b gymnasiou a tomos 14 15
DESCRIPTION
ÂTRANSCRIPT
B A
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76
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86
91
3.
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112
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2. < , = , >, +12 … … 17 4 … … 15 9 … … + 9 0 … … 2 | 4| … … | 15| | 9| … … 0 ( 2) ( 6) … … 4 + 16 24 ( 4) … … 3
3. 19 + 24 32 15 (+8) + (+13) 4 + ( 11)( 10) (+13) 5 ( 5) ( 9) ( 8) (+25) ( 5) 8 (+3) 0 ( 8) 5 2 2 + 1 3 1 2 1
10
4. : 2 + = 4 ( 2) = 4 ( 2) = 4 ( 2) = 4 ( 2 + 6) = 4 ( 1) 24 (+12) = 4
5. 5 + 9 6 + 2 +2 6 + 8 + 0 4 + 5
25 + ( 4) ( 9) 7 + ( 12) (+9) ( 4) 9 + ( 7) (+3) + 16 (1 15): (19 12) 2 ( 4 + 7) 7 + 6 (+4 4) 5 (+2) 18 ( 3) 7 ( 4) (+12) + 8
6. 32, 48 80
7. : ( 2) (+4) ( 3) ( 6 + 5)
+
8.
4 + 5 = 9 3 + 12 = 0 3(2 ) + = 6 10 7 3(2 5) = 20 2( 1) 3 = 16 3( + 2) = + + 1 = = 3 14
9. ,
98° 0° 270° 90° 180° 225° 81° 360°
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11. , , .
i. ii. iii.
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,
12
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( , )
13
16. = 2 1. N
(0,0) .
17. = 3 .
N = 3 .
18. , , 30% . . €80
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19. €40000 25%
20. €15000 €13200.
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. ,
i. , ii. .
1
16
1: 1
…
, > 1 .
3 3 3 3 = 3 3 = 813 4 .
= , = , 2 = 1 5 = 5
17
1:
0,333 … = 0, 3 334,23181818 … = 34,2318 18 .
, = / , , 0
. A
2,666 … = 20,25 =
18
1: 1
20
10
3
10000
(10 ) = 10
1 8 10
10
-
-1940 -
Kasner
-
19
1:
(1)
2
2
2
2 2 2
2 2 2 2 2 2 2 2 2 2
(3,1) (3,1) ( 3) ( 3) ( 3)
= 2 2 2 2 2
5 5 :
(7 ) (7 ) · (7 ) · (7 ) (7 ) 7 [ ] ( )
20
1: 1
(2)
10
10
10
(7 ) 7 7 7 7 = 7 7 7 7 7 (5 ) (3 ) ( )
· · · = · · ·· · ·
21
1:
· = , (+3) (+3) = (+3) = (+3)
= , > 0
= , > 0
( 2) ( 2) = ( 2) = ( 2) ( )( ) = ( 2) = ( 2)
( ) = · ,
(2 ) = 2 = 2
, ( ) = · , 0
(2 3) = 2 3
= , 0
=
= = , 0 ±1(+5) = (+5) = 2
22
1: 1
5 · 5 : 3 · 9 81( 3) · (+3) 3 + 7 3 3 + 2 3
5 · 5 = 5 · 5 = 5 = 5
: = =
3 · 9 · 81 = 3 · 3 · 3 = 3 = 3
( 3) · (+3) = (+3) · (+3)= (+3)= (+3)
( 3) = (+3) ( 3)(+3) -
3 + 7 3 3 + 2 3 = (1 + 7 1 + 2) 3 = 9 3 = 3 3= 3
(5 ) 125
(5 ) = 5 = 5
= · =
125 = (5 ) = 5 · = 5 125 = 5
23
1:
5
( ) ( )
5 = 5 = ( )= ( 1)= +1( ) ( ) = [( ) ( )] = ( )
== ( ) = 6 6 = 6 = 6 = 36
( 2) + ( 2) = ( 2) ( 5) ( 5) = ( 5) 5 3 = 5 2 5 = 10 3 9 = 3
24
1: 1
5 · 5 y ( 7) ( 7) · (0,1) (0,1) (0,1) ( 2) ( 2) ( 2) (11 11 ) 11 5 · (5 5 ) ( ) ( ) ( )
pH pH 10 pH 7 pH 3
2 2 = 2 (+2) (+2) = (+2)2 2 = 2 7 7 = 7 7
( 2) ( 2) = ( 2) ( 3,1) ( 3,1) = 3,1 ( 3) ( 3) ( 3) = ( 3) = (+2) (+2) = 16 5 5 = 25
3 4 ;
( 3 ) ( ) ( ) (2 ) ( ) ( ) ( ) ( )
25
1:
7 ( )
( 2) ( 5)
( ) ( ) ( ) 3 ( ) ( )
25 5 4 16 (+2) 9 (+3) 27 ( 3) (+3) ( 2) (+2) ( 7) (+7) ( 7) ( 8) 16 4 ( 3) 9 ( 27)
2 + 2 + 2 + 2 3 7 + 5 7 7 2 2 + 2 2 + 7 2 2 2 3 3 + (3 ) + 2 3 3 3 3
= 3 ( 7) = 2 3 ( 7) = 2 ( 7) ( )
2 = 2 25 = 5
26
1: 1
(1)
19
19 1 . .21 128 . .
