oefe aalg 2006-2014 problems (without logos).pdf
22
ΑΛΓΕΒΡΑ Α’ ΛΥΚΕΙΟΥ ΕΠΑΝΑΛΗΠΤΙΚΑ ΘΕΜΑΤΑ (ΠΡΟΣΟΜΟΙΩΣΗΣ ΕΞΕΤΑΣΕΩΝ) Ο.Ε.Φ.Ε. 2006-2014 οι εκφωνήσεις (χωρίς υδατογραφήματα) επιμέλεια: parmenides51 δημιουργός των μαθηματικών ιστοσελίδων: για την αγάπη των μαθηματικών για τους ρομαντικούς της Γεωμετρίας
-
Upload
evaggelia-lamproysi -
Category
Documents
-
view
12 -
download
3
Transcript of oefe aalg 2006-2014 problems (without logos).pdf
-
( )
.... 2006-2014
( )
: parmenides51
:
-
1A()
1
. >0 xx 13
. x1 x2 x2+x+=0, 0. :i) x1+x2=-
ii) xx 21 =
12 2
(1): 4x2y ++=(2): 15x12y +=
. (1) (2) 12
. =3 :i) (1) yy (2) xx
M 8ii)
5
-
2 3
2x3x1x)x(f 2
2
+
=
. f 7
. 9
. : 20032006
220053200512005
2
2=
+
9
4
(+2)x2-2x-1=0 2 R (1) . 2 .
8 . x1,x2 (1) :
i) x1+x2 x1:x2 4
ii) (1) 6
iii) x1+x2
-
11
: .. =
!" #$ % = .......................
&
'(
)
(*+,- .,-
" + = + % -x * Rx/ 0,12%-3 4/3135& a2 = * Ra 3"36,"4-,"46-,46-6#$
"$
-
22
/
-
33
8%" 6),-2,-
&% 6 (
/
R *@
: ( ) 12 += xxxf , Rx .
&
' 8$ 6*@ 6
(A(.
&
-
11
A'
1
. x1 x2 2 0, 0ax x + + = . :
i. x1 + x2 = - ii. x1 x2 =
(9 ). :
i. 1 :y =2x+5 2: y=x+2008 :. =5. =2008. = - 1
2. =2
ii. x25x+=0 2 :. =6. =0. = 2. = -6
iii. D=0 Dx=Dy=5 :. . . (x,y) = (0,0). (x,y) = (5,5)
(6 )
-
22
. () () :i. x 0 |x|=xii. x2+x1 =0 IRiii. 2 = 2( ) , IRiv. - = , > > 0v. xy = x2 x = y , x, y IR
(10 )
2
:f(x) =
3
2
42
x x
x x
+
A. .(10 )
. := (3) (1)(4) 2
f ff
(8 ). |f(4) x 1| = |2 f(3) x|
(7 )
3
x2 (+1)x + = 0i. .
(8 )ii. x1, x2 (x1+x2)2 - 2x1x2 =10
(8 )iii. =3, 2 2x1 2x2.
(9 )
-
1
1
'
1 . .
( 5) . 0, a , : =
( 10) . ()
().
) , : +=+ . ) f
.
) x2 + x + = 0, 0a , 0a .
) 0 , >>a . ( 10)
-
2
2
3 : 01)1(2 =++ xx , x1 x2. ) 21
( 7) ) .
( 6) )
21 xx += , 21 xx = , 21
11xx
+= ( 6)
) : 533 21221221 =+++ xxxxxx
( 6) 4
+= 52,
25,52)(
xxxaxxf ,a
: )4()2( ff = )1()2( = ff ) = 1 = 5.
( 7) )
(1): y = (4 +2) x + f(1) (2): y = f(-3) + (132 -34) x ,
( 8)
) f :
f(x) = 1
( 10)
-
1
1
' .
1 . > 0 |x| < < x < .
10 . (x1 , y1)
(x2, y2). , . 5
. , , . ) , R , : | | = | |.
2 ) < 0 , x2 + x +
x2 + x+ = (x x1)(x x2) , x1, x2 . 2
) = , R . 2
) > 0 , : = . 2
) x > 0, 2x 1
x= .
2 2 1 2
1: y = ( 2)x + 1, 2: y = 2 x 14
) 1 2
. 10
) 1 2 .
15
-
2
2
3 f f(x) = x4 x2 + 2, x R ,
2 1 2 12 1 2 1+
= + +
. ) = 6.
8 ) N f(1).
2 ) : f(x) = f(1).
8 ) : f(x) f(1) 0.
