M6math2552
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M6math2552 M6math2552 Document Transcript

  • ÃËÑÊÇÔªÒ 04 ¤³ÔµÈÒʵÃì ˹éÒ 2 ÇѹàÊÒÃì·Õ‹ 20 ¡ØÁÀҾѹ¸ì 2553 àÇÅÒ 11.30 - 13.30 ¹. Êèǹ·Õ‹ 1 ẺÃкÒµÑÇàÅ×Í¡ áµèÅТéÍÁդӵͺ·Õ‹¶Ù¡µéͧ·Õ‹ÊØ´à¾Õ§¤ÓµÍºà´ÕÂÇ ¨Ó¹Ç¹ 36 ¢éÍ (¢éÍ 1–36) ¢éÍÅÐ 1 ¤Ðá¹¹ 1. ãËé A = {1, 2, 3, . . .} áÅÐ B = {{1, 2}, {3, 4, 5}, 6, 7, 8, . . .} ¢éÍã´à»š¹à·ç¨ 1. A − B ÁÕÊÁÒªÔ¡ 5 µÑÇ 2. ¨Ó¹Ç¹ÊÁÒªÔ¡¢Í§à¾ÒàÇÍÃì૵¢Í§ B − A à·èҡѺ 4 3. ¨Ó¹Ç¹ÊÁÒªÔ¡¢Í§ (A − B) ∪ (B − A) ໚¹¨Ó¹Ç¹¤Ùè 4. A ∩ B ¤×Í૵¢Í§¨Ó¹Ç¹¹Ñº·Õ‹ÁÕ¤èÒÁÒ¡¡ÇèÒ 5 2. ¾Ô¨ÒóҡÒÃãËéà˵ؼŵèÍ仹Ռ à赯 1) A 2) àËç´à»š¹¾×ªÁÕ´Í¡ ¼Å àËç´à»š¹¾×ªªÑŒ¹ÊÙ§ ¢éÍÊÃØ»¢éÒ§µé¹ÊÁà˵ØÊÁ¼Å ¶éÒ A á·¹¢éͤÇÒÁã´ 1. ¾×ªªÑŒ¹ÊÙ§·Ø¡ª¹Ô´ÁÕ´Í¡ 2. ¾×ªªÑŒ¹ÊÙ§ºÒ§ª¹Ô´ÁÕ´Í¡ 3. ¾×ªÁÕ´Í¡·Ø¡ª¹Ô´à»š¹¾×ªªÑŒ¹ÊÙ§ 4. ¾×ªÁÕ´Í¡ºÒ§ª¹Ô´à»š¹¾×ªªÑŒ¹ÊÙ§
  • ÃËÑÊÇÔªÒ 04 ¤³ÔµÈÒʵÃì ˹éÒ 3 ÇѹàÊÒÃì·Õ‹ 20 ¡ØÁÀҾѹ¸ì 2553 àÇÅÒ 11.30 - 13.30 ¹. 3. ¾Ô¨ÒóҢéͤÇÒÁµèÍ仹Ռ ¡. ¨Ó¹Ç¹·Õ‹à»š¹·È¹ÔÂÁäÁèÃÙ騺ºÒ§¨Ó¹Ç¹à»š¹¨Ó¹Ç¹ÍµÃáÂÐ ¢. ¨Ó¹Ç¹·Õ‹à»š¹·È¹ÔÂÁäÁèÃÙ騺ºÒ§¨Ó¹Ç¹à»š¹¨Ó¹Ç¹µÃáÂÐ ¢éÍã´¶Ù¡µéͧ 1. ¢éÍ ¡. áÅТéÍ ¢. 2. ¢éÍ ¡. à·èҹь¹ 3. ¢éÍ ¢. à·èҹь¹ 4. ¢éÍ ¡. áÅТéÍ ¢. ¼Ô´ 4. ¡Ó˹´ãËé s, t, u áÅÐ v ໚¹¨Ó¹Ç¹¨ÃÔ§ «Ö‹§ s < t áÅÐ u < v ¾Ô¨ÒóҢéͤÇÒÁµèÍ仹Ռ ¡. s − u < t − v ¢. s − v < t − u ¢éÍã´¶Ù¡µéͧ 1. ¢éÍ ¡. áÅТéÍ ¢. 2. ¢éÍ ¡. à·èҹь¹ 3. ¢éÍ ¢. à·èҹь¹ 4. ¢éÍ ¡. áÅТéÍ ¢. ¼Ô´
  • ÃËÑÊÇÔªÒ 04 ¤³ÔµÈÒʵÃì ˹éÒ 4 ÇѹàÊÒÃì·Õ‹ 20 ¡ØÁÀҾѹ¸ì 2553 àÇÅÒ 11.30 - 13.30 ¹. 5. ¼Åà©Å¢ͧÊÁ¡Òà 2|5 − x| = 1 ÍÂÙè㹪èǧ㴠1. (−10, −5) 2. (−6, −4) 3. (−4, 5) 4. (−3, 6) 6. ¶éÒ 3 4 ໚¹¼Åà©ÅÂ˹֋§¢Í§ÊÁ¡Òà 4x2 + bx − 6 = 0 àÁ×‹Í b ໚¹¨Ó¹Ç¹¨ÃÔ§áÅéÇ ÍÕ¡¼Å à©ÅÂ˹֋§¢Í§ÊÁ¡ÒùՌÁÕ¤èҵç¡Ñº¢éÍã´ 1. −2 2. − 1 2 3. 1 2 4. 2 7. ¢éÍã´ÁÕ¤èÒµèÒ§¨Ò¡¢éÍÍ׋¹ 1. (−1)0 2. (−1)0.2 3. (−1)0.4 4. (−1)0.8 8. |4 √ 3 − 5 √ 2| − |3 √ 5 − 5 √ 2| + |4 √ 3 − 3 √ 5| 2 à·èҡѺ¢éÍã´ 1. 0 2. 180 3. 192 4. 200
  • ÃËÑÊÇÔªÒ 04 ¤³ÔµÈÒʵÃì ˹éÒ 5 ÇѹàÊÒÃì·Õ‹ 20 ¡ØÁÀҾѹ¸ì 2553 àÇÅÒ 11.30 - 13.30 ¹. 9. ¡Ó˹´ãËé a ໚¹¨Ó¹Ç¹¨ÃÔ§ºÇ¡ áÅÐ n ໚¹¨Ó¹Ç¹¤ÙèºÇ¡ ¾Ô¨ÒóҢéͤÇÒÁµèÍ仹Ռ ¡. n √ a n = |a| ¢. n √ an = |a| ¢éÍã´¶Ù¡µéͧ 1. ¢éÍ ¡. áÅТéÍ ¢. 2. ¢éÍ ¡. à·èҹь¹ 3. ¢éÍ ¢. à·èҹь¹ 4. ¢éÍ ¡. áÅТéÍ ¢. ¼Ô´ 10. ¶éÒ f(x) = −x2 + x + 2 áÅéÇ ¢éÍÊÃػ㴶١µéͧ 1. f(x) ≥ 0 àÁ×‹Í −1 ≤ x ≤ 2 2. ¨Ø´Ç¡¡ÅѺ¢Í§¡ÃÒ¿¢Í§¿˜§¡ìªÑ¹ f ÍÂÙè㹨µØÀÒ¤·Õ‹Êͧ 3. ¿˜§¡ìªÑ¹ f ÁÕ¤èÒÊÙ§ÊØ´à·èҡѺ 2 4. ¿˜§¡ìªÑ¹ f ÁÕ¤èÒµ‹ÓÊØ´à·èҡѺ 2
