ᠠᠷᠪᠠᠨ ᠲᠦᠷᠪᠡᠳᠤᠭᠡᠷ ᠪᠦᠯᠦᠭ ᠪᠦᠬᠦᠯᠢ ᠢᠯᠡᠷᠭᠡᠢᠯᠡᠯ ᠤᠨ ᠦᠷᠡᠵᠢᠬᠦᠯᠬᠦ ᠤᠢᠯᠡᠳᠤᠯ ᠬᠢᠭᠡᠳ ᠦᠷᠡᠵᠢᠭᠳᠡᠬᠦᠨ ᠪᠤᠯᠭᠠᠨ ᠵᠠᠳᠠᠯᠬᠤ （ᠨᠢᠭᠡᠴᠢ ᠶᠢᠨ ᠪᠠᠢᠴᠠᠭᠠᠯᠲᠠ 2） .pdf

ᠠᠷᠪ ᠠᠨ ᠲ ᠦᠷᠪ ᠡᠳᠦᠭ ᠡᠷ ᠪᠥᠯ ᠥᠭ ᠦ ᠨ 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 ᠂ ᠨ ᠠᠢᠲ ᠡ 30 ᠬᠤ ᠪᠢ ︶ 1.ᠲ ᠤᠤᠷᠠᠬ ᠢ ᠪᠤ ᠳᠤᠯᠭ ᠠ ᠳᠤ ᠵᠦ᠋ᠪ ᠪᠤᠯ ᠬᠤ ᠨ ᠢ ( ) A. ( 𝑚 + 𝑛) 2= 𝑚 2+ 𝑛 2 B. ( 𝑚𝑛 ) 3 = 𝑚 3𝑛 3 C. ( 𝑚 3) 2= 𝑚 5 D. 𝑚 ∙𝑚 2= 𝑚 2 2. ᠬᠡᠷᠪᠡ 𝑎 2− 2𝑎 − 3= 0 ᠪᠤᠯ᠂ ᠢᠯ ᠡᠷᠬᠡᠢ ᠯᠡᠯ 𝑎2 ∙ 2− 𝑎3 ᠦᠨ ᠬ ᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠨ ᠢ ( ) A. 0 B. − 𝑎 23 C. 2 D. − 12 3. ᠵᠢᠷᠤ ᠭ ᠮ ᠡᠳᠤ᠂ ᠤᠷᠳ ᠤ ᠥ ᠷᠭᠡᠨ ᠨᠢ ᠲᠤᠰ ᠲᠤ ᠰ 𝑎 , b ᠪᠤᠯᠬᠤ ᠤᠷᠳ ᠤ ᠲᠦ᠋ ᠷᠪᠡᠯ ᠵᠢᠨ ᠦ ᠬᠦᠷ ᠢᠶᠡᠨ ᠦ ᠤ ᠷᠳᠤ ᠨᠢ 14᠂ ᠲᠠᠯ ᠠᠪᠠᠢ ᠨᠢ 10᠂ ᠲᠡᠭᠡᠪᠡᠯ 𝑎 3 𝑏+ 𝑎𝑏 3+ 2𝑎 2𝑏 2 ᠦᠨ ᠬᠡᠮᠵ ᠢᠭᠳᠡᠯ ᠨ ᠢ ( ) A. 70 B. 140 C. 2560 D. 490 4. ᠲᠤᠤ ᠷᠠᠬᠢ ᠦᠷ ᠡᠵᠢᠭ ᠳᠡᠭᠦ ᠨ ᠪᠤ ᠯᠭᠠᠨ ᠵᠠᠳ ᠠᠯᠤᠭᠰᠠᠨ ᠳᠤ ᠵᠦ᠋ᠪ ᠪᠤᠯ ᠬᠤ ᠨ ᠢ ( ) ① − 4𝑚 3+ 12 𝑚 2= −𝑚 2(4 