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Real problem2 p

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ข้อสอบเข้ามหาวิทยาลัยระดับชั้นมัธยมปลาย เรื่องจำนวนจริง
Onet,คณิต กข.,คณิต1,Anet,Pat1

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Real problem2 p

  1. 1. 1 F ก F 1. 2 ( 2 8 18 32)+ + + F F ก F F [O-net ʾก ก 2548] 1. 60 2. 60 2 3. 100 2 4. 200
  2. 2. 2 2. 65 33 2 32 2 27 (64) − + F F ก F F [O-net ʾก ก 2548] 1. 13 24 − 2. 5 6 − 3. 2 3 4. 19 24
  3. 3. 3 3. ก F a,b ˈ F F ก [O-net ʾก ก 2548] 1. F a b< F F 2 2 a b< 2. F 0a b< < F F 2 ab a< 3. F a b< F F a b< 4. F 2 2 a b< F F a b<
  4. 4. 4 4. F sin 65x = ° F ก F F ˈ [O-net ʾก ก 2548] 1. 2 1 x x x x < < + 2. 2 2 1 1 x x x x x < < + + 3. 2 2 2 1 x x x x < < + 4. 2 2 2 1 x x x x < < +
  5. 5. 5 5. ก F I ˈ 1 1 2 { | } 1 3 x A x I x − − = ∈ ≤ − ก A F ก F F [O-net ʾก ก 2548] 1. 4 2. 5 3. 6 4. 7
  6. 6. 6 6. 1 1 2 2 2 2 − − − F F ก F F [O-net ʾก ก 2549] 1. 3 2 2 2 − 2. 2 3 2 2 − 3. 5 3 2 2 2 − 4. 3 2 5 2 2 −
  7. 7. 7 7. 2 2 3 3 (1 2) (2 8) (1 2) (2 8)− + + − F F ก F F [O-net ʾก ก 2549] 1. 32− 2. 24− 3. 32 16 2− − 4. 24 16 2− −
  8. 8. 8 8. F 5x ≤ F F F ก [O-net ʾก ก 2549] 1. 2 25x ≤ 2. 5x ≤ 3. 25x x ≤ 4. 2 ( ) 25x x− ≤
  9. 9. 9 9. F 1 2 x = − ˈ ก ก 2 3 1 0ax x+ − = F ก ก ก ก F F ก F F [O-net ʾก ก 2549] 1. 5− 2. 1 5 − 3. 1 5 4. 5
  10. 10. 10 10. m F ก 2 4 0x mx− + = ก ˈ ˈ F [O-net ʾก ก 2549] 1. ( 5,5)− 2. ( , 4) [3, )−∞ − ∪ ∞ 3. ( ,0) [5, )−∞ ∪ ∞ 4. ( , 3) [4, )−∞ − ∪ ∞
  11. 11. 11 11. 25 2 ( ) 6 15 − F F ก F F [O-net ʾก ก 2550] 1. 3 10 2. 7 10 3. 5 2− 4. 6 2−
  12. 12. 12 12. 43 ( 18 2 125 3 4)+ − − F F ก F F [O-net ʾก ก 2550] 1. 1000− 2. 1000 3. 2 5 5 2− 4. 5 2 2 5−
  13. 13. 13 13. ก 1 2 1 1 2 x − ≤ + ≤ − F F [O-net ʾก ก 2550] 1. [ 2 1,1]− 2. [ 2 1,2]− 3. [3 2 2,1]− 4. [3 2 2,2]−
  14. 14. 14 14. ก F F ˈ กก F 2 [O-net ʾก ก 2550] 1. 2 ( 2) 1 0x − + = 2. 2 2 ( 2) ( 1) 0x x+ − = 3. 2 2 ( 1) ( 2) 0x x− + = 4. 2 2 ( 1)( 2) 0x x− + =
  15. 15. 15 15. ก ก ก 3 2x x x− = F ก F F [O-net ʾก ก 2550] 1. 0 2. 3 3. 3 1− 4. 3 1+
  16. 16. 16 16. F F (ก) ก F กก F 0 ( ) ก F กก F 0 F F ก F [O-net ʾก ก 2551] 1. (ก) ก ( ) ก 2. (ก) ก ( ) 3. (ก) ( ) ก 4. (ก) ( )
  17. 17. 17 17. F 1 2 2 8 2 2 ( 2) 32   + − +       F ก F F [O-net ʾก ก 2551] 1. 1− 2. 1 3. 3 4. 5
  18. 18. 18 18. ก F F ก F F 3 3 5 1.732 2.236 F F (ก) 2.235 1.731 5 3 2.237 1.733+ ≤ + ≤ + ( ) 2.235 1.731 5 3 2.237 1.733− ≤ − ≤ − F F ก F [O-net ʾก ก 2551] 1. (ก) ก ( ) ก 2. (ก) ก ( ) 3. (ก) ( ) ก 4. (ก) ( )
  19. 19. 19 19. F F (ก) ก F ก ก ก F F a b 0b a a b+ = = + ( ) ก F ก ก F F a b 1ba ab= = F F ก F [O-net ʾก ก 2551] 1. (ก) ก ( ) ก 2. (ก) ก ( ) 3. (ก) ( ) ก 4. (ก) ( )
  20. 20. 20 20. F a b ˈ ก ก F ก F c d ˈ ก ก F ก F F (ก) a-b ˈ ก ( ) c-d ˈ ก F F ก F [O-net ʾก ก 2551] 1. (ก) ก ( ) ก 2. (ก) ก ( ) 3. (ก) ( ) ก 4. (ก) ( )
  21. 21. 21 21. ก 7 6x − = F F ˈ [O-net ʾก ก 2551] 1. ก F F 10 15 2. ก ก F F ก 14 3. ก กก F 2 4. ก F F F F ก F 3
  22. 22. 22 22. F F ก. ˈ F F ˈ ก . ˈ F F ˈ ก F ก F [O-net ʾก ก 2552] 1. F ก 2. F ก F 3. F F 4. F ก
  23. 23. 23 23. ก F s,t,u v ˈ s t< u v< F F ก. s u t v− < − . s v t u− < − F ก F [O-net ʾก ก 2552] 1. F ก F 2. F ก F 3. F F 4. F ก F
  24. 24. 24 24. ก 2 5 1x− = F F [O-net ʾก ก 2552] 1. ( 10, 5)− − 2. ( 6, 4)− − 3. ( 4,5)− 4. ( 3,6)−
  25. 25. 25 25. F 3 4 ˈ ก 2 4 6 0x bx+ − = b ˈ F ก ก F ก F [O-net ʾก ก 2552] 1. 2− 2. 1 2 − 3. 1 2 4. 2
  26. 26. 26 26. 2 ( 4 3 5 2 3 5 5 2 4 3 3 5 )− − − + − F ก F [O-net ʾก ก 2552] 1. 0 2. 180 3. 192 4. 200
