Διαγωνίσματα Μαθηματικών Α γυμνασίου

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Transcript of Διαγωνίσματα Μαθηματικών Α γυμνασίου

2007

: T: : Copyright Copyright 2007: EKOTIKO OPANIMO IBANH ABE 98 106 80 A. T.: 210 3661200, Fax: 210 3617791 http://www.livanis.gr A , , , , , , , , , . N 2121/1993 E. : E O ISBN 978-960-14-1481-2

.

.....................................................................................................................9 1: .......................................................................13 2: .................................................................................17 3: ....................................................................21 4: . ......................................25 5: ...........................29 6: .......................................................35 7: .............................................................41 8: ......................................................45 9: .....................................................................................53 10: - - ...................................59 11: ..................................65 12: ..................................71 1: .......................................................................77 2: .................................................................................81 3: ....................................................................85 4: . ......................................89 5: ...........................93 6: .......................................................99 7: ...........................................................105 8: ....................................................109 9: ...................................................................................115 10: - - .................................121 11: ................................125 12: ................................131 7

8

. 12 , 10 2 , . , , , . . , . .

9

10

11

12

1 1 : 1

1

. : 1. ............. , .................. ................., ............................ , ............................ .............................., = ....... + .........., < ... .... . = 0 = .......... ........................ . (1,7)

2. : 1 = ..... . ..... : = 0. : ..... . ............................ ........................... . .................................... . ......... . , ... (, ) = ......... . ............................................. ... (, ) = .............. . (1,7)

13

. : + . + = .... + .... + ( + ) = (.... + ....) + .... + 0 = ....... ............................. = .... .... ( ) = ........... .... = 0 = .... ( + ) = ........ + ........ ( ) = ........ ..... (1,6)

2

. , : 254+4=2 254+4=6 73+80=0 5+7:43=0 9+9:9+1=3 (2,5)

. : 418 31 100 23 7 1 12 (2,5)

14

3

. : 1. , = 1, : . 1 = 1 . = 1 1 . = 1 = 1 . = 0 = 1 2. : 7 104 + 3 102 + 1 : . 7.301 . 731 . 70.301 . 73.100 3. ; . 5 + 0 + 0 + 4 . (5 + 0) (0 + 4) . (5 0) + (0 + 4) . 5 0 0 4 4. : x + x + x x x x x : . x3 4 x 3 4 B. x x . 3 x 4 x . 3 x x4 5. : x x x x = 0 : . 2 . 3 . 4 . 5 6. 10 30; . 4 . 5 . 6 . 7

15

7. 5 : . 5, 6, 7, 8 . 0, 2, 4, 6 . 0, 1, 2, 3, 4, 5 . 0, 1, 2, 3, 4 8. ; . 0 = 3 0 + 0 . 40 = 6 6 + 4 . 83 = 10 8 + 3 . 91 = 10 8 + 11 2,4(8x0,3)

. , . 3_ 6 _ _ 2 _1 96__ _36_ 2_ _ 4 3___ 2 3 3, 10 5, 9 2, 3, 4 2, 3, 4, 5, 9, 10 (2,6)

4

. : x = (23 10) 2, y = 14 + 24 + 34 = (23 + 5 2) : 32 + (15 3 5)5. x + y + , . (2,5)

. = ... (6, 8) = ... (45, 60, 75) : = ( + )3 = 3 + 3 2 + 3 2 + 3 (2,5)

16

2 2 : 1

1

. : i)

........ = ........ ................... . .................................. , ............................. .

ii)

7

>

4 7

1

0

5. 1 (5x0,2)

36

2

. :

+ 12,64

+ 104,23

16 3 0 2 (4x0,5)

. : 1. 2. 3. 4. 5. 6. ( + 2) + ( 3) ( + 3) (2) (3) + ( + 3) ( + 2) ( + 3) + (3) (2 + 3 + 3) (2 + 3 3) + ( 2 + 3 3) 1 2 3 . . . . . . 4 2 3 3 2+3+3 233 2 + 3 + 3 2 + 3 3 23+3 5 6 3 (6x0,5)

3

. : 1. = 2, : . + 2 . 2 . + 2 2 . 37

2. = 3, : . + 3 . 3 . + 3 3 . 3. : 3 1 . 8,64 < < 1000 < 0 < < + 4 < 100 2 5 3 1 . 1000 < 8,64 < < 0 < < + 4 < 100 2 5 3 1 . 1000 < < 8,64 < 0 < < + 4 < 100 2 5 3 1 . 1000 < < 8,64 < < 0 < + 4 < 100 2 5 4. . 5 3 3 : 5 3 . + 5

. +

5 3

.

3 5

5. , = : . = . > . + = 0 . = 1 6. 5 1, ... . 2 .. . .. . 2. 1 = . = ., 0 = , 0 3. > 0 ..0 < 0 < 0 < 0 > 0 4. = .. 5. : = , 0 6. = .

41

7. ..... ......... = 8. ...... .....

9. . ..... . 10. .....

( )

.....

= ........

2,5 (10x0,25)

. , . 1. x : x = x 2. x5 = x5 3. x3 x4 = x12 4. (x3)2 = x5 5. (x)4 = x4 2,5 (5x0,5)5 3 2

2

. N : 1. , ( + )2008 : . 1 . + . 0 . 2008 2. ( )3 < 0, : . , . , . , . = = 0 3. =( 1)0 + ( 1)1 + ( 1)2 + ( 1)3 : . 0 . 1 . 2 . 342

4. x : 1014 = 1010 : . 102 . 1014 . 1024 . 102 5. : 0,00001 x = 107 : . 1012 . 1013 . 1014 . 1011 2,5 (5x0,5)

. : 1. 2 = 1 1 2. 2 = 4 3. ( 2) = 8 4. ( 2) = 16 5. (2)3 = 64 2,5 (5x0,5)

. = 2 . = 0 . = 2 . = 3 . = 4 2 3 4 5

1

3

N : 1 2 3 4 3 34 32 + ( 3)2 23 2 84 43 4 4 233

1 3

3 1 23 8

9 0 24 12

3 2 4

43

5 6 7 83

( 3)02 3 4

1 81 16 1 512

116 81 1 54

381 16 1 512

(56)266 6 2 ( 3)8 1 3 4 2 2 3 3 3

2627 64

28

6864 27

9

1

10

( 1) + 1

2

0

2 5 (10x0,5)

4

. : x = 24 + 2 ( 3)2 [23 ( 3)3] : 7 B. : = 9 xx + 1 2 x2 + 18 xx + 3 5 (2x2,5)

44

8 1 : 1 30

1

. : 1. . 2. . . . .. . . 3. . . . . 4. .... . . . 5. ... . 2,5 (5x0,5)

45

. : 1.

2.

3. 4. 5. 6. 7. 8. 9. 10. 2,5 (10x0,25)

46

2

. :

. 1.

.

2.

. 3.

.

4.

. 5.

47

.

6.

.

7.

.

8.

.

9.

1

2

3

4

5

6

7

8

9

1,8 (9x0,2)

48

B. N x : 1. = 12cm x = 2.

x= ....

3. x= .... 4.

x= .... 3,2 (4x0,8)

3

A. N : 1. , : . . . . 49

2. 3/10 : . 98 . 102 . 108 . 54 3. : . . . . ,

4. , : . 65 . 90 . 120 . 45

5. 2/5 . : . 18 . 72 . 108 . 8 2,5 (5x0,5)

. xOy . , , .

2,5

4

. : 1. ....... . 2. . . 0,5 (2x0,25)

50

. : 1. 70 2. 0,5 (2x0,25)

. (,2cm). x . : x 2 cm 1 E 1

.

. x :

1.

x =

2.

x =

51

3.

x =

4.

x =

5.

x =

1,5 (5x0,3)

E. = 50 , A . ;

1,5

52

9 2 B : 1 30

1

. : 1. , , ....... . 2. ........................... .................. . ............................... . 3. , ............... ................................................................... . 1,5 (3x0,5)

53

. : 1. . 2. . .

2

1

3,5

2

. (1,3), (2,0), (0,4), M(4,4) : xOy (1,3) (2,0) (0,4) M(4,4) 2,5

54

. 3 cm = 4cm. ( ). ) . ) , . ) ; ) A = 40 BOK . 2,5

3

. :

1. .............. . 1

2. .............................. . .............................. . : ......., ....... ..., ......... . // , =............. // , + = ............ = , ............. . 1

55

. . :

120

2

. :

1

4

. //. .

1,5

56

. , B, .

1,5

. // , , , .

.

2

57

58

10 3 B - - : 1 30

1

. :

1. ...................................................... .................................................... . 2. ............ . 3. ............................. . 4. ...... ................. . ...................... . ......... .................................. .

5. ................... . ........... . .......................... ............ . 2 (5x0,4)

59

. :

1

. :

1. 2. 90 3. = 90 + 4. = + 2 (4x0,5)

2

. , B, , . ;

B 40

80 110 60 92 45 52

60 40 45 100

75 30 60

40 52 1,4

60

. x.

1.

x = .... 2.

x = .... 3.

x = .... 4.

x = .... 5.

x = ....

61

6.

x = .... 3,6 (6x0,6)

3

. 2/5 B . . 2,5

. . , , .

2,5

. .

. . .

. .

2,5 (5x0,5)

62

. (,2cm) (,4cm). . , , , . ; . 2,5

63

64

11 : 2

1

. :

1. 0 .................................. . 2. ................... . ........................................... . 3. ....................... . 4. .................. . 5. ........................................... ............................ . 6. ............................. ......................................... : = + .......... . 7. ............................. ......................................... : : = . 8. 0 ...................... ...................... . : = ....... . 2,4 (8x0,3)

65

. :

1. ;

2.

+ = ;

3.

+ + = ; +

4. ;

5. ;

6. ; 7. ; 8. , ;

9. ;

10. o + ;

11. , , 2 = 2; 12. 12345 3, 5 9; 13. x = 4, |x 4| + |2 x| + |x + 2| = 6; 2,6 (13x0,2)

66

2

. :

1. , = 1 : . = 1 = 1 . = 0,5 = 2 . = +1 = +1 = 1 = 1 . 2. 120 350 : 120 = 23 3 5 350 = 2 52 7 ...(120,350) : . 2 5 . 23 3 5 . 22 3 52 ...(120,350) : . 2 5 . 23 3 5 . 22 3 52

. 23 3 5 7

. 23 3 52 7

3. : . (12 5) 2 . 3 (4 5 6 : 7) . 2 (4 + 7 32) . 7 + (2 5)+(3 : 4) 4. 5cm. , (%) ; . 100% . 200% . 300% . 400% 5. 3 + 5 = 2 , : . 3 . 5 . 8 . 11

6. , x :

. x = 2 B. x = 3 . x = 4 . x = 5

67

7. 1; ( 1) ( 2) ( 3) . 6 2 + ( 1) ( 3) . ( 3) 2 + 5 6 +4 8 . + 3 2 4 . 52 4 5 30 : 15 8. 36 12, : . = 2 = 12 . = 2 = 3 . = 1 = 3 . = 2 = 4

9. (2) (3) (+4) (2)> 0, : . > 0 . < 0 . = 0

. .