  • ÃËÑÊÇÔªÒ 04 ¤³ÔµÈÒʵÃì ˹éÒ 6 ÇѹàÊÒÃì·Õ‹ 20 ¡ØÁÀҾѹ¸ì 2553 àÇÅÒ 11.30 - 13.30 ¹. 11. ¤ÇÒÁÊÑÁ¾Ñ¹¸ìã¹¢éÍã´à»š¹¿˜§¡ìªÑ¹ 1. {(1, 2), (2, 3), (3, 2), (2, 4)} 2. {(1, 2), (2, 3), (3, 1), (3, 3)} 3. {(1, 3), (1, 2), (1, 1), (1, 4)} 4. {(1, 3), (2, 1), (3, 3), (4, 1)} 12. ¶éÒ f(x) = √ 3 − x áÅÐ g(x) = −2 + |x − 4| áÅéÇ Df ∪ Rg ¤×Í¢éÍã´ 1. (−∞, 3] 2. [−2, ∞) 3. [−2, 3] 4. (−∞, ∞)
  • ÃËÑÊÇÔªÒ 04 ¤³ÔµÈÒʵÃì ˹éÒ 7 ÇѹàÊÒÃì·Õ‹ 20 ¡ØÁÀҾѹ¸ì 2553 àÇÅÒ 11.30 - 13.30 ¹. 13. ¡Ó˹´ãËé¡ÃÒ¿¢Í§¿˜§¡ìªÑ¹ f ໚¹´Ñ§¹ÕŒ ßß½¼ ¼ ß ¤èҢͧ 11f(−11) − 3f(−3)f(3) ¤×Í¢éÍã´ 1. 57 2. 68 3. 75 4. 86
  • ÃËÑÊÇÔªÒ 04 ¤³ÔµÈÒʵÃì ˹éÒ 8 ÇѹàÊÒÃì·Õ‹ 20 ¡ØÁÀҾѹ¸ì 2553 àÇÅÒ 11.30 - 13.30 ¹. 14. ÃÙ»ÊÒÁàËÅՋÂÁÁØÁ©Ò¡Ãٻ˹֋§ Áվ׌¹·Õ‹ 600 µÒÃҧૹµÔàÁµÃ ¶éÒ´éÒ¹»ÃСͺÁØÁ ©Ò¡´éҹ˹֋§ÂÒÇ໚¹ 75% ¢Í§´éÒ¹»ÃСͺÁØÁ©Ò¡ÍÕ¡´éҹ˹֋§áÅéÇ àÊé¹ÃͺÃÙ»ÊÒÁ àËÅՋÂÁÁØÁ©Ò¡ÃÙ»¹ÕŒ ÂÒǡՋૹµÔàÁµÃ 1. 120 2. 40 3. 60 √ 2 4. 20 √ 2 15. ¢ºÇ¹¾ÒàËôÃÙ»ÊՋàËÅՋÂÁ¼×¹¼éÒ¢ºÇ¹Ë¹Ö‹§ »ÃСͺ´éǼÙéà´Ô¹à»š¹á¶Ç á¶ÇÅÐà·èÒæ ¡Ñ¹ (ÁÒ¡¡ÇèÒ 1 á¶Ç áÅÐá¶ÇÅÐÁÒ¡¡ÇèÒ 1 ¤¹) â´ÂÁÕ੾ÒмÙéÍÂÙèÃÔÁ´éÒ¹¹Í¡·ÑŒ§ÊՋ´éÒ¹¢Í§ ¢ºÇ¹à·èҹь¹ ·Õ‹ÊÇÁªØ´ÊÕá´§ «Ö‹§Áշь§ËÁ´ 50 ¤¹ ¶éÒ x ¤×ͨӹǹá¶Ç¢Í§¢ºÇ¹ ¾ÒàËô áÅÐ N ¤×ͨӹǹ¤¹·Õ‹ÍÂÙèã¹¢ºÇ¹¾ÒàËôáÅéÇ ¢éÍã´¶Ù¡µéͧ 1. 31x − x2 = N 2. 29x − x2 = N 3. 27x − x2 = N 4. 25x − x2 = N
