الفصل الأول تمارين

‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ ﺍﻟﺜﺎﻟﺜﺔ‬ ‫)‪(1‬‬ ‫‪y‬‬ ‫=‬ ‫‪1‬‬ ‫‪x4‬‬ ‫‪(2) y = (x + 1) 3‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-4‬ﺃﻭﺟﺪ ﻣﻌﻜﻮﺱ ﻛﻞ ﺩﺍﻟﺔ ﻣﻤﺎ ﻳﻠﻲ‪:‬‬ ‫‪2‬‬ ‫‪(3) y = (x + 1) 2 - 3 (4) y = x + 5‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(5-7‬ﺍﻛﺘﺐ ﻛﻞ ﺩﺍﻟﺔ ﻛﺜﻴﺮﺓ ﺣﺪﻭﺩ ﻓﻲ ﺍﻟﺼﻮﺭﺓ ﺍﻟﻌﺎﻣﺔ‪ ،‬ﺛﻢ ﺻﻨّﻔﻬﺎ ﺑﺤﺴﺐ ﻋﺪﺩ ﺍﻟﺤﺪﻭﺩ ﻭﺑﺤﺴﺐ ﺍﻟﺪﺭﺟﺔ‪.‬‬ ‫‪(5) f (x) = 3x2 - 7x4 + 9 - x4 (6) f (x) = 11x2 + 8x - 3x2‬‬ ‫‪(7) f (x) = 2x^x - 3h^x + 2h‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺮﻳﻨﻴﻦ )‪ ،(8-9‬ﺃﻭﺟﺪ ﺃﺻﻔﺎﺭ ﺍﻟﺪﺍﻟﺔ ﺛﻢ ﺍﺭﺳﻢ ﺑﻴﺎﻧًﺎ ﺗﻘﺮﻳﺒﻴًّﺎ ﻟﻬﺎ ﻣﺮﺍﻋ ًﻴﺎ ﺳﻠﻮﻙ ﺍﻟﻨﻬﺎﻳﺔ‪) .‬ﻗ ّﺮﺏ ﺇﻟﻰ ﺃﻗﺮﺏ ﺟﺰﺀ ﻣﻦ ﻋﺸﺮﺓ ﻋﻨﺪ‬ ‫ﺍﻟﻀﺮﻭﺭﺓ(‪.‬‬ ‫‪(8) f (x) = x^x - 3h^x + 2h‬‬ ‫‪(9) f (x) = ^x - 2h2 ^x - 1h‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(10-13‬ﺣ ّﻞ ﻛﻞ ﻣﻌﺎﺩﻟﺔ‪ .‬ﺃﻋﻂ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺪﻗﻴﻘﺔ ﺃﻭ ﻗ ّﺮﺏ ﺇﺟﺎﺑﺘﻚ ﺇﻟﻰ ﺃﻗﺮﺏ ﺟﺰﺀ ﻣﻦ ﻋﺸﺮﺓ‪.‬‬ ‫‪(10) ^x - 3h^x2 + 3x - 4h = 0‬‬ ‫‪(11) ^x + 2h^x2 + 5x + 1h = 0‬‬ ‫‪(12) x3 - 2x2 - x + 2 = 0‬‬ ‫‪(13) x4 - 2x2 - x + 2 = 0‬‬ ‫‪(14) 0 , 4 , - 2‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺮﻳﻨﻴﻦ )‪ ،(14-15‬ﺍﻛﺘﺐ ﺩﺍﻟﺔ ﻛﺜﻴﺮﺓ ﺣﺪﻭﺩ ﻓﻲ ﺍﻟﺼﻮﺭﺓ ﺍﻟﻌﺎﻣﺔ ﺇﺫﺍ ﻋﻠﻤﺖ ﺃﺻﻔﺎﺭﻫﺎ‪:‬‬ ‫)ﻣﻜﺮﺭ ﻣﺮﺗﻴﻦ( ‪(15) 2 , - 1‬‬ ‫‪(16) ^x3 + 7x2 - 36h ' ^x + 3h‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺮﻳﻨﻴﻦ )‪ ،(16-17‬ﺍﻗﺴﻢ ﻣﺴﺘﺨﺪ ًﻣﺎ ﻗﺴﻤﺔ ﻛﺜﻴﺮﺓ ﺍﻟﺤﺪﻭﺩ ﺍﻟﻤﻄﻮﻟﺔ‪.‬‬ ‫‪(17) ^x3 + 7x2 - 5x - 6h ' ^x + 2h‬‬ ‫‪(18) ^x3 + x2 + x - 14h ' ^x - 3h‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺮﻳﻨﻴﻦ )‪ ،(18-19‬ﺍﻗﺴﻢ ﻣﺴﺘﺨﺪ ًﻣﺎ ﺍﻟﻘﺴﻤﺔ ﺍﻟﺘﺮﻛﻴﺒﻴﺔ‪.‬‬ ‫‪(19) ^x4 - 5x2 + 4x + 12h ' ^x + 1h‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺮﻳﻨﻴﻦ )‪ ،(20-21‬ﺍﺳﺘﺨﺪﻡ ﺍﻟﻘﺴﻤﺔ ﺍﻟﺘﺮﻛﻴﺒﻴﺔ ﻭﻧﻈﺮﻳﺔ ﺍﻟﺒﺎﻗﻲ ﻹﻳﺠﺎﺩ )‪f(a‬‬ ‫)‪(20‬‬ ‫‪f (x) = 2x4 + 19x3 - 2x2 - 44x - 24 ,‬‬ ‫‪a‬‬ ‫=‬ ‫‪-2‬‬ ‫‪3‬‬ ‫‪(21) f (x) = - x3 - x2 + x , a = 0‬‬ ‫‪51‬‬

‫ﺗﻤﺎﺭﻳﻦ ﺇﺛﺮﺍﺋﻴﺔ‬ ‫)‪ (1‬ﻟﺘﻜﻦ‪g(x) = (m + 1)x3 + 11x2 + 4x - 4 :‬‬ ‫ﺍﻟﺤﺪﻭﺩ‪.‬‬ ‫ﺃﺣﺪ ﺃﺻﻔﺎﺭ ﻛﺜﻴﺮﺓ‬ ‫‪1‬‬ ‫ﻗﻴﻤﺔ ‪ m‬ﺑﺤﻴﺚ ﻳﻜﻮﻥ‬ ‫ﺃﻭﺟﺪ‬ ‫‪2‬‬ ‫)‪ (2‬ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ‪:‬‬ ‫‪(a) 2x4 + x3 - 11x2 + 11x - 3 = 0‬‬ ‫‪(b) 4x4 - x2 + 6x - 9 = 0‬‬ ‫)‪ (3‬ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ a‬ﺑﺤﻴﺚ ﺗﻜﻮﻥ‪ f(x) = x5 + x4 - 6x3 - 14x2 - ^a + 5hx - (a - 3) :‬ﻗﺎﺑﻠﺔ ﻟﻠﻘﺴﻤﺔ ﻋﻠﻰ ‪^x + 1h2‬‬ ‫‪x3 - 7x + 6‬‬ ‫ﺑ ّﺴﻂ ﻣﺎ ﻳﻠﻲ‪:‬‬ ‫)‪(4‬‬ ‫‪x4 + x3 - 5x2 + x - 6‬‬ ‫)‪g (x) = 4x4 - 11x3 - 2x2 + 23x - 14 (5‬‬ ‫)‪ (a‬ﺣﻠّﻞ )‪ g(x‬ﺇﻟﻰ ﻋﻮﺍﻣﻞ‪.‬‬ ‫)‪ (b‬ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ‪ .g(x) = 0 :‬ﻗ ّﺮﺏ ﺇﺟﺎﺑﺘﻚ ﺇﻟﻰ ﺃﻗﺮﺏ ﺟﺰﺀ ﻣﻦ ﻣﺌﺔ‪.‬‬ ‫)‪ (6‬ﻟﺘﻜﻦ‪f (x) = x3 - ^3a + 2bhx2 + ^a + bhx :‬‬ ‫)‪ (a‬ﺃﻭﺟﺪ ﻗﻴﻢ ‪ a, b‬ﺑﺤﻴﺚ ﺗﻜﻮﻥ ‪ ^x - 1h,^x - 2h‬ﻣﻦ ﻋﻮﺍﻣﻞ )‪f(x‬‬ ‫)‪ (b‬ﺣﻠّﻞ ﻓﻲ ﻫﺬﻩ ﺍﻟﺤﺎﻟﺔ )‪ f(x‬ﺇﻟﻰ ﻋﻮﺍﻣﻞ‪.‬‬ ‫)‪ (7‬ﺃﻭﺟﺪ ﺩﺍﻟﺔ ﻛﺜﻴﺮﺓ ﺍﻟﺤﺪﻭﺩ ﻣﻦ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ ﺗﻘﺒﻞ ﺍﻟﻘﺴﻤﺔ ﻋﻠﻰ ‪ ^x + 5h,^2x - 1h‬ﻭﺑﺎﻗﻲ ﻗﺴﻤﺘﻬﺎ ﻋﻠﻰ ‪^x - 3h‬‬ ‫ﻳﺴﺎﻭﻱ ‪40‬‬ ‫)‪ (8‬ﻟﺘﻜﻦ‪g(x) = x3 + 8 :‬‬ ‫)‪ (a‬ﺃﻭﺟﺪ ﺻﻔ ًﺮﺍ ﻟﻜﺜﻴﺮﺓ ﺍﻟﺤﺪﻭﺩ‪.‬‬ ‫)‪ (b‬ﺣﻠّﻞ )‪ g(x‬ﺇﻟﻰ ﻋﻮﺍﻣﻞ‪.‬‬ ‫)‪ (a) (9‬ﺍﻛﺘﺐ ‪ V(x) = ^x2 + ax + bh2‬ﻓﻲ ﺍﻟﺼﻮﺭﺓ ﺍﻟﻌﺎﻣﺔ‪.‬‬ ‫)‪ (b‬ﺃﺛﺒﺖ ﺃﻥ‪ f(x) = x4 + 6x3 + 7x2 - 6x + 1 :‬ﻫﻲ ﻣﺮﺑﻊ ﻟﻜﺜﻴﺮﺓ ﺣﺪﻭﺩ ﻣﻦ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ‪.‬‬ ‫)‪ (10‬ﺃﻭﺟﺪ ﻧﻤﻮﺫ ًﺟﺎ ﺗﻜﻌﻴﺒﻴًّﺎ ﻟﻠﺪﺍﻟﺔ ﺍﻟﺘﻲ ﺗﻤﺮ ﻓﻲ‪ ،^-1, - 3h, (0,0), (1, - 1), (2,0) :‬ﺛﻢ ﺍﺳﺘﺨﺪﻡ ﻫﺬﺍ ﺍﻟﻨﻤﻮﺫﺝ‬ ‫ﻟﺘﻘﺪﻳﺮ ﻗﻴﻤﺔ ‪ y‬ﻋﻨﺪﻣﺎ ‪x = 17‬‬ ‫‪52‬‬

‫‪d = 3.8 # 102 cm‬‬ ‫ﺣﺠﻢ‬ ‫ﻹﻳﺠﺎﺩ‬ ‫‪،V‬‬ ‫=‬ ‫‪πh‬‬ ‫‪^R‬‬ ‫‪2‬‬ ‫‪+‬‬ ‫‪Rd‬‬ ‫‪+‬‬ ‫‪d2h‬‬ ‫ﺍﻟﻌﻼﻗﺔ‪:‬‬ ‫ﺍﺳﺘﺨﺪﻡ‬ ‫ﺍﻟﻬﻨﺪﺳﺔ‪:‬‬ ‫)‪(11‬‬ ‫‪h = 3.5 # 102 cm‬‬ ‫‪3‬‬ ‫ﺍﻟﻤﺨﺮﻭﻁ ﺍﻟﻨﺎﻗﺺ ﺍﻟﻤﻮﺿﺢ ﻓﻲ ﺍﻟﺸﻜﻞ‪.‬‬ ‫‪R = 5.6 # 102 cm‬‬ ‫ﺍﻛﺘﺐ ﺇﺟﺎﺑﺘﻚ ﻓﻲ ﺍﻟﺼﻮﺭﺓ ﺍﻟﻌﻠﻤﻴﺔ‪.‬‬ ‫)‪ (12‬ﺍﻟﻬﻨﺪﺳﺔ‪ :‬ﺻﻨﺪﻭﻕ ﻳﻘﻞ ﻋﺮﺿﻪ ‪ 2 m‬ﻋﻦ ﻃﻮﻟﻪ‪ ،‬ﻭ ﻳﻘﻞ ﺍﺭﺗﻔﺎﻋﻪ ‪ 1 m‬ﻋﻦ ﻃﻮﻟﻪ‪.‬‬ ‫ﺃﻭﺟﺪ ﻃﻮﻝ ﺍﻟﺼﻨﺪﻭﻕ ﻋﻨﺪﻣﺎ ﻳﻜﻮﻥ ﺣﺠﻤﻪ ‪60 m3‬‬ ‫)‪ (13‬ﺗﺮﻳﺪ ﺷﺮﻛﺔ ﻟﻠﺘﺨﺰﻳﻦ ﺻﻨﻊ ﺻﻨﺪﻭﻕ ﻟﻠﺘﺨﺰﻳﻦ ﺣﺠﻤﻪ ﻣﺜﻠﻲ ﺣﺠﻢ ﺃﻛﺒﺮ ﺻﻨﺪﻭﻕ ﺗﺨﺰﻳﻦ ﻟﺪﻳﻬﺎ‪ ،‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺃﺑﻌﺎﺩ‬ ‫ﺃﻛﺒﺮ ﺻﻨﺪﻭﻕ ﺗﺨﺰﻳﻦ ﻟﺪﻳﻬﺎ ﻫﻲ ‪ 120 cm‬ﻃﻮ ًﻻ‪ 100 cm ،‬ﻋﺮ ًﺿﺎ‪،‬‬ ‫‪ 90 cm‬ﺍﺭﺗﻔﺎ ًﻋﺎ‪ ،‬ﻭﻳﺮﺍﺩ ﺻﻨﻊ ﺍﻟﺼﻨﺪﻭﻕ ﺍﻟﺠﺪﻳﺪ ﺑﺰﻳﺎﺩﺓ ﻛﻞ ﺑﻌﺪ ﺍﻟﻤﻘﺪﺍﺭ ﻧﻔﺴﻪ‪،‬‬ ‫ﻓﺄﻭﺟﺪ ﺍﻟﺰﻳﺎﺩﺓ ﻓﻲ ﻛﻞ ﺑﻌﺪ‪.‬‬ ‫)‪ (14‬ﺍﻟﺤﺴﺎﺏ ﺍﻟﺬﻫﻨﻲ‪ :‬ﺇﺫﺍ ﻛﺎﻥ ﻧﺎﺗﺞ ﺿﺮﺏ ﺛﻼﺛﺔ ﺃﻋﺪﺍﺩ ﺻﺤﻴﺤﺔ ﻣﺘﺘﺎﻟﻴﺔ‪ ^n - 1h, n , ^n + 1h :‬ﻫﻮ ‪ ،210‬ﻓﺎﻛﺘﺐ‬ ‫ﻣﻌﺎﺩﻟﺔ ﻭﺃﻭﺟﺪ ﺣﻠﻬﺎ ﻹﻳﺠﺎﺩ ﺍﻷﻋﺪﺍﺩ‪.‬‬ ‫)‪ (15‬ﺍﻟﻬﻨﺪﺳﺔ‪ :‬ﺣﺠﻢ ﺧ ّﺰﺍﻥ )‪ (V‬ﻳﻤﺜّﻞ ﺑﺎﻟﺪﺍﻟﺔ‪ .V(x) = x3 + 8x2 + 15x :‬ﻟﻨﻔﺮﺽ ﺃﻥ ‪ x‬ﺗﻤﺜﻞ ﺍﻟﻌﺮﺽ‪ x + 3 ،‬ﺗﻤﺜّﻞ‬ ‫ﺍﻟﻄﻮﻝ‪ x + 5 ،‬ﺗﻤﺜﻞ ﺍﻻﺭﺗﻔﺎﻉ‪ ،‬ﺣﺠﻢ ﺍﻟﺨ ّﺰﺍﻥ ‪ ،70 m3‬ﻓﻤﺎ ﺃﺑﻌﺎﺩﻩ؟‬ ‫‪53‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺍﺳﺘﻜﺸﺎﻑ ﺍﻟﻨﻤﺎﺫﺝ ﺍﻷﺳﻴﺔ ‪4-1‬‬ ‫‪Exploring Exponential Models‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-5‬ﺍﺫﻛﺮ ﻣﺎ ﺇﺫﺍ ﻛﺎﻧﺖ ﻛﻞ ﺩﺍﻟﺔ ﺗﻤﺜّﻞ ﻧﻤ ًّﻮﺍ ﺃﺳﻴًّﺎ ﺃﻭ ﺗﻀﺎﺅ ًﻻ ﺃﺳﻴًّﺎ‪ .‬ﻣﺎ ﺍﻟﻨﺴﺒﺔ ﺍﻟﻤﺌﻮﻳﺔ ﻟﺰﻳﺎﺩﺓ ﺍﻟﺪﺍﻟﺔ ﺃﻭ ﻧﻘﺼﺎﻧﻬﺎ؟‬ ‫‪(1) y = 1298(1 . 63) x‬‬ ‫‪(2) y = 0 . 65(1.3) x‬‬ ‫‪(3) f(x) = 2 (0 . 65) x‬‬ ‫)‪(4‬‬ ‫‪f‬‬ ‫)‪(t‬‬ ‫=‬ ‫‪0‬‬ ‫‪.‬‬ ‫`‪8‬‬ ‫‪1‬‬ ‫‪t‬‬ ‫‪(5) y = 5(6) x‬‬ ‫‪8‬‬ ‫‪j‬‬ ‫)‪ (6‬ﺍﻟﺪﺭﺍﺳﺎﺕ ﺍﻻﺟﺘﻤﺎﻋﻴﺔ‪ :‬ﻳﻌﺮﺽ ﺍﻟﺠﺪﻭﻝ ﺍﻟﺘﺎﻟﻲ ﻣﻌﻠﻮﻣﺎﺕ ﻋﻦ ﻋﺪﺩ ﺍﻟﺴﻜﺎﻥ ﻓﻲ ﺃﻛﺒﺮ ﺃﺭﺑﻊ ﻣﺪﻥ ﻓﻲ ﺍﻟﻌﺎﻟﻢ ﻓﻲ‬ ‫ﺳﻨﺔ ‪.1994‬‬ ‫ﻋﺪﺩ ﺍﻟﺴﻜﺎﻥ ﻓﻲ ﺳﻨﺔ ‪ 1994‬ﻣﺘﻮﺳﻂ ﻣﻌﺪﻝ ﺍﻟﻨﻤﻮ ﺍﻟﺴﻨﻮﻱ‬ ‫ﺍﻟﻤﺪﻳﻨﺔ )ﺍﻟﺪﻭﻟﺔ(‬ ‫ﺍﻟﺘﺮﺗﻴﺐ ﻓﻲ ﺳﻨﺔ ‪1994‬‬ ‫)‪(Ι‬‬ ‫‪1.4%‬‬ ‫‪26 518 000‬‬ ‫ﻃﻮﻛﻴﻮ )ﺍﻟﻴﺎﺑﺎﻥ(‬ ‫‪1‬‬ ‫‪0.3%‬‬ ‫ﻧﻴﻮﻳﻮﺭﻙ )ﺍﻟﻮﻻﻳﺎﺕ ﺍﻟﻤﺘﺤﺪﺓ( ‪16 271 000‬‬ ‫‪2‬‬ ‫‪2.0%‬‬ ‫‪16 110 000‬‬ ‫ﺳﺎﻭﺑﺎﻭﻟﻮ )ﺍﻟﺒﺮﺍﺯﻳﻞ(‬ ‫‪3‬‬ ‫‪0.7%‬‬ ‫‪15 525 000‬‬ ‫ﻣﻜﺴﻴﻜﻮ )ﺍﻟﻤﻜﺴﻴﻚ(‬ ‫‪4‬‬ ‫)‪ (a‬ﻟﻨﻔﺘﺮﺽ ﺍﺳﺘﻤﺮﺍﺭ ﻫﺬﻩ ﺍﻟﻤﻌﺪﻻﺕ ﻟﻠﻨﻤﻮ‪ ،‬ﺍﻛﺘﺐ ﻣﻌﺎﺩﻟﺔ ﺗﻤﺜّﻞ ﺍﻟﻨﻤﻮ ﺍﻟﻤﺴﺘﻘﺒﻠﻲ ﻟﻌﺪﺩ ﺍﻟﺴﻜﺎﻥ ﻓﻲ ﻛﻞ ﻣﺪﻳﻨﺔ‪.‬‬ ‫)‪ (b‬ﺍﺳﺘﺨﺪﻡ ﻣﻌﺎﺩﻻﺗﻚ ﻛﻲ ﺗﺘﻮﻗﻊ ﻋﺪﺩ ﺳﻜﺎﻥ ﻛﻞ ﻣﺪﻳﻨﺔ ﻓﻲ ﺳﻨﺔ ‪ .2004‬ﻫﻞ ﺗﻐﻴﺮ ﺍﻟﺘﺮﺗﻴﺐ؟‬ ‫ﻓﻲ ﺍﻟﺘﻤﺮﻳﻨﻴﻦ )‪ ،(7-8‬ﻣﺜّﻞ ﻛﻞ ﺩﺍﻟﺔ ﺑﻴﺎﻧﻴًّﺎ‪ .‬ﺑﻴّﻦ ﻣﺎ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺪﺍﻟﺔ ﺗﻤﺜﻞ ﻧﻤ ًّﻮﺍ ﺃﺳﻴًّﺎ ﺃﻭ ﺗﻀﺎﺅ ًﻻ ﺃﺳﻴًّﺎ ﻣﺤﺪ ًﺩﺍ ﺍﻟﻌﺎﻣﻞ‪.‬‬ ‫‪(7) y = 100(0 . 5) x‬‬ ‫‪(8) f(x) = 2x‬‬ ‫)‪ (9‬ﺍﻟﺴﺆﺍﻝ ﺍﻟﻤﻔﺘﻮﺡ‪ :‬ﺍﻛﺘﺐ ﻣﺴﺄﻟﺔ ﺣﻴﺎﺗﻴﺔ ﺗﻤﺜّﻞ ﻧﻤ ًّﻮﺍ ﺃﺳﻴًّﺎ ﺃﻭ ﺗﻀﺎﺅ ًﻻ ﺃﺳﻴًّﺎ ﻟﻜﻞ ﺩﺍﻟﺔ ﻓﻲ ﺍﻟﺘﻤﺮﻳﻨﻴﻦ )‪ (7‬ﻭ)‪.(8‬‬ ‫)‪ (10‬ﺍﻻﻗﺘﺼﺎﺩ‪ :‬ﺍﻓﺘﺮﺽ ﺃﻧﻚ ﺗﺮﻳﺪ ﺷﺮﺍﺀ ﺳﻴﺎﺭﺓ ﺛﻤﻨﻬﺎ ‪ 4 500‬ﺩﻳﻨﺎﺭ‪ .‬ﻣﻦ ﺍﻟﻤﺘﻮﻗﻊ ﺃﻥ ﺗﻨﺨﻔﺾ ﻗﻴﻤﺘﻬﺎ ﺑﻤﻌﺪﻝ ‪20%‬‬ ‫ﺳﻨﻮﻳًّﺎ‪ ،‬ﺇﺫﺍ ﺃﺧﺬﺕ ﻗﺮ ًﺿﺎ ﻣﺪﺗﻪ ﺃﺭﺑﻊ ﺳﻨﻮﺍﺕ ﻟﺸﺮﺍﺀ ﺍﻟﺴﻴﺎﺭﺓ‪ ،‬ﻓﻜﻢ ﺳﺘﻜﻮﻥ ﻗﻴﻤﺔ ﺍﻟﺴﻴﺎﺭﺓ ﺑﻌﺪ ﺃﻥ ﺗﺴﺪﺩ ﺍﻟﻘﺮﺽ‬ ‫ﻓﻲ ﺃﺭﺑﻊ ﺳﻨﻮﺍﺕ؟‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(11-14‬ﺍﻛﺘﺐ ﺩﺍﻟﺔ ﺃﺳﻴّﺔ ﻟﺘﻤﺜﻴﻞ )ﻧﻤﺬﺟﺔ( ﻛﻞ ﻣﻮﻗﻒ ﻣﻤﺎ ﻳﻠﻲ‪ .‬ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﺍﻟﺪﺍﻟﺔ ﺑﻌﺪ ﺧﻤﺲ ﺳﻨﻮﺍﺕ‪.‬‬ ‫)‪ (11‬ﺗﺠ ّﻤﻊ ﻣﻦ ﺍﻟﻀﻔﺎﺩﻉ ﻣﺆﻟﻒ ﻣﻦ ‪ 250‬ﺿﻔﺪﻋﺔ‪ ،‬ﻳﺘﺰﺍﻳﺪ ﺑﻤﻌﺪﻝ ‪ 22%‬ﺳﻨﻮﻳًّﺎ‪.‬‬ ‫)‪ (12‬ﻣﺠﻤﻮﻋﺔ ﻃﻮﺍﺑﻊ ﺛﻤﻨﻬﺎ ‪ 35‬ﺩﻳﻨﺎ ًﺭﺍ‪ ،‬ﻳﺘﺰﺍﻳﺪ ﺛﻤﻨﻬﺎ ﺑﻤﻌﺪﻝ ‪ 7.5%‬ﺳﻨﻮﻳًّﺎ‪.‬‬ ‫‪54‬‬

‫)‪ (13‬ﺳﻴﺎﺭﺓ ﺷﺤﻦ ﺻﻐﻴﺮﺓ ﺛﻤﻨﻬﺎ ‪ 1 750‬ﺩﻳﻨﺎ ًﺭﺍ ﺗﻨﺨﻔﺾ ﻗﻴﻤﺘﻬﺎ ﺑﻤﻌﺪﻝ ‪ 11%‬ﺳﻨﻮﻳًّﺎ‪.‬‬ ‫)‪ (14‬ﻗﻄﻴﻊ ﻣﻦ ﺍﻟﻤﺎﻋﺰ ﻋﺪﺩﻩ ‪ 115‬ﻳﺘﻨﺎﻗﺺ ﺑﻤﻌﺪﻝ ‪ 1.25%‬ﺳﻨﻮﻳًّﺎ‪.‬‬ ‫)‪ (15‬ﻟﻨﻔﺘﺮﺽ ﺃﻧﻚ ﺗﺸﺘﺮﻱ ﺳﻴﺎﺭﺓ ﺟﺪﻳﺪﺓ‪ ،‬ﻭﺗﺮﻳﺪ ﺃﻥ ﻳﻜﻮﻥ ﻟﻬﺬﻩ ﺍﻟﺴﻴﺎﺭﺓ ﺃﻋﻠﻰ ﻗﻴﻤﺔ ﺑﻌﺪ ﻣﺮﻭﺭ ﺧﻤﺲ ﺳﻨﻮﺍﺕ ﻋﻠﻰ‬ ‫ﺷﺮﺍﺋﻬﺎ‪ ،‬ﺃﻱ ﺍﺧﺘﻴﺎﺭ ﻣﻦ ﺍﻻﺧﺘﻴﺎﺭﺍﺕ ﺍﻟﺜﻼﺛﺔ ﺍﻟﻤﻮﺿﺤﺔ ﻓﻲ ﺍﻟﺠﺪﻭﻝ ﺍﻟﺘﺎﻟﻲ ﺳﻮﻑ ﺗﺨﺘﺎﺭ؟‬ ‫ﻗﻴﻤﺔ ﺍﻻﻧﺨﻔﺎﺽ ﺍﻟﻤﺘﻮﻗﻊ‬ ‫ﺍﻟﺴﻌﺮ ﺍﻷﺳﺎﺳﻲ‬ ‫ﺍﻟﺴﻴﺎﺭﺓ‬ ‫‪10%‬‬ ‫‪ 4 275‬ﺩﻳﻨﺎ ًﺭﺍ‬ ‫‪1‬‬ ‫‪12%‬‬ ‫‪ 4 500‬ﺩﻳﻨﺎﺭ‬ ‫‪2‬‬ ‫‪15%‬‬ ‫‪ 4 850‬ﺩﻳﻨﺎ ًﺭﺍ‬ ‫‪3‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-4‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ab‬‬ ‫)‪ (1‬ﺍﻟﺪﺍﻟﺔ ‪ y = 3(2) x‬ﺗﻤﺜﻞ ﺗﻀﺎﺅ ًﻻ ﺃﺳﻴًّﺎ‪.‬‬ ‫‪ab‬‬ ‫‪ab‬‬ ‫ﺗﻤﺜﻞ ﻧﻤ ًﻮﺍ ﺃﺳﻴًّﺎ‪.‬‬ ‫‪2a‬‬ ‫‪1‬‬ ‫‪-‬‬ ‫‪x‬‬ ‫ﺍﻟﺪﺍﻟﺔ‬ ‫)‪(2‬‬ ‫‪ab‬‬ ‫‪3‬‬ ‫‪y‬‬ ‫‪y‬‬ ‫=‬ ‫‪k‬‬ ‫ﻫﻮ ‪2‬‬ ‫=‪y‬‬ ‫‪1‬‬ ‫)‪(2‬‬ ‫‪2x‬‬ ‫ﻋﺎﻣﻞ ﺍﻟﻨﻤﻮ ﻟﻠﺪﺍﻟﺔ‬ ‫)‪(3‬‬ ‫‪3‬‬ ‫)‪ (4‬ﺇﺫﺍ ﻛﺎﻥ ﺑﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ ‪ y = bx‬ﻛﻤﺎ ﻓﻲ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ ﻓﺈﻥ ‪x b 2 1‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(5-8‬ﻇﻠّﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫ﻫﻮ‪:‬‬ ‫‪y‬‬ ‫=‬ ‫‪aa‬‬ ‫‪1‬‬ ‫‪-2 x‬‬ ‫ﻋﺎﻣﻞ ﺍﻟﻨﻤﻮ ﻟﻠﺪﺍﻟﺔ‬ ‫)‪(5‬‬ ‫‪3‬‬ ‫‪kk‬‬ ‫‪a‬‬ ‫‪1‬‬ ‫‪b‬‬ ‫‪1‬‬ ‫‪c3 d9‬‬ ‫‪3‬‬ ‫‪9‬‬ ‫‪y‬‬ ‫)‪ (6‬ﻟﻴﻜﻦ ﺑﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ‪ y = 2bx :‬ﻛﻤﺎ ﻓﻲ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ‪:‬‬ ‫ﻓﺈﻥ ‪ b‬ﻳﻤﻜﻦ ﺃﻥ ﺗﺴﺎﻭﻱ‪:‬‬ ‫‪x‬‬ ‫‪a -2‬‬ ‫‪b0‬‬ ‫‪c‬‬ ‫‪1‬‬ ‫‪d2‬‬ ‫‪2‬‬ ‫)‪ (7‬ﺍﻟﺪﺍﻟﺔ ﺍﻷﺳﻴﺔ ‪ y = abx‬ﺗﻨﻤﺬﺝ ﺍﻟﺘﺰﺍﻳﺪ ﺍﻟﺴﻜﺎﻧﻲ‪ ،‬ﺇﺫﺍ ﻛﺎﻥ ﻣﻌﺪﻝ ﺍﻟﺘﺰﺍﻳﺪ ﺍﻟﺴﻜﺎﻧﻲ ﻓﻲ ﻣﺪﻳﻨﺔ ﻣﺎ ﻫﻮ ‪ 2.5%‬ﻓﺈﻥ‬ ‫ﻋﺎﻣﻞ ﺍﻟﻨﻤﻮ ﻳﺴﺎﻭﻱ‪:‬‬ ‫‪a 0.025‬‬ ‫‪b 1.25‬‬ ‫‪c 1.025‬‬ ‫‪d 3.5‬‬ ‫‪55‬‬

