: 2009 2010
- 2010 : : A :28 2010 : 2:30 -: 7.45- 10.15
____________________________________________________________________________ 3 : () . () ( ). () . 15 12. 1 .
:
(1) 2x 2 3x 5 = 0 . (2) (-1,2) (3,-4). (3) y =2
4x . 2x 1
(4) 2x 3x + 4 = 0 , . (5) :
x y = 6. 2x y = 3
2
(6) : 2 x 1 + 2 x + 2 x 1 + 2 x = 2 .
(7) =35, =25. . .(8) =8cm, =2cm, =4cm =10cm. //, . . (9) 3x 4 7 x 2 + 4 = 0 . (10)
. x .
2
(11) 3x 2 ( 3 ) x = 0 , . (12) : 3x y 5 = 0 : x 2y = 0 : 2x y + 8 = 0 .1 2 3
(13) 3 + 2 = 0 90 < < 180 , :
+
1 = 9
(14) . , , . , , . ()()=()(). (15) . =. =.
:
6 4. 2 .2
(1) 3x 4x + 3 = 0 , x1 x2. , (i) . () x + x , () x x ,1 2 1 2
x2 () 1
+x
2 2
x x1
2
(ii) 1, 2, 1 = 2 =
2 x1
2 . x2
(2) () : () :
(x
2
5x 14 2x 2 + x 62 2
)( ( x 9) ( x
+1
)
) 0.
- =
3
(3) f(x)=x2 + x + , ( 0 ) .
, : () , () , () , () x f (x ) = 0 , () , , () x f (x ) > 0 ,()
f ( 2) f (0) . f (- 1)
(4) (6,4), (-1,-2) (3,-1). () , () , , () 1
( )
, () 1 /.
( )
(5) () :
(180 + ) 90 + (-) 90 + + = 1. 180 - (- ) 90 -
(
)
(
)
(
(
)
)
() (2 ) x 2 3x + + = 0 .
, , 0 < < 90 .(6) (,), . = 2 , : () , () ()()=()(), () .
( )
2009-2010
2010 : : : 28/05/2010 : 2,5 : 07.30-10.00
(4) . ( ). . . 15 12. 1 . 1. 2. : x 2 11x + 10 = 0.
, : 10 7 ) ) 3 2 4+ 2 :y= 3x 1 x+5
3. 4.
(2, 3) (): 4 x + y 1 = 0. = 3 270o < < 360o , : 5 6 + 10 + 4 = x + y + = 3 2x + 3y + 4 = 0 x + y 2 = 4
5.
6.
:
7.
:
x x = x . 1 x
1/4
8.
R, : (1): ( 3)x 2y + 10 = 0 (2): 8 x + 4y 3 = 0 . :x 3 6x 2 + 9x 2x 2 5x 3
9. 10.
x1 x 2 3x 2 2x + 9 = 0 , , : 1 1 3 3 + 3x1 x 2 + 3x1x 2 . ) ) x1 x 2 ( = 900 ). : 2 () = ()() .
11.
12. 13.
:
x+y=2 2 x xy = 12
(,R). , = 450 , 0 = 30 = . : ) ) ) .
14.
: (900 + ) (1800 ) (900 + ) ( ) = . (1800 ) (900 ) (3600 + ) R, x 2 x + 2 3 x R.
15.
2/4
B 6 4. 2 . 1.
f(x) = x 2 + x + . , ( ): ) f(x). ) , . ) f(x) = 0 . ) f(x) 0 . ) = 2
y
x
2.
) : 1 1 = 22 x 1. 2 2 1 + x 1 + x ) , : 2 x 1 1 1 = 2 2 x + 1 1 + x 1 + 2 x , : () : 2x y + 1 = 0 , () : 3x + y 11 = 0 , (0,1), (3,2) (4, 1) AB , . : ) . ) . ) . ) x, 2x y + 6 = 0 .y
3.
x
3/4
4.
x 2 2 x + 2 = 0 x1 , x 2 1800 < < 2700 . : ) . ) x1 , x 2 . ) 1 = 1 x1 2 = 1 x 2 : x 2 2(1 )x + ( 2) = 0 . ) ( + 1)x 2 2( 2)x + 3 = 0 R {1} . : ) . ) 3 . ) 2x1 + 3x1x 2 + 2x 2 0 . ) :f(x) = x(x 2 16)( x 2 3x + 4) .