  • ÃËÑÊÇÔªÒ 04 ¤³ÔµÈÒʵÃì ˹éÒ 9 ÇѹàÊÒÃì·Õ‹ 20 ¡ØÁÀҾѹ¸ì 2553 àÇÅÒ 11.30 - 13.30 ¹. 16. ÃÙ»ÊՋàËÅՋÂÁ¼×¹¼éÒÊͧÃÙ» ÁÕ¢¹Ò´à·èҡѹ â´ÂÁÕàÊé¹·á§ÁØÁÂÒÇ໚¹Êͧà·èҢͧ´éÒ¹ ¡ÇéÒ§ ¶éÒ¹ÓÃÙ»ÊՋàËÅՋÂÁ¼×¹¼éҷь§ÊͧÁÒÇÒ§µè͡ѹ´Ñ§ÃÙ» ¨Ø´ A áÅШش B ÍÂÙèËèÒ§¡Ñ¹à»š¹ ÃÐÂСՋà·èҢͧ´éÒ¹¡ÇéÒ§ 1. 1.5 2. 3 3. √ 2 4. 2 √ 2 17. â´Â¡ÒÃãªéµÒÃÒ§ËÒÍѵÃÒÊèǹµÃÕ⡳ÁԵԢͧÁØÁ¢¹Ò´µèÒ§æ ·Õ‹¡Ó˹´ãËéµèÍ仹Ռ θ sin θ cos θ 72◦ 0.951 0.309 73◦ 0.956 0.292 74◦ 0.961 0.276 75◦ 0.966 0.259 ÁØÁÀÒÂ㹷ՋÁÕ¢¹Ò´àÅ硷ՋÊØ´¢Í§ÃÙ»ÊÒÁàËÅՋÂÁ·Õ‹ÁÕ´éÒ¹·ÑŒ§ÊÒÁÂÒÇ 7, 24 áÅÐ 25 ˹èÇ ÁÕ¢¹Ò´ã¡Åéà¤Õ§¡Ñº¢éÍã´ÁÒ¡·Õ‹ÊØ´ 1. 15◦ 2. 16◦ 3. 17◦ 4. 18◦
  • ÃËÑÊÇÔªÒ 04 ¤³ÔµÈÒʵÃì ˹éÒ 10 ÇѹàÊÒÃì·Õ‹ 20 ¡ØÁÀҾѹ¸ì 2553 àÇÅÒ 11.30 - 13.30 ¹. 18. ÁØÁÁØÁ˹֋§¢Í§ÃÙ»ÊÒÁàËÅՋÂÁÁØÁ©Ò¡ÁÕ¢¹Ò´à·èҡѺ 60 ͧÈÒ ¶éÒàÊé¹ÃͺÃÙ»¢Í§ÃÙ» ÊÒÁàËÅՋÂÁ¹ÕŒÂÒÇ 3 − √ 3 ¿ØµáÅéÇ ´éÒ¹·Õ‹ÂÒÇ໚¹Íѹ´ÑºÊͧÁÕ¤ÇÒÁÂÒÇà·èҡѺ¢éÍã´ 1. 2 − √ 3 ¿Øµ 2. 2 + √ 3 ¿Øµ 3. 2 √ 3 − 3 ¿Øµ 4. 2 √ 3 + 3 ¿Øµ 19. ¡Åéͧǧ¨Ã»´«Ö‹§¶Ù¡µÔ´µÑŒ§ÍÂÙèÊÙ§¨Ò¡¾×Œ¹¶¹¹ 2 àÁµÃ ÊÒÁÒö¨ÑºÀÒ¾ä´éµ‹Ó·Õ‹ÊØ´·Õ‹ÁØÁ ¡éÁ 45◦ áÅÐÊÙ§·Õ‹ÊØ´·Õ‹ÁØÁ¡éÁ 30◦ ÃÐÂзҧº¹¾×Œ¹¶¹¹ã¹á¹Ç¡Åéͧ ·Õ‹¡Åéͧ¹ÕŒÊÒÁÒö ¨ÑºÀÒ¾ä´é¤×Íà·èÒã´ (¡Ó˹´ãËé √ 3 ≈ 1.73) 1. 