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‫ﻭﺯﺍﺭﺓ ﺍﻟﺘﺮﺑﻴﺔ‬ ‫‪»ª∏Y öûY …OÉ◊G ∞q °üdG‬‬ ‫‪∫hC’G »°SGQódG π°üØdG‬‬ ‫‪øjQɪàdG á°SGôq c‬‬ ‫ﺍﻟﻠﺠﻨﺔ ﺍﻹﺷﺮﺍﻓﻴﺔ ﻟﺪﺭﺍﺳﺔ ﻭﻣﻮﺍﺀﻣﺔ ﺳﻠﺴﻠﺔ ﻛﺘﺐ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ‬ ‫ﺃ‪ .‬ﺇﺑﺮﺍﻫﻴﻢ ﺣﺴﲔ ﺍﻟﻘﻄﺎﻥ )ﺭﺋﻴ ﹰﺴﺎ(‬ ‫ﺃ‪ .‬ﻓﺘﺤﻴﺔ ﻣﺤﻤﻮﺩ ﺃﺑﻮ ﺯﻭﺭ‬ ‫ﺃ‪ .‬ﺣﺼﺔ ﻳﻮﻧﺲ ﻣﺤﻤﺪ ﻋﻠﻲ‬ ‫ﺍﻟﻄﺒﻌﺔ ﺍﻟﺜﺎﻧﻴﺔ‬ ‫‪ ١٤٣٧ - ١٤٣٦‬ﻫـ‬ ‫‪ ٢٠١٦ - ٢٠١٥‬ﻡ‬

‫ﳉﻨﺔ ﺩﺭﺍﺳﺔ ﻭﻣﻮﺍﺀﻣﺔ ﻛﺘﺐ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ ﻟﻠﺼﻒ ﺍﳊﺎﺩﻱ ﻋﺸﺮ ﻋﻠﻤﻲ‬ ‫ﺃ‪ .‬ﺣﺴﻦ ﻧﻮﺡ ﻋﻠﻲ ﺍﳌﻬﻨﺎ )ﺭﺋﻴ ﹰﺴﺎ(‬ ‫ﺃ‪ .‬ﻣﺼﻄﻔﻰ ﻣﺤﻤﺪ ﺷﻌﺒﺎﻥ ﻣﺤﻤﻮﺩ‬ ‫ﺃ‪ .‬ﺣﺴﲔ ﺍﻟﻴﻤﺎﻧﻲ ﺍﻟﺸﺎﻣﻲ‬ ‫ﺃ‪ .‬ﺷﻴﺨﺔ ﻓﻼﺡ ﻣﺒﺎﺭﻙ ﺍﳊﺠﺮﻑ‬ ‫ﺃ‪ .‬ﺻﺪﻳﻘﺔ ﺃﺣﻤﺪ ﺻﺎﻟﺢ ﺍﻻﻧﺼﺎﺭﻱ‬ ‫ﺃ‪ .‬ﻣﻨﻰ ﻋﻠﻲ ﻋﻴﺴﻰ ﺍﳌﺴﺮﻱ‬ ‫ﺩﺍﺭ ﺍﻟ ﱠﺘﺮﺑﹶﻮ ﹼﻳﻮﻥ ‪ House of Education‬ﺵ‪.‬ﻡ‪.‬ﻡ‪ .‬ﻭﺑﻴﺮﺳﻮﻥ ﺇﺩﻳﻮﻛﻴﺸﻦ ‪٢٠١٣‬‬ ‫© ﹶﺟﻤﻴﻊ ﺍﳊﻘﻮﻕ ﹶﻣﺤﻔﻮﻇﺔ‪ :‬ﻻ ﻳﹶﺠﻮﺯ ﻧ ﹾﺸﺮ ﺃ ﹼﻱ ﹸﺟﺰﺀ ﻣﻦ ﻫﺬﺍ ﺍﻟ ﹺﻜﺘﺎﺏ ﺃﻭ ﺗﹶﺼﻮﻳﺮﻩ ﺃﻭ ﺗﹶﺨﺰﻳﻨﻪ ﺃﻭ ﺗﹶﺴﺠﻴﻠﻪ ﺑﺄ ﹼﻱ‬ ‫ﻭﹶﺳﻴﻠﹶﺔ ﺩﹸﻭﻥ ﹸﻣ ﹶﻮﺍﻓ ﹶﻘﺔ ﺧ ﹼﻄ ﱠﻴﺔ ﹺﻣ ﹶﻦ ﺍﻟ ﹼﻨﺎ ﹺﺷﺮ‪.‬‬ ‫ﺍﻟﻄﺒﻌﺔ ﺍﻷﻭﻟﻰ ‪٢٠١٣‬‬ ‫ﺍﻟﻄﺒﻌﺔ ﺍﻟﺜﺎﻧﻴﺔ ‪٢٠١٥‬‬









‫ﺍﻟﻤﺤﺘﻮﻳﺎﺕ‬ ‫ﺍﻟﻮﺣﺪﺓ ﺍﻷﻭﻟﻰ‪ :‬ﺍﻷﻋﺪﺍﺩ ﺍﻟﺤﻘﻴﻘﻴﺔ‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪9 .............................................................................................................................................1-1‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪12 .............................................................................................................................................1-2‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪15 .............................................................................................................................................1-3‬‬ ‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ ﺍﻷﻭﻟﻰ‪17 .................................................................................................................................................‬‬ ‫ﺗﻤﺎﺭﻳﻦ ﺇﺛﺮﺍﺋﻴّﺔ ‪19 ............................................................................................................................................................‬‬ ‫ﺍﻟﻮﺣﺪﺓ ﺍﻟﺜﺎﻧﻴﺔ‪ :‬ﺍﻟﺪﻭﺍﻝ ﺍﻟﺤﻘﻴﻘﻴﺔ‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪20 .............................................................................................................................................2-1‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪22 ............................................................................................................................................ 2-2‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪24 .............................................................................................................................................2-3‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪27 ............................................................................................................................................ 2-4‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪30 ........................................................................................................................................... 2-5‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪32 ............................................................................................................................................ 2-6‬‬ ‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ ﺍﻟﺜﺎﻧﻴﺔ ‪34 ..................................................................................................................................................‬‬ ‫ﺗﻤﺎﺭﻳﻦ ﺇﺛﺮﺍﺋﻴّﺔ ‪36 ...........................................................................................................................................................‬‬ ‫ﺍﻟﻮﺣﺪﺓ ﺍﻟﺜﺎﻟﺜﺔ‪ :‬ﻛﺜﻴﺮﺍﺕ ﺍﻟﺤﺪﻭﺩ‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪38 .............................................................................................................................................3-1‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪41 .............................................................................................................................................3-2‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪43 .............................................................................................................................................3-3‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪46 .............................................................................................................................................3-4‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪49 .............................................................................................................................................3-5‬‬ ‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ ﺍﻟﺜﺎﻟﺜﺔ ‪51 ..................................................................................................................................................‬‬ ‫ﺗﻤﺎﺭﻳﻦ ﺇﺛﺮﺍﺋﻴّﺔ ‪52 ...........................................................................................................................................................‬‬

‫ﺍﻟﻮﺣﺪﺓ ﺍﻟﺮﺍﺑﻌﺔ‪ :‬ﺍﻟﺪﻭﺍﻝ ﺍﻷﺳﻴﺔ ﻭﺍﻟﺪﻭﺍﻝ ﺍﻟﻠﻮﻏﺎﺭﻳﺘﻤﻴﺔ‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪54 .............................................................................................................................................4-1‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪57 ............................................................................................................................................ 4-2‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪59 .............................................................................................................................................4-3‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪62 .............................................................................................................................................4-4‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪65 .............................................................................................................................................4-5‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪67 .............................................................................................................................................4-6‬‬ ‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ ﺍﻟﺮﺍﺑﻌﺔ ‪69 .................................................................................................................................................‬‬ ‫ﺗﻤﺎﺭﻳﻦ ﺇﺛﺮﺍﺋﻴّﺔ ‪71 ............................................................................................................................................................‬‬ ‫ﺍﻟﻮﺣﺪﺓ ﺍﻟﺨﺎﻣﺴﺔ‪ :‬ﺍﻟﻤﺘﺠﻬﺎﺕ‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪72 .............................................................................................................................................5-1‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪74 .............................................................................................................................................5-2‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪77 .............................................................................................................................................5-3‬‬ ‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ ﺍﻟﺨﺎﻣﺴﺔ‪81 ..............................................................................................................................................‬‬ ‫ﺗﻤﺎﺭﻳﻦ ﺇﺛﺮﺍﺋﻴّﺔ ‪84 ............................................................................................................................................................‬‬ ‫ﺍﻟﻮﺣﺪﺓ ﺍﻟﺴﺎﺩﺳﺔ‪ :‬ﺍﻟﺠﺒﺮ ﺍﻟﻤﺘﻘﻄﻊ )ﺍﻹﺣﺼﺎﺀ(‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪85 .............................................................................................................................................6-1‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪87 ............................................................................................................................................ 6-2‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪89 .............................................................................................................................................6-3‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪91 .............................................................................................................................................6-4‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪93 .............................................................................................................................................6-5‬‬ ‫ﺗَ َﻤ ﱠﺮ ْﻥ ‪95 .............................................................................................................................................6-6‬‬ ‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ ﺍﻟﺴﺎﺩﺳﺔ ‪97 .............................................................................................................................................‬‬ ‫ﺗﻤﺎﺭﻳﻦ ﺇﺛﺮﺍﺋﻴّﺔ ‪99 ...........................................................................................................................................................‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺍﻟﺠﺬﻭﺭ ﻭﺍﻟﺘﻌﺒﻴﺮﺍﺕ ﺍﻟﺠﺬﺭﻳﺔ ‪1-1‬‬ ‫‪Roots and Radical Expressions‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫)‪ (1‬ﺑﺎﺳﺘﺨﺪﺍﻡ ﻗﻮﺍﻧﻴﻦ ﺍﻟﺠﺬﻭﺭ ﺃﻭﺟﺪ ﺇﻥ ﺃﻣﻜﻦ‪:‬‬ ‫‪(a) 400‬‬ ‫‪(b) 1600‬‬ ‫‪(c) 104‬‬ ‫‪(d) 0.01‬‬ ‫‪(f) 0.0064‬‬ ‫‪(e) 0.25‬‬ ‫‪(j) 36 # 25‬‬ ‫)‪(g‬‬ ‫‪- 16‬‬ ‫)‪(h‬‬ ‫‪2‬‬ ‫‪49‬‬ ‫‪50‬‬ ‫)‪(i‬‬ ‫‪12‬‬ ‫)‪(k‬‬ ‫‪-1‬‬ ‫‪(l) 75 # 300‬‬ ‫‪147‬‬ ‫‪121‬‬ ‫)‪ (2‬ﺑﺎﺳﺘﺨﺪﺍﻡ ﻗﻮﺍﻧﻴﻦ ﺍﻟﺠﺬﻭﺭ ﺃﻭﺟﺪ‪:‬‬ ‫‪(a) 3 27‬‬ ‫‪(b) 3 1000‬‬ ‫‪(c) 3 -64‬‬ ‫‪(d) 3 0.125‬‬ ‫‪(f) 3 216 # 343‬‬ ‫‪(h) 3 0‬‬ ‫)‪(e‬‬ ‫‪3‬‬ ‫‪8‬‬ ‫)‪(g‬‬ ‫‪3‬‬ ‫‪-‬‬ ‫‪375‬‬ ‫‪125‬‬ ‫‪24‬‬ ‫‪(i) 3 60 # 90‬‬ ‫)‪ (3‬ﺑ ّﺴﻂ ﻛ ًّﻼ ﻣﻦ ﺍﻟﺘﻌﺒﻴﺮﺍﺕ ﺍﻟﺠﺬﺭﻳﺔ ﺍﻟﺘﺎﻟﻴﺔ ﻣﺴﺘﺨﺪ ًﻣﺎ ﻗﻮﺍﻧﻴﻦ ﺍﻟﺠﺬﻭﺭ‪:‬‬ ‫‪(a) 16x2‬‬ ‫‪(b) 0 . 25x6‬‬ ‫‪(c) x8 y18‬‬ ‫‪(d) 8x3 , x $ 0‬‬ ‫‪(g) 3 -125y6‬‬ ‫)‪(e‬‬ ‫‪x3y5‬‬ ‫‪,‬‬ ‫‪y‬‬ ‫‪$‬‬ ‫‪0,‬‬ ‫‪x‬‬ ‫‪2‬‬ ‫‪0‬‬ ‫‪(f) 5 216x2 + 23 64x4, x 2 0‬‬ ‫‪25x‬‬ ‫‪(h) 3 81x2‬‬ ‫‪(i) 3 -250x6 y5‬‬ ‫‪(j) 3 49x2 # 3 56xy3‬‬ ‫‪(k) 3 256u5 v ' 3 4u2 v10 , u ! 0 , v ! 0‬‬ ‫‪(a) 5 # 40‬‬ ‫)‪ (4‬ﺑ ّﺴﻂ ﻛ ًّﻼ ﻣﻦ ﺍﻟﺘﻌﺒﻴﺮﺍﺕ ﺍﻟﺘﺎﻟﻴﺔ ﻣﺴﺘﺨﺪ ًﻣﺎ ﻗﻮﺍﻧﻴﻦ ﺍﻟﺠﺬﻭﺭ‪:‬‬ ‫‪(d) 5 # ^ 5 + 15h‬‬ ‫‪(g) ^5 + 2 11h2‬‬ ‫‪(b) 3 4 # 3 80‬‬ ‫)‪(c‬‬ ‫‪3 640‬‬ ‫‪(j) 75 - 4 18 + 2 32‬‬ ‫‪3 270‬‬ ‫‪(e) ^ 3 - 2h2‬‬ ‫‪(f) 2 # ^ 50 + 7h‬‬ ‫)‪(h‬‬ ‫‪3.6 # 108‬‬ ‫‪(i) 3 3 16 - 4 3 54 + 3 128‬‬ ‫‪4 # 103‬‬ ‫‪(k) 4 3 81 - 3 3 54‬‬ ‫‪(l) 3 -18 # 3 -12‬‬ ‫‪(m) ^2 7 + 1h2 - ^ 3 - 1h^ 3 + 1h‬‬ ‫‪9‬‬

‫)‪ (5‬ﺣﺪﻳﻘﺔ ﻣﺴﺘﻄﻴﻠﺔ ﺍﻟﺸﻜﻞ ﻃﻮﻟﻬﺎ ‪ 5 21 m‬ﻭﻋﺮﺿﻬﺎ ‪2 7 m‬‬ ‫)‪ (a‬ﺃﻭﺟﺪ ﻣﺤﻴﻂ ﺍﻟﺤﺪﻳﻘﺔ‪.‬‬ ‫)‪ (b‬ﺃﻭﺟﺪ ﻣﺴﺎﺣﺔ ﺍﻟﺤﺪﻳﻘﺔ‪.‬‬ ‫)‪ (6‬ﺍﻛﺘﺐ ﻛ ًّﻼ ﻣﻤﺎ ﻳﻠﻲ ﺑﺤﻴﺚ ﻳﻜﻮﻥ ﺍﻟﻤﻘﺎﻡ ﻋﺪ ًﺩﺍ ﻧﺴﺒﻴًّﺎ‪:‬‬ ‫)‪(a‬‬ ‫‪21 7‬‬ ‫)‪(b‬‬ ‫‪3‬‬ ‫‪(c) 4‬‬ ‫‪4 # 27‬‬ ‫‪32‬‬ ‫‪3 3-2‬‬ ‫‪(d) 3 + 8‬‬ ‫‪(e) 5 + 5‬‬ ‫‪(f) 5 - 2 - ^9 - 4 5h‬‬ ‫‪2-2 8‬‬ ‫‪4-3 5‬‬ ‫‪5+2‬‬ ‫)‪(g‬‬ ‫‪22‬‬ ‫‪(h) 3 - 1‬‬ ‫‪(i) x + 1 , x d Z+ , x ! 1‬‬ ‫‪-‬‬ ‫‪2 2 2- 3‬‬ ‫‪x-1‬‬ ‫‪3- 2 3+ 2‬‬ ‫)‪(j‬‬ ‫‪x‬‬ ‫‪+‬‬ ‫‪y+2‬‬ ‫‪xy‬‬ ‫‪,‬‬ ‫‪x,‬‬ ‫‪y‬‬ ‫‪d‬‬ ‫‪Z+‬‬ ‫‪x+‬‬ ‫‪y‬‬ ‫)‪ (7‬ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺘﻌﺒﻴﺮ‪ ، x2 - 6 :‬ﺇﺫﺍ ﻛﺎﻥ ‪x = 4‬‬ ‫‪5-1‬‬ ‫=‪x‬‬ ‫‪1+‬‬ ‫‪5‬‬ ‫ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺘﻌﺒﻴﺮ‪ ، x2 - x + 1 :‬ﺇﺫﺍ ﻛﺎﻥ‬ ‫)‪(8‬‬ ‫‪2‬‬ ‫)‪ (9‬ﺍﻛﺘﺐ ﻛ ًّﻼ ﻣﻦ ﺍﻟﺘﻌﺒﻴﺮﻳﻦ ﺍﻟﺘﺎﻟﻴﻴﻦ ﻋﻠﻰ ﺍﻟﺼﻮﺭﺓ ‪a + b 2 , a , b d Z‬‬ ‫‪E = 5 + 6 2^3 2 + 4h‬‬ ‫‪F = ^7 2 - 4h2‬‬ ‫)‪ (10‬ﺍﻟﺤﺴﺎﺏ ﺍﻟﺬﻫﻨﻲ‪ .‬ﺑ ّﺴﻂ‪1 + 5 + 11 + 21 + 13 + 7 + 3 + 1 :‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-5‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪(1) 3 -64x3 + 4x = 0‬‬ ‫‪ab‬‬ ‫‪ab‬‬ ‫)‪(2‬‬ ‫‪8-‬‬ ‫‪7‬‬ ‫‪+‬‬ ‫‪3‬‬ ‫‪dZ‬‬ ‫‪ab‬‬ ‫‪3‬‬ ‫‪4-‬‬ ‫‪ab‬‬ ‫‪7‬‬ ‫‪ab‬‬ ‫‪(3) ^3 - 2 2h27 # ^3 + 2 2h27 = 1‬‬ ‫‪(4) 3 2 + 3 3 = 3 5‬‬ ‫‪(5) m # m2 = m2 , 6m d R‬‬ ‫‪10‬‬

‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(6-12‬ﻇﻠّﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫)‪ (6‬ﺍﻟﺘﻌﺒﻴﺮ ﺍﻟﺠﺬﺭﻱ ﺍﻟﺬﻱ ﻓﻲ ﺃﺑﺴﻂ ﺻﻮﺭﺓ ﻫﻮ‪:‬‬ ‫‪a 3 216‬‬ ‫‪b‬‬ ‫‪2‬‬ ‫‪c 39‬‬ ‫‪d‬‬ ‫‪2‬‬ ‫‪a2‬‬ ‫‪32‬‬ ‫‪3‬‬ ‫‪a 2- 3‬‬ ‫‪a ϕ2 + ϕ = 1‬‬ ‫ﻓﻲ ﺃﺑﺴﻂ ﺻﻮﺭﺓ ﻧﻀﺮﺏ ﻛ ًّﻼ ﻣﻦ ﺍﻟﺒﺴﻂ ﻭﺍﻟﻤﻘﺎﻡ ﻓﻲ‪:‬‬ ‫‪3‬‬ ‫‪5‬‬ ‫ﻟﻮﺿﻊ ﺍﻟﺘﻌﺒﻴﺮ ﺍﻟﺠﺬﺭﻱ‬ ‫)‪(7‬‬ ‫‪a -1‬‬ ‫‪3‬‬ ‫‪4‬‬ ‫‪b 32‬‬ ‫‪c2‬‬ ‫‪d4‬‬ ‫)‪ 7 + 4 3 (8‬ﻳﺴﺎﻭﻱ‪:‬‬ ‫‪b 2+ 3‬‬ ‫‪c 3- 2‬‬ ‫‪d 3+ 2‬‬ ‫‪b ϕ2 = ϕ + 1‬‬ ‫‪b -x‬‬ ‫ﻓﺈﻥ‪:‬‬ ‫=‪ϕ‬‬ ‫‪1+‬‬ ‫‪5‬‬ ‫ﺇﺫﺍ ﻛﺎﻥ‬ ‫)‪(9‬‬ ‫‪2‬‬ ‫‪c ϕ + ϕ2 + 1 = 0‬‬ ‫‪d ϕ2 + 1 = ϕ‬‬ ‫ﻳﺴﺎﻭﻱ‪:‬‬ ‫‪1‬‬ ‫‪:‬‬ ‫‪x‬‬ ‫ﻓﺈﻥ‬ ‫ﺇﺫﺍ ﻛﺎﻥ ‪x d R-‬‬ ‫)‪(10‬‬ ‫‪x‬‬ ‫‪c1 d x‬‬ ‫)‪ (11‬ﺇﺫﺍ ﻛﺎﻥ ﺣﺠﻢ ﺷﺒﻪ ﺍﻟﻤﻜﻌﺐ ﺍﻟﻤﻘﺎﺑﻞ ﻳﺴﺎﻭﻱ ‪ ،40 cm3‬ﻓﺈﻥ ‪ x‬ﺗﺴﺎﻭﻱ‪:‬‬ ‫‪x‬‬ ‫‪5x‬‬ ‫‪a 2 cm‬‬ ‫‪b 2 2 cm‬‬ ‫‪c -2 2 cm‬‬ ‫‪d 4 cm‬‬ ‫)‪ (12‬ﺇﺫﺍ ﻛﺎﻥ ﺣﺠﻢ ﺃﺳﻄﻮﺍﻧﺔ ﺍﺭﺗﻔﺎﻋﻬﺎ ‪ h‬ﻭﻃﻮﻝ ﻧﺼﻒ ﻗﻄﺮﻫﺎ ‪ r‬ﻳﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ‪ V = π r2h :‬ﺣﻴﺚ ﺍﻟﺤﺠﻢ )‪(V‬‬ ‫ﺑﺪﻻﻟﺔ ﻛﻞ ﻣﻦ ﺍﺭﺗﻔﺎﻉ ﻭﻧﺼﻒ ﻗﻄﺮ ﺍﻷﺳﻄﻮﺍﻧﺔ‪ ،‬ﻓﺄﻱ ﻣﻦ ﺍﻟﻌﻼﻗﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ ﺻﺤﻴﺤﺔ؟‬ ‫‪a h = π r2V‬‬ ‫‪b‬‬ ‫‪h‬‬ ‫=‬ ‫‪π‬‬ ‫‪:V‬‬ ‫‪c r = π hV‬‬ ‫‪d‬‬ ‫=‪r‬‬ ‫‪V‬‬ ‫‪r2‬‬ ‫‪πh‬‬ ‫‪11‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺍﻷﺳﺲ ﺍﻟﻨﺴﺒﻴﺔ ‪1-2‬‬ ‫‪Rational Exponents‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫‪(a) -4 81‬‬ ‫‪(b) 4 -81‬‬ ‫)‪ (1‬ﺑ ّﺴﻂ ﻛ ًّﻼ ﻣﻦ ﺍﻟﺘﻌﺒﻴﺮﺍﺕ ﺍﻟﺠﺬﺭﻳﺔ ﺍﻟﺘﺎﻟﻴﺔ ﺇﻥ ﺃﻣﻜﻦ‪:‬‬ ‫‪(c) 4 36 # 108‬‬ ‫)‪(d‬‬ ‫‪5 256‬‬ ‫‪(e) 5 32y10‬‬ ‫‪(f) 5 -x20‬‬ ‫‪58‬‬ ‫‪(h) 4 81 + 4 729‬‬ ‫‪16x25‬‬ ‫‪(g) 5 0.01024‬‬ ‫)‪(i‬‬ ‫‪4‬‬ ‫‪y12‬‬ ‫‪:‬‬ ‫‪x,‬‬ ‫‪y‬‬ ‫‪2‬‬ ‫‪0‬‬ ‫)‪ (2‬ﺍﻛﺘﺐ ﻛﻞ ﻋﺪﺩ ﻣﻤﺎ ﻳﻠﻲ ﺑﺎﻟﺼﻮﺭﺓ ﺍﻟﺠﺬﺭﻳﺔ‪:‬‬ ‫)‪(a‬‬ ‫‪x‬‬ ‫‪1‬‬ ‫‪,‬‬ ‫‪x‬‬ ‫‪$‬‬ ‫‪0‬‬ ‫)‪(b‬‬ ‫‪x‬‬ ‫‪2‬‬ ‫)‪(c‬‬ ‫‪y-‬‬ ‫‪9‬‬ ‫‪,‬‬ ‫‪y‬‬ ‫‪2‬‬ ‫‪0‬‬ ‫‪6‬‬ ‫‪7‬‬ ‫‪8‬‬ ‫‪(d) x1.5 , x $ 0‬‬ ‫)‪(e‬‬ ‫‪x‬‬ ‫‪3‬‬ ‫‪,‬‬ ‫‪x‬‬ ‫‪$‬‬ ‫‪0‬‬ ‫)‪(f‬‬ ‫‪2‬‬ ‫‪4‬‬ ‫‪73‬‬ ‫‪(g) y3.2‬‬ ‫)‪(h‬‬ ‫‪x-‬‬ ‫‪2‬‬ ‫|‬ ‫‪x‬‬ ‫!‬ ‫‪0‬‬ ‫‪3‬‬ ‫)‪ (3‬ﺑ ّﺴﻂ ﻛﻞ ﻋﺪﺩ ﻣﻦ ﺍﻷﻋﺪﺍﺩ ﺍﻟﺘﺎﻟﻴﺔ )ﺩﻭﻥ ﺍﺳﺘﺨﺪﺍﻡ ﺍﻵﻟﺔ ﺍﻟﺤﺎﺳﺒﺔ(‪:‬‬ ‫)‪(a‬‬ ‫‪2‬‬ ‫)‪(b‬‬ ‫‪^-‬‬ ‫‪32h-‬‬ ‫‪4‬‬ ‫‪(c) 41.5‬‬ ‫‪5‬‬ ‫‪64 3‬‬ ‫)‪ (4‬ﺍﻛﺘﺐ ﻛﻞ ﻋﺪﺩ ﺑﺎﻟﺼﻮﺭﺓ ﺍﻷﺳﻴﺔ‪:‬‬ ‫‪(a) 7x3 , x $ 0‬‬ ‫‪(b) ^7xh3 , x H 0‬‬ ‫‪(c) ^ 7xh3 , x H 0‬‬ ‫‪(d) 3 ^5xyh6‬‬ ‫‪(g) 5 ^1024h3‬‬ ‫‪(e) 4 81x3 , x H 0‬‬ ‫‪(f) 0.0049t52‬‬ ‫)‪ (5‬ﺑ ّﺴﻂ ﻛ ًّﻼ ﻣﻤﺎ ﻳﻠﻲ )ﺩﻭﻥ ﺍﺳﺘﺨﺪﺍﻡ ﺍﻵﻟﺔ ﺍﻟﺤﺎﺳﺒﺔ(‪:‬‬ ‫‪(a) 2 4 163‬‬ ‫‪(b) 3 ^-27h-4‬‬ ‫‪(c) 5 -243‬‬ ‫‪23‬‬ ‫‪31‬‬ ‫‪2‬‬ ‫‪:‬‬ ‫‪y‬‬ ‫‪-‬‬ ‫‪1‬‬ ‫‪:‬‬ ‫‪4‬‬ ‫)‪(d‬‬ ‫‪x 7 : x 14 , x H 0‬‬ ‫)‪(e‬‬ ‫‪x 5 ' x 10 , x 2 0‬‬ ‫)‪(f‬‬ ‫‪x3‬‬ ‫‪1‬‬ ‫‪,‬‬ ‫‪x‬‬ ‫‪2‬‬ ‫‪0, y‬‬ ‫‪2‬‬ ‫‪0‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪y‬‬ ‫‪-‬‬ ‫‪x2‬‬ ‫)‪(g‬‬ ‫‪x‬‬ ‫‪1‬‬ ‫‪:‬‬ ‫‪y-‬‬ ‫‪1‬‬ ‫)‪(h‬‬ ‫‪3‬‬ ‫‪-‬‬ ‫‪1‬‬ ‫‪i2‬‬ ‫‪1‬‬ ‫)‪(i‬‬ ‫‪9t‬‬ ‫‪-12‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪2‬‬ ‫‪27t2‬‬ ‫‪`_3 2‬‬ ‫‪j3‬‬ ‫‪o,‬‬ ‫‪3‬‬ ‫‪1‬‬ ‫‪,‬‬ ‫‪x‬‬ ‫‪2‬‬ ‫‪0, y‬‬ ‫‪2‬‬ ‫‪0‬‬ ‫‪x‬‬ ‫‪, x20‬‬ ‫‪e3‬‬ ‫‪t‬‬ ‫‪2‬‬ ‫‪0‬‬ ‫‪4‬‬ ‫‪2‬‬ ‫‪x‬‬ ‫‪-‬‬ ‫‪:‬‬ ‫‪y‬‬ ‫‪-‬‬ ‫‪12‬‬