17
1 2
27
1:
(2)
7 7 = …
:
7 7 = 7 = 7
7 7 = = = =
,
= , 0
2 = == ( 4) = 64
0
-
-
= =
7 7 =
28
1: 1
= = , 0
3 = = =
10
10 = == = +
2 2 2 ( 7) ( 7) ( 2) ( 2)
29
1:
2 2 2 = 2 ( ) 2= 2 2= 2 = 2 --
( 7) ( 7) = ( 7) ( )= ( 7) = ( 7) --
( 2) ( 2) = ( 2) ( 2) ( 2) ( 2) = ( 2) ( ) ( 2)= ( 2) ( 2)= ( 2)= ( 2)= ( 2) ( 2) (8 9) + ( 2) ( 2) ( 3) + (+1)
( 2) (8 9) += ( 2) ( 1) + = ( 1) + = ( 1) + += + 1 += +1= +3.
30
1: 1
( 2) ( 2) ( 3) + (+1)
= ( 2) ( 3) + (+1) = ( 8) ( 27) + (+1) = 8 + 27 + 1 = 8 + 28 = +20 3 1 ( 5)– ( )– [( 2) ]–
0 1
1 1
4 ( 4) 8 16 2 3 2 3
3 ( 3) 6 ( 2 + 1) + 3 ( 1) + + ( 3) ( 3) + ( 2) + 5 ( 2) ( 5) +2 ( 3)
31
1:
= ( 1) + ( 1) + ( 1) + + ( 1) + ( 1) = 2 =
=
( 1756) ( 1) ( 1) (+1) , 0
3 9 1 27
( 3)
4 (+2) ( 4)– ( 2) 0 1
5 5 5 · 5 6 6 ( 5) ( 5) ( 5) · 3 (+2) + 2 · 8 · 16 ( 3) ( 3)
,
( 3) · ( 3) = ( 3) ( 0,2) · ( 0,2) = ( 0,2) = · =
– ( )
32
1: 1
100;
169
5 = 1 + 3 + 5 + 7 + 9 = 25
33
1:
)
12 8 225
)
27 1
64
216 300 343 (
).
( 0)
= = 0 , 0
169 = 13, 13 = 169 = , =0,25 = 0,5 , (0,5) = 0,25
H
( 9) = 81 81 9
-
34
1: 1
= | |.
3 = |3| = 3 ( 3) = | 3| = 3 0, =
4 = 4
= = 0 , 0
8 = 2 2 = 8= =0,064 = 0,4 (0,4) = 0,064
0, =
8 = 8
36 1210,36
, .36 = 6 6 = 36121 = 11 11 = 1210,36 = 0,6 0,6 = 0,36
-
35
1:
1 64 0,001
1 = 1 1 = 164 = 4 4 = 640,001 = 0,1 0,1 = 0,001 8 4
4 = 28 4 = 8 2 = 16 = 4
64 25 8 0,36 343 2,25 0,09 0,0001 0,16
900
216
36
1: 1
36 = 6 25 = 5 ( 3) = 3 64 = 4 = 64 = 8
13 ( 11) 26 26 8 8 819 + 8 ( 3) + ( 9)
2 3
16 = 8 16 + 9 = 5 8 = 2 36 = 36 ( 3) = 3
49 81 121
50 + 50 4 + 2 16 3 9 4 8 1 + 3 + 4 + 64
37
1:
8 = 4
( 3 + 4) = (9 2 )
52 900
173
12. > 0,
+ 16 + 92 121 = 4
100 - -
-
38
1: 1
+ +
.
+ + 9
16
36
39
1:
= , , 0.
= , 0, > 0.
= · , , 0. = , 0, > 0. ( ):
0 0 (i) = = , 0 = = = , 0
= =
= , , 0 ( ): 0 0 (ii)
= = 0, > 0
= = = 0, > 0
= =
= , 0, > 0
( ) ( )
= =
40
1: 1
18 2 81 3
= = =.18 2 = 18 2 = 36 = 6
= = 4 = 2
81 3 = = = 27 = 3 3 7 + 5 7 5(2 4 5) 2 2 + 16 2 (2 3 2)(5 + 2)
3 7 + 5 7 = 2 7 + 5
41
1:
5 2 4 5 = 2 5 4 5 5= 2 5 4 5 = 2 5 20 4 2 + 16 2 = + = 4 + = 4+ 8 = 4+2 = 6
2 3 2 5 + 2 = 10 + 2 2 15 2 + 3 2 2 = 10 13 2 + 3 2 = 10 13 2 + 6 = 16 13 2
4 = 2 , > 0 36 + = 6 + , > 0 = = , > 0
3 2 + 5 2 2 14 5 + 2 2 53 + 3 + 3 10 + 10 + 10
42
1: 1
44 11 + 12 3 14 2 8 32 + 9 3 3 10 5 10 3 27 3 ·
1 2
25 6 6 5
ii
iii 36 5
iv 5 6
= 9 = 16 = + + = +
= 16 9
43
1:
2 ( 8 + 32) 27 48 3 2 2 ( 2 + 3) 3 10 3 + 10 ( / )
= 16 , ( )
4 9 25 u = ,
N
44
1: 1
20 1955
572 497
45
1:
(1)
1500
24 30
5
4 ,
, ,
46
1: 1
(2)
13 12
1
;
2
1
47
1:
( = 90°)
+ =
= 5 = 3 = 4 ,= +5 = 4 + 3
0 < < < + =
, 2 + , , 0 >
= 2 = 1= 2 1 = 3= 2 2 1 = 4= 2 + 1 = 53, 4, 5
48
1: 1
( + )
( ) 1 2 = + + + = + + + = + +
= + + + + = + + + + = + + + = + + =
1852 1940 ” 367
1452 1519 201831 1881(~325 – 265
100
1900 . ., --
1679 2400 2929.
49
1:
= 8 + 6 = 64 + 36= 100= 100 = 10 13 = + 5169 = + 25= 169 25= 144 = 144= 12
3, 4, 5 1,5 , 2, 2,5
3,4,5 3 < 4 < 55 = 25 3 +4 = 9 + 16 = 25 5 = 3 + 4 ,3, 4, 5
= 100
= 10 = 10
-
50
1: 1
(
(
;
;
;
51
1:
, , (3,0), (6,0), (3,4) ( 3, 2), (4, 2), ( 3, 5)
3 4 5
= 7 , = 24 , = 25 = 2 , = 2 , = 4
= 15 , = 13 = 9
52
1: 1
N
3
90
: = , = 2 = + , > .
