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1 C
1
1 2
1 1
1
5
3
3 4
1
1
6 10
1 1
4 10
5
1
.
. 1398
-----------------------------------------------------: : : : : : : 1398 : : : #,++’#,(,)!)*$ &*- "% : ------------------------------------------------------
. .
"
.
"
.
1397
. . . . .
. .
.
. .
3 5 9 13 17 19 23 27 29 33 39 43 46
51 57 61 65 71 77 81 91 93
97 99 103 105 107 109 111 113 115
119 121 129 133 137 141 143 147 149
153 157 159 163 167 171
n n
175 179 183 185 189 193 197 199
205 209 213 215 217 221 223 227
2 2
229 231 235 239 243 245
249 251 253 255 259
(Gouse)
261 265 275 277
281 283 285 287 289 291 295 299 301
305 309 311 313 317 319 322 323
1
2
Conic of section
v
D D
v
V
D D
D
. .
.
3
.
.
(Circle) .
. .
(Ellipse)
(Parabola)
(Hyperbola) .
-1 -2 -3 .
4
Ellipse
F F . FF ' 2c
. : F F
.
A' , F , F ' , M , M A
3 M ' ( 1, ) 2
| AA' |
MF , MF '
. MF
.
5
MF ' | AA' |
M( 1,
M
| MF | | MF '|
| M ' F | | M ' F '|
.
3 ) 2
. : ( 2a )
. A'
.(
A ).
F F
AA' 2a
| M ' F | | M ' F '| 2a , | MF | | MF '| 2a | M ' F | M ' F ' | | MF | | MF ' |
:
A Major
(
A'
)
axis M
Minor
FF '
BB
AA
axis
.
AA'
AA' 2a
. BB
BB ' 2b
. B'
B
M .
MF
MF '
: MF
MF ' 2a
2MF MF
2a a
6
:
: : a2
b2
c2
c2
a2 b2 a2 b2
c
AA' a
BB'
c
. (O)
. OA OA' a OB OB' b OF
OF ' c :(Eccentricity)
.
e e
2c 2a
0 e 1
e
c a
e
1
c a 0 c
a
b2 a2
.
7
e
:
. 1
e
.
–1 2 3
–2 .
8
Equation of Ellipse
x
. M ( x , y)
. M
. MF ' MF
. : MF
MF '
( x c)2
2a
( y 0) 2
( ( x c) 2 ( x c) 2 x2
cx
( x c) 2
y 2 )2 y2
4a 2
(cx a 2 ) 2
x2
2cx c 2
4 a ( x c) 2 y2
a2
( x c) 2
y2
4a 2
4a ( x c) 2
a ( x c) 2
y 2 )2
2a 2cx a 4
a 2 [( x c) 2
c2 x2
2a 2cx a 4
a2 x2
a2 x2
c2 x2
y2]
2a 2cx a 2c 2
a 2c 2
a4
x 2 (a2
c2 ) a2 y 2
a4
x 2 (a2
c2 ) a2 y 2
a 2 (a 2
y2
y 2 / ( 4)
c2 x2
a2 y2
y2
cx a 2
(a ( x c) 2
2a
y 2 )2
4 a ( x c) 2
y2
a ( x c) 2
( y 0) 2
( x c) 2
(2 a
4a 2
2cx c 2
4cx
:
a2 y2
0
a 2c 2 c2 )
9
y2
y2
b2
a2 c2
a2
b2 c2
. x 2b 2
a2 y2
a 2b 2 / a 2b 2
x 2b 2 a 2 y 2 a 2b 2 a 2b 2 a 2b 2 a 2b 2 x2 y2 1 , a b a2 b2
b
a
:
x
. B( 0 b ) B ' (0 b )
A( a 0 ) A' ( a , 0)
F ( c 0) F ' ( c 0)
y
:
y2 a2
x2 b2
1
:
. y
| AA' | 6
| BB' | 4
:1 . :
AA'
2a
6
BB'
2b
4
2
x b2
2
y a2
a
3
b
2
2
1
x 4
y2 9
1
b
.
| AA' | 2a 10
a
:2
x
| BB' | 2b 8
. : | A A' | 2a 10
a
5
| B B ' | 2b 8
b
4
10
:
A' ( 5 , 0) A(5 , 0) : B' (0 , 4) , B(0 , 4) :
c
: o
a2
b2
c2
c2
25 16
(5) 2 c2
(4) 2
c2
9c
3
F' (-3 , 0) , F (3 , 0) e
4x2
c a
3 5
y2
16
: : :3 . 16
: 4x2 16
y2 16
x2 4 y
16 16
.
y2 16
:
1 a
b
: a2 b
2
16 4
a b
4 2
A(0 4) A (0
4)
B ( 2 0) B ( 2 0)
: c2
a 2 b2
c2
( 4) 2 ( 2) 2
c2
16 4 12
F (0, 12 ) , F ' (0,
c
12
12 )
11
:4
P (2 , 4)
.
F ' ( 3 2 0)
F (3 2 , 0 )
:
. | PF | | PF '| 2a....I
PF
( 2 3 2)
2
42
PF PF
.
PF '
. (2 3 2)
2
42
( 2 3 2)
4 12 2 18 16
2
38 12 2
76 2 1444 288 4a 2 76 2 34 4a 2 144 144 4a 2 36 a 6 a2 4 AA 2a 2 6 12
4a 2
76 68 4a 2
a2 b2
(3 2 ) 2 b2
2a
( 2a ) 2
38 12 2 2 (38 12 2 )(38 12 2
c2
42
2a
4 12 2 18 16
38 12 2 ) 2
( 38 12 2
42
(2 3 2 ) 2
(6 ) 2 b 2
18
BB
b
2b
9 2 36 b 2 9 2
2 3 2
b 3 2
6 2
–1
. x2 a) 36
y2 16
x2 b) 100
1
.
0.8
y2 36
1
–2
12
x
.
(h , k )
F' F
.
P ( x, y )
F' F
B' , B A ' , A
( h, k )
. :
: PF '
:
PF
PF '
x2
2a
PF
[ x (h c )]2
( y k )2
PF '
[ x (h c)]2
( y k)2
[ x (h c)]2
( y k )2
[( x h) c ]2
( y k)2
[ x (h c)]2
PF
( y k)2
2 x (h c ) ( h c ) 2
4a 2 4a 2
[ x (h c)]2 2a
( y k)2
[ x (h c)]2
4a [ x (h c)]2 4a [ x (h c)]2
13
2a
( y k)2
( y k )2 ( y k )2
/( ) 2
[ x (h c)]2 x2
( y k)2
2 x(h c) (h c ) 2
x2
2hx
4( a 2
4hc 4cx a2
hc cx x)
a2
c ( x h)
a2
c(h
h2
2cx
2hc
c2
4a 2
4a [ x (h c)] 2
a [ x ( h c)] 2
a [ x (h c)] 2
x2
2hx
h2
2cx
c2
2hc
k) 2
(y
a [ x (h c)] 2
k)2
k)2 / 4
(y
(y
a [ x (h c)] 2
(y
k)2
/ ( 1)
k) 2
(y
: c 2 ( h x) 2
2ca 2 ( x h) a 4
a 2 [{x (h c)}2 ( y k ) 2 ]
c 2 ( x h) 2
2ca 2 ( x h) a 4
a 2 [( x h) c]2
c 2 ( x h) 2
2ca 2 ( x h) a 4
a 2 ( x h) 2
c 2 ( x h) 2 a 2 ( x h) 2 a 2 ( y k ) 2 ( x h ) 2 (c 2 a 2 ) a 2 ( y k ) 2
a2 ( y k )2
2 a 2 c ( x h) a 2 c 2
a 2c 2 a 4
a 2 (c 2 a 2 )
( x h) 2 ( a 2 c 2 ) a 2 ( y k ) 2
a 2 (a 2 c 2 ) a2 c2
: b 2 ( x h) 2
a2 ( y k )2
a 2 ( y k )2
a 2b 2 /
( a 2b 2 )
( x h) 2 a2
( y k )2 b2
b2
1
:1 ( x 6) 2 36
.
( y 4) 2 16
1
: a2
36
a
6
b 2 16
b
4
c
a2 b2
36 16
20
(6 , 4)
x
.
2 5
: A(h a k )
A(6 6
A' (h a , k ) (6 6
4) 4)
A(12 A' (0
4) 4)
: B (h, K b)
B (6
B (h, K b)
B ' ( 6 , 4 4)
4 4)
B(6 ,0) B ' (6
8)
14
: F (h c k )
(6 2 5
F ' (h c k )
(6 2 5
y
.
4) 4)
: 2
( x h) 2 ( y k) 1 2 b a2 A(h , k a) A (h , k a )
B (h b , k ) , B (h b , k ) F (h , k c) F (h , k c)
A
Ax 2 Cy 2
C
A 0 C
16 x 2
.
0
25 y 2
Dx Ey F A 0 ,C
0 0
64 x 50 y 311 0
25 y 2
64 x 50 y
311
16( x 2
4 x) 25( y 2
16( x 2
4 x 4 4) 25( y 2
16[( x 2) 2
2 y ) 311
16( x 2)
2 y 1 1) 311
4] 25[( y 1) 2 1] 311
16( x 2) 2 64 25( y 1) 2 2
:2 :
. 16 x 2
:
25( y 1)
2
16( x 2) 2
25 311
400
15
25( y 1) 2
311 64 25
( x 2) 25
2
( y 1) 2 16
1
400
.
(2 –1)
.
:3
. x2 9 y 2
4 x 18 y 23 0
:
: x2
9y2
4 x 18 y 23 0
4 x 9( y 2
2 y ) 23 0
x2
4 x ( 2) 2
(2) 2
9[ y 2
2 (1) 2
x2
4 x ( 2) 2
(2) 2
9( y2
2 y (1) 2 ) 9 23 0
x
2
( x 2) 2
9( y 1) 2
9 4 23 0
( x 2) 2
9( y 1) 2
36 0
2
2
( x 2)
9( y 1)
36
2)2
(x
36 ( x 2) 2 62
(1) 2 ] 23 0
9 ( y 1) 2 36 2 ( y 1) 1 22
36 36
36
:
–1
. a)
( x 3) 2
2
( y 1) 9
( 6 , 2)
2
1
b) x 2
2 y2
4 x 12 y 20 0
(1 , 2) .
-2 (4 6) -3
. a) 9 x 2
25 y 2
b) 16 x 2
36 x 150 y 36 0
16
4 y2
96 x 8 y 84 0
Parabola
K F M KM FM
. .
D
(F)
. MF
(Directrix )
MK .
D . . S
FH (e 1)
17
S
. BB '
. LL '
.
FH
18
.
.
y
. .
x . M
F .
M ( x , y)
K
(
)
K F
. K ,M F,M MF
MK
: :
| MF | | MK | x
:
y2
p x
2
p MK
MF
y2
MK
( p x) 2
x
( y 2 ( p x))2 y2
p2 y2
2 px x 2
MF
p p)2
(x x2
2 px
p2
4 px
x .
19
x
.
F ( p , 0)
p
.
p
0
p
0
.
:1
F ( 2 , 0)
P
2
0
: P
: y2
4 px
y2
4 2x
y2
2
x
. .
8x y2
8x
x
2
: y2 y
8 2 4
y 2 16
: M ( 2 , 4) , M ' ( 2 , 4)
20
x
D
y
(F )
:
. M ( x , y)
:
:
| MF | | MK | | MF |
x 0
2
y
| MK |
x x
2
[ y
p
x2
p)2 2
x2
y
x2
y 2 2 py
x2
2 py 2 py
p
y
p ]2
x2 ( y
x2
2
y
(y y p2
p
2
p p
2
p ) 2 /( ) 2
2
y2
p2
2 py
4 py
y
. .
y
F (0 , p )
p
.
p
0
.
p
0
21
x2
.
:2
12 y
y p
: 4 P 12
x2
4 py
P 3 P
. x2 0 3y 0
x 0 y 0
S 0,0
p
–2
(
–3
3:
y
–1
) x2
.
0
–1
:
F (0 , 3) : y
:
2y
y2 4x 0
–2
: a) S (0 , 0)
F (0 , 5)
b) S (0 , 0)
F ( 2 , 0)
22
(h , k ) x
. M
F
M ( x , y)
.
N
(
N,M F,M
: S (h , k )
. N
:
M F
:
:
MF
.
p)]2 ( y k ) 2
[ x (h
: x2
2(h
p)]2
( y k )2
p ) x (h
p)2
p )]2
(y
2ky k 2
x2
[ x (h y2
(h
p, y )
MN
[( x (h
[( x (h
)
23
y)2 2(h
p ) x (h
p) 2
p)]2 ( y
y)2
: y2
2ky k 2
4 px 4 ph
2
y k 4p x h S (h , k ) y
.
k
x
p h
F (h
. .
.
F ( h, k
p
0
p
0
(h , k )
y
:
p , k)
p)
( x h) 2
s(h , k )
.
y
k
p
x
. .
24
4 p( y k )
h
p
0
p
0
x 1
2
:1
12 y 2
. k=2 h=1: 4 p 12
p
2
x h 12 4
4p y k
3
S(1,2)
.
F (h k
:
p ) F (1 2 3) F (1 5 ) :
y k p
y 2 3
x h
1
:
x 1
y
:
y 5 x 1
2
12 5 2
2
2
12 3 36 x 1 x 1 x 1 6 6 1 5 x1 6 1 7 , x2 L7 5
L'
5 5
y 4
2
:2
6x 3
. k 4p
4 h
3
S
6
p
3 2
3 4
:
3 2
P
. F h x
h
p
p k x
3 : 2 y k
F(
9 4) 2
y
4 x
.
25
0
9 2
y 4
2
y 4
2
y 4
2
M1(
Ax 2
A
y2
3 4 7
y1
C
6 x 3 9 6( 3) 2 9 3 y 4 3 4 1
9 9 M 2 ( ,1) ,7) 2 2 2 Cy Dx Ey F 0
. (C 0 A
0
C
0A
x h
. y2
:
0 ) 2
4p y k
: :3
2 y 8 x 25 0
. y
:
A=0
:
y 2 2 y (1) 2 (1) 2 8 x 25 0 ( y 1) 2 8 x 24 0 ( y 1) 2
8( x 3)
k 1, h F (h
x h y
4p
3
p , k) p
( y 1) 2 8( x 3) 0
S
8
2 0
3 ,1
:
F ( 3 2 1) x
p
3 1
F ( 5 1)
:
4
:
y 1
k
:
–1
: S (1,3), F ( 1,3) ( y 1) 2
. 6 y 8 x 41 0
–2 –3
. a) y 2
12( x 4)
b) x 2
26
2 x 6 y 53 0
Hyperbola
M ' M , F ' , F , A' , A
| AA' |
.
| MF '| , | MF | | MF '| | MF |
. | AA' |
.
| M ' F ' | | MF |
M'
.
| M F ' | | M F | | MF ' | MF
2a
.
. : (Hyperbola)
| MF '| | MF | | AA'| 2a :
M' M
.
F' F FF '
a
.
c FF ' 2c
27
AA' 2a
FF '
.
.
FF '
.
A
(Transverse axis)
:
FF '
.
A
.
| AA' | 2a
. FF '
B B
(Conjagate axix)
.
OB OB' b | BB' | 2b
. c2
a2
b2
c b,a
. c
a
:
c, b, a
e 1
e
. .
.
28
1
b2 a2 e
c a
. F F
.
P( x , y)
F F‘P
.
.
PF
. : PF ' | | PF x c
x c
2
2a y2
2
x c
y2
2a
2
y2
2a 2
x c
y2
: x2
2cx c 2
y2
4cx 4a 2
cx a 2 c2 x2
c2 x2
4a
4a ( x c ) 2
a ( x c) 2
2a 2 cx a 4
(c 2
4a 2
a2 )x2 a2 x2
a2 x2
a2 y2
x c y2
2
y2
x2
2cx c 2
/ 4
y2
:
2a 2 cx a 2 c 2
a 2 (c 2
a2 y2
a2 )
a 2 y 2 a 2c 2 a 4 c 2 a2 b2
c2
a2
b2
.
29
y2
PF
b2 x2
a2 y2
x2 a2
a 2b 2 / a 2b 2
y2 b2
1
x
. y
: y2 a2
x2 b2
AA'
:
1
. y
x c a
e
.
y
a e
a2 b2 : a
c a
.
a2 c
a e
.
x
a e
y x
(
)
. . y 2a 2
x 2b 2 a 2b 2
y 2a 2
b2 (x2 a2 )
y
b a2 x 1 2 a x
y2
b2 2 a2 x ( 1 ) a2 x2
30
x2 a2
y2 b2
1
a2 x2
x
b x a
y
(1
.
a2 ) x2
x
.
a x b
y
y x2 16
y2 4
:1
1
. a 2 16 b2
c
a
4
b
a2
b2
4
A ( 4 0)
2
A' ( 4 0) :
B ( 0 2)
16 4
B ' (0
:
2)
2 5
:
F (2 5 0) F ' ( 2 5 0) x
a e
4 5 2
8 5: 5
x
: : y x
b x a 2y
2 x 4 x 2y 0 y
y2 4
.
x2 9
x 2y
:2
1
. : .
31
y
0
a2
4
b2
c
a
9
2
a
2
b 2
b
A(0 , 2) A' (0 , 2)
3 2
B (3 , 0) , B ' ( 3 ,0)
c
13 ) F ' (0 ,
F (0
:
2
4 9 13
c
13
:
13 )
y
: :
a2 c
a e
y
4 13 , 13
y1
4 13 y2
4 13 13
4 13 13 y
a x b
y
3y
y
y
2x
y
:
2 x 3
3y 2x
: 0 ,
3y 2x
0
4 13
4 13
4x 2
.
