الرياضيــــــــــــات للصف الرابع العلمي
Transcript of الرياضيــــــــــــات للصف الرابع العلمي
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1 : [1-1]
[2-1] ....
[3-1]
[4-1]
[5-1]
[6-1]
[7-1]
...
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5
Mathematical logic :
. .
) (
()
()
.
Logical statement 11
:
) .
) ( ) .
.
.
( 1 - 1 )
(T) (True) P P
P (F) (False) P
P
( 1 - 1 ) .
P P
F T
T F
-
6
() ():
(2-1) (1-3)
If then ( 21
( ... )
.( Compound Statement )
:
:
(( )) (( ))
. (( .... )) .
( ) ()
. :
( .... ) .
( )
( ) .
P Q P Q
T T T
T F F
F T F
F F F
P Q P Q
T T T
T F T
F T T
F F F
-
7
:
: ( )
:
1)
2)
3)
4)
.
( .... )
. P Q : Q P
(( Q P ((
P Q :
(1-4)
P Q .
P QQP
TTT
FFT
TTF
TFF
-
8
1 :
1) 3 >2
2) 5+7 = 12 6+2= 7
3) 5+7 =11 6+2=8
4) =1 3
1) .
2) .
3) .
4) .
If and only if (31]
:
(Q P) (P Q)
:
. (
) PQ
PQ (Q P) (P Q)
(Q P) :
PQ : ( 5 - 1 )
-2 R
-
9
(1-5)
P Q :
.
2
X=-1,X=4 X2-3X-4=0 ( X5=-32 X=-2 (
Implication 41]
:
:
P X=3 :
Q X2= 9 :
X2=9 X = 3
P Q :
X = 3 X2=9
Q P :
P Q P Q Q P (p Q)(Q p)
T T T T T
T F F T F
F T T F F
F F T T T
PQ
-
10
:
P X =3
Q X3=27
X3=27 X = 3
P Q
X = 3 X3 = 27
Q P
P Q (Q P) (P Q)
3
.
X3 =8 (
X>2 , X>5 (
X20 , X0 (
) P : Q :
X3=8 X=2 (
X>5 X>2 (
X0 X2 0(
Q P (
X = 2 ,
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11
Equivalent Statements
(1-1)
P Q Q P
4
PQ~ PQ
:
P Q ~ P P Q ~P Q
T T F T T
T F F F F
F T T T T
F F T T T
-
12
( 1 - 1 )
1
:
) 5 25 7 25 .
) 5 25 7 25 .
) 7 4 .
) .
) .
2
:
P Q
a.b=0 , a,bR a=0 b=0
X = -3X2 = 9
X2 = 25X = 5
X3 = -125X = -5
X=1 X=2(X-1)(X-2)=0
-
13
3
:
PQ ~Q ~P ( 1
( 2
4
P Q S
( 1
( 2
( 3
( 4
5
: -
S P :
P ~P( 1
~P P ( ~P ( ~ P P ( P P (
S S ( 2
) ) ) )
3 ) S 9 > 5 + 3~ : -
~S 9 < 5 + 3 ( ~S 9 5 + 3 (
S 9 5 + 3 ( ~S 9 5 + 3 (
~(PQ) P~Q
(PQ) S
(PS) P
(SQ) P
(SS) S
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14
Open Sentences 51]
( ) .
:
) X
)
) a b
) . . . . .
.
() 9 X (9 )
(Y) () . a b 3
(3+3=6) . ()
.
(1-2)
(1
.
2)
.
61]
:
(Z)
{2} {2}.
.
P(X)
Y+1=3Q(Y)
a+b=6G(a,b)
P(X): 2X=4Q(x): X-1=1
Q(X) P(X)
Q(X),P(X)
-
15
5
. Z
{2}
{ } {2} { }
.
(1-3)
.
6
. Z
Q(X):X2=4
P(X):X=2
P(X),Q(X)
P(X)
Q(X)2,-22,-2P(X)
Q(X)
P(X)P(X)
~ P(X)P(X)
X2-4=0
X
X=4 X+16
X2-40
X
X4 X+1=6
~ P(X)P(X)
-
16
(X-3)(X-4)=0
( 2 - 1 )
1
:
N XX0
{10, 6 , 5 , 3} X2-11X+30=0
N (X-1)(X-5)=0 X>4
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17
Quantifiered Propositions 71]
[1-7-1] :
:
: A :
a A
a A
:
.
( )
.
:
X :
: : A
A
:
bA ( )
bA
X+1=2 Z :
X+1=2 XZ
:
XZ X+1=2 .
F(X)
F(a)
F(a)
aAF(a)
aAF(a)
(X+1)2=X2+2X+1
(X+1)2=X2+2X+1 XN
G(x)
G(x)
G(b)
G(b)
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18
[2-7-1] : :
:
.
- :
:
.
- :
2 6 :
2 6.
- :
.
.
:
~[ P(x) xX ] ~ P(x) xX
~[ P(x) xX ] ~ P(x) xX
7
:
X (1 :
X > 0 X :
X (2 :
: X
(3
P(X)
P(X)
P(X)
P(X)
P [XR : X+35 ]
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19
1) ( ) ~
( ) P ~ :
: .
~ (X) 2)
( )P~ : X : X
X .
( ) (3
: Tautology 3-7-1] ]
P P
.
8
P P~ P
.
. ( Contradiction ) :
P ~ P P~ P
T
F
F
T
T
T
P X
XX
XX P X
P
XX
P X+3 < 5 : X R
~ [ (X) X ] X
~ [ P ( ) ]
~
X 0
-
20
(1-3)
1
:
) .
) .
) .
) .
) .
)
)
2
:
) X ( ) P :
X2 = X X : (X)
) X (X) :
X2 = X : (X)
) X (X) :
(X) : X X2 .
) Q P : .
) P : PP ~
) P Q : ( P Q ) ( P Q ) .
( XR : X < 8 ) P
X
P
P
PX
P
P
Q P Q
Q: X N : X2 = 25
-
212121
2 :
Y =|X| 1-2] ]
[2-2]
[3-2]
[4-2]
[5-2] ( )
[6-2]
:
-
-
-
-
-
-
22
Absolute Value 1 2 ]
( 15 - 2 )
X |X| :
X, X> 0 | | = , X = 0
-X , X < 0
1
:
X R | X-3 | ( )
X-3, > 3 (
| | = X = 3 )
-X+ 3,
( 15 - 2 ) :
(1
(2
(3
(4
(5
Y 0 = | |
(6
(7
]
]
XY
|X||Y|
:
X Y
.
| X |0 XR
| -X |= | X |XR
-| X |X | X | XR
| X . Y |=| X | . | Y | XR
|X|2= X2 , XR
| X + Y || X | + | Y | X,YR-aXa | X | a a0 XR
x
>
0
X-3
X
0,
X 0
3= 9 < 10
3 10 = 10 3 > 010 3( ) > 03= 9 < 10
3 10 = 10 3 > 010 3( ) > 0
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23
2
Y=| X |
(2-15)
Y = X :
: :
]X Y ( X , Y )
0 0 ( 0, 0 )
1 1 ( 1 , 1 )
2 2 ( 2 , 2 )
X Y ( X , Y )
0 0 ( 0 , 0 )
-1 1 ( -1 , 1 )
-2 2 ( -2 , 2 )
Y=| X |
( 1 , 1 )
( 2 , 2 )
Y = XY=-X
Y
( -2 , 2 )
(- 1, 1 )
( 0 , 0 )X
X , X>0
Y= 0 , X=0
-X , X
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24
3
(2-15)
:
:
:
X Y ( Y , X )
1 3 ( 1 , 3 )
3 5 ( 3 , 5 )
X Y ( X, Y )
1 3 ( 3 , 1 )
0 4 ( 0 , 4 )
Y
X
( 3,5 )
( 1,3 )
( 0,4)
Y=| X - 1 |+3
]]
(X-1)+3 ,X1
(-X+1)+3 , X
-
25
2 2 ]
4
. X R :
:
= |3X+6|
:
....... { X: X -2 }3 X+6=9
........... { X: X< -2 } X- 6 = 9
x y . y .
:
S1 = { 1 } S
2 = { -5}
. S= S1 S
2 {5- , 1 } =
}
}
|3X+6|=9
-2X 03X+6 3X+6
-2>X 0>3X+6 -(3X+6)
(2)
(1)
-3
-
26
5
: .
:
X3 - 8 = 0 , X 0 X3 = 8 X = 2 S
1 = { 2}
- X3 - 8 = 0 , X
-
27
3 2 ]
S 2S
1
S = S . 1 S
2
.S = S1 S
2
7
Y X :
X - 2Y = 5 . . . . . (1)
2X + Y = 0 . . . . . (2)
: (2) 2 :
X - 2Y = 5 . . . . . (1)
4X + 2Y = 0 . . . . . (2)
(1):
= {( 2 -, 1 )}. .
