Θέματα Πανελληνιών Εξετάσεων 2012

TPITH 29 MAOY 2012 17

AANTHEI TO MAHMA TN MAHMATIKN ETIKH KAI TEXNOOIKH KATEYYNH - 2012

2012

1. . 253. 2. . 191. 3. . 258.4. ) ) ) ) ) 1. : z x iy= + , x, y R : ( ) ( )2 2 2 2 22 2x 1 y x 1 y 4 x y 1 + + + + = + = : ( )z , z ( )1,0B , ( )A 1,0 (1) :

2 2 2B A AB + = , oBMA 90= . , , . 2 2 2x y 1+ = . 2.

. B2

K2

= , 2

2 2 2OK 1 OK2 2 1

1 12 4 2

= = = 2OK O

1 2OK 2

2 2 = = = .

3. w x iy= + , x, y R . ( )x iy 5 x iy 12 x iy 5x 5yi 12+ = + + =

2 216x 36y 144+ = 2 2x y

19 4

+ = 2 9 = , 2 4 = , 2 5 =

w 3 w 2. 4.z w z w z max w 1 3 4 + + = + =( )z w z w z min w 1 2 1 = A .

1. ( ) ( )f x x 1 ln x 1= , x 0> f : ( ) ( )x ln x x 1

x1

f x ln x (x 1)x

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

x 1> : ln x 0 x ln x 0> > x 1 0 > ( )f x 0 > f 0 x 1< < : x 1 0 < ln x 0< ( )x 1x ln x 0+ < :

2012


2 2 2B A AB + = , oBMA 90= . , , . 2 2 2x y 1+ = . 2.

. B2

K2

= , 2

2 2 2OK 1 OK2 2 1

1 12 4 2

= = = 2OK O

1 2OK 2

2 2 = = = .


2 216x 36y 144+ = 2 2x y

19 4

+ = 2 9 = , 2 4 = , 2 5 =


1. ( ) ( )f x x 1 ln x 1= , x 0> f : ( ) ( )x ln x x 1

x1

f x ln x (x 1)x

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


2012


2 2 2B A AB + = , oBMA 90= . , , . 2 2 2x y 1+ = . 2.

. B2

K2

= , 2

2 2 2OK 1 OK2 2 1

1 12 4 2

= = = 2OK O

1 2OK 2

2 2 = = = .


2 216x 36y 144+ = 2 2x y

19 4

+ = 2 9 = , 2 4 = , 2 5 =


1. ( ) ( )f x x 1 ln x 1= , x 0> f : ( ) ( )x ln x x 1

x1

f x ln x (x 1)x

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


2012


2 2 2B A AB + = , oBMA 90= . , , . 2 2 2x y 1+ = . 2.

. B2

K2

= , 2

2 2 2OK 1 OK2 2 1

1 12 4 2

= = = 2OK O

1 2OK 2

2 2 = = = .


2 216x 36y 144+ = 2 2x y

19 4

+ = 2 9 = , 2 4 = , 2 5 =


1. ( ) ( )f x x 1 ln x 1= , x 0> f : ( ) ( )x ln x x 1

x1

f x ln x (x 1)x

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


2012


2 2 2B A AB + = , oBMA 90= . , , . 2 2 2x y 1+ = . 2.

. B2

K2

= , 2

2 2 2OK 1 OK2 2 1

1 12 4 2

= = = 2OK O

1 2OK 2

2 2 = = = .


2 216x 36y 144+ = 2 2x y

19 4

+ = 2 9 = , 2 4 = , 2 5 =


1. ( ) ( )f x x 1 ln x 1= , x 0> f : ( ) ( )x ln x x 1

x1

f x ln x (x 1)x

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


2012


2 2 2B A AB + = , oBMA 90= . , , . 2 2 2x y 1+ = . 2.

. B2

K2

= , 2

2 2 2OK 1 OK2 2 1

1 12 4 2

= = = 2OK O

1 2OK 2

2 2 = = = .