10. = 10 232 : . 10 2 9 = 8 9 = 72 . 10 62 = 10 36 = 26 . 10 2 9 = 10 18 = 8 . (10 2 3)2 = (10 6)2 = 42 = 16 2,4 (10x0,24)

B. 1. 5 . :

1,3

5

5

5

5

5

68

2. x :

15 2 3 7 3 4 25

1,3

3

. : x =

7 2 5 5 + 4 y = (3 ) 3 3 4 2 x . 2

. y2 : x2. 1

. 20 x% y , . ( x y x . ). : i. . ii. (%) . 2

69

4

// . : A = (2 3 2 7) 2 [(2 3 2 2 ) : 2 6] 5 . , x .

5

70

12 : 2

1

. x :2 3x = 3 , 7

1.

x = .....

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

x = .....4. (x 3) : 3 = 1, 3 6. = 1, 7x

3. (6 x) : 9 = 0 ,x5 =5, 5. 3

x = .....

x = .....

x = .....

x = .....

7. 10 x = 0,0001 , 10 104 1 = , x 10 10

x = .....

8. 5 102 x 10 6 = 10 7 , x = .....9 = 91 , 2 x

9.

x = .....

10.

x = ..... x = ..... 2,6 (10x0,26)

71

. :

1. . 2. . 3. , . 4. . 5. .

6. . 7. 180. 8. . 2,4 (8x0,3)

2 . =40:1. AOB ...................................... . 2. ................... 3. = ....... . 4. = ........... . 5. ......... , ........................... .......................... ........................ . 1,5 (5x0,3)

72

. + : / 1,5 (5x0,3)

+

. //.

: . . . . . , . . x. 2 (4x0,5)

3

= = 45 . cm 30:(x-2)=5, cm : 1 2 2 = ... (2, 3, 10) + ... (2, 6, 10). 2y

2

73

1. . 2. .

5 (22,5)

4

16 cm = x + 2 = x cm. = 140 .

1. . 2,8

2. , , , , , , , , , . 2,2

74

76

1

1

. 1. , , , = + , < 0. = 0 = . 2. : 1 = 0:=0 : 0 . . 2. ... (, ) = 1. ... (, ) = .

77

. +=+ + ( + ) = ( + ) + +0=0 ( + )= + = ( ) = ( ) 1 = 0 = 0 ( ) =

2.

2 5 (4 + 4) = 2, : 2 5 8 = 10 8 = 2 2 (5 4) + 4 = 6, : 2 1 + 4 = 2 + 4 = 6 (7 3 + 8) 0 = 0, : (4 + 8) 0 = 12 0 = 0 (5 + 7) : 4 3 = 0, : 12 : 4 3 = 3 3 = 0 (9 + 9) : 9 + 1 = 3, : 18 : 9 + 1 = 2 + 1 = 3 . 418 31 13 15 162 23 7 1 100 8 12 4

1. 418 =5 31 13 =

2. = 23 7 + 1 = 161 + 1 = 162 78

3. = + , < 100 = 12 + , < 100 =4 12 8=

3.

1. 2. , 7 10.000 + 3 100 + 1 = 70.000 + 300 + 1 = 70.301. 3. , : ) 5 + 4 = 9, ) 5 4 = 20, ) 0 + 4 = 4, ) 0. 4. 3 3 5. , : x x = 0 x = x , = 3. 6. , : 11, 13, 17, 19, 23, 29. 7. , . . : 3062 6201 9600 4365 2004 3600 . .

4.

x = 13 2 = 26 y = 1 + 16 + 81 = 98 = (8 + 10) : 32 + (15 15)5 = 18 : 9 + 05 = 18 : 9 + 0 = 2 + 0 = 2 x + y + = 26 + 98 + 2 = 126 = 2 32 7 79

.3 3 6 = 2 3, 8 = 2 ... (6, 8) = 3 2 = 3 8 = 24, = 24 2 45 = 3 5, 60 = 22 3 5, 2 75 = 3 5 , ... (45, 60, 75) = 3 5 = 15, = 15 = (24 + 15)3 = 393 = 59.319 3 2 2 3 = 24 + 3 24 15 + 3 24 15 + 15 = 13.824 + 3 24 225+3 576 15 + 3.375 = = 13.824 + 16.200 + 25.920 + 3.375 = 59.319 = .

80

2

1.

i)

= . , .

ii) : 8 4 1 1 > < 7 7 4 3 iii) . 1 .

. i. ii. iii. iv. v.

:+ = + = + 1 = 1+

vi.

81

2.

61 48 4 5 41 24

+

1. + =

5 7 (6,8)=24 5 4 7 3 20 21 41 + = + = + = 6 8 6 4 8 3 24 24 24

2. =

22 2 20 20 : 5 4 = = = 25 25 25 25 : 5 5 27 5 (16,12)=48 27 3 5 4 81 20 61 = = = 16 12 16 3 12 4 48 48 48

3. = .

1, 2 =

2 1

6,

20 20 : 4 5 = = 24 24 : 4 6 21 21: 7 3 = = 28 28 : 7 4

3,

4 4 3 , x = x 1 = x 3 3 4 6 5 6 7 , < < 11 11 11 11

: 1 1 4 4 2 24 8 : = = = = , 5 5 4 20 5 5 4 20 7 . : 20

82

31. :

1 1 1 1 (2,4)=4 2 1 3 + = + = + = 2 22 2 4 4 4 4

1 7 2. : 2 3 = 3 = 7 3 3

3. :3 4

3 5 3 5 9 20 29 + (4,3)=12 + + 4 3 4 3 = 12 12 = 12 = 29 6 = 29 = 1 1 (2,6)=6 3 3 1 4 12 4 8 + + 2 1 1 2 6 6 6 6 + 2 6

4. 5. , x + 5 < x + 6 6. , 2 (x + 3) = 2x + 6 7. : 3 1 7 2 + : = 4 4 6 3 =

3 1 7 22 3 1 7 4 + : = + : = 4 4 6 32 4 4 6 6 1

3 1 3 3 1 1 = + : = + : = 4 4 6 4 4 22

=

3 1 2 3 2 5 + = + = 4 4 1 4 4 4

4

. 2/3 18 1/3 18 : 2 = 9 3/3 3 9 = 27 .

83

.2 7 1 1 13 A = [ + (1 ) ] : = 7 5 2 2 15 2 7 2 1 1 13 2 7 1 1 13 = [ + ( ) ]: = ( + ): = 7 5 2 2 2 15 7 5 2 2 15 2 1 13 . ..(5,4)=20 2 4 1 5 13 = ( + ): = ( 5 4 + 4 5 ) : 15 = 5 4 15 8 5 13 = ( + ): = 20 20 15 13 15 15 15 : 5 3 = = = = 20 13 20 20 : 5 4

1 1 22 1 2 22 = ( + 2 ) : = ( + ) : = 5 5 5 5 5 5 3 4 3 5 3 = : = = 5 5 5 4 4 = .

84

3

1.

1. 123 : 100 = 1,23 2. 0,0574 1.000 = 57,4 3. 75 0,001 = 0,075 4. 32 : 0,01 = 3.200 5. 0,00475 1.000.000 = 4.750 6. 302 : 10.000 = 0,0302 7. 70.034 0,0001 = 7,0034 8. 5 : 10 = 0,5 9. 12 : 1.000 = 0,012 10. 11.092 : 103 = 11,092 . 17,024 2.749 0,9156 304,9999 724.058 476,302 50.325 17,02 2.750 0,9 305 724.100 476 50.000 85

2

. = 2 ( + ) = 2 + 2. 1 : = 2 (3,2+8) = 2 11,2 = 22,4dm 2 : = 20cm = 2( + ), : 160 = 2(20 + ) 160 : 2 = 20 + 80 = 20 + = 80 20 = 60cm 3 : = 180dm = 180:10 = 18 m = 2( + ), + = : 2, + = 18 : 2 = 9, + = 9 + 5 = 9, = 4 m. 4 : = 400mm = 400 : 1000 = 0,4m = 80cm = 80 : 100 = 0,8m = 2 (0,4 + 0,8) = 2 1,2 = 2,4m 1 2 3 4

8dm 60cm 5m 80cm

22,4dm 160cm 180dm 2,4m

3,2dm 20cm 4m 400mm

B. . 0,052m < 0,7003m < 0,703m < 2m < 2,023m < 5,23m . .. m : 0,023km = 0,023 1000 = 23m 12,3m = 12,3m 86

134dm = 134 : 10 = 13,4m 1004cm = 1004 : 100 = 10,04m 11011mm = 11011 : 1000 = 11,011m : 10,04m < 11,011m < 12,3m < 13,4m < 23m 1004cm < 11011mm < 12,3m < 134dm < 0,023km. . . ()160 4 523 100 41 6 523 10000

() 6,83333....

:160 = 160 : 4 = 40 4 523 = 523 : 100 = 5,23 100 41 = 41: 6 = 6,8333...... 6

40

0,0523

5,23

523 = 523 : 10000 = 10000 = 0,0523

. 3 5 21 25 3 800

3 : 5 = 0,6 21 : 25 = 0,84 3 : 800 = 0,00375

6 10 84 100 375 100000

87

3

x = 15 + 0,02 103 = 15 + 0,02 1000 = 15 + 20 = 35 cm y = 0,2 2 100 = 0,4 100 = 40 cm = (0,1 + 0,01 + 0,001) 1000 = 0,111 1000 = 111 cm = (x + y) = 111 (40 + 35) = 111 75 = 36 cm.

4

= 9,62 (23 : 0,8 + 0,52 : 0,01) = 9,62 (8 : 0,8 + 0,25 : 0,01) = 9,62 (10 + 25) = = 9,62 35 = 92,16 35 = 57,16. = (23 0,8 0,5) 2+9,6:0,01 (23:24)2 = (8 0,8 0,5) 2+9,6:0,01 (8:16)2 = = (8 0,4) 2 + 9,6 : 0,01 0,52 = 7,6 2 + 9,6 : 0,01 0,25 = = 15,2 + 960 0,25 = 975,2 0,25 = 974,95.

88

41. 2.