𝑚 − 12 ) ② 𝑥 4− 1= (𝑥 2+ 1) (𝑥 2− 1) ③ 𝑥 2+ 2𝑥 + 4= (𝑥 + 2) 2 ④ ( 𝑎 2 + 𝑏 2 ) 2 − 4𝑎 2𝑏 2= (𝑎 + 𝑏) 2(𝑎 − 𝑏) 2 A. ① B. ② C. ③ D. ④ 5.ᠤ ᠯᠠᠨᠬᠡᠰ ᠢᠬᠦᠨᠳ ᠤ ( 𝑥 2 + 𝑚𝑥 + 8) ᠪᠠ ( 𝑥 2 − 3𝑥 + 𝑛) ᠤ ᠦᠷᠡᠵ ᠢᠪᠦᠷᠢ ᠳᠦ 𝑥 2 ᠪᠠ 𝑥 3 ᠠ ᠭᠤᠯᠬᠤ ᠬᠡᠰᠢ ᠬᠦᠨ ᠪᠠᠢ ᠬᠤ ᠦᠭᠡᠢ ᠂ ᠲ ᠡᠭᠡᠪᠡᠯ m , n ᠤ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠨ ᠢ ( ) A. − 1 , 1 B . 2 , − 1 C. 2 , 3 D. 3 , 1 6. ᠬᠡᠷᠪᠡ m ᠨᠢ 0 ᠡ ᠴᠡ ᠶᠡᠬᠡ ᠪᠤ ᠯᠬᠤ ᠪᠦᠬᠦ ᠯᠢ ᠲᠤᠭ ᠠ ᠪᠤᠯ᠂ (𝑚 + 1) 2− (𝑚 − 1) 2 ᠨᠢ ᠯ ᠠᠪᠳᠠᠢ ( ) A. 2 ᠤᠨ ᠲᠠᠬ ᠢᠨ ᠲ ᠤᠭ ᠠ B. 4 ᠶᠢᠨ ᠲᠠᠬᠢ ᠨ ᠲ ᠤᠭ ᠠ C. 6 ᠶᠢᠨ ᠲ ᠠᠬᠢᠨ ᠲᠤ ᠭ ᠠ D. 16 ᠶᠢᠨ ᠲ ᠠᠬᠢᠨ ᠲᠤ ᠭ ᠠ 7. ᠬᠡᠷᠪᠡ 𝑎 ᠨᠢ ᠶᠤᠰᠤ ᠳᠠᠢ ᠲᠤᠭ ᠠ ᠪ ᠤᠯ ᠂ ᠪ ᠦᠬᠦᠯ ᠢ ᠢᠯ ᠡᠷᠭᠡᠢ ᠯᠡᠯ 𝑎 2 (𝑎 2− 2) − 2𝑎 2+ 4 ᠶᠢᠨ ᠬ ᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠨ ᠢ ( ) A. ᠰᠦ᠋ ᠭᠡᠷᠬᠦ ᠪᠤᠰ ᠤ ᠲ ᠤᠭ ᠠ B. ᠬᠡᠵᠢ ᠶᠡᠳᠡ ᠡ ᠶᠡᠷᠬᠦ ᠲᠤ ᠭ ᠠ C. ᠬᠡᠵᠢ ᠶᠡᠳᠡ ᠰᠦ᠋ ᠭᠡᠷᠬᠦ ᠲᠤ ᠭ ᠠ D. 0 ᠲ ᠡᠢ ᠲ ᠡᠩᠴ ᠡᠭᠦᠦ ᠪ ᠤᠰᠤ 8. ᠦᠭ ᠭᠦᠭᠳ ᠡᠭᠰᠡᠨ ᠨᠢ᠂ 𝑥 = 3y + 5 ᠪ ᠦᠭᠡᠳ 𝑥 2 − 7𝑥𝑦 + 9𝑦 2= 24 ᠂ ᠲᠡᠭᠡᠪᠡᠯ 𝑥 2𝑦 − 3𝑥 𝑦 2 ᠦᠨ ᠬ ᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠨ ᠢ ( ) A. 0 B . 1 C . 5 D. 12 9. ᠦᠭ ᠭᠦᠭᠳ ᠡᠭᠰᠡᠨ ᠨᠢ 2 𝑎 = 3 , 2 𝑏= 6 , 2 𝑐= 12 ᠲᠡᠭᠡᠪᠡᠯ 𝑎 , b , c ᠶᠢᠨ ᠬᠤ ᠯᠪᠤᠭᠳ ᠠᠯ ᠤ ᠨ ᠢᠯ ᠡᠷᠬᠡᠢ ᠯᠡᠯ᠄ ① b = 𝑎+ 1 ② c = 𝑎+ 2 ③ 𝑎 + c= 2b ④ b + c= 2𝑎 + 3 ᠵ ᠦ᠋ᠪ ᠪ ᠤᠯᠬᠤ ᠶᠢᠨ ᠪ ᠤᠳᠠᠭᠠᠯ ᠢ ᠲ ᠤᠭ ᠠ ᠨᠢ ( ) A. 1 B. 2 C. 3 D. 4 10. ᠵᠢᠷᠤ ᠭ ᠮ ᠡᠳᠤ᠂ ᠲᠠᠯ ᠠ ᠶᠢ ᠨ ᠤ ᠷᠳᠤ ᠨᠢ 𝑎 cm ᠪᠤᠯᠬᠤ ᠶᠡᠬᠡ ᠲᠡᠭ ᠰᠢ ᠲ ᠦ᠋ᠷᠪᠡᠯ ᠵᠢᠨ ᠦ ᠲ ᠤᠳᠤᠷ ᠠ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠤᠷ ᠳᠤ ᠨ ᠢ bc m( 𝑎 b) ᠪᠤᠯᠬᠤ ᠭᠤᠷᠪᠠ ᠨ ᠵᠢ ᠵᠢᠭ ᠲᠡᠭᠰ ᠢ ᠲᠦ᠋ ᠷᠪᠡᠯ ᠵᠢᠨ ᠢ ᠲ ᠠᠯᠪᠢ ᠵᠠᠢ᠂ ᠵᠢᠵᠢ ᠭ ᠲᠡᠭᠰ ᠢ ᠲ ᠦ᠋ᠷᠪᠡᠯ ᠵᠢᠨ ᠦ ᠪ ᠥᠷᠬᠥ ᠵᠤ ᠴᠢᠳ ᠠᠭᠰᠠᠨ ᠦᠭᠡᠢ ᠬᠡᠰ ᠡᠭ ᠦᠨ ᠲᠠᠯᠠᠪᠠ ᠢ ᠨ ᠢ 4 𝑐𝑚 2᠂ ᠲᠡᠭᠡᠪᠡᠯ 𝑎 2− 2𝑎𝑏 + 𝑏 2 ᠦᠨ ᠬ ᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠨ ᠢ ( ) A. 2 B. 4 