32
y2
16
2
(h , k )
. . B,B' A,A'
x
F' F
P( x , y)
F' F
(h , k )
PF '
.
:
PF
2a
:
P( x , y)
33
| PF '| | PF | 2a x (h c)
2
( y k )2
x (h c)
2
( y k )2
x (h c) 2a
2
( y k )2
x (h c)
2
2a
( y k )2
.
x ( h c) x (h c) x2
2
2
( y k )2
( y k )2
2 x(h c) (h c) 2
2
2a
4a 2
4a
4a 2
4a
x (h c x (h c)
2
2
x ( h c)
( y k )2 ( y k )2
2
2
x (h c) 2
( y k )2
x2
( y k )2
2 x(h c) (h c) 2
: cx (ch a 2 )
a
x (h c)
2
( y k )2
c 2 x 2 2cx(ch a 2 ) (ch a 2 ) 2
a 2 x ( h c)
2
a2 ( y k)2
: c 2 x 2 a 2 x 2 2c 2 hx 2a 2 hx c 2 h 2 a 2 h 2 a 2 ( y k ) 2 x 2 (c 2 a 2 ) 2hx(c 2 a 2 ) h 2 (c 2 a 2 ) a 2 ( y k ) 2 (c 2 a 2 )( x 2 2hx h 2 ) a 2 ( y k ) 2 (c 2 a 2 )( x h) 2 a 2 ( y k ) 2
a 2 (c 2 a 2 )
a 2 (c 2 a 2 )
a 2 (c 2 a 2 ) c2
: b 2 ( x h) 2 a 2 ( y k ) 2
a 2c 2 a 4
a 2b 2
34
a2
b2
a 2b 2
: b 2 ( x h) 2 a 2b 2 ( x h) 2 a2
a 2 ( y k )2 a 2b 2 ( y k )2 b2
a 2b 2 a 2b 2
1
A(h a , k ) ,
A' (h a , k )
B ( h , k b) ,
B ' ( h , k b) :
F (h c , k ) , y
:
F ' (h c , k )
:
b ( x h) k a a x h e
: :
y
: : ( y k )2 a2
( x h) 2 b2
1
. : Ax 2 A
B
A
B
By 2 A, B
Dx Ey F
0
0
. 9( x 3) 2 4( y 1) 2
144
:1
. 9( x 3) 2 144 ( x 3) 2 16
4( y 1) 2 144 ( y 1) 2 36
1
.
:
1
35
a 2 16
a
A(3 4
1) A 7
A' 3 4
1
4
A'
: 1 1
1
b 2 36 b 6 B 3 6 1 B (3 5) B ' (3
6 1) (3
7)
: c2
a2 b2
c2
16 36 52
c
52
2 13
F ( 52 3 1) F ' ( 52 3 1) b 6 y ( x h) k y ( x 3) 1 a 4 y 2y 2y
:
3 ( x 3) 1 / 2 2 y 3( x 3) 2 2 3( x 3) 2 2 y 3x 11 3( x 3) 2 2y 3x 7 2 x 2 8 x 3 y 2 18 y 31 0
:2
. : 2( x 2 4 x) 3( y 2 6 y ) 31 0 2 x 2 4 x 22 22 2 ( x 2)
2
4
3 y 2 6 y 32 3 2
3 ( y 3)
2
2( x 2) 2 8 3( y 3) 2
9
31 0
31 0
27 31 0
2( x 2) 2 3( y 3) 2 12 0 2( x 2) 2 3( y 3) 2 12 12
: 2( x 2) 2 12
3( y 3) 2 12
12 12
( x 2) 2 6
36
( y 3) 2 4
1
h
2
O(2 , 3)
. b2
4
c
a 2 b2
b
a2
2 , 6 4
a
6
6
10
: F ( h c, k )
F (2
10 , 3) ,
F ' (2
10 , 3)
: A(h a, k )
A(2
6 , 3)
,
A' (2
6 , 3)
: B (h, k b)
B(2 , 1)
,
B ' (2 , 5)
: x h
y k
y
a e
a e
x
b ( x h) a
2 x 2 6
3/. 6
6 y 2x 4 3 6
2
x
b ( x h) k a
y
6 y 2x 4 3 6
6 10 10
h
6y
2( x 2) 3 6
0
0
37
3 10 5
2
:
k
3
.
9x2
38
4 y2
54 x 16 y 79 0
–1
O
. .
: : . y
.
x2
x 3
y2
9
:1
(-3,0) (0 3)
: y
. .
y
x 3
x2
y2
9
2x 2 6x
x 2 ( x 3) 2 0
x2
9
0 , x2
x1
3
y
. y y2
0 3 3 3
y1
3
y2
0
x2 6x 9 9 0 y
x 3
x
(-3,0) (0 3)
.
39
y
x .
. Y
.
0
-1
0
-2
0
-3
.
(0 , 3)
X
( 3 , 0)
.
. x2
3 y 4 x 15 0
y2
2x 4 y 4
0
:2
. :
. x2 x
2
y2
2x 4 y 4 0
2 x (1) 2 (1) 2
( x 1) 2
( y 2) 2
y2
4 y ( 2) 2 ( 2) 2
4 0
9 , c (1, 2) x
.
(1,-2)
3
. 3 y 4 x 15 0
3y
4 x 15
y
4 x 5 3 x
y .
4 ( x 1) 2 ( x 5 2) 2 9 3 4 x 2 2 x 1 ( x 7) 2 9 3
40
Y
16 2 4 x 14 x 49 9 9 3 25 2 50 9 x 9 x 9 41 0 9 3 25 x 2 150 x 369 0
3 y 4 x 15 0
x2
5
X (1, 2)
2x 1
b2
4ac
22500 36900
14400 0 0
. y x2 1 0
.
0
y
:3
x 1
: y
x 1
y x2 1 0
( x 1) x 2 1 0
x 1 x2 1 0
x2
b 2 4ac
x
0
( 1) 2 4(1) 0
:
1 0
1 ,
y
. y x2 1 0
.
0
y
x 1
0
. x2
x1, 2
x
0 b
1 1 2
2a
x1 1
x2
0
: (0, 1) (1,0)
. ( x 2) 2 9
y2 4
1
x
:4
5
. 41
x
.
5
:
(5 2 ) 2 y 2 9 4 2 y 1 1 4
y
y2 4
1
9 9
y2
0
1
b 2 4ac 0
x
0
. :
. Ax 2 By 2 Dx Ey F A, B, D, E , F IR
0
: . . A
.
B
A
B
-1
A
B
-2
A
B
-3 -4
. Ax 2
Bx Cy D
Ay 2
0 ,
By Cx D
0
-1
. a) y 2 2 y x 3 0
b) 9 x 2 9 y 2
c) 25 x 2 16 y 2
d ) x2
e)
400
y2
27 0
y2 6 y x 2 0 y
. .
y
x
9x 2
3
x2 2 y 2
42
4
4 y2
36
-2 -3
: . :
F' F
. | AA' | 2a :
.
x2 y2 a2 b2 a b
1
x2 b2
2
3
( x h) 2 a2
y2 a2
1
( y k )2 b2
(0, a), (0, a)
4
( x h) 2 b2
Cy 2
x
.
( h b, k ) ( h, k
x
y
Dx Ey F
( h c, k )
y
(h, k )
Ax 2
(0, c), (0, c)
( h, k b )
( h, k a ) .
A
x .
(h, k )
1
(b,0), ( b,0)
y
.
( h a, k )
1
(c,0), ( c,0)
.
.
( y k)2 a2
y
x
.
(0,0)
1
(0, b), (0, b)
(a,0), ( a,0)
(0,0)
c)
.
0 C
. : .
43
D
.
C
(F)
1
y2
4 px
S (0,0)
F (P,0)
x
p
y
0
2
x2
4 py
S (0,0)
F (0, p)
y
p
x
0
3
( y k )2
4 p ( x h)
S ( h, k )
4
( x h) 2
4 p( y k )
S ( h, k )
Ax 2
A
.
cy 2
e 1
44
F (h
p, k )
F ( h, k
Dx Ey
p)
F
x
h
p
y
k
y
k
p
x
h
: .
x2
y2
a2
b2
y2
x2
a2
b2
( x h )2
1
S ( 0 ,0 )
1
x
( 0 ,b ) ,( 0 , b ) y .
( 0 ,a ),( 0, a ) y
( y k )2 2
( y k )2
( x h )2
a2
b2
1
S ( h ,k )
1
S ( h ,k )
A( h ,k
.
e 1
a)
By 2
F ( c ,0 )
x
a c
y
b x a
y
a c
y
a x b
x h
a c
y
y
a c
y k
F' ( c ,0 ) x .
F ( 0, c ) y
.
A( h a ,k )
Ax 2
B
( b ,0 ),( b ,0 ) x
.
b
A
( a ,0 ),( a ,0 ) .
2
a
S ( 0 ,0 )
B( h ,k b )
.
F ( h c ,k )
B( h b ,k )
F ( h ,k
Dx Ey F
0
.
45
c)
k
k
b (x h) a
a (x h) b
A
B
. -1
: -d
-c
-b
-a -2
: -b
.
-d
.
-c
.
2a
.
-a
. F
F
M
-3
: a) MF
MF
2a
b) MF
MF
a
c) MF
MF
2a
d ) MF
MF
0
-4 a) e
a c
c a
b) e
c) e
b c
d) e
c b
-5
: a) a 2
b2 e2
b) a 2 b 2
c2
c) a 2
b2
d ) a2
c2
e2
.
b2 P
-b
.
( x 1) 2
:
b) F ( 1, 4)
-a -c
. 8 ( y 2)
c) F ( 1, 2)
-7
d ) F (4, 1)
p
F ,F
. 46
-6
4 p ( x h)
.
-d
. a ) F ( 1 2)
( y k )2
0
-8
a) PF
PF
a
b) PF
PF
2a
c) PF
PF
0
d ) PF
PF
2a x2
y x
-9
-b
y
-a
-d
y x
-c
-10 a) e 1
b) e
1
c) e 1
d) e 1
x2 4
: x
y2
-11
1
-b
y
y
-d
-a
x
-c -12
: a ) (2,4)
y2
-b
-a
-d
-c -13
4( x 2)
b) (4,2)
c) (2,0)
d ) ( 2,0)
4x2
:
4 y2
-14
8y 3 0
-b
-a
-d
-c . -1 :
a) x 2 c) 16 x
4 y2 2
b) 9 x 2
4
96 x 9 y
2
90 y 225 0
d) x
47
2
2 y2
15
12 x 120 y 288 0
-2
: .
y
.
x
2 , (0,0)
-a
64 , (0,0)
-b
e 0.5 , a e
0.5 , b
-3 . a) 4( x 1)
2
y
2
4
x2 b) 4
y2 9
1
-4
. a) x 2 11y
b) y 2 4 y 4 x 2 0
0
-5
. a) 4 x 2
y 2 8 y 32 0
b) 2 y 2
4 y x 2 10 x 25 0
-6
(4,0) ( 4,0) 5 x 4
y
.
-7
(1,3) ( 1,3)
4
. .
( x 1)2 4
( y 2) 2 9
48
1
y
2x
-8
49
3 miles 50
Law of sine
B c A
a
a
124.3
C
41.6
C
b
ABC
. . sinC
CH 3
sinB
BH 2 , AH 1 AH 1
ACH 1 ABH1
. sinA
BH 2
BCH 2
ABH 2
sinC
. . ABH1
:
ACH 1
sin B AH 1 sin C AH 1 c sin B b sin C / bc
sin B b
:
sin C ........ c
51
: AH1 AH1 c AB c sin B............. (1) AH1 AH1 b AC b sin C .......... (2)
( 2) (1)
BCH 2
: sin A sin C
BH 2 c BH 2 a
BH 2
c sin A ........ (3)
BH 2
a sin C ........ (4)
ABH 2
c sin A a sin C / ac :
4 3
sin A sin C ........ II a c sin A sin B sin C : a b c a b c sin A sin B sin C
II I
(law of sine)
. (A,B,C)
b ,a )
ABC
: (c
: a sin A
b sin B
c sin C C
c
CE
b 180
a
ABC AD
. .
C
sin(180
AD b
C)
:
ADC
: sin(180 AD ........ (1) b AD sin B ........ (2) c sin C c : sin B b
sin C
C ) sin C
: :
ADB 2 1
52
sinC c
sin A sin B sin A sin B sinA a
sinB ........ I b CE ........ (3) b CE ........ (4) a a : b sinB ........ II b
: AEC
:
BCE
:
4 3 : II
: sinA a
sinB b
sinC c
a sin A
I
b sin B
c sin C
. :1
. .
:
: c sin C
b sin B
6 3 sin C
9 sin 60
sin C
3 3 9
sin C 9 9
1
6 3 sin 60 9
sin C
6 3
3 2
9
1
.
53
C
90 :
sin 90
1 :
A B C 180
:
A 180
(B C)
A 180
(60
A 180
150
90 ) A 30 a
. sin A a
sin B b
a
b sin A sin B
9
9.sin 30 sin 60
1 2 3 2
9 3
9 3 3
3 3
a 3 3cm B
:2
A
.
sin A
:
sin C
: sin C sin A c a a.sin C 422 ft sin 110 c sin A sin 30
sin 30
: c
422 0.9396 ft 0.5
c 793. 0224 ft
54
0.5
sin 110
0.9396
:3 . .
180
: .
A 180
(31
20 )
A 180
51
A 129 a
: sin A a a
sin B sin 129 b a sin 129 210cm sin 20
sin 20 210
sin 129
: a a
0.7771 210cm 0.342 477.166cm
163.191cm 0.342
sin C c
sin 20 210
sin 31 c
c
: c
210cm 0.5150 0.342
sin 20
0.342
477.166cm
: sin B b
0.7771
108.1500cm 0.342
sin B sin C b c 210cm sin 31 sin 20 sin 31
316.228cm
55
c
0.5150
: ( AAS ) :
(SSA) : Side
.
c 10m
S
Angle
A
-1
b 6 m , a 8m
. ..
B
A
56
-2
Law of cosine x
4cm
60
(x)
a
A
ABC
c
.
C
b H
.
B
AC
B
. Aˆ
ABC
AH
. . . ABC
BH
. CH
x
b
x
:
AH
x BH
:
HBC
b
c
BC a
a2
2
CH
2
BH
(b x) 2
h 2 ........
h
: h
2
c
2
(b
x) 2
a2
b2
2bx
a2
b2
c2
c2 x2
x2 c2
x
HAB
2
h2
: a2
2
h:
x2
2bx
57
I
AHB
:
x c
cos A
a2
b2
c2
a2
x
.
b2 c 2 a 2 cos A a2 2bc 2 b c2 a2 2bc cos A cos A 2bc :
b2
a 2 c 2 2ac cos B
c2
b2
cos B
a 2 2ba cos C
b2
c2
2bc cos A ...I
:
...III
.
III II
.
:
: (SAS)(
). (SSS) (
).
:
(
3
) (SSS)
(
) (SAA)
(
) (ASA) (
) (SAS)
(
) (AAA)
.
.
58
c cos A
b 2 c 2 2bx
a 2 c 2 b2 ...II 2ac a2 b2 c2 2ab
cos C
x
A
.
:1
ABC
. 28 , b
a a
b
2
c
2
4 , c
6 , A ?
2bc cos A
( 28 ) 2 (4) 2 (6) 2 2 4 6 cos A 28 16 36 48 cos A 28 52 48 cos A 48 cos A 52 28 48 cos A 24 24 1 cos A 48 2 Aˆ 60 b 10 ft , a 16 ft ABC :2
c
.
C 110
: . c2
b2
c2
(10) 2 (16) 2
c
2
a 2 2ab cos C 2(16)(10) cos110
100 256 320 cos110
cos110
0.342 :
c2
356 320( 0.342)
c2
356 109.44
c 2 465.44 c 21.57
c
465.44
100m
.
:3 60
. : cos 60
OL OH
x 100 cos 60
x 100 x 100
1 2
50m
59
OHL
:
: HL HL HL HL HL
OH
2
sin C
: a2 2
c2 b2 (8)
cos 60
2
2
(100)
2
2OH OL cos 60 1 (50) 2 100 50 2
OL 2
2
2
10000 2500 5000
2
7500 m 2
HL
(5) 1 2
Aˆ 60
a2
a2
89 40
sin C
c
:4
.
sin C
a
:
49
a
7
sinC
.
sin C
ABC
2 8 5 cos 60
1 2
sin C
50 3 m
2bc cos A 2
a 2 64 25 80
c sin C
7500m
50 3 m
b 5, c 8
a
a
2
a sin A
sin C
c sin A a
8 sin 60 7 3 8 2 7 4 3 7
sin 60
:
Aˆ 45
.
b
4 ft , a 5 ft
ABC
3 2
-1
. c
.
60
b
60
9cm , a
3cm
-2
Law of Tangent
D
.
sin a sin A
D
.
b sin B
. . . . . a b a b
A B 2 A B tan 2 tan
: a sin A
b sin B
:
D
:
D
a b D , sin A sin B a D sin A ... I b
:
D
D sin B ... II
61
: a b
D (sin A sin B)
a b D (sin A sin B)
: a b a b
sin A sin B sin A sin B
: A B A B cos 2 2 A B A B sin A sin B 2 cos sin 2 2 A B A B 2 sin cos a b sin A sin B 2 2 a b sin A sin B 2 cos A B sin A B 2 2 a b A B A B tan cot a b 2 2 A B tan a b 2 : a b tan A B 2 sin A sin B 2 sin
A B A B cos 2 2 A B A B sin cos 2 2 sin
cot
A B 2
1 A B tan 2
. c a c a
tan tan
C C
A 2 2
A
,
b c b c
B C 2 B C tan 2 tan
. C B
A 90
b c b c
1 ABC 3
:1 .