‫‪y‬‬ ‫)‪ (8‬ﺃﻱ ﻣﻦ ﺍﻟﺪﻭﺍﻝ ﺍﻷﺳﻴﺔ ﺍﻟﺘﺎﻟﻴﺔ ﻳﻤﻜﻦ ﺃﻥ ﻳﻤﺜﻠﻬﺎ ﺍﻟﺮﺳﻢ ﺍﻟﺒﻴﺎﻧﻲ ﺍﻟﻤﻘﺎﺑﻞ‪:‬‬ ‫‪4‬‬ ‫‪3‬‬ ‫‪x‬‬ ‫‪2‬‬ ‫‪1‬‬ ‫‪0‬‬ ‫‪1‬‬ ‫‪a‬‬ ‫‪y‬‬ ‫=‬ ‫‪1‬‬ ‫‪(2) x‬‬ ‫‪b‬‬ ‫‪y‬‬ ‫=‬ ‫‪2a‬‬ ‫‪1‬‬ ‫‪x‬‬ ‫‪c y = - 3 (2) x‬‬ ‫‪d y = - 2 (3) x‬‬ ‫‪3‬‬ ‫‪3‬‬ ‫‪k‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(9-11‬ﻟﺪﻳﻚ ﻗﺎﺋﻤﺘﺎﻥ ﺍﺧﺘﺮ ﻣﻦ ﺍﻟﻘﺎﺋﻤﺔ )‪ (2‬ﻣﺎ ﻳﻨﺎﺳﺐ ﻛﻞ ﺗﻤﺮﻳﻦ ﻣﻦ ﺍﻟﻘﺎﺋﻤﺔ )‪ (1‬ﻟﻠﺤﺼﻮﻝ ﻋﻠﻰ ﺇﺟﺎﺑﺔ ﺻﺤﻴﺤﺔ‪.‬‬ ‫ﻳﺒﻴّﻦ ﺍﻟﺘﻤﺜﻴﻞ ﺍﻟﺒﻴﺎﻧﻲ ﺍﻷﺳﻲ ﺍﻟﻤﻘﺎﺑﻞ ﺍﻻﻧﺨﻔﺎﺽ ﻓﻲ ﻗﻴﻤﺔ ﺳﻴﺎﺭﺓ ﺧﻼﻝ ﺍﻟﺴﻨﺔ ﺍﻷﻭﻟﻰ‪.‬‬ ‫ﺍﻟﻘﻴﻤﺔ )ﺑﺂﻻﻑ ﺍﻟﺪﻧﺎﻧﻴﺮ(‬ ‫‪y‬‬ ‫‪20‬‬ ‫‪16‬‬ ‫‪13.500‬‬ ‫‪12‬‬ ‫‪8‬‬ ‫‪4‬‬ ‫‪1 2 3 4 5 6x‬‬ ‫ﻋﺪﺩ ﺍﻟﺴﻨﻮﺍﺕ‬ ‫ﺍﻟﻘﺎﺋﻤﺔ )‪(2‬‬ ‫ﺍﻟﻘﺎﺋﻤﺔ )‪(1‬‬ ‫‪a -0.325‬‬ ‫)‪ (9‬ﻣﻘﺪﺍﺭ ﺍﻻﻧﺨﻔﺎﺽ )ﺑﺎﻵﻻﻑ(=‬ ‫‪b 0.675‬‬ ‫)‪ (10‬ﻧﺴﺒﺔ ﺍﻻﻧﺨﻔﺎﺽ =‬ ‫‪c 0.325‬‬ ‫)‪ (11‬ﻋﺎﻣﻞ ﺍﻻﻧﺨﻔﺎﺽ =‬ ‫‪d -6.5‬‬ ‫‪56‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺍﻟﺪﻭﺍﻝ ﺍﻷﺳﻴﺔ ﻭﺗﻤﺜﻴﻠﻬﺎ ﺑﻴﺎﻧﻴًّﺎ ‪4-2‬‬ ‫‪Exponential Functions and their Graphs‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-4‬ﻣﺜّﻞ ﺑﻴﺎﻧﻴًّﺎ ﻛ ًّﻼ ﻣﻦ ﺍﻟﺪﻭﺍﻝ ﺍﻷﺳﻴﺔ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫‪(1) y = 4x‬‬ ‫‪(2) y = 6x + 3‬‬ ‫‪(3) y = 2-x‬‬ ‫‪(4) y = - 3x+4‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(5-8‬ﻣﺜّﻞ ﺑﻴﺎﻧﻴًّﺎ ﻛ ًّﻼ ﻣﻦ ﺍﻟﺪﻭﺍﻝ ﺍﻷﺳﻴﺔ ﺍﻟﺘﺎﻟﻴﺔ ﻣﺴﺘﺨﺪ ًﻣﺎ ﺩﺍﻟﺔ ﺍﻟﻤﺮﺟﻊ‪:‬‬ ‫‪(5) y = (5) x - 1‬‬ ‫)‪(6‬‬ ‫‪y‬‬ ‫=‬ ‫`‬ ‫‪1‬‬ ‫‪x +2‬‬ ‫‪(7) y = (4) x-2 + 3‬‬ ‫‪(8) y = - 2 (3) 2x + 1‬‬ ‫‪3‬‬ ‫‪j‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(9-13‬ﺍﺳﺘﺨﺪﻡ ﺁﻟﺘﻚ ﺍﻟﺤﺎﺳﺒﺔ ﻹﻳﺠﺎﺩ ﻧﺎﺗﺞ ﻛﻞ ﻣﻘﺪﺍﺭ ﻣﻘ ّﺮ ًﺑﺎ ﺍﻟﻨﺎﺗﺞ ﺇﻟﻰ ﺃﺭﺑﻌﺔ ﺃﺭﻗﺎﻡ ﻋﺸﺮﻳﺔ‪.‬‬ ‫‪(9) e3‬‬ ‫‪(10) 5e6‬‬ ‫)‪(11‬‬ ‫`‬ ‫‪5‬‬ ‫‪je‬‬ ‫‪1‬‬ ‫‪4‬‬ ‫‪2‬‬ ‫)‪(12‬‬ ‫‪4‬‬ ‫‪(13) ee‬‬ ‫‪e6‬‬ ‫)‪ (14‬ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ a‬ﺍﻟﺘﻲ ﻳﺼﺒﺢ ﻋﻨﺪﻫﺎ ﺍﻟﺮﺳﻢ ﺍﻟﺒﻴﺎﻧﻲ ﻟﻠﺪﺍﻟﺔ‪ y = abx :‬ﺧﻄ ًّﺎ ﺃﻓﻘﻴًّﺎ‪.‬‬ ‫)‪ (a) (15‬ﺍﻟﻜﻴﻤﻴﺎﺀ‪ :‬ﺗﻌﻄﻲ ﺍﻟﻌﻼﻗﺔ‪ A = Pe-0.0001t :‬ﺍﻟﻜﻤﻴﺔ ﺍﻟﻤﺘﺒﻘﻴﺔ »‪ «A‬ﺑﺎﻟﻤﻴﻜﺮﻭﺟﺮﺍﻡ ﻣﻦ ﻣﺎﺩﺓ ﺇﺷﻌﺎﻋﻴﺔ ﻣﻌﻴﻨﺔ ﺑﻌﺪ‬ ‫»‪ «t‬ﺳﻨﺔ ﻣﻦ ﺍﻟﺘﻀﺎﺅﻝ؛ »‪ «P‬ﻫﻲ ﺍﻟﻜﻤﻴﺔ ﺍﻷﻭﻟﻴﺔ ﻟﻠﻤﺎﺩﺓ ﺍﻟﻤﺸﻌﺔ‪ .‬ﺍﺳﺘﺨﺪﻡ ﺍﻟﻌﻼﻗﺔ ﻹﻛﻤﺎﻝ ﺍﻟﺠﺪﻭﻝ ﺍﻟﺘﺎﻟﻲ‪:‬‬ ‫ﺍﻟﻜﻤﻴﺔ ﺍﻟﻤﺘﺒﻘﻴﺔ ﻣﻦ ﺍﻟﻤﺎﺩﺓ )‪(A‬‬ ‫ﺍﻟﺴﻨﻮﺍﺕ )‪(t‬‬ ‫ﺍﻟﻜﻤﻴﺔ ﺍﻷﻭﻟﻴﺔ ﻣﻦ ﺍﻟﻤﺎﺩﺓ )‪(P‬‬ ‫‪5‬‬ ‫‪10 000‬‬ ‫‪5‬‬ ‫‪7 500‬‬ ‫‪5‬‬ ‫‪6 000‬‬ ‫‪5‬‬ ‫‪5 000‬‬ ‫‪5‬‬ ‫‪2 500‬‬ ‫‪5‬‬ ‫‪2 000‬‬ ‫)‪ (b‬ﻗﺎﺭﻥ ﺑﻴﻦ ﻗﻴﻢ ﻛﻞ ﻣﻦ ‪ .A , P‬ﻣﺎﺫﺍ ﺗﻼﺣﻆ؟‬ ‫)‪ (16‬ﻋﻠﻢ ﺍﻟﻤﺤﻴﻄﺎﺕ‪ :‬ﻛﻠﻤﺎ ﻏﺼﻨﺎ ﻓﻲ ﺃﻋﻤﺎﻕ ﺍﻟﻤﺤﻴﻂ‪ ،‬ﻗﻠﺖ ﺷﺪﺓ ﺃﺷﻌﺔ ﺍﻟﺸﻤﺲ‪ .‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺷﺪﺓ ﺃﺷﻌﺔ ﺍﻟﺸﻤﺲ ﻋﻠﻰ‬ ‫ﺳﻄﺢ ﺍﻟﻤﺤﻴﻂ ﻫﻲ ‪ ،y‬ﻓﺈﻥ ﺍﻟﻨﺴﺒﺔ ﺍﻟﻤﺌﻮﻳﺔ ﻣﻦ ‪ y‬ﺍﻟﺘﻲ ﺗﺼﻞ ﺇﻟﻰ ﻋﻤﻖ ‪ x m‬ﺗﻌﻄﻰ ﺑﺎﻟﻌﻼﻗﺔ‪y = 20 # (0 . 92) x :‬‬ ‫)ﻳﻌﺪ ﻫﺬﺍ ﺍﻟﻨﻤﻮﺫﺝ ﻣﻨﺎﺳﺒًﺎ ﻟﻸﻋﻤﺎﻕ ﻣﻦ ‪ 6 m‬ﺇﻟﻰ ‪ 180 m‬ﺗﺤﺖ ﻣﺴﺘﻮﻯ ﺳﻄﺢ ﺍﻟﺒﺤﺮ(‪.‬‬ ‫)‪ (a‬ﺃﻭﺟﺪ ﺍﻟﻨﺴﺒﺔ ﺍﻟﻤﺌﻮﻳﺔ ﻷﺷﻌﺔ ﺍﻟﺸﻤﺲ ﺍﻟﻤﻮﺟﻮﺩﺓ ﻋﻠﻰ ﻋﻤﻖ ‪ 15 m‬ﺗﺤﺖ ﻣﺴﺘﻮﻯ ﺳﻄﺢ ﺍﻟﺒﺤﺮ‪.‬‬ ‫)‪ (b‬ﺇﺫﺍ ﻛﺎﻥ ﺃﻗﺼﻰ ﻋﻤﻖ ﻣﺴﺠﻞ ﻟﺮﻳﺎﺿﺔ ﺍﻟﻐﻄﺲ ﻫﻮ ‪ 107 m‬ﺗﺤﺖ ﻣﺴﺘﻮﻯ ﺳﻄﺢ ﺍﻟﺒﺤﺮ‪ ،‬ﻓﺄﻭﺟﺪ ﺍﻟﻨﺴﺒﺔ‬ ‫ﺍﻟﻤﺌﻮﻳﺔ ﻷﺷﻌﺔ ﺍﻟﺸﻤﺲ ﻋﻨﺪ ﻫﺬﺍ ﺍﻟﻌﻤﻖ‪.‬‬ ‫‪57‬‬

‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-5‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ‪ ،‬ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ab‬‬ ‫)‪ (1‬ﺟﻤﻴﻊ ﺍﻟﺪﻭﺍﻝ ﺍﻷﺳﻴﺔ ﻋﻠﻰ ﺍﻟﺼﻮﺭﺓ‪ y = abx a ! 0 , b 2 0 , b ! 1 :‬ﻣﺘﻘﺎﻃﻌﺔ‪.‬‬ ‫‪ab‬‬ ‫)‪ (2‬ﺑﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ ‪ y = - 2x‬ﻫﻮ ﺍﻧﻌﻜﺎﺱ ﻓﻲ ﻣﺤﻮﺭ ﺍﻟﺴﻴﻨﺎﺕ ﻟﺒﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ ‪y = 2x‬‬ ‫)‪ (3‬ﺑﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ ‪ y = - (3) x‬ﻫﻮ ﺍﻧﻌﻜﺎﺱ ﻓﻲ ﻣﺤﻮﺭ ﺍﻟﺼﺎﺩﺍﺕ ﻟﺒﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ ‪a b y = - (3)-x‬‬ ‫‪ab‬‬ ‫)‪ (4‬ﺑﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ ‪ y = 3(5) x-2‬ﻫﻮ ﺍﻧﺴﺤﺎﺏ ﻟﺒﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ ‪y = 3(5) x‬‬ ‫‪ab‬‬ ‫ﺑﻤﻘﺪﺍﺭ ﻭﺣﺪﺗﻴﻦ ﺟﻬﺔ ﺍﻟﻴﻤﻴﻦ‪.‬‬ ‫‪a y = 3 (2) x‬‬ ‫)‪ (5‬ﺑﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ ‪ y = 3(2)x‬ﻳﻘﻄﻊ ﺟﺰﺀًﺍ ﻣﻦ ﻣﺤﻮﺭ ﺍﻟﺼﺎﺩﺍﺕ ﻗﺪﺭﻩ ‪.3‬‬ ‫ﻓﻲ ﺍﻟﺒﻨﻮﺩ )‪ ،(6-12‬ﻇﻠّﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫ﻓﺈﻥ ﺩﺍﻟﺔ ﺍﻟﻤﺮﺟﻊ ﻟﻬﺎ ﻳﻤﻜﻦ ﺃﻥ ﺗﻜﻮﻥ‪:‬‬ ‫‪y‬‬ ‫=‬ ‫‪3a‬‬ ‫‪1‬‬ ‫‪x+1‬‬ ‫‪+‬‬ ‫‪5‬‬ ‫ﻟﺘﻜﻦ‬ ‫)‪(6‬‬ ‫‪2‬‬ ‫‪k‬‬ ‫‪b y = 3 (2) -x‬‬ ‫‪c‬‬ ‫‪y‬‬ ‫=‬ ‫‪3a‬‬ ‫‪1‬‬ ‫‪x+1‬‬ ‫‪d‬‬ ‫‪y‬‬ ‫=‬ ‫‪a‬‬ ‫‪1‬‬ ‫‪x‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪k‬‬ ‫‪k‬‬ ‫ﻛﺪﺍﻟﺔ ﻣﺮﺟﻊ ﻳﻤﻜﻦ ﺭﺳﻢ ﺑﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ‪:‬‬ ‫‪y‬‬ ‫=‬ ‫‪1‬‬ ‫‪(4) x‬‬ ‫ﺍﻟﺪﺍﻟﺔ‬ ‫ﺑﻴﺎﻥ‬ ‫ﺑﺎﺳﺘﺨﺪﺍﻡ‬ ‫)‪(7‬‬ ‫‪3‬‬ ‫‪a y = 3 (4) x‬‬ ‫‪b y = 3 (4) -x‬‬ ‫‪c‬‬ ‫‪y‬‬ ‫=‬ ‫‪1‬‬ ‫‪(2) 2x + 1‬‬ ‫‪d‬‬ ‫‪y‬‬ ‫=‬ ‫‪1‬‬ ‫)‪(2‬‬ ‫‪3x‬‬ ‫‪a -3‬‬ ‫‪3‬‬ ‫‪3‬‬ ‫ﺧﻄًّﺎ ﺃﻓﻘﻴًّﺎ ﻫﻲ‪:‬‬ ‫`‪8‬‬ ‫‪1‬‬ ‫‪^α +‬‬ ‫‪2hx‬‬ ‫‪ α‬ﺍﻟﺘﻲ ﺗﺠﻌﻞ ﺑﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ‬ ‫ﻗﻴﻤﺔ‬ ‫)‪(8‬‬ ‫‪2‬‬ ‫=‪y‬‬ ‫‪j‬‬ ‫‪+‬‬ ‫‪3‬‬ ‫‪b -2‬‬ ‫‪c -8‬‬ ‫‪d0‬‬ ‫)‪ (9‬ﺑﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ‪ f(x) = 3(5) x - 1 :‬ﻫﻮ ﺍﻧﻌﻜﺎﺱ ﻓﻲ ﻣﺤﻮﺭ ﺍﻟﺼﺎﺩﺍﺕ ﻟﺒﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ‪g(x) = :‬‬ ‫‪a 3 (5) x + 1‬‬ ‫‪b 3 (5) -x - 1‬‬ ‫‪c -3 (5) x + 1‬‬ ‫‪d 3 (5) -x + 1‬‬ ‫ﺑﺎﻧﺴﺤﺎﺏ‪:‬‬ ‫‪y‬‬ ‫=‬ ‫‪1‬‬ ‫‪(5) x‬‬ ‫ﺍﻟﺪﺍﻟﺔ‬ ‫ﺑﻴﺎﻥ‬ ‫ﺑﺎﺳﺘﺨﺪﺍﻡ‬ ‫‪y‬‬ ‫=‬ ‫‪1‬‬ ‫‪(5) x+2‬‬ ‫‪-‬‬ ‫‪3‬‬ ‫ﺍﻟﺪﺍﻟﺔ‬ ‫ﺑﻴﺎﻥ‬ ‫ﺭﺳﻢ‬ ‫ﻳﻤﻜﻦ‬ ‫)‪(10‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪ a‬ﻭﺣﺪﺗﻴﻦ ﺟﻬﺔ ﺍﻟﻴﺴﺎﺭ ﻭ‪ 3‬ﻭﺣﺪﺍﺕ ﻷﺳﻔﻞ ‪ b‬ﻭﺣﺪﺗﻴﻦ ﺟﻬﺔ ﺍﻟﻴﻤﻴﻦ ﻭ‪ 3‬ﻭﺣﺪﺍﺕ ﻷﺳﻔﻞ‬ ‫‪ 3 c‬ﻭﺣﺪﺍﺕ ﺟﻬﺔ ﺍﻟﻴﻤﻴﻦ ﻭﻭﺣﺪﺗﻴﻦ ﻷﻋﻠﻰ ‪ d‬ﻭﺣﺪﺗﻴﻦ ﺟﻬﺔ ﺍﻟﻴﻤﻴﻦ ﻭ‪ 3‬ﻭﺣﺪﺍﺕ ﻷﻋﻠﻰ‬ ‫)‪ (11‬ﻣﻌﺎﺩﻟﺔ ﺍﻟﺪﺍﻟﺔ ﺍﻷﺳﻴﺔ ﺍﻟﺘﻲ ﻋﻠﻰ ﺍﻟﺼﻮﺭﺓ ‪ y = a(b)x‬ﺣﻴﺚ ﺍﻷﺳﺎﺱ ﻳﺴﺎﻭﻱ ‪ 0.6‬ﻭﻳﻤﺮ ﺭﺳﻤﻬﺎ ﺍﻟﺒﻴﺎﻧﻲ ﺑﺎﻟﻨﻘﻄﺔ‬ ‫)‪ (2 , 1.8‬ﻫﻲ‪:‬‬ ‫‪a y = 1.8 (2) x‬‬ ‫‪b y = 0.2 (1.8) x‬‬ ‫‪c y = 2 (0.6) x‬‬ ‫‪d y = 5 (0.6) x‬‬ ‫‪x0 1 2 3‬‬ ‫)‪ (12‬ﺃﻱ ﻣﻦ ﺍﻟﺪﻭﺍﻝ ﺍﻟﺘﺎﻟﻴﺔ ﺗﻨﻤﺬﺝ ﺑﻴﺎﻧﺎﺕ ﺍﻟﺠﺪﻭﻝ ﺍﻟﻤﻘﺎﺑﻞ‪:‬‬ ‫‪y 4 5.2 6.76 8.79‬‬ ‫‪a‬‬ ‫‪y‬‬ ‫=‬ ‫‪x2‬‬ ‫‪+‬‬ ‫‪1‬‬ ‫‪x‬‬ ‫‪+‬‬ ‫‪4‬‬ ‫‪b y = 4 (1.3) x‬‬ ‫‪c y = 1.6 (4) x‬‬ ‫‪d y = 4 (0.6) x + 2.8‬‬ ‫‪2‬‬ ‫‪58‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺍﻟﺪﻭﺍﻝ ﺍﻟﻠﻮﻏﺎﺭﻳﺘﻤﻴﺔ ﻭﺗﻤﺜﻴﻠﻬﺎ ﺑﻴﺎﻧﻴًّﺎ ‪4-3‬‬ ‫‪Logarithmic Functions and their Graphs‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-8‬ﺍﻛﺘﺐ ﻛﻞ ﻣﻌﺎﺩﻟﺔ ﻣﻤﺎ ﻳﻠﻲ ﻓﻲ ﺍﻟﺼﻮﺭﺓ ﺍﻟﻠﻮﻏﺎﺭﻳﺘﻤﻴﺔ‪:‬‬ ‫‪(1) 42 = 16‬‬ ‫‪(2) 73 = 343‬‬ ‫)‪(3‬‬ ‫`‬ ‫‪1‬‬ ‫‪-2‬‬ ‫=‬ ‫‪4‬‬ ‫)‪(4‬‬ ‫‪8‬‬ ‫‪-‬‬ ‫‪2‬‬ ‫=‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪4‬‬ ‫‪j‬‬ ‫)‪(5‬‬ ‫`‬ ‫‪1‬‬ ‫‪3‬‬ ‫=‬ ‫‪1‬‬ ‫‪(6) 10-2 = 0 . 01‬‬ ‫‪3‬‬ ‫)‪(8‬‬ ‫‪5-3‬‬ ‫=‬ ‫‪1‬‬ ‫‪3‬‬ ‫‪27‬‬ ‫‪125‬‬ ‫‪j‬‬ ‫‪(7) 62 = 6 6‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(9-14‬ﺍﻛﺘﺐ ﻛﻞ ﻣﻌﺎﺩﻟﺔ ﻣﻤﺎ ﻳﻠﻲ ﻓﻲ ﺍﻟﺼﻮﺭﺓ ﺍﻷﺳﻴﺔ‪:‬‬ ‫‪(9) log2128 = 7‬‬ ‫‪(10) log464 = 3‬‬ ‫‪(11) log 100 = 2‬‬ ‫‪(13) log 0.0001 = - 4‬‬ ‫)‪(12‬‬ ‫‪log3‬‬ ‫‪1‬‬ ‫‪=-‬‬ ‫‪2‬‬ ‫)‪(14‬‬ ‫‪log3‬‬ ‫‪1‬‬ ‫‪=-‬‬ ‫‪5‬‬ ‫‪9‬‬ ‫‪243‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(15-20‬ﺃﻭﺟﺪ ﻗﻴﻤﺔ ﻛﻞ ﻟﻮﻏﺎﺭﻳﺘﻢ ﻣﻤﺎ ﻳﻠﻲ‪:‬‬ ‫‪(15) log2 4‬‬ ‫‪(16) log2 8‬‬ ‫‪(17) log8 8‬‬ ‫‪(18) log2 25‬‬ ‫)‪(19‬‬ ‫‪log 1‬‬ ‫‪1‬‬ ‫‪(20) log 0.01‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(21-23‬ﺃﻭﺟﺪ ﻣﺠﺎﻝ ﺍﻟﺘﻌﺮﻳﻒ ﻟﻜﻞ ﺩﺍﻟﺔ ﻣﻤﺎ ﻳﻠﻲ‪:‬‬ ‫)‪(21) y = log (x + 1‬‬ ‫)‪(22‬‬ ‫‪y‬‬ ‫=‬ ‫‪log‬‬ ‫)‪(x‬‬ ‫‪-‬‬ ‫‪2‬‬ ‫)‪(23) y = log(x2 - 4‬‬ ‫‪6‬‬ ‫‪8‬‬ ‫)‪ (24‬ﻳﺴﺎﻭﻱ ﺗﺮﻛﻴﺰ ﺃﻳﻮﻥ ﺍﻟﻬﻴﺪﺭﻭﺟﻴﻦ @‪ 6H +‬ﻓﻲ ﺍﻟﻠﻴﻢ )ﻧﻮﻉ ﻣﻦ ﺍﻟﻠﻴﻤﻮﻥ( ﺣﻮﺍﻟﻰ ‪1 . 26 # 10-2‬‬ ‫ﺃﻭﺟﺪ ﺭﻗﻤﻪ ﺍﻟﻬﻴﺪﺭﻭﺟﻴﻨﻲ )‪ (pH‬ﻋﻠ ًﻤﺎ ﺃﻥ @‪.pH = - log6H+‬‬ ‫)‪ (25‬ﻳﺴﺎﻭﻱ ﺍﻟﺮﻗﻢ ﺍﻟﻬﻴﺪﺭﻭﺟﻴﻨﻲ ﻟﻌﺼﻴﺮ ﺧﻞ ﺍﻟﺘﻔﺎﺡ )‪ (Cider Vinegar‬ﺣﻮﺍﻟﻰ ‪3.1‬‬ ‫ﺃﻭﺟﺪ ﺗﺮﻛﻴﺰ ﺃﻳﻮﻧﻪ ﺍﻟﻬﻴﺪﺭﻭﺟﻴﻨﻲ @‪.6H+‬‬ ‫)‪(26) y = log3 (x‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺮﻳﻨﻴﻦ )‪ ،(26-27‬ﻣﺜّﻞ ﺑﻴﺎﻧﻴًّﺎ ﻛﻞ ﺩﺍﻟﺔ ﻟﻮﻏﺎﺭﻳﺘﻤﻴﺔ ﻣﻌﻴﻨًﺎ ﺍﻟﻤﺠﺎﻝ ﻭﺍﻟﻤﺪﻯ‪.‬‬ ‫‪(27) y = log3 (x - 1) + 2‬‬ ‫)‪ (28‬ﺍﺷﺮﺡ ﻟﻤﺎﺫﺍ ‪ b‬ﻻ ﺗﺴﺘﻄﻴﻊ ﺃﻥ ﺗﺄﺧﺬ ﻗﻴﻤﺔ ‪ 1‬ﻓﻲ ﺍﻟﺪﺍﻟﺔ‪y = logb (x) :‬‬ ‫‪59‬‬

‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫‪ab‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-5‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ab‬‬ ‫)‪ (1‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ y = 3x‬ﻓﺈﻥ ‪x = log y‬‬ ‫‪ab‬‬ ‫‪ab‬‬ ‫)‪ (2‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ log2(- y) = x‬ﻓﺈﻥ ‪y = 2-x‬‬ ‫‪ab‬‬ ‫)‪ (3‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ 4x = 5‬ﻓﺈﻥ ‪. 2x = log2 5‬‬ ‫)‪ (4‬ﻣﺠﺎﻝ ﺍﻟﺪﺍﻟﺔ ‪ f(x) = log^x2h‬ﻫﻮ ‪R‬‬ ‫)‪ (5‬ﺑﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ ‪ y = log3 x‬ﻫﻮ ﺍﻧﻌﻜﺎﺱ ﻓﻲ ﺍﻟﻤﺴﺘﻘﻴﻢ ‪ y - x = 0‬ﻟﺒﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ ‪y = 3x‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(6-11‬ﻇﻠّﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫)‪ (6‬ﻣﻌﻜﻮﺱ ﺍﻟﺪﺍﻟﺔ ‪ y = log2x‬ﻫﻮ‪:‬‬ ‫‪a y = logx 2‬‬ ‫‪b y = x2‬‬ ‫‪c y = 2x‬‬ ‫‪d y = log 2x‬‬ ‫)‪ (7‬ﻣﺠﺎﻝ ﺍﻟﺪﺍﻟﺔ ‪ y = log x - 1‬ﻫﻮ‪:‬‬ ‫‪aR‬‬ ‫‪b R+‬‬ ‫‪c ^1, 3h‬‬ ‫‪d R/\"1,‬‬ ‫)‪ (8‬ﻣﺠﺎﻝ ﺍﻟﺪﺍﻟﺔ ‪ y = log^x2 + 1h‬ﻫﻮ‪:‬‬ ‫‪aR‬‬ ‫‪b R+‬‬ ‫‪c 61, 3h‬‬ ‫‪d ^1, 3h‬‬ ‫‪a y = log^x - 1h - 1‬‬ ‫)‪ (9‬ﺑﺎﺳﺘﺨﺪﺍﻡ ﺩﺍﻟﺔ ﺍﻟﻤﺮﺟﻊ ‪ y = log5x‬ﻳﻤﻜﻦ ﺗﻤﺜﻴﻞ ﺍﻟﺪﺍﻟﺔ‪:‬‬ ‫)‪b y = log5 (5x‬‬ ‫‪c y = log5 ^x - 1h - 1‬‬ ‫‪d y = log5 ^x2 + 1h‬‬ ‫)‪ (10‬ﻳﻤﻜﻦ ﺭﺳﻢ ﺑﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ ‪ y = log(x + 1) - 2‬ﻣﻌﺘﺒ ًﺮﺍ ﺩﺍﻟﺔ ﺍﻟﻤﺮﺟﻊ ‪ y = log x‬ﺑﺎﻧﺴﺤﺎﺏ‪:‬‬ ‫‪ b‬ﻭﺣﺪﺓ ﺇﻟﻰ ﺍﻟﻴﻤﻴﻦ ﻭﻭﺣﺪﺗﻴﻦ ﻷﺳﻔﻞ‬ ‫‪ a‬ﻭﺣﺪﺓ ﺇﻟﻰ ﺍﻟﻴﺴﺎﺭ ﻭﻭﺣﺪﺗﻴﻦ ﻷﺳﻔﻞ‬ ‫‪ d‬ﻭﺣﺪﺗﻴﻦ ﺇﻟﻰ ﺍﻟﻴﺴﺎﺭ ﻭﻭﺣﺪﺓ ﻷﻋﻠﻰ‬ ‫‪ c‬ﻭﺣﺪﺗﻴﻦ ﺇﻟﻰ ﺍﻟﻴﻤﻴﻦ ﻭﻭﺣﺪﺓ ﻷﻋﻠﻰ‬ ‫)‪ (11‬ﻳﻌﻄﻰ ﺍﻟﺮﻗﻢ ﺍﻟﻬﻴﺪﺭﻭﺟﻴﻨﻲ )‪ (pH‬ﺑﺎﻟﻌﻼﻗﺔ‪ pH = - log6H+@ :‬ﺇﺫﺍ ﻛﺎﻥ ﺗﺮﻛﻴﺰ ﺃﻳﻮﻥ ﺍﻟﻬﻴﺪﺭﻭﺟﻴﻨﻲ @‪ 6H+‬ﻓﻲ‬ ‫ﺍﻟﺴﺒﺎﻧﺦ ﻫﻮ ‪ 4 # 10-6‬ﻓﺈﻥ ﺍﻟﺮﻗﻢ ﺍﻟﻬﻴﺪﺭﻭﺟﻴﻨﻲ ﻟﻠﺴﺒﺎﻧﺦ ﻫﻮ‪:‬‬ ‫‪a -6.6‬‬ ‫‪b 6.6‬‬ ‫‪c -5.4‬‬ ‫‪d 5.4‬‬ ‫‪60‬‬