1

. : 1 2 3 4 5 6

. 1. 0 0, : x 4 = 0, x = 4. (). 2. 1 , : x + 12 = 20, x = 20 12 = 8. (). 3. : 2x 6 = 8 . 2x = 8 + 6 2x = 14 x = 14 : 2 = 7. (). 4. x = 3. x = 3 : 12 2 3 = 7 315 12 6 = 21 15 6 = 6 . (). 5. 2 : 2 1 4 1 4 5 + = 1 , . + = 1 , . = 1, . (). 5 5 5 5 5

89

6. : ().

3 1 = 1 x = 3 3 x = 9 . x 3

7. : 0x = 0 . (). 8. 0x = 2, . (). : A B 1 2 3 4 5 6 7 8

2

. 1. . . 2. 1 , : 6 x = 60 3. x = 60 : 6 = 10 cm. 10 cm 20 cm. 4. = 10 20 = 200 cm2. B. = 30 10 = 20 = (4 + 2) 5 = 6 5 = 30 = (8 4)3 : 4 + 2 17 = 43 : 4 + 2 17 = 64 : 4 + 2 17 = 16 + 34 = 50 : + + = 20 + 30 + 50 = 100 2 + 2 + 2 = 400 + 900 + 2500 = 3800 : 100 x = 3800 x = 3800 : 100 = 38.

31.

:

314 314 : 10 31,4 = = = 31,4 % 1000 1000 : 10 100

90

2.

3. 4. 5.

: 200% 200 200 200 = 2 200 = 400 100 4 4 20 80 : = = = 80 % 5 5 20 100

1m = 100cm 3% 100cm

3 100 = 3 cm 100

6.

30 : 30 30 2 60 = = = 60 % 50 50 2 100 :14 43,75 = 14 : 32 = 0,4375 = = 43,75 % 32 100

7.

4

10 : 500 500 + 50 = 550.

10 5000 = = 50 . 100 100

12 8% 550 : 8 4400 550 = = 44 550 + 44 = 594, . 100 100 594 500 = 94. :94 94 : 5 18,8 = = = 18,8 % 18%. 500 500 : 5 100

91

92

5

1

. x y x , y . x y . x y y = x (x, y) .

. 1. , :4 12 6 15 60 120 180 = = =6 10 20 30

2. , :

3. , x , 4 4x : = = 4 .

93

4. , x , : 1, 2, 3, 4, ..., x2, : 12 = 1, 22 = 4, 32 = 9, 42 = 16, ... : 1 1 1 2 4 2 3 9 3 4 16 4 ... ... ...

5. , y = x . 6. , . 7. , y = x x y. 8. , x y . 9. , . 10. , .

2

. x y x y . x y y = x (x, y) x, y.

94

. y x = 6, y =

6 x

. 1. , 10 6 , 20 3 .

2. , = x y

= x y = 12 = .

95

:

x y E=xy

12 1 12

6 2 12

4 3 12

3 4 12

2 6 12

1 12 12

3. , . 4. , . 5. , x y 1000. 6. , x, y y = , 0 . x 7. , x, y y = , 0 x .

3

) x , y y = 3x.

x = y =

2 6

3 9

3,5 10,5

4 12

) : 6 9 10,5 12 = 3, = 3, = 3, = 3, 2 3 3,5 4 x y : y = 3 y = 3x. x

96

) :

4

. ) x . 120+x . : (./) () 120 50 120 + x 40

A : 40 (120 + x) = 50 120 4800 + 40x = 6000 40x = 6000 4800 40x = 1200 x = 1200 : 40 = 30 97

30 . 40 . ) : (./) () 60 (120 x) = 50 120 7200 60x = 6000 60x = 7200 6000 60x = 1200 x = 1200 : 60 x = 20 20 . . 120 50 120 x 60

98

6

1

. , , . , . . , . +. . 1. . 2. . , . 3. : - . - . - .

99

. 1 2 3

. 1. . . 2. . x = x = : = 3. . 4. . , . 5. .

2.

+ 12,64 104,2316 3

12,64 + 104,23+ 16 3

12,64 104,2316 3

12,64 104,2316 3

0

0

0

0

. : . +, , +, ( ), . . . 1. (+ 2) + ( 3) (+ 3) = 2 3 3 2. ( 2) ( 3) + (+ 3) = 2 + 3 + 3 3. (+ 2) (+ 3) + ( 3) = 2 3 3 4. ( 2 + 3 + 3) = 2 + 3 + 3 5. ( 2 + 3 3) = 2 3 + 3 6. + ( 2 + 3 3) = 2 + 3 3 100 . . . . . .

1

2

3

4

5

6

3

1. . || = 2, = + 2 = 2. 2. . , .3 1 3. . = 1,5 = 0,2 . 2 5

5 5 3 5 4. . , + 3 3 5 3 5. . 0. 6. . 5 8, 6 7. : |x| = 6 x = + 6 x = 6 |x| = 7 x = + 7 x = 7. 4 : 6, + 6, 7, + 7. . 1. = 5, = + 7 + = ( 5) + (+ 7) = 5 + 7 = 2 = 5 (+ 7) = 5 7 = 12 = ( 5)(+ 7) = (5 7) = 35 5 : = ( 5) : (+ 7) = (5 : 7) = 7 1 1 2. = , = 3 3 1 1 + = + ( ) = 0 3 3 1 1 1 1 2 = ( ) = + = 3 3 3 3 3 1 1 1 1 1 = ( ) = ( ) = 3 3 3 3 9 1 1 1 1 : = : ( ) = ( : ) = 1 3 3 3 3 101

