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Νέο Βιβλίο Άλγεβρας Β λυκείου 2012-2013

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: . .

: 1991 : 1992, 1993, 1994, 1995, 1996, 1997, 1998, 2012 .

. , . , . . , .

. (2010) , . , . , . , . . . : , (2010), . , . , . , . . . (2010), (2010). : 1 , , . , . 2 , , , . 3 . , , . . 4 . 5 , .

1 : 1.1 .......................................................................9 1.2 ..............................................................24

2: 2.1 -- ............................30 2.2 - ...........................40 3: 3.1 ..................................................49 3.2 ...........................................60 3.3 1 ......................................................65 3.4 ..................................................73 3.5 ............................................83 3.6 ...........................89 3.7 2.......................................97 3.8 ...................103 3.9 f(x)=x+x ..................................................108 3.10 ....................................................................... 114 4: - 4.1 ..................................................................................128 4.2 ................................................................132 4.3 .....................................140 4.4 .......150 5: 5.1 ....................................................................160 5.2 ...................................................................................173 5.3 ..............................................................181 .........................................................193 - ...................................195

1

1.1 x + y = x + y = , 0 0, , . : : 0, , :

x + y = y = x + y= x+ , = y'y . y 0 x+y=, =0 x

y

0

x+y=, 0 x

: 9 0, (. ),

10

1:

x'x y'y (. ). y = 0 ( 0 ), x+y=, =0 x = x = . 0 x y'y x'x . y y : 1 y= x1 2 9 x 2y = 2 1 x+y=, =0 y x 1 = 2 1 y y = 0 2 1 2 0 x y= x1 2 -1 y'y 1 . 9 = 0 , y =

x

1 y= x1 0 2 x 2 9 y=2 -1 x'x y'y 2 . 0 2 x -1

y

2 y

y=2

0 2

y=2

x

0 y

x

9 x=2 y'y x'x 2.

y 0

x=2

2 x=2

x

0 2 .

x

1.1

11

, (4,-1) x 2y = 6, 4 2(1) = 4 + 2 = 6. , , (16,5), (10, 8) 1 k, k 3 , k . 2 2 x 2 x+y= x+y= , , , 2 x 2 x + y = ' x + ' y = ' . . . . : . () (') , .. (), () + ' ( ') , () 0, (') '. () + ' ( ') () ('). 1o : 9 0, : = = 9 = = , + = +. , , : 6 x 2y =(1) 8 3x + 4y =(2)

12

1:

, ( ) , .. (1) x. x = 2y + 6 8 3x + 4y = x 3(2y + 6) + 4y = 6y + 18 + 4y = 8 8 10y = 10 y =1 x = 2y + 6 y = 1 y : x = 2(1) + 6 = 4 (4, 1). , . , .

, x = 4 y = 1, : 1 : 4 2(1) = 6 2 : 3 4 + 4(1) = 12 4 = 8 ( ) , :

1.1

13

= x 2y 6 (3) 3x + 6y =18 8 3x + 4y = 8 1 3x + 4y = , , : 3x + 6y + 3x + 4y = 18 + 8 10y = 10 y = 1 . : x 2(1) = 6 x + 2 = 6 x = 4 . (4, 1) ( ). 2 x 2 6 x 2y = 8 3x + 4y = . , .y 3x+4y=8 O x2y=6 A(4, 1) x

, x + y = ' x + ' y = ' , , . 2 x 2 . , 2 x 2 , , , .

14

1:

2 x 2 . : 2 y = x +1 3 2x + 3y = 3 , 9 4x 9y = 1 4 1 y x = 9 9 , . y , 2 1 1 (1, 0,3) . , x 1 0 (1, 1 ) . 3 3 x + 2y = 9 5 2x + 4y = 1 3 y= x+ 2 2 , , 1 5 y = x 2 4 . 2x y + 1 = 9 4x 2y = 2 y = 2x 1 , y = 2x 1 , . (k, 2k 1), k .y 1 0 x 2

y

1 2

0

x

1.1

15

, 2 x 2 : 9 9 9 . 2 x 2 2 x 2 . x + y = ' x + ' y = ' 0 0. : y = x + (1 ) y =' x + ' ( ) 2 ' ' 1 2 ' 1 = 2 = ' ' , ' ' 0 , 1 2 ' , ' ' ' ' x + = x + x = ' ' ' '

( ' ' )x = ' ' ' ' x= ' ' : ' ' '+ '+ ' ' y = + = ' ' ( ' ' ) = ( ' ' ) ( ' ' )

16

1:

y =

' ' ' '

, , ' ' ' ' (x, y) = , ' ' ' ' ' = , - =0 , 1 2 ' , . . =0 ' =0. x + y = ' x + ' y = ' : ' ' 0 ' ' ' ' (x, y) = , ' ' ' ' ' ' = 0 . ' ' , ' ' : D= = ' ' . ' ' D= D, x , : = ' ' . ' ' , D, y , : Dx = Dy = = ' ' . ' '

1.1

17

, , : x + y = ' x + ' y = '

D Dx y = y D D D=0, . D0, , (x,y) x = : 6 x 2y = 9 8 3x + 4y = =1 4 3(2) = 4 + 6 =10 0, 3 4 . Dx = : = x 1 6 6 2 8 10 = 24 + 16 = 40 D y = = 18 = 3 8 8 4 D y 10 D x 40 = = 1 . = = 4 y = D 10 D 10 D= 1 2

, (x, y) (4, 1) . = 40 2x 3y = 9 80 4x 6y = 2 3 D= = 6) 4(3) =12 + 12 = 2( 0 4 6 . 2, 40 2x 3y = , 40 2x 3y = 40. 2x 3y = 2x 40 2x 3y = 40 y = . 3

18

1:

2k 40 k, , k 3 11 3x + y = 9 6 9x + 3y = 3 1 = 99 = 0 9 3 . 11 3x + y = 2 3x + y = . D= 1 x y = 1 2 x 2y =

, . , . , . , . D, Dx, Dy. : D= 2

1 2

= 2 + 2 = ( 2)

Dx = Dy =

1 1 = 2( 1) + = 2 2 1 2

= 2 2 ( 1) = 2 (1 + 1) = 2 (2 )

1.1

19

, D = 0 . :D = 0 ( 2)= 0 =0 =2

: 9 D0, 0 2, (x, y), : Dx ( 2) 2 1 = = = = x D ( 2) ( 2) y= Dy D = 2 (2 ) 2 ( 2) = = . ( 2) ( 2)

, 0 2 , 1 , . 9 D = 0 , = 0 = 2 , . : = 0 , 1 0x y = 1 y = 2y 0x = 0 = 0 y . = 2 , = = 2x y 1 2x y 1 = = 4x 2y 2 2x y 1 . , 2x y =1 y = 2x 1 ,

( k, 2k 1) ,

k .

3 x 3 x+y+z=0 , , , , . . 2x+3y+z=6 (2, 1,5) , 22+3(1)+5=6.

20

1:

: 1x + 1 y + 1z = 1 , 2 x + 2 y + 2 z = 2 3 x + 3 y + 3 z = 3 , , , 3 x 3 1x + 1 y + 1z = 1 2 x + 2 y + 2 z = 2 x + y + z = 3 3 3 3 2 x 2 . (1) 2x y + 3z = 9 10 (2) x + 3y z = 3x + y z = 8 (3) . , .. (3) z, . : 3x + y z = 8 z = 3x + y 8 (4) (1) (2) : 9 2x y + 3z = 9 2x y + 3(3x + y 8) = 9 11x + 2y = 15 9 x + 3y z = 10 x + 3y (3x + y 8) = 10 x + y = 1 (5), (6) 15 11x + 2y = , 1 x + y = x=1 y=2 . x y (4) z = 3. (1, 2, 3). 3 x 3, , 2 x 2, 3 x 3 .

(5) (6)

1.1

21

4 x y = 1. 2 x + y = i) ii) . 2. x y = i) 7 8 x + y = 45 3. : x 5 2y + 1 0 2 + 7 +2= i) x + 6 y 6 = 8 3 2 4. : 3 x 3y = i) x 3 y =2 y+2 2x 1 3 = 4 4 ii) x x +3 3 = y 2 3 2y= x + 2 ii) 1 0 2 x y +1 =

x 1 y 2 = ii) 3 4 4x + 3y = 8

5. : 7 2x + y = i) 4 3x 5y = = 2y 3x 8 ii) 0 x + 3y + 1 =

6. , . 4 2x 5y = i) ii) 100 6x + 7y = 7. : 40 2x 3y = 80 4x 6y = 11 3x + y = iii) 2 9x 3y =

2 ( 3 1)x + 2y = i) x + ( 3 + 1)y =1 3

( 3 + 1)x + 4y = 7 ii) 1 1 x + ( 3 1)y = 2

22

1:

8. : 3x 2y =11 i) 2x 5y 2 =3 5x + y 2 =33

y x + 2 = 3 2 5x y + 3 = 4 ii) x 3y + = 2 iii) 3x + y + = 5 3x 2y + 2 = 2 2 5x + 3y 2 =16 y 1 2 O 1 1 4 1

1. i) 1 2 . ii) 1 2 ;

x

2. 26 , 68 . ; 3. 1,5 4. 2200 5050 . . 4. R T R =T + . 20 C 0,4 80 C 0,5 , . 5. , 50% 80% . 100 ml 68% ; 3 , 6. 1 : 2x + 4y = 2 : x + 2y = . i) 1 2 . ii) ; iii) ;

1.1

23

7. : i) 1 : x + y = 2 ii) 3 : x y = 2 : x + y =1. 2 : x + y =1. ( 2)x + 5y =5 ii) , x + ( + 2)y =53

8. : ( 1)x 2y =1 i) , 4x ( + 1)y = 2

9. 1 2 =6 , O1O3 = 5 O2O3 = 7 . . 10. , . N = x , B = y = z , , .