  27. 27. 27 27. ก F a ˈ ก n ˈ F ก F F ก. ( )nn a a= . n n a a= F ก F [O-net ʾก ก 2552] 1. F ก F 2. F ก F 3. F F 4. F ก F
  28. 28. 28 28. F 2 ( 3 1)− − ˈ F F [O-net ʾก ก 2553] 1. ˈ ก F ก F 1.8 2. ˈ ก กก F 1.8 3. ˈ ก F ก F 1.8 4. ˈ ก กก F 1.8
  29. 29. 29 29. F F ก. F a b ˈ a b< F 3 3 a b< . F a , b c ˈ ac bc= F a b= F F ก F [O-net ʾก ก 2553] 1. ก ก ก 2. ก ก 3. ก ก 4. ก
  30. 30. 30 30. ก F a,b c ˈ 3 0a b c > F F ก. 0ac > . 0bc > F F ก F [O-net ʾก ก 2553] 1. ก ก ก 2. ก ก 3. ก ก 4. ก
  31. 31. 31 31. F ก 2 2 ( 1)(2 6 ) 0x x x c+ − + = ก ˈ 1 ก F c F F F [O-net ʾก ก 2553] 1. (0,3) 2. (3,6) 3. (6,9) 4. (9,12)
  32. 32. 32 32. F 2 3 2 3 x + = − 2 3 2 3 y − = + F 2 2 4x xy y− + F ก F [O-net ʾก ก 2553]
  33. 33. 33 33. F a,b,c d ˈ 2 3 ( 1) ( ) 4x ax b cx dx− + = + + ก x F a+b+c+d F ก F [O-net ʾก ก 2553]
  34. 34. 34 34. F 2 ( 2) 25p − = 2 ( 1) 81q + = F F ก ˈ F 2p q− F ก F [O-net ʾก ก 2553]
  35. 35. 35 35. F F ʽ ( , )a b ˈ ก 1 6 3 17x x− + − < 2x > F a b+ F ก F [O-net ʾก ก 2553]
  36. 36. 36 36. F 6 3 6 3 x + = − 6 3 6 3 y − = + F F 2 2 4x xy y− + F F ก [Entrance ก . ʾ 2520] ก. -2 . -4 . -6 . 30 . 34
  37. 37. 37 37. F y ก ก 2 2 5 2 3 2 5 7 2y y y y− + − + + + − = F F ก F y ก F [Entrance ก . ʾ 2520] ก. 2 5 35y + = . 3 2 14y + = . 4 3 39y + = . 2 5 35y − = . 2 7 1y − =
  38. 38. 38 38. ก 2 11 2 12 5 8 x ≤ + F ก [Entrance ก . ʾ 2520] ก. 5 3 2 11x< + < . 17 3 2 20x≤ + < . 7 2 3 11x< + < . 11 2 3 13x≤ + < . F F ก
  39. 39. 39 39. ก 2 4 13 4 1x x− + < F กF [Entrance ก . ʾ 2520] ก. (1,3) . 1 ( ,3) 4 . 1 { ,3} 4 . {1,3} . F F ก
  40. 40. 40 40. (2 3)− (2 3)+ ˈ ก ก [Entrance ก . ʾ 2521] ก. 2 2 0x x+ − = . 2 2 3 0x x+ − = . 2 2 0x x− − = . 2 2 3 1 0x x+ + = . 2 4 1 0x x− + =
  41. 41. 41 41. F x F ก 3 1 1 x x x − < − − [Entrance ก . ʾ 2521] ก. 3x > 1x < . 3 1x− < < . 1x < . 3x < − . F F ก
  42. 42. 42 42. x F F F ก 16 11 5x x x− + − = + ˈ [Entrance ก . ʾ 2521] ก. 16 3 − . 20 . 27 . 31 . F F ก
  43. 43. 43 43. F 10 100x< < 1 5y< < (1) 2 100 x y < < (2) 2 1 1 10 4 y x < < (3) 2 11 125x y< + < (4) 9 95x y< − < F ก F [Entrance ก . ʾ 2523] ก. F (1) F . F (4) F . F (2),(3) F ก . F (1),(3) F ก . F (3),(4) F ก
  44. 44. 44 44. a b ˈ F ก F [Entrance ก . ʾ 2523] ก. F a ˈ F 2 a ˈ F . F 0a ≥ F 2 a a≥ . F n n a b= ก F n F a=b . F a ˈ ก b ˈ ก F ab ˈ ก . ก ก F
  45. 45. 45 45. ก F x,y,z ˈ 3 ( )x y xy x y= + +△ F F ก F [Entrance ก . ʾ 2524] ก. x y△ ˈ ก . x y y x≠△ △ . ( ) ( )x y z z y x=△ △ △ △ . F a x a x=△ . F F ก
  46. 46. 46 46. ก F x,y,z ˈ F F ก F [Entrance ก . ʾ 2524] ก. F 2x y− ˈ F F ก F 1 F 1 ( 1) 2 y x≥ − . F x y< 0z ≠ F xz yz yz< ≤ . 2 2 2 x y x y+ < + . ก 2 5 4x x− > F (5, )∞ . F ,a b ˈ 0a > x b a− ≤ F b a x a b− ≤ ≤ +
  47. 47. 47 47. ก F x,y,z ˈ F ก x y z< < F ก x,y,z F ก F 57 F F x ก F ก ก F F ก F [Entrance ก . ʾ 2524] ก. 19 . 11 . 13 . 17 . 15
  48. 48. 48 48. ก F I ˈ * ˈ ก * 2a b a b= + + ,a b I∈ ˈ F 4 F * [Entrance ก . ʾ 2524] ก. 0 . -2 . -4 . -6 . -8
  49. 49. 49 49. F F ˈ [Entrance ก . ʾ 2525] ก. F a ˈ ก F a ˈ ก . { | 2 ,A x x n n= = ˈ } ʽ Fก F ˈ F . F a b ˈ F 0ax b+ = . F a c< b d< a,b,c,d ˈ F a bi c di+ < + 2 1i = − . ก 2 3 4 0z z− + = F F ก
  50. 50. 50 50. F S ˈ F ˈ F ก F △ F ˈ b a b a=△ a,b ˈ F ˈ F S F (1) ก ก F ˈ 1 (2) F F ก ˈ 0 (3) ก F [Entrance ก . ʾ 2525] ก. F (1) F ก . F (2) F ก . F (3) F ก . F (1) (2) F ก . F (1),(2),(3) F
  51. 51. 51 51. F x F ก ก ( 1)( 5) 0 ( 1) x x x + − < − ˈ ก F [Entrance ก . ʾ 2525] ก. 2 6 5 0x x− + > . 2 1 1x − > . 1 2x< < . 1x < − 1 4x< < . 2 1 0x − <