7
-
1
1
'
1 . 2x x 0,,, R, 0+ + =
1 2x ,x , : 1 2 x x = . 10
. f , ;
5 . ,
, , , . i) 0> x x < < < . ii) 0 , : = . iii) < , : 2( ) = . iv) , , = , :
( = = 0). v) f f(x) = (xc), c > 0,
c .
10 2 ) : 2x 4x 3 0 + = .
5 ) : 2x 6x 8 0 + < .
8 ) : 10 2 2(x 1)(x 6x 8)(x 4x 3) 0+ + + .
12
-
2
2
3 2x - x + 3 = 0 , R , x1, x2. ) < 0 > 12.
8 ) = 4 :
i) x1, x2 . 4
ii) x2 , 2x 2011 x+ .
6 iii) x1 , 3 1 1x x 2= .
7
4 f(x) = 1 x + 3
2 , , x ,
y = 1 x + 32
. )
y = 1 x + 32
xx 45 .
8 ) = 3
2:
i) f xx, yy .
8 ii) f .
5 iii) , 2f ( ) f ( 1)> .
4
-
A.1. ,
: P(A) P(B). 10
.2. . 5
.3. , . . ,
: =
. , IR : ( )2 = ( )2. . ,
. . x = , < 0 ,
| | . x2 +x + > 0 >0
-
.2. x f(x) = 3. 9
.3. IR : ( )18 3x f 2012 8
.1. 2x 1 | | x+ = , IR 1. IR ,
x .
8 2. IR :
x 1= , , , 3 2.
7 2.
1: y = (2 4) x + + 1, IR 2: y = (2 + 4 3) x +2, IR IR , 1,2 xx, .
10
(), * , = 2 : 7 = 11 f(x) = 1x2 + 4x + 1, 1 4, . .1. 1 4.
8
-
.2. 1 = 1 4 = 5 x1, x2 f(x) = 0, : ) 2 21 2 2 1x x x x = +
4 ) 1 2
2 1
x xBx x= +
4 ) 3 1 2 1 2400 (x x ) 2012 x x 12 = + +
4 .3. : 2| x B 2 | | x A | + = , , ,
.2. 5
-
1. pi , , pi :
| | = | | | |
9
2. . > 0, , pi
;
3
. pi ;
3
3. pi pi ,
pi pi , pi , , pi .
. > >0, > .
2
. pi :2a =.
2
. = 0 0, x + = 0 .
2
. x IR , : x x .
2
. x2 + x + = 0, 0, :
x1, x2, , x2 + x + = (x x1)(x x2).
2
-
pi: 3 3A 4 2 2= 1 1
B = +
2+ 2 22
.
1. pi = 2.
10 2. pi = 2.
8
3. 31 1
x = +
A+ A A.
7
: ( ) ( )2y = | 3| 1 x + +2|| 3 , IR . pi :
1. pi y = x;
7 2. xx;
8 3. pi (0, 0) ;
10
24x 4x+5 , IR .
1. pi .
7 2. pi:
. .
3
. 2f (x)= 4x 4x+5 pi IR .
4 3. pi , pi
x1, x2 x1+ x2= x1 x21.
5
4. pi , pi IRx :
2 2 24x 4P(A)x+5P(A) 4x 4P(A)x+5P(A) 4x 4P()x+5P() 0
6
-
1. pipi pi pi () () .
) , , < 0. 2
) , + + .
2 ) y = x xy x'y' .
2
) (,) pipi, pi ' (,).
2 ) pi , .
2
2. , 0, pi = .
15
, , pi pi
pi, ( )1
2 = , ( )
2B'
3 = ( )
1A B
3 = .
1. pi pi ( )B ( 4) ( )A B ( 5). 9
2. pi pi ( )A B . 8
-
3. ( )1
3 = N(B) = 40, pi ().
8
pi 31
( 1) 8 = 6
13 = .
1. pi 1 ( 5) pi ( 4).
9
1=3 = 2, :
2. pi pi pi, pi pi, pi 440.
8
3. 2 2 22 3 5
x , x , 2x ,
pi, 1 , x pi.
8
( )2x x 0 1 + = , pi . 1. ( 5) pi (1)
( 3). 8
= 5,
( ) ( )2 1 2 1 2g x x 2 x x x 5(x x )= + + , ( )2
2x 3x 1f x
x 1
+=
,
pi 1 2
x , x ( )1 .
2. ) pi ( )g x x 5= . 5
) pi f ( 3) pipi pi ( 4).
7 )
fC
gC .
5
exofullo200620072008200920102011201220132014