1

2

11. . 50% , 10% 30% . 52 lit 32% , . . 12. , y = x 2 + x + . .y 3

yy=f(x)

K(1,4) y=g(x)4

y

y=h(x)

O

K(2, 1)

x

1

O

x

O

2

4

x

24

1:

1.2 , . , 13 89. x, y , x + y = x 2 + y 2 =. 13 89 , : 13 (1) x + y = 2 2 89 x + y = (2) : (1), , .. x , (2). x + y = 13 y = 13 x x 2 + (13 x) 2 = 89 x 2 + 169 26x + x 2 = 89 0 2x 2 26x + 80 = 0. x 2 13x + 40 = 2 = 9 . : = x 13 3 = 28 5

(3).

(3), x=8 y=5 , x=5 y=8. (8, 5) (5, 8). 5 8. , , . 5 x + y = (1) 1 . (2) xy = 6

1.2

25

(1) x (2). : x + y =5 y =5 x (3). xy =6 x(5 x) =6 5x x 2 = 6 x 2 5x + 6 = 0 x 2 = 3. = x (3) x=2 y=3 , x=3 y=2 . (2, 3) (3, 2) . 5 6. , Vieta 2 5 + 6 = 0 . 2 3 (2,3) (3,2). x+y=5 , xy=6 6 y = . x .

y

A 1 O 1

B x

26

1:

(2,3) (3,2) . (2,3) (3,2) . 2 (1) xy = 6 2 2 13 x + y = (2)

(1) x (2). xy = 6 y = (2) :2 2

6 x2 2

6 x + y = 13 x + = 13 x 36 x2 + 2 = 13 x x 4 + 36 = 2 13x x 4 13x 2 + 36 = 0

. x 2 = , 2 13 + 36 = 0 , =9 =4 . 9 =9 2 x =9 x =3 x = 3 (1) x=3 y=2 x = 3 . 9 = 4 x 2 = 4 x = 2 x = 2 (1) x=2 y=3 x = 2 y = 3 . (3,2), (3, 2), (2,3) (2, 3).

6 , x 2 2 x + y = (0,0) = 13 . 13

xy = 6 y =

1.2

27

. y

A

B x

1 B A

x 2 + y 2 + xy = 3 1. : . 1 x + y = 2. : y = 3x 2 i) 4 12x 3y = x 2 + y2 = 9 ii) 0 x y = x 2 + y2 = 5 iii) xy = 2

. 1 3. S = 0 t + t 2 = 0 + t , 2 + 0 = S t . 2 x 2 2y + 10 = 1. o 2 2 25 x + y = .

28

1:

2xy y 2 5y = 0 2. : 2 y = x 4x + 3 3. 120cm2 . 3cm , 2cm , . . 2x 4. y = x 2 y = + k, k . k . 2y = x 2 5. o y= x + . 1 I. :1 x y = 1 1 2 x + 2y = x + y = 2x + 4y = ( 4 ) : (1 ) : ( 3 ) : ( 2 ) : 2 1, 4, 0, x + 2y = x + 2y = 2x + 4y = x + y =1

, , . ) , ) , ) . (1) (2) (3) (4) II. , , . 1. , . A 2. D = 0 , . xy = 1 . 3. 0 x + y = 4. x 2 + y 2 = = x 2 + 1 1 y . A

2 . . 2.1 -- T = f (t) t (t =0) (t =24).11 T(C)

T=f(t) 5 3

4

16

24 t(h)

) [4,16] .11 T(C)

T=f(t) 5 f(t1) f(t2)

4

t1

t2

16

24 t(h)

2.1 --

31

, , , t1 , t 2 [4,16] t1 < t 2 : f (t1 ) < f (t 2 ) T = f (t ) [4,16] . : f , x1 , x 2 x1 < x 2 : f (x1 ) < f (x 2 ) f f . , f (x) 2x 3 . = x1 , x 2 R , x1 < x 2 . : x1 < x 2 2x1 < 2x 2 2x1 3 < 2x 2 3 : f (x1 ) < f (x 2 )

f (x) = x + , > 0 . ) , [16,24] .T(C)

T=f(t) 5 f(t1) f(t2)

4

16

t1

t2

24 t(h)

32

2:

, , , t1 , t 2 [16, 24] t1 < t 2 : f (t1 ) > f (t 2 ) T = f (t ) [16,24]. : f , x1 , x 2 x1 < x 2 : f (x1 ) > f (x 2 ) f f , f (x) = + 5 . 2x x1 , x 2 , x1 < x 2 . : x1 < x 2 2x1 > 2x 2 2x1 + 5 > 2x 2 + 5 f (x1 ) > f (x 2 ) : f (x) = x + , < 0 . . T = f (t).T(C)

11

T=f(t) 5 3

O

4

16

24 t(h)

2.1 --

33

: ) t1 = 4 , f (4) = 3 . : f (t) f (4) = t [0, 24] 3 , T = f (t) t = 4 , f (4) = 3 . : f, , x 0 A () : f (x) f (x 0 ) , x A x 0 A , f (x 0 ) f min f (x) . , f (x) 3x 4 + 1 . = : x 4 0 , x , 3x 4 0 , x , 3x 4 + 1 1 , x . f (x) f (0) , x . , f x 0 = 0 , f (0) = 1 ) t 2 = 16 , T(16)=11 . : f (t) f (16) = t [0, 24] 11 , T =f (t) t =16 , f (16)=11. : f, , x 0 A () : f (x) f (x 0 ) , x A

34

2:

x 0 A , f (x 0 ) f max f ( x) . , f (x) = 4 + 1 . 3x : x 4 0 , x , 3x 4 0 , x , 3x 4 + 1 1 , x . f (x) f (0) , x

, f x 0 = 0 , f (0) = 1 . () () . (. ) (. ) (. ) (. ). y y=f(x) 1 1 x 1 1

y y=f(x)

x

y 1 y=f(x) 1 x

y 1 y=f(x) 1 x

2.1 --

35

) Cf f . Cf y'y, Cf y'y Cf .M

y yf(x) f(x)

M x x

x OCf

, , M(x, y) Cf y'y M '( x, y) M(x, y) M '( x, y) Cf , y = f (x) y = f (x) , : f ( x) = f (x) f . : f, , , x A : f (x) x A f ( x) = yy. , f (x) = 2x 4 x 2 + 1 , x : f ( x) = 2( x) 4 ( x) 2 + 1 = 2x 4 x 2 + 1 = f (x) , y'y . ) Cf f . Cf , Cf Cf .y y xf(x) Cf

Mf(x)

O y

x

x

M

36

2:

, , M(x, y) Cf M '( x, y) M(x, y) M '( x, y) Cf , y = f ( x) y = f ( x), : f ( x) =f (x) f . : f, , , x A : x A f ( x) =f (x) . = , f (x) 2x 3 x , x : f ( x) = x)3 ( x) =2x 3 + x =f (x) 2( , . : y = x 2 , y = x 4 , y = x 6 ., , y'y , , y = x , y = x 3 , y = x 5 ., , , . f [6,6]. f :

2.1 --

37

) f : i) , ii) , iii) . ) f , .y

1

x

f , y'y . , y'y f, f, ().A A

4

B

6 5

2

2

5

6

x

: ) f : i) [0,2] [5,6], ii) [2, 0] [6, 5], [0,2] [5,6] f . iii) [5, 2] [2,5] .

38

2:

) f 4 x 6 6. : max f (x) = f (6) = f (6) = 4 f 0 x 0. : min f (x) = f (0) = 0.

