2555 3
(
(
2557 ) )
KRM-MATH-56-12-B1-6
© 2014 © 2014 © 2014
2537
Math Lite Ebook
... 6 ...15
...2 7
... 64
... 50
…
...93
72
79
…
...
…
92
... 1 39
101...
130
... 110
1
... 159
71
…
… 187
1
Set
Set
6
Set)
(
Element ( )
{ }
(,)
{}
(Relative Universe)
1. {a,e,i,o,u} 2. {x | x
}
1
EX
1. 2. 3.
R N Q
1.{1,2,3,4}
I
2.{x | x
Q
{}
{ }
I-
3.
I+
EX
1. 2.
{{ }}
4
}3
1.
A
B
A
B
2.
A B
Ex A B
A
A = {1,2,3} B = {x | x
B A
A
3.
B
A
A
A
A
3 A
B
}
B A
A
B
B
B
B
{ 1,2,3}
B
Ex
P(A)
A
A = {1,2,3}
} }} } } }} } } } } } }
P(A) = {
}
P(A)
{ }
P(A)
A
n
-
2n
2
3
4
Ex 1.
= {1,2,3,4,5} A={1,2} B={2,3,4}
A B B
{1,2,3,4}
2. A
B
A
4.
A
A – B = A
(A
(A
A
B) = A
B
B) = A
B
(B
C) = (A
B)
(A
A
{1}
{ 3,4,5}
2 3
B =B -A
A
A
n(A
n(A B C)
B) = n(A) + n(B)
n(A B)
= n(A) + n(B) + n(C) C)
A
B
B
A-B
{2}
B
3.
A
A
A B
B
n(A B)
n(A C)
+ n(A B C)
n(B C)
{A}
B
A
{A} P(B)
A
{A} P(B)
A
B (B) (B)
1 A = {1,2,3,{1,2,3}}
1. A B = {{1,2}}
1 A 2 A
2.A
3 A – B ={{1,2,3},3}
1. B
A
B
1 3 4
B
A
2
A
B
A
B – A = {{1,2}}
(
B B
3. A
B=A
4. A
A
A
)
B=B
B
A=B 2
{1,2,3}
{{1,2}}
2. A
(Ent)
2
3
2
A
{{1,2}}
B = {{1,2},{1,2,3}}
B = {{1,2},{1,2,3}} 3. A – B = {{1,2,3},3} 4.B – A =
B = {{1,2}}
4 B – A =
B = {1,2,{1,2}}
B
A
=
B
A
B
A
B=B
3
A ={ ,1,2,3, … } B={{1},{2,3},{4,5,6},7,8,9,… }
(A-B)
A – B ={
(B-A)
,1,2,3,4,5,6}; n(A – B) = 7
B – A = {{1}, {2, 3}, {4, 5, 6}}; n (B – A) = 3 (A-B)
(B-A) ={ ,1,2,3,4,5,6,{1},{2,3},{4,5,6}}
n((A-B)
(B-A)) = 10
n((A-B)
(B-A)) = n(A – B) + n(B – A) =7+3 = 10
A−B) (
4 1.(A 3.(A
B)
(B−A)
1 0
A,B,C,D
(D C)
B)
(D C)
(A
C) – (B
= (A
D)
= (A = (A
2. 4. (A
,{ , , , …}
5 A= {x | x = 1 −
C) – (B
(A B)
(C
D)
(A B)
(D
)
(B
D)
C)
(B
D )
B)
(D
C)
(D
)
n
}
B = {0,1, , , } C ={-2,- ,0,1, (A
C) – B
}
D)
C)
= (A B)
4
{-1,-
A
A
C
A
C
(A
6
1
A = {0, 1, {1}}
P(A)
B ={0, {1}, {0,1}}
P(B)
P(A
{ 0}
B
}
P{B}
{{1}}
1.
,0,
{0, }
C) – B = { }
. A
, , , …
2.
P(A)
2
B) = 2
3. 1
4.
P(B)
.
P(A) = { , {0}, {1}, {{1}}, {0,1}, {0,{1}}, {1,{1}}, {0,1,{1}}}
P(B) = { , {0}, {{1}} ,{{0, 1}}, {0, {1}}, {0, {{0,1}}}, {{1}, {0,1}}, {0,{1}, {0,1}}}
{0, 1} P(A)
P(A)
P(B) = { , {0} , {{1}} , {0,{1}}}
{{1}} A
P(A)
P(B)
B ={0, {1}}
4
3
2
22
7
300 2
80 140 100
x
2
80 + 140 – x +100 = 300 x = 20
20 = 60 80 – 140 20 =120 60 + 120 = 180 –
8
6 40 30 20 3
3
82
10
30 5
10
x
85 – x = 150 85 – 150 = x
3
30 + 5 + 5 + 10 +10 +5 – x + x + 20 – x = 150
x=3
3
2
/
((
(
)
) )
)
(
(
)
(
)
(proposition)
= 10 1+5 ( Sentence) Open
x + 1 = 5 p q p T
T
T
F
F
T
F
F
q
q
p
T
q
p
q
T
T
p
1. p
F
2.
q
p
q
(p
q)
p
q)
p
q
q)
F
F
F
F
F
T
F
T
3. p
(q
r)
(p
F
F
T
T
T
p
(q
r)
(p
4. p
q
(True)
T
(False)
F
(Tautology)
1.
1.
and)
2.
(or)
b
(p
p
q
(p
q)
q)
(q
(p
r) r)
p)
(
(… if and only if ..)
2.
(if… then …)
4.
4
q
(p
T
3.
(compound proposition)
p
T
T
3. a
a
b
1 p, q, r, s p s 1. p q 3. r s p s
[p
2. q 4. s
(q
r)]
(s
r)
(Ent 45
)
r
p
p
p
p q
p
q
T
T T
s
r
1
s
Ex
P1,P2 ,P3,…,Pn
[P1 P ,P3 … Pn]
C
C
1. P
Q
2. Q P
1. P Q T 2. Q T
1 P
F
Q
2
x
Ex 1 x[x+1 > 3]
3
x = 1
2. Existential Quantifier x[P(x)]
4 ,5
6
x
1
= I+
}
Ex 2 x[3x+1 > 5 ]
x[P(x)]
1. Universal Quantifier
(Quantifier)
x y
x y
x
y
x
y
2
1
x y
x y
x y
x y
x x
1 x
y
y
x y
x y
y
2 y[x > y] y[x > y]
x
x
x y[x > y]
x y[x > y]
={0,1,2,3 } x = 0 y = 1 x = 2 y =0
x
~ x[P(x)]
x[~P(x)]
~ x[P(x)]
x[~P(x)]
{ -1,1,2}
(1)2 – 1 + 6 = 6 (2)2 – 2 + 6 = 8
x[x2 – x + 6= 0]
(-1) + 1 + 6 = 7
x2 – x + 6= 0
2
3
y
4
p,q,r
(p q)
p
1. (p 3. (p
r
s (r
s)
(q
r)
2.q
s)
(r
q)
4. (r
(p q)
T r
F
2
(r
s)
(PAT1
q)
q
[p
(q ~r)]
s)
[q
p
(p r)]
F
53
)
r
s
p r
5
1.p v (q~r)
2
[~r (p
3.(p q) v (q
3
p,q,r q)]
r) v (~p
~q)
(q v ~r) 3.p v (~ q
~q) v ~ (~p v r)]
5
q)
2.p
[~ (p
1.(p
~[(p
(~p v r)]
r)
4.p
[~(~p v ~q) v (p [(p [(p
q) v (p
(~q v r)
~r)]
~r)]
(q v ~r)]
2.(~p v q) ~q)
4.[(p
(~q
q) v r)]
~p)
[(~p v q) v (~r
~p)]
(
)
1. 2.
2
(Inductive reasoning)
2. 3.
1. 2.
3.
1
1. 2. 3.
1.
( reasoning) Deductive
2.
1. 2. 3.
1. 2. 3.
–
1.
3.
1 1. 2.
2 1. 2. 3.
7 17
3 .
3
2 2
− ± 2−
≤ ≤≤ ≥≥ ≥ ≥ ≤
|x| |x| |x|
a
-a
x
a
x
a
x
(x)2
(y)2
|y|
a
-a
a|b
a
b
(
( Number ) Real
)
1,2,3,...
a
b
(a + b) + c = a + (b + c)
a
4 , 3.67
b
b)
c=a
(b
a x b
c)
1
-a
-1
(a )
a (b+c) = a b + a c
axb=bxa (a
0
a + b R
a + b = b + a
2 2
6.5123...,e,
3.4848..
-1,-2, -3, ...
-1
R
Operation
Operators) (
1
*
a
b
a
b
a * b = 2a + b a*a=1 b*b=0 5 * (10 * 5)
5 * (10 * 5) =
35 2
5 * (2(10)+5)
=
5 * 25
=
2(5) + 25
=
35
*
a * b = 4a* + 2b – 1
e
4a + 2e −1
=
4a + 2e –1
a*e =
1 – 3a 2
a
2e
=
1 – 3a
e
=
1 – 3a 2
a * e = a
2 (Quadratic equation)
2
ax + bx + c = 0
x
a,b,c
a
0
1.