  52. 52. 52 52. 3 2 5 1 1 x x − > + − [Entrance ก . ʾ 2526] ก. 1 ( 6, 2) (0, ) 4 − − ∪ . 1 ( 6, 2) ( 1, ) 4 − − ∪ − . 1 ( 6, 1) (0, ) 4 − − ∪ . ( 6, 1) ( 1, )− − ∪ − ∞
  53. 53. 53 53. F F [Entrance ก . ʾ 2526] ก. F , 0a b > a b≠ F 2 a b b a + < . F , 0a b > a b≠ F 2 2 1 1a b b a a b + > + . F 2 2 1a b+ = 2 2 1c d+ = F 1ac bd+ ≤ . ก ก ก ˈ ก F
  54. 54. 54 54. 3 2 4 2 x x − < + [Entrance ก . ʾ 2526] ก. 11 5 5 ( , ) ( , ) 2 6 6 − − ∪ − ∞ . 5 ( , 11) ( , ) 6 −∞ − ∪ − ∞ . 11 5 ( , ) ( , ) 2 6 −∞ − ∪ − ∞ . 3 5 ( , ) ( , ) 2 6 −∞ − ∪ − ∞
  55. 55. 55 55. ,x y x y x y+ ≤ + F M F 3 2 2 3 4x x x M− + − ≤ ก x [ ]3,2− [Entrance ก . ʾ 2526] ก. 4 . 4− . 2 . 58
  56. 56. 56 56. ก F F ก [Entrance ก . ʾ 2526] ก. 1 2x x+ − = . 12 2x x+ + = . 4 8 2x x− + − = . 3 3x x− = −
  57. 57. 57 57. ( 32 243) ( 72 27) ( 12 3 8) ( 75 48) − + + + − − F ก F F [Entrance ก . ʾ 2527] ก. 2 ( 3 2) 3 − . 2 3 ( 3 2) 3 − . 2 ( 2 6) 3 − . 2 ( 6 2) 3 −
  58. 58. 58 58. F F F [Entrance ก . ʾ 2528] ก. F x ˈ ก F F x F F 9x < . F a ˈ F ˈ F F p q , 0p q ≠ p a q = . F a ˈ F ˈ ก F a F F . F a ˈ F n n a a= 2,4,6,...n =
  59. 59. 59 59. ก 1 2 1 22 x x + < + [Entrance ก . ʾ 2528] ก. { }| 2x x > − . { }| 0x x > . { }| 0x x ≥ . 1 | 0 2 x x   − ≤ ≤   
  60. 60. 60 60. ก 2 55 ( 2) (2 3) 2 1 3x x x+ − + − + = F F [Entrance ก . ʾ 2528] ก. [ ]10,300− . [ ]400,600 . [ ]64, 32− − . [ ]250,350
  61. 61. 61 61. F F (1) F F { | , , 0n A x x a a R a= = ∈ > n ˈ } F A ʽ ก (2) F F { | ,A x x ab a= = ˈ ก b ˈ ก } F A ˈ ก (3) F F A ˈ ก * A *x y xy= − ,x y A∈ F A ก ก F F * ˈ -1 (4) F F A ˈ ก ก ∆ A ( )x y y x y∆ = − ,x y A∈ F ∆ ก F F ก [Entrance ก . ʾ 2529] ก. F (1) F (3) ˈ . F (1) F (4) ˈ . F (2) F (4) ˈ . F (2) F (3) ˈ
  62. 62. 62 62. F A ˈ ก 4 2 2 1x x ≥ − + F F ก [Entrance ก . ʾ 2529] ก. A = ∅ . ( 2,10]A ⊂ − . { 1, 2} (2, )A = − ∪ ∞ . F F ก
  63. 63. 63 63. F F ก [Entrance ก . ʾ 2530] ก. ก ก ก ก ก F F ก F F . ก ก ก ก ก F ก ก F . ก ,a b R∈ ก F * (2 )(2 )a b a b = R ก F * F ก ก F . ก a ก b a b+ ˈ ก
  64. 64. 64 64. F A ˈ ก 4 3 1x x− + − = F A F ก F [Entrance ก . ʾ 2530] ก. {3, 4} . 7 1 { | } 2 2 x R x∈ − ≤ . ( , 4)−∞ . [3, )∞
  65. 65. 65 65. F A ˈ ก 2 3 5 2 0x x+ + < B ˈ ก 2 1 0 3 x x + ≥ − F ( ) 'A B∪ F [Entrance ก . ʾ 2530] ก. ∅ . 2 [ 1, ) 3 − − . 1 ( ,3] 2 − . 2 1 ( , 1] [ , ) [3, ) 3 2 −∞ − ∪ − − ∪ ∞
  66. 66. 66 66. F , ,x y z ˈ F F ก [Entrance ก . ʾ 2530] ก. F x y< F xz yz< xz yz> . F 1 x y< ≤ n ˈ F ( 1) ( 1)n n x y− ≤ − . 2 2 2( ) ( ) 2 ( ) 2 2 x y x y xy x y xy + + − ≤ ≤ + − . F 1 2x − < F 3 3 1 1 2 2 2 2 x < < −
  67. 67. 67 67. ก F R ˈ { | 5 2} { | 2 5} A x R x x B x R x = ∈ + − ≤ = ∈ − < F F ก [Entrance ก . ʾ 2531] ก. { | 3 7}A B x R x∪ = ∈ − < < . 1 { | 3 } 16 A B x R x∩ = ∈ − < ≤ . { | 7}A B x R x− = ∈ > . 1 { | 3 } 16 B A x R x− = ∈ − < <
  68. 68. 68 68. F 2 2 { | 2 6 11 2 3 5 25}S x U x x x x= ∈ − + + − + = F ก ก S F F ก F F [Entrance ก . ʾ 2531] ก. 3 . 4 . 5 . 6
  69. 69. 69 69. ก F * 8, ,a b a b a b I= + − ∀ ∈ I = F F F ก [Entrance ก . ʾ 2531] ก. (2 *3)* 4 2 *(3* 4)≠ . ก ก F “*” I 8 . F a “*” I a− . “*” F ก
  70. 70. 70 70. , ,a b c F F F F “ F a bc< F F a b< a c< ” F ˈ [Entrance ก . ʾ 2532] ก. 1, 4, 1a b c= = = . 1, 2, 0a b c= − = − = . 1, 1, 1a b c= = − = − . 1, 1, 2a b c= − = − = −
  71. 71. 71 71. F F ก [Entrance ก . ʾ 2532] ก. ก 0a ≠ ก b ab ˈ ก . F ,a b ˈ ก ก F b a ˈ ก . ก ,a b a b≠ − a b+ ˈ ก . F ,a b ˈ ก 1 b a ≠ F ab ˈ ก