2 216x 36y 144+ = 2 2x y

19 4

+ = 2 9 = , 2 4 = , 2 5 =


1. ( ) ( )f x x 1 ln x 1= , x 0> f : ( ) ( )x ln x x 1

x1

f x ln x (x 1)x

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


2012


2 2 2B A AB + = , oBMA 90= . , , . 2 2 2x y 1+ = . 2.

. B2

K2

= , 2

2 2 2OK 1 OK2 2 1

1 12 4 2

= = = 2OK O

1 2OK 2

2 2 = = = .


2 216x 36y 144+ = 2 2x y

19 4

+ = 2 9 = , 2 4 = , 2 5 =


1. ( ) ( )f x x 1 ln x 1= , x 0> f : ( ) ( )x ln x x 1

x1

f x ln x (x 1)x

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


2012


2 2 2B A AB + = , oBMA 90= . , , . 2 2 2x y 1+ = . 2.

. B2

K2

= , 2

2 2 2OK 1 OK2 2 1

1 12 4 2

= = = 2OK O

1 2OK 2

2 2 = = = .


2 216x 36y 144+ = 2 2x y

19 4

+ = 2 9 = , 2 4 = , 2 5 =


1. ( ) ( )f x x 1 ln x 1= , x 0> f : ( ) ( )x ln x x 1

x1

f x ln x (x 1)x

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


2012


2 2 2B A AB + = , oBMA 90= . , , . 2 2 2x y 1+ = . 2.

. B2

K2

= , 2

2 2 2OK 1 OK2 2 1

1 12 4 2

= = = 2OK O

1 2OK 2

2 2 = = = .


2 216x 36y 144+ = 2 2x y

19 4

+ = 2 9 = , 2 4 = , 2 5 =


1. ( ) ( )f x x 1 ln x 1= , x 0> f : ( ) ( )x ln x x 1

x1

f x ln x (x 1)x

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


2012


2 2 2B A AB + = , oBMA 90= . , , . 2 2 2x y 1+ = . 2.

. B2

K2

= , 2

2 2 2OK 1 OK2 2 1

1 12 4 2

= = = 2OK O

1 2OK 2

2 2 = = = .


2 216x 36y 144+ = 2 2x y

19 4

+ = 2 9 = , 2 4 = , 2 5 =


1. ( ) ( )f x x 1 ln x 1= , x 0> f : ( ) ( )x ln x x 1

x1

f x ln x (x 1)x

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


2012


2 2 2B A AB + = , oBMA 90= . , , . 2 2 2x y 1+ = . 2.

. B2

K2

= , 2

2 2 2OK 1 OK2 2 1

1 12 4 2

= = = 2OK O

1 2OK 2

2 2 = = = .


2 216x 36y 144+ = 2 2x y

19 4

+ = 2 9 = , 2 4 = , 2 5 =


1. ( ) ( )f x x 1 ln x 1= , x 0> f : ( ) ( )x ln x x 1

x1

f x ln x (x 1)x

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


( )1f , ( )2f . ( ) ( ) ( )1

x 0f 1 , lim f xf

+

= , ( )f 1 1=

( ) ( )x 0 x 0

ln x 1lim f x lim x 1+ + = +=

( )1 1f = .

x 0lim ln x+

= ( )x 0

x 1lim 1 = . ( ) [ )1 1,f = + . ( ) ( ) ( )2

xf 1 , lim f xf

+

= ( ) ( )[ ]

x xlim f x lim x 1 ln x 1+ +

= += ( )xlim x 1+

= + , xlim ln x+

= + ( ) [ )2 1,f = + . f ( ) ( ) [ )1 2 1,f f = + .

2. x 1 2013x e = (1) ( ) ( )x 1 2013 x 1 2013x e ln x ln e x 1 ln x 2013 x 1 ln x 1 2012 = = = =

( )f x 2012 = (2) ( )12012 f ( )111x x 2012: f = ( )22012 f ( )222x x 2012: f = f , (2) 2 . (1) 2 .

3. 0x . ( ) ( )f x f x 2012 0 + = ( ) ( )x x xf x e f x e 2012e 0 + = .

( )1f x 2012= ( )2f x 2012= . ( ) ( ) x xh x f x e 2012e= , [ ]1 2x x ,x . h . h

( ) ( ) ( )x x xh x f x e f x e 2012e = + ( ) ( ) 1 1x x1 1h x f x e 2012e 0= = , ( ) ( ) 2 2x x2 2h x f x e 2012e 0= =

Rolle , h [ ]1 2x ,x , ( )1 20x x , x : ( ) ( ) ( ) ( ) ( )0 0 0x x x0 0 0 0 0x 0 f x e f x e 2012e 0 f x f x 2012h = + = + =

4. , :

( ) ( )e

e e e e2 2 2

11 1 1 1

x x x 1g x dx x 1 ln xdx x ln xdx x ln x x dx2 2 2 x

= = = = e2 2 2 2e2 2

1 1

e xe 0 1 dx2 2

e 2e x e 2e e 1x e 1

42 4 2 4

= = =

=2 2 2e 2e e 4e 33

42 4 4e + =

: .

( ) ( )1

g x dx E M

= 0 M 1< < ( )

M 0E Mlim+

1. ( )f x 0 .

( ) ( )2x x 1 2

1

x xg x f t dt 0e

+ = (1)

( )1f , ( )2f . ( ) ( ) ( )1

x 0f 1 , lim f xf

+

= , ( )f 1 1=

( ) ( )x 0 x 0


( )1 1f = .

x 0lim ln x+

= ( )x 0

x 1lim 1 = . ( ) [ )1 1,f = + . ( ) ( ) ( )2

xf 1 , lim f xf

+

= ( ) ( )[ ]


= += ( )xlim x 1+

= + , xlim ln x+

= + ( ) [ )2 1,f = + . f ( ) ( ) [ )1 2 1,f f = + .


( )f x 2012 = (2) ( )12012 f ( )111x x 2012: f = ( )22012 f ( )222x x 2012: f = f , (2) 2 . (1) 2 .





4. , :

( ) ( )e

e e e e2 2 2

11 1 1 1


= = = = e2 2 2 2e2 2

1 1

e xe 0 1 dx2 2


42 4 2 4

= = =

=2 2 2e 2e e 4e 33

42 4 4e + =

: .

( ) ( )1

g x dx E M

= 0 M 1< < ( )

M 0E Mlim+

1. ( )f x 0 .

( ) ( )2x x 1 2

1

x xg x f t dt 0e

+ = (1)

( )1f , ( )2f . ( ) ( ) ( )1

x 0f 1 , lim f xf

+

= , ( )f 1 1=

( ) ( )x 0 x 0


( )1 1f = .

x 0lim ln x+

= ( )x 0

x 1lim 1 = . ( ) [ )1 1,f = + . ( ) ( ) ( )2

xf 1 , lim f xf

+

= ( ) ( )[ ]


= += ( )xlim x 1+

= + , xlim ln x+

= + ( ) [ )2 1,f = + . f ( ) ( ) [ )1 2 1,f f = + .


( )f x 2012 = (2) ( )12012 f ( )111x x 2012: f = ( )22012 f ( )222x x 2012: f = f , (2) 2 . (1) 2 .





4. , :

( ) ( )e

e e e e2 2 2

11 1 1 1


= = = = e2 2 2 2e2 2

1 1

e xe 0 1 dx2 2


42 4 2 4

= = =

=2 2 2e 2e e 4e 33

42 4 4e + =

: .

( ) ( )1

g x dx E M

= 0 M 1< < ( )

M 0E Mlim+

1. ( )f x 0 .

( ) ( )2x x 1 2

1

x xg x f t dt 0e

+ = (1)

g ( ) ( )2x x 1

1f t dtx

+ =

2x xe .

( )x ( )x1

f t dt 2x x 1 + .

g ( ) ( ) ( )2 1 2xg x f x x 1 2x 1e = + .

( )g 0 0= ( )g 1 0= . (1): ( ) ( )g x g 0 ( ) ( )g x g 1 . g 0x 0= , 0x 1=

Fermat ( )0 0g = ( )1 0g = . ( )( ) 1f 1 1 0e

= ( ) 1f 1 1 0e =

( ) 1f 1e

= ( ) 1f 1e

= . ( )f x 0< .

