2018 ᠣᠨ ᠤ ᠲᠦᠩᠯᠢᠶᠣᠣ ᠬᠤᠲᠠ ᠶᠢᠨ ᠲᠣᠭᠠᠨ ᠤ ᠤᠬᠠᠭᠠᠨ ᠤ ᠳᠤᠮᠳᠠᠳᠤ ᠰᠢᠯᠭᠠᠯᠲᠠ ᠶᠢᠨ ᠰᠡᠲᠤᠪ

20 18 ᠤᠨ ᠤ ᠲᠦ ᠩᠯᠢᠶ ᠤᠤ ᠬ ᠤᠲᠠ ᠢ ᠢᠨ ᠠ ᠩᠬ ᠠ ᠲᠤᠮᠳ ᠠᠳᠤ ᠶᠢᠨ ᠰᠤᠷᠤᠭᠴᠢ ᠳ ᠦ ᠨ ᠰ ᠤᠷᠤᠯ ᠭ ᠠ ᠲᠠᠪᠰᠢᠬ ᠤ ᠰᠢ ᠯᠭᠠᠯᠲ ᠠ ᠢᠢᠨ ᠰᠡᠳᠦᠪ ᠲᠤᠭ ᠠᠨ ᠦ ᠤᠬᠠᠭᠠ ᠨ ᠨᠢᠭᠡ᠂ ᠰᠤᠩᠭᠤ ᠬᠤ ᠰᠡ ᠳᠦᠪ ᠃ (ᠰᠡ ᠳᠦᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪᠢ᠂ ᠨᠡᠶᠢᠲᠡ 30 ᠬᠤᠪᠢ) 1. 12018 ᠢᠢᠨ ᠲᠤᠩᠭ ᠤᠷᠤᠤ ᠲᠤᠭ ᠠ ᠨᠢ ( ) A. 2018 B. −2018 C. − 12018 D. 12018 2. ᠴᠠᠭᠠᠰᠤᠨ ᠬᠠᠶᠢ ᠴᠢᠯᠠᠮ ᠠᠯ ᠨᠢ ᠮᠠᠨ ᠤ ᠤᠯ ᠤᠰ ᠦ ᠨ ᠠ ᠷᠠᠳ ᠦᠨ ᠤᠯᠠᠮ ᠵᠢᠯᠠᠯ ᠲᠤ ᠤᠷᠠᠯᠢᠭ ᠶᠤᠮ ᠃ ᠲᠠ ᠷᠠᠭᠠᠬ ᠢ ᠴᠠ ᠭᠠᠰᠤᠨ ᠬᠠᠶᠢ ᠴᠢᠯᠠᠮ ᠠᠯ ᠳ᠋ᠤ ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠲᠦ ᠲᠡᠭᠰᠢ ᠬᠡᠮᠲᠡᠢ ᠪᠢᠰ ᠢ ᠮᠦ ᠷᠲᠡᠭᠡᠨ ᠪᠠᠰᠠ ᠲᠦ ᠪ ᠲᠦ ᠲᠡᠭ ᠰᠢ ᠬᠡ ᠮᠲᠡᠢ ᠪᠢᠰ ᠢ ᠵᠢᠷ ᠤᠭ ᠲᠦᠷᠰᠦ ᠨᠢ ( ) A. B. C. D . 3. ᠲᠠᠷᠠᠭ ᠠᠬᠢ ᠬᠡᠯᠡᠯᠭᠡ ᠳ᠋ᠤ ᠪᠤᠷᠤᠭ ᠤ ᠪᠤᠯ ᠬᠤ ᠨᠢ ( ) A. ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠵ ᠢᠯᠲᠡ ᠪᠤᠶ ᠤ ᠡᠷ ᠭᠢᠯᠲᠡ ᠡᠴᠠ ᠤᠯᠤᠭ ᠰᠠᠨ ᠵᠢᠷᠤᠭ ᠲᠦᠷᠰᠦ ᠨᠢ ᠤᠭ ᠵᠢᠷᠤᠭ ᠲᠠᠢ ᠪᠠᠨ ᠪᠦᠷᠢᠨ ᠲᠡᠩ ᠴᠡᠭᠦᠤ B. ︽ᠮᠦᠷᠭᠦᠯᠲ ᠦᠭᠡ ᠦᠨᠴ ᠦᠭ ᠲᠡᠩ ᠴᠡᠭᠦᠤ ︾ ᠶᠢᠨ ᠤᠷᠪᠠ ᠭᠤ ᠦ ᠭᠦᠯᠡᠯ ᠨᠢ ᠵᠢᠩᠭᠢᠨᠢ ᠦᠭᠦᠯ ᠡᠯ ᠮ ᠦᠨ C. ᠲᠤᠭ ᠤᠷᠢᠭ ᠲᠦ ᠪᠠ ᠭᠲᠠᠭᠰᠠᠨ ᠵᠦᠪ ᠵ ᠢᠷᠭᠤᠭᠠ ᠨ ᠦᠨᠴ ᠦᠭᠲᠦ ᠢᠢᠨ ᠲᠠᠯ ᠠ ᠢᠢᠨ ᠤᠷ ᠲᠤ ᠨᠢ ᠲᠤᠰ ᠲᠤᠭᠤᠷ ᠢᠭ ᠦ ᠨ ᠷᠠ ᠳᠢᠤᠰ ᠲᠠᠢ ᠲᠡᠩ ᠴᠡᠭᠦᠤ D. ︽ᠵᠠᠮ ᠬᠠᠷ ᠢᠯᠴᠠᠭ ᠠᠨ ᠤ ᠲᠤᠬᠢᠶ ᠠᠨ ᠳ᠋ᠧᠩ ᠲᠠ ᠢ ᠵᠠ ᠮ ᠢᠢᠠᠷ ᠦᠩ ᠭᠡᠷᠡᠭᠦ ᠳ᠋ᠥ ᠤᠯᠠᠭ ᠠᠨ ᠳ᠋ᠧᠩ ᠲᠤᠬ ᠢᠶᠠᠯᠳᠤᠬ ᠤ ︾ᠪᠤᠯ ᠲᠠᠰᠢᠷᠠ ᠮ ᠦᠨ ᠶᠠᠪᠤᠳᠠᠯ ᠶᠤᠮ 4. ᠴᠤᠭᠲᠤ ᠭᠡᠷ ᠡᠴᠠ ᠪᠠᠨ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠳ᠋ᠥ ᠤᠴᠢ ᠬᠤ ᠳ᠋ᠥ ᠤᠷᠢᠳ ᠢᠢᠠ ᠷ ᠵᠢᠭ ᠳᠡ ᠬᠤᠷᠳᠤᠴᠠ ᠪᠠᠷ ᠠᠯ ᠬᠤᠨ ᠨᠡᠶᠢᠲᠡ ᠢᠢᠨ ᠮᠠᠰᠢᠨ ᠲᠡ ᠷᠭᠡᠨ ᠦᠷᠲᠡᠭᠡ ᠨ ᠳ᠋ᠥ ᠭᠦᠷᠴᠡᠢ᠂ ᠬᠡᠳᠦ ᠨ ᠮᠢ ᠨᠦᠲ ᠲᠡᠷᠭᠡ ᠬᠦᠯᠢ ᠶᠡᠭᠰ ᠡᠨ ᠤ ᠲᠠᠷᠠᠭ ᠠ ᠨᠡ ᠶᠢᠲᠡ ᠢᠢᠨ ᠮᠠᠰᠢᠨ ᠲᠡ ᠷᠭᠡᠨ ᠳ᠋ᠥ ᠰᠠᠭ ᠤᠵᠤ᠂ ᠵᠢᠭᠳᠡ ᠬᠤᠷᠳᠤᠴᠠ ᠪᠠᠷ ᠬᠡᠰᠡᠭ ᠬᠤᠭᠤᠴᠠᠭ ᠠ ᠠ ᠶᠠᠯᠠᠭ ᠠᠳ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠳ᠋ᠥ ᠪᠠᠨ ᠬᠦᠷ ᠴᠡᠢ᠃ ᠴᠤᠭ ᠲᠤ ᠢᠢᠨ ᠭᠡᠷ ᠡᠴᠠ ᠪᠠᠨ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠳ᠋ᠥ ᠭᠦᠷᠬ ᠦ ᠳ᠋ᠥ ᠶ ᠠᠪᠤᠭᠰ ᠠᠨ ᠵᠠ ᠮ ᠦᠨ ᠤᠷᠲᠤ s (ᠨᠢᠭᠡ ᠴᠢ᠄ m) ᠬᠢᠭᠡᠳ ᠬᠡᠷ ᠡᠭᠯᠡᠭ ᠰᠡᠨ ᠬᠤᠭᠤᠴᠠᠭ ᠠ t (ᠨᠢᠭᠡ ᠴᠢ᠄ mi n) ᠦ ᠨ ᠬ ᠤᠭᠤᠷ ᠤᠨᠳᠤᠬ ᠢ ᠹ ᠦᠩᠺᠼ ᠦ ᠨ ᠬ ᠤᠯᠪᠤᠭᠳᠠᠯ ᠦᠨ ᠶᠡᠬᠡ ᠲᠦ ᠯᠦᠪ ᠵᠢᠷᠤᠮᠠᠭ ᠨᠢ ( ) A. B . C. D. 5. ᠨᠢᠭᠡᠨ ᠭᠧᠤᠮᠧᠲ᠋ᠷ ᠢ ᠢᠢᠨ ᠪᠡᠶ ᠡ ᠢᠢᠨ ᠭᠤᠷᠪ ᠠᠨ ᠬᠠᠷ ᠠᠭᠠᠨ ᠵᠢᠷᠤᠭ ᠨᠢ ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠦ ᠂ ᠭᠤᠤ ᠯ ᠬᠠᠷᠠᠭ ᠠᠨ ᠵᠢᠷᠤᠭ ᠪᠠ ᠵᠡᠭᠦᠨ ᠬᠠᠷ ᠠᠭᠠᠨ ᠵᠢᠷᠤᠭ ᠨᠢ ᠴᠦᠮ ᠲᠠ ᠯ ᠠ ᠢᠢᠨ ᠤ ᠷᠲᠤ ᠨᠢ 6 ᠪᠤᠯᠬ ᠤ ᠲᠡᠩ ᠴᠡᠭᠦᠤ ᠲᠠᠯᠠᠳᠤ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ᠂ ᠲᠡᠭᠡ ᠷ ᠡ ᠬᠠᠷᠠᠭ ᠠᠨ ᠵᠢᠷ ᠤᠭ ᠨᠢ ᠳ᠋ᠢᠶ ᠠᠮᠧᠲ᠋ᠷ ᠨᠢ 6 ᠪ ᠤᠯᠬᠤ ᠲᠤᠭ ᠤᠷᠢᠭ ᠪᠤᠯᠤᠨ ᠠ᠃ ᠲᠡᠭᠡᠪ ᠡᠯ ᠲᠤᠰ ᠭᠧᠤᠮᠧᠲ᠋ᠷ ᠢ ᠢᠢᠨ ᠪᠡᠶ ᠡ ᠢᠢᠨ ᠪᠦᠬᠦ ᠲᠠᠯ ᠠᠪᠠᠢ ᠨᠢ ( ) A. 18π B. 24π C. 27π D. 42π 6. ᠨᠢᠭᠡᠨ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠨᠤᠮ ᠦᠨ ᠦᠨ ᠦᠷ ᠲᠠ ᠢ ᠰ ᠤᠷᠭᠠᠭ ᠤᠯᠢ ᠢᠢᠨ ᠬᠦ ᠷᠢᠶ ᠡ ᠢᠢ ᠪᠦᠲ ᠦᠭᠡᠨ ᠪᠠᠶᠢᠭ ᠤᠯᠬᠤ ᠢᠢᠨ ᠲᠦ ᠯᠦᠭᠡ ᠨᠢᠭᠡ ᠪᠦᠯᠦᠭ ᠨᠤᠮ ᠲᠡᠪᠲᠡ ᠷ ᠬᠤᠳᠠᠯ ᠳᠤᠨ ᠠᠪᠴᠠᠢ᠃ ᠰᠢᠨᠵ ᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠ ᠨ ᠤ ᠲᠦᠷᠦᠯ ᠦᠨ ᠨᠤᠮ ᠢ ᠬᠤᠳᠠᠯ ᠳᠤᠨ ᠠᠪᠬ ᠤ ᠳ᠋ᠥ 10 00 0 ᠲᠦᠭ ᠦᠷᠢᠭ ᠵᠠᠷ ᠤᠴᠠᠭ ᠤᠯᠵᠤ᠂ ᠤᠳᠬ ᠠ ᠵᠤᠬᠢᠶᠠᠯ ᠦ ᠨ ᠲᠦᠷᠦᠯ ᠦᠨ ᠨᠤᠮ ᠢ ᠬᠤᠳ ᠠᠯᠳᠤᠨ ᠠᠪᠬᠤ ᠳ᠋ᠥ 90 00 ᠲᠦᠭ ᠦᠷᠢᠭ ᠵᠠᠷ ᠤᠴᠠᠭ ᠤᠯᠵᠠᠢ᠃ ᠡᠭᠦᠨ ᠳ᠋ᠥ ᠰᠢᠨᠵ ᠢᠯᠡᠬᠦ ᠤᠬᠠᠭᠠ ᠨ ᠤ ᠲᠦᠷᠦᠯ ᠦᠨ ᠨᠤᠮ ᠦᠨ ᠲᠡᠪᠲᠡ ᠷ ᠪᠦᠷᠢ ᠢᠢᠨ ᠲᠤᠮᠳ ᠠᠴᠢ ᠦᠨ ᠡ ᠨᠢ ᠤᠳᠬ ᠠ ᠵᠤᠬᠢᠶᠠᠯ ᠦᠨ ᠲᠦ ᠷᠦᠯ ᠦ ᠨ ᠨᠤ ᠮ ᠦᠨ ᠲᠡ ᠪᠲᠡᠷ ᠪᠦᠷᠢ ᠢᠢ ᠨ ᠲᠤᠮᠳ ᠠᠴᠢ ᠦᠨ ᠡ ᠡᠴᠠ 5 ᠲᠦᠭ ᠦᠷᠢᠭ ᠦᠨ ᠡ ᠲᠠ ᠢ ᠪᠦᠭᠡᠳ ᠬᠤᠳᠠᠯᠳᠤ ᠨ ᠠ ᠪᠤᠭᠰᠠᠨ ᠰᠢᠨᠵᠢᠯ ᠡᠬᠦ ᠤᠬᠠᠭᠠ ᠨ ᠤ ᠲᠦᠷᠦᠯ ᠦ ᠨ ᠨᠤ ᠮ ᠦᠨ ᠲᠤᠭ ᠠ ᠨᠢ ᠤᠳᠬ ᠠ ᠵᠤᠬᠢᠶᠠᠯ ᠦ ᠨ ᠲᠦᠷᠦᠯ ᠦᠨ ᠨᠤᠮ ᠦᠨ ᠲᠤᠭ ᠠ ᠡᠴᠠ 1 00 ᠲᠡ ᠪᠲᠡᠷ ᠲᠤᠲᠠᠭ ᠤ᠃ ᠬᠡᠷᠪᠡ ᠰᠢᠨᠵᠢᠯ ᠡᠬᠦ ᠤᠠᠬᠠᠭ ᠠᠨ ᠤ ᠲᠦ ᠷᠦᠯ ᠦᠨ ᠨᠤᠮ ᠦᠨ ᠲᠡᠪᠲᠡ ᠷ ᠪᠦᠷᠢ ᠢᠢ ᠨ ᠲᠤᠮᠳ ᠠᠴᠢ ᠦ ᠨ ᠡ ᠢᠢ 𝑥 ᠲᠦᠭ ᠦᠷᠢᠭ ᠭᠡᠵ ᠦ ᠵᠢᠱᠢᠪᠡᠯ ᠂ ᠲᠠᠷ ᠠᠭᠠᠬ ᠢ ᠪᠠᠶᠢᠭ ᠤᠯᠤᠭᠰ ᠠᠨ ᠲᠡ ᠭᠰᠢᠳ ᠭᠡᠯ ᠳ᠋ᠥ ᠵ ᠦᠪ ᠨᠢ ( ) A. 10000 𝑥 − 9000𝑥; 5 = 100 B . 9000𝑥; 5 − 10000 𝑥 = 100 C. 10000𝑥 ;5 − 9000𝑥 = 100 D. 9000 𝑥 − 10000𝑥 ;5 = 100 7. ᠦᠭ ᠭᠦᠭᠳᠡᠭ ᠰᠡᠨ ᠨᠢ ⨀O ᠢᠢᠨ ᠷᠠᠳᠢᠤᠰ ᠨᠢ 10 ᠂ ᠲᠤᠭ ᠤᠷᠢᠭ ᠦ ᠨ ᠲᠦᠪ O ᠡᠴᠠ ᠬᠦᠪᠴᠢ AB ᠬᠦ ᠷᠬᠦ ᠵᠠ ᠢ ᠨᠢ 5 ᠪᠤᠯ᠂ ᠬᠦᠪ ᠴᠢ AB ᠢᠢᠨ ᠡᠰᠡᠷ ᠭᠦ ᠤᠷ ᠤᠰᠢᠬ ᠤ ᠪᠠ ᠭᠲᠠᠭ ᠰᠠᠨ ᠦᠨ ᠴᠦᠭ ᠦᠨ ᠭᠷ ᠠᠳᠦᠰ ᠦᠨ ᠲᠤᠭ ᠠ ᠨᠢ ( ) A. 30° B. 60° C. 30° ᠪᠤᠶ ᠤ 150° D . 60° ᠪᠤᠶ ᠤ 120° 8. ᠨᠢᠭᠡᠨ ᠲᠡᠯᠭᠡᠭᠦᠷ ᠲᠠᠧᠠ ᠷ ᠪᠦᠷᠢ 150 ᠲᠦᠭ ᠦᠷᠢᠭ ᠢᠢᠠᠷ ᠠᠳᠠᠯᠢ ᠪᠤᠰᠤ ᠬᠤᠶ ᠠᠷ ᠵᠦᠢᠯ ᠦᠨ ᠲᠠᠧᠠ ᠷ ᠬᠤᠳᠠᠯ ᠳᠤᠨ ᠭᠠ ᠷᠭᠠᠵᠠᠢ᠃ ᠡᠭᠦᠨ ᠳ᠋ᠥ ᠨᠢᠭᠡ ᠲᠠᠧᠠ ᠷ ᠨᠢ 25% ᠪᠠᠷ ᠠᠰᠢᠭ ᠤᠯᠵ ᠤ᠂ ᠨᠢ ᠭᠡ ᠲᠠᠧᠠ ᠷ ᠨᠢ 25% ᠪᠠᠷ ᠪᠡᠯ ᠡ ᠪ ᠠᠨ ᠠ ᠯᠳᠠᠵᠠᠢ ᠃ ᠲᠡ ᠭᠡᠪᠡᠯ ᠲᠤᠰ ᠲᠡᠯᠭᠡᠭᠦᠷ ᠡᠨᠡ ᠬᠤᠶ ᠠᠷ ᠭᠣᠣᠯ ᠬᠠᠷᠠ ᠭᠠᠨ ᠵᠢᠷᠤᠭ ᠵᠡᠭᠦᠨ ᠬᠠᠷᠠ ᠭᠠᠨ ᠵᠢᠷᠤᠭ ᠳᠡᠭᠡᠷ ᠡ ᠬᠠᠷᠠ ᠭᠠᠨ ᠵᠢᠷᠤᠭ ᠲᠠᠧᠠᠷ ᠢ ᠬᠤᠳᠠᠯ ᠳᠤᠭᠠ ᠳ ᠶᠡᠷ ᠥᠩᠬᠡ ᠢ ᠤ ᠯᠵᠤ ᠠ ᠯᠳᠠᠭᠰᠠᠨ ᠪᠠᠶᠢᠳ ᠠᠯ ᠨᠢ ( ) A. 20 ᠲᠦᠭ ᠦᠷᠢᠭ ᠦᠨ ᠪᠠᠯᠡ ᠠᠯ ᠳᠠᠵᠠᠢ B. 30 ᠲᠦᠭ ᠦᠷᠢᠭ ᠦᠨ ᠠᠰᠢᠭ ᠤᠯ ᠵᠠᠢ C. 50 ᠲᠦᠭ ᠦᠷᠢᠭ ᠦᠨ ᠪᠠᠯᠡ ᠠᠯᠳᠠ ᠵᠠᠢ D. ᠠᠰᠢᠭ ᠤᠯᠤᠭ ᠰᠠᠨ ᠦᠭᠡᠢ ᠠᠯᠳᠠᠭ ᠰᠠᠨ ᠦᠭᠡᠢ 9. ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ y = 𝑥 2 + 2𝑥 + 𝑘+ 1 ᠶᠢᠨ ᠵᠢᠷ ᠤᠮᠠᠭ ᠨᠢ 𝑥 ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠲᠠᠢ ᠠᠳᠠᠯᠢ ᠪᠤᠰ ᠤ ᠬᠤᠶ ᠠᠷ ᠴᠡ ᠭ ᠲᠦ ᠤᠭᠲᠤᠯ ᠤᠯᠴᠠᠨ ᠠ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ᠨᠢ ᠭᠡᠳᠦᠭᠡᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠹᠦᠩᠺᠼ y= 𝑘 𝑥 − 𝑘 ᠪᠠ ᠤᠷᠪᠠ ᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠲᠤ ᠹᠦᠩᠺᠼ y= 𝑘𝑥 ᠤᠨ ᠢᠵ ᠢᠯᠢ ᠨᠢᠭᠡᠨ ᠲᠡᠭᠰᠢ ᠦᠨᠴ ᠦᠭᠲᠦ ᠪᠠᠶᠢᠷ ᠢᠰᠢᠯ ᠦᠨ ᠰᠢᠰᠲ᠋ᠧ ᠮ ᠳ᠋ᠥ ᠬᠢ ᠶᠡᠬᠡ ᠲᠦ ᠯᠦᠪ ᠵᠢᠷᠤᠮᠠᠭ ᠨᠢ ( ) A. B. C. D . 10. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠦ᠂ ᠵᠢᠰᠢᠭᠦ ᠯᠵᠢᠨ AB CD ᠢᠢᠨ ᠲᠠᠰᠢᠭᠤ ᠪᠴᠢ AC , BD ᠨᠢ ᠴᠡᠭ O ᠳ᠋ᠥ ᠤᠭᠲᠤ ᠯᠤᠯᠴᠠᠵ ᠤ ∠ADC ᠢᠢᠨ ᠲᠠᠯᠠᠯᠠᠨ ᠬᠤᠪᠢᠶ ᠠᠭᠴᠢ DE ᠨᠢ AB ᠲ ᠠᠢ ᠴᠡ ᠭ E ᠳ᠋ᠥ ᠤᠭᠲ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ OE ᠢᠢ ᠬᠤᠯ ᠪᠤᠨ ᠠ᠃ ᠬᠡᠷ ᠪᠡ ∠B CD = 60° , AD = 12 AB ᠪᠤᠯ᠂ ᠲᠠᠷᠠᠭ ᠠᠬᠢ ᠳ᠋ᠦᠩᠨᠡ ᠯᠲᠡ᠄ ① ᠵᠢᠰᠢᠭᠦᠯ ᠵᠢᠨ AB CD ᠢᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ = AD ∙BD ② DB ᠨᠢ ∠CDE ᠢᠢᠨ ᠲᠠᠯᠠᠯᠠᠨ ᠬᠤᠪᠢ ᠶᠠᠭᠴᠢ ③ AO = DE ④ S∆ ADE = 5S ∆O FE ᠢᠢᠨ ᠵᠦᠪ ᠪᠤ ᠯᠬᠤ ᠪᠤᠳᠠᠭ ᠠᠯᠢ ᠲᠤᠭ ᠠ ᠨᠢ ( ) A. 1 ᠪᠠᠶᠢᠨ ᠠ B . 2 ᠪᠠᠶᠢᠨ ᠠ C. 3 ᠪᠠᠶᠢᠨ ᠠ D. 4 ᠪᠠᠶᠢᠨ ᠠ ᠬᠤᠶ ᠠᠷ᠂ ᠬᠤ ᠭᠤᠰ ᠤᠨ ᠲᠠᠭᠯ ᠠᠬᠤ ᠰᠡᠳ ᠦᠪ᠃ ( ᠰᠡᠳᠦᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪᠢ᠂ ᠨᠡᠶᠢᠲᠡ 21 ᠬᠤᠪᠢ ) 11 . 2 01 8 ᠤᠨ ᠤ 5 ᠰᠠᠷ ᠠ ᠢᠢᠨ 13 ᠤ ᠡᠳᠦ ᠷ ᠮᠠ ᠨ ᠤ ᠤᠯᠤᠰ ᠦᠨ ᠦᠢᠯ ᠡᠳᠪᠦᠷᠢᠯᠡᠭ ᠰᠡᠨ ᠠᠩᠬ ᠠᠳᠤᠭ ᠠᠷ ᠨᠢ ᠰᠭᠡᠯ ᠦ ᠨ ᠰ ᠠᠭᠤᠷᠢ ᠤᠩᠭ ᠤᠴᠠ ᠲᠠᠯᠠᠢ ᠳ᠋ᠥ ᠭᠠᠷ ᠴᠤ ᠲᠤᠷ ᠰᠢᠨ ᠠᠶᠠᠯᠠ ᠵᠠᠢ᠃ ᠡᠨ ᠡ ᠨᠢ ᠮᠠᠨ ᠤ ᠤᠯᠤ ᠰ ᠬᠤᠤ ᠰ ᠨᠢ ᠰᠭᠡᠯ ᠦᠨ ᠰᠠᠭ ᠤᠷᠢ ᠤᠩᠭ ᠤᠴᠠ ᠢᠢᠨ ᠴᠠ ᠭ ᠦᠶ ᠡ ᠳ᠋ᠥ ᠤᠷᠤᠭᠰ ᠠᠨ ᠤ ᠪᠡᠯᠭᠡ ᠲᠡᠮᠲᠡᠭ ᠪᠤᠯ ᠵᠠᠢ᠃ ᠪᠠᠷ ᠤᠭᠴ ᠠᠭᠠᠯᠠᠪᠠ ᠯ ᠢ ᠦᠨᠳᠦ ᠰᠦᠯᠡᠪ ᠡᠯ ᠲᠤᠰ ᠨᠢᠰᠭᠡᠯ ᠦ ᠨ ᠰ ᠠᠭᠤᠷᠢ ᠤᠩᠭ ᠤᠴᠠ ᠢᠢ ᠨ ᠲᠦᠭ ᠦᠷᠡᠩ ᠠᠴᠢ ᠶᠠᠨ ᠤ ᠤᠰᠤ ᠲᠦᠷᠢᠬ ᠦ ᠬᠡᠮᠵᠢᠶ ᠡ ᠨᠢ ᠯᠢᠶ ᠤᠤᠨᠢᠩ ᠬᠦ ᠯᠭᠡ ᠣᠩᠭᠣ ᠴᠠᠨ ᠤ ᠬᠢ ᠲᠠᠢ ᠠᠳᠠ ᠯᠢ ᠪ ᠠᠶᠢᠵᠤ᠂ ᠪᠠᠷᠤ ᠭ 67 50 0 ᠲᠤᠨ᠂ ᠲᠤᠭ ᠠ 6 75 00 ᠢᠢ ᠰᠢᠨᠵᠢ ᠯᠡᠬᠦ ᠤᠬᠠᠭᠠ ᠨᠴᠢ ᠲᠤᠭ ᠠ ᠲᠡ ᠮᠲᠡᠭᠯᠡᠬ ᠦ ᠠ ᠷᠭ ᠠ ᠪᠠᠷ ᠲᠡᠮᠲᠡᠭᠯᠡᠪ ᠡᠯ ( ) ᠃ 12. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠦ᠂ ∠A OB ᠢᠢᠨ ᠨᠢᠭᠡ ᠲᠠᠯ ᠠ OA ᠨᠢ ᠬᠠᠪ ᠲᠠᠭᠠᠢ ᠲᠤᠯ ᠢ᠂ ∠A OB = 37 °45 ′᠂ OB ᠲᠠ ᠯ ᠠ ᠲᠡᠭᠡᠷ ᠡᠬᠢ ᠨᠢᠭᠡᠨ ᠴᠡᠭ E ᠡᠴᠠ ᠰᠠᠴᠤᠷᠠ ᠭᠰᠠᠨ ᠨᠢ ᠭᠡᠨ ᠪᠠᠭᠴᠠ ᠭᠡᠷᠡᠯ ᠱᠤᠭᠤ ᠮ ᠨᠢ ᠬᠠᠪ ᠲᠠᠭᠠᠢ ᠲᠤᠯ ᠢ ᠲᠡ ᠭᠡᠷ ᠡ ᠤᠶᠢᠭᠰᠠᠨ ᠤ ᠲᠠᠷᠠᠭ ᠠᠬᠢ ᠤᠶᠢᠮ ᠠᠯ ᠭᠡᠷ ᠡᠯ ᠱᠤᠭᠤᠮ DC ᠨᠢ ᠶᠠᠭ ᠰᠠᠶᠢᠬ ᠠᠨ OB ᠲᠠ ᠢ ᠵᠡ ᠷᠭᠡᠴᠡᠵ ᠡᠢ᠃ ᠲᠡᠭᠡᠪ ᠡᠯ ∠D EB ᠢᠢᠨ ᠭᠷᠠᠳᠦᠰ ᠦ ᠨ ᠲᠤᠭ ᠠ ᠨᠢ ( )᠃ 13. ᠨᠢᠭᠡ ᠪᠦᠯᠦᠭ ᠲᠤᠭ ᠠ ᠪ ᠠᠷᠢᠮᠲᠠ 2, 𝑥, 1, 3 , 5 , 4 ᠢᠢᠨ ᠲᠤᠮᠳ ᠠ ᠪ ᠠᠶᠢᠷᠢ ᠢᠢᠨ ᠲᠤᠭ ᠠ ᠨᠢ 3 ᠪᠤᠯ᠂ ᠡᠨᠡ ᠪᠦᠯᠦ ᠭ ᠲᠤᠭ ᠠ ᠪᠠᠷᠢᠮᠲᠠ ᠢᠢᠨ ᠰᠠᠷᠨᠢᠴᠠ ᠨᠢ ( )᠃ 14 . ᠲᠠ ᠷᠠᠭᠠᠬ ᠢ ᠵᠢᠷ ᠤᠭ ᠪᠤᠯ ᠭᠤᠷᠪ ᠠᠳᠤᠭ ᠠᠷ ᠵᠠ ᠭᠤᠨ ᠤ ᠦ ᠶ ᠡ ᠢᠢᠨ ᠮᠠᠨ ᠤ ᠤᠯᠤ ᠰ ᠦᠨ ᠬᠠᠨ ᠤᠯᠤᠰ ᠦᠨ ᠦ ᠶ ᠡ ᠢᠢᠨ ᠲᠤᠭᠠ ᠨ ᠤ ᠤᠬᠠᠭᠠ ᠨᠲᠠᠨ ᠵᠤᠤ ᠱᠤᠧᠠ ᠩ ︽ᠵᠧᠤ ᠪᠢ ᠵᠢᠷᠤᠬ ᠠᠢ ︾ᠳᠤ ᠲᠠᠶᠢᠯᠪ ᠤᠷᠢ ᠬᠢᠬᠦ ᠦᠶ ᠡᠰ ᠵᠢᠷ ᠤᠭᠰᠠᠨ ᠵᠢᠷᠤᠭ ᠶᠤᠮ ᠃ ᠬ ᠦᠮᠦ ᠰ ︽ᠵᠤᠤ ᠱᠤᠧᠠᠩ ᠰᠢᠶᠠᠨ ᠵᠢᠷ ᠤᠭ ︾ ᠭᠡᠳᠡ ᠭ᠃ ᠮᠡᠳᠡᠬ ᠦ ᠨᠢ AE = 3, BE = 2 ᠬᠡᠷᠪᠡ ᠲᠡᠭᠰᠢ ᠲᠦᠷᠪ ᠡᠯᠵᠢᠨ AB CD ᠢᠢᠨ ᠲᠤᠲᠤᠷ ᠠ ᠲᠤᠷ ᠠ ᠪᠠᠷ ᠬᠠᠨᠠᠭ ᠤᠷ ᠵᠠᠪᠠ ᠰᠢᠳᠡ ᠪᠡᠯ ︵ᠤᠳᠠ ᠭ ᠠ ᠪᠦᠷᠢ ᠴᠦᠮ ᠲᠡ ᠭᠰᠢ ᠲᠦᠷᠪ ᠡᠯᠵᠢᠨ AB CD ᠲᠤᠲᠤᠷ ᠠ ᠲᠤᠰᠬ ᠤ ᠪᠦᠭ ᠡᠳ ᠲᠡᠭᠰᠢ ᠲᠦ ᠷᠪᠡᠯ ᠵᠢᠨ AB CD ᠲᠤᠲᠤᠷᠠ ᠬᠢ ᠡᠯ ᠡ ᠴᠡᠭ ᠲᠦ ᠲᠤᠰᠬ ᠤ ᠪᠤ ᠯᠤᠯᠴᠠ ᠭᠠᠲᠤ ᠴᠢᠨ ᠠᠷ ᠨᠢ ᠠᠳᠠᠯᠢ ) ᠂ ᠶᠠᠭ ᠰᠠᠶ ᠢᠬᠠᠨ ᠲᠡ ᠭᠰᠢ ᠲᠦ ᠷᠪᠡᠯ ᠵᠢᠨ EF GH ᠲᠤᠲᠤᠷ ᠠ ᠲᠤᠰᠬ ᠤ ᠲ ᠤᠬᠢᠶ ᠠᠯᠳᠤᠮ ᠵᠢ ᠨᠢ ( )᠃ 15. ᠰᠤᠷᠤᠭᠴᠢᠳ ᠦ ᠨ ᠪ ᠡᠶ ᠡ ᠮᠠᠬᠠᠪᠤ ᠳ ᠢ ᠰ ᠠᠶᠢᠵ ᠢᠷᠠᠭ ᠤᠯᠵᠤ᠂ ᠬᠦ ᠯ ᠪᠦᠮ ᠪᠦᠭᠡ ᠢᠰᠭ ᠦᠯᠬᠦ ᠮᠡᠷᠭᠡᠵ ᠢᠯ ᠢ ᠲᠡᠭᠡᠭᠰ ᠢᠯᠡᠭᠦ ᠯᠬᠦ ᠢᠢ ᠨ ᠲᠦ ᠯᠦᠭᠡ᠂ ᠮᠠᠨ ᠤ ᠬᠤᠲ ᠠ ᠡ ᠴᠠ ︽ᠬᠤᠲ ᠠ ᠢᠢᠨ ᠲᠠᠷᠤᠭ ᠠ ᠬᠤᠨᠳᠠ ᠭᠠᠲᠤ ︾ᠬᠦᠯ ᠪᠦᠮᠪᠦᠭ ᠡ ᠢᠢᠨ ᠤᠷ ᠤᠯᠳᠤᠭᠠ ᠨ ᠦᠷᠨ ᠢᠭᠦᠯᠵ ᠡᠢ᠃ ᠤᠷ ᠤᠯᠳᠤᠭᠠ ᠨ ᠨᠢ ᠲᠡ ᠩ ᠡᠭᠡᠯ ᠵᠢᠲᠦ ᠤᠷᠤᠯ ᠳᠤᠭᠠ ᠨ ( ᠬᠤᠶ ᠠᠷ ᠠ ᠩᠭᠢ ᠪᠦᠷ ᠢ ᠢᠢᠨ ᠬᠤᠭᠤᠷ ᠤᠨᠳᠤ ᠴᠦ ᠮ ᠨᠢ ᠭᠡ ᠤᠳᠠᠭ ᠠ ᠤᠷ ᠤᠯᠳᠤᠨ ᠠ ) ᠤ ᠬᠡᠯᠪᠡ ᠷᠢ ᠪ ᠠᠷ ᠶᠠᠪᠤ ᠭᠳᠠᠨ ᠠ᠃ ᠦᠨᠦ 21 ᠲᠠᠯᠠᠪᠤᠷ ᠤᠷᠤᠯ ᠳᠤᠬ ᠤ ᠪᠠ ᠷ ᠲᠤᠯ ᠤᠪᠯᠡᠭ ᠰᠡᠨ ᠪᠤᠯ᠂ ᠬᠡᠳᠦ ᠨ ᠬᠦᠯ ᠪᠦᠮᠪᠦᠭᠡ ᠢᠢᠨ ᠠᠩᠭ ᠢ ᠢᠢ ᠤᠷᠢᠵᠤ ᠤᠷᠤ ᠯᠳᠤᠭᠠ ᠨ ᠳ᠋ᠥ ᠤᠷᠤ ᠯᠴᠠᠭ ᠤᠯᠬᠤ ᠪᠤᠢ ︖ 𝑥 ᠬᠦᠯ ᠪᠦᠮᠪᠦᠭᠡ ᠢᠢᠨ ᠠ ᠩᠭᠢ ᠢᠢ ᠤᠷ ᠢᠵᠤ ᠤᠷᠤᠯ ᠳᠤᠭᠠ ᠨ ᠳ᠋ᠥ ᠤᠷᠤ ᠯᠴᠠᠭ ᠤᠯᠤᠨ ᠠ ᠬᠡ ᠵᠦ ᠵᠢᠱᠢᠭᠡᠳ᠂ ᠰᠡᠳᠦᠪ ᠦᠨ ᠤᠳᠬ ᠠ ᠢᠢ ᠦᠨᠳ ᠦᠰᠦᠯ ᠡᠨ ᠲᠡᠭᠰᠢᠳ ᠭᠡᠯ ᠪᠠᠶᠢᠭ ᠤᠯᠪᠠ ᠯ ( )᠃ 16. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠦ᠂ ∆AB C ᠳ᠋ᠥ ᠲᠠᠷᠠᠭ ᠠᠬᠢ ᠮᠡᠲᠦ ᠠᠯᠬ ᠤᠮ ᠲᠠᠷᠠᠭ ᠠ ᠪ ᠠᠷ ᠵᠢᠷ ᠤᠭ ᠪᠠᠶᠢᠭ ᠤᠯᠤᠨ ᠠ᠄ ① ᠴᠡᠭ A ᠪᠠ C ᠪᠠᠷ ᠲᠤᠰ ᠲᠤᠰ ᠲᠤᠭ ᠤᠷᠢᠭ ᠦ ᠨ ᠲᠦ ᠪ ᠪᠤ ᠯᠭᠠᠵᠤ᠂ 12 AC ᠡᠴᠠ ᠶᠡᠬᠡ ᠤᠷᠲᠤ ᠪᠠᠷ ᠷᠠᠳᠢᠤᠰ ᠪᠤᠯᠭ ᠠᠨᠨ ᠨᠤᠮᠤ ᠪᠠᠶᠢᠭ ᠤᠯᠪᠠ ᠯ ᠬᠤᠶ ᠠᠷ ᠨᠤᠮ ᠤ ᠨᠢ ᠲᠤ ᠰ ᠲᠤᠰ M, N ᠳ᠋ᠥ ᠤᠭᠲ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ ② ᠰᠢᠯ ᠤᠭᠤᠨ MN ᠢ ᠪᠠᠶᠢᠭ ᠤᠯᠪᠠ ᠯ BC ᠲᠠ ᠢ D ᠳ᠋ᠥ ᠤᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠨ ᠠ᠂ AD ᠢᠢ ᠬᠤᠯ ᠪᠤᠨ ᠠ᠃ ᠬᠡᠷᠪᠡ AB = BD ,AB = 6, ∠C = 30° ᠪᠤᠯ ∆A CD ᠢᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ ( )᠃ G FH B C D E A N M DB C A 17. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠦ᠂ ᠬᠠᠪᠲᠠᠭ ᠠᠢ ᠢᠢᠨ ᠲᠡᠭᠰᠢ ᠦᠨ ᠴᠦᠭᠲᠦ ᠪᠠᠶᠢᠷ ᠢᠰᠢᠯ ᠦ ᠨ ᠰ ᠢᠰᠲ᠋ ᠧᠮ ᠳ᠋ᠥ ᠤᠷᠪᠠ ᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠲᠤ ᠹᠦᠩᠺᠼ y= 𝑘𝑥 ( 𝑘 0) ᠵᠢᠨ ᠵᠢᠷᠤᠮᠠ ᠭ ᠨᠢ ᠷᠠᠳᠢᠤᠰ ᠨᠢ 5 ᠪ ᠤᠯᠬᠤ ⨀ O ᠲᠠᠢ ᠲᠤᠰ ᠲᠤᠰ M, N ᠬᠤᠶ ᠠᠷ ᠴᠡ ᠭ ᠲᠦ ᠤᠭᠲᠤᠯ ᠤᠯᠴᠠ ᠨ ᠠ᠃ ∆MON ᠤ ᠲᠠ ᠯᠠᠪᠠᠢ ᠨᠢ 3 .5᠂ ᠬᠡᠷᠪᠡ ᠴᠡᠭ P ᠨᠢ 𝑥 ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠲᠡᠭᠡᠷ ᠡᠬᠢ ᠨᠢᠭᠡᠨ ᠬᠦ ᠳᠡᠯᠬᠦ ᠴᠡ ᠭ ᠪ ᠤᠯ PM + PN ᠤ ᠬᠠᠮᠤᠭ ᠪᠠᠭ ᠠ ᠬᠡ ᠮᠵᠢᠭᠳᠡᠯ ᠨᠢ ( ) ᠃ ᠭᠤᠷᠪ ᠠ᠂ ᠪᠤᠳᠤᠵ ᠤ ᠬᠠᠷ ᠢᠭᠤᠯ ᠬᠤ ᠰᠡ ᠳᠦᠪ ᠃ ( ᠨᠡ ᠶᠢᠲᠡ 