A
–
–
©
– 9
1.
18
26
2. ( + ) ( ) 42 ( + )( ) 48 ( + ) ( ) 52
3. –
65
– 70 –
76
– 80
85
94
– 98
– 103
4.
122
126
131
139
155
… a A
8
9
1. ( 2) ( 3) ( 2) + ( 3) ( 2) ( 3) 2 + 5 3 3 + 4 | 5| ( 3 5) ( 4) ( 3) ( 5) (+4) ( 3) ( 1) + ( 5) (+4) ( 3) + ( 5) ( 1) ( 5) ( 4 + 3)
2. ,
21, , 8, , , 9 , 2, 9
3. 81 27 5 16 81 16
4. 8 2 2 + 3 5 2 7 + 4 ( 5 ) ( 4 ) ( 2 ) (2 ) ( 6 ) (30 ) ( 5 ) ( 18 ) ( 6 ) ( 2 ) ( ) + 5 6 ( 2 ) (3 )
5. 2 (3 + 5) 3 (2 1) ( 2)( + 2) (2 + 3)( + 2) (4 5)(2 1) ( + )( ) (3 1)( 2) 10 3 ( 2 ) 3 (2 ) 3 ( 2 + 3) (2 ) 2(3 + 4 2) + (2 1)
10
6. = 3( 2 ) 2 + 5( + 1).
= 2 = 1. = 15.
7. 20
4 .
8. (8 10 ) (2 ) (9 6 ) ( 3 ) (9 24 ) ( 3 ) ( 4 ) ( 4) ( 12) ( 4) (25 + 10 + 4) (5 + 1)
9. = 4 + 7, = + 5 5 = + 1 + ( + ) 2
10. ( ) = 7 5 + 4 ( ) = 3 + 5 9. ( 1) (+2) ( ) = ( ) + ( ) ( 2)
11
11. 2 6 = 4 2 3( 1) + 3 = 33 5( + 1) = 2 3 5( + 2) 3(1 ) = 8 + 6 = + ( ) = 1 +
12. 3 46 c ,
13. = 1 2 + = 4
+ 2 = 3 3 + 4 = 7
14. « ».
-
90°
-
-
-
12
15. , y, z
16. .
, .
17. = 2 . (2,1) ( 2, 4) .
. = 2 = 1.
18. y = + (0, 4) (1, 2).
.
19. 5 + 6 = 30.
, .
.
13
20..
. . = 6 3 .
.
21. 10
15 , 11 , 10 , 10 , 14 , 16 , 19 , 18 , 13 , 17 10 10
22. To
14
23. 6, 5,
, 4, : .
1
:
16
1:
E-82
3
-
000
17
1:
-
–
E- = =
-
PISA 2012
=
250
m250 m
18
1:
19
1:
( , 7)
( , 4)
( , 3).
( , 6)
20
1:
21
1:
, ,
, ,
.
-
.
, ,
22
1:
, =
=
= .
= =
=
23
1:
24
1:
1 .
, , , , , , , ,
= = = = =
25
1:
| | = 3 = 3=
= =
2
4 .
26
1:
+ =
27
1:
+ + =
= =
+ = + =
= = .
.
-
,
.
28
1:
.
= + ( )
+ + +
: + = + = + = + =
. = + = + = = + = + = 2. , .
+
29
1:
+
+ ( ) =
30
1:
.
+ = + = + + = + = + + + =
+ + + + + + +
31
1:
= =
+
32
1:
+ + + +
.+ = 2
= = , ,
33
1:
8 80 80 /
5 1,5
4
( 1)
5 , 6 , 8 2 .( 2)
=
1 = 1852
(nm
34
1:
. =
, ,
+ + + + + +
, ,
+ + = ,
35
1:
= . =
, , , , , , ,= = , , , ,
36
1:
0 0.
+ = +
= = = =
,
= 6 + 7
2
:
: ( + ) = + + ( ) = + ( + )( ) = ( + ) = + 3 + + ( ) = 3 +
38
2:
O Jean
29% 32% 41% 71%
20%
= 10000 (1,5 0,2) = 10000 (1,5 0,8) = 10000 (1,2 0,2) = 10000 (1,2 0,8)
39
2:
:
/
/
/
/
PISA 2012
2000 2001 2002 2003 2004 2005 2006 2007 2008 0
0,8
0,2
0,4
0,6
1,0
1,2
40
2:
…
· = = ( ) = · ( ) = · = , 0 = = , 0
= , 0 = , 0 = , , 0 = · , , 0 = , 0, > 0 = , 0, > 0
.
: 3 5
: 2 2 2 2 13
.
.
8 2) 2 +2
41
2:
.
3 7 = (3 7) = 4
( 2 ) (3 ) = ( 2 3) = 6 = 6
: (8 ) (4 ) = = = 2 =
2 5 + 2: 2 , 5 2
3 + 2 2 3 2 + 3 3
(2 5 + 2) ( 4 + 1) = 2 5 + 2 + 4 1 = 2 5 + 4 + 2 1 = + 1
( 2) ( + 4) = ( + 4) 2 ( + 4)= + 4 2 8= + 2 8
42
2:
( ± )
: 41 101 999 105
( + ) ( )
( + ) ( )
«C_En _ »
,
41 =101 =999 =105 =
43
2:
:
. . 45 , 105 , 567 , . . ».
