c‬‬ ‫ب ﺇﺫﺍ ﻛﺎﻧﺖ ﺟﺘﺎ ‪ ٤ - = i‬ﺣﻴﺚ ٠٩‪ c١٨٠ > i > c‬ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻣﻦ:‬ ‫٥‬ ‫ً‬ ‫: ﻇﺎ )‪(c١٨٠- i‬‬ ‫ﺃﻭﻻ: ﺟﺎ )٠٨١‪( i -c‬‬ ‫............................................................‬ ‫............................................................‬ ‫7 ﺃﻭﺟﺪ ﻗﻴﺎﺱ ﺍﻟﺰﻭﺍﻳﺎ ﺑﺎﻟﺪﺭﺟﺎﺕ ﻓﻰ ﺍﻟﻔﺘﺮﺓ ٠‪ c٣٦٠ H iHc‬ﻟﻜﻞ ﻣﻤﺎ ﻳﺄﺗﻰ:‬ ‫٣‬ ‫ﺟ ﺟﺘﺎ-١ )  ٢ (‬ ‫ب ﺟﺎ-١ )- ١ (‬ ‫أ ﻇﺎ-١ ١‬ ‫٢‬ ‫.............................................‬ ‫.............................................‬ ‫.............................................‬ ‫د ﻇﺎ-١)- ٣ (‬ ‫.............................................‬ ‫8 ﻣﻨﺤﺪﺭﺍ ﻃﻮﻟﻪ ٤٢ ﻣﺘﺮﺍ، ﻭﺍﺭﺗﻔﺎﻋﻪ ﻋﻦ ﺳﻄﺢ ﺍﻷﺭﺽ ٩ ﺃﻣﺘﺎﺭ، ﺍﻛﺘﺐ ﺩﺍﻟﺔ ﻣﺜﻠﺜﻴﺔ ﻳﻤﻜﻦ ﺍﺳﺘﺨﺪﺍﻣﻬﺎ ﻹﻳﺠﺎﺩ ﻗﻴﺎﺱ‬ ‫ً‬ ‫ً‬ ‫ﺯﺍﻭﻳﺔ ﻣﻴﻞ ﺍﻟﻤﻨﺤﺪﺭ ﻣﻊ ﺍﻷﺭﺽ ﺍﻷﻓﻘﻴﺔ، ﺛﻢ ﺃﻭﺟﺪ ﻗﻴﺎﺳﻬﺎ. ..........................................................................................................‬ ‫‪ïM‬‬ ‫−‬
‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ‬ ‫‪.√É£©ªdG äÉHÉLE’G ø«H øe áë«ë°üdG áHÉL’G ôàNG‬‬ ‫1 ﺍﻟﺰﺍﻭﻳﺔ ٥٨٥‪ c‬ﺗﻜﺎﻓﻲﺀ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻲ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ:‬ ‫ﺟ ٥٢٢‪c‬‬ ‫ب ٥٣١‪c‬‬ ‫أ ٥٤‪c‬‬ ‫د ٥١٣‪c‬‬ ‫2 ﺇﺫﺍ ﻛﺎﻥ ﺟﺎ‪ ،٠ > i‬ﻇﺎ ‪ ٠ < i‬ﻓﺈﻥ ﺯﺍﻭﻳﺔ ﺗﻘﻊ ‪ i‬ﻓﻰ ﺍﻟﺮﺑﻊ:‬ ‫ﺟ ﺍﻟﺜﺎﻟﺚ‬ ‫ب ﺍﻟﺜﺎﻧﻰ‬ ‫أ ﺍﻷﻭﻝ‬ ‫د ﺍﻟﺮﺍﺑﻊ‬ ‫3 ﺇﺫﺍ ﻛﺎﻧﺖ ‪ i‬ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ ﻛﺎﻥ ﺟﺎ )‪ = (c٢٠ + i‬ﺟﺘﺎ ٠٣‪ c‬ﻓﺈﻥ ﻕ )‪ (i c‬ﺗﺴﺎﻭﻯ:‬ ‫ﻭ‬ ‫د ٠٥‪c‬‬ ‫ﺟ ٠٤‪c‬‬ ‫ب ٠٣‪c‬‬ ‫أ ٠٢‪c‬‬ ‫4 ﺍﻟﺰﺍﻭﻳﺔ )-٠٥٨‪ (c‬ﺗﻘﻊ ﻓﻰ ﺍﻟﺮﺑﻊ:‬ ‫ب ﺍﻟﺜﺎﻧﻰ‬ ‫أ ﺍﻷﻭﻝ‬ ‫ﺟ ﺍﻟﺜﺎﻟﺚ‬ ‫د ﺍﻟﺮﺍﺑﻊ‬ ‫5 ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺑﺎﻟﺪﺭﺟﺎﺕ ﺍﻟﺘﻰ ﺗﻘﺎﺑﻞ ﻗﻮﺳﺎ ﻃﻮﻟﻪ ٦‪ r‬ﻓﻰ ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٩ﺳﻢ ﺗﺴﺎﻭﻯ:‬ ‫ً‬ ‫د ٠٥١‪c‬‬ ‫ﺟ ٠٢١‪c‬‬ ‫ب ٠٦‪c‬‬ ‫أ ٠٣‪c‬‬ ‫6 ﺃﺑﺴﻂ ﺻﻮﺭﺓ ﻟﻠﻤﻘﺪﺍﺭ: ﺟﺘﺎ )٠٨١‪ + (i + c‬ﺟﺎ )٠٩‪ (i + c‬ﻳﺴﺎﻭﻯ:‬ ‫ﺟ ٢ ﺟﺘﺎ ‪i‬‬ ‫ب ٢‬ ‫أ ٠‬ ‫7 ﻇﺎ )-٠٣‪ (c‬ﺗﺴﺎﻭﻯ:‬ ‫أ - ٣‬ ‫ب - ١‬ ‫٣‬ ‫ﺟ‬ ‫د ٢ ﺟﺎ ‪i‬‬ ‫١‬ ‫٣‬ ‫د‬ ‫٣‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG‬‬ ‫8 ‪ C‬ﺏ ﻗﻮﺱ ﻓﻰ ﺩﺍﺋﺮﺓ ﻛﺰﻫﺎ ﻭ ﻭﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٠١ ﺳﻢ ، ‪ C‬ﺏ = ٦١ ﺳﻢ.‬ ‫ﻣﺮ‬ ‫ﺃﻭﺟﺪ ‪ i‬ﺑﺎﻟﻘﻴﺎﺱ ﺍﻟﺪﺍﺋﺮﻯ ﺛﻢ ﺃﻭﺟﺪ ﻃﻮﻝ ﺍﻟﻘﻮﺱ ‪ C‬ﺏ :‬ ‫‪C‬‬ ‫‪i‬‬ ‫9 ﺇﺫﺍ ﻛﺎﻥ ٥ ﺟﺎ ‪ ٤ = C‬ﺣﻴﺚ ٠٩‪c١٨٠ > C > c‬‬ ‫ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﻤﻘﺪﺍﺭ ﺟﺎ )٠٨١‪+ (C - c‬ﻇﺎ )٠٦٣‪٢+ (C - c‬ﺟﺎ )٠٧٢‪(C - c‬‬ ‫01 ﺃﻭﺟﺪ ﻓﻰ ﺃﺑﺴﻂ ﺻﻮﺭﺓ ﻗﻴﻤﺔ ﺍﻟﻤﻘﺪﺍﺭ: ﺟﺎ ٠٢١‪ c‬ﺟﺘﺎ٠٣٣‪ - c‬ﺟﺘﺎ ٠٢٤‪ c‬ﺟﺎ )-٠٣‪.(c‬‬ ‫11 ﺍﻭﺟﺪ ﺑﺎﻟﺮﺩﻳﺎﻥ ‪ (C c) X‬ﺇﺫﺍ ﻛﺎﻥ ٢ ﺟﺘﺎ ‪ ٠ = ٢ + C‬ﺣﻴﺚ ‪ C‬ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ.‬ ‫٣‬ ‫21 ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻲ ﻟﻠﺰﺍﻭﻳﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻲ ﻳﻘﻄﻊ ﺩﺍﺋﺮﺓ ﺍﻟﻮﺣﺪﺓ ﻋﻨﺪ ﺍﻟﻨﻘﻄﺔ )- ٢ ، ١ ( ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ‬ ‫٢‬ ‫ﻛﻞ ﻣﻦ: ﻃﺎ‪ ، i‬ﻗﺎ‪i‬‬ ‫31 ﺃﻭﺟﺪ ﺍﻟﺪﻭﺍﻝ ﺍﻟﻤﺜﻠﺜﻴﺔ ﺍﻷﺳﺎﺳﻴﺔ ﻟﻠﺰﺍﻭﻳﺔ ‪ i‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻲ ﻣﺮﺳﻮﻣﺎ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻲ ﻭﻳﻤﺮ ﺑﺎﻟﻨﻘﻄﺔ‬ ‫ً‬ ‫)٦، -٨(‬ ‫¯‬ ‫−‬ ‫¯‬
‫ﺍﺧﺘﺒﺎﺭ ﺗﺮﺍﻛﻤﻲ‬ ‫‪k‬‬ ‫‪Oó©àe øe QÉ«àN’G á∏Ä°SCG :’hCG‬‬ ‫1 ﺃﻯ ﻣﻦ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻵﺗﻴﺔ ﻳﻜﻮﻥ ﺍﻟﺠﻴﺐ ﻭﺟﻴﺐ ﺍﻟﺘﻤﺎﻡ ﻟﻬﺎ ﺳﺎﻟﺒﻴﻦ :‬ ‫ﺟ ٠٢٢‪c‬‬ ‫ب ٠٤١‪c‬‬ ‫أ ٠٤‪c‬‬ ‫.............................................‬ ‫.............................................‬ ‫د ٠٢٣‪c‬‬ ‫.............................................‬ ‫.............................................‬ ‫2 ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﻛﺰﻳﺔ ﺍﻟﺘﻰ ﺗﻘﺎﺑﻞ ﻗﻮﺳﺎ ﻃﻮﻟﻪ ٢‪ r‬ﻓﻰ ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٦ ﺳﻢ ﻳﺴﺎﻭﻯ :‬ ‫ﺍﻟﻤﺮ‬ ‫ً‬ ‫د ‪r‬‬ ‫ﺟ ‪r‬‬ ‫ب ‪r‬‬ ‫‪r‬‬ ‫أ‬ ‫٤‬ ‫٦‬ ‫.............................................‬ ‫٣‬ ‫.............................................‬ ‫٢‬ ‫.............................................‬ ‫.............................................‬ ‫3 ﺇﺫﺍ ﻛﺎﻥ ﻇﺎ ٤‪ = i‬ﻇﺘﺎ٢‪ i‬ﺣﻴﺚ ‪ i‬ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ ﻣﻮﺟﺒﺔ ﻓﺈﻥ ﺟﺎ)٠٩ ْ - ‪ (i‬ﺗﺴﺎﻭﻯ :‬ ‫ﺟ ٣‬ ‫أ ١‬ ‫ب ١‬ ‫د ١‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫.............................................‬ ‫.............................................‬ ‫.............................................‬ ‫.............................................‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫٣‬ ‫4 ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻟﻠﺰﺍﻭﻳﺔ ‪ i‬ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﻳﻘﻄﻊ ﺩﺍﺋﺮﺓ ﺍﻟﻮﺣﺪﺓ ﻋﻨﺪ ﺍﻟﻨﻘﻄﺔ ) ١ ، ٢ ( ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ‬ ‫٢‬ ‫ﻛﻞ ﻣﻦ ﻇﺘﺎ ‪ ،i‬ﻗﺘﺎ‪...................................................................................................................................................................................... .i‬‬ ‫5 ﺑﺪﻭﻥ ﺍﺳﺘﺨﺪﺍﻡ ﺍﻵﻟﺔ ﺍﻟﺤﺎﺳﺒﺔ ﺃﻭﺟﺪ )ﺇﻥ ﺃﻣﻜﻦ ﺫﻟﻚ( ﻗﻴﻤﺔ ﻛﻞ ﻣﻦ :‬ ‫ﺟ ﻗﺎ ٣‪r‬‬ ‫ب ﺟﺎ )- ٥٣١‪(c‬‬ ‫أ ﺟﺘﺎ ٠١٢‪c‬‬ ‫د ﻇﺘﺎ )- ٢‪( r‬‬ ‫٣‬ ‫٢‬ ‫.............................................‬ ‫.............................................‬ ‫.............................................‬ ‫.............................................‬ ‫6 ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻟﻠﺰﺍﻭﻳﺔ )٠٩‪ (i - c‬ﺣﻴﺚ ‪ i‬ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ ﻣﻮﺟﺒﺔ، ﻳﻘﻄﻊ ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٥‬ ‫ﻭﺣﺪﺍﺕ ﻃﻮﻝ ﻓﻰ ﺍﻟﻨﻘﻄﺔ )٤ ، ﻙ( ﻓﺄﻭﺟﺪ :‬ ‫د ﻕ)‪(i c‬‬ ‫ﺟ ﺟﺘﺎ )٠٩‪(i - c‬‬ ‫ب ﺟﺎ )٠٩‪(i - c‬‬ ‫أ ﻗﻴﻤﺔ ﻙ‬ ‫.............................................‬ ‫.............................................‬ ‫.............................................‬ ‫.............................................‬ ‫7 ﺩ ﺍﺟﺎﺕ: ﻳﺼﻌﺪ ﻛﺮﻳﻢ ﺑﺪﺭﺍﺟﺘﻪ ﻣﻨﺤﺪﺭﺍ ﻳﻤﻴﻞ ﻋﻠﻰ ﺍﻷﻓﻘﻰ ﺑﺰﺍﻭﻳﺔ ﻗﻴﺎﺳﻬﺎ ٥٥١‪ c‬ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ‬ ‫ً‬ ‫أ ﺍﻛﺘﺐ ﺩﺍﻟﺔ ﻣﺜﻠﺜﻴﺔ ﺗﺒﻴﻦ ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ‪ C‬ﻭﻃﻮﻝ ﺍﻟﻤﻨﺤﺪﺭ.‬ ‫ب ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ C‬ﻷﻗﺮﺏ ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ.‬ ‫¯‬ ‫‪ï‬‬ ‫¯‬ ‫١‬ ‫٢‬ ‫٣‬ ‫٤‬ ‫٥‬ ‫٦‬ ‫٧‬ ‫٤- ٣‬ ‫٤-٢‬ ‫٤-٤‬ ‫٤-٣‬ ‫٤- ٤‬ ‫٤-٤‬ ‫٤-٤‬ ‫‪ïM‬‬ ‫−‬
‫ﺍﺧﺘﺒﺎﺭﺍﺕ ﻋﺎﻣﺔ‬ ‫)اﻟﺠﺒﺮ وﺣﺴﺎب اﻟﻤﺜﻠﺜﺎت(‬ ‫اﻻﺧﺘﺒﺎر ا ول‬ ‫‪k‬‬ ‫‪≈`JCÉjÉe πªcCG :’hCG‬‬ ‫1 ﺇﺫﺍ ﻛﺎﻥ ﺱ = -١ ﻫﻰ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ‪ C‬ﺱ – ٢ = ٠ ﻓﺈﻥ ‪= C‬‬ ‫2 ﺇﺷﺎﺭﺓ ﺍﻟﺪﺍﻟﺔ ﺩ ﺣﻴﺚ ﺩ)ﺱ( = ﺱ٢ + ٣ ﺗﻜﻮﻥ‬ ‫...............................................................................‬ ‫..................................................................................................................................‬ ‫3 ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﻓﻰ ﻣﺠﻤﻮﻋﺔ ﺍﻷﻋﺪﺍﺩ ﻛﺒﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ – ﺕ، ﺕ ﻫﻰ‬ ‫ﺍﻟﻤﺮ‬ ‫4 ﻣﺪﻯ ﺍﻟﺪﺍﻟﺔ ﺩ ﺣﻴﺚ ﺩ)‪ ٣ = (i‬ﺟﺎ‪ i‬ﻫﻮ‬ ‫...................................................................‬ ‫................................................................................................................................................‬ ‫5 ﺃﺻﻐﺮ ﺯﺍﻭﻳﺔ ﻣﻮﺟﺒﺔ ﻣﻜﺎﻓﺌﺔ ﻟﻠﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ )-٠٤٨‪ (c‬ﻗﻴﺎﺳﻬﺎ ............................ ﻭﺗﻘﻊ ﻓﻰ ﺍﻟﺮﺑﻊ‬ ‫................................‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫1 أ ﺃﺛﺒﺖ ﺃﻥ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ - ٥ﺱ + ٣ = ٠ ﺣﻘﻴﻘﻴﺎﻥ ﻣﺨﺘﻠﻔﺎﻥ، ﺛﻢ ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺍﻟﺤﻞ ﻓﻰ ﺡ ﻣﻘﺮﺑﺎ‬ ‫ً‬ ‫ﺍﻟﻨﺎﺗﺞ ﻟﺮﻗﻢ ﻋﺸﺮﻯ ﻭﺍﺣﺪ. ................................................................................................................................................................‬ ‫ب ﺃﻭﺟﺪ ﻓﻰ ﺃﺑﺴﻂ ﺻﻮﺭﺓ ﻗﻴﻤﺔ ﺍﻟﻤﻘﺪﺍﺭ:ﺟﺎ )- ٠٣˚( ﺟﺘﺎ ٠٢٤˚ + ﻇﺎ٥٢‪c‬‬ ‫....................................................................‬ ‫ﻇﺘﺎ٥٦‪c‬‬ ‫2 أ ﻓﻰ ﺍﻟﻤﻌﺎﺩﻟﺔ )‪ (٥ – C‬ﺱ٢ + )‪ (١٠ – C‬ﺱ – ٥ = ٠ ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ C‬ﻓﻰ ﺍﻟﺤﺎﻻﺕ ﺍﻵﺗﻴﺔ:‬ ‫: ﺇﺫﺍ ﻛﺎﻥ ﻣﺠﻤﻮﻉ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ = ٤ ...........................................................................................................................‬ ‫: ﺇﺫﺍ ﻛﺎﻥ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﻫﻮ ﺍﻟﻤﻌﻜﻮﺱ ﺍﻟﻀﺮﺑﻰ ﻟﻠﺠﺬﺭ ﺍﻵﺧﺮ. ...........................................................‬ ‫ب ﺍﺑﺤﺚ ﺇﺷﺎﺭﺓ ﺍﻟﺪﺍﻟﺔ ﺩ ﺣﻴﺚ ﺩ)ﺱ( = ﺱ٢ + ٢ ﺱ – ٥١ ﻣﻊ ﺗﻮﺿﻴﺢ ﺫﻟﻚ ﻋﻠﻰ ﺧﻂ ﺍﻷﻋﺪﺍﺩ.‬ ‫.......................................................................................................................................................................................................................‬ ‫3‬ ‫أ ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺔ: ٥ﺱ٢ + ٢١ﺱ ‪٤٤ G‬‬ ‫............................................................................................................‬ ‫.......................................................................................................................................................................................................................‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ﺟﺎ ‪ ٣ = i‬ﺣﻴﺚ ٠٩‪ ،c١٨٠ > i > c‬ﺃﻭﺟﺪ ﻗﻴﻤﺔ: ﺟﺘﺎ )٠٧٢‪،(i – c‬ﻇﺎ )٠٨١‪(i + c‬‬ ‫٥‬ ‫.........................‬ ‫.......................................................................................................................................................................................................................‬ ‫4 أ ﺿﻊ ﺍﻟﻌﺪﺩ ﻛﺐ ﺍﻵﺗﻰ ﻓﻰ ﺃﺑﺴﻂ ﺻﻮﺭﺓ )٦٢ – ٤ﺕ( – )٩ – ٠٢ ﺕ( ﺣﻴﺚ ﺕ٢ = -١‬ ‫ﺍﻟﻤﺮ‬ ‫ب ﺍﻟﺮﺑﻂ ﺑﺎﻟ ﺎﺿﺔ: ﻛﻞ ﻻﻋﺐ ﻛﺮﺓ ﺍﻟﻘﺪﻡ ﺍﻟﻜﺮﺓ ﻧﺤﻮ ﺍﻟﻬﺪﻑ ﻣﻦ ﻣﺴﺎﻓﺔ ﺱ ﻣﺘﺮﺍ ﻋﻦ ﺣﺎﺭﺱ ﺍﻟﻤﺮﻣﻰ،‬ ‫ﻳﺮ‬ ‫ﻓﻴﻘﻔﺰ ﺍﻟﺤﺎﺭﺱ ﻭﻳﻤﺴﻚ ﺍﻟﻜﺮﺓ ﻋﻠﻰ ﺍﺭﺗﻔﺎﻉ ١٫٢ ﻣﺘﺮﺍ ﻋﻦ ﺳﻄﺢ ﺍﻷﺭﺽ ﻓﺈﺫﺍ ﻛﺎﻥ‬ ‫ً‬ ‫ﻣﺴﺎﺭ ﺍﻟﻜﺮﺓ ﻳﻤﻴﻞ ﺑﺰﺍﻭﻳﺔ ﻗﻴﺎﺳﻬﺎ ٠٣‪ c‬ﻣﻊ ﺍﻷﻓﻘﻰ. ﻓﺄﻭﺟﺪ ﻷﻗﺮﺏ ﺭﻗﻢ ﻋﺸﺮﻯ‬ ‫‪c‬‬ ‫ﻭﺍﺣﺪ ﺍﻟﻤﺴﺎﻓﺔ ﺑﻴﻦ ﺍﻟﻼﻋﺐ ﻭﺣﺎﺭﺱ ﺍﻟﻤﺮﻣﻰ ﻋﻨﺪﻣﺎ ﺭﻛﻞ ﺍﻟﻼﻋﺐ ﺍﻟﻜﺮﺓ.‬ ‫.................................‬ ‫.............................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
‫ﺍﺧﺘﺒﺎﺭﺍﺕ ﻋﺎﻣﺔ‬ ‫)اﻟﺠﺒﺮ وﺣﺴﺎب اﻟﻤﺜﻠﺜﺎت(‬ ‫اﻻﺧﺘﺒﺎر اﻟﺜﺎﻧﻰ‬ ‫‪k‬‬ ‫‪:IÉ£©ªdG äÉHÉLE’G ø«H øe áë«ë°üdG áHÉLE’G ôàNG :’hCG‬‬ ‫1 ﺃﺑﺴﻂ ﺻﻮﺭﺓ ﻟﻠﻌﺪﺩ ﺍﻟﺘﺨﻴﻠﻰ ﺕ٣٧ ﻫﻮ:‬ ‫ب ١‬ ‫أ -١‬ ‫...................................................................................................................................................‬ ‫ﺟ -ﺕ‬ ‫د ﺕ‬ ‫2 ﺍﻟﺪﺍﻟﺔ ﺩ: ]- ٤، ٧[ # ﺡ ﺣﻴﺚ ﺩ)ﺱ( = ٦ - ٢ﺱ ﺗﻜﻮﻥ ﺇﺷﺎﺭﺗﻬﺎ ﻣﻮﺟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ:‬ ‫د [ ٣، ٧ ]‬ ‫ﺟ ]- ٤، ٧[‬ ‫ب [ ٣، ٧ ]‬ ‫أ ]- ٤، ٣ ]‬ ‫.............................................‬ ‫3 ﺇﺫﺍ ﻛﺎﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ٤ ﺱ٢ – ٢١ ﺱ + ﺟـ = ٠ ﻣﺘﺴﺎﻭﻳﻴﻦ ﻓﺈﻥ ﺟـ ﺗﺴﺎﻭﻯ:‬ ‫ﺟ ٩‬ ‫ب ٤‬ ‫أ ٣‬ ‫4 ﻇﺎ ‪ ` r -j‬ﺗﺴﺎﻭﻯ:‬ ‫٦‬ ‫أ - ٣‬ ‫...................................................................‬ ‫د ٦١‬ ‫.........................................................................................................................................................................................‬ ‫ب - ١‬ ‫٣‬ ‫ﺟ‬ ‫١‬ ‫٣‬ ‫د‬ ‫٣‬ ‫5 ﺍﻟﻘﻴﺎﺱ ﺍﻟﺪﺍﺋﺮﻯ ﻟﺰﺍﻭﻳﺔ ﻛﺰﻳﺔ ﺗﺤﺼﺮ ﻗﻮﺳﺎ ﻃﻮﻟﻪ ٣ﺳﻢ ﻣﻦ ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻗﻄﺮﻫﺎ ٤ﺳﻢ ﻫﻮ:‬ ‫ﻣﺮ‬ ‫ً‬ ‫‪E‬‬ ‫‪E‬‬ ‫٣ ‪E‬‬ ‫٢ ‪E‬‬ ‫د ٦‬ ‫ﺟ ٥‬ ‫ب )٢(‬ ‫أ )٣(‬ ‫....................................‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫1 أ ﺑﻴﻦ ﻧﻮﻉ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ٩ = ٦ ﺱ، ﺛﻢ ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺍﻟﺤﻞ.‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ: ٧ ﻗﺘﺎ ‪ ٢٥ = C‬ﺣﻴﺚ ‪ .r > C > r‬ﻓﺄﻭﺟﺪ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻌﺪﺩﻳﺔ ﻟﻠﻤﻘﺪﺍﺭ: ﻇﺎ ) ‪ - ( C + r‬ﻇﺘﺎ )‪( r - C‬‬ ‫٢‬ ‫٢‬ ‫.........................................................................‬ ‫.......................................................................................................................................................................................................................‬ ‫2‬ ‫أ ﺃﻭﺟﺪ ﻗﻴﻤﺘﻰ ‪ ،C‬ﺏ ﺍﻟﺤﻘﻴﻘﻴﺘﻴﻦ ﺍﻟﻠﺘﻴﻦ ﺗﺤﻘﻘﺎﻥ ﺍﻟﻤﻌﺎﺩﻟﺔ: )‪) – (٣ + C‬ﺏ – ١( ﺕ = ٧ – ٩ ﺕ ﺣﻴﺚ ﺕ٢ = -١‬ ‫.......................................................................................................................................................................................................................‬ ‫ب ﺣﻮﻝ ﻗﻴﺎﺱ ﻛﻞ ﻣﻦ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﻤﻜﺘﻮﺑﺔ ﺑﺎﻟﺪﺭﺟﺎﺕ ﺇﻟﻰ ﺭﺍﺩﻳﺎﻥ ﻭﺍﻟﻤﻜﺘﻮﺑﺔ ﺑﺎﻟﺮﺍﺩﻳﺎﻥ ﺇﻟﻰ ﺩﺭﺟﺎﺕ‬ ‫: ٨‪r‬‬ ‫  ..........................................................................‬ ‫: ٥١٢‪................................................................. c‬‬ ‫٦‬ ‫3 أ ﺍﺑﺤﺚ ﺇﺷﺎﺭﺓ ﺍﻟﺪﺍﻟﺔ ﺩ ﺣﻴﺚ ﺩ)ﺱ( = ٢ﺱ٢ – ٣ ﺱ + ٤ ﻣﻊ ﺗﻮﺿﻴﺢ ﺫﻟﻚ ﻋﻠﻰ ﺧﻂ ﺍﻷﻋﺪﺍﺩ ﺍﻟﺤﻘﻴﻘﻴﺔ‬ ‫ب ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺔ ‪ i‬ﻣﺮﺳﻮﻣﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ، ﺣﻴﺚ ﻳﻤﺮ ﺿﻠﻌﻬﺎ ﺍﻟﻨﻬﺎﺋﻰ ﺑﺎﻟﻨﻘﻄﺔ )٤، - ٣(‬ ‫ﻓﺄﻭﺟﺪ ﺟﺎ‪ ،i‬ﻇﺘﺎ‪.......................................................................................................................................................................... .i‬‬ ‫4‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ )ﺱ + ٢(٢ + )ﺱ + ١( )ﺱ – ٤( > ٠‬ ‫: ﺍﻛﺘﺐ ﺍﻟﻤﺘﺒﺎﻳﻨﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﻓﻰ ﺃﺑﺴﻂ ﺻﻮﺭﺓ.‬ ‫: ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺔ.‬ ‫.......................................................................................................................................................................................................................‬ ‫٢‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ﻝ ، ٢ ﻫﻤﺎ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٦ ﺱ + ٤ = ٠ ﻓﺄﻭﺟﺪ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ )ﻝ + ﻡ(، ﻝ ﻡ.‬ ‫ﻡ‬ ‫.......................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
‫ﺍﺧﺘﺒﺎﺭﺍﺕ ﻋﺎﻣﺔ‬ ‫)اﻟﺠﺒﺮ وﺣﺴﺎب اﻟﻤﺜﻠﺜﺎت(‬ ‫اﻻﺧﺘﺒﺎر اﻟﺜﺎﻟﺚ‬ ‫‪k‬‬ ‫‪IÉ£©ªdG äÉHÉLE’G ø«H øe áë«ë°üdG áHÉLE’G ôàNG :’hCG‬‬ ‫1 ﺇﺫﺍ ﻛﺎﻥ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪C‬ﺱ٢ + ٢ ﺱ + ٥ = ٠ ﻣﻌﻜﻮﺳﺎ ﺿﺮﺑﻴﺎ ﻟﻠﺠﺬﺭ ﺍﻵﺧﺮ ﻓﺈﻥ ‪ C‬ﺗﺴﺎﻭﻯ:‬ ‫ً‬ ‫ًّ‬ ‫د ٥‬ ‫ﺟ ٢‬ ‫ب -٢‬ ‫أ -٥‬ ‫2 ﺇﺷﺎﺭﺓ ﺍﻟﺪﺍﻟﺔ ﺩ ﺣﻴﺚ ﺩ)ﺱ( = ٦ – ٢ ﺱ ﺗﻜﻮﻥ ﻣﻮﺟﺒﺔ ﺇﺫﺍ ﻛﺎﻧﺖ:‬ ‫ﺟ ﺱ>٣‬ ‫ب ﺱ‪٣G‬‬ ‫أ ﺱ<٣‬ ‫..........................‬ ‫..........................................................................................‬ ‫د ﺱ‪٣H‬‬ ‫3 ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ ١ + ﺕ ، ١ – ﺕ ﺣﻴﺚ ﺕ٢ = -١ ﻫﻰ:‬ ‫أ ﺱ٢ + ٢ﺱ + ٢ = ٠ ب ﺱ٢ – ٢ﺱ + ٢ = ٠ ﺟ ﺱ٢ + ٢ﺱ – ٢ = ٠‬ ‫...............................................................................‬ ‫د ﺱ٢ – ٢ﺱ – ٢ = ٠‬ ‫4 ﺇﺫﺍ ﻛﺎﻧﺖ ‪ i‬ﺯﺍﻭﻳﺔ ﻣﺮﺳﻮﻣﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﺑﺤﻴﺚ ﺟﺘﺎ ‪ ،٠ < i‬ﻓﻰ ﺃﻯ ﺭﺑﻊ ﻳﻘﻊ ﺿﻠﻊ ﺍﻟﻨﻬﺎﻳﺔ ﻟﻠﺰﺍﻭﻳﺔ ‪:i‬‬ ‫د ﺍﻷﻭﻝ ﺃﻭ ﺍﻟﺮﺍﺑﻊ‬ ‫ﺟ ﺍﻷﻭﻝ ﺃﻭ ﺍﻟﺜﺎﻟﺚ‬ ‫ب ﺍﻷﻭﻝ ﺃﻭ ﺍﻟﺜﺎﻧﻰ‬ ‫أ ﺍﻷﻭﻝ‬ ‫5 ﺇﺫﺍ ﻛﺎﻧﺖ ٢ ﺟﺘﺎ ‪ ٢ - = C‬ﻓﺈﻥ ﺃﻗﻞ ﺯﺍﻭﻳﺔ ﻣﻮﺟﺒﺔ ﺗﺤﻘﻖ ﻫﺬه ﺍﻟﺪﺍﻟﺔ ﺍﻟﻤﺜﻠﺜﻴﺔ ﻫﻰ:‬ ‫د ٥١٣‪c‬‬ ‫ﺟ ٥٢٢‪c‬‬ ‫ب ٥٣١‪c‬‬ ‫أ ٥٤‪c‬‬ ‫.........................................................‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫1‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ ﻝ، ﻡ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ )٢ ﺱ + ٣( = ٥ ﻓﺄﻭﺟﺪ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ ﻝ + ١ ، ﻡ + ١.‬ ‫.................‬ ‫.......................................................................................................................................................................................................................‬ ‫ب ﺯﺍﻭﻳﺔ ﻛﺰﻳﺔ ﻗﻴﺎﺳﻬﺎ ٠٦‪ c‬ﻭﺗﻘﺎﺑﻞ ﻗﻮﺳﺎ ﻃﻮﻟﻪ ٧‪ r‬ﺳﻢ، ﺍﺣﺴﺐ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮ ﺩﺍﺋﺮﺗﻬﺎ.‬ ‫ﻣﺮ‬ ‫ً‬ ‫٣‬ ‫...........................‬ ‫.......................................................................................................................................................................................................................‬ ‫2 أ ﺿﻊ ﺍﻟﻌﺪﺩ ٢ - ٣ﺕ ﻓﻰ ﺻﻮﺭﺓ ﻋﺪﺩ ﻛﺐ. ﺣﻴﺚ ﺕ٢ = -١.‬ ‫ﻣﺮ‬ ‫٣ + ٢ﺕ‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ٤ ﺟﺎ ‪ ٠ = ٣ - C‬ﺃﻭﺟﺪ ‪ ( Cc) X‬ﺣﻴﺚ ‪ ،٠ [ ∋ C‬ﻁ‬ ‫] ........................................................................................‬ ‫٢‬ ‫.........................................................................................‬ ‫3‬ ‫أ ﺇﺫﺍ ﻛﺎﻧﺖ ﺩ : ﺡ # ﺡ ﺣﻴﺚ ﺩ)ﺱ( = - ﺱ٢ + ٨ ﺱ – ٥١‬ ‫: ﻋﻴﻦ ﻣﻦ ﺍﻟﺮﺳﻢ ﺇﺷﺎﺭﺓ ﻫﺬه ﺍﻟﺪﺍﻟﺔ.‬ ‫: ﺍﺭﺳﻢ ﻣﻨﺤﻨﻰ ﺍﻟﺪﺍﻟﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ ] ١، ٧ [‬ ‫.......................................................................................................................................................................................................................‬ ‫‬‫ب ﺇﺫﺍ ﻛﺎﻥ ﺱ = ٣ + ٢ﺕ، ﺹ = ٤١ -٢ﺕ ﻓﺄﻭﺟﺪ ﺱ + ﺹ ﻓﻰ ﺻﻮﺭﺓ ﻋﺪﺩ ﻛﺐ.‬ ‫ﻣﺮ‬ ‫ﺕ‬ ‫..................................................‬ ‫4 أ ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﺘﺒﻴﺎﻳﻨﺔ ﺱ٢ + ٣ﺱ – ٤ ‪٠ H‬‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ﻇﺎ ﺏ = ٣ ﺣﻴﺚ ٠٨١‪ > c‬ﺏ > ٠٧٢‪ c‬ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ: ﺟﺘﺎ )٠٦٣‪ – c‬ﺏ( - ﺟﺘﺎ )٠٩‪ – c‬ﺏ(‬ ‫٤‬ ‫..............................................................................................................‬ ‫.......................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
