math mama57
DESCRIPTION
เอกสารติวคณิตศาสตร์มาม่าปี57TRANSCRIPT
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4
4
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. 7
GAT PAT O-NET
...
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(O-Net + PAT 1 + ) . , .
3
8 13
16
. ( We By The Brain)
23
35
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0, , , , , , 3 2
0 3 2
sin 0
1 2 3 1 0 - 1 0
cos 1
3 2 1 0 - 1 0 1
tan 0 1 1 3 0 0
cot 3 1 1 0 0
sec 1 2 2 2 - 1 1
csc 2 2 2 1 - 1
2
3
3
6 4 3 2 2
030 45 60 90 270180
3606 4 3 2
2
2
2
2
3
3
2
2
1. sin . csc = 1 5. cot = cos , sin 0 2. cos . sec = 1
3. tan . cot = 1 6. cos2+ sin2= 1 4. tan = sin , cos 0 7. 1 + tan
2= sec
2
8. cot2+ 1 = csc2cos
sin
. , . ,
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sin (A + B) = sin A cos B + cos A sin B sin (A - B) = sin A cos B - cos A sin B cos (A + B) = cos A cos B - sin A sin B cos (A - B) = cos A cos B + sin A sin B
tan (A + B) = tan A + tan B tan (A - B) = tan A - tan B
sin 2A = 2 sin A cos A cos 2A = cos2A - sin2A = 2 cos2A - 1 = 1 - 2 sin2A tan 2A = 2 tan A
sin 3A = 3 sin A - 4 sin3A
cos 3A = 4 cos3A - 3 cos A tan 3A = 3 tan A - tan
3A
sinA = 1 - cos A
cosA = 1 + cos A
tanA = 1 - cos A
2 sin A cos B = sin (A + B) + sin (A - B) 2 cos A sin B = sin (A + B) - sin (A - B) 2 cos A cos B = cos (A + B) + cos (A - B) 2 sin A sin B = cos (A - B) - cos (A + B) sin A + sin B = 2 sinA + B cosA - B
sin A - sin B = 2 cosA + B sinA - B
cos A + cos B = 2 cosA + B
cos
A - B
cos A - cos B = - 2 sinA + B sinA - B
1 - tan A tan B
1 - tan2A
1 - 3 tan2A
1 + tan A tan B
2
2
2
2
1 + cos A
2
2
2
2
2
2
2
2
2
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1. sin (arcsin x) = x, - 1 x 1 cos (arccos x) = x, - 1 x 1 tan (arctan x) = x, x R 2. arcsin (sin x) = x, -x
arccos (cos x) = x, 0 x arctan (tan x) = x, -< x <
3. arcsin (- x) = - arcsin x arccos (- x) = - arccos x arctan (- x) = - arctan x
ABC A + B + C = 180 A + B = 180- C
sin (A + B) = sin C A + B = 90-C
sinA + B = sin (90- C) = cosC
1. ABC = 1 ab sin C = 1 bc sin A = 1 ca sin B
2. : a = b = c
3. :a2 = b2+ c2- 2bc cos A
b
2
= c
2
+ a
2
- 2ca cos B c2 = a2+ b2- 2ab cos C 4. a = b cos C + c cos B b = c cos A + a cos C c = a cos B + b cos A
1. sin A - cos A = 0 csc A
2. 1
3. cos - sin = 5 sin 2
2
2
2
2 2
2
3
2
2 2
2
2
sin A sin B sin C
2
2
2
2
2
1 - xy
1 + xy
4. arcsin x + arccos x =
arctan x + arccot x =
arcsec x + arccsc x =
5. arctan x + arctan y = arctan x + y
arctan x - arctan y = arctan x - y
BA
C
2 sin x + 3 cos x
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1. ABC C sin B = 12 tan(B - A)
1. 1 2. 119 3. 13 4. 169
2. ABC B cosec (B - C) = 3 cot A + cot B + cot C
1. 5 2. 2 5 3. 9 5 4. 1
3. 1> 0,
2> 0
1+
2p sin
1= 3 cos
1= 5 sin (
1+
2)
1. 43 2. 54 3. 63 4. 64
4. tan 15+ tan 75 1. 1 2. 2 3. 3 4. 4
5. [sin (2 arccos 3 )] [cos (2 arctan 2)]
1. 96 2. 72 3. - 24 4. - 72
6. cos (arcsin (- 5))
1. 12 2. - 12 3. 5 4. - 5
13
12
59
2 5 13
65 6565 65
12
4. 1sin 10sin 50sin 70
5.