1 2 33510501
44
2:
( + ) = +
( + ) = + +
( + 3) = + 2 3 + 3= + 6 + 932 = (30 + 2) = 30 + 2 30 2 + 2= 900 + 120 + 4= 1024
:
= ( + )
= ( + )( + ) = + + + = + + + = + + = B
( + ) = + + ( + )2 = 2 + + 2
+
-
( + )
+ 2 +
= … = = =
,
= = =
45
2:
:
( ) = +( 3) = 2 3 + 3= 6 + 919 = (20 1) = 20 2 20 1 + 1= 400 40 + 1= 361
: = ( )
= ( )( ) = + = + = + = B
( ) = +
N ( + 2) (2 3 ) 2 5 + 2
( + 2) = + 2 2 + 2 = + 4 + 4 (2 3 ) = (2 ) 2 2 3 + (3 ) = 4 12 + 9 2 = 2 2 + 2 = 2 + 4 5 + 2 = 5 + 2 5 2 + 2 = 5 + 4 5 + 4 = 9 + 4 5 ( + 1) 1 = ( + 2)
= ( + 1) 1= ( + 2 1 + 1 ) 1 = + 2 + 1 1= + 2= ( + 2)=
=
46
2:
= ( + 1) 1= ( + 2 1 + 1 ) 1 - = + 2 + 1 1= + 2 = ( + 2) = + 2 -
= ( + 1) 1 = ( + 2)
+ 2 + + 3,
+ = 0+ 2 + + 2 + + 3 = ( + ) + 3 = 0 + 3 = 3
,
= 56 = 4 + 2 0 = 0= 1 += 0 2( + ) = 2 +
N ( + 1) ( 3) (7 ) (6 + ) (4 1) (3 + 2) (5 ) 5 + 2 (2 + ) 3 +
47
2:
( 1) + ( + 1) (2 + 1) 2 (2 ) (2 + 3) (1 2 ) (2 3) + ( 1)( 1)
N , ( + ) = + 4 + 4 ( + ) = 9 + 6 +( + ) = 25 + + 10 ( ) = 20 + 4 + 25(5 ) = + 16 40 ( + ) = + +
4 + 4 9 + 6 + 14 + 4 + 25 + 8 166 + 3 2 + 1
:
+ 2 + + = 5 2 + = 2 + 6 + 9 + 2 = 1 :
( 2) ( + 2) = 8 ( + ) 4 = ( ) ( + ) ( ) = 4 + 2 = + 2 ( + + ) = + + + 2 + 2 + 2
= 90
+ +
-( ± )
48
2:
( + ) ( )
–
,
«C_En _ »,
( )( + ) = ,
( )( + ) =
49
2:
: ( )( + ) =
( 3)( + 3) = 3 = 932 28 = (30 + 2)(30 2)= 30 2 = 900 4= 896
:
= ( )( + ) = + = = B
, ( )( + ) =
N : ( 5)( + 5) (2 3)(2 + 3) (2 )( + 2) 2 ( 1)( + 1)
( 5)( + 5) = 5 = 25
2 3 2 + 3 = 2 3 = 4 3 = 1(2 )( + 2) = (2 )(2 + ) = 2= 4
50
2:
2 ( 1)( + 1) = 2 ( 1) = 2 + 1= 1 -
N
( 3)( + 3) (2 1)(2 + 1) ( 1)( + 1) ( 2 )( + 2 ) + (5 2)(5 + 2) (3 1)(1 + 3 ) (3 + 5)(5 3 )
N ,
4 + 1 = 1 + 8 = 2 + = 64 + 5 =
N : ( 3)( + 3)( + 9) ( + 2)( 2) + 3 5 ( 1)( + 1)( + 1) 1 (4 1)(4 + 1) + (4 + 2)2 (2 + 3)(2 3) ( 1)(1 + )
,
103 29 29101 99 198 202
51
2:
: ( 1) + 2 = ( 1)( + 1) + 2 ( )( + ) + 2 = ( ) + 2 ( 1)( + 1)( + 1)( + 1) = 1 [( ) + ]( 9 ) = 81
( + )( ) = :
1 2 + 1 2 1
= (3 + 1) + ( 3) + 10(1 )(1 + )
y = 1 , :
= ( ) ( 3)( + 3)
52
2:
( + ) ( )
( + ) ( + )
«C_En , ( + )
( )
( + ) = + + +
(2 + ) = 2 + 3 2 + 3 2 + = 8 + 12 + 6 +
53
2:
= ( + ) = ( + ) ( + )
= ( + ) ( + + ) = + 2 + + + + = + 3 + + = ( + ) = + 3 + +
( ) = +
( 5) = 3 5 + 3 5 5 = 15 + 75 125
A’ = ( ) = ( ) ( + )
= ( ) ( + ) = 2 + + = 3 + = ’ = ( ) = + ( ) = + 3 ( ) + ( ) + ( ) = 3 + = ( ) = 3 +
( + 3 )2
( + 3 ) = ( ) + 3 ( ) (3 ) + 3 ( ) (3 ) + (3 ) = + 3 (3 ) + 3 ( ) (9 ) + 27 = + 9 + 27 + 27
54
2:
:2 = (2 ) 3 (2 ) + 3 (2 )
= 8 3 (4 ) + 3 (2 ) = 8 4 + ( ) = 5 1 ( ) = 4 5,
( ) + 1 ( + 1)
( ) + 1 = 5 1 + 1 = 5 ( ) = 5 1 ( + ) = ( + ) 5( + ) 1
: = + 3 + 3 + 1 5( + 2 + 1) 1 = + 3 + 3 + 1 5 10 5 1 = 2 7 5
+ = 5,
+ +
+ = 5, + = 5 + 2 + = 25 + 2 + = 25 + = 25 2 + = 23
+ = 5,3 ( + ) = 3 53 + = 3 + = 15
+ = 5+ 3 + 3 + = 125+ 3 + + = 125+ 15 + = 125 + = 125 15 = 110
= = ,
55
2:
( + 3) (2 ) ( + 3) ( 3 ) (2 1)
( 2) ( 3)( + 3) + 4 (1 + ) 2 (2 + 1) ( 2) + (2 3)(3 + 2 )
( + 2) 2 = 6( + 1) + ( 1) + (3 7) = ( 2)( + 2) 1
+ = ( + ) 2 + = ( + ) 3 ( + ) + + + = 2 = 8
+ = 3 = 2 + + ( )
5 3
6410
:
GeoGebra CAS
(Calculate Algebra System) -
”
CAS”
-
!