‫ﺍﺧﺘﺒﺎﺭﺍﺕ ﻋﺎﻣﺔ‬ ‫)اﻟﻬﻨﺪﺳﺔ(‬ ‫اﻻﺧﺘﺒﺎر اﻟﺮاﺑﻊ‬ ‫‪k‬‬ ‫‪πªcCG :’hCG‬‬ ‫1 ﺇﺫﺍ ﻗﻄﻊ ﻣﺴﺘﻘﻴﻤﺎﻥ ﻋﺪﺓ ﻣﺴﺘﻘﻴﻤﺎﺕ ﻣﺘﻮﺍﺯﻳﺔ، ﻓﺈﻥ ﺃﻃﻮﺍﻝ ﺍﻟﻘﻄﻊ ﺍﻟﻨﺎﺗﺠﺔ ﻋﻠﻰ ﺃﺣﺪ ﺍﻟﻘﺎﻃﻌﻴﻦ ﺗﻜﻮﻥ‬ ‫....................‬ ‫2 ﺍﻟﻨﺴﺒﺔ ﺑﻴﻦ ﻣﺴﺎﺣﺘﻰ ﺳﻄﺤﻰ ﻣﺜﻠﺜﻴﻦ ﻣﺘﺸﺎﺑﻬﻴﻦ ﻫﻲ ٣ : ٥، ﺇﺫﺍ ﻛﺎﻧﺖ ﻣﺴﺎﺣﺔ ﺳﻄﺢ ﺍﻟﻤﺜﻠﺚ ﺍﻷﻭﻝ ٦٣ ﺳﻢ٢ ﻓﺈﻥ‬ ‫ﻣﺴﺎﺣﺔ ﺳﻄﺢ ﺍﻟﻤﺜﻠﺚ ﺍﻟﺜﺎﻧﻰ ﺗﺴﺎﻭﻯ .....................................................................................................................................................‬ ‫‪C‬‬ ‫3‬ ‫: ﺇﺫﺍ ﻛﺎﻥ ﺱ ﺹ // ﺏ ﺟـ ، ﺱ ﺹ : ﺏ ﺟـ = ٣ : ٨ ﻓﺈﻥ:‬ ‫أ ‪C‬ﺱ : ﺱ ﺏ = ........................... : ...........................‬ ‫ب‬ ‫ﻣﺤﻴﻂ 9‪C‬ﺱ ﺹ : ﻣﺤﻴﻂ 9‪C‬ﺏ ﺟـ = ......................... : ..........................‬ ‫4‬ ‫: ﺇﺫﺍ ﻛﺎﻥ ﺟـ ‪ E‬ﻳﻨﺼﻒ )‪c‬ﺟـ(،‬ ‫‪ C‬ﺟـ = ٣ ﺳﻢ، ﺏ ﺟـ = ٥٫٧ ﺳﻢ، ﻓﺈﻥ ‪ : E C‬ﺏ ‪..................................................... = E‬‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫1 أ ﺃﻭﺟﺪ ﻗﻮﺓ ﺍﻟﻨﻘﻄﺔ ‪ C‬ﺑﺎﻟﻨﺴﺒﺔ ﺇﻟﻰ ﺍﻟﺪﺍﺋﺮﺓ ﻡ ﺍﻟﺘﻰ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٣ ﺳﻢ ، ‪C‬ﻡ = ٤ ﺳﻢ.‬ ‫ب ﺭﺳﻢ ﻣﻬﻨﺪﺱ ﻣﻌﻤﺎﺭﻯ ﻣﺨﻄﻄًﺎ ﻟﻘﻄﻌﺔ ﺃﺭﺽ ﻣﺴﺘﻄﻴﻠﺔ ﺍﻟﺸﻜﻞ، ﻃﻮﻟﻬﺎ ﺿﻌﻒ ﻋﺮﺿﻬﺎ، ﻭﻣﺴﺎﺣﺘﻬﺎ ٠٠٢‬ ‫ﻣﺘﺮ٢ ﺑﻤﻘﻴﺎﺱ ﺭﺳﻢ ١ : ٠٠٢، ﺃﻭﺟﺪ ﻃﻮﻝ ﻗﻄﻌﺔ ﺍﻷﺭﺽ ﻓﻰ ﺍﻟﻤﺨﻄﻂ.‬ ‫: ﺱ ﺹ // ‪ E‬ﻫـ // ﻝ ﻉ ﺃﻭﺟﺪ:‬ ‫2‬ ‫: ﻃﻮﻝ ﻫـ ﻡ‬ ‫‪E‬‬ ‫: ﻃﻮﻝ ﻡ ﻉ‬ ‫?‬ ‫?‬ ‫3‬ ‫: ‪ C‬ﺏ ﻗﻄﺮ ﻓﻰ ﺍﻟﺪﺍﺋﺮﺓ،‬ ‫ﺟـ ‪ E‬ﻣﻤﺎﺱ ﻟﻠﺪﺍﺋﺮﺓ ﻋﻨﺪ ﺟـ ، ‪C‬ﺟـ = ٢١ ﺳﻢ، ﺍﺏ = ٣١ ﺳﻢ. ﺃﺛﺒﺖ ﺃﻥ:‬ ‫أ 9 ‪ E‬ﺟـ ﺏ + 9 ‪ C E‬ﺟـ‬ ‫ب ﺃﻭﺟﺪ ﻃﻮﻝ ﺟـ ‪ E‬ﻷﻗﺮﺏ ﺳﻢ‬ ‫ﺟ ﺃﻭﺟﺪ ﻣﺴﺎﺣﺔ 9 ‪C‬ﺏ ﺟـ‬ ‫‪C‬‬ ‫‪E‬‬ ‫4 ‪C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ‪ ،C‬ﻓﻴﻪ ‪ C‬ﺏ = ٠٢ ﺳﻢ، ‪C‬ﺟـ = ٥١ ﺳﻢ، ‪ ∋ E‬ﺏ ﺟـ ﺑﺤﻴﺚ ﻛﺎﻥ ﺏ ‪ ١٠ = E‬ﺳﻢ،‬ ‫ﺭﺳﻢ ‪ C‬ﻫـ = ﺏ ﺟـ ﻭﻳﻘﻄﻊ ﺏ ﺟـ ﻓﻰ ﻫـ ، ﻭﻣﻦ ‪ E‬ﺭﺳﻢ ‪ E‬ﻭ // ﺏ ‪ C‬ﻭﻳﻘﻄﻊ ‪ C‬ﻫـ ﻓﻰ ﻭ.‬ ‫ﺃﺛﺒﺖ ﺃﻥ ﺟـ ﻭ ﻳﻨﺼﻒ ‪c‬ﺟـ.‬ ‫‪ïM‬‬ ‫−‬
‫ﺍﺧﺘﺒﺎﺭﺍﺕ ﻋﺎﻣﺔ‬ ‫)اﻟﻬﻨﺪﺳﺔ(‬ ‫اﻻﺧﺘﺒﺎر اﻟﺨﺎﻣﺲ‬ ‫‪k‬‬ ‫‪:πªcCG :’hCG‬‬ ‫1 ﺍﻟﻨﺴﺒﺔ ﺑﻴﻦ ﻣﺴﺎﺣﺘﻰ ﺳﻄﺤﻰ ﻣﺜﻠﺜﻴﻦ ﻣﺘﺸﺎﺑﻬﻴﻦ ﻛﺎﻟﻨﺴﺒﺔ ﺑﻴﻦ‬ ‫.....................................................................................................‬ ‫2 ﻳﺘﺸﺎﺑﻪ ﺍﻟﻤﻀﻠﻌﺎﻥ ﺇﺫﺍ ﻛﺎﻥ‬ ‫................................................................... ...................................................................‬ ‫3‬ ‫:‬ ‫،‬ ‫‪E‬‬ ‫أ )‪= ٢(E C‬‬ ‫ب‬ ‫‪ E‬ﻥ * ﻥ ﻫـ = ....................................................‬ ‫ﺟ‬ ‫9 ‪ E C‬ﺟـ + 9 ..............................................‬ ‫....................................................................‬ ‫‪C‬‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫1 أ ﺃﻭﺟﺪ ﻗﻮﺓ ﺍﻟﻨﻘﻄﺔ ﺏ ﺑﺎﻟﻨﺴﺒﺔ ﺇﻟﻰ ﺍﻟﺪﺍﺋﺮﺓ ﻡ، ﺍﻟﺘﻰ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٨ ﺳﻢ، ﺏ ﻡ = ٥ ﺳﻢ‬ ‫ب‬ ‫:‬ ‫: ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻤﻀﻠﻊ ‪ C‬ﺏ ﺟـ ‪ + E‬ﺍﻟﻤﻀﻠﻊ ﺱ ﺏ ﻉ ﺹ‬ ‫‪C‬‬ ‫   ﻓﺎﺛﺒﺖ ﺃﻥ: ﺱ ﺹ // ‪. E C‬‬ ‫: ﺇﺫﺍ ﻛﺎﻥ ﻣﺤﻴﻂ ﺍﻟﻤﻀﻠﻊ ‪ C‬ﺏ ﺟـ ‪ ١٤ = E‬ﺳﻢ،‬ ‫   ﻣﺤﻴﻂ ﺍﻟﻤﻀﻠﻊ ﺱ ﺏ ﻉ ﺹ = ٠١ ﺳﻢ،‬ ‫   ﻃﻮﻝ ﺱ ﺏ = ٢ ﺳﻢ، ﻓﺄﻭﺟﺪ ﻃﻮﻝ ‪ C‬ﺏ‬ ‫2‬ ‫: ‪ C‬ﺏ = ٦ ﺳﻢ، ﺏ ﺟـ = ٢١ ﺳﻢ،‬ ‫ﺟـ ‪ ٨ = C‬ﺳﻢ، ﻭﺟـ = ٣ ﺳﻢ ، ‪ E‬ﺏ = ٥٫٤ ﺳﻢ ، ‪ E‬ﻭ = ٦ ﺳﻢ.‬ ‫ﺃﺛﺒﺖ ﺃﻥ:‬ ‫أ 9 ‪ C‬ﺏ ﺟـ + 9 ‪ E‬ﺏ ﻭ‬ ‫ب 9 ﻫـ ﻭ ﺟـ ﻣﺘﺴﺎﻭﻯ ﺍﻟﺴﺎﻗﻴﻦ.‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪E‬‬ ‫3 ﺱ ﺹ ﻉ ﻣﺜﻠﺚ، ﻧﺼﻔﺖ ﺯﺍﻭﻳﺔ ﺹ ﺑﻤﻨﺼﻒ ﻗﻄﻊ ﺱ ﻉ ﻓﻰ ﻡ، ﺛﻢ ﺭﺳﻢ ﻥ ﻡ // ﺹ ﻉ ﻓﻘﻄﻊ ﺱ ﺹ ﻓﻰ ﻥ.‬ ‫ﺱﻥ‬ ‫ﺱﺹ‬ ‫ﺃﺛﺒﺖ ﺃﻥ: ﺹ ﻉ = ﺹ ﻥ ، ﻭﺇﺫﺍ ﻛﺎﻥ ﺱ ﺹ = ٦ ﺳﻢ ، ﺹ ﻉ = ٤ ﺳﻢ، ﻓﺄﻭﺟﺪ ﻃﻮﻝ ﺱ ﻥ .‬ ‫4 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ‪ .C‬ﺭﺳﻢ ‪ = E C‬ﺏ ﺟـ ﻓﻘﻄﻌﻬﺎ ﻓﻰ ‪.E‬‬ ‫ﺭﺳﻢ ﺍﻟﻤﺜﻠﺜﺎﻥ ﺍﻟﻤﺘﺴﺎﻭﻳﺎ ﺍﻷﺿﻼﻉ ‪ C‬ﺏ ﻫـ ، ﺟـ ‪ C‬ﻭ ﺧﺎﺭﺝ ﺍﻟﻤﺜﻠﺚ ‪ C‬ﺏ ﺟـ‬ ‫ﺃﺛﺒﺖ ﺃﻥ:‬ ‫أ ﺍﻟﺸﻜﻞ ﺍﻟﺮﺑﺎﻋﻰ ‪ E C‬ﺏ ﻫـ + ﺍﻟﺸﻜﻞ ﺍﻟﺮﺑﺎﻋﻰ ﺟـ ‪ C E‬ﻭ.‬ ‫ب ﻣﺴﺎﺣﺔ ﺳﻄﺢ ﺍﻟﺸﻜﻞ ‪ E C‬ﺏ ﻫـ ﺏ ‪E‬‬ ‫=‬ ‫ﻣﺴﺎﺣﺔ ﺳﻄﺢ ﺍﻟﺸﻜﻞ ﺟـ ‪ C E‬ﻭ ﺟـ ‪E‬‬ ‫¯‬ ‫−‬ ‫¯‬
‫ﺍﺧﺘﺒﺎﺭﺍﺕ ﻋﺎﻣﺔ‬ ‫)اﻟﻬﻨﺪﺳﺔ(‬ ‫اﻻﺧﺘﺒﺎر اﻟﺴﺎدس‬ ‫‪k‬‬ ‫‪:πªcCG :’hCG‬‬ ‫1 أ ﺇﺫﺍ ﺭﺳﻢ ﻣﺴﺘﻘﻴﻢ ﻳﻮﺍﺯﻯ ﺃﺣﺪ ﺃﺿﻼﻉ ﻣﺜﻠﺚ، ﻭ ﻳﻘﻄﻊ ﺍﻟﻀﻠﻌﻴﻦ ﺍﻵﺧﺮﻳﻦ ﻓﺈﻧﻪ‬ ‫ُ‬ ‫ب‬ ‫‪ E C‬ﻣﻤﺎﺱ ﻟﻠﺪﺍﺋﺮﺓ ﻋﻨﺪ ‪ ،E‬ﻓﺈﻥ:‬ ‫:‬ ‫: ‪ C‬ﺟـ * ‪ C‬ﺏ = .........................................‬ ‫: ﺇﺫﺍ ﻛﺎﻥ ‪ C‬ﺟـ = ٨ ﺳﻢ، ‪ C‬ﺏ = ٢ﺳﻢ، ﻓﺈﻥ ‪.................................................... = E C‬‬ ‫٢ ﺳﻢ ﻓﺈﻥ، ‪ C‬ﺟـ = .................................‬ ‫: ﺇﺫﺍ ﻛﺎﻥ ‪ C‬ﺏ = ﺏ ﺟـ ، ‪٣ = E C‬‬ ‫........................................................‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫1 أ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻨﺴﺒﺔ ﺑﻴﻦ ﻣﺴﺎﺣﺘﻰ ﺳﻄﺤﻰ ﻣﻀﻠﻌﻴﻦ ﻣﺘﺸﺎﺑﻬﻴﻦ ﺗﺴﺎﻭﻯ ٦١ : ٩٤، ﻓﻤﺎ ﺍﻟﻨﺴﺒﺔ ﺑﻴﻦ ﻃﻮﻟﻰ ﺿﻠﻌﻴﻦ‬ ‫ﻣﺘﻨﺎﻇﺮﻳﻦ ﻓﻴﻬﻤﺎ? ﻭﻣﺎ ﺍﻟﻨﺴﺒﺔ ﺑﻴﻦ ﻣﺤﻴﻄﻴﻬﻤﺎ?‬ ‫ب ﺩﺍﺋﺮﺗﺎﻥ ﻣﺘﻘﺎﻃﻌﺘﺎﻥ ﻓﻰ ‪ ،C‬ﺏ ﺭﺳﻢ ﻣﻤﺎﺱ ﻣﺸﺘﺮﻙ ﻳﻤﺴﺎﻧﻬﻤﺎ ﻓﻰ ﺱ، ﺹ.‬ ‫ﺇﺫﺍ ﻛﺎﻥ ‪ C‬ﺏ ∩ ﺱ ﺹ = }ﺟـ{ ﺍﺛﺒﺖ ﺃﻥ ﺟـ ﻣﻨﺘﺼﻒ ﺱ ﺹ .‬ ‫2‬ ‫أ‬ ‫: ‪ C‬ﺱ // ﺏ ﺹ // ﺟـ ﻉ ،‬ ‫ﻭ ‪ ٦ = C‬ﺳﻢ ، ﻭ ﺱ = ٤ ﺳﻢ ، ﺱ ﺹ = ٣ ﺳﻢ،‬ ‫ﺏ ﺟـ = ٥٫٧ ﺳﻢ. ﺃﻭﺟﺪ ﻃﻮﻝ ﻛﻞ ﻣﻦ ‪ C‬ﺏ ، ﻉ ﺹ‬ ‫‪C‬‬ ‫ب‬ ‫:‬ ‫9 ﺟـ ‪ E‬ﻫـ + 9 ﺟـ ﺏ ‪C‬‬ ‫ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻷﻃﻮﺍﻝ ﺍﻟﻤﻮﺿﺤﺔ ﻋﻠﻰ ﺍﻟﺮﺳﻢ‬ ‫ﺃﻭﺟﺪ ﻃﻮﻝ ﻛﻞ ﻣﻦ ﺏ ﻫـ ، ‪ E‬ﻫـ .‬ ‫3‬ ‫‪E‬‬ ‫‪C‬‬ ‫?‬ ‫أ ﺃﻭﺟﺪ ﻗﻮﺓ ﺍﻟﻨﻘﻄﺔ ﺟـ ﺑﺎﻟﻨﺴﺒﺔ ﺇﻟﻰ ﺍﻟﺪﺍﺋﺮﺓ ﻡ ﺍﻟﺘﻰ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٦ ﺳﻢ، ﺟـ ﻡ = ٦ ﺳﻢ‬ ‫ب‬ ‫: ‪ C‬ﺏ ∩ ‪ E‬ﻫـ = }ﺟـ{،‬ ‫ﺟـ ‪ = C‬ﺟـ ﺏ ، ﺟـ ‪٢ = E‬ﺳﻢ ، ﺟـ ﻫـ = ٨ ﺳﻢ،‬ ‫ﻡ ‪ E‬ﻣﻤﺎﺳﺔ ﻟﻠﺪﺍﺋﺮﺓ. ﻡ ﺏ = ١ ‪ C‬ﺏ. ﺃﻭﺟﺪ ﻃﻮﻝ ﻡ ‪. E‬‬ ‫٢‬ ‫‪E‬‬ ‫‪C‬‬ ‫4‬ ‫: ‪C‬ﺏ ﺟـ ﻣﺜﻠﺚ، ﻓﻴﻪ ﺱ ∋ ‪ C‬ﺏ ﺑﺤﻴﺚ ﻛﺎﻥ ‪ C‬ﺱ = ٤ ﺳﻢ،‬ ‫ﺱ ﺏ = ٦ ﺳﻢ، ﺹ ∋ ‪ C‬ﺟـ ﺑﺤﻴﺚ ﻛﺎﻥ ‪ C‬ﺹ = ٥ ﺳﻢ، ﺹ ﺟـ = ٣ ﺳﻢ.‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ: 9 ‪ C‬ﺱ ﺹ + 9 ‪ C‬ﺟـ ﺏ‬ ‫ب ﺍﻟﺸﻜﻞ ﺱ ﺏ ﺟـ ﺹ ﺭﺑﺎﻋﻰ ﺩﺍﺋﺮﻯ.‬ ‫ﺟ ﺇﺫﺍ ﻛﺎﻧﺖ ﻣـ)9 ‪ C‬ﺱ ﺹ( = ٨ ﺳﻢ٢. ﺃﻭﺟﺪ ﻣﺴﺎﺣﺔ ﺳﻄﺢ ﺍﻟﻤﻀﻠﻊ ﺱ ﺏ ﺟـ ﺹ.‬ ‫‪ïM‬‬ ‫−‬ ‫‪C‬‬
‫‪z‬‬ ‫ﺇﺟﺎﺑﺎﺕ ﺑﻌﺾ ﺍﻟﺘﻤﺎﺭﻳﻦ‬ ‫−‬ ‫:‬ ‫¯‬ ‫1 ﺏ    2 ﺩ‬ ‫7 أ }-٢{‬ ‫د ﺱ٢ + ٠١ ﺱ - ١٢ = ٠‬ ‫ﺟ ٣ﺱ٢ + ٤١ﺱ + ٤ = ٠‬ ‫22 ٤٥ * ٢ = )ﺱ + ٦()ﺱ + ٩( ﻭﻣﻨﻬﺎ ﺱ٢ +٥١ﺱ -٤٥ = ٠ ﻭﻣﻨﻬﺎ ﺱ = ٣‬ ‫   3 ﺩ‬ ‫ب‬ ‫   4 ﺩ‬ ‫42 ﺣﻞ ﻳﻮﺳﻒ ﺻﺤﻴﺢ‬ ‫‪z‬‬ ‫ﺟ }-٣، ١{‬ ‫8 أ } -٥، ٨{‬ ‫د }٤٧٫٦، -٤٧٫٠{‬ ‫9‬ ‫5 ﺃ‬ ‫−‬ ‫ﺟ }- ٣ ، ٢ {‬ ‫٢ ٣‬ ‫1 ﺳﺎﻟﺒﺔ ، ﺡ‬ ‫5 [٣، ∞]‬ ‫ﻫ }١٦٫٢ ، -١٦٫٤{‬ ‫و }٤١٫٢، -٤٩٫٠{‬ ‫8 [٢، ∞]، [- ∞، ٢]‬ ‫9 }-١، ٣{، ﺡ -]-١، ٣[، [-١، ٣]‬ ‫ب }٤٫٤، ٦٫١‬ ‫ﺟ }-٤، -٢{‬ ‫ﺟ ﻥ = ٢٢‬ ‫ب ﻥ = ٨١‬ ‫01 أ ﻥ = ٢١‬ ‫٢‬ ‫٢‬ ‫11 أ ﺩ)ﺱ( = ﺱ٢ + ﺱ - ٦ ب ﺩ)ﺱ( = -ﺱ - ٣ﺱ ﺟ ﺩ)ﺱ( = ﺱ - ٧ﺱ‬ ‫01 أ ﻣﻮﺟﺒﺔ ﻓﻰ ﺡ‬ ‫ب ﻣﻮﺟﺒﺔ ﻓﻰ [ ٠، ∞ ] ﺳﺎﻟﺒﺔ ﻓﻰ [- ∞،٠ ]، ﺻﻔﺮ ﻋﻨﺪﻣﺎ ﺱ =٠‬ ‫ﺟ ﻣﻮﺟﺒﺔ ﻓﻰ[- ∞، ٠ ] ﺳﺎﻟﺒﺔ ﻓﻰ [ ٠،∞ ]، ﺻﻔﺮ ﻋﻨﺪﻣﺎ ﺱ =٠‬ ‫د ﻣﻮﺟﺒﺔ ﻋﻨﺪﻣﺎ ﺱ < -٢ ، ﺳﺎﻟﺒﺔ ﻋﻨﺪﻣﺎ ﺱ > -٢، ﺻﻔﺮ ﻋﻨﺪﻣﺎ ﺱ = -٢‬ ‫21 ﺇﺟﺎﺑﺔ ﺯﻳﺎﺩ ﺧﻄﺄ؛ ﻷﻧﻪ ﻗﺴﻢ ﺍﻟﻄﺮﻓﻴﻦ ﻋﻠﻰ ﻣﺘﻐﻴﺮ ﻭﻫﻮ )ﺱ - ٣(‬ ‫31 ﻥ = ٢ ﺃﻭ ﻥ = ٤‬ ‫ﻫ ﻣﻮﺟﺒﺔ ﻋﻨﺪﻣﺎ ﺱ > ٣ ، ﺳﺎﻟﺒﺔ ﻋﻨﺪﻣﺎ ﺱ < ٣ ، ﺻﻔﺮ ﻋﻨﺪﻣﺎ ﺱ = ٣‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫ح ﻣﻮﺟﺒﺔ ﻓﻰ ﺡ - ]-٢ ،٢[] ﺳﺎﻟﺒﺔ ﻓﻰ [ - ٢، ٢ ]، ﺻﻔﺮ ﻋﻨﺪﻣﺎ ﺱ∋ } -٢، ٢{‬ ‫−‬ ‫1‬ ‫3‬ ‫4‬ ‫أ ١ + ٥ﺕ‬ ‫5‬ ‫أ ١-ﺕ‬ ‫ﺟ !٣ ٢ ﺕ‬ ‫6‬ ‫7 ٧ - ٢ﺕ‬ ‫ﺟ‬ ‫ب -ﺕ‬ ‫ب ٧١ + ٦١ ﺕ ﺟ‬ ‫ب‬ ‫ﺟ‬ ‫ب ١ - ٤ﺕ‬ ‫د‬ ‫د -ﺕ‬ ‫١‬‫١١ + ٥٤ ﺕ‬ ‫٤ + ٧ﺕ‬ ‫٣‬ ‫٠١ - ١١ ﺕ د ٦ + ٨ ﺕ‬ ‫٥ ٥‬ ‫٠١‬ ‫!٥ﺕ‬ ‫8 ﺣﻞ ﺃﺣﻤﺪ ﺻﺤﻴﺢ.‬ ‫2 ﺏ‬ ‫3 ﺃ‬ ‫أ ﺟﺬﺭﺍﻥ ﻛﺒﺎﻥ.‬ ‫ﻣﺮ‬ ‫ب ﺟﺬﺭﺍﻥ ﺣﻘﻴﻘﻴﺎﻥ ﻏﻴﺮ ﻧﺴﺒﻴﻴﻦ‬ ‫د ﺟﺬﺭﺍﻥ ﻛﺒﺎﻥ‬ ‫ﻣﺮ‬ ‫ب - ٣ + ١ ﺕ،- ٣ - ١‬ ‫ﺕ‬ ‫5‬ ‫أ ٢ + ﺕ، ٢ - ﺕ‬ ‫6‬ ‫أ ٦١ - ٤ ﻙ < ٠ ﺃﻯ ﻙ > ٤‬ ‫٢‬ ‫٢‬ ‫ب ٩ - ٤ * )٢ +‬ ‫ﺟ ٤٦ - ٤ * ٦١ ﻙ > ٠ ﺃﻯ ﺃﻥ: ﻙ < ١‬ ‫٢‬ ‫٢‬ ‫١‬ ‫ﻙ ( = ٠ ﺃﻯ ﺃﻥ ﻙ = ٤‬ ‫7 ﺍﻟﻤﻤﻴﺰ = ) ﻝ - ﻡ(٢ + ٤ ﻝ ﻡ = ) ﻝ - ﻡ (٢ ﺃﻱ ﻣﺮﺑﻊ ﻛﺎﻣﻞ، ﻟﺬﻟﻚ ﻓﺈﻥ ﺟﺬﺭﻯ‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ ﻋﺪﺩﺍﻥ ﻧﺴﺒﻴﺎﻥ.‬ ‫9 ﺇﺟﺎﺑﺔ ﺃﺣﻤﺪ ﺧﻄﺄ؛ ﻷﻥ ﺍﻟﺤﺪ ﺍﻟﻤﻄﻠﻖ = -٥ ﻓﻰ ﺍﻟﻤﻌﺎﺩﻟﺔ.‬ ‫11 ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﻫﻮ }٣ ﺕ ، - ٢ﺕ{‬ ‫−‬ ‫٦، -٩‬ ‫1‬ ‫5 ﺟـ‬ ‫3 ﺱ٢ - ٥ﺱ + ٦ =٠‬ ‫7 ﺟـ‬ ‫2 ٨‬ ‫6 ﺟـ‬ ‫8‬ ‫أ -٩١ ، -٤١‬ ‫٣‬ ‫٣‬ ‫9‬ ‫أ -٣، ٣‬ ‫01‬ ‫أ ﺗﻜﻮﻥ ‪ ، ٧- = C‬ﺏ = ٠١‬ ‫11‬ ‫ب -١، -٥٣‬ ‫٤‬ ‫ب ٢، ١‬ ‫٢‬ ‫أ ﺣﻘﻴﻘﻴﺎﻥ ﻧﺴﺒﻴﺎﻥ ، }-٧، ٥{‬ ‫ب ‪ ،١ = C‬ﺏ = ٤‬ ‫ب ﻛﺒﺎﻥ ، } - ٣ + ٧٫١ﺕ ، - ٣ - ٧٫١ ﺕ{‬ ‫ﻣﺮ‬ ‫٤‬ ‫٤‬ ‫ﺟ ﻛﺒﺎﻥ }٢ + ﺕ ، ٢ - ﺕ{‬ ‫ﻣﺮ‬ ‫21 ﺟـ = ٤‬ ‫51 ﻙ = ١‬ ‫ﺩ)ﺱ( <٠ ﻋﻨﺪﻣﺎ ﺱ∋ ]-٢٫١ ، ٢٫٣]‬ ‫1 ]-٣، ٣[‬ ‫5 [٢، ٥]‬ ‫1 ﺏ‬ ‫71‬ ‫1‬ ‫٣‬ ‫41 ﺟـ = ٥٢ ، ﺍﻟﺤﻞ ﻫﻮ } ٥ {‬ ‫٦‬ ‫٢١‬ ‫ب‬ ‫71 أ ﺱ٢ - ٢ﺱ -٨ = ٠‬ ‫ﺟ ٦ﺱ٢ - ٣١ﺱ + ٦ = ٠‬ ‫81 ﺱ٢ - ٨ﺱ + ٥ = ٠‬ ‫12 أ ﺱ٢ +٤١ﺱ +٢١ =٠‬ ‫ﻫـ‬ ‫91‬ ‫ب‬ ‫−‬ ‫ﺱ٢ + ٥٢ = ٠‬ ‫ﺱ٢ + ٧١ = ٠‬ ‫ﺱ٢ - ٩ﺱ -١ = ٠‬ ‫ﺱ٢ - ١١ﺱ + ١٢ = ٠‬ ‫¯‬ ‫2 ]-١، ١[‬ ‫6 ]-٣، ١[‬ ‫7 ‪z‬‬ ‫8 ‪z‬‬ ‫2 ﺩ‬ ‫ب ١١، -٣١‬ ‫أ ٥، -٤‬ ‫3 ﺃ‬ ‫ﺟ ٤، -٢‬ ‫4 ﺏ‬ ‫٤‬ ‫أ ﻙ= ٣‬ ‫3 ﺡ - ]٠، ٢[ 4 ‪z‬‬ ‫¯‬ ‫2‬ ‫3‬ ‫أ ﻙ=- ٣ ،ﻙ=٦‬ ‫٢‬ ‫٣‬ ‫ﺟ‬ ‫ﻙ< ٢‬ ‫٧‬ ‫ﺟ‬ ‫ﻙ= ٢‬ ‫٤‬ ‫ب‬ ‫ﻙ> ٣‬ ‫ب ﻙ=٦‬ ‫أ ﺱ٢ - ٩ﺱ + ٨١ = ٠‬ ‫ب ﺱ٢ - ٥ﺱ + ٦ = ٠‬ ‫6 ﺩ)ﺱ(= ٠ ﻋﻨﺪﻣﺎ ﺱ = -٢ ، ﺱ = ٣ ، ﺩ)ﺱ( ﻣﻮﺟﺒﺔ ﻓﻰ [-٢ ، ٣ ]‬ ‫٤‬ ‫٤‬ ‫ﺩ)ﺱ( ﺳﺎﻟﺒﺔ ﻓﻰ [-٣، -٢]∪[ ٣ ، ٢]‬ ‫٤‬ ‫ب ﺡ - ]١، ٥[‬ ‫7 أ ‪z‬‬ ‫ﻫ }٥{‬ ‫د [-٣ ، ١]‬ ‫2 ﺏ‬ ‫1 ﺏ‬ ‫٢‬ ‫6 أ ﺱ - ٣ﺱ +١ = ٠‬ ‫ب ﺩ)ﺱ( = ٠ ﻋﻨﺪﻣﺎ ﺱ = }-٤، ٢{‬ ‫3 ﺏ‬ ‫ﺟ ‪z‬‬ ‫و ﺡ - [- ٣ ، ٥]‬ ‫٢‬ ‫4 ‪C‬‬ ‫ﺩ)ﺱ( = ٠ ﻋﻨﺪﻣﺎ ﺱ = [-٤، ٢]‬ ‫د ﺣﻘﻴﻘﻴﺎﻥ ﻣﺘﺴﺎﻭﻳﺎﻥ } ٤ {‬ ‫31 ‪٤- = C‬‬ ‫61 ﻙ = ٢‬ ‫¯‬ ‫21 ﺩ)ﺱ( = ٠ ﻋﻨﺪﻣﺎ ﺱ∋ } -٢٫١، ٢٫٣{‬ ‫ﺩ)ﺱ( >٠ ﻋﻨﺪﻣﺎ ﺱ∋ ]-٣ ، -٢٫١] ∪ [٢٫٣ ، ٥[‬ ‫−‬ ‫ﺟ ﺟﺬﺭﺍﻥ ﻣﺘﺴﺎﻭﻳﺎﻥ.‬ ‫4‬ ‫11 ﻣﻦ ﺍﻟﺮﺳﻢ ﻧﺠﺪ ﺃﻥ : ﺩ)ﺱ( = ٠ ﻋﻨﺪﻣﺎ ﺱ∋ } -٣، ٣{ ، ﺩ)ﺱ( <٠ ﻓﻰ [٣، ٤[‬ ‫ﺩ)ﺱ( > ٠ ﻓﻰ [-٣، ٣]‬ ‫41 ﺇﺷﺎﺭﺓ ﺍﻟﺪﺍﻟﺔ ﻣﻮﺟﺒﺔ ﻟﺠﻤﻴﻊ ﻗﻴﻢ ﻥ ﺍﻟﺤﻘﻴﻘﻴﺔ ، ﻳﺘﻨﺎﻗﺺ ﺍﻹﻧﺘﺎﺝ ﻣﻦ ﻋﺎﻡ ٠٩٩١ ﺣﺘﻰ‬ ‫ﻋﺎﻡ ٠٠٠٢، ﺛﻢ ﻳﺒﺪﺃ ﺍﻹﻧﺘﺎﺝ ﻓﻰ ﺍﻟﺰﻳﺎﺩﺓ ﻣﻦ ﻋﺎﻡ ٠٠٠٢ ﺣﺘﻰ ﻋﺎﻡ ٠١٠٢.‬ ‫−‬ ‫1 ﺏ‬ ‫3 ﺡ - }٣{‬ ‫7 [ -٥، ١]‬ ‫2 ﻣﻮﺟﺒﺔ ، ﺡ‬ ‫6 [ -٢ ، ١ ]‬ ‫د ﻥ = ٠٣‬ ‫أ -١‬ ‫4 [٢ ،∞ ]‬ ‫ب } -٣ ، ١ {‬ ‫٢‬ ‫أ }٧٫٤، -٧٫٤{‬ ‫أ ٥ - ٣ﺕ‬ ‫٨‬ ‫52 ﻙ = ٠ ﺃﻭ ﻙ = - ٣‬ ‫ﺩ)ﺱ( = ٠ ﻋﻨﺪﻣﺎ ﺱ = ﺡ - ]-٤، ٢[‬ ‫7‬ ‫أ }٧٩٦٫٠، ٣٠٣٫٤{‬ ‫:‬ ‫ب ]-٢، ٧[‬ ‫‪ï‬‬ ‫−‬ ‫1 ب ﺍﻟﺸﻜﻞ ‪ C‬ﺏ ﺟـ ‪ + E‬ﺍﻟﺸﻜﻞ ﺹ ﺱ ﻝ ﻉ‬ ‫ﺟ 9 ‪ C‬ﺏ ﺟـ + 9ﻫـ ‪ E‬ﻭ‬ ‫، ٠١‬ ‫٧‬ ‫، ٧‬ ‫٢١‬
‫ﺇﺟﺎﺑﺎﺕ ﺑﻌﺾ ﺍﻟﺘﻤﺎﺭﻳﻦ‬ ‫د ﺍﻟﺸﻜﻞ ‪ C‬ﺏ ﺟـ ‪ + E‬ﺍﻟﺸﻜﻞ ﻉ ﻝ ﺱ ﺹ‬ ‫ﺟ ‪CE‬‬ ‫ب ﺟـ ‪E‬‬ ‫2‬ ‫أ ﺱﺹ‬ ‫4‬ ‫أ ٦٩ﺳﻢ، ٠٤٥ﺳﻢ‬ ‫5‬ ‫4 ﺟـ ﻫـ = ٥٫٤ﺳﻢ‬ ‫،٥‬ ‫٤‬ ‫أ ﻣﻌﺎﻣﻞ ﺗﺸﺎﺑﻪ ﺍﻟﻤﻀﻠﻊ ﻡ١ ﻟﻠﻤﻀﻠﻊ ﻡ٣ = ٤‬ ‫6‬ ‫ب ٨٫٢١ﺳﻢ، ٦٫٩ﺳﻢ‬ ‫٢‬ ‫٢‬ ‫7‬ ‫5 ﻉ ﻡ = ٥٫٣١‬ ‫ب ٥‬ ‫أ ٣‬ ‫ب ﻳﻮﺍﺯﻯ‬ ‫أ ﻻ ﻳﻮﺍﺯﻯ‬ ‫8 ﺱ ﻝ = ٢، ﺱ ﻡ = ٢‬ ‫ﺱﺹ ٥ ﺱﻉ ٥‬ ‫ﻣﻌﺎﻣﻞ ﺗﺸﺎﺑﻪ ﺍﻟﻤﻀﻠﻊ ﻡ٢ ﻟﻠﻤﻀﻠﻊ ﻡ٣ = ٣‬ ‫51‬ ‫أ ﻫـ ﻭ‬ ‫ﻣﻌﺎﻣﻞ ﺗﺸﺎﺑﻪ ﺍﻟﻤﻀﻠﻊ ﻡ ﻟﻠﻤﻀﻠﻊ ﻡ = ٣‬ ‫٢‬ ‫٣ ٢‬ ‫71‬ ‫أ ﻡ ﻭ = ٠١ﺳﻢ‬ ‫91‬ ‫6 ﺱ = ٠١١، ﺹ = ٠٠١، ﻉ = ٠٧‬ ‫7 ٠١ﺳﻢ ﺗﻘﺮﻳﺒﺎ.‬ ‫ً‬ ‫أ ٤٫٨ ﻣﺘﺮ، ١٫٥ ﻣﺘﺮ‬ ‫ﺟ ٤٤٫٩١ ﻣﺘﺮ ﻣﺮﺑﻊ‬ ‫−‬ ‫٢‬ ‫8 ٠٦ﺳﻢ، ٠٠٤٢ﺳﻢ‬ ‫ب ١٫٥ ﻣﺘﺮ، ٩٫٣ ﻣﺘﺮ‬ ‫1‬ ‫د ٥٢٫٠١١ ﻣﺘﺮ ﻣﺮﺑﻊ.‬ ‫د ﺍﻷﺿﻼﻉ ﺍﻟﻤﺘﻨﺎﻇﺮﺓ ﻣﺘﻨﺎﺳﺒﺔ.‬ ‫ﻫ ﺗﻄﺎﺑﻖ ﺯﺍﻭﻳﺔ ﻣﻦ ﻣﺜﻠﺚ ﻟﺰﺍﻭﻳﺔ ﻣﻦ ﻣﺜﻠﺚ ﺁﺧﺮ ﻭﺗﻨﺎﺳﺐ ﺃﻃﻮﺍﻝ ﺍﻷﺿﻼﻉ‬ ‫ﺍﻟﺘﻰ ﺗﺤﺘﻮﻳﻬﺎ.‬ ‫و ﺗﻄﺎﺑﻖ ﺯﺍﻭﻳﺔ ﻣﻦ ﻣﺜﻠﺚ ﻟﺰﺍﻭﻳﺔ ﻣﻦ ﻣﺜﻠﺚ ﺁﺧﺮ ﻭﺗﻨﺎﺳﺐ ﺃﻃﻮﺍﻝ ﺍﻷﺿﻼﻉ‬ ‫ﺍﻟﺘﻰ ﺗﺤﺘﻮﻳﻬﺎ.‬ ‫2‬ ‫أ ﺱ = ٦٣‬ ‫ﺟ ﺱ = ٣، ﺹ = ٤، ﻉ = ٤٫٨‬ ‫61‬ ‫−‬ ‫٥‬ ‫−‬ ‫1‬ ‫أ ٦٩٢١ﺳﻢ‬ ‫2‬ ‫3 ٥٧ﺳﻢ‬ ‫٢‬ ‫2‬ ‫٢‬ ‫ب -١٦١‬ ‫أ ٣٦‬ ‫ﺟ ﺻﻔﺮ‬ ‫د ١‬ ‫6 ٨١ ٢ ﺳﻢ‬ ‫7 ب ﺱ ﺟـ = ٦ ٦ ، ﺱ ﻭ = ٦ﺳﻢ.‬ ‫٢‬ ‫−‬ ‫21 ٨ ﺃﻣﺘﺎﺭ‬ ‫9‬ ‫31 ٩٤ﺳﻢ ﺗﻘﺮﻳﺒﺎ‬ ‫1 ‪ ،C‬ﺏ، ‪E‬‬ ‫5 ب ٤ ﺳﻢ‬ ‫7 ٥٫٤ ﻣﺘﺮ‬ ‫8‬ ‫2 ٢ ﺳﻢ‬ ‫4 ب ٩ : ٦١‬ ‫6 ﺱ = ١١ﺳﻢ، ﺹ = ٥٫٦١ﺳﻢ‬ ‫8 أ ٤ ، ٠٤ﺳﻢ ب - ١ ، ٤١ﺳﻢ‬ ‫أ ٦‬ ‫3 ‪C‬‬ ‫ﺟ ٩ : ٥٢‬ ‫2 ٤ﺳﻢ‬ ‫د ٤‬ ‫: ¯‬ ‫‪C‬ﺏ‬ ‫1‬ ‫3‬ ‫أ ٥٫٤‬ ‫` ﻫـ ﻭ = ٤٢ﺳﻢ‬ ‫‪ C a‬ﺏ = ٠٢١ﺳﻢ‬ ‫ﻫـ ٠٢ﺳﻢ‬ ‫4 ‪E‬‬ ‫ب ٥‬ ‫أ ﺱ=٣ ٢‬ ‫ﺟ ﺱ = ٠٦‬ ‫4‬ ‫5 ٢١ﺳﻢ‬ ‫أ ‪ C‬ﺏ = ٦ﺳﻢ، ‪ C‬ﻫـ = ٣ﺳﻢ، ﺟـ ‪٥ = E‬ﺳﻢ‬ ‫ب ‪X‬ﻡ)ﺱ( = -٣ * ٢ = -٦ ، ‪X‬ﻡ)ﺱ( = ٠‬ ‫¯‬ ‫1 ﺟـ‬ ‫5 ﺏ‬ ‫¯‬ ‫ب ٣، ٣‬ ‫٤ ٧‬ ‫ب ﺹ = ٩١، ﻉ = ٦ ٥١‬ ‫1‬ ‫3 ﺏ‬ ‫2 ﺟـ‬ ‫6 ﺱ = ٣ﺳﻢ، ﺹ = ٨١ ﺳﻢ‬ ‫01 ﺏ ﻫـ = ٨ﺳﻢ، ﺏ ﺟـ = ٢١ﺳﻢ‬ ‫د ﺱ = ١٣‬ ‫ﺟ ﺱ = ٥٫٤، ﺹ = ١١‬ ‫2 أ ﺱ=٦‬ ‫4 ﻟﺘﻜﻦ ﻡ ﻧﻘﻄﺔ ﺗﻘﺎﻃﻊ ﺍﻟﺪﻋﺎﻣﺘﻴﻦ‬ ‫` 9ﻡ ‪ C‬ﺏ + 9ﻡ ﻭ ﻫـ ، ﻫـ ﻭ = ١‬ ‫٥‬ ‫−‬ ‫أ ٨، ٣‬ ‫٣ ٥‬ ‫11 ٠٠١‪c‬‬ ‫ب ﺱ = ٤١‬ ‫¯‬ ‫5 ﺟـ‬ ‫8 ٦ ٣ ، ٩، ٣‬ ‫٣‬ ‫أ ٦٢‪c‬‬ ‫ب ٤٧‪c‬‬ ‫ﺟ ٠٢‪c‬‬ ‫٢‬ ‫3 ٩ﺳﻢ‬ ‫1 ‪E‬‬ ‫أ ﺱ = ٠١١‬ ‫ب ﺹ = ٠١‬ ‫ﺟ ﻉ = ٥٤‬ ‫01 ٣٤٫٤٢ﺳﻢ ﺗﻘﺮﻳﺒﺎ.‬ ‫ً‬ ‫1‬ ‫2 ‪ ،C‬ﺏ‬ ‫11 ٥ﺳﻢ‬ ‫ب ٣‬ ‫ﺟ ٠١‬ ‫01 ب ٥٫٤ﺳﻢ‬ ‫1‬ ‫أ ﺍﻟﻨﻘﻄﺔ ‪ C‬ﺗﻘﻊ ﺩﺍﺧﻞ ﺍﻟﺪﺍﺋﺮﺓ، ‪ C‬ﻡ = ٨ﺳﻢ.‬ ‫ب ﺍﻟﻨﻘﻄﺔ ﺏ ﺗﻘﻊ ﺧﺎﺭﺝ ﺍﻟﺪﺍﺋﺮﺓ، ﺏ ﻡ = ٤١ﺳﻢ‬ ‫ب ٢١ﺳﻢ‬ ‫ب ٠٠٥ﺳﻢ‬ ‫ﺟ ٦، ٥٢‬ ‫‪ C‬ﻭ ⊃ ‪ C‬ﺟـ ، ﺟـ ﻭ ⊃ ‪ C‬ﺟـ‬ ‫` ﻟﻠﻤﺜﻠﺜﺎﻥ ﻧﻔﺲ ﺍﻻﺭﺗﻔﺎﻉ ﻭﻳﻜﻮﻥ: ‪ C 9) W‬ﺏ ﻭ( = ‪ C‬ﻭ = ٦ = ٢‬ ‫‪) W‬ﺟـ ﺏ ﻭ( ﺟـ ﻭ ٣ ١‬ ‫1‬ ‫أ ١‬ ‫٩‬ ‫ب ٤‬ ‫ﺟ ٢‬ ‫د ٢‬ ‫ﻛﺎﻥ ﻓﻰ ﺍﻟﺮﺃﺱ ﺏ،‬ ‫‪ 99 a‬ﺏ ‪ C‬ﻭ ، ﺏ ﺟـ ﻭ ﻣﺸﺘﺮ‬ ‫3‬ ‫4 ﺟـ ﻫـ = ٥ﺳﻢ.‬ ‫١‬ ‫21 9‪ C‬ﺏ ﺟـ + 9‪ E C‬ﻫـ ، ﻣﻌﺎﻣﻞ ﺍﻟﺘﺸﺎﺑﻪ =‬ ‫أ ٤ﻛﻢ‬ ‫ﺟ ‪ E‬ﺟـ‬ ‫‪ C‬ﺟـ‬ ‫د ﺏ ‪ C * E‬ﺟـ‬ ‫` 9‪ C‬ﺏ ﻭ ﻣﺘﺴﺎﻭﻯ ﺍﻟﺴﺎﻗﻴﻦ، ‪ C‬ﻭ = ٦ﺳﻢ‬ ‫د ﻥ، ﻡ‬ ‫ب ﺹ‬ ‫7 ﺏ ‪٦ = E‬ﺳﻢ، ‪ C‬ﺏ = ٦ ٣ ﺳﻢ، ‪ C‬ﺟـ = ٦ ٦ ﺳﻢ‬ ‫ب ٤ ٥ ﻛﻢ‬ ‫ب ‪ C‬ﻡ = ٨٫٠١ﺳﻢ‬ ‫ب ﺱ = ٤، ﺹ = ٣‬ ‫5 ‪ E‬ﻫـ = ٨ﺳﻢ ، ‪ ١٥ ٢ = E C‬ﺳﻢ ، ‪ C‬ﻫـ = ٢ ٠١‬ ‫01 ﻓﻰ 9‪ C‬ﺏ ﺟـ : ﺏ ﺟـ = ٠١ - ٤ = ٦ﺳﻢ‬ ‫‪ a‬ﺏ ‪ ،٢ = C‬ﺏ ‪٢ = E‬‬ ‫` ‪ E C‬ﻳﻨﺼﻒ ‪Cc‬‬ ‫‪ C‬ﺟـ ٣ ‪ E‬ﺟـ ٣‬ ‫ﻓﻰ 9‪ C‬ﺏ ﻭ: ‪ C a‬ﻫـ ﻳﻨﺼﻒ ‪ C ،Cc‬ﻫـ = ﺏ ﻭ‬ ‫ﺟ ﺹ‬ ‫أ ﺱ‬ ‫أ‬ ‫د ‪E‬ﻭ‬ ‫2‬ ‫3 ‪ C‬ﺏ = ٨ﺳﻢ، ﺏ ﺟـ = ٠١ﺳﻢ‬ ‫ب ٥٢، ٥١ )٥ + ٥ (‬ ‫4 أ ٤ ، ٢٢‬ ‫أ ﻗﻴﺎﺳﺎﺕ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﻤﺘﻨﺎﻇﺮﺓ ﻣﺘﺴﺎﻭﻳﺔ.‬ ‫ب ﺱ = ٠٢، ﺹ = ٥١‬ ‫ﺏ‪C‬‬ ‫‪ C‬ﺟـ‬ ‫ﺟ ‪ E‬ﻫـ‬ ‫ب ﺟـ ‪E‬‬ ‫‪E‬ﺏ‬ ‫أ ١١‬ ‫−‬ ‫1‬ ‫` ﻝ ﻡ // ﺹ ﻉ‬ ‫أ ﺱ = ٨، ﺹ = ٣‬ ‫ﻝ = ١٢، ﻡ = ٨٢، ﻥ = ٠٣‬ ‫ﺟ ﻳﻮﺍﺯﻯ‬ ‫ب ‪E‬ﻭ‬ ‫ب ﻣﻌﺎﻣﻞ ﺗﺸﺎﺑﻪ ﺍﻟﻤﻀﻠﻊ ﻡ١ ﻟﻠﻤﻀﻠﻊ ﻡ٣ = ١‬ ‫9‬ ‫ﺟ ٩‬ ‫د ٣‬ ‫:‬ ‫2 ﺏ‬ ‫6‬ ‫¯‬ ‫‪E‬‬ ‫3 ﺟـ‬ ‫7 ﺟـ‬ ‫4 ‪C‬‬ ‫8 ٦ﺳﻢ‬ ‫:‬ ‫−‬ ‫ﺟ ٥‬ ‫٢‬ ‫1‬ ‫‪ïM‬‬ ‫−‬ ‫و ٠٧١‪c‬‬ ‫ز ﺍﻟﺜﺎﻟﺚ‬ ‫ح ٠٣‪c‬‬ ‫2 ﺟ‬
‫ﺇﺟﺎﺑﺎﺕ ﺑﻌﺾ ﺍﻟﺘﻤﺎﺭﻳﻦ‬ ‫3‬ ‫أ -٦٠٣‪c‬‬ ‫ب ٠٧٢‪c‬‬ ‫ﺟ ٥٣٢‪c‬‬ ‫د -١٠٣‪c‬‬ ‫4‬ ‫أ ﺍﻷﻭﻝ‬ ‫ب ﺍﻟﺜﺎﻟﺚ‬ ‫د ﺍﻟﺜﺎﻧﻰ‬ ‫3‬ ‫7‬ ‫أ ٧٧١‪c‬‬ ‫ب ٣٤١‪c‬‬ ‫ﺟ ﺍﻟﺮﺍﺑﻊ‬ ‫ﺟ ٥٤‪c‬‬ ‫د ٠٥١‪c‬‬ ‫4‬ ‫−‬ ‫2 ﺟ‬ ‫6 ب‬ ‫3‬ ‫7‬ ‫أ‬ ‫٤‬ ‫ب ٣‪r‬‬ ‫٣‬ ‫ﺟ -٤‪r‬‬ ‫1 د‬ ‫5 ﺟ‬ ‫01‬ ‫2‬ ‫أ ٥‪r‬‬ ‫٤‬ ‫‪E‬‬ ‫أ ٨٨٩٫٠‬ ‫ب ٢٤٤٫٠‬ ‫11‬ ‫41 ٢٫٤ﺳﻢ‬ ‫61 ٥٧‪١٫٣٠٩ ، c‬‬ ‫‪E‬‬ ‫ﺟ ٧٠٨٫٢‬ ‫51 ٥٧١٫٢ ، ٦ ً٧٣ َ٤٢١‪c‬‬ ‫81 ٧٥٫٨٢ﺳﻢ‬ ‫71 ٥ ‪r‬‬ ‫٣‬ ‫12 ٢١٧٤ ﻛﻢ/ﺱ‬ ‫ب ١‬ ‫أ - ٣‬ ‫٢‬ ‫٢‬ ‫أ ﺟﺘﺎ‪ ،  ٤ = i‬ﺟﺎ‪ ،  ٣ = i‬ﻇﺎ‪٣ = i‬‬ ‫٤‬ ‫٥‬ ‫٥‬ ‫ب ﺟﺘﺎ‪ ،  ٥ = i‬ﺟﺎ‪ ،  ١٢- =i‬ﻇﺎ‪١٢- = i‬‬ ‫٥‬ ‫٣١‬ ‫٣١‬ ‫02 ٢‪r‬ﺳﻢ‬ ‫32 أ ‪r‬‬ ‫٣‬ ‫6‬ ‫¯‬ ‫:‬ ‫ﺟ‬ ‫1‬ ‫22 ٩٢ﺳﻢ‬ ‫ب ٨ ﺳﺎﻋﺎﺕ‬ ‫ﺟ ٠٢‪r‬‬ ‫42 ﺹ = ٣ ﺱ‬ ‫2 أ‬ ‫6 ﺟ‬ ‫3 ﺟ‬ ‫7 أ‬ ‫د‬ ‫١‬ ‫٣‬ ‫ب ٠١٢‪ c٣٣٠ ،c‬ﺟ ٠٣‪ c٣٣٠ ،c‬د ٠١‪c٣٠٠ ،c‬‬ ‫أ ٥٤‪c٢٢٥ ،c‬‬ ‫‪E‬‬ ‫91 ٦٧٫٦١ﺳﻢ‬ ‫٣‬ ‫ﺟ‬ ‫4 ﺟ‬ ‫8 ب‬ ‫٥‬ ‫د ٣‪r‬‬ ‫ب ٨٣ ً٥٤ َ٢٨‪c‬‬ ‫أ ٦٫٩ﺳﻢ‬ ‫5‬ ‫د‬ ‫ب ٠٩‪c‬‬ ‫أ ٠٠٣‪c‬‬ ‫ﺟ ٧١ ً٠١ َ٤٦‪c‬‬ ‫أ‬ ‫2‬ ‫6‬ ‫أ ٣‬ ‫ب ٤‬ ‫٥‬ ‫7‬ ‫أ ٥٢‬ ‫ب‬ ‫4 ب‬ ‫8 أ‬ ‫3‬ ‫ﺟ‬ ‫ﺟ ٣‬ ‫٥‬ ‫د ٢١ ً٢٥ َ٦٣‬ ‫ﺟ ٤١٫٠١ﻣﺘﺮ‬ ‫ﺟﺎ٥٢ = ‪C‬‬ ‫٥٢‬ ‫−‬ ‫1 ﺟ‬ ‫5 ﺟ‬ ‫9‬ ‫‪C‬‬ ‫٢‬ ‫٣‬ ‫ﺟﺘﺎ‪i‬‬ ‫ﺏ‬ ‫٢‬ ‫٥‬ ‫ﺟﺎ‪i‬‬ ‫٣‬ ‫٢‬ ‫-‬ ‫١‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫:‬ ‫11 أ )-(‬ ‫21 أ -١‬ ‫31 ٠٣‪c‬‬ ‫أ ٣٫٤، ٧٫٠‬ ‫4‬ ‫٣‬ ‫٤‬ ‫ب )+(‬ ‫ب ٤‬ ‫أ ٧١ + ٦١ﺕ ب‬ ‫:‬ ‫41 ﺇﺟﺎﺑﺔ ﺃﺣﻤﺪ‬ ‫ﺟ )+(‬ ‫51 ﺻﺤﻴﺤﺔ‬ ‫1 -ﺟﺘﺎ‪i‬‬ ‫2 - ﻇﺎ‪i‬‬ ‫3 - ﻗﺘﺎ‪i‬‬ ‫4 ﺟﺎ‪i‬‬ ‫5 ﺟﺘﺎ‪i‬‬ ‫6 - ﻇﺎ‪i‬‬ ‫7 ﻗﺘﺎ‪i‬‬ ‫91 ب‬ ‫ب ٥٢‪c‬‬ ‫02 د‬ ‫ﺟ ٠١‪c‬‬ ‫12 ﺟ‬ ‫د ٠٦‪c‬‬ ‫−‬ ‫2 ]-٢، ٢[‬ ‫1 ]-١، ١[‬ ‫ﺷﻜﻞ )١( ١ﺟﺎ ‪i‬‬ ‫4 -٣‬ ‫3 ٤‬ ‫  ﺷﻜﻞ )٢( ﺟﺘﺎ ‪i‬‬ ‫1‬ ‫أ ٤، ٠١‬ ‫أ ﻣﻮﺟﺒﺔ ﻟﻜﻞ ﺱ = ﺡ‬ ‫أ ١، -١، ]-١، ١[‬ ‫ب ]-٣، ٣[‬ ‫6‬ ‫1‬ ‫−‬ ‫2 ﺟ‬ ‫أ‬ ‫أ ١، ٣‬ ‫ب ١ ، - ١ ﺟ -٣، ٤‬ ‫1‬ ‫3‬ ‫أ -٣، - ١‬ ‫٣‬ ‫٢‬ ‫ب -٥، -٣‬ ‫٣ ٥‬ ‫ﺟ ٣، ٤‬ ‫٤ ٣‬ ‫4‬ ‫أ ٠٣‪c‬‬ ‫ب ٥٣١‪c‬‬ ‫5‬ ‫أ ٢١ ً٢٥ َ٦٣‪c‬‬ ‫ب ٤ ً٩ َ٤٦‪c‬‬ ‫ﺟ ٣١ ً٠٣ ٥٥‪c‬‬ ‫6‬ ‫ب ٨٢ ً٦٥ َ٩٢١‪c٢٣٠َ ٣ً ٣٢ ، c‬‬ ‫أ ٧٣ ً٧٣ َ٣١‪c١٦٦َ ٢٢ً ٢٣ ، c‬‬ ‫أ ٤٤ ً١٣ َ٠٦١ ب -٨٢٤٩٫٠ ، -٦٣٥٣٫٠ ، -٧٠٦٠٫١‬ ‫1‬ ‫أ ٩٠٫٢‬ ‫‪E‬‬ ‫‪E‬‬ ‫ﺟ ٥٨٫٣‬ ‫‪E‬‬ ‫¯‬ ‫−‬ ‫أ ٢ﺱ٢ + ﺱ > ٠، [-١، ٠]‬ ‫4‬ ‫٢‬ ‫:‬ ‫4 ‪E‬‬ ‫3 ﺏ‬ ‫¯ :‬ ‫5 ﺏ‬ ‫أ ٧‬ ‫ب ٧ﺳﻢ‬ ‫ب‬ ‫٢‬ ‫أ‬ ‫3‬ ‫٣:٥ ب‬ ‫ً‬ ‫ب ٠١ﺳﻢ 2 ﺃﻭﻻ: ٦ﺳﻢ‬ ‫:‬ ‫أ ‪ C‬ﺏ * ‪ C‬ﺟـ‬ ‫ً‬ ‫1 ب ﺃﻭﻻ: )‪(E C‬‬ ‫٢‬ ‫٣:٨‬ ‫ﺛﺎﻧﻴﺎ : ١٢ﺳﻢ‬ ‫ً‬ ‫:‬ ‫ب ﺏ ﻥ * ﻥ ﺟـ‬ ‫3 ﺱ ﻥ = ٦٫٣ﺳﻢ‬ ‫:‬ ‫2‬ ‫١‬‫٥‬ ‫:‬ ‫1 ب ﺛﺎﻧﻴﺎ: ٨٫٢ﺳﻢ‬ ‫ً‬ ‫3‬ ‫¯‬ ‫ب ٧‬ ‫٢١‬ ‫ب ٣٤ ، ٠٤٢‪c‬‬ ‫٦٣‬ ‫1 أ ٢ﺱ٢ - ﺱ - ٨ = ٠‬ ‫ب ٨٤٨٫٠‬ ‫2 أ -ﺕ‬ ‫4 أ ]-٤، ١[‬ ‫3 ب ٦ + ٣ﺕ‬ ‫1‬ ‫ب ٣١٫١‬ ‫5 ﺏ‬ ‫:‬ ‫3‬ ‫ﺟ ٢١ ً ٢٥ َ ٦٠٣‪c‬‬ ‫7‬ ‫1 ‪E‬‬ ‫1‬ ‫٥ ٥‬ ‫٢‬ ‫:‬ ‫2 ﺟـ‬ ‫2 ٠٠١ﺳﻢ‬ ‫أ ]-٤، ٤[‬ ‫٢ ٢‬ ‫3‬ ‫ب ٣، -٣ ]-٣، ٣[‬ ‫ﺟ ٣ ، -٣، ]-٣، ٣ [‬ ‫٢ ٢ ٢ ٢‬ ‫4 ﺏ‬ ‫3 ﺟـ‬ ‫أ ﻣﺘﺴﺎﻭﻳﺎﻥ، }٣{‬ ‫8 - ﺟﺎ‪i‬‬ ‫5‬ ‫٢٫٤ﻣﺘﺮ‬ ‫2‬ ‫81 أ‬ ‫2‬ ‫:‬ ‫٣‬ ‫أ ٦١‪c٨٠ ،c‬‬ ‫أ ٦، ﺻﻔﺮ‬ ‫:‬ ‫2‪C‬‬ ‫1 ‪E‬‬ ‫−‬ ‫32‬ ‫ب ﺻﻔﺮ‬ ‫1‬ ‫٣‬‫٥‬ ‫١‬‫٣‬ ‫٢‬ ‫-١‬ ‫٥‬ ‫ﻇﺎ‪i‬‬ ‫٢‬ ‫3 ﺱ٢ + ١ = ٠‬ ‫2 ﻣﻮﺟﺒﺔ ﻟﻜﻞ ﺱ ∋ ﺡ‬ ‫1 ١‬ ‫‪E‬‬ ‫ﺟـ‬ ‫-‬ ‫:‬ ‫٤‬‫٥‬ ‫٣‬ ‫:‬ ‫ﺛﺎﻧﻴﺎ: ٤ﺳﻢ‬ ‫ً‬ ‫ﺛﺎﻟﺜﺎ: ٦‬ ‫ً‬ ‫أ ٤ : ٧، ٤ : ٧‬ ‫أ ‪ C‬ﺏ = ٥٫٤ﺳﻢ، ﻉ ﺹ = ٥ﺳﻢ‬ ‫ب ٤ ٣ ﺳﻢ‬ ‫أ ﺻﻔﺮ‬ ‫ﺟ 9‪ C‬ﺏ ‪E‬‬
كتاب الانشطه - مصر- ترم اول - 2014