  72. 72. 72 72. ก F 1 { | 0} 2 x A x R x − = ∈ ≤ − { |1 3}B x R x= ∈ ≤ ≤ R ˈ 'A B∪ F F [Entrance ก . ʾ 2532] ก. [ 3, 1] [1,3]− − ∪ . [ , 2] [2, ]−∞ − ∪ ∞ . [ 3,3]− . ( , )−∞ ∞
  73. 73. 73 73. F ก F ( , )a b ( , )c d F ก F F F [Entrance ก . ʾ 2533] ก. F a c< b d< F c b< . F a c< d b< F c b< . F a c> b c< F d a< . F a c> b d< F b c>
  74. 74. 74 74. ,A B {( ) | , }A B a b a A b B+ = + ∈ ∈ F { | 2 1 3 2}A x x x= + − − = 1 1 2 4 { | 6 0}B x x x= − − = F A B+ F F [Entrance ก . ʾ 2533] ก. {97} . {85,93} . {20, 28} . {20, 28,85,93}
  75. 75. 75 75. x ก F ก ก 2 22 4 2 3 x x − ≥ ˈ ก F [Entrance ก . ʾ 2533] ก. [ 1,0.5)− . [0.5,1) . [1,1.5) . [1.5, 2)
  76. 76. 76 76. F a ˈ F F 2 4 3x x a− + ≤ ก F x 4 11 5x − ≤ F a F ก ก F [Entrance ก . ʾ 2533] ก. 2 5 6 0x x− + = . 2 2 3 0x x+ − = . 2 3 2 0x x− + = . 2 5 4 0x x+ + =
  77. 77. 77 77. ก F a b ˈ F F a x b< < F F F [Entrance ก . ʾ 2534] ก. 0x a+ > . 0x b+ < . 1 1 x b < . 1 1 x a <
  78. 78. 78 78. x ˈ ก 15 22 2 105x− = − F F ก F [Entrance ก . ʾ 2534]
  79. 79. 79 79. F R ˈ 2 { | 3 2 0}A x R x x= ∈ + − > { | 3 2 4}B x R x= ∈ − ≤ F F 1 2 (1) [ , ) 2 3 1 2 (2) ' ( , ) ( , ) 2 3 B A A B − = − ∪ = −∞ − ∪ ∞ F F [Entrance ก . ʾ 2535] ก. (1) ก (2) ก . (1) ก (2) . (1) (2) ก . (1) (2)
  80. 80. 80 80. F F F ก [Entrance ก . ʾ 2535] ก. 1x xy y − = x y ˈ . F xy zy> F x z> ,x y z ˈ . F 0x > 0y > F n n nx y xy= n ˈ ก . n n x x= x ˈ n ˈ
  81. 81. 81 81. F R ˈ 3 2 { | 6 9 1k R x x x k∈ − + + < ก [0,3]}x ∈ F F [Entrance ก . ʾ 2535] ก. 1 ( , ) 2 ∞ . (1, )∞ . 31 ( , ) 8 ∞ . (5, )∞
  82. 82. 82 82. ก ก ก 1 1 2 1 6 x x x x − + = − F F [Entrance ก . ʾ 2535]
  83. 83. 83 83. F a b ˈ ก a b< 2 2 3( ) 10a b ab+ = F 3 a b a b +    −  F F ก F [Entrance ก . ʾ 2536] ก. -2 . -4 . -6 . -8
  84. 84. 84 84. F p ˈ ก ,m n ˈ F 3x + 3 2 x mx nx p+ + + 1x − 3 2 x mx nx p+ + + 4 F m n F ก F [Entrance ก . ʾ 2537] ก. 4, 4m n= = − . 2, 2m n= = − . 4, 4m n= − = . 2, 2m n= − =
  85. 85. 85 85. F , ,m x y z ˈ F F F F 0 x z y > > F F F ˈ [Entrance ก . ʾ 2538] ก. 1y x z < . x yz> . my mz x < . mx mz y >
  86. 86. 86 86. ก F S ˈ ก 1 2 2 x x − > + a ˈ F F S F 2 1a + F ก F F [Entrance ก . ʾ 2538] ก. 2 . 5 . 10 . 26
  87. 87. 87 87. F a ˈ F x a− 3 2 2 5 2x x x+ − − 4 F ก F a F ก ก F F ก F F [Entrance ก . ʾ 2538] ก. -6 . -2 . 2 . 6
  88. 88. 88 88. ก F ก F { |x x ˈ F F 0 100 100}x− ≤ ≤ F { |A x= . . . x ก 21 ˈ 3 } ก A F ก F F [Entrance ก . ʾ 2538] ก. 29 . 34 . 68 . 58
  89. 89. 89 89. F x y ˈ ก 80 200x< < x pq= p q ˈ p q≠ F x y ˈ F . . . x y F ก 15015 F ก F y F ก F F ก F [Entrance ก . ʾ 2538]
  90. 90. 90 90. ก F A ˈ ก 3 0 2 x x − ≥ + B ˈ ก 1 1 2 2 x − ≤ ( ) 'A B− F ก F F [Entrance ก . ʾ 2540] ก. ( , 2) ( 1, )−∞ − ∪ − ∞ . ( , 2) [ 1, )−∞ − ∪ − ∞ . ( , 2] ( 1, )−∞ − ∪ − ∞ . ( , 2] [ 1, )−∞ − ∪ − ∞
  91. 91. 91 91. F 1 500 F 3 5 F ก F F [Entrance ก . ʾ 2540] ก. 167 . 200 . 233 . 266
  92. 92. 92 92. F n ˈ ก . . . n 42 F ก 6 F 0 0 042 ,0nq r r n= + < < 0 1 1 02 ,0n r r r r= + < < 0 12r r= 0 0 1, ,q r r ˈ F . . . n 42 F F ก F [Entrance ก . ʾ 2540]
  93. 93. 93 93. 1 2 2 x x − > − F F F [Entrance ก . ʾ 2541] ก. ∅ . (2,3) . ( 1,2) (2,7)− ∪ . 5 ( ,2) (2,3) 3 ∪
  94. 94. 94 94. F ,a b ˈ ก a b< , 5 a 3 b F ,a b ˈ F . . . ,a b F ก 165 F a b F ก F F [Entrance ก . ʾ 2541] ก. 1 . 2 . 3 . 4
  95. 95. 95 95. ก 210 ก [Entrance 1 , 2541] ก. 14 . 15 . 16 . 17
  96. 96. 96 96. ก F A B ˈ ก 2 3 0 2 x x − ≥ + 2 2 2x− ≤ F ˈ B A− [Entrance 1 , 2541] ก. { 1.6,1,6}− . { 1.7,1,7}− . { 1.8,1,8}− . { 1.8,1,7}−
  97. 97. 97 97. a b F ( , )a b = . . . a b F {1,2,3,...,400}A = ก { | ( ,40) 5}x A x∈ = F F ก F F [Entrance 1 , 2542] ก. 30 . 40 . 60 . 80