R
5X=5 X = 1
Y = -2
1 - 2Y = 5
-
28
X - 2Y = 5 : :
2X + Y = 0 :
8
R x.y
x y
X= 1 + Y
2
:
X Y ( X , Y )
0 -5/2 ( 0 , -5/2 )
1 -2 ( 1 , -2 )
5 0 ( 5 , 0 )
:
X, YYX
(0,0 )00
(1 ,-2)-21
(-1 ,2)2-1
X - Y = 1
X2 + Y2 = 13
L2
L1
(5,0)
(1,-2)
(-1,2) L1
L2
(1+y)2 + y2 = 13 2 y2 + 2y-12 = 0y2 + y-6 = 0 (y+3)(y-2)=0y + 3 = 0 y = -3 x = -2 (-2,3)y - 2 = 0 y = 2 x = 3 (3,2)S= (-2,-3),(3,2){ }
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29
9
R x,y
2x2 3y2 = -46 , x2 + y2 = 17
(
)
x2 + y2 = 17....12x2 3y2 = -46....2---------------------------3x2 + 3y2 = 512x2 3y2 = -46----------------------5x2 = 5 x2 = 1 x= m1x=1 (1)2 + y2 = 17 y2 = 16 y = m4 (1, 4), (1,4)x=-1 (-1)2 + y2 = 17 y2 = 16 y = m4 (1, 4), (1,4)S = (1, 4), (1,4), (1, 4), (1,4){ }
( 3
1) ( )
* *
2)
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30
Intervals 4 2 ]
a b R a < b
1 ) :
{ bX : X R a X } Closed Interval a b
[ a , b ] ( 1 - 2 )
) a )
( b ) ( )
a b [a , b ]
a b ( 1 - 2 )
2 )
Open Interval ( a ) ( b )
( 2 - 2 )
a b ( 2 - 2 )
a , b
.
b (a , b) , a ( a , b)
(a,b) ={X:XR,a
-
31
3 ) :
( a , b ] = { X : X R a < X b }[ a , b ) = { X : X R a X < b } ( Half Open ) a < b
( 3 - 2 )
a b
( 3 - 2 )
( 4 - 2 )
a b
( 4 - 2 )
4 ) ( a ) :
{ X : X R X a } ( 5 - 2 ) { X : XR X > a } ( 6 - 2 )
a a
( 5 - 2 ) ( 6 - 2 )
( a ) 5 )
{ X : X R X a } ( 7 - 2 )
{ X : X R X -5 }
2 ) { X : X -3 } ( -5 , 2 ] = [-3 , 2 ]
-
33
5 2 ] ()
g(X) f(X ) g(X) < f(X ): ( X )
. ( X ) Inequality
( X )
.
.
( 16 - 2 )
.
.
1
3X +1< X +5 :
R .
3X +1< X + 5
2X +1 < 5
2X < 4 .
f(X) < g(X) h(X) < I(X)
g(X) f(X)
2X + 1 + (-1) < 5 +(-1)
3X + 1+(-X)
-
34
(2X) 4 >
X < 2 .
{ X : X R X < 2 } =
2 X
S2S
1
S : : 1 S
2 .
:
2
( R ) :
5X + 11 < 1 2X + 3 < 6 .
= { X : X
-
35
32
32
32
32
32
{ X : X < -2 } { X : X < }
S1 S
2 =S
1 =
S = S1 S
2 =
2-
S 1S
1 S
2
{ X : X < -2 X R }
3
:
2X + 3 < 6 5 x + 11 < 1 :
2-
S2S
2S
1
S= { X : < -2 > X }32
S2 S
1 = {X: X < X < -2}
S = {X:X R,X 5
( R )
X -2 , X2 =
2- X , X < 2}X-2 -2-X > 5 X-2 > 5 |X-2| > 5
:
S1 S2 = { X : X R X > 7 } { X : X R X < -3 }
S2 -3 7 S
1
5
x R x+1 2
(7) 52
(1-)
-2+ -1( ) x+1+ 1( ) 2+ 1( )3 x 1
s = 3,1[ ]
x+1 2 -2 x+1 2
-
37
6 2 ]
( a) :
[- a , a ] X2 a2 1)
(- a , a ) X2 < a 2 2)
2) :
(b 0 )
: a . b < (b >0 a 0] [(X - a ) > 0 (X+ a ) 9 R / [-3 ,3 ]
/ X2
-
38
7
: 5
2X+5| 5| :
[ - (2X+5) 5 ] ] 2 X + 55 ]
[ [0 2 2] [
7 > |2X+5|
2X +5 , X 2X +5 =
-( 2X+ 5) , X < }- 52
- 52
7 >7 >
7 >X12 > - 2X 10
[ -6 < X -5] [ 1> X 0]( -6,-5] [ 0,1)=
>
:
*
* :
(0) (
(1))
R *
-
39
(2-4)
1 /
A B AB A - B B - A
2 /
) Y =| X + 2 | - 5 )
3 /
:
) )
) )
X2 -2 |X| - 15= 0 (
)
4 /
:
) 2X + Y = 4 X-Y = -1 ( )
) 4X + 3Y= 17 2X + 3Y= 13 ()
X - Y = 1 5X2 + 2Y2 = 53 (
2X2 - Y2 = 34 3X2 + 2Y2= 107 (
5 /
:
) )
2X2 8 ( )
3X2 -27 >0 (
A= [-2 , 5)
B = { X: X 1 }
|4X + 3| =1
X |X| +4= 0
|X2 +4| = 29
|X-6| 12 |X+1| 4
-9 < |2X -3 |-12 -3
y = 3 x+1
x x+2 = 3
2x+1 = x
-
404040
3 : [1-3]
[2-3] [3-3]
a + b 4-3] ] [5-3]
-
a x
a + b
-
fa (x)=ax
-
41
Indices and Roots :
:
1 )
2 )
3 )
.
:
( 1044 - 1122 ) = ( - 435 515 )
.
( 235 - 164 ) = ( 850 - 781 ) ( )
.
. :
( 1855 - 1777 ) ( 1829 - 1802 )
. Abelian Group
-
42
am an
1 an
ab
.
.
1 3 ]
Indices
( 1 - 3 ) a R n Z
a0 = 1 2 )
( Z
) b 0 a 0 :
an am= am+n ( 1 [
]
a-n = 2 )
am-n (3 = [
]
[ ] ( 4
( 5
( 6
:
an
a a
n
. n a
( 1an = a a . . . . . a( a n )
1 a
a -n = (a-1)n , a-1 = a 0 ( 3
a b R n m Z
(am)n = a mn (a . b)n = an. bn
an bn
.
.
1 3 ]
Indices
( 1 - 3 ) a R n Z
a0 = 1 2 )
( Z
) b 0 a 0 :
an am= am+n ( 1 [
]
a-n = 2 )
am-n (3 = [
]
[ ] ( 4
( 5
6 ) = n ( )
-
43
n N n > 1 n 0 = 0
Xn
Xn =
Xn =
Roots
( 2 - 3 )
aR nN n > 1 X : Xn = a (a) n a
:
( 1
2 ) ( n ) ( a )
a X = - n a X = n a
3 ) ( n ) ( a )
( XR Xn ) a
4 ) ( n ) ( a )
= a
( 1 - 3 )
a b R n N n > 1
( b 0 a 0 ( n ) )n a. b = n a . n b ( 1
}n n abn a
b=
a1n
bR /{0} ,aR
n
n
0 < b , 0a
-
44
1
= =
2
m n Z :
=
= =
=
=
8 -3 18 2
81 16 -2
8 -3 18 2
81 16 -2( 2 3)- 3 ( 3 2 2 ) 2
3 4 ( 2 4 )-2 2 -9 2 2 3 4
3 4 2 -8
59
5 3 ( 5 3 )m-2 ( 5 2 )m+n
( 3 5 2 )m 5 2n+m
5 3 5 m-2 3 m-2 5 2m+2n
3 m 5 2m 5 2n+m
59
13 2
:
m ...
.
a - ... m .m
(- a)m = a m
(-1) 25 =-1
(-1) 64 = 1
125 15m-2 25m+n
75m 5 2n+ m
5 3+m-2+2m+2n-2m-2n-m 3 m-2-m
= = 5 = 3-2 5 =
3 4-4 2-9+2+8 = 3 0 2 1= 1 2 =2
15m-2 25m+n
75m 5 2n+ m125
-
45
(3-1)
1 /
:
) 0(8)+0(9) ) 1-(2)+ (3) ) 1 -(16) + 16 ) 64 3
3a 0 ( ) ) ) ( 27 )
) ) ) 3-( )
2 /
:
) 2 ( ) ) 2 [ ]
c 0 ) x 0 )
3 /
( 1 )
:
) d 0 ) b 0 ) 5
x 4 x x 0 ( ) b 0 (
2 -3 4 -5
6 -1 3 3 10 3 4 7
10 -5 2 5
34
20 a 3
45
( -a )3 6 729 3 a
3x-5 . y2
2-1 y-2
b cd
1b 5
4b2
b2 c-31
b2 + c2
-1
53
( 3 a )0 ( a + b )0
a( - a )4
25 b2 c-8
x
3
5 -32
a 0( )a+b 0( )
a 0( )a+b 0( )
-
46
4 /
a R m
a m 0 ( a m 0 ( a m < 0 ( a m > 0 (
5 /
a Ra ,
a m 0 ( a m 0 ( a m < 0 ( a m > 0 (
6 /
a (x-y)z . a (z-x)y . a (y-z)x = 1 ( :
)
7 /
: 1= +
8 /
: =
9 /
:
10 /
: 27= [ ]
11 + a b - c
11 + a c - b
5 3 2n - 4 3 2 n - 1
2 3 2 n + 1 -3 2 n1115
3 2 +n + 3 n + 1
3 n -3 n - 1
6 4 n - 1 27 2 n
2 n + 1 8 n - 1 9 n + 2
( 9 n + ) 3 3 n
3 3 -n
14
m
1n
xn2 1 xn1
1n = xn1
-
47
2 3 ]
Exponential Equation .