56
2:
(2 + 5) + (2 ) 2 7 ( + 3)( + 3) ( 5 1)( 5 + 1) ( + )( ) ( )( + ) ( 5) (2 3 ) (1 + )( + 1)( 1)(1 + )
+ + = + 25 + 1 + = + + + 25y = 4 + + 9 = 4 = + 3 + = + = y + 8y + = + + + = + 27 + 1 + + = +
: ( + ) ( + ) = 2 ( ) + ( )( + ) ( + ) ( ) = 2 ( + 3 ) = ( + 1) = ( ) + 2( ) + 1
57
2:
:
4
N = 2015 2014 = 1,61 1,6
P( ) = 2 + 2 + 80 ( 1) P( ) P( 1) = 4 4 .
P(100) P(99) + = 2 = 3 + = 10 + = 26 ( ) = 16
N , + , , , 0
-
(0 < < < ) + = .
58
2:
= ( 1) ( 4 2) + 7, = + 5 + 6. = 5 + 7 = ( 1)( + 1)
= 9
= (3 ) (3 2)(3 + 2)
+ = 3 ( + 1) + ( + 1) ( ) 2
: ( + 5) = (3 ) + 24 3( 3) + 2( 3)( + 3) = 5(2 1) 15
59
2:
= 2 15 = 6 10: ,
2.
( )( ) ( ) ( )
3. : = 2015 + (2 2016)
Blaise Pascal (1623 – 1662).
:
( + ) , ( + ) = ( + )1 = + ( + )2 = + + ( + )3 = + + + ( + )4 = + + + + ( + )5 = … + … + … + … + … + … ( + )6 = … + … + … + … + … + … + …
( + ) ( + ) ( )
60
2:
(S)
(F) (E)
(T)
Ca 3 3
Sp 3 3 3 3 3
KK 3 3
= x + + + «Ca»
«Ca»
«Ca»
= x + x + x + x
= 140
m 70 0,80
:
3
62
3:
= : : :
:
PISA 2012
63
3:
(9 6 ) 3 = (9 3 ) (6 3 ) = 3 2
84 = 2 3 7
( ) ( ) ( ) 0 ( ) ( ) ( ) ( )
( ) = ( ) ( ) + ( ) ( ) ( )
6 + 5 + 3 6 3 +2 + 3 2 1 +2
2 + 1
6 + 5 + 3 = (2 + 1)(3 + 1) + 2( ) = 3 + 1 ( ) = 2
3 + 1
84 120120 = 2 3 5 (84,120) = 2 3 = 12
64
3:
6 = 2 3
6 88 = 2 (6,8) = 2 3 = 8 3 = 24
4 + 3 = 5 5 3 4 + 3 = 5 5 3 4 5 + 3 = 5 3 2 = 8 = = 4
65
3:
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
2
2.3 + 2 = ( 2)( 1) 2 4 + 3 = 2 ( 2) + 3 3 12 + 12 = 3( 2) 4 = ( 2)( 2) + 2 9 = ( 2)( 2)
2 6 + 12 8
66
3:
(12 , 24 , 36 ) = 12 (4( )( + ), 6( + ) , 2( + )) = 2( + )( ) ( ) ( ) 0 ( ) ( )( ) ( ) ( ) = ( ) ( ) + ( )
( ) ( ) ( ) = 0 ( ) ( ) ( ) ( )
6 + 8+ 42 + 82 804
( ) = 0
2 ( ) = ( ) ( ) + ( ) ( 6 + 8) = ( 4)( 2)46 + 8
9 + 2 7 3 2
9 + 2 7 3 2
-
67
3:
9 + 0 7 + 2
3 2 -
= 33
9 + 0 7 + 29 + 6 +6 7 + 2
3 2 3(3 2)3 (3 2) = 9 6 A
-9 + 6 -
3
-
9 + 0 7 + 2 9 + 6 +6 7 + 2 6 + 4 3
3 2 = +2+2 (3 2) = 6 4
-6 + 4-
3 + 2
-
9 + 0 7 + 2 9 + 6 +6 7 + 2 6 + 4 3 + 2 +3 2 0
3 2 = 11(3 2) = 3 + 2-+3 2
3 + 2 1
9 + 2 7 = (3 2)(3 + 2 1)9 + 2 7 3 + 2 1 .
68
3:
2 4
2 4
4
2=
+ 0 4 + 2 + 0 +2 4
2 2( 2 ) = 2-+ 2-
+ 0 4 + 2 + 0 +2 4 2 + 4
2 = +2+2 ( 2 ) = +2 4
-2 + 4+ 2
2+ 2 44 = ( 2 )( + 2 )
69
3:
16 , 36 y 30( 1)( + y) , 18( 1) , 42( 1)
(16 , 36 ) = 4[30( )( + ) , 18( ) , 42( ) ] = 6( 1)
: ( 2) ( + 1) 5 + 2 4 6
: + 6 2 + 13 27 + 4 8 3 9 6 +
2 + 3 + 4 1 4 22 + 32 8 2 4 + 2 3 + 12 4 3
24 , 15 , 6 16 , 4 , 12 9 ( ) , 6 ( ) ( + )
: + = + 3 (2 5) = 6 15 ( 2) = 3 6 = 3 15
70
3:
6 12 = 2(3 6 ) 6 12 = 3(2 4 )
6 12 = (6 12) 6 12 = 3 (2 4)6 12 = 6 ( 2)
–
1
H
3:
, ,
+ 5 ( + 5)
+ + ( + + 1)
+ + 5+ 5 ( + )+ 5( + ) ( + )( + 5)
«C_E _ exe» 5
72
3:
, - .
