ᠬᠤᠷᠢᠨ ᠬᠤᠶᠠᠳᠤᠭᠠᠷ ᠪᠦᠯᠦᠭ ᠬᠤᠶᠠᠳᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠶᠢᠨ ᠹᠦᠩᠺᠼ （ᠨᠢᠭᠡᠴᠢ ᠶᠢᠨ ᠪᠠᠢᠴᠠᠭᠠᠯᠲᠠ 2） .pdf

ᠬᠣᠷᠢ ᠨ ᠬᠣᠶᠠ ᠳ᠋ᠤᠭᠠ ᠷ ᠪᠥᠯ ᠥᠭ ᠦ ᠨ ᠪᠠ ᠢᠴᠠ ᠭᠠᠯᠲ ᠠ ᠠᠩᠭᠢ ᠪᠥᠯᠥᠭ᠄ ᠨᠡᠷ ᠡ ᠄ ︵90 ᠮ ᠢᠨᠦ᠋ᠲ ᠂ ᠨᠠᠢ ᠲᠡ 120 ᠬᠤ ᠪᠢ ︶ ᠨᠢᠭᠡ ᠂ ᠰ ᠤᠩᠭᠤ ᠬᠤ ᠰ ᠡᠲᠤᠪ ᠃ ︵ ᠰᠡᠲ ᠤᠪ ᠪ ᠦᠷᠢ 3 ᠬᠤᠪᠢ ᠂ ᠨᠠᠢ ᠲᠡ 30 ᠬᠤ ᠪᠢ ︶ 1. ᠹᠦᠩ ᠺᠽ y= (𝑚 + 2) 𝑥 𝑚 2− 2 ᠨᠢ ᠬᠣᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦ ᠩᠺᠽ ᠪᠣᠯ ᠂ m ᠢᠢᠨ ᠬᠡᠮᠵ ᠢᠭᠳ᠋ᠡᠯ ᠨ ᠢ ( ) A. ± 2 B. 2 C. − 2 D. ± 1 2. ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ y = −( 𝑥− 1) 2− 2 ᠦᠨ ᠣᠷᠣᠢ ᠢᠢᠨ ᠴ ᠡᠭ ᠦᠨ ᠪᠠᠶᠢ ᠷᠢᠰᠢ ᠯ ᠨᠢ ( ) A. (1 ,2 ) B. (1 ,− 2) C. ( −1 ,2 ) D. (− 1 , −2 ) 3. ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ y = 𝑥 2 + 𝑝𝑥 + 𝑞 ᠦᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠦᠨ ᠣᠷᠣᠢ ᠢᠢᠨ ᠴᠡᠭ ᠦᠨ ᠪᠠᠶᠢ ᠷᠢᠰᠢ ᠯ ᠨᠢ (3 ,2 ) ᠪᠣᠯ ᠂ ᠡᠨᠡ ᠬ ᠠᠶᠠᠯᠲ ᠠ ᠢ ᠢᠨ ᠱᠤ ᠭᠤᠮ ᠦᠨ ᠠ ᠨᠠᠯᠢᠰ ᠢᠯ ᠡᠷᠬᠡᠶ ᠢᠯᠡᠯ ᠨᠢ ( ) A. y = 𝑥 2 + 6𝑥 + 11 B. y = 𝑥 2 − 6𝑥 − 11 C. y = 𝑥 2 − 6𝑥 + 11 D. y = 𝑥 2 − 6𝑥 + 7 4. ᠵᠢᠷᠤᠭ 1 ᠮᠡᠲᠦ ᠂ ᠬᠣᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦᠩ ᠺᠽ y= 𝑎𝑥 2+ 𝑏𝑥 + 𝑐 ( 𝑎≠ 0) ᠦᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠨᠢ 𝑥 ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠲᠠᠢ A , B ᠬᠣ ᠶᠠᠷ ᠴᠡᠭ ᠲᠦ ᠣᠭᠲᠣᠯ ᠣᠯᠴᠠᠵᠤ ᠂ y ᠲᠡ ᠩᠬᠡᠯ ᠢᠭ ᠲᠠᠢ C ᠴᠡᠭ ᠲᠦ ᠣᠭᠲᠣᠯ ᠣᠯᠴᠠᠬᠤ ᠪᠦᠭᠡᠳ ᠳᠡᠭ ᠰᠢ ᠬ ᠡᠮ ᠦᠨ ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠨᠢ ᠰ ᠢᠯᠤᠭᠤ ᠨ 𝑥= 1 B ᠴᠡᠭ ᠦᠨ ᠪᠠᠶᠢ ᠷᠢᠰᠢ ᠯ ᠨᠢ (− 1, 0) ᠲᠡᠭᠡ ᠪᠡᠯ ᠳᠣᠣ ᠷᠠᠬᠢ ᠳ᠋ᠦ᠋ ᠩᠨᠡᠯ ᠲᠡ ᠡᠴᠡ ᠵᠦ᠋ᠪ ᠪᠣᠯ ᠬᠤ ᠨᠢ ( ) ① 2 𝑎+ b= 0 ② 4 𝑎− 2𝑏 + 𝑐 0 ④ 𝑦 2 A. ① ③ B. ② ③ C. ① ② D. ③ ④ 5. ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤ ᠭᠤᠮ y = −2 (𝑥 + 3) 2+ 1 ᠢ ᠢᠨ ᠳᠡᠭ ᠰᠢ ᠬ ᠡᠮ ᠦᠨ ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠨᠢ ( ) A. ᠰᠢᠯ ᠤᠭᠤᠨ 𝑥 = 3 B. ᠰᠢᠯ ᠤᠭᠤᠨ 𝑥 = 1 C. ᠰᠢᠯ ᠤᠭᠤᠨ 𝑥 = −1 D. ᠰᠢᠯ ᠤᠭᠤᠨ 𝑥 = −3 6. ᠬᠣᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦᠩ ᠺᠽ 𝑦= 𝑎𝑥 2+ 𝑏𝑥 + 𝑐 ᠢ ᠢᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠨᠢ ᠵᠢᠷᠤᠭ 2 ᠮ ᠡᠲᠦ ᠪ ᠣᠯ ᠂ ᠳᠣᠣ ᠷᠠᠬᠢ ᠳ᠋ᠦᠩ ᠨᠡᠯᠲᠡ ᠡᠴᠡ ᠪ ᠤᠷᠤᠭᠤ ᠪᠣᠯ ᠬᠤ ᠨ ᠢ ( ) A. ᠵᠢᠷᠤᠮᠠ ᠭ ᠦᠨ ᠳᠡᠭ ᠰᠢ ᠬ ᠡᠮ ᠦᠨ ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠨᠢ ᠰ ᠢᠯᠤᠭᠤ ᠨ 𝑥 = 1 B. 𝑥 1 ᠦᠶ ᠡᠰ ᠂ y ᠨᠢ 𝑥 ᠦᠨ ᠨᠡᠮᠡᠭ ᠳ᠋ᠡᠬᠦ ᠢᠢ ᠳᠠᠭᠠᠯ ᠳ᠋ᠤᠨ ᠬᠣᠷᠤᠭ ᠳ᠋ᠠᠨ ᠠ C. ᠨ ᠢᠭᠡ ᠦᠯ ᠦ ᠮ ᠡᠳ᠋ᠡᠭᠳ᠋ ᠡᠭᠴᠢ ᠲᠡᠢ ᠬᠣᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠳᠡᠭ ᠰᠢᠳ ᠭᠡᠯ 𝑎𝑥 2+ 𝑏𝑥 + 𝑐= 0 ᠦᠨ ᠬ ᠣᠶᠠᠷ ᠢᠵᠠᠭᠤᠷ ᠨ ᠢ − 1 , 3 D. − 1 0 B. 3 𝑎+ c 2 ④ 4𝑎 − 2b + c 0 ᠳᠡᠭᠡ ᠷᠡᠬᠢ ᠳ᠋ᠦ᠋ ᠩᠨᠡᠯ ᠲᠡ ᠡᠴᠡ ᠵᠦ᠋ᠪ ᠪᠣᠯ ᠬᠤ ᠨᠢ ( ) ᠪᠠᠶᠢ ᠨ ᠠ ᠃ A. 1 B. 2 C. 3 D. 4 9. ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ y = 𝑎𝑥 2+ 𝑏𝑥 + 𝑐 ᠳ᠋ᠤ ᠂ 4 𝑎− b= 0 , 𝑎− b+ c 0 ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ 𝑥 ᠲ ᠡᠩᠬᠡᠯ ᠢᠭ ᠲᠠᠢ ᠬᠣᠶ ᠠᠷ ᠣ ᠭᠲᠣᠯ ᠣᠯᠴᠠᠯ ᠦᠨ ᠴᠡᠭ ᠲᠠᠢ ᠪ ᠦᠭᠡᠳ ᠡᠨᠡ ᠬᠣ ᠶᠠᠷ ᠴᠡᠭ ᠦᠨ ᠬ ᠣᠭᠣᠷᠤᠨ ᠳ᠋ᠤᠬᠢ ᠵᠠᠢ ᠨᠢ 2 ᠡ ᠴᠡ ᠪ ᠠᠭ ᠠ ᠂ ᠲ ᠡᠭᠡᠪᠡᠯ ᠳᠣᠣ ᠷᠠᠬᠢ ᠳ᠋ᠦ᠋ ᠩᠨᠡᠯ ᠲᠡ ᠡᠴᠡ ᠵᠦ᠋ᠪ ᠪᠣᠯ ᠬᠤ ᠨᠢ ( ) ① 𝑎 𝑏𝑐 0 ③ 𝑎 + b+ c 0 ④ 4 𝑎 c A. ① ② B. ② ③ ④ C. ① ② ③ D. ① ② ③ ④ 10. ᠹᠦᠩ ᠺᠽ y= 𝑎𝑥 2+ 𝑏𝑥 + 𝑐 ᠢ ᠢᠨ ᠵᠢᠷᠤᠮᠠ ᠭ ᠨ ᠢ ᠵᠢ ᠷᠤᠭ 5 ᠮᠡᠲᠦ ᠂ ᠲ ᠡᠭᠡᠪᠡᠯ ᠨᠢ ᠭᠡ ᠦᠯ ᠦ ᠮ ᠡᠳ᠋ᠡᠭᠳ᠋ ᠡᠭᠴᠢ ᠲᠡᠢ ᠬᠣᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠳᠡᠭᠰ ᠢᠳᠭᠡᠯ 𝑎𝑥 2+ 𝑏𝑥 + 𝑐− 2= 0 ᠦᠨ ᠢᠵᠠᠭᠤᠷ ᠦᠨ ᠪᠠᠶᠢ ᠳ᠋ᠠᠯ ᠨᠢ ( ) A. ᠬᠣ ᠶᠠᠷ ᠲᠡᠩ ᠴᠡᠭᠦ ᠦ ᠪᠤ ᠰᠤ ᠪ ᠣᠳ᠋ᠠᠲ ᠤ ᠲ ᠣᠭᠠᠨ ᠢᠵ ᠠᠭᠤᠷ ᠲᠠᠢ B. ᠣᠨᠳ᠋ ᠣᠤ ᠲ ᠡᠮᠲᠡᠭ ᠲᠠᠢ ᠬᠣ ᠶᠠᠷ ᠪᠣᠳ᠋ᠠ ᠲᠤ ᠲ ᠣᠭᠠᠨ ᠢᠵ ᠠᠭᠤᠷ ᠲᠠᠢ C. ᠲᠡᠩ ᠴᠡᠭᠦ ᠦ ᠬᠣ ᠶᠠᠷ ᠪᠣᠳ᠋ᠠ ᠲᠤ ᠲ ᠣᠭᠠᠨ ᠢᠵ ᠠᠭᠤᠷ ᠲᠠᠢ D. ᠪᠣᠳ᠋ ᠠᠲᠤ ᠲᠣᠭᠠᠨ ᠢᠵ ᠠᠭᠤᠷ ᠦᠭᠡᠢ ᠬᠣᠶ ᠠᠷ᠂ ᠬᠠᠭᠠ ᠯᠲᠠ ᠳ᠋ ᠤ ᠲ ᠠᠭᠯᠠᠬᠤ ᠰᠡᠲᠤ ᠪ᠃ ︵ᠰᠡᠲ ᠤᠪ ᠪ ᠦᠷᠢ 3 ᠬᠤᠪᠢ ᠂ ᠨᠠᠢ ᠲᠡ 24 ᠬᠤᠪᠢ ︶ 11. ᠬᠣ ᠶᠠᠳ᠋ᠤ ᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦᠩ ᠺᠽ y= 𝑎𝑥 2+ 𝑏𝑥 + 𝑐 ᠦᠨ ᠵᠢᠷᠤᠮ ᠠᠭ ᠨᠢ 𝑥 ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠲᠠᠢ (− 1 , 0) , (5 ,0 ) ᠬᠣᠶ ᠠᠷ ᠴ ᠡᠭ ᠲᠦ ᠣᠭᠲᠣᠯ ᠣᠯᠴᠠ ᠪᠠᠯ ᠂ ᠡ ᠨᠡ ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ ᠦᠨ ᠳᠡᠭ ᠰᠢ ᠬ ᠡᠮ ᠦᠨ ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠨᠢ ( ) 12. ᠬᠠᠶ ᠠᠯᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ ᠦᠨ y ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠲᠠᠢ ᠣᠭᠲᠣᠯ ᠣᠯᠴᠠᠯ ᠦᠨ ᠴᠡᠭ ᠦᠨ ᠪᠠᠶᠢ ᠷᠢᠰᠢ ᠯ ᠨᠢ (0 ,4 ) ᠂ 𝑥 ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠲ ᠠᠢ ᠣᠭᠲᠣᠯ ᠣᠯᠴᠠᠯ ᠦᠨ ᠴᠡᠭ ᠦᠨ ᠪᠠᠶᠢ ᠷᠢᠰᠢ ᠯ ᠨᠢ (− 1 , 0) , (2 ,0 ) ᠪᠣᠯ ᠂ ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤ ᠭᠤᠮ ᠦᠨ ᠠ ᠨᠠᠯᠢᠰ ᠢᠯ ᠡᠷᠬᠡᠶ ᠢᠯᠡᠯ ᠨ ᠢ ( ) 13. ᠨᠢᠭᠡ ᠳᠡᠭ ᠰᠢ ᠳᠦ᠋ ᠷᠪᠡᠯ ᠵᠢᠨ ᠦ ᠲ ᠠᠯ ᠠ ᠢᠢᠨ ᠤᠷᠲ ᠤ ᠨ ᠢ 𝑥 cm ᠡᠨᠡ ᠳᠡᠭᠰ ᠢ ᠳᠦ᠋ ᠷᠪᠡᠯ ᠵᠢᠨ ᠦ ᠲᠠᠯ ᠠ ᠢᠢᠨ ᠤ ᠷᠲᠤ ᠢᠢ 2c m ᠨᠡᠮᠡᠭ ᠳ᠋ᠡᠭᠦ ᠯᠪᠡᠯ᠂ ᠳᠡᠭᠰ ᠢ ᠳᠦ᠋ ᠷᠪᠡᠯ ᠵᠢᠨ ᠦ ᠲᠠᠯᠠ ᠪᠡᠢ ᠨᠢ S 𝑐𝑚 2᠂ S ᠨᠢ 𝑥 ᠦᠨ ᠬᠣᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦ ᠩᠺᠽ ᠪᠣᠯ ᠂ ᠠ ᠨᠠᠯᠢᠰ ᠢᠯ ᠡᠷᠬᠡᠶ ᠢᠯᠡᠯ ᠨᠢ ( ) ᠂ ᠬ ᠣᠶᠠᠳ᠋ ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠭᠡᠰ ᠢᠭᠦᠨ ᠦ ᠢᠯ ᠡᠳᠭᠡᠮᠵᠢ ᠨᠢ ( ) ᠨᠢᠭᠡᠳ᠋ ᠦᠭᠡᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠭᠡᠰ ᠢᠭᠦᠨ ᠤ ᠢᠯ ᠡᠳᠭᠡᠮᠵᠢ ᠨᠢ ( ) ᠰ ᠤᠯᠠ ᠭ ᠡᠰᠢᠭ ᠦᠨ ᠨ ᠢ ( ) 14. ᠥᠭ ᠭᠦᠭᠳ᠋ ᠡᠭᠰᠡᠨ ᠨᠢ᠂ ᠬᠣᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦᠩ ᠺᠼ 𝑦= −𝑥 2+ 6𝑥 + 𝑘 ᠦᠨ ᠵᠢᠷᠤᠮᠠ ᠭ ᠳᠡᠭᠡ ᠷᠡᠬᠢ ᠬᠣ ᠶᠠᠷ ᠴᠡᠭ ᠦᠨ ᠪᠠᠶᠢ ᠷᠢᠰᠢ ᠯ ᠨᠢ ᠲᠣᠰ ᠲᠣ ᠰ ( 2, 𝑦1 ) , (5 ,𝑦 2 ) ᠪᠣᠯᠣ ᠨ ᠠ ᠂ (4) (5) ᠲᠡᠭᠡᠪᠡᠯ 𝑦 1 , 𝑦 2 ᠤ ᠨ ᠶ ᠡᠬᠡ ᠪᠠᠭ ᠠ ᠢᠢᠨ ᠬᠣ ᠯᠪᠣᠭᠳ᠋ ᠠᠯ ᠨ ᠢ 𝑦 1 _ ___ _ 𝑦 2 ︵ , 0 ᠪᠣᠯᠬᠤ ᠢᠢ ᠪᠠ ᠲᠤᠯᠠ ᠃ (3) 𝑥 ᠶᠠᠮᠠᠷ ᠬᠡᠮᠵᠢ ᠭᠳ᠋ᠡᠯ ᠠᠪᠬᠤ ᠦᠶ ᠡᠰ y 0 𝑥 ᠶᠠᠮᠠᠷ ᠬᠡᠮᠵ ᠢᠭᠳ᠋ᠡᠯ ᠠ ᠪᠬᠤ ᠦᠶ ᠡᠰ y 0 ᠪᠣᠯᠬᠤ ᠪᠤᠢ ︖ 20. ( 8 ᠬ ᠤᠪᠢ ) ᠨ ᠢᠭᠡᠨ ᠲᠠᠧᠠᠷ ᠦᠨ ᠤᠷᠤᠭᠤ ᠯᠤᠭᠰᠠᠨ ᠦᠨ ᠡ ᠨ ᠢ ᠨ ᠢᠭᠡ ᠪᠦᠷᠢ ᠳ᠋ᠥ ᠪᠠᠨ 8 ᠲᠦ᠋ ᠭᠦᠷᠢᠭ ᠂ ᠵ ᠠᠬ ᠠ ᠲᠡᠯ ᠭᠡᠬᠦᠷ ᠦᠨ ᠪᠠᠶᠢ ᠴᠠᠭᠠᠯ ᠲᠠ ᠡᠴᠡ ᠦᠵᠡᠪᠡᠯ ᠲᠠᠧᠠᠷ ᠪᠦᠷ ᠢ ᠢᠢ 10 ᠲᠦ᠋ ᠭᠦᠷᠢ ᠭ ᠢᠢ ᠠᠷ ᠬᠤᠳ᠋ ᠠᠯᠳ᠋ᠤ ᠪᠠᠯ ᠂ ᠭ ᠠᠷᠠᠭ ᠪᠦᠷ ᠢ 55 ᠬᠤᠳ᠋ ᠠᠯᠳ᠋ᠤ ᠨ ᠠ ᠃ ᠬ ᠡᠷᠪᠡ ᠲᠠᠧᠠᠷ ᠪᠦᠷᠢ ᠢᠢᠨ ᠦᠨ ᠡ ᠢᠢ 1 ᠲᠦ᠋ ᠭᠦᠷᠢ ᠭ ᠨᠡᠮᠡᠭ ᠳ᠋ᠡᠭᠦ ᠯᠪᠡᠯ ᠭᠠᠷᠠ ᠭ ᠪᠦᠷ ᠢ 5 ᠳᠤ ᠲᠠᠭᠤ ᠬᠣᠳ᠋ ᠠᠯᠳ᠋ᠤ ᠨ ᠠ ᠃ ᠬ ᠡᠷᠪᠡ ᠲᠠᠧᠠᠷ ᠪᠦᠷ ᠢ ᠢᠢ ᠨ ᠦᠨ ᠡ ᠢᠢ 1 ᠲᠦ᠋ ᠭᠦᠷᠢ ᠭ ᠪᠠᠭᠤ ᠷᠠᠭᠤᠯ ᠪᠠᠯ ᠭᠠᠷᠠᠭ ᠪᠦᠷ ᠢ 5 ᠢᠯ ᠡᠭᠦᠦ ᠬᠣᠳ᠋ ᠠᠯᠳ᠋ᠤ ᠨ ᠠ ᠃ (1) ᠲ ᠤᠰ ᠲᠠᠧᠠᠷ ᠢ ᠬᠤᠳ᠋ ᠠᠯᠳ᠋ᠤ ᠭᠠᠳ ᠣᠯᠬᠤ ᠠᠰᠢ ᠭ ᠬ ᠣᠨᠵᠢ ᠪᠦᠷᠢ y ( ᠲ ᠦ᠋ᠭᠦ ᠷᠢᠭ ) ᠪᠠ 𝑥 ᠲᠥ ᠭᠥᠷᠢᠭ ᠨᠡᠮᠡᠭ ᠳ᠋ᠡᠭᠦ ᠯᠦᠭᠰᠡᠨ ᠦ ᠬᠣᠶ ᠢᠨᠠᠬᠢ 𝑥 ᠪᠠ y ᠢᠢᠨ ᠬ ᠣᠭᠣᠷᠤᠨᠳ᠋ ᠤᠬᠢ ᠹᠦᠩ ᠺᠽ ᠦᠨ ᠬᠣᠯ ᠪᠤᠭᠳ᠋ ᠠᠯ ᠢᠯᠡᠷᠬ ᠡᠶᠢᠯᠡᠯ ᠢ ᠢᠯ ᠡᠷᠬᠡᠶ ᠢᠯᠡᠪᠡᠯ ︖ (2) ᠬᠡᠳ᠋ ᠦᠢ ᠲ ᠦ᠋ᠭᠦ ᠷᠢᠭ ᠨᠡᠮᠡᠭ ᠳ᠋ᠡᠭᠦ ᠯᠬᠦ ᠦᠶ ᠡᠰ ᠠ ᠰᠢᠭ ᠬᠣ ᠨᠵᠢᠪᠦ ᠷᠢ ᠨ ᠢ ᠬ ᠠᠮᠤᠭ ᠶᠡᠬ ᠡ ᠪᠤ ᠢ ︖ ᠬ ᠠᠮᠤᠭ ᠶᠡᠬᠡ ᠠᠰ ᠢᠭ ᠬᠣ ᠨᠵᠢᠪᠦ ᠷᠢ ᠨ ᠢ ᠬᠡᠳ᠋ ᠦᠢ ᠪ ᠤᠢ ︖ 21.( 10 ᠬᠤᠪᠢ ) ᠵ ᠢᠷᠤᠭ ᠪᠣᠯ ᠨᠢᠭᠡᠨ ᠳᠤ ᠮᠳ᠋ᠠᠳ᠋ ᠤ ᠰ ᠤᠷᠭᠠᠭᠤᠯ ᠢ ᠢᠢ ᠨ ᠰ ᠤᠷᠤᠭᠴ ᠢ ᠢᠢ ᠨ ᠰ ᠢᠷᠡᠮᠦᠨ ᠪᠦᠮᠪ ᠦᠭᠡ ᠰᠢᠷᠲ ᠡᠭᠰᠡᠨ ᠵᠢᠷᠤᠭ ᠪᠣᠯᠤ ᠨ ᠠ ᠂ ᠂ ᠰᠢᠷᠡᠮᠦ ᠨ ᠪ ᠦᠮᠪᠦ ᠭᠡ ᠨᠢ A ᠴᠡᠭ ᠦᠨ ᠦᠶ ᠡ ᠳ᠋ ᠦ ᠭ ᠠᠷ ᠡ ᠴᠡ ᠰᠠᠯᠵᠤ ᠂ B ᠴᠡᠭ ᠳᠡᠭᠡ ᠷ ᠡ ᠭᠠᠵᠠ ᠷ ᠲ ᠦ ᠤ ᠨᠠᠨ ᠠ ᠃ ᠰᠢ ᠷᠡᠮᠦᠨ ᠪᠦᠮ ᠪᠦᠭᠡ ᠢᠢᠨ ᠳᠡᠭᠡ ᠭᠰᠢᠯ ᠡᠭᠥ ᠦᠨ ᠳ᠋ᠦᠷ y( ᠨᠢᠭᠡᠴ ᠢ ᠄ m ) ᠪᠣ ᠯᠤᠨ ᠳᠦ᠋ ᠪᠰᠢᠨ ᠵᠠᠢ 𝑥( ᠨᠢᠭᠡᠴ ᠢ ᠄ m) ᠦᠨ ᠬᠣᠭ ᠣᠷᠤᠨᠳ᠋ ᠤᠬᠢ ᠬᠣᠯ ᠪᠤᠭᠳ᠋ ᠠᠯ ᠨ ᠢ y= − 112 𝑥 2+ 23 𝑥 + 53 ᠃ (1) ᠺᠦ ᠧᠠᠳ᠋ᠯ ᠠᠬᠤ ᠠᠷᠭ ᠠ ᠪᠠ ᠷ y = − 112 𝑥 2+ 23 𝑥 + 53 ᠢᠢ ᠬᠤᠪᠢ ᠷᠠᠭᠠᠭ ᠠᠳ y = 𝑎 (𝑥 − ℎ) 2 + 𝑘 ᠢᠢᠨ ᠬᠡᠯᠪᠡᠷᠢ ᠳ᠋ᠡᠢ ᠪᠣᠯᠭ ᠠ ᠃ (2) ᠰᠢ ᠷᠡᠮᠦᠨ ᠪᠦᠮ ᠪᠦᠭᠡ ᠰ ᠢᠲᠡᠭ ᠳ᠋ᠡᠭᠰ ᠡᠨ ᠦ ᠰᠡᠭᠦ ᠯᠡᠷ ᠭᠠᠵᠠ ᠷ ᠦᠨ ᠭᠠᠳ᠋ ᠠᠷᠭᠤ ᠡᠴ ᠠ ᠵ ᠠᠶᠢᠳ᠋ ᠠᠭᠰᠠᠨ ᠵᠠᠢ ᠢ ᠢᠨ ᠬᠠᠮᠤ ᠭ ᠦᠨ ᠶ ᠡᠬᠡ ᠬ ᠡᠮᠵᠢᠭ ᠳ᠋ᠡᠯ ᠨ ᠢ ᠬᠡ ᠳ᠋ᠦᠢ ᠪᠤᠢ ︖ 22. (10 ᠬᠤᠪᠢ ) ᠵ ᠢᠷᠤᠭ ᠮᠡᠲ ᠦ ᠂ ᠳᠡᠭᠰ ᠢ ᠬᠡᠮ ᠦᠨ ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠨᠢ ᠰᠢᠯᠤ ᠭᠤᠨ 𝑥 = −1 ᠪᠣᠯᠬᠤ ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ y = a𝑥 2+ 𝑏𝑥 + 𝑐 ( 𝑎≠ 0) ᠨᠢ 𝑥 ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠲᠠᠢ A , B ᠬᠣ ᠶᠠᠷ ᠴᠡᠭ ᠲᠦ ᠣᠭᠲᠣᠯ ᠣᠯᠴᠠᠨ ᠠ ᠂ ᠡᠭᠦᠨ ᠳ᠋ᠦ A ᠴᠡᠭ ᠦᠨ ᠪᠠᠶᠢ ᠷᠢᠰᠢ ᠯ ᠨ ᠢ ( −3 ,0 ) ᠪᠦᠭᠡᠳ ᠴᠡᠭ (2 ,5 ) ᠨᠢ ᠬᠠᠶ ᠠᠯᠲᠠ ᠢᠢᠨ ᠱᠤ ᠭᠤᠮ ᠳᠡᠭᠡᠷ ᠡ ᠣ ᠷᠣᠰᠢ ᠨ ᠠ ᠃ (1) ᠬᠠᠶ ᠠᠯᠲᠠ ᠢᠢᠨ ᠱᠤ ᠭᠤᠮ ᠦᠨ ᠠ ᠨᠠᠯᠢᠰ ᠢᠯ ᠡᠷᠬᠡᠶ ᠢᠯᠡᠯ ᠢ ᠣ ᠯ ᠃ (2) ᠴᠡᠭ C ᠨᠢ ᠬ ᠠᠶᠠᠯᠲ ᠠ ᠢ ᠢᠨ ᠱᠤᠭᠤ ᠮ ᠪᠠ y ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠦᠨ ᠣᠭᠲᠣᠯ ᠣᠯᠴᠠᠯ ᠦᠨ ᠴᠡᠭ ᠪᠣᠯ ᠂ ① ᠬ ᠠᠶᠠᠯᠲ ᠠ ᠢᠢ ᠨ ᠱ ᠤᠭᠤᠮ ᠳᠡᠭᠡ ᠷ ᠡ ᠨᠢᠭᠡ ᠴᠡᠭ P ᠣ ᠷᠣᠰᠢ ᠬᠤ ᠪᠦᠭᠡᠳ 𝑆 ∆𝑃 𝑂𝐶 = 4𝑆 ∆𝐵 𝑂𝐶 ᠪᠣᠯᠤ ᠨ ᠠ ᠂ P ᠴᠡᠭ ᠦᠨ ᠪᠠᠶᠢ ᠷᠢᠰᠢ ᠯ ᠢ ᠣᠯ ᠃ ② ᠴ ᠡᠭ Q ᠨᠢ ᠬᠡᠷᠴ ᠢᠮ AC ᠳᠡᠭᠡ ᠷᠡᠬᠢ ᠨᠢ ᠭᠡᠨ ᠬᠡᠦᠳ᠋ ᠡᠶᠢᠯᠡᠬ ᠦ ᠴᠡᠭ ᠂ QD ⊥ 𝑥 ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠢᠢᠡᠷ ᠪᠠᠶᠢ ᠭᠤᠯᠪᠠ ᠯ ᠬ ᠠᠶᠠᠯᠲ ᠠ ᠢ ᠢᠨ ᠱᠤᠭᠤ ᠮ ᠲ ᠠᠢ D ᠴᠡᠭ ᠲᠦ ᠣᠭᠲᠣᠯ ᠣᠯᠴᠠᠨ ᠠ ᠃ ᠬᠡᠷᠴ ᠢᠮ QD ᠢᠢᠨ ᠬᠠᠮᠤ ᠭ ᠦᠨ ᠶ ᠠᠬᠡ ᠬ ᠡᠮᠵᠢᠭ ᠳ᠋ᠡᠯ ᠢ ᠣ ᠯ ᠃ 23. (10 ᠬᠤᠪᠢ ) ᠵ ᠢᠷᠤᠭ ᠮᠡᠲ ᠦ ᠂ ᠬᠣᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦ ᠩᠺᠽ ᠦᠨ ᠵᠢ ᠷᠤᠮᠠᠭ ᠨᠢ 𝑥 ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠲᠠᠢ A( −3 ,0 ) , 𝐵( 1 , 0) ᠬ ᠣᠶᠠᠷ ᠴᠡᠭ ᠲᠦ ᠣᠭᠲᠣᠯ ᠣᠯᠴᠠᠵᠤ ᠂ y ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠲᠠᠢ C( 0 , 3) ᠳ᠋ᠤ ᠣᠭᠲᠤᠯ ᠤᠯᠴᠠᠨ ᠠ ᠃ ᠴᠡᠭ C ᠪᠠ ᠴᠡᠭ D ᠨᠢ ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ ᠦᠨ ᠵᠢᠷᠤᠮᠠ ᠭ ᠳᠡᠭᠡ ᠷᠡᠬᠢ ᠬᠠᠶ ᠠᠯᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ ᠦᠨ ᠳᠡᠭᠰ ᠢ ᠬᠡᠮ ᠦᠨ ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠦᠨ ᠬᠤᠪ ᠢ ᠳ᠋ ᠤ ᠳᠡᠭ ᠰᠢ ᠬᠡᠮᠲ ᠦ ᠴ ᠡᠭ ᠂ ᠨᠢᠭᠡᠳ᠋ ᠦᠭᠡᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦ ᠩᠺᠽ ᠦᠨ ᠵᠢᠷᠤᠮ ᠠᠭ ᠨ ᠢ ᠴᠡᠭ B , D ᠢᠢ ᠳᠠᠭᠠ ᠷᠢᠨ ᠠ ᠃ (1) ᠴᠡᠭ D ᠦᠨ ᠪᠠ ᠶᠢᠷᠢ ᠰᠢᠯ ᠢ ᠰ ᠢᠭᠤᠳ ᠪᠢᠴ ᠢ ᠃ (2) ᠬᠣ ᠶᠠᠳ᠋ᠤ ᠭᠠᠷ ᠵ ᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦᠩ ᠺᠽ ᠦᠨ ᠠ ᠨᠠᠯᠢᠰ ᠢᠯ ᠡᠷᠬᠡᠶ ᠢᠯᠡᠯ ᠢ ᠣ ᠯ ᠃ (3) ᠵᠢᠷᠤᠮᠠᠭ ᠢ ᠦᠨᠳ᠋ ᠦᠰᠦᠯ ᠡᠵᠤ ᠂ ᠨᠢ ᠭᠡᠳ᠋ᠦᠭᠡᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦ ᠩᠺᠽ ᠦᠨ ᠬᠡᠮᠵᠢ ᠭᠳ᠋ᠡᠯ ᠨᠢ ᠬ ᠣᠶᠠᠳ᠋ ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦ ᠩᠺᠽ ᠦᠨ ᠬᠡᠮᠵ ᠢᠭᠳ᠋ᠡᠯ ᠡᠴᠡ ᠶ ᠡᠬᠡ ᠪ ᠣᠯᠬᠤ x ᠦᠨ ᠬ ᠡᠮᠵᠢᠭ ᠳ᠋ᠡᠯ ᠠ ᠪᠬᠤ ᠬ ᠡᠪᠴᠢ ᠶ ᠡ ᠢ ᠢ ᠪᠢ ᠴᠢ ᠃ 24. (10 ᠬᠤᠪᠢ ) ᠵ ᠢᠷᠤᠭ ᠮᠡᠳ᠋ ᠤ ᠂ ᠬᠤᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠠᠷᠭᠡ ᠢᠢᠨ ᠹᠥ ᠩᠺᠼ y ＝𝑥 2 ＋b𝑥 ＋c ᠨ ᠢ ᠴᠡᠭ A( 1,0) , C (0 , －3) ᠵᠢ ᠲᠠᠭᠠᠷ ᠢᠨ ᠠ ᠃ (1) ᠲᠤ ᠰ ᠬᠤ ᠶᠠᠳ᠋ᠤ ᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠥᠩ ᠺᠼ ᠦᠨ ᠠ ᠨᠠᠯᠢᠰ ᠢᠯ ᠡᠷᠭᠡᠶ ᠢᠯᠡᠯ ᠢ ᠤᠯ ᠃ (2) ᠤᠷᠤᠢ ᠢᠢᠨ ᠴᠡᠭ ᠦᠨ ᠪᠠ ᠶᠢᠷᠢ ᠰᠢᠯ ᠢ ᠣ ᠯ ᠃ (3) ᠬᠠᠶ ᠠᠯᠳ᠋ᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ ᠲ ᠡᠭᠡᠷ ᠡ ᠨᠢᠭᠡᠨ ᠴᠡᠭ P ᠤ ᠷᠤᠰᠢ ᠵᠤ ΔAB P ᠢ ᠢᠨ ᠲᠠᠯᠠᠪᠡᠢ ᠵᠢ 10 ᠪᠤᠯ ᠭᠠᠨ ᠠ ᠂ ᠴᠡᠭ P ᠢ ᠢᠨ ᠪ ᠠᠶᠢᠷ ᠢᠰᠢᠯ ᠢ ᠤᠯ ᠃ 25. (10 ᠬᠤᠪᠢ ) ᠵᠢ ᠷᠤᠭ ᠮᠡᠲ ᠦ ᠂ ᠬᠠᠪᠲ ᠠᠭᠠᠢ ᠢᠢᠨ ᠳᠡᠭᠰ ᠢ ᠦᠨᠴ ᠦᠭᠲᠦ ᠪᠠᠶ ᠢᠷᠢᠰ ᠢᠯ ᠦᠨ ᠰᠢᠰ ᠲ᠋ᠧᠮ ᠳᠡᠭᠡ ᠷ ᠡ ᠱᠠᠲᠤ ᠯᠵᠢᠨ AO BC (O ᠨᠢ ᠡᠬᠢᠨ ᠦ ᠴᠡᠭ ) ᠳ᠋ᠤ ᠂ AC ∥ OB , AO ⊥ OB , AC = 1 , OA = 2 , OB = 5᠃ (1) O , C , B ᠭᠤᠷᠪᠠ ᠨ ᠴ ᠡᠭ ᠢ ᠳᠠᠭᠠᠷᠢ ᠬᠤ ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ ᠦᠨ ᠠᠨᠠᠯ ᠢᠰ ᠢᠯᠡᠷᠬ ᠡᠶᠢᠯᠡᠯ ᠢ ᠣ ᠯ ᠃ (2) AC ᠢᠢ ᠦᠷᠭᠦ ᠯᠵᠢᠯ ᠡᠭᠦᠯᠵᠤ᠂ ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ ᠲᠠᠢ D ᠴᠡᠭ ᠲᠦ ᠣᠭᠲᠣᠯ ᠣᠯᠴᠠᠭᠤ ᠯᠪᠠᠯ ᠂ CD ᠬᠡᠷᠴ ᠢᠮ ᠦᠨ ᠤ ᠷᠲᠤ ᠢᠢ ᠣ ᠯ ᠃ (3) (2) ᠦᠨ ᠨᠦ᠋ᠬ ᠦᠴᠡᠯ ᠳᠣᠣ ᠷ ᠠ ᠂ ᠬᠦᠳ᠋ ᠡᠯᠬᠦ ᠴᠡᠭ P , Q ᠨᠢ ᠲ ᠤᠰ ᠪᠦᠷᠢ O , D ᠴᠡᠭ ᠡᠴᠡ ᠰᠧᠺᠦ ᠨᠳ᠋ ᠪᠦᠷᠢ 1 ᠨ ᠢᠭᠡᠴ ᠢ ᠢᠢ ᠨ ᠬ ᠤᠷᠳ᠋ᠤ ᠴᠠ ᠪᠠᠷ ᠬᠦᠳ᠋ ᠡᠯᠭᠡᠭ ᠡᠨ ᠬ ᠢᠨ ᠡ ᠂ ᠡ ᠭᠦᠨ ᠳ᠋ᠦ ᠴᠡᠭ P ᠨᠢ O → B ᠲᠡᠶᠢ ᠰᠢ ᠬᠦᠳ᠋ ᠡᠯᠭᠡᠭ ᠡᠨ ᠬᠢᠵᠤ ᠂ ᠴ ᠡᠭ Q ᠨᠢ D → C ᠲᠡᠶᠢ ᠰᠢ ᠬᠦᠳ᠋ ᠡᠯᠭᠡᠭ ᠡᠨ ᠬᠢᠨ ᠡ ( ᠨᠢᠭᠡ ᠴᠡᠭ ᠨᠢ ᠡᠴᠦᠰ ᠦᠨ ᠴᠡᠭ ᠳᠡᠭᠡ ᠷ ᠡ ᠬᠦᠷᠪᠡᠯ ᠨᠦ᠋ ᠭᠦᠭᠡ ᠴᠡᠭ ᠨᠢ ᠬᠦᠳ᠋ ᠡᠯᠭᠡᠭ ᠡᠨ ᠬᠢᠬᠦ ᠪᠡᠨ ᠵᠣᠭᠰᠤ ᠨ ᠠ ) ᠴ ᠡᠭ Q ᠢᠢ ᠳᠠᠭᠠᠷᠢ ᠭᠤᠯᠤ ᠨ QM ⊥ CD ᠪᠠ ᠷ ᠪᠠ ᠶᠢᠭᠤ ᠯᠪᠠᠯ ᠂ BC ᠲ ᠠᠢ M ᠴᠡᠭ ᠲ ᠦ ᠣ ᠭᠲᠣ ᠯᠣᠯᠴ ᠠᠵᠤ ᠂ PM ᠢᠢ ᠬᠣᠯ ᠪᠤᠨ ᠠ ᠃ ᠬᠦᠳ᠋ ᠡᠯᠬᠦ ᠴᠡᠭ ᠦᠨ ᠬᠦᠳ᠋ ᠡᠯᠭᠡᠭ ᠡᠨ ᠬᠢᠭᠰ ᠡᠨ ᠬᠤᠭᠤ ᠴᠠᠭ ᠠ ᠢᠢ t ᠰ ᠧᠺᠦ ᠨᠳ᠋ ᠭᠡᠪᠡᠯ ᠂ t ᠶᠠᠮᠠᠷ ᠬᠡᠮᠵ ᠢᠭᠳ᠋ᠡᠯ ᠢ ᠠ ᠪᠬᠤ ᠦᠶ ᠡᠰ ∆ PM B ᠳ᠋ᠤ ᠨ ᠢᠭᠡ ᠦᠨ ᠴᠦᠭ ᠨᠢ ᠳᠡᠭ ᠰᠢ ᠦᠨᠴ ᠦᠭ ᠪ ᠣᠯᠬᠤ ᠪᠤᠢ ︖