‫)‪ (6‬ﺃﻭﺟﺪ ﻧﺎﺗﺞ ﻛ ّﻞ ﻣﻤﺎ ﻳﻠﻲ ﻓﻲ ﺃﺑﺴﻂ ﺻﻮﺭﺓ‪:‬‬ ‫‪(a) 3 64x6‬‬ ‫)‪(b‬‬ ‫‪2‬‬ ‫‪#‬‬ ‫‪25‬‬ ‫‪-‬‬ ‫‪1‬‬ ‫)‪(c‬‬ ‫‪3 82 # 4 32‬‬ ‫‪(d) 10 1024 - 2 6 26‬‬ ‫‪3‬‬ ‫‪88 4‬‬ ‫‪53‬‬ ‫‪11‬‬ ‫‪^32h2 # ^16h- 3‬‬ ‫)‪(e‬‬ ‫‪6 64‬‬ ‫‪(f) ^2 - 3 8h^2 + 3 8h‬‬ ‫)‪ (7‬ﺃﻭﺟﺪ ﻋﺪ ًﺩﺍ ‪ x‬ﺑﺤﻴﺚ ﻳﻜﻮﻥ ‪ ^4 + 5h # x‬ﻋﺪ ًﺩﺍ ﻧﺴﺒﻴًّﺎ‪.‬‬ ‫ﻏﺎﺯ‪.‬‬ ‫ﻣﻦ‬ ‫ﻋﻴﻨﺔ‬ ‫‪ V‬ﻳﻤﺜﻞ ﺣﺠﻢ‬ ‫ﺍﻟﻀﻐﻂ‪،‬‬ ‫ﻳﻤﺜﻞ‬ ‫‪P‬‬ ‫ﺣﻴﺚ‬ ‫‪، PV‬‬ ‫‪7‬‬ ‫ﺍﻟﺘﻌﺒﻴﺮ‬ ‫ﻓﻲ‬ ‫)‪(8‬‬ ‫‪5‬‬ ‫ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺘﻌﺒﻴﺮ ﺇﺫﺍ ﻛﺎﻥ‪P = 6, V = 32 :‬‬ ‫‪5 #_4‬‬ ‫‪-‬‬ ‫‪5‬‬ ‫‪1‬‬ ‫‪i‬‬ ‫=‬ ‫‪5#‬‬ ‫‪1‬‬ ‫=‬ ‫‪20 -‬‬ ‫‪25‬‬ ‫‪1‬‬ ‫=‬ ‫‪15‬‬ ‫ﺗﺤﻠﻴﻞ ﺍﻟﺨﻄﺄ‪ :‬ﺃﻭﺟﺪ ﺍﻟﺨﻄﺄ ﻓﻲ ﺍﻟﺤﻞ ﺍﻟﺘﺎﻟﻲ‪:‬‬ ‫)‪(9‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪4 - 5 # 52‬‬ ‫‪m‬‬ ‫=‬ ‫‪46‬‬ ‫‪#‬‬ ‫‪104‬‬ ‫ﻛﺎﻥ‬ ‫ﺇﺫﺍ‬ ‫ﺍﻟﺘﻌﺒﻴﺮ‪،‬‬ ‫ﻗﻴﻤﺔ‬ ‫ﺃﻭﺟﺪ‬ ‫ﺍﻟﺴﻮﺍﺋﻞ‪.‬‬ ‫ﻟﺪﺭﺍﺳﺔ‬ ‫‪3‬‬ ‫ﻋﻠﻢ ﺍﻷﺣﻴﺎﺀ‪ :‬ﻳﺴﺘﺨﺪﻡ ﺍﻟﺘﻌﺒﻴﺮ‪:‬‬ ‫)‪(10‬‬ ‫‪0.036 m 4‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫‪33‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-5‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ab‬‬ ‫‪(1) 16- 4 = 32- 5‬‬ ‫‪ab‬‬ ‫‪ab‬‬ ‫‪1 32‬‬ ‫‪ab‬‬ ‫‪ab‬‬ ‫‪(2) x 2 ' x 4 = x 3‬‬ ‫)‪(3‬‬ ‫‪x‬‬ ‫‪-‬‬ ‫‪1‬‬ ‫‪1‬‬ ‫=‬ ‫‪x-‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪6‬‬ ‫‪:‬‬ ‫‪x3‬‬ ‫‪(4) 4 x = x , x 2 0‬‬ ‫‪(5) 32 # 16-1 = 4‬‬ ‫ﻓﻲ ﺍﻟﺒﻨﻮﺩ )‪ ،(6-12‬ﻇﻠّﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫)‪ (6‬ﺇﺫﺍ ﻛﺎﻥ ‪ ، n 2 0‬ﻓﺈﻥ ﺍﻟﺘﻌﺒﻴﺮ ﺍﻟﺬﻱ ﻻ ﻳﻜﺎﻓﺊ ‪ 4 4n2‬ﻫﻮ‪:‬‬ ‫‪a ^4n2h41‬‬ ‫‪b‬‬ ‫‪2n‬‬ ‫‪1‬‬ ‫‪c ^2nh21‬‬ ‫‪d 2n‬‬ ‫‪2‬‬ ‫‪a 14y‬‬ ‫= ‪(8) _4 x-2 y4 i-2‬‬ ‫‪15‬‬ ‫ﻳﺴﺎﻭﻱ‪:‬‬ ‫‪56 3 # y 3‬‬ ‫)‪ (7‬ﺇﺫﺍ ﻛﺎﻥ‪ ، y 2 0 :‬ﻓﺈﻥ ﺍﻟﺘﻌﺒﻴﺮ‬ ‫‪1‬‬ ‫‪^7y2h3‬‬ ‫‪b‬‬ ‫‪1‬‬ ‫‪y‬‬ ‫‪c 2y‬‬ ‫‪d‬‬ ‫‪8‬‬ ‫‪y‬‬ ‫‪7‬‬ ‫‪7‬‬ ‫‪:x!0 , y!0‬‬ ‫‪a x-1 y2‬‬ ‫‪b x y-2‬‬ ‫‪c xy2‬‬ ‫‪d x-2 y2‬‬ ‫)‪(9‬‬ ‫‪11‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫= ‪3 5 # 3 52‬‬ ‫‪c 52‬‬ ‫‪d 53‬‬ ‫‪a5‬‬ ‫‪-‬‬ ‫‪1‬‬ ‫‪b‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪5‬‬ ‫‪13‬‬

‫)‪ (10‬ﺇﺫﺍ ﻛﺎﻥ ‪ x2 - xy + y2 = 4 , x + y = 2‬ﻓﺈﻥ ‪ 6 x3 + y3‬ﻳﺴﺎﻭﻱ‪:‬‬ ‫‪a2‬‬ ‫‪b 32‬‬ ‫‪c 36‬‬ ‫‪d2‬‬ ‫=‪V‬‬ ‫‪243‬‬ ‫=‪, P‬‬ ‫‪32‬‬ ‫‪7‬‬ ‫)‪(11‬‬ ‫‪32‬‬ ‫‪27‬‬ ‫ﻓﻲ ﺍﻟﺘﻌﺒﻴﺮ ‪ P.V 5‬ﺣﻴﺚ ‪ P‬ﻳﻤﺜﻞ ﺍﻟﻀﻐﻂ‪ V ،‬ﻳﻤﺜﻞ ﺣﺠﻢ ﻋﻴﻨﺔ ﻣﻦ ﻏﺎﺯ ﻓﺈﻥ ﻗﻴﻤﺘﻪ ﻋﻨﺪﻣﺎ‬ ‫ﻳﺴﺎﻭﻱ‪:‬‬ ‫‪a‬‬ ‫‪4‬‬ ‫‪b4‬‬ ‫‪c‬‬ ‫‪81‬‬ ‫‪d‬‬ ‫‪243‬‬ ‫‪81‬‬ ‫‪4‬‬ ‫‪4‬‬ ‫ﺗﺴﺎﻭﻱ‪:‬‬ ‫‪3 x6 : 4 x5‬‬ ‫)‪ (12‬ﺇﻥ ﻗﻴﻤﺔ ﺍﻟﺘﻌﺒﻴﺮ ‪, x 2 0‬‬ ‫‪x3 : 8 x2‬‬ ‫‪ax‬‬ ‫‪b‬‬ ‫‪1‬‬ ‫‪c1 d x‬‬ ‫‪x‬‬ ‫‪14‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫‪1-3‬‬ ‫ﺣﻞ ﺍﻟﻤﻌﺎﺩﻻﺕ‬ ‫‪Solving Equations‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫)‪ (1‬ﺣﻞ ﻛ ًّﻼ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫‪(a) 3 x + 3 = 15‬‬ ‫‪(b) x + 3 = 5‬‬ ‫)‪(c‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪^x + 5h3 = 4‬‬ ‫‪(d) ^x + 1h2 - 2 = 25‬‬ ‫‪(e) 3 - 4x - 2 = 0‬‬ ‫‪3‬‬ ‫‪3‬‬ ‫‪(f) 2^2x + 4h4 = 16‬‬ ‫‪(g) ^5 - 3xh2 + 4 = 3‬‬ ‫)‪ (a) (2‬ﺍﻟﺤﺠﻢ‪ :‬ﻳﺘﺴﻊ ﺧ ّﺰﺍﻥ ﻛﺮﻭﻱ ﺍﻟﺸﻜﻞ ِﻟـ ‪ 424 . 75 m3‬ﺃﻭﺟﺪ ﻃﻮﻝ ﻗﻄﺮ ﻫﺬﺍ ﺍﻟﺨ ّﺰﺍﻥ‪.‬‬ ‫‪ d‬ﻃﻮﻝ ﻗﻄﺮ ﺍﻟﻜﺮﺓ(‪.‬‬ ‫ﺣﻴﺚ‬ ‫‪π # d3‬‬ ‫=‬ ‫)ﻣﺴﺎﻋﺪﺓ‪ :‬ﺣﺠﻢ ﺍﻟﻜﺮﺓ‬ ‫‪6‬‬ ‫)‪ (b‬ﺗﺮﺍﺑﻂ ﺣﻴﺎﺗﻲ‪ :‬ﺗﻘﺎﺱ ﺍﻟﻜﻤﻴﺔ ﺍﻟﻘﺼﻮﻯ ‪ K‬ﻟﺘﺪﻓّﻖ ﺍﻟﻤﻴﺎﻩ ﻓﻲ ﺃﻧﺒﻮﺏ‪ ،‬ﺑﺎﻟﻘﺎﻧﻮﻥ‪ ،K = m # V :‬ﺣﻴﺚ ‪ m‬ﻫﻲ‬ ‫ﻣﺴﺎﺣﺔ ﺍﻟﻤﻘﻄﻊ ﺍﻟﻌﺮﺿﻲ ﻟﻸﻧﺒﻮﺏ‪ V ،‬ﻫﻲ ﺍﻟﺴﺮﻋﺔ ﺍﻟﻤﺘﺠﻬﺔ ﻟﻠﻤﻴﺎﻩ‪ .‬ﺃﻭﺟﺪ ﻃﻮﻝ ﻗﻄﺮ ﺍﻷﻧﺒﻮﺏ ﺍﻟﺬﻱ ﻳﺴﻤﺢ‬ ‫ﺑﺘﺪﻓﻖ ‪ 1.48 m3/min‬ﺑﺴﺮﻋﺔ ‪183 m/min‬‬ ‫)‪ (3‬ﺣﻞ ﻛ ًّﻼ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫‪(a) 11x + 3 - 2x = 0‬‬ ‫‪(b) 3x + 13 - 5 = x‬‬ ‫‪(c) -3x - 5 = x + 3‬‬ ‫‪1‬‬ ‫‪1‬‬ ‫‪(f) 10x - 2 5x - 25 = 0‬‬ ‫‪(d) ^x + 3h2 - 1 = x‬‬ ‫‪(e) x + 8 = ^x2 + 16h2‬‬ ‫‪11‬‬ ‫)‪(h‬‬ ‫‪11‬‬ ‫‪3‬‬ ‫‪(g) ^3x + 2h2 - ^2x + 7h2 = 0‬‬ ‫‪^x - 9h2 + 1 = x 2‬‬ ‫‪(i) ^2x + 3h4 - 3 = 5‬‬ ‫‪4‬‬ ‫)‪(k‬‬ ‫‪^3x‬‬ ‫‪+‬‬ ‫‪1‬‬ ‫=‬ ‫‪8^3x‬‬ ‫‪+‬‬ ‫‪2h-‬‬ ‫‪1‬‬ ‫‪11‬‬ ‫‪2‬‬ ‫‪(j) 2^x - 1h3 + 4 = 36‬‬ ‫‪2h2‬‬ ‫‪(l) ^2x + 1h3 = ^3x + 2h3‬‬ ‫)‪(m‬‬ ‫‪11‬‬ ‫)ﻣﺴﺎﻋﺪﺓ‪ :‬ﺭﻓﻊ ﻃﺮﻓﻲ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺇﻟﻰ ﺍﻟﻘﻮﺓ ‪(6‬‬ ‫‪(n) ^x + 5h21 - ^5 - 2xh41 = 0‬‬ ‫‪^2x - 1h3 = ^x + 1h6‬‬ ‫= ‪ ،S‬ﺣﻴﺚ ‪ x‬ﻫﻲ ﻃﻮﻝ ﺍﻟﻀﻠﻊ‪.‬‬ ‫‪3‬‬ ‫‪3 x2‬‬ ‫ﺍﻟﻬﻨﺪﺳﺔ‪ :‬ﻗﺎﻧﻮﻥ ﻣﺴﺎﺣﺔ ﻣﻀﻠّﻊ ﺳﺪﺍﺳﻲ ﻣﻨﺘﻈﻢ ﻫﻮ‪:‬‬ ‫)‪(4‬‬ ‫‪2‬‬ ‫)‪ (a‬ﺃﻭﺟﺪ ﻃﻮﻝ ﺍﻟﻀﻠﻊ ‪ x‬ﺑﺪﻻﻟﺔ ﺍﻟﻤﺴﺎﺣﺔ ‪S‬‬ ‫)‪ (b‬ﺃﺭﺍﺩ ﺃﺣﺪ ﺍﻷﺷﺨﺎﺹ ﺻﻨﻊ ﺻﻨﺪﻭﻕ ﻗﺎﻋﺪﺗﻪ ﻣﻀﻠﻊ ﺳﺪﺍﺳﻲ ﻣﻨﺘﻈﻢ ﻭﻣﺴﺎﺣﺘﻪ ‪x‬‬ ‫ﺗﺴﺎﻭﻱ ‪ 200 cm2‬ﺃﻭﺟﺪ ﻃﻮﻝ ﺍﻟﻤﻀﻠﻊ‪ .‬ﺛﻢ ﺃﻭﺟﺪ ﺍﻟﺒﻌﺪ ﺑﻴﻦ ﺿﻠﻌﻴﻦ ﻣﺘﻮﺍﺯﻳﻴﻦ‪.‬‬ ‫‪x‬‬ ‫)‪ (5‬ﺻﻨﺪﻭﻕ ﻣﻜﻌﺐ ﺍﻟﺸﻜﻞ ﺳﻌﺘﻪ ‪ 150 m3‬ﺃﻭﺟﺪ ﻃﻮﻝ ﺿﻠﻌﻪ‪.‬‬ ‫‪3x‬‬ ‫)‪ x, y (6‬ﻫﻤﺎ ﻋﺪﺩﺍﻥ ﺣﻘﻴﻘﻴﺎﻥ‪.‬‬ ‫‪2‬‬ ‫)‪ (a‬ﺃﻭﺟﺪ ﺍﻟﻨﺎﺗﺞ‪^x - yh^x2 + xy + y2h :‬‬ ‫‪ ، 3‬ﺑﺤﻴﺚ ﻳﻜﻮﻥ ﺍﻟﻤﻘﺎﻡ ﻋﺪ ًﺩﺍ ﻧﺴﺒﻴًّﺎ‪.‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﺼﻴﻐﺔ ﺍﻟﺴﺎﺑﻘﺔ‪ ،‬ﺍﻛﺘﺐ ﺍﻟﻜﺴﺮ‬ ‫)‪(b‬‬ ‫‪3-3‬‬ ‫‪15‬‬

‫)‪ (7‬ﺣﻞ ﻛ ًّﻼ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻷﺳﻴﺔ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫‪(a) 52x-3 = 125‬‬ ‫‪(b) 3x+1 = 1‬‬ ‫‪(c) 3x2+5 = 39‬‬ ‫‪(e) 4x = 2x‬‬ ‫)‪(d‬‬ ‫= ‪3x2-5x‬‬ ‫‪1‬‬ ‫‪(h) 5x2-3x = 1‬‬ ‫)‪(f‬‬ ‫‪a‬‬ ‫‪1‬‬ ‫‪n‬‬ ‫=‬ ‫‪0.25‬‬ ‫‪92‬‬ ‫‪2‬‬ ‫‪k‬‬ ‫‪(g) 5x = 125 5‬‬ ‫‪(i) ^3x - 27h^2x - 1h = 0‬‬ ‫)‪(j‬‬ ‫‪a‬‬ ‫‪2‬‬ ‫‪x-1‬‬ ‫=‬ ‫‪a‬‬ ‫‪125‬‬ ‫‪x‬‬ ‫‪5‬‬ ‫‪8‬‬ ‫‪k‬‬ ‫‪k‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-5‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪a‬‬ ‫)‪ (1‬ﻣﺠﻤﻮﻋﺔ ﺣﻞ ‪ 73-x = 1‬ﻫﻲ ‪b \"3,‬‬ ‫‪a‬‬ ‫‪a‬‬ ‫)‪ (2‬ﻣﺠﻤﻮﻋﺔ ﺣﻞ ‪ x - 1 = 1 - x‬ﻫﻲ ‪b \"0,‬‬ ‫‪a‬‬ ‫‪a‬‬ ‫)‪ (3‬ﺇﺫﺍ ﻛﺎﻥ ‪ 3 9 + x2 = 3‬ﻓﺈﻥ ‪b x = 3 2‬‬ ‫‪b‬‬ ‫‪2x2-4‬‬ ‫=‬ ‫‪1‬‬ ‫‪ x = - 1‬ﺣ ًّﻼ ﻟﻠﻤﻌﺎﺩﻟﺔ‬ ‫)‪(4‬‬ ‫‪32‬‬ ‫‪b‬‬ ‫‪R-‬‬ ‫ﻫﻲ‬ ‫‪25‬‬ ‫‪|x‬‬ ‫‪|+‬‬ ‫‪1‬‬ ‫‪5 1 - 2x‬‬ ‫ﻣﺠﻤﻮﻋﺔ ﺣﻞ‬ ‫)‪(5‬‬ ‫‪2‬‬ ‫=‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(6-10‬ﻇﻠّﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪:‬‬ ‫‪1‬‬ ‫)‪ (6‬ﻣﺠﻤﻮﻋﺔ ﺣﻞ ‪ _ x20i5 - x2 = 0‬ﻫﻲ‪:‬‬ ‫‪a \"0,‬‬ ‫‪b R+‬‬ ‫‪c R-‬‬ ‫‪dR‬‬ ‫)‪ (7‬ﻣﺠﻤﻮﻋﺔ ﺣﻞ ‪ 3 x - 2 = x - 2‬ﻫﻲ‪:‬‬ ‫‪a \"2,‬‬ ‫‪b \"1,2,‬‬ ‫‪c \"1,2,3,‬‬ ‫‪d \"2,3,‬‬ ‫)‪ (8‬ﻣﺠﻤﻮﻋﺔ ﺣﻞ ‪ 3 2x2 + 2 = 3 3 - x‬ﻫﻲ‪:‬‬ ‫‪a‬‬ ‫‪&-‬‬ ‫‪1,‬‬ ‫‪1‬‬ ‫‪0‬‬ ‫‪b‬‬ ‫&‬ ‫‪1‬‬ ‫‪0‬‬ ‫‪c‬‬ ‫‪&-‬‬ ‫‪1,‬‬ ‫‪-1‬‬ ‫‪0‬‬ ‫‪d‬‬ ‫‪&1,‬‬ ‫‪1‬‬ ‫‪0‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫)‪ (9‬ﻣﺠﻤﻮﻋﺔ ﺣﻞ ‪ x2 = x‬ﻫﻲ‪:‬‬ ‫‪a \"-1, 0, 1,‬‬ ‫‪b \"0,1,‬‬ ‫‪c \"0,‬‬ ‫‪d \"1,‬‬ ‫ﺗﺴﺎﻭﻱ‪:‬‬ ‫‪x‬‬ ‫ﻓﺈﻥ‬ ‫‪a‬‬ ‫‪1‬‬ ‫‪x+1‬‬ ‫=‬ ‫‪32-x‬‬ ‫ﻛﺎﻥ‬ ‫ﺇﺫﺍ‬ ‫)‪(10‬‬ ‫‪9‬‬ ‫‪k‬‬ ‫‪a -2‬‬ ‫‪b2‬‬ ‫‪c -4‬‬ ‫‪d4‬‬ ‫‪16‬‬

‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ ﺍﻷﻭﻟﻰ‬ ‫‪(a) 121x90‬‬ ‫‪(b) 3 -64y81‬‬ ‫)‪ (1‬ﺑ ّﺴﻂ ﻛ ًّﻼ ﻣﻦ ﺍﻟﺘﻌﺒﻴﺮﺍﺕ ﺍﻟﺠﺬﺭﻳﺔ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫‪(c) 5 32y25‬‬ ‫‪(d) 0 . 0081x60‬‬ ‫‪(e) 16x36 y96‬‬ ‫‪(f) 8^ 24 + 3 8h‬‬ ‫‪(h) 32‬‬ ‫)ﺣﻴﺚ ‪ y‬ﻋﺪﺩ ﺣﻘﻴﻘﻲ ‪(g) 2 5x3 # 3 28x3y2 , ^x $ 0,‬‬ ‫‪2‬‬ ‫‪(i) 3 2x2 # 3 4x‬‬ ‫)‪ (2‬ﺍﻛﺘﺐ ﻛﻞ ﻛﺴﺮ ﻣﻤﺎ ﻳﻠﻲ ﺑﺤﻴﺚ ﻳﻜﻮﻥ ﻣﻘﺎﻣﻪ ﻋﺪ ًﺩﺍ ﻧﺴﺒﻴًّﺎ‪:‬‬ ‫)‪(a‬‬ ‫^‬ ‫‪5+2‬‬ ‫‪1‬‬ ‫‪5-2‬‬ ‫‪3h‬‬ ‫‪(b) 5‬‬ ‫^‪3 h‬‬ ‫‪4 7+5‬‬ ‫‪(c) 2 + 10‬‬ ‫‪(d) -2 + 8‬‬ ‫‪2-3 5‬‬ ‫‪-3 - 2‬‬ ‫)‪ (3‬ﺑ ّﺴﻂ ﻛ ًّﻼ ﻣﻦ ﺍﻟﺘﻌﺎﺑﻴﺮ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫‪2‬‬ ‫‪(b) 251.5‬‬ ‫)‪(c‬‬ ‫‪11‬‬ ‫‪(a) 64 3‬‬ ‫‪6 2 # 12 2‬‬ ‫‪(d) 81- 0.25‬‬ ‫)‪(e) 8 # 2 - 2 75 + 5 12 (f‬‬ ‫‪22‬‬ ‫‪-‬‬ ‫‪3- 2 3+ 2‬‬ ‫)‪ (4‬ﻟﻴﻜﻦ ‪ x‬ﺍﻟﻌﺪﺩ ﺍﻟﺤﻘﻴﻘﻲ‪x = 6 - 2 5 - 6 + 2 5 ،‬‬ ‫)‪ (a‬ﺍﺣﺴﺐ ‪x2‬‬ ‫)‪ (b‬ﺃﺛﺒﺖ ﺃﻥ ﻗﻴﻤﺔ ‪ x‬ﺗﺴﺎﻭﻱ ‪-2‬‬ ‫)‪(a‬‬ ‫‪5‬‬ ‫)‪(b‬‬ ‫‪y‬‬ ‫‪-‬‬ ‫‪2‬‬ ‫‪,‬‬ ‫‪y‬‬ ‫!‬ ‫‪0‬‬ ‫)‪ (5‬ﺍﻛﺘﺐ ﻛﻞ ﺗﻌﺒﻴﺮ ﻣﻤﺎ ﻳﻠﻲ ﺑﺎﻟﺼﻮﺭﺓ ﺍﻟﺠﺬﺭﻳﺔ‪:‬‬ ‫‪9‬‬ ‫‪(c) ^5 xh2‬‬ ‫‪x7‬‬ ‫‪(d) 3 4 64‬‬ ‫‪(e) 2 3 # 54 3‬‬ ‫‪(f) 3 x # 2 3 x , x $ 0‬‬ ‫‪(g) 2 3 3 ' 4 3‬‬ ‫‪(h) 5 10 # 2 4 10 # 3 10‬‬ ‫‪(i) 2 ' 36 8‬‬ ‫)‪ (6‬ﺑ ّﺴﻂ ﻛ ًّﻼ ﻣﻦ ﺍﻟﺘﻌﺒﻴﺮﺍﺕ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫)‪(a‬‬ ‫‪^8‬‬ ‫‪-3‬‬ ‫‪y‬‬ ‫‪-6h-‬‬ ‫‪2‬‬ ‫)‪(b‬‬ ‫‪d‬‬ ‫‪16x14‬‬ ‫‪1‬‬ ‫‪y‬‬ ‫!‬ ‫‪0‬‬ ‫‪3‬‬ ‫‪81y18‬‬ ‫‪n2 ,‬‬ ‫)‪(c‬‬ ‫__‬ ‫‪-‬‬ ‫‪1‬‬ ‫‪i2‬‬ ‫‪1‬‬ ‫)‪(d‬‬ ‫‪x‬‬ ‫‪1‬‬ ‫‪:‬‬ ‫‪y-‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪2‬‬ ‫‪i3‬‬ ‫‪x‬‬ ‫‪,‬‬ ‫‪x‬‬ ‫‪2‬‬ ‫‪0‬‬ ‫‪3 3 ,x 2 0, y 2 0‬‬ ‫‪x6 :y4‬‬ ‫‪17‬‬