5.
6.
, : (,R), = 900 . : ) ()2 = ()() ) ()() = ()() .
4/4
2009 2010
2010: : 26 / 05 / 2010: ) ( ). ) . ) . ) 4 . (15) (12). (5) .
: : 07:30 10:00 (: 2:30 )
1) : 6 x 2 7 x + 1 = 0
2) 3 x 2 5 x 6 = 0 , : () () 3 x +1
3) =
4) (3, 4) (1, 0). 5 00 < < 900 , 12 26 + 12 A= 15
5) =
6) .
( 2 ) x 2 + 3 x + + 2 = 0
,
-1-
7)
AB = 120 = 60 : () () =
8) :
( 2 + 3 ) + (1 2 3 )(1 + 2 3 ) 2
48 = 4
9) :
3 x x 2 = 5 x =1
10) OX 60 x 2 = 8 2x + 3 = 5. 11) : 3 ( x 1)( x 2 ) ( x 2 + 9 ) 02
12) :
= 2 2 1 +
13) (, R) , ( ) . .
14) :
360o . 180o + + ( ).(270 )
(
) ( ) ( 90 + ) . (180
)
= 2
15) ( = 90 ) . . : () () ().() = ().()-2-
(6) (4) (10)
1 1) : ( 2 1) x 2 ( + 2 ) x + 2 9 = 0 , R , x1 x 2 . 2 : ) 1 ) ) P = 4S + 1 ,S
2) ( // = = 90 ) (1, 1) , (3, 1) (0, 2). : () , , . () .
3) :
x+7 9 8 x 1 x x 1
4) )
+ = 2 1 1 + 2 = 1 1 +
) 1 =
) S (900 ). (1800 + ). (2700 ) S = 3 , 00 < < 900 2 2 ( ) + .
5) (=) . . . : ) ) ( ) ) = -3
6) = ( 2 3 4 ) x 2 + ( 2 + 3) x + 2 9 , R {1, 4} . : ) (max) ) = 16 ) x = 0.
-4-
- 2010 : 19.05.10
2009-2010
: ____________________________ : _______________ : ___________ : _______ : 2.30 : _______________ : _______________
) ( ). ) . ) .
:
12 15 . .
1. : 102 + 7 3 = 0
2. x1 = 3 x2 = 8.
1 11
.
2009-2010
3.
(-3, 2) (1, 14).
4. N . ( 62 =
15 8 , 62 = , 17 17
62 =
15 ). 8
34 62o 5.
x+3 < 0. x5
6. = = , . 10 12 8 x
2 11
.
2009-2010
7. 4 22 8 = 0
8. : =
3x 2x 4
9. , , , , . , , .
2+5 B + K
55 A
3 11
.
2009-2010
10. : 42 + 3 8 = 0. , 8 8 : () 1 + 2 () 12 () + 1 2
11. = 22 8 + 1. : (i) ( ), (ii) , (iii) .
12. : = 17888 182358
4 11
.
2009-2010
13.
:
4 4 + 1 2 2 + 5 32
14. :
1 +1 2 + = 2 +1 1 2
15.
22 2 + 22 = 0, 90, , .
5 11
.
2009-2010
: 4 6 . .1. 2 ( + 3) + 4 3 = 0 1, 2. :
() , () , () , () 2, () (1 + 21)(1 + 22) 4.
6 11
.
2009-2010
2. f() = 2 + + . :
( ), (ii) , (iii) f(), (iv) f(), (v) 2 + + = 0, (vi) , , , (vii) f(2), f(2010).
(i)
13 12 11 10 9 8 7 6 5 4 3 2 1 -3 -2 -1-1 -2 -3 -4 -5
y
x1 2 3 4 5 6 7 8
7 11
.
2009-2010
3. = ( + ) + 5 3 + 8 = 0 2 (2 ) 1 = 0 , .
