3
1 1(MM ) (NN )
(OM) (ON)= 1 1
(OM ) (ON )
(OM) (ON)= 1 1
1 1
(MM ) (NN )
(OM ) (ON )=
3.1
. - - MM1 NN1 , 1O M M
1O N N
, :
,
,
, 1(MM )
(OM)1(OM )
(OM)1
1
(MM )
(OM )
, - . , , , , , .
, 1M O M
, :
1(MM )
(OM) = ( )
11
50 3:
, - ,
, 00 3600
Oxy , Ot Ox - Ot (. , ). Ox , Ot .
(x, y) M1 x'x (. ). (. ), , , :
, ,
1( ) x = , 1( M) y = 2 2(OM) x y 0= + = > . ,
:
, , , .
1(MM )
(OM) = 1
( M )
(OM)
= 1
1
(MM )
(OM ) = 1
1
( M )
(M )
=
y =
x
=
y
x =
x
y = 2 2x y 0 = + >
1( M )
(OM)
=
1
1
(MM )
(OM ) =
1
1
( M )
(M )
=
( ) ( )
( )
2
1 1
2
x xx
t
y
t
(x,y)(x,y)
yy
3.1 51
1
1
( M )
(M )
=
2 2x y 0 = + >
, - ( ).
:
(x, y) ( )
2 2x y 0 = + >
.
360
Ox - Oxy - . 300 , Ox
360 30 390 . = + =
360 , - :
360 , = + N* 00 < 3600 Ox, , - 30 , Ox - 360 30 390+ = :
, ( x0) ,
, ( y0)
y =
y
x =
x =
x
y =
2 2x y 0 = + >
(360 30 ) 390 . = + =
y
x
390
30
0
0
0
y
t
t
x 30
390
0
52 3:
- :
( 360 ), = + N 0 360 .
( ) Ox .
Ox , , - , 360 , + . Ox , , , 360 , + - .
, k 360 , + k , .
, k :
, ( x0) ,
, ( y0)
y =
y
x =
x =
x
y =
2 2x y 0 = + >
(k 360 ) , + = (k 360 ) + =
(k 360 ) , + = (k 360 ) + =
3.1 53
(0,0) - =1 - . - .
- , .. - 35 , = (, ).
35 =
=1
35 0,57. =
, , , . - , - .
, 35 =
=1, 35 0,82. =
, (x, y), :
x'x , y'y .
:
1. ' , 1. :
= x = = y =
1 1 1 1
7060
5040
20
10
350
340
330320
310300
290280270260
250240
220210
200
190
180170
160
150
140130
120110
100 90 80
y
x
(x,y)
N(,)
x
y
35
0,1
0,1 0
o
o
o
o
o
o
o
o o
o
o o o o
o
o
o
o
o
o
o
o
o
o
o o
o o
o
o
o o
o
54 3:
2. , , .
10 20 30 40
+ + + + + + + +
- M(x, y). - . 1 ,
( ) ( )( )
( ) 1
= = =
yE , (AE)= yE ,
= yE.
.
:
= yE =
, x =1 , .
AO
B
B
M
A
t
x
y
E(1,y )E
3.1 55
. , AB (, ) ( 1rad), . , ( rad) S .= - :
( 1 rad ) , -, ( 1 rad).
- , :
, rad . - 2,
360 2 rad.
,
1 rad 3602
,
,
rad 180
.
, 180 =
, :
180 =
:9 60 ,
180 =
= 600
180 3 = =
603
= rad.
A
1rad
B
O
56 3:
9 56
rad ,
180 =
56
=
556 150
180 6 180
= = =
56
rad=150 .
- .
rad
0 0 0 1 0
30 6 1
23
23
33
45 4 2
22
21 1
60 3 3
212
3 33
90 2 1 0
0
, x rad x,
(x rad), ( x rad), (x rad) (x rad),
x, x, x x.
, .. rad3
3
(100rad ) 100 .
3.1 57
1 h , . .
:
h48 , =
h48
=
h70 , =
h70
=
20m = =
h h 20,48 70
=
20 70 48
h .70 48
=
70 2,75 48 1,11.
61,05h 371,64
37m .
2 750o .
750 360 2 30,
1750 (2 360 30 ) 30
2 = + = =
3750 302
= =
3750 303
= = 750 30 3 = =
750 2 360 30= +
(1) :
58 3:
3 793
rad.
79 79
2 .3 6=
79 6
13 1. 79 79 12 13 2 13 2 ,
3 6 6 3 = = + = +
:
79 313 23 3 3 2 = + = =
79 13 3 2
= =
79 33 3
= =
79 33 3 3
= =
1. x, y .
2. .
3. S = 6cm. , :
i) = 1cm ii) = 2cm iii) = 3cm .
4. rad i) 30 ii) 120 iii) 1260 iv) 1485 .
5. :
i) 10
rad ii) 5
6
rad iii) 91
3
rad iv) 100rad .
6. i) 1830 ii) 2940 iii) 1980 iv) 3600 .
A
x
260
3
B
B
A
y
0
6
300
300
4
3.1 59
1. - , - 1000 - (1 0,3 m) .
.
i) h =30 ,45 60 .
ii) , h=1000 ;
2. :
i) : () = () = 2 45 = 2.
ii) ( ) 4 22,5 . =
iii) ().
iv) ,
, ( ) 2 2 2 . = v) 22,5 . vi)
;3. .
4. 1mm . - .
1.000 pi
N (N)
() (pi)
h
700
A B
1 1
22,5o 3 345o
A
B
6
300
300
60 3:
3.2
. .
:
1. 2 2 1 + =
M (x, y) , :
x = y =
,(OM) =1
2 22 2 2( ) x y x y = + = + :
2 2x y 1,+ =
:
2 2 1 + =
2. = =
:
y
x
= =
( x 0= )
x
y
= =
( y 0= ).
AA
t
y
y
xx
1
|x|
|y|
(x,y)
O
3.2 61
(1) (2), - .
3. . = 1
:
=
=
( 0 0)
:
1 = =
.
4.
22 2
2 2
11 1
= =
+ +
i) 2 2 1 + = 2 0 :
2 22 2
2 2 2 2 2
1 1 111
+ = + = =
+ .
2 21
.1
=+
ii) 2 2 1 + = 2 21 ,
1 =
+
:2
2 2 22 2 2
1 11 1 .
1 1 1
+ = = =+ + +
2
22 .1
=
+
62 3:
1 513
= 90 180 ,<
3.2 63
1. 3x5
= x ,2< < -
x rad.
2. 2x3
= 3
x ,2
< < -
x rad.
3. 3x3
= 3
x 2 ,2< < -
x rad.
4. 2 5x5
= 0 x ,
2
< < -
x rad.
5. x 2 = 3 x 2 ,2< < -
2 x x
.1 x
+
6. , x :
i) x = 0 x = 0.
ii) x = 1 x = 1.
iii) 3
x5
=
4x .
5 =
7. , M ( x, y) x = 3 y = 3, O(0,0) = 3.
8. x = 2 y = 3 , 9x2 + 4y2=36.
9. x = r , y = r z = r , x2 + y2 + z2 = r2.
10. :
i) 11
=+
ii) 4 4 22 1 = .
64 3:
11. :
i) 1 21 +
+ =+
ii) x x 2 .1 x 1 x x
+ =
+
12. :
i) + =+
ii) 2 2 2 2 . =
13. :
i) x x x x1 x 1 x
+ = +
ii) 1(1 x) 1 x x
x + =
iii) 1 x xx x
= +
iv) 1 1x x x x.x x
=
B
1. x + x = , -:
i) x x ii) 1 1x x+
iii) x x + iv) 3 3x x. +
2. :
i) 4 4 2 2x x 1 2 x x + = ii) 6 6 2 2x x 1 3 x x. + =
iii) 6 6 4 42( x x) 3( x x) + + - x , .
3. x ,2 2
< < 1 x 1 x
2 x.1 x 1 x
+ =
+
4. 0 x ,2
< 1 x 1 x 1 x x
.x 1 x1 x 1 x
+ + + = =
+
3.3 1 65
3.3 10
, , 0 90 .
' ' .