:
1) : (( 1))
:
2) xn = yn x = y n
x = + y n
3) m = n =
:
)
x + 2 = 3 x = 1
= {1}
)
-
a x = a y x = y , a 1
xn = ym
15 27
35
35
35( x + 2)- = ( x + 2)- = 3 -
x23 = 3-2
x23 =
132
(x13 )2 = (1
3)2
x 13 = 1
3
x13 )3 = ( 1
3)3
x= 133
x= 127
x23 = 32
(x23 ) = m(32 )
32
x= m 3-3
x= 133
=127
or x= -133
=127
= { }
(
-
48
3
2x -2x+1 = 4x+3 :
x2 - 2x +1 = 2x + 6
x2 -4x - 5 = 0
= { 5 , 1-}
.
4
32x+1 - 43x+2 = -81 :
32x 3 - 4 3x 32 +81 = 0 3
(3x - 3) ( 3x - 9) = 0
3x = 9 3x = 3 2 x = 2
3x = 3 x = 1
= { 2 , 1 } .
2x -2x+1 = 2 2(x+3)
( x-5) ( x+1) = 0 x = 5 x = -1
32x - 12 3x+ 27 = 0
2
2
-
49
5
x : -
) 3x-1 = 5x-1 ) )
) 3
3x-1 = 5x-1 x - 1 = 0 x = 1
) 2
(x+3)5 = 45 x+ 3 = 4 x = 1
) 2
(x-1)6 = 26 x - 1 = + 2 x = 3
x =-1
6
R
-
(x+3)5 = 54 ( x - 1 )6 = 2 6
8x2 +8
x2+
13 + 8
x2 + 8
x2+
23 =14
8x2 +8
x2 8
13 + 8
x2 8
23 =14
8x2 1+ 8
13 + 8
23
=14
8x2 1+ 2+ 4( ) =14
8x2 7 =14 8
x2 = 2 23( )
x2 = 2 8
3 x2 = 21 3x
2= 1 x =
23
8x2 +8
x2+
13 +8
x2 + 8
x2+
23 =14
8x2 +8
x2 8
13 + 8
x2 8
23 =14
8x2 1+8
13 +8
23
=14
8x2 1+2+ 4( ) =14
8x2 7 =14 8
x2 = 2 23( )
x2 = 2 8
3 x2 = 21 3x
2= 1 x =
23
8x2 +8
x2+
13 + 8
x2 + 8
x2+
23 =14
8x2 +8
x2 8
13 + 8
x2 8
23 =14
8x2 1+ 8
13 + 8
23
=14
8x2 1+ 2+ 4( ) =14
8x2 7 =14 8
x2 = 2 23( )
x2 = 2 8
3 x2 = 21 3x
2= 1 x =
23
8x2 +8
x2+
13 + 8
x2 + 8
x2+
23 =14
8x2 +8
x2 8
13 + 8
x2 8
23 =14
8x2 1+ 8
13 + 8
23
=14
8x2 1+ 2+ 4( ) =14
8x2 7 =14 8
x2 = 2 23( )
x2 = 2 8
3 x2 = 21 3x
2= 1 x =
23
8x2 +8
x2+
13 + 8
x2 + 8
x2+
23 =14
8x2 +8
x2 8
13 + 8
x2 8
23 =14
8x2 1+8
13 + 8
23
=14
8x2 1+2+ 4( ) =14
8x2 7 =14 8
x2 = 2 23( )
x2 = 2 8
3 x2 = 21 3x
2= 1 x =
23
-
50
( 2 - 3 )
1 /
:
) = x3 ) )
) ) )
) xx-5x+6 = 1 ) )
2 /
R
3 / :
81 =
4 /
x R :
3x-1+3x+3x ) 39 = 1+
)
12
3x
127
12
12
(243)x-1(27)x-2
(729) x12
5 ( 5 243 )2 = (x- )2 (x+2) = 3
10(x-4)(x-5) = 100 6x -3x-2 = 36 -65x+25x+5 = 0
22x+3-57 = 65(2x-1) 5 (5x+5-x)=26
3 x+1 9 x - 9 3 = 0
2
2
2 2 2
4x + 4 2x( )+34x +2x
= 25
-
51
21
3
35
n n n
55
21
3
3x
2y = 3
3 3 x3 2 y
5 3 12 6 147
4 4 4
x
y
x
y
n
n
n
3 3 ]
:
61 10 2
.
x y = x y .1 .
6 12 = 72 :
5 3 x3 = 4 15x3
.y o =
: 7 = =
7
:
6 147 5 3 12
3 12 = 6 12 2 = 6 144
5 = 6 53 = 6 125
6 147 = 6 147
:
5
-
52
Conjugate Numbers 4 3 ]
.
3 2 3
3 3 3 32 = 3 33 = 3 3 3 32 3
19 = 6 - 25 = ( ) ( )
( 3 2 - 2 5 ) ( 3 2 + 2 5 ) = 9 2 - 4 5 = -2
53 1 ( 3 5 - 1) ( 3 25 + 3 5 + 1) = 3 125 - 1 = 5 - 1 = 4
( )
8
:
+ +
+ +
+ + =
= 2 + 1 + 3 - 2 + 2 - 3 = 3
-
1
2 - 1
1
2 + 3
1
3 + 2
a + b
2 3 3 = 2 3 = 6
5 + 65 - 6
5 + 6
5 2 - 2 3 5 2 + 2 3
5 - 6
3 52 + 3 5 +1
1
2 - 1
2 + 1
2 + 1
1
3 + 2
3 - 2
3 - 2
1
2 + 3
2 - 3
2 - 3
2 + 1
2 - 1
2 - 3
4 - 3
3 - 2
3 - 2
-
53
Real Functions 5 3 ]
: .
R. :
y = f(x) , A,B R y = f(x) , A,B R ) y) f: A B , x A y B f: A B , x A y B f: A B , x A y B
1 5 3 ] :
* :
f(x) = x3 + 2x2 +x-1 , g(x) = x2 5x + 9 , f(x) = 3x - 1 : f(x) = x3 + 2x2 +x-1 , g(x) = x2 5x + 9 , f(x) = 3x - 1 : f(x) = x3 + 2x2 +x-1 , g(x) = x2 5x + 9 , f(x) = 3x - 13) h(x) =
x + 7x2 - 3x
x2 - 3x = 0 x(x-3) = 0 x = 0, x = 3 R\{0, : {3 R
.R f(x) = x3 + 2x2 +x-1 , g(x) = x2 5x + 9 , f(x) = 3x - 1:x : = *
{ R\{x. :1) f(x) = 2x-1
x+5 x + 5 = 0 x = 5 R\{-5} 1 ) f(x) = 2x-1
x+5 x + 5 = 0 x = 5 R\{-5} 1) f(x) =
2x-1x+5
x + 5 = 0 x = 5 R\{-5}
2) g(x) = 2x2 4
x2 -4 = 0 x = m 2 R\{ m {2
2) g(x) = 2x2 4
x2 -4 = 0 x = m 2 R\{ m 2}
2) g(x) = 2x2 4
x2 -4 = 0 x = m 2 R\{ m 2}
3) h(x) = x + 7x2 - 3x
x2 - 3x = 0 x(x-3) = 0 x = 0, x = 3 R\{0, = (h(x (3 {3x + 7
x2 - 3x x2 - 3x = 0 x(x-3) = 0 x = 0, x = 3 R\{0, 3}
3) h(x) = x + 7x2 - 3x
x2 - 3x = 0 x(x-3) = 0 x = 0, x = 3 R\{0, 3} * ( ): x
:
1) f(x) = x-7 , x-7 0 x 7 {x R:x 7}
2) g(x) = 3x+5, 3x + 5 0 x -53
{x R: x -53
}
3) h(x) = 1-2x , 1-2 0 2x -1 x -12
{x R:x -12
}
1) f(x) = x-7 , x-7 0 x 7 {x R:x 7}
2) g(x) = 3x+5, 3x + 5 0 x -53
{x R: x -53
}
3) h(x) = 1-2x , 1-2 0 2x -1 x -12
{x R:x -12
}
f(x) = x-7 (1 , x-7 0 x 7 {x R:x 7}
2) g(x) = 3x+5, 3x + 5 0 x -53
{x R: x -53
}
3) h(x) = 1-2x , 1-2 0 2x -1 x -12
{x R:x -12
}
1) f(x) = x-7 , x-7 0 x 7 {x R:x 7}
2) g(x) = 3x+5, 3x + 5 0 x -53
{x R: x -53
}
3) h(x) = 1-2x , 1-2 0 2x -1 x -12
{x R:x -12
}
1) f(x) = x-7 , x-7 0 x 7 {x R:x 7}
2) g(x) = 3x+5, 3x + 5 0 x -53
{x R: x -53
}
3) h(x) = 1-2x , 1-2 0 2x -1 x -12
{x R:x -12
}
1) f(x) = x-7 , x-7 0 x 7 {x R:x 7}
2) g(x) = 3x+5, 3x + 5 0 x -53
{x R: x -53
}
3) h(x) = 1-2x , 1-2 0 2x -1 x -12
{x R:x -12
}
1) f(x) = x-7 , x-7 0 x 7 {x R:x 7}
2) g(x) = 3x+5, 3x + 5 0 x -53
{x R: x -53
}
3) h(x) = 1-2x , 1-2 0 2x -1 x -12
{x R:x -12
}x
1) f(x) = x-7 , x-7 0 x 7 {x R:x 7}
2) g(x) = 3x+5, 3x + 5 0 x -53
{x R: x -53
}
3) h(x) = 1-2x , 1-2 0 2x -1 x -12
{x R:x -12
}- 12
12
}
1) f(x) = x-7 , x-7 0 x 7 {x R:x 7}
2) g(x) = 3x+5, 3x + 5 0 x -53
{x R: x -53
}
3) h(x) = 1-2x , 1-2 0 2x -1 x -12
{x R:x -12
}
1) f(x) = x-7 , x-7 0 x 7 {x R:x 7}
2) g(x) = 3x+5, 3x + 5 0 x -53
{x R: x -53
}
3) h(x) = 1-2x , 1-2 0 2x -1 x -12
{x R:x -12
}
x R a x a R+ \{1} fa (x)=ax * f1
2
(x) = ( 12
)x , h 5 (x) = ( 5 )x , g3(x)=3x , f2 (x) = 2x :
f(x) = 1x a=1 : 1 =
f(x) = anxn + an-1xn-1 + ...+ a0 an , a1, ... a0 Ran ,an-1,...a0 R
-
54
2 5 3 ] a 0 , a,b R f(x) = ax + b :
f(x) = 2x + 3 , x R :
x 1 0 -1y 5 3 1
a,b R, a 0 , f(x) = ax2 +b : a > 0, b 0 f(x) = 2x2 : 3+
x -1 0 1y 5 3 5
U
a < 0 f(x) = -4x2 : x 1 0 -1y -4 0 -4
I
:
a,b R , a 0, f(x) = ax3 +bf(x) = x3 +2 : :
x 1 0 -1y 3 2 1
f(x) = -x3 : x 1 0 -1y -1 0 1
y
0 x
y
x
y
x
y
x
y
x
y = 2x + 3
f(x) = -x3f(x) = x3 +2
-
55
fa (x)=ax :
:
) 3- 2- 1- 0 1 2 3
.