More Related Content

PDF
كتاب الانشطة مصر - ترم ثانى -2013-2014
byمنتدى الرياضيات المتقدمة
 
PDF
دليل المعلم لمادة التّوحيد للصّف الرّابع الإبتدائي في المملكة العربية السعو...
byد.فداء الشنيقات
 
PDF
كتاب اللغة العربية الثامن - الجزء الثاني
bymustafa002
 
PDF
دليل المعلم لمادة الفقة للصف الخامس الفصل الاول والثاني
byد.فداء الشنيقات
 
PDF
دليل المعلم للصف الخامس
bymustafa002
 
PDF
ج1 دليل الصف 12
bymustafa002
 
PDF
Collection taybet zaman
byMay Haddad MD.MPH
 
PDF
دليل المعلم الثامن - الجزء الثاني
bymustafa002
 
كتاب الانشطة مصر - ترم ثانى -2013-2014
byمنتدى الرياضيات المتقدمة
 
دليل المعلم لمادة التّوحيد للصّف الرّابع الإبتدائي في المملكة العربية السعو...
byد.فداء الشنيقات
 
كتاب اللغة العربية الثامن - الجزء الثاني
bymustafa002
 
دليل المعلم لمادة الفقة للصف الخامس الفصل الاول والثاني
byد.فداء الشنيقات
 
دليل المعلم للصف الخامس
bymustafa002
 
ج1 دليل الصف 12
bymustafa002
 
Collection taybet zaman
byMay Haddad MD.MPH
 
دليل المعلم الثامن - الجزء الثاني
bymustafa002
 

Viewers also liked

PPTX
Discipline
byJoan Lopes
 
PDF
طرق مجربة لجمع المال والثروة pdf
byosmanabdelrhman
 
PPTX
Materi Aqidak Akhlak kelas 2
byrifqi_sahabat
 
DOC
Md. Hafizur Rahman
byMd Hafizur Rahman
 
DOCX
Cloths
byNirmala Samaj Kalyan Sangstha
 
PDF
Executive Pulse 14 FINAL
byTom Jones
 
PPTX
Infrastruktura i środowisko
bymateuszprzybysz
 
PDF
U.S. vs. Terry L. Loewen Criminal Complaint Report
byDylan Hock
 
PPTX
Nuestras islas powerpoint
byCeipbuenavista1
 
PDF
If pure oxygen gets you high how is air legal?
by5oxygen
 
PPTX
Blog
byKarlitaAvila
 
PDF
ES169 02 - Especial Ropa del Hogar - Ropa Dormitorio
byBint
 
Discipline
byJoan Lopes
 
طرق مجربة لجمع المال والثروة pdf
byosmanabdelrhman
 
Materi Aqidak Akhlak kelas 2
byrifqi_sahabat
 
Md. Hafizur Rahman
byMd Hafizur Rahman
 
Cloths
byNirmala Samaj Kalyan Sangstha
 
Executive Pulse 14 FINAL
byTom Jones
 
Infrastruktura i środowisko
bymateuszprzybysz
 
U.S. vs. Terry L. Loewen Criminal Complaint Report
byDylan Hock
 
Nuestras islas powerpoint
byCeipbuenavista1
 
If pure oxygen gets you high how is air legal?
by5oxygen
 
Blog
byKarlitaAvila
 
ES169 02 - Especial Ropa del Hogar - Ropa Dormitorio
byBint
 

كتاب الانشطه - مصر- ترم اول - 2014

  • 2.
    äÉ«°VÉjôdG äÉÑjQóàdG h ᣰûfC’GÜÉàc ∫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 OGDƒaôªY /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
  • 4.
  • 5.
    ‫ﺍﻟﻤﻘﺪﻣﺔ‬ ‫بسم الل ّٰهالرحمن 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 ﻓﻰﺍﻟﻤﺜﻠﺚ‬ ‫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 اﻟﺪرﺟﺔاﻟﺜﺎﻧﻴﺔ 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 ﻛﻞﺷﻜﻞ 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 ﻛﻞﺷﻜﻞ 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 اﻋﺪاد 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 ﺍﻟﺒﺮﺍﻣﺞﺍﻟﺮﺳﻮﻣﻴﺔ 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 ﺟﺬرىاﻟﻤﻌﺎدﻟﺔ 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 ﻛﺎﻥﺟﺬﺭﺍ 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 ﺟﺬرىﻣﻌﺎدﻟﺔ 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 ﻧﻮﻉﺍﻟﺠﺬﺭﻳﻦ 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: ﻗﻄﻌﺔﺃﺭﺽ 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 ofa 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 øYÖLCG :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 ‫‪QuadraticInequalities‬‬ ‫¯‬ ‫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’Gø«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 ﺍﻛﺘﺐﻗﺎﻋﺪﺓ 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©àeøe 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 ﺃﻭﺟﺪﻗﻴﻤﺔ 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 ofPolygons‬‬ ‫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 ﻛﻞﻣﻦ 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 OfTriangles‬‬ ‫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
  • 30.
    ‫‪ï‬‬ ‫4 ‪ C‬ﺏ، ‪ E‬ﺟـ ﻭﺗﺮﺍﻥ ﻓﻰ ﺩﺍﺋﺮﺓ، ‪ C‬ﺏ ∩ ‪ E‬ﺟـ = }ﻫـ{ ﺣﻴﺚ ﻫـ ﺧﺎﺭﺝ ﺍﻟﺪﺍﺋﺮﺓ، ‪ C‬ﺏ = ٤ﺳﻢ، ‪ E‬ﺟـ = ٧ﺳﻢ،‬ ‫ﺏ ﻫـ = ٦ﺳﻢ. ﺃﺛﺒﺖ ﺃﻥ 9‪ E C‬ﻫـ + 9ﺟـ ﺏ ﻫـ، ﺛﻢ ﺃﻭﺟﺪ ﻃﻮﻝ ﺟـ ﻫـ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫5 ‪ C‬ﺏ ﺟـ، ‪ E‬ﻫـ ﻭ ﻣﺜﻠﺜﺎﻥ ﻣﺘﺸﺎﺑﻬﺎﻥ. ﺭﺳﻢ ‪ C‬ﺱ = ﺏ ﺟـ ﻟﻴﻘﻄﻌﻪ ﻓﻰ ﺱ، ﻭﺭﺳﻢ ‪ E‬ﺹ = ﻫـ ﻭ ﻟﻴﻘﻄﻌﻪ ﻓﻰ ﺹ.‬ ‫ﺃﺛﺒﺖ ﺃﻥ ﺏ ﺱ * ﺹ ﻭ = ﺟـ ﺱ * ﺹ ﻫـ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫6 ﻓﻰ ﺍﻟﻤﺜﻠﺚ ‪ C‬ﺏ ﺟـ، ‪ C‬ﺟـ < ‪ C‬ﺏ، ﻡ ∋ ‪ C‬ﺟـ ﺣﻴﺚ ‪ Cc)X‬ﺏ ﻡ( = ‪c) X‬ﺟـ( ﺃﺛﺒﺖ ﺃﻥ )‪ C‬ﺏ(٢ = ‪ C‬ﻡ * ‪ C‬ﺟـ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫7 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ‪ ،C‬ﺭﺳﻢ‬ ‫‪EC‬‬ ‫ﺏ‪E‬‬ ‫= ﺏ ﺟـ ﻟﻴﻘﻄﻌﻪ ﻓﻰ ‪ .E‬ﺇﺫﺍ ﻛﺎﻥ ‪ E‬ﺟـ = ١ ، ‪ ٢ ٦ = E C‬ﺳﻢ‬ ‫٢‬ ‫ﺃﻭﺟﺪ ﻃﻮﻝ ﻛﻞ ﻣﻦ ﺏ ‪ C ، E‬ﺏ ، ‪ C‬ﺟـ .‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫8‬ ‫:‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ‪،C‬‬ ‫‪C‬‬ ‫‪ = E C‬ﺏ ﺟـ ، ‪ E‬ﻫـ = ‪ C‬ﺏ ، ‪ E‬ﻭ = ‪ C‬ﺟـ‬ ‫ﺃﺛﺒﺖ ﺃﻥ:‬ ‫أ 9‪ E C‬ﻫـ + 9ﺟـ ‪ E‬ﻭ‬ ‫ب ﻣﺴﺎﺣﺔ ﺍﻟﻤﺴﺘﻄﻴﻞ ‪ C‬ﻫـ ‪ E‬ﻭ = ‪C‬ﻫـ * ﻫـ ﺏ * ‪ C‬ﻭ * ﻭ ﺟـ‬ ‫‪E‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
  • 31.
    ‫: ‪ C‬ﺏﺟـ ﻣﺜﻠﺚ ﻣﻨﻔﺮﺝ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ‪،C‬‬ ‫9‬ ‫‪ C‬ﺏ = ‪ C‬ﺟـ. ﺭﺳﻢ‬ ‫‪EC‬‬ ‫‪C‬‬ ‫= ‪ C‬ﺏ ﻭﻳﻘﻄﻊ ﺏ ﺟـ ﻓﻰ ‪.E‬‬ ‫ﺃﺛﺒﺖ ﺃﻥ: ٢)‪ C‬ﺏ(٢ = ﺏ ‪ * E‬ﺏ ﺟـ‬ ‫‪E‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫01 ﺗﻌﺒﺮ ﺍﻟﻤﺠﻤﻮﻋﺘﺎﻥ ‪ ،C‬ﺏ ﻋﻦ ﺃﻃﻮﺍﻝ ﺃﺿﻼﻉ ﻣﺜﻠﺜﺎﺕ ﻣﺨﺘﻠﻔﺔ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ.‬ ‫ﺍﻛﺘﺐ ﺃﻣﺎﻡ ﻛﻞ ﻣﺜﻠﺚ ﻣﻦ ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ C‬ﺭﻣﺰ ﺍﻟﻤﺜﻠﺚ ﺍﻟﺬﻯ ﻳﺸﺎﺑﻬﻪ ﻣﻦ ﺍﻟﻤﺠﻤﻮﻋﺔ ﺏ‬ ‫ﻣﺠﻤﻮﻋﺔ )ﺏ(‬ ‫ﻣﺠﻤﻮﻋﺔ )‪(C‬‬ ‫‪٥ ، ٤ ، ٢٫٥ C‬‬ ‫١ ٦ ، ٦ ، ٦‬ ‫ﺏ ٨ ، ٥٫٣١ ، ٤١‬ ‫٢ ٥ ، ٧ ، ١١‬ ‫ﺟـ ٥٢ ، ٥٣ ، ٥٥‬ ‫٣ ٥ ، ٨ ، ٠١‬ ‫‪١١ ، ١١ ، ١١ E‬‬ ‫٤ ٧ ، ٨ ، ٢١‬ ‫ﻫـ ٥٫٣ ، ٤ ، ٦‬ ‫٥ ٦١ ، ٧٢ ، ٨٢‬ ‫ﻭ ٨ ، ٦ ، ٠١‬ ‫ﺯ ٢٣ ، ٤٥ ، ٢٤‬ ‫11‬ ‫: ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻓﻴﻪ ‪ C‬ﺏ = ٦ﺳﻢ ، ﺏ ﺟـ = ٩ﺳﻢ ،‬ ‫‪E‬‬ ‫‪ C‬ﺟـ = ٥٫٧ﺳﻢ ، ‪ E‬ﻧﻘﻄﺔ ﺧﺎﺭﺟﺔ ﻋﻦ ﺍﻟﻤﺜﻠﺚ ‪ C‬ﺏ ﺟـ‬ ‫ﺣﻴﺚ ‪ E‬ﺏ = ٤ﺳﻢ ، ‪٥ = C E‬ﺳﻢ. ﺃﺛﺒﺖ ﺃﻥ:‬ ‫أ 9‪ C‬ﺏ ﺟـ + 9‪ E‬ﺏ ‪C‬‬ ‫ب ﺏ ‪ C‬ﻳﻨﺼﻒ ‪ E c‬ﺏ ﺟـ‬ ‫‪C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫ﺃﻛﻤﻞ:‬ ‫9 ‪ C‬ﺏ ﺟـ + 9‬ ‫ﻭﻣﻌﺎﻣﻞ ﺍﻟﺘﺸﺎﺑﻪ = .............................‬ ‫ﺳﻢ‬ ‫21‬ ‫‪C‬‬ ‫...............................‬ ‫ﺳﻢ ‪E‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
  • 32.
    ‫‪ï‬‬ ‫: ‪ C‬ﺏﺟـ + ﺱ ﺹ ﻉ، ﻫـ ﻣﻨﺘﺼﻒ ﺏ ﺟـ ،‬ ‫31‬ ‫‪C‬‬ ‫ﻡ ﻣﻨﺘﺼﻒ ﺹ ﻉ ، ﺟـ ‪ C = E‬ﺏ ، ﻉ ﻝ = ﺱ ﺹ ﺃﺛﺒﺖ ﺃﻥ:‬ ‫‪E‬‬ ‫أ 9‪ C‬ﻫـ ﺟـ + 9ﺱ ﻡ ﻉ‬ ‫ب ﺟـ ‪E‬‬ ‫‪ C‬ﻫـ‬ ‫ﻉﻝ = ﺱﻡ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫41 ‪ C‬ﺏ ﺟـ، ﺱ ﺹ ﻉ ﻣﺜﻠﺜﺎﻥ ﻣﺘﺸﺎﺑﻬﺎﻥ، ﺣﻴﺚ ‪ C‬ﺏ < ‪ C‬ﺟـ، ﺱ ﺹ < ﺱ ﻉ.‬ ‫ﻫـ، ﻝ ﻣﻨﺘﺼﻔﻰ ﺏ ﺟـ ، ﺹ ﻉ ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ، ﺭﺳﻢ ‪ C‬ﻭ = ﺏ ﺟـ ، ﺱ ﻡ = ﺹ ﻉ‬ ‫ﺃﺛﺒﺖ ﺃﻥ 9 ‪ C‬ﻫـ ﻭ + 9 ﺱ ﻝ ﻡ‬ ‫..................................................................................................................................................................................................................................‬ ‫51 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ، ‪ ∋ E‬ﺏ ﺟـ ﺣﻴﺚ )‪ = ٢(E C‬ﺏ ‪ E * E‬ﺟـ ، ﺏ ‪ = E C * C‬ﺏ ‪ C * E‬ﺟـ ﺃﺛﺒﺖ ﺃﻥ:‬ ‫ﺟ ‪ c) X‬ﺏ ‪ C‬ﺟـ( = ٠٩‪c‬‬ ‫أ 9 ‪ C‬ﺏ ‪9 + E‬ﺟـ ‪E C‬‬ ‫ب ‪ = E C‬ﺏ ﺟـ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫61 ﻳﺒﻴﻦ ﺍﻟﻤﺨﻄﻂ ﺍﻟﻤﻘﺎﺑﻞ ﻣﻮﻗﻊ ﻣﺤﻄﺔ ﺧﺪﻣﺔ ﻭﺗﻤﻮﻳﻦ ﺳﻴﺎﺭﺍﺕ ﻳﺮﺍﺩ‬ ‫ﺇﻗﺎﻣﺘﻬﺎ ﻋﻠﻰ ﺍﻟﻄﺮﻳﻖ ﺍﻟﺴﺮﻳﻊ ﻋﻨﺪ ﺗﻘﺎﻃﻊ ﻃﺮﻳﻖ ﺟﺎﻧﺒﻰ ﻳﺆﺩﻯ ﺇﻟﻰ‬ ‫ﺍﻟﻤﺪﻳﻨﺔ ﺟـ ﻭﻋﻤﻮﺩﻳﺎ ﻋﻠﻰ ﺍﻟﻄﺮﻳﻖ ﺍﻟﺴﺮﻳﻊ ﺑﻴﻦ ﺍﻟﻤﺪﻳﻨﺘﻴﻦ ‪ ،C‬ﺏ.‬ ‫ًّ‬ ‫أ ﻛﻢ ﻳﻨﺒﻐﻰ ﺃﻥ ﺗﺒﻌﺪ ﺍﻟﻤﺤﻄﺔ ﻋﻦ ﺍﻟﻤﺪﻳﻨﺔ ﺟـ?‬ ‫ب ﻣﺎ ﺍﻟﺒﻌﺪ ﺑﻴﻦ ﺍﻟﻤﺪﻳﻨﺘﻴﻦ ﺏ، ﺟـ?‬ ‫‪C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫ﻧﺸﺎط‬ ‫ﺍﺳﺘﺨﺪﺍﻡ ﺑﺮﻧﺎﻣﺞ ﺧﺮﺍﺋﻂ)‪ (Google Earth‬ﻟﺤﺴﺎﺏ ﺃﻗﺼﺮ ﺑﻌﺪ ﺑﻴﻦ ﻋﻮﺍﺻﻢ ﻣﺤﺎﻓﻈﺎﺕ ﺟﻤﻬﻮﺭﻳﺔ ﻣﺼﺮ ﺍﻟﻌﺮﺑﻴﺔ‬ ‫‪ïM‬‬ ‫−‬
  • 33.
    ‫اﻟﻌﻼﻗﺔ ﺑﻴﻦ ﻣﺴﺎﺣﺘﻰﺳﻄﺤﻰ ﻣﻀﻠﻌﻴﻦ ﻣﺘﺸﺎﺑﻬﻴﻦ‬ ‫2-3‬ ‫‪The Relation between the Area of two Similar Polygons‬‬ ‫1 ﺃﻛﻤﻞ:‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ 9‪ C‬ﺏ ﺟـ + 9ﺱ ﺹ ﻉ، ﻛﺎﻥ ‪ C‬ﺏ = ٣ ﺱ ﺹ ﻓﺈﻥ ‪9) W‬ﺱ ﺹ ﻉ( = ...............................‬ ‫ﻭ‬ ‫‪ C9) W‬ﺏ ﺟـ(‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ 9 ‪ C‬ﺏ ﺟـ + 9 ‪ E‬ﻫـ ﻭ، ‪ C 9) W‬ﺏ ﺟـ( = ٩ ‪ E 9)W‬ﻫـ ﻭ( ﻛﺎﻥ ‪ E‬ﻫـ = ٤ﺳﻢ ﻓﺈﻥ:‬ ‫ﻭ‬ ‫‪ C‬ﺏ = .............................. ﺳﻢ‬ ‫2 ﺍﺩﺭﺱ ﻛﻼ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ، ﺣﻴﺚ ﻙ ﺛﺎﺑﺖ ﺗﻨﺎﺳﺐ، ﺛﻢ ﺃﻛﻤﻞ:‬ ‫ًّ‬ ‫ب‬ ‫‪C‬‬ ‫أ‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫‪ C‬ﺏ ∩ ﺟـ ‪} = E‬ﻫـ{‬ ‫٢‬ ‫‪ C 9)W‬ﺟـ ﻫـ( = ٠٠٩ ﺳﻢ‬ ‫٢‬ ‫ﻓﺈﻥ: ‪ E 9)W‬ﻫـ ﺏ( = ............................... ﺳﻢ‬ ‫‪c)X‬ﺏ ‪ C‬ﺟـ( = ٠٩‪ = E C ،c‬ﺏ ﺟـ‬ ‫‪ E C 9)W‬ﺟـ( = ٠٨١ ﺳﻢ٢ ﻓﺈﻥ:‬ ‫٢‬ ‫‪ C 9)W‬ﺏ ﺟـ( = ............................... ﺳﻢ‬ ‫3 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ، ‪ C ∋E‬ﺏ ﺣﻴﺚ ‪ ٢ = E C‬ﺏ ‪ ،E‬ﻫـ ∋ ‪ C‬ﺟـ ﺣﻴﺚ ‪ E‬ﻫـ // ﺏ ﺟـ‬ ‫ﺇﺫﺍ ﻛﺎﻧﺖ ﻣﺴﺎﺣﺔ 9 ‪ E C‬ﻫـ = ٠٦ﺳﻢ٢. ﺃﻭﺟﺪ ﻣﺴﺎﺣﺔ ﺷﺒﻪ ﺍﻟﻤﻨﺤﺮﻑ ‪ E‬ﺏ ﺟـ ﻫـ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫4 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ﺏ، ﺭﺳﻤﺖ ﺍﻟﻤﺜﻠﺜﺎﺙ ﺍﻟﻤﺘﺴﺎﻭﻳﺔ ﺍﻷﺿﻼﻉ ‪ C‬ﺏ ﺱ، ﺏ ﺟـ ﺹ، ‪ C‬ﺟـ ﻉ‬ ‫ﺃﺛﺒﺖ ﺃﻥ: ‪ C9) W‬ﺏ ﺱ( + ‪9) W‬ﺏ ﺟـ ﺹ( = ‪ C9) W‬ﺟـ ﻉ(.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪C‬ﺏ‬ ‫5 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻓﻴﻪ ﺏ ﺟـ = ٤ ، ﺭﺳﻤﺖ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﻤﺎﺭﺓ ﺑﺮﺅﻭﺳﻪ. ﻣﻦ ﻧﻘﻄﺔ ﺏ ﺭﺳﻢ ﺍﻟﻤﻤﺎﺱ ﻟﻬﺬه ﺍﻟﺪﺋﺮﺓ ﻓﻘﻄﻊ‬ ‫٣‬ ‫‪ C 9) W‬ﺏ ﺟـ( ٧‬ ‫‪ C‬ﺟـ ﻓﻰ ﻫـ. ﺃﺛﺒﺖ ﺃﻥ: ‪ C 9) W‬ﺏ ﻫـ( = ٦١‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
  • 34.
    ‫‪M‬‬ ‫6 ‪ C‬ﺏﺟـ ‪ E‬ﻣﺘﻮﺍﺯﻯ ﺃﺿﻼﻉ ﺱ ∋ ‪ C‬ﺏ ، ﺱ ∌ ‪ C‬ﺏ ﺣﻴﺚ ﺏ ﺱ = ٢ ‪ C‬ﺏ، ﺹ ∋ ﺟـ ﺏ ، ﺹ ∌ ﺟـ ﺏ‬ ‫ﺣﻴﺚ ﺏ ﺹ = ٢ ﺏ ﺟـ ، ﺭﺳﻢ ﻣﺘﻮﺍﺯﻯ ﺍﻷﺿﻼﻉ ﺏ ﺱ ﻉ ﺹ ﺃﺛﺒﺖ ﺃﻥ: ‪ C) W‬ﺏ ﺟـ ‪١ = (E‬‬ ‫‪) W‬ﺱ ﺏ ﺹ ﻉ( ٤‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫7 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ﺏ، ﺏ ‪ C = E‬ﺟـ ﻳﻘﻄﻌﺔ ﻓﻰ ‪ ،E‬ﺭﺳﻢ ﻋﻠﻰ ‪ C‬ﺏ ، ﺏ ﺟـ ﺍﻟﻤﺮﺑﻌﺎﻥ‬ ‫ُ‬ ‫‪ C‬ﺱ ﺹ ﺏ، ﺏ ﻡ ﻥ ﺟـ ﺧﺎﺭﺝ ﺍﻟﻤﺜﻠﺚ ‪ C‬ﺏ ﺟـ.‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﻤﻀﻠﻊ ‪ C E‬ﺱ ﺹ ﺏ = ﺍﻟﻤﻀﻠﻊ ‪ E‬ﺏ ﻡ ﻥ ﺟـ‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ‪ C‬ﺏ = ٦ﺳﻢ، ‪ C‬ﺟـ = ٠١ﺳﻢ. ﺃﻭﺟﺪ ﺍﻟﻨﺴﺒﺔ ﺑﻴﻦ ﻣﺴﺎﺣﺘﻰ ﺳﻄﺤﻰ ﺍﻟﻤﻀﻠﻌﻴﻦ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫8 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ، ‪ C‬ﺏ ، ﺏ ﺟـ ، ‪ C‬ﺟـ ﺃﺿﻼﻉ ﻣﺘﻨﺎﻇﺮﺓ ﻟﺜﻼﺛﺔ ﻣﻀﻠﻌﺎﺕ ﻣﺘﺸﺎﺑﻬﺔ ﻣﺮﺳﻮﻣﺔ ﺧﺎﺭﺝ ﺍﻟﻤﺜﻠﺚ، ﻭﻫﻰ‬ ‫ﺍﻟﻤﻀﻠﻌﺎﺕ ﺑﻴﻦ ‪ ،N ،M‬ﻉ ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ.‬ ‫ﻓﺈﺫﺍ ﻛﺎﻧﺖ ﻣﺴﺎﺣﺔ ﺍﻟﻤﻀﻠﻊ ‪٤٠ = M‬ﺳﻢ٢، ﻭﻣﺴﺎﺣﺔ ﺍﻟﻤﻀﻠﻊ ‪ ٨٥= N‬ﺳﻢ٢، ﻭﻣﺴﺎﺣﺔ ﺍﻟﻤﻀﻠﻊ ﻉ = ٥٢١ﺳﻢ٢.‬ ‫ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﻤﺜﻠﺚ ‪ C‬ﺏ ﺟـ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫9 ‪ C‬ﺏ ﺟـ ‪ E‬ﻣﺮﺑﻊ ﻗﺴﻤﺖ ‪ C‬ﺏ ، ﺏ ﺟـ ، ﺟـ ‪ C E ، E‬ﺑﺎﻟﻨﻘﺎﻁ ﺱ، ﺹ، ﻉ، ﻝ ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ ﺑﻨﺴﺒﺔ ١ : ٣‬ ‫ﺃﺛﺒﺖ ﺃﻥ:‬ ‫‪ W‬ﺍﻟﻤﺮﺑﻊ ﺱ ﺹ ﻉ ﻝ‬ ‫=٥‬ ‫ب‬ ‫أ ﺍﻟﺸﻜﻞ ﺱ ﺹ ﻉ ﻝ ﻣﺮﺑﻊ‬ ‫٨‬ ‫‪ W‬ﺍﻟﻤﺮﺑﻊ ‪ C‬ﺏ ﺟـ ‪E‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫01 ﺻﺎﻟﺔ ﺃﻟﻌﺎﺏ ﻣﺴﺘﻄﻴﻠﺔ ﺍﻟﺸﻜﻞ ﺃﺑﻌﺎﺩﻫﺎ ٨ ﻣﺘﺮ، ٢١ ﻣﺘﺮ، ﺗﻢ ﺗﻐﻄﻴﺔ ﺃﺭﺿﻴﺘﻬﺎ ﺑﺎﻟﺨﺸﺐ، ﻓﻜﻠﻔﺖ ٠٠٢٣ ﺟﻨﻴﻪ.‬ ‫ﺍﺣﺴﺐ )ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﺘﺸﺎﺑﻪ( ﺗﻜﺎﻟﻴﻒ ﺗﻐﻄﻴﺔ ﺃﺭﺿﻴﺔ ﺻﺎﻟﺔ ﻣﺴﺘﻄﻴﻠﺔ ﺃﻛﺒﺮ ﺑﻨﻔﺲ ﻧﻮﻉ ﺍﻟﺨﺸﺐ ﻭﺑﻨﻔﺲ‬ ‫ﺍﻷﺳﻌﺎﺭ، ﺇﺫﺍ ﻛﺎﻥ ﺃﺑﻌﺎﺩﻫﺎ ٤١، ١٢ ﻣﻦ ﺍﻷﻣﺘﺎﺭ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
  • 35.
    ‫ﺗﻄﺒﻴﻘﺎت اﻟﺘﺸﺎﺑﻪ ﻓﻰاﻟﺪاﺋﺮة‬ ‫2-4‬ ‫‪Applications of similarity in the circle‬‬ ‫1 ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻵﻟﺔ ﺍﻟﺤﺎﺳﺒﺔ ﺃﻭ ﺍﻟﺤﺴﺎﺏ ﺍﻟﻌﻘﻠﻰ، ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺱ ﺍﻟﻌﺪﺩﻳﺔ ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ.‬ ‫    )ﺍﻷﻃﻮﺍﻝ ﻣﻘﺪﺭﺓ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ(‬ ‫ﺟ‬ ‫ب‬ ‫أ‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫‪E‬‬ ‫................................................................................‬ ‫د‬ ‫‪E‬‬ ‫................................................................................‬ ‫................................................................................‬ ‫و‬ ‫ﻫ‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫‪C‬‬ ‫+‬ ‫‪E‬‬ ‫‪C‬‬ ‫................................................................................‬ ‫ز‬ ‫................................................................................‬ ‫ح‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫................................................................................‬ ‫ى‬ ‫ط‬ ‫‪E‬‬ ‫‪E‬‬ ‫................................................................................‬ ‫‪C‬‬ ‫................................................................................‬ ‫ك‬ ‫................................................................................‬ ‫ل‬ ‫‪E‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬ ‫................................................................................‬ ‫................................................................................‬
  • 36.
    ‫‪M‬‬ ‫¯‬ ‫‪ï‬‬ ‫2 ﻓﻰ ﺃﻯﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ ﺗﻘﻊ ﺍﻟﻨﻘﻂ ‪ ،C‬ﺏ، ﺟـ، ‪ E‬ﻋﻠﻰ ﺩﺍﺋﺮﺓ ﻭﺍﺣﺪﺓ? ﻓﺴﺮ ﺇﺟﺎﺑﺘﻚ.‬ ‫ِّ‬ ‫ٍّ‬ ‫)ﺍﻷﻃﻮﺍﻝ ﻣﻘﺪﺭﺓ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ(‬ ‫ﺟ‬ ‫ب‬ ‫‪C‬‬ ‫‪C‬‬ ‫أ‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫....................................................................................................................................................................................................................................................................................‬ ‫3 ﻓﻰ ﺃﻯ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ ‪ C‬ﺏ ﻣﻤﺎﺱ ﻟﻠﺪﺍﺋﺮﺓ ﺍﻟﻤﺎﺭﺓ ﺑﺎﻟﻨﻘﻂ ﺏ، ﺟـ، ‪.E‬‬ ‫ٍّ‬ ‫ﺟ‬ ‫‪ C‬ب‬ ‫‪C‬‬ ‫أ‬ ‫‪E‬‬ ‫٩٢‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫..............................................................................................................................................................................................................................................................................................‬ ‫ِ‬ ‫4 ﺩﺍﺋﺮﺗﺎﻥ ﻣﺘﻘﺎﻃﻌﺘﺎﻥ ﻓﻰ ‪ ،C‬ﺏ . ﺟـ ∋ ‪ C‬ﺏ ، ﺟـ ∌ ‪ C‬ﺏ ﺭﺳﻢ ﻣﻦ ﺟـ ﺍﻟﻘﻄﻌﺘﺎﻥ ﺟـ ﺱ ، ﺟـ ﺹ ﻣﻤﺎﺳﺘﺎﻥ‬ ‫ُ َ‬ ‫ﻟﻠﺪﺍﺋﺮﺗﻴﻦ ﻋﻨﺪ ﺱ، ﺹ. ﺃﺛﺒﺖ ﺃﻥ ﺟـ ﺱ = ﺟـ ﺹ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫.‬ ‫5‬ ‫: ﺍﻟﺪﺍﺋﺮﺗﺎﻥ ﻡ، ﻥ ﻣﺘﻤﺎﺳﺘﺎﻥ ﻋﻨﺪ ﻫـ‬ ‫‪ C‬ﺟـ ﻳﻤﺲ ﺍﻟﺪﺍﺋﺮﺓ ﻡ ﻋﻨﺪ ﺏ، ﻭﻳﻤﺲ ﺍﻟﺪﺍﺋﺮﺓ ﻥ ﻋﻨﺪ ﺟـ،‬ ‫‪ C‬ﻫـ ﻳﻘﻄﻊ ﺍﻟﺪﺍﺋﺮﺗﻴﻦ ﻋﻨﺪ ﻭ، ‪ E‬ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ‬ ‫ﺣﻴﺚ ‪ C‬ﻭ = ٤ﺳﻢ، ﻭ ﻫـ = ٥ﺳﻢ، ﻫـ ‪٧ = E‬ﺳﻢ.‬ ‫ﺃﺛﺒﺖ ﺃﻥ ﺏ ﻣﻨﺘﺼﻒ ‪ C‬ﺟـ‬ ‫‪E‬‬ ‫‪C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
  • 37.
    ‫6‬ ‫: ﻝ ∋ﺱ ﺹ ﺣﻴﺚ ﺱ ﻝ = ٤ﺳﻢ،‬ ‫ﺹ ﻝ = ٨ﺳﻢ ، ﻡ ∋ ﺱ ﻉ ﺣﻴﺚ ﺱ ﻡ = ٦ﺳﻢ ، ﻉ ﻡ = ٢ﺳﻢ‬ ‫ﺃﺛﺒﺖ ﺃﻥ:‬ ‫أ 9ﺱ ﻝ ﻡ + 9ﺱ ﻉ ﺹ‬ ‫ب ﺍﻟﺸﻜﻞ ﻝ ﺹ ﻉ ﻡ ﺭﺑﺎﻋﻰ ﺩﺍﺋﺮﻯ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫٥‬ ‫7 ‪ C‬ﺏ ∩ ﺟـ ‪} = E‬ﻫـ{، ‪ C‬ﻫـ = ٢١ ﺏ ﻫـ، ‪ E‬ﻫـ = ٣ ﻫـ ﺟـ، ﺇﺫﺍ ﻛﺎﻥ ﺏ ﻫـ = ٦ﺳﻢ، ﺟـ ﻫـ = ٥ﺳﻢ.‬ ‫٥‬ ‫ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﻨﻘﻂ ‪ ،C‬ﺏ، ﺟـ، ‪ E‬ﺗﻘﻊ ﻋﻠﻰ ﺩﺍﺋﺮﺓ ﻭﺍﺣﺪﺓ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫8 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ، ‪ ∋ E‬ﺏ ﺟـ ﺣﻴﺚ ‪ E‬ﺏ = ٥ﺳﻢ، ‪ E‬ﺟـ = ٤ﺳﻢ. ﺇﺫﺍ ﻛﺎﻥ ‪ C‬ﺟـ = ٦ﺳﻢ. ﺃﺛﺒﺖ ﺃﻥ:‬ ‫أ ‪ C‬ﺟـ ﻣﻤﺎﺳﺔ ﻟﻠﺪﺍﺋﺮﺓ ﺍﻟﺘﻰ ﺗﻤﺮ ﺑﺎﻟﻨﻘﻂ ‪ ،C‬ﺏ، ‪.E‬‬ ‫ب 9‪ C‬ﺟـ ‪9 + E‬ﺏ ﺟـ ‪C‬‬ ‫ﺟ ‪ C9) W‬ﺏ ‪ C9) W : (E‬ﺏ ﺟـ( = ٥ : ٩‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫9 ﺩﺍﺋﺮﺗﺎﻥ ﻣﺘﺤﺪﺗﺎ ﻛﺰ ﻡ، ﻃﻮﻻ ﻧﺼﻔﻰ ﻗﻄﺮﻳﻬﻤﺎ ٢١ﺳﻢ، ٧ﺳﻢ، ﺭﺳﻢ ﺍﻟﻮﺗﺮ ‪ E C‬ﻓﻰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﻜﺒﺮﻯ ﻟﻴﻘﻄﻊ‬ ‫ﺍﻟﻤﺮ‬ ‫ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺼﻐﺮﻯ ﻓﻰ ﺏ ، ﺟـ ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ. ﺃﺛﺒﺖ ﺃﻥ: ‪ C‬ﺏ * ﺏ ‪٩٥ = E‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫01 ‪ C‬ﺏ ﺟـ ‪ E‬ﻣﺴﺘﻄﻴﻞ ﻓﻴﻪ ‪ C‬ﺏ = ٦ﺳﻢ، ﺏ ﺟـ = ٨ﺳﻢ. ﺭﺳﻢ ﺏ ﻫـ = ‪ C‬ﺟـ ﻓﻘﻄﻊ ‪ C‬ﺟـ ﻓﻰ ﻫـ، ‪ E C‬ﻓﻰ ﻭ.‬ ‫ب ﺃﻭﺟﺪ ﻃﻮﻝ ‪ C‬ﻭ .‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ )‪ C‬ﺏ(٢ = ‪ C‬ﻭ * ‪.E C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
  • 38.
    ‫‪M‬‬ ‫¯‬ ‫‪ï‬‬ ‫11 ﺍﻟﺮﺑﻂ ﻣﻊﺍﻟﺼﻨﺎﻋﺔ: ﻛُﺴﺮ ﺃﺣﺪ ﺗﺮﻭﺱ ﺁﻟﺔ ﻭﻻﺳﺘﺒﺪﺍﻟﻪ ﻣﻄﻠﻮﺏ‬ ‫ﻣﻌﺮﻓﺔ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮ ﺩﺍﺋﺮﺗﻪ. ﻳﺒﻴﻦ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ ﺟﺰﺀﺍ ﻣﻦ‬ ‫ً‬ ‫ﻫﺬﺍ ﺍﻟﺘﺮﺱ، ﻭﺍﻟﻤﻄﻠﻮﺏ ﺗﻌﻴﻴﻦ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮ ﺩﺍﺋ ﺗﻪ ................... .‬ ‫ﺮ‬ ‫‪C‬‬ ‫¯‬ ‫21 ﺍﻟﺮﺑﻂ ﻣﻊ ﺍﻟ ﻴﺌﺔ: ﻳﺒﻴﻦ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ ﻣﺨﻄﻄًّﺎ ﻟﺤﺪﻳﻘﺔ ﻋﻠﻰ‬ ‫ﺷﻜﻞ ﺩﺍﺋﺮﺓ ﺑﻬﺎ ﻃﺮﻳﻘﺎﻥ ﻳﺘﻘﺎﻃﻌﺎﻥ ﻋﻨﺪ ﻧﺎﻓﻮﺭﺓ ﺍﻟﻤﻴﺎه. ﺃﻭﺟﺪ ﺑﻌﺪ‬ ‫ُْ‬ ‫ﻧﺎﻓﻮﺭﺓ ﺍﻟﻤﻴﺎه ﻋﻨﺪ ﺍﻟﻤﺪﺧﻞ ﺟـ.‬ ‫‪E‬‬ ‫¯‬ ‫¯‬ ‫31 ﺍﻟﺮﺑﻂ ﻣﻊ ﺍﻟﻤﻨ ﻝ: ﺗﺴﺘﺨﺪﻡ ﻫﺪﻯ ﺷﺒﻜﺔ ﻟﺸﻰ ﺍﻟﻠﺤﻮﻡ ﻋﻠﻰ ﺷﻜﻞ‬ ‫ﺩﺍﺋﺮﺓ ﻣﻦ ﺍﻟﺴﻠﻚ، ﻃﻮﻝ ﻗﻄﺮﻫﺎ ٠٥ﺳﻢ، ﻳﺪﻋﻤﻬﺎ ﻣﻦ ﺍﻟﻮﺳﻂ ﺳﻠﻜﺎﻥ‬ ‫ﻣﺘﻮﺍﺯﻳﺎﻥ ﻭﻣﺘﺴﺎﻭﻳﺎﻥ ﻓﻰ ﺍﻟﻄﻮﻝ ﻛﻤﺎ ﻓﻰ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ، ﻭﺍﻟﺒﻌﺪ‬ ‫ﺑﻴﻨﻬﻤﺎ ٠١ﺳﻢ.‬ ‫ﺍﺣﺴﺐ ﻃﻮﻝ ﻛﻞ ﻣﻦ ﺳﻠﻜﻰ ﺍﻟﺪﻋﺎﻣﺔ. .........................................................‬ ‫‪C‬‬ ‫41 ﺍﻟﺮﺑﻂ ﻣﻊ ﺍﻻ ﺼﺎﻝ: ﺗﻨﻘﻞ ﺍﻷﻗﻤﺎﺭ ﺍﻟﺼﻨﺎﻋﻴﺔ ﺍﻟﺒﺮﺍﻣﺞ ﺍﻟﺘﻠﻴﻔﺰﻳﻮﻧﻴﺔ‬ ‫ﺇﻟﻰ ﻛﺎﻓﺔ ﻣﻨﺎﻃﻖ ﺍﻷﺭﺽ، ﻭﺗﺴﺘﺨﺪﻡ ﺃﻃﺒﺎﻕ ﺧﺎﺻﺔ ﻻﺳﺘﻘﺒﺎﻝ‬ ‫ﺇﺷﺎﺭﺍﺕ ﺍﻟﺒﺚ ﺍﻟﺘﻠﻴﻔﺰﻳﻮﻧﻰ، ﻭﻫﻰ ﺃﻃﺒﺎﻕ ﻣﻘﻌﺮﺓ ﻋﻠﻰ ﺷﻜﻞ ﺟﺰﺀ‬ ‫ﻣﻦ ﺳﻄﺢ ﻛﺮﺓ.‬ ‫ﻳﺒﻴﻦ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ ﻣﻘﻄﻌﺎ ﻓﻰ ﺃﺣﺪ ﻫﺬه ﺍﻷﻃﺒﺎﻕ، ﻃﻮﻝ ﻗﻄﺮه‬ ‫ً‬ ‫٠٨١ﺳﻢ، ﻭﺍﻟﻤﻄﻠﻮﺏ ﺣﺴﺎﺏ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮ ﻛﺮﺓ ﺗﻘﻌﺮه ﻡ ‪. C‬‬ ‫.......................................................................................................................................‬ ‫.......................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
  • 39.
    ‫ﺗﻤﺎﺭﻳﻦ ﻋﺎﻣﺔ‬ ‫1‬ ‫أ‬ ‫ب‬ ‫ﺟ‬ ‫د‬ ‫: ﺃﻯﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺘﺎﻟﻴﺔ ﻏﻴﺮ ﺻﺤﻴﺤﺔ:‬ ‫)‪ E = ٢(E C‬ﺏ * ‪ E‬ﺟـ‬ ‫)‪ C‬ﺏ(٢ = ﺏ ‪ * E‬ﺏ ﺟـ‬ ‫‪ C‬ﺟـ * ﺏ ﺟـ = ‪ C‬ﺏ * ‪E C‬‬ ‫‪ C‬ﺏ * ‪ C‬ﺟـ = ‪ * E C‬ﺏ ﺟـ‬ ‫‪C‬‬ ‫‪E‬‬ ‫..................................................................................................................................................................................................................................‬ ‫2‬ ‫: ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ‪ C ∋ E‬ﺏ ، ﻫـ ∋ ‪ C‬ﺟـ .‬ ‫ﺃﺛﺒﺖ ﺃﻥ 9‪ E C‬ﻫـ + 9‪ C‬ﺟـ ﺏ‬ ‫ﺛﻢ ﺃﻭﺟﺪ ﻃﻮﻝ ﻫـ ‪E‬‬ ‫‪C‬‬ ‫‪E‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫: ‪ C‬ﺏ ﻗﻄﺮ ﻓﻰ ﺍﻟﺪﺍﺋﺮﺓ ﻡ، ﻃﻮﻟﻪ ٢١ﺳﻢ‬ ‫3‬ ‫‪ C ∋ E‬ﺏ ﺣﻴﺚ ‪١٦ = E C‬ﺳﻢ، ﺟـ ﺗﻘﻊ ﻋﻠﻰ ﺍﻟﺪﺍﺋﺮﺓ‬ ‫ﺣﻴﺚ ﺟـ ‪٨ = E‬ﺳﻢ. ﺟـ ﻫـ = ‪ C‬ﺏ . ﺃﺛﺒﺖ ﺃﻥ:‬ ‫‪C‬‬ ‫‪E‬‬ ‫أ ﺟـ ‪ E‬ﻣﻤﺎﺳﺔ ﻟﻠﺪﺍﺋﺮﺓ ﻡ.‬ ‫ب 9‪ E‬ﺟـ ﺏ + 9‪ C E‬ﺟـ‬ ‫ﺟ ﺟـ ﻫـ = ٨٫٤ﺳﻢ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫4 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ﺏ. ﺏ ‪ C = E‬ﺟـ ، ‪ C‬ﺏ = ٥١ﺳﻢ، ‪٩ = E C‬ﺳﻢ. ﺭﺳﻢ ﻋﻠﻰ ‪ C‬ﺏ ، ﺏ ﺟـ ﻣﻦ‬ ‫ﺍﻟﺨﺎﺭﺝ ﺍﻟﻤﺮﺑﻌﺎﻥ ‪ C‬ﺏ ﺹ ﺱ، ﺏ ﺟـ ﻫـ ﻭ.‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﻤﻀﻠﻊ ‪ C E‬ﺱ ﺹ ﺏ + ﺍﻟﻤﻀﻠﻊ ‪ E‬ﺏ ﻭ ﻫـ ﺟـ.‬ ‫ب ﺃﻭﺟﺪ ‪) W‬ﺍﻟﻤﻀﻠﻊ ‪ C E‬ﺱ ﺹ ﺏ( : ‪) W‬ﺍﻟﻤﻀﻠﻊ ‪ E‬ﺏ ﻭ ﻫـ ﺟـ(‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
  • 40.