4 2 = 2 (2 1)
5 5 + 5 = += ( + ) - 5512 8 = 3 2= (3 2 ) - 4
+ + + =-
+ + + =( + ) + ( + ) = ( + )( + ) ( + )
-( 1) -
-’
-
-
73
3:
10 + 10 3 2 ( + ) + 5y( + ) 10( y) + 2 (y )
10 + 10 = 10 ( + 1) --103 9 =3 ( 3) --32 ( + ) + 5 ( + ) = ( + )(2 + 5 ) 2 ( + ) 5 ( + )-( + )10( ) + 2 ( ) =
10( ) 2 ( ) = 2( )(5 ) 10( ) 2 ( )
( ) -2 ( )
+ 21 + 7 + + 12 +
+ 21 + 7 + 3 =
3 -
= ( + 7) + 3( + 7)-( + 7) = ( + 7)( + 3)
+ 21 + 7 + 3 = ( + 7)( + 3)
( ) = +( + )
= ( )
74
3:
+ = = ( ) + ( ) ( )
= ( )( + ) + = ( )( + )
12 8 9 + 6 = 43 = 4 (3 2 ) 3 (3 2 )
(3 2 ) = (3 2 )(4 3 ) 12 8 9 + 6 = (3 2 )(4 3 )
+ 7 + 7 7 3 3 + 6 7 14
7 + 7 6 612 6 8 6 4 8 5 + 10 4 18 + 16 15 + 30 10
75
3:
2 ( + ) + 5 ( + ) 3( ) + ( )( + 1) 2(1 + ) 4 (2 5) + 9 (5 2 )( 2 ) 5(2 ) ( + ) + 3( + )( + ) 3 ( + ) ( ) +
+ 5 + 2 + 10 3 + 3 + 8 12 10 + 15 5 + 5 2 + 3 2 3 + = 4 = 3
: = 4 + 4 = 2 + 2 = + + + = +
+
( )
: = ( 3) 4 2
:
76
3:
–
115.
15
= ( )( + )
25 = = ( + 5)( 5)(2 1) = (2 1 + )(2 1 )–
= ( )( + + ) + = ( + )( + )
27 = 3 = ( 3)( + 3 + 3 )= ( 3)( + 3 + 9)8 + = (2 ) + = (2 + )[(2 ) 2 + ] = (2 + )(4 2 + )
.
x
x
y
y
77
3:
4 9 8 + 27 3 24 + +
4 9= (2 ) (3 )= (2 3 )(2 + 3 )
8 + 27= (2 ) + (3 )= (2 + 3 )(4 6 + 9 )
3 24 = 3( 8) = 3( 2 )= 3( 2)( + 2 + 4)+ + = ( )( + ) + ( + )
= ( + )( + 1)
+ = 5 = 4+
( + ) = + 3 + 3 ++ = ( + ) 3 3+ = ( + ) 3 ( + )+ = 5 3 4 5 + = 125 60 = 65
78
3:
+ = ( + )( + )
( + ) = + 2 + + = ( + ) 2 + = 5 2 4 + = 25 8 + = 17
+ = ( + )( + ) + = 5 (17 4) + = 5 13 + = 65
4 49 94 9 81 + 16 49 3 3 2 3225 1 81 16 18 50
(2 1) (5 + 1) 4 4 ( ) ( + ) 16 ( + ) ( + )( ) + ( ) ( ) ( 2 )
5 5 4 + 4 ( 3) ( 3) 16( + 4) ( + 4) ( 3 ) + 2( 9 ) ( )(2 + ) ( )(4 )
79
3:
– 8 125 + + + 2 54 3 24 + 1
( ) + ( ) 2( )+ 3( + ) ( 1) 2( 1)
: 100 99 805 795
= 2 + 2 = 2 2
( )
80
3:
( –
1
«C_E ggb»
: ( ( + 4 + 3 ( + 3)( + 1) + 3 + 2 + 4 + 4 + 5 + 6
+ + = ( + )( + )
3:
= 4 + 4 + 1 = + 5 + 4
+ + 0
= 1+ + , = + =
+ + = ( + )( + ) + 3 10 = ( 2)( + 5) +3 10 2+5
22( ) )
( )(x xx x xx x
82
3:
+ 2 + = ( + ) 2 + = ( )
4 + 12 + 9 = (2 ) + 2 2 3 + 3 = (2 + 3) 12 + 36 = 2 6 + (6 ) = ( 6 )
----
.