(f (x :)
.
:
f (x) = 2x (
-3-2-10123x
12482x12
14
18
R Y )
(x , 2 x) = (-x , 2x)
(1 - 3)
(1 - 3)
12
f (x) = 2x x =
f (x)
12
g (x) = f (x) =( )x = (2-1)x = 2- x = f (-x)
Ry : (x, y) = (-x, y)
12g(x)=( )
x
f (x) = 2x
12
)
1 2 3
2 x2 -x
y
x
x
x
x
x
x
x
x
x x xxx
-3 -2 -1
-
56
12
x 1( ) x( )5
13
x 1( )4
x( ) .......
o
y
x
(0,1)
2x3x
4x
12
x( )
13
x( )14
x( )
:
f (x) = ax
1. :
2x 3x 4x 5x ......
:
:
. x ax a > 1 :
. x ax : 0 < 1 0 < < 1 a
a (1 , 0)
. R a 0 ax 2.
( 2 - 3 )
a
a
-
57
(3-3)
1 /
3 /
4/
:
)
)
)
)
) x
5 /
:
6 / : .
2x a - b
18x3
( a - b )5 .
5 - 1
- 15 x - 6 + x
524
( + 3 ) 2
Ans :
32
- 8 27
3x - 2
3 0 = - + x +3
1 + 5 - 5 - 1
5 + 1
3a - b
a - b x
Ans :
13
Ans : 1
a+ 3b
x+ 3x = 8
y =15
x
a+ 3b
x+ 3x = 8
y =15
x
- -1
) a 2- b2
a + b ]
] a - ba + b
a + ba - b
)
y = 43 2 +1, x = 23 +1x y = 3x y =
:
a) f(x) = x2 5+9 b) f(x) = x-1x+9
c) f(x) = x-9
d) f(x) = 3-5x e) f(x) = 1x2 9
2:/ :
a) f(x) = -4x2 + 5 b) f(x) = x - 8 c) f(x) = 2 - x3
-
585858
4 :
[1-4] [2-4]
[3-4] [4-4]
[5-4] [6-4]
[7-4] [8-4]
[9-4]
( B A , B C ) D Q
Sin x x Cos x x Tan x x
( Cos x , Sin x )
:
- -
- -
- -
- -
- -
-
-
59
: Trigonometry :
. .
:
(440 - 362) = (1048 - 973 ) :
: = r
r: a : b : x :
: (388 - 328) = (988 - 940) : 959 .
:
Sin x =2Sin Cos
Sin(x+y)= Sin2x -Sin2x Siny + Sin2y - Sin2y Sin2x
Tanx = Cotx =
Secx = 1+Tan 2x cecx = 1+ Cot2x
1492) ) = ) 899)
( ) 2
6.28318571795865
:
....)
(1617 -
1550 )
.
b Cos xa - Cos x
x2
Sin xCos x
Cos xSin x
x2
2Sin2 = Cos x 1-x2
2 =
-
60
1 4]
( 1 - 4)
Directed Angle : B C B A
B ( B A, B C ) B A
. A B C ( B A B C ) B B C
( 2 - 4 )
:
( 1 - 4 ) .
( 1 - 4 ) ( 2 - 4 )
>
A
C
B
C
A
B
O
Y
XX
( )
Y
( )
XX
L= Q . r
-
61
2 4 ]
Degree Measure : :
360 360
1
: 1 = 60 = 60 1 = 60 = 360
: Radian Measure
.
(3 - 4)
Arc .
( 3 - 4) A O B (L)
A O B L = r r =
= 1 .
A O B (4 - 4) L = 2r
= 2 .
(3 - 4) r :
| Q | .
| Q | = =
A
B
L
r or
B
L
r
o
r
( 4
- 3 )
( 4
- 4 )
Lr
m >m >
A
L= Q . r
-
62
3 4 ]
2r =
=
2 = 360
= 180
1 =
1 = .
: ) = Q
Q =
) = D :
D = ( ) :
: =
.
Lr
2rr
180
180
D
180
180QD
180
| Q | = = 2
Q
-
63
1
A O B 10cm
. 12cm
) A O B :
.
) m A O B :
L = 10 cm r = 12 cm
| Q | = 833 .0 = = = )
) :
| Q | = 833 .0 = = =
Q = - 0. 833 ( ).
2
A O B
=
= D = 180 =
Lr
1012
56
1012
56
34
QD
180
3 4
D34
cm
cm
cm
-
67
4 4]
C (5 - 4)
( 5 - 4 )
(4 - 4)
:
( Q ) ( Sine ) 1.
(Q) ( Cosine ) 2.
(Q) ( Tangent ) 3.
A
B C
Hypot
enuse
Opp
osite
Adjacent
Q
ACAB
OPP.HYP.
ACBC
ABC
m < ABC = Q
AC = = Sin QAB
BCAB
ADJ.HYP.
Cos Q = = BCAB
OPP.ADJ.
tan Q = = ACBC
-
68
AB
AB
(AC)2+(BC)2 = (AB)2
(AB)2
2
Sin2 Q + Cos2 Q = 1
tan Q = ACBC
tan Q = BC AB
AC AB
tan Q = Sin Q
Cos Q
AC
AB
BC
AB
( 5 - 4): ( )
:
= +
( 4 - 4 )
:
(4-4) :
( AB ) :
2 2
-
69
Trigonometric Ratio 5 4]
1 ) 45
(45) (45) . B ABC
AB = BC = L
:
(AC)2 = L2+L2 = 2L2
2 ) 30 60
2L =
= 60
ADBC
CD = DB = L
m BAD = 30
=
A
CB 60L
2L
L
L60
30 30
2L
D
(AC)2 = (AB)2 + (BC)2
AC = 2 L
Sin 45 = = Sin 45= L2L
1
21
2
Cos 45= = Cos 45= L2L
1
21
2
tan 45= = tan 45 = 1LL
1
AD 3 L
Sin 30 = = Sin 30= L2L
1
21
23L
2Lsin 60 = = sin 60= 3
2
3
L
L 1
A
CB
45
L
2
45
L
L
3
2
-
70
:
.
Q
) : (
:
60 , 30 :
60 + 300 = 90
Cos 30= = Cos 30= 3L2L
3
232
Cos 60= = Cos 60= L2L
1
212
tan 30 = = tan 30= L3L
1
31
3tan 60 = = tan 60= 3L
L33
sin 30 = 1
2cos 60 =
3
2cos 30 = sin 60 =
sin (90 -Q ) = cos Q
cos (90 -Q ) = sin Q
90 - Q
* Sin2Q + cos2Q = 1 , tan Q = sincos Q
* sin (90o Q) = cos Q , cos (90o Q) = sinQ
* sin30o = cos 60o = 12
, cos 30o = sin 60o = 32
* sin 45o = cos 45o = 12
* sin Q = , cos Q = , tan Q =
sinQcosQ
-
71
4 ] 6
(5 - 4)
: .
m B O A = Q
B
sin Q = sin
cos Q = cos
(6 - 4)
Trigonometric Point
.
y1
A y
xO
y
B(x,y )
x
:
: sin (180-Q ) = sin Q cos (180-Q ) = -cos Q tan (180-Q ) = -tan Q
x1
B(x,y) = (cosQ,sin Q)
B(x,y) O B
Q
Q= Y
Q= X
-
72
Q AOB B
. x = = (x , y)
7
tan Q
0 90 180 .