8 11
.
2009-2010
4. (-2, 3) (2, 9). 4 : (i) , (ii) , (iii) , (iv) , (v) .y
10
5
Ax-5 5
9 11
.
2009-2010
5.
(90 + ) (180 + ) (90 ) = 2 0 0 :
=
4 5
: 3 . 6
6
+ (
7
2 )
7
1
6. . . = 80 , : , , , , ( )
7. f(x) = 3x 2 - 6x + 7 . : ) ) . ) .
8. :
2x + = 5 2 x + x = 6
9. = -
8 > 0 . : 17
=
34 +17 8 -15x 3x
10. :
(
2
+ 11x - 4 > 0
)
11 . ( -1)x 2 - 4x + 5 + 6 = 0 , 1 . : ) -3 ) 12. :2x x + = x x x.2x
13. = ( x + 3)cm , = (2 x 1)cm , = 65 + = 10 + 2
:
) ) )
2
14. (= 90o ) , =. . 15. : (180 + ) + 2 90 + = 5 270 - - 6o
(
o
)
(
o
)
(1 ) : y = x - 2
x + ( + 1)x + 5 = 0 ( 2 ) : x - = 0 .
2
6 4 2 . 1 . ( B = 90 )o
(3,2) , (-1,-2)
3x - -11 = 0 . : ) ) ) ) .
2.
f ( x ) = x 2 + x + . :
( ) ) , ) ) x 2 + x + = 02 - 6 2 ) x x + x + 0
) A =
) B =
f ( 2 ) f (1) f ( 0)
3
3. )
( x 2 2 x + 3) ( 2 x + 1) f ( x) = 2(3 x )
) =
270 + . 360 + . 90 + o o
(
) ( ) ( ( 90 - ) . (180 - )o o
o
)
= -
(
2
2
)
2
+ 4 + 90 + 180 +
2
2
(
o
)
(
o
)
2 =
4. . ( ) . : ) ) )
( ) = ( ) .( ) ( ) . ( ) = ( ) . ( )x1 x , 2 = 2 x2 x1
2
5. x 2 -3x + = 0 x1 , x2 , ) 1 =
) 1 , 2 x 2 - (9 - 2)x + = 0 , 6. 1 1 + 1 21
x 2 + 2 (1+ ) x + 2 + = 0
) ) x1 , x2 (x1 + x 2 ) 2 + 2x1 x 2 -12 = 6
: ( .) ( )
:
4
5
2009-2010
: : :08/06/2010 :2:30 5
. . . .
1512.5. 1. : x 2 + 7 x + 6 = 0 . 2. (..)(..) y = 3. (3,4)(1,2). 4. 25.
x2 . x 1
3x 2 10 x + 3 5. : . x2 96. (,R) = 74 .
, .
7. x =
5 x 4 x 3 , 0 < x < 90 , A = . 6 x 5
8. : 2 x 2 + 7 x + 3 0 .
1
9. :
2009-2010
x + x = 1 . x x
10. : x 4 15 x 2 16 = 0 . 11. x1 , x2 x 2 5 x + 3 = 0 , :)
x1 x2 2 + ) 5 x1 x2 + 5 x12 x2 x2 x1
12. :
x 2 + y 2 = 10 2x y = 5
.
13. .(1):y=2x2 (5,0) .
14. :
(180 ) ( 360 )
( 90 ) ( 360 + )
= 2
15. ., : )()()=()() )()()=()()
2
2009-2010
64.10. 1. 2 f ( x ) = x + x + .
: ) )2 ) x + x + = 0
) ) )
)
) ) f ( 2 ) ) f ( ) f ( ) ( 1,3) , ( 3, + ) .
2. ) , 0 < < 90 :
2 ( 90 ) (180 ) ( 270 ) = 3 ( ) ( 360 )
):
3 = 3
3
2009-2010
2 3. ( 1) x + ( + 2 ) x + 2 ( + 1) = 0 , 1 . : ) ) ) )1. 4. (5,1)(13,5). (7,1). ). ). )(9,3) . 2 2 5. ): x + ( 2 1) x + 4 = 0 . x1 , x2 ,
,:
x1 x2 + 2 . x2 x1
)(..) f : f ( x ) =
5 x 2 8 x + 4 x2 + 2 x
4
2009-2010
6. )(,R).,
.:()()=()() ).
