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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èǧã´
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»ÊÒÁàËÅÕ‹ÂÁÁØÁ©Ò¡Ãٻ˹֋§ Áվ׌¹·Õ‹ 600 µÒÃҧૹµÔàÁµÃ ¶éÒ´éÒ¹»ÃСͺÁØÁ
©Ò¡´éҹ˹֋§ÂÒÇ໚¹ 75% ¢Í§´éÒ¹»ÃСͺÁØÁ©Ò¡ÍÕ¡´éҹ˹֋§áÅéÇ àÊé¹ÃͺÃÙ»ÊÒÁ
àËÅÕ‹ÂÁÁØÁ©Ò¡ÃÙ»¹ÕŒ ÂÒǡՋૹµÔàÁµÃ
1. 120 2. 40
3. 60
√
2 4. 20
√
2
15. ¢ºÇ¹¾ÒàËôÃÙ»ÊÕ‹àËÅÕ‹ÂÁ¼×¹¼éÒ¢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§ÊÕ‹´éÒ¹¢Í§
¢ºÇ¹à·èҹь¹ ·Õ‹ÊÇÁªØ´ÊÕá´§ «Ö‹§ÁÕ·ÑŒ§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»ÊÕ‹àËÅÕ‹ÂÁ¼×¹¼éÒÊͧÃÙ» ÁÕ¢¹Ò´à·èҡѹ â´ÂÁÕàÊé¹·á§ÁØÁÂÒÇ໚¹Êͧà·èҢͧ´éÒ¹
¡ÇéÒ§ ¶éÒ¹ÓÃÙ»ÊÕ‹àËÅÕ‹ÂÁ¼×¹¼éÒ·ÑŒ§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
ÁØÁÀÒÂã¹·Õ‹ÁÕ¢¹Ò´àÅç¡·Õ‹ÊØ´¢Í§ÃÙ»ÊÒÁàËÅÕ‹ÂÁ·Õ‹ÁÕ´éÒ¹·ÑŒ§ÊÒÁÂÒÇ 7, 24 áÅÐ 25 ˹èÇÂ
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»ÊÒÁàËÅÕ‹ÂÁÁØÁ©Ò¡ÁÕ¢¹Ò´à·èҡѺ 60 ͧÈÒ ¶éÒàÊé¹ÃͺÃÙ»¢Í§ÃÙ»
ÊÒÁàËÅÕ‹ÂÁ¹ÕŒÂÒÇ 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
ÁÕ¡Õ‹¾¨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¡ÒÃ
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
´ÕàÂÕ‹ÂÁ 3
´Õ 20
¾Íãªé 12
µéͧá¡éä¢ 5
¢éÍÁÙÅ·Õ‹à¡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
ÃËÑÊÇÔªÒ 04 ¤³ÔµÈÒʵÃì ˹éÒ 14
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28. ¤èÒà©ÅÕ‹ÂàÅ¢¤³Ôµ¢Í§¹ŒÓ˹ѡ¢Í§¾¹Ñ¡§Ò¹¢Í§ºÃÔÉѷ˹֋§ à·èҡѺ 48.01 ¡ÔâÅ¡ÃÑÁ ºÃÔÉÑ·
¹ÕŒÁÕ¾¹Ñ¡§Ò¹ªÒ 43 ¤¹ áÅо¹Ñ¡§Ò¹Ë­Ô§ 57 ¤¹ ¶éÒ¤èÒà©ÅÕ‹ÂàÅ¢¤³Ôµ¢Í§¹ŒÓ˹ѡ
¾¹Ñ¡§Ò¹Ë­Ô§à·èҡѺ 45 ¡ÔâÅ¡ÃÑÁ áÅéÇ ¹ŒÓ˹ѡ¢Í§¾¹Ñ¡§Ò¹ªÒ·ь§ËÁ´ÃÇÁ¡Ñ¹à·èҡѺ
¢éÍã´
1. 2, 236 ¡ÔâÅ¡ÃÑÁ
2. 2, 279 ¡ÔâÅ¡ÃÑÁ
3. 2, 322 ¡ÔâÅ¡ÃÑÁ
