ᠵᠢᠷᠭᠤᠳᠤᠭᠠᠷ ᠪᠥᠯᠥᠭ ᠪᠣᠳᠠᠳᠤ ᠲᠤᠭᠠ （ᠨᠢᠭᠡᠴᠢ ᠶᠢᠨ ᠪᠠᠢᠴᠠᠭᠠᠯᠲᠠ 2）
ᠵᠢᠷᠭᠤᠳᠤ ᠭᠠᠷ ᠪᠥᠯ ᠥᠭ ᠦᠨ ᠪ ᠠᠢᠴ ᠠᠭᠠᠯᠲ ᠠ ᠠᠩᠭᠢ ᠪᠥᠯ ᠥᠭ᠄ ᠨᠡᠷ ᠡ᠄ ︵90 ᠮᠢᠨᠦ᠋ ᠲ᠂ ᠨᠠᠢᠲᠡ 12 0 ᠬ ᠤᠪᠢ ︶ ᠨᠢᠭᠡ᠂ ᠰᠤᠩᠭᠤᠬ ᠤ ᠰᠡ ᠲᠤᠪ ᠃ ︵ ᠰᠡᠲᠤᠪ ᠪᠦ ᠷᠢ 3 ᠬᠤᠪᠢ᠂ ᠨᠠᠢᠲᠡ 30 ᠬᠤᠪᠢ ︶ 1. ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ 35 , 𝜋3 , 0.12 122 122 1᠁ ,3 .14 159 26 ,√ 4 3 , −√ 81 ᠳᠦ᠂ ᠶᠤᠰᠤ ᠬᠦᠢ ᠲᠣᠭ ᠠ ᠨᠢ ( ) ᠪᠠᠢᠨ ᠠ᠃ A. 2 B. 3 C. 4 D. 5 2. ᠬᠡᠷᠪᠡ ᠨᠢᠭᠡ ᠲᠣᠭ ᠠᠨ ᠤ ᠺᠦᠸ ᠠᠳ᠋ ᠢᠵᠠᠭ ᠤᠷ ᠪ ᠠ ᠲᠡ ᠭᠦᠨ ᠦ ᠺᠦᠪ ᠢᠵᠠᠭ ᠤᠷ ᠨᠢ ᠪᠦᠷᠢᠮ ᠦᠰᠦᠨ ᠠᠳᠠᠯᠢ ᠪᠣᠯ᠂ ᠡᠨᠡ ᠲ ᠣᠭ ᠠ ᠨᠢ ( ) A. 1 B. − 1 C. 0 D. ± 1 , 0 3. √16 ᠶᠢᠨ ᠺᠦᠸᠠᠳ᠋ ᠢᠵ ᠠᠭᠤᠷ ᠨᠢ ( ) A. − 4 B. ±2 C. ±4 D. 4 4. ᠬᠡᠷᠪᠡ 𝑎 ᠨᠢ ( −3 ) 2 ᠶᠢᠨ ᠺᠦᠸ ᠠᠳ᠋ ᠢ ᠵᠠᠭᠤᠷ ᠪᠣᠯ᠂ √ 𝑎 3 ᠲᠡ ᠩᠴᠡᠭᠦᠦ ( ) A. −3 B. √3 3 C. √3 3 ᠪᠠ −√ 3 3 D. 3 ᠪᠠ − 3 5. ᠳᠣᠣᠷ ᠠᠬᠢ ᠦᠭ ᠦᠯᠡᠯ ᠄ ① ᠬᠤᠶ ᠠᠷ ᠰ ᠢᠯᠤᠭᠤᠨ ᠨᠢ ᠭᠤᠷᠪ ᠠᠳᠤᠭ ᠠᠷ ᠰ ᠢᠯᠤᠭᠤᠨ ᠳᠤ ᠤᠭᠲᠤᠯ ᠤᠭᠳᠠᠪ ᠠᠯ᠂ ᠲᠣᠬ ᠢᠷᠠᠭᠰᠠᠨ ᠦᠨᠴᠦ ᠭ ᠲᠡ ᠩᠴᠡᠭᠦᠦ ② 0. 1 ᠦ ᠨ ᠲᠣᠭ ᠠ ᠪ ᠣᠳᠣᠯ ᠭ ᠠ ᠶᠢᠨ ᠺᠦᠸ ᠠᠳ᠋ ᠢᠵᠠᠭ ᠤᠷ ᠨᠢ 0. 