: ( ) ( )x

1

ln t tln x x dt e f xf t

= + (3)

ln x x 1 x < (3) , (3) ( )

( )x

1

ln x xf x

ln t t dt ef t

= +.

f

. (3): ( ) ( )x

1

ln x xf x

ln t tdt e

f t = + . ( ) ( )

x

1

ln t th x dt

f t= ,

( )h 1 0= : ( ) ( ) ( ) ( ) ( ) ( )x x 1 xe e e e eh x h x h x h x h x h x + = = =( )( ) ( )x 1 xh x e e = ( ) x 1 xh x e e c = + ( ) 1 x x xxh e e ce= + ( ) xxh e ce= +( )h 1 0 e ce 0 c 1= + = =

( ) ( ) ( ) ( ) ( )xx x x x

1x t x

ln x xln t th e e d e e e f e ln x xf t f x

= = = =

2. ( ) ( )xx 0 x 0lim f x lim e ln x x+ +

= =

( )f x = x 0+

( ) ( ) ( )2 2

x 0

11 1

lim f x f x lim lim1f x+

= =

t1 = : 2t 0 t 0 t 0 t 0

t t t 11 t 1 tlim lim lim lim 0

t t t 2t 2t = = = =

3. ( ) ( )x f xF = ( ) ( )( ) ( )x xx 1x e ln x x e e 1x

F ln x x = = = +

x x1 1e ln x x 1 e x 1 ln x 0x x

= + + = + > ( )x 0F > F x 0> . [ ]x,2x , [ ]2x,3x F , ( ) ( ) ( ) ( )11

F 2x F xx,2x : F

x =

g ( ) ( )2x x 1

1f t dtx

+ =

2x xe .

( )x ( )x1

f t dt 2x x 1 + .

g ( ) ( ) ( )2 1 2xg x f x x 1 2x 1e = + .

( )g 0 0= ( )g 1 0= . (1): ( ) ( )g x g 0 ( ) ( )g x g 1 . g 0x 0= , 0x 1=

Fermat ( )0 0g = ( )1 0g = . ( )( ) 1f 1 1 0e

= ( ) 1f 1 1 0e =

( ) 1f 1e

= ( ) 1f 1e

= . ( )f x 0< .

: ( ) ( )x

1


= + (3)

ln x x 1 x < (3) , (3) ( )

( )x

1

ln x xf x

ln t t dt ef t

= +.

f

. (3): ( ) ( )x

1

ln x xf x

ln t tdt e

f t = + . ( ) ( )

x

1

ln t th x dt

f t= ,


( ) ( ) ( ) ( ) ( )xx x x x

1x t x


= = = =


= =

( )f x = x 0+

( ) ( ) ( )2 2

x 0

11 1


= =

t1 = : 2t 0 t 0 t 0 t 0


t t t 2t 2t = = = =


F ln x x = = = +


= + + = + > ( )x 0F > F x 0> . [ ]x,2x , [ ]2x,3x F , ( ) ( ) ( ) ( )11

F 2x F xx,2x : F

x =

g ( ) ( )2x x 1

1f t dtx

+ =

2x xe .

( )x ( )x1

f t dt 2x x 1 + .

g ( ) ( ) ( )2 1 2xg x f x x 1 2x 1e = + .

( )g 0 0= ( )g 1 0= . (1): ( ) ( )g x g 0 ( ) ( )g x g 1 . g 0x 0= , 0x 1=

Fermat ( )0 0g = ( )1 0g = . ( )( ) 1f 1 1 0e

= ( ) 1f 1 1 0e =

( ) 1f 1e

= ( ) 1f 1e

= . ( )f x 0< .

: ( ) ( )x

1


= + (3)

ln x x 1 x < (3) , (3) ( )

( )x

1

ln x xf x

ln t t dt ef t

= +.

f

. (3): ( ) ( )x

1

ln x xf x

ln t tdt e

f t = + . ( ) ( )

x

1

ln t th x dt

f t= ,


( ) ( ) ( ) ( ) ( )xx x x x

1x t x


= = = =


= =

( )f x = x 0+

( ) ( ) ( )2 2

x 0

11 1


= =

t1 = : 2t 0 t 0 t 0 t 0


t t t 2t 2t = = = =


F ln x x = = = +


= + + = + > ( )x 0F > F x 0> . [ ]x,2x , [ ]2x,3x F , ( ) ( ) ( ) ( )11

F 2x F xx,2x : F

x =

g ( ) ( )2x x 1

1f t dtx

+ =

2x xe .

