POSN5708BKK
of 21
-
Upload
kanchit-saeho -
Category
Documents
-
view
25 -
download
1
description
004
Transcript of POSN5708BKK
-
(.. 2557) 1
( .) 31 2557 9.00 12.00 .
2 : , ( ) =
3 : , ( ) =
4 : , ( ) = 0
8
9 [( ) ] [( ) ( ) ] [( ) ( ) ( ) ]
10 d( )
= {1, 2, 3} = { } 1 6 1. = { | | | } | |
2.
16 Oct 2014
-
2 (.. 2557)
3.
4.
5.
-
(.. 2557) 3
6. = { | : } = { | 1 1 } | |
7. (1) 10 10 (2)
8. 1. 1 2. 3
-
4 (.. 2557)
9. (1) 10 10 , (2) |d( )| = 6 (3) d( ) = d(2 ) = d(3 ) = d(4 ) = d(5 ) = d(6 )
10. (1) 10 10 , (2) |d( )| = |d(2 )| = |d(3 )| (3) d( ) d( ) d(3 ) = {1, 2, 3, 4, 5, 6, 7, 8, 9}
-
(.. 2557) 5
11. (1 )(1 )(1 ) (1 ) = , ,
12.
=
13. =
(10 1) (
100
1m
)
-
6 (.. 2557)
14. = 3 ( ) = 3 ( )
15. = { | 100 } = { | [ | ] } | |
16. = { | 3 = 0 } | |
-
(.. 2557) 7
17. : Z ( ) = 10 100
1i
( )
18. 17 30 10 ( )
19. 1 = 0 > 0 100
1n
( )
-
8 (.. 2557)
20. = { 1 1 | } = { | < < }
21. ABC G G B C AP, BQ, CR P, Q, R BQ = 7 CR = 5 AP
22. 2 , 4 , P, Q, R PQR
-
(.. 2557) 9
23. ABC AC = BC = 18 AD, CE BC, AB D, E
sin
=
DE
24. ABC B AB, BC 5, 12 AB B AC AB D AD
25. AD, BE, CF ABC 9, 12, 15 AB
-
10 (.. 2557)
26. ABCD 132 AB CD AB, CD 14, 8 E ABCD BEC = 90 E AB
27. ABCDE A(0, 0) , B(7, 0) , C(7,
) , D(5, 3) , E(0, 8) = ABCDE =
28. ABC AC = BC CD AB D CAB CD E BC F AF CD AEC
-
(.. 2557) 11
29. ABCD AB ABC ACD 150 120 ABCD ... ( ) = 1
30. ABCD ADC X DC AX = AD ,
BX = BA CX = CB
=
... ( ) = 1
-
12 (.. 2557)
1. 675 9. 142857 16. 2 24. 0.2 2. 4096 10. 192 , 219 , 17. 1124 25. 10 3. 1024 273 , 327 18. 8 26. 8.83 4. 64 11. 1302 19. 5050 27. 4 5. 5 12. 3.83 20. 28. 108 6. 24 13. 415 21. 12 29. 9 7. 1023 14. 20 22. 1.2 30. 2 8. 142857 15. 8 23. 7.5
1. 675 1: ( ) 2: 1 ( ) 2: 1 = ( )(
1)( 1) = (3)(15)(15) = 675
2. 4096 [( ) ] ( ) ( ) ( ) ( ) 4 4 = 16 4 12 = 4096
3. 1024 4 4 = 16 ( ) ( ), ( ), ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 , ) = 4 ( ) ( ) ( ) ( ) ( ) + 4 =
+ 4 = 10
10 = 1024
4. 64 = ( ) ( ) =
= 6
= = 64
5. 5
-
(.. 2557) 13
3 3 = 9 (1, 1), (2, 2), (3, 3) (1, 2)&(2, 1) (1, 3)&(3, 1) (2, 3)&(3, 2) (1, 2)&(2, 1) (2, 3)&(3, 2) = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)} (1, 2) (2, 3) (1, 3) ( ) 3 1 1 3 + 1 + 1 = 5
