ᠬᠤᠶᠠᠳᠤᠭᠠᠷ ᠪᠥᠯᠥᠭ ᠪᠥᠬᠦᠯᠢ ᠢᠯᠡᠷᠭᠡᠢᠯᠡᠯ （ᠨᠢᠭᠡᠴᠢ ᠶᠢᠨ ᠪᠠᠢᠴᠠᠭᠠᠯᠲᠠ 1） .pdf

ᠬᠤᠶ ᠠᠳᠤᠭᠠᠷ ᠪᠥᠯ ᠥᠭ ᠦᠨ ᠪᠠᠢ ᠴᠠᠭᠠᠯᠲ ᠠ ᠠᠩᠭᠢ ᠪᠥᠯ ᠥᠭ᠄ ᠨᠡᠷ ᠡ᠄ ︵10 0 ᠮᠢ ᠨᠦ᠋ᠲ᠂ ᠨᠠᠢᠲᠡ 120 ᠬᠤᠪᠢ ︶ ᠨᠢᠭᠡ᠂ ᠰᠤᠩᠭᠤᠬ ᠤ ᠰᠡ ᠲᠤᠪ ᠃ (ᠰᠡᠲᠤ ᠪ ᠪᠦᠷᠢ 3 ᠬ ᠤᠪᠢ᠂ ᠨᠠᠢᠲᠡ 30 ᠬᠤ ᠪᠢ) 1. 𝜋𝑝 22 ᠦᠨ ᠢᠯᠡᠳᠬᠡᠮ ᠵᠢ ᠨᠢ ( ) A. 12 B. π C. 2 D. 𝜋2 2. ᠲᠤᠤᠷᠠ ᠬᠢ ᠬᠡ ᠯᠡᠯᠬᠡ ︽ 4𝑎 ︾ ᠵᠢ ᠢᠯᠡᠷ ᠬᠡᠢᠯ ᠡᠵᠤ ᠪᠤᠯᠬ ᠤ ᠦᠬᠡᠢ ᠨᠢ ( ) A. 𝑎 ᠦᠨ 4 ᠲᠠ ᠬᠢᠨ B. 4 ᠶᠢᠨ 𝑎 ᠲᠠᠬᠢᠨ C. ᠲᠦ᠋ᠷᠪ ᠡᠨ 𝑎 ᠦᠨ ᠨᠡᠢ ᠯᠡᠪᠦᠷ ᠢ D. ᠲᠦ᠋ᠷᠪ ᠡᠨ𝑎 ᠦᠨ ᠦᠷᠡ ᠵᠢᠪᠦᠷᠢ ᠃ 3.ᠲᠤᠤᠷᠠ ᠬᠢ ᠲᠦ᠋ᠷᠪ ᠡᠨ ᠢ ᠯᠡᠷᠬᠡᠢᠯ ᠡᠯ ᠳᠦ ᠬᠠᠭ ᠠᠯᠳᠠ ᠠᠴ ᠠ ᠵᠦ᠋ ᠪ ᠲᠠ ᠢᠯᠤᠭᠰ ᠠᠨ ᠨᠢ ( ) A. 𝑥 2− 2( 2𝑥 − 𝑦+ 2) = 𝑥 2 − 4𝑥 − 2𝑦 + 4 B. −3 (𝑚 + 𝑛) − 𝑚𝑛 = −3 𝑚 + 3𝑛 − 𝑚𝑛 C. − (5 𝑥− 3𝑦 )+ 4( 2𝑥𝑦 − 𝑦 2 )= −5 𝑥+ 3𝑦 + 8𝑥𝑦 − 4𝑦 2 D. 𝑎 b− 5( −𝑎 + 3) = 𝑎𝑏 + 5𝑎 − 3 4. ᠦᠭ ᠬᠦᠭᠳᠡ ᠭᠰᠡᠨ ᠨᠢ A , B , C ᠪᠤᠯ ᠤᠯᠠᠨ ᠬᠡᠰᠢᠬᠦᠨ ᠳᠤ ᠂ ᠵᠤᠯ ᠠ ︽A − B ︾ᠵᠢ ᠪᠤᠳᠤᠬ ᠤ ᠦ ᠶᠡᠰ᠂ ᠰᠡᠳ ᠦᠪ ᠢᠶᠡ ᠨ ᠪᠤᠷᠤᠭ ᠤ ᠬᠠᠭ ᠤᠯᠵᠤ ︽A + B ︾ᠬᠡᠵᠤ ᠪᠣᠳᠤᠭᠠᠳ ᠳᠦ᠋ ᠩ ᠢ ᠨᠢ C ᠲᠠᠢ ᠲᠡᠨᠭ ᠴᠡᠬᠦ ᠯᠵᠡᠢ᠂ ᠡᠬᠦᠨ ᠳ ᠦ A = 12 𝑥 2+ 𝑥− 1 , C = 𝑥 2 + 2𝑥 ᠪᠣ ᠯᠤᠨ ᠠ᠂ ᠲᠠᠭᠡᠪ ᠡᠯ ︽A − B ︾ ᠦᠨ ᠳᠦ᠋ ᠩ ᠨᠢ ( ) A. 𝑥 2− 2𝑥 B. 𝑥 2+ 2𝑥 C. − 2 D. − 2𝑥 5. 8𝑏 2− 2𝑏 2 ᠦ ᠨ ᠳ ᠦ᠋ᠩ ᠨᠢ ( ) A. 5b 2 B. 10b 2 C. 6b 2 D. 6 6. ᠬᠡᠷᠪᠡ 𝑥= 1 ᠦᠶ ᠡᠰ 𝑎𝑥 + b+ 1 ᠦᠨ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠨᠢ −2 ᠪᠤᠯ᠂ (𝑎 + 𝑏− 1) (1 − 𝑎− 𝑏) ᠦᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠨᠢ ( ) A. − 16 B. −8 C. 8 D. 16 7. ᠵᠢᠷᠤᠭ ᠮᠡᠳᠤ ᠂ ᠲᠣᠭ ᠠ 𝑥 , 𝑦 ᠨᠢ ᠲᠤᠭ ᠠᠨ ᠤ ᠲᠡᠨᠭᠬᠡ ᠯᠢᠭ ᠲᠡᠭᠡᠷ ᠡᠬᠢ ᠴᠡᠭ ᠲᠠᠢ ᠬᠠᠷᠠᠭ ᠠᠯᠵᠠᠨ ᠠ᠂ |𝑥 + 𝑦| − |𝑦 − 𝑥| ᠢ ᠬᠢᠯ ᠪᠠᠷᠴᠢ ᠯᠠᠭᠰᠠᠨ ᠳᠦ᠋ ᠩ ᠨᠢ ( ) A. 0 B. 2𝑥 C. 2𝑦 D. 2 𝑥− 2𝑦 8.ᠨᠢ ᠬᠡ ᠤᠯᠠᠨ ᠬᠡᠰᠢᠬᠦᠨ ᠳᠤ ᠳᠡᠭᠡᠷ ᠡ 3y 2− 2y − 5 ᠢ ᠨᠡ ᠮᠡᠪᠡᠯ ᠤᠯ ᠠᠨᠬᠡᠰᠢᠬᠦᠨ ᠳᠤ 5y 3− 4y − 6 ᠢ ᠣᠯᠤᠨ ᠠ᠂ ᠤᠤᠯ ᠣᠯ ᠠᠨᠭᠡᠰᠢᠭ ᠦᠨᠲᠦ ᠨᠢ ( ) A. 5y 3+ 3y 2+ 2y − 1 B. 5y 3− 3y 2− 2y − 6 C. 5y 3+ 3y 2− 2y − 1 D. 5y 3− 3y 2− 2y − 1 9.