2019 ᠣᠨ ᠤ ᠪᠣᠭᠤᠲᠤ ᠬᠤᠲᠠ ᠶᠢᠨ ᠳᠤᠮᠳᠠᠳᠤ ᠰᠢᠯᠭᠠᠯᠲᠠ ᠶᠢᠨ ᠰᠡᠲᠤᠪ

20 19 ᠣᠨ ᠤ ᠪᠣᠭᠣᠳᠤ᠂ ᠰᠢᠯ ᠢᠩᠬᠣ ᠲᠠ᠂ ᠠᠯ ᠠᠱᠨ ᠂ ᠦᠬᠠᠢ᠂ ᠪ ᠠᠶᠠᠨᠨ ᠠᠭᠤᠷ᠂ ᠤᠯᠠᠭᠠᠨᠴᠠᠪ ᠬᠣ ᠲᠠ ᠶᠢᠨ ᠠᠩ ᠬ ᠠ ᠲᠣᠮᠳ ᠠᠳᠤ ᠢᠢᠨ ᠰᠣ ᠷᠭᠠᠭ ᠣᠯ ᠢ ᠲᠠᠪᠰᠢᠬ ᠤ ᠰᠢᠯ ᠭᠠᠯᠳᠠ ᠲᠣᠭᠠᠨ ᠤ ᠣᠬ ᠠᠭᠠᠨ ᠨᠢᠬᠡ ᠂ ᠰᠣᠩ ᠭᠣᠬ ᠤ ᠰᠡᠳᠣ ᠪ ᠃ (ᠰᠡᠳᠣ ᠪ ᠪᠦᠷ ᠢ 3 ᠬᠣᠪᠢ ᠂ ᠨᠡᠶᠢᠳᠡ 36 ᠬᠣ ᠪᠢ) 1. | －√9 |+( 13 ) ; 1 ᠵ ᠢ ᠪᠣᠳᠣᠭ ᠰᠠᠨ ᠳ᠋ᠦ ᠩ ᠨ ᠢ ( ) A. 0 B. 83 C. 103 D. 6 2. ᠪᠣᠳᠠᠳᠤ ᠲᠣᠭ ᠠ 𝑎 ,b ᠦᠨ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠲᠡᠭᠡᠷᠡᠬᠢ ᠬᠠᠷᠠ ᠭᠠᠯᠵᠠᠭ ᠰᠠᠨ ᠴᠡᠭ ᠦ ᠨ ᠪᠠᠶᠢᠷ ᠢ ᠶᠢ ᠵᠢᠷᠣ ᠭ ᠳᠤ ᠢᠯᠡᠷᠬᠡᠶᠢᠯ ᠡᠭᠰᠡᠨ ᠮᠡᠳᠤ ᠂ ᠲᠠᠷᠠᠭᠠᠬᠢ ᠳ᠋ᠦ ᠩᠨᠡᠯᠲ ᠡ ᠡᠴᠡ ᠵᠦᠪ ᠪᠣᠯᠬᠤ ᠨ ᠢ ( ) A. 𝑎 b B. 𝑎 −b C. − 𝑎 b D. −𝑎 −1 C. 𝑥 −1 ᠪᠦᠬᠡᠳ 𝑥≠ 2 B. 𝑥 ≥ −1 D. 𝑥 ≥ −1 ᠪᠦᠬᠡᠳ 𝑥≠ 2 6. ᠲᠠᠷᠠᠭᠠ ᠬᠢ ᠬᠡᠯᠡᠯᠬ ᠡ ᠢ ᠢᠨ ᠵᠦᠪ ᠪᠣᠯᠬᠤ ᠨ ᠢ ( ) A. ᠺᠦᠪ ᠢᠵ ᠠᠭᠣᠷ ᠨ ᠢ ᠲᠡᠬᠦᠨ ᠤ ᠲᠣᠰ ᠪ ᠡᠶ ᠡ ᠪᠣᠯᠬᠤ ᠲᠣᠭ ᠠ ᠨ ᠢ ᠯᠠ ᠪᠳᠠ ᠢ 1 ᠪᠠ 0 B. ᠷᠣᠮᠪ ᠢ ᠢᠨ ᠲᠦᠷ ᠪᠡᠨ ᠲᠠᠯ ᠠ ᠢᠢᠨ ᠲᠣᠮᠳᠠ ᠴᠡᠭ ᠢ ᠲᠡᠰ ᠲᠠᠷᠠ ᠭᠠᠯᠠᠨ ᠬᠣᠯᠪᠣᠵᠦ ᠣᠯᠣ ᠭᠰᠠᠨ ᠲᠦᠷᠪᠡᠯᠵ ᠢᠨ ᠨ ᠢ ᠲ ᠡᠭᠰ ᠢ ᠦᠨ ᠴᠦᠭᠳᠤ C. ᠹᠦᠩᠺ ᠼ 𝑦= 𝑘𝑥 + 𝑏( 𝑘≠ 0) ᠳ᠋ᠣ᠋᠂ 𝑦 ᠦ ᠨ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠨᠢ 𝑥 ᠤᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠦᠨ ᠨᠡᠮᠡᠭᠳ ᠡᠬᠦ ᠶᠢ ᠲᠠ ᠭᠠᠯᠳᠣᠨ ᠨᠡᠮᠡᠭᠳᠡᠨ ᠡ D. ᠭᠡᠷ ᠪᠡ ᠬᠣᠶᠠᠷ ᠪᠠᠭᠳ ᠠᠭᠰᠠᠨ ᠦ ᠨᠴᠦᠭ ᠨᠢ ᠲᠡᠩᠴᠡᠬᠦᠦ ᠪᠣᠯ᠂ ᠲᠡᠳ ᠡᠬᠡᠷ ᠦᠨ ᠬᠠᠷᠠ ᠭᠠᠯᠵᠠᠭᠰᠠᠨ ᠡᠰᠡᠷᠬᠦ ᠨᠣᠮᠣ᠋ ᠶᠢᠨ ᠣᠷ ᠳᠣ᠋ ᠨᠢ ᠯᠠᠪᠳᠠ ᠢ ᠲ ᠡᠩᠴ ᠡᠬᠦᠦ 7. ᠵᠢᠷᠣ ᠭ ᠮᠡᠳᠣ᠋᠂ Rt ∆ ABC ᠳ᠋ᠣ᠋ ∠B = 90° ᠂ A ᠴᠡᠭ ᠢᠶᠠᠷ ᠲᠣᠭᠣ ᠷᠢᠭ ᠤ ᠨ ᠲᠦᠪ ᠪᠣᠯᠭᠠᠨ᠂ ᠵᠣᠬᠢᠬᠣ᠋ ᠣᠷᠳ ᠣ᠋ ᠪᠠᠷ ᠷᠠᠳᠢᠣᠰ ᠪᠣᠯᠭᠠᠵᠦ ᠨᠣᠮ ᠣ᠋ ᠪᠠ ᠢᠭᠣᠯ ᠪᠠᠯ᠂ A B , A C ᠲᠠᠢ ᠲᠣᠰ ᠲᠣᠰ D , E ᠴᠡᠭ ᠲᠠᠢ ᠣᠭ ᠳᠣᠯᠣ ᠯᠴᠠᠵᠣ᠋᠂ ᠲᠠᠬᠢᠨ D , E ᠴᠡᠭ ᠢᠶᠠᠷ ᠲᠣᠰ ᠲᠣᠰ ᠲᠣᠭᠣ ᠷᠢᠭ ᠤᠨ ᠲᠦᠪ ᠪᠣᠯᠭ ᠠᠨ᠂ 12 DE ᠶᠢᠨ ᠣᠷᠳ ᠣ᠋ ᠡ ᠴᠠ ᠶᠡᠬᠡ ᠪᠡᠷ ᠷᠠᠳᠢᠣᠰ ᠪᠣᠯᠭᠠᠵᠦ ᠨᠣᠮᠣ᠋ ᠪᠠᠠᠢᠭᠣᠯ ᠪᠠᠯ᠂ ᠬᠣᠶᠠᠷ ᠨᠣᠮᠣ᠋ ᠨᠢ F ᠴᠡᠭ ᠲᠦ ᠣᠭᠳ ᠣᠯᠣᠯ ᠴᠠᠵᠣ᠋᠂ ᠰᠠᠴᠣᠷ ᠠᠭ AF ᠢ ᠪᠠᠢᠭᠣᠯᠪᠠᠯ ᠲᠠᠯ ᠠ BC ᠲᠠᠢ G ᠴᠡᠭ ᠲᠦ ᠣᠭᠳ ᠣᠯᠣᠯᠴᠠᠨ ᠠ᠂ ᠬᠡᠷᠪᠡ BG = 1 , AC = 4 ᠪᠣᠯᠪᠠᠯ᠂ ∆A CG ᠶᠢᠨ ᠲᠠᠯᠠ ᠪᠠᠢ ᠨᠢ ( ) 6 4 ᠭᠣᠣ ᠯ ᠬᠠᠷᠠ ᠭᠠᠨ ᠵᠢᠷᠤ ᠭ ᠵᠡᠭᠦᠨ ᠬᠠᠷᠠᠭᠠᠨ ᠵᠢᠷᠤᠭ ᠳᠡᠭᠡᠷ ᠡ ᠬᠠᠷᠠ ᠭᠠᠨ ᠵᠢᠷᠤ ᠭ G E D A B C F A. 1 B. 32 C. 2 D. 52 8. ᠵᠢᠷᠣᠭ ᠮᠡᠳᠣ᠋ ᠂ Rt ∆ABC ᠳ᠋ᠣ᠋᠂ ∠A CB = 90° , AC = BC = 2√ 2᠂ BC ᠪᠠᠷ ᠳ᠋ᠢᠶᠠᠮᠧᠲᠷ ᠪᠣᠯᠭ ᠠᠵᠦ ᠬᠠᠭᠠᠰ ᠲᠣᠭᠣ ᠷᠢᠭ ᠢ ᠪᠠᠢ ᠭᠣᠯᠪᠠᠯ᠂ AB ᠲᠠᠢ D ᠴᠡᠭ ᠲ ᠦ ᠣᠭᠳ ᠣᠯᠣᠯ ᠴᠠᠨ ᠠ᠂ ᠲᠡᠭᠡᠪᠡᠯ ᠰᠡᠬᠦᠳᠡᠷ ᠲᠠᠢ ᠬᠡᠰᠡ ᠭ ᠦᠨ ᠲᠠᠯ ᠠᠪᠠᠢ ᠨᠢ ( ) A. π − 1 B. 4 − π C. √ 2 D. 2 9.ᠲᠠᠷᠠᠭᠠ ᠬᠢ ᠦᠬ ᠦᠯᠡᠯ᠄ ① ᠬᠡᠷᠪᠡ 𝑥 2 + 𝑘 𝑥 + 14 ᠨᠢ ᠬᠦᠢᠴᠡ ᠳ ᠺᠦᠸᠠᠳ᠋ ᠤ ᠨ ᠬᠡᠯᠪᠡᠷᠢ ᠳᠠᠢ ᠪᠣᠯ᠂ 𝑘 = 1 ② ᠬᠡᠷ ᠪᠡ A(2,6 ) , B (0,4) , P (1,m) ᠭᠣᠷᠪᠠᠨ ᠴᠡᠭ ᠨᠢ ᠢᠵ ᠢᠯ ᠨᠢᠬᠡᠨ ᠰᠢᠯᠣᠭᠣ ᠨ ᠲᠡᠭᠡᠷ ᠡ ᠣᠷᠣ ᠰᠢᠪᠠᠯ᠂ 𝑚 = 5 ③ ᠠᠳᠠᠯᠢ ᠬᠠᠵᠠᠭᠣᠳᠣ᠋ ᠭᠣᠷᠪᠠᠯᠵ ᠢᠨ ᠣ᠋ ᠨᠢᠬᠡᠨ ᠲᠠᠯ ᠠ ᠲᠣᠬ ᠡᠷᠡᠬᠢ ᠲᠣᠮᠳᠠ ᠱᠣᠭᠣᠮ ᠤᠨ ᠣᠷᠣ ᠰᠢᠬᠣ᠋ ᠰᠢᠯᠣᠭᠣ ᠨ ᠨᠢ ᠲᠡᠬᠦᠨ ᠣ᠋ ᠲᠡᠭᠰᠢ ᠬᠡᠮ ᠦ ᠨ ᠲᠡᠩᠬᠡᠯ ᠢᠭ ④ ᠨᠢᠬᠡᠨ ᠣᠯᠠᠨ ᠦᠨᠴᠦᠭᠳ ᠦ ᠶᠢᠨ ᠲᠣᠳ ᠣᠭᠠᠳᠣ᠋ ᠦᠨᠴᠦᠭ ᠦᠨ ᠨᠡᠢᠯᠡᠪᠦᠷᠢ ᠨᠢ ᠲᠡᠬᠦ ᠨ ᠣ᠋ ᠭᠠ ᠳᠠᠭᠠᠳᠣ᠋ ᠦᠨᠴᠦᠭ ᠦᠨ ᠨᠡᠢ ᠯᠡᠪᠦᠷᠢ ᠶᠢᠨ 2 ᠲᠠᠬᠢᠨ ᠪᠣᠯᠪᠠᠯ᠂ ᠡᠨᠡ ᠣᠯᠠᠨ ᠦᠨᠴᠦᠭᠳ ᠦ ᠨᠢ ᠵᠢᠷᠭᠣᠭᠠ ᠨ ᠦᠨᠴᠦᠭᠳᠦ ᠪ ᠣᠯᠣᠨ ᠠ᠃ ᠲᠣᠳ ᠣᠷ ᠠ ᠵᠢᠩᠬ ᠢᠨᠢ ᠦᠬ ᠦᠯᠡᠯ ᠦᠨ ᠪᠣᠳ ᠠᠭᠠᠯᠢ ᠲᠣ ᠭ ᠠ ᠨᠢ ( ) A.1 B. 2 C. 3 D. 4 10. ᠦᠭᠬᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ ᠠᠳᠠᠯᠢ ᠬᠠ ᠵᠠᠭᠣᠳᠣ᠋ ᠭᠣᠷᠪᠠᠯᠵᠢᠨ ᠣ᠋ ᠭᠣᠷ ᠪᠠᠨ ᠲᠠᠯ ᠠ ᠨᠢ ᠲᠣᠰ ᠲᠣᠰ 𝑎 , b , 4 ᠪᠦᠬᠡᠳ 𝑎 , b ᠨᠢ 𝑥 ᠤᠨ ᠨᠢᠬᠡ ᠦᠯᠦ ᠮᠡᠳᠡᠭᠳᠡᠭᠴᠢ ᠲᠡᠢ ᠬᠣᠶᠠᠳᠣ ᠭᠠᠷ ᠵᠡᠷᠬᠡ ᠶᠢᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ 𝑥 2 − 12 𝑥+ m + 2= 0 ᠦᠨ ᠬᠣᠶᠠᠷ ᠢᠵᠠᠭᠣᠷ ᠪᠣᠯ᠂ 𝑚 ᠦ ᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠨᠢ ( ) A. 34 B. 3 0 C. 30 ᠪᠣᠶᠣ᠋ 34 D. 3 0 ᠪᠣᠶᠣ᠋ 36 11 .