ᠬᠤᠷᠢᠨ ᠨᠢᠭᠡᠳᠤᠭᠡᠷ ᠪᠦᠯᠦᠭ ᠨᠢᠭᠡ ᠦᠯᠦ ᠮᠡᠳᠡᠭᠳᠡᠭᠴᠢᠲᠡᠢ ᠬᠤᠶᠠᠳᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠶᠢᠨ ᠲᠡᠭᠰᠢᠳᠭᠡᠯ （ᠨᠢᠭᠡᠴᠢ ᠶᠢᠨ ᠪᠠᠢᠴᠠᠭᠠᠯᠲᠠ 2） .pdf

ᠬᠣ ᠷᠢᠨ ᠨᠢᠭᠡᠳ ᠤᠭᠡᠷ ᠪ ᠥᠯᠥ ᠭ ᠦᠨ ᠪᠠ ᠢᠴᠠ ᠭᠠᠯᠲ ᠠ ᠠᠩᠭᠢ ᠪᠥᠯᠥᠭ᠄ ᠨᠡᠷ ᠡ ᠄ ︵90 ᠮ ᠢᠨᠦ᠋ᠲ ᠂ ᠨᠠᠢ ᠲᠡ 120 ᠬᠤ ᠪᠢ ︶ ᠨᠢᠭᠡ ᠂ ᠰ ᠤᠩᠭᠤ ᠬᠤ ᠰ ᠡᠲᠤᠪ ᠃ ︵ ᠰᠡᠲ ᠤᠪ ᠪ ᠦᠷᠢ 3 ᠬᠤᠪᠢ ᠂ ᠨᠠᠢ ᠲᠡ 30 ᠬᠤ ᠪᠢ ︶ 1. 𝑥= 1 ᠨᠢ ᠬᠣᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠳᠡᠭ ᠰᠢᠳ ᠭᠡᠯ (𝑚 2− 1) 𝑥 2 − 𝑚𝑥 + 𝑚 2= 0 ᠦᠨ ᠨ ᠢᠭᠡ ᠢᠵ ᠠᠭᠤᠷ ᠪᠣᠯ ᠂ 𝑚 ᠢᠢᠨ ᠬᠡᠮᠵ ᠢᠭᠳ᠋ᠡᠯ ᠨ ᠢ ( ) A. 12 ᠪᠤᠶᠤ −1 B. − 12 ᠪ ᠤᠶᠤ 1 C. 12 ᠪᠤᠶᠤ 1 D. − 12 2. 𝑥 2= −2 ᠦᠨ ᠳᠣ ᠬᠠᠢ ᠬᠡᠯᠡᠯ ᠭᠡ ᠡ ᠴᠡ ᠵᠦ᠋ᠪ ᠪᠣᠯ ᠬᠤ ᠨ ᠢ ( ) A. 𝑥 2≥ 0 ᠪᠣᠯᠬᠤ ᠤᠴᠢ ᠷ ᠡ ᠴᠠ 𝑥 2 ᠨᠢ −2 ᠲ ᠠᠢ ᠲ ᠡᠩᠴ ᠡᠬᠦ ᠦᠭᠡᠢ ᠂ ᠡᠶᠢᠮᠦ ᠡᠴᠡ ᠡᠨᠡ ᠨ ᠢ ᠳᠡᠭ ᠰᠢᠳ ᠭᠡᠯ ᠪᠢᠰᠢ B. 𝑥 2= −2 ᠨᠢ ᠳᠡᠭᠰ ᠢᠳᠭᠡᠯ ᠂ ᠭᠡᠲᠡᠯ ᠡ ᠨ ᠢᠭᠡᠳ᠋ ᠦᠭᠡᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠬᠡᠰ ᠢᠭᠦᠨ ᠦᠭᠡᠢ ᠣᠴ ᠢᠷ ᠡᠴᠠ ᠨ ᠢᠭᠡ ᠦᠯ ᠦ ᠮᠡᠳ᠋ ᠡᠭᠳ᠋ᠡᠭᠴᠢ ᠲᠡᠢ ᠬᠣ ᠶᠠᠳ᠋ᠤ ᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠳᠡᠭ ᠰᠢᠳ ᠭᠡᠯ ᠪ ᠢᠰᠢ C. 