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???? 2555 (???? ???????????? 7 ???? ?? 55) ?????? ?????????????????? ??? ?. Sila Sookrasamee ???? 1 ??? 18
??????? Boonchuay Pataralertsiri (?????????????????? ?? ?????????, ??????? ,??????????) http://www.facebook.com/groups/HighSchoolMath /
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????????????????????? x ???
3 7
3 2 0 , 3 7 0
2 3
x x x x? = → = ? = → = ???? 3 ????
1.
3
: | 3 2 | | 3 7 | 0 3 2 (3 7) 0 4
2
x x x x x x< ? ? ? ≥ → ? + ? ≥ → ≥ ????????????????????????????
2.
3 7 7
: | 3 2 | | 3 7 | 0 (3 2 ) (3 7) 0 2 2
2 3 3
x x x x x x x≤ < ? ? ? ≥ → ? ? + ? ≥ → ≥ → ≤ <
3.
7 7
: | 3 2 | | 3 7 | 0 2 3 (3 7) 0 4 4
3 3
x x x x x x x≥ ? ? ? ≥ → ? ? ? ≥ → ≤ → ≤ ≤
??????? 2. ??? 3. ?????????????? x ?????? union ??????????????????????? [2,4] ???????
[ , ] [2,4] 2 4 6a b a b= + = + = ??? 6
???????
?.?.?. ?????? 2 ?????? n ??? 720 = 10800
4 3 2
1080 8 1350 8 25 54 8 25 2 27 2 3 5= × = × × = × × × = × ×
2 4 4 2
720 9 80 9 16 5 3 2 5 2 3 5= × = × × = × × = × × ???????
3 2 2
3 5 3 15 3 225 675n = × = × = × =
??? 675
( )
2
22
2 1 2 1
22
2tan 2 2 2 2
sec (2tan 2) 1 tan (2tan 2) 1 1 1 1 8 9
1 tan 1 21 2
θ
θ
? ?
? ?
? ?? ? ? ?= + = + = + = + = + =? ?? ? ? ?? ?? ?? ? ? ? ? ??
? ?
??? 9

????????? C ????????? OB ??????? OC ?? vector ?????????????? ?????????????? ?????????????????? C =(3m,-6m,2m) ???
OC ?????????? AC ????????????
2 2 2 2
0 . (3 6 2 ) [(3 1) ( 6 4) (2 2) ]
3 (3 1) ( 6 )( 6 4) (2 )(2 3)
(9 3 ) (36 24 ) (4 6 ) 49 21 7 (7 3)
OC AC m OB AC mi mj mk m i m j m k
m m m m m m
m m m m m m m m m m
= ? = ? = ? + ? ? + ? + + ?
= ? + ? ? + + +
= ? + ? + + = ? = ?
???????
3
7
m = . ??????? 2 2 23 3
| | | | 3 ( 6) 2 9 36 4 3
7 7
OC m OB= = + ? + = + +=
2 2 2
2
1 3
2 2
3 3 4 3 3 3 4 3 3
3 4 3 3 9 0
(3 3 3)(3 3) 0 3 3,3 3 3 ,3
x x x x
x x
x x x
?
+ = → + = × →
? × + =
? ? = → = =
1 3
,
2 2
x = ???
1 3
2
2 2
+ =
???????
33log 2 log 23 31 33 3log 2 log 2
log( 27 10 27 3 3 2 8 2) 1 log10x x x x x x x+ → = + = + = + = + = + → == =
??? 2

