ᠣᠷᠳᠤᠰ ᠬᠤᠲᠠ ᠶᠢᠨ 2020 ᠣᠨ ᠤ ᠰᠤᠷᠤᠯᠭ᠎ᠠ ᠲᠡᠭᠦᠰᠴᠤ ᠰᠤᠷᠭᠠᠭᠤᠯᠢ ᠳᠠᠪᠰᠢᠬᠤ ᠰᠢᠯᠭᠠᠯᠲᠠ ᠶᠢᠨ ᠰᠡᠳᠦᠪ 2020中考题.docx

ᠨᠢᠭᠡ᠂ ᠭᠠᠭᠴᠠ ᠬᠠᠷ ᠢᠭᠤᠯ ᠲᠠ ᠰᠤᠩᠭᠤ ᠬᠤ ᠰᠡᠲᠤᠪ᠃ ︵ᠲᠤᠰ ᠶᠡᠭᠡ ᠰᠡᠲᠤ ᠤ ᠲ᠊ ᠤ 1 0 ᠪᠠᠭ ᠠ ᠰᠡᠲᠤ ᠤ ᠲᠠᠢ᠂ ᠪᠠᠭ ᠠ ᠰᠡᠲᠤᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪ ᠢ᠂ ᠨᠠᠢᠲᠡ 30 ᠬᠤᠪᠢ︶ 1. ᠪᠣᠳᠠᠳᠤ ᠲᠤᠭ 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 ᠲᠡ ᠭᠦᠨ ᠲᠠᠢ ᠬᠠᠷᠠᠭ ᠠᠯᠵᠠᠭᠰᠠᠨ ᠪᠡᠶᠡᠳᠤ ᠵᠢᠷ ᠤᠭ ᠲᠥᠷᠰ ᠤ ᠨᠢ ︵︶ A. B. C. D. 3. ᠹᠦᠩᠺᠼ 𝑦 = √𝑥 + 3 ᠲᠤ᠂ ᠬᠤᠪᠢ ᠰᠬᠤ ᠨ᠊ 𝑥 ᠤᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠠᠪᠬ ᠤ ᠬᠡ ᠪᠴᠢᠶ ᠡ ᠶᠢ ᠲᠤᠭ 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 ᠠᠯᠵᠢ ᠨ᠊ ᠤᠤ ᠢ ᠶᠢ ᠨᠢᠭᠡ ᠬᠠᠭ ᠤᠳᠠᠰᠤ ᠲᠡᠭᠰᠢ ᠥᠨᠴᠥᠭᠳᠥ ᠬᠡᠯᠪᠡ ᠷᠢᠲᠠᠢ ᠴᠠᠭᠠᠰᠤᠨ ᠬᠠᠪ ᠲᠠᠰᠤ ᠨ᠊ ᠳᠡᠭᠡᠷ ᠡ ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ ᠲᠠᠯᠪᠢᠵᠠᠢ᠂ ∠𝐸𝐺𝐹 = 90° , ∠ 𝐹𝐸𝐺 = 30° , ∠1 = 125° ᠪᠣᠯ ∠𝐵𝐹𝐺 ᠶᠢᠨ ᠶᠡᠭᠡ ᠪᠠ ᠬ᠋᠊ ᠠ ᠨᠢ ︵︶ A. 125° B. 115° C. 110° D. 120° 6. ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠬᠥᠰᠥ ᠨᠥᠭᠳᠥ ᠪᠣᠯ ᠨᠢᠭᠡ ᠨ᠊ ᠠ ᠩᠭᠢ ᠶᠢᠨ 5 ᠰ᠊ ᠤᠷᠤᠭᠴᠢ ᠶᠢᠨ ᠲᠤᠭ ᠠᠨ ᠤ ᠰᠢᠯᠭ ᠠᠯᠲᠠ ᠶᠢᠨ ᠬᠢᠷᠢ ᠪᠦᠳ ᠦᠮᠵᠢ ᠶᠢᠨ ᠪᠦᠷᠢᠳᠭᠡᠯ ᠪᠣᠯ ᠣᠨ ᠠ᠄ ᠲᠡᠭᠡᠪ ᠡᠯ 𝑎 ,𝑏 ᠬᠤᠶ ᠠᠷ ᠲᠤᠭ ᠠ ᠪᠠᠷ ᠢᠮᠲᠠ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ ︵︶ A. 81 ,80 B. 80 ,2 C. 81 ,2 D. 80 ,80 7. ᠲᠥᠷᠪ ᠡᠯᠵᠢᠨ 𝐴𝐵𝐶𝐷 ᠲᠤ᠂ 𝐴𝐷 // 𝐵𝐶᠂ ∠𝐷 = 90°᠂ 𝐴𝐵 = 8᠂ 𝐵𝐶 = 6᠂ ᠴᠡᠭ 𝐴 ᠪᠠ ᠴᠡᠭ 𝐶 ᠪᠠ ᠷ᠊ ᠲᠤᠰ ᠲᠤᠰ ᠲᠤᠭ ᠣᠷᠢᠭ ᠤᠨ ᠲᠥᠪ ᠪᠣᠯ ᠭᠠᠨ᠂ 12 𝐴𝐶 ᠠᠴᠠ ᠶᠡᠭᠡ ᠪᠣᠯᠬᠤ ᠤᠷᠳᠤ ᠪᠠᠷ ᠷᠠᠳᠢᠤᠰ ᠪᠣᠯᠭ ᠠᠨ ᠨᠤᠮᠤ ᠪᠠᠢᠭ ᠤᠯᠪᠠ ᠯ᠊᠂ ᠬᠤᠶ ᠠᠷ ᠨᠤᠮᠤ ᠨᠢ ᠴᠡᠭ 𝐸 ᠲᠤ ᠣ ᠭᠲᠣᠯ ᠣᠯᠴᠠ ᠨ᠊ ᠠ᠂ ᠰ᠊ ᠠᠴᠤᠷᠠ ᠬ᠋᠊ 𝐵𝐸 ᠶᠢ 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 F O DA B C E ᠤᠷᠳᠤ ᠰ᠊ ᠬᠤᠳᠠ ᠢᠢᠨ 20 20 ᠣ ᠨ᠊ ᠤ ᠠᠩᠬ ᠠ ᠲᠤ ᠮᠳᠠᠳᠤ ᠢᠢ ᠨ᠊ ᠰᠤᠷ ᠤᠯᠭ ᠠ ᠲᠡᠭ ᠦᠰᠴ ᠤ ᠰᠤᠷ ᠭᠠᠭ ᠤᠯᠢ ᠲᠠᠪᠰ ᠢᠬᠤ ᠰᠢᠯ ᠭᠠᠯᠳᠠ ᠢᠢᠨ ᠰᠡᠳᠤ ᠤ ᠪᠠᠢᠭ ᠤᠯᠪᠠ ᠯ᠊ 𝐴𝐷 ᠲᠠᠢ ᠴᠡᠭ 𝐹 ᠳ᠋᠊ ᠦ ᠣ ᠭᠳᠣ ᠯᠣᠯᠴᠠᠵ ᠤ᠂ 𝐴𝐶 ᠲᠠᠢ ᠴᠡᠭ 𝑂 ᠲᠤ ᠣᠭᠳᠣ ᠯᠣᠯᠴᠠ ᠨ᠊ ᠠ᠂ ᠬᠡᠷ ᠪᠡ ᠴᠡᠭ 𝑂 ᠨᠢ 𝐴𝐶 ᠶᠢᠨ ᠳᠤᠮ ᠳᠠ ᠴᠡᠭ ᠪᠣᠯ᠂ 𝐶𝐷 ᠦᠨ ᠤᠷᠳ ᠤ ᠨᠢ A. 4√ 2 B. 2 √10 C. 6 D. 8 8. ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠬᠡᠯᠡᠯᠡᠭᠡ ᠠᠴᠠ ᠵᠥ ᠤ ᠪᠤ ᠯᠬᠤ ᠨᠢ ︵︶ ① √ 5− 1 2 12 ② ᠵᠥᠪ ᠵᠢᠷ ᠭᠤᠭᠠ ᠨ᠊ ᠥᠨᠴ ᠥᠭᠳᠥ ᠶᠢᠨ ᠲᠣᠳᠣᠭᠠ ᠲᠤ ᠥᠨᠴ ᠥᠭ ᠦᠨ ᠨᠠᠢᠯ ᠡᠪᠤᠷᠢ ᠨᠢ 72 0°᠂ ᠲᠡ ᠭᠦᠨ ᠦ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠤᠷ ᠲᠤ ᠨᠢ ᠷᠠᠳᠢᠤᠰ ᠲᠠᠢ ᠪᠠ ᠨ᠊ ᠲᠡ ᠩᠴᠡᠬᠦ ᠦ᠃ ③ ᠨᠢ ᠭᠡ ᠢ ᠶᠢ ᠡᠩ ᠦᠨ ᠬᠦᠵ ᠦᠷ ᠲᠣᠲᠣᠷ ᠡ ᠠᠴᠠ ᠳᠠᠰᠢᠷᠠ ᠮ᠊ ᠪᠠ ᠷ᠊ ᠨᠢ ᠭᠡ ᠬᠠᠭ ᠤᠳᠠᠰᠤ ᠶᠢ ᠰᠤᠭᠤᠯ ᠤᠨ ᠠᠪᠬᠤ ᠲᠤ᠂ ᠲᠡᠷᠡ ᠨᠢ ᠬᠠᠷ ᠠ ᠲᠤᠭ ᠣᠷ ᠪᠣᠯᠬᠤ ᠶᠢᠨ ᠲᠤᠬ ᠢᠶᠠᠯᠳᠤᠮᠵᠢ ᠨᠢ 14᠃ ④ ᠠ ᠡ ᠬᠤᠶ ᠠᠷ ᠬᠦᠮᠦ ᠨ᠊ ᠲᠤᠰ ᠲᠤᠰ 10 ᠤᠳᠠ ᠭᠠᠨ ᠤ ᠬᠠᠷᠪᠤ ᠯᠲᠠ ᠶᠢᠨ ᠪᠣᠯ ᠪᠠᠰᠤ ᠷᠠᠯ ᠬᠢᠭᠰ ᠠ ᠳ᠋᠊ 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 ᠪᠣᠯ ᠣᠨ ᠠ᠃ ᠲᠡ ᠭᠡᠪᠡᠯ ᠡ ᠶᠢ ᠨ᠊ ᠬᠠᠷ ᠪᠤᠯᠲᠠ ᠶᠢᠨ ᠬᠢᠷᠢ ᠪᠦᠳᠦ 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 ᠲᠤ ᠷᠪᠡᠯ ᠶᠢᠨ 𝑂𝐴 𝐴1 𝐵1 ᠲᠤ᠂ ᠲᠠᠰᠢᠭᠤᠪ ᠴᠢ ᠪᠠᠷ ᠲᠠ ᠯ᠊ ᠠ ᠪᠣᠯᠭ ᠠᠨ ᠬᠤᠶ ᠠᠳᠤᠭ ᠠᠷ ᠲᠡᠭᠰᠢ ᠲᠥᠷᠪ ᠡᠯᠵᠢᠨ 𝑂 𝐴1 𝐴2 𝐵2 ᠶᠢ ᠪᠠᠢᠭ ᠤᠯᠤᠨ ᠠ᠂ 𝐴1 𝐴2 ᠶᠢ ᠬᠤᠯᠪᠣᠪᠠᠯ ∆𝐴 𝐴1 𝐴2 ᠢ ᠣ ᠯᠣᠨ ᠠ᠃ ᠳᠠ ᠬᠢᠨ ᠲᠠᠰᠢᠭᠤᠪ ᠴᠢ 𝑂 𝐴2 ᠪᠠᠷ ᠲᠠᠯ ᠠ ᠪᠣᠯᠭ ᠠᠨ ᠭᠤᠷᠪ ᠠᠳᠤᠭ ᠠᠷ ᠲᠡᠭᠰᠢ ᠲᠤ ᠷᠪᠡᠯ ᠶᠢᠨ 𝑂 𝐴2 𝐴3 𝐵3 ᠶᠢ ᠪᠠᠢᠭ ᠤᠯᠤᠨ ᠠ᠂ 𝐴1 𝐴3 ᠶᠢ ᠬᠣ ᠯᠪᠣᠪ ᠠᠯ ∆𝐴 1 𝐴 2 𝐴 3 ᠢ ᠣᠯᠣᠨ ᠠ᠃ ᠤ ᠠᠰᠠ ᠳᠠ ᠬᠢᠨ ᠳᠠᠰᠢ ᠭᠤᠪ ᠴᠢ 𝑂𝐴 3 ᠪᠠ ᠷ᠊ ᠲᠠ ᠯ᠊ ᠠ ᠪᠣᠯᠭ ᠠᠨ ᠲᠥᠷᠪ ᠡᠳᠤᠭᠡᠷ ᠲᠡᠭᠰᠢ ᠲᠤ ᠷᠪᠡᠯ ᠶᠢᠨ 𝑂 𝐴3 𝐴4 𝐵4 ᠶᠢ ᠪᠠᠢᠭ ᠤᠯᠤᠨ ᠠ᠂ 𝐴2 𝐴4 ᠶᠢ ᠬᠤᠯ ᠪᠣᠪᠠ ᠯ᠊ ∆ 𝐴2 𝐴3 𝐴4 ᠢ ᠣᠯᠣ ᠨ᠊ ᠠ ᠁ ᠬᠡᠷᠪᠡ ∆𝐴 𝐴1 𝐴2 , ∆ 𝐴1 𝐴2 𝐴3 , ∆ 𝐴2 𝐴3 𝐴4 ᠶᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠶᠢ ᠲᠤᠰ ᠲᠤᠰ 𝑆1 ,𝑆 2 ,𝑆 3 ᠁ ᠭᠡᠵ ᠤ ᠶᠢ ᠱᠢᠵᠤ᠂ ᠡᠨᠡ ᠮᠡᠲᠤ ᠤ ᠠᠷ ᠦᠷᠬ ᠦᠯᠵᠢᠯ ᠠ ᠤ ᠠᠢᠭ ᠤᠯᠪᠠ ᠯ᠊ 𝑆 2020 ᠶᠢᠨ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ 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 ᠠᠴᠠ ᠡᠬᠢᠯ ᠠ ᠮᠣᠷᠳᠠᠵᠤ᠂ ᠲᠤᠰ ᠠ ᠶᠠᠯᠠᠯ ᠤᠨ ᠵᠠᠮ ᠢᠶᠠᠷ ᠶᠠᠪᠤᠵ ᠤ ᠵᠠ ᠭᠠᠨ ᠤ ᠣᠷᠳᠤᠨ ᠲᠤ ᠬᠦᠷᠦ ᠨ᠊ ᠡ᠂ ᠠᠶ ᠠᠯᠠᠯ ᠤᠨ ᠶᠠᠪᠤᠴᠠ ᠲᠤ ᠰᠢᠪᠠ ᠭᠤᠨ ᠤ ᠬᠦ ᠷᠢᠶᠡᠯ ᠡᠩ ᠳ᠋᠊ ᠦ ᠵᠣᠭᠰ ᠣᠨ ᠠ ︵ᠲᠡ ᠷᠭᠡᠨ ᠳ᠋᠊ ᠦ ᠭᠠᠷ ᠴᠤ ᠪᠠᠭᠤᠬ ᠤ ᠴᠠ ᠬ᠋᠊ ᠢ ᠪᠣᠳᠣ ᠯᠬᠢᠯ ᠠᠬᠤ ᠦᠭᠡᠢ︶᠃ ᠨᠢ ᠭᠡᠳᠤᠭᠡᠷ ᠡᠭᠡᠯᠵ ᠢ ᠶᠢ ᠨ᠊ ᠮᠠ ᠰᠢᠨ ᠲᠡ ᠷᠭᠡ ᠨᠢ ᠦᠳᠡ ᠶᠢᠨ ᠡᠮᠤᠨ ᠡ ᠶᠢᠨ 9:2 0 ᠡ ᠴᠡ ᠡᠬᠢᠯ ᠠ ᠮᠣᠷᠳᠠᠵᠤ᠂ ᠳᠠ ᠷᠠᠭ ᠠ ᠨᠢ 10 ᠮᠢᠨᠦ᠋ ᠲ᠊ ᠪᠦᠷᠢ ᠲᠤ ᠨᠢᠭᠡ ᠡᠭᠡᠯᠵ ᠢ ᠶᠢ ᠨ᠊ ᠮᠠ ᠰᠢᠨ ᠲᠡ ᠷᠭᠡ ᠣᠷᠤᠬ ᠤ ᠡᠭᠦᠳᠡ ᠨ᠊ ᠡ ᠴᠡ ᠡᠬᠢᠯ ᠠ ᠮᠣᠷᠳᠠᠬᠤ ᠪᠥᠭᠡᠳ ᠬ᠊ ᠤᠷᠳᠤᠴᠠ ᠨᠢ ᠴ᠊ ᠥᠮ ᠠᠳᠠᠯᠢ᠃ ᠰ᠊ ᠣᠶᠣᠯ ᠭᠠᠷ ᠠᠭ ᠤ ᠨ᠊ ᠡᠳᠤᠷ ᠠᠮᠢᠳᠠ ᠨ᠊ ᠤ ᠬᠦᠷᠢᠶ ᠡᠩ ᠲᠤ ᠵᠤᠭᠠ ᠴᠠᠬᠤ ᠪᠠᠷ ᠦᠳᠡ ᠶᠢᠨ ᠡᠮ ᠤᠨ ᠡ ᠶᠢᠨ 9 ᠴᠠᠭ ᠲᠤ ᠣᠷᠤᠬ ᠤ ᠡᠭᠦᠳᠡ ᠨ᠊ ᠲᠤ ᠬᠦᠷᠴᠡᠢ᠂ ᠨᠠ ᠢᠲᠡ ᠶᠢᠨ ᠮᠠᠰᠢ ᠨ᠊ ᠲᠡ ᠷᠭᠡ ᠶᠢᠨ ᠶᠠᠪᠤᠬ ᠤ ᠴᠠᠭ ᠨᠢ ᠪᠣᠯᠤᠭ ᠠ ᠦᠭᠡᠢ ᠤᠴᠢᠷ ᠠᠴᠠ ᠰᠤᠶ ᠤᠯ ᠣᠷᠤᠬ ᠤ ᠡᠭᠦᠳᠡ ᠨ᠊ ᠡᠴ ᠡ ᠡᠬ ᠢᠯᠡᠨ ᠠᠶᠠᠯ ᠠᠯ ᠤ ᠨ᠊ ᠵᠠ ᠮ᠊ ᠳᠠ ᠭᠠᠤ 25 ᠮᠢ ᠨᠦ᠋ᠲ ᠶᠠᠪᠠᠭ ᠠᠨ A 3 A 4 A 2 A 1 AO B 1B 2 B 3 B 4 ᠶᠠᠪᠤᠵ ᠤ᠂ ᠰᠢᠪᠠ ᠭᠤᠨ ᠤ ᠬᠦ ᠷᠢᠶᠡᠯ ᠡᠩ ᠳ᠋᠊ ᠦ ᠬᠦ ᠷᠴᠡᠢ᠃ ᠣᠷ ᠣᠬᠤ ᠡᠭᠦᠳᠡ ᠨ᠊ ᠡᠴ ᠡ ᠵ᠊ ᠠᠢᠳᠠ ᠭᠰᠠᠨ ᠵᠠᠮ ᠤᠨ ᠤᠷᠳᠤ 𝑦 ︵ᠮᠧ ᠲᠷ︶ ᠪᠣᠯ ᠣᠨ ᠬᠤᠭᠤ ᠴᠠᠭ ᠠ 𝑥 ︵ᠮᠢ ᠨᠦ᠋ᠲ︶ ᠦᠨ ᠬᠤᠭᠤᠷ ᠣᠨᠳᠤᠬ ᠢ ᠹᠦᠩᠺᠼ ᠦᠨ ᠬᠤᠯᠪᠣᠭᠳᠠᠯ ᠨᠢ ᠵᠢᠷᠤᠭ 2 ᠮᠡᠲᠤ᠃ ᠲᠡᠭᠡᠪ ᠡᠯ ᠳᠠ ᠷᠠᠭᠠᠬ ᠢ ᠬᠡ ᠯᠡᠯᠭᠡ ᠠᠴᠠ ᠪᠤᠷᠤᠭ ᠤ ᠪᠤ ᠯᠬᠤ ᠨᠢ᠄ A. ᠨᠢᠭᠡᠳᠤᠭᠡᠷ ᠡᠭᠡ ᠯᠵᠢ ᠶᠢᠨ ᠮᠠᠰᠢᠨ ᠲᠡ ᠷᠭᠡ ᠶᠢᠨ ᠣᠷᠤᠬ ᠤ ᠡ ᠭᠦᠳᠡᠨ ᠠᠴᠠ ᠵᠶᠢᠳᠠᠭᠰᠠᠨ ᠵᠠᠮ ᠤᠨ ᠤᠷᠳᠤ 𝑦 ︵ᠮᠧ ᠲᠷ︶ ᠪᠣᠯ ᠣᠨ ᠬᠤᠭᠤᠴᠠᠭ ᠠ 𝑥 ︵ᠮᠢ ᠨᠦ᠋ᠲ︶ ᠦᠨ ᠬᠤᠭᠤᠷ ᠣᠨᠳᠤᠬ ᠢ ᠹ᠊ ᠦᠩᠺᠼ ᠦᠨ ᠬᠤᠯ ᠪᠣᠭᠳᠠᠯ ᠤᠨ ᠠᠨᠠᠯ ᠢᠰ ᠢ ᠯᠡᠷᠭᠡᠢ ᠯᠡᠯ 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 ᠠ ᠡ᠃ C. ᠰᠣᠶ ᠣᠯ ᠰ᠊ ᠢᠪᠠᠭ ᠤᠨ ᠤ ᠬᠦ ᠷᠢᠶᠡᠯ ᠡᠩ ᠲᠤ 40 ᠮᠢᠨᠦ᠋ ᠲ᠊ ᠲᠤᠭ ᠣᠷᠢᠭ ᠰᠠᠨ ᠤ ᠳᠠ ᠷᠠᠭ ᠠ᠂ ᠨᠠᠢᠲᠡ ᠶᠢᠨ ᠮᠠᠰᠢᠨ ᠲᠡ ᠷᠭᠡᠨ ᠳ᠋᠊ ᠦ ᠰ᠊ ᠠᠭᠤᠭᠠ ᠳ᠋᠊ ᠵᠠᠭᠠᠨ ᠤ ᠣᠷᠳᠤ ᠨ᠊ ᠳ᠋᠊ ᠤ ᠬᠦ ᠷᠬᠦ ᠭᠡᠪᠡᠯ᠂ ᠬᠠᠮᠤᠭ ᠡᠷ ᠲᠡ ᠳ᠋᠊ ᠦ ᠪᠠᠨ ᠲᠥᠷᠪ ᠡᠳᠤᠭᠡᠷ ᠡᠭᠡᠯᠵ ᠢ ᠶᠢ ᠨ᠊ ᠮᠠ ᠰᠢᠨ ᠲᠡᠷ ᠭᠡᠨ ᠳ᠋᠊ ᠦ ᠰᠠ ᠭᠤᠵᠤ ᠳᠠᠢᠯ ᠦᠨ ᠡ᠃ D. ᠰᠤᠶᠤᠯ ᠰᠢᠪᠠ ᠭᠤᠨ ᠤ ᠬ᠊ ᠦᠷᠢᠶᠡᠯ ᠡᠩ ᠲᠤ 40 ᠮᠢ ᠨᠦ᠋ᠲ ᠲᠤᠭ ᠣᠷᠢᠵᠤ ᠪᠠᠷᠠᠭ ᠠᠳ᠂ ᠲᠠ ᠪᠤᠳᠤ ᠭᠠᠷ ᠡᠭᠡᠯᠵ ᠢ ᠶᠢ ᠨ᠊ ᠮᠠ ᠰᠢᠨ ᠲᠡ ᠷᠭᠡᠨ ᠳ᠋᠊ ᠦ ᠰ᠊ ᠠᠭᠤᠭᠠ ᠳ᠋᠊ ᠵᠠᠭᠠᠨ ᠤ ᠤᠷᠳᠤᠨ ᠲᠤ ᠬᠦᠷᠬ ᠦ ᠴᠠ ᠬ᠋᠊ ᠨᠢ ᠲᠡ ᠭᠦᠨ ᠦ ᠰ᠊ ᠢᠪᠠᠭ ᠤᠨ ᠤ ᠬᠦᠷᠢᠶ ᠡᠯᠡᠩ ᠲᠤ ᠲᠤᠭ ᠣᠷᠢᠵᠤ ᠪᠠᠷᠠᠮ ᠠᠭᠴᠠ ᠰᠢᠭ ᠤᠳ ᠶᠠᠪᠠᠭ ᠠᠨ ᠶᠠᠪᠤ ᠵᠤ ᠵᠠ ᠭᠠᠨ ᠤ ᠤᠷᠳᠤᠨ ᠲᠤ ᠬᠦ ᠷᠬᠦ ᠴᠠ ᠬ᠋᠊ ᠠᠴᠠ 7 ᠮᠢᠨᠦ᠋ᠲ ᠤᠷ ᠢᠳᠴᠢᠯ ᠠᠨ ᠠ ︵ᠰᠣ ᠶᠣᠯ ᠤ ᠨ᠊ ᠶᠠᠪ ᠠᠭᠠᠨ ᠶᠠᠪᠤᠬ ᠤ ᠳᠤᠮᠳᠠ ᠴᠢ ᠬᠤᠷᠳᠤᠴ ᠠ ᠨ᠊ ᠢ ᠬᠤᠪᠢᠷ ᠠᠬᠤ ᠦᠭᠡᠢ᠃︶ ᠬᠤᠶ ᠠᠷ᠂ ᠬᠤᠭᠤᠰ ᠤᠨ ᠲᠤ ᠲᠠᠭ ᠯᠠᠬᠤ ᠰᠡᠲ ᠤᠪ᠃ ︵ᠰ᠊ ᠡᠲᠤᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪᠢ᠂ ᠨᠠ ᠢᠲᠡ 18 ᠬ᠊ ᠤᠪᠢ︶ 11. 20 20 ᠣᠨ ᠤ 7 ᠰ᠊ ᠠᠷ ᠠ ᠶᠢᠨ 2 ᠦ ᠡᠳᠤᠷ ᠪᠣᠯᠲᠠ ᠯ᠊ ᠠ᠂ ᠪᠦ ᠬᠦ ᠳᠡ ᠯᠡᠭᠡᠢ ᠲᠤ ᠰᠢᠨ ᠡ ᠲᠢᠳ ᠢᠮᠯᠢᠭ ᠬᠡᠯᠪᠡ ᠷᠢ ᠶᠢᠨ ᠠᠭᠤᠰ ᠬᠢᠨ ᠥᠷᠪ ᠢᠰᠤᠯ ᠲᠤ ᠨᠡᠷᠪᠡ ᠭᠳᠡᠭᠰᠡᠨ ᠬᠦᠮᠦ ᠨ᠊ ᠦ ᠲᠤᠭ ᠠ ᠨᠢ ᠭᠡᠨᠲᠡ 10 51 ᠲᠤ ᠮᠡᠨ ᠢ ᠳᠠᠪᠠ ᠵᠠᠢ᠂ ᠲᠡ ᠭᠡᠪᠡᠯ 10 51 ᠲᠤ ᠮᠡ ᠶ᠊ ᠢ ᠰᠢᠨᠵᠢᠯ ᠡᠬᠦ ᠤ ᠬᠠᠭᠠᠨ ᠴᠢ ᠲᠤᠭ ᠠ ᠲᠡ ᠮᠳᠡᠭᠯᠡᠬᠦ ᠠᠷᠭ ᠠ ᠪᠠᠷ ᠢᠯ ᠡᠷᠭᠡᠢᠯ ᠡᠪᠡᠯ ︵︶ 12. ᠪᠣᠳᠣ᠃ √ 27 + ︵13︶ − 2− 3𝑡𝑎𝑛 60 °+ ︵𝜋 − √2︶ 0 = ︵︶ 13. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ 𝐴𝐵 ᠪᠣᠯ ⊙ 𝑜 ᠶᠢᠨ ᠳᠢᠶᠠᠮ ᠧᠳ᠋ᠷ᠂ ᠬᠥᠪ ᠴᠢ 𝐶𝐷 ⊥ 𝐴𝐵 ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ 𝐸᠂ ∠ 𝐵𝐶 𝐷 = 30°᠂ 𝐶𝐷 = 2√ 3 ᠤ ᠣᠯ ᠰ᠊ ᠡᠭᠦᠳᠡᠷ ᠲᠠᠢ ᠭᠡᠰᠡᠭ ᠦᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ ︵︶ 14. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠬᠠᠪ ᠲᠠᠭᠠᠢ ᠶ᠊ ᠶᠢᠨ ᠲᠡᠭᠰᠢ ᠥᠨᠴ ᠥᠭᠳᠥ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤ ᠨ᠊ ᠰ᠊ ᠢᠰᠲᠧ ᠮ᠊ ᠲᠤ᠂ ᠷᠤᠮᠪ 𝐴𝐵𝐶𝐷 ᠨᠢ ᠨᠢᠭᠡᠳᠤᠭᠡᠷ ᠮᠥ ᠴᠡ ᠳ᠋᠊ ᠦ ᠣᠷᠣᠰ ᠶᠢᠨ ᠠ᠂ ᠲᠠ ᠯ᠊ ᠠ 𝐵𝐶 ᠪᠣᠯᠣᠨ 𝑥 ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠬᠠᠷᠢᠯ ᠴᠠᠨ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ᠂ 𝐴 , 𝐵 ᠬᠤᠶ ᠠᠷ ᠴᠡ ᠬ᠋᠊ ᠦᠨ ᠭᠤᠯ ᠲᠤ ᠤ ᠠᠢᠷᠢᠰᠢᠯ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ 6 ,4 ᠪᠣᠯ ᠣᠨ ᠠ᠂ ᠤᠷᠪᠠ ᠭᠤ ᠬᠠᠷ ᠢᠴᠠᠯ ᠲᠤ ᠹᠦᠩᠺᠼ 𝑦= 𝑘𝑥 ︵𝑥 0︶ ᠤᠨ ᠵᠢᠷ ᠤᠮᠠᠭ ᠨᠢ ᠴᠡ ᠬ᠋᠊ 𝐴 , 𝐵 ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠨ ᠠ᠂ ᠬᠡᠷ ᠪᠡ ᠷᠤᠮᠪ 𝐴𝐵𝐶𝐷 ᠶᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ 2√ 5 ᠪᠣᠯ᠂ 𝑘 ᠶᠢᠨ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠨᠢ ︵︶ 15. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠲᠡ ᠩᠴᠡᠬᠦ ᠦ ᠲᠠ ᠯᠠᠳᠤ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ 𝐴𝐵𝐶 ᠲᠤ 𝐴𝐵 = 6᠂ ᠴᠡ ᠬ᠋᠊ 𝐷 ,𝐸 ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ ᠲᠠ ᠯ᠊ ᠠ 𝐵𝐶 ,𝐴𝐶 ᠳᠡᠭᠡᠷ ᠡ ᠣᠷᠣᠰ ᠢᠬᠤ ᠪᠥᠭᠡᠳ 𝐵𝐷 = 𝐶𝐸 ᠪᠣᠯ ᠣᠨ ᠠ᠂ 𝐴𝐷 ,𝐵𝐸 ᠶᠢ ᠬᠤᠯ ᠪᠣᠪᠠᠯ ᠴᠡᠭ 𝐹 ᠲᠤ ᠣᠭᠳᠣ ᠯᠣᠯᠴᠠ ᠨ᠊ ᠠ᠂ 𝐶𝐹 ᠶᠢ ᠬᠤᠯᠪᠣᠨ ᠠ᠂ ᠲᠡ ᠭᠡᠪᠡᠯ 𝐶𝐹 ᠶᠢᠨ ᠬᠠᠮᠤᠭ ᠤᠨ ᠪᠠᠭ ᠠ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠨᠢ ︵︶ 16. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠲᠡᠭᠰᠢ ᠲᠥᠷᠪ ᠡᠯᠵᠢᠨ 𝐴𝐵𝐶𝐷 ᠲᠤ᠂ ᠴᠡᠭ 𝑀 ᠪᠣᠯ ᠲᠠᠯ ᠠ 𝐵𝐴 ᠶᠢᠨ ᠥᠷᠬ ᠥᠯᠵᠢ ᠯᠡᠯ ᠱᠤᠭᠤ ᠮ᠊ ᠳᠡ ᠭᠡᠷᠠᠬ ᠢ ᠨᠢ ᠭᠡᠨ ᠬᠦᠳᠡ ᠯᠬᠦ ᠴᠡᠭ 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 ︵ᠴᠡᠭ 𝐴 ᠲᠠᠢ ᠳᠠᠪᠬᠤᠴ ᠠᠬᠤ ᠦᠭᠡᠢ︶ ᠤ ᠥᠭᠡᠳ 𝐴𝑀 𝐴𝐵᠂ ∆ 𝐶𝐸𝐵 ᠪᠣᠯ ∆ 𝐷𝐴𝑀 ᠶᠢ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠵ ᠢᠬᠦᠯᠦ ᠭᠡᠳ ᠣ ᠯᠣᠭᠰ ᠠᠨ ᠵᠢᠷᠤᠭ ᠲᠥᠷᠰᠤ᠂ ᠬᠡᠷᠪᠡ ᠴᠡᠭ 𝐸 ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠭ ᠤᠯᠤᠨ 𝐸𝐻 ⊥ 𝐴𝐶 ᠪᠠᠷ ᠪᠠᠢᠭ ᠤᠯᠪᠠ ᠯ᠊᠂ ᠡᠭ ᠴᠡ ᠶᠢᠨ ᠰ᠊ ᠠᠭᠤᠷᠢ ᠨᠢ 𝐻 ᠪᠣᠯ ᠣᠨ ᠠ᠃ ᠲᠡᠭᠡᠪ ᠡᠯ ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠭᠡᠯᠡᠯᠭᠡ ᠠᠴᠠ ᠵᠥ ᠤ ᠪᠤ ᠯᠬᠤ ᠵᠣᠷᠪ ᠣᠰ ᠤᠳ ᠨᠢ ︵︶ ︵ᠪᠤᠢ ᠪᠦ ᠬᠦᠢ ᠵᠥᠪ ᠪᠣᠯᠬᠤ ᠬᠡᠯ ᠡᠯᠭᠡ ᠶᠢᠨ ᠨᠣᠮ ᠧᠷ ᠢ ᠲᠠᠭᠯ ᠠ᠃︶ ① ᠴᠡᠭ 𝑀 ᠶᠢᠨ ᠤ ᠠᠢᠷᠢ ᠨᠢ ᠬᠤᠪᠢᠷ ᠠᠭᠠᠳ ∠ 𝐷𝐻𝐶 = 60° ᠪᠣᠯ ᠬᠤ ᠦᠶ ᠡᠰ᠂ 2𝐵𝐸 = 𝐶𝑀 ᠪᠣᠯ ᠣᠨ ᠠ᠂ ② ᠴᠡᠭ 𝑀 ᠨᠢ ᠬᠥᠳᠡ ᠯᠭᠡᠭᠡ ᠨ᠊ ᠬᠢᠭᠡᠳ ᠶᠠᠮᠠ ᠷ᠊ ᠪᠠᠢᠷ ᠢ ᠲᠤ ᠬᠦᠷᠦ ᠭᠰᠡᠨ ᠴᠤ᠂ ᠬᠡᠵᠢᠶ ᠡᠳᠡ 𝐷𝑀 = √2 𝐻𝑀 ᠪᠣᠯ ᠣᠨ ᠠ᠃ ③ ᠴᠡᠭ 𝑀 ᠨᠢ ᠬᠦᠳᠡ ᠯᠭᠡᠭᠡᠨ ᠬᠢᠬ ᠦ ᠶᠠᠪᠤ ᠴᠠ ᠳ᠋᠊ ᠤ᠂ ᠳᠥᠷᠪ ᠡᠯᠵᠢᠨ 𝐶𝐸𝑀 𝐷 ᠨᠢ ᠷᠤᠮᠪ ᠪᠣᠯᠬ ᠤ ᠪᠤ ᠯᠣᠯᠴᠠᠭ ᠠ ᠲᠠ ᠢ᠃ ④ ᠴᠡᠭ 𝑀 ᠨᠢ ᠬᠦᠳᠡ ᠯᠭᠡᠭᠡᠨ ᠬᠢᠭᠡᠳ ᠶᠠᠮᠠᠷ ᠪᠠ ᠢᠷᠢᠨ ᠳ᠋᠊ ᠤ ᠬᠦᠷᠦ ᠭᠰᠡᠨ ᠴᠤ ∠𝐶𝐻𝑀 ᠨᠢ ᠯᠠᠪᠲᠠ ᠢ 135° ᠠᠴᠠ ᠶᠡᠭᠡ ᠭᠤᠷᠪ ᠠ᠂ ᠪᠣᠳᠣ ᠵᠤ ᠬᠠᠷ ᠢᠭᠤᠯᠬᠤ ᠰᠡ ᠲᠤᠪ 17. ᠲᠤᠰ ᠪᠠ ᠬ᠋᠊ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ ᠬᠤᠪᠢ ︵1︶ ᠲᠡᠩ ᠴᠡᠳ ᠭᠡᠯ ᠪᠤᠰᠤ ᠶᠢ ᠨ᠊ ᠪᠥᠯᠥ ᠬ᠋᠊ { 3 ︵𝑥− 1︶ 5𝑥 + 2 𝑥− 22 ≤ 7− 32 𝑥 ᠶᠢ ᠪᠣᠳ ᠣᠭᠠᠳ᠂ ᠲᠡᠭᠦᠨ ᠦ ᠬᠠᠮᠤᠭ ᠤ ᠨ᠊ ᠤ ᠠᠭ ᠠ ᠪᠥᠬᠦ ᠯᠢ ᠲᠣ ᠭᠠᠨ ᠤ ᠬᠠᠷ ᠢᠭᠤ ᠶᠢ ᠣᠯ᠃ ︵2︶ ᠤᠷᠢᠳᠠ ᠪᠠᠷ ᠬᠢᠯ ᠪᠠᠷᠴᠢᠯ ᠠᠵᠤ᠂ ᠳᠠᠷ ᠠᠭ ᠠ ᠨᠢ ᠬᠡ ᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠨᠢ ᠣ ᠯ᠊᠃ ︵𝑎 2 −1 𝑎 2 −2 𝑎+ 1︶ ÷ 2 𝑎 2 −𝑎᠂ ᠡᠭᠦᠨ ᠳ᠋᠊ ᠦ 𝑎 ᠨᠢ 𝑎 2 + 2𝑎 − 15 = 0 ᠢ ᠬᠠᠩᠭ ᠠᠨ ᠠ᠃ 18. ︵ᠲᠤᠰ ᠪᠠ ᠬ᠋᠊ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 9 ᠬᠤᠪᠢ︶ ᠨᠢᠭᠡᠨ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠨᠢ ᠰᠤᠷᠤᠭᠴᠢᠳ ᠤᠨ ᠬᠢᠴᠢ ᠶᠡᠯ ᠦᠨ ᠭᠠᠳᠠ ᠨᠠᠬᠢ ᠰᠤᠯᠠ ᠴᠠᠭ ᠢᠶᠠᠨ ᠠᠰᠢᠭ ᠯᠠᠨ ᠪᠡᠶ ᠡ ᠪᠠᠨ 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 ᠠ ᠴᠠᠭ ᠤ ᠨ᠊ ᠪᠠᠢᠳᠠ ᠯ᠊ ᠲᠤ ᠪᠠᠢᠴᠠᠭ ᠠᠯᠲᠠ ᠬᠢᠵᠠᠢ᠃ ᠪᠠᠢᠴᠠᠭ ᠠᠯᠲᠠ ᠶ᠊ ᠶᠢᠨ ᠳᠦ᠋ ᠩ᠊ ᠲᠤ᠂ ᠲᠤᠰ ᠠᠩᠭᠢ ᠶᠢᠨ ᠰᠤᠷ ᠤᠭᠴ ᠢᠳ ᠤᠨ ᠨᠢᠭᠡ ᠭᠠᠷ ᠠᠭ ᠤᠨ ᠳᠠᠪᠳ ᠠᠯᠭ ᠠ ᠬᠢᠭᠰ ᠠ ᠴᠠ ᠬ᠋᠊ ᠨᠢ ᠵᠥ ᠪᠬᠡᠨ 1 ᠴᠠᠭ᠂ 2 ᠴᠠ ᠬ᠋᠊᠂ 3 ᠴᠠᠭ᠂ 4 ᠴᠠᠭ ᠭᠡᠬᠦ ᠲᠤ ᠷᠪᠡᠨ ᠵᠦᠯ ᠦᠨ ᠪᠠᠢᠳᠠ ᠯ᠊ ᠤ ᠠᠢᠨ ᠠ ︵ ᠲᠤ ᠷᠪᠡ ᠶᠢ ᠬᠠᠶᠠᠵ ᠤ ᠲᠠ ᠪᠤ ᠶᠢ ᠳᠡ ᠪᠰᠢᠭ ᠤᠯᠵᠠᠢ︶᠃ ᠦᠭ ᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ ᠲᠤᠰ ᠠᠩᠭᠢ ᠲᠤ ᠨᠠᠢᠲᠡ 50 ᠰᠤᠷᠤᠭ ᠴᠢ ᠲᠠᠢ᠂ ᠪᠠᠢ ᠴᠠᠭᠠᠯᠲ ᠠ ᠶᠢᠨ ᠳᠦ᠋ ᠩ᠊ ᠢᠶᠡ ᠨ᠊ ᠦᠨᠳᠤᠰᠤ ᠯᠡᠨ᠂ ᠳᠣᠣᠷ ᠠᠬᠢ ᠮᠡᠲᠤ ᠬᠤᠶ ᠠᠷ ᠪᠦᠷᠢᠨ ᠪᠤᠰᠤ ᠪᠦᠷᠢᠳᠭᠡᠯ ᠦᠨ ᠬᠦᠰ ᠦᠨᠥᠭ ᠲᠤ ᠪᠠ ᠵᠢᠷᠤᠭ ᠢ ᠵᠢᠷ ᠤᠵᠠᠢ᠃ ᠡᠭᠦᠨ ᠲᠤ ᠲᠤᠰ ᠠᠩᠭᠢ ᠶᠢᠨ ᠡᠮᠡᠭᠳᠡᠢ ᠰᠤᠷᠤᠭᠴᠢ ᠶ᠊ ᠶᠢᠨ ᠨᠢ ᠭᠡ ᠭᠠᠷ ᠠᠭ ᠤᠨ ᠳᠠᠪᠳ ᠠᠯᠭ ᠠ ᠬᠢᠭᠰ ᠠ ᠴᠠ ᠬ᠋᠊ ᠨᠢ ᠳᠣ ᠣᠷᠠᠬ ᠢ ᠮᠡ ᠲᠤ ︵ᠨᠢᠭᠡᠴᠢ᠄ ᠴᠠᠭ︶᠄ 1 1 1 2 2 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 ᠶᠢᠰᠤᠳᠤᠭᠡᠷ ᠣᠨ ᠤ ᠨᠢᠭᠡᠳᠤᠭᠡᠷ ᠠᠩᠭᠢ ᠶᠢᠨ ᠡᠮᠡᠭᠳᠡᠢ ᠰᠤᠷᠤᠭᠴᠢ ᠶᠢᠨ ᠨᠢᠭᠡ ᠭᠠᠷᠠᠭ ᠤᠨ ᠳᠠᠪᠳ ᠠᠯᠭ ᠠ ᠬᠢᠭᠰᠡᠨ ᠴᠠᠭ ᠤ ᠨ᠊ ᠳᠠ ᠪᠳᠠᠭᠳᠠ ᠭᠰᠠᠨ ᠲᠤᠭ ᠠᠨ ᠤ ᠲᠠᠷ ᠬᠠᠯᠲ ᠠ ᠶᠢᠨ ᠬᠥᠰᠥ ᠨᠥᠭᠳᠥ ᠪᠠ ᠡᠷᠡᠭᠳᠡᠢ ᠰᠤᠷᠤᠭᠴᠢ ᠶᠢᠨ ᠨᠢᠭᠡ ᠭᠠᠷᠠᠭ ᠤᠨ ᠳᠠᠪᠳ ᠠᠯᠭ ᠠ ᠬᠢᠭᠰ ᠠ ᠴᠠ ᠬ᠋᠊ ᠤᠨ ᠲᠤᠭ ᠣᠷᠢᠭ ᠪᠦᠷᠢᠳᠭᠡᠯ ᠦᠨ ᠵᠢᠷᠤᠭ ᠨᠢ ᠳᠣᠣᠷ ᠠᠬᠢ ᠮᠡᠲᠤ᠃ ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠮᠡᠳᠡᠭᠡ ᠵᠠᠩᠭᠢ ᠶᠢ ᠦᠨᠳᠤ ᠰᠤᠯᠡᠨ᠂ ᠳᠠ ᠷᠠᠭᠠᠬ ᠢ ᠠ ᠰᠠᠭᠤᠯᠲ ᠠ ᠶᠢ ᠬᠠᠷᠢᠭ ᠤᠯ᠃ ︵1︶ ᠪᠦᠷᠢᠳᠭᠡᠯ ᠦᠨ ᠬᠦᠰᠥ ᠨᠥᠭᠳᠥ ᠳᠠᠬ ᠢ 𝑎 = ︵︶᠂ ᠲᠤᠰ ᠳᠠᠪᠳᠠᠯᠭ ᠠ ᠬᠢᠭᠰᠡ ᠨ᠊ ᠴᠠᠭ ᠳᠠᠪ ᠳᠠᠭᠳᠠᠭ ᠰᠠᠨ ᠲᠤᠭ ᠠ ︵ᠰᠤᠷ ᠤᠭᠴ ᠢ ᠶᠢᠨ ᠲᠤᠭ ᠠ︶ 1 ᠴᠠᠭ 3 2 ᠴᠠᠭ 𝑎 3 ᠴᠠᠭ 4 4 ᠴᠠᠭ 6 1 ᠴᠠᠭ 2 ᠴᠠᠭ 3 ᠴᠠᠭ 4 ᠴᠠᠭ d % 50% 20% 10% ᠠᠩᠭᠢ ᠶᠢᠨ ᠡᠮᠡᠭᠳᠡᠢ ᠰᠤᠷᠤᠭᠴᠢ ᠶᠢᠨ ᠨᠢᠭᠡ ᠭᠠᠷᠠᠭ ᠤᠨ ᠳᠠᠪᠳ ᠠᠯᠭ ᠠ ᠬᠢᠭᠰᠡᠨ ᠴᠠ ᠬ᠋᠊ ᠤᠨ ᠳᠤᠮᠳᠠ ᠪᠠᠢᠷ ᠢ ᠶᠢᠨ ᠲᠤᠭ ᠠ ᠨᠢ ︵︶ ᠴᠠᠭ ︵2︶ ᠲᠤᠭ ᠣᠷᠢᠭ ᠪᠦᠷᠢᠳᠭᠡᠯ ᠦᠨ ᠵᠢᠷᠤ ᠬ᠋᠊ ᠳᠡ ᠬᠢ ᠲᠤᠰ ᠠᠩᠭᠢ ᠶᠢᠨ ᠡᠷᠡᠭᠳᠡᠢ ᠰᠤᠷᠤᠭᠴᠢ ᠶᠢᠨ ᠨᠢᠭᠡ ᠭᠠᠷᠠᠭ ᠤᠨ ᠳᠠᠪᠳ ᠠᠯᠭ ᠠ ᠬᠢᠭᠰᠡᠨ ᠴᠠᠭ ᠨᠢ 4 ᠴᠠ ᠬ᠋᠊ ᠪᠤ ᠯᠬᠤ ᠲᠠᠢ ᠬᠠᠷᠠᠭ ᠠᠯᠵᠠᠭ ᠰᠠᠨ ᠲᠥᠪ ᠥᠨᠴᠥ ᠬ᠋᠊ ᠨᠢ ︵︶ ᠭᠷᠠᠳᠦ᠋ᠰ᠃ ︵3︶ ᠬᠡᠷᠪᠡ ᠲᠤᠰ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠶᠢᠨ ᠶᠢᠰᠤᠳᠤᠭᠡᠷ ᠣ ᠨ᠊ ᠳ᠋᠊ ᠤ ᠨᠠ ᠢᠲᠡ 60 0 ᠰ᠊ ᠤᠷᠤᠭᠴᠢ ᠲᠠᠢ ᠪᠣᠯ᠂ ᠡᠭᠦᠨ ᠦ ᠲᠣᠲᠣᠷ ᠡ ᠨᠢ ᠭᠡ ᠭᠠᠷᠠᠭ ᠤᠨ ᠳᠠ ᠪᠳᠠᠯᠭ ᠠ ᠬᠢᠭᠰ ᠠ ᠴᠠᠭ ᠨᠢ 4 ᠴᠠ ᠬ᠋᠊ ᠤ ᠣᠯᠬᠤ ᠰᠤᠷᠤᠭᠴᠢ ᠨᠢ ᠬᠡᠳᠤᠢ ᠪᠠᠢᠬᠤ ᠶᠢ ᠪᠠᠷ ᠤᠭᠴ ᠠᠭᠠᠯᠠᠨ ᠪᠣᠳᠣ᠃ ︵4︶ ᠲᠤᠰ ᠠᠩᠭᠢ ᠶᠢᠨ ᠳᠠᠪᠳ ᠠᠯᠭ ᠠ ᠬᠢᠭᠰᠡᠨ ᠴᠠᠭ ᠨᠢ 4 ᠴᠠᠭ ᠪᠣᠯᠬᠤ ᠲᠤ ᠷᠪᠡᠨ ᠡᠮᠡᠭᠳᠡᠢ ᠰᠤᠷᠤᠭᠴᠢ ᠣᠶᠣ ᠨ᠊᠂ ᠤᠯ ᠠᠭᠠᠨ᠂ ᠨᠣᠮ ᠶᠢᠨ᠂ ᠵᠢᠮ ᠢᠰ ᠦ ᠨ᠊ ᠳᠣᠲᠣᠷ ᠠ ᠠᠴᠠ ᠳᠠᠰᠢᠷᠠ ᠮ᠊ ᠢᠶᠠ ᠷ᠊ ᠬᠤᠶ 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 ᠬᠤᠶ ᠠᠷ ᠰᠤᠷᠤᠭᠴᠢ ᠨᠢ ᠶᠠᠭ ᠰᠠᠢᠬᠠᠨ ᠤᠯ ᠠᠭᠠᠨ ᠪᠠ ᠵᠢᠮ ᠢᠰ ᠪᠣᠯ ᠬᠤ ᠶᠢ ᠨ᠊ ᠲ᠊ ᠣᠬᠢᠶ ᠠᠯᠳᠤᠮ ᠶᠢ ᠶ᠊ ᠢ ᠣ ᠯ᠊᠃ 19. ︵ᠲᠤᠰ ᠪᠠ ᠬ᠋᠊ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 8 ᠬᠤᠪᠢ︶ ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ᠂ ᠨᠢᠭᠡᠳᠤᠭᠡᠷ ᠵᠡᠷ ᠭᠡ ᠶᠢ ᠨ᠊ ᠹᠦᠩᠺᠼ 𝑦 = 𝑘𝑥 + 𝑏 ᠦ ᠨ᠊ ᠵᠢᠷ ᠤᠮᠠᠭ ᠨᠢ ᠤᠷ ᠪᠠᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠲᠤ ᠹᠦᠩᠺᠼ 𝑦 = 𝑎𝑥 ᠦ ᠨ᠊ ᠵᠢᠷᠤᠮᠠᠭ ᠲᠠᠢ ᠨᠢᠭᠡᠳᠤᠭᠡᠷ ᠮᠥ ᠴᠡ ᠳ᠋᠊ ᠦ ᠴᠡᠭ 𝐴 ︵4 ,3︶ ᠲᠤ ᠣᠭᠳᠣᠯ ᠣᠯᠴᠠ ᠵᠤ᠂ 𝑦 ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠦ ᠨ᠊ ᠰ᠊ ᠥᠭᠡᠷ ᠬᠦ ᠬᠠᠭ ᠠᠰ ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠲᠠᠢ ᠴᠡᠭ 𝐵 ᠲᠤ ᠣᠭᠳᠣᠯ ᠣᠯᠴᠠ ᠬᠤ ᠪᠥᠭᠡ ᠳ᠋᠊ 𝑂𝐴 = 𝑂𝐵 ᠪᠣᠯ ᠣᠨ ᠠ᠃ ︵1︶ ᠹᠦᠩᠺᠼ 𝑦 = 𝑘𝑥 + 𝑏 ᠪᠠ 𝑦 = 𝑎𝑥 ᠶᠢᠨ ᠠᠨᠠᠯ ᠢᠰ ᠢ ᠯᠡᠷᠭᠡᠢᠯ ᠡᠯ ᠢ ᠣᠯ᠃ ︵2︶ ᠦᠭ ᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ ᠴᠡᠭ 𝐶 ︵0,5︶᠂ ᠨᠢᠭᠡᠳᠤᠭᠡᠷ ᠵᠡᠷᠭᠡ ᠶᠢᠨ ᠹᠦᠩᠺᠼ ᠦᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠳᠡᠭᠡ ᠷ᠊ ᠡ ᠨᠢᠭᠡᠨ ᠴᠡᠭ 𝑀 ᠶᠢ ᠳᠣᠭᠲᠠᠭᠠ ᠪᠠ᠂ ᠭᠡᠬᠦ ᠲᠤ ᠤ ᠠ 𝑀𝐵 = 𝑀𝐶 ᠪᠣᠯ ᠭᠠᠨ ᠠ᠃ ᠠ ᠡ ᠦᠶ ᠡᠰ ᠦ ᠨ᠊ ᠴᠡ ᠬ᠋᠊ 𝑀 ᠦ ᠨ᠊ ᠤ ᠠᠢᠷᠢᠰᠢᠯ ᠢ ᠣᠯ᠃ x y 4 3 C B A O 20. ︵ᠲᠤᠰ ᠪᠠ ᠬ᠋᠊ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 8 ᠬᠤᠪᠢ︶ ᠵᠢᠷᠤᠭ 1 ᠪᠣᠯ᠂ ᠬᠠᠨ ᠠᠨ ᠲᠤ ᠡᠯᠬᠦ ᠬᠦ ᠪᠡᠶ ᠡ ᠣ ᠬᠢᠶᠠᠯ ᠬ᠋᠊ ᠠ ᠶᠢᠨ ᠰᠦ᠋ᠷ ᠴᠢᠬᠦᠷ ᠦᠨ ᠪᠣᠳᠠᠳᠠ ᠢ ᠵᠢᠷ ᠤᠭ ᠪᠣᠯᠣ ᠨ᠊ ᠠ᠂ ᠵᠢᠷᠤᠭ 2 ᠪᠣᠯ ᠵᠢᠷᠤᠭ ᠶᠢ ᠦᠨᠳᠤ ᠰᠤᠯᠡ ᠨ᠊ ᠵᠢᠷ ᠤᠭᠰ ᠠᠨ ᠭᠧᠣᠮᠧᠳ᠋ᠷ ᠢ ᠶᠢᠨ ᠵᠢᠷᠤᠭ ᠲᠥᠷᠰᠤ ᠪᠣᠯᠣ ᠨ᠊ ᠠ᠃ ᠡᠩ ᠦᠨ ᠪᠠᠢᠳᠠ ᠯ᠊ ᠲᠤ᠂ ᠰ᠊ ᠦᠷᠴᠢᠬᠦᠷ ᠦᠨ ᠬᠠᠮᠤ ᠬ᠋᠊ ᠳᠡᠭᠡᠳᠤ ᠴᠡ ᠬ᠋᠊ 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 ᠳᠤᠬ ᠢᠷᠠᠮᠵᠢᠳᠠᠢ ᠪᠠᠢᠳᠠ ᠬ᠋᠊᠃ 𝑂 ᠪᠣᠯ ᠰᠦ᠋ᠷ ᠴᠢᠬᠦᠷ ᠢ ᠬᠠᠨ ᠠᠨ ᠲᠤ ᠳᠣᠭᠲᠠᠭ ᠠᠵᠤ ᠪᠠᠢᠬᠤ ᠴᠡ ᠬ᠋᠊᠂ 𝑂𝐴 = 10 𝑐𝑚 ᠰᠦ᠋ ᠷᠴᠢᠬᠦᠷ ᠦᠨ ᠲᠣᠯᠤᠭ ᠠᠢ ᠶᠢᠨ ᠳᠢᠶᠠᠮ ᠧᠳ᠋ᠷ 𝐴𝐵 = 8𝑐𝑚᠂ ∠ 𝐶𝑂𝐴 = 26°᠂ ∠ 𝑂𝐴𝐵 = 146°᠂ ᠬᠡ ᠷᠪᠡ ᠪᠡᠶ ᠡ ᠶᠢᠨ ᠦᠨᠳᠤᠷ ᠨᠢ 175 𝑐𝑚 ᠪᠣᠯᠬᠤ ᠬᠦᠮᠦ ᠨ᠊ ᠣ ᠬᠢᠶᠠᠯ ᠬ᠋᠊ ᠠ ᠬᠢᠬᠦ ᠭᠡᠪᠡᠯ᠂ ᠴᠡ ᠬ᠋᠊ 𝑂 ᠨᠢ ᠭᠠᠵᠠᠷ ᠤᠨ ᠨᠢᠭᠤᠷ ᠠ ᠴᠠ ᠬᠡᠳᠤᠢ ᠵᠠᠢ ᠲᠠᠢ ᠪᠠᠢᠪ ᠠᠯ ᠵᠣᠬᠢᠬᠤ ᠪᠣᠢ︖ ︵ᠯᠠᠪᠯᠠᠬᠤ ᠲᠤᠭ ᠠ ᠤ ᠠᠷᠢᠮᠲᠠ᠄ 𝑠𝑖𝑛 26° ≈ 0.44 , 𝑐𝑜𝑠 26° ≈ 0.90 , 𝑡𝑎𝑛 26° ≈ 0.49 ᠪᠣᠳᠣ ᠯᠭ ᠠ ᠶᠢᠨ ᠳᠦ᠋ᠩ ᠢ 1𝑐𝑚 ᠬᠦᠷᠲᠡᠯ ᠡ ᠨᠠᠷᠢᠪ ᠴᠢᠯᠠ︶ 21. ︵ᠲᠤᠰ ᠪᠠ ᠬ᠋᠊ 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᠊ ᠤ ᠢᠳᠠ ᠪᠠᠰᠠ ᠲᠤᠭ ᠣᠷᠢᠭ ᠤ ᠨ᠊ ᠲ᠊ ᠥᠪ ᠦᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ︵𝑎 ,𝑏︶᠂ ᠷᠠᠳᠢᠤᠰ ᠨᠢ 𝑟 ᠪᠣᠯ ᠬᠤ ᠲᠤᠭ ᠣᠷᠢᠭ ᠤᠨ ᠲᠡᠭᠰᠢᠳ ᠭᠡᠯ ᠨᠢ ︵𝑥− 𝑎︶ 2+ ︵𝑦 − 𝑏︶ 2= 𝑟 2 ᠪᠣᠯ ᠬᠤ ᠶᠢ ᠰ᠊ ᠤᠷᠤᠯ ᠴᠠᠨ ᠠ᠃ ᠵᠢᠱᠢᠶ ᠡᠯᠡᠪᠡ ᠯ᠊᠄ ᠲᠤᠭ ᠣᠷᠢᠭ ᠤᠨ ᠲᠤ ᠤ ᠨᠢ 𝑃 ︵−2 ,1︶᠂ ᠷᠠ ᠳᠢᠤᠰ ᠨᠢ 3 ᠪᠣᠯᠬᠤ ᠲᠤᠭ ᠣᠷᠢᠭ ᠤᠨ ᠲᠡᠭᠰᠢ ᠲᠭᠡᠯ ᠨᠢ ︵𝑥+ 2︶ 2+ ︵𝑦 − 1︶ 2= 9 ᠤ ᠣᠯᠣᠨ ᠠ᠃ ︵1︶ ᠴᠡᠭ 𝑀 ︵− 3 , −1︶ ᠢᠶᠡᠷ ᠲᠤᠭ ᠣᠷᠢᠭ ᠤᠨ ᠲᠥᠪ ᠪᠤ ᠯᠭᠠᠨ᠂ √3 ᠪᠠᠷ ᠷᠠᠳᠢᠤᠰ ᠪᠣᠯᠭ ᠠᠭᠰᠠ ᠨ᠊ ᠲᠤᠭ ᠣᠷᠢᠭ ᠤ ᠨ᠊ ᠲᠡ ᠭᠰᠢᠳ ᠭᠡᠯ ᠨᠢ _ __ __ __ __ __ __ __ ︵2︶ ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠴᠡᠭ 𝑀 ︵− 3 , 0︶ ᠢᠶᠡᠷ ᠲᠤᠭ ᠣᠷᠢᠭ ᠤᠨ ᠲᠤ ᠤ ᠪᠤ ᠯᠭᠠᠭᠰᠠᠨ ᠲᠤᠭ ᠣᠷᠢᠭ ᠨᠢ 𝑦 ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠲᠠ ᠢ ᠡᠬ ᠶᠢᠨ ᠦ ᠴᠡᠭ ᠲᠤ ᠰᠢᠬ ᠠᠯᠴᠠᠨ ᠠ᠂ ᠴᠡᠭ 𝐶 ᠪᠣᠯ ⨀𝑂 ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠨᠢᠭᠡ ᠴᠡᠭ᠂ 𝑂𝐶 ᠶᠢ ᠬᠤᠯ 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 ᠬ᠋᠊ 𝐸 ᠲᠤ ᠣᠭᠳᠣ ᠯᠣᠯᠴᠠ ᠨ᠊ ᠠ᠂ 𝑠𝑖𝑛 ∠𝐴𝑂𝐶 = 35᠃ ① 𝐸𝐶 ᠶᠢ ᠬᠣ ᠯᠪᠣᠨ ᠠ᠂ 𝐸𝐶 ᠨᠢ ⨀𝐵 ᠶᠢᠨ ᠰᠢᠬ ᠠᠭᠴᠢ ᠰᠢᠯᠤᠭᠤ ᠨ᠊ ᠪᠣᠯᠬᠤ ᠶᠢ ᠪᠠ ᠳᠤᠯᠠ᠃ ② 𝐵𝐸 ᠳᠡᠭᠡᠷ ᠡ 𝑄𝐵 = 𝑄𝐶 = 𝑄𝐸 = 𝑄𝑂 ᠪᠣᠯᠭ ᠠᠬᠤ ᠴᠡᠭ 𝑄 ᠣᠷ ᠣᠰᠢᠨ ᠠ ᠤᠤ︖ ᠬᠡᠷᠪᠡ ᠣᠷᠣᠰ ᠢᠪᠠᠯ ᠴᠡᠭ 𝑄 ᠦᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠢ ᠣᠯᠵ ᠤ᠂ ᠪᠠᠰᠠ ᠴᠡᠭ 𝑄 ᠪᠠᠷ ᠲᠤ ᠤ ᠤ ᠣᠯᠭᠠᠨ᠂ 𝑄𝐵 ᠪᠠᠷ ᠷᠠᠳᠢᠤᠰ ᠪᠣᠯᠭ ᠠᠭᠰᠠᠨ ⨀𝑄 ᠶᠢᠨ ᠲᠡᠭᠰᠢᠳ ᠭᠡᠯ ᠢ ᠪᠢᠴᠢᠵ ᠤ ᠭᠠᠷ ᠬ᠋᠊ ᠠ᠃ ᠬᠡᠷᠪᠡ ᠣᠷᠣᠰ ᠢᠬᠤ ᠦᠭᠡᠢ ᠪᠣᠯ ᠰᠢᠯᠲᠠᠭ ᠠᠨ ᠢ ᠨᠢ ᠲᠣᠳ ᠣᠷᠬᠠᠢᠯ ᠠ᠃ B A O C 22. ︵ᠲᠤᠰ ᠪᠠ ᠬ᠋᠊ ᠠ ᠨᠠᠢᠲᠡ 8 ᠬᠤᠪᠢ︶ ᠨᠢᠭᠡᠨ ᠵᠢᠮᠢᠰ ᠦ ᠨ᠊ ᠳᠡ ᠯᠭᠡᠭᠦᠷ ᠬᠠᠳᠠ ᠭᠰᠠᠨ ᠦᠨ ᠡ ᠨᠢ ᠵᠢᠩ ᠪᠦᠷᠢ ᠲᠤ ᠤ ᠠ 10 ᠲᠤ ᠬᠥᠷᠢᠭ ᠪᠠᠢᠭᠰᠠ ᠨ᠊ ᠨᠢᠭᠡ ᠵᠦᠯ ᠦᠨ ᠵᠢᠮᠢᠰ ᠦᠨ ᠦᠨ ᠡ ᠶᠢ ᠬᠤᠶ ᠠᠷ ᠤᠳᠠ ᠬ᠋᠊ ᠠ ᠪᠠᠭᠤᠷᠠ ᠭᠤᠯᠤᠭᠠ ᠳ᠋᠊ ᠵᠢᠩ ᠪᠦᠷᠢ ᠲᠤ ᠤ ᠠ 8 .1 ᠲᠥᠬᠦ‍ ᠷᠢᠭ ᠪᠣᠯᠭ ᠠᠵᠠᠢ᠃ ᠡᠭᠦᠨ ᠲᠤ ᠵᠢᠮᠢᠰ ᠦ ᠨ᠊ ᠦ ᠨ᠊ ᠡ ᠶᠢᠨ ᠬᠤᠶ ᠠᠷ ᠤᠳᠠ ᠬ᠋᠊ ᠠ ᠶ᠊ ᠶᠢᠨ ᠳᠤᠮᠳᠠᠴᠢ ᠪᠠᠭ ᠤᠷᠠᠯ ᠲᠠ ᠶ᠊ ᠶᠢᠨ ᠨᠣᠷᠮ ᠠ ᠨᠢ ᠠᠳᠠᠯᠢ᠃ ︵1︶ ᠲᠤᠰ ᠵᠦᠯ ᠦ ᠨ᠊ ᠵᠢᠮ ᠢᠰ ᠦ ᠨ᠊ ᠦᠨ ᠡ ᠶᠢᠨ ᠤᠳᠠ ᠬ᠋᠊ ᠠ ᠪᠦᠷᠢ ᠶᠢ ᠨ᠊ ᠲᠤ ᠮᠳᠠᠴᠢ ᠪᠠᠭᠤᠷᠠ ᠯᠲᠠ ᠶᠢᠨ ᠨᠣᠷᠮ ᠠ ᠶᠢ ᠣᠯ᠃ ︵2︶ ᠵᠢᠮ ᠢᠰ ᠶᠢᠨ ᠦᠨ ᠡ ᠨᠢ ᠬᠤᠶ ᠠᠳᠤᠭ ᠠᠷ ᠤ ᠳᠠᠭ ᠠ ᠪᠠᠭ ᠤᠷᠠᠭᠰᠠᠨ ᠤ ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠨᠢᠭᠡᠳᠤᠭᠡᠷ ᠡᠳᠤᠷ ᠠᠴᠠ ᠡᠬᠢᠯ ᠠ 𝑥 ᠳᠤᠭᠠᠷ ᠡᠳᠤᠷ ︵𝑥 ᠪᠣᠯ ᠪᠥᠬᠦ ᠯᠢ ᠲᠤᠭ ᠠ︶ ᠲᠠᠬᠢ ᠪᠣᠷᠣᠯ ᠠᠭᠤᠯᠬᠤ ᠬᠡᠮᠵᠢ ᠶ᠊ ᠡ ᠵᠢᠴᠢ ᠬᠠᠳᠠ ᠭᠠᠯᠠᠬ ᠤ ᠤ ᠠ ᠬᠤᠷᠣᠭᠳᠠ ᠯ᠊ ᠤᠨ ᠵᠠᠷᠤᠳᠠᠯ ᠤᠨ ᠪᠠᠢᠳᠠ ᠯ᠊ ᠨᠢ ᠳᠠᠷ ᠠᠭᠠᠬ ᠢ ᠬᠥᠰᠥ ᠨᠥᠭᠳᠥ ᠲᠤ ᠦ ᠵᠡᠬᠦᠯ ᠦᠭᠰᠡᠨ ᠮᠡᠲᠤ᠃ ᠦᠭ ᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ ᠲᠤᠰ ᠵᠦᠯ ᠦᠨ ᠵᠢᠮᠢᠰ ᠦᠨ ᠣᠷᠣᠭᠤᠯ ᠤᠭᠰ ᠠᠨ ᠦᠨ ᠡ ᠨᠢ ᠵᠢᠩ ᠪᠦᠷᠢ ᠲᠤ ᠤ ᠠ 4.1 ᠲᠥᠬᠦ‍ ᠷᠢᠭ᠂ ᠲᠤᠰ ᠵᠦᠯ ᠦᠨ ᠵᠢᠮᠢᠰ ᠢ ᠪᠣᠷᠣᠯ ᠠᠭᠤᠯ ᠤᠭᠰᠠᠨ 𝑥 ᠳᠤᠭᠠᠷ ᠡᠳ ᠤᠷ ᠲᠠᠬᠢ ᠠ ᠰᠢᠭ ᠬᠤᠨᠵ ᠢᠪᠤᠷᠢ ᠶᠢ 𝑦 ᠲᠥᠬᠦ‍ ᠷᠢᠭ ᠭᠡᠪᠡᠯ᠂ 𝑦 ᠪᠠ 𝑥 ︵1≤ 𝑥≤ 10︶ ᠤᠨ ᠬᠤᠭᠤᠷ ᠣᠨᠳᠤᠬ ᠢ ᠹᠦᠩᠺᠼ ᠦᠨ ᠬᠤᠯ ᠪᠣᠭᠳᠠᠯ ᠤᠨ ᠠᠨᠠᠯ ᠢᠰ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡᠯ ᠢ ᠣᠯ᠃ ᠪᠠᠰᠠ ᠬᠡᠳᠤᠳᠤᠭᠡᠷ ᠡᠳᠤᠷ ᠲᠠᠬ ᠢ ᠪᠣᠷᠣᠯ ᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠠᠰᠢᠭ ᠬᠤᠨᠵ ᠢᠪᠤᠷᠢ ᠨᠢ ᠬᠠᠮᠤᠭ ᠤᠨ ᠶᠡᠭᠡ ᠪᠣᠯᠬ ᠤ ᠵᠢᠴᠢ ᠬᠠᠮᠤᠭ ᠤᠨ ᠶᠡᠭᠡ ᠪᠣᠷᠣᠯ ᠠᠭᠤᠯ ᠤᠭᠰ ᠠᠨ ᠠ ᠰᠢᠭ ᠬᠤᠨ ᠵᠢᠪᠤᠷᠢ ᠨᠢ ᠬᠡᠳᠤ ᠢ ᠪᠣᠯᠬᠤ ᠶᠢ ᠣᠯ᠃ ᠬᠤᠭᠤᠴᠠᠭ ᠠ ︵ᠡᠳᠤᠷ︶ ᠪᠣᠷᠣ ᠯᠠᠭᠤᠯᠬᠤ ᠬᠡᠮ ᠵᠢᠶ ᠡ ︵ᠡᠳᠤᠷ︶ ᠬᠠᠳᠠ ᠭᠠᠯᠠᠬ ᠤ ᠪᠠ ᠬᠤᠷᠣᠭᠳᠠ ᠯ᠊ ᠤᠨ ᠵᠠᠷ ᠤᠳᠠᠯ ︵ᠡᠳᠤᠷ︶ 𝑥 120 − 𝑥 3𝑥 2− 64 𝑥+ 400 x y A D E B O C 23. ︵ᠲᠤᠰ ᠪᠠ ᠬ᠋᠊ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 10 ᠬᠤ ᠪᠢ︶ ᠣᠷᠣᠯ ᠳᠤᠯᠭ ᠠ ᠠᠴ ᠠ ᠠ ᠵᠢᠭᠯ ᠠᠵᠤ ᠣᠯᠬ ᠤ ᠵᠢᠷᠤᠭ 1 ᠮᠡ ᠲᠤ᠂ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ 1 ᠨᠢ ᠭᠡᠴᠢ ᠤᠷᠳᠤ ᠪᠣᠯ ᠬᠤ ᠶᠢ 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 ᠰᠤᠭᠠ ᠷ᠊ ᠵᠢᠷ ᠤᠭ ᠪᠠᠢᠭ ᠤᠯ᠃ ∆𝐴 𝐵𝐶 ᠶᠢ ᠴᠡᠭ 𝐴 ᠪᠠ ᠷ᠊ ᠲᠥᠪ ᠪᠣᠯ ᠭᠠᠨ᠂ ᠴᠠ ᠬ᠋᠊ ᠤᠨ ᠵᠡᠭᠦᠦ ᠶᠢᠨ ᠳᠠ ᠭᠠᠤ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ 90° ᠡᠷᠬᠢᠭ ᠦᠯ᠃ ᠴᠡ ᠬ᠋᠊ 𝐵 ᠦᠨ ᠬᠠᠷᠠᠭ ᠠᠯᠵᠠᠭᠰ ᠠᠨ ᠴᠡᠭ ᠨᠢ 𝐵 ′᠂ ᠴᠡ ᠬ᠋᠊ 𝐶 ᠶᠢᠨ ᠬᠠᠷᠠᠭ ᠠᠯᠵᠠᠭᠰᠠᠨ ᠴᠡ ᠬ᠋᠊ ᠨᠢ 𝐶 ′ ᠪᠣᠯᠣᠨ ᠠ᠂ 𝐵𝐵 ′ ᠶᠢ ᠬᠣ ᠯᠪᠤ᠃ ② ᠰᠡᠲᠤᠪ ① ᠲᠤ ᠪᠠᠢᠭ ᠤᠯᠤᠭ ᠰᠠᠨ ᠵᠢᠷᠤᠭ ᠲᠠᠬᠢ ∠𝐴 𝐵 ′𝐵 = ︵︶ °᠃ ᠠᠰᠠᠭᠤᠳᠠ ᠯ᠊ ᠢ ᠰᠢᠢᠳ ᠪᠤᠷᠢᠯ ᠡᠬᠦ ᠵᠢᠷᠤᠭ 2 ᠮᠡ ᠲᠤ᠂ 𝑅𝑡 ∆𝐴𝐵𝐶 ᠲᠤ 𝐵𝐶 = 1 , ∠𝐶 = 90°᠂ 𝐶𝐴 ᠶᠢ ᠦᠷᠬ ᠦᠯᠵᠢᠯ ᠡᠬᠦᠯᠵ ᠤ ᠴᠡ ᠬ᠋᠊ 𝐷 ᠬᠦᠷ ᠭᠡᠬᠦ ᠲᠤ ᠤ ᠠ 𝐶𝐷 = 1 ᠤ ᠣᠯᠭᠠᠨ ᠠ᠂ ᠵᠢᠰᠢᠬᠦᠦ ᠲᠠ ᠯ᠊ ᠠ 𝐴𝐵 ᠶᠢ ᠴᠡᠭ 𝐴 ᠪᠠ ᠷ᠊ ᠲᠥᠪ ᠪᠣᠯ ᠭᠠᠨ᠂ ᠴᠠ ᠬ᠋᠊ ᠤᠨ ᠵᠡᠭᠦᠦ ᠶᠢ ᠨ᠊ ᠳᠠ ᠭᠠᠤ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ 90° ᠡᠷᠬ ᠢᠬᠦᠯᠦᠨ 𝐴𝐸 ᠶᠢ ᠣᠯᠵ ᠠᠢ᠂ 𝐷𝐸 ᠶᠢ ᠬᠣ ᠯᠪᠣᠨ ᠠ᠂ ∠𝐴𝐷𝐸 ᠶᠢᠨ ᠭᠷᠠᠳᠦ᠋ᠰ ᠤᠨ ᠲᠤᠭ ᠠ ᠶᠢ ᠣᠯ᠃ ᠥᠷᠭᠡᠳᠭᠡᠨ ᠬᠡᠷᠡᠭᠯᠠᠬ ᠦ ᠵᠢᠷᠤᠭ 3 ᠮᠡ ᠲᠤ᠂ ᠲ᠊ ᠥᠷᠪᠡ ᠯᠵᠢᠨ 𝐴𝐵𝐶𝐷 ᠲᠤ 𝐴𝐸 ⊥ 𝐵𝐶᠂ ᠡᠭ ᠴᠡ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ 𝐸᠂ ∠𝐵𝐴𝐸 = ∠𝐴𝐷𝐶 , 𝐵𝐸 = 𝐶𝐸 = 1, 𝐶𝐷 = 3, 𝐴𝐷 = 𝑘𝐴𝐵 ︵𝑘 ᠨᠢ ᠪᠠᠢᠩᠭ ᠤ ᠲᠤᠭ ᠠ︶᠂ 𝐵𝐷 ᠶᠢᠨ ᠤᠷᠳᠤ ᠶᠢ ᠣᠯ᠃ ︵𝑘 ᠠᠭ ᠤᠯᠤᠭᠰ ᠠᠨ ᠢ ᠯᠡᠷᠭᠡᠢᠯ ᠡᠯ ᠢᠶᠡᠷ ᠢᠯ ᠡᠷᠭᠡᠢᠯ ᠡ᠃︶ C A B E D B C A E A B C D 24. ︵ᠲᠤᠰ ᠪᠠ ᠬ᠋᠊