:
5
2009-2010
2010 : : , 21/05/2010 : 7.45 .. : 2,5
: ( ). . 5 . .
:
15 12. (1) .
1. 2 x 2 9 x + 10 = 0
2. =
x+2 x3
3. (2, 1) 3 4. 1 = 3 2 = 10 5. : =
3 30 4 60 + 3 60 2 45
6. : 3 2 + 5 2 0
1
7. : = 3 16 + 7. 3 1 3 1000 , = 7 +2 . 7 2
(
)(
)
8. 2 2 8 6 = 0 1 , 2 . , : ) ) : = 1 + 2 + 1. 2 , = 312 + 3 2 2 9. ( , ) = 140 .
. ( ).
140
10. 1 : = (2 + 1) + 5 2 : 3 + = 1 . , .
11. =
5 270 < < 360 , : 13 = 13 24 12
12. : (4 2 + 1).( 2 9).( 2 12) < 0
13. ( 2 ) 2 2( ) + 2 = 0 .
2
14. :
2 + = 4
2 + 2 = 515. . = 90 , .
:
6 4. (2) .
1. f ( ) = 2 + + . :
)
y
3) ) ) 2 + + = 0
2 1x
-4
-3
-2
-1 -1 -2 -3
1
2
3
4
) )
-4) f (1) f (4) ) f ( ) 0
-5 -6
3
2. :
) 4 3 2 4 = 0
)
2 .( 4 3 2 4) 0 ( 1).( 2 2 + 4)
3. 2 + ( 1) + 1 = 0 .
: ) 1 ) ) 6 )
4. ) 0 < < 90
(180 ). (270 + ) + (90 + ). (360 ) = 2, . .
) :1 2 = 1 2 2 2 1 +
5. , = 3 4 (3,5) (7, ). ) -2, ) ) , = 3 )
4
6. . . , : ) , ) .
: B.
E
5
: 2009-10 MAIOY- 2010 : 21/ 5/ 2010 : 2.30
: : :
. . . 4 .
: 15 12. 5 . 1. N : 2 2 3 + 1 = 0 2. : () 5 5 () 2 6 1
3. : 1 = 3 + 3 2 = 3 3 4. ( 1 ) (4,2)
( 2 ) : 4 2 1 = 05. : =
1 2 +2
6. , , .
7.
(180 ) (90 ) (180 + ) 1 = , 0 < < 90 , . (180 ) (180 ) 2 . 3 = . 3
8. :
9. =
3 180 < < 270 , : 4 10 8 = 3
10. 1 , 2 3 2 7 + 2 = 0 , ,
: () () = ( 1 2 ) 2
11. :
+ 3 = 4 + 2 = 2
12. ( = 90) = 30 .
( ). () () , = .
2 4 13. : 2 0 3 + 4
14. ( = 90) .
( ). : ( ).() = ( ).()
15. , ( )
( ). .
: 6 4. 10 . 1. f ( ) = 2 + + , 0.
: () f ( )
() f ( ) () ()
2 + + () () f ( ) = 0 () , , () () () f (7) () f ( ) < 0
2. () :
. = (1 + ) 1
() .
3. (, R). ,
. , ( ) , . : () ()()=()() () ()2 = ()()
4. () 2 + ( + 5) + = 0 R . 1 = 2 2 () : 1 +
5. 2 1 2 (2 ) + 1 = 0 1 2 .
(
)
: () . () 1 + 2 = 2. () 1 . 2 = 2 ()1 1 + = 2 1 2 2
6. (5,12), (2,5), (1,7). :
() () () ()
:
:
:
..