ᠦᠭ ᠬᠦᠭᠳᠡ ᠭᠰᠡᠨ ᠨᠢ᠄ 6x − 1 ᠶᠢᠨ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠨᠢ 2 ᠂ 𝑦 2 ᠦ ᠨ ᠬᠡ ᠮᠵᠢᠭᠳᠡᠯ ᠨᠢ 4 ᠂ ᠲᠡᠬᠡᠪ ᠡᠯ (5 𝑥 2 𝑦+ 5𝑥𝑦 − 7𝑥 )− (4 𝑥 2 𝑦+ 5𝑥𝑦 − 7𝑥 ) ᠦᠨ ᠵᠦ᠋ᠪ ᠳᠦ᠋ᠩ ᠨᠢ ( ) A. − 12 B. 12 C. ± 12 D. 2 ᠪᠤ ᠶᠤ 12 10 .ᠲᠤᠤᠷᠠ ᠬᠢ ᠵᠢᠷᠤᠭ ᠲᠦ᠋ᠷᠰᠦ ᠪᠤᠯ ᠲᠤᠭᠳᠠ ᠭᠰᠠᠨ ᠵᠢᠮ ᠡ ᠶᠤ ᠰᠤ ᠲᠠᠢ ᠪᠠᠷ ᠪᠦᠷᠢᠯᠳᠦ ᠭᠰᠡᠨ ᠶᠤ ᠮ᠂ ᠨᠢᠬᠡᠳᠦᠭᠡᠷ ᠵᠢᠷᠤᠭ ᠲᠤ 2 ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠲᠠᠢ ᠪᠠᠢᠵᠤ᠂ ᠬᠤᠶ ᠠᠳᠤᠭ ᠠᠷ ᠵᠢᠷ ᠤᠭ ᠲᠤ 8 ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠲᠠᠢ ᠪᠠᠢᠵᠤ ᠂ ᠭᠤᠷᠪ ᠠᠳᠤᠭ ᠠᠷ ᠵᠢᠷᠤᠭ ᠲᠤ 14 ᠭᠤᠷ ᠪᠠᠯᠵ ᠢᠨ ᠲᠠᠢ ᠂ ᠲᠡ ᠭᠡᠪᠡᠯ 5 ᠳᠤᠭᠠ ᠷ ᠵᠢᠷ ᠤᠭ ᠲᠡ ᠬᠢ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠤ ᠪᠤᠳᠠᠭ ᠠᠯᠢ ᠶᠢᠨ ᠲᠣᠭ ᠠ ᠨᠢ ( ) A. 22 B. 2 4 C. 26 D. 2 8 ᠬᠣᠶ ᠠᠷ᠂ ᠬᠠᠭ ᠠᠯᠲᠠ ᠳᠤ ᠲᠠᠭᠯ ᠠᠬᠤ ᠰᠡᠲᠤ ᠪ᠃ (ᠰᠡᠲᠤᠪ ᠪᠦᠷᠢ 3 ᠬ ᠤᠪᠢ᠂ ᠨᠠᠢᠲ ᠡ 2 4 ᠬᠤᠪᠢ ) 11. ᠨᠢᠬᠡᠨ ᠭᠡᠰᠢᠬᠦᠨ ᠳᠤ 9 𝑎 4 𝑏 ᠦ ᠨ ᠵᠡ ᠷᠬᠡ ᠶᠢᠨ ᠲᠣᠭ ᠠ ᠨᠢ᠄ __ __ __ __ __ __ 12. ᠪᠤᠳᠤ᠄ 9 𝑎− 2𝑎 =_ __ __ __ __ _ 13 .ᠨᠢᠬ ᠡ ᠣ ᠯᠠᠨᠭᠡᠰᠢᠭ ᠦᠨᠲᠦ ᠳᠡ ᠭᠡᠷ ᠡ 5𝑥 2− 4𝑥 − 3 ᠨᠡ ᠮᠡᠭᠡᠳ −𝑥 2− 3𝑥 ᠤᠯ ᠵᠡᠢ᠂ ᠡᠨᠡᠬᠦ ᠣᠯ ᠠᠨᠭᠡᠰᠢᠭ ᠦᠨᠲᠦ ᠨᠢ ᠄__ __ __ __ __ __ __ _ 14. ᠬᠡᠷᠪᠡ ᠢ ᠯᠡᠷᠬᠡᠢᠯ ᠡᠯ ( 2𝑥 2+ 3𝑎 𝑥− 𝑦) − 2( 𝑏𝑥 2− 3𝑥 + 2𝑦 − 1) ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠨᠢ ᠦᠯ ᠦ ᠮᠡ ᠳᠡᠬᠦ ᠲᠣᠭ ᠠ 𝑥 ᠲᠠᠢ ᠬᠤᠯ ᠪᠤᠭᠳᠠᠯ ᠦᠬᠡᠢ ᠂ ᠲᠡ ᠭᠡᠪᠡᠯ 3( 𝑎− 𝑏) − 2( 𝑎+ 𝑏) ᠦᠨ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠨᠢ᠄_ __ __ __ __ __ _ 15 .ᠨᠢᠬ ᠡ ᠨᠡ ᠷᠡᠳᠤ ᠲᠡ ᠮᠳᠡᠭᠳᠤ ᠬᠤᠪᠴᠠᠰᠤᠨ ᠲᠡ ᠯᠬᠡᠬᠦᠷ ᠨᠢ ᠬᠡ ᠵᠠᠬ ᠠ ᠬᠤᠪᠴᠠᠰ ᠤ ᠶᠢ ᠤᠷ ᠤᠭᠤᠯ ᠵᠤ᠂ 𝑎 ᠲᠥᠭ ᠦᠷᠢᠭ ᠵᠤᠭᠤᠰ ᠨᠡᠮᠡ ᠵᠤ ᠬᠠᠳᠠ ᠭᠰᠠᠨ ᠤ ᠳᠠ ᠷᠠᠭ ᠠ 7 ᠨᠤᠭ ᠤᠯᠵᠤ b ᠲᠦ᠋ᠬᠦᠷ ᠢᠭ ᠢᠶᠡᠷ ᠬᠤᠳᠠᠯ ᠳᠤᠪᠠ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ᠡᠨᠡᠬᠦ ᠬᠤᠪᠴᠠᠰᠤᠨ ᠤ ᠤᠷ ᠤᠭᠤᠯ ᠤᠭᠰᠠᠨ ᠦᠨ ᠡ ᠨᠢ᠄_ __ __ __ __ __ __ __ _ 