    ‫ﺗﻤﺎﺭﻳﻦ ﻋﺎﻣﺔ‬ ‫5‬ ‫: ﺍﻟﺪﺍﺋﺮﺗﺎﻥﻡ، ﻥ ﻣﺘﻘﺎﻃﻌﺘﺎﻥ ﻓﻰ ‪ ،C‬ﺏ‬ ‫‪ C‬ﺏ ∩ ﺟـ ‪ ∩ E‬ﻫـ ﻭ = }ﺱ{ ﺣﻴﺚ‬ ‫ﺱ ‪ E ٢ = E‬ﺟـ، ﻫـ ﻭ = ٠١ﺳﻢ، ﻭ ﺱ = ٦ ﺳﻢ‬ ‫‪E‬‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﺸﻜﻞ ﺟـ ‪ E‬ﻭ ﻫـ ﺭﺑﺎﻋﻰ ﺩﺍﺋﺮﻯ.‬ ‫ب ﺃﻭﺟﺪ ﻃﻮﻝ ﺟـ ‪E‬‬ ‫.................................................................................................................................‬ ‫‪C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫6‬ ‫: ﺩﺍﺋﺮﺗﺎﻥ ﻣﺘﺤﺪﺗﺎ ﻛﺰ،‬ ‫ﺍﻟﻤﺮ‬ ‫ﻭﺍﻷﻃﻮﺍﻝ ﺍﻟﻤﺒﻴﻨﺔ ﻟﻠﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ.‬ ‫ﺃﻭﺟﺪ ﻗﻴﻢ ﺱ، ﺹ ﺍﻟﻌﺪﺩﻳﺔ.‬ ‫............................................................................................................................................‬ ‫............................................................................................................................................‬ ‫¯‬ ‫7 ﺣﺪﻳﻘﺔ ﺣﻴﻮﺍﻥ: ﻓﻰ ﺭﺣﻠﺔ ﻣﺪﺭﺳﻴﺔ ﺇﻟﻰ ﺣﺪﻳﻘﺔ ﺍﻟﺤﻴﻮﺍﻥ ﺃﺭﺍﺩ‬ ‫ﺣﺴﺎﻡ ﺃﻥ ﻳﻌﺮﻑ ﺍﺭﺗﻔﺎﻉ ﺣﻴﻮﺍﻥ ﺍﻟﺰﺭﺍﻓﺔ. ﻭﺿﻊ ﺣﺴﺎﻡ ﻣﺮﺁﺓ‬ ‫ﻣﺴﺘﻮﻳﺔ ﻋﻠﻰ ﺍﻷﺭﺽ ﺗﺒﻌﺪ ﻋﻨﻪ ﻣﺘﺮﺍﻥ ﻭﻋﻦ ﺍﻟﺰﺭﺍﻓﺔ ٦ ﺃﻣﺘﺎﺭ،‬ ‫ﻓﺈﺫﺍ ﻛﺎﻥ ﺣﺴﺎﻡ ﻭﺍﻟﻤﺮﺁﺓ ﻭﺍﻟﺰﺭﺍﻓﺔ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ‬ ‫ﻭﺍﺭﺗﻔﺎﻉ ﺣﺴﺎﻡ ٥٫١ ﻣﺘﺮﺍ . ﻛﻢ ﻳﺒﻠﻎ ﺍﺭﺗﻔﺎﻉ ﺍﻟﺰﺭﺍﻓﺔ.‬ ‫ً‬ ‫¯‬ ‫¯‬ ‫.................................................................................................................................‬ ‫.................................................................................................................................‬ ‫8 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻔﻴ ﺎﺀ: ﺍﺣﺴﺐ ﻣﻌﺎﻣﻞ ﻣﻐﻴﺮ ﺍﻟﺒﻌﺪ، ﻭﺍﺣﺴﺐ ﻗﻴﻤﺔ ﺱ ﺍﻟﻌﺪﺩﻳﺔ ﻓﻰ ﻛﻞ ﺷﻜﻞ ﻣﻤﺎ ﻳﻠﻰ.‬ ‫ب‬ ‫أ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
  • 41.
    ‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ‬ ‫1 ﺃﻛﻤﻞﻣﺎ ﻳﺄﺗﻲ:‬ ‫أ ﺍﻟﻤﻀﻠﻌﺎﻥ ﺍﻟﻤﺸﺎﺑﻬﺎﻥ ﻟﺜﺎﻟﺚ .............................................................................................................................................................‬ ‫ب‬ ‫ﺇﺫﺍ ﺗﻨﺎﺳﺒﺖ ﺃﻃﻮﺍﻝ ﺍﻷﺿﻼﻉ ﺍﻟﻤﺘﻨﺎﻇﺮﺓ ﻓﻰ ﻣﺜﻠﺜﻴﻦ ﻓﺈﻧﻬﻤﺎ ..................................................................................................‬ ‫ﺟ‬ ‫ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻨﺴﺒﺔ ﺑﻴﻦ ﻣﺤﻴﻄﻰ ﻣﻀﻠﻌﻴﻦ ﻣﺘﺸﺎﺑﻬﻴﻦ ٣ : ٥ ﻓﺈﻥ ﺍﻟﻨﺴﺒﺔ ﺑﻴﻦ ﻣﺴﺎﺣﺘﻴﻬﻤﺎ ...........................................‬ ‫د ﺇﺫﺍ ﺗﻘﺎﻃﻊ ﻭﺗﺮﺍﻥ ‪ C‬ﺏ ، ﺟـ ‪ E‬ﻟﺪﺍﺋﺮﺓ ﻓﻰ ﻧﻘﻄﺔ ﺱ ﻓﺈﻥ:‬ ‫................... * ................... = ................... *‬ ‫‪E‬‬ ‫...................‬ ‫ﻫ ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻤﺴﺘﻄﻴﻞ ‪ C‬ﺏ ﺟـ ‪ + E‬ﺍﻟﻤﺴﺘﻄﻴﻞ ﺱ ﺏ ﻉ ﺹ،‬ ‫‪١٥ = E C‬ﺳﻢ ، ﺟـ ‪٢٠ = E‬ﺳﻢ، ﺹ ﻉ = ٦١ﺳﻢ‬ ‫ﻓﺈﻥ: ﺱ ﻉ = ...........................................................‬ ‫‪C‬‬ ‫‪C‬‬ ‫2‬ ‫: ‪ C‬ﻫـ // ‪ E‬ﺟـ ، ‪ C‬ﺟـ = ﻫـ ‪} ∩ E‬ﺏ{،‬ ‫‪ C‬ﺏ = ٣ﺳﻢ، ﺏ ﺟـ = ٦ﺳﻢ، ﻫـ ‪١٢ = E‬ﺳﻢ‬ ‫ﻓﺄﻭﺟﺪ ﻃﻮﻝ ﻫـ ﺏ‬ ‫.......................................................................................................................................‬ ‫‪E‬‬ ‫.......................................................................................................................................‬ ‫‪E‬‬ ‫: ﺍﻟﻤﻀﻠﻊ ‪ C‬ﺏ ﺟـ ‪ + E‬ﺍﻟﻤﻀﻠﻊ ﺱ ﻫـ ﺟـ ﻭ‬ ‫3‬ ‫‪C‬‬ ‫ﺃﺛﺒﺖ ﺃﻥ ‪ C‬ﺏ // ﺱ ﻫـ‬ ‫١‬ ‫ﻭﺇﺫﺍ ﻛﺎﻧﺖ ﺱ ﻫـ = ٢ ‪C‬ﺏ، ﺟـ ﻭ = ٩ﺳﻢ ﻓﺄﻭﺟﺪ ﻃﻮﻝ ﻭ ‪E‬‬ ‫.......................................................................................................................................‬ ‫.......................................................................................................................................‬ ‫4 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻓﻴﻪ ﺱ ∋ ‪ C‬ﺏ ﺑﺤﻴﺚ ﻛﺎﻥ ‪ C‬ﺱ = ٨ﺳﻢ، ﺱ ﺏ = ٢١ﺳﻢ‬ ‫ﺹ ∋ ‪ C‬ﺟـ ، ﺑﺤﻴﺚ ﻛﺎﻥ ‪ C‬ﺹ = ٠١ﺳﻢ، ﺹ ﺟـ = ٦ﺳﻢ.‬ ‫ﺃﺛﺒﺖ ﺃﻥ:‬ ‫أ 9 ‪ C‬ﺏ ﺟـ + 9 ‪ C‬ﺹ ﺱ‬ ‫ب ﺍﻟﺸﻜﻞ ﺱ ﺏ ﺟـ ﺹ ﺭﺑﺎﻋﻰ ﺩﺍﺋﺮﻯ.‬ ‫‪C‬‬ ‫.......................................................................................................................................‬ ‫.......................................................................................................................................‬ ‫5 ‪ C‬ﺏ ، ﺟـ ‪ E‬ﻭﺗﺮﺍﻥ ﻓﻰ ﺩﺍﺋﺮﺓ ﻣﺘﻘﺎﻃﻌﺎﻥ، ﻓﻰ ﻫـ ﻓﺈﺫﺍ ﻛﺎﻥ ﻫـ ﻣﻨﺘﺼﻒ ‪ C‬ﺏ ، ﺟـ ﻫـ = ٤ﺳﻢ، ﻫـ ‪٩ = E‬ﺳﻢ‬ ‫..........................................................................................................................................................................‬ ‫ﻓﺄﻭﺟﺪ ﻃﻮﻝ ‪ C‬ﺏ .‬ ‫¯‬ ‫−‬ ‫¯‬
  • 42.
    ‫ﺍﺧﺘﺒﺎﺭ ﺗﺮﺍﻛﻤﻰ‬ ‫‪Oó©àe øeQÉ«àN’G á∏Ä°SCG‬‬ ‫1 ﺇﺫﺍ ﻛﺎﻥ ٢ﺱ +١١ = ٣ ﻓﺈﻥ ١١ - ﺱ ﺗﺴﺎﻭﻯ:‬ ‫٢‬ ‫ﺱ+‬ ‫ب‬ ‫ﺻﻔﺮﺍ‬ ‫أ -٠١‬ ‫ً‬ ‫ﺟ ٥‬ ‫2 ﻣﺴﺘﻌﻴﻨﺎ ﺑﻤﻌﻄﻴﺎﺕ ﺍﻟﺸﻜﻞ، ﻓﺈﻥ ﺱ ﺗﺴﺎﻭﻯ:‬ ‫ً‬ ‫ب‬ ‫٨١‬ ‫أ ٢٣‬ ‫د ١٥‬ ‫ﺟ ٧٢‬ ‫د ٠١‬ ‫+‬ ‫‪c‬‬ ‫− ‪c‬‬ ‫‪c‬‬ ‫−‬ ‫3 ﻣﺴﺘﻌﻴﻨﺎ ﺑﻤﻌﻄﻴﺎﺕ ﺍﻟﺸﻜﻞ، ﻓﺈﻥ ﺱ ﺗﺴﺎﻭﻯ:‬ ‫ً‬ ‫ب ١١‬ ‫أ ٥‬ ‫د ٤١‬ ‫ﺟ ٢١‬ ‫‪E‬‬ ‫4‬ ‫أ ٥ ﺳﻢ‬ ‫ﺟ ٨ ﺳﻢ‬ ‫: ‪ C‬ﺏ = ٢١ﺳﻢ، ﺟـ ﻫـ = ٤ ﺳﻢ، ﻓﺈﻥ ﻫـ ‪ E‬ﺗﺴﺎﻭﻯ:‬ ‫ب ٦ ﺳﻢ‬ ‫د ٩ ﺳﻢ‬ ‫‪C‬‬ ‫5 ﻣﺴﺘﻄﻴﻼﻥ ﻣﺘﺸﺎﺑﻬﺎﻥ ﺑﻌﺪﺍ ﺍﻷﻭﻝ ٠١ ﺳﻢ، ٨ ﺳﻢ، ﻭﻣﺤﻴﻂ ﺍﻟﺜﺎﻧﻰ ٨٠١ ﺳﻢ ﻓﺈﻥ ﻃﻮﻝ ﺍﻟﻤﺴﺘﻄﻴﻞ ﺍﻟﺜﺎﻧﻰ ﻳﺴﺎﻭﻯ:‬ ‫د ٦٣ ﺳﻢ‬ ‫ﺟ ٠٣ ﺳﻢ‬ ‫ب ٤٢ ﺳﻢ‬ ‫أ ٨١ ﺳﻢ‬ ‫‪C‬‬ ‫‪:Iô«°ü≤dG äÉHÉLE’G äGP á∏Ä°SC’G‬‬ ‫6‬ ‫: ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻣﻦ ﺱ، ﺹ‬ ‫ﺍﻷﻃﻮﺍﻝ ﻣﻘﺪﺭﺓ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ.‬ ‫+‬ ‫‪E‬‬ ‫−‬ ‫7 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻓﻴﻪ ‪ C‬ﺏ = ‪ C‬ﺟـ، ‪ ∋ E‬ﺏ ﺟـ . ﺭﺳﻢ ‪ E‬ﻫـ = ‪ C‬ﺏ ، ‪ E‬ﻭ = ‪ C‬ﺟـ .‬ ‫ﺏ ﻫـ ‪ E‬ﻫـ‬ ‫ﺃﺛﺒﺖ ﺃﻥ: ﺟـ ﻭ =‬ ‫‪E‬ﻭ‬ ‫‪ïM‬‬ ‫−‬
  • 43.
    ‫ﺍﺧﺘﺒﺎﺭ ﺗﺮﺍﻛﻤﻰ‬ ‫8‬ ‫: ‪C‬ﺏ = ‪ C‬ﺟـ ، ‪ = E C‬ﺏ ﺟـ‬ ‫‪ c) X‬ﺏ( = ٠٣‪ C ، c‬ﺟـ = ٦ ﺳﻢ‬ ‫‪C‬‬ ‫ﺃﻭﺟﺪ ﻃﻮﻝ ﻛﻞ ﻣﻦ: ‪ C‬ﺏ ، ﺏ ‪E C ، E‬‬ ‫‪c‬‬ ‫‪E‬‬ ‫‪:á∏jƒ£dG äÉHÉLE’G äGP øjQɪàdG‬‬ ‫‪ E‬ﻫـ‬ ‫‪ C‬ﻫـ‬ ‫9 ‪ C‬ﺏ ﺟـ ‪ E‬ﺷﺒﻪ ﻣﻨﺤﺮﻑ ﺗﻘﺎﻃﻊ ﻗﻄﺮﺍه ﻓﻰ ﻫـ، ﺇﺫﺍ ﻛﺎﻥ ‪ // E C‬ﺏ ﺟـ ﺃﺛﺒﺖ ﺃﻥ: ﻫـ ﺟـ = ﻫـ ﺏ‬ ‫: ‪ C‬ﺏ ﺟـ ‪ E‬ﻣﺴﺘﻄﻴﻞ، ﻡ ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٦ ﺳﻢ‬ ‫01‬ ‫‪E‬‬ ‫ﻭﺗﻤﺲ ‪ C‬ﺏ ﻋﻨﺪ ﻫـ، ﺟـ ‪ E‬ﻋﻨﺪ ﻭ.‬ ‫ﺭﺳﻢ ﻡ ﺹ // ‪ C‬ﺏ ﻭﻳﻘﻄﻊ ﺍﻟﺪﺍﺋﺮﺓ ﻓﻰ ﺱ، ‪ E C‬ﻓﻰ ﺹ.‬ ‫ﺇﺫﺍ ﻛﺎﻥ: ﺱ ﺹ = ٢ﺳﻢ، ‪ C 9) W‬ﻫـ ﻡ( = ١‬ ‫‪ C 9) W‬ﺏ ﺟـ( ٤‬ ‫ﺃﻭﺟﺪ ﻃﻮﻝ ﺏ ﻫـ ، ﺏ ﺟـ‬ ‫‪C‬‬ ‫‪?á«aÉ°VCG IóYÉ°ùªd êÉàëJ πg‬‬ ‫ﺇﻥ ﻟﻢ ﺗﺴﺘﻄﻊ ﺍﻹﺟﺎﺑﺔ ﻋﻦ ﺃﻯ ﻣﻦ ﺍﻷﺳﺌﻠﺔ ﺍﻟﺴﺎﺑﻘﺔ ﻓﺎﺭﺟﻊ ﻟﻠﺠﺪﻭﻝ ﺍﻟﺘﺎﻟﻰ:‬ ‫١‬ ‫¯‬ ‫٢‬ ‫−‬ ‫٣‬ ‫¯‬ ‫٤‬ ‫٥‬ ‫٦‬ ‫٧‬ ‫٨‬ ‫٩‬ ‫٠١‬
  • 44.
    ‫-‬ ‫‪IóMƒdG‬‬ ‫3‬ ‫ﻧﻈﺮﻳﺎﺕ ﺍﻟﺘﻨﺎﺳﺐ ﻓﻰﺍﻟﻤﺜﻠﺚ‬ ‫‪The Triangle Proportionality Theorems‬‬ ‫ﻣﻌﺒﺪ ﺣﺘﺸﺒﺴﻮت )اﻷﻗﺼﺮ(‬ ‫دروس اﻟﻮﺣﺪة‬ ‫ﺍﻟﺪﺭﺱ )٣ - ١(: ﺍﻟﻤﺴﺘﻘﻴﻤﺎﺕ ﺍﻟﻤﺘﻮﺍﺯﻳﺔ ﻭﺍﻷﺟﺰﺍﺀ ﺍﻟﻤﺘﻨﺎﺳﺒﺔ.‬ ‫ﺍﻟﺪﺭﺱ )٣ - ٢(: ﻣﻨﺼﻔﺎ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ﺍﻟﻤﺜﻠﺚ ﻭﺍﻷﺟﺰﺍﺀ ﺍﻟﻤﺘﻨﺎﺳﺒﺔ.‬ ‫ﺍﻟﺪﺭﺱ )٣ - ٣(: ﺗﻄﺒﻴﻘﺎﺕ ﺍﻟﺘﻨﺎﺳﺐ ﻓﻰ ﺍﻟﺪﺍﺋﺮﺓ.‬ ‫‪ïM‬‬ ‫−‬
  • 45.
    ‫اﻟﻤﺴﺘﻘﻴﻤﺎت اﻟﻤﺘﻮازﻳﺔ واﺟﺰاء اﻟﻤﺘﻨﺎﺳﺒﺔ‬ ‫3-1‬ ‫‪Parallel lines and proportional parts‬‬ ‫1‬ ‫‪C‬‬ ‫‪ E‬ﻫـ // ﺏ ﺟـ ﺃﻛﻤﻞ:‬ ‫ﺏ‬ ‫ﺟـ ﻫـ‬ ‫‪C‬‬ ‫‪EC‬‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ ‪ E‬ﺏ = ٥ ﻓﺈﻥ : ﺏ ‪ ،  ............. = E‬‬ ‫= .............‬ ‫٣‬ ‫ﻫـ ‪C‬‬ ‫.............‬ ‫.............‬ ‫ﺟـ ﻫـ‬ ‫ﺏ‪E‬‬ ‫ﻫـ‬ ‫= .............  ، ‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ‪ ٤ = C‬ﻓﺈﻥ :‬ ‫= .............‬ ‫‪ C‬ﺟـ‬ ‫2‬ ‫.............‬ ‫ﻫـ ‪C‬‬ ‫٧‬ ‫.............‬ ‫‪E‬‬ ‫‪C‬ﺏ‬ ‫‪ E‬ﻫـ // ﺏ ﺟـ . ﺣﺪﺩ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺼﺤﻴﺤﺔ ﻣﻦ ﻣﺎ ﻳﻠﻲ:‬ ‫‪C‬ﺏ‬ ‫أ ‪E‬ﺏ‬ ‫ﺟ ‪C‬ﺏ‬ ‫ﺏ‪E‬‬ ‫ﻫ ‪ C‬ﺟـ‬ ‫‪EC‬‬ ‫ﺏ‪E‬‬ ‫ب ‪EC‬‬ ‫‪ C‬ﻫـ = ﻫـ ﺟـ‬ ‫‪C‬ﺏ‬ ‫‪ C‬ﺟـ‬ ‫د ﺏ‪= E‬‬ ‫ﺟـ ﻫـ‬ ‫و ﺟـ ﻫـ ‪ C‬ﺟـ‬ ‫ﺏ‪C = E‬ﺏ‬ ‫‪ C‬ﻫـ‬ ‫= ﻫـ ﺟـ‬ ‫‪ C‬ﺟـ‬ ‫= ‪ C‬ﻫـ‬ ‫‪C‬ﺏ‬ ‫= ‪ C‬ﻫـ‬ ‫3‬ ‫‪E‬‬ ‫‪ E‬ﻫـ // ﺏ ﺟـ . ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺱ ﺍﻟﻌﺪﺩﻳﺔ )ﺍﻷﻃﻮﺍﻝ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ(.‬ ‫ﺟ‬ ‫ب‬ ‫‪C‬‬ ‫أ‬ ‫‪E‬‬ ‫‪E‬‬ ‫+‬ ‫‪E‬‬ ‫د‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫ﻫ‬ ‫‪C‬‬ ‫و‬ ‫+‬ ‫+‬ ‫‪E‬‬ ‫+‬ ‫4‬ ‫: ‪ C‬ﺏ // ‪ E‬ﻫـ ، ‪ C‬ﻫـ ∩ ﺏ ‪} = E‬ﺟـ{‬ ‫‪ C‬ﺟـ = ٦ﺳﻢ، ﺏ ﺟـ = ٤ﺳﻢ، ﺟـ ‪٣ = E‬ﺳﻢ‬ ‫ﺃﻭﺟﺪ ﻃﻮﻝ ﺟـ ﻫـ‬ ‫¯‬ ‫−‬ ‫¯‬ ‫‪C‬‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪E‬‬
  • 46.
    ‫5 ﺱ ﺹ∩ ﻉ ﻝ = }ﻡ{، ﺣﻴﺚ ﺱ ﻉ // ﻝ ﺹ ، ﻓﺈﺫﺍ ﻛﺎﻥ ﺱ ﻡ = ٩ﺳﻢ، ﺹ ﻡ = ٥١ﺳﻢ، ﻉ ﻝ = ٦٣ ﺳﻢ.‬ ‫ﺃﻭﺟﺪ ﻃﻮﻝ ﻉ ﻡ .‬ ‫6 ﻟﻜﻞ ﻣﻤﺎ ﻳﺄﺗﻰ: ﺍﺳﺘﺨﺪﻡ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ ﻭﺍﻟﺒﻴﺎﻧﺎﺕ ﺍﻟﻤﻌﻄﺎﺓ ﻹﻳﺠﺎﺩ ﻗﻴﻤﺔ ﺱ:‬ ‫أ ‪ ، ٤ = E C‬ﺏ ‪ ،  ٨ = E‬ﺟـ ﻫـ = ٦ ، ‪ C‬ﻫـ = ﺱ.‬ ‫ب ‪ C‬ﻫـ = ﺱ ، ﻫـ ﺟـ = ٥ ، ‪ = E C‬ﺱ - ٢ ، ‪ E‬ﺏ = ٣.‬ ‫ﺟ ‪ C‬ﺏ = ١٢ ، ﺏ ﻭ = ٨ ، ﻭ ﺟـ = ٦ ، ‪ = E C‬ﺱ.‬ ‫‪C‬‬ ‫‪E‬‬ ‫د ‪ = E C‬ﺱ  ، ﺏ ﻭ = ﺱ + ٥ ، ٢‪ E‬ﺏ = ٣ﻭ ﺟـ = ٢١.‬ ‫7 ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ، ﺣﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﺱ ﺹ // ﺏ ﺟـ‬ ‫ب‬ ‫أ‬ ‫ﺟ‬ ‫‪C‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫8 ﺱ ﺹ ﻉ ﻣﺜﻠﺚ ﻓﻴﻪ ﺱ ﺹ = ٤١ﺳﻢ، ﺱ ﻉ = ١٢ﺳﻢ، ﻝ ∋ ﺱ ﺹ ﺑﺤﻴﺚ ﺱ ﻝ = ٦٫٥ﺳﻢ،‬ ‫ﻡ ∋ ﺱ ﻉ ﺣﻴﺚ ﺱ ﻡ = ٤٫٨ﺳﻢ. ﺃﺛﺒﺖ ﺃﻥ ﻝ ﻡ // ﺹ ﻉ‬ ‫9 ﻓﻰ ﺍﻟﻤﺜﻠﺚ ‪ C‬ﺏ ﺟـ، ‪ C ∋ E‬ﺏ ، ﻫـ ∋ ‪ C‬ﺟـ ، ٥‪ C‬ﻫـ = ٤ ﻫـ ﺟـ.‬ ‫ﺇﺫﺍ ﻛﺎﻥ ‪ ١٠ = E C‬ﺳﻢ، ‪ E‬ﺏ = ٨ﺳﻢ. ﺣﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ‪ E‬ﻫـ // ﺏ ﺟـ . ﻓﺴﺮ ﺇﺟﺎﺑﺘﻚ.‬ ‫01 ‪ C‬ﺏ ﺟـ ‪ E‬ﺷﻜﻞ ﺭﺑﺎﻋﻰ ﺗﻘﺎﻃﻊ ﻗﻄﺮﺍه ﻓﻰ ﻫـ. ﻓﺈﺫﺍ ﻛﺎﻥ ‪ C‬ﻫـ = ٦ﺳﻢ، ﺏ ﻫـ = ٣١ﺳﻢ، ﻫـ ﻭ = ٠١ﺳﻢ،‬ ‫ﻫـ ‪٧٫٨ = E‬ﺳﻢ. ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﺸﻜﻞ ‪ C‬ﺏ ﺟـ ‪ E‬ﺷﺒﻪ ﻣﻨﺤﺮﻑ.‬ ‫11 ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﻘﻄﻌﺔ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺍﻟﻤﺮﺳﻮﻣﺔ ﺑﻴﻦ ﻣﻨﺘﺼﻔﻰ ﺿﻠﻌﻴﻦ ﻓﻰ ﻣﺜﻠﺚ ﻳﻮﺍﺯﻯ ﺿﻠﻌﻪ ﺍﻟﺜﺎﻟﺚ، ﻭﻃﻮﻟﻬﺎ ﻳﺴﺎﻭﻯ‬ ‫ﻧﺼﻒ ﻃﻮﻝ ﻫﺬﺍ ﺍﻟﻀﻠﻊ.‬ ‫21 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ، ‪ C ∋ E‬ﺏ ﺣﻴﺚ ٣‪ E ٢ = E C‬ﺏ، ﻫـ ∋ ‪ C‬ﺟـ ﺣﻴﺚ ٥ ﺟـ ﻫـ = ٣ ‪ C‬ﺟـ، ﺭﺳﻢ ‪ C‬ﺱ ﻳﻘﻄﻊ ﺏ ﺟـ‬ ‫ﻓﻰ ﺱ. ﺇﺫﺍ ﻛﺎﻥ ‪ C‬ﻭ = ٨ﺳﻢ، ‪ C‬ﺱ = ٠٢ﺳﻢ، ﺣﻴﺚ ﻭ ∋ ‪ C‬ﺱ . ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﻨﻘﻂ ‪ ،E‬ﻭ، ﻫـ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ.‬ ‫ﻫـ‬ ‫‪E‬‬ ‫31 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ، ‪ ∋ E‬ﺏ ﺟـ ، ﺑﺤﻴﺚ ﺏﺟـ = ٣ ، ﻫـ ∋ ‪ ، E C‬ﺑﺤﻴﺚ ‪ ، ٣ = E C‬ﺭﺳﻢ ﺟـ ﻫـ ﻓﻘﻄﻊ ‪ C‬ﺏ ﻓﻰ ﺱ،‬ ‫٧‬ ‫٤‬ ‫‪E‬‬ ‫‪C‬‬ ‫ﺭﺳﻢ ‪ E‬ﺹ // ﺟـ ﺱ ﻓﻘﻄﻊ ‪ C‬ﺏ ﻓﻰ ﺹ. ﺃﺛﺒﺖ ﺃﻥ ‪ C‬ﺱ = ﺏ ﺹ.‬ ‫41 ‪ C‬ﺏ ﺟـ ‪ E‬ﻣﺴﺘﻄﻴﻞ ﺗﻘﺎﻃﻊ ﻗﻄﺮﺍه ﻓﻰ ﻡ. ﻫـ ﻣﻨﺘﺼﻒ ‪ C‬ﻡ ، ﻭ ﻣﻨﺘﺼﻒ ﻡ ﺟـ . ﺭﺳﻢ ‪ E‬ﻫـ ﻳﻘﻄﻊ ‪ C‬ﺏ ﻓﻰ ﺱ،‬ ‫ﻭﺭﺳﻢ ‪ E‬ﻭ ﻳﻘﻄﻊ ﺏ ﺟـ ﻓﻰ ﺹ. ﺃﺛﺒﺖ ﺃﻥ: ﺱ ﺹ // ‪ C‬ﺟـ .‬ ‫‪ïM‬‬ ‫−‬
  • 47.
    ‫51 ﺍﻛﺘﺐ ﻣﺎﺗﺴﺎﻭﻳﻪ ﻛﻞ ﻣﻦ ﺍﻟﻨﺴﺐ ﺍﻟﺘﺎﻟﻴﺔ ﻣﺴﺘﺨﺪﻣﺎ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ:‬ ‫ً‬ ‫أ‬ ‫ﺟ‬ ‫ﻫ‬ ‫ز‬ ‫‪ C‬ﺏ = ‪ E‬ﻫـ‬ ‫ﺏ ﺟـ‬ ‫ﻡ‪E‬‬ ‫ﻡ‪C‬‬ ‫= ................‬ ‫‪C‬ﺏ‬ ‫................‬ ‫ﻡﺏ‬ ‫=‬ ‫‪ C‬ﺏ ‪ E‬ﻫـ‬ ‫ﺏ ﺟـ ﻫـ ﻭ‬ ‫ﻡ ﺏ = ................‬ ‫................‬ ‫ب ‪ C‬ﺟـ‬ ‫ﺏ ﺟـ = ﻫـ ﻭ‬ ‫................‬ ‫‪C‬‬ ‫................‬ ‫ﺟـ‬ ‫د ‪= C‬‬ ‫‪ C‬ﺏ ‪ E‬ﻫـ‬ ‫و ﻡ ﺟـ ﻡ ﻭ‬ ‫= ................‬ ‫‪E‬‬ ‫‪ C‬ﺟـ‬ ‫ح ‪ E‬ﻭ = ‪ C‬ﺟـ‬ ‫................‬ ‫ﻡﻭ‬ ‫61 ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ، ﺍﺣﺴﺐ ﻗﻴﻢ ﺱ، ﺹ ﺍﻟﻌﺪﺩﻳﺔ )ﺍﻷﻃﻮﺍﻝ ﻣﻘﺪﺭﺓ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ(‬ ‫ﺟ‬ ‫ب‬ ‫أ‬ ‫+‬ ‫+‬ ‫−‬ ‫−‬ ‫+‬ ‫+‬ ‫:‬ ‫71‬ ‫‪E‬‬ ‫‪ C‬ﺏ ∩ ﺟـ ‪} = E‬ﻡ{، ﻫـ ∋ ﻡ ﺏ ،‬ ‫‪C‬‬ ‫ﻭ ∋ ﻡ ‪ C ، E‬ﺟـ // ﻭ ﻫـ // ‪ E‬ﺏ‬ ‫ﺃﻭﺟﺪ:‬ ‫أ ﻃﻮﻝ ﻡ ﻭ‬ ‫ب ﻃﻮﻝ ‪ C‬ﻡ‬ ‫ﻭ‬ ‫81 ‪ C‬ﺏ ∩ ﺟـ ‪} = E‬ﻫـ{ ، ﺱ ∋ ‪ C‬ﺏ ، ﺹ ∋ ﺟـ ‪ ، E‬ﻛﺎﻥ ﺱ ﺹ // ﺏ ‪ C // E‬ﺟـ‬ ‫ﺃﺛﺒﺖ ﺃﻥ: ‪ C‬ﺱ * ﻫـ ‪ = E‬ﺟـ ﺹ * ﻫـ ﺏ‬ ‫91 ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ، ﺍﺣﺴﺐ ﻗﻴﻢ ﺱ، ﺹ ﺍﻟﻌﺪﺩﻳﺔ:‬ ‫ب‬ ‫‪C‬‬ ‫أ‬ ‫−‬ ‫‪E‬‬ ‫+‬ ‫+‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫02 ‪ C‬ﺏ ﺟـ ‪ E‬ﺷﻜﻞ ﺭﺑﺎﻋﻰ ﻓﻴﻪ ‪ C‬ﺏ // ﺟـ ‪ ، E‬ﺗﻘﺎﻃﻊ ﻗﻄﺮﺍه ﻓﻰ ﻡ، ﻧﺼﻒ ﺏ ﺟـ ﻓﻰ ﻫـ،‬ ‫ﻭﺭﺳﻢ ﻫـ ﻭ // ﺏ ‪ ، C‬ﻭﻳﻘﻄﻊ ﺏ ‪ E‬ﻓﻰ ﺱ ، ‪ C‬ﺟـ ﻓﻰ ﺹ ، ‪ E C‬ﻓﻰ ﻭ.‬ ‫ﺃﺛﺒﺖ ﺃﻥ:‬ ‫أ ﻫـ ﺹ = ١ ‪ C‬ﺏ.‬ ‫٢‬ ‫¯‬ ‫−‬ ‫+‬ ‫+‬ ‫ﺟ‬ ‫+‬ ‫ب ‪C‬ﺹ ﺏﺱ‬ ‫=‬ ‫ﺟـ ﻡ ‪ E‬ﻡ‬ ‫−‬ ‫¯‬ ‫−‬
  • 48.
    ‫ﻣﻨﺼﻔﺎ اﻟﺰواﻳﺔ ﻓﻰاﻟﻤﺜﻠﺚ وا ﺟﺰاء اﻟﻤﺘﻨﺎﺳﺒﺔ‬ ‫3-2‬ ‫‪Angle Bisectors and Proportional Parts‬‬ ‫: ‪ E C‬ﻳﻨﺼﻒ ‪ . Cc‬ﺃﻛﻤﻞ:‬ ‫1‬ ‫‪E‬‬ ‫أ ﺏﺟـ =‬ ‫‪E‬‬ ‫ب ‪ C‬ﺟـ =‬ ‫.....................................‬ ‫‪C‬ﺏ‬ ‫ﺟ ﺏ‪E‬‬ ‫= ........................................‬ ‫ﺏ‪C‬‬ ‫‪C‬‬ ‫........................................‬ ‫د ‪ C‬ﺏ * ﺟـ ‪= E‬‬ ‫‪E‬‬ ‫........................‬ ‫2 ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ، ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺱ )ﺍﻷﻃﻮﺍﻝ ﻣﻘﺪﺭﺓ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ(‬ ‫ب‬ ‫‪C‬‬ ‫أ ‪C‬‬ ‫+‬ ‫−‬ ‫‪E‬‬ ‫‪E‬‬ ‫......................................................................................................‬ ‫‪C‬‬ ‫ﺟ‬ ‫......................................................................................................‬ ‫د‬ ‫+‬ ‫‪C‬‬ ‫+‬ ‫+‬ ‫‪E‬‬ ‫‪E‬‬ ‫......................................................................................................‬ ‫......................................................................................................‬ ‫3 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻣﺤﻴﻄﻪ ٧٢ﺳﻢ، ﺭﺳﻢ ﺏ ‪ E‬ﻳﻨﺼﻒ ‪ c‬ﺏ ﻭﻳﻘﻄﻊ ‪ C‬ﺟـ ﻓﻰ ‪.E‬‬ ‫ﺇﺫﺍ ﻛﺎﻥ ‪٤ = E C‬ﺳﻢ، ﺟـ ‪٥ = E‬ﺳﻢ، ﺃﻭﺟﺪ ﻃﻮﻝ ﻛﻞ ﻣﻦ ‪ C‬ﺏ ، ﺏ ﺟـ ، ‪E C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫4 ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺱ، ﺛﻢ ﺃﻭﺟﺪ ﻣﺤﻴﻂ 9‪ C‬ﺏ ﺟـ.‬ ‫ﺟ‬ ‫ب‬ ‫‪C‬‬ ‫‪C‬‬ ‫أ‬ ‫+‬ ‫‪E‬‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪E‬‬ ‫5 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻓﻴﻪ ‪ C‬ﺏ = ٨ﺳﻢ، ‪ C‬ﺟـ = ٤ﺳﻢ، ﺏ ﺟـ = ٦ﺳﻢ، ﺭﺳﻢ ‪ E C‬ﻳﻨﺼﻒ ‪ C‬ﻭﻳﻘﻄﻊ ﺏ ﺟـ ﻓﻰ ‪ ،E‬ﻭﺭﺳﻢ‬ ‫‪ C‬ﻫـ ﻳﻨﺼﻒ ‪ Cc‬ﺍﻟﺨﺎﺭﺟﺔ ﻭﻳﻘﻄﻊ ﺏ ﺟـ ﻓﻰ ﻫـ ﺃﻭﺟﺪ ﻃﻮﻝ ﻛﻞ ﻣﻦ ‪ E‬ﻫـ ، ‪ C ، E C‬ﻫـ .‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
  • 49.
    ‫6 ﻓﻰ ﻛﻞﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ: ﺃﺛﺒﺖ ﺃﻥ ﺱ ﺹ // ﺏ ﺟـ‬ ‫‪E‬‬ ‫أ‬ ‫ب‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫......................................................................................................‬ ‫......................................................................................................‬ ‫......................................................................................................‬ ‫......................................................................................................‬ ‫7 ﻓﻰ ﻛﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ، ﺃﺛﺒﺖ ﺃﻥ ﺏ ﻫـ ﻳﻨﺼﻒ ‪ Cc‬ﺏ ﺟـ.‬ ‫ب‬ ‫‪C‬‬ ‫أ‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫......................................................................................................‬ ‫......................................................................................................‬ ‫......................................................................................................‬ ‫......................................................................................................‬ ‫‪C‬‬ ‫8‬ ‫: ﻫـ ‪ // E‬ﺱ ﺹ // ﺏ ﺟـ ،‬ ‫‪ * E C‬ﺏ ﺱ = ‪ C‬ﺟـ * ﻫـ ﺱ.‬ ‫ﺃﺛﺒﺖ ﺃﻥ ‪ C‬ﺹ ﻳﻨﺼﻒ ‪c‬ﺟـ ‪.E C‬‬ ‫‪E‬‬ ‫.................................................................................................................................‬ ‫.................................................................................................................................‬ ‫9 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ‪ ∋ E‬ﺏ ﺟـ ، ‪ ∌ E‬ﺏ ﺟـ ﺣﻴﺚ ﺟـ ‪ C = E‬ﺏ. ﺭﺳﻢ ﺟـ ﻫـ // ‪ C E‬ﻭﻳﻘﻄﻊ ‪ C‬ﺏ ﻓﻰ ﻫـ، ﻭﺭﺳﻢ‬ ‫ﻫـ ﻭ // ﺏ ﺟـ ﻭﻳﻘﻄﻊ ‪ C‬ﺟـ ﻓﻰ ﻭ ﺃﺛﺒﺖ ﺃﻥ ﺏ ﻭ ﻳﻨﺼﻒ ‪ Cc‬ﺏ ﺟـ‬ ‫..................................................................................................................................................................................................................................‬ ‫01‬ ‫: ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻓﻴﻪ ‪ C‬ﺏ = ٦ﺳﻢ، ‪ C‬ﺟـ = ٩ﺳﻢ،‬ ‫ﺏ ﺟـ = ٠١ﺳﻢ. ‪ ∋ E‬ﺏ ﺟـ ﺑﺤﻴﺚ ﺏ ‪٤ = E‬ﺳﻢ .‬ ‫ﺭﺳﻢ ﺏ ﻫـ = ‪ E C‬ﻭﻳﻘﻄﻊ ‪ C ، E C‬ﺏ ﻓﻰ ﻫـ، ﻭ ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ.‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ ‪ E C‬ﻳﻨﺼﻒ ‪.Cc‬‬ ‫ب ﺃﻭﺟﺪ ‪ C9) W‬ﺏ ﻭ( : ‪9) W‬ﺟـ ﺏ ﻭ(‬ ‫‪C‬‬ ‫‪E‬‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
  • 50.
    ‫ﺗﻄﺒﻴﻘﺎت اﻟﺘﻨﺎﺳﺐ ﻓﻰاﻟﺪاﺋﺮة‬ ‫3-3‬ ‫‪Applications of Proportionality in the Circle‬‬ ‫1 ﺣﺪﺩ ﻣﻮﻗﻊ ﻛﻞ ﻣﻦ ﺍﻟﻨﻘﻂ ﺍﻟﺘﺎﻟﻴﺔ ﺑﺎﻟﻨﺴﺒﺔ ﺇﻟﻰ ﺍﻟﺪﺍﺋﺮﺓ ﻡ، ﻭﺍﻟﺘﻰ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٠١ﺳﻢ، ﺛﻢ ﺍﺣﺴﺐ ﺑﻌﺪ ﻛﻞ‬ ‫ُ َ‬ ‫ﻧﻘﻄﺔ ﻋﻦ ﻛﺰ ﺍﻟﺪﺍﺋﺮﺓ.‬ ‫ﻣﺮ‬ ‫ﺟ ‪X‬ﻡ)ﺟـ( = ﺻﻔﺮ‬ ‫ب ‪X‬ﻡ)ﺏ( = ٦٩‬ ‫أ ‪X‬ﻡ) ‪٣٦ - = ( C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫2 ﺃﻭﺟﺪ ﻗﻮﺓ ﺍﻟﻨﻘﻄﺔ ﺍﻟﻤﻌﻄﺎﺓ ﺑﺎﻟﻨﺴﺒﺔ ﺇﻟﻰ ﺍﻟﺪﺍﺋﺮﺓ ﻡ، ﻭﺍﻟﺘﻰ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ‪:H‬‬ ‫.................................................................................................................‬ ‫أ ﺍﻟﻨﻘﻄﺔ ‪ C‬ﺣﻴﺚ ‪ C‬ﻡ = ٢١ﺳﻢ ، ‪ ٩ = H‬ﺳﻢ‬ ‫.................................................................................................................‬ ‫ب ﺍﻟﻨﻘﻄﺔ ﺏ ﺣﻴﺚ ﺏ ﻡ = ٨ ﺳﻢ ، ‪ ١٥ = H‬ﺳﻢ‬ ‫.................................................................................................................‬ ‫ﺟ ﺍﻟﻨﻘﻄﺔ ﺟـ ﺣﻴﺚ ﺟـ ﻡ = ٧ ﺳﻢ ، ‪ ٧ = H‬ﺳﻢ‬ ‫.................................................................................................................‬ ‫د ﺍﻟﻨﻘﻄﺔ ‪ E‬ﺣﻴﺚ ‪ E‬ﻡ = ٧١ ﺳﻢ، ‪ ٤ = H‬ﺳﻢ‬ ‫3 ﺇﺫﺍﻛﺎﻥ ﺑﻌﺪ ﻧﻘﻄﺔ ﻋﻦ ﻛﺰ ﺩﺍﺋﺮﺓ ﻳﺴﺎﻭﻯ ٥٢ﺳﻢ ﻭﻗﻮﺓ ﻫﺬه ﺍﻟﻨﻘﻄﺔ ﺑﺎﻟﻨﺴﺒﺔ ﺇﻟﻰ ﺍﻟﺪﺍﺋﺮﺓ ﻳﺴﺎﻭﻯ ٠٠٤.‬ ‫ﻣﺮ‬ ‫ﺃﻭﺟﺪ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮ ﻫﺬه ﺍﻟﺪﺍﺋﺮﺓ.  ..............................................................................................................................................‬ ‫ﻣﺮ‬ ‫4 ﺍﻟﺪﺍﺋﺮﺓ ﻡ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٠٢ﺳﻢ. ‪ C‬ﻧﻘﻄﺔ ﺗﺒﻌﺪ ﻋﻦ ﻛﺰ ﺍﻟﺪﺍﺋﺮﺓ ﻣﺴﺎﻓﺔ ٦١ﺳﻢ، ﺭﺳﻢ ﺍﻟﻮﺗﺮ ﺏ ﺟـ‬ ‫ﺣﻴﺚ ‪ ∋ C‬ﺏ ﺟـ ، ‪ C‬ﺏ = ٢ ‪ C‬ﺟـ. ﺇﺣﺴﺐ ﻃﻮﻝ ﺍﻟﻮﺗﺮ ﺏ ﺟـ .‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫5‬ ‫: ﺍﻟﺪﺍﺋﺮﺗﺎﻥ ﻡ، ﻥ ﻣﺘﻘﺎﻃﻌﺘﺎﻥ ﻓﻰ ‪ ،C‬ﺏ‬ ‫ﺣﻴﺚ ‪ C‬ﺏ ∩ ﺟـ ‪ ∩ E‬ﻫـ ﻭ = }ﺱ{، ﺱ ‪ E ٢ = E‬ﺟـ ، ﻫـ ﻭ = ٠١ﺳﻢ،‬ ‫‪X‬ﻥ )ﺱ( = ٤٤١.‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ ‪ C‬ﺏ ﻣﺤﻮﺭ ﺃﺳﺎﺳﻰ ﻟﻠﺪﺍﺋﺮﺗﻴﻦ ﻡ، ﻥ.‬ ‫ب ﺃﻭﺟﺪ ﻃﻮﻝ ﻛﻞ ﻣﻦ ﺱ ﺟـ ، ﺱ ﻭ‬ ‫ﺟ ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﺸﻜﻞ ﺟـ ‪ E‬ﻭ ﻫـ ﺭﺑﺎﻋﻰ ﺩﺍﺋﺮﻯ.‬ ‫‪E‬‬ ‫‪C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
  • 51.
    ‫6 ﻣﺴﺘﻌﻴﻨﺎ ﺑﻤﻌﻄﻴﺎﺕﺍﻟﺸﻜﻞ، ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺮﻣﺰ ﺍﻟﻤﺴﺘﺨﺪﻡ ﻓﻰ ﺍﻟﻘﻴﺎﺱ.‬ ‫ً‬ ‫ب‬ ‫‪E‬‬ ‫‪C‬‬ ‫أ‬ ‫‪c‬‬ ‫‪E‬‬ ‫‪E‬‬ ‫ﺟ‬ ‫‪c‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫................................................................‬ ‫د‬ ‫................................................................‬ ‫‪E‬‬ ‫ﻫ‬ ‫و‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫‪c‬‬ ‫ز‬ ‫................................................................‬ ‫ح‬ ‫‪E‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪C‬‬ ‫‪C‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫ط‬ ‫‪E‬‬ ‫− ‪c‬‬ ‫‪C‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫+‬ ‫‪C‬‬ ‫................................................................‬ ‫− ‪c‬‬ ‫‪c‬‬ ‫................................................................‬ ‫+‬ ‫‪E‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪E c‬‬ ‫................................................................‬ ‫+‬ ‫‪c‬‬ ‫................................................................‬ ‫‪c‬‬ ‫................................................................‬ ‫‪c‬‬ ‫7‬ ‫: ‪c)X‬ﺏ ‪ C‬ﺟـ( = ٣٣‪c)X ،c‬ﺏ ‪ E‬ﺟـ( = ٠٧‪،c‬‬ ‫‪ C )X‬ﺏ ( = ٤٩‪ )X ، c‬ﺟـ ﺹ ( = ٠٠١‪ c‬ﺃﻭﺟﺪ ﻗﻴﺎﺱ ﻛﻞ ﻣﻦ:‬ ‫أ ﺱﺹ‬ ‫ب ‪C‬ﺱ‬ ‫ﺟ ‪c‬ﺏ ﻫـ ﺟـ‬ ‫‪c‬‬ ‫‪E‬‬ ‫‪c‬‬ ‫‪C‬‬ ‫‪c‬‬ ‫..................................................................................................................................................................................................................................‬ ‫8 ﺍﻟﺮﺑﻂ ﻣﻊ ﺍﻟﺼﻨﺎﻋﺔ: ﻣﻨﺸﺎﺭ ﺩﺍﺋﺮﻯ ﻟﻘﻄﻊ ﺍﻟﺨﺸﺐ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮ‬ ‫ﺩﺍﺋﺮﺗﻪ ٠١ﺳﻢ. ﻳﺪﻭﺭ ﺩﺍﺧﻞ ﺣﺎﻓﻈﺔ ﺣﻤﺎﻳﺔ، ﻓﺈﺫﺍ ﻛﺎﻥ ‪c)X‬ﺏ ‪= (E C‬‬ ‫٥٤‪ )X ،c‬ﺏ ‪ c١٥٥ = ( E‬ﺃﻭﺟﺪ ﻃﻮﻝ ﻗﻮﺱ ﻗﺮﺹ ﺍﻟﻤﻨﺸﺎﺭ ﺧﺎﺭﺝ ﺣﺎﻓﻈﺔ‬ ‫ﺍﻟﺤﻤﺎﻳﺔ.‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪E‬‬ ‫‪C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫‪C‬‬ ‫9 ﺍ ﺼﺎﻻﺕ: ﺗﺘﺒﻊ ﺍﻹﺷﺎﺭﺍﺕ ﺍﻟﺘﻰ ﺗﺼﺪﺭ ﻋﻦ ﺑﺮﺝ ﺍﻻﺗﺼﺎﻻﺕ ﻓﻰ ﻣﺴﺎﺭﻫﺎ‬ ‫ﺷﻌﺎﻋﺎ، ﻧﻘﻄﺔ ﺑﺪﺍﻳﺘﻪ ﻋﻠﻰ ﻗﻤﺔ ﺍﻟﺒﺮﺝ، ﻭﻳﻜﻮﻥ ﻣﻤﺎﺳﺎ ﻟﺴﻄﺢ ﺍﻷﺭﺽ،‬ ‫ً‬ ‫ً‬ ‫ﻛﻤﺎ ﻓﻰ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ. ﺣﺪﺩ ﻗﻴﺎﺱ ﺍﻟﻘﻮﺱ ﺍﻟﻤﺤﺼﻮﺭ ﺑﺎﻟﻤﻤﺎﺳﻴﻦ‬ ‫ﺑﻔﺮﺽ ﺃﻥ ﺍﻟﺒﺮﺝ ﻳﻘﻊ ﻋﻠﻰ ﻣﺴﺘﻮﻯ ﺳﻄﺢ ﺍﻟﺒﺤﺮ، ‪c)X‬ﺟـ ‪ C‬ﺏ( = ٠٨‪c‬‬ ‫¯‬ ‫−‬ ‫¯‬