5 6 + 6 + 9 3 + 6 24 10 + 25
5 6 = ( 6)( + 1)
3 + 6 24 = 3( + 2 8)= 3( + 4)( 2)
+ 6 + 9 = ( + 3) 10 + 25 = 10 + 25= ( 5)
= ( 5 )( 5 + )
83
3:
+ ( + ) + + ( + )( + ) + 2 8 2 + 8 2 8 ( 2) ( + 8)+ + + + + +
«C_E
9 + 20 2 15+ 5 + 6 6 + 8 + 8 6 ( 15) + 56 + 90 y 5y 150y + 2 2 14 + 205 4 5 6
18 + 81 64 16 +25 + 30 + 9 4 + 12 + 9 6 + 9 16 + 24 + 9
84
3:
= 4 = 2 8 + 4 25 30 + 9
992 + 2 8 992 + 8 0,58 + 0,42 + 2 0,58 0,42
2 + 16 + 8 1+ + + 2 + 1 1 2
( + )( + ), , + 7 + + 10 8 + , > 0 5 + , > 0
85
3:
1, 3, 6, 10, …
1 4 6
- = ( + )
55 120 150 200
86
3:
:
+ 4 + 4+ 4 + 3+ 4 + 5 + 4 + 4 = 0+ 4 + 3 = 0+ 4 + 5 = 0
( ) ( ) = 0 ( ) = 0 ( ) = 0
( ) ( ) … ( ) = 0 ( ) = 0 ( ) = 0 … = 0
( ) ( ) … ( ) = 0
( 2) (2 + 3) = 0 2 = 0 2 + 3 = 0
= 2 = ( 2) ( + 5) = 0 = 0 2 = 0 + 5 = 0
= 0 = 2 = 5 + + = 0 0
2 1 = 0
+ = 0 , 0
=
-
87
3:
4 0 + + = 0 , 0 = =
, = ±
2 1 = 0 = 2 , = 1, = 1 , = ±= ( )± ( ) ( ) = ±
= == 1
== =
+ + = 0, 0 4 + 4 + 4 = 0 4 + 4 + 4 = 0 (4 + 4 + ) + 4 = 0 , . (2 + ) = 4
4 0
2 + = 42 = + 4=2 + = 42 = 4=
+ + = 0, 0 , = ±
88
3:
(2 1)( 1)( + 2) = 0 2 + 2 = 0 3 = 64 ( 18) + 81 = 0
: H ( ) ( ) … ( ) = 0
(2 1)( 1) = 0 = 0 2 1 = 0 1 = 0 = 0 2 = 1 = 1 = 0 = = 1
H ( ) ( ) … ( ) = 02 + 2 = 0 2 = 0 + 2 = 0
= 2 = 2 ( ) ( ) … ( ) = 0 3 = 64 64 = 0 -
( 64) = 0 -
( 8)( + 8) = 0
-
-
= 0 8 = 0 + 8 = 0
= 0 = 8 = 8
89
3:
( ) ( ) … ( ) = 0 ( 18) + 81 = 0 18 + 81 = 0 -
+ 2 + = ( + ) ( 9) = 0
9 = 0 = 9 -9
1 15
== + 1. = = 15 ( +1) = 15
( + 1) = 30 2 + 30 = 0 ( 5)( + 6) = 0 ( 5) = 0 ( + 6) = 0
= 5 = 6
5 6 (2 5) = 25 4 + 4 + 10 = 02 6 = 0 + 3 + 1 = 0
90
3:
: (2 5) = 25 (2 5) = (+5) (2 5) = ( 5)2 5 = 5 2 5 = 52 = 5 + 5 2 = 5 + 52 = 0 2 = 10= 0 = 5 4 + 4 + 10 = 0
4 + 4 + 10 = 0= 4 , = 4 , = 10
H
, = ± = ±= ± 244 + 4 + 10 = 0
4 + 4 + 10 = 04 + 4 + 1 1 + 10 = 04 + 4 + 1 + 9 = 0(2 + 1) + 9 = 0(2 + 1) = 9(2 + 1) 0(2 + 1) = 9
3:
2 6 = 0 = 2 , = 1 , = 6 H, = ± = ( )± ( ) ( )= ±= ±
= = = 2 = = =
+ 3 + 1 = 0 , = 1 , = 3 , = 1 H , = ± = ± ( ) = ±
= = = =
( + 5) = 0 ( 7)( + 5)(2 3) = 0 5 10 = 0 (2 5)( 6 + 5) = 0 4 = 0 25 10 + 1 = 0 5 + 6 = 0 ( 9)(9 6 + 1) = 0
92
3:
= 25 = 12 = 35 2 = 2 + = 2 ( 6) = 9 ( + 4)( 3) = 3 ( 10) + 25 = 0
2 5 + 10 = 0 5 4 1 = 02 2 3 = 0 2 + 1 = 0 3 + 2 = 0 ( 7 + 12)(2 5 + 2) = 0
: + 1 = 0 (2 ) = 0 ( 2) = 4 (3 1) + 9 = 0 4 + 12 + 8 = 0 9 6 + 2 = 0 25 20 + 12 = 0 16 + 40 + 20 = 0
( ) = ( + 1) + ( 1) + 7: ( ) = 2 + 5 + 7 ( ) = 0
9 6 + 2 = 0
93
3:
74
8 4
:
( 3,14)
3 cm 16
100 5 = 0)
«C_ ggb»= 0)
94
3:
2 1!!!
, ( )( ) ,
95
3:
H ( )( )( 1)( + 1) 0, 1 1. , ,
.
3 3( 1)( + 1) = 3( 1) ( 1)( + 1) = 3+ 1
1 1
0 0
+ 5 05
0
-
0 0
--
, 0, =
= 0
96
3:
4 0 ( 2)( + 2) 0 2 2= ( )( ) =
( )( )
= =( )( ) =
= ( )( )( ) =
97
3:
, 0
( )
( )( )
( )( ) ( )( ) ( )( )
= { 6, 3}
98
3:
–
40
kwh
3
20
8 3
, , , = , =
:
1 1
99
3:
: = =
= : = === ( )( )( )( )
= ( )( )( )( )( )( )= : ==
= 3
:
=
3:
: = ( ) ( )( )
= ( ) ( )( )( )( ) = ( )
= ( )( ) : ( )( )
= ( )( ) ( )( ) =
= ( )( )( )( ) = ( )( )( )( )( ) =
20 75 1 1 60
(1 2 ) = (2 1) (2 ) = ( 2)
= , , 0
3:
10 / / 72 / /
( ) ( )
: : : :
: :
22 32 3 2
2
x xx x x
x 32 ( 2)
xx x ( 1)
3( 2)( 1)
xx
x x
x 2x 2
3 x2 3x x
2
2
12x
3:
= 2013
( )
A = :
3:
–
(2 , 6 , ) = 6 ( )( + ), 5( + ) = 5( + ) ( )
3:
+ = + , =
= ( ) = =
(2 , + 1) = 2 ( + 1)+ = ( )( ) = ( ) = ( )
( )( ) = 0 ( ) = 0 ( ) 0. ( )( ) = 0 ,
( ), ( ).