‫ﻛﺘﺐ ﺃﺣﺪ ﺍﻟﻄﻼﺏ ﻣﺎ ﻳﻠﻲ‪:‬‬ ‫‪1‬‬ ‫‪2 h2‬‬ ‫ﺗﺤﻠﻴﻞ ﺍﻟﺨﻄﺄ‪ :‬ﻓﻲ ﺳﺒﻴﻞ ﺗﺒﺴﻴﻂ ﺍﻟﻜﺴﺮ‬ ‫)‪(7‬‬ ‫‪^1 -‬‬ ‫‪1‬‬ ‫‪2 h2‬‬ ‫‪= ^1 -‬‬ ‫‪2 h-2‬‬ ‫‪^1 -‬‬ ‫‪= 1-2 - ^ 2 h-2‬‬ ‫^ ‪= 1-‬‬ ‫‪1‬‬ ‫‪2 h2‬‬ ‫=‬ ‫‪1‬‬ ‫‪-‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪1‬‬ ‫‪=2‬‬ ‫ﻣﺎ ﺍﻟﺨﻄﺄ ﺍﻟﺬﻱ ﺍﺭﺗﻜﺒﻪ ﺍﻟﻄﺎﻟﺐ؟‬ ‫)‪ (8‬ﺣﻞ ﻛ ًّﻼ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫‪(a) 5 x + 7 = 8‬‬ ‫‪(b) x + 2 = x‬‬ ‫‪(c) 4x - 23 - 3 = 2‬‬ ‫)ﻣﺴﺎﻋﺪﺓ‪ :‬ﺗﺮﺑﻴﻊ ﻃﺮﻓﻲ ﺍﻟﻤﻌﺎﺩﻟﺔ ﻣﺮﺗﻴﻦ ﻣﺘﺘﺎﻟﻴﺘﻴﻦ( ‪(d) 2x + 1 - x + 11 = 0 (e) x - x - 5 = 2‬‬ ‫‪(f) 3x - 9 = 2x + 4‬‬ ‫)‪ (9‬ﺍﻟﻔﻴﺰﻳﺎﺀ‪ :‬ﺍﻟﺴﺮﻋﺔ ‪ V‬ﻟﺠﺴﻢ ﻣﺎ ﺃﺳﻘﻂ ﻋﻦ ﺳﻄﺢ ﻣﺒﻨﻰ ﻋﺎﻝ ﻣﻌﻄﺎﺓ ﺑﺎﻟﻘﺎﻧﻮﻥ‪ ،V = 8 m :‬ﺣﻴﺚ ‪ m‬ﻫﻲ ﺍﺭﺗﻔﺎﻉ‬ ‫ﺍﻟﻤﺒﻨﻰ‪ .‬ﺃﻭﺟﺪ ﺍﻻﺭﺗﻔﺎﻉ ‪ m‬ﺑﺪﻻﻟﺔ ﺍﻟﺴﺮﻋﺔ ‪V‬‬ ‫)‪ (10‬ﺇﺫﺍ ﻛﺎﻥ ‪ ، x = 2‬ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ ‪x2 ^3 - xh‬‬ ‫‪3-1‬‬ ‫)‪ (11‬ﺣﻞ ﻛ ًّﻼ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫‪(a) 2x2 = 512‬‬ ‫‪(b) 4x2-x = 16‬‬ ‫‪18‬‬

‫ﺗﻤﺎﺭﻳﻦ ﺇﺛﺮﺍﺋﻴﺔ‬ ‫)‪ (1‬ﺑ ّﺴﻂ ﻛ ًّﻼ ﻣﻤﺎ ﻳﻠﻲ ﺩﻭﻥ ﺍﺳﺘﺨﺪﺍﻡ ﺍﻵﻟﺔ ﺍﻟﺤﺎﺳﺒﺔ‪:‬‬ ‫‪(a) 3 -343‬‬ ‫‪(b) 4 810 000‬‬ ‫‪(c) _4 3 i8‬‬ ‫‪(d) -4 6 561‬‬ ‫‪(e) 5 -0.00001‬‬ ‫‪(f) 9^ 3 - 2h2 - 4^1 - 3h2‬‬ ‫)‪ (2‬ﺃﻭﺟﺪ ﻧﺎﺗﺞ ﻣﺎ ﻳﻠﻲ‪:‬‬ ‫)‪(g‬‬ ‫‪27-2 # 45-3‬‬ ‫)‪(h‬‬ ‫‪123 # 18-2‬‬ ‫‪36-5 # 454‬‬ ‫‪6-2 # 3-5‬‬ ‫‪(a) 4 ^3 4 - 4h4 - 3 -8^3 2 + 1h6‬‬ ‫‪(b) _5 32 + 3i^3 - 6 8h‬‬ ‫)‪(c‬‬ ‫‪3 132 # 13‬‬ ‫‪3‬‬ ‫‪1‬‬ ‫‪13 2‬‬ ‫‪8x9 y3‬‬ ‫‪2‬‬ ‫)‪ (3‬ﺑ ّﺴﻂ ﻛ ًّﻼ ﻣﻦ ﺍﻟﺘﻌﺎﺑﻴﺮ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫‪27x2 y12‬‬ ‫)‪(a‬‬ ‫‪e‬‬ ‫‪3‬‬ ‫‪, x ! 0, y ! 0‬‬ ‫)‪(b‬‬ ‫_‬ ‫‪x‬‬ ‫‪-3‬‬ ‫‪:‬‬ ‫‪y‬‬ ‫‪1‬‬ ‫‪i16‬‬ ‫‪,‬‬ ‫‪x‬‬ ‫‪2‬‬ ‫‪0,‬‬ ‫‪y‬‬ ‫‪$‬‬ ‫‪0‬‬ ‫‪8‬‬ ‫‪4‬‬ ‫‪o‬‬ ‫)‪(c‬‬ ‫‪y h_ x‬‬ ‫‪1‬‬ ‫‪1‬‬ ‫‪1‬‬ ‫‪1‬‬ ‫‪i‬‬ ‫)‪(d‬‬ ‫‪3‬‬ ‫‪x2 #‬‬ ‫‪x, x20‬‬ ‫‪3‬‬ ‫‪3‬‬ ‫‪6‬‬ ‫‪6‬‬ ‫‪^3 x + 3‬‬ ‫‪y -6‬‬ ‫‪x‬‬ ‫‪:‬‬ ‫‪+‬‬ ‫‪y‬‬ ‫‪+‬‬ ‫‪x‬‬ ‫‪:‬‬ ‫‪y‬‬ ‫‪1‬‬ ‫‪3‬‬ ‫‪x2‬‬ ‫‪B‬‬ ‫)‪ ABC (4‬ﻣﺜﻠﺚ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ‪A‬‬ ‫‪M‬‬ ‫‪AN = 2 + 3 AM = 2 3 - 1‬‬ ‫‪MN // BC‬‬ ‫‪MB = 1‬‬ ‫‪A‬‬ ‫‪(a) CN‬‬ ‫ﺃﻭﺟﺪ‪(b) MN :‬‬ ‫‪NC‬‬ ‫)‪ (5‬ﺍﻛﺘﺐ ﻛﻞ ﻛﺴﺮ ﻣﻤﺎ ﻳﻠﻲ ﺑﺤﻴﺚ ﻳﻜﻮﻥ ﻣﻘﺎﻣﻪ ﻋﺪ ًﺩﺍ ﻧﺴﺒﻴًّﺎ ﺩﻭﻥ ﺍﺳﺘﺨﺪﺍﻡ ﺍﻵﻟﺔ ﺍﻟﺤﺎﺳﺒﺔ‪:‬‬ ‫‪(a) 2 6‬‬ ‫‪1‬‬ ‫‪3+ 2- 5‬‬ ‫)‪(b‬‬ ‫‪1‬‬ ‫)‪(c‬‬ ‫‪x2 +1‬‬ ‫‪,‬‬ ‫‪x d Z+‬‬ ‫‪,‬‬ ‫‪x!1‬‬ ‫‪3 3-3 2‬‬ ‫‪1‬‬ ‫‪x2 -1‬‬ ‫)‪ (6‬ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ x‬ﻟﻴﻜﻮﻥ ﺍﻟﻌﺪﺩ ‪ x # -x‬ﻋﺪ ًﺩﺍ ﺣﻘﻴﻘﻴًّﺎ‪.‬‬ ‫)‪ (7‬ﺗﺤﻠﻴﻞ ﺍﻟﺨﻄﺄ‪ :‬ﺃﻭﺟﺪ ﺍﻟﺨﻄﺄ ‪16 = ^-2h # ^-8h = -2 # -8‬‬ ‫)‪ (8‬ﻣﺎ ﻗﻴﻤﺔ ‪ ،x‬ﺇﺫﺍ ‪320.8 # x = 1‬؟‬ ‫‪1‬‬ ‫)‪ (9‬ﺑ ّﺴﻂ ﻛ ًّﻼ ﻣﻤﺎ ﻳﻠﻲ‪:‬‬ ‫‪a-b‬‬ ‫)‪(a‬‬ ‫‪d‬‬ ‫‪xa2‬‬ ‫)‪(b‬‬ ‫‪2 # 3x+2 - 8 # 3x‬‬ ‫)‪(c‬‬ ‫_‬ ‫‪x‬‬ ‫‪1‬‬ ‫‪y‬‬ ‫‪-‬‬ ‫‪1‬‬ ‫‪i‬‬ ‫‪,‬‬ ‫‪x‬‬ ‫‪$‬‬ ‫‪0‬‬ ‫‪,‬‬ ‫‪y‬‬ ‫!‬ ‫‪0‬‬ ‫‪xb2‬‬ ‫‪n‬‬ ‫‪3x+1 + 2 # 3x‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪#‬‬ ‫)‪ (10‬ﺣﻞ ﻛ ًّﻼ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫‪(a) ^0 . 01hx = 0 . 000001‬‬ ‫‪(b) 2 1‬‬ ‫‪^x+3h‬‬ ‫‪23‬‬ ‫‪(c) ^32x - 9h^2x - 16h = 0‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪(d) ^3xh2 - 10 # 3x + 9 = 0‬‬ ‫=‬ ‫‪19‬‬ ‫)ﻣﺴﺎﻋﺪﺓ‪ :‬ﻟﻴﻜﻦ ‪(3x = y‬‬ ‫‪(e) 4x-1 - 9 # 2x-1 + 8 = 0‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫‪2-1‬‬ ‫ﻣﺠﺎﻝ ﺍﻟﺪﺍﻟﺔ‬ ‫‪Domain of the Function‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-6‬ﺍﺳﺘﺨﺪﻡ ﺍﺧﺘﺒﺎﺭ ﺍﻟﻤﺴﺘﻘﻴﻢ ﺍﻟﺮﺃﺳﻲ ﻟﺘﺤﺪﻳﺪ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﺑﻴﺎﻥ ﻛﻞ ﻋﻼﻗﺔ ﻣﻤﺎ ﻳﻠﻲ ﻳﻤﺜﻞ ﺑﻴﺎﻥ ﺩﺍﻟﺔ ﺃﻡ ﻻ‪.‬‬ ‫‪(1) y‬‬ ‫‪(2) y‬‬ ‫‪(3) y‬‬ ‫‪xx‬‬ ‫‪x‬‬ ‫‪(4) y‬‬ ‫‪(5) y‬‬ ‫‪(6) y‬‬ ‫‪xx x‬‬ ‫)‪(7‬‬ ‫)‪f (x‬‬ ‫‪=-‬‬ ‫‪1‬‬ ‫‪x2‬‬ ‫‪+‬‬ ‫‪x2‬‬ ‫‪-‬‬ ‫‪1‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(7-16‬ﺣ ّﺪﺩ ﻣﺠﺎﻝ ﻛ ّﻞ ﻣﻦ ﺍﻟﺪﻭﺍﻝ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫‪2‬‬ ‫‪(8) g(x) = 3x - 7 + 2‬‬ ‫= )‪(9) t(x‬‬ ‫‪-2x + 3‬‬ ‫)‪(10‬‬ ‫)‪h (x‬‬ ‫‪=-‬‬ ‫‪3x - 1‬‬ ‫‪x-1‬‬ ‫‪5 - 2x‬‬ ‫‪(11) u(x) = 3 7 - 5x‬‬ ‫)‪(12‬‬ ‫= )‪v (x‬‬ ‫‪2x - 1‬‬ ‫‪3+x‬‬ ‫)‪(13‬‬ ‫)‪h (x‬‬ ‫=‬ ‫‪5+‬‬ ‫‪x-2‬‬ ‫= )‪(14) u(x‬‬ ‫‪3 + 4x - 3‬‬ ‫‪2x - 1‬‬ ‫‪25 - 9x2‬‬ ‫)‪(15‬‬ ‫= )‪v (x‬‬ ‫‪3‬‬ ‫‪-‬‬ ‫‪2‬‬ ‫‪(16) w (x) = 3 x2 - 2^ 2x - 3h‬‬ ‫‪x+1‬‬ ‫‪x2 - 1‬‬ ‫‪20‬‬

‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-5‬ﻇﻠﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪a‬‬ ‫)‪ (1‬ﻣﺠﺎﻝ ﺍﻟﺪﺍﻟﺔ ‪ f(x) = ^x - 2h2‬ﻫﻮ ‪b R‬‬ ‫‪a‬‬ ‫‪a‬‬ ‫‪b‬‬ ‫= )‪ f(x‬ﻫﻮ ‪63, 3h‬‬ ‫)‪ (2‬ﻣﺠﺎﻝ ﺍﻟﺪﺍﻟﺔ ‪3‬‬ ‫‪a‬‬ ‫‪a‬‬ ‫‪2x - 6‬‬ ‫‪ay‬‬ ‫)‪ (3‬ﻣﺠﺎﻝ ﺍﻟﺪﺍﻟﺔ ‪ f(x) = -x‬ﻫﻮ @‪b ^-3,0‬‬ ‫‪b‬‬ ‫ﻫﻮ ‪6-3,3h‬‬ ‫= )‪f (x‬‬ ‫‪1‬‬ ‫)‪ (4‬ﻣﺠﺎﻝ ﺍﻟﺪﺍﻟﺔ ‪x + 3‬‬ ‫‪x2‬‬ ‫)‪ (5‬ﻣﺠﺎﻝ ﺍﻟﺪﺍﻟﺔ ‪ f(x) = x - 2‬ﻫﻮ ‪b R‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(6-11‬ﻇﻠﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻟﺔ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫)‪ (6‬ﺃﻳًّﺎ ﻣﻤﺎ ﻳﻠﻲ ﻻ ﻳﻤﺜﻞ ﺑﻴﺎﻥ ﺩﺍﻟﺔ‪:‬‬ ‫‪by‬‬ ‫‪cy‬‬ ‫‪dy‬‬ ‫‪xx x x‬‬ ‫ﻫﻮ‪:‬‬ ‫= )‪f (x‬‬ ‫‪x2 - 1‬‬ ‫ﻣﺠﺎﻝ ﺍﻟﺪﺍﻟﺔ‬ ‫)‪(7‬‬ ‫‪x2 + 2x + 1‬‬ ‫‪aR‬‬ ‫‪b R /\"1,‬‬ ‫‪c R/\"-1, 1,‬‬ ‫‪d R/\"-1,‬‬ ‫‪b 60, 3h‬‬ ‫‪a R /\"0,‬‬ ‫‪b R / \"0,1,‬‬ ‫= )‪ f(x‬ﻫﻮ‪:‬‬ ‫‪x2‬‬ ‫ﻣﺠﺎﻝ ﺍﻟﺪﺍﻟﺔ‬ ‫)‪(8‬‬ ‫‪x‬‬ ‫‪a R /\"1,‬‬ ‫‪c ^-3, 0h‬‬ ‫‪d ^0, 3h‬‬ ‫‪a ^0, 3h‬‬ ‫@‪a 6-2, 2‬‬ ‫ﻫﻮ‪:‬‬ ‫= )‪f (x‬‬ ‫‪x-1‬‬ ‫ﻣﺠﺎﻝ ﺍﻟﺪﺍﻟﺔ‬ ‫)‪(9‬‬ ‫‪c ^0, 2h‬‬ ‫‪x- x‬‬ ‫‪21‬‬ ‫‪c R - \"0,‬‬ ‫‪d ^0, 3h/\"1,‬‬ ‫= )‪ f(x‬ﻫﻮ‪:‬‬ ‫‪x‬‬ ‫ﻣﺠﺎﻝ ﺍﻟﺪﺍﻟﺔ‬ ‫)‪(10‬‬ ‫‪x+1-1‬‬ ‫‪b 61, 3h‬‬ ‫‪c ^-1, 3h‬‬ ‫‪d 6-1, 3h/\"0,‬‬ ‫)‪ (11‬ﻟﺘﻜﻦ ‪ . f(x) = x x , g :6-2, 2@ \" R , g(x) = x2‬ﻓﺈﻥ ﻣﺠﺎﻝ ﺍﻟﺪﺍﻟﺔ ‪ f : g‬ﻫﻮ‪:‬‬ ‫@‪b 60, 2‬‬ ‫ﻟﻴﺲ ﺃﻳًّﺎ ﻣﻤﺎ ﺳﺒﻖ ﺻﺤﻴ ًﺤﺎ ‪d‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫‪2-2‬‬ ‫ﺍﻟﺪﻭﺍﻝ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﻭﻧﻤﺬﺟﺘﻬﺎ‬ ‫‪Quadratic Functions and their Modelling‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-8‬ﺃﻱ ﻣﻦ ﺍﻟﺪﻭﺍﻝ ﺍﻟﺘﺎﻟﻴﺔ ﺧﻄﻴﺔ؟ ﻭﺃﻳﻬﺎ ﺗﺮﺑﻴﻌﻴﺔ؟‬ ‫‪(1) y = x + 4‬‬ ‫‪(2) f (x) = x2 - 7‬‬ ‫‪(3) y = 3^x - 1h2 + 4‬‬ ‫‪(4) r(x) = - 7x‬‬ ‫)‪(5‬‬ ‫)‪f (x‬‬ ‫=‬ ‫‪1‬‬ ‫‪^4x + 10h‬‬ ‫‪2‬‬ ‫‪(6) y = 3x^x - 2h‬‬ ‫‪(7) y = (2x + 1) (x - 2) + 4 - 2x2‬‬ ‫‪(8) y = (3x + 7) 2 - ^9x2 - 49h‬‬ ‫)‪ (9‬ﺍﻟﺘﻔﻜﻴﺮ ﺍﻟﻨﺎﻗﺪ‪ :‬ﻣﺎ ﺍﻟﺤﺪ ﺍﻷﺩﻧﻰ ﻟﻌﺪﺩ ﺃﺯﻭﺍﺝ ﺍﻟﺒﻴﺎﻧﺎﺕ ﺍﻟﻤﻄﻠﻮﺑﺔ ﻹﻳﺠﺎﺩ ﻧﻤﻮﺫﺝ ﺗﺮﺑﻴﻌﻲ ﻟﻤﺠﻤﻮﻋﺔ ﻣﺎ ﻣﻦ ﺍﻟﺒﻴﺎﻧﺎﺕ؟‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(10-12‬ﺃﻭﺟﺪ ﺩﺍﻟﺔ ﺗﺮﺑﻴﻌﻴﺔ ﻟﻜﻞ ﻣﺠﻤﻮﻋﺔ ﻣﻦ ﺍﻟﺒﻴﺎﻧﺎﺕ‪.‬‬ ‫‪(10) x -1 0 1 2 3‬‬ ‫‪f(x) 4 -3 -6 -5 0‬‬ ‫‪(11) x -1 0 1 2 3‬‬ ‫‪f(x) -1 0 3 8 15‬‬ ‫‪(12) x -1 0 1 2 3‬‬ ‫‪f(x) 17 20 17 8 -7‬‬ ‫‪22‬‬

‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-5‬ﻇﻠﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭﻇﻠﻞ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ab‬‬ ‫)‪ (1‬ﺍﻟﺪﺍﻟﺔ ‪ f(x) = kx2 + x - 3 , k d Z‬ﻳﻤﻜﻦ ﺃﻥ ﺗﻜﻮﻥ ﺩﺍﻟﺔ ﺧﻄﻴﺔ‪.‬‬ ‫‪ab‬‬ ‫‪ab‬‬ ‫‪ f(x) = x +‬ﻫﻲ ﺩﺍﻟﺔ ﺧﻄﻴﺔ‪.‬‬ ‫‪x‬‬ ‫)‪ (2‬ﺍﻟﺪﺍﻟﺔ‬ ‫‪x‬‬ ‫)‪ (3‬ﺍﻟﻨﻘﻄﺔ )‪ A(1, 6‬ﺗﻨﺘﻤﻲ ﺇﻟﻰ ﻣﻨﺤﻨﻰ ﺍﻟﺪﺍﻟﺔ‪f(x) = ^3xh^2xh + 6 :‬‬ ‫‪ab‬‬ ‫)‪ (4‬ﺍﻟﺪﺍﻟﺔ ‪ y = x^1 - xh - ^1 - x2h‬ﻫﻲ ﺩﺍﻟﺔ ﺧﻄﻴﺔ‪.‬‬ ‫‪ab‬‬ ‫)‪ (5‬ﺍﻟﺪﺍﻟﺔ ‪ f(x) = π2 - x‬ﻫﻲ ﺩﺍﻟﺔ ﺗﺮﺑﻴﻌﻴﺔ‪.‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(6-10‬ﻇﻠﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫‪a y = ^3x + 1h^-x - 3h‬‬ ‫)‪ (6‬ﺍﻟﺪﺍﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺍﻟﺘﻲ ﺣﺪﻫﺎ ﺍﻟﺜﺎﺑﺖ ﻳﺴﺎﻭﻱ ‪ -3‬ﻓﻴﻤﺎ ﻳﻠﻲ ﻫﻲ‪:‬‬ ‫‪c f (x) = ^x - 3h^x - 3h‬‬ ‫‪b y = x2 - 3x + 3‬‬ ‫‪d y = - 3x2 + 3x + 9‬‬ ‫‪a y = ^x - 1h^x - 2h‬‬ ‫)‪ (7‬ﺃﻱ ﺩﺍﻟﺔ ﻣﻤﺎ ﻳﻠﻲ ﻟﻴﺴﺖ ﺩﺍﻟﺔ ﺗﺮﺑﻴﻌﻴﺔ‪:‬‬ ‫‪c y = 3x - x2‬‬ ‫‪b y = x2 + 2x - 3‬‬ ‫)‪d y = - x2 + x (x - 3‬‬ ‫)‪ (8‬ﺃﻱ ﻧﻘﻄﺔ ﻣﻤﺎ ﻳﻠﻲ ﺗﻨﺘﻤﻲ ﺇﻟﻰ ﻣﻨﺤﻨﻰ ﺩﺍﻟﺔ ‪ f(x) = 3x2 - 5x + 1‬؟‬ ‫)‪a (3, 12‬‬ ‫‪b ^-1, - 1h‬‬ ‫)‪c (2, 3‬‬ ‫)‪d (-2, 22‬‬ ‫)‪ (9‬ﺗﻜﻮﻥ ﺍﻟﺪﺍﻟﺔ ‪ f(x) = ^a2 - 4hx2 - ^a - 2hx + 5‬ﺩﺍﻟﺔ ﺗﺮﺑﻴﻌﻴﺔ ﻟﻜﻞ ‪ a‬ﺗﻨﺘﻤﻲ ﺇﻟﻰ‪:‬‬ ‫‪aR‬‬ ‫‪b R - \"-2, 2,‬‬ ‫‪c R - \"2,‬‬ ‫‪d R - \"-2,‬‬ ‫)‪ (10‬ﻳﻤﻜﻦ ﻧﻤﺬﺟﺔ ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ‪ x, y‬ﻓﻲ ﺍﻟﺠﺪﻭﻝ ﺍﻟﺘﺎﻟﻲ ﺑﺎﻟﺪﺍﻟﺔ‪:‬‬ ‫‪x -1 1 2‬‬ ‫‪y -1 3 8‬‬ ‫‪a f (x) = x2 + x + 1‬‬ ‫‪b f (x) = x2 + 2x - 1‬‬ ‫‪c f (x) = - x2 + 2x + 2‬‬ ‫‪d f (x) = x2 + 2x‬‬ ‫‪23‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺍﻟﺪﻭﺍﻝ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﻭﺍﻟﻘﻄﻮﻉ ﺍﻟﻤﻜﺎﻓﺌﺔ ‪2-3‬‬ ‫‪Quadratic Functions and Parabolas‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-4‬ﻛﻞ ﻧﻘﻄﺔ ﺗﻘﻊ ﻋﻠﻰ ﻗﻄﻊ ﻣﻜﺎﻓﺊ ﺭﺃﺳﻪ ﻧﻘﻄﺔ ﺍﻷﺻﻞ‪ .‬ﺍﻛﺘﺐ ﻣﻌﺎﺩﻟﺔ ﻫﺬﺍ ﺍﻟﻘﻄﻊ ﺍﻟﻤﻜﺎﻓﺊ‪ ،‬ﻭﺍﺫﻛﺮ ﻣﺎ ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺮﺳﻢ‬ ‫ﺍﻟﺒﻴﺎﻧﻲ ﻣﻔﺘﻮ ًﺣﺎ ﺇﻟﻰ ﺃﻋﻠﻰ ﺃﻡ ﺇﻟﻰ ﺃﺳﻔﻞ‪.‬‬ ‫)‪(1) F(3, 2‬‬ ‫)‪(2) F(8, -12‬‬ ‫)‪(3) H(-6, -2) (4) G(-2, 5‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(5-10‬ﺍﻛﺘﺐ ﻣﻌﺎﺩﻟﺔ ﻛﻞ ﻗﻄﻊ ﻣﻜﺎﻓﺊ ﺑﺪﻻﻟﺔ ﺇﺣﺪﺍﺛﻴﺎﺕ ﺭﺃﺳﻪ‪.‬‬ ‫‪(5) y‬‬ ‫‪(6) y‬‬ ‫)‪(0, 4‬‬ ‫‪3‬‬ ‫)‪3 (1, 3‬‬ ‫‪1‬‬ ‫)‪1 (2, 1‬‬ ‫‪-3 -1 1‬‬ ‫‪3x‬‬ ‫‪-3 -2 -1‬‬ ‫‪1 2 3x‬‬ ‫‪(7) y‬‬ ‫‪(8) y‬‬ ‫‪-1 1 3 5 x‬‬ ‫)‪(-2, 0‬‬ ‫‪-2‬‬ ‫‪-4 -2-1 1 2 3 x‬‬ ‫‪-4‬‬ ‫‪(9) y x = 1‬‬ ‫‪(10) y‬‬ ‫‪2‬‬ ‫‪(-4, 4) (-2, 4)4‬‬ ‫‪2‬‬ ‫‪-2‬‬ ‫‪2 4x‬‬ ‫‪-5 -2‬‬ ‫‪2x‬‬ ‫‪-2‬‬ ‫‪(-3, -2) -2‬‬ ‫‪24‬‬

‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(11-18‬ﺍﺭﺳﻢ ﻣﻨﺤﻨﻰ ﻛﻞ ﺩﺍﻟﺔ ﻣﻦ ﺍﻟﺪﻭﺍﻝ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫‪(11) y = ^x + 3h2‬‬ ‫‪(12) y = ^x - 2h2‬‬ ‫‪(13) y = - ^x + 1h2‬‬ ‫‪(14) y = - x2 + 3‬‬ ‫‪(15) y = ^x + 4h2 + 1‬‬ ‫‪(16) y = 3^x - 2h2 + 4‬‬ ‫‪(17) y = - 4^x + 3h2‬‬ ‫‪(18) y = - 2^x + 1h2 - 4‬‬ ‫)‪ (19‬ﺍﻟﻜﺘﺎﺑﺔ‪ :‬ﺻﻒ ﺍﻟﺨﻄﻮﺍﺕ ﺍﻟﺘﻲ ﺳﻮﻑ ﺗﺴﺘﺨﺪﻣﻬﺎ ﻟﺮﺳﻢ ﺍﻟﺪﺍﻟﺔ‪ y = - 2^x - 3h2 + 4 :‬ﺑﻴﺎﻧﻴًّﺎ‪.‬‬ ‫)‪ (20‬ﺍﻟﺴﺆﺍﻝ ﺍﻟﻤﻔﺘﻮﺡ‪ :‬ﺍﻛﺘﺐ ﻣﻌﺎﺩﻟﺔ ﻟﺪﺍﻟﺔ ﻳﻤﺜﻠﻬﺎ ﺑﻴﺎﻧﻴًّﺎ ﻗﻄﻊ ﻣﻜﺎﻓﺊ ﻟﻪ ﻣﺤﻮﺭ ﺍﻟﺘﻤﺎﺛﻞ ﺍﻟﺘﺎﻟﻲ‪x = - 2 :‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(21-25‬ﺍﺭﺳﻢ ﻛﻞ ﻗﻄﻊ ﻣﻜﺎﻓﺊ ﻣﺴﺘﺨﺪ ًﻣﺎ ﺍﻟﻤﻌﻠﻮﻣﺎﺕ ﺍﻟﻤﻌﻄﺎﺓ‪ .‬ﺛﻢ ﺍﻛﺘﺐ ﻣﻌﺎﺩﻟﺘﻪ ﺑﺪﻻﻟﺔ ﺇﺣﺪﺍﺛﻴﺎﺕ ﺍﻟﺮﺃﺱ‪.‬‬ ‫)‪ (21‬ﺍﻟﺮﺃﺱ )‪ V(0, 0‬ﻭﻳﻤﺮ ﺑﺎﻟﻨﻘﻄﺔ )‪P(2, 10‬‬ ‫)‪ (22‬ﺍﻟﺮﺃﺱ )‪ V(0, 0‬ﻭﻳﻤﺮ ﺑﺎﻟﻨﻘﻄﺔ )‪P(-2, -10‬‬ ‫)‪ (23‬ﺍﻟﺮﺃﺱ )‪ V(0, 5‬ﻭﻳﻤﺮ ﺑﺎﻟﻨﻘﻄﺔ )‪P(1, -2‬‬ ‫)‪ (24‬ﺍﻟﺮﺃﺱ )‪ V(3, 1‬ﻭﺍﻟﺠﺰﺀ ﺍﻟﻤﻘﻄﻮﻉ ﻣﻦ ﻣﺤﻮﺭ ﺍﻟﺼﺎﺩﺍﺕ ‪-2‬‬ ‫)‪ (25‬ﺍﻟﺮﺃﺱ )‪ V(-2, 6‬ﻭﺍﻟﺠﺰﺀ ﺍﻟﻤﻘﻄﻮﻉ ﻣﻦ ﻣﺤﻮﺭ ﺍﻟﺴﻴﻨﺎﺕ ‪2‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-5‬ﻇﻠﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭﻇﻠﻞ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ab‬‬ ‫)‪ (1‬ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ y = 2x2 - 2(3 - x)2‬ﺗﻤﺜﻞ ﻣﻌﺎﺩﻟﺔ ﻗﻄﻊ ﻣﻜﺎﻓﺊ‪.‬‬ ‫‪ab‬‬ ‫ﻓﺘﺤﺘﻪ ﺇﻟﻰ ﺍﻷﻋﻠﻰ‪.‬‬ ‫‪y =-‬‬ ‫‪1‬‬ ‫‪(x‬‬ ‫‪+‬‬ ‫)‪2‬‬ ‫‪2‬‬ ‫‪-‬‬ ‫‪3‬‬ ‫ﺍﻟﻘﻄﻊ ﺍﻟﻤﻜﺎﻓﺊ‬ ‫)‪(2‬‬ ‫‪3‬‬ ‫‪a‬‬ ‫‪b‬‬ ‫=‪y‬‬ ‫‪1‬‬ ‫‪x2‬‬ ‫‪-‬‬ ‫‪2‬‬ ‫ﺍﻟﺪﺍﻟﺔ‬ ‫ﻳﻜﻮﻥ ﺑﻴﺎﻧﻬﺎ ﺃﻛﺜﺮ ﺍﺗﺴﺎ ًﻋﺎ ﻣﻦ ﺑﻴﺎﻥ‬ ‫‪y = 2 (x - 1) 2 + 2‬‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫)‪(3‬‬ ‫‪2‬‬ ‫‪ab‬‬ ‫)‪ (4‬ﺗﻮﺟﺪ ﻋﻨﺪ ﺭﺃﺱ ﻣﻨﺤﻨﻰ ﺍﻟﺪﺍﻟﺔ ‪ y = - (x - 3)2 - 2‬ﻗﻴﻤﺔ ﻋﻈﻤﻰ‪.‬‬ ‫‪ab‬‬ ‫)‪ (5‬ﻣﻨﺤﻨﻰ ﺍﻟﻘﻄﻊ ﺍﻟﻤﻜﺎﻓﺊ ‪ y = (- x + 2)2 + 3‬ﻳﻤﺮ ﺑﺎﻟﻨﻘﻄﺔ )‪P(2, 3‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(6-11‬ﻇﻠﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻟﺔ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫)‪ (6‬ﺍﻟﺪﺍﻟﺔ ‪ y = a(3 - x)2 - 2‬ﻳﻜﻮﻥ ﺭﺳﻤﻬﺎ ﺃﻭﺳﻊ ﻣﻦ ﺭﺳﻢ ﺑﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ ‪ y = - 2x2‬ﺇﺫﺍ ﻛﺎﻥ‪:‬‬ ‫‪a a =2‬‬ ‫‪b a 22‬‬ ‫‪c a12‬‬ ‫‪d a 12‬‬ ‫)‪ (7‬ﻣﻌﺎﺩﻟﺔ ﺍﻟﻘﻄﻊ ﺍﻟﻤﻜﺎﻓﺊ ‪ y = 2x2‬ﺍﻟﺬﻱ ﺗﻢ ﺇﺯﺍﺣﺔ ﺭﺃﺳﻪ ﻭﺣﺪﺗﻴﻦ ﻳﺴﺎ ًﺭﺍ ﻭ‪ 4‬ﻭﺣﺪﺍﺕ ﻷﻋﻠﻰ ﻫﻲ‪:‬‬ ‫‪a y = (2x + 2) 2 + 4‬‬ ‫‪b y = 2 (x - 2) 2 + 4‬‬ ‫‪c y = 2 (x + 2) 2 + 4‬‬ ‫‪d y = 2 (x + 2) 2 - 4‬‬ ‫‪25‬‬

‫)‪ (8‬ﺍﻟﺸﻜﻞ ﺃﺩﻧﺎﻩ ﻳﻤﺜﻞ ﻣﻨﺤﻨﻰ ﻗﻄﻊ ﻣﻜﺎﻓﺊ ﻣﻌﺎﺩﻟﺘﻪ ﻫﻲ‪:‬‬ ‫‪y‬‬ ‫‪12345 6x‬‬ ‫‪2‬‬ ‫‪1‬‬ ‫‪0‬‬ ‫‪-4 -3 -2 --1 1‬‬ ‫‪-2‬‬ ‫‪a y = (x - 2) 2 + 2‬‬ ‫‪b‬‬ ‫‪y‬‬ ‫=‬ ‫‪1‬‬ ‫‪(x - 2) 2 + 2‬‬ ‫‪2‬‬ ‫‪c‬‬ ‫‪y‬‬ ‫=‬ ‫‪-‬‬ ‫‪1‬‬ ‫‪(x‬‬ ‫‪-‬‬ ‫‪2) 2‬‬ ‫‪-‬‬ ‫‪2‬‬ ‫‪d‬‬ ‫‪y‬‬ ‫‪=-‬‬ ‫‪1‬‬ ‫‪(x‬‬ ‫‪-‬‬ ‫‪2) 2‬‬ ‫‪+‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫)‪ (9‬ﺍﻟﻘﻄﻊ ﺍﻟﻤﻜﺎﻓﺊ ‪ y = a(x - h)2 + k‬ﻳﻘﻄﻊ ﺍﻟﻤﺤﻮﺭﻳﻦ ﻋﻠﻰ ﺍﻷﻛﺜﺮ ﻓﻲ‪:‬‬ ‫ﻧﻘﻄﺔ ‪a‬‬ ‫ﻧﻘﻄﺘﻴﻦ ‪b‬‬ ‫‪ 3‬ﻧﻘﺎﻁ ‪c‬‬ ‫‪ 4‬ﻧﻘﺎﻁ ‪d‬‬ ‫ﺍﻟﻨﻘﻄﺔ‪:‬‬ ‫ﻋﻨﺪ‬ ‫ﻫﻲ‬ ‫‪y‬‬ ‫=‬ ‫‪1‬‬ ‫‪(3‬‬ ‫‪- x) 2 - 2‬‬ ‫ﻟﻠﺪﺍﻟﺔ‬ ‫ﺍﻟﺼﻐﺮﻯ‬ ‫ﺍﻟﻘﻴﻤﺔ‬ ‫)‪(10‬‬ ‫‪3‬‬ ‫)‪a (3, - 2‬‬ ‫)‪b (- 3, 2‬‬ ‫)‪c (- 3, - 2‬‬ ‫)‪d (3, 2‬‬ ‫)‪ (11‬ﻳﻘﻊ ﺟﺴﺮ ﻋﻠﻰ ﺷﻜﻞ ﻗﻄﻊ ﻣﻜﺎﻓﺊ ﻓﻮﻕ ﻧﻬﺮ‪ .‬ﻳﺒﻠﻎ ﺍﻟﺒﻌﺪ ﺑﻴﻦ ﻗﺎﻋﺪﺗﻴﻪ ‪ 20 m‬ﻭﺍﺭﺗﻔﺎﻋﻪ ﺍﻷﻗﺼﻰ ‪ 8 m‬ﻣﻌﺎﺩﻟﺔ‬ ‫ﺍﻟﻘﻄﻊ ﺍﻟﻤﻜﺎﻓﺊ ﻫﻲ‪:‬‬ ‫‪y‬‬ ‫‪8‬‬ ‫‪4‬‬ ‫‪0‬‬ ‫‪4 8 12 16 20 x‬‬ ‫‪a y = 0.08 (x - 10) 2 + 8‬‬ ‫‪b y = - 0.08 (x - 10) 2 + 8‬‬ ‫‪c y = - 0.08 (x - 20) 2 + 8‬‬ ‫‪d y = 0.08 (x + 10) 2 + 8‬‬ ‫‪26‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫‪2-4‬‬ ‫ﻣﻘﺎﺭﻧﺔ ﺑﻴﻦ ﺻﻮﺭﺓ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﺑﺪﻻﻟﺔ ﺇﺣﺪﺍﺛﻴﺎﺕ ﺭﺃﺱ ﺍﻟﻤﻨﺤﻨﻰ ﻭﺍﻟﺼﻮﺭﺓ ﺍﻟﻌﺎﻣﺔ‬ ‫‪Comparing Vertex and General Form Equation of Quadratic‬‬ ‫‪Functions‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-7‬ﺍﻛﺘﺐ ﻛ ًّﻼ ﻣﻦ ﺍﻟﺪﻭﺍﻝ ﺍﻟﺘﺎﻟﻴﺔ ﺑﺪﻻﻟﺔ ﺇﺣﺪﺍﺛﻴﺎﺕ ﺍﻟﺮﺃﺱ‪:‬‬ ‫‪(1) y = x2 - 4x + 6‬‬ ‫‪(2) y = x2 + 2x + 5‬‬ ‫‪(3) y = 4x2 + 7x‬‬ ‫‪(4) f (x) = - 2x2 + 35‬‬ ‫‪(7) y = - 3x2 - 2x + 1‬‬ ‫‪(5) y = - 8x2‬‬ ‫‪(6) f (x) = 2x2 + x‬‬ ‫‪(8) y = ^x + 3h2 - 4‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(8-13‬ﺍﻛﺘﺐ ﻣﻌﺎﺩﻟﺔ ﻛﻞ ﻗﻄﻊ ﻣﻜﺎﻓﺊ ﻓﻲ ﺍﻟﺼﻮﺭﺓ ﺍﻟﻌﺎﻣﺔ‪.‬‬ ‫‪(11) y = ^5x + 6h2 - 9‬‬ ‫‪(9) f (x) = 2^x - 2h2 + 5‬‬ ‫‪(10) f (x) = - ^x - 7h2 + 10‬‬ ‫‪(12) f (x) = - ^3x - 4h2 + 6‬‬ ‫‪(13) f (x) = - 2x^x + 7h + 8x‬‬ ‫)‪ (14‬ﺍﻟﺘﻔﻜﻴﺮ ﺍﻟﻨﺎﻗﺪ‪ :‬ﻣﻌﺎﺩﻟﺔ ﺃﺣﺪ ﺍﻟﺮﺳﻤﻴﻦ ﺍﻟﺒﻴﺎﻧﻴﻴﻦ ﺃﺩﻧﺎﻩ ﻫﻲ‪y = x2 - 8x + 18 :‬‬ ‫ﺍﻛﺘﺐ ﻣﻌﺎﺩﻟﺔ ﺍﻟﺮﺳﻢ ﺍﻟﺒﻴﺎﻧﻲ ﺍﻵﺧﺮ ﻓﻲ ﺍﻟﺼﻮﺭﺓ ﺍﻟﻌﺎﻣﺔ‪.‬‬ ‫‪y‬‬ ‫‪5‬‬ ‫‪3‬‬ ‫‪1‬‬ ‫‪1 3 5x‬‬ ‫)‪ (15‬ﻣﻨﺤﻨﻰ ﺍﻟﺪﺍﻟﺔ‪ ،y = 2x2 - 12x + c :‬ﻟﻪ ﺭﺃﺱ ﻋﻨﺪ ﺍﻟﻨﻘﻄﺔ )‪ .(3, 5‬ﻓﻤﺎ ﻗﻴﻤﺔ ‪c‬؟‬ ‫)‪ (16‬ﻣﻨﺤﻨﻰ ﺍﻟﺪﺍﻟﺔ‪ ، y = ax2 + bx + 8 :‬ﻟﻪ ﺭﺃﺱ ﻋﻨﺪ ﺍﻟﻨﻘﻄﺔ )‪ .(2, - 4‬ﻓﻤﺎ ﻗﻴﻢ ‪a , b‬؟‬ ‫‪27‬‬

‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-4‬ﻇﻠﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫)‪ (1‬ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ y = - 2(x + 3)2 + 4‬ﻫﻲ ﻣﻌﺎﺩﻟﺔ ﻗﻄﻊ ﻣﻜﺎﻓﺊ ﺑﺪﻻﻟﺔ ﺇﺣﺪﺍﺛﻴﺎﺕ ﺭﺃﺱ ﺍﻟﻤﻨﺤﻨﻰ‪a b .‬‬ ‫)‪ (2‬ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ y = 3(x - 2)2 + 4(x - 2) + 1‬ﻫﻲ ﻣﻌﺎﺩﻟﺔ ﻗﻄﻊ ﻣﻜﺎﻓﺊ ﻓﻲ ﺍﻟﺼﻮﺭﺓ ﺍﻟﻌﺎﻣﺔ‪a b .‬‬ ‫‪ab‬‬ ‫)‪ (3‬ﺭﺃﺱ ﺍﻟﻘﻄﻊ ﺍﻟﻤﻜﺎﻓﺊ ﺍﻟﺬﻱ ﻣﻌﺎﺩﻟﺘﻪ ‪ y = x2 - 2x - 3‬ﻫﻮ ‪V^1, - 4h‬‬ ‫‪ab‬‬ ‫)‪ (4‬ﻣﻌﺎﺩﻟﺔ ﻣﺤﻮﺭ ﺍﻟﺘﻤﺎﺛﻞ ﻟﻠﻘﻄﻊ ﺍﻟﻤﻜﺎﻓﺊ‪ y = 3x2 + 12x + 8 :‬ﻫﻲ ‪y = - 4‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(5-12‬ﻇﻠﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻟﺔ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫)‪ (5‬ﺭﺃﺱ ﺍﻟﻘﻄﻊ ﺍﻟﻤﻜﺎﻓﺊ ﺍﻟﺬﻱ ﻣﻌﺎﺩﻟﺘﻪ ‪ y = ax2 + 2ax + 5, a ≠ 0‬ﻳﻤﻜﻦ ﺃﻥ ﻳﻜﻮﻥ‪:‬‬ ‫)‪a (1, 1‬‬ ‫)‪b (-1, 1‬‬ ‫)‪c (1, 5‬‬ ‫)‪d (-1, 5‬‬ ‫)‪ (6‬ﻣﻌﺎﺩﻟﺔ ﺍﻟﻘﻄﻊ ﺍﻟﻤﻜﺎﻓﺊ ﺍﻟﻤﺎﺭ ﺑﺎﻟﻨﻘﻄﺔ )‪ (-3, 10‬ﻭﺭﺃﺳﻪ )‪ (0, 1‬ﻫﻲ‪:‬‬ ‫‪a y = 5x2 + 1‬‬ ‫‪b y = - 3x2 + 10‬‬ ‫‪c y = x2 + 1‬‬ ‫‪d y = - x2 - 1‬‬ ‫)‪ (7‬ﻣﻨﺤﻨﻰ ﺍﻟﺪﺍﻟﺔ ‪ y = - 2x2 + 4x - 5‬ﻟﻪ ﺭﺃﺱ ﻋﻨﺪ ﺍﻟﻨﻘﻄﺔ‪:‬‬ ‫)‪a (-2, -3‬‬ ‫)‪b (1, -3‬‬ ‫)‪c (1, -1‬‬ ‫)‪d (-1, -3‬‬ ‫)‪ (8‬ﻳﻘﻊ ﺭﺃﺱ ﻣﻨﺤﻨﻰ ‪ y = - x2 - 16x - 62‬ﻓﻲ ﺍﻟﺮﺑﻊ‪:‬‬ ‫ﺍﻷ ّﻭﻝ ‪a‬‬ ‫ﺍﻟﺜﺎﻧﻲ ‪b‬‬ ‫ﺍﻟﺜﺎﻟﺚ ‪c‬‬ ‫ﺍﻟﺮﺍﺑﻊ ‪d‬‬ ‫‪a x = 12‬‬ ‫‪b x=6‬‬ ‫)‪ (9‬ﻣﻌﺎﺩﻟﺔ ﻣﺤﻮﺭ ﺍﻟﺘﻤﺎﺛﻞ ﻟﻠﻘﻄﻊ ﺍﻟﻤﻜﺎﻓﺊ ‪ y = x2 - 6x + 2‬ﻫﻲ‪:‬‬ ‫‪c x=3‬‬ ‫‪d x=2‬‬ ‫)‪ (10‬ﺍﻟﻤﺴﺎﺣﺔ ﺍﻟﻌﻈﻤﻰ ﺑﺎﻟﻮﺣﺪﺍﺕ ﺍﻟﻤﺮﺑﻌﺔ ﻟﻤﺴﺘﻄﻴﻞ ﻣﺤﻴﻄﻪ ‪ 128 m‬ﻫﻲ‪:‬‬ ‫‪a 4 096‬‬ ‫‪b 1 024‬‬ ‫‪c 256‬‬ ‫‪d 32‬‬ ‫)‪ (11‬ﻳﻨﻤﺬﺝ ﻣﺪﺧﻮﻝ ﺇﺣﺪﻯ ﺍﻟﺸﺮﻛﺎﺕ ﺑﺎﻟﻌﻼﻗﺔ ‪ R = - 15 p2 + 300 p + 12 000‬ﺣﻴﺚ ‪) p‬ﺑﺎﻟﺪﻳﻨﺎﺭ( ﻫﻮ ﺳﻌﺮ‬ ‫ﻣﺒﻴﻊ ﺇﺣﺪﻯ ﺍﻟﻘﻄﻊ ﺍﻟﻤﻨﺘﺠﺔ‪ .‬ﻗﻴﻤﺔ ‪ p‬ﺍﻟﺘﻲ ﺗﻌﻄﻲ ﺃﻋﻠﻰ ﻣﺪﺧﻮﻝ ﻫﻲ‪:‬‬ ‫‪a 30‬‬ ‫‪b 10‬‬ ‫‪c 15‬‬ ‫‪d 12‬‬ ‫)‪ (12‬ﺃﻱ ﻣﻨﺤﻨﻰ ﻣﻦ ﺍﻟﺪﻭﺍﻝ ﺃﺩﻧﺎﻩ ﻟﻪ ﺧﻂ ﺗﻤﺎﺛﻞ ‪x = 3‬؟‬ ‫‪a y = 2^x + 3h2‬‬ ‫‪b y = x2 - 6x + 9‬‬ ‫‪c y = x2 + 3x + 6‬‬ ‫‪d y = 4^x + 3h2‬‬ ‫‪28‬‬

‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ (13-15‬ﻟﺪﻳﻚ ﻗﺎﺋﻤﺘﺎﻥ‪ ،‬ﺍﺧﺘﺮ ﻣﻦ ﺍﻟﻘﺎﺋﻤﺔ )‪ (2‬ﻣﺎ ﻳﻨﺎﺳﺒﻪ ﻓﻲ ﺍﻟﻘﺎﺋﻤﺔ )‪ (1‬ﻟﺘﺤﺼﻞ ﻋﻠﻰ ﺇﺟﺎﺑﺔ ﺻﺤﻴﺤﺔ‪.‬‬ ‫ﺍﻟﻘﺎﺋﻤﺔ )‪(2‬‬ ‫ﺍﻟﻘﺎﺋﻤﺔ )‪(1‬‬ ‫‪ay‬‬ ‫ﺍﻟﺘﻤﺜﻴﻞ ﺍﻟﺒﻴﺎﻧﻲ ﻟﻠﺪﺍﻟﺔ‪:‬‬ ‫‪2‬‬ ‫)‪ y = x2 + 4x + 1 (13‬ﻫﻮ‪:‬‬ ‫‪-5 -3 -1 x‬‬ ‫‪by‬‬ ‫)‪ y = - x2 - 4x + 1 (14‬ﻫﻮ‪:‬‬ ‫‪2‬‬ ‫‪-4 -2‬‬ ‫‪x‬‬ ‫‪-2‬‬ ‫‪-4‬‬ ‫‪cy‬‬ ‫ﻫﻮ‪:‬‬ ‫‪y‬‬ ‫‪=-‬‬ ‫‪1‬‬ ‫‪x2‬‬ ‫‪-‬‬ ‫‪2x‬‬ ‫‪+‬‬ ‫‪1‬‬ ‫)‪(15‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪-4 -2‬‬ ‫‪x‬‬ ‫‪-2‬‬ ‫‪-4‬‬ ‫‪dy‬‬ ‫‪2‬‬ ‫‪-4 -2‬‬ ‫‪x‬‬ ‫‪-2‬‬ ‫‪-4‬‬ ‫‪29‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫‪2-5‬‬ ‫ﺍﻟﻤﻌﻜﻮﺳﺎﺕ ﻭﺩﻭﺍﻝ ﺍﻟﺠﺬﺭ ﺍﻟﺘﺮﺑﻴﻌﻲ‬ ‫‪Inverses and Square Root Functions‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-3‬ﺍﺭﺳﻢ ﺑﻴﺎﻧﻴًّﺎ ﺍﻟﺪﺍﻟﺔ ﺍﻟﻤﻌﻄﺎﺓ ﻭﻣﻌﻜﻮﺳﻬﺎ ﻋﻠﻰ ﻣﺤﺎﻭﺭ ﺍﻹﺣﺪﺍﺛﻴﺎﺕ ﻧﻔﺴﻬﺎ‪ .‬ﺛﻢ ﺍﻛﺘﺐ ﻣﻌﺎﺩﻟﺔ ﺍﻟﻤﻌﻜﻮﺱ‪.‬‬ ‫)‪(1‬‬ ‫‪y‬‬ ‫=‬ ‫‪1‬‬ ‫‪x‬‬ ‫)‪(2‬‬ ‫‪y‬‬ ‫=‬ ‫‪x+1‬‬ ‫‪(3) y = 5x + 3‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(4-10‬ﺍﻛﺘﺐ ﻣﻌﺎﺩﻟﺔ ﺍﻟﻤﻌﻜﻮﺱ ﻟﻜﻞ ﺩﺍﻟﺔ ﻣﻤﺎ ﻳﻠﻲ‪:‬‬ ‫)‪(4‬‬ ‫‪y‬‬ ‫=‬ ‫‪1‬‬ ‫‪x2‬‬ ‫‪(5) y = x2 - 1‬‬ ‫‪(6) y = ^x - 2h2 + 1‬‬ ‫)‪(7‬‬ ‫‪y‬‬ ‫=‬ ‫‪x+5‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪(8) y = 6x + 2‬‬ ‫‪(9) y = x2 - 3‬‬ ‫‪(10) y = (x + 5)2 + 2‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(11-14‬ﺍﺭﺳﻢ ﻛﻞ ﺩﺍﻟﺔ ﺟﺬﺭ ﺗﺮﺑﻴﻌﻲ‪ .‬ﺛﻢ ﺍﺫﻛﺮ ﺍﻟﻤﺠﺎﻝ ﻭﺍﻟﻤﺪﻯ‪.‬‬ ‫‪(11) y = - x - 1‬‬ ‫‪(12) y = - x + 2‬‬ ‫‪(13) y = x - 4 + 2 (14) y = - x + 3 - 2‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺮﻳﻨﻴﻦ )‪ ،(15-16‬ﺍﺭﺳﻢ ﺑﻴﺎﻧًﺎ ﻟﻤﻌﻜﻮﺱ ﺍﻟﺮﺳﻢ ﺍﻟﺒﻴﺎﻧﻲ‪ ،‬ﺛﻢ ﺍﻛﺘﺐ ﻣﻌﺎﺩﻟﺔ ﻛﻞ ﺭﺳﻢ ﺑﻴﺎﻧﻲ‪ ،‬ﻭﻣﻌﺎﺩﻟﺔ ﻣﻌﻜﻮﺳﻪ‪.‬‬ ‫‪(15) y‬‬ ‫‪(16) y‬‬ ‫‪4‬‬ ‫‪2‬‬ ‫‪x‬‬ ‫‪2‬‬ ‫‪0‬‬ ‫‪0‬‬ ‫‪24‬‬ ‫‪2 4x‬‬ ‫)‪ (a) (17‬ﺍﻟﺮﻳﺎﺿﻴﺎﺕ ﻓﻲ ﺍﻹﻋﻼﻧﺎﺕ ﺍﻟﺘﺠﺎﺭﻳﺔ‪ :‬ﺍﻛﺘﺐ ﺩﺍﻟﺔ ﺗﻌﻄﻲ ﺛﻤﻦ ﺍﻟﺒﻴﻊ ‪ y‬ﻟﻠﺜﻤﻦ ﺍﻷﺻﻠﻲ ‪ x‬ﺑﺎﻟﻨﺴﺒﺔ ﺇﻟﻰ ﺍﻟﺴﻠﻊ ﻓﻲ‬ ‫ﺍﻹﻋﻼﻥ ﺍﻟﻤﺠﺎﻭﺭ‪.‬‬ ‫ﺣﺴﻮﻣﺎﺕ ﺃﺳﺮﻉ!‬ ‫)‪ (b‬ﺃﻭﺟﺪ ﻣﻌﻜﻮﺱ ﺍﻟﺪﺍﻟﺔ ﺍﻟﺘﻲ ﺃﻭﺟﺪﺗﻬﺎ ﻓﻲ ﺍﻟﻔﻘﺮﺓ )‪.(a‬‬ ‫ﺳﻮﻑ ﺗﻨﺘﻬﻲ ﺍﻟﺤﺴﻮﻣﺎﺕ ﻓﻲ ‪ 31‬ﻳﻨﺎﻳﺮ‬ ‫)‪ (c‬ﺍﻟﻜﺘﺎﺑﺔ‪ :‬ﻣﺎﺫﺍ ﺗﻤﺜﻞ ﺍﻟﺪﺍﻟﺔ ﺍﻟﺘﻲ ﻛﺘﺒﺘﻬﺎ ﻓﻲ ﺍﻟﺴﺆﺍﻝ )‪(b‬؟‬ ‫ﻭﻓّﺮ ‪20 %‬‬ ‫‪30‬‬

‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫‪ab‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-5‬ﻇﻠﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ab‬‬ ‫)‪ (1‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻨﻘﻄﺔ )‪ M(x, y‬ﺗﻨﺘﻤﻲ ﻟﺒﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ ‪ f‬ﻓﺈﻥ ﺍﻟﻨﻘﻄﺔ )‪N(y, x‬‬ ‫‪ab‬‬ ‫ﺗﻨﺘﻤﻲ ﻟﺒﻴﺎﻥ ﻣﻌﻜﻮﺱ ﻫﺬﻩ ﺍﻟﺪﺍﻟﺔ‪.‬‬ ‫‪ab‬‬ ‫‪ab‬‬ ‫)‪ (2‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ f(x) = x + 1, g(x) = x - 1‬ﻓﺈﻥ ﺍﻟﺪﺍﻟﺘﻴﻦ ﻛﻞ ﻣﻨﻬﻤﺎ ﻣﻌﻜﻮﺱ ﻟﻸﺧﺮﻯ‪.‬‬ ‫)‪ (3‬ﺍﻟﻤﺴﺘﻘﻴﻢ ‪ y = x‬ﻫﻮ ﺧﻂ ﺍﻧﻌﻜﺎﺱ ﻟﺒﻴﺎﻥ ﺩﺍﻟﺔ ‪ f‬ﻭﺑﻴﺎﻥ ﻣﻌﻜﻮﺳﻬﺎ‪.‬‬ ‫)‪ (4‬ﺇﺫﺍ ﻣﺮ ﺑﻴﺎﻥ ﺩﺍﻟﺔ ﺑﻨﻘﻄﺔ ﺍﻷﺻﻞ ﻓﺈﻥ ﺑﻴﺎﻥ ﻣﻌﻜﻮﺳﻬﺎ ﻳﻤﺮ ﺃﻳ ًﻀﺎ ﺑﻨﻘﻄﺔ ﺍﻷﺻﻞ‪.‬‬ ‫)‪ (5‬ﻻ ﻳﺘﻐﻴﺮ ﻣﺠﺎﻝ ﺩﺍﻟﺔ ﺍﻟﺠﺬﺭ ﺍﻟﺘﺮﺑﻴﻌﻲ ﺑﻌﺪ ﺇﺯﺍﺣﺔ ﺑﻴﺎﻧﻬﺎ ‪ 3‬ﻭﺣﺪﺍﺕ ﻳﻤﻴﻨًﺎ‪.‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(6-10‬ﻇﻠﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻟﺔ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪:‬‬ ‫)‪ (6‬ﺇﺫﺍ ﺍﻧﺘﻤﺖ ﺍﻟﻨﻘﻄﺔ )‪ A(2, 3‬ﺇﻟﻰ ﺑﻴﺎﻥ ﺩﺍﻟﺔ ﻓﺈﻥ ﺍﻟﻨﻘﻄﺔ ﺍﻟﺘﻲ ﺗﻨﺘﻤﻲ ﺇﻟﻰ ﺑﻴﺎﻥ ﻣﻌﻜﻮﺱ ﺗﻠﻚ ﺍﻟﺪﺍﻟﺔ ﻫﻲ‪:‬‬ ‫)‪a (-2, 3‬‬ ‫)‪b (2, -3‬‬ ‫)‪c (3, -2‬‬ ‫)‪d (3, 2‬‬ ‫)‪ (7‬ﺑﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ ‪ y = x + 2 - 2‬ﻫﻮ ﺍﻧﺴﺤﺎﺏ ﻟﺒﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ ‪:y = x‬‬ ‫ﻭﺣﺪﺗﻴﻦ ﺇﻟﻰ ﺍﻟﻴﺴﺎﺭ ﻭﻭﺣﺪﺗﻴﻦ ﻟﻸﻋﻠﻰ ‪a‬‬ ‫ﻭﺣﺪﺗﻴﻦ ﺇﻟﻰ ﺍﻟﻴﺴﺎﺭ ﻭﻭﺣﺪﺗﻴﻦ ﻟﻸﺳﻔﻞ ‪b‬‬ ‫ﻭﺣﺪﺗﻴﻦ ﺇﻟﻰ ﺍﻟﻴﻤﻴﻦ ﻭﻭﺣﺪﺗﻴﻦ ﻟﻸﻋﻠﻰ ‪c‬‬ ‫ﻭﺣﺪﺗﻴﻦ ﺇﻟﻰ ﺍﻟﻴﻤﻴﻦ ﻭﻭﺣﺪﺗﻴﻦ ﻟﻸﺳﻔﻞ ‪d‬‬ ‫)‪ (8‬ﻣﻌﻜﻮﺱ ﺍﻟﺪﺍﻟﺔ ‪ y = x2 + 2‬ﻫﻮ‪:‬‬ ‫‪a y= x-2‬‬ ‫‪b y =- x-2‬‬ ‫‪c y =! x-2‬‬ ‫ﻟﻴﺲ ﺃﻳًّﺎ ﻣﻤﺎ ﺳﺒﻖ ﺻﺤﻴ ًﺤﺎ ‪d‬‬ ‫)‪ (9‬ﻣﻌﻜﻮﺱ ﺍﻟﺪﺍﻟﺔ ‪ y = 5x - 1‬ﻫﻮ‪:‬‬ ‫‪a y = 5x + 1‬‬ ‫‪b‬‬ ‫‪y‬‬ ‫=‬ ‫‪x+1‬‬ ‫‪5‬‬ ‫‪c‬‬ ‫‪y‬‬ ‫=‬ ‫‪x‬‬ ‫‪+1‬‬ ‫‪d‬‬ ‫‪y‬‬ ‫=‬ ‫‪x‬‬ ‫‪-1‬‬ ‫‪5‬‬ ‫‪5‬‬ ‫)‪ (10‬ﻣﺠﺎﻝ ﻣﻌﻜﻮﺱ ﺍﻟﺪﺍﻟﺔ ‪ y = x + 3 - 1‬ﻫﻮ‪:‬‬ ‫‪aR‬‬ ‫‪b ^-1, 3h‬‬ ‫‪c ^-3, 1h‬‬ ‫‪d 6-1, 3h‬‬ ‫‪31‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺣﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ ‪2-6‬‬ ‫‪Solving Inequalities‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫)‪ (1‬ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﻛﻞ ﻣﻦ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫‪(a) ^x - 3h^2x + 5h 1 0‬‬ ‫‪(b) 2x2 - 3x - 5 $ 0‬‬ ‫‪(c) -3x2 + 2x 1 - 1‬‬ ‫‪(d) 4x2 + 12x + 9 $ 0‬‬ ‫‪(e) -9x2 + 6x 1 1‬‬ ‫‪(f) 21 + 4x 2 x2‬‬ ‫)‪ (2‬ﻟﻨﻌﺘﺒﺮ ﻋﺮﺽ ﻣﺴﺘﻄﻴﻞ ‪ ^x - 2h cm‬ﻭﻃﻮﻟﻪ ‪2x cm‬‬ ‫)‪ (a‬ﻭ ّﺿﺢ ﻟﻤﺎﺫﺍ ﻳﺠﺐ ﺃﻥ ﺗﻜﻮﻥ ﻗﻴﻤﺔ ‪ x‬ﺃﻛﺒﺮ ﻣﻦ ‪2‬‬ ‫)‪ (b‬ﺍﻛﺘﺐ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﻲ ﺗﻌﻄﻲ ﻣﺴﺎﺣﺔ ﻫﺬﺍ ﺍﻟﻤﺴﺘﻄﻴﻞ‪.‬‬ ‫)‪ (c‬ﻋﻠ ًﻤﺎ ﺃ ّﻥ ‪ x‬ﻋﺪﺩ ﺻﺤﻴﺢ‪ ،‬ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ x‬ﻟﺘﻜﻮﻥ ﻣﺴﺎﺣﺔ ﺍﻟﻤﺴﺘﻄﻴﻞ ﺑﻴﻦ ‪ 90 cm2‬ﻭ‪ ،100 cm2‬ﺛﻢ ﺍﺳﺘﻨﺘﺞ‬ ‫ﻃﻮﻝ ﺍﻟﻤﺴﺘﻄﻴﻞ ﻭﻋﺮﺿﻪ‪.‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(3-9‬ﺣ ّﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫)‪(3‬‬ ‫‪x-1‬‬ ‫‪1‬‬ ‫‪0‬‬ ‫)‪(4‬‬ ‫‪x2 - 1‬‬ ‫‪#‬‬ ‫‪0‬‬ ‫)‪(5‬‬ ‫‪x2 + x - 12‬‬ ‫‪2‬‬ ‫‪0‬‬ ‫‪x2 - 4‬‬ ‫‪x2 + 1‬‬ ‫‪x2 - 4x + 4‬‬ ‫)‪(6‬‬ ‫‪1‬‬ ‫‪+‬‬ ‫‪1‬‬ ‫‪#‬‬ ‫‪0‬‬ ‫)‪(7‬‬ ‫‪1‬‬ ‫‪-‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪0‬‬ ‫)‪(8‬‬ ‫‪x‬‬ ‫‪+‬‬ ‫‪2‬‬ ‫‪H‬‬ ‫‪0‬‬ ‫‪x+1‬‬ ‫‪x-3‬‬ ‫‪x+2‬‬ ‫‪x-1‬‬ ‫‪x+1‬‬ ‫‪x-1‬‬ ‫)‪* (9‬‬ ‫‪2x + 1‬‬ ‫‪+‬‬ ‫‪3x‬‬ ‫‪G‬‬ ‫‪0‬‬ ‫‪x‬‬ ‫‪1 - 2x‬‬ ‫)‪ (10‬ﻋﻤﺮ ﺟ ّﺪ ﺃﺣﻤﺪ ﻳﺴﺎﻭﻱ ‪ 8‬ﺃﺿﻌﺎﻑ ﻋﻤﺮ ﺃﺣﻤﺪ‪ .‬ﺑﻌﺪ ‪ 3‬ﺳﻨﻮﺍﺕ‪ ،‬ﺳﻴﺘﺨﻄﻰ ﺗﺮﺑﻴﻊ ﻋﻤﺮ ﺃﺣﻤﺪ ﺿﻌﻒ ﻋﻤﺮ ﺟ ّﺪﻩ‬ ‫)ﻟﻠﻤ ّﺮﺓ ﺍﻷﻭﻟﻰ(‪ .‬ﺃﻭﺟﺪ ﻋﻤﺮ ﺃﺣﻤﺪ ﻭﻋﻤﺮ ﺟ ّﺪﻩ ﺍﻵﻥ‪.‬‬ ‫)‪ (11‬ﻟﻨﻌﺘﺒﺮ ﻣﻌﺎﺩﻟﺔ ﺍﻟﻤﺴﺘﻘﻴﻢ ‪ ،(d) : y = - 1‬ﺃﻭﺟﺪ ﺑﻴﺎﻧﻴًّﺎ ﺍﻟﺤﻞ ﻟـ ‪ f(x) = y , f(x) 2 y , f(x) 1 y‬ﻓﻲ ﻛ ّﻞ ﻣﻦ‬ ‫ﺍﻟﺤﺎﻻﺕ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫‪(a) f (x) = 2x2 + 4x - 1‬‬ ‫‪(b) f (x) = x2 + 1‬‬ ‫‪(c) f (x) = - x2 + 4x - 1‬‬ ‫)‪ (12‬ﻟﻨﻌﺘﺒﺮ ﻣﻌﺎﺩﻟﺔ ﺍﻟﻤﺴﺘﻘﻴﻢ ‪ ،(d): y = 2‬ﺃﻭﺟﺪ ﺑﻴﺎﻧﻴًّﺎ ﺍﻟﺤﻞ ﻟـ ‪ f(x) $ y , f(x) 1 y‬ﻓﻲ ﻛ ّﻞ ﻣﻦ ﺍﻟﺤﺎﻟﺘﻴﻦ ﺍﻟﺘﺎﻟﻴﺘﻴﻦ‪:‬‬ ‫‪(a) f (x) = 3x2 + 2‬‬ ‫‪(b) f (x) = x2 - x - 2‬‬ ‫‪32‬‬

‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1 - 5‬ﻇﻠﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪:‬‬ ‫‪ab‬‬ ‫)‪ (1‬ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺔ ‪ ^x + 3h2 2 0‬ﻫﻲ ‪R‬‬ ‫‪ab‬‬ ‫‪x-1‬‬ ‫‪$0‬‬ ‫ﻫﻮ ﺣﻞ ﻟﻠﻤﺘﺒﺎﻳﻨﺔ‬ ‫‪^0,‬‬ ‫‪3h‬‬ ‫ﻛﻞ ‪ x‬ﻳﻨﺘﻤﻲ ﻟﻠﻔﺘﺮﺓ‬ ‫)‪(2‬‬ ‫‪ab‬‬ ‫‪x2 - x‬‬ ‫)‪ (3‬ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺔ ‪ ^x + 3h2 + 2 1 1‬ﻫﻲ ﺍﻟﻤﺠﻤﻮﻋﺔ ﺍﻟﺨﺎﻟﻴﺔ ‪φ‬‬ ‫‪ab‬‬ ‫ﻫﻲ ‪^-1,3h‬‬ ‫‪x+2‬‬ ‫‪$1‬‬ ‫ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺔ‬ ‫)‪(4‬‬ ‫‪ab‬‬ ‫‪x+1‬‬ ‫)‪ (5‬ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺔ ‪ (- x - 3)2 1 0‬ﻫﻲ }‪{3‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(6-13‬ﻇﻠﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻟﺔ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫ﻫﻲ‪:‬‬ ‫‪-3 (x + 1)ax +‬‬ ‫‪1‬‬ ‫‪k‬‬ ‫‪#‬‬ ‫‪2‬‬ ‫ﻟﻠﻤﺘﺒﺎﻳﻨﺔ‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﻤﻨﺎﻇﺮﺓ‬ ‫)‪(6‬‬ ‫‪3‬‬ ‫‪a‬‬ ‫‪-3x2 + 2x -‬‬ ‫‪5‬‬ ‫=‬ ‫‪0‬‬ ‫‪b‬‬ ‫‪x2‬‬ ‫‪+‬‬ ‫‪4‬‬ ‫‪x‬‬ ‫‪+‬‬ ‫‪1‬‬ ‫=‬ ‫‪0‬‬ ‫‪c -3x2 + 4x - 3 = 0 d -3x2 + 2x + 1 = 0‬‬ ‫‪3‬‬ ‫‪3‬‬ ‫)‪ (7‬ﺇﻥ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺔ ‪ (1 - 2x)(4 + 5x) 1 0‬ﻫﻲ‪:‬‬ ‫‪a‬‬ ‫‪`-‬‬ ‫‪4‬‬ ‫‪,‬‬ ‫‪1‬‬ ‫‪j‬‬ ‫‪b‬‬ ‫‪` - 3,‬‬ ‫‪-‬‬ ‫‪4‬‬ ‫‪j‬‬ ‫‪,‬‬ ‫`‬ ‫‪1‬‬ ‫‪,‬‬ ‫‪3j‬‬ ‫‪5‬‬ ‫‪2‬‬ ‫‪5‬‬ ‫‪2‬‬ ‫‪c‬‬ ‫‪` - 3,‬‬ ‫‪-‬‬ ‫‪1‬‬ ‫‪j‬‬ ‫‪,‬‬ ‫`‬ ‫‪4‬‬ ‫‪,‬‬ ‫‪3j‬‬ ‫‪d‬‬ ‫‪` - 3,‬‬ ‫‪-‬‬ ‫‪4‬‬ ‫‪j‬‬ ‫‪,‬‬ ‫‪`-‬‬ ‫‪1‬‬ ‫‪,‬‬ ‫‪3j‬‬ ‫‪2‬‬ ‫‪5‬‬ ‫‪5‬‬ ‫‪2‬‬ ‫ﻫﻲ‪:‬‬ ‫)‪(x2 + 1) (x - 3‬‬ ‫‪20‬‬ ‫ﺇﻥ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺔ‬ ‫)‪(8‬‬ ‫‪x-3‬‬ ‫‪aR‬‬ ‫*‪b R‬‬ ‫‪c R - \"3,‬‬ ‫‪d R - \"0, 3,‬‬ ‫)‪ (9‬ﺍﻟﻤﺘﺒﺎﻳﻨﺔ ﺍﻟﺘﻲ ﻣﺠﻤﻮﻋﺔ ﺣﻠﻬﺎ ]‪ [-2, 3‬ﻫﻲ‪:‬‬ ‫‪a x2 - x - 6 1 0‬‬ ‫‪b x2 - x - 6 # 0‬‬ ‫‪c x2 - x - 6 2 0‬‬ ‫‪d x2 - x - 6 $ 0‬‬ ‫)‪ (10‬ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﺘﺒﺎﻳﻨﺔ ‪ x2 + x 2 0‬ﻫﻲ‪:‬‬ ‫‪aR‬‬ ‫‪b ^0, 3h‬‬ ‫‪c R - \"0,‬‬ ‫ﻟﻴﺲ ﺃﻳًّﺎ ﻣﻤﺎ ﺳﺒﻖ ﺻﺤﻴ ًﺤﺎ ‪d‬‬ ‫ﻓﺈﻥ ﻗﻴﻢ ‪ x‬ﺍﻟﺘﻲ ﺗﺠﻌﻞ ‪ f‬ﻏﻴﺮ ﻣﻌ ّﺮﻓﺔ ﻫﻲ‪:‬‬ ‫)‪f (x‬‬ ‫=‬ ‫)‪x (x + 1‬‬ ‫ﺇﺫﺍ ﻛﺎﻧﺖ‬ ‫)‪(11‬‬ ‫)‪(2x - 3) (3x + 2‬‬ ‫‪a‬‬ ‫‪$‬‬ ‫‪2‬‬ ‫‪,‬‬ ‫‪-‬‬ ‫‪3‬‬ ‫‪.‬‬ ‫‪b‬‬ ‫‪$‬‬ ‫‪-2‬‬ ‫‪,‬‬ ‫‪3‬‬ ‫‪.‬‬ ‫‪c‬‬ ‫‪$‬‬ ‫‪2‬‬ ‫‪,‬‬ ‫‪3‬‬ ‫‪.‬‬ ‫‪d‬‬ ‫‪$‬‬ ‫‪-2‬‬ ‫‪,‬‬ ‫‪-‬‬ ‫‪3‬‬ ‫‪.‬‬ ‫‪3‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪2‬‬ ‫)‪ (12‬ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ x2 + x - 2 = 0‬ﻫﻲ‪:‬‬ ‫‪a \"1, - 2,‬‬ ‫‪b \"-1, 2,‬‬ ‫‪c \"-1,1,‬‬ ‫‪d \"-2, 2,‬‬ ‫ﻏﻴﺮ ﻣﻮﺟﺒﺔ ﻭﻻ ﺗﺴﺎﻭﻱ ﺍﻟﺼﻔﺮ ﻫﻲ‪:‬‬ ‫ﻓﺈﻥ ﻗﻴﻢ ‪ x‬ﺍﻟﺘﻲ ﺗﺠﻌﻞ )‪f(x‬‬ ‫)‪f (x‬‬ ‫‪=-‬‬ ‫‪3x2‬‬ ‫‪+‬‬ ‫‪x‬‬ ‫‪-‬‬ ‫‪1‬‬ ‫ﺇﺫﺍ ﻛﺎﻧﺖ‬ ‫)‪(13‬‬ ‫‪12‬‬ ‫‪a ^-3, 0h‬‬ ‫‪b ^0, 3h‬‬ ‫‪c‬‬ ‫‪$‬‬ ‫‪1‬‬ ‫‪.‬‬ ‫‪d‬‬ ‫‪R‬‬ ‫‪-‬‬ ‫‪$‬‬ ‫‪1‬‬ ‫‪.‬‬ ‫‪6‬‬ ‫‪6‬‬ ‫‪33‬‬

‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ ﺍﻟﺜﺎﻧﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺮﻳﻨﻴﻦ )‪ ،(1-2‬ﺃﻭﺟﺪ ﻣﺠﺎﻝ ﻛ ّﻞ ﻣﻦ ﺍﻟﺪﻭﺍﻝ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫= )‪(1) f(x‬‬ ‫‪9x2 - 4 + 2‬‬ ‫= )‪(2) g(x‬‬ ‫‪-x + 2 - 3‬‬ ‫‪2x - 3‬‬ ‫‪x2 - 4‬‬ ‫)‪ (3‬ﻳﺒﻴّﻦ ﺍﻟﺠﺪﻭﻝ ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﺭﺑﺢ ﺇﺣﺪﻯ ﺍﻟﺸﺮﻛﺎﺕ ‪ y‬ﺑﺂﻻﻑ ﺍﻟﺪﻧﺎﻧﻴﺮ ﻭﻋﺪﺩ ﺍﻟﻘﻄﻊ ﺍﻟﻤﻨﺘﺠﺔ ‪x‬‬ ‫‪x1 2 3 4 5‬‬ ‫ﺍﻛﺘﺐ ﺩﺍﻟﺔ ﺗﺮﺑﻴﻌﻴﺔ ﺗﻨﻤﺬﺝ ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ‪x, y‬‬ ‫‪y 0 -1 0 3 8‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺮﻳﻨﻴﻦ )‪ ،(4-5‬ﺍﺭﺳﻢ ﻛﻞ ﻣﺠﻤﻮﻋﺔ ﺑﻴﺎﻧﺎﺕ ﻣﻤﺎ ﻳﻠﻲ‪ ،‬ﺛﻢ ﺍﻛﺘﺐ ﻣﻌﺎﺩﻟﺔ ﻛ ّﻞ ﻣﻨﻬﺎ‪:‬‬ ‫‪(4) x -1 0 1 2 3 4‬‬ ‫‪(5) x -1 0 1 2 3‬‬ ‫‪f(x) -1 -3 -1 5 15 29‬‬ ‫‪f(x) -2 1 6 13 22‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺮﻳﻨﻴﻦ )‪ ،(6-7‬ﺍﺭﺳﻢ ﻣﻨﺤﻨﻰ ﺍﻟﻘﻄﻊ ﺍﻟﻤﻜﺎﻓﺊ ﺇﺫﺍ ﻋﺮﻓﺖ ﺇﺣﺪﺍﺛﻴﺎﺕ ﺍﻟﺮﺃﺱ ﻭﻧﻘﻄﺔ ﺇﺿﺎﻓﻴﺔ ﻳﻤﺮ ﺑﻬﺎ‪.‬‬ ‫)‪ (7‬ﺍﻟﺮﺃﺱ )‪A(2,11) , V(1,5‬‬ ‫)‪ (6‬ﺍﻟﺮﺃﺱ )‪A(- 3,3) , V(0,0‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(8-11‬ﺍﺭﺳﻢ ﻛﻞ ﺩﺍﻟﺔ ﺗﺮﺑﻴﻌﻴﺔ‪ .‬ﺛﻢ ﺣ ّﺪﺩ ﺇﺣﺪﺍﺛﻴﺎﺕ ﺍﻟﺮﺃﺱ‪.‬‬ ‫‪(8) f (x) = x2 - 7‬‬ ‫‪(9) f (x) = x2 + 2x + 6‬‬ ‫‪(10) f (x) = - x2 + 5x - 3‬‬ ‫)‪(11‬‬ ‫)‪f (x‬‬ ‫=‬ ‫‪-‬‬ ‫‪1‬‬ ‫‪x2‬‬ ‫‪-‬‬ ‫‪8‬‬ ‫‪2‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(12-15‬ﺃﻭﺟﺪ ﻣﻌﻜﻮﺱ ﻛﻞ ﺩﺍﻟﺔ ﻣﻤﺎ ﻳﻠﻲ‪:‬‬ ‫‪(12) y = 4x + 1‬‬ ‫)‪(13‬‬ ‫‪y‬‬ ‫=‬ ‫‪2‬‬ ‫‪x-6‬‬ ‫‪(14) y = x2 - 10‬‬ ‫‪(15) y = ^x + 2h2 - 3‬‬ ‫‪3‬‬ ‫)‪ (16‬ﺳﺆﺍﻝ ﻣﻔﺘﻮﺡ‪ :‬ﺍﻛﺘﺐ ﻣﻌﺎﺩﻟﺔ ﺩﺍﻟﺔ‪ ،‬ﺣﻴﺚ ﻣﻨﺤﻨﻰ ﻣﻌﻜﻮﺳﻬﺎ ﻫﻮ ﻗﻄﻊ ﻣﻜﺎﻓﺊ‪.‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(17-20‬ﺍﻛﺘﺐ ﻛﻞ ﺩﺍﻟﺔ ﺑﺪﻻﻟﺔ ﺇﺣﺪﺍﺛﻴﺎﺕ ﺍﻟﺮﺃﺱ‪ .‬ﺛﻢ ﺍﺭﺳﻢ ﻣﻨﺤﻨﻰ ﺍﻟﻘﻄﻊ ﺍﻟﻤﻜﺎﻓﺊ ﻭﺣ ّﺪﺩ ﺇﺣﺪﺍﺛﻴﺎﺕ ﺍﻟﺮﺃﺱ‪.‬‬ ‫‪(17) y = x2 - 6x + 5‬‬ ‫‪(18) y = - x2 + 8x - 10‬‬ ‫‪(19) y = 2x2 - 3x + 1‬‬ ‫)‪(20‬‬ ‫‪y‬‬ ‫‪=-‬‬ ‫‪1‬‬ ‫‪x2‬‬ ‫‪+‬‬ ‫‪4x‬‬ ‫‪-‬‬ ‫‪9‬‬ ‫‪2‬‬ ‫)‪ (21‬ﺃﻭﺟﺪ ﺃﻛﺒﺮ ﻣﺴﺎﺣﺔ ﻟﺤﺪﻳﻘﺔ ﻣﻜﻮﻧﺔ ﻣﻦ ﻣﺴﺘﻄﻴﻠﻴﻦ ﻟﻬﻤﺎ ﺿﻠﻊ ﻣﺸﺘﺮﻙ ﻭﻳﻤﻜﻦ ﺇﺣﺎﻃﺘﻬﻤﺎ ﺑﺸﺮﻳﻂ ﻃﻮﻟﻪ‬ ‫‪) .120 m‬ﺍﻧﻈﺮ ﺍﻟﺼﻮﺭﺓ ﺍﻟﻤﻘﺎﺑﻠﺔ(‪.‬‬ ‫‪x xx‬‬ ‫‪1‬‬ ‫‪^120‬‬ ‫‪-‬‬ ‫‪3xh‬‬ ‫‪2‬‬ ‫‪34‬‬