1) : ) ) .y y=f(x) y y=g(x) x y y=h(x) x 1 2

1

2

2

1

x

2) , . 3) : i) f (x) = x 2 6x + 10 x=3. 2x x=1. ii) g(x) = 2 x +1 4) : i) f1 (x) 3x 2 + 5x 4 = iv) f 4 (x) x 3x =3 5

ii) f 2 (x) 3 x + 1 = x2 v) f 5 (x) = 1+ x

iii) f 3 (x) x + 1 = vi) f 6 (x) = 2x . x2 +1

2.1 --

39

5) : i) f1 (x) = 1 x x+ ii) f 2 (x) = x2 iii) f 3 (x) = x 1 x + 1

1 iv) f 4 (x) = 2 x x +1

v) f 5 (x) =

x

vi) f 6 (x) =

1 x2 .

6) .y y=f(x) y y=g(x) x y y=h(x)

x

x

7) y y=f(x) y y=g(x) y y=h(x)

x

x

x

8) ) y C1

y C2

) .y C3 x

x

x

40

2:

2.2 - = ) f (x) x + 1 . x + 1, x < 0 f (x) = , x + 1, x 0 y f (x) x + 1 , = y=|x|+1 1 9 y = x + 1 , x 0 9 y= x + 1 , x 0 , 1 y'y x' 1 y=|x| 1 O 1 x 1

1

x 'Oy xOy , , (x) =(). x , (x) = x (1) 1 , f (x) x + 1 . , , , = : f (x) = + 1 , x , (x) x f (x) 1 (x) . : f , : f (x) = + c , c > 0 , (x) c ( )(1)

y'y .

2.2 -

41

y c c c O

y=(x)+c c y=(x) c x

= ) f (x) x 1 . x 1, x < 0 f (x) = , x 1, x 0 f (x) x 1 , = 9 y = x 1 , x 0 y=|x| 1 1 1

y 1

9 y= x 1 , x 0 , O 1 y=|x|1 x 1 1 y'y x 'Oy xOy (x) =(). x x , (x) = 1 , = f (x) x 1. , , , : f (x) = 1 , x , (x) x f (x) 1 (x). : f , : f (x) = c , c > 0 , (x) c ( )

42

2:

y y=(x) O c c c y=(x)c c c x

) f (x) x 1 . = x + 1, x < 1 f (x) = , x 1, x 1 f (x) x 1 , = 9 y = x + 1 , x 1 9 y= x 1 , x 1 ,y 1 1 y=|x| 1 1 y=|x|1

1 x'x O x x 'Oy xOy (x) =(). x , (x) = x (2) 1 , f (x) x 1 . , , , = f (x) =(x 1) , x , = f (x) x 1 x (x) = x 1. x(2)

x'x .

2.2 -

43

: f : f (x) =(x c) , c > 0 , c ( ). f (x) =(x c) , f x x c , c x. , f c ( ).y c C (x-c) f(x) c c Cf

c xc x O x

= ) f (x) x + 1 . x 1, x < 1 f (x) = , x + 1, x 1 f (x) x + 1 , = y=|x| 1 9 y = x 1 , x 1 1 9 y= x + 1 , x 1 , y=|x+1| 1 x'x 1 x 'Oy xOy

y 1 1

O

1

x

(x) =(). x , (x) = x 1 , -

44

2:

f (x) x + 1 . , , , = f (x) =(x + 1) , x , f (x) x + 1 x = (x) = x +1. x : f , : f (x) =(x + c) , c > 0 , c ( ). f (x) =(x + c) , f x x + c , c x. , f c ( ).c y c

Cfc

C f(x) =(x+c)

c O x x+c x

f (x) = x + 3 + 2 .

= y x + 3 , y = x 3 . y = x + 3 + 2 , y x + 3 2 = .

2.2 -

45

, f (x) = x + 3 + 2 y = x , 3 2 ().

y

y=|x+3|+2

y=|x+3|

y=|x| 2 3 O 1 x

: , : f (x) =(x c) d , c,d > 0 , .

A 1. : (x) = x, f (x) x + 2 = g(x) x 2 . =

2. : (x) = x, h(x) x + 2 = q(x) x 2 . =

3. : (x) = x, F(x) = x + 2 + 1 G(x) = x 2 1 .

46

2:

4. i) f (x) = 2x 2 4x + 5 f (x) =(x p) 2 + q g(x) = 2x 2 f. ii) f (x) = 2 + 8x 9 , 2x 2 g g(x) = 2x . 5. 1 O(0,0) A(2,0).y C

O

x

: (x) (x) i) f (x) = + 2 g(x) = 2 ii) h(x) =(x + 3) q(x) =(x 3) iii) F(x) =(x + 3) + 2 G(x) =(x 3) 2 . 6. (x) = 2x 2 1 . f : i) 2 1 . ii) 3 2 . iii) 2 1 . iv) 3 2 .

2.2 -

47

2 I) , , . 1. f , f . 2. . 3. (1,2) , (2,1) (3,3) . 4. f 1, f (0) < 0 . 5. f (1,2) (2,5) , f . 6. f 1, f (x) = 2 . 7. f :[1, 2] f (x) = 3x 2 . 8. , . 9. f , f . 10. f , f .

II) f. f, (x) = , 1 3x 4 2 , : ) f (x) = 3(x 1) 4 + 2 ) f (x) = 3(x + 1) 4 + 2 ) f (x) = 3(x 1) 4 2 ) f (x) = 3(x + 1) 4 2

3

3.1 . MM1 NN1 , O M M1 O N N1 , : (MM1 ) (NN1 ) (OM1 ) (ON1 ) , = = (OM) (ON) (OM) (ON) , (MM1 ) (OM1 ) , (OM) (OM) (MM1 ) (OM1 ) 1 1

(MM1 ) (NN1 ) = (OM1 ) (ON1 )

, . , , , , , . , M1 O M , :(MM1 ) (OM)

=

(

)

50

3:

= =

(M1 ) (OM)

(MM1 ) (OM1 )

( ( (

) ) )

, , (M1 ) = (M )1

, 00 3600 Oxy , Ot Ox Ot (. , ). Ox , Ot .y t 2 (x,y) (x,y) y x 1 x 1 2 x t y

(x, y) M1 x'x (. ). (. ), , , : = (MM1 ) (M1 ) (MM1 ) (M1 ) , = , = = (OM) (OM) (OM1 ) (M1 )

y (1 ) = x , (1M) = (OM) = x 2 + y 2 = > 0 . , : = y x x y , = , = = , = y xx 2 + y2 > 0 .

3.1

51

, ( ). : = y, x ,

= =

y xx y

( x0) ( y0)

,=

x 2 + y2 > 0

=

(x, y) ( ) = . 360 Ox Oxy . 300 , Ox = 360 + 30 390. = 360 , : = 360 + , N* 00 < 3600 Ox, , 30 , Ox 390 360 + 30 = : = (360 + 30 ) = 390.y 390 0 300 x t y t 300

x 2 + y 2 > 0

x

390 0

52

3:

: = ( 360 + ), N 0 < 360 360 , , 0 360. , , , : = y, x ,

= =

y xx y

( x0) ( y0)

,=

x 2 + y2 > 0

=

(x, y) ( ) = .x 2 + y 2 > 0

( ) Ox . Ox , , , 360 + , . Ox , , , 360 + , . , k 360 + , k , . , k : (k 360 + ) = , (k 360 + ) = , (k 360 + ) = (k 360 + ) =

3.1

53

, , , . , . (0,0) =1 . . , .. =35 , (, ). =1 35 = 0,57. 35 = , 35 =110 120 o 130 o 140o 150o 160o 170o 180o 190 o 200o 210 o 220 oo 100o 90 80 o o

y

70 o

60 o

50o 40o

N(,)20 o

x

0,1

3500,1

10 o

x350 o 340o

(x,y)

330 o 320 o y 310o 300o 240o o 290o 250 o o 280o 260 270

=1, 35 = 0,82. , (x, y), : = x = = y = x'x , y'y . : 1. ' , 1. :1 1 1 1

54

3:

2. , , .10 20 30 40

+ + + +

+

+ +

+

M(x, y). . 1 , ( ) ( ) = = ( ) () 1y B

t E(1,yE )

MA O

A

x

B

=

yE , (AE)= yE , = yE. . : = yE = , x =1 , .

3.1

55

. , AB (, ) ( 1rad), . , ( rad) S = . :

360 , 2 180 . 180 , :

, =

: 9 60 , = 180 = 600 = = 180 3 60 = rad. 3

= 180

56

3:

9

5 = = 6 180 5 5 6 = = = 150 180 6 180 5 rad=150. 6 . 0 30 45 60 90 rad 0 0 13 2 2 2

03 3

3 13 3

6 4 3 2

1 22 2 3 2

1 3

1 20

1

0

3.1

57

1 h , . .

:

48 = 70 =

h h , = 48 h h , = 70

= = 20mh h 2070 48 = 20, h = . 48 70 70 48 70 2, 75 48 1,11. (1) : 61, 05 37 1, 64 37m . h 2 750o .

750 360 2 30, 750 =360 + 30 2

750 = (2 360 + 30 ) = 30 =750 = 30 = 3 3

1 2

750 = 30 =750 = 30 = 3

3 2

58

3:

3

79 79 = 2. 79 6 3 6 79 79 1 = 2 13 + 2 13 2 + , = = 13 1. 3 6 6 3 : 79 3 = 13 2 + = = 3 3 3 2 79 = = 3 3 3

79 1 = = 3 3 2

79 3 = = 3 3 3

1. x, y .B 6 300

A x y 3

4A 600

2. .B

2

30

0

3. S = 6cm. , : i) = 1cm ii) = 2cm iii) = 3cm . 4. rad i) 30 ii) 120 iii) 1260 iv) 1485.

5. : i)

ii)

iii)

6. i) 1830 ii) 2940 iii) 1980 iv) 3600.

3.1

59

1. N (N ) h , 70 ( ) ( ) 1000 1.000 (1 0,3 m) . . i) h = 30 , 45 60. ii) , h=1000 ;0

2. : i) : () = () = 2 45 = 2. ii) ()= 4 22,5. iii) (). iv) ,

, () 2 2 2 . = v) 22,5. vi) ; A 3. 30 . 60

4. 1mm . .

B

3 330 0

A

22,5 o 1

45o 1

B

60

3:

3.2 . . : 1. M (x, y) , : x = y = t y

2 + 2 = 1

, 2 2 2 (OM) =1 ( )= x + y= x 2 + y 2 : x +y = 1, : 2 + 2 = 1 2. = 2 2

(x,y) |y| A 1

y

x |x| O

A

x

=

:

= =

y = ( x = 0 ) x x = ( y = 0 ). y

3.2

61

(1) (2), . 3. : . = 1

=:

= ( 0 0) = 1.

=

4.

= 2

1 2 2 = 1 + 2 1 + 2

i) 2 + 2 = 1 2 0 : 2 2 1 1 1 2 + = = +1 2 = . 2 2 2 2 1 + 2 2 = 1 . 1 + 2 1 , 1 + 2

ii) 2 + 2 = 1 2 = : 2 + 2 =

1 1 2 = 1 2 = 1 2 = . 1 + 2 1 + 2 1 + 2

2 . 1 + 2

62

3:

5 1 = 90o < < 180o, 13 .

2 + 2 = 1 2 = 1 2 . 5 : 13 2 25 169 25 144 5 2 = 1 = 1 = = . 169 169 169 13 90 < < 180 , < 0 , : 144 12 = 169 13 = = , : = 5 12 5 12 = 13 = = 13 = . 12 5 12 5 13 13 2 4 4 2 2 i) + = 1 2 4 4 2 ii) = 2 1

i) : 4 + 4 = ( 2 ) 2 + ( 2 ) 2= ( 2 + 2 ) 2 2 2 2

= 1 2 2 2 , ii) : 4 4 = ( 2 ) 2 ( 2 ) 2

( 2 + 2 = 1 )

= ( 2 + 2 )( 2 2 ) ( 2 + 2 = 1 ) = 2 2 = 2 (1 2 ) = 2 2 1.

3.2

63

3 < x < , 5 2 x rad. 2 3 2. x = < x < , 3 2 x rad.1. x = 3. x =

3 3 < x < 2, 2 3 x rad. 2 5 0 < x < , 2 5 x rad. 3 < x < 2, 2 2xx . 1 + xi) x = 0 x = 0.

4. x =

5. x = 2

6. , x : ii) x = 1 x = 1. 3 4 iii) x = x = . 5 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. : 1 i) = 1 +

4 4 2 ii) = 2 1 .

64

3:

11. : i) 1 + 2 x x 2 + =ii) + = . 1 x 1 + x x 1 +

12. : i) + = + ii) 2 2 = 2 2 .

13. : x x 1 + = x + x ii) (1 x) 1 + i) = x x 1 x 1 x x iii) 1 1 1 = x x iv) x x = x x. x + x x x

B 1. x + x = , : 1 1 i) x x ii) + x x iii) x + x 2. : i) 4 x + 4 x = 1 2 2 x 2 x ii) 6 x + 6 x = 1 3 2 x 2 x. iii) 2( 6 x + 6 x) 3( 4 x + 4 x) x , . 3. iv) 3 x + 3 x.

< x < , 2 2

1 + x 1 x = 2x. 1 x 1 + x

4. 0 x < , 2

1 + x + 1 x 1 + x x = = . 1 + x 1 x x 1 x

3.3 1

65

3.3 10 , , 0 90. ' ' . ' , ' = , , , x'x. . , :y (x,y) t

A

A

x

(x, y) t'

() = () =

() = () =

: . : 9 :

1 2 3 (30 ) = (30 ) = 3 (30 ) = (30 ) = 9 , : 2 = = 4 2 4 = = 1 4 4

(30 ) = (30 ) =

3 2

(30 ) = (30 ) = 3

2 = = 4 2 4 = = 1 4 4

66

3:

180 ' 180 , = 180 , ' t , , ' (x,y) y'y . A . , : (180 ) = (180 ) = : 180o . : = 9 150 180 30 , :y B t 180 0

(x,y)

A x

B

(180 ) = (180 ) =

150 = (180 30 ) = 30 =

1 23 2

150 = (180 30 ) = 30 = 150 = (180 30 ) = 30 = 3 3

150 = (180 30 ) = 30 = 3

9

2 = , : 3 3

3 2 = = = 3 3 2 3

3.3 1

67

1 2 = = = 3 3 2 3 2 = = = 3 3 3 3

3 2 = = = 3 3 3 3 180 ' 180 , = 180 + , , ' , ' . . , : (180 + ) = (180 + ) = y B 1800+ A (x,y) t t (x,y) A x

(180 + ) = (180 + ) =

: 180o , . : 9 210 180 + 30 , : =

210 = (180 + 30 ) = 30 =

1 23 2

210 = (180 + 30 ) = 30 = 210 = (180 + 30 ) = 30 =210 = (180 + 30 ) = 30 =

3 33

68

3:

9

4 = + , : 3 3

3 4 = + = = 3 3 2 3 1 4 = + = = 3 3 2 3 4 = + = = 3 3 3 3

3 4 = + = = 3 3 3 3 90 ' 90 , ' 90 , , = , ' xOy . . , : (90 ) = (90 ) = , 90o , . , 60 90 30 , : =60 = 30 =60 = 30 =y B t (y,x) y=x t (x,y) A O0 90

A

x

B

(90 ) = (90 ) =

3 , 23

1 , 2 3 60 = 30 = 3 60 = 30 =

3.3 1

69

, 0 90.

1 36o = 54o.

1+ 5 . 4

54 90 36 , =54 = 36 =

1+ 5 4

2 + 2 = 1 2 54 + 2 54 = 1, : 2 1+ 5 6 + 2 5 10 2 5 2 2 54 = 1 54 = 1 = , = 1 4 16 16 : 10 2 5 54 = 4 : 54 = = 54 541+ 5 10 2 5

54 =

54 = 54

10 2 5 . 1+ 5

2 : ) 90 + ,

) 270

) 270 +

i) 90 += 90 (), : (90 + ) = (90 ()) = () = . 90 + .

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)

ii)

3. : i) = ( + ) iii) ii) + ( + ) = 0 iv)

+ = 2 2

+ = . 2 2

4. () (180 + ) . () (90 + )

9 ( x) (2 + x) + x 2 = 1. 5. : 21 (13 + x) ( x) x 2 6. : 2 ( x) + ( x)(2 x) + 2 2 x . 2

3.3 1

71

1. : 495 120 + 495 (120 ) . (120 ) + 495 2. : 5 7 (5 + ) (7 ) + 2 2 = 2 1. 5 7 (5 + ) (7 ) + 2 2

3. x + + x = 5, 3 6 : 2 x + 2 + x . 3 6

4. : 00 , x A : i) ii)

x + T A, x T A f (x + T) = f (x T) = f (x)

f.

. .

1 1. -

, ,

, , , ., f (t) = t, t . . : x (x rad) .

x = (x rad) ( + 360 ) = ( 360 ) = , x :

(x + 2) = (x 2) = x

2. .

3.4

75

x = (x rad). 2. , : x x = x :1 =

{x x 0}

x 1 (x + ) = (x ) = x, . , , : x x = x : 2 =

{x x 0}

. f(x) = x f(x) = x 2, 2, . [0,2]. x xrad . , , . : y B x 0 M , . 2 M x , x rad O A f(x) = x 0, . 2 f(x) = x :

76

3:

, 2 3 , 2 3 , 2 2

, = 1 2 2 3 3 = 1. x = , 2 2 : x = x x 0 0 2 1 . 0 3 2 1 . 2 0

. : x x 0 0 4 2 0, 71 2 2 1 3 4 0,71 0 5 4 3 2 7 4 0,71 2 0

0,71 1

. [0, 2]:y 1 O 1 4 3 2 4 y= x 5 3 7 4 2 4 2 0x2 x

3.4

77

f(x) = x , 2, [2, 4], [4, 6] . [2, 0],[4, 2] . , .y 1.5 1 0.5 /2 0 0.5 1 1.5 /2 3/2 2 5/2 x

. x ( x) = x. f(x) = x 0(0,0) . f(x) = x f(x) = x 2, 2, .. [0, 2]. : x x 0 1 . 2 0 1 . 3 2 0 2 1 .