4. 2, 365 ¡ÔâÅ¡ÃÑÁ
ÃËÑÊÇÔªÒ 04 ¤³ÔµÈÒʵÃì ˹éÒ 15
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29. á¼¹ÀÒ¾µé¹-㺢ͧ¹ŒÓ˹ѡã¹Ë¹èÇ¡ÃÑÁ¢Í§ä¢èä¡è 10 ¿Í§ ໚¹´Ñ§¹ÕŒ
5 7 8
6 7 8 9
7 0 4 4 7
8 1
¢éÍÊÃØ»ã´à»š¹à·ç¨
1. °Ò¹¹ÔÂÁ¢Í§¹ŒÓ˹ѡ¢Í§ä¢èä¡èÁÕà¾Õ§¤èÒà´ÕÂÇ
2. ¤èÒà©ÅÕ‹ÂàÅ¢¤³ÔµáÅÐÁѸ°ҹ¢Í§¹ŒÓ˹ѡ¢Í§ä¢èä¡èÁÕ¤èÒà·èҡѹ
3. ÁÕä¢èä¡è 5 ¿Í§·Õ‹ÁÕ¹ŒÓ˹ѡ¹éÍ¡ÇèÒ 70 ¡ÃÑÁ
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Bahare Shariat Jild 2 By SadurshSharia Mufti Amjad Ali Azmi
 

คณิต 52

  • 1. 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§
  • 2. 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´
  • 3. 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
  • 4. 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
  • 5. 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. (−∞, ∞)
  • 6. 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
  • 7. ÃËÑÊÇÔªÒ 04 ¤³ÔµÈÒʵÃì ˹éÒ 8 ÇѹàÊÒÃì·Õ‹ 20 ¡ØÁÀҾѹ¸ì 2553 àÇÅÒ 11.30 - 13.30 ¹. 14. ÃÙ»ÊÒÁàËÅÕ‹ÂÁÁØÁ©Ò¡Ãٻ˹֋§ Áվ׌¹·Õ‹ 600 µÒÃҧૹµÔàÁµÃ ¶éÒ´éÒ¹»ÃСͺÁØÁ ©Ò¡´éҹ˹֋§ÂÒÇ໚¹ 75% ¢Í§´éÒ¹»ÃСͺÁØÁ©Ò¡ÍÕ¡´éҹ˹֋§áÅéÇ àÊé¹ÃͺÃÙ»ÊÒÁ àËÅÕ‹ÂÁÁØÁ©Ò¡ÃÙ»¹ÕŒ ÂÒǡՋૹµÔàÁµÃ 1. 120 2. 40 3. 60 √ 2 4. 20 √ 2 15. ¢ºÇ¹¾ÒàËôÃÙ»ÊÕ‹àËÅÕ‹ÂÁ¼×¹¼éÒ¢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§ÊÕ‹´éÒ¹¢Í§ ¢ºÇ¹à·èҹь¹ ·Õ‹ÊÇÁªØ´ÊÕá´§ «Ö‹§ÁÕ·ÑŒ§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
  • 8. ÃËÑÊÇÔªÒ 04 ¤³ÔµÈÒʵÃì ˹éÒ 9 ÇѹàÊÒÃì·Õ‹ 20 ¡ØÁÀҾѹ¸ì 2553 àÇÅÒ 11.30 - 13.30 ¹. 16. ÃÙ»ÊÕ‹àËÅÕ‹ÂÁ¼×¹¼éÒÊͧÃÙ» ÁÕ¢¹Ò´à·èҡѹ â´ÂÁÕàÊé¹·á§ÁØÁÂÒÇ໚¹Êͧà·èҢͧ´éÒ¹ ¡ÇéÒ§ ¶éÒ¹ÓÃÙ»ÊÕ‹àËÅÕ‹ÂÁ¼×¹¼éÒ·ÑŒ§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 ÁØÁÀÒÂã¹·Õ‹ÁÕ¢¹Ò´àÅç¡·Õ‹ÊØ´¢Í§ÃÙ»ÊÒÁàËÅÕ‹ÂÁ·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◦