− ±2−
2.
b 2 4ac > 0
b
2
4ac = 0
b
2
4ac < 0
2 2
1
3
Ex x3 – 2x2 – 6x + 12 = 0 x2(x – 2) – 6(x – 2) = 0 (x2 – 6)(x – 2) = 0
(x – x=–
)(x +
)(x – 2) = 0
,2,
3
2
Cubic 3 ax 4 + bx3 + cx2+ d = 0 equation 4 ax + bx + cx + dx +e = 0 Quartic equation
2
Ex x3 + 4x2 – 7x – 10 = 0
2
(x - 2)(x + 5)(x + 1)=0 x = 2 , –5 , –1
(x - 2)(x2 + 6x + 5) = 0
(Interval) a (Opened Interval) (Closed Interval)
a
b b
a b
a
[ a,b]={x
≤≤ R|a
x
b}
(Interval half open on the right) b
a
≤ ≤
[a,b) = {x
R|a
x
(Interval half open on the left) b ( a,b] = {x R | a x b }
a
1.
2. 3. 4.
Ex
0
(
0)
x2 – 6x + 2 > -7 x2 – 6x + 9 > 0
+
3
(x – 3)(x – 3) > 0
+
R – {3}
3
+
3
2.(x – 6)2(x – 2)(x + 1) < 0
1.(x – 3)(5 – x) > 0
(3,5)
5
3
-1,2)
(
5
3.(7 – x)2(x + 3)
(∞,-3]
3
≤
3 5
2 −−− − ≥
0
4.
{7}
(-∞,-2)
{2}
2
(1,3]
0
-
1
0
−
4
−
+1<0
− − −−
<–
1
<0
<0
(1,2)
a,
a− a < 0 |a| |x| = |–x|
≤ ≤ ≤
|x|
a
-a
x
a
a
a |a| < 0
|x + y|
|x| + |y|
|x – y|
≤
|xn| = |x|n
;y≠0
|x2|=x2
a
|x|
-2 < x < 8
(4x + 3)2 > (3x – 1)2
≥ ≥ x
a
x
≤
(4x + 3 )2 - (3x – 1)2 > 0
-a
(4x+3-3x+1)(4x+3+3x-1) > 0
|x–2|>7
(x + 4)(7x + 2) > 0 x – 2 < -7
x–2>7
x < -5
x>9
-4
|x-3|+|x-1|
≤ ≤≤ ≤ ≥ x
-2x+4
x
-(x-3)
1
x
3
+ (x-1)
2
-2
4
≤≤ ≤ ≤ ≤ ≤ 1
2
-
-
2
2
-2x
≤
1
- (x-3) - (x-1)
|a|
| 4x + 3 | > | 3x - 1 |
a
≥
-5 < x - 3 < 5 |x|
|x| – |y|
|x-3|<5
a ≥ 0
|a|
|x – y| =| y –x |
|x • y| = |x||y| | | =
2
x
2
[1,3]
3 2
2x -4
2
2x
6
x
3
≥ ≤≤≤ ≤ x
(x-3)+(x-1)
3
Ex
n
n|m
n
m
m
m
n
4|20
20 = 4×5 + 0 0 4
20
n
m = n(q) + r
m
n
q
(a,b)
a
b
a
2
0 0 a a
15
14 58 = 14(4) + 2 14 = 2(7) + 0 (14,58) = 2
,)
b b
Ex
4•6= 24 = 2 •
a
b
a•b = (a,b) • [a,b]
d|a
d|b
6
•
4
= 12
a,b
Ex
a,b
[a,b]
5 25
(15,25) = 5
b
1) 2) (a,b) = ( 3) 4)
r
(a,b) =1
d
a a,b
b
2
a
b
(m,n)
0
58
146
Ex
192
192 = 146(1) + 48 146 = 48(3) + 2 48 = 2(24) + 0
2
a|d
146
2192
a
d
b|d
a,b
146
b a
b
34
112
a • b = (a,b) • [a,b]
2×17×56 = 1904 34|1,904
1,904 34 112 |1,904
2 0
[m,n]
Ex
192
112 1,904
5
S=
1.(-∞,-3)
,2− ≥ 2−
−−
2.(-1,0.5)
≥
(-1,0.5)
6
S ={x |
1. {x| x3=1}
3.(-0.5,2)
−
≥ ≥ ≥ 2
4.(1,∞)
(-∞,-4)
(-1,1)
=1}
2.{x|x2=1}
∞)
S (PAT1)
3.{x|x3=-1}
4.{x4=x}
{1,-1}
{x| x3=1}1.
4
4.{x =x}
1
3.{x|x3=-1}
x 4
{1} {1,-1}
2.{x|x2=1}
(2,
S
S
2
S (PAT1)
{-1} {0,1} x
x = x
4
7 (A-NET
|x2 + x - 2| < (x + 2)
(
a+b
– (x + 2) < x2 + x – 2 < x + 2
– (x + 2) < x2 + x – 2
x2 + x – 2 < x + 2
0 < x2 + 2x
x2 < 4
0 < x(x + 2)
x < -2,2
(0,2)
0+2=2
2
4
A = {x | (2x+1)(x-1) < 2} B = {x | |2x-10| < 2} C (A
B)
C
A = (2x+1)(x-1) < 2 2
2x – x – 3 < 0 (2x-3)(x+1) < 0
B = -2< 2x - 10 < 2
(A
8<
2x
< 12
4<
x
<6
A
B=
) a,b
50)
C
B)
C = (A A
B)
B = (-1,1.5)
(4,6
A
B
4
A x B = {(x,y) | x
Ay
B}
x
y
x
y
(x,y) (x,y)
y = f(x)
A
B (into function)
A
x
y
function) B (onto (1-1 function)
gof(x) = g(f(x)
f(x)
g(x)
fof-1(x) = x
(f+g)(x) = f(x) + g(x) ; D f+g = Df
Dg
(f-g)(x) = f(x) – g(x) ; Df-g = Df Dg (f g)(x) = f(x) g(x) ; Df•g = Df Dg
⋅ ≠0 f(g⋅)(x) = gf ; g(x)
D = Df
Dg
( Pair) Order
a, b)(
(
a a,) ( b) = (c, d)
A
A
a, b
b=d
A
b
a,( b)
a
B
B
b
a=c
B
A x B = {(x,y) | x
A ={1,2,3} B = {a,b}
Ay
AxB
B}
A x B = {(1,a),(1,b),(2,a),(2,b),(3,a),(3,b)}
=3x2=6 n(A) x n(B)
r
1.
2. 3. 4.
B
A
B
B A
5. x r y
x
6. x r y
x
A x B
y
r
r
9 r 3
Dr = {x | (x, y)
(Range) Rr
Rr = {y | (x, y)
Dr
r r
r
(x,y)
r
3
”
“3
(x,y)
C
RxR
9
(
Domain)
AxB
A
y
r
AxB
2 n(A x B)
B
r
B
B
C
A
2.
A
A
(9,3)
1.
r
r}
r}
r
y
y
1 r =
0
,
0
r= r= r=
y
, , − -
y≠0
0 0
(-∞,-2) ( -∞, )
(-2, ∞)
2
R – {-2}
R – {0}
( , ∞)
x ≠ -2 x
x+2 ≠0
2.
x
x
1.
2 r = {(x,y)| y = (x−3)2 } 3.
{x | x
R}
0{y | y ≥ 0 }
A ≥0
A
3 r = {(x,y)| y=
}
2x – 8 ≥ 0 2x ≥ 8 x ≥4
4.
{x | x ≥ 4}
0{y | y ≥ 0 }
4 r = {(x,y)| y = |5 – 2x|}
{x | x
R}
0{y | y ≥ 0 }
5.