: x x 4 2 2 0,71 2 0 3 2 0,71 5 4 0,71 3 2 0 7 4 0,71

0 1

1

2 1

78

3:

y = x 0 x 2p .y 1 0 1 4 2 y = x 3 5 4 4 3 7 2 2 4 0x2 x

f(x) = x 2, :y 1 2 0 1 2 3 2 2 5 2 y = x x 3 7 2 4

. x (x) = x. f(x) = x y'y. f(x) = x f(x) = x , , .. , . 2 2 ( , , ). 2 2 x rad . x . : y x 2 2 O x ' , . f(x) = x , . 2 2

3.4

79

x . 2 x = 2 f. x 2 x +. x = 2 f. x y y B B A A O O B B A A x M x M E E A A y yB B

O O B B

A A x x

f(x)=x , , :x x 2 3 4 6 0 6 4 3 2

3 0, 6 0 3 1, 7 1 3

3 0, 6 1 3

3 1, 7

. f(x)=x .y = x

2

3 2

2

x 2 3 2 2

f(x)=x , ( 5.3: (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.x x 3x 0 0 0 2 1

0 0

3 2 1

2 0 02

y 3 2 1

y = 3 x

y= x

3

3

1 2 3

2

3

x 4

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.x 2x 0 0 4 1 2 0 3 4 1y

03 2y= x

1 2 1 2 3 2

y = 2x

x 2

3 f(x)=32x .

3, 3 .

3.4

81

f(x)=32x :x 32x 0 0 4 3 2 0 3 4 3y

0 2

y = 3 2x

3 x 0 3 4 2 2 3 2 2

.

3

, f(x)= x, ,>0: (i) , . (ii) 2 . f(x)= x, ,>0

1. , = i) f (x) = 2x, g(x) 0,5 x, h(x) = 2 x, 0 x 2 = ii) f (x) = 2x, 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 . x 5. f (x) = 2 . 2 ; ; f .

82

3:

x . 2 7. 6. f (x) = 2 i) f (x) = x ii) g(x) = 1 + x . iii) h(x) = 1 + x

8. f (x) = 2x. 9. f (x) = x. B 1. :y 1 1 1 y y 2 x 2 2 x x

1 3

y y 2

x

x y y 2.5 2 x 2 2 x

0.5

2. y= 3 t , y 6 t . i) . ii) 0 t 12.

3.5

83

3. 1m. , 1 h = 1 + 3t, 3 t . i) . 1m 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

1 x = . 2 y = x 1 y = . 2y y= 1 2 6 5 6 2 y= x x

x=

O

x , f (x) = x 1 . 2, 2 x, (. ), 2 . 2, .

84

3:

1 5 [0, 2], = , x = 2 6 6 6 5 1 = = . 6 6 2 y

x =

1 2 x= 2 + 6 , 5 x= 2 + 6

M

1 2 O5/6 /6

M x A

, x = , = , :

x= 2 + , x = 2 + ( ) 1 x =

3 2

3 3 = , ( ) = . 3 2 3 2 x = ( ), : 3 x= 2 3 , x = 2 + + 3 1 2 2x + = 4 2

1 = , 2x + = , 6 2 4 6

3.5

85

2x + 4 = 2 + 6 , 2x + = 2 + 4 6

2x + = 2 + 2x= 2 + x= 4 6 6 4 242x + 7 = 2 + 2x = 2 + x = + 4 6 6 4 24

x = 24 , 7 x = + 24 x = , 1 .. x = . 2 x = = 3

1 [, ] , 2 3 3 1 = . 3 2y M 3 1 2 3 M x

1 x = 2 x= 2 + 3 , x= 2 3

86

3:

, x = , = , x= 2 + , x = 2

1 x =

2 2

2 = , x = , 4 4 2 : x= 2 + 4 , x= 2 4 2 2x =

3 2

3 3 5 3 = , = = . 6 2 6 2 6 2 5 2x = , 6 5 2x= 2 + 6 , 5 2x= 2 6 5 x = + 12 , 5 x = 12

3.5

87

x = x = 3. . , , , .. , 2 2 , .y 3 x

2

2

, x = 3 .

, = 3. 3 3 x = 3 : x = + , . 3 , x = , x = , :

x = + , x = . 1 x = 1

= 1, = 1. x = , 4 4 4 x = , 42 x = 3

= 3, x = , 6 6 x = + , 6

88

3:

1. i) x = 0 ii) x =

2. 1 i) x = ii) x = 1 2 3. 3 i) x = 0 ii) x = 3 4. 3 i) x = 3 5. i) (1 x)(2x 3) = 0 6.

2 2

iii) x = 0 iii) x = iii) x = 1 ii) x = 3 3 2 2

iv) x =

2 2

iv) x = 1 iv) x = 3

ii) (2x + 2)(1 x) = 0

i) ( 3 + x)(1 x) = 0 ii) (2x + 1)(2 x 3)x = 0 7. ii) x = 0,809 iii) x = 28,636 i) x = 0,951 8. x 2x ii) + 1 = 0 iii) 3 i) 23x = 3 3=0 5 7 9. i) x + = 1 ii) 2 3x = 1 iii) 5x = 3 3 4 4 10. i) 2 2 + 1 = 0 ii) 2 2 x + 3x 2 = 0 iii) 32 t= 3 + 2 3t 11. ii) x 2x = 1 i) 2 x + 5 2 x = 4 12. x, : i) f (x) = 3 x , 0 x < 2, ii) g(x) = 7 x , 0 x < 2 2 2

3.6

89

13. ( t ) = 75 + 50 , S 6 t o t = 1 . i) 100000 , ii) . 1. i) x + x = 0 ii) 2x + 3x = 0 3 4 2. 1 i) x x + 1 = x + x ii) 2x = 4 2 x 3. x = 1 (3,4). * 4. 1+x=x [0,2). 5. : x = + x [0, 2). 3

3.6 , M1 , M 2 (. 1). , . (. 2).y M 1 (, )-

M 2 (, )

y

M((-), (-))-

x (1,0) O

x (1,0)

O

. (1)

. (2)

90

3:

, M1 , M 2 , : 1: 2: : : 1 ( ) 0 ( )

M 2 OM1 = AOM = , (M 2 M1 ) = (AM). (M 2 M1 ) 2 = (AM) 2

:

(P1P2 ) =

(x 2 x1 ) 2 + (y 2 y1 ) 2 ,

P1 (x1 , y1 ) P2 (x 2 , y 2 ), : (M 2 M1 )2 = ( ) 2 + ( ) 2 = 2 + 2 2 + 2 + 2 2= 2 2( + )

( )2

= [( ) 1]2 + [( ) 0]2 = 2 ( ) + 1 2( ) + 2 ( )= 2 2( ).

( 2 1 ) 2 = ( ) 2 2 2( + )= 2 2( )

:

+ = ( ) ( ) = +

(1)

, , 0 < < 3600 , , . , :( ()) = () + () = : ( + ) =

(2)

3.6

91

(1) (2) , . , : 15 = (45 30 ) = 45 30 + 45 30 = 2 3 2 1 + = 2 2 2 2 2 3 2 1 = 2 2 2 2 2( 3 + 1) 4 2( 3 1) 4

75 = (45 + 30 ) = 45 30 45 30 =

(1), , . x = x x = x, : 2 2 ( + ) = ( + ) = ( ) 2 2 = + = + 2 2

:( + ) = +

(3)

:( ) =

, : 15 = (45 30 ) = 45 30 45 30 = 2 3 2 1 = 2 2 2 2 2( 3 1) 4

75 = (45 + 30 ) = 45 30 + 45 30 = 2 3 2 1 + = 2 2 2 2 2( 3 + 1) 4

92

3:

+ , , . , : ( + ), , ( + ) 0, 0 0. :

= ( + )

( + ) + = 0 ( + ) + + = = 1 : ( + ) = + 1 (5)

, : ( ) = 1 + (6)

: ( + ) = 1 + (7) ( ) = + 1 (8)

, :45 30 15 = (45 30 ) = = 1 + 45 30 = 3 3 = 3 3 3 3+ 3 1+ 3 1

(3 3)(3 3) 12 6 3 = = 2 3 6 (3 + 3)(3 3)

3.6

93

3 45 + 30 3 = 3+ 3 75 = (45 + 30 ) = = 1 45 30 3 3 3 1 3

1+

=

(3 + 3)(3 + 3) 12 + 6 3 = = 2+ 3 6 (3 3)(3 + 3)

- 3 3 12 3 1 = , < < 2 = , < < , 5 2 13 2 +.

(+) = + ( + ) = . : 2 = 1 2 = 1 2 = 1 2 = 1 9 16 = , = 25 25

16 4 3 < < 2 = , 2 25 5

25 5 144 25 3 = , < < = , = 169 13 169 169 2

3 12 4 5 16 ( + ) = + = 5 13 5 13 65 4 12 3 5 63 = , 5 13 5 13 6516 63

( + ) =

:( + ) =

( + ) =

63 16

2 : ( + ) ( ) = 2 2

( + )( ) = ( + )( ) = 2 2 2 2 = 2 (1 2) (1 2 ) 2 = 2 2 2 2 + 2 2 = 2 2

94

3:

3 : 2x = x + 6

2x = x + 2x = x + x 6 6 6 2x = x 4= 3= x x = 3 3 x = + , 3 x = 3 1 x + x 2 2 3x + x 3x

[ x 0 ]

4 : + + =

, , , . + = , ( + ) : 2( + ) = ( )

+ = 1 + = + + + =

5 2 4 1 = t, 2 = (t + ) 3 = (t + ). 3 3 = 1 + 2 + 3 .

3.6

95

= t + (t + 2 ) + (t + 4 )

3 3 2 2 4 4 = t + t + t + t + t 3 3 3 3 3 3 1 1 = t + t + t 2 + t 2 + t 2 2 1 1 = t t t = 0 2 2

1. , , : i) 12 4 12 4 ii) 170 50 + 170 50

iii) 110 70 110 70

iv)

2. : i) 3x(2x) 3x(2x) 3. :) i) (x + ) + (x = 4 4 2x

7 7 + 12 12 12 12 4 4

ii) (x + )x + (x + )x ii) 2 (x ) 2 (x + ) = 2xx 4 4

4. , , : i) 17 4 17 4 18 9 18 9

ii) iv)

70 20 + 70 20165 + 15 1 165 15

5. : i) 2xx + 2xx iii)x 2x 1 + x2x

7 12 4 iii) 7 1 + 12 4

ii) x + x x + x 6 6

+ 2x + x 3 6 iv) 1 + 2x x 3 6

96

3:

6. : i) x + + x = x 3 3 ii) ( + )( + ) = ( + ) + ( )

7. , , 105 195. 8. : i) + =( + )

ii) + =

( + )

9. , : i) ( + ) = ii)63 655 13 5

16 ( + ) = 65 3 5

3

10. +, : i) = , ii) = ,3 5 = 5 , 13 4 = , 5 6 0 2 , x < 3 1< x < 2 . x = 3, x =1 x = 2 . A(x) . B(x). ... . (x)>0 ( 0 (< 0) , (x 1)(x 2 + x 6)(2x 2 + x + 1) < 0 x P(x) = (x 1)(x 2 + x 6)(2x 2 + x + 1) . P(x) x (, 3) (1, 2). x 3 3x 2 + x + 2 > 0

1, 1 5 1+ 5 (x 2) x x > 0. 2 2 P(x) = x 3 3x 2 + x + 2 :

(x 2)(x 2 x 1) > 0

1 (2, +) (x) ,

4.3

145

1+ 5 . (x) , 2 ,2 , x2, . , (x) .

1 5 2

+ 1+ 5 2

2

+ +

x , 1 5 < x < 1 + 5 x>2.2 2

, . , , :

f (x) = x + 1x 1 + ...... + 1x + 0 , 0 (< 0) . : A(x) A(x) > 0 A(x) B(x) > 0 < 0 A(x) B(x) < 0, B(x) B(x) , A(x) B(x) > 0 A(x) B(x) < 0 (x). , (x 1)(2x 2 + x + 1) > 0. x2 + x 6 (x 1)(x 2 + x 6)(2x 2 + x + 1) > 0 , P(x)> 0 , , P(x) x (3,1) (2, +).

4.4

153

A(x) 0 B(x) x

A(x) B(x) 0 B(x) 0.

x 2 4x + 3 2 0 (x 2 4x + 3)(x 2 + 3x 4) 0 x + 3x 4 0. 2 x + 3x 4 x 2 4x + 3 1 3 ,

x 2 4x + 3 0. : x 2 + 3x 4

x 2 + 3x 4 1 4 . :P(x) = (x 2 4x + 3)(x 2 + 3x 4) x x - 4x + 3 x2 + 3x-4 P(x)2

-

-4

+ + +

+ 0 0-

1 0 0 0

-

3 0 0

+ + + +

+-

x (, 4) [3, +).

1. : 3x 2 1 2 x 2 3x + 2 x2 2 4 i) 2 = ii) = 2 x 1 x +1 x 1 x 1 x x x 3 2 2. : 2 x + x + 2x 2 = 0 3. : i) iv)

x 3 = 4x

ii) v)

3x 2 = 4 x +3 =

iii)

5x 1 = 4 x + x 20 = 10

x + 3 = x +1

10 x + 1 vi)

x 8 + 3 viii) 1 + 2 x = x + 1 2 x 4. : 2x + 1 x2 0 ii) i) >0 x 3 x +1 x vii) =

iii)

x2 x 2 0 x2 + x 2

154

4: -

5. : 2x + 3 x2 i) ii) >4 4 x 1 3x + 5 x 2 x 3 iv) v) 2x 1 x + 2 3x 5 x 16. : x 2 + B 1. : i) 2x + 3 < 1 3x 2. : ii) ii) ii) ii)3

iii)

x 2 3x 10 +20 x 1

2 1 2x 1 x(2x 1)

x 3 > x 5x2 + 3 x 6 = 0

0 i) x + 3 x 10 = 3. :i) x 2 + x 4 = i)

x2 + x 2

x 1 + x 4=4x 2 + = 2x 1

x+4

4. :

x 1 =

5. 2 4 x 33 x 3 2 x 3x + 4 = 0 ( )

1. (x 2 + x + 1)(x 1) = x 3 1, P(x) = 3 + x 3+1 + x 3+ 2 x2+x+1 (, x , ). 2. : i) f (x) = x +1 ( + 1)x + 1 (x 1) 2 . . ii) g(x) = (n - 2)x n -nx n-1 + nx -n + 2 (x 1)3 . 3. :

0 i) x 4 + 2x 3 + 2x + 1 =

0 ii) x 4 + x 3 4x 2 + x + 1 =

( x 4 + x 3 + x 2 + x + = 0, 0. ).

4.4

155

: x2 x +

1 = y. x

4.

0 i) x 4 + x 3 16x 2 2x + 4 =: x 2 + 2x 1 = y.

0 ii) x 4 + 8x 3 + 13x 2 8x + 1 =

5. (x 2 + 2x 1) 2 3(x 2 + 2x + 3) + 14 = 0 6. , x 5 + 3x 2 + x + , x 2 2, 5x+8. 7. P(x) =17 12x16 + 12x15 12x14 + ... + 12x 1, x (11) (13). 8. . C , x = 10x 3 100x 2 + 270x 180. i) ; ii) ;

156

4: -

2 = 1, 414213,

,

= 2. 2

2

2,

157

x1,2 =

2 4 2

x 3 + x = (*) + + + 3 + x= 2 2 2 2 2 2 3 3 2 3 2

* 3 : x 3 + x 2 + x + = 0 x= y x 3 + Ax = B. 3

158

4: -

-

4 5

5.1 . : , : .... >1 , = , N* =1 , : 3 1 1 1 1 1 = = 8 2 2 2 2

: 1 1 0 = = 1 = , 0 N* . :9 2 3 = = 4 3 2 1 1 2 2 , 5 4 2 2

, 0, . . O , . T 2 .. 35 ; Av ( p )q = pq , :(3 = 3= 32 )2 5 5 2 5 5

5.1 2 5

161

3 x 5 = 32 , 5 32 . 3 5 = 5 32 . Av >0, , : =

2

, , , , 0 = 0. ..= 83 4 32 3

= 823

3

= 4 643

27 =

27 4 =

1 = 27 4

3

1 1 = 4 4 3 27

, . To . ..4

3 = 4 3 = 4

1

1

1 1 + 3

= 12 = 12 7

7

: 21,4 1,43,421 57 3

x 1.4y y

2 1.4 5

= = =

x 3.21 +/ xy

( 7 3 )

: x x , ; 3 2 ; ( B' . 104) 2 , , . 1, 1,4, 1,41, 1,414, 1,4142, 1,41421, 1,414213,.... (1)

162

5:

2 3: 3 1, 31,4, 31,41, 31,414, 31,4142, 31,41421, 31,414213, (2) : 31 = 3 31,4 4, 6555367 31,41 4, 7069650 31,414 4, 7276950 31,4142 4, 7287339 31,41421 4, 7287839 31,414213 4, 7288015 Av : (1) , (2) , . 3 2 . 3 2 4, 7288. : Av >0, x x , + , ( ) , ( ). To x x. : x lim =