: :
2009-2010 : 04/06/2010 : 2,5
2010
. ( ) 4 15 12 . 1 . 1. : 5 x 2 + 3 x 2 = 0 2. : y =
x x2
3. 2 ( 1, 2 ) 4. ) x > 0 , : A = 64 x 2 3 3 x 6 39 x3
)
3 2 6
,
5. : 2 x 2 + 5 x 2 > 0 6. x1 x2 2 x 2 3 x 6 = 0 . ,
: ) x1 + x27. =
) x1 x2
)
2 x1 2 x2 + x1 x2
2 , 1800 < < 2700 , : 2
A=
4 3
8. ) 2 x1 = 2 x2 = 1 .
) R 0 .
1/ 4
9. :
x4 1 x x21 1 + 1 1 + = 2
10. 00 < < 1800 , :
11. = 900 , = + 800 , = 3 = 2 500 .
: ) x 90
+80
) , , , .
2-50
3
x 2 + y 2 = 10 12. : 2 x y = 1
13. = 900 , = 5 cm = 13 cm .
(
)
. )
.
) : ( ) = ( ) ( )2
14. ( 3 + 1, 2 ) ( 2,1)
( ) : x 2 y = 3 , R . :) ) , . ( 2 )
2/ 4
15. R , ( x ) = ( 2 ) x 2 2 x + 3
x R . 6 4 . 2 . 1. f ( x ) = x 2 + x + .
. .
6
4
32
-5
-1-2
1
2
5
10
) ) f ) )
) ) ) f ( x ) = 0 ) f ( x )
2. ) : x3 2 x 2 x 2 2 x 3 0 .
(
)(
)
) k R x 2 4k 4 5k 2 9 x 2k + 3 = 0 , .3. ( 2,1) B (1, 4 ) .
(
)
2 x + y + 5 = 0 , :) ) ) , 2 x 5 y + 9 = 0 . 3/ 4
4. : x 2 + 2 x + 3 2 = 0 , 0 .
) ; ) 1 () , ( 1 ) : y = 1 x + 2 ( 2 ) : y = ( 1 2 ) x k + 1 . ) , yy .
5. ) x1 =
3600 900 1800
(
) ( ) + ) ( 90 + ) ( 2700
(
0
+
)
x2 =
2 ( )
2 , : x 2 2 x + 2 = 0 .) 00 ,900 , , . ) =
(
)
4
, : ( 6 + x ) + ( 2 + x ) =
1 . x x
6. .
. : ) ( ) = ( ) ( )2
) . ) ( ) ( ) = ( ) ( )
4/ 4
2009 2010
2010 : : : 2,5
: 14/6/2010
: 1. .( ) 2. . 3. . 4. . 5. 4 15 12. 5 . 1. 2. 3. : 32 + 5 + 2 = 0 2 1 = 1 2 = 5.
: ) = 3 + 5 ) = -1 2 3 + 2 > 0
4.
:
5.
: 10 2 ) ) 5 7 3
6. 7.
(-1, 3)
( 2, -3)
= 4 cm A =3 cm.
4cm
:3cm
, , , .
-1-
8.
= 38 , . ^
9. :
4 2 + 5 = 2 + 5 6
10. 11.
: ( + )2 . ) 1 + 2 = ) 12=
2 =1 2: 6 3 = 2
1: = (3+1) +5
12.
32 2 + 5 = 0, : ) 212 + 31 +2 +32 = 3 6 +2
13. 14.
: =
= 2 7 . ) . ) . . 6 4. 10 . 1. : 2 +( +1) + 1 = 0 , 0 : (i) . (ii) (iii) . 7.-2-
15.
2. : = 3. ) =
( + 3)2 ( 2 3 4) ( 2 + 3 + 5)( + 5)
3 2703 +2 ) f -1() ) g() = 2+3 f-1 0 g .
-2-
' 6 4 . 2 .
1. (2,3), : +7 = 18 : - = 2. ) . ( 4,2) (1,-4) ) () . ) () . 2 5 , .
2. P() : P(1) = 0 P(-2) - P() 2- 4 R ) P() ) : log(4 p(x)-x + 9) = log5 + log(2 p(x)-x +1)2 2
3.
(=) =5cm , =11cm () B =30 // () . (). .
()
-3-
4.
, 4 68 , -36 o ( +1)2010 + ( +2)2010 + ( +3)2010 + ... + ( +10)2010 . 6 = lim x 2010 + 102010 ) 6 = 10 ) .