???????
??????? r+1 =
( ) ( ) ( )
( )
10
10 10 1 10
2 2 2 2
3 3 3
10
2 20 5 4
3
102 2 2
10 ..
10 10 102 10 9 8 7
2 20 5 0 4 2 16 3360
4 24
r
r
r
r
r r
x x x x
rx x x
x x r r
r rx
? ?
?
?
? ?? ? ? ? ? ?
+ = + + + ? ?? ? ? ? ? ?
? ? ? ? ? ?? ?
? ? ? ? ? ? × × ×? ?
= → ? = → = → = × =? ? ? ? ? ?? ?
? ?? ? ? ? ? ?
?????????????? 5 ???? 0
x ???????? 3360 ??? 3360
????????????????? 5 ??????? x %
????????????????????
86% 4 2 % 540 344
90% 90 6 540 344 2 98%
4 2 2
x
x x
× + ?
= → × = = + → = =
+
??? 98%
????????????????? =
4
3
=
8
2
3
dy
x
dx
= ? 2
4 8
8 73 3 2 (2) 2 (2) 1
2 3 3
x y
+
∴ = = → = ? + = ??????????????????????(2,1)
???????????????????????? ??? ???????????????? 2 2 2 2
8 3 10| | | 4(2) 3(1) 10 |
3
54 3
ax by c
d
a b
? ++ + ? +
= = = =
+ +
??? 3 ?????
2 2
2 2 3 2
00 0
6 | 2 | (12 6 ) 6 2 | 6(4) 2(8) 8x x dx x x dx x x? = ? = ? = ? =∫ ∫

???????
( )
3 2
3 2
3 2
3 2
( ) 1 ( 1)( 2)( 3) ( 6 11 6)
( ) ( 6 11 6) 1
1
(4) 0 4 6 4 11 4 6 1 (64 96 44 6) 1 6 1
6
1 1 24
( ) ( 6 11 6) 1 (5) (125 150 55 6) 1 1 3
6 6 6
p x A x x x A x x x
p x A x x x
p A A A A
p x x x x p
?= ? ? ? = ? + ?
= ? + ? +
? ?
= = ? × + × ? + = ? + ? + = + → =?? ?
? ?
=? ? + ? + → =? ? + ? + =? + =?
?????? 1
2
3 1 3 1
2 4 2 2
z z i
? ?
+ =? → + =±? ?? ?
? ?
??????????? Im( ) 0Z > ???????
3 1 5
1
2 2 6
z i
π
=? + =∠
8 40 2 1 3
1 1 6
6 3 2 2
z i
π π
π=∠ =∠ + =? + ?????? 5.

???????
5 ???? ?.?.?. ?????? 2 ?????? a,b ????????? 5 , 5a k b l= = ?????
1575 25 25 25 125 125 63 5 5 ( 5)( 5) 25ab a b kl k l kl k l k l= ? ? = ? ? → = ? ? = ? ? ? →
( 5)( 5) 63 25 88 11 8 16, 13 80, 65k l k l a b? ? = + = = × → = = → = =
??????????
25 25 80 65 25 80 25 65 1575ab a b ok? ? = × ? × ? × =
80 65 15a b? = ? = ?????? 1
23 4
| | | || | sin 1 5sin 5sin 3 sin cos 1 sin
5 5
u v u v θ θ θ θ θ θ× = =× = =→ = → =? ? =?
? ?
?????????? cosθ ??????
2 2 2 24
(2 ) ( ) 2 . . . 2 | | | || | cos | | 2(1) 1(5) 5 2 4 25 19
5
u v u v u v u v v v u u v vθ
? ?
+ ? ? = ? ? = ? ? = ? ? ? = + ? =?? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ?
?????? 2

2 2
2 2
2 2
2 2
2 2
9 72 16 32 16
9( 8 16) 16( 2 1) 16 9 16 16 1 16 144 16
( 4) ( 1)
9( 4) 16( 1) 144 1
4 3
x x y y
x x y y
x y
x y
? ? ? =
? + ? + + = + × ? × = + ?
? +
? ? + = → ? =
??????? H=Hyperbola ?????????????????? (4,-1) ?????????????????? X ????????? a=4, ??? b=3
???????????? 2 2 2 2
4 3 5c a b= + = + =
E ???????? ?????????????????????????? H ??????? 5a = ????????????????????????????? ?????? (4,-1)
???????? E:
2 2 2 2
2 2 2 2
( 4) ( 1) ( 4) ( 1)
1
5
x y x y
a b b
? + ? +
+ = + =
??????????????? ??? E ???
2 2 2 2 2
2
2 2
1 1 5 1 4
5 5 55
c a b a b b
e e
a a a a
? ? ?
= = = → = = → = =
2 2 24 4
5 20
5 5
b a= = = ??????????????? E:
2 2
( 4) ( 1)
1
25 20
x y? +
+ =?????? 3