𝑥 2= −2 ᠨᠢ ᠨ ᠢᠭᠡ ᠦᠯ ᠦ ᠮᠡ ᠳ᠋ᠡᠭᠳ᠋ ᠡᠭᠴᠢ ᠲᠡᠢ ᠬᠣ ᠶᠠᠳ᠋ᠤ ᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠳᠡᠭᠰ ᠢᠳᠭᠡᠯ ᠪᠦᠭᠡ ᠳ ᠬᠠᠷᠢ ᠭᠤ ᠲ ᠠᠢ ᠃ D. 𝑥 2= −2 ᠨᠢ ᠨᠢᠭᠡ ᠦᠯ ᠦ ᠮᠡᠳ᠋ ᠡᠭᠳ᠋ᠡᠭᠴᠢ ᠲᠡᠢ ᠬᠣᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠳᠡᠭ ᠰᠢᠳ ᠭᠡᠯ ᠭᠡᠪᠡᠴ ᠦ ᠬᠠᠷᠢ ᠭᠤ ᠦᠭᠡᠢ 3. ᠲᠣᠮᠢ ᠶᠠᠨ ᠤ ᠠ ᠷᠭ ᠠ ᠪᠠᠷ ᠨᠢᠭᠡ ᠦᠯ ᠦ ᠮᠡᠳ᠋ ᠡᠭᠳ᠋ᠡᠭᠴᠢ ᠲᠡᠢ ᠬᠣᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠳᠡᠭ ᠰᠢᠳ ᠭᠡᠯ ᠦᠨ ᠬᠠᠷᠢ ᠭᠤ ᠢ ᠢ ᠣ ᠯᠬᠤ ᠦᠶ ᠡᠰ ᠂ ᠤ ᠷᠢᠳ᠋ᠠᠪᠡᠷ 𝑎 , b, c ᠢ ᠢᠨ ᠬᠡᠮᠵᠢ ᠭᠳ᠋ᠡᠯ ᠢ ᠳᠣᠭᠲᠠ ᠭᠠᠨ ᠠ ᠂ ᠳᠡᠭᠰ ᠢᠳᠭᠡᠯ −4 𝑥 2 + 3= 5𝑥 ᠦᠨ ᠳᠣ ᠬᠠᠢ ᠳᠣᠣ ᠷᠠᠬᠢ ᠬᠡᠯᠡᠯ ᠭᠡ ᠡ ᠴᠡ ᠵᠦ᠋ᠪ ᠪᠣᠯ ᠬᠤ ᠨ ᠢ ( ) A. 𝑎 = −4 , b = 5 , c = 3 B. 𝑎 = −4 , b = −5 , c = 3 C. 𝑎 = 4 , b = 5 , c = 3 D. 𝑎 = 4 , b= −5 , c = −3 4. ᠳᠡᠭᠰ ᠢᠳᠭᠡᠯ (𝑥 2− 3) 2− 5( 3− 𝑥 2 )+ 2= 0 ᠳ᠋ᠤ ᠂ ᠬᠡᠷᠪᠡ 𝑥 2 − 3= y ᠂ ᠤᠭ ᠳᠡᠭ ᠰᠢᠳ ᠭᠡᠯ ᠢ ᠬᠤ ᠪᠢᠷᠠᠭᠠᠪ ᠠᠯ ( ) A. 𝑦 2− 5𝑦 + 2= 0 B. 𝑦 2+ 5𝑦 − 2= 0 C. 𝑦 2− 5𝑦 − 2= 0 D. 