1.00 àÁµÃ 2. 1.46 àÁµÃ 3. 2.00 àÁµÃ 4. 3.46 àÁµÃ 20. ¡Ó˹´ãËé 3 2 , 1, 1 2 , . . . ໚¹ÅӴѺàÅ¢¤³Ôµ ¼ÅºÇ¡¢Í§¾¨¹ì·Õ‹ 40 áÅо¨¹ì·Õ‹ 42 à·èҡѺ ¢éÍã´ 1. −18 2. −19 3. −37 4. −38
  • ÃËÑÊÇÔªÒ 04 ¤³ÔµÈÒʵÃì ˹éÒ 11 ÇѹàÊÒÃì·Õ‹ 20 ¡ØÁÀҾѹ¸ì 2553 àÇÅÒ 11.30 - 13.30 ¹. 21. ã¹ 40 ¾¨¹ìáá¢Í§ÅӴѺ an = 3 + (−1)n ÁաՋ¾¨¹ì ·Õ‹ÁÕ¤èÒà·èҡѺ¾¨¹ì·Õ‹ 40 1. 10 2. 20 3. 30 4. 40 22. ¡Ó˹´ãËé a1, a2, a3, . . . ໚¹ÅӴѺàâҤ³Ôµ ¶éÒ a2 = 8 áÅÐ a5 = −64 áÅéÇ ¼ÅºÇ¡ ¢Í§ 10 ¾¨¹ìáá¢Í§ÅӴѺ¹ÕŒà·èҡѺ¢éÍã´ 1. 2, 048 2. 1, 512 3. 1, 364 4. 1, 024 23. ·ÒÊÕàËÃÕ­ÊÒÁÍѹ´Ñ§¹ÕŒ àËÃÕ­áá´éҹ˹֋§·ÒÊÕ¢ÒÇ ÍÕ¡´éҹ˹֋§·ÒÊÕá´§ àËÃÕ­·Õ‹ Êͧ´éҹ˹֋§·ÒÊÕá´§ ÍÕ¡´éҹ˹֋§·ÒÊÕ¿‡Ò àËÃÕ­·Õ‹ÊÒÁ´éҹ˹֋§·ÒÊÕ¿‡Ò ÍÕ¡´éҹ˹֋§ ·ÒÊÕ¢ÒÇ â¹àËÃÕ­·ÑŒ§ÊÒÁ¢ÖŒ¹¾ÃéÍÁ¡Ñ¹ ¤ÇÒÁ¹èÒ¨Ð໚¹·Õ‹àËÃÕ­¨Ð¢ÖŒ¹Ë¹éÒµèÒ§Êաѹ ·ÑŒ§ËÁ´à»š¹´Ñ§¢éÍã´ 1. 1 2 2. 1 4 3. 1 8 4. 1 16
  • ÃËÑÊÇÔªÒ 04 ¤³ÔµÈÒʵÃì ˹éÒ 12 ÇѹàÊÒÃì·Õ‹ 20 ¡ØÁÀҾѹ¸ì 2553 àÇÅÒ 11.30 - 13.30 ¹. 24. ¡ÅèͧãºË¹Ö‹§ºÃèØÊÅÒ¡ËÁÒÂàÅ¢ 1–10 ËÁÒÂàÅ¢ÅÐ 1 㺠¶éÒÊØèÁËÂÔºÊÅÒ¡¨Ó¹Ç¹Êͧ 㺠â´ÂËÂÔº·ÕÅÐãºáººäÁèãÊè¤×¹ ¤ÇÒÁ¹èÒ¨Ð໚¹·Õ‹¨ÐËÂÔºä´éÊÅÒ¡ËÁÒÂàÅ¢µ‹Ó¡ÇèÒ 5 à¾Õ§˹֋§ãºà·èҹь¹ à·èҡѺ¢éÍã´ 1. 