( 6 - 4 ) :
tan 0 = = tan0 = 0
(cos 90 , sin 90 ) = (0,1 ) cos 90= 0 sin 90=1
= tan90
(cos 180 , sin 180) = ( -1, 0) cos 180= -1 sin 180= 0
tan 180= 0
(6 - 4)
sin0cos0
01
sin90cos90
(
y sin Qcos Q
cos Qsin Q Q = 0 90 180
(cos 0 , sin 0) = (1 , 0) cos 0 = 1
sin 0 = 0
C (-1,0)
Y
(cos0 , sin0)=(1,0)
X
B (cos90, sin90)=(0,1)
O
A
=0
-
73
7 4]
[1-7-4] :
.
A C A
C A ( Angle of Elevation C
A) CAB (7 - 4).
C
C A
( C A
Angle of Depression A C )
A C D ( 7 - 4) .
(7 - 4) 8
30m ( )
45 .
= L
. B A B C
(8 - 4)
A
D
A C
C
B
12
OPP.Hyp.
L30sin 45 = =
302
L= = 21.21m
30m
A
C B
45
45
L
A C
-
74
C
9
8m 60
: B tan 60=
= 3
3 8 = .
(9 - 4)
10
2350m 70
.sin 700 = 0.9396
=
B ABC
sin 70 =
= 0.9396
(10 - 4)
AB8C
A
B60
ABAC
2350AC
OPP.ADJ.
AB
A B C
(
A
B70
C
70
2350
m
23500.9396
AC = 2500m
C
-
75
11
7 60
30 .
DAC = AC B
[ = ]
: B ABC
tan 30 =
. (11 - 4)
: D E A D tan 60 =
X = 21 m = 3
X+ 7 =
28m =21+7=
7Y
13
7Y
X 7 3
XY
A
C
E
X
D
7
Y6030
B 30Y
7m
Y = 7 3 =
-
76
D
xy
12
30
1000 45 .
: B ABC
tan45=
= 1
x = y . . . . . 1
tan 30 = . . . . .
3 y = y + 1000
1.7 y - y = 1000
y =
x = 1429 .
xy + 1000
13
D
tan 30 =
yy + 1000
10000.7
A
BC
X
Y301000
45
45
xy
2
(12 - 4 ) =
= 1428.6
-
77
: Circular Sector 2-7-4] ]
(7 - 4)
.
(13-4) AOB Central Angle
180.
AB =
= . . . . . 1
Q =
L = Q r = Q
1 :
= . . . . . 2 ( 13 - 4 )
= 1 :
2 : = =
= D .
= =
=
12
12
Lr
12
AB
C
O
rrQ
L
12 Qr
2
r2
Q2
Q2
D360
Q2
D360
360
:
= r + r + L = 2 r + L L r
.
)L r
Q r 2
2
r
12 ( 2 ) r
2 = r 2
-
78
13
.8cm 60
Q r2 =
= 64
= 64
: =
64 =
= 14. 3 64
14
15cm2 6cm
.
L r = 1 -
r + L = 2 -
= 2 5 + 6 = 16
Q | - 3 | = Q = = 1.2
= =
D = = 7898 . 68
12
12
60 180
12
3.14 3
D36060360
16
12
12
Lr
65
QD
180
1.2D
3.14180
180 1.23.14
33.49 cm2
15 = 6 r r =5
cm
2
33 . 49 cm2
-
79
: Circular Segment 3-7-4] ]
(8 - 4)
.
AOB (14 - 4)
180
:
(14 -4)
Q
OAB - ( OACB) = ACB
Q r2 = (OACB (
OA OB sin Q = OAB
r r sin Q = OAB
r2 sin Q = AC -
Q r .
12
)
)
12
12
12
12
12
A B
C
O
rrQ
r2 (Q - sin Q) = ACB
Q r2
-
80
15
12cm 30.
Q = 0.5236 = =
=
144 ( 0.5236 -0.5 ) =
1.7cm 2 = 144 (0.0236) =
16
O 6cm 6cm cm2
.
AOB (15 -4)
= = Q = = = 1.047
=
=
18 ( 1.047 -0.865) =
3.276cm2 = 18 (0.182) =
180
12
QD
Q30
12
12
180
180
QD
180
Q60
3
22 21
12
= = = 1.047
B
O6cm
60
6cm
(r 2 ( Q - sin 30
m AOB = 60
12
(r2 (Q - sinQ
36
A
(1.047-sin60 )
-
81
( 2 - 4 )
1 /
70 50
tan 700 = 2. 8 tan 500 = 1.2 30 m
14.28m/
2 /
(50m) 300
28.9m /
3 /
.3.2cm 8 cm
12.8cm2/
4 /
.10cm 100
87.3cm2 /
5 /
37.68cm2 6cm .
12.56cm /
6 /
10cm. 450.
3.98cm2 /
7 /
.8cm 60
5.81cm2 /
-
82
84]
[2-4] :
DEG
(DEGREE).
RAD ( RADIAN) .
DRG DEG RAD .
.
.(sine) sin
.(cosine) cos
.(tangent) tan
. (DRG) (RAD) (DEG)
.
.
:
17
(1) * : DEG .
* 30.
* (sin) = 0.5 .
.1
.2
.3
:
(1) sin 30 (2) cos 120 (3) tan 350
sin( Q) = - sin Q
cos (- Q) = cos Q
tan (- Q) = - tan Q
-
83
DEG 2) * : )
* 120.
* (cos) = 0.5
DEG 3) * : )
* 350 (tan) ~ 1763 . 0
.( ) ( )
18
RAD : 2ndf
( . . . ) .
: = (1)
RAD *
* 2ndf 3.141592654 5 = 15.70796327
4 = 3.926990817 0.707106781 -
(2)
cos( - Q) = cos Q ( ).
.RAD *
* 2ndf = 3.141592654 3= 424777961 .9
cos
- 1 =
-
-
tan - 350 ~ - 0.1763
5 4
(2) sin 7(1) 5
tan 3) ) )cos (- 3
5 4
sin
= sin
)cos (-3
tan ( - Q ) = - tan Q
INV
-
84
(3)
.RAD *
* 2ndf = 3.141592654 7 = 9114858. 21
5 = 398229715. 4 = 3.07763537 .
:
(1) (2) (400-)
0.5 (1)
0.766044443 (2)
- 0.267949192 (3)
- 0.588 (4)
- 0.5 (5)
-3.077683537 (6)
6
7 5
tan
tan
sincostan (-15 ) (3)tan (-36 ) (4)
2 3
cos (5) 8 5
tan (6)
-
85
Solution of Right Angled Triangle 94]
] [
.
19
tan 22 = 0. 4 :
sin 22 cos 22 (1)
cos 68 sin 68 (2)
tan 22 = = =
2k =
5k =
. . . . . .
4K2 + 25K2 = (Ac)2
AC = 29 K
(1) = = = 22
= = =22
68 = (90 -22) = 22 = (2)
25
410
BCAC
ABAC
( AB )2 + ( BC )2 =( AC )2
2k29k
5k29k
229
529
sin
cos
cossin
5 KB
C
A
68
22
2 K
sin
cos 68 = cos( 90 - 22) = sin 22= 229
29 k
529
-
86
K 2
20
tan C sinA, .B ABC cos C =
.cosA
: B ABC
cos C = =
()
K 2 = (AB)2 + 25
(AB)2= 144 K 2 AB = 12K
tan C = =
cosA = =
21
ABC A AB = 7 cm AC = 24 cm :
sin C sin B tan C cos B
( BC )2 =( A B)2 +(A C)2
(B C )2= (7) 2 +(24) 2 = 49 + 576 = 625
BC = 25 cm
513
5k13k
12k5k
125
12k13k
1213
(AC) 2 = (AB)2 + (BC) 2
169
C
A
B
12K13K
5K
5k13k
513
sin A=
=
C
B
A
25cm
24cm
7cm
-
87
sin C = sin B =
tan C= cos B =
22
AC = 6 cm AB = 3 cm . B ABC
(AC)2 = (AB)2 + (BC)2
36 = 9 + (BC)2
BC = 3 3
tan C = =
m 0 A .
(13 - 5) K < 0 () K A A
. || A || K
. A A K
(13 - 5)
A = (x , y) K A = A K = ( K x , K y )
A = ( x , y )
KyKx
Ky
X
Y
Kx
yX
,Ky)(Kx
,y)(x
k > 0
k < 0
,
-
103
8
2C C -3C
2C = 2(3,-1) = (6 , -2 )
C = (3,-1) = ( , )
-3C = -3 (3 , -1) = (-9,3)
9
K = 3 L = -2
(1) A+ B = ( 3 + 4 , -2 + 3) = (7 , 1)
(2) K A = 3 (3 , -2) = (9 , -6)
(3)L B = -2 (4 , 3) = ( -8 , -6 )
(4) K A + L B = (9 , - 6) + ( -8 , -6 )
= (1 , -12)
445]
(1) : A B K :
K ( A+ B) = K A+ K B
(2) : A K L R : (K L) A = K ( L A) = L (K A)
C = (3, -1)12
32
12
-12
A = (3 , -2) B = (4 , 3)(1) A + B (2) K A (3) L B (4) K A + L B
( A + B) K = A K + B K
12
-
104
10
(3) : A B K R K K A = K B A= B .
A = A 1 = A (4)
A = A 0 = 0 (5)
545]
(7 - 5) A B A - B
10
A - B
A - B = A + (-B) = (3 , 4) + (1 , -3) = ( 4 , 1)
:
: A - B
.B A
(14 - 5)
.B
A + (- B)
A = (3 , 4) B = (-1 , 3)
(3 ,4)
(4 ,1)
(1 ,-3)
(-1 ,3)
B A
A-B
B
X
Y
-
-
105
11
KA - LB ( 2 ) A - B (1)
(1) A - B = (2 , 3)-(-2 , -1)
= (2 , 3)+(2 , 1)=(4 , 4)
(15 -5) :
( 15 - 5 )
(2) K A- L B = 2 (2 ,3) - (-1)(-2,-1)
= (4 , 6) + (-2,- 1)
= (2 , 5)
:
( 16 - 5 )
- B B A A
B = B - 1-
A+ B B
.
