دليل معلم مقرر رياضيات 1

‫‪‬‬ ‫ﺍﻛﺘﺐ ﻛﻞ ﻋﺒﺎﺭﺓ ﺷﺮﻃﻴﺔ ﻣﻤﺎ ﻳﺄﺗﻲ ﻋﻠﻰ ﺻﻮﺭﺓ )ﺇﺫﺍ‪ ...‬ﻓﺈﻥ‪.(...‬‬ ‫‪2 ‬‬ ‫‪ (5‬ﺍﻟﺸﺨﺺ ﺍﻟﺬﻱ ﺗﺠﺎﻭﺯ ﻋﻤﺮﻩ ‪ 18‬ﻋﺎ ﹰﻣﺎ ﻳﻤﻜﻨﻪ ﺍﺳﺘﺨﺮﺍﺝ ﺭﺧﺼﺔ ﻗﻴﺎﺩﺓ‪.‬‬ ‫‪ (10‬ﺧﺎﻃﺌﺔ؛ ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ، x = -4‬ﻓﺈﻥ‬ ‫‪ (6‬ﻳﺤﺘﻮﻱ ﺍﻟﺠﺒﻦ ﻋﻠﻰ ﻋﻨﺼﺮ ﺍﻟﻜﺎﻟﺴﻴﻮﻡ‪ .‬ﺇﺫﺍ ﻛﺎﻧﺖ ﻫﺬﻩ ﺟﺒﻨﺔ‪ ،‬ﻓﺈﻧﻬﺎ ﺗﺤﺘﻮﻱ ﻋﻠﻰ ﻋﻨﺼﺮ ﺍﻟﻜﺎﻟﺴﻴﻮﻡ‪.‬‬ ‫‪ (5‬ﺇﺫﺍ ﺗﺠﺎﻭﺯ ﻋﻤﺮ ﺍﻟﺸﺨﺺ ‪18‬‬ ‫‪ .(-4)2 = 16‬ﻫﺬﺍ ﺍﻟﻤﺜﺎﻝ ﺍﻟﻤﻀﺎﺩ‬ ‫ﻋﺎ ﹰﻣﺎ‪ ،‬ﻓﺈﻧﻪ ﻳﻤﻜﻨﻪ ﺍﺳﺘﺨﺮﺍﺝ ﺭﺧﺼﺔ‬ ‫‪ (7‬ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺤﺎﺩﺓ ﺑﻴﻦ ‪ 0°‬ﻭ ‪90°‬‬ ‫ﻳﺜﺒﺖ ﺃﻥ ﺍﻟﻨﺘﻴﺠﺔ ﺧﺎﻃﺌﺔ‪ ،‬ﺃﻱ ﺃﻥ ﺍﻟﻌﺒﺎﺭﺓ‬ ‫‪ (8‬ﺍﻟﻤﺜﻠﺚ ﺍﻟﻤﺘﻄﺎﺑﻖ ﺍﻷﺿﻼﻉ ﻣﺘﻄﺎﺑﻖ ﺍﻟﺰﻭﺍﻳﺎ‪ .‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻤﺜﻠﺚ ﻣﺘﻄﺎﺑﻖ ﺍﻷﺿﻼﻉ‪ ،‬ﻓﺈﻧﻪ ﻳﻜﻮﻥ ﻣﺘﻄﺎﺑﻖ ﺍﻟﺰﻭﺍﻳﺎ‪.‬‬ ‫ﻗﻴﺎﺩﺓ‪.‬‬ ‫‪ (7‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺔ ﺣﺎﺩﺓ‪ ،‬ﻓﺈﻥ‬ ‫ﺍﻟﺸﺮﻃﻴﺔ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪  (9‬ﻫﻨﺎﻙ ﺃﻧﻮﺍﻉ ﻣﺨﺘﻠﻔﺔ ﻣﻦ ﻫﻄﻞ ﺍﻟﻤﻄﺮ‪ ،‬ﺗﺘﺸﻜﻞ ﻓﻲ ﻇﺮﻭﻑ ﻣﺨﺘﻠﻔﺔ‪ .‬ﺍﻛﺘﺐ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﺜﻼﺙ‬ ‫‪ (11‬ﺧﺎﻃﺌﺔ؛ ﺍﻟﻔﺮﺽ ﺻﺤﻴﺢ‪ ،‬ﺃﻣﺎ ﺍﻟﻨﺘﻴﺠﺔ‬ ‫ﻗﻴﺎﺳﻬﺎ ﺑﻴﻦ ‪ 0°‬ﻭ‪.90°‬‬ ‫ﺍﻵﺗﻴﺔ ﻋﻠﻰ ﺻﻮﺭﺓ )ﺇﺫﺍ‪ ...‬ﻓﺈﻥ‪.(...‬‬ ‫‪ (9a‬ﺇﺫﺍ ﺗﻜﺎﺛﻒ ﺑﺨﺎﺭ ﺍﻟﻤﺎﺀ ﻓﻲ‬ ‫ﻓﻬﻲ ﺧﺎﻃﺌﺔ؛ ﻷﻥ ﺍﻟﺮﻳﺎﺽ ﻻ ﺗﻘﻊ ﻓﻲ‬ ‫‪ (a‬ﻳﺘﻜﺎﺛﻒ ﺑﺨﺎﺭ ﺍﻟﻤﺎﺀ ﻓﻲ ﺍﻟﻐﻼﻑ ﺍﻟﺠﻮﻱ ﻓﻴﺴﻘﻂ ﻋﻠﻰ ﺷﻜﻞ ﻣﻄﺮ‪.‬‬ ‫ﺍﻟﻐﻼﻑ ﺍﻟﺠﻮﻱ‪ ،‬ﻓﺈﻧﻪ ﻳﺴﻘﻂ ﻋﻠﻰ‬ ‫‪ (b‬ﻳﺘﺠﻤﺪ ﺑﺨﺎﺭ ﺍﻟﻤﺎﺀ ﺍﻟﺸﺪﻳﺪ ﺍﻟﺒﺮﻭﺩﺓ ﻓﻲ ﺍﻟﻐﻴﻮﻡ ﺍﻟﺮﻛﺎﻣﻴﺔ ﻓﻴﺴﻘﻂ ﻋﻠﻰ ﺷﻜﻞ ﹶﺑ ﹶﺮﺩ‪.‬‬ ‫ﺍﻟﻜﻮﻳﺖ؛ ﺇﺫﻥ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ (c‬ﻳﻜﻮﻥ ﺍﻟﻬﻄﻞ ﻋﻠﻰ ﺷﻜﻞ ﺛﻠ ﹴﺞ‪ ،‬ﻋﻨﺪﻣﺎ ﺗﻜﻮﻥ ﺩﺭﺟﺔ ﺍﻟﺤﺮﺍﺭﺓ ﻣﺘﺪﻧﻴ ﹰﺔ ﺟ ﹼﹰﺪﺍ ﺇﻟﻰ ﺣ ﱢﺪ ﺍﻟﺘﺠﻤﺪ ﻓﻲ ﺍﻟﻐﻼﻑ ﺍﻟﺠﻮﻱ‪.‬‬ ‫ﺷﻜﻞ ﺃﻣﻄﺎﺭ‪.‬‬ ‫‪ (9b‬ﺇﺫﺍ ﺗﺠﻤﺪ ﺑﺨﺎﺭ ﺍﻟﻤﺎﺀ ﺍﻟﺸﺪﻳﺪ‬ ‫ﻣﺜﺎﻝ ﻣﻀﺎﺩ‪ :‬ﺃﻧﺎ ﺃﻋﻴﺶ ﻓﻲ ﺍﻟﺮﻳﺎﺽ‬ ‫ﺣ ﱢﺪﺩ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻜ ﱢﻞ ﻋﺒﺎﺭﺓ ﺷﺮﻃﻴﺔ ﻓﻴﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﻭﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺎﺋﺒﺔ‪ ،‬ﻓﻔ ﱢﺴﺮ ﺗﺒﺮﻳﺮﻙ‪ ،‬ﺃﻣﺎ ﺇﺫﺍ ﻛﺎﻧﺖ ﺧﺎﻃﺌﺔ‪،‬‬ ‫ﻓﺄﻋﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹼﹰﺩﺍ‪ (10–16 .‬ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‪.‬‬ ‫ﺍﻟﺒﺮﻭﺩﺓ ﻓﻲ ﺍﻟﻐﻴﻮﻡ ﺍﻟﺮﻛﺎﻣﻴﺔ‪ ،‬ﻓﺈﻧﻪ‬ ‫ﻟﻜﻨﻨﻲ ﻻ ﺃﻋﻴﺶ ﻓﻲ ﺍﻟﻜﻮﻳﺖ‪.‬‬ ‫‪ (10‬ﺇﺫﺍ ﻛﺎﻥ ‪ ،x2 = 16‬ﻓﺈﻥ ‪x = 4‬‬ ‫ﻳﺴﻘﻂ ﻋﻠﻰ ﺷﻜﻞ ﺑﺮﺩ‪.‬‬ ‫‪ (12‬ﺻﺎﺋﺒﺔ؛ ﻋﻨﺪﻣﺎ ﻳﻜﻮﻥ ﺍﻟﻔﺮﺽ ﺻﺤﻴ ﹰﺤﺎ‪،‬‬ ‫ﻓﺈﻥ ﺍﻟﻨﺘﻴﺠﺔ ﺗﻜﻮﻥ ﺻﺎﺋﺒﺔ ﺃﻳ ﹰﻀﺎ؛ ﻷﻥ ﻳﻮﻡ‬ ‫‪ (9c‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺩﺭﺟﺔ ﺍﻟﺤﺮﺍﺭﺓ‬ ‫ﻣﺘﺪﻧﻴﺔ ﺟ ﹼﹰﺪﺍ ﺇﻟﻰ ﺣ ﱢﺪ ﺍﻟﺘﺠﻤﺪ ﻓﻲ‬ ‫ﺍﻟﺠﻤﻌﺔ ﺑﻌﺪ ﻳﻮﻡ ﺍﻟﺨﻤﻴﺲ؛ ﻟﺬﺍ ﻓﺈﻥ‬ ‫ﺍﻟﻐﻼﻑ ﺍﻟﺠﻮﻱ‪ ،‬ﻓﺈﻥ ﺍﻟﻬﻄﻞ ﻳﻜﻮﻥ‬ ‫ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺻﺎﺋﺒﺔ ﺃﻳ ﹰﻀﺎ‪.‬‬ ‫ﻋﻠﻰ ﺷﻜﻞ ﺛﻠﺞ‪3 .‬‬ ‫‪ (13‬ﺧﺎﻃﺌﺔ؛ ﺍﻟﺤﻴﻮﺍﻥ ﺛﻮ ﹲﺭ ﻟﻪ ﻗﺮﻧﺎﻥ‪ ،‬ﻫﺬﺍ‬ ‫ﺍﻟﻤﺜﺎﻝ ﺍﻟﻤﻀﺎﺩ ﻳﺜﺒﺖ ﺃﻥ ﺍﻟﻨﺘﻴﺠﺔ ﺧﺎﻃﺌﺔ؛‬ ‫‪ (11‬ﺇﺫﺍ ﻛﻨﺖ ﺗﻌﻴﺶ ﻓﻲ ﺍﻟﺮﻳﺎﺽ‪ ،‬ﻓﺈﻧﻚ ﺗﻌﻴﺶ ﻓﻲ ﺍﻟﻜﻮﻳﺖ‪.‬‬ ‫ﺃﻱ ﺃﻥ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ (12‬ﺇﺫﺍ ﻛﺎﻥ ﻳﻮﻡ ﻏﺪ ﻫﻮ ﺍﻟﺠﻤﻌﺔ‪ ،‬ﻓﺈﻥ ﺍﻟﻴﻮﻡ ﻫﻮ ﺍﻟﺨﻤﻴﺲ‪.‬‬ ‫‪ (13‬ﺇﺫﺍ ﻛﺎﻥ ﻟﻠﺤﻴﻮﺍﻥ ﻗﺮﻧﺎﻥ‪ ،‬ﻓﺈﻧﻪ ﻛﺒﺶ‪.‬‬ ‫‪ (14‬ﺇﺫﺍ ﻛﺎﻥ ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻘﺎﺋﻤﺔ ‪ ،95°‬ﻓﺈﻥ ﺍﻟﻨﺤﻠﺔ ﺗﻜﻮﻥ ﺳﺤﻠﻴﺔ‪.‬‬ ‫‪ (14‬ﺻﺎﺋﺒﺔ؛ ﺍﻟ ﹶﻔﺮﺽ ﺧﺎﻃﺊ؛ ﻷﻥ ﻗﻴﺎﺱ‬ ‫ﺃﻭﺟﺪ ﻗﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻟﻜﻞ ﻋﺒﺎﺭﺗﻴﻦ ﻓﻴﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﺛﻢ ﻗ ﱢﺮﺭ ﻫﻞ ﻫﻤﺎ ﻣﻜﺎﻓﺌﺘﺎﻥ ﻣﻨﻄﻘ ﹼﹰﻴﺎ ﺃﻡ ﻻ؟‬ ‫‪4 ‬‬ ‫ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻘﺎﺋﻤﺔ ˚‪ ،90‬ﻭﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‬ ‫‪∼p q, ∼(p q) (15‬‬ ‫ﺍﻟﺘﻲ ﻳﻜﻮﻥ ﻓﻴﻬﺎ ﺍﻟ ﹶﻔﺮﺽ ﺧﺎﻃ ﹰﺌﺎ ﺗﻜﻮﻥ‬ ‫ﺻﺎﺋﺒﺔ ﺩﺍﺋ ﹰﻤﺎ؛ ﻟﺬﺍ ﻓﻬﺬﻩ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‬ ‫‪∼p ∨ ∼q, ∼(p ∨ q) (16‬‬ ‫ﺻﺎﺋﺒﺔ‪.‬‬ ‫‪5 ‬‬ ‫‪p q ∼p p q ∼(p q) ∼p q (15‬‬ ‫ﺍﻛﺘﺐ ﺍﻟﻌﻜﺲ ﻭﺍﻟﻤﻌﻜﻮﺱ ﻭﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ ﻟﻜ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺗﻴﻦ ﺍﻟﺸﺮﻃﻴﺘﻴﻦ ﺍﻵﺗﻴﺘﻴﻦ‪ .‬ﺛﻢ ﺣﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﺃ ﱞﻱ ﻣﻨﻬﺎ‬ ‫ﺻﺎﺋ ﹰﺒﺎ ﺃﻡ ﺧﺎﻃ ﹰﺌﺎ‪ ،‬ﻭﺇﺫﺍ ﻛﺎﻥ ﺧﺎﻃ ﹰﺌﺎ ﻓﺄﻋﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹼﹰﺩﺍ‪ (17, 18 .‬ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‪.‬‬ ‫‪TTF T‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫‪ (17‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻌﺪﺩ ﻳﻘﺒﻞ ﺍﻟﻘﺴﻤﺔ ﻋﻠﻰ ‪ ، 2‬ﻓﺈﻧﻪ ﻳﻘﺒﻞ ﺍﻟﻘﺴﻤﺔ ﻋﻠﻰ ‪4‬‬ ‫‪TFF F‬‬ ‫‪T‬‬ ‫‪F‬‬ ‫‪ (18‬ﺟﻤﻴﻊ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻜﻠﻴﺔ ﺃﻋﺪﺍﺩ ﺻﺎﺋﺒﺔ‪.