62
.
:
:
A B C 180 B C 180
A 180
90
B C 45 2 B C B C tan tan b c 3 2 2 b c tan B C 3 tan 45 2 3 B C tan 3 2
B C
90
: 3 B C 2 3 B C 30 2 B C 60 .....
tan
150
tan
B C 75
( 2) 75
B
B
60 60
75 C
C
C
60
C 15
a 925 cm , c 432cm , B 42 30
ABC
. A B C 180 A C 179 60 A C 2 a c a c
3 3
tan 30
B C 2
(1 )
.
1
(1)
90 .....
B C 2B
tan 30
tan 45
:
137 30 2 A C tan 2 A C tan 2
A C 180 4 2 30 A C 2
A C 180
B
42 30
A C 137 30 136 90 2
925 432 925 432
68 45
tan 68 45 A B tan 2
63
1357 493
tan 68 45 A B tan 2
:2
1357 tan
A C 2
tan 68 45
2.571
493 tan 68 45
A C 0.363 2.571 2 A C 43 2 A C 86 . A C 86
tan
tan
tan
A C 2
A C 2
493 . tan 68 45 1357 :
0.933
A C 137 30
A
A C 86 A C 137 30 2 A 223 30
A C
2 A 222 90
137 30
C 136 90 C
136
A 111 45
90
C
.
A 136 90 111 45
25 45
b
. b sin B
c sin C
b 432
sin 42 30 sin 25 45
b c.
sin 25 45
0.434
sin 42 30
0.676
b 432
0.676 0.434
sin B sin C
672 885
. , a 35 ft
-a
B 75 , b 37cm , A 45
-b
C
. .
75 , B 60
64
Trigonometric Identities ( a b) 2
a2
2ab b 2
b a . sin 2
cos 2
1
B A : cot csc
A
B
1
csc cot
1
0 30 45 60 90
B A
. .
: . cot csc
. .
65
1
csc cot
1
:
Y
sin 2
. P r
O
=1
C ( o, r )
y x
cos 2
OM P
:
X
M
:
y r
sin
x r
cos
: x2
cos 2
x2 y2 r 2 r2 r2 r2 x y ( )2 ( )2 1 r r x r
1:
r2
r2
:
sin 2
y2
cosa
y r
sina
: sin 2
cos 2
1
,
1 tan 2
sec 2
1 cot 2
,
csc 2
: cos . sec
cot cot
1
,
sin . sec
1 tan cos sin
, ,
.
1
,
tan . cot 1 sec cos sin tan cos 2 1 tan sec 2
x2 x2 x2
y2 x2
r2 x2 : r ( )2 x
y 1 ( )2 x 1 tan 2
sec 2
:
y2
y x
66
tan
1
,
r2: x2
r x
sec
csc
1 sin
:
: 1 cot 2
csc 2
,
1 tan
cot
,
1 cos
sec
.
.
. .
. . sin cos tan cot 1 sin 2
.
tan a
:1
:
:
sin cos cos sin cos 2 sin cos sin tan 2 cos cos
sin
sin cos tan cot 1 sin 2
cos
sin 2
.
cot 2
cos 2
tan 2
tan 2
:2 :
sin
2
cot
sin 2 sin 2 1 tan
2
cos 2
2
cos sin 2
cos 2
cos 2
1
tan
2
tan 2
sin cos 2
2
tan 2
2
67
(1 sin 2 )(1 sec 2 )
cos
.
:3 :
(1 sin cos 2
2
2
)(1 sec ) 1 (1 ) cos 2
cos 2 ( cos 2
cos 2 cos 2
1
)
1 cos ) 2
(sin
(sin
cos ) 2
:4
2:
:
. (sin
cos ) 2 (sin
sin 2
2 sin cos
2 sin
2
2 cos
2
cos ) 2 cos 2
2(sin
2
2
sin 2 cos
2
2 sin cos )
2
21 2
:5
. sin A 1 cos A 1 cos A sin A sin A 1 cos A 1 cos A sin A
cos 2
2 csc A
sin 2 A 1 2 cos A cos 2 A sin A(1 cos A) sin 2 A cos 2 A 1 2 cos A 1 1 2 cos A sin(1 cos A) sin A(1 cos A) 2 2 cos A 2(1 cos A) 1 2 2 csc A sin(1 cos A) sin A(1 cos A) sin A
. 1 tan 2 A 1 cot 2 A
tan 2 A
68
:6
:
: 1 tan 2 A sec 2 A 1 cot 2 A csc 2 A 2
sec A csc 2 A
1 cos 2 A 1 sin 2 A
2
tan A
sin 2 A cos 2 A
.
(sin
cos )(cot
.
cos
sin
(
sin A 2 ) cos A
tan )
tan 2 A
1 cos
1 sin tan
(sin
cos )
cos sin
sin cos
1 cos
(sin
cos )
cos 2 sin 2 sin cos
1 cos
(sin
cos )
1 sin cos
sin
(sin
cos )(cot
sin
:
1 sin cos sin cos
cos
1 cos
1 sin
1 sin
:
cos( x y ) cos x sin y
.
cot
1 sin
1 cos
tan )
:7
tan x cot y
:8 :
cos( x y ) cos x sin y
cos x cos y sin x sin y cos x sin y cos x cos y sin x sin y cos y cos x sin y cos x sin y sin y cot y tan x tan x cot y
sin 2
x tan x sin x 2 2 tan x 1 cos x 2 1 cos x ) : 2 2
sin 2
. x 2
(
69
sin x cos x
:9 :
sin 2
x 2
tan x sin x 2 tan x
1 cos x 2
tan x sin x 2 tan x tan x : tan x tan x tan x cos x 2 tan x
tan x 1 cos x tan x 2 sin x tan x ( ) cos x cos x 2 tan x
.
tan x sin x 2 tan x 1 sin x cos x
cos x 1 sin x
2 sec x
:10 :
(1 sin x) 2 cos 2 x cos x(1 sin x)
1 sin x cos x cos x 1 sin x
1 2 sin x sin 2 x cos 2 x cos x(1 sin x) 1 2 sin x 1 2 2 sin x cos x(1 sin x) cos x(1 sin x ) 2(1 sin x) 2 1 2 cos x(1 sin x) cos x cos x 2 sec x
2 sec x
-1
: a)
sin 25 0
sec 2
b)
1
c)
cos 25 0
cos 8 0 sin
: a) cot cos
,
1
b) cot 2
: a)
csc
cos
cot tan sin sin 2 c) 1 cos cos 2
tan x cot x 1 2 cos 2 x tan x cot x 1 tan 2 x cos 2 x d) 1 tan 2 x
b) tan
-2
70
-3
y
Trigonometric Equations sin 2 r
cos 2
y
o
.
x
x
1 tan 2
1
sin
sec 2
cos
1
1 2 sin
0
: 1 2 sin
1 tan 2
0
sec 2
0 30 60 90 0 1 tan 2
1 2 sin
.
sec 2 1 tan 2 1 2 sin
sec 2 0
: : . . .
71
a sin x b
0 :
:
. Y
sin x
.
1
2 sin x 1 0
1 2
6
:1
2 sin x 1 0
.
:
2 sin x 1
sin x
X
O
1 2
sin
.
1 2
6
Y
1 2
sin x
6
6
X
. x
1 2
. x
6
,3
6
6
,2
6
6
A2 A
, 2
6
6 , 3
x / x 2k
,5
6
x/ x
2k
6
,...
,. . . sin x
,4
6
6 ,5
, ...
6
, ...
x 2k
6
6
k Z
: A
sin
sin
: A1
,4
1 2
x
2k
k
72
Z
1 2
x
( 1) n
n
n
0 ,1, 2 , 3 , .....
: :2
2 sin x 3 0
.
2 sin x 3 2
.
3
,
2 3 2
.
sin x
3 : 2
2 1 sin x 1
sin a cos x b 0 :
:
. 2 cos x
.
3
cos x
. 2 cos x
3
0
2 cos x
3
cos x cos
y
6
.
x
x
0,
6
6
3 2
cos
6
x
6
x x
A
, 2.
6 2n
6
, 2
x/ x
,4
6
...
... 6 n 0 ,1, 2 , 3 , ... 6
2k
,4
6
x 2k
:
: 6
k
73
:
3 2
cos x
3 2
cos x
:1
0
Z
: A
x/ x
2k
x
2k
, k
ZI
0,2
.
2 cos x
2
:2
0
: 2 cos x
2
2 cos x
0 2
2 2
cos x
3 : 4
x
A
x / x 2k
cos 3 4
3 , k 4
x 2k
2k
x
2k
3 4 3 4
k 0
x1
k 1
x2
2 2
:
0,2
. x
Z
3 4
3 4 5 4 a tan x b
:3
0:
. tan x
. tan x tan x
3
.
3 :
tan
:1
0
tan x
3
tan x
3
(
x
2
: ,
2
)
60
3
3
Y
x 3
3
3
X
x
,2
3
74
3 ,3
,4
3
3 ,5
, ...
3
,...
: A
x/ x k
(2k 1)
3
3
,k
Z
: A
x/x k
,k
Z
:2
: 1 3 1 3
tan x tan x
A
0,2
x/ x k
tan (2 x
4
)
x
6
tan( x
:
6
, k
3
Z
:3
)
. : tan( 2 x 2x x
4
)
tan( x
k
3
3
)
2x
x
4
k
4 7 12
k
(x
3
)
k
0,2
. x
k
A
x/x
k 1
k
7 ,k 12
7 12
x1
k 0
7 12
x2
Z
7 12
:
75
19 12
:4
a cot x b 0 :
. :1
cot x 1 0
.
: cot x 1 0 cot x 1
x
4
. x
4
,2
x
4
4 , 3
, 4 4
4 ,5
, ... 4
. A1 A2
4
, 2
4
4 , 3
, 4
4
4
, ...
, 5
, ...
4
: A
x/ x k
A
x/x k
4
, (2k 1)
4
, k Z
k Z cot 3x
. cot 3x cot x
3x
k
x
2x k
. a) 3 cos x 5 0
b ) tan x
3
76
:2
cot x x
k 2
,
k Z :
, ...
.
d
:
. a sin 2 x b cos 2 x c sin x cos x d d
.
6 sin 2 x 5 sin x 1 0
.
y
: 6 y2
5y b
2
( 5) 2
4ac
y2
4(6)(1)
1
5 1 5 1 6 , y1 12 12 12 5 1 4 1 12 12 3
y1
1 2
: sin x
y1
sin x
y2
1 2 1 3
x
. x
2k
A
n
( 1) n
sin x
6
6
x (2k 1) x
6
x 19 30'
:
1 2
k Z 6 x
n
( 1) n
n
6
77
0 , 1 , 2...
:1 :
1 0
25 24 y1 , 2
sin x
c ,b, a
:
19 30'
.
sin x
13 rad 120
1 3
1 3
19 30
. 13 120
x 2k A
k Z
13 120
x (2k 1)
cos 2 x sin x
.
:2
0
cos 2 x 1 2 sin 2 x
: 1 2 sin 2 x sin x
0
2 sin 2 x sin x 1 0 y sin x
. 2 y2
y 1 0
2y 1 0 y 1 0
(2 y 1)( y 1) 2y
y2
1
y1
0 1 2
1 y sin x
. sin x
y1
sin x
y2 1
1 2
78
:
sin x
«
sin x 1 A
1 2
sin x A1
, 2
2
2
(
1 ) 2
2 7 x 6
, 4
2
7 , 4 6
2
A2
x
7 6
x
.
1 2
sin x
»
, ...
7 ,6 6
7 , ... 6
: A
x
n
( 1) n
,
2
x 2n
7 6
, n 0 ,1, 2 , 3 ...
2 sin 2 x
2 sin x
0 :3
: sin x(2 sin x sin x
0
2) 0
2 sin x
2
2 sin x
2
0
2 2
sin x
0
x1
x2
4
: A1 A2
0 , , 2 , 3 , 4 , 5 , ... 4
,
4
,2
4
,3
4
, ...
79
. x -1 2 3 cos 2 x 2 cos x 5 0 - 2 cos 2 x 1 2 sin 2
sin 2 x (1
80
3 ) sin x cos x
3 cos 2 x
0 -3
: : sin x sin y x y
:
a
cos x cos y x y
a
y x
.
a
: sin x siny a ... I x y ............. II
I 2 sin x
x
y 2
y
cos
x
y
a ... I
2
.... II
. 2 sin
cos
2 x
cos
x 2
y 2
y
I
II
:
2 sin
x
y
a ... I
a 2 sin
2
I
2
: 1
a 2 sin
81
1 2
:
a2 4 sin
4 sin 2
a2 2
4 sin
a2
4 sin 2
a2
4 sin 2
1 4 sin 2
1:
2
2
2
2
2
2 2
0
:1
. sin x sin y 1 x
y
2
.
(
a2
a
: 1 4 sin 2 2 0 2 2 1 4( ) 2 0 2
2
1 4 sin 2 1 4
2 4
4
0
a 1
4 sin 2
2
: 0
0 1 2 0
1 0
)
. 2 sin
x
y 2
cos
x
y 2
:
1
. x
.
y 2
x y 2 sin cos 1 4 2 x y x y 2 cos 1 2 cos 1 2 2 2 2 x y 1 x y cos cos cos 2 4 2 2 x y ,x y 2 4 2
82
4
x
y
2
x y x
2
y
2
2x
........ II
2
y
2
......... I
2 y
2
2 2
2x
x
2
y
:
2 2 y 0
x
. x y sin x cos y
: x y sin x sin y
a
x y cos x cos y
a
y x
. cos 2
a
2
sin 2
a a
:
2
:2
. x y .......I sin x sin y 1.....II
:
a 1
.
cos 2
a sin 2
2
cos 2 sin 2 cos 2
:
2
2 2
2 cos 2
sin 2
2
a sin 2
2
0
:
1
:
2
0 a 1 II
: 2 sin x sin y
cos( x
y ) cos( x
.
y)
.
83
sin x sin y 1 :
cos( x y ) ( 1)
cos( x cos( x x
x
y) 2 1
y)
cos( x
y ) cos
2:
x
cos( x y ) 1 2
:
cos
cos( x
2
: y
: 1:
y) 1
y ) cos 0
y 0 x y 2
2
cos( x y ) cos( x
x
y x x
,y
2x
x
:
2
: x y sin x sin y
x y cos x cos y
a
: a
y x
.
a
:3
. x
y
sin x sin y
2 3
:
: . sinx siny sinx siny sin x sin y
3 1 3 1 x 2 sin
y 2
cos
x
y 2
sin x sin y
2 cos
x 2
. 2 cos 2 sin
x x
y 2 2
y
sin cos
x x
y 2
y
3 1 3 1
2
84
y
sin
x
y 2
cot
4
tan
x
3 1 3 1
y 2
x
:
y 2
x
4
cot
. tan
3 1 tan 15 3 1 x y tan tan 15o 2 30
x y x y 2x x
x
3
2
y
15o
6
6 6
4 12
3
6
y
2
2
y
1
4
:
4 6
y
. x
2
3 1 3 1
x y 2
x y
y
6 y
y
3 2
6
x
6 y
6
:
: x y tan x tan y
x y cot x cot y
a
y
. a2
4 4a cot
x .
0:
85
a a
:4
. x
y
3 tan x tan y tan( x
y)
2 3 tan
:
:
3
: tan( x
y)
tan x tan y 1 tan x tan y tan x tan y 1 tan x tan y
3
( 2 3)
: 2 3 1 tan x tan y
tan x tan y
3
. 2 1 tan x tan y
1
1 tan x tan y
2
tan x tan y
tan x tan y
3..............I
tan x tan y
2 3.......II
3
II
I
. tan x
3
tan x
2 3 tan y
( 2 3 tan y ) tan y
3
tan 2 y 2 3 tan y 3 0 3) 2
(tan y
0
tan y
3
tan x
3
tan y
I
. tan x tan y
0
x
3 3
tan x
3 3
3 3 3
3
86
3
y
3
y
tan x
3
2 3
x
:
: x y tan x tan y
a y
.
x
1 a cos 1 a
1
a
1
:5
. x
y
7
6 tan x tan y
0
. sin x.sin y cos x. cos y 1 sin x sin y cos(x y) cos(x y) ........ I 2 1 cos x cos y cos( x y ) cos( x y ) ........ II 2 1 [cos(x y) cos(x y)] : tan x. tan y 2 1 [cos(x y) cos(x y)] 2 tan x. tan y
I
. cos( x
y ) cos 7
cos( x
y ) cos 7
tan x tan y
y ) cos 7
cos( x
y ) cos 7
6
x
6 6
: cos( x
:
0
6
87
II
y
cos( x
y ) cos 7
0:
6
cos 7
: cos( x
y)
cos( x
y)
3 2
6
3 2
0
3 2
x
5 6
y
: x y
5
x
7
y 5
2x
6 6 7
...I ...II
6 12 6
2x
x
12 , x 12 y
: x
y 5
5 y 6 5 y 6 6 5 y 6 y
y 5
6
6
6
6
88
I
x
:
: x y tan x tan y
a
a 1 sin a 1
1
1:
:6 x
y
2
tan x tan y
.....I 3.....II
II . tan x tan y tan x tan y
3 1 3 1
2
. sin( x y ) sin( x y ) cos x cos y 2 sin( x y ) sin( x y ) cos x cos y 2 sin( x y ) sin( x y )
2
: 2 sin( x sin( x
y ) sin y)
: x
y
x
y
1 2
2 x
6
1 y
6
:
6 2
89
x
y
2
: x
y
x
y
2x
6 2
6
.....II
2 3 6
2x x
.....I
4 12
4 6
3
x y y
y
6 2 6
y
3
I
y
. 6
y
6
3
6
6
. a)
x
x y
y
b)
4 tan x tan y 1
90
sin x cos x
3 2
x
ABC
: :
sin A a
sin B b
sin C c
. ABC
b a
c
:
. a2
b 2 c 2 2bc cos A
b2
a 2 c 2 2ac cos B
c2
a 2 b 2 2ab cos C
. c
ABC
b a
:
. a b a b
A B 2 A B tan 2 tan
c a c a
,
tan tan
C C
A 2
A
,
B C 2 B C tan 2
tan
b c b c
2
: . : .