  • 52.
    ‫ﺗﻤﺎﺭﻳﻦ ﻋﺎﻣﺔ‬ ‫1 ﺃﻛﻤﻞﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺘﺎﻟﻴﺔ:‬ ‫أ‬ ‫ب‬ ‫ﺟ‬ ‫د‬ ‫ﻫ‬ ‫ﺍﻟﻤﻨﺼﻔﺎﻥ ﺍﻟﺪﺍﺧﻠﻰ ﻭﺍﻟﺨﺎﺭﺟﻰ ﻟﺰﺍﻭﻳﺔ ﻭﺍﺣﺪﺓ‬ ‫ﻣﻨﺼﻔﺎﺕ ﺯﻭﺍﻳﺎ ﺍﻟﻤﺜﻠﺚ ﺗﺘﻘﺎﻃﻊ ﻓﻰ ..............................................................................................................................................‬ ‫ﺇﺫﺍ ﺭﺳﻢ ﻣﺴﺘﻘﻴﻢ ﻳﻮﺍﺯﻯ ﺃﺣﺪ ﺃﺿﻼﻉ ﻣﺜﻠﺚ، ﻭﻳﻘﻄﻊ ﺍﻟﻀﻠﻌﻴﻦ ﺍﻵﺧﺮﻳﻦ ﻓﺈﻧﻪ .........................................................‬ ‫ﺍﻟﻤﻨﺼﻒ ﺍﻟﺨﺎﺭﺟﻰ ﻟﺰﺍﻭﻳﺔ ﺭﺃﺱ ﺍﻟﻤﺜﻠﺚ ﺍﻟﻤﺘﺴﺎﻭﻯ ﺍﻟﺴﺎﻗﻴﻦ .................................... ﻗﺎﻋﺪﺓ ﺍﻟﻤﺜﻠﺚ.‬ ‫ﺇﺫﺍ ﻛﺎﻧﺖ ﻗﻮﺓ ﺍﻟﻨﻘﻄﺔ ‪ C‬ﺑﺎﻟﻨﺴﺒﺔ ﻟﻠﺪﺍﺋﺮﺓ ﻡ ﻛﻤﻴﺔ ﺳﺎﻟﺒﺔ، ﻓﺈﻥ ﻧﻘﻄﺔ ‪ C‬ﺗﻘﻊ .....................................................................‬ ‫......................................................................................................................‬ ‫2 ﻣﺴﺘﻌﻴﻨﺎ ﺑﻤﻌﻄﻴﺎﺕ ﺍﻟﺸﻜﻞ، ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺮﻣﺰ ﺍﻟﻤﺴﺘﺨﺪﻡ ﻓﻰ ﺍﻟﻘﻴﺎﺱ.‬ ‫ً‬ ‫ب‬ ‫+‬ ‫أ‬ ‫+‬ ‫‪E‬‬ ‫‪c‬‬ ‫−‬ ‫‪c‬‬ ‫ﺟ‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪c‬‬ ‫‪C‬‬ ‫−‬ ‫3 ﺩﺍﺋﺮﺗﺎﻥ ﻡ، ﻥ ﻣﺘﻘﺎﻃﻌﺘﺎﻥ ﻓﻰ ‪ ،C‬ﺏ.‬ ‫ﻫـ ‪ E‬ﻣﻤﺎﺱ ﻣﺸﺘﺮﻙ ﻟﻠﺪﺍﺋﺮﺗﻴﻦ ﻡ، ﻥ ﻋﻨﺪ ‪ ،E‬ﻫـ ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ،‬ ‫ﺏ ‪ E ∩ C‬ﻫـ = }ﺟـ{‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ: ﺏ ﺟـ ﻣﺤﻮﺭ ﺃﺳﺎﺳﻰ ﻟﻠﺪﺍﺋﺮﺗﻴﻦ.‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ‪ C‬ﺏ = ٩ﺳﻢ، ‪) X‬ﺟـ( = ٦٣، ﺃﻭﺟﺪ ﻃﻮﻝ ﺟـ ‪ ، C‬ﺟـ ‪E‬‬ ‫‪E‬‬ ‫‪C‬‬ ‫ﻥ‬ ‫4‬ ‫ﺃﺣﺪ ﺍﻟﺤﻮﺍﺟﺰ ﺍﻟﻤﺮﻭﺭﻳﺔ ‪ C‬ﺏ ﺟـ ‪ E‬ﻋﻠﻰ ﺷﻜﻞ‬ ‫ﻣﺴﺘﻄﻴﻞ ﻭﻣﻜﻮﻥ ﻣﻦ ﻣﺘﻮﺍﺯﻳﺔ ﻭﻣﺘﻄﺎﺑﻘﺔ، ﻭﻋﻠﻰ ﺃﺑﻌﺎﺩ ﻣﺘﺴﺎﻭﻳﺔ،‬ ‫ﻭﻣﺜﺒﺖ ﺑﻪ ﺩﻋﺎﻣﺘﺎﻥ ‪ C‬ﺟـ ، ﺏ ‪ ، E‬ﺗﻘﻄﻌﺎﻥ ﺃﺣﺪ ﺍﻟﻘﻀﺒﺎﻥ ﺍﻟﺮﺃﺳﻴﺔ ﻓﻰ‬ ‫ﻭ، ﻫـ ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ ﻓﺈﺫﺍ ﻛﺎﻥ ‪ C‬ﺏ = ٠٢١ﺳﻢ ﺃﻭﺟﺪ ﻃﻮﻝ ﻫـ ﻭ .‬ ‫5 ﻫﻨﺪﺳﺔ ﻣﻌﻤﺎ ﺔ: ﻣﻦ ﻧﻘﻄﺔ ‪ C‬ﻭﺍﻟﺘﻲ ﺗﺒﻌﺪ ٦٫١ ﻣﺘﺮﺍ ﻋﻦ ﻗﺎﻋﺪﺓ ﻗﻨﻄﺮﺓ‬ ‫ً‬ ‫ﺗﻌﻠﻮ ﺑﺎﺏ ﻣﻨﺰﻝ، ﻭﺟﺪ ﺃﻥ ﻗﻮﺓ ﺍﻟﻨﻘﻄﺔ ‪ C‬ﺑﺎﻟﻨﺴﺒﺔ ﻟﺪﺍﺋﺮﺓ ﻗﻮﺱ ﺍﻟﻘﻨﻄﺮﺓ‬ ‫ﻳﺴﺎﻭﻯ ٤٫٦ ﻣﺘﺮ ﻣﺮﺑﻊ.‬ ‫أ ﺃﻭﺟﺪ ﻃﻮﻝ ﻗﺎﻋﺪﺓ ﺍﻟﻘﻨﻄﺮﺓ )ﺏ ﺟـ(.‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ﺍﺭﺗﻔﺎﻉ ﺍﻟﻘﻨﻄﺮﺓ ﻳﺴﺎﻭﻯ ٠٨ﺳﻢ، ﻓﺄﻭﺟﺪ ﻗﻮﺓ ﺍﻟﻨﻘﻄﺔ ‪E‬‬ ‫ﺑﺎﻟﻨﺴﺒﺔ ﻟﺪﺍﺋﺮﺓ ﺍﻟﻘﻨﻄﺮﺓ ﻭﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ.‬ ‫‪ïM‬‬ ‫−‬ ‫‪E‬‬ ‫‪C‬‬ ‫¯‬ ‫‪E‬‬ ‫‪C‬‬
  • 53.
    ‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ‬ ‫1 ﻣﺴﺘﺨﺪﻣﺎﻣﻌﻄﻴﺎﺕ ﺍﻟﺸﻜﻞ، ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺮﻣﺰ ﺍﻟﻤﺴﺘﺨﺪﻡ ﻓﻰ ﺍﻟﻘﻴﺎﺱ.‬ ‫ً‬ ‫ب‬ ‫‪E‬‬ ‫‪C‬‬ ‫أ‬ ‫‪C‬‬ ‫ﺟ‬ ‫−‬ ‫‪E‬‬ ‫‪E‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫‪C‬‬ ‫+‬ ‫‪c‬‬ ‫: ‪ C c‬ﺟـ ﺏ ﻗﺎﺋﻤﺔ، ﺏ ﺟـ // ‪ E‬ﻫـ‬ ‫2‬ ‫ﺟـ ‪ // E‬ﻫـ ﻭ . ﺃﺛﺒﺖ ﺃﻥ:‬ ‫٢‬ ‫‪ C‬ﻭ * ‪ C‬ﺏ = )‪ C‬ﻫـ(٢ + )ﻫـ ‪(E‬‬ ‫‪E‬‬ ‫‪C‬‬ ‫3 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ، ﻥ ﻧﻘﻄﺔ ﺩﺍﺧﻞ ﺍﻟﻤﺜﻠﺚ. ﻧﺼﻔﺖ ﺍﻟﺰﻭﺍﻳﺎ ‪ C‬ﻥ ﺏ، ﺏ ﻥ ﺟـ ، ﺟـ ﻥ ‪C‬‬ ‫ﺑﻤﻨﺼﻔﺎﺕ ﻻﻗﺖ ‪ C‬ﺏ ، ﺏ ﺟـ ، ﺟـ ‪ C‬ﻓﻰ ‪ ،E‬ﻫـ، ﻭ ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ.‬ ‫‪ E C‬ﺏ ﻫـ ﺟـ ﻭ‬ ‫*‬ ‫*‬ ‫ﺃﺛﺒﺖ ﺃﻥ:‬ ‫‪ E‬ﺏ ﻫـ ﺟـ ﻭ ‪C‬‬ ‫=١‬ ‫4 ‪ C‬ﻧﻘﻄﺔ ﺧﺎﺭﺝ ﺍﻟﺪﺍﺋﺮﺓ ﻡ، ‪ C‬ﺏ ﻣﻤﺎﺱ ﻟﻠﺪﺍﺋﺮﺓ ﻋﻨﺪ ﺏ.‬ ‫ﺭﺳﻢ ‪ C‬ﺟـ ، ‪ C‬ﻫـ ﻳﻘﻄﻌﺎﻥ ﺍﻟﺪﺍﺋﺮﺓ ﻓﻰ ﺟـ، ‪ ،E‬ﻫـ، ﻭ ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ،‬ ‫‪ C‬ﺟـ = ٤ﺳﻢ، ﻫـ ﻭ = ٩ﺳﻢ.‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ ‪X‬ﻡ) ‪ ٣٦ = ( C‬ﺃﻭﺟﺪ ﻃﻮﻝ ﻛﻞ ﻣﻦ ‪ C‬ﺏ ، ‪ C‬ﻫـ ، ﺟـ ‪E‬‬ ‫ب ﺇﺫﺍ ﻛﺎﻧﺖ ﺱ ∋ ﺟـ ‪ E‬ﺣﻴﺚ ﺟـ ﺱ = ٢ﺳﻢ ﺃﻭﺟﺪ ‪X‬ﻡ)ﺱ(، ‪X‬ﻡ )‪.(E‬‬ ‫‪E‬‬ ‫‪C‬‬ ‫5 ‪ E C‬ﻣﺘﻮﺳﻂ ﻓﻰ 9‪ C‬ﺏ ﺟـ، ﺟـ ﺱ ﻳﻨﺼﻒ ‪ E C c‬ﺏ ﻭﻳﻘﻄﻊ ‪ C‬ﺏ ﻓﻰ ﺱ، ‪ E‬ﺹ ﻳﻨﺼﻒ ‪ E Cc‬ﺟـ ﻭﻳﻘﻄﻊ‬ ‫‪ C‬ﺟـ ﻓﻰ ﺹ.‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ: ﺱ ﺹ // ﺏ ﺟـ‬ ‫ب ﺇﺫﺍ ﺭﺳﻢ ‪ E‬ﻉ = ﺱ ﺹ ﻭ ﻳﻘﻄﻌﻪ ﻓﻰ ﻉ، ﻛﺎﻥ ﺱ ﻉ = ٩ﺳﻢ، ﻉ ﺹ = ٦١ﺳﻢ‬ ‫ﻭ‬ ‫ﺃﻭﺟﺪ ﻃﻮﻝ ﻛﻞ ﻣﻦ : ‪ E‬ﺱ ، ‪ E‬ﺹ .‬ ‫¯‬ ‫−‬ ‫¯‬
  • 54.
    ‫ﺍﺧﺘﺒﺎﺭ ﺗﺮﺍﻛﻤﻲ‬ ‫‪Oó©àe øeQÉ«àN’G á∏Ä°SCG‬‬ ‫ﺱ‬ ‫1 ﺇﺫﺍ ﻛﺎﻥ ٦ = ٩ ﻓﺈﻥ ﺱ ﺗﺴﺎﻭﻯ:‬ ‫٢‬ ‫ب ٦١‬ ‫أ ٢١‬ ‫ﺟ ٧٢‬ ‫2 ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ﺱ - ٠٢ = ﺻﻔﺮ ﻫﻤﺎ:‬ ‫ب ٤، -٥‬ ‫أ ٢، -٠١‬ ‫ﺟ ٥، -٤‬ ‫د ١٨‬ ‫د -٤، ٥‬ ‫‪C‬‬ ‫3 ﺇﺫﺍ ﻛﺎﻥ ‪ E‬ﻫـ // ﺏ ﺟـ ﻓﺈﻥ ‪ C‬ﺟـ ﻳﺴﺎﻭﻯ:‬ ‫ب ٤ﺳﻢ‬ ‫أ ٣ﺳﻢ‬ ‫د ٠١ﺳﻢ‬ ‫ﺟ ٦ﺳﻢ‬ ‫‪E‬‬ ‫4 ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻤﺴﺘﻘﻴﻤﺎﺕ ﻝ١، ﻝ٢، ﻝ٣ ﻣﺘﻮﺍﺯﻳﺔ، ﻳﻘﻄﻌﻬﺎ ﺍﻟﻤﺴﺘﻘﻴﻤﺎﻥ‬ ‫ﻡ، ﻡ/ ﻭﺍﻷﻃﻮﺍﻝ ﻣﻘﺪﺭﺓ ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮﺍﺕ ﻓﺈﻥ ﺱ ﺗﺴﺎﻭﻯ:‬ ‫ب ٣‬ ‫أ ٥‬ ‫د ٢‬ ‫ﺟ ٧‬ ‫−‬ ‫+‬ ‫5‬ ‫‪ E C‬ﻳﻨﺼﻒ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺨﺎﺭﺟﺔ‬ ‫ﻋﻨﺪ ‪ C‬ﻓﺈﻥ ﻃﻮﻝ ﺟـ ‪ E‬ﻳﺴﺎﻭﻯ:‬ ‫ب ٠١ﺳﻢ‬ ‫أ ٥ﺳﻢ‬ ‫د ٨١ﺳﻢ‬ ‫ﺟ ٢١ﺳﻢ‬ ‫‪C‬‬ ‫‪E‬‬ ‫6 ﺍﻟﺪﺍﺋﺮﺓ ﻡ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٥ﺳﻢ، ‪ E C‬ﻣﻤﺎﺱ ﻟﻠﺪﺍﺋﺮﺓ ﻋﻨﺪ ‪،E‬‬ ‫‪١٢ = E C‬ﺳﻢ ﻓﺈﻥ ﻃﻮﻝ ‪ C‬ﺟـ ﻳﺴﺎﻭﻯ:‬ ‫ب ٢١ﺳﻢ‬ ‫أ ٧ﺳﻢ‬ ‫د ٨١ﺳﻢ‬ ‫ﺟ ٥١ﺳﻢ‬ ‫7 ﺇﺫﺍ ﻛﺎﻧﺖ ﻣﺴﺎﺣﺔ ﺳﻄﺢ 9‪ E C‬ﻫـ = ٦١ﺳﻢ‬ ‫ﻓﺈﻥ ﻣﺴﺎﺣﺔ ﺳﻄﺢ ﺍﻟﻤﺜﻠﺚ ‪ C‬ﺏ ﺟـ = .......................... ﺳﻢ٢.‬ ‫أ ٦١‬ ‫ب ٢٣‬ ‫ﺟ ٤٦‬ ‫د ٨٢١‬ ‫‪E‬‬ ‫‪C‬‬ ‫٢‬ ‫‪ïM‬‬ ‫−‬ ‫‪C‬‬ ‫‪E‬‬
  • 55.
    ‫ﺍﺧﺘﺒﺎﺭ ﺗﺮﺍﻛﻤﻰ‬ ‫‪:Iô«°ü≤dG äÉHÉLE’GäGP á∏Ä°SC’G‬‬ ‫8‬ ‫:‬ ‫‪ C‬ﺏ // ﺟـ ‪ ، E‬ﺏ ﻫـ = ٢ﺳﻢ، ﺟـ ﻫـ = ٣ﺳﻢ،‬ ‫‪١٠ = E C‬ﺳﻢ. ﺃﻭﺟﺪ ﻃﻮﻝ ﻫـ ‪E‬‬ ‫‪C‬‬ ‫‪E‬‬ ‫: ﺏ ﻫـ ﻳﻨﺼﻒ ‪c‬ﺏ،‬ ‫9‬ ‫ﻭﻳﻘﻄﻊ ‪ C‬ﺟـ ﻓﻰ ﻫـ. ‪ C‬ﺏ = ٦ﺳﻢ، ﺟـ ‪٥ = E‬ﺳﻢ، ‪٧٫٥ = C E‬ﺳﻢ‬ ‫ﺏ ﺟـ = ٤ﺳﻢ . ﺃﺛﺒﺖ ﺃﻥ ‪ E‬ﻫـ ﻳﻨﺼﻒ ‪ E C c‬ﺟـ.‬ ‫‪E‬‬ ‫‪C‬‬ ‫:‬ ‫01‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪ C‬ﺏ ، ﺟـ ‪ E‬ﻭﺗﺮﺍﻥ ﻓﻰ ﺍﻟﺪﺍﺋﺮﺓ، ‪ C‬ﺏ ∩ ﺟـ ‪} = E‬ﻫـ{‬ ‫ﺃﺛﺒﺖ ﺃﻥ 9‪ C‬ﻫـ ﺟـ + 9‪ E‬ﻫـ ﺏ‬ ‫‪á∏jƒ£dG äÉHÉLE’G äGP øjQɪàdG‬‬ ‫11‬ ‫: ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻓﻴﻪ ‪ C‬ﺏ = ٢ ﺏ ﺟـ = ٢١ﺳﻢ،‬ ‫‪ C‬ﺟـ = ٩ﺳﻢ، ‪ C ∋ E‬ﺏ ﺣﻴﺚ ‪٣ = E C‬ﺳﻢ،‬ ‫ﻫـ ∋ ‪ C‬ﺟـ ﺣﻴﺚ ‪ C‬ﻫـ = ٤ﺳﻢ.‬ ‫ﺃﺛﺒﺖ ﺃﻥ 9 ‪ C‬ﺏ ﺟـ + 9 ‪ C‬ﻫـ ‪E‬‬ ‫ﺛﻢ ﺃﻭﺟﺪ ﻃﻮﻝ ﻫـ ‪. E‬‬ ‫‪C‬‬ ‫‪E‬‬ ‫21 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ، ‪ ∋ E‬ﺏ ﺟـ ، ‪ ∌ E‬ﺏ ﺟـ ، ﺭﺳﻢ ‪ E‬ﻭ ﻓﻘﻄﻊ ‪ C‬ﺟـ ، ‪ C‬ﺏ ﻓﻰ ﻫـ، ﻭ ﻋﻠﻰ ﺍﻟﺘﺮﺗﻴﺐ‬ ‫ﻭﺏ‬ ‫ﻓﺈﺫﺍ ﻛﺎﻥ ﺍﻟﺸﻜﻞ ﺏ ﺟـ ﻫـ ﻭ ﺭﺑﺎﻋﻴﺎ ﺩﺍﺋﺮ ﻳﺎ ﺃﺛﺒﺖ ﺃﻥ ﺏ ‪ = E‬ﺟـ ﻫـ .‬ ‫ًّ‬ ‫ًّ‬ ‫‪ E‬ﻫـ‬ ‫‪?á«aÉ°VEG IóYÉ°ùªd êÉàëJ πg‬‬ ‫ﺃﻥ ﻟﻢ ﺗﺴﺘﻄﻊ ﺇﺟﺎﺑﺔ ﺃﻯ ﻣﻦ ﺍﻷﺳﺌﻠﺔ ﺍﻟﺴﺎﺑﻘﺔ ﻓﺎﺭﺟﻊ ﻟﻠﺠﺪﻭﻝ ﺍﻟﺘﺎﻟﻲ:‬ ‫١‬ ‫−‬ ‫¯‬ ‫٢‬ ‫٣‬ ‫٤‬ ‫٥‬ ‫٦‬ ‫٧‬ ‫٨‬ ‫٩‬ ‫٠١‬ ‫١١‬ ‫٢١‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫−‬ ‫¯‬
  • 56.
    ‫‪IóMƒdG‬‬ ‫4‬ ‫ﺣﺴﺎﺏ ﺍﻟﻤﺜﻠﺜﺎﺕ‬ ‫‪Trigonometry‬‬ ‫دروس اﻟﻮﺣﺪة‬ ‫ﺍﻟﺪﺭﺱ)٤ - ١(: ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﻮﺟﻬﺔ.‬ ‫ﺍﻟﺪﺭﺱ )٤ - ٢(: ﻃﺮﻕ ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ.‬ ‫ﺍﻟﺪﺭﺱ )٤ - ٣(: ﺍﻟﺪﻭﺍﻝ ﺍﻟﻤﺜﻠﺜﻴﺔ.‬ ‫ﺍﻟﺪﺭﺱ )٤ - ٤(: ﺍﻟﻌﻼﻗﺎﺕ ﺑﻴﻦ ﺍﻟﺪﻭﺍﻝ ﺍﻟﻤﺜﻠﺜﻴﺔ.‬ ‫ﺍﻟﺪﺭﺱ )٤ - ٥(: ﺍﻟﺘﻤﺜﻴﻞ ﺍﻟﺒﻴﺎﻧﻰ ﻟﻠﺪﻭﺍﻝ ﺍﻟﻤﺜﻠﺜﻴﺔ.‬ ‫ﺍﻟﺪﺭﺱ )٤ - ٦(: ﺇﻳﺠﺎﺩ ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ﺑﻤﻌﻠﻮﻣﻴﺔ ﺩﺍﻟﺔ ﻣﺜﻠﺜﻴﺔ.‬
  • 57.
    ‫اﻟﺰاوﻳﺔ اﻟﻤﻮﺟﻬﺔ‬ ‫4-1‬ ‫‪Directed Angle‬‬ ‫:‬ ‫1‬ ‫أﺗﻜﻮﻥ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﻮﺟﻬﺔ ﻓﻰ ﻭﺿﻊ ﻗﻴﺎﺳﻰ ﺇﺫﺍ ﻛﺎﻥ‬ ‫ب‬ ‫ﻳﻘﺎﻝ ﻟﻠﺰﺍﻭﻳﺔ ﺍﻟﻤﻮﺟﻬﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﺃﻧﻬﺎ ﻣﺘﻜﺎﻓﺌﺔ ﺇﺫﺍ ﻛﺎﻥ .............................................................................‬ ‫.................................................................................................‬ ‫ﺟ ﺗﻜﻮﻥ ﺍﻟﺰﺍﻭﻳﺔ ﻣﻮﺟﺒﺔ ﺇﺫﺍ ﻛﺎﻥ ﺩﻭﺭﺍﻥ ﺍﻟﺰﺍﻭﻳﺔ.................................. ﻭﺗﻜﻮﻥ ﺳﺎﻟﺒﺔ ﺇﺫﺍ ﻛﺎﻥ ﺩﻭﺭﺍﻥ ﺍﻟﺰﺍﻭﻳﺔ‬ ‫د ﺇﺫﺍ ﻭﻗﻊ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻟﻠﺰﺍﻭﻳﺔ ﺍﻟﻤﻮﺟﻬﺔ ﻋﻠﻰ ﺃﺣﺪ ﻣﺤﺎﻭﺭ ﺍﻹﺣﺪﺍﺛﻴﺎﺕ ﺗﺴﻤﻰ ..................................‬ ‫ﻫ ﺇﺫﺍ ﻛﺎﻥ )‪ (i‬ﺯﺍﻭﻳﺔ ﻣﻮﺟﻬﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ، ﻥ∋ ‪ N‬ﻓﺈﻥ )‪ + i‬ﻥ * ٠٦٣‪ (c‬ﺗﺴﻤﻰ ﺑﺎﻟﺰﻭﺍﻳﺎ‬ ‫و‬ ‫ﺃﺻﻐﺮ ﻗﻴﺎﺱ ﻣﻮﺟﺐ ﻟﻠﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ ٠٣٥‪ c‬ﻫﻮ ..................................‬ ‫ز ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ ٠٣٩‪ c‬ﺗﻘﻊ ﻓﻰ ﺍﻟﺮﺑﻊ‬ ‫2 ﺃﻱ ﻣﻦ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﻤﻮﺟﻬﺔ ﺍﻵﺗﻴﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻲ ‬ ‫..................................‬ ‫......................................................................................................................‬ ‫ب‬ ‫ﺟ‬ ‫د‬ ‫3 ﺃﻭﺟﺪ ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﻮﺟﻬﺔ ‪ i‬ﺍﻟﻤﺸﺎﺭ ﺇﻟﻴﻬﺎ ﻓﻰ ﻛﻞ ﺷﻜﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ:‬ ‫ﺟ‬ ‫ب‬ ‫د‬ ‫أ‬ ‫‪i‬‬ ‫‪i‬‬ ‫‪c‬‬ ‫‪c‬‬ ‫..................................‬ ‫..................................‬ ‫ح ﺃﺻﻐﺮ ﻗﻴﺎﺱ ﻣﻮﺟﺐ ﻟﻠﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ –٠٩٦‪ c‬ﻫﻮ‬ ‫أ‬ ‫...............................‬ ‫‪c‬‬ ‫‪i‬‬ ‫..............................................................‬ ‫..............................................................‬ ‫..............................................................‬ ‫.............................................................‬ ‫4 ﻋﻴﻦ ﺍﻟﺮﺑﻊ ﺍﻟﺬﻯ ﺗﻘﻊ ﻓﻴﻪ ﻛﻞ ﻣﻦ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﺘﻰ ﻗﻴﺎﺳﺎﺗﻬﺎ ﻛﺎﻵﺗﻰ:‬ ‫أ ٤٢‪       c‬ب ٥١٢‪       c‬ﺟ - ٠٤‪       c‬د -٠٢٢‪       c‬ﻫ ٠٤٦‪c‬‬ ‫........................................  ........................................    ........................................    ........................................     ........................................‬ ‫¯‬ ‫‪i‬‬ ‫−‬ ‫¯‬
  • 58.
    ‫5 ﺿﻊ ﻛﻼﻣﻦ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻵﺗﻴﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ، ﻣﻮﺿﺤﺎ ﺫﻟﻚ ﺑﺎﻟﺮﺳﻢ:‬ ‫ًّ‬ ‫ً‬ ‫أ ٢٣‪       c‬ب ٠٤١‪       c‬ﺟ - ٠٨‪       c‬د -٠١١‪       c‬ﻫ -٥١٣‪c‬‬ ‫6 ﻋﻴﻦ ﺃﺣﺪ ﺍﻟﻘﻴﺎﺳﺎﺕ ﺍﻟﺴﺎﻟﺒﺔ ﻟﻜﻞ ﺯﺍﻭﻳﺔ ﻣﻦ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻵﺗﻴﺔ:‬ ‫ﺟ ٠٩‪c‬‬ ‫ب ٦٣١‪c‬‬ ‫أ ٣٨‪c‬‬ ‫........................................‬ ‫د ٤٦٢‪c‬‬ ‫........................................‬ ‫........................................‬ ‫و ٠٧٠١‪c‬‬ ‫ﻫ ٤٦٩‪c‬‬ ‫........................................‬ ‫........................................‬ ‫........................................‬ ‫7 ﻋﻴﻦ ﺃﺻﻐﺮ ﻗﻴﺎﺱ ﻣﻮﺟﺐ ﻟﻜﻞ ﺯﺍﻭﻳﺔ ﻣﻦ ﺍﻟﺰﺍﻭﻳﺎ ﺍﻵﺗﻴﺔ:‬ ‫ﺟ -٥١٣‪c‬‬ ‫ب -٧١٢‪c‬‬ ‫أ -٣٨١‪c‬‬ ‫8 ﺃﻯ ﻣﻦ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﻤﻮﺟﻬﺔ ﻓﻰ ﺍﻷﺯﻭﺍﺝ ﺍﻟﻤﺮﺗﺒﺔ ﺍﻵﺗﻴﺔ ﻓﻰ ﺍﻟﺸﻜﻞ‬ ‫ﺍﻟﻤﻘﺎﺑﻞ ﻓﻰ ﻭﺿﻊ ﻗﻴﺎﺳﻰ? ﻟﻤﺎﺫﺍ?‬ ‫أ ) ﻭ‪ ، C‬ﻭ ‪( E‬‬ ‫ب ) ﻭ ﺯ ، ﻭ ﺟـ (‬ ‫ﺟ ) ‪C‬ﺏ ، ‪ C‬ﺟـ (‬ ‫‪E‬‬ ‫د ) ﻭ ﻫـ ، ﻭ ‪( E‬‬ ‫ﻫ ) ﻭ‪ ، E‬ﻭ ﺯ (‬ ‫د -٠٧٥‪c‬‬ ‫و ) ﻭﺏ ، ﻭﺯ (‬ ‫‪C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫9 ﻳﺪﻭﺭ ﺃﺣﺪ ﻻﻋﺒﻰ ﺍﻟﺠﻤﺒﺎﺯ ﻋﻠﻰ ﺟﻬﺎﺯ ﺍﻷﻟﻌﺎﺏ ﺑﺰﺍﻭﻳﺔ ﻗﻴﺎﺳﻬﺎ ٠٠٢‪ c‬ﺍﺭﺳﻢ ﻫﺬه ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ‬ ‫ﻛﺎﻥ ﻣﻊ‬ ‫01 ﺍﻛﺘﺸﻒ ﺍﻟﺨﻄﺄ: ﺍﻛﺘﺐ ﻗﻴﺎﺱ ﺃﺻﻐﺮ ﺯﺍﻭﻳﺔ ﺑﻘﻴﺎﺱ ﻣﻮﺟﺐ ﻭﺯﺍﻭﻳﺔ ﺃﺧﺮﻯ ﺑﻘﻴﺎﺱ ﺳﺎﻟﺐ ﺗﺸﺘﺮ‬ ‫ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻟﻠﺰﺍﻭﻳﺔ )-٥٣١‪(c‬‬ ‫¯‬ ‫ﺃﺻﻐﺮ ﺯﺍﻭﻳﺔ ﺑﻘﻴﺎﺱ ﻣﻮﺟﺐ = -٥٣١‪ c٤٥ = c١٨٠+ c‬ﺃﺻﻐﺮ ﺯﺍﻭﻳﺔ ﺑﻘﻴﺎﺱ ﻣﻮﺟﺐ = -٥٣١‪c٢٢٥ = c٣٦٠+ c‬‬ ‫ﺃﺻﻐﺮ ﺯﺍﻭﻳﺔ ﺑﻘﻴﺎﺱ ﺳﺎﻟﺐ = -٥٣١‪ c٣١٥- = c١٨٠ - c‬ﺃﺻﻐﺮ ﺯﺍﻭﻳﺔ ﺑﻘﻴﺎﺱ ﺳﺎﻟﺐ = -٥٣١‪c٤٩٥- = c٣٦٠ - c‬‬ ‫ﺃﻯ ﺍﻹﺟﺎﺑﺘﻴﻦ ﺻﺤﻴﺢ ? ﻓﺴﺮ ﺇﺟﺎﺑﺘﻚ.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
  • 59.
    ‫ﻃﺮق ﻗﻴﺎس اﻟﺰاوﻳﺔ‬ ‫4-2‬ ‫‪Methodsof measuring the angle‬‬ ‫‪k‬‬ ‫‪:Oó©àe øe QÉ«àNG :’hCG‬‬ ‫1 ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ ٠٦‪ c‬ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﺗﻜﺎﻓﺊ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ:‬ ‫ب ٠٤٢‪c‬‬ ‫أ ٠٢١‪c‬‬ ‫ﺟ ٠٠٣‪c‬‬ ‫١٣‬ ‫2 ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ ٦‪ r‬ﺗﻘﻊ ﻓﻰ ﺍﻟﺮﺑﻊ:‬ ‫٩‬‫3 ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ ٤‪ r‬ﺗﻘﻊ ﻓﻰ ﺍﻟﺮﺑﻊ:‬ ‫ﺟ ﺍﻟﺜﺎﻟﺚ‬ ‫د ﺍﻟﺮﺍﺑﻊ‬ ‫.............................................................................................................................................‬ ‫ب ﺍﻟﺜﺎﻧﻰ‬ ‫أ ﺍﻷﻭﻝ‬ ‫د ٠٢٤‪c‬‬ ‫....................................................................................................................................‬ ‫ب ﺍﻟﺜﺎﻧﻰ‬ ‫أ ﺍﻷﻭﻝ‬ ‫........................................................................‬ ‫ﺟ ﺍﻟﺜﺎﻟﺚ‬ ‫د ﺍﻟﺮﺍﺑﻊ‬ ‫4 ﺇﺫﺍ ﻛﺎﻥ ﻣﺠﻤﻮﻉ ﻗﻴﺎﺳﺎﺕ ﺯﻭﺍﻳﺎ ﺃﻯ ﻣﻀﻠﻊ ﻣﻨﺘﻈﻢ ﺗﺴﺎﻭﻯ ٠٨١ ْ)ﻥ – ٢( ﺣﻴﺚ ﻥ ﻋﺪﺩ ﺍﻷﺿﻼﻉ، ﻓﺈﻥ ﻗﻴﺎﺱ‬ ‫ﺯﺍﻭﻳﺔ ﺍﻟﻤﺨﻤﺲ ﺍﻟﻤﻨﺘﻈﻢ ﺑﺎﻟﻘﻴﺎﺱ ﺍﻟﺪﺍﺋﺮﻯ ﺗﺴﺎﻭﻯ: .....................................................................................................................‬ ‫أ‬ ‫ب ٧‪r‬‬ ‫٢‬ ‫‪r‬‬ ‫٣‬ ‫ﺟ‬ ‫5 ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ ٧‪ r‬ﻗﻴﺎﺳﻬﺎ ﺍﻟﺴﺘﻴﻨﻰ ﻳﺴﺎﻭﻯ:‬ ‫٣‬ ‫ب ٠١٢‪c‬‬ ‫أ ٥٠١‪c‬‬ ‫٣‪r‬‬ ‫٥‬ ‫........................................................................................................................‬ ‫ﺟ ٠٢٤‪c‬‬ ‫6 ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻘﻴﺎﺱ ﺍﻟﺴﺘﻴﻨﻰ ﻟﺰﺍﻭﻳﺔ ﻫﻮ ٨٤ َ ٤٦ ْ ﻓﺈﻥ ﻗﻴﺎﺳﻬﺎ ﺍﻟﺪﺍﺋﺮﻯ ﻳﺴﺎﻭﻯ:‬ ‫أ ٨١٫٠‬ ‫ب ٦٣٫٠‬ ‫‪E‬‬ ‫‪E‬‬ ‫د‬ ‫٢‪r‬‬ ‫٣‬ ‫ﺟ ٨١٫٠ ‪r‬‬ ‫د ٠٤٨‪c‬‬ ‫.......................................................................‬ ‫د ٦٣٫٠ ‪r‬‬ ‫7 ﻃﻮﻝ ﺍﻟﻘﻮﺱ ﻓﻰ ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻗﻄﺮﻫﺎ ٤٢ ﺳﻢ ﻭﻳﻘﺎﺑﻞ ﺯﺍﻭﻳﺔ ﻛﺰﻳﺔ ﻗﻴﺎﺳﻬﺎ ٠٣‪ c‬ﻳﺴﺎﻭﻯ:‬ ‫ﻣﺮ‬ ‫د ٥‪ r‬ﺳﻢ‬ ‫ﺟ ٤‪ r‬ﺳﻢ‬ ‫ب ٣‪ r‬ﺳﻢ‬ ‫أ ٢‪ r‬ﺳﻢ‬ ‫............................................‬ ‫8 ﺍﻟﻘﻮﺱ ﺍﻟﺬﻯ ﻃﻮﻟﻪ ٥‪r‬ﺳﻢ ﻓﻰ ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٥١ﺳﻢ ﻳﻘﺎﺑﻞ ﺯﺍﻭﻳﺔ ﻛﺰﻳﺔ ﻗﻴﺎﺳﻬﺎ ﻳﺴﺎﻭﻯ:‬ ‫ﻣﺮ‬ ‫ب ٠٦‪c‬‬ ‫أ ٠٣‪c‬‬ ‫ﺟ ٠٩‪c‬‬ ‫.................‬ ‫د ٠٨١‪c‬‬ ‫9 ﺇﺫﺍ ﻛﺎﻥ ﻗﻴﺎﺱ ﺇﺣﺪﻯ ﺯﺍﻭﻳﺎ ﻣﺜﻠﺚ ٥٧‪ c‬ﻭﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ﺃﺧﺮﻯ ﻓﻴﻪ ‪ r‬ﻓﺈﻥ ﺍﻟﻘﻴﺎﺱ ﺍﻟﺪﺍﺋﺮﻯ ﻟﻠﺰﺍﻭﻳﺔ ﺍﻟﺜﺎﻟﺜﺔ‬ ‫٤‬ ‫ﻳﺴﺎﻭﻯ: .................................................................................................................................................................................................................‬ ‫د ٥‪r‬‬ ‫ﺟ ‪r‬‬ ‫ب ‪r‬‬ ‫أ ‪r‬‬ ‫٣‬ ‫٤‬ ‫٦‬ ‫٢١‬ ‫¯‬ ‫−‬ ‫¯‬
  • 60.
    ‫‪¯I‬‬ ‫‪:á«JB’G á∏Ä°SC’G øYÖLCG :Ék«fÉK‬‬ ‫01 ﺃﻭﺟﺪ ﺑﺪﻻﻟﺔ ‪ r‬ﺍﻟﻘﻴﺎﺱ ﺍﻟﺪﺍﺋﺮﻯ ﻟﻠﺰﻭﺍﻳﺎ ﺍﻟﺘﻰ ﻗﻴﺎﺳﺎﺗﻬﺎ ﻛﺎﻵﺗﻰ:‬ ‫.........................................‬ ‫ب  ٠٤٢‪c‬‬ ‫.........................................‬ ‫أ ٥٢٢‪c‬‬ ‫ﺟ‬ ‫.........................................‬ ‫د  ٠٠٣‪c‬‬ ‫٥٣١‪......................................... c‬‬‫.........................................‬ ‫و  ٠٨٧‪c‬‬ ‫.........................................‬ ‫ﻫ ٠٩٣‪c‬‬ ‫11 ﺃﻭﺟﺪ ﺑﺎﻟﺮﺍﺩﻳﺎﻥ ﺍﻟﻘﻴﺎﺱ ﺍﻟﺪﺍﺋﺮﻯ ﻟﻠﺰﻭﺍﻳﺎ ﺍﻟﺘﻰ ﻗﻴﺎﺳﺎﺗﻬﺎ ﻛﺎﻵﺗﻰ، ﻣﻘﺮﺑﺎ ﺍﻟﻨﺎﺗﺞ ﻟﺜﻼﺛﺔ ﺃﺭﻗﺎﻡ ﻋﺸﺮﻳﺔ:‬ ‫ً‬ ‫ﺟ‬ ‫٨٤ ً ٠٥ َ ٠٦١‪c‬‬ ‫ب ٨١ َ ٥٢‪c‬‬ ‫أ ٦٫٦٥‪c‬‬ ‫.................................................‬ ‫.................................................‬ ‫21 ﺃﻭﺟﺪ ﺍﻟﻘﻴﺎﺱ ﺍﻟﺴﺘﻴﻨﻰ ﻟﻠﺰﻭﺍﻳﺎ ﺍﻟﺘﻰ ﻗﻴﺎﺳﺎﺗﻬﺎ ﻛﺎﻵﺗﻰ، ﻣﻘﺮﺑﺎ ﺍﻟﻨﺎﺗﺞ ﻷﻗﺮﺏ ﺛﺎﻧﻴﺔ:‬ ‫ً‬ ‫‪E‬‬ ‫‪E‬‬ ‫ب ٧٢٫٢‬ ‫أ ٩٤٫٠‬ ‫.................................................‬ ‫.................................................‬ ‫.................................................‬ ‫١ ‪E‬‬ ‫ﺟ -٢٣‬ ‫.................................................‬ ‫31 ﺇﺫﺍ ﻛﺎﻧﺖ ‪ i‬ﺯﺍﻭﻳﺔ ﻛﺰﻳﺔ ﻓﻰ ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ‪ H‬ﻭﺗﺤﺼﺮ ﻗﻮﺳﺎ ﻃﻮﻟﻪ ﻝ :‬ ‫ﻣﺮ‬ ‫ً‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ ‪ ٢٠ = H‬ﺳﻢ، ‪ c٧٨ َ ١٥ ً ٢٠ = i‬ﺃﻭﺟﺪ ﻝ. .......................................................... )ﻷﻗﺮﺏ ﺟﺰﺀ ﻣﻦ ﻋﺸﺮﺓ(‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ﻝ = ٣٫٧٢ ﺳﻢ، ‪ c٧٨ َ ٠ ً ٢٤ = i‬ﺃﻭﺟﺪ ‪) ......................................................... .H‬ﻷﻗﺮﺏ ﺟﺰﺀ ﻣﻦ ﻋﺸﺮﺓ(‬ ‫41 ﺯﺍﻭﻳﺔ ﻛﺰﻳﺔ ﻗﻴﺎﺳﻬﺎ ٠٥١‪ c‬ﻭﺗﺤﺼﺮ ﻗﻮﺳﺎ ﻃﻮﻟﻪ ١١ ﺳﻢ، ﺍﺣﺴﺐ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮ ﺩﺍﺋﺮﺗﻬﺎ )ﻷﻗﺮﺏ ﺟﺰﺀ ﻣﻦ ﻋﺸﺮﺓ(‬ ‫ﻣﺮ‬ ‫ً‬ ‫..................................................................................................................................................................................................................................‬ ‫51 ﺃﻭﺟﺪ ﺍﻟﻘﻴﺎﺱ ﺍﻟﺪﺍﺋﺮﻯ ﻭﺍﻟﻘﻴﺎﺱ ﺍﻟﺴﺘﻴﻨﻰ ﻟﻠﺰﺍﻭﻳﺔ ﻛﺰﻳﺔ ﺍﻟﺘﻰ ﺗﻘﺎﺑﻞ ﻗﻮﺳﺎ ﻃﻮﻟﻪ ٧٫٨ ﺳﻢ ﻓﻰ ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻧﺼﻒ‬ ‫ﺍﻟﻤﺮ‬ ‫ً‬ ‫ﻗﻄﺮﻫﺎ ٤ ﺳﻢ. ................................................................................................................................................................................................................‬ ‫61 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻬﻨﺪﺳﺔ: ﻣﺜﻠﺚ ﻗﻴﺎﺱ ﺇﺣﺪﻯ ﺯﻭﺍﻳﺎه ٠٦‪ c‬ﻭﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ﺃﺧﺮﻯ ﻣﻨﻪ ﻳﺴﺎﻭﻯ ‪ r‬ﺃﻭﺟﺪ ﺍﻟﻘﻴﺎﺱ‬ ‫٤‬ ‫ﺍﻟﺪﺍﺋﺮﻯ ﻭﺍﻟﻘﻴﺎﺱ ﺍﻟﺴﺘﻴﻨﻰ ﻟﺰﺍﻭﻳﺘﻪ ﺍﻟﺜﺎﻟﺜﺔ. .........................................................................................................................................‬ ‫71 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻬﻨﺪﺳﺔ: ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٤ ﺳﻢ، ﺭﺳﻤﺖ ‪ Cc‬ﺏ ﺟـ ﺍﻟﻤﺤﻴﻄﻴﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ ٠٣‪ c‬ﺃﻭﺟﺪ‬ ‫ﻃﻮﻝ ﺍﻟﻘﻮﺱ ﺍﻷﺻﻐﺮ ‪ C‬ﺟـ .......................................................................................................................................................................‬ ‫81 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻬﻨﺪﺳﻴﺔ: ﻓﻰ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ ﺇﺫﺍ ﻛﺎﻥ ﻣﺴﺎﺣﺔ ﺍﻟﻤﺜﻠﺚ ﻡ ‪ C‬ﺏ‬ ‫ﺍﻟﻘﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ﻡ = ٢٣ ﺳﻢ٢ ﻓﺄﻭﺟﺪ ﻣﺤﻴﻂ ﺍﻟﺸﻜﻞ ﻣﻘﺮﺑﺎ ﺍﻟﻨﺎﺗﺞ ﻷﻗﺮﺏ‬ ‫ً‬ ‫ﺭﻗﻤﻴﻦ ﻋﺸﺮﻳﻴﻦ ............................................................................................................................‬ ‫‪C‬‬ ‫‪ïM‬‬ ‫−‬
  • 61.
    ‫91 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻬﻨﺪﺳﺔ:‪ C‬ﺏ ﻗﻄﺮ ﻓﻰ ﺩﺍﺋﺮﺓ ﻃﻮﻟﻪ ٤٢ ﺳﻢ ، ﺭﺳﻢ ﺍﻟﻮﺗﺮ ‪ C‬ﺟـ ﺑﺤﻴﺚ ﻛﺎﻥ ﻕ)‪c‬ﺏ ‪C‬ﺟـ( = ٠٥‪c‬‬ ‫ﺃﻭﺟﺪ ﻃﻮﻝ ﺍﻟﻘﻮﺱ ﺍﻷﺻﻐﺮ ‪ C‬ﺟـ ﻣﻘﺮ ﺑﺎ ﺍﻟﻨﺎﺗﺞ ﻷﻗﺮﺏ ﺭﻗﻤﻴﻴﻦ ﻋﺸﺮﻳﻴﻦ. ..........................................................................‬ ‫ً‬ ‫02 ﻣﺴﺎﻓﺎﺕ: ﻛﻢ ﺍﻟﻤﺴﺎﻓﺔ ﺍﻟﺘﻰ ﺗﻘﻄﻌﻬﺎ ﻧﻘﻄﺔ ﻋﻠﻰ ﻃﺮﻑ ﻋﻘﺮﺏ ﺍﻟﺪﻗﺎﺋﻖ ﺧﻼﻝ ٠١ ﺩﻗﺎﺋﻖ ﺇﺫﺍ ﻛﺎﻥ ﻃﻮﻝ ﻫﺬﺍ‬ ‫ﺍﻟﻌﻘﺮﺏ ٦ ﺳﻢ?‬ ‫..................................................................................................................................................................................................................................‬ ‫12 ﻓﻠﻚ: ﻗﻤﺮ ﺻﻨﺎﻋﻰ ﻳﺪﻭﺭ ﺣﻮﻝ ﺍﻷﺭﺽ ﻓﻰ ﻣﺴﺎﺭ ﺩﺍﺋﺮﻯ ﺩﻭﺭﺓ ﻛﺎﻣﻠﺔ ﻛﻞ ٦ ﺳﺎﻋﺎﺕ، ﻓﺈﺫﺍ ﻛﺎﻥ ﻃﻮﻝ ﻧﺼﻒ‬ ‫ﻗﻄﺮ ﻣﺴﺎﺭه ﻋﻦ ﻛﺰ ﺍﻷﺭﺽ ٠٠٠٩ ﻛﻢ، ﻓﺄﻭﺟﺪ ﺳﺮﻋﺘﻪ ﺑﺎﻟﻜﻴﻠﻮﻣﺘﺮ ﻓﻰ ﺍﻟﺴﺎﻋﺔ.‬ ‫ﻣﺮ‬ ‫..................................................................................................................................................................................................................................‬ ‫22 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻬﻨﺪﺳﺔ: ﻓﻰ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ:‬ ‫‪ C‬ﺏ ، ‪ C‬ﺟـ ﻣﻤﺎﺳﺎﻥ ﻟﻠﺪﺍﺋﺮﺓ ﻡ، ‪ c) X‬ﺟـ‪ C‬ﺏ ( = ٠٦‪ C ،c‬ﺏ = ٢١ ﺳﻢ.‬ ‫ﺃﻭﺟﺪ ﻷﻗﺮﺏ ﻋﺪﺩ ﺻﺤﻴﺢ ﻃﻮﻝ ﺍﻟﻘﻮﺱ ﺍﻷﻛﺒﺮ ﺏ ﺟـ .‬ ‫‪C‬‬ ‫‪c‬‬ ‫.....................................................................................................................................................‬ ‫32 ﺍﻟﺮﺑﻂ ﺑﺎﻟﺰﻣﻦ: ﺗﺴﺘﺨﺪﻡ ﺍﻟﻤﺰﻭﻟﺔ ﺍﻟﺸﻤﺴﻴﺔ ﻟﺘﺤﺪﻳﺪ ﺍﻟﻮﻗﺖ ﺃﺛﻨﺎﺀ ﺍﻟﻨﻬﺎﺭ ﻣﻦ‬ ‫ﺧﻼﻝ ﻃﻮﻝ ﺍﻟﻈﻞ ﺍﻟﺬﻯ ﻳﺴﻘﻂ ﻋﻠﻰ ﺳﻄﺢ ﻣﺪﺭﺝ ﻹﻇﻬﺎﺭ ﺍﻟﺴﺎﻋﺔ ﻭﺃﺟﺰﺍﺋﻬﺎ،‬ ‫ﻓﺈﺫﺍ ﻛﺎﻥ ﺍﻟﻈﻞ ﻳﺪﻭﺭ ﻋﻠﻰ ﺍﻟﻘﺮﺹ ﺑﻤﻌﺪﻝ ٥١‪ c‬ﻟﻜﻞ ﺳﺎﻋﺔ.‬ ‫أ ﺃﻭﺟﺪ ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺑﺎﻟﺮﺍﺩﻳﺎﻥ ﺍﻟﺘﻰ ﻳﺪﻭﺭ ﺍﻟﻈﻞ ﻋﻨﻬﺎ ﺑﻌﺪ ﻣﺮﻭﺭ ٤ ﺳﺎﻋﺎﺕ.‬ ‫......................................................................................................................................................................‬ ‫ب ﺑﻌﺪ ﻛﻢ ﺳﺎﻋﺔ ﻳﺪﻭﺭ ﺍﻟﻈﻞ ﺑﺰﺍﻭﻳﺔ ﻗﻴﺎﺳﻬﺎ ٢‪ r‬ﺭﺍﺩﻳﺎﻥ?‬ ‫٣‬ ‫....................................‬ ‫ﺟ ﻣﺰﻭﻟﺔ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٤٢ ﺳﻢ، ﺃﻭﺟﺪ ﺑﺪﻻﻟﺔ ‪ r‬ﻃﻮﻝ ﺍﻟﻘﻮﺱ ﺍﻟﺬﻯ ﻳﺼﻨﻌﻪ ﺩﻭﺭﺍﻥ ﺍﻟﻈﻞ ﻋﻠﻰ ﺣﺎﻓﺔ‬ ‫ﺍﻟﻘﺮﺹ ﺑﻌﺪ ﻣﺮﻭﺭ ٠١ ﺳﺎﻋﺎﺕ. ......................................................................................................................................................‬ ‫42 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ: ﻣﺴﺘﻘﻴﻢ ﻳﺼﻨﻊ ﺯﺍﻭﻳﺔ ﻗﻴﺎﺳﻬﺎ ‪ r‬ﺭﺍﺩﻳﺎﻥ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﻟﺪﺍﺋﺮﺓ ﺍﻟﻮﺣﺪﺓ ﻣﻊ ﺍﻻﺗﺠﺎه ﺍﻟﻤﻮﺟﺐ‬ ‫٣‬ ‫ﻟﻤﺤﻮﺭ ﺍﻟﺴﻴﻨﺎﺕ. ﺃﻭﺟﺪ ﻣﻌﺎﺩﻟﺔ ﻫﺬﺍ ﺍﻟﻤﺴﺘﻘﻴﻢ. .................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