= 0 2 = + 3 03 = 2
+
3:
+ = + =
=
---
= ( )( )= ( )( )( )= ( )( )
-
= ( + 1)( 1) = ( )( )
=
+ = 2 3 + 2 4 = 25 + 6
2 3 + 2( 2) = 2( 2)( 3)
2( 2) 2 3 + 3 2( 2) = 2 2 ( 2)( 3)
4( 2) + ( 3)2( 2)( 3) = 42( 2)( 3)
4( 2) + ( 3) = 4 4 8 + 3 4 = 0 + 12 = 0 ( + 4)( 3) = 0 + 4 = 0 3 = 0 = 4 = 3
H = 4 = 3
-
= 2( 2)( 3)2( 2)( 3) 02 0 3 0
-
E
-
2 3
3:
+ + + + + +
+ 2 1 0
3:
+ = ( )
+ + + +
(2 ): 3 ( + 2)
1 + + : 1 + :
= 0 = 0= = 2 = 1 + = 2 + = + =
= 0 1) = 0 = 3 ( ) = 0 2) = 2
= 0 = 3
= 1 4) = 1 5) = 0 = 1 = 1
3:
3 + 3 5 + 10 15 7 + 10 12 8 4 2 2 + + 5 + 5 6 + 8 27 16 8 +
( + 1) 9 3 3 + 2 2 (3 + 2 ) 16 4 ( 1) + 4 (1 ) 3 + 3 3 3 10 + 25 + 4 20 3 3 + 3 3 10 + 25 3 + 2 2 + + + ( )2 ( + 1)2 (3 9)( 9) ( 3) 2 + 4 36 12 + 2
=
+
, + = +
3:
( )
+
:
: ( ) ( ) ( ) ( )
.
3:
( ) : + ( ) = ( ) = ( ) = ( ) ( ) =
( 2)( + 8) = 0 3 (5 2) = 0 3( + 5) = 0 = 9 ( + 3) = 4 16 8 + 1 = 0 ( + 1) = 3 ( + 1) ( 3)( + 2) = +6
5 24 2 10
20
= 0 ( ) = = 2 = = 2 + =
( ) = + =
3:
:
+ = 1 ( + 2 ) + (2 ) = 5
3:
, , 0 + + = 0
= ( ) = + 1 = 1
= ( )= + 1 ( 1,0) = 1
= = ( )( ) 2013 + 1 2014
= 1 1 1 … 1 1
= ( ) :
= + + + . . . +
2 1
3:
( =
1 1 8 4
4
3
3:
= 7,0 x 12 12
16 35
35
3
4
:
,
116
4: –
117
3: –
1,5
PISA 2012
118
4: –
…
=
=
+ = °
, , , ,
180 . + + = .
= + .
=
119
3: –
=
=
= ,
, =
120
4: –
-
-
121
3: –
122
4: –
, ,
(cm)
, , , , , , , ,
123
3: –
«C n4_KataskeuasimoTrig »
,
,
| | < < +| | < < +| | < < +
,
.
124
4: –
> >
, ,
.
.
= 1 , = 2 , = 4 = 8 , = 6 , = 12
+ = 1 + 2 = 3 = 4 + <
+ = 8 + 6 = 14 = 12 + > . 0,8 3,9 , ,
| | < < +3,9 0,8 < < 3,9 + 0,83,1 < < 4,7 = 4 3,1 4,7 .
3,1 m < < 4,7 m
-
125
3: –
,
= 1 , = 2 , = 3 = 10 , = 5 , = 7 = 3 , = 4 , = 5
= 2,7 = 10
( = ) = 70°.
> >
126
4: –
–
«C_En4_IsaSsxhmataTrig »
«C_En4_IsaSsximataTetrapleyra
, ,
, ,
127
3: –
(COPY)(PASTE
–
,
,
128
4: –
= , = , = , = , = = , = , = , = , = .
= , = , == , = = .
, = , = ,=
-
-
-
129
3: –
= = ,= = = =
= =
= = = = = = = = =
= =
130
4: –
,
,
,
«
,
131
3: –
«C n4kritirio_IsoTrig »
(
.
2
132
4: –
3
,
«C_En4 kritirio_IsoTrig »,
,
.
«C_En4_PeriexomeniGonia »
, ,
4
«C n4 kritirio_IsoTrig »
,
.
133
3: –
1 ( )
= = ) == ( )
2
= )= == ( )
3 ( )
= ( )= == ( )
134
4: –
o = ( ( )= ( ) ( )= ) ( )
,
= , = =
= =
+ + = 180° = 180° , (1) + + = 180° = 180° (2)= ==
=
-
-
135
3: –
=
= ( ( ) = ( )= ( )
=
= = =
=,
= , = , = , = 100° = 50°
=
=
136
4: –
= , = , = = , = = , = , = = , = , = = , = , =
= , = _______ = , = _______ = , = _______ = , _______ _______
4.
= ___________________
= ___________________
= ___________________
137
3: –
, , = , = =
, , , , , , = =
( = ), =
138
4: –
( = )
K
(3,1), (8,1) (8,8)
-. . .
, , =
, =
=
-
--
-
139
3: –
2500
( ) ( ) ( = ). ( ) ( )
( ) ( ).
140
4: –
o «C_En4_KritirioOrthTrig »,
.
,
141
3: –
,
= = 90° )= ) == ( )
( ).
= = 90° )= ) == ( )
))
))
))
))
142
4: –
,
,
,
:
=
= ) ( ) = ( ) ( )
, = )
= .
= ( ) = ( ) ( )
,
= ) = ,
=
=
143
3: –
144
4: –
: ( = ), .