‫)‪ (22‬ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﻛﻞ ﻣﺘﺒﺎﻳﻨﺔ ﻣﻤﺎ ﻳﻠﻲ‪:‬‬ ‫‪(a) x2 - 8x + 15 # 0‬‬ ‫‪(b) -x2 + 7x - 120 1 0‬‬ ‫)‪(c‬‬ ‫‪3x - 4‬‬ ‫‪$ -1^x‬‬ ‫!‬ ‫‪2h‬‬ ‫‪x-2‬‬ ‫)‪ (a) (23‬ﺍﺭﺳﻢ ﻣﻨﺤﻨﻰ ﺍﻟﺪﺍﻟﺔ‪ ، f(x) = - x2 + 4x - 3 :‬ﻭﺍﻟﺨﻂ ﺍﻟﻤﺴﺘﻘﻴﻢ ‪ y = - 8‬ﻋﻠﻰ ﺷﺒﻜﺔ ﺇﺣﺪﺍﺛﻴﺎﺕ ﻭﺍﺣﺪﺓ‪.‬‬ ‫)‪ (b‬ﺍﺩﺭﺱ ﺑﻴﺎﻧﻴًّﺎ‪f(x) = - 8 , f(x) 1-8 , f(x) 2-8 :‬‬ ‫)‪ (c‬ﺗﺤ ّﻘﻖ ﺣﺴﺎﺑﻴًّﺎ ﻣﻦ ﺍﻟﻨﺘﺎﺋﺞ ﺍﻟﺘﻲ ﺣﺼﻠﺖ ﻋﻠﻴﻬﺎ ﻓﻲ ﺍﻟﻔﻘﺮﺓ )‪.(b‬‬ ‫‪35‬‬

‫ﺗﻤﺎﺭﻳﻦ ﺇﺛﺮﺍﺋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺮﻳﻨﻴﻦ )‪ ،(1-2‬ﺃﻭﺟﺪ ﻣﺠﺎﻝ ﻛ ّﻞ ﻣﻦ ﺍﻟﺪﻭﺍﻝ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫)‪(1‬‬ ‫)‪f (x‬‬ ‫=‬ ‫‪2x‬‬ ‫‪-‬‬ ‫‪x‬‬ ‫‪+‬‬ ‫‪2‬‬ ‫‪x2 + 1‬‬ ‫= )‪(2) f(x‬‬ ‫‪x2 - 4x + 4‬‬ ‫‪x+1‬‬ ‫‪2+x‬‬ ‫‪9 - x2‬‬ ‫‪x2 + 7 - 4‬‬ ‫)‪ (3‬ﻓﻲ ﺇﺣﺪﻯ ﻣﺒﺎﺭﻳﺎﺕ ﻛﺮﺓ ﺍﻟﻘﺪﻡ‪ ،‬ﺗﻮﺍﺟﺪ ﺃﺣﺪ ﺍﻟﻼﻋﺒﻴﻦ ﻣﻨﻔﺮ ًﺩﺍ ﻭﺟ ًﻬﺎ ﻟﻮﺟﻪ ﻣﻊ ﺣﺎﺭﺱ ﻣﺮﻣﻰ ﺍﻟﻔﺮﻳﻖ ﺍﻟﻤﻨﺎﻓﺲ‬ ‫ﻓﻘﺮﺭ ﺭﻓﻊ ﺍﻟﻜﺮﺓ ﻓﻮﻕ ﺍﻟﺤﺎﺭﺱ ﺁﻣ ًﻼ ﺃﻻ ﺗﻌﻠﻮ ﻣﺮﻣﻰ ﺍﻟﻔﺮﻳﻖ ﺍﻟﻤﻨﺎﻓﺲ‪ ،‬ﻭﻛﺎﻥ ﻫﺬﺍ ﺍﻟﻼﻋﺐ ﻋﻠﻰ ﺑﻌﺪ ‪ 16 m‬ﻣﻦ‬ ‫ﺧﻂ ﺍﻟﻤﺮﻣﻰ‪ ،‬ﺑﻴﻨﻤﺎ ﺍﻟﺤﺎﺭﺱ ﻳﻘﻒ ﻋﻠﻰ ﺑﻌﺪ ‪ 7 m‬ﻣﻦ ﺍﻟﻼﻋﺐ‪ .‬ﻳﻨﻤﺬﺝ ﻣﺴﺎﺭ ﺍﻟﻜﺮﺓ ﺍﻟﻤﻨﻄﻠﻘﺔ ﻣﻦ ﺍﻷﺭﺽ ﻋﺒﺮ‬ ‫ﺗﺴﺪﻳﺪﺓ ﺍﻟﻼﻋﺐ ﻋﻠﻰ ﺷﻜﻞ ﻗﻄﻊ ﻣﻜﺎﻓﺊ ﻣﻌﺎﺩﻟﺘﻪ‪y = a^x - 10h2 + 3 :‬‬ ‫)‪ (a‬ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ a‬ﻣﻌﺘﺒ ًﺮﺍ ﻧﻘﻄﺔ ﺍﻧﻄﻼﻕ ﺗﺴﺪﻳﺪﺓ ﺍﻟﻼﻋﺐ ﻫﻲ ﻧﻘﻄﺔ ﺍﻷﺻﻞ‪.‬‬ ‫)‪ (b‬ﻋﻠ ًﻤﺎ ﺃﻥ ﺍﻟﺤﺎﺭﺱ ﻋﻨﺪ ﺍﺳﺘﺨﺪﺍﻡ ﻳﺪﻳﻪ ﻳﺼﻞ ﺇﻟﻰ ﺍﺭﺗﻔﺎﻉ ‪ 2.53 m‬ﻭﺃﻥ ﺍﺭﺗﻔﺎﻉ ﺍﻟﻤﺮﻣﻰ ﻫﻮ ‪2.44 m‬‬ ‫ﻓﻬﻞ ﺳﺘﺘﺨﻄﻰ ﺍﻟﻜﺮﺓ ﺍﻟﺤﺎﺭﺱ؟ ﻭﻫﻞ ﺳﻴﺴﺠﻞ ﺍﻟﻼﻋﺐ ﻫﺪﻓًﺎ؟‬ ‫)‪ (4‬ﻓﻲ ﺇﺣﺪﻯ ﺩﻭﺭﺍﺕ ﻛﺮﺓ ﺍﻟﻤﻀﺮﺏ‪ ،‬ﺗﻮﺍﺟﺪ ﺃﺣﺪ ﺍﻟﻼﻋﺒﻴﻦ ﻋﻠﻰ ﺑﻌﺪ ‪ 3 m‬ﻣﻦ ﺍﻟﺸﺒﻜﺔ‪ ،‬ﻓﻘﺮﺭ ﺍﻟﻼﻋﺐ ﺍﻟﺜﺎﻧﻲ‬ ‫ﺍﻟﻤﺘﻮﺍﺟﺪ ﻋﻠﻰ ﺍﻟﺨﻂ ﺍﻟﺨﻠﻔﻲ ﻣﻦ ﺍﻟﻤﻠﻌﺐ ﺭﻓﻊ ﺍﻟﻜﺮﺓ ﻓﻮﻕ ﻣﻨﺎﻓﺴﻪ ﻋﻠﻰ ﺃﻥ ﺗﺄﺗﻲ ﺍﻟﻜﺮﺓ ﺩﺍﺧﻞ ﻣﻠﻌﺐ ﻣﻨﺎﻓﺴﻪ‪.‬‬ ‫ﻋﻠ ًﻤﺎ ﺃﻥ ﻃﻮﻝ ﻣﻠﻌﺐ ﻛﺮﺓ ﺍﻟﻤﻀﺮﺏ ‪ 23.8 m‬ﺗﺘﻮﺳﻄﻪ ﺍﻟﺸﺒﻜﺔ ﺍﻟﺘﻲ ﺗﻘﺴﻢ ﺍﻟﻤﻠﻌﺐ ﺇﻟﻰ ﻗﺴﻤﻴﻦ ﻣﺘﺴﺎﻭﻳﻴﻦ‪.‬‬ ‫)‪ (a‬ﺇﺫﺍ ﺍﻋﺘﺒﺮﻧﺎ ﺃﻥ ﻣﺴﺎﺭ ﺍﻟﻜﺮﺓ ﻣﻦ ﻣﻀﺮﺏ ﺍﻟﻼﻋﺐ ﻋﻠﻰ ﺍﺭﺗﻔﺎﻉ ‪ 1 m‬ﻋﻠﻰ ﺷﻜﻞ ﻗﻄﻊ ﻣﻜﺎﻓﺊ ﻣﻌﺎﺩﻟﺘﻪ‪:‬‬ ‫‪ y = - 0.08^x - 9h2 + k‬ﻓﻤﺎ ﻗﻴﻤﺔ ‪k‬؟‬ ‫)‪ (b‬ﻣﺎ ﺍﻻﺭﺗﻔﺎﻉ ﺍﻷﻗﺼﻰ ﻟﻠﻜﺮﺓ ﻋﻦ ﺃﺭﺽ ﺍﻟﻤﻠﻌﺐ؟‬ ‫)‪ (c‬ﻫﻞ ﺳﺘﺘﺨﻄﻰ ﺍﻟﻜﺮﺓ ﺍﻟﻼﻋﺐ ﺍﻟﻤﻨﺎﻓﺲ ﺇﺫﺍ ﻛﺎﻥ ﺃﻗﺼﻰ ﺍﺭﺗﻔﺎﻉ ﻳﻤﻜﻦ ﺍﻟﻮﺻﻮﻝ ﺇﻟﻴﻪ ﺑﺎﺳﺘﺨﺪﺍﻡ ﻣﻀﺮﺑﻪ ﻫﻮ‬ ‫‪3.3 m‬؟‬ ‫)‪ (d‬ﻫﻞ ﺳﺘﺴﻘﻂ ﺍﻟﻜﺮﺓ ﺩﺍﺧﻞ ﻣﻠﻌﺐ ﺍﻟﻼﻋﺐ ﺍﻟﻤﻨﺎﻓﺲ؟ ﺇﺫﺍ ﻛﺎﻧﺖ ﺇﺟﺎﺑﺘﻚ ﻧﻌﻢ‪ ،‬ﺃﻭﺟﺪ ﺑﻌﺪﻫﺎ ﻋﻦ ﺧﻂ ﺍﻟﻤﻠﻌﺐ‪.‬‬ ‫)‪ (a) (5‬ﺍﺭﺳﻢ ﺑﻴﺎﻧﻴًّﺎ ﻣﻨﺤﻨﻰ ﺍﻟﺪﺍﻟﺔ‪y = x2 - 4x :‬‬ ‫)‪ (b‬ﺃﻭﺟﺪ ﻣﻌﻜﻮﺱ ﺍﻟﺪﺍﻟﺔ‪ ،‬ﺛﻢ ﺍﺭﺳﻤﻪ ﻋﻠﻰ ﺍﻟﻤﺴﺘﻮﻯ ﺍﻹﺣﺪﺍﺛﻲ ﻧﻔﺴﻪ‪.‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(6-10‬ﺣ ّﻞ ﻛ ًّﻼ ﻣﻦ ﺍﻟﻤﺘﺒﺎﻳﻨﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫)‪(6) (x - 3) (x + 2) 2 (x - 3) (2x - 1‬‬ ‫)‪(7) 4x2 - 9 # (3 - 2x) (x + 1‬‬ ‫‪(8) x2 (x - 3) 2 0‬‬ ‫‪(9) (x - 6) 2 (x - 5) 2 0‬‬ ‫)‪(10‬‬ ‫‪3x - 1‬‬ ‫‪$‬‬ ‫‪0‬‬ ‫‪(2x - 7) 2‬‬ ‫‪36‬‬

‫)‪ (a) (11‬ﺃﻛﻤﻞ ﺍﻟﺠﺪﻭﻟﻴﻦ ﺍﻟﺘﺎﻟﻴﻴﻦ‪ .‬ﺍﻛﺘﺐ ﻓﻲ ﺍﻟﺼﻒ ﺍﻷﺧﻴﺮ ﻣﻦ ﻛﻞ ﻣﻨﻬﻤﺎ ﺍﻟﻔﺮﻕ ﺑﻴﻦ ﻗﻴﻢ ‪ y‬ﺍﻟﻤﺘﺘﺎﻟﻴﺔ‪.‬‬ ‫ﺟﺪﻭﻝ )‪(2‬‬ ‫ﺟﺪﻭﻝ )‪(1‬‬ ‫‪543210‬‬ ‫‪x‬‬ ‫‪543210‬‬ ‫‪x‬‬ ‫‪50 32 18 8 2 0‬‬ ‫‪10 8 6 4 2 0‬‬ ‫‪y = 2x2‬‬ ‫‪y = 2x‬‬ ‫‪62‬‬ ‫‪22‬‬ ‫ﺍﻟﻔﺮﻕ‬ ‫ﺍﻟﻔﺮﻕ‬ ‫)‪ (b‬ﺃﻱ ﻣﻦ ﺍﻟﺪﺍﻟﺘﻴﻦ ﺩﺍﻟﺔ ﺗﺮﺑﻴﻌﻴﺔ؟‬ ‫)‪ (c‬ﺃﻱ ﻧﻤﻂ ﺗﺮﺍﻩ ﻓﻲ ﺍﻟﺼﻒ ﺍﻷﺧﻴﺮ ﻣﻦ ﺍﻟﺠﺪﻭﻝ )‪(1‬؟ ﻭﻣﻦ ﺍﻟﺠﺪﻭﻝ )‪(2‬؟‬ ‫)‪ (d‬ﻛ ّﻮﻥ ﺟﺪﻭ ًﻻ ﻟﻜ ّﻞ ﻣﻦ ﺍﻟﺪﺍﻟﺘﻴﻦ‪ y = - x + 4 , y = - x2 + 4 :‬ﻣﺴﺘﺨﺪ ًﻣﺎ ﻗﻴﻢ ‪ x‬ﻧﻔﺴﻬﺎ ﻓﻲ ﺍﻟﻔﻘﺮﺓ )‪.(a‬‬ ‫ﻫﻞ ﺗﺮﻯ ﺍﻷﻧﻤﺎﻁ ﻧﻔﺴﻬﺎ ﻛﻤﺎ ﻓﻲ ﺍﻟﻔﻘﺮﺓ )‪(c‬؟‬ ‫)‪ (e‬ﻛﻴﻒ ﺗﺴﺎﻋﺪﻙ ﻗﻴﻢ ‪ y‬ﻟﻤﺠﻤﻮﻋﺔ ﺍﻟﺒﻴﺎﻧﺎﺕ ﻓﻲ ﺗﻮﻗﻊ ﻣﺎ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺪﺍﻟﺔ ﺍﻟﺨﻄﻴﺔ ﺃﻭ ﺍﻟﺪﺍﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ ﻫﻲ‬ ‫ﺍﻟﻨﻤﻮﺫﺝ ﺍﻷﻓﻀﻞ؟‬ ‫)‪ (12‬ﻳﺒﻴّﻦ ﺍﻟﺠﺪﻭﻝ ﺍﻟﺘﺎﻟﻲ ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ﻋﻤﻖ ﺍﻟﻤﻴﺎﻩ ﻓﻲ ﺍﻟﻤﺤﻴﻂ ‪ y‬ﺑﺎﻷﻣﺘﺎﺭ )‪ (m‬ﻭﺳﺮﻋﺔ ﺍﻟﺘﺴﻮﻧﺎﻣﻲ ‪) x‬ﻣﺘﺮ ﻓﻲ‬ ‫ﺍﻟﺜﺎﻧﻴﺔ ‪.(m/s‬‬ ‫‪x 52 58 61 65 71 76 82 98‬‬ ‫‪y 270.40 336.40 372.10 422.50 504.10 577.60 672.40 960.40‬‬ ‫ﺍﺳﺘﺨﺪﻡ ﺍﻟﺒﻴﺎﻧﺎﺕ ﺍﻟﻤﺪﻭﻧﺔ ﻓﻲ ﺍﻟﺠﺪﻭﻝ ﻹﻳﺠﺎﺩ ﻣﻌﺎﺩﻟﺔ ﺗﺮﺑﻴﻌﻴﺔ ﺗﻨﻤﺬﺝ ﺍﻟﻌﻼﻗﺔ ﺑﻴﻦ ‪ x, y‬ﺛﻢ ﺗﺤﻘﻖ‪.‬‬ ‫)ﺍﺳﺘﺨﺪﺍﻡ ﺍﻵﻟﺔ ﺍﻟﺤﺎﺳﺒﺔ(‬ ‫‪37‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫‪3-1‬‬ ‫ﺩﻭﺍﻝ ﺍﻟﻘﻮﻯ ﻭﻣﻌﻜﻮﺳﺎﺗﻬﺎ‬ ‫‪Power Functions and their Inverses‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-4‬ﺍﻷﺷﻜﺎﻝ ﺍﻟﺘﺎﻟﻴﺔ ﺗﻤﺜﻞ ﺩﻭﺍﻝ‪ .‬ﺻﻒ ﺗﻤﺎﺛﻞ ﻛﻞ ﺩﺍﻟﺔ ﺛﻢ ﻭ ّﺿﺢ ﻫﻞ ﻫﻲ ﺯﻭﺟﻴﺔ ﺃﻡ ﻓﺮﺩﻳﺔ ﺃﻡ ﻟﻴﺴﺖ ﺯﻭﺟﻴﺔ ﻭﻟﻴﺴﺖ‬ ‫ﻓﺮﺩﻳﺔ‪.‬‬ ‫‪(1) y = - x2 + 1 6x ! R‬‬ ‫‪(2) y = 3 x 6x ! R‬‬ ‫‪y‬‬ ‫‪y‬‬ ‫‪1‬‬ ‫‪1‬‬ ‫‪-1‬‬ ‫‪12‬‬ ‫‪x‬‬ ‫‪-1‬‬ ‫‪1‬‬ ‫‪x‬‬ ‫‪-1‬‬ ‫‪-1‬‬ ‫‪(3) y = x2 - 2x + 2 6x ! 6-1, 3h‬‬ ‫)‪(4‬‬ ‫=‪y‬‬ ‫‪x‬‬ ‫‪6x ! R/\"1,‬‬ ‫‪x-1‬‬ ‫‪y‬‬ ‫‪y‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪x‬‬ ‫‪1‬‬ ‫‪1‬‬ ‫‪-2 -1 1 2‬‬ ‫‪-2 --11 1 2‬‬ ‫‪-1‬‬ ‫‪-2‬‬ ‫‪x‬‬ ‫‪38‬‬

‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(5-9‬ﺍﺫﻛﺮ ﻣﺎ ﺇﺫﺍ ﻛﺎﻧﺖ ﻛﻞ ﻣﻦ ﺍﻟﺪﻭﺍﻝ ﺍﻟﺘﺎﻟﻴﺔ ﻓﺮﺩﻳﺔ ﺃﻡ ﺯﻭﺟﻴﺔ ﺃﻡ ﻟﻴﺴﺖ ﻓﺮﺩﻳﺔ ﻭﻟﻴﺴﺖ ﺯﻭﺟﻴﺔ‪.‬‬ ‫‪(5) y = x3‬‬ ‫‪(6) y = (x - 1) 3 + 2‬‬ ‫‪(7) y = x4‬‬ ‫‪(8) y = - x4 + 3‬‬ ‫‪(9) y = - 4 x‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(10-15‬ﺃﻭﺟﺪ ﻣﻌﻜﻮﺱ ﻛﻞ ﺩﺍﻟﺔ ﻣﻤﺎ ﻳﻠﻲ‪:‬‬ ‫)‪(10‬‬ ‫‪y‬‬ ‫=‬ ‫‪1‬‬ ‫‪x3‬‬ ‫‪(11) y = 24 x‬‬ ‫)‪(12‬‬ ‫‪y‬‬ ‫=‬ ‫‪1‬‬ ‫‪x4‬‬ ‫‪3‬‬ ‫‪(14) y = 3 x - 1‬‬ ‫‪3‬‬ ‫)‪(13‬‬ ‫=‪y‬‬ ‫‪13‬‬ ‫‪x‬‬ ‫‪(15) y = ^x + 2h4 - 3‬‬ ‫‪3‬‬ ‫)‪ (a) (16‬ﺍﻟﻌﻼﻗﺔ‪ ،M = 0 . 008p3 :‬ﻭﺯﻥ ﺑﻄﻴﺨﺔ »‪ «M‬ﺑﺎﻟﺠﺮﺍﻡ ﺣﻴﺚ ﻣﺤﻴﻄﻬﺎ »‪ «p‬ﺑﺎﻟﺴﻨﺘﻴﻤﺘﺮ )‪.(cm‬‬ ‫ﻗ ّﺪﺭ ﻭﺯﻥ ﺑﻄﻴﺨﺔ ﻣﺤﻴﻄﻬﺎ ‪80 cm‬‬ ‫)‪ (b‬ﻣﻦ ﺍﻟﻌﻼﻗﺔ‪ M = 0 . 008p3 :‬ﺍﻛﺘﺐ ‪ p‬ﺑﺪﻻﻟﺔ ‪.M‬‬ ‫)‪ (c‬ﺃﻭﺟﺪ ﻣﺤﻴﻂ ﺍﻟﺒﻄﻴﺨﺔ ﺍﻟﺘﻲ ﻭﺯﻧﻬﺎ ‪3.250 kg‬‬ ‫)‪ (17‬ﺍﻟﺴﺆﺍﻝ ﺍﻟﻤﻔﺘﻮﺡ‪ :‬ﺍﻛﺘﺐ ﺩﺍﻟﺔ ﻗﻮﻯ ﻳﻘﻊ ﺭﺳﻤﻬﺎ ﺍﻟﺒﻴﺎﻧﻲ ﻓﻲ ﺍﻟﺮﺑﻊ ﺍﻟﺜﺎﻧﻲ ﻭﺍﻟﺮﺑﻊ ﺍﻟﺮﺍﺑﻊ‪.‬‬ ‫)‪ (18‬ﻋﻨﺪﻣﺎ ﺗﺪﻭﺭ ﺩﺍﺋﺮﺓ ﺣﻮﻝ ﺧﻂ ﻣﺜﻞ ﺍﻟﺨﻂ ﺍﻟﻤﻮﺿﺢ ﻓﻲ ﺍﻟﺸﻜﻞ ﺃﺩﻧﺎﻩ‪ ،‬ﻓﺈﻥ ﺍﻟﺴﻄﺢ ﺍﻟﻨﺎﺗﺞ ﻳﺴﻤﻰ ﻧﺘﻮﺀًﺍ ﻣﺴﺘﺪﻳ ًﺮﺍ‬ ‫ﺑﺎﻟﻌﻼﻗﺔ‪:‬‬ ‫ﺣﺠﻤﻪ‬ ‫ﻭﻳﻌﻄﻰ‬ ‫‪(torus‬‬ ‫‪or‬‬ ‫)‪donut‬‬ ‫‪R2‬‬ ‫‪V‬‬ ‫=‬ ‫‪2π2‬‬ ‫‪R1‬‬ ‫‪R‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪R1‬‬ ‫‪V‬‬ ‫=‬ ‫‪6π2‬‬ ‫‪R‬‬ ‫‪3‬‬ ‫ﺃﻥ‪:‬‬ ‫ﺗﺤ ّﻘﻖ‬ ‫‪، R1‬‬ ‫=‬ ‫‪3R2‬‬ ‫ﺃﻥ‪:‬‬ ‫ﺍﻓﺮﺽ‬ ‫)‪(a‬‬ ‫‪2‬‬ ‫)‪ (b‬ﺃﻭﺟﺪ ‪ V‬ﺇﺫﺍ ‪ ،R1 = 3R2‬ﺣﻴﺚ ‪ .R2 = 1 . 27 cm‬ﻗ ّﺮﺏ ﺍﻟﻨﺎﺗﺞ ﺇﻟﻰ ﺃﻗﺮﺏ ﺟﺰﺀ ﻣﻦ ‪10‬‬ ‫‪11‬‬ ‫)‪ (19‬ﻭ ّﺿﺢ ﻛﻴﻒ ﺃﻥ ﺍﻟﻤﻘﺪﺍﺭ ‪ ^- 64h2‬ﻻ ﻳﻤﺜّﻞ ﻋﺪ ًﺩﺍ ﺣﻘﻴﻘﻴًّﺎ‪ ،‬ﻓﻲ ﺣﻴﻦ ﺃﻥ ﺍﻟﻤﻘﺪﺍﺭ ‪ ^- 64h3‬ﻳﻤﺜّﻞ ﻋﺪ ًﺩﺍ ﺣﻘﻴﻘﻴًّﺎ‪.‬‬ ‫)‪ (20‬ﺍﻟﺘﻔﻜﻴﺮ ﺍﻟﻨﺎﻗﺪ‪ :‬ﺻﻒ ﺑﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ ‪ f(x) = axn‬ﺑﺤﺴﺐ ﺍﻟﺸﺮﻭﻁ ﺍﻟﻤﻮﺿﻮﻋﺔ ﻋﻠﻰ ‪.a , n‬‬ ‫)‪ n (b‬ﻋﺪﺩ ﺻﺤﻴﺢ ﺯﻭﺟﻲ‪a 1 0 ،‬‬ ‫)‪ n (a‬ﻋﺪﺩ ﺻﺤﻴﺢ ﺯﻭﺟﻲ‪a 2 0 ،‬‬ ‫)‪ n (d‬ﻋﺪﺩ ﺻﺤﻴﺢ ﻓﺮﺩﻱ‪a 1 0 ،‬‬ ‫)‪ n (c‬ﻋﺪﺩ ﺻﺤﻴﺢ ﻓﺮﺩﻱ‪a 2 0 ،‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-5‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ‪ ،‬ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ab‬‬ ‫)‪ y = x4 (1‬ﺩﺍﻟﺔ ﻗﻮﻯ‬ ‫‪ab‬‬ ‫)‪ f :6-3,3@ $ R , f(x) = x5 (2‬ﺩﺍﻟﺔ ﻓﺮﺩﻳﺔ‬ ‫‪ab‬‬ ‫)‪ y = x x (3‬ﺩﺍﻟﺔ ﺯﻭﺟﻴﺔ‬ ‫‪ab‬‬ ‫)‪ y = ^x + 4h2 (4‬ﺩﺍﻟﺔ ﺯﻭﺟﻴﺔ‬ ‫‪39‬‬