3. =

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

4 4 4 = = ( + ) = 3 3 3 3 2

1 4 1 4 3 8 11 + = + ( ) = = = 2 3 2 3 6 6 63 2

1 4 1 4 3 8 5 = ( ) = + = + = 2 3 2 3 6 6 61 4 1 4 4 2 = ( ) ( ) = +( ) = = 2 3 2 3 6 3 1 4 1 4 1 3 3 : = ( ) : ( ) = +( : ) = = 2 3 2 3 2 4 8

4

1. (+ 8)( 2)( 1) = + (8 2 1) = 16 2. [( 5)( 3)] : [(+ 1)( 1)] = (+ 15) : ( 1) = 15 3. ( 4) ( 5) = 4 + 5 = + 1 4. (+ 3) 0 ( 1) 5 = 0 5. 5 8 + 2 x = 7 8 x = 1 x 1 x = 20 x = 1 + 20 = 19 : 5 8 + 2 19 = 5 + 2 8 19 = 7 27 = 20 6. 7 10 = 3 3x = 12 x = ( 12) : ( 3) = + 4 : (7 10) 4 = 3 4 = 12 2 7. x: ( ) = 5 3 2 2 10 x = 5 ( ) = (5 ) = 3 3 3 10 2 10 2 10 3 10 =5 : : = + : = = 3 3 3 3 3 2 2

102

.4 4 1. = 3 + 4 + 10,2 + 3 10,2 3 3 =33+4 =6+4=2

. - .

=

( 1)( +3)( 4) +12 = = +2 = 2 6 6

=

18 18 18 = = = +3 = 3 4 ( +10) 4 10 6

2. =

( + 2) 2 2( 2 + 2 3) = = ( ) ( 2) [2 ( 2)] ( 2 2 2) 2 2( 2 + 6) 2 2 4 2 8 10 5 = = = = = (2 + 2) ( 2 4) 4 ( 6) 24 24 12

3. : x = || | | | | : x = |2| |+ 2| |+ 3| A : x = 223 A : x = 3 5 :x = 3 12 5 5 5 5 1 x= : ( 3 ) = :3 = = 12 36 12 12 3

103

104

7

1

. 1. , > 1, , . 2 . 3. 2. 1 = = 1, 0 1 = , 0 3. > 0 > 0 < 0 < 0 . < 0 > 0 . 4. = + 5. : = , 0 6. = () 7. . 105

8. =

9. 10. () = . 1. x5 : x3 = x53 = x2 1 2. x5 = 5 x 3 4 3. x x = x3+4 = x7 4. (x3)2 = x32 = x6 5. ( x)4 = x4

2

. 1. . , + = 0 ( + )2008 = 02008 = 0 2. . , < 0. . 3. . = ( 1)0 + ( 1)1 + ( 1)2 + ( 1)3 = 1 + ( 1) + 1 + ( 1) = 1 1 + 1 1 = 0 4. . x = 1010 1014 = 1010 + 14 = 1024 5. . 0,00001 x = 107 105 x = 107 x = 107 : 105 = 107 ( 5) = 107 + 5 = 1012 . 1. 2 = 1, = 0 2. 2 =1 4 2 = 22, = 2

3. ( 2) = 8 ( 2) = ( 2)3, = 3 4. ( 2) = 16 ( 2) = 24 ( 2) = ( 2)4, = 4 106

5. (2)3 = 64 2 3 = 26 3=6 =6:3=2 1 2 3 4 5

3

1. 2. 3. 4. 5. 6. 7. 8. 1 2 3

. 334 = 31 = . 9 + 9 = 0 . 23 21 = 23+1 = 248 4 . = 24 23 = 16 8 = 8 4 24 3

33 34 32 + (3)2 23 2 84 43 4 4 23 ( 3)02 3 4

1 3

. 134 81 3 . = 4 = 2 16 24

(56)266 6 2 ( 3)8

. 5(6) 2 = 512 = .4 33

1 5128

66 6 2 66 + 2 68 6 = 8 = 8 = = 28 8 3 3 3 3 3 3

9.

3 2

3

1 3 4 12 . = = 13 = 1 2 2 3 12

10.

( 1) + 1

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

107

4

. x = 24 + 2 ( 3)2 [23 ( 3)3] : 7 = 24 + 2 ( 3)2 [8 ( 27)] : 7 = = 24 + 2 ( 3)2 (8 + 27) : 7 = 24 + 2 ( 3)2 35 : 7 = = 16 + 2 9 35 : 7 = = 16 + 18 5 = = 16 5 + 18 = , = 21 + 18 = 3 , x=3 .

. x+1 2 x+3 3+1 2 3+3 = 9 x 2 x + 18 x = 9 ( 3) 2 ( 3) + 18 ( 3) = = 9 ( 3)2 2 9 + 18 ( 3)0 = 1 1 18 + 18 1 = 9 18 + 18 = 1 = 9 2 9 ( 3)

108

8

1

. 1. . 2. . . 90 180 . 3. . . 4. . . 5. . . 1. , . 2. , , , .

109

3. :

, .

4. , x x :

5. ,

6. , 90, 90 . 7. , = 180 = 180 . 8. , 90 180, 180. 9. , . 10. , .

110

2.

.

1

2

3

4

5

6

7

8

9

1. , = = 12 : 2 = 6cm. , = 6 : 2 = 3cm. = + = 3cm + 6cm = 9cm. x = 9cm. 2. x = 360 35 = 325 .

3. x + x + 110 = 180 2x = 180 110 2x = 70 x = 70 : 2 = 35. 4. : x + 5 = 70 x = 70 5 = 65.

3. 1. ,

. 3 1080o o 360 = = 108o 10 10

2. , 3. 4. ,

180 45 = 135, 90 45 = 45. : 3 45 = 135. 2 360o o 180 = = 72o. : 90 72 = 18. 5 5

5. ,

111

. xy , = 70 . , 70 : + + 70 = 180 + 70 + 70 = 180 + 140 = 180 = 180 140 = 40. 70 : = 70. 70 : = 180 70 = 110.

4

. 1. . 2. . .

1. , :

= 70 , = 70 .

2. .

112

. T E E . 2 1 0

x 1) 2 cm 2) 2 cm 3) 2 m

1)

2)

3)

. 1. , x = 180 . 2. , x = 90. 3. = 50 360. x = 360 50 = 310.

113

4. ( ) . . x = 40. 5. , , = 90. x = 90 55 = 35.

. = 50 . = 50 : 2 = 25. : 180 = 180 50 = 130. = 130 : 2 = 65. : = + = 25o + 65o = 90o, . = 90.

114

9

1

. 1. , , . 2. . . 3. , . . 1

1. -

2

. 2. .

.

1

.

1

115

3

.

1

.

2 2 . 2 . 1

2

1

2 4 2 . 1

.

.

1

.

2A.

(1,3) (2,0) (0,4) M(4,4)

(7,5) (6,8) (8,4) (4,4)

116

(1,3) (2,0) (0,4) M(4,4)

(3,1) (0,2) (4,0) (4,4)

.

) = = 3cm, . ) , . = 2 = 2 3 = 6cm . ) , , = . ) AO , 40. -

AO , = 40. = + = 40 + 40 = 80. = 180 80 = 100 ( AOB ).117

3.

1. . 2. , . , . : , , . // = . // + = 180. , . . 1. : = 180 120 = 60 . 2. , = 120 . 3. , = 60 . 4. , = 60 . 5. , = 120 . 6. , : = 60 . 7. , = 180 60 = 120 .

120 60 120 60 60 120 60 120

118

. 300 = 360 300 = 60. : + = 60 + 120 = 180. //.

4

. 50 : = 180 50 = 130. 140 : = 180 140 = 40. , : = = 40. x 50 = 360 x = 360 50 = 310. . = 180 75 = 105 . , = 180 105 = 75 . + = 180 , = 180 75 = 105 + = 180 , = 180 105 = 75 . . 72 : = 72. : = 180 (90 + ) = 180 (90 + 72 ) = 180 162 = 18 .

= 18 .

119

120

10 - -

1

. 1. , , o . 2. 180 . 3. , 90. 4. 2 . . . 5. . 60 . . . . . . . 1) . 2) , 90. 3) , . 4) , , : = 180 . 121

2

. 180. : ; , 110

80

40

60

75

180.

60 60 92 45 40 52

90 60 40 45 100 76

30 60 48 90 40 52

. 1) 2x + 3x + 4x = 180, 9x = 180 x = 180 : 9 = 20. 2) : x + x + 50 = 180, 2x = 180 50 = 130, x = 130 : 2 = 65. 3) T . 30 + 30 = 60, x = 180 60 = 120. 4) : x + 2x = 90 3x = 90 x = 90 : 3 = 30. 5) x : x = 40 + 70 = 110. ( : 180 (40 + 70 ) = 180 110 = 70. x 70 x = 180 70 = 110). 6) 72 + 48 = 120. 180 120 = 60. 2x = 60 x = 60 : 2 = 30.

122

3. =

2 2 90 180 90 = = = 36 . 5 5 5 + + = 180 , + = 180 36 = 144 144 : 4 : = 144 : 4 = 36 . = . = 3 = 3 36 = 108 > 90 . . = 70 = 180 120 = 60 + + = 180 , = 180 (70 + 60 ) = 180 130 = 50 = , = 50.

4.

()

()

. . .

.

.

.

: = = 2cm = = 4cm, , . 123

124

11

1

. 1. 0 .

2. . . 3. . 4. . 5. . 6. . = + ( ) 7. 1 . : = 8. 0 1 1 . =

125

. 1. . . 2. . . : = . 3. . . + : + = . 4. .

5. . . : = 6. . . . 7. . , . 8. . . .. ( 2)4 = + 16. 9. . () = . 10. . + . 11. . 3 +3 , ( 3)2 = (+ 3)2 = 9. 12. . 12.345 3 1 + 2 + 3 + 4 + 5 = 15 3, 5 5, 9 15 9. 13. . |x 4|+|2 x |+|x + 2|= =|4 4|+|2 4|+|4 + 2|= 0 +| 2|+|6|= 0 + 2 + 6 = 8. 126

2

. 1. . : (+ 1) (+ 1) = 1 ( 1) ( 1) = 1 , = 1. 2. ... : . ... : . 3. . 7+(2 5) + (3 : 4) = 7 + 2 5 + 3 : 4 = 7 + 10 + 0,75 = 17,75. 4. 5 cm 4 5 = 20 cm. 2 5 = 10 cm 40 40 5 200 = = = 200% . 4 10 = 40 cm. 20 20 5 100 . 5. 3 + 5 3 5 = + = 3 + 5 = 3 2 + 5 = 6 + 5 = 11. .