16. ᠦᠭ ᠬᠦᠭᠳᠡᠭ ᠰᠡᠨ ᠨᠢ᠄ 𝑎 2 + 2𝑎 𝑏= 10 ,𝑏 2+ 2𝑎 𝑏= 16 ᠂ ᠲᠡᠬᠡ ᠪᠡᠯ 𝑎 2 + 4𝑎 𝑏+ 𝑏 2 = ___ ___ ___ __ ᠂ 𝑎 2 − 𝑏 2 = ___ ___ ___ ___ ᠃ 17. ᠬᠡᠷᠪᠡ 3 𝑎 𝑚 +2 𝑏 4 ᠪᠠ − 𝑎 5 𝑏 𝑛 −1 ᠶᠢᠨ ᠨᠡᠢ ᠯᠡᠪᠦᠷ ᠢ ᠨᠢ ᠨᠢᠬᠡᠨ ᠬᠡᠰᠢᠬᠦᠨ ᠳᠦ ᠪᠤᠯ᠂ m + n= ___ ___ ___ ___ ___ ___ _᠃ 18. ᠨᠢᠬᠡ ᠬᠠᠷ ᠢᠮᠠᠭ ᠵᠢᠨᠭᠯᠡᠭᠦᠷ ᠦᠨ ᠤᠤᠯ ᠤᠷᠳᠤ ᠨᠢ 12 cm ᠂ ᠪᠤᠳᠠᠰ ᠵᠢᠨᠭᠯᠡᠬ ᠦ ᠦ ᠶᠡᠰ ᠬᠠᠷᠢᠮᠠ ᠭ ᠵᠢᠨᠭᠯᠡᠬᠦᠷ ᠰᠤᠨᠥ ᠨ ᠠ᠂ ᠬᠠᠷ ᠢᠮᠠᠭ ᠵᠢᠨᠭᠯᠡᠭᠦ ᠷ ᠦᠨ ᠤᠷᠳᠤ ᠶᠢ 𝑦 (cm) ᠬᠡᠵ ᠤ᠂ ᠵᠢᠨᠭᠯᠡᠬ ᠦ ᠪᠡᠶ ᠡ ᠶᠢᠨ ᠭᠦᠨᠳᠤ ᠶᠢ 𝑥k g ᠬᠡ ᠪᠡᠯ᠂ ᠬᠠᠷ ᠢᠮᠠᠭ ᠦᠨ ᠤᠷᠳᠤ ᠶᠢᠨ ᠬᠤᠪᠢᠷ ᠠᠯᠳᠠ ᠨᠢ ᠬᠦᠰᠦ ᠨᠦᠭᠳᠤ ᠮᠡᠳᠤ ᠄ ᠲᠡᠭᠡᠪ ᠡᠯ 𝑥 kg ᠬᠦ ᠨᠳᠤ ᠪᠡᠶ ᠡ ᠶᠢ ᠵᠢᠨᠭᠯᠡᠬᠦ ᠦᠶ ᠡᠰ ᠦ ᠨ ᠬᠠᠷ ᠢᠮᠠᠭ ᠵᠢᠨᠭᠯᠡᠬᠦ ᠷ ᠦ ᠨ ᠤᠷ ᠳᠤ y = ___ ___ ___ cm ᠃ ᠭᠤᠷᠪ ᠠ᠂ ᠪᠣᠳᠣᠵ ᠤ ᠬᠠᠷ ᠢᠭᠤᠯ ᠬᠤ ᠰᠡ ᠲᠤᠪ᠃ ︵ ᠨᠠᠢᠲᠡ 66 ᠬᠤᠪᠢ ︶ 19. ︵16 ᠬᠤᠪᠢ ︶ᠬᠢᠯᠪᠠ ᠷᠴᠢᠯ ᠠ ᠃ (1) 5 𝑎 ＋(4b －c) －3( 𝑎 ＋3b －2c ) (2) ( 9𝑎 －2b ) －[8 𝑎 －(5b －2𝑎 )] ＋2c ᠵᠢᠨᠭᠯᠡᠬᠦ ᠪᠤᠳᠠᠰ ᠤᠨ ᠭᠦᠨᠳᠤ 𝑥(k g ) 1 2 3 4 ᠬᠠᠷᠢᠮᠠ ᠭ ᠤᠨ ᠤᠷᠳᠤ 𝑦 (cm ) 12 ＋0.5 12 ＋1 12 ＋1.5 12 ＋2 (3) 3 𝑥 2 －[5 𝑥 －( 12 𝑥 − 3) ＋3𝑥 2] (4) ( 19 4 𝑎 2+ 52 𝑎𝑏 − 183 𝑏 2 )− (− 54 𝑎 2− 236 𝑎𝑏 + 34 𝑏 2) 20. (10 ᠬᠤᠪᠢ ) ᠤᠷ ᠢᠳ ᠢᠶᠠ ᠷ ᠬᠢ ᠯᠪᠠᠷ ᠴᠢᠯᠠᠵᠤ᠂ ᠳᠠᠷ ᠠᠭ ᠠ ᠨᠢ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠤᠯ ᠃ (1) ( 4𝑎 2 －2𝑎 －6) －2( 2𝑎 2 －2𝑎 －5) ᠡᠭᠦᠨ ᠳᠦ 𝑎 ＝ －1 (2) 2 (𝑥 + 𝑥 2 𝑦) − 23 ( 6𝑥 2𝑦 + 3𝑥 )− 𝑦 ᠡᠭᠦᠨ ᠳ ᠦ 𝑥= 1 , 𝑦= 3 21. ( 8 ᠬᠤᠪᠢ ) ᠤᠷ ᠢᠳ ᠢᠶᠠ ᠷ ᠬᠢᠯ ᠪᠠᠷᠴᠢᠯ ᠠᠵᠤ᠂ ᠳᠠᠷ ᠠᠭ ᠠ ᠨᠢ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠤᠯ︕ 5𝑎 2 ＋3b 2 ＋2( 𝑎 2 －b 2) －(5 𝑎 2 ＋3b 2) ᠡᠭᠦᠨ ᠳ ᠦ 𝑎 ᠨᠢ ᠬᠠᠮᠤᠭ ᠶᠡᠬᠡ ᠰᠦ᠋ᠬ ᠡᠷᠬᠦ ᠪᠦᠬᠦᠯᠢ ᠲᠣᠭ ᠠ᠂ b ᠨᠢ 2 ᠦᠨ ᠲᠤᠨᠭᠭᠤᠷ ᠤᠤ ᠲᠣᠭ ᠠ ᠃ 22. (10 ᠬᠤᠪ ᠢ) ᠦᠭ ᠬᠦᠭᠳᠡᠭ ᠰᠡᠨ ᠨᠢ᠄ ᠲᠣᠭ ᠠ 𝑎 ,b ,c ,d ᠪ ᠤᠯ ᠶᠤᠰᠤ ᠳᠠᠢ ᠲᠣᠭ ᠠ ᠂ ᠬᠡᠷᠪᠡ 𝑎 ,b ,c ,d ᠨᠢ ᠲᠤᠭ ᠠᠨ ᠤ ᠲᠡ ᠨᠭᠬᠡᠯ ᠢᠭ ᠲᠦ ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠤ ᠪᠠᠢᠷ ᠢ ᠳᠤ ᠪᠠᠢᠬᠤ ᠪᠦᠭᠡᠳ |𝑐| = |𝑑 |− 7ᠪ ᠤᠯ᠂ ᠢᠯᠡᠷ ᠬᠡᠢᠯ ᠡᠯ| 𝑐− 𝑎− 𝑏| − |𝑎 + 𝑐− 𝑑| − |𝑐 − 𝑏| ᠢ ᠬᠢᠯᠪᠠ ᠷᠴᠢᠯ ᠡ᠃ 23. (10 ᠬᠤᠪᠢ ) ᠴᠤᠪᠤᠭᠤ ᠪᠤᠯ ᠲᠡ ᠷᠢᠬᠢ ᠪᠡᠨ ᠬᠦᠳᠡᠯ ᠬᠡᠬᠦ ᠲᠤᠷ ᠠᠳᠠᠢ ᠪᠢᠴᠢᠬ ᠠᠨ ᠶᠤ ᠮ᠂ ᠲᠡᠷᠡ ᠨᠢᠬ ᠡ ᠲᠤᠭ ᠠᠨ ᠤ ᠵᠢᠮ ᠡ ᠶᠤᠰᠤ ᠶᠢ ᠢ ᠯᠡᠷᠡᠬ ᠦᠯᠵᠡᠢ᠂ ᠵᠢᠷᠤᠭ ᠮᠡᠳᠤ ᠵᠢᠭ ᠰᠠᠭᠠ ᠭᠳᠠᠭ ᠰᠠᠨ ᠲᠡᠭᠰᠢ ᠲᠣᠭ ᠠ ᠶᠢ ︽+ ︾ ᠲᠦ᠋ᠷᠪ ᠡᠯᠵᠢ ᠯᠡᠭᠡᠰᠦ ᠪᠠᠷ ᠰᠢᠯᠵ ᠢᠬᠦᠯᠪᠠ ᠯ᠂ᠳ ᠥᠷᠪᠡ ᠯᠵᠢᠯ ᠡᠭᠡᠰᠦ ᠲᠤᠲᠤ ᠷᠠᠬᠢ 5 ᠲ ᠣᠭ ᠠ ᠶᠢ ᠠᠵᠢᠭ ᠯᠠᠵᠤ ᠂ ᠲᠤᠤᠷᠠ ᠬᠢ ᠠᠰᠠᠭᠤᠯᠳᠠ ᠶᠢ ᠬᠠᠷᠢᠭ ᠤᠯ︕ (1) ᠡᠨᠡᠬ ᠦ 5 ᠲᠣᠭ ᠠᠨ ᠤ ᠨᠡᠢ ᠯᠡᠪᠦᠷ ᠢ ᠪᠠ ᠲᠤᠮᠳ ᠠᠬᠢ ᠲᠣᠭ ᠠ 16 ᠨᠢ ᠶᠠᠮᠠᠷ ᠬᠤᠯ ᠪᠤᠭᠳᠠᠯ ᠲᠠ ᠢ ᠪᠤᠢ ︖ (2) ᠲᠤᠮᠳ ᠠᠬᠢ ᠲᠣᠭ ᠠ ᠶᠢ 𝑎 ᠬᠡ ᠵᠤ ᠵᠢᠱᠢᠭᠡᠳ ᠡᠨᠡᠬᠦ ᠳᠥᠷᠪ ᠡᠯᠵᠢᠯ ᠡᠭᠡᠰᠦ ᠲᠤᠲ ᠤᠷᠠᠬ ᠢ 5 ᠲᠤᠭ ᠠᠨ ᠤ ᠨᠡᠢᠯ ᠡᠪᠦᠷᠢ ᠶᠢ 𝑎 ᠵᠢᠡ ᠷ ᠢ ᠯᠡᠷᠬᠡᠢᠯ ᠡ︕ (3) ᠬᠡᠷ ᠪᠡ ︽+ ︾ᠳᠥᠷᠪ ᠡᠯᠵᠢᠯ ᠡᠭᠡᠰ ᠦ ᠶᠢ ᠵᠡᠬᠦ ᠨᠰᠢ ᠪᠠᠷᠠᠭ ᠤᠨᠰᠢ᠂ ᠲᠡᠭᠡᠭᠰ ᠢ ᠲ ᠤᠷᠤᠭᠰ ᠢ ᠰᠢᠯᠵ ᠢᠬᠦᠯᠬᠦ ᠳᠦ᠂ ᠳᠥᠷᠪ ᠡᠯᠵᠢᠯ ᠡᠭᠡᠰᠦ ᠲᠤᠳ ᠤᠷᠠᠬ ᠢ 5 ᠲᠤᠭ ᠠᠨ ᠤ ᠨᠡᠢᠯ ᠡᠪᠦᠷᠢ ᠨᠢ 20 18 ᠲᠠᠢ ᠲᠡᠨᠭ ᠴᠡᠭᠦᠦ ᠪᠣᠯᠤᠨ ᠠ ᠤᠤ ︖ᠬᠡ ᠷᠪᠡ ᠪᠤᠯᠪᠠ ᠯ ᠡᠨ ᠡᠬᠦ 5 ᠲᠣᠭ ᠠ ᠵᠢ ᠪᠢ ᠴᠢ᠂ ᠪᠤᠯᠬ ᠤ ᠦᠬᠡᠢ ᠪᠤᠯ ᠰᠢᠯ ᠳᠠᠭᠠᠨ ᠢ ᠨᠢ ᠶ ᠠᠷᠢ︕ 24. (10 ᠬᠤᠪᠢ ) ᠨᠢ ᠬᠡᠨ ᠦᠢᠯ ᠡᠳᠪᠦᠷ ᠢ A , B ᠬᠤᠶ ᠠᠷ ᠵᠦ᠋ᠢᠯ ᠦᠨ ᠪᠤᠬᠢᠷ ᠳᠤᠯ ᠦᠬᠡᠢ ᠴᠠᠭ ᠠᠰᠤᠨ ᠤᠭ ᠤᠳᠠ ᠵᠢ ᠡᠳᠦᠷ ᠪᠦᠷᠢ 45 00 ᠬᠢᠨ ᠡ᠂ ᠬᠤᠶ ᠠᠷ ᠵᠦ᠋ᠢᠯ ᠦᠨ ᠤᠭ ᠤᠳᠠ ᠬᠢᠬᠦ ᠦᠪᠡᠷ ᠦᠨ ᠦᠷᠳᠡ ᠭ ᠪ ᠠ ᠪᠤᠷᠤᠯ ᠠᠭᠤᠯᠬᠤ ᠦᠨ ᠡ ᠨᠢ ᠬᠦᠰᠦ ᠨᠦᠭᠳᠦ ᠮᠡᠳᠤ ︵ᠨᠢᠬ ᠡᠴᠢ᠄ ᠲᠦ᠋ᠬᠦᠷ ᠢᠭ ︶᠂ ᠬ ᠡᠷᠪᠡ A ᠵᠦ᠋ᠢᠯ ᠦᠨ ᠴᠠᠭᠠᠰᠤ ᠨ ᠤᠭ ᠤᠳᠠ ᠶᠢ ᠡᠳᠦ ᠷ ᠪᠦᠷᠢ 𝑥 ᠦᠢᠯ ᠡᠳᠪᠦᠷᠢᠯᠡᠨ ᠡ ᠬᠡ ᠵᠤ ᠵᠢ ᠱᠢᠪᠡᠯ᠂ (1) ᠲᠤᠰ ᠦᠢ ᠯᠡᠳᠪᠦᠷᠢ ᠶ ᠢᠨ ᠡᠳᠦᠷ ᠪᠦᠷᠢ ᠶᠢ ᠨ ᠦᠢ ᠯᠡᠳᠪᠦᠷᠢ ᠯᠡᠬᠦ ᠴᠠᠭᠠᠰᠤᠨ ᠤᠭ ᠤᠳᠠᠨ ᠤ ᠦᠪᠡᠷ ᠦᠨ ᠦᠷᠳᠡ ᠭ ᠢ 𝑥 ᠠ ᠭᠤᠯᠤᠭ ᠰᠠᠨ ᠢᠯᠡᠷ ᠬᠡᠢᠯ ᠡᠯ ᠢᠶᠡ ᠷ ᠢ ᠯᠡᠷᠬᠡᠢᠯ ᠡ︕ (2) ᠲᠤᠰ ᠦᠢ ᠯᠡᠳᠪᠦᠷᠢ ᠶ ᠢᠨ ᠡᠳᠦᠷ ᠪᠦᠷᠢ ᠶᠢ ᠨ ᠶᠡᠷ ᠤᠨᠭᠬᠡᠢ ᠠᠰᠢᠭ ᠢ 𝑥 ᠠᠭ ᠤᠯᠤᠭᠰ ᠠᠨ ᠢᠯᠡᠷ ᠬᠡᠢᠯ ᠡᠯ ᠢᠶᠡᠷ ᠢᠯ ᠡᠷᠬᠡᠢᠯ ᠡ︕ ︵ᠠᠰᠢᠭ = ᠬᠤ ᠳᠠᠯᠳᠤᠬ ᠤ ᠦᠨ ᠡ − ᠦᠪᠡᠷ ᠦᠨ ᠥᠷᠲᠡᠭ ︶ (3) 𝑥 = 1500 ᠦᠶ ᠡᠰ᠂ ᠡᠳᠦ ᠷ ᠪᠦᠷᠢ ᠦᠢ ᠯᠡᠳᠪᠦᠷᠢᠯᠡᠬᠦ ᠦᠪᠡᠷ ᠦᠨ ᠦᠷᠳᠡ ᠭ ᠪ ᠠ ᠠ ᠰᠢᠭ ᠢ ᠤᠯ︕ A B ᠦᠪᠡᠷ ᠦ ᠨ ᠦᠷ ᠳᠡᠭ 2 3 ᠬᠤᠳᠠ ᠯᠳᠤᠬ ᠤ ᠦᠨ ᠡ 2.3 3.5