  98. 98. 98 98. F { | 2 4}A x x= − < 2 1 { |15 8 1 0}B x x x− − = − + > F A B∩ F F [Entrance 1 , 2542] ก. ( 2,3) (5,6)− ∪ . (0,3) (5,6)∪ . (0,3) (3,5) (5,6)∪ ∪ . ( 2,0) (0,3) (5,6)− ∪ ∪
  99. 99. 99 99. F {0,1,2,...,7}S = *a b = กก ab F 6 ก ,a b S∈ F F (1) *1x x= ก x S∈ (2) {4* | } {0,2,4}x x S∈ = F F ˈ [Entrance 1 , 2542] ก. (1) (2) ก . (1) ก F (2) . (1) F (2) ก . (1) (2)
  100. 100. 100 100. F , ,x y z ˈ ก F ก ก F ก F y ˈ ก F F F 3 x y z+ + ˈ ก F y F F [Entrance 1 , 2543]
  101. 101. 101 101. ก F 1x + 1x − ˈ ก 3 2 ( ) 3p x x x ax b= + − + ,a b ˈ F F กก ( )p x F x a b− − F ก F F [Entrance 1 , 2544] ก. 15 . 17 . 19 . 21
  102. 102. 102 102. ก F { | 1 2A x x= − < 1 1 } 1 2x > + 2 { | 2 0}B x x x= + < A B∩ F F F [Entrance 1 , 2544] ก. ( 1,0)− . [ 1,0)− . (0,1) . (0,1]
  103. 103. 103 103. ก F 3 2 ( ) 2P x x ax bx= + + + a b ˈ F 1x − 3x + F ( )P x F 5 2a b+ F F ก F [Entrance 1 , 2544] ก. -11 . -1 . 1 . 9
  104. 104. 104 104. ก F A ˈ ก 2 12 0x x+ − < B ˈ ก 3 1x− < A B∩ ˈ F F [Entrance 1 , 2545] ก. ( 5, 3)− − . ( 3, 1)− − . (1,3) . (3,5)
  105. 105. 105 105. F S ˈ ก 3 2 2 1 x x − ≥ − F F (1) ( 1,0] (1, )S = − ∪ ∞ (2) [ ( 2) ]x x S x S∃ ∈ ∧ + ∉ F F ก [Entrance 1 , 2545] ก. (1) ก (2) ก . (1) ก (2) . (1) (2) ก . (1) (2)
  106. 106. 106 106. ก F A ˈ ก 1x x> − B ˈ ก 5 0 ( 1)( 3) x x x − ≥ + + F A B− F ( , )a b F a b+ F F ก F [Entrance 1 , 2546]
  107. 107. 107 107. ก F I { | 1 1 1 1 50}S x x x= − − − + <i ก S I∩ F ก F F [Entrance 1 , 2546] ก. 13 . 14 . 15 . 16
  108. 108. 108 108. ก F 3 2 ( ) 4f x x kx mx= + + + k m ˈ F F 2x − ˈ ก ( )f x 1x + ( )f x F 3 F F F k m+ F ก F [Entrance 1 , 2546]
  109. 109. 109 109. F F (1) F ,a b c ˈ | (2 )a b c− 2 | ( )a b c+ F | 3a c (2) F 2 2 2 { | 1} 2 x x A x R x − + = ∈ < − 3 2 { | 2 0}B x R x x= ∈ − < F A B= F F ก [Entrance 1 , 2546] ก. (1) ก (2) ก . (1) ก (2) . (1) (2) ก . (1) (2)
  110. 110. 110 110. F S ˈ ก 3 2 0 1 1 x x − ≥ − − { | 0x x > }x S≠ ˈ F F [Entrance 1- ʾ 2547] ก. [0,1] . 1 3 [ , ] 4 2 . 1 [ ,2] 2 . 3 [ ,3] 4
  111. 111. 111 111. F a b ˈ F 2 x ax b+ + 3 2 3 5 7x x x− + + F ก 10 F F a b+ F ก F F [Entrance 1- ʾ 2547] ก. 1 . 2 . 3 . 4
  112. 112. 112 112. ก F m ˈ ก n ˈ F m 777 910 F n F m n− F ก F [Entrance 1- ʾ 2547]
  113. 113. 113 113. F F F [Entrance 1- ʾ 2547] ก. F , ,a b n ˈ ก |n a |n b F F F n . . . ,a b F . F , ,a b n ˈ ก |a n |b n F F F . . . ,a b n F . F , ,a m n ˈ ก |a mn F F F |a m |a n . F d c ˈ . . . . . . ก ,m n F F F dc mn=
  114. 114. 114 114. ˈ ก 2 6 5 1 x x − − ≤ ≤ F ก F F [Entrance 1- ʾ 2547] ก. 8 . 9 . 10 . 11
  115. 115. 115 115. F a ˈ F ก b ˈ ก F F ก [A-net ก F ʾ 2549] 1. a b ˈ F 2. a b+ ˈ 3. . . . a b F ก . . . a 2b 4. . . . a b F ก . . . a 2b
  116. 116. 116 116. ก F I ˈ F 2 { | 2 9 26 0S x I x x= ∈ − − ≤ 1 2 3}x− ≥ F ก ก S F ก F [A-net ก F ʾ 2549]
  117. 117. 117 117. F x ˈ ก F 9,12 15 x F 11 x 7 F x F F ก F [A-net ก F ʾ 2549]
  118. 118. 118 118. ก F { | (2 1)( 1) 2}A x x x= + − < 2 { |16 9 0}B x x= − > A B∩ ˈ F F F [A-net ก F ʾ 2550] 1. 2 7 ( , ) 3 3 − 2. 5 ( 1, ) 3 − 3. 4 5 ( , ) 3 4 − 4. 5 ( ,1) 3 −
  119. 119. 119 119. ก F n ˈ ก F F F 7 F F ก 4 F 9 11 F ก ( 2)n − F n [A-net ก F ʾ 2550]
  120. 120. 120 120. F ก 2 2 ( 2)x x x+ − < − F ( , )a b F a b+ F F ก F [A-net ก F ʾ 2550]
  121. 121. 121 121. ก F A ˈ ก 2 2 2 4 3x x x x+ − ≤ − + {1}B A= − F a ˈ ก B 0a b− ≥ ก b B∈ F F F ก. 4 3 a ˈ F . 5 a ˈ F F F ก [A-net ก F ʾ 2551] 1. ก. ก . ก 2. ก. ก . 3. ก. . ก 4. ก. .