5. 2 = + = - : ) + = 2 ) + = 2(1 +2) 1 -22
6. f() =
ln (2 -2) + 2 , R -1 ) Na f. ) f =
3 2 + 4 = 16. f ( 2 , ln2 ), a , .
) = 1 =0 :
2[( -1)f() ] -1 = 2ln(x-1) .e 2+(-1).f(x)
(...)
-4-
-
. 2009-2010
- 2010 : : ' : 31/05/2010 : 07:45 . -10:15 . : 2.30' ------------------------------------------------------------------------------------------------------------------------------------: 1) . 2) ( ) 3) . ------------------------------------------------------------------------------------------------------------------------------------ ( 15 12. 5/100.) 1.
12 ( + 3 )( 2 )lim 2 4 x 2 2
.
2.
A3. :
)
3 = +3 3
) = e4
4. 12m 384m3. . 5. 29 -2. . 6. A(2,1) 3-4=12 B(2,). 7. =2 . . 8. :
5 5 = 8 2 4
9. = ln +
+ 6 (1,5) 4. .1/3
10. f ( ) = 2 + 3 , g ( ) = : f
5 , R { 4} . +4 g g , f
. A11. (3,6) (7,2) (3,-2). ) . ) 12. 1 1 1 1 + + + + = 1 2 2 3 3 4 ( + 1) + 1 13. .. (a ) 1 = = 0, 7 = 192 . ) = 1000 = 100 ) = .., .
1,
1, 3 = 12
14. :
, ,
= 2 2
1 + 2 2
15. . 8cm 6cm. 12 . 2cm, ( 6 4. 10/100) 1. 3 1 , 9 11 . ) . ) 8, 17 . 2. ) 2 = 4 - 1000 . ) 1 1 1 1 + + + = 1 , . 2 3 4 5
2/3
3. )
1 2 = 2 . 1 + 2
) : 1 + 2 = + 4 2 )
1 + 2 = 4 2
4. . 2+=8. +2=7 (1,5) : ) ) ) ) ) B5. ( = 90 ) = 2 . // . .
6. ) :
( ) = + , .
. : .
( ) + ( ) = 0
= 6 = 4 , . 2 2
) e = ln (0,1). 1 + 3
3/3
2009-2010
/ 2010 : : 28 / 05 / 2010 : 2.30 : 3
: . , . -, . , . :
, . . .
:
15 12. 5/100 . = 4 52 1
1. : 2. : 2-1 = 8-1
3. : 1 1 1 1, , , ,... 3 9 27 4. : log + log(-1) = log12 . 5. 20 : 3, 5, 7, 6. : f () =2 2 2 2 4
log 2
, .
7. : lim
8. f : f(x) = 3 + 5 = 0. 9. : 2 = (-)2 10. R , (): (2+1) 2 + 3 4 = 0 (1): 2 + 3 5 = 0 (2): 4 3 =0. 11. , , , (-)=1. 1
12. : log
log log 2010
= 2010 .
13. f () = 1 + + 3 . f(). , . 14. : + . = . + 2 2
15. : x + log2 (9-2x) = 3 .
:
6 4. 10/100 .
1. : f() =
3 . 2 1 i. : ) f(x). ) lim f ()x
ii.
) f(x) . ) f (x).
2. (2, 1) . ) . ) ( ) . ) ( ) . . ) . () 3. A = 90o =4cm, =3cm, () . 48 cm3. : () (). () .
2
4. : = x2 . lnx , x>0 . ) : x (d 2 d + 1) = 3x + 2 dx dx
) =x2 . lnx . 5. , . 10 30 . . 6. ) : = . ) : 5.3 2 = 5 6 . + +
.. .
. - (.)
. (.. ) .. . - .. .
3
: 2009-2010 - 2010 : : 31.5.2010 :
: 2,5 _______________________________________________________________________________ 4 : 1) 2) ( ). 3) . 15 12. 5 . 5x 4 ( x 2)( x + 1)
:
1) :
2) :
(3x 60) =
1 2
3) : log 2 (3x 2) = log 3 9 4) : ) = 2 x 5 3 x + 3e x ) = x ln 2 x + 1 x
5) 64 m 2 5 m . . 6) 6, 3, 9 .
