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