‫ﻓﻲ ﺍﻟﺒﻨﻮﺩ )‪ ،(12-15‬ﻟﺪﻳﻚ ﻗﺎﺋﻤﺘﺎﻥ ﺍﺧﺘﺮ ﻣﻦ ﺍﻟﻘﺎﺋﻤﺔ )‪ (2‬ﻣﺎ ﻳﻨﺎﺳﺐ ﻛﻞ ﺗﻤﺮﻳﻦ ﻓﻲ ﺍﻟﻘﺎﺋﻤﺔ )‪ (1‬ﻟﺘﺤﺼﻞ ﻋﻠﻰ ﺇﺟﺎﺑﺔ ﺻﺤﻴﺤﺔ‪.‬‬ ‫ﺍﻟﻘﺎﺋﻤﺔ )‪(2‬‬ ‫ﺍﻟﻘﺎﺋﻤﺔ )‪(1‬‬ ‫‪a y = 4x‬‬ ‫ﻣﻌﻜﻮﺱ ﺍﻟﺪﺍﻟﺔ‪:‬‬ ‫‪b‬‬ ‫‪y‬‬ ‫=‬ ‫‪a‬‬ ‫‪-1‬‬ ‫‪-x‬‬ ‫)‪ y = - log 1 x (12‬ﻫﻮ‬ ‫‪4‬‬ ‫‪k‬‬ ‫‪4‬‬ ‫‪c‬‬ ‫‪y‬‬ ‫=‬ ‫‪a‬‬ ‫‪1‬‬ ‫‪x‬‬ ‫)‪ y = - log4x (13‬ﻫﻮ‬ ‫‪4‬‬ ‫‪k‬‬ ‫‪d y = (- 4) -x‬‬ ‫ﺍﻟﻘﺎﺋﻤﺔ )‪(2‬‬ ‫ﺍﻟﻘﺎﺋﻤﺔ )‪(1‬‬ ‫ﺑﻴﺎﻥ ﻣﻌﻜﻮﺱ ﻛﻞ ﺩﺍﻟﺔ ﻣﻤﺎ ﻳﻠﻲ ﻫﻮ‪:‬‬ ‫‪a‬‬ ‫‪y‬‬ ‫‪8‬‬ ‫)‪y = log3(x) (14‬‬ ‫‪7‬‬ ‫)‪y = log2(4x) (15‬‬ ‫‪6‬‬ ‫‪5‬‬ ‫‪4‬‬ ‫‪3‬‬ ‫‪2‬‬ ‫‪1‬‬ ‫‪-6 -5 -4 -3 -2 -1‬‬ ‫‪1 2 3 4 5 6x‬‬ ‫‪-1‬‬ ‫‪-2‬‬ ‫‪-3‬‬ ‫‪-4‬‬ ‫‪b‬‬ ‫‪y‬‬ ‫‪5‬‬ ‫‪4‬‬ ‫‪3‬‬ ‫‪2‬‬ ‫‪1‬‬ ‫‪-3 -2 -1‬‬ ‫‪1 23 4 56 7 8 9‬‬ ‫‪x‬‬ ‫‪-1‬‬ ‫‪-2‬‬ ‫‪-3‬‬ ‫‪-4‬‬ ‫‪-5‬‬ ‫‪-6‬‬ ‫‪-7‬‬ ‫‪c‬‬ ‫‪y‬‬ ‫‪5‬‬ ‫‪4‬‬ ‫‪3‬‬ ‫‪2‬‬ ‫‪1‬‬ ‫‪-3 -2 -1‬‬ ‫‪1 23 4 56 7 8 9‬‬ ‫‪x‬‬ ‫‪-1‬‬ ‫‪-2‬‬ ‫‪-3‬‬ ‫‪-4‬‬ ‫‪-5‬‬ ‫‪-6‬‬ ‫‪-7‬‬ ‫‪61‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺧﻮﺍﺹ ﺍﻟﻠﻮﻏﺎﺭﻳﺘﻤﺎﺕ ‪4-4‬‬ ‫‪Properties of Logarithms‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-8‬ﺍﻛﺘﺐ ﻛﻞ ﻣﻘﺪﺍﺭ ﻟﻮﻏﺎﺭﻳﺘﻤﻲ ﻓﻲ ﺻﻮﺭﺓ ﻟﻮﻏﺎﺭﻳﺘﻢ ﻭﺍﺣﺪ‪.‬‬ ‫‪(1) log 7 + log 2‬‬ ‫)‪(2‬‬ ‫‪1‬‬ ‫‪log4 y - log4 x‬‬ ‫‪,‬‬ ‫‪^x‬‬ ‫‪2‬‬ ‫‪0‬‬ ‫‪,‬‬ ‫‪y 2 0h‬‬ ‫‪2‬‬ ‫)‪(3) 4 log M - log N , (M 2 0 , N 2 0‬‬ ‫)‪(4) log x + log y + log z , (x 2 0 , y 2 0 , z 2 0‬‬ ‫)‪(5‬‬ ‫‪log‬‬ ‫‪a‬‬ ‫‪+‬‬ ‫‪log‬‬ ‫‪b‬‬ ‫‪-‬‬ ‫‪log‬‬ ‫‪c‬‬ ‫‪, (a 2 0 ,‬‬ ‫‪b 2 0,‬‬ ‫)‪c 2 0‬‬ ‫‪4‬‬ ‫‪3‬‬ ‫‪2‬‬ ‫)‪(6) log a + 3 log b , (a 2 0 , b 2 0‬‬ ‫)‪(7‬‬ ‫‪1‬‬ ‫‪^log‬‬ ‫‪7‬‬ ‫‪x‬‬ ‫‪+‬‬ ‫‪log7 yh -‬‬ ‫‪3 log7 a‬‬ ‫‪,‬‬ ‫‪^x‬‬ ‫‪2‬‬ ‫‪0,‬‬ ‫‪y‬‬ ‫‪2‬‬ ‫‪0,a‬‬ ‫‪2‬‬ ‫‪0h‬‬ ‫‪2‬‬ ‫)‪(8) 7 log r - log x + log n , (r 2 0 , x 2 0 , n 2 0‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(9-16‬ﺃﻭﺟﺪ ﻣﻔﻜﻮﻙ ﻛﻞ ﻟﻮﻏﺎﺭﻳﺘﻢ ﻣﻤﺎ ﻳﻠﻲ‪:‬‬ ‫)‪(9‬‬ ‫‪y‬‬ ‫‪, ^x 2 0, y 2 0h‬‬ ‫‪(10) log x3 + y5 , ^x 2 0 , y 2 0h‬‬ ‫‪log5 x‬‬ ‫)‪(11‬‬ ‫‪log3 7 (2x - 3) 2‬‬ ‫‪,‬‬ ‫‪`x‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪j‬‬ ‫)‪(12‬‬ ‫‪log‬‬ ‫‪a2b3‬‬ ‫‪,‬‬ ‫)‪(a 2 0 , b 2 0 , c 2 0‬‬ ‫‪2‬‬ ‫‪c4‬‬ ‫)‪(13) log 3M 4N-2 , (M 2 0 , N 2 0‬‬ ‫)‪(14) log45 x , (x 2 0‬‬ ‫)‪(15) log(2(x + 1))3 , (x 2-1‬‬ ‫‪(16) log‬‬ ‫‪2x‬‬ ‫)‪, (x 2 0 , y 2 0‬‬ ‫‪y‬‬ ‫)‪ (17‬ﺍﻟﺴﺆﺍﻝ ﺍﻟﻤﻔﺘﻮﺡ‪ :‬ﺍﺳﺘﺨﺪﻡ ﺧﻮﺍﺹ ﺍﻟﻠﻮﻏﺎﺭﻳﺘﻤﺎﺕ ﻹﻋﺎﺩﺓ ﻛﺘﺎﺑﺔ ‪ log 64‬ﺑﺄﺭﺑﻊ ﻃﺮﺍﺋﻖ ﻣﺨﺘﻠﻔﺔ‪.‬‬ ‫)‪ (18‬ﺍﻟﻜﺘﺎﺑﺔ‪ :‬ﺍﺷﺮﺡ ﻟﻤﺎﺫﺍ ‪log^5 # 2h ! log 5 # log 2‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(19-23‬ﺍﺳﺘﺨﺪﻡ ﺧﻮﺍﺹ ﺍﻟﻠﻮﻏﺎﺭﻳﺘﻤﺎﺕ ﻹﻳﺠﺎﺩ ﻗﻴﻤﺔ ﻛﻞ ﻣﻘﺪﺍﺭ‪.‬‬ ‫‪(19) log24 - log216‬‬ ‫‪(20) log55 - log5125‬‬ ‫‪(21) 3 log22 - log24‬‬ ‫‪(22) log 1 + log 100‬‬ ‫‪(23) log 5 + log 8 - 2 log 2‬‬ ‫‪62‬‬

‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(24-28‬ﻟﻨﻔﺘﺮﺽ ﺃﻥ ‪ .log 4 - 0.6021 , log 5 - 0.6990 , log 6 - 0.7782‬ﺍﺳﺘﺨﺪﻡ ﺧﻮﺍﺹ‬ ‫ﺍﻟﻠﻮﻏﺎﺭﻳﺘﻤﺎﺕ ﻹﻳﺠﺎﺩ ﻗﻴﻤﺔ ﻛﻞ ﻣﻘﺪﺍﺭ‪ .‬ﺩﻭﻥ ﺍﺳﺘﺨﺪﺍﻡ ﺁﻟﺘﻚ ﺍﻟﺤﺎﺳﺒﺔ ﻗ ّﺮﺏ ﺇﺟﺎﺑﺎﺗﻚ ﺇﻟﻰ ﺃﻗﺮﺏ ﺟﺰﺀ ﻣﻦ ﺃﻟﻒ‪.‬‬ ‫‪(24) log20‬‬ ‫‪(25) log16‬‬ ‫‪(26) log1.25‬‬ ‫‪(27) log125‬‬ ‫)‪(28‬‬ ‫‪log‬‬ ‫‪1‬‬ ‫‪36‬‬ ‫)‪ (29‬ﺍﻟﻌﻠﻮﻡ‪ :‬ﻳﺴﺘﻄﻴﻊ ﺍﻹﻧﺴﺎﻥ ﺳﻤﺎﻉ ﻣﺪﻯ ﻭﺍﺳﻊ ﻣﻦ ﺷﺪﺓ ﺍﻟﺼﻮﺕ‪ ،‬ﻭﻫﺬﺍ ﻣﺎ ﻳﻮﺿﺤﻪ ﺍﻟﺠﺪﻭﻝ ﺍﻟﺘﺎﻟﻲ‪ .‬ﺷﺪﺓ ﺍﻟﺼﻮﺕ‬ ‫ﻫﻲ ﻗﻴﺎﺱ ﻛﻤﻴﺔ ﺍﻟﻄﺎﻗﺔ ﺍﻟﻨﺎﺗﺠﺔ ﻋﻦ ﻣﺼﺪﺭ ﺍﻟﺼﻮﺕ‪ ،‬ﻭﻳﻌﺘﻤﺪ ﻣﺴﺘﻮﻯ ﺷﺪﺓ ﺍﻟﺼﻮﺕ ﻋﻠﻰ ﺷﺪﺓ ﺍﻟﺼﻮﺕ‪ ،‬ﻭﻋﻠﻰ‬ ‫ﺍﻟﻤﺴﺎﻓﺔ ﺑﻴﻦ ﻣﺼﺪﺭ ﺍﻟﺼﻮﺕ ﻭﺍﻟﺸﺨﺺ ﺍﻟﺬﻱ ﻳﺴﻤﻌﻪ‪ .‬ﻭﻳﻌﺮﻑ ﻣﺴﺘﻮﻯ ﺷﺪﺓ ﺍﻟﺼﻮﺕ ﺍﻟﻤﻘﺎﺱ ﺑﺎﻟﺪﻳﺴﻴﺒﻞ‬ ‫ﺑﺎﻟﻜﺎﺩ‬ ‫ﺍﻟﺼﻮﺕ‬ ‫ﺷﺪﺓ‬ ‫‪I0‬‬ ‫ﺍﻟﺼﻮﺕ‪،‬‬ ‫ﺷﺪﺓ‬ ‫‪I‬‬ ‫ﺣﻴﺚ‬ ‫‪،10 log‬‬ ‫‪I‬‬ ‫=‬ ‫ﺍﻟﺼﻮﺕ‬ ‫ﺷﺪﺓ‬ ‫ﻣﺴﺘﻮﻯ‬ ‫ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫ﺑﺎﻟﻤﻌﺎﺩﻟﺔ‬ ‫)‪(dB‬‬ ‫‪I0‬‬ ‫ﻣﺴﻤﻮﻉ‪.‬‬ ‫ﺃﻛﻤﻞ ﺍﻟﺠﺪﻭﻝ ﺍﻟﺘﺎﻟﻲ‪:‬‬ ‫ﻣﺴﺘﻮﻯ ﺷﺪﺓ ﺍﻟﺼﻮﺕ )ﺩﻳﺴﻴﺒﻞ ‪(dB‬‬ ‫ﺍﻟﺸﺪﺓ ‪W/m2‬‬ ‫ﻧﻮﻉ ﺍﻟﺼﻮﺕ‬ ‫‪120‬‬ ‫‪1‬‬ ‫ﺻﻮﺕ ﻋﺎ ٍﻝ‬ ‫‪0‬‬ ‫‪10-2‬‬ ‫ﺻﻮﺕ ﺁﻟﺔ ﺛﻘﺐ‬ ‫‪10-5‬‬ ‫ﺻﻮﺕ ﺷﺎﺭﻉ ﻣﺰﺩﺣﻢ‬ ‫‪10- 6‬‬ ‫ﺻﻮﺕ ﻣﺤﺎﺩﺛﺔ‬ ‫‪10- 10‬‬ ‫‪10-11‬‬ ‫ﺻﻮﺕ ﻫﻤﺲ‬ ‫‪10-12‬‬ ‫ﺣﻔﻴﻒ ﺃﻭﺭﺍﻕ ﺍﻷﺷﺠﺎﺭ‬ ‫ﺻﻮﺕ ﺑﺎﻟﻜﺎﺩ ﻣﺴﻤﻮﻉ‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-6‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ab‬‬ ‫)‪log (x - 1)2 = 2 log x - 1 (1‬‬ ‫‪ab‬‬ ‫‪ab‬‬ ‫‪log‬‬ ‫‪1‬‬ ‫‪= - 2 log‬‬ ‫‪x,x‬‬ ‫‪2‬‬ ‫‪0‬‬ ‫)‪(2‬‬ ‫‪ab‬‬ ‫‪x2‬‬ ‫‪ab‬‬ ‫‪m‬‬ ‫‪ab‬‬ ‫‪log c‬‬ ‫‪n‬‬ ‫=‪m‬‬ ‫‪1‬‬ ‫‪log m - log n ,‬‬ ‫‪m‬‬ ‫‪2‬‬ ‫‪0,n‬‬ ‫‪2‬‬ ‫‪0‬‬ ‫)‪(3‬‬ ‫‪2‬‬ ‫‪log‬‬ ‫‪16‬‬ ‫‪-‬‬ ‫‪log2‬‬ ‫‪2‬‬ ‫=‬ ‫‪log‬‬ ‫‪8‬‬ ‫)‪(4‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫= )‪log (x - y‬‬ ‫‪log x‬‬ ‫‪,‬‬ ‫‪x,‬‬ ‫‪y ! R+/\"1,‬‬ ‫)‪(5‬‬ ‫‪log y‬‬ ‫)‪log64 + log69 = 2 (6‬‬ ‫‪63‬‬

‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(7-13‬ﻇﻠّﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫)‪ (7‬ﺍﻟﻤﻘﺪﺍﺭ ‪ 2 log4 8 + log5 125‬ﻳﺴﺎﻭﻱ‪:‬‬ ‫‪a 4 b 5 c 6 d 15‬‬ ‫)‪ (8‬ﺇﺫﺍ ﻛﺎﻥ ‪ log 3 = x , log 5 = y‬ﻓﺈﻥ ‪ log 45‬ﺗﺴﺎﻭﻱ‪:‬‬ ‫‪a x+y‬‬ ‫‪b 2x + y‬‬ ‫‪c 2y + x‬‬ ‫‪d x2y‬‬ ‫‪a1‬‬ ‫ﻳﺴﺎﻭﻱ‪:‬‬ ‫‪log2 x + log2 2x + log2‬‬ ‫‪1‬‬ ‫‪,x‬‬ ‫‪2‬‬ ‫‪0‬‬ ‫)‪(9‬‬ ‫‪x2‬‬ ‫‪b2‬‬ ‫‪c x d 2x‬‬ ‫)‪ (10‬ﺇﺫﺍ ﻛﺎﻥ ‪ log 2 = m , log 3 = n‬ﻓﺈﻥ ﺍﻟﻤﻘﺪﺍﺭ ‪ m + n - 1‬ﻳﺴﺎﻭﻱ‪:‬‬ ‫‪a log 0.06‬‬ ‫‪b log 0.6‬‬ ‫‪c log 6‬‬ ‫‪d log 60‬‬ ‫)‪ (11‬ﻋﻨﺪﻣﺎ ‪ m = 3 , n = 2‬ﻓﺈﻥ ﺍﻟﻤﻘﺪﺍﺭ ﺍﻷﻛﺒﺮ ﻗﻴﻤﺔ ﻓﻴﻤﺎ ﻳﻠﻲ ﻫﻮ‪:‬‬ ‫‪a log n2 - log m3‬‬ ‫‪b log m2 - log n2‬‬ ‫‪c 3 log n - 2 log m d 2 log m - 3 log n‬‬ ‫‪ logc3‬ﻫﻮ‪:‬‬ ‫‪8‬‬ ‫‪m‬‬ ‫ﺍﻟﻤﻘﺪﺍﺭ‬ ‫ﻣﻔﻜﻮﻙ‬ ‫)‪(12‬‬ ‫‪x3‬‬ ‫‪a‬‬ ‫‪3‬‬ ‫‪log‬‬ ‫‪8‬‬ ‫‪b‬‬ ‫‪1‬‬ ‫‪^log‬‬ ‫‪^8‬‬ ‫‪-‬‬ ‫‪x3hh‬‬ ‫‪c log 2 - log x‬‬ ‫‪d log 2 - 3 log x‬‬ ‫‪x3‬‬ ‫‪3‬‬ ‫‪L‬‬ ‫=‬ ‫‪10‬‬ ‫‪log‬‬ ‫‪I‬‬ ‫ﺇﺫﺍ ﻛﺎﻥ ﻣﺴﺘﻮﻯ ﺷﺪﺓ ﺻﻮﺕ ﺻﻔﺎﺭﺓ ﺇﻧﺬﺍﺭ )‪ (L‬ﺗﺴﺎﻭﻱ ‪ 140 dB‬ﻭﺍﻟﺘﻲ ﺗﻘﺎﺱ ﺑﺎﻟﻌﻼﻗﺔ‪:‬‬ ‫)‪(13‬‬ ‫‪10-12‬‬ ‫ﻓﺈﻥ ﺷﺪﺓ ﺻﻮﺗﻬﺎ ‪ I‬ﺗﺴﺎﻭﻱ‪:‬‬ ‫‪a1‬‬ ‫‪b 1000‬‬ ‫‪c 10‬‬ ‫‪d 100‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺮﻳﻨﻴﻦ )‪ ،(14-15‬ﻟﺪﻳﻚ ﻗﺎﺋﻤﺘﺎﻥ ﺍﺧﺘﺮ ﻣﻦ ﺍﻟﻘﺎﺋﻤﺔ )‪ (2‬ﻣﺎ ﻳﻨﺎﺳﺐ ﻛﻞ ﺗﻤﺮﻳﻦ ﻓﻲ ﺍﻟﻘﺎﺋﻤﺔ )‪ (1‬ﻟﺘﺤﺼﻞ ﻋﻠﻰ ﺇﺟﺎﺑﺔ ﺻﺤﻴﺤﺔ‪.‬‬ ‫ﺳﻠﻢ ﺗﺪﺭﺝ ﺍﻟﻀﺠﻴﺞ‬ ‫ﺍﻟﻤﻘﺎﺑﻞ‪.‬‬ ‫ﻭﺍﻟﺸﻜﻞ‬ ‫‪L‬‬ ‫=‬ ‫‪10 log‬‬ ‫‪I‬‬ ‫ﺍﻟﻌﻼﻗﺔ‪:‬‬ ‫ﺍﺳﺘﺨﺪﻡ‬ ‫‪dB‬‬ ‫‪10-12‬‬ ‫‪160‬‬ ‫ﺍﻟﻘﺎﺋﻤﺔ )‪(2‬‬ ‫ﺍﻟﻘﺎﺋﻤﺔ )‪(1‬‬ ‫ﻣﺆﻟﻢ‬ ‫‪ a‬ﻫﺎﺩﺋﺔ‬ ‫ﺇﺫﺍ ﻛﺎﻧﺖ ﺷﺪﺓ ﺻﻮﺕ ﻣﺎ )‪ (I‬ﻫﻲ‪:‬‬ ‫ﺟﱟﺪﺍ ‪140‬‬ ‫ﻣﺆﻟﻢ‬ ‫‪120‬‬ ‫ﻋﺎ ٍﻝ‬ ‫‪ b‬ﻣﺆﻟﻤﺔ‬ ‫)‪ 10-5 (14‬ﻓﺈﻥ ﻗﻮﺗﻪ ﺗﻜﻮﻥ‪:‬‬ ‫‪ c‬ﻋﺎﻟﻴﺔ‬ ‫)‪ 1.65 # 10-2 (15‬ﻓﺈﻥ ﻗﻮﺗﻪ ﺗﻜﻮﻥ‪:‬‬ ‫ﺟ ًّﺪﺍ ‪90‬‬ ‫‪ d‬ﻋﺎﻟﻴﺔ ﺟ ًّﺪﺍ‬ ‫ﻋﺎ ٍﻝ‬ ‫‪70‬‬ ‫ﻣﻌﺘﺪﻝ‬ ‫‪50‬‬ ‫ﻫﺎﺩﺉ‬ ‫‪30‬‬ ‫ﻫﺎﺩﺉ ﺟ ًّﺪﺍ‬ ‫‪0‬‬ ‫ﻋﺘﺒﺔ ﺍﻟﺴﻤﻊ‬ ‫‪64‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻷﺳﻴﺔ ﻭﺍﻟﻠﻮﻏﺎﺭﻳﺘﻴﻤﻴﺔ ‪4-5‬‬ ‫‪Exponential and Logarithmic Equations‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-8‬ﺣ ّﻞ ﻛﻞ ﻣﻌﺎﺩﻟﺔ ﻣﻤﺎ ﻳﻠﻲ‪ .‬ﺍﺧﺘﺒﺮ ﺻﺤﺔ ﻛﻞ ﺣﻞ‪:‬‬ ‫‪(1) 92y = 66‬‬ ‫‪(2) 12y-2 = 20‬‬ ‫‪(3) 5 - 3x = - 40‬‬ ‫‪(4) 252x+1 = 144‬‬ ‫‪(7) 7 n2 - 12 = 5‬‬ ‫‪(8) -3 + 2 4 x3 = 33‬‬ ‫‪3‬‬ ‫‪5‬‬ ‫‪(5) 3x 2 = 27 , x 2 0‬‬ ‫‪(6) 2 + 8r 3 = 26‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(9-13‬ﺍﺳﺘﺨﺪﻡ ﻗﺎﻋﺪﺓ ﺗﻐﻴﻴﺮ ﺍﻷﺳﺎﺱ ﻹﻳﺠﺎﺩ ﻗﻴﻤﺔ ﻛﻞ ﻟﻮﻏﺎﺭﻳﺘﻢ ﻣﻤﺎ ﻳﻠﻲ‪:‬‬ ‫‪(9) log27‬‬ ‫‪(10) log333‬‬ ‫‪(11) log21 0.085‬‬ ‫‪(12) log5 510‬‬ ‫‪(13) log4 1.116‬‬ ‫‪x‬‬ ‫=‬ ‫‪80‬‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ‪:‬‬ ‫ﺑﺎﻋﺘﺒﺎﺭ‬ ‫)‪(14‬‬ ‫‪23‬‬ ‫)‪ (a‬ﺣ ّﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺑﺄﺧﺬ ﺍﻟﻠﻮﻏﺎﺭﻳﺘﻢ ﺑﺄﺳﺎﺱ ‪ 2‬ﻟﻜﻞ ﻃﺮﻑ‪.‬‬ ‫)‪ (b‬ﺣ ّﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺑﺄﺧﺬ ﺍﻟﻠﻮﻏﺎﺭﻳﺘﻢ ﺑﺄﺳﺎﺱ ‪ 10‬ﻟﻜﻞ ﻃﺮﻑ‪.‬‬ ‫)‪ (c‬ﻗﺎﺭﻥ ﺑﻴﻦ ﺇﺟﺎﺑﺎﺗﻚ ﻓﻲ ﺍﻟﻔﻘﺮﺗﻴﻦ )‪ .(a), (b‬ﺃﻱ ﻃﺮﻳﻘﺔ ﺗﻔﻀﻠﻬﺎ؟ ﻭﻟﻤﺎﺫﺍ؟‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(15-20‬ﺣﻞ ﻛﻞ ﻣﻌﺎﺩﻟﺔ ﻟﻮﻏﺎﺭﻳﺘﻤﻴﺔ ﻣﻤﺎ ﻳﻠﻲ‪:‬‬ ‫‪(15) log 6x - 3 = - 4‬‬ ‫‪(16) log x - log 3 = 8‬‬ ‫‪(17) log2(3x - 5) = 1‬‬ ‫‪(18) log^2xh + log^x - 3h = log 8‬‬ ‫‪(19) log^3xh - log^x + 20h = - log 2‬‬ ‫‪(20) log(2x-1) 49 = 2‬‬ ‫‪(21) log 64 = log 4‬‬ ‫)‪(5x - 3‬‬ ‫)‪ (22‬ﺍﻷﺣﻴﺎﺀ ﺍﻟﺒﺮﻳﺔ‪ :‬ﻟﻨﻔﺮﺽ ﺃﻥ ﻓﺼﻴﻠﺔ ﻣﻌﻴﻨﺔ ﻣﻦ ﺍﻟﺤﻴﻮﺍﻧﺎﺕ ﺍﻟﺒﺮﻳﺔ ﺍﻟﻤﻌﺮﺿﺔ ﻟﺨﻄﺮ ﺍﻻﻧﻘﺮﺍﺽ ﺗﺘﻨﺎﻗﺺ ﺃﻋﺪﺩﺍﻫﺎ‬ ‫ﺑﻤﻌﺪﻝ ‪ 3.5%‬ﺳﻨﻮﻳًّﺎ ﻭﻗﺪ ﺃﺣﺼﻴﺖ ‪ 80‬ﺣﻴﻮﺍﻧًﺎ ﻣﻦ ﻫﺬﻩ ﺍﻟﻔﺼﻴﻠﺔ ﻓﻲ ﻣﻮﻃﻨﻬﺎ ﺍﻟﺬﻱ ﺗﻘﻮﻡ ﺑﺪﺭﺍﺳﺘﻪ‪.‬‬ ‫)‪ (a‬ﺗﻮﻗﻊ ﻋﺪﺩ ﺣﻴﻮﺍﻧﺎﺕ ﻫﺬﻩ ﺍﻟﻔﺼﻴﻠﺔ ﺍﻟﺬﻱ ﺳﻴﺒﻘﻰ ﺑﻌﺪ ‪ 10‬ﺳﻨﻮﺍﺕ‪.‬‬ ‫)‪ (b‬ﺑﻌﺪ ﻛﻢ ﺳﻨﺔ ﺳﻮﻑ ﻳﺘﻨﺎﻗﺺ ﻋﺪﺩ ﺣﻴﻮﺍﻧﺎﺕ ﻫﺬﻩ ﺍﻟﻔﺼﻴﻠﺔ ﻷﻭﻝ ﻣﺮﺓ ﺇﻟﻰ ﺃﻗﻞ ﻣﻦ ‪ 15‬ﺣﻴﻮﺍﻧًﺎ‪ ،‬ﺑﺎﻟﻤﻌﺪﻝ‬ ‫ﻧﻔﺴﻪ؟‬ ‫‪65‬‬

‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-5‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ab‬‬ ‫‪x‬‬ ‫=‬ ‫‪1‬‬ ‫ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ 9x = 3‬ﻫﻮ‬ ‫)‪(1‬‬ ‫‪ab‬‬ ‫‪2‬‬ ‫‪ab‬‬ ‫‪ab‬‬ ‫)‪ (2‬ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ 2 log x = - 1‬ﻫﻮ ‪x = 10-0.5‬‬ ‫‪ab‬‬ ‫)‪ (3‬ﺇﺫﺍ ﻛﺎﻥ ‪ log(x + 6) = 0‬ﻓﺈﻥ ‪x =-5‬‬ ‫‪a x . 15‬‬ ‫‪a x=6‬‬ ‫‪x‬‬ ‫=‬ ‫‪log 146‬‬ ‫ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ 149x = 146‬ﻫﻮ‬ ‫)‪(4‬‬ ‫‪a x=3‬‬ ‫‪log 14‬‬ ‫‪a x = 10-1‬‬ ‫‪a \"2,‬‬ ‫)‪ (5‬ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ 3 log x - log 6 + log 2.4 = 9‬ﻫﻮ ‪5 # 104‬‬ ‫‪a \"-1,‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(6-14‬ﻇﻠّﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫‪a4‬‬ ‫)‪ (6‬ﺇﺫﺍ ﻛﺎﻥ ‪ (1.5) x = 356‬ﻓﺈ ّﻥ‪:‬‬ ‫‪a log (6 - x2) = 1‬‬ ‫‪b x . 14.5‬‬ ‫‪c x . 15.3‬‬ ‫‪d x . 16.3‬‬ ‫‪3‬‬ ‫)‪ (7‬ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ 8 + 10x = 1008‬ﻫﻮ‪:‬‬ ‫‪a -3‬‬ ‫‪b x . 3.5‬‬ ‫‪c x=3‬‬ ‫‪d x=2‬‬ ‫)‪ (8‬ﺇﺫﺍ ﻛﺎﻥ ‪ 2x2 = 512‬ﻓﺈ ّﻥ‪:‬‬ ‫‪b x=9‬‬ ‫‪c x = 3 , x =-3‬‬ ‫‪d x =-9‬‬ ‫)‪ (9‬ﺇﺫﺍ ﻛﺎﻥ ‪ 2 log x = - 2‬ﻓﺈ ّﻥ‪:‬‬ ‫‪b x = 100.5‬‬ ‫‪c x = 10-2‬‬ ‫‪d x = 10-0.5‬‬ ‫)‪ (10‬ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ‪ log(x2 + 2) = log(5x - 4) :‬ﻫﻲ‪:‬‬ ‫‪b \"3,‬‬ ‫‪c \"2, 3,‬‬ ‫‪d \"-2, - 3,‬‬ ‫)‪ (11‬ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ‪ log (x2 - x) = 1 :‬ﻫﻲ‪:‬‬ ‫‪2‬‬ ‫‪b \"1, 2,‬‬ ‫‪c \"-1, 2,‬‬ ‫‪d \"-1, - 2,‬‬ ‫)‪ (12‬ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ log(x + 21) + log x = 2‬ﻫﻮ‪:‬‬ ‫‪b -25 , 4‬‬ ‫‪c 25‬‬ ‫‪d 4 , 25‬‬ ‫)‪ (13‬ﻳﻜﻮﻥ ‪ x = 3‬ﺣ ًﻼ ﻟﻠﻤﻌﺎﺩﻟﺔ‪:‬‬ ‫‪b‬‬ ‫‪log 9‬‬ ‫=‬ ‫‪2‬‬ ‫‪c log (x2 + 1) = 2‬‬ ‫‪d log3 x3 + log3 x = 4‬‬ ‫‪x‬‬ ‫‪3‬‬ ‫‪3‬‬ ‫)‪ (14‬ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ log 81 - log 9 = 2‬ﻫﻮ‪:‬‬ ‫‪xx‬‬ ‫‪b‬‬ ‫‪1‬‬ ‫‪c3 d9‬‬ ‫‪3‬‬ ‫‪66‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺍﻟﻠﻮﻏﺎﺭﻳﺘﻢ ﺍﻟﻄﺒﻴﻌﻲ ‪4-6‬‬ ‫‪Natural Logarithm‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-8‬ﺍﻛﺘﺐ ﻛﻞ ﺗﻌﺒﻴﺮ ﻣﻤﺎ ﻳﻠﻲ ﻛﻠﻮﻏﺎﺭﻳﺘﻢ ﻃﺒﻴﻌﻲ ﻭﺍﺣﺪ‪:‬‬ ‫‪(1) 3 ln 5‬‬ ‫‪(2) ln 24 - ln 6‬‬ ‫‪(3) ln 3 - 5 ln 3‬‬ ‫‪(4) 5 ln m + 3 ln n , ^m 2 0 , n 2 0h‬‬ ‫‪(5) 2 ln 8 - 3 ln 4‬‬ ‫‪(6) 7‬‬ ‫)‪(7‬‬ ‫‪ln a‬‬ ‫‪-‬‬ ‫‪2 ln b‬‬ ‫‪+‬‬ ‫‪1‬‬ ‫‪ln c‬‬ ‫‪,‬‬ ‫‪(a‬‬ ‫‪2‬‬ ‫‪0,‬‬ ‫‪b‬‬ ‫‪2‬‬ ‫‪0,‬‬ ‫‪c‬‬ ‫‪2‬‬ ‫)‪0‬‬ ‫)‪(8‬‬ ‫‪1‬‬ ‫‪(ln‬‬ ‫‪x‬‬ ‫‪+‬‬ ‫‪ln‬‬ ‫)‪y‬‬ ‫‪-‬‬ ‫‪4‬‬ ‫‪ln‬‬ ‫‪c‬‬ ‫‪,‬‬ ‫^‬ ‫‪x‬‬ ‫‪2‬‬ ‫‪0,‬‬ ‫‪y‬‬ ‫‪2‬‬ ‫‪0,‬‬ ‫‪c2‬‬ ‫‪0h‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫)‪ (9‬ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ y‬ﻓﻲ‪y = 15 + 3 ln 7.2 :‬‬ ‫)‪ (10‬ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ y‬ﻓﻲ‪y = 0.05 - 10 ln x , x = 0.09 :‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺮﻳﻨﻴﻦ )‪ ،(11-12‬ﺍﺳﺘﺨﺪﻡ ﺍﻟﻌﻼﻗﺔ‪ ،V = - 0.0098t + C lnR :‬ﺣﻴﺚ ‪ R‬ﻧﺴﺒﺔ ﻛﺘﻠﺔ ﺍﻟﺼﺎﺭﻭﺥ‪ t ،‬ﺯﻣﻦ ﺍﺷﺘﻌﺎﻟﻪ‪ C ،‬ﺳﺮﻋﺔ‬ ‫ﺍﻧﻄﻼﻕ ﺍﻟﺒﺨﺎﺭ‪ V ،‬ﺳﺮﻋﺔ ﺍﻟﺼﺎﺭﻭﺥ‪.‬‬ ‫)‪ (11‬ﺃﻭﺟﺪ ﺃﻗﺼﻰ ﺳﺮﻋﺔ ﻟﺼﺎﺭﻭﺥ ﻧﺴﺒﺔ ﻛﺘﻠﺘﻪ ‪ 20‬ﻭﺳﺮﻋﺔ ﺍﻧﻄﻼﻕ ﺑﺨﺎﺭﻩ ‪ 2.7 km/s‬ﻭﺯﻣﻦ ﺍﺷﺘﻌﺎﻟﻪ ‪30 s‬‬ ‫)‪ (12‬ﺃﻭﺟﺪ ﻧﺴﺒﺔ ﻛﺘﻠﺔ ﺻﺎﺭﻭﺥ ﺳﺮﻋﺔ ﺍﻧﻄﻼﻕ ﺑﺨﺎﺭﻩ ‪ 3.15 km/s‬ﻭﺯﻣﻦ ﺍﺷﺘﻌﺎﻟﻪ ‪ 50 s‬ﻭﻟﻪ ﺃﻗﺼﻰ ﺳﺮﻋﺔ ‪6.9 km‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(13-18‬ﺍﺳﺘﺨﺪﻡ ﺍﻟﻠﻮﻏﺎﺭﻳﺘﻢ ﺍﻟﻄﺒﻴﻌﻲ ﻟﺤﻞ ﻛﻞ ﻣﻌﺎﺩﻟﺔ ﻣﻤﺎ ﻳﻠﻲ‪:‬‬ ‫‪(13) 3e2x = 12‬‬ ‫‪(14) ex+1 = 30‬‬ ‫)‪(15‬‬ ‫‪x‬‬ ‫‪e9 -8 = 6‬‬ ‫‪(16) 4ex+2 = 32‬‬ ‫‪(17) 2e3x-2 + 4 = 16‬‬ ‫‪(18) 2e2x = ex + 6‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(19-28‬ﺣ ّﻞ ﻛﻞ ﻣﻌﺎﺩﻟﺔ ﻣﻤﺎ ﻳﻠﻲ‪:‬‬ ‫‪(19) ln 3x = 6‬‬ ‫‪(20) ln^4x - 1h = 36‬‬ ‫‪(21) ln(x - 1) 2 = 3‬‬ ‫)‪(22‬‬ ‫` ‪ln‬‬ ‫‪x‬‬ ‫‪-‬‬ ‫‪1‬‬ ‫‪j‬‬ ‫=‬ ‫‪4‬‬ ‫‪(23) 2 ln 2x2 = 1‬‬ ‫‪(24) ln x - 3 ln 3 = 3‬‬ ‫‪(26) 1.1 + ln x2 = 6‬‬ ‫‪(27) ln^2x - 1h = 0‬‬ ‫‪2‬‬ ‫)‪(25‬‬ ‫‪1‬‬ ‫‪ln‬‬ ‫‪x‬‬ ‫‪+‬‬ ‫‪ln‬‬ ‫‪2‬‬ ‫‪-‬‬ ‫‪ln‬‬ ‫‪3‬‬ ‫=‬ ‫‪3‬‬ ‫‪2‬‬ ‫)‪(28‬‬ ‫‪1‬‬ ‫‪ln (5x - 3) 3 = 2‬‬ ‫)‪ (29‬ﺍﻟﺘﻔﻜﻴﺮ ﺍﻟﻨﺎﻗﺪ‪ :‬ﻫﻞ ﻳﻤﻜﻦ ﻛﺘﺎﺑﺔ ‪ ln 5 + log210‬ﻋﻠﻰ ﺷﻜﻞ ﻟﻮﻏﺎﺭﻳﺘﻢ ﻭﺍﺣﺪ؟ ﺍﺷﺮﺡ‪.‬‬ ‫ﺍﻟﻘﻤﺮ‬ ‫ﺗﺸﻐﻴﻞ‬ ‫ﻣﺪﺓ‬ ‫ﻓﻤﺎ‬ ‫ﻳﻮﻡ‪،‬‬ ‫‪n‬‬ ‫ﺑﻌﺪ‬ ‫ﺻﻨﺎﻋﻲ‬ ‫ﻟﻘﻤﺮ‬ ‫)‪(W‬‬ ‫ﺑﺎﻟﻮﺍﻁ‬ ‫)‪(b‬‬ ‫ﺍﻟﺨﺎﺭﺟﺔ‬ ‫ﺍﻟﻘﻮﺓ‬ ‫‪b‬‬ ‫=‬ ‫‪40‬‬ ‫‪-n‬‬ ‫ﺗﻌﻄﻲ ﺍﻟﻌﻼﻗﺔ‪:‬‬ ‫)‪(30‬‬ ‫‪e 300‬‬ ‫ﺍﻟﺼﻨﺎﻋﻲ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻘﻮﺓ ﺍﻟﺨﺎﺭﺟﺔ ‪15 W‬؟‬ ‫‪67‬‬

‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-5‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ab‬‬ ‫)‪log4 (ln e4) = 1 (1‬‬ ‫‪ab‬‬ ‫)‪4ln8 + ln10 = 4ln80 (2‬‬ ‫‪ab‬‬ ‫‪ab‬‬ ‫)‪lne2 = 2 (3‬‬ ‫‪ab‬‬ ‫)‪ (4‬ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ‪ lnx = - 2 :‬ﻫﻮ ‪e2‬‬ ‫‪x‬‬ ‫)‪ (5‬ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ‪ e5 + 4 = 7 :‬ﻫﻮ ‪5ln3‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(6-14‬ﻇﻠّﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫)‪ 3 ln 4 - 5 ln 2 (6‬ﻋﻠﻰ ﺷﻜﻞ ﻟﻮﻏﺎﺭﻳﺘﻢ ﻭﺍﺣﺪ ﺗﻜﺘﺐ‪:‬‬ ‫)‪a ln (- 18‬‬ ‫‪b‬‬ ‫‪lna‬‬ ‫‪6‬‬ ‫‪k‬‬ ‫‪c ln 2‬‬ ‫‪d ln 32‬‬ ‫‪5‬‬ ‫)‪ eln10 (7‬ﺗﺴﺎﻭﻱ‪:‬‬ ‫‪a 10‬‬ ‫‪b e10‬‬ ‫‪c0‬‬ ‫‪d‬‬ ‫‪1‬‬ ‫‪10‬‬ ‫)‪ (8‬ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ ln(2m + 3) = 8‬ﻫﻮ‪:‬‬ ‫‪a e8 - 3‬‬ ‫‪b‬‬ ‫‪e8‬‬ ‫‪-‬‬ ‫‪3‬‬ ‫‪c‬‬ ‫‪e8 - 3‬‬ ‫‪d e4 - 3‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫)‪ (9‬ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ ln4r2 = 3‬ﻫﻮ‪:‬‬ ‫‪3‬‬ ‫‪33‬‬ ‫‪-‬‬ ‫‪3‬‬ ‫‪2‬‬ ‫‪a‬‬ ‫‪e2‬‬ ‫‪b‬‬ ‫‪e2‬‬ ‫‪-e2‬‬ ‫‪c‬‬ ‫‪e‬‬ ‫‪d‬‬ ‫‪33‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪,‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪e2 ,- e2‬‬ ‫)‪ (10‬ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ e2x = 10‬ﻫﻮ‪:‬‬ ‫‪a‬‬ ‫‪x‬‬ ‫=‬ ‫‪ln 10‬‬ ‫‪b ln 5‬‬ ‫‪c‬‬ ‫‪5‬‬ ‫‪d 2 ln 10‬‬ ‫‪2‬‬ ‫‪e‬‬ ‫)‪ \"e2, (11‬ﻫﻲ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ‪:‬‬ ‫‪a ln x = 2‬‬ ‫‪b ln x2 = 2‬‬ ‫‪c ln x2 = 4‬‬ ‫‪d ln x = 4‬‬ ‫)‪ (12‬ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ ex+1 = 13‬ﻫﻮ‪:‬‬ ‫‪a x = ln 13 + 1‬‬ ‫‪b x = ln 13 - 1‬‬ ‫‪c x = ln 13‬‬ ‫‪d x = ln 12‬‬ ‫)‪ (13‬ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ‪ ln(x - 2)2 = 6‬ﻫﻮ‪:‬‬ ‫‪a 2 + e3‬‬ ‫‪b 2 - e3‬‬ ‫‪c 2 ! e3‬‬ ‫‪d 2 ! e6‬‬ ‫‪b x = 2 ln 5 - 2‬‬ ‫ﻫﻮ‪:‬‬ ‫‪e‬‬ ‫‪x‬‬ ‫‪+‬‬ ‫‪1‬‬ ‫‪+‬‬ ‫‪3‬‬ ‫=‬ ‫‪8‬‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﺣﻞ‬ ‫)‪(14‬‬ ‫‪2‬‬ ‫‪a x = 2 ln 5 - 1‬‬ ‫‪c x = 2 ln 4‬‬ ‫‪d‬‬ ‫‪x‬‬ ‫=‬ ‫‪1‬‬ ‫‪^ln‬‬ ‫‪5‬‬ ‫‪-‬‬ ‫‪1h‬‬ ‫‪2‬‬ ‫‪68‬‬

‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ ﺍﻟﺮﺍﺑﻌﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-4‬ﺍﺭﺳﻢ ﻛ ًّﻼ ﻣﻦ ﺍﻟﺪﻭﺍﻝ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫‪(1) y = - 3 (0 . 25) x‬‬ ‫)‪(2‬‬ ‫)‪f (x‬‬ ‫=‬ ‫‪1‬‬ ‫‪(6) - x‬‬ ‫‪(3) y = 0.1 (10) x-2‬‬ ‫‪(4) f (x) = (2) x+1 + 3‬‬ ‫‪2‬‬ ‫)‪ (5‬ﺍﻟﻜﺘﺎﺑﺔ‪ :‬ﻭ ّﺿﺢ ﻛﻴﻒ ﻳﻤﻜﻨﻚ ﺗﺤﺪﻳﺪ ﻣﺎ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺪﺍﻟﺔ ﺍﻷﺳﻴﺔ ﺗﻤﺜّﻞ ﻧﻤ ًّﻮﺍ ﺃﺳﻴًّﺎ ﺃﻡ ﺗﻀﺎﺅ ًﻻ ﺃﺳﻴًّﺎ‪.‬‬ ‫ﺍﻋﺮﺽ ﻣﺜﺎ ًﻻ ﻟﻜﻞ ﻣﻨﻬﺎ‪.‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(6-8‬ﺍﻛﺘﺐ ﻣﻌﺎﺩﻟﺔ ﺗﺼﻒ ﺍﻟﺪﺍﻟﺔ ﺍﻷﺳﻴﺔ ﺍﻟﺘﻲ ﻋﻠﻰ ﺍﻟﺼﻮﺭﺓ‪ ، y = abx :‬ﺑﻤﻌﻠﻮﻣﻴﺔ ﺍﻷﺳﺎﺱ ﺍﻟﻤﻌﻄﻰ ﻭﺍﻟﺘﻲ ﻳﻤﺮ‬ ‫ﺭﺳﻤﻬﺎ ﺍﻟﺒﻴﺎﻧﻲ ﺑﺎﻟﻨﻘﻄﺔ ﺍﻟﻤﻌﻄﺎﺓ‪.‬‬ ‫)‪ (6‬ﺍﻷﺳﺎﺱ ‪ ،3‬ﺍﻟﻨﻘﻄﺔ )‪(2, 3‬‬ ‫)‪ (7‬ﺍﻷﺳﺎﺱ ‪ ،4‬ﺍﻟﻨﻘﻄﺔ )‪(-1, 1‬‬ ‫)‪ (8‬ﺍﻷﺳﺎﺱ ‪ ،2‬ﺍﻟﻨﻘﻄﺔ )‪(0, 3‬‬ ‫)‪ (9‬ﻋﻠﻢ ﺍﻟﺰﻻﺯﻝ‪ :‬ﻛﻢ ﻣﺮﺓ ﻳﻜﻮﻥ ﺯﻟﺰﺍﻝ ﻗﻮﺗﻪ ‪ 5.2‬ﺑﻤﻘﻴﺎﺱ ﺭﻳﺨﺘﺮ ﺃﻗﻮﻯ ﻣﻦ ﺯﻟﺰﺍﻝ ﻗﻮﺗﻪ ‪ 3‬ﻋﻠ ًﻤﺎ ﺑﺄﻥ ﺍﻟﻄﺎﻗﺔ ﺍﻟﻤﻨﻄﻠﻘﺔ‬ ‫ﺗﺴﺎﻭﻱ ‪ x ،E # 30x‬ﻫﻲ ﺩﺭﺟﺔ ﻗﻮﺓ ﺍﻟﺰﻟﺰﺍﻝ ﺑﻤﻘﻴﺎﺱ ﺭﻳﺨﺘﺮ‪.‬‬ ‫)‪ (10‬ﺍﺭﺳﻢ ﺑﻴﺎﻥ ﺍﻟﺪﺍﻟﺔ ‪ y = log8x‬ﺛﻢ ﺍﺳﺘﺨﺪﻣﻬﺎ ﻛﺪﺍﻟﺔ ﻣﺮﺟﻊ ﻟﺮﺳﻢ ﺑﻴﺎﻥ ﻛﻞ ﻣﻦ ﺍﻟﺪﻭﺍﻝ ﺍﻟﻠﻮﻏﺎﺭﻳﺘﻤﻴﺔ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫)‪(a) y = log8 (x + 2‬‬ ‫‪(b) y = log8x - 1‬‬ ‫‪(c) y = log8 (x + 2) - 1‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(11-14‬ﺃﻭﺟﺪ ﻣﻔﻜﻮﻙ ﻛﻞ ﻣﻦ ﺍﻟﻠﻮﻏﺎﺭﻳﺘﻤﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫)‪(11) log4r2 n , (r 2 0 , n 2 0‬‬ ‫‪(12) log2 ^x + 1h2 , ^x 2-1h‬‬ ‫)‪(14) log 3x3y2 , (x 2 0 , y 2 0‬‬ ‫)‪(13‬‬ ‫‪log7‬‬ ‫‪a‬‬ ‫‪,‬‬ ‫‪^a 2 0‬‬ ‫‪,‬‬ ‫‪b 2 0h‬‬ ‫‪b‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(15-18‬ﺍﺳﺘﺨﺪﻡ ﺧﻮﺍﺹ ﺍﻟﻠﻮﻏﺎﺭﻳﺘﻤﺎﺕ ﻹﻳﺠﺎﺩ ﻧﺎﺗﺞ ﻛﻞ ﻣﻦ ﺍﻟﻤﻘﺎﺩﻳﺮ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫‪(15) log327 - log39‬‬ ‫‪(16) 2 log 264 - log 22‬‬ ‫)‪(17‬‬ ‫‪-log 4‬‬ ‫‪1‬‬ ‫‪-‬‬ ‫‪log 464‬‬ ‫‪(18) 2 log 5 + log 40‬‬ ‫‪16‬‬ ‫)‪ (19‬ﺳﺆﺍﻝ ﻣﻔﺘﻮﺡ‪ :‬ﺍﻛﺘﺐ ﻣﻘﺪﺍﺭﻳﻦ ﻟﻮﻏﺎﺭﻳﺘﻤﻴﻴﻦ‪ .‬ﺃﻱ ﻣﻨﻬﻤﺎ ﻟﻪ ﺍﻟﻘﻴﻤﺔ ﺍﻷﻛﺒﺮ؟ ﺍﺷﺮﺡ‪.‬‬ ‫‪3‬‬ ‫‪3‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(20-30‬ﺣﻞ ﻛ ًّﻼ ﻣﻦ ﺍﻟﻤﻌﺎﺩﻻﺕ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫‪(22) log 4x = 3‬‬ ‫‪(20) x4 = 81‬‬ ‫‪(21) 3k2 = 24‬‬ ‫‪(23) 2 log x = - 4‬‬ ‫‪(24) log 2x + log x = 1‬‬ ‫‪(25) log x - log(x - 1) = 1‬‬ ‫‪69‬‬

‫‪(26) logx(3x + 4) = 2‬‬ ‫‪(27) ln(x - 2) + ln x = 1‬‬ ‫‪(28) ln (x + 1) + ln (x - 1) = 4‬‬ ‫‪(29) ln x + ln(2x - 1) = 7‬‬ ‫‪(30) 3 ln x - ln 2 = 4‬‬ ‫)‪ (31‬ﻟﻨﻔﺘﺮﺽ ﺃﻥ ﺛﻤﻦ ﺁﻟﺔ ﺗﺴﺘﺨﺪﻡ ﻓﻲ ﺻﻨﺎﻋﺔ ﺳﻠﻌﺔ ﻣﺎ ﻟﻬﺎ ﻋﺎﻣﻞ ﺗﻀﺎﺅﻝ ﺳﻨﻮﻱ ﻗﻴﻤﺘﻪ ‪ .0.75‬ﺇﺫﺍ ﺑﻠﻎ ﺛﻤﻦ ﺍﻵﻟﺔ‬ ‫‪ 10 000‬ﺩﻳﻨﺎﺭ ﺑﻌﺪ ‪ 5‬ﺳﻨﻮﺍﺕ ﻣﻦ ﺍﻻﺳﺘﺨﺪﺍﻡ‪ ،‬ﻓﻤﺎ ﻗﻴﻤﺘﻬﺎ ﺍﻷﺳﺎﺳﻴﺔ؟‬ ‫)‪ (32‬ﺍﻟﺪﺭﺍﺳﺎﺕ ﺍﻻﺟﺘﻤﺎﻋﻴﺔ‪ :‬ﻋﺎﻡ ‪ 1991‬ﻛﺎﻥ ﻋﺪﺩ ﺳﻜﺎﻥ ﻛﺎﺭﺍﺗﺸﻲ ﻓﻲ ﺑﺎﻛﺴﺘﺎﻥ ﺣﻮﺍﻟﻰ ‪ 8‬ﻣﻼﻳﻴﻦ ﻧﺴﻤﺔ‪ ،‬ﻭﻛﺎﻥ‬ ‫ﻋﺎﻣﻞ ﺍﻟﻨﻤﻮ ﺍﻟﺴﻨﻮﻱ ﻓﻲ ﻫﺬﺍ ﺍﻟﻮﻗﺖ ‪.1.039‬‬ ‫)‪ (a‬ﻣﺎ ﻋﺪﺩ ﺍﻟﺴﻜﺎﻥ ﺍﻟﻤﺘﻮﻗﻊ ﻓﻲ ﻋﺎﻡ ‪2010‬؟‬ ‫)‪ (b‬ﻣﺎ ﻣﻌﺪﻝ ﺍﻟﺰﻳﺎﺩﺓ ﺍﻟﺴﻨﻮﻳﺔ ﺍﻟﻤﺘﻮﻗﻊ؟‬ ‫)‪ (c‬ﻣﺘﻰ ﻳﺼﻞ ﻋﺪﺩ ﺍﻟﺴﻜﺎﻥ ﺇﻟﻰ ‪ 10‬ﻣﻼﻳﻴﻦ ﻧﺴﻤﺔ؟‬ ‫)‪ (33‬ﺳﻜﺎﻥ ﺍﻟﻌﺎﻟﻢ‪ :‬ﺑﻠﻎ ﻋﺪﺩ ﺳﻜﺎﻥ ﺍﻟﻌﺎﻟﻢ ﻓﻲ ﻋﺎﻡ ‪ 1994‬ﺣﻮﺍﻟﻰ ‪ 5.63‬ﺑﻼﻳﻴﻦ ﻧﺴﻤﺔ‪ ،‬ﻭﻳﻘﺎﻝ ﺇﻧﻪ ﻳﻨﻤﻮ‬ ‫ﺑﻤﻌﺪﻝ ‪ 2%‬ﺳﻨﻮﻳًّﺎ‪.‬‬ ‫)‪ (a‬ﺍﻛﺘﺐ ﻣﻌﺎﺩﻟﺔ ﺃﺳﻴﺔ ﻟﻮﺻﻒ ﻫﺬﺍ ﺍﻟﻨﻤﻮ‪.‬‬ ‫)‪ (b‬ﺻﻒ ﻧﻤﻮ ﻋﺪﺩ ﺍﻟﺴﻜﺎﻥ ﻛﻞ ‪ 35‬ﺳﻨﺔ‪.‬‬ ‫)‪ (c‬ﺻﻒ ﻧﻤﻮ ﻋﺪﺩ ﺍﻟﺴﻜﺎﻥ ﻓﻲ ﻧﺼﻒ ﺍﻟﻤﺪﺓ ﺍﻟﺰﻣﻨﻴﺔ ﺍﻟﻤﺤﺪﺩﺓ ﻓﻲ ﺍﻟﺠﺰﺀ )‪.(b‬‬ ‫‪70‬‬

‫ﺗﻤﺎﺭﻳﻦ ﺇﺛﺮﺍﺋﻴﺔ‬ ‫)‪ (1‬ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ‪(ex - 1)ex = 3ex - 3 :‬‬ ‫)‪ (2‬ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ‪3(ex ) 2 - ex - 4 = 0 :‬‬ ‫)‪ (3‬ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ‪ln 3x - 1 + ln x - 1 = ln x - 2 :‬‬ ‫)‪ (4‬ﻫﻞ ﺻﺤﻴﺢ ﺃﻥ‪aln b = bln a , a 2 0 , b 2 0 :‬؟‬ ‫‪4e2x‬‬ ‫=‬ ‫‪4‬‬ ‫ﺃﺛﺒﺖ ﺃﻥ‪:‬‬ ‫)‪(5‬‬ ‫‪e2x + 3‬‬ ‫‪1 + 3e-2x‬‬ ‫)‪ (6‬ﺣ ّﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ‪ex + 2e- x = 3 :‬‬ ‫)‪ (7‬ﺃﻭﺟﺪ ﻣﺠﻤﻮﻋﺔ ﺣﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ‪2(ln x) 2 - 5 ln x - 3 = 0 :‬‬ ‫)‪ (8‬ﺍﻟﺼﻨﺎﻋﺎﺕ‪ :‬ﻟﻨﻔﺮﺽ ﺃﻧﻚ ﺗﻌﻤﻞ ﻓﻲ ﻣﺼﻨﻊ ﻟﻠﻤﻜﺎﻧﺲ ﺍﻟﻜﻬﺮﺑﺎﺋﻴﺔ‪ ،‬ﻭﻗﺪ ﺳﺎﻫﻤﺖ ﻓﻲ ﺻﻨﻊ ﺗﺼﻤﻴﻢ ﻣﺴﺘﺨﺪ ًﻣﺎ‬ ‫ﻣﻜﻮﻧﺎﺕ ﺟﺪﻳﺪﺓ ﺗﻌﻤﻞ ﻋﻠﻰ ﺗﺨﻔﻴﺾ ﺷﺪﺓ ﺻﻮﺕ ﻃﺮﺍﺯ ﻣﻌﻴﻦ ﻣﻦ ‪ 10- 4 w/m2‬ﺇﻟﻰ ‪،6 . 31 # 10- 6 w/m2‬‬ ‫ﻣﺎ ﺍﻟﻨﺴﺒﺔ ﺍﻟﻤﺌﻮﻳﺔ ﻻﻧﺨﻔﺎﺽ ﺍﻟﺼﻮﺕ ﺍﻟﺬﻱ ﺣﻘﻘﻪ ﺍﺳﺘﺨﺪﺍﻡ ﻫﺬﻩ ﺍﻟﻤﻜﻮﻧﺎﺕ ﺍﻟﺠﺪﻳﺪﺓ؟‬ ‫)ﺍﺳﺘﺨﺪﻡ ‪.^I0 = 10-12 w/m2‬‬ ‫)‪ (9‬ﺇﺫﺍ ﻛﺎﻥ ﻛﻞ ﻣﻦ ﺍﻟﺪﺍﻟﺘﻴﻦ‪ ، y = logbx , y = bx :‬ﻣﻌﻜﻮﺱ ﻟﻸﺧﺮﻯ‪ ،‬ﺍﺳﺘﺨﺪﻡ ﺍﻟﺨﺎﺻﻴﺔ ‪ logbbx = x‬ﻭﺑﺮﻫﺎﻥ‬ ‫ﺍﻟﺨﻄﻮﺓ ﺍﻟﻮﺍﺣﺪﺓ ﻟﺨﺎﺻﻴﺔ ﻧﺎﺗﺞ ﺍﻟﻀﺮﺏ ﻓﻲ ﺍﻟﻠﻮﻏﺎﺭﻳﺘﻤﺎﺕ‪ ،‬ﻟﻤﺴﺎﻋﺪﺗﻚ ﻋﻠﻰ ﺑﺮﻫﻨﺔ ﻛﻞ ﻣﻦ ﺧﺎﺻﻴﺔ ﺍﻟﻘﺴﻤﺔ‬ ‫ﻭﺧﺎﺻﻴﺔ ﺍﻟﻘﻮﻯ‪.‬‬ ‫)‪ (10‬ﺑﺎﻋﺘﺒﺎﺭ ﺍﻟﻤﻌﺎﺩﻟﺔ‪ax = b :‬‬ ‫)‪ (a‬ﺣ ّﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻠﻮﻏﺎﺭﻳﺘﻢ ﻟﻸﺳﺎﺱ ‪10‬‬ ‫)‪ (b‬ﺣ ّﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻠﻮﻏﺎﺭﻳﺘﻢ ﻟﻸﺳﺎﺱ ‪a‬‬ ‫)‪ (c‬ﺍﺳﺘﺨﺪﻡ ﻧﺘﺎﺋﺠﻚ ﻓﻲ ﺍﻟﻔﻘﺮﺗﻴﻦ )‪ (a), (b‬ﻟﺘﺤﻘﻖ ﻗﺎﻋﺪﺓ ﺗﻐﻴﻴﺮ ﺍﻷﺳﺎﺱ‪.‬‬ ‫)‪ (11‬ﺍﻟﻬﻨﺪﺳﺔ‪ :‬ﺗﺄﺧﺬ ﺑﻌﺾ ﻗﻄﺮﺍﺕ ﺍﻟﻤﻄﺮ ﺷﻜ ًﻼ ﻛﺮﻭﻳًّﺎ‪ .‬ﻟﻨﻔﺮﺽ ﺃﻥ ﻧﺼﻒ ﻗﻄﺮ ﻗﻄﺮﺓ ﻣﻄﺮ ﻣﺘﺴﺎﻗﻄﺔ ﻳﺘﻨﺎﻗﺺ‬ ‫ﺑﻤﻘﺪﺍﺭ ‪ 0.02 mm‬ﻧﺘﻴﺠﺔ ﺍﻟﺘﺒﺨﺮ‪ ،‬ﺇﺫﺍ ﻛﺎﻥ ﺣﺠﻢ ﻗﻄﺮﺓ ﺍﻟﻤﻄﺮ ﺍﻵﻥ ‪ ،7 mm3‬ﻓﻤﺎ ﻃﻮﻝ ﻧﺼﻒ ﺍﻟﻘﻄﺮ ﺍﻷﺻﻠﻲ‬ ‫ﻟﻘﻄﺮﺓ ﺍﻟﻤﻄﺮ؟‬ ‫)‪ (12‬ﺗﺼﻒ ﺍﻟﺪﺍﻟﺔ‪ ، f(x) = 1 . 31 e0.548x :‬ﺍﻟﺘﺰﺍﻳﺪ ﺍﻷﺳﻲ ﻟﻌﺪﺩ ﻣﺴﺘﺨﺪﻣﻲ ﺍﻟﺸﺒﻜﺔ ﺍﻟﺪﻭﻟﻴﺔ ﻟﻠﻤﻌﻠﻮﻣﺎﺕ )ﺍﻹﻧﺘﺮﻧﺖ(‬ ‫ﺑﺎﻟﻤﻠﻴﻮﻥ ﻣﻦ ﻋﺎﻡ ‪ 1990‬ﺇﻟﻰ ﻋﺎﻡ ‪ .1995‬ﻟﻨﻔﺮﺽ ﺃﻥ ‪ x‬ﺗﻤﺜّﻞ ﺍﻟﺰﻣﻦ ﺑﺎﻟﺴﻨﻮﺍﺕ ﻣﻨﺬ ﻋﺎﻡ ‪.1990‬‬ ‫)‪ (a‬ﻣﺎ ﺃﻭﻝ ﻋﺎﻡ ﻛﺎﻥ ﻋﺪﺩ ﻣﺴﺘﺨﺪﻣﻲ ﺍﻹﻧﺘﺮﻧﺖ ﻓﻴﻪ ‪ 13‬ﻣﻠﻴﻮﻥ ﻣﺴﺘﺨﺪﻡ؟‬ ‫)‪ (b‬ﻣﺎ ﺍﻟﻤﺪﺓ ﺍﻟﻤﺴﺘﻐﺮﻗﺔ ﻟﺘﻀﺎﻋﻒ ﻋﺪﺩ ﻣﺴﺘﺨﺪﻣﻲ ﺍﻹﻧﺘﺮﻧﺖ ﻣﻨﺬ ﻋﺎﻡ ‪1990‬؟‬ ‫)‪ (c‬ﺣ ّﻞ ﺍﻟﻤﻌﺎﺩﻟﺔ‪ f(x) = 1 . 31 e0.548x :‬ﻓﻲ ‪x‬‬ ‫)‪ (d‬ﺍﻟﻜﺘﺎﺑﺔ‪ :‬ﺍﺷﺮﺡ ﻛﻴﻒ ﻳﻤﻜﻨﻚ ﺍﺳﺘﺨﺪﺍﻡ ﻣﻌﺎﺩﻟﺘﻚ ﻣﻦ ﺍﻟﻔﻘﺮﺓ )‪ (c‬ﻟﺘﺘﺤﻘﻖ ﻣﻦ ﺇﺟﺎﺑﺎﺗﻚ ﻋﻦ ﺍﻟﻔﻘﺮﺗﻴﻦ‬ ‫)‪ .(a), (b‬ﻣﺎ ﺍﻟﻨﺎﺗﺞ ﺍﻟﺬﻱ ﺣﺼﻠﺖ ﻋﻠﻴﻪ؟‬ ‫‪71‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺍﻟﻤﺘﺠﻪ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ ‪5-1‬‬ ‫‪The Vector in the Plane‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫)‪ (1‬ﻟﻨﺄﺧﺬ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ ﺍﻹﺣﺪﺍﺛﻲ ﺍﻟﻨﻘﺎﻁ‪A^-3,4h, B^2, - 1h, C^3,5h :‬‬ ‫)‪ (a‬ﻋﻴّﻦ ﺍﻟﺰﻭﺝ ﺍﻟﻤﺮﺗﺐ ﺍﻟﺬﻱ ﻳﻤﺜﻞ ﻣﺘﺠﻪ ﺍﻟﻤﻮﺿﻊ ﻟﻜ ّﻞ ﻣﻦ‪1 AB 2,1 BC 2,1 CA 2 :‬‬ ‫)‪ (b‬ﺇﺫﺍ ﻛﺎﻥ ﻣﺘﺠﻪ ﺍﻟﻤﻮﺿﻊ ‪ OM‬ﺣﻴﺚ ‪ M^4,3h‬ﻳﻤﺜّﻞ ﺍﻟﻘﻄﻌﺔ ﺍﻟﻤﻮﺟﻬﺔ ‪BE‬‬ ‫ﻓﺄﻭﺟﺪ ﺇﺣﺪﺍﺛﻴﺎﺕ ‪ E‬ﺑﻔﺮﺽ ﺃﻥ ‪E^x , yh‬‬ ‫)‪ (2‬ﻟﻨﺄﺧﺬ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ ﺍﻹﺣﺪﺍﺛﻲ ﺍﻟﻨﻘﺎﻁ‪E^-3,2h, F^2, - 1h, G^4, - 2h :‬‬ ‫ﺃﻭﺟﺪ ﻣﺮ ّﻛﺒﺎﺕ ﻛﻞ ﻣﻦ ﺍﻟﻤﺘﺠﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ‪1 EF 2,1 GF 2,1 EG 2 :‬‬ ‫)‪ (a) (3‬ﻟﻜﻞ ﻣﻦ ﺍﻟﻤﺘﺠﻬﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ‪u =1 3, 2 2, v =1-2, 4 2, w =1-3, - 2 2, t =1 2, - 3 2 :‬‬ ‫ﺍﺭﺳﻢ ﻣﺘﺠﻪ ﺍﻟﻤﻮﺿﻊ‪.‬‬ ‫)‪ (b‬ﺃﻭﺟﺪ ﻃﻮﻝ ﻛﻞ ﻣﺘﺠﻪ ﻭﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺘﻲ ﻳﺼﻨﻌﻬﺎ ﻣﻊ ﺍﻻﺗﺠﺎﻩ ﺍﻟﻤﻮﺟﺐ ﻟﻤﺤﻮﺭ ﺍﻟﺴﻴﻨﺎﺕ‪.‬‬ ‫‪ u‬ﻣﺘﺠﻪ ﻭﺣﺪﺓ‪.‬‬ ‫ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ ‪ x‬ﺑﺤﻴﺚ ﻳﺼﺒﺢ‬ ‫‪u‬‬ ‫‪=1‬‬ ‫‪x‬‬ ‫‪,‬‬ ‫‪3‬‬ ‫‪2‬‬ ‫ﺇﺫﺍ ﻛﺎﻥ‬ ‫)‪(4‬‬ ‫‪5‬‬ ‫)‪ (5‬ﻟﻨﺄﺧﺬ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ ﺍﻹﺣﺪﺍﺛﻲ ﺍﻟﻨﻘﺎﻁ‪A^3, - 1h, B^5, - 4h, C^2 ,4h, D^4 ,1h :‬‬ ‫ﺃﺛﺒﺖ ﺃﻥ‪1 AB 2=1 CD 2 :‬‬ ‫)‪ (6‬ﻟﻴﻜﻦ‪ A =1 4 , - 3 2, B =1 3x - 2 , 4y + 1 2 :‬ﺃﻭﺟﺪ ﻗﻴﻤﺘﻲ ‪ x, y‬ﺑﺤﻴﺚ ﻳﻜﻮﻥ‪A = B :‬‬ ‫)‪ (7‬ﻟﻨﺄﺧﺬ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ ﺍﻹﺣﺪﺍﺛﻲ‪A^5,2h, B^-2,6h, C^-3,3h, D^4, - 1h :‬‬ ‫ﺃﺛﺒﺖ ﺃﻥ‪ 1 AB 2 :‬ﻣﻌﺎﻛﺲ ﻟـ ‪1 CD 2‬‬ ‫)‪ (8‬ﻟﻨﺄﺧﺬ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ ﺍﻹﺣﺪﺍﺛﻲ ﺍﻟﻨﻘﺎﻁ‪A^2, - 3h, B^-1,3h, C^1, - 1h :‬‬ ‫ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﻨﻘﺎﻁ ﺍﻟﺜﻼﺙ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ‪.‬‬ ‫)‪ ABC (9‬ﻣﺜﻠﺚ ‪A‬‬ ‫‪1 AE 2 = -‬‬ ‫‪1‬‬ ‫‪1‬‬ ‫‪AB‬‬ ‫‪2‬‬ ‫ﺣﻴﺚ‪:‬‬ ‫‪1‬‬ ‫‪AE 2‬‬ ‫ﺍﺭﺳﻢ‬ ‫)‪(a‬‬ ‫‪2‬‬ ‫= ‪1 BD 2‬‬ ‫‪3‬‬ ‫‪1‬‬ ‫‪BC‬‬ ‫‪2‬‬ ‫ﺣﻴﺚ‪:‬‬ ‫‪1‬‬ ‫‪BD 2‬‬ ‫ﺍﺭﺳﻢ‬ ‫)‪(b‬‬ ‫‪2‬‬ ‫‪BC‬‬ ‫‪72‬‬

