2019 ᠣᠨ ᠤ ᠣᠷᠳᠣᠰ ᠬᠤᠲᠠ ᠶᠢᠨ ᠳᠤᠮᠳᠠᠳᠤ ᠰᠢᠯᠭᠠᠯᠲᠠ ᠶᠢᠨ ᠰᠡᠲᠤᠪ

ᠨᠢᠭᠡ᠂ ᠭᠠᠭᠴᠠ ᠬᠠᠷᠢ ᠭᠤᠯᠲᠠ ᠰᠤᠩᠭᠤᠬ ᠤ ᠰ ᠡᠲᠤᠪ᠃ ( ᠲᠤᠰ ᠶᠡᠭᠡ ᠰᠡᠲᠤᠪ ᠲᠤ 10 ᠪᠠᠭ ᠠ ᠰ ᠡᠲᠤᠪ ᠲᠠᠢ᠂ ᠪᠠᠭ ᠠ ᠰᠡᠲᠤᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪ ᠢ᠂ ᠨᠠᠢᠲᠡ 30 ᠬᠤᠪᠢ ) 1. ᠶᠤᠰᠤᠲᠠᠢ ᠲᠣᠭ ᠠ − 13 ᠶᠢᠨ ᠡᠰᠡᠷ ᠬᠦ ᠲ ᠣᠭ ᠠ ᠨᠢ ( ) A. −3 B. − 13 C. 13 D. 3 2. ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠲᠥᠷᠪ ᠡᠨ ᠵᠢᠷ ᠤᠭ ᠲᠥ ᠷᠰᠤ ᠳᠤ᠂ ᠨᠤᠭ ᠤᠯᠬᠤ ᠪ ᠠᠷ ᠲᠠ ᠮᠵᠢᠨ ᠵᠢᠷ ᠤᠭ ᠳᠠᠬᠢ ᠭᠧᠣᠮᠧᠲᠷ ᠢ ᠶᠢᠨ ᠪᠡᠶ ᠡ ᠶᠢ ᠬᠦᠷᠢᠶᠡᠯ ᠡᠨ ᠪ ᠦᠳᠥ ᠭᠡᠵᠤ ᠲᠠᠢ ᠯᠬᠦ ᠨᠢ ( ) A. B. C. D. 3. ᠵᠢᠬᠦᠷ ᠲᠡᠨ ᠦ ᠬᠢ ᠵᠢᠭ ᠦᠨ ᠸᠢᠷᠦ᠋ ᠰ ᠦᠨ ᠷᠠ ᠳᠢᠤᠰ ᠨᠢ ᠪᠠᠷ ᠤᠭ 0.00 0 0 00 45 m᠂ ᠲᠡᠭᠦᠨ ᠦ ᠳᠢ ᠶᠠᠮᠧᠲᠷ ᠢ ᠰ ᠢᠨᠵᠢ ᠯᠡᠬᠦ ᠤᠬ ᠠᠭᠠᠨᠴᠢ ᠲᠣᠭ ᠠ ᠲᠡ ᠮᠳᠡᠭᠯᠡᠬᠦ ᠠᠷᠭ ᠠ ᠪᠠᠷ ᠢᠯ ᠡᠷᠭᠡᠢᠯ ᠡᠪᠡᠯ ( ) A. 0.9 × 10 −7 𝑚 B. 9× 10 −7 𝑚 C. 9× 10 −6 𝑚 D. 9× 10 7𝑚 4. ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ᠂ ᠲᠡᠭᠰᠢ ᠲᠥ ᠷᠪᠡᠯ ᠵᠢᠨ AB CD ᠶᠢᠨ ᠭᠠᠳᠠ ᠨ ᠠ ᠲᠠᠯ ᠠ ᠳ ᠤ ᠲᠡ ᠩᠴᠡᠬᠦ ᠦ ᠲᠠ ᠯᠠᠳᠤ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ AB E ᠶᠢ ᠪᠠᠢᠭ ᠤᠯᠤᠨ ᠠ᠂ ᠲᠡ ᠭᠡᠪᠡᠯ ∠B ED ᠨᠢ ( ) A . 15° B. 35° C. 45° D. 55° 5. ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠪᠣᠳᠣᠯ ᠭ ᠠ ① √9 = ±3 ② 3 𝑎 2 − 2𝑎 = 𝑎 ③ (2 𝑎 2 ) 3 = 6𝑎 6④ 𝑎 8 ÷ 𝑎 4 = 𝑎 2 ⑤ √ −27 3 = −3 ᠡᠭᠦᠨ ᠦ ᠳᠣᠲᠣᠷ ᠠ ᠠᠴᠠ ᠳᠤᠷ ᠠ ᠶᠢᠨ ᠨᠢᠭᠡ ᠪᠣᠳᠣᠯ ᠭ ᠠ ᠶᠢ ᠰᠤᠭᠤᠯ ᠵᠤ ᠠᠪᠬᠤ ᠳᠤ᠂ ᠪᠣᠳᠣᠯ ᠭ ᠠ ᠶᠢᠨ ᠳᠦ᠋ᠩ ᠨᠢ ᠵᠥᠪ ᠪᠣᠯᠬᠤ ᠶᠢᠨ ᠲᠣᠬ ᠢᠶᠠᠯᠳᠤᠮᠵᠢ ᠨᠢ ( ) A. 15 B. 25 C. 35 D. 45 6. ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠬᠥᠰ ᠥᠨᠥᠭ ᠳᠦ ᠪᠣᠯ ᠨᠢᠭᠡᠨ ᠠᠩᠭᠢ ᠶᠢᠨ10 ᠰᠤᠷᠤᠭ ᠴᠢ ᠶᠢᠨ ᠳᠤᠮ ᠳᠠᠳᠤ ᠰᠢᠯᠭ ᠠᠯᠲᠠ ᠶᠢᠨ ᠲᠠᠮᠢᠷ ᠤᠨ ᠬᠢᠷᠢ ᠪᠦᠳᠦ ᠮᠵᠢ ᠶᠢᠨ ᠪᠦᠷᠢ ᠳᠭᠡᠯ ᠪ ᠣᠯᠣᠨ ᠠ᠄ ᠬᠡᠷᠪᠡ ᠬᠢᠷᠢ ᠪᠦᠳ ᠦᠮᠵᠢ ᠶᠢᠨ ᠳᠤᠮ ᠳᠠᠴᠢ ᠲᠣᠭ ᠠ ᠨᠢ 23 ᠂ ᠳᠤᠮᠳᠠ ᠪᠠᠢᠷ ᠢ ᠶᠢᠨ ᠲᠣᠭ ᠠ ᠨᠢ 𝑎᠂ ᠮᠣᠳ᠋ ᠨᠢ 𝑏 ᠪ ᠣᠯ᠂ 𝑎− 𝑏 ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠨᠢ ( ) A. −5 B. −2 .5 C. 2.5 D. 5 7. ᠵᠢᠷᠤᠭ ᠮᠡᠳᠤ ᠂ ᠵᠢᠰᠢᠬᠦᠯᠵ ᠢᠨ AB CD ᠳᠦ᠂ ∠B DC = 47 °42 ′᠂ ᠣᠣ ᠢ ᠭᠣᠷᠳᠢᠭ ᠢᠶᠠ ᠷ ᠵᠢᠷ ᠤᠭ ᠪᠠᠢᠭ ᠤᠯᠤᠭᠰᠠᠨ ᠣᠷᠣ ᠮ ᠢ ᠦᠨᠳᠤᠰᠤ ᠯᠡᠪᠡᠯ᠂ α ᠶᠢᠨ ᠭᠷᠠᠳᠦ᠋ᠰ ᠤᠨ ᠲᠣᠭ ᠠ ᠨᠢ ( ) A. 67 °29 ′ B. 67°9′ C. 66 °2 9′ D. 66°9′ 8. ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠬᠡᠯᠡᠯᠭᠡ ᠡᠴᠡ ᠵᠥᠪ ᠪᠣ ᠯᠬᠤ ᠨᠢ ( ) ᠬᠢᠷᠢ ᠪᠦᠳᠦ ᠮᠵᠢ ( ᠬᠤᠪ ᠢ) ᠬᠦᠮᠦᠨ ᠢ ᠲᠣᠭ ᠠ 30 2 25 𝑥 20 𝑦 15 1 A E C D B ᠤᠷᠳ᠋ ᠤᠰ ᠬᠤᠳ᠋ᠠ ᠢᠢᠨ 20 19 ᠣᠨ ᠤ ᠠᠩᠬ ᠠ ᠲᠤ ᠮᠳ᠋ᠠᠳ᠋ ᠤ ᠢᠢ ᠨ ᠰᠤᠷ ᠤᠯᠭ ᠠ ᠲᠡᠭ ᠦᠰᠴ ᠤ ᠰᠤᠷ ᠭᠠᠭ ᠤᠯᠢ ᠲᠠᠪᠰ ᠢᠬᠤ ᠰᠢᠯ ᠭᠠᠯᠳ᠋ᠠ ᠢᠢᠨ ᠰᠡᠳ᠋ᠤ ᠪ ① ᠹᠦᠩᠺᠼ y = √ 1 3𝑥 +1 ᠳᠦ ᠬᠤᠪᠢ ᠰᠬᠤ ᠨ𝑥 ᠤᠨ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠠ ᠪᠬᠤ ᠬᠡ ᠪᠴᠢᠶ ᠡ ᠨᠢ 𝑥 ≥ − 13 ᠃ ② ᠬᠡᠷ ᠪᠡ ᠠᠳᠠᠯᠢ ᠬᠠᠵᠠᠭᠤᠲ ᠤ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠤ ᠬᠤᠶ ᠠᠷ ᠲᠠ ᠯ ᠠ ᠶᠢᠨ ᠤᠷᠲᠤ ᠨᠢ ᠲᠣᠰ ᠲᠣᠰ 3 ᠪᠠ 7 ᠪ ᠣᠯ᠂ ᠭᠤᠷᠪ ᠠᠳᠤᠭ ᠠᠷ ᠲᠠ ᠯ ᠠ ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ 3 ᠪᠤᠶ ᠤ 7 ᠪᠣᠯ ᠣᠨ ᠠ᠃ ③ ᠨᠢ ᠭᠡᠨ ᠵᠥᠪ ᠵᠢᠷᠭ ᠤᠭᠠᠨ ᠥᠨᠴᠥ ᠭᠳᠥ ᠶᠢᠨ ᠲᠣᠳ ᠣᠭᠠᠳᠤ ᠥᠨᠴᠥ ᠭ ᠦᠨ ᠨᠠᠢᠯ ᠡᠪᠤᠷᠢ ᠨᠢ ᠭᠠᠳᠠ ᠭᠠᠳᠤ ᠥᠨᠴᠥ ᠭ ᠦᠨ ᠨᠠᠢᠯ ᠡᠪᠤᠷᠢ ᠶᠢᠨ 2 ᠲᠠᠬᠢᠨ ᠪᠣᠯᠣᠨ ᠠ᠃ ④ ︽ ᠲᠣᠳᠣᠭᠠ ᠳᠤ ᠥᠷᠥᠭᠡᠯ ᠥᠨᠴᠥ ᠭ ᠨᠢ ᠬᠠᠷ ᠢᠯᠴᠠᠨ ᠨᠥᠬᠥ ᠪᠤᠷᠢ ᠥᠨᠴ ᠥᠭ ᠪᠣᠯᠣᠨ ᠠ ︾ ᠭᠡᠬᠦ ᠦᠬᠦᠯᠡᠯ ᠨᠢ ᠵᠢ ᠩᠭᠬᠢᠨᠢ ᠦᠬᠦᠯᠡᠯ ᠪᠣᠯᠣᠨ ᠠ᠃ ⑤ 𝑥 ᠦ ᠨ ᠲᠤᠬᠠᠢ ᠨᠢ ᠭᠡ ᠥᠯ ᠥ ᠮᠡᠳᠡᠭᠳᠡᠭ ᠴᠢᠳᠡᠢ ᠬᠣᠶ ᠠᠳᠤᠭ ᠠᠷ ᠵᠡᠷᠭᠡ ᠶᠢᠨ ᠲᠡᠭᠰᠢ ᠳᠭᠡᠯ 𝑥 2− ( 𝑘+ 3) 𝑥+ 𝑘= 0 ᠲ ᠤ ᠲᠡᠩ ᠴᠡᠬᠦ ᠦ ᠪᠤᠰ ᠤ ᠬ ᠣᠶᠠᠷ ᠪᠣᠳᠠᠳᠤ ᠲᠣᠭ ᠠᠨ ᠢ ᠵᠠᠭᠤᠷ ᠲᠠᠢ᠃ A. ①②③ B. ①④ ⑤ C. ②④ D. ③⑤ 9. ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ᠂ ᠲᠡ ᠭᠰᠢ ᠥᠨᠴᠥ ᠭᠳᠥ 𝐴𝐵𝐶𝐷 ᠪᠣᠯ ᠣᠨ ᠷᠤᠮᠪ 𝐸𝐹 𝐺𝐻 ᠶᠢᠨ ᠲᠠᠰᠢᠭᠤᠪ ᠴᠢ ᠨᠢ ᠴᠥᠮ ᠴᠡ ᠭ 𝑂 ᠳᠤ ᠣᠭᠳᠣᠯ ᠣᠯᠴᠠ ᠬᠤ ᠪᠥᠭᠡᠳ 𝐸𝐺 ∕∕ 𝐵𝐶 ᠂ ᠲᠡ ᠭᠰᠢ ᠥᠨᠴ ᠥᠭᠳᠥ ᠶᠢ ᠨᠤᠭ ᠤᠯᠵᠤ᠂ ᠴᠡᠭ 𝐶 ᠪᠣᠯᠣ ᠨ ᠴᠡᠭ 𝑂 ᠶᠢ ᠲᠠᠪᠬᠤᠴ ᠠᠭᠤᠯ ᠪᠠᠯ᠂ ᠨᠤᠭ ᠤᠯᠤᠭᠠ ᠰᠤ𝑀𝑁 ᠨᠢ ᠴᠡᠭ 𝐺 ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠨ ᠠ᠃ ᠬᠡ ᠷᠪᠡ 𝐴𝐵 = √6 , 𝐸𝐹 = 2 , ∠𝐻 = 120 ° ᠪᠣᠯ 𝐷𝑁 ᠤ ᠤᠷ ᠳᠤ ᠨᠢ ( ) A. √6 − √3 B. √ 6+ √32 C. √ 32 D. 2√ 3− √6 10. ︽ ᠢ ᠷᠤᠭᠡᠳᠤᠢ ᠵᠦ᠋ ᠩ ᠴᠢᠷ ᠮᠠᠢ ︾ ᠭᠡᠬᠦ ᠲᠧᠯ ᠧᠢ ᠰ ᠬᠣᠷᠢᠶ ᠠ ᠫᠷ ᠣᠭᠷᠠᠮ ᠳᠤ᠂ ᠬᠦᠰ ᠡᠯᠡᠩ ᠪᠠ ᠴᠡᠩᠭᠡᠯ ᠳᠤ ᠵᠢᠯᠣ ᠭᠣᠴᠢ ᠦᠭᠡᠢ ᠮᠠᠰᠢᠨ ᠲᠡ ᠷᠭᠡ ᠪᠡᠷ ᠪᠠᠷ ᠠᠭ ᠠ ᠲᠡᠭᠡᠭᠡᠪᠤᠷᠢᠯᠡᠨ ᠬᠦᠷᠭᠡᠬ ᠦ ᠲᠣᠭᠯ ᠠᠯᠳᠠ ᠭᠠᠷᠭ ᠠᠵᠠᠢ᠃ ᠬᠦᠰᠡᠯ ᠡᠩ ᠦ ᠨ ᠡᠵᠡᠮᠰᠢ ᠭᠰᠡᠨ ᠬᠤᠷᠳᠤᠨ ᠲᠡᠷᠭᠡ ᠪᠠ ᠴᠡᠩᠭᠡᠯ ᠳᠤ ᠶᠢᠨ ᠡᠵᠡᠮᠰᠢᠭ ᠰᠡᠨ ᠣᠳᠠᠭ ᠠᠨ ᠲᠡᠷᠭᠡ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ 𝐴 ,𝐵 ᠬᠣᠶ ᠠᠷ ᠭᠠᠵᠠᠷ ᠠ ᠴᠠ ᠵᠡᠷᠭᠡ ᠪᠡᠷ ᠭᠠᠷ ᠴᠤ᠂ ᠥᠭᠡᠳᠡ ᠥᠭᠡᠳᠡ ᠡᠴᠡ ᠪᠡ ᠨ ᠶᠠᠪᠤ ᠪᠠ᠃ ᠬ ᠤᠷᠳᠤᠨ ᠲᠡᠷᠭᠡ ᠨᠢ 𝐵 ᠭᠠᠵᠠᠷ ᠲᠤ ᠬᠦ ᠷᠦᠭᠡᠳ 3 𝑠 ᠵᠣᠭᠰᠣ ᠵᠤ ᠪ ᠠᠷᠠᠭ ᠠ ᠪᠠᠨ ᠪᠠᠭᠤᠯ ᠭᠠᠭᠰᠠᠨ ᠤ ᠳᠠ ᠷᠠᠭ ᠠ᠂ ᠤᠤᠯ ᠤᠨ ᠵᠠᠮ ᠢᠶᠠᠷ ᠢᠶᠡ ᠨ𝐴 ᠭᠠᠵᠠᠷ ᠲᠤ ᠪᠤᠴᠠᠨ ᠠ᠃ ᠤᠳᠠ ᠭᠠᠨ ᠲᠡᠷᠭᠡ ᠨᠢ 𝐴 ᠭᠠᠵᠠᠷ ᠲᠤ ᠬᠦ ᠷᠦᠭᠡᠳ ᠠ ᠮᠠᠷᠠᠵᠠᠢ᠃ ᠳᠣᠣᠷ ᠠᠬᠢ ᠵᠢᠷᠤᠭ ᠪᠣᠯ ᠬᠣᠶ ᠠᠷ ᠲᠡ ᠭᠡᠨ ᠦ ᠬᠣᠭᠣᠷ ᠣᠨᠳᠣᠬ ᠢ ᠵᠠ ᠢ 𝑦 (ᠨᠢᠭᠡ ᠴᠢ:𝑚 ) ᠪᠠ ᠠᠶᠠᠯ ᠠᠭᠰᠠᠨ ᠬᠤᠭᠤ ᠴᠠᠭ ᠠ 𝑥(ᠨᠢᠭᠡ ᠴᠢ᠄𝑠 ) ᠤᠨ ᠹᠦᠩᠺᠼ ᠦ ᠨ ᠵᠢᠷ ᠤᠮᠠᠭ ᠪᠣᠯᠣᠨ ᠠ᠃ ᠵᠢᠷᠤᠭ ᠲᠠᠬ ᠢ ᠮᠡ ᠳᠡᠭᠡ ᠵᠠᠩᠭᠢ ᠶᠢ ᠦᠨᠳᠤ ᠰᠤᠯᠡᠪ ᠡᠯ᠂ 𝑎 , 𝑏 ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠨᠢ ᠲᠤᠰ ᠲ ᠤᠰ ( ) A. 39 , 26 B. 39 , 2 6.4 C. 38 , 2 6 D. 38 , 2 6.4 ᠬᠣᠶ ᠠᠷ᠂ ᠬᠣᠭᠣᠰᠤ ᠨ ᠳ ᠤ ᠲᠠ ᠭᠯᠠᠬ ᠤ ᠰ ᠡᠲᠤᠪ᠃ ( ᠪᠠᠭ ᠠ ᠰᠡᠲᠤᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪᠢ᠂ ᠨᠠᠢᠲᠡ 18 ᠬᠤᠪᠢ) 11. ᠪᠣᠳᠣ᠃ (π + 1) 0+ |√ 3− 2| − ( 12 ) −2 =( ) 12. ᠨᠢᠭᠡ ᠪᠥᠯᠥ ᠭ ᠲᠣᠭ ᠠ ᠪᠠᠷ ᠢᠮᠳᠠ −1 ,0,1, 2,3 ᠶᠢᠨ ᠰᠠᠷᠨᠢᠴᠡ ᠨᠢ ( ) 13. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ∆𝐴𝐵𝐶 ᠳᠦ᠂ 𝐴𝐵 = 𝐴𝐶 , 𝐴𝐵 ᠪᠠᠷ ᠳᠢᠶ ᠠᠮᠧᠲᠷ ᠪᠣ ᠯᠭᠠᠭᠰᠠᠨ ⊙ O ᠨᠢ 𝐵𝐶 , 𝐴𝐶 ᠲᠠᠢ ᠲᠤᠰ ᠲᠤᠰ ᠴᠡᠭ 𝐷 , 𝐸 ᠳᠦ ᠣᠭᠳᠣᠯ ᠣᠯᠴᠠ ᠨ ᠠ᠃ 𝐷𝐸 ᠶᠢ ᠬᠣᠯ ᠪᠣᠵᠤ᠂ ᠴᠡᠭ 𝐷 ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠭ ᠤᠯᠤᠨ 𝐷𝐹 ⊥ 𝐴𝐶 ᠪᠠᠷ ᠪᠠᠢᠭ ᠤᠯᠪᠠ ᠯ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ 𝐹 ᠃ ᠬᠡᠷᠪᠡ 𝐴𝐵 = 6 , ∠𝐶𝐷𝐹 = 15 ° ᠪᠣᠯ᠂ ᠰᠡᠭᠦᠳᠡ ᠷ ᠲᠠᠢ ᠭᠡᠰᠡᠭ ᠦᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ ( )᠃ E D' D C A O H G F MB 14. ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠤ ᠨᠢᠭᠡ ᠲᠠ ᠯ ᠠ ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠳᠤᠮᠲᠠ ᠱᠤᠭᠤ ᠮ ᠤᠨ ᠤᠷᠳᠤ ᠨᠢ ᠡᠨ ᠡ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠤᠷᠳᠤ ᠲᠠ ᠢ ᠪᠠᠨ ᠲᠡᠩ ᠴᠡᠬᠦ ᠦ ᠪ ᠣᠯ᠂ ᠡᠨᠡ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠢ ︽ᠰᠤᠨᠢᠷ ᠬᠠᠯᠲ ᠠᠢ ᠭᠤᠷ ᠪᠠᠯᠵ ᠢᠨ ︾ ᠭᠡᠨ ᠡ᠃ ᠬᠡᠷᠪᠡ 𝑅𝑡 ∆𝐴𝐵𝐶 ᠨᠢ ︽ᠰᠤᠨᠢᠷ ᠬᠠᠯᠲ ᠠᠢ ᠭᠤᠷᠪ ᠠᠯᠵᠢ ᠨ ︾ᠪᠣᠯᠬ ᠤ ᠪᠥᠭᠡᠳ ᠂ ∠𝐴 = 90 ° ᠪᠣᠯ᠂ 𝑡𝑎𝑛 ∠𝐴𝐵𝐶 =( ) 15. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠨᠢᠭᠡᠨ ᠲᠠᠬᠢᠷ ᠱᠤᠭᠤᠮ A 1 B 1 A 2 B 2 A 3 B 3 A 4 B 4 ᠁ ᠪᠣᠢ᠂ ᠲᠡᠷᠡ ᠨᠢ A 1 (0 ,0) ,B 1 (4 ,4) A 2 (8 ,0) ᠢ ᠲᠠᠭ ᠠᠷᠢᠵ ᠤ ᠪᠦᠳ ᠦᠭᠰᠡᠨ ᠲᠠᠬᠢᠷ ᠱᠤᠭᠤ ᠮ ᠢ ᠲᠡᠰ ᠳᠠ ᠷᠠᠭᠠᠯᠠᠨ 8,16 ,24 , ᠁ ᠨᠢ ᠭᠡᠴᠢ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯ ᠵᠢᠬᠦᠯ ᠵᠤ ᠣᠯᠵ ᠠᠢ᠃ ᠰᠢᠯ ᠤᠭᠤᠨ 𝑦 = 𝑘𝑥 + 2 ᠪ ᠣᠯᠣᠨ ᠲᠤᠰ ᠲᠠ ᠬᠢᠷ ᠱᠤᠭᠤ ᠮ ᠨᠢ 2 𝑛( 𝑛≥ 1 ᠪ ᠥᠭᠡᠳ 𝑛 ᠨᠢ ᠪᠥᠬᠥ ᠯᠢ ᠲᠣᠭ ᠠ) ᠣᠭᠳᠣᠯ ᠣᠯᠴᠠᠯ ᠤᠨ ᠴᠡᠭ ᠲᠠ ᠢ ᠪ ᠣᠯ᠂ 𝑘 ᠶᠢᠨ ᠬᠡ ᠮᠵᠢᠭ ᠳᠡᠯ ᠨᠢ ( ) 16. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠲᠥ ᠪ ᠥᠨᠴᠥ ᠭ ᠨᠢ 90 ° ᠪᠣ ᠯᠬᠤ ᠳᠡᠪᠢᠭᠦᠷ ᠬᠡᠯᠪᠡ ᠷᠢᠳᠤ 𝑂𝐴𝐵 ᠳᠤ᠂ 𝑂𝐵 = 2 , 𝑃 ᠨᠢ 𝐴𝐵̂ ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠳᠤᠷ ᠠ ᠶᠢᠨ ᠨᠢᠭᠡᠨ ᠴᠡᠭ ᠂ ᠴᠡᠭ 𝑃 ᠢ ᠳᠠᠭ ᠠᠷᠢᠭ ᠤᠯᠤᠨ 𝑃𝐸 ⊥ 𝑂𝐵 ᠢᠶᠠᠷ ᠪ ᠠᠢᠭ ᠤᠯᠪᠠ ᠯ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ 𝐸 ᠃ 𝑀 ᠨᠢ ∆𝑂𝑃 𝐸 ᠶᠢᠨ ᠲᠣᠳᠣᠭᠠ ᠳᠤ ᠲᠥ ᠪ ᠬᠡᠵᠤ ᠵᠢᠱᠢᠪᠡᠯ ᠂ ᠴᠡᠭ 𝑃 ᠨᠢ ᠴᠡᠭ 𝐴 ᠡᠴᠡ ᠬᠦᠳᠡᠯ ᠭᠡᠭᠡᠨ ᠬᠢ ᠵᠤ ᠴᠡᠭ 𝐵 ᠬᠦᠷᠬᠦ ᠦᠶ ᠡᠰ᠂ ᠲᠣᠳᠣᠭᠠ ᠳᠤ ᠲᠥ ᠪ 𝑀 ᠶᠢᠨ ᠶᠠᠪᠤᠵ ᠤ ᠥ ᠩᠭᠡᠷ ᠡᠭᠰᠡᠨ ᠵᠠᠮ ᠤᠨ ᠤᠷᠳᠤ ᠨᠢ ( ) ᠭᠤᠷᠪ ᠠ᠂ ᠪᠣᠳᠣ ᠵᠤ ᠬᠠᠷ ᠢᠭᠤᠯᠬᠤ ᠰᠡ ᠲᠤᠪ᠃ ( ᠨᠠ ᠢᠲᠡ 72 ᠬᠤᠪᠢ ) 17. (ᠲᠤᠰ ᠪᠠ ᠭ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 8 ᠬᠤᠪᠢ ) ⑴ ᠤᠷ ᠢᠳᠠᠪᠠ ᠷ ᠬᠢᠯᠪᠠ ᠷᠴᠢᠯ ᠠ᠄ 𝑥 2 −4 𝑥 2 −4 𝑥+ 4 + 𝑥 𝑥 2 −𝑥 ÷ 𝑥− 2 𝑥− 1 ᠳᠠᠷᠠᠭ ᠠ ᠨᠢ 𝑥 ᠲᠤ −1 ≤ 𝑥≤ 3 ᠬᠡᠪᠴᠢᠶ ᠡᠨ ᠳᠣᠲᠣᠷ ᠠ ᠠᠴᠠ ᠳᠣᠬ ᠢᠷᠠᠮᠵᠢᠳᠠᠢ ᠨᠢᠭᠡ ᠪᠥ ᠬᠥᠯᠢ ᠲᠣᠭ ᠠ ᠶᠢ ᠰᠤᠩᠭᠤᠵ ᠤ᠂ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠨᠢ ᠣ ᠯ᠃ ⑵ ᠲᠡ ᠩᠴᠡᠳᠭᠡᠯ ᠪᠤᠰᠤ ᠶᠢ ᠨ ᠪᠥᠯᠥ ᠭ { − (2 𝑥+ 1) 2OA ᠂ ᠴᠡ ᠭ C ᠨᠢ OB ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠨᠢᠭᠡᠨ ᠴᠡᠭ ᠂ ∠ CA D = 60 ° ᠪᠣᠯ ᠂ OC ᠶᠢᠨ ᠤᠷᠳᠤ ᠶᠢ ᠣ ᠯ᠃ C O D B A ᠵᠢᠷ ᠤᠭ 3 E F O B DA C ᠵᠢᠷ ᠤᠭ 1 FE O D A B C ᠵᠢᠷ ᠤᠭ 2 24 . (ᠲᠤᠰ ᠪᠠ ᠭ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 12 ᠬᠤᠪᠢ ) ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ᠂ ᠬᠠᠶᠠᠯ ᠲᠠ ᠶᠢᠨ ᠱ ᠤᠭᠤᠮ 𝑦 = 𝑎𝑥 2+ 𝑏𝑥 − 2( 𝑎≠ 0) ᠨᠢ 𝑥 ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠲᠠᠢ 𝐴( −3 ,0) , 𝐵 (1 ,0) ᠬᠣᠶ ᠠᠷ ᠴᠡᠭ ᠲᠦ ᠣᠭᠳᠣᠯ ᠣᠯᠴᠠ ᠵᠤ᠂ 𝑦ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠲᠠᠢ ᠴᠡᠭ 𝐶 ᠲᠦ ᠣᠭᠳ ᠣᠯᠣᠯ ᠴᠠᠨ ᠠ᠃ ᠰᠢᠯᠤᠭᠤᠨ 𝑦 = −𝑥 ᠨᠢ ᠲᠤᠰ ᠬᠠᠶᠠ ᠯᠲᠠ ᠶᠢᠨ ᠱᠤᠭᠤ ᠮ ᠲ ᠠᠢ 𝐸 , 𝐹 ᠬᠣᠶ ᠠᠷ ᠴᠡ ᠭ ᠲᠦ ᠣ ᠭᠳᠣᠯ ᠣᠯᠴᠠ ᠨ ᠠ᠃ ⑴ ᠬᠠᠶᠠᠯ ᠲᠠ ᠶᠢᠨ ᠱᠤᠭᠤ ᠮ ᠤᠨ ᠠᠨᠠ ᠯᠢᠰ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡᠯ ᠢ ᠣᠯ᠃ ⑵ 𝑃 ᠪᠣᠯ ᠰᠢᠯ ᠤᠭᠤᠨ 𝐸𝐹 ᠶᠢᠨ ᠳᠣᠣᠷ ᠠᠳᠤ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠬᠠᠶᠠᠯᠲ ᠠ ᠶᠢᠨ ᠱᠤᠭᠤ ᠮ ᠳᠡ ᠭᠡᠷᠡᠬ ᠢ ᠨᠢ ᠭᠡᠨ ᠬᠦᠳᠡᠯᠬᠦ ᠴᠡᠭ ᠂ 𝑃𝐻 ⊥ 𝐸𝐹 ᠢᠶᠡᠷ ᠪᠠᠢᠭ ᠤᠯᠪᠠ ᠯ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ 𝐻 , 𝑃𝐻 ᠶᠢᠨ ᠬᠠᠮᠤᠭ ᠤᠨ ᠶᠡᠭᠡ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠢ ᠣ ᠯ᠃ ⑶ ᠴᠡᠭ 𝐶 ᠢᠶᠡᠷ ᠲᠣᠭ ᠣᠷᠢᠭ ᠤᠨ ᠲ ᠥᠪ ᠪᠣᠯᠭ ᠠᠵᠤ᠂ 1 ᠪᠠᠷ ᠷᠠ ᠳᠢᠤᠰ ᠪᠣᠯ ᠭᠠᠨ ᠲᠣᠭ ᠣᠷᠢᠭ ᠪᠠᠢᠭ ᠤᠯᠪᠠ ᠯ᠂ ⊙ C ᠳᠡᠭᠡᠷ ᠡ ᠨᠢ ᠭᠡᠨ ᠴᠡᠭ 𝑀 ᠣᠷ ᠣᠰᠢᠵ ᠤ᠂ ∆ 𝐵𝐶 𝑀 ᠶᠢ 𝐶𝑀 ᠪᠡᠷ ᠡᠭᠴᠡ ᠲᠠᠯ ᠠ ᠪᠠᠨ ᠪᠣᠯᠭ ᠠᠭᠰᠠ ᠨ ᠲᠡ ᠭᠰᠢ ᠥᠨᠴᠥᠭᠳᠥ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠪᠣᠯ ᠭᠠᠵᠤ ᠲᠠ ᠢᠯᠤᠨ ᠡ ᠤᠤ ︖ ᠬᠡᠷᠪᠡ ᠣᠷᠣᠰ ᠢᠪᠠᠯ 𝑀 ᠴᠡᠭ ᠦᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠢ ᠰᠢᠭ ᠤᠳ ᠪᠢᠴᠢᠵ ᠤ ᠭᠠᠷ ᠭ ᠠ᠃ ᠬᠡᠷ ᠪᠡ ᠣ ᠷᠣᠰᠢ ᠬᠤ ᠦ ᠭᠡᠢ ᠪ ᠣᠯ ᠰ ᠢᠯᠲᠠᠭ ᠠᠨ ᠢ ᠨᠢ ᠲᠣᠳᠣᠷ ᠬᠠᠢ ᠯᠠ᠃ ᠪᠡᠯᠡᠳᠭᠡᠯ ᠵᠢᠷᠤᠭ