Algebra a Theoria

8/12/2019 Algebra a Theoria

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= ,..., 21 ,..., 21 1,2,

f1,f2,...f : f1+f2+...f=1

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2

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( )()()B :

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: )()(

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:(-)=()-( ).BA

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,

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, IR : |||||| :

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, IR : | + | || +| |:

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11

, 0 :

) ) 0,

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x+=0.. ;

. ;

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14

x2+x+=0 0 =2-4.

:

. >0

2

x 2,1

. =0 =

2


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: x+>0

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>0 : |x|


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1

:

=1+(-1)

:...

.........

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.

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, ,

2=+

:

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.......

21

1 :

=1-1

:

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22

, , :

2=

:

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.......


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.......

23

1

:

1

1S 1

:

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.

24

(x1,y1) (x2,y2) ,

()= 2122

12 )yy()xx(

:


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(x1,y1) (x2,y2) y=x+

: =12

12

xx

yy

:

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1,2 1: y=1x+1 2: y=2x+2.

: 1//2 1=2.:

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6. - ( ) ;

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7. , IR , , 2 ;.........................................................................................................................................

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12. f f(x)=(x)+c, c>0

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13. f f(x)=(x)-c, c>0

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14 f f(x)=(x-c) , c>0

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15. f f(x)=(x+c) , c>0

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16. ;


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18. f

. f ;

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.........................................................................................................................................19. f . f;

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20. x


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21. y=x+;.........................................................................................................................................

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22. x ,>0 ;.........................................................................................................................................

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23. x ,>0 ;

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24. x ,


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25. x ,


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29. 2

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30. 2

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31. f(x)=x2+x+ , 0

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32. 0)x(B

)x(A 0

)x(B

)x(A

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33. 0)x(...)x(B)x( (< 0 )

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34. f(x)=|x|;

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35. ;

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;:

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44. ;

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45. ;

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46. ;

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47. ;

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48. ;

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49. ;

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50. ;

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......

51. ;

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.......

52. ,;

:.

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53.

:.

.......


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54. ;

:

......

55. ,;

:

......

56. =1 ;

:

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57. , , , .

;

:

......

58. ;

:

......


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59. () =S-S-1

:

60.

;

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......

61.

;:

......

62. ;

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63. ;


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64. ;

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65. ;

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66. ;

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......


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. :

1. (2010

+1)x=2009 , IR .

2. 0,8

3. : 22

4. 2=

5. f(x)=x2+x+ , 0

2

2x)x(f

6. x2-2x+>0 IRx >1

7. >0 : |x|0

x

10. IR ||2

11. x=


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16. x+1

33. () +1=+5 .

34. , , +=

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34


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35. , , +=

36. -4, -1, 2

.

37. :2

)(S 1

38. , , -=-

39. =1

40. 0.

41 , ,

42. 1 : S=1

43. () +1=5

44 2=+ , , .

45. , :

. )BA(P)A(P . )A(P)BA(P 46. : 1)A(P)A(P

47. : )B(P)A(P)BA(P)BA(P

48. BA )B(P)A(P

49. , : P(A-B)+P(B)=P(B-A)+P(A)

50. )BA(P)BA(P

51. .

52.

.

53.

.

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53


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. 1

A.I)

. 0 x+=0 x=-

.

. , , >0 >0

. > > +0

II) :

. >0 =....... x

2+x+=0, 0 >0 =...... x1,2=......

. >0 x = x=...... x=......

. =......

.

.

-------------------------------------------------------------------------------------------------------

2

A. x =2 x2-4=0.

B. x 1 >4;

.

-------------------------------------------------------------------------------------------------------

3

x2+5x+10=0,

. ;. ;

. .-------------------------------------------------------------------------------------------------------

4

. f(x)=-x2+2x-5.

. -x2+2x-53x-1-------------------------------------------------------------------------------------------------------

5

: 2

2

x-3 -x -x+6

x -5x-6


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x2-4x++1=0.

. ;

. x1,x2 , ,

1 2

2 1

x x

x x

=2.

-------------------------------------------------------------------------------------------------------

7

3 x-1 +1 x-1 +11

- x-1 =2 7

-------------------------------------------------------------------------------------------------------

8

x2-2x+(+1)=0 (1) R.

. S, P (1)

S+P=4, R.. A=3x1+5x1x2+3x2 x1,x2 (1).

-------------------------------------------------------------------------------------------------------

9

. x1,x2 x2+x+=0, 0

S=x1+x2=

P=x1x2=

.

B. x2+3x-4=0. x1,x2 ,

3x1-2, 3x2-2.

-------------------------------------------------------------------------------------------------------

10

: x2+x-2=0 (1)

.. (1)

. =1.

-------------------------------------------------------------------------------------------------------

11

: x2-2x-5=0, x1, x2.

:

. x1+x2 . x1x2 . x12x2+x1x22-------------------------------------------------------------------------------------------------------

12

. : 3 1 = 5

. :

3 (x -5) x < 5 2 3

x >1.

-------------------------------------------------------------------------------------------------------


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.. : 2x2+2x-12=0.

. : x2- 2x -5 = 0. x1, x2

:

i. x1+x2 ii. x1x2 iii. x12+x2 2 ( ).

------------------------------------------------------------------------------------------------------- 14

x2-x+-1=0, R.

i. .

ii. (1) .

iii. x1,x2 2x1+x2=3.

iv. : 1=3x1-2 2=3x2-2.

-------------------------------------------------------------------------------------------------------

15

f(x)=| x-1| - , IR

. (-2, 1) =2

. =2

.

2008

2

xg(x) 2 | x 1|

x 4x 3

-------------------------------------------------------------------------------------------------------

16

. ( )

-x2+2x+15

. f(x)=|-x2+2x+15|+1990, x [-3,5]

1. f 2. f(1+ 2 )3. f(x) -2001x+1

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.

1

=22+5-3 =2-9 , R . =0. .

. x=

,

2

k 2x2+kx+161 (2-3k)>0 x R ;

3

. : 23x

4x3

. x

=|x+2|-3|x-3|-x+7

4

: x2-(3-2)x ++1=0 R . 1,2

. : . 1+2 . 12

.

: 111

21

5

x2-3x+-1=0. (1)

. (1)

. 1,2 (1)

2121

2


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|x-3| 2 |2y+3| 5.

x y

.

=x-2y

7

. : 2x2+x-3=0

. : 3-|x-1| 0

. f(x)=3xx2

|1x|32

8

f(x)=xx

x2

. .

. : |f(x)|

10

: x2-x-=0 , 0 (1)

.

0 . x1,x2 (1)

x1+x2 x1x2

. x12+x2

2>3


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:12

|3x|5

3

1

4

1|3x|

12

x2-3x+|-1|=0 (1)

.

. x1,x2 (1) x1=2x2

x1,x2 .

13

x2

-x-=0, 0 (1). 0 . x1, x2 (1)

i) x1+x2 ii) x1x2

) 1xx

xx

21

21

14

|x|x

|x|)x(f

2

) )

) )2(f

15

2

2|3x|

3

4

3

|3x|2

x(3-x)+2x>2(2x-5)-5x-6

16

4|1x|

x25)x(f

2

. f(3)=-2. . x

4+x

2+f(3)=0


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f(x)=-3x2+11x+4

) f(x) .

) 4)x(f)x(g

) :16)8x(x

)x(f)x(K

) x : K(x) 1 .

18

25x10x)x(f 2 |5x|)x(g 2

. f g. x IR f

y=2

. 1


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:

3

1)A(P ,

6

1)BA(P

6

5)BA(P

: P(A) P(B)

22

:

7,0)A(P , 6,0)B(P 4,0)BA(P : )BA(P P(B-)

23

:

3,0)A(P , 4,0)B(P 6,0)BA(P :.

. ,

. . ,

. ,

24

:

12

5)A(P 4

3)B(P

.

. :12

5)BA(P

6

1

25

: P(A)=0,3

P(B)=0,4. : 7,0)BA(P4,0

26

: P(A)=0,3

P(B)=0,2. : 3,0)BA(P1,0

27

12 15 . 3

2

5

3

. .

.


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.

10

1 x1, x2 x2+x+=0 0, S=x1+x2 =x1x2 :

S

10

2. x2+x+=0 0 =2-4.

.

5

3.

.

) , , 0 2

.

) : = .

) : +=0 ==0.

) x + = 0 , IR, =0 =0.

) x

= 53

x8

M 15

2.

.

M 10


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: x2-x-6=0, (1) x

2-(

2+1)x+

2-2=0, IR, (2).

1. (1) (2)

, .

10

2. (1) (2) .

7

3. x1 x2 (2) 1xx

xx

21

21

,

.

8

:4|1x|

|x2|5)x(f

1. f.

8

2. f xx yy

8

3. f(3) = -1 :

0)3(f2|x|x2 9


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1i.

1 :

=1+(-1) 10

ii. ; 7

2. () ()

:

1. x = -3 x = 3 x = -3.

2. 1 2

1 2x

2- 3x + 1 = 0.

4. x : 23x = 3 x.

5. 5x = x + 5, x > 0. 8

: (x 1)2 (2x 3) = 0, (1) R.

x1, x2 (1) K=

2

3

2

321 xx

.

25

x2 6x + 3 1 = 0, R.

R :1. . 5

2. . 10

3. x1, x2 , R :

5

2

x

1

x

1

21

, 3

1.

10

1. : x2

+ ( - 1)x + 1 = 0 (1)1. . 5

2. R (1) . 5)5xx2)(1x( 2

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Algebra a Theoria