???????
2 2 2 2 2
2
2 cos2 3cos2 (2cos 1) 3(2cos 1) (4cos 1) 6cos 3) 10cos 4
1 2
10cos 2 0 cos cos 2 cos
55
cos cos[180 ( )] cos( ) cos cos sin sin
A B A B B B B
B B A B
C A B A B A B A B
?= + = ? + ?= ? + ?= ? →
? =→ = → = =
= ? + =? + =? +
2 1 5 2 5 1 2 2 3
5 5 5 5 55
? ?
=? + =? + ?????? 3

2 1 2
24 2 2 4 (4 4 4 ) (4 8 2 ) 12 6
2 1 2
1 1 1
3 2 2
2 1 2
| | 1 1 1 ( 4 3 4) ( 6 4 2) 3
3
2
2 2
4c a b a b c b c
a b c
x a
AX B y b
z
A
b
c
c
? ?? ?
? ?= = ? ? + ? + =? + →? ?
? ?? ?
? ? ? ? ? ?
? ? ? ? ? ?== ? =? ? ? ? ? ?
? ? ? ? ? ??? ? ? ? ? ?
= ? = ? ? + + ? +
=
+
+
= ?
?
?
???????????? x ???????????? column ?????? A ?????????
1 2
1 1
2 2 2 4 ( 2 2 2 ) 6 3
| |
4
3 3
2
a
b
c a c b c a
b c
b b c
x
A
?
? ? ? + + ? + + ?
= = ?
?
+= =
?
?????? 5

1 1
1 0 0
1
0 1 0 | | | |
| |
0 0 1
AX B
AX I A X A X
X
? ?
=
? ?
? ?= = → = → = =? ?
? ?? ?
1
1 1 1
1 1
0 2 3 2 15 0 ( 10 15 0) 8 | | | |
| | 8
5 5 1
X A X
X
?
= = + + + ? ? + =? → = = =? ?????? 2
2
2
1 1 1 1 11
2 4 2 2 22
1
1 1 1 1 1
2 2 2 2 2
log ( 1) 2log ( 2) log (9 3) log ( 1) log ( 2) log (9 3)
( 1)( 2) 1
log ( 1) log ( 2) log (9 3) log 0 log
9 3 2
x x x x x x
x x
x x x
x
? ?
? ?
? ?
≥
+ + + ? ?= + + + ? ?
+ + ? ?
= + + + ? ?= ≤= ? ?
? ? ?
???????
( 1)( 2)
1
9 3
x x
x
+ +
≥
?
????? 2 2( 1)( 2)
1 3 2 9 3 6 5 ( 1)( 5) 0
9 3
x x
x x x x x x x
x
+ +
≥ + + ≥ ? → ? + = ? ? ≥
?
??????? 1 ( ,1] [5, )S = ?∞ ∞? ???????????????? log y ??? 0y > ??????? 1 0, 2 0, 9 3 0x x x+ > + > ? > ???????
1
3
x >
??????? ?????????????????????????? 1
1
( ,1] [5, )
3
S= ∞? ????? 1 2 {1,5,6,7,8,9,10}S S∩ = ???????? 7 ?????? ?????? 3