6. 24 : , 1 1 1 4 2 1 3 3 1 2 4 3 2 = 24
7. 1023 10 10 + 9 2 = 10 10 9 ( ) ( ) ( 10 10 + 9 ) 10 10 1 1000 ( ) 10 10
10 1000
< 1 = 10 (1000 10 + 1)(1) = 991 9 1
< 2 = 10, 11 (9 5 + 1)(2) = 10
= 4 = 10, 11, 12 3 = 3 = 10, 11, 12, 13 = 2 = 10, 11, 12, 13, 14 = 1 10 11 1 19 10 991 + 10 + 3 + 4 + 5 + 10 = 1023
8. 142857 1xxx x 1 ( xxx x) = 10
{(1, 1), (2, 2), (3, 3)} {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} {(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)} {(1, 1), (2, 2), (3, 3), (2, 3), (3, 2)} {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2)}
-
14 (.. 2557)
xxx x1 10 + 1 10 + 1 = 3(10 ) 3 7 3 10 1 999 9 =
999 9 7 9
= 42857 = 142857
9. 142857 d( ) d(2557) = {2, 5, 7} = 10 2 1 100,000 999 999 6 2 6 2 4 = 10abcd : 2, 3, 4, 5, 6 2, 3, 4,
5, 6 ( 0 ) 3 2, 3, 4, 5, 6 4 2, 3, 4, 5, 6
= 11abcd : 1 2 = 12abcd : 2, 3, 4, 5, 6 ()
2, 3, (4 5), 6, 7 3 2, 3, (4 5), 6, 7 2 3, (4 5), 6, 7 d 3 3 9 9 , d 4 2 8 , 5 2 0 , 6 3 8 , 7 8 d
= 13abcd : 2, 3, 4, 5, 6 2, (3 4), 5, 6, (7 8) 3 2, 5, 6, (7 8) d 2, 6, 8 5 0 , 5 2 0 , 7 d
= 14abcd : 2, 3, 4, 5, 6 2, 4, 5, 7, 8 4 2, 5, 7, 8 d 2, 8 0 , 5 0 , 7 2 6 d = 7 = 14abc7 3 2 8 3 = 5 = 14ab57 = 142857 , 3 285714 , 428571 , 571428 , 714285 , 857142
10. 192, 219, 273, 327 = 1000 3 1 100 999 |d( )| 3 2 |d(2 )|, |d(3 )| 3 3 d( ) d( ) d(3 ) 9 , 2 , 3 3 ( 0)
7 2999 9 42857
-
(.. 2557) 15
3 3 3 98 3 9 = abc , 2 = def , 3 = ghi a = 1, 2, 3 a = 3bc : 2 3 3 3 2 = 6ef , 3 = 9hi () 3, 6, 9 3, 6, 9 , 2 , 3 3, 6, 9 (4, 8, 2) , (7, 4, 1)
= 2bc : , 2 , 3 2 (3, 6, 9) , (7, 4, 1) , (8, 6, 4) , (9, 8, 7)
= 1bc : , 2 , 3 1 (2, 4, 6) , (3, 6, 9) , (4, 8, 2) , (6, 2, 8) , (8, 6, 4) , (9, 8, 7) 6
= 192 , 219 , 273 , 327 4
11. 1302 1
= 3b4 2 = 6e8 3 = 9h2
1, 5, 7
327 654 981
= 3b7 2 = 6e4 3 = 9h1
2, 5, 8
= 2b3 2 = de6 3 = gh9
1, 4, 5, 7, 8 b 2 , 3 b = 7
273 546 819
2b7 5e4 8h1
= 2b7 2 = de4 3 = gh1
3, 5, 6, 8, 9 d = 5 g = 8
3, 6, 9
= 2b8 2 = de6 3 = gh4
1, 3, 5, 7, 9 d g 5, 7
2b8 5e6 7h4
1, 3, 9
