Θέματα Πανελληνιών Εξετάσεων 2012

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TPITH 29 MAΪOY 2012 17 AΠANTHΣEIΣ ΣTO MAΘHMA TΩN MAΘHMATIKΩN ΘETIKHΣ KAI TEXNOΛOΓIKHΣ KATEYΘYNΣHΣ - 2012 ΘΕΜΑ Α Α1. Θεωρία σχολικού βιβλίου σελ. 253. Α2. Θεωρία σχολικού βιβλίου σελ. 191. ΘΕΜΑ Β Β1.α΄ τρόπος: Αν z x iy = + , x, y R τότε: ( ) ( ) 2 2 2 2 2 2 2 x1 y x1 y 4 x y 1 + + + + = + = β΄ τρόπος: Έστω ( ) z Μ , η εικόνα του z και ( ) 1,0 B , ( ) A 1,0 η (1) γράφεται: 2 2 2 B A AB Μ = η γωνία του τριγώνου ΑΒΜ που είναι απέναντι από την ΑΒ είναι ορθή, άρα o BMA 90 = . Εποένως το ΑΒ φαίνεται από το Μ, υπό ορθή γωνία, οπότε ο γεωετρικός τόπος του Μ είναι ο κύκλος διαέτρου ΑΒ. Άρα 2 2 2 x y 1 + = . Β2. Το ΟΑΓΒ είναι ρόβος οπότε οι διαγώνιες τένονται κάθετα στο Κ και διχοτοούνται. Το B 2 K 2 = , άρα 2 2 2 2 OK 1 OK 2 2 1 1 1 2 4 2 = = = 2 OK O 1 2 OK 2 2 2 Γ= = = . Β3. Έστω w x iy = + , x, y R . ( ) x iy 5 x iy 12 x iy 5x 5yi 12 + = + + = 2 2 16x 36y 144 + = 2 2 x y 1 9 4 + = 2 9 α = , 2 4 β = , 2 5 γ = Η έγιστη τιή του w είναι 3 και η ελάχιστη τιή του w είναι 2. Β4. Στο διπλανό σχήα φαίνεται ότι η έγιστη απόσταση A′Λ ή ΚΑ και ελάχιστη ΣΒ ή ΣΒ . ΘΕΜΑ Γ Γ1. ( ) ( ) f x x 1 lnx 1 = , x 0 > οπότε: () ( ) x ln x x1 x 1 f x ln x (x 1) x + = + = . Είναι ( ) f 1 0 = . Αν x 1 > τότε : ln x 0 x ln x 0 > > και x 1 0 > άρα ( ) f x 0 > οπότε η f Αν 0 x 1 < < τότε: x 1 0 < και ln x 0 < άρα ( ) x1 x ln x 0 + < Τα παραπάνω φαίνονται στον παρακάτω πίνακα: Α3. Θεωρία σχολικού βιβλίου σελ. 258. Α4. α) Σ β) Σ γ) Λ δ) Λ ε) Λ ΘΕΜΑ Β Για το πεδίο τιών θα βρούε τα ( ) 1 f , ( ) 2 f . ( ) () ( ) 1 x 0 f 1, lim f x f + = , ( ) f1 1 =− ( ) ( ) x 0 x 0 ln x 1 lim f x lim x 1 + + = +∞ = αφού Ολικό ελάχιστο το ( ) 1 1 f =− . x 0 lim ln x + = −∞ και ( ) x 0 x1 lim 1 =− .Άρα ( ) [ ) 1 1, f = +∞ . Το ( ) () ( ) 2 x f 1 , lim f x f →+∞ = ( ) ( ) [ ] x x lim f x lim x 1 ln x 1 →+∞ →+∞ = +∞ = αφού ( ) x lim x 1 →+∞ = +∞ , x lim ln x →+∞ = +∞ άρα ( ) [ ) 2 1, f =− +∞ . Οπότε το σύνολο τιών της f είναι ( ) ( ) [ ) 1 2 1, f f = +∞ . Γ2. x1 2013 x e = (1) ( ) ( ) x1 2013 x1 2013 x e ln x ln e x 1 lnx 2013 x 1 lnx 1 2012 = = = = ( ) f x 2012 = (2) Το ( ) 1 2012 f οπότε υπάρχει ( ) 1 1 1 x x 2012 :f = Το ( ) 2 2012 f οπότε υπάρχει ( ) 2 2 2 x x 2012 :f = Η f είναι ονότονη στα διαστήατα αυτά, άρα η (2) έχει 2 ακριβώς θετικές ρίζες. Εποένως και η ισοδύναή της (1) έχει ακριβώς 2 ρίζες θετικές. Γ3. Το 0 x ρίζα της εξίσωσης. ( ) ( ) f x f x 2012 0 + = ή ( ) ( ) x x x f xe f xe 2012e 0 + = . Επίσης ( ) 1 f x 2012 = και ( ) 2 f x 2012 = . Έστω ( ) ( ) x x hx f xe 2012e = , [ ] 1 2 x x ,x . Η h συνεχής ως αποτέλεσα πράξεων συνεχών συναρτήσεων. Η h παραγωγίσιη ε παράγωγο ( ) ( ) ( ) x x x h x f xe f xe 2012e = + ( ) ( ) 1 1 x x 1 1 hx f x e 2012e 0 = = , ( ) ( ) 2 2 x x 2 2 hx f x e 2012e 0 = = Άρα ικανοποιούνται οι προϋποθέσεις