ΤΡΑΠΕΖΑ ΘΕΜΑΤΩΝ ΑΛΓΕΒΡΑΣ Β ΛΥΚΕΙΟΥ

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Transcript of ΤΡΑΠΕΖΑ ΘΕΜΑΤΩΝ ΑΛΓΕΒΡΑΣ Β ΛΥΚΕΙΟΥ

  • 1

    1

    1. >0 1 1, 2>0 :

    log(12) = log1 + log2. 9

    2. ; 6

    3. ,

    :

    . f(x) = x .

    . .

    . f(x) = x *1 , 1].

    . x>0 elnx = x.

    . f(x) = x 0

  • 2

    3

    . P(x) x - ,

    P(x) , x = , = P(). ( 13)

    . ,

    . 10 log ,x x 0

    . 1 1

    2 2

    lnln

    ln

    , 1 , 2 > 0

    . , ( )x , ( ) 0

    . ( ) ( )f x x 2

    ( 04)

    . :

    . x x

    . log ...xa a 0 , 1 Rx

    . H ( ) xf x e , x .

    . loga . > 0 , 1 , R 0

    ( 08)

    4

    . , P x x

    x , .P 8

    . , .

    1. .......... ..........,

    2. .......... ..........,

    3. ..........,

    x x x k

    x x x k

    x x k

    8

    . : ( ) , .xf x e x

  • 3

    ,

    .

    . f 0, . . f . . 1,0 f .

    9

    5

    1. P(x) x

    x = , = (). 10

    3. .

    . f(x) = x > 1 R.

    . log , > 0 , 1 > 0.

    . 0.

    6

    . . >0 1, >0 R log=log ( 9)

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

    1. >1 ...........................

    2. 0

  • 4

    2)

    (6)

    3) () () () (), ,

    . (7)

    8

    . : ( )x x

    x . ( )P . (12 )

    . ; ( loga ) (5 )

    . .

    . ( ) xf x a R.

    . ln 0e .

    . ln 0 , 0

  • 5

    10

    . > 0 0, 1,2> 0, log 1 2 = log1 + log2

    * 12]

    . f ;

    * 5]

    . () ().

    1. > 0 , 1 > 0 log =

    2.

    2 =

    3. .

    4. f(x) = x 0 < < 1 R

    * 8]

    11

    ) : ( )P x x

    ( )P x . ( 07)

    ) ,

    ,

    .

    . (180 ) .

    . .

    . ( )x

    f x a >0 .

    . 1

    ( )

    x

    f xe

    x .

    . ( )P x x ( )P .

    ( 12)

    )

    . A x = x= .. k

  • 6

    . A x = x=... . . . . . . . . . . . k

    . A x = x=.. k

    ( 6)

    12

    1. P(x) x x = , : = () 15

    2. , . (i). D 2x2 D 0

    (ii). 3

    2

    < 0

    (iv). 0

    13

    1. > 0 1 1,2 >0 : log(12) = log1+log2 (. 10)

    2. > 0 1 > 0. ; (. 5)

    3. .

    ) 00 : log =.

    ) (x) x-

    x=. =().

    ) f(x) = logx >1 (0, )

    ) 0

  • 7

    14

    1. +1

    1 +1 + 0=0, .

    0 , 0.

    ( 9 )

    2.

    .

    3.

    :

    ) .

    ) .

    ) (x) x- ()=0 .

    ) f(x)= >1 .

    ) >0 1, 1 , 2>0 : log(12) = log 1 + log 2

    ( 5x2=10)

    15

    )

    1) f(x) = x 2) g(x)=x 3) h(x)=lnx 4) t(x)= ex ( 10)

    ) :

    1) f(0) = 2) g() = 3) h(e) = .. 4) t(0) =.. 5) h(1) =. ( 5)

    16

    A. >0 1 , 1, 2 >0 log(1.2)=log1+log2 (. 8)

    . :

    log=.... log1=. log (1 : 2 )= ...

    log

    =... (. 5)

    ) x= 1) x=+ ,

    ) x= 2)

    ) x= 3)

    ( 6)

  • 8

    . ,

    .

    i) P(x) - ( ) 0P

    ii) ( ) xf x e (0, )

    iii) ( ) logaf x x 00 1, 1, 2 >0

    log (1. 2 )=log 1+log 2 (10 )

    2. ,

    (10 )

    1. f(x) = x *- 1, 1]

    2. log 1 1a ( >0, 1)

    3. x= x=+

    4 . f(x)=lnx , R

    5 . 0

    3. (>0) ( 5 )

    18

    ) 1 2, 0 : 1 2 1 2log log log (15 )

    ) ()

    ():

    i) 2 2 1 :.

    ii) x=2 .

    iii) f(x)=lnx,x>0 (1,0).

    iv) .

    v) : log 10xx . (10 )

  • 9

    19

    A1. 0 < 1 1, 2 > 0 :

    1 2log ( ) = 1log + 2log . (10)

    2. P x ; ( 5)

    3. , , , , .

    1. ( ) xf x 2.

    2. 0 0 1 log

    log .

    3. P(x) x- , 1 . 4.

    .

    5. ( ) lnf x x .

    ( 2x5)

    20

    . : P(x) x- P(x),

    P() = 0.

    . ; (>0, 00 0

  • 10

    2. 0 1 0 log log

    .

    3. ln 0e .

    4.

    .

    5. xf x 0 1 . 10

    22

    . : 2+ 2 =1. ( 11)

    . ,

    .

    ) elnx =x x > 0 .

    ) f(x)= x >1 R.

    ) ( ) lnf x x (0,+ ).

    ) 0 1 1 2, 0 : 1

    1 2

    2

    loglog log

    log

    ( 08)

    .

    .

    ) (-)= ..

    ) (90-)=

    ) (-)= .. ( 06)

    23

    1. > 0 1, > 0 R , log loga a

    10

    2. f .

    5

  • 11

    3.

    .

    . f , x A :

    -x A ( ) ( )f x f x .

    . ( ) xf x a > 1 R.

    . x = , x = 2 + x = 2 .

    . .

    . >0 1 log 1a a .

    5x2=10

    24

    ) () (),

    i) .

    ii) log(1 2) = log1 log2

    iii) 0 .

    iv) 300 = 1

    ( 10)

    )

    1) x= i) x = +

    2) x = ii) x = 2 x= 2 +

    3) x = iii) x = 2 ( 9)

    ) P(x) x

    P(x), P() = 0. ( 6)

  • 12

    25

    1. .

    2. ;

    3. P(x) (x )

    x = , = P().

    4. ,

    , .

    1. ( )2

    , .

    2. f(x) = (x) , 2

    (>0).

    3. f(x) = x, 0 < < 1 .

    26

    1) : 2+2=1 ( 11)

    2) :

    ) (-)=-

    ) P(x) - P(x) =

    ) log(12)=log1log2

    3) :

    ) x= x= ) y=logx y=10x

    .

    ) logx=.. ) loga = .. ) log1=. ( 10)

    27

    ) 1 , 2 , 1 >0 2 >0, : ln(1.2) = ln1 + ln2

    10)

    ) :

    1) f(x) =ex g(x)= lnx, :

    ) xx ) yy ) y=1 ) y=x ) y = -x

  • 13

    2) f(x)=ex :

    ) xx ) yy ) y=1 ) y=x ) y = -x

    3) g(x)= lnx, :

    ) xx ) yy ) y=1 ) y=x ) y = -x

    4) g(x)=lnx, :

    ) (0, + ) ) (- , 0) ) R ) (0, 1) ) (1, + )

    5) f(x) =ex :

    ) R ex >0 x R

    ) (0, + ) x>0

    ) (0, + ) ex >0 x R

    ) (0, + ) ex >0, x>0

    ( 5x2 =10

  • 14

    1

    (x)=2x2-4x-6

    . :

    X -2 -1 1 2 3 4

    (x)

    . (x) .

    . (x)=0 (x)>0.

    . .