6. : x + 3 = 9 x, x + x = 9 3, 2x = 6, x = 6:2 = 3. . 7. .2 + ( 1) ( 3) 2 + 3 1 6 = = = 1. = = 1, 6 + 5 1 ( 3) 2 + 5 6 = 2 + 2 + 2 = 2, = 25 20 2 = 3.1 36 36 = = 12 . 3 3

8. .

9. . 3 , . < 0. 10. . , .

. 1. : = + , < . 5 : 0, 1, 2, 3, 4. = 0, = 5 ( = ), = 5 5 + 0 = 25, = 1 = 5 = 5 5 + 1 = 25 + 1 = 26, ...

127

0 5 5 25

1 5 5 26

2 5 5 27

3 5 5 28

4 5 5 29

2. 1 : A x x + 1 : x + x + 1 = 15 2x + 1 = 15 2x = 15 1, 2x = 14, x = 14 : 2 = 7. H x = 7. 2 : x : 2 7 7 2 9 5x = , 5x = + , 5x = , 5x = 3 , x = 3 : 5 = 0,6 . 3 3 3 3 3 3 : 4 : x = 25 4 : x =1 , 32 4 1 = , x 32 x = 4 32 = 128 .

15

2x + 1 = 15

x=7

2 3 7 3

5x

2 7 = 3 3

x = 0,6

4 25

4:x=

1 32

x = 128

128

3.x=

7 8 15 + = =5 3 3 32

5 3 5 5 6 5 5 1 =4 5 2 3 y = ( ) = ( ) = = = 4 1 2 4 2 2 4 2 4 4 4

.3 32 1 9 1 9 25 225 y 2 : x 2 = ( )2 : 52 = 2 : 2 = : = = 4 4 5 16 25 16 1 16

.i. 5% , 5 5 20 100 20 = = = 1 . 100 100 100

3 y 3 1 3 35 15 ii. = 4 = = = = = 15% . x 5 4 5 20 20 5 100

4

A = (8 14)2 [(8 4) : 2 6] 5 = ( 6)2 [(8 4) : 2 6] 5 = = 36 (4 : 2 6) 5 = 36 (2 6) 5 = 36 ( 4) 5 = = 36 ( 20) = 36 + 20 = 56o

: 56 + B + = 180 , + = 180 56 = 124 . : = = , + = 124 , 2 = 124 , = 124 : 2 = 62 . x A //, x = 56. //, = 62.( , x, , -

180o + x + = 180o , x + = 56o + 62o = 118o , = 180o 118o = 62o ).

129

130

12

1.1.

2 2 7 3 x = 3, x , . x = 7 7 2

2. ( 1)( 2)( 3) x ( 1)( 2)( 3)( +5) , x= 5 3. 4.6x = 0, 6x 0, . x=0 9 x3 = 1, x= 1 3

5. 5 x= 3

6. x = 0 7. 10x = 10 4 x = 4 8. 5 x 10 -2 10 6 = 10 7 5 x 10 -2 +(-6) = 10 7 , 5 x 10 -8 = 10 7 , 5 x = 10 -7 : 10 85x = 10 -7-(-8) , 5x = 10 -7+8

5x = 1015x = 10x = 10 : 5 = 2

131

9.

101+ 4 1 105 1 , = = x x 10 10 10 10

10 x = 10 105 , 10x = 101+ 5 , 10 x = 106x=6

10. x = 9 :

9 1 = 912 = 91 = 2 9 99

x = 9

( 9 )

2

=

9 1 = 912 = 91 = 2 9 9

B. 1. . . 2. . 3. . , 180. 4. . 60, , 90. 5. . 6. . , , , 180. 7. . : , , :180 , 180 , 180 : (180 ) + (180 ) + (180 ) = 3 180 ( + + ) = 540 180 = 360

8. . .

2.

1. AOB . 2. 3. = 40 , . 4. = 140 , : 180 40 = 140. 132

5. 40 , .

. / + + + + + + + + + + + +

. . : , . . : . . : . : , , , , , . // : , , . . //, : = = x . Z = =2x . + + = 180 , : 39 + x + 2x = 180 39 + 3x = 180 3x = 180 39 = 141 x = 141 : 3 = 47.

3

x = 8 30 : (8 2) = 30 : 6 = 5 .133

: : 5 (x2) = 30 5x10 = 30 5x = 30+10 5x = 40 x = 40 : 5 = 8 = 8cm. ...(2,3,10) = 30 ...(2,6,10) = 22 : 2 2 = 30 + 2 1 2 2y 22 = 32 2 4 2y = 32 8 2y = 32 2y = 32 : 8 2y = 4 y = 2, = 2cm.y

2

1. : = = 45 //. = 180 = 180 45 = 135 .

= 135 .2. , . 90 = 45 = 45 , . = = 2cm. = = 2cm. = = 8 2 2 = 4 cm. , = 2cm.

4

1. + + + = x + 2 + x + x + 2 + x = 4x + 4. 134

16cm, 4x + 4 = 16 4x = 16 4 4x = 12 x = 12 : 4 = 3cm = 3 + 2 = 5cm, = 5cm = 3cm, = 3cm.2. , = 140 . . , = .

+ + = 180 + = 180 = 180 140 = 40 = = 40 : 2 = 20.

// = 20 . = = 20 . ( = 90 ), = 90 20 = 70. = = = 70. () : = = 140. + = 180 () = 180 = 180 140 = 40 .

135