, x>0, 0x=0. O : 3 2xy xy

3

2

2

xexp

= =

2

5.1

163

, ' , . : , x, x1 , x 2 , : x1 x 2 = x1 + x 2

x1 : x 2 = x1 x 2 ( x1 ) x 2 =2 x1x x ( )x = x

( ) x = x x

. x x . x x, :

f : f (x) = x , , 1, . = 1, f(x) = 1. f(x) = 2x . : x y=2x-5

1 32

-4

1 16

-3

1 8

-2

1 4

-1

1 2

0 1

1 2 y

2 4

3 8

4 16

5 32

(x, y) . , f (x) = x >1, :

y=2x

1 0 1

x

164

5:

. (0, +) . . x1 , x 2 : x1 < x 2 , x < x y'y (0,1) x.1 2

y x2

x1 A(0,1) x

y=x , >1x x1 x2

1 g(x) = . 2 :

x -5 x 1 32 2

-4

-3

-2

-1

0 1

16

8

4

2

1 1 2

2 1 4

3 1 8y

4 1 16

5 1 32

(x, y) . ,

y=( 1 ) 21

x

f (x) = x 0 0 1 > 0 , f(x) = x . log . , > 0 1 > 0, : a x = q x = loga q : O log o :

log 2 8 = 3 1 log 4 2 = 2 log10 0, 001 = 3log 0,5 0, 25 = 2

, , , ,

8 = 232=4

0, 001 = 103 0, 25 = 0,52

, > 0 1 , x > 0 : log x = log = x , 1 = 0 = 1, : log 1 = 0 log = 1 = , . , , , , .

5.2

175

>0 1, 1 , 2 , > 0 k : = 1. log (12 ) log 1 + log 2

2 3. log = k log k

1 2. log = log 1 log 2

1. : log 1 = 1 x x1 =1 x1 x 2 = 12 ,

log 2 = 2 x x 2 =2

(1)

x1 + x 2 = 12 , log (12 ) x1 + x 2 = , (1), :

log (12 ) log 1 + log 2 =2. . 3. : (2) log = x x = : kx =k , log k = kx , (2), : log k = k log > 0

.

= , 1 log = log = log 1

1

176

5:

:A = 1 log 2 256 + 2 log 2 3 log 2 18 2 1 log 2 256 + 2 log 2 3 log 2 18 2

:A =

= log 2 256 + log 2 32 log 2 18

[ 3]

= log 2 16 + log 2 9 log 2 18 16 9 = log 2 18

[ 1,2]

= log 2 8 log 2 23 3 = =

, 10. . , log log10. : log = x 10 x = , : log 213 log 0,325 213 0.325 log log = = 2.3283796030.488116639

e . e. . , ln, loge. : ln = x e x =

5.2

177

, , : ln 325 ln0,37 325 0,37 ln ln = = 5.7838251720.994252273

10 e, . , . ,>0 , 1 , >0 : log =log log

*

log = x. = x , :log= log = x log= log log x

= ( x log )log log

: log log= log , log =

: log = log log log = ln ln

log log log, ln ln. :

log 2 17 =

log17 = 4.087462841 log 2

log > 0 1 > 0, , > 0 1 > 0 .

178

5:

1 Richter R I I R = log I 0 . i) R I = 1000I 0 ii) I R I 0 iii) 1 Richter.

I0

= i) 1000 0 R = log

I : I0

1000I0 = log= log1000 3 R = I0ii)

R = log

I I = 10R I = I0 10R I0 I0

(1)

iii) I , I ' R, R' . ' R= R + 1 (1) :

I ' I 10R ' 10R +1 ' = 0 = = 10, I= 10 I I I0 10R 10R I ' 10 I 1 Richter. 2 p . ' pH = log[H + ] , [H + ] H + . i) pH : ~ : [H + ] - 6, 3 103 ~ : [H + ] - 5,0 109 ii) [H + ] : ~ : pH- 4, 2 ~ : pH - 6,6

5.2

179

i) pH log(6,3 103 ) 2, 2 pH log(5, 0 109 ) 8,3 ii) pH 4, 2 , 4, 2 = + ] log[H + ] = [H + ] = [H + ] =105 log[H 4, 2 104,2 6,3 pH 6, 6 , 6, 6 = + ] log[H + ] = [H + ] = [H + ] =107 log[H 6, 6 106,6 2,5 3o 32 N(t) N 0 e 0,0495t , t , = 32.

t , N(t) =N 1 N 0 e 0,0495t =0 e 0,0495t = 2 2

N0 . : 2

1 2 0, 0495t = 0, 69314718 0, 0495t =ln

t = 14

1. , , : ii) log 1 10 iii) log 1 32 i) log10 0, 00110 2

iv) log 9

2. x : 1 ii) log 4 x = i) log10 x = 3 3. : 3 ii) log 8 = i) log 16 = 4

27 3

v) log 2 16

vi) log 32

8 272

2

iii) log

x=

2 3

4. : 2 ii) 3log 1 i) log 2 3 + 2 log 2 4 log 2 12 = 10 2 + log10 5 log10 4 = iii)

2

iii) log 0,1 = 3

1 1 1 log10 25 + log10 8 log10 32 = 2 1 log10 2 3 5

iv) 2log 2 6 2log 2

3

=2

180

5:

v) 2 log 2 (2 + 2) + log 2 (6 4 2) = 2 5. t Q(t) = Q0 e0,34t , Q0 . ; 6. , p ( Pascals), h ( ) i) k, 3050m 68900 Pascals. ii) 1000m; 7. () . L0 6. L m : L

= 101300 e kh p

m = 2,5 log 6

i) m L 5 100 L0 . = ii) 1 6 ; 8. S(t) ( ) t S(t) = 100(1 e kt ). i) k, 15000 . ii) 5 ;

L0

1. : i) 41 1 log 2 3 2

ii) 9 2

1

log3 18 1

2. 1, 2, 3, ... , log1, log2, log3, ... . 3. log2 log8. - . 2

= log 2

10 log log 10 10 ....10 10 = 4. :

5.3

181

5. :

* 6. x> 0 : *7. :

1 1 1 1 log(1 ) + log(1 ) + log(1 ) + .... + log(1 ) = log 2 3 4 log x = log 2 x 2ii) log 2 log 3 = 6

i) log log =1 iii) log log log =1 *8. : i) log + log 1 = 0

ii) log () + log (= log () log () )

5.3 . 4.2, x>0 log x. , x (0, +) log x,

f : (0, +) f (x) = log x . , , f (x) = log x. log x = y y = x, (,) y = log x, (,) y = x . , , (,) (,) xOy x 'Oy '. :

y = log x y = x

xOy x 'Oy '.

182

5:

y (,) y=x 1 y=x

y= x

y

(,)

1 1 (,) x 1

(,) x

>1 y=logx y=x

y=logx 0 0, 0= 103 + 102 + 10 + . f (x) = x 3 + x 2 + x + f (x) = (x 1)(x) + f (1) x=1 x=10. 5. ( ). . : x + 1x 1 + .... + 1x + 0 = 0 . 0 , 0 . : i) : 2x 3 + x 2 + x 1 = 0 6x 4 + 29x 3 + 27x 2 + 9x + 1 = 0 ii) 2 12 . 6. 3x + 4 x = . 5x 7.

2 4 2x + = 3 3

x

194

(' )

8. 2 log(x + 3) = log(x) ; 9. x + log x = 2 (0,) 4 . 10. : log 3 (16x 2 12x ) 2x + 1 11. i) f (x) = ln x g(x) = 1 x ii) f (x) = ln x g(x) =x 2 + 1 2 ln x x + 1. 12. , , , , : = = , , > 0, ++ = : 2 = 2 + 2 2ln x 1 x.

2 = 2 + 2 2 B 2 = 2 + 2 2 13. , , , , : = = , , > 0, ++ = () = , (KM) = , () = , = , = , =

14. , , , , : 2 = 2 + 2 2 2 2 2 0 < , , < = + 2 2 = 2 + 2 2 : = =

++ = .

15. , , , , : 2 = 2 + 2 2 2 2 2 0 < , , < = + 2 2 = 2 + 2 2 , (M) = , (KM) = , () = , = A, = , = .

-

195

-

(3, 1) (3, 4)

(21, 24) (5, 2).

3 5 , . 4 3

ii)

3, = 3, 3 = 3,

+2 , , 2 (2, 1).

( 3 + 1)( + 1), ,

)

10. x = , y = , z = . 11. f (x) = x 2 4x + 3, g(x) =x 2 + 2x + 3, h(x) 0,5x 2 3x + 4. =

(4,3, 5)(10 + 2, 16 + 2, ), 1 x 1 : y = x + 2, 2 : y = 1 2

1 11 R = T + . 600 301 1 = , 2 2 = 1 2

(1, 2), (2, 1).

