2020 ᠣᠨ ᠤ ᠣᠷᠳᠣᠰ ᠬᠤᠲᠠ ᠶᠢᠨ ᠳᠤᠮᠳᠠᠳᠤ ᠰᠢᠯᠭᠠᠯᠲᠠ ᠶᠢᠨ ᠰᠡᠲᠤᠪ

ᠨᠢᠭᠡ᠂ ᠭᠠᠭᠴᠠ ᠬᠠᠷᠢ ᠭᠤᠯᠲᠠ ᠰᠤᠩᠭᠤ ᠬᠤ ᠰᠡᠲᠤᠪ ᠃ ( ᠲᠤᠰ ᠶᠡᠭᠡ ᠰᠡᠲᠤ ᠪ ᠲ ᠤ 10 ᠪᠠᠭ ᠠ ᠰ ᠡᠲᠤᠪ ᠲᠠᠢ᠂ ᠪᠠᠭ ᠠ ᠰᠡᠲᠤᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪ ᠢ᠂ ᠨᠠᠢᠲᠡ 30 ᠬᠤᠪᠢ ) 1. ᠪᠣᠳᠠᠳᠤ ᠲᠣᠭ ᠠ − √3 ᠶᠢᠨ ᠬᠠᠷᠢᠴᠠᠯ ᠠᠰᠢᠬᠦ ᠢ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠨᠢ ( ) A. √3 B. − √33 C. −√ 3 D. √33 2. ᠨᠢᠭᠡᠨ ᠪᠡᠶᠡᠳᠤ ᠵᠢᠷ ᠤᠭ ᠲᠥᠷᠰᠤ ᠶᠢ ᠨ ᠭᠤᠷᠪ ᠠᠨ ᠬᠠᠷᠠᠭ ᠠᠨ ᠵᠢᠷᠤᠭ ᠨᠢ ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠮᠡᠲᠤ ᠦᠭ ᠭᠦᠭᠳᠡᠵᠡᠢ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ᠲᠡᠭᠦᠨ ᠲᠠᠢ ᠬᠠᠷᠠᠭ ᠠᠯᠵᠠᠭᠰᠠᠨ ᠪᠡᠶᠡᠳᠤ ᠵᠢᠷ ᠤᠭ ᠲᠥᠷᠰ ᠤ ᠨᠢ ( ) A. B. C. D. 3. ᠹᠦᠩᠺᠼ 𝑦 = √𝑥 + 3 ᠳᠦ᠂ ᠬᠤᠪᠢᠰ ᠬᠤᠨ 𝑥 ᠤ ᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠠᠪᠬ ᠤ ᠬᠡ ᠪᠴᠢᠶ ᠡ ᠶᠢ ᠲᠣᠭ ᠠᠨ ᠤ ᠲᠡᠩᠭᠡᠯᠢᠭ ᠳᠡᠭᠡ ᠷ ᠡ ᠵᠥᠪ ᠢᠯ ᠡᠷᠭᠡᠢᠯ ᠡᠭᠰᠡᠨ ᠨᠢ ( ) A. B. C. D. 4. ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠪᠣᠳᠣ ᠯᠭ ᠠ ᠠᠴ ᠠ ᠪᠤᠷᠤᠭ ᠤ ᠪᠣᠯ ᠬᠤ ᠨᠢ ( ) A. (− 3𝑎 𝑏 2 ) 2 = 9𝑎 2𝑏 4 B. −6 𝑎 3 𝑏÷ 3𝑎𝑏 = −2 𝑎 2 C. (𝑎 2) 3− (− 𝑎 3 ) 2 = 0 D. (𝑥 + 1) 2= 𝑥 2 + 1 5. ᠨᠢᠭᠡᠨ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠤᠤ ᠢ ᠶᠢ ᠨᠢᠭᠡ ᠬᠠᠭ ᠤᠳᠠᠰᠤ ᠲᠡ ᠭᠰᠢ ᠥᠨᠴᠥ ᠭᠳᠥ ᠬᠡᠯᠪᠡ ᠷᠢᠲᠠᠢ ᠴᠠᠭᠠᠰᠤᠨ ᠬᠠᠪ ᠲᠠᠰᠤ ᠨ ᠳᠡᠭᠡᠷ ᠡ ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ ᠲᠠᠯᠪᠢᠵᠠᠢ᠂ ∠ 𝐸𝐺𝐹 = 90° , ∠𝐹 𝐸𝐺 = 30° , ∠1 = 125° ᠪᠣᠯ ∠𝐵𝐹𝐺 ᠶᠢᠨ ᠶᠡᠬᠡ ᠪᠠ ᠭ ᠠ ᠨᠢ ( ) A. 125° B. 115° C. 110° D. 120° 6. ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠬᠥᠰᠥ ᠨᠥᠭᠳᠥ ᠪ ᠣᠯ ᠨᠢᠭᠡᠨ ᠠᠩᠭᠢ ᠶᠢᠨ 5 ᠰᠤᠷᠤᠭᠴᠢ ᠶᠢᠨ ᠲᠣᠭ ᠠᠨ ᠤ ᠰᠢᠯ ᠭᠠᠯᠲ ᠠ ᠶᠢᠨ ᠬᠢᠷᠢ ᠪᠦᠳ ᠦᠮᠵᠢ ᠶᠢᠨ ᠪᠦᠷᠢᠳᠭᠡᠯ ᠪᠣᠯᠣᠨ ᠠ ᠄ ᠲᠡᠭᠡᠪ ᠡᠯ 𝑎 , 𝑏 ᠬᠣᠶ ᠠᠷ ᠲ ᠣᠭ ᠠ ᠪᠠᠷᠢᠮᠲᠠ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ ( ) A. 81 , 80 B. 80 , 2 C. 81 , 2 D. 80 , 80 7. ᠲᠥᠷᠪ ᠡᠯᠵᠢᠨ 𝐴𝐵𝐶𝐷 ᠳᠦ᠂ 𝐴𝐷 // 𝐵𝐶 ᠂ ∠𝐷 = 90° ᠂ 𝐴𝐷 = 8᠂ ᠭᠡᠰᠢᠬ ᠦᠨ ᠬᠢᠷᠢ ᠪᠦᠳᠦ ᠮᠵᠢ ( ᠬᠤᠪ ᠢ) ᠠ 77 ᠡ 81 ᠢ 𝑎 ᠣ 80 ᠦ 82 ᠳᠤᠮᠳᠠᠴ ᠢ ᠲ ᠣᠭ ᠠ 80 ᠮᠤᠳ᠋ 𝑏 1 F DA B G E C ᠤᠷᠳ᠋ᠤ ᠰ ᠬᠤᠳ᠋ᠠ ᠢᠢᠨ 20 20 ᠣ ᠨ ᠤ ᠠᠩᠬ ᠠ ᠲᠤ ᠮᠳ᠋ᠠᠳ᠋ᠤ ᠢᠢ ᠨ ᠰᠤᠷ ᠤᠯᠭ ᠠ ᠲᠡᠭ ᠦᠰᠴ ᠤ ᠰᠤᠷ ᠭᠠᠭ ᠤᠯᠢ ᠲᠠᠪᠰ ᠢᠬᠤ ᠰᠢᠯ ᠭᠠᠯᠳ᠋ᠠ ᠢᠢᠨ ᠰᠡᠳ᠋ᠤ ᠪ 𝐵𝐶 = 6᠂ ᠴᠡᠭ 𝐴 ᠪᠠ ᠴᠡᠭ 𝐶 ᠪᠠᠷ ᠲᠤᠰ ᠲᠤᠰ ᠲᠣᠭ ᠣᠷᠢᠭ ᠤᠨ ᠲᠥ ᠪ ᠪᠣᠯᠭ ᠠᠨ᠂ 12 𝐴𝐶 ᠠᠴᠠ ᠶᠡᠭᠡ ᠪᠣᠯᠬ ᠤ ᠤᠷᠳᠤ ᠪᠠᠷ ᠷᠠᠳᠢᠤᠰ ᠪᠣᠯᠭ ᠠᠨ ᠨᠤᠮᠤ ᠪᠠᠢᠭ ᠤᠯᠪᠠ ᠯ᠂ ᠬᠣᠶ ᠠᠷ ᠨᠤᠮ ᠤ ᠨᠢ ᠴᠡᠭ 𝐸 ᠳᠦ ᠣ ᠭᠲᠣᠯ ᠣᠯᠴᠠᠨ ᠠ᠂ ᠰᠠᠴᠤ ᠷᠠᠭ 𝐵𝐸 ᠶᠢ ᠪᠠᠢᠭ ᠤᠯᠪᠠ ᠯ 𝐴𝐷 ᠲᠠᠢ ᠴᠡᠭ 𝐹 ᠳ ᠦ ᠣ ᠭᠳᠣᠯ ᠣᠯᠴᠠ ᠵᠤ᠂ 𝐴𝐶 ᠲᠠᠢ ᠴᠡᠭ 𝑂 ᠳᠦ ᠣᠭᠳᠣᠯ ᠣᠯᠴᠠ ᠨ ᠠ᠂ ᠬᠡᠷ ᠪᠡ ᠴᠡᠭ 𝑂 ᠨᠢ 𝐴𝐶 ᠶᠢᠨ ᠳᠤᠮᠳᠠ ᠴᠡᠭ ᠪᠣᠯ᠂ 𝐶𝐷 ᠦᠨ ᠤᠷᠳ ᠤ ᠨᠢ ( ) A. 4√ 2 B. 2 √10 C. 6 D. 8 8. ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠬᠡᠯᠡᠯᠡᠭᠡ ᠡᠴᠡ ᠵᠥ ᠪ ᠪᠣ ᠯᠬᠤ ᠨᠢ ( ) ① √ 5− 1 2 12 ② ᠵᠥᠪ ᠵᠢᠷ ᠭᠤᠭᠠ ᠨ ᠥᠨᠴ ᠥᠭᠳᠥ ᠶᠢᠨ ᠲᠣᠳᠣᠭᠠ ᠳᠤ ᠥᠨᠴ ᠥᠭ ᠦ ᠨ ᠨᠠ ᠢᠯᠡᠪᠤᠷ ᠢ ᠨᠢ 72 0° ᠂ ᠲᠡ ᠭᠦᠨ ᠦ ᠲᠠ ᠯ ᠠ ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ ᠷᠠᠳᠢᠤᠰ ᠲᠠᠢ ᠪᠠ ᠨ ᠲᠡ ᠩᠴᠡᠬᠦ ᠦ᠃ ③ ᠨᠢ ᠭᠡ ᠢ ᠵᠢ ᠡᠩ ᠦᠨ ᠬᠦᠵ ᠦᠷ ᠲᠣᠲᠣᠷ ᠡ ᠠᠴᠠ ᠳᠠᠰᠢᠷᠠ ᠮ ᠪᠠ ᠷ ᠨᠢ ᠭᠡ ᠬᠠᠭ ᠤᠳᠠᠰᠤ ᠶᠢ ᠰᠤᠭᠤᠯ ᠤᠨ ᠠᠪᠬᠤ ᠳᠤ᠂ ᠲᠡᠷᠡ ᠨᠢ ᠬᠠᠷ ᠠ ᠲᠣᠭ ᠣᠷ ᠪᠣᠯᠬᠤ ᠶᠢᠨ ᠲᠣᠬᠢᠶ ᠠᠯᠳᠤᠮ ᠵᠢ ᠨᠢ 14 ᠃ ④ ᠠ ᠡ ᠬᠣ ᠶᠠᠷ ᠬᠦᠮᠦ ᠨ ᠲᠤᠰ ᠲᠤᠰ 10 ᠤᠳᠠ ᠭᠠᠨ ᠤ ᠬᠠᠷᠪᠣ ᠯᠲᠠ ᠶᠢᠨ ᠪᠣᠯ ᠪᠠᠰᠤ ᠷᠠᠯ ᠬᠢᠭᠰ ᠡᠨ ᠳ ᠦ᠂ ᠳᠤᠮᠳᠠ ᠴᠢ ᠬᠢᠷ ᠢ ᠪᠦᠳᠦ ᠮᠵᠢ ᠨᠢ ᠠ ᠳᠠᠯᠢ᠂ ᠰ ᠠᠷᠨᠢᠴᠠ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ 𝑆 ᠠ 2= 1.3 , 𝑆 ᠡ 2= 1.1 ᠪᠣᠯᠣᠨ ᠠ᠃ ᠲᠡ ᠭᠡᠪᠡᠯ ᠡ ᠶᠢᠨ ᠬᠠᠷᠪᠤᠯ ᠲᠠ ᠶᠢᠨ ᠬᠢ ᠷᠢ ᠪ ᠦᠳᠦ ᠮᠵᠢ ᠨᠢ ᠠ ᠠᠴᠠ ᠳᠣᠭᠲᠠᠪ ᠤᠷᠢᠳ ᠠᠢ᠃ A. ① ② ③ ④ B. ① ② ④ C. ① ④ D. ② ③ 9. ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ᠂ ᠲᠠ ᠯ ᠠ ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ 1 ᠪᠣᠯᠬᠤ ᠲᠡᠭᠰᠢ ᠲᠥ ᠷᠪᠡᠯ ᠵᠢᠨ 𝑂𝐴 𝐴1 𝐵1 ᠳᠦ᠂ ᠲᠠᠰᠢᠭᠤᠪ ᠴᠢ ᠪᠠᠷ ᠲᠠ ᠯ ᠠ ᠪᠣᠯᠭ ᠠᠨ ᠬᠣᠶ ᠠᠳᠤᠭ ᠠᠷ ᠲᠡᠭᠰᠢ ᠲᠥᠷᠪ ᠡᠯᠵᠢᠨ 𝑂 𝐴1 𝐴2 𝐵2 ᠶᠢ ᠪᠠᠢᠭ ᠤᠯᠤᠨ ᠠ᠂ 𝐴1 𝐴2 ᠶᠢ ᠬᠣᠯᠪᠣᠪᠠᠯ ∆𝐴 𝐴1 𝐴2 ᠢ ᠣ ᠯᠣᠨ ᠠ᠃ ᠳᠠ ᠬᠢᠨ ᠲᠠᠰᠢᠭᠤᠪ ᠴᠢ 𝑂 𝐴2 ᠪᠠᠷ ᠲᠠᠯ ᠠ ᠪᠣᠯᠭ ᠠᠨ ᠭᠤᠷᠪ ᠠᠳᠤᠭ ᠠᠷ ᠲᠡᠭᠰᠢ ᠲᠥᠷᠪ ᠡᠯᠵᠢ ᠨ 𝑂 𝐴2 𝐴3 𝐵3 ᠶᠢ ᠪᠠᠢᠭ ᠤᠯᠤᠨ ᠠ᠂ 𝐴1 𝐴3 ᠶᠢ ᠬᠣ ᠯᠪᠣᠪ ᠠᠯ ∆𝐴 1 𝐴 2 𝐴 3 ᠢ ᠣᠯᠣᠨ ᠠ᠃ ᠪ ᠠᠰᠠ ᠳᠠ ᠬᠢᠨ ᠳᠠᠰᠢ ᠭᠤᠪ ᠴᠢ 𝑂𝐴 3 ᠪᠠ ᠷ ᠲᠠ ᠯ ᠠ ᠪᠣᠯᠭ ᠠᠨ ᠲᠥᠷᠪ ᠡᠳᠤᠭᠡᠷ ᠲᠡᠭᠰᠢ ᠲᠥᠷᠪ ᠡᠯᠵᠢ ᠨ 𝑂𝐴 3 𝐴 4 𝐵 4 ᠶᠢ ᠪᠠᠢᠭ ᠤᠯᠤᠨ ᠠ᠂ 𝐴2 𝐴4 ᠶᠢ ᠬᠣᠯ ᠪᠣᠪᠠ ᠯ ∆ 𝐴2 𝐴3 𝐴4 ᠶᠢ ᠣᠯᠣ ᠨ ᠠ᠁ ᠬᠡᠷᠪᠡ ∆𝐴 𝐴1 𝐴2 , ∆ 𝐴1 𝐴2 𝐴3 , ∆ 𝐴2 𝐴3 𝐴4 ᠶᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠶᠢ ᠲᠤᠰ ᠲᠤᠰ 𝑆1 , 𝑆2 , 𝑆3 ᠁ ᠭᠡᠵ ᠤ ᠵᠢᠱ ᠢᠵᠤ᠂ ᠡᠨ ᠡ ᠮᠡᠲᠤ ᠪ ᠠᠷ ᠦᠷᠬ ᠦᠯᠵᠢᠯ ᠡᠨ ᠪᠠᠢᠭ ᠤᠯᠪᠠ ᠯ 𝑆 2020 ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠨᠢ ( ) A. 1 2 2020 B. 2 2018 C. 2 2018 + 12 D. 1010 10. ᠣᠷ ᠳᠣᠰ ᠤᠨ ᠠᠮᠢ ᠲᠠᠨ ᠤ ᠬᠦᠷᠢᠶ ᠡᠯᠡᠩ ᠲᠣᠳᠣᠷᠠ ᠬᠢ ᠨᠢ ᠭᠡ ᠭᠡᠰᠡᠭ ᠠᠶ ᠠᠯᠠᠯ ᠤᠨ ᠵᠠᠮ ᠨᠢ ᠵᠢᠷᠤᠭ 1 ᠮᠡᠲᠤ᠂ ᠠᠮᠢᠲᠠ ᠨ ᠤ ᠬᠦᠷᠢ ᠶᠡᠯᠡᠩ ᠲᠣᠳᠣ ᠷᠠᠬᠢ ᠨᠠᠢᠲᠡ ᠶᠢᠨ ᠮᠠᠰᠢᠨ ᠲᠡ ᠷᠭᠡ ᠨᠢ ᠣ ᠷᠣᠬᠤ ᠡᠭᠦᠳᠡ ᠡᠴᠡ ᠡᠬᠢᠯ ᠡᠨ F O DA B C E A 3 A 4 A 2 A 1 AO B 1B 2 B 3 B 4 ᠮᠣᠷᠳᠠᠵᠤ᠂ ᠲᠤᠰ ᠠᠶ ᠠᠯᠠᠯ ᠤᠨ ᠵᠠᠮ ᠢᠶᠠᠷ ᠶᠠᠪᠤᠵ ᠤ ᠵᠠ ᠭᠠᠨ ᠤ ᠣᠷᠳᠤᠨ ᠳᠤ ᠬᠦᠷᠦ ᠨ ᠡ᠂ ᠠᠶ ᠠᠯᠠᠯ ᠤ ᠨ ᠶᠠᠪᠤ ᠴᠠ ᠳᠤ ᠰᠢᠪᠠ ᠭᠤᠨ ᠤ ᠬᠦ ᠷᠢᠶᠡᠯᠡ ᠩ ᠳ ᠦ ᠵᠣᠭ ᠰᠣᠨ ᠠ (ᠲᠡ ᠷᠭᠡᠨ ᠳ ᠦ ᠭᠠᠷ ᠴᠤ ᠪᠠᠭᠤᠬ ᠤ ᠴᠠ ᠭ ᠢ ᠪᠣᠳᠣᠯ ᠬᠢᠯᠠᠬᠤ ᠦᠭᠡᠢ)᠃ ᠨᠢᠭᠡᠳᠤᠭᠡᠷ ᠡᠭᠡᠯ ᠵᠢ ᠶ ᠢᠨ ᠮᠠᠰᠢᠨ ᠲᠡ ᠷᠭᠡ ᠨᠢ ᠦᠳᠡ ᠶᠢᠨ ᠡᠮ ᠤᠨ ᠡ ᠶᠢᠨ 9:2 0 ᠡ ᠴᠡ ᠡᠬᠢᠯ ᠡᠨ ᠮᠣᠷᠳᠠᠵᠤ᠂ ᠳᠠᠷᠠᠭ ᠠ ᠨᠢ 10 ᠮᠢ ᠨᠦ᠋ᠲ ᠪᠦᠷᠢ ᠳᠦ ᠨᠢ ᠭᠡ ᠡᠭᠡᠯ ᠵᠢ ᠶᠢ ᠨ ᠮᠠ ᠰᠢᠨ ᠲᠡ ᠷᠭᠡ ᠣᠷᠣᠬ ᠤ ᠡᠭᠦᠳᠡ ᠨ ᠡᠴᠡ ᠡᠬᠢᠯ ᠡᠨ ᠮᠣᠷᠳᠠᠬᠤ ᠪᠥᠭᠡᠳ ᠬᠤᠷᠳᠤᠴ ᠠ ᠨᠢ ᠴᠥ ᠮ ᠠ ᠳᠠᠯᠢ᠃ ᠰᠣᠶᠣᠯ ᠭᠠᠷᠠᠭ ᠤᠨ ᠡᠳᠤᠷ ᠠᠮᠢᠳᠠ ᠨ ᠤ ᠬᠦᠷᠢ ᠶᠡᠩ ᠳ ᠦ ᠵᠤᠭᠠ ᠴᠠᠬᠤ ᠪᠠᠷ ᠦᠳᠡ ᠶᠢᠨ ᠡᠮᠤᠨ ᠡ ᠶᠢᠨ 9 ᠴᠠᠭ ᠲᠤ ᠣᠷᠣᠬ ᠤ ᠡ ᠭᠦᠳᠡᠨ ᠳᠦ ᠬᠦᠷᠴᠡᠢ ᠂ ᠨᠠ ᠢᠲᠡ ᠶᠢᠨ ᠮᠠᠰᠢᠨ ᠲᠡ ᠷᠭᠡ ᠶᠢᠨ ᠶᠠᠪᠤᠬ ᠤ ᠴᠠᠭ ᠨᠢ ᠪᠣᠯᠣᠭ ᠠ ᠦᠭᠡ ᠢ ᠤᠴᠢᠷ ᠠᠴᠠ ᠰ ᠤᠶᠤᠯ ᠣ ᠷᠣᠬᠤ ᠡᠭᠦᠳᠡ ᠨ ᠡᠴ ᠡ ᠡᠬ ᠢᠯᠡᠨ ᠠᠶᠠᠯᠠᠯ ᠤᠨ ᠵᠠᠮ ᠳᠠ ᠭᠠᠤ 25 ᠮᠢ ᠨᠦ᠋ᠲ ᠶᠠᠪᠠᠭ ᠠᠨ ᠶᠠᠪᠤᠵ ᠤ᠂ ᠰᠢᠪᠠ ᠭᠤᠨ ᠤ ᠬᠦᠷᠢᠶᠡᠯ ᠡᠩ ᠳᠦ ᠬᠦᠷᠴᠡᠢ᠃ ᠣᠷ ᠣᠬᠤ ᠡᠭᠦᠳᠡ ᠨ ᠡ ᠴᠡ ᠵᠠᠢᠳᠠᠭᠰᠠᠨ ᠵᠠᠮ ᠤ ᠨ ᠤᠷᠳᠤ 𝑦 (ᠮᠧ ᠲᠷ) ᠪᠣᠯᠣᠨ ᠬᠤᠭᠤᠴᠠᠭ ᠠ 𝑥 (ᠮᠢᠨᠦ᠋ᠲ ) ᠦᠨ ᠬᠣᠭᠣᠷ ᠣᠨᠳᠣᠬ ᠢ ᠹᠦᠩᠺ ᠼ ᠦ ᠨ ᠬᠣ ᠯᠪᠣᠭᠳᠠᠯ ᠨᠢ ᠵᠢᠷᠤᠭ 2 ᠮᠡᠲᠤ᠃ ᠲᠡ ᠭᠡᠪᠡᠯ ᠳᠠᠷᠠ ᠭᠠᠬᠢ ᠬᠡᠯᠡᠯᠭᠡ ᠡᠴᠡ ᠪᠤᠷᠤᠭ ᠤ ᠪ ᠣᠯᠬᠤ ᠨᠢ᠄ ( ) A . ᠨᠢ ᠭᠡᠳᠤᠭᠡᠷ ᠡᠭᠡ ᠯᠵᠢ ᠶᠢᠨ ᠮᠠ ᠰᠢᠨ ᠲᠡᠷᠭᠡ ᠶᠢᠨ ᠣᠷᠣ ᠬᠤ ᠡᠭᠦᠳᠡ ᠨ ᠡᠴ ᠡ ᠵᠠᠢᠳᠠᠭᠰᠠᠨ ᠵᠠ ᠮ ᠤᠨ ᠤᠷ ᠳᠤ 𝑦 (ᠮᠧ ᠲᠷ) ᠪ ᠣᠯᠣᠨ ᠬᠤᠭᠤᠴᠠᠭ ᠠ 𝑥 (ᠮᠢ ᠨᠦ᠋ᠲ ) ᠦᠨ ᠬᠣᠭᠣᠷ ᠣᠨᠳᠣᠬ ᠢ ᠹᠦ ᠩᠺᠼ ᠦᠨ ᠬᠣᠯᠪᠣᠭᠳᠠᠯ ᠤᠨ ᠠ ᠨᠠᠯᠢᠰ ᠢ ᠯᠡᠷᠭᠡᠢᠯ ᠡᠯ ᠨᠢ 𝑦= 200 𝑥− 400 (20 ≤ 𝑥≤ 38 ) B. ᠨᠢᠭᠡᠳᠤᠭᠡᠷ ᠡᠭᠡᠯᠵ ᠢ ᠶᠢᠨ ᠮᠠᠰᠢᠨ ᠲᠡ ᠷᠭᠡ ᠨᠢ ᠣ ᠷᠣᠬᠤ ᠡᠭᠦᠳᠡ ᠨ ᠡᠴ ᠡ ᠰ ᠢᠪᠠᠭ ᠤᠨ ᠤ ᠬᠦᠷᠢᠶ ᠡᠯᠡᠩ ᠬᠦᠷᠲᠡᠯ ᠡ 1 0 ᠮᠢ ᠨᠦ᠋ᠲ ᠬᠡᠷᠡᠭᠰ ᠡᠨ ᠡ᠃ C. ᠰᠣᠶ ᠣᠯ ᠰ ᠢᠪᠠᠭ ᠤᠨ ᠤ ᠬᠦ ᠷᠢᠶᠡᠯ ᠡᠩ ᠳᠦ 40 ᠮᠢᠨᠦ᠋ ᠲ ᠲᠣᠭ ᠣᠷᠢᠭ ᠰᠠᠨ ᠤ ᠳᠠ ᠷᠠᠭ ᠠ᠂ ᠨᠠᠢᠲᠡ ᠶᠢᠨ ᠮᠠᠰᠢᠨ ᠲᠡ ᠷᠭᠡᠨ ᠳ ᠦ ᠰ ᠠᠭᠤᠭᠠ ᠳ ᠵᠠᠭᠠᠨ ᠤ ᠣᠷᠳᠤ ᠨ ᠳ ᠤ ᠬᠦ ᠷᠬᠦ ᠭᠡᠪᠡᠯ᠂ ᠬᠠᠮᠤᠭ ᠡᠷ ᠲᠡ ᠳ ᠦ ᠪᠡᠨ ᠲᠥᠷᠪ ᠡᠳᠤᠭᠡᠷ ᠡᠭᠡᠯᠵ ᠢ ᠶᠢ ᠨ ᠮᠠ ᠰᠢᠨ ᠲᠡᠷ ᠭᠡᠨ ᠳ ᠦ ᠰᠠ ᠭᠤᠵᠤ ᠳᠠᠢᠯ ᠦᠨ ᠡ᠃ D. ᠰᠤᠶᠤᠯ ᠰᠢᠪᠠ ᠭᠤᠨ ᠤ ᠬ ᠦᠷᠢᠶᠡᠯ ᠡᠩ ᠳᠦ 40 ᠮᠢ ᠨᠦ᠋ᠲ ᠲᠣᠭ ᠣᠷᠢᠵᠤ ᠪᠠᠷᠠᠭ ᠠᠳ᠂ ᠲᠠ ᠪᠤᠳᠤ ᠭᠠᠷ ᠡᠭᠡᠯᠵ ᠢ ᠶᠢ ᠨ ᠮᠠ ᠰᠢᠨ ᠲᠡ ᠷᠭᠡᠨ ᠳ ᠦ ᠰ ᠠᠭᠤᠭᠠ ᠳ ᠵᠠᠭᠠᠨ ᠤ ᠤᠷᠳᠤᠨ ᠳᠤ ᠬᠦᠷᠬ ᠦ ᠴᠠ ᠭ ᠨᠢ ᠲᠡ ᠭᠦᠨ ᠦ ᠰ ᠢᠪᠠᠭ ᠤᠨ ᠤ ᠬᠦᠷᠢᠶ ᠡᠯᠡᠩ ᠳᠦ ᠲᠣᠭ ᠣᠷᠢᠵᠤ ᠪᠠᠷᠠᠮ ᠠᠭᠴᠠ ᠰᠢᠭ ᠤᠳ ᠶᠠᠪᠠᠭ ᠠᠨ ᠶᠠᠪᠤ ᠵᠤ ᠵᠠ ᠭᠠᠨ ᠤ ᠤᠷᠳᠤᠨ ᠳᠤ ᠬᠦ ᠷᠬᠦ ᠴᠠ ᠭ ᠠᠴᠠ 7 ᠮᠢᠨᠦ᠋ᠲ ᠤᠷ ᠢᠳᠴᠢᠯ ᠠᠨ ᠠ (ᠰᠣ ᠶᠣᠯ ᠤ ᠨ ᠶᠠᠪ ᠠᠭᠠᠨ ᠶᠠᠪᠤᠬ ᠤ ᠳᠤᠮᠳᠠ ᠴᠢ ᠬᠤᠷᠳᠤᠴᠠ ᠨᠢ ᠬᠤᠪᠢᠷ ᠠᠬᠤ ᠦᠭᠡᠢ᠃ ) ᠬᠣᠶ ᠠᠷ᠂ ᠬᠣᠭᠣᠰ ᠤᠨ ᠳᠤ ᠲᠠᠭ ᠯᠠᠬᠤ ᠰᠡᠲ ᠤᠪ᠃ 11. 20 20 ᠣᠨ ᠤ 7 ᠰ ᠠᠷ ᠠ ᠶᠢᠨ 2 ᠦ ᠡᠳᠤᠷ ᠪᠣᠯ ᠲᠠᠯ ᠠ᠂ ᠪᠥ ᠬᠥ ᠳᠡ ᠯᠡᠭᠡᠢ ᠳᠦ ᠰᠢᠨ ᠡ ᠲᠢᠳ ᠢᠮᠯᠢᠭ ᠬᠡᠯᠪᠡ ᠷᠢ ᠶᠢᠨ ᠠᠭᠤᠰ ᠬᠢᠨ ᠥᠷᠪ ᠢᠰᠤᠯ ᠳᠤ ᠨᠡᠷᠪᠡ ᠭᠳᠡᠭᠰᠡᠨ ᠬᠦᠮᠦ ᠨ ᠦ ᠲᠣᠭ ᠠ ᠨᠢ ᠭᠡᠨᠲᠡ 10 51 ᠲᠥ ᠮᠡᠨ ᠢ ᠳᠠᠪᠠ ᠵᠠᠢ᠂ ᠲᠡ ᠭᠡᠪᠡᠯ 10 51 ᠲᠥ ᠮᠡ ᠶ ᠢ ᠰᠢᠨᠵᠢᠯ ᠡᠬᠦ ᠤ ᠬᠠᠭᠠᠨ ᠴᠢ ᠲᠣᠭ ᠠ ᠲᠡ ᠮᠳᠡᠭᠯᠡᠬᠦ ᠠᠷᠭ ᠠ ᠪᠠᠷ ᠢᠯ ᠡᠷᠭᠡᠢᠯ ᠡᠪᠡᠯ ( ) ① ② 12. ᠪᠣᠳᠣ ᠃ √ 27 + ( 13 ) − 2− 3𝑡𝑎𝑛 60 °+ (𝜋 − √2 ) 0 =( ) 13. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ 𝐴𝐵 ᠪᠣᠯ ⊙ 𝑜 ᠶᠢᠨ ᠳᠢᠶᠠᠮ ᠧᠲᠷ ᠂ ᠬᠥᠪᠴ ᠢ 𝐶𝐷 ⊥ 𝐴𝐵 ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ 𝐸 ᠂ ∠ 𝐵𝐶 𝐷 = 30° ᠂ 𝐶𝐷 = 2√ 3 ᠪ ᠣᠯ ᠰ ᠡᠭᠦᠳᠡ ᠷ ᠲᠠᠢ ᠭᠡᠰᠡᠭ ᠦᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ ( ) 14. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠬᠠᠪᠲᠠᠭ ᠠᠢ ᠶᠢᠨ ᠲᠡᠭᠰᠢ ᠥᠨᠴᠥᠭᠳᠥ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤ ᠨ ᠰ ᠢᠰᠲᠧᠮ ᠳᠦ᠂ ᠷᠤᠮᠪ 𝐴𝐵𝐶𝐷 ᠨᠢ ᠨᠢᠭᠡᠳᠤᠭᠡᠷ ᠮᠥᠴᠡ ᠳᠦ ᠣᠷᠣᠰ ᠢᠨ ᠠ᠂ ᠲᠠ ᠯ ᠠ 𝐵𝐶 ᠪᠣᠯᠣᠨ 𝑥 ᠲᠡᠩᠭᠡ ᠯᠢᠭ ᠬᠠᠷᠢᠯ ᠴᠠᠨ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ᠂ 𝐴 , 𝐵 ᠬᠣ ᠶᠠᠷ ᠴᠡᠭ ᠦᠨ ᠭᠣᠯ ᠳᠤ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ 6 , 4 ᠪᠣᠯᠣᠨ ᠠ᠂ ᠤᠷ ᠪᠠᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠳᠤ ᠹᠦᠩᠺᠼ 𝑦 = 𝑘𝑥 ( 𝑥 0) ᠤᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠨᠢ ᠴᠡ ᠭ 𝐴 ,𝐵 ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠨ ᠠ᠂ ᠬᠡᠷ ᠪᠡ ᠷᠤᠮᠪ 𝐴𝐵𝐶𝐷 ᠶᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ 2√ 5 ᠪ ᠣᠯ᠂ 𝑘 ᠶᠢᠨ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠨᠢ ( ) 15. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠲᠡ ᠩᠴᠡᠬᠦ ᠦ ᠲᠠ ᠯᠠᠳᠤ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ 𝐴𝐵𝐶 ᠳᠤ 𝐴𝐵 = 6᠂ ᠴᠡ ᠭ 𝐷 , 𝐸 ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ ᠲᠠ ᠯ ᠠ 𝐵𝐶 , 𝐴𝐶 ᠳᠡᠭᠡᠷ ᠡ ᠣ ᠷᠣᠰᠢᠬ ᠤ ᠪᠥ ᠭᠡᠳ 𝐵𝐷 = 𝐶𝐸 ᠪᠣᠯ ᠣᠨ ᠠ᠂ 𝐴𝐷 , 𝐵𝐸 ᠶᠢ ᠬᠣᠯ ᠪᠣᠪᠠᠯ ᠴᠡ ᠭ 𝐹 ᠳᠤ ᠣᠭᠳᠣᠯ ᠣᠯᠴᠠᠨ ᠠ᠂ 𝐶𝐹 ᠶᠢ ᠬᠣᠯ ᠪᠣᠨ ᠠ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ 𝐶𝐹 ᠶᠢᠨ ᠬᠠᠮᠤᠭ ᠤᠨ ᠪᠠᠭ ᠠ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠨᠢ ( ) 16. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠲᠡᠭᠰᠢ ᠲᠥ ᠷᠪᠡᠯ ᠵᠢᠨ 𝐴𝐵𝐶𝐷 ᠳᠦ᠂ ᠴᠡ ᠭ 𝑀 ᠪᠣᠯ ᠲᠠᠯ ᠠ 𝐵𝐴 ᠶᠢᠨ ᠥᠷᠬ ᠥᠯᠵᠢᠯ ᠡᠯ ᠱᠤᠭᠤᠮ ᠳᠡ ᠭᠡᠷᠡᠬ ᠢ ᠨᠢ ᠭᠡᠨ ᠬᠦᠳᠡᠯᠬ ᠦ ᠴᠡ ᠭ ( ᠴᠡᠭ 𝐴 ᠲᠠᠢ ᠳᠠᠪᠬᠤᠴ ᠠᠬᠤ ᠦᠭᠡᠢ ) ᠪ ᠥᠭᠡᠳ 𝐴𝑀 𝐴𝐵 ᠂ ∆ 𝐶𝐸𝐵 ᠪᠣᠯ ∆ 𝐷𝐴𝑀 ᠶᠢ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠵ ᠢᠬᠦᠯᠦ ᠭᠡᠳ ᠣ ᠯᠣᠭᠰ ᠠᠨ ᠵᠢᠷᠤ ᠭ ᠲᠥᠷᠰᠤ᠂ ᠬᠡ ᠷᠪᠡ ᠴᠡᠭ 𝐸 ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠭ ᠤᠯᠤᠨ 𝐸𝐻 ⊥ 𝐴𝐶 ᠪᠠᠷ ᠪᠠᠢᠭ ᠤᠯᠪᠠ ᠯ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ 𝐻 ᠪᠣᠯᠣᠨ ᠠ᠃ ᠲᠡ ᠭᠡᠪᠡᠯ ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠭᠡᠯᠡᠯᠭᠡ ᠡᠴᠡ ᠵᠥᠪ ᠪᠣᠯᠬ ᠤ ᠵᠣᠷ ᠪᠣᠰ ᠤᠳ ᠨᠢ ( ) ( ᠪᠤᠢ ᠪᠥᠬᠦ ᠢ ᠵᠥᠪ ᠪᠣᠯᠬᠤ ᠬᠡ ᠯᠡᠯᠭᠡ ᠶᠢᠨ ᠨᠣᠮᠧᠷ ᠢ ᠲᠠ ᠭᠯᠠ ᠃ ) ① ᠴᠡᠭ 𝑀 ᠶᠢᠨ ᠪᠠᠢᠷ ᠢ ᠨᠢ ᠬᠤᠪᠢᠷ ᠠᠭᠠᠳ ∠ 𝐷𝐻𝐶 = 60° ᠪᠣᠯᠬᠤ ᠦ ᠶᠡᠰ᠂ 2𝐵𝐸 = 𝐷𝑀 ᠪᠣᠯᠣᠨ ᠠ᠂ ② ᠴᠡᠭ 𝑀 ᠨᠢ ᠬᠥᠳᠡ ᠯᠭᠡᠭᠡᠨ ᠬᠢᠭᠡᠳ ᠶᠠᠮᠠ ᠷ ᠪᠠᠢᠷ ᠢ ᠳᠤ ᠬᠦᠷᠦ ᠭᠰᠡᠨ ᠴᠤ᠂ ᠬᠡᠵ ᠢᠶᠡᠳᠡ 𝐷𝑀 = √2 𝐻𝑀 ᠪᠣᠯᠣᠨ ᠠ᠃ ③ ᠴᠡᠭ 𝑀 ᠨᠢ ᠬᠦᠳᠡᠯ ᠭᠡᠭᠡᠨ ᠬᠢᠬᠦ ᠶᠠᠪᠤᠴᠠ ᠳᠤ᠂ ᠳᠥ ᠷᠪᠡᠯ ᠵᠢᠨ 𝐶𝐸𝑀 𝐷 ᠨᠢ ᠷᠤᠮᠪ ᠪᠣ ᠯᠬᠤ ᠪᠣᠯᠣᠯ ᠴᠠᠭ ᠠ ᠲᠠᠢ᠃ ④ ᠴᠡᠭ 𝑀 ᠨᠢ ᠬᠦᠳᠡᠯᠭᠡᠭᠡᠨ ᠬᠢᠭᠡᠳ ᠶᠠᠮᠠ ᠷ ᠪᠠᠢᠷ ᠢᠨ ᠳ ᠤ ᠬᠦ ᠷᠦᠭᠰᠡᠨ ᠴᠤ ∠ 𝐶𝐻𝑀 ᠨᠢ ᠯᠠᠪᠲᠠ ᠢ 135° ᠠᠴᠠ ᠶᠡᠭᠡ ᠭᠤᠷᠪ ᠠ᠂ ᠪᠣᠳᠣᠵ ᠤ ᠬᠠᠷ ᠢᠭᠤᠯ ᠬᠤ ᠰ ᠡᠲᠤᠪ ᠃ ( ᠲᠤᠰ ᠶᠡᠬᠡ ᠰᠡᠲᠤ ᠪ ᠨᠢ 8 ᠪᠠᠭ ᠠ ᠰᠡᠲᠤᠪ ᠲᠠᠢ᠂ ᠨᠠᠢᠲᠡ 72 ᠬᠤᠪᠢ᠂ ᠪᠣᠳᠣᠵ ᠤ ᠬᠠᠷ ᠢᠭᠤᠯ ᠬᠤ ᠳᠤ ᠪ ᠠᠨ ᠴᠢᠬᠤᠯ ᠠ ᠪᠣ ᠯᠬᠤ ᠦᠰ ᠦᠭ ᠪᠢᠴᠢᠭ ᠦᠨ ᠲ ᠠᠢᠯᠪ ᠤᠷᠢᠯᠠ ᠯᠲᠠ᠂ ᠪᠣ ᠳᠣᠯᠭ ᠠ ᠶᠢᠨ ᠠᠯᠬ ᠤᠮ ᠳᠠ ᠷᠠᠭ ᠠ ᠪᠤᠶᠤ ᠪᠠ ᠳᠤᠯᠠᠬᠤ ᠶᠠᠪᠤᠴᠠ ᠪᠠᠨ ᠪᠢᠴᠢᠨ ᠡ᠃ ) E D B O A C x y 6 4 C D B A O F E A CB D H MAB C D E 17. ᠲᠤᠰ ᠪᠠ ᠭ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 8 ᠬᠤᠪ ᠢ ⑴ ᠲᠡ ᠩᠴᠡᠳᠭᠡᠯ ᠪᠤᠰᠤ ᠶᠢᠨ ᠪᠥ ᠯᠥᠭ { 3 (𝑥 − 1) 5𝑥 + 2 𝑥− 22 ≤ 7− 32 𝑥 ᠶᠢ ᠪᠣᠳᠣᠭᠠᠳ ᠂ ᠲᠡᠭᠦᠨ ᠦ ᠬᠠᠮᠤᠭ ᠤ ᠨ ᠪ ᠠᠭ ᠠ ᠪᠥᠬᠥ ᠯᠢ ᠲᠣ ᠭᠠᠨ ᠤ ᠬᠠᠷ ᠢᠭᠤ ᠶᠢ ᠣᠯ᠃ ⑵ ᠤᠷ ᠢᠳᠠᠪᠠ ᠷ ᠬᠢᠯ ᠪᠠᠷᠴᠢᠯ ᠠᠵᠤ᠂ ᠳᠠ ᠷᠠᠭ ᠠ ᠨᠢ ᠬᠡ ᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠨᠢ ᠣ ᠯ᠃ ( 𝑎 2 −1 𝑎 2 −2 𝑎+ 1 − 11−𝑎 )÷ 2 𝑎 2 −𝑎 ᠂ ᠡᠭᠦᠨ ᠳ ᠦ 𝑎 ᠨᠢ 𝑎 2 + 2𝑎 − 15 = 0 ᠢ ᠬᠠᠩᠭ ᠠᠨ ᠠ᠃ 18. ( ᠲᠤᠰ ᠪᠠ ᠭ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 9 ᠬᠤᠪᠢ ) ᠨᠢᠭᠡᠨ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠨᠢ ᠰᠤᠷᠤᠭᠴᠢᠳ ᠤᠨ ᠬᠢᠴᠢ ᠶᠡᠯ ᠦᠨ ᠭᠠᠳᠠ ᠨᠠᠬᠢ ᠰᠤᠯᠠ ᠴᠠᠭ ᠢᠶᠠᠨ ᠠᠰᠢᠭ ᠯᠠᠨ ᠪᠡᠶ ᠡ ᠪᠡᠨ ᠳᠠᠭ ᠠᠭᠠᠨ ᠳᠠᠪᠳ ᠠᠯᠭ ᠠ ᠬᠢᠵᠤ ᠪᠠᠢᠬᠤ ᠪᠠᠢᠳᠠ ᠯ ᠢ ᠣᠢᠯ ᠠᠭᠠᠬᠤ ᠶᠢ ᠨ ᠲᠥᠯᠥᠭᠡ᠂ ᠶᠢᠰ ᠤᠳᠤᠭᠡᠷ ᠣᠨ ᠤ ᠨᠢᠭᠡᠳᠤᠭᠡᠷ ᠠ ᠩᠭᠢ ᠶᠢᠨ ᠰᠤᠷᠤᠭ ᠴᠢᠳ ᠤᠨ ᠨᠢᠭᠡ ᠭᠠᠷ ᠠᠭ ᠤᠨ ᠳᠠ ᠪᠳᠠᠯᠭ ᠠ ᠬᠢᠭᠰ ᠡᠨ ᠴᠠᠭ ᠤ ᠨ ᠪᠠᠢᠳᠠ ᠯ ᠳᠦ ᠪᠠᠢᠴᠠᠭ ᠠᠯᠲᠠ ᠬᠢᠵᠠᠢ᠃ ᠪᠠᠢᠴᠠᠭ ᠠᠯᠲᠠ ᠶ ᠢᠨ ᠳᠦ᠋ ᠩ ᠳᠦ᠂ ᠲᠤᠰ ᠠᠩᠭᠢ ᠶᠢᠨ ᠰᠤᠷ ᠤᠭᠴ ᠢᠳ ᠤᠨ ᠨᠢᠭᠡ ᠭᠠᠷ ᠠᠭ ᠤᠨ ᠳᠠᠪᠳ ᠠᠯᠭ ᠠ ᠬᠢᠭᠰ ᠡᠨ ᠴᠠ ᠭ ᠨᠢ ᠵᠥ ᠪᠬᠡᠨ 1 ᠴᠠᠭ᠂ 2 ᠴᠠ ᠭ᠂ 3 ᠴᠠᠭ᠂ 4 ᠴᠠᠭ ᠭᠡᠬᠦ ᠲᠥ ᠷᠪᠡᠨ ᠵᠦ᠋ᠢᠯ ᠦᠨ ᠪᠠᠢᠳᠠ ᠯ ᠪ ᠠᠢᠨ ᠠ ( ᠲᠥ ᠷᠪᠡ ᠶᠢ ᠬᠠᠶᠠᠵ ᠤ ᠲᠠ ᠪᠤ ᠶᠢ ᠳᠡ ᠪᠰᠢᠭ ᠤᠯᠵᠠᠢ )᠃ ᠦᠭ ᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ ᠲᠤᠰ ᠠᠩᠭᠢ ᠳᠤ ᠨᠠᠢᠲᠡ 50 ᠰᠤᠷᠤᠭ ᠴᠢ ᠲᠠᠢ᠂ ᠪᠠᠢ ᠴᠠᠭᠠᠯᠲ ᠠ ᠶᠢᠨ ᠳᠦ᠋ ᠩ ᠢᠶᠡ ᠨ ᠦᠨᠳᠤᠰᠤ ᠯᠡᠨ᠂ ᠳᠣᠣᠷ ᠠᠬᠢ ᠮᠡᠲᠤ ᠬᠣᠶ ᠠᠷ ᠪᠦᠷᠢᠨ ᠪᠤᠰᠤ ᠪᠦᠷᠢᠳᠭᠡᠯ ᠦᠨ ᠬᠦᠰ ᠦᠨᠥᠭ ᠳᠥ ᠪᠠ ᠵᠢᠷᠤᠭ ᠢ ᠵᠢᠷ ᠤᠵᠠᠢ᠃ ᠡᠭᠦᠨ ᠳᠦ ᠲᠤᠰ ᠠᠩᠭᠢ ᠶᠢᠨ ᠡᠮᠡᠭᠳᠡᠢ ᠰᠤᠷᠤᠭᠴᠢ ᠶ ᠢᠨ ᠨᠢ ᠭᠡ ᠭᠠᠷ ᠠᠭ ᠤᠨ ᠳᠠᠪᠳ ᠠᠯᠭ ᠠ ᠬᠢᠭᠰ ᠡᠨ ᠴᠠ ᠭ ᠨᠢ ᠳᠣ ᠣᠷᠠᠬ ᠢ ᠮᠡ ᠲᠤ ( ᠨᠢᠭᠡᠴᠢ᠄ ᠴᠠ ᠭ ) ᠄ 1 1 1 2 2 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 ᠶᠢᠰᠤᠳᠤᠭᠡᠷ ᠣᠨ ᠤ ᠨᠢᠭᠡᠳᠤ ᠭᠡᠷ ᠠᠩᠭ ᠢ ᠶᠢᠨ ᠡᠮ ᠡᠭᠳᠡᠢ ᠰᠤᠷᠤᠭᠴᠢ ᠶᠢᠨ ᠨᠢᠭᠡ ᠭᠠᠷᠠᠭ ᠤᠨ ᠳᠠᠪᠳ ᠠᠯᠭ ᠠ ᠬᠢᠭᠰᠡᠨ ᠴᠠᠭ ᠤ ᠨ ᠳᠠ ᠪᠳᠠᠭᠳᠠ ᠭᠰᠠᠨ ᠲᠣᠭ ᠠᠨ ᠤ ᠲᠠᠷ ᠬᠠᠯᠲ ᠠ ᠶᠢᠨ ᠬᠥᠰᠥ ᠨᠥᠭᠳᠥ ᠪᠠ ᠡᠷᠡᠭᠳᠡᠢ ᠰᠤᠷᠤᠭᠴᠢ ᠶᠢᠨ ᠨᠢᠭᠡ ᠭᠠᠷᠠᠭ ᠤᠨ ᠳᠠᠪᠳ ᠠᠯᠭ ᠠ ᠬᠢᠭᠰ ᠡᠨ ᠴᠠ ᠭ ᠤᠨ ᠲᠣᠭ ᠣᠷᠢᠭ ᠪᠦᠷᠢᠳᠭᠡᠯ ᠦᠨ ᠵᠢᠷᠤᠭ ᠨᠢ ᠳᠣᠣᠷ ᠠᠬᠢ ᠮᠡᠲᠤ᠃ ᠳᠠᠪᠳᠠᠯᠭ ᠠ ᠬᠢᠭᠰᠡ ᠨ ᠴᠠᠭ ᠳᠠᠪ ᠳᠠᠭᠳᠠᠭ ᠰᠠᠨ ᠲᠣᠭ ᠠ( ᠰᠤᠷ ᠤᠭᠴ ᠢ ᠶᠢᠨ ᠲᠣᠭ ᠠ) 1 ᠴᠠᠭ 3 2 ᠴᠠᠭ 𝑎 3 ᠴᠠᠭ 4 4 ᠴᠠᠭ 6 ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠮᠡᠳᠡᠭᠡ ᠵᠠᠩᠭᠢ ᠶᠢ ᠦᠨ ᠳᠤᠰ ᠤᠯᠡᠨ᠂ ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠠᠰᠠᠭᠤᠯᠲ ᠠ ᠶᠢ ᠬᠠᠷᠢ ᠭᠤᠯ᠃ ⑴ ᠪᠦᠷᠢᠳᠭᠡᠯ ᠦᠨ ᠬᠦᠰ ᠥᠨᠥᠭ ᠳᠥ ᠳᠠᠬᠢ 𝑎 = ( ) ᠂ ᠲᠤᠰ ᠠᠩᠭᠢ ᠶᠢᠨ ᠡᠮᠡᠭᠳᠡᠢ ᠰᠤᠷᠤᠭᠴᠢ ᠶᠢᠨ ᠨᠢᠭᠡ ᠭᠠᠷᠠᠭ ᠤᠨ ᠳᠠᠪᠳ ᠠᠯᠭ ᠠ ᠬᠢᠭᠰ ᠡᠨ ᠴᠠ ᠭ ᠤᠨ ᠳᠤᠮᠳᠠ ᠪᠠ ᠢᠷᠢ ᠶᠢᠨ ᠲᠣᠭ ᠠ ᠨᠢ ( ) ᠴᠠᠭ ⑵ ᠲᠣᠭ ᠣᠷᠢᠭ ᠪᠦᠷᠢᠳᠭᠡᠯ ᠦᠨ ᠵᠢᠷᠤᠭ ᠳᠡᠬᠢ ᠲᠤᠰ ᠠᠩᠭᠢ ᠶᠢᠨ ᠡᠷᠡᠭᠳᠡᠢ ᠰᠤᠷᠤᠭᠴᠢ ᠶᠢᠨ ᠨᠢᠭᠡ ᠭᠠᠷᠠᠭ ᠤᠨ ᠳᠠᠪᠳ ᠠᠯᠭ ᠠ ᠬᠢᠭᠰᠡᠨ ᠴᠠᠭ ᠨᠢ 4 ᠴᠠ ᠭ ᠪᠣ ᠯᠬᠤ ᠲᠠᠢ ᠬᠠᠷᠠᠭ ᠠᠯᠵᠠᠭ ᠰᠠᠨ ᠲᠥᠪ ᠥᠨᠴᠥ ᠭ ᠨᠢ ( ) ᠭᠷᠠᠳᠦ᠋ᠰ᠃ ⑶ ᠬᠡ ᠷᠪᠡ ᠲᠤᠰ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠶᠢᠨ ᠶᠢᠰ ᠤᠳᠤᠭᠡᠷ ᠣ ᠨ ᠳ ᠤ ᠨᠠ ᠢᠲᠡ 60 0 ᠰ ᠤᠷᠤᠭᠴᠢ ᠲᠠ ᠢ ᠪᠣᠯ᠂ ᠡᠭᠦᠨ ᠦ ᠲᠣᠲᠣᠷ ᠡ ᠨᠢ ᠭᠡ ᠭᠠᠷᠠᠭ ᠤᠨ ᠳᠠ ᠪᠳᠠᠯᠭ ᠠ ᠬᠢᠭᠰ ᠡᠨ ᠴᠠᠭ ᠨᠢ 4 ᠴᠠ ᠭ ᠪ ᠣᠯᠬᠤ ᠰᠤᠷᠤᠭᠴᠢ ᠨᠢ ᠬᠡᠳᠤᠢ ᠪᠠᠢᠬᠤ ᠶᠢ ᠪᠠᠷ ᠤᠭᠴ ᠠᠭᠠᠯᠠᠨ ᠪᠣᠳᠣ᠃ ⑷ ᠲᠤᠰ ᠠᠩᠭᠢ ᠶᠢᠨ ᠳᠠᠪᠳ ᠠᠯᠭ ᠠ ᠬᠢᠭᠰᠡᠨ ᠴᠠᠭ ᠨᠢ 4 ᠴᠠ ᠭ ᠪᠣ ᠯᠬᠤ ᠲᠥ ᠷᠪᠡᠨ ᠡᠮᠡᠭᠳᠡᠢ ᠰᠤᠷᠤᠭᠴᠢ ᠣᠶᠣ ᠨ᠂ ᠤᠯ ᠠᠭᠠᠨ᠂ ᠨᠣᠮ ᠢᠨ᠂ ᠵᠢᠮ ᠢᠰ ᠦ ᠨ ᠳᠣᠲᠣᠷ ᠠ ᠠᠴᠠ ᠳᠠᠰᠢᠷᠠ ᠮ ᠢᠶᠠ ᠷ ᠬᠣᠶ ᠠᠷ ᠰ ᠤᠷᠤᠭᠴᠢ ᠶᠢ ᠰᠤᠩᠭᠤ ᠵᠤ ᠶᠠᠷᠢᠶ ᠠ ᠢ ᠯᠡᠳᠭᠡᠯ ᠬᠢᠯ ᠭᠡᠨ ᠡ᠃ ᠮᠤᠳᠤ ᠯᠵᠢᠨ ᠵᠢᠷᠤᠭ ᠪᠤᠶ ᠤ ᠬᠥᠰ ᠥᠨᠥᠭᠵᠢᠬᠦ ᠯᠬᠦ ᠠ ᠷᠭ ᠠ ᠪᠠᠷ ᠰᠤᠩᠭᠤᠭᠰᠠᠨ ᠬᠣᠶ ᠠᠷ ᠰᠤᠷᠤᠭᠴᠢ ᠨᠢ ᠶᠠᠭ ᠰᠠᠢᠬᠠᠨ ᠤᠯ ᠠᠭᠠᠨ ᠪᠠ ᠵᠢᠮ ᠢᠰ ᠪᠣᠯ ᠬᠤ ᠶᠢ ᠨ ᠲ ᠣᠬᠢᠶ ᠠᠯᠳᠤᠮ ᠵᠢ ᠶ ᠢ ᠣ ᠯ᠃ 1ᠴᠠᠭ 2 ᠴᠠᠭ 3 ᠴᠠᠭ 4ᠴᠠᠭ d % 50% 20% 10% 19. ( ᠲᠤᠰ ᠪᠠ ᠭ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 8 ᠬᠤᠪᠢ ) ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ᠂ ᠨᠢᠭᠡᠳᠤᠭᠡᠷ ᠵᠡᠷ ᠭᠡ ᠶᠢ ᠨ ᠹᠦᠩᠺᠼ 𝑦 = 𝑘𝑥 + 𝑏 ᠦ ᠨ ᠵᠢᠷ ᠤᠮᠠᠭ ᠨᠢ ᠤᠷ ᠪᠠᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠳᠤ ᠹᠦᠩᠺᠼ 𝑦 = 𝑎𝑥 ᠦ ᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠲᠠ ᠢ ᠨᠢ ᠭᠡᠳᠤᠭᠡᠷ ᠮᠥᠴᠡ ᠳᠦ ᠴᠡᠭ 𝐴 (4 ,3) ᠳᠤ ᠣᠭᠳᠣᠯ ᠣᠯᠴᠠ ᠵᠤ᠂ 𝑦 ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠦ ᠨ ᠰ ᠥᠭᠡᠷ ᠬᠦ ᠬᠠᠭ ᠠᠰ ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠲᠠᠢ ᠴᠡᠭ 𝐵 ᠳᠤ ᠣᠭᠳᠣᠯ ᠣᠯᠴᠠ ᠬᠤ ᠪᠥᠭᠡ ᠳ 𝑂𝐴 = 𝑂𝐵 ᠪᠣᠯ ᠣᠨ ᠠ᠃ ⑴ ᠹᠦᠩᠺᠼ 𝑦 = 𝑘𝑥 + 𝑏 ᠪᠠ 𝑦 = 𝑎𝑥 ᠶᠢᠨ ᠠᠨᠠᠯ ᠢᠰ ᠢ ᠯᠡᠷᠭᠡᠢᠯ ᠡᠯ ᠢ ᠣᠯ᠃ ⑵ ᠦᠭ ᠭᠦᠭᠳᠡᠭᠰᠡ ᠨ ᠨᠢ ᠴᠡᠭ 𝐶( 0,5 )᠂ ᠨᠢᠭᠡᠳ ᠤᠭᠡᠷ ᠵᠡᠷ ᠭᠡ ᠶᠢᠨ ᠹᠦᠩᠺᠼ ᠦ ᠨ ᠵᠢᠷ ᠤᠮᠠᠭ ᠳᠡᠭᠡᠷ ᠡ ᠨᠢᠭᠡᠨ ᠴᠡᠭ 𝑀 ᠶᠢ ᠳᠣᠭᠲᠠᠭᠠ ᠪᠠ᠂ ᠭᠡᠬᠦ ᠳᠦ ᠪ ᠡᠨ 𝑀𝐵 = 𝑀𝐶 ᠪᠣᠯ ᠭᠠᠨ ᠠ᠃ ᠡᠨ ᠡ ᠦᠶ ᠡᠰ ᠦ ᠨ ᠴᠡ ᠭ 𝑀 ᠦ ᠨ ᠪ ᠠᠢᠷᠢᠰᠢᠯ ᠢ ᠣᠯ᠃ x y 4 3 C B A O 20. (ᠲᠤᠰ ᠪᠠ ᠭ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 8 ᠬᠤᠪᠢ ) ᠵᠢᠷᠤᠭ 1 ᠪᠣᠯ᠂ ᠬᠠᠨ ᠠᠨ ᠳᠤ ᠡᠯᠬᠦ ᠬᠦ ᠪᠡᠶ ᠡ ᠣ ᠬᠢᠶᠠᠯ ᠭ ᠠ ᠶᠢᠨ ᠰᠦ᠋ᠷ ᠴᠢᠬᠦᠷ ᠦᠨ ᠪᠣᠳᠠᠳᠠ ᠢ ᠵᠢᠷ ᠤᠭ ᠪᠣᠯᠣ ᠨ ᠠ᠂ ᠵᠢᠷᠤᠭ 2 ᠪᠣᠯ ᠵᠢᠷᠤᠭ ᠶᠢ ᠦᠨᠳᠤ ᠰᠤᠯᠡ ᠨ ᠵᠢᠷ ᠤᠭᠰ ᠠᠨ ᠭᠧᠣᠮᠧᠲᠷ ᠢ ᠶᠢᠨ ᠵᠢᠷᠤᠭ ᠲᠥᠷᠰᠤ ᠪᠣᠯᠣ ᠨ ᠠ᠃ ᠡᠩ ᠦᠨ ᠪᠠᠢᠳᠠ ᠯ ᠳᠤ᠂ ᠰ ᠦ᠋ᠷᠴᠢᠬᠦᠷ ᠦᠨ ᠬᠠᠮᠤ ᠭ ᠳᠡᠭᠡᠳᠤ ᠴᠡ ᠭ 𝐵 ᠨᠢ ᠪᠡᠶ ᠡ ᠣᠬᠢᠶ ᠠᠬᠤ ᠬᠦᠮᠦ ᠨ ᠦ ᠪᠡᠶ ᠡ ᠶᠢᠨ ᠦᠨᠳᠤᠷ ᠠᠴᠠ 15 𝑐𝑚 ᠡᠭᠴᠡᠯ ᠡᠬᠦᠷ ᠦᠨᠳᠤᠷ ᠵᠠᠢ ᠲᠠᠢ ᠪᠠᠢᠪ ᠠᠯ ᠬᠠᠮ ᠤᠭ ᠳᠣᠬ ᠢᠷᠠᠮᠵᠢᠳᠠᠢ ᠪᠠᠢᠳᠠ ᠭ᠃ 𝑂 ᠪᠣᠯ ᠰᠦ᠋ᠷ ᠴᠢᠬᠦᠷ ᠢ ᠬᠠᠨ ᠠᠨ ᠳᠤ ᠳᠣᠭᠲᠠᠭ ᠠᠵᠤ ᠪᠠᠢᠬᠤ ᠴᠡ ᠭ᠂ 𝑂𝐴 = 10 𝑐𝑚 ᠰᠦ᠋ ᠷᠴᠢᠬᠦᠷ ᠦᠨ ᠲᠣᠯᠣᠭ ᠠᠢ ᠶᠢᠨ ᠳᠢᠶᠠᠮ ᠧᠲᠷ 𝐴𝐵 = 8𝑐𝑚 ᠂ ∠ 𝐶𝑂𝐴 = 26° ᠂ ∠ 𝑂𝐴𝐵 = 146° ᠂ ᠬᠡ ᠷᠪᠡ ᠪᠡᠶ ᠡ ᠶᠢᠨ ᠦᠨᠳᠤᠷ ᠨᠢ 175 𝑐𝑚 ᠪᠣᠯᠬᠤ ᠬᠦᠮᠦ ᠨ ᠣ ᠬᠢᠶᠠᠯ ᠭ ᠠ ᠬᠢᠬᠦ ᠭᠡᠪᠡᠯ᠂ ᠴᠡ ᠭ 𝑂 ᠨᠢ ᠭᠠᠵᠠᠷ ᠤᠨ ᠨᠢᠭᠤᠷ ᠠ ᠴᠠ ᠬᠡᠳᠤᠢ ᠵᠠᠢ ᠲᠠᠢ ᠪᠠᠢᠪ ᠠᠯ ᠵᠣᠬᠢ ᠬᠤ ᠪᠣᠢ︖ ( ᠯᠠᠪᠯᠠᠬᠤ ᠲᠣᠭ ᠠ ᠪ ᠠᠷᠢᠮᠲᠠ᠄ 𝑠𝑖𝑛 26° ≈ 0.44 , 𝑐𝑜𝑠 26° ≈ 0.90 , 𝑡𝑎𝑛 26° ≈ 0.49 ᠪᠣᠳ ᠣᠯᠭ ᠠ ᠶᠢᠨ ᠳ ᠦ᠋ᠩ ᠢ 1 𝑐𝑚 ᠬᠦᠷᠲᠡᠯ ᠡ ᠨᠠᠷᠢᠪ ᠴᠢᠯᠠ ) B A O C ① ② 21. (ᠲᠤᠰ ᠪᠠ ᠭ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 9 ᠬᠤᠪᠢ ) ᠪᠢᠳᠠ ᠨᠢᠭᠡᠨᠲᠡ ᠣᠷᠣᠢ ᠶᠢᠨ ᠴᠡᠭ ᠦᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ( ℎ, 𝑘) ᠪᠣᠯ ᠬᠤ ᠬᠠᠶᠠᠯ ᠲᠠ ᠶᠢᠨ ᠱᠤᠭᠤᠮ ᠤᠨ ᠠᠨᠠᠯ ᠢᠰ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡᠯ ᠨᠢ 𝑦= 𝑎( 𝑥− ℎ) 2+ 𝑘( 𝑎≠ 0) ᠪᠣᠯᠬᠤ ᠶᠢ ᠰᠤᠷᠤᠯ ᠴᠠᠵᠠᠢ᠃ ᠳᠠᠷᠠᠭ ᠠᠴᠢ ᠪᠠ ᠷ ᠪ ᠢᠳᠠ ᠪᠠᠰᠠ ᠲᠣᠭ ᠣᠷᠢᠭ ᠤ ᠨ ᠲ ᠥᠪ ᠦᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ (𝑎 ,𝑏 )᠂ ᠷᠠᠳᠢᠤᠰ ᠨᠢ 𝑟 ᠪᠣᠯ ᠬᠤ ᠲᠣᠭ ᠣᠷᠢᠭ ᠤᠨ ᠲᠡᠭᠰᠢ ᠳᠭᠡᠯ ᠨᠢ ( 𝑥− 𝑎) 2+ (𝑦 − 𝑏) 2= 𝑟 2 ᠪᠣᠯᠬ ᠤ ᠶᠢ ᠰ ᠤᠷᠤᠯ ᠴᠠᠨ ᠠ᠃ ᠵᠢᠱᠢᠶ ᠡᠯᠡᠪᠡ ᠯ᠄ ᠲᠣᠭ ᠣᠷᠢᠭ ᠤᠨ ᠲᠥ ᠪ ᠨᠢ 𝑃( −2 ,1) ᠂ ᠷᠠ ᠳᠢᠤᠰ ᠨᠢ 3 ᠪᠣᠯᠬᠤ ᠲᠣᠭ ᠣᠷᠢᠭ ᠤᠨ ᠲᠡᠭᠰᠢ ᠳᠭᠡᠯ ᠨᠢ ( 𝑥+ 2) 2+ (𝑦 − 1) 2= 9 ᠪ ᠣᠯᠣᠨ ᠠ᠃ ⑴ ᠴᠡᠭ 𝑀 (− 3 , −1 ) ᠢᠶᠡᠷ ᠲᠣᠭ ᠣᠷᠢᠭ ᠤᠨ ᠲᠥᠪ ᠪᠣ ᠯᠭᠠᠨ᠂ √3 ᠪᠠᠷ ᠷᠠᠳᠢᠤᠰ ᠪᠣᠯᠭ ᠠᠭᠰᠠ ᠨ ᠲᠣᠭ ᠣᠷᠢᠭ ᠤ ᠨ ᠲᠡ ᠭᠰᠢᠳ ᠭᠡᠯ ᠨᠢ _ __ __ __ __ __ __ __ ⑵ ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠴᠡᠭ 𝐵 (− 3 , 0) ᠢᠶᠡᠷ ᠲᠣᠭ ᠣᠷᠢᠭ ᠤᠨ ᠲᠥ ᠪ ᠪᠣᠯ ᠭᠠᠭᠰᠠᠨ ᠲᠣᠭ ᠣᠷᠢᠭ ᠨᠢ 𝑦 ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠲᠠ ᠢ ᠡᠬ ᠢᠨ ᠦ ᠴᠡᠭ ᠲᠦ ᠰᠢᠬ ᠠᠯᠴᠠᠨ ᠠ᠂ ᠴᠡᠭ 𝐶 ᠪᠣᠯ ⨀𝑂 ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠨᠢᠭᠡ ᠴᠡᠭ ᠂ 𝑂𝐶 ᠶᠢ ᠬᠣᠯ ᠪᠣᠨ ᠠ᠂ 𝐵𝐷 ⊥ 𝑂𝐶 ᠪᠠᠷ ᠪᠠᠢᠭ ᠤᠯᠪᠠ ᠯ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ 𝐷 ᠪᠣᠯᠣᠨ ᠠ᠂ 𝐵𝐷 ᠶᠢ ᠦᠷᠬ ᠦᠯᠵᠢᠯ ᠡᠬᠦᠯᠪᠡ ᠯ 𝑦 ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠲᠠ ᠢ ᠴᠡ ᠭ 𝐸 ᠳᠦ ᠣᠭᠳᠣ ᠯᠣᠯᠴᠠ ᠨ ᠠ᠂ 𝑠𝑖𝑛 ∠𝐴𝑂𝐶 = 35 ᠃ ①𝐸𝐶 ᠶᠢ ᠬᠣ ᠯᠪᠣᠨ ᠠ᠂ 𝐸𝐶 ᠨᠢ ⨀𝐵 ᠶᠢᠨ ᠰᠢᠬ ᠠᠭᠴᠢ ᠰᠢᠯᠤᠭᠤ ᠨ ᠪᠣᠯᠬᠤ ᠶᠢ ᠪᠠ ᠳᠤᠯᠠ᠃ ②𝐵𝐸 ᠳᠡᠭᠡᠷ ᠡ 𝑄𝐵 = 𝑄𝐶 = 𝑄𝐸 = 𝑄𝑂 ᠪᠣᠯ ᠭᠠᠬᠤ ᠴᠡᠭ 𝑄 ᠣᠷ ᠣᠰᠢᠨ ᠠ ᠤᠤ ︖ ᠬᠡᠷᠪᠡ ᠣᠷᠣᠰ ᠢᠪᠠᠯ ᠴᠡᠭ 𝑄 ᠦᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠢ ᠣᠯᠵ ᠤ᠂ ᠪᠠᠰᠠ ᠴᠡᠭ 𝑄 ᠪᠠᠷ ᠲᠥ ᠪ ᠪ ᠣᠯᠭᠠᠨ᠂ 𝑄𝐵 ᠪᠠᠷ ᠷᠠᠳᠢᠤᠰ ᠪᠣᠯᠭ ᠠᠭᠰᠠᠨ ⨀𝑄 ᠶᠢᠨ ᠲᠡ ᠭᠰᠢᠳᠭ ᠡᠯ ᠢ ᠪᠢᠴᠢ ᠵᠤ ᠭᠠᠷᠭ ᠠ᠃ ᠬᠡᠷᠪᠡ ᠣᠷᠣᠰ ᠢᠬᠤ ᠦᠭᠡᠢ ᠪᠣᠯ ᠰᠢᠯᠲᠠᠭ ᠠᠨ ᠢ ᠨᠢ ᠲᠣᠳ ᠣᠷᠬᠠᠢᠯ ᠠ᠃ x y A D E B O C 22. (ᠲᠤᠰ ᠪᠠ ᠭ ᠠ ᠨᠠᠢᠲᠡ 8 ᠬᠤᠪᠢ ) ᠨᠢᠭᠡᠨ ᠵᠢᠮᠢᠰ ᠦ ᠨ ᠳᠡ ᠯᠭᠡᠭᠦᠷ ᠬᠠᠳᠠ ᠭᠰᠠᠨ ᠦᠨ ᠡ ᠨᠢ ᠵᠢᠩ ᠪᠦᠷᠢ ᠳᠦ ᠪ ᠡᠨ 10 ᠲᠥ ᠬᠥᠷᠢᠭ ᠪᠠᠢᠭᠰᠠ ᠨ ᠨᠢᠭᠡ ᠵᠦ᠋ᠢᠯ ᠦᠨ ᠵᠢᠮᠢᠰ ᠦᠨ ᠦᠨ ᠡ ᠶᠢ ᠬᠣᠶ ᠠᠷ ᠤᠳᠠ ᠭ ᠠ ᠪᠠᠭᠤᠷᠠ ᠭᠤᠯᠤᠭᠠ ᠳ ᠵᠢᠩ ᠪᠦᠷᠢ ᠳᠦ ᠪ ᠡᠨ 8 .1 ᠲᠥᠬᠥ ᠷᠢᠭ ᠪᠣᠯᠭ ᠠᠵᠠᠢ᠃ ᠡᠭᠦᠨ ᠳᠦ ᠵᠢᠮᠢᠰ ᠦ ᠨ ᠦ ᠨ ᠡ ᠶᠢᠨ ᠬᠣᠶ ᠠᠷ ᠤᠳᠠ ᠭ ᠠ ᠶ ᠢᠨ ᠳᠤᠮᠳᠠᠴᠢ ᠪᠠᠭ ᠤᠷᠠᠯ ᠲᠠ ᠶ ᠢᠨ ᠨᠣᠷᠮ ᠠ ᠨᠢ ᠠᠳᠠᠯᠢ ᠃ ⑴ ᠲᠤᠰ ᠵᠦ᠋ᠢᠯ ᠦᠨ ᠵᠢᠮᠢᠰ ᠦᠨ ᠦᠨ ᠡ ᠶᠢᠨ ᠤᠳᠠ ᠭ ᠠ ᠪᠦᠷᠢ ᠶᠢ ᠨ ᠳᠤ ᠮᠳᠠᠴᠢ ᠪᠠᠭᠤᠷᠠ ᠯᠲᠠ ᠶᠢᠨ ᠨᠣᠷᠮ ᠠ ᠶᠢ ᠣᠯ᠃ ⑵ ᠵᠢᠮ ᠢᠰ ᠶᠢᠨ ᠦ ᠨ ᠡ ᠨᠢ ᠬᠣ ᠶᠠᠳᠤᠭ ᠠᠷ ᠤᠳᠠ ᠭ ᠠ ᠪᠠᠭᠤᠷᠠ ᠭᠰᠠᠨ ᠤ ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠨᠢᠭᠡᠳᠤᠭᠡᠷ ᠡᠳᠤᠷ ᠠᠴᠠ ᠡᠬᠢᠯ ᠡᠨ 𝑥 ᠳᠤᠭᠠᠷ ᠡᠳᠤᠷ (𝑥 ᠪᠣᠯ ᠪᠥᠬᠥ ᠯᠢ ᠲᠣᠭ ᠠ ) ᠳᠠᠬᠢ ᠪᠣᠷᠣᠯ ᠠᠭᠤᠯᠬᠤ ᠬᠡᠮᠵᠢ ᠶ ᠡ ᠵᠢᠴᠢ ᠬᠠᠳᠠ ᠭᠠᠯᠠᠬ ᠤ ᠪ ᠠ ᠬᠣᠷᠣᠭᠳᠠ ᠯ ᠤᠨ ᠵᠠᠷᠤᠳᠠᠯ ᠤᠨ ᠪᠠᠢᠳᠠ ᠯ ᠨᠢ ᠳᠠᠷ ᠠᠭᠠᠬ ᠢ ᠬᠥᠰᠥ ᠨᠥᠭᠳᠥ ᠳᠦ ᠦ ᠵᠡᠬᠦᠯ ᠦᠭᠰᠡᠨ ᠮᠡᠲᠤ᠃ ᠦᠭ ᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ ᠲᠤᠰ ᠵᠦ᠋ᠢᠯ ᠦᠨ ᠵᠢᠮᠢᠰ ᠦᠨ ᠣᠷᠣᠭᠤᠯ ᠤᠭᠰ ᠠᠨ ᠦᠨ ᠡ ᠨᠢ ᠵᠢᠩ ᠪᠦᠷᠢ ᠳᠦ ᠪ ᠡᠨ 4.1 ᠲᠥᠬᠥ ᠷᠢᠭ᠂ ᠲᠤᠰ ᠵᠦ᠋ ᠢᠯ ᠦ ᠨ ᠵᠢᠮᠢᠰ ᠢ ᠪᠣᠷᠣᠯ ᠠᠭᠤᠯ ᠤᠭᠰᠠᠨ 𝑥 ᠳᠤᠭᠠᠷ ᠡᠳᠤᠷ ᠲᠠ ᠬᠢ ᠠ ᠰᠢᠭ ᠬᠣᠨᠵ ᠢᠪᠤᠷᠢ ᠶᠢ 𝑦 ᠲᠥᠬᠥ ᠷᠢᠭ ᠭᠡᠪᠡᠯ᠂ 𝑦 ᠪᠠ 𝑥( 1≤ 𝑥≤ 10 ) ᠤᠨ ᠬᠣᠭᠣᠷ ᠣᠨᠳᠣᠬ ᠢ ᠹᠦᠩᠺᠼ ᠦᠨ ᠬᠣᠯ ᠪᠣᠭᠳᠠᠯ ᠤᠨ ᠠᠨᠠᠯ ᠢᠰ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡᠯ ᠢ ᠣᠯ᠃ ᠪᠠᠰᠠ ᠬᠡᠳᠤᠳᠤᠭᠡᠷ ᠡᠳᠤᠷ ᠲᠡᠬ ᠢ ᠪᠣᠷᠣᠯ ᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠠᠰᠢᠭ ᠬᠣᠨᠵ ᠢᠪᠤᠷᠢ ᠨᠢ ᠬᠠᠮ ᠤᠭ ᠤ ᠨ ᠶᠡᠭᠡ ᠪᠣᠯᠬᠤ ᠵᠢᠴᠢ ᠬᠠᠮ ᠤᠭ ᠤᠨ ᠶᠡᠭᠡ ᠪᠣᠷᠣᠯ ᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠠᠰᠢᠭ ᠬᠣᠨ ᠵᠢᠪᠤᠷᠢ ᠨᠢ ᠬᠡᠳᠤᠢ ᠪᠣᠯᠬ ᠤ ᠶᠢ ᠣᠯ᠃ ᠬᠤᠭᠤᠴᠠᠭ ᠠ( ᠡᠳᠤᠷ ) ᠪᠣᠷᠣ ᠯᠠᠭᠤᠯᠬᠤ ᠬᠡᠮ ᠵᠢᠶ ᠡ ( ᠡᠳᠤᠷ ) ᠬᠠᠳᠠ ᠭᠠᠯᠠᠬ ᠤ ᠪᠠ ᠬ ᠣᠷᠣᠭᠳᠠ ᠯ ᠤᠨ ᠵᠠᠷᠤᠳᠠ ᠯ ( ᠲᠥ ᠬᠥᠷᠢᠭ ) 𝑥 120 − 𝑥 3𝑥 2− 64 𝑥+ 400 23. (ᠲᠤᠰ ᠪᠠ ᠭ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 10 ᠬᠤ ᠪᠢ) ⑴ 【ᠣ ᠷᠣᠯᠳᠤ ᠯᠭ ᠠ ᠠᠴ ᠠ ᠠ ᠵᠢᠭᠯ ᠠᠵᠤ ᠣᠯᠬ ᠤ 】 ᠵᠢᠷᠤᠭ 1 ᠮᠡ ᠲᠤ᠂ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ 1 ᠨᠢ ᠭᠡᠴᠢ ᠤᠷᠳᠤ ᠪᠣᠯ ᠬᠤ ᠵᠢ ᠵᠢᠭ ᠲᠡᠭ ᠰᠢ ᠲ ᠥᠷᠪᠡ ᠯᠵᠢᠨ ᠢᠶᠡᠷ ᠪᠦᠷᠢᠯᠳ ᠥᠭᠰᠡ ᠨ ᠲᠥᠷᠪ ᠡᠯᠵᠢᠭᠡᠰᠤ ᠲᠣᠳᠣᠷ ᠡ᠂ ∆𝐴𝐵𝐶 ᠶᠢᠨ ᠭᠤᠷᠪ ᠠᠨ ᠣᠷᠣᠢ ᠶᠢᠨ ᠴᠡᠭ ᠨᠢ ᠴᠥᠮ ᠬᠠᠨ ᠠᠯᠵᠢ ᠴᠡᠭ ᠳᠡᠭᠡᠷ ᠡ ᠣ ᠷᠣᠰᠢᠨ ᠠ᠃ ① ᠱᠠᠭ ᠠᠷᠳᠠᠯᠭ ᠠ ᠶᠤ ᠰᠤᠭ ᠠᠷ ᠵᠢᠷ ᠤᠭ ᠪᠠᠢᠭ ᠤᠯ᠃ ∆𝐴𝐵𝐶 ᠶᠢ ᠴᠡᠭ 𝐴 ᠪᠠ ᠷ ᠲᠥᠪ ᠪᠣᠯ ᠭᠠᠨ ᠂ ᠴᠠ ᠭ ᠤᠨ ᠵᠡᠭᠦᠦ ᠶᠢᠨ ᠳᠠ ᠭᠠᠤ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ 90° ᠡᠷᠬᠢᠭ ᠦᠯ᠃ ᠴᠡ ᠭ 𝐵 ᠦᠨ ᠬᠠᠷᠠᠭ ᠠᠯᠵᠠᠭᠰ ᠠᠨ ᠴᠡᠭ ᠨᠢ 𝐵 ′᠂ ᠴᠡ ᠭ 𝐶 ᠶᠢᠨ ᠬᠠᠷᠠᠭ ᠠᠯᠵᠠᠭᠰᠠᠨ ᠴᠡ ᠭ ᠨ ᠢ 𝐶 ′ᠪᠣᠯᠣᠨ ᠠ᠂ 𝐵𝐵 ′ ᠶᠢ ᠬᠣ ᠯᠪᠣ᠃ ② ᠰᠡᠲᠤᠪ ① ᠳᠤ ᠪᠠᠢᠭ ᠤᠯᠤᠭᠰᠠᠨ ᠵᠢᠷ ᠤᠭ ᠲᠡᠬᠢ ∠𝐴 𝐵 ′𝐵 = ( )° ᠃ ⑵【ᠠ ᠰᠠᠭᠤᠳᠠ ᠯ ᠢ ᠰᠢᠢᠳᠪ ᠤᠷᠢᠯ ᠡᠬᠦ 】 ᠵᠢᠷᠤᠭ 2 ᠮᠡ ᠲᠤ᠂ 𝑅𝑡 ∆𝐴𝐵𝐶 ᠳᠦ 𝐵𝐶 = 1 , ∠𝐶 = 90° ᠂ 𝐶𝐴 ᠶᠢ ᠦᠷᠬ ᠦᠯᠵᠢᠯ ᠡᠬᠦᠯᠵ ᠤ ᠴᠡ ᠭ 𝐷 ᠬᠦᠷ ᠭᠡᠬᠦ ᠳᠦ ᠪᠡᠨ 𝐶𝐷 = 1 ᠪ ᠣᠯᠭᠠᠨ ᠠ᠂ ᠵᠢᠰᠢᠬᠦᠦ ᠲᠠ ᠯ ᠠ 𝐴𝐵 ᠶᠢ ᠴᠡᠭ 𝐴 ᠪᠠ ᠷ ᠲᠥᠪ ᠪᠣᠯ ᠭᠠᠨ᠂ ᠴᠠ ᠭ ᠤᠨ ᠵᠡᠭᠦᠦ ᠶᠢ ᠨ ᠳᠠ ᠭᠠᠤ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ 90° ᠡᠷᠬ ᠢᠬᠦᠯᠦᠨ 𝐴𝐸 ᠶᠢ ᠣᠯᠵ ᠠᠢ᠂ 𝐷𝐸 ᠶᠢ ᠬᠣ ᠯᠪᠣᠨ ᠠ᠂ ∠𝐴𝐷𝐸 ᠶᠢᠨ ᠭᠷᠠᠳᠦ᠋ᠰ ᠤᠨ ᠲᠣᠭ ᠠ ᠶᠢ ᠣᠯ᠃ ⑶【ᠥᠷᠭᠡᠳᠭᠡᠨ ᠬᠡᠷ ᠡᠭᠯᠡᠬ ᠦ 】 ᠵᠢᠷᠤᠭ 3 ᠮᠡᠲᠤ᠂ ᠲᠥ ᠷᠪᠡᠯ ᠵᠢᠨ 𝐴𝐵𝐶𝐷 ᠳᠦ 𝐴𝐸 ⊥ 𝐵𝐶 ᠂ ᠡᠭ ᠴᠡ ᠶ ᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ 𝐸 ᠂ ∠