7) :
5 5 = 4 8 2
8) : = 8 3 m 2 , = 60 = 8 m .
../2
-29) = f ( x) . (0, 5)
) , , x f ( x) ) , , f ( x) = 0 ) f (0)
10) : )im
x +
(
x +1 x
)
( x 2 4 x + 3) x ) im x 0 x3 9 x 9 x2 g ( x) = 3 + x . 2x 5 f g1
11) f ( x) =
)
f ) . g 12) 4 cm 12 cm . 36 cm 2 36 5 cm 2 . . . 13) ( , 2 ), > 0 1 , (2, 3) (4,5) ) = 3 ) . 14) ) log 2 [log 3 36 log 3 4 (log 5 5)4 ]3 =3 1 log1000 + ln 2 e log 2 [log 3 36 log 3 4 (log 5 5)4 ]3 ) : log 3 x log x 81 = 1 log1000 + ln 2 e ../3
-315)
2. . .
:
6 4. 10 .
1) S1 = 8 + 4 + 2 + 1 + ... S 2 = 3 1 + 1 + ... + 27 S ) S1 = 2 12 ) S1 (5 x 60 )
12 = S2
x
, x [0,180] .
2) 2 = f ( x) = x 2 + x + 2, (0, 2) f (1) = 6 . ) , . 2x 5 ) f ( x) = x 2 + 3x 2 , im . x 2 f ( x) 1 ) g ( x) = , g f . x+2 3) 120
, //, = = 90 , = 120 = 2 3 cm . ) 64 cm3 , . ) 6 cm , .
../4
-44) f ( x) = e . ) f ( x) + f ( x) + f ( x) + f ( x) + ... = 3e 32x2x 3
) : f ( x) + f ( x) + f ( x) + f ( x ) + ... = ln f (0) + 2 f ( x) ) = f ( x) 3x + 8 = 5 . i) x = ln 8 . ii) = f ( x) .
5) x 2 + 2 4 x 4 + 3 = 0 . (2, 2) . x = 1 . .
) (0,3) (0,1) ) : ) ) ) 2 x = 3 5 ) . . 2 )
6) : ) 6 log( ) + log16 = log 2 2 ) f ( x) = 2(1 ) ln x + ln x 2 2
1 ln 2 x = e 3. 2 x
.
2009 2010
- 2010: : .: 25 / 5 / 2010 : 2:30'
*** 4 *** : ) ( ) ) . ) . ' 15 12. 5 . 1) :
3x -11 (x +3) (x -1)
2)
3 x + 40 = 3 [0 , 360 ] .
(
)
3) 3 -96. : ) , ) . 4) 1280cm3 15cm .
.5) :
)
y = e4x 3x ,
) y =2x3 + 3x 1 22 5 7 = 5 4
6) :
7) = 3. 2 cm ,
.8) : )
x
lim
3x3 2x + 7 5 2x2
,
) lim+x 3
9 x2
5
.
. 1 4
9) 6cm .
. 96 cm3 , .10) (5, 3). H 4 + 3 -15 = 0 .
) . ) . ) K.11) f(x) =
2x +1 , x 2 . x +3
) . ) f ( ).12) f, g, h f(x) =
R . ) fog ( ). ) , (fog)(2) = h(2) .13) f(x) =
10a x +1 , g(x) = h(x) = 5x + 20 , x x +3
) (1) f x = 3 2x 32 = 0 . ) (1) f (2) g B (2, 3) , .
x2 15 g(x) = . x +3 3
14) (-1, 2), (3, 0) 8 .. =2+3. 15) 4x2 2 2 = 3 2 . ( , ) 2 4
. 2 4
' 6 4 . 10 . 1) () () . -1 . .. 2) (-2, 1), (-15, 2) +3+9=0. ) , ) (9, -6), : i. . ii. .
3) = = 90 , =4 cm, = 13 cm , = 9 cm = 90o , =15 cm. () 3 cm , ().