  122. 122. 122 122. ก F n ˈ F ก F n 551 731 r F ก n 1093 2r + F 1r n − F F ก F F [A-net ก F ʾ 2551] 1. 1 17 2. 1 18 3. 1 19 4. 1 20
  123. 123. 123 123. ก F 2 { | 2 3 0}A x x x= + − < { | 1 2 }B x x x= + ≥ F ( , )A B a b− = F 3 a b+ F F [A-net ก F ʾ 2551]
  124. 124. 124 124. F 3 2 ( ) 10P x x ax bx= + + + ,a b ˈ 2 ( ) 9Q x x= + F ( )Q x ( )P x 1 F ( ) ( )P a P b+ F F [A-net ก F ʾ 2551]
  125. 125. 125 125. ก F 3 { | 1}S x x= = F F F ก S [PAT1 ʾ 2552] 1. 3 { | 1}x x = 2. 2 { | 1}x x = 3. 3 { | 1}x x = − 4. 4 { | }x x x=
  126. 126. 126 126. ก F S ˈ ก 3 2 2 7 7 2 0x x x− + − = ก ก S F ก F F [PAT1 ʾ 2552] 1. 2.1 2. 2.2 3. 3.3 4. 3.5
  127. 127. 127 127. ก F { | 1 3 }A x x x= − ≤ − a ˈ ก F ก A F a F F F [PAT1 ʾ 2552] 1. (0,0.5] 2. (0.5,1] 3. (1,1.5] 4. (1.5,2]
  128. 128. 128 128. ก F n ˈ r ˈ กก 2 n F 11 F F ˈ F r F F [PAT1 ʾ 2552] 1. 1 2. 3 3. 5 4. 7
  129. 129. 129 129. ก F ( )P x ( )Q x ˈ ก 2551 F ก ( ) ( )P n Q n= 1,2,...,2551n = (2552) (2552) 1P Q= + F (0) (0)P Q− F ก F F [PAT1 ʾ 2552] 1. 0 2. 1 3. -1 4. F F F F
  130. 130. 130 130. ก F A ˈ ก (2 1)( 1) 0 2 x x x + − ≥ − B ˈ ก 2 2 7 3 0x x− + < F [ , )A B c d∩ = F 6c d− F ก F F [PAT1 ก ก ʾ 2552] 1. 4 2. 5 3. 6 4. 7
  131. 131. 131 131. ก F 2 2 { | ( 1)( 3) 15}A x x x= − − ≤ F a ˈ ก F F A b ˈ ก F ก A F 2 ( )b a− F ก F F [PAT1 ก ก ʾ 2552] 1. 24 2. 16 3. 8 4. 4
  132. 132. 132 132. ก F S ˈ ก 4 2 2 13 36 0 5 6 x x x x − + ≥ + + F a ˈ F F (2, )S ∩ ∞ b ˈ F ก b S∉ F 2 2 a b− F ก F F [PAT1 ก ก ʾ 2552] 1. -9 2. -5 3. 5 4. 9
  133. 133. 133 133. F F 100 999 F 2 F F 3 F F ก F F [PAT1 ก ก ʾ 2552] 1. 250 2. 283 3. 300 4. 303
  134. 134. 134 134. ก F A ˈ ก 3 2 27 27 0x x x+ − − = B ˈ ก 3 2 (1 3) (36 3) 36 0x x x+ − − + − = A B∩ ˈ F F F [PAT1 ʾ 2552] 1. [ 3 5, 0.9]− − 2. [ 1.1,0]− 3. [0,3 5] 4. [1,5 3]
  135. 135. 135 135. ก F 2 2 2 { | } 3 2 1 x x S x x x x + = ≥ − + − F F F ˈ S [PAT1 ʾ 2552] 1. ( , 3)−∞ − 2. ( 1,0.5)− 3. ( 0.5,2)− 4. (1, )∞
  136. 136. 136 136. ก F A ˈ F ก F ก. 1 A∈ . F x A∈ F 1 A x ∈ . x A∉ ก F 2x A∈ F F ˈ ก A [PAT1 ʾ 2552] 1. 1 2 2. 1 8 3. 1 16 4. 1 32
  137. 137. 137 137. F a ˈ . . . 403 465 b ˈ . . . 431 465 F a b− F F [PAT1 ʾ 2552]
  138. 138. 138 138. ก F 1 2 1 (0,1) ( ,2) ( ,3) ... ( , ) 2 3 n n I n n − = ∩ ∩ ∩ ∩ n ˈ F n F F 2551 2553 ( , ] 2554 2552 nI ⊆ F ก F F [PAT1 ʾ 2552] 1. 2554 2. 2552 3. 1277 4. 1276
  139. 139. 139 139. ก F 2 { | 6 9 4}A x R x x= ∈ − + ≤ R F F ก F [PAT1 ʾ 2553] 1. ' { | 3 4}A x R x= ∈ − > 2. ' ( 1, )A ⊂ − ∞ 3. { | 7}A x R x= ∈ ≤ 4. { | 2 3 7}A x R x⊂ ∈ − <
  140. 140. 140 140. F N ก F b a b a∗ = ,a b N∈ F F , ,a b c N∈ ก. a b b a∗ = ∗ . ( ) ( )a b c a b c∗ ∗ = ∗ ∗ . ( ) ( ) ( )a b c a b a c∗ + = ∗ + ∗ . ( ) ( ) ( )a b c a c b c+ ∗ = ∗ + ∗ F F ก F [PAT1 ʾ 2553] 1. ก 2 F . . 2. ก 2 F . . 3. ก 1 F . 4. ก. . . . ก F
  141. 141. 141 141. F { | 3 1 1 7 1}S x R x x x= ∈ + + − = + R F ก ก S F ก F [PAT1 ʾ 2553]
  142. 142. 142 142. F R F 1 2 { | 1} 3 x A x R x x − − = ∈ > + − F [0,1)A ∩ F ก F F [PAT1 ก ก ʾ 2553] 1. 1 2 { | } 3 3 x x< < 2. 1 { | 1} 3 x x< < 3. 2 { | 1} 3 x x< < 4. 2 3 { | } 3 2 x x< <
  143. 143. 143 143. F R F { | 1 3 1 7 1}S x R x x x= ∈ + + − = − { | 3 1, }T y R y x x S= ∈ = + ∈ F ก ก T F ก F [PAT1 ก ก ʾ 2553]
  144. 144. 144 144. a b ˈ ก ก F a b⊗ ˈ F (ก) 4a a a⊗ = + ( ) a b b a⊗ = ⊗ ( ) ( )a a b a b a b b ⊗ + + = ⊗ F (8 5) 100⊗ ⊗ F ก F [PAT1 ก ก ʾ 2553]
  145. 145. 145 145. F N ก F a b a b∗ = + ,a b N∈ F F ก. ( ) ( )a b c a b c∗ ∗ = ∗ ∗ , ,a b c N∈ . ( ) ( ) ( )a b c a b a c∗ + = ∗ + ∗ , ,a b c N∈ F F ก F [PAT1 ʾ 2553] 1. ก. ก . ก 2. ก. ก F . 3. ก. F . ก 4. ก. .