.
91
: sin x sin y x y
a
cos x cos y x y
a
: x y sin x cos y
a
x y sin x sin y
a
x y cos x cos y
a
: x y sin x sin y
x y cos x cos y
a
a
: x y tan x tan x
a
x y cot x cot y
a
: x y tan x tan y
a
: x y tan x tan y
a
92
. a
:
c
.
a) 16.4cm
7cm
b) 16cm
c) 15.9cm c 10 ft
B
: a) 28.5
b) 29.4
a) 8 ft
b) 2.52 ft
b) IR
c) IR
.
b)
1 tan x
sin C a c 10cm
B
–4 –5
d ) tan 2 x
c) cot x
: b 5 ft c 8 ft b 5cm , a 8cm
A 30 b
C B
–3
9.5 ft
0 d ) IR sec x (sec x cos x)
: a) tan x
d ) 28 22 , A 48
c) 9 ft d) x arc sec y
: a) IR
d ) 4.176cm b 5 ft , a 8 ft –2
c) 29 a 5 ft
b
:
A 20 -1
b 10cm
c b
A 30
-1 -2 B A
3 2
ABC
-3
. A . 30 60km
cos
.
93
-4
40km
sin
cot 2
-5
-6
: a)
sin 2 A 1 cos 2 A
1 1 1 d) 1
tan A
b)
c) tan A cot A 2 csc 2 A e)
cos A 1 sin A
A ) 2
tan(45
cos 2 A tan 2 A cos 2 A cos A cos B cos( A B ) cos A cos B cos( A B )
f ) cos cos(6 0
) cos(6 0
)
tan
1 cos 3 4
-7
:
2 tan 15 1 tan 2 15 c) cos 4 x 2 sin 2 2 x
b) 1 2 sin 2
a)
A B cot 2 2
cos 2
d ) (cos 2 x 2 sin x cos x sin 2 x) 2 2 sin 2 x cos x
2 cos 2 x sin x
-8
. :
a) cos 2 x cos 4 x c) 4 cos
2
b) tan 2 x 4 tan x 3 0
0
0
d ) cos x
2
e) cos x 3 sin x cos x
3 sin x 1
1
. tan x tan y 1 a)
cos x cos y
2 2
-9
b)
sin( x
y ) cos( x y ) tan x tan y 1
94
-10
95
.
96
.
. : :Postulates .
. :Logical Reason . :Theorem
. .
(
:
) :
. (
) . .
: : . :
.
:
. . :
.
:
.
97
:
.
. .
. . .
. : : . :
. : . .
(
) .
.
: : :
–1 –2
.
–3
.
98
.
: . . . . –1 . p
O
.
:
–2 . p
d
: .
99
–3 d
.
p
:
: -1
.
-2 .
-3 .
:
100
–1
: : . P'
–2
:
. P'
.
P
P
AB
A
B
: . P'
.
101
P
-3
RT
p
p
p
R T
–2
p
K M
p
–3
AB
AB
p
P'
C
B, A
C
P P'
102
–1
P
B, A
-4
: .
: .
A
.
p
d1
A A
d1
. : . p
A
d1
: . A
d2
p
.
d2 d2
P
A
103
d1
d1
p
d1
. : . : Q
d2
d1
P A B
d2
Q
.
d2
A
.
d1
d1
Q p
d2
B .
–1 . F
L2
E
F E
L1
. F E
L2 // L1
p
F E
p F
p E
104
–2 –3
.
X ' O' Y '
. O'Y '
O' A'
OY
OA
O' X '
OX
O' B'
.
XOY
OB
OA A'O '
. O ' A' B'
OAB
. :
. XOY
X ' O' Y ' OY // O'Y ' O' X '
:
OX // O' X '
OX
. O ' A'
OA
. OO '
OA A'O '
BB '
. OB O' B'
AB OA
A' B' A'O ' AB
AB B' A' .
A' B'
OAB
O ' A' B'
AOB A' O' B'
105
.
180
.
: .
–1 –2 .
d2
.
106
d1
-3
.
p
p
d d
p Q p
Q p d
( d2
)
d1
. . :
. . p d1
d
: p
p
. . p
d
. p
d .
p
p 107
: . Q p
: d
O
Q p
d Q p d
. d2
d1
(O)
:
. d1'
d '2
(O)
d2
P
d '1
2
1
3
d1 d2
d2
3
(O)
d2
. d '2
d1
d1'
d '2
.
:
2
,
d1
1
d1
–1 –2 p
p
. 108
P
(O) (O)
.
OX , OY (O)
. OA'
.
B
OY
p
OZ
. OA OX
M
.
C
OZ
(O)
: (O) OY
.
OX
:
(O) P
P
.
OZ
. OA'
OA
P
OX OZ
C
OY
. 109
B M
:
OY
AA'
OX
BA BA' CA CA'
M C B
A' BC
. MA'
: MA
MA
MO
.
MAA
OZ
Q P
C B
x Q P
A
.
.
.
A'
MA'
AA'
BC
ABC
Q P
. x
BC
d2 L
P
L
P
110
d1
–1 –2
A
.
2
1
A 1
B B
2
2
d2
1
d1
. A
2
1
B
d2
d1
. : . A )B
2
d1
p p
(
d1
2
. 111
:
1
. d2
1
2
Q
1
d2
Q p d1
1
p
d1 d2
2
Q
.
Q p
1
d2
2
Q p
d2
E
d1
d1
Q p
.
L2
d2
.
L1
F
L2
112
L1
E F
-1 :Postulates .
.
:Logical Reason . :Theorem
. .
(
:
) :
. .
(
)
.
: : . :
.
:
. . :
* .
:
.
*
: .
* .
.
(
:
) .
: :
.
113
* .
: * * * * * * * * * *
. . . O
O
114
P
. P
B A
B A
P P
: AB AB
-b
AB
-d
AB
–1 -a
AB
–c
–2
p
: p
-a
p
-b p
-c -d
p
–3
. -b
-a
-d
-c -4 -a
p d
'
. '
.
-b
'
-c
'
-d
. . p
. a) 0
b) 1
c)
–5
p
2
d)
115
p
. E
L2
F E
L1
–1 F
L1 // L2
–2 L3
.
P2
p
P1
–3
L2 , L1
P2
–4
P1
–5 AD
-a
C
-b ABCD
. DFC
-d
116
EDˆ F
-c
117
118
Sequences .
a1, a2 , a3 ,..., an
.
. an
-n
.
a1
2 , 4 , 6 , 8 , ... , 2n 1, 3 , 5 , 7 , ... , 2n 1 5 ,10 ,15 , 20 , ... , 5n n
: an
2n
,
n 1, 2 , 3 , ...
bn
2n 1 ,
n 1, 2 , 3 , ...
cn
5n
n 1, 2 , 3 , ...
,
: .
{a n } {
. . . 119
n 1 } n ( 1) n an n
1
1 1 1 1 , , , ... , 5 10 15 5n
. 3 n
bn
. bn
an
:1
n2
an
: .
an
n 2 1, 4 , 9 ,16 , 25 , 36 , ...
bn
3 3 3 3 3 , ,1, , , ... n 2 4 5
-n
.
2
5
5
-n
2n 1 :
an
.
:2
1 , 2 ,4 , 8 ,...
n n 1
an
: :3
5
n 1, 2 , 3 , 4 , 5
:
. an
n2 n 1
n 1
,
a1
n 2
,
a2
n 3
,
a3
n 4
,
a4
n 5
,
a5
1 1 1 22 2 1 32 3 1 42 4 1 52 5 1
1 2 4 3 9 4 16 5 25 6
-1
n
. 1, 3 , 5 , 7 , . . . 1 1 1 1 , , , , ... 3 5 7
.
6
an
120
( 1) n n
1
-2
Arithmetic Sequence
5 , 8 ,11,14 ,17 , 20
:
. : : . d
(Common difference) (d 0)
d
. (d 0)
: 2 , 5 , 8 ,11,14 ,17 , ... d d d
5 2 3 8 5 3 11 8 3
d
3 0
.
121
d
0
. 4 , 0 , 4 , 8 , 12 , 16 , 20 , ... d d d d d
0 4 4 4 0 4 8 ( 4) 4 12 ( 8) 4 16 ( 12)
4 0
d 4
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2
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2
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a2
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a2
d
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a4
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d
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d
a1
d
d
d
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a1 , a2 , a3 , ... , an
: a1 , (a1 d ) , (a1 2d )
d
, (a1 3d ) , ...
3 3 3 3 ,( 2) , ( 2 2) , ( 2 2 2) , ... 2 2 2 2 3 7 11 15 , , , , ... 2 2 2 2
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(a (b
122
:a
. 3 5 7 1, , 2 , , 3 , , 4 2 2 2 3 1 3 1 d d 2 1 2 2 2 2 5 1 7 1 d 3 d 3 2 2 2 2 7 1 d 4 4 2 1 2
. :b 1
2 d
2 1 1
4 d
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2
8 d
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4
d
16 16 8 8
. -n d
-n
a
. . 5 , 7 , 9 ,11, ... d
7 5
2
5 , (5 2) , (5 4) , (5 6) , ... 5 , (5 2) , (5 2 2) , (5 2 2 2) , ... 5 , (5 2) , (5 2 2) , (5 2 3) , ... a1 , a2 , a3 , ... a1 , (a1 d ), (a1 2d ), (a1 3d ) , ...
: 123
: -n a
a d
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.
a (n 1)d
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an an
n , d , a1
2 , 5 ,12 , ...
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30
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2
a1
d 5 ( 2) n 30 a30 ?
7
an a (n 1)d 2 (30 1)7 a30 2 29 7 a30 2 203 a30 a30
201
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. 35 , 40 , 45 , ... , 2000
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35
d
40 35 5
an n ?
2000
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124
:
a10
10
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3
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.
1
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1
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1
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1
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11
an 1 , an , an
n 2 , 3 , 4 , ...
. an
a2
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[2a1 2nd 2d ] 2[a1 (n 1)d ] 2an
1
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1
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125
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1
:
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1
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2 5x 4 2
2 x 1 3x 3 2
4 x 8 5x 4
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12 x 12 , x 2( 12) 1 , 2( 12) 4 , 3( 12) 3 24 1 , 24 4 , 36 3 23 ,
28 ,
33 ,
38 43...
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126
7 , 12 , 17 , 22 , 27 , 32 , 37 , 42 , 47
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d
2
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bn
2 , 4 , 6 , 8 ,10 , ...
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1
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n
2 , 3 , 4 , ...
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1 an 1
1 an
1 an 1 2
a n 1 an 1 ( an 1 )(an 1 ) 2
an 1 an 1 1 (an 1 )( an 1 ) 2
1 an
: an
1 an
an 1 an 1 2(an 1 )(an 1 )
2(an 1 )(an 1 ) an (an
1
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an
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.
an
2(2 8) 2 8
2 16 10
32 10
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16 2 , 2 2
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128
Geometric Sequences
64
) (
3 , 6 , 12 , 24 , 48 , ...
.
. : q an 1 an
q
an
: an q, n 1, 2 , 3 , ...
1
: q
a1
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96 , 48 , 24 , 12 , 6 , ...
. q 96
48 q
48 96
1 2
a2 a1
:
24 q
24 48
1 2
12 12 24
q
. 129
1 2
6 q
6 12 q
1 2 1 2
a3
a4
.
a2
q
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3
2 q 1
.
q 1
.
q 1
.
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2700 , 900 , 300 , 100 , ...
. a1
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q
900 2700
9 27
1 3 q 1
. -n q
n
a
. a1 , a2 , ... , an
:
-n a1
aq
aq 2
aq 3
a1
a2
a3
a4
a1
a1
q
a2 a1
q
a3
a2 q
a1q q
.. q... .
a3 a2 a4 a3
a4
a3 q
a1q 2 q
q
an an 1
a2
a1 q
an
an
1
q
a1q 2 a1q 3
(a1q n 2 ) q
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130
1
aq 4 ......... aq n 1 a5
......... an
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.
an
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q n 6 a6
: a1q n
an
1
2 a6 5 ( 2) 6 1 a6 5( 2) 5 a6
5 ( 32)
160
a6
?
1
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a qn
8, 4 , 2 ,.. .
.
:2 :
n 12 a 8
a12
1 2
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q
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an
1
1 1 1 8( )12 1 8( )11 8 11 2 2 2 3 8 2 1 2 3 11 2 8 11 11 2 2 28
M a
b M M b
.
b
a
M2
1 256
a b
ab
a
(Geometric Mean)
131
M
3 , 12
.
:3
M
:
ab
:
a 3 b 12
M
3 12
36 6
M
2,
. a1
2
a5
32
a5 q4
q ? n 5 a2 a1 q
6
a q n 1 32 2q 5 32 16 q 4 16 2
1
,
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32 2q 4 q4
a2
2 2 4 , a3 a1 q 2
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24
q
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2
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2 , 4 , 8 , 16 , 32 :
q
n -n
an
q
.
x :5
. x 1 , x 3 , x 1 a
M
M 2
a b
x2 6 x 9
. a)
2 , 3
b
( x 3) 2 x2 1
( x 1)( x 1) 6 x 10 0 , x
5 16 2 2 , , 3 3
10 6
x
5 3
5 2 , ... 3
5 -1 -2 b)
.
4 , 2 , 0 , 2 , 4 , ...
5,
5 5 , , ... 2 4
. .
1 , ? , ? , ? , 27 3
132
-3 3 4
3
-4 -5
.
2 , 4 , 6 , 8 , ...
. :
.
2 4 6 8 10 12 14 16 18 20 110
4 6 10
n
n
100 10000
. n
a1 , a2 , a3 , . . . an
.
n
a1
a2
a3 ... an
ai i 1
. 1 n
. i
. n
n
2k k 1
n
2i i 1
n,k , j
2j
21 2 2 2 3
2n
:
j 1
7
. i 1
133
1 i
:1
: 7 i 1
1 1 1 i 1 2
1 3
1 4
1 5
1 6
1 7
1089 420
:2
. a) 1 3 5 7 b) 1 4 9
...
( 2n - 1 ) n2
...
: n
a) 1 3 5 7
...
(2n 1)
(2i 1) i 1
b) 1 4 9
...
n
n
2
i2
i 1
:3
. n
i(i 2) ? i 4
: 4(4 2) 5(5 2) 6(6 2) 7(7 2) 24 35 48 63
...
...
n ( n 2)
n ( n 2) 10
.
n 7
n 1 n 1
:4 :
10 n
n 1 7 n 1
7 1 7 1 8 9 6 7 15396 3024
8 1 9 1 10 1 8 1 9 1 10 1 10 11 4032 3888 3780 3696 8 9 3024 5132 108
134
n (an )
: a1
a2
a3 ... an ...
ai i 1
i
. (series)
. -n
.
ai i 1
an
a1 a2 a3 ,...
n . ak
a1
a2
n
a3 ... an ...
k 1
Sn
: S1
a1
S2
a1 a2
S3
a1 a2
a3
Sn
a1 a2
a3 ... an ...
-n
.
ak k 1
.
1 2 3 .... n ...
s8
s6 :5
: S 6 1 2 3 4 5 6 21 S8 1 2 3 4 5 6 7 8 36
135
c
bk
an
k 1
n 1
: n
c c ... c
nc
k 1 n
n
cak k 1 n k 1
c
ak k 1
( ak
bk )
n
n
ak
k 1
bk
k 1
-1
. 3
( 4k 2
6
-c
3k )
k 1
i 1
1
6
-b
i 1
i
-2
. 1 4 9 ... n 2
1 2
2 3 19 ... 3 4 20 1 3 5 7 ... (2n 1)
(b
n
3i 2 i 1
136
(c
n
n
(c
i 1
(a
-3
. (2 5i )
-a
i 1
i (i 2)
(b i 4
(a
n a1 , a2 , a3 , ..., an
.
a1 a2
a3 ... an
. an
d
a
: S
a (a
d ) (a 2d ) (a 3d ) ... (an 2d ) (an
S
an ( a n d ) ( an
d ) an ......... I
2d ) (an 3d ) ..... (a 2d ) (a d ) a .......... II
II I
. 2S
(a an ) (a an ) (a an ) (a an ) ... (a an ) (a an ) a an n ( a an )
2S
n(a an )
S
n (a an )..........I 2
. an
25 , a
:1
4
.
137
8
:
: a 4 an 25
S
n 8
S
n (a1 an ) 2 8 (4 25) 4
S
4(29) 116
1
an
: S an S
n (a1 an )......1 2 a1 (n 1)d n [a1 a1 (n 1)d ] 2
S
n [2a1 (n 1)d ] .......... III 2
n
.
201
. a1
III
7
7 11 15 ... d 4 n n 201 S [2a1 (n 1)d ] 2 Sn ?
S S S
201 [2 7 (201 1)4] 2 201 (14 200 4) S 2 81807
201 (14 800) 2
138
201 814 201 407 81807 2
:2
-n . n
:3
2 4 6 8 .....
. : 2 4 6 8 ... d
a d Sn
2 2 ?
4 2
2
n [ 2 a ( n 1) d ] 2 n [ 2 2 ( n 1) 2] Sn 2 n n Sn [ 4 2n 2] (2 2n) 2 2 S n n ( n 1) Sn
2 4 ...