( )x ( )x1

f t dt 2x x 1 + .

g ( ) ( ) ( )2 1 2xg x f x x 1 2x 1e = + .

( )g 0 0= ( )g 1 0= . (1): ( ) ( )g x g 0 ( ) ( )g x g 1 . g 0x 0= , 0x 1=

Fermat ( )0 0g = ( )1 0g = . ( )( ) 1f 1 1 0e

= ( ) 1f 1 1 0e =

( ) 1f 1e

= ( ) 1f 1e

= . ( )f x 0< .

: ( ) ( )x

1


= + (3)

ln x x 1 x < (3) , (3) ( )

( )x

1

ln x xf x

ln t t dt ef t

= +.

f

. (3): ( ) ( )x

1

ln x xf x

ln t tdt e

f t = + . ( ) ( )

x

1

ln t th x dt

f t= ,


( ) ( ) ( ) ( ) ( )xx x x x

1x t x


= = = =


= =

( )f x = x 0+

( ) ( ) ( )2 2

x 0

11 1


= =

t1 = : 2t 0 t 0 t 0 t 0


t t t 2t 2t = = = =


F ln x x = = = +


= + + = + > ( )x 0F > F x 0> . [ ]x,2x , [ ]2x,3x F , ( ) ( ) ( ) ( )11

F 2x F xx,2x : F

x =

g ( ) ( )2x x 1

1f t dtx

+ =

2x xe .

( )x ( )x1

f t dt 2x x 1 + .

g ( ) ( ) ( )2 1 2xg x f x x 1 2x 1e = + .

( )g 0 0= ( )g 1 0= . (1): ( ) ( )g x g 0 ( ) ( )g x g 1 . g 0x 0= , 0x 1=

Fermat ( )0 0g = ( )1 0g = . ( )( ) 1f 1 1 0e

= ( ) 1f 1 1 0e =

( ) 1f 1e

= ( ) 1f 1e

= . ( )f x 0< .

: ( ) ( )x

1


= + (3)

ln x x 1 x < (3) , (3) ( )

( )x

1

ln x xf x

ln t t dt ef t

= +.

f

. (3): ( ) ( )x

1

ln x xf x

ln t tdt e

f t = + . ( ) ( )

x

1

ln t th x dt

f t= ,


( ) ( ) ( ) ( ) ( )xx x x x

1x t x


= = = =


= =

( )f x = x 0+

( ) ( ) ( )2 2

x 0

11 1


= =

t1 = : 2t 0 t 0 t 0 t 0


t t t 2t 2t = = = =


F ln x x = = = +


= + + = + > ( )x 0F > F x 0> . [ ]x,2x , [ ]2x,3x F , ( ) ( ) ( ) ( )11

F 2x F xx,2x : F

x =

g ( ) ( )2x x 1

1f t dtx

+ =

2x xe .

( )x ( )x1

f t dt 2x x 1 + .

g ( ) ( ) ( )2 1 2xg x f x x 1 2x 1e = + .

( )g 0 0= ( )g 1 0= . (1): ( ) ( )g x g 0 ( ) ( )g x g 1 . g 0x 0= , 0x 1=

Fermat ( )0 0g = ( )1 0g = . ( )( ) 1f 1 1 0e

= ( ) 1f 1 1 0e =

( ) 1f 1e

= ( ) 1f 1e

= . ( )f x 0< .

