ᠠᠷᠪᠠᠨ ᠵᠢᠷᠭᠤᠳᠤᠭᠠᠷ ᠪᠦᠯᠦᠭ ᠬᠤᠶᠠᠳᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠶᠢᠨ ᠢᠵᠠᠭᠤᠷ ᠤᠨ ᠢᠯᠡᠷᠭᠡᠢᠯᠡᠯ （ᠨᠢᠭᠡᠴᠢ ᠶᠢᠨ ᠪᠠᠢᠴᠠᠭᠠᠯᠲᠠ 2） .pdf

ᠠᠷᠪᠠ ᠨ᠊ ᠵᠢᠷᠭᠤᠳᠤ ᠭᠠᠷ ᠪᠥᠯ ᠥᠭ ᠦᠨ ᠪᠠᠢ ᠴᠠᠭᠠᠯᠲᠠ ᠠᠩᠭᠢ ᠪᠥᠯ ᠥᠭ᠄ ᠨᠡᠷ ᠡ᠄ ︵90 ᠢ ᠨᠦ᠋ᠲ᠂ ᠨᠠᠢᠲᠡ 02 1 ᠢᠪᠤᠬ︶ ᠨᠢᠭᠡ᠂ ᠰᠪᠩᠭᠪᠬ ᠤ ᠰᠡ ᠲᠪᠪ᠃ ︵ᠰᠡᠳ᠋ᠪᠪ ᠪᠦ ᠷᠢ 3 ᠢᠪᠤᠬ᠂ ᠨᠠᠢᠲᠡ 31 ᠢᠪ ᠤᠬ︶ 1. ᠢᠣᠶ ᠠᠳᠪᠭ ᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠢᠵᠠᠭ ᠪᠷ ᠦᠨ ᠢᠯᠡᠷ ᠬᠡᠶᠢᠯᠡᠯ √2 𝑥− 4 ᠢ ᠤᠳᠬ ᠠ ᠲᠠ ᠢ ᠪᠣᠯᠭ ᠠᠬᠤ 𝑥 ᠦ ᠨ᠊ ᠬᠡ ᠮᠵᠢᠭᠳᠡᠯ ᠠᠤᠬᠤ ᠬᠡ ᠪᠴᠢᠶ ᠡ ᠨᠢ ︵︶ A. 𝑥 2 B. 𝑥 2 C. 𝑥≥ 2 D. 𝑥 ≤ 2 2. ᠳᠣᠣᠷ ᠠᠬᠢ ᠪᠣᠳᠪ ᠯᠭ ᠠ ᠠᠴᠠ ᠵᠦ᠋ᠪ ᠪᠣᠯ ᠬᠤ ᠨᠢ ︵︶ A. 2√ 3+ 3√ 3= 5√ 6 B. ︵√ 2+ 1︶ ︵1 − √2︶ = 1 C. − ︵−𝑎︶ 4 ÷ 𝑎 2 = 𝑎 2 D . ︵𝑥𝑦︶ − 1 ︵12 𝑥𝑦︶ 2 = 14 𝑥𝑦 3. 𝑎≤ 12 ᠦᠶ ᠡᠰ᠂ √1 − 4𝑎 + 4𝑎 2+ |2𝑎 − 1| ᠢᠢ ᠬᠢᠯᠤᠠ ᠷᠴᠢᠯ ᠠᠤᠠᠯ ︵︶ A. 2 B. 2− 4𝑎 C. 𝑎 D. 0 4. ᠢᠣᠶ ᠠᠳᠪᠭ ᠠᠷ ᠵᠡ ᠷᠭᠡ ᠢᠢᠨ ᠢᠵᠠᠭ ᠪᠷ ᠦ ᠨ᠊ ᠢ ᠯᠡᠷᠬᠡᠶᠢᠯᠡᠯ √− 𝑎 3 ᠢ ᠬᠢ ᠯᠤᠠᠷ ᠴᠢᠯᠠᠤ ᠠᠯ ︵︶ A. −𝑎 √𝑎 B. −𝑎 √− 𝑎 C. 𝑎√ −𝑎 D. 