  • 9. ÃËÑÊÇÔªÒ 04 ¤³ÔµÈÒʵÃì ˹éÒ 10 ÇѹàÊÒÃì·Õ‹ 20 ¡ØÁÀҾѹ¸ì 2553 àÇÅÒ 11.30 - 13.30 ¹. 18. ÁØÁÁØÁ˹֋§¢Í§ÃÙ»ÊÒÁàËÅÕ‹ÂÁÁØÁ©Ò¡ÁÕ¢¹Ò´à·èҡѺ 60 ͧÈÒ ¶éÒàÊé¹ÃͺÃÙ»¢Í§ÃÙ» ÊÒÁàËÅÕ‹ÂÁ¹ÕŒÂÒÇ 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
  • 10. 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 ÁÕ¡Õ‹¾¨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
  • 11. 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§
  • 12. 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 ´ÕàÂÕ‹ÂÁ 3 ´Õ 20 ¾Íãªé 12 µéͧá¡éä¢ 5 ¢éÍÁÙÅ·Õ‹à¡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
  • 13. ÃËÑÊÇÔªÒ 04 ¤³ÔµÈÒʵÃì ˹éÒ 14 ÇѹàÊÒÃì·Õ‹ 20 ¡ØÁÀҾѹ¸ì 2553 àÇÅÒ 11.30 - 13.30 ¹. 28. ¤èÒà©ÅÕ‹ÂàÅ¢¤³Ôµ¢Í§¹ŒÓ˹ѡ¢Í§¾¹Ñ¡§Ò¹¢Í§ºÃÔÉѷ˹֋§ à·èҡѺ 48.01 ¡ÔâÅ¡ÃÑÁ ºÃÔÉÑ· ¹ÕŒÁÕ¾¹Ñ¡§Ò¹ªÒ 43 ¤¹ áÅо¹Ñ¡§Ò¹Ë­Ô§ 57 ¤¹ ¶éÒ¤èÒà©ÅÕ‹ÂàÅ¢¤³Ôµ¢Í§¹ŒÓ˹ѡ ¾¹Ñ¡§Ò¹Ë­Ô§à·èҡѺ 45 ¡ÔâÅ¡ÃÑÁ áÅéÇ ¹ŒÓ˹ѡ¢Í§¾¹Ñ¡§Ò¹ªÒ·ь§ËÁ´ÃÇÁ¡Ñ¹à·èҡѺ ¢éÍã´ 1. 2, 236 ¡ÔâÅ¡ÃÑÁ 2. 2, 279 ¡ÔâÅ¡ÃÑÁ 3. 2, 322 ¡ÔâÅ¡ÃÑÁ 4. 2, 365 ¡ÔâÅ¡ÃÑÁ
  • 14. ÃËÑÊÇÔªÒ 04 ¤³ÔµÈÒʵÃì ˹éÒ 15 ÇѹàÊÒÃì·Õ‹ 20 ¡ØÁÀҾѹ¸ì 2553 àÇÅÒ 11.30 - 13.30 ¹. 29. á¼¹ÀÒ¾µé¹-㺢ͧ¹ŒÓ˹ѡã¹Ë¹èÇ¡ÃÑÁ¢Í§ä¢èä¡è 10 ¿Í§ ໚¹´Ñ§¹ÕŒ 5 7 8 6 7 8 9 7 0 4 4 7 8 1 ¢éÍÊÃØ»ã´à»š¹à·ç¨ 1. °Ò¹¹ÔÂÁ¢Í§¹ŒÓ˹ѡ¢Í§ä¢èä¡èÁÕà¾Õ§¤èÒà´ÕÂÇ 2. ¤èÒà©ÅÕ‹ÂàÅ¢¤³ÔµáÅÐÁѸ°ҹ¢Í§¹ŒÓ˹ѡ¢Í§ä¢èä¡èÁÕ¤èÒà·èҡѹ 3. ÁÕä¢èä¡è 5 ¿Í§·Õ‹ÁÕ¹ŒÓ˹ѡ¹éÍ¡ÇèÒ 70 ¡ÃÑÁ 4. ä¢èä¡è·Õ‹ÁÕ¹ŒÓ˹ѡÊÙ§¡ÇèÒ°Ò¹¹ÔÂÁ ÁըӹǹÁÒ¡¡ÇèÒ ä¢èä¡è·Õ‹ÁÕ¹ŒÓ˹ѡà·èҡѺ°Ò¹ ¹ÔÂÁ
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