𝑦 2+ 5𝑦 + 2= 0 5. ᠨᠢᠭᠡ ᠬᠣ ᠶᠠᠷ ᠣᠷᠤᠨᠲ ᠤ ᠲ ᠣᠭ ᠠ ᠨᠢ 𝑎 ᠂ ᠨᠢ ᠭᠡ ᠭᠤᠷᠪᠠ ᠨ ᠤ ᠷᠤᠨᠲ ᠤ ᠲ ᠣᠭ ᠠ ᠨᠢ b ᠪᠠ ᠶᠢᠨ ᠠ ᠂ ᠬᠡᠷᠪᠡ ᠬᠣᠶ ᠠᠷ ᠤ ᠷᠤᠨᠲ ᠤ ᠲ ᠣᠭ ᠠ ᠢᠢ ᠭᠤᠷᠪᠠ ᠨ ᠤ ᠷᠤᠨᠲ ᠤ ᠲ ᠣᠭᠠᠨ ᠤ ᠡᠮᠦᠨ ᠡ ᠨ ᠢ ᠲ ᠠᠯᠪᠢ ᠪᠠᠯ ᠨᠢᠭᠡᠨ ᠲᠠᠪ ᠤᠨ ᠤᠷᠤᠨᠲ ᠤ ᠲ ᠣᠭ ᠠ ᠪᠣᠯ ᠤᠨ ᠠ ᠃ ᠲ ᠡᠭᠡᠪᠡᠯ ᠡᠨᠡ ᠲᠠᠪᠤ ᠨ ᠤᠷᠤ ᠨᠲᠤ ᠲᠣᠭ ᠠ ᠢᠢ ( ) ᠢᠯ ᠡᠷᠬᠡᠶ ᠢᠯᠡᠨ ᠡ ᠃ A. 10 𝑎+ b B. 100 𝑎+ b C. 1000 𝑎+ b D. 𝑎 + b 6. ᠬᠡᠷᠪᠡ 𝑡 ᠨᠢ ᠨ ᠢᠭᠡ ᠦᠯ ᠦ ᠮ ᠡᠳ᠋ᠡᠭᠳ᠋ ᠡᠭᠴᠢ ᠲᠡᠢ ᠬᠣᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠳᠡᠭ ᠰᠢᠳ ᠭᠡᠯ 𝑎𝑥 2+ 𝑏𝑥 + 𝑐= 0 ( 𝑎≠ 0) ᠦᠨ ᠢᠵᠠᠭᠤᠷ ᠪᠣᠯ ᠂ ᠢᠵᠠᠭᠤᠷ ᠦᠨ ᠢᠯᠭᠠᠭᠴ ᠢ ᠢᠯ ᠡᠷᠬᠡᠶ ᠢᠯᠡᠯ ∆= 𝑏 2 − 4𝑎𝑐 ᠪᠠ ᠭᠦᠢᠴ ᠡᠳ ᠺᠦᠧᠠᠳ᠋ ᠦᠨ ᠢᠯᠡᠷᠬ ᠡᠶᠢᠯᠡᠯ M = (2 𝑎+ 𝑏) 2 ᠦᠨ ᠬᠣᠭᠣ ᠷᠤᠨᠳ᠋ ᠤᠬᠢ ᠬᠣᠯ ᠪᠤᠭᠳ᠋ ᠠᠯ ᠨ ᠢ ( ) A. ∆ M B. ∆ － 74 B. k ≥ － 74 ᠪᠦᠭᠡᠳ k ≠ 0 C. k ≥ － 74 D. k 74 10. ᠳᠡᠭ ᠰᠢᠳ ᠭᠡᠯ 𝑎 3 − 5𝑎 2+ 3𝑎 = 0 ᠦᠨ ᠭᠤᠷᠪᠠ ᠨ ᠢᠵᠠᠭᠤᠷ ᠨᠢ ᠲᠤᠰ ᠪᠦ ᠷᠢ 𝑎1 ,𝑎 2 , 𝑎3 ᠪᠣᠯ᠂ 𝑎1 (𝑎 2 + 𝑎3 )+ 𝑎2 (𝑎 1 + 𝑎3 )+ 𝑎3 (𝑎 1 + 𝑎2 )ᠦᠨ ᠬᠡᠮᠵᠢ ᠭᠳ᠋ᠡᠯ ᠨ ᠢ ( ) A. − 5 B. 6 C. − 6 D. 3 ᠬᠣᠶ ᠠᠷ᠂ ᠬᠠᠭᠠ ᠯᠲᠠ ᠳ᠋ ᠤ ᠲ ᠠᠭᠯᠠᠬᠤ ᠰᠡᠲᠤ ᠪ᠃ ︵ᠰᠡᠲ ᠤᠪ ᠪ ᠦᠷᠢ 3 ᠬᠤᠪᠢ ᠂ ᠨᠠᠢ ᠲᠡ 24 ᠬᠤᠪᠢ ︶ 11. ᠭᠡᠷᠪᠡ 𝑥 ᠦᠨ ᠲᠤᠬᠠᠢ ᠨᠢ ᠭᠡ ᠦᠯ ᠦ ᠮ ᠡᠳ᠋ᠡᠭᠳ᠋ ᠡᠭᠴᠢ ᠳ᠋ᠡᠢ ᠬᠤᠶ ᠠᠳ᠋ᠤᠭ ᠠᠷ ᠵ ᠡᠷᠭᠡ ᠢᠢᠨ ᠲᠡᠭ ᠰᠢᠳ ᠭᠡᠯ (𝑚 − 1) 𝑥 2 + 𝑥+ m 2− 1= 0 ᠦᠨ ᠨ ᠢᠭᠡ ᠢᠵ ᠠᠭᠤᠷ ᠨᠢ 0 ᠪ ᠤᠯ 𝑚 = ( ) 12. 𝑥 ᠦᠨ ᠳᠣᠬᠠᠢ ᠳᠡᠭᠰ ᠢᠳᠭᠡᠯ (𝑎 + 3) 𝑥 | 𝑎| −1 − 3𝑥 + 2= 0 ᠨᠢ ᠨ ᠢᠭᠡ ᠦᠯ ᠦ ᠮᠡᠳ᠋ ᠡᠭᠳ᠋ᠡᠭᠴᠢ ᠲᠡᠢ ᠬᠣ ᠶᠠᠳ᠋ᠤ ᠭᠠᠷ ᠵ ᠡᠷᠭᠡ ᠢᠢᠨ ᠳᠡᠭᠰᠢ ᠳᠭᠡᠯ ᠪᠣᠯ ᠂ 𝑎 ᠢᠢᠨ ᠬᠡᠮᠵ ᠢᠭᠳ᠋ᠡᠯ ᠨ ᠢ ( ) 13. ᠢᠯ ᠡᠷᠬᠡᠶ ᠢᠯᠡᠯ 𝑚 2+ 2𝑘𝑚 + 16 ᠨᠢ ᠭᠦᠢᠴ ᠡᠳ ᠺᠦᠧᠠᠳ᠋ ᠦᠨ ᠢᠯᠡᠷᠬ ᠡᠶᠢᠯᠡᠯ ᠪᠣᠯ ᠂ 𝑘= ( ) 14. 𝑎 ,b ᠨᠢ ᠲᠡᠭᠰ ᠢᠳᠭᠡᠯ 𝑥 2+ 𝑥− 2015 = 0 ᠦᠨ ᠲᠡᠩ ᠴᠡᠬᠦ ᠦ ᠪᠤ ᠰᠤ ᠬ ᠣᠶᠠᠷ ᠪᠣᠳ᠋ᠠᠳ᠋ ᠤ ᠲ ᠣᠭᠠᠨ ᠢᠵᠠᠭᠤᠷ ᠪ ᠣᠯ᠂ 𝑎 2 + 2𝑎 + b ᠦᠨ ᠬᠡᠮᠵ ᠢᠭᠳ᠋ᠡᠯ ᠨ ᠢ ( ) 15. 𝑥 ᠦᠨ ᠳᠣᠬᠠᠢ ᠳᠡᠭ ᠰᠢᠳ ᠭᠡᠯ 𝑥 2 + 2𝑘 𝑥+ 𝑘 2 + 𝑘+ 3= 0 ᠦᠨ ᠬᠣᠶ ᠠᠷ ᠢᠵᠠᠭᠤᠷ ᠨᠢ ᠲᠤᠰ ᠪᠦᠷ ᠢ 𝑥 1 , 𝑥 2 ᠪᠣᠯ ᠂ ( 𝑥1 − 1) 2+ (𝑥 2 − 1) 2 ᠦᠨ ᠬᠠᠮᠤᠭ ᠦᠨ ᠪᠠ ᠭ ᠠ ᠬᠡᠮᠵ ᠢᠭᠳ᠋ᠡᠯ ᠨ ᠢ ( ) 16. ᠭᠡᠷᠪᠠ 𝑥= 1 ᠨ ᠢ 𝑥 ᠶᠢᠨ ᠲᠤ ᠬᠠᠢ ᠨᠢᠭᠡ ᠦᠯ ᠤ ᠮ ᠡᠳ᠋ᠡᠭᠳ᠋ ᠡᠭᠴᠢ ᠲᠡᠢ ᠬᠤᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠶᠢᠨ ᠲᠡᠭᠰ ᠢᠳᠭᠡᠯ 𝑥 