ᠬᠤᠷᠢᠨ ᠬᠤᠶᠠᠳᠤᠭᠠᠷ ᠪᠦᠯᠦᠭ ᠬᠤᠶᠠᠳᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠶᠢᠨ ᠹᠦᠩᠺᠼ （ᠨᠢᠭᠡᠴᠢ ᠶᠢᠨ ᠪᠠᠢᠴᠠᠭᠠᠯᠲᠠ 1） .pdf

ᠬ ᠣᠷᠢᠨ ᠬᠣᠶᠠ ᠳ᠋ᠤᠭ ᠠᠷ ᠪᠥ ᠯᠥᠭ ᠦᠨ 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. ᠬᠣᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦ ᠩᠺᠽ 𝑦 = 𝑎𝑥 2+ 𝑏𝑥 + 𝑐 ᠢᠢᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠨᠢ ᠵᠢᠷᠤᠭ 1 ᠮ ᠡᠲᠦ ᠪᠣᠯ ᠂ ᠨᠢ ᠭᠡᠳ᠋ᠦᠭᠡᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦᠩ ᠺᠽ 𝑦= b𝑥 + 𝑎 ᠢ ᠢᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠨᠢ ( ) ᠮᠦ᠋ ᠴᠡ ᠢᠢ ᠳᠠᠭᠠᠷ ᠢᠬᠤ ᠦᠭᠡᠢ ᠃ A. ᠨᠢᠭᠡᠳ᠋ ᠦᠭᠡᠷ ᠮ ᠦ᠋ᠴᠡ B. ᠬᠣ ᠶᠠᠳ᠋ᠤ ᠭᠠᠷ ᠮᠦ᠋ᠴ ᠡ C. ᠭᠤ ᠷᠪᠠᠳ᠋ ᠤᠭᠠᠷ ᠮᠦ᠋ᠴ ᠡ D. ᠳᠦ᠋ ᠷᠪᠡᠳ᠋ ᠦᠭᠡᠷ ᠮ ᠦ᠋ᠴᠡ 2. ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ y = (𝑥 + 2) 2+ 6 ᠦᠨ ᠳᠣᠬᠠᠢ ᠬᠡᠯᠡᠯ ᠭᠡ ᠡᠴᠡ ᠵᠦ᠋ᠪ ᠪᠣᠯᠬᠤ ᠨᠢ ( ) A. ᠬᠠᠶ ᠠᠯᠲᠠ ᠢᠢᠨ ᠱᠤ ᠭᠤᠮ ᠦᠨ ᠠ ᠩᠭᠠ ᠶᠢᠯᠲ ᠠ ᠢ ᠢᠨ ᠴ ᠢᠭᠯᠡᠯ ᠨᠢ ᠳᠣ ᠷᠤᠭᠰ ᠢ B. ᠬᠠᠶ ᠠᠯᠲᠠ ᠢᠢᠨ ᠱᠤ ᠭᠤᠮ ᠦᠨ ᠣ ᠷᠣᠢ ᠢᠢᠨ ᠴᠡᠭ ᠦᠨ ᠪᠠ ᠶᠢᠷᠢ ᠰᠢᠯ ᠨ ᠢ ( 2 , 6) C. ᠬᠠᠶ ᠠᠯᠲᠠ ᠢᠢᠨ ᠱᠤ ᠭᠤᠮ ᠦᠨ ᠳᠡᠭ ᠰᠢ ᠬ ᠡᠮ ᠦᠨ ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠨᠢ ᠰ ᠢᠯᠤᠭᠤ ᠨ 𝑥 = 6 D. ᠬᠠᠶ ᠠᠯᠲᠠ ᠢᠢᠨ ᠱᠤ ᠭᠤᠮ ᠨᠢ ᠴ ᠡᠭ ( 0 , 10 ) ᠢᠢ ᠳᠠᠭᠠᠷ ᠢᠨ ᠠ 3. ᠹᠦᠩ ᠺᠽ y= 𝑥 2 + 𝑥 ᠦᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠢ ᠪᠠᠷᠠᠭᠤᠨ ᠰᠢ 𝑎 ( 𝑎 0) ᠨᠢ ᠭᠡᠴᠢ ᠵᠡᠷᠭᠡᠴ ᠡᠭᠡ ᠰᠢᠯᠵ ᠢᠭᠦᠯᠪᠡᠯ ᠂ ᠹᠦᠩ ᠺᠽ 𝑦= 𝑥 2 − 3𝑥 + 2 ᠦᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠢ ᠣ ᠯᠣᠨ ᠠ ᠃ ᠲᠡᠭᠡᠪᠡᠯ 𝑎 ᠢ ᠢᠨ ᠬ ᠡᠮᠵᠢᠭ ᠳ᠋ᠡᠯ ᠨ ᠢ ( ) A. 1 B. 2 C. 3 D. 4 4. ᠨᠢᠭᠡᠨ ᠬᠣ ᠶᠠᠳ᠋ᠤ ᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦᠩ ᠺᠽ ᠦᠨ ᠵᠢᠷᠤᠮ ᠠᠭ ᠦᠨ ᠣᠷᠣᠢ ᠢᠢᠨ ᠴᠡᠭ ᠦᠨ ᠪᠠᠶᠢ ᠷᠢᠰᠢ ᠯ ᠨᠢ (2 ,4 ) ᠪᠦᠭᠡᠳ ᠴ ᠡᠭ ( 0 , −4 ) ᠢᠢ ᠳᠠᠭᠠᠷ ᠢᠨ ᠠ ᠂ ᠲ ᠡᠭᠡᠪᠡᠯ ᠡ ᠨᠡ ᠬᠣᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦᠩ ᠺᠽ ᠦᠨ ᠠ ᠨᠠᠯᠢᠰ ᠢᠯ ᠡᠷᠬᠡᠶ ᠢᠯᠡᠯ ᠨ ᠢ ( ) A. y = −2 (𝑥 + 2) 2+ 4 B. y = −2 (𝑥 − 2) 2+ 4 C. y = 2( 𝑥+ 2) 2− 4 D. y = 2( 𝑥− 2) 2− 4 5. ᠢᠵᠢ ᠯᠢ ᠨ ᠢᠭᠡᠨ ᠬᠠᠪᠲ ᠠᠭᠠᠢ ᠢᠢᠨ ᠳᠡᠭᠰᠢ ᠦᠨ ᠴᠦᠭᠲᠦ ᠪᠠᠶ ᠢᠷᠢᠰ ᠢᠯ ᠦᠨ ᠰ ᠢᠰᠲ᠋ ᠧᠮ ᠳᠡ ᠭᠡᠷ ᠡ ᠨᠢᠭᠡᠳ᠋ ᠦᠭᠡᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦ ᠩᠺᠽ y = −𝑚𝑥 + 𝑛 2 ᠪᠠ ᠬᠣ ᠶᠠᠳ᠋ᠤ ᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦ ᠩᠺᠽ 𝑦 = 𝑥 2 + 𝑚 ᠦᠨ ᠵ ᠢᠷᠤᠮᠠᠭ ᠢ ᠢᠯ ᠡᠷᠬᠡᠶ ᠢᠯᠡᠭᠰ ᠡᠨ ᠡ ᠴᠡ ᠵᠦ᠋ᠪ ᠪᠣᠯ ᠬᠤ ᠨ ᠢ ( ) A. B. C. D. 6. ᠵᠢᠷᠤᠭ 2 ᠮᠡᠲ ᠦ ᠂ ᠣᠷᠣᠢ ᠢᠢᠨ ᠴ ᠡᠭ ᠦᠨ ᠪᠠᠶᠢ ᠷᠢᠰᠢ ᠯ ᠨ ᠢ (− 3 , −6 ) ᠪᠣᠯᠬᠤ ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ y= 𝑎𝑥 2+ 𝑏𝑥 + 𝑐 ᠨᠢ ᠴᠡᠭ (− 1 , −4 ) ᠢ ᠳᠠᠭᠠᠷᠢ ᠨ ᠠ ᠃ ᠲᠡᠭᠡᠪᠡᠯ ᠳᠣᠣ ᠷᠠᠬᠢ ᠬᠡᠯᠡᠯ ᠭᠡ ᠡ ᠴᠡ ᠪ ᠤᠷᠤᠭᠤ ᠪ ᠣᠯᠬᠤ ᠨᠢ ( ) A. 𝑏 2 4𝑎 c B. 𝑎 𝑥 2 + 𝑏𝑥 + 𝑐≥ −6 (2) (1) C. ᠬᠡᠷᠪᠡ ᠴᠡᠭ (− 2 , m) ᠪᠠ ᠴᠡᠭ (− 5 , 𝑛) ᠨᠢ ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ ᠳᠡᠭᠡ ᠷ ᠡ ᠣᠷᠣᠰᠢ ᠪᠠᠯ ᠂ m n D. 