???????
????????
???????? 7 ?? ????????????? ??????? 6 ???? 7 ??? ??? 2 ????
???????? 6 ?? ????????????? ??????? 5,6 ???? 7 ??? ?????????? 6 ???? 7 ???????? 1 ??? ???????? ????? 2 ????
???????? 5 ?? ????????????? ??????? 4,5,6 ???? 7 ??? ?????????? 5 - 7 ???????? 2 ??? ???????? ????? 2 ????
???????? 4 ?? ????????????? ??????? 3,4,5,6 ???? 7 ??? ?????????? 4 - 7 ???????? 3 ??? ???????? ????? 2 ????
???????? 3 ?? ????????????? ??????? 2,3,4,5,6 ???? 7 ??? ?????????? 3 - 7 ???????? 4 ??? ???????? ????? 2 ????
???????? 2 ?? ????????????? ??????? 1,2,3,4,5,6 ???? 7 ??? ?????????? 2 - 7 ???????? 5 ??? ???????? ????? 2 ????
???????? 1 ?? ????????????? ??????? 1,2,3,4,5,6 ???? 7 ??? ?????????? 1 - 7 ???????? 6 ??? ???????? ????? 1 ????
??????????????????????????????????? 7 ??????????? 6
2 1 64= × = ???? ??? ???3

1. ???????????????????
1 1
1 1
( 3) ( 3)
n n
i i
i i
X X X X
n n= =
= + ? += ?∑ ∑ ???????? ???
2. ????????????????????? =
_ max _ min max min max min max min
_ max _ min max min max min max min
( 3) ( 3)
( 3) ( 3) 6
new new
new new
X X X X X X X X
X X X X X X X X
? + ? + ? ?
= = <
+ + + + + + +
???
3. ?????????
1 1
1 1
( 3) 3 3
n n
new i i
i i
X X X X
n n= =
= + = + = +∑ ∑ ????????? ???
4. ??????? ????????????????????????????????????????????????? ?????????? ???
5. 2 2 2 2
1 1
1 1
[( 3) ( 3)] ( )
n n
new i i
i i
S X X X X S
n n=
= + ? += ?=∑ ∑ S→ ?????? ???
?????? 2.

117.8
0.44 (1)
126.7
1.34 (2)
126.7 117.8
1.78 (2) (1)
8.9
(2) 5 126.7 1.34 5 120
1.78
X
S
X
S
S
S X
?
= ?
?
=
?
= ?
→ = = → = ? × =
?????????????? 125 ?????????
125 125 120
1
5
X
S
? ?
= = =Z ????????????????? ??????? 0.3413 ??????????????????????
0.5 0.3413 0.8413 84.13%+ = = ?????? 1

???????????????????? ?????????????????????????????? Y ???
2 2 2
( ) 4 ( ) ( 3) 4 ( 9)x h p y k x p y? = ? = ? = ? ?????????????(1,5)?????
2 1
(1 3) 4 (5 9) 4 4 ( 4)
4
p p p? = ? = = ? → =? ??????????????? 2
( 3) ( 9)x y? =? ? ????????????? 2
6y x x=? +
????? 0y = ??? 0,6x = ?????????????????? x ???????????????????
???????
6 6
2 3 2 6 3 2
00 0
1 1
( 6 ) 3 | 6 3(6 ) 36
3 3
ydx x x dx x x= ? + =? + =? + =∫ ∫ ?????? 4.

2
2
( ) 2 ( ( ) ( )
(2) 2 2 (2) 2 (2) 4 ( 1) 4 (0) 4
d d
f x cx g x cx g x
dx dx
d d
f c g c g c c c
dx dx
= +
= + = ? + =?
??????????? C ??????????? ( )f x ????????????????????? x=1 ??? 2
( ) 2 ( ( ) ( )
d d
f x cx g x cx g x
dx dx
= +
2
(2) 2 2 (2) 2 (2) 4 ( 1) 4 (0) 4
d d
f c g c g c c c
dx dx
= + = ? + =?
2 12
(1) ( (1) 1) (1) 2(1) 10 ( 1)4 12 1 2
4
f c g c c= + = + = + = → = ? =
(2) 4 4(2) 8
d
f c
dx
=? =? =? ?????? 1
40
1
(1 3 5 .. 39) 2(2 4 6 .. 40)n
n
a
=
= + + + + + + + + +∑
1 39 2 40
20 2 20
2 2
+ +
= × + × × ????????????????????? = ?????????? x ?????????????????????????????????
20 20 40 21 400 840 1240= × + × = + = ?????? 4