‫‪ab‬‬ ‫)‪ (5‬ﺍﻟﻤﺴﺘﻘﻴﻢ ﺍﻟﺬﻱ ﻣﻌﺎﺩﻟﺘﻪ ‪ y = x‬ﻫﻮ ﺧﻂ ﺗﻨﺎﻇﺮ ﺑﻴﻦ ﺍﻟﻨﻘﺎﻁ ﺍﻟﺘﻲ ﺗﻤﺜﻞ‬ ‫ﺍﻟﻌﻼﻗﺔ ‪ r‬ﻭﺍﻟﻨﻘﺎﻁ ﺍﻟﺘﻲ ﺗﻤﺜﻞ ﻣﻌﻜﻮﺳﻬﺎ‪.‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(6-10‬ﻇﻠّﻞ ﺩﺍﺋﺮﺓ ﺍﻟﺮﻣﺰ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫)‪ (6‬ﻣﻌﻜﻮﺱ ﺩﺍﻟﺔ ﺍﻟﻘﻮﻯ ‪ y = 0.2x4‬ﻫﻮ‪:‬‬ ‫‪a‬‬ ‫‪y=4‬‬ ‫‪x‬‬ ‫‪b‬‬ ‫‪y =!4‬‬ ‫‪x‬‬ ‫‪c‬‬ ‫‪y =!4‬‬ ‫‪x‬‬ ‫‪d y = - 4 5x‬‬ ‫‪0.2‬‬ ‫‪0.2‬‬ ‫‪2‬‬ ‫)‪ (7‬ﺃﻱ ﻣﻤﺎ ﻳﻠﻲ ﺗﻤﺜﻞ ﺩﺍﻟﺔ ﺯﻭﺟﻴﺔ‪.‬‬ ‫‪ay‬‬ ‫‪by‬‬ ‫‪cy‬‬ ‫‪dy‬‬ ‫‪xx‬‬ ‫‪xx‬‬ ‫)‪ (8‬ﺍﻟﺪﺍﻟﺔ ‪ y = 4.9t2‬ﺩﺍﻟﺔ ﺯﻭﺟﻴﺔ ﺇﺫﺍ ﻛﺎﻥ ﻣﺠﺎﻟﻬﺎ‪:‬‬ ‫‪a 6-4, 4h‬‬ ‫‪b 6-4, 2h‬‬ ‫@‪c 6-2, 2‬‬ ‫‪d 60,3h‬‬ ‫‪aR‬‬ ‫ﻫﻮ‪:‬‬ ‫‪f -1‬‬ ‫ﻓﺈﻥ ﻣﺠﺎﻝ‬ ‫= )‪f : 6-4, 4@ $ R , f (x‬‬ ‫‪x3‬‬ ‫ﺇﺫﺍ ﻛﺎﻧﺖ‬ ‫)‪(9‬‬ ‫‪64‬‬ ‫‪y‬‬ ‫‪1‬‬ ‫‪b R+‬‬ ‫@‪c 6-4, 4‬‬ ‫@‪d 6-1, 1‬‬ ‫‪x‬‬ ‫)‪ (10‬ﻟﻴﻜﻦ ﺑﻴﺎﻥ ‪ f -1‬ﻛﻤﺎ ﻫﻮ ﻣﻮﺿﺢ ﻓﻲ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ‪ .‬ﺑﻴﺎﻥ ‪ f‬ﻳﻤﻜﻦ ﺃﻥ ﻳﻜﻮﻥ‪:‬‬ ‫‪ay‬‬ ‫‪b‬‬ ‫‪yc‬‬ ‫‪yd‬‬ ‫‪y‬‬ ‫‪1‬‬ ‫‪x‬‬ ‫‪1x‬‬ ‫‪1x‬‬ ‫‪1‬‬ ‫‪x‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺮﻳﻨﻴﻦ )‪ ،(11-12‬ﻟﺪﻳﻚ ﻗﺎﺋﻤﺘﺎﻥ ﺍﺧﺘﺮ ﻣﻦ ﺍﻟﻘﺎﺋﻤﺔ )‪ (2‬ﻣﺎ ﻳﻨﺎﺳﺐ ﺍﻟﺴﺆﺍﻝ ﻓﻲ ﺍﻟﻘﺎﺋﻤﺔ )‪ (1‬ﻟﺘﺤﺼﻞ ﻋﻠﻰ ﺇﺟﺎﺑﺔ ﺻﺤﻴﺤﺔ‪.‬‬ ‫ﺍﻟﻘﺎﺋﻤﺔ )‪(2‬‬ ‫ﺍﻟﻘﺎﺋﻤﺔ )‪(1‬‬ ‫ﺍﻟﻤﺴﺘﻘﻴﻢ ﺍﻟﺬﻱ ﻣﻌﺎﺩﻟﺘﻪ ‪x = 0‬‬ ‫‪a‬‬ ‫)‪ (11‬ﺑﻴﺎﻥ ﺩﺍﻟﺔ ﺯﻭﺟﻴﺔ ﻣﺘﻤﺎﺛﻞ ﺣﻮﻝ‪:‬‬ ‫ﺍﻟﻤﺴﺘﻘﻴﻢ ﺍﻟﺬﻱ ﻣﻌﺎﺩﻟﺘﻪ ‪y = 0‬‬ ‫‪b‬‬ ‫)‪ (12‬ﺑﻴﺎﻥ ﺩﺍﻟﺔ ﻓﺮﺩﻳﺔ ﻣﺘﻤﺎﺛﻞ ﺣﻮﻝ‪:‬‬ ‫ﺍﻟﻤﺴﺘﻘﻴﻢ ﺍﻟﺬﻱ ﻣﻌﺎﺩﻟﺘﻪ ‪y = x‬‬ ‫‪c‬‬ ‫‪d‬‬ ‫ﻧﻘﻄﺔ ﺍﻷﺻﻞ‬ ‫‪40‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺍﻟﺪﻭﺍﻝ ﺍﻟﺤﺪﻭﺩﻳﺔ ‪3-2‬‬ ‫‪Polynomial Functions‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-9‬ﺍﻛﺘﺐ ﻛﻞ ﻛﺜﻴﺮﺓ ﺣﺪﻭﺩ ﻣﻤﺎ ﻳﻠﻲ ﺑﺎﻟﺼﻮﺭﺓ ﺍﻟﻌﺎﻣﺔ ﺛﻢ ﺻﻨﻔﻬﺎ ﺗﺒ ًﻌﺎ ﻟﻠﺪﺭﺟﺔ ﻭﻋﺪﺩ ﺍﻟﺤﺪﻭﺩ‪.‬‬ ‫‪(1) ^2x2 + 9h - ^3x2 - 7h‬‬ ‫‪(2) ^7x2 + 8x - 5h + ^9x2 - 9xh‬‬ ‫‪(3) ^7x3 + 9x2 + 8x + 11h - ^5x3 - 13x - 16h‬‬ ‫‪(4) ^30x3 - 49x2 + 7xh + ^50x3 - 75x - 60x2h‬‬ ‫)‪(5‬‬ ‫‪3x5 + 4x‬‬ ‫‪6‬‬ ‫‪(6) 5x2 ^6x - 2h‬‬ ‫‪(7) ^x2 + 1h2‬‬ ‫‪(8) ^2c - 3h^2c + 4h^2c - 1h‬‬ ‫‪(9) ^w - 1h4‬‬ ‫)‪ (10‬ﺗﺼﻤﻴﻢ ﺍﻟﻌﺒﻮﺍﺕ‪ :‬ﺍﻟﺸﻜﻞ ﺃﺩﻧﺎﻩ ﻳﻮ ّﺿﺢ ﺯﺟﺎﺟﺔ ﻋﻄﺮ ﺗﺘﻜ ّﻮﻥ ﻣﻦ ﻗﺎﻋﺪﺓ ﺃﺳﻄﻮﺍﻧﻴﺔ ﻭﻏﻄﺎﺀ ﻧﺼﻒ ﻛﺮﻭ ّﻱ‪.‬‬ ‫)‪ (a‬ﺍﻛﺘﺐ ﻣﻘﺪﺍ ًﺭﺍ ﻳﻌﺒّﺮ ﻋﻦ ﺣﺠﻢ ﺍﻷﺳﻄﻮﺍﻧﺔ‪.‬‬ ‫‪h = 10 cm‬‬ ‫)‪ (b‬ﺍﻛﺘﺐ ﻣﻘﺪﺍ ًﺭﺍ ﻳﻌﺒّﺮ ﻋﻦ ﺣﺠﻢ ﺍﻟﻐﻄﺎﺀ ﻧﺼﻒ ﺍﻟﻜﺮﻭ ّﻱ‪.‬‬ ‫)‪ (c‬ﺍﻛﺘﺐ ﻛﺜﻴﺮﺓ ﺣﺪﻭﺩ ﺗﻤﺜّﻞ ﺍﻟﺤﺠﻢ ﺍﻟﻜﻠ ّﻲ‪.‬‬ ‫‪R‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ (11-15‬ﻋﻴّﻦ ﺳﻠﻮﻙ ﺍﻟﻨﻬﺎﻳﺔ ﻟﺒﻴﺎﻥ ﻛﻞ ﺩﺍﻟﺔ‪.‬‬ ‫‪(11) y = 3x + 2‬‬ ‫‪(12) f (x) = - x2 + x‬‬ ‫)‪(13‬‬ ‫)‪f (x‬‬ ‫=‬ ‫‪1‬‬ ‫‪x‬‬ ‫‪4‬‬ ‫‪-‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪(14) y = - 4x4 + 5x5‬‬ ‫)‪(15‬‬ ‫)‪f (x‬‬ ‫=‬ ‫‪-‬‬ ‫‪1‬‬ ‫‪x3‬‬ ‫‪-‬‬ ‫‪4x2‬‬ ‫‪+‬‬ ‫‪x‬‬ ‫‪-‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪41‬‬

‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-4‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫)‪ (1‬ﻛﺜﻴﺮﺓ ﺍﻟﺤﺪﻭﺩ‪ f(x) = ax3 + (a + 2)x2 + 5 , 6a ! R ،‬ﻫﻲ ﻣﻦ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻟﺜﺔ‪a b .‬‬ ‫‪ab‬‬ ‫)‪ (2‬ﺍﻟﻤﻌﺎﻣﻞ ﺍﻟﺮﺋﻴﺴﻲ ﻟﻜﺜﻴﺮﺓ ﺍﻟﺤﺪﻭﺩ ‪ f(x) = 2x5 - 3x3 ^1 - x2h‬ﻫﻮ ‪2‬‬ ‫‪ab‬‬ ‫)‪ (3‬ﻛﺜﻴﺮﺓ ﺍﻟﺤﺪﻭﺩ ‪ ^1 - x2h3 ^x + 1h‬ﻫﻲ ﻣﻦ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺴﺎﺑﻌﺔ‪.‬‬ ‫‪ab‬‬ ‫)‪ (4‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺪﺍﻟﺔ ﺍﻟﺤﺪﻭﺩﻳﺔ ﻣﻦ ﺍﻟﺪﺭﺟﺔ ‪ n‬ﻓﺈﻥ ﻟﻬﺎ ‪ n‬ﺣ ًّﺪﺍ‪.‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(5-7‬ﻇﻠّﻞ ﺩﺍﺋﺮﺓ ﺍﻟﺮﻣﺰ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫)‪ ^x + 1h3 (5‬ﻳﺴﺎﻭﻱ‪:‬‬ ‫‪a x3 + 1‬‬ ‫‪b ^x + 1h^x2 + x + 1h‬‬ ‫‪c x3 + 3x2 + 3x + 1‬‬ ‫‪d x3 + x2 + x + 1‬‬ ‫‪a ^x4 - 2x2 + 3h - ^x4 - x2 - 9h‬‬ ‫)‪ (6‬ﺃﻱ ﻣﻤﺎ ﻳﻠﻲ ﻳﺴﺎﻭﻱ ‪2x4 - 3x + 6‬؟‬ ‫‪c ^3x4 - x + 3h + ^3 - 2x - x4h‬‬ ‫‪b 2x4 - 3^x + 6h‬‬ ‫‪d x^2x3 - 3xh + 6‬‬ ‫‪y‬‬ ‫)‪ (7‬ﺳﻠﻮﻙ ﻧﻬﺎﻳﺔ ﺍﻟﺪﺍﻟﺔ ‪ x‬ﻫﻮ‪:‬‬ ‫‪a ^6 , 3h‬‬ ‫‪b ^5 , 4h‬‬ ‫‪c ^5 , 3h‬‬ ‫‪d ^6 , 4h‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ (8-11‬ﻟﺪﻳﻚ ﻗﺎﺋﻤﺘﺎﻥ‪ ،‬ﺍﺧﺘﺮ ﻣﻦ ﺍﻟﻘﺎﺋﻤﺔ )‪ (2‬ﻣﺎ ﻳﻨﺎﺳﺐ ﻛﻞ ﺗﻤﺮﻳﻦ ﻓﻲ ﻣﻦ ﺍﻟﻘﺎﺋﻤﺔ )‪ (1‬ﻟﺘﺤﺼﻞ ﻋﻠﻰ ﺇﺟﺎﺑﺔ ﺻﺤﻴﺤﺔ‪.‬‬ ‫ﺍﻟﻘﺎﺋﻤﺔ )‪(2‬‬ ‫ﺍﻟﻘﺎﺋﻤﺔ )‪(1‬‬ ‫ﺳﻠﻮﻙ ﻧﻬﺎﻳﺔ ﺍﻟﺪﺍﻟﺔ‪:‬‬ ‫‪a ^6 , 3h‬‬ ‫‪b ^5 , 4h‬‬ ‫)‪f (x) = x4 - 2x5 (8‬‬ ‫‪c ^5 , 3h‬‬ ‫)‪g (x) = 2x + x3 + 5 (9‬‬ ‫‪d ^6 , 4h‬‬ ‫‪a ^6 , 3h‬‬ ‫ﺳﻠﻮﻙ ﻧﻬﺎﻳﺔ ﺍﻟﺪﺍﻟﺔ‪:‬‬ ‫‪b ^5 , 4h‬‬ ‫)‪f (x) = - x6 + 7x (10‬‬ ‫‪c ^5 , 3h‬‬ ‫‪d ^6 , 4h‬‬ ‫)‪g (x‬‬ ‫=‬ ‫‪1‬‬ ‫‪x‬‬ ‫‪4‬‬ ‫‪-‬‬ ‫‪2‬‬ ‫)‪(11‬‬ ‫‪2‬‬ ‫‪42‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺍﻟﻌﻮﺍﻣﻞ ﺍﻟﺨﻄﻴﺔ ﻟﻜﺜﻴﺮﺍﺕ ﺍﻟﺤﺪﻭﺩ ‪3-3‬‬ ‫‪Linear Factors of Polynomials‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-3‬ﺍﻛﺘﺐ ﻛﻞ ﺩﺍﻟﺔ ﻛﺜﻴﺮﺓ ﺣﺪﻭﺩ ﻓﻲ ﺍﻟﺼﻮﺭﺓ ﺍﻟﻌﺎﻣﺔ ﻭﺍﺫﻛﺮ ﺩﺭﺟﺘﻬﺎ‪.‬‬ ‫‪(1) y = ^x + 3h^x + 4h^x + 5h (2) y = ^x - 3h2 ^x - 1h‬‬ ‫‪(3) y = x^x - 1h^x + 1h‬‬ ‫)‪ (4‬ﺍﻟﻬﻨﺪﺳﺔ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ﻃﻮﻝ ﺻﻨﺪﻭﻕ ‪ 2x + 1‬ﻣﻦ ﺍﻟﻮﺣﺪﺍﺕ‪ ،‬ﻭﻋﺮﺿﻪ ‪ x + 4‬ﻣﻦ ﺍﻟﻮﺣﺪﺍﺕ‪ ،‬ﻭﺍﺭﺗﻔﺎﻋﻪ ‪ x + 3‬ﻣﻦ‬ ‫ﺍﻟﻮﺣﺪﺍﺕ‪ ،‬ﻭﻗﺪ ﻛﻮﻧﺘﻪ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻜﺘﻞ ﺍﻟﺨﺸﺒﻴﺔ ‪ ،x ،x2 ،x3‬ﺍﻟﻮﺣﺪﺓ )‪.(1‬‬ ‫‪1 x x2‬‬ ‫‪x3‬‬ ‫‪2x + 1‬‬ ‫ﻓﺈﻟﻰ ﻛﻢ ﻛﺘﻠﺔ ﺗﺤﺘﺎﺝ ﻣﻦ ﻛﻞ ﻣﻨﻬﺎ؟‬ ‫‪x+3‬‬ ‫‪x+4‬‬ ‫)‪ (5‬ﺍﻟﻬﻨﺪﺳﺔ‪ :‬ﺻﻨﺪﻭﻕ ﻋﻠﻰ ﺷﻜﻞ ﺷﺒﻪ ﻣﻜﻌﺐ ﻃﻮﻟﻪ‪ 2x + 3 :‬ﻣﻦ ﺍﻟﻮﺣﺪﺍﺕ‪ ،‬ﻋﺮﺿﻪ ‪ 2x - 3‬ﻣﻦ ﺍﻟﻮﺣﺪﺍﺕ‪ ،‬ﺍﺭﺗﻔﺎﻋﻪ‬ ‫‪ 3x‬ﻣﻦ ﺍﻟﻮﺣﺪﺍﺕ‪ .‬ﻋﺒّﺮ ﻋﻦ ﺣﺠﻢ ﺍﻟﺼﻨﺪﻭﻕ ﻓﻲ ﺻﻮﺭﺓ ﻛﺜﻴﺮﺓ ﺣﺪﻭﺩ‪.‬‬ ‫‪(6) y = ^x - 1h^x + 2h‬‬ ‫‪(7) y = ^x + 3h3‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(6-8‬ﻋﻴّﻦ ﺃﺻﻔﺎﺭ ﻛﻞ ﺩﺍﻟﺔ ﻭﺗﻜﺮﺍﺭﻫﺎ‪.‬‬ ‫‪(8) y = x^x - 2h2 ^x + 9h‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(9-12‬ﺃﻭﺟﺪ ﺃﺻﻔﺎﺭ ﻛﻞ ﺩﺍﻟﺔ ﻣﻤﺎ ﻳﻠﻲ ﺛﻢ ﺍﺭﺳﻢ ﺑﻴﺎﻧًﺎ ﺗﻘﺮﻳﺒﻴًّﺎ ﻟﻜﻞ ﻣﻨﻬﺎ ﻣﺮﺍﻋ ًﻴﺎ ﺳﻠﻮﻙ ﺍﻟﻨﻬﺎﻳﺔ ﻟﺒﻴﺎﻥ ﻛﻞ ﺩﺍﻟﺔ‪.‬‬ ‫‪(9) y = ^x - 2h^x + 2h‬‬ ‫‪(10) y = ^x + 1h^x - 2h^x - 3h‬‬ ‫‪(11) y = x^x + 2h2‬‬ ‫‪(12) y = ^x + 1h2 ^x - 2h^x - 1h‬‬ ‫)‪ (13‬ﺍﻟﺘﻔﻜﻴﺮ ﺍﻟﻨﺎﻗﺪ‪ :‬ﻛﻴﻒ ﺗﻌﺮﻑ ﻧﻘﺎﻁ ﺗﻘﺎﻃﻊ ﺍﻟﺮﺳﻢ ﺍﻟﺒﻴﺎﻧﻲ ﻟﺪﺍﻟﺔ ﻛﺜﻴﺮﺓ ﺍﻟﺤﺪﻭﺩ ﻣﻊ ﻣﺤﻮﺭ ﺍﻟﺼﺎﺩﺍﺕ ﺩﻭﻥ ﺭﺳﻤﻬﺎ‬ ‫ﺑﻴﺎﻧﻴًّﺎ؟‬ ‫)‪ (14‬ﺍﻟﻬﻨﺪﺳﺔ ﺍﻟﺘﺤﻠﻴﻠﻴﺔ‪ :‬ﻳﻮﺿﺢ ﺍﻟﺸﻜﻞ ﺃﺩﻧﺎﻩ ﻣﻨﻄﻘﺔ ﻣﺴﺘﻄﻴﻠﺔ ﺍﻟﺸﻜﻞ‪ ،‬ﺃﺣﺪ ﺃﺭﻛﺎﻧﻬﺎ ﻳﻘﻊ ﻋﻠﻰ ﺍﻟﺮﺳﻢ ﺍﻟﺒﻴﺎﻧﻲ ﻟﻠﺪﺍﻟﺔ‪:‬‬ ‫‪y y = - x2 + 2x + 4‬‬ ‫)‪ (a‬ﺍﻛﺘﺐ ﻣﺴﺎﺣﺔ ﺍﻟﻤﻨﻄﻘﺔ ﺍﻟﻤﺴﺘﻄﻴﻠﺔ )‪ (A‬ﻛﺪﺍﻟﺔ ﻛﺜﻴﺮﺓ ﺣﺪﻭﺩ ﻓﻲ ﺍﻟﺼﻮﺭﺓ ﺍﻟﻌﺎﻣﺔ‪(A) (x, y) .‬‬ ‫‪x‬‬ ‫‪x‬‬ ‫=‬ ‫‪2‬‬ ‫‪1‬‬ ‫ﺃﻭﺟﺪ ﻣﺴﺎﺣﺔ ﺍﻟﻤﻨﻄﻘﺔ ﺍﻟﻤﺴﺘﻄﻴﻠﺔ ﺇﺫﺍ ﻛﺎﻧﺖ‬ ‫)‪(b‬‬ ‫‪2‬‬ ‫)‪ (15‬ﺍﻟﺴﺆﺍﻝ ﺍﻟﻤﻔﺘﻮﺡ‪ :‬ﺍﻛﺘﺐ ﺩﺍﻟﺔ ﻛﺜﻴﺮﺓ ﺣﺪﻭﺩ ﻟﻬﺎ ﺍﻟﻤﻤﻴﺰﺍﺕ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫ﺛﻼﺛﺔ ﺃﺻﻔﺎﺭ ﻣﺨﺘﻠﻔﺔ‪ ،‬ﺃﺣﺪ ﺃﺻﻔﺎﺭﻫﺎ ﻫﻮ ﺍﻟﻌﺪﺩ ‪ ،1‬ﻭﺻﻔﺮ ﺁﺧﺮ ﻣﻦ ﺃﺻﻔﺎﺭﻫﺎ ﻣﻜﺮﺭ ﻣﺮﺗﻴﻦ‪.‬‬ ‫‪43‬‬

‫‪2x + 1‬‬ ‫)‪ (16‬ﺍﻟﺼﻨﺎﻋﺎﺕ ﺍﻟﺨﺸﺒﻴﺔ‪ :‬ﺑﺪﺃ ﻧﺠﺎﺭ ﻋﻤﻠﻪ ﺑﺎﺳﺘﺨﺪﺍﻡ ﻛﺘﻠﺔ ﺧﺸﺒﻴﺔ ﻛﺎﻟﻤﻮﺿﺤﺔ ﻓﻲ ﺍﻟﺸﻜﻞ‪.‬‬ ‫‪x+3‬‬ ‫‪x+2‬‬ ‫)‪ (a‬ﻋﺒّﺮ ﻋﻦ ﺣﺠﻢ ﺍﻟﻜﺘﻠﺔ ﺍﻟﺨﺸﺒﻴﺔ ﺍﻷﺻﻠﻴﺔ ﻭﺣﺠﻢ ﺍﻟﺘﺠﻮﻳﻒ ﻓﻲ ﺷﻜﻞ‬ ‫ﻛﺜﻴﺮﺗﻲ ﺣﺪﻭﺩ ﻓﻲ ﺍﻟﺼﻮﺭﺓ ﺍﻟﻌﺎﻣﺔ‪.‬‬ ‫‪x+1‬‬ ‫)‪ (b‬ﺍﻛﺘﺐ ﻛﺜﻴﺮﺓ ﺣﺪﻭﺩ ﻟﺤﺠﻢ ﺍﻟﺨﺸﺐ ﺍﻟﻤﺘﺒﻘﻲ‪.‬‬ ‫‪x+4‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(17-20‬ﺍﻛﺘﺐ ﺩﺍﻟﺔ ﻛﺜﻴﺮﺓ ﺍﻟﺤﺪﻭﺩ ﻓﻲ ﺍﻟﺼﻮﺭﺓ ﺍﻟﻌﺎﻣﺔ ﻣﺴﺘﺨﺪ ًﻣﺎ ﺍﻷﺻﻔﺎﺭ ﺍﻟﻤﻌﻄﺎﺓ‪:‬‬ ‫‪(17) 1, - 1‬‬ ‫‪(18) 0, 1, 2‬‬ ‫‪(19) -4 , - 1, 3‬‬ ‫)‪(20‬‬ ‫‪1‬‬ ‫‪,‬‬ ‫‪-1‬‬ ‫‪,2‬‬ ‫ﻣﺮﺗﻴﻦ(‬ ‫)ﻣﻜﺮﺭ‬ ‫‪2‬‬ ‫‪2‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-5‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ab‬‬ ‫‪f‬‬ ‫‪a‬‬ ‫‪3‬‬ ‫‪k‬‬ ‫=‬ ‫‪0‬‬ ‫ﺇﺫﺍ ﻛﺎﻧﺖ ‪ f‬ﺗﻘﺒﻞ ﺍﻟﻘﺴﻤﺔ ﻋﻠﻰ ‪ ^2x + 3h‬ﻓﺈﻥ‬ ‫)‪(1‬‬ ‫‪ab‬‬ ‫‪2‬‬ ‫)‪ (2‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ^x + 2h‬ﻋﺎﻣﻞ ﻣﻦ ﻋﻮﺍﻣﻞ ﺍﻟﺤﺪﻭﺩﻳﺔ ‪ g‬ﻓﺈﻥ ‪g^-2h = 0‬‬ ‫‪ab‬‬ ‫)‪ (3‬ﺇﺫﺍ ﻗﺒﻠﺖ ‪ f(x) = x4 - 2x2 + k + 1‬ﺍﻟﻘﺴﻤﺔ ﻋﻠﻰ ‪ x‬ﻓﺈﻥ ‪k = - 1‬‬ ‫‪ab‬‬ ‫)‪ (4‬ﺑﺎﻗﻲ ﻗﺴﻤﺔ ﺣﺪﻭﺩﻳﺔ ﻣﻦ ﺍﻟﺪﺭﺟﺔ ‪ n‬ﻋﻠﻰ ﺣﺪﻭﺩﻳﺔ ﻣﻦ ﺍﻟﺪﺭﺟﺔ‬ ‫ﺍﻷﻭﻟﻰ ﻫﻮ ﻋﺪﺩ ﺛﺎﺑﺖ‪.‬‬ ‫‪ab‬‬ ‫)‪ ^x + 1h (5‬ﻋﺎﻣﻞ ﻣﻦ ﻋﻮﺍﻣﻞ ﺍﻟﺤﺪﻭﺩﻳﺔ‪p(x) = x3 - x2 - 2x :‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(6-13‬ﻇﻠّﻞ ﺩﺍﺋﺮﺓ ﺍﻟﺮﻣﺰ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫)‪ (6‬ﺇﺫﺍ ﻛﺎﻥ ‪ x = - 2a‬ﺻﻔﺮ ﻣﻦ ﺃﺻﻔﺎﺭ ﻛﺜﻴﺮﺓ ﺣﺪﻭﺩ ﻓﺈﻥ ﺃﺣﺪ ﻋﻮﺍﻣﻠﻬﺎ ﻫﻮ‪:‬‬ ‫‪a ^x - 2ah‬‬ ‫‪b ^2x + ah‬‬ ‫‪c ^2x - ah‬‬ ‫‪d ^x + 2ah‬‬ ‫)‪ (7‬ﺃﻱ ﻣﻦ ﺍﻟﻤﻘﺎﺩﻳﺮ ﺍﻟﺘﺎﻟﻴﺔ ﺇﺫﺍ ﺿﺮﺏ ﻓﻲ ‪ ^x - 1h‬ﻳﺼﺒﺢ ﺍﻟﻨﺎﺗﺞ ﻛﺜﻴﺮﺓ ﺣﺪﻭﺩ ﺗﻜﻌﻴﺒﻴﺔ ﺛﻼﺛﻴﺔ‪:‬‬ ‫‪a ^x - 1h2‬‬ ‫‪b x2 - x‬‬ ‫‪c x2 - 1‬‬ ‫‪d x2 + 1‬‬ ‫‪y‬‬ ‫)‪ (8‬ﻟﻴﻜﻦ ﺑﻴﺎﻥ ‪ f‬ﻛﻤﺎ ﻓﻲ ﺍﻟﺸﻜﻞ ﺍﻟﻤﺮﺳﻮﻡ ﻓﺈﻥ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ f(x) = 0‬ﻫﻲ‪:‬‬ ‫‪2‬‬ ‫‪1‬‬ ‫‪a \"-1, 2, 3,‬‬ ‫‪b \"1, - 2, - 3,‬‬ ‫‪-1 1 2 3 x‬‬ ‫‪c \"-1, 0, 2, 3,‬‬ ‫‪d \"0,‬‬ ‫‪a 4x2 - 9‬‬ ‫)‪ (9‬ﺷﺒﻪ ﻣﻜﻌﺐ ﺃﺑﻌﺎﺩﻩ ‪ 2x + 3, 2x - 3, 3x‬ﻓﺘﻜﻮﻥ ﺩﺍﻟﺔ ﺍﻟﺤﺠﻢ )‪ f(x‬ﺗﺴﺎﻭﻱ‪:‬‬ ‫‪b 3x^4x2 + 9h‬‬ ‫‪c 12x2 - 9x‬‬ ‫‪d 12x3 - 27x‬‬ ‫)‪ (10‬ﻗﻴﻤﺔ ‪ k‬ﺍﻟﺘﻲ ﺗﺠﻌﻞ ‪ ^x - 1h‬ﻋﺎﻣ ًﻼ ﻣﻦ ﻋﻮﺍﻣﻞ ‪ f(x) = ^x2 + x - 2h + 2k‬ﻫﻲ‪:‬‬ ‫‪a1‬‬ ‫‪b2‬‬ ‫‪c0‬‬ ‫‪d‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪44‬‬

‫)‪ f(x) = x3 - x (11‬ﺗﻘﺒﻞ ﺍﻟﻘﺴﻤﺔ ﻋﻠﻰ ‪ x - k‬ﺇﺫﺍ ﻛﺎﻥ ‪ k‬ﻳﻨﺘﻤﻲ ﺇﻟﻰ ﺍﻟﻤﺠﻤﻮﻋﺔ‪:‬‬ ‫‪a \"0,‬‬ ‫‪b \"-1,‬‬ ‫‪c \"1,‬‬ ‫‪d \"0, - 1,1,‬‬ ‫‪ x = 2 b‬ﺻﻔﺮ ﻣﻜﺮﺭ ﻣﻦ ﺃﺻﻔﺎﺭ ﺍﻟﺪﺍﻟﺔ ‪f‬‬ ‫)‪ (12‬ﺇﺫﺍ ﻛﺎﻧﺖ )‪ f(x‬ﺗﻘﺒﻞ ﺍﻟﻘﺴﻤﺔ ﻋﻠﻰ ‪ ^x - 2h2‬ﻓﺈﻥ‪:‬‬ ‫‪ x = - 2 d‬ﺻﻔﺮ ﻣﻜﺮﺭ ﻣﻦ ﺃﺻﻔﺎﺭ ﺍﻟﺪﺍﻟﺔ ‪f‬‬ ‫‪ x = 2 a‬ﺻﻔﺮ ﻣﻦ ﺃﺻﻔﺎﺭ ﺍﻟﺪﺍﻟﺔ ‪f‬‬ ‫‪ x = - 2 c‬ﺻﻔﺮ ﻣﻦ ﺃﺻﻔﺎﺭ ﺍﻟﺪﺍﻟﺔ ‪f‬‬ ‫‪a f (x) = x2 + m‬‬ ‫)‪ x + m (13‬ﻋﺎﻣﻞ ﻣﻦ ﻋﻮﺍﻣﻞ‪:‬‬ ‫‪c f (x) = x3 + mx2‬‬ ‫‪b f (x) = x3 + mx‬‬ ‫‪d f (x) = x2 + m2‬‬ ‫‪45‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﻗﺴﻤﺔ ﻛﺜﻴﺮﺍﺕ ﺍﻟﺤﺪﻭﺩ ‪3-4‬‬ ‫‪Dividing Polynomials‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫‪(1) ^x2 - 3x - 40h ' ^x + 5h‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-4‬ﺍﻗﺴﻢ ﻣﺴﺘﺨﺪ ًﻣﺎ ﻗﺴﻤﺔ ﻛﺜﻴﺮﺓ ﺍﻟﺤﺪﻭﺩ ﺍﻟﻤﻄﻮﻟﺔ‪.‬‬ ‫‪(3) ^x3 - 13x - 12h ' ^x - 4h‬‬ ‫)‪(2) ^x3 + 3x2 - x + 2h ' (x - 1‬‬ ‫)‪(4) (9x3 - 18x2 - x + 2) ' (3x + 1‬‬ ‫‪(5) x - 3‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺮﻳﻨﻴﻦ )‪ ،(5-6‬ﺑﻴّﻦ ﻣﺎ ﺇﺫﺍ ﻛﺎﻧﺖ ﻛﻞ ﺛﻨﺎﺋﻴﺔ ﺣﺪ ﻋﺎﻣ ًﻼ ﻣﻦ ﻋﻮﺍﻣﻞ ‪x3 + 4x2 + x - 6‬‬ ‫‪(6) x + 2‬‬ ‫)‪(7) ^x3 + 3x2 - x - 3h ' (x - 1‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(7-11‬ﺍﻗﺴﻢ ﻣﺴﺘﺨﺪ ًﻣﺎ ﺍﻟﻘﺴﻤﺔ ﺍﻟﺘﺮﻛﻴﺒﻴﺔ‪.‬‬ ‫)‪(9) ^2x4 + 6x3 + 5x2 - 45h ' (x + 3‬‬ ‫)‪(8) ^-2x3 + 5x2 - x + 2h ' (x + 2‬‬ ‫‪(11) ^2x3 + 4x2 - 10x - 9h ' ^x - 3h‬‬ ‫)‪(10) ^x3 - 3x2 - 5x - 25h ' (x - 5‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺮﻳﻨﻴﻦ )‪ ،(12-13‬ﺍﺳﺘﺨﺪﻡ ﺍﻟﻘﺴﻤﺔ ﺍﻟﺘﺮﻛﻴﺒﻴﺔ ﻭﺍﻟﻌﺎﻣﻞ ﺍﻟﻤﻌﻄﻰ ﻟﺘﺤﻠﻴﻞ ﻛﻞ ﺩﺍﻟﺔ ﻛﺜﻴﺮﺓ ﺣﺪﻭﺩ ﺑﺎﻟﻜﺎﻣﻞ‪.‬‬ ‫‪(12) y = x3 + 2x2 - 5x - 6 ; x + 1‬‬ ‫‪(13) y = x3 - 4x2 - 9x + 36 ; x + 3‬‬ ‫)‪ (14‬ﺍﻟﻬﻨﺪﺳﺔ‪ :‬ﻳﻌﻄﻰ ﺣﺠﻢ ﺻﻨﺪﻭﻕ ﺑﺎﻟﻤﻌﺎﺩﻟﺔ‪ V(x) = x3 + x2 - 6x :‬ﺑﺎﻷﻣﺘﺎﺭ ﺍﻟﻤﻜﻌﺒﺔ ‪x 2 2 : ^m3h‬‬ ‫ﻣﺎ ﺍﻷﺑﻌﺎﺩ ﺍﻟﻤﻤﻜﻨﺔ ﻟﻬﺬﺍ ﺍﻟﺼﻨﺪﻭﻕ؟‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(15-18‬ﺍﺳﺘﺨﺪﻡ ﺍﻟﻘﺴﻤﺔ ﺍﻟﺘﺮﻛﻴﺒﻴﺔ ﻭﻧﻈﺮﻳﺔ ﺍﻟﺒﺎﻗﻲ ﻹﻳﺠﺎﺩ )‪f(a‬‬ ‫‪(15) f (x) = x3 + 4x2 - 8x - 6 ; a = - 2‬‬ ‫‪(16) f (x) = x3 - 7x2 + 15x - 9 ; a = 3‬‬ ‫)‪(17‬‬ ‫)‪f (x‬‬ ‫=‬ ‫‪2x3 - x2 + 10x + 5‬‬ ‫;‬ ‫‪a‬‬ ‫=‬ ‫‪1‬‬ ‫‪(18) f (x) = 2x4 + 6x3 + 5x2 - 45 ; a = - 3‬‬ ‫‪2‬‬ ‫)‪ (a) (19‬ﺍﻟﺘﻔﻜﻴﺮ ﺍﻟﻤﻨﻄﻘﻲ‪ :‬ﻛﺜﻴﺮﺓ ﺣﺪﻭﺩ )‪ f(x‬ﻗﺴﻤﺖ ﻋﻠﻰ ﺛﻨﺎﺋﻴﺔ ﺍﻟﺤ ّﺪ ‪ ^x - ah‬ﻭﺍﻟﺒﺎﻗﻲ ﺻﻔﺮ‪.‬‬ ‫ﻣﺎﺫﺍ ﻳﻤﻜﻨﻚ ﺃﻥ ﺗﺴﺘﻨﺘﺞ؟ ﻓ ّﺴﺮ‪.‬‬ ‫)‪ (b‬ﺗﻔﻜﻴﺮ ﻧﺎﻗﺪ‪ :‬ﻭ ّﺿﺢ ﻟﻤﺎﺫﺍ ‪ x2 + 1‬ﻻ ﻳﻤﻜﻦ ﺗﺤﻠﻴﻠﻬﺎ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺃﻋﺪﺍﺩ ﺣﻘﻴﻘﻴﺔ؟‬ ‫)‪ (c‬ﺍﻛﺘﺸﺎﻑ ﺍﻟﺨﻄﺄ‪ :‬ﺣﻠّﻞ ﻃﺎﻟﺐ ﻛﺜﻴﺮﺓ ﺍﻟﺤﺪﻭﺩ‪ x3 - x2 - 2x :‬ﺇﻟﻰ ﺛﻼﺛﺔ ﻋﻮﺍﻣﻞ‪ ،‬ﻭﻛﺎﻥ ‪ ^x - 1h‬ﺃﺣﺪ ﻫﺬﻩ‬ ‫ﺍﻟﻌﻮﺍﻣﻞ‪ .‬ﺍﺳﺘﺨﺪﻡ ﺍﻟﻘﺴﻤﺔ ﻟﺘﺜﺒﺖ ﺃﻥ ﺍﻟﻄﺎﻟﺐ ﺍﺭﺗﻜﺐ ﺧﻄﺄ‪.‬‬ ‫‪46‬‬

‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(20-22‬ﺍﻗﺴﻢ ﻣﺎ ﻳﻠﻲ‪:‬‬ ‫‪(20) ^2x3 + 9x2 + 14x + 5h ' ^2x + 1h‬‬ ‫‪(21) ^x5 + 1h ' ^x + 1h‬‬ ‫)‪(22) ^3x4 - 5x3 + 2x2 + 3x - 2h ' (3x - 2‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(23-25‬ﺍﻗﺴﻢ ﺛ ّﻢ ﺃﻭﺟﺪ ﻧﻤﻄًﺎ ﻓﻲ ﺍﻹﺟﺎﺑﺎﺕ‪.‬‬ ‫‪(23) ^x2 - 1h ' ^x - 1h‬‬ ‫‪(24) ^x3 - 1h ' ^x - 1h‬‬ ‫‪(25) ^x4 - 1h ' ^x - 1h‬‬ ‫)‪ (26‬ﻣﺴﺘﺨﺪ ًﻣﺎ ﺍﻷﻧﻤﺎﻁ‪ ،‬ﺍﻗﺴﻢ ‪^x5 - 1h ' ^x - 1h‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(27-29‬ﺍﻗﺴﻢ ﺛ ّﻢ ﺃﻭﺟﺪ ﻧﻤﻄًﺎ ﻓﻲ ﺍﻹﺟﺎﺑﺎﺕ‪.‬‬ ‫‪(27) ^x3 + 1h ' ^x + 1h‬‬ ‫‪(28) ^x5 + 1h ' ^x + 1h‬‬ ‫‪(29) ^x7 + 1h ' ^x + 1h‬‬ ‫)‪ (30‬ﻣﺴﺘﺨﺪ ًﻣﺎ ﺍﻷﻧﻤﺎﻁ‪ ،‬ﺃﻭﺟﺪ ‪^x9 + 1h ' ^x + 1h‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-5‬ﻇﻠّﻞ ﺍﻟﺪﺍﺋﺮﺓ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻹﺟﺎﺑﺔ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ab‬‬ ‫)‪ (1‬ﺇﺫﺍ ﻛﺎﻥ ﺑﺎﻗﻲ ﻗﺴﻤﺔ ﻛﺜﻴﺮﺓ ﺍﻟﺤﺪﻭﺩ )‪ f(x‬ﻋﻠﻰ ‪ ^x + αh‬ﻳﺴﺎﻭﻱ ﺻﻔ ًﺮﺍ ﻓﺈﻥ ‪α‬‬ ‫‪ab‬‬ ‫ﻋﺎﻣﻞ ﻣﻦ ﻋﻮﺍﻣﻞ ‪f‬‬ ‫‪ab‬‬ ‫)‪ (2‬ﺍﻟﺪﺍﻟﺔ ‪ f(x) = ^x - 2h2 - 1‬ﺗﻘﺒﻞ ﺍﻟﻘﺴﻤﺔ ﻋﻠﻰ )‪(x - 1‬‬ ‫)‪ (3‬ﺑﺎﻗﻲ ﻗﺴﻤﺔ )‪ (x3 + a3‬ﻋﻠﻰ )‪ (x - a‬ﻫﻮ ‪2a3‬‬ ‫‪ab‬‬ ‫)‪ (4‬ﻧﺎﺗﺞ ﻗﺴﻤﺔ ﺣﺪﻭﺩﻳﺔ ﻣﻦ ﺍﻟﺪﺭﺟﺔ ‪ n‬ﺣﻴﺚ ‪ n H 2‬ﻋﻠﻰ ﺣﺪﻭﺩﻳﺔ ﻣﻦ‬ ‫ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ ﺗﻜﻮﻥ ﺣﺪﻭﺩﻳﺔ ﻣﻦ ﺍﻟﺪﺭﺟﺔ ‪^n - 2h‬‬ ‫‪ab‬‬ ‫)‪ (5‬ﻧﺎﺗﺞ ﻗﺴﻤﺔ ﺣﺪﻭﺩﻳﺔ ﻣﻦ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺴﺎﺩﺳﺔ ﻋﻠﻰ ﺣﺪﻭﺩﻳﺔ ﻣﻦ‬ ‫ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻟﺜﺔ ﺗﻜﻮﻥ ﺣﺪﻭﺩﻳﺔ ﻣﻦ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ‪.‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ ﻣﻦ )‪ ،(6-11‬ﻇﻠّﻞ ﺩﺍﺋﺮﺓ ﺍﻟﺮﻣﺰ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫)‪ (6‬ﺑﺎﻗﻲ ﻗﺴﻤﺔ )‪ f(x‬ﻋﻠﻰ ‪ g(x) = x - k‬ﻫﻮ‪:‬‬ ‫)‪a g(k‬‬ ‫)‪b f(k‬‬ ‫)‪c f (- k‬‬ ‫‪d -k‬‬ ‫)‪ (7‬ﺑﺎﻗﻲ ﻗﺴﻤﺔ )‪ (x4 + 2‬ﻋﻠﻰ )‪ (x - 3‬ﻫﻮ‪:‬‬ ‫‪a3‬‬ ‫‪b 27‬‬ ‫‪c 81‬‬ ‫‪d 83‬‬ ‫‪47‬‬

‫)‪ (8‬ﻧﺎﺗﺞ ﻗﺴﻤﺔ )‪ (2x4 - 8x2‬ﻋﻠﻰ )‪ (x + 2‬ﻳﺴﺎﻭﻱ‪:‬‬ ‫‪a 2x3 - 4x2‬‬ ‫‪b 2x3 - 8x2‬‬ ‫‪c x3 - 4x2‬‬ ‫‪d 2x3 - 4x2 + 2x‬‬ ‫)‪ (9‬ﺇﺫﺍ ﻛﺎﻥ ‪ 0‬ﻫﻮ ﺑﺎﻗﻲ ﻗﺴﻤﺔ ‪ f(x) = 2x3 - 4x2 + kx - 1‬ﻋﻠﻰ )‪ (x + 1‬ﻓﺈﻥ ‪ k‬ﺗﺴﺎﻭﻱ‪:‬‬ ‫‪a7‬‬ ‫‪b -7‬‬ ‫‪c -3‬‬ ‫‪d3‬‬ ‫)‪ (10‬ﺇﺫﺍ ﻛﺎﻥ ﺑﺎﻗﻲ ﻗﺴﻤﺔ ‪ f(x) = x4 - kx2 + x - k‬ﻋﻠﻰ )‪ (x - 1‬ﻫﻮ ‪ 3‬ﻓﺈﻥ ‪ k‬ﺗﺴﺎﻭﻱ‪:‬‬ ‫‪a‬‬ ‫‪1‬‬ ‫‪b3‬‬ ‫‪c‬‬ ‫‪-‬‬ ‫‪1‬‬ ‫‪d‬‬ ‫‪5‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫)‪ (11‬ﺇﺫﺍ ﻛﺎﻥ ‪ f(- 1) = f(0) = f(3) = - 2‬ﻓﺈﻥ )‪ f(x‬ﻳﻤﻜﻦ ﺃﻥ ﺗﻜﻮﻥ‪:‬‬ ‫‪a x3 - x2 + 3x - 2‬‬ ‫‪b x3 - 2x2 - 3x‬‬ ‫‪c 2x3 - 2x2 - 3x - 2‬‬ ‫‪d 2x3 - 4x2 - 6x - 2‬‬ ‫‪48‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺣﻞ ﻣﻌﺎﺩﻻﺕ ﻛﺜﻴﺮﺍﺕ ﺍﻟﺤﺪﻭﺩ ‪3-5‬‬ ‫‪Solving Polynomial Equations‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-9‬ﺣﻞ ﻛﻞ ﻣﻌﺎﺩﻟﺔ ﻣﻤﺎ ﻳﺄﺗﻲ ﻭﻗ ّﺮﺏ ﺇﺟﺎﺑﺘﻚ ﻷﻗﺮﺏ ﺟﺰﺀ ﻣﻦ ﻣﺌﺔ ﻋﻨﺪﻣﺎ ﻳﻜﻮﻥ ﺫﻟﻚ ﺿﺮﻭﺭﻳًّﺎ‪.‬‬ ‫‪(1) 6y2 = 48y‬‬ ‫‪(2) 3x3 - 6x2 - 9x = 0‬‬ ‫‪(3) 12x3 - 60x2 + 75x = 0‬‬ ‫‪(4) 4x3 = 4x2 + 3x‬‬ ‫‪(5) 2a4 - 5a3 - 3a2 = 0‬‬ ‫‪(6) 2d4 + 18d3 = 0‬‬ ‫‪(7) x3 - 6x2 + 6x = 0‬‬ ‫‪(8) x3 + 13x = 10x2‬‬ ‫‪(9) 2x3 - 5x2 = 12x‬‬ ‫‪(10) x3 - 2x2 - 3 = x - 5‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(10-12‬ﺍﺳﺘﺨﺪﻡ ﺍﻟﺘﻘﺴﻴﻢ ﻟﺤﻞ ﻛﻞ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫‪(11) x3 + 3x2 - 4x - 12 = 0 (12) x3 + 2x^x - 1h = 1‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(13-17‬ﺍﺳﺘﺨﺪﻡ ﺍﻷﺻﻔﺎﺭ ﺍﻟﻨﺴﺒﻴﺔ ﺍﻟﻤﻤﻜﻨﺔ ﻟﺤﻞ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫‪(13) x4 + 2x3 + x2 = 4x2 + 8x + 4‬‬ ‫‪(14) x3 - 3x + 2 = 0‬‬ ‫‪(15) x3 + x2 - 8x - 12 = 0‬‬ ‫‪(16) x3 - 7x + 6 = 0‬‬ ‫‪(17) x4 + x3 - 6x2 - 4x + 8 = 0‬‬ ‫)‪ (18‬ﺍﻟﺴﺆﺍﻝ ﺍﻟﻤﻔﺘﻮﺡ‪ :‬ﻟﺤﻞ ﻣﻌﺎﺩﻟﺔ ﻛﺜﻴﺮﺓ ﺣﺪﻭﺩ‪ ،‬ﻳﻤﻜﻨﻚ ﺍﺳﺘﺨﺪﺍﻡ ﻃﺮﻳﻘﺔ ﺃﻭ ﺃﻛﺜﺮ ﻣﻦ ﺍﻟﻄﺮﻕ ﺍﻟﺘﺎﻟﻴﺔ‪ :‬ﺍﻟﺮﺳﻢ ﺍﻟﺒﻴﺎﻧﻲ‪،‬‬ ‫ﺍﻟﺘﺤﻠﻴﻞ ﺇﻟﻰ ﻋﻮﺍﻣﻞ‪ ،‬ﺍﻟﻘﺎﻧﻮﻥ ﺍﻟﻌﺎﻡ ﻟﺤﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺍﻟﺘﺮﺑﻴﻌﻴﺔ‪ .‬ﺍﻛﺘﺐ ﻣﻌﺎﺩﻟﺔ ﻭﺣﻠﻬﺎ ﻟﺘﻮﺿﺢ ﻛﻞ ﻃﺮﻳﻘﺔ‪.‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-5‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ab‬‬ ‫‪&-‬‬ ‫‪4‬‬ ‫‪,‬‬ ‫‪4‬‬ ‫‪0‬‬ ‫ﻫﻲ‬ ‫‪9x2 + 16 = 0‬‬ ‫ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫)‪(1‬‬ ‫‪ab‬‬ ‫‪3‬‬ ‫‪3‬‬ ‫‪ab‬‬ ‫)‪ (2‬ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ x ! R ،2x3 + 2 = 0‬ﻫﻲ ﻣﺠﻤﻮﻋﺔ ﺃﺣﺎﺩﻳﺔ‪.‬‬ ‫‪^4x2‬‬ ‫‪+‬‬ ‫‪1ha‬‬ ‫‪x2‬‬ ‫‪- 1k = 0‬‬ ‫ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﺇﺫﺍ ﻛﺎﻧﺖ ‪ 2k‬ﺗﻨﺘﻤﻲ ﺇﻟﻰ‬ ‫)‪(3‬‬ ‫‪4‬‬ ‫ﻓﺈﻥ ‪k ! \"-1,1,‬‬ ‫‪ab‬‬ ‫)‪ (4‬ﺇﻥ ‪ \"1,‬ﻫﻲ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪3x4 + 12x2 - 15 = 0‬‬ ‫‪a‬‬ ‫‪b‬‬ ‫ﺣﻴﺚ ‪b , c ! R‬‬ ‫‪f (x) = 2x3 + bx2 + cx - 3‬‬ ‫ﻳﻤﻜﻦ ﺃﻥ ﻳﻜﻮﻥ ﺻﻔ ًﺮﺍ ﻟﻠﺤﺪﻭﺩﻳﺔ‬ ‫‪2‬‬ ‫)‪(5‬‬ ‫‪3‬‬ ‫‪49‬‬

‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(6-8‬ﻇﻠّﻞ ﺩﺍﺋﺮﺓ ﺍﻟﺮﻣﺰ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫)‪ 5 (6‬ﻳﻤﻜﻦ ﺃﻥ ﻳﻜﻮﻥ ﺻﻔ ًﺮﺍ ﻣﻦ ﺃﺻﻔﺎﺭ ﺍﻟﺤﺪﻭﺩﻳﺔ )‪ f (x‬ﺗﺴﺎﻭﻱ‪:‬‬ ‫‪a ax3 + x4 + 5‬‬ ‫‪b x5 - 1‬‬ ‫‪c 5x3 + 6x - 1‬‬ ‫‪d ^x + 5h^x2 + 25h‬‬ ‫‪a -1‬‬ ‫‪b -3‬‬ ‫)‪ (7‬ﺃﻱ ﻗﻴﻤﺔ ﻣﻤﺎ ﻳﻠﻲ ﻟﻴﺴﺖ ﺣ ًّﻼ ﻟﻠﻤﻌﺎﺩﻟﺔ‪x4 - 10x2 + 9 = 0 :‬‬ ‫‪c3 d2‬‬ ‫)‪a f (x) = (x - 1) (x + m) (x + n‬‬ ‫)‪ (8‬ﺇﺫﺍ ﻛﺎﻥ ‪ f(m) = f(n) = f(- 1) = 0‬ﻓﺈﻥ ‪ f‬ﻣﻤﻜﻦ ﺃﻥ ﺗﻜﻮﻥ‪:‬‬ ‫‪c f (x) = (x + 1) (x - m) (x - n) 2‬‬ ‫)‪b f (x) = (x - 1) (x - m) 2 (x - n‬‬ ‫)‪d f (x) = (x + 1) (x - mn‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(9-11‬ﻟﺪﻳﻚ ﻗﺎﺋﻤﺘﺎﻥ ﺍﺧﺘﺮ ﻣﻦ ﺍﻟﻘﺎﺋﻤﺔ )‪ (2‬ﻣﺎ ﻳﻨﺎﺳﺐ ﻛﻞ ﺗﻤﺮﻳﻦ ﻓﻲ ﺍﻟﻘﺎﺋﻤﺔ )‪ (1‬ﻟﺘﺤﺼﻞ ﻋﻠﻰ ﺇﺟﺎﺑﺔ ﺻﺤﻴﺤﺔ‪.‬‬ ‫ﺍﻟﻘﺎﺋﻤﺔ )‪(2‬‬ ‫ﺍﻟﻘﺎﺋﻤﺔ )‪(1‬‬ ‫)‪ (9‬ﻣﺠﻤﻮﻋﺔ ﺣﻞ ‪ f(x) = 0‬ﻫﻲ ‪\"-1, 2,3,‬‬ ‫‪a‬‬ ‫‪y‬‬ ‫‪1‬‬ ‫` ﺑﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ ‪ f‬ﻳﻤﻜﻦ ﺃﻥ ﻳﻜﻮﻥ‪:‬‬ ‫‪1x‬‬ ‫)‪ (10‬ﻣﺠﻤﻮﻋﺔ ﺣﻞ ‪ f(x) = 0‬ﻫﻲ ‪\"-1,2,‬‬ ‫‪y‬‬ ‫` ﺑﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ ‪ f‬ﻳﻤﻜﻦ ﺃﻥ ﻳﻜﻮﻥ‪:‬‬ ‫‪b‬‬ ‫‪-1 2 3 x‬‬ ‫)‪ (11‬ﻣﺠﻤﻮﻋﺔ ﺣﻞ ‪ f(x) = 0‬ﻫﻲ ‪\"1, - 2, - 3,‬‬ ‫‪y‬‬ ‫` ﺑﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ ‪ f‬ﻳﻤﻜﻦ ﺃﻥ ﻳﻜﻮﻥ‪:‬‬ ‫‪c‬‬ ‫‪-1 2 x‬‬ ‫‪y‬‬ ‫‪d‬‬ ‫‪-3 -2‬‬ ‫‪1x‬‬ ‫‪50‬‬