sin (arctan 2 + arctan 3)
6. cos (arctan 7 + arcsin 4 )
7. sec [ 1(arcsin 3+ arccos 3 )] + tan [ 1(arcsin 4+ arccos 4 )]
2 2
2
5
120 60
55
5
5
8
5
125 125 12525
13
13 13 13 13
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7. () arccos (- 1) - arcsin 1 = arctan (- 1)
() arctan 3 - arccos 3= arcsin 1
1. () () 2. () () 3. () () 4. () ()
8. arctan x = 2 arctan 1- arctan 1 cos (270- arctan x)
1. - 13 2. - 9 3. 9 4. 13
9. ABC a = 16 , b = 10 c = 14 a
1. 3 2. 3 3 3. 5 3 4. 7 3
10. ABC C tan B = 2.4 D E AC BC AB DE 13 12 DE AB AD
1. 1 2. 5 3. 5 4. 12
2
2
5 10 5 10 5 10 5 10
2
2 3
13 12 13
1. 2 2. 3 3. 3 4. 4 5. 4
6. 1 7. 3 8. 2 9. 3 10. 1
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{1, 2, 3, , n} a1, a
2, a
3,
1. n + 1 n n a
n + 1- a
n d
a
1 d
, a - 3d, a - d, a + d, a + 3d,
, a - 2d, a - d, a, a + d, a + 2d,
2. n + 1 n n an + 1 r
a1, a
1+ d, a
1+ 2d, a
1+ 3d, , a
1+ (n - 1) d,
n an= a
1+ (n - 1) d
a1, a
1r, a
1r2, a
1r3, , a
1rn - 1,
n an= a
1rn - 1
an
an b
n lim a
n= A lim b
n= B
1. lim c = c, c
2. lim kan= kA, k
3. lim (anb
n) = A B
4. lim (anb
n) = AB
5. bn0 n B 0 lim an = A
6. an0 n m > 1 lim a
n = A
7. k 7.1 lim nk 7.2 lim 1 = 0
n
n
n
n
n
n
n
n n
n
bn
nk
B
m m
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a
1, a
2, a
3, , a
k
an= a
1+ a
2+ a
3+ ... + a
kn= 1
k
a1, a
2, a
3, , a
k,
an= a
1+ a
2+ a
3+ ... + a
k+ ...
S
n= a
1+ a
2+ a
3+ ... + a
n n
n
Sn =n(a1+ an)
Sn = n[2a
1+ (n - 1) d]
n S
n =
a1- ra
n, r 1
Sn =
a1(1 - rn), r 1
a
1
, a2
, a3
, S
1 = a
1
S2 = a
1+ a
2
Sn = a
1+ a
2+ a
3+ ... + a
n
Sn n S
1, S
2, S
3,
1. c c = ck
2. (anb
n) = a
n b
n
3. (can) = c a
n
4. 1 + 2 + 3 + ... + n = n(n + 1)
5. 12+ 22+ 32+ ... + n2 = n(n + 1) (2n + 1)
6. 13+ 23+ 33+ ... + n3 = n2(n + 1)2 = (1 + 2 + 3 + ... + n)2
n= 1
n= 1
n= 1
n= 1 n= 1
n= 1 n= 1
k
k
k k
k k
2
6
4
2
2
1 - r
1 - r
.
.
.
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1. S
n lim S
n= S S
S
2. Sn lim S
n
1. : a
n= a
1+ (n - 1) d
a1= d = 0
2. : a
n= a
1rn - 1
|r| 1 |r| < 1 S = a1
3. 1 = 1 + 1 + 1 + ... p
p > 1 p 1
n
n
1. an lim a
n= 0
lim an= 0 a
n
2. lim an0 an
3. an b
n (a
n+ b
n)
4. an b
n (a
n+ b
n)
1 - r
n
n
n
n= 1
np 2p 3p
n= 1
n= 1
n= 1
n= 1
n= 1 n= 1 n= 1
n= 1 n= 1
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1. a1, a
2, a
3, ... a
2+ a
3+ ... + a
9= 100 S
10= a
1+ a
2+ ... + a
10
2. {100, 101, 102, ..., 600} 6 12
3. 1 + 6 + 11 + 16 + ... + 101
4. 10 2, 5, 10, 17, 26, ...