𝑥 ᠦᠨ ᠳᠣ ᠬᠠᠢ ᠨᠢᠭᠡ ᠦᠯ ᠦ ᠮᠡ ᠳ᠋ᠡᠭᠳ᠋ ᠡᠭᠴᠢ ᠲᠡᠢ ᠬᠣ ᠶᠠᠳ᠋ᠤ ᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠳᠡᠭᠰᠢ ᠳᠭᠡᠯ 𝑎𝑥 2+ 𝑏𝑥 + 𝑐= −4 ᠦᠨ ᠬ ᠣᠶᠠᠷ ᠢᠵᠠᠭᠤᠷ ᠨ ᠢ − 5 ᠪᠠ −1 7. ᠬᠡᠷᠪᠡ ᠨᠢᠭᠡ ᠬᠣᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦᠩ ᠺᠽ y= 𝑎𝑥 2+ 𝑏𝑥 + 𝑐 (𝑎 0) ᠦᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠨᠢ A( −1 ,𝑛 ) , 𝐵 (3 ,𝑛 ) , 𝐶( 𝑚 + 1 , 𝑦1 ) , 𝐷( 1− 𝑚 ,𝑦 2 ) , 𝐸( 1 , 𝑦3 ) ᠲᠠᠪᠤ ᠨ ᠴ ᠡᠭ ᠢ ᠳᠠᠭᠠᠷ ᠢᠪᠠᠯ ᠂ ᠳᠣᠣ ᠷᠠᠬᠢ ᠡᠴ ᠠ ᠵ ᠦ᠋ᠪ ᠪ ᠣᠯᠬᠤ ᠨᠢ ( ) A. 𝑦 1 𝑦2 𝑦3 B. 𝑦 1 = 𝑦2 𝑦3 C. 𝑦 1 𝑦1 𝑦2 8. ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ y = 𝑥 2 + 2𝑥 + 3 ᠢᠢ ᠵᠡᠷᠭᠡᠴ ᠡᠭᠡ ᠰᠢᠯᠵᠢ ᠭᠦᠯᠦᠭᠰᠡᠨ ᠦ ᠰᠡᠭᠦ ᠯᠡᠷ ᠣᠯᠤ ᠭᠰᠠᠨ ᠬᠠᠶ ᠠᠯᠲᠠ ᠢᠢᠨ ᠱᠤ ᠭᠤᠮ ᠨᠢ y= 𝑥 2 ᠪᠣᠯ᠂ ᠳᠣ ᠣᠷᠠᠬ ᠢ ᠬᠡᠯ ᠡᠯᠭᠡ ᠡᠴ ᠡ ᠵ ᠥᠪ ᠨ ᠢ ( ) A. ᠵᠡᠭᠦ ᠨᠰᠢ 1 ᠨ ᠢᠭᠡᠴ ᠢ ᠰᠢ ᠯᠵᠢᠭ ᠦᠯᠵᠤ ᠂ ᠳᠠᠬ ᠢᠨ ᠳᠡᠭᠡᠭ ᠰᠢ 2 ᠨᠢᠭᠡᠴ ᠢ ᠵᠡᠷᠭᠡᠴ ᠡᠭᠡ ᠰ ᠢᠯᠵᠢ ᠭᠦᠯᠬᠦ B. ᠵᠡᠭᠦ ᠨᠰᠢ 1 ᠨ ᠢᠭᠡᠴ ᠢ ᠰᠢ ᠯᠵᠢᠭ ᠦᠯᠵᠤ ᠂ ᠳᠠᠬ ᠢᠨ ᠳᠣᠷᠣ ᠭᠰᠢ 2 ᠨᠢᠭᠡᠴ ᠢ ᠵᠡᠷᠭᠡᠴ ᠡᠭᠡ ᠰ ᠢᠯᠵᠢ ᠭᠦᠯᠬᠦ C. ᠪᠠᠷᠠᠭᠤᠨᠰᠢ 1 ᠨ ᠢᠭᠡᠴ ᠢ ᠰᠢ ᠯᠵᠢᠭ ᠦᠯᠵᠤ ᠂ ᠳᠠᠬ ᠢᠨ ᠳᠡᠭᠡᠭ ᠰᠢ 2 ᠨᠢᠭᠡᠴ ᠢ ᠵᠡᠷᠭᠡᠴ ᠡᠭᠡ ᠰ ᠢᠯᠵᠢ ᠭᠦᠯᠬᠦ D. ᠪᠠᠷᠠᠭᠤᠨᠰᠢ 1 ᠨ ᠢᠭᠡᠴ ᠢ ᠰᠢ ᠯᠵᠢᠭ ᠦᠯᠵᠤ ᠂ ᠳᠠᠬ ᠢᠨ ᠳᠣᠷᠣ ᠭᠰᠢ 2 ᠨ ᠢᠭᠡᠴ ᠢ ᠵᠡᠷᠭᠡ ᠴᠡᠭᠡ ᠰᠢᠯ ᠵᠢᠭᠦ ᠯᠬᠦ 9. ᠬᠣᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠥ ᠩᠺᠽ 𝑦 = 𝑘𝑥 2− 6𝑥 − 9 ᠦᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠨ ᠢ 𝑥 ᠲ ᠡᠩᠬᠡᠯ ᠢᠭ ᠲᠠᠢ ᠠᠳ᠋ᠠᠯ ᠢ ᠪᠤ ᠰᠤ ᠬᠣᠶ ᠠᠷ ᠣ ᠭᠲᠣᠯ ᠣᠯᠴᠠᠯ ᠦᠨ ᠴᠡᠭ ᠲᠠᠢ ᠪᠣᠯ ᠂ 𝑘 ᠢ ᠢᠨ ᠬᠡᠮᠵ ᠢᠭᠳ᠋ᠡᠯ ᠠᠪᠬᠤ ᠬᠡᠪᠴ ᠢᠶ ᠡ ᠨ ᠢ ( ) A. 𝑘 −1 B. 𝑘 −1 ᠪᠦᠭᠡᠳ 𝑘 ≠ 0 C. 𝑘 ≥ −1 D. 𝑘 ≥ −1 ᠪᠦᠭᠡᠳ 𝑘 ≠ 0 10. ᠬᠣ ᠶᠠᠳ᠋ᠤ ᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦᠩ ᠺᠽ y= 𝑎𝑥 2+ 𝑏𝑥 + 𝑐 ᠦᠨ ᠵᠢᠷᠤᠮᠠ ᠭ ᠨ ᠢ ᠵᠢ ᠷᠤᠭ 3 ᠮ ᠡᠲᠦ ᠂ ᠵ ᠢᠷᠤᠮᠠᠭ ᠨᠢ ᠴᠡᠭ A( −3 ,0 ) ᠢ ᠳᠠᠭᠠᠷᠢ ᠬᠤ ᠪᠦᠭᠡᠳ ᠳᠡᠭ ᠰᠢ ᠬᠡᠮ ᠦᠨ ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠨᠢ ᠰᠢᠯᠤ ᠭᠤᠨ 𝑥 = −1 ᠪᠣ ᠯ ᠂ ᠳ᠋ᠦ᠋ᠩ ᠨᠡᠯᠲᠡ ① 𝑐 0 ②2𝑎 − 𝑏= 0 ③ ᠬᠡ ᠷᠪᠡ ᠴᠡᠭ B( − 32 , 𝑦1 ) , 𝐶( − 52 , 𝑦2 ) ᠨᠢ ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ ᠳᠡᠭᠡ ᠷᠡᠬᠢ ᠬᠣ ᠶᠠᠷ ᠴᠡᠭ ᠪᠣᠯ ᠂ 𝑦1 0 ② ᠳᠡᠭ ᠰᠢᠳ ᠭᠡᠯ 𝑎𝑥 2+ 𝑏𝑥 + 𝑐= 0 ᠦᠨ ᠬᠣᠶ ᠠᠷ ᠢᠵᠠᠭᠤᠷ ᠨᠢ 𝑥1 = −1 , 𝑥 2 = 3 ③ 2 𝑎+ 𝑏= 0 ④ 𝑥 0 ᠦᠶ ᠡᠰ᠂ y ᠨ ᠢ 𝑥 ᠦᠨ ᠨᠡᠮᠡᠭ ᠳ᠋ᠡᠬᠦ ᠢᠢ ᠳᠠᠭᠠᠯ ᠳ᠋ᠤᠨ ᠬᠣ ᠷᠤᠭᠳ᠋ ᠠᠨ ᠠ ᠃ ᠳᠡᠭᠡ ᠷᠡᠬᠢ ᠭᠡᠯ ᠡᠯᠭᠡ ᠡᠴᠡ ᠵᠦ᠋ᠪ ᠪᠣᠯᠬᠤ ᠨᠢ ( ) 18. ᠵᠢᠷᠤᠭ 5 ᠮᠡᠲᠦ ᠂ ᠬ ᠠᠶᠠᠯᠲ ᠠ ᠢ ᠢᠨ ᠱᠤᠭᠤ ᠮ y = −𝑥 2+ 2𝑥 + 3 ᠨ ᠢ y ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠲᠠᠢ ᠴᠡᠭ C ᠳ᠋ᠤ ᠣᠭᠲᠣᠯ ᠣᠯᠴᠠᠵᠤ ᠂ ᠴ ᠡᠭ D ᠢᠢᠨ ᠪᠠᠶᠢ ᠷᠢᠰᠢ ᠯ ᠨᠢ (0 ,1 )᠂ ᠴᠡᠭ P ᠨ ᠢ ᠬ ᠠᠶᠠᠯᠲ ᠠ ᠢᠢ ᠨ ᠱ ᠤᠭᠤᠮ ᠳᠡᠭᠡ ᠷᠡᠬᠢ ᠨᠢᠭᠡᠨ ᠬᠥᠳ᠋ ᠡᠯᠬᠦ ᠴᠡᠭ ᠂ ᠬᠡᠷᠪᠡ ∆PC D ᠨᠢ CD ᠪᠡᠷ ᠰᠠᠭᠤ ᠷᠢ ᠪᠠ ᠨ ᠪ ᠣᠯᠭᠠᠭᠰᠠᠨ ᠠᠳ᠋ᠠᠯ ᠢ ᠬ ᠠᠵᠠᠭᠤᠲ ᠤ ᠭ ᠤᠷᠪᠠ ᠯᠵᠢᠨ ᠪᠣᠯ ᠂ ᠴ ᠡᠭ P