-
-
53
1:
(1)
, , , : 0,25 . . : 1 . . : 2 . . : 4 . .
54
1: 1
(2)
2,10 1 2
B n2
2 , 10 , 8
55
1:
0
2 = 1,414213562 … 2 = 1,25992105 … 3 = 1,732050807 …
2, 3, , 5, 7, 8, … 2 , 3, 4, 5 , …
, ,
-
56
1: 1
,
= 2
1 < 2 < 41 < 2 < 41 < 2 < 2
1,96 = 1,4 < 2 < 1,5 = 2,25 1,4 < 2 < 1,5
7289
1800 1600
1900
30 1, 245110 42, 2535 1900
57
1:
5.5 … 54 < 5 < 9 2 < 5 < 32 < 5 < 3 (2,1) , (2,2) , … , 3 .5., , , , , , … , , , , , , … 54,84 < 5 < 5,29 2,2 < 5 < 2,32,2 < 5 < 2,32,2 < 5 < 2,3 52,2 2,3 54,84 5,29, a 2,2. 5 2,2
, = , = 1,6
-
=
--
-
58
1: 1
7 3,5 125 3
13 8,4
25
8, 17 6 5 7 36 27 20 0,44444
28 21 52 135 38,7 79,2
7 < < < 8 < , > , =
7 … 1, 5 … 2,25 17 … 17 6,25 …
59
1:
7, 9, 50, 85 91, 7, 5, 38 0,47 , 0,474 , 0,47 , 0,23
6 < 7 < 8 2 < 5 < 3 3 < 15 < 9 5 < 5 < 5 3 < 29 < 4
55
6 1,7
3 1
3,5 4,3
60
1: 1
20 12 2,51 ).
5
100 280
12 5 8
61
1:
3 (+1) ( 2) ( 2) ( 6 1)– +
+
< , > , =
( 1) … … ( 2016) ( 5) … … ( 2) … … (+2) 2 4 … … 4 2
81 243 19683 9 6561 81 3
5 10
3 3 3 9 3 27 3 81 3 243 3 729 3 2187 3 6561 3 196833 590493 1771473 531441
62
1: 1
( 2) ( 2) ( 2) = ( 2) (3 ) = 3(+5) (+5) (+5) = (+5) 5 5 5 = 1 (+2) (+2) = (+2) = 2 ( 3) · ( 3) = 3 3 3 = 3
11 11 11 5 5 5
25 ( 5) 9 3 27 2 ( 2)
2 9 3 27 + 3 3 + (3 ) 3 27 + 2 3 3 + 7 (3 ) ( 3) ( 3) + 3 (7 8)
( ) ( ) ( ) ( )( ) ( )
= +2 = 1 = + 4( + )
63
1:
64 1,21 ( 5)
125 5 2 4 4 0,027
= 2 + 5
= 6 + 7 8 = 20 125 = 2 = 4 = 9
+ 1 + 3 + +
<, >, =
5 ___ 7 30 ___ 6 8 ___ 4 9 ___ 9 28 ___ 8 5 ___ 1,2
+ = + , > 0 9 = 3 8 = 4 = 5, = 10 12, 9 , 13
4 = 2 0 = , 0
64
1: 1
8 . . 10 . . 4
5 5 8 29 29
8 9
65
1:
2
= 8 = 15
66
1: 1
1 . = 0,3 (0,01 9)
2 . ,
= [ ( ) ( ) ]
3 . 16 = 2 ( 25) = 5( )( ) = 10 4 . A = 11 12 + 12 12 12
= 3 3
= 5 .
( 1) + (+1) ( 1) + 1 + + ( )
6 . 2 + 2 + 2 7
7 .
22 + 7 + 1 + 9 = 5 + 9 = 2 8 .
= , , 0.= , 0, 0 0.9 . , , > 1 = 1 0 < < 1 = 0
67
1:
10 . =
16 3
11 .
12 .
13 .
0
+ 4
18 . 3 . 13 . 20 . 17 .
5 . 6 .
68
1: 1
14 .
15 .
20
2
:
70
2:
71
2:
o
72
2:
.
3 , , 5 , , ,
2
2 2 1 3
.
8 2 , ( ).
.
2 2 2 +2
--
1 =
-
-
-
73
2:
) 4 + 3 3
5 2
: 4 + 3
=
5
3 3 0
5 2 = 10 10
2
» +2 3 + 3 + 2 ( + 1) + 1234
-
74
2:
7 + 3
-
3
+ 4
3
3
3 , 2 4 , + 6
2 2 5 ».
75
2:
4
8
8 5
9
2 +2
76
2:
.
.
.
3 + 7 = (3 + 7) = 10
3 ( 7 ) = 3 + 7 = 11
--
77
2:
( 2 ) (3 ) = ( 2 3) = 6 = 6
12 ( 4 ) = = 3 = 3 (8 ) (4 ) = = = 2 =
3 , 4 3 , 2 , 3
. 3 + ( 4 ) = (3 4) = 3 + ( 2 ) + ( 3 ) = 3 3 2 = 2
(4 ) ( 3 y) ( )
(4 y) ( 3 y) = ( 4 3) = 12( ) = ( 3 ) ( 3 ) = +9
--
-= .
78
2:
2 (2 ) ( 12 ) ( 6 )
2 (2 ) = = ( ) =( 12 ) ( 6 ) = =
o
+ 3 5 2 4 3 9 4 7 4 + 5 7 3 2 7
8
2 5 : 2 4
3 16 8x
8 32
5 + 3 2 3 5 + 6
2 + 3 3 12 + 4 3 + 4 6 2 +
-= .
79
2:
2 ( 3 ) ( 4 ) ( 3 ) ( ) 2 y (3 )( 2y) ( 3y ) 5 ( 2 ) 4 ( 3 ) ( 2 ) (2 )
(2 ) (2 ) + 3 4 + (3 )(5 ) (6 ) ( 2 ) + ( 3 )( 2 ) 2 + (30 ) (6 )
7 + = 3 4 y = 5 y ( 2 ) = 6 8 2 + = 3 = 25 18 y = 3 y = 9 ( 2y ) = 3 y T ( = )
8 24 2 20
(12 ) (2 ) ( 12 ) ( 6 ) ( 6 y ) (3 ) (21 ) ( 3 ) ( 18 y ) ( 3 ) ( 2 ) ( 4 )
2
3
80
2:
1)
+1
1
( + 2) + (2 2 + 3)
+
=
(2 + + 1) + ( 2 ) + (3 2)
En Athroisma Polionyma.ggb»
( + 2) + (2 2 + 3) = 3 + 1
81
2:
O (2 + 3 3) ( + 2 3)
=
(2 + 3 3) ( + 2 3) = +
(2 + 3 3) ( + 2 3) = (2 + 3 3) + ( 2 + 3)
+
=
(2 + 3 3) ( + 2 3) = (2 + 3 3) + ( 2 + 3)
= +
82
2:
2 5 + 2
2 , 5 2 .
2 + 2 5
.
+ 2 3 2 + 3
3 + 2 2 3 2 + 3 3
(2 5 + 2) + ( 4 + 1) = 2 5 + 2 + 4 + 1
= 2 + 5 4 + 2 + 1 = 3 9 + 3
(2 5 + 2) ( 4 + 1) = 2 5 + 2 + 4 1 = 2 5 + 4 + 2 1 = + 1
p(2) p( ) = 2. p(2) = 3 2 2 2 + 3 = 3 . 4 2 . 2 + 3
= 12 4 + 3 = 11
-
-
-
--
p( ) = 3 2 + 3
( ) = + 2 = + 2
83
2:
= 4 5 + 3 = 3 2 +
+ = (4 5 + 3) + (3 2)
= 4 5 + 3 + 3 2
= 4 5 + 3 + 3 2 = 4 2 + 1 .
4 5 + 3 (+) + 3 2 4 2 + 1
= (4 5 + 3) (3 2) = 4 5 + 3 + ( 3 + 2)
-
= 4 5 + 3 3 + 2 = 4 8 + 5
2. ( ) = 2 + 5 ( 1)
3
( 1) 1 . ( 1) = 2( 1) ( 1) + 5
( 1) = 2( 1) (+1) + 5 ( 1) = 2 1 + 5 ( 1) = +2
84
2:
= + 3 2 = 4 2 + 1
5 + (4 + 7) ( 2 + 3) + (2 + 3 + 3) ( + 2) ( 2 + 3 )
(3 + 2) ( 2 + 2 + 1)( + 1) (2 7 + 1) ( 4 + 3) + ( + 3 )
= 5 2 + 7 = 5 2 9 + 4 +
( ) = 2 + 4 ( ) = 3 5 ( ) = + + 1
(2) ( 1) ( ) + ( ) ( ) ( )
: 8 + 9 + 8 = 12 9 + 4 ( + 3) + + 3 = + ( + + 3 ) ( + 2 + ) = 2 + 7 + 7 + + = 4 + 14 + 5 10
2 + 5 + 7 8 + 4 5
85
2:
+
+
= 15
86
2:
1)
6 12 ( )
€20
€35
2)
1 1
87
2:
( + 2)
( 2 ) ( + 1) = ( 2 ) ( ) + ( 2 ) 1 = 2 2
( 2) ( + 4) = ( + 4) 2 ( + 4) = + 4 2 8 = + 2 8
.
( + ) = +
( + 2) = + 2
( + 2) ( + 2)..
--
.
= … = = =
--
( )
( ).
-
- = = =
88
2:
( + 3) 2 ( 1) ( + 1) ( + 1)( 3)
( + 3) = + 3
2 ( 1) = 2 2
( + 1) =
( + 1)( 3) = 3 + 3 = 2 3
( + 1)( + 1) 1 = ( + 2)
= ( + 1)( + 1) 1 = + 1 + 1 + 1 1 = + 2 = ( + 2) =
.
= ( + 1)( + 1) 1
= + + + 1 1 = + 2 = ( + 2) = + 2
= = + 2
89
2:
( + 1) ( + 5) 5 ( 3 ) 6 ( 5) ( ) 2 (3 + )
( + 1)( + 2) ( + 4)( + 3) ( 4)( + 4) ( + 3) ( 2)(4 ) ( 1)
: 3 2 + = + 12 ( + 3) = + 12 4 2 = 20 ( + 1)( + 3)= + + 3 + 3 ( 2) = +
( + 2)( 2) (3 5) ( 1)(1 + ) 2 ( + 3)( 3) (4 1)(4 + 2) (4 + 3) ( ) = 4 5 ( ) = 1 2 ( ) ( ) ( ) [ ( )] 2 ( ) 3 ( ) ( ) + ( ) ( )
O
5
90
2:
( 1) + 2 = ( 1)( + 1) + 2 ( )( + ) + 2 = ( ) + 2 ( + )( + ) (2 + ) = ( )( + ) +
y = 1 = ( )( ) ( 9)
20 + 4
( ) 2 ( ) 3 ( ) ( ) 6
91
2:
=x x-
= 4x3- x
=x x-
= x -x+8x-4= x + 7x - 4
E = += (4x3 - x + x + 7x -= 4x3 - x + x + 7x - 4= 4x3 + 7x - 4
21
2
+ 4
=
92
2:
(12 6 ) (2 ) = (12 2 ) (6 2 ) = 6 3
( ) ( ) ( ) 0 ( ) ( ) ( ) ( ) ( ) = ( ) ( ) + ( )
( ) ( ).