5. U = {1, 2, 3, ..., 500} 3
6. U = {200, 201, 202, ..., 2543} 5
7. 1 + 3 + 5 + 7 + ... + 209
8. 2 + 6 +18 + 54 + ... + 4347
9. 2 + 6 + 18 + 54 + 162 + 468
10. 1 + 1 + 1 + L + 1
8 .9 9 .10 10 .11 119 .120
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11. n 7 + 77 + 777 + 7777 + ...
12.
12
- 22+ 3
2
- 42
+ ... + (2n - 1)2
- (2n)2
1. an
an + 2= 2 n
an= 31 a
n
1. 21275- 1 2. 21276- 1 3. 22551- 1 4. 22552- 1
2. 4, a, 9, b, c 5 a > 0 a + 8c1. 156 2. 160 3. 164 4. 168
3. k x3- 3x2- kx + 3 = 0 1. 1 2. 2 3. 3 4. 4
4. x + 2, 9, x + 10 3 d xdn - 1
1. 410- 1 2. 49- 1 3. 3 (410) - 1 4. 3 (4
9) - 1
5. 2,000 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, ... 1. 10 2. 11 3. 12 4. 13
6. lim (3n + 1) - n2- 4n
7. an= n
3
- n2- 2
8. an=
1 + 2 + 3 + ... + n ( 3)
1. 121 2. 120 3. 111 4. 110
9. 4 + 1+ 2+ 3+ ... + n+ ...
1. 5 2. 6 3. 7 4. 8
an
3
n - 8n3+ 6n
2n2- 1
an
12+ 22+ 32+ ... + n2
2n + 1
3
n= 1
n= 1
n= 1
n= 1
10
10
10
2552
n 3
2 22 23 2n
10. n + 2 - n - 1n= 1
n2+ 3n + 2
1. 2 2. 4 3. 1 4. 1 5. 2 6. 2 7. 0.25 8. 2 9. 2 10. 2
2
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R
Q Q
I Q - I
I- {0} N
: p(x) = anxn+ a
n - 1xn - 1+ ... + a
1x + a
0
an, a
n - 1, , a
1, a
0 a
n0 n
p(x) x - c p(c)
: p(x) = anxn+ a
n - 1xn - 1+ ... + a
1x + a
0
an, a
n - 1, , a
1, a
0 a
n0 n
x - c p(x) p(c) = 0
a1
a2
an
an - 1
a1 a2 an - 2 anan - 1
+ +-
1 2 (x - a
1) (x - a
2) ... (x - a
n -1) (x - a
n) a
1< a
2< < a
n
3 x (x - a1) (x - a
2) ... (x - a
n - 1) (x - a
n) = 0
x = a1, a
2, , a
n - 1, a
n
4 x
5 +
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: a b 1. |a| 0 2. |a| = |- a| 3. |a2| = |a|2= a2 4. a2 = |a| 5. |ab| = |a||b| 6. a = |a|, b 0
7. |a - b| = |b - a| 8. |a + b| |a| + |b| 9. |a - b| ||a| - |b|| |a| - |b| 10. |a| = |b| a = b 11. a 12. a 11.1 |x| < a - a < x < a 12.1 |x| > a x < - a x > a
11.2 |x| a - a x a 12.2 |x| a x - a x a
6 < 0 > 0 =
a |a|
|a| = a, a 0
- a, a < 0
b |b|
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+
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+ >
1. 3 2. 3 3. 1 4. 6 5. 1
6. 4 7. 1 8. 9. 3 10. 193
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( ) ( )
( )
( )
( ) ( )
( ) ( )
+
+ +
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1. 720 2. 2. 3. 18 4. 135 5. 3 6. 6 7. 3 8. 5 9. 0.14 10. 4 11. 4 12. 2
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(O-Net + PAT 1 + )
.
