4. F 7 ( )
F 7 . . 2555
1 10 F F 2
1. F ก F3 − 2x − 3x − 7 ≥ 0 [a,b]
F a + b F F ก F
6
ก F 2x − 3 − 3x − 7 ≥ 0
2x − 3 ≥ 3x − 7
[(2x − 3) − (3x − 7)][(2x − 3) + (3x − 7)] ≥ 0
(−x + 4)(5x − 10) ≥ 0
(x − 4)(5)(x − 2) ≤ 0
∴ [a,b] = [2,4]
a + b = 2 + 4 = 6
2. F S ˈ n . . . 720 n F F ก 10800
F ก S F F F F ก F
675
ก F
720 = 24 × 32 × 5
n = 2a × 3b × 5c
[720, n] = 24 × 33 × 52
F F n = ก [720, n] = F2a × 3b × 5c 24 × 33 × 52
a = 0, 1, 2, 3, 4 b = 3 c = 2
n F = 20 × 33 × 52 = 675
+ +-
42
1

5. 3. F F ก Fsec2(2 tan−1 2 )
9
F A = tan−1 2 → tan A = 2
=sec2(2 tan−1 2 ) sec22A = 1
cos22A
= 1



1− tan2A
1+ tan2A



2
= 1


1− 2
1+ 2


2
= 9
4. ก F O ˈ ก A = (1,−4,−3) B = (3,−6,2)
F C ˈ OB F AC กก OB F OC F
3
OA = i − 4j − 3k
OB = 3i − 6j + 2k
= ProjectionOC OA OB = Pr ojOBOA
=OC (OA ⋅ OB) OB
OB
2
=OC OA ⋅ OB
OB
OB
2
=
OA⋅OB
OB
=
(1)(3) +(−4)(−6)+ (−3)(2)
32 +(−6)2 + 22
= 3
5. ก ก F F ก F3x + 32−x = 4 3
2
= =3x + 32
3x 4 3 3x 3 ,3 3
= =32x + 9 4 3 (3x) 3x 3
1
2 ,3
3
2
= 0 x =32x − 4 3 (3x) + 9 1
2
, 3
2
= 0 ∴ ก(3x − 3 )(3x − 3 3 ) = 1
2
+ 3
2
= 2
A(1, -4, -3)
B(3, -6, 2)O(0, 0, 0)
C
2

6. 6. F F x F F ก Flog 
x + 27log32
 = 1
2
= 1log[x + 33 log32
]
= 1log[x + 3log323
]
= 1log [x + 8]
=x + 8 101
∴ x = 2 F ˈ
7. ก ก F

x2 + 2
x3


10
F F F F F F ก F
3,360
กก ก F(a + b)n Tr+1 = 

n
r

 an −r br
ก F Tr+1 = 

10
r

 (x2)10 −r

2
x3


r
= 

10
r

 (2r)(x20 −2r)(x−3r)
F F ก F F F ก x ˈ 0
∴r = 420 − 2r − 3r = 0
∴ F F T4+ 1 = 

10
4

 (24) = 3,360
8. ก F ก 5 F F F ก
ก F ˈ F ก ก ก
ก F 86 F F F F ก ก ˈ
90 F F F F F ก 5 F ก ก F F
98%
µw =
w1x1 + w2x2 +w3x3 +w4x4 +w5x5
w1 +w2 +w3 +w4 +w5
µw =
1⋅x1 +1⋅ x2 + 1⋅x3 +1⋅ x4 +2 ⋅x5
1+ 1+1 +1+ 2
µw =
(x1 +x2 +x3 +x4)+ 2x5
6
90 =
86× 4+2x5
6
x5 = 98
3