( ) = 0 ( ) ( ) ( ) ( ).
+ 3 4
0
1 ( )= + 3 4 ( ) = 1
( ) = + 4 ( ) = 0
+ 4
1 + 3 4.
( ) ( ) ( )
( ): ( ) ( ) ( ):
93
2:
(15 3 ) (3 )
(15 3 ) (3 ) =
= 5 1
-
=
= 4 + 5
-
( 4 ) ( 4) (6 + 3 + 5 ) (2 + 1)
.
4
4 -
= ,
4 + 4 4 4 4
+ 4 .
4 + 4 0
4 ,
.
0. ( 4) = 4
4
94
2:
.
6 + 5 + 3
2 + 1 --= 3 ,
3
6 + 5 + 36 3 2 + 1 3 2 + 1
6 + 3
6 3 .
3
6 + 5 + 3 6 3 +2 + 3
2 + 1
.
3
6 + 5 + 3 6 3 +2 + 3 2 1 +2
2 + 1 = 1
1(2 + 1) = 2 + 1 2 1 -
3 + 1
3 + 1 2.(2 + 1)(3 + 1) + 2 = 6 + 5 + 3
(4 12 ) ( 2 ) ( 12 + 24 ) (6 ) ( 12 + 6 ) (6 ) ( 12 + 15 ) (3 ) ( + ) ( ) ( 4 + 5 + 6 ) ( 2 )
95
2:
( 4) ( 2) ( 3 ) ( 3) ( 3 ) ( 3) (6 ) ( 1)
( 5 + 4) ( 1) ( 6) ( 3) ( 2 8) ( 4) ( + 3 + 1) ( + 1) (2 3 + 3) (2 + 1) (9 + 6 + 1) (3 + 1)
=2
= 3 + +
+ 2 + +4 + 8 0
2 + 2 + 3 +5 + 5 +1
1 + 4 3 +
( + 5) (3 1) 3
( + 1) (3 + 5) 1
( 4) ( 5 + 4)
( + ) ( 4)
96
2:
2 + 4 7 + 5 3 (4 )(2 ) (+3 )( 5 ) ( 3 )( 2 )(4 ) ( 33 ) ( 11 ) (14 ) ( 2 )
8 + 1) + (3 + 7) ( 3 + ) (3 + ) 3 ( 2 ) 2 ( 2 + 3) (10 12 ) ( 2 ) ( 2 ) ( 2 ) ( + 2)( 2) ( + )( + ) 3 + (3 + 7 ) ( 3)( 3)
( ) = 2 + 3 ( ) = 5 ( ) = + 1 r( ) p( ) r( ) + p( ) + q( ) p( ) q( ) p( ) q( ) q(3) p( 1)
( 10 + 20) ( 5) ( 8 16 + 6) (2 3) ( 5 + 2 + 6) ( 2)
( ) = 2 + 2 9 ( 3) = (2) 3 (1) + (3) = 0
97
2:
12 + + = 3 + = + 7 ( 2 ) = 2 3 = 2 = ( 24 ) = ( 3 ) = 4 ( ) = + 7 3 = 27 12 + = 6 + 1
4 . + 9 2 2 3
3 3 6 + 3 7 + 6 7 4 3 87 8
= 1 = 3 = 3 = 2
(3 + 1) ( 2) 5
( 3 4) ( 1)
98
2:
(5 + 2) 25 + 20 + 4
(3 + 1) ( 2 )
( 2)( + 2) ( + 8) = 4(1 + 2 ) 2 + ( 2)( + 3) = ( + 3) 6 (2 3) ( + 1) = ( 2)( + 2) ( + ) 4 = ( )
= 80 , = 40 = 10
2 + 4
1 2 + 1 2 1
99
2:
3 + 6 = 7
: 4 + = 10: + : 2 + = 11= = 9 = 2 25
( 1) = (+1) ( ) = 5 + 3 + ( 2)
= + 1 = + 2
. 9 .
( + 1) 2 5
100
2:
€600 2000 5
6. ( )( ) ( )
3 + 3
16
( + + ) = + + + 2 + 2 + 2 = ( )[( ) + 2 ] = ( + 2) 6( + 1) = + 2
+ +
( ) = 2010 + 2 (2 2009)
, , ( ) = ( 5 + 4 3) ( 2 + 1) + (3 + ) ( ) = + +
.
3
102
3:
=
= =
…
, = = . + = 360
(
103
3:
, , , , , , ;
(1
( )
(2
«B_En3_Symetria ggb»
«B_En3_Symetria ggb»
104
3:
=
A
,
105
3:
, )
.
= = .
.
. 1 ( . 2 (
. 1 . 2 .
106
3:
.
107
3:
3,
« _En3_symmetria_kath
108
3:
4
109
3:
= = 5 = 40°
,
110
3:
…
,
=
+ = 180°
–
=
180°
+ + = 180
4 360°
+ + + = 360
360°
,
111
3:
= =
= + = ( + )
=
= +
= ( + )
=
= + +
=
= +
112
3:
7
«B_En3_tagram ggb
«B n3_Parallilogrammo ggb»
113
3:
#
= =
= =
= =
, ,
114
3:
.
= 5 8 = 3 + 4 2 = 12 = 6
. = 2 7 = + 7 = 14
. = + 22° = 60° = 60° 22° = 38° = + 59° = + 22° = 59° 22° = 37° = 37°
= = 6 3 = 18 = 4 18 = 4
4 = 18 = 18 4 = 4,5
115
3:
= 8 = = 18 = = 101° = = 80° =
=
116
3:
120 15
26 8 20
( 1,1) (2,1) (0, 4);
117
3:
;
(2
.