01 ③ ᠲᠣᠭ ᠠ ᠪᠣᠳᠣ ᠯᠭ ᠠ ᠶᠢᠨ ᠺᠦᠸ ᠠᠳ᠋ ᠢᠵᠠᠭ ᠤᠷ ᠨᠢ ᠲᠡ ᠭᠦᠨ ᠦ ᠲᠣᠰ ᠪᠡᠶ ᠡ ᠲᠠ ᠢ ᠪᠠᠨ ᠲᠡᠩ ᠴᠡᠬᠦ ᠲᠣᠭ ᠠ ᠨᠢ 1 ④ ᠬᠡ ᠷᠪᠡ ᠴᠡᠭ P (3 − 2n ,1 ) ᠡᠴᠡ ᠬᠤᠶ ᠠᠷ ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠬᠦᠷᠬᠦ ᠵᠠ ᠢ ᠨᠢ ᠲᠡ ᠩᠴᠡᠭᠦᠦ᠂ ᠲᠡᠭᠡᠪ ᠡᠯn = 1 ⑤ ᠬᠡᠷᠪᠡ 𝑎 2= 𝑏 2 ᠂ ᠲᠡ ᠭᠡᠪᠡᠯ 𝑎= 𝑏⑥ ᠬᠡᠷᠪᠡ √𝑎 3 = √𝑏 3 ᠲᠡᠭᠡᠪ ᠡᠯ 𝑎= 𝑏 ᠳᠤᠮ ᠳᠠ ᠬᠠᠭᠤᠷ ᠮᠠᠭ ᠦᠭ ᠦᠯᠡᠯ ᠦᠨ ᠪᠤᠳᠠ ᠭᠠᠯᠢ ᠲᠣᠭ ᠠ ᠨᠢ ( ) A. 3 B. 4 C. 5 D. 6 6. ᠳᠣᠣᠷ ᠠᠬᠢ ᠢᠯᠡᠷ ᠬᠡᠢᠯ ᠡᠯ ᠡᠴᠡ ᠵᠥ ᠪ ᠨᠢ ( ) A. √16 = ±4 B. ± √16 = 4 C. √( −4 ) 2 = −4 D. √− 27 3 = −3 7. 2 + √7 ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠪᠠᠷ ᠤᠭᠴ ᠠᠭᠠᠯᠠᠪᠠ ᠯ ( ) A. 2 ᠠᠴᠠ 3 ᠶᠢᠨ ᠬᠤᠭᠤᠷ ᠤᠨᠳᠤ B. 3 ᠠᠴᠠ 4 ᠶᠢᠨ ᠬᠤᠭᠤᠷ ᠤᠨᠳᠤ C. 4 ᠠᠴᠠ 5 ᠶᠢᠨ ᠬᠤᠭᠤᠷ ᠤᠨᠳᠤ D. 5 ᠠᠴᠠ 6 ᠶᠢᠨ ᠬᠤᠭᠤᠷ ᠤᠨᠳᠤ 8. ᠳᠣᠣᠷ ᠠᠬᠢ ᠬᠡᠯᠡᠯᠭᠡ ᠡᠴᠡ ᠪᠤᠷᠤᠭ ᠤ ᠪᠣ ᠯᠬᠤ ᠨᠢ ( ) A. ᠰᠦ᠋ ᠭᠡᠷᠭᠦ ᠪᠤᠰ ᠤ ᠲᠣᠭ ᠠ ᠨᠢ ᠲᠣᠭ ᠠ ᠪᠣᠳ ᠤᠯᠭ ᠠ ᠶᠢᠨ ᠺᠦᠸ ᠠᠳ᠋ ᠢᠵᠠᠭ ᠤᠷ ᠲᠠᠢ B. √5 ᠨᠢ √25 ᠶᠢᠨ ᠲᠣᠭ ᠠ ᠪᠣᠳᠤᠯ ᠭ ᠠ ᠶᠢᠨ ᠺᠦᠸ ᠠᠳ᠋ ᠢᠵᠠᠭ ᠤᠷ C. √− 1 ᠨᠢ ᠤᠳᠬ ᠠ ᠦᠭᠡᠢ D. ᠰᠤᠩᠭᠤ ᠯᠲᠠ ᠦᠭᠡᠢ 9. ᠳᠣᠣᠷ ᠠᠬᠢ ᠳᠦ᠋ᠷ ᠪᠡᠨ ᠲᠣᠭ ᠠ ᠠᠴ ᠠ ᠰ ᠦ᠋ᠭᠡᠷ ᠭᠦ ᠲᠣᠭ ᠠ ᠪᠣ ᠯᠬᠤ ᠨᠢ ( ) A. −2 2 B. √ (− 1) 2 C. (− 2) 2 D. |− 2| 10 . ᠬᠡ ᠳᠦᠨ ᠲᠣᠭ ᠠ ᠶᠢ ᠳᠤᠮᠳᠠ ᠬᠠᠭ ᠠᠯᠲᠠᠨ ᠳᠤ ᠲᠠᠯᠪᠢᠭ ᠠᠳ ᠳᠠᠩ ᠴᠡᠭ ᠢᠶᠡᠷ ᠰ ᠠᠯᠭᠠ ᠨ ᠠ᠂ ᠵᠢᠱᠢᠶ ᠡᠯᠡᠪᠡ ᠯ᠄ { 1,2 ,3} , { −2 ,7, 8,19 }᠂ ᠪᠢᠳᠠ ᠡᠭᠦᠨ ᠢ ᠴᠤ ᠭᠯᠠᠷ ᠠᠯ ᠭᠡᠵ ᠦ᠂ ᠡ ᠭᠦᠨ ᠤ ᠳᠣ ᠲᠣᠷᠠᠬ ᠢ ᠲᠣᠭ ᠠ ᠶᠢ ᠴᠣᠭ ᠯᠠᠷᠠᠯ ᠤᠨ ᠡᠯᠧᠮ ᠧᠨᠲ ᠭᠡᠨ ᠡ᠃ ᠬᠡᠷᠪᠡ ᠨᠢᠭᠡ ᠴᠤᠭ ᠯᠠᠷᠠᠯ ᠨᠢ ᠄ ᠪ ᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ 𝑎 ᠨᠢ ᠴᠤᠭᠯ ᠠᠷᠠᠯ ᠤ ᠨ ᠡ ᠯᠧᠮᠧᠨ ᠲ ᠦ ᠶᠡᠰ᠂ ᠪᠣᠳᠠ ᠲᠤ ᠲᠣᠭ ᠠ 8 − 𝑎 ᠨᠢ ᠪ ᠠᠰᠠ ᠯᠠᠪ ᠡᠨᠡ ᠴᠤᠭ ᠯᠠᠷᠠᠯ ᠤᠨ ᠡᠯᠧᠮ ᠧᠨᠲ ᠪᠣᠯᠤᠨ ᠠ᠂ ᠪᠢᠳᠠ ᠡᠢᠮ ᠦ ᠴᠤᠭ ᠯᠠᠷᠠᠯ ᠢ ︽ᠰᠠᠢᠨ ᠴᠤᠭᠯ ᠠᠷᠠᠯ ︾ᠭᠡᠨ ᠡ᠃ ᠳᠣ ᠣᠷᠠᠬ ᠢ ᠴᠤᠭᠯ ᠠᠷᠠᠯ ᠠᠴᠠ ︽ᠰᠠᠢᠨ ᠴᠤᠭᠯ ᠠᠷᠠᠯ ︾ ᠪᠣᠯᠬᠤ ᠨᠢ ( ) A. {1 ,2} B. {1 ,4, 7} C. {1 ,7, 8} D. {− 2,6 } ᠬᠣᠶ ᠠᠷ᠂ ᠬᠠᠭ ᠠᠯᠲᠠ ᠳᠤ ᠲᠠᠭᠯ ᠠᠬᠤ ᠰᠡᠲᠤᠪ᠃ ︵ᠰᠡᠲ ᠤᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪ ᠢ᠂ ᠨᠠ ᠢᠲᠡ 24 ᠬᠤᠪᠢ ︶ 11. ᠺᠢᠨᠣ᠋ ᠶᠢᠨ ᠫᠢᠯᠧᠲ ᠳᠡᠭᠡᠷ ᠡ ︽4 ᠳᠦᠭᠡᠷ ᠮᠦ᠋ᠷ ᠦᠨ 5 ᠳ ᠤᠭᠠᠷ ᠨᠣᠮᠧᠷ ︾ ᠢ ( 4 ,5 ) ᠭᠡᠵᠤ ᠲᠡᠮ ᠲᠡᠭᠯᠡᠨ ᠡ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ 5 ᠳᠦᠭᠡᠷ ᠮᠦ᠋ᠷ ᠦ ᠨ4 ᠳᠤᠭᠠ ᠷ ᠨᠣᠮᠧ ᠷ ᠢ ( ) ᠭᠡ ᠵᠦ ᠲᠡᠮ ᠲᠡᠭᠯᠡᠨ ᠡ᠃ 12. ᠴᠡᠭ (− 2 , 3) ᠢ ᠤᠷᠢᠳ ᠢᠶᠠ ᠷ ᠪ ᠠᠷᠠᠭ ᠤᠨᠰᠢ 2 ᠨᠢ ᠭᠡᠴᠢ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠵ ᠢᠭᠦᠯᠵ ᠦ᠂ ᠳᠠᠷᠠᠭ ᠠ ᠨᠢ ᠳᠣᠷ ᠣᠭᠰ ᠢ ᠪᠠᠨ 3 ᠨᠢ ᠭᠡᠴᠢ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠵ ᠢᠭᠦᠯᠦ ᠨ ᠡ᠃ ᠡᠨᠡ ᠦᠶ ᠡᠰ ᠦ ᠨ ᠪ ᠠᠢᠷᠢᠰᠢᠯ ᠨᠢ ( ) 13 . ᠬᠠᠪ ᠲᠠᠭᠠᠢ ᠶ ᠢᠨ ᠲᠡᠭᠰᠢ ᠦᠨᠴᠦᠭ ᠲᠦ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤ ᠨ ᠰ ᠢᠰᠲ᠋ ᠧᠮ ᠳᠦ᠂ ᠴᠡᠭ ( 3, −5 ) ᠨᠢ ( ) ᠮᠦ᠋ᠴᠡ ᠳᠡᠭᠡᠷ ᠡ ᠪᠠᠢ ᠨ ᠠ᠃ 14. ᠦ ᠭᠭᠦᠭᠳᠡ ᠭᠰᠡᠨ ᠨᠢ 𝑎 𝑏 0᠂ ᠲᠡ ᠭᠡᠪᠡᠯ ᠴᠡᠭ ( 𝑎− 𝑏, 𝑏) ᠨᠢ ( ) ᠮᠦ᠋ ᠴᠡ ᠳᠡᠭᠡᠷ ᠡ ᠪᠠᠢᠨ ᠠ᠃ 15. ∆AB O ᠳᠤ᠂ OA = OB = 5᠂ OA ᠦᠨ ᠲᠠᠯ ᠠ ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠦᠨᠳᠦ ᠷ ᠨᠢ 4᠂ ∆AB O ᠢ ᠬᠠᠪᠲᠠᠭ ᠠᠢ ᠶᠢᠨ ᠲᠡᠭᠰᠢ ᠦᠨᠴᠦᠭᠲᠦ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠳ ᠤ ᠲᠠ ᠯᠪᠢᠪ ᠠᠯ᠂ ᠴᠡᠭ O ᠪᠠ ᠡᠬᠢᠨ ᠦ ᠴᠡ ᠭ ᠲᠠ ᠢ ᠲᠠᠪᠬᠤᠴ ᠠᠭᠤᠯ ᠪᠠᠯ᠂ ᠴᠡᠭ A ᠨᠢ 𝑥 ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠦᠨ ᠡᠶᠡᠷᠭ ᠦ ᠬᠠᠭ ᠠᠰ ᠲᠡ ᠩᠬᠡᠯ ᠢᠭ ᠳᠡ ᠭᠡᠷ ᠡ ᠪᠠᠢᠨ ᠠ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ᠴᠡ ᠭ B ᠦ ᠨ ᠪ ᠠᠢᠷᠢᠰᠢᠯ ᠨᠢ ( ) 16. ᠦ ᠭᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ ᠴᠡᠭ P ᠨᠢ ᠬᠤᠶ ᠠᠳᠤᠭ ᠠᠷ ᠮ ᠦ᠋ᠴᠡ ᠳᠡ ᠭᠡᠷ ᠡ ᠤᠷᠤᠰ ᠢᠬᠤ ᠮᠦ᠋ᠷ ᠲᠡᠭᠡᠨ 𝑥 ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠬᠦᠷᠬᠦ ᠵᠠᠢ ᠨᠢ 2᠂ 𝑦 ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠬᠦᠷᠬᠦ ᠵᠠᠢ ᠨᠢ 3᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ᠴᠡ ᠭ P ᠦᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ( ) 17 . ᠷᠤᠮᠪ ᠶᠢᠨ ᠳᠦ᠋ᠷᠪ ᠡᠨ ᠤᠷ ᠤᠢ ᠶᠢᠨ ᠴᠡᠭ ᠨᠢ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤ ᠨ ᠲᠡ ᠩᠬᠡᠯ ᠢᠭ ᠳ ᠡᠭᠡᠷ ᠡ ᠤᠷᠤᠰ ᠢᠨ ᠠ᠂ ᠦᠭ ᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ ᠬᠤᠶ ᠠᠷ ᠤᠷ ᠤᠢ ᠶᠢᠨ ᠴᠡᠭ ᠦᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ᠲᠣᠰ ᠲᠣᠰ( 3,0 ),( 0,4 )᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ᠨᠦ᠋ᠭᠦᠭᠡ ᠬᠤᠶ ᠠᠷ ᠤᠷ ᠤᠢ ᠶᠢᠨ ᠴᠡᠭ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ( ) 18. ᠳᠣᠣᠷ ᠠᠬᠢ ᠢᠯᠡᠷ ᠬᠡᠢᠯ ᠡᠯ ᠢ ᠠᠵᠢᠭᠯ ᠠ᠄ √1 + 13 = 2√ 13 , √2 + 14 = 3√ 14 , √3 + 15 = 4√ 15 ᠁ ᠬᠡ ᠷᠪᠡ √𝑎 + 1𝑏 = 8√ 1𝑏 (𝑎 ,𝑏 ᠨᠢ ᠡᠶᠡᠷᠭᠦ ᠪᠦᠬᠦᠯᠢ ᠲᠣᠭ ᠠ) ᠨᠢ ᠳᠡ ᠭᠡᠷᠡᠬ ᠢ ᠵᠢᠮ ᠡ ᠶᠤᠰ ᠤ ᠶᠢ ᠬᠠᠩᠭ ᠠᠪᠠᠯ᠂ √𝑎 + 𝑏= ( ) ᠭᠤᠷᠪ ᠠ᠂ ᠪᠣᠳᠣᠵ ᠤ ᠬᠠᠷ ᠢᠭᠤᠯ ᠬᠤ ᠰᠡ ᠲᠤᠪ᠃ ︵ ᠨᠠᠢᠲᠡ 66 ᠬᠤᠪᠢ ︶ 19. ( 8 ᠬᠤᠪᠢ ) ᠪᠤᠳᠤ᠄ (1) √ 