  • 62.
    ‫اﻟﺪوال اﻟﻤﺜﻠﺜﻴﺔ‬ ‫4-3‬ ‫‪Trigonometric Functions‬‬ ‫‪k‬‬ ‫‪:Oó©àeøe QÉ«àN’G :’hCG‬‬ ‫٣‬ ‫1 ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺔ ‪ i‬ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﻟﺪﺍﺋﺮﺓ ﺍﻟﻮﺣﺪﺓ ﻳﻤﺮ ﺿﻠﻌﻬﺎ ﺍﻟﻨﻬﺎﺋﻰ ﺑﺎﻟﻨﻘﻄﺔ ) ١ ، ٢ (‬ ‫٢‬ ‫ﻓﺈﻥ ﺟﺎ ‪ i‬ﺗﺴﺎﻭﻯ: ...........................................................................................................................................................................................‬ ‫ﺟ ٣‬ ‫أ ١‬ ‫د ٢‬ ‫ب ١‬ ‫٢‬ ‫٢‬ ‫٣‬ ‫٣‬ ‫2 ﺇﺫﺍ ﻛﺎﻧﺖ ﺟﺎ ‪ ١ = i‬ﺣﻴﺚ ‪ i‬ﺯﺍﻭﻳﺔﺣﺎﺩﺓ ﻓﺈﻥ ‪ (ic)X‬ﺗﺴﺎﻭﻯ‬ ‫٢‬ ‫ﺟ ٠٦‪c‬‬ ‫ب ٥٤‪c‬‬ ‫أ ٠٣‪c‬‬ ‫3 ﺇﺫﺍ ﻛﺎﻧﺖ ﺟﺎ ‪ ،١ - = i‬ﺟﺘﺎ ‪ ٠ = i‬ﻓﺈﻥ ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ‪ i‬ﺗﺴﺎﻭﻯ‬ ‫ﺟ ٣‪r‬‬ ‫أ ‪r‬‬ ‫ب ‪r‬‬ ‫...........................................................................................‬ ‫د ٠٩‪c‬‬ ‫..............................................................................................‬ ‫د ٢‪r‬‬ ‫٢‬ ‫٢‬ ‫4 ﺇﺫﺍ ﻛﺎﻧﺖ ﻗﺘﺎ ‪ ٢ = i‬ﺣﻴﺚ ‪ i‬ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ ﻓﺈﻥ ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ‪ i‬ﺗﺴﺎﻭﻯ‬ ‫ﺟ ٥٤‪c‬‬ ‫ب ٠٣‪c‬‬ ‫أ ٥١‪c‬‬ ‫٣‬ ‫5 ﺇﺫﺍ ﻛﺎﻧﺖ ﺟﺘﺎ ‪ ، ١ = i‬ﺟﺎ ‪ ٢ - = i‬ﻓﺈﻥ ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ‪ i‬ﺗﺴﺎﻭﻯ‬ ‫٢‬ ‫ﺟ ٥‪r‬‬ ‫ب ٥‪r‬‬ ‫٢‪r‬‬ ‫أ‬ ‫٦‬ ‫٣‬ ‫....................................................................‬ ‫.........................................................................................‬ ‫٣‬ ‫6 ﺇﺫﺍ ﻛﺎﻧﺖ ﻇﺎ ‪ ١ = i‬ﺣﻴﺚ ‪ i‬ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ ﻣﻮﺟﺒﺔ ﻓﺈﻥ ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ‪ i‬ﺗﺴﺎﻭﻯ‬ ‫ﺟ ٥٤‪c‬‬ ‫ب ٠٣‪c‬‬ ‫أ ٠١‪c‬‬ ‫7 ﻇﺎ ٥٤‪ + c‬ﻇﺘﺎ ٥٤‪ - c‬ﻗﺎ ٠٦‪ c‬ﺗﺴﺎﻭﻯ‬ ‫ب ١‬ ‫أ ﺻﻔﺮﺍ‬ ‫ً‬ ‫٢‬ ‫د‬ ‫١١‪r‬‬ ‫٦‬ ‫..................................................................‬ ‫د ٠٦‪c‬‬ ‫........................................................................................................................................................‬ ‫ﺟ‬ ‫٣‬ ‫د ١‬ ‫٢‬ ‫٣‬ ‫8 ﺇﺫﺍ ﻛﺎﻧﺖ ﺟﺘﺎ ‪ ٢ = i‬ﺣﻴﺚ ‪ i‬ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ ﻓﺈﻥ ﺟﺎ ‪ i‬ﺗﺴﺎﻭﻯ‬ ‫أ ١‬ ‫ﺟ ٢‬ ‫ب ١‬ ‫٢‬ ‫د ٠٦‪c‬‬ ‫٣‬ ‫.............................................................................................‬ ‫٣‬ ‫د‬ ‫٣‬ ‫٢‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫9 ﺃﻭﺟﺪ ﺟﻤﻴﻊ ﺍﻟﺪﻭﺍﻝ ﺍﻟﻤﺜﻠﺜﻴﺔ ﻟﻠﺰﺍﻭﻳﺔ ‪ i‬ﺍﻟﻤﺮﺳﻮﻣﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ، ﻭﺍﻟﺘﻰ ﻳﻤﺮ ﺿﻠﻌﻬﺎ ﺍﻟﻨﻬﺎﺋﻰ ﺑﺎﻟﻨﻘﺎﻁ ﺍﻵﺗﻴﺔ.‬ ‫٥‬ ‫أ )٢، ٣ (‬ ‫٣‬ ‫٣‬ ‫ﺟ )- ٢ ، ١ (‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫ب ) ٢ ،- ٢ (‬ ‫د )- ٣ ، - ٤ (‬ ‫٥ ٥‬ ‫..................................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
  • 63.
    ‫01 ﺇﺫﺍ ﻛﺎﻥ‪ i‬ﻫﻮ ﻗﻴﺎﺱ ﺯﺍﻭﻳﻪ ﻣﻮﺟﻬﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ، ﻭﺍﻟﺘﻰ ﻳﻤﺮ ﺿﻠﻌﻬﺎ ﺍﻟﻨﻬﺎﺋﻰ ﺑﺪﺍﺋﺮﺓ ﺍﻟﻮﺣﺪﺓ ﻓﺄﻭﺟﺪ‬ ‫ﺟﻤﻴﻊ ﺍﻟﺪﻭﺍﻝ ﺍﻟﻤﺜﻠﺜﻴﺔ ﻟﻠﺰﺍﻭﻳﺔ ‪ i‬ﻓﻰ ﺍﻟﺤﺎﻻﺕ ﺍﻵﺗﻴﺔ:‬ ‫ﺣﻴﺚ ‪٠ < C‬‬ ‫أ )٣ ‪(C٤ - ،C‬‬ ‫ب ) ٣ ‪(C٢- ،C‬‬ ‫٣‪r‬‬ ‫ﺣﻴﺚ ٢ > ‪r٢ > i‬‬ ‫٢‬ ‫.................................................................................................................‬ ‫.................................................................................................................‬ ‫11 ﺍﻛﺘﺐ ﺇﺷﺎﺭﺍﺕ ﺍﻟﺪﻭﺍﻝ ﺍﻟﻤﺜﻠﺜﻴﺔ ﺍﻵﺗﻴﺔ:‬ ‫ب ﻇﺎ ٥٦٣‪c‬‬ ‫أ ﺟﺎ ٠٤٢‪c‬‬ ‫....................................‬ ‫ﺟ ﻗﺘﺎ ٠١٤‪c‬‬ ‫....................................‬ ‫٩‪r‬‬ ‫....................................‬ ‫ﻫ ﻗﺎ - ٩‪r‬‬ ‫٤‬ ‫د‬ ‫ﻇﺘﺎ ٤‬ ‫....................................‬ ‫و ﻇﺎ‬ ‫....................................‬ ‫-٠٢‪r‬‬ ‫٩‬ ‫...................................‬ ‫:‬ ‫21‬ ‫‪r‬‬ ‫٣‪r‬‬ ‫‪r‬‬ ‫أ ﺟﺘﺎ ٢ * ﺟﺘﺎ ٠ + ﺟﺎ ٢ * ﺟﺎ ٢‬ ‫ب ﻇﺎ٢ ٠٣‪ ٢ + c‬ﺟﺎ٢ ٥٤‪ + c‬ﺟﺘﺎ٢ ٠٩‪c‬‬ ‫......................................................................................................‬ ‫......................................................................................................‬ ‫31 ﺍﻟﺮﺑﻂ ﺑﺎﻟﻔﻴ ﺎﺀ: ﻋﻨﺪ ﺳﻘﻮﻁ ﺃﺷﻌﺔ ﺍﻟﻀﻮﺀ ﻋﻠﻰ ﺳﻄﺢ‬ ‫ﺷﺒﻪ ﺷﻔﺎﻑ، ﻓﺈﻧﻬﺎ ﺗﻨﻌﻜﺲ ﺑﻨﻔﺲ ﺯﺍﻭﻳﺔ ﺍﻟﺴﻘﻮﻁ ﻭﻟﻜﻦ‬ ‫ﺍﻟﺒﻌﺾ ﻣﻨﻬﺎ ﻳﻨﻜﺴﺮ ﻋﻨﺪ ﻣﺮﻭﺭه ﺧﻼﻝ ﻫﺬﺍ ﺍﻟﺴﻄﺢ. ﻛﻤﺎ‬ ‫ﻓﻰ ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ:‬ ‫ﺇﺫﺍ ﻛﺎﻥ ﺟﺎ ‪ = ١i‬ﻙ ﺟﺎ‪،٢i‬ﻛﺎﻧﺖ ﻙ = ٣ ، ‪c٦٠ = ١i‬‬ ‫ﻓﺄﻭﺟﺪ ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ‪................................................................................ . i‬‬ ‫٢‬ ‫‪U‬‬ ‫‪i‬‬ ‫‪F‬‬ ‫‪i‬‬ ‫‪i‬‬ ‫41 ﺍﻛﺘﺸﻒ ﺍﻟﺨﻄﺄ: ﻃﻠﺐ ﺍﻟﻤﻌﻠﻢ ﻣﻦ ﻃﻼﺏ ﺍﻟﻔﺼﻞ ﺇﻳﺠﺎﺩ ﻧﺎﺗﺞ ٢ ﺟﺎ ٥٤‪.c‬‬ ‫¯‬ ‫٢ ﺟﺎ ٥٤‪ = c‬ﺟﺎ ٢ * ٥٤‪c‬‬ ‫  = ﺟﺎ ٠٩‪١ = c‬‬ ‫ﺃﻯ ﺍﻹﺟﺎﺑﺘﻴﻦ ﺻﺤﻴﺢ?ﻭﻟﻤﺎﺫﺍ?‬ ‫٢ ﺟﺎ ٥٤ = ٢ * ١ = ٢ *‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫         = ٢‬ ‫......................................................................................................................................................................‬ ‫51 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ: ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺔ ‪ i‬ﻣﺮﺳﻮﻣﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ، ﺣﻴﺚ ﻇﺘﺎ ‪ ،١ - = i‬ﻗﺘﺎ ‪ . ٢ = i‬ﻫﻞ ﻣﻦ‬ ‫٣‪r‬‬ ‫? ﻓﺴﺮ ﺇﺟﺎﺑﺘﻚ. .......................................................................................................................‬ ‫ﺍﻟﻤﻤﻜﻦ ﺃﻥ ﻳﻜﻮﻥ ﻕ)‪٤ = (ic‬‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
  • 64.
    ‫اﻟﻌﻼﻗﺎت ﺑﻴﻦ اﻟﺪوالاﻟﻤﺜﻠﺜﻴﺔ‬ ‫4-4‬ ‫‪Relations between trigonometric functions‬‬ ‫‪k‬‬ ‫‪:≈JCÉjÉe πªcCG :’hCG‬‬ ‫2 ﻇﺎ ) ٠٨١‪= (i - c‬‬ ‫...............................‬ ‫1 ﺟﺘﺎ )٠٨١ ‪= (i +c‬‬ ‫4 ﺟﺎ )٠٦٣‪= ( i + c‬‬ ‫...............................‬ ‫5 ﺟﺎ )٠٩ ‪= (i +c‬‬ ‫...............................‬ ‫6 ﻇﺘﺎ ) ٠٩‪= (i - c‬‬ ‫...............................‬ ‫7 ﻗﺎ ) ٠٧٢‪= (i+ c‬‬ ‫...............................‬ ‫8 ﺟﺘﺎ )٠٧٢‪= (i - c‬‬ ‫3 ﻗﺘﺎ )٠٦٣‪= (i - c‬‬ ‫...............................‬ ‫...............................‬ ‫...............................‬ ‫‪v‬‬ ‫‪IOÉM ájhGR ¢SÉ«≤H ≈JCÉj ɪe Óc πªcCG :Ék«fÉK‬‬ ‫9 ﺟﺎ ٥٢‪ = c‬ﺟﺘﺎ ...............................‪c‬‬ ‫01 ﺟﺘﺎ ٧٦‪ = c‬ﺟﺎ ...............................‪c‬‬ ‫11 ﻇﺎ ٢٤‪ = c‬ﻇﺘﺎ ...............................‪c‬‬ ‫21 ﻗﺘﺎ ٣١‪ = c‬ﻗﺎ ...............................‪c‬‬ ‫31 ﺇﺫﺍ ﻛﺎﻥ ﻇﺘﺎ٢‪ = i‬ﻃﺎ‪ i‬ﺣﻴﺚ ٠‪ c٩٠ >i>c‬ﻓﺈﻥ ‪=(i c) X‬‬ ‫...............................‬ ‫41 ﺇﺫﺍ ﻛﺎﻥ ﺟﺎ ٥‪ = i‬ﺟﺘﺎ٤‪ i‬ﺣﻴﺚ ‪ i‬ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ ﻣﻮﺟﺒﺔ ﻓﺈﻥ ‪c............................... = i‬‬ ‫51 ﺇﺫﺍ ﻛﺎﻥ ﻗﺎ ‪ = i‬ﻗﺎ)٠٩‪ (i - c‬ﻓﺈﻥ ﻇﺘﺎ ‪= i‬‬ ‫...............................‬ ‫61 ﺇﺫﺍ ﻛﺎﻥ ﻇﺎ ٢‪ = i‬ﻇﺘﺎ٣‪ i‬ﺣﻴﺚ ‪ ] r ، ٠[∋ i‬ﻓﺈﻥ ‪= =(i c) X‬‬ ‫٢‬ ‫71 ﺇﺫﺍ ﻛﺎﻥ ﺟﺘﺎ ‪ = i‬ﺟﺎ٢‪ i‬ﺣﻴﺚ ‪ i‬ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ ﻣﻮﺟﺒﺔ ﻓﺈﻥ ﺟﺎ٣‪= i‬‬ ‫...............................‬ ‫‪E‬‬ ‫...............................‬ ‫‪k‬‬ ‫‪:Oó©àe øe QÉ«àN’G :ÉãdÉK‬‬ ‫81 ﺇﺫﺍ ﻛﺎﻧﺖ ﻇﺎ )٠٨١‪ ١ = (i + c‬ﺣﻴﺚ ‪ i‬ﻗﻴﺎﺱ ﺃﺻﻐﺮ ﺯﺍﻭﻳﺔ ﻣﻮﺟﺒﺔ ﻓﺈﻥ ﻗﻴﺎﺱ ‪ i‬ﻳﺴﺎﻭﻯ‬ ‫د ٥٣١‪c‬‬ ‫ﺟ ٠٦‪c‬‬ ‫ب ٠٣‪c‬‬ ‫أ ٥٤‪c‬‬ ‫91 ﺇﺫﺍ ﻛﺎﻥ ﺟﺘﺎ ٢‪ = i‬ﺟﺎ‪ i‬ﺣﻴﺚ ‪ ] r ،٠[∋ i‬ﻓﺈﻥ ﺟﺘﺎ ٢‪ i‬ﺗﺴﺎﻭﻯ‬ ‫٢‬ ‫ﺟ ٣‬ ‫ب ١‬ ‫أ ١‬ ‫٢‬ ‫٢‬ ‫............................................................................................‬ ‫٢‬ ‫02 ﺇﺫﺍ ﻛﺎﻥ ﺟﺎ ‪ = a‬ﺟﺘﺎ ‪ ،b‬ﺣﻴﺚ ‪ b ،a‬ﺯﺍﻭﻳﺘﺎﻥ ﺣﺎﺩﺗﺎﻥ ﻓﺈﻥ ﻇﺎ)‪ (b + a‬ﺗﺴﺎﻭﻯ‬ ‫أ ١‬ ‫ب ١‬ ‫ﺟ ٣‬ ‫٣‬ ‫د ١‬ ‫.............................................................‬ ‫د ﻏﻴﺮ ﻣﻌﺮﻭﻑ‬ ‫12 ﺇﺫﺍ ﻛﺎﻥ ﺟﺎ ٢‪ = i‬ﺟﺘﺎ ٤‪ i‬ﺣﻴﺚ ‪ i‬ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ ﻣﻮﺟﺒﺔ ﻓﺈﻥ ﻇﺎ)٠٩‪ (i٣ - c‬ﺗﺴﺎﻭﻯ‬ ‫ب ١‬ ‫د‬ ‫ﺟ ١‬ ‫أ -١‬ ‫......................................................‬ ‫٣‬ ‫‪ïM‬‬ ‫−‬ ‫٣‬
  • 65.
    ‫22 ﺇﺫﺍ ﻛﺎﻥﺟﺘﺎ)٠٩‪ ١ = (i + c‬ﺣﻴﺚ ‪ i‬ﻗﻴﺎﺱ ﺃﺻﻐﺮ ﺯﺍﻭﻳﺔ ﻣﻮﺟﺒﺔ ﻓﺈﻥ ﻗﻴﺎﺱ ‪ i‬ﻳﺴﺎﻭﻯ‬ ‫٢‬ ‫ﺟ‬ ‫ب‬ ‫د‬ ‫٠٣٣‪c‬‬ ‫٠٤٢‪c‬‬ ‫٠١٢‪c‬‬ ‫أ ٠٥١‪c‬‬ ‫..................................................‬ ‫‪á«JB’G á∏Ä°SC’G øY ÖLCG :Ék©HGQ‬‬ ‫32 ﺃﻭﺟﺪ ﺇﺣﺪﻯ ﻗﻴﻢ ‪ i‬ﺣﻴﺚ٠‪c٩٠ > i H‬ﺍﻟﺘﻰ ﺗﺤﻘﻖ ﻛﻼ ﻣﻦ ﺍﻵﺗﻰ:‬ ‫ًّ‬ ‫.................................................................................................................‬ ‫أ ﺟﺎ)٣‪ = (c١٥ + i‬ﺟﺘﺎ)٢‪(c٥ - i‬‬ ‫ب ﻗﺎ)‪ =       (c٢٥ + i‬ﻗﺘﺎ)‪(c١٥ + i‬‬ ‫.................................................................................................................‬ ‫ﺟ ﻇﺎ)‪ =     (c٢٠ + i‬ﻇﺘﺎ )٣‪(c٣٠ + i‬‬ ‫.................................................................................................................‬ ‫د ﺟﺘﺎ ‪ =       c٢٠ + i‬ﺟﺎ ‪c٤٠ + i‬‬ ‫.................................................................................................................‬ ‫٢‬ ‫٢‬ ‫42 ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻣﻤﺎ ﻳﺄﺗﻰ:‬ ‫ب ﻗﺘﺎ ٥٢٢‬ ‫أ ﺟﺎ ٠٥١‪c‬‬ ‫.....................................‬ ‫ﻫ ﻗﺘﺎ‬ ‫.....................................‬ ‫١١‪r‬‬ ‫و ﺟﺎ‬ ‫٦‬ ‫.....................................‬ ‫٧‪r‬‬ ‫٤‬ ‫.....................................‬ ‫ﺟ ﻗﺎ٠٠٣‪c‬‬ ‫.....................................‬ ‫ز ﻇﺘﺎ‬ ‫-٢‪r‬‬ ‫٣‬ ‫.....................................‬ ‫د ﻇﺎ ٠٨٧‪c‬‬ ‫.....................................‬ ‫ح ﺟﺘﺎ‬ ‫-٧‪r‬‬ ‫٤‬ ‫.....................................‬ ‫52 ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻟﻠﺰﺍﻭﻳﺔ ‪ i‬ﺍﻟﻤﺮﺳﻮﻣﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻲ ﻳﻘﻄﻊ ﺩﺍﺋﺮﺓ ﺍﻟﻮﺣﺪﺓ ﻓﻰ ﺍﻟﻨﻘﻄﺔ ﺏ )- ٣ ، ٤ (‬ ‫٥ ٥‬ ‫ﻓﺄﻭﺟﺪ:‬ ‫ب ﺟﺘﺎ ) ‪(i - r‬‬ ‫أ ﺟﺎ)٠٨١‪(i + c‬‬ ‫٢‬ ‫.................................................‬ ‫.................................................‬ ‫د ﻗﺘﺎ ) ٣‪(i - r‬‬ ‫٢‬ ‫ﺟ ﻇﺎ )٠٦٣‪(i -c‬‬ ‫.................................................‬ ‫.................................................‬ ‫62 ﺍﻛﺘﺸﻒ ﺍﻟﺨﻄﺄ: ﺟﻤﻴﻊ ﺍﻹﺟﺎﺑﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ ﺻﺤﻴﺤﺔ ﻣﺎﻋﺪﺍ ﺇﺟﺎﺑﺔ ﻭﺍﺣﺪﺓ ﻓﻘﻂ ﺧﻄﺄ، ﻓﻤﺎ ﻫﻰ:‬ ‫١- ﺟﺘﺎ ‪ i‬ﺗﺴﺎﻭﻯ‬ ‫.................................................................................................................................................................................................‬ ‫ب ﺟﺎ ) ٠٧٢‪(i - c‬‬ ‫أ ﺟﺎ )‪(c٢٧٠ - i‬‬ ‫٢- ﺟﺎ ‪ i‬ﺗﺴﺎﻭﻯ‬ ‫أ ﺟﺘﺎ ) ‪( i - r‬‬ ‫٢‬ ‫ﺟ ﺟﺘﺎ )٠٦٣‪(i - c‬‬ ‫د ﺟﺘﺎ ) ٠٦٣ ‪(i +c‬‬ ‫...................................................................................................................................................................................................‬ ‫٣- ﻇﺎ ‪ i‬ﺗﺴﺎﻭﻯ‬ ‫ب ﺟﺎ ) ‪(i - r‬‬ ‫٣‪r‬‬ ‫ﺟ ﺟﺘﺎ ) ٢ + ‪(i‬‬ ‫‪r‬‬ ‫د ﺟﺎ ) ٢ + ‪(i‬‬ ‫.....................................................................................................................................................................................................‬ ‫ب ﻇﺘﺎ ) ٠٧٢‪(i - c‬‬ ‫أ ﻇﺘﺎ ) ٠٩‪(i-c‬‬ ‫¯‬ ‫−‬ ‫¯‬ ‫ﺟ ﻇﺎ )٠٧٢‪(i - c‬‬ ‫د ﻇﺎ ) ٠٨١ ‪(i +c‬‬
  • 66.
    ‫72 ﺍﻟﺮﺑﻂ ﺑﺎﻟﺘﻜﻨﻮﻟﻮﺟﻴﺎ:ﻋﻨﺪ ﺍﺳﺘﺨﺪﺍﻡ ﻛﺮﻳﻢ ﺣﺎﺳﻮﺑﻪ ﺍﻟﻤﺤﻤﻮﻝ‬ ‫ﻛﺎﻧﺖ ﺯﺍﻭﻳﺔ ﻣﻴﻠﻪ ﻣﻊ ﺍﻷﻓﻘﻰ ٢٣١‪ c‬ﻛﻤﺎ ﻫﻮ ﻣﻮﺿﺢ ﺑﺎﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ.‬ ‫أ ﺍﺭﺳﻢ ﺍﻟﺸﻜﻞ ﺍﻟﺴﺎﺑﻖ ﻓﻰ ﺍﻟﻤﺴﺘﻮﻯ ﺍﻹﺣﺪﺍﺛﻰ، ﺑﺤﻴﺚ ﺗﻜﻮﻥ‬ ‫ﺍﻟﺰﺍﻭﻳﺔ ٢٣١‪ c‬ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﺛﻢ ﺃﻭﺟﺪ ﺯﺍﻭﻳﺘﻬﺎ ﺍﻟﻤﻨﺘﺴﺒﺔ.‬ ‫............................................................................................................................................‬ ‫‪c‬‬ ‫ب ﺍﻛﺘﺐ ﺩﺍﻟﺔ ﻣﺜﻠﺜﻴﺔ ﻳﻤﻜﻦ ﺍﺳﺘﺨﺪﺍﻣﻬﺎ ﻓﻰ ﺇﻳﺠﺎﺩ ﻗﻴﻢ ‪ ،C‬ﺛﻢ ﺃﻭﺟﺪ‬ ‫ﻗﻴﻤﺔ ‪ C‬ﻷﻗﺮﺏ ﺳﻨﺘﻴﻤﺘﺮ.‬ ‫‪C‬‬ ‫..................................................................................................................................................................................................................................‬ ‫ﺃﻟﻌﺎﺏ: ﺗﻨﺘﺸﺮ ﻟﻌﺒﺔ ﺍﻟﻌﺠﻠﺔ ﺍﻟﺪﻭﺍﺭﺓ ﻓﻰ ﻣﺪﻳﻨﺔ ﺍﻟﻤﻼﻫﻰ، ﻭﻫﻰ‬ ‫ﻋﺒﺎﺭﺓ ﻋﻦ ﻋﺪﺩ ﻣﻦ ﺍﻟﺼﻨﺎﺩﻳﻖ ﺗﺪﻭﺭ ﻓﻰ ﻗﻮﺱ ﺩﺍﺋﺮﻯ ﻳﺒﻠﻎ‬ ‫ﻛﺔ‬ ‫ﻧﺼﻒ ﻗﻄﺮه ٢١ ﻣﺘﺮﺍ، ﻓﺈﺫﺍ ﻛﺎﻥ ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﺸﺘﺮ‬ ‫ً‬ ‫ﻣﻊ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ٥‪r‬‬ ‫٤ .‬ ‫أ ﺍﺭﺳﻢ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ ٥‪ r‬ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ.‬ ‫٤‬ ‫‪C‬‬ ‫...........................................................................................................‬ ‫ب ﺍﻛﺘﺐ ﺩﺍﻟﺔ ﻣﺜﻠﺜﻴﺔ ﻳﻤﻜﻦ ﺍﺳﺘﺨﺪﺍﻣﻬﺎ ﻓﻰ ﺇﻳﺠﺎﺩ ﻗﻴﻤﺔ‬ ‫‪ C‬ﺛﻢ ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ C‬ﺑﺎﻟﻤﺘﺮ ﻷﻗﺮﺏ ﺭﻗﻤﻴﻦ ﻋﺸﺮﻳﻴﻦ.‬ ‫...........................................................................................................‬ ‫82 ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ:‬ ‫أ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺔ ‪ i‬ﻣﺮﺳﻮﻣﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ، ﺣﻴﺚ ﻇﺘﺎ ‪ ، ١- = i‬ﻗﺘﺎ ‪ . ٢ = i‬ﻓﻬﻞ ﻳﻤﻜﻦ ﺃﻥ‬ ‫٣‪r‬‬ ‫ﻳﻜﻮﻥ ‪ ? ٤ = (ic) X‬ﻓﺴﺮ ﺇﺟﺎﺑﺘﻚ?‬ ‫.......................................................................................................................................................................................................................‬ ‫٣‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ﺟﺘﺎ ) ٣‪ ، ٢ = (i - r‬ﺟﺎ ) ‪ ١ = (i + r‬ﻓﺄﻭﺟﺪ ﺃﺻﻐﺮ ﻗﻴﺎﺱ ﻣﻮﺟﺐ ﻟﻠﺰﺍﻭﻳﺔ ‪.i‬‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫.......................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
  • 67.
    ‫اﻟﺘﻤﺜﻴﻞ اﻟﺒﻴﺎﻧﻰ ﻟﻠﺪوالاﻟﻤﺜﻠﺜﻴﺔ‬ ‫4-5‬ ‫‪Graphing trigonometric functions‬‬ ‫‪k‬‬ ‫‪:≈JCÉjÉe πªcCG :’hCG‬‬ ‫1 ﻣﺪﻯ ﺍﻟﺪﺍﻟﺔ ﺩ ﺣﻴﺚ ﺩ)‪ = (i‬ﺟﺎ‪ i‬ﻫﻮ‬ ‫................................‬ ‫2 ﻣﺪﻯ ﺍﻟﺪﺍﻟﺔ ﺩ ﺣﻴﺚ ﺩ)‪ ٢ = (i‬ﺟﺎ‪ i‬ﻫﻮ‬ ‫................................‬ ‫3 ﺍﻟﻘﻴﻤﺔ ﺍﻟﻌﻈﻤﻰ ﻟﻠﺪﺍﻟﺔ ﻉ ﺣﻴﺚ ﻉ)‪٤ = (i‬ﺟﺎ‪ i‬ﻫﻰ‬ ‫................................‬ ‫4 ﺍﻟﻘﻴﻤﺔ ﺍﻟﺼﻐﺮﻯ ﻟﻠﺪﺍﻟﺔ ﻫـ ﺣﻴﺚ ﻫـ)‪٣ = (i‬ﺟﺘﺎ‪ i‬ﻫﻰ‬ ‫................................‬ ‫‪.É¡d ôXÉæªdG πμ°ûdG QGƒéH á«ã∏ãe ádGO πc IóYÉb ÖàcG :Ék«fÉK‬‬ ‫‪r‬‬ ‫‪r‬‬ ‫‪r‬‬ ‫‪r‬‬ ‫−‬ ‫−‪r− r‬‬ ‫− ‪r −r‬‬ ‫ﺷﻜﻞ )١( ﺍﻟﻘﺎﻋﺪﺓ ﻫﻰ:‬ ‫‪r‬‬ ‫‪r‬‬ ‫‪r‬‬ ‫‪r‬‬ ‫−‬ ‫−‪r− r‬‬ ‫− ‪r − r‬‬ ‫ﺷﻜﻞ )٢( ﺍﻟﻘﺎﻋﺪﺓ ﻫﻲ:‬ ‫........................................................................................................‬ ‫........................................................................................................‬ ‫‪k‬‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :ÉãdÉK‬‬ ‫5 ﺃﻭﺟﺪ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻌﻈﻤﻰ ﻭﺍﻟﻘﻴﻤﺔ ﺍﻟﺼﻐﺮﻯ، ﺛﻢ ﺍﺣﺴﺐ ﺍﻟﻤﺪﻯ ﻟﻜﻞ ﺩﺍﻟﺔ ﻣﻦ ﺍﻟﺪﻭﺍﻝ ﺍﻵﺗﻴﺔ :‬ ‫أ ﺹ = ﺟﺎ‪i‬‬ ‫.......................................................................................................................................................................................................................‬ ‫ب ﺹ = ٣ ﺟﺘﺎ‪i‬‬ ‫.......................................................................................................................................................................................................................‬ ‫ﺟ ﺹ = ٣ ﺟﺎ‪i‬‬ ‫٢‬ ‫.......................................................................................................................................................................................................................‬ ‫6 ﻣﺜﻞ ﻛﻞ ﻣﻦ ﺍﻟﺪﻭﺍﻝ ﺹ = ٤ ﺟﺘﺎ‪ ، i‬ﺹ = ٣ ﺟﺎ‪ i‬ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻵﻟﺔ ﺍﻟﺤﺎﺳﺒﺔ ﺍﻟﺮﺳﻮﻣﻴﺔ ﺃﻭ ﺑﺄﺣﺪ ﺑﺮﺍﻣﺞ ﺍﻟﺤﺎﺳﻮﺏ‬ ‫ﺍﻟﺮﺳﻮﻣﻴﺔ ﻭﻣﻦ ﺍﻟﺮﺳﻢ ﺃﻭﺟﺪ :‬ ‫ب ﺍﻟﻘﻴﻢ ﺍﻟﻌﻈﻤﻰ ﻭﺍﻟﻘﻴﻢ ﺍﻟﺼﻐﺮﻯ ﻟﻠﺪﺍﻟﺔ.‬ ‫أ ﻣﺪﻯ ﺍﻟﺪﺍﻟﺔ.‬ ‫.‬ ‫.................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬ ‫.................................................................................................‬
  • 68.
    ‫إﻳﺠﺎد ﻗﻴﺎس زاوﻳﺔﺑﻤﻌﻠﻮﻣﻴﺔ داﻟﺔ ﻣﺜﻠﺜﻴﺔ‬ ‫‪Finding the measure of an angle given the‬‬ ‫4-6‬ ‫‪value of one of its functions‬‬ ‫‪k‬‬ ‫‪:Oó©àe øe QÉ«àN’G :’hCG‬‬ ‫1 ﺇﺫﺍ ﻛﺎﻥ ﺟﺎ ‪ ٠٫٤٣٢٥ = i‬ﺣﻴﺚ ‪ i‬ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ ﻣﻮﺟﺒﺔ ﻓﺈﻥ ‪ (i) c X‬ﺗﺴﺎﻭﻯ‬ ‫ﺟ ٨٨٣٫٢٣‪c‬‬ ‫ب ٧٤٣٫٤٦‪c‬‬ ‫أ ٦٢٦٫٥٢‪c‬‬ ‫2 ﺇﺫﺍ ﻛﺎﻥ ﻇﺎ ‪ ١٫٨ = i‬ﻛﺎﻧﺖ ٠٩‪ c٣٦٠ H iHc‬ﻓﺈﻥ ‪ (i) c X‬ﺗﺴﺎﻭﻯ‬ ‫ﻭ‬ ‫ﺟ ٥٤٩٫٠٤٢‪c‬‬ ‫ب ٥٥٠٫٩١١‪c‬‬ ‫أ ٥٤٩٫٠٦‪c‬‬ ‫.............................................................‬ ‫د ٦١٣٫٦٤‪c‬‬ ‫............................................................................‬ ‫د ٥٥٠٫٩٩٢‪c‬‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫1 ﺇﺫﺍ ﻗﻄﻊ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻟﻠﺰﺍﻭﻳﺔ ‪ i‬ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﺩﺍﺋﺮﺓ ﺍﻟﻮﺣﺪﺓ ﻓﻰ ﺍﻟﻨﻘﻄﺔ ﺏ، ﻓﺄﻭﺟﺪ ﻛﻼ ﻣﻦ‬ ‫ًّ‬ ‫ﺟﺘﺎ ‪ ،i‬ﺟﺎ ‪ i‬ﻓﻰ ﺍﻟﺤﺎﻻﺕ ﺍﻵﺗﻴﺔ:‬ ‫٣‬ ‫أ ﺏ )١، ٢ (‬ ‫٢‬ ‫ب ﺏ) ١ ، - ١ (‬ ‫٢‬ ‫٢‬ ‫٦ ٨‬ ‫ﺟ ﺏ )- ٠١ ، ٠١ (‬ ‫..................................................................‬ ‫..................................................................‬ ‫..................................................................‬ ‫..................................................................‬ ‫..................................................................‬ ‫................................................................‬ ‫2 ﺇﺫﺍ ﻗﻄﻊ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻟﻠﺰﺍﻭﻳﺔ ‪ i‬ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﺩﺍﺋﺮﺓ ﺍﻟﻮﺣﺪﺓ ﻓﻰ ﺍﻟﻨﻘﻄﺔ ﺏ، ﻓﺄﻭﺟﺪ ﻛﻼ ﻣﻦ‬ ‫ًّ‬ ‫ﻗﺎ‪ ،i‬ﻗﺘﺎ‪ i‬ﻓﻰ ﺍﻟﺤﺎﻻﺕ ﺍﻵﺗﻴﺔ:‬ ‫٢‬ ‫٢‬ ‫ﺟ ﺏ )- ٥ ، - ٢١‬ ‫ب ﺏ)- ١ ، - ٢‬ ‫٣١ ٣١ (‬ ‫(‬ ‫أ ﺏ) ٢ ،- ٢ (‬ ‫٥‬ ‫٥‬ ‫..................................................................‬ ‫..................................................................‬ ‫..................................................................‬ ‫..................................................................‬ ‫..................................................................‬ ‫................................................................‬ ‫3 ﺇﺫﺍ ﻗﻄﻊ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻟﻠﺰﺍﻭﻳﺔ ‪ i‬ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﺩﺍﺋﺮﺓ ﺍﻟﻮﺣﺪﺓ ﻓﻰ ﺍﻟﻨﻘﻄﺔ ﺏ، ﻓﺄﻭﺟﺪ ﻛﻼ ﻣﻦ‬ ‫ًّ‬ ‫ﻇﺎ ‪ ،i‬ﻇﺘﺎ ‪ i‬ﻓﻰ ﺍﻟﺤﺎﻻﺕ ﺍﻵﺗﻴﺔ:‬ ‫ﺟ ﺏ )- ٤ ، - ٣ (‬ ‫ب ﺏ) ٣ ،- ٥ (‬ ‫أ ﺏ) ١ ،- ٣ (‬ ‫٥‬ ‫٥‬ ‫٠١‬ ‫٠١‬ ‫٤٣‬ ‫٤٣‬ ‫..................................................................‬ ‫..................................................................‬ ‫..................................................................‬ ‫..................................................................‬ ‫..................................................................‬ ‫................................................................‬ ‫4 ﺇﺫﺍ ﻗﻄﻊ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻟﻠﺰﺍﻭﻳﺔ ‪ i‬ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﺩﺍﺋﺮﺓ ﺍﻟﻮﺣﺪﺓ ﻓﻰ ﺍﻟﻨﻘﻄﺔ ﺏ‬ ‫ﻓﺄﻭﺟﺪ: ‪ (ic)X‬ﺣﻴﺚ ٠ ْ > ‪ ْ ٣٦٠ > i‬ﻋﻨﺪﻣﺎ:‬ ‫٣‬ ‫أ ﺏ ) ٢ ، ١(‬ ‫٢‬ ‫ب ﺏ)- ١ ، ١ (‬ ‫٢‬ ‫..................................................................‬ ‫٢‬ ‫..................................................................‬ ‫‪ïM‬‬ ‫−‬ ‫٦‬ ‫ﺟ ﺏ ) ٠١ ، -٨ (‬ ‫٠١‬ ‫..................................................................‬