«B_En3_Orthogonio ggb»
118
3:
=
= 4 = 3 = 65°.
, ,
( ) = ( ) + ( ) ( ) = 4 + 3 ( ) = 16 + 9 ( ) = 25 = 25 = 5
-
= = = .
119
3:
= = 5 = = = = = 2,5 = 65°
= : = = = 65°
= 2 = 130° ( )
= (30 ) = (4 60)
= 30°
(1,1), (5,1), (5,3), (1,3)
120
3:
72
6 4 5
24 8
-
-
121
3:
(1)
.
(2)
, , ,
«B_En3_Romvos.ggb»,
;
;
122
3:
(3)
«B_En3_EmvadoRomvou.ggb».
.
:
.
, = =
= = = = = = .
O - -
123
3:
=
= ( = = )
=
= 40°
+ = 180° = 180° 40° = 140° = = 40° = = 140°
= = = 70° = = = 20° + + = 180° = 180° (70° + 20°) = 90° .
124
3:
5 = 8
= = = 4
, ( ) = ( ) ( ) ( ) = 5 4 ( ) = 25 16 ( ) = 9 = 9 = 3
= 6 = 2 = 2 = 8 62 = 24
= 4 = 4 5 = 20
= (2 + 3) , = 10 = 8
, , ,
= 2 = 10 ,
, , , ,
125
3:
24 10 24 8
12 16
128 .
56 cm 16 cm,
126
3:
(1)
«B_En3_diastavrosi.ggb»
(2)
127
3:
«B_En3_Tetragono.ggb».
.
:
= 0 = =
= = = = = = = = 45°.
128
3:
10 ,
= = 10 . ( ) = ( ) + ( ) ( ) = 10 + 10 ( ) = 100 + 100 ( ) = 200 ( ) = 200 = 100 2 = 10 2
= = = = 5 2 30
, . = .
,
. = 30 2( + ) = 30 + = 15 2 + = 15 3 = 15 = 5 = 2 = 10 . = . = 5 10 = = 50 = 50 = 25 2 ( = 5 2
129
3:
2 90° 45° 2 8 2
= 18
– , – , –
.
= 90° =
130
3:
)
B
A
B
A
B
A
7 = 4 .
131
3:
(1)
)
«B_En3_Trapezio.ggb».
(2)
«B_En3_EmvadoTrapeziou.ggb».
132
3:
.
.
: ( = = ).
( = ).
= ( + ) = ( + )
= + + ( ) + ( )
133
3:
:
, = = 125° ,
350 . . ),
134
3:
+ = 180°
= 180° = 180° 125° = 55° = = 55° = = 125° (
13 , 20 99
= ( ) 99 = ( ) 99 = 99 2 = 33 = 99 233 = 3 2 = 6
4 6 20
= 4 = 20 = 20 4 = 5
= = 5
= 25 = ( ) = 25 ( ) = 25 = 25 5 = 25 = 25 5 = 5
135
3:
, ,
1 2 3 4
-
1 2 3
136
3:
5.
36 8 4
( ) = 12 , = 10 , = 20 = 90°
( = = 90° = 45°, = 4 = 28
137
3:
( , )
AB
AB
AB
138
3:
21 127,20°
–5000252000 39690 40048
N «B_En3_Eratoshenes ggb»
.
www ea gr/ep/mobile/era/index html
139
3:
50 160 x 220 . ,
180 , 50 .
3 20
,
« 3_ ikoskyklou ».
»
. ;
.
140
3:
«B_En3_Emv ggb»
= 10
R 10 180° 10 90° 10 60° 10 30° 10 36°
.
= =3,1415926535897932384626433
-
--
141
3:
, , .
= =
(
= 2 = =
-
-
--
-
-
1900 = 31700 = 3287 212
96 3 3
-
2424 67 890
142
3:
2,40 = 2,40 = 1,20
= 2 = 2 1,2 = 2,4 7,536
, , = = 0,8 = 0,6 = = 0,3 = 22 = 0,3 = 0,3 0,942 = + + + = 0,8 + 0,6 + 0,8 + 0,942 3,142 45° 2 .
.
= 2360
2 = 2 45360 21 = 4 = 8
= 2 = 2 8 = 16
16 .
2,40 8060
143
3:
10 .
3. 10 .
25,12 16 . 5. 3,45 .
)
6. 40
, 1,256 .
7. 6400 . N
8.
3000 25 12 1,60
, 12 .
9. 120° 2 .
144
3:
.
20 , 1,50 . ,
€12,75
= 50 .
30 20 / ;
145
3:
, 50
, =25 .
«B_En3_kyklos ggb
.
(0 25 ) (0 360°) .
146
3:
«B n3_Emv ggb»
= 10
R
10 180° 10 90° 10 60° 10 30° 10 36°
.
147
3:
=
( . .
. . = 360 =
= 9 = 3
.
= = 9 = 81 = 2 = 2 9 = 18
= 2360 3 = 3 = 3 = = 60°
. . = 360 =
=
=
- -
148
3:
, = 16
2
= = 16 = =
= = 8 = = 8 = 8 = 4 = . . + . . = ( ) ( ) = 8 16 = 128
. = = = = 16 = = = = 8 = . . + . = 128 (16 + 8 ) = (128 24 ) (128 24 3,14) = 52,64
= + + + = + + .+ . . = = = = 4 . = 22 = 2 42 = 4 = + + .+ .
= 16 + 8 + (4 + 4 )
= (24 + 8 )
(24 + 8 3,14)
= 49,12
3,14
149
3:
10 , 10 , 12,56 , .