    . x ;

    . ; ;

    25

    2

    (): 1 + 2 = 2

    1 + = 3( 2) R

    1. N D, Dx, Dy 9 2. 16

    3

    5( 1) (2 )

    5 ( 2)

    1. 3 -3 . (.12)

    2. =-3 . (. 6)

    3. =3

    . (. 7)

  • 15

    4

    ) (1) D = 2 - 1 , Dx = +1 Dy =-1. (1)

    . ( 5)

    ) () : 1

    2 0

    ax y

    x y

    1) , ,x yD D D (). ( 9)

    2) ; ( 2)

    3) (x0, y0), . ( 4)

    4) , (x0, y0) : 3x0 + y0 = 1

    ( 5)

    5

    . , :

    i)

    2 3 30

    4 6 40 ii)

    2 5 12

    3 4 41

    . 2,5 6 .

    1.400 5.600 .

    . ( : 10 + 15)

    6

    f *-3,3+.

    [-3,3].

    1. f(-2)=15, f(2). 5

    2. f x = -3 f(-3) = 35 f x = 3

    . 8

    3. f(-2x) < f(2). 7

    4. f(x) = -x3 x + 5 , g Cf

    2 5 . 5

  • 16

    1

    1. : (1 + )2 + (1 )2 = 2

    2 13

    2. N : (1 + )2 + (1 )2 = 4 12

    2

    . :

    . 330 =

    . (-300) =

    . (-210) =

    . 240 =

    . 1. :

    . 3

    2 . . .

    1

    2 .

    2. x:

    . x 1 , . -1 x 1 , . x , .

    .

    3. 2x + 52x=4 (25 )

    3

    ( )f x x .

    . 2

    ( )2

    f x . (9 )

    . , ( ) 2 (4 )g x f x .

    (6 )

    . x , ( ) 1g x . (10 )

    1

    2

    2

    2

    3

    2

  • 17

    4

    f(x)= + .2x >0, 4

    (3

    ,-5).

    1. .

    2. =-2 =6:

    i. f.

    ii. f .

    iii. x f .

    vi. f y=1.

    5

    f()=.(

    ) , . f

    (0,-2) (

    ,) :

    . =-2 =2 ( 8)

    . f ( 9)

    . f()=1 ( 8)

    6

    3

    5x

    2x

    , :

    1. x rad. ( 8)

    2. : 10 12

    5

    x x

    x

    . ( 4 )

    3. : 2 22

    ( ) 25

    x x K . ( 13)

  • 18

    7

    4 4 2

    1

    x x xA

    x

    1. =1+ x (v 10)

    2. 4

    5 2x x

    (v 7)

    3. 5 1x (v 8)

    8

    ) :

    i) 1

    3 2x

    (8 )

    ii) 22 2x x (12 )

    ) 2

    x

    .

    (5 )

    9

    ( ) 2 4x , x [0,]f x

    1. , ( ) 5f ( 6)

    = 3:

    2. f ;

    ( 6)

    3. f(x) = 4 ( 13)

    10

    . ) :

    i) 7

    6

    rad ii)

    5

    4

    rad iii)

    4

    3

    rad

    ) rad :

    i) 120 ii) 135 iii) 150 ( : 3 + 3 )

  • 19

    . :

    ) (1+) ( + 3 ) = 0 ) 42 4 3 = 0 ( : 8 + 11 )

    11

    72

    2

    2113

    2

    22 0x x (1)

    . . 12

    . 1 , (1). 13

    12

    : =1-2x =1-22x

    . :

    2 2 = 3 4x ( 12)

    . : = 0 ( 13)

    13

    ( ) ( ) , 0 , 0f x x = *0, 2+,

    .

  • 20

    1. ( )f x x

    .

    2. ( )f x , .

    3. ( ) 4 (2 )f x x ( ) 2f x *0,+.

    4. ( ) , ( )9 7

    f f

    , .

    8 + 6 + 6 + 5 = 25

    14

    f(x)= ( )*(3 + 4)x+ , IR , (3 + 4) > 0 ( ) > 0. f

    3 = .

    1. , .

    = -1 = 2, :

    B2. f *0, + f

    *0, +.

    B3. f 3

    2y .

    8+10+7=25

    15

    f(x)=+ .2x >0, 4

    (3

    ,-5).