x
5 r = {(x,y)| |x|+|y| = 4}
y
{x | −4 ≤ x ≤
4} {y | − 4 ≤ y ≤ 4 }
Df = A Rf
Df = A Rf x
B
B
f: A
f: A 1-1 B
y) (
y
B
B
f: A
(1-1 function)
(onto function)
1
(into function)
1
x
f(x1 ) < f(x2)
f
f
1
2 x1
y = f(x)
(x,y)
f
f(x1 ) > f(x2) f
x1 < x2
1 f={(1,2),(3,4),(5,6)}
2 f(x) =x 2+3x-1
f(1)
=2
f(2)
= (2) +3(2)-1
f(5)
=6
f(2)
=9
3 f(2 x+6 ) = 3x-4
f(2x-1) = 4x + 5
f(2)
= 3(−2) − 4
f(2)
f(x) = 4[
= −10
4
]+5
= 2(x+1)+5 = 2x+7
f( ) =
2x1
2x+6 = 2
– 1 = x2
2x1 = x2 + 1
2x = -4
x1 =
x =-2
f(2) 2
2
(r-1)
Ex
r = {(1,2),(3,4),(5,6)}
r-1={(2,1),(4,3),(6,5)}
f(x)
f-1(x)
f(x)
Ex f={(x,y)
I+ x I | y = 2x+1}
f-1={(y,x)
I x I+ | y = 2x+1}
f-1={(x,y)
f-1={(x,y)
f(x)
–1 1
1. f-1(x)
f = R 3. R f = D 2. D
f f
-1
Ex f = {(1,2),(3,4),(5,6)} f-1={(2,1),(4,3),(6,5)} f-1 (4) = 3
f (4)
I x I+ | y =
f-1(x)
I x I+ | x = 2y+1}
}
−
Ex
f-1 (2)
f(x) = 3x +1
Ex f(x) =
y = 3x +1
x
y
x
y= f
-1
− (2) =
f-1 (2)
y=
x = 3y +1
y
−
f-1 (x) =
− − − – x=
x(6y+5) = 4y
1
x(6y+5) +1 = 4y
6xy – 4y + 5x +1 = 0 y(6x-4) = - 5x – 1
−−− −−− − y=
f-1 (x) =
Ex f(x) =
f-1 (2) =
±± − ±−±
f-1(x) =
Ex f(4x+1) = 2x – 2 -1
f (2x – 2) = 4x +1
f (x) = 4[ ] +1 -1
f-1(x) = 2(x+2) +1 f-1(x) = 4x+5 f-1(-2) = 4(-2) +5 = –3
f( f-1(-2)
f-1
) =
( )
=
A
x2
2x – 2 = A x=
(Composite Function) x
gof(x) = g(f(x)) Rf
≠ Dg
Dgof
Ex f(x) = x 2+3 g(x) = -x +5 gof(1)
g(f(1)) = g(12+3) = g(4) = -4 + 5 =1
-1
(fof )(x) = x
(f-1of)(x) =x
f(x) = 3x+7
(fof-1)(5)
5
(f-1ogof)(x) ≠x
-1
-1
-1
(fog) (x) = g of (x)
(f+g)(x) = f(x) + g(x) ; Df+g = Df
Dg
Ex 1
(f-g)(x) = f(x) – g(x) ; Df-g = Df
Dg
f = {(1,-1),(3,2)} g(x)={(1,2),(7,3)}
⋅ ≠0 fg⋅ gf ; g(x)
(f g)(x) = f(x) g(x) ; Df•g = Df ( )(x) =
(f+g)(x) = {(1,1)}
Dg
D = Df
Dg
(f-g)(x) = {(1,-3)}
f(g⋅)(x) ={(1, )}
(f g)(x) = {(1,-2)}
f(x)= -x+5 g(x) = 2x+6 (f+g)(x) = (-x+5) + (2x+6) = x+11 Ex2
(f-g)(x) = (-x+5) - (2x+6) = -3x-1 (f g)(x) = (-x+5)(2x+6) =-2x2+x+30
f(g⋅)(x) = −
5 .
f ={(1,2),(2,7),(3,4),(4,1),(5,6)} f-1(f(3))
f(f(1))
f(f(1)) f(2)
f-1 ={(2,1),(7,2),(4,3),(1,4),(6,5)}
= f(2)
f-1(f(3)) = f-1(4)
=7
7
f-1(4)
=3
3 6
f(0) = 10
f(x+1) = f(x) + 5
f(1)
( f(0)) +5 = 15 (
15 + 0(5))
f(2)
( f(1)) +5 = 20 (
15 + 1(5))
f(3)
( f(2)) +5 = 25 (
15 + 2(5))
f(20)
15 + 19(5) = 110
110
f(20)
5
n
(Coordinate System)
x)
(
( y)
O (Origin)
(0,0)
y y x x
|x| |x| |y| |y|
x
x y
y
(x,y)
2
Ex
√ 22−− 2 2
|AB| =
mab = tan( ) =
P=(
)
√ − −
|AB| =
=
mab = P
AB
3
( 2, 2)
2.
(Centroid)
n
Ex
1.
=P
=
− − 1.
= (2,1)
2.
n
0 4
5
2 25 0
|2+25+0+0+4+5| =18
( )
x1.y1 ,
1.
m
y - y1 = m(x - x1)
2.
y = mx + c
1
2
m1 =m2
3
m1 × m2 = -1
−22
Ex d
d
d=
22
d= =
Ex
−− 22
d
.
d=
2−22
d
d=
−2− 2
=d=
tan =
X(
Y)
X
P(x, y)
X y
X Y
Y
x
conic)
(conic section
200
(Circle)
2
+ 6y - 12 = 0 x2 +y2– 4x
x2 – 4x +4 + y2 +6y + 9 = 12 + 4 + 9 (x2–2(2)x+22) + (y2+2(3)y+32) = 12+22+32
Ex
(x–2) 2 + (y+3)2 = 52
(x–2)2 + (y+3)2 = 25
-3)
(2, 5
x2 +y2– 4x + 6y = 12
x + y + ax + by + c = 0 2
r
) center
(x - h)2+(y - k)2=r2 (h,k)
(radius )
Ex
-3)
7
(5,
1.
(m
×m
= -1)
2.
(ellipse)
(focus)
2
−2 2 −2 2
−2 2 −2 2
=1
a
b
c2 = a2 - b2
V, V’ C 2
a
0
2
0 1
e e
e=
b
2
(Parabola)
(x - h)2 = 4c(y-k)
(
)
(y - k)2 = 4c(x - h)
V F (LR) = 4C
(
)
(Hyperbola)
= 2a
C V, V
’
F, F
’
2b
2a
2
(LR) =
1. 2.
x
−2 2 −2 2 −2 2 −2 2
y – k = ± (x - h)
x - h = ± (y – k)
a=b
y
1 A (-3,1) B(8,3) C(8,3) D(2,-3) (PAT 1 . 53) 1. AB BC 2.
AB
3. 4.
10√2 C C
B
− = 1 −− −− = − = 1
DC
A
D D
ABCD
mAB = mDC
|AB| = |DC| =
1
AB // DC
√ √ =
=
=
|AB| +|DC| =
+
=
2
C
D
C
CD
x–y –5=0
−−− – −−−− A
CD
CD
√2−2 =
=
3
−− √2−2– −
A
=
=
4
4
y – 3 =1(x – 8)
2 1. (3,1)
(1, -3 )
x + 2y = 5
2. (-1,3)
(- )
3. (1,2)
(1, -3 )
4.(5,0)
2
y + 3 = 2(x - 1) 2x – y – 5 =0
2
2
x + 2y = 5 ------------------- (1)
2x – y = 5 (1)+ 2x(2)
----------------- (2)
5x = 15 x = 3
y=1
(3,1)
1 3 1. y + 2x = -3
2.2y – x = 9
B(-1,4) A(-5,2), 3.2x+3y =6
4.2x + y = 3
A B − = = (-3,3) − AB = −− =
(y+3)=-2(x-3) 2x + y = -3
1
-2
6
A+B
A-B
(k)
kA
×B
A
t
A
det det = 0
det ≠ 0
det
A-1 =
detA
[Cij]t
AA-1 = I
I
(
+
–
=
=
At=
)
A=
[ ]
m×n
m×n
t
n×m
A=[a] m x n
k
KA=[ka] mxn
A=
5A =
A
B
1. 2.
1.
A
2 2
A
B
m×p
B
q × n
"
A=
p = q AB ( i j
2.
× n m
)
j
* + * + * +
i "
B=
AB = AB =
☺
AB
☺
BA
x1
n
n×n
1
A = [a] detA =a
2
x2 A=
* + x3
detA = ad - bc
3
A=
detA =
aei + bfg + cdh – gec – hfa - idb
x3
3i
detA = ai1Ci1 + ai2Ci2 + … ainCin
j
detA = a1jC1j + a2jC2j + … anjCnj
i
Cij = (-1)i+j • Mij(A)
j A=
C23 = (-1)2+3 = (-1)(2) = -2
C23
Singular Matrix() Non Singular Matrix ( )
1.
2.
1.
2. 3.
det
det = 0
det
det
detAm = (detA)m
3 2 det A
Ex A =
detAt = detA
AA
-1
=I
Matrix)
det 3At
9 x 2 = 18
A [A | I] ~ [I | A-1] A Ex A= I
A-1
detkA = k × detA n × n
detA = 2
n
det
detAB = detA × det B
* +
0
det
÷
(Non – Singular 0
R2-R3
R
R1 2
A-1 =
A-1
1×1
Ex
A = [a] A -1 =[ ]
=
-1
A=
B-1=
B=
2×2
* + * +
A = [3] A-1 =[ ]
A =
* + * + ( )
C=
3×3
A-1 =
[adjA] ; adjA { adjoint
}
detC = 7
C-1=
t
adjA = [Cij]
= =
=
t t
(A ) = A
(A )
(A×B) t= Bt × At
(A×B) -1= B-1 × A-1
(A±B) t = At ± Bt
(A±B) -1
t
(kA) = kA
t
(An) t = (A t) n .
-1 -1
(kA)
-1
=A
-1 -1
=k A
(An) -1 = (A -1) n
adjoint (
)
adj(AB) = adj(A) × adj(B)
adj(kA) = kn-1 × adj(A) ; A
A adj(A) = (adjA)A = (detA)I
det(adjA) = (detA)n-1 ***
adj(At) = (adjA) t
n×n ;k
***
A
* +* + *+
Ex x + y = 4
A
x-y=8
=
X
B
detA detA 2 detA detA detA detA 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
X1, =
X2 =
, … ,Xn =
a1x + b1y + c1z = d1 a2x + b2y + c2z = d2 a3x + b3y + c3z = d3
x , y ,z
x=
n-2
adj(adjA) = (detA)
1.
y=
2. -1 X= A B
3.