‬‬ ‫‪FTT F T T‬‬ ‫‪FFT F‬‬ ‫‪T‬‬ ‫‪F‬‬ ‫ﻏﻴﺮ ﻣﺘﻜﺎﻓﺌﺘﻴﻦ ﻣﻨﻄﻘ ﹼﹰﻴﺎ‪.‬‬ ‫‪‬‬ ‫‪p q ∼p ∼q p q ∼p ∼q ∼(p q) (16‬‬ ‫ﺣ ﱢﺪﺩ ﺍﻟﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ ﻓﻲ ﻛ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻵﺗﻴﺔ‪:‬‬ ‫‪1 ‬‬ ‫‪TT F F T‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫‪ (19‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﻣﺘﺠﺎﻭﺭﺗﻴﻦ‪ ،‬ﻓﺈﻥ ﻟﻬﻤﺎ ﺿﻠ ﹰﻌﺎ ﻣﺸﺘﺮ ﹰﻛﺎ‪.‬‬ ‫‪ (19‬ﺍﻟﻔﺮﺽ‪ :‬ﺍﻟﺰﺍﻭﻳﺘﺎﻥ‬ ‫‪ (20‬ﺇﺫﺍ ﻛﻨﺖ ﻗﺎﺋﺪ ﻣﺠﻤﻮﻋﺘﻨﺎ‪ ،‬ﻓﺈﻧﻨﻲ ﺳﺄﺗﺒﻌﻚ‪ .‬ﺍﻟﻔﺮﺽ‪ :‬ﺃﻧﺖ ﻗﺎﺋﺪ ﻣﺠﻤﻮﻋﺘﻨﺎ؛ ﺍﻟﻨﺘﻴﺠﺔ‪ :‬ﺳﻮﻑ ﺃﺗﺒﻌﻚ‪.‬‬ ‫ﻣﺘﺠﺎﻭﺭﺗﺎﻥ؛ ﺍﻟﻨﺘﻴﺠﺔ‪:‬‬ ‫‪TF F T T‬‬ ‫‪T‬‬ ‫‪F‬‬ ‫‪31  1-3‬‬ ‫ﻟﻠﺰﺍﻭﻳﺘﻴﻦ ﺿﻠﻊ ﻣﺸﺘﺮﻙ‪.‬‬ ‫‪FT T F T‬‬ ‫‪T‬‬ ‫‪F‬‬ ‫‪FF T T F‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫ﻏﻴﺮ ﻣﺘﻜﺎﻓﺌﺘﻴﻦ ﻣﻨﻄﻘ ﹰﹼﻴﺎ‪.‬‬ ‫‪‬‬ ‫‪ (17‬ﺍﻟﻌﻜﺲ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻌﺪﺩ ﻳﻘﺒﻞ ﺍﻟﻘﺴﻤﺔ‬ ‫‪ ‬‬ ‫‪‬‬ ‫ﻋﻠﻰ ‪ ، 4‬ﻓﺈﻧﻪ ﻳﻘﺒﻞ ﺍﻟﻘﺴﻤﺔ ﻋﻠﻰ ‪،2‬‬ ‫ﺻﺎﺋﺒﺔ‪.‬‬ ‫ﺍﻟﻤﻌﻜﻮﺱ‪ :‬ﺇﺫﺍ ﻟﻢ ﻳﻜﻦ ﺍﻟﻌﺪﺩ ﻳﻘﺒﻞ‬ ‫‪60-705719-28‬‬ ‫‪‬‬ ‫ﺍﻟﻘﺴﻤﺔ ﻋﻠﻰ ‪ ،2‬ﻓﺈﻧﻪ ﻻ ﻳﻘﺒﻞ ﺍﻟﻘﺴﻤﺔ‬ ‫‪60-7053-5719-48‬‬ ‫‪‬‬ ‫ﻋﻠﻰ ‪ ،4‬ﺻﺎﺋﺒﺔ‪.‬‬ ‫ﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ‪ :‬ﺇﺫﺍ ﻟﻢ ﻳﻜﻦ ﺍﻟﻌﺪﺩ‬ ‫‪30-70 ‬‬ ‫ﻳﻘﺒﻞ ﺍﻟﻘﺴﻤﺔ ﻋﻠﻰ ‪ ، 4‬ﻓﺈﻧﻪ ﻻ ﻳﻘﺒﻞ‬ ‫ﺍﻟﻘﺴﻤﺔ ﻋﻠﻰ ‪ ،2‬ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ (18‬ﺍﻟﻌﻜﺲ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻌﺪﺩ ﺻﺤﻴ ﹰﺤﺎ‪ ،‬ﻓﺈﻧﻪ ﻋﺪﺩ ﻛﻠﻲ‪،‬‬ ‫ﺧﺎﻃﺌﺔ‪ .‬ﻣﺜﺎﻝ ﻣﻀﺎﺩ‪-3 :‬‬ ‫ﻣﺜﺎﻝ ﻣﻀﺎﺩ‪ :‬ﺍﻟﻌﺪﺩ ‪ 6‬ﻻ ﻳﻘﺒﻞ ﺍﻟﻘﺴﻤﺔ‬ ‫ﺍﻟﻤﻌﻜﻮﺱ‪ :‬ﺇﺫﺍ ﻟﻢ ﻳﻜﻦ ﺍﻟﻌﺪﺩ ﻛﻠ ﹼﹰﻴﺎ‪ ،‬ﻓﺈﻧﻪ ﻟﻴﺲ ﻋﺪ ﹰﺩﺍ‬ ‫ﻋﻠﻰ ‪ ،4‬ﻭﻟﻜﻨﻪ ﻳﻘﺒﻞ ﺍﻟﻘﺴﻤﺔ ﻋﻠﻰ ‪2‬‬ ‫ﺻﺤﻴ ﹰﺤﺎ‪ ،‬ﺧﺎﻃﺌﺔ‪ .‬ﻣﺜﺎﻝ ﻣﻀﺎﺩ‪-3 :‬‬ ‫ﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ‪ :‬ﺇﺫﺍ ﻟﻢ ﻳﻜﻦ ﺍﻟﻌﺪﺩ ﺻﺤﻴ ﹰﺤﺎ‪ ،‬ﻓﺈﻧﻪ‬ ‫ﻟﻴﺲ ﻋﺪ ﹰﺩﺍ ﻃﺒﻴﻌ ﹼﹰﻴﺎ‪ ،‬ﺻﺎﺋﺒﺔ‪.‬‬ ‫‪31  1-3‬‬

‫‪ (21‬ﺇﺫﺍ ﻛﺎﻥ ‪ ،3x – 4 = 11‬ﻓﺈﻥ ‪ x = 5‬ﺍﻟﻔﺮﺽ ‪3x – 4 = 11‬؛ ﺍﻟﻨﺘﻴﺠﺔ‪x = 5 :‬‬ ‫‪ (22‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﻣﺘﻘﺎﺑﻠﺘﻴﻦ ﺑﺎﻟﺮﺃﺱ‪ ،‬ﻓﺈﻧﻬﻤﺎ ﻣﺘﻄﺎﺑﻘﺘﺎﻥ‪ .‬ﺍﻟﻔﺮﺽ‪ :‬ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﻣﺘﻘﺎﺑﻠﺘﺎﻥ ﺑﺎﻟﺮﺃﺱ‪ .‬ﺍﻟﻨﺘﻴﺠﺔ‪ :‬ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﻣﺘﻄﺎﺑﻘﺘﺎﻥ‪.‬‬ ‫ﺍﻛﺘﺐ ﻛﻞ ﻋﺒﺎﺭﺓ ﺷﺮﻃﻴﺔ ﻣﻤﺎ ﻳﺄﺗﻲ ﻋﻠﻰ ﺻﻮﺭﺓ )ﺇﺫﺍ ‪ ...‬ﻓﺈﻥ ‪.(...‬‬ ‫‪2 ‬‬ ‫‪ (23‬ﺍﺣﺼﻞ ﻋﻠﻰ ﻗﺎﺭﻭﺭﺓ ﻣﺎﺀ ﻣﺠﺎ ﹰﻧﺎ ﻋﻨﺪ ﺷﺮﺍﺋﻚ ﺧﻤﺲ ﻗﻮﺍﺭﻳﺮ‪.‬‬ ‫‪ (23‬ﺇﺫﺍ ﺍﺷﺘﺮﻳﺖ ﺧﻤﺲ ﻗﻮﺍﺭﻳﺮ ﻣﺎﺀ‪ ،‬ﻓﺈﻧﻚ ﺗﺤﺼﻞ‬ ‫ﻋﻠﻰ ﻗﺎﺭﻭﺭﺓ ﻣﺠﺎﻧﻴﺔ‪.‬‬ ‫‪ (24‬ﻛﻞ ﻣﻦ ﺣﻀﺮ ﺍﻟﺤﻔﻞ ﺳﻴﺤﺼﻞ ﻋﻠﻰ ﻫﺪﻳﺔ‪ .‬ﺇﺫﺍ ﺣﻀﺮﺕ ﺍﻟﺤﻔﻞ‪ ،‬ﻓﺈﻧﻚ ﺳﺘﺤﺼﻞ ﻋﻠﻰ ﻫﺪﻳﺔ‪.‬‬ ‫‪ (25‬ﺗﻘﺎﻃﻊ ﻣﺴﺘﻮﻳﻴﻦ ﻳﻤﺜﻞ ﻣﺴﺘﻘﻴ ﹰﻤﺎ‪ .‬ﺇﺫﺍ ﺗﻘﺎﻃﻊ ﻣﺴﺘﻮﻳﺎﻥ‪ ،‬ﻓﺈﻥ ﺗﻘﺎﻃﻌﻬﻤﺎ ﻣﺴﺘﻘﻴﻢ‪.‬‬ ‫‪ (26‬ﻣﺴﺎﺣﺔ ﺍﻟﺪﺍﺋﺮﺓ ﺗﺴﺎﻭﻱ ‪ πr2‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺸﻜﻞ ﺩﺍﺋﺮﺓ‪ ،‬ﻓﺈﻥ ﻣﺴﺎﺣﺘﻪ ﺗﺴﺎﻭﻱ ‪πr2‬‬ ‫‪ (27‬ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻘﺎﺋﻤﺔ ‪ 90°‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺔ ﻗﺎﺋﻤﺔ‪ ،‬ﻓﺈﻥ ﻗﻴﺎﺳﻬﺎ ‪90°‬‬ ‫ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻤﺎﺩﺓ ﻓﻮﺳﻔﻮ ﹰﺭﺍ‪ ،‬ﻓﺈﻧﻬﺎ‬ ‫‪  (28‬ﺍﻛﺘﺐ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻵﺗﻴﺔ ﻋﻠﻰ ﺻﻮﺭﺓ )ﺇﺫﺍ‪ ...‬ﻓﺈﻥ‪.(...‬‬ ‫ﺗﻨﺼﻬﺮ ﻋﻨﺪ ﺩﺭﺟﺔ ˚‪ 44‬ﺳﻴﻠﻴﺰﻳﺔ‪.‬‬ ‫ﻳﻨﺼﻬﺮ ﺍﻟﻔﻮﺳﻔﻮﺭ ﻋﻨﺪ ﺩﺭﺟﺔ ‪ 44°‬ﺳﻴﻠﻴﺰﻳﺔ‪.‬‬ ‫‪  (29‬ﻳﺘﻐﻴﺮ ﺍﻟﻤﺎﺀ ﻋﻠﻰ ﺍﻷﺭﺽ ﺑﺎﺳﺘﻤﺮﺍﺭ ﻋﺒﺮ ﻋﻤﻠﻴﺔ ﹸﺗﺴ ﹼﻤﻰ ﺩﻭﺭﺓ ﺍﻟﻤﺎﺀ‪ .‬ﺍﻛﺘﺐ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﺜﻼﺙ‬ ‫ﺃﺩﻧﻰ ﺍﻟﺸﻜﻞ ﻋﻠﻰ ﺻﻮﺭﺓ )ﺇﺫﺍ‪ ...‬ﻓﺈﻥ‪.(...‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ (29a‬ﺇﺫﺍ ﺟﺮ￯ ﺍﻟﻤﺎﺀ ﻋﻠﻰ ﺳﻄﺢ‬ ‫ﺍﻷﺭﺽ‪ ،‬ﻓﺈﻧﻪ ﻳﺼﺐ ﻓﻲ ﺍﻟﻤﺴﻄﺤﺎﺕ ‪‬‬ ‫‪‬‬ ‫ﺍﻟﻤﺎﺋﻴﺔ‪.‬‬ ‫‪ (29b‬ﺇﺫﺍ ﺃﻋﺎﺩﺕ ﺍﻟﻨﺒﺎﺗﺎﺕ ﺍﻟﻤﺎﺀ ﺇﻟﻰ ‪ (a‬ﺟﺮﻳﺎﻥ ﺍﻟﻤﺎﺀ ﺍﻟﺴﻄﺤﻲ ﻳﺼﺐ ﻓﻲ ﺍﻟﻤﺴﻄﺤﺎﺕ ﺍﻟﻤﺎﺋﻴﺔ‪.‬‬ ‫ﺍﻟﻬﻮﺍﺀ‪ ،‬ﻓﺈﻥ ﺫﻟﻚ ﻳﺘﻢ ﻋﻦ ﻃﺮﻳﻖ ﺍﻟﻨﺘﺢ‪.‬‬ ‫‪ (b‬ﺗﻌﻴﺪ ﺍﻟﻨﺒﺎﺗﺎﺕ ﺍﻟﻤﺎﺀ ﺇﻟﻰ ﺍﻟﻬﻮﺍﺀ ﻣﻦ ﺧﻼﻝ ﻋﻤﻠﻴﺔ ﺍﻟﻨﺘﺢ‪.‬‬ ‫‪ (29c‬ﺇﺫﺍ ﺃﻋﺎﺩﺕ ﺍﻟﻤﺴﻄﺤﺎﺕ ﺍﻟﻤﺎﺋﻴﺔ‬ ‫ﺍﻟﻤﺎﺀ ﺇﻟﻰ ﺍﻟﻬﻮﺍﺀ‪ ،‬ﻓﺈﻥ ﺫﻟﻚ ﻳﺘﻢ ﻋﻦ ‪ (c‬ﺗﻌﻴﺪ ﺍﻟﻤﺴﻄﺤﺎﺕ ﺍﻟﻤﺎﺋﻴﺔ ﺍﻟﻤﺎﺀ ﺇﻟﻰ ﺍﻟﻬﻮﺍﺀ ﻋﻦ ﻃﺮﻳﻖ ﺍﻟﺘﺒﺨﺮ‪.‬‬ ‫ﻃﺮﻳﻖ ﺍﻟﺘﺒﺨﺮ‪.‬‬ ‫‪ 3‬ﺣﺪﺩ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻜﻞ ﻋﺒﺎﺭﺓ ﺷﺮﻃﻴﺔ ﻓﻴﻤﺎ ﻳﺄﺗﻲ‪ .‬ﻭﺇﺫﺍ ﻛﺎﻧﺖ ﺻﺎﺋﺒﺔ‪ ،‬ﻓﻔ ﱢﺴﺮ ﺗﺒﺮﻳﺮﻙ‪ ،‬ﺃﻣﺎ ﺇﺫﺍ ﻛﺎﻧﺖ ﺧﺎﻃﺌﺔ ﻓﺄﻋﻂ ﻣﺜﺎ ﹰﻻ‬ ‫ﻣﻀﺎ ﹰﹼﺩﺍ‪ (30-36 :‬ﺍﻧﻈﺮ ﻣﻠﺤﻖ ﺍﻹﺟﺎﺑﺎﺕ‬ ‫‪ (30‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻌﺪﺩ ﻓﺮﺩ ﹼﹰﻳﺎ‪ ،‬ﻓﺈﻧﻪ ﻳﻘﺒﻞ ﺍﻟﻘﺴﻤﺔ ﻋﻠﻰ ‪5‬‬ ‫‪ (31‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻷﺭﻧﺐ ﺣﻴﻮﺍ ﹰﻧﺎ ﺑﺮﻣﺎﺋ ﹼﹰﻴﺎ‪ ،‬ﻓﺈﻥ ﻫﺬﺍ ﺍﻟﻔﺼﻞ ﻫﻮ ﻓﺼﻞ ﺍﻟﺼﻴﻒ‪.‬‬ ‫‪ (32‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺟﺪﺓ ﻓﻲ ﺍﻟﻴﻤﻦ‪ ،‬ﻓﺈﻥ ﺻﻨﻌﺎﺀ ﻫﻲ ﻋﺎﺻﻤﺔ ﺍﻟﻤﻤﻠﻜﺔ ﺍﻟﻌﺮﺑﻴﺔ ﺍﻟﺴﻌﻮﺩﻳﺔ‪.‬‬ ‫‪ (33‬ﺇﺫﺍ ﻧﺘﺞ ﺍﻟﻠﻮﻥ ﺍﻷﺑﻴﺾ ﻋﻦ ﻣﺰﺝ ﺍﻟﻠﻮﻧﻴﻦ ﺍﻷﺯﺭﻕ ﻭﺍﻷﺣﻤﺮ‪ ،‬ﻓﺈﻥ ‪3 – 2 = 0‬‬ ‫‪ (34‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﻣﺘﻄﺎﺑﻘﺘﻴﻦ‪ ،‬ﻓﺈﻧﻬﻤﺎ ﻣﺘﻘﺎﺑﻠﺘﺎﻥ ﺑﺎﻟﺮﺃﺱ‪.‬‬ ‫‪ (35‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺤﻴﻮﺍﻥ ﻃﺎﺋ ﹰﺮﺍ‪ ،‬ﻓﺈﻧﻪ ﻳﻜﻮﻥ ﻧﺴ ﹰﺮﺍ‪.‬‬ ‫‪ (36‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻤﻮﺯ ﺃﺯﺭﻕ‪ ،‬ﻓﺈﻥ ﺍﻟﺘﻔﺎﺡ ﻣﻦ ﺍﻟﺨﻀﺮﺍﻭﺍﺕ‪.‬‬ ‫‪ 1 32‬‬ ‫‪ 1 32‬‬

‫‪ ‬ﺍﺳﺘﻌﻤﻞ ﺍﻟﻌﺒﺎﺭﺓ ﺃﺩﻧﺎﻩ ﻟﻜﺘﺎﺑﺔ ﻛ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻵﺗﻴﺔ‪ ،‬ﺛﻢ ﺍﺳﺘﻌﻤﻞ ﻣﻌﻠﻮﻣﺎﺕ ﺍﻟﺮﺑﻂ ﻣﻊ ﺍﻟﺤﻴﺎﺓ‬ ‫ﻟﺘﺤﺪﻳﺪ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻜ ﱟﻞ ﻣﻨﻬﺎ‪ ،‬ﻭﺇﺫﺍ ﻛﺎﻧﺖ ﺃ ﱞﻱ ﻣﻨﻬﺎ ﺧﺎﻃﺌﺔ‪ ،‬ﻓﺄﻋﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪ (37-40 .‬ﺍﻧﻈﺮ ﻣﻠﺤﻖ ﺍﻹﺟﺎﺑﺎﺕ‬ ‫”ﺍﻟﺤﻴﻮﺍﻥ ﺍﻟﺬﻱ ﺗﻈﻬﺮ ﻋﻠﻰ ﺟﺴﻤﻪ ﺧﻄﻮﻁ ﻫﻮ ﺍﻟﺤﻤﺎﺭ ﺍﻟﻮﺣﺸﻲ“‪.‬‬ ‫‪ ‬ﻓﻲ ﺍﻟﺴﺆﺍﻝ ‪،52‬‬ ‫‪ (38‬ﻋﻜﺲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‬ ‫‪ (37‬ﻋﺒﺎﺭﺓ ﺷﺮﻃﻴﺔ‬ ‫‪ (40‬ﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ ﻟﻠﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‬ ‫‪ (39‬ﻣﻌﻜﻮﺱ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‬ ‫ﻳﺴﺘﻌﻤﻞ ﺍﻟﻄﻼﺏ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﻤﻨﻄﻘﻴﺔ ﻭﺍﻟﻠﻔﻈﻴﺔ‬ ‫ﻭﺃﺷﻜﺎﻝ ﭬﻦ ﻻﺳﺘﻘﺼﺎﺀ ﺧﺎﺻﻴﺔ ﺍﻟﺘﻌﺪﻱ‬ ‫ﺃﻭﺟﺪ ﻗﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻟﻜﻞ ﻋﺒﺎﺭﺗﻴﻦ ﻓﻴﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﺛﻢ ﻗ ﱢﺮﺭ ﻫﻞ ﻫﻤﺎ ﻣﺘﻜﺎﻓﺌﺎﻥ ﻣﻨﻄﻘ ﹼﹰﻴﺎ ﺃﻡ ﻻ؟ ‪ (41-48‬ﺍﻧﻈﺮ ﻣﻠﺤﻖ ﺍﻹﺟﺎﺑﺎﺕ‬ ‫‪4 ‬‬ ‫ﻟﻠﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ‪.‬‬ ‫‪5 ‬‬ ‫‪∼(p → q) , ∼ p → ∼ q (41‬‬ ‫‪‬‬ ‫‪∼(p → q) , ∼(∼ q → ∼ p) (42‬‬ ‫‪ (52a‬ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪:‬‬ ‫‪(p q) ∨ r , p (q ∨ r) (43‬‬ ‫ﺇﺫﺍ ﻛﻨﺖ ﺗﺴﻜﻦ ﻣﺪﻳﻨﺔ ﺟﺪﺓ‪ ،‬ﻓﺄﻧﺖ‬ ‫ﺗﺴﻜﻦ ﻣﻨﻄﻘﺔ ﻣﻜﺔ ﺍﻟﻤﻜﺮﻣﺔ‪ ،‬ﻭﺇﺫﺍ‬ ‫ﺍﻛﺘﺐ ﺍﻟﻌﻜﺲ ﻭﺍﻟﻤﻌﻜﻮﺱ ﻭﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ ﻟﻜ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻵﺗﻴﺔ‪ ،‬ﺛﻢ ﺣ ﱢﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﺃ ﱞﻱ ﻣﻨﻬﺎ‬ ‫ﻛﻨﺖ ﺗﺴﻜﻦ ﻣﻨﻄﻘﺔ ﻣﻜﺔ ﺍﻟﻤﻜﺮﻣﺔ‬ ‫ﺻﺎﺋ ﹰﺒﺎ ﺃﻡ ﺧﺎﻃ ﹰﺌﺎ‪ .‬ﻭﺇﺫﺍ ﻛﺎﻥ ﺧﺎﻃ ﹰﺌﺎ‪ ،‬ﻓﺄﻋﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪ (44-48 .‬ﺍﻧﻈﺮ ﻣﻠﺤﻖ ﺍﻹﺟﺎﺑﺎﺕ‬ ‫ﻓﺈﻧﻚ ﺗﺴﻜﻦ ﺍﻟﻤﻤﻠﻜﺔ ﺍﻟﻌﺮﺑﻴﺔ‬ ‫‪ (44‬ﺇﺫﺍ ﻛﻨﺖ ﺗﻌﻴﺶ ﻓﻲ ﺍﻟﺪﻣﺎﻡ‪ ،‬ﻓﺈﻧﻚ ﺗﻌﻴﺶ ﻓﻲ ﺍﻟﻤﻤﻠﻜﺔ ﺍﻟﻌﺮﺑﻴﺔ ﺍﻟﺴﻌﻮﺩﻳﺔ‪.‬‬ ‫ﺍﻟﺴﻌﻮﺩﻳﺔ‪ ،‬ﻭﺇﺫﺍ ﻛﻨﺖ ﺗﺴﻜﻦ ﺍﻟﻤﻤﻠﻜﺔ‬ ‫‪ (45‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻄﺎﺋﺮ ﻧﻌﺎﻣﺔ‪ ،‬ﻓﺈﻧﻪ ﻻ ﻳﺴﺘﻄﻴﻊ ﺃﻥ ﻳﻄﻴﺮ‪.‬‬ ‫ﺍﻟﻌﺮﺑﻴﺔ ﺍﻟﺴﻌﻮﺩﻳﺔ‪ ،‬ﻓﺈﻧﻚ ﺗﺴﻜﻦ ﻗﺎﺭﺓ‬ ‫‪ (46‬ﺟﻤﻴﻊ ﺍﻟﻤﺮﺑﻌﺎﺕ ﻣﺴﺘﻄﻴﻼﺕ‪.‬‬ ‫ﺁﺳﻴﺎ‪.‬‬ ‫‪ (47‬ﺟﻤﻴﻊ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺍﻟﻤﺘﻄﺎﺑﻘﺔ ﻟﻬﺎ ﺍﻟﻄﻮﻝ ﻧﻔﺴﻪ‪.‬‬ ‫‪ (48‬ﺍﻟﻤﺜﻠﺚ ﺍﻟﻘﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻳﺤﻮﻱ ﺯﺍﻭﻳﺔ ﻗﻴﺎﺳﻬﺎ ‪90°‬‬ ‫‪(52b‬‬ ‫ﺍﺳﺘﻌﻤﻞ ﺃﺷﻜﺎﻝ ﭬﻦ ﺃﺩﻧﺎﻩ؛ ﻟﺘﺤﺪﻳﺪ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻜ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻵﺗﻴﺔ‪ .‬ﻭﻓ ﱢﺴﺮ ﺗﺒﺮﻳﺮﻙ‪.‬‬ ‫‪   ‬‬ ‫ﻣﺘﺴﺎﻗﻄﺔ‬ ‫ﺍﻟﺜﺪﻳﻴﺎﺕ‬ ‫ﺍﻟﺪﻭﺍﻝ‬ ‫‪‬‬ ‫‪ (52c‬ﺇﺫﺍ ﻛﻨﺖ ﺗﺴﻜﻦ ﻓﻲ ﻣﺪﻳﻨﺔ ﺟﺪﺓ ﻓﺈﻧﻚ‬ ‫ﺍﻷﻭﺭﺍﻕ‬ ‫ﺍﳊﻴﻮﺍﻧﺎﺕ ﺍﻟﺒﺤﺮﻳﺔ‬ ‫ﻏﻴﺮ ﺍﳋﻄﻴﺔ‬ ‫‪‬‬ ‫ﺗﺴﻜﻦ ﻓﻲ ﻗﺎﺭﺓ ﺁﺳﻴﺎ‪ .‬ﻧﻌﻢ ﺻﺤﻴﺤﺔ‪.‬‬ ‫ﺩﺍﺋﻤﺔ ﺍﳋﻀﺮﺓ‬ ‫‪‬‬ ‫ﺍﻟﺪﻭﺍﻝ‬ ‫‪ ‬‬ ‫‪ (52d‬ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ :‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ a‬ﺻﺤﻴﺤﺔ‪،‬‬ ‫ﺍﻟﺘﺮﺑﻴﻌﻴﺔ‬ ‫‪ ‬‬ ‫ﻓﺈﻥ ‪ c‬ﺻﺤﻴﺤﺔ‪ .‬ﺇﺫﺍ ﻛﻨﺎ ﻧﻌﻠﻢ ﺃﻥ ‪a‬‬ ‫‪‬‬ ‫ﺻﺤﻴﺤﺔ‪ ،‬ﻓﺈﻧﻨﺎ ﻧﻌﻠﻢ ﺃﻥ ‪ b‬ﺻﺤﻴﺤﺔ‪،‬‬ ‫‪ ‬‬ ‫ﻭﺇﺫﺍ ﻛﻨﺎ ﻧﻌﻠﻢ ﺃﻥ ‪ b‬ﺻﺤﻴﺤﺔ‪ ،‬ﻓﺈﻥ ‪c‬‬ ‫ﺻﺤﻴﺤﺔ ﺃﻳ ﹰﻀﺎ؛ ﺇﺫﻥ ﻋﻨﺪﻣﺎ ﺗﻜﻮﻥ ‪a‬‬ ‫‪ (49‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺪﺍﻟﺔ ﻏﻴﺮ ﺧﻄﻴﺔ‪ ،‬ﻓﺈﻧﻬﺎ ﺗﻜﻮﻥ ﺩﺍﻟﺔ ﺗﺮﺑﻴﻌﻴﺔ‪.‬‬ ‫ﺻﺤﻴﺤ ﹰﺔ‪ ،‬ﻓﺈﻥ ‪ c‬ﺗﻜﻮﻥ ﺻﺤﻴﺤ ﹰﺔ‪.‬‬ ‫‪ (49‬ﺧﺎﻃﺌﺔ؛ ﺍﻟﻤﻨﻄﻘﺔ ﺍﻟﺰﺭﻗﺎﺀ ‪ (50‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺤﻴﻮﺍﻥ ﻣﻦ ﺍﻟﺜﺪﻳﻴﺎﺕ‪ ،‬ﻓﺈﻧﻪ ﻻ ﻳﻜﻮﻥ ﺣﻴﻮﺍ ﹰﻧﺎ ﺑﺤﺮ ﹰﹼﻳﺎ‪.‬‬ ‫ﻓﻲ ﺷﻜﻞ ﭬﻦ ﺗﺤﺘﻮﻱ ﺍﻟﺪﻭﺍﻝ ‪ (51‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺸﺠﺮﺓ ﻣﺘﺴﺎﻗﻄﺔ ﺍﻷﻭﺭﺍﻕ‪ ،‬ﻓﺈﻧﻬﺎ ﻻ ﺗﻜﻮﻥ ﺩﺍﺋﻤﺔ ﺍﻟﺨﻀﺮﺓ‪.‬‬ ‫ﻏﻴﺮ ﺍﻟﺨﻄﻴﺔ ﻭﻏﻴﺮ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ‪.‬‬ ‫‪  (52‬ﻓﻲ ﻫﺬﻩ ﺍﻟﻤﺴﺄﻟﺔ ﺳﻮﻑ ﺗﺴﺘﻘﺼﻲ ﺃﺣﺪ ﻗﻮﺍﻧﻴﻦ ﺍﻟﻤﻨﻄﻖ ﺑﺎﺳﺘﻌﻤﺎﻝ ﺍﻟﻌﺒﺎﺭﺍﺕ‬ ‫‪ (50‬ﺧﺎﻃﺌﺔ؛ ﺗﺤﺘﻮﻱ ﺍﻟﻤﻨﻄﻘﺔ‬ ‫ﺍﻟﺸﺮﻃﻴﺔ‪ (a–d .‬ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‪.‬‬ ‫ﺍﻟﺨﻀﺮﺍﺀ ﻓﻲ ﺷﻜﻞ ﭬﻦ ﺣﻴﻮﺍﻧﺎﺕ‬ ‫‪   (a‬ﺍﻛﺘﺐ ﺛﻼﺙ ﻋﺒﺎﺭﺍﺕ ﺷﺮﻃﻴﺔ ﺻﺎﺋﺒﺔ‪ ،‬ﺑﺤﻴﺚ ﺗﻜﻮﻥ ﻧﺘﻴﺠﺔ ﻛﻞ ﻋﺒﺎﺭﺓ ﻓﺮ ﹰﺿﺎ ﻟﻠﻌﺒﺎﺭﺓ ﺍﻟﺘﻲ ﺗﻠﻴﻬﺎ‪.‬‬ ‫ﺛﺪﻳﻴﺔ ﻭﺑﺤﺮﻳﺔ ﻓﻲ ﺍﻟﻮﻗﺖ ﻧﻔﺴﻪ‪.‬‬ ‫‪   (b‬ﺍﺭﺳﻢ ﺷﻜﻞ ﭬﻦ ﻳﻮﺿﺢ ﻫﺬﻩ ﺍﻟﺴﻠﺴﻠﺔ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ‪.‬‬ ‫‪ (51‬ﺻﺎﺋﺒﺔ؛ ﻻ ﻳﻮﺟﺪ ﻣﻨﻄﻘﺔ‬ ‫‪   (c‬ﺍﻛﺘﺐ ﻋﺒﺎﺭ ﹰﺓ ﺷﺮﻃﻴ ﹰﺔ ﻣﺴﺘﻌﻤ ﹰﻼ ﻓﺮﺽ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻷﻭﻟﻰ‪ ،‬ﻭﻧﺘﻴﺠﺔ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺜﺎﻟﺜﺔ‪ .‬ﺇﺫﺍ ﻛﺎﻥ ﻓﺮﺽ‬ ‫ﻣﺸﺘﺮﻛﺔ ﺑﻴﻦ ﺍﻟﻤﻨﻄﻘﺘﻴﻦ ﺍﻟﻠﺘﻴﻦ‬ ‫ﺗﻤﺜﻼﻥ ﺍﻷﺷﺠﺎﺭ ﺍﻟﻤﺘﺴﺎﻗﻄﺔ‬ ‫ﺍﻟﻌﺒﺎﺭﺓ ﺍﻷﻭﻟﻰ ﺻﺎﺋ ﹰﺒﺎ‪ .‬ﻓﻬﻞ ﺗﻜﻮﻥ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﻨﺎﺗﺠﺔ ﺻﺎﺋﺒ ﹰﺔ؟‬ ‫ﺍﻷﻭﺭﺍﻕ ﻭﺍﻷﺷﺠﺎﺭ ﺍﻟﺪﺍﺋﻤﺔ‬ ‫ﺍﻟﺨﻀﺮﺓ‪.‬‬ ‫‪   (d‬ﺇﺫﺍ ﹸﺃﻋﻄﻴﺖ ﺍﻟﻌﺒﺎﺭﺗﻴﻦ ﺍﻟﺸﺮﻃﻴﺘﻴﻦ ﺍﻟﺼﺎﺋﺒﺘﻴﻦ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ‪ ، a‬ﻓﺈﻥ ‪ ، b‬ﻭﺇﺫﺍ ﻛﺎﻥ ‪ ،b‬ﻓﺈﻥ ‪ ،c‬ﻓﺎﻛﺘﺐ‬ ‫ﺗﺨﻤﻴﻨﹰﺎ ﺣﻮﻝ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺓ ‪ c‬ﻋﻨﺪﻣﺎ ﺗﻜﻮﻥ ﺍﻟﻌﺒﺎﺭﺓ ‪ a‬ﺻﺎﺋﺒﺔ‪ .‬ﻓ ﱢﺴﺮ ﺗﺒﺮﻳﺮﻙ‪.‬‬ ‫‪33  1-3‬‬ ‫‪‬‬ ‫‪ ‬ﻧﺎﻗﺶ ﻣﻊ ﺍﻟﻄﻼﺏ ﻣﻌﻨﻰ ﺍﻟﺸﻜﻞ ﺍﻟﺮﺑﺎﻋﻲ ﻭﺍﻟﻤﻌﻴﻦ‪ ،‬ﺛﻢ ﺍﻃﻠﺐ ﺇﻟﻴﻬﻢ ﻛﺘﺎﺑﺔ ﻋﻜﺲ ﺍﻟﻌﺒﺎﺭﺓ \"ﻛﻞ ﺍﻷﺷﻜﺎﻝ‬ ‫ﺍﻟﺮﺑﺎﻋﻴﺔ ﻣﻌﻴﻨﺎﺕ\"‪ .‬ﻭﺣ ﱢﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﺃﻡ ﺧﺎﻃﺌﺔ؟‬ ‫ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺸﻜﻞ ﺭﺑﺎﻋ ﹰﹼﻴﺎ‪ ،‬ﻓﺈﻧﻪ ﻣﻌﻴﻦ )ﺧﺎﻃﺌﺔ(‪.‬‬ ‫ﺍﻟﻌﻜﺲ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺸﻜﻞ ﻣﻌﻴﻨﹰﺎ‪ ،‬ﻓﺈﻧﻪ ﺷﻜﻞ ﺭﺑﺎﻋﻲ )ﺻﺤﻴﺤﺔ(‪.‬‬ ‫‪33  1-3‬‬

‫‪‬‬ ‫‪  (53‬ﺣ ﱢﺪﺩ ﻛ ﱞﻞ ﻣﻦ ﺃﺣﻤﺪ ﻭﻣﺎﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ \"ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻌﺪﺩ ‪ 15‬ﺃﻭﻟ ﹼﹰﻴﺎ‪ ،‬ﻓﺈﻥ‬ ‫‪!‬‬ ‫‪53 ‬‬ ‫ﺍﻟﻌﺪﺩ ‪ 20‬ﻳﻘﺒﻞ ﺍﻟﻘﺴﻤﺔ ﻋﻠﻰ ‪ . \"4‬ﻛﻼﻫﻤﺎ ﻳﻌﺘﻘﺪ ﺃﻥ ﻫﺬﻩ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺎﺋﺒﺔ‪ ،‬ﻭﻟﻜﻨﻬﻤﺎ ﺑ ﱠﺮﺭﺍ ﺫﻟﻚ ﺑﺘﺒﺮﻳﺮﻳﻦ ﻣﺨﺘﻠﻔﻴﻦ‪.‬‬ ‫‪‬‬ ‫ﺃﻳﱡﻬﻤﺎ ﻛﺎﻥ ﻣﺼﻴ ﹰﺒﺎ؟ ﻓ ﱢﺴﺮ ﺗﺒﺮﻳﺮﻙ‪ .‬ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‪.‬‬ ‫‪  ‬‬ ‫‪  ‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ‬؛‪ 15 ‬‬ ‫‪‬؛‪20 ‬‬ ‫‪ ‬‬ ‫‪ ‬؛‪‬‬ ‫‪4‬؛‪‬‬ ‫‪.‬‬ ‫‪ 4‬‬ ‫‪.‬‬ ‫‪ ‬ﺍﻃﻠﺐ ﺇﻟﻰ ﺍﻟﻄﻼﺏ ﺃﻥ ﻳﻜﺘﺒﻮﺍ‬ ‫‪  (54‬ﻋﺒﺎﺭﺓ ﺷﺮﻃﻴﺔ ﻓﺮﺿﻬﺎ ﺻﺎﺋﺐ‪ ،‬ﻭﻧﺘﻴﺠﺘﻬﺎ ﺧﺎﻃﺌﺔ‪ .‬ﻫﻞ ﻳﻜﻮﻥ ﻣﻌﻜﻮﺳﻬﺎ ﺻﺎﺋ ﹰﺒﺎ؟ ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‬ ‫ﻋﻦ ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﻭﻣﺎ‬ ‫‪  (55‬ﺍﻛﺘﺐ ﻋﺒﺎﺭﺓ ﺷﺮﻃﻴﺔ‪ ،‬ﺑﺤﻴﺚ ﻳﻜﻮﻥ ﺍﻟﻌﻜﺲ ﻭﺍﻟﻤﻌﻜﻮﺱ ﻭﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ ﻟﻬﺎ ﺟﻤﻴﻌﻬﺎ‬ ‫ﺳﻴﺘﻌﻠﻤﻮﻧﻪ ﻓﻲ ﺍﻟﺪﺭﺱ ﺍﻟﻘﺎﺩﻡ ﺣﻮﻝ ﺍﻟﺘﺒﺮﻳﺮ‬ ‫ﺻﺎﺋﺒﺔ‪ .‬ﻓ ﹼﺴﺮ ﺗﺒﺮﻳﺮﻙ‪.‬‬ ‫ﺍﻻﺳﺘﻨﺘﺎﺟﻲ‪.‬‬ ‫‪ (55‬ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻌﺪﺩ ‪   (56‬ﺗﺠﺪ ﺃﺩﻧﺎﻩ ﻣﻌﻜﻮﺱ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ‪ .A‬ﺍﻛﺘﺐ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ‪ A‬ﻭﻋﻜﺴﻬﺎ ﻭﻣﻌﺎﻛﺴﻬﺎ ﺍﻹﻳﺠﺎﺑﻲ‪.‬‬ ‫‪‬‬ ‫‪ 4‬ﻳﻘﺒﻞ ﺍﻟﻘﺴﻤﺔ ﻋﻠﻰ ﺍﻟﻌﺪﺩ ‪، 2‬‬ ‫ﻓ ﹼﺴﺮ ﺗﺒﺮﻳﺮﻙ‪ .‬ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‪.‬‬ ‫ﻓﺈﻥ ﻟﻠﻄﻴﻮﺭ ﺭﻳ ﹰﺸﺎ‪ .‬ﺣﺘﻰ ﻳﻜﻮﻥ‬ ‫‪ (53‬ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‪ ،‬ﻣﺎﺟﺪ؛ ﻋﻨﺪﻣﺎ ﻳﻜﻮﻥ‬ ‫ﺍﻟ ﹶﻔﺮﺽ ﺧﺎﻃ ﹰﺌﺎ ﻓﻲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪ ،‬ﻓﺈﻥ‬ ‫\"ﺇﺫﺍ ﻟﻢ ﺗﺪﺭﻙ ﺗﻜﺒﻴﺮﺓ ﺍﻹﺣﺮﺍﻡ ﻣﻊ ﺍﻹﻣﺎﻡ‪ ،‬ﻓﺈﻧﻚ ﺫﻫﺒﺖ ﺇﻟﻰ ﺍﻟﻤﺴﺠﺪ ﻣﺘﺄﺧ ﹰﺮﺍ‪\".‬‬ ‫ﺍﻟﻌﻜﺲ ﻭﺍﻟﻤﻌﻜﻮﺱ ﻭﺍﻟﻤﻌﺎﻛﺲ‬ ‫ﺍﻹﻳﺠﺎﺑﻲ ﺟﻤﻴﻌﻬﺎ ﺻﺎﺋﺒﺔ ﻳﺠﺐ‬ ‫ﺍﻟﻌﺒﺎﺭﺓ ﺗﻜﻮﻥ ﺻﺎﺋﺒﺔ ﺩﺍﺋ ﹰﻤﺎ‪.‬‬ ‫ﺃﻥ ﻳﻜﻮﻥ ﺍﻟﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ‬ ‫‪ (54‬ﻧﻌﻢ؛ ﺑﻤﺎ ﺃﻥ ﺍﻟﻨﺘﻴﺠﺔ ﺧﺎﻃﺌﺔ‪ ،‬ﻓﻴﺠﺐ ﺃﻥ‬ ‫ﻳﻜﻮﻥ ﻋﻜﺲ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺎﺋ ﹰﺒﺎ‪.‬‬ ‫‪   (57‬ﹺﺻ ﹺﻒ ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﻭﻋﻜﺴﻬﺎ ﻭﻣﻌﻜﻮﺳﻬﺎ ﻭﻣﻌﺎﻛﺴﻬﺎ ﺍﻹﻳﺠﺎﺑﻲ‪.‬ﺍﻧﻈﺮ ﻣﻠﺤﻖ ﺍﻹﺟﺎﺑﺎﺕ‬ ‫ﺻﺎﺋﺒﻴﻦ ﺃﻭ ﺧﺎﻃﺌﻴﻦ ﻣ ﹰﻌﺎ‪.‬‬ ‫ﻭﺍﻟﻌﻜﺲ ﻭﺍﻟﻤﻌﻜﻮﺱ ﻣﺘﻜﺎﻓﺌﺎﻥ ﻣﻨﻄﻘ ﹰﹼﻴﺎ‪،‬‬ ‫‪B‬‬ ‫؟‬ ‫‪_10a2 -_15ab‬‬ ‫‪‬ﻣﺎ ﺃﺑﺴﻂ ﺻﻮﺭﺓ ﻟﻠﻌﺒﺎﺭﺓ‬ ‫‪‬‬ ‫‪(59‬‬ ‫‪‬‬ ‫ﻭﻋﻠﻴﻪ ﻳﻜﻮﻥ ﺍﻟﻤﻌﻜﻮﺱ ﺻﺎﺋ ﹰﺒﺎ‪.‬‬ ‫‪4a2 - 9b2‬‬ ‫‪ (58‬ﺇﺫﺍ ﻛﺎﻥ ﻣﺠﻤﻮﻉ ﻗﻴﺎ ﹶﺳﻲ ﺯﺍﻭﻳﺘﻴﻦ ﻳﺴﺎﻭﻱ ˚‪ 90‬ﻓﺈﻧﻬﻤﺎ ﻣﺘﺘﺎﻣﺘﺎﻥ‪.‬‬ ‫‪ (56‬ﺍﻟﻔﺮﺽ ﻟﻠﻤﻌﻜﻮﺱ ﻫﻮ ‪ : ~p‬ﻟﻢ ﺗﺪﺭﻙ‬ ‫ﺃ ﱡﻱ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻵﺗﻴﺔ ﻫﻲ ﻋﻜﺲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺃﻋﻼﻩ؟ ‪A‬‬ ‫ﺗﻜﺒﻴﺮﺓ ﺍﻹﺣﺮﺍﻡ ﻣﻊ ﺍﻹﻣﺎﻡ‪.‬‬ ‫‪_a‬‬ ‫‪C‬‬ ‫‪_5a‬‬ ‫‪A‬‬ ‫‪ A‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﻣﺘﺘﺎﻣﺘﻴﻦ‪ ،‬ﻓﺈﻥ ﻣﺠﻤﻮﻉ ﻗﻴﺎﺳﻴﻬﻤﺎ ˚‪90‬‬ ‫ﺍﻟﻨﺘﻴﺠﺔ ﻟﻠﻤﻌﻜﻮﺱ ﻫﻲ ‪ : ~q‬ﺫﻫﺒﺖ‬ ‫‪2a + 3b‬‬ ‫‪2a - 3b‬‬ ‫ﺇﻟﻰ ﺍﻟﻤﺴﺠﺪ ﻣﺘﺄﺧ ﹰﺮﺍ‪.‬‬ ‫‪ B‬ﺇﺫﺍﻛﺎﻧﺖﺍﻟﺰﺍﻭﻳﺘﺎﻥﻏﻴﺮﻣﺘﺘﺎﻣﺘﻴﻦ‪،‬ﻓﺈﻥﻣﺠﻤﻮﻉﻗﻴﺎﺳﻴﻬﻤﺎ ˚‪90‬‬ ‫‪_a‬‬ ‫‪D‬‬ ‫‪_5a‬‬ ‫‪B‬‬ ‫ﺇﺫﻥ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ‪ A‬ﻫﻲ ‪: p→q‬‬ ‫‪ C‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﻣﺘﺘﺎﻣﺘﻴﻦ‪ ،‬ﻓﺈﻥ ﻣﺠﻤﻮﻉ ﻗﻴﺎﺳﻴﻬﻤﺎ ﻻ‬ ‫ﺇﺫﺍ ﻛﻨﺖ ﻗﺪ ﺃﺩﺭﻛﺖ ﺗﻜﺒﻴﺮﺓ ﺍﻹﺣﺮﺍﻡ‬ ‫‪2a - 3b‬‬ ‫‪2a + 3b‬‬ ‫ﻳﺴﺎﻭﻱ ˚‪90‬‬ ‫ﻣﻊ ﺍﻹﻣﺎﻡ‪ ،‬ﻓﺈﻧﻚ ﺫﻫﺒﺖ ﺇﻟﻰ ﺍﻟﻤﺴﺠﺪ‬ ‫‪ D‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﻏﻴﺮ ﻣﺘﺘﺎﻣﺘﻴﻦ‪ ،‬ﻓﺈﻥ ﻣﺠﻤﻮﻉ ﻗﻴﺎﺳﻴﻬﻤﺎ ﻻ‬ ‫ﻣﺒﻜ ﹰﺮﺍ‪.‬‬ ‫ﻳﺴﺎﻭﻱ ˚‪90‬‬ ‫ﻭﻋﻜﺲ ﺍﻟﻌﺒﺎﺭﺓ ‪ A‬ﻫﻮ ‪ : q →p‬ﺇﺫﺍ‬ ‫ﺫﻫﺒﺖ ﺇﻟﻰ ﺍﻟﻤﺴﺠﺪ ﻣﺒﻜ ﹰﺮﺍ‪ ،‬ﻓﺈﻧﻚ‬ ‫‪ 1 34‬‬ ‫ﺳﺘﺪﺭﻙ ﺗﻜﺒﻴﺮﺓ ﺍﻹﺣﺮﺍﻡ ﻣﻊ ﺍﻹﻣﺎﻡ‪.‬‬ ‫‪‬‬ ‫ﻭﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ ﻟﻠﻌﺒﺎﺭﺓ ‪A‬‬ ‫ﻫﻮ ‪ : ~q →~p‬ﺇﺫﺍ ﻟﻢ ﺗﺬﻫﺐ ﺇﻟﻰ‬ ‫‪   ‬‬ ‫ﺍﻟﻤﺴﺠﺪ ﻣﺒﻜ ﹰﺮﺍ‪ ،‬ﻓﺈﻧﻚ ﻟﻦ ﹸﺗﺪﺭﻙ‬ ‫‪      ‬‬ ‫‪     ‬‬ ‫ﺗﻜﺒﻴﺮﺓ ﺍﻹﺣﺮﺍﻡ ﻣﻊ ﺍﻹﻣﺎﻡ‪.‬‬ ‫‪   ‬‬ ‫‪ 1 34‬‬