.
200
:4 :
a
2
S 200(200 1) S 200 200(200 1)
d 2 n 200 S 200 S 200 ? S 200
200(201) 40200
.
139
-n
.
-1
i) 2 , 0 , 2 , 4 , ... ii) 1, 5 , 9 ,13, ... iii) 2 , 1, 0 ,1, 2 , ...
2,5,8,11,...
. a ) S8
b) S10
140
-2
n
.
q
a
q
.
n
. .
: {S n }
.
. Sn
: Sn
a1 (1 q n ) 1 q
a1 a1q a1q
Sn
a1q a1q 2 Sn q
S n (1 q)
Sn
2
0
... a1q n
1
.......... I
:
II
: Sn q
n
a2 ...an
q 1 1 q
Sn
a1
q
I
a1q 3 .... a1q n .......... II
a1 aq n n
a1 (1 q )
a1 (1 q n ) Sn
a1 (1 q n ) (1 q)
a1 (q n 1) q 1
II
: ,
q 1
:
(-1) n
.
141
I
5.
1 2
q
a1
:1
2
10
.
: a1 q S
1 2( ) 0 2 2 1 2( ) 1 2 1 31 1 2( ) 2 2 4
a1
2 1 2 ?
a2 a3
Sn
(1 q n ) a1 1 q
S10
1023 1024 2 1 2
a4 a4 1 2
a5
S10
1 1 ( )10 2 2 1 1 2
1023 4 1024 1
4092 1024
1 S10
2
1 2( ) 4 1 2 1 1 2( ) 8 4 1 1 2( )5 1 2( ) 4 2 2
1 1024 2 1 2
2
1 16
1 8
1024 1 2 1024 1 2
3.99609375
:2
80 2 , 6 , 18 , ...
:
. a ( q n 1) q 1
S a
2
6 2 n ? q
S
80
2[(3) n 1] 3 1 80 (3) n 1
80 3
(3) 4
81 3n n
80 1 (3) n 3n
4
10
. . 484
142
2 2 , , ... 3 9 3,6,12, ... ,384
-2
4,12,36, ...
-3
2,
-1
.
(q
1)
. Divergent series
. Convergent
.
q
S
:
1 qn 1 q
a(
q
n
aq a q 1
S
a
aq a q 1
a q 1
1
qn 1 ) q 1
:
1
S 1 3 9 27 ...
.
r
:
3 1
S
:
: (q n
q n (n
0)
)
: S
a
1 qn 1 q
S
a aq n 1 q
a a 0 1 q
a 1 q
143
n
S
a
1 qn 1 q
(q a
S 1
. 1 2
q
1 2
1 4
1 1 2
1)
1 q
:
1 1 ... 8 16 1 ,a 1 q 2
:1 : .
1
1 2
1 4
a 1 1 ... 8 16 1 q
q
1 1 1 2
1 3
1 2 1
1 1 2
a1
2
:2
27
. q
. a a1 a 2 ...
1 3
1 1 3
:
a
1 q 1 1 1 27 27 9 3 1 ... 1 3 9 27 1 3 3 81 27 40 .5 2 2
27 3 1 3
27 2 3
:3
0.623
. : 0.623 0.6232323... 0.6 0.023 0.00023 0.0000023 ... 6 23 23 23 ... 10 1000 100000 10000000 6 23 1 1 [1 ( ) ...] 10 1000 100 10000 6 23 1 1 2 [1 ( ) ...] 10 1000 100 100
144
1 1 100
. a
S
1 1 100 1 99 1 100 100 100 6 23 1 1 1 10 1000 99 1 100 100 100 6 23 594 23 99 10 990 990
1
1 q
1
6 23 10 1000 6 23 10 1000 617 0.623 990
617 990
:4
0.3
.
: 0.3 0.3333 0.3 0.03 0.003 0.0003 ... 3 3 3 3 ... 10 100 1000 10000
q
. 3 10 S 1 q 1 1 10 3 10 3 1 10 9 10 9 10 1 0.3 3 a
1 10
1 1 10
3 1 10 10 1 10 3 1 9 3
145
a
0 .3
:
-1
. 1 1 ... 2 3 33 1 1 5 1 ... 5 52
1
1 3
(ii
-2
. a ) 0. 5
146
(i
b) 0.24
: a1 , a2 , a3 ,..., an
a1 .
-n
.
an
: . an 1 , a n , a n
:
:
1
an
an
.
an
1
1
2
q
: . aq n 1 :
an an
1
an
an
n :
1
an
.
(an 1 )(an 1 ) :
2,3,4,...
n
: a1 a2
a3
...
an
ai 1 i
Series
.
n
n
.
a1
n
n [2a (n 1)d ] 2
n (a1 2
an )
: Sn
ak k 1
: S
a3 ...
a2
a1 (1 q n ) 1 q
q 1
147
n
a
n
1 q
aq k
.
k 1
1
q
1
a 1 q
n
q S
:
148
1
. : a)
n 1 n
n 3 n 2 11 7
b)
: a) 3
c)
n n c
c) 5
d) 6 9, 5, 1,3,...
: b) 38
c) 35 d) 0.1 , 0.4 , 0.7 , 1 , 1.3,...
: a) 0.3
a)
b ) 0 .1 c ) 0 .3 96, 48 ,24, 12 ,6,... :
1 2
1 2
b)
3 512
1 qn a 1 2
S
c) S
a
0
-2
38
-4 -5
n b) S
1 qn 1 q
-6
-7
qn 1 a q 1
d)
: a) q
3n 1 2n 1
d ) 0 .2
: a)
-1
-3
2 2 d) 3 3 5 5 5 5 5, , , , , ..... 2 4 8 16 3 5 c) d) 512 512
5 512
b)
-n
c)
: a)
an
n
b) 4
a) 35
3 4 5 2, , , ,... 2 3 4 n 1 d) n
q
-8
b) | q | 1
c) | q | 1
149
d)
. -1 31 21
. 21
3720
n
,
-2
,
,31
124
-3 100
.
-4
3,5,7,9,11,...
192
.
6
-5
17
8
-6
. -7
. 0.1 0.01 0.001 0.0001 ...
1 9
9
-8 .
.
4 3
2
.
2 3
,
-9
,
,96
8 .
d
3 ,a
-10 4
-11 -12
. a) 2. 8
,
2 ... 9
n 12
.
96 3
b) 3.57
150
151
152
y
Exponential functions
y
x2
x 2 , f ( x)
f ( x) 2
x
0
x2
2x
x
x x
0
f ( x)
2
x
2x
, f ( x)
x2
a
f ( x)
a
. IR
{1} x
R f : IR f ( x)
.
.
f ( x)
2x
ax
a 1
a
IR ax
2
f ( x)
x ZI (
2
x
, f ( x)
2x
. f ( x)
.
1 ( )x 2
: .
f ( x) 2 x
x ZI
153
x
, f ( x)
ZI )
y f ( x)
2
f ( x)
2x
f ( x)
2x
2x
y y
. a
. a
y 2
2
x
x
y
2
y
2x
y
2 1:
2x
y
2 0
y
2
2x
y
y
. x
2x
y
2
x
x
x y
3 1 8
2 1 4
: y
2x
y
2x
1 0 1 2 3 1 1 2 4 8 2
(0 ,1)
x
y
x y
f ( x)
.
3 8
2 4
1 0 2
1
2
x
1 1 2
2 1 4
3 2x
3 1 8
:
f ( x) 3 2 x
: a
y
x
.
154
2
y
y 3 2x x 3 2 1 0 1 2 3 f ( x) 0.375 0.750 1.5 3 6 12 24
x
f ( x) a x
x, y
: f (x
y)
f ( x) f ( y )
f (x
y)
f ( x) f ( y)
f (a x)
( f ( x))a
: -1 . x1
(injective)
x2
-2 f ( x1 )
.
. .
y
a 1
a 1
(0,1)
g ( x) a x , f ( x) a x
155
f ( x2 )
-3 -4 -5
.
-6
log a x y
.
. a ) f ( x ) 2 .3 x b ) f ( x ) 2 .3 x 1 c) f ( x) ( ) x 2 d ) f ( x ) ( 4) x
156
log a x
y
ax
Logarithm
: y(
)
ax (
) x(
0.0001
0.001
10
)
0.01
100
1000
10000
3
104
4
2
10
3
(1 ) : : . y
ax
log a y
x
y (Base)
.
a
. . 157
10
: log10 0.001 log10 10
3
3 a 0,a 1
. (
:
) .
log 2 8 3 log10 1000 3
: 23 8
log 2 8 3
103 1000
log10 1000 3
-1
. a)
log10 N
b)
log 1 36
x 2
6
c)
log 9 81 2
d)
log 5 5 1
-2
. a)
43
256
b)
25
32
c)
10 4
d)
10
e)
y
21
f)
y
3x
10000 10 y
1
158
.
f : IR
1
IR , f 1 ( x) log a x
: IR
ax a
0,a 1
a
: f
R , f ( x)
,
a IR , a 1
: f ( x) f 1 ( x)
.
y
ax x ay
log a x
y y
log a x
x log x
159
f ( x)
0
ax
1
a
0
1
x1 , x2
:
IR
a 1
log a x1
. y
.
log3 x
y
log a x2 , x1
2 1 9
y
y 3x
x 1 y log3 1 1 3
y
log3
x y
1 3 1
:
1 0 1 2 1 1 3 9 3
y
:
x
:1
3x
. x
x2
x 3 y log3 3
(1, 0)
1 3
y
0
log3 3 1
1
3
1 0
1
1
log3 x
(3 ,1)
y
1 ( , 1) 3
y 3x
y log3 x x
1 ( )x , 2x 2
x 1, 2
.
log a x
: log1a
0 , log aa
1 , a IR , a 0 , a 1
160
:
f (1) , f (3 2 ) , f (9) , f (3)
f ( x) log 3 x
:2 .
x
: f ( x)
log3 x
f (3)
log3 3 1
f ( x)
log3 9
f (9)
log3 32
2
f (3 ) f (1)
log3 3
2
:
2
2
log3 1 0
x
.
log 3 x
4
:3 :
: log 3 x
34
4
x
x 81
: -1
.
-2
log a 1 0 x0
.
(1.0)
. x1
1
(injective )
x2
-3 .
.
f ( x) log 2 x
x 16 ,
.
f ( x1 )
f ( x) log 2 x
x
f ( x) log 2 x
f (16) log 2 16
log 2 2 4
f ( x) log 2 x
1 1 f ( ) log 2 ( ) 8 8
log 2 2
161
4 3
3
f ( x2 )
:4 :
x
.
f (32) , f (
.
28 ,
2
1 ) , f (1) , f (2) 32 1 f ( ) , f (1) 81
162
f ( x) log 2 x
f ( x)
log 2 x
-1
f ( x)
log3 x
-2
Common logarithm and Natural logarithm
3 2
1
e
.
10
Briggs)
:
10
-1
: (Briggs)
.
(
log
. : IR log10x , f ( x) log10x
f : IR
log x
100 101 102
.
103
10
1
: :
log 10 0
y
10 y
10 0
y
log 10 1
y
10 y
10 1
y
log 10 2
y
10 y
10 2
y
2
log 10 3
y
10 y
10 3
y
3
log 10
1
log 10 n
y
10 y
y
10 y
10
1
10 n
0 1
y y
1 n
163
x
:
y
y log a x
1 0 1
x
10
...10 3 10 2 10 1 100 101 102 103 log x ... 3 2 1 0 1 2 3 x
Natural
e .
f : IR
-2
: ln
.
logarithms
IR , f ( x) loge x ln x
e
e 2.718281828
e
x f ( x) e x
e
.
ex
y
1 x ) x
.
Exp( x) e x
. x
(1
y
ax
. y
ex
. x y
ex
2 1 7 .3
1 1 2.71
0 1
1
2
2.71
7.34
164
x y
e
x
2
1
0
7.34
2.71
1
y
.
x
e y
e
y
x
. 1 1 2.7
y
ex
x
y
e
x
2 1 7.3
y
ex
x
y
:
ln x
. ln e1 , ln e 2 , ln e3 , ln e0 ln e 1 , ln e
. y
ln e1
y
ey
e1
2
y
e
y
2
y
2
ln e3
y
ey
e3
y
3
ln e0
y
ey
e0
y
0
ln e
e
ln x
y 1
ln e
1
y
ey
e
1
ln e
2
y
ey
e
2
y
1
y
2
165
loge x :
2
: :
y
:
ln x
y y
2 1
1 e
1
1
e
ln x
x
e2
. .
: 1 e3
a) loge e8
b) ln
c) log 0,01
d ) log
1 10 2
166
1 e7 log 0.0001 y
ln
Law of logarithm
. . . . .
I
:
a IR , a 1, log a a 1 a IR I {1} a1
.
a
:
log a a 1
log5 5 1
I
.
51
5
: :
a IR I {1} , a 0 1
log a 1 0
log 5 1 0
( 5 )0
1
: : : .
log a ( x y ) log a x log a y
167
:
:
:
x a p ........... I a q .......... II
y
x y
a p aq
ap
II I
: qp
: loga ( x y)
q
p q
loga ( x y) loga x loga y
:1
50
. log 50 log(5 10)
log 5 log10 log 5 1
:
log 4 2 log 4 8 ? :2
: log 4 2 log 4 8 log 4 (2 8)
log(2 2 4)
log 4 (4 4)
log 4 4 log 4 4 1 1 2
. log a ( x
y ) log a x log a y
log a ( x y ) log a x log a y
: log a
x y
. x ap
........... I
log x
p
y aq
.......... II
log y
q
x y
ap aq
ap
q
x log a ( ) y
log a x log a y
y
aq
: x ap
:
II I
p q :
168
x log a ( ) y
.
q p
log a x log a y .
log 2 0.3010 , log 5 0.6900 log
5 2
log
5 :1 2
log 5 log 2 0.6900 0.3010 0.3890
: :2
log y (10 y 2 x) log y (2 xy)
:
: log y (10 y x) log y (2 xy )
10 y 2 x log y 2 xy
log y (5 y )
log y 5 log y y
log y 5 1
: log a x n
n log a x
: :
log a x n
log a ( x x x ... x)
log a x n
log a x log a x ... log a x n log a x
log a x n 1
log a n x
log a ( x) n
1 log a x n
:
n log a x :
5 log 625 ? :1
log 625 log 54
4 log 5 4(0.6990)
2.7960
: log 3 3 9
2
log 3 3 9
log 3 3 32
log 3 (3) 3
2 log 3 3 3
169
2 2 1 3 3
? :2
:
: log 3 (0.12) ?
log 5 8
?
-1 . a) log 4 (5 x 2 ) ?
b) log10 (10 x 2 y ) ?
c) log10 5 log10 20 ?
d ) log12 36 log12 14 ?
-2 . 63 ? 49 c) loga ( x 2 a ) log a x 2 a) log7
125 ? 80 d ) log10 1000 log10 100 ?
b) log ?
-3
: a) log10 (0.0001)
b) log 2 ( 8)
170
1 3
.
: log a m log a b
:
log b m m by
log a m
log b m log a b y
log a m
y log a b
y
:
a
:
y
: log b m log b m log a b log b m
log a m log a b
log b m. log a b log b b
log b m log a b
log b m
log 9 b log9 27 ? :
. log 9 27
log 3 27 log 3 9
log 3 (3) 3 log 3 (3) 2
3 log 3 3 2 log 3 3
31 2 1
3 : 2
: log3 75 ? :2
. log 3 75
log 5 75 log 5 3
log 5 (3 52 ) log 5 3
log 5 3 2 log 5 5 log 5 3
log 5 3 2 log 5 3
: :
(clog) Co log arithm
: log a
1 M
log a M
co log a M
171
log 2 log 2
1 32
log 2 1 log 2 32 log 2 1 log 2 25
log a M
1 log M a
0 5 log 2 2
x
: :
log M a x
x log M a 1
?:
5
: 1 log M a
: 1 log M a
51
1 32
x
:
1 .......... I
1
:
I
1 log M M log M a x
log M M
ax M log a M
x
log a M
a
1 log M a
x
:
log125 5 1
log125 5
log125 (5) 2
1 log125 5 2
1 2 log 5 125
1 2 log 5 53
1 6 log 5 5
? :1
1 6
:
. log 64 2 ?
log 4 256
?
: log a n x
:
1 log a x n
log a x
. n
log a x
x am
m
x (a m ) n
m
x (a n ) n
172
m
:
m log a n x n 1 log a n x log a x n
log a n x
1 m n
: m
: :
1) log a n x m
m log a x n
2) log 1 n
1 x
3) log a n x n
log n x
log a x
. log 25 125 ? :1
log 25 125 log 5 2 53 : 3 log 5 5 2
3 1 2
3 2
log a n x m
:
m log a x n
log
1 3
(27) 2
? :2
3
: log
1
(27) 2
log
1
(33 ) 2
3
1 (3) 3
3
log
1 3
(3) 6
6 log 3 3 6( 3) log 3 3 1 3
. a)
log 3 6 ?
b) log8 3 4
173
?
18 1
18
e 10 b,a
.
b,a
x
log b x log a x log a b b 10
: log
log e x log e 10
10
log e x
a e
log10 x. log e 10
:
ln x log e 10. log x ln x
2,326. log x log10 x log10 x log10 e log10 x log x log e x. log10 e
b e
a 10
:
log x log e. ln x log10 e 0,4343 :
:
log x
0 , 4343 . ln x
ln 4.69 ?
.
:
ln x 2.3026 log x ln 4.69 2.3026 log 4.69 ln 4.69 2.3026 0.6712 1.5455 ln 6.73 1,9066 .
log 6 , 73
log x 6,4343. ln x log 6,73 0,4343. ln 6,73 0,4343.1,9066 0,8280
. 4
?
b) log 9 27 ?
c) log 8 4 ?