By P'Golf We By The Brain
1. (a + b)3 = a3 + 3a2b+ 3ab2 + b3
2. (a b)3 = a3 3a2b+ 3ab2 b3
3. a3 + b3 = (a + b)(a2 ab + b2)
4. a3 b3 = (a b)(a2 +ab + b2)
a, b, am, an, bn a, b 0 m, n
1. am an = am + n
2. am
an = am n
3. (am)n = amn
4. (ab)n = anbn
5. a
b
n
= an
bn b 0
6. an = 1an
a 0
7. a1n = n a
8. a0 = 1 a 0
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Exponential Function
f = {(x, y) R R+ / y = ax ; a> 0 a 1}
1. 1
2. R
3. R+
4.
5. , a> 1 0< a< 1
Logarithmic Function
g = {(x, y) R+ R / y = logax ; a> 0 a 1}
1. 1
2. R+
3. R
4.
5. , a> 1 0 < a< 1
log
1. 6.loga1 = 0 logbna = 1
n logba
2. 7.logaa = 1 alogam = m
3. 8.loga(mn) = logam + logan alogbm = mlogba
4. 9. loga mn = logam logan logba =
log
m
a
logmb m> 0 m1
5. 10.logbam = m logba logba = 1
logab
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Problems
1. A = 7 3 5 , B = 5 3 7 , C = 3 5 7 D = 3 7 5
1. D > C > A > B 2. A > C > B > D
3. A > B > D > C 4. C > A > D > B
2. a = 248 , b = 336 c = 524
1. 2.1b> 1c>
1a
1a>
1
b> 1c
3. 4.1b> 1a>
1c
1a>
1c>
1
b
3. a = 7 + 4 3 , b = 2 2 2 2... c = 2 + 3
1. 2.1c> 1a> 1b
1c>
1
b> 1a
3. 4.1b> 1a>
1c
1
b> 1c>
1a
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4. x, y z x + y+ z = 16
xyz yx+ z = x2(x+ z) 3y = 3(9z)
5. R
C = xR (3x2 11x + 7)(3x2 + 4x+ 1) = 1
C
6. x2 + 5x +5 (x5) = 1
1. 2. 3. 0 4.5 52
5
2
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7. A 31+ 2x + 92x = 244
A
1. 2. 3. 4.(1,4) (2,0.5) (0,5) (3,0)
8. x
(2x 4)3 + (4x 2)3 = (4x + 2x 6)3
1. 2 2. 2.5 3. 3 4. 3.5
9. A
5(1+ x2 4x1) +5
( 5+ 4xx2
2+ x2 4x1)
= 126
A
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10. x, y, z 81x = 27
1y = 36
1z
x+yz
11. A x [0, 2)
2(1 + 3sinx) 5 22sinx + 2(2 + sinx) = 1 A
12. . (log2557)log2557
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13. (log2x)3 + 12
logx2 = 7(log 1
2
x)2
1. 8 2. 16 3. 24 4. 25
14. Suppose the number a satisfiesaloga +log (loga) log (log (log2) log (loga)) = 0
What is the value of aaaa
1. 1 2. 2 3. 4 4. 85. 16
15. x y x + y 5(x2A)2yA = (16)64 A = logy
logx
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16. log2x+6logx2 5 = 0
1. 8 2. 10 3. 12 4. 145. 16
17. If x and y are real numbers such that 2log (2y 3x) = logx+ log y , find xy
18. A log3(3(2x2 + 2x) +9) = x2 + x + 1
log3
B B = {x2 xA}
19. If and then the value ofx = 10 + 2
2 y = 10 2
2 , log2(x2
+ xy + y2
)is equal to1. 0 2. 2 3. 3 4. 4
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20. a b
log(1 + a2) loga 2log2 = 1 log(100+ b2) +log b 2ab
21. R A = {xR log
3(x 1) log 3
3(x 1) = 1}
B = {xR x +1 + x 1 = 2}
A B
22. A log( x+1 + 5) = logx B log2(3x) +log4(9x) + log8(27x) = 3 + 2log64(x)
A B
1. 2. 3. 4.129169
329
969
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23. 2x 2x+1 2x+ 2 = 4x + 4x+ 1 + 4x+ 2
1. 2. 3. 4.log221
10 log2
21
8 log2
21
6 log2
21
4
5. log2212
24. Positive number x, y and z satisfy xyz= 1081
and (log10x)(log10yz) + (log10y)(log10z) = 468
Find (log10x)2 + (log10y)2 + (log10z)2
25. A log6(3 4x +2 9x) = x + log65
B x + 1 x2 = 1+ 2x 1 x2
A B
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26. xy = 256 x + y logyx+ logxy = 10
3
27. R
A
3
5
(5x2 23x+ 3)>
5
3
(x + 5)
A
1. 2.{xR (5x 1)(x 3)
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29. 2log(y x) =log(2y) + log(y x 1)
1. 2.