[A | B] ~ [I | X]
z=
(Row echelon form matrix)
0
1
* + * + * +
[A | B] ~ [
| X]
7
xn
Expo = {(x,y)
Log ={(x,y)
R+ x R | y =logax
a>0,a
1}
R x R+ | y =ax, a > 0 , a
1}
a×…×a( an = a × a ×
1. a 2.(a ) = a 3. 4. ,5. - 6.
m+n-p
=
m n
n
2 ± 2 ±
a+b±2√ab
(a+b)+2
0
a
mn
0
a =1 -n
a =
)
=(
)2+2
=(
+
)2
(
+
)
|
= +
+(
)2
+
|
=
0
0
{
()
=
=
=
×
() =
×
=
2 =2
2
4
a
2 2
a
2
2
4 2 42 – 2 4
a
2
Expo = {(x,y)
R x R+ | y =a x, a > 0 , a
a>1
1}
1.
Ex
0
3y • 4 =2 16y-3
y
3y+2 2 = 24(y-3)
3y+2= 4y-12 y = 14
2.Ex
x
21 X-3 = 63x-9
3x-93
= 63x-9
3x−9 = 0
3x = 9 x= 3 Log = {(x,y)
R+ x R | y =logax, a > 0 , a
a>1
0
3.
1}
2
Ex
take log
2x=3x+1
x log 2 = (x+1) log 3 X log 2 = x log 3 + log 3 x (log 2 – log 3) = log3 x=
log
g
g g log
Ex
g−gg
6 log (x − 2y) = log x3 +log y3 (x − 2y)6 = x3 y3 (x − 2y)2= xy x2 − 5xy + 4y2=0 (x − 4y)(x − y) = 0 =4
1
2
g g 2 g g =
take log
x
=
log
(logx b)(log x a) = (logx a)(logx b)
log 2 =
A
10
A
n
A = N × 10 ; 1 ≤ N < 10
log A = log N + log 10 n
N+n log A = log
N
(Characteristic)
log N
(Mantissa)
log A
log A
0
1
e (e
2.71828
)
ln x = logex
log x = A
antilog A = x log 5 = 0.699 5 = 100.699
log
5
antilog 0.699
1
x
x −4 (3) 6 x −12=0 +3(2)
A =2x
B =3x
AB + 3A - 4B-12=0 A(B + 3) - 4(B + 3)=0 (A - 4)(B + 3)=0 A = 4 B = -3 x
x
3 =-3
2 =4
x=2
2 ( 7 )
3x A +
A
3x+ 32−x =
A A2 + 9 = A −
x=
(A − A=
3x =
,
A + 9 =0
)(A − ,
2
2.3
log50 5 + log50 2
B
=1 = 50
5A 2B
= 52 21
A = 2 B =1
3
=1
B
log50 (5 2 ) 5A 2B
A+B=
2
)=0
A log50 5 +Blog 50 2 = 1 3.4
A
A
A
3 A B (B - PAT) 1.2
=
3x =
A
2
A+B 4.5
4
log2 log 3 log5 (2x +3) = 0
log2 256 – A
B
A
2A – 3B
log2 log3 log5(2x +3)
log 3 log5 (2x +3) log5(2x +3) 2x + 3 x
log2 256A–
=0 = 20 =3
1
= 125 = 61 = log
2
256 – 61
= log2 28 – 61 = 8 – 61 = – 53 2A – 3B
= 122 +159 = 281
281
log Expo, >1
Ex
0<
<1
2
2 2 >
>
2
x <4 |x| <2 −2 < x <2
Ex log (5x-3) > log (4x+6)
log 5x-3 > 4x+6 x>9
1
8
°°°
30 45 60
sin θ + cos
2
2
2
θ
=1
= 2sinAcosA
cos2A
= cos2A
tan2A
=
3
tan3A =
A
=
– sin2A = 2cos2A – 1 = 1- 2sin2A
− 2
3sinA
cos3A =
– 4sin3A 4cos3A – 3cosA 3 tanA – tan3 A 1–3tan2A
sin3A =
sin
sin2A
−cs
cos
A
=
cs
tan
A
−cs cs
=
sin(A±B) = sinAcosB ± cosAsinB cos(A±B) = cosAcosB sinAsinB tan(A±B) =
∓ tA ±tB ∓
sinA+sinB=2sin sinA s–
inB 2cos =
cosA+cosB=2cos cosA
− cosB =
2sinAcosB
=
sin(A+B)+sin(A-B)
2cosAsinB
=
sin(A+B)-sin(A-B)
2cosAcosB
=
cos(A+B)+cos(A-B)
2sinAsinB
=
cos(A-B)-cos(A+B)
( 3 metro ) trigonon ,
cosec cot
cos
sin sec tan
°
° ° °
°
°
1
sin(90 − A) = cos A sec(90 − A) = cosec A tan(90 − A) = cot A
1.
1 1 1 1 1
x 2 +y2=1
2 Ex 360 1 360 1) ( 1 60 ) (1 1 60
°
2.
Ex 2
(
n
f (
n
)=
1 2 sin 3 tan 4 cos
° °
°° ° °° °
(
)
= sin(60 )
sec(280 ) = sec(270 +10 ) = sec(10 )
sin(120 ) = sin(180 − 60 )
± A) = co f (A)
Ex
± A) = f (A)
n
f
n
f ( n
2
(
)
° ° °
°° °° °° °
°° °° °° °
(
1
)
sin(0 + n(360 )) = 0 cos(0 + n(360 )) = 1 sin(180 + n(360 )) = 0 cos(180 + n(360 )) = -1
1
sin(90 + n(360 )) = 1 cos(90 + n(360 )) = 0 sin(270 + n(360 )) = -1 cos(270 + n(360 )) = 0
°
15
° ° 75(
°
)
n 72, 18
° °
° ° ° ° √ √ √ ° √ ° √ √ √ √ 36, 54 (
54
36
)
1
– + 1
°
°
sin60 – cos30 + tan45
°
=1
1 2
A
°
< 90 0 < A
°
2 – 2sinA = 1
2 – 2sinA = 1 2 –1
A = 30
= 2sinA
°
3
= sinA
sin A =
sin
°
12sin30 (tanA) 13
132 = 52 + x2 x
5
= 12
°
x
tan A =
6sin30 (tanA) = 6(
)
=
×
= 180
°° °
= 60 = 30
= 90
°
= 45
sin
sec
2
+ cos
2
2
– tan
cosec
– cot
2
sin2A
cos2A
tA ±tB∓ ∓ 3
cos(A±B) = cosAcosB
=1
2
=1
sin3A =
3sinA – 4sin3A
=
cos3A = tan3A =
4cos3 A – 3cosA 3 tanA – tan3 A 1–3tan2A
cot3A =
cot3A – 3cotA 3cot2A – 1
2 tan A 1 + tan2A
=1 – 2sin2A
sin cos tan
A A A
=
sinAsinB
= 2sinAcosA
= 2cos2A - 1
=
1 – tan2A 1 + tan2A
=
2 tan A 1 – tan2A
tan(A±B) =
= cos2A – sin2A
tan2A
sin(A±B) = sinAcosB ± cosAsinB
=1
2
2
−cs cs −cs cs
= =
sinA + sinB = 2sin(A+B) cos(A−B) 2 2
2sinAcosB
=
sin(A + B) + sin(A – B)
sinA – sinB = 2cos(A+B) sin(A−B) 2 2
2cosAsinB
=
sin(A + B) – sin(A – B)
2cosAcosB
=
cos(A + B) + cos(A – B)
2sinAsinB
=
cos(A – B) – cos(A + B)
cosA + cosB = 2cos(A+B) cos (A−B) 2 2 cosA - cosB = −2sin(A+B) 2 sin(A−B) 2
4
sec
A
B
cos
cosA =
cosAcosB sinAsinB
sec A =
sec B =
cos(A+B)
cosB =
5
3
= ( )( ) – ( )( )
4
=
− −
=
17 15 8
− 5
°
°
cosec(75 ) × tan(75 )
× s° ° cs° ° scs°
cosec(75 )
× tan(75 ) =
°
=
°
cos 75
= cos(30
° ° ° ° ° ° + 45 )
= cos30 cos45 = =
−
=
− cs°
−
sin30 sin45
− −−
arcsin (-x) = −arcsin (x)
arctan x + arctan y = arctan
; xy < 1
arctan(-x) = −arctan(x)
arctan x + arctan y = arctan
; xy > 1
arcos (-x) =
π
–arcos(x)
arctan x - arctan y = arctan
C a
b
A
B
c
ABC
= = 2
2
=
2R, R
2
1. a = b +c - 2bc cos A 2. b2 = a2 + c2 - 2ac cos B 3. c2 = a2 + b2 - 2ab cos C cos A =
2−c2 c2
ab sinC =
bc sinA =
ac sinB
y = A sin Bx
|A|
y = A cos Bx
y = A tan Bx
sin
cos
tan
|| ||
cosec
sec
cot
sin
[− , ]
1
arcsin
[-1,1]
[− 1 ,1]
[-1,-1] [0, ]
[− , ] cos
[0, ]
arccos
[-1,-1]
tan
(− , )
arctan
R
(− , R
cosec
[− , ] – {0}
)
arccosec
R – {-1,1}
R – {-1,1}
[− , ] – {0}
sec
[0, ] – { }
arcsec
R – {-1,1}
R – {-1,1}
[0, ] – { }
cot
(0, ) R
arccot
R (0, )
9
|
u ×v
|
u ⋅ v×r
|
(
)|
(vector)
scalar )
(
B
A
A B
(initial point) point) (terminal AB AB | |
B
D
A
C
=
(negative of )
3
0 360
+ “”
(zero vector)
1.
2.
=
+
+
=
=
P
-
a
a
a
a
a=0
a
= +(
)
-
u
a
a =
v 1u a 23u b 2 u =-
=
=
1
x
y
a
1
1
1
| |=
m = tan
=
=
12 22 32 , a , a | |=
cos , cos , cos
cos
2
+ cos2 + cos2 =1
,
x
(a1,a2,a3)
a ⌈1⌉12 22 a *22 11+
a1 + a2
a
1
m
0 = m( )
m
y
z
1.
2.
2 22 111
A(x 1,y ,z ) 1
1
2
1
B(x 2,y ,z ) 2
a ̂ 1 a a
2
=
*+ * +
=
a = c
b=d
0 0 0 0 a=d
+ *+ *+ * + *+ *+ * * + * + *+ k
+
=
-
=
=
k
b=e
c=f
k
=
k
(Dot Product)
⋅ ⋅ u ⋅v = a1b1+a2b2
= a1b1+a2b2+a3b3
= | || |cos ;
= | || |cos ;
=2 + 3
= -3 + 4
= (2 • (-3 )) + (3 • 4 )
= -6 + 12 =6
(Cross Product)
1 2 u 1 v 2 1 2 u ×v =
=| || |sin
uu⋅⋅vv wv⋅u u ⋅v u ⋅w u ⋅v u ⋅v uu ⋅u⋅ u2u⋅v uu ⋅v⋅vu v 0 u v uu⋅vv u v u ⋅ v uu vv uuv v uu⋅v⋅v u v u v u v u ×v u v u ⋅v =
=
=
=
=
)
=0
>0
=90
=0 <0
|
|2 = | |2 + | |2 + 2| | | |cos 2
2
2
|
| = | | + | | - 2| | | |cos
|
|2 - |
|2 = 4(|
2
2
| =2(| | + | |2)
|
| +|
|
| = | | +| | –(
2
|)
2
2
2
2
)
uu××vv wv×uu ×v u ×w u ×v u ×v uu ×u× u×v uu v v 0 u ×v u⋅u ×v
=
=
=
= =0
)
=
=
=0
1.
(
) × u ×v
= =|
|
= | || |sin
2.
(
=
)
u v×r u ⋅ v×r
= | || =|
3.
(
v u = au v⋅ v Proj v u = v Proj
v v ⋅ uav
v u v
|cos
(
)|
)
-a
=0
vv ⋅u⋅uava v⋅2v u ⋅ vuv⋅v u 4. v u u 12 u ⋅ v ⋅ 1
=0
=
a=
Proj
=
× | | | | sin
2
= ×
×
(
)
×
10
i
n
a b
a
b
z
x2+1=0
Re(z) z Im(z)
(Imaginary part)
(Real part)
√√ √√ √ √
1. 2.
1. 2.
(Purely imaginary number)
: 1. 2. 3. 4.
4 4 4 4
i2 = −1
i3 =−i
z = (a,b) = a + bi
a
i4 = 1
i5 = i
Re(z) b
b
a
1. 2.
i6 = −1
183 1384 4 i7 = −i
Im(z)
i8 = 1
4 + 2i
1 2 cd 11 21 1 2 112 : ; 1. 2.
i
3.
i
4. 5. 6.
(a+bi)(c+di)
=
1 : ; ×
z1= 4 – 6i z2= -3 + 2i
5−i
z1 + z2 = (4 - 3) + (-6+2)i = 1 – 4i 3(z1 + z2) = 3 – 12i
3 – 12i
3(z1 + z2)
2
4 + 3i =(3a+b) + (a+2b)i 3a + b = 4 --------- (1) a + 2b = 3 --------- (2)
(1) x 2
6a + 2b = 8 --------- (3)
(3) (2) –
5a = 5 a=1 a
(2) 1 + 2b = 3 2b = 2 B=1
a+b = 2
2:5 2;3 2:5 2;3 2:3 2:3 2:54;92:3 4:14:2113:4:91:15 3
a
b
a+b
4
x
y
:11 :3
y–x
=
1 – 4i
y + 11i = (1 - 4i)(x+3i) = x + 3i – 4xi –12i2
y + 11i = (x + 12) + (3 - 4x)i 11 = 3 - 4x 4x = -8 x = -2 y = (x + 12) = (-2+12) = 10 y –x
= 10 – (-2) = 12
12
̿ ̅ ̅ 1̅ 22122 12 1 2 ̅ ̅ 1. 2. 3. 4. 5. 6. 7.
√2 2 122̅ 1 22̅ √2 2 2 1. 2. 3. 4. 5.
112 22 |11|22
6. 7.
:2 ;2 . 2 2 / . :2 ;2/ ;1 ;: ̅ 5
;1 1
|(5 - 4i)(5 + 12i)(-3i)|
|(5 - 4i)|(5 + 12i)|(-3i)| =
√2 2√2 2√2 √ √
=
= 39
√
39
(13)(3)
6
z
|(7 - 24i)||(3 + 4i)||z6|
√2 2√2 2
(X)
(Ent 42)
1251 15 =
z
= |z2| =
(Y)
c c c
=1
|z6|
a
z
=1
|z2|
6 |(7-24i)(3+4i)z |=1
|z6| = 1
(25)(5)|z6|
15
ab rc
(Polar Co-Ordinate System)
abr c
z1=r1cis
1
z2=r2cis
z1 × z2 = r1 × r2 cis ( 1+ z1 ÷ z2 = r1 ÷ r2 cis ( 1n
n
z1 = (r1) cis (
z
1
= r1cis -
2
2)
2)
1)
1
c c 7
[3(cos35 +i sin35 )][5(cos55 +I sin55 )]
[3cis35 ][5cis55 ] = [3×5 cis (35 +45 ) = 15 cis (90 )
= 15(cos90 + i sin 90 ) = 15 (0 + 1i) = 15i
15
i
;√;4
2
;√ ;4
2
√ [ ]
axn + bxn-1 +cxn-2 + … + s = 0
8
(-1)n
–
2 |z| =
√2 2
=5
2
z
z
z = 3
+ 4i
. / . /
=
(2 + i)
11
(
AAAAA
n!
1
1
Star and
n
Cr
n!
52
Bar
MMCAI
)
ABCDE
) 5! = 120
(
1. 2.
1 2
1
1 2
k
k
k
2
2
1 2
+ +...+
1
3 3 3
2
3 = 6 3 + 3 3
3
3
2
3 6 9
k
1
=18 3+6+9
1
k
1
1
1 2
2 k
1 2 3
1 2 ...
6
3
1 2 3
3
6 5 4 x 5 x 4 =6 120
5
4
2
10
(Ent
( 9
48)
)
9 8
(10
1
)
x 9 x 8 = 9 684
3 (
A-F
1,260 1
(B-PAT
1
9
)
5 2
1 3
1 5
10 x 10 x 6
1
2 6
110
600 1 6 2
600
2
4
10 x 6 x 10
6 x 6 x 10 600 1 6 2 10
6 x 10 x 10
3,780
1 6 2 10
3 0 51
– )
5,040
2,520
(Code)
360 2 6 110 10 x 6 x 6
2
10
360
= 2,520 600+600+600+360+360
1
n
n
n!
n n! = n(n-1)(n-2)…(3)(2)(1)
0! = 1 1! = 1 4! = 24 5! = 120 6! = 720 7! = 5,040 ABCDE AAAAA 1 n! ( ) 5! = 120
(
n
Cr
MMCAI
n!
52
Star and Bar
1
)
n!
n
1. 2.
n
Pr = Pn,r =
;
r
n
1 n
2
x4x3x2x1 5 = 5!
1. n
Ex Pn,r =
10 1 1;7 13
=
= 60480
7
2.
n
n
n1 n2 n3 … nk
Pr = Pn,r =
1
4
3
4
4!
3
2
5
2
7
3
!
420
2
2
1
24x (5x4) = 480
352
= 10
2
3
Pn,r =
4
Ex
7!
2
!
2
3
2
3
A
D
C
2
A
C
C
24 242
C
C
D
2 4 6 2 2 4
D
A
x
90
41
C
–
90
D
A
120
C
10
4
6
4
5 C 2 = 10 4C2 = 6
1 2
10x 6
=
60
2
r! x
()
c
n n
ab , bc r 0 ( r
r
n
n
ca n)
r
r!