‫)‪ (10‬ﻟﻨﺄﺧﺬ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ ﺍﻹﺣﺪﺍﺛﻲ ﺍﻟﻨﻘﺎﻁ‪A^3,2h, B^1,5h, C^7, 4h :‬‬ ‫‪1‬‬ ‫‪BD‬‬ ‫‪2=-‬‬ ‫‪1‬‬ ‫‪1‬‬ ‫‪BA‬‬ ‫‪2‬‬ ‫ﺣﻴﺚ‪:‬‬ ‫‪D‬‬ ‫ﺍﻟﻨﻘﻄﺔ‬ ‫ﺇﺣﺪﺍﺛﻴﺎﺕ‬ ‫ﺃﻭﺟﺪ‬ ‫)‪(a‬‬ ‫‪2‬‬ ‫‪1‬‬ ‫‪AE‬‬ ‫=‪2‬‬ ‫‪3‬‬ ‫‪1‬‬ ‫‪AC‬‬ ‫‪2‬‬ ‫ﺣﻴﺚ‪:‬‬ ‫‪E‬‬ ‫ﺍﻟﻨﻘﻄﺔ‬ ‫ﺇﺣﺪﺍﺛﻴﺎﺕ‬ ‫ﺃﻭﺟﺪ‬ ‫)‪(b‬‬ ‫‪2‬‬ ‫)‪ (c‬ﺃﺛﺒﺖ ﺃﻥ‪ 1 DE 2, 1 BC 2 :‬ﻟﻬﻤﺎ ﺍﻻﺗﺠﺎﻩ ﻧﻔﺴﻪ‪.‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-4‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫ﻟﻨﺄﺧﺬ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ ﺍﻹﺣﺪﺍﺛﻲ ﺍﻟﻨﻘﺎﻁ ﺍﻟﺘﺎﻟﻴﺔ‪A^2,1h, B^-3,0h, C^3, - 4h, D^x , yh :‬‬ ‫‪ab‬‬ ‫)‪ (1‬ﺍﻟﺰﻭﺝ ﺍﻟﻤﺮﺗﺐ ﺍﻟﺬﻱ ﻳﻤﺜﻞ ﻣﺘﺠﻪ ﺍﻟﻤﻮﺿﻊ ﻟـ ‪ :BA‬ﻫﻮ ‪^-5, - 1h‬‬ ‫‪ab‬‬ ‫)‪ (2‬ﻣﺮ ّﻛﺒﺎﺕ ‪ BC‬ﻫﻲ ‪1 6, 4 2‬‬ ‫‪ab‬‬ ‫)‪ (3‬ﺍﻟﻤﺜﻠﺚ ‪ ABC‬ﻫﻮ ﻣﺘﻄﺎﺑﻖ ﺍﻟﻀﻠﻌﻴﻦ‪.‬‬ ‫‪ab‬‬ ‫)‪ (4‬ﺇﺫﺍ ﻛﺎﻥ ‪ 1 AB 2=1 CD 2‬ﻓﺈﻥ‪x = - 2 , y = - 5 :‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(5-8‬ﻇﻠّﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫)‪ (5‬ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ ﺍﻹﺣﺪﺍﺛﻲ ﺇﺫﺍ ﻛﺎﻥ ‪u =1-2,2 2‬‬ ‫ﻓﺈﻥ ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺘﻲ ﻳﺼﻨﻌﻬﺎ ‪ u‬ﻣﻊ ﺍﻻﺗﺠﺎﻩ ﺍﻟﻤﻮﺟﺐ ﻟﻤﺤﻮﺭ ﺍﻟﺴﻴﻨﺎﺕ ﻳﺴﺎﻭﻱ‪:‬‬ ‫‪a 45º‬‬ ‫‪b -45º‬‬ ‫‪c 135º‬‬ ‫‪d 225º‬‬ ‫ﻣﺘﺠﻪ ﻭﺣﺪﺓ ﻓﺈﻥ ‪ y‬ﻳﺴﺎﻭﻱ‪:‬‬ ‫‪u‬‬ ‫‪ .u =1‬ﺇﺫﺍ ﻛﺎﻥ‬ ‫‪12‬‬ ‫‪,‬‬ ‫‪y2‬‬ ‫ﻟﻨﺄﺧﺬ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ ﺍﻹﺣﺪﺍﺛﻲ‬ ‫)‪(6‬‬ ‫‪13‬‬ ‫‪a‬‬ ‫‪1‬‬ ‫‪b‬‬ ‫‪13‬‬ ‫‪c‬‬ ‫‪5‬‬ ‫‪d‬‬ ‫‪5‬‬ ‫‪13‬‬ ‫‪13‬‬ ‫‪13‬‬ ‫‪! 13‬‬ ‫)‪ (7‬ﻟﺘﻜﻦ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ ﺍﻹﺣﺪﺍﺛﻲ ﺍﻟﻨﻘﺎﻁ‪ A^1,3h, B^3,2h, C^0, - 1h, D^-4,1h :‬ﻓﻴﻜﻮﻥ‪:‬‬ ‫‪a 1 AB 2= 1 CD 2‬‬ ‫‪b 1 AB 2 = - 1 CD 2‬‬ ‫‪c 1 CD 2= - 2 1 AB 2‬‬ ‫‪d 1 AB 2 = - 2 1 CD 2‬‬ ‫)‪ (8‬ﻟﻨﺄﺧﺬ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ ﺍﻹﺣﺪﺍﺛﻲ ﺍﻟﻨﻘﺎﻁ‪ E^2,4h, F^-1, - 5h, G^x , yh :‬ﺇﺫﺍ ﻛﺎﻥ‪ 1 EF 2=1 EG 2 :‬ﻓﺈﻥ‬ ‫)‪ (x, y‬ﻳﺴﺎﻭﻱ‪:‬‬ ‫)‪a (- 1, - 5‬‬ ‫)‪b (- 5, - 13‬‬ ‫)‪c (5,13‬‬ ‫)‪d (1,5‬‬ ‫‪73‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺟﻤﻊ ﺍﻟﻤﺘﺠﻬﺎﺕ ﻭﻃﺮﺣﻬﺎ ‪5-2‬‬ ‫‪Addition and Subtraction of Vectors‬‬ ‫‪A‬‬ ‫‪C‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫‪E‬‬ ‫‪A‬‬ ‫‪B‬‬ ‫)‪ (1‬ﻓﻲ ﺍﻟﻤﺜﻠﺚ ‪ ABC‬ﺍﻟﻤﻘﺎﺑﻞ ‪ E‬ﻣﻨﺘﺼﻒ ‪ AB‬ﻭ‪ F‬ﻣﻨﺘﺼﻒ ‪BC‬‬ ‫)‪ (a‬ﻋﻴّﻦ ﺍﻟﻨﻘﻄﺔ ‪ M‬ﺣﻴﺚ‪1 BM 2=1 BE 2+1 BF 2 :‬‬ ‫‪F‬‬ ‫)‪ (b‬ﻋﻴّﻦ ﺍﻟﻨﻘﻄﺔ ‪ N‬ﺣﻴﺚ‪1 AN 2=1 AE 2+1 AF 2 :‬‬ ‫)‪ (c‬ﺃﺛﺒﺖ ﺃﻥ‪1 AB 2=1 MN 2 :‬‬ ‫)‪ (2‬ﻓﻲ ﺍﻟﻤﺜﻠﺚ ‪ ABC‬ﺍﻟﻤﻘﺎﺑﻞ‪ M ،‬ﻣﻨﺘﺼﻒ ‪BC‬‬ ‫‪BMC‬‬ ‫)‪ (a‬ﻋﻴّﻦ ﺍﻟﻨﻘﻄﺔ ‪ P‬ﺣﻴﺚ‪1 BP 2=1 MA 2+1 MC 2 :‬‬ ‫‪A ED‬‬ ‫)‪ (b‬ﻋﻴّﻦ ﺍﻟﻨﻘﻄﺔ ‪ Q‬ﺣﻴﺚ‪1 BQ 2=1 AC 2+1 MB 2 :‬‬ ‫)‪ (3‬ﻓﻲ ﺍﻟﺸﻜﻞ ﺍﻟﺮﺑﺎﻋﻲ ‪ ABCD‬ﺍﻟﻤﻘﺎﺑﻞ ‪ E‬ﻣﻨﺘﺼﻒ ‪ AD‬ﻭ‪ F‬ﻣﻨﺘﺼﻒ ‪BC‬‬ ‫)‪ (a‬ﻋﻴّﻦ ﺍﻟﻨﻘﻄﺔ ‪ P‬ﺣﻴﺚ‪1 CP 2=1 CD 2+1 BA 2 :‬‬ ‫)‪ (b‬ﺃﺛﺒﺖ ﺃﻥ‪1 CP 2=1 CE 2+1 BE 2 :‬‬ ‫‪B‬‬ ‫)‪ (c‬ﺃﺛﺒﺖ ﺃﻥ‪2 1 EF 2=1 AB 2+1 DC 2 :‬‬ ‫‪F‬‬ ‫)‪ A, B, C, D (4‬ﻧﻘﺎﻁ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ‪ ،‬ﺑ ّﺴﻂ‪:‬‬ ‫‪C‬‬ ‫)‪2 1 AB 2+4 1 BC 2+2 1 CD 2+2 1 DA 2 (a‬‬ ‫)‪2 1 AB 2- 3 1 AC 2+1 AD 2+2 1 BD 2 (b‬‬ ‫)‪ (5‬ﺍﻧﻄﻠﻖ ﻣﺮﻛﺐ ﺻﻴﺪ ﻣﻦ ﺍﻟﻤﻴﻨﺎﺀ ﻧﺎﺣﻴﺔ ﺍﻟﺸﺮﻕ ﻭﺍﺟﺘﺎﺯ ﻣﺴﺎﻓﺔ ‪ ،250 km‬ﺛﻢ ﺍﻧﺤﺮﻑ ﻋﻤﻮﺩﻳًّﺎ ﺑﺎﺗﺠﺎﻩ ﺍﻟﺸﻤﺎﻝ‬ ‫ﻟﻴﺠﺘﺎﺯ ﻣﺴﺎﻓﺔ ‪ ،40 km‬ﺛﻢ ﻋﺎﺩ ﻣﺒﺎﺷﺮﺓ ﺑﺨﻂ ﻣﺴﺘﻘﻴﻢ ﺇﻟﻰ ﺍﻟﻨﻘﻄﺔ ﺍﻟﺘﻲ ﺍﻧﻄﻠﻖ ﻣﻨﻬﺎ ﻓﻲ ﺍﻟﻤﻴﻨﺎﺀ ﺑﻤﺘﻮﺳﻂ ﺳﺮﻋﺔ‬ ‫ﻳﺴﺎﻭﻱ ‪50 km/h‬‬ ‫)‪ (a‬ﺍﺳﺘﺨﺪﻡ ﺍﻟﻤﺘﺠﻬﺎﺕ ﻟﺘﻨﻤﺬﺝ ﻣﺴﺎﺭ ﺍﻟﻤﺮﻛﺐ ﻓﻲ ﺭﺣﻠﺘﻪ‪.‬‬ ‫)‪ (b‬ﻣﺎ ﺍﻟﻮﻗﺖ ﺍﻟﺬﻱ ﺍﺳﺘﻐﺮﻗﻪ ﺍﻟﻤﺮﻛﺐ ﻟﻠﻌﻮﺩﺓ ﺇﻟﻰ ﺍﻟﻤﻴﻨﺎﺀ؟‬ ‫)‪ (6‬ﻳﺴﺒﺢ ﺧﺎﻟﺪ ﻣﻦ ﺿﻔﺔ ﺍﻟﻨﻬﺮ ﺍﻟﺠﻨﻮﺑﻴﺔ ﺇﻟﻰ ﺍﻟﻀﻔﺔ ﺍﻟﺸﻤﺎﻟﻴﺔ ﺍﻟﻤﻘﺎﺑﻠﺔ ﺑﻤﺘﻮﺳﻂ ﺳﺮﻋﺔ ﻳﺴﺎﻭﻱ ‪ 35 km/h‬ﻭﺗﺘﺤﺮﻙ‬ ‫ﺍﻟﻤﻴﺎﻩ ﺑﺎﺗﺠﺎﻩ ﺍﻟﺸﺮﻕ ﺑﻤﺘﻮﺳﻂ ﺳﺮﻋﺔ ﻳﺴﺎﻭﻱ ‪.12 km/h‬‬ ‫)‪ (a‬ﺍﺳﺘﺨﺪﻡ ﺍﻟﻤﺘﺠﻬﺎﺕ ﻟﺘﻨﻤﺬﺝ ﻣﻌﻄﻴﺎﺕ ﺍﻟﻤﺴﺄﻟﺔ‪.‬‬ ‫)‪ (b‬ﺃﻭﺟﺪ ﻣﺘﻮﺳﻂ ﺍﻟﺴﺮﻋﺔ ﺍﻟﻨﺎﺗﺠﺔ ﺍﻟﺘﻲ ﻳﻨﺘﻘﻞ ﺑﻬﺎ ﺧﺎﻟﺪ ﻣﻦ ﺿﻔﺔ ﺍﻟﻨﻬﺮ ﺍﻟﺠﻨﻮﺑﻴﺔ ﺇﻟﻰ ﺍﻟﻀﻔﺔ ﺍﻟﺸﻤﺎﻟﻴﺔ ﺍﻟﻤﻘﺎﺑﻠﺔ‪.‬‬ ‫‪74‬‬

‫)‪ (7‬ﻣﺜّﻞ ﺍﻟﻨﻘﺎﻁ ﺍﻟﺘﺎﻟﻴﺔ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ ﺍﻹﺣﺪﺍﺛﻲ ﺣﻴﺚ ‪ O‬ﻧﻘﻄﺔ ﺍﻷﺻﻞ‪ i , j ،‬ﻣﺘﺠﻬﻲ ﺍﻟﻮﺣﺪﺓ ﺍﻷﺳﺎﺳﻴﺎﻥ‬ ‫‪OA = 3 i - 4j , OB = - 2 i + 3j ،OC = - 4 i - j‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫‪ab‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-5‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ‪ ،‬ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ab‬‬ ‫)‪ (1‬ﺇﺫﺍ ﻛﺎﻥ ‪ 1 AB 2+1 BC 2=1 AC 2‬ﻓﺈﻥ‪AB + BC = AC :‬‬ ‫)‪1 AC 2+1 BA 2+1 CB 2= 0 (2‬‬ ‫)‪ ABCF (3‬ﻣﺘﻮﺍﺯﻱ ﺃﺿﻼﻉ ﺣﻴﺚ‪BA =1-2,3 2 ،BF =1 1, 4 2 :‬‬ ‫‪ab‬‬ ‫` ‪1 BC 2 = 1 3,1 2‬‬ ‫)‪ (4‬ﻓﻲ ﺍﻟﻤﺴﺘﻄﻴﻞ ‪ 1 AE 2=1 BD 2 :ABCD‬ﺇﺫًﺍ ‪a b 1 AC 2+1 AD 2=1 AE 2‬‬ ‫‪E‬‬ ‫‪AD‬‬ ‫‪ab‬‬ ‫‪BC‬‬ ‫)‪ (5‬ﻓﻲ ﺍﻟﻤﺜﻠﺚ ‪1 AB 2-1 AC 2+1 BC 2-1 BA 2=1 AB 2 :ABC‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(6-9‬ﻇﻠّﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫)‪ (6‬ﺇﺫﺍ ﻛﺎﻥ ‪ L =1 AC 2+ 2 1 AB 2-1 BC 2‬ﻓﺈﻥ‪:‬‬ ‫‪a‬‬ ‫‪L‬‬ ‫=‬ ‫‪1‬‬ ‫‪1 AB 2‬‬ ‫‪b‬‬ ‫‪L‬‬ ‫‪=-‬‬ ‫‪1‬‬ ‫‪1 AB‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪c L = 3 1 AB 2‬‬ ‫‪d L = - 3 1 AB 2‬‬ ‫)‪ (7‬ﺇﺫﺍ ﻛﺎﻥ ‪ ،1 AM 2= 2(3 i - j) + 3(- 2 i ) - 2j‬ﻓﺈﻥ ‪ 1 AM 2‬ﻳﺴﺎﻭﻱ‪:‬‬ ‫‪a 2 i - 3j‬‬ ‫‪b 3 i - 2j‬‬ ‫‪c -4j‬‬ ‫‪d 6 i - 6j‬‬ ‫)‪ ABCD (8‬ﻣﺘﻮﺍﺯﻱ ﺃﺿﻼﻉ ﺣﻴﺚ‪ . A^-2,1h, B^0, - 2h, C^3, - 1h :‬ﺇﺫًﺍ ﺇﺣﺪﺍﺛﻴﺎﺕ ‪ D‬ﻫﻲ‪:‬‬ ‫‪a ^2 ,2h‬‬ ‫‪b ^-1,2h‬‬ ‫‪c ^1,2h‬‬ ‫‪d ^1, - 2h‬‬ ‫‪75‬‬

‫)‪ U = 4 i - 2j , V = x i - j (9‬ﻫﻤﺎ ﻣﺘﺠﻬﺎﻥ ﻣﺘﻮﺍﺯﻳﺎﻥ‪ .‬ﻗﻴﻤﺔ ‪ x‬ﻫﻲ‪:‬‬ ‫‪a2‬‬ ‫‪b -2‬‬ ‫‪c8‬‬ ‫‪d -8‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ (10-13‬ﻟﺪﻳﻚ ﻗﺎﺋﻤﺘﺎﻥ‪ ،‬ﺍﺧﺘﺮ ﻣﻦ ﺍﻟﻘﺎﺋﻤﺔ )‪ (2‬ﻣﺎ ﻳﻨﺎﺳﺐ ﻛﻞ ﺗﻤﺮﻳﻦ ﻓﻲ ﺍﻟﻘﺎﺋﻤﺔ )‪ (1‬ﻟﺘﺤﺼﻞ ﻋﻠﻰ ﺇﺟﺎﺑﺔ ﺻﺤﻴﺤﺔ‪.‬‬ ‫‪E‬‬ ‫‪B‬‬ ‫‪AC‬‬ ‫‪O‬‬ ‫‪D‬‬ ‫‪F‬‬ ‫ﻣﻦ ﺍﻟﺸﻜﻞ ﺃﻋﻼﻩ‬ ‫ﺍﻟﻘﺎﺋﻤﺔ )‪(2‬‬ ‫ﺍﻟﻘﺎﺋﻤﺔ )‪(1‬‬ ‫‪a BD‬‬ ‫)‪AB + AD = (10‬‬ ‫‪b AC‬‬ ‫)‪CE + CF = (11‬‬ ‫‪c0‬‬ ‫‪d DB‬‬ ‫ﺍﻟﻘﺎﺋﻤﺔ )‪(2‬‬ ‫ﺍﻟﻘﺎﺋﻤﺔ )‪(1‬‬ ‫‪a 2 BA‬‬ ‫)‪EA = (12‬‬ ‫‪b 2 BE‬‬ ‫)‪2 OC = (13‬‬ ‫‪c - CA‬‬ ‫‪d CA‬‬ ‫‪76‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺍﻟﻀﺮﺏ ﺍﻟﺪﺍﺧﻠﻲ ‪5-3‬‬ ‫‪Scalar Product‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫)‪ (1‬ﻓﻲ ﻛﻞ ﺷﻜﻞ ﻣﻤﺎ ﻳﻠﻲ ﺃﻭﺟﺪ‪u : v :‬‬ ‫)‪(a‬‬ ‫)‪(b‬‬ ‫‪y‬‬ ‫‪3‬‬ ‫‪v‬‬ ‫‪u‬‬ ‫‪2‬‬ ‫‪60c‬‬ ‫‪1v‬‬ ‫‪u‬‬ ‫‪-3 -2 --1 1‬‬ ‫‪1 2 3 4x‬‬ ‫‪u = 4 units‬‬ ‫‪-2‬‬ ‫‪v = 3 units‬‬ ‫‪-3‬‬ ‫)‪ (2‬ﻟﻨﺄﺧﺬ‪ u =1 2, - 1 2, v =1-3, 2 2, w =1 1, 2 2 :‬ﺃﻭﺟﺪ‪:‬‬ ‫‪(a) u : v‬‬ ‫‪(b) u : w‬‬ ‫‪(c) v : w‬‬ ‫)‪(d) (3u) : (- 2v‬‬ ‫)‪(e) (- 4u) : (3v‬‬ ‫)‪ u , v (3‬ﻣﺘﺠﻬﺎﻥ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ ﺍﻹﺣﺪﺍﺛﻲ ﺣﻴﺚ‪ . u = 4 , v = 5, u : v = - 6 :‬ﺃﻭﺟﺪ‪:‬‬ ‫‪(a) ^2u + 3v h2‬‬ ‫‪(b) ^3u - 2v h : ^-2u + v h‬‬ ‫)‪ (4‬ﻟﻨﺄﺧﺬ ‪u =1 x , 4 2, v =1 2 , - 3 2‬‬ ‫)‪ (a‬ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ x‬ﺑﺤﻴﺚ ﻳﻜﻮﻥ ‪ u‬ﻣﺘﻌﺎﻣﺪ ﻣﻊ ‪.v‬‬ ‫)‪ (b‬ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ x‬ﺑﺤﻴﺚ ﻳﻜﻮﻥ ‪. u = 5 units‬‬ ‫)‪ (5‬ﻟﻨﺄﺧﺬ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ ﺍﻹﺣﺪﺍﺛﻲ ‪u =1 2 , - 2 2, v =1- 2,0 2‬‬ ‫ﺃﻭﺟﺪ ‪m^u , v h‬‬ ‫)‪ A^-1,3h, B^-3,1h, C^3, - 1h (6‬ﺛﻼﺙ ﻧﻘﺎﻁ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ ﺍﻹﺣﺪﺍﺛﻲ‪.‬‬ ‫)‪ (a‬ﺃﻭﺟﺪ‪AB , AC , BC :‬‬ ‫)‪ (b‬ﺃﻭﺟﺪ‪ AB : AC :‬ﺛﻢ ﺍﺳﺘﻨﺘﺞ ﻧﻮﻉ ﺍﻟﻤﺜﻠﺚ ‪.ABC‬‬ ‫‪77‬‬

‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(7-10‬ﺃﻭﺟﺪ ‪.u : v‬‬ ‫)‪u = 2 , v = 5 , m^u, v h = 135º (8‬‬ ‫)‪u = 2 , v = 3 , m^v, u h = 30º (7‬‬ ‫)‪u = 4 2 , v = 7 6 , m^u, v h = 90º (10‬‬ ‫)‪u = 3 , v = 4 , m^u, v h = 180º (9‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(11-14‬ﺍﺳﺘﺨﺪﻡ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ ﻹﻳﺠﺎﺩ‪:‬‬ ‫‪AB‬‬ ‫‪30º 4‬‬ ‫‪4‬‬ ‫)‪DE : BC (12‬‬ ‫)‪CF : DE (11‬‬ ‫‪E2 D‬‬ ‫‪C4‬‬ ‫)‪AD : BF (14‬‬ ‫)‪BF : CF (13‬‬ ‫‪5‬‬ ‫‪F‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-6‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ab‬‬ ‫)‪ (1‬ﺇﺫﺍ ﻛﺎﻥ ‪ ،u : v = 0‬ﻓﺈﻥ ‪u = v‬‬ ‫‪ab‬‬ ‫‪ab‬‬ ‫)‪ (2‬ﺇﺫﺍ ﻛﺎﻥ ‪ ،u =1-2, x 2, v =1 5,1 2, u = v‬ﻓﺈﻥ ‪x =-10‬‬ ‫‪ab‬‬ ‫‪ab‬‬ ‫)‪ (3‬ﺇﺫﺍ ﻛﺎﻥ ‪ ،u : w = - 5 , v : w = 3‬ﻓﺈﻥ ‪^u - v h : w = - 8‬‬ ‫‪ab‬‬ ‫)‪ (4‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ، A^-1,2h, B^2,3h, C^-4,5h‬ﻓﺈﻥ ‪AB : AC = - 6‬‬ ‫)‪ (5‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ ،L^-3,4h, M^0,5h‬ﻓﺈﻥ ‪LM = 10‬‬ ‫)‪ A , B (6‬ﻣﺘﺠﻬﺎﻥ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ ﺣﻴﺚ ‪A =1 2, - 3 2, B =1 1,0 2‬‬ ‫`‬ ‫‪cos (A, B) = 2‬‬ ‫‪13‬‬ ‫‪13‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(7-14‬ﻇﻠّﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪:‬‬ ‫)‪ (7‬ﺇﺫﺍ ﻛﺎﻥ ‪ ،u =1 2 , - 2 2, v =1-1, m 2, u : v = 3‬ﻓﺈﻥ ‪ m‬ﺗﺴﺎﻭﻱ‪:‬‬ ‫‪a‬‬ ‫‪5‬‬ ‫‪b‬‬ ‫‪5‬‬ ‫‪c‬‬ ‫‪1‬‬ ‫‪d‬‬ ‫‪1‬‬ ‫‪-2‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪-2‬‬ ‫‪C‬‬ ‫)‪ (8‬ﻓﻲ ﻣﺜﻠﺚ ‪ H ،ABC‬ﻫﻮ ﺍﻟﻤﺴﻘﻂ ﺍﻟﻌﻤﻮﺩﻱ ﻟـ ‪ C‬ﻋﻠﻰ ‪. AB‬‬ ‫= ‪AB : AC‬‬ ‫‪4‬‬ ‫‪A2 H‬‬ ‫‪4‬‬ ‫‪B‬‬ ‫‪a -6‬‬ ‫‪b 12‬‬ ‫‪c -12‬‬ ‫‪d6‬‬ ‫‪78‬‬

A AB = AC = 3 cm , m(BC , BA) = 70% ‫( ﻓﻲ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ‬9) 3 :‫ ﻳﺴﺎﻭﻱ ﺗﻘﺮﻳﺒًﺎ‬AB : AC \\ = 70c b 6.89 c3 d - 2.3 BC B :‫ ﺷﺒﻪ ﻣﻨﺤﺮﻑ ﻗﺎﺋﻢ )ﺍﻧﻈﺮ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ( ﺣﻴﺚ‬ABCD (10) a 2.3 b -11 AB = 5 cm , AO = 2 cm , OD = 2 cm , CD = 3 cm D 3C :‫ ﻳﺴﺎﻭﻱ‬OB : OC 2 O c 12 d -12 2 A5 a 11 (11) AB : AC = y B 3 A2 1 -3 -2 --11 1 2 3 4 5 6 x -2 -3 C a2 b -2 c 18 d0 cos (AB, AC) = ،‫( ﻓﻲ ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ‬12) y 6C 5 4 3 A2 1 B -3-2-1 1 2 3 4 5 6 x a0 b 3 c 1 d 1 79 5 2 10

‫)‪ (13‬ﺇﺫﺍ ﻛﺎﻥ ‪ u =1-5, m 2, v =1 2,3 2 ،u = v‬ﻓﺈﻥ ‪ m‬ﺗﺴﺎﻭﻱ‪:‬‬ ‫‪a‬‬ ‫‪10‬‬ ‫‪b‬‬ ‫‪3‬‬ ‫‪c‬‬ ‫‪- 10‬‬ ‫‪d‬‬ ‫‪15‬‬ ‫‪3‬‬ ‫‪- 10‬‬ ‫‪3‬‬ ‫‪2‬‬ ‫)‪ (14‬ﺇﺫﺍ ﻛﺎﻥ ‪ AB : BC = - 2‬ﻓﺈﻥ ) ‪ m(BA , BC‬ﻻ ﻳﻤﻜﻦ ﺃﻥ ﻳﺴﺎﻭﻱ‪:‬‬ ‫‪a 60º‬‬ ‫‪b 28º‬‬ ‫‪c 122º‬‬ ‫‪d 50º‬‬ ‫‪80‬‬

‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ ﺍﻟﺨﺎﻣﺴﺔ‬ ‫)‪ (1‬ﻟﻴﻜﻦ ‪A^2, 3h, B^-1, 5h, C^3, - 4h‬‬ ‫)‪ (a‬ﻋﻴّﻦ ﺍﻟﺰﻭﺝ ﺍﻟﻤﺮﺗﺐ ﺍﻟﺬﻱ ﻳﻤﺜﻞ ﻣﺘﺠﻪ ﺍﻟﻤﻮﺿﻊ ﻟـ ‪BA‬‬ ‫‪y‬‬ ‫)‪ (b‬ﺇﺫﺍ ﻛﺎﻥ ﻣﺘﺠﻪ ﺍﻟﻤﻮﺿﻊ ‪ OM‬ﻳﻤﺜﻞ ﺍﻟﻘﻄﻌﺔ ﺍﻟﻤﻮﺟﻬﺔ ‪ ،AC‬ﻓﺄﻭﺟﺪ ﺇﺣﺪﺍﺛﻴﺎﺕ ‪.M‬‬ ‫)‪ (2‬ﺇﺫﺍ ﻛﺎﻥ ‪u =1 2, - 2 2‬‬ ‫‪2‬‬ ‫‪x‬‬ ‫ﻓﺎﺭﺳﻢ ﻣﺘﺠﻪ ﺍﻟﻤﻮﺿﻊ‪ ،‬ﺛﻢ ﺃﻭﺟﺪ ﺍﻟﻤﻌﻴﺎﺭ‪ ،‬ﻭﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ‪ θ‬ﺍﻟﺘﻲ ﻳﺼﻨﻌﻬﺎ ﻣﻊ ﺍﻻﺗﺠﺎﻩ‬ ‫ﺍﻟﻤﻮﺟﺐ ﻟﻤﺤﻮﺭ ﺍﻟﺴﻴﻨﺎﺕ‪.‬‬ ‫‪2u‬‬ ‫ﻣﺘﺠﻪ ﻭﺣﺪﺓ‪.‬‬ ‫‪u‬‬ ‫‪ ،u =1‬ﻓﺄﻭﺟﺪ ﻗﻴﻤﺔ ‪ y‬ﺑﺤﻴﺚ ﻳﺼﺒﺢ‬ ‫‪22‬‬ ‫‪,y 2‬‬ ‫ﺇﺫﺍ ﻛﺎﻥ‬ ‫)‪(3‬‬ ‫‪3‬‬ ‫)‪ A, B, C, D (4‬ﺃﺭﺑﻊ ﻧﻘﺎﻁ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ ﻣﺨﺘﻠﻔﺔ ﻭﻟﻴﺴﺖ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ‪ .‬ﻟﺘﻜﻦ ﺍﻟﻨﻘﻄﺔ ‪ N‬ﺑﺤﻴﺚ‪:‬‬ ‫‪1 AN 2 = 1 AD 2+1 AB 2+1 DC 2‬‬ ‫)‪ (a‬ﺍﻛﺘﺐ ﺍﻟﻤﺘﺠﻪ ‪ 1 AN 2‬ﺑﺪﻻﻟﺔ ‪1 AC 2, 1 AB 2‬‬ ‫)‪ (b‬ﺍﺳﺘﻨﺘﺞ ﺃﻥ ﺍﻟﻤﻀﻠﻊ ‪ ABNC‬ﻫﻮ ﻣﺘﻮﺍﺯﻱ ﺃﺿﻼﻉ‪.‬‬ ‫‪M‬‬ ‫)‪ (5‬ﺍﺳﺘﺨﺪﻡ ﺍﻟﺮﺳﻢ ﺍﻟﻤﻘﺎﺑﻞ‪:‬‬ ‫)‪ (a‬ﺃﻭﺟﺪ ‪ 1 AM 2‬ﺑﺪﻻﻟﺔ ‪1 NA 2, 1 NM 2‬‬ ‫‪AM : AB‬‬ ‫=‬ ‫‪AN : AB‬‬ ‫=‬ ‫‪1‬‬ ‫‪AB‬‬ ‫ﺃﺛﺒﺖ ﺃﻥ‪2 :‬‬ ‫)‪(b‬‬ ‫‪2‬‬ ‫‪BN‬‬ ‫)‪ ABC (6‬ﻣﺜﻠﺚ ﺑﺤﻴﺚ‪A . AC = 2 3 , AB = 6, AB : AC = 18 :‬‬ ‫ﺃﻭﺟﺪ ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ) ‪m(AB , AC‬‬ ‫)‪ (7‬ﻟﻴﻜﻦ‪ A =1 x - 5, x - 5 2 , B =1 1,1 - x 2 :‬ﺃﻭﺟﺪ‪:‬‬ ‫)‪ (a‬ﻗﻴﻤﺔ ‪ x‬ﺑﺤﻴﺚ ﻳﻜﻮﻥ ﺍﻟﻤﺘﺠﻪ ‪ A‬ﻟﻪ ﺍﺗﺠﺎﻩ ‪B‬‬ ‫)‪ (b‬ﻗﻴﻤﺔ ‪ x‬ﺑﺤﻴﺚ ﻳﻜﻮﻥ ﺍﻟﻤﺘﺠﻪ ‪ A‬ﻣﺘﻌﺎﻣ ًﺪﺍ ﻣﻊ ﺍﻟﻤﺘﺠﻪ ‪B‬‬ ‫)‪ (8‬ﻟﻴﻜﻦ‪ A = 2, - 1 , B = 1,2 :‬ﻣﺘﺠﻬﻴﻦ ﻓﻲ ﻣﺴﺘﻮﻯ ﺇﺣﺪﺍﺛﻲ‪ .‬ﺃﻭﺟﺪ‪:‬‬ ‫‪(a) A : B‬‬ ‫‪(b) B 2‬‬ ‫‪(c) 1 3A + B 2:1 A + B 2‬‬ ‫‪(d) 1 A + 2B 2:1 2A - B 2‬‬ ‫‪81‬‬