= 2b9 2 = de8 3 = gh7
1, 3, 4, 5, 6 d g 4, 6
2b9 4e8 6h7
1, 3, 5
219 438 657
= 1b2 2 = de4 3 = gh6
3, 5, 7, 8, 9 d g 3, 5
1b2 3e4 5h6
7, 8, 9
192 384 576
= 1b3 2 = de6 3 = gh9
2, 4, 5, 7, 8 d g 2, 4
1b3 2e6 4h9
5, 7, 8
= 1b4 2 = de8 3 = gh2
3, 5, 6, 7, 9 d g 3, 5
1b4 3e8 5h2
6, 7, 9
= 1b8 2 = de6 3 = gh4
2, 3, 5, 7, 9 b 2 , 3
= 1b6 2 = de2 3 = gh8
3, 4, 5, 7, 9 d = 3
1b6 3e2 gh8
4, 5, 7, 9 b 2 , 3
= 1b9 2 = de8 3 = gh7
2, 3, 4, 5, 6 b 2 , 3
-
16 (.. 2557)
= ( ) ( ) ( )
= ( )
( )
= 1022 + 248 + 32 = 1302
12. 3.83 = 3 , = , = 13
3 = 0 ( 13) = 0
= 0 + ( )
=
3.83
13. 415 100
1m
= 100
1m
(10 1) =
100
1m
(10 1) =
(
100
1m
10 100
1m
1 )
=
( 111 1 0
100) =
( 111 1 110
100) =
( 111 1 010
)
1 9
= 12345679 = ( )
8 = 37
111 1
9 10 = 10(37) = 370
010 9
1 3 890 = ( )
= 45
(100
1m
) = 370 + 45 = 415
14. 20 ( ) = =
2 1+2+3+4+5 = 15 5 (5)(15) = 75 3 1+2+3+4 = 10 6 (6)(10) = 60 ( ) = 3 = 3 = 75 , 0 = (75)(60) = 4500
( )( ) ( )( ) ( )( ) ( )( ) ( )( ) + ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) + ( )( ) ( )( ) + ( )( )
=
= = 0 ( ) = 0
= 0 = 0
(3 ) = 0
= 0 ,
18 = 0
=
=
(1)( )( )( )( )( ) (3 )( 3 )( 3 )( 3 )( 3 )( 3 ) (3 )( 3 )( 3 )( 3 )( 3 )( 3 ) (3 )( 3 )( 3 )( 3 )( 3 )( 3 ) (3 )( 3 )( 3 )( 3 )( 3 )( 3 )
-
(.. 2557) 17
(4500) = (4)(5) = 20
15. 8 = 1, 2, 3 1 1 1, 2, 3 = 4, 5, 6, 7, 8 1 ... 1, 2 = 2
2 1, 2 4, 6, 8 9 10 11 1 1 3 ... 1, 2, 3 = 6
6 1, 2, 3 12 1 1 18 1 3 ... 1, 2, 3, 4 = 12
12 1, 2, 3, 4 24 3 ... 1, 2, 3, 4, 5 = 60 60 = 36, 37, 38 8 ... 1, 2, 3, 4, 5, 6 = 60 60 ... 1 0 100 = {1, 2, 3, 4, 6, 8, 12, 24} 8
16. 2 = 3 = 0 = 1 1 3 = 0
= = 2 = 2 | | = 2
17. 1124
10 = 25
2 5 (1)( )(3) ( ) 2 5 ( 2 ) 5
5
5 5 5 ()
25 5 25
5 =
( 125 = 100)
( ) =
100
1i
( ) = 100
1i
(
) =
100
1i
+
100
1i
( )
100
1i
: = 1, 2, 3, 4
= 0 5
1
= 5 9 = 1 = 10 14
= 2 = 95 99
= 19
= 100 0
100
1i
(0) (1) ( ) (3) (19) 0
= 0 + 5( ( )
) + 20 = 970
1 1 0 3 2 1 1 2
1 1 2 0 ( 1)( ) = 0 ( 1)( 1)( ) = 0
-
18 (.. 2557)
100
1i
100
1i
= 24(0) + 25(1) + 25(2) + 25(3) + 4
= 0 + 25(6) + 4 = 154
( ) 100
1i
( ) = 870 + 154 = 1124