' , ' = , , - , x'x. . , - :
1( 30 ) (30 )
2 = =
3( 30 ) (30 )
2 = =
3( 30 ) (30 )
3 = =
( 30 ) (30 ) 3 = =
9 , :
24 4 2 = =
24 4 2 = =
14 4 = =
1
4 4 = =
: . :9 :
( ) ( )
( ) ( )
= = = =
A
(x,y)
(x,y)
A
y
x
t'
t
66 3:
180
' -
180 , ' 180 , = , , ' y'y . . , - :
:
180o . :
9 150 180 30 ,= :
9 2 ,3 3 = :
2 33 3 3 2 = = =
(180 ) (180 )
(180 ) (180 )
= =
= =
1150 (180 30 ) 30
2 = = =
3150 (180 30 ) 30
2 = = =
3150 (180 30 ) 30
3 = = =
150 (180 30 ) 30 3 = = =
tt
y
x
A A
(x,y) (x,y)180
B
B
0
3.3 1 67
180
'
180 , ' 180 , = + , , - ' . - . - , :
2 13 3 3 2 = = =
2 33 3 3 = = =
2 33 3 3 3 = = =
:
180o -, . :
9 210 180 30 ,= + : 1
210 (180 30 ) 302
= + = =
3210 (180 30 ) 30
2 = + = =
3210 (180 30 ) 30
3 = + = =
210 (180 30 ) 30 3 = + = =
(180 ) (180 )
(180 ) (180 )
+ = + =
+ = + =
y
A A
B
(x,y)
(x,y)
t
t
y
x
180 +0
68 3:
9 4 ,3 3 = + :
4 33 3 3 2 = + = =
4 13 3 3 2 = + = =
4 33 3 3 = + = =
4 33 3 3 3 = + = =
90
' - 90 , ' 90 , = , , ' xOy . . , - :
,
90o , - .
, 60 90 30 ,= :
360 30 ,2
= =
160 30 ,2
= =
60 30 3 = = 360 30
3 = =
(90 ) (90 )
(90 ) (90 )
= =
= =
t
t
(x,y)
(y,x)B
A
B
A90
y
xO
y=x
0
3.3 1 69
, 0 90 .
1 1 536 .4+
=o
54 .o
54 90 36 ,=
2 2 1 + =
2 254 54 1, + = : 22 2 1 5 6 2 5 10 2 554 1 54 1 1 ,
4 16 16 + +
= = = =
:
:
2 - :
) 90 ,+ ) 270 ) 270 +
i) 90 90 ( ),+ = :
(90 ) (90 ( )) ( ) . + = = =
90 .+
54 10 2 554 .
54 1 5
= = +
1 554 364+
= =
10 2 5544
=
54 1 55454 10 2 5
+ = =
70 3:
ii) 270 180 (90 ) = + , : (270 ) (180 (90 )) (90 ) = + = = .
270 .
iii) 270 360 90 360 ( 90 ),+ = + = + : (270 ) ( 90 ) (90 ) + = = = .
270 .+
1. : i) 1200 ii) 2850 .
2.
i) 1876
rad ii) 21
4
rad.
3. : i) ( ) = + ii) ( ) 0+ + =
iii) 2 2 +
= iv) .2 2 +
=
4.
5. :
9( x) (2 x) x
21.
21(13 x) ( x) x
2
+ + = +
6. :2 2( x) ( x) (2 x) 2 x .
2 + +
( ) (180 ).
( ) (90 )
+ +
3.3 1 71
1. :
2. :
2
5 7(5 ) (7 )
2 21.
5 7(5 ) (7 )
2 2
+ + =
+ +
3. x x 5,3 6 + + =
-
:
2 2x x .
3 6 + +
4. :( x)
0 1.x ( x)
+< < + +
I) - , , - .
1. =1, = 0 . 2. = 0 , = 1. 3. + =2 . 4. 21 = 5. 2 220 70 1 + =
6. x (x ) x =
7. x 2 2x x =
8. (x ) x 0,2
+ = x 0 =
9. x (x ) ( x) 06 3
+ =
495 120 495 ( 120 ).
( 120 ) 495
+ +
72 3:
II. .
III. .
1. ( =900 ) , :
) 2 2 1, + = ) 2 2 1, + = ) =1. 2. : ) ( + ) = , ) ( + ) = , ) ( + ) = . 3. :
) ( ) ,2 2
+ = ) ( ) ,2 2
+ = ) ( ) .
2 2+
=
1 1200
2 1500
3 2100
4 3000
5 2100
6 3000
7 3000
8 2100
3
32
33
12
12
33
32
3
3.4 73
- -
112
- -
. t,
1
t 12
+ 1
t 12
. -
, t, -
t, 1
t 12
+ , 1t 12
.
112
.
t.
, t, t + 2 sec t2 sec.
3.4
74 3:
( t) 2 sec.
:
, , 1 1. -
. .
- , , , .,
f (t) t,= t .
- .
:
x (x rad) .
x (x rad) =
( 360 ) ( 360 ) , + = = x :(x 2 ) (x 2 ) x + = =
2. .
f , >0 , x A :
i) x T A,+ x T A
ii) f (x T) f (x T) f (x)+ = =
f.
3.4 75
x (x rad). =
2.
, : x
xx
=
:
{ }1 x x 0= x 1
(x ) (x ) x, + = = .
, , :
xx
x
=
:
{ }2 x x 0= .
f(x) = x
f(x) = x 2, 2, . [0,2]. x xrad . - , , . :
x 0
,2
.
, - f(x) = x
0, .2
f(x) = x :
y
xx rad
B
A
M
M
O
76 3:
,2
3,2
3 ,22
x ,2
= 12
=
3
x ,2
= 3
1.2
=
:
. [0, 2]:
. :
x 0 2
x 0 1 0,71 0 0,71 1 0,71 02 0,712
4
2 3
4 5
4 3
2 7
4
x
x
2 3
2
20
0 1 0 1 0. .
y
x
0x2pi
O
1
12pipi
4
3pi2
7pi4
y = x
43pi4
pi 2pi5pi
3.4 77
f(x) = x , 2, [2, 4], [4, 6] . [ 2 ,0],[ 4 , 2 ] . , .
. - x ( x) x. = f(x) = x 0(0,0) .
f(x) = x
f(x) = x 2, 2, .. [0, 2]. :
- :
x
x
2 3
2
20
1 0 1 0 1. . .
x 0 2
x 1 0,71 0 0,71 1 0,71 0 0,71 1
4
2 5
43
2 3
2 7
4
22
pi/2pi/2
1.5
0.5
0.5
1
1.5
0
1
pi 2pi3pi/2 5pi/2
x
y
78 3:
y = x
0 x 2 p .
f(x) = x 2, :
y
x
pipi 0
1
1
2pi 3pi 4pipi2
pi2
5pi2
7pi2
y = x
3pi2
. x (x) = x. f(x) = x - y'y.
f(x) = x
f(x) = x ,
, .. , .2 2
( , ,2 2
).
x rad - . x .: x
2
2
' , . f(x) = x
, .2 2
y
x
0x2pi
0
1
1
pi
2pipi
43pi2
7pi4
y = x
5pi4
3pi4
2pi
y
O
x
3.4 79
x 2
x .
x2
= -
f. x 2
x .+ x2
= f.
f(x)=x , - , :
- . f(x)=x .
f(x)=x , ( 5.3: (x)=x .
x 0
x 1 0 1
2
3
4
6
6
4
3
2
3 1,7 3 1,73 0,6
3
3 0,63
x2pipi pi 2pi
23pi
2pi
23pi
2pi
y = x
yy
O O
BE
MB
B
A AA A
B
x x
yy
O O
BE
MB
B
A AA A
B
x x
80 3:
1 f(x)=3x .
f(x)=3x - (x)=x. 2, :
f (x 2 ) 3 (x 2 ) 3 x f (x)+ = + = = , x . f (x 2 ) 3 (x 2 ) 3 x f (x), = = = x .
- f(x)=3x.
2 f(x)=2x
f(x)=2x , 2x 2, , x - . , f(x)=2x . :
f (x ) 2(x ) (2x 2 ) 2x f (x),+ = + = + = = x f (x ) 2(x ) (2x 2 ) 2x f (x), = = = = x. , - f(x)=2x.