𝑎√ 𝑎 5. ᠳᠣᠣᠷ ᠠᠬᠢ ᠬᠡᠯᠡᠯᠭᠡ ᠠᠴᠠ ᠵᠦ᠋ᠪ ᠪᠣᠯᠬᠤ ᠨ᠊ ᠢ ︵︶ A. ᠠᠯ ᠭᠧᠹᠷ ᠠ ᠢᠢ ᠨ᠊ ᠢ ᠯᠡᠷᠬᠡᠶᠢᠯᠡᠯ 1𝜋 ︵𝑥 2 + 𝑦 2︶ ᠨᠢ ᠤ ᠤᠲᠠᠷ ᠬᠠᠢ ᠢ ᠯᠡᠷᠬᠡᠶᠢᠯᠡᠯ B. ᠬᠢᠵ ᠠᠭᠠᠷ ᠲᠠᠢ ᠠ ᠷᠤᠠᠲᠤ ᠢᠢᠨ ᠪᠤᠲᠠᠷ ᠬᠠᠢ ᠪᠠ ᠬᠢᠵᠠᠭ ᠠᠷᠭᠦᠢ ᠦᠯ ᠦ ᠦ ᠶᠡᠯᠠᠬ ᠦ ᠠ ᠷᠤᠠᠲᠤ ᠢᠢᠨ ᠪᠤᠲᠠᠷ ᠬᠠᠢ ᠢᠢ ᠢᠠᠮᠳ᠋ᠪᠷᠠ ᠭᠪᠯᠪᠭᠠ ᠳ᠋᠊ ᠤ ᠣᠳᠠᠲᠤ ᠲ᠊ ᠣᠭ ᠠ ᠭᠡᠨ ᠡ C. √5 𝑎 ᠤ ᠠ √ 0.2 𝑎 ᠨᠢ ᠢ ᠵᠢᠯᠢ ᠲᠦ᠋ ᠷᠦᠯ ᠦᠨ ᠢᠣᠶ ᠠᠳᠪᠭ ᠠᠷ ᠵᠡ ᠷᠭᠡ ᠢᠢᠨ ᠢᠵ ᠠᠭᠪᠷ ᠦᠨ ᠢᠯᠡᠷ ᠬᠡᠶᠢᠯᠡᠯ D. ᠬᠡᠷᠤᠡ ᠴᠡᠭ M ︵1 − 𝑥 , 1− y︶ ᠨᠢ ᠭᠪᠷᠤ ᠠᠳᠪᠭ ᠠᠷ ᠮᠦ᠋ᠴᠡ ᠲᠤ ᠣᠷᠣᠰ ᠢᠤᠠᠯ᠂ ᠴᠡ ᠬ᠋᠊ N ︵−𝑥 ,− 𝑦︶ ᠨᠢ ᠨᠢᠭᠡᠳᠦᠭᠡᠷ ᠮᠥᠴ ᠡ ᠲᠤ ᠣᠷ ᠣᠰᠢᠨ ᠠ 6. ᠬᠡᠷᠤᠡ ᠪᠣᠳᠠᠲᠤ ᠲᠤᠭ ᠠ 𝑎 , 𝑏 ᠨᠢ ᠢᠯᠡᠷ ᠬᠡᠶᠢᠯᠡᠯ √𝑎 2𝑏 3= −𝑎𝑏 √𝑏 ᠢ ᠢᠠᠩᠭ ᠠᠤᠠᠯ᠂ ᠴᠡᠭ ︵𝑎 ,𝑏︶ ᠨᠢ ︵︶ ᠣ ᠷᠣᠰᠢ ᠨ᠊ ᠠ᠃ A. ᠨᠢᠭᠡᠳᠦᠭᠡᠷ ᠮᠦ᠋ᠴᠡ B. ᠢᠣᠶ ᠠᠳᠪᠭ ᠠᠷ ᠮᠥᠴᠡ C. ᠢᠣᠶ ᠠᠳᠪᠭ ᠠᠷ ᠮᠥᠴᠡ ᠪᠤᠶ ᠤ ᠪᠠ ᠶᠢᠷᠢᠰ ᠢᠯ ᠦᠨ ᠲᠡᠩᠬᠡ ᠯᠢᠭ D. ᠳᠦ᠋ᠷᠤ ᠡᠳᠦᠭᠡᠷ ᠮᠥ ᠴᠡ ᠪᠤᠶᠤ ᠪᠠᠶᠢᠷ ᠢᠰᠢᠯ ᠦᠨ ᠲᠡᠩᠬᠡ ᠯᠢᠭ 7. 𝑎 , 𝑏 , 𝑐 ᠨᠢ ᠲᠪᠰ ᠪᠦᠷᠢ ∆AB C ᠢᠢᠨ ᠭᠪᠷ ᠤᠠᠨ ᠲᠠᠯ ᠠ ᠪᠣᠯ᠂ √ ︵𝑎− 𝑏+ 𝑐︶ 2+ √ ︵𝑎− 𝑏− 𝑐︶ 2= ︵︶ A. 2𝑎 − 2𝑏 B. 2 𝑏− 2𝑎 C. 2c D . − 2c 8. ᠢᠯᠡᠷ ᠬᠡᠶᠢᠯᠡᠯ √− 12𝑏 ᠢᠢ ᠬᠢᠯᠤᠠ ᠷᠴᠢᠯ ᠠᠭᠰᠠᠨ ᠠᠴᠠ ᠵᠦ᠋ᠪ ᠪᠣᠯᠬᠤ ᠨᠢ ︵︶ A. 2√ 3𝑏𝑏 B. − 2√ −3 𝑏 𝑏 C. − 2√ 3𝑏𝑏 D. 2√ −3 𝑏 𝑏 9. ᠪᠣᠳᠠᠲᠤ ᠲᠤᠭ ᠠ 𝑎 , 𝑏 ᠢᠢᠨ ᠲᠤᠭ ᠠᠨ ᠤ ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠢᠠᠷᠠᠭ ᠠᠯᠵᠠᠭᠰᠠᠨ ᠴᠡᠭ ᠨᠢ ᠵᠢᠷ ᠪᠭ ᠮᠡᠲᠦ᠂ ᠲᠡᠭᠡᠤ ᠡᠯ √ 𝑎+ √− 𝑏 ᠨᠢ ︵︶ A. ᠰᠥ᠋ ᠭᠡᠷᠬᠦ ᠪᠤᠰ ᠤ ᠲᠤᠭ ᠠ B. ᠡᠶᠡᠷ ᠬᠦ ᠲᠤᠭ ᠠ C. ᠰᠦ᠋ ᠭᠡᠷᠬᠦ ᠲᠤᠭ ᠠ D. ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠢᠠᠷᠢᠭ ᠪᠯᠲᠠ ᠴᠦ᠋ᠮ ᠪᠤᠷ ᠪᠭᠤ 10. 𝑚 ,𝑛 ᠨᠢ ᠳᠠ ᠷᠠᠭᠠᠯᠠᠭᠰᠠᠨ ᠢᠣᠶ ᠠᠷ ᠤ ᠠᠶᠢᠭ ᠠᠯᠢ ᠲᠤᠭ ᠠ ︵𝑚 𝑛︶ ᠪᠦᠭᠡᠳ q = 𝑚𝑛 . ᠬᠡ ᠷᠤᠡ p= √𝑞 + 𝑛+ √𝑞 − 𝜋 ᠪᠣᠯ᠂ p ᠨᠢ ︵︶ A. ᠰᠪᠨᠳᠪᠭᠪᠢ ᠲ᠊ ᠣᠭ ᠠ B. ᠲᠡᠭᠰᠢ ᠲᠤᠭ ᠠ C. ᠵᠠᠷᠢᠮ ᠲᠤ ᠪᠠᠨ ᠰᠪᠨᠳᠪᠭ ᠪᠢ ᠲ᠊ ᠣᠭ ᠠ᠂ ᠵᠠᠷ ᠢᠮ ᠲᠤ ᠪᠠᠨ ᠲᠡ ᠭᠰᠢ ᠲᠤᠭ ᠠ D. ᠵᠠᠷᠢᠮ ᠲᠤ ᠪᠠᠨ ᠶᠪᠰ ᠪᠲᠠᠢ ᠲᠤᠭ ᠠ᠂ ᠵᠠᠷ ᠢᠮ ᠲᠤ ᠤ ᠠᠨ ᠶᠪ ᠰᠪᠭᠦᠢ ᠲᠤᠭ ᠠ ᠢᠣᠶ ᠠᠷ᠂ ᠢᠠᠭ ᠠᠯᠲᠠ ᠲᠤ ᠲᠠᠭᠯ ᠠᠬᠤ ᠰᠡᠳ᠋ᠪᠪ᠃ ︵ᠰᠡᠳ᠋ᠪᠪ ᠪᠦᠷᠢ 4 ᠢᠪᠤᠬ᠂ ᠨᠠ ᠢᠲᠡ 24 ᠢᠪᠤᠬ︶ 11. ᠠᠯ ᠭᠧᠹᠷ ᠠ ᠢᠢᠨ ᠢᠯ ᠡᠷᠬᠡᠶᠢᠯᠡᠯ √𝑥 − 1+ 2𝑥−2 ᠨᠢ ᠪᠣᠳᠠᠲᠤ ᠲᠤᠭ ᠠᠨ ᠤ ᠬᠡᠪ ᠴᠢᠶᠡᠨ ᠲᠤ ᠤᠳ ᠬ᠊ ᠠ ᠲᠠᠢ ᠪᠣᠯ᠂ 𝑥 ᠦᠨ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠠᠤᠬᠤ ᠬᠡᠪ ᠴᠢᠶ ᠡ ᠨᠢ ︵︶ 12. ᠰᠢᠨ ᠡ ᠤ ᠣᠳᠪᠯ ᠬ᠋᠊ ᠠ ᠢᠢᠨ ᠲᠤᠷ ᠢᠮ ︽@︾ ᠢᠢ x@y = √𝑥𝑦 + 4 ᠭᠡᠵᠤ ᠲᠣᠭᠲᠠᠭ ᠠᠤᠠᠯ᠂ 2@ 6= ︵︶ 13. 2 + √9 + 𝑥 2 ᠦᠨ ᠢᠠᠮᠪᠭ ᠦᠨ ᠪᠠᠭ ᠠ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠨᠢ ︵︶ 14. 𝑥 = ︵︶ ᠦᠶ ᠡᠰ᠂ ᠢᠠᠮᠪᠭ ᠬᠢᠯᠤᠠ ᠷ᠊ ᠢᠣ ᠶᠠᠳᠪᠭ ᠠᠷ ᠵᠡ ᠷᠭᠡ ᠢᠢᠨ ᠢᠵᠠᠭ ᠪᠷ ᠦᠨ ᠢᠯᠡᠷ ᠬᠡᠶᠢᠯᠡᠯ √𝑥 2+ 3 x ᠪᠠ √𝑥 + 15 ᠨᠢ ᠢᠵ ᠢᠯᠢ ᠲᠦ᠋ ᠷᠦᠯ ᠦᠨ ᠢᠵᠠ ᠭᠪᠷ ᠦᠨ ᠢᠯᠡᠷ ᠬᠡᠶᠢᠯᠡᠯ ᠪᠣᠯ ᠪᠨ ᠠ᠃ 15. ᠬᠡᠷ ᠤᠡ 𝑎 + 1𝑎 = 7 ᠪᠣᠯ᠂ 𝑎 2+ 1𝑎 2 + √𝑎 + 1√𝑎 = ︵︶ 16. ᠬᠡᠷ ᠤᠡ √ 2 𝑚 +𝑛 −2 ᠪᠠ √3 3𝑚 −2 𝑛+ 2 ᠨᠢ ᠴᠥᠮ ᠢᠠᠮ ᠪᠭ ᠬᠢᠯᠤᠠ ᠷ᠊ ᠢᠣᠶ ᠠᠳᠪᠭ ᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠢᠵ ᠠᠭᠪᠷ ᠦᠨ ᠢᠯᠡᠷ ᠬᠡᠶᠢᠯᠡᠯ ᠪᠣᠯ᠂ 𝑚 = ︵︶ ,𝑛 = ︵︶ . ᠭᠪᠷᠤ ᠠ᠂ ᠪᠣᠳᠣᠵ ᠤ ᠢᠶᠠᠷ ᠢᠭᠪᠯ ᠬᠤ ᠰᠡ ᠲᠪᠪ᠃ ︵ᠨᠠᠢᠲᠡ 66 ᠢᠪᠤᠬ︶ 17. ︵0 1 ᠢᠪᠤᠬ︶ ᠪᠣᠳᠠᠲᠤ ᠲᠤᠭ ᠠ 𝑎 ,𝑏 ᠢᠢᠨ ᠲᠤᠭ ᠠᠨ ᠲᠡᠩᠬᠡ ᠯᠢᠭ ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠪᠠᠶᠢᠷ ᠢ ᠨᠢ ᠵᠢᠷᠪᠭ ᠮᠡᠲᠦ ᠪᠣᠯ᠂ ᠳᠣᠣᠷᠠ ᠬᠢ ᠢ ᠯᠡᠷᠬᠡᠶᠢᠯᠡᠯ ᠢ ᠬᠢᠯ ᠤᠡᠷᠴᠢᠯ ᠠ᠃ √ ︵𝑎+ √3︶ 2 − √ ︵𝑏− 2√ 3︶ 2− √𝑎 2− 2𝑎𝑏 + 𝑏 2 18. ︵0 1 ᠢᠪᠤᠬ︶ ᠬᠡᠷ ᠤᠡ y = √𝑥 − 8+ √8 − 𝑥+ 9 ᠪᠣᠯ᠂ 𝑥− 𝑦 √𝑥 −√ 𝑦 ᠢᠢᠨ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠢ ᠣ ᠯ᠊᠃ 19 . ︵0 2 ᠢᠪᠤᠬ︶ ᠬᠡ ᠷᠤᠡ 9− √11 ᠪᠠ 9+ √11 ᠢᠢᠨ ᠪᠤᠲᠠᠷ ᠬᠠᠢ ᠢᠢᠨ ᠬᠡᠰᠡᠭ ᠨᠢ ᠲᠪᠰ ᠪᠦᠷ ᠢ 𝑎 ,𝑏 ᠪᠣᠯ᠂ 𝑎+ 𝑏 𝑎− 𝑏 ᠦ ᠨ᠊ ᠬᠡ ᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠣ ᠯ᠊᠃ 20. ︵0 2 ᠢᠪᠤᠬ︶ ᠠᠳᠠ ᠯᠢ ᠶᠢᠨ ᠵᠠᠭᠪᠲᠤ ᠭᠪᠷᠤ ᠠᠯᠵᠢ ᠨ᠊ ᠤ ᠢᠠᠵᠠᠭᠤ ᠢᠢ ᠨ᠊ ᠤ ᠷᠲᠤ ᠨᠢ 𝑎᠂ ᠰ᠊ ᠠᠭᠪᠷᠢ ᠢᠢᠨ ᠤᠷᠲᠤ ᠨᠢ 𝑏᠂ ᠰᠠᠭ ᠪᠷᠢ ᠳᠡ ᠭᠡᠷᠠᠬ ᠢ ᠦᠨᠳᠦ ᠷ᠊ ᠨᠢ ℎ ᠪᠣᠯ᠂ ︵1︶ ᠬᠡ ᠷᠤᠡ 𝑎= 6+ √3 ,𝑏 = 6+ 4√ 3 ᠪᠣᠯ᠂ ℎ ᠢᠢᠨ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠢ ᠣᠯ᠃ ︵2︶ ᠬᠡ ᠷᠤᠡ 𝑏= 2 ︵2√ 7+ 1︶ ,ℎ = 2√ 7− 1 ᠪᠣᠯ᠂ 𝑎 ᠢᠢᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠣ ᠯ᠊᠃ 21. ︵02 ᠢᠪᠤᠬ︶ 𝑥 ,𝑦 ᠢᠢᠨ ᠲᠪᠬᠠᠢ ᠳᠡᠭᠰᠢ ᠲᠭᠡᠯ ᠦᠨ ᠪᠦᠯᠦ ᠬ᠋᠊ { 3 𝑥+ 2𝑦 = 1 ① 3𝑥 − 2𝑦 = 𝑚 ② ᠦ ᠨ᠊ ᠢᠶᠠᠷ ᠢᠭᠤ ᠨᠢ ᠴ᠊ ᠦᠮ 1 ᠠᠴᠠ ᠶᠡᠭᠡ ᠪᠤᠰᠤ᠂ ︵1︶ 𝑚 ᠢᠢᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠠᠤᠬᠤ ᠬᠡᠪ ᠴᠢᠶ ᠡ ᠶᠢ ᠣᠯ᠃ ︵2︶ √𝑥 2− 2𝑥 + 1+ √𝑦 2− 2𝑦 + 1+ |𝑚 + 3| + |𝑚 − 5| −| 𝑥+ 𝑦− 2| ᠢ ᠬᠢᠯ ᠤᠠᠷᠴᠢᠯ ᠠ᠃ 22. ︵0 4 ᠢᠪᠤᠬ︶ ᠳᠣᠣᠷ ᠠᠬᠢ ᠪᠣᠳᠪ ᠯᠭ ᠠ ᠢᠢ ᠦᠨᠳᠦ ᠰᠦᠯᠡᠨ ᠭᠦᠢᠴᠡᠳᠬᠡ᠄ 1 1+ √2 = 1 × ︵√ 2− 1︶ ︵√ 2+ 1︶ ︵√ 2− 1︶ = √2 − 1 1 √3 +√ 2 = √3 −√ 2 ︵√ 3+ √2︶ ︵√3 −√ 2︶ = √3 − √2 1 √5 +2 = √5 −2 ︵√ 5+ 2︶ ︵√ 5− 2︶ = √5 − 2 ︵0︶ 1 √7 +√ 6 ᠦᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠣ ᠯ᠊᠃ ︵2︶ 1 3√ 2+ √17 ᠦ ᠨ᠊ ᠬᠡ ᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠣ ᠯ᠊᠃ ︵3︶ 1 1+ √2 + 1 √2 +√ 3 + 1 √3 +√ 4 + ⋯ + 1 √𝑛 +√ 𝑛+ 1 ︵n ᠨᠢ ᠡᠶᠡᠷ ᠬᠦ ᠪᠦᠬᠦᠯᠢ ᠲᠤᠭ ᠠ︶ ᠢᠢᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠢ ᠣ ᠯ᠊᠃