‫‪‬‬ ‫ﺃﻧﺸﺊ ﺟﺪﻭﻝ ﺍﻟﺼﻮﺍﺏ ﻟﻜ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﻤﺮﻛﺒﺔ ﺍﻵﺗﻴﺔ‪ (60-63 1-2 .‬ﺍﻧﻈﺮ ﻣﻠﺤﻖ ﺍﻹﺟﺎﺑﺎﺕ‬ ‫‪~p ~q (63‬‬ ‫‪~p q (62‬‬ ‫‪~q p (61‬‬ ‫‪q p (60‬‬ ‫ﺍﻛﺘﺐ ﺗﺨﻤﻴ ﹰﻨﺎ ﻣﻌﺘﻤ ﹰﺪﺍ ﻋﻠﻰ ﺍﻟﻤﻌﻠﻮﻣﺎﺕ ﺍﻟﻤﻌﻄﺎﺓ ﻓﻲ ﻛ ﱟﻞ ﻣﻤﺎ ﻳﺄﺗﻲ‪ .‬ﻭﺍﺭﺳﻢ ﺷﻜ ﹰﻼ ﻳﻮﺿﺢ ﺗﺨﻤﻴﻨﻚ ‪1-1‬‬ ‫‪ (64‬ﺗﻘﻊ ﺍﻟﻨﻘﺎﻁ ‪ J, H , K‬ﻋﻠﻰ ﺃﺿﻼﻉ ﻣﺨﺘﻠﻔﺔ ﻟﻤﺜﻠﺚ‪ .‬ﺍﻟﻨﻘﺎﻁ ‪ J, H, K‬ﻟﻴﺴﺖ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ‪.‬‬ ‫‪B‬‬ ‫‪ R , S ,T . R(3, -4), S(-2, -4) , T(0, -4) (65‬ﺗﻘﻊ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ‪.‬‬ ‫‪10‬‬ ‫‪ ABCD A(-1, -7), B(4, -7), C(4, -3) , D(-1, -3) (66‬ﻣﺴﺘﻄﻴﻞ‪.‬‬ ‫‪14.4‬‬ ‫‪8‬‬ ‫‪C‬‬ ‫‪12‬‬ ‫‪6‬‬ ‫‪10‬‬ ‫‪  (67‬ﺗﺼﻨﻊ ﺍﻟﻄﺎﺋﺮﺍﺕ ﺍﻟﻮﺭﻗﻴﺔ ﺑﺸﻜﻞ ﻳﺸﺒﻪ ﺍﻟﻤﺎﺳﺔ؛ ﻟﺬﻟﻚ ﺗﺴﻤﻰ ﺍﻟﻄﺎﺋﺮﺓ ﺍﻟﻤﺎﺳﻴﺔ‪.‬‬ ‫‪E‬‬ ‫ﺳ ﹼﻢ ﺟﻤﻴﻊ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺍﻟﻤﺘﻄﺎﺑﻘﺔ ﻓﻲ ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ‪ .‬‬ ‫‪8‬‬ ‫‪BC CD, BE ED, BA DA‬‬ ‫‪A 14.4‬‬ ‫‪D‬‬ ‫‪‬‬ ‫‪ ‬ﺣﺪﺩ ﺍﻟﻌﻤﻠﻴﺔ ﺍﻟﺘﻲ ﺍﺳﺘﻌﻤﻠﺘﻬﺎ ﻟﺘﺤﻮﻳﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ )‪ (1‬ﺇﻟﻰ ﺍﻟﻤﻌﺎﺩﻟﺔ )‪ (2‬ﻓﻲ ﻛ ﱟﻞ ﻣﻤﺎ ﻳﺄﺗﻲ‪.‬‬ ‫ﺿﺮﺏ ﻛﻼ‬ ‫‪_1‬‬ ‫‪m‬‬ ‫=‬ ‫‪2‬‬ ‫)‪(1‬‬ ‫‪(70‬‬ ‫‪ x + 9 = 4 - 3x (1) (69‬ﺇﺿﺎﻓﺔ ‪ 3x‬ﻟﻜ ﱟﻞ‬ ‫‪ 8(y - 11) = 32 (1) (68‬ﻗﺴﻤﺔ ﻛﻼ‬ ‫‪3‬‬ ‫)‪ m = 6 (2‬ﺍﻟﻄﺮﻓﻴﻦ ﻓﻲ ‪3‬‬ ‫)‪ 4x + 9 = 4 (2‬ﻣﻦ ﺍﻟﻄﺮﻓﻴﻦ‪.‬‬ ‫)‪ y - 11 = 4 (2‬ﺍﻟﻄﺮﻓﻴﻦ ﻋﻠﻰ ‪8‬‬ ‫‪35  1-3‬‬ ‫‪35  1-3‬‬

‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪Biconditinal Statments‬‬ ‫ﹸﻳﻌ ﱡﺪ ﺳﻌﺪ ﺃﻓﻀﻞ ﻃﻼﺏ ﺍﻟﻤﺪﺭﺳﺔ ﻓﻲ ﻟﻌﺒﺔ ﻛﺮﺓ ﺍﻟﻘﺪﻡ‪ .‬ﻭﺇﺫﺍ ﺍﻧ ﹸﺘﺨﺐ ﻣﻦ ﻗﺒﻞ ﺃﻋﻀﺎﺀ ﻓﺮﻳﻖ ﻛﺮﺓ ﺍﻟﻘﺪﻡ ﺍﻟﻤﺪﺭﺳﻲ‪،‬‬ ‫‪ 1‬‬ ‫ﻓﺈﻧﻪ ﺳﻴﻤﺜﻞ ﺍﻟﻤﺪﺭﺳﺔ ﻓﻲ ﻓﺮﻳﻖ ﺍﻟﻤﻨﻄﻘﺔ ﺍﻟﺘﻌﻠﻴﻤﻴﺔ‪ .‬ﺇﺫﺍ ﻣ ﹼﺜﻞ ﺍﻟﻤﺪﺭﺳﺔ ﻓﻲ ﻓﺮﻳﻖ ﺍﻟﻤﻨﻄﻘﺔ ﺍﻟﺘﻌﻠﻴﻤﻴﺔ‪ ،‬ﻓﺈﻧﻪ ﻳﻜﻮﻥ ﻗﺪ ﺍﻧ ﹸﺘﺨﺐ ﻣﻦ‬ ‫‪‬‬ ‫ﹺﻗ ﹶﺒﻞ ﺃﻋﻀﺎﺀ ﻓﺮﻳﻖ ﻛﺮﺓ ﺍﻟﻘﺪﻡ ﺍﻟﻤﺪﺭﺳﻲ‪.‬‬ ‫ﺗﺤﺪﻳﺪ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﺜﻨﺎﺋﻴﺔ‪،‬‬ ‫‪ :p‬ﺍﻧ ﹸﺘﺨﺐ ﺳﻌ ﹲﺪ ﻣﻦ ﹺﻗ ﹶﺒ ﹺﻞ ﺃﻋﻀﺎﺀ ﻓﺮﻳﻖ ﻛﺮﺓ ﺍﻟﻘﺪﻡ ﺍﻟﻤﺪﺭﺳﻲ‪.‬‬ ‫ﻭﺍﺳﺘﻌﻤﺎﻟﻬﺎ‪ ،‬ﻭﺇﻳﺠﺎﺩ ﻗﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻟﻬﺎ‪.‬‬ ‫‪ :q‬ﻣ ﱠﺜﻞ ﺳﻌﺪ ﺍﻟﻤﺪﺭﺳﺔ ﻓﻲ ﻓﺮﻳﻖ ﺍﻟﻤﻨﻄﻘﺔ ﺍﻟﺘﻌﻠﻴﻤﻴﺔ‪.‬‬ ‫‪‬‬ ‫‪ :p → q‬ﺇﺫﺍ ﺍﻧ ﹸﺘﺨﺐ ﺳﻌﺪ ﻣﻦ ﻗﺒﻞ ﻓﺮﻳﻖ ﻛﺮﺓ ﺍﻟﻘﺪﻡ ﺍﻟﻤﺪﺭﺳﻲ‪ ،‬ﻓﺈﻧﻪ ﺳﻴﻤﺜﻞ ﺍﻟﻤﺪﺭﺳﺔ ﻓﻲ ﻓﺮﻳﻖ ﺍﻟﻤﻨﻄﻘﺔ ﺍﻟﺘﻌﻠﻴﻤﻴﺔ‪.‬‬ ‫ﻫﻴﺊ ﺍﻟﻄﻼﺏ ﺑﺄﻥ ﺗﻄﻠﺐ ﺇﻟﻴﻬﻢ ﺇﻋﻄﺎﺀ ﺃﻣﺜﻠﺔ‬ ‫‪ :q → p‬ﺇﺫﺍ ﻣ ﱠﺜﻞ ﺳﻌﺪ ﺍﻟﻤﺪﺭﺳﺔ ﻓﻲ ﻓﺮﻳﻖ ﺍﻟﻤﻨﻄﻘﺔ ﺍﻟﺘﻌﻠﻴﻤﻴﺔ‪ ،‬ﻓﺈﻧﻪ ﻗﺪ ﺍﻧ ﹸﺘﺨﺐ ﻣﻦ ﻗﺒﻞ ﺃﻋﻀﺎﺀ ﻓﺮﻳﻖ ﻛﺮﺓ ﺍﻟﻘﺪﻡ ﺍﻟﻤﺪﺭﺳﻲ‪.‬‬ ‫ﻋﻠﻰ ﻋﺒﺎﺭﺍﺕ ﺷﺮﻃﻴﺔ ﻭﻋﻜﺴﻬﺎ‪ ،‬ﺛﻢ ﺍﻃﻠﺐ‬ ‫ﺇﻟﻴﻬﻢ ﺃﻥ ﻳﻔﻜﺮﻭﺍ ﻓﻲ ﻣﻌﻨﻰ ﻛ ﱟﻞ ﻣﻨﻬﺎ‪ ،‬ﻭﺯ ﱢﻭﺩ‬ ‫ﻓﻲ ﻫﺬﻩ ﺍﻟﺤﺎﻟﺔ‪ ،‬ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ‪ p → q‬ﻭﻋﻜﺴﻬﺎ ‪ q → p‬ﻛﻼﻫﻤﺎ ﺻﺎﺋﺐ‪ .‬ﻭﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﻤﺮﻛﺒﺔ ﺍﻟﻨﺎﺗﺠﺔ ﻋﻦ ﻭﺻﻞ ﻫﺎﺗﻴﻦ‬ ‫ﺍﻟﻌﺒﺎﺭﺗﻴﻦ ﺑﺎﺳﺘﻌﻤﺎﻝ )ﻭ( ﺗﺴﻤﻰ ﻋﺒﺎﺭﺓ ﺷﺮﻃﻴﺔ ﺛﻨﺎﺋﻴﺔ‪.‬‬ ‫ﺍﻟﻄﻼﺏ ﺑﺄﻣﺜﻠﺔ ﹸﺗﺜﺒﺖ ﺃﻥ ﺻﺤﺔ ﺍﻟﻌﺒﺎﺭﺓ‬ ‫ﺍﻟﺸﺮﻃﻴﺔ ﻻ ﺗﻌﺘﻤﺪ ﻋﻠﻰ ﺻﺤﺔ ﻋﻜﺴﻬﺎ‪.‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ 2‬‬ ‫‪  ‬‬ ‫‪qp  (p ↔ q)  (p → q) (q → p)   ‬‬ ‫‪‬‬ ‫ﺇﺫﻥ ﹸﺗﻜﺘﺐ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﺜﻨﺎﺋﻴﺔ ﺍﻟﺴﺎﺑﻘﺔ ﻋﻠﻰ ﺍﻟﻨﺤﻮ ﺍﻟﺘﺎﻟﻲ‪:‬‬ ‫ﻭ ﱢﺯﻉ ﺍﻟﻄﻼﺏ ﻣﺠﻤﻮﻋﺎﺕ ﺛﻨﺎﺋﻴﺔ ﻣﺘﻔﺎﻭﺗﺔ‬ ‫‪ :p ↔ q‬ﹸﻳﻨﺘﺨﺐ ﺳﻌﺪ ﻣﻦ ﻗﺒﻞ ﺃﻋﻀﺎﺀ ﻓﺮﻳﻖ ﻛﺮﺓ ﺍﻟﻘﺪﻡ ﺍﻟﻤﺪﺭﺳﻲ ﺇﺫﺍ ﻭﻓﻘﻂ ﺇﺫﺍ ﻣ ﹼﺜﻞ ﺍﻟﻤﺪﺭﺳﺔ ﻓﻲ ﻓﺮﻳﻖ ﺍﻟﻤﻨﻄﻘﺔ ﺍﻟﺘﻌﻠﻴﻤﻴﺔ‪.‬‬ ‫ﺍﻟﻘﺪﺭﺍﺕ‪ ،‬ﻭﺍﻃﻠﺐ ﺇﻟﻴﻬﻢ ﻗﺮﺍﺀﺓ ﺍﻷﻣﺜﻠﺔ‬ ‫ﻭﺍﻟﺘﺤﻘﻖ ﻣﻨﻬﺎ‪ ،‬ﻭﺇﻧﺸﺎﺀ ﺟﺪﻭﻝ ﺻﻮﺍﺏ‬ ‫‪‬‬ ‫ﻟﻠﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﻭﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ‬ ‫ﺍﻟﺜﻨﺎﺋﻴﺔ ﺍﻟﻤﺘﻀﻤﻨﺔ‪ ،‬ﻭﺑﺬﻟﻚ ﻳﻤﻜﻨﻬﻢ ﺃﻥ ﻳﺮﺑﻄﻮﺍ‬ ‫ﺍﻛﺘﺐ ﻛ ﹼﹰﻼ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺗﻴﻦ ﺍﻟﺸﺮﻃﻴﺘﻴﻦ ﺍﻟﺜﻨﺎﺋﻴﺘﻴﻦ ﺍﻵﺗﻴﺘﻴﻦ ﻋﻠﻰ ﺻﻮﺭﺓ ﻋﺒﺎﺭﺓ ﺷﺮﻃﻴﺔ ﻭﻋﻜﺴﻬﺎ‪ ،‬ﺛﻢ ﺣﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﺜﻨﺎﺋﻴﺔ ﺻﺎﺋﺒﺔ ﺃﻡ‬ ‫ﻋﺒﺎﺭﺍﺕ ﺍﻷﻣﺜﻠﺔ ‪ 1-4‬ﺑﻘﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻓﻲ‬ ‫ﺧﺎﻃﺌﺔ‪ .‬ﻭﺇﺫﺍ ﻛﺎﻧﺖ ﺧﺎﻃﺌﺔ ﻓﺄﻋﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪.‬‬ ‫ﺍﻟﺠﺪﻭﻝ‪.