    1. .

    2. =-2 =6:

    i. f.

    ii. f .

    iii. x f .

    vi. f y=1.

  • 21

    16

    ( )f x x .

    . 2

    ( )2

    f x .

    . , ( ) 2 (4 )g x f x .

    . x , ( ) 1g x .

    17

    :12

    25)(t

    tf

    ,

    t .

    :

    ) .

    ) .

    )

    24.

    ) 24 6

    1

    P(x) = x4 +x3 +x2 2x + 4, , IR. P(x) x 1

    P(x) x + 1 6.

    1. = 1 = 2. 7

    2. P(x) = 0. 6

    3. P(x) > 0. 5

    4. 4x 3x + 22x 2x + 2 = 0 *

    2 , ).

  • 22

    2

    P(x), 2v , :

    3 28(x 1).P(x) x.P(2x 3) 52x 8x 6x 16 , x .

    P(x) x-1 2:

    1. P(x) 2x 6x 5 .

    2. P(x) 2x 6x 5 x x 4 :

    3

    ) P(x):

    i. yy . ii. y=2

    ) P(x)

    y=2.

    4

    : 3 2( ) 6 11 , .x x x x

    . ( )x

    1x 24. 7

    . 6 ( )x

    1.x 8

    . : ( ) 0.x

    5

    P(x) = x3 + x2 -2x + + 7. x + 2 P(x)

    P(x) (x + 1) 8.

    1. = 1 = -1. 8

  • 23

    2. P(x):(x2 2x)

    . 8

    3. P(x) < 8. 9

    6

    P()=x3+2++4 , R +1,-2

    . =-3 =0 ( 7)

    . , P()=0 ( 7)

    . C f()=P() =-3 =0

    :(i) C yy ( 4)

    (ii) C ( 7)

    7

    3 2( ) 2 2x x x x .

    . ( )x 2x . (7 )

    . 5 , ( ) 0x . (10 )

    . 5 , 2( ) ( 1)P x x

    . (8 )

    8

    P(x)=(x 2 +) + 2 +5 -1 P(-2)=3

    A) B , 9

    ) =-1 5

    1) P(x)=0 MONAE 8

    2) P(x)8 MONAE 8

  • 24

    9

    o P(x) = 2x3 + x2 + x + 6 , x .

    ) P(x) x = 3 30 x1

    P(x), = 1 = 7 * 13]

    ) , ), P(x) 0

    * 12+

    10

    (x) = 2x3 + x2 + x 20, , R

    1. (x) x + 2 x + 1

    16 : = 12 = 6 10

    2. : (x) = 0 10

    3. x (x)

    xx ; 5

    11

    3 3 3P x x x x , R 1.

    . . 8

    . 1 ,

    i) 0P x . 9

    ii) P x 2x 4

    iii) 2013 P x .

    12

    P(x)= x3+(2+1)x2-(+)x-9 , R.

    . -3 P(x) P(x) x+1 -8,

    . ( 08)

  • 25

    . =2 = -5 :

    ) : (x)=0 ( 08)

    ) : ( )

    02

    P x

    x

    ( 09)

    13

    2x .

    1. 3 .

    2. ( )P x 2 2 1x x .

    3. ( ) 0P x . 8 + 8 + 9 = 25

    14

    P(x) = x3 + x2 5x + , , R.

    1. P(x) x 1 x + 3, .

    = 1 = 3

    2. P(x) = 0.

    3. P(x): (x2 + 3x + 4).

    4. : P(x) Q(x) > 0, 2

    1( )

    9Q x

    x

    6+5+6+8=25

    15

    2 4 3 21

    ( ) 1 1 1 32

    P x x x x x , , R .

    1. ( )P x 3 ( ) : ( 1)P x x 4

    .

    2. 1 2 :

  • 26

    . ( ) : ( 1)P x x .

    . ( ) 0P x .

    . ( ) 4P x

    16

    (x) = (+1) x3 + ( x -1)2 + x -1, R .

    ) (x) x-1 2.

    ) R , (x) .

    ) =0 : (x) = 1.