του εωρήατος Rolle , για την h στο [ ] 1 2 x ,x , οπότε υπάρχει ένα ( ) 1 2 0 x x ,x : ( ) ( ) ( ) ( ) ( ) 0 0 0 x x x 0 0 0 0 0 x 0 f x e f x e 2012e 0 f x f x 2012 h = + = + = Γ4. Όπως διατυπώνεται, το ζητούενο εβαδόν είναι: ( ) ( ) e e e e e 2 2 2 1 1 1 1 1 x x x 1 g x dx x 1 lnxdx x ln xdx x lnx x dx 2 2 2 x = = = = e e 2 2 2 2 e 2 2 1 1 e x e 0 1 dx 2 2 e 2e x e 2e e 1 x e 1 4 2 4 2 4 = = = = 2 2 2 e 2e e 4e 3 3 4 2 4 4 e + = Σχόλιο: Η εκφώνηση δεν περιγράφει σωστά το εβαδόν του ζητούενου χωρίου. Υπάρχει περίπτωση να υπολογίσουε () ( ) 1 g x dx EM Μ = ε 0 M 1 < < και να πάρουε ( ) M 0 EM lim + Η ( ) x φ είναι η σύνθεση των παραγωγίσιων συναρτήσεων () x 1 f t dt και 2 x x 1 + . Οπότε η g είναι παραγωγίσιη ε παράγωγο ( ) ( ) ( ) 2 1 2x g x f x x1 2x 1 e = −+ . Είναι () g0 0 = και () g1 0 = . Άρα η (1): ( ) () gx g0 και ( ) ( ) gx g1 . Η g παρουσιάζει τοπικά ακρότατα στα 0 x 0 = , 0 x 1 = και από Θεώρηα Fermat ( ) 0 0 g = και ( ) 1 0 g = . Οπότε ( )( ) 1 f1 1 0 e = και () 1 f11 0 e = Είναι ln x x 1 x < οπότε το πρώτο έλος της (3) δεν ηδενίζεται, άρα και κάθε όρος του δεύτερου έλους της (3) είναι η ηδενικός άρα ( ) () x 1 ln x x f x ln t t dt e ft = + . Η συνάρτηση f είναι παραγωγίσιη αφού το δεύτερο έλος αποτελείται από παραγωγίσιες συναρτήσεις. Από την (3): ( ) () x 1 ln x x fx ln t t dt e f t = + . Αν θέσουε () () x 1 ln t t h x dt f t = , ( ) h1 0 = η παραπάνω ισότητα γράφεται: () () () () () () x x 1x e e e e e h x h x h x h x h x h x + = = = ( ) ( ) ( ) x 1x hxe e = () x 1x h xe e c =− + ( ) 1x x x x h e e ce = + ( ) x x h e ce = + ( ) h1 0 e ce 0 c 1 = ⇒− + = = Άρα ( ) () () ( ) ( ) x x x x x 1 x t x ln x x ln t t h e e d e e e f e ln x x ft f x = = = = 2. ( ) ( ) x x 0 x 0 lim f x lim e ln x x + + = = −∞ Έστω ( ) f x αν x 0 + τότε υ → −∞ () () () 2 2 x 0 1 1 1 lim f x f x lim lim 1 f x + υ→−∞ υ→−∞ υ υ −υ υ −υ υ υ = = Θέτω t 1 = υ : 2 t 0 t 0 t 0 t 0 t t t 1 1 t 1 t lim lim lim lim 0 t t t 2 t 2t υ − = = = = 3. ( ) ( ) x f x F = ενώ ( ) ( ) ( ) ( ) x x x 1 x e ln x x e e 1 x F ln x x = =− = ′′ + x x 1 1 e ln x x 1 e x 1 lnx 0 x x = ++ = −− + > Άρα η ( ) x 0 F > ′′ άρα η F στρέφει τα κοίλα προς τα άνω για x 0 > . Στα [ ] x,2x , [ ] 2x,3x η F ικανοποιεί τις προϋποθέσεις του Θεωρήατος Μέσης ιής, οπότε υπάρχει ( ) ( ) ( ) ( ) 1 1 F 2x Fx x,2x :F x ξ∈ ξ = και υπάρχει () 1 f1 e = και () 1 f1 e =− . Άρα ( ) f x 0 < . Η δοσένη σχέση: () ( ) x 1 ln t t ln x x dt e f x ft = + (3) Είναι οπότε το πρώτο έλος της (3) Η g είναι η διαφορά των () () 2 x x1 1 f t dt x + φ = και 2 x x e . ( ) ( ) ( ) ( ) 2 2 F 3x F 2x 2x,3x :F x ξ ξ = . Το 1 2 ξ , η Fγνησίως αύξουσα. ποένως, ( ) ( ) 1 2 F F ξ < ξ ( ) ( ) ( ) ( ) ( ) () ( ) F 2x Fx F 3x F 2x x x 2F 2x Fx F 3x < > + 4. Έστω ( ) ( ) ( ) ( ) x F F3 2F x h = β+ β− , [ ] x ,2 ∈β β . Η h συνεχής ως διαφορά συνεχών