3
d ) log121 14641 ?
e) ln 672000
f ) ln 0.00927
g ) ln 0.235
174
: :
:
a) log 1 3
:
:
log 0,501 log 5,01
Characteristic and Mantissa : log10 1000 3 , log10 100
2 , log10 1 0
log 50,1 log 501 x
S 10 n
x
.
n, 1
.
x
. log x log( s.10 n ) log s log10
n
n
log s n log10 log s n 1
log x
:
s 10
s 10
log s
log x
.
0 log s 1
.
. 0.001
10
3
log 0.001 10 3
0.01
10
2
1
10
0
1000
10
3
4
10
0.602
log 1 10 2
0.001 0 .01
7
10
0.845
10
1 10
log10 7 3
0.602
100 10
175
log 1
10 0
20 101.390 10
20
1
.
( 1
.
10
2
.
100
. 10
1
)
x 10 x 100 x 1000
:
. . Scientific notation N
10
: n
.
N
1 a 10
:1
. a) 2573
a 10 n
b) 573216
c) 0.0028
: a) 2573 2.573 103 b) 573216 5.73216 105 c) 0.0028
28 10,000
28 104
28.10
4
2,8.10.10
176
4
2,8.10
3
: . log526.9 .
:2
3 1 2
:
. log0.002
:3 :
log 0,002 log(2.10 3 ) log 2 log10 3 log 2 log10 log 2 3
. .
log 89435
5 1
4
log 56.784
2 1
1
log 0.995
0 1
1
log 0.0789
1 1
2
177
-3
a) log 0.9560
b) log 956.0
c) log 9560
d ) log 0.0009560
e) log 3.875
f ) log 2345
178
(
) .
10 10
.
.
. :1
log 765 ? log 765 log(7.65.102 )
:
log 7.65 log 102 log 7.65 2
179
76
2
.
0.8837
5
0.8837
765
. N
0
1
2
3
4
5
6
7
8
9
0.8808
0.8814
0.8820
0.8825
0.8831
0.8837
0.8842
0.8848
0.8854
0.8859
74 75 76 77 78 79
log 765 0.8837 2
2.8837 log 70.9 ? :2
. log 70.9
log(7.09 10) log 7.09 log101
:
log 7.09 1
N
0
70
8451
.. .
1 8457
2
3
4
5
6
7
8
8463
8470
8476
8482
8488
8494
8500
9 8506
79
.. .
8506
9 70.9
. log 70.9
70
70.9 0.8506
0.8506 1 1.8506
.
log 0.0247 ? :3
: log 0.0247 log(2.47 10 2 ) log 2.47 log10 log 2.47 2 log 0.00247 log 2.47 3
2
180
0.3927
.
log 0.0247 log 2,24 2 0.3927 2
7
24
2,3927
: . log 0.0247 3927 2
2.3927
log 9280 ?
. 0.007 ,
3 , 900 ,105 , 15 4
:4 .
log 15 (3 5) log 3 log 5 0.47712 0.69897 1.17609 log(105) log(5 3 7) log 5 log 3 log 7 0.69897 0.47712 0.84570 2.01079 log(900) log(9 102 ) log 9 log 102 0.95424 2 2.95424
0.95424 2 log 10
3 log( ) log 3 log 4 0.47712 0.60206 4 0.12494 log(0.007) log(7 10 3 ) log 7 log 10
3
3 .84510
181
0.84510 3
10 1.0000
9 0.95424
8 0.90309
7 0.84570
6 0.77815
5 0.69897
4 0.60206
3 0.47712
2 0.30103
0.0000
1
-1 . a) log 222
b) log 0.921
c) log 928
d ) log 527
e) log 0.024
f ) log 2400
g ) log 0.00024
h) log 24
. a) log(2.73)3
b) log 5 0.0762
182
-2
Anti Logarithm
y
x
anti log x :
.
34
y
log a y
.
x
log 34 1.5315
1.5315
N
log N
:
2.8779
. 0. 8779
. . 2 .
: 5
75.5
N
2.9939 :1
log N
N
.
75
0.9939
: .
.
0.9939
9.86
9.86
3
2 N
986
log 986 2.9939 anti log 2.9939 986 9.5 9.6 9.7 9.8 N
0.9912
0.9917
0.9921
0.9926
0.9930
0.9934
0.9939
0.9943
0.9948
0.9952
0
1
2
3
4
5
6
7
8
9
183
log N
0.9791 :2
9791
:
953
: N
9.53
log 9.53 0.9791 anti log 0.9791 9.53 N
.
.
log N
3.0531 :3
: .
1
.
. N
:
0.9469
8
885 N
0.000885
anti log( 3.0531) 0.000885
: :4
. a) 2
b) 0.2
c) 0.02
d ) 0.0002
: a) log 2 0.3010 b) log 0.2 0.3010 1 1 .3010 c) log 0.02 0.3010 2
2 .3010
d ) log 0.0002 0.3010 4
4.3010 0.3010
.
a) anti log( 5.0521)
b) anti log 4.9479
184
B
30
Linear Interpolation A
C B
.
A
C
log C
B
B
. C
A
log B b , log A a
c
log C
log C
b,a a
b
. : . 1.2345
. log 5.235 :1
: 5.240
. 185
5.230
5.235
. log 5.230 0.7185 log 5.240 0.7193 log 5.230 log 5.235 log 5.240
5.230 5.235 5.240 :
.
0.7185 log 5.235 0.7193 0.7185 x 0.7193
: log 5.235
:
x
. 5.240 0.010
0.005
0.7193 0.0008 x d 0.7185
5.235 5.230
: d 0.0008
0.005 0.010 5,235
d
0.005 0.0008 0.010
d
0.0004
d 0.0004 0.7185 0.7189
.
log 5.235 07189
: log 0.0007957 :2
: log 0.0007957 log(7.957 10 4 ) log 7.957 log10 4 log 7.957 4 log10 log 7.957 4
.
186
-4
:
7.96
7.95
7.957
. log 7.96 0.9009 log 7.95 0.9004
log 7.950 log 7.957 log 7.960
7.950 7.957 7.960 :
log 7.957
.
0.01
0.007
7.96
0.9009
7.957
x
7.950
0.9004
0.007 0.01 0.007 0.0005 0.01
x
0.0005
d
d 0.0005 d
0.00035 0.0004
d
. 0.9004 0.0004 0.9008
.
0.0007957
log 0.0007957 0.9008 4 4.9008
x
.
:3
4.5544
. x
anti log 4.5544
: :
log x
4.5544 4 0.5544 log( t 104 )
log t log104
log t 4
187
0.5551
d
0.5544
x
.
0.01
0.5539
3.59
0.5551
t 3.58
0.5544 0.5539
0.0005 0.0012 0.0005 0.01 0.0012 0,0042
0.0005
0.0012
d 0.01 d d
0,000005 0,0012
0,0041667
d
. t
3.58 d 3.5842
log x log x
3.58 0.0042
log(3.5842 104 ) log 35842
: x 35842
x
a) z
log 0.001582
b) x log 6.289
188
t
Exponential and logarithm equations
.
log 2 ( x
2
1) 3 5
x
5
1 x 2 2
x
. (
.
) 2x
1
32
:1
. 2x
1
25
x 1 5 ,
x
:
6 83 x 1
83 x
1
24
2 3( 3 x
1)
24
:2
24
.
:
3(3 x 1) 4 9x 3 4
9x
9x 7
x
8 8
3
7 1 9
7 3 3
2
2 2
4
4
4
2
8 8
4 3 7 9
7 1 3 4 3
24 2
4
4 3 3
(2 )
24
:
4
189
(16) x
x
1
64 x
2
:
. : log 2 ( x 2 1) 3 :
:1
x
log 2 ( x 2 1) 3 x 2 1 23 x2 x2
1 8 9
x2
9
x
3
:
.
log 3 ( x 2)
2 log3 9
:
x
:2 :
log3 ( x 2) 2 log3 9 log3 ( x 2) log3 9 2 x 2 92 x 81 2 79
x 2 81 x 79
log
5
x log 5 3 log 5 5 log
190
5
4 0
x
:3
:
. log
x
5
log
5
3 log
5
5 log
4
5
3.5 4
log
5
15 4
x
15 4
log3 (32 x
x
2)
x 1
:4
: log 3 (32 x
2)
x 1
32 x
32 x 3 x 1 2 0 32 x (3 x ) 2 3.3 x 2 0
2 3x
3 3x
:
1
2 0
:
3x
t
t 2 3t 2 0 (t 1)(t 2) 0 t 1 0 t 1 t 2 0 t 2 3x 1 3
x
3x
2
x2
30
x1
0
log3 2 x
log( x 2
36) 2 log( x) 1
:5 :
log( x
2
36) 2 log( x) 1
log( x
2
36) log( x) 2
log x2
1
x 2 36 x 2 36 log 10 10 x2 x2 36 10 x 2 10 x 2 x 2 36 0
9 x 2 36 0 9x 2 x2
36 4
191
x
x12
2,
a) 113 x c)
1
x1 11
log x 3 4
2,
x2
2 b) 7 2 x
1
d ) log5
3x
3
x 1 x 2
192
2
.
log M N
log M
log N
. 3.17 88.2
.
:
.
:
log(3.17 88.2) log 3.17 log 88.2 0.5011 1.9455 2.4466 0.4466
.
0.4472 , 0.4456
: anti log 0.4472 2.80 anti log 0.4456 2.79
0.01
2.79
0.4456
t
0.4466
2.80
0.4472
d
0.0006
0.0016
193
d 0.0006 0.01 0.0016 0.01 0.0006 0.0006 d 0.00375 0.0016 0.0016 t 2.79 0.00375 2.79375
:
log x log(2.79375 102 ) x 279.375 anti log 2.4466 279.375 3.17 88.2 279.375
.
. 74.2 62.0 ?
log
M N
log M
log N
. 8750 3.49
. log
8750 3.49
:1
log 8750 log 3.49
: :
log 8750 3.9420 log 3.49 05428 log 8750 log 3.49 3.9420 0.5428 3.3992
194
.
8750 3.49
anti log 3.3992
2507.16
2507
: 374 16,2
.
. (1.05)6
.
: :
log(1.05)6
6 log1.05 6 (0.0212) 0.1272
anti log 0.1272 1.340
.
.
2
(694) 3
195
?
:
-1
:
-2
0.097 7.78 ?
a)
32.2 25.1
?
b)
8 737 2
:
196
(964) 3
-3
f ( x) a x
a
.
a 1
a
:
: . f ( x)
. x1
0,x
(injective)
x2
f ( x1 ) a 1
a 1
.
(0,1)
. x
g ( x) a , f ( x) a x
y
.
f ( x2 )
: y
ax
log a y
x
.
.
: -1
. log a 1 0
.
x0
1
(1.0)
. f ( x1 )
.
f ( x2 )
x1
x
.
g ( x)
-2
-3
x2 log 1 x
-4
f ( x) log a x
a
10 .
:(
log
Briggs system e
. log e x
)
ln x
197
log a a 1 : log a 1 0 : log a ( x. y ) log a x log a y : x y
log a
log a x log a y :
log a x n
n log a x : 1 : log m a
log a m
log a m log a b
log b m
1 log a x : n
log a n x
n
1 S 10 , log x
:
n log s
:
. (logs)
.
:
. x
y
log a y
x
: y
anti log x :
: .
. .
: . : . -
198
. 1 log 2 ( ) 4 d) 3
: a) 4
b)
4
c) 3
b
: a)
1 4
b) 81
c)
b) 4
c) 6
b) 3
4
c) 4
b) 3
-4
x
d ) 13.5
log 2 16 ? -5
: a) 4
-3
d) 9 log 18 log 2 x log 3
: a) 2
-2
log3 81 log 0,01 ?
: a) 0
1 4
logb 4 81 d)
81
-1
c) 5
d)
4
log 1 125 -6
:
5
a) 3
b)
3
c) 4
d) 5
log 1
: 1 a) 2
2
1 2
b)
c) 1
d)
a) x
3
b) x 9
c) x
1
3x
: 9
b) 1
c) 2
1
x
9
d) x 3 log 234,21
: a) 0
1 log a m
b) log a m
-8 -9
d) 3
-10
: a) log a m
1 -7 2
1 log a m
199
c) log a m
1 log a m
d)
33 x
a) 3 x
2
b) 32 x
94 x 1
1
64 x 2 b
c) log 3 ( x 2) 2 log 3 9
d ) 16 x
e)152 x
f ) log x 1 1
1
7x
-1
x
:
1 log x 2 h) log 5 ( x 1) log 5 ( x 2) log 5 2
1
g ) log(4 x 3) 2 log 20
-2
: a) log 3 (12 x 2 ) log 3 (8 x 3 y 2 ) log 3 (2 xy 2 ) ? b) log 5 ( c) log
4ab x ) log 5 ( )b ? x 100ab
2
43 2
?
-1
: a) log 8 3 4
1 ? 243 8 d ) log( ) ? 128
?
c) log10 4 100
b) log 3
?
3
e) log10
10 0,1
-2
: a) 1.7300
b) 0.8954
c) 4.5682
d ) 2 .1987
-3
: a) 89500
b) 91
c) 65.3
d ) log 0.002
-4
: a ) 2,01 52 99
b) (0,0062) ( 34,8)
-5
: a) 0.888 256
b) 17.3 7.47
-6
: a) (7.42) 3
?
b) ( 84.7) 2
c)
418
200
d)
0.21
201
202
203
204
Matrixes
(5 5 25)
.
M ( x , y)
.
M ' ( x' , y ' )
x
.
M ( x , y)
. ) S ', S
y
P
P'
.
(
:
x
x'
1x
0 y
x'
y
y'
0x
1y
y'
.
1
0
0
1
M
M '
y
y'
x
M'
x'
1 0 0 1
M x' y'
.
205
x y
1 0
. S ', S
1 0
P' , P
0 1
1 0 1 , 0 1 0
0 1
0 1
. 1 0
0 1
1 0 0 1
1
0 1
0
: Matrix
. ... C , B , A
.
.
. . 2 3 4
A
1 7 2
4 5 1
B
1 2 3
C
i
j
0
4 7 3
2
a
aiJ
j i
i 1, 2 , 3 ... , j 1, 2 , 3 ... .
A
a11 a12
a13 ...a1n
a21 a22
a23 ...a2 n
am 1 am 2
am 3 ...a m n
n :
m n
m
A ( m n)
A
A
. 206
ai j
m n
. 2 3 4 1 5 6
C
1 3 5
B
A
( x) 1
2 3 1 5 1 2
A
D
a c
b d
A
1
. :
. a) (a i j ) 2
2
(i
j) 2
b) (aij ) 3
2
(i j ) 3
2
2
: :
2 2
a
a11 a12 a21 a22 aij
i
j
a11 1 1 2 , a12 1 2 3 , a21 2 1 3 , a22 2 2 4 2 3 : 3 4
2
: a11
a12
a21
a22
a31
a32
3
3 2
a11 1 1 1 , a21 2 1 2 , a31 3 1 3 a12 1 2 2 , a22 2 2 4 , a32 3 2 6
a11
a12
a21
a22
a31
a32
1 2 2 4 3 6
:
.
207
(b
b
2
a
a b
1
2 3
1 2
2 : 3
. 1 , 1 2 2
: A
1 2 4 2 5 6 3 6 0
1 3 1 0
B
2 4
–2
: a) ( 2 i 3 j ) 3
3
b)
i j
–1
3 3
208
:Row Matrix A ( 4 5 9 0 )1
–1
4
:Column Matrix 5 3 2
A
: 3 1
:Zero Matrix 0m
. 02
4
0 0 0 0 0 0 0 0
03
2
2 4
A
0 0 0 0 0 0
A
–3
n
3 2
:( square Matrix ) (m
: 1 3 4 5 7 9 1 0 2
–2
–4
n)
3 3
a11 , a22 , ... , ann
Main Diagonal
a1n , ... , a n1
. 209
Minion Diagonal
a11 a12
...a1n
a21 a22
...a2 n
an1 an
...an n
4 1 1 3
:Diagonal Matrix
:5
: B
1 0 0 5
1 0 0 0 3 0 0 0 3
A 2 2
3 3
:Scalar Matrix
–6
: K 0 0 K A
0 :
0 :
0 ... 0 ... K ... :
0
0
0 ...
0 0 0 : K
n n
K
.
8 0 0
8
:Unit Matrix I
. I2
1 0 0 1
,
I3
1 0 0 0 1 0 0 0 1
,
In
1 0 0 0
210
0 1 0 0
0 ... 0 0 ... 0 1... 0 0 ...1
0 8 0
0 0 8
: –7
3 3
. 3 3
. . -1 Triangular Matrix
.
Upper Triangular Matrix Lower Triangular Matrix
. B
. A
1 0 0 2 4 0 3 7 9
,
A
1 2 3 0 7 9 0 0 4
B
A
( A)
A aij
.
: ( A)
. A (a i j ) m
n
A ( a i j )m
)
A
.
m n
(
n
:
A
2 1 2
4 2 3
5 0 4
A
2 1 2
4 2 3
5 0 4
211
a) A
3 1 0
4 2 1
5 3 0
b) B
d) D
3 0 0
0 3 0
0 0 3
e) E
g) G
3 2 1 0
h) H
0 0 0
0 0 0
5
0 0 0
c) C
1 0 0 1 0 0
6 7 8
f) F
1
3 1
212
1 0
2
0 0 1
Addition and subtraction of Matrix .