A = (2 , 3) B = (-2 , -1) K = 2 L = -1
2
2
Y
X
(4 ,4)(2 ,3)
(2 ,1)
(-2 , -1)
A
B
(4 ,6)
(2 ,5)
Y
XB
(-2 ,-1)
KA
KA
-L B
-
106
55]
Unit Vector 1 5 5 ]
(8 - 5)
U 1(1)
.
U 2(2)
.
:
C = (x , 0) + (0 , y)
C = x (1 , 0) + y ( 0 , 1)
U 1 U
2
C C = x U
1 + y U
2
U : 1 U
2 (0 , 9 ) (0 , 3-) (2- , 0) (6 , 0)
9U1 (-3 , 0) = -3U
1 (0 ,-2) = -2U
2 (0 , 6) = 6 U
2
12
A + B .
A + B = ( 4 , 7 ) + ( -5 , 3 ) = ( -1, 10 ) = -(1, 0 ) + 10(0 , 1) = - U 1 + 10 U
2
U 1 = (1 , 0 )
U 2 = (0 , 1)
C = (x , y)
A = (4 , 7) B = (-5 , 3)
(9,0)=
-
107
U :1 U
2
(2, 5) = 2U1 + 5U
2
(-4, 2) = -4 U 1 + 2U
2
( -2 , -3 ) = -2 U1 - 3U
2
A = 4 U 1 + 5U
2
. B=- 2U 1 + 3U
2
13
A+ B A = U1 - 3U
2 B = 2U
1 + U
2
A+ B = ( U1 - 3U
2) + (2U
1 + U
2) = U
1 (1+2) + U
2 (-3 + 1) = 3U
1 - 2U
2
14
K= 2 L = 3 K A - L B
.
K A - L B = 2 (5, -3) - 3 (-3, 4)
= (10,-6) + (9 , -12)
= (19 ,-18)
= 19 U1 - 18 U
2
A = (4,5)
B = (-2 , 3)
A = (5 , -3) B = (-3,4)
=(3,-2)
-
108
( 2 - 5 )
1 / :
(-2 , -2) (3, 0) 3 U 1 + U
2 -U
1 - 2U
2
2 / :
4(1 ,-1) 2(1 , -1) -7(1,5) 3 (2,-1 )+ 4(-1 , 5) 7(3U1 + 2U
2) -4(2U
1-U
2
3 /:U
1 U
2
(-1 ,4) (-3 , -5) (0 , -1) ( 5 , 3) (2 , 0) (2 , 3)
4 / x y R A
A + E = E + A = A
5 /A = -B
6 / A = ( 3 , 1) B = ( 2 , 3) K = 3 L = -2
:
7 /U
1 U
2 6
8 /U
1 U
2
( ) 3 () 10
() 5 ( )
9 /. 2A + 3x = 5B : x
E = (0 , 0)
K B L A A +B K A + B K A - B K A + L B
K A - L B K (A + B) (L + K) A (L + K) (A + B)
K (L A + K B) K L ( A - B)
A + B = B + A = (0,0)
A = (5 , 2) B = (2 ,-4)
E = (x , y)
34
34
6
)
-
109109109
6 : [1-6] .
[2-6] . [3-6] ( ) .
[4-6] . [5-6] [6-6] .
[7-6] . [8-6] .
n1
n2
n1x
2+n
2x
1n
1+n
2
n1y
2+n
2y
1n
1+n
2
|ax1 + by
1 +c|
a2+b2
L = (x-x
1)2 + (y
2_y
1)2
( , )
L1 // L
2 m
1 = m
2L
1,L
2
L1
L2
m1 m
2 = -1 L
1,L
2
ax + by + c = 0
D = D
:
- -
- -
- -
- -
-
-
110
Analgtic Geometrey : :
16]
(O) x x y y
(R) O
x x y y
A
A B A C (1 - 6)
.
.
(1 - 6 )
A ( 3 , 2)
Y
X
A( 3,2)C( 0 ,2)
O( 0 ,0) B( 3 ,0)
-
111
Distance Between Two Points 2 6]
:
:
C :A B C
+ = L2 ............
L = ( x2- x
1 ) 2+(y
2- y
1)2
.
:
A B = B -A
A B = (x2 , y
2) - ( x
1 , y
1)
x) = (2 - 6)2 -x
1 y
2 - y
1)
..
1
.
:
A B = B -A = (-3 , 4) - (-2, 7) = (-1 , -3)
AC = C -A = (1 , 16) - (-2 , 7) = (3 , 9) = -3 (-1 ,-3)
A C = -3A B
A B C .
:
............
A (-2 , 7) B (-3, 4) C (1, 16)
|| A B ||= ( x2- x
1 ) 2+(y
2- y
1)2
A (x1 , y
1) B ( x
2 , y
2)
= (AC)2 ( BC)2
B( x2,y
2)
L
A( x1,y
1)
y2-y
1
x2-x
1 C
O
y
x
-
112
:
AB = ( -2 + 3 )2 + ( 7 - 4 )2 = 1 + 9 = 10
BC = ( -3 - 1 )2 + ( 4 - 16 )2 = 16 + 144 = 160 = 4 10
AC = ( -2-1)2 + (7-16)2 = 9 +81 = 90 = 3 10
BC = AB + AC
A B C
.
2
AB = ( 2- 1)2+(2-1)2 = 1+1 = 2
AC = ( 5- 1)2+(-1-1)2 = 16+4 = 20
BC = ( 5- 2)2+(-1-2)2 = 9+9 = 18
AC2 = AB2 + BC2 : 2 ( 18 ) + 2 ( 2 ) = 2 ( 20 )
18 + 2 = 20
. B ABC
A (1 , 1) B (2 , 2) C (5 , -1)
-
113
3
.
AB = ( -3 -1 )2 + ( -1 + 4 )2 = 16 + 9 = 25 =5
BC= ( 1 -10 )2 + ( -4 + 5 ) 2 = 81 +1 = 82
CD = ( 10 - 6 )2 + ( -5 + 2 )2 = 16 + 9 = 25 =5
AD = ( 6 + 3 )2 + (-2 +1 )2 = 81 + 1 = 82
AB = CD BC = AD
ABCD ( ).
4
. a R AB=AC
AB=AC
( )
A(-3, -1) B (1 , -4) C (10, -5) D (6 , -2)
C(4, 1) B (a,1) A (3, 2a)
B
A
C
3 a( )2 + 2a 1( )2 = 3 4( )2 + 2a 1( )2
3 a( )2 + 2a 1( )2 =1+ 2a 1( )2
3 a( )2 =1 3 a = 1 : 3 a =1 a = 2 : 3 a = 1 a = 4
(3 , 2a)
(a,1) (4,1)
-
114
( 1 - 6 )
1 /
:
) (4 , 3) (0 , 0). ) (4 , 6) (2 , 1) .
) (5 - , 3- ) ( 1- , 5 ) ) (4 , 1-) (3 , 2-).
2 /
.A(5 , 7) B( 1, 10) C (-3 , -8)
3 /
.
4 /
(A (3 , -2) B ( -5 , 0) C (0 , -7) D (8 ,-9
.
5 /
ABCD
.D
6 /
.
7 /
(0 , 0) (8 , 6) (4- , 3-) .
A (2 , 3) B (-1 , -1) C (3 , -4)
A(-2 , 5) B (3 , 3) C (-4 , 2)
A(4 , -3) B( 7 , 10) C (-8 ,2) D (-1 , -5)
-
115
36] ( )
.
C A B
n : 1 : n
2
=
X = Y =
C ( , )
4
x = 1 = = =
y = 2- = = =
(2- , 1 )
X = X = X
A = (x1 , y
1) B = (x
2 , y
2)
C = (x , y)
n1x
2+n
2x
1n
1+n
2
n1y
2+n
2y
1n
1+n
2
n1
n2
ACCB
n1x
2+n
2x
1n
1+n
2
n1y
2+n
2y
1n
1+n
2
12
A (4 , -3 ) B (-5, 0)
1 (-5)+2(4)1+2
n1x
2+n
2x
1n
1+n
2
n1y
2+n
2y
1n
1+n
2
1 (0)+2(-3)1+2
-63
-5+83
B(x 2
,y 2)
C(x ,y)
A(x 1 ,y 1)
n2
n1
-
116
:
AB M
( , ) = M
n .1 = n
2 = n
5
C AB
C
C = ( , )
( , ) =
C = ( 2 , -3 )
x1+x
22
A(-3 , 2) B (7 , -8)
A ( x1 , y
1) B ( (x
2 , y
2)
y1+y
22
x1+x
22
y1+y
22
2 + (-8)2
-3+72
B(x 2,y
2) A(x
1,y
1 (C(x,y (
n1 :n2
(x1 + x22
, y1 + y22
) :
C( n1 + n2x1n1 + n2
, n1y2 + n2y1n1 + n2
)C(n1 + n2x1n1 + n2
, n1y2 + n2y1n1 + n2
)C(n1 + n2x1n1 + n2
, n1y2 + n2y1n1 + n2
)x2C( n1 + n2x1n1 + n2
, n1y2 + n2y1n1 + n2
)C( n1 + n2x1n1 + n2
, n1y2 + n2y1n1 + n2
)C( n1 + n2x1n1 + n2
, n1y2 + n2y1n1 + n2
)
-
117
A (2 , 1) B (1, -3)
( 2 - 6 )
1 /
A B
.
2 /
. AB
3 /
AB C
4 /
C A B
B (4 , -4)
5 /
A B C :
.
6 /
(8- , 5-) ( 3- , 3- ) (3 , 1) (2-, 1-)
.
A(4 , 0) B (5 , 2) C (2 , -3)
A (2 , -4) B (-3 , -6)
A (1 , 3) B (4 , 6)
A(2 , 6)
21
35
-
118
Slope of The Line 4 6]
(1 - 6)
B (x 2 , y
2) A ( x
1 , y
1 (
. x
1 x
2 AB =
:
AB = y2 - y
11) 0 =
AB // .
= = .
AB x2 - x
12) 0 =
AB // .
= .
3) Q AB
. Q [ 0 , 180 ) / { 90 } tanQ
6
A (2 , 3) B (5 , 1)
m AB = = =
y2 - y
1x
2 - x
1
y2 - y
1
x2 - x
1
1 - 35 - 2
- 2 3
AB
-
119
Parallel Condition 5 6]
. m
1 = m
2L
1 // L
2
7
( A(4 , 3) B (2 , 1) C (1 , 0 .
m AB = = =
m BC = = =
m AB = mBC
C B A .
Perpendicular Condition 6 6]
L 1 L
2 = 1-
m
1 m
2 = -1
m
1 =
=
= .
1
1
- 2 - 2
- 1 - 1
1 - 32 - 4
0 - 31 - 2
- 1 m
2 - 3
4 4 - 3
3 4
-
120
8
(A (3 , -1) B (10, 4) C (5 , 11
B
mAB = = , mBC = =
mAB mBC = 1- =
AB BC
. B ABC
9
(C (-2 , b-4) B (-1, 2) A (0 , b
. b R
A,B,C
:
4 - (-1)5 - 10
5 7 10 - 3
11 - 4 7 -5
7 7 5
-5
mAB =mBC
2b1o
=b 4( ) 221
2b1
=b 61
2b= b6
b= 4
m=VyVx
mAB =mBC
2b1o
=b 4( ) 221
2b1
=b 61
2b= b6
b= 4
m=VyVx
-
121
( 3 - 6 )
1 /
(1) ( 0 , 2) (2- , 0).
(2) (6 , 7- ) (4 , 1-) , (3 , 2) .
AB = h A(2 , 3) B ( -3 , h )3) )
A(1 , 6) B (-2 , -8) C (7 , -2)
(ABC (4
. B ABC
2 /
:
1) H L H (3 , 2) (5 , 1) L
) ) 2- ) )
2) H L H (2- , 3) (2 , 3- ) L
) ) - ) )
(3 , 4) (x , 6) H (-1 , 3) (-1 , 5) L 3)
x ) 3- ) 3 ) 1 ) .
1 2
1 2
2 3 3
- 2
3 2
2 3 3
- 2 -3 2
L H
m
-
122
3 /
1) (A(5, 2) B (-2 , 1) C (2 , -2 .
2) (A(-1, 5) B (5 , 1) C (6 ,-2) D(0 , 2 ABCD .
3) (A(5 , 2) B (2 ,-1) C (-1 , 2) D (2 , 5 ABCD .
ABC (4 (A (2 , 4) B (6 , 0) C (-2 , -3 :
. BC A )
. AC B )
5) ( A (-2 , 2) B (2 , -2) C (4 , 2) D (2 , 4
.
6) x (x , 4) (-2 , -9)
(3 , 0) (1 , 4).
-
123
Equation of The Line 76]
x y ( x , y)
.
a x + b y + c = 0 :
x = 0 1.
y =
x = y = 0
2. b = 0 ax + c = 0
x = 0 .
3. a = 0 b y + c = 0
y = 0 .
4. c = 0 ax + by = 0 .
: .1
: A (x1, y
1) B (x
2, y
2 AB (
C (x , y ) AB :
= .
2. :
m =
y y . . . . . . . . . . . . . .1 = m (x x
1 )
-c b
-c a
y - y1
x - x1
y2 - y
1
x2 - x
1
y2 - y
1
x2 - x
1
-
124
9
(5 , 4) (3- , 2).
=
=
=
= 4x - 8
4x y 11 = 0 ......... .
10
(0,3) (7,1) (4 , 3)
=
2x -14 = -7y + 7
2x + 7y ....... . 21 = 0
. (4 ,3) x = 3 y = 4
2( 3) + 7( 4 ) - 21 = 0
6 + 28 - 21 = 0
y + 3 x - 2
y - y1
x - x1
y2 - y
1
x2 - x
1
4 - 25 + 3
1 4 y + 3
x - 2
3 - 10 - 7
y - 1 x - 7
y+3
??
(4,5 )
Y
X
(2, -3)
(3,4)
Y
X
-
125
13 0
(4 ,3) .
11
(3- , 1) .
y - y1 = m (x - x
1)
y + 3 = (x - 1 )
2y + 6 = x - 1
x - 2 y - 7 = 0 . . . . . . . . .
12
A(-2 , 5 (
. B (4, -1) C (-2 ,3 )
D = ( , ) = B C D
AD : =
3y 15 = -4x - 8
4x + 3 y 7 = 0 . . . . . . . . .
. . . . . . . . .
D = ( , ) =
2
1 2
1 2
1 - 5
-1 + 32
4 + (-2) (1,1)
1 + 2y - 5 x + 2
Y
X
(1 ,-3 )
D X
A
C
B
Y
-
126
13
(5 ,3-) .
. O (0 ,0) A (-3 ,5 )
O A : =
=
5x + 3y = 0 ........ .
:
ax + by + c = 0 :
= x y
b 0
= -
=
(5 ,3-) .
y - 0 : = x - 0
5 - 0-3 - 0
y 5 -3 x
y x
-a b
x -
y
Y
X
A(-3,5)
O
-
127
14
:
3x 4 y 12 = 0
m = = =
:
15
150 (4-, 1) .
y - y1 = m (x - x
1)
y + 4 = -1 (x - 1)
x + 3 y + 4 3 1 = 0 . . . . . . . . . .
-a -3 3b -4 4
x = 0
:
m = tan 150
= tan (180-30)
= - tan 30
m = - 13
3
-4y - 12 =0 y=-3
-
128
-a -2 2
b -3 3 -3
2
-3
2
16
(2,1-)
2x - 3y -7 = 0
m = : = =
= ( ) .
y y1 = m (x x
1)
y 1 = (x + 2)
x + 2 y + 4 = 0 3 ......
2x - 3y -7 = 0
1)
2) 0 =
3)
4)
5) m
x2 ,y2( ), x1,y1( )m= y2 -y1x2 -x1ax+by+c
m = tan
y-y1x-x1
= y2 -y1x2 -x1
x2 ,y2( ), x1,y1( )
x2 ,y2( ), x1,y1( )y-y1 =m x-x1( )
m= ab
m= ab
-
129
( 4 - 6 )
1 /1. = ( 0 , 4 -) .
2. (1- , 2) .
3. (1- , 2 ) .
4. (3 ,1-) (5 ,1-) .
L = .15. L (1-, 2)
6. (2-, 0) = .
7. (4- , 3 ) (2- ,2)
(0 ,3)
AB 8.
2 /1. = 3- 7
2. = 2 6
3. :
L . 1: 2 x -3y + 5 = 0
L . 2: 8 y = 4x + 16
L . 3: 3 y = -4
4. (5- , 2 ) :
2 x - y + 3 = 0
- 12
A (4,-2) B (1, 2)
23
-35
-
130
5. L 4
. 2y = 4 x -1
6. L : x + y -2= 0
L
7. L (2- ,2)
x + y = 0 L .
H : 3x + 6y = -3 L : 2x - y = 3 8.
L H .
. L H .
9. 135
.
10. (2 ,1) :
a R
L 2 : a+1( )+y = 2L1 :2y = ax+6
)
L )
)
L : 2y = ax +1
-
131
8 6 ]
(2 - 6)
L: ax +b y + c = 0 N
L (D) N L :
= D . . .
17
. 2y + x = 2 :
x + 2y 2 =0 a = 1, b = 2, c = -2 :
= =
L :1: a
1x + b
1 y + c
1 = 0 L
2:a
2 x + b
2 y + c
2 = 0
= L1 L
2
N (x1,y
1)
A(1,3)
| a x1 + b y
1 +c |
a2 + b2| (1)(1)+(2)(3)-2|
(1)2+(2)2
55
D = = 5 unit
| C2-C
1|
a2+b2
| a x1 + b y
1 +c |
a2 + b2
L:ax+by+c=0
D=NM
N (x 1
,y 1)
M
D
X
Y
0
-
132
18
:
L1: x - 3y = 1 L
2: x - 3y = 4
.
L1: y = 0 x = 1 :
(0 ,1)
D =
D = =
:
D = =
:
L
| (1)(1) - 3(0)- 4|
1 + 9
| a x1 + b y
1 + c|
a2 + b2
| 4-1 |
1 + 9
310
310
L1
L2
D
-
133
19
:AB
=
= =
3 x 2 y + 1 = 0
ABC AB
=
AB = (3-1)2 + (5-2)2 = 4 + 9 = 13
Area = (AB) . D
unit2 = . (13 ) =
:AB
y2 - y
1
x2 - x
1
A (1,2) B (3 ,5) C ( 1 ,3)
y - y1
x - x1
C (-1,3)
D = 813
32
y - 2x -1
5 - 23 -1
y - 2x -1
| 3(-1) -2(3)+1 |
9 + 4
4
12
12
813
-
unit
X
Y
B
C D
A
-
134
( 5 - 6 )
1 /
() () :
1. : y = 3 3 .
