2020 ᠣᠨ ᠤ ᠪᠣᠭᠤᠲᠤ ᠬᠤᠲᠠ ᠶᠢᠨ ᠳᠤᠮᠳᠠᠳᠤ ᠰᠢᠯᠭᠠᠯᠲᠠ ᠶᠢᠨ ᠰᠡᠲᠤᠪ

20 20 ᠣᠨ ᠤ ᠪᠣᠭᠣᠳᠤ᠂ ᠰᠢᠯᠢᠩ ᠬᠣᠲᠠ ᠂ ᠠᠯᠠᠱᠨ ᠂ ᠦᠬᠠᠢ᠂ ᠪ ᠠᠶᠠᠨᠨᠠᠭᠤᠷ᠂ ᠤᠯᠠᠭᠠᠨᠴᠠᠪ ᠬᠣ ᠲᠠ ᠶᠢᠨ ᠠᠩ ᠬ ᠠ ᠲᠣᠮᠳ ᠠᠳᠤ ᠢᠢᠨ ᠰᠣ ᠷᠭᠠᠭ ᠣᠯ ᠢ ᠲᠠᠪᠰᠢᠬ ᠤ ᠰᠢᠯ ᠭᠠᠯᠳᠠ ᠲᠣᠭᠠᠨ ᠤ ᠣᠬᠠᠭᠠᠨ ᠨᠢᠬᠡ ᠂ ᠰᠣᠩᠭᠣᠬᠤ ᠰᠡᠳᠣᠪ ᠃ (ᠰᠡᠳ ᠣᠪ ᠪ ᠦᠷ ᠢ 3 ᠬᠣᠪ ᠢ ᠂ ᠨᠡᠶᠢᠳᠡ 36 ᠬᠣᠪᠢ) 1. √8 + √2 ᠢ ᠵᠥᠪ ᠪᠣᠳᠣ ᠭᠰᠠᠨ ᠳ᠋ᠦᠩ ᠨᠢ ( ) A. 5 B. √ 10 C. 3 √2 D. 4 + √2 2. 2020 ᠣᠨ ᠤ ᠡᠬᠢ ᠨ ᠳ᠋ ᠦ ᠤᠯᠤ ᠰ ᠤᠨ ᠲᠣᠭ ᠠ ᠪᠦᠷᠢᠳᠭ ᠡᠯ ᠦᠨ ᠳᠣᠪ ᠴᠢᠶ ᠡ ᠨᠢ ᠣᠳᠣ ᠬᠢ ᠬᠡᠷᠡᠭᠵᠢᠬᠦᠯᠵᠦ ᠪᠠᠢ ᠭ ᠠ ᠬᠦᠦᠳᠡᠯᠭᠡᠷ ᠳᠤᠰᠬᠤ ᠨ ᠤ ᠶᠠᠳᠠᠭᠤ ᠬᠦᠴᠢᠷᠳᠡᠯ ᠦᠨ ᠪᠠᠷᠢ ᠮᠵᠢᠶ ᠠ ᠶᠢ ᠦᠨ ᠳᠤᠰᠤᠯ ᠡᠨ 20 19 ᠣᠨ ᠤ ᠡᠴᠤᠰ ᠪᠣᠭᠯᠠᠭᠳᠠᠯ ᠠ ᠪᠥ ᠬᠥ ᠤᠯᠠ ᠨ ᠤᠯᠤ ᠰ ᠤᠨ ᠬᠦᠦᠳᠡᠯᠭᠡᠷ ᠳᠤᠰᠬᠤ ᠨ ᠤ ᠶᠠᠳᠠᠭᠤ ᠬᠦᠢᠴᠷᠢᠳᠡ ᠯ ᠲᠠᠢ ᠬᠦᠮᠦᠨ ᠠᠮᠠ ᠶᠢᠨ ᠲᠣᠭ ᠠ ᠨᠢ ᠬᠠᠰᠤᠭ ᠳᠠᠭᠠᠳ 551 ᠲᠥᠪᠮᠡᠨ ᠳᠦ ᠪᠣᠯᠵᠤ ᠨᠠᠢ ᠲᠡ 93 48 ᠲᠥᠪᠮᠡᠨ ᠳᠦ ᠬᠦᠮᠦᠨ ᠶᠠᠳᠠᠭᠤ ᠠᠴᠠ ᠠᠩᠬᠢ ᠵᠢᠷᠠᠪᠠ ᠭᠡᠵᠦ ᠨᠠᠢ ᠲᠡᠯᠡᠪᠡ᠃ 9348 ᠲᠥᠮᠡ ᠶᠢ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬ ᠠᠭᠠᠨᠴᠢ ᠲᠣᠭ ᠠ ᠲᠡᠮᠳᠡ ᠭᠯᠡᠬᠦ ᠠᠷ ᠭ ᠠ ᠪᠠᠷ ᠳᠡ ᠮᠳᠡᠭᠯᠡᠪ ᠡᠯ ( ) A. 0.93 48× 10 8 B. 9.3 48× 10 7 C. 9.34 8× 10 8 D. 93 .48× 10 6 3. ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠭᠡᠯᠢ ᠭ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠨᠢᠭᠡᠨ ᠴᠡᠭ ᠶᠢᠨ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠭᠰ ᠠᠨ ᠲᠣᠭ ᠠ ᠶᠢ 2 𝑎+ 1 ᠪ ᠡᠷ ᠢᠯ ᠡᠷᠭᠡᠢᠯᠡᠬ ᠦ ᠪᠥᠭᠡᠳ ᠴᠡᠭ A ᠡᠴᠡ ᠡᠬᠢᠨ ᠦ ᠴᠡᠭ ᠬᠦᠷᠬᠦ ᠵᠠᠢ ᠨᠢ 3 ᠪᠣᠯᠣ ᠨ ᠠ᠃ ᠲᠡᠭᠡᠪᠡᠯ 𝑎 ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠨᠢ ( ) A. − 2 ᠪᠤᠶ ᠤ 1 B. − 2 ᠪᠤᠶᠤ 2 C. − 2 D. 1 4. ᠳᠠᠷᠠᠭᠠᠬᠢ ᠪᠣᠳᠣ ᠯᠭ ᠠ ᠶᠢᠨ ᠳ᠋ᠦ ᠩ ᠦ ᠨ ᠵᠥᠪ ᠪᠣᠯᠬᠤ ᠨᠢ ( ) A. ( 𝑎 3 ) 2 = 𝑎 5 B. ( −𝑏𝑐 ) 5 ÷ (− 𝑏𝑐 ) 2 = −𝑏 2𝑐 2 C. 1 + 1𝑎 = 2𝑎 D. 𝑎 ÷ b∙ 1𝑏 = 1𝑏 2 5. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠤ ∠A CD ᠪᠣᠯ △ ABC ᠶᠢᠨ ᠭᠠᠳᠠᠭᠠᠳᠤ ᠥᠨᠴᠥᠭ ᠂᠂ ᠬᠡᠷᠪᠡ CE \\ AB , ∠A CB = 75° , ∠ECD = 50° ᠪᠣᠯ ∠A ᠶᠢᠨ ᠭᠷ ᠠᠳᠦᠰ ᠤᠨ ᠲᠣ ᠭ ᠠ ᠨᠢ ( ) A. 50° B. 55° C. 70° D. 75° 6. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠤ᠂ ᠶᠡᠬᠡ ᠪᠠᠭ ᠠ ᠨᠢ ᠠᠳᠠᠯ ᠢᠬᠠᠨ ᠵᠢᠷᠭᠤ ᠭᠠᠨ ᠵᠢᠵᠢᠭ ᠲᠡᠭᠰᠢ ᠳᠥᠷᠪᠡᠯᠵᠢ ᠨ ᠪᠡᠶ ᠡ ᠪᠡᠷ ᠦᠷᠦ ᠭᠰᠡᠨ ᠭᠧᠣ ᠮᠧᠲᠷᠢ ᠶᠢᠨ ᠵᠢᠷᠤ ᠭ ᠲᠥᠷᠰᠤ ᠠᠴᠠ ① ᠵᠢᠵᠢᠭ ᠲᠡᠭᠰᠢ ᠳᠥᠷᠪᠡᠯᠵᠢ ᠨ ᠪᠡᠶ ᠡ ᠶᠢ ᠰᠠᠯᠭᠠᠵᠦ ᠠᠪᠤᠪᠠᠯ᠂ ᠣᠯᠣ ᠭᠰᠠᠨ ᠭᠧᠣᠮᠧ ᠲᠷᠢ ᠶᠢ ᠨ ᠵᠢᠷᠤ ᠭ ᠲᠥᠷᠰᠤ ᠶᠢᠨ ( ) A. ᠭᠣᠣ ᠯ ᠬᠠ ᠷᠠᠭᠠᠨ ᠵᠢᠷᠤ ᠭ ᠨᠢ ᠬᠤᠪᠢᠷ ᠠᠬᠤ ᠦᠭᠡᠢ ᠂ ᠵᠡᠭᠦᠨ ᠬᠠᠷᠠ ᠭᠠᠨ ᠵᠢᠷᠤ ᠭ ᠨᠢ ᠬᠤᠪᠢᠷᠠᠨ ᠠ᠃ B. ᠳᠡᠭᠡᠷ ᠡ ᠬᠠ ᠷᠠᠭᠠᠨ ᠵᠢᠷᠤ ᠭ ᠨᠢ ᠬᠤᠪᠢᠷ ᠠᠬᠤ ᠦᠭᠡᠢ᠂ ᠵᠡᠭᠦᠨ ᠬᠠᠷᠠ ᠭᠠᠨ ᠵᠢᠷᠤ ᠭ ᠨᠢ ᠬᠤᠪᠢᠷᠠᠨ ᠠ C. ᠳᠡᠭᠡᠷ ᠡ ᠬᠠ ᠷᠠᠭᠠᠨ ᠵᠢᠷᠤ ᠭ ᠨᠢ ᠬᠤ ᠪᠢᠷᠠᠨ ᠠ᠂ ᠵᠡᠭᠦᠨ ᠬᠠᠷᠠ ᠭᠠᠨ ᠵᠢᠷᠤ ᠭ ᠨᠢ ᠬᠤᠪᠢᠷ ᠠᠨ ᠠ D. ᠭᠣᠣ ᠯ ᠬᠠ ᠷᠠᠭᠠᠨ ᠵᠢᠷ ᠤᠭ ᠬᠤᠪᠢᠷ ᠠᠬᠤ ᠦᠭᠡᠢ᠂ ᠵᠡᠭᠦ ᠨ ᠬᠠ ᠷᠠᠭᠠᠨ ᠵᠢᠷᠤ ᠭ ᠨᠢ ᠬᠤᠪᠢᠷ ᠠᠬᠤ ᠦᠭᠡᠢ᠃ 7. ᠬᠣᠶᠠᠷ ᠪᠥᠯᠥᠭ ᠲᠣᠭ ᠠ ᠪᠠᠷᠢ ᠮᠲᠠ 3 , 𝑎 ,𝑏 , 5 ᠪᠠ 𝑎 , 4 , 2𝑏 ᠶᠢ ᠨ ᠳᠣᠮᠳᠠᠴᠢ ᠲᠣᠭ ᠠ ᠨᠢ ᠴᠥᠮ 3᠂ ᠬᠡᠷᠪᠡ ᠡᠨᠡ ᠬᠣᠶᠠ ᠷ ᠪᠥᠯᠥᠭ ᠲᠣᠭ ᠠ ᠪ ᠠᠷᠢᠮᠲ ᠠ ᠶᠢ ᠨᠠᠢ ᠯᠡᠭᠦᠯ ᠦᠭᠡᠳ ᠨᠢᠭᠡ ᠪᠥᠯᠥᠭ ᠰᠢᠨ ᠡ ᠲᠣ ᠭ ᠠ ᠪᠠᠷᠢᠮᠳᠠ ᠪᠣᠯᠭᠠᠪᠠᠯ᠂ ᠡᠨᠡ ᠪ ᠥᠯᠥᠭ ᠰᠢᠨ ᠡ ᠲᠣᠭ ᠠ ᠪᠠᠷ ᠢᠮᠲᠠ ᠶᠢᠨ ᠮᠣᠳ᠋ ᠨᠢ ( ) A. 2 B. 3 C. 4 D. 5 8. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠤ᠂ 𝑅𝑡 ∆𝐴𝐵𝐶 ᠳ᠋ᠦ᠂ ∠A CB = 90° ᠪᠣᠯ ᠂ AB ᠶᠢᠨ ᠳᠤᠮᠲ ᠠ ᠴᠡᠭ C᠂ BE ⊥ CD ᠂ CD ᠶᠢᠨ ᠦᠷᠬᠦᠯᠵᠢ ᠯᠡᠯ ᠱᠤᠭᠤᠮ ᠲᠠᠢ ᠴᠡᠭ E ᠳ᠋ ᠦ ᠣ ᠭᠲᠣᠯᠣᠯᠴᠠᠨ ᠠ᠃ ᠬᠡᠷᠪᠡ AC = 2, BC = 2√ 2 ᠪᠣᠯ᠂ BE ᠶᠢᠨ ᠤᠷ ᠳᠤ ᠨᠢ ( ) A. 2 √63 B. √ 62 C. √ 3 D. √ 2 9. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠤ᠂ AB ᠪᠣᠯ ⊙ O ᠶᠢᠨ ᠳ᠋ᠢᠶᠠᠮᠧᠲ ᠷ᠂ CD ᠪᠣᠯ ᠬᠥᠪᠴᠢ᠂ ᠴᠡᠭ C,D ᠨᠢ ᠳ᠋ᠢ ᠶᠠᠮᠧᠲ ᠷ ᠶᠢᠨ ᠬᠣ ᠶᠠᠷ ᠲᠠᠯ ᠠ ᠳ᠋ᠤᠨᠢ ᠪᠠᠢᠨ ᠠ᠃ ᠬᠡᠷᠪ ᠡ ∠A OC :∠A OD :∠DOB = 2:7 :11 , CD = 4 ᠪᠣᠯ ᠂ CD ᠶᠢᠨ ᠤᠷ ᠳᠤ ᠨᠢ ( ) A. 2π B. 4π C. √ 2𝜋2 D. √ 2𝜋 10. ᠳᠠᠷᠠᠭᠠᠬᠢ ᠦᠬᠦᠯ ᠡᠯ ᠦᠨ ᠵᠥᠪ ᠪᠣᠯᠬᠤ ᠨ ᠢ ( ) A. ᠬᠡᠷᠪᠡ ᠪᠤᠲᠠᠷᠬᠠᠢ ᠢᠯᠡᠷᠭᠡᠢᠯᠡᠯ 𝑥 2− 4 𝑥− 2 ᠤᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠨᠢ 0 ᠪᠣᠯᠪᠠᠯ᠂ 𝑥 ᠦ ᠨ ᠬᠡᠮᠵᠢᠭᠡᠯ ᠨᠢ ±2 B. ᠨᠢᠭᠡᠨ ᠡᠶᠡᠷᠬᠦ ᠲᠣᠭᠠ ᠨ ᠤ ᠲᠣᠭ ᠠ ᠪᠣᠳᠣᠯ ᠭ ᠠ ᠶᠢᠨ ᠺᠦᠸᠠᠳ᠋ ᠢᠵᠠᠭᠤ ᠷ ᠨᠢ ᠯᠠᠪᠲᠠᠢ ᠡᠨᠡ ᠲᠣᠭ ᠠ ᠠᠴᠠ ᠪ ᠠᠭ ᠠ C. ᠬᠡᠷᠪᠡ 𝑏 𝑎 0 ᠪᠣᠯ 𝑎𝑏 𝑎+ 1 𝑏+ 1 D. ᠬᠡᠷᠪᠡ 𝑐≥ 2 ᠪᠣᠯ ᠨᠢᠭᠡ ᠥᠯᠥ ᠮᠡᠳᠡ ᠭᠳᠡᠭᠴᠢᠳᠡᠢ ᠬᠣᠶᠠ ᠳᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠶᠢᠨ ᠲᠡᠭᠰᠢᠳᠭᠡᠯ 𝑥 2 + 2𝑥 + 3= 𝑐 ᠨᠢ ᠪᠣᠳᠠᠳᠤ ᠲᠣ ᠭᠠᠨ ᠤ ᠢ ᠵᠠᠭᠤᠷ ᠲᠠᠢ ᠃ 11. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠤ᠂ ᠬᠠᠪᠲᠠᠭ ᠠᠢ ᠶᠢᠨ ᠲᠡᠭᠰᠢ ᠥᠨᠴᠥᠭᠳᠥ ᠪᠠᠢᠷᠢ ᠰᠢᠯ ᠤ ᠨ ᠰᠢᠰᠲ ᠧᠮ ᠳ᠋ᠦ᠂ ᠰᠢᠯᠤᠭᠤ ᠨ y= − 32 𝑥 + 3 ᠨᠢ 𝑥 ᠲᠡᠩᠭᠡᠯᠢ ᠭ y ᠲᠡᠩᠭᠡᠯᠢ ᠭ ᠲᠠᠢ ᠲᠤᠰ ᠲᠤᠰ ᠴᠡᠭ A ,B ᠳ᠋ᠤ ᠣᠭᠲ ᠣᠯᠣᠯᠴᠠᠨ ᠠ᠂ 𝐶 ᠪᠣᠯ ᠬᠡᠷᠴᠢᠮ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠨᠢᠭᠡᠨ ᠴᠡᠭ᠂ ᠴᠡᠭ 𝐶 ᠢ ᠳᠠᠭ ᠠᠷᠢᠭᠤᠯᠤ ᠨ 𝐶𝐷 ⊥ 𝑥 ᠲᠡᠩᠭᠡᠯᠢ ᠭ ᠢᠶᠡᠷ ᠪᠠᠢᠭᠤᠯᠤ ᠨ ᠠ᠂ ᠡᠭᠴ ᠡ ᠶᠢᠨ ᠰᠠᠭᠤᠷ ᠢ ᠨᠢ ᠴᠡᠭ 𝐷᠂ 𝐶𝐸 ⊥ 𝑦 ᠲᠡᠩᠭᠡᠯᠢ ᠭ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭᠤᠷ ᠢ ᠨᠢ 𝐸 ᠂ 𝑠∆ 𝐵𝐸𝐶 :𝑠 𝐶𝐷𝐴 = 4: 1᠂ ᠬᠡᠷᠪᠡ ᠬᠣᠣ ᠰ ᠮᠤᠷᠤ ᠢ ᠱᠤᠭᠤᠮ 𝑦 = 𝑘𝑥 ( 𝑥 0) ᠨᠢ ᠴᠡᠭ 𝐶 ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠪᠠᠯ᠂ 𝑘 ᠦ ᠨ ᠬᠡᠮ ᠵᠢᠭᠳᠡᠯ ᠨᠢ ( ) 12. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠤ᠂ 𝑅𝑡 ∆𝐴𝐵𝐶 ᠳ᠋ᠦ ∠𝐴𝐶𝐵 = 90° , 𝐵𝐶 𝐴𝐶 ᠂ ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠠᠯᠬᠤ ᠮ ᠢᠶᠠᠷ ᠵᠢᠷᠤ ᠭ ᠪᠠᠢᠭᠤᠯᠪᠠᠯ᠄ ᠴᠡᠭ 𝐴, 𝐵 ᠪᠠᠷ ᠲᠤᠰ ᠲᠤᠰ ᠲᠣᠭᠣ ᠷᠢᠭ ᠤ ᠨ ᠲᠥᠪ ᠪᠣᠯᠭᠠᠨ᠂ 12 𝐴𝐵 ᠡᠴᠡ ᠶᠡᠭᠡ ᠤᠷᠳᠤ ᠪ ᠠᠷ ᠷᠠᠳᠢᠤᠰ ᠪᠣᠯᠭᠠᠨ ᠨᠤᠮᠤ ᠪᠠᠢᠭ ᠤᠯᠪᠠᠯ᠂ ᠬᠣᠶᠠᠷ ᠨᠤᠮᠤ ᠨᠢ ᠴᠡᠭ 𝑀 ,𝑁 ᠳ᠋ᠤ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠᠨ ᠠ ( ᠴᠡᠭ 𝑀 ᠨᠢ 𝐴𝐵 ᠶᠢᠨ ᠳᠡᠭᠡᠷ ᠡ ᠲᠠᠯ ᠠ ᠳ᠋ᠤᠨᠢ ᠪᠠᠢᠨ ᠠ)᠂ ᠰᠢᠯᠤᠭᠤ ᠨ𝑀𝑁 ᠢ ᠪᠠᠢᠭᠤᠯᠪᠠᠯ 𝐴𝐵 ᠲᠠᠢ ᠴᠡᠭ 𝑂 ᠳ᠋ᠦ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠᠵᠤ᠂ 𝐵𝐶 ᠲᠠᠢ ᠴᠡᠭ 𝐷 ᠳ᠋ ᠤ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠᠨ ᠠ᠃ ᠭᠣᠷᠲ ᠢᠭ ᠢᠶᠠᠷ ᠰᠠ ᠴᠤᠷᠠᠭ 𝑂𝑀 ᠳᠡᠭᠡᠷ ᠡ𝑂𝐸 = 𝑂𝐷 ᠪᠡᠷ ᠣᠭᠲ ᠣᠯᠵᠤ ᠂𝐴𝐷 ,𝐴𝐸 ,𝐵𝐸 ᠶᠢ ᠬᠣᠯᠪᠣᠨ ᠠ᠃ ᠴ ᠡᠭ 𝑂 ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠭ ᠤᠯᠤᠨ 𝑂𝐹 ⊥ 𝐴𝐶 ᠪᠠᠷ ᠪᠠᠢᠭᠤ ᠯᠪᠠᠯ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ 𝐹, 𝐴𝐷 ᠲᠠᠢ ᠴᠡᠭ 𝐺 ᠳ᠋ᠤ ᠣᠭᠲ ᠣᠯᠣᠯᠴᠠᠨ ᠠ᠃ ᠳᠣᠣ ᠷᠠᠬᠢ ᠳ᠋ᠦᠩᠨᠡᠯᠲ ᠡ᠄① 𝐶𝐷 = 2𝐺𝐹 ;② 𝐵𝐷 2− 𝐶𝐷 2= 𝐴𝐶 2; ③ 𝑆∆ 𝐵𝑂𝐸 = 2𝑆 ∆𝐴𝑂𝐺 ; ④ ᠬᠡᠷ ᠪᠡ 𝐴𝐶 = 6, 𝑂𝐹 + 𝑂𝐴 = 9 ᠪᠣᠯ ᠲᠥᠷᠪᠡᠯᠵ ᠢᠨ𝐴𝐷𝐵𝐸 ᠶᠢᠨ ᠬᠦᠷᠢᠶᠡ ᠨ ᠦ ᠤᠷᠳᠤ ᠨᠢ 25 ᠃ ᠡᠭᠦ ᠨ ᠳ᠋ ᠦ ᠵᠥᠪ ᠳ᠋ ᠦᠩᠨᠡᠯᠲᠡ ᠨᠢ ( ) A. 1 ᠪᠠᠢᠨ ᠠ B. 2 ᠪᠠᠢᠨ ᠠ C. 3 ᠪᠠᠢᠨ ᠠ D. 4 ᠪᠠᠢᠨ ᠠ ᠬᠤᠶᠠᠷ᠂ ᠬᠠᠭᠠᠯᠲ ᠠ ᠲ ᠠᠭᠯᠠᠬ ᠤ ᠰᠡᠲᠥᠪ ᠃ ( ᠰᠡᠲᠥ ᠪ ᠪᠦᠷ ᠢ 3 ᠬᠤ ᠪᠢ᠂ ᠨᠡᠢᠲ ᠡ 24 ᠬᠤᠪᠢ) 13. ᠹᠦᠩᠺ ᠼ𝑦 = 𝑥𝑥−3 ᠳ᠋ᠦ ᠬᠤᠪᠢᠰᠬᠤ ᠨ 𝑥 ᠤᠨ ᠬᠡᠮ ᠵᠢᠭᠳ ᠡᠯ ᠠᠪᠬᠤ ᠬᠡᠪ ᠴᠢᠶ ᠡ ᠨᠢ ( ) 14. ᠪᠤᠲ ᠠᠷᠬᠠᠢ ᠢᠯ ᠡᠷᠭᠡᠢ ᠯᠡᠯᠲ ᠤ ᠲᠡᠭᠰᠢᠳᠭ ᠡᠯ 3 −𝑥𝑥−2 + 𝑥2−𝑥 = 1 ᠦ ᠨ ᠬᠠ ᠷᠢᠭᠤ ᠨᠢ ( ) 15. ᠪᠣᠳᠣ᠄ (√ 3+ √2 )( √3 − √2 ) 2 = ( ) ᠃ 16. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠤ᠂ ᠲᠡᠭᠰᠢ ᠲᠥᠷᠪᠡᠯᠵ ᠢᠨ𝐴𝐵𝐶𝐷 ᠳ᠋ᠦ᠂ 𝐸 ᠪᠣᠯ ᠳᠠᠰᠢᠭᠤ ᠪᠴᠢ 𝐵𝐷 ᠳᠡᠭᠡᠷᠡᠬᠢ ᠨᠢᠭᠡᠨ ᠴᠡᠭ᠂ 𝐴𝐸 ᠶᠢᠨ ᠦᠷᠬᠦᠯᠵᠢ ᠯᠡᠯ ᠱᠤᠭᠤᠮ ᠨᠢ 𝐶𝐷 ᠲᠠᠢ ᠴᠡᠭ 𝐹 ᠳ᠋ᠤ ᠣᠭᠲ ᠣᠯᠣᠯᠴᠠᠨ ᠠ᠂𝐶𝐸 ᠶᠢ ᠬᠣᠯᠪᠣᠨ ᠠ᠂ ᠬᠡᠷᠪᠡ ∠B AE = 56 ° ᠪᠣᠯ ∠CEF = ( ) 17. ᠬᠠ ᠲᠠᠭᠤ ᠴᠠᠭᠠᠰᠤ ᠨ ᠬᠠᠢᠷᠴᠠᠭ ᠲᠤ ᠪᠦᠷᠢᠮᠥᠰᠥᠨ ᠠᠳᠠᠯᠢ ᠬᠠᠨ ᠭᠤᠷᠪᠠ ᠨ ᠺᠠᠷ ᠲ ᠢ ᠬᠢᠵᠦ ᠳᠡᠭᠡᠷ ᠡ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ 1, 2,3 ᠭᠡᠰᠡᠨ ᠨᠣᠮᠧ ᠷ ᠢ ᠪᠢᠴᠢ ᠪᠡ᠃ ᠬᠠᠢᠷᠴᠠᠭ ᠳᠣᠲ ᠣᠷ ᠠ ᠠᠴᠠ ᠳᠠᠰᠢᠷ ᠠᠮ ᠢ ᠶᠠᠷ ᠳᠤᠷ ᠠ ᠶᠢ ᠨ ᠨᠢᠭᠡ ᠺᠠᠷ ᠲ ᠢ ᠰᠤᠭᠤᠯᠵ ᠤ ᠪᠤᠴᠠᠭᠠᠭᠰᠠ ᠨ ᠤ ᠳᠠᠷᠠᠭ ᠠ᠂ ᠳᠠᠬᠢᠨ ᠳᠠᠰᠢᠷᠠᠮ ᠢ ᠶᠠᠷ ᠨᠢᠭᠡ ᠺᠠᠷ ᠲ ᠢ ᠰᠤᠭᠤᠯ ᠤᠨ ᠠ᠃ ᠬᠣᠶᠠᠳᠤ ᠭᠠᠷ ᠤᠳᠠᠭ ᠠᠨ ᠤ ᠰᠤᠭᠤᠯ ᠤᠭᠰᠠᠨ ᠺᠠᠷ ᠲ ᠤᠨ ᠨᠣᠮᠧᠷ ᠨᠢ ᠨᠢᠭᠡᠳᠤ ᠭᠡᠷ ᠤᠳᠠᠭ ᠠ ᠰᠤᠭᠤᠯ ᠤᠭᠰᠠᠨ ᠺᠠᠷ ᠲ ᠤᠨ ᠨᠣᠮᠧᠷ ᠠᠴᠠ ᠶᠡᠬᠡ ᠪᠣᠯᠬᠤ ᠶᠢᠨ ᠳᠣᠬ ᠢᠶᠠᠯᠳᠤᠮᠵᠢ ᠨᠢ ( ) 18. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠤ᠂ □ 𝐴𝐵𝐶𝐷 ᠳ᠋ᠦ 𝐴𝐵 = 2 , ∠ABC ᠶᠢᠨ ᠲᠠᠯᠠᠯᠠᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠪᠠ ∠BCD ᠶᠢᠨ ᠲᠠᠯ ᠠᠯᠠᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ ᠨᠢ ᠴᠡᠭ 𝐸 ᠲᠦ ᠣᠭᠲ ᠣᠯᠣᠯᠴᠠᠨ ᠠ᠂ ᠬᠡᠷᠪᠡ ᠴᠡᠭ 𝐸 ᠨᠢ ᠶᠠᠭ ᠰᠠᠢᠬᠠᠨ AD ᠳᠡᠭᠡᠷ ᠡ ᠪᠠᠢᠪ ᠠᠯ᠂ BE 2+ CE 2 ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠨᠢ ( ) 19. ᠬᠠ ᠪᠲᠠᠭᠠᠢ ᠶᠢᠨ ᠲᠡᠭᠰᠢ ᠥᠨᠴᠥᠭ ᠳᠥ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠤᠨ ᠰᠢᠰᠲ ᠧᠮ ᠳ᠋ᠦ᠂ ᠦᠭᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢA (− 1, m) ᠪᠠ 𝐵( 5, 𝑚) ᠨᠢ ᠬᠠᠶᠠ ᠯᠲᠠ ᠶᠢᠨ ᠱᠤᠭᠤᠮ 𝑦= 𝑥 2 + 𝑏𝑥 + 1 ᠤ ᠨ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠬᠣᠶᠠᠷ ᠴᠡᠭ᠂ ᠬᠠᠶᠠᠯᠲ ᠠ ᠶᠢᠨ ᠱᠤᠭᠤᠮ 𝑦 = 𝑥 2 + 𝑏𝑥 + 1 ᠤ ᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠢ ᠳᠡᠭᠡᠭᠰᠢ 𝑛 (𝑛 ᠪᠣᠯ ᠡᠶᠡᠷᠬᠦ ᠪᠥᠬᠥᠯᠢ ᠲᠣᠭ ᠠ) ᠨᠢᠭᠡᠴᠢ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠵᠢᠬᠦᠯᠦᠭᠰᠡᠨ ᠦ ᠳᠠᠷᠠᠭᠠ ᠬᠢ ᠵᠢᠷᠤ ᠭ ᠲᠥᠷᠰᠤ ᠨᠢ 𝑥 ᠲᠡᠩᠭᠡᠯᠢ ᠭ ᠲᠠᠢ ᠣᠭᠲ ᠣᠯᠣᠯᠴᠠᠯ ᠤ ᠨ ᠴᠡᠭ ᠦᠭᠡᠢ᠂ ᠲᠡᠭᠡᠪᠡᠯ 𝑛 ᠤ ᠬᠠᠮᠤᠭ ᠪᠠᠭ ᠠ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠨᠢ ( ) 20. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠤ᠂ ᠲᠡᠭᠰᠢ ᠥᠨ ᠴᠥᠭᠳᠥ 𝐴𝐵𝐶𝐷 ᠳ᠋ᠦ᠂ 𝐵𝐷 ᠪᠣᠯ ᠳᠠᠰᠢᠭ ᠤᠪᠴᠢ᠂ 𝐴𝐸 ⊥ 𝐵𝐷 ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭᠤᠷ ᠢ ᠨᠢ 𝐸 ᠂ 𝐶𝐸 ᠶᠢ ᠬᠣᠯᠪᠣᠨ ᠠ᠂ ᠬᠡᠷᠪᠡ ∠ADB = 30 ° ᠪᠣᠯ tan ∠DEC ᠤᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠨᠢ ( ). ᠭᠤᠷᠪᠠ᠂ ᠪᠤᠳᠤᠵ ᠤ ᠬᠠᠷᠢᠭᠤᠯᠬᠤ ᠰᠡᠲ ᠥᠪ᠃ ( ᠨᠡ ᠢᠲᠡ6 0 ᠬᠤ ᠪᠢ) 21. (ᠲᠤᠰ ᠰᠡᠲᠤᠪ ᠤᠨ ᠪᠦᠷᠢᠨ ᠬᠤᠪ ᠢ ᠨᠢ 8 ᠬᠤᠪ ᠢ) ᠮᠠᠨ ᠤ ᠤᠯᠤ ᠰ ᠤᠨ 5𝐺 ᠤᠨ ᠮᠡᠷᠭᠡᠵᠢᠯ ᠨᠢ ᠲᠦᠷᠭ ᠡᠨ ᠬᠦᠭᠵᠢᠵᠦ᠂ ᠪᠥᠬᠥ ᠳᠡᠯᠭᠡᠢ ᠶᠢᠨ ᠲᠡᠷᠢᠬᠦᠨ ᠦ ᠬᠡᠮᠵ ᠢᠶᠡᠨ ᠳ᠋ ᠦ ᠬᠦᠷᠪᠡ᠃ ᠨᠢᠭᠡᠨ ᠺᠣᠮ ᠫᠠᠨᠢ ᠠ ᠴᠠ ᠨᠢᠭᠡ ᠵᠦᠢᠯ ᠦᠨ5 𝐺 ᠶᠢᠨ ᠰᠢᠨ ᠡ ᠪᠦᠳᠦᠭ ᠡᠭᠳᠡᠭᠦᠨ ᠢ ᠦᠢᠯᠡᠳᠪᠤᠷ ᠢᠯᠡᠪᠡ ᠃ ᠬᠡᠷᠡᠭᠯᠡᠭᠴᠢᠳ ᠦᠨ ᠲᠤᠰ ᠪᠦᠳᠦᠭ ᠡᠭᠳᠡᠬ ᠦᠨ ᠳ᠋ ᠦ ᠥᠭᠬᠦ ᠬᠠᠨᠤᠮ ᠵᠢ ᠶᠢᠨ ᠪᠠᠢᠳᠠᠯ ᠢ ᠣᠢᠯᠠᠭᠠᠬᠤ ᠶᠢᠨ ᠲᠥᠯᠥ ᠭᠡ 30 ᠡᠷᠤᠭᠡ ᠳ᠋ ᠤ ᠳᠠᠰᠢᠷ ᠠᠮ ᠤ ᠨ ᠪᠠᠢᠴᠠ ᠭᠠᠯᠲ ᠠ ᠬᠢᠵᠦ᠂ ᠣᠯᠣ ᠭᠰᠠᠨ ᠦᠨᠡᠯᠡᠯᠭᠡ ᠶᠢᠨ ᠬᠤᠪᠢ ᠨᠢ ᠳᠣᠣ ᠷᠠᠬᠢ ᠮᠡᠲᠤ ( ᠨᠢᠭᠡᠴᠢ᠄ ᠬᠤ ᠪᠢ)᠄ 83 9 2 68 55 7 7 71 73 6 2 7 3 95 92 94 72 64 59 66 7 1 75 69 8 6 87 79 8 1 7 7 68 82 62 77 61 88 ᠳᠡᠭᠡᠷᠡᠬᠢ ᠲᠣᠭ ᠠ ᠪᠠᠷᠢ ᠮᠲᠠ ᠶᠢ ᠡᠮᠬᠢ ᠯᠡᠭᠰᠡᠨ ᠦ ᠳᠠᠷᠠᠭ ᠠ ᠨᠢᠭᠡᠨ ᠪᠦᠷᠢᠨ ᠪᠤᠰᠤ ᠳᠠᠪᠳᠠᠭᠳᠠ ᠭᠰᠠᠨ ᠲᠣᠭᠠᠨ ᠤ ᠰᠢᠯᠤᠭᠤ ᠨ ᠳᠥᠷᠪᠡᠯᠵᠢ ᠨ ᠵ ᠢᠷᠤᠭ ᠢ ᠣᠯᠪᠠ ᠃ ᠦᠭᠭᠦᠭᠳ ᠡᠭᠰᠡᠨ ᠮᠡᠳᠡ ᠭᠡ ᠵᠠᠩᠭᠢ ᠢ ᠦᠨᠳᠤᠰᠤ ᠯᠡᠨ ᠳᠣᠣ ᠷᠠᠬᠢ ᠠ ᠰᠠᠭᠤᠯ ᠲᠠ ᠶᠢ ᠬᠠᠷᠢᠭᠤᠯ᠃ (1) ᠳᠠᠪ ᠲᠠᠭᠳ ᠠᠭᠰᠠᠨ ᠲᠣ ᠭᠠᠨ ᠤ ᠰᠢᠯᠤ ᠭᠤᠨ ᠲᠥᠷ ᠪᠡᠯᠵᠢᠨ ᠵᠢᠷᠤ ᠭ ᠢ ᠪᠦᠷ ᠢᠨ ᠪᠣᠯᠭ ᠠ᠃ (2) ᠪᠠᠢᠴᠠ ᠭᠠᠯᠲ ᠠ ᠳ᠋ᠤ ᠣᠷᠣ ᠯᠴᠠᠭᠰᠠᠨ ᠨᠢᠭᠡᠨ ᠡᠷᠤᠭᠡ ᠠᠴᠠ᠄ ︽ ᠮᠢᠨᠤ ᠦᠨᠡᠯᠡᠯᠭ ᠡ ᠶᠢᠨ ᠬᠤᠪᠢ ᠨᠢ ᠡᠨᠡ 30 ᠡᠷᠤᠭᠡ ᠶᠢᠨ ᠲᠤᠲ ᠤᠷᠠᠬᠢ ᠳᠤᠮᠲ ᠠ ᠪᠠᠢᠷᠢ ᠶᠢᠨ ᠲᠣᠭ ᠠ ᠪᠣᠯᠣ ᠨ ᠠ︾ ᠭᠡᠵᠠᠢ᠃ ᠲᠡᠭᠡᠪᠡᠯ ᠲᠤᠰ ᠡᠷᠤᠭᠡ ᠶᠢᠨ ᠦᠨᠡᠯᠡ ᠯᠭᠡ ᠶᠢᠨ ᠬᠤᠪᠢ ᠨᠢ ( ) ᠬᠤᠪᠢ᠃ (3) ᠪᠠᠢᠴᠠ ᠭᠠᠭᠳ ᠠᠬᠤ ᠡᠷᠤᠭᠡ ᠶᠢᠨ ᠦᠨᠡᠯᠡᠯᠡᠭᠡ ᠶᠢᠨ ᠬᠤᠪᠢ ᠠᠴᠠ ᠦᠵᠡᠪᠡᠯ ᠬᠡᠷᠡᠭᠯᠡᠭ ᠴᠢᠳ ᠦ ᠨ ᠦᠨᠡᠯᠡᠯᠭᠡ ᠶᠢᠨ ᠳᠡᠰ ᠢ ᠭᠤᠷ ᠪᠠᠨ ᠳᠡᠰ ᠬᠤᠪᠢᠶᠠᠵᠠᠢ ᠄ ᠲᠤᠰ ᠺᠣᠮᠫᠠᠨᠢ ᠶᠢᠨ 5𝐺 ᠪᠦᠳᠦᠭ ᠡᠭᠳᠡᠭᠦᠨ ᠢ ᠬᠡᠷᠡᠭᠯᠡᠵᠦ ᠪᠠᠢᠬ ᠤ 1500 ᠡᠷᠤ ᠭᠡ ᠶ ᠢᠨ ᠦᠨᠡᠯᠡᠯᠭᠡ ᠶᠢᠨ ᠳᠡᠰ ᠨᠢ ︽ᠲᠤ ᠢᠯ ᠤᠨ ᠬᠠᠨᠤᠮᠵᠢᠲ ᠠᠢ︾ ᠪᠣᠯᠬᠤ ᠬᠦᠮᠦᠨ ᠦ ᠲ ᠣᠭ ᠠ ᠶᠢ ᠪᠠᠷ ᠤᠭᠴᠠᠭ ᠠᠯᠠᠨ ᠪᠣᠳᠣ ᠃ ᠦᠨᠡ ᠯᠡᠭᠡ ᠶᠢᠨ ᠬᠤᠪᠢ ᠦᠨᠡ ᠯᠡᠭᠡ ᠶᠢᠨ ᠳᠡᠰ 60 ᠬᠤᠪ ᠢ ᠠᠴᠠ ᠪ ᠠᠭ ᠠ ᠬᠠᠨᠤ ᠮᠵᠢ ᠦᠭᠡᠢ 60 ᠬᠤᠪ ᠢ ᠠᠴᠠ 89 ᠬᠤᠪᠢ ᠬᠠᠨᠤ ᠮᠵᠢᠲᠠᠢ 90 ᠬᠤᠪ ᠢ ᠠᠴᠠ ᠪ ᠠᠭ ᠠ ᠪᠤᠰᠤ ᠲᠤᠢᠯ ᠤ ᠨ ᠬᠠ ᠨᠤᠮᠵᠢᠲᠠ ᠢ 22. (ᠲᠤᠰ ᠰᠡᠲᠤᠪ ᠤᠨ ᠪᠦᠷᠢᠨ ᠬᠤᠪᠢ ᠨᠢ 8 ᠬᠤᠪᠢ ) ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠤ᠂ ᠨᠢᠭᠡ ᠨ ᠬᠦᠮᠦᠨ ᠳᠤᠭᠤᠢ ᠤᠨᠤᠭᠠᠳ 𝐴 ᠭᠠᠵᠠ ᠷ ᠠᠴᠠ 𝐶 ᠭᠠᠵᠠᠷ ᠬᠦᠷᠬᠦ ᠳ᠋ ᠦ 𝐵 ᠭᠠᠵᠠᠷ ᠢᠶᠠᠷ ᠳᠠᠭᠠᠷᠢᠨ ᠠ᠃ ᠲᠡᠷᠡ 𝐴 ᠭᠠᠵᠠᠷ ᠠᠴ ᠠ ᠭᠠᠷᠬᠤ ᠳ᠋ᠤ ᠲᠡᠭ ᠦᠨ ᠦ ᠤᠮᠠᠷ ᠠ ᠠᠴᠠ ᠵᠡᠭᠦᠨᠰᠢ 45° ᠤᠨ ᠴᠢᠭᠯᠡᠯ ᠳ᠋ ᠦ ᠨᠢᠭᠡᠨ ᠲᠧᠯᠸᠢ ᠰ ᠰᠣᠪᠣᠷᠭ ᠠ 𝑃 ᠪᠠᠢᠬᠤ ᠶᠢ ᠣᠯᠵ ᠤ ᠬᠠ ᠷᠠᠵᠠᠢ᠃ ᠲᠡᠷᠡ 𝐴 ᠭᠠ ᠵᠠᠷ ᠠᠴᠠ ᠠᠰ ᠬᠣᠢᠰᠢ 3√ 2𝑘𝑚 ᠶᠠᠪᠤᠭᠠᠳ 𝐵 ᠭᠠᠵᠠᠷ ᠬᠦᠷᠬᠦ ᠦᠶᠡᠰ᠂ ᠲᠧᠯᠸᠢᠰ ᠰᠣᠪᠣᠷᠭ ᠠ 𝑃 ᠨᠢ ᠲᠡᠭᠦᠨ ᠦ ᠤᠮᠠᠷ ᠠ ᠠᠴᠠ ᠵᠡᠭ ᠦᠨᠰᠢ 75° ᠤᠨ ᠴᠢᠭᠯᠡᠯ ᠳ᠋ ᠦ ᠪᠠᠢᠬᠤ ᠶᠢ ᠮᠡᠳᠡᠪ ᠡ᠃ ᠲᠡᠭᠡᠭᠡᠳ ᠲᠡᠷᠡ 𝐵 ᠭᠠᠵᠠ ᠷ ᠠᠴᠠ ᠭᠠᠷᠴ ᠤ ᠤᠮᠠ ᠷ ᠠ ᠠᠴᠠ ᠵᠡᠭᠦᠨ ᠰᠢ 15° ᠤᠨ ᠴᠢᠭᠯᠡ ᠯ ᠢᠶᠡᠷ ᠪᠠᠰᠠ 6𝑘𝑚 ᠶᠠᠪᠤᠭ ᠠᠳ 𝐶 ᠭᠠ ᠵᠠᠷ ᠬᠦᠷ ᠦᠭᠰᠡᠨ ᠪᠠ ᠢᠨ ᠠ᠃ (1) 𝐴 ᠭᠠᠵᠠ ᠷ ᠠᠴᠠ ᠲᠧᠯᠸᠢᠰ ᠰᠣᠪᠣᠷᠭ ᠠ 𝑃 ᠬᠦᠷᠬᠦ ᠵᠠᠢ ᠶᠢ ᠣᠯ ᠃ (2) 𝐶 ᠭᠠᠵᠠ ᠷ ᠠᠴᠠ ᠲᠧᠯᠸᠢᠰ ᠰᠣᠪᠣᠷᠭ ᠠ 𝑃 ᠬᠦᠷᠬᠦ ᠵᠠᠢ ᠶᠢ ᠣᠯ ᠃ 23. (ᠲᠤ ᠰ ᠰᠡᠲᠤᠪ ᠤᠨ ᠪᠦᠷᠢᠨ ᠬᠤᠪᠢ ᠨᠢ10 ᠬᠤᠪᠢ) ᠨᠢᠭᠡᠨ ᠳᠡᠯᠭᠡᠭᠦᠷ 𝐴, 𝐵 ᠬᠣᠶᠠᠷ ᠵᠦᠢᠯ ᠦᠨ ᠪᠠᠷᠠᠭ ᠠ ᠶᠢ ᠪᠣᠷᠣ ᠯᠠᠭᠤᠯᠪᠠ᠃ 𝐴ᠪᠠᠷᠠᠭ ᠠ ᠶᠢᠨ ᠪᠣᠷᠣ ᠯᠠᠭᠤᠯᠭ ᠠ ᠶᠢᠨ ᠨᠢᠭᠡᠴᠢ ᠦᠨ ᠡ ᠨᠢ𝐵 ᠪᠠᠷᠠᠭ ᠠ ᠶᠢᠨ ᠪᠣᠷᠣ ᠯᠠᠭᠤᠯᠭ ᠠ ᠶᠢᠨ ᠨᠢᠭᠡᠴᠢ ᠦᠨ ᠡ ᠡᠴ ᠡ 40 ᠲᠥᠬᠥᠷᠢ ᠭ ᠬᠢᠮᠳᠠ᠂ 2 ᠬᠤᠪᠢ 𝐴 ᠪᠠᠷᠠᠭ ᠠ ᠪᠠ 3 ᠬᠤᠪᠢ 𝐵 ᠪᠠᠷᠠᠭ ᠠ ᠶᠢᠨ ᠪᠣᠷᠣ ᠯᠠᠭᠤᠯᠤ ᠭᠰᠠᠨ ᠪᠥᠬᠥ ᠦᠨ ᠡ ᠨᠢ 82 0 ᠲᠥ ᠬᠥᠷᠢᠭ᠃ (1) 𝐴 ᠪᠠᠷᠠᠭ ᠠ ᠪᠠ 𝐵 ᠪᠠᠷᠠᠭ ᠠ ᠶᠢᠨ ᠪᠤᠷᠤᠯᠠᠭ ᠤᠯᠭ ᠠ ᠶᠢᠨ ᠨᠢᠭᠡ ᠴᠢ ᠦᠨ ᠡ ᠨᠢ ᠲᠣᠰ ᠲᠣᠰ ᠬᠡᠳᠦᠢ ᠲᠥᠭᠦᠷᠢᠭ ᠪᠣᠢ ︖ (2) ᠲᠤᠰ ᠳᠡᠯᠭᠡᠭᠦᠷ ᠡᠴᠡ 𝐴, 𝐵 ᠬᠣᠶᠠᠷ ᠵᠦᠢᠯ ᠦᠨ ᠪᠠᠷᠠᠭ ᠠ ᠶᠢ ᠨᠠᠢᠲᠡ 60 ᠬᠤᠪᠢ ᠣᠷᠣ ᠭᠤᠯᠬᠤ ᠪᠠᠷ ᠲᠥᠯᠥ ᠪᠯᠡᠪᠡ᠃ 𝐴, 𝐵ᠪᠠ ᠷᠠᠭ ᠠ ᠶᠢᠨ ᠣᠷᠣ ᠭᠤᠯᠤ ᠭᠰᠠᠨ ᠪᠥᠬᠥ ᠦᠨ ᠡ ᠨᠢ 7800 ᠲᠥᠬᠥᠷᠢᠭ ᠡᠴᠡ ᠳᠠᠪᠠᠵᠤ ᠪᠣᠯᠬᠤ ᠦᠭ ᠡᠢ᠂ 𝐴 ᠪᠠᠷᠠᠭ ᠠ ᠪᠠ 𝐵 ᠪᠠᠷᠠᠭ ᠠ ᠬᠤᠪᠢ ᠪᠦᠷᠢ ᠶᠢᠨ ᠣᠷᠣ ᠭᠤᠯᠤ ᠭᠰᠠᠨ ᠦᠨ ᠡ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ 110 ᠲᠥᠬᠥᠷᠢᠭ ᠪᠠ 140 ᠲᠥᠬᠥᠷᠢᠭ᠃ ᠪᠠᠷᠠᠭ ᠠ ᠶᠢ ᠬᠡᠷᠬᠢ ᠨ ᠣᠷᠣ ᠭᠤᠯᠪᠠᠯ ᠡᠨᠡ ᠬᠣᠶᠠᠷ ᠵᠦᠢᠯ ᠦᠨ ᠪᠠᠷᠠᠭ ᠠ ᠶᠢ ᠪᠥᠬᠥᠨ ᠢ ᠬᠤᠳᠠᠯᠳᠤ ᠵᠤ ᠪᠠᠷᠠᠭᠰᠠᠨ ᠤ ᠳᠠᠷᠠᠭ ᠠ ᠶᠡᠷᠤᠩ ᠬᠡᠢ ᠠᠰᠢᠭ ᠬᠣᠨᠵᠢᠪᠤᠷ ᠢ ᠨᠢ ᠬᠠᠮᠤᠭ ᠤᠨ ᠶᠡᠬᠡ ᠪ ᠣᠯᠬᠤ ᠪᠣᠢ ︖ ᠬᠣᠢᠨ ᠠ ᠠ ᠵᠡᠭᠦᠨ 24. (ᠲᠤ ᠰ ᠰᠡᠲᠤᠪ ᠤᠨ ᠪᠦᠷᠢᠨ ᠬᠤᠪ ᠢ ᠨ ᠢ 10 ᠬᠤᠪᠢ) ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠤ᠂ 𝐴𝐵 ᠪᠣᠯ ᠶᠢᠨ ⊙ 𝑂 ᠳ᠋ᠢᠶᠠᠮᠧᠲ ᠷ᠂ ᠷᠠᠳᠢᠤᠰ 𝑂𝐶 ⊥ 𝐴𝐵 ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ 𝑂 ᠂ ᠰᠢᠯᠤ ᠭᠤᠨ 𝑙 ᠪᠣᠯ ⊙ 𝑂 ᠶᠢᠨ ᠰᠢᠬᠠᠭᠴᠢ ᠰᠢᠯᠤᠭᠤ ᠨ᠂ 𝐴 ᠪᠣᠯ ᠰᠢᠬᠠᠯᠲ ᠠ ᠶᠢᠨ ᠴᠡᠭ᠂ 𝐷 ᠨᠢ 𝑂𝐴 ᠳᠡᠭᠡᠷᠡᠬᠢ ᠨᠢᠭᠡ ᠨ ᠴᠡᠭ᠂ 𝐶𝐷 ᠦᠨ ᠦᠷᠭᠦᠯᠵᠢ ᠯᠡᠯ ᠱᠤᠭ ᠤᠮ ᠨᠢ ᠰᠢᠯᠤᠭᠤ ᠨ 𝑙 ᠲᠠᠢ ᠴᠡᠭ 𝐸 ᠳ᠋ᠦ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠᠨ ᠠ᠂ 𝐹 ᠪᠣᠯ 𝑂𝐵 ᠳᠡᠭᠡᠷᠡᠬᠢ ᠨᠢᠭᠡᠨ ᠴᠡᠭ᠂ 𝐶𝐷 ᠦᠨ ᠦᠷᠭᠦᠯᠵᠢ ᠯᠡᠯ ᠱᠤᠭᠤᠮ ᠨᠢ ⊙ 𝑂 ᠲᠠᠢ ᠴᠡᠭ 𝐺 ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠᠨ ᠠ᠂ 𝐴𝐶 , 𝐴𝐺 ᠢ ᠬᠣᠯᠪᠤᠨ ᠠ᠃ ᠦᠭ ᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ⊙ 𝑂 ᠶᠢᠨ ᠷᠠᠳᠢᠤᠰ ᠨᠢ 3᠂ 𝐶𝐸 = √34 , 5𝐵𝐹 − 5𝐴𝐷 = 4᠃ (1) 𝐴𝐸 ᠶᠢᠨ ᠤᠷ ᠳᠤ ᠶᠢ ᠣ ᠯ᠃ (2) cos ∠𝐶𝐴𝐺 ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠪᠠ 𝐶𝐺 ᠶᠢᠨ ᠤᠷ ᠳᠤ ᠶᠢ ᠣᠯ᠃ 25. (ᠲᠤ ᠰ ᠰᠡᠲᠤᠪ ᠤᠨ ᠪᠦᠷᠢᠨ ᠬᠤᠪ ᠢ ᠨᠢ 1 2 ᠬᠤᠪᠢ) ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠤ᠂ Rt ∆ABC ᠳ᠋ᠦ᠂ ∠𝐴𝐶𝐵 = 90° ,𝐴𝐶 = 4, 𝐵𝐶 = 2, 𝑅𝑡 ∆𝐴𝐵𝐶 ᠶᠢ ᠴᠡᠭ 𝐶 ᠶᠢ ᠲᠣᠭᠣ ᠷᠢᠭᠤ ᠯᠤᠨ ᠴᠠᠭ ᠤ ᠨ ᠵᠡᠭ ᠦᠦ ᠶᠢᠨ ᠳᠠᠭᠠᠤ ᠴᠢᠭᠯᠡ ᠯ ᠢᠶᠡ ᠷ ᠡᠷᠬ ᠢᠭᠦᠯᠦᠭ ᠡᠳ 𝑅𝑡 ∆𝐴 ′𝐵 ′𝐶′ ᠶᠢ ᠣᠯᠣ ᠨ ᠠ᠂ 𝐴 ′ 𝐶ᠨᠢ 𝐴𝐵 ᠲᠠᠢ ᠴᠡᠭ 𝐷 ᠳ᠋ᠤ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠᠨ ᠠ᠃ (1) ᠵᠢ ᠷᠤᠭ ᠮᠡᠲᠤ᠂ 𝐴 ′ 𝐵 ′// 𝐴𝐶 ᠦᠶᠡᠰ᠂ ᠴᠡᠭ 𝐵 ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠭᠤᠯᠤ ᠨ 𝐵𝐸 ⊥ 𝐴′𝐶 ᠪᠠᠷ ᠪᠠᠢᠭ ᠤᠯᠤᠨ ᠠ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭᠤᠷ ᠢ ᠨᠢ 𝐸 ᠂ 𝐴𝐸 ᠶᠢ ᠬᠣᠯ ᠪᠣᠨ ᠠ᠃ ① ᠪᠠᠳᠤ ᠯᠠᠭᠳᠠᠬᠤ ᠨᠢ᠄ 𝐴𝐷 = 𝐵𝐷 ② 𝑆∆ 𝐴𝐶𝐸 𝑆∆ 𝐴𝐵𝐸 ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭᠳ ᠡᠯ ᠢ ᠣᠯ᠃ (2) ᠵᠢ ᠷᠤᠭ ᠮᠡᠲᠤ᠂ 𝐴′𝐶 ⊥ 𝐴𝐵 ᠦᠶᠡᠰ᠂ ᠴᠡᠭ 𝐷 ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠭᠤᠯᠤ ᠨ𝐷𝑀 // 𝐴 ′ 𝐵 ′ ᠪ ᠠᠷ ᠪᠠᠢᠭᠤᠯᠤ ᠨ ᠠ᠂ 𝐵 ′𝐶 ᠲᠠᠢ ᠴᠡᠭ 𝑁 ᠳ᠋ᠤ ᠣᠭᠲ ᠣᠯᠣᠯᠴᠠᠵᠤ᠂ 𝐴𝐶 ᠶᠢᠨ ᠦᠷᠬᠦᠯᠵᠢ ᠯᠡᠯ ᠱᠤᠭᠤᠮ ᠲᠠᠢ ᠴᠡᠭ 𝑀 ᠳ᠋ᠤ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠᠨ ᠠ᠂ 𝐷𝑁𝑁𝑀 ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠢ ᠣ ᠯ᠃ 26. (ᠲᠤ ᠰ ᠰᠡᠲᠤᠪ ᠤᠨ ᠪᠦᠷᠢᠨ ᠬᠤᠪᠢ ᠨᠢ 12 ᠬᠤᠪᠢ) ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ᠂ ᠬᠠ ᠪᠲᠠᠭᠠ ᠢ ᠶᠢᠨ ᠲᠡᠭᠰ ᠢ ᠥᠨᠴᠥᠭᠳᠥ ᠪᠠᠢᠷᠢ ᠰᠢᠯ ᠤ ᠨ ᠰᠢᠰᠲ ᠧᠮ ᠳ᠋ ᠦ᠂ ᠬᠠ ᠶᠠᠯᠲ ᠠ ᠶᠢᠨ ᠱᠤᠭᠤᠮ 𝑦 = 13 𝑥 2− 2𝑥 ᠨᠢ ᠡᠬᠢᠨ ᠦ ᠴᠡᠭ ᠢ ᠳᠠᠭᠠᠷᠢᠵᠤ᠂ 𝑥 ᠲᠡᠩᠭᠡᠯᠢ ᠭ ᠦᠨ ᠡᠶᠡᠷᠬᠦ ᠬᠠᠭ ᠠᠰ ᠲᠡᠩᠭ ᠡᠯᠢᠭ ᠲᠠᠢ ᠴᠡᠭ 𝐴 ᠳ᠋ᠤ ᠣᠭᠲ ᠣᠯᠣᠯ ᠴᠠᠨ ᠠ᠂ ᠲᠤᠰ ᠬᠠᠶᠠᠯᠲ ᠠ ᠶᠢᠨ ᠱᠤᠭᠤᠮ ᠤᠨ ᠣᠷᠣᠢ ᠶᠢᠨ ᠴᠡᠭ ᠨᠢ𝑀 ᠂ ᠰᠢᠯᠤᠭᠤ ᠨ 𝑦 = − 12 𝑥 + 𝑏 ᠨᠢ ᠴᠡᠭ 𝐴 ᠶᠢ ᠳᠠᠭᠠᠷᠢᠵᠤ᠂ 𝑦 ᠲᠡᠩᠭᠡᠯᠢ ᠭ ᠲᠠᠢ ᠴᠡᠭ 𝐵 ᠳ᠋ᠤ ᠣᠭᠲ ᠣᠯᠣᠯᠴᠠᠨ ᠠ᠂ 𝑂𝑀 ᠶᠢ ᠬ ᠣᠯᠪᠣᠨ ᠠ᠃ (1) 𝑏 ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠪᠠ ᠴᠡᠭ 𝑀 ᠶᠢᠨ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠢ ᠣᠯ᠃ (2) ᠰᠢᠯᠤᠭᠤ ᠨ𝐴𝐵 ᠶᠢ ᠳᠣᠷ ᠣᠭᠰᠢ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠵᠢᠬᠦᠯᠦᠭ ᠡᠳ ᠴᠡᠭ 𝑀 ᠶᠢ ᠳᠠᠭᠠᠷᠢᠭᠤᠯ ᠤᠨ ᠰᠢᠯᠤᠭᠤ ᠨ 𝑦 = 𝑚𝑥 + 𝑛 ᠢ ᠣᠯᠵ ᠤ᠂ ᠲᠡᠷᠡ ᠨᠢ 𝑥 ᠲᠡᠩ ᠭᠡᠯᠢᠭ ᠦ ᠨ ᠰᠥᠭᠡᠷᠬᠦ ᠬᠠᠭᠠᠰ ᠲᠡᠩᠭᠡᠯᠢ ᠭ ᠲᠠᠢ ᠴᠡᠭ 𝐶 ᠳ᠋ ᠤ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠᠨ ᠠ᠂ ᠴᠡᠭ 𝐷 (2 ,0) ᠶᠢ ᠠᠪᠴᠤ᠂ 𝐷𝑀 ᠶᠢ ᠬᠣᠯᠪᠣᠨ ᠠ᠂ ᠪᠠᠳᠤ ᠯᠠᠭᠳᠠᠬᠤ ᠨᠢ᠄ ∠𝐴𝐷𝑀 − ∠𝐴𝐶𝑀 = 45 ° (3) ᠴᠡᠭ 𝐸 ᠨᠢ ᠬᠡᠷᠴᠢᠮ 𝐴𝐵 ᠳᠡᠭᠡᠷᠡᠬᠢ ᠨᠢᠭᠡᠨ ᠬᠦᠳᠡᠯᠬᠦ ᠴᠡᠭ ᠂ ᠴᠡᠭ 𝐹 ᠨᠢ ᠬᠡᠷᠴᠢᠮ 𝑂𝐴 ᠳᠡᠭᠡᠷᠡᠬᠢ ᠨᠢᠭᠡᠨ ᠬᠦᠳᠡᠯᠬ ᠦ ᠴᠡᠭ᠂ 𝐸𝐹 ᠶᠢ ᠬᠣᠯᠪᠣᠨ ᠠ᠂ ᠬᠡᠷᠴᠢᠮ 𝐸𝐹 ᠦᠨ ᠦᠷᠬᠦᠯᠵᠢ ᠯᠡᠯ ᠱᠤᠭᠤᠮ ᠨᠢ ᠬᠡᠷᠴᠢᠮ 𝑂𝑀 ᠲᠠᠢ ᠴᠡᠭ 𝐺 ᠳ᠋ᠤ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠᠨ ᠠ᠂ ∠𝐵𝐸𝐹 = 2∠ 𝐵𝐴𝑂 ᠦᠶᠡᠰ᠂ 3𝐺𝐹 = 4𝐸𝐹 ᠪᠣᠯᠭᠠᠬᠤ ᠴᠡᠭ 𝐸 ᠣᠷᠣ ᠰᠢᠨ ᠠ ᠤᠤ ︖ ᠬᠡᠷᠪᠡ ᠣᠷᠣ ᠰᠢᠪᠠᠯ ᠴᠡᠭ 𝐸 ᠦᠨ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠢ ᠣᠯ᠃ ᠬᠡᠷᠪᠡ ᠣᠷ ᠣᠰᠢᠬ ᠤ ᠦᠭᠡᠢ ᠪ ᠣᠯ ᠰᠢᠯᠲᠠᠭᠠᠨ ᠢ ᠨᠢ ᠲ ᠣᠳᠣᠷ ᠬᠠᠢ ᠯᠠ᠃