: ( ) ( )x

1


= + (3)

ln x x 1 x < (3) , (3) ( )

( )x

1

ln x xf x

ln t t dt ef t

= +.

f

. (3): ( ) ( )x

1

ln x xf x

ln t tdt e

f t = + . ( ) ( )

x

1

ln t th x dt

f t= ,


( ) ( ) ( ) ( ) ( )xx x x x

1x t x


= = = =


= =

( )f x = x 0+

( ) ( ) ( )2 2

x 0

11 1


= =

t1 = : 2t 0 t 0 t 0 t 0


t t t 2t 2t = = = =


F ln x x = = = +


= + + = + > ( )x 0F > F x 0> . [ ]x,2x , [ ]2x,3x F , ( ) ( ) ( ) ( )11

F 2x F xx,2x : F

x =

( ) ( ) ( ) ( )22F 3x F 2x

2x,3x : Fx = . 1 2 < , F . ,

( ) ( )1 2F F < ( ) ( ) ( ) ( ) ( ) ( ) ( )F 2x F x F 3x F 2xx x 2F 2x F x F 3x < > +

4. ( ) ( ) ( ) ( )x F F 3 2F xh = + , [ ]x ,2 . h . ( ) ( ) ( ) ( ) ( ) ( )F F 3 2F F 3 F 0h = + = < ( ) ( )F x f x 0 = < F . ( ) ( ) ( ) ( )2 F F 3 2F 2 0h = + > Bolzano

( ), 2 : ( ) ( ) ( ) ( )0 F F 3 2Fh = + = . h , ( ) ( )h x 2f x = . ,

( ) ( ) ( ) ( )22F 3x F 2x

2x,3x : Fx = . 1 2 < , F . ,

( ) ( )1 2F F < ( ) ( ) ( ) ( ) ( ) ( ) ( )F 2x F x F 3x F 2xx x 2F 2x F x F 3x < > +

4. ( ) ( ) ( ) ( )x F F 3 2F xh = + , [ ]x ,2 . h . ( ) ( ) ( ) ( ) ( ) ( )F F 3 2F F 3 F 0h = + = < ( ) ( )F x f x 0 = < F . ( ) ( ) ( ) ( )2 F F 3 2F 2 0h = + > Bolzano

( ), 2 : ( ) ( ) ( ) ( )0 F F 3 2Fh = + = . h .

,

2012


2 2 2B A AB + = , oBMA 90= . , , . 2 2 2x y 1+ = . 2.

. B2

K2

= , 2

2 2 2OK 1 OK2 2 1

1 12 4 2

= = = 2OK O

1 2OK 2

2 2 = = = .


2 216x 36y 144+ = 2 2x y

19 4

+ = 2 9 = , 2 4 = , 2 5 =


1. ( ) ( )f x x 1 ln x 1= , x 0> f : ( ) ( )x ln x x 1

x1

f x ln x (x 1)x

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


2012


2 2 2B A AB + = , oBMA 90= . , , . 2 2 2x y 1+ = . 2.

. B2

K2

= , 2

2 2 2OK 1 OK2 2 1

1 12 4 2

= = = 2OK O

1 2OK 2

2 2 = = = .


2 216x 36y 144+ = 2 2x y

19 4

+ = 2 9 = , 2 4 = , 2 5 =


1. ( ) ( )f x x 1 ln x 1= , x 0> f : ( ) ( )x ln x x 1

x1

f x ln x (x 1)x

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


( )1f , ( )2f . ( ) ( ) ( )1

x 0f 1 , lim f xf

+

= , ( )f 1 1=

( ) ( )x 0 x 0


( )1 1f = .

x 0lim ln x+

= ( )x 0

x 1lim 1 = . ( ) [ )1 1,f = + . ( ) ( ) ( )2

xf 1 , lim f xf

+

= ( ) ( )[ ]


= += ( )xlim x 1+

= + , xlim ln x+

= + ( ) [ )2 1,f = + . f ( ) ( ) [ )1 2 1,f f = + .


( )f x 2012 = (2) ( )12012 f ( )111x x 2012: f = ( )22012 f ( )222x x 2012: f = f , (2) 2 . (1) 2 .





4. , :

( ) ( )e

e e e e2 2 2

11 1 1 1


= = = = e2 2 2 2e2 2

1 1

e xe 0 1 dx2 2


42 4 2 4

= = =

=2 2 2e 2e e 4e 33

42 4 4e + =

: .

