2021 ᠣᠨ ᠤ ᠣᠷᠳᠣᠰ ᠬᠤᠲᠠ ᠶᠢᠨ ᠳᠤᠮᠳᠠᠳᠤ ᠰᠢᠯᠭᠠᠯᠲᠠ ᠶᠢᠨ ᠰᠡᠲᠤᠪ

ᠡᠳᠦ ᠷ ᠬᠡᠷᠡᠭᠯ ᠡᠬᠦ ᠣᠰᠤ ᠨ ᠤ ᠬᠡᠮᠵ ᠢᠶ ᠡ / ᠲ ᠣᠨ ᠨᠢᠭᠡ᠂ ᠭᠠᠭᠴᠠ ᠬᠠᠷ ᠢᠭᠤᠯ ᠲᠠ ᠰᠤᠩᠭᠤ ᠬᠤ ᠰᠡᠲᠤᠪ ᠃ ( ᠲᠤᠰ ᠶᠡᠭᠡ ᠰᠡᠲᠤᠪ ᠲᠤ 10 ᠪᠠᠭ ᠠ ᠰ ᠡᠲᠤᠪ ᠲᠠᠢ᠂ ᠪᠠᠭ ᠠ ᠰᠡᠲᠤᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪ ᠢ᠂ ᠨᠠᠢᠲᠡ 30 ᠬᠤᠪᠢ ) 1. ᠪᠣᠳᠠᠳᠤ ᠲ ᠣᠭ ᠠ 0 , 𝜋 , |− 2| ,− 1 ᠲᠣᠲ ᠣᠷ ᠠ ᠂ ᠬᠠ ᠮᠤᠭ ᠤ ᠨ ᠪ ᠠᠭ ᠠ ᠲᠣᠭ ᠠ ᠨᠢ A. |− 2| B. 0 C. −1 D. 𝜋 2. ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠵᠢᠷᠤᠭ ᠲᠤ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡᠭᠰᠡᠨ ᠭᠧᠣᠮᠧᠲᠷ ᠢ ᠶᠢᠨ ᠪᠡᠶ ᠡ ᠪᠣᠯ ᠪᠦᠷᠢᠮ ᠥᠰᠥᠨ ᠠᠳᠠᠯᠢ ᠲᠠ ᠪᠤᠨ ᠪᠠᠭ ᠠ ᠲᠡᠭᠰᠢ ᠳᠥᠷ ᠪᠡᠯᠵ ᠢᠨ ᠪ ᠡᠶ ᠡ ᠪᠡᠷ ᠪᠦᠷᠢᠯᠳᠥᠵᠠᠢ᠂ ᠲᠡᠭᠦ ᠨ ᠦ ᠵᠡᠭᠦᠨ ᠬᠠᠷᠠᠭ ᠠᠨ ᠵᠢᠷ ᠤᠭ ᠨᠢ A. A . B. C. D. 3. ᠳᠡᠯᠡᠭᠡᠢ ᠶᠢᠨ ᠡᠷᠡᠭᠦᠯ ᠬᠠᠮᠠᠭ ᠠᠯᠠᠯ ᠤᠨ ᠵᠣᠬᠢᠶᠠᠨ ᠪᠠᠢᠭ ᠤᠯᠤᠯᠲᠠ ᠠᠴᠠ ᠲᠢᠳ ᠢᠮᠯᠢᠭ ᠸᠢᠷ ᠦ᠋ᠰ ᠦᠨ ᠬᠠᠮᠤᠭ ᠤᠨ ᠶᠡᠬᠡ ᠳ᠋ᠢᠶᠠᠮ ᠧᠲᠷ ᠨᠢ ᠪᠠᠷᠤᠭ 0 .00 00 0 0 12 𝑚 ᠭᠡᠵ ᠦ ᠨᠠᠢᠲᠡ ᠯᠡᠵᠠᠢ᠃ 0.00 000 01 2𝑚 ᠢ ᠰᠢᠨᠵᠢ ᠯᠡᠬᠦ ᠤᠬᠠᠭᠠ ᠨᠴᠢ ᠲᠣᠭ ᠠ ᠲᠡ ᠮᠳᠡᠭᠯᠡᠬᠦ ᠠᠷᠭ ᠠ ᠪᠠᠷ ᠢᠯ ᠡᠷᠭᠡᠢᠯ ᠡᠪᠡᠯ A. 1.2 × 10 ;7 B. 0.12 × 10 ;6 C. 12 × 10 ;8 D. 1.2 × 10 ;6 4. ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠪᠣᠳᠣ ᠯᠭ ᠠ ᠠᠴ ᠠ ᠵᠥ ᠪ ᠪᠣ ᠯᠬᠤ ᠨᠢ A. 𝑎 2+ 𝑎 2 = 2𝑎 4 B. 𝑎 6÷ 𝑎 2 = 𝑎 3 C. ( 𝑎+ 3) (𝑎 − 3) = 𝑎 2 − 6𝑎 + 9 D. (− 3𝑎 3) 2= 9𝑎 6 5. ᠨᠢᠭᠡ ᠥᠨ ᠴᠥᠭ ᠨᠢ 30 0ᠪᠣᠯᠬᠤ ᠲᠡ ᠭᠰᠢ ᠥᠨᠴᠥ ᠭᠳᠥ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠤᠤᠢ ᠪᠠ ᠠᠰ ᠣᠣᠢ ᠶᠢ ᠳᠣᠣᠷᠠ ᠬᠢ ᠵᠢᠷᠤᠭ ᠮᠡ ᠲᠤ ᠲᠠᠯᠪᠢ ᠵᠠᠢ᠂ ᠬᠡ ᠷᠪᠡ ∠1 = 16 6°3 3 ′ ᠪᠣᠯ᠂ ∠2 ᠤᠨ ᠭᠷᠠᠳᠦ᠋ᠰ ᠤᠨ ᠲᠣᠭ ᠠ ᠨᠢ A. 66°27 ′ B. 63°27 ′ C. 64 °3 3′ D. 63°33′ 6. ᠰᠠᠷᠠᠭ ᠤᠯ᠂ ᠣᠷᠳᠣᠰ ᠬᠣᠲ ᠠ ᠶᠢᠨ ᠨᠢᠭᠡᠨ ᠵᠣᠴᠢᠳ ᠤ ᠨ ᠪᠠᠭᠤᠳᠠᠯ ᠤᠨ 2 02 1 ᠣᠨ ᠤ 3 ᠰᠠᠷ ᠠ ᠶᠢᠨ 1 ᠦ ᠡᠳ ᠦᠷ ᠡᠴᠡ 3 ᠰᠠᠷ ᠠ ᠶᠢᠨ 6 ᠦ ᠡᠳᠦᠷ ᠦᠨ ᠬᠣᠭᠣᠷ ᠣᠨᠳᠤ ᠡᠳᠦᠷ ᠪᠦᠷᠢ ᠬᠡᠷᠡᠭᠯᠡᠬ ᠦ ᠤ ᠰᠤᠨ ᠤ ᠬᠡᠮᠵᠢᠶ ᠡ ︵ ᠨᠢᠭᠡᠴᠢ᠄ ᠲᠣᠨ ︶ ᠶᠢ ᠪᠦᠷᠢᠳᠭᠡᠨ ᠡᠮ ᠬᠢᠳ ᠭᠡᠵᠦ᠂ ᠳᠣ ᠣᠷᠠᠬ ᠢ ᠮᠡ ᠲᠤ ᠳᠠᠬᠢᠷ ᠱᠤᠭᠤᠮᠤᠨ ᠪᠦᠷᠢᠳᠭᠡᠯ ᠦ ᠨ ᠵᠢᠷᠤᠭ ᠢ ᠵᠢᠷ ᠤᠵᠠᠢ᠃ ᠳᠠ ᠷᠠᠭᠠᠬ ᠢ ᠳ᠋ᠦ᠋ ᠩᠨᠡᠯ ᠲᠡ ᠡᠴᠡ ᠵᠥ ᠪ ᠨᠢ A . ᠳᠤᠮ ᠳᠠᠴᠢ ᠲᠣᠭ ᠠ ᠨᠢ 234 B. ᠮᠣᠳ᠋ ᠨᠢ 10 C. ᠳᠤᠮᠳᠠ ᠪᠠᠢᠷ ᠢ ᠶᠢᠨ ᠲᠣᠭ ᠠ ᠨᠢ 8. 5 D. ᠰ ᠠᠷᠨᠢᠴᠡ ᠨᠢ 253 7. ᠵᠢᠰᠢᠭᠦ ᠯᠵᠢᠨ 𝐴𝑂𝐶𝐷 ᠶᠢᠨ ᠣᠷᠣᠢ ᠶᠢᠨ ᠴᠡ ᠭ 𝑂 (0 ,0) ᠂ ᠴᠡ ᠭ 𝐶 ᠨᠢ 𝑥 ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠦ ᠨ ᠡᠶᠡᠷ ᠬᠦ ᠬᠠᠭ ᠠᠰ ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠳᠡᠭᠡᠷ ᠡ ᠣᠷᠣᠰ ᠢᠨ ᠠ᠃ ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠠᠯᠬ ᠤᠮ ᠢᠶᠠ ᠷ ᠵᠢᠷᠤᠭ ᠪᠠᠢᠭ ᠤᠯᠤᠨ ᠠ᠃ ① ᠴᠡᠭ 𝑂 ᠢᠶᠡᠷ ᠲᠣᠭ ᠣᠷᠢᠭ ᠤᠨ ᠲᠥᠪ ᠪᠣᠯ ᠭᠠᠵᠦ᠂ ᠵᠣᠬ ᠢᠰᠲᠠᠢ ᠤᠷᠳᠤ ᠪᠠᠷ ᠷᠠᠳᠢᠤᠰ ᠪᠣᠯᠭ ᠠᠨ ᠨᠤᠮᠤ ᠪᠠᠢᠭ ᠤᠯᠪᠠ ᠯ᠂ 𝑂𝐴 , 𝑂𝐶 ᠲᠠᠢ ᠲᠤᠰ ᠲᠤᠰ ᠴᠡᠭ 𝑀 , 𝑁 ᠳ᠋ᠦ ᠤᠷᠳ᠋ᠤ ᠰ ᠬᠤᠳ᠋ᠠ ᠢᠢᠨ 20 21 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 ᠵᠢᠷ ᠤᠭ ① ᠵᠢᠷ ᠤᠭ ② ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠ ᠨ ᠠ᠃ ② ᠲᠤᠰ ᠲᠤᠰ ᠴᠡᠭ 𝑀 , 𝑁 ᠢᠶᠡᠷ ᠲᠣᠭ ᠣᠷᠢᠭ ᠤᠨ ᠲᠥ ᠪ ᠪᠣᠯ ᠭᠠᠨ᠂ 12 𝑀𝑁 ᠡᠴᠡ ᠶᠡᠭᠡ ᠪᠣᠯᠬᠤ ᠤᠷᠳᠤ ᠪᠠᠷ ᠷᠠᠳᠢᠤᠰ ᠪᠣᠯᠭ ᠠᠨ ᠨᠤᠮ ᠤ ᠪᠠᠢᠭ ᠤᠯᠪᠠ ᠯ᠂ ᠬᠣᠶ ᠠᠷ ᠨᠤᠮ ᠤ ᠨᠢ ∠𝐴𝑂𝐶 ᠶᠢᠨ ᠲᠣᠳᠣᠭᠠ ᠲᠤ ᠭᠡᠰᠡᠭ ᠦᠨ ᠴᠡᠭ 𝐸 ᠳ᠋ᠦ ᠣ ᠭᠲᠣ ᠯᠣᠯᠴᠠ ᠨ ᠠ᠃ ③ ᠰ ᠠᠴᠤᠷᠠ ᠭ 𝑂𝐸 ᠶᠢ ᠪᠠᠢᠭ ᠤᠯᠪᠠᠯ ᠂ 𝐴𝐷 ᠲᠠᠢ ᠴᠡᠭ 𝐹( 2,3 ) ᠳ᠋ᠤ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠᠨ ᠠ᠃ ᠲᠡᠭᠡ ᠪᠡᠯ ᠴᠡᠭ 𝐴 ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ A. (− 54 , 3) B. (3 − √13 ,3 ) C. (− 45 , 3) D. (2 − √13 ,3 ) 8. 20 20 ᠣᠨ ᠤ ᠬᠢᠵᠢᠭ ᠰᠡᠷ ᠭᠡᠢᠯ ᠡᠬᠦ ᠬ ᠤᠭᠤᠴᠠᠭ ᠠᠨ ᠳ᠋ᠤ᠂ ᠣᠷ ᠳᠣᠰ ᠬᠣᠲ ᠠ ᠶᠢ ᠨ ᠨᠢ ᠭᠡᠨ ᠴᠠᠬᠢᠯ ᠭᠠᠨ ᠬᠣᠯ ᠪᠣᠭᠠ ᠨ ᠤ ᠺᠣᠮᠫ ᠨᠠᠢ ᠨᠢ ᠪᠥᠬᠥ ᠠᠵ ᠢᠯᠲᠠᠨ ᠤ ᠪᠠ ᠨ ᠬᠡ ᠷᠡᠭᠴᠡᠭᠡ ᠶᠢ ᠬᠠᠩᠭ ᠠᠬᠤ ᠶ ᠢᠨ ᠲᠥ ᠯᠥᠭᠡ᠂ 1 ᠲᠥᠮᠡᠨ ᠲᠥᠬᠥ ᠷᠢᠭ ᠢᠶᠡᠷ ᠨᠢ ᠭᠡ ᠪᠥᠯᠥ ᠭ ᠠ ᠮᠠᠪᠴ ᠢ ᠬᠤᠳᠠ ᠯᠳᠤᠨ ᠠ ᠪᠴᠠᠢ᠃ 2 02 1 ᠣᠨ ᠳ᠋ ᠤ ᠬᠢᠵ ᠢᠭ ᠦ ᠨ ᠪ ᠠᠢᠳᠠ ᠯ ᠨᠠᠮᠳᠠ ᠬᠤ ᠪᠠ ᠬᠢᠵᠢᠭ ᠢ ᠰ ᠡᠷᠭᠡᠢᠯ ᠡᠬᠦ ᠡᠯ ᠡ ᠵᠦ᠋ᠢᠯ ᠦᠨ ᠡᠳ᠋ ᠪᠣᠳᠠᠰ ᠬᠦ ᠢᠴᠡᠳ ᠬᠠᠩᠭ ᠠᠯᠲᠠ ᠲᠠ ᠢ ᠪᠣ ᠯᠬᠤ ᠶᠢ ᠳᠠᠭ ᠠᠯᠳᠤᠨ ᠠᠮᠠ ᠪᠴᠢ ᠶᠢᠨ ᠦᠨ ᠡ ᠨᠢ ᠪᠠᠭᠯᠠᠭ ᠠ ᠨᠢ ᠭᠡ ᠪ ᠦᠷᠢ ᠳ᠋ᠦ ᠪ ᠡᠨ1 0 ᠲᠥ ᠬᠥᠷᠢᠭ ᠪᠠᠭ ᠤᠷᠠᠵᠠᠢ᠃ ᠲᠡᠭᠡᠭᠡᠳ ᠲᠤᠰ ᠴᠠᠬ ᠢᠯᠭᠠᠨ ᠬᠣᠯᠪᠣᠭᠠ ᠨ ᠤ ᠺᠣᠮᠫ ᠠᠨᠢ ᠨᠢ 60 00 ᠲᠥᠬᠥ ᠷᠢᠭ ᠢᠶᠡᠷ ᠪᠠᠰᠠ ᠨᠢᠭᠡ ᠪᠥᠯᠥ ᠭ ᠠ ᠮᠠᠪᠴᠢ ᠬᠤᠳ ᠤᠯᠳᠤ ᠨ ᠠᠪᠬᠤ ᠳ᠋ᠤ 20 20 ᠣ ᠨ ᠤ ᠬᠤᠳᠠᠯ ᠳᠤᠨ ᠠᠪᠤᠭ ᠰᠠᠨ ᠠᠴᠠ 10 0 ᠪ ᠠᠭᠯᠠᠭ ᠠ ᠢ ᠯᠡᠭᠦᠦ ᠠᠪ ᠴᠠᠢ᠃ 2 02 0 ᠣᠨ ᠤ ᠠᠮᠠᠪ ᠴᠢ ᠪ ᠠᠭᠯᠠᠭ ᠠ ᠨᠢ ᠭᠡ ᠪ ᠦᠷᠢ ᠳ᠋ᠦ ᠪ ᠡᠨ 𝑥 ᠲᠥᠬᠥ ᠷᠢᠭ ᠭᠡᠪᠡᠯ᠂ ᠣᠯᠬ ᠤ ᠲ ᠡᠭᠰᠢᠳᠭᠡᠯ ᠨᠢ A. 1𝑥 + 100 = 6000𝑥; 10 B. 10000 𝑥 − 100 = 6000𝑥: 10 C. 10000 𝑥 = 6000𝑥: 0 − 100 D. 10000 𝑥 − 100 = 6000𝑥; 10 9. ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ᠂ 𝑅𝑡 ∆𝐴𝐵 𝐶 ᠳ᠋ᠦ᠂ ∠𝐴𝐶𝐵 = 90° , 𝐴𝐶 = 8 , 𝐵𝐶 = 6 , ᠲᠠᠯ ᠠ 𝐵𝐶 ᠶᠢ 𝐶𝑁 ᠤ ᠳᠠ ᠭᠠᠤ ᠨᠤᠭ ᠤᠯᠵᠦ᠂ ᠴᠡ ᠭ 𝐵 ᠶᠢ 𝐴𝐵 ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠴᠡᠭ 𝐵′ ᠲᠦ ᠳᠤᠰᠬᠠ ᠵᠦ᠂ ᠳᠠᠷᠠᠭ ᠠ ᠨᠢ ᠲᠠᠯ ᠠ 𝐴𝐶 ᠶᠢ 𝐶𝑀 ᠳᠠᠭ ᠠᠤ ᠨᠤᠭ ᠤᠯᠵᠦ᠂ ᠴᠡᠭ 𝐴 ᠶᠢ 𝐶𝐵 ′ ᠤᠨ ᠦᠷᠬ ᠦᠯᠵᠢᠯ ᠡᠯ ᠱᠤᠭᠤᠮ ᠳᠡ ᠭᠡᠷᠡᠬ ᠢ ᠴᠡᠭ 𝐴 ′ ᠲᠦ ᠳᠤᠰᠬ ᠠᠪᠠᠯ᠂ ᠬᠣᠶ ᠠᠷ ᠨᠤᠭ ᠤᠯᠤᠭᠠ ᠰᠤ ᠨᠢ ᠵᠢᠰᠢᠬᠦᠦ ᠲᠠ ᠯ ᠠ 𝐴𝐵 ᠲᠠᠢ ᠲᠤᠰ ᠲᠤᠰ ᠴᠡᠭ 𝑁 , 𝑀 ᠳ᠋ᠤ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠ ᠨ ᠠ᠃ 𝐴′𝑀 ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ A. 95 B. 85 C. 75 D. 65 10. ᠵᠢᠷ ᠤᠭ ① ᠮᠡᠲᠤ᠂ ᠲᠡ ᠭᠰᠢ ᠥᠨᠴ ᠥᠭᠳᠥ 𝐴𝐵𝐶𝐷 ᠳ᠋ᠦ᠂ 𝐻 ᠪᠣᠯ 𝐶𝐷 ᠲᠠᠯ ᠠ ᠳᠡ ᠭᠡᠷᠡᠬ ᠢ ᠨᠢ ᠭᠡ ᠴᠡᠭ ᠪᠣᠯᠣᠨ ᠠ᠃ ᠴᠡᠭ 𝑀 ᠨᠢ ᠴᠡᠭ 𝐴 ᠡᠴᠡ ᠭᠠᠷᠴᠤ ᠳᠠᠬᠢᠷ ᠱᠤᠭᠤ ᠮ 𝐴𝐻 − 𝐻𝐶 − 𝐶𝐵 ᠶᠢᠨ ᠳᠠᠭ ᠠᠤ ᠬᠥᠳᠡ ᠯᠭᠡᠭᠡᠨ ᠬᠢᠭᠡᠳ ᠴᠡ ᠭ 𝐵 ᠳ᠋ᠦ ᠬᠦ ᠷᠦᠨ ᠵᠣᠭᠰ ᠣᠵᠤ᠂ ᠴᠡ ᠭ 𝑁 ᠨᠢ ᠴᠡᠭ 𝐴 ᠡᠴᠡ ᠭᠠᠷᠴᠤ 𝐴𝐵 ᠶᠢᠨ ᠳᠠᠭ ᠠᠤ ᠬᠥᠳᠡ ᠯᠭᠡᠭᠡᠨ ᠬᠢᠵᠦ ᠴᠡᠭ 𝐵 ᠳ᠋ᠤ ᠬᠦ ᠷᠴᠦ ᠵᠣᠭᠰᠣ ᠨ ᠠ᠃ ᠲᠡ ᠳᠡᠭᠡᠷ ᠦ ᠨ ᠬᠥᠳᠡ ᠯᠭᠡᠭᠡᠨ ᠬᠢᠬᠦ ᠬᠤᠷᠳᠤ ᠴᠠ ᠨᠢ ᠴᠥᠮ 1𝑐𝑚 /𝑠 ᠂ ᠬᠡᠷᠪᠡ ᠴᠡᠭ 𝑀 , 𝑁 ᠨᠢ ᠵᠡᠷᠭᠡ ᠪᠡᠷ ᠡᠬ ᠢᠯᠡᠨ ᠬᠥᠳᠡ ᠯᠵᠦ᠂ ᠬᠥᠳᠡ ᠯᠭᠡᠭᠡᠨ ᠬᠢᠬᠦ ᠬᠤᠭᠤᠴᠠᠭ ᠠ 𝑡 (𝑠) , ∆ 𝐴𝑀 𝑁 ᠤ ᠲᠠᠯᠠᠪ ᠠᠢ ᠶᠢ 𝑆( 𝑐𝑚 2) ᠭᠡᠵᠦ ᠵᠢᠱᠢᠪᠡᠯ᠂ 𝑆 ᠪᠠ 𝑡 ᠶᠢᠨ ᠬᠣᠭᠣᠷ ᠣᠨᠳᠣᠬ ᠢ ᠹᠦᠩᠺᠼ ᠦ ᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠨᠢ ᠵᠢᠷᠤᠭ ② ᠮᠡᠲᠤ᠃ ᠳᠠᠷ ᠠᠭᠠᠬ ᠢ ᠭᠡᠯ ᠡᠯᠭᠡ ᠡᠴᠡ ᠵᠥᠪ ᠨᠢ ① 0 𝑥: 12 − 1 ᠢ ᠪ ᠣᠳᠣᠭᠠᠳ ᠂ ᠬᠠᠷᠢᠭ ᠤ ᠴᠤᠭᠯ ᠠᠷᠠᠯ ᠢ ᠨᠢ ᠲᠣᠭ ᠠᠨ ᠤ ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠳᠡ ᠭᠡᠷ ᠡ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡᠵᠦ ᠭᠠᠷᠭ ᠠ᠃ ︵2 ︶ ᠤᠷ ᠢᠳᠠᠪᠠ ᠷ ᠬᠢᠯᠪᠠ ᠷᠴᠢᠯ ᠠ᠃ 𝑥 2; 4𝑥 :4 2𝑥 ;𝑥 2 ÷ (2 𝑥− 4: 𝑥 2𝑥 ) ᠂ ᠳᠠᠷᠠᠭ ᠠ ᠨᠢ 𝑥 ᠲᠤ −2 , 0 , 1 , 2 ᠠᠴᠠ ᠳᠣᠬ ᠢᠷᠠᠮᠵᠢᠲ ᠠᠢ ᠲᠣᠭ ᠠ ᠶᠢ ᠨᠢ ᠰᠤᠩᠭᠤ ᠵᠦ ᠣᠷᠣᠭᠤᠯ ᠤᠭᠠᠳ ᠂ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠨᠢ ᠣ ᠯ᠃ 18. ︵ᠲᠤᠰ ᠪᠠ ᠭ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 9 ᠬᠤᠪᠢ ︶ ᠨᠢᠭᠡᠨ ᠳᠤᠮᠳᠠᠳ ᠤ ᠰ ᠤᠷᠭᠠᠭ ᠤᠯᠢ ᠨᠢ ᠶᠢᠰᠤ ᠳᠤᠭᠡᠷ ᠣ ᠨ ᠤ ᠰᠤᠷᠤᠭ ᠴᠢᠳ ᠠᠴᠠ ︽ ᠮᠢᠨᠤ ᠬᠠᠮ ᠤᠭ ᠤ ᠨ ᠪᠠᠬᠠᠷ ᠬᠠᠳᠠ ᠭ ᠣ ᠷᠳᠣᠰ ᠤᠨ ᠦ ᠵᠡᠮᠵ ᠢ ᠶᠢ ᠨ ᠣ ᠷᠣᠨ ︾ ᠤ ᠲᠤ ᠬᠠᠢ ᠶᠠᠩᠵ ᠤ ᠰ ᠤᠭᠤᠯ ᠤᠨ ᠪᠠᠢᠴᠠᠭ ᠠᠯᠲᠠ ᠬᠢᠵᠠᠢ ᠃ ︵ ᠬᠦ ᠮᠦᠨ ᠪᠦᠷᠢ ᠵᠥᠪᠭᠡ ᠨ ᠨᠢ ᠭᠡ ᠶ ᠢ ᠰ ᠤᠩᠭᠤᠨ ᠠ ︶᠄ A — ᠠᠮᠢᠲᠠ ᠨ ᠤ ᠬᠦᠷᠢᠶ ᠡᠯᠡᠩ ᠂ B— ᠴᠢ ᠰᠢᠩ ᠬᠤ ᠨᠠᠭ ᠤᠷ᠂ C— ᠣᠷ ᠳᠣᠰ ᠲᠠ ᠯ ᠠ ᠨᠤᠲᠤᠭ᠂ D— ᠺᠠᠩ ᠵᠧᠨ ᠪᠠᠯᠭᠠᠰᠤ᠂ E — ᠡᠷᠳᠡᠨ ᠢ ᠶᠢᠨ ᠳᠤᠪ ᠴᠢ ᠰᠤᠶ ᠤᠯ ᠵᠢᠭ ᠤᠯᠴᠢᠯ ᠠᠯ ᠤᠨ ᠭᠠᠵᠠ ᠷ᠃ ᠡᠮᠬ ᠢᠳᠭᠡᠨ ᠪᠦᠷᠢᠳᠭᠡᠭ ᠰᠡᠨ ᠲᠣᠭ ᠠ ᠪ ᠠᠷᠢᠮᠲᠠ ᠶᠢ ᠦᠨᠳᠤᠰᠤ ᠯᠡᠵᠦ᠂ ᠬᠣᠶ ᠠᠷ ᠪ ᠦᠷᠢᠨ ᠪᠤᠰ ᠤ ᠪᠦᠷᠢᠳ ᠭᠡᠯ ᠦ ᠨ ᠵᠢ ᠷᠤᠭ ᠢ ᠵᠢᠷ ᠤᠵᠠᠢ᠂ ᠡᠭᠦᠨ ᠳ᠋ᠦ B ᠲᠠᠢ ᠬᠠᠷᠠᠭ ᠠᠯᠵᠠᠭᠰᠠᠨ ᠲᠥ ᠪ ᠥᠨᠴ ᠥᠭ ᠨᠢ 90° ᠪᠣᠯ ᠣᠨ ᠠ᠃ ᠵᠢᠷ ᠤᠭ ᠲᠠᠬᠢ ᠮᠡᠳᠡᠭᠡ ᠵᠠᠩᠭᠢ ᠶᠢ ᠦᠨᠳᠤ ᠰᠤᠯᠡ ᠵᠦ᠂ ᠳ ᠠᠷᠠᠭ ᠠᠬᠢ ᠠᠰᠠᠭᠤᠯᠲ ᠠ ᠶᠢ ᠬᠠᠷᠢᠭ ᠤᠯ᠃ (1) ᠰ ᠤᠩᠭᠤ ᠵᠦ ᠠᠪᠤ ᠭᠰᠠᠨ ᠶᠢᠰᠤ ᠳᠤᠭᠡ ᠷ ᠣ ᠨ ᠤ ᠰᠤᠷᠤ ᠭᠴᠢ ᠨᠠᠢᠲᠡ ᠬᠡᠳᠤᠢ ᠬᠦᠮᠦ ᠨ ᠪ ᠠᠢᠬᠤ ᠪᠣᠢ ︖ ᠰᠠᠪᠠᠭ ᠠᠨ ᠪ ᠦᠷᠢᠳᠭᠡᠯ ᠦ ᠨ ᠵᠢᠷ ᠤᠭ ᠢ ᠪᠦᠷᠢᠨ ᠪᠣᠯᠭ ᠠ᠃ (2) ᠲᠣᠭ ᠣᠷᠢᠭ ᠪᠦᠷᠢᠳᠭᠡᠯ ᠦᠨ ᠵᠢᠷᠤ ᠭ ᠲᠤ 𝑚 = ( )᠂ 𝐷 ᠶᠢ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡᠭᠰᠡᠨ ᠳᠡ ᠪᠢᠬᠦᠷ ᠬᠡᠯᠪᠡ ᠷᠢᠳᠥ ᠶᠢᠨ ᠲᠥᠪ ᠥᠨᠴ ᠥᠭ ᠨᠢ ︵ ︶ ᠭᠷᠠᠳᠦ᠋ᠰ᠃ (3) ᠶᠢᠰᠤ ᠳᠤᠭᠡᠷ ᠣ ᠨ ᠨᠢ ᠦᠵ ᠡᠮᠵᠢ ᠶᠢᠨ ᠣᠷᠣᠨ 𝐴 ᠶᠢ ᠬᠠᠮᠤᠭ ᠤᠨ ᠪᠠᠬᠠᠷ ᠬᠠᠳᠠ ᠭ 5 ᠰᠤᠷᠤᠭᠴᠢ ᠠ ᠴᠠ ᠳᠠᠰᠢᠷᠠ ᠮ ᠢᠶᠠᠷ 2 ᠰᠤᠷᠤᠭᠴᠢ ᠶᠢ ᠰᠤᠩᠭ ᠤᠨ ᠠ ᠪᠣᠳᠠᠳᠠ ᠢ ᠪᠠᠢᠴᠠᠭ ᠠᠯᠲᠠ ᠬᠢᠬᠦ ᠪᠡᠷ ᠪᠣᠯᠪᠠ᠂ ᠡᠨ ᠡ 5 ᠰᠤᠷᠤᠭᠴᠢ ᠶᠢᠨ ᠲᠣᠲᠣᠷ ᠠ 2 ᠡᠷᠡᠭᠲᠡᠢ ᠰᠤᠷᠤᠭᠴᠢ ᠪᠠ 3 ᠡᠮᠡᠭᠳᠡᠢ ᠰᠤᠷ ᠤᠭᠴ ᠢ ᠪᠠᠢᠨ ᠠ᠂ ᠮᠤᠳᠤᠯ ᠵᠢᠨ ᠵᠢᠷᠤᠭ ᠪᠤᠶᠤ ᠬ ᠥᠰᠥ ᠨᠥᠭᠵ ᠢᠬᠦᠯᠬᠦ ᠠᠷᠭ ᠠ ᠪᠠᠷ ᠰᠤᠩᠭᠤᠵ ᠦ ᠭᠠᠷ ᠭᠠᠭᠰᠠᠨ 2 ᠰ ᠤᠷᠤᠭᠴᠢ ᠨᠢ ᠴᠥ ᠮ ᠡᠮ ᠡᠭᠲᠡᠢ ᠰᠤᠷᠤᠭᠴ ᠢ ᠪᠣ ᠯᠬᠤ ᠶᠢ ᠨ ᠳᠣᠬ ᠢᠶᠠᠯᠳᠤᠮ ᠵᠢ ᠶ ᠢ ᠣᠯ᠃ 19. ︵ ᠲᠤᠰ ᠪᠠ ᠭ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 8 ᠬᠤᠪᠢ ︶ ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ᠂ ᠲᠡᠭᠰᠢ ᠥᠨᠴ ᠥᠭᠳᠥ 𝐴𝐵𝐶𝐷 ᠶᠢᠨ ᠲᠠᠯ ᠠ 𝐴𝐵 , 𝐵𝐶 ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ 3 , 8 ᠪᠣᠯ ᠣᠨ ᠠ᠂ 𝐶 , 𝐷 ᠨᠢ 𝑦 ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠳᠡᠭᠡᠷ ᠡ ᠣ ᠷᠣᠰᠢ ᠨ ᠠ᠂ 𝐸 ᠪᠣᠯ 𝐴𝐷 ᠶᠢᠨ ᠳᠤᠮᠲᠠ ᠴᠡᠭ ᠂ ᠤᠷ ᠪᠠᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠲᠦ ᠹᠦᠩᠺᠼ 𝑦 = 𝑘𝑥 ( 𝑘≠ 0) ᠶᠢᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠨᠢ ᠴᠡ ᠭ 𝐸 ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠵ ᠦ᠂ 𝐵𝐶 ᠲᠠᠢ ᠴᠡᠭ 𝐹 ᠳ᠋ᠤ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠᠬ ᠤ ᠪᠥ ᠭᠡᠳ 𝐶𝐷 − 𝐵𝐸 = 1 ᠪ ᠣᠯᠣᠨ ᠠ᠃ (1) ᠲᠤᠰ ᠤᠷᠪᠠ ᠭᠤ ᠬᠠᠷ ᠢᠴᠠᠯ ᠲᠦ ᠹᠦᠩ ᠺᠼ ᠦ ᠨ ᠠ ᠨᠠᠯᠢᠰ ᠢᠯ ᠡᠷᠬᠡᠢᠯ ᠡᠯ ᠢ ᠣᠯ᠃ (2) 𝑦 ᠲᠡᠩᠭ ᠡᠯᠢᠭ ᠳᠡᠭᠡᠷ ᠡ ᠨᠢᠭᠡ ᠨ ᠴᠡᠭ 𝑃 ᠶᠢ ᠣᠯᠵ ᠦ᠂ 𝑆∆ 𝐶𝐸𝑃 = 23 𝑆 ᠲᠡᠭᠰ ᠢ ᠥ ᠨᠴᠥᠭ ᠲᠥ 𝐴𝐵𝐶 𝐷 ᠪᠣᠯᠭ ᠠᠨ ᠠ᠂ ᠡᠨᠡ ᠦ ᠶᠡᠰ ᠦᠨ ᠴᠡᠭ 𝑃 ᠦ ᠨ ᠪ ᠠᠢᠷᠢᠰᠢᠯ ᠢ ᠣᠯ ᠃ ᠬᠦᠮᠦ ᠨ ᠦ ᠲᠣᠭ ᠠ ᠦᠵ ᠡᠮᠵᠢ ᠶᠢᠨ ᠣᠷᠣ ᠨ 20. ( ᠲᠤᠰ ᠪᠠᠭ ᠠ ᠰ ᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 8 ᠬᠤᠪᠢ) ᠵᠢᠷᠤᠭ ① ᠪᠣᠯ ᠲᠠᠯ ᠪᠢᠭᠤᠷ ᠬᠠᠪᠳᠠᠰᠤ᠂ ᠲᠤᠯ ᠤᠭᠴᠢ ᠬᠠᠪᠳᠠᠰᠤ ᠪᠠ ᠰᠠᠭ ᠤᠷᠢ ᠭᠤᠷᠪ ᠠ ᠪᠠᠷ ᠪᠦᠷᠢᠯᠳ ᠦᠭᠰᠡᠨ ᠨᠢᠭᠡᠨ ᠭᠠᠷ ᠤᠲᠠᠰᠤ ᠲᠠᠯᠪᠢ ᠬᠤ ᠬᠠᠪᠳᠠᠭ ᠠᠢ ᠵᠢᠶᠠᠰ ᠠ ᠪ ᠣᠯᠣᠨ ᠠ᠂ ᠭᠠᠷ ᠤᠲᠠᠰᠤ ᠪᠠᠨ ᠲᠠᠯᠪᠢᠭ ᠤᠷ ᠬᠠᠪ ᠳᠠᠰᠤ ᠨ ᠳ᠋ ᠤ ᠲᠠᠯᠪᠢᠨ ᠠ᠃ ᠵᠢᠷᠤᠭ ② ᠪᠣᠯ ᠬᠠᠵᠠᠭᠤ ᠠᠴᠠ ᠨᠢ ᠬᠠᠷ ᠠᠭᠰᠠᠨ ᠪᠦᠲ ᠦᠴᠡ ᠶᠢᠨ ᠲᠠ ᠢᠯᠪᠤᠷᠢ ᠵᠢᠷᠤᠭ᠂ ᠲᠠ ᠯᠪᠢᠭ ᠤᠷ ᠬᠠᠪᠳᠠᠰᠤᠨ ᠤ ᠤᠷᠲᠤ 𝐴 𝐵 = 115 𝑚𝑚 ᠂ ᠲᠤᠯ ᠤᠭᠴ ᠢ ᠬᠠᠪᠲᠠᠰᠤ ᠶᠢᠨ ᠤᠷ ᠳᠤ 𝐶𝐷 = 70 𝑚𝑚 ᠂ ᠲᠠᠯᠪᠢᠭ ᠤᠷ ᠬᠠᠪᠲᠠᠰᠤ 𝐴𝐵 ᠶᠢ ᠲᠤᠯ ᠤᠭᠴ ᠢ ᠬᠠᠪ ᠲᠠᠰᠤ ᠶᠢ ᠨ ᠣᠷᠣᠢ ᠶᠢᠨ ᠴᠡᠭ 𝐶 ᠳ᠋ᠤ ᠲᠣᠭᠲᠠᠭ ᠠᠬᠤ ᠪᠥᠭᠡᠳ 𝐶𝐵 = 35 𝑚𝑚 ᠂ ᠲᠠᠯᠪᠢᠭ ᠤᠷ ᠬᠠᠪᠲᠠᠰᠤ 𝐴𝐵 ᠨᠢ ᠴᠡᠭ 𝐶 ᠶᠢ ᠲᠣᠭ ᠣᠷᠢᠨ ᠡᠷᠬᠢᠯ ᠳᠦᠵᠦ᠂ ᠲᠤ ᠯᠤᠭᠴᠢ ᠬᠠᠪᠲᠠᠰᠤ 𝐶𝐷 ᠨᠢ ᠴᠡᠭ 𝐷 ᠶᠢ ᠲᠣᠭ ᠣᠷᠢᠨ ᠡᠷ ᠬᠢᠯᠳᠦ ᠵᠦ ᠪᠣᠯᠣ ᠨ ᠠ᠂ ∠𝐶𝐷𝐸 = 60° (1) ᠬᠡᠷ ᠪᠡ ∠ 𝐷𝐶𝐵 = 70° ᠪᠣᠯ᠂ ᠴᠡ ᠭ 𝐴 ᠡᠴᠡ ᠰᠢᠯᠤᠭᠤᠨ 𝐷𝐸 ᠬᠦᠷᠬ ᠦ ᠵᠠ ᠢ ᠶᠢ ᠣ ᠯ ( ᠳ᠋ᠦ᠋ ᠩ ᠢ ᠨᠢ ᠨᠢᠭᠡᠴᠢ ᠶᠢ ᠨ ᠣ ᠷᠣᠨ ᠬᠦ ᠷᠲᠡᠯ ᠡ ᠨᠠ ᠷᠢᠪᠴᠢᠯ ᠠ ) (2) ᠦᠵ ᠡᠬᠦ ᠳ᠋ᠦ ᠲᠠᠭ ᠠᠲᠠᠢ ᠪᠣᠯᠭ ᠠᠬᠤ ᠶᠢᠨ ᠲᠥ ᠯᠥᠭᠡ ᠵᠢᠷ ᠤᠭ③ ᠳ᠋ᠤ᠂ (1 ) ᠶᠢᠨ ∠𝐷𝐶𝐵 = 70° ᠢ 90° ᠪᠣᠯ ᠭᠠᠨ ᠲᠣᠬ ᠢᠷᠠᠭ ᠤᠯᠵᠦ᠂ ᠳᠠ ᠬᠢᠭᠠ ᠳ 𝐶𝐷 ᠶᠢ ᠴᠡᠭ 𝐷 ᠶᠢ ᠲᠣᠭ ᠣᠷᠢᠭ ᠤᠯᠤᠨ ᠴᠠᠭ ᠦ ᠨ ᠵᠡ ᠭᠦᠦ ᠶᠢᠨ ᠡᠰᠡᠷ ᠬᠦ ᠴᠢᠭᠯᠡᠯ ᠲᠠᠭ ᠠᠤ ᠡᠷ ᠬᠢᠭᠦᠯ ᠵᠦ᠂ ᠴᠡᠭ 𝐵 ᠶᠢ ᠰᠢᠯᠤᠭᠤᠨ 𝐷𝐸 ᠳ᠋ᠤ ᠳᠤᠰᠬ ᠠᠨ ᠠ᠃ 𝐶𝐷 ᠶᠢᠨ ᠡᠷᠬᠢᠯ ᠲᠡ ᠶᠢᠨ ᠥᠨᠴᠥ ᠭ ᠨᠢ ᠬᠡᠳᠤᠢ ᠭᠷᠠᠳᠦ᠋ᠰ ᠪᠤᠢ︖ ( ᠯ ᠠᠪᠯᠠᠬ ᠤ ᠲᠣᠭ ᠠ ᠪ ᠠᠷᠢᠮᠲᠠ : 𝑠𝑖𝑛 50 0≈ 0.8 , 𝑐𝑜𝑠 50 0≈ 0.6 , 𝑡𝑎 𝑛50 0≈ 1.2 , 𝑠𝑖𝑛 26 .6 0≈ 0.4 , 𝑐𝑜𝑠 26 .6 0≈ 0.9 , 𝑡𝑎𝑛 26 .6 ≈ 0.5 , √3 ≈ 1.7 ) 21. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ∆𝐴𝐵𝐶 ᠳ᠋ᠦ 𝐴𝐵 = 𝐴𝐶 ᠪᠡᠷ ᠳ᠋ᠢᠶ ᠠᠮᠧᠲᠷ ᠪᠣ ᠯᠭᠠᠭᠰᠠᠨ ⊙ 𝑜 ᠨᠢ 𝐴𝐶 , 𝐵𝐶 ᠲᠠᠢ ᠲᠤᠰ ᠲᠤᠰ ᠴᠡᠭ 𝐷 , 𝐸 ᠳ᠋ᠤ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠᠨ ᠠ᠃ ᠰᠢᠯᠤᠭᠤ ᠨ 𝐸𝐹 ⊥ 𝐴𝐶 ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ 𝐹᠂ ᠰᠢᠯᠤᠭᠤᠨ 𝐸𝐹 ᠨᠢ 𝐴𝐵 ᠶᠢᠨ ᠦᠷᠬ ᠦᠯᠵᠢᠯ ᠡᠯ ᠱᠤᠭᠤ ᠮ ᠲᠠ ᠢ ᠴᠡ ᠭ 𝐻 ᠳ᠋ᠦ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠ ᠨ ᠠ᠃ (1) ᠪᠠᠲᠤ ᠯᠠᠭᠳᠠ ᠬᠤ ᠨᠢ ᠄ 𝐻𝐹 ᠪᠣᠯ ⊙ 𝑜 ᠶᠢᠨ ᠰᠢᠬ ᠠᠭᠴᠢ ᠰᠢᠯ ᠤᠭᠤᠨ (2) 𝐸𝐵 = 6 , 𝑐𝑜𝑠 ∠𝐴𝐵𝐸 = 13 ᠦ ᠶᠡᠰ᠂ 𝑡𝑎𝑛𝐻 ᠶᠢᠨ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠢ ᠣ ᠯ᠃ 22. ᠣᠷ ᠳᠣᠰ ᠬᠣᠲ ᠠ ᠶᠢ ᠨ ᠨᠢᠭᠡᠨ ᠵᠣᠴᠢ ᠳ ᠤᠨ ᠪᠠᠭᠤᠳᠠᠯ ᠨᠢ 50 ᠥᠷᠥᠭᠡ ᠭᠠᠷ ᠭᠠᠵᠦ ᠵᠤᠭᠠ ᠴᠠᠭᠰᠠᠳ ᠢ ᠰᠠᠭ ᠤᠯᠭᠠᠨ ᠠ᠃ ᠥ ᠷᠥᠭᠡ ᠨᠢᠭᠡ ᠪᠦᠷᠢ ᠶᠢ ᠨ ᠦᠨ ᠡ ᠨᠢ 20 0 ᠲᠥᠬᠥ ᠷᠢᠭ ᠡᠴᠡ ᠪᠠᠭ ᠠ ᠪᠠᠢᠵᠦ ᠪᠣᠯᠬ ᠤ ᠦᠭᠡᠢ ᠪᠥᠭᠡᠳ 32 0 ᠲᠥᠬᠥ ᠷᠢᠭ ᠡᠴᠡ ᠬᠡᠲᠤᠷᠡ ᠬᠦ ᠦᠭᠡᠢ᠃ ᠬᠡᠷ ᠪᠡ ᠵᠤᠭᠠ ᠴᠠᠭᠰᠠᠳ ᠰᠠᠭ ᠤᠪᠠᠯ ᠂ ᠵᠣᠴᠢᠳ ᠤᠨ ᠪᠠᠭ ᠤᠳᠠᠯ ᠠᠴᠠ ᠡᠳᠤᠷ ᠪᠦ ᠷᠢ ᠥᠷᠥᠭᠡ ᠨᠢ ᠭᠡ ᠪᠦᠷᠢ ᠳ᠋ᠦ 20 ᠲᠥᠭ ᠥᠷᠢᠭ ᠦᠨ ᠡᠯ ᠡ ᠵᠦ᠋ᠢᠯ ᠦᠨ ᠵᠠᠷᠤᠳᠠ ᠯ ᠭᠠᠷ ᠭᠠᠨ ᠠ᠃ ᠵᠤᠭᠠ ᠴᠠᠭᠰᠠᠳ ᠤ ᠨ ᠰ ᠠᠭᠤᠬ ᠤ ᠥᠷᠥ ᠭᠡᠨ ᠦ ᠲᠣᠭ ᠠ 𝑦 (ᠥᠷᠥᠭᠡ ) ᠨᠢ ᠥᠷᠥᠭ ᠡ ᠨᠢ ᠭᠡ ᠪᠦᠷᠢ ᠶᠢ ᠨ ᠦ ᠨ ᠡ 𝑥 ( ᠲᠥ ᠭᠥᠷᠢᠭ ) ᠦᠨ ᠨᠢᠭᠡᠳᠦ ᠭᠡᠷ ᠵᠡᠷᠭᠡ ᠶᠢᠨ ᠹᠦᠩᠺᠼ ᠪᠣᠯ ᠣᠨ ᠠ᠃ ᠲᠡ ᠭᠦᠨ ᠦ ᠵᠢᠷ ᠤᠮᠠᠭ ᠨᠢ ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ᠃ (1) 𝑦 ᠪᠠ 𝑥 ᠤ ᠨ ᠬᠣᠭᠣᠷ ᠣᠨᠳᠣᠬ ᠢ ᠹᠦᠩᠺᠼ ᠦᠨ ᠠᠨᠠᠯ ᠢᠰ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡᠯ ᠢ ᠣᠯᠵ ᠦ᠂ ᠬᠤᠪᠢᠰ ᠬᠤᠨ 𝑥 ᠤ ᠨ ᠬᠡ ᠮᠵᠢᠭ ᠳᠡᠯ ᠠᠪᠬ ᠤ ᠬᠡᠪ ᠴᠢᠶ ᠡ ᠶᠢ ᠪᠢᠴᠢ ᠵᠦ ᠭᠠᠷ ᠭ ᠠ᠃ (2) ᠥᠷᠥᠭᠡ ᠶᠢᠨ ᠦᠨ ᠡ ᠶᠢ ᠬᠡᠳᠤᠢ ᠲᠥᠬᠥ ᠷᠢᠭ ᠢᠶᠡᠷ ᠳᠣᠭᠲᠠᠭ ᠠᠪᠠᠯ᠂ ᠵᠣᠴ ᠢᠳ ᠤᠨ ᠪᠠᠭᠤᠳᠠᠯ ᠤᠨ ᠣᠯᠬ ᠤ ᠠ ᠰᠢᠭ ᠬᠣᠨᠵ ᠢᠪᠤᠷᠢ ᠨᠢ ᠬᠠᠮ ᠤᠭ ᠤ ᠨ ᠶᠡᠬᠡ ᠪᠣᠯ ᠬᠤ ᠪᠣᠢ︖ ᠬᠠᠮ ᠤᠭ ᠤ ᠨ ᠶᠡᠬᠡ ᠠᠰᠢᠭ ᠬᠣᠨᠵ ᠢᠪᠤᠷᠢ ᠨᠢ ᠬᠡᠳᠤᠢ ᠲᠥᠬᠥ ᠷᠢᠭ ᠪᠣᠢ︖ 23. ( ᠲᠤᠰ ᠪᠠ ᠭ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 11 ᠬᠤᠪᠢ ) ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ᠂ ᠬᠠᠶᠠᠯᠲ ᠠ ᠶᠢ ᠨ ᠱᠤᠭᠤ ᠮ 𝑦 = 𝑥 2 + 2𝑥 − 8 ᠨᠢ 𝑥 ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠲᠠᠢ 𝐴 , 𝐵 ( 𝐴 ᠨᠢ 𝐵 ᠶᠢᠨ ᠵᠡᠭᠦᠨ ᠲᠠ ᠯ ᠠ ᠳ᠋ ᠤ ᠣ ᠷᠣᠰᠢᠨ ᠠ ) ᠬᠣ ᠶᠠᠷ ᠴᠡᠭ ᠲᠦ ᠣ ᠭᠲᠣᠯ ᠣᠯᠴᠠᠵ ᠤ᠂ 𝑦 ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠲᠠᠢ ᠴᠡᠭ 𝐶 ᠳ᠋ᠦ ᠣᠭᠲᠣᠯᠣᠯ ᠴᠠᠨ ᠠ᠃ (1) ᠴᠡᠭ 𝐴 , 𝐵 , 𝐶 ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠢ ᠣᠯ᠃ (2) 𝐴𝐶 ᠶᠢ ᠬᠣᠯᠪᠣᠨ ᠠ᠃ ᠰ ᠢᠯᠤᠭᠤᠨ 𝑥 = 𝑚( −4 𝑚 0) ᠨᠢ ᠲᠤᠰ ᠬᠠᠶᠠᠯ ᠲᠠ ᠶᠢ ᠨ ᠱᠤᠭᠤᠮ ᠲᠠ ᠢ ᠴᠡ ᠭ 𝐸 ᠳ᠋ᠤ ᠣᠭᠲ ᠣᠯᠣᠯ ᠴᠠᠵᠤ᠂ 𝐴𝐶 ᠲᠠᠢ ᠴᠡᠭ 𝐷 ᠳ᠋ᠦ ᠣ ᠭᠲᠣᠯ ᠣᠯᠴᠠᠨ ᠠ᠃ 𝑂𝐷 ᠶᠢ ᠬᠣ ᠯᠪᠣᠨ ᠠ᠃ 𝑂𝐷 ⊥ 𝐴𝐶 ᠦᠶ ᠡᠰ᠂ ᠬᠡᠷ ᠴᠢᠮ 𝐷𝐸 ᠶᠢᠨ ᠤᠷᠲᠤ ᠶᠢ ᠣᠯ᠃ ᠥᠷ ᠥᠭᠡ ᠲᠥᠭᠥᠷ ᠢᠭ (3) ᠴᠡᠭ 𝑀 ᠨᠢ 𝑦 ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠳᠡᠭᠡᠷ ᠡ ᠣ ᠷᠣᠰᠢᠵ ᠦ᠂ ᠴᠡ ᠭ 𝑁 ᠨᠢ ᠰᠢᠯᠤᠭᠤ ᠨ 𝐴𝐶 ᠳᠡᠭᠡᠷ ᠡ ᠣᠷᠣᠰ ᠢᠨ ᠠ᠃ 𝑃 ᠨᠢ ᠬᠠᠶᠠ ᠯᠲᠠ ᠶᠢᠨ ᠱᠤᠭ ᠤᠮ ᠤ ᠨ ᠲᠡ ᠭᠰᠢ ᠬᠡᠮ ᠦᠨ ᠲᠡᠩᠭᠡᠯᠢᠭ ᠳᠡᠭ ᠡᠷᠡᠬᠢ ᠨᠢᠭᠡ ᠴᠡᠭ ᠂ ᠲᠡ ᠭᠡᠪᠡᠯ 𝐶 , 𝑁 , 𝑀 , 𝑃 ᠪᠠᠷ ᠣᠷᠣᠢ ᠶᠢᠨ ᠴᠡᠭ ᠢᠶᠡᠷ ᠪᠣ ᠯᠭᠠᠭᠰᠠᠨ ᠲᠥᠷᠪ ᠡᠯᠵᠢᠨ ᠢ ᠷᠤᠮᠪ ᠪᠣᠯᠭ ᠠᠬᠤ ᠴᠡᠭ 𝑀 ᠣᠷᠣᠰ ᠢᠨ ᠠ ᠤᠤ︖ ᠬᠡ ᠷᠪᠡ ᠣᠷᠣᠰ ᠢᠪᠠᠯ 𝑀 ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠢ ᠰᠢᠭ ᠤᠳ ᠪᠢᠴᠢᠭᠡᠷ ᠡᠢ᠃ ᠣᠷ ᠣᠰᠢᠬ ᠤ ᠦ ᠭᠡᠢ ᠪ ᠣᠯ ᠰᠢᠯᠲᠠᠭ ᠠᠨ ᠢ ᠨᠢ ᠲᠣᠳ ᠣᠷᠬᠠᠢᠯ ᠠ᠃ 24. ( ᠲᠤᠰ ᠪᠠ ᠭ ᠠ ᠰᠡᠲᠤᠪ 11 ᠬᠤᠪᠢ ) ᠡᠷᠬᠢᠯ ᠲᠡ ᠪᠣᠯ ᠨᠢᠭᠡᠨ ᠴᠢᠬᠤ ᠯᠠ ᠪᠣᠯᠬ ᠤ ᠵᠢᠷ ᠤᠭ ᠲᠥᠷᠰᠤ ᠶᠢ ᠨ ᠬᠤᠪᠢᠷ ᠠᠭᠠᠯᠲ ᠠ ᠪ ᠣᠯᠣᠨ ᠠ᠃ ᠵᠢᠷ ᠤᠭ ᠲᠥᠷᠰᠤ ᠳ᠋ᠤ ᠨᠢᠭᠡ ᠬᠣ ᠣᠰ ᠵᠡᠷᠭᠡᠯ ᠡᠳᠡᠭᠡ ᠲᠠᠯ ᠠ ᠨᠢ ᠲᠡᠩ ᠴᠡᠭᠦᠦ ᠦᠶ ᠡᠰ ᠦᠷ ᠬᠦᠯᠵ ᠢ ᠡᠷ ᠬᠢᠯᠲᠡ ᠶᠢ ᠠᠰᠢᠭ ᠯᠠᠨ ᠠᠰᠠᠭᠤᠳᠠ ᠯ ᠢ ᠰᠢᠢᠳ ᠪᠤᠷᠢᠯ ᠡᠵᠦ ᠪᠣᠯ ᠣᠨ ᠠ᠃ (1) ᠳᠤᠷ ᠰᠢᠨ ᠰᠢᠢᠳᠪ ᠤᠷᠢᠯ ᠡᠬᠦ᠄ ᠵᠢᠷᠤᠭ ① ᠮᠡᠲᠤ᠂ ᠠᠳᠠ ᠯᠢ ᠬᠠᠵᠠᠭᠤᠳᠤ 𝑅𝑡 ∆𝐴𝐵𝐶 ᠳ᠋ᠤ᠂ ∠𝐵𝐴𝐶 = 90° , 𝐴𝐵 = 𝐴𝐶 , 𝑀 ᠪᠣᠯ 𝐵𝐶 ᠲᠠᠯ ᠠ ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠨᠢᠭᠡ ᠴᠡᠭ ᠂ 𝐵𝑀 = 1𝑐𝑚 , 𝐶𝑀 = 2𝑐𝑚 , ∆ 𝐴𝐵𝑀 ᠶᠢ ᠴᠡᠭ 𝐴 ᠶᠢ ᠲᠣᠭ ᠣᠷᠢᠭ ᠤᠯᠤᠨ ᠡᠷ ᠬᠢᠭᠦᠯ ᠵᠦ ∆𝐴𝐶𝑁 ᠢ ᠣ ᠯᠣᠨ ᠠ᠃ 𝑀𝑁 ᠢ ᠬᠣᠯ ᠪᠣᠪᠠ ᠯ᠂ 𝐴𝑀 = ( ) (2) ᠠ ᠳᠠᠯᠢᠳᠬ ᠠᠨ ᠰᠥᠪᠡ ᠭᠴᠢᠯ ᠡᠬᠦ᠄ ᠵᠢᠷᠤᠭ ② ᠮᠡ ᠲᠤ᠂ ᠳᠥᠷᠪ ᠡᠯᠵᠢᠨ 𝐴𝐵𝐶𝐷 ᠳ᠋ᠦ᠂ 𝐴𝐵 = 𝐴𝐷 = 𝑎 , 𝐶𝐵 = 𝐶𝐷 , 𝐴𝐵 ⊥ 𝐵𝐶 , 𝐴𝐷 ⊥ 𝐶𝐷 , ᠡᠭᠴᠡ ᠶᠢ ᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ 𝐵 , 𝐷 ᠪᠣᠯ ᠣᠨ ᠠ᠂ 𝑃 , 𝑄 ᠪᠣᠯ ᠲᠤᠰ ᠲᠤᠰ 𝐴𝐵 , 𝐴𝐷 ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠴᠡᠭ ᠪᠥᠭᠡᠳ ∠ 𝑃𝐶𝐵 + ∠𝑄𝐶𝐷 = ∠𝑃 𝐶𝑄 , ᠲᠡᠭᠡᠪ ᠡᠯ ∆ 𝐴𝑃 𝑄 ᠶᠢᠨ ᠬᠦᠷᠢᠶᠡᠨ ᠦ ᠤᠷ ᠳᠤ ᠶᠢ ᠣᠯ᠃ ( ᠳ᠋ ᠦ᠋ᠩ ᠢ ᠨᠢ 𝑎 ᠪᠡᠷ ᠢᠯ ᠡᠷᠬᠡᠢᠯ ᠡ ) (3) ᠥᠷᠭᠡᠳ ᠭᠡᠨ ᠬᠡᠷᠡᠭᠯᠡᠬ ᠦ᠄ ᠵᠢᠷᠤᠭ ③ ᠮᠡ ᠲᠤ᠂ ᠳ ᠥᠷᠪᠡ ᠯᠵᠢᠨ 𝐴𝐵𝐶𝐷 ᠳ᠋ᠦ᠂ 𝐴𝐷 = 𝐶𝐷 , ∠𝐴𝐷𝐶 = 60° , ∠𝐴𝐵𝐶 = 75° , 𝐴𝐵 = 2√ 2 , 𝐵𝐶 = 2 ᠪ ᠣᠯ᠂ ᠳᠥ ᠷᠪᠡᠯ ᠵᠢᠨ 𝐴𝐵𝐶𝐷 ᠶᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠶᠢ ᠣᠯ᠃ ᠵᠢᠷ ᠤᠭ ① ᠵᠢᠷ ᠤᠭ ② ᠵᠢᠷ ᠤᠭ ③