  146. 146. 146 146. F N ,a b N∈ , , , a a b a b a a b b a b >  ⊗ = =  < , , , b a b a b a a b a a b >  ∆ = =  < F F , ,a b c N∈ ก. a b b a⊗ = ⊗ . ( ) ( )a b c a b c⊗ ⊗ = ⊗ ⊗ . ( ) ( ) ( )a b c a b a c∆ ⊗ = ∆ ⊗ ∆ F F ก F [PAT1 ʾ 2553] 1. ก 1 F F ก. 2. ก 2 F F ก. F . 3. ก 2 F F ก. F . 4. ก 3 F F ก. . .
  147. 147. 147 147. a b ˈ ก a b∗ a kb= ก k F ,x y z ˈ ก F F F ˈ [PAT1 ʾ 2553] 1. F x y∗ y z∗ F ( )x y z+ ∗ 2. F x y∗ x z∗ F ( )x yz∗ 3. F x y∗ x z∗ F ( )x y z∗ + 4. F x y∗ F y x∗
  148. 148. 148 148. F R F 2 2 { | 2 2 9 2 3 15}A x R x x x x= ∈ − + − − + = F ก ก ก A F ก F [PAT1 ʾ 2553]
  149. 149. 149 149. ก F ,x y z ˈ ก F ก ก 1 1 2, 32, 81xyz x y z x = + = + = 1 p z y q + = p q ˈ ก . . . p q F ก 1 F F p q− F ก F F [PAT1 ʾ 2554] 1. 3,925 2. 4,832 3. 4,951 4. 5,182
  150. 150. 150 150. ก F I F 4 2 2 5 2 2 75 ( ) 270 x x a x f x x b x − + − = + − ,a b I∈ F {( , ) | (30) 0}A a b I I f= ∈ × = 2 2 {( , ) | 2 3}B a b I I a ab b= ∈ × − + < F ก A B∩ F ก F [PAT1 ʾ 2554]
  151. 151. 151 151. F d ˈ ก กก F 1 3456, 2561 1308 F d F ก r F d r+ F ก F [PAT1 ʾ 2554]
  152. 152. 152 152. ก F , ,a b c ˈ 2 2 x y ax by cxy∗ = + + ,x y F 1 2 3, 2 3 4∗ = ∗ = 0d > x d x∗ = ก x F F 2 3 4a b c d+ + + F ก F [PAT1 ʾ 2554]
  153. 153. 153 153. ก F ( 1)( 1) 1x y x y∗ = + + − F F [PAT1 ʾ 2554] 1. ( 1) ( 1) ( ) 1x x x x− ∗ + = ∗ − 2. ( 2) ( ) ( 2)x y x y x∗ + = ∗ + ∗ 3. ( 2) ( ) 2x y x y∗ ∗ = ∗ ∗ 4. ( ) ( 1)( )x x y x x y x∗ ∗ = + ∗ +
  154. 154. 154 154. F A F ก ก 3 1 2 2 3 1x x x− − > + B ก 2 ( 2)( 1) 0x x x+ + < F F F ก F [PAT1 ʾ 2555] 1. A B− ก 5 2. A B A∪ = 3. A B∩ ก 1 4. ( ) ( )A B B A B− ∪ − =
  155. 155. 155 155. b a b a∗ = a b ˈ ก F ,a b c ˈ ก F F F ก F [PAT1 ʾ 2555] 1. ( ) ( )a b c a c b∗ ∗ = ∗ ∗ 2. ( ) ( )a b c a bc∗ ∗ = ∗ 3. ( ) ( )a b c a b c∗ ∗ = ∗ ∗ 4. ( ) ( ) ( )a b c a c b c+ ∗ = ∗ + ∗
  156. 156. 156 156. ก F 7 4 3 , 2 2 2 2...a b= + = 2 3c = + F F ก F [PAT1 ʾ 2555] 1. 1 1 1 c a b > > 2. 1 1 1 c b a > > 3. 1 1 1 b a c > > 4. 1 1 1 b c a > >
  157. 157. 157 157. F a b ˈ F 5 4ax bx+ + F 2 ( 1)x − F a b− F ก F [PAT1 ʾ 2555]
  158. 158. 158 158. F 5 4 3 2 ( )f x x ax bx cx dx e= + + + + + , , , ,a b c d e ˈ F ก ( )y f x= ก ก 3 2y x= + 1,0,1,2x = − F F (3) ( 2)f f− − F ก F [PAT1 ʾ 2555]
  159. 159. 159 159. F d ˈ ก กก F 1 1059 , 1417 2312 F d F F ก r F F d r+ F ก F [PAT1 ʾ 2555]
  160. 160. 160 160. ก F ab ˈ ก , {1,2,...,9}a b∈ a F ก F b F (310 ) (465 ) 2790ab ba× − × = F a b+ F ก F [PAT1 ʾ 2555]
  161. 161. 161 161. ก S ˈ ( , , , , , )a b c d e f , , , , , {0,1,2,...,9}a b c d e f ∈ F ก 3 2 2 4 , 2 7b a c d− = − = 3 2 1e f− = − ก S F ก F [PAT1 ʾ 2555]
  162. 162. 162 162. ก F I F { | 2 7 9}A x I x= ∈ + ≤ 2 { | 1 1}B x I x x= ∈ − − > F F (ก) ก A B∩ F ก 7 ( ) A B− ˈ F F F ก F [PAT1 ʾ 2555] 1. (ก) ก ( ) ก 2. (ก) ก F ( ) 3. (ก) F ( ) ก 4. (ก) ( )
  163. 163. 163 163. F 0,1,2,3 4 4× F ( F 1 ) a b 2 3 0 c d 1 2 0 1 3 F F 0,1,2 3 F ก F 0,1,2 3 F F (ก) F a c< F b d< ( ) F a b> F c d< ( ) F b d< F c d< ( ) a b c d+ = + F F F ก F [PAT1 ʾ 2555] 1. (ก)-( ) ก 1 F 2. (ก)-( ) ก 2 F 3. (ก)-( ) ก 3 F 4. (ก)-( ) ก ก F
  164. 164. 164 164. F A ˈ ก 3 2 2 3 1 3 10 6 3 1 14x x x x+ + + + + + + = F B ˈ ก ก ก A B∪ F ก F [PAT1 ʾ 2555]
  165. 165. 165 165. ก F {1,2,3,..., }A k= k ˈ ก F {( , ) | 0 7}B a b A A b a= ∈ × < − ≤ F k F ก F F ก B F ก 714 [PAT1 ʾ 2555]
  166. 166. 166 166. F x ก abc y ก cba , , {1,2,3,...,9}a b c ∈ , ,a b c ก F ก F S ˈ x x y− F ก F ก ก S F ก F [PAT1 ʾ 2555]