( ) ( )1

g x dx E M

= 0 M 1< < ( )

M 0E Mlim+

1. ( )f x 0 .

( ) ( )2x x 1 2

1

x xg x f t dt 0e

+ = (1)

TPITH 29 MAOY 201216 2012 ( )

1. , , () . , , , . ( ) , (). , , , , . . , . 1. , () . . , , . , ( ) : , . , . ( ). , , : ( ). , , , , ( ). , ( ). , , , : , ( ). , , , . , , , . 2.. , , . : ( - < h -

TPITH 22 MAOY 2012 15

ENEIKTIKE AANTHEI TO MAHMA TH NEOEHNIKH A ENIKH AIEIA & EA.. B

, EMPO MEOIKO , . O .

1

& ..

1.

. , , ,

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EMTH 24 MAOY 2012 15

AANTHEI TO MAHMA TN MAHMATIKN & TOIXEIN TATITIKH ENIKH AIEIA - 2012

AANTHEI TO MAHMA TH BIOOIA ENIKH AIEIA - YKEIOY 2012

2012

A 1. , . 31 A2. , . 148A3. , . 96A4. ) ) ) ) )

1. ,

50% y y x x .

25. 25 = .

2. 50-50 . 1 2 3 4 + = + 4 3 6 2 8 2 + + = + + 8 2 4 6 8 = + = .

: () ix iv if % iN iF %

[ )5,15 10 12 20 12 20[ )15,25 20 18 30 30 50[ )25,35 30 24 40 54 90[ )35,45 40 6 10 60 100 60 100

3. 1 1 2 2 3 3 4 41 2 3 4

x x x x 10 12 20 18 30 24 40 6x

60 + + + + + + = = = + + +

12 36 72 242 6 12 4 246

+ + + = + + + ==

( ) ( ) ( ) ( )2 2 222 1 1 2 2 3 3 4 4S x x x x x x x x+ + +=

( ) ( ) ( ) ( )2 2 2 22 10 60 20 60 30 60 40 60S 6012 18 24 6 + + + =

2S 84 S 9,17= =

4. . x %

45 35 10 x 845 37 x = =

2012

A 1. , . 31 A2. , . 148A3. , . 96A4. ) ) ) ) )

1. ,

50% y y x x .

25. 25 = .

2. 50-50 . 1 2 3 4 + = + 4 3 6 2 8 2 + + = + + 8 2 4 6 8 = + = .


[ )5,15 10 12 20 12 20[ )15,25 20 18 30 30 50[ )25,35 30 24 40 54 90[ )35,45 40 6 10 60 100 60 100

3. 1 1 2 2 3 3 4 41 2 3 4

x x x x 10 12 20 18 30 24 40 6x

60 + + + + + + = = = + + +

12 36 72 242 6 12 4 246

+ + + = + + + ==

( ) ( ) ( ) ( )2 2 222 1 1 2 2 3 3 4 4S x x x x x x x x+ + +=

( ) ( ) ( ) ( )2 2 2 22 10 60 20 60 30 60 40 60S 6012 18 24 6 + + + =

2S 84 S 9,17= =

4. . x %

45 35 10 x 845 37 x = =

2012

A 1. , . 31 A2. , . 148A3. , . 96A4. ) ) ) ) )

1. ,

50% y y x x .

25. 25 = .

2. 50-50 . 1 2 3 4 + = + 4 3 6 2 8 2 + + = + + 8 2 4 6 8 = + = .


[ )5,15 10 12 20 12 20[ )15,25 20 18 30 30 50[ )25,35 30 24 40 54 90[ )35,45 40 6 10 60 100 60 100

3. 1 1 2 2 3 3 4 41 2 3 4

x x x x 10 12 20 18 30 24 40 6x

60 + + + + + + = = = + + +

12 36 72 242 6 12 4 246

+ + + = + + + ==

( ) ( ) ( ) ( )2 2 222 1 1 2 2 3 3 4 4S x x x x x x x x+ + +=

( ) ( ) ( ) ( )2 2 2 22 10 60 20 60 30 60 40 60S 6012 18 24 6 + + + =

2S 84 S 9,17= =

4. . x %

45 35 10 x 845 37 x = =

2012

A 1. , . 31 A2. , . 148A3. , . 96A4. ) ) ) ) )

1. ,

50% y y x x .