ᠦᠨ ᠪᠠ ᠶᠢᠷᠢ ᠰᠢᠯ ᠨ ᠢ ( ) ᠭᠤᠷᠪᠠ᠂ ᠪᠣᠳ᠋ ᠣᠵᠤ ᠬᠠᠷᠢ ᠭᠤᠯᠬᠤ ᠰᠡᠲᠤ ᠪ᠃ ︵ ᠨᠠᠢ ᠲᠡ 66 ᠬᠤ ᠪᠢ ︶ 19. ( 8 ᠬᠤᠪᠢ ) ᠬ ᠣᠶᠠᠳ᠋ ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦ ᠩᠺᠽ y = 34 𝑥 2+ 𝑏𝑥 + 𝑐 ᠦᠨ ᠳᠡᠭᠰ ᠢ ᠬᠡᠮ ᠦᠨ ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠨᠢ ᠰᠢᠯᠤ ᠭᠤᠨ 𝑥 = 1 ᠪᠦᠭᠡᠳ ᠴᠡᠭ (2 ,− 94 ) ᠢᠢ ᠳᠠᠭᠠᠷ ᠢᠨ ᠠ ᠃ ᠲᠤᠰ ᠬᠣ ᠶᠠᠳ᠋ᠤ ᠭᠠᠷ ᠵ ᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦᠩ ᠺᠽ ᠦᠨ ᠠ ᠨᠠᠯᠢᠰ ᠢᠯ ᠡᠷᠬᠡᠶ ᠢᠯᠡᠯ ᠢ ᠣ ᠯ ᠃ 20. ( 8 ᠬᠤ ᠪᠢ) ᠨᠢᠭᠡ ᠵᠦ᠋ ᠢᠯ ᠦᠨ ᠬᠠᠨᠳ᠋ ᠠᠭᠠᠰ ᠤ ᠢᠢ ᠨ ᠣ ᠷᠣᠭᠤ ᠯᠬᠤ ᠦᠨ ᠡ ᠨ ᠢ 40 ᠲᠦ᠋ ᠭᠦᠷᠢ ᠭ ᠂ 60 ᠲᠥ ᠭᠥᠷᠢ ᠭ ᠢᠢ ᠡᠷ ᠬ ᠤᠳ᠋ᠠᠯ ᠳ᠋ᠤᠪᠠ ᠯ ᠨ ᠢᠭᠡ ᠭᠠᠷᠠ ᠭ ᠲ ᠦ 300 ᠵᠠᠬ ᠠ ᠬ ᠤᠳ᠋ᠠᠯ ᠳ᠋ᠤᠵᠤ ᠳᠡᠶᠢ ᠯᠦᠨ ᠡ ᠃ ᠠᠰᠢ ᠭ ᠬᠣᠨᠵᠢ ᠪᠦᠷᠢ ᠪᠠ ᠨ ᠨᠡᠮᠡᠭ ᠳ᠋ᠡᠭᠦ ᠯᠬᠦ ᠢ ᠢᠨ ᠲᠦ᠋ ᠯᠦᠭᠡ ᠬ ᠠᠨᠳ᠋ᠠᠭᠠ ᠰᠤ ᠢ ᠢᠨ ᠬᠤᠳ᠋ ᠠᠯᠳ᠋ᠤ ᠬᠤ ᠦᠨ ᠡ ᠢᠢ ᠦᠰ ᠭᠡᠨ ᠬᠤᠳ᠋ ᠠᠯᠳ᠋ᠤ ᠵᠠᠢ ᠃ ᠵ ᠠᠬ ᠠ ᠳᠡᠯ ᠭᠡᠭᠦᠷ ᠦᠨ ᠪᠠᠶᠢ ᠴᠠᠭᠠᠯ ᠲᠠ ᠢᠢ ᠦᠨ ᠳ᠋ᠦᠰ ᠦᠯᠡᠪᠡᠯ 1 ᠲ ᠥᠭᠥ ᠷᠢᠭ ᠥᠰ ᠭᠡᠬᠦ ᠪᠦᠷᠢ 10 ᠵᠠᠬ ᠠ ᠳᠤ ᠲᠠᠭᠤ ᠬ ᠤᠳ᠋ᠠᠯ ᠳ᠋ᠤᠨ ᠠ ᠃ ᠲᠤᠰ ᠬ ᠠᠨᠳ᠋ᠠᠭᠠᠰ ᠤ ᠢᠢ ᠬ ᠤᠳ᠋ᠠᠯ ᠳ᠋ᠤᠵᠤ ᠣᠯᠤ ᠭᠰᠠᠨ ᠭ ᠠᠷᠠᠭ ᠦᠨ ᠠᠰᠢ ᠭ (4) (5) ᠬᠣᠨᠵᠢ ᠪᠦᠷᠢ ᠢᠢ y ( ᠲ ᠦ᠋ᠭᠦ ᠷᠢᠭ ) ᠪᠠ ᠬᠤ ᠳ᠋ᠠᠯᠳ᠋ ᠤᠬᠤ ᠨᠢᠭᠡᠴ ᠢ ᠦᠨ ᠡ 𝑥 ( ᠲᠦ᠋ ᠭᠦᠷᠢ ᠭ ) ᠦᠨ ᠬᠣᠭᠣ ᠷᠤᠨᠳ᠋ ᠤᠬᠢ ᠹᠦᠩ ᠺᠽ ᠦᠨ ᠬᠣᠯ ᠪᠤᠭᠳ᠋ ᠠᠯ ᠢᠯ ᠡᠷᠬᠡᠶ ᠢᠯᠡᠯ ᠢ ᠣ ᠯᠵᠤ ᠂ ᠬ ᠤᠳ᠋ᠠᠯ ᠳ᠋ᠤᠬᠤ ᠨᠢ ᠭᠡᠴᠢ ᠦᠨ ᠡ ᠨ ᠢ ᠬᠡ ᠳ᠋ᠦᠢ ᠲᠦ᠋ ᠭᠦᠷᠢ ᠭ ᠦᠶ ᠡᠰ ᠠ ᠰᠢᠭ ᠬᠣ ᠨᠵᠢᠪᠦ ᠷᠢ ᠨ ᠢ ᠬᠠᠮᠤ ᠭ ᠶᠡᠬ ᠡ ᠪᠣ ᠯᠬᠤ ᠪᠤᠢ ︖ᠬᠠᠮᠤᠭ ᠶᠡᠬᠡ ᠠᠰ ᠢᠭ ᠬᠣ ᠨᠵᠢᠪᠦ ᠷᠢ ᠨ ᠢ ᠬᠡ ᠳ᠋ᠦᠢ ︖ 21. (10 