‬‬ ‫‪ (a‬ﺗﻜﻮﻥ ﺍﻟﺰﺍﻭﻳﺔ ﻗﺎﺋﻤﺔ ﺇﺫﺍ ﻭﻓﻘﻂ ﺇﺫﺍ ﻛﺎﻥ ﻗﻴﺎﺳﻬﺎ ‪90°‬‬ ‫ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪ :‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺔ ﻗﺎﺋﻤﺔ‪ ،‬ﻓﺈﻥ ﻗﻴﺎﺳﻬﺎ ‪90°‬‬ ‫‪ ‬ﺍﻃﻠﺐ ﺇﻟﻰ ﺍﻟﻄﻼﺏ ﺣﻞ ﺍﻷﺳﺌﻠﺔ‬ ‫ﺍﻟﻌﻜﺲ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ‪ ،90°‬ﻓﺈﻧﻬﺎ ﺯﺍﻭﻳﺔ ﻗﺎﺋﻤﺔ‪.‬‬ ‫‪. 5-1‬‬ ‫ﻛ ﱞﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﻭﻋﻜﺴﻬﺎ ﺻﺎﺋﺒﺎﻥ؛ ﺇﺫﻥ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﺜﻨﺎﺋﻴﺔ ﺻﺎﺋﺒﺔ‪.‬‬ ‫‪ 3‬‬ ‫‪ x (b‬ﻋﺪ ﹲﺩ ﻣﻮﺟ ﹲﺐ ﺇﺫﺍ ﻭﻓﻘﻂ ﺇﺫﺍ ﻛﺎﻥ ‪x > -2‬‬ ‫✓ ‪‬‬ ‫ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ‪ x‬ﻋﺪ ﹰﺩﺍ ﻣﻮﺟ ﹰﺒﺎ‪ ،‬ﻓﺈﻥ ‪ . x > -2‬ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺻﺎﺋﺒﺔ‪.‬‬ ‫ﺍﻟﻌﻜﺲ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ‪ ،x > -2‬ﻓﺈﻥ ‪ x‬ﻋﺪﺩ ﻣﻮﺟﺐ‪ .‬ﺍﻓﺘﺮﺽ ﺃﻥ ‪ x = -1‬؛ ﺇﺫﻥ ‪ ،-1 > -2‬ﻟﻜﻦ ‪ -1‬ﻟﻴﺲ ﻋﺪ ﹰﺩﺍ ﻣﻮﺟ ﹰﺒﺎ؛ ﺇﺫﻥ ﻋﻜﺲ ﺍﻟﻌﺒﺎﺭﺓ‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻷﺳﺌﻠﺔ ‪1-4‬؛ ﻟﻠﺘﺤﻘﻖ ﻣﻦ ﻓﻬﻢ‬ ‫ﺍﻟﻄﻼﺏ ﻣﻜﻮﻧﺎﺕ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﺜﻨﺎﺋﻴﺔ‬ ‫ﺍﻟﺸﺮﻃﻴﺔ ﺧﺎﻃﺊ‪ ،‬ﻭﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﺜﻨﺎﺋﻴﺔ ﺧﺎﻃﺌﺔ‪.‬‬ ‫ﻭﻛﻴﻔﻴﺔ ﺗﺤﺪﻳﺪ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻬﺎ‪.‬‬ ‫‪ (1–4 ‬ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‪.‬‬ ‫‪‬‬ ‫ﺍﻛﺘﺐ ﻛﻞ ﻋﺒﺎﺭﺓ ﺷﺮﻃﻴﺔ ﺛﻨﺎﺋﻴﺔ ﻣﻤﺎ ﻳﺄﺗﻲ ﻋﻠﻰ ﺻﻮﺭﺓ ﻋﺒﺎﺭﺓ ﺷﺮﻃﻴﺔ ﻭﻋﻜﺴﻬﺎ‪ .‬ﺛﻢ ﺣﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﺜﻨﺎﺋﻴﺔ ﺻﺎﺋﺒﺔ ﺃﻡ ﺧﺎﻃﺌﺔ‪ .‬ﻭﺇﺫﺍ ﻛﺎﻧﺖ‬ ‫ﺍﺳﺘﻌﻤﻞ ﺍﻟﺴﺆﺍﻝ ‪4‬؛ ﻟﺘﺤﺪﻳﺪ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ‬ ‫ﺧﺎﻃﺌﺔ ﻓﺄﻋﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪.‬‬ ‫ﺑﻤﻘﺪﻭﺭ ﺍﻟﻄﻼﺏ ﺍﺳﺘﻌﻤﺎﻝ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‬ ‫‪ (1‬ﺗﻜﻮﻥ ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﻣﺘﺘﺎﻣﺘﻴﻦ ﺇﺫﺍ ﻭﻓﻘﻂ ﺇﺫﺍ ﻛﺎﻥ ﻣﺠﻤﻮﻉ ﻗﻴﺎﺳﻴﻬﻤﺎ ‪ (2 90°‬ﻻ ﺩﻭﺍﻡ ﻓﻲ ﺍﻟﻤﺪﺍﺭﺱ ﺇﺫﺍ ﻭﻓﻘﻂ ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻴﻮﻡ ﻫﻮ ﺍﻟﺠﻤﻌﺔ‪.‬‬ ‫ﺍﻟﺜﻨﺎﺋﻴﺔ ﻓﻲ ﺳﻴﺎ ﹴﻕ ﺟﺒﺮ ﱟﻱ‪.‬‬ ‫‪ |2x| = 4 (4‬ﺇﺫﺍ ﻭﻓﻘﻂ ﺇﺫﺍ ﻛﺎﻥ ‪x = 2‬‬ ‫‪ (3‬ﻳﺘﻘﺎﻃﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺎﻥ ﺇﺫﺍ ﻭﻓﻘﻂ ﺇﺫﺍ ﻛﺎﻧﺎ ﻏﻴﺮ ﺃﻓﻘﻴﻴﻦ‪.‬‬ ‫‪ 1 36‬‬ ‫‪ 1 36‬‬ ‫‪ (3‬ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪ :‬ﺇﺫﺍ ﺗﻘﺎﻃﻊ ﻣﺴﺘﻘﻴﻤﺎﻥ‪ ،‬ﻓﺈﻧﻬﻤﺎ ﻏﻴﺮ‬ ‫‪‬‬ ‫ﺃﻓﻘ ﱠﻴﻴﻦ‪ .‬ﺻﺎﺋﺒﺔ‪.‬‬ ‫‪ (1‬ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪ :‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﻣﺘﺘﺎﻣﺘﻴﻦ‪ ،‬ﻓﺈﻥ‬ ‫ﺍﻟﻌﻜﺲ‪ :‬ﺇﺫﺍ ﻟﻢ ﻳﻜﻦ ﺍﻟﻤﺴﺘﻘﻴﻤﺎﻥ ﺃﻓﻘ ﱠﻴﻴﻦ‪ ،‬ﻓﺈﻧﻬﻤﺎ‬ ‫ﻣﺠﻤﻮﻉ ﻗﻴﺎﺳﻴﻬﻤﺎ ˚‪ 90‬ﺻﺎﺋﺒﺔ‪.‬‬ ‫ﻳﺘﻘﺎﻃﻌﺎﻥ‪.‬‬ ‫ﺍﻟﻌﻜﺲ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ﻣﺠﻤﻮﻉ ﻗﻴﺎ ﹶﺳﻲ ﺯﺍﻭﻳﺘﻴﻦ ˚‪، 90‬‬ ‫ﺧﺎﻃﺌﺔ‪ :‬ﺍﻟﻤﺴﺘﻘﻴﻤﺎﻥ ﺍﻟﺮﺃﺳﻴﺎﻥ ﺍﻟﻤﺘﻮﺍﺯﻳﺎﻥ ﻻ‬ ‫ﻓﺈﻧﻬﻤﺎ ﻣﺘﺘﺎﻣﺘﺎﻥ‪ .‬ﺻﺎﺋﺒﺔ‪.‬‬ ‫ﻳﺘﻘﺎﻃﻌﺎﻥ‪.‬‬ ‫ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﺜﻨﺎﺋﻴﺔ ﺻﺎﺋﺒﺔ‪.‬‬ ‫ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﺜﻨﺎﺋﻴﺔ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ (4‬ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ‪ ، x = 2‬ﻓﺈﻥ ‪،|2x| = 4‬‬ ‫‪ (2‬ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻴﻮﻡ ﻫﻮ ﺍﻟﺠﻤﻌﺔ‪ ،‬ﻓﺈﻧﻪ ﻻ‬ ‫ﻳﻮﺟﺪ ﺩﻭﺍﻡ ﻓﻲ ﺍﻟﻤﺪﺍﺭﺱ‪ .‬ﺻﺎﺋﺒﺔ‪.‬‬ ‫ﺻﺎﺋﺒﺔ‪.‬‬ ‫ﺍﻟﻌﻜﺲ‪ :‬ﺇﺫﺍ ﻟﻢ ﻳﻜﻦ ﻫﻨﺎﻙ ﺩﻭﺍﻡ ﻓﻲ ﺍﻟﻤﺪﺍﺭﺱ‪،‬‬ ‫ﺍﻟﻌﻜﺲ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ‪ ، |2x| = 4‬ﻓﺈﻥ ‪x = 2‬‬ ‫ﻓﺈﻥ ﺍﻟﻴﻮﻡ ﻫﻮ ﺍﻟﺠﻤﻌﺔ‪ .‬ﺧﺎﻃﺌﺔ؛ ﻷﻧﻪ ﻻ ﺩﻭﺍﻡ ﻓﻲ‬ ‫ﺧﺎﻃﺌﺔ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ‪ ، x = -2‬ﻓﺈﻥ ‪|2x| = 4‬‬ ‫ﺍﻟﻤﺪﺍﺭﺱ ﻳﻮﻡ ﺍﻟﺴﺒﺖ ﺃﻳ ﹰﻀﺎ‪.‬‬ ‫ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﺜﻨﺎﺋﻴﺔ ﺧﺎﻃﺌﺔ‪.‬‬ ‫ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﺜﻨﺎﺋﻴﺔ ﺧﺎﻃﺌﺔ‪.‬‬

‫‪  ‬‬ ‫‪1 -3 ‬‬ ‫‪          ‬‬ ‫‪ (17)  ‬‬ ‫‪( 1 6 )‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫‪    1-3‬‬ ‫‪ 1-3‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﺇﺫﺍ ﻏ ﹼﻴﺮﺕ ﺍﻟﻔﺮﺽ ﺃﻭ ﺍﻟﻨﺘﻴﺠﺔ ﻓﻲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪ ،‬ﻓﺈﻧﻚ ﺳﺘﺤﺼﻞ ﻋﻠﻰ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﻤﺘﺮﺍﺑﻄﺔ‪ .‬ﻭﻳﺒ ﹼﻴﻦ ﺍﻟﺠﺪﻭﻝ ﺃﺩﻧﺎﻩ ﺛﻼﺛﺔ‬ ‫ﻋﺒﺎﺭﺓ )ﺇﺫﺍ‪...‬ﻓﺈﻥ‪ (...‬ﻫﻲ ﻋﺒﺎﺭﺓ ﻣﺜﻞ \"ﺇﺫﺍ ﻛﻨﺖ ﺗﻘﺮﺃ ﻫﺬﻩ ﺍﻟﺼﻔﺤﺔ‪ ،‬ﻓﺈﻧﻚ ﺗﺪﺭﺱ ﺭﻳﺎﺿﻴﺎﺕ\"‪.‬ﻭﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺘﻲ ﻳﻤﻜﻦ ﻛﺘﺎﺑﺘﻬﺎ ﻋﻠﻰ ﺍﻟﺼﻮﺭﺓ‬ ‫ﺃﻧﻮﺍﻉ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﻤﺘﺮﺍﺑﻄﺔ ﻫﻲ‪ :‬ﺍﻟﻌﻜﺲ ﻭﺍﻟﻤﻌﻜﻮﺱ ﻭﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ‪ ،‬ﻭﻛﻴﻔﻴﺔ ﺍﺭﺗﺒﺎﻃﻬﺎ ﺑﺎﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪.‬‬ ‫)ﺇﺫﺍ ‪ ...‬ﻓﺈﻥ‪ (...‬ﹸﺗﺴ ﹼﻤﻰ ﻋﺒﺎﺭ ﹰﺓ ﺷﺮﻃﻴ ﹰﺔ‪ ،‬ﻭﺍﻟﺠﻤﻠﺔ ﺍﻟﺘﻲ ﺗﻠﻲ ﻛﻠﻤﺔ \"ﺇﺫﺍ\" ﻣﺒﺎﺷﺮﺓ ﹸﺗﺴ ﹼﻤﻰ ﺍﻟﻔﺮﺽ‪ ،‬ﻭﺍﻟﺠﻤﻠﺔ ﺍﻟﺘﻲ ﺗﻠﻲ ﻛﻠﻤﺔ \"ﻓﺈﻥ\" ﻣﺒﺎﺷﺮﺓ‬ ‫‪‬‬ ‫‪  ‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﹸﺗﺴ ﹼﻤﻰ ﺍﻟﻨﺘﻴﺠﺔ‪.‬‬ ‫ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﻣﺘﻘﺎﺑﻠﺘﻴﻦ ﺑﺎﻟﺮﺃﺱ‪ ،‬ﻓﺈﻧﻬﻤﺎ ﻣﺘﻄﺎﺑﻘﺘﺎﻥ‪.‬‬ ‫‪ p → q  ‬ﻓﺮﺽ ﹸﻣﻌ ﹰﻄﻰ ﻭﻧﺘﻴﺠﺔ‬ ‫ﻭﻳﻤﻜﻦ ﺗﻤﺜﻴﻞ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺑﺎﻟﺮﻣﻮﺯ ﻋﻠﻰ ﺍﻟﻨﺤﻮ ﺍﻵﺗﻲ ‪ ، p → q:‬ﻭﺗﻘﺮﺃ \"‪ p‬ﺗﺆﺩﻱ ﺇﻟﻰ ‪ \"، q‬ﺃﻭ \"ﺇﺫﺍ ﻛﺎﻥ ‪ ،p‬ﻓﺈﻥ ‪.\"q‬‬ ‫ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﻣﺘﻄﺎﺑﻘﺘﻴﻦ‪ ،‬ﻓﺈﻧﻬﻤﺎ ﻣﺘﻘﺎﺑﻠﺘﺎﻥ ﺑﺎﻟﺮﺃﺱ‪.‬‬ ‫‪ q → p‬ﺗﺒﺪﻳﻞ ﺍﻟﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ‬ ‫‪‬‬ ‫ﺣ ﹼﺪﺩ ﺍﻟﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ‪ ،‬ﻓﻲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻵﺗﻴﺔ‪ ،‬ﻭﺍﻛﺘﺒﻬﺎ ﻓﻲ ﺻﻮﺭﺓ )ﺇﺫﺍ‪ ...‬ﻓﺈﻥ‪(...‬‬ ‫‪1‬‬ ‫ﺇﺫﺍ ﻟﻢ ﺗﻜﻦ ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﻣﺘﻘﺎﺑﻠﺘﻴﻦ ﺑﺎﻟﺮﺃﺱ‪ ،‬ﻓﺈﻧﻬﻤﺎ ﻏﻴﺮ‬ ‫‪ ∼p → ∼q‬ﻧﻔﻲ ﻛ ﱟﻞ ﻣﻦ ﺍﻟﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ ﻓﻲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪.‬‬ ‫‪‬‬ ‫ﺳﺘﺤﺼﻞ ﻋﲆ ﻓﻄﲑﺓ ﳎﺎﻧﻴﺔ ﺇﺫﺍ ﺍﺷﱰﻳﺖ ‪ 3‬ﻓﻄﺎﺋﺮ‪.‬‬ ‫ﻣﺘﻄﺎﺑﻘﺘﻴﻦ‪.‬‬ ‫ﺇﺫﺍ ﻟﻢ ﺗﻜﻦ ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﻣﺘﻄﺎﺑﻘﺘﻴﻦ‪ ،‬ﻓﺈﻧﻬﻤﺎ ﻏﻴﺮ ﻣﺘﻘﺎﺑﻠﺘﻴﻦ‬ ‫ﻧﻔﻲ ﻛ ﱟﻞ ﻣﻦ ﺍﻟﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ ﻓﻲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ‪،‬‬ ‫‪∼q → ∼p‬‬ ‫‪ ‬‬ ‫‪ ‬ﺍﺷﱰﻳﺖ ‪ 3‬ﻓﻄﺎﺋﺮ‪.‬‬ ‫ﺑﺎﻟﺮﺃﺱ‪.‬‬ ‫ﻭﻣﻦ ﺛﻢ ﺗﺒﺪﻳﻞ ﻣﻮﻗ ﹶﻌﻴ ﹺﻬﻤﺎ‪.‬‬ ‫‪ ‬ﺳﺘﺤﺼﻞ ﻋﲆ ﻓﻄﲑﺓ ﳎﺎﻧﻴﺔ‪.‬‬ ‫ﻭﻳﻤﻜﻦ ﲢﺪﻳﺪ ﻗﻴﻢ ﺻﻮﺍﺏ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﴩﻃﻴﺔ ﺍﳌﱰﺍﺑﻄﺔ )‪ (T‬ﺃﻭ )‪:(F‬‬ ‫‪ ‬ﺇﺫﺍ ﺍﺷﱰﻳﺖ ‪ 3‬ﻓﻄﺎﺋﺮ‪ ،‬ﻓﺈﻧﻚ ﺳﺘﺤﺼﻞ ﻋﲆ ﻓﻄﲑ ﹴﺓ ﳎﺎﻧﻴﺔ‪.‬‬ ‫ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﳌﺘﻜﺎﻓﺌﺔ ﻣﻨﻄﻘ ﹼﹰﻴﺎ‪ :‬ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺘﻲ ﳍﺎ ﻗﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻧﻔﺴﻬﺎ‪ ،‬ﹸﺗﺴ ﱠﻤﻰ ﻋﺒﺎﺭﺍﺕ ﻣﺘﻜﺎﻓﺌﺔ ﻣﻨﻄﻘ ﹼﹰﻴﺎ‪.‬‬ ‫ﺣ ﹼﺪﺩ ﺍﻟﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ ﻓﻲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻵﺗﻴﺔ‪:‬‬ ‫‪2‬‬ ‫‪‬‬ ‫ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ، ∠S ∠R , ∠R ∠X‬ﻓﺈﻥ ‪∠S ∠X‬‬ ‫‪ (1‬ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﻭﻣﻌﺎﻛﺴﻬﺎ ﺍﻹﻳﺠﺎﺑﻲ ﻣﺘﻜﺎﻓﺌﺘﺎﻥ ﻣﻨﻄﻘ ﹼﹰﻴﺎ‪.‬‬ ‫‪∠S ∠R , ∠R ∠X ‬‬ ‫‪ (2‬ﻋﻜﺲ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﻭﻣﻌﻜﻮﺳﻬﺎ ﻣﺘﻜﺎﻓﺌﺘﺎﻥ ﻣﻨﻄﻘ ﹰﹼﻴﺎ‪.‬‬ ‫‪.∠S ∠X ‬‬ ‫‪ ∼(p q) (3‬ﺗﻜﺎﻓﺊ ﻣﻨﻄﻘ ﹼﹰﻴﺎ ‪∼p ∼q‬‬ ‫‪ ∼(p q) (4‬ﺗﻜﺎﻓﺊ ﻣﻨﻄﻘ ﹰﹼﻴﺎ ‪∼p ∼q‬‬ ‫‪‬‬ ‫‪‬‬ ‫ﺣ ﹼﺪﺩ ﺍﻟﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ ﻓﻲ ﻛ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻵﺗﻴﺔ‪:‬‬ ‫ﺍﻛﺘﺐ ﺍﻟﻌﻜﺲ ﻭﺍﻟﻤﻌﻜﻮﺱ ﻭﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ ﻟﻜ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻵﺗﻴﺔ‪ ،‬ﻭﺣ ﹼﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻟﻤﺮﺗﺒﻄﺔ ﺻﺎﺋﺒﺔ‬ ‫‪ (1‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻴﻮﻡ ﻫﻮ ﺍﻟﺠﻤﻌﺔ‪ ،‬ﻓﺈﻧﻪ ﻻ ﻳﻮﺟﺪ ﺩﻭﺍﻡ ﻣﺪﺭﺳﻲ‪         .‬‬ ‫ﺃﻡ ﺧﺎﻃﺌﺔ‪ ،‬ﻭﺇﺫﺍ ﻛﺎﻧﺖ ﺧﺎﻃﺌﺔ‪ ،‬ﻓﺄﻋ ﹺﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪.‬‬ ‫‪ (2‬ﺇﺫﺍ ﻛﺎﻥ ‪ ،x-8 = 32‬ﻓﺈﻥ ‪ x-8 = 32 : . x = 40‬؛ ‪x = 40 : ‬‬ ‫‪ (1‬ﺇﺫﺍ ﻛﻨﺖ ﺗﻘﻴﻢ ﻓﻲ ﺍﻟﺮﻳﺎﺽ‪ ،‬ﻓﺈﻧﻚ ﺗﻘﻴﻢ ﻓﻲ ﺍﻟﻤﻤﻠﻜﺔ ﺍﻟﻌﺮﺑﻴﺔ ﺍﻟﺴﻌﻮﺩﻳﺔ‪.‬‬ ‫‪ (3‬ﺇﺫﺍ ﻛﺎﻥ ﻟﻤﻀﻠﻊ ﺃﺭﺑﻊ ﺯﻭﺍﻳﺎ ﻗﺎﺋﻤﺔ‪ ،‬ﻓﺈﻥ ﺍﻟﻤﻀﻠﻊ ﻣﺴﺘﻄﻴﻞ‪        .‬‬ ‫‪                         ‬‬ ‫ﺍﻛﺘﺐ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻵﺗﻴﺔ ﻋﻠﻰ ﺻﻮﺭﺓ )ﺇﺫﺍ‪ ...‬ﻓﺈﻥ‪:(..‬‬ ‫‪                     ‬‬ ‫‪ (4‬ﻛﻞ ﺍﻟ ﹺﻘﺮﺩﺓ ﹸﺗﺤﺐ ﺍﻟﻤﻮﺯ‪        .‬‬ ‫‪               ‬‬ ‫‪ (5‬ﻣﺠﻤﻮﻉ ﻗﻴﺎ ﹶﺳﻲ ﺍﻟﺰﺍﻭﻳﺘﻴﻦ ﺍﻟﻤﺘﺘﺎﻣﺘﻴﻦ ﻳﺴﺎﻭﻱ ‪90°         .90°‬‬ ‫‪ (2‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﻣﺘﺘﺎﻣﺘﻴﻦ‪ ،‬ﻓﺈﻥ ﻣﺠﻤﻮﻉ ﻗﻴﺎ ﹶﺳﻴﻬﻤﺎ ﻳﺴﺎﻭﻱ ‪.90°‬‬ ‫‪    90°       ‬‬ ‫‪ (6‬ﺍﻟﻨﻘﺎﻁ ﺍﻟﻮﺍﻗﻌﺔ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣ ﹴﺔ ﻭﺍﺣﺪ ﹴﺓ ﻫﻲ ﺍﻟﻨﻘﺎﻁ ﺍﻟﺘﻲ ﺗﻘﻊ ﻋﻠﻰ ﺍﻟﻤﺴﺘﻘﻴﻢ ﻧﻔﺴﻪ‪.‬‬ ‫‪ 90°           ‬‬ ‫‪            ‬‬ ‫‪      90°         ‬‬ ‫ﺣ ﹼﺪﺩ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻜﻞ ﻋﺒﺎﺭ ﹴﺓ ﺷﺮﻃﻴ ﹴﺔ ﻓﻴﻤﺎ ﻳﺄﺗﻲ‪ ،‬ﻭﺇﺫﺍ ﻛﺎﻧﺖ ﺻﺎﺋﺒﺔ‪ ،‬ﻓﻮ ﹼﺿﺢ ﺗﺒﺮﻳﺮﻙ‪ ،‬ﻭﺇﻥ ﻟﻢ ﺗﻜﻦ ﻛﺬﻟﻚ‪ ،‬ﻓﺄﻋ ﹺﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪:‬‬ ‫‪p q p qp q‬‬ ‫)‪(p q‬‬ ‫‪pq‬‬ ‫‪ (3‬ﺃﻭﺟﺪ ﻗﻴﻢ ﺍﻟﺼﻮﺍﺏ ﻟﻠﻌﺒﺎﺭﺗﻴﻦ ‪،∼(p ∼q), ∼p q‬‬ ‫ﺛﻢ ﻗ ﹼﺮﺭ ﻫﻞ ﻫﻤﺎ ﻣﺘﻜﺎﻓﺌﺘﺎﻥ ﻣﻨﻄﻘ ﹼﹰﻴﺎ ﺃﻡ ﻻ؟