  167. 167. 167 167. ก F R F { | 2 5 7}A x R x x= ∈ − + ≤ 2 { | 12 }B x R x x= ∈ < + F F (ก) { |1 4}A B x R x∩ ⊂ ∈ ≤ < ( ) A B− ˈ ก (finite set) F F ก F [PAT1 ʾ 2556] 1. (ก) ก ( ) ก 2. (ก) ก F ( ) 3. (ก) F ( ) ก 4. (ก) ( )
  168. 168. 168 168. ก F 33 3 7 5 , 5 7 , 5 7A B C= = = 3 7 5D = F F ก F [PAT1 ʾ 2556] 1. D C A B> > > 2. A C B D> > > 3. A B D C> > > 4. C A D B> > >
  169. 169. 169 169. ก F , ,a b c d ˈ ก 2 , 5 , 6a b b c c d< < < 100d < F a F ก F ก F [PAT1 ʾ 2556]
  170. 170. 170 170. ก F , , {1,2,3,...,9}a b c ∈ 3 ก abc F ก F ก ก abc ab ba ac ca bc cb= + + + + + ( abc 3 ก , , , , ,ab ba ac ca bc cb 2 ก) [PAT1 ʾ 2556]
  171. 171. 171 171. x y ˈ ก ก F x y∗ ˈ ก F (1) ( ) ( )x xy x x y∗ = ∗ (2) (1 ) 1x x x∗ ∗ = ∗ (3) 1 1 1∗ = F 2 (5 (5 6))∗ ∗ ∗ F ก F [PAT1 ʾ 2556]
  172. 172. 172 172. F R F 2 2 { | 3 4 3 2}A x R x x x x= ∈ + − + > + F A ˈ F F [PAT1 ʾ 2557] 1. ( ,2) (3,4)−∞ ∪ 2. ( ,0) (3, )−∞ ∪ ∞ 3. ( , 1) (4, )−∞ − ∪ ∞ 4. ( 1, )− ∞
  173. 173. 173 173. ก F a b ˈ ก a b< ก x a x b b a− − − = − F ก F F [PAT1 ʾ 2557] 1. { }b 2. ( , ]a b 3. [ , )b ∞ 4. ( , ) 2 a b+ ∞
  174. 174. 174 174. F , , , ,a b c d e ˈ ก 5 4 3 2a b c d e= = = = 2 3 4 5a b c d e+ + + + ˈ ก F F F 4 3 4a b c d e+ + + + F ก F F [PAT1 ʾ 2557] 1. 52 2. 120 3. 262 4. 312
  175. 175. 175 175. F ʽ F ˈ 10 ก ABCDEFGHIJ (ก) , , , , , , , , , {0,1, 2,...,9}A B C D E F G H I J ∈ , , , , , , , , ,A B C D E F G H I J ˈ ก F ก ( ) , , ,A B C D ˈ ก A B C D> > > ( ) , ,E F G ˈ F ก E F G> > ( ) H I J> > 15H I J+ + = F C F I+ + F ก F F [PAT1 ʾ 2557] 1. 10 2. 13 3. 15 4. 17
  176. 176. 176 176. F x ˈ ก ˈ ก 2 2 14 3 9 5 1x x x x+ − − + − = F F 1 2 2 1 4 12 9 3 2 x x x x − − − − − + − F ก F [PAT1 ʾ 2557]
  177. 177. 177 177. F A ก 2 2 2 4x x x− + + = − F A ˈ F F [PAT1 ʾ 2557] 1. ( 4,0)− 2. ( 1,1)− 3. (0, 4) 4. ( 3, 2)−
  178. 178. 178 178. F A x F ก ก 2 2 4 3 1 4 8 7 4 10 7 x x x x x x + = − + − + F B x F ก ก 2 2 2 4x x x− + > F F (ก) A B⊂ ( ) ก F A B∩ F ก 2 F F ก F [PAT1 ʾ 2557] 1. (ก) ก ( ) ก 2. (ก) ก F ( ) 3. (ก) F ( ) ก 4. (ก) ( )
  179. 179. 179 179. ก F ,a b c ˈ ก a b< F F (ก) 2 3 4 2 3 3 2 3 3 2 a b c a b a b c a b + + + > + + + ( ) 3 2 3 2 2 3 2 3 a b c a b a b c a b + + + > + + + F F ก F [PAT1 ʾ 2557] 1. (ก) ก ( ) ก 2. (ก) ก F ( ) 3. (ก) F ( ) ก 4. (ก) ( )
  180. 180. 180 180. F ก ก ก ABCDEF , , , , , {0,1, 2,...,9}A B C D E F ∈ 14A B+ = 0C D D E E F− > − > − > F F ก [PAT1 ʾ 2557]
  181. 181. 181 181. F A 2 2 2 2 a b c d+ + + , , ,a b c d ˈ ก (ก) a b d= + ( ) ( ) ( )a b c d b a c d+ + + = − ( ) 2 ( 1)cd a c+ = − F M F ก A m F F A F F M m− F ก F [PAT1 ʾ 2557]
  182. 182. 182 182. F a b ˈ ก aRb a F b F F (ก) F xRy yRz F ( )xR y z+ ก ก ,x y z ( ) F wRx yRz F ( ) ( )wy R xz ก ก , ,w x y z F F ก F [PAT1 ก ʾ 2557] 1. (ก) ก ( ) ก 2. (ก) ก F ( ) 3. (ก) F ( ) ก 4. (ก) ( )
  183. 183. 183 183. F , , ,a b c d x ˈ ก F F (ก) F a c b d < F a x c x b d + + < ( ) a a x b b x + < + F F ก F [PAT1 ก ʾ 2557] 1. (ก) ก ( ) ก 2. (ก) ก F ( ) 3. (ก) F ( ) ก 4. (ก) ( )
  184. 184. 184 184. ก F , ,A B C D ˈ ก F ก ,B C D D A C B= + = + − 2A C B= − F F ก F [PAT1 ก ʾ 2557] 1. D A C B< < < 2. A D C B< < < 3. D C A B< < < 4. C A D B< < <
  185. 185. 185 185. F S ก 2 3 2 6 2 4 4 10 3x x x x+ − − + − = − F ก ก S F ก a b . . . a b F ก 1 F a b+ F ก F [PAT1 ก ʾ 2557]

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