συναρτήσεων. ( ) ( ) ( ) ( ) ( ) ( ) F F3 2F F3 F 0 h β = β+ β− β = β− β < διότι ( ) ( ) F x f x 0 = < οπότε η F γνησίως φθίνουσα. ( ) ( ) ( ) ( ) 2 F F3 2F 2 0 h β = β+ β− β > Ικανοποιούνται οι προϋποθέσεις του θεωρήατος Bolzano οπότε υπάρχει ( ) ,2 : ξ∈ β β () () ( ) () 0 F F3 2F h ξ = β+ β = ξ . Η h είναι γνησίως ονότονη άρα το ξ είναι οναδικό, διότι ( ) ( ) h x 2f x =− . α τω Κωστής Στρατής, αατς z w z w z max w 1 3 4 + + = + = ( ) z w z w z min w 1 2 1 = Η f αποτελείται από παραγωγίσιες συναρτήσεις ΘΕΜΑ 1. Η ( ) f x 0 και ως συνεχής διατηρεί σταθερό πρόσηο. Αν ( ) () 2 x x1 2 1 x x gx f t dt 0 e + = (1)

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Θέματα Πανελληνιών Εξετάσεων 2012

Transcript of 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

    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

    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

    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

    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

    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

    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

    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

    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

    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

    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+ < :

    ( )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

    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

    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)

    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

    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

    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

    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

    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

    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 =

    ( ) ( ) ( ) ( )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

    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

    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+ < :

    ( )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)

  • TPITH 29 MAOY 201216 2012 ( )

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  • TPITH 22 MAOY 2012 15

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

    , EMPO MEOIKO , . O .

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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 = + = .

    : () 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 = + = .

    : () 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 = + = .

    : () 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 = + = .

    : () 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 = =

    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