ΜΑΘΗΜΑΤΙΚΑ ΚΑΤΕΥΘΥΝΣΗΣ Γ ΛΥΚΕΙΟΥ ΠΑΝΕΛΛΗΝΙΕΣ ΕΞΕΤΑΣΕΙΣ ΕΝΙΑΙΟΥ ΛΥΚΕΙΟΥ ΕΤΟΥΣ 2007 ΛΥΣΕΙΣ – ΑΠΑΝΤΗΣΕΙΣ | www.lazaridi.info | tel. 6977.385.358 | www.lazaridi.info Page 1 of 3 | ΜΑΘΗΜΑΤΙΚΑ ΚΑΤΕΥΘΥΝΣΗΣ Γ ΛΥΚΕΙΟΥ | ΕΝΙΑΙΟ 2007 | ΛΥΣΕΙΣ | |ΘΕΜΑ 1ο Α1. θεωρία A2. θεωρία A3. θεωρία B. Λ / Λ / Λ / Σ / Σ |ΘΕΜΑ 2ο i a ai z 2 2 + + = Α. Αρκεί νδο 1 | | = z Πράγματι, 1 4 4 | 2 | | 2 | 2 2 | | 2 2 = + + = + + = + + = a a i a ai i a ai z Β. για 0 = a είναι i i i i z - = - = = = 1 1 2 2 1 και για 2 = a είναι 1 2 2 2 2 2 = + + = i i z Ι) 2 | 1 | | | 2 1 = - - = - i z z ΙΙ) ( ) v v v v v v z i i i z ) ( ) 1 ( ) ( ) ( ) ( ) ( 2 2 2 2 2 1 - = - = = - = - = |ΘΕΜΑ 3ο θ ημ 2 3 2 3 ) ( - - = x x x f Α. • 3 3 ) ( 2 - = ′ x x f • 1 1 0 3 3 0 ) ( 2 2 ± = ⇒ = ⇒ = - ⇒ = ′ x x x x f • x x f 6 ) ( = ′ ′ • 0 0 6 0 ) ( = ⇒ = ⇒ = ′ ′ x x x f - + 1 x () ) 1 ( , 1 min f 1 - + ( ) ) 1 ( , 1 max - - f ) ( x f ′ ) ( x f - + 0 ) ( x f ′ ′ ) ( x f x ( ) ) 0 ( , 0 . . f Κ Σ

Μαθηματικά Κατεύθυνσης Απαντήσεις Θέματων Πανελληνίων 2007

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2007

| www.lazaridi.info | tel. 6977.385.358 |

www.lazaridi.info Page 1 of 3

| | 2007 | |

| 1

1. A2. A3. B. / / / /

| 2

iaai

z2

2++

=

.

1|| =z ,

14

4|2||2|

22||

2

2=

+

+=

++

=++

=a

a

iaai

iaai

z

.

0=a iiii

z ====

11

22

1 2=a 12222

2 =++

=ii

z

) 2|1||| 21 == izz ) ( ) vvvvvv ziiiz )()1()()()()( 222221 =====

| 3

23 23)( = xxxf

.

33)( 2 = xxf 110330)( 22 ==== xxxxf

xxf 6)( = 0060)( === xxxf

+

1x

( ))1(,1min

f

1

+

( ))1(,1max

f

)( xf

)( xf

+

0)( xf

)( xf

x

( ))0(,0..

f
2007

| www.lazaridi.info | tel. 6977.385.358 |

www.lazaridi.info Page 2 of 3

.

f )1,(1 =A ( ) )2,()22,()(lim),(lim 22

11 ==

=

xfxfAf

xx

( )10 Af f 1A 0)( =xf 1A

f ]1,1[2 =A ( ) [ ] ]2),1(2[]22,22[)1(),1( 22222 +=== ffAf

( )20 Af f 2A 0)( =xf 2A

f ),1(3 +=A ( ) )),1(2(),22()(lim),(lim 22

13++=+=

=

+xfxfAf

xx

( )30 Af f 3A 0)( =xf 3A

0)( =xf pi

.

( ))1(,1 fA ( ))1(,1 fB ( ))0(,0 f

( ) 222,1 A ( ) 222,1 B ( ) 22,0

, , pi 222 = xy

.

1,00)1(02223)( 23223 ====== xxxxxxxxxyxf

21

.....)()(1

0

30

1

3 ==+=

dxxxdxxxE ..
2007

| www.lazaridi.info | tel. 6977.385.358 |

www.lazaridi.info Page 3 of 3

| 4

.

0)( >xg , ]1,0[x f ]1,0[ 0)()()0(0)()0(0 >