ᠬᠤᠷᠢᠨ ᠵᠢᠷᠭᠤᠳᠤᠭᠠᠷ ᠪᠦᠯᠦᠭ ᠤᠷᠪᠠᠭᠤ ᠬᠠᠷᠢᠴᠠᠯᠳᠤ ᠹᠦᠩᠺᠼ （ᠨᠢᠭᠡᠴᠢ ᠶᠢᠨ ᠪᠠᠢᠴᠠᠭᠠᠯᠲᠠ 1） .pdf

ᠬᠣᠷ ᠢᠨ ᠵᠢᠷᠭᠤᠳᠤ ᠭᠠᠷ ᠪᠥᠯᠥ ᠭ ᠦᠨ ᠨᠢᠭᠡ ᠴᠢ ᠶᠢ ᠨ ᠪᠠᠢ ᠴᠠᠭᠠᠯᠲ ᠠ (90 ᠮᠢᠨᠦ᠋ ᠲ᠂ 12 0 ᠬ ᠤᠪᠢ᠃ ) ᠨᠢᠭᠡ᠂ ᠰᠤᠩᠭᠤ ᠬᠤ ᠰᠡ ᠲᠤᠪ᠃ (ᠰ ᠡᠳᠦᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪᠢ᠂ ᠨᠡᠢᠳᠡ 30 ᠬᠤᠪᠢ ) 1. ᠴᠡᠭ A (1, ) ,B （3, ）ᠨᠢ ᠹᠦᠩᠺᠼ y ᠶ ᠢᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠲᠡᠭᠡᠷ ᠠ ᠣᠷᠤᠰ ᠢᠨ ᠠ᠃ ᠲᠡᠭᠡᠪ ᠡᠯ , ᠶ ᠢᠨ ᠶᠡ ᠬᠡ ᠪ ᠠᠭ ᠠ ᠶᠢᠨ ᠬᠣᠯ ᠪᠤᠭᠳᠠᠯ ᠨᠢ ( ) A. B . C. D. ᠲᠤᠭᠳᠠ ᠭᠠᠬᠤ ᠠ ᠷᠭ ᠠ ᠦᠭᠡᠢ᠃ 2. ᠴᠡᠭ P( 3 ,1) ᠨᠢ ᠬᠣᠣ ᠰ ᠮᠤᠷ ᠤᠢ ᠱᠤᠭᠤᠮ y ᠲᠡᠭᠡᠷ ᠠ ᠣ ᠷᠤᠰᠢᠪᠠ ᠯ ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠨᠢ ( ) A. 3 B. 3 C. D. 3. ᠵᠢᠷᠤᠭ ᠮᠡᠲᠦ᠂ ᠬᠠᠪᠲᠠᠭ ᠠᠢ ᠶ ᠢᠨ ᠲᠡᠭᠰ ᠢ ᠦᠨᠴᠦᠭᠳᠦ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠳ᠋ᠤ ᠹᠦᠩᠺᠼ 𝑥 + b （k≠ 0 ） ᠪᠠ M (𝑀 ≠ 0) ᠶᠢᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠨᠢ ᠴᠡ ᠭ A (2,3 ) B ( 6 , 1) ᠳ᠋ᠤ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠᠨ ᠠ᠃ ᠲᠡᠭᠡᠪ ᠡᠯ ᠲᠡᠩ ᠴᠡᠳᠭᠡᠯ ᠪᠤᠰᠤ 𝑥+ b 𝑀 ᠶᠢᠨ ᠬᠠᠷ ᠢᠭᠤ ᠴᠤᠭᠯ ᠠᠷᠠᠯ ᠨᠢ ( ) A. 𝑥 6 B. 6 𝑥 0 ᠪ ᠤᠶᠤ 𝑥 2 C. 𝑥 2 D . 𝑥 6 ᠪ ᠤᠶᠤ 0 𝑥 2 4. ᠢᠵᠢᠯ ᠨᠢᠭᠡ ᠨ ᠲᠡᠭᠰᠢ ᠦᠨᠴ ᠦᠭᠳᠦ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤᠨ ᠰᠢᠰ ᠲ᠋ᠧᠮ ᠳ᠋ᠤ ᠹᠦᠩᠺᠼ 𝑥 + ᠪᠠ ᠶᠢᠨ ᠶᠡᠬᠡ ᠲᠥ ᠯᠦᠪ ᠵᠢᠷᠤᠭ ᠢ ᠵᠥ ᠪ ᠵᠢᠷᠤᠭ ᠰᠠᠨ ᠨᠢ ( ) A. B ． C ． D ． 5. ᠵᠢᠷᠤᠭ ᠮᠡᠲᠦ᠂ ᠴᠡ ᠭ ( ) ᠪᠣᠯ ᠤᠷᠪᠠ ᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠳᠤ ᠹᠦᠩᠺᠼ y= ᠶᠢᠨ ᠨᠢᠭᠡᠳᠦᠭᠡᠷ ᠮᠥᠴᠡ ᠳ᠋ᠡᠬᠢ ᠵᠢᠷᠤᠮᠠᠭ ᠲᠡᠭᠡᠷ ᠡᠬᠢ ᠨᠢᠭᠡᠨ ᠴᠡᠭ ᠪᠣᠯ ᠤᠨ ᠠ᠂ ᠴᠡᠭ p ᠢᠶᠡᠷ ᠨᠢᠭᠡ ᠣᠷᠤᠢ ᠶᠢᠨ ᠴᠡᠭ ᠪᠣᠯᠭ ᠠᠵᠤ ᠲᠡᠩ ᠴᠡᠭᠦᠦ ᠲᠠᠯᠠᠳᠤ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ P ᠪ ᠠᠢᠭ ᠤᠯᠤᠨ ᠠ᠂ ᠴᠡᠭ A ᠂ B ᠨᠢ 𝑥 ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠳᠡᠭᠡᠷ ᠡ ᠣ ᠷᠤᠰᠢ ᠨ ᠠ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ P ᠶᠢᠨ ᠲᠠᠯ ᠠᠪᠠᠢ ᠨᠢ ( ) A. B. 3√ 3 C . + √ 4 ． D . + √ 6. ᠤᠷᠪᠠ ᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠳᠤ ᠹᠦᠩᠺᠼ ᠤᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠳᠡ ᠭᠡᠷᠡᠬ ᠢ ᠭᠤᠷᠪ ᠠᠨ ᠴᠡᠭ ᠤᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ︵𝑥 ) , ︵𝑥 ) , ︵𝑥 ) ᠬᠡ ᠷᠪᠡ 𝑥 𝑥 0 𝑥 ᠪᠣᠯ ᠶ ᠢᠨ ᠶᠡᠬᠡ ᠪᠠᠭ ᠠ ᠶ ᠢᠨ ᠬᠣ ᠯᠪᠤᠭᠳᠠᠯ ᠨᠢ ( ) A. B. C. D. 7. ᠵᠢᠷᠤᠭ ᠮᠡ ᠲᠦ᠂ O ᠶᠢᠨ ᠲᠠᠯᠠ ᠪᠠᠢ ᠨᠢ 3᠂ O ᠬᠣᠣ ᠰ ᠮᠤᠷᠤᠢ ᠱᠤᠭᠤ ᠮ y k ᠴᠡᠭ A ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠨ ᠠ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ k ᠶᠢᠨ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠨᠢ ) ( A. B. 3 C. 6 D . 9 8. ᠬᠠᠪᠲᠠᠭ ᠠᠢ ᠶᠢᠨ ᠲᠡᠭᠰᠢ ᠦᠨᠴ ᠦᠭᠳᠦ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠦ ᠨ ᠰ ᠢᠰᠲ᠋ᠧᠮ ᠳᠡ ᠭᠡᠷ ᠡ ᠰᠢᠯᠤᠭᠤᠨ AB ᠨᠢ 𝑥 ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠲᠤ ᠡᠭᠴᠡᠯ ᠡᠨ ᠡ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ ᠴᠡᠭ C ( ᠴᠡᠭ C ᠡᠬ ᠢᠨ ᠤ ᠴᠡᠭ ᠤᠨ ᠪᠠᠷᠠᠭ ᠤᠨ ᠲᠠᠯ ᠠ ᠳ᠋ ᠤ ᠣ ᠷᠤᠰᠢᠨ ᠠ) ᠮᠥᠷᠲᠡ ᠭᠡᠨ ᠰᠢᠯᠤᠭᠤᠨ y 𝑥 ᠪ ᠠ ᠬᠣᠣ ᠰ ᠮᠤᠷ ᠤᠢ ᠱᠤᠭᠤ ᠮ y 1 ᠲᠠᠢ ᠲᠤ ᠰ ᠲᠤᠰ ᠴᠡᠭ A , B ᠳ᠋ᠤ ᠣᠭᠲᠤ ᠯᠤᠯᠴ ᠠᠨ ᠠ᠂ C + C 4᠂ ᠲᠡᠭᠡᠪ ᠡᠯ O ᠶᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ ( ) A. 2 √3 +3 ᠪᠤᠶ ᠤ 2 √3 3 B. √2 +1 ᠪᠤᠶ ᠤ √ 2 1 C. 2√ 3 3 D. √2 1 9. ᠵᠢᠷᠤᠭ ᠮᠡ ᠲᠦ᠂ ᠴᠡᠭ A (𝑎 3 ) ᠂ B( b 1) ᠴᠥ ᠮ ᠬᠣᠣ ᠰ ᠮᠤᠷ ᠤᠢ ᠱᠤᠭᠤ ᠮ ᠲᠡ ᠭᠡᠷ ᠠ ᠣᠷᠤᠰ ᠢᠨ ᠠ᠂ ᠴᠡᠭ C ᠂ D ᠲᠤᠰᠪ ᠤᠷᠢ 𝑥 ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠪᠠ ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠲᠡᠭᠡᠷ ᠡᠬᠢ ᠬᠥᠳᠡ ᠯᠬᠦ ᠴᠡᠭ ᠂ ᠲᠦ᠋ᠷᠪ ᠡᠯᠵᠢᠨ ABC D ᠶᠢᠨ ᠬᠦᠷᠢᠶ ᠡᠨ ᠤ ᠤᠷᠲᠦ ᠶᠢᠨ ᠬᠠᠮ ᠤᠭ ᠪᠠᠭ ᠠ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠨᠢ᠄ ( ) A. 5√ 2 B. 6√ 2 C . 2 √10 + 2√ 2 D. 8√ 2 10. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠦ᠂ A ᠂ B ᠬᠤ ᠶᠠᠷ ᠴᠡᠭ ᠤᠷ ᠪᠠᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠲᠤ ᠪᠦᠩᠺᠽ y 1 ᠶᠢᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠳᠡᠭᠡᠷ ᠡ ᠣ ᠷᠤᠰᠢᠵ ᠤ᠂ ᠴᠡᠭ C ᠂ D ᠤᠷᠪᠠ ᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠲᠤ ᠹᠦᠩᠺᠼ y 2 ᠦ ᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠳᠡᠭᠡᠷ ᠡ ᠪᠠᠢᠨ ᠠ᠃ AC ⊥ y ᠲᠡ ᠩᠬᠡᠯ ᠢᠭ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ ᠴᠡᠭ E ᠂ BD ⊥ y ᠲᠡ ᠩᠬᠡᠯ ᠢᠭ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ ᠴᠡ ᠭ F ᠂ C 2᠂ D 1᠂ EF 3 ᠂ ᠲᠡ ᠭᠡᠪᠡᠯ ᠶᠢᠨ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠨᠢ ( ) A. 6 B. 4 C . 3 D. 2 ᠬᠤᠶ ᠠᠷ᠄ ᠬᠠᠭ ᠠᠯᠲᠠ ᠳ᠋ᠤ ᠲᠠᠭᠯ ᠠ᠃ ( ᠨᠡ ᠢᠲᠡ2 4 ᠬ ᠤᠪᠢ) 11 . ᠤᠷ ᠪᠠᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠲᠤ ᠹᠦᠩᠺ ᠼ ᠦᠨ ᠵᠢᠷ ᠤᠮᠠᠭ ᠨᠢ ᠴᠡᠭ (1 , 2) ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠨ ᠠ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ k ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠨᠢ ( ) 12 . ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠦ᠂ ᠰᠢᠯᠤᠭᠤᠨ 𝑎𝑥 ᠪᠠ ᠬᠣᠣ ᠰ ᠮᠤᠷ ᠤᠢ ᠱᠤᠭᠤᠮ ( 𝑥 0) ᠴᠡ ᠭ A( 1 , 2 ) ᠳ᠋ᠤ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠨ ᠠ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ᠲᠡ ᠩᠴᠡᠳᠭᠡᠯ ᠪᠤᠰᠤ 𝑎𝑥 ᠶᠢᠨ ᠬᠠᠷᠢᠭ ᠤ ᠴᠤᠭᠯ ᠠᠷᠠᠯ ᠨᠢ ( ) 13. ᠤᠷᠪᠠ ᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠲᠤ ᠹᠦᠩᠺᠼ ᠳ᠋ᠤ᠂ 𝑥 1 ᠦᠶ ᠡᠰ ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠠᠪᠬᠤ ᠭᠡᠪ ᠴᠢᠶ ᠡ ᠨᠢ ( ) 14. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠦ᠂ ᠴᠡᠭ A , B ᠪᠣᠯ ᠲᠤᠰᠪ ᠤᠷᠢ ᠤᠷᠪᠠ ᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠲᠦ ᠹᠦᠩᠺᠼ ᠪᠠ ᠶ ᠢᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠲᠡᠭᠡᠷ ᠡᠬᠢ ᠴᠡᠭ ᠂ ᠬᠡ ᠷᠪᠡ ᠴᠡᠭ A ᠨᠢ ᠭᠡ ᠷᠴᠢᠮ O B ᠶᠢᠨ ᠳᠤᠮ ᠳᠠ ᠴᠡᠭ ᠪᠣᠯ k ᠶ ᠢᠨ ᠬᠡ ᠮᠵᠢᠭ ᠳᠡᠯ ᠨᠢ ( ) 1· 世 纪* 教 育 网 15 . ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠦ᠂ ᠴᠡᠭ C ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ (2 , 1) ᠂ C ∥𝑥 ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠂ C ∥ ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠂ ᠴᠡ ᠭ A , B ᠨᠢ ᠴᠥᠮ ᠰ ᠢᠯᠤᠭᠤᠨ 𝑥 + 6 ᠳᠡᠭᠡᠷ ᠡ ᠣᠷᠤᠰ ᠢᠨ ᠠ᠃ ᠬᠡ ᠷᠪᠡ ᠬᠣᠣ ᠰ ᠮᠤᠷᠤᠢ ᠱᠤᠭᠤ ᠮ y= ( 𝑥 0) ᠪᠠ C ᠶᠡᠷ ᠦᠩᠬᠡ ᠢ ᠴᠡ ᠭ ᠲᠠ ᠢ ᠪᠣᠯ᠂ k ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠠ ᠪᠬᠤ ᠬᠡᠪ ᠴᠢᠶ ᠡ ᠨᠢ ( ) 16. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠦ᠂ ᠲᠡᠭᠰᠢ ᠦᠨᠴᠦᠭᠳ ᠦ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤᠨ ᠰᠢᠰᠲ᠋ ᠧᠮ ᠳᠡᠭᠡᠷ ᠡ᠂ ᠷᠣᠮᠪ OA BC ᠶᠢᠨ ᠲᠠ ᠯ ᠠ OA ᠨᠢ 𝑥 ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠳᠡᠭᠡᠷ ᠡ ᠣ ᠷᠤᠰᠢᠨ ᠠ᠂ ᠲᠠᠰᠢᠭᠤᠪ ᠴᠢ OB ᠪᠠ AC ᠨᠢ ᠴᠡ ᠭ D (8 4) ᠳ᠋ᠤ ᠣᠭᠲᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠃ ᠤᠷᠪᠠ ᠭᠤ ᠬᠠᠷ ᠢᠴᠠᠯ ᠲᠤ ᠹᠦ ᠩᠺᠼ ᠶ ᠢᠨ ᠵᠢᠷ ᠤᠮᠠᠭ ᠨᠢ ᠴᠡᠭ D ᠶᠢ ᠳᠠ ᠭᠠᠷᠢᠨ ᠠ᠂ ᠬᠡᠷᠪᠡ ᠷᠣᠮᠪ OA BC ᠶᠢ ᠠᠰ ᠵᠡᠭᠦᠨᠰᠢ ᠨᠢᠭᠡᠴᠢ ᠵᠡᠷ ᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠵ ᠢᠭᠦᠯᠵ ᠤ ᠴᠡᠭ C ᠤᠷᠪᠠ ᠭᠤ ᠬᠠᠷ ᠢᠴᠠᠯ ᠳᠤ ᠹᠦᠩᠺᠼ ᠤᠨ ᠵᠢᠷ ᠤᠮᠠᠭ ᠳᠡᠭᠡᠷ ᠡ ᠣ ᠷᠤᠰᠢᠪᠠᠯ ᠦ ᠬᠡ ᠮᠵᠢᠭ ᠳᠡᠯ ᠨᠢ ( ) 17 . ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠦ᠂ ᠲᠡᠭᠰᠢ ᠦᠨ ᠴᠦᠭᠳᠦ A BO C ᠶ ᠢᠨ ᠣᠷᠤᠢ ᠶᠢᠨ ᠴᠡᠭ O ᠪᠣᠯ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤᠨ ᠡᠬᠢᠨ ᠤ ᠴᠡ ᠭ᠂ ᠴᠡᠭ B , C ᠲᠤᠰ ᠪᠤᠷᠢ 𝑥 , ᠲᠡ ᠩᠬᠡᠯ ᠢᠭ ᠤᠨ ᠡᠶᠡᠷ ᠭᠦ ᠬᠠᠭ ᠠᠰ ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠳᠡᠭᠡᠷ ᠡ ᠣ ᠷᠤᠰᠢ ᠨ ᠠ᠂ ᠣᠷᠤᠢ ᠶᠢᠨ ᠴᠡᠭ A ᠨᠢ ᠤᠷ ᠪᠠᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠲᠤ ᠹᠦᠩᠺᠼ ( ᠨᠢ ᠪ ᠠᠢᠩᠭ ᠤ ᠲᠣᠭ ᠠ᠂ 0 , 𝑥 0) ᠶᠢᠨ ᠵᠢᠷ ᠤᠮᠠᠭ ᠳᠡᠭᠡᠷ ᠡ ᠪ ᠠᠢᠨ ᠠ᠃ ᠬᠡᠷᠪᠡ ᠲᠡᠭᠰᠢ ᠥᠨᠴ ᠦᠭᠲᠦ AB OC ᠶᠢ ᠴᠡᠭ A ᠶᠢ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠯᠤᠨ ᠴᠠᠭ ᠤᠨ ᠵᠡᠭᠦᠦ ᠶ ᠢᠨ ᠡᠰᠡᠷ ᠭᠦ ᠴᠢᠭᠯᠡᠯ ᠢ ᠢᠠᠷ 90 ᠡᠷᠭᠢᠭ ᠦᠯᠪᠡ ᠯ ᠲᠡᠭᠰᠢ ᠥᠨᠴᠦ ᠭᠲᠦA ᠂ ᠶᠢ ᠣᠯᠤᠨ ᠠ᠃ O ᠶᠢᠨ ᠬᠠᠷ ᠠᠭᠠᠯᠵ ᠠᠭᠰᠠᠨ ᠴᠡ ᠭ ᠨᠢ ᠶᠠᠭ ᠰᠠᠢᠬᠠ ᠨ ᠤᠷ ᠪᠠᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠲᠤ ᠹᠦᠩᠺᠼ ᠤᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠳᠡᠭᠡᠷ ᠡ ᠪᠠᠢᠪ ᠠᠯ ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠨᠢ ( ) 1 8. ᠵᠢᠷᠤᠭ ᠮᠡᠲᠦ᠂ ᠬᠠᠪ ᠲᠠᠭᠠᠢ ᠶ ᠢᠨ ᠲᠡᠭᠰᠢ ᠦᠨᠴ ᠦᠭᠳᠦ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠳ᠋ᠤ᠂ ᠲ ᠡᠭᠰᠢ ᠥᠨᠴ ᠦᠭᠲᠦ OA BC ᠶᠢᠨ ᠣᠷᠤᠢ ᠶᠢᠨ ᠴᠡᠭ A , C ᠨᠢ ᠲᠤᠰᠪ ᠤᠷᠢ 𝑥 ᠲᠡ ᠩᠬᠡᠯ ᠢᠭ ᠤᠨ ᠰᠥᠭᠡᠷ ᠭᠦ ᠬᠠᠭ ᠠᠰ ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠪᠠ y ᠲᠡ ᠩᠬᠡᠯ ᠢᠭ ᠤᠨ ᠡᠶᠡᠷ ᠭᠦ ᠬᠠᠭ ᠠᠰ ᠲᠡ ᠩᠬᠡᠯ ᠢᠭ ᠳᠡᠭᠡᠷ ᠡ ᠣ ᠷᠤᠰᠢᠨ ᠠ᠃ ᠴᠡᠭ B ᠬᠣᠶ ᠠᠳᠤᠭ ᠠᠷ ᠮᠥᠴᠡ ᠶᠢᠨ ᠴᠡᠭ ᠂ ᠲᠡ ᠭᠰᠢ ᠦᠨ ᠴᠦᠭᠳᠦ OA BC ᠶᠢ ᠴᠡ ᠭ O ᠶᠢ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠯᠤᠨ ᠴᠠᠭ ᠤᠨ ᠵᠡᠭᠦᠦ ᠶᠢᠨ ᠴᠢᠭᠯᠡᠯ ᠳᠠᠭ ᠠᠤ ᠡᠷ ᠭᠢᠭᠦᠯ ᠵᠤ ᠴᠡ ᠭ B ᠶᠢ y ᠲᠡ ᠩᠬᠡᠯ ᠢᠭ ᠳᠡ ᠭᠡᠷ ᠡ ᠤᠨᠠ ᠭᠠᠭᠠᠳ ᠲᠡᠭ ᠰᠢ ᠦᠨ ᠴᠦᠭᠳᠦ O DE F ᠶᠢ ᠣᠯᠤᠨ ᠠ᠃ BC ᠪᠠ OD ᠴᠡᠭ M ᠳ᠋ᠤ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠨ ᠠ᠃ ᠬᠡᠷᠪᠡ ᠴᠡ ᠭ M ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠭᠰᠠᠨ ᠤᠷᠪᠠ ᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠲᠤ ᠹᠦᠩᠺᠼ (𝑥 0 ) ᠤᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠨᠢ A B ᠲᠠᠢ ᠴᠡᠭ N ᠳ᠋ᠤ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠬᠤ ᠪᠥᠭᠡᠳ ᠲᠡ ᠭᠰᠢ ᠦᠨᠴᠦ ᠭᠳᠦ OA BC ᠶᠢᠨ ᠲᠠ ᠯᠠᠪᠠᠢ ᠨᠢ 32 ᠂ tan ∠ DO E ᠪᠣᠯ BN ᠤ ᠤᠷᠲᠤ ᠨᠢ ( ) ᠭᠤᠷᠪ ᠠ᠄ ᠪᠣᠳᠤ ᠯᠭ ᠠ ᠶᠢᠨ ᠰᠡᠲᠦ ᠪ᠃ ( ᠨᠡ ᠢᠲᠡ 66 ᠬᠤᠪᠢ ) 19 . ( 10 ᠬᠤᠪᠢ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠦ᠂ ᠰᠢᠯᠤᠭᠤᠨ 𝑥 + b ᠶ ᠢᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠨᠢ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤᠨ ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠲᠠᠢ ᠲᠤᠰ ᠪᠤᠷᠢ A , B ᠳ᠋ᠤ ᠣᠭᠲᠤᠯᠤᠯ ᠴᠠᠬᠤ ᠪᠥᠭᠡᠳ ᠤᠷ ᠪᠠᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠲᠤ ᠹᠦᠩᠺᠼ ᠤ ᠨᠢᠭᠡᠳᠦ ᠭᠡᠷ ᠮᠥᠴᠡ ᠳ᠋ ᠡᠬᠢ ᠵᠢᠷᠤᠮᠠᠭ ᠲᠠᠢ ᠴᠡᠭ C ᠳ᠋ᠤ ᠣ ᠭᠲᠤᠯ ᠤᠯᠴᠠ ᠨ ᠠ᠃ CD ᠨᠢ 𝑥 ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠲᠠᠢ ᠡᠭᠴᠡ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ D ᠂ O 3 , OD 6 , O ᠶᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ 3᠃ (1) ᠨᠢ ᠭᠡᠳᠦᠭᠡᠷ ᠵᠡ ᠷᠭᠡ ᠤᠨ ᠹᠦᠩᠺ ᠼ ᠪᠠ ᠤᠷᠪᠠ ᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠲᠤ ᠹᠦᠩᠺᠼ ᠤ ᠨ ᠠ ᠨᠠᠯᠢᠰ ᠢᠯᠡᠷ ᠬᠡᠢᠯ ᠡᠯ ᠢ ᠤᠯ᠃ (2) 𝑥 0ᠦᠶ ᠡᠰ᠂ ᠲᠡ ᠩᠴᠡᠳᠭᠡᠯ ᠪᠤᠰᠤ 𝑥 + b 0 ᠶ ᠢᠨ ᠬᠠᠷ ᠢᠭᠤ ᠴᠤᠭ ᠯᠠᠷᠠᠯ ᠢ ᠰ ᠢᠭᠤᠳ ᠪᠢᠴᠢᠵ ᠤ ᠭᠠᠷ ᠭ ᠠ᠃ 20 . ( 10 ᠬᠤᠪᠢ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠦ᠂ ᠤᠷ ᠪᠠᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠲᠤ ᠹᠦᠩᠺᠼ ᠪ ᠠ ᠨᠢ ᠭᠡᠳᠦ ᠭᠡᠷ ᠵᠡᠷᠭᠡ ᠶᠢᠨ ᠹᠦᠩᠺᠼ 𝑎𝑥 + b ᠶᠢᠨ ᠵᠢᠷ ᠤᠮᠠᠭ ᠨᠢ ᠴᠡᠭ ( 1 4 ) ᠪ ᠠ ( 2) ᠳ᠋ᠤ ᠣ ᠭᠲᠤᠯ ᠤᠯᠴᠠᠨ ᠠ᠃ (1) ᠡᠨᠡ ᠬᠤ ᠶᠠᠷ ᠹᠦᠩᠺᠼ ᠤᠨ ᠢᠯ ᠡᠷᠬᠡᠢᠯ ᠡᠯ ᠢ ᠣᠯ᠃ (2) ᠵᠢᠷ ᠤᠭ ᠢ ᠦ ᠵᠡᠵᠤ ᠨᠢᠭᠡᠳᠦ ᠭᠡᠷ ᠵᠡᠷᠭᠡ ᠶᠢᠨ ᠹᠦᠩᠺᠼ ᠤ ᠨ ᠬᠡ ᠮᠵᠢᠭ ᠳᠡᠯ ᠨᠢ ᠤᠷ ᠪᠠᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠲᠤ ᠹᠦᠩᠺᠼ ᠤ ᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠠ᠋ ᠴᠠ ᠶᠡᠬᠡ ᠪᠣᠯᠬ ᠤ 𝑥 ᠤᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠠᠪᠬᠤ ᠭᠡ ᠪᠴᠢᠶ ᠡ ᠶᠢ ᠰᠢᠭ ᠤᠳ ᠪᠢᠴᠢ ᠵᠤ ᠭᠠᠷᠭ ᠠ᠃ 21 . ( 10 ᠬᠤᠪᠢ ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠦ ᠂ ᠤᠷ ᠪᠠᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠲᠤ ᠹᠦᠩᠺᠼ y = ( ≠ 0) ᠤ ᠨ ᠵᠢᠷ ᠤᠮᠠᠭ ᠨᠢ ᠴᠡ ᠭ B(3 , 2) ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠨ ᠠ᠃ ᠴᠡ ᠭ B ᠪᠠ C ᠨᠢ ᠡᠬᠢᠨ ᠤ ᠴᠡ ᠭ ᠤ ᠨ ᠬᠤᠪᠢ ᠳ᠋ ᠤ ᠲᠡ ᠭᠰᠢ ᠬᠡᠮᠳᠡ ᠢ᠂ BA ᠨᠢ 𝑥 ᠲᠡ ᠩᠬᠡᠯ ᠢᠭ ᠲᠤ ᠡᠭᠴᠡ ᠪᠣᠯᠵᠤ ᠡᠭᠴᠡ ᠶᠢ ᠨ ᠰ ᠠᠭᠤᠷᠢ ᠨᠢ A ᠂ CD ᠨᠢ 𝑥 ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠲᠤ ᠡᠭᠴᠡᠯ ᠡᠨ ᠡ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ D ᠃ (1) ᠤᠷᠪᠠ ᠭᠤ ᠬᠠᠷ ᠢᠴᠠᠯ ᠲᠤ ᠹᠦ ᠩᠺᠼ ᠤᠨ ᠢᠯ ᠡᠷᠬᠡᠢᠯ ᠡᠯ ᠢ ᠣᠯ᠃ (2) CD ᠶᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠶᠢ ᠣᠯ᠃ 22. (1 2 ᠬᠤᠪᠢ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠦ ᠂ ᠲᠡ ᠭᠰᠢ ᠥᠨᠴ ᠦᠭᠲᠦ OA BC ᠳ᠋ᠤ OA 3 , AB 4᠂ ᠬᠣᠣ ᠰ ᠮᠤᠷ ᠤᠢ ᠱᠤᠭᠤ ᠮ (k 0) ᠶᠢᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠨᠢ ᠲᠡᠭᠰᠢ ᠥᠨᠴᠦᠭᠲᠦ ᠶᠢᠨ ᠲᠠᠯ ᠠ AB , BC ᠲᠠᠢ D , E ᠳ᠋ᠤ ᠣᠭᠲ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ ᠮᠦ᠋ᠷᠳ ᠡᠭᠡᠨ D 2AD ᠃ (1) ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠪᠠ ᠴᠡ ᠭ E ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠢ ᠣᠯ᠃ (2) ᠴᠡᠭ P ᠪᠣᠯ ᠭᠡᠷᠴᠢᠮ OC ᠳᠡ ᠭᠡᠷᠡᠬ ᠢ ᠬᠥᠳᠡ ᠯᠬᠦ ᠴᠡ ᠭ᠂∠ APE =9 0 ᠪᠣᠯᠭ ᠠᠬᠤ ᠴᠡᠭ P ᠣᠷᠤᠰ ᠢᠨ ᠠ ᠤ ᠤ ? ᠣᠷᠤᠰ ᠢᠪᠠᠯ P ᠴᠡ ᠭ ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠢ ᠣ ᠯᠵᠤ᠂ ᠦᠭᠡᠢ ᠪᠣᠯ ᠰ ᠢᠯᠲᠠᠭ ᠠᠨ ᠢ ᠲᠣᠳᠤᠷ ᠬᠠᠢᠯᠠ᠃ 1 世 纪 教 育 网 版 权 所 有 23. (1 2 ᠬᠤᠪᠢ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠦ᠂ ᠰᠢᠯ ᠤᠭᠤᠨ 2 + 6 ᠨᠢ ᠤᠷᠪᠠ ᠭᠤ ᠬᠠᠷᠢᠴᠠ ᠯᠲᠤ ᠹᠦᠩᠺᠼ ( 0) ᠶᠢᠨ ᠵᠢᠷ ᠤᠮᠠᠭ ᠲᠠ ᠢ ᠴ ᠡᠭ A (1, m) ᠳ᠋ᠤ ᠣ ᠭᠲᠤᠯᠤᠯ ᠴᠠᠵᠤ᠂ 𝑥 ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠲᠠᠢ B ᠳ᠋ᠤ ᠣ ᠭᠲᠤᠯ ᠤᠯᠴᠠ ᠨ ᠠ᠃ 𝑥 ᠲᠡ ᠩᠬᠡᠯ ᠢᠭ ᠲᠠᠢ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠪᠣᠯ ᠬᠤ ᠰᠢᠯᠤᠭᠤᠨ (0 6) ᠨᠢ ᠤᠷᠪᠠ ᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠲᠤ ᠹᠦᠩᠺᠼ ᠤᠨ ᠵᠢᠷ ᠤᠮᠠᠭ ᠲᠠᠢ M ᠳ᠋ᠤ ᠣ ᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠨ ᠠ᠂ AB ᠲᠠᠢ ᠴᠡᠭ N ᠳ᠋ᠤ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠨ ᠠ᠂ BM ᠶᠢ ᠬᠣ ᠯᠪᠤᠨ ᠠ᠃ (1) m ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠪᠠ ᠤᠷ ᠪᠠᠭᠤ ᠬᠠᠷ ᠢᠴᠠᠯ ᠲᠤ ᠹᠦᠩᠺᠼ ᠤ ᠨ ᠢ ᠯᠡᠷᠬᠡᠢᠯ ᠡᠯ ᠢ ᠣᠯ᠃ (2) ᠰᠢᠯ ᠤᠭᠤᠨ ᠢ y ᠲᠡ ᠩᠬᠡᠯ ᠢᠭ ᠤᠨ ᠴᠢᠭᠯᠡᠯ ᠲᠠᠭ ᠠᠤ ᠵᠡ ᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠵ ᠢᠭᠦᠯᠦ ᠨ ᠡ᠂ n ᠨᠢ ᠶᠠᠮᠠ ᠷ ᠬᠡᠮ ᠵᠢᠭᠳᠡᠯ ᠠᠪ ᠬᠤ ᠦ ᠶᠡᠰ MN ᠤ ᠲᠠᠯᠠᠪ ᠠᠢ ᠬᠠᠮᠤᠭ ᠶᠡᠬᠡ ᠪᠣᠯᠬᠤ ᠪᠤᠢ︖ 24. ( 12 ᠬᠤᠪᠢ ) ᠨᠢᠭᠡᠨ ᠺᠣ ᠮᠫᠠᠨᠢ ᠥᠪᠡᠷ ᠤᠨ ᠦᠢ ᠯᠡᠳᠪᠦᠷᠢᠯᠡᠯ ᠤᠨ ᠮᠡᠷᠭᠡᠵ ᠢᠯ ᠢ ᠶᠡᠨ ᠰᠠᠢᠵᠢᠷᠠ ᠭᠤᠯᠤᠭᠰ ᠠᠨ ᠡᠴᠡ ᠲᠡ ᠳᠡᠨ ᠤ ᠪᠦᠳᠦ ᠭᠡᠭᠳᠡᠭᠦ ᠨ ᠤ ᠥᠪᠡᠷ ᠤᠨ ᠥᠷᠲᠡᠭ ᠨᠢ ᠳᠠ ᠰᠤᠷᠠᠯᠲ ᠠ ᠦᠭᠡᠢ ᠳᠤᠷ ᠤᠭᠰ ᠢᠯᠠᠵᠠᠢ᠂ ᠪᠤᠳᠠᠳᠠ ᠢ ᠲ ᠣᠭ ᠠ ᠪᠠᠷᠢᠮ ᠳᠠ ᠨᠢ ᠭᠦᠰ ᠦᠨᠦᠭ ᠮᠡᠲᠦ ᠃ (1) ᠭᠦ ᠰᠦᠨᠦ ᠭᠲᠦ ᠳ᠋ ᠡᠬᠢ ᠲᠣᠭ ᠠ ᠪ ᠠᠷᠢᠮᠲᠠ ᠶᠢᠨ ᠬᠤᠪᠢᠷ ᠠᠯᠲᠠ ᠶᠢ ᠨᠢ ᠭᠡᠳᠦᠭᠡᠷ ᠵᠡᠷᠭᠡ ᠶᠢᠨ ᠹᠦᠩᠺᠼ ᠪᠤᠶᠤ ᠤᠷᠪᠠ ᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠲᠤ ᠹᠦᠩᠺᠼ ᠂ ᠠᠯᠢ ᠪᠡ ᠷ ᠨᠢ ᠢ ᠯᠡᠷᠬᠡᠢᠯ ᠡᠵᠤ ᠪᠣᠯᠬ ᠤ ᠪ ᠤᠢ︖ ᠹᠦᠩᠺᠼ ᠤᠨ ᠢᠯᠡᠷ ᠬᠡᠢᠯ ᠡᠯ ᠢ ᠪᠢᠴᠢᠵ ᠤ ᠭᠠᠷ ᠭ ᠠ᠃ 源 ：21 · 世 纪· 教 育· 网 】 (2) ᠲᠡᠭᠡᠷ ᠡᠬᠢ ᠬᠤᠪᠢᠷ ᠠᠯᠲᠠ ᠶᠢ ᠨ ᠵᠢᠮ ᠠ ᠶᠣᠰᠤ ᠪᠡ ᠷ 20 17 ᠣ ᠨ ᠳ᠋ ᠤ ᠮᠡᠷᠭᠡᠵ ᠢᠯ ᠢ ᠶᠡᠨ ᠰᠠᠢᠵᠢᠷᠠ ᠭᠤᠯᠬ ᠤ ᠳᠤ ᠵᠠ ᠷᠤᠭᠰ ᠠᠨ ᠪᠡᠯᠡ ᠮᠥᠩᠭ ᠦ 5 ᠲ ᠦ᠋ᠮᠡᠨ ᠲᠥ ᠭᠦᠷᠢ ᠭ ᠪ ᠣᠯ᠂ ① ᠪᠦᠳ ᠦᠭᠡᠭᠳᠡᠭᠦᠨ ᠤ ᠥᠪᠡᠷ ᠤᠨ ᠥᠷᠳᠡ ᠭ ᠨᠢ 20 16 ᠣᠨ ᠤ ᠬᠢ ᠡᠴᠡ ᠬᠡᠳᠦ ᠢ ᠲᠦ᠋ᠮᠡᠨ ᠲᠥᠭ ᠦᠷᠢᠭ ᠪᠠᠭ ᠤᠷᠠᠬ ᠤ ᠶᠢ ᠮᠠ ᠭᠠᠳᠯ ᠠ᠃ ② 20 17 ᠣᠨ ᠳ᠋ ᠡᠬᠢ ᠪᠦᠳ ᠦᠭᠡᠭᠳᠡᠭ ᠦᠨ ᠨᠢᠭᠡᠪᠦᠷ ᠢ ᠶᠢᠨ ᠥᠪᠡᠷ ᠤᠨ ᠥᠷᠳᠡ ᠭ ᠢ 3.2 ᠲᠦ᠋ᠮᠡᠨ ᠲᠥᠭ ᠦᠷᠢᠭ ᠡᠴᠡ ᠲᠤᠤᠷ ᠠ ᠪ ᠠᠢᠯᠭᠠᠬᠤ ᠭᠡᠪᠡᠯ ᠮᠡᠷᠭᠡᠵ ᠢᠯ ᠤ ᠨ ᠰ ᠠᠢᠵ ᠢᠷᠠᠭ ᠤᠯᠤᠯᠲ ᠠ ᠳ᠋ᠤ ᠪᠠᠰᠠ ᠬᠡᠳᠦ ᠢ ᠲᠦ᠋ᠮᠡᠨ ᠲᠥᠭ ᠦᠷᠢᠭ ᠨᠥᠬᠦᠪᠦᠷ ᠢᠯᠡᠬᠦ ᠴᠢᠬᠤ ᠯᠠᠲᠠᠢ ᠪᠤᠢ ︖ (0.0 1 ᠲ ᠦ᠋ᠮᠡᠨ ᠲᠥ ᠭᠦᠷᠢ ᠭ ᠬᠦ ᠷᠳᠡᠯ ᠠ ᠨᠠ ᠷᠢᠪᠴᠢᠯ ᠠ᠃) ᠣᠨ ᠪᠦᠳ ᠦᠭᠡᠭᠳ ᠡᠭᠦᠨ ᠤ ᠥᠪᠡᠷ ᠤᠨ ᠥᠷᠲ ᠡᠭ (ᠲᠦ᠋ ᠮᠡᠨ ᠲᠦ᠋ ᠭᠦᠷᠡᠭ/ ᠲᠣ ᠨᠤᠭ ᠨᠢᠭᠡ ᠪᠦᠷᠢ ) ᠮᠡᠷᠭᠡ ᠵᠢᠯ ᠢ ᠶᠡᠨ ᠰᠠᠢ ᠵᠢᠷᠠ ᠭᠤᠯᠬᠤ ᠳ᠋ᠤ ᠵᠠᠷᠤ ᠭᠰᠠᠨ ᠪᠡᠯᠡ ᠮᠥᠩᠭᠦ 𝑥 ( ᠲ ᠦ᠋ᠮᠡᠨ ᠲᠦ᠋ ᠭᠦᠷᠡᠭ ) 2013 7.2 2.5 2014 6 3 2015 4.5 4 2016 4 4.5