C. 6 D. 8 ᠬᠣᠶ ᠠᠷ᠂ ᠬᠠᠭᠠᠯ ᠲᠠ ᠳ ᠤ ᠲ ᠠᠭᠯᠠᠬᠤ ᠰᠡᠲᠤ ᠪ᠃ ︵ᠰᠡᠲ ᠤᠪ ᠪ ᠦᠷᠢ 3 ᠬ ᠤᠪᠢ ᠂ ᠨᠠᠢ ᠲᠡ 24 ᠬᠤ ᠪᠢ 11. ᠬᠢᠯ ᠪᠠᠷᠴᠢ ᠯᠠ᠄ (2 𝑎− 3) (2 𝑎+ 3) − (𝑎 − 1) 2= ______ _ 12. ᠬᠡᠷᠪᠡ 𝑎 3 𝑚 +𝑛 = 54 , 𝑎 𝑚 = 3 ᠂ ᠲᠡᠭᠡᠪᠡᠯ 𝑎 𝑛= ___ ___ ___ _ 13. ᠦᠭ ᠭᠦᠭᠳ ᠡᠭᠰᠡᠨ ᠨᠢ 𝑎− 2b = −2 ᠂ ᠲᠡᠭᠡᠪᠡᠯ ᠠ ᠯᠭᠧᠪᠷᠠ ᠶᠢᠨ ᠢᠯᠡᠷᠬ ᠡᠢᠯᠡᠯ −2 𝑎 2 + 8𝑎𝑏 − 8𝑏 2 ᠦᠨ ᠬ ᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠨ ᠢ _____ __ 14. ᠦᠷᠡᠵ ᠢᠭᠳᠡᠬ ᠦᠨ ᠪ ᠤᠯᠭᠠᠨ ᠵᠠᠳ ᠠᠯ᠄ 2𝑥 3− 6𝑥 2+ 4𝑥 = ___ ___ ___ ___ 15. ᠬᠡᠷᠪᠡ 𝑥 ᠤᠨ ᠲᠤᠬᠠᠢ ᠬᠤᠶ ᠠᠳᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠶᠢᠨ ᠭᠤᠷᠪᠠ ᠨᠭᠡᠰᠢ ᠭᠦᠨᠲ ᠦ 9 𝑥 2 + 2( 𝑎− 4) 𝑥+ 16 ᠪᠤᠯ ᠨᠢᠭᠡᠨ ᠬᠦᠢ ᠴᠡᠳ ᠺᠦᠸᠠᠳ᠋ ᠤᠨ ᠢᠯᠡᠷᠬ ᠡᠢᠯᠡᠯ᠂ ᠲᠡᠭᠡᠪᠡᠯ 𝑎 ᠦᠨ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠨᠢ _______ 16. ᠬᠡᠷᠪᠡ 𝑎− b= 2 , a − c= 1 ᠂ ᠲ ᠡᠭᠡᠪᠡᠯ (2 𝑎− 𝑏− 𝑐) 2+ (𝑐 − 𝑎) 2= ___ ___ ___ 17. ᠦᠭ ᠭᠦᠭᠳ ᠡᠭᠰᠡᠨ ᠨ ᠢ ᠄𝑎 2+ 𝑏 2 + 𝑐 2 − 𝑎𝑏 − 𝑏𝑐 − 𝑐𝑎 = 0 ᠂ ᠲᠡᠭᠡᠪᠡᠯ 𝑎 , b , c ᠦᠨ ᠶ ᠡᠬᠡ ᠪ ᠠᠭ ᠠ ᠶᠢᠨ ᠬᠤ ᠯᠪᠤᠭᠳ ᠠᠯ ᠨ ᠢ ____ 18. 𝑥 , y ᠨᠢ ᠲᠤᠷ ᠠ ᠶᠢ ᠨ ᠪ ᠤᠳᠠᠳ ᠤ ᠲ ᠤᠭ ᠠ ᠪᠦᠭᠡᠳ ᠪᠤᠳ ᠤᠯᠭ ᠠ ᠶᠢ ᠨ ᠲ ᠦ᠋ᠷᠢ ᠮ᠄ 𝑥∗ y= (𝑥 + 1) (𝑦 + 1) − 1 ᠭ ᠡᠵᠦ ᠲᠤᠭᠳ ᠠᠭᠠᠵᠠᠢ᠂ ᠲᠡᠭᠡᠪᠡᠯ ᠳᠠᠷᠠᠭᠠᠬ ᠢ ᠲ ᠠᠪᠤᠨ ᠳᠦ᠋ ᠩᠨᠡᠯ ᠳᠡ᠄ ① 𝑥 ∗𝑦 = 𝑦∗ 𝑥 ② 𝑥 ∗( 𝑦+ 𝑧) = 𝑥∗ 𝑦+ 𝑥∗ 𝑧 ③ (𝑥 + 1) ∗( 𝑥− 1) = (𝑥 ∗𝑥 )− 1 ④ 𝑥 ∗0 = 0 ⑤ ( 𝑥+ 1) ∗( 𝑥+ 1) = 𝑥∗ 𝑥+ 2∗ 𝑥+ 1 ᠠᠴᠠ ᠵᠦ᠋ᠪ ᠪ ᠤᠯᠬᠤ ᠶᠢᠨ ᠨᠣᠮᠧᠷ ᠢ ᠪᠢᠴ ᠢᠪᠡᠯ ______ ᠭᠤᠷᠪᠠ᠂ ᠪᠣᠳ ᠣᠵᠤ ᠬᠠᠷᠢ ᠭᠤᠯᠬᠤ ᠰᠡᠲᠤ ᠪ᠃ ( ᠨᠠᠢ ᠲᠡ 66 ᠬᠤ ᠪᠢ) 19. ( 10 ᠬᠤ ᠪᠢ) (1 )(− 2𝑥 − 1) 2− 4( 𝑥− 1) (𝑥 + 2) (2) [( 𝑥− 2𝑦 ) 2 − 𝑥( 𝑥− 4𝑦 )− 8𝑥𝑦 ]÷ 4𝑦 20. ( 10 ᠬᠤ ᠪᠢ ) ᠦᠷᠡᠵ ᠢᠭᠳᠡᠭᠦ ᠨ ᠪᠣ ᠯᠭᠠᠨ ᠵᠠᠳ ᠠᠯ (1) 𝑚 3(𝑥 − 2) + 𝑚( 2− 𝑥) (2) ( 3𝑥 + 𝑦) 2− (𝑥 − 3𝑦 ) 2 21. ( 10 ᠬᠤ ᠪᠢ) ᠬᠢᠯᠪᠠ ᠷᠴᠢᠯ ᠠᠵᠤ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠢ ᠨᠢ ᠣ ᠯ᠃ (3 𝑎 5 𝑏 3 + 𝑎 4 𝑏 2 )÷ (− 𝑎 2 𝑏) 2− (2 + 𝑎) (2 − 𝑎) − 