    17

    () = 3+2+7-3 Q() = (+1 -)2+(-1)+-3

    1. , , ()= Q() -1 ().

    2. ():( -1)

    3. ()=0

    4. Q()>=0

    18

    () = 4 + 3 -3 2 +4 -4 Q(x) 3 .

    . (. 20)

    . N , ().

    . +2 ()

    . = -1, : (. 50)

    . 1 , 2 , -1 () ;

    . -1 () ;

    . ():(-3) Horner

    iv. ():(2 -4) ... .

    v. ()=0

  • 27

    . Q(x) , 2-+3 2-1 -17. (. 6)

    19

    P(x) x x x .

    1) P(x) ;

    2) 1 P(x) ;

    3) ;

    4) P(0) = 2013 ;

    -

    1

    f(x) = log(4x 2).

    1. f f

    y = 1 log5. 6

    2. x-

    x- 1

    124 6

    4 2 05 5

    . 6

    3. f(x) + f(x

    2) = 1 + log

    x-x-

    112

    4 64 2

    5 5. 8

    4. 2x = f(5

    2) + f(1) f(

    3

    2). 5

    2

    f(x) = k+log(x 2-3) , k .

    1. f.

    2. k f (2)= log100.

  • 28

    3. k=2 :

    )N :

    1y log

    1000

    ) N : f(x) >2.

    3

    ( )1

    x

    f x

    x .

    . , f . ( 10)

    . f (2) 4f ,

    i. N . ( 07)

    ii. 2a ( 1) 8f x (08)

    4

    log 3 2 log50f x x 1

    log 2 22

    g x x .

    1. f g . 10

    2. : f x g x .

    5

    : 2 1

    ln .1

    x

    x

    ef x

    e

    . f 6

    . : 0.f x

    9

  • 29

    . x f

    ' .x x 10

    6

    f()=ln(3e2xe-2)

    . f ( 10)

    . ln2

    ( 5)

    . f()=3 ( 10)

    7

    1

    ( ) ,3

    x

    f x

    x .

    i) ( )f x . (.7)

    ii) ( )f x . (.8)

    iii) 7 , ( ) (2 ) 2f x f x (.10)

    8

    2( ) ln( 2 3)f x x x ln 1xg x e .

    . . (8 )

    . (2) 3 (1) (3) ln 4f f f (7 )

    . ( ) ln3 ( )xf e g x . (10 )

  • 30

    9

    f(x)=ln(e -2)-ln(e -1)

    ) f 7

    ) f(x)=ln (1-1

    1xe ) 7

    ) f(x)=-ln2 MONA 6

    ) N f =

    MON 5

    10

    f(x) = + ln(ex 2), .

    ) f * 8+

    ) f (ln3,1)

    * 9+

    ) = 1 f(x) = 0 [ 8+

    11

    3

    ( )3

    xa

    f xa

    . . ( 08)

    . , f . ( 08)

    . =2 : ( 1) ( ) 4f x f x ( 09)

    12

    f(x) = 2x , xR

    1. . (. 3)

    2. . (. 3)

    3. : 2f(2x) 5f(x) + 2 = 0 (. 10)

    4. : f(x2+x-2) < 1 (. 9)

  • 31

    13

    f(x)=ln(3x-5) .

    1. f. ( 5 )

    2. f xx ( 8)

    3. x f

    xx. ( 13 )

    14

    ( ) log(9 3)xf x

    . (. 7)

    . : log6 2log (1)a f (. 8)

    . ( ) log2 log3xf x (. 10

    15

    f(x)= 2ln( 2 3)x xe e g(x)= ln( 1)xe

    1. f(x) g(x) (v 10)

    2. f(x)=g(x) (v 15)

    16

    f(x) = x + ln(xe 5).

    1. . ( 5)

    2. f(ln6) < f(ln10). ( 8)

    ln( ) ln3 ( )xg x e f x .

    3. g ( 8)

    4. g (,0);(

    ) ( 4)

  • 32

    17

    log 2f x x 1log 4 2x xg x .