=
;
=
; =
=
; (;;) ; ; ) ; () (; = =
=
0 (
r
; =
r
=
3
n
2
3
r
Cr
r
3a , b
n
n)
5 12
12
12
12
12C 2
2 2 12
12C – 12 = 54 2
4 12 2
12 12 C2 – 12( =
12112
54
2 )
12
– 12
= 66 -12 = 54
54
6
6
1.20
(Ent)
2.35
3 4 5 6
3.42
4.65 6
6C3 = 20 6C = 15 4 6C = 6 5
6C = 1 6
20 + 15 + 6 + 1 = 42
3 42 7 4
1.1,920
6
4
2.2,400
3.2,520 4.2,880
6C 2 7C 1 1 4! 6C x 7C x 1 x 4! 2 1 3 2,520
2
(Ent 46
2
8
)
4
8 10
720
1.
2. 640
10 C
3
–
4 112
4.112
1x 1
120 – 8 = 112
3.102
10
6/4
x
3 8
3
Ex
1 3322
10
10 10! n n
3
2 3
2
2
r
n-1
Cr-1
n-1+r
http://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics) http://jhyun95.hubpages.com/hub/Stars-and-Bars-Combinatorics
2
Stars and bars r
Cr-1
3
2
ABC
BCA
ACB
ABC , BCA
→
CBA
C B
1
C
A
C
3! = 6
C B
BAC
B
B →
CAB
A
A
C
CAB
3 A ,B
→
C
→
B
C A
A
ACB BAC
→
A
→
B
1
CBA
3
2
A
A
B
C
C
B
n n-1 ( )(n -2)(n-3 )... 3x2x1 = (n-1)!
n-1
(n-1)!
n
82
2
9
2
!28 !
!
) 2! ( ) 8!
(
( Probability )
1
2
1
(Random Experiment)
( Sample space)
1
1
1
1
Head ( )
Tail)(
1,2,3,4,5 ,6
(event)
n(E)
n(s)
P(E) =
B) = P(A) + P(B) – P(A
P(A )
B=
P(E)
1
B)
B)
P(A
9
3
B) P(A) + P(B)
4
( Sample space)
= 1 – P(A)
P(A – B) = P(A) – P(A
A
0
sample space
P(A
7
4×37× 27 =
7
2
x 43
x7 6
2
4
6
xy
x , y
n
xy 1 23 23 2 2 2 3 xy xy45 45 34 223 23 243 4 5 xy x xy xyxyxy xy ()x (1)x;1y ( )x;y ()y ()
=x+y = =
+2xy + + +
=
+
=
+
+
+
+
+
+
+
n
=
x,y
n
=
an
Tr+1 =
n+1
rbr
(a+b)n
n
(a+b) n
2 n
11
7
1
T7 = T6+1 =10C6(x2)4(- )6 10
1
C4 x8 ( )6
210x2
2 x -
1)
10
(
12
2279)
(
1736 Seven ( Bridges 7 7
7
of Königsberg)
C
A e2
e1
1
G
B
1
G Vertex)
(
V(G)
V(G) = {A,B,C}
Edge) (
E(G)={e1, e2}
E(G)={AB , BC}
E(G)
(loop) uu
e1
e2
e3
e 2
1
multiple edges) (
{u,u}
(parallel edges)
e 1
e4
2
Ex
Ex
Ex
Handshaking Lemma
4
K4
2
G
U
V
U – V
)
(Cycle)
(Circuit)
Ex
Ex
(
Ex
G
G
1
1
G G
G 6 G . 2 G
.
G
1,1,4,4,6 7 5,4,2,2,2,3 3
8
3
16 1. 2. 3. 4.
4 5 6 7
G
3
1.6
14
A-Z
2.7
1
3.8
4.22
6
4
1
2
3
4
1 3 4
2
5
1
2
3
4
1 2
3
4
6 G
G G
15 6
10 1,1,3,3,3,5, x-5,x-3,x-2
4 30
G
x
2 1+1+1+3+3+3+5+ x-5+x-3+x-2+x = 3
30
6+4x = 30 4x = 24 x=6
1,1,1,3,3,3,3,4,5,6
2
13
(Statistic)
population)
1. 2.
S(
(
x
(
)
30-34 35-39 40-44 45-49 50-54
10 15 25 6 20
35– 39
Frequency distribution ( )
2)
1)
(Finite population)
)
1.
(Infinite population) (parameter) ( ) ( ) Sample ( )
30-34 34.5
35:342
1)
35– 39
2)
39:42
39.5
34 5:39 25
35– 39
37 5)
7)
10)
=
9)
=
8)
39.5 -34.5 = 5
35– 39
–
6)
x 100%
x 100%
1.
(Histogram)
2.
(frequency polygon)
3.
(frequency curve)
4.
–
(Stem – leaf Diagram)
0 2 4 11
3 4 5 0 1 2 0 6 8 –
} {3,4,5,20,41,42,110,116,118
1.
wi
=
x= f
∑∑ ∑ =
∑ ∑
xi
wi
=
∑
=
=
x= f
∑∑ ∑
x ∑ =
=
∑∑̅
xi
=
x x<1 Nx x <1 x x <1 M2
1.xmax 2. 3.
4.
xmin
2.
Me = Med =
Me =
2
x
=a +b
M =
x
y i = axi + b
5.
;N
:12
Me
=
Me f(M) Me I fm Me I 3. Mo Mod Mo = Mo = L
Med
1
68 68 85
8:8:85:95 4 :12 4:12 =
2
95 = 79
68
= 2.5
50 -- 70 60 60 70 - 80 80 - 90
:2 85
= 76.5
1
28 16 38 6
x (
)
y = x - 65
(f)
50 - 60 60 - 70 70 - 80 80 - 90
28 16 38 6 88
y ;1::1:2 y 4 y
y = x – 65
=
28 44 82 88 242 – 65
=
=5
=
+ 65 = 5 + 65 = 70
70
55 65 75 85 280
-10 0 10 20
20
75
2 882 =
= 44
59.5 + 1044;28 1
= 59.5 +10 = 69.5
4.
(geometric mean) x 1, x2, x3, …, xn N G.M.
1 2 4
k
N f<1 lgX
N l gX < 1
log G.M. =
fi
N
log G.M. =
Xi
i i
5.
harmonic ( mean) x 1, x2, x3, …, xn N
::1::} N
H.M.
=
∑1
Xi
::::}
H.M.
=
<1 f
1.
(Qr)
(Dr)
3.
D 91, D2, D2, ..., D9
Q r=
4
k
10
100 99 P 1, P2, P P99
Qr=
D r= P r=
Pr=
+
4 11
Q r, D r, Pr (
3,…
:14 :1 :111
(P r)
Q31, Q2, Q3
2.
fi
i i
D r=
)
2
1.
=
–
;2
2.
2.
=
3.
1.
(QD)
(MD)
∑;̅ ∑;̅
4.
SD)
∑ 2 ∑;∑; ;1 ̅ ∑;1;̅ =
4. x
=
SD = =
:–
33 11
MD =
MD =
= 3.
2.
1.
(
̅
∑; ∑ 2 ;1 ̅ ∑;1; 2 ∑; 2 =
=
SD
SD =
=
(Variance)
1.
3.
S
2.
4.
= | k | S
(k)
(k)
0
( )
Box - plot
25
(standard value :Z)
; ̅ ; ̅
1.
=0
2.
∑ ∑2
3. 4.
z z
1
0
∑2
N
N–1
68.27%
95.45%
99.73%
% 50
z 0
1
0.3413
z 0
2
0.4772
z 0
3
0.4987
Ent
1.
3
= ax +b (
y) ----- (1)
̂ ∑ ∑∑Nb ∑2 ∑ (1) ∑∑ ∑ ∑ ∑ ∑ ∑ (2) ∑∑ ∑∑ ∑ x̂ () ̂ 0 ̂ ∑ 2 ∑2 3b∑ N ∑∑2 ∑ 4b∑23∑ 2 ∑ b∑ ∑ – (2)
y=
y=a
x+
b
y=a
x + N(b)
xy =
(ax + b)
xy = a
x2 + b
= ay+b (
x )
x)
y
=
x
x
x
2.
(∑)
(ax + b)
y = ax + b
x
y = ax + b
a
------------- (1)
------ (2) --- (3)
(∑
y = ax2 +(1)bx + c
∑∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ y=
y=a
x2 + b 2
x +b
c
x + Nc 2
∑ ∑∑2 ∑ ∑ (2) ∑∑ ∑∑ ∑ ∑ x3 + b
x2 + c
y = ax + bx + c
y=ab
x ) (∑
x 2)
xy =
(ax + bx + c)
xy = a
x4 + b
x3 + c
∑ lg l̂ ga ∑ Nl2 g b ∑∑ lg l∑ ga ∑ lg b (1)∑ ∑∑ ∑∑ ∑ ∑∑ ∑ ∑ ∑ (2) log
log y =
= log a + log b
x+
Log y = log a
x + N(log b)
log b
y = ax + b
x ) (∑
(log a + log b)
xlog y = log a
(∑)
(log a + log b)
log y = log a
xy =
(∑
x2
log y = log a + log b
2
x
2
x
(∑)
(ax + bx + c)
xy = a
3.