18. 8 30! 10 10 30! 10 = 25 5 30! 5, 10, 15, 20, 25, 30 25 5 5 2 10, 20, 30 4 2 3 5 7 = 8 8 30! 8 10 1 0 30 9 10 10 9 10 ( 5 )
(1 3 9) (8) = ( ) (8) = 2 9 10 8
19. 5050 1 = 0 = 1 = 1 = ( 1) = = ( ) =
100
1n
( ) = 100
1n
= ( )
= 5050
20. = ( 1, 0.874032049] (1 2.288245611] < <
21. 12 DE BCRQ CR // DE // BQ CD = DB
ED =
= 6 ( BR )
AGP ~ DGE
=
=
ED = 6
=
AP 1
1 2 3 4 6 7 9 11 12 13 14 16 17 18 19 21 22 23 24 26 27 28 29
=
=
A
C B
G P R
Q
D
E
7 5
-
(.. 2557) 19
22. 1.2 = 2 = 1, 2, 3 C C C 1 1 3 3 3 C C C C =
3 = 6
PQR C C C 3
= 6 (
1 1) (
C ) (
3 3 C )
=
(
) (
3 3
)
=
(
) (
3 3
) = 0 1 = 1.2
23. 7.5
ABC CE C
= ACE ACD ACE = ADE CAD = C D DFE ~ CFA
=
= sin = sin
=
CA = 18 DE = 18 = 7.5
24. 0.2 AC E BOC EOC (--) OC C
=
=
=
AD ( ) = 0.2
25. 10 Apollonius
(1) + (2) + (3) : 3 3 3 0 1 0 ( ) AB ( ) (3) 3 AB 10
1 1
2
2
3
3
C
C C P
Q R
18
C
A B
D
E
F
B
A
C 12
D
O
E
A
B C D
E F
AB + AC = 2(AD + BD ) ( ) + ( ) = (9 ) + = 9 + 81 (1)
+ 1 ( ) + (3)
AB + BC = 2(B + A ) BC + AC = 2(CF + AF )
-
20 (.. 2557)
26. 8.83 FG, CH 132 =
(8 1 ) = 12 = FG = CH
EG = FE = 1 EGB EFB B = C = (1 ) = 1 0
ECB CB = B C = 1 0 = 09 HCB CB = C B = 1 3 = 153
09 = 153 1 8 0 =
=
= 6 + 2 ~ 6 + 2(1.414) = 8.828 ~ 8.83
27. 4 = D = DE F DE
=
= 8
F ( , 8) =
(8 ( 8))( )
= (( 8) 3)( ) +
(3
) ( )
2
1 = 1 1 1 3 0 =
= 8 B = 8 = 8 + 2( 2) = 4
28. 108
AED CFB
1 ( )
AD = DB
= 2 AF = 2CD
=
( )
1 F C
CEF CFE = CFA 2 + + 180 (1) CDA + 90 + 2 180 ( ) 2(1) (2) 5 0 A C = ECF + CFE = = 108
29. 9 A B AD // BC AB // CD ABCD CE AB
AB = 150
CD = 120
AB =
CD =
ACB = 90 ACB ~ AEC ~ CEB
=
( )
7 G H 4 3 A B
C D
E
4 4 F
B(7, 0) A(0, 0)
C(7,
)
D(5, 3)
E(8, 0)
=
F
A B
C
D
F
E
A B
C D
E
-
(.. 2557) 21
EB =
=
=
AE = AB EB =
=
( )
=
8100 = 90 = 310
AB =
= 1010 , CD =
= 810 , AD = BC = B = (310)
(
)
= 90 10 = 10
= ( )
= 910 10 = 9 10 10 = 9
30. 2 3 3 3 DC // AB D A = XAB C B = XBA = XBA = D = B = 2 X = 180
= = 2
180 = 72
DXA D = = 72 A = 180 2(72) = 36 DF D DXA AXD AX = AD = DFX DFA DX = DF = AF = XF = AX AF =
AXD ~ DFX
=
=
= 0
( )
1 0
=
( ) ( ) ( )( )
( ) =
=
=
=
=
= 5(1) + 5 4(2) = 2
Thitipong Hopetrungraung
A B
C D
X
A
D X
36
36
36
72
72
F