7. 9. ก F ˈ F ก ˈL1 4x − 3y + 10 = 0
ˈ F F FL2 y = x2 − 8
3
x + 7
3
F ก F F F F F ก FL2 L1 L1 L2
3
y = x2 − 8
3
x + 7
3
. .. mL1
= 4
3
L1 //L2
∴mL2
= m = 4
3
. m = m F
(a, b) 4
3
= 2x − 8
3
4x − 3y + 10 = 0 4
3
= 2a − 8
3
∴ a = 2
(a, b) F F F y = x2 − 8
3
x + 7
3
→ b = 22 − 8
3
(2) + 7
3
= 1
∴ (2, 1)
F F ก F ก ก ก (a, b)L1 L2 L1
ก d =
Ax1 +By1+C
A2 +B2
=
4(2) −3(1)+10
42 +(−3)2
= 15
5
= 3
10. F F ก F
2
0
∫ 6x x − 2 dx
8
F Fx ∈ [0,2] x − 2 = − (x − 2) = 2 − x
=
2
0
∫ 6x x − 2 dx
2
0
∫ 6x(2 − x)dx =
2
0
∫ (12x − 6x2)dx
= 6x2 − 2x3 2
0
= 24 − 16 = 8
(a,b)
L1
L2
d
m = 4
3
m = 4
3
4

8. 2 20 F F 4
11. ก F P(x) ˈ ก 3 F F ก P(x)x − 1,x − 2 x − 3
F 1 P(x) F P(5) F F ก F Fx − 4
1. 2.−3 −1
3. 0 4. 2
5. 3
1
ก F F F
P(x) = a(x − 1)(x − 2)(x − 3) + 1
P(4) = 0
P(4) = a(4 − 1)(4 − 2)(4 − 3) + 1
0 = 6a + 1 → a = − 1
6
P(x) = −1
6
(x − 1)(x − 2)(x − 3) + 1
∴ P(5) = −1
6
(4)(3)(2) + 1 = − 3
12. F z ˈ F F ก กIm(z) > 0
F F ก F F

z +
3
2



2
= − 1
4
z8
1. 2.−
3
2
− 1
2
i −
3
2
+ 1
2
i
3. 4.1
2
−1
2
−
3
2
i
5. −1
2
+
3
2
i
5
=

z +
3
2



2
−1
4
=z +
3
2
1
2
i,−1
2
i
z = −
3
2
+ 1
2
i , −
3
2
− 1
2
i
∴ =z8 

−
3
2
+ 1
2
i



8
= (cis5π
6
)8 = cis20π
3
= cis(6π + 2π
3
) = cis2π
3
= cos 2π
3
+ i sin 2π
3
= − 1
2
+
3
2
i
F F F Im(z) > 0
5

9. 13. ก F a, b ˈ ก ab − 25a − 25b = 1575
F . . . F F F ก F F(a,b) = 5 a − b
1. 15 2. 45
3. 90 4. 210
5. 435
1
ก F = 1575a(b − 25) − 25b
= 1575 + 625a(b − 25) − 25b + 625
= 2200a(b − 25) − 25(b − 25)
= 2200(a − 25)(b − 25)
ก (a, b) = 5 F a = 5k1 , b = 5k2 k1,k2 ∈ I+ (k1,k2) = 1
∴ = 2200(5k1 − 25)(5k2 − 25)
= 2200(5)(k1 − 5)(5)(k2 − 5)
= 88(k1 − 5)(k2 − 5)
1
k1- 5 k2- 5 k1 k2 (k1, k2) = 1
88
44
22
11
1
2
4
8
93
49
27
16
6
7
9
13
×
×
×
Fa = 5(16) = 80 , b = 5(13) = 65 a − b = 15
1 ก กk1 > k2 (a > b)
F ˈ F ก F Fk1 < k2 (a < b)
a = 65 , b = 80 F F F กa − b
6

10. 14. ก F ˈ ก F ˂ F กu v
F F ก ˈ F F ก 3 Fu v
F 1 5 F Fu v
F F ก F F(2u + v) ⋅ (u − v)
1. 2.−27 −19
3. 0 4. 19
5. 27
2
ก F u = 1 , v = 5
F = u × v
3 = u v sin θ
3 = (1)(5)sin θ
=sin θ 3
5
F = ( Fcos θ −4
5
cos θ
ก ˂ F ก )u v
∴ =(2u + v) ⋅ (u − v) 2u ⋅ u − 2u ⋅ v + v ⋅ u − v ⋅ v
= 2 u 2 − u ⋅ v − v 2 u v cos θ
= 2(1)2 − (1)(5)
−4
5

 − 52
= −19
u
v
0
7

11. 15. ก F H ˈ F ก ˈ 9x2 − 72x − 16y2 − 32y = 16
F E ˈ F ก H Fก F ก
F E ก F F1
5
1. 2.( x−4)2
25
+
(y+ 1)2
16
= 1
(x +4)2
25
+
(y−1)2
16
= 1
3. 4.(x −4)2
25
+
(y+1)2
20
= 1
(x +4)2
25
+
(y−1)2
20
= 1
5. (x −4)2
16
+
(y+1)2
9
= 1
3
ก ก HYPER = 169x2 − 72x − 16y2 − 32y
=9(x2 − 8x + 42) − 16(y2 + 2y + 12) 16 + 144 − 16
= 1449(x − 4)2 − 16(y + 1)2
= 1(x −4)2
16
−
(y+1)2
9
= 5c = + = 16 + 9
*** ก ก HYPER F F F
F F ***
F F ก HYPER F
e = 1
5
→ c
a = 1
5
→ c
5
= 1
5
→ c = 5
ก a2 = b2 + c2
F 52 = b2 + ( 5 )2 → b2 = 20
∴ ก (x −4)2
25
+
(y+1)2
20
= 1
ก F
(9, -1)(4, -1)(-1, -1)
55
y
x
'
'
8

12. 16. ก F ABC A B ˈ
F cos 2A + 3 cos 2B = − 2 cos A − 2 cos B = 0
F F F ก F Fcos C
1. 2.1
5
( 3 − 2 ) 1
5
( 3 + 2 )
3. 4.1
5
(2 3 − 2 ) 1
5
( 2 + 2 3 )
5. 1
5
(2 2 − 3 )
3
cos A = 2 cos B (1)
2 cos2A − 1 + 3(2 cos2B − 1) = − 2
2 cos2A + 6 cos2B = 2
2( 2 cos B)2 + 6 cos2B = 2
B ˈcos2B = 1
5
→ cos B = 1
5
. .
.
(1)cos B ,cos A =
2
5
=cos C cos [180 − (A + B)] = − cos (A + B) = − [cos A cos B − sin A sin B]
= −



2
5
⋅ 1
5
−
3
5
⋅ 2
5


 = 1
5

2 3 − 2 

A
5 3
2
5 2
1
9

13. 17. F x, y, z F ก ก = a2x + y + 2z
= bx + y − z
= c3x + 2y − 2z
F F
2 −1 −2
2 2 4
a b c
= 24
F x F F ก F F
1. 2. 3. 0 4.−4 −4
5
4
5
5. 4
5
ก F
2 −1 −2
2 2 4
a b c
= 24
1 ก 3 :
a b c
2 2 4
2 −1 −2
= − 24
2 ก ก 2 : 2
a b c
1 1 2
2 −1 −2
= − 24 →
a b c
1 1 2
2 −1 −2
= − 12
ก Fdet A = det At
a 1 2
b 1 −1
c 2 −2
= − 12
กก F
x =
a 1 2
b 1 −1
c 2 −2
2 1 2
1 1 −1
3 2 −2
= −12
−3
= 4
10

14. 18. ก F A ˈ ก F i = 1, 2, 33 × 3 AXi = Bi
F X1 =





1
0
5





, X2 =





1
2
5





, X3 =





1
3
1





B1 =





1
0
0





, B2 =





0
1
0





, B3 =





0
0
1





F F F ก F Fdet (A)
1. 2. 3. 4. 1−8 −1
8
1
8
5. 8
2
ก F i = 1, 2, 3AXi = Bi
F AX1 = B1 → A





1
0
5





=





1
0
0





AX2 = B2 → A





1
2
5





=





0
1
0





AX3 = B3 → A





1
3
1





=





0
0
1





A





1 1 1
0 2 3
5 5 1





=





1 0 0
0 1 0
0 0 1





take det 2 F ก
F =(det A)
1 1 1
0 2 3
5 5 1
det I
= 1(det A)(−8)
=det A −1
8
11

15. 19. F S1 = {x log 1
2
(x + 1) + 2 log1
4
(x + 2) − log1
2
(9x − 3) ≤ 0}
ˈS2 = {x x −10 ≤ x ≤ 10}
F ก F ก F FS1 ∩ S2
1. 5 2. 6 3. 7 4. 8
5. 9
3
log1
2
(x + 1) + 2 log


1
2


2(x + 2) − log1
2
(9x − 3) ≤ 0
log1
2
(x + 1) + log1
2
(x + 2) − log1
2
(9x − 3) ≤ 0
log1
2


(x+ 1)(x +2)
9x−3


≤ 0 →
(x+1)(x+ 2)
9x −3
≤ 

1
2


0
(x + 1)(x + 2) ≤ 9x − 3 → x2 + 3x + 2 ≤ 9x − 3
x2 − 6x + 5 ≥ 0 → (x − 1)(x − 5) ≥ 0
log : x + 1 > 0 x + 2 > 0 9x − 3 > 0
x > − 1 x > − 2 x > 1
3
x > 1
3
∴ S1 = (1
3
,1] ∪ [5,∞)
∴ S1 ∩ S2 = {1,5,6,7,8,9,10}
+ +-
51
513
1
12

16. 20. ก ก 7 1, 2, 3, 4, 5, 6, 7 กF 7
1, 2, 3, 4, 5, 6, 7 ก F ก กF k F กก F
F ก ก F ก F Fk − 1
1. 32 2. 60 3. 64 4. 120
5. 128
3
กF ก F
7 6, 7 2
6 5, 6, 7 2 ( F ก ก 7 กF F )
5 4, 5, 6, 7 2 ( F ก ก 7, 6 กF F )
4 3, 4, 5, 6, 7 2 ( F ก ก 7, 6, 5 กF F )
3 2, 3, 4, 5, 6, 7 2 ( F ก ก 7, 6, 5, 4 กF F )
2 1, 2, 3, 4, 5, 6, 7 2 ( F ก ก 7, 6, 5, 4, 3 กF F )
1 1, 2, 3, 4, 5, 6, 7 1 ( ก 1 )
∴ = 2 × 2 × 2 × 2 × 2 × 2 × 1 = 64
13

17. 21. F ˈ กก F ก ก F
F F ก ก 3 F F F F
F F
1. F 2.
3. F 4. F
5. F
2
ก ก 3 F 3µ,Med
F F FM.D.,σ
=
xmax −xmin
xmax +xmin
F F F 6xmax − xmin xmax + xmin
22. ก F F ก ก ก
F ก ก 117.8 ก F 67% ก ก 126.7 ก
F 9% F F F ก F ก F 125 ก
F ก F F ก F F F ก
Z 0.17 0.44 1 1.1 1.2 1.34
F F F 0.4554 0.1700 0.3413 0.3643 0.3849 0.41
1. 84.13 2. 86.43 3. 88.49 4. 89.25
5. 90
1
ก F
17%(A = 0.17)
41%(A = 0.41)
x1= 117.8 x2= 126.7
14

18. Z1 = − 0.44 Z2 = 1.34
ก ∆Z = ∆x
σ → σ = ∆x
∆Z
= 126.7−117.8
1.34−(−0.44)
= 5
ก Z1 =
x1 −µ
σ → − 0.44 =
117.8 −µ
5
→ µ = 120
x3 = 125 → Z3 = 125− 120
5
= 1 → A3 = 0.3413
A = 0.5 + 0.3413 = 0.8413
ˈ 84.13%
23. ก ก ก Y F (3, 9)
F (1, 5) ʽ F F ก X
F ก F F
1. 9 F 2. 18 F
3. 27 F 4. 36 F
5. 54 F
4
ก , =(x − h)2 4c(y − k)
=(x − 3)2 4c(y − 9)
F (1, 5) : =(1 − 3)2 4c(5 − 9)
∴ 4c = −1
ก =(x − 3)2 −(y − 9)
y = 0 : = 9(x − 3)2
=x − 3 3,−3
∴ x = 6, 0
∴ ʽ F F= 2
3
(6)(9) = 36
z3= 1
y
x
(1,5)
2
v(3,2)
0 6
9
6
15