ᠬᠤᠪᠢ ) ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ y= −𝑥 2+ 𝑏𝑥 + 𝑐 ᠨᠢ x ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠲᠠᠢ A( −1 ,0 ) , B( 3 , 0) ᠬ ᠣᠶᠠᠷ ᠴᠡᠭ ᠲᠦ ᠣᠭᠲᠣᠯ ᠣᠯᠴᠠᠵᠤ ᠂ 𝑦 ᠲ ᠡᠩᠬᠡᠯ ᠢᠭ ᠲᠠᠢ ᠴᠡᠭ C ᠳ᠋ᠤ ᠣᠭᠲᠣᠯ ᠣᠯᠴᠠᠨ ᠠ ᠃ ᠴᠡᠭ C ᠢᠢ ᠳᠠᠭᠠᠷ ᠢᠭᠤᠯ ᠤᠨ CD ∥ 𝑥 ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠢᠢᠡᠷ ᠪᠠᠶᠢ ᠭᠤᠯᠪᠠ ᠯ ᠬᠠᠶ ᠠᠯᠲᠠ ᠢᠢᠨ ᠱᠤ ᠭᠤᠮ ᠲᠠᠢ D ᠴᠡᠭ ᠲᠦ ᠣ ᠭᠲᠣᠯ ᠣᠯᠴᠠᠨ ᠠ ᠃ (1) ᠬᠠᠶ ᠠᠯᠲᠠ ᠢᠢᠨ ᠱᠤ ᠭᠤᠮ ᠦᠨ ᠠ ᠨᠠᠯᠢᠰ ᠢᠯ ᠡᠷᠬᠡᠶ ᠢᠯᠡᠯ ᠢ ᠣ ᠯ ᠃ (2) ᠬᠡᠷ ᠪᠡ ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ ᠢ ᠳᠣᠷᠤᠭ ᠰᠢ 𝑚 ᠨᠢ ᠭᠡᠴᠢ ᠵᠡᠷᠭᠡᠴ ᠡᠭᠡ ᠰᠢᠯᠵᠢ ᠭᠦᠯᠵᠤ ᠂ ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ ᠦᠨ ᠣᠷᠣᠢ ᠢᠢᠨ ᠴᠡᠭ ᠢ ᠴ ᠡᠭ D ᠲᠠᠢ ᠳᠠᠪᠬᠤᠴ ᠠᠭᠤᠯ ᠪᠠᠯ ᠂ 𝑚 ᠢᠢᠨ ᠬᠡᠮᠵ ᠢᠭᠳ᠋ᠡᠯ ᠢ ᠣᠯ ᠃ 22. (10 ᠬᠤᠪᠢ ) ᠬᠣᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦᠩ ᠺᠽ y= −𝑥 2+ 2𝑥 + 𝑚 ᠳ᠋ᠦ ᠂ ( 1) ᠬᠡᠷᠪᠡ ᠬᠣᠶ ᠠᠳ᠋ᠣᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦ ᠩᠺᠽ ᠦᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠨᠢ 𝑥 ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠲᠠᠢ ᠬᠣᠶ ᠠᠷ ᠣᠭᠲᠣᠯ ᠣᠯᠴᠠᠯ ᠦᠨ ᠴ ᠡᠭ ᠲ ᠠᠢ ᠪᠣ ᠯ ᠂ 𝑚 ᠢᠢᠨ ᠬ ᠡᠮᠵᠢᠭ ᠳ᠋ᠡᠯ ᠠ ᠪᠬᠤ ᠬᠡᠪᠴ ᠢᠶ ᠡ ᠢᠢ ᠣ ᠯ ᠃ (2) ᠵᠢᠷᠤ ᠭ ᠮ ᠡᠲᠦ ᠬᠣᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦᠩ ᠺᠽ ᠦᠨ ᠵᠢᠷᠤᠮᠠ ᠭ ᠨ ᠢ ᠴ ᠡᠭ A (3 ,0 ) ᠢ ᠳᠠᠭᠠᠷᠢ ᠵᠤ ᠂ y ᠲᠡ ᠩᠬᠡᠯ ᠢᠭ ᠲᠠᠢ B ᠴᠡᠭ ᠲᠦ ᠣᠭᠲᠣᠯ ᠣᠯᠴᠠᠪᠠ ᠯ ᠰ ᠢᠯᠤᠭ ᠤᠨ AB ᠪᠠ ᠬᠣ ᠶᠠᠳ᠋ᠤ ᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦᠩ ᠺᠽ ᠦᠨ ᠳᠡᠭᠰ ᠢ ᠬᠡᠮ ᠦᠨ ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠦᠨ ᠣᠭᠲᠣᠯ ᠣᠯᠴᠠᠯ ᠦᠨ ᠴᠡᠭ ᠢ P ᠭᠡᠪᠡᠯ ᠂ P ᠴᠡᠭ ᠦᠨ ᠪᠠᠶᠢ ᠷᠢᠰᠢ ᠯ ᠢ ᠣ ᠯ ᠃ 23. ︵ 10 ᠬᠤᠪᠢ ︶ ᠨᠢ ᠭᠡᠨ ᠬᠤᠪᠴᠠᠰᠤᠨ ᠳᠡᠯ ᠭᠡᠭᠦᠷ ᠦᠨ ᠰᠠᠷ ᠠ ᠪᠦᠷ ᠢ ᠢᠢ ᠨ ᠵ ᠠᠬ ᠠ ᠪᠦᠷ ᠢ ᠢᠢ ᠨ ᠬᠤᠳ᠋ ᠠᠯᠳ᠋ᠤ ᠬᠤ ᠦ ᠨ ᠡ ᠪ ᠠ ᠬᠤ ᠳ᠋ᠠᠯᠳ᠋ ᠤᠭᠰᠠᠨ ᠬᠡᠮᠵᠢ ᠶ ᠡ ᠨ ᠢ ᠬᠦᠰ ᠦᠨᠦᠭᠲᠦ ᠳ᠋ᠦ ᠦᠵ ᠡᠭᠦᠯᠦᠭᠰᠡᠨ ᠮᠡᠲ ᠦ ᠄ ᠨᠢᠭᠡᠨᠲᠡ ᠮᠡᠳ᠋ᠡᠭ ᠰᠡᠨ ᠨᠢ ᠲᠤᠰ ᠬᠤ ᠪᠴᠠᠰᠤᠨ ᠤ ᠣ ᠷᠣᠭᠤ ᠯᠬᠤ ᠦᠨ ᠡ ᠨ ᠢ ᠵᠠᠬ ᠠ ᠪ ᠦᠷᠢ ᠳ᠋ᠦ ᠪᠡᠨ 60 ᠲᠦ᠋ ᠭᠦᠷᠢᠭ ᠂ ᠬ ᠤᠳ᠋ᠠᠯ ᠳ᠋ᠤᠬᠤ ᠦᠨ ᠡ ᠢᠢ 𝑥 ᠲᠥ ᠭᠥᠷᠢ ᠭ ᠭᠡᠪᠡᠯ ᠄ (1) 𝑥 ᠠ ᠭᠤᠯᠤ ᠭᠰᠠᠨ ᠢᠯ ᠡᠷᠬᠡᠶ ᠢᠯᠡᠯ ᠢᠢᠡᠷ ᠲᠤᠰ ᠬ ᠤᠪᠴᠠ ᠰᠤᠨ ᠤ ᠵᠠᠬ ᠠ ᠪᠦᠷ ᠢ ᠢᠢ ᠨ ᠠᠰᠢ ᠭ ᠬᠣᠨᠵᠢ ᠪᠦᠷᠢ ᠢᠢ ᠢᠯ ᠡᠷᠬᠡᠶ ᠢᠯᠡᠪᠡᠯ ( ) ᠲᠦ᠋ ᠭᠦᠷᠢ ᠭ ᠂ ᠰᠠᠷ ᠠ ᠢᠢ ᠨ ᠬ ᠤᠳ᠋ᠠᠯ ᠳ᠋ᠤᠬᠤ ᠬᠡᠮᠵᠢ ᠶ ᠡ ᠨ ᠢ ( ) ᠵᠠᠬ ᠠ ᠃ (2) ᠲᠤ ᠰ ᠬ ᠤᠪᠴᠠᠰᠤᠨ ᠳᠡᠯ ᠭᠡᠭᠦᠷ ᠦᠨ ᠰᠠᠷ ᠠ ᠢᠢᠨ ᠠᠰ ᠢᠭ ᠬᠣᠨᠵᠢ ᠪᠦᠷᠢ ᠢᠢ y ᠲᠥ ᠭᠥᠷᠢ ᠭ ᠭ ᠡᠪᠡᠯ ᠂ ᠬᠤᠳ᠋ ᠠᠯᠳ᠋ᠤ ᠬᠤ ᠦᠨ ᠡ ᠨ ᠢ ᠬᠡ ᠳ᠋ᠦᠢ ᠲᠥ ᠭᠥᠷᠢ ᠭ ᠦᠶ ᠡᠰ ᠠ ᠰᠢᠭ ᠬᠣ ᠨᠵᠢᠪᠦ ᠷᠢ ᠨ ᠢ ᠬ ᠠᠮᠤᠭ ᠶᠠᠬᠡ ᠪᠤᠢ ︖ ᠬ ᠠᠮᠤᠭ ᠶᠡᠬᠡ ᠠᠰ ᠢᠭ ᠬᠣ ᠨᠵᠢᠪᠦ ᠷᠢ ᠨ ᠢ ᠬᠡᠳ᠋ ᠦᠢ ᠲ ᠥᠭᠥ ᠷᠢᠭ ᠪᠤᠢ ︖ 24. ︵10 ᠬᠤ ᠪᠢ ︶ ᠬᠣ ᠶᠠᠳ᠋ᠤ ᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦ ᠩᠺᠽ y = 2𝑥 2+ m ᠳ᠋ᠦ ᠂ (1) ᠬᠡᠷᠪᠡ ᠴᠡᠭ (− 2 , 𝑦1 ) , ( 3 , 𝑦2 ) ᠨᠢ ᠲ ᠤᠰ ᠬᠣᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦ ᠩᠺᠽ ᠳᠡᠭᠡ ᠷ ᠡ ᠣ ᠷᠣᠰᠢ ᠪᠠᠯ ᠂ 𝑦1 __ _ 𝑦 2 ( ︽ ︾ ᠲᠡᠮ ᠳ᠋ᠡᠭ ᠢ ᠲ ᠠᠭᠯᠠ ) (2) ᠵ ᠢᠷᠤᠭ ᠮᠡᠲᠦ ᠲᠤᠰ ᠬ ᠣᠶᠠᠳ᠋ ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹ ᠦᠩᠺᠽ ᠦᠨ ᠵᠢᠷᠤᠮᠠ ᠭ ᠨᠢ ᠴᠡᠭ (0 ,− 4) ᠢ ᠳᠠᠭᠠᠷᠢ ᠵᠤ ᠂ ᠳᠡᠭ ᠰᠢ ᠳᠦ᠋ ᠷᠪᠡᠯ ᠵᠢᠨ AB CD ᠢᠢᠨ ᠣ ᠷᠣᠢ ᠢᠢᠨ ᠴᠡᠭ C , D ᠨᠢ ᠬᠤᠳ᠋ ᠠᠯᠳ᠋ᠤ ᠬᠤ ᠦ ᠨ ᠡ ᠬᠤᠳ᠋ ᠠᠯᠳ᠋ᠤ ᠬᠤ ᠬ ᠡᠮᠵᠢᠶ ᠡ 100 200 110 180 120 160 130 140 ᠂ ᠂ ᠂ ᠂ ᠂ ᠂ 𝑥 ᠲ ᠡᠩᠬᠡᠯ ᠢᠭ ᠳᠡᠭᠡ ᠷ ᠡ ᠣᠷᠣᠰᠢ ᠬᠤ ᠪᠦᠭᠡᠳ ᠴᠡᠭ A , B ᠨᠢ ᠶᠠᠭ ᠰᠠᠶᠢ ᠬᠠᠨ ᠬᠣ ᠶᠠᠳ᠋ᠤ ᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦᠩ ᠺᠽ ᠦᠨ ᠵᠢᠷᠤᠮ ᠠᠭ ᠳᠡᠭᠡᠷ ᠡ ᠣ ᠷᠣᠰᠢ ᠪᠠᠯ ᠂ ᠵᠢᠷᠤ ᠭ ᠳ᠋ ᠠᠬᠢ ᠰ ᠡᠭᠦᠳ᠋ ᠡᠷ ᠲ ᠠᠢ ᠭ ᠡᠰᠡᠭ ᠦᠨ ᠲ ᠠᠯᠠᠪᠡᠢ ᠢᠢ ᠣ ᠯ ᠃ 25. (10 ᠬᠤᠪᠢ ) 𝑥 ᠦᠨ ᠳᠣᠬᠠᠢ ᠳᠡᠭᠰ ᠢᠳᠭᠡᠯ 𝑘𝑥 2+ (2 𝑘+ 1) 𝑥+ 2= 0 ᠳ᠋ᠦ ᠂ ( 1) 𝑘 ᠶᠠᠮᠠᠷ ᠬᠡᠮᠵᠢ ᠭᠳ᠋ᠡᠯ ᠠᠪᠤ ᠭᠰᠠᠨ ᠴᠤ ᠳᠡᠭᠰ ᠢᠳᠭᠡᠯ ᠬᠡᠵᠢ ᠶᠡᠳ᠋ᠡ ᠪᠣᠳ᠋ᠠᠲ ᠤ ᠲ ᠣᠭᠠᠨ ᠬᠠᠷᠢ ᠭᠤ ᠲᠠᠢ ᠪᠣᠯᠬᠤ ᠢᠢ ᠪᠠ ᠲᠤᠯᠠ ᠃ (2) ᠬᠠᠶ ᠠᠯᠲᠠ ᠢᠢᠨ ᠱᠤ ᠭᠤᠮ y = 𝑘𝑥 2+ (2 𝑘+ 1) 𝑥+ 2 ᠦᠨ ᠵ ᠢᠷᠤᠮᠠᠭ ᠪᠠ 𝑥 ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠦᠨ ᠬᠣᠶ ᠠᠷ ᠣ ᠭᠲᠣᠯ ᠣᠯᠴᠠᠯ ᠦᠨ ᠴᠡᠭ ᠦᠨ ᠪᠠᠶᠢ ᠷᠢᠰᠢ ᠯ ᠦᠨ ᠬᠦᠨᠳ᠋ ᠡᠯᠡᠨ ᠪᠠᠶ ᠢᠷᠢᠰ ᠢᠯ ᠨ ᠢ ᠴ ᠦ᠋ᠮ ᠪᠦᠬ ᠦᠯᠢ ᠲᠣᠭ ᠠ ᠪ ᠦᠭᠡᠳ 𝑘 ᠨᠢ ᠡᠶᠡᠷᠬ ᠦ ᠪᠦᠬ ᠦᠯᠢ ᠲᠣᠭ ᠠ ᠂ ᠬᠡᠷᠪᠡ ᠴᠡᠭ P( 𝑎 , 𝑦1 ) , 𝑄( 1 , 𝑦2 ) ᠨ ᠢ ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ ᠳᠡᠭᠡ ᠷᠡᠬᠢ ᠬᠣ ᠶᠠᠷ ᠴᠡᠭ ᠪᠦᠭᠡᠳ 𝑦 1 𝑦2 ᠪᠣ ᠯ ᠂ ᠪᠣᠳ᠋ᠠᠲ ᠤ ᠲ ᠣᠭ ᠠ 𝑎 ᠢᠢᠨ ᠬᠡᠮᠵᠢ ᠭᠳ᠋ᠡᠯ ᠠ ᠪᠬᠤ ᠬᠡᠪᠴ ᠢᠶ ᠡ ᠢᠢ ᠣ ᠯ ᠃