= ( ) ( )= ( ) ( )= ( )
= . = ) = == , + = 180 = = = 90
) =
145
3: –
, ,
, ( 4,3), (4,3), (0,0) ( 3,3), ( 2, 2), (2, 2)
( 1,1), (1,5) ( 4,5) ( 2,4), ( 7,4) ( 4,6) ( 3,2), ( 1,3) ( 3,1) ( 7,7), ( 7,9) ( 3,7)
= 90° = 90° =
146
4: –
( = ) , , ( = ).
,
( = 90° , = 90°). = = ,
( = ) .
147
3: –
148
4: –
9 , 4 6
= = ,
15 100
= 10, = 23, = 11.
, = 60 AB =
( = )
= = ,
= , , =
149
3: –
= =
( )
( = = ). ( = ).
.
150
4: –
( = ) , , , = = =
+ < +
= , = = ,
151
3: –
, , = = = ,
> = 2
= =
, = .
=
al-Wafa
«
»
- – 998
-
-
152
4: –
)
)
“C_En4
PISA 2003
A
154 :
: 155
: –
9
1. 6 5 10 4 217 17 352. , 8, , 9, 2 21, 93. 6 1
364. 6 5 5
20 4 12 6 3 7 12
5. 6 + 10 6 + 3 4 2 + 7 + 6 8 4 10 + 5 3 17 + 2 3 3 6 6 + 9 + 2 6 6 + 3
6. + 5 8 207. 0 < < 10 4 80 + 400 576 8. 4 5 3 + 2 3 + 8
2 + 1 + 1 + 3 = 5 + 1 = 3
9. 5 4 + 13 3 + 310. 16 11 4 5
1111. = 2 =
A = 5 = 012. = 130 13. = 1 , = 2 = 1, = 1
156 :
14.
-
90°--
-
-
15. = = 119, = 61 = 90, = 37, = 5316. = 4 = 2
17.( 2, 4) (0,0)
18. = 2 ,= 4 = 2
: 157
19. ( 6,0)(0,5) = 15 . .
20. = 3 = 3 + 6
21. 14,3 14,5 50%22. 2 2 123. ( ) = ( ) = ( ) =
( ) = ( ) =
:
23
1. 2. , , . .
158 :
3. = 1, = 2, = 4, = 1, = 1, = 2, = 4, = 2, = 4
= , = = = == == =
4.
8. = = , ,
30
1.
3.
= = = += 2 + 2 = 2 = 4= 2 2 = 4 24.
= = += + = +5. + + = + + = 2 = =7.
= == 2 == = = 2= 2 =
: 159
33
1.
2. 22 3 2 5. | | = 5 , = 8 | | = 107. = 58.
= 2== +=
36
1.
+ + =0
2. = = = ( )
160 :
:
( ± ) 6
1.
2. + 2 + 1 6 + 9 14 + 49 + 12 + 36 16 8 + 1 9 + 12 + 4 25 10 + 10 2 + 27 + 4 + 4 + 9 + 2 + 1
3. 4 + 4 8 + 16 10 + 25
4. 2 + 2 6 + 1 16 + 8 5 14 + 10
5. ( + 2 ) = + 4 + 4 ( + 3 ) = 9 + 6 + 5 + = 25 + + 10 ( 2 5 ) = 20 + 4 + 25 (5 4 ) = 25 + 16 40 ( + ) = + + 2
6. 4 + 4 9 + 6 + 1 2 + 17. 25 4
19. 10. 4 4
( + )( ) 50
1. 9 4 1 1 4
: 161
239 1 25 92. 4 1 4 + 1 = 16 1 8 + 8 = 64
2 + 8 8 2 = 64 4 5 + 5 = 25
3. 81 + 3 9 2 32 + 16 + 3 2 + 9
4. 10609 841 9999 39996
7. = 5 28. = 209. = 7
( ± ) 55
1. + 9 + 27 + 27 8 12 + 6 + 9 + 27 + 27 9 + 27 27 8 12 + 6 1 3 +
2. 3 + 12 + 1 7 5 + + 1 5 6 + 3 8
4. + = 20 + = 565. 5 9
16. 13 7. 188. 2015
56
1. 4 + 20 + 25 + 2 + 4 + 4 11 4 7
162 :
+ 6 + 9 4 15 + 75 125 8 36 + 54 27 1
2. + 2 + = + 25 + 1 + 10 = 5 + 1 16 + 40 + 25y = 4 + 5 2 + 9 2 9 = 4 81 = + 3 3 + = 9 + = y + 8y + 16 = + 4 + + = + 27 + 1 + 27 + 9 = 3 + 1
5. = 4029 = 0,03217. 2 + 6 + 76 39611. 20 2
412. = = 2
9
1. 33. 17 4.
( + ) = + 5 + 10 + 10 + 5 + ( + ) = + 6 + 15 + 20 + 15 + 6 + ( ) = 6 + 15 20 6 + 15 + 5. 15
6. 0,5 89,6 5,38
: 163
3: –
9
1. 2 5 18 38 122.
3.
2 + 3 + 4 1 2 44 22 + 32 8 4 14 + 4 24 + 2 23 + 12 4 + 4 34. 3 4 3 ( )5. 66. + 3 = + 3 3 (2 5) = 6 15
3 2 = 3 6 3 5 = 3 15 - 7
1.