‫)‪ (9‬ﻟﺘﻜﻦ ﺍﻟﻨﻘﺎﻁ‪ A^1,2h, B^4,0h, C^3,3h :‬ﻓﻲ ﻣﺴﺘﻮﻯ ﺇﺣﺪﺍﺛﻲ‪.‬‬ ‫ﺍﻟﻤﺴﺘﻘﻴﻢ ﺍﻟﻤﺘﻌﺎﻣﺪ ﻣﻊ ‪ AB‬ﺍﻟﻤﺎﺭ ﺑﺎﻟﻨﻘﻄﺔ ‪ C‬ﻳﻘﻄﻊ ﻣﺤﻮﺭ ﺍﻟﺼﺎﺩﺍﺕ ﺑﺎﻟﻨﻘﻄﺔ ‪.D‬‬ ‫ﺃﻭﺟﺪ ﺇﺣﺪﺍﺛﻴﺎﺕ ﺍﻟﻨﻘﻄﺔ ‪.D‬‬ ‫‪y‬‬ ‫‪C‬‬ ‫‪3‬‬ ‫‪2A‬‬ ‫‪j‬‬ ‫‪B‬‬ ‫‪x‬‬ ‫‪34‬‬ ‫‪i1‬‬ ‫‪D‬‬ ‫‪A‬‬ ‫)‪ ABC (10‬ﻣﺜﻠﺚ ﻣﺘﻄﺎﺑﻖ ﺍﻷﺿﻼﻉ‪ ،‬ﻃﻮﻝ ﺿﻠﻌﻪ ‪4 cm‬‬ ‫ﻟﻴﻜﻦ‪a = 1 AB 2, b = 1 AC 2 :‬‬ ‫)‪ (a‬ﺃﻭﺟﺪ ‪ 1 CB 2‬ﺑﺪﻻﻟﺔ ‪ a , b‬ﻭﺍﺳﺘﻨﺘﺞ ‪a - b‬‬ ‫‪C NB‬‬ ‫)‪ (b‬ﺃﻧﺸﺊ ﺍﻟﻨﻘﻄﺔ ‪ D‬ﺑﺤﻴﺚ ‪1 AD 2= a + b‬‬ ‫)‪ (c‬ﻣﺎ ﻧﻮﻉ ﺍﻟﺮﺑﺎﻋﻲ ‪ABDC‬؟‬ ‫)‪ (d‬ﺃﻭﺟﺪ ‪a + b‬‬ ‫‪N‬‬ ‫)‪ ABCD (11‬ﻣﺘﻮﺍﺯﻱ ﺃﺿﻼﻉ‪ ،‬ﻣﺮﻛﺰﻩ ‪.O‬‬ ‫‪CD‬‬ ‫‪ M‬ﻣﻨﺘﺼﻒ ‪ ،1 AB 2‬ﺍﻟﻨﻘﻄﺔ ‪ N‬ﺣﻴﺚ‪1 DN 2=1 OC 2 :‬‬ ‫)‪ (a‬ﺃﻭﺟﺪ ‪ 1 ON 2‬ﺑﺪﻻﻟﺔ ‪1 BC 2‬‬ ‫‪O‬‬ ‫)‪ (b‬ﺃﺛﺒﺖ ﺃﻥ‪1 BC 2 = 1 OD 2 +1 OC 2 :‬‬ ‫)‪ (c‬ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﻨﻘﺎﻁ ‪ M, N, O‬ﺗﻘﻊ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ‪.‬‬ ‫‪BM‬‬ ‫‪A‬‬ ‫)‪ (12‬ﺃﻭﺟﺪ ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ‪ θ‬ﺍﻟﻤﺤﺪﺩﺓ ﺑﺎﻟﻤﺘﺠﻬﻴﻦ ‪1 AB 2,1 AC 2‬‬ ‫‪y‬‬ ‫‪8B‬‬ ‫‪7‬‬ ‫‪6‬‬ ‫‪5‬‬ ‫‪4‬‬ ‫‪3‬‬ ‫‪2θ‬‬ ‫‪1A‬‬ ‫‪C‬‬ ‫‪1 2 3 4 5 6x‬‬ ‫‪82‬‬

‫)‪ (13‬ﺇﺫﺍ ﻛﺎﻧﺖ ‪ A^-4,1h, B^-1,2h, C^1, - 4h‬ﺭﺅﻭﺱ ﺍﻟﻤﺜﻠﺚ ‪ABC‬‬ ‫ﻓﺄﺛﺒﺖ ﺃﻥ ﺍﻟﻤﺜﻠﺚ ﻗﺎﺋﻢ ﻓﻲ ‪.B‬‬ ‫‪y‬‬ ‫‪4‬‬ ‫‪3‬‬ ‫‪B2‬‬ ‫‪A1‬‬ ‫‪-5 -4 -3 -2 -1-1‬‬ ‫‪1x‬‬ ‫‪C‬‬ ‫‪-2‬‬ ‫‪-3‬‬ ‫‪-4‬‬ ‫)‪ (14‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻤﺘﺠﻬﺎﺕ‪A = - 4 i - 2j , B = - i - 3j , C =1-5,5 2 ،‬‬ ‫)‪ (a‬ﺃﺛﺒﺖ ﺃﻥ‪B ! C :‬‬ ‫)‪ (b‬ﺃﻭﺟﺪ‪A : B , A : C :‬‬ ‫)‪ (c‬ﻣﺎﺫﺍ ﻧﺴﺘﻨﺘﺞ؟‬ ‫ﻓﻲ ﺍﻟﺘﻤﺮﻳﻦ )‪ ،(15‬ﺍﺧﺘﺮ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫)‪ (15‬ﻟﻴﻜﻦ‪ ، A =1-4,3 2 :‬ﻓﺈﻥ ﺍﻟﻤﺘّﺠﻪ ﺍﻟﻤﺘﻌﺎﻣﺪ ﻣﻊ ‪ A‬ﻣﻤﺎ ﻳﻠﻲ ﻫﻮ‪:‬‬ ‫)‪(a‬‬ ‫‪1‬‬ ‫‪2, -‬‬ ‫‪3‬‬ ‫‪2‬‬ ‫‪(b) 1 3, - 4 2‬‬ ‫)‪(c‬‬ ‫‪1‬‬ ‫‪3‬‬ ‫‪,2 2‬‬ ‫‪(d) 1 4 ,3 2‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪83‬‬

‫‪D‬‬ ‫‪C‬‬ ‫ﺗﻤﺎﺭﻳﻦ ﺇﺛﺮﺍﺋﻴﺔ‬ ‫‪RQ‬‬ ‫‪5‬‬ ‫‪B‬‬ ‫)‪ (1‬ﻟﻨﺄﺧﺬ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ ﺍﻹﺣﺪﺍﺛﻲ ﺍﻟﻤﻨﺘﻈﻢ ﺍﻟﻤﺘﻌﺎﻣﺪ ﺍﻟﻨﻘﺎﻁ‪:‬‬ ‫‪A xP‬‬ ‫‪ A^2, 2h, B^4, 5h, C^4 - m, 0h‬ﺣﻴﺚ ‪ m‬ﻋﺪﺩ ﺣﻘﻴﻘﻲ‪.‬‬ ‫)‪ (a‬ﺃﻭﺟﺪ ﻗﻴﻤﺔ ‪ m‬ﺑﺤﻴﺚ ﻳﻜﻮﻥ ﺍﻟﻤﺜﻠﺚ ‪ ABC‬ﻗﺎﺋﻢ ‪.A‬‬ ‫)‪ (b‬ﻟﻘﻴﻤﺔ ‪ m‬ﺍﻟﺘﻲ ﻭﺟﺪﺗﻬﺎ‪ ،‬ﺃﺛﺒﺖ ﺃﻥ ‪ ABC‬ﻣﺜﻠﺚ ﻣﺘﻄﺎﺑﻖ ﺍﻟﻀﻠﻌﻴﻦ‪.‬‬ ‫)‪ (2‬ﺍﻟﺸﻜﻞ ﺍﻟﻤﻘﺎﺑﻞ ﻳﻤﺜﻞ ﻣﺮﺑ ًﻌﺎ ﺭﺳﻢ ﻓﻲ ﺩﺍﺧﻠﻪ ﻣﺴﺘﻄﻴﻞ‪.‬‬ ‫ﺃﺛﺒﺖ ﺃﻥ ﺍﻟﻤﺴﺘﻘﻴﻤﻴﻦ‪:‬‬ ‫‪ CQ , PR‬ﻣﺘﻌﺎﻣﺪﻳﻦ‪.‬‬ ‫)ﻣﺴﺎﻋﺪﺓ‪ :‬ﺍﺳﺘﺨﺪﻡ ﻋﻼﻗﺔ ﺷﺎﻝ(‬ ‫)‪ (3‬ﻓﻲ ﺍﻟﻤﺜﻠﺚ ‪ MAB‬ﺍﻷﺩﻧﺎﻩ ﺃﺛﺒﺖ ﺃﻥ‪:‬‬ ‫‪MA : MB = MI 2 - a2‬‬ ‫‪M‬‬ ‫‪D‬‬ ‫‪5 cm‬‬ ‫‪C‬‬ ‫‪Aa I a B‬‬ ‫‪2 cm‬‬ ‫‪B‬‬ ‫‪θ‬‬ ‫)‪ (4‬ﺇﺫﺍ ﻛﺎﻥ‪ ، A + B = w , A - 2B = - w :‬ﻓﺄﺛﺒﺖ ﺃﻥ‪:‬‬ ‫‪A‬‬ ‫‪ A , B‬ﻟﻬﻤﺎ ﺍﻻﺗﺠﺎﻩ ﻧﻔﺴﻪ‪.‬‬ ‫‪E‬‬ ‫)‪ (5‬ﻓﻲ ﺍﻟﻤﺴﺘﻄﻴﻞ ﺍﻟﻤﻘﺎﺑﻞ ‪ E‬ﻣﻨﺘﺼﻒ ‪. AB‬‬ ‫ﺃﻭﺟﺪ ‪) θ‬ﺍﺳﺘﺨﺪﻡ ﺍﻵﻟﺔ ﺍﻟﺤﺎﺳﺒﺔ(‪.‬‬ ‫‪84‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺍﻟﻤﺠﺘﻤﻊ ﺍﻹﺣﺼﺎﺋﻲ ﻭﺍﻟﻤﻌﺎﻳﻨﺔ ‪6-1‬‬ ‫‪Statistical Population and Sampling‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫)‪ (1‬ﺃﺫﻛﺮ ﻣﺮﺍﺣﻞ ﺍﻟﺒﺤﺚ ﺍﻹﺣﺼﺎﺋﻲ ﺍﻷﺭﺑﻌﺔ ﻣﺮﺗﺒﺔ‪.‬‬ ‫)‪ (2‬ﻣﺎ ﻫﻲ ﺃﺳﺎﻟﻴﺐ ﺟﻤﻊ ﺍﻟﺒﻴﺎﻧﺎﺕ‪.‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺮﻳﻨﻴﻦ )‪ ،(3-4‬ﺍﺫﻛﺮ ﻣﺎ ﻧﻮﻉ ﺍﻟﺒﻴﺎﻧﺎﺕ ﺍﻟﺘﻲ ﺗﺼﻒ ﻛ ًّﻼ ﻣﻦ ﺍﻟﺤﺎﻻﺕ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫)‪ (3‬ﻋﺪﺩ ﺍﻟﺘﺬﺍﻛﺮ ﺍﻟﻤﺒﺎﻋﺔ ﻹﺣﺪﻯ ﺍﻟﻤﺴﺮﺣﻴﺎﺕ‪.‬‬ ‫)‪ (4‬ﺃﻧﻮﺍﻉ ﻣﻨﺘﺠﺎﺕ ﻣﻌﺠﻮﻥ ﺍﻷﺳﻨﺎﻥ ﺍﻟﻤﺒﺎﻋﺔ ﻟﻠﻤﺴﺘﻬﻠﻚ‪.‬‬ ‫)‪ (b‬ﺃﻧﻮﺍﻉ ﺍﻟﻜﺘﺐ ﻓﻲ ﻣﻜﺘﺒﺔ ﺍﻟﻤﺪﺭﺳﺔ‪.‬‬ ‫)‪ (5‬ﺣ ّﺪﺩ ﻧﻮﻉ ﺍﻟﺒﻴﺎﻧﺎﺕ ﻟﻜﻞ ﻣﻤﺎ ﻳﻠﻲ‪:‬‬ ‫)‪ (d‬ﺃﻟﻮﺍﻥ ﺃﺣﺬﻳﺔ ﺍﻟﻄ ّﻼﺏ ﻓﻲ ﺻﻔﻚ‪.‬‬ ‫)‪ (a‬ﺃﻭﺯﺍﻥ ﻃ ّﻼﺏ ﺍﻟﺼﻒ ﺍﻟﺤﺎﺩﻱ ﻋﺸﺮ ﻓﻲ ﻣﺪﺭﺳﺘﻚ‪.‬‬ ‫)‪ (c‬ﺍﻟﺪﺧﻞ ﺍﻟﺸﻬﺮﻱ ﻟﻸﺳﺮﺓ ﻓﻲ ﺩﻭﻟﺔ ﻣﺎ‪.‬‬ ‫)‪ (6‬ﻋﺮﻑ ﺍﻟﻤﺠﺘﻤﻊ ﺍﻟﻤﻨﺘﻬﻲ ﻭﺍﻟﻤﺠﺘﻤﻊ ﻏﻴﺮ ﺍﻟﻤﻨﺘﻬﻲ‪.‬‬ ‫)‪ (7‬ﻋ ّﺮﻑ ﻛ ًّﻼ ﻣﻦ‪:‬‬ ‫)‪ (c‬ﺍﻟﺤﺼﺮ ﺍﻟﺸﺎﻣﻞ‪.‬‬ ‫)‪ (b‬ﺍﻟﻤﺠﺘﻤﻊ ﺍﻹﺣﺼﺎﺋﻲ‪.‬‬ ‫)‪ (a‬ﻋﻠﻢ ﺍﻹﺣﺼﺎﺀ‪.‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫‪ab‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-5‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ab‬‬ ‫)‪ (1‬ﺍﻟﻤﻮﺍﻟﻴﺪ ﻓﻲ ﺍﻟﻌﺎﻟﻢ ﺳﻨﺔ ‪ 2010‬ﻋﺒﺎﺭﺓ ﻋﻦ ﻣﺠﺘﻤﻊ ﻏﻴﺮ ﻣﻨﺘﻪ‪.‬‬ ‫‪ab‬‬ ‫)‪ (2‬ﻭﺣﺪﺓ ﺍﻟﺪﺭﺍﺳﺔ ﻟﻌﺪﺩ ﺯﻭﺍﺭ ﻣﺮﻛﺰ ﻋﻠﻤﻲ ﻓﻲ ﻳﻮﻡ ﻭﺍﺣﺪ ﻫﻲ ﺃﻱ ﺯﺍﺋﺮ‪.‬‬ ‫‪ab‬‬ ‫)‪ (3‬ﻳﻤﻜﻦ ﺍﺳﺘﺨﺪﺍﻡ ﺍﻟﺤﺼﺮ ﺍﻟﺸﺎﻣﻞ ﻓﻲ ﺩﺭﺍﺳﺔ ﺃﻧﻮﺍﻉ ﺍﻟﺴﻤﻚ ﺍﻟﻤﻮﺟﻮﺩﺓ‬ ‫‪ab‬‬ ‫ﻓﻲ ﺃﺣﺪ ﺍﻟﻤﺤﻴﻄﺎﺕ‪.‬‬ ‫)‪ (4‬ﻋﺪﺩ ﺍﻟﺼﻔﺤﺎﺕ ﻓﻲ ﻛﺘﺎﺏ ﻣﺎ ﻫﻮ ﺑﻴﺎﻧﺎﺕ ﻛﻤﻴﺔ ﻣﺴﺘﻤﺮﺓ‪.‬‬ ‫)‪ (5‬ﻋﻨﺪ ﺗﺮﺗﻴﺐ ﺍﻷﺷﻴﺎﺀ ﻧﺴﺘﺨﺪﻡ ﺑﻴﺎﻧﺎﺕ ﻛﻴﻔﻴﺔ ﻣﺮﺗﺒﺔ‪.‬‬ ‫‪85‬‬

‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(6-10‬ﻇﻠّﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫)‪ (6‬ﺍﻟﺒﻴﺎﻧﺎﺕ ﺍﻟﻜﻴﻔﻴﺔ ﺗﻜﻮﻥ‪:‬‬ ‫‪ b‬ﻣﺮﺗﺒﺔ ﻓﻘﻂ‬ ‫‪ a‬ﺍﺳﻤﻴﺔ ﺃﻭ ﻣﺮﺗﺒﺔ‬ ‫‪ d‬ﺍﺳﻤﻴﺔ ﻓﻘﻂ‬ ‫‪ c‬ﻣﺘﻘﻄﻌﺔ‬ ‫)‪ (7‬ﺍﻟﺒﻴﺎﻧﺎﺕ ﺍﻟﻤﺴﺘﻤﺮﺓ ﻫﻲ ﺑﻴﺎﻧﺎﺕ‪:‬‬ ‫‪ d‬ﻛﻴﻔﻴﺔ‬ ‫‪ c‬ﻛﻤﻴﺔ‬ ‫‪ b‬ﻣﺮﺗﺒﺔ‬ ‫‪ a‬ﺍﺳﻤﻴﺔ‬ ‫)‪ (8‬ﻋﻨﺪ ﺇﺟﺮﺍﺀ ﺗﺤﺎﻟﻴﻞ ﺍﻟﺪﻡ ﻧﺴﺘﺨﺪﻡ‪:‬‬ ‫‪ b‬ﺍﻟﻤﻌﺎﻳﻨﺔ‬ ‫‪ a‬ﺍﻟﺤﺼﺮ ﺍﻟﺸﺎﻣﻞ‬ ‫‪ d‬ﻟﻴﺲ ﺃﻳًّﺎ ﻣﻤﺎ ﺳﺒﻖ‬ ‫‪ c‬ﺍﻟﺤﺼﺮ ﺍﻟﺸﺎﻣﻞ ﻭﺍﻟﻤﻌﺎﻳﻨﺔ‬ ‫)‪ (9‬ﺍﻟﺒﻴﺎﻧﺎﺕ ﺍﻟﻜﻤﻴﺔ ﺗﻜﻮﻥ‪:‬‬ ‫‪ b‬ﻣﺮﺗﺒﺔ ﻓﻘﻂ‬ ‫‪ a‬ﺍﺳﻤﻴﺔ ﺃﻭ ﻣﺮﺗﺒﺔ‬ ‫‪ d‬ﻣﺴﺘﻤﺮﺓ ﻓﻘﻂ‬ ‫‪ c‬ﻣﺘﻘﻄﻌﺔ ﺃﻭ ﻣﺴﺘﻤﺮﺓ‬ ‫)‪ (10‬ﻋﺪﺩ ﺍﻟﻤﺸﺎﻫﺪﻳﻦ ﻓﻲ ﻣﺒﺎﺭﺍﺓ ﻛﺮﺓ ﻗﺪﻡ ﻫﻮ ﻋﺒﺎﺭﺓ ﻋﻦ ﺑﻴﺎﻧﺎﺕ‪:‬‬ ‫‪ b‬ﻛﻴﻔﻴﺔ ﻣﺮﺗﺒﺔ‬ ‫‪ a‬ﻛﻴﻔﻴﺔ ﺍﺳﻤﻴﺔ‬ ‫‪ d‬ﻛﻤﻴﺔ ﻣﺴﺘﻤﺮﺓ‬ ‫‪ c‬ﻛﻤﻴﺔ ﻣﺘﻘﻄﻌﺔ‬ ‫‪86‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺍﻟﻌﻴﻨﺎﺕ ‪6-2‬‬ ‫‪Samples‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫)‪ (1‬ﺃﻭﺟﺪ ﻛﺴﺮ ﺍﻟﻤﻌﺎﻳﻨﺔ ﻋﻨﺪﻣﺎ ﻳﻜﻮﻥ ﺣﺠﻢ ﺍﻟﻌﻴﻨﺔ ‪ 8‬ﻭﺣﺠﻢ ﺍﻟﻤﺠﺘﻤﻊ ‪.100‬‬ ‫)‪ (2‬ﺃﻭﺟﺪ ﺣﺠﻢ ﺍﻟﻤﺠﺘﻤﻊ ﺍﻹﺣﺼﺎﺋﻲ ﺇﺫﺍ ﻛﺎﻥ ﻃﻮﻝ ﺍﻟﻔﺘﺮﺓ ‪ 5‬ﻭﺣﺠﻢ ﺍﻟﻌﻴﻨﺔ ‪.100‬‬ ‫)‪ (3‬ﻣﺎ ﺍﻟﻔﺮﻕ ﺑﻴﻦ ﺍﻟﻌﻴﻨﺔ ﺍﻟﻌﺸﻮﺍﺋﻴﺔ ﺍﻟﺒﺴﻴﻄﺔ ﻭﺍﻟﻌﻴﻨﺔ ﺍﻟﻌﺸﻮﺍﺋﻴﺔ ﺍﻟﻄﺒﻘﻴﺔ؟‬ ‫)‪ (4‬ﺷﺮﻛﺔ ﺩﺭﺍﺳﺎﺕ ﺗﺮﻳﺪ ﺍﺳﺘﻔﺘﺎﺀ ﺍﻟﻌ ّﻤﺎﻝ ﻭﺃﺻﺤﺎﺏ ﺍﻟﻌﻤﻞ ﻓﻲ ﻣﻨﻄﻘﺔ ﻣﻌﻴّﻨﺔ‪ .‬ﻳﺒﻠﻎ ﻋﺪﺩ ﺍﻟﻌ ّﻤﺎﻝ ‪ 200‬ﻋﺎﻣﻞ ﻭﺃﺻﺤﺎﺏ‬ ‫ﺍﻟﻌﻤﻞ ‪.40‬‬ ‫)‪ (a‬ﺃﻱ ﻧﻮﻉ ﻋﻴﻨﺔ ﻋﺸﻮﺍﺋﻴﺔ ﺗﺴﺘﺨﺪﻡ ﻓﻲ ﻫﺬﻩ ﺍﻟﺤﺎﻟﺔ؟‬ ‫)‪ (b‬ﻛﻢ ﻳﺴﺎﻭﻱ ﻛﺴﺮ ﺍﻟﻤﻌﺎﻳﻨﺔ ﺇﺫﺍ ﻛﻨﺎ ﻧﺮﻳﺪ ﻋﻴﻨﺔ ﻣﻦ ‪ 60‬ﺷﺨﺺ؟‬ ‫)‪ (c‬ﻫﻞ ﻧﺴﺘﺨﺪﻡ ﺟﺪﻭﻝ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻌﺸﻮﺍﺋﻴﺔ ﻓﻲ ﻫﺬﻩ ﺍﻟﺪﺭﺍﺳﺔ؟‬ ‫)‪ (d‬ﻧﺮﻗّﻢ ﺍﻟﻌ ّﻤﺎﻝ ﻣﻦ ‪ 1‬ﺇﻟﻰ ‪ 200‬ﻭﺃﺻﺤﺎﺏ ﺍﻟﻌﻤﻞ ﻣﻦ ‪ 201‬ﺇﻟﻰ ‪.240‬‬ ‫ﺍﺳﺘﺨﺪﻡ ﺍﻟﺼﻒ ﺍﻟﺴﺎﺩﺱ ﻭﺍﻟﻌﻤﻮﺩ ﺍﻟﺴﺎﺩﺱ ﻭﻋ ّﺪﺩ ﺃ ّﻭﻝ ‪ 5‬ﺃﻋﺪﺍﺩ ﻟﻠﺴﺤﺐ ﺍﻟﻌﺸﻮﺍﺋﻲ ﻣﻦ ﻛﻞ ﻃﺒﻘﺔ‪.‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-5‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ab‬‬ ‫)‪ (1‬ﻟﻠﺤﺼﻮﻝ ﻋﻠﻰ ﺃﻓﻀﻞ ﺗﻤﺜﻴﻞ ﻟﻠﻤﺠﺘﻤﻊ ﻧﺨﺘﺎﺭ ﺍﻟﻌﻴﻨﺔ ﺑﻄﺮﻳﻘﺔ ﻋﺸﻮﺍﺋﻴﺔ‪.‬‬ ‫‪ab‬‬ ‫‪ab‬‬ ‫)‪ (2‬ﻻ ﻳﻮﺟﺪ ﻓﺮﻕ ﺑﻴﻦ ﺍﻟﻌﻴﻨﺔ ﺍﻟﻌﺸﻮﺍﺋﻴﺔ ﺍﻟﺒﺴﻴﻄﺔ ﻭﺍﻟﻌﻴﻨﺔ ﺍﻟﻌﺸﻮﺍﺋﻴﺔ ﺍﻟﻄﺒﻘﻴﺔ‪.‬‬ ‫‪ab‬‬ ‫ﻛﺴﺮ ﺍﻟﻤﻌﺎﻳﻨﺔ‬ ‫=‬ ‫ﺍﻟﻤﺠﺘﻤﻊ‬ ‫ﺣﺠﻢ‬ ‫)‪(3‬‬ ‫‪ab‬‬ ‫ﺣﺠﻢ ﺍﻟﻌﻴﻨﺔ‬ ‫)‪ (4‬ﺣﺠﻢ ﺍﻟﻤﺠﺘﻤﻊ ﺍﻹﺣﺼﺎﺋﻲ = ﻃﻮﻝ ﺍﻟﻔﺘﺮﺓ ‪ #‬ﺣﺠﻢ ﺍﻟﻌﻴﻨﺔ‬ ‫)‪ (5‬ﺇﺫﺍ ﻛﺎﻥ ﻃﻮﻝ ﺍﻟﻔﺘﺮﺓ ﻳﺴﺎﻭﻱ ‪ ،70‬ﻭﺍﻟﻤﻔﺮﺩﺓ ﺍﻷﻭﻟﻰ ﺗﺴﺎﻭﻱ ‪،43‬‬ ‫ﻓﺎﻟﻤﻔﺮﺩﺓ ﺍﻟﺨﺎﻣﺴﺔ ﺗﺴﺎﻭﻱ ‪322‬‬ ‫‪87‬‬

‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(6-10‬ﻇﻠّﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪:‬‬ ‫)‪ (6‬ﻳﺘﻮﺍﻓﺮ ﻓﻲ ﺍﻟﻌﻴﻨﺔ ﺍﻟﻌﺸﻮﺍﺋﻴﺔ ﺍﻟﺒﺴﻴﻄﺔ‪:‬‬ ‫‪ b‬ﺍﻹﺗﺎﺣﺔ ﻟﻜﻞ ﻋﻨﺼﺮ ﻓﻴﻬﺎ ﺍﻟﻔﺮﺻﺔ ﻧﻔﺴﻬﺎ ﻓﻲ ﺍﻟﻈﻬﻮﺭ‬ ‫‪ a‬ﺷﺮﻁ ﺍﻟﺘﺤﻴﺰ‬ ‫‪ d‬ﻛﻞ ﻣﻤﺎ ﺳﺒﻖ‪.‬‬ ‫‪ c‬ﺷﺮﻁ ﺍﻟﻌﺸﻮﺍﺋﻴﺔ ﻭﺍﻻﻧﺘﻈﺎﻡ‬ ‫)‪ (7‬ﻳﺘﻮﻓﺮ ﻓﻲ ﺍﻟﻌﻴﻨﺔ ﺍﻟﻤﻨﺘﻈﻤﺔ‪:‬‬ ‫‪ b‬ﺷﺮﻁ ﺍﻻﻧﺘﻈﺎﻡ ﻓﻘﻂ‬ ‫‪ a‬ﺷﺮﻁ ﺍﻟﻌﺸﻮﺍﺋﻴﺔ ﻭﺍﻻﻧﺘﻈﺎﻡ‬ ‫‪ d‬ﻟﻴﺲ ﺃﻳًّﺎ ﻣﻤﺎ ﺳﺒﻖ‬ ‫‪ c‬ﺷﺮﻁ ﺍﻟﻌﺸﻮﺍﺋﻴﺔ ﻓﻘﻂ‬ ‫)‪ (8‬ﻋﻨﺪ ﺍﺳﺘﺨﺪﺍﻡ ﺍﻟﻌﻴﻨﺔ ﺍﻟﻄﺒﻘﻴﺔ ﻳﻔﻀﻞ ﺃﻥ‪:‬‬ ‫‪ b‬ﺗﻜﻮﻥ ﻃﺒﻘﺎﺕ ﺍﻟﻤﺠﺘﻤﻊ ﻣﺘﺠﺎﻧﺴﺔ ﺑﺪﺍﺧﻠﻬﺎ ﻣﺨﺘﻠﻔﺔ ﻓﻲ ﻣﺎ ﺑﻴﻨﻬﺎ‬ ‫‪ a‬ﺗﻜﻮﻥ ﻋﺸﻮﺍﺋﻴﺔ ﻭﻣﻨﺘﻈﻤﺔ‬ ‫‪ d‬ﻟﻴﺲ ﺃﻳًّﺎ ﻣﻤﺎ ﺳﺒﻖ‬ ‫‪ c‬ﻻ ﺗﺘﻴﺢ ﻟﻜﻞ ﻋﻨﺼﺮ ﻓﻴﻬﺎ ﺍﻟﻔﺮﺻﺔ ﻧﻔﺴﻬﺎ ﻓﻲ ﺍﻟﻈﻬﻮﺭ‬ ‫)‪ (9‬ﺇﺫﺍ ﻛﺎﻥ ﺣﺠﻢ ﺍﻟﻌﻴﻨﺔ ﻳﺴﺎﻭﻱ ‪ 100‬ﻭﺣﺠﻢ ﺍﻟﻤﺠﺘﻤﻊ ﺍﻹﺣﺼﺎﺋﻲ ﻳﺴﺎﻭﻱ ‪ ،2 000‬ﻓﻜﺴﺮ ﺍﻟﻤﻌﺎﻳﻨﺔ ﻳﺴﺎﻭﻱ‪:‬‬ ‫‪a 0.3‬‬ ‫‪b 0.5‬‬ ‫‪c 0.05‬‬ ‫‪d 0.02‬‬ ‫)‪ (10‬ﺇﺫﺍ ﻛﺎﻥ ﻃﻮﻝ ﺍﻟﻔﺘﺮﺓ ﻳﺴﺎﻭﻱ ‪ 40‬ﻭﺣﺠﻢ ﺍﻟﻤﺠﺘﻤﻊ ﺍﻹﺣﺼﺎﺋﻲ ﻳﺴﺎﻭﻱ ‪ ،1000‬ﻓﺤﺠﻢ ﺍﻟﻌﻴﻨﺔ ﻳﺴﺎﻭﻱ‪:‬‬ ‫‪a 35‬‬ ‫‪b 25‬‬ ‫‪c 40‬‬ ‫‪d 30‬‬ ‫‪88‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺃﺳﺎﻟﻴﺐ ﻋﺮﺽ ﺍﻟﺒﻴﺎﻧﺎﺕ ‪6-3‬‬ ‫‪Ways to Display Data‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫)‪ (1‬ﺃﺛﻨﺎﺀ ﻋﻤﻞ ﺍﻟﻄﻼﺏ ﻓﻲ ﻣﺠﻤﻮﻋﺎﺕ ﻋﻠﻰ ﻧﺸﺎﻁ ﻣﻌﻴﻦ ﻓﻲ ﺍﻟﺼﻒ ﺳﺠﻞ ﺍﻟﻤﻌﻠﻢ ﺍﻟﻤﻼﺣﻈﺎﺕ ﺍﻟﻤﺒﻴّﻨﺔ ﻓﻲ ﺍﻟﺠﺪﻭﻝ‬ ‫ﺍﻟﺘﺎﻟﻲ‪:‬‬ ‫ﻳﺴﺘﻤﻊ ﻓﻘﻂ ﻳﺤﺎﻭﺭ ﻭﻳﻨﺎﻗﺶ ﺍﻟﻔﺌﺔ‬ ‫ﻳﺘﺨﺬ ﻗﺮﺍ ًﺭﺍ‬ ‫ﺍﻟﻤﺠﻤﻮﻉ ﻏﻴﺮ ﻣﺸﺎﺭﻙ‬ ‫‪4‬‬ ‫‪6 22‬‬ ‫ﺍﻟﺘﻜﺮﺍﺭ‬ ‫‪5‬‬ ‫‪7‬‬ ‫)‪ (a‬ﺃﻭﺟﺪ ﺍﻟﺘﻜﺮﺍﺭ ﺍﻟﻨﺴﺒﻲ ﻭﺍﻟﺘﻜﺮﺍﺭ ﺍﻟﻤﺌﻮﻱ ﻟﻜﻞ ﻓﺌﺔ‪.‬‬ ‫)‪ (b‬ﺍﻋﺮﺽ ﻫﺬﻩ ﺍﻟﺒﻴﺎﻧﺎﺕ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺍﻟﻘﻄﺎﻋﺎﺕ ﺍﻟﺪﺍﺋﺮﻳﺔ‪.‬‬ ‫)‪ (2‬ﻳﺒﻴّﻦ ﺍﻟﺠﺪﻭﻝ ﺍﻟﺘﺎﻟﻲ ﻭﻗﺖ ﺧﺮﻭﺝ ﺍﻟﺴﻴﺎﺭﺍﺕ ﻣﻦ ﺃﺣﺪ ﺍﻟﻤﻨﺘﺠﺎﺕ ﺍﻟﺴﻴﺎﺣﻴﺔ ﺑﻌﺪ ﻇﻬﺮ ﺃﺣﺪ ﺍﻷﻳﺎﻡ‪.‬‬ ‫ﺍﻟﻤﺠﻤﻮﻉ ‪ 4- 5- 6- 7- 8- 9-‬ﺍﻟﻔﺌﺔ‬ ‫‪ 17 31 25 14 7 6 100‬ﺍﻟﺘﻜﺮﺍﺭ‬ ‫)‪ (a‬ﺃﻛﻤﻞ ﺍﻟﺠﺪﻭﻝ ﺑﺈﺿﺎﻓﺔ ﻣﺮﺍﻛﺰ ﺍﻟﻔﺌﺎﺕ‪.‬‬ ‫)‪ (b‬ﺍﺭﺳﻢ ﺍﻟﻤﻨﺤﻨﻰ ﺍﻟﺘﻜﺮﺍﺭﻱ‪.‬‬ ‫)‪ (c‬ﺍﺭﺳﻢ ﺍﻟﻤﺪﺭﺝ ﺍﻟﺘﻜﺮﺍﺭﻱ ﻭﻣﻨﻪ ﺍﻟﻤﻨﺤﻨﻰ ﺍﻟﺘﻜﺮﺍﺭﻱ‪.‬‬ ‫)‪ (3‬ﻳﻌﺮﺽ ﻣﺪﻳﺮ ﺃﺣﺪ ﻣﻄﺎﻋﻢ ﺍﻟﻮﺟﺒﺎﺕ ﺍﻟﺴﺮﻳﻌﺔ ﻓﻲ ﺍﻟﺠﺪﻭﻝ ﺍﻟﺘﺎﻟﻲ ﻋﺪﺩ ﺍﻟﻮﺟﺒﺎﺕ ﺍﻟﻤﺮﺳﻠﺔ ﺇﻟﻰ ﺍﻟﻤﻨﺎﺯﻝ ﺧﻼﻝ ﺃﺣﺪ‬ ‫ﺍﻷﺳﺎﺑﻴﻊ‪ ،‬ﻭﺑُﻌﺪ ﻫﺬﻩ ﺍﻟﻤﻨﺎﺯﻝ ﻋﻦ ﺍﻟﻤﻄﻌﻢ‪.‬‬ ‫ﺍﻟﻤﺠﻤﻮﻉ ‪ 0- 4- 8- 12- 16- 20- 24-‬ﺍﻟﺒﻌﺪ )‪(km‬‬ ‫‪ 12 25 21 20 12 8 4 102‬ﺍﻟﺘﻜﺮﺍﺭ‬ ‫)‪ (a‬ﺃﻛﻤﻞ ﺍﻟﺠﺪﻭﻝ ﺑﺈﺿﺎﻓﺔ ﻣﺮﺍﻛﺰ ﺍﻟﻔﺌﺎﺕ‪.‬‬ ‫)‪ (b‬ﺍﺭﺳﻢ ﺍﻟﻤﻨﺤﻨﻰ ﺍﻟﺘﻜﺮﺍﺭﻱ‪.‬‬ ‫)‪ (c‬ﺍﺭﺳﻢ ﺍﻟﻤﺪﺭﺝ ﺍﻟﺘﻜﺮﺍﺭﻱ ﻭﻣﻨﻪ ﺍﻟﻤﻨﺤﻨﻰ ﺍﻟﺘﻜﺮﺍﺭﻱ‪.‬‬ ‫‪89‬‬

‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫‪ab‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-5‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫)‪ (1‬ﺍﻟﺘﻜﺮﺍﺭ ﺍﻟﻨﺴﺒﻲ ﻳﺴﺎﻭﻱ‪ :‬ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﺮﻛﺰﻳﺔ ﻟﻘﻄﺎﻉ ‪360º #‬‬ ‫‪ab‬‬ ‫)‪ (2‬ﺍﻟﺘﻜﺮﺍﺭ ﺍﻟﻨﺴﺒﻲ = ﻣﺠﻤﻮﻉ ﺍﻟﺘﻜﺮﺍﺭﺍﺕ‬ ‫ﺗﻜﺮﺍﺭ ﺍﻟﻘﻴﻤﺔ‬ ‫‪ab‬‬ ‫‪ab‬‬ ‫)‪ (3‬ﻣﺮﻛﺰ ﻓﺌﺔ ‪ 20-‬ﻃﻮﻟﻬﺎ ‪ 10‬ﻳﺴﺎﻭﻱ ‪30‬‬ ‫)‪ (4‬ﻻ ﻳﻤﻜﻦ ﺭﺳﻢ ﺍﻟﻤﻨﺤﻨﻰ ﺍﻟﺘﻜﺮﺍﺭﻱ ﻗﺒﻞ ﺍﻟﻤﺪﺭﺝ ﺍﻟﺘﻜﺮﺍﺭﻱ‪.‬‬ ‫)‪ (5‬ﻳﻤﻜﻦ ﺗﻤﺜﻴﻞ ﺑﻴﺎﻧﺎﺕ ﻛﻤﻴﺔ ﻣﺴﺘﻤﺮﺓ ﺑﺎﻟﻘﻄﺎﻋﺎﺕ ﺍﻟﺪﺍﺋﺮﻳﺔ‪a b .‬‬ ‫‪y‬‬ ‫ﺃﻋﺪﺍﺩ ﺍﻟﻤﺮﺍﺟﻌﻴﻦ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(6-10‬ﻇﻠّﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪40 .‬‬ ‫‪30‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺮﻳﻨﻴﻦ )‪ (6-7‬ﺍﺳﺘﺨﺪﻡ ﺍﻟﻤﺪﺭﺝ ﺍﻟﺘﻜﺮﺍﺭﻱ ﺍﻟﻤﻘﺎﺑﻞ ﺍﻟﺬﻱ ﻳﻤﺜﻞ ﺃﻋﺪﺍﺩ ‪20‬‬ ‫‪15‬‬ ‫‪10‬‬ ‫ﺍﻟﻤﺮﺍﺟﻌﻴﻦ ﻓﻲ ﺇﺣﺪﻯ ﺍﻟﻮﺯﺍﺭﺍﺕ ﺧﻼﻝ ﺳﺎﻋﺎﺕ ﺍﻟﺪﻭﺍﻡ ﺍﻟﻴﻮﻣﻲ ﻓﻲ ﺩﻭﻟﺔ ﻣﺎ‪.‬‬ ‫‪7 9 11 13 15 x‬‬ ‫)‪ (6‬ﺇﺟﻤﺎﻟﻲ ﻋﺪﺩ ﺍﻟﻤﺮﺍﺟﻌﻴﻦ ﻫﻮ‪:‬‬ ‫ﺳﺎﻋﺎﺕ ﺍﻟﺪﻭﺍﻡ‬ ‫‪a 80‬‬ ‫‪b 65‬‬ ‫‪c 70‬‬ ‫‪d 75‬‬ ‫‪a4‬‬ ‫)‪ (7‬ﻃﻮﻝ ﺍﻟﻔﺘﺮﺓ ﻳﺴﺎﻭﻱ‪:‬‬ ‫ﺍﻟﺘﺮﺑﻴﺔ ﺍﻟﺒﺪﻧﻴﺔ‬ ‫‪b3 c2 d1‬‬ ‫ﻟﻐﺔ ﻓﺮﻧﺴﻴﺔ ﺍﻟﺘﺮﺑﻴﺔ ﺍﻟﻔﻨﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ (8-10‬ﺍﺳﺘﺨﺪﻡ ﺍﻟﺸﻜﻞ ﺍﻟﺒﻴﺎﻧﻲ ﺍﻟﻤﻘﺎﺑﻞ ﺍﻟﺬﻱ ﻳﻤﺜﻞ ﺍﻟﻤﻮﺍﺩ ﺍﻻﺧﺘﻴﺎﺭﻳﺔ ﺍﻟﻤﻔﻀﻠّﺔ‬ ‫‪a 120º‬‬ ‫ﻟﺪﻯ ﻃ ّﻼﺏ ﺇﺣﺪﻯ ﺍﻟﻤﺪﺍﺭﺱ ﺍﻟﺒﺎﻟﻎ ﻋﺪﺩﻫﻢ ‪ 200‬ﻃﺎﻟﺐ‪.‬‬ ‫‪a 30‬‬ ‫‪a 50‬‬ ‫ﻟﻐﺔ ﺇﻧﺠﻠﻴﺰﻳﺔ‬ ‫)‪ (8‬ﻛﻢ ﻳﺴﺎﻭﻱ ﻗﻴﺎﺱ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﺮﻛﺰﻳﺔ ﻟﻘﻄﺎﻉ ﺍﻟﺘﺮﺑﻴﺔ ﺍﻟﺒﺪﻧﻴﺔ؟‬ ‫‪b 45º‬‬ ‫‪c 180º‬‬ ‫‪d 90º‬‬ ‫‪b 25‬‬ ‫‪b 40‬‬ ‫)‪ (9‬ﻛﻢ ﻳﺒﻠﻎ ﻋﺪﺩ ﺍﻟﻄ ّﻼﺏ ﺍﻟﻤﺴ ّﺠﻠﻴﻦ ﺑﺎﻟّﻠﻐﺔ ﺍﻹﻧﺠﻠﻴﺰﻳﺔ؟‬ ‫‪c 35‬‬ ‫‪d 40‬‬ ‫)‪ (10‬ﻛﻢ ﻳﺒﻠﻎ ﻋﺪﺩ ﺍﻟﻄ ّﻼﺏ ﺍﻟﻤﺴﺠﻠﻴﻦ ﺑﺎﻟﻤﻮﺍﺩ ﺍﻟﻠﻐﻮﻳّﺔ؟‬ ‫‪c 55‬‬ ‫‪d 60‬‬ ‫‪90‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ ‪6-4‬‬ ‫‪Standard Deviation‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫ﺍﻟﻔﺌﺔ )ﺑﺎﻟﺪﻳﻨﺎﺭ(‬ ‫)‪ (1‬ﺃﻭﺟﺪ ﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ ﻟﻠﺒﻴﺎﻧﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ‪ .5 , 5 , 5 , 5 :‬ﻓ ّﺴﺮ ﺇﺟﺎﺑﺘﻚ‪.‬‬ ‫ﺍﻟﺘﻜﺮﺍﺭ‬ ‫)‪ (2‬ﺳ ّﺠﻞ ﺻﺎﺣﺐ ﻣﺘﺠﺮ ﺃﻥ ﻣﺒﻴﻊ ﺍﻟﺴﻠﻊ ﺑﺤﺴﺐ ﺃﺳﻌﺎﺭﻫﺎ ﻫﻮ ﻛﻤﺎ ﻳﻠﻲ‪:‬‬ ‫ﺍﻟﻤﺠﻤﻮﻉ ‪0- 10- 20- 30- 40- 50-‬‬ ‫‪190 300 470 280 260 100 1 600‬‬ ‫)‪ (a‬ﺃﻭﺟﺪ ﺍﻟﻤﺘﻮ ّﺳﻂ ﺍﻟﺤﺴﺎﺑﻲ‪.‬‬ ‫)‪ (b‬ﺃﻭﺟﺪ ﺍﻟﺘﺒﺎﻳﻦ ﻭﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ ﻷﺳﻌﺎﺭ ﺍﻟﺴﻠﻊ‪.‬‬ ‫)‪ (3‬ﺗﺼﻨﻊ ﻣﺆﺳﺴﺔ ﻋﺒﻮﺍﺕ ﻟﺤﻔﻆ ﺍﻷﺟﺒﺎﻥ ﻋﻠﻰ ﺃﻥ ﺗﺤﺘﻮﻱ ﺍﻟﻌﻠﺒﺔ ﺍﻟﻮﺍﺣﺪﺓ ﻋﻠﻰ ‪ 170 g‬ﻣﻦ ﺍﻟﺠﺒﻨﺔ‪ .‬ﻭﻟﻜﻦ ﻋﻨﺪ ﻭﺯﻥ‬ ‫‪ 200‬ﻋﻠﺒﺔ‪ ،‬ﺟﺎﺀﺕ ﺍﻷﻭﺯﺍﻥ ﻛﻤﺎ ﻳﺒﻴﻦ ﺍﻟﺠﺪﻭﻝ ﺍﻟﺘﻜﺮﺍﺭﻱ ﺍﻟﺘﺎﻟﻲ‪:‬‬ ‫ﺍﻟﻮﺯﻥ‬ ‫ﺍﻟﻤﺠﻤﻮﻉ ‪167 168 169 170 171 172 173 174‬‬ ‫‪g‬‬ ‫‪ 10 15 24 55 48 34 8 6 200‬ﺍﻟﺘﻜﺮﺍﺭ‬ ‫)‪ (a‬ﺃﻭﺟﺪ ﺍﻟﻤﺘﻮ ّﺳﻂ ﺍﻟﺤﺴﺎﺑﻲ ﻟﻬﺬﻩ ﺍﻷﻭﺯﺍﻥ‪.‬‬ ‫)‪ (b‬ﺃﻭﺟﺪ ﺍﻟﺘﺒﺎﻳﻦ ﻭﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ ﻟﻬﺬﻩ ﺍﻷﻭﺯﺍﻥ‪.‬‬ ‫‪ab‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫‪ab‬‬ ‫‪ab‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-4‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ab‬‬ ‫)‪ (1‬ﺇﺫﺍ ﺃﺿﻔﻨﺎ ﺍﻟﻌﺪﺩ ﻧﻔﺴﻪ ﻋﻠﻰ ﺟﻤﻴﻊ ﺍﻷﻋﺪﺍﺩ ﻓﻲ ﺍﻟﺒﻴﺎﻧﺎﺕ‪ ،‬ﻧﺤﺼﻞ ﻋﻠﻰ‬ ‫ﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ ﻧﻔﺴﻪ‪.‬‬ ‫‪91‬‬ ‫)‪ (2‬ﺇﺫﺍ ﺿﺮﺑﻨﺎ ﺍﻷﻋﺪﺍﺩ ﻓﻲ ﺍﻟﺒﻴﺎﻧﺎﺕ ﺑﺎﻟﻌﺪﺩ ﻧﻔﺴﻪ‪ ،‬ﻻ ﻳﺘﻐﻴّﺮ ﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ‪.‬‬ ‫)‪ (3‬ﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ ﻳﻜﻮﻥ ﺩﺍﺋ ًﻤﺎ ﺃﺻﻐﺮ ﻣﻦ ﺍﻟﻤﺘﻮ ّﺳﻂ ﺍﻟﺤﺴﺎﺑﻲ‪.‬‬ ‫)‪ (4‬ﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ ﻳﻜﻮﻥ ﺩﺍﺋ ًﻤﺎ ﻣﻮﺟﺒًﺎ‪.‬‬

‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(5-9‬ﻇﻠّﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫)‪ (5‬ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺘﺒﺎﻳﻦ ﻳﺴﺎﻭﻱ ‪ ،100‬ﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ ﻳﺴﺎﻭﻱ‪:‬‬ ‫‪a !10‬‬ ‫‪b -10‬‬ ‫‪c 10‬‬ ‫ﻟﻴﺲ ﺃﻳًﺎ ﻣﻤﺎ ﺳﺒﻖ ‪d‬‬ ‫)‪ (6‬ﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ ﻟﻠﺒﻴﺎﻧﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ‪ 1 , 2 , 3 , 4 , 4 , 5 , 5 , 6 :‬ﻳﺴﺎﻭﻱ‪:‬‬ ‫‪a 0.78‬‬ ‫‪b 1.56‬‬ ‫‪c 2.78‬‬ ‫‪d 3.78‬‬ ‫)‪ (7‬ﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ ﻳﺴﺎﻭﻱ ﺻﻔ ًﺮﺍ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﺒﻴﺎﻧﺎﺕ‪:‬‬ ‫‪ b‬ﻧﺼﻔﻬﺎ ﻫﻮ ﺍﻟﻤﻌﻜﻮﺱ ﺍﻟﻀﺮﺑﻲ ﻟﻠﻨﺼﻒ ﺍﻵﺧﺮ‬ ‫‪ a‬ﻣﺘﺴﺎﻭﻳﺔ‬ ‫‪ c‬ﻧﺼﻔﻬﺎ ﻫﻮ ﺍﻟﻤﻌﻜﻮﺱ ﺍﻟﺠﻤﻌﻲ ﻟﻠﻨﺼﻒ ﺍﻵﺧﺮ ‪ d‬ﻻ ﻳﻤﻜﻦ ﺃﻥ ﻳﺴﺎﻭﻱ ﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ ﺻﻔ ًﺮﺍ‪.‬‬ ‫)‪ (8‬ﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ ﻫﻮ ﻣﻘﻴﺎﺱ‪:‬‬ ‫‪ b‬ﺗﺸﺘﺖ ﺍﻟﻘﻴﻢ ﻓﻲ ﺍﻟﺒﻴﺎﻧﺎﺕ‬ ‫‪ a‬ﺗﻤﺮﻛﺰ ﺍﻟﻘﻴﻢ ﻓﻲ ﺍﻟﺒﻴﺎﻧﺎﺕ‬ ‫‪ d‬ﻟﻴﺲ ﺃﻳًّﺎ ﻣﻤﺎ ﺳﺒﻖ‬ ‫‪ c‬ﺍﻧﺤﺮﺍﻑ ﺍﻟﻘﻴﻢ ﻓﻲ ﺍﻟﺒﻴﺎﻧﺎﺕ‬ ‫)‪ (9‬ﻳﺴﺎﻭﻱ ﺍﻧﺤﺮﺍﻑ ﻣﻌﻴﺎﺭﻱ ﻟﺒﻴﺎﻧﺎﺕ ﻣﻌﻴّﻨﺔ ‪ .4‬ﺑﻌﺪ ﺿﺮﺏ ﺍﻟﺒﻴﺎﻧﺎﺕ ﻓﻲ ﺍﻟﻌﺪﺩ ‪ ،3‬ﻳﺼﺒﺢ ﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ‪:‬‬ ‫‪a 13‬‬ ‫‪b 12‬‬ ‫‪c 11‬‬ ‫‪d 10‬‬ ‫‪92‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺍﻟﻘﺎﻋﺪﺓ ﺍﻟﺘﺠﺮﻳﺒﻴﺔ ‪6-5‬‬ ‫‪Empirical Rule‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫)‪ (1‬ﻣﺎ ﻫﻮ ﺍﻟﺘﻮﺯﻳﻊ ﺍﻟﻄﺒﻴﻌ ّﻲ؟‬ ‫)‪ (2‬ﻣﺎ ﻫﻲ ﺧﺼﺎﺋﺺ ﺍﻟﺘﻮﺯﻳﻊ ﺍﻟﻄﺒﻴﻌ ّﻲ؟‬ ‫)‪ (3‬ﻣﺎ ﺍﻟﺸﻜﻞ ﺍﻟﺬﻱ ﻳﺄﺧﺬﻩ ﺍﻟﺘﻮﺯﻳﻊ ﺍﻟﻄﺒﻴﻌ ّﻲ؟‬ ‫)‪ (4‬ﺃﻛﻤﻞ ﺍﻟﺮﺳﻢ ﺃﺩﻧﺎﻩ‪:‬‬ ‫‪99.7%‬‬ ‫‪68%‬‬ ‫‪13.5%‬‬ ‫‪13.5%‬‬ ‫‪x - 3σ‬‬ ‫‪x-σ‬‬ ‫‪x‬‬ ‫‪x+σ‬‬ ‫‪x + 3σ‬‬ ‫)‪ (5‬ﺗﺒﻴﻦ ﻹﺣﺪﻯ ﺍﻟﻤﺆﺳﺴﺎﺕ ﺍﻟﺼﻨﺎﻋﻴﺔ ﺃﻥ ﺍﻟﻤﺘﻮﺳﻂ ﺍﻟﺤﺴﺎﺑﻲ ﻷﺭﺑﺎﺣﻬﺎ ﺍﻟﺸﻬﺮﻳﺔ ‪ 1 250‬ﺩﻳﻨﺎ ًﺭﺍ ﺑﺎﻧﺤﺮﺍﻑ ﻣﻌﻴﺎﺭﻱ‬ ‫‪ 225‬ﺩﻳﻨﺎ ًﺭﺍ ﻭﺃﻥ ﺍﻟﻤﻨﺤﻨﻰ ﺍﻟﺘﻜﺮﺍﺭﻱ ﻟﻬﺬﻩ ﺍﻷﺭﺑﺎﺡ ﻫﻮ ﻋﻠﻰ ﺷﻜﻞ ﺍﻟﺠﺮﺱ )ﺗﻮﺯﻳﻊ ﻃﺒﻴﻌﻲ(‪.‬‬ ‫)‪ (a‬ﻃﺒﻖ ﺍﻟﻘﺎﻋﺪﺓ ﺍﻟﺘﺠﺮﻳﺒﻴﺔ‪.‬‬ ‫)‪ (b‬ﻫﻞ ﻭﺻﻠﺖ ﺃﺭﺑﺎﺡ ﻫﺬﻩ ﺍﻟﻤﺆﺳﺴﺔ ﺇﻟﻰ ‪ 2 000‬ﺩﻳﻨﺎﺭ؟‬ ‫)‪ (6‬ﻳﻌﻠﻦ ﻣﺼﻨﻊ ﻹﻧﺘﺎﺝ ﺍﻷﺳﻼﻙ ﺍﻟﻤﻌﺪﻧﻴﺔ ﺃﻥ ﻣﺘﻮﺳﻂ ﺗﺤﻤﻞ ﺍﻟﺴﻠﻚ ﻫﻮ ‪ 1 400 kg‬ﺑﺎﻧﺤﺮﺍﻑ ﻣﻌﻴﺎﺭﻱ ‪.200 kg‬‬ ‫ﻋﻠﻰ ﺍﻓﺘﺮﺍﺽ ﺃﻥ ﺍﻟﻤﻨﺤﻨﻰ ﺍﻟﻤﻤﺜﻞ ﻟﺘﻮﺯﻳﻊ ﺗﺤﻤﻞ ﺍﻷﺳﻼﻙ ﺍﻟﻤﻌﺪﻧﻴﺔ ﻳﻘﺘﺮﺏ ﻛﺜﻴ ًﺮﺍ ﻣﻦ ﺍﻟﺘﻮﺯﻳﻊ ﺍﻟﻄﺒﻴﻌﻲ‪:‬‬ ‫)‪ (a‬ﻃﺒﻖ ﺍﻟﻘﺎﻋﺪﺓ ﺍﻟﺘﺠﺮﻳﺒﻴﺔ‪.‬‬ ‫)‪ (b‬ﺃﻭﺟﺪ ﺍﻟﻨﺴﺒﺔ ﺍﻟﻤﺌﻮﻳﺔ ﻟﻸﺳﻼﻙ ﺍﻟﻤﻌﺪﻧﻴﺔ ﺍﻟﺘﻲ ﻳﺰﻳﺪ ﻣﺘﻮﺳﻂ ﺗﺤﻤﻠﻬﺎ ﻋﻦ ‪.1 000 kg‬‬ ‫‪93‬‬

‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-5‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ab‬‬ ‫)‪ (1‬ﻳﻤﻜﻦ ﺃﻥ ﻳﻜﻮﻥ ﺷﻜﻞ ﺍﻟﺘﻮﺯﻳﻊ ﺍﻟﻄﺒﻴﻌ ّﻲ ﺟﺮ ًﺳﺎ ﻏﻴﺮ ﻣﺘﻤﺎﺛﻞ‪.‬‬ ‫‪ab‬‬ ‫)‪ (2‬ﻓﻲ ﺍﻟﺘﻮﺯﻳﻊ ﺍﻟﻄﺒﻴﻌ ّﻲ ﺍﻟﻤﻨﻮﺍﻝ ﻭﺍﻟﻮﺳﻴﻂ ﻏﻴﺮ ﻣﺘﺴﺎﻭﻳﻴﻦ‪.‬‬ ‫‪ab‬‬ ‫)‪ (3‬ﻓﻲ ﺍﻟﺘﻮﺯﻳﻊ ﺍﻟﻄﺒﻴﻌ ّﻲ ﺍﻟﻔﺘﺮﺓ @‪ 6x - σ , x + σ‬ﺗﺤﺘﻮﻱ ﻋﻠﻰ ‪ 95%‬ﻣﻦ ﺍﻟﺒﻴﺎﻧﺎﺕ‪.‬‬ ‫‪ab‬‬ ‫)‪ (4‬ﻓﻲ ﺍﻟﺘﻮﺯﻳﻊ ﺍﻟﻄﺒﻴﻌ ّﻲ ‪ 99.7%‬ﻣﻦ ﺍﻟﺒﻴﺎﻧﺎﺕ ﺗﻮﺟﺪ ﻓﻲ ﺍﻟﻔﺘﺮﺓ @‪.6x - 3σ , x + 3σ‬‬ ‫)‪ (5‬ﺗﺴﺘﺨﺪﻡ ﺍﻟﻘﺎﻋﺪﺓ ﺍﻟﺘﺠﺮﻳﺒﻴﺔ ﻟﺪﺭﺍﺳﺔ ﺍﻟﺠﻮﺩﺓ ﻓﻲ ﻣﻮﺍﻗﻒ ﺇﺣﺼﺎﺋﻴﺔ ﻣﺘﻌﺪﺩﺓ ﻟﻌﻴﻨﺎﺕ ﺫﺍﺕ ﻗﻴﻢ ﻣﻔﺮﺩﺓ‪a b .‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(6-8‬ﻇﻠّﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫)‪ (6‬ﺗﺰﻋﻢ ﺷﺮﻛﺔ ﺃﻥ ﻣﺘﻮﺳﻂ ﻋﻤﺮ ﻣﻨﺘﺠﻬﺎ ﻫﻮ ‪ 50‬ﺷﻬ ًﺮﺍ ﻣﻊ ﺍﻧﺤﺮﺍﻑ ﻣﻌﻴﺎﺭﻱ ‪ 5‬ﺃﺷﻬﺮ‪ .‬ﺍﻟﻨﺴﺒﺔ ﺍﻟﻤﺌﻮﻳﺔ ﻟﻠﻤﻨﺘﺠﺎﺕ‬ ‫ﺍﻟﺘﻲ ﻳﺰﻳﺪ ﻋﻤﺮﻫﺎ ﻋﻦ ‪ 50‬ﺷﻬ ًﺮﺍ ﻫﻲ‪:‬‬ ‫‪a 50%‬‬ ‫‪b 55%‬‬ ‫‪c 45%‬‬ ‫‪d 40%‬‬ ‫)‪ (7‬ﺍﻟﺘﻤﺜﻴﻞ ﺍﻷﻓﻀﻞ ﻟﻠﺘﻮﺯﻳﻊ ﺍﻟﻄﺒﻴﻌ ّﻲ ﻫﻮ‪:‬‬ ‫‪ab c d‬‬ ‫‪ 99.7% b‬ﻣﻦ ﺍﻟﺒﻴﺎﻧﺎﺕ‬ ‫)‪ (8‬ﺍﻟﻔﺘﺮﺓ @‪ 6x - 2σ , x + 2σ‬ﺗﺤﺘﻮﻱ ﻋﻠﻰ‪:‬‬ ‫‪ 95% d‬ﻣﻦ ﺍﻟﺒﻴﺎﻧﺎﺕ‬ ‫‪ 68% a‬ﻣﻦ ﺍﻟﺒﻴﺎﻧﺎﺕ‬ ‫‪ 90% c‬ﻣﻦ ﺍﻟﺒﻴﺎﻧﺎﺕ‬ ‫‪94‬‬