2516 + √− 8 3 − ( 12 ) 2 (2) √ 3+ √2 − 2√ 3+ 3√ 2 20. ( 8 ᠬᠤᠪᠢ ) ᠳᠣᠣᠷ ᠠᠬᠢ ᠨᠦ᠋ᠬᠦ ᠴᠡᠯ ᠳᠡᠬ ᠢ ᠦ ᠯᠦ ᠮᠡ ᠳᠡᠬᠦ ᠲᠣᠭ ᠠᠨ ᠤ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠣᠯ᠃ (1) 125 𝑥 3 = 8 (2) 4 𝑦 2 − 36 = 0 21. (10 ᠬ ᠤᠪᠢ ) ᠦᠭ ᠭᠦᠭᠳᠡᠭ ᠰᠡᠨ ᠨᠢ 𝑥 ᠶᠢᠨ ᠺᠦ ᠪ ᠢ ᠵᠠᠭᠤᠷ ᠨᠢ 3᠂ 2𝑥 + 10 ᠦᠨ ᠲᠣᠭ ᠠ ᠪᠣᠳᠤᠯ ᠭ ᠠ ᠶᠢᠨ ᠺᠦᠸ ᠠᠳ᠋ ᠢ ᠵᠠᠭᠤ ᠷ ᠢ ᠣ ᠯ᠃ 22 .(10 ᠬᠤᠪᠢ ) ᠦᠭ ᠭᠦᠭᠳᠡᠭ ᠰᠡᠨ ᠨᠢ ᠨᠢ ᠭᠡ ᠲᠣᠭ ᠠᠨ ᠤ ᠺᠦᠸᠠᠳ᠋ ᠢ ᠵᠠᠭᠤᠷ ᠨᠢ ± (𝑎 + 4) ᠂ ᠲᠣᠭ ᠠ ᠪᠣᠳᠤ ᠯᠭ ᠠ ᠶᠢᠨ ᠺᠦᠸ ᠠᠳ᠋ ᠢ ᠵᠠᠭᠤᠷ ᠨᠢ 2𝑎 − 1᠂ ᠡᠨᠡ ᠲᠣᠭ ᠠ ᠶᠢ ᠣᠯ᠃ 23 .(10 ᠬᠤᠪᠢ ) ᠳᠣ ᠣᠷᠠᠬ ᠢ ᠮᠠ ᠲ᠋ᠧᠷ ᠢᠶᠠᠯ ᠢ ᠤᠩ ᠰᠢᠵᠤ ᠠᠰᠠᠭᠤᠯᠲ ᠠ ᠶᠢ ᠬᠠᠷ ᠢᠭᠤᠯ᠄ ᠣᠯᠠᠨ ᠤ ᠮᠡᠳᠡᠬ ᠦ ᠪᠡᠷ √2 ᠨᠢ ᠶᠤᠰᠤ ᠭᠦᠢ ᠲᠣᠭ ᠠ᠂ ᠮᠦ᠋ᠷᠲᠡ ᠭᠡᠨ ᠶᠤᠰᠤᠭ ᠦᠢ ᠲᠣᠭ ᠠ ᠨᠢ ᠬᠢᠵ ᠠᠭᠠᠷ ᠭᠦᠢ ᠦᠯ ᠦ ᠦᠶ ᠡᠯᠡᠬᠦ ᠪᠤᠲᠠᠷ ᠬᠠᠢ᠃ ᠢ ᠢ ᠮᠦ ᠡᠴᠡ √2 ᠶᠢᠨ ᠪᠤᠲᠠᠷ ᠬᠠᠢ ᠶ ᠢᠨ ᠬᠡᠰᠡᠭ ᠢ ᠪ ᠦᠭᠦᠳᠡ ᠶ ᠢ ᠪᠦᠷ ᠪᠢᠴᠢᠵ ᠦ ᠳᠡ ᠢᠯᠬᠦ ᠦᠭᠡᠢ᠂ ᠭᠡᠲᠡᠯ ᠡ √ 2− 1 ᠪᠡᠷ √2 ᠶᠢᠨ ᠪᠤᠲᠠᠷ ᠬᠠᠢ ᠶᠢᠨ ᠬᠡᠰᠡᠭ ᠢ ᠢᠯᠡᠷ ᠬᠡᠢᠯ ᠡᠵᠦ ᠳᠡᠢᠯ ᠤᠨ ᠡ᠃ ᠰᠢᠯᠲᠠᠭ ᠠᠨ ᠨᠢ ᠄ ᠤᠴᠢᠷ ᠨᠢ᠄ √2 ᠶᠢᠨ ᠪᠦᠬ ᠦᠯᠢ ᠲᠣᠭ ᠠᠨ ᠤ ᠬᠡᠰᠡᠭ ᠨᠢ 1᠂ ᠢᠢ ᠮᠦ ᠡᠴᠡ ᠲᠣᠰ ᠲᠣᠭ ᠠᠨ ᠠ ᠴᠠ ᠪᠦᠬᠦᠯᠢ ᠲᠣᠭ ᠠᠨ ᠤ ᠬᠡᠰᠡᠭ ᠢ ᠨᠢ ᠬᠠᠰ ᠤᠵᠤ ᠬᠠᠶᠠᠪᠠ ᠯ ᠪᠤᠲᠠᠷ ᠬᠠᠢ ᠶ ᠢᠨ ᠬᠡ ᠰᠡᠭ ᠨᠢ ᠦ ᠯᠡᠳᠡᠨ ᠡ᠃ ᠦᠭ ᠭᠦᠭᠳᠡᠭ ᠰᠡᠨ ᠨᠢ ᠄ 2 + √6 ᠦᠨ ᠪᠤᠲᠠᠷ ᠬᠠᠢ ᠶᠢᠨ ᠬᠡᠰᠡᠭ ᠨᠢ 𝑎 ᠂ 5 − √6 ᠦᠨ ᠪᠤᠲᠠᠷ ᠬᠠᠢ ᠶ ᠢᠨ ᠬᠡ ᠰᠡᠭ ᠨᠢ 𝑏 ᠂ 𝑎 + 𝑏 ᠦ ᠨ ᠬᠡ ᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠣᠯ᠃ 24.( 10 ᠬ ᠤᠪᠢ ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠦ᠂ ᠲᠠᠪᠤᠨ ᠲᠠ ᠯ ᠠ ᠶᠢᠨ ᠤᠷ ᠲᠤ ᠨᠢ 1 ᠪᠣᠯ ᠬᠤ ᠵᠢᠵᠢᠭ ᠲᠡ ᠭᠰᠢ ᠳᠦ᠋ᠷᠪ ᠡᠯᠵᠢᠨ ᠢᠶᠡᠷ ᠪᠦᠷᠢᠯᠳᠦᠭᠰᠡᠨ ᠴᠠ ᠭᠠᠰᠤᠨ ᠬᠠᠪ ᠲᠠᠰᠤ ᠪᠠᠢᠨ ᠠ᠂ ᠲᠡᠭᠦ ᠨ ᠢ ᠬᠠᠴ ᠢᠯᠠᠵᠤ ᠨᠢ ᠭᠡ ᠲᠡᠭᠰᠢ ᠳᠦ᠋ᠷᠪ ᠡᠯᠵᠢᠨ ᠢ ᠪᠦᠷᠢᠳᠦᠭ ᠦᠯᠦᠨ ᠡ᠃ (1) ᠪᠦᠷᠢᠯᠳ ᠦᠭᠦᠯ ᠦᠭᠰᠡᠨ ᠲᠡᠭᠰᠢ ᠳᠦ᠋ᠷ ᠪᠡᠯᠵ ᠢᠨ ᠦ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ __ __ __ __ _᠂ ᠬᠦᠷᠢᠶᠡᠨ ᠦ ᠤᠷᠲᠤ ᠨᠢ_ __ __ __ __ _᠃ (2) ᠲᠣᠭᠠ ᠨ ᠤ ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠳᠡᠭᠡᠷ ᠡ √ 5, −2 √5 ᠶᠢ ᠢᠯᠡᠷ ᠬᠡᠢᠯ ᠡ᠃ (3) ᠠᠷ ᠪᠠᠨ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠤᠷᠲᠤ ᠨᠢ 1 ᠪᠣᠯᠬᠤ ᠵᠢᠵ ᠢᠭ ᠲᠡᠭᠰᠢ ᠳᠦ᠋ᠷᠪ ᠡᠯᠵᠢᠨ ᠢᠶᠡᠷ ᠨᠢᠭᠡ ᠶᠡᠬᠡ ᠲᠡᠭᠰᠢ ᠳᠦ᠋ᠷᠪ ᠡᠯᠵᠢᠨ ᠢ ᠪᠦᠷᠢ ᠯᠳᠦᠭᠦ ᠯᠵᠦ ᠳᠡᠢ ᠯᠦᠨ ᠡ ᠦ ᠦ︖ ᠬᠡᠷᠪᠡ ᠳᠡᠢᠯ ᠦᠪᠡᠯ ᠰᠡᠲᠥ ᠪ ᠦᠨ ᠵᠢᠷ ᠤᠭ ᠤᠨ ᠠᠷ ᠤ ᠳᠤ ᠵᠢᠷ ᠤ᠂ ᠬᠡᠷᠪᠡ ᠳᠡᠢᠯᠦ ᠬᠦ ᠦᠭᠡᠢ ᠪᠣᠯ ᠰᠢᠯ ᠲᠠᠭᠠᠨ ᠢ ᠨᠢ ᠲᠣᠳᠤᠷ ᠬᠠᠢᠯ ᠠ᠃ 25. (10 ᠬᠤᠪᠢ ) ᠦᠭ ᠭᠦᠭᠳᠡᠭ ᠰᠡᠨ ᠨᠢ ᠲᠣᠭ ᠠᠨ ᠤ ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠳᠡ ᠭᠡᠷ ᠡ ᠴᠡᠭ O ᠨᠢ ᠡᠬ ᠢᠨ ᠦ ᠴᠡᠭ ᠂ ᠴᠡᠭ A ᠶᠢ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡᠬᠦ ᠲᠣᠭ ᠠ ᠨᠢ 9 ᠂ ᠬᠦᠳᠡ ᠯᠬᠦ ᠴᠡᠭ B , C ᠨᠢ ᠲᠣᠭ ᠠᠨ ᠤ ᠲᠡ ᠩᠬᠡᠯ ᠢᠭ ᠳᠡᠭᠡᠷ ᠡ ᠬᠦᠳᠡᠯᠭᠡᠭᠡᠨ ᠬᠢᠬᠦ ᠮᠦ᠋ᠷᠲᠡ ᠭᠡᠨ BC = 2 ᠢ ᠪᠠᠷᠢᠮᠲᠠ ᠯᠠᠨ ᠠ ( ᠴᠡᠭ C ᠨᠢ ᠴᠡᠭ B ᠦᠨ ᠪᠠᠷᠠᠭ ᠤᠨ ᠲᠠᠯ ᠠ ᠳ ᠤ ᠪᠠ ᠢᠨ ᠠ )᠃ ᠴᠡᠭ B ᠦᠨ ᠢᠯᠡᠷ ᠬᠡᠢᠯ ᠡᠬᠦ ᠲᠣᠭ ᠠ ᠶᠢ 𝑚 ᠭᠡᠪᠡ ᠯ᠄ (1) ᠵᠢᠷ ᠤᠭ1 ᠮᠡᠲᠦ᠂ ᠴᠡᠭ B , C ᠨᠢ ᠬᠡᠷᠴᠢᠮ OA ᠳᠡᠭᠡᠷ ᠡ ᠬᠦ ᠳᠡᠯᠭᠡᠭᠡᠨ ᠬᠢᠬᠦ ᠦᠶ ᠡᠰ᠄ ① ᠬᠡᠷ ᠪᠡ ᠴᠡᠭ B ᠨᠢ OA ᠶᠢᠨ ᠳᠤᠮᠳᠠ ᠴᠡᠭ ᠪᠣᠯ᠂ AC = ___ ___ ___ ② ᠬᠡ ᠷᠪᠡ ᠴᠡᠭ B , C ᠨᠢ ᠬᠦᠳᠡᠯ ᠭᠡᠭᠡᠨ ᠬᠢᠭᠡᠳ ᠶᠠᠮᠠ ᠷ ᠨᠢ ᠭᠡ ᠪᠠᠢᠷ ᠢᠨ ᠳ ᠤ ᠬᠦᠷᠬ ᠦ ᠦᠶ ᠡᠰ᠂ ᠶᠠᠭ ᠰᠠᠢᠬᠠᠨ AC = OB ᠪᠣᠯ ᠪᠠᠯ᠂ 𝑚 ᠦᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠣᠯ᠃ (2) ᠬᠡ ᠷᠴᠢᠮ BC ᠨᠢ ᠰᠠᠴᠤᠷᠠ ᠭAO ᠤᠨ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ ᠬ ᠦᠳᠡᠯ ᠭᠡᠭᠡᠨ ᠬᠢᠬᠦ ᠦᠶ ᠡᠰ᠂ ᠬᠡᠷᠪᠡ AC − OB = 13 AB ᠨᠢ ᠪᠠᠢᠭ ᠤᠯᠤᠭᠳᠠᠪ ᠠᠯ᠂ ᠨᠦ᠋ ᠬᠦᠴᠡᠯ ᠳᠦ ᠨᠡᠢᠴᠡᠬ ᠦ 𝑚 ᠶᠢ ᠣᠯ᠃ ᠵᠢᠷᠤᠭ1 ᠪᠡᠯ ᠡᠳᠬᠡ ᠯ ᠵᠢᠷᠤ ᠭ
ᠮᠣᠩᠭᠣᠯ ᠳ᠋ᠣᠺ ᠤᠨ ᠬᠥᠮᠥᠷᠭᠡ