2
xy =
X
x+
y = ax +(2)bx + c
(ax + bx + c)
y=a
2
y
x2 + log b
x
a, b,c, m
1.
y
x
7,000
y = 2x+5
100
200 4,000 1. 2. 3. 4. x x = 200 y = 2(200) +5
2
y = 405
4 2 2 y = 0.5(x) + 1000x − 3000 x = 10 y
2.
3x3 + 6x2 − 12x − 24 =0
1.
√
32
x = 5 y 2012.5 2. 3. 3 4. 2
x = 10
y = 0.5(10)2 +1000(10) – 3000 = 50 + 10000 – 3000 = 7050
1
3x3 + 6x2
12x 24
=0
x2(3x + 6) −4(3x + 6)
=0
2
(x −4)(3x + 6)
=0
(x −2)(x +2)(3x + 6)
=0
x = 2, −2
32x - 4 x
√
32
-4
√
= −128 x 32
= −4096
2
= 0.5(5) +1000(5) – 3000
y
= 12.5 + 5000 – 3000 = 2012.5
x=5
2
14
1. 2.
(
)
Sequence
1
3
3
6
15
5
10
{1,3,6,10,15}
{1,2,3,4,5}
{1, 2, 3, 4, … , n}
3
4
5
...
n
…
3
5
7
9
…
2n - 1
...
2
4
2
2. sin 30 + sin 60 + sin 90 + …
1
1 1
1. 2,4,6,8,10,…,100
Ex
finite ( sequence) (infinite sequence)
(Term)
5(
5
n2- 1)
2 (5) – 1 =9
a1 , a2, a3, … , an
a
a1, a 2, a3, … , an , ...
a 1
a2
a3
an
1 2 3
n
1)
7, 14 , 21 , 28 , 35 , 42
3) an = 2n + 3 4) an = (n+1)2
*2
2) 3, 6 , 12 , 24 , …
n +1
2
n
2 * 2
7
an()
an = an+b ;
n
n (1) (2) –
(1 )
R
n = 2
a,b
n = 1
1
,3,5,7, …
a(1)+b = 1 ------- (1) a(2)+b = 3 ------- (2)
a=2 a
(1)
2+b =1 b = -1
an = 2n – 1
an = ar +b ; a,b n
n
n = 1
n
n = 2
(1)(2) –
r) (
,8,16,32, 4 … 2
(1 )
R
1
a(2) +b = 4 ------- (1) a(2)2+b = 8 ------- (2)
2a = 4 a
a=2
2(2)1+b = 4
(1)
b=0
an = 2(2)n
2n+1
an = an2+bn+c ;
a,b,c
R
5
, 18, 35, 81, …
2
n
n = 1
n
n = 2
n
n = 3
a(1)2+b(1)+c = 5 ------- (1) a(2)2+b(2)+c = 18 ------- (2) a(3)2+b(3)+c = 35 ------- (3)
(1) (2) –
3a+b = 13 ------- (4)
(2) (3) –
5a+b = 17
(4) (5) –
2a = 4
------- (5) a=2
a
(4)
a
b
(1)
2(1) + 7 + c = 5 c = -4
an = 2n +7n – 4 2
1 2 3 4
b=7
( n +1 , 4, 8, 16 2 , …
2 4 = 2 x 2 = 22 8 = 4 x 2 = 23 16 = 8 x 2 = 24 5
n2
n n 2
n)
10
Ex
(arithmetic sequence) 1. an = a1+(n-1)d d
1.1,3,5,…
a
a10 = 1(4)10-1
l 6 33 2
l 1.
1.
c
2.
l ( + l ( l ( ⋅ l ()
4. 5. 6.
l
can = c
divergent (
d )
l 1
c an
l)l ) l l ll)
(
2.
c = c
l l l ⋅ l l ≠ 0
an +
bn
an - bn =
an -
bn
an bn =
an
bn
bn
0
an bn =
=
|r|
2 3 6 l 2 2
l
3
3
3. 4.
3.
| r | < 1
l
Ex 1. 1. (Convergent sequence) 1. d ) ( 0 2. r) ( 1
3. 2. sequence)
2
4 262144
2. 1,4,16, …
sequence 2. geometric ( ) n-1 an=a1r r (ratio)
d
19
10 = 1+(10-1)2
(common difference)
4.
1 10
(Ent)
1. 54
2.38
200, 182, 164, 146, ...
3.22
– 18 10 a10 = 200 + (9)(-18)
4.20
an = a 1 + (n-1)d
= 200 – 163 = 38
38, 20, 2, -16 38 – (-16) 1 2 57
= 54
3
-1
(a1r )(a1)(a1r) = 343
a13
= 343
a1
=7
a1 + a2 + a 3 = 57 a1 + 7 + a3 = 57 7r-1+ 7r
r
7 + 7 r2
= 50 = 50r
7r2 – 50r +7 = 0 (7r - 1)(r - 7) = 0 r = 7,
1,7,49
49
17
49,7,1
343 3 PAT 1 (
53)
l 32:2;1 :5;3 ; 3: 3 :2;1 l 2:5;3 l 2:;
3 n
=
0
0
l3 3:2:; 2 32 4 l 52 :3 ( ) :5 5 5 :3 3 l 22:5 5 =
0
0
=
Ent
:55: l =
= 25 25
5
–1
1
l 4 2
0
>
>
(Series)
6
Sn
1 + 2 + 4 + 8 +... + a7
a1(r)n-1
2
6 1(2)
1+2+4+8+16+32+64
∑
a1+ a2 +a3+… +an
<1 1. 2 3 4
<551 2 <<311 <1
ai
i
1
n
1+2+3+4+5 2 + 221+ 32 + 42 + 52
(2+1) + (4+1) + (6 + 1) 3 + 5 + 7 (2+1) + (4+1) + (6+ +(8+1) 1) + …
7
1. 2. 3.
<1 2 2 <1 2 <1 < 1 < 1 <1 2 <1
Sn =
∑ ∑ ∑
2 <1
20
. / .6 6 / . 6 / 6
20
= 1+2+3+…+n =
:1 2:12:1 2 * :1 2+
2
= 1+22+32+…+n2 =
3
= 1+23+33+…+n3 =
16,180
<;1
-1
n
1.
2.
3.
∑
<1 c cf f <1 <1 <1 f g f<1 g <1
2
2
2 2
Ex
1 + 3 + 5 n+... -1) +(2
Sn
Sn = [1+(2n-1)]
Sn = (a1 + an) = (2a1 - (n-1)d)
= [2n] = n2
() () 1;;1
Sn
= =
Ex
r < 1
Sn =
r > 1
=
( ) () 3;12
2 + 6 + 18 + ...
Sn
= 2(3n-1)
[Infinite Series]
n
S∞ =
n
l <1
S ∞ S ∞
S∞
S∞ =
8
S∞ = 8 + 4 + 2 + 1 +
1;
12
1;
|r|<1
= =
8
4
6 + 6.5 +7.25 +8.125 + ...
1 5 + 6 + 7 + 8+... +12
2
82
S8
12 1;8
r=
1
+…
a= 8
9
= [5+12] = 68
2 1 + 0.5 + 0.25 +0.125 + ... S8
=
= =
1[1;] 1;
*255 + 128
2
69
127128
15
(lim)
y
x
(Calculus)
Differential ( Calculus) , . Integral ( Calculus) ( )
1
f(x) = 2x
x
x
2
f(x) 3
6
1.5
3
2.5
5
1.8
3.6
2.1
4.2
1.9
3.8
2.05
4.1
1.99
3.98
2.01
4.02
1.999
3.998
2.001
4.002
2 f(x)
4
f(x)
2
x
x
x≠2
1.0
l fx l fx
(
a) a (x > a) a) a (x < a)
x
x
(
2 1
=2 a
llffxx
a
-
a
l fx
a = f(a)
l fx
l fx
= f(x)
(a,b) [a,b]
b =f(b)
f(x)
1 – 3 = -2 =
= -2
f(1) -2(1) = -2
x x x x 5;;2 x l1 fx l1 fx 5;1;2 ;24
l1 fx
=1
(Continuity of function)
l1 fx
+
(a,b) [a,b]
2
1)
f(x)
2) 3)
ll1 ffxx l1fx 1
4) f(1) 5)
1)
x
2)
f(x)
l1 fx x
2 2
2. 3. 4. 5. 6. 7.
=4
4) f(1) = 4 5)
l l
(x>1)
l1 fx l1 f x1 1 fx l fxl
1
f(x)
1.