19. 24. ก F g ˈ ˆ กF ˈ F(2,−1)
ก g F (1, 4) F c ˈ F F ˆ กF f
f(x) =





(cx2 + 1)g(x)
2x + 10
x ≥ 1
x < 1
F x = 1 F F F ก F Ff (2)
1. 2. 3. 0 4. 4−8 −4
5. 8
1
ก g F (2,−1) → g (2) = 0 , g(2) = − 1
ก g F (1,4) → g(1) = 4
f F x = 1 : 1( ) = 1( F )
= 12(c + 1)g(1)
= 12 ∴c = 2(c + 1)(4)
x = 1 f(x) = (2x2 + 1)g(x)
=f (x) (2x2 + 1)g (x) + g(x)(4x)
∴ =f (2) (9)g (2) + g(2)(8)
= (9)(0) + (−1)(8) = − 8
25. F an =





n
2n
F F F ก F F
40
k= 1
Σ ak
1. 860 2. 1060 3. 1080 4. 1240
5. 1440
4
=
40
k= 1
Σ ak a1 + a2 + a3 + a4 + ..... + a39 + a40
= (a1 + a3 + a5 + ..... + a39) + (a2 + a4 + a6 + ..... + a40)
= (1 + 3 + 5 + ......... + 39) + (4 + 8 + 12 + ......... + 80)
= 20
2
(1 + 39) + 20
2
(4 + 80) = 400 + 840 = 1240
n ˈ
n ˈ F
20 F 20 F
16

20. 26. F a ˈ FA =



a 1 − a
1 + a −a


 I =



1 0
0 1



F F ก F Fdet(A − 2 I)(A − 3 I)(A − 5 I)(A − 7 I)
1. 2.48 − 13a (a − 2 )(a − 3 )(a − 5 )(a − 7 )
3. 17a 4. 17
5. 48
5
=A − xI



a 1 − a
1 + a −a


 − x



1 0
0 1


 =



a − x 1 − a
1 + a −(a + x)



=A − xI −(a − x)(a + x) − (1 + a)(1 − a)
= −(a2 − x2) − (1 − a2) = − a2 + x2 − 1 + a2 = x2 − 1
∴det(A − 2 I)(A − 3 I)(A − 5 I)(A − 7 I)
= [( 2 )2 − 1][( 3 )2 − 1][( 5 )2 − 1][( 7 )2 − 1] = (1)(2)(4)(6) = 48
27. ก F ˈ ก ˈEn
x2
an
2
+
y2
bn
2
= 1
n = 1, 2, 3, .....an = 2 bn ≥ 0
F ˈ ก กa1 = 2 En En−1 n ≥ 2
F F F ก F F
∞
n= 1
Σ an
1. 2. 3. 4. 156 + 4 3 8 + 4 3 10 + 4 3
5. 17
2
ก F กbn =
an
2
c = a2 − b2
a1 = 2 , b1 = 1 , c1 = a1
2
− b1
2
= 3
a2 = c1 = 3 , b2 =
3
2
, c2 = a2
2
− b2
2
= 3
2
a3 = c2 = 3
2
∴ =
∞
n= 1
Σ an a1 + a2 + a3 + .....
= 2 + 3 + 3
2
+ ..... r =
3
2
= 2
1 −
3
2
= 4
2− 3
= 4(2 + 3 ) = 8 + 4 3
17

21. 28. F F
1. F x = 0f(x) = x x + 1
2. F x = 0f(x) = x
x+1
3. F x = 0f(x) = x (x + 1)
4. F x = 0f(x) = x2 x + 1
5. F x = 0f(x) = x x
3
F f F x = 0 F F ก F F ก0− 0+
ก 1 : Ff(x) = x x + 1 x = 0− x = 0+
f(x) F F ก f(x) = x(x + 1) = x2 + x → f (x) = 2x + 1
∴ F f(x) F x = 0f (0−) = f (0+) = 2(0) + 1 = 1
ก 2 : Ff(x) = x
x+1
x = 0− x = 0+
f(x) F F ก f(x) = x
x+1
→ f (x) =
(x+1)(1)− x(1)
(x+1)2
= 1
(x+1)2
∴ F f(x) F x = 0f (0−) = f (0+) = 1
(0 +1)2
= 1
ก 3 : f(x) = x (x + 1)
: f(x) =x = 0− (−x)(x + 1) = − x2 − x
=f (x) −2x − 1 → f (0−) = − 2(0) − 1 = − 1
: f(x) =x = 0+ x(x + 1) = x2 + x
=f (x) 2x + 1 → f (0+) = 2(0) + 1 = 1
F ∴f(x) F F x = 0f (0−) ≠ f (0+)
ก 4 : Ff(x) = x2 x + 1 x = 0− x = 0+
f(x) F F ก f(x) = x2(x + 1) = x3 + x2 → f (x) = 3x2 + 2x
∴ F f(x) F x = 0f (0−) = f (0+) = 0
ก 5 f(x) = x x
:x = 0− f(x) = x(−x) = − x2 → f (x) = − 2x → f (0−) = 0
:x = 0+ f(x) = x(x) = x2 → f (x) = 2x → f (0+) = 0
F ∴ f(x) F x = 0f (0−) = f (0+)
18

22. 29. ก F F ก F a1,a2,...,a91
an =





n
3 + 4n
F F F ก F F
1. 63 2. 68 3. 71 4. 74
5. 76
4
ก F F F
a1 = 7 , a3 = 15 , a5 = 23 , a7 = 31 , a9 = 39 , .....
a2 = 2 , a4 = 4 , a6 = 6 , a8 = 8 , a10 = 10 , a12 = 12 , a14 = 14
a16 = 16 , a18 = 18 , a20 = 20 , a22 = 22 , a24 = 24 , .....
F ก F ก
2, 4, 6, 7 , 8, 10, 12, 14, 15, 16, 18, 20, 22, 23, 24, 26, 28, 30, 31
32, 34, 36, 38, 39, 40, ....., 70, 71, 72, 74, ....., 90, 367
F a1 = x4 = 7 , a3 = x9 = 15 , a5 = x14 = 23 , a7 = x19 = 31
a9 = x24 = 39 , a11 = x29 = 47 , a13 = x34 = 55 , a15 = x39 = 63
a17 = x44 = 71 , a19 = x49 = 79
∴Med = x46 = 74
n ˈ ก F
n ˈ ก
x4 x9 x14 x19
x24 x44 x46 x91
19

23. 30. ก F M =






a b
c d


 a,b,c,d ∈ {−1,0,1}



F F ก ก F ก F ก F F ˈ F ก FM
F ก F F ก F F
1. 2. 3. 4.24
81
31
81
33
81
48
81
5. 50
81
4
F A =



a b
c d


 a,b,c,d ∈ {−1,0,1}
(a, b, c, d ก F 3 , 0, 1)n(S) = 3 × 3 × 3 × 3 = 81 −1
E : F ก F F ก Non-Singular Matrix→
∴det A ≠ 0
:nnnn((((EEEE )))) det A = 0
det A =
a b
c d ad
−bc
= ad − bc = 0 → ad = bc
F F ad bc ˈ Fa,b,c,d ∈ {−1,0,1} −1,1,0
ก ad = bc 3 ก
ก 1 2 (1)(1) กad = bc = 1 → ad = 1 (−1)(−1)
2 (1)(1) กbc = 1 (−1)(−1)
∴ก 1 2 × 2 = 4
ก 2 2 กad = bc = − 1 → ad = − 1 (1)(−1) (−1)(1)
2 กbc = − 1 (1)(−1) (−1)(1)
∴ก 2 2 × 2 = 4
ก 3 5ad = bc = 0 → ad = 0 (0)(0),(0)(1),(0)(−1),
(−1)(0),(1)(0)
5bc = 0 (0)(0),(0)(1),(0)(−1),
(−1)(0),(1)(0)
∴ก 3 5 × 5 = 25
∴ 3 ก 4 + 4 + 25 = 33
20

24. n(E) = n(S) − n(E ) = 81 − 33 = 48
∴ P(E) = 48
81
************************
21