2. : y = -5 5 .
3. : x = -5 5 .
4. : y = 4 y = -1 3 .
2 /
6x + 8y 21 = 0 :1. (1, 2-)
2. =
4 .
3. :
L1: 8x - 6y + 4 = 0
L2: 4x - 3y - 1 = 0
. 4. (2-, 0)
. 5. ABC
13
A(-4, 6) B(-3, -1) C (5, -2)
A (1,-1) B (3, 5)
7 Statistics : [1-7] .
[2-7] .
[3-7] .
[4-7] .
[5-7]
X
ME
MO
R
S
r
-
135135135
7 Statistics : [1-7] .
[2-7] .
[3-7] .
[4-7] .
[5-7]
X
ME
MO
R
S
r
- :
-
-
-
-
-
-
-
-
-
-
-
136
: Statistics :
Measures of Central Tendency 1 7 ]
.
.
.
:
.
.
.
. .
-
137
Arithmatic Mean 2 7 ]
( 1 7)
.
.
:
1) ( ) :
=
:
1
: 12,11,9,8,5
.
=
9
x1 + x
2 + x
3 + .......... + x
n
nX =
X =x
1 + x
2 + x
3 + .......... + x
n
n12
+ 11
+ 9
+ 8+5
5455
=
-
138
2) :
:
=
2
(3) 8 (5) 9 (4)
11 12 :
( ) ( )
.
x
f
:
9.786 = ( )
.
( ) ( )
121198
2453
(x) (f) (x f)
8 3 8 3 = 24
9 5 9 5 = 45
11 4 11 4 = 44
12 2 12 2 = 24
14 137
X =x
1 f1+ x
2 f2+ x
3 f3+ .......... + x
nfn
f1+f
2+........+f
n
X =137
14
-
139
3
.
(x) : = = 35
= 35 + 10 = 45........... .
:
.(x) 1)
(f) (x) 2)
3) :
61.1
) :
(f) (x) x f
30- 9 35 315
40- 15 45 675
50- 22 55 1210
60- 25 65 1625
70- 18 75 1350
80- 90 11 85 935
100 6110
X =
X =
6110
100
30 + 402
-30-40-50-60-9070-80
10011182522159
x1 f1+ x
2 f2+ x
3 f3+ .......... + x
nfn
f1+f
2+........+f
n
X =
-
140
4
:
13.3
:
( )
:
= +
X 0 X =
0 +
X-X = f = . 0 f = ,
-8-10-12-14-2016-18
60461020155
(f) (x) x f
8- 5 9 45
10- 15 11 165
12- 20 13 260
14- 10 15 150
16- 6 17 102
18- 20 4 19 76
60 798
X =798
60X =
( )
f . E f
X =
E
-
141
5
100 .
1) .
) (21) . 2) (
- = ) (3
).
4) (f) .
5) (f.E)
:
(f)
(X)
f.E
18- 20 19 19 - 21 = -2 20 -2 = -40
20- 44 21=X0
21 - 21 = 0 44 0 = 0
22- 18 23 23 - 21 = 2 18 2 = 36
24- 13 25 25 - 21 = 4 13 4 = 52
26- 3 27 27 - 21 = 6 3 6 = 18
28- 30 2 29 29 - 21 = 8 2 8 = 16
100 82
X 0
= X-X0 E
= X-X0 E
f . E f
X = X 0 +
82100
X = 0.82 + 21 = + 21
X = 21 .82
) .
) (X ) (X) ( ) (21) .
18 20 22 24 26 28-30
20 44 18 13 3 2 100
-
142
:
:
(1) .
(2) .
:
(1) .
(2) .
Median 3 7]
(7-2)
.
:
) :
.
.
.
-
143
6
: 52 58 50 63 55 .
:
=55
7
57 63 50 58 52 :
55.
:
= = = 3 ()
= + 1 = 3 + 1 = 4 ()
= = = 56
50 52 55 63
50 52 58 63
62
n2n2
+ 2
57 + 552
58
55 57
-
144
30-99
40-1524
50-2246
60-2571
70-1889
80 - 9011100
100
+
+
+
+
+
-
2
2 :
: :
1) .
2) =
3)
.
-
= +
W . + ME = L = fb ME
fm : W: L:
.
8
:
= = 50
= ( 70 - 60 )
ME = L + . W
ME = 60 + 10
ME = 60 + 1.6 = 61.6 + 60 =
f 2 - f b
f m
100 2
f 2 - f b
f m 50 46 -
25 8 5
-
145
: :
(1)
(2) .
:
(1) .
(2) .
Mode 47]
(3 - 7) .
1)
9
:
) 4 7 9 4 3 8 7 4 2 4
= 4 .
) 6 5 1 8 6 5 10 18
= 5 6 .
) 8 5 4 3 7 10 11 12
=
MO
-
146
2 ) : ) ( ) :
= +
d1 = - .
d2 = - .
. .
10
:
d1 = 25 - 22 = 3
d2 = 25 - 18 = 7
= 60 - 70 = 10
= +
= 60 + 10
= 60 + 3
= 63
) () :
1) .
=
x =2)
3) ( = ).
4) x .
= +
9 30-
15 40-
22 50-
25 60-
18 70-
11 80-90
d1d1 + d2
d1d1+ d2
3 + 7 3
-
147
11
:
= (70-60)
= = 10
=
(10 -x) (37)= x (38)
370 - 37x = 38x
75x = 370
x = 4.9 =
= 60 + 4.9 = 64.9
:
:
(1)
(2) .
:
(1)
(2) .
(3)
-10080-70-60-50-40- 90
283759386
= 37
= 38
= = 10 10 -x x
370 75
-
148
( 1 - 7 )
1 / .2 / . 15 17 16 18 16 15 17 18 17
19 : ) ) )
3 / (40000) .
90 / 4 :
: ) . ) . ) .
5 / 60 :
6 /
( ) :
-20-24-28-32-36-4840-44
9079152318108
-170-160-150 -200-210190-180
372015105
-24-2820-16-12-8-4
6122015108
-
149
Measusres of Variation 57]
.
: 30 40 50 60 70 50
: 10 20 90 100 30 50
.
: :
. Range 1 -
. Standard Deviation 2 -
157] .
.
) :
12
: 12 35 68 24 98
R = 98 - 12 = 86
-
150
) :
13
:
= - 55-5
R = 50
257]
. n
. x x 1 x
2 .... xn
x 1 x
2 x x , ....,
.
.
(7-4)
:
. (S)
:
S = - ( x )2
-5 -45-5535-25-15
7141583
x2 n
-
151
:
(15) :
x2 n
25 5
x1 + x
2 ...
+ x
n
n
165 5
20 5 5
8 + 6 + 4 + 2 + 0
x2 n 120 5
14
: 1 3 5 7 9
X = 5 = =
S = - ( x )2
S = 25 - 33 = 25 -
2 2 = 8 ....
15
1 1 3 5 7 9 .
(14)
1 3 5 7 9
1 : 0 2 4 6 8
X = 4 = =
S = - ( x )2
S = 16 - 24 = 16 -
2 2 = 8 ....
(1) (14)
x x2
1 1
3 9
5 25
7 49
9 81
25 165
x x2
0 0
2 4
4 16
6 36
8 64
20 120
-
152
: Standard Degree (4 - 7)
:
.
:
Correlation 3 5 7 ]
(5 - 7) :
.
: x y ( ) Correlation Cofficient
: r
x = x
y = y
x =S x
y =S y
: (r) (r (1 () .
(r=1 (2 .
(r (3 () .
(r= -1 (4 .
(r= 0 (5 .
r
1 1
.
+-
[ -1 , 1 ]
r= x y
n- x y
Sx S
y
X - X S
SD=
r
-
153
16
x y x=5 :
S x =
S y =
x yy2x2yx
24121
816442
1836963
32641684
5010025105
110220553015
54321x
108642y
155
X =
Y = 305
555 -9 = 2
2205
-36 = 8 = 2 2
=3
=6
r = = x y
n- x y
Sx S
y
= 1= 44
- (3)(6)( 2) (2 2)
22 - 18 4
r=
110
5
SD =x xSx
SD =5 3
2=
22= 2
-
154
17
x y :
1 =
3 =
S x =
S y =
r = =
=r
741-2-5x
-30369y
55
X =
Y = 155
955 - (1)
2 = 19 -1 = 18 = 3 2
1355 - (1)
2 = 27 -9 = 18 = 3 2
x yn
- x y
Sx S
y
-7515 - (1) (3)
3 2 3 2
= -1=-15 -3 -18
(9) (2) (18)
xyy2x2yx
-4581259- 5
-123646-2
39131
001604
-21949-37
-7513595155
-
155
( 2 - 7 )
1 / ) : 12 9 7 8 0 3
) :
2 / : 2 4 6 8 10
3 / : 3 6 2 1 7 5 5
.
4 / x y
5 / x 4
.
6 / x y
322826242220 - 30
251020105
321x
642y
1284x
642y
31-5-9-13-x
-5-3-1+1+3y
_ _ _ _ _
-
156
:
:
: :
:
:
:
3
4
21
40
58
89
109
135
156
