ΤΡΑΠΕΖΑ ΘΕΜΑΤΩΝ ΑΛΓΕΒΡΑΣ Β ΛΥΚΕΙΟΥ
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Transcript of ΤΡΑΠΕΖΑ ΘΕΜΑΤΩΝ ΑΛΓΕΒΡΑΣ Β ΛΥΚΕΙΟΥ
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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. ()