–1 B (b i j ) m
A B C
bij
aij
A (a i j ) m
n
n
Ci J
Cij
. Am
n
Bm
C
n
( a i j )m
m n
n
( b i j )m
( a ij
n
b i j )m
n
(cij ) m
n
: A
1 2 2 0 1 7
B 3 2
1 2 1
A B
2 0 7
3 1 0
2 2 4
3 1 0
2 2 4
3 2
1 3 2 1 1 0
2 2 0 2 7 ( 4)
4 1 1
4 2 3
C3
2
–2 . B ( bi j ) m
. Am
Bm
n
.
n
( a i j )m
n
( b i j )m
n
B
A B.
213
( a ij 2 3
b i j )m 1 4
n
A ( ai j ) m
n
(cij ) m n C m n
A
2 3
1 1
n
-1 A B
B
A
A B B A ( A B) C A ( B C )
-2
. ( Identity Element) A 0 0 A
A B
?.
11 1
5
0
3
0
2
5
6
B
A
-3
A. 1
2
3
2
5
4
6
0
1
:1
. A B
: 1 2 3 2 5 4 6 0 1
11 1 5 0 3 0 2 5 6
A B
.
1 2 6
A B.
11 0 2
2 1 5 3 0 5
.
A B
B
3 1 3 2 5 1
3 5 4 0 1 6
B
10 1 2 2 4 5
A 1 2 3 4 0 6
A
:2
B A B
3 2
.
: A
2 3
. a) A
1 4
b) A
2 1
2 1 2 1
3 0
,
,
B
B
3 0 2 1
1 1 2 1
0 1 , c) A
1 1 2 0 1 3
214
,
2 4 5
B
1 3 4 1 0 1
a b c d
A
K
.
KA
a b c d e f g h i
A
K
a b c d e f , KA g h i
A
. 5 0 0 5
B
.
. . KA .
ai j
A (ai j )
K IR
K
C
ci j
ci j
KA
:
K
A
2
1 3 6 2 1 0 1 0 1
K ( ai j )
: :
1 3 6 KA 2 2 1 0 1 0 1
21 2 2 12
3 2 12 0 2
6 2 0 2 12
215
2 4 2
6 2 0
12 0 2
(
) B A
: a)
( A B)
b) ( c)
(
A
B
)A
A A
A) (
)A
( A)
2,
3 6 3 9
3, A (
.
A) (
)A
: ( A)
: ( A) (
3 2
3 6 3 9
3
23 62 32 92
) A (3 2)
3 6 3 9
6
3 6 3 9
A) 2 3
3 6 3 9
2
33 36 33 93
A) (
)A
( A)
( (
3
1
3
6 12 6 18
63 66 3 6 96 2
9 9
18 36 18 54 18 18
18 27
18 36 18 54
1 0 , A 1 0
2 , B
36 54
2 1
0 1
–1
. .
1 A K
KA
216
K 3
A
5 1 1 0 1 2 0 0 1
–2
Multiplication of two Matrixes B A A B B A
B ( bi j ) n
. C ( ai j ) m
p
A ( ai j ) m
p
n
.
.
. Am
n
Bn
C
p
m p
:
. . . n
( ai j ) m
n
( bi j ) n
ai j bi j ( Ci j ) m
p
:
p
i, j 1
.
A B
B
217
1 0 1
A
1 2 0 1 0 1
:1
:
: 1 0 1
1 2 0 1 0 1
A B
11 2 0 0 1 1 1 0 0 1 ( 1)
1 0 0 1 0 1
1 2
. . A B
1 3 2 0 12
B
.
A
2
3 2 1
1 2
:2 . :
2 2
A B
3 1
2 3
1 3 2 0
1 2
1
A B 2 1 2
2 3
1 2
1 2
3 2
3 1 0
2 1 3 2 ( 1)( 1)
2 3 3 0 ( 1) 2
1
2
( 2)(1) 1 2 2( 1)
2(3) 1 0 2 2
1
3
2 6 1 2 2 2
2 1
2
3
2 1
2
0 2
A B
.
B
6 0 2 6 0 4
3 2 0 6 1 7
A
9 2
1 3 5 2
4 2
2 2
:3 :
AB
1 3 5 2
3 2 0 6 1 7
13 3 6 53 26
1 2 31 5 2 21
3 18
2 3
0 21
21 5
15 12
10 2
0 14
27 12 14
218
21
10 37 50 27
BA AB
B
4 0
2 1
0 3
2 0
A
1 1
. DC CD
3 4
D
4 3
C
2 1
1 2
:4 . :
CD
DC
2 1
1 2
3 4
4 3
2 ( 3) ( 1)( 4) 1 ( 3) 2( 4)
6 4 8 3
2
11
3 8 4 6
11
2
3
4 2
1
3 2 41
4
3 1
2
4 2 1 ( 3)
6 4
3 8
2
11
8 3
4 6
11
2
2 4 ( 1)( 3) 1 4 2 ( 3)
3 4 2 4( 1) ( 3) 2
CD DC
.
A
:1 AB
C B,A
:2
.
( AB) C A ( BC )
B
BA
: :3 :
a) A ( B C) AB AC b) ( A B) C AC BC c) K ( AB) ( KA) B A (KB) K IR d ) IA AI A
219
: a)
0 2
c) 3
1 0
5 0 0 1 2
1
? 2 1 1
2 1
b)
d)
2 4
3 2 3 2
3 4
?
3 2
?
220
Transpose of a Matrix
1 2 1
A
1 0 3 1 0 2
. . . (m
n
)
:
. . .
T
A
1
0
2
1
3
4
A :
AT
.
n m
A
1 2 3 0 1 4
A 3 2
1 4 5 4 2 3 5 3 0
A : 2 3
AT
Symmetric Matrix
:
1 4 5 T
A
a b c b c f
: .
c f d
221
A
T
A
4 2 3 5 3 0
:
. ( AT )T
T
( ai j )T
A
( a ji )T
ai j
A
A :
( AT )
aij
(a j i )
: T
A B
AT
BT
A B C ...
T
AT
BT
CT
...
AB
T
B T AT
A
T
AT
( A)T
AT
2 5
B
B T , ( A B )T
AT
IR
:
AT
: ( A B )T
:
3 2
A
3 2
C
5 2
7 1
0
6
3
B
,
:
BT
:
. 1 4
2 1
3 1
2
1 0
3
2 3
3 3
: B
3 1 2 1 0 1
C
3 5
4 0
2 6
BT 2 3
7 1 3
C
T
3 3
3
1
1
0
2
3
3 2
1
4
0
5 7
2 1
6 3
3 3
B A
. A
1 0
2 1
3 5
0
2
3
, 3 3
.
B
0 4
4 5
2 3
2
3
0
-1
3 3
4
222
3
–2
Determinant ad cb
.
A
. a11
a12
a21
a22
A
detA
|A|
a11 a12
A
a21 a22
. n
n
n n
A
. a11
a12 ... a1 n
| A | det A a12
a 22 ...a 2 n
an1
a n 2 ...a
nn
n
ai j
.
n n
n
n
2 2
| A|
a c
b d
a b c d
A
: a d bc 3 7
A
. | A|
3 7
4 2
A
.
223
4 2
3 2 7 4 6 28
1 6
4 3
: 22 :
A3
: a11
a12 ... a1 n
| A | det A a12
a22 ...a2 n
an1
an 2 ...an n
:3 3
3
n n
A
.
:
2 2
: .
a11
a12
a13
a21
a22
a23
a31
a32
a33
(a12 a23 a13 a22 )a31 2 2
: .
a11 a12 a13 a21 a22 a23
(a11 a23 a21 a13 )a32
a31 a32 a33 2 2
: .
a11 a12 a13 a21 a22 a23
(a11 a22 a12 a21) a33
a31 a32 a33
A
3,2,1
: .
a11 a12 a13 | A | a 21 a22 a23
(a12 a23 a13 a22 ) a31 (a11 a 23 a21 a13 ) a32 ( a11 a22 a12 a21 ) a33
a31 a31 a33 a12 a23 a31 a13 a22 a31 a11 a23 a32 a 21 a13 a32 a11 a22 a33 a12 a21 a33
224
B
2 5 4
6 1 1
3 2 . 7
: :
2
6
3
I) 5
1
2
4
1
7
6 2 1( 3) 4
2
6
3
II) 5
1
2
4
1
7
2
6
3
III) 5
1
2
4
1
7
12 3 4 15 4 60
(2 2 5( 3))( 1)
(2 1 5(6)) 7
I II III 60 19 196
4 15 19
(2 30) 7
28 7
196
117
A
.
3 4 1
0 3 2 1 1 0
:
: : a11 a12 a13 a11 a12 A a21 a22 a23 a21 a22 a31 a32 a33 a31 a32
:
A
| A | (a11 a22 a33 a12 a23 a31 a13 a21 a32 ) (a13 a22 a31 a11 a23 a32 a12 a21 a33 )
.
225
a11 a12 a13 a21 a22 a23 a31 a33 a32 a11 a12 a13 a21 a22 a23
:2
: |M |
3 2 1 4 3 0 5
|M |
3 4 5
2 1 3 3 0 4 2 6 5
2 6
2 3 2
3 3 6 2 0 5 ( 1)( 4)( 2) ( 1) 3 5 3 0( 2) 2( 4) 6 : (54 0 8) ( 15 0 48) 46 63 109
2 1 0 1 2 3 . 1 2 1
| A|
| A|
-1
. a)
c)
1 2
0 4
b)
1 1 0 2 1 3 1 5 6
5 3
2 7 0 4 2 5 1 3 2 1 5
d)
-2
. a)
3 5 1 2 4 1
6 0 7
b)
226
1 5 1
2 1 2
3 7 3
.
| A|
1 2 3 1 2 3
0 0 0 1 2 3
| A|
2 1 4
2 1 4
2 1 3 T
|A |
| A|
| AT | | A |
0
0
0
a d
b c e f
0 ,
| A|
-1
m
0
a
b
0 0
c d e f
0
A A
A
n m
| A |n
. | A|
.
| A|
.
4 2 6 5 4 1
A
.
-2
| A |mxn a a
b c b c
d
e
| A|
f
a a
b c b c
d
e
.
0
f
–3
A nxn
. a | A|
b a
d
c b
e
c f
a a
b c b c
d
e f
( 0) 0
.
| A| 0
. | AT |
-4
| A|
. 227
. .
2 | A| 1
1 2
4 2
3
1
0
:1 :
. 2 A
B
1 4
1 2 3 1
2 0
(0 6 4) (24 4 0)
1 2 2 1
4 2
1 3
0
10
(0 24 4) (4 6 0)
20 10 10
A A
B
: k . . 16 3 22 8 2 21 20 1 25
A
4
. | A|
| A|
.
16 3 22 8 2 21 20 1 25
:
4 3 22 4 2 2 21 5 1 25
.
: 0 1 2 a) 0 3 4 0 5 6
:2
1 2 4 b) 7 9 11 0 0 0
3 5 8 c) 0 4 2 0 0 1
228
2 2
Multiplicative Inverse of 2 2 Matrix
d
a .
.
2 1 3 2 2 1 3 2
A
A
.
B
. B
A (ai J ) n
:
n
AB BA I n : A
1
A AA
1
B
A 1 A In
.
(singular) A
| A| 0
:
non – singular
| A| 0
. . B
7 2
3 ,A 1
1 2
3 7
:1 :
A B
1 2
7 6 14 14
3 3 6 7 7 2
B A
.
3 7
3 1
7 2
3 1
( 1) ( 7) 3( 2) 2( 7) ( 7) ( 2)
( 1) ( 3) 3( 1) 2( 3) 7( 1)
1 0 0 1 1 3 2 7
7 6 2 2
21 21 6 7
B A
1 0 0 1
AB BA
229
I .
:Ad joint of Matrix
(
)
Ad joint = Adj : A
a b c d
K
1 2 5 7
d c
Adj A
b a
7 5
Adj K
2 1
| A| 0 . a b
. A
1 Adj A | A|
1
a b c d
A c d
|A|
0 2 2
: | A|
3
2
5
6
6 8 5 8
2 8 3 8
3 5
A
2 6
:1
18 10 8 0
:
. A1
1 Adj A | A|
3
2
5
6
1 8
3 4 5 8
6
2
5
3
1 4 3 8
9 4
5 4 15 15 4 4
3 4 5 4 a b
A
A
.
1
3 4 5 8 3 4 9 4
1 4 3 8
1
0
0
1
:
I
c d
1 | A|
d
b
c
a
| A| 0
-1
. a) A
5
1
10
2
b) B
5
19
4
15
-2
. 1) A
1 2 3 1
2) B
1 1
2 3
3) C
230
1 1
3 1
a1 x b1 y
:
a2 x b2
c1 c2
. . . . A
a11
b12
a 21
b22
x
X
,
B
,
y
x
B
IX A
1
A
A B
1
AX
A x y
X
1
B
A
1
(A
1
A) X
A
1
1
A .
B
B x 2y 5 x 3y 7
. 5 7
B
X
x y
1 2
x
5
1 3
y
7
A
C2
B .
AX
C1
1 2 1 3
,A
1 2 1 3
:1 :
3 2 1 0
231
| A| 0
: 3 1
Adj A A
1
2 1 3 1
1 | A|
X
A 1B
X
x y
2 1
1
A
3 1
X
2 1
3.5 2.7 1.5 1.7
1 ,x 1 , 2
3 1
1 1
2 1
3 1
2 1
5 7 15 14 5 7
2
y
5x 2 y 2 3x y 3
.
:2 :
B | A|
2 3
, X
5 3
2 1
,
A
5 3
5 6 1
A
0
x y
2 3
: A
1 1 Adj A | A| 1
1
X
A 1B
X
x y
x 4,
1 3
2 5
1.5 2.3 3.2 5.3
1 3
2 5
1 2 3 5
2 3 2 6
2 15
4 6
y 9
232
:
y x 2x
3y
4
4x
6y
1
A
2 4
2 4
3 6
| A|
2 4
3 6 x y 3 6
4 1
B
4 1
x 12
X
x y
A1B 12
12 12 0
A .
233
:3
. a)
2x y 1 5x 2 y 2
b)
3 p 5q 7 2 p 4q 6
c)
a b 11 4a b 9
234
Crammer's Rule x, y, z
A
.
a11 x a12 y a13 z d1 a21 x a22 y a12 z d 2 a31 x a32 y a33 z d 3 A
a11
a12
a13
a21
a22 a23
a31
a32 a33
| A| 0
! x y z
z
| Ax | | A| | A y| | A| | A z| | A| | Az|
y,x
z
y,x
.
| A y| , | Ax|
. a11
a12
a13 d1
a21
a22
a23 d 2
a31
a32
a33 d 3
235
(
(x
)
|Ax|
)
| A y|
3 3
. )
| A z|
(y
) 3 3
3 3
.
| Az|
| A y | , | A x| x 3y 3 2x y 2
.
:1 :
A 1 2
x
y
1 2
3 1
1 ( 6) 1 6 7 0
3 3 1 2 | A x| | A| | A y| | A|
3 2
3 1 | A|
1 3 2 2 7
3 ( 6) 7 2 6 7
3 6 7
9 7
4 7
:2
. 3x 2 y 2 z 4 x 3y z 5 2 x 2 y z 11
236
: 3 | A| 1 2
2 3 2
2 3 1 1 1 2
2 3 2
9 4 4 12 6 2 21 8 29 0 | A| 0
. 4
2 2
4
2
5
3 1 5
3
11
2 1 11
2
| Ax| | A x| A
X
58 29
4
23
4
| A y| 1
5
1 1
3
2
11
1 2 11
| A y|
58 29
A
| Az |
10 68
58
2
3
y
12 22 20 (66 8 10)
15 8 22 ( 20 33 4) 23 22 57
58
2
3
2
4 3
2
1
3
5
1
3
2
2
11 2
2
99 20 8 ( 24 30 22)
71 16 87 z
| Az | A
87 29
3
z
. 3)(2) 2(2) 2( 3) 4
4
2 3(2) 3
6 4 6
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k k
0
a1b2 k a1c2 j b1a2 k b1c2 i c1a2 j c1b2 i (b1c2 i c1a2 j a1b2 k ) (c1b2 i a1c2 j b1a2 k )
269
(b1c2 c1b2 ) i (c1a 2 a1c2 ) j ( a1b2 b1a 2 ) k (b1c2 c1b2 ) i (a1c2 c1a 2 ) j ( a1b2 b1a 2 ) k u v
i
j
k
a1
b1
c1
a2
b2
c2
(b1c2 c1b2 ) i (a1c2 c1a 2 ) j (a1b2 b1a 2 ) k
b
a b
4i 2 j k
a
2i
:2
j k a b
.
i 6 j 8k
:
: a b
(2 i
j k ) (4 i 2 j k ) 8( i i ) 4( i
j ) 2( i k )
4( j i ) 2( j j ) ( j k ) 4( k i ) 2( k j ) ( k k ) 0 4k 2 j 4k 0 i 4 j 2 i 0
3i 6 j
Triple Product(
) : . (i
(a b ) c b, a c c
. ( a b)c
.
b
(b c ) a
(c a ) b
(c a )b
(b c ) a
a a1 a 2 a3 a (b c )
b1 b2 b3
(ii
c1 c2 c3 a b c
b (c a )
a ( a b)
0
270
c (b a )
(iii (iv
a(b c)
z
c b,a h a
e
e c
b c
o
b
:
|b c |
: V | b c | (a e ) b (a c ) e
y
V
b (a c ) | b c | h
x
b
2i
j 2k
a
-1
4i 3 j k
a b
: :
a b
i
j
k
4 2
3 1 1 2
7 i 6 j 10 k
b
a b
a
b
: b a
i 2 4
j k 1 2 3 1
7 i 6 j 10 k
7 i 6 j 10 k a
(7 i 6 j 10k )
b a
a b
-2 sin(
y
) sin
cos
A
sin
OB
OA
: x
x
0
sos
B
271
y
x
. AOB OB
cos(
) i sin(
)j
OA cos
i sin
ˆ :
ˆ
j
: OA OB
(cos i
j ) (cos i
sin
k ( sin cos OA
sin cos )
OB
j)
sin
k sin(
i cos cos
k sin( )
c
b sin B
c sin C
b a
: ABC
: A
A
c
B
a b c b
b c C
B
C
a
c
: (b c ) c
a c
(b c ) ( c c ) c c
0
b c
a c c a
-3
ABC
:
CA BC
k 0 0
)
) sin(
a sin A
j sin sin
c a b c
c a
272
AB
0
a
...(i )
: b c Sin(
A)
b c SinA b SinB
c a Sin(
c a SinB a SinA
B) bSinA aSinB
.......... (ii )
SinB b
SinA a
.......... (ii )
b
: (b c ) b
i
a b b a
(b b ) ( c b ) b a
:b b 0 c b sin A
(c b ) b a b a sin C
c sin A a sin C / ac a c sin A sin C
...iii ,
sin A a
sin C c
...iii
(iii) (ii)
: a sin A
v v
b sin B
u
c sin C
-4
:
u
. :
u
h
u sin
h
u sin
v
:
v
u v sin
273
u v
. : : 1 ( 2
1 (u v ) 2
)
V
u
a3
3t 2
2t
2
a2
2t 2 t , a1
t2 t 2
-1
. b
4i 6 j 8k , a
-2
2i 3 j 4k
. -3
B (2,1, 1) A(1, 1,1)
C ( 1,1, 2) ABC
.