  • 69.
    ‫5 ﺃﻭﺟﺪ ﺑﺎﻟﻘﻴﺎﺱﺍﻟﺴﺘﻴﻨﻰ ﺃﺻﻐﺮ ﺯﺍﻭﻳﺔ ﻣﻮﺟﺒﺔ ﺗﺤﻘﻖ ﻛﻼ ﻣﻦ:‬ ‫ًّ‬ ‫ب ﺟﺘﺎ-١ ٦٣٤٫٠‬ ‫أ ﺟﺎ-١ ٦٫٠‬ ‫..................................................................‬ ‫د ﻗﺎ-١ )- ٤٦٣٢٫٢(‬ ‫ﺟ ﻇﺎ-١ ٢٥٥٤٫١‬ ‫..................................................................‬ ‫ﻫ ﻇﺘﺎ-١ ٨١٢٦٫٣‬ ‫..................................................................‬ ‫..................................................................‬ ‫و ﻗﺘﺎ-١ )-٤٠٠٦٫١(‬ ‫..................................................................‬ ‫6 ﺇﺫﺍ ﻛﺎﻧﺖ ٠‪ c٣٦٠ H iHc‬ﻓﺄﻭﺟﺪ ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ‪ i‬ﻟﻜﻞ ﻣﻤﺎ ﻳﺄﺗﻰ:‬ ‫ب ﺟﺘﺎ-١ )- ٢٤٦٫٠(‬ ‫أ ﺟﺎ-١ )٦٥٣٢٫٠(‬ ‫..................................................................‬ ‫..................................................................‬ ‫ﺟ ﻇﺎ-١ )- ٦٥٤١٫٢(‬ ‫..................................................................‬ ‫..................................................................‬ ‫7 ﺇﺫﺍ ﻛﺎﻥ ﺟﺎ ‪ ١ = i‬ﻛﺎﻧﺖ ٠٩‪c١٨٠ H iHc‬‬ ‫٣ﻭ‬ ‫أ ﺍﺣﺴﺐ ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ‪ i‬ﻷﻗﺮﺏ ﺛﺎﻧﻴﺔ‬ ‫.......................................................................................................................................................................................................................‬ ‫ب ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻣﻦ: ﺟﺘﺎ‪ ، i‬ﻇﺎ‪ ، i‬ﻗﺎ‪. i‬‬ ‫ٍّ‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫8 ﺳﻼﻟﻢ: ﺳﻠﻢ ﻃﻮﻟﻪ ٥ ﺃﻣﺘﺎﺭ ﻳﺴﺘﻨﺪ ﻋﻠﻰ ﺟﺪﺍﺭ ﻓﺈﺫﺍ ﻛﺎﻥ ﺍﺭﺗﻔﺎﻉ ﺍﻟﺴﻠﻢ ﻋﻦ‬ ‫ﺳﻄﺢ ﺍﻷﺭﺽ ﻳﺴﺎﻭﻯ ٣ ﺃﻣﺘﺎﺭ ﻓﺄﻭﺟﺪ ﺑﺎﻟﺮﺍﺩﻳﺎﻥ ﺯﺍﻭﻳﺔ ﻣﻴﻞ ﺍﻟﺴﻠﻢ ﻋﻠﻰ ﺍﻷﻓﻘﻰ.‬ ‫‪ci‬‬ ‫.‬ ‫..................................................................................................................................................................................................................................‬ ‫..................................................................................................................................................................................................................................‬ ‫9 ﺃﻭﺟﺪ ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ‪ i‬ﺑﺎﻟﻘﻴﺎﺱ ﺍﻟﺴﺘﻴﻨﻰ ﻓﻰ ﻛﻞ ﺷﻜﻞ ﻣﻦ ﺍﻷﺷﻜﺎﻝ ﺍﻵﺗﻴﺔ:‬ ‫ب‬ ‫أ‬ ‫ﺟ‬ ‫‪i‬‬ ‫‪i‬‬ ‫............................................................‬ ‫¯‬ ‫−‬ ‫............................................................‬ ‫¯‬ ‫‪i‬‬ ‫............................................................‬
  • 70.
    ‫ﲤﺎﺭﻳﻦ ﻋﺎﻣﺔ‬ ‫‪:ø«jô°ûY ø«ªbQÜôbC’ èJÉædG ÉkHô≤e á«JB’G á∏Ä°SC’G øY ÖLCG‬‬ ‫1 ﺣﻮﻝ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻵﺗﻴﺔ ﻣﻦ ﺩﺭﺟﺎﺕ ﺇﻟﻰ ﺭﺍﺩﻳﺎﻥ:‬ ‫ِّ‬ ‫أ ٠٢١‪c‬‬ ‫ب ٨٫٤٦‪c‬‬ ‫   ...............................‬ ‫               ...............................‬ ‫ﺟ ٦٣ َ ٠٢٢‪c‬‬ ‫................................‬ ‫2 ﺣﻮﻝ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻵﺗﻴﺔ ﻣﻦ ﺭﺍﺩﻳﺎﻥ ﺇﻟﻰ ﺩﺭﺟﺎﺕ:‬ ‫أ ٥‪r‬‬ ‫٣‬ ‫ب - ٣‪r‬‬ ‫٢‬ ‫       ...............................‬ ‫      ...............................‬ ‫ﺟ ٢١٫١‬ ‫‪E‬‬ ‫      ...............................‬ ‫3 ‪ i‬ﺯﺍﻭﻳﺔ ﻛﺰﻳﺔ ﻓﻰ ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ‪ H‬ﻭﺗﺤﺼﺮ ﻗﻮﺳﺎ ﻃﻮﻟﻪ ﻝ:‬ ‫ﻣﺮ‬ ‫ً‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ ‪ ٨ = H‬ﺳﻢ،  ‪ E١٫٢ = i‬ﺃﻭﺟﺪ ﻝ.‬ ‫    ‬ ‫................................................................................................‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ﻝ = ٦٢ ﺳﻢ، ‪ ١٨ = H‬ﺳﻢ ﺃﻭﺟﺪ ‪ i‬ﺑﺎﻟﺪﺭﺟﺎﺕ.‬ ‫...............................................................................................‬ ‫4 ﺑﺪﻭﻥ ﺍﺳﺘﺨﺪﺍﻡ ﺍﻵﻟﺔ ﺍﻟﺤﺎﺳﺒﺔ ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻣﻤﺎ ﻳﺄﺗﻰ:‬ ‫٣١‬ ‫أ ﻇﺎ ٠٢١‪    c‬ب ﺟﺎ ) ٦‪    ( r‬ﺟ ﺟﺘﺎ ٠٣٣‪    c‬د ﻇﺘﺎ )- ٠٠٣‪    (c‬ﻫ ﻗﺘﺎ )- ‪( r‬‬ ‫٣‬ ‫..............................        ..............................            ..............................            ..............................               ..............................‬ ‫5 ﺃﻭﺟﺪ ﺟﻤﻴﻊ ﺍﻟﺪﻭﺍﻝ ﺍﻟﻤﺜﻠﺜﻴﺔ ﻟﻠﺰﺍﻭﻳﺔ ‪ i‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻣﺮﺳﻮﻣﺎ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﻭﻳﻤﺮ ﺑﻜﻞ ﻧﻘﻄﺔ‬ ‫ً‬ ‫ﻣﻦ ﺍﻟﻨﻘﺎﻁ ﺍﻵﺗﻴﺔ:‬ ‫ﺟ )- ٣‬ ‫د )- ٥ ، ٢(‬ ‫٢ ، - ٢(‬ ‫ب - )٥، - ٢١(‬ ‫أ )٤، ٣(‬ ‫.............................................‬ ‫6‬ ‫.............................................‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ:‬ ‫: ﺟﺎ ٠٦ = ٢ ﺟﺎ ٠٣‪ c‬ﺟﺘﺎ ٠٣‪c‬‬ ‫.............................................‬ ‫.............................................‬ ‫: ﺟﺘﺎ ٠٠٣‪ ٢ = c‬ﺟﺎ٢ ٠٦‪١- c‬‬ ‫ب ﺇﺫﺍ ﻛﺎﻧﺖ ﺟﺘﺎ ‪ ٤ - = i‬ﺣﻴﺚ ٠٩‪ c١٨٠ > i > c‬ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻣﻦ:‬ ‫٥‬ ‫ً‬ ‫: ﻇﺎ )‪(c١٨٠- i‬‬ ‫ﺃﻭﻻ: ﺟﺎ )٠٨١‪( i -c‬‬ ‫............................................................‬ ‫............................................................‬ ‫7 ﺃﻭﺟﺪ ﻗﻴﺎﺱ ﺍﻟﺰﻭﺍﻳﺎ ﺑﺎﻟﺪﺭﺟﺎﺕ ﻓﻰ ﺍﻟﻔﺘﺮﺓ ٠‪ c٣٦٠ H iHc‬ﻟﻜﻞ ﻣﻤﺎ ﻳﺄﺗﻰ:‬ ‫٣‬ ‫ﺟ ﺟﺘﺎ-١ )  ٢ (‬ ‫ب ﺟﺎ-١ )- ١ (‬ ‫أ ﻇﺎ-١ ١‬ ‫٢‬ ‫.............................................‬ ‫.............................................‬ ‫.............................................‬ ‫د ﻇﺎ-١)- ٣ (‬ ‫.............................................‬ ‫8 ﻣﻨﺤﺪﺭﺍ ﻃﻮﻟﻪ ٤٢ ﻣﺘﺮﺍ، ﻭﺍﺭﺗﻔﺎﻋﻪ ﻋﻦ ﺳﻄﺢ ﺍﻷﺭﺽ ٩ ﺃﻣﺘﺎﺭ، ﺍﻛﺘﺐ ﺩﺍﻟﺔ ﻣﺜﻠﺜﻴﺔ ﻳﻤﻜﻦ ﺍﺳﺘﺨﺪﺍﻣﻬﺎ ﻹﻳﺠﺎﺩ ﻗﻴﺎﺱ‬ ‫ً‬ ‫ً‬ ‫ﺯﺍﻭﻳﺔ ﻣﻴﻞ ﺍﻟﻤﻨﺤﺪﺭ ﻣﻊ ﺍﻷﺭﺽ ﺍﻷﻓﻘﻴﺔ، ﺛﻢ ﺃﻭﺟﺪ ﻗﻴﺎﺳﻬﺎ. ..........................................................................................................‬ ‫‪ïM‬‬ ‫−‬
  • 71.
    ‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ‬ ‫‪.√É£©ªdG äÉHÉLE’Gø«H øe áë«ë°üdG áHÉL’G ôàNG‬‬ ‫1 ﺍﻟﺰﺍﻭﻳﺔ ٥٨٥‪ c‬ﺗﻜﺎﻓﻲﺀ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻲ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ:‬ ‫ﺟ ٥٢٢‪c‬‬ ‫ب ٥٣١‪c‬‬ ‫أ ٥٤‪c‬‬ ‫د ٥١٣‪c‬‬ ‫2 ﺇﺫﺍ ﻛﺎﻥ ﺟﺎ‪ ،٠ > i‬ﻇﺎ ‪ ٠ < i‬ﻓﺈﻥ ﺯﺍﻭﻳﺔ ﺗﻘﻊ ‪ i‬ﻓﻰ ﺍﻟﺮﺑﻊ:‬ ‫ﺟ ﺍﻟﺜﺎﻟﺚ‬ ‫ب ﺍﻟﺜﺎﻧﻰ‬ ‫أ ﺍﻷﻭﻝ‬ ‫د ﺍﻟﺮﺍﺑﻊ‬ ‫3 ﺇﺫﺍ ﻛﺎﻧﺖ ‪ i‬ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ ﻛﺎﻥ ﺟﺎ )‪ = (c٢٠ + i‬ﺟﺘﺎ ٠٣‪ c‬ﻓﺈﻥ ﻕ )‪ (i c‬ﺗﺴﺎﻭﻯ:‬ ‫ﻭ‬ ‫د ٠٥‪c‬‬ ‫ﺟ ٠٤‪c‬‬ ‫ب ٠٣‪c‬‬ ‫أ ٠٢‪c‬‬ ‫4 ﺍﻟﺰﺍﻭﻳﺔ )-٠٥٨‪ (c‬ﺗﻘﻊ ﻓﻰ ﺍﻟﺮﺑﻊ:‬ ‫ب ﺍﻟﺜﺎﻧﻰ‬ ‫أ ﺍﻷﻭﻝ‬ ‫ﺟ ﺍﻟﺜﺎﻟﺚ‬ ‫د ﺍﻟﺮﺍﺑﻊ‬ ‫5 ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺑﺎﻟﺪﺭﺟﺎﺕ ﺍﻟﺘﻰ ﺗﻘﺎﺑﻞ ﻗﻮﺳﺎ ﻃﻮﻟﻪ ٦‪ r‬ﻓﻰ ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٩ﺳﻢ ﺗﺴﺎﻭﻯ:‬ ‫ً‬ ‫د ٠٥١‪c‬‬ ‫ﺟ ٠٢١‪c‬‬ ‫ب ٠٦‪c‬‬ ‫أ ٠٣‪c‬‬ ‫6 ﺃﺑﺴﻂ ﺻﻮﺭﺓ ﻟﻠﻤﻘﺪﺍﺭ: ﺟﺘﺎ )٠٨١‪ + (i + c‬ﺟﺎ )٠٩‪ (i + c‬ﻳﺴﺎﻭﻯ:‬ ‫ﺟ ٢ ﺟﺘﺎ ‪i‬‬ ‫ب ٢‬ ‫أ ٠‬ ‫7 ﻇﺎ )-٠٣‪ (c‬ﺗﺴﺎﻭﻯ:‬ ‫أ - ٣‬ ‫ب - ١‬ ‫٣‬ ‫ﺟ‬ ‫د ٢ ﺟﺎ ‪i‬‬ ‫١‬ ‫٣‬ ‫د‬ ‫٣‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG‬‬ ‫8 ‪ C‬ﺏ ﻗﻮﺱ ﻓﻰ ﺩﺍﺋﺮﺓ ﻛﺰﻫﺎ ﻭ ﻭﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٠١ ﺳﻢ ، ‪ C‬ﺏ = ٦١ ﺳﻢ.‬ ‫ﻣﺮ‬ ‫ﺃﻭﺟﺪ ‪ i‬ﺑﺎﻟﻘﻴﺎﺱ ﺍﻟﺪﺍﺋﺮﻯ ﺛﻢ ﺃﻭﺟﺪ ﻃﻮﻝ ﺍﻟﻘﻮﺱ ‪ C‬ﺏ :‬ ‫‪C‬‬ ‫‪i‬‬ ‫9 ﺇﺫﺍ ﻛﺎﻥ ٥ ﺟﺎ ‪ ٤ = C‬ﺣﻴﺚ ٠٩‪c١٨٠ > C > c‬‬ ‫ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﻤﻘﺪﺍﺭ ﺟﺎ )٠٨١‪+ (C - c‬ﻇﺎ )٠٦٣‪٢+ (C - c‬ﺟﺎ )٠٧٢‪(C - c‬‬ ‫01 ﺃﻭﺟﺪ ﻓﻰ ﺃﺑﺴﻂ ﺻﻮﺭﺓ ﻗﻴﻤﺔ ﺍﻟﻤﻘﺪﺍﺭ: ﺟﺎ ٠٢١‪ c‬ﺟﺘﺎ٠٣٣‪ - c‬ﺟﺘﺎ ٠٢٤‪ c‬ﺟﺎ )-٠٣‪.(c‬‬ ‫11 ﺍﻭﺟﺪ ﺑﺎﻟﺮﺩﻳﺎﻥ ‪ (C c) X‬ﺇﺫﺍ ﻛﺎﻥ ٢ ﺟﺘﺎ ‪ ٠ = ٢ + C‬ﺣﻴﺚ ‪ C‬ﻗﻴﺎﺱ ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ.‬ ‫٣‬ ‫21 ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻲ ﻟﻠﺰﺍﻭﻳﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻲ ﻳﻘﻄﻊ ﺩﺍﺋﺮﺓ ﺍﻟﻮﺣﺪﺓ ﻋﻨﺪ ﺍﻟﻨﻘﻄﺔ )- ٢ ، ١ ( ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ‬ ‫٢‬ ‫ﻛﻞ ﻣﻦ: ﻃﺎ‪ ، i‬ﻗﺎ‪i‬‬ ‫31 ﺃﻭﺟﺪ ﺍﻟﺪﻭﺍﻝ ﺍﻟﻤﺜﻠﺜﻴﺔ ﺍﻷﺳﺎﺳﻴﺔ ﻟﻠﺰﺍﻭﻳﺔ ‪ i‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻲ ﻣﺮﺳﻮﻣﺎ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻲ ﻭﻳﻤﺮ ﺑﺎﻟﻨﻘﻄﺔ‬ ‫ً‬ ‫)٦، -٨(‬ ‫¯‬ ‫−‬ ‫¯‬
  • 72.
    ‫ﺍﺧﺘﺒﺎﺭ ﺗﺮﺍﻛﻤﻲ‬ ‫‪k‬‬ ‫‪Oó©àe øeQÉ«àN’G á∏Ä°SCG :’hCG‬‬ ‫1 ﺃﻯ ﻣﻦ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻵﺗﻴﺔ ﻳﻜﻮﻥ ﺍﻟﺠﻴﺐ ﻭﺟﻴﺐ ﺍﻟﺘﻤﺎﻡ ﻟﻬﺎ ﺳﺎﻟﺒﻴﻦ :‬ ‫ﺟ ٠٢٢‪c‬‬ ‫ب ٠٤١‪c‬‬ ‫أ ٠٤‪c‬‬ ‫.............................................‬ ‫.............................................‬ ‫د ٠٢٣‪c‬‬ ‫.............................................‬ ‫.............................................‬ ‫2 ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﻛﺰﻳﺔ ﺍﻟﺘﻰ ﺗﻘﺎﺑﻞ ﻗﻮﺳﺎ ﻃﻮﻟﻪ ٢‪ r‬ﻓﻰ ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٦ ﺳﻢ ﻳﺴﺎﻭﻯ :‬ ‫ﺍﻟﻤﺮ‬ ‫ً‬ ‫د ‪r‬‬ ‫ﺟ ‪r‬‬ ‫ب ‪r‬‬ ‫‪r‬‬ ‫أ‬ ‫٤‬ ‫٦‬ ‫.............................................‬ ‫٣‬ ‫.............................................‬ ‫٢‬ ‫.............................................‬ ‫.............................................‬ ‫3 ﺇﺫﺍ ﻛﺎﻥ ﻇﺎ ٤‪ = i‬ﻇﺘﺎ٢‪ i‬ﺣﻴﺚ ‪ i‬ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ ﻣﻮﺟﺒﺔ ﻓﺈﻥ ﺟﺎ)٠٩ ْ - ‪ (i‬ﺗﺴﺎﻭﻯ :‬ ‫ﺟ ٣‬ ‫أ ١‬ ‫ب ١‬ ‫د ١‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫.............................................‬ ‫.............................................‬ ‫.............................................‬ ‫.............................................‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫٣‬ ‫4 ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻟﻠﺰﺍﻭﻳﺔ ‪ i‬ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﻳﻘﻄﻊ ﺩﺍﺋﺮﺓ ﺍﻟﻮﺣﺪﺓ ﻋﻨﺪ ﺍﻟﻨﻘﻄﺔ ) ١ ، ٢ ( ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ‬ ‫٢‬ ‫ﻛﻞ ﻣﻦ ﻇﺘﺎ ‪ ،i‬ﻗﺘﺎ‪...................................................................................................................................................................................... .i‬‬ ‫5 ﺑﺪﻭﻥ ﺍﺳﺘﺨﺪﺍﻡ ﺍﻵﻟﺔ ﺍﻟﺤﺎﺳﺒﺔ ﺃﻭﺟﺪ )ﺇﻥ ﺃﻣﻜﻦ ﺫﻟﻚ( ﻗﻴﻤﺔ ﻛﻞ ﻣﻦ :‬ ‫ﺟ ﻗﺎ ٣‪r‬‬ ‫ب ﺟﺎ )- ٥٣١‪(c‬‬ ‫أ ﺟﺘﺎ ٠١٢‪c‬‬ ‫د ﻇﺘﺎ )- ٢‪( r‬‬ ‫٣‬ ‫٢‬ ‫.............................................‬ ‫.............................................‬ ‫.............................................‬ ‫.............................................‬ ‫6 ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻀﻠﻊ ﺍﻟﻨﻬﺎﺋﻰ ﻟﻠﺰﺍﻭﻳﺔ )٠٩‪ (i - c‬ﺣﻴﺚ ‪ i‬ﺯﺍﻭﻳﺔ ﺣﺎﺩﺓ ﻣﻮﺟﺒﺔ، ﻳﻘﻄﻊ ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٥‬ ‫ﻭﺣﺪﺍﺕ ﻃﻮﻝ ﻓﻰ ﺍﻟﻨﻘﻄﺔ )٤ ، ﻙ( ﻓﺄﻭﺟﺪ :‬ ‫د ﻕ)‪(i c‬‬ ‫ﺟ ﺟﺘﺎ )٠٩‪(i - c‬‬ ‫ب ﺟﺎ )٠٩‪(i - c‬‬ ‫أ ﻗﻴﻤﺔ ﻙ‬ ‫.............................................‬ ‫.............................................‬ ‫.............................................‬ ‫.............................................‬ ‫7 ﺩ ﺍﺟﺎﺕ: ﻳﺼﻌﺪ ﻛﺮﻳﻢ ﺑﺪﺭﺍﺟﺘﻪ ﻣﻨﺤﺪﺭﺍ ﻳﻤﻴﻞ ﻋﻠﻰ ﺍﻷﻓﻘﻰ ﺑﺰﺍﻭﻳﺔ ﻗﻴﺎﺳﻬﺎ ٥٥١‪ c‬ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ‬ ‫ً‬ ‫أ ﺍﻛﺘﺐ ﺩﺍﻟﺔ ﻣﺜﻠﺜﻴﺔ ﺗﺒﻴﻦ ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ‪ C‬ﻭﻃﻮﻝ ﺍﻟﻤﻨﺤﺪﺭ.‬ ‫ب ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ C‬ﻷﻗﺮﺏ ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ.‬ ‫¯‬ ‫‪ï‬‬ ‫¯‬ ‫١‬ ‫٢‬ ‫٣‬ ‫٤‬ ‫٥‬ ‫٦‬ ‫٧‬ ‫٤- ٣‬ ‫٤-٢‬ ‫٤-٤‬ ‫٤-٣‬ ‫٤- ٤‬ ‫٤-٤‬ ‫٤-٤‬ ‫‪ïM‬‬ ‫−‬
  • 73.
    ‫ﺍﺧﺘﺒﺎﺭﺍﺕ ﻋﺎﻣﺔ‬ ‫)اﻟﺠﺒﺮ وﺣﺴﺎباﻟﻤﺜﻠﺜﺎت(‬ ‫اﻻﺧﺘﺒﺎر ا ول‬ ‫‪k‬‬ ‫‪≈`JCÉjÉe πªcCG :’hCG‬‬ ‫1 ﺇﺫﺍ ﻛﺎﻥ ﺱ = -١ ﻫﻰ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ‪ C‬ﺱ – ٢ = ٠ ﻓﺈﻥ ‪= C‬‬ ‫2 ﺇﺷﺎﺭﺓ ﺍﻟﺪﺍﻟﺔ ﺩ ﺣﻴﺚ ﺩ)ﺱ( = ﺱ٢ + ٣ ﺗﻜﻮﻥ‬ ‫...............................................................................‬ ‫..................................................................................................................................‬ ‫3 ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﻓﻰ ﻣﺠﻤﻮﻋﺔ ﺍﻷﻋﺪﺍﺩ ﻛﺒﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ – ﺕ، ﺕ ﻫﻰ‬ ‫ﺍﻟﻤﺮ‬ ‫4 ﻣﺪﻯ ﺍﻟﺪﺍﻟﺔ ﺩ ﺣﻴﺚ ﺩ)‪ ٣ = (i‬ﺟﺎ‪ i‬ﻫﻮ‬ ‫...................................................................‬ ‫................................................................................................................................................‬ ‫5 ﺃﺻﻐﺮ ﺯﺍﻭﻳﺔ ﻣﻮﺟﺒﺔ ﻣﻜﺎﻓﺌﺔ ﻟﻠﺰﺍﻭﻳﺔ ﺍﻟﺘﻰ ﻗﻴﺎﺳﻬﺎ )-٠٤٨‪ (c‬ﻗﻴﺎﺳﻬﺎ ............................ ﻭﺗﻘﻊ ﻓﻰ ﺍﻟﺮﺑﻊ‬ ‫................................‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫1 أ ﺃﺛﺒﺖ ﺃﻥ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ - ٥ﺱ + ٣ = ٠ ﺣﻘﻴﻘﻴﺎﻥ ﻣﺨﺘﻠﻔﺎﻥ، ﺛﻢ ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺍﻟﺤﻞ ﻓﻰ ﺡ ﻣﻘﺮﺑﺎ‬ ‫ً‬ ‫ﺍﻟﻨﺎﺗﺞ ﻟﺮﻗﻢ ﻋﺸﺮﻯ ﻭﺍﺣﺪ. ................................................................................................................................................................‬ ‫ب ﺃﻭﺟﺪ ﻓﻰ ﺃﺑﺴﻂ ﺻﻮﺭﺓ ﻗﻴﻤﺔ ﺍﻟﻤﻘﺪﺍﺭ:ﺟﺎ )- ٠٣˚( ﺟﺘﺎ ٠٢٤˚ + ﻇﺎ٥٢‪c‬‬ ‫....................................................................‬ ‫ﻇﺘﺎ٥٦‪c‬‬ ‫2 أ ﻓﻰ ﺍﻟﻤﻌﺎﺩﻟﺔ )‪ (٥ – C‬ﺱ٢ + )‪ (١٠ – C‬ﺱ – ٥ = ٠ ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ C‬ﻓﻰ ﺍﻟﺤﺎﻻﺕ ﺍﻵﺗﻴﺔ:‬ ‫: ﺇﺫﺍ ﻛﺎﻥ ﻣﺠﻤﻮﻉ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ = ٤ ...........................................................................................................................‬ ‫: ﺇﺫﺍ ﻛﺎﻥ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﻫﻮ ﺍﻟﻤﻌﻜﻮﺱ ﺍﻟﻀﺮﺑﻰ ﻟﻠﺠﺬﺭ ﺍﻵﺧﺮ. ...........................................................‬ ‫ب ﺍﺑﺤﺚ ﺇﺷﺎﺭﺓ ﺍﻟﺪﺍﻟﺔ ﺩ ﺣﻴﺚ ﺩ)ﺱ( = ﺱ٢ + ٢ ﺱ – ٥١ ﻣﻊ ﺗﻮﺿﻴﺢ ﺫﻟﻚ ﻋﻠﻰ ﺧﻂ ﺍﻷﻋﺪﺍﺩ.‬ ‫.......................................................................................................................................................................................................................‬ ‫3‬ ‫أ ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺔ: ٥ﺱ٢ + ٢١ﺱ ‪٤٤ G‬‬ ‫............................................................................................................‬ ‫.......................................................................................................................................................................................................................‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ﺟﺎ ‪ ٣ = i‬ﺣﻴﺚ ٠٩‪ ،c١٨٠ > i > c‬ﺃﻭﺟﺪ ﻗﻴﻤﺔ: ﺟﺘﺎ )٠٧٢‪،(i – c‬ﻇﺎ )٠٨١‪(i + c‬‬ ‫٥‬ ‫.........................‬ ‫.......................................................................................................................................................................................................................‬ ‫4 أ ﺿﻊ ﺍﻟﻌﺪﺩ ﻛﺐ ﺍﻵﺗﻰ ﻓﻰ ﺃﺑﺴﻂ ﺻﻮﺭﺓ )٦٢ – ٤ﺕ( – )٩ – ٠٢ ﺕ( ﺣﻴﺚ ﺕ٢ = -١‬ ‫ﺍﻟﻤﺮ‬ ‫ب ﺍﻟﺮﺑﻂ ﺑﺎﻟ ﺎﺿﺔ: ﻛﻞ ﻻﻋﺐ ﻛﺮﺓ ﺍﻟﻘﺪﻡ ﺍﻟﻜﺮﺓ ﻧﺤﻮ ﺍﻟﻬﺪﻑ ﻣﻦ ﻣﺴﺎﻓﺔ ﺱ ﻣﺘﺮﺍ ﻋﻦ ﺣﺎﺭﺱ ﺍﻟﻤﺮﻣﻰ،‬ ‫ﻳﺮ‬ ‫ﻓﻴﻘﻔﺰ ﺍﻟﺤﺎﺭﺱ ﻭﻳﻤﺴﻚ ﺍﻟﻜﺮﺓ ﻋﻠﻰ ﺍﺭﺗﻔﺎﻉ ١٫٢ ﻣﺘﺮﺍ ﻋﻦ ﺳﻄﺢ ﺍﻷﺭﺽ ﻓﺈﺫﺍ ﻛﺎﻥ‬ ‫ً‬ ‫ﻣﺴﺎﺭ ﺍﻟﻜﺮﺓ ﻳﻤﻴﻞ ﺑﺰﺍﻭﻳﺔ ﻗﻴﺎﺳﻬﺎ ٠٣‪ c‬ﻣﻊ ﺍﻷﻓﻘﻰ. ﻓﺄﻭﺟﺪ ﻷﻗﺮﺏ ﺭﻗﻢ ﻋﺸﺮﻯ‬ ‫‪c‬‬ ‫ﻭﺍﺣﺪ ﺍﻟﻤﺴﺎﻓﺔ ﺑﻴﻦ ﺍﻟﻼﻋﺐ ﻭﺣﺎﺭﺱ ﺍﻟﻤﺮﻣﻰ ﻋﻨﺪﻣﺎ ﺭﻛﻞ ﺍﻟﻼﻋﺐ ﺍﻟﻜﺮﺓ.‬ ‫.................................‬ ‫.............................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
  • 74.
    ‫ﺍﺧﺘﺒﺎﺭﺍﺕ ﻋﺎﻣﺔ‬ ‫)اﻟﺠﺒﺮ وﺣﺴﺎباﻟﻤﺜﻠﺜﺎت(‬ ‫اﻻﺧﺘﺒﺎر اﻟﺜﺎﻧﻰ‬ ‫‪k‬‬ ‫‪:IÉ£©ªdG äÉHÉLE’G ø«H øe áë«ë°üdG áHÉLE’G ôàNG :’hCG‬‬ ‫1 ﺃﺑﺴﻂ ﺻﻮﺭﺓ ﻟﻠﻌﺪﺩ ﺍﻟﺘﺨﻴﻠﻰ ﺕ٣٧ ﻫﻮ:‬ ‫ب ١‬ ‫أ -١‬ ‫...................................................................................................................................................‬ ‫ﺟ -ﺕ‬ ‫د ﺕ‬ ‫2 ﺍﻟﺪﺍﻟﺔ ﺩ: ]- ٤، ٧[ # ﺡ ﺣﻴﺚ ﺩ)ﺱ( = ٦ - ٢ﺱ ﺗﻜﻮﻥ ﺇﺷﺎﺭﺗﻬﺎ ﻣﻮﺟﺒﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ:‬ ‫د [ ٣، ٧ ]‬ ‫ﺟ ]- ٤، ٧[‬ ‫ب [ ٣، ٧ ]‬ ‫أ ]- ٤، ٣ ]‬ ‫.............................................‬ ‫3 ﺇﺫﺍ ﻛﺎﻥ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ٤ ﺱ٢ – ٢١ ﺱ + ﺟـ = ٠ ﻣﺘﺴﺎﻭﻳﻴﻦ ﻓﺈﻥ ﺟـ ﺗﺴﺎﻭﻯ:‬ ‫ﺟ ٩‬ ‫ب ٤‬ ‫أ ٣‬ ‫4 ﻇﺎ ‪ ` r -j‬ﺗﺴﺎﻭﻯ:‬ ‫٦‬ ‫أ - ٣‬ ‫...................................................................‬ ‫د ٦١‬ ‫.........................................................................................................................................................................................‬ ‫ب - ١‬ ‫٣‬ ‫ﺟ‬ ‫١‬ ‫٣‬ ‫د‬ ‫٣‬ ‫5 ﺍﻟﻘﻴﺎﺱ ﺍﻟﺪﺍﺋﺮﻯ ﻟﺰﺍﻭﻳﺔ ﻛﺰﻳﺔ ﺗﺤﺼﺮ ﻗﻮﺳﺎ ﻃﻮﻟﻪ ٣ﺳﻢ ﻣﻦ ﺩﺍﺋﺮﺓ ﻃﻮﻝ ﻗﻄﺮﻫﺎ ٤ﺳﻢ ﻫﻮ:‬ ‫ﻣﺮ‬ ‫ً‬ ‫‪E‬‬ ‫‪E‬‬ ‫٣ ‪E‬‬ ‫٢ ‪E‬‬ ‫د ٦‬ ‫ﺟ ٥‬ ‫ب )٢(‬ ‫أ )٣(‬ ‫....................................‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫1 أ ﺑﻴﻦ ﻧﻮﻉ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ + ٩ = ٦ ﺱ، ﺛﻢ ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺍﻟﺤﻞ.‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ: ٧ ﻗﺘﺎ ‪ ٢٥ = C‬ﺣﻴﺚ ‪ .r > C > r‬ﻓﺄﻭﺟﺪ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻌﺪﺩﻳﺔ ﻟﻠﻤﻘﺪﺍﺭ: ﻇﺎ ) ‪ - ( C + r‬ﻇﺘﺎ )‪( r - C‬‬ ‫٢‬ ‫٢‬ ‫.........................................................................‬ ‫.......................................................................................................................................................................................................................‬ ‫2‬ ‫أ ﺃﻭﺟﺪ ﻗﻴﻤﺘﻰ ‪ ،C‬ﺏ ﺍﻟﺤﻘﻴﻘﻴﺘﻴﻦ ﺍﻟﻠﺘﻴﻦ ﺗﺤﻘﻘﺎﻥ ﺍﻟﻤﻌﺎﺩﻟﺔ: )‪) – (٣ + C‬ﺏ – ١( ﺕ = ٧ – ٩ ﺕ ﺣﻴﺚ ﺕ٢ = -١‬ ‫.......................................................................................................................................................................................................................‬ ‫ب ﺣﻮﻝ ﻗﻴﺎﺱ ﻛﻞ ﻣﻦ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﻤﻜﺘﻮﺑﺔ ﺑﺎﻟﺪﺭﺟﺎﺕ ﺇﻟﻰ ﺭﺍﺩﻳﺎﻥ ﻭﺍﻟﻤﻜﺘﻮﺑﺔ ﺑﺎﻟﺮﺍﺩﻳﺎﻥ ﺇﻟﻰ ﺩﺭﺟﺎﺕ‬ ‫: ٨‪r‬‬ ‫  ..........................................................................‬ ‫: ٥١٢‪................................................................. c‬‬ ‫٦‬ ‫3 أ ﺍﺑﺤﺚ ﺇﺷﺎﺭﺓ ﺍﻟﺪﺍﻟﺔ ﺩ ﺣﻴﺚ ﺩ)ﺱ( = ٢ﺱ٢ – ٣ ﺱ + ٤ ﻣﻊ ﺗﻮﺿﻴﺢ ﺫﻟﻚ ﻋﻠﻰ ﺧﻂ ﺍﻷﻋﺪﺍﺩ ﺍﻟﺤﻘﻴﻘﻴﺔ‬ ‫ب ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺔ ‪ i‬ﻣﺮﺳﻮﻣﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ، ﺣﻴﺚ ﻳﻤﺮ ﺿﻠﻌﻬﺎ ﺍﻟﻨﻬﺎﺋﻰ ﺑﺎﻟﻨﻘﻄﺔ )٤، - ٣(‬ ‫ﻓﺄﻭﺟﺪ ﺟﺎ‪ ،i‬ﻇﺘﺎ‪.......................................................................................................................................................................... .i‬‬ ‫4‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ )ﺱ + ٢(٢ + )ﺱ + ١( )ﺱ – ٤( > ٠‬ ‫: ﺍﻛﺘﺐ ﺍﻟﻤﺘﺒﺎﻳﻨﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﻓﻰ ﺃﺑﺴﻂ ﺻﻮﺭﺓ.‬ ‫: ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺔ.‬ ‫.......................................................................................................................................................................................................................‬ ‫٢‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ﻝ ، ٢ ﻫﻤﺎ ﺟﺬﺭﺍ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ٢ – ٦ ﺱ + ٤ = ٠ ﻓﺄﻭﺟﺪ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ )ﻝ + ﻡ(، ﻝ ﻡ.‬ ‫ﻡ‬ ‫.......................................................................................................................................................................................................................‬ ‫‪ïM‬‬ ‫−‬
  • 75.
    ‫ﺍﺧﺘﺒﺎﺭﺍﺕ ﻋﺎﻣﺔ‬ ‫)اﻟﺠﺒﺮ وﺣﺴﺎباﻟﻤﺜﻠﺜﺎت(‬ ‫اﻻﺧﺘﺒﺎر اﻟﺜﺎﻟﺚ‬ ‫‪k‬‬ ‫‪IÉ£©ªdG äÉHÉLE’G ø«H øe áë«ë°üdG áHÉLE’G ôàNG :’hCG‬‬ ‫1 ﺇﺫﺍ ﻛﺎﻥ ﺃﺣﺪ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪C‬ﺱ٢ + ٢ ﺱ + ٥ = ٠ ﻣﻌﻜﻮﺳﺎ ﺿﺮﺑﻴﺎ ﻟﻠﺠﺬﺭ ﺍﻵﺧﺮ ﻓﺈﻥ ‪ C‬ﺗﺴﺎﻭﻯ:‬ ‫ً‬ ‫ًّ‬ ‫د ٥‬ ‫ﺟ ٢‬ ‫ب -٢‬ ‫أ -٥‬ ‫2 ﺇﺷﺎﺭﺓ ﺍﻟﺪﺍﻟﺔ ﺩ ﺣﻴﺚ ﺩ)ﺱ( = ٦ – ٢ ﺱ ﺗﻜﻮﻥ ﻣﻮﺟﺒﺔ ﺇﺫﺍ ﻛﺎﻧﺖ:‬ ‫ﺟ ﺱ>٣‬ ‫ب ﺱ‪٣G‬‬ ‫أ ﺱ<٣‬ ‫..........................‬ ‫..........................................................................................‬ ‫د ﺱ‪٣H‬‬ ‫3 ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ ١ + ﺕ ، ١ – ﺕ ﺣﻴﺚ ﺕ٢ = -١ ﻫﻰ:‬ ‫أ ﺱ٢ + ٢ﺱ + ٢ = ٠ ب ﺱ٢ – ٢ﺱ + ٢ = ٠ ﺟ ﺱ٢ + ٢ﺱ – ٢ = ٠‬ ‫...............................................................................‬ ‫د ﺱ٢ – ٢ﺱ – ٢ = ٠‬ ‫4 ﺇﺫﺍ ﻛﺎﻧﺖ ‪ i‬ﺯﺍﻭﻳﺔ ﻣﺮﺳﻮﻣﺔ ﻓﻰ ﺍﻟﻮﺿﻊ ﺍﻟﻘﻴﺎﺳﻰ ﺑﺤﻴﺚ ﺟﺘﺎ ‪ ،٠ < i‬ﻓﻰ ﺃﻯ ﺭﺑﻊ ﻳﻘﻊ ﺿﻠﻊ ﺍﻟﻨﻬﺎﻳﺔ ﻟﻠﺰﺍﻭﻳﺔ ‪:i‬‬ ‫د ﺍﻷﻭﻝ ﺃﻭ ﺍﻟﺮﺍﺑﻊ‬ ‫ﺟ ﺍﻷﻭﻝ ﺃﻭ ﺍﻟﺜﺎﻟﺚ‬ ‫ب ﺍﻷﻭﻝ ﺃﻭ ﺍﻟﺜﺎﻧﻰ‬ ‫أ ﺍﻷﻭﻝ‬ ‫5 ﺇﺫﺍ ﻛﺎﻧﺖ ٢ ﺟﺘﺎ ‪ ٢ - = C‬ﻓﺈﻥ ﺃﻗﻞ ﺯﺍﻭﻳﺔ ﻣﻮﺟﺒﺔ ﺗﺤﻘﻖ ﻫﺬه ﺍﻟﺪﺍﻟﺔ ﺍﻟﻤﺜﻠﺜﻴﺔ ﻫﻰ:‬ ‫د ٥١٣‪c‬‬ ‫ﺟ ٥٢٢‪c‬‬ ‫ب ٥٣١‪c‬‬ ‫أ ٥٤‪c‬‬ ‫.........................................................‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫1‬ ‫أ ﺇﺫﺍ ﻛﺎﻥ ﻝ، ﻡ ﺟﺬﺭﻯ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺱ )٢ ﺱ + ٣( = ٥ ﻓﺄﻭﺟﺪ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﻰ ﺟﺬﺭﺍﻫﺎ ﻝ + ١ ، ﻡ + ١.‬ ‫.................‬ ‫.......................................................................................................................................................................................................................‬ ‫ب ﺯﺍﻭﻳﺔ ﻛﺰﻳﺔ ﻗﻴﺎﺳﻬﺎ ٠٦‪ c‬ﻭﺗﻘﺎﺑﻞ ﻗﻮﺳﺎ ﻃﻮﻟﻪ ٧‪ r‬ﺳﻢ، ﺍﺣﺴﺐ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮ ﺩﺍﺋﺮﺗﻬﺎ.‬ ‫ﻣﺮ‬ ‫ً‬ ‫٣‬ ‫...........................‬ ‫.......................................................................................................................................................................................................................‬ ‫2 أ ﺿﻊ ﺍﻟﻌﺪﺩ ٢ - ٣ﺕ ﻓﻰ ﺻﻮﺭﺓ ﻋﺪﺩ ﻛﺐ. ﺣﻴﺚ ﺕ٢ = -١.‬ ‫ﻣﺮ‬ ‫٣ + ٢ﺕ‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ٤ ﺟﺎ ‪ ٠ = ٣ - C‬ﺃﻭﺟﺪ ‪ ( Cc) X‬ﺣﻴﺚ ‪ ،٠ [ ∋ C‬ﻁ‬ ‫] ........................................................................................‬ ‫٢‬ ‫.........................................................................................‬ ‫3‬ ‫أ ﺇﺫﺍ ﻛﺎﻧﺖ ﺩ : ﺡ # ﺡ ﺣﻴﺚ ﺩ)ﺱ( = - ﺱ٢ + ٨ ﺱ – ٥١‬ ‫: ﻋﻴﻦ ﻣﻦ ﺍﻟﺮﺳﻢ ﺇﺷﺎﺭﺓ ﻫﺬه ﺍﻟﺪﺍﻟﺔ.‬ ‫: ﺍﺭﺳﻢ ﻣﻨﺤﻨﻰ ﺍﻟﺪﺍﻟﺔ ﻓﻰ ﺍﻟﻔﺘﺮﺓ ] ١، ٧ [‬ ‫.......................................................................................................................................................................................................................‬ ‫‬‫ب ﺇﺫﺍ ﻛﺎﻥ ﺱ = ٣ + ٢ﺕ، ﺹ = ٤١ -٢ﺕ ﻓﺄﻭﺟﺪ ﺱ + ﺹ ﻓﻰ ﺻﻮﺭﺓ ﻋﺪﺩ ﻛﺐ.‬ ‫ﻣﺮ‬ ‫ﺕ‬ ‫..................................................‬ ‫4 أ ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﺘﺒﻴﺎﻳﻨﺔ ﺱ٢ + ٣ﺱ – ٤ ‪٠ H‬‬ ‫ب ﺇﺫﺍ ﻛﺎﻥ ﻇﺎ ﺏ = ٣ ﺣﻴﺚ ٠٨١‪ > c‬ﺏ > ٠٧٢‪ c‬ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ: ﺟﺘﺎ )٠٦٣‪ – c‬ﺏ( - ﺟﺘﺎ )٠٩‪ – c‬ﺏ(‬ ‫٤‬ ‫..............................................................................................................‬ ‫.......................................................................................................................................................................................................................‬ ‫¯‬ ‫−‬ ‫¯‬
  • 76.
    ‫ﺍﺧﺘﺒﺎﺭﺍﺕ ﻋﺎﻣﺔ‬ ‫)اﻟﻬﻨﺪﺳﺔ(‬ ‫اﻻﺧﺘﺒﺎر اﻟﺮاﺑﻊ‬ ‫‪k‬‬ ‫‪πªcCG:’hCG‬‬ ‫1 ﺇﺫﺍ ﻗﻄﻊ ﻣﺴﺘﻘﻴﻤﺎﻥ ﻋﺪﺓ ﻣﺴﺘﻘﻴﻤﺎﺕ ﻣﺘﻮﺍﺯﻳﺔ، ﻓﺈﻥ ﺃﻃﻮﺍﻝ ﺍﻟﻘﻄﻊ ﺍﻟﻨﺎﺗﺠﺔ ﻋﻠﻰ ﺃﺣﺪ ﺍﻟﻘﺎﻃﻌﻴﻦ ﺗﻜﻮﻥ‬ ‫....................‬ ‫2 ﺍﻟﻨﺴﺒﺔ ﺑﻴﻦ ﻣﺴﺎﺣﺘﻰ ﺳﻄﺤﻰ ﻣﺜﻠﺜﻴﻦ ﻣﺘﺸﺎﺑﻬﻴﻦ ﻫﻲ ٣ : ٥، ﺇﺫﺍ ﻛﺎﻧﺖ ﻣﺴﺎﺣﺔ ﺳﻄﺢ ﺍﻟﻤﺜﻠﺚ ﺍﻷﻭﻝ ٦٣ ﺳﻢ٢ ﻓﺈﻥ‬ ‫ﻣﺴﺎﺣﺔ ﺳﻄﺢ ﺍﻟﻤﺜﻠﺚ ﺍﻟﺜﺎﻧﻰ ﺗﺴﺎﻭﻯ .....................................................................................................................................................‬ ‫‪C‬‬ ‫3‬ ‫: ﺇﺫﺍ ﻛﺎﻥ ﺱ ﺹ // ﺏ ﺟـ ، ﺱ ﺹ : ﺏ ﺟـ = ٣ : ٨ ﻓﺈﻥ:‬ ‫أ ‪C‬ﺱ : ﺱ ﺏ = ........................... : ...........................‬ ‫ب‬ ‫ﻣﺤﻴﻂ 9‪C‬ﺱ ﺹ : ﻣﺤﻴﻂ 9‪C‬ﺏ ﺟـ = ......................... : ..........................‬ ‫4‬ ‫: ﺇﺫﺍ ﻛﺎﻥ ﺟـ ‪ E‬ﻳﻨﺼﻒ )‪c‬ﺟـ(،‬ ‫‪ C‬ﺟـ = ٣ ﺳﻢ، ﺏ ﺟـ = ٥٫٧ ﺳﻢ، ﻓﺈﻥ ‪ : E C‬ﺏ ‪..................................................... = E‬‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫1 أ ﺃﻭﺟﺪ ﻗﻮﺓ ﺍﻟﻨﻘﻄﺔ ‪ C‬ﺑﺎﻟﻨﺴﺒﺔ ﺇﻟﻰ ﺍﻟﺪﺍﺋﺮﺓ ﻡ ﺍﻟﺘﻰ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٣ ﺳﻢ ، ‪C‬ﻡ = ٤ ﺳﻢ.‬ ‫ب ﺭﺳﻢ ﻣﻬﻨﺪﺱ ﻣﻌﻤﺎﺭﻯ ﻣﺨﻄﻄًﺎ ﻟﻘﻄﻌﺔ ﺃﺭﺽ ﻣﺴﺘﻄﻴﻠﺔ ﺍﻟﺸﻜﻞ، ﻃﻮﻟﻬﺎ ﺿﻌﻒ ﻋﺮﺿﻬﺎ، ﻭﻣﺴﺎﺣﺘﻬﺎ ٠٠٢‬ ‫ﻣﺘﺮ٢ ﺑﻤﻘﻴﺎﺱ ﺭﺳﻢ ١ : ٠٠٢، ﺃﻭﺟﺪ ﻃﻮﻝ ﻗﻄﻌﺔ ﺍﻷﺭﺽ ﻓﻰ ﺍﻟﻤﺨﻄﻂ.‬ ‫: ﺱ ﺹ // ‪ E‬ﻫـ // ﻝ ﻉ ﺃﻭﺟﺪ:‬ ‫2‬ ‫: ﻃﻮﻝ ﻫـ ﻡ‬ ‫‪E‬‬ ‫: ﻃﻮﻝ ﻡ ﻉ‬ ‫?‬ ‫?‬ ‫3‬ ‫: ‪ C‬ﺏ ﻗﻄﺮ ﻓﻰ ﺍﻟﺪﺍﺋﺮﺓ،‬ ‫ﺟـ ‪ E‬ﻣﻤﺎﺱ ﻟﻠﺪﺍﺋﺮﺓ ﻋﻨﺪ ﺟـ ، ‪C‬ﺟـ = ٢١ ﺳﻢ، ﺍﺏ = ٣١ ﺳﻢ. ﺃﺛﺒﺖ ﺃﻥ:‬ ‫أ 9 ‪ E‬ﺟـ ﺏ + 9 ‪ C E‬ﺟـ‬ ‫ب ﺃﻭﺟﺪ ﻃﻮﻝ ﺟـ ‪ E‬ﻷﻗﺮﺏ ﺳﻢ‬ ‫ﺟ ﺃﻭﺟﺪ ﻣﺴﺎﺣﺔ 9 ‪C‬ﺏ ﺟـ‬ ‫‪C‬‬ ‫‪E‬‬ ‫4 ‪C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ‪ ،C‬ﻓﻴﻪ ‪ C‬ﺏ = ٠٢ ﺳﻢ، ‪C‬ﺟـ = ٥١ ﺳﻢ، ‪ ∋ E‬ﺏ ﺟـ ﺑﺤﻴﺚ ﻛﺎﻥ ﺏ ‪ ١٠ = E‬ﺳﻢ،‬ ‫ﺭﺳﻢ ‪ C‬ﻫـ = ﺏ ﺟـ ﻭﻳﻘﻄﻊ ﺏ ﺟـ ﻓﻰ ﻫـ ، ﻭﻣﻦ ‪ E‬ﺭﺳﻢ ‪ E‬ﻭ // ﺏ ‪ C‬ﻭﻳﻘﻄﻊ ‪ C‬ﻫـ ﻓﻰ ﻭ.‬ ‫ﺃﺛﺒﺖ ﺃﻥ ﺟـ ﻭ ﻳﻨﺼﻒ ‪c‬ﺟـ.‬ ‫‪ïM‬‬ ‫−‬
  • 77.
    ‫ﺍﺧﺘﺒﺎﺭﺍﺕ ﻋﺎﻣﺔ‬ ‫)اﻟﻬﻨﺪﺳﺔ(‬ ‫اﻻﺧﺘﺒﺎر اﻟﺨﺎﻣﺲ‬ ‫‪k‬‬ ‫‪:πªcCG:’hCG‬‬ ‫1 ﺍﻟﻨﺴﺒﺔ ﺑﻴﻦ ﻣﺴﺎﺣﺘﻰ ﺳﻄﺤﻰ ﻣﺜﻠﺜﻴﻦ ﻣﺘﺸﺎﺑﻬﻴﻦ ﻛﺎﻟﻨﺴﺒﺔ ﺑﻴﻦ‬ ‫.....................................................................................................‬ ‫2 ﻳﺘﺸﺎﺑﻪ ﺍﻟﻤﻀﻠﻌﺎﻥ ﺇﺫﺍ ﻛﺎﻥ‬ ‫................................................................... ...................................................................‬ ‫3‬ ‫:‬ ‫،‬ ‫‪E‬‬ ‫أ )‪= ٢(E C‬‬ ‫ب‬ ‫‪ E‬ﻥ * ﻥ ﻫـ = ....................................................‬ ‫ﺟ‬ ‫9 ‪ E C‬ﺟـ + 9 ..............................................‬ ‫....................................................................‬ ‫‪C‬‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫1 أ ﺃﻭﺟﺪ ﻗﻮﺓ ﺍﻟﻨﻘﻄﺔ ﺏ ﺑﺎﻟﻨﺴﺒﺔ ﺇﻟﻰ ﺍﻟﺪﺍﺋﺮﺓ ﻡ، ﺍﻟﺘﻰ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٨ ﺳﻢ، ﺏ ﻡ = ٥ ﺳﻢ‬ ‫ب‬ ‫:‬ ‫: ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻤﻀﻠﻊ ‪ C‬ﺏ ﺟـ ‪ + E‬ﺍﻟﻤﻀﻠﻊ ﺱ ﺏ ﻉ ﺹ‬ ‫‪C‬‬ ‫   ﻓﺎﺛﺒﺖ ﺃﻥ: ﺱ ﺹ // ‪. E C‬‬ ‫: ﺇﺫﺍ ﻛﺎﻥ ﻣﺤﻴﻂ ﺍﻟﻤﻀﻠﻊ ‪ C‬ﺏ ﺟـ ‪ ١٤ = E‬ﺳﻢ،‬ ‫   ﻣﺤﻴﻂ ﺍﻟﻤﻀﻠﻊ ﺱ ﺏ ﻉ ﺹ = ٠١ ﺳﻢ،‬ ‫   ﻃﻮﻝ ﺱ ﺏ = ٢ ﺳﻢ، ﻓﺄﻭﺟﺪ ﻃﻮﻝ ‪ C‬ﺏ‬ ‫2‬ ‫: ‪ C‬ﺏ = ٦ ﺳﻢ، ﺏ ﺟـ = ٢١ ﺳﻢ،‬ ‫ﺟـ ‪ ٨ = C‬ﺳﻢ، ﻭﺟـ = ٣ ﺳﻢ ، ‪ E‬ﺏ = ٥٫٤ ﺳﻢ ، ‪ E‬ﻭ = ٦ ﺳﻢ.‬ ‫ﺃﺛﺒﺖ ﺃﻥ:‬ ‫أ 9 ‪ C‬ﺏ ﺟـ + 9 ‪ E‬ﺏ ﻭ‬ ‫ب 9 ﻫـ ﻭ ﺟـ ﻣﺘﺴﺎﻭﻯ ﺍﻟﺴﺎﻗﻴﻦ.‬ ‫‪E‬‬ ‫‪C‬‬ ‫‪E‬‬ ‫3 ﺱ ﺹ ﻉ ﻣﺜﻠﺚ، ﻧﺼﻔﺖ ﺯﺍﻭﻳﺔ ﺹ ﺑﻤﻨﺼﻒ ﻗﻄﻊ ﺱ ﻉ ﻓﻰ ﻡ، ﺛﻢ ﺭﺳﻢ ﻥ ﻡ // ﺹ ﻉ ﻓﻘﻄﻊ ﺱ ﺹ ﻓﻰ ﻥ.‬ ‫ﺱﻥ‬ ‫ﺱﺹ‬ ‫ﺃﺛﺒﺖ ﺃﻥ: ﺹ ﻉ = ﺹ ﻥ ، ﻭﺇﺫﺍ ﻛﺎﻥ ﺱ ﺹ = ٦ ﺳﻢ ، ﺹ ﻉ = ٤ ﺳﻢ، ﻓﺄﻭﺟﺪ ﻃﻮﻝ ﺱ ﻥ .‬ ‫4 ‪ C‬ﺏ ﺟـ ﻣﺜﻠﺚ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻰ ‪ .C‬ﺭﺳﻢ ‪ = E C‬ﺏ ﺟـ ﻓﻘﻄﻌﻬﺎ ﻓﻰ ‪.E‬‬ ‫ﺭﺳﻢ ﺍﻟﻤﺜﻠﺜﺎﻥ ﺍﻟﻤﺘﺴﺎﻭﻳﺎ ﺍﻷﺿﻼﻉ ‪ C‬ﺏ ﻫـ ، ﺟـ ‪ C‬ﻭ ﺧﺎﺭﺝ ﺍﻟﻤﺜﻠﺚ ‪ C‬ﺏ ﺟـ‬ ‫ﺃﺛﺒﺖ ﺃﻥ:‬ ‫أ ﺍﻟﺸﻜﻞ ﺍﻟﺮﺑﺎﻋﻰ ‪ E C‬ﺏ ﻫـ + ﺍﻟﺸﻜﻞ ﺍﻟﺮﺑﺎﻋﻰ ﺟـ ‪ C E‬ﻭ.‬ ‫ب ﻣﺴﺎﺣﺔ ﺳﻄﺢ ﺍﻟﺸﻜﻞ ‪ E C‬ﺏ ﻫـ ﺏ ‪E‬‬ ‫=‬ ‫ﻣﺴﺎﺣﺔ ﺳﻄﺢ ﺍﻟﺸﻜﻞ ﺟـ ‪ C E‬ﻭ ﺟـ ‪E‬‬ ‫¯‬ ‫−‬ ‫¯‬
  • 78.
    ‫ﺍﺧﺘﺒﺎﺭﺍﺕ ﻋﺎﻣﺔ‬ ‫)اﻟﻬﻨﺪﺳﺔ(‬ ‫اﻻﺧﺘﺒﺎر اﻟﺴﺎدس‬ ‫‪k‬‬ ‫‪:πªcCG:’hCG‬‬ ‫1 أ ﺇﺫﺍ ﺭﺳﻢ ﻣﺴﺘﻘﻴﻢ ﻳﻮﺍﺯﻯ ﺃﺣﺪ ﺃﺿﻼﻉ ﻣﺜﻠﺚ، ﻭ ﻳﻘﻄﻊ ﺍﻟﻀﻠﻌﻴﻦ ﺍﻵﺧﺮﻳﻦ ﻓﺈﻧﻪ‬ ‫ُ‬ ‫ب‬ ‫‪ E C‬ﻣﻤﺎﺱ ﻟﻠﺪﺍﺋﺮﺓ ﻋﻨﺪ ‪ ،E‬ﻓﺈﻥ:‬ ‫:‬ ‫: ‪ C‬ﺟـ * ‪ C‬ﺏ = .........................................‬ ‫: ﺇﺫﺍ ﻛﺎﻥ ‪ C‬ﺟـ = ٨ ﺳﻢ، ‪ C‬ﺏ = ٢ﺳﻢ، ﻓﺈﻥ ‪.................................................... = E C‬‬ ‫٢ ﺳﻢ ﻓﺈﻥ، ‪ C‬ﺟـ = .................................‬ ‫: ﺇﺫﺍ ﻛﺎﻥ ‪ C‬ﺏ = ﺏ ﺟـ ، ‪٣ = E C‬‬ ‫........................................................‬ ‫‪C‬‬ ‫‪E‬‬ ‫‪:á«JB’G á∏Ä°SC’G øY ÖLCG :Ék«fÉK‬‬ ‫1 أ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻨﺴﺒﺔ ﺑﻴﻦ ﻣﺴﺎﺣﺘﻰ ﺳﻄﺤﻰ ﻣﻀﻠﻌﻴﻦ ﻣﺘﺸﺎﺑﻬﻴﻦ ﺗﺴﺎﻭﻯ ٦١ : ٩٤، ﻓﻤﺎ ﺍﻟﻨﺴﺒﺔ ﺑﻴﻦ ﻃﻮﻟﻰ ﺿﻠﻌﻴﻦ‬ ‫ﻣﺘﻨﺎﻇﺮﻳﻦ ﻓﻴﻬﻤﺎ? ﻭﻣﺎ ﺍﻟﻨﺴﺒﺔ ﺑﻴﻦ ﻣﺤﻴﻄﻴﻬﻤﺎ?‬ ‫ب ﺩﺍﺋﺮﺗﺎﻥ ﻣﺘﻘﺎﻃﻌﺘﺎﻥ ﻓﻰ ‪ ،C‬ﺏ ﺭﺳﻢ ﻣﻤﺎﺱ ﻣﺸﺘﺮﻙ ﻳﻤﺴﺎﻧﻬﻤﺎ ﻓﻰ ﺱ، ﺹ.‬ ‫ﺇﺫﺍ ﻛﺎﻥ ‪ C‬ﺏ ∩ ﺱ ﺹ = }ﺟـ{ ﺍﺛﺒﺖ ﺃﻥ ﺟـ ﻣﻨﺘﺼﻒ ﺱ ﺹ .‬ ‫2‬ ‫أ‬ ‫: ‪ C‬ﺱ // ﺏ ﺹ // ﺟـ ﻉ ،‬ ‫ﻭ ‪ ٦ = C‬ﺳﻢ ، ﻭ ﺱ = ٤ ﺳﻢ ، ﺱ ﺹ = ٣ ﺳﻢ،‬ ‫ﺏ ﺟـ = ٥٫٧ ﺳﻢ. ﺃﻭﺟﺪ ﻃﻮﻝ ﻛﻞ ﻣﻦ ‪ C‬ﺏ ، ﻉ ﺹ‬ ‫‪C‬‬ ‫ب‬ ‫:‬ ‫9 ﺟـ ‪ E‬ﻫـ + 9 ﺟـ ﺏ ‪C‬‬ ‫ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻷﻃﻮﺍﻝ ﺍﻟﻤﻮﺿﺤﺔ ﻋﻠﻰ ﺍﻟﺮﺳﻢ‬ ‫ﺃﻭﺟﺪ ﻃﻮﻝ ﻛﻞ ﻣﻦ ﺏ ﻫـ ، ‪ E‬ﻫـ .‬ ‫3‬ ‫‪E‬‬ ‫‪C‬‬ ‫?‬ ‫أ ﺃﻭﺟﺪ ﻗﻮﺓ ﺍﻟﻨﻘﻄﺔ ﺟـ ﺑﺎﻟﻨﺴﺒﺔ ﺇﻟﻰ ﺍﻟﺪﺍﺋﺮﺓ ﻡ ﺍﻟﺘﻰ ﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ٦ ﺳﻢ، ﺟـ ﻡ = ٦ ﺳﻢ‬ ‫ب‬ ‫: ‪ C‬ﺏ ∩ ‪ E‬ﻫـ = }ﺟـ{،‬ ‫ﺟـ ‪ = C‬ﺟـ ﺏ ، ﺟـ ‪٢ = E‬ﺳﻢ ، ﺟـ ﻫـ = ٨ ﺳﻢ،‬ ‫ﻡ ‪ E‬ﻣﻤﺎﺳﺔ ﻟﻠﺪﺍﺋﺮﺓ. ﻡ ﺏ = ١ ‪ C‬ﺏ. ﺃﻭﺟﺪ ﻃﻮﻝ ﻡ ‪. E‬‬ ‫٢‬ ‫‪E‬‬ ‫‪C‬‬ ‫4‬ ‫: ‪C‬ﺏ ﺟـ ﻣﺜﻠﺚ، ﻓﻴﻪ ﺱ ∋ ‪ C‬ﺏ ﺑﺤﻴﺚ ﻛﺎﻥ ‪ C‬ﺱ = ٤ ﺳﻢ،‬ ‫ﺱ ﺏ = ٦ ﺳﻢ، ﺹ ∋ ‪ C‬ﺟـ ﺑﺤﻴﺚ ﻛﺎﻥ ‪ C‬ﺹ = ٥ ﺳﻢ، ﺹ ﺟـ = ٣ ﺳﻢ.‬ ‫أ ﺃﺛﺒﺖ ﺃﻥ: 9 ‪ C‬ﺱ ﺹ + 9 ‪ C‬ﺟـ ﺏ‬ ‫ب ﺍﻟﺸﻜﻞ ﺱ ﺏ ﺟـ ﺹ ﺭﺑﺎﻋﻰ ﺩﺍﺋﺮﻯ.‬ ‫ﺟ ﺇﺫﺍ ﻛﺎﻧﺖ ﻣـ)9 ‪ C‬ﺱ ﺹ( = ٨ ﺳﻢ٢. ﺃﻭﺟﺪ ﻣﺴﺎﺣﺔ ﺳﻄﺢ ﺍﻟﻤﻀﻠﻊ ﺱ ﺏ ﺟـ ﺹ.‬ ‫‪ïM‬‬ ‫−‬ ‫‪C‬‬
  • 79.
    ‫‪z‬‬ ‫ﺇﺟﺎﺑﺎﺕ ﺑﻌﺾ ﺍﻟﺘﻤﺎﺭﻳﻦ‬ ‫−‬ ‫:‬ ‫¯‬ ‫1ﺏ    2 ﺩ‬ ‫7 أ }-٢{‬ ‫د ﺱ٢ + ٠١ ﺱ - ١٢ = ٠‬ ‫ﺟ ٣ﺱ٢ + ٤١ﺱ + ٤ = ٠‬ ‫22 ٤٥ * ٢ = )ﺱ + ٦()ﺱ + ٩( ﻭﻣﻨﻬﺎ ﺱ٢ +٥١ﺱ -٤٥ = ٠ ﻭﻣﻨﻬﺎ ﺱ = ٣‬ ‫   3 ﺩ‬ ‫ب‬ ‫   4 ﺩ‬ ‫42 ﺣﻞ ﻳﻮﺳﻒ ﺻﺤﻴﺢ‬ ‫‪z‬‬ ‫ﺟ }-٣، ١{‬ ‫8 أ } -٥، ٨{‬ ‫د }٤٧٫٦، -٤٧٫٠{‬ ‫9‬ ‫5 ﺃ‬ ‫−‬ ‫ﺟ }- ٣ ، ٢ {‬ ‫٢ ٣‬ ‫1 ﺳﺎﻟﺒﺔ ، ﺡ‬ ‫5 [٣، ∞]‬ ‫ﻫ }١٦٫٢ ، -١٦٫٤{‬ ‫و }٤١٫٢، -٤٩٫٠{‬ ‫8 [٢، ∞]، [- ∞، ٢]‬ ‫9 }-١، ٣{، ﺡ -]-١، ٣[، [-١، ٣]‬ ‫ب }٤٫٤، ٦٫١‬ ‫ﺟ }-٤، -٢{‬ ‫ﺟ ﻥ = ٢٢‬ ‫ب ﻥ = ٨١‬ ‫01 أ ﻥ = ٢١‬ ‫٢‬ ‫٢‬ ‫11 أ ﺩ)ﺱ( = ﺱ٢ + ﺱ - ٦ ب ﺩ)ﺱ( = -ﺱ - ٣ﺱ ﺟ ﺩ)ﺱ( = ﺱ - ٧ﺱ‬ ‫01 أ ﻣﻮﺟﺒﺔ ﻓﻰ ﺡ‬ ‫ب ﻣﻮﺟﺒﺔ ﻓﻰ [ ٠، ∞ ] ﺳﺎﻟﺒﺔ ﻓﻰ [- ∞،٠ ]، ﺻﻔﺮ ﻋﻨﺪﻣﺎ ﺱ =٠‬ ‫ﺟ ﻣﻮﺟﺒﺔ ﻓﻰ[- ∞، ٠ ] ﺳﺎﻟﺒﺔ ﻓﻰ [ ٠،∞ ]، ﺻﻔﺮ ﻋﻨﺪﻣﺎ ﺱ =٠‬ ‫د ﻣﻮﺟﺒﺔ ﻋﻨﺪﻣﺎ ﺱ < -٢ ، ﺳﺎﻟﺒﺔ ﻋﻨﺪﻣﺎ ﺱ > -٢، ﺻﻔﺮ ﻋﻨﺪﻣﺎ ﺱ = -٢‬ ‫21 ﺇﺟﺎﺑﺔ ﺯﻳﺎﺩ ﺧﻄﺄ؛ ﻷﻧﻪ ﻗﺴﻢ ﺍﻟﻄﺮﻓﻴﻦ ﻋﻠﻰ ﻣﺘﻐﻴﺮ ﻭﻫﻮ )ﺱ - ٣(‬ ‫31 ﻥ = ٢ ﺃﻭ ﻥ = ٤‬ ‫ﻫ ﻣﻮﺟﺒﺔ ﻋﻨﺪﻣﺎ ﺱ > ٣ ، ﺳﺎﻟﺒﺔ ﻋﻨﺪﻣﺎ ﺱ < ٣ ، ﺻﻔﺮ ﻋﻨﺪﻣﺎ ﺱ = ٣‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫ح ﻣﻮﺟﺒﺔ ﻓﻰ ﺡ - ]-٢ ،٢[] ﺳﺎﻟﺒﺔ ﻓﻰ [ - ٢، ٢ ]، ﺻﻔﺮ ﻋﻨﺪﻣﺎ ﺱ∋ } -٢، ٢{‬ ‫−‬ ‫1‬ ‫3‬ ‫4‬ ‫أ ١ + ٥ﺕ‬ ‫5‬ ‫أ ١-ﺕ‬ ‫ﺟ !٣ ٢ ﺕ‬ ‫6‬ ‫7 ٧ - ٢ﺕ‬ ‫ﺟ‬ ‫ب -ﺕ‬ ‫ب ٧١ + ٦١ ﺕ ﺟ‬ ‫ب‬ ‫ﺟ‬ ‫ب ١ - ٤ﺕ‬ ‫د‬ ‫د -ﺕ‬ ‫١‬‫١١ + ٥٤ ﺕ‬ ‫٤ + ٧ﺕ‬ ‫٣‬ ‫٠١ - ١١ ﺕ د ٦ + ٨ ﺕ‬ ‫٥ ٥‬ ‫٠١‬ ‫!٥ﺕ‬ ‫8 ﺣﻞ ﺃﺣﻤﺪ ﺻﺤﻴﺢ.‬ ‫2 ﺏ‬ ‫3 ﺃ‬ ‫أ ﺟﺬﺭﺍﻥ ﻛﺒﺎﻥ.‬ ‫ﻣﺮ‬ ‫ب ﺟﺬﺭﺍﻥ ﺣﻘﻴﻘﻴﺎﻥ ﻏﻴﺮ ﻧﺴﺒﻴﻴﻦ‬ ‫د ﺟﺬﺭﺍﻥ ﻛﺒﺎﻥ‬ ‫ﻣﺮ‬ ‫ب - ٣ + ١ ﺕ،- ٣ - ١‬ ‫ﺕ‬ ‫5‬ ‫أ ٢ + ﺕ، ٢ - ﺕ‬ ‫6‬ ‫أ ٦١ - ٤ ﻙ < ٠ ﺃﻯ ﻙ > ٤‬ ‫٢‬ ‫٢‬ ‫ب ٩ - ٤ * )٢ +‬ ‫ﺟ ٤٦ - ٤ * ٦١ ﻙ > ٠ ﺃﻯ ﺃﻥ: ﻙ < ١‬ ‫٢‬ ‫٢‬ ‫١‬ ‫ﻙ ( = ٠ ﺃﻯ ﺃﻥ ﻙ = ٤‬ ‫7 ﺍﻟﻤﻤﻴﺰ = ) ﻝ - ﻡ(٢ + ٤ ﻝ ﻡ = ) ﻝ - ﻡ (٢ ﺃﻱ ﻣﺮﺑﻊ ﻛﺎﻣﻞ، ﻟﺬﻟﻚ ﻓﺈﻥ ﺟﺬﺭﻯ‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ ﻋﺪﺩﺍﻥ ﻧﺴﺒﻴﺎﻥ.‬ ‫9 ﺇﺟﺎﺑﺔ ﺃﺣﻤﺪ ﺧﻄﺄ؛ ﻷﻥ ﺍﻟﺤﺪ ﺍﻟﻤﻄﻠﻖ = -٥ ﻓﻰ ﺍﻟﻤﻌﺎﺩﻟﺔ.‬ ‫11 ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﻫﻮ }٣ ﺕ ، - ٢ﺕ{‬ ‫−‬ ‫٦، -٩‬ ‫1‬ ‫5 ﺟـ‬ ‫3 ﺱ٢ - ٥ﺱ + ٦ =٠‬ ‫7 ﺟـ‬ ‫2 ٨‬ ‫6 ﺟـ‬ ‫8‬ ‫أ -٩١ ، -٤١‬ ‫٣‬ ‫٣‬ ‫9‬ ‫أ -٣، ٣‬ ‫01‬ ‫أ ﺗﻜﻮﻥ ‪ ، ٧- = C‬ﺏ = ٠١‬ ‫11‬ ‫ب -١، -٥٣‬ ‫٤‬ ‫ب ٢، ١‬ ‫٢‬ ‫أ ﺣﻘﻴﻘﻴﺎﻥ ﻧﺴﺒﻴﺎﻥ ، }-٧، ٥{‬ ‫ب ‪ ،١ = C‬ﺏ = ٤‬ ‫ب ﻛﺒﺎﻥ ، } - ٣ + ٧٫١ﺕ ، - ٣ - ٧٫١ ﺕ{‬ ‫ﻣﺮ‬ ‫٤‬ ‫٤‬ ‫ﺟ ﻛﺒﺎﻥ }٢ + ﺕ ، ٢ - ﺕ{‬ ‫ﻣﺮ‬ ‫21 ﺟـ = ٤‬ ‫51 ﻙ = ١‬ ‫ﺩ)ﺱ( <٠ ﻋﻨﺪﻣﺎ ﺱ∋ ]-٢٫١ ، ٢٫٣]‬ ‫1 ]-٣، ٣[‬ ‫5 [٢، ٥]‬ ‫1 ﺏ‬ ‫71‬ ‫1‬ ‫٣‬ ‫41 ﺟـ = ٥٢ ، ﺍﻟﺤﻞ ﻫﻮ } ٥ {‬ ‫٦‬ ‫٢١‬ ‫ب‬ ‫71 أ ﺱ٢ - ٢ﺱ -٨ = ٠‬ ‫ﺟ ٦ﺱ٢ - ٣١ﺱ + ٦ = ٠‬ ‫81 ﺱ٢ - ٨ﺱ + ٥ = ٠‬ ‫12 أ ﺱ٢ +٤١ﺱ +٢١ =٠‬ ‫ﻫـ‬ ‫91‬ ‫ب‬ ‫−‬ ‫ﺱ٢ + ٥٢ = ٠‬ ‫ﺱ٢ + ٧١ = ٠‬ ‫ﺱ٢ - ٩ﺱ -١ = ٠‬ ‫ﺱ٢ - ١١ﺱ + ١٢ = ٠‬ ‫¯‬ ‫2 ]-١، ١[‬ ‫6 ]-٣، ١[‬ ‫7 ‪z‬‬ ‫8 ‪z‬‬ ‫2 ﺩ‬ ‫ب ١١، -٣١‬ ‫أ ٥، -٤‬ ‫3 ﺃ‬ ‫ﺟ ٤، -٢‬ ‫4 ﺏ‬ ‫٤‬ ‫أ ﻙ= ٣‬ ‫3 ﺡ - ]٠، ٢[ 4 ‪z‬‬ ‫¯‬ ‫2‬ ‫3‬ ‫أ ﻙ=- ٣ ،ﻙ=٦‬ ‫٢‬ ‫٣‬ ‫ﺟ‬ ‫ﻙ< ٢‬ ‫٧‬ ‫ﺟ‬ ‫ﻙ= ٢‬ ‫٤‬ ‫ب‬ ‫ﻙ> ٣‬ ‫ب ﻙ=٦‬ ‫أ ﺱ٢ - ٩ﺱ + ٨١ = ٠‬ ‫ب ﺱ٢ - ٥ﺱ + ٦ = ٠‬ ‫6 ﺩ)ﺱ(= ٠ ﻋﻨﺪﻣﺎ ﺱ = -٢ ، ﺱ = ٣ ، ﺩ)ﺱ( ﻣﻮﺟﺒﺔ ﻓﻰ [-٢ ، ٣ ]‬ ‫٤‬ ‫٤‬ ‫ﺩ)ﺱ( ﺳﺎﻟﺒﺔ ﻓﻰ [-٣، -٢]∪[ ٣ ، ٢]‬ ‫٤‬ ‫ب ﺡ - ]١، ٥[‬ ‫7 أ ‪z‬‬ ‫ﻫ }٥{‬ ‫د [-٣ ، ١]‬ ‫2 ﺏ‬ ‫1 ﺏ‬ ‫٢‬ ‫6 أ ﺱ - ٣ﺱ +١ = ٠‬ ‫ب ﺩ)ﺱ( = ٠ ﻋﻨﺪﻣﺎ ﺱ = }-٤، ٢{‬ ‫3 ﺏ‬ ‫ﺟ ‪z‬‬ ‫و ﺡ - [- ٣ ، ٥]‬ ‫٢‬ ‫4 ‪C‬‬ ‫ﺩ)ﺱ( = ٠ ﻋﻨﺪﻣﺎ ﺱ = [-٤، ٢]‬ ‫د ﺣﻘﻴﻘﻴﺎﻥ ﻣﺘﺴﺎﻭﻳﺎﻥ } ٤ {‬ ‫31 ‪٤- = C‬‬ ‫61 ﻙ = ٢‬ ‫¯‬ ‫21 ﺩ)ﺱ( = ٠ ﻋﻨﺪﻣﺎ ﺱ∋ } -٢٫١، ٢٫٣{‬ ‫ﺩ)ﺱ( >٠ ﻋﻨﺪﻣﺎ ﺱ∋ ]-٣ ، -٢٫١] ∪ [٢٫٣ ، ٥[‬ ‫−‬ ‫ﺟ ﺟﺬﺭﺍﻥ ﻣﺘﺴﺎﻭﻳﺎﻥ.‬ ‫4‬ ‫11 ﻣﻦ ﺍﻟﺮﺳﻢ ﻧﺠﺪ ﺃﻥ : ﺩ)ﺱ( = ٠ ﻋﻨﺪﻣﺎ ﺱ∋ } -٣، ٣{ ، ﺩ)ﺱ( <٠ ﻓﻰ [٣، ٤[‬ ‫ﺩ)ﺱ( > ٠ ﻓﻰ [-٣، ٣]‬ ‫41 ﺇﺷﺎﺭﺓ ﺍﻟﺪﺍﻟﺔ ﻣﻮﺟﺒﺔ ﻟﺠﻤﻴﻊ ﻗﻴﻢ ﻥ ﺍﻟﺤﻘﻴﻘﻴﺔ ، ﻳﺘﻨﺎﻗﺺ ﺍﻹﻧﺘﺎﺝ ﻣﻦ ﻋﺎﻡ ٠٩٩١ ﺣﺘﻰ‬ ‫ﻋﺎﻡ ٠٠٠٢، ﺛﻢ ﻳﺒﺪﺃ ﺍﻹﻧﺘﺎﺝ ﻓﻰ ﺍﻟﺰﻳﺎﺩﺓ ﻣﻦ ﻋﺎﻡ ٠٠٠٢ ﺣﺘﻰ ﻋﺎﻡ ٠١٠٢.‬ ‫−‬ ‫1 ﺏ‬ ‫3 ﺡ - }٣{‬ ‫7 [ -٥، ١]‬ ‫2 ﻣﻮﺟﺒﺔ ، ﺡ‬ ‫6 [ -٢ ، ١ ]‬ ‫د ﻥ = ٠٣‬ ‫أ -١‬ ‫4 [٢ ،∞ ]‬ ‫ب } -٣ ، ١ {‬ ‫٢‬ ‫أ }٧٫٤، -٧٫٤{‬ ‫أ ٥ - ٣ﺕ‬ ‫٨‬ ‫52 ﻙ = ٠ ﺃﻭ ﻙ = - ٣‬ ‫ﺩ)ﺱ( = ٠ ﻋﻨﺪﻣﺎ ﺱ = ﺡ - ]-٤، ٢[‬ ‫7‬ ‫أ }٧٩٦٫٠، ٣٠٣٫٤{‬ ‫:‬ ‫ب ]-٢، ٧[‬ ‫‪ï‬‬ ‫−‬ ‫1 ب ﺍﻟﺸﻜﻞ ‪ C‬ﺏ ﺟـ ‪ + E‬ﺍﻟﺸﻜﻞ ﺹ ﺱ ﻝ ﻉ‬ ‫ﺟ 9 ‪ C‬ﺏ ﺟـ + 9ﻫـ ‪ E‬ﻭ‬ ‫، ٠١‬ ‫٧‬ ‫، ٧‬ ‫٢١‬
  • 80.
    ‫ﺇﺟﺎﺑﺎﺕ ﺑﻌﺾ ﺍﻟﺘﻤﺎﺭﻳﻦ‬ ‫دﺍﻟﺸﻜﻞ ‪ C‬ﺏ ﺟـ ‪ + E‬ﺍﻟﺸﻜﻞ ﻉ ﻝ ﺱ ﺹ‬ ‫ﺟ ‪CE‬‬ ‫ب ﺟـ ‪E‬‬ ‫2‬ ‫أ ﺱﺹ‬ ‫4‬ ‫أ ٦٩ﺳﻢ، ٠٤٥ﺳﻢ‬ ‫5‬ ‫4 ﺟـ ﻫـ = ٥٫٤ﺳﻢ‬ ‫،٥‬ ‫٤‬ ‫أ ﻣﻌﺎﻣﻞ ﺗﺸﺎﺑﻪ ﺍﻟﻤﻀﻠﻊ ﻡ١ ﻟﻠﻤﻀﻠﻊ ﻡ٣ = ٤‬ ‫6‬ ‫ب ٨٫٢١ﺳﻢ، ٦٫٩ﺳﻢ‬ ‫٢‬ ‫٢‬ ‫7‬ ‫5 ﻉ ﻡ = ٥٫٣١‬ ‫ب ٥‬ ‫أ ٣‬ ‫ب ﻳﻮﺍﺯﻯ‬ ‫أ ﻻ ﻳﻮﺍﺯﻯ‬ ‫8 ﺱ ﻝ = ٢، ﺱ ﻡ = ٢‬ ‫ﺱﺹ ٥ ﺱﻉ ٥‬ ‫ﻣﻌﺎﻣﻞ ﺗﺸﺎﺑﻪ ﺍﻟﻤﻀﻠﻊ ﻡ٢ ﻟﻠﻤﻀﻠﻊ ﻡ٣ = ٣‬ ‫51‬ ‫أ ﻫـ ﻭ‬ ‫ﻣﻌﺎﻣﻞ ﺗﺸﺎﺑﻪ ﺍﻟﻤﻀﻠﻊ ﻡ ﻟﻠﻤﻀﻠﻊ ﻡ = ٣‬ ‫٢‬ ‫٣ ٢‬ ‫71‬ ‫أ ﻡ ﻭ = ٠١ﺳﻢ‬ ‫91‬ ‫6 ﺱ = ٠١١، ﺹ = ٠٠١، ﻉ = ٠٧‬ ‫7 ٠١ﺳﻢ ﺗﻘﺮﻳﺒﺎ.‬ ‫ً‬ ‫أ ٤٫٨ ﻣﺘﺮ، ١٫٥ ﻣﺘﺮ‬ ‫ﺟ ٤٤٫٩١ ﻣﺘﺮ ﻣﺮﺑﻊ‬ ‫−‬ ‫٢‬ ‫8 ٠٦ﺳﻢ، ٠٠٤٢ﺳﻢ‬ ‫ب ١٫٥ ﻣﺘﺮ، ٩٫٣ ﻣﺘﺮ‬ ‫1‬ ‫د ٥٢٫٠١١ ﻣﺘﺮ ﻣﺮﺑﻊ.‬ ‫د ﺍﻷﺿﻼﻉ ﺍﻟﻤﺘﻨﺎﻇﺮﺓ ﻣﺘﻨﺎﺳﺒﺔ.‬ ‫ﻫ ﺗﻄﺎﺑﻖ ﺯﺍﻭﻳﺔ ﻣﻦ ﻣﺜﻠﺚ ﻟﺰﺍﻭﻳﺔ ﻣﻦ ﻣﺜﻠﺚ ﺁﺧﺮ ﻭﺗﻨﺎﺳﺐ ﺃﻃﻮﺍﻝ ﺍﻷﺿﻼﻉ‬ ‫ﺍﻟﺘﻰ ﺗﺤﺘﻮﻳﻬﺎ.‬ ‫و ﺗﻄﺎﺑﻖ ﺯﺍﻭﻳﺔ ﻣﻦ ﻣﺜﻠﺚ ﻟﺰﺍﻭﻳﺔ ﻣﻦ ﻣﺜﻠﺚ ﺁﺧﺮ ﻭﺗﻨﺎﺳﺐ ﺃﻃﻮﺍﻝ ﺍﻷﺿﻼﻉ‬ ‫ﺍﻟﺘﻰ ﺗﺤﺘﻮﻳﻬﺎ.‬ ‫2‬ ‫أ ﺱ = ٦٣‬ ‫ﺟ ﺱ = ٣، ﺹ = ٤، ﻉ = ٤٫٨‬ ‫61‬ ‫−‬ ‫٥‬ ‫−‬ ‫1‬ ‫أ ٦٩٢١ﺳﻢ‬ ‫2‬ ‫3 ٥٧ﺳﻢ‬ ‫٢‬ ‫2‬ ‫٢‬ ‫ب -١٦١‬ ‫أ ٣٦‬ ‫ﺟ ﺻﻔﺮ‬ ‫د ١‬ ‫6 ٨١ ٢ ﺳﻢ‬ ‫7 ب ﺱ ﺟـ = ٦ ٦ ، ﺱ ﻭ = ٦ﺳﻢ.‬ ‫٢‬ ‫−‬ ‫21 ٨ ﺃﻣﺘﺎﺭ‬ ‫9‬ ‫31 ٩٤ﺳﻢ ﺗﻘﺮﻳﺒﺎ‬ ‫1 ‪ ،C‬ﺏ، ‪E‬‬ ‫5 ب ٤ ﺳﻢ‬ ‫7 ٥٫٤ ﻣﺘﺮ‬ ‫8‬ ‫2 ٢ ﺳﻢ‬ ‫4 ب ٩ : ٦١‬ ‫6 ﺱ = ١١ﺳﻢ، ﺹ = ٥٫٦١ﺳﻢ‬ ‫8 أ ٤ ، ٠٤ﺳﻢ ب - ١ ، ٤١ﺳﻢ‬ ‫أ ٦‬ ‫3 ‪C‬‬ ‫ﺟ ٩ : ٥٢‬ ‫2 ٤ﺳﻢ‬ ‫د ٤‬ ‫: ¯‬ ‫‪C‬ﺏ‬ ‫1‬ ‫3‬ ‫أ ٥٫٤‬ ‫` ﻫـ ﻭ = ٤٢ﺳﻢ‬ ‫‪ C a‬ﺏ = ٠٢١ﺳﻢ‬ ‫ﻫـ ٠٢ﺳﻢ‬ ‫4 ‪E‬‬ ‫ب ٥‬ ‫أ ﺱ=٣ ٢‬ ‫ﺟ ﺱ = ٠٦‬ ‫4‬ ‫5 ٢١ﺳﻢ‬ ‫أ ‪ C‬ﺏ = ٦ﺳﻢ، ‪ C‬ﻫـ = ٣ﺳﻢ، ﺟـ ‪٥ = E‬ﺳﻢ‬ ‫ب ‪X‬ﻡ)ﺱ( = -٣ * ٢ = -٦ ، ‪X‬ﻡ)ﺱ( = ٠‬ ‫¯‬ ‫1 ﺟـ‬ ‫5 ﺏ‬ ‫¯‬ ‫ب ٣، ٣‬ ‫٤ ٧‬ ‫ب ﺹ = ٩١، ﻉ = ٦ ٥١‬ ‫1‬ ‫3 ﺏ‬ ‫2 ﺟـ‬ ‫6 ﺱ = ٣ﺳﻢ، ﺹ = ٨١ ﺳﻢ‬ ‫01 ﺏ ﻫـ = ٨ﺳﻢ، ﺏ ﺟـ = ٢١ﺳﻢ‬ ‫د ﺱ = ١٣‬ ‫ﺟ ﺱ = ٥٫٤، ﺹ = ١١‬ ‫2 أ ﺱ=٦‬ ‫4 ﻟﺘﻜﻦ ﻡ ﻧﻘﻄﺔ ﺗﻘﺎﻃﻊ ﺍﻟﺪﻋﺎﻣﺘﻴﻦ‬ ‫` 9ﻡ ‪ C‬ﺏ + 9ﻡ ﻭ ﻫـ ، ﻫـ ﻭ = ١‬ ‫٥‬ ‫−‬ ‫أ ٨، ٣‬ ‫٣ ٥‬ ‫11 ٠٠١‪c‬‬ ‫ب ﺱ = ٤١‬ ‫¯‬ ‫5 ﺟـ‬ ‫8 ٦ ٣ ، ٩، ٣‬ ‫٣‬ ‫أ ٦٢‪c‬‬ ‫ب ٤٧‪c‬‬ ‫ﺟ ٠٢‪c‬‬ ‫٢‬ ‫3 ٩ﺳﻢ‬ ‫1 ‪E‬‬ ‫أ ﺱ = ٠١١‬ ‫ب ﺹ = ٠١‬ ‫ﺟ ﻉ = ٥٤‬ ‫01 ٣٤٫٤٢ﺳﻢ ﺗﻘﺮﻳﺒﺎ.‬ ‫ً‬ ‫1‬ ‫2 ‪ ،C‬ﺏ‬ ‫11 ٥ﺳﻢ‬ ‫ب ٣‬ ‫ﺟ ٠١‬ ‫01 ب ٥٫٤ﺳﻢ‬ ‫1‬ ‫أ ﺍﻟﻨﻘﻄﺔ ‪ C‬ﺗﻘﻊ ﺩﺍﺧﻞ ﺍﻟﺪﺍﺋﺮﺓ، ‪ C‬ﻡ = ٨ﺳﻢ.‬ ‫ب ﺍﻟﻨﻘﻄﺔ ﺏ ﺗﻘﻊ ﺧﺎﺭﺝ ﺍﻟﺪﺍﺋﺮﺓ، ﺏ ﻡ = ٤١ﺳﻢ‬ ‫ب ٢١ﺳﻢ‬ ‫ب ٠٠٥ﺳﻢ‬ ‫ﺟ ٦، ٥٢‬ ‫‪ C‬ﻭ ⊃ ‪ C‬ﺟـ ، ﺟـ ﻭ ⊃ ‪ C‬ﺟـ‬ ‫` ﻟﻠﻤﺜﻠﺜﺎﻥ ﻧﻔﺲ ﺍﻻﺭﺗﻔﺎﻉ ﻭﻳﻜﻮﻥ: ‪ C 9) W‬ﺏ ﻭ( = ‪ C‬ﻭ = ٦ = ٢‬ ‫‪) W‬ﺟـ ﺏ ﻭ( ﺟـ ﻭ ٣ ١‬ ‫1‬ ‫أ ١‬ ‫٩‬ ‫ب ٤‬ ‫ﺟ ٢‬ ‫د ٢‬ ‫ﻛﺎﻥ ﻓﻰ ﺍﻟﺮﺃﺱ ﺏ،‬ ‫‪ 99 a‬ﺏ ‪ C‬ﻭ ، ﺏ ﺟـ ﻭ ﻣﺸﺘﺮ‬ ‫3‬ ‫4 ﺟـ ﻫـ = ٥ﺳﻢ.‬ ‫١‬ ‫21 9‪ C‬ﺏ ﺟـ + 9‪ E C‬ﻫـ ، ﻣﻌﺎﻣﻞ ﺍﻟﺘﺸﺎﺑﻪ =‬ ‫أ ٤ﻛﻢ‬ ‫ﺟ ‪ E‬ﺟـ‬ ‫‪ C‬ﺟـ‬ ‫د ﺏ ‪ C * E‬ﺟـ‬ ‫` 9‪ C‬ﺏ ﻭ ﻣﺘﺴﺎﻭﻯ ﺍﻟﺴﺎﻗﻴﻦ، ‪ C‬ﻭ = ٦ﺳﻢ‬ ‫د ﻥ، ﻡ‬ ‫ب ﺹ‬ ‫7 ﺏ ‪٦ = E‬ﺳﻢ، ‪ C‬ﺏ = ٦ ٣ ﺳﻢ، ‪ C‬ﺟـ = ٦ ٦ ﺳﻢ‬ ‫ب ٤ ٥ ﻛﻢ‬ ‫ب ‪ C‬ﻡ = ٨٫٠١ﺳﻢ‬ ‫ب ﺱ = ٤، ﺹ = ٣‬ ‫5 ‪ E‬ﻫـ = ٨ﺳﻢ ، ‪ ١٥ ٢ = E C‬ﺳﻢ ، ‪ C‬ﻫـ = ٢ ٠١‬ ‫01 ﻓﻰ 9‪ C‬ﺏ ﺟـ : ﺏ ﺟـ = ٠١ - ٤ = ٦ﺳﻢ‬ ‫‪ a‬ﺏ ‪ ،٢ = C‬ﺏ ‪٢ = E‬‬ ‫` ‪ E C‬ﻳﻨﺼﻒ ‪Cc‬‬ ‫‪ C‬ﺟـ ٣ ‪ E‬ﺟـ ٣‬ ‫ﻓﻰ 9‪ C‬ﺏ ﻭ: ‪ C a‬ﻫـ ﻳﻨﺼﻒ ‪ C ،Cc‬ﻫـ = ﺏ ﻭ‬ ‫ﺟ ﺹ‬ ‫أ ﺱ‬ ‫أ‬ ‫د ‪E‬ﻭ‬ ‫2‬ ‫3 ‪ C‬ﺏ = ٨ﺳﻢ، ﺏ ﺟـ = ٠١ﺳﻢ‬ ‫ب ٥٢، ٥١ )٥ + ٥ (‬ ‫4 أ ٤ ، ٢٢‬ ‫أ ﻗﻴﺎﺳﺎﺕ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﻤﺘﻨﺎﻇﺮﺓ ﻣﺘﺴﺎﻭﻳﺔ.‬ ‫ب ﺱ = ٠٢، ﺹ = ٥١‬ ‫ﺏ‪C‬‬ ‫‪ C‬ﺟـ‬ ‫ﺟ ‪ E‬ﻫـ‬ ‫ب ﺟـ ‪E‬‬ ‫‪E‬ﺏ‬ ‫أ ١١‬ ‫−‬ ‫1‬ ‫` ﻝ ﻡ // ﺹ ﻉ‬ ‫أ ﺱ = ٨، ﺹ = ٣‬ ‫ﻝ = ١٢، ﻡ = ٨٢، ﻥ = ٠٣‬ ‫ﺟ ﻳﻮﺍﺯﻯ‬ ‫ب ‪E‬ﻭ‬ ‫ب ﻣﻌﺎﻣﻞ ﺗﺸﺎﺑﻪ ﺍﻟﻤﻀﻠﻊ ﻡ١ ﻟﻠﻤﻀﻠﻊ ﻡ٣ = ١‬ ‫9‬ ‫ﺟ ٩‬ ‫د ٣‬ ‫:‬ ‫2 ﺏ‬ ‫6‬ ‫¯‬ ‫‪E‬‬ ‫3 ﺟـ‬ ‫7 ﺟـ‬ ‫4 ‪C‬‬ ‫8 ٦ﺳﻢ‬ ‫:‬ ‫−‬ ‫ﺟ ٥‬ ‫٢‬ ‫1‬ ‫‪ïM‬‬ ‫−‬ ‫و ٠٧١‪c‬‬ ‫ز ﺍﻟﺜﺎﻟﺚ‬ ‫ح ٠٣‪c‬‬ ‫2 ﺟ‬
  • 81.
    ‫ﺇﺟﺎﺑﺎﺕ ﺑﻌﺾ ﺍﻟﺘﻤﺎﺭﻳﻦ‬ ‫3‬ ‫أ-٦٠٣‪c‬‬ ‫ب ٠٧٢‪c‬‬ ‫ﺟ ٥٣٢‪c‬‬ ‫د -١٠٣‪c‬‬ ‫4‬ ‫أ ﺍﻷﻭﻝ‬ ‫ب ﺍﻟﺜﺎﻟﺚ‬ ‫د ﺍﻟﺜﺎﻧﻰ‬ ‫3‬ ‫7‬ ‫أ ٧٧١‪c‬‬ ‫ب ٣٤١‪c‬‬ ‫ﺟ ﺍﻟﺮﺍﺑﻊ‬ ‫ﺟ ٥٤‪c‬‬ ‫د ٠٥١‪c‬‬ ‫4‬ ‫−‬ ‫2 ﺟ‬ ‫6 ب‬ ‫3‬ ‫7‬ ‫أ‬ ‫٤‬ ‫ب ٣‪r‬‬ ‫٣‬ ‫ﺟ -٤‪r‬‬ ‫1 د‬ ‫5 ﺟ‬ ‫01‬ ‫2‬ ‫أ ٥‪r‬‬ ‫٤‬ ‫‪E‬‬ ‫أ ٨٨٩٫٠‬ ‫ب ٢٤٤٫٠‬ ‫11‬ ‫41 ٢٫٤ﺳﻢ‬ ‫61 ٥٧‪١٫٣٠٩ ، c‬‬ ‫‪E‬‬ ‫ﺟ ٧٠٨٫٢‬ ‫51 ٥٧١٫٢ ، ٦ ً٧٣ َ٤٢١‪c‬‬ ‫81 ٧٥٫٨٢ﺳﻢ‬ ‫71 ٥ ‪r‬‬ ‫٣‬ ‫12 ٢١٧٤ ﻛﻢ/ﺱ‬ ‫ب ١‬ ‫أ - ٣‬ ‫٢‬ ‫٢‬ ‫أ ﺟﺘﺎ‪ ،  ٤ = i‬ﺟﺎ‪ ،  ٣ = i‬ﻇﺎ‪٣ = i‬‬ ‫٤‬ ‫٥‬ ‫٥‬ ‫ب ﺟﺘﺎ‪ ،  ٥ = i‬ﺟﺎ‪ ،  ١٢- =i‬ﻇﺎ‪١٢- = i‬‬ ‫٥‬ ‫٣١‬ ‫٣١‬ ‫02 ٢‪r‬ﺳﻢ‬ ‫32 أ ‪r‬‬ ‫٣‬ ‫6‬ ‫¯‬ ‫:‬ ‫ﺟ‬ ‫1‬ ‫22 ٩٢ﺳﻢ‬ ‫ب ٨ ﺳﺎﻋﺎﺕ‬ ‫ﺟ ٠٢‪r‬‬ ‫42 ﺹ = ٣ ﺱ‬ ‫2 أ‬ ‫6 ﺟ‬ ‫3 ﺟ‬ ‫7 أ‬ ‫د‬ ‫١‬ ‫٣‬ ‫ب ٠١٢‪ c٣٣٠ ،c‬ﺟ ٠٣‪ c٣٣٠ ،c‬د ٠١‪c٣٠٠ ،c‬‬ ‫أ ٥٤‪c٢٢٥ ،c‬‬ ‫‪E‬‬ ‫91 ٦٧٫٦١ﺳﻢ‬ ‫٣‬ ‫ﺟ‬ ‫4 ﺟ‬ ‫8 ب‬ ‫٥‬ ‫د ٣‪r‬‬ ‫ب ٨٣ ً٥٤ َ٢٨‪c‬‬ ‫أ ٦٫٩ﺳﻢ‬ ‫5‬ ‫د‬ ‫ب ٠٩‪c‬‬ ‫أ ٠٠٣‪c‬‬ ‫ﺟ ٧١ ً٠١ َ٤٦‪c‬‬ ‫أ‬ ‫2‬ ‫6‬ ‫أ ٣‬ ‫ب ٤‬ ‫٥‬ ‫7‬ ‫أ ٥٢‬ ‫ب‬ ‫4 ب‬ ‫8 أ‬ ‫3‬ ‫ﺟ‬ ‫ﺟ ٣‬ ‫٥‬ ‫د ٢١ ً٢٥ َ٦٣‬ ‫ﺟ ٤١٫٠١ﻣﺘﺮ‬ ‫ﺟﺎ٥٢ = ‪C‬‬ ‫٥٢‬ ‫−‬ ‫1 ﺟ‬ ‫5 ﺟ‬ ‫9‬ ‫‪C‬‬ ‫٢‬ ‫٣‬ ‫ﺟﺘﺎ‪i‬‬ ‫ﺏ‬ ‫٢‬ ‫٥‬ ‫ﺟﺎ‪i‬‬ ‫٣‬ ‫٢‬ ‫-‬ ‫١‬ ‫٢‬ ‫٢‬ ‫٢‬ ‫:‬ ‫11 أ )-(‬ ‫21 أ -١‬ ‫31 ٠٣‪c‬‬ ‫أ ٣٫٤، ٧٫٠‬ ‫4‬ ‫٣‬ ‫٤‬ ‫ب )+(‬ ‫ب ٤‬ ‫أ ٧١ + ٦١ﺕ ب‬ ‫:‬ ‫41 ﺇﺟﺎﺑﺔ ﺃﺣﻤﺪ‬ ‫ﺟ )+(‬ ‫51 ﺻﺤﻴﺤﺔ‬ ‫1 -ﺟﺘﺎ‪i‬‬ ‫2 - ﻇﺎ‪i‬‬ ‫3 - ﻗﺘﺎ‪i‬‬ ‫4 ﺟﺎ‪i‬‬ ‫5 ﺟﺘﺎ‪i‬‬ ‫6 - ﻇﺎ‪i‬‬ ‫7 ﻗﺘﺎ‪i‬‬ ‫91 ب‬ ‫ب ٥٢‪c‬‬ ‫02 د‬ ‫ﺟ ٠١‪c‬‬ ‫12 ﺟ‬ ‫د ٠٦‪c‬‬ ‫−‬ ‫2 ]-٢، ٢[‬ ‫1 ]-١، ١[‬ ‫ﺷﻜﻞ )١( ١ﺟﺎ ‪i‬‬ ‫4 -٣‬ ‫3 ٤‬ ‫  ﺷﻜﻞ )٢( ﺟﺘﺎ ‪i‬‬ ‫1‬ ‫أ ٤، ٠١‬ ‫أ ﻣﻮﺟﺒﺔ ﻟﻜﻞ ﺱ = ﺡ‬ ‫أ ١، -١، ]-١، ١[‬ ‫ب ]-٣، ٣[‬ ‫6‬ ‫1‬ ‫−‬ ‫2 ﺟ‬ ‫أ‬ ‫أ ١، ٣‬ ‫ب ١ ، - ١ ﺟ -٣، ٤‬ ‫1‬ ‫3‬ ‫أ -٣، - ١‬ ‫٣‬ ‫٢‬ ‫ب -٥، -٣‬ ‫٣ ٥‬ ‫ﺟ ٣، ٤‬ ‫٤ ٣‬ ‫4‬ ‫أ ٠٣‪c‬‬ ‫ب ٥٣١‪c‬‬ ‫5‬ ‫أ ٢١ ً٢٥ َ٦٣‪c‬‬ ‫ب ٤ ً٩ َ٤٦‪c‬‬ ‫ﺟ ٣١ ً٠٣ ٥٥‪c‬‬ ‫6‬ ‫ب ٨٢ ً٦٥ َ٩٢١‪c٢٣٠َ ٣ً ٣٢ ، c‬‬ ‫أ ٧٣ ً٧٣ َ٣١‪c١٦٦َ ٢٢ً ٢٣ ، c‬‬ ‫أ ٤٤ ً١٣ َ٠٦١ ب -٨٢٤٩٫٠ ، -٦٣٥٣٫٠ ، -٧٠٦٠٫١‬ ‫1‬ ‫أ ٩٠٫٢‬ ‫‪E‬‬ ‫‪E‬‬ ‫ﺟ ٥٨٫٣‬ ‫‪E‬‬ ‫¯‬ ‫−‬ ‫أ ٢ﺱ٢ + ﺱ > ٠، [-١، ٠]‬ ‫4‬ ‫٢‬ ‫:‬ ‫4 ‪E‬‬ ‫3 ﺏ‬ ‫¯ :‬ ‫5 ﺏ‬ ‫أ ٧‬ ‫ب ٧ﺳﻢ‬ ‫ب‬ ‫٢‬ ‫أ‬ ‫3‬ ‫٣:٥ ب‬ ‫ً‬ ‫ب ٠١ﺳﻢ 2 ﺃﻭﻻ: ٦ﺳﻢ‬ ‫:‬ ‫أ ‪ C‬ﺏ * ‪ C‬ﺟـ‬ ‫ً‬ ‫1 ب ﺃﻭﻻ: )‪(E C‬‬ ‫٢‬ ‫٣:٨‬ ‫ﺛﺎﻧﻴﺎ : ١٢ﺳﻢ‬ ‫ً‬ ‫:‬ ‫ب ﺏ ﻥ * ﻥ ﺟـ‬ ‫3 ﺱ ﻥ = ٦٫٣ﺳﻢ‬ ‫:‬ ‫2‬ ‫١‬‫٥‬ ‫:‬ ‫1 ب ﺛﺎﻧﻴﺎ: ٨٫٢ﺳﻢ‬ ‫ً‬ ‫3‬ ‫¯‬ ‫ب ٧‬ ‫٢١‬ ‫ب ٣٤ ، ٠٤٢‪c‬‬ ‫٦٣‬ ‫1 أ ٢ﺱ٢ - ﺱ - ٨ = ٠‬ ‫ب ٨٤٨٫٠‬ ‫2 أ -ﺕ‬ ‫4 أ ]-٤، ١[‬ ‫3 ب ٦ + ٣ﺕ‬ ‫1‬ ‫ب ٣١٫١‬ ‫5 ﺏ‬ ‫:‬ ‫3‬ ‫ﺟ ٢١ ً ٢٥ َ ٦٠٣‪c‬‬ ‫7‬ ‫1 ‪E‬‬ ‫1‬ ‫٥ ٥‬ ‫٢‬ ‫:‬ ‫2 ﺟـ‬ ‫2 ٠٠١ﺳﻢ‬ ‫أ ]-٤، ٤[‬ ‫٢ ٢‬ ‫3‬ ‫ب ٣، -٣ ]-٣، ٣[‬ ‫ﺟ ٣ ، -٣، ]-٣، ٣ [‬ ‫٢ ٢ ٢ ٢‬ ‫4 ﺏ‬ ‫3 ﺟـ‬ ‫أ ﻣﺘﺴﺎﻭﻳﺎﻥ، }٣{‬ ‫8 - ﺟﺎ‪i‬‬ ‫5‬ ‫٢٫٤ﻣﺘﺮ‬ ‫2‬ ‫81 أ‬ ‫2‬ ‫:‬ ‫٣‬ ‫أ ٦١‪c٨٠ ،c‬‬ ‫أ ٦، ﺻﻔﺮ‬ ‫:‬ ‫2‪C‬‬ ‫1 ‪E‬‬ ‫−‬ ‫32‬ ‫ب ﺻﻔﺮ‬ ‫1‬ ‫٣‬‫٥‬ ‫١‬‫٣‬ ‫٢‬ ‫-١‬ ‫٥‬ ‫ﻇﺎ‪i‬‬ ‫٢‬ ‫3 ﺱ٢ + ١ = ٠‬ ‫2 ﻣﻮﺟﺒﺔ ﻟﻜﻞ ﺱ ∋ ﺡ‬ ‫1 ١‬ ‫‪E‬‬ ‫ﺟـ‬ ‫-‬ ‫:‬ ‫٤‬‫٥‬ ‫٣‬ ‫:‬ ‫ﺛﺎﻧﻴﺎ: ٤ﺳﻢ‬ ‫ً‬ ‫ﺛﺎﻟﺜﺎ: ٦‬ ‫ً‬ ‫أ ٤ : ٧، ٤ : ٧‬ ‫أ ‪ C‬ﺏ = ٥٫٤ﺳﻢ، ﻉ ﺹ = ٥ﺳﻢ‬ ‫ب ٤ ٣ ﺳﻢ‬ ‫أ ﺻﻔﺮ‬ ‫ﺟ 9‪ C‬ﺏ ‪E‬‬