2 9 2. 8 15 3. 2 35 4. 11 156 25. 㹡ÒÃÇÑ´ÊèǹÊÙ§¹Ñ¡àÃÕ¹áµèÅФ¹ã¹ªÑŒ¹ ¾ºÇèҹѡàÃÕ¹·Õ‹ÊÙ§·Õ‹ÊØ´ÊÙ§ 177 ૹµÔàÁµÃ áÅйѡàÃÕ¹·Õ‹àµÕŒÂ·Õ‹ÊØ´ÊÙ§ 145 ૹµÔàÁµÃ ¾Ô¨ÒóÒ૵¢Í§ÊèǹÊÙ§µèÍ仹Ռ S = { H | H ໚¹ÊèǹÊÙ§ã¹Ë¹èÇÂૹµÔàÁµÃ¢Í§¹Ñ¡àÃÕ¹㹪ь¹} T = { H | 145 ≤ H ≤ 177 } ૵㴶×Í໚¹»ÃÔÀÙÁÔµÑÇÍÂèÒ§ (á«ÁແÅÊ໫) ÊÓËÃѺ¡Ò÷´ÅͧÊØèÁ¹ÕŒ 1. S áÅÐ T 2. S à·èҹь¹ 3. T à·èҹь¹ 4. ·ÑŒ§ S áÅÐ T äÁè໚¹»ÃÔÀÙÁÔµÑÇÍÂèÒ§
  • ÃËÑÊÇÔªÒ 04 ¤³ÔµÈÒʵÃì ˹éÒ 13 ÇѹàÊÒÃì·Õ‹ 20 ¡ØÁÀҾѹ¸ì 2553 àÇÅÒ 11.30 - 13.30 ¹. 26. 㹡ÒÃàÅ×Í¡¤³Ð¡ÃÃÁ¡Òêش˹֋§ «Ö‹§»ÃСͺ´éÇ »Ãиҹ Ãͧ»Ãиҹ áÅÐ àŢҹءÒÃÍÂèÒ§ÅÐ 1 ¤¹ ¨Ò¡Ë­Ô§ 6 ¤¹ áÅЪÒ 4 ¤¹ ¤ÇÒÁ¹èÒ¨Ð໚¹·Õ‹¤³Ð¡ÃÃÁ¡Òà ªØ´¹ÕŒ ¨ÐÁÕ»ÃиҹáÅÐÃͧ»Ãиҹ໚¹Ë­Ô§à·èҡѺ¢éÍã´ 1. 1 18 2. 1 12 3. 1 9 4. 1 3 27. ¤ÃÙÊ͹ÇÔ·ÂÒÈÒʵÃìÁͺËÁÒÂãËé¹Ñ¡àÃÕ¹ 40 ¤¹ ·Óâ¤Ã§§Ò¹µÒÁ¤ÇÒÁʹ㨠ËÅѧ¨Ò¡ µÃǨÃÒ§ҹâ¤Ã§§Ò¹¢Í§·Ø¡¤¹áÅéÇ ¼ÅÊÃػ໚¹´Ñ§¹ÕŒ ¼Å¡ÒûÃÐàÁÔ¹ ¨Ó¹Ç¹â¤Ã§§Ò¹ ´ÕàÂՋÂÁ 3 ´Õ 20 ¾Íãªé 12 µéͧá¡éä¢ 5 ¢éÍÁÙŷՋà¡çºÃǺÃÇÁ à¾×‹ÍãËéä´é¼ÅÊÃØ»¢éÒ§µé¹à»š¹¢éÍÁÙŪ¹Ô´ã´ 1. ¢éÍÁÙÅ»°ÁÀÙÁÔ àªÔ§»ÃÔÁÒ³ 2. ¢éÍÁÙŷصÔÂÀÙÁÔ àªÔ§»ÃÔÁÒ³ 3. ¢éÍÁÙÅ»°ÁÀÙÁÔ àªÔ§¤Ø³ÀÒ¾ 4. ¢éÍÁÙŷصÔÂÀÙÁÔ àªÔ§¤Ø³ÀÒ¾
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