ᠵᠢᠷᠣ ᠭ ᠮᠡᠳᠣ᠋ ᠂ ᠲᠡᠭᠰᠢ ᠲᠦ ᠷᠪᠡᠯᠵᠢᠨ A BCD ᠳ᠋ᠣ᠋᠂ AB = 1, E , F ᠴᠡᠭ ᠨᠢ ᠲᠣᠰ ᠲᠣᠰ ᠲᠠᠯ ᠠ BC ᠪᠠ CD ᠲᠡᠭᠡᠷ ᠡ ᠣᠷᠣ ᠰᠢᠵᠦ᠂ AE = AF , ∠E AF = 60° ᠪᠣᠯᠣ ᠨ ᠠ᠂ ᠲᠡᠭᠡᠪᠡᠯ CF ᠦᠨ ᠣᠷᠳᠣ᠋ ᠨ ᠢ ( ) A. √ 3: 1 4 B. √ 32 C. √3 − 1 D. 23 12 . ᠵᠢᠷᠣᠭ ᠮᠡᠳᠣ᠋ ᠂ ᠲᠡᠭᠰᠢ ᠦᠨ ᠴᠦᠭᠳᠦ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠤᠨ ᠰᠢᠰᠲ ᠧᠮ ᠳ᠋ᠣ᠋᠂ ᠦᠭᠬᠦᠭᠳ ᠡᠭᠰᠡᠨ ᠨᠢ A( −3 ,− 2) , B (0 ,− 2) , C. (− 3, 0) , M ᠪᠣᠯ ᠬᠡᠷᠴᠢᠮ AB ᠲᠡᠭᠡᠷᠡᠬᠢ ᠨᠢᠬᠡᠨ ᠬᠦᠳᠡᠯᠬᠦ ᠴᠡᠭ ᠂ CM ᠢ ᠬᠣᠯᠪᠣᠨ ᠠ᠂ M ᠴᠡᠭ ᠢ ᠲᠠᠭ ᠠᠷᠢᠭᠣᠯᠣ ᠨ MN ⊥ MC ᠪᠠᠷ ᠪᠠᠢᠭᠣᠯᠪᠠᠯ y ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠳᠠᠢ N ᠴᠡᠭ ᠲᠦ ᠣᠭᠳ ᠣᠯᠣᠯᠴᠠ ᠨ ᠠ᠂ ᠬᠡᠷᠪᠡ M , N ᠴᠡᠭ ᠨᠢ ᠰᠢᠯᠣᠭ ᠣᠨ 𝑦= 𝑘 𝑥 + 𝑏 ᠲᠡᠭᠡᠷ ᠡ ᠣᠷ ᠣᠰᠢᠪᠠᠯ᠂ 𝑏 ᠦ ᠨ ᠬᠠᠮᠣᠭ ᠤᠨ ᠶᠡᠬᠡ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠨᠢ ( ) A. − 78 B. − 34 C. − 1 D. 0 ᠬᠣᠶᠠᠷ᠂ ᠬᠠᠭᠠᠯᠳᠠ ᠲᠠᠭᠯᠠᠬᠣ᠋ ᠰᠡᠳᠣ ᠪ᠃ ( ᠰᠡᠳᠣᠪ ᠪᠦᠷᠢ 3 ᠬᠣᠪᠢ᠂ ᠨᠡᠢᠳᠡ 24 ᠰᠡᠳᠣᠪ ) 13. 2018 ᠣᠨ ᠣ᠋ ᠮᠠᠨ ᠣ᠋ ᠣᠯᠣ ᠰ ᠤᠨ ᠲᠣᠳ ᠣᠭᠠᠳᠣ᠋ ᠳ᠋ᠡᠬᠢ ᠦᠢᠯᠡᠳᠪᠦᠷᠢᠯ ᠡᠯ ᠦᠨ ᠶᠡᠷᠣᠩ ᠬᠡᠢ ᠦ ᠷᠳᠡᠭ (GDP) ᠨᠢ 90 0309 ᠳ᠋ᠦᠩᠰᠢᠭᠣ ᠷ ᠲᠥᠬᠥᠷᠢ ᠭ ᠬᠦᠷᠴᠣ᠋᠂ ᠠᠩᠬᠠᠳᠣᠭᠠᠷ ᠣᠳᠠᠭ ᠠ 90 ᠲᠦᠮᠡᠨ ᠳ᠋ᠦᠩᠰᠢᠭᠣ ᠷ ᠲᠦᠬᠦᠷᠢᠭ ᠣᠨ ᠪᠣᠭᠣᠮᠳᠠ ᠶᠢ ᠦᠩ ᠬᠡᠷᠡᠪᠡ᠃ 90 ᠲᠦᠮᠡᠨ ᠳ᠋ᠦᠩᠰᠢᠭᠣ ᠷ ᠢ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠣᠬᠠᠭᠠᠨ ᠣ᠋ ᠲᠣᠭ ᠠ ᠲᠡᠮ ᠳᠡᠭᠯᠡᠬ ᠦ ᠠᠷ ᠭ ᠠ ᠪᠠᠷ ᠢ ᠯᠡᠷᠬᠡ ᠢᠯᠡᠪ ᠡᠯ ( ) D A C B B C A D E F x y B O A C M N 14 .ᠦᠭᠬᠦᠭᠳᠡ ᠭᠰᠡᠨ ᠨᠢ ᠲᠡᠩ ᠴᠡᠳᠭᠡᠯ ᠪᠣᠰᠣ᠋ ᠶᠢᠨ ᠪᠦᠯᠦᠭ { 2 𝑥+ 9 −6x + 1 𝑥− 𝑘 1 ᠶᠢᠨ ᠬᠠᠷᠢᠭᠣ᠋ ᠴᠣᠭ ᠯᠠᠷᠠᠯ ᠨᠢ 𝑥 −1 ᠲᠡᠭᠡᠪᠡᠯ 𝑘 ᠦ ᠨ ᠬᠡᠮ ᠵᠢᠭᠳᠡᠯ ᠠᠪᠬ ᠣ᠋ ᠬᠡᠪᠴᠢᠶ ᠡ ᠨᠢ ( ) 15. ᠬᠢᠯᠪᠠᠷᠴᠢᠯᠠ ᠄ 1 − 𝛼; 1 𝛼: 2 ÷ 𝛼 2 ;1 𝛼 2 :4 𝛼: 4 = ( ) 16 . ᠠ᠂ ᠡ ᠬᠣᠶᠠᠷ ᠠᠩᠬᠢ ᠲᠣᠭᠠᠨ ᠣ᠋ ᠣᠬᠠᠭ ᠠᠨ ᠣ᠋ ᠮᠡᠳᠡ ᠯᠬᠡ ᠶᠢᠨ ᠣᠷᠣ ᠯᠳᠣᠭᠠᠨ ᠬᠢᠪᠡ ᠂ ᠣᠷᠣ ᠯᠳᠣᠭᠠᠨ ᠳ᠋ᠣ᠋ ᠣᠷᠣ ᠯᠴᠠᠭᠰᠠᠨ ᠰᠣ ᠷᠣᠭᠴᠢ ᠶᠢᠨ ᠣᠷᠣ ᠯᠳᠣᠭᠠᠨ ᠳ᠋ᠣ᠋ ᠣᠯᠣ ᠭᠰᠠᠨ ᠬᠣᠪᠢ ᠶᠢ ᠪᠦᠷᠢᠳᠬ ᠡᠭᠰᠡᠨ ᠳ᠋ᠦᠩᠨᠡᠯᠳᠡ ᠨᠢ ᠲᠠᠷᠠᠭᠠᠬᠢ ᠬᠦᠰᠦᠨᠦᠭᠲᠦ ᠮᠡᠳᠣ᠋ ᠄ ᠨᠢᠬᠡᠨ ᠰᠣᠷ ᠣᠭᠴᠢ ᠲᠡᠬᠡᠷᠡᠬᠢ ᠬᠦᠰᠦᠨᠦᠭᠲᠦ ᠳ᠋ᠣ᠋ ᠵᠠᠳᠠᠯᠣᠯᠳᠠ ᠬᠢᠵ ᠦ᠂ ᠲᠠᠷᠠᠭᠠᠬᠢ ᠳ᠋ ᠦᠩᠨᠡᠯᠳᠡ ᠶᠢ ᠣᠯᠪᠠ ᠄ ① ᠠ᠂ ᠡ ᠬᠣᠶᠠ ᠷ ᠠᠩᠬ ᠢ ᠶᠢᠨ ᠰᠣᠷ ᠣᠭᠴᠢ ᠶᠢᠨ ᠲᠡᠭᠰᠢᠳ ᠬᠡᠭᠰᠡᠨ ᠬᠢᠷᠢ ᠪ ᠦᠳᠣᠮᠵᠢ ᠨᠢ ᠠᠳᠠᠯᠢ᠃ ② ᠡ ᠠᠩᠬᠢ ᠶᠢᠨ ᠰᠢᠯᠢᠳᠡᠭ ᠦᠨ ᠬᠦᠮᠦᠨ ᠣ᠋ ᠲᠣᠭ ᠠ ᠨᠢ ᠠ ᠠᠩᠬᠢ ᠶᠢᠨ ᠰᠢᠯᠢᠳᠡᠭ ᠦᠨ ᠬᠦᠮᠦᠨ ᠣ᠋ ᠲᠣᠭ ᠠ ᠡᠴᠡ ᠴᠦᠬᠡ ᠨ ( ᠣᠷᠣ ᠯᠳᠣᠭᠠᠨ ᠳ᠋ᠣ᠋ ≥ 85 ᠬᠣᠪᠢ ᠣᠯᠬᠣ᠋ ᠶᠢ ᠰᠢᠯᠢᠳᠡᠭ ᠬᠡᠨ ᠡ )᠃ ③ ᠠ ᠠᠩ ᠬᠢ ᠶᠢ ᠨ ᠬᠢᠷᠢ ᠪᠦᠳᠣ ᠮᠵᠢ ᠶᠢᠨ ᠬᠡᠯᠪᠡᠯᠵᠡ ᠬᠦ ᠨᠢ ᠡ ᠠ ᠩᠬᠢ ᠡᠴᠠ ᠪᠠᠭ ᠠ᠃ ᠲᠡᠬᠡᠷᠡᠬᠢ ᠳ᠋ᠦᠩᠨᠡᠯᠳᠡ ᠲᠣᠳ ᠣᠷ ᠠ ᠵᠦᠪ ᠪᠣᠯᠬᠣ᠋ ᠨᠢ ( ) ( ᠵᠦᠪ ᠳ᠋ᠦ ᠩᠨᠡᠯᠳᠡ ᠶᠢᠨ ᠳ᠋ᠣᠭᠠᠷ ᠢ ᠪᠦᠷ ᠪᠢ ᠴᠢ) 17. ᠵᠢᠷᠣᠭ ᠮᠡᠳᠣ᠋ ᠂ ∆ ABC ᠳ᠋ᠣ᠋᠂ ∠CAB = 55° , ∠ABC = 25° ᠃ ᠢᠵᠢᠯ ᠨᠢᠬᠡᠨ ᠬᠠᠪᠳᠠ ᠭᠠᠢ ᠲᠣᠳ ᠣᠷ ᠠ᠂ ∆ABC ᠶᠢ A ᠴᠡᠭ ᠢ ᠲᠣᠭᠣ ᠷᠢᠭᠣ ᠯᠣᠨ ᠴᠠᠭ ᠤ ᠨ ᠵᠡᠬ ᠦᠦ ᠶᠢᠨ ᠡᠰᠡᠷᠬᠦ ᠴᠢᠭᠯᠡᠯ ᠲᠠᠭ ᠠᠣ᠋ 70° ᠡᠷᠬ ᠢᠬᠦᠯᠦᠬ ᠡᠳ∆ ADE ᠶᠢ ᠣᠯᠵ ᠦ᠂ 𝐸𝐶 ᠶᠢ ᠬᠣᠯᠪᠣᠨ ᠠ᠂ ᠲᠡᠭᠡᠪᠡᠯ tan∠DEC ᠶᠢᠨ ᠬᠡᠮᠵ ᠢᠭᠳᠡᠯ ᠨᠢ ( ) 18 . ᠵᠢᠷᠣ ᠭ ᠮᠡᠳᠣ᠋ ᠂ BD ᠪᠣᠯ ⊙ O ᠤᠨ ᠳ᠋ᠢᠶ ᠠᠮᠧᠲᠷ᠂ A ᠴᠡᠭ ᠪᠣᠯ ⊙ O ᠤᠨ ᠭᠠᠳ ᠠᠨᠠᠬᠢ ᠨᠢᠬᠡᠨ ᠴᠡᠭ᠂ C ᠴᠡᠭ ᠨᠢ ⊙ O ᠲᠡᠬᠡᠷᠡᠬᠢ ᠴᠡᠭ᠂ AC ᠨᠢ ⊙ O ᠲᠠᠢ C ᠴᠡᠭ ᠲᠦ ᠰᠢᠬ ᠠᠯᠴᠠᠵᠦ᠂ ∠CAB = 90° ᠪᠣᠯᠣ ᠨ ᠠ᠃ ᠬᠡᠷᠪᠡ BD = 6,𝐴𝐵 = 4 , ∠ABC = ∠CBD ᠪᠣᠯᠪᠠᠯ᠂ ᠬᠦᠪ ᠴᠢ BC ᠶᠢᠨ ᠣᠷᠳ ᠣ᠋ ᠨᠢ ( ) 19. ᠵᠢᠷᠣᠭ ᠮᠡᠳᠣ᠋ ᠂ ᠲᠡᠭᠰᠢ ᠦᠨᠴᠦᠭᠳ ᠦ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠤᠨ ᠰᠢᠰᠲ ᠧᠮ ᠳ᠋ᠣ᠋᠂ A (− 1, 0 ) , B (0 ,2 ) ᠬᠡᠵᠦ ᠦᠭᠬᠦᠭᠳᠡᠪᠡ ᠃ ∆ABO ᠶᠢ ᠰᠢᠯᠣᠭᠣ ᠨ AB ᠲᠠᠭᠠ ᠣ᠋ ᠨᠣᠭᠣᠯ ᠵᠦ ∆ABC ᠶᠢ ᠣᠯᠣ ᠨ ᠠ᠃ ᠬᠡᠷᠪᠡ ᠣᠷᠪᠠᠭᠣ᠋ ᠬᠠᠷᠢ ᠴᠠᠯᠳᠣ᠋ ᠹᠦᠩᠺ ᠼ y= 𝑘𝑥 ( 𝑥 0) ᠦᠨ ᠵᠢᠷᠣ ᠮᠠᠭ ᠨᠢ C ᠴᠡᠭ ᠢ ᠲᠠᠭᠠᠷᠢᠪᠠᠯ 𝑘 = ( ) 20. ᠵᠢᠷᠣ ᠭ ᠮᠡᠳᠣ᠋ ᠂ Rt ∆ABC ᠳ᠋ᠣ᠋ ∠ABC = 90° ,BC = 3, D ᠨᠢ ᠵᠢᠰᠢᠬᠦᠦ ᠲᠠᠯ ᠠ AC ᠶᠢᠨ ᠲᠣᠮᠳᠠ ᠴᠡᠭ᠂ BC ᠶᠢ ᠬᠣᠯᠪᠣᠨ ᠠ᠂ F ᠨᠢ BC ᠲᠠᠯ ᠠ ᠲᠡᠭᠡᠷᠡᠬᠢ ᠬᠦᠳ ᠡᠯᠬᠦ ᠴᠡᠭ ( B ᠂ C ᠴᠡᠭ ᠲᠠᠢ ᠲᠠᠪᠬᠣ ᠴᠠᠬᠣ᠋ ᠦᠬᠡᠢ )᠂ B ᠴᠡᠭ ᠢ ᠲᠠᠭ ᠠᠷᠢᠭᠣᠯᠣ ᠨ BE ⊥ BD ᠪᠠᠷ ᠪᠠᠢᠭᠣᠯᠪ ᠠᠯ DF ᠤᠨ ᠦᠷᠬᠦᠯᠵᠢ ᠯᠡᠯ ᠱᠣᠭᠣᠮ ᠳ ᠠᠢ E ᠴᠡᠭ ᠲᠦ ᠣᠭᠳ ᠣᠯᠣᠯ ᠴᠠᠵᠣ᠋᠂ CE ᠶᠢ ᠬᠣᠯ ᠪᠣᠨ ᠠ᠂ ᠲᠠᠷᠠᠭᠠᠬᠢ ᠳ᠋ᠦᠩᠨᠡᠯᠳᠡ᠄ ᠠᠩᠭᠢ ᠠ ᠡ ᠤᠷᠤ ᠯᠳᠤᠭ ᠠᠨ ᠤ ᠣᠷᠣᠯ ᠴᠠᠭᠰᠠᠨ ᠬᠦᠮᠦᠨ ᠦ ᠲᠣᠭ ᠠ 45 45 ᠲᠡᠭᠰᠢ ᠳᠭᠡᠭ ᠰᠡᠨ ᠲᠣᠭ ᠠ 83 83 ᠳᠤᠮᠲ ᠠ ᠪᠠ ᠢᠷᠢ ᠶᠢᠨ ᠲ ᠣᠭ ᠠ 86 84 ᠰᠠᠷ ᠨᠢᠴᠡ 82 135 E D B A C O C A B D x y B A O C F A B E C D ① ᠬᠡᠷᠪᠡ BF = CF ᠪᠣᠯ CE 2+AD 2=DE 2 ᠃ ② ᠬᠡᠷᠪᠡ ∠BDE = ∠B AC , AB = 4 ᠪᠣᠯ᠂ CE = 158 ᠃ ③ ∆ABD ᠪᠠ ∆CBE ᠨᠢ ᠯᠠᠪᠳᠠᠢ ᠲᠦᠰᠳᠠᠢ ᠃ ④ ᠬᠡᠷᠪᠡ ∠A = 30° , ∠BCE = 90° ᠪᠣᠯ᠂ DE = √21 ᠃ ᠲᠣᠳ ᠣᠷ ᠠ ᠵᠦᠪ ᠪᠣᠯ ᠬᠣ᠋ ᠨᠢ ( ) ᠵᠦᠪ ᠳ᠋ᠦ ᠩᠨᠡᠯᠳᠡ ᠶᠢᠨ ᠳ᠋ᠣᠭᠠᠷ ᠢ ᠪᠦᠷ ᠪᠢᠴᠢ ᠭᠣᠷᠪᠠ ᠂ ᠪᠣᠳᠣ ᠵᠦ ᠬᠠᠷᠢᠭᠤ ᠯᠬᠤ ᠰᠡᠳᠣ ᠪ ( 6 ᠪᠠᠭ ᠠ ᠰᠡ ᠳᠣᠪ ᠳᠠᠢ᠂ ᠨᠡ ᠢᠳᠡ 60 ᠬᠣᠪᠢ ) 21. (8 ᠬᠣ ᠪᠢ) ᠨᠢᠬᠡᠨ ᠰᠣᠷᠭᠠ ᠭᠣᠯᠢ ᠶᠢᠰᠣᠳᠣ ᠬᠡᠷ ᠣᠨ ᠣ᠋ ᠰᠣᠷᠣ ᠭᠴᠢ ᠶᠢᠨ ᠲᠠᠮᠢᠷ ᠤᠨ ᠪᠠᠷᠢ ᠮᠵᠢᠶ ᠠ ᠳ᠋ᠣ᠋ ᠬᠦᠷᠬᠦ ᠪᠠᠢᠳᠠᠯ ᠢ ᠣᠢᠯᠭᠠ ᠬᠣ᠋ ᠶᠢᠨ ᠲᠦᠯᠦ ᠬᠡ᠂ ᠲᠠᠰᠢᠷᠠᠮ ᠢᠶᠡᠷ ᠶᠢᠰᠦᠳᠣ ᠬᠡᠷ ᠣᠨ ᠣ᠋ ᠰᠣᠷᠣ ᠭᠴᠢ ᠡᠴᠡ 50 ᠰᠣᠷᠣᠭᠴᠢ ᠶᠢ ᠰᠣᠭᠣᠯ ᠣᠨ ᠠᠪᠴᠣ᠋᠂ ᠲᠠᠮᠢᠷ ᠤᠨ ᠪᠠᠷᠢ ᠮᠵᠢᠶ ᠠ ᠳ᠋ᠣ᠋ ᠬᠦᠷᠬᠦ ᠲᠦᠷ ᠣᠯ ᠵᠦᠢᠯ ᠦᠨ ᠰᠣᠷᠢ ᠯᠳᠠ ᠶᠢ ᠬᠢᠪᠠ᠃ ᠰᠣᠷᠢ ᠯᠳᠡ ᠶᠢᠨ ᠬᠢᠷᠢ ᠪᠦᠳᠦᠮᠵᠢ ᠨᠢ ᠲᠠᠷᠠ ᠭᠠᠬᠢ ᠬᠦᠰᠦᠨ ᠦᠭᠲᠦ ᠮᠡ ᠳᠣ᠋᠄ ᠬᠦᠰᠦᠨᠦᠭᠲᠦ ᠳ᠋ᠠᠬ ᠢ ᠲᠣᠭ ᠠ ᠪᠠᠷᠢᠮᠳᠡ ᠶᠢ ᠦ ᠨᠳᠣᠰᠣ ᠯᠡᠨ᠂ ᠲᠠᠷ ᠠᠭᠠᠬᠢ ᠠᠰᠠ ᠭᠣᠯᠳᠠ ᠶᠢ ᠬᠠᠷᠢᠭᠣ ᠯ᠄ (1) ᠡᠨᠡ ᠰᠣᠷᠭᠠᠭᠣᠯᠢ ᠶᠢᠨ ᠶᠢᠰᠣᠳᠣ ᠬᠡᠷ ᠣᠨ ᠣ᠋ 450 ᠰᠣᠷᠣ ᠭᠴᠢ ᠶᠢᠨ ᠲᠠᠮᠢᠷ ᠤᠨ ᠰᠣᠷᠢ ᠯᠳᠠ ᠳ᠋ᠣ᠋ 25 ᠬᠣᠪᠢ ᠣᠯᠬᠣ᠋ ᠰᠣᠷ ᠣᠭᠴᠢ ᠶᠢᠨ ᠲᠣᠭ ᠠ ᠶᠢ ᠪᠠᠷ ᠣᠭᠴᠠᠭ ᠠᠯᠠ᠃ (2) ᠡᠨᠡ ᠰᠣᠷᠭᠠᠭᠣᠯᠢ ᠶᠢᠨ ᠲᠠᠮᠢᠷ ᠤᠨ ᠪᠠᠭᠰᠢ ᠨᠢ ᠡᠨᠡ ᠣᠳᠠᠭᠠᠨ ᠣ᠋ ᠰᠣᠷᠢ ᠯᠳᠠ ᠳ᠋ᠣ᠋ 23 ᠬᠣᠪᠢ ᠣᠯᠣ ᠭᠰᠠᠨ ᠠ᠂ ᠡ᠂ ᠢ᠂ ᠣ 4 ᠰᠣ ᠷᠣᠭᠴᠢ ᠶᠢ ᠲᠣᠭᠣᠢ ᠯᠠᠩ ᠬᠣᠪᠢᠶᠠᠵᠦ ᠴᠢᠩᠭᠠᠳᠬᠠᠬ ᠣ᠋ ᠪᠣᠯᠪ ᠠᠰᠣᠷ ᠠᠯ ᠬᠢᠨ ᠡ᠂ ᠰᠠᠭᠠᠷᠳᠠᠯᠭ ᠠ ᠨᠢ ᠬᠣᠶᠠᠷ ᠰᠣᠷᠣ ᠭᠴᠢ ᠨᠢᠬᠡ ᠲᠣᠭᠣ ᠢᠯᠠᠩ ᠪᠣᠯᠣ ᠨ ᠠ᠃ ᠠ᠂ ᠡ ᠶᠠᠭ ᠰᠠᠢ ᠬᠠᠨ ᠨᠢᠬᠡ ᠲᠣᠭᠣ ᠢᠯᠠᠩ ᠳ᠋ᠣ᠋ ᠬᠣᠪᠢᠶᠠᠭᠳ ᠠᠬᠣ᠋ ᠲᠣᠬ ᠢᠶᠠᠯᠳᠣᠮᠵᠢ ᠶᠢ ᠣᠯ᠃ ( ᠬᠦᠰᠦᠨᠦᠭᠲᠦ ᠪᠠᠢᠭᠣᠯᠬᠣ᠋ ᠪᠣᠶᠣ᠋ ᠮᠦᠴᠢᠷᠯᠢᠭ ᠵᠢᠷᠣᠭ ᠤᠨ ᠠᠷᠭ ᠠ ᠶᠢ ᠬᠡᠷᠡᠭᠯᠡᠨ ᠬ ᠠᠷᠢᠭᠣᠯ ) 22. (8 ᠬᠣᠪᠢ ) ᠵᠢᠷᠣ ᠭ ᠮᠡᠳᠣ᠋ ᠂ ᠲᠦᠷᠪᠡᠯᠵ ᠢᠨ ABCD ᠳ᠋ᠣ᠋ AD ∥BC , AB = BC , ∠B AD = 90° , AC ᠨᠢ BC ᠳᠠᠢ E ᠴᠡᠭ ᠲᠦ ᠣᠭᠳ ᠣᠯᠣᠯ ᠴᠠᠵᠣ᠋᠂ ∠ABD = 30° , AD = √3 , ᠬᠡᠷᠴᠢᠮ AC ᠪᠠ DE ᠶᠢᠨ ᠣᠷ ᠳᠣ᠋ ᠶᠢ ᠣ ᠯ᠃ ( ᠠᠩ ᠬᠠᠷ᠄ 1 √𝛼 :√ 𝑏 = √𝛼 ;√ 𝑏 (√ 𝛼: √𝑏 )(√ 𝛼; √𝑏 ) = √𝛼 ;√ 𝑏 𝛼; 𝑏 ) ᠰᠣᠷ ᠢᠯᠲᠠ ᠶᠢᠨ ᠬ ᠢᠷᠢ ᠪᠦᠳᠦ ᠮᠵᠢ ᠬᠦᠮᠦᠨ ᠦ ᠲᠣᠭ ᠠ 23 4 25 18 26 15 28 8 30 5 23 . (10 ᠬᠣᠪ ᠢ) ᠨᠢᠬᠡᠨ ᠲᠦᠷᠢ ᠶᠡᠰᠣᠨ ᠺᠣᠮᠫᠠᠨ ᠢ ᠨᠢ ᠠᠳᠠᠯᠢ ᠬᠡᠯᠪᠡᠷᠢ ᠮᠠᠶᠢᠭ ᠤᠨ ᠬᠡᠳᠣᠨ ᠠᠴᠢᠶᠠᠨ ᠲᠡᠷᠭᠡ ᠶᠢ ᠭᠠᠳᠠᠭᠰ ᠢ ᠲᠦᠷᠢ ᠶᠡᠰᠣᠯᠡᠨ ᠡ᠂ ᠠᠴᠢᠶᠠ ᠨ ᠲᠡᠷᠭ ᠡ ᠪᠦᠷᠢ ᠶᠢᠨ ᠡᠳᠣ ᠷ ᠤᠨ ᠲᠦᠷᠢ ᠶᠡᠰᠣᠨ ᠮᠦᠩᠬᠦ ᠶᠢ ᠬᠦᠭᠵᠢᠩ ᠬᠦᠢ ᠪᠣᠰᠣ᠋ ᠴᠠᠭ᠂ ᠬᠦᠭᠵᠢᠩ ᠬᠦᠢ ᠴᠠᠭ ᠬᠡᠵᠦ ᠬᠣᠶᠠᠷ ᠵᠦᠢᠯ ᠦᠨ ᠦᠨ ᠡ ᠬᠠ ᠩᠰᠢ ᠶᠢᠨ ᠪᠠᠷᠢ ᠮᠵᠢᠶ ᠠ ᠬᠡᠷᠡᠭᠵᠢᠬᠦᠯᠦᠨ ᠡ᠂ ᠬᠦᠭᠵᠢᠩ ᠬᠦᠢ ᠴᠠᠭ ᠲᠤ ᠠᠴᠢᠶᠠᠨ ᠲᠡᠷᠬᠡ ᠪᠦᠷᠢ ᠶᠢᠨ ᠡᠳᠣ ᠷ ᠤᠨ ᠲᠦᠷᠢ ᠶᠡᠰᠦ ᠨ ᠮᠦᠩᠬᠦ ᠨᠢ ᠬᠦᠭᠵᠢᠩ ᠬᠦᠢ ᠪᠣᠰᠣ᠋ ᠴᠠᠭ ᠤ ᠨ ᠦᠨ ᠡ ᠡ ᠴᠡ 13 ᠪᠡᠷ ᠦᠰᠦᠨ ᠡ᠃ ᠪᠦᠷᠢᠳᠬ ᠡᠯ ᠢ ᠦᠨᠳᠦᠰᠦᠯᠡᠪᠡ ᠯ ᠬᠦᠭ ᠵᠢᠩ ᠬᠦᠢ ᠪᠣᠰᠣ᠋ ᠴᠠᠭ ᠲᠤ ᠺᠣᠮᠫᠠᠨᠢ ᠨᠢ ᠡᠳᠣ ᠷ ᠪᠦᠷᠢ ᠲᠡᠭᠰᠢᠳᠬᠡᠬᠡᠳ 10 ᠠᠴᠢᠶᠠᠨ ᠲᠡᠷᠭᠡ ᠲᠦᠷᠢ ᠶᠡᠰᠦᠯᠡᠵᠦ ᠭᠠᠷᠬ ᠣ᠋ ᠦᠬᠡᠢ᠂ ᠡᠳᠣ ᠷ ᠤᠨ ᠲᠦᠷᠢ ᠶᠡᠰᠣᠨ ᠮᠦᠩᠬᠦ ᠶᠢᠨ ᠶᠡᠷᠣᠩ ᠬᠡᠢ ᠣᠷᠣ ᠯᠭ ᠠ ᠨᠢ 15 00 ᠲᠦᠬᠦᠷᠢᠭ ᠂ ᠬᠦᠭᠵᠢᠩ ᠬᠦᠢ ᠴᠠᠭ ᠲᠤ ᠺᠣᠮᠫᠠᠨ ᠢ ᠶᠢᠨ ᠭᠠᠳᠠᠭ ᠰᠢ ᠲᠦᠷᠢ ᠶᠡᠰᠦᠯᠡᠬᠦ ᠠᠴᠢᠶᠠᠨ ᠲᠡᠷᠬᠡ ᠨᠢ ᠬᠡᠷᠡᠭᠴᠡᠬᠡ ᠶᠢ ᠬᠠ ᠩᠭᠠᠵᠦ ᠲᠡᠢᠯᠬᠦ ᠦᠬᠡᠢ ᠣᠴᠢᠷ ᠡᠴᠠ ᠪᠦᠬᠦ ᠠᠴᠢᠶ ᠠᠨ ᠲᠡᠷᠬᠡ ᠨᠢ ᠡᠳᠣ ᠷ ᠪᠦᠷᠢ ᠴᠦᠮ ᠲᠦᠷᠢ ᠶᠡᠰᠦᠯᠡᠭᠳᠡᠨ ᠡ᠂ ᠡᠳᠣ ᠷ ᠤᠨ ᠲᠦᠷᠢ ᠶᠡᠰᠣᠨ ᠮᠦᠩᠬᠦ ᠶᠢᠨ ᠶᠡᠷᠣ ᠩᠬᠡᠢ ᠣᠷ ᠣᠯᠭ ᠠ ᠨᠢ 40 00 ᠲᠦᠬᠦᠷᠢᠭ᠃ (1) ᠡᠨᠡ ᠲᠦᠷᠢᠶ ᠡᠰᠣᠨ ᠺᠣᠮᠫᠠᠨᠢ ᠶᠢᠨ ᠭᠠᠳᠠᠭᠰᠢ ᠲᠦᠷᠢ ᠶᠡᠰᠦᠯᠡᠬᠦ ᠠᠴᠢᠶᠠ ᠨ ᠲᠡᠷᠭᠡ ᠣ᠋ ᠬᠡᠳᠣᠢ ᠪᠠᠢᠬᠣ᠋ ᠪᠣᠢ︖ ᠬᠦᠭᠵᠢᠩ ᠬᠦᠢ ᠪᠣᠰᠣ᠋ ᠴᠠᠭ ᠲᠤ ᠠᠴᠢᠶᠠᠨ ᠲᠡᠷᠬᠡ ᠪᠦᠷᠢ ᠶᠢᠨ ᠡᠳᠣ ᠷ ᠤᠨ ᠲᠦᠷᠢ ᠶᠡᠰᠣᠨ ᠦᠨ ᠡ ᠨᠢ ᠬᠡᠳᠣᠢ ᠲᠦᠬᠦᠷᠢᠭ ᠪᠣᠢ ︖ (2) ᠵᠠᠬ ᠠ ᠲᠡᠯᠬᠡ ᠬᠦᠷ ᠦ ᠨ ᠪᠠᠢ ᠴᠠᠭᠠᠯᠳ ᠠ ᠡᠴᠡ ᠮᠡᠳᠡᠬᠦ ᠨᠢ᠂ ᠠᠴᠢᠶᠠᠨ ᠲᠡᠷᠭᠡ ᠪᠦᠷᠢ ᠶᠢᠨ ᠡᠳᠣ ᠷ ᠤᠨ ᠲᠦᠷᠢ ᠶᠡᠰᠣᠨ ᠮᠦᠩᠬᠦ ᠨᠢ 20 ᠲ ᠦᠬᠦᠷᠢ ᠭ ᠦᠰᠬᠡᠬᠦ ᠪᠦᠷᠢ ᠲᠦᠷᠢᠶᠠᠰᠣᠯᠡᠵᠦ ᠭᠠ ᠷᠬᠣ᠋ ᠦᠬᠡᠢ ᠠᠴᠢᠶᠠᠨ ᠲᠡᠷᠬᠡ ᠶᠢᠨ ᠲᠣᠭ ᠠ ᠨᠢ ᠡᠳᠣ ᠷ ᠪᠦᠷᠢ 1 ᠲᠡᠷᠭᠡ ᠨᠡᠮ ᠡᠭᠳᠡᠨ ᠡ᠃ ᠪᠣᠰᠣᠳ ᠰᠢᠯᠳᠠᠭᠠᠨ ᠢ ᠪᠣᠳᠣᠯ ᠬᠢᠯᠠᠬᠣ᠋ ᠦᠬᠡᠢ ᠪᠣᠯ᠂ ᠠᠴᠢᠶᠠᠨ ᠲᠡᠷᠬᠡ ᠪᠦᠷᠢ ᠶᠢᠨ ᠡᠳᠣᠷ ᠤᠨ ᠲᠦᠷᠢ ᠶᠡᠰᠣᠨ ᠮᠦᠩᠬᠦ ᠶᠢ ᠬᠡᠳᠣᠢ ᠦᠰᠬᠡᠪᠡᠯ᠂ ᠡᠨᠡ ᠲᠦᠷᠢ ᠶᠡᠰᠣᠨ ᠺᠣᠮᠫᠠᠨ ᠢ ᠶᠢᠨ ᠡᠳᠣ ᠷ ᠤᠨ ᠲᠦᠷᠢᠶ ᠡᠰᠦᠨ ᠮᠦᠩ ᠬᠦ ᠶᠢᠨ ᠶᠡᠷᠣᠩ ᠬᠡᠢ ᠣ ᠷᠣᠯᠭ ᠠ ᠨᠢ ᠬᠠ ᠮᠣᠭ ᠤ ᠨ ᠦᠨᠳᠣᠷ ᠪᠣ ᠢ︖ 24. ( 10 ᠬᠣᠪᠢ ) ᠵᠢᠷᠣ ᠭ ᠮᠡᠳᠣ᠋ ᠂ ⊙ O ᠳ᠋ᠣ᠋᠂ B ᠪᠣᠯ ⊙ O ᠲᠡᠬᠡᠷᠡᠬᠢ ᠨᠢᠬᠡᠨ ᠴᠡᠭ᠂ ∠ABC = 120° ᠂ ᠬᠦᠪᠴᠢ AC = 2√ 3 ᠂ ᠬᠦᠪᠴᠢ BM ᠨᠢ ∠ABC ᠶᠢ ᠲᠠᠯᠠᠯᠠᠨ ᠬᠣᠪᠢᠶ ᠠᠵᠦ AC ᠳᠠᠢ D ᠴᠡᠭ ᠲᠦ ᠣᠭᠳᠣ ᠯᠣᠯᠴ ᠠᠨ ᠠ᠂ MA , MC ᠶᠢ ᠬᠣᠯᠪᠣ ᠨ ᠠ᠃ (1) ⊙ O ᠣᠨ ᠷᠠᠳᠢᠣ ᠰ ᠤᠨ ᠣᠷᠳᠣ᠋ ᠶᠢ ᠣᠯ ᠃ (2) ᠪᠠᠳᠣᠯᠠᠭᠳᠠᠬᠣ᠋ ᠨᠢ᠄ AB + BC = BM ᠃ D O M C B A 25. (12 ᠬᠣᠪᠢ ) ᠵᠢᠷᠣ ᠭ ᠮᠡᠳᠣ᠋ ᠂ ᠲᠡᠭᠰᠢ ᠲᠦᠷᠪᠡᠯᠵ ᠢᠨ ABCD ᠳ᠋ᠣ᠋ AB= 6 , M ᠪᠣᠯ ᠲᠠᠰᠢᠭᠣᠪᠴᠢ BD ᠲᠡᠬᠡᠷᠡᠬᠢ ᠨᠢᠬᠡᠨ ᠬᠦᠳᠡᠯᠬᠦ ᠴᠡᠭ (0 𝐷𝑀 12 𝐵𝐷 ) AM ᠢ ᠬᠣᠯᠪᠣᠨ ᠠ᠂ M ᠴᠡᠭ ᠢ ᠲᠠᠭᠠᠷᠢᠭᠣ ᠯᠣᠨ MN ⊥ AM ᠢᠶᠠᠷ ᠪᠠᠢᠭᠣᠯᠪᠠ ᠯ BC ᠲᠠᠯ ᠠ ᠲᠠᠢ N ᠳ᠋ᠣ᠋ ᠣᠭᠳ ᠣᠯᠣᠯ ᠴᠠᠨ ᠠ᠃ (1) ᠵᠢᠷᠣᠭ ① ᠮᠡᠳᠣ᠋ ᠂ ᠪᠠᠳᠣ ᠯᠠᠭᠳᠠᠬ ᠣ᠋ ᠨᠢ᠄ MA = MN (2) ᠵᠢᠷᠣ ᠭ ② ᠮᠡᠳᠣ᠋ ᠂ AN ᠢ ᠬᠣᠯᠪᠣᠨ ᠠ᠂ O ᠨᠢ AN ᠤᠨ ᠲᠣᠮᠳᠠ ᠴᠡᠭ᠂ MO ᠦᠨ ᠦᠷᠬᠦᠯᠵᠢ ᠯᠡᠯ ᠱᠣᠭ ᠣᠮ ᠨᠢ AB ᠲᠠᠯ ᠠ ᠳᠠᠢ P ᠴᠡᠭ ᠲᠦ ᠣᠭᠳ ᠣᠯᠣᠯ ᠴᠠᠨ ᠠ᠂ 𝑆∆ 𝐴𝑀𝑁 𝑆∆ 𝐵𝐶 𝐷 = 1318 ᠦᠶᠡᠰ᠂ AN ᠪᠠ PM ᠤᠨ ᠣᠷᠳᠣ᠋ ᠶᠢ ᠣᠯ᠃ (13 ) ᠵᠢ ᠷᠣᠭ ③ ᠮᠡᠳᠣ᠋ ᠂ N ᠴᠡᠭ ᠢ ᠲᠠᠭ ᠠᠷᠢᠭᠣᠯᠣ ᠨ NH ⊥ BD ᠢᠶᠡᠷ ᠪᠠᠢ ᠭᠣᠯᠪᠠ ᠯ BD ᠳᠠᠢ H ᠴᠡᠭ ᠲᠦ ᠣᠭᠳ ᠣᠯᠣᠯ ᠴᠠᠨ ᠠ᠂ AM= 2√ 5 ᠦᠶᠡᠰ᠂ ∆HMN ᠦᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠶᠢ ᠣ ᠯ᠃ 26 . ( 12 ᠬ ᠣᠪᠢ ) ᠵᠢᠷᠣ ᠭ ᠮᠡᠳᠣ᠋ ᠂ ᠲᠡᠭᠰᠢ ᠦᠨᠴᠦᠭᠳᠦ ᠪᠠᠢᠷᠢᠰᠢ ᠯ ᠤᠨ ᠰᠢᠰᠲ ᠧᠮ ᠳ᠋ᠣ᠋᠂ ᠦᠭᠬᠦᠭᠳᠡᠭᠰᠡ ᠨ ᠨᠢ ᠬᠠᠶᠠᠯᠳᠠ ᠶᠢᠨ ᠱᠣᠭᠣᠮ y = α𝑥 2+ 𝑏𝑥 + 2 ( 𝛼 ≠ 0) ᠨᠢ 𝑥 ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠲᠠᠢ A (− 1, 0) ᠪᠠ B( 3 , 0 ) ᠬᠣᠶᠠᠷ ᠴᠡᠭ ᠲᠤ ᠣᠭᠳᠣᠯᠣ ᠯᠴᠠᠵᠣ᠋᠂ y ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠳᠠᠢ C ᠴᠡᠭ ᠲᠦ ᠣᠭᠳ ᠣᠯᠣᠯ ᠴᠠᠨ ᠠ᠂ 𝐵𝐶 ᠶᠢ ᠬᠣᠯ ᠪᠣᠨ ᠠ᠃ (1) ᠡᠨᠡ ᠬᠠᠶᠠᠯᠳᠠ ᠶᠢᠨ ᠱᠣᠭᠣᠮ ᠤᠨ ᠠᠨᠠᠯᠢᠰ ᠢᠯᠡᠷᠬᠡᠢᠯᠡᠯ ᠢ ᠣᠯᠵ ᠦ᠂ ᠪᠠᠰᠠ ᠲᠡᠬᠦ ᠨ ᠣ᠋ ᠲᠡᠭᠰᠢ ᠬᠡᠮ ᠦ ᠨ ᠲᠡᠩᠬᠡᠯᠢ ᠭ ᠢ ᠪ ᠢᠴᠢ᠃ (2) ᠴᠡᠭ D ᠨᠢ ᠬᠠ ᠶᠠᠯᠳᠠ ᠶᠢᠨ ᠱᠣᠭ ᠣᠮ ᠤᠨ ᠲᠡᠭᠰᠢ ᠬᠡᠮ ᠦ ᠨ ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠲᠡᠬᠡᠷᠡᠬᠢ ᠨᠢᠬᠡᠨ ᠴᠡᠭ ᠂ CD , DB ᠶᠢ ᠬᠣᠯ ᠪᠣᠨ ᠠ᠂ ᠬᠡᠷᠪ ᠡ ∠DCB = ∠CBD ᠪᠣᠯ᠂ D ᠴᠡᠭ ᠦ ᠨ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠢ ᠣᠯ᠃ (3) ᠦᠭᠬᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ ᠴᠡᠭ 𝐹( 1, 1) ᠂ ᠬᠡᠷᠪᠡ 𝐸( 𝑥 ,𝑦 ) ᠴᠡᠭ ᠪᠣᠯ ᠬᠠᠶᠠᠯᠳᠠ ᠶᠢᠨ ᠱᠣᠭᠣᠮ ᠲᠡᠭᠡᠷᠡᠬᠢ ᠨᠢᠬᠡᠨ ᠬᠦᠳ ᠡᠯᠬᠦ ᠴᠡᠭ (1 𝑥 2) ᠂ CE , CF , EF ᠢ ᠬᠣᠯᠪᠣᠨ ᠠ᠂ ∆ CEF ᠦᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠶᠢᠨ ᠬᠠᠮᠣᠭ ᠤᠨ ᠶᠡᠬᠡ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠵᠢᠴᠢ ᠡᠨᠡ ᠦᠶᠡᠰ ᠲᠡᠬᠢ E ᠴᠡᠭ ᠦ ᠨ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠢ ᠣᠯ᠃ NB C A D M O NB C DA M P H NB C DA M (4) ᠬᠡᠷᠪᠡ N ᠴᠡᠭ ᠨᠢ ᠬᠠ ᠶᠠᠯᠳᠠ ᠶᠢᠨ ᠱᠣᠭᠣᠮ ᠤᠨ ᠲᠡᠭᠰᠢ ᠬᠡᠮ ᠦ ᠨ ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠲᠡᠬᠡᠷᠡᠬᠢ ᠨᠢᠬᠡᠨ ᠴᠡᠭ᠂ ᠬᠠᠶᠠᠯᠳᠠ ᠶᠢᠨ ᠱᠣᠭᠣᠮ ᠲᠡᠬᠡᠷ ᠡ᠂ B , C , M , N ᠢᠶᠡᠷ ᠣᠷᠣ ᠢ ᠶᠢᠨ ᠴᠡᠭ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠲᠦᠷᠪᠡᠯᠵ ᠢᠨ ᠢ ᠵᠢᠰᠢᠬᠦᠯᠵᠢᠨ ᠪᠣᠯ ᠭᠠᠵᠦ ᠲᠡᠢᠯᠬᠦ M ᠴᠡᠭ ᠣᠷᠣ ᠰᠢᠨ ᠠ ᠣᠢᠮᠣᠰᠣᠨ ᠣ᠋ ᠤᠤ ︖ ᠬᠡᠷᠪᠡ ᠣᠷᠣ ᠰᠢᠪᠠᠯ᠂ ᠨᠦᠬᠦᠴᠡᠯ ᠢ ᠬᠠᠩ ᠭᠠᠬᠣ᠋ M ᠴᠡᠭ ᠦᠨ ᠪᠣᠢ ᠪᠦᠬᠦᠢ ᠪᠠᠢᠷᠢᠰᠢ ᠯ ᠢ ᠰᠢᠭᠣᠳ ᠪᠢᠴᠢ᠃ ᠬᠡᠷ ᠪᠡ ᠣᠷᠣ ᠰᠢᠬᠣ᠋ ᠦᠬ ᠡᠢ ᠪᠣᠯ᠂ ᠰᠢᠯᠳᠠ ᠭᠠᠨ ᠢ ᠲᠣᠳ ᠣᠷᠬᠠᠢ ᠯᠠ᠃ x y C BA O