‬ ‫‪TTFF‬‬ ‫‪F‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪  ‬‬ ‫ﻛﺎﻥ ﺍﻟﺠﻤﻌﺔ‪.‬‬ ‫ﺍ‪ ‬ﻷ‪ ‬ﺭﺑ‪‬ﻌ‪‬ﺎﺀ‪،‬ﻓ‪‬ﺈ ‪‬ﻥ‪‬ﺃ‪‬ﻣ‪‬ﺲ‬ ‫ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﻴﻮﻡ ﻫﻮ‬ ‫‪(7‬‬ ‫‪F‬‬ ‫‪F‬‬ ‫‪ ‬‬ ‫‪  ‬‬ ‫‪TFFT‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪  ‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ ‬‬ ‫‪T‬‬ ‫‪T‬‬ ‫‪FTTF‬‬ ‫‪F‬‬ ‫‪ (8‬ﺇﺫﺍ ﻛﺎﻥ ‪ a‬ﻋﺪ ﹰﺩﺍ ﻣﻮﺟ ﹰﺒﺎ‪ ،‬ﻓﺈﻥ ‪ 10a‬ﺃﻛﺒﺮ ﻣﻦ ‪.a‬‬ ‫‪  a   10a    a                ‬‬ ‫‪FFTT‬‬ ‫‪F‬‬ ‫‪1‬‬ ‫‪17‬‬ ‫‪ ‬‬ ‫‪1‬‬ ‫‪16‬‬ ‫‪ ‬‬ ‫‪ ‬‬ ‫‪(19)  ‬‬ ‫‪( 1 8 )‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪  1-3‬‬ ‫‪ ‬‬ ‫‪1-3‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪   (5‬ﺭﺳﻢ ﻋﺎﺻﻢ ﺷﻜﻞ ﭬﻦ؛ ﻟﻴﻮﺿﺢ ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ‬ ‫ﺍﻷﺷﻜﺎﻝ ﺍﻟﺮﺑﺎﻋﻴﺔ ﻭﺍﻟﻤﺴﺘﻄﻴﻼﺕ ﻭﺍﻟﻤﺮﺑﻌﺎﺕ ﻭﺍﻟﻤﻌ ﱠﻴﻨﺎﺕ‪.‬‬ ‫‪    (1‬ﻗﺮﺃ ﻋﻠ ﹼﻲ ﻓﻲ ﺃﺣﺪ ﺍﻷﺑﺤﺎﺙ ﺃﻥ ﺍﻷﺷﺨﺎﺹ‬ ‫ﺣ ﹼﺪﺩ ﺍﻟﻔﺮﺽ ﻭﺍﻟﻨﺘﻴﺠﺔ ﻓﻲ ﻛ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻵﺗﻴﺔ‪:‬‬ ‫ﺍﻟﺬﻳﻦ ﻳﺘﻌﺮﺿﻮﻥ ﻷﺷﻌﺔ ﺍﻟﺸﻤﺲ ﻓﺘﺮﺍﺕ ﻃﻮﻳﻠﺔ ﻳ ﹸﻜﻮﻧﻮﻥ ﺃﻛﺜﺮ‬ ‫‪ (1‬ﺇﺫﺍ ﺍﺷﺘﺮﻳﺖ ﺛﻼﺟﺔ‪ ،‬ﻓﺈﻧﻚ ﺳﺘﺤﺼﻞ ﻋﻠﻰ ﺧﻼﻁ ﻛﻬﺮﺑﺎﺋﻲ ﻣﺠﺎﻧﻲ‪.‬‬ ‫)ﺇﺭﺷﺎ ﹲﺩ‪ :‬ﺍﻟ ﹸﻤﻌﻴﻦ ﺷﻜﻞ ﺭﺑﺎﻋﻲ ﻟﻪ ﺃﺭﺑﻌﺔ ﺃﺿﻼﻉ ﻣﺘﻄﺎﺑﻘﺔ(‪.‬‬ ‫ﻋﺮﺿﺔ ﻟﻺﺻﺎﺑﺔ ﺑﺴﺮﻃﺎﻥ ﺍﻟﺠﻠﺪ‪ ،‬ﻓﻬﻞ ﻳﻤﻜﻦ ﺃﻥ ﻳﺴﺘﻨﺘﺞ ﻋﻠ ﱞﻲ‬ ‫ﻣﻦ ﻫﺬﻩ ﺍﻟﻤﻌﻠﻮﻣﺔ ﺃﻧﻪ ﺳﻴﻘﻠﻞ ﺍﺣﺘﻤﺎﻝ ﺇﺻﺎﺑﺘﻪ ﺑﺴﺮﻃﺎﻥ ﺍﻟﺠﻠﺪ‪،‬‬ ‫‪  ‬‬ ‫‪ ‬‬ ‫ﺇﺫﺍ ﺍﻣﺘﻨﻊ ﻋﻦ ﺍﻟﺘﻌﺮﺽ ﻷﺷﻌﺔ ﺍﻟﺸﻤﺲ ﻓﺘﺮﺍ ﹴﺕ ﻃﻮﻳﻠﺔ؟‬ ‫‪     ‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪         ‬‬ ‫‪ (2‬ﺇﺫﺍ ﻛﺎﻥ ‪ ، x + 8 = 4‬ﻓﺈﻥ ‪.x = -4‬‬ ‫‪      ‬‬ ‫‪x = -4   x + 8 = 4 ‬‬ ‫‪  (2‬ﻳﻘﻮﻝ ﺯﻳﺪ‪ :‬ﺇﻥ ﻧﺼﻴﺒﻲ ﻣﻦ ﻣﻴﺮﺍﺙ ﺃﺑﻲ ﺳﻴﻜﻮﻥ ﻣﺜﻞ ﻧﺼﻴﺐ‬ ‫‪ (3‬ﺇﺫﺍ ﺗﻔ ﹼﻮﻕ ﺃﺣﻤﺪ ﻓﻲ ﺍﻟﺪﺭﺍﺳﺔ‪ ،‬ﻓﺈﻧﻪ ﺳ ﹸﻴﻜﺎﻓﺄ ﺑﺮﺣﻠﺔ ﺳﻴﺎﺣﻴﺔ ﺇﻟﻰ ﺃﺑﻬﺎ‪.‬‬ ‫ﺃﺧﻲ ﺑﺤﺴﺐ ﺍﻟﻘﻮﺍﻋﺪ ﺍﻟﺸﺮﻋﻴﺔ‪ ،‬ﺍﻛﺘﺐ ﻫﺬﻩ ﺍﻟﻌﺒﺎﺭﺓ ﻓﻲ ﺻﻮﺭﺓ‬ ‫‪          ‬‬ ‫ﺇﺫﺍ ﻛﺎﻥ ‪ Q‬ﻳﺮﻣﺰ ﺇﱃ ﺷﻜﻞ ﺭﺑﺎﻋﻲ‪ ،‬ﻓﺤ ﹼﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻧﺖ ﻛﻞ‬ ‫)ﺇﺫﺍ‪ ...‬ﻓﺈﻥ‪.(...‬‬ ‫ﺍﻛﺘﺐ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻵﺗﻴﺔ ﻓﻲ ﺻﻮﺭﺓ )ﺇﺫﺍ‪ ...‬ﻓﺈﻥ‪:(...‬‬ ‫ﻋﺒﺎﺭﺓ ﳑﹼﺎ ﻳﺄﰐ ﺻﺎﺋﺒﺔ ﺃﻡ ﺧﺎﻃﺌﺔ‪ ،‬ﻭﺇﺫﺍ ﻛﺎﻧﺖ ﺧﺎﻃﺌﺔ‪ ،‬ﻓﺄﻋ ﹺﻂ‬ ‫‪ (4‬ﺍﻟﻤﻀﻠﻊ ﺫﻭ ﺍﻷﺿﻼﻉ ﺍﻷﺭﺑﻌﺔ ﺷﻜﻞ ﺭﺑﺎﻋﻲ‪         .‬‬ ‫‪          ‬‬ ‫ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪.‬‬ ‫‪  ‬‬ ‫‪ (5‬ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺤﺎﺩﺓ ﺃﻗﻞ ﻣﻦ ‪90°          .90°‬‬ ‫‪ (a‬ﺇﺫﺍ ﻛﺎﻥ ‪ Q‬ﻣﺮﺑ ﹰﻌﺎ‪ ،‬ﻓﺈﻥ ‪ Q‬ﻣﺴﺘﻄﻴﻞ‪ .‬‬ ‫‪    (3‬ﻳﺘﻀﻤﻦ ﻣﻮﻗﻊ ﺇﻟﻜﺘﺮﻭﻧﻲ ﻟﺤﺠﺰ ﺗﺬﺍﻛﺮ‬ ‫ﺣ ﹼﺪﺩ ﻗﻴﻤﺔ ﺍﻟﺼﻮﺍﺏ ﻟﻜ ﱟﻞ ﻣﻦ ﺍﻟﻌﺒﺎﺭﺍﺕ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻵﺗﻴﺔ‪ ،‬ﻓﺈﺫﺍ ﻛﺎﻧﺖ ﺻﺤﻴﺤﺔ ﻓﻮ ﹼﺿﺢ ﺗﺒﺮﻳﺮﻙ‪ .‬ﻭﺇﺫﺍ ﻟﻢ ﺗﻜﻦ ﻛﺬﻟﻚ ﻓﺄﻋ ﹺﻂ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹼﹰﺩﺍ‪.‬‬ ‫‪ (b‬ﺇﺫﺍ ﱂ ﻳﻜﻦ ‪ Q‬ﻣﺴﺘﻄﻴ ﹰﻼ‪ ،‬ﻓﺈﻥ ‪ Q‬ﻟﻴﺲ ﻣﻌ ﱠﻴﻨﹰﺎ‪.‬‬ ‫ﺍﻟﻄﻴﺮﺍﻥ ﺍﻟﺨﻴﺎﺭ‪:‬‬ ‫‪ (6‬ﺇﺫﺍ ﻛﺎﻥ ﻟﺪﻳﻚ ‪ 5‬ﺭﻳﺎﻻ ﹴﺕ‪ ،‬ﻓﺈﻥ ﻟﺪﻳﻚ ‪ 5‬ﺃﻭﺭﺍﻕ ﻧﻘﺪﻳﺔ ﻣﻦ ﻓﺌﺔ ﺍﻟﺮﻳﺎﻝ ﺍﻟﻮﺍﺣﺪ‪.‬‬ ‫\"ﺇﺫﺍ ﻛﻨﺖ ﺗﺮﻳﺪ ﺗﺤﺪﻳﺪ ﻣﻘﻌﺪﻙ ﻓﻲ ﺍﻟﻄﺎﺋﺮﺓ‪ ،‬ﻓﺎﺩﻓﻊ ﺭﺳﻮ ﹰﻣﺎ‬ ‫‪                      ‬‬ ‫ﺇﺿﺎﻓﻴ ﹰﺔ ﻣﻘﺪﺍﺭﻫﺎ ‪ 20‬ﺭﻳﺎ ﹰﻻ\"‪.‬‬ ‫‪ (c‬ﺇﺫﺍ ﻛﺎﻥ ‪ Q‬ﻣﺴﺘﻄﻴ ﹰﻼ ﻭﻟﻜﻦ ﻟﻴﺲ ﻣﺮﺑ ﹰﻌﺎ‪ ،‬ﻓﺈﻥ ‪ Q‬ﻟﻴﺲ‬ ‫‪ (7‬ﺇﺫﺍ ﹸﺭﻣﻲ ﻣﻜﻌﺒﺎﻥ ﻋﺪﺩﻳﺎﻥ‪ ،‬ﻭﻛﺎﻥ ﻣﺠﻤﻮﻉ ﺍﻟﻌﺪﺩﻳﻦ ﺍﻟﻈﺎﻫﺮﻳﻦ ﻋﻠﻰ ﺍﻟﻮﺟﻬﻴﻦ ﺍﻟﻌﻠﻮﻳﻴﻦ ﻳﺴﺎﻭﻱ ‪ ،11‬ﻓﺈﻥ ﺃﺣﺪ ﻫﺬﻳﻦ ﺍﻟﻌﺪﺩﻳﻦ ﻳﻜﻮﻥ ‪.5‬‬ ‫ﻣﻌﻴﻨﹰﺎ‪.‬‬ ‫ﺍﻛﺘﺐ ﺍﻟﻌﻜﺲ ﻭﺍﻟﻤﻌﻜﻮﺱ ﻭﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ ﻟﻬﺬﻩ ﺍﻟﻌﺒﺎﺭﺓ‪.‬‬ ‫‪5   6      11              ‬‬ ‫‪‬‬ ‫‪    20       ‬‬ ‫‪   ‬‬ ‫‪ (d‬ﺇﺫﺍ ﱂ ﻳﻜﻦ ‪ Q‬ﻣﻌ ﱠﻴﻨﹰﺎ‪ ،‬ﻓﺈﻥ ‪ Q‬ﻟﻴﺲ ﻣﺮﺑ ﹰﻌﺎ‪.‬‬ ‫‪    ‬‬ ‫‪ (8‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﻣﺘﻜﺎﻣﻠﺘﻴﻦ‪ ،‬ﻓﺈﻥ ﺇﺣﺪﺍﻫﻤﺎ ﺗﻜﻮﻥ ﺯﺍﻭﻳ ﹰﺔ ﺣﺎﺩ ﹰﺓ‪.‬‬ ‫‪‬‬ ‫‪          ‬‬ ‫‪                    ‬‬ ‫‪     20‬‬ ‫‪ (9‬ﺍﻛﺘﺐ ﺍﻟﻌﻜﺲ ﻭﺍﻟﻤﻌﻜﻮﺱ ﻭﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ ﻟﻠﻌﺒﺎﺭﺓ ﺍﻟﺸﺮﻃﻴﺔ ﺍﻵﺗﻴﺔ‪ ،‬ﻭﺣﺪﺩ ﻣﺎ ﺇﺫﺍ ﻛﺎﻧﺖ ﻛﻞ ﻋﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﺃﻡ ﺧﺎﻃﺌﺔ‪ .‬ﻭﺇﺫﺍ‬ ‫‪  20          ‬‬ ‫ﻛﺎﻧﺖ ﺧﺎﻃﺌﺔ‪ ،‬ﻓﺎﻛﺘﺐ ﻣﺜﺎ ﹰﻻ ﻣﻀﺎ ﹰﹼﺩﺍ‪\" .‬ﺇﺫﺍ ﻛﺎﻥ ‪ 89‬ﻳﻘﺒﻞ ﺍﻟﻘﺴﻤﺔ ﻋﻠﻰ ‪ ،2‬ﻓﺈﻥ ‪ 89‬ﻋﺪﺩ ﺯﻭﺟ ﱞﻲ\"‪.‬‬ ‫‪ 2    89       89   ‬‬ ‫‪       ‬‬ ‫‪      89  2   89    ‬‬ ‫‪ 2     89      89     ‬‬ ‫‪  (4‬ﹸﻛﺘﺐ ﻋﻠﻰ ﻗﺎﺭﻭﺭﺓ ﺩﻭﺍﺀ ﺍﻟﻌﺒﺎﺭﺓ ﺍﻵﺗﻴﺔ \"ﺇﺫﺍ ﻛﻨﺖ ﺳﺘﻘﻮﺩ‬ ‫ﺍﻟﺴﻴﺎﺭﺓ‪ ،‬ﻓﻴﺠﺐ ﺃﻻ ﺗﺘﻨﺎﻭﻝ ﻫﺬﺍ ﺍﻟﺪﻭﺍﺀ\"‪ ،‬ﺍﻛﺘﺐ ﺍﻟﻌﻜﺲ‬ ‫ﻭﺍﻟﻤﻌﻜﻮﺱ ﻭﺍﻟﻤﻌﺎﻛﺲ ﺍﻹﻳﺠﺎﺑﻲ ﻟﻬﺬﻩ ﺍﻟﻌﺒﺎﺭﺓ‪.‬‬ ‫‪          ‬‬ ‫‪         ‬‬ ‫‪          ‬‬ ‫‪        ‬‬ ‫‪1‬‬ ‫‪18‬‬ ‫‪  ‬‬ ‫‪1‬‬ ‫‪19‬‬ ‫‪  ‬‬ ‫‪36A    1‬‬