‫ﺗﻤ ﱠﺮ ْﻥ‬ ‫ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻌﻴﺎﺭﻳﺔ ‪6-6‬‬ ‫‪Standarized Value‬‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ A‬ﺗﻤﺎﺭﻳﻦ ﻣﻘﺎﻟﻴﺔ‬ ‫)‪ (1‬ﺃﻛﻤﻞ ﺍﻟﺠﻤﻠﺔ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬ ‫ﻗﻴﻤﺔ ﻣﻔﺮﺩﺓ ﻣﻦ ﺑﻴﺎﻧﺎﺕ ﻋﻦ ‪.‬‬ ‫ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻌﻴﺎﺭﻳﺔ ﻫﻲ ﻣﺆﺷﺮ ﻳﺪﻝ ﻋﻠﻰ‬ ‫ﻟﻘﻴﻢ ﻫﺬﻩ ﺍﻟﺒﻴﺎﻧﺎﺕ‪.‬‬ ‫ﻭﺫﻟﻚ ﺑﺎﺳﺘﺨﺪﺍﻡ‬ ‫)‪ (2‬ﻓﻲ ﺃﺣﺪ ﺍﻻﺧﺘﺒﺎﺭﺍﺕ ﺣﻴﺚ ﺍﻟﺪﺭﺟﺔ ﺍﻟﻌﻈﻤﻰ ‪ ،20‬ﺟﺎﺀﺕ ﺩﺭﺟﺔ ﺃﺣﺪ ﺍﻟﻄﻼﺏ ‪ 15‬ﻣﻊ ﻣﺘﻮﺳﻂ ﺣﺴﺎﺑﻲ ‪14‬‬ ‫ﻭﺍﻧﺤﺮﺍﻑ ﻣﻌﻴﺎﺭﻱ ‪ .4‬ﻣﺎ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻌﻴﺎﺭﻳﺔ ﻟﻠﺪﺭﺟﺔ ‪ 15‬ﻣﻘﺎﺭﻧﺔ ﺑﺒﻘﻴﺔ ﺩﺭﺟﺎﺕ ﻫﺬﺍ ﺍﻻﺧﺘﺒﺎﺭ؟‬ ‫)‪ (3‬ﻟﻨﺄﺧﺬ ﺍﻟﺒﻴﺎﻧﺎﺕ‪.5 ،5 ،6 ،7 ،7 :‬‬ ‫)‪ (a‬ﺃﻭﺟﺪ‪ :‬ﺍﻟﻤﺘﻮﺳﻂ ﺍﻟﺤﺴﺎﺑﻲ ‪ ، x‬ﻭﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ ‪ σ‬ﻟﻬﺬﻩ ﺍﻟﺒﻴﺎﻧﺎﺕ‪.‬‬ ‫)‪ (b‬ﺃﻭﺟﺪ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻌﻴﺎﺭﻳﺔ ﻟﻬﺬﻩ ﺍﻟﺒﻴﺎﻧﺎﺕ‪.‬‬ ‫)‪ (4‬ﻓﻲ ﺍﻟﻤﺪﻳﻨﺔ ‪ A‬ﻳﺰﻥ ﺃﺣﺪ ﺍﻟﺮﺟﺎﻝ ‪ 75 kg‬ﻣﻊ ﻣﺘﻮﺳﻂ ﺣﺴﺎﺑﻲ ﻟﻠﺮﺟﺎﻝ ‪ 70 kg‬ﻭﺍﻧﺤﺮﺍﻑ ﻣﻌﻴﺎﺭﻱ ‪.5 kg‬‬ ‫ﻭﻓﻲ ﺍﻟﻤﺪﻳﻨﺔ ‪ B‬ﻳﺰﻥ ﺃﺣﺪ ﺍﻟﺮﺟﺎﻝ ‪ 80 kg‬ﻣﻊ ﻣﺘﻮﺳﻂ ﺣﺴﺎﺑﻲ ﻟﻠﺮﺟﺎﻝ ‪ 76 kg‬ﻭﺍﻧﺤﺮﺍﻑ ﻣﻌﻴﺎﺭﻱ ‪.8 kg‬‬ ‫ﺃﻭﺟﺪ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻌﻴﺎﺭﻳﺔ ‪ z1‬ﻟﻮﺯﻥ ‪ 75 kg‬ﻓﻲ ﺍﻟﻤﺪﻳﻨﺔ ‪ A‬ﻭﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻌﻴﺎﺭﻳﺔ ‪ z2‬ﻟﻮﺯﻥ ‪ 80 kg‬ﻓﻲ ﺍﻟﻤﺪﻳﻨﺔ ‪.B‬‬ ‫)‪ (5‬ﻓﻲ ﺍﺧﺘﺒﺎﺭﺍﺕ ﻣﺎﺩﺓ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ ﻧﺎﻝ ﺧﺎﻟﺪ ﺍﻟﺪﺭﺟﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ ﻣﻦ ‪.12 ،15 ،16 ،17 :20‬‬ ‫ﺃﻣﺎ ﻓﻲ ﺍﺧﺘﺒﺎﺭﺍﺕ ﻣﺎﺩﺓ ﺍﻟﻜﻴﻤﻴﺎﺀ ﻓﻘﺪ ﻧﺎﻝ ﺍﻟﺪﺭﺟﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ ﻣﻦ ‪.11 ،13 ،15 ،9 :20‬‬ ‫)‪ (a‬ﺃﻭﺟﺪ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻌﻴﺎﺭﻳﺔ ‪ z1‬ﻟﻠﺪﺭﺟﺔ ‪ 15‬ﻓﻲ ﻣﺎﺩﺓ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ ﻭﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻌﻴﺎﺭﻳﺔ ‪ z2‬ﻟﻠﺪﺭﺟﺔ ‪ 15‬ﻓﻲ ﻣﺎﺩﺓ‬ ‫ﺍﻟﻜﻴﻤﻴﺎﺀ‪.‬‬ ‫)‪ (b‬ﻓﻲ ﺃ ّﻱ ﻣﺎﺩﺓ ﻛﺎﻧﺖ ﺍﻟﺪﺭﺟﺔ ‪ 15‬ﻫﻲ ﺃﻓﻀﻞ ﻣﻘﺎﺭﻧﺔ ﺑﺒﻘﻴﺔ ﺍﻟﺪﺭﺟﺎﺕ؟‬ ‫ﺍﻟﻤﺠﻤﻮﻋﺔ ‪ B‬ﺗﻤﺎﺭﻳﻦ ﻣﻮﺿﻮﻋﻴﺔ‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(1-4‬ﻇﻠّﻞ ‪ a‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺻﺤﻴﺤﺔ ﻭ ‪ b‬ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻌﺒﺎﺭﺓ ﺧﺎﻃﺌﺔ‪.‬‬ ‫‪ab‬‬ ‫‪x-x‬‬ ‫ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻌﻴﺎﺭﻳﺔ =‬ ‫)‪(1‬‬ ‫‪ab‬‬ ‫‪σ‬‬ ‫‪ab‬‬ ‫)‪ (2‬ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻌﻴﺎﺭﻳﺔ ﺗﺆﺷﺮ ﺇﻟﻰ ﺗﺸﺘﺖ ﻗﻴﻤﺔ ﻋﻦ ﺑﻘﻴﺔ ﻗﻴﻢ ﺍﻟﺒﻴﺎﻧﺎﺕ‪.‬‬ ‫‪95‬‬ ‫)‪ (3‬ﻓﻲ ﺑﻴﺎﻧﺎﺕ ﺣﻴﺚ ﺍﻟﻤﺘﻮﺳﻂ ﺍﻟﺤﺴﺎﺑﻲ ‪ x = 14‬ﻭﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ ‪σ = 4‬‬ ‫ﻓﺈﻥ ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻌﻴﺎﺭﻳﺔ ﻟﻠﻤﻔﺮﺩﺓ ‪ x = 16‬ﻫﻲ‪z = 0.5 :‬‬

‫)‪ (4‬ﻓﻲ ﺑﻴﺎﻧﺎﺕ ﺣﻴﺚ ﺍﻟﻤﺘﻮﺳﻂ ﺍﻟﺤﺴﺎﺑﻲ ‪ x = 12‬ﻭﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻌﻴﺎﺭﻳﺔ ﻟﻠﻤﻔﺮﺩﺓ ‪x = 15‬‬ ‫‪ab‬‬ ‫ﻫﻲ‪ ، z = 0.4 :‬ﻓﺈﻥ ﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ‪σ = 7.5 :‬‬ ‫ﻓﻲ ﺍﻟﺘﻤﺎﺭﻳﻦ )‪ ،(5-8‬ﻇﻠّﻞ ﺭﻣﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﺪﺍﻝ ﻋﻠﻰ ﺍﻹﺟﺎﺑﺔ ﺍﻟﺼﺤﻴﺤﺔ‪.‬‬ ‫)‪ (5‬ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻌﻴﺎﺭﻳﺔ ﻟﻠﻤﻔﺮﺩﺓ ‪ 14‬ﻣﻘﺎﺭﻧﺔ ﺑﻘﻴﻢ ﺑﻴﺎﻧﺎﺕ ﺣﻴﺚ ﺍﻟﻤﺘﻮﺳﻂ ﺍﻟﺤﺴﺎﺑﻲ ‪ 12.5‬ﻭﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ ‪6‬‬ ‫ﻫﻲ‪:‬‬ ‫‪a -0.25‬‬ ‫‪b 0.25‬‬ ‫‪c 2.5‬‬ ‫‪d -2.5‬‬ ‫)‪ (6‬ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻌﻴﺎﺭﻳﺔ ﻟﻤﻔﺮﺩﺓ ﻣﻦ ﺑﻴﺎﻧﺎﺕ ﻫﻲ ‪ 0.625‬ﻭﺍﻟﻤﺘﻮﺳﻂ ﺍﻟﺤﺴﺎﺑﻲ ‪ 12‬ﻭﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ ‪ 8‬ﻓﺈﻥ ﻫﺬﻩ‬ ‫ﺍﻟﻤﻔﺮﺩﺓ ﺗﺴﺎﻭﻱ‪:‬‬ ‫‪a7‬‬ ‫‪b -7‬‬ ‫‪c 17‬‬ ‫‪d -17‬‬ ‫)‪ (7‬ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻌﻴﺎﺭﻳﺔ ﻟﻠﻤﻔﺮﺩﺓ ‪ 14‬ﻣﻦ ﺑﻴﺎﻧﺎﺕ ﻫﻲ ‪ 0.6‬ﻭﺍﻟﻤﺘﻮﺳﻂ ﺍﻟﺤﺴﺎﺑﻲ ‪ 11‬ﻓﺈﻥ ﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ ﻟﻘﻴﻢ ﻫﺬﻩ‬ ‫ﺍﻟﺒﻴﺎﻧﺎﺕ ﻫﻮ‪:‬‬ ‫‪a 0.2‬‬ ‫‪b -0.2‬‬ ‫‪c -5‬‬ ‫‪d5‬‬ ‫)‪ (8‬ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻌﻴﺎﺭﻳﺔ ﻟﻠﻤﻔﺮﺩﺓ ‪ 18‬ﻣﻦ ﺑﻴﺎﻧﺎﺕ ﻫﻲ ‪ 0.75‬ﻭﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ ‪ 8‬ﻓﺈﻥ ﺍﻟﻤﺘﻮﺳﻂ ﺍﻟﺤﺴﺎﺑﻲ ﻫﻮ‪:‬‬ ‫‪a 24‬‬ ‫‪b 12‬‬ ‫‪c -12‬‬ ‫‪d -24‬‬ ‫‪96‬‬

‫ﺍﺧﺘﺒﺎﺭ ﺍﻟﻮﺣﺪﺓ ﺍﻟﺴﺎﺩﺳﺔ‬ ‫)‪ (1‬ﻫﻞ ﻳﻤﻜﻦ ﺍﺳﺘﺨﺪﺍﻡ ﺍﻟﺤﺼﺮ ﺍﻟﺸﺎﻣﻞ ﻓﻲ ﺩﺭﺍﺳﺔ ﺍﻟﻤﺠﺘﻤﻌﺎﺕ ﺍﻹﺣﺼﺎﺋﻴﺔ ﺍﻟﺘﺎﻟﻴﺔ ﺃﻡ ﻻ؟ ﺍﺷﺮﺡ ﺍﻟﺴﺒﺐ‪.‬‬ ‫)‪ (a‬ﺩﺭﺍﺳﺔ ﻛﻤﻴﺔ ﺍﻟﺴﻜﺮ ﺍﻟﻤﻮﺟﻮﺩﺓ ﻓﻲ ﺍﻟﺪﻡ ﻋﻨﺪ ﺃﺣﺪ ﺍﻷﺷﺨﺎﺹ‪.‬‬ ‫)‪ (b‬ﺇﻳﺠﺎﺩ ﺍﻟﻤﺘﻮﺳﻂ ﺍﻟﺤﺴﺎﺑﻲ ﻷﻭﺯﺍﻥ ﻃﻼﺏ ﺻﻔﻚ‪.‬‬ ‫)‪ (2‬ﻓﻲ ﺇﺣﺪﻯ ﺍﻟﻤﺆﺳﺴﺎﺕ ﺗ ّﻢ ﺳﺤﺐ ﻋﻴﻨﺔ ﻋﺸﻮﺍﺋﻴﺔ ﻃﺒﻘﻴﺔ ﻣﻜﻮﻧﺔ ﻣﻦ ‪ 70‬ﻓﺮ ًﺩﺍ ﻭﻛﺴﺮ ﺍﻟﻤﻌﺎﻳﻨﺔ ﻟﻬﺬﻩ ﺍﻟﻌﻴﻨﺔ ‪0.08‬‬ ‫)‪ (a‬ﺃﻭﺟﺪ ﻋﺪﺩ ﺍﻷﻓﺮﺍﺩ ﺍﻟﻌﺎﻣﻠﻴﻦ ﻓﻲ ﻫﺬﻩ ﺍﻟﻤﺆﺳﺴﺔ )ﺍﻟﻤﺠﺘﻤﻊ ﺍﻹﺣﺼﺎﺋﻲ(‪.‬‬ ‫)‪ (b‬ﻋﻠ ًﻤﺎ ﺃﻥ ﺍﻟﻤﺆﺳﺴﺔ ﻣﻜﻮﻧﺔ ﻣﻦ ﺛﻼﺙ ﻓﺌﺎﺕ‪ :‬ﺍﻟﻔﺌﺔ ‪ A‬ﺣﻴﺚ ﺣﺠﻢ ﺍﻟﻌﻴﻨﺔ ﺍﻟﻄﺒﻘﻴﺔ ‪ ، 30‬ﺍﻟﻔﺌﺔ ‪ B‬ﺣﻴﺚ ﺣﺠﻢ‬ ‫ﺍﻟﻌﻴﻨﺔ ﺍﻟﻄﺒﻘﻴﺔ ‪ ،30‬ﺍﻟﻔﺌﺔ ‪ C‬ﺣﻴﺚ ﺣﺠﻢ ﺍﻟﻌﻴﻨﺔ ﺍﻟﻄﺒﻘﻴﺔ ‪ ،10‬ﺃﻭﺟﺪ ﺣﺠﻢ ﺍﻟﻌﻴﻨﺔ ﺍﻟﻤﻨﺎﻇﺮﺓ ﻟﻜ ّﻞ ﻓﺌﺔ‪.‬‬ ‫)‪ (3‬ﻓﻲ ﺇﺣﺪﻯ ﺍﻟﺸﺮﻛﺎﺕ ﺗ ّﻢ ﺳﺤﺐ ﻋﻴﻨﺔ ﻋﺸﻮﺍﺋﻴﺔ ﻣﻨﺘﻈﻤﺔ ﻣﻜﻮﻧﺔ ﻣﻦ ‪ 25‬ﻓﺮ ًﺩﺍ ﺑﺤﻴﺚ ﺇﻥ ﻃﻮﻝ ﺍﻟﻔﺘﺮﺓ ‪،50‬‬ ‫ﺃﻭﺟﺪ ﺣﺠﻢ ﺍﻟﻤﺠﺘﻤﻊ ﺍﻻﺣﺼﺎﺋﻲ )ﻋﺪﺩ ﺃﻓﺮﺍﺩ ﺍﻟﻌﺎﻣﻠﻴﻦ ﻓﻲ ﺍﻟﺸﺮﻛﺔ(‪.‬‬ ‫)‪ (4‬ﻓﻲ ﺍﺳﺘﻄﻼﻉ ﺃﺟﺮﻱ ﻋﻠﻰ ﺍﻟﺼﻒ ﺍﻟﺜﺎﻧﻲ ﻋﺸﺮ ﻋﻠﻤﻲ ﻟﻤﻌﺮﻓﺔ ﺁﺭﺍﺋﻬﻢ ﺣﻮﻝ ﻣﻬﻨﺔ ﺍﻟﻤﺴﺘﻘﺒﻞ ﺟﺎﺀﺕ ﺍﻹﺟﺎﺑﺎﺕ ﻛﻤﺎ‬ ‫ﻳﺒﻴّﻦ ﺍﻟﺠﺪﻭﻝ ﺍﻟﺘﺎﻟﻲ‪:‬‬ ‫ﺍﻟﻤﺠﻤﻮﻉ ﺭﺟﻞ ﺃﻋﻤﺎﻝ ﻣﺤﺎﻡ ﻃﺒﻴﺐ ﻣﻬﻨﺪﺱ ﺿﺎﺑﻂ ﻣﻌﻠﻢ ﺍﻟﻤﻬﻨﺔ‬ ‫‪ 2 3 6 7 5 2 25‬ﺍﻟﺘﻜﺮﺍﺭ‬ ‫)‪ (a‬ﺃﻛﻤﻞ ﺍﻟﺠﺪﻭﻝ ﻹﻳﺠﺎﺩ ﺍﻟﺘﻜﺮﺍﺭ ﺍﻟﻨﺴﺒﻲ ﻭﺍﻟﻨﺴﺒﺔ ﺍﻟﻤﺌﻮﻳﺔ ﻟﻠﺘﻜﺮﺍﺭ‪.‬‬ ‫)‪ (b‬ﻣﺜّﻞ ﻫﺬﻩ ﺍﻟﺒﻴﺎﻧﺎﺕ ﺑﺎﻟﻘﻄﺎﻋﺎﺕ ﺍﻟﺪﺍﺋﺮﻳﺔ‪.‬‬ ‫)‪ (5‬ﻓﻲ ﺍﻟﺒﻴﺎﻧﺎﺕ ﺍﻟﺘﺎﻟﻴﺔ‪ .7 ،8 ،6 ،5 ،4 ،9 ،3 :‬ﺃﻭﺟﺪ ﺍﻟﻤﺘﻮﺳﻂ ﺍﻟﺤﺴﺎﺑﻲ ‪ ، x‬ﺍﻟﺘﺒﺎﻳﻦ ‪ v‬ﻭﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ ‪σ‬‬ ‫)‪ (6‬ﻋﻠﻰ ﺍﻓﺘﺮﺍﺽ ﺃﻥ ﺍﻟﻤﺘﻮﺳﻂ ﺍﻟﺤﺴﺎﺑﻲ ﻷﺭﺑﺎﺡ ﺇﺣﺪﻯ ﺍﻟﺸﺮﻛﺎﺕ ﻫﻮ ‪ 850‬ﺩﻳﻨﺎ ًﺭﺍ ﻭﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ‬ ‫‪ 175‬ﺩﻳﻨﺎ ًﺭﺍ ﻭﺍﻟﻤﻨﺤﻨﻰ ﺍﻟﺘﻜﺮﺍﺭﻱ ﻷﺭﺑﺎﺡ ﻫﺬﻩ ﺍﻟﺸﺮﻛﺔ ﻫﻮ ﻋﻠﻰ ﺷﻜﻞ ﺟﺮﺱ )ﺗﻮﺯﻳﻊ ﻃﺒﻴﻌﻲ(‪.‬‬ ‫)‪ (a‬ﻃﺒّﻖ ﺍﻟﻘﺎﻋﺪﺓ ﺍﻟﺘﺠﺮﻳﺒﻴﺔ ﻋﻠﻰ ﺍﻟﻤﺘﻮﺳﻂ ﺍﻟﺤﺴﺎﺑﻲ ﻷﺭﺑﺎﺡ ﻫﺬﻩ ﺍﻟﺸﺮﻛﺔ‪.‬‬ ‫)‪ (b‬ﻫﻞ ﺍﻧﺨﻔﻀﺖ ﺃﺭﺑﺎﺡ ﻫﺬﻩ ﺍﻟﺸﺮﻛﺔ ﺇﻟﻰ ‪ 300‬ﺩﻳﻨﺎﺭ؟ ﺍﺷﺮﺡ ﺫﻟﻚ‪.‬‬ ‫)‪ (c‬ﻫﻞ ﻭﺻﻠﺖ ﺃﺭﺑﺎﺡ ﻫﺬﻩ ﺍﻟﺸﺮﻛﺔ ﺇﻟﻰ ‪ 1400‬ﺩﻳﻨﺎﺭ؟ ﺍﺷﺮﺡ ﺫﻟﻚ‪.‬‬ ‫‪97‬‬

‫)‪ (7‬ﻧﺎﻝ ﺍﻟﻄﺎﻟﺐ ﺳﺎﻟﻢ ‪ 15‬ﻣﻦ ‪ 20‬ﻓﻲ ﺍﺧﺘﺒﺎﺭ ﻣﺎﺩﺓ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ ﺣﻴﺚ ﺍﻟﻤﺘﻮﺳﻂ ﺍﻟﺤﺴﺎﺑﻲ ﻟﻠﺪﺭﺟﺎﺕ ‪ 13‬ﻭﺍﻻﻧﺤﺮﺍﻑ‬ ‫ﺍﻟﻤﻌﻴﺎﺭﻱ ‪ 2.5‬ﻭﻗﺪ ﻧﺎﻝ ﺃﻳ ًﻀﺎ ‪ 13‬ﻣﻦ ‪ 20‬ﻓﻲ ﺍﺧﺘﺒﺎﺭ ﻣﺎﺩﺓ ﺍﻟﻔﻴﺰﻳﺎﺀ ﺣﻴﺚ ﺍﻟﻤﺘﻮﺳﻂ ﺍﻟﺤﺴﺎﺑﻲ ﻟﻠﺪﺭﺟﺎﺕ ‪11.5‬‬ ‫ﻭﺍﻻﻧﺤﺮﺍﻑ ﺍﻟﻤﻌﻴﺎﺭﻱ ‪2.4‬‬ ‫ﻓﻲ ﺃﻱ ﻣﺎﺩﺓ ﺗﻌﺘﺒﺮ ﺩﺭﺟﺔ ﺳﺎﻟﻢ ﻫﻲ ﺍﻷﻓﻀﻞ ﻣﻘﺎﺭﻧﺔ ﺑﺪﺭﺟﺎﺕ ﻛﻞ ﻣﺎﺩﺓ؟ ﺍﺷﺮﺡ‪.‬‬ ‫)‪ (8‬ﻳﺒﻴّﻦ ﺍﻟﺠﺪﻭﻝ ﺍﻟﺘﺎﻟﻲ ﺍﻟﺘﻮﺯﻳﻊ ﺍﻟﺘﻜﺮﺍﺭﻱ ﻷﻭﺯﺍﻥ ﻃ ّﻼﺏ ﺍﻟﺼﻒ ﺍﻟﺤﺎﺩﻱ ﻋﺸﺮ ﺑﺎﻟﻜﻴﻠﻮﺟﺮﺍﻡ )‪.(kg‬‬ ‫ﺍﻟﻤﺠﻤﻮﻉ ‪ 64- 68- 72- 76- 80-‬ﺍﻟﻔﺌﺔ‬ ‫‪ 4 5 7 6 3 25‬ﺍﻟﺘﻜﺮﺍﺭ‬ ‫)‪ (a‬ﺃﻛﻤﻞ ﺍﻟﺠﺪﻭﻝ ﻹﻳﺠﺎﺩ ﻣﺮﺍﻛﺰ ﺍﻟﻔﺌﺎﺕ‪.‬‬ ‫)‪ (b‬ﻣﺜّﻞ ﻫﺬﻩ ﺍﻟﺒﻴﺎﻧﺎﺕ ﺑﺎﻟﻤﺪﺭﺝ ﺍﻟﺘﻜﺮﺍﺭﻱ ﻭﺍﻟﻤﻀﻠﻊ ﺍﻟﺘﻜﺮﺍﺭﻱ‪.‬‬ ‫‪98‬‬

‫ﺗﻤﺎﺭﻳﻦ ﺇﺛﺮﺍﺋﻴﺔ‬ ‫)‪ (1‬ﻫﻞ ﻳﻤﻜﻦ ﺍﺳﺘﺨﺪﺍﻡ ﺍﻟﺤﺼﺮ ﺍﻟﺸﺎﻣﻞ ﻓﻲ ﺩﺭﺍﺳﺔ ﺍﻟﻤﺠﺘﻤﻌﺎﺕ ﺍﻹﺣﺼﺎﺋﻴﺔ ﺍﻟﺘﺎﻟﻴﺔ‪ ،‬ﺃﻡ ﻻ؟ ﻣﻊ ﺫﻛﺮ ﺍﻟﺴﺒﺐ‪.‬‬ ‫)‪ (a‬ﺩﺭﺍﺳﺔ ﺃﻧﻮﺍﻉ ﺍﻟﺤﺸﺮﺍﺕ ﻓﻲ ﺩﻭﻟﺔ ﺍﻟﻜﻮﻳﺖ‪.‬‬ ‫)‪ (b‬ﺩﺭﺍﺳﺔ ﻧﺴﺒﺔ ﻋﺪﺩ ﺍﻹﻧﺎﺙ ﺇﻟﻰ ﻋﺪﺩ ﺍﻟﺬﻛﻮﺭ ﺍﻟﻌﺎﻣﻠﻴﻦ ﻓﻲ ﺃﺣﺪ ﺍﻟﻤﺼﺎﺭﻑ ﻓﻲ ﺩﻭﻟﺔ ﺍﻟﻜﻮﻳﺖ‪.‬‬ ‫)‪ (2‬ﺍﻟﻜﺘﺎﺑﺔ ﻓﻲ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ‪ :‬ﺍﺫﻛﺮ ﺃﻣﺜﻠﺔ ﺗﺘﻀﻤﻦ ﻣﺎ ﻳﻠﻲ‪:‬‬ ‫)‪ (a‬ﻣﺠﺘﻤﻊ ﺇﺣﺼﺎﺋﻲ ﻣﻨﺘﻪ ‪ -‬ﻭﺣﺪﺓ ﺍﻟﺪﺭﺍﺳﺔ ‪ -‬ﺍﻟﻤﺘﻐﻴﺮ ﺍﻟﻤﺮﺍﺩ ﺩﺭﺍﺳﺘﻪ‪.‬‬ ‫)‪ (b‬ﻣﺠﺘﻤﻊ ﺇﺣﺼﺎﺋﻲ ﻏﻴﺮ ﻣﻨﺘﻪ ‪ -‬ﻭﺣﺪﺓ ﺍﻟﺪﺭﺍﺳﺔ ‪ -‬ﺍﻟﻤﺘﻐﻴﺮ ﺍﻟﻤﺮﺍﺩ ﺩﺭﺍﺳﺘﻪ‪.‬‬ ‫)‪ (3‬ﻓﻲ ﺃﺣﺪ ﻣﺼﺎﻧﻊ ﻏﺰﻝ ﺍﻟﻨﺴﻴﺞ‪ ،‬ﺍﻟﺬﻱ ﻳﺤﻮﻱ ‪ 600‬ﻋﺎﻣﻞ ﻣﺮﻗﻤﻴﻦ ﻣﻦ ‪ 1‬ﺇﻟﻰ ‪ .600‬ﺃﺭﺍﺩ ﺻﺎﺣﺐ ﺍﻟﻤﺼﻨﻊ ﻣﻨﺎﻗﺸﺔ‬ ‫ﻋﺪﺩ ﻣﻦ ﺍﻟﻌﻤﺎﻝ ﻓﻲ ﻛﻴﻔﻴﺔ ﺗﺤﺴﻴﻦ ﺍﻹﻧﺘﺎﺝ‪ .‬ﺍﻟﻤﻄﻠﻮﺏ ﺳﺤﺐ ﻋﻴﻨﺔ ﻋﺸﻮﺍﺋﻴﺔ ﺑﺴﻴﻄﺔ ﻣﻜﻮﻧﺔ ﻣﻦ ‪ 15‬ﻋﺎﻣ ًﻼ‬ ‫ﺑﺎﺳﺘﺨﺪﺍﻡ ﺟﺪﻭﻝ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻌﺸﻮﺍﺋﻴﺔ‪.‬‬ ‫)‪ (4‬ﺃﺭﺍﺩ ﻣﺪﻳﺮ ﻋﺎﻡ ﺷﺮﻛﺔ ﻛﺒﺮﻯ ﻹﻧﺘﺎﺝ ﻣﻮﺍﺩ ﺍﻟﺪﻫﺎﻥ ﺗﻘﻴﻴﻢ ﺃﺩﺍﺀ ﻛﺎﻓﺔ ﺍﻟﻤﻮﻇﻔﻴﻦ‪ ،‬ﻋﻠ ًﻤﺎ ﺃﻥ ﺍﻟﺸﺮﻛﺔ ﺗﻀﻢ ‪80‬‬ ‫ﻣﻬﻨﺪ ًﺳﺎ ﺗ ّﻢ ﺗﺮﻗﻴﻤﻬﻢ ﻣﻦ ‪ 201‬ﺇﻟﻰ ‪ 120 ،280‬ﺍﺧﺘﺼﺎﺻﻲ ﻣﺨﺘﺒﺮ ﺗ ّﻢ ﺗﺮﻗﻴﻤﻬﻢ ﻣﻦ ‪ 301‬ﺇﻟﻰ ‪ ،420‬ﻭﺃﺧﻴ ًﺮﺍ‬ ‫‪ 220‬ﻋﺎﻣ ًﻼ ﺗ ّﻢ ﺗﺮﻗﻴﻤﻬﻢ ﻣﻦ ‪ 501‬ﺇﻟﻰ ‪ .720‬ﺍﻟﻤﻄﻠﻮﺏ ﺳﺤﺐ ﻋﻴﻨﺔ ﻋﺸﻮﺍﺋﻴﺔ ﻃﺒﻘﻴﺔ ﻣﻜﻮﻧﺔ ﻣﻦ ‪ 21‬ﻓﺮ ًﺩﺍ‬ ‫ﺗﻤﺜﻞ ﺟﻤﻴﻊ ﺍﻟﻌﺎﻣﻠﻴﻦ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺟﺪﻭﻝ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻌﺸﻮﺍﺋﻴﺔ ﺍﺑﺘﺪﺍﺀً ﻣﻦ ﺍﻟﺼﻒ ﺍﻟﺴﺎﺑﻊ ﻭﺍﻟﻌﻤﻮﺩ ﺍﻷ ّﻭﻝ‪.‬‬ ‫)‪ (5‬ﺃﺭﺍﺩ ﻣﻌﻠﻢ ﻓﻲ ﺃﺻﻮﻝ ﺗﻌﻠﻴﻢ ﺍﻟﻘﺮﺁﻥ ﺍﻟﻜﺮﻳﻢ ﺗﺸﻜﻴﻞ ﻣﺠﻤﻮﻋﺎﺕ ﻓﻲ ﺍﻟﺼﻔﻮﻑ ﺍﻟﺜﺎﻧﻮﻳﺔ ﻹﺣﺪﻯ ﺍﻟﻤﺪﺍﺭﺱ ﺍﻟﺘﻲ‬ ‫ﺗﺤﻮﻱ ‪ 144‬ﻃﺎﻟﺒًﺎ ﻣﺮﻗﻤﻴﻦ ﻣﻦ ‪ 1‬ﺇﻟﻰ ‪ .144‬ﺍﻟﻤﻄﻠﻮﺏ ﺳﺤﺐ ﻋﻴﻨﺔ ﻋﺸﻮﺍﺋﻴﺔ ﻣﻨﺘﻈﻤﺔ ﻣﻜﻮﻧﺔ ﻣﻦ ‪ 16‬ﻃﺎﻟﺒًﺎ‬ ‫ﺑﺎﺳﺘﺨﺪﺍﻡ ﺟﺪﻭﻝ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻌﺸﻮﺍﺋﻴﺔ ﺍﺑﺘﺪﺍﺀً ﻣﻦ ﺍﻟﺼﻒ ﺍﻟﺜﺎﻟﺚ ﻭﺍﻟﻌﻤﻮﺩ ﺍﻟﺜﺎﻟﺚ‪.‬‬ ‫)‪ (6‬ﻳﺘﺄﻟﻒ ﻓﺮﻳﻖ ﺍﻟﻌﻤﻞ ﻓﻲ ﺇﺣﺪﻯ ﺍﻟﺸﺮﻛﺎﺕ ﻣﻦ ‪ 360‬ﻣﻮﻇ ًﻔﺎ ﻭﻫﻢ ﻣﻦ ﺍﻟﺠﻨﺴﻴﻦ ﺃﻱ ﺫﻛﻮﺭ ﻭﺇﻧﺎﺙ ﻭﻳﻌﻤﻠﻮﻥ ﺇﻣﺎ‬ ‫ﺑﺪﻭﺍﻡ ﻛﺎﻣﻞ ﺃﻭ ﺑﺪﻭﺍﻡ ﺟﺰﺋﻲ ﻛﻤﺎ ﻫﻮ ﻣﺒﻴّﻦ ﻓﻲ ﺍﻟﺠﺪﻭﻝ ﺍﻟﺘﺎﻟﻲ‪:‬‬ ‫‪ 180‬ﻣﺮﻗﻤﻴﻦ ﻣﻦ ‪ 1‬ﺇﻟﻰ ‪180‬‬ ‫ﺫﻛﻮﺭ‪/‬ﺩﻭﺍﻡ ﻛﺎﻣﻞ‬ ‫‪ 36‬ﻣﺮﻗﻤﻴﻦ ﻣﻦ ‪ 181‬ﺇﻟﻰ ‪217‬‬ ‫ﺫﻛﻮﺭ‪/‬ﺩﻭﺍﻡ ﺟﺰﺋﻲ‬ ‫‪ 18‬ﻣﺮﻗﻤﻴﻦ ﻣﻦ ‪ 218‬ﺇﻟﻰ ‪236‬‬ ‫ﺇﻧﺎﺙ‪/‬ﺩﻭﺍﻡ ﻛﺎﻣﻞ‬ ‫‪ 126‬ﻣﺮﻗﻤﻴﻦ ﻣﻦ ‪ 237‬ﺇﻟﻰ ‪363‬‬ ‫ﺇﻧﺎﺙ‪/‬ﺩﻭﺍﻡ ﺟﺰﺋﻲ‬ ‫ﺍﻟﻤﻄﻠﻮﺏ ﺃﺧﺬ ﻋﻴﻨﺔ ﻃﺒﻘﻴﺔ ﺣﺠﻤﻬﺎ ‪ 40‬ﻣﻮﻇ ًﻔﺎ‪ ،‬ﻭﻓ ًﻘﺎ ﻟﻠﻔﺌﺎﺕ ﺃﻋﻼﻩ ﺑﺎﺳﺘﺨﺪﺍﻡ ﺑﺮﻧﺎﻣﺞ ﺇﺣﺼﺎﺋﻲ‪.‬‬ ‫‪99‬‬