2.3 . 2

2 4 , 3 3

3 2 3 2 2 , 2 , 3 2 3 2 , 2 2 (1, 2), (1, 2), (2, 1), (2,1).

1,

2 + + 1 , , +1 +1

196

-

(4,3), (4,3), (0, 5). (1, 0), (3, 0), (2, 1), (4,3). 1. x = 3, y = 3 2, = 450.

2. = 1, = 3. 3. 1.(,1],

[1, +),[2, +),

(, 0],(, 1],

4.

6

2 3 33 4

[1, 0], f (1) = 1 h(1) = 2, h(1) = 2 f (x) f (3) g(x) g(1)

[1, +).

5460 18000 . 6.1 3 3 , , , 3 2 2 3 3 1 3 , , 3, 2 2 20, 1, 0, 0,1, 0,

478, 733, 1062

58.

4.

f (x) =2 (x 1) 2 + 3 f (x) =2 (x 2) 2 1. 2(x 2) 2 2(x + 2) 2 2(x 3) 2 3 2(x + 3) 2 3

2 212 + 8 3, 12 3

2 2 . 2

197

3. + ( + ) =180 ,4 3 4 x = , x = , x = . 5 4 3 5 5 2 5 , x = , x = . x = 3 2 5 1 3 x = 3, x = , x = . 2 2

+ + 90. = 2 2

2. 3.

4. x =5

5 , x = 2

5 2 , x = . 3 3

5. 8 5 20 . 6. 2 1 2 2 2 1 (3 2 ) . 2 3.31200 = 3 1 , 1200 = , 2 2

2.

2 2 1

2.x y = x, y = , y = 3x 2 y = x, y = 3x, = 0,5x, yy = 2,5x.

3 1200 = 1200 = . 3, 3

3 2 3

2 3

ii)

1 3 (2850 ) = , (2850 ) = , 2 2 3 (2850 ) = , (2850 ) = 3. 3

187 1 = , 6 2

187 3 = 6 2

x = , x= 2 +

187 3 = , 6 3

187 = 3. 6

21 2 21 2 = , = 4 2 4 2

x = + , 2 x= 2 , 4x= 2 6 x= 2 +

4

x= 2 +

3 , 2

21 21 = = 1. 4 4

7 , 6

198

-

x = x = , = x x = + , 6 x = + , 4 x= x = + , . 6 x = , 6 x = x = , 3 = x x= 2 + x= 2 + 3 2 10. = 2 x= 2 + , 3 5 =x= 2 + 4 4 x= 2 , x = x = + , 3 4 2 x = 2 x = 3 3 x = + , 2 x= 2

x= 2 , 2 3 x= 2 , 4 x= 2 + , .

22 , . 45 6 + 6 + 2 = x = x , 9 9 x = +x = 6 + 2 , 9

10 + 5, 42 + 7 , 12 2 5 , 6 24 + 7 24

24 , 36

12 , . 60 2 2 2 + 6 55 2 == 2 , , 66 x= 2 , 3

t = , . 6

2 x= 2 x= 2 , 2 3 3 3 x= 2 + x= 2 + , 5 5 2 x= 2 , 3 3 x= 2 + , 54 , 512.

x= 2

3.

199

x=

, 2

g = 7 2

3x

3 g = 7. 2

3 x= 2

x 7. 105 =

1 2

x

2(1 + 3) , 4 2(1 3) 105 = , 4 105 = 2 3,

105 = 2 + 3, x = , 2(1 3) 8 195 = , 4 = x + , 5 30 2(1 + 3) 195 = , 4 x = + , 4 195= 2 3, 2 x = x = + , 4 5 195= 2 + 3. 2 x = + , 33 ( + ) = , 5 65 13 . 56 4 ( + ) = , 65 , . 33 2 ( + ) = , 56 7 13 19 56 , , , ( + ) = . 12 2 12 12 33 x = + , 6 x = , 1 1 3 2 2 x = + , 4 2 x 2 ( )

3

=

200

-

( + 2) = (( + ) + ) = ....

3. x = 4.

4 =

x=

5 . 4

2 2+ 2 = , 16 2+ 2+ 2 2+ 2+ 2 = . 16 2 2+ 2 9 4 , = , = 2 13 2 13 2 3 = , = . 2 3 2 2

, 4 = = .... 4 6 x= 2 . 3

8. x =

1 4 = , = , 2 5 2 5

1

3 3 4 6

3 2 2

1 = , = 2. 2 2 2 2 x= 2 , 2 3

x= 2

x= 2

2 =

24 7 , 2 = , 25 25 24 7 2 = , 2 = . 7 24 16 13

, 3 x= 2 , 2 x= 2 + ,

3.

( + 2) =

3 + = , 8 8 2 3 = ... 8 86 6 x = , x = + 3 3 2

x= 2 x= 2 + , x= 2 2 x= 2 + , 3 2

6. x= 2 x= 2 + x= 2 + 5 ,

2 2+ 2 = , 16 2 = 16 2+ 2+ 2 , 2

1 , 4

3+2 4

201

1 4

1 4

3x x

++ =

2x 6x 2x + 8x 1 (8x 4x) 2 1 2x 2

2 + x = = x , 3 10= x6 2

2 + 2 x = , 8 2

2 22 4xx

5.

23xx

x 2 2 + 4 22 2

2 2x 2x

11 f (x) = 2 x + 6 3 f (x) = 2 x + 4 4 f (x) = 2 x + 3 f (x) = 2 x 4 4 , 3 x= 2 2 x= 2 x= 2 ,

x . 2 x = x = 2 + 4 6 5 x= 2 + , 6

x= 2

x = = xx=

, 3

7 x = + 12 12

7 12 11 x= 2 + , 12

6 + 2 x= 9 6 6 2 x , = 9

=

5 = 12 12 = () + () 2 2 + 4

2.

= , 4 () + () = 2 2.

202

-

3.

10 2 2 + 2, 2 2 + 2. 7 x = + , 24 24

4. x = +

6 5 x= 2 + , 6 x= 2 + 13 3 (10 + 2 2)m, 7

5. = , 4 = 40( 2 1) h

6.

6.

= , 8 P= 1+ 2

(10 2 2)m

(t + 1) 10 + 4 4 2 6+ 2 , 2 4 12. x= 2 + 4 2 = x + , 3 4

9. 1. AB 284 m 2. B 1,58 m 5. 1150 6. 51, 2 m 7. 60, 25 1.-5. 6.-7. 8.-10.1 2 11. = 300. =

6 2 , 4

13. = 200 2 2 + + 200 S 4

= = 200( 2 + 1) , Smax 8

15. B = 60 , =45

1. 2.1 2 4. =1

3. 4. x= 2 , 3

3. =

5.

1

1,

203

5. P(11) = 4840 6. 1. 7. = 10,

P(x) x y =P( y) = 0

2. = 2, = 19 3. = 3, = 5 4.2 2 0, , 3 3 2 = 3

8. 9. 10.= 2 3

P(x) x + 1 =

P(1) = 0.

5. P(x) = x 2 5x + 1.

1. 2. 3.

x x =P(x) = (x + )(x) +

= 0 = =

1.

(x) = 3x 3x 8, (x) = x + 3x + 9x + 27,(x) = 4x 3 16 5 , (x) = 6 3 2 (x) = 2x + 1,3 2

2

x 1

x2

4.

P(0) = 0, P(1) = 0, 1 P( ) = 0 2 5. = , = 1

(x) = + 3, (x) = 6x 2 8x + 3 x

(x) = (x) = x 2 + 7 x2 , 2. 3. 4.k = 4(x) = x 2 + 10x 25, = 0

1.

0,

6 6 , 5 5 5 0,1, 1, 3 1, 2, 32 13 ,1, 3 22

2,3, 3 2, 22

(x) x 2 8x + 64, 0 = =(x) x 4 + x 3 + x 2 + x + 1, 2 = = (x) = 3x 3 6x 2 12x 24, = 48 (x) 4x 2 + 14x 30, = 0 =

2,9 + 4 5,9 4 5 2, 2 , 1, 2

204

-

2. 3. 6.

1, 4 1, 2 1, 5

2

1

x > 2x 1

x =1

x < 2

1.

x = 3

2,

2

x2

2. x = 2 + 3.

x = (2 + 1) , 6 6

1 ( , 0) 2 8. 2 5 < x < 0 1 < x < 2 + 5

7.

596. x 1

1

9.

2, 2

3 ,2 5

x>0

1. 1.1, 2, 4 3, 2, 15

2. 1, 2, 2, 3 3. 3,1, 7, 11 4. 5. 1, 1, 2 + 3 2 , 6. 7. t 4, 6 c 0, 45 8. 9. 10. 11. y + 4x 108 12.= 5x 3 y

3 2 x

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