-4
R(2, 1,4) Q( 1,2.4), P(0,0,0)
S (1,1, 8)
v
4i
v u (iii
274
2j
k
u
2i
u v (ii
j
k
-5
u u (i
: . Position Vector
. ay
y
. Q ( x2 , y 2 )
a
ax ay
P( x1 , y1 )
x2 y2
a
a
:
x1 y1
ax2
a
a
:
ax ay
a
ax .
x
a
PQ
1 0
:
PQN
a y2 , ( x2
x1 ) 2 ( y2
Q
x1 x2 2 y1 y2 2
xm
M
PQ
y1 ) 2
ym
P
Q
P
: . j
x
k
0 0 1
0 1 0
j
z
.
i
0 1
i
1 0 0
y
y x
: u v
u v cos
275
:
v
u
v
u u v
: u v v
u
v
u v sin
n
u
u v
. : u v u V u v
v
u
a
.
a1 i b1 j c1 k
a b
b
: b
a
a2 i b2 j c2 k
: a b
i a1 a2
j b1 b2
k c1 c2
b1 b2
c1 i c2
a1 a2
c1 a j 1 c2 a2
b1 k b2
: b
a
: u
h
u sin
v
u v
276
b
:
4 i 3 j 2k
a
3i
a b :(b
OQ
OP
a b :(a
Q(61 2) xi
CD
B(2,0) A(1, 1)
:(2
P(2,3)
PQ
yj
:(3
AB D ( 2,2
C
: 2 CB 2 CA
w
5i
j 3k
(2, 6)
? :(iii)
v
:(1
j 5k
B
( 1,1)
3 i 2 j 2k
u
:(4
(2,5)
A
? :(ii)
2 AB CB
C ( 1,3
AB ? :(i)
:(5
i 2j k
: i) : u
2v
ii ) : v 3 w
w
w b a
v
iii ) : 3 v w
?
u
: (iv
a b
b
:(6
a b a
a
a b b
: a b
4i 2i
j 2k j k
:(iv)
a
3i 2 j k
b
i
j
:(iii)
a
i
j
b
i
j
:(ii)
a
2i
b
i
j k j k
:(i) :(7
: i ) : P(0,0,0), Q (2,3,2), R ( 1,1,4) i i ) : P(1, 1, 1), Q(2,0, 1), R (0,2,1)
277
:(8
: i ) : A(0,0,0), B(1,2,3), C (2, 1,1), D(3,1,4) ii ) : A(1,2, 1), B(4,2, 3), C (6, 5,2), D (9, 5,0) iii ) : A(1, 1,1), B( 1,2,2), C ( 3,4, 5), D ( 3,5, 4)
:(9 i) : u
5i
j k, v
ii ) : u
i 2 j k, v
j 5k , w i
j k, w
15i 3 j 3k 2
278
i
2
j
279
150kg ? 170cm 280
Coefficient Variations
.
150kg ? 170cm
. . .
.
. CV : CV
(
)
S x
. 100
. CV % 100
S x
data
.
data
.
data
. . 281
data data
data
. 1
3
X
S2
:1
5
xi
1 3 5 9 3 n 3 3 ( xi x ) 2 (1 3) 2 (3 3) 2 n 3
S
2.67
CV
S x
2.67 3
(5 3) 2
4 4 3
8 3
2.67
0.543
A
:2 1875
B
1495
B
310 280
. (
A
) : :A
C D SA
SA XA
280 1495
CV A % 0.87 100 18.7%
0.187
:B C D SB
SB xB
310 1875
0.165
CVB % 0.165 100 16.5%
A
.
CV A
-1
: 1
3
CVB
4
5 6
10
282
6 4
-2 -3
Normal Carve ": x
3S x
2S
x
S
x
x
S x
2S x
".
3S
.
B A
.
A B
. . S
x
68%
(x S , x S)
. 96%
( x 2S , x 2S )
. 283
( x 3S , x 3S )
99%
. 3S
2S
data
data .
3S
. 12500
: 700
: (
. 1400
x 3S
.
(
x 2S , x S
(
S
1500 700
X S
11800-13200
68%
X 2S
11100-13900
96%
X 3S
10400-14600
99.6%
1400
1500
:
S
1400 x
: 1500
.
2 .1 x
.
99.73%
95.45%
2S
1400
( x s , x s)
.
284
68.28%
A B XA
XB data data
.
data .
: -1
:(skewness) .
.
.
:
(
. 3
1 n
x )3
( xi S3
. 3
3
0
3
0
3
0
. SK
P
3( x med ) S
:
:
285
(
. -2
:(kurtosis) . : 1 n 4
( xi
x)
S4
1 n
x )4
f i ( xi S4
S
.
x
xi
fi
. . :
4
3 (
.
)
: 3
4
3
.
.
4
. 3 .
.
4
(
)
-1
. 4
6
10
4
4
2
40-50
50-60
60-70
70-80
80-90
90-100
. 286
.
. data
. . . . . . . . .
287
: :
.
(1, 8) , (2 ,3) , (1,7) , (3 , 5) , (2 , 4)
. data -
.
data
:
: 1
2
3
4
5
1
2
1
3
2
8
3
7
5
4
:
1 2 3 4 5
1 2 1 3 2
8 3 7 5 4 .
data
-1
288
y
Scater Diagram
x
.
1,2 , 2,3 , 3,4 , 4,5
. . .
data
. y
. y
. x
y Y
: 1, 8 , 2 , 3 , 1, 7 , 3 , 5 , 2 , 4
1
2
X
3
. . data . . 289
:
:
( )
(
( )
)
()
y
( )
x
(
x
)
( ) y
. x
y
x
( )
.
y
x
() y
. y
(5000)
.
( )
y ( ) (
. (
)
290
)
y
D C B A
y4
y3
y2 y1 x1
x2
x3
x
x4
.
(
)
( )
( )
()
y y x
( ) ( ( ) (
)
) () ( )
y x y
() ( )
() ( ) ( ) (
) x
291
y
x
.
y
. r
. y
x
( x
n
r
( x
)( y
)( y
xy xy n sx s y
)
)
data : . (x)
(y)
y x
1 2 1 3 2
1 2 3 4 5
8 3 7 5 4
9
8 6 7 15 8
27
44
. y
n 5
y
: x
S 2x
S y2
9 1 .8 5
x
y
x
x
28 5 .6 5 (1 1.8) 2 (2 1.8) 2 (1 1.8) 2 (3 1.8) 2 (2 1.8) 2 5 0.64 0.04 0.64 1.44 0.04 2.8 0.56 5 5 (8 5.4) 2 (3 5.4) 2 (7 5.4) 2 (5 5.4) 2 (4 5.4) 2 5 6.76 5.76 2.56 0.16 1.96 17.2 3.44 5 5 y
292
:
y
x
x
data
y n
(1 8) (2 3) (1 7) (3 5) ( 2 4) 5
10.2 (1.8)(5.6) 0.74 1.86
r
0.12 1.376
44 5
8.8
0.09 :
0.09
y x
. :
: x
y
x
y
x
1 3
1
2
1 2 .5
2 5
2
6
2 5 .5
3 7
3
6
3 6.5
4 9
4 10
4 8.5
y
y
y
x
(
y
x
x
)
( )
( )
.
( (
)
)
( ) . ( )
( ) ( )
: x
S 2x
10 4
. 2 .5
x
y
(1.5) 2 (0.5) 2
24 6 4 (0.5) 2 (1.5) 2
4
293
:(
(
)
)
2.25 0.25 2.25 4
5 1.25 4
32 12 12 32 4
y y
x 70
x
2.5
x
=1.25 y
x 72
x
2 .5
x
=1.25 y
9 1 1 9 20 5 4 4 (1 3) (2 5) (3 7) (9 4) 3 10 21 36
(2.5)(6) 17.5 15 2.5 4 1 2.5 1.25 5 6.25 24 :( ) 6 y 4 16 0 0 16 y 4 2 12 18 40 72 (2.5)(6) 3 4 0.9486 1.25 8 10 23 y 5.75 4 y 4.6875
x
8
:( )
-
16.75
4
16.75 (2.5)(5.75) 1.25 4.6875
x y
(.
70
2.375 5.858
0.9812
y
)
-1 1
.
.
data -1
x 1 2 3 4 5 y 4 3 2 1 0
. -2
294
Regression : y
ax b
.
y y
(
ax b
.
Dy
a, y
b
Dx
b
) ax b
x
y x y
. x
.
y y
0
0
a
ax b
ax b
y a
a
y
ax b ax b
0
.
y
d3 1
.
-
d1 1 d2
2 x
d1
. a
2
d2
2
y
d3
2
d1 d 2
d3
ax b
a
a
295
.
y
ax b
( x , y)
.
. - 1 +1 . x
y :
y ax b
:
( x4 , y4 ) ( x3 , y3 )
( x2 , y2 )
( x1 , y1 )
y
( .
ax b
)
.
(Minimum) :
data
x 1 5 9 y 6 5 7
: y
y
0.5
3
0.5 2 1
1
x
x
( )
(
296
)
(
)
( ) .
( ) ( ( 2) ( 1) 3 0 : (0.5) ( 1) 0.5 0 :
. 2 . ( 2) 2 (0.5) 2
:
(
)
( 1) 2
( 1) 2
(3) 2 14
(0.5) 2 1.5
( ) .
(
)
y
y
: (y
y )2
y (ax b)
2
( y b ax) 2 y1 ) 2
(ax1 b
(ax2 b
.
y2 ) 2 ...
b a
y x
b a . a
b
r
Sy
y
Sx
x
y ax
,
b
x y nx y x
2
nx
xy
2
x2
. 297
b
a
)
: . data :
. x 1 2 3 4 5 y 4 3 2 1 0
x
.
y :
: xi
1 2 3 4 5 5 n yi 4 3 2 1 0 5 n 2 ( xi x) 4 1 0 2 5 2 ( yi y ) 4 1 0
x y Sx
2
Sy
2
1 4
n 1 n
2
Sx
2
2
Sy
2
1 4
n 5 4 6 6 4 0 20 4 5 5 1 (4 (6)) xy ( x)( y ) 5 sx s y 2 2
xy
r
15 3 5 10 2 5
4 6 2 2
2 2
1
: a b
y
sy
( 1) 2 1 1 a sx 2 y a x 2 ( 1)3 2 3 5
r
. y ax b
y
2
x
x
x 5:
y
298
y
2x 3
:
: C V
.
S x
CV
data
100
data .
.
: . . data
: . SK P
3
: 3( x med ) S
Sk p
3
1 n
( xi S3
. 1 n
4
( xi
x) 3
4
x )4
S4
. : .
. . . .
299
S : x
: . . ( ( ( -
: . . y
x
.
(
)
.
r
1 n
xy ( x )( y) sx sy Sx
y
y
y
x
x
xy
x
y
. (
x
Sy
:
) .
b a .
y
x
x
a
r
b
y
Sy Sx ax
r x
Sx y
Sy
.
300
y
S2
y
64
x
-1
50
2 x 10
y
x
-2
20%
-3
.
x
S x x S
x
2S x
x
2S
x 3S
x
x
3S
-4 . (
( ( -5
20%
: (2,10) , (3,10) , (3,14) , ( 4,10) , (4,14) (5,14) , (5,16) , (6,12) , (6,16) , (6,18) (7,14) , (7,18) , (7,20) , (8,16) , (8,18)
. .
(
) Data -6 x 1 1 2 3
.
y 1 5 4 2 y
-7
. - 1 +1
y
301
-8
12
-9
B A
. A: B:
65 68
63 66
67 68
64 65
62 69
70 66
66 68
68 65
67 71
78 67
69 68
71 70
: .
(
b a .
. x
y
Sx
1 x b 2 3) y x b
1) y
( ( y x
Sy 2) y 4) y
20
1 x b 2 x b 20%
-10
-11 .
11
10
9
8
7
6
5
4
3
2
1
12
10
16
6
10
6
16
18
12
8
18
10
14
10
6
10
10
14
18
8
10
16
20
19
18
17
16
15
14
13
12
12
14
14
6
12
18
16
10
12
16
14
12
8
12
12
16
12
6
-
. 1.5
-12 .
0
5
10
20
30
40
50
90
10
118
122
126
132
136
. .
302
(
) (mm)
303
304
Permutation
.
: n
:
:
n (n 1) ... 2 1
nn
n n ... n n
n
(1 2 3 ….n)
) n! .
1! 1
n
0! 1
-1 (
n (permutation)
n
-2 ) Pn
(. Pn
305
n! :
n
k
Pn(k )
: Pn( k )
n! , k k!
n
: 8! 5! 3!
-1
n! (n 1) n !
-2
n! 1 2 3 ... n ! :
:1 :
3! 1 2 3 6 5! 1 2 3 4 5 120 8! 1 2 3 4 5 6 7 8 40320 n! 1 2 3 4...n (1 2 3....n 1).(n)
n(n 1)!.
:2
: 16
:2 .
16 . 16! k Pn( k )
n! k!
n ,
(k
n)
: 16 4
: P16( 4)
16! 16! 13 14 15 16 (16 4)! 12!
306
43680
:
pn
( k1 , k 2 ,...k n )
n
k1 , k 2 ,...k n n! k1!, k2!, k3! , ... , kn !
p ( k1 , k 2 , ... , k n )
: 55544
: P3 (2 , 3)
5! 10 2! 3!
:3 :
: :
55544 55454 54554 45554 45545 45455 44555 45455 54455 55445
5
"
-
"
:4 3
5
:
3 5 3 15
(
:
5 278
)
:5 :
:
3
4
2
4 4 3
307
3
2
.
: P (43)
4
4! 24 (4 3)! 4 3 2 1! 24 1!
2
.1 . 10
.2
. 4
2 .(
308
3
.3 )
Combination 2 1 2 1
. .
. k
n k
n
: k
: n k
n! k! (n k )!
, k , n IN ,
2 k
0 k
n
( a b) 2 k
a b
2
Ck2 a 2 k b k
a2
ab
b2
k
n IN n
n 0
IN
2 2
n n
. .
n k
0 ,1, 2
0 ,1, 2 ,
2 0
1
1
( a b) n k
0 ,1, 2 , 3 , 4
4 k
. . 309
2 k
0
:
nk n
k
n 0 n r
n
:
n n
1 , n r 1
r
n :
1
( I)
n 1 r
(II)
r
:( III ) C(nr )
7
.
:
4 7 7 7 4
C(74 )
7! 4!(7 4)!
4
:
4 7 6 5 4! 35 6 4!
: 4
7
:2
. B
A B
ABCD A
C CABD
D
C D
. 7 4
. P( 4)
7
7! 4!
7 6 5 4 3! (7 4)!
7 6 5 4 3! 840 3!
4
-1
G,F,E,D,C,B,A
-2 3
310
2 .
6
4
-3
Combination
3
5 . 3
5
. . n n
k
:
k
. : n
C(nk )
k C(nk )
k n k
n n k
n! , k!(n k )!
311
0 k
n :
:
4
30 4
C430
30
4
30! 30 29 28 27 26! 27405 4!(30 4)! 26! 4!
3 : C35
5 3
:1
27405
. :
30
5! 1 2 3 4 5 3!(5 3)! 1 2 1 2 3
A
:2
1 2 3 4 5
5 3
:
2 5 10
7
10
-1 10
7 5 -2
. n
312
Pn( 2 ) Cn( 2 )
36
-3
Variation 10
k
-n k
....
: n
k
: .
vkn
4
k! Ckn
variation
vkn
: k!
n! k!(n k )!.
k! Ckn
n! (n k )!
vkn
n! (n k )!
30
:
30
4
: :
313
v430
30! (30 4)!
30 29 28 27 26! 657520 26!
k n
k
permutation
n
Pn
combination
Ckn
Variation
vkn
k
n! , n
k!
n k
Pkn
k
n! k!(n k )!
n k
n! (n k )!
n
n! k!
n k 1
Ckn
k
vkn
nk
: 1 2
. .
T=
STSS
S=
T
SSTT
314
):
4 1, 2 , 3 .
0 0
1
2
1
0
1
1
3
4
2
1 0
1
1
1
1 16
1
4
2
6
3
4
4
1 16
8
3
4
3
1
16
16
8
2
2
0
4
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