36 .
3. 9 .
30° 3 ²
5. = 2 = 4 .
6. 144 ². 60°.
7.
8. 55 15 . 135°
9. 10
. ;
= 36 50°.
150
3:
O
HK
K(O, OP)
€20.50 cm
151
3:
1. , ,
.
3. = = = = = =
" ",
-
-
-
-
90°
152
3:
5.
6.
7.
.
8.
9. 16 8 . 24 10 . 150 10 .
153
3:
100 . 60 .
45° 20,
, ,
, , , ,
.
.
10
.
154
3:
3. + .
155
3:
40 ,
5. « _En3 ggb
.
6. ( 1,3), (1,4) (1,2).
.
(0,0).
.
156
3:
7.« _En3_Symetria ggb»
.
1 2.
8.
En pi. », .
9.
16
– 14
158 :
: 159
1.
2. > <> >> >= <3. 5 47 21 723 0 13 7224 0 7 15 14. 2 2 8 21 25. 0 5 0 2014 2 6 74 246. (32,48,80) = 16 (32,48,80) = 4807. 4 4 271 1 8. = 1 = 4 = 2 = 5= 6 = 4 = 0 = 79.
10. = 55° = 35°, = 145° = 22°= 20° = 65°, = 115°11. = 60°, = 60°
i. ii. iii.
160 :
1
1.
2. 5 ( 7) (0,1) ( 2) 11 5
3. 104. 14 8 10
6 6, 4 3 5 5 1 5
5. 276. 3
2
7. 7 i.ii. 27-
12. = 4, = 3 = 8, = 2 = 72°, = = 18° 13.
= 72° = 108° = 45° = 45° = 135°14. = 40° = = 30° = 30° = 30°15. (0,3) ( 3,0) (0,0)= 4,5 . . = + 316. (0,0)18. = = €400 = €5619. €3200020. 12%21. ( ) = 1 ( ) = , ( ) = , ( ) = , ( ) = 022. 40 30 i. ii.
: 161
8. 64 100000100000 1 649. 5 (+2) 3
( 3) ( 2) ( 7) ( 2) ( 3)
10. 2 2 7 3
11. 2 3 ( 7) 2 3 ( 7) 3 ( 7)12. = 9 = 2
1. 1 125
18 1162.
4. < = < >
5. +7 +34 11
6. 07. +8. =
=9. 3 3 3
3 3 310.11. 5 6
5 ( 5) 3 3
+ 2 ( 3)12. = 13 = 12
= 7 = 1
162 :
1. 8 5 2 0,6 7 1,5 0,3 0,01 0,4 18 23 13
2. 303. = 6 4.
5. 13 11 26 8 27 12
53 96.
7. = 28 = 36 = 448. 10 6 3 2 2 3
10. 230 11.
5,8,13,2013 (13 + 4 = 173) 41
1.
2. 7 2 12 5 + 2 3 + 2 3 2 10 + 10
3. 8 10 35 150 9 6 9 12
4. i. ii. iii. iv.
5. = 19 = 3 23 = 1
: 163
6. = 45 = 12 5 = 10 6 = 4 3 + 10 2 7. = 48. 12 1 4 2 19. i. = 64 / ii. = 96 / iii. = 160 /
1. 25 . . 100 . .2. = 15 = 20
= 8 = 24 3. = 50 4.
= 48 = 36 = 432 = 84
5.
6. 5 = 3 + 47. = 90
8.
9. = 5 = 4
10. = 136,5
1. 2,65 1,87 11,18 1,44
164 :
2.
138,425
8, 176573627200,444443. 5 5 7
12 6 94. = 51 = 595. < >
> =6. 25 = 57. 7, 50, 9, 85
5, 38, 7, 91 0,474 47 7 0,23
8.
9. 5,8 2,5 2,8 11. = 50 = 4112. 23 13. 297,32 14. = 13 = 15
: 165
61
1. + +1 64 +25 + + 0 +16 32
2. = > < =
3. 19683 177147 81 9 19683 1
5. 2 6 6 3 12 2 6 2
6. 11 5 5 3 2 6 2 3 3
7. 9 32 10000 0
8. +10 +10 29.
10. 8 1,1 5 5 10 2 0,3
11. 7 5 1012. 2 9 13
5, 2513. < < =
< > >15. 5 , 8 5 , 29 2916. = 70 = 8,3
166 :
66
1. (0,3)2. 14. = 12 = 2 = 65. 411. = 13 . = 17 . = 3 .
= 18 . = 20 .12. = 8 13. = 8 14. , , = =
, , = =15. = 34 16.
17. 21, 20, 1
2
-
1. +2 , 3 , , , 1234
17. = 48 = 1184 = 46418. 25 19. = 1320. = 17
: 167
3.
-
21 4
3 3 8+ 4
31
4. 3 2 , 3 45. = 2 = 4, = 66. 27.
1. v vi ii i
2.
2 5 : 2 43 6 15 16 8 4
8 16 40 32 16 83. 6 2 5 7 6
4. 2 12 3 6 6 10 12 44
5. 6 2 2 7
168 :
66.
60 2 45 2 37. 4 19 3 78. 10 9 8 95 6 69. = 910. 3 20
–
1. + 3 + 2 + 4 2 + 12. + 7 3 + + 6 3 3 + 2 5 2 + 1 + 6 2 + 33. 10 2 11 + 11 5 + 2 7 2 + 7 + 34. 4 1+ + 5 5 + 95.
3 8 4 + (9 + 8) = 12 9 + 4( + 3) + 2 + 3 = + 6( + 3 + 3) ( 2 + 3 ) =2 + 21 7 2 + 7 + 5 + 3= 4 + 14 + 5 10
6. 6 127.