2 2 2 2 2 2 2 2
2 3 5 7
[(2 ) ( 1)][(3 ) ( 1)][(5 ) ( 1)][(7 ) ( 1)]
A I A I A I A I
a a a a a a a a
? ? ? ?
= ? + ? ? + ? ? + ? ? + ?
1 2 4 6 48= × × × = ??? ???5
???????
1
1 1
2 2 2 2
2 1 1 1
2
2 2 2
3 2 2 2
1
1
2, 1
2
2 1 3
3 3 3
( 3) 3
2 4 2
3 1 1 2 2 4 4(2 3)
2 3 ... 2 4(2 3)
2 1 1 1 4 33 2 3
1
2
n
i
i
a
a b
a c a b
a c a b
r r
a a
r r r
∞∞
=
= = =
= = ? = ? =
? ?
= = ? = ? = ? =? ?? ?
? ?
? ? +
= + + + = × = × = = = = = +
? ? ? ??
?
∑
?????? 2

2 2
2 2 2 2
2 2
( ), '( )
1. ( 1), 2 1 ( 1), 2 1
1 1
2. , ,
( 1) ( 1) ( 1) ( 1)
3. ( 1), 2 1 ( 1), 2 1
4. ( 1)
( ), '
, 3 2 ( 1),
( ) '(0 ) '
3 2
1
1
1
0
0
(0 )
1
1
1
0
0
5. , 2 , 2
x x oo
f x f x
x x x x x x
x x
x x x x
x x x x
f x f x f
x x
x x x x x x x x
x
f
x x x
?
= ?
+
= +
=
+ += + ?=
= =
+ + + +
+ += ? + ? ?=
+ += ? + ? ?=
= ? ?
=
=
?
????????????????????????????????????????????????????????? '(0 )f +
??? '(0 )f ?
???????????????????? 3 ??? 1 ???-1 ????????????
????????????????????????? 0x = ?????? 3

??????? ?????????????????????????????????????? 7 ????????????????????? ?????????????????? 1 ????????? 3 4 1 7= + × =
(??????????????????????????????????? 7 ???? 2,4,6,1,3,5,7 ????????? 2,4,6,7,15,31 ?????????)
?????????????????????????????????? 10 ???????????????????? ????????????? ??????????? 3, 5,7,.. ?????????
???????????????????????? 7 10 8 87+ × = ???????????????????? 1 8 2 17+ × = ????????? 3 4 17 71= + × =
????????????????????? 2, ???? 4 ???? 87 2 89+ = ???? 87 4 91+ = ??????????????????????????
?????????????????? 88,89 ???? ,88,89,90 ??? 91 ????????????? 88, 3+4x89, 90, 3+4x91???????????????? ??????? 17(71)
?????????????????????????????????????????????????????????????????????????????????????? ?????????????????????????????????????????
????????????????????????????? 17(71) ??????????????? ??????????????????? 70,17,72,74 ??????????
?????????????? 85,89,??? 91 ???????????????????????? ??????? 70 ????????? 70,72, ??? 74 ?????????
?????????????? 1 2 91, ,..,a a a ?????????? 74a ?????? 4

?????? 4 ??? ??? a,b,c,d ???????????????? 3 ??? ??????? ????????????????? =3x3x3x3 = 81 ????
??? 0M ad bc= ? = ???????????????? ????????? ?????? ?? 3 ???? ??? ????? 1,0,1ad bc= = ?
???? 1ad bc= = ? ??????????????????? -1,1 ???? 1,-1 ???? 2 ???? ?????????? 2 2 4= × = ????
???? 0ad bc= = ??????????????????? 0,0 0,-1 ???? 0,1 -1,0 1,0 ???? 5???? ??????? ??? 5 5 25× = ????
???? 1ad bc= = ? ??????????????????? 1,1 ???? -1,-1 ???? 2 ???? ?????????? 2 2 4= × = ????
??? 3 ??????? 0M ad bc= ? = ??? 4 25 4 33+ + = ???? ??????? 0M ad bc= ? ≠ ???? 81 33 48? = ????
???????????????? M ?????????? inverse ??? ???????? 0M ad bc= ? ≠ ??????????
48
81
?????? 4.

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