3 f(x)=32x .
3, 3 .
x 0 2
x 0 1 0 1 03x 0 3 0 3 0
2 3
2
x 0
2x 0 1 0 1 0
4
2 3
4
y = xy = 3x
y
2pi pi123
321 2pi 3pi 4pi
xpi
y = x
y = 2x
x3pi2 2 2 2
pi 3pi 2pipi pi
1
1
y
3.4 81
, f(x)= x, ,>0: (i) ,
. (ii) 2 .
f(x)= x, ,>0
1. ,
i) f (x) 2 x,= g(x) 0,5 x,= h(x) 2 x,= 0 x 2 ii) f (x) 2 x,= g(x) 0,5 x,= h(x) 2 x,= 0 x 2 .2.
f (x) x=
g(x) 1 x= + h(x) 1 x= + 3.
f (x) x= g(x) 3x,= 0 x 2 .4.
f (x) x= g(x) 3x,= 0 x 2 .
5. xf (x) 2 .2
= -
; ; f .
x 0
32x 0 3 0 3 0
2
4 3
4
f(x)=32x :
- .
y
x
3
3
pi 2pi3pi3pi0pi pi pi2
pi4 2 2 2
y = 32x
82 3:
6. xf (x) 22
= .
7.
i) f (x) x= ii) g(x) 1 x= + iii) h(x) 1 x= + .
8. f (x) 2x.=
9. f (x) x.=
B
1. :
2.
y 3 t ,6 =
y
t .i) -
.ii) 0 t 12.
pi
pi
pi
pi
pi
pi
pi
2pipi
pi pi 2pi
pi pi 2pi
pi pi 2pi
2pi
2pi
2pi1
1
1
1
3
2.5
0.5
x
x
x
x
x
x
x
y
y
y
y
y
y
y
3.5 83
3. 1m. ,
1
h 1 3t,3
= +
t . i) -
. ii) iii) -
0 t 2 .4. H x
- x(t) =0,1+0,1.3t, t . i) .ii) 0 t 2 . 0,15m;
3.5
x=
1
x .2
=
y = x
1
y .2
=
x, f (x) x= -
1
.2
2,
x, (. ), 2 . 2, .
xy=x
pipiO
2pi
yy=
pi6
12
5pi6
pi1m
84 3:
-
1
x2
= [0, 2], 6
5 ,
6 6
=
5 1.
6 6 2
= =
1x
2 =
, x = , = , :
,
x 2= +
x 2 ( )= +
x 23
,
x 23
= = + +
1 3x2
=
3 ,
3 2
= 3
( ) .3 2
=
x ( ),3
= :
2 12x4 2 + =
1 ,
6 2
= 2x4 6 + =
,
y
M M
x5pi/6
pi/6
12
AOx 2 6,
5x 2
6
= + = +
3.5 85
2x 2 2x 2 x4 6 6 4 24
+ = + = + =
72x 2 2x 2 x
4 6 6 4 24 + = + = + = +
x =
,
.. 1
x .2
=
-
1
x2
= [ , ] 3
,3
1.
3 3 2 = =
1
x2
=
,
x24
,7
x24
= = +
2x 24 6
,
2x 24 6
+ = + + = +
x 23
x 23
= + =
y
x
M
M
pi3pi3
12
86 3:
,
, x = , = ,
1 2x2
=
2 ,
4 2
= x ,4
=
:
2 32x2
=
3 ,
6 2
= 3
6 2 =
5 3.
6 2
=
5
2x ,6
=
x 2
x 2
= +
=
x 24
,
x 24
= + =
52x 2
6,
52x 2
6
= + =
5x
12,
5x
12
= + =
3.5 87
x =
x 3. = - . , , , ..
,2 2
, .
, x 3 =
. ,3
3.3
=
x 3 = : x ,3
= + .