𝑎( 𝑎− 5𝑏 ) ᠂ ᠡᠭᠦᠨ ᠳᠦ 𝑎𝑏 = − 12 22. ( 1 2 ᠬ ᠤᠪᠢ ) ᠤ ᠩᠰᠢ ᠨ ᠵ ᠠᠳᠠᠯᠬ ᠤ ᠰ ᠡᠲᠥᠪ ᠄ ᠤᠬᠠᠭᠳ ᠠᠬᠤᠨ ᠄ ᠬ ᠡᠷᠪᠡ ᠨᠢᠭᠡ ᠲᠣ ᠭᠠᠨ ᠤ ᠺ ᠦᠸᠠ ᠳ᠋ ᠨᠢ − 1 ᠲ ᠠᠢ ᠲ ᠡᠩᠴ ᠡᠭᠦᠦ ᠪᠣᠯ᠂ 𝑖 2 = −1 ᠭᠡᠵᠦ ᠲᠡ ᠮᠲᠡᠭᠯᠡ ᠨ ᠡ᠂ ᠡᠨᠡ ᠲ ᠣᠭ ᠠ 𝑖 ᠶᠢ ᠬᠠᠭᠤ ᠷᠮᠠᠭ ᠲ ᠣᠭᠠᠨ ᠤ ᠨᠢᠭᠡᠴ ᠢ ᠭ ᠡᠨ ᠡ ᠃ ᠲᠡᠭᠡᠪᠡᠯ 𝑎 + 𝑏𝑖 (𝑎 ,𝑏 ᠨᠢ ᠪᠣᠳ ᠠᠲᠤ ᠲᠣᠭ ᠠ) ᠮᠡᠲᠦ ᠶᠢᠨ ᠲᠣᠭ ᠠ ᠶᠢ ᠬᠠᠮ ᠲᠣ ᠭ ᠠ ᠭᠡᠨ ᠡ ᠂ 𝑎 ᠶᠢ ᠡᠨᠡ ᠬᠠᠮ ᠲᠣ ᠭᠠᠨ ᠤ ᠪᠣᠳᠠᠲ ᠣ ᠬ ᠡᠰᠡᠭ᠂ 𝑏 ᠶᠢ ᠡᠨᠡ ᠬᠠᠮ ᠲᠣ ᠭᠠᠨ ᠤ ᠬᠠᠭᠤ ᠷᠮᠠᠭ ᠬᠡᠰ ᠡᠭ ᠭᠡᠨ ᠡ ᠃ ᠡᠭᠦᠨ ᠦ ᠨ ᠡᠮᠡᠬᠦ᠂ ᠬᠠᠰ ᠤᠬᠤ᠂ ᠦᠷᠡᠵ ᠢᠭᠦᠯᠬ ᠦ ᠪᠣᠳ ᠤᠯᠭ ᠠ ᠨᠢ ᠪᠦᠬ ᠦᠯᠢ ᠢᠯᠡᠷᠬ ᠡᠢᠯᠡᠯ ᠦᠨ ᠨᠡᠮᠡᠬ ᠦ᠂ ᠬ ᠠᠰᠤᠬᠤ᠂ ᠦ ᠷᠡᠵᠢ ᠭᠦᠯᠬᠦ ᠪᠣᠳ ᠤᠯᠭ ᠠ ᠲ ᠠᠢ ᠲ ᠦ᠋ᠰᠲ ᠡᠢ᠃ ᠵᠢᠱᠢ ᠶᠡᠯᠡᠪᠡᠯ᠄ ᠪᠤᠳ ᠤ᠄ (2 + 𝑖) + (3 − 4𝑖) = 5− 3𝑖 (1) ᠬᠠᠭᠠᠯ ᠲᠠ ᠲᠠᠭᠯ ᠠ᠄ 𝑖 3 = ___ ___ , 2𝑖 4= ___ ___ __ (2) ᠪᠤᠳ ᠤ᠄ ① (2 + 𝑖)( 2− 𝑖) ② (2 + 𝑖) 2 (3) ᠬᠡᠷᠪᠡ ᠬᠤ ᠶᠠᠷ ᠬᠠᠮ ᠲᠣ ᠭ ᠠ ᠨᠢ ᠲ ᠡᠩᠴ ᠡᠭᠦᠦ᠂ ᠲᠡᠭᠡᠪᠡᠯ ᠲᠡᠳ ᠡᠭᠡᠷ ᠦᠨ ᠪᠣᠳᠠ ᠲᠤ ᠲ ᠣᠭᠠᠨ ᠤ ᠬ ᠡᠰᠡᠭ ᠪᠠ ᠬ ᠠᠭᠤᠷᠮᠠᠭ ᠬᠡᠰ ᠡᠭ ᠨᠢ ᠲᠣ ᠰ ᠲ ᠣᠰ ᠲᠡᠩ ᠴᠡᠭᠦ ᠦ᠂ ᠲ ᠣᠣᠷᠠᠬ ᠢ ᠠ ᠰᠠᠭᠤ ᠯᠲᠠ ᠶᠢ ᠬ ᠠᠷᠢᠭᠤ ᠯ᠄ ᠦᠭ ᠭᠦᠭᠳ ᠡᠭᠰᠡᠨ ᠨᠢ᠄ (𝑥 + 3𝑦 )+ 3𝑖 = (1 − 𝑥) − 𝑦𝑖 ( 𝑥 ,𝑦 ᠨᠢ ᠪᠣᠳᠠᠲ ᠤ ᠲᠣᠭ ᠠ ) ᠂ 𝑥 ,𝑦 ᠦᠨ ᠬᠡᠮᠵ ᠢᠭᠳᠡᠯ ᠢ ᠣ ᠯ᠃ (4) ᠳᠤ ᠷᠰᠢᠵᠤ ᠦᠵᠡ᠄ 𝑖 2 = −1 ᠲᠣ ᠰ ᠮ ᠡᠳᠡᠯᠭᠡ ᠶᠢ ᠠᠰᠢ ᠭᠯᠠᠵᠤ᠂ 𝑚 2+ 25 (𝑚 ᠨᠢ ᠪᠣᠳᠠᠲ ᠤ ᠲ ᠣᠭ ᠠ ) ᠶᠢ ᠬᠤ ᠶᠠᠷ ᠬᠠᠮ ᠲᠣ ᠭᠠᠨ ᠤ ᠦᠷᠡ ᠵᠢᠪᠦ ᠷᠢ ᠶ ᠢᠨ ᠬ ᠡᠯᠪᠡᠷᠢ ᠲ ᠠᠢ ᠦ ᠷᠡᠵᠢ ᠭᠳᠡᠬᠦ ᠨ ᠪᠣ ᠯᠭᠠᠨ ᠵᠠᠳ ᠠᠯ᠃ 23. (1 2 ᠬ ᠤᠪᠢ ) ᠵᠢ ᠷᠤᠭ ᠮᠡᠲ ᠦ᠂ 𝑎 × 𝑎 , 𝑏 × 𝑏( 𝑎 𝑏) ᠦᠨ ᠲᠡᠭᠰ ᠢ ᠳᠦ᠋ ᠷᠪᠡᠯ ᠵᠢᠨ ᠬᠠᠪᠲ ᠠᠰᠤ ᠪᠠ 𝑎× 𝑏 ᠶᠢᠨ ᠤᠷᠲ ᠤ ᠳᠦ᠋ ᠷᠪᠡᠯ ᠵᠢᠨ ᠦ ᠬᠠᠪᠲ ᠠᠰᠤ ᠬᠡᠳᠦᠨ ᠬᠡᠰ ᠡᠭ ᠪᠠᠢ ᠨ ᠠ᠃ ᠡᠳ ᠡᠭᠡᠷ ᠬᠠᠪᠲ ᠠᠰᠤ( ᠨᠢᠭᠡ ᠵᠦ᠋ ᠢᠯ ᠦᠨ ᠬᠠᠪᠲ ᠠᠰᠤ ᠶᠢ ᠠ ᠳᠠᠭ ᠲᠤ ᠪᠠᠨ ᠨᠢᠭᠡ ᠤᠳ ᠠᠭ ᠠ ᠬᠡᠷᠡᠭᠯ ᠡᠨ ᠡ ) ᠶᠢ ᠬᠡᠷᠡᠭ ᠯᠡᠵᠦ ᠨᠢᠭᠡ ᠤᠷᠲ ᠤ ᠳᠦ᠋ ᠷᠪᠡᠯ ᠵᠢᠨ ᠢ ᠵᠦ᠋ᠢ ᠵᠦ ᠭ ᠠᠷᠭ ᠠ ᠃ ( ᠬᠤᠶ ᠠᠷ ᠬ ᠠᠪᠲᠠ ᠰᠤ ᠶ ᠢᠨ ᠬᠤᠭᠤ ᠷᠤᠨᠳ ᠤ ᠳᠠᠪᠬᠤᠴ ᠠᠬᠤ ᠦᠭᠡᠢ ᠂ ᠵᠠᠢ ᠵᠠᠪᠰ ᠠᠷ ᠦᠭᠡᠢ ᠂ ᠵᠦ᠋ᠢ ᠵᠤ ᠭᠠᠷᠭᠠᠭ ᠰᠠᠨ ᠵᠢᠷᠤ ᠭ ᠤᠨ ᠤᠷᠤ ᠮᠦ᠋ ᠷ ᠢᠶ ᠡᠨ ᠦᠯ ᠡᠳᠡᠭᠡ )᠃ ᠵᠦ᠋ ᠢᠵᠤ ᠭᠠᠷᠭᠠᠭᠰ ᠠᠨ ᠤᠷᠲ ᠤ ᠳᠦ᠋ ᠷᠪᠡᠯ ᠵᠢᠨ ᠦ ᠲ ᠠᠯᠠᠪᠠ ᠢ ᠶᠢ 2 𝑎 2 + 5𝑎𝑏 + 2𝑏 2 ᠪᠠ ᠢᠯᠭ ᠠ ᠪᠠ ᠰᠠ ᠵᠦ᠋ᠢ ᠵᠤ ᠭᠠᠷᠭᠠᠭ ᠰᠠᠨ ᠤᠷᠲ ᠤ ᠳᠦ᠋ ᠷᠪᠡᠯ ᠵᠢᠨ ᠦ ᠤ ᠷᠲᠤ ᠦᠷᠭᠡᠨ ᠢ ᠨᠢ ᠪᠢᠴ ᠢᠵᠦ ᠭᠠᠷᠭ ᠠ᠃ 24. (1 2 ᠬ ᠤᠪᠢ ) ᠰ ᠤᠷᠴᠤ ᠦ ᠩᠭᠡᠷᠡᠭ ᠰᠡᠨ ᠮᠡᠳ ᠡᠯᠭᠡ ᠪᠡᠨ ᠠᠰ ᠢᠭᠯᠠᠵᠤ ᠬᠡᠯ ᠪᠡᠷᠢ ᠨᠢ ᠭᠤᠶ ᠤ ᠰ ᠠᠢᠬᠠᠨ ᠲᠡᠩ ᠴᠡᠳ ᠬᠡᠯ ᠢ ᠭᠠᠷᠭᠠ ᠵᠤ ᠳᠡᠢ ᠯᠦᠨ ᠡ᠄ 𝑎 2 + 𝑏 2 + 𝑐 2 − 𝑎𝑏 − 𝑏𝑐 − 𝑎𝑐 = 12 [ (𝑎 − 𝑏) 2+ (𝑏 − 𝑐) 2+ (𝑎 − 𝑐) 2] ᠂ ᠲ ᠣᠰ ᠲᠡᠩ ᠴᠡᠳ ᠬᠡᠯ ᠦᠨ ᠵᠡᠭᠦᠨ ᠡᠴᠡ ᠪᠠᠷᠠᠭᠤᠨᠰᠢ ᠬᠢ ᠬ ᠤᠪᠢᠷᠠᠭᠠᠯ ᠲᠠ ᠨᠢ ᠲ ᠡᠭᠰᠢ ᠬᠡᠮᠲ ᠦ ᠣ ᠨᠴᠠᠯᠢ ᠭ ᠢ ᠪᠠᠷᠢ ᠮᠲᠠᠯᠠᠵ ᠤ᠂ ᠪᠠ ᠰᠠ ᠲᠣᠭᠠᠨ ᠤ ᠭ ᠤᠶᠤ ᠰᠠᠢ ᠬᠠᠨ ᠢ ᠢᠯᠡᠷᠡᠭᠦ ᠯᠵᠦ ᠪᠠᠢ ᠨ ᠠ᠃ (1) ᠡᠨᠡ ᠲᠡᠩ ᠴᠡᠳ ᠬᠡᠯ ᠦᠨ ᠵᠥ ᠪ ᠪᠣ ᠯᠬᠤ ᠡᠰᠡᠬ ᠦ ᠶᠢ ᠰᠢᠯᠭᠠᠨ ᠲᠣ ᠳᠣᠷᠬ ᠠᠢᠯᠠ᠃ (2) ᠬᠡᠷᠪᠡ 𝑎 = 2011 , 𝑏= 2012 , 𝑐= 2013 ᠪᠣ ᠯ᠂ 𝑎 2 + 𝑏 2 + 𝑐 2 − 𝑎𝑏 − 𝑏𝑐 − 𝑎𝑐 ᠤᠨ ᠬ ᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠣ ᠯᠵᠤ ᠴᠢᠳ ᠠᠨ ᠠ ᠦᠦ ︖ (3) ᠬ ᠡᠷᠪᠡ 𝑎− 𝑏= 35 , 𝑏− 𝑐= 35 , 𝑎 2 + 𝑏 2 + 𝑐 2 = 1 ᠪᠣᠯ᠂ 𝑎𝑏 + 𝑏𝑐 + 𝑎𝑐 ᠦᠨ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠢ ᠣ ᠯ᠃