(x<1)
=2
3)
1
x = 1
l
(an+ bn) =
an +
bn
an - b n ) =
an -
bn
an b n =
an
bn
c can = c
l (( ⋅ l () l ( )
=
l
an
c=c
l ⋅ l ) l l
f(x) n = [
f(x) ]n
bn ≠ 0
1l fxl1 fx
1. 2. 3.
x
c
3
x
0
3)
22
4)
-1
l2 ,
-1
-1
- 1
l2 ,;:2;4- 2 l2 ,:2;2 - l2 ,:2;1;1;2 l2
=
=
2
-4
l fx l fx =
= f(x)
3
f
g
x
l fx
a
x = a
l gx
=0
=
=0
l l l fx l gx l l f
g
=∞
a
=∞
=
4 4.1)
2 x fx6x x l 4 fx x x
= 12
l1 fx
= 13
l1 fx l1 f =
4l fx
= 6x – 5
= 6x – 5
= 6(3) -5
= 6(4) – 5
= 13
= 19
4l fx
= 3x = 3(4) = 12
4l flx fx4l fx ≠
4
g (x)
4.2)
a,b,c
f
a√ x f bcx xx x
x = 1
f
;√;15;
a-
√
b
=
= = x = 1
= =
2(
58 52
= -2[2+ ] = -4 - 5
-9
0
=0;a=2
2;;1√5; 2;;1√5; 2:2:√√5;5; 4;5; ;12:√ 2:√ 5; 5; 1 2:1 √5;1 4
– a + b-1c) = -2[(2) + 4(
= -9
√
a -
=
2(
– a + b-1c)
=0
x = 1
x = 1
)]
cx 145 =
x = 1 2c(1)
=
c
=
45 8
0
x+h 2 f(x) = x +1
y = (x) x
y
f(x)
=
= =
6
x
:;
x+h
x
x+h
;;
x = 2
4
x
4;2 4;2 17;5 4;2 Ex
The ( derivative of a function)
f (x)
x
y
f′(x)
P(x,y)
differentiation (
1. y = c (
)
y = 0
f′ =
y =1 n
y = n(x)
=
n-1
; ×
4. y = f g
5. y = cf(x)
y = f (x) g (x) y = cf (x)
6. y = fg(x) y =f(x)g (x)+g(x) f (x) 7. y =
8. y = (gof)(x)
y =
y = g (f(x))
f (x)
= =
l :;
f(x) = x2
)
; l :; l : :2: l 2: ; l
= 2x
2. y = x 3. y = x
x
y = f(x)
x 0
f (x0)
Ex
y
x = 1 = 2y
x = 1
= 5x 7 + 3x 4 – log sin0 tan45 2.f(x) 1.f(x) = x 35–3x +7 1. f(x) = x 3 – 3x +7
2. 3.
f (x) = 3x2 – 3
f(x) = 5x 7 + 3x4 – log45 tan45
f (x) = 35x6 + 12x3 (
f(x) = (2x+3)
y2+3
3.f(x) = (2x+3)5
log45 tan45
, g(x) = f (x)
(gof)(x)
5
(1)2 = 2
) 0
f (x) = 5(2x+3)4(2)
= 10(2x+3)4
6 f(x) = x3 – x2+ 2x – 1
g(x) = f (x)
f ( )x 3 x22 2 x f ( x) 6 x2
(g
g ( x)
(f ) 6x
g x231 x)2
f)( ) x ( (g)) f (x
3
2
2
2
6 x 6 x 6 6 x2
6 x36
x
x
6( x x 2 x1) 3
2
x 212 8x 3
( g f )(1)
6(1)
6(1)
2
6 6 12 8
4
12(1) 8
(gof)(1)
s(t)
s (t) = f(t) t
t t
(t)
s (t) s
t
7
F(x) = f(g(x)) ; g(3) =5 ; g (3) = 7 ; f (3) = -4 ; f (5) = 9 ;F (3) =
A; B = A + (A - 2) + (A - 4) + (A - 6) + … + -1 ; C = log2sin30 + 4tan60 sin60 B – 4AC
F
(x) = f (g(x))(g (x))
F (3) = f (g(3))(g (3)) = (f (5))(7) = (9)(7)
= 63 = A
B
A + (A - 2) + (A - 4) +(A - 6) = 63 + 61 + 59 +57 + … -1
-1
n
= (n-1)(-2)
33
=n
=5
332 (63 1) 2
= (a1+an)
S33
=
-
= 1023 B = 1023
B – 4AC = 1023 – (4)(63)(5) = 1023 – 1260 = - 37
– 37
√3 √ 2
= -1 + (2)3
Sn
C
= log22-1 + 4
= 63 + (n-1)(-2)
-64
log2sin30 + 4tan60 sin60
C =5
(
)
x x
x1
f (x1)
2.
f′(x) < 0
f (x) = 0
f (x) < 0
f (x) (
2)
2
[a,b]
f
[a,b]
2
2 2
f f f(a) f(b)
f (x) > 0
f (x) = 0
1. 2. 3. 4.
f (x1) = 0
f′(x) > 0
f (x1)
1. f (x) = 0
y y
3 3
3
f f
8 [0,2]
f
f(x) = x3 - 3x+2
f(x) = x3-3x+2
f (x)
= 3x2 - 3 = 3(x2 - 1) = 3(x-1)(x+1)
2 x = 1
x = -1
-1
-1
[0,2]
F
x = 1 x = 2
f(1) = 13 - 3(1) + 2 f(2) = 23 - 3(2) + 2
x = 2 f x =1 f(1) =0 f
f(1) =0 f(2) =4
f(2) =4
1. 2.
(Integration)
= kx+c
2.
=
3.
∫8x3dx 4
Ex
∫dx ∫f∫∫fxdx dxxgx:1∫ fx∫fdxxdx ∫gx ∫fx×gx
1.
=
+c
=
= 2(x)4 + c
=
4.
Ex
3 3 ;2
f(x) = f′ (x) x = a
f
x=b
32
= 2x4 + x2-z+c
a
b
4
32
3 -2
32
2
= [2(3) + (3) -(3)+c ]–[2(-2)4 + (-2)2-(-2)+c ] = 130 + 7.5 - 5 = 132.5
x
x
A=
2
f(x) =
=
=
2
2
=
∫∫22 2
= [-4] – [0] A = -4
–
4
x= 0
F (x)
∫5 x
= F(6) - F(5)
12
= 4 - F(5)
F(5)
= -16
F(0) = 4 F(6) = 4
F(5)
16
(Linear Programming ) , ,
ax+ by = c ; a
1
-
53
x
-
x
5x + 3y = 15
b
(3,0)
y
(0,5)
(
)
P = ax +by
(
C = ax +by (
)
)
P = 5x + 12y
2.
3.
)
1.
(
2x + y < 100
P ,
x>0
Ex
P = 3x – 2y P
(0,5) B ,10 (0 )
A
y
5, x
0
x = 0
x+y=10
0 + y = 10 y = 10 y = 10 x = 0 C
(5,5)
y = 5
x+y=10
x + 5 = 10 x=5
1
x = 5
y = 5
P = 3x – 2y
A(0,5)
3(0) – 2(5) = -10
B(0,10)
3(0) – 2(10) = -20
C(5,5)
3(5) - 2(5) = 5
P
x = 5 y = 5
P=5
y
5, x
0, x+y 10
2 1
P = 3x – 2y
2y = 3x – P
3 2 2
y= x-
2 2 H (-4,5)
2
P = 3(-4) – 2(5) = -22
O (-3,-6)
P = 3(-3) – 2(-6) = 3
3
C
C
2
Ex
P = 3x – 2y
B
P
y
5, x
0, x+y 10
1 2x + y ≥ 50 x+2 ≥70 x ≥ 0 , y≥0 P 1.1
P = ax + 2y
100
a
2.2
(Ent 44
3.4
)
4.6
2
2x+y=50 ----- (1) x+2y=70 ----- (2) (1)×2
4x+2y=100 -- (3)
(3) –(2)
3x = 30 x = 10 y=3
(x,y)
a
P = ax + 2y
P
(0,0)
a(0)+2(0)
0
(0,35)
a(0)+2(35)
70
(25,0)
a(25)+2(0)
25a
(10,30)
a(10)+2(30)
10a+60
P
( 10,30)
P = 10a + 60 100 = 10a + 60 40 = 10a a =4
2
2
C = 40x +32y 6x + 2y 12 2x + 2y 8 4x +12y 24
C 108 (
(Ent)
2.112
1.
)
3.136
4.152
6x + 2y = 12 2x + 2y 8 4x +12y 24
(0,6) (2,0) (0,4) (4,0) (0,2) (6,0)
X
Y
x+2y = 12 6 --- (1) 2x+2y
(1) (2) -
4
8
--- (2)
x=4
(2) 2(1)+2y = 8 2y = 6
2x = 6
y=3
x=3
(1,3)
y =
432
C = x+
Z
2x+2y 8 --- (1) x+2y = 126 --- (1) x+12y 4 24 --- (2) x+12y 4 = 24 --- (2) (2) –2(1) 8 y = 8 (2) – 6(1) -32x = -48 y =1 x= x = y x+2(1)=8 (1) 2 x (1))+26(y = 12
x=1 x
321
C
3,(1)
40x +32y
m=
4832 32 32 23 32,(32) 432 54
9 + 2y = 12 y=
Z C=
40 32 3 32
( ) + 2( )
= 108
1
Z
Z
081-2-76272-9
[email protected]
MATH KIT EBOOK ( )
MATH KIT EBOOK
[email protected]