, x = , x = , :
x ,= +
x . =
1 x 1 =
1,4
= 1.4 =
x ,
4 =
x ,
4
=
2 x 3 =
x ,6
= +
3,6
= x ,6
=
pi xpi pi
y
pi22
3
88 3:
1.
i) x 0 = ii) 2x2
= iii) x 0 = iv) 2
x2
=
2.
i) 1
x2
=
ii) x 1 = iii) 2
x2
=
iv) x 1 =
3.
i) x 0 =
ii) 3
x3
= iii) x 1 = iv) x 3 =
4.
i) 3
x3
=
ii) 3
x3
=
5. i) (1 x)(2 x 3) 0 =
ii) (2 x 2)(1 x) 0 + =
6.
i) ( 3 x)(1 x) 0+ =
ii) 2(2 x 1)( x 3) x 0 + = 7. -
i) x 0,951 =
ii) x 0,809 = iii) x 28,636 =
8. i) 2 3x 3 =
ii)
x 1 05
+ = iii) 2x3 3 07
=
9.
i) x 13 + =
ii) 2 3x 1
4 =
iii) 5x 3
4 =
10. i) 22 1 0 + = ii)
22 x 3 x 2 0 + = iii) 23 t 3 2 3 t = +
11. i) 2 2x 5 x 4 + = ii) x 2x 1 = 12. x,
:
i) f (x) 3 x ,2 =
0 x 2 , < ii) g(x) 7 x ,2
=
0 x 2 <
3.6 89
13. ( -
) tS 75 50 ,
6
= +
t o t = 1 . i) 100000 , ii)
. 1.
i) x x 04 + =
ii) 2x 3x 0
3 + =
2. i) x x 1 x x + = + ii) 2
12 x 4
x =
3. x 1 = (3,4).
* 4. 1+x=x [0,2).
5. : x x3 = +
[0, 2).
3.6
, - 1 2M , M (. 1). , - . (. 2).
O O
y y
x x -
-
M (, )1M (, ) M((-), (-))2
(1,0)
. (1) . (2)
(1,0)
90 3:
, 1 2M ,M , :
1: 2: : 1 0 : ( ) ( )
2 1 M OM AOM ,= = 2 1(M M ) (AM).= 2 2
2 1(M M ) (AM)=
:2 2
1 2 2 1 2 1(P P ) (x x ) (y y ) ,= +
1 1 1P (x , y ) 2 2 2P (x , y ), :
2 2( )= +
(2)
2 2 ( ).=
2 22 1( ) ( ) =
2 2( ) 2 2 ( ) + =
( )+ =
:
:
(1)( ) = +
, , 00 360 , < < , .
, :( ( )) ( ) ( ) = + =
( ) + =
22 1(M M )
2 2
2 2 2 2
( ) ( )
2 2= +
= + + +
2( ) 2 2
2 2
[ ( ) 1] [ ( ) 0]
( ) 1 2 ( ) ( )
= +
= + +
3.6 91
(1) (2) - , - . , :
(1), , - .
, :
x x2 =
x x,
2 =
:
( ) ( ) ( )2 2
2 2
+ = + = = + = +
:
(3)
:
( ) + = +
( ) =
15 (45 30 ) 45 30 45 30
2 3 2 1 2( 3 1)
2 2 2 2 4
= = +
+= + =
75 (45 30 ) 45 30 45 30
2 3 2 1 2( 3 1)
2 2 2 2 4
= + =
= =
15 (45 30 ) 45 30 45 30
2 3 2 1 2( 3 1)
2 2 2 2 4
= =
= =
75 (45 30 ) 45 30 45 30
2 3 2 1 2( 3 1)
2 2 2 2 4
= + = +
+= + =
92 3:
: (5)
, :
(6)
:
(7) (8)
( )1 +
+ =
( )1
=+
1( )
+ =
+1
( )+
=
, :
3145 30 3 3315 (45 30 )1 45 30 3 3 31
3(3 3)(3 3) 12 6 3 2 3
6(3 3)(3 3)
= = = =
+ ++
= = =
+
( )( )
( )
1
+ + + = =
+
+ + = =
- + , , .
, : ( ), + , ( ) 0, + 0 0. :
0
3.6 93
-
1 3 ,5
= 3 22< <
12 ,13
= 3 ,2
< <
+.
(+) = + ( ) + = . :
2 2 9 161 1 ,25 25
= = = 3 22< <
2 2 144 251 1 ,169 169
= = =25 5 ,
169 13 = = 3
2
Top Related