25. 25 = .

2. 50-50 . 1 2 3 4 + = + 4 3 6 2 8 2 + + = + + 8 2 4 6 8 = + = .


[ )5,15 10 12 20 12 20[ )15,25 20 18 30 30 50[ )25,35 30 24 40 54 90[ )35,45 40 6 10 60 100 60 100

3. 1 1 2 2 3 3 4 41 2 3 4

x x x x 10 12 20 18 30 24 40 6x

60 + + + + + + = = = + + +

12 36 72 242 6 12 4 246

+ + + = + + + ==

( ) ( ) ( ) ( )2 2 222 1 1 2 2 3 3 4 4S x x x x x x x x+ + +=

( ) ( ) ( ) ( )2 2 2 22 10 60 20 60 30 60 40 60S 6012 18 24 6 + + + =

2S 84 S 9,17= =

4. . x %

45 35 10 x 845 37 x = =

2012

A 1. , . 31 A2. , . 148A3. , . 96A4. ) ) ) ) )

1. ,

50% y y x x .

25. 25 = .

2. 50-50 . 1 2 3 4 + = + 4 3 6 2 8 2 + + = + + 8 2 4 6 8 = + = .


[ )5,15 10 12 20 12 20[ )15,25 20 18 30 30 50[ )25,35 30 24 40 54 90[ )35,45 40 6 10 60 100 60 100

3. 1 1 2 2 3 3 4 41 2 3 4

x x x x 10 12 20 18 30 24 40 6x

60 + + + + + + = = = + + +

12 36 72 242 6 12 4 246

+ + + = + + + ==

( ) ( ) ( ) ( )2 2 222 1 1 2 2 3 3 4 4S x x x x x x x x+ + +=

( ) ( ) ( ) ( )2 2 2 22 10 60 20 60 30 60 40 60S 6012 18 24 6 + + + =

2S 84 S 9,17= =

4. . x %

45 35 10 x 845 37 x = =

8%

1. : : : ( )

23

P1

= +

, ( )2

2P

1+ = +

, ( )2

1P

1+ = +

:

( ) ( ) [ ]( ) ( )

2 2

2x 1 x 1 2P

x 3 2

2 x 3 2 2 x 3 4lim lim

x x x x 1 =

+ ++ + = =+ +

( ) ( )( ) ( )

( )( )x 1 x 12 2

x 1 x 1 x 1 411 4x 3 2 x 3 2

2 2lim lim

x x 1 x + = + + + +

= = =+

.

2. ( ) ( ) ( ) ( )P 1 P P I P 1 = + =

22 2 2 23 2 1 3 1

1 11 1 1 1

3 3 + + ++ = = + + + +

= =3. ( ) ( ) :

( ) ( )( ) ( ) ( )P P P I = + ( ) ( ) ( ) ( )P P P I P I= + =( ) ( ) ( )P P I 2P I + =

2 2 23 2 1

21 1 1

+ += + + + +

3 = ( ) ( )( ) 3P 5 =

4. ( ) 4 2P 10 5 = =

( ) ( )( ) ( ) ( ) ( )2 2

P 25 532 160 80 = = = = =

1. ( )( )

( ) ( )2

22

2 2 22ln x ln x 1 ln x 1

0x x

12 ln x x 1 ln x 1

xf x f xx

= +

= =

( )f 0, +2. ( ) ( ) ( ) 2 xE x xf x E x 1 ln= = + , x 0>( ) 2ln x

xE x = , ( )E x 0 ln x 0 x 1 = = =( )E x 0 x 1 > > ( )E x 0 x 1 < ( ) 2ln x

xE x = , ( )E x 0 ln x 0 x 1 = = =( )E x 0 x 1 > > ( )E x 0 x 1 < ( ) 2ln x

xE x = , ( )E x 0 ln x 0 x 1 = = =( )E x 0 x 1 > > ( )E x 0 x 1

Θέματα Πανελληνιών Εξετάσεων 2012

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