    . f g . 6

    . 1

    4g x f

    . 7

    . i) log5 1 log2 4

    ii) 5f x

    x , 0x 8

    18

    f(x) = ln(3 x) ln(3 + x)

    . f . ( 06)

    . : f(x) < (1) ( 09)

    . : (2 ) ( 1)) (0) (1)xf f f f x . ( 10)

    19

    1

    ( ) ln2

    xf x

    x

    .

    1. .

    2. 1

    (3) (4) ( ) 02

    f f f .

    3. ln2( ) ln( 2) ln1f x x e . 8 + 8 + 9 = 25

    20

    1 2

    2 2

    2 2 24( ) ln( )

    2 3 2 1

    x x

    x xf x

    1. : 2+1 + 2+2 24 = 0 22+2 3 2 1 = 0.

    2. f.

    3. f xx yy.

    4. f(x) = ln3 (x + 1)ln2. 8+5+5+7=25

  • 33

    1

    P(x) = x4 +( ) x3 +(2 2)x + ,

    *0,+.

    ) P(x) 1, . * 12+

    ) = 1

    3x P(1) , x 3 * 13]

    2

    4 2 2 3 2P(x) x ( ) x 3 x 2x 1

    P( )x 1x :

    . 0, ( 15)

    . : ( ) P(2 7)Q x x 4x . ( 10)

    3

    3 2( ) ln( ) ln(1 2 ) 8P x x x x ,

    0,2

    .

    i. 2 ( )x . ( 13)

    ii. 2

    , ( ) 0x ( 12)

    4

    :

    4 3 22ln 1 2 , 0, 0, , .x x x x x x

    . x 3 -1, , . 8

  • 34

    . 3 2 1,x x x x x x

    : 0.x

    7

    . :

    ) 2 0, 0, . 5

    ) 2 ln 0. 5

    5

    log5 log 2 log 4 1

    2 log5 log 2 log8 2

    : 2

    ,3 3

    x yR

    x y

    i) : 2

    3 . (7 )

    ii) 2

    3 , ,x yD D D

    (6 )

    iii) 2

    3 . (12 )

    6

    3( ) 3 3P x x x ax ( ) log(2 )xf x a ,

    .

    P(x) x=1, :

    i) =1. (4 )

    ii) P(x)=0 x . (7 )

    iii) : P(x)

  • 35

    7

    = log381 + log51 - log44 = 2log23 + log26 log227 P(x) = -2x3 +

    (3+2)x2 5x + 1 .

    ) =3 =1

    ) =3 =1 , P(x) : (2x +1)

    7 .

    ) P(x) > 7

    ) P[ - ( 8 ) ] = 7

    8

    .

    . 0 ().

    . :

    ) .

    )

    9

    ) (1) : log(x-1) + 1 =2 log 8x

    ) (1) , ,R P ()=3 + 2

    + 3 + 5 , - 5.

    ) =4-1 , =2-1 R.

    10

    :

    4 3 22ln 1 2 , 0, 0, , .x x x x x x

    . () 3 1x , , .

    . ()=3-2+-1 x : 0.x

  • 36

    . :

    ) 2 0, 0, .

    ) 2 ln 0.

    11

    (x) = (+1) x3 + ( x - 1)2 + x - 1, R .

    ) (x) x - 1 2.

    ) R , (x) .

    ) = 0 : (x) = 1.

    12

    2 4 3 21

    ( ) 1 1 1 32

    P x x x x x , , R .

    ) ( )P x 3 ( ) : ( 1)P x x 4

    .

    ) 1 2 :

    . ( ) : ( 1)P x x .

    . ( ) 0P x .

    . ( ) 4P x .

    13

    3 2 2P(x) 2 (x x ) , 0, 2

    .

    . x P( )x .

    . P( )x x .

    . P( )x 1x 1, .

    14

    3 2( ) 5 8 , .P x x x x

    . x 1 P ( x ) , .

  • 37

    . = -4 , P ( x ) = 0 .

    . 3 25 8 4x x x .

    15

    P(x)=x3 +x2 +8x +

    ) () -1 ():(+1) -18, ,.

    ) = -5 = -4

    1. ():(+2)

    2. ()=0

    3. ()