8 1 3 4 + 4 5+ 5 6 5 + 2 9 3 + 26 2 + 1 7 + 3 4 2 + 2 38. = ( 7 + 181)° = 76°, = 31°, = 73°
: 169
1. 2 + 6 2 + 4 2 + 1 + + 2 3 + 3 2 + 1 3
2. 6 + 33 18 4
3. 2 34. 4 + 2 + 2 + 2, = 1
2, = 5 3 + 1
5.
=2= 3 + + 1
+ 2 8+ 2+4 84 + 802 3 + 2 43 + 3+5 45 + 5+1
1+ 4 3 + 56. 3 + 14 27. 3 + 5 + 3 + 48.
9. = 12
1. 5 + 9 3 8 15 24 3 7
2. 4 7 6 6 3 6 2 4 + 6 5 + 6 +4 + 2 + 7
170 :
6 + 93. 2 + 4 2 + 3 7 2 9 8 + 15
2 + 11, = 52 2 24. 2 2 3 5, = 5
4 2, 2 1, = 46. 14 8
8 12 , 7 9 , 3 2 , 2
7.
8. 3 5 79. 2, = 611. 3 5 214. 4 280015.
. .1, 2 + 42, + 2, 2 + 416. 5 2
2. = 33. = 5 . , = 4 .4. 1200000 + 1000 56. = 5 , = 8 8. 49.
10. = 300 500011. = 3 = 7 = 4
: 171
1.
4. = 6, = 0 = +6 +35. 3 1,8,0 2 ,4, 5, 6, 7, 99.
11. 1 3
1. 8 9 101° 100°2. = = 63 =109°
=71°3. = 9 = 24 6. = 24 = 22 = 84 = 42 = 90 = 48 7. = 8 8. = 4 = 2,5 9. (3, 4) ( 3, 4) (1,6)
1. = 18 2.
= = = = 30°= = = = 60°= = 60°= = 120°3. = 8 . . (3,2)4.
= 6 .= 52 . = 20 .5.
6. = 6 = 12
172 :
7. = 4,8 = 4,8 8. = 16 = 49. = 10 50 25 = 50 2 25
1. 90° = 4,5 = 10 , = 8 ,= 12 2. 60°, 120°, 30°, 60°, 90° = 40 3. = 120 = 524. = 6 5. = 40 = 96 6. = 8 = 32 7. = 12 = 40 8. = 192 = 216
1. = 7 = 45°2. = = = = = = = = 90°
= = = = = = = 45° = = = = 2 = = 8 = = = = 2
4. = = = = 3 6.
8.
9. = 5 10.
11. = = 4 2 + 4 , = 4 = = 2 2 + 2 , = 1 = 4 2 , = 2
: 173
= 4 + 2 2 , = 2 = 4 + 2 2 , = 2
1. = = 120°= = 60° = = 85°= = 95° = = 100°= = 80°
2.
3.
1 2 3 4
-
5.
6. = 25 = 246 7. = 35 . . = 25 . .8. = 6 9. = 96 10. = 28 = 24 = 112 = 5611. = 5 = 9
1. = 20 62,8 3. 20 31,4 4. = 4 = 8 5. = 0,55 6. 10007. = 40192 8. 24
174 :
9. = 6 10. = = 11. = 6 + = (20 + 6 )
= (8 + 2 ) = (24 + 3 ) 12. 72,22 . €920,8113. 31
1. = 100 314 = 25 78,5 = 4 12,56 = 36 113,04
2. = 6 3. = 6 4. = 6 5. = 126. = 4 7.
8. = 1050 = 3297 9.
10. = 1296 = 1116 11. = 1296 = 1256 12. = (36 8 ) = 2
= 60 = (57 9 ) = (40 6 ) = (64 96)
1. = = 119= 61 = 53= 37= 902. = 30 .3. 5 8 3
: 175
42° 84° 96°
4.
- - -
90°
5.
6.
8. = 120 9. = 4 = 40 10. = 120 = 52 11. = 10 = 20 12. = 10= 20 = = 13.
= 16214. 3 915. = 216. = (22 + 2 ) = (16 + 2 ) 17. = 100= (30 + 5 ) = (100 25 ) = (20 + 5 )
= (100 25 )= 10 = = 10 +
176 :
1.
= = 45= = 20= = 65
= = = = 20= 5= = 2 10 = 18 . = 20 .
4. = 400 10. = 8
{ } =
{1,2,3,4, . . . }
{0,1,2,3,4, . . . }
{0, ±1, ±2, ±3, ±4, . . . }
{1,2,3,4, . . . } { 1, 2, 3, 4, . . . }
{ / : ,
/
o
=
·
Ans
^
SHIFT 2nd F
AC
DEL
Abs
(-)
a b/c 23
S D a b/c
CLR
M+
M-
M
3 + 4 3+4 7 2,34 – 1,1 2.34 – 1.1 1.243 2 3 2 6 2 2 2 32 5 10 5 3 5 103 5000
3 4 2
2
2 3 4 4 4 2 5 5 25 2 (7 3) 2 (7 3) 8 2 (7 3) 2 8 2 ( 3) 2 ( 3) 5 | 3| | 3| 3 0,25
9 6 2
9 – 6 2 + 6
3 (9 6: 2) 6: (9 6: 2) 3 – 6 17
+ = 3 4
· – 2 3 4 1 · 1 =
3 x 2 =
2 5 =
5 P 3 =
3 a b/c 4 =
3 a b/c 4 =
SHIFT 23 2 3 4 =
2 a b/c 3 a b/c 4 =
4 =
5 X² =
= 3 – 7 ( x 2 )
x 2 = 3 – = 7 Ans
2 ( (-) 3 ) =–
= 3 (-) Abs
S D
a b/c SHIFT
= 2 ÷ 6 – 9
3 – 6 ÷ =