2+ 3𝑚𝑥 + 𝑛= 0 ᠤᠨ ᠬᠠᠷ ᠢᠭᠤ ᠪ ᠣᠯ 6 𝑚 + 2𝑛 = ( ) 17. ᠨᠢᠭᠡ ᠭᠤ ᠷᠪᠠᠯ ᠵᠢᠨ ᠤ ᠬ ᠤᠶᠠᠷ ᠲᠠᠯ ᠠ ᠨᠢ ᠲᠤᠰ ᠲᠤ ᠰ 3 ᠪᠠ 6᠂ ᠭᠤ ᠷᠪᠠᠳ᠋ ᠤᠭᠠᠷ ᠲᠠᠯ ᠠ ᠨᠢ ᠲᠡᠭᠰ ᠢᠳᠭᠡᠯ 𝑥 2− 6𝑥 + 8= 0 ᠶᠢᠨ ᠬᠠᠷᠢ ᠭᠤ ᠪᠣᠯᠤ ᠨ ᠠ᠃ ᠲ ᠡᠭᠡᠪᠠ ᠯ ᠡ ᠨᠡ ᠭᠤᠷᠪᠠ ᠯᠵᠢᠨ ᠤ ᠬᠦᠷᠢ ᠶᠠᠠ ᠦ ᠤ ᠷᠳ᠋ᠤ ᠨᠢ ( ) 18. ᠦᠭ ᠬᠦᠭᠳ᠋ ᠡᠭᠰᠠᠠ ᠨᠢ 𝑥1 , 𝑥2 ᠨᠢ ᠲᠡᠭᠰ ᠢᠳᠭᠡᠯ 2 𝑥 2 − 3𝑥 − 8= 0 ᠤᠨ ᠬᠤᠶ ᠠᠷ ᠢᠵᠠᠭᠤᠷ᠂ ᠲᠡᠭᠡ ᠪᠠᠯ 𝑥1 2+ 𝑥 2 2= ( ) ᠭᠤᠷᠪᠠ᠂ ᠪᠣᠳ᠋ ᠣᠵᠤ ᠬᠠᠷᠢ ᠭᠤᠯᠬᠤ ᠰᠡᠲᠤ ᠪ᠃ ︵ ᠨᠠᠢ ᠲᠡ 66 ᠬᠤ ᠪᠢ ︶ 19. ( 8 ᠬᠤ ᠪᠢ) ᠲᠤᠤ ᠷᠠᠬᠢ ᠲᠡᠭ ᠰᠢᠳ ᠭᠡᠯ ᠢ ᠪ ᠤᠳ᠋ᠤ (1) ( 𝑥− 5) 2= 16 (2) 𝑥 2+ 8𝑥 − 9= 0 (3) 2 𝑥 2 − 4𝑥 − 5= 0 (4) 2 𝑥 2 + 10 𝑥= 0 20. ( 8 ᠬᠤᠪ ᠢ) y ᠢ ᠢᠨ ᠲᠤᠬᠠᠢ ᠨᠢᠭᠡ ᠥᠯ ᠥ ᠮ ᠡᠳ᠋ᠡᠭᠳ᠋ ᠡᠭᠴᠢ ᠳ᠋ᠡᠢ ᠬᠤᠶ ᠠᠳ᠋ᠤᠭᠠ ᠷ ᠵ ᠡᠷᠭᠡ ᠢᠢᠨ ᠲᠡᠭᠰ ᠢᠳᠭᠡᠯ y 2+ 𝑘 y+ 6 ＝0 ᠦᠨ ᠨ ᠢᠭᠡ ᠢᠵ ᠠᠭᠤᠷ ᠨᠢ 2᠃ (1) 𝑘 ᠢᠢᠨ ᠬᠡᠮᠵ ᠢᠭᠳ᠋ᠡᠯ ᠢ ᠣ ᠯ ᠃ (2) ᠡᠨᠡ ᠨᠢᠭᠡ ᠥᠯ ᠥ ᠮ ᠡᠳ᠋ᠡᠭᠳ᠋ ᠡᠭᠴᠢ ᠳ᠋ᠡᠢ ᠬᠤᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠢᠢᠨ ᠲ ᠡᠭᠰᠢ ᠳᠭᠡᠯ ᠢᠢᠨ ᠨᠦ᠋ ᠭᠦᠭᠡ ᠢᠵᠠᠭᠤᠷ ᠢ ᠣ ᠯ ᠃ 21. (10 ᠬᠤᠪᠢ ) 𝑥 ᠶᠢᠨ ᠲ ᠤᠬᠠᠢ ᠨᠢᠭᠡ ᠦᠯ ᠤ ᠮ ᠡᠳ᠋ᠡᠭᠳ᠋ ᠡᠭᠴᠢ ᠲᠡᠢ ᠬᠤᠶ ᠠᠳ᠋ᠤᠭ ᠠᠷ ᠵᠡᠷᠭᠡ ᠶᠢᠨ ᠲᠡᠭᠰ ᠢᠳᠭᠡᠯ ( 𝑥− 𝑚) 2− 2( 𝑥− 𝑚) = 0( 𝑚 ᠨᠢ ᠪᠠ ᠢᠩᠭᠤ ᠲᠣ ᠭ ᠠ ) (1) ᠪᠠ ᠳ᠋ᠤᠯᠨ ᠄ 𝑚 ᠳ᠋ᠤ ᠶᠠᠮᠠᠷ ᠭᠡᠮᠵ ᠢᠭᠳ᠋ᠡᠯ ᠠᠪᠤ ᠭᠰᠠᠠ ᠴᠤ ᠲᠡᠭᠡᠷᠠᠬ ᠢ ᠲ ᠡᠭᠰᠢ ᠳᠭᠡᠯ ᠳ᠋ᠦ ᠬᠤᠶ ᠠᠷ ᠲᠡᠨᠭ ᠴᠡᠭᠦ ᠦ ᠪᠤ ᠰᠤ ᠪ ᠤᠳ᠋ᠠᠳ᠋ ᠤ ᠲ ᠤᠭᠠᠨ ᠤ ᠢᠵ ᠠᠭᠤᠷ ᠲᠡᠢ ᠃ (2) ᠭᠡᠷᠪᠠ ᠨ ᠢᠭᠡ ᠢ ᠵᠠᠭᠤᠷ ᠨᠢ 3 ᠪ ᠤᠯ 𝑚 ᠢᠢᠨ ᠭᠡᠮᠵᠢ ᠭᠳ᠋ᠡᠯ ᠢ ᠤᠯ ᠃ 22. (10 ᠬᠤᠪᠢ ) ᠦᠭ ᠬᠦᠭᠳ᠋ ᠡᠭᠰᠠᠠ ᠨᠢ 𝑥 ᠪᠤ ᠯ ᠲ ᠡᠭᠰᠢ ᠳᠭᠡᠯ 𝑥 2+ 3𝑥 − 1= 0 ᠦᠨ ᠢᠵᠠᠭᠤᠷ᠂ ᠠ ᠯᠭᠧᠪᠷᠠ ᠶᠢᠨ ᠢᠯᠡᠷᠭᠡ ᠢᠯᠠᠯ 𝑥− 3 3𝑥 2− 6𝑥 ÷ (𝑥 + 2− 5𝑥−2 ) ᠶ ᠢᠨ ᠭᠡᠮᠵᠢ ᠭᠳ᠋ᠡᠯ ᠢ ᠤᠯ ᠃ 23. (10 ᠬᠤᠪᠢ ) ᠢᠯᠡᠯ ᠳ᠋ᠠ ᠳᠤᠰ ᠬᠤᠨ ᠤ 201 1 ᠤᠨ ᠳ᠋ ᠡᠭᠢ ᠬᠦᠮᠦᠨ ᠪᠦᠷ ᠢ ᠵ ᠢᠨ ᠳᠡᠭᠰ ᠢᠳᠭᠡᠭ ᠰᠡᠨ ᠤᠷᠤᠯ ᠭ ᠠ ᠨᠢ 1200 ᠳᠦ᠋ ᠭᠦᠷᠢ ᠭ ᠪ ᠤᠯᠤᠨ ᠠ ᠃ 2013 ᠤ ᠨ ᠳ᠋ ᠡᠭᠢ ᠬᠦᠮᠦᠨ ᠪᠦᠷᠢ ᠵᠢᠨ ᠳᠡᠭᠰ ᠢᠳᠭᠡᠭ ᠰᠡᠨ ᠤ ᠷᠤᠯᠭ ᠠ ᠨ ᠢ 1452 ᠳᠦ᠋ ᠭᠦᠷᠢ ᠭ ᠪ ᠤᠯᠤᠨ ᠠ ᠃ ᠬᠦᠮᠦᠨ ᠪᠦᠷ ᠢ ᠵ ᠢᠨ ᠳᠡᠭᠰ ᠢᠳᠭᠡᠭ ᠰᠡᠨ ᠤᠷᠤᠯ ᠭ ᠠ ᠵ ᠢᠨ ᠵ ᠢᠯ ᠦᠨ ᠳᠤ ᠮᠳ᠋ᠠᠴ ᠢ ᠦᠰ ᠦᠯᠳ᠋ᠠ ᠵᠢᠨ ᠨ ᠤᠷᠮ ᠠ ᠵᠢ ᠤᠯ ᠃ 24. (10 ᠬᠤᠪ ᠢ) ᠨᠢᠭᠡ ᠬᠤ ᠪᠴᠠᠰ ᠤᠨ ᠳᠡᠯᠭᠡᠭ ᠦᠷ ᠵᠠᠬ ᠠ ᠪᠦᠷ ᠢ ᠳ᠋ ᠦ ᠪᠡᠨ 80 ᠲᠥ ᠭᠥᠷᠢᠭ ᠢᠢᠡᠷ ᠣᠷᠣᠭᠤ ᠯᠵᠤ ᠂ 120 ᠲᠥ ᠭᠥᠷᠢᠭ ᠢᠢᠡᠷ ᠬᠤᠳ᠋ ᠠᠯᠳ᠋ᠤ ᠪᠠᠯ ᠂ ᠡ ᠳ᠋ᠦᠷ ᠪᠦᠷᠢ 20 ᠵᠠᠬ ᠠ ᠬ ᠤᠳ᠋ᠠᠯ ᠳ᠋ᠤᠨ ᠠ ᠃ ᠠᠷᠪᠠ ᠨ ᠰᠠᠷ ᠠ ᠢᠢᠨ ᠨᠢᠭᠡᠨ ᠦ ᠪᠠ ᠶᠠᠷ ᠦᠨ ᠡᠳ᠋ᠦᠷ ᠢ ᠳᠣ ᠬᠢᠶᠠᠯ ᠳ᠋ᠠᠭᠤ ᠯᠤᠨ ᠲᠤ ᠰ ᠳᠡᠯ ᠭᠡᠭᠦᠷ ᠡᠴᠡ ᠬᠦᠳ᠋ ᠡᠯᠭᠡᠭ ᠡᠨ ᠦᠷᠨᠢ ᠭᠦᠯᠵᠤ ᠠᠰᠢ ᠭ ᠬ ᠣᠨᠵᠢ ᠪᠦᠷᠢ ᠪᠠᠨ ᠨᠡᠮᠡᠭ ᠳ᠋ᠡᠭᠦ ᠯᠬᠦ ᠪᠡᠷ ᠳᠣᠭᠲᠠᠪᠠ ᠂ ᠵᠠᠬ ᠠ ᠳᠡᠯ ᠭᠡᠭᠦᠷ ᠦᠨ ᠪᠠᠶᠢ ᠴᠠᠭᠠᠯ ᠲᠠ ᠢᠢ ᠦ ᠨᠳ᠋ᠦᠰ ᠦᠯᠡᠪᠡᠯ 1 ᠲ ᠥᠭᠥ ᠷᠢᠭ ᠪᠠᠭᠤ ᠷᠠᠭᠤᠯ ᠬᠤ ᠪ ᠦᠷᠢ 2 ᠵ ᠠᠬ ᠠ ᠢᠯ ᠡᠭᠦᠦ ᠬᠤ ᠳ᠋ᠠᠯᠳ᠋ ᠤᠨ ᠠ ᠃ (1) ᠬᠡᠷᠪᠡ 𝑥 ᠲᠥ ᠭᠥᠷᠢᠭ ᠪᠠᠭᠤ ᠷᠠᠭᠤᠯ ᠪᠠᠯ ᠂ ᠡ ᠳ᠋ᠦᠷ ᠪᠦᠷᠢ ( ) ᠵᠠᠬ ᠠ ᠬ ᠤᠳ᠋ᠠᠯ ᠳ᠋ᠤᠨ ᠠ ᠃ ᠵᠠᠬ ᠠ ᠪᠦᠷ ᠢ ᠢᠢ ᠨ ᠠ ᠰᠢᠭ ᠬᠣ ᠨᠵᠢᠪᠦ ᠷᠢ ᠨ ᠢ ( ) ( 𝑥 ᠠ ᠭᠤᠯᠤ ᠭᠰᠠᠨ ᠢᠯ ᠡᠷᠬᠡᠶ ᠢᠯᠡᠯ ᠢᠢᠡᠷ ᠢᠯᠡᠷᠬ ᠡᠶᠢᠯᠡ ) (2) ᠵ ᠠᠬ ᠠ ᠪᠦᠷ ᠢ ᠢᠢ ᠨ ᠬ ᠤᠳ᠋ᠠᠯ ᠳ᠋ᠤᠬᠤ ᠦ ᠨ ᠡ ᠢᠢ ᠬᠡᠳ᠋ᠦᠢ ᠲ ᠥᠭᠥ ᠷᠢᠭ ᠪᠠᠭᠤ ᠷᠠᠭᠤᠯ ᠪᠠᠯ ᠂ ᠡ ᠳ᠋ᠦᠷ ᠪᠦᠷᠢ ᠢᠢᠨ ᠳᠤᠮᠳ᠋ ᠠᠴᠢ ᠠ ᠰᠢᠭ ᠬᠣ ᠨᠵᠢᠪᠦ ᠷᠢ ᠨ ᠢ 1200 ᠲᠥ ᠭᠥᠷᠢ ᠭ ᠪᠣ ᠯᠬᠤ ᠪᠤᠢ ︖ 25. ( 10 ᠬ ᠤᠪᠢ ) ᠺᠦ ᠸᠠᠳ᠋ᠯ ᠠᠬᠤ ᠠᠷᠭ ᠠ ᠪᠠ ᠷ ᠨ ᠢᠭᠡ ᠦᠯ ᠤ ᠮ ᠡᠳ᠋ᠡᠭᠳ᠋ ᠡᠭᠴᠢ ᠲᠡᠢ ᠬᠤᠶ ᠠᠳ᠋ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠶᠢᠨ ᠲᠡᠭ ᠰᠢᠳ ᠭᠡᠯ ᠢ ᠪᠤᠳ᠋ᠤ ᠵᠤ ᠪᠤᠯᠬᠤ ᠪᠦᠭᠡᠳ ᠪᠠ ᠰᠠ ᠮᠠᠰᠢ ᠠᠷᠪᠢᠨ ᠠ ᠰᠠᠭᠤ ᠳ᠋ᠠᠯ ᠢ ᠰ ᠢᠢᠳᠪ ᠤᠷᠢᠯ ᠡᠵᠤ ᠪᠣᠯᠤ ᠨ ᠠ᠃ ᠵᠢ ᠱᠢᠶᠠᠯ ᠠᠪᠠᠯ ᠄ 3 𝑎 2 ≥ 0 ᠡ ᠢᠮᠤ ᠠᠴᠠ 3𝑎 2+ 1 ᠬ ᠠᠮᠤᠭ ᠪᠠᠭ ᠠ ᠭᠡᠮᠵ ᠢᠭᠳ᠋ᠡᠯ ᠨᠢ 1 ᠪᠣᠯᠤ ᠨ ᠠ ᠂ 3 𝑎 2 + 1≥ 1 ᠂ ᠵᠦ᠋ ᠪᠭᠡᠨ a = 0 ᠦᠶ ᠠᠰ ᠰ ᠠᠶᠢ ᠡᠨᠡ ᠢᠯᠡᠷᠭᠡ ᠢᠯᠠᠯ ᠤᠨ ᠬᠠᠮᠤᠭ ᠤᠨ ᠪᠠᠭ ᠠ ᠭᠡ ᠮᠵᠢᠭ ᠳ᠋ᠡᠯ 1 ᠢ ᠣᠯᠤ ᠨ ᠠ᠃ ᠠ ᠳ᠋ᠠᠯᠢ ᠪᠠᠷ− 3𝑎 2≤ 0 ᠡ ᠢᠮᠤ ᠠᠴᠠ −3 𝑎 2 + 1 ᠨ ᠢ ᠬᠠᠮᠤᠭ ᠤᠨ ᠶᠡᠭᠡ ᠭᠡᠮᠵ ᠢᠭᠳ᠋ᠡᠯ ᠨᠢ 1 ᠪ ᠣᠯᠤᠨ ᠠ ᠂ ᠲᠡᠷᠤᠢ − 3𝑎 2+ 1≤ 1 ᠳ᠋ ᠦ ᠵᠦ᠋ ᠪᠭᠡᠨ a = 0 ᠦᠶ ᠠᠰ ᠰ ᠠᠶᠢ ᠡ ᠨᠡ ᠢᠯ ᠡᠷᠭᠡᠢ ᠯᠠᠯ ᠤ ᠨ ᠬᠠᠮᠤ ᠭ ᠶᠡᠭᠡ ᠭᠡᠮᠵᠢ ᠭᠳ᠋ᠡᠯ ᠨᠢ 1 ᠪᠣᠯ ᠤᠨ ᠠ ᠃ (1) 𝑥 =( ) ᠦᠶ ᠡᠰ ᠠ ᠯᠭᠧᠪᠷᠠ ᠶᠢᠨ ᠢᠯᠡᠷᠭᠡᠢ ᠯᠠᠯ 3( 𝑥+ 3) 2+4 ᠨᠢ ᠬᠠᠮᠤᠭ ᠤᠨ ( ) ᠲᠡᠢ ᠭᠡᠮᠵ ᠢᠭᠳ᠋ᠡᠯ ᠨᠢ ( ) (2) 𝑥 = ( ) ᠦᠶ ᠡᠰ ᠠ ᠯᠭᠧᠪᠷᠠ ᠶᠢᠨ ᠢᠯᠡᠷᠭᠡ ᠢᠯᠠᠯ − 2𝑥 2+ 4𝑥 + 3 ᠨᠢ ᠬᠠᠮᠤᠭ ᠤᠨ ( ) ᠲᠡᠢ ᠭ ᠡᠮᠵᠢᠭ ᠳ᠋ᠡᠯ ᠨᠢ ( ) (3) ᠲᠡᠭ ᠰᠢ ᠦᠠᠴ ᠦᠭᠳ᠋ᠤ ᠴᠡᠴᠡᠷᠯ ᠢᠭ ᠦᠨ ᠨᠢᠭᠡ ᠲᠠᠯ ᠠ ᠨᠢ ᠭᠡᠷᠡᠮ ᠲᠡᠢ ᠨᠠᠭᠠᠯ ᠳ᠋ᠤᠵᠤ᠂ ᠪᠤᠰ ᠤᠤᠨ ᠭᠤ ᠷᠪᠠᠨ ᠲᠠᠯ ᠠ ᠨᠢ ᠤᠷᠳ᠋ ᠤ ᠨᠢ 16 m ᠪᠤᠯᠬᠤ ᠭᠡᠷ ᠡᠰᠭᠡ ᠪᠠᠷ ᠬᠢᠪᠠ ᠂ ᠲ ᠡᠭᠡᠪᠠ ᠯ ᠭ ᠡᠷᠡᠮ ᠲᠡᠢ ᠵᠡᠷᠭᠡᠯ ᠡᠳ᠋ᠡᠭᠡ ᠲᠠᠯ ᠠ ᠶᠢ ᠨ ᠤᠷᠳ᠋ ᠤ ᠨᠢ ᠭᠡᠳ᠋ᠤ ᠢ ᠪᠠᠷ ᠠᠪᠬᠤ ᠦᠶ ᠠᠰ ᠲ ᠠᠯᠠᠪᠠ ᠢ ᠨᠢ ᠬᠠᠮᠤᠭ ᠶᠡᠭᠡ ᠪᠤᠯ ᠬᠤ ᠪ ᠤᠢ ︖ ᠬ ᠠᠮᠤᠭ ᠤᠨ ᠶᠡᠭᠡ ᠲᠡᠯ ᠠᠪᠠᠢ ᠨᠢ ᠭᠡᠳ᠋ᠤ ᠢ ᠪᠤ ᠢ ︖