+ ( + )7 + 7 7 7 7 ( + )3 3 + 6 3 3( + 2)7 14 7 7 ( 2)2. 7( + ) 6( 1) 6(2 )
( ) 2 (4 3) 4 ( 2 ) 5 (1 + 2 ) 2 (2 + 9 8) 5 (3 + 6 2 ) (1 2 )
3. ( + )(2 + 5 ) ( )(3 + ) ( + 1)( 2) (2 5)(4 9 ) ( 2 )( + 5) ( + )( + + 3) ( + )( + 3 ) ( )( 1)
4. ( + 5)( + 2) ( 3 )( + ) ( )( + )
164 :
(2 3 )(4 5) ( 5)( 1) (2 + 3 )( )5. = 16 = 72
= 16 = 307. = ( 3) 35 8. = 2 (4 ) = 4( + + 4)
- 78
1. ( 2)( + 2) ( 7)( + 7) (3 )(3 + ) (2 3 )(2 + 3 ) ( 9)( + 9) + 7 + 7 3 ( 1)( + 1) (5 1)(5 + 1) 2( 4)( + 4) (3 2 )(3 + )(9 + 4 ) 2(3 5 )(3 + 5 )
2. (2 1 )(2 1 + ) (5 1)(5 + 3) ( + )(3 ) ( + 2 4)( + 2 + 4) ( )( + + ) ( )( + + ) ( )( + )( + ) (2 3 )
3. ( + )( 5) ( )( + 4) ( 3)( )( + ) ( + 4) (4 ) ( 3 )( 2 + 6 ) ( )(2 + )(2 )
4. ( 2)( + 2 + 4) (5 + )(25 5 + ) ( )( + + ) + + + + ( )( + + ) 2( 3 )( + 3 + 9 ) 3( 2)( + 2 + 4) ( + 1)( + 1)
5. ( )( + ) ( + ) ( )( + + 2 2 ) ( + )( + 3) ( 1) ( + 1)
6. 199 160007. 2 = 4
: 165
- 3
1.
+ ( + ) + + ( + )( + )+ 2 8 2 + 8 2 8 ( 2) ( + 8)+ + 2 8 2 + 8 2 8 ( + 2) ( + 8)
8 2 8 + 2 8 2 ( 8) ( + 2)+ ( 8) ( 2) 8 2 8 2 ( 8) ( 2)+ ( 4) ( 4) 4 4 4 4 ( 4)+ + 16 1 16 + 1 16 1 ( + 16)( + 1)3. ( 5) ( 4) ( 5)( + 3) ( + 2) ( + 3)
( 4) ( 2) ( 4) ( 2) ( 7)( 8) ( 9) ( + 10) ( 15) ( + 10) ( 1) ( + 2) 2 ( 5) ( 2) ( + 1) ( + 4) ( 6) ( + 1)
4. ( 9) ( 8) (5 + 3) (2 + 3 ) ( 3) (4 + 3 )
5. = ( 2)( + 2)= (2 1)( 2)( + 2)= ( 2)( + 2)( 1)
6. = 4(5 3 )7. 1 000000 18. ( )( + ) ( 4 1)( + 4 + 1)
( + 1)( + + 1) (1 )(1 + + )9. = ±6 = ±7 = ±11
= 16 , = 15, = 7, = 12 = 6 = 4 91
1. = 0 = 5 = 7 = = 5 = 0 = 2 = = 1 = 5= 0 = 2 = 2 = = 2 = 3 = 3 = 3
2. = 0 = 5 = 5 = 3 = 4
166 :
= 7 = 5 = 0 = 2= 0 = 2 = 1 = 3= 6 = 2 = 0 = 5 3. = = 1= = = 1 = = 1 = = = 2 = 3 = 4 4. = 2= 0 = 4 = 2 = 1 = =5. = 1 =6.
7. 5 + 6 = 0= 2 = 38. = 49. 5 ,7 5, 710. 10 , 6 11. = 10 = 7 12. 3, 13. = 514. = 20
96
1. {±5} { 7,2} {3}2. , .3. .4. ( )
( )5. ( )6. 2019
: 167
- 100
1. 75
2. 36 / 20 /3. ( ) ( ) 4. 1
( ) ( ) 5. 6. 2013 7. 9
( ) -
106
1. 3
2. .
3.
13004. ( ) ( )( ) ( )
( )( )( ) ( )
5. + , + + , +
6.
7. 2 6 ( )
8. = 7 = 3
168 :
y = 1 y = 5 = 1 = 3 = 2 = 0 = 2 = 8 = 2
9. 3) 1) 2) 4)
108
1. 3 ( + ) 5 (2 + 3 ) ( 5)( 2) ( 4) ( 2) 4 (2 ) 2 ( 1) ( + 1)( + )( + 5) ( 4) ( + 3) ( 3) ( + 3 + 9) (4 )
2. ( 2) ( + 4) ( )(3 + + ) (3 2 ) (3 + 6 ) 4( )( + )( 1)( + 1) ( + )( + 3) ( 5) ( 1) ( + 1)( )( + + ) ( 5 )( 5 + ) ( + 1)( + + ) ( + 1)(2 + 1) (3 + 8) ( 3) ( + 6)( + 6 2)
3. ( ) 364. ( + )( 1)5. 3
6. ( )1 ( ) ( )( )( )
: 169
( ) ( ) ( )7.
( ) ( )( )8.
9.
( ) 10. = 2 = 8 = 0 =
= 5 = 0 = 9= 4 = 1 = = 0 = 1 = 3 = 411. = 3 = 8 12. = 8 = 6 13. = 0 = 4 = 514. = 2 == 3 == 14 = 3= 10 = 415. To 116.
= 3 .= 4 .= 5 .
112
1. ( + )[ ( ) + + ]
170 :
3. = 2 , 0
= 2 3, 2
4. + 1
5.
7.
8.
1 1 82 4 16
3 9 24
4 16 32
5 25 40
= 89. 14 37 75
: 171
4:
A T 125
1. , , , , = , = , =( ) ( ) ,,2.
3. = 1,2,3, … ,11,12= 8,9,10, …4. >
- 129
1. = , = , =4.
5.= , = , = , == , = , = , == 20°
135
1. = = 50°, = = 100°, = = 30°2.
172 :
3.= = == = = = = =
4. - - - - - -
5. )6.
- -
- -
1
1.
2.
3.
4.
1 7
1.
2.
4. 7 49 6. < 33 < 44
{ } =
{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 ÷ =
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