3. 4.
5.
y
x
2
y
x
y
x
2
y
x
6
4
2
y
x
6
4
2
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3. A B det(A) = 23 3
x y B =
1 3 2
0 1 x0 2 y
AB + 3A = 2I I 3 3 x + y 1. 0 2. 3. 4.1 2 2.5
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4. x det A = 3A =
2x 1
x x
B I 2 2 BA + BA1 = 2I
det B 2 21. [1, 2] 2. 3. [0, 1] 4.[1, 0] [2, 1]
5. A B 2 2
2A B =
4 45 6
A 2B =
5 84 0
det(A4B1)
6. A =
3 1 0
2 4 35 4 2
B =
1 1 0
0 2 11 2 3
(A1B) C(BtA)
t = [det(2A)]I C1
1. 2.
1 12
3
2
1
2 9
2 2
3
2 2 5
28
1
8
3
8
1
8 9
8 4
8
3
8 4
8 10
8
3. 4.
1 12
32
12
9
2 2
32
2 5
2
8 1
8 3
8
18
9
8
4
8
38
4
8
10
8
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7. S x
4 2 7x 1 32 0 x
y S A=
y 1
1 y
det
A
t
1
t
1
8. a A =
a 1 a1 + a a
I =
1 0
0 1
det(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
9. a, b, c, d, x y
A =
1 x
y 1
, B =
a b
c d
, C =
1 0
0 1
I =
1 0
0 1
AB = 2CA2 = I
det(B1)1. 0.25 2. 0.5 3. 2 4. 4
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10. 3 1 A =
1 2 4
3 8 01 2 1
A1
11. A =
1 20 1
, I =
1 0
0 1
B 2 2 x det(A2 + xI) = 0() det(A +xI) = 0
() det(A2 + xI B) = det(Bt)1. () () 2. () () 3. () () 4. () ()
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12. S = {3, 2, 1,1,2,3}
M=
a1 a2 a3
0 a4 a5
0 0 a6
ai S, 1i6
M 1 27 271. 2. 3. 4. 5.2
634
636
638
6310
63
13.
. A =
100 sin2a cos2a
200 2sin2b 2cos2b
300 3sin2c 3cos2c
det(A) = 0
. A = 2 2
3 2
det(3A4(A1)t(A At) ) = 72
1. . . 2. . . 3. . . 4. . .
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14. x A =
2 x 1
1 0 11 x 2 2 x
C22(A) = 14 det (adjA)
15. A 3 3 det (A) 0() (det(A))3 = det(adj(A))
() A2 = 2A det(A) = 21. () () 2. () () 3. () () 4. () ()
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17. A B det(A) 0 det(B) 0 det(A1 + B1) 0 det(A +B) 0
(A + B)1
1. 2.B1(A1 + B1)A1 B1(A1 + B1)1A1
3. 4.B(A1 +B1)A B(A1 +B1)1A
16. A i = 1, 2, 33 3 AXi = Bi
X1 =
1
0
5
, X2 =
1
2
5
, X3 =
1
3
1
B1 =
1
0
0
, B2 =
0
1
0
, B3 =
0
0
1
det(A)
1. 2. 3. 4. 1 5. 88 18
1
8
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18. a b A =
1 a
b 4 , I =
1 0
0 1
A ab 0 2(A I)1 = 4I A
() ab = 2() det(3A2AtA1) = 324
1. () () 2. () () 3. () () 4. () ()
19. x, y, z
= ax 2y +3z = bx 3y
= c2x 5y +5z
1 2 31 3 02 5 5
a
b
c
1 2 30 1 3
0 0 1
9
5
2
c
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20. x, y, z 2x 2y z = 1
x 3y+ z = 7
x + y z = 5
1x+ 2y+ 3z1. 0 2. 2 3. 5 4. 8
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