69 ᠬᠤᠪᠢ ) 18. ( 5 ᠬᠤᠪᠢ ) ᠪᠤᠳᠤ᠄ −| 4− √12 |− (𝜋 − 3.14 ) 0 + (1 − cos 30 °) × ( 12 ) ; 2 19. ( 6 ᠬᠤᠪᠢ ) ᠤᠷ ᠢᠳ ᠢᠢᠠ ᠷ ( 1− 3𝑥 :2 )÷ 𝑥 2; 2𝑥 :1 𝑥 2; 4 ᠶᠢ ᠬᠢᠯᠪᠠ ᠷᠴᠢᠯ ᠠᠵᠤ᠂ ᠲᠠ ᠷᠠᠭ ᠠ ᠨᠢ ᠲᠡᠩ ᠴᠡᠳᠭᠡᠯ ᠪᠤᠰᠤ 2𝑥 − 6 0 ᠦ ᠨ ᠰ ᠦᠭᠡᠷ ᠬᠦ ᠪᠤᠰ ᠤ ᠪᠦᠬᠦᠯᠢ ᠲᠤᠭ ᠠᠨ ᠤ ᠬᠠᠷ ᠢᠭᠤ ᠲᠤᠲ ᠤᠷ ᠠ ᠡᠴᠠ ᠲᠤᠬ ᠢᠷᠠᠮᠵᠢᠲ ᠠᠢ ᠲᠤᠭ ᠠ ᠢᠢ ᠰᠤᠩᠭᠤᠨ 𝑥 ᠲᠦ ᠤᠷᠤᠯ ᠠᠭᠤᠯ ᠵᠤ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠤᠯ᠃ 20. ( 6 ᠬᠤᠪᠢ ) ᠮᠠ ᠨ ᠤ ᠬᠤᠲ ᠠ 3 04 ᠤᠯ ᠤᠰ ᠦᠨ ᠵᠠᠮ ᠲᠦ ᠩᠯᠢᠶ ᠤᠤ ᠡᠴ ᠠ ᠬᠤᠤ ᠯᠢᠩᠭᠤᠯ ᠭᠦᠷᠬ ᠦ ᠬᠡᠰᠡᠭ ᠢ ᠵᠠ ᠰᠠᠬᠤ ᠶᠠᠪᠤᠴᠠ ᠳ᠋ᠥ ᠨᠢᠭᠡ ᠠᠭᠤᠯ ᠠ ᠢᠢ ᠦᠩᠭᠡᠷᠡ ᠬᠦ ᠪᠤ ᠯᠵᠠᠢ ᠃ ᠵᠢᠷ ᠤᠭ ᠲᠦ ᠢᠯᠡᠷ ᠬᠡᠶᠢᠯᠡᠭᠰ ᠡᠨ ᠮᠡᠲᠦ᠂ ᠡᠭᠦᠨ ᠳ᠋ᠥ ᠠᠭᠤ ᠯᠠᠨ ᠬᠤᠷᠮ ᠤᠢ ᠳ᠋ᠥ ᠬᠢ A , C ᠬᠤᠶ ᠠᠷ ᠭᠠᠵᠠᠷ ᠨᠢ ᠴᠦ ᠮ ᠲᠠᠯᠠᠢ ᠢᠢᠨ ᠲᠦᠪᠰᠢ ᠨ ᠡ ᠴᠠ ᠪᠠᠷᠤᠭ 10 00 ᠮᠧ ᠲ᠋ᠷ ᠦᠨᠳᠦ ᠷ᠂ ᠠᠭᠤ ᠯᠠᠨ ᠤᠷᠤᠢ ᠳ᠋ᠥ ᠬᠢ B ᠭᠠᠵᠠᠷ ᠨᠢ ᠲᠠ ᠯᠠᠢ ᠢᠢᠨ ᠲᠦ ᠪᠰᠢᠨ ᠡᠴᠠ ᠪᠠᠷ ᠤᠭ 14 00 ᠮᠧᠲ᠋ᠷ ᠦᠨᠳᠦ ᠷ᠂ B ᠭᠠᠵᠠᠷ ᠡᠴᠠ ᠠᠭᠤ ᠯᠠᠨ ᠬᠤᠷᠮ ᠤᠢ ᠢᠢᠨ A ᠭᠠᠵᠠᠷ ᠢ ᠬᠠᠷ ᠠᠬᠤ ᠳ᠋ᠥ ᠰᠢᠷᠭ ᠤᠭᠰ ᠠᠨ ᠦᠨ ᠴᠦᠭ ᠨᠢ 30 °᠂ B ᠭᠠᠵᠠᠷ ᠡᠴᠠ ᠠᠭᠤᠯ ᠠᠨ ᠬᠤᠷᠮ ᠤᠢ ᠢᠢᠨ C ᠭᠠᠵᠠᠷ ᠢ ᠬᠠᠷᠠᠬᠤ ᠳ᠋ᠥ ᠰᠢᠷᠭ ᠤᠭᠰ ᠠᠨ ᠦᠨᠴᠦᠭ ᠨᠢ 45 °᠃ ᠬᠡᠷᠪᠡ A , C ᠬᠤᠶ ᠠᠷ ᠭᠠ ᠵᠠᠷ ᠦ ᠨ ᠬ ᠤᠭᠤᠷ ᠤᠨᠳᠤ ᠨᠦᠬᠡ ᠨ ᠵᠠ ᠮ ᠨᠡ ᠪᠲᠡᠷ ᠡᠭᠦᠯᠬᠦ ᠭᠡᠪᠡ ᠯ᠂ ᠨᠦᠬᠡ ᠨ ᠵᠠ ᠮ ᠦᠨ ᠬᠠᠮᠤᠭ ᠤᠬ ᠤᠷ ᠵᠠᠮ ᠦᠨ ᠤᠷᠲᠤ ᠨᠢ ᠬᠡᠳᠦ ᠢ ᠮᠧ ᠲ᠋ᠷ ᠪ ᠤᠢ ︖ ( ᠪ ᠤᠳᠤᠯ ᠭ ᠠ ᠢᠢᠨ ᠳ᠋ᠦ ᠩ ᠳ᠋ᠥ ᠪᠦᠬᠦ ᠯᠢ ᠲᠤᠭ ᠠ ᠦ ᠯᠡᠳᠡᠭᠡ᠂ ᠯᠠᠪᠯᠠᠯᠲ ᠠ ᠪᠤᠯ ᠭᠠᠬᠤ ᠲᠤᠭ ᠠ √ 3≈ 1.732 ) x y M N O P 21. (6 ᠬᠤᠪᠢ ) ᠨᠢ ᠭᠡᠨ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠢᠢᠨ ᠶᠢᠰᠦ ᠳᠦᠭᠡᠷ ᠤᠨ ᠤ ᠰ ᠤᠷᠤᠭᠴᠢᠳ ᠦᠨ ᠵᠤᠭᠰᠤ ᠭ ᠠ ᠪᠠᠷ ᠢᠢᠠᠨ ᠬᠤᠯ ᠠ ᠬᠠᠷ ᠠᠶᠢᠬ ᠤ ᠬᠢᠷ ᠢ ᠬᠡ ᠮᠵᠢᠶ ᠡ ᠢᠢ ᠤᠶᠢᠯᠠᠭ ᠠᠬᠤ ᠢᠢᠨ ᠲᠦ ᠯᠦᠭᠡ ᠲᠠᠰᠢᠷᠠ ᠮ ᠢᠢᠠᠷ ᠲᠤᠰ ᠤᠨ ᠡᠴᠠ 50 ᠨᠡᠷ ᠡᠰ ᠰ ᠤᠷᠤᠭᠴ ᠢ ᠢᠢ ᠰᠤᠭᠤᠯ ᠤᠨ ᠠᠪᠴᠤ ᠵᠤᠭ ᠰᠤᠭ ᠠ ᠪ ᠠᠷ ᠢᠢᠠᠨ ᠬᠤ ᠯᠠ ᠬᠠᠷᠠᠶ ᠢᠬᠤ ᠰᠢᠯᠭ ᠠᠯᠲᠠ ᠶᠠᠪᠤᠭᠳᠠᠭ ᠤᠯᠵᠠᠢ᠃ ᠰᠢᠯᠭ ᠠᠯᠲᠠ ᠢᠢᠨ ᠬᠢᠷᠢ ᠪᠦᠲᠦ ᠮᠵᠢ ( ᠨᠢ ᠭᠡᠴᠢ ᠄ m ) ᠶᠢ ᠲᠠᠷᠠᠭ ᠠᠬᠢ ᠵᠢᠷᠤᠭ ᠮᠡᠲᠦ ᠬᠤᠶ ᠠᠷ ᠪᠦᠷᠢᠨ ᠪᠤᠰᠤ ᠲᠠᠪᠲᠠ ᠭᠲᠠᠭᠰᠠᠨ ᠲᠤᠭ ᠠᠨ ᠤ ᠲᠠ ᠷᠬᠠᠯᠲ ᠠ ᠢᠢᠨ ᠬᠦᠰ ᠦᠨᠦᠭ ᠲᠦ ᠪ ᠠ ᠲᠠ ᠪᠲᠠᠭᠲᠠᠭᠰᠠ ᠨ ᠲᠤᠭ ᠠᠨ ᠤ ᠲᠠᠷᠬ ᠠᠯᠲᠠ ᠢᠢᠨ ᠰᠢᠯᠤᠭᠤᠨ ᠲᠦᠷᠪ ᠡᠯᠵᠢᠨ ᠵᠢᠷ ᠤᠭ ᠪᠤᠯᠭ ᠠᠨ ᠵᠢᠷᠤᠵ ᠠᠢ᠃ ᠵᠢᠷᠤᠭ ᠲᠦ ᠬᠢ ᠰᠤᠷᠠ ᠭ ᠵᠠ ᠩᠬᠢ ᠢᠢ ᠦᠨᠳ ᠦᠰᠦᠯ ᠡᠵᠦ ᠲᠠᠷ ᠠᠭᠠᠬ ᠢ ᠠ ᠰᠠᠭᠤ ᠯᠲᠠ ᠢᠢ ᠬᠠᠷ ᠢᠭᠤᠯ᠄ (1) ᠬᠦᠰᠦ ᠨᠦᠭᠳᠦ ᠳ᠋ᠥ ᠬᠢ 𝑎 = ( ) 𝑏= ( ) ᠶᠠᠩ ᠵᠤ ᠢᠢᠨ ᠬᠢᠷ ᠢ ᠪᠦ ᠲᠦᠮᠵ ᠢ ᠢᠢᠨ ᠲᠤᠮ ᠳᠠ ᠪᠠᠶᠢᠷ ᠢ ᠢᠢᠨ ᠲᠤᠭ ᠠ ᠨᠢ ( ) ᠬᠡ ᠪᠴᠢᠶ ᠡᠨ ᠳ᠋ᠥ ᠪᠠᠶᠢᠨ ᠠ᠃ (2 ) ᠲᠠ ᠪᠳᠠᠭᠳᠠ ᠭᠰᠠᠨ ᠲᠤᠭ ᠠᠨ ᠤ ᠰᠢᠯᠤᠭᠤᠨ ᠲᠦ ᠷᠪᠡᠯ ᠵᠢᠨ ( ) ᠵᠢᠷ ᠤᠭ ᠢ ᠨᠦᠬᠦᠵᠦ ᠪᠦᠷᠢᠨ ᠪᠤᠯᠭ ᠠ᠃ (3 ) ᠶᠢᠰᠦ ᠳᠦᠭᠡᠷ ᠤᠨ ᠳ᠋ᠥ ᠨᠡ ᠶᠢᠲᠠ 10 00 ᠰᠤᠷᠤᠭᠴᠢ ᠲᠠᠢ ᠪᠤᠯ᠂ ᠲᠤᠰ ᠤᠨ ᠤ ᠰᠤᠷᠤᠭᠴᠢᠳ ᠦᠨ ᠵᠤᠭᠰᠤ ᠭ ᠠ ᠪᠠᠷ ᠢᠢᠠᠨ ᠬᠤᠯ ᠠ ᠬᠠᠷ ᠠᠶᠢᠭᠰᠠᠨ ᠬᠢᠷᠢ ᠪᠦᠲᠦ ᠮᠵᠢ ᠨᠢ 2 .4 ≤ 𝑥 2.8 ᠢᠢᠨ ᠬᠡᠪᠴᠢ ᠶᠡᠨ ᠳ᠋ᠥ ᠪᠠᠶ ᠢᠬᠤ ᠰᠤᠷᠤᠭᠴᠢ ᠬᠡᠳᠦ ᠢ ᠬᠦ ᠮᠦᠨ ᠪᠠᠶᠢᠬᠤ ᠪᠤᠢ ︖ 22. (7 ᠬᠤᠪᠢ ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠦ᠂ ∆AB C ᠳ᠋ᠥ ᠴᠡᠭ D ᠨᠢ BC ᠲᠠᠯ ᠠ ᠲᠡ ᠭᠡᠷᠡᠬ ᠢ ᠨᠢ ᠭᠡᠨ ᠴᠡᠭ ᠂ ᠴᠡᠭ E ᠨᠢ AD ᠢᠢᠨ ᠲ ᠤᠮᠳᠠ ᠴᠡᠭ ᠂ ᠴᠡ ᠭ A ᠢᠢ ᠲᠠᠭ ᠠᠷᠢᠭ ᠤᠯᠤᠨ BC ᠢᠢᠨ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯ ᠤᠭᠤᠨ ᠢ ᠪᠠᠶᠢᠭ ᠤᠯᠪᠠ ᠯ BE ᠢᠢᠨ ᠦᠷᠭᠦ ᠯᠵᠢᠯ ᠡᠯ ᠱᠤᠭᠤᠮ ᠲᠠ ᠢ ᠴᠡ ᠭ F ᠳ᠋ᠥ ᠤᠭᠲᠤᠯ ᠤᠯᠴᠠ ᠬᠤ ᠪᠦᠭᠡᠳ AF = CD ,CF ᠢᠢ ᠬᠤᠯ ᠪᠤᠨ ᠠ᠃ (1) ᠪᠠᠲᠤᠯ ᠠᠭᠳᠠ ᠬᠤ ᠨᠢ᠄ ∆AEF ≌ ∆D EB (2 ) ᠬᠡ ᠷᠪᠡ AB = AC ᠪᠤᠯ ᠲᠦ ᠷᠪᠡᠯ ᠵᠢᠨ A DCF ᠦᠨ ᠬᠡᠯᠪᠡ ᠷᠢ ᠲᠦᠷᠰᠦ ᠢᠢ ᠮᠠᠭ ᠠᠳᠯᠠᠵᠤ ᠪᠠᠰᠠ ᠰᠢᠯ ᠲᠠᠭᠠᠨ ᠢ ᠨᠢ ᠪᠠᠲᠤᠯ ᠠᠵᠤ ᠲᠤᠳᠤᠷ ᠬᠠᠶᠢᠯ ᠠ᠃ ᠭᠡᠰ ᠡᠭᠯᠡᠪᠡᠯ 1.2 ≤ 𝑥 1.6 1.6 ≤ 𝑥 2.0 2.0 ≤ 𝑥 2.4 2.4 ≤ 𝑥 2.8 ᠳᠠᠪ ᠳᠠᠭᠳᠠᠭ ᠰᠠᠨ ᠲᠣᠭ ᠠ m ᠰᠤᠷ ᠤᠭᠴ ᠢ ᠶᠢᠨ ᠵᠣᠭᠰᠣ ᠭ ᠠ ᠪᠠᠷ ᠢᠶᠠᠨ ᠬᠣᠯ ᠠ ᠬᠠᠷᠢ ᠠᠬᠤ ᠰᠢᠯᠭ ᠠᠯᠲᠠ ᠶᠢᠨ ᠬᠢᠷᠢ ᠪᠦᠳᠦ ᠮᠵᠢ ᠶᠢᠨ ᠲᠠᠪᠳᠠᠭᠳᠠᠭ ᠰᠠᠨ ᠲᠣᠭ ᠠᠨ ᠤ ᠲᠠᠷᠬᠠᠯᠲ ᠠ ᠶᠢᠨ ᠬᠦᠰᠦᠨᠦᠭᠳᠦ ᠰᠤᠷ ᠤᠭᠴ ᠢ ᠶᠢᠨ ᠵᠣᠭᠰᠣ ᠭ ᠠ ᠪᠠᠷ ᠢᠶᠠᠨ ᠬᠣᠯ ᠠ ᠬᠠᠷᠠ ᠢᠬᠤ ᠰᠢᠯᠭ ᠠᠯᠲᠠ ᠶᠢᠨ ᠬᠢᠷᠢ ᠪᠦᠳᠦ ᠮᠵᠢ ᠶᠢᠨ ᠲᠠᠪᠳᠠᠭᠳᠠᠭ ᠰᠠᠨ ᠲᠣᠭ ᠠᠨ ᠤ ᠲᠠᠷᠠᠬᠠᠯᠲ ᠠ ᠶᠢᠨ ᠰᠢᠯᠤᠭ ᠤᠨ ᠲᠥ ᠷᠪᠡᠯᠵ ᠢᠨ ᠵᠢᠷ ᠤᠭ ᠲᠠᠪᠳᠠᠭᠳᠠᠭ ᠰᠠᠨ ᠲᠣᠭ ᠠ (ᠰᠤᠷ ᠤᠭᠴ ᠢ ᠶᠢᠨ ᠲᠣᠭ ᠠ) ᠬᠢᠷᠢ ᠪᠦᠳᠦ ᠮᠵᠢ E C FA B D 23. (8 ᠬᠤᠪᠢ ) ᠨᠢ ᠭᠡᠨ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠰ ᠤᠷᠤᠭᠴᠢ ᠢᠢᠨ ᠤᠷᠠᠯᠢᠭ ᠦᠨ ᠬᠡᠪ ᠪᠤᠯᠪᠠ ᠰᠤᠷᠠ ᠯ ᠢ ᠲᠡᠭᠡᠭ ᠰᠢᠯᠡᠭᠦ ᠯᠬᠦ ᠢᠢ ᠨ ᠲ ᠦᠯᠦ ᠭᠡ A ᠤᠷᠠᠨ ᠪᠢᠴᠢᠯ ᠭᠡ B ᠤᠷᠠᠨ ᠵᠢᠷᠤᠭ C ᠭᠦᠭᠵᠢᠮ D ᠪᠦᠵ ᠢᠭ ᠵᠡᠷᠭᠡ ᠲᠦ ᠷᠪᠡᠨ ᠲᠦ ᠷᠦᠯ ᠦ ᠨ ᠤᠷ ᠠᠯᠢᠭ ᠦᠨ ᠰᠤᠩᠭᠤᠨ ᠰᠤ ᠷᠬᠤ ᠬᠢᠴᠢ ᠶᠡᠯ ᠨᠡᠭᠡᠭᠡᠬ ᠦ ᠪᠠᠷ ᠲᠦᠯᠦᠪᠯᠡᠵ ᠡᠢ᠃ ᠰᠤᠷᠤᠭᠴᠢᠳ ᠦᠨ ᠲᠤᠰ ᠲᠦᠷᠪ ᠡᠨ ᠲᠦᠷᠦᠯ ᠦᠨ ᠬᠢᠴᠢᠶ ᠡᠯ ᠢ ᠰᠤᠷᠬᠤ ᠤᠷᠮ ᠠ ᠪ ᠠᠬ ᠠ ᠢᠢ ᠤᠶᠢ ᠯᠠᠭᠠᠬᠤ ᠢ ᠢᠨ ᠲᠦ ᠯᠦᠭᠡ ᠤᠷ ᠢᠳ ᠢᠢᠠ ᠷ ᠪᠦᠬᠦ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠢᠢᠨ ᠬᠡᠪᠴᠢᠶ ᠡᠨ ᠡᠴᠠ ᠲᠠᠰᠢᠷᠠ ᠮ ᠢᠢᠠᠷ ᠬᠡᠰᠡᠭ ᠰᠤᠷᠤᠭᠴᠢ ᠢ ᠢ ᠰ ᠤᠩᠭᠤᠵ ᠤ ᠠ ᠰᠠᠭᠤᠯ ᠲᠠ ᠢᠢᠨ ᠬᠠᠭ ᠤᠳᠠᠰᠤᠨ ᠪᠠᠶᠢᠴᠠ ᠭᠠᠯᠲ ᠠ ᠬᠢ ᠵᠡᠢ ( ᠪ ᠠᠶᠢᠴᠠ ᠭᠠᠭᠳᠠ ᠭᠰᠠᠨ ᠰᠤᠷᠤᠭᠴᠢ ᠪᠦᠷᠢ ᠵᠠᠪᠠ ᠯ ᠰ ᠤᠩᠭᠤᠬ ᠤ ᠪᠦᠭᠡᠳ ᠵᠦᠪᠬ ᠡᠨ ᠨᠢᠭᠡ ᠲᠦ ᠷᠦᠯ ᠦ ᠨ ᠬᠢ ᠴᠢᠶᠡᠯ ᠢ ᠰᠤᠩ ᠭᠤᠨ ᠠ ) ᠃ ᠲᠠ ᠷᠠᠭ ᠠ ᠨᠢ ᠤᠯᠤᠭᠰ ᠠᠨ ᠲᠤᠭ ᠠ ᠪ ᠠᠷᠢᠮᠲᠠ ᠪᠠᠨ ᠡᠮᠬᠢᠯ ᠡᠵᠦ ᠲᠠᠷᠠᠭ ᠠᠬᠢ ᠮᠡᠲᠦ ᠬᠤᠶ ᠠᠷ ᠵ ᠦᠢᠯ ᠦ ᠨ ᠪ ᠦᠷᠢᠨ ᠪᠤᠰᠤ ᠪᠦᠷᠢᠳᠭᠡᠯ ᠦᠨ ᠵᠢᠷᠤᠭ ᠪᠤᠯᠭ ᠠᠨ ᠵᠢᠷ ᠤᠵᠠᠢ ᠃ ᠵᠢᠷ ᠤᠭ ᠲᠦ ᠬᠠᠩᠭ ᠠᠭᠰᠠᠨ ᠰᠤᠷᠠ ᠭ ᠵᠠ ᠩᠭᠢ ᠢᠢ ᠦᠨᠳᠦ ᠰᠦᠯᠡᠨ ᠲᠠᠷ ᠠᠭᠠᠬ ᠢ ᠠ ᠰᠠᠭᠤᠯᠲ ᠠ ᠢᠢ ᠬᠠᠷᠢᠭ ᠤᠯ᠄ (1 ) ᠲᠤᠰ ᠤᠳᠠ ᠭᠠᠨ ᠤ ᠪᠠᠶᠢᠴᠠ ᠭᠠᠯᠲ ᠠ ᠳ᠋ᠥ ᠨᠡᠶᠢᠲᠡ ᠬᠡᠳᠦ ᠢ ᠰ ᠤᠷᠤᠭ ᠴᠢ ᠤᠷᠤᠯ ᠴᠠᠭᠰᠠᠨ ᠪᠤᠢ ︖ ᠲᠡᠪᠢᠭᠦᠷ ᠬᠡᠯᠪᠡ ᠷᠢᠲᠦ ᠪᠦᠷᠢᠳᠭᠡᠯ ᠦᠨ ᠵᠢᠷᠤᠭ ᠲᠦ ᠬᠢ ∠ α ᠢᠢᠨ ᠭᠷᠠᠳᠦᠰ ᠦ ᠨ ᠲᠤ ᠭ ᠠ ᠨᠢ ᠬᠡ ᠳᠦᠢ ᠪᠤᠢ ︖ (2) ᠰᠠᠪ ᠠᠭᠠᠨ ᠪᠦᠷᠢᠳᠭᠡᠯ ᠦ ᠨ ᠵᠢᠷ ᠤᠭ ᠢ ᠪᠦᠲ ᠦᠨ ᠪᠤᠯᠭ ᠠᠨ ᠨᠦᠬᠦᠨ ᠵᠢᠷᠤ ᠵᠤ ᠦᠭ (3 ) ᠲᠤᠰ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ 2 01 8 ᠤᠨ ᠤ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠢᠢᠨ ᠬᠦᠷᠢᠶᠡᠨ ᠤᠷ ᠠᠯᠢᠭ ᠦᠨ ᠨᠠᠶᠢᠷ ᠬᠢᠬ ᠦ ᠢᠢᠨ ᠲᠦ ᠯᠦᠭᠡ A ᠤᠷᠠ ᠨ ᠪ ᠢᠴᠢᠯ ᠭᠡ B ᠤᠷᠠᠨ ᠵᠢᠷᠤᠭ C ᠭᠦᠭᠵᠢᠮ D ᠪᠦᠵ ᠢᠭ ᠲᠦᠷᠪ ᠡᠨ ᠤᠷᠠᠯᠢᠭ ᠦᠨ ᠬᠡᠯᠪᠡ ᠷᠢ ᠡᠴᠠ ᠬᠤᠶ ᠠᠷ ᠲ ᠦᠷᠦ ᠯ ᠢ ᠰᠤᠩᠭᠤᠨ ᠰᠢᠨ ᠡ ᠨᠢ ᠭᠡ ᠲᠤᠭᠯ ᠠᠯᠲᠠ ᠢᠢᠨ ᠬᠡᠯᠪᠡ ᠷᠢ ᠪ ᠤᠯᠭᠠᠬᠤ ᠪᠠᠷ ᠲᠤᠭᠲᠠᠵ ᠠᠢ᠃ ᠬᠦᠰ ᠦᠨᠦᠭᠵᠢ ᠭᠦᠯᠬᠦ ᠠᠷᠭ ᠠ ᠪᠤᠶ ᠤ ᠮᠤᠳᠤᠯ ᠵᠢᠨ ᠵᠢᠷ ᠤᠭ ᠦ ᠨ ᠠ ᠷᠭ ᠠ ᠪᠠᠷ ᠤᠷᠠᠨ ᠪᠢᠴᠢᠯ ᠭᠡ ᠪ ᠠ ᠬᠦ ᠭᠵᠢᠮ ᠬᠠᠮ ᠲᠤ ᠪᠠᠷ ᠰᠤᠩᠭᠤᠭᠳᠠᠬᠤ ᠲᠤᠬ ᠢᠶᠠᠯᠳ ᠤᠮᠵᠢ ᠢᠢ ᠤᠯ᠃ 24. (9 ᠬᠤᠪᠢ ) ᠰ ᠦᠯᠵ ᠢᠶᠡᠨ ᠲᠡᠯᠭ ᠡᠭᠦᠷ ᠲᠦ ᠠ ᠡ ᠬᠤ ᠶᠠᠷ ᠵᠦᠢᠯ ᠦᠨ ᠦᠳᠦ ᠨ ᠪ ᠦᠮᠪᠦ ᠭᠡ ᠬᠤᠳᠠ ᠯᠳᠤᠵ ᠤ ᠪ ᠠᠶᠢᠨ ᠠ᠃ ᠠ ᠵᠦᠢᠯ ᠦᠨ ᠦᠳᠦ ᠨ ᠪ ᠦᠮᠪᠦᠭ ᠡ ᠢᠢᠨ ᠲᠦ ᠩᠰᠠ ᠪᠦᠷ ᠢ ᠢᠢᠨ ᠬᠤᠳᠠᠯ ᠳᠤᠬ ᠤ ᠦ ᠨ ᠡ ᠨᠢ ᠡ ᠵᠦ ᠢᠯ ᠦ ᠨ ᠦᠳᠦ ᠨ ᠪ ᠦᠮᠪᠦᠭᠡ ᠢᠢᠨ ᠲᠦ ᠩᠰᠠ ᠪᠦᠷᠢ ᠢᠢ ᠨ ᠬᠤᠳᠠᠯ ᠳᠤᠬ ᠤ ᠦᠨ ᠡ ᠡᠴᠠ 15 ᠲᠦᠭ ᠦᠷᠢᠭ ᠦᠨᠡᠲᠡ ᠢ᠂ ᠪᠠ ᠪ ᠠᠭᠰᠢ ᠲᠤᠰ ᠰᠦᠯ ᠵᠢᠶᠡᠨ ᠲᠡᠯᠭ ᠡᠭᠦᠷ ᠡᠴᠠ 2 ᠲᠦᠩᠰ ᠠ ᠠ ᠵᠦ ᠢᠯ ᠦ ᠨ ᠦᠳᠦ ᠨ ᠪ ᠦᠮᠪᠦᠭᠡ ᠪᠠ 3 ᠲᠦᠩᠰ ᠠ ᠡ ᠵᠦᠢᠯ ᠦᠨ ᠦᠳᠦ ᠨ ᠪᠦᠮᠪᠦ ᠭᠡ ᠬᠤᠳᠠ ᠯᠳᠤᠨ ᠠᠪ ᠤᠭᠠᠳ ᠨᠡ ᠶᠢᠲᠡ 25 5 ᠲᠦᠭ ᠦᠷᠢᠭ ᠵᠠᠷ ᠤᠴᠠᠭ ᠤᠯᠵᠠᠢ᠃ (1) ᠲᠤᠰ ᠰᠦᠯ ᠵᠢᠶᠡᠨ ᠲᠡᠯ ᠭᠡᠭᠦᠷ ᠦᠨ ᠠ᠂ ᠡ ᠬᠤ ᠶᠠᠷ ᠵᠦᠢᠯ ᠦᠨ ᠦᠳ ᠦᠨ ᠪᠦᠮᠪᠦᠭᠡ ᠢᠢᠨ ᠲᠦ ᠩᠰᠠ ᠪᠦᠷᠢ ᠢᠢᠨ ᠬᠤ ᠳᠠᠯᠳᠤᠬ ᠤ ᠦᠨ ᠡ ᠨᠢ ᠲᠤᠰ ᠪᠦᠷ ᠢ ᠬᠡ ᠳᠦᠢ ᠲᠦᠭ ᠦᠷᠢᠭ ᠪᠤᠢ ︖ (2) ᠲᠤᠰ ᠲᠡᠯᠭᠡᠭᠦᠷ ᠬᠡᠷᠡᠭᠯᠡᠭ ᠴᠢᠳ ᠦᠨ ᠱᠠᠭ ᠠᠷᠳᠠᠯᠭ ᠠ ᠢᠢ ᠦᠨᠳᠦ ᠰᠦᠯᠡᠨ ᠠ᠂ ᠡ ᠬᠤ ᠶᠠᠷ ᠵᠦᠢᠯ ᠦᠨ ᠦᠳᠦ ᠨ ᠪᠦᠮᠪᠦᠭᠡ ᠢᠢ ᠨᠡᠶᠢᠲᠡ 20 0 ᠲᠦᠩᠰ ᠠ ᠤᠷ ᠤᠭᠤᠯ ᠬᠤ ᠦᠶ ᠡ ᠳ᠋ᠥ ᠵᠠᠷᠤᠴ ᠠᠭᠤᠯ ᠬᠤ ᠰᠦᠢᠳᠭᠡᠯ ᠢ 8 78 0 ᠲᠦᠭ ᠦᠷᠢᠭ ᠡᠴᠠ ᠬᠡᠳᠦ ᠷᠡᠭᠦᠯ ᠬᠦ ᠦᠭᠡᠢ ᠪᠦᠭᠡᠳ ᠠ ᠵᠦᠢᠯ ᠦᠨ ᠦᠳ ᠦᠨ ᠪᠦᠮᠪᠦᠭᠡ ᠢᠢᠨ ᠲᠤᠭ ᠠ ᠨᠢ ᠡ ᠵᠦᠢᠯ ᠦᠨ ᠦᠳ ᠦᠨ ᠪᠦᠮᠪᠦᠭᠡ ᠢᠢᠨ ᠲᠤᠭ ᠠᠨ ᠤ 35 ᠡᠴᠠ ᠶᠡᠬᠡ ᠪᠠᠶᠢᠬᠤ ᠬᠡᠷᠡᠭᠲᠡᠢ ᠭᠡᠵ ᠦ ᠲᠤ ᠭᠲᠠᠭ ᠠᠵᠠᠢ᠃ ᠦᠭ ᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ᠂ ᠠ ᠵᠦᠢᠯ ᠦ ᠨ ᠦᠳ ᠦᠨ ᠪᠦᠮᠪᠦᠭᠡ ᠢᠢᠨ ᠲᠦᠩ ᠰᠠ ᠪᠦᠷᠢ ᠢᠢᠨ ᠤᠷᠤᠭᠤ ᠯᠬᠤ ᠦᠨ ᠡ ᠨᠢ 50 ᠲᠦᠭ ᠦᠷᠢᠭ ᠂ ᠡ ᠵᠦᠢᠯ ᠦᠨ ᠦᠳᠦ ᠨ ᠪᠦᠮᠪᠦᠭᠡ ᠢᠢᠨ ᠲᠦᠩᠰ ᠠ ᠪᠦᠷᠢ ᠢᠢ ᠨ ᠤᠷ ᠤᠭᠤᠯ ᠬᠤ ᠦᠨ ᠡ ᠨᠢ 40 ᠲᠦᠭ ᠦᠷᠢᠭ ᠃ ① ᠠ ᠵᠦᠢᠯ ᠦᠨ ᠦᠳ ᠦᠨ ᠪᠦᠮᠪᠦ ᠭᠡ ᠢᠢ 𝑚 ᠲᠦᠩᠰ ᠠ ᠤᠷ ᠤᠭᠤᠯ ᠤᠨ ᠠ ᠭᠡᠵᠦ ᠵᠢᠱᠢᠪᠡᠯ ᠂ ᠲᠤᠰ ᠲᠡᠯᠭᠡᠭᠦᠷ ᠶᠠᠮᠠᠷ ᠬᠡᠳᠦ ᠨ ᠵ ᠦᠢᠯ ᠦ ᠨ ᠪᠠᠷᠠᠭ ᠠ ᠤᠷ ᠤᠭᠤᠯ ᠬᠤ ᠲᠦ ᠰᠦᠯ ᠪᠠᠶᠢᠬᠤ ᠪᠤᠢ ︖ ② ᠬᠡᠷᠪᠡ ᠤᠷᠤᠭᠤᠯ ᠤᠭᠰᠠᠨ ᠦᠳ ᠦᠨ ᠪᠦᠮᠪᠦᠭᠡ ᠪᠠᠨ ᠪᠦᠬᠦᠨ ᠢ ᠬᠤᠳᠠ ᠯᠳᠤᠭ ᠰᠠᠨ ᠪᠤᠯ᠂ ᠰᠦ ᠯᠵᠢᠶ ᠡᠨ ᠲᠡᠯᠭᠡ ᠭᠦᠷ ᠦ ᠨ ᠤᠯ ᠤᠭᠰ ᠠᠨ ᠠᠰᠢ ᠭ ᠬᠤᠨᠵ ᠢᠪᠤᠷᠢ W (ᠲᠦᠭ ᠦᠷᠢᠭ ) ᠪ ᠠ ᠠ ᠵᠦᠢᠯ ᠦᠨ ᠦᠳ ᠦᠨ ᠪᠦᠮᠪᠦᠭ ᠡ ᠢᠢᠨ ᠪᠠᠷᠠᠭ ᠠ ᠤᠷᠤᠭᠤᠯ ᠤᠭᠰᠠᠨ ᠬᠡᠮᠵᠢᠶ ᠡ 𝑚 (ᠲᠦᠩᠰ ᠠ) ᠦ ᠨ ᠬᠤᠭᠤᠷ ᠤᠨᠳᠤᠬ ᠢ ᠹᠦᠩᠺᠼ ᠦᠨ ᠬᠤᠯ ᠪᠤᠭᠳᠠᠯ ᠦᠨ ᠢᠯᠡᠷ ᠬᠡᠶᠢᠯᠡᠯ ᠢ ᠤᠯ ᠃ ᠬᠤᠪᠢᠰ ᠬᠤᠨ 𝑚 ᠶᠠᠮᠠᠷ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠠᠪᠬᠤ ᠦᠶ ᠡᠰ ᠤᠯ ᠬᠤ ᠠ ᠰᠭ ᠬᠤᠨᠵ ᠢᠪᠤᠷᠢ ᠨᠢ ᠬᠠᠮ ᠤᠭ ᠦ ᠨ ᠶᠡᠭ ᠡ ᠪᠤᠯ ᠬᠤ ᠪ ᠤᠢ ︖ ᠬᠠᠮ ᠤᠭ ᠦ ᠨ ᠶᠡᠬᠡ ᠠᠰᠢᠭ ᠬᠤᠨᠵ ᠢᠪᠤᠷᠢ ᠨᠢ ᠬᠡᠳᠦ ᠢ ᠪᠤᠢ ︖ 20% 10% 40% D C B A α x y 16 8 4 A B C D 4 8 12 16 O ᠰᠤᠷ ᠤᠭᠴ ᠢ ᠶᠢᠨ ᠬᠢᠴᠢᠶᠡᠯ ᠰᠤᠩ ᠭᠤᠭ ᠰᠠᠨ ᠳᠡᠪᠢᠭᠦ ᠷ ᠬᠡᠯᠪ ᠡᠷᠢᠳᠤ ᠪᠦᠷᠢᠳᠭᠡ ᠯ ᠦᠨ ᠵᠢᠷᠤᠭ ᠰᠤᠷ ᠤᠭᠴ ᠢ ᠶᠢᠨ ᠬᠢᠴᠢᠶᠡᠯ ᠰᠤᠩ ᠭᠤᠭ ᠰᠠᠨ ᠰᠠᠪᠠᠭ ᠠᠨ ᠪᠦᠷᠢᠳᠭᠡ ᠯ ᠦᠨ ᠵᠢᠷᠤᠭ ᠬᠦᠮᠦᠨ ᠦ ᠲᠣᠭ ᠠ ᠬᠢᠴᠢᠶᠡᠯ ᠲᠥ ᠷᠥᠯ 25 . (10 ᠬᠤᠪᠢ ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠦ᠂ ⨀O ᠨᠢ ∆AB C ᠢᠢ ᠪᠠᠭᠲᠠᠭ ᠠᠭᠰᠠᠨ ᠲᠤᠭ ᠤᠷᠢᠭ ᠪᠦᠭᠡᠳ ᠴᠡᠭ O ᠨᠢ BC ᠲᠠᠯ ᠠ ᠲᠡ ᠭᠡᠷ ᠡ ᠤᠷᠤᠰ ᠢᠨ ᠠ᠃ ∠B AC ᠢᠢᠨ ᠲᠠᠯᠠᠯᠠᠨ ᠬᠤᠪᠢ ᠶᠠᠭᠴᠢ AD ᠨᠢ ⨀O ᠲᠠ ᠢ ᠴᠡ ᠭ D ᠳ᠋ᠥ ᠤᠭᠲᠤᠯ ᠤᠯᠴᠠᠨ ᠠ᠂ BD ,CD ᠢᠢ ᠬᠤᠯ ᠪᠤᠨ ᠠ᠃ ᠴᠡᠭ D ᠢᠢ ᠲᠠᠭ ᠠᠷᠢᠭ ᠤᠯᠤᠨ BC ᠲᠠᠢ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠤᠭᠤ ᠨ ᠪ ᠠᠶᠢᠭ ᠤᠯᠪᠠ ᠯ᠂ AC ᠢᠢᠨ ᠦᠷᠭᠦ ᠯᠵᠢᠯ ᠡᠯ ᠱ ᠤᠭᠤᠮ ᠲᠠᠢ ᠴᠡᠭ P ᠳ᠋ᠥ ᠤᠭᠳᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠃ (1) ᠪᠠᠳᠤᠯ ᠠᠭᠳᠠ ᠬᠤ ᠨᠢ ᠄ PD ᠨᠢ ⨀ O ᠢᠢᠨ ᠰᠢᠬ ᠠᠭᠴᠢ ᠰᠢᠯ ᠤᠭᠤᠨ ᠃ (2) ᠪᠠᠲᠤᠯ ᠠᠭᠳᠠ ᠬᠤ ᠨᠢ ᠄ ∆ AB D ∽ ∆D CP ᠃ (3) AB = 5c m ,AC = 12 cm ᠦᠶ ᠡᠰ᠂ ᠬᠡᠷ ᠴᠢᠮ PC ᠢᠢᠨ ᠤᠷ ᠳᠤ ᠢᠢ ᠤᠯ᠃ 26. (12 ᠬᠤᠪᠢ ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠦ᠂ ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤᠭᠤ ᠮ y = 𝑎𝑥 2+ 𝑏𝑥 − 5 ᠨᠢ ᠪᠠᠶᠢᠷ ᠢᠰᠢᠯ ᠦ ᠨ ᠲᠡ ᠩᠬᠡᠯ ᠢᠭ ᠲᠠᠢ A( −1 ,0) , B( 5,0 ) , C ( 0, −5 ) ᠭᠤᠷᠪ ᠠᠨ ᠴᠡ ᠭ ᠲᠦ ᠤᠭᠳᠤ ᠯᠤᠯᠴ ᠠᠵᠤ᠂ ᠤᠷᠤᠢ ᠢᠢᠨ ᠴᠡᠭ ᠨᠢ D᠃ (1) ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢ ᠨ ᠱᠤᠭᠤ ᠮ ᠦᠨ ᠠᠨᠠ ᠯᠢᠰ ᠢᠯᠡᠷ ᠬᠡᠶᠢᠯᠡᠯ ᠪᠠ ᠬᠠᠶᠠᠯᠲ ᠠ ᠢᠢᠨ ᠱᠤ ᠭᠤᠮ ᠦᠨ ᠤᠷᠤᠢ ᠢᠢᠨ ᠴᠡᠭ D ᠢᠢᠨ ᠪᠠᠶᠢᠷ ᠢᠰᠢᠯ ᠢ ᠰ ᠢᠭᠤᠳ ᠪᠢᠴᠢ ᠃ (2) BC ᠢᠢ ᠬᠤᠯ ᠪᠤᠪᠠᠯ ᠬᠠᠶᠠ ᠯᠲᠠ ᠢ ᠢᠨ ᠱᠤᠭᠤᠮ ᠦᠨ ᠲᠡᠭᠰᠢ ᠬᠡᠮ ᠦᠨ ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠲᠠᠢ ᠴᠡᠭ E ᠳ᠋ᠥ ᠤᠭᠳᠤᠯ ᠤᠯᠴᠠ ᠨ ᠠ᠂ ᠴᠡᠭ P ᠨᠢ ᠬᠡᠷᠴᠢᠮ BC ᠲᠡ ᠭᠡᠷᠡᠬ ᠢ ᠨᠢ ᠭᠡᠨ ᠬᠦᠳ ᠡᠯᠬᠦ ᠴᠡᠭ (ᠴᠡ ᠭ P ᠨᠢ B, C ᠬᠤᠶ ᠠᠷ ᠴᠡ ᠭ ᠲᠠ ᠢ ᠲᠠ ᠪᠬᠤᠴ ᠠᠬᠤ ᠦᠭᠡᠢ ) ᠂ ᠴᠡ ᠭ P ᠢᠢ ᠲᠠᠭ ᠠᠷᠢᠭ ᠤᠯᠤᠨ PF ∥DE ᠪᠠᠷ ᠪᠠᠶᠢᠭ ᠤᠯᠪᠠ ᠯ᠂ ᠬᠠᠶᠠᠯ ᠲᠠ ᠢᠢᠨ ᠱᠤᠭ ᠤᠮ ᠲᠠᠢ ᠴᠡᠭ F ᠳ᠋ᠥ ᠤᠭᠳᠤ ᠯᠤᠯᠴᠠ ᠨ ᠠ᠃ ᠬᠡᠷᠪᠡ ᠴᠡᠭ P ᠢᠢᠨ ᠬᠦ ᠨᠳᠡᠯᠡᠨ ᠪᠠᠶᠢᠷ ᠢᠰᠢᠯ ᠢ 𝑚 ᠭᠡᠵᠦ ᠵᠢᠱᠢᠪᠡᠯ᠄ ① ᠲᠦᠷᠪ ᠡᠯᠵᠢᠨ PED F ᠢ ᠵᠢᠰᠢ ᠭᠦᠯᠵ ᠢᠨ ᠪᠤᠯᠭ ᠠᠬᠤ ᠴᠡᠭ P ᠤᠷᠤᠰ ᠢᠨ ᠠ ᠤᠤ︖ ᠤᠷ ᠤᠰᠢᠪᠠ ᠯ ᠴᠡ ᠭ P ᠢᠢᠨ ᠪᠠᠶᠢᠷ ᠢᠰᠢᠯ ᠢ ᠤᠯ ᠵᠤ᠂ ᠤᠷᠤᠰ ᠢᠬᠤ ᠦᠭᠡᠢ ᠪᠤᠯ ᠰᠢᠯᠲᠠᠭ ᠠᠨ ᠢᠢᠠᠨ ᠲᠤᠳᠤᠷ ᠬᠠᠶᠢᠯ ᠠ ᠃ ② ᠴᠡᠭ F ᠢᠢ ᠲᠠᠭ ᠠᠷᠢᠭ ᠤᠯᠤᠨ FH ⊥ BC ᠪᠠᠷ ᠪᠠᠶᠢᠭ ᠤᠯᠪᠠ ᠯ᠂ ᠡᠭ ᠴᠡ ᠢᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ H ᠂ ∆ PFH ᠦᠨ ᠬᠦᠷᠢᠶᠡᠨ ᠤᠷ ᠲᠤ ᠢᠢᠨ ᠬᠠᠮᠤᠭ ᠦᠨ ᠶᠡᠬᠡ ᠬᠡᠮ ᠵᠢᠭᠳᠡᠯ ᠢ ᠤᠯ᠃ PD B O C A x y PE H F C D BA O