() 4) f ( x ) f (x) +2 x2 + f ' (x +2)
) f (x) .
) f ( x ) = x2 +2x + 4 , log 9f(x)-x 1 =1 + log 3f(x)-x 1 .
(
2
)
(
2
)
. 3 4
5) f ( x ) = ln (3 - e x ) x < ln3 . 2, f(), ln25 . ) = -ln2 . ) . ln10 .f x ) e ( ) + e2x = 5 .
) : f'(x) e f(x) f''(x) =
3 ex
2e x
.
6)
f(x) =
8x(1 + 8x) , 3 4x
g ( x ) = 2 5x -2x +6x 4x .
) f ( ) = 44 . ) g ( ) = 2 . ) g'(x) = 2 2 x . 3x f( ) ) lim 2 4 . x 0 x 2x
. 4 4
. -
2009 2010
2010 : : 27/5/2010:
: : 2.30
) . ) ( ). ) . ) .
: 15 12. 1 . 1. 3 96. . 2. : log x + log( x 1) = 1 + log 3
3.
d : dx
) = x 3x
) = x 4 x 2 + 1
4. (2 , 2) , (4 , 2) (2 , 4). ) . ) . 5. : (2 x + 20 0 ) = 50 0 6. . 48 cm2 . 7. :
x ) lim 2 0 x + 3 x
1 2 + 3x ) lim 0
1
8. :
x+2 ( x 1)( x 2 + x + 1)
9. :
1 4 a 2 a = a 1 + 2 a 4 a
10. ( = 90 0 ) = 60 0 = 4 cm. . .
11. 36 m3. 45 . : ) . ) .
12. ( 3 1 ) 2, .13. : (196 2 x + 98 2 x + 49 2 x + ...) (2 x + 5 x + 8 x + ... + 47 x) = 0 14. : f ( x) = ln( x 4) ) f g
g ( x) = x 2
) ( f g ) (3) 15. = x 3 ax + 2
. =1 a = 1 .2
: 6 4. 2 .
1. ) limx0
2 x2x + 9 3
) log 2 (9 x 1 + 11) = 2 + 2 log 4 (3 x 2 + 4)
2. ) 1 < x < 1 x 0 , = 1 2 x + 4 x 6 x + 8 x 2 x .
) x 0, 4 , 2 3 , 3 2 x . x2 . ex
3. =
2 x2 ) x1 = 1 .
) + 2 + =
) = x + 3
45 , , < 0 .
4. ) :
2 3 2 2 = 2 3
) :
x x = 8 2 x 4 5 x 5 x
) : 30 30 150 150 60 60 300 300 = 4
3
5. (1, 1) , (0, 0 ) (x , ) . ) x + 2 = 0 9 cm2
( > 0 ).
) ( 12, 6 ) , . ) .
6. = cm , =
2
cm
= cm . . ) 32 cm 2 , . ) .
30
45
4
.
2009-2010
- 2010 : : 2.30 : : 26/05/2010 : ) .) . ) . ( ). ) . : 15 12. 5/100. 1. o lim x2 + 5x + 2 x + 3x 2 + 4 2 + 3 5
2. : y =
3. A =
7 x 27 . x 6x + 52
4. : ) f ( x) = e x . x ) y = (4 x 2 3)3 5. : 22 x +1 3 2 x 2 = 0 6. 2 -8. 15 . 7. 10cm 12cm . .1/4
8. : 5 5 = 4 8 2
9. 2 x 5 x + 2 = 0 0 x 360o .
10. (3,1) (6,5). ) . ) (-6 ,8) .
11. y 2 + x 3 y 3x 2 + 3 = 0 dy ) dx ) x = 1 y < 0 .
12. . 9 cm 13 cm , .
1 13. log x = x (log 5 1) . 2 + x 1
d 2 d 14. = ln( ) : + 1 = 2 dx 2 dx 15. (1, 2) , (3,3) (4,1) . : ) ) .
2
2/4
: 6 4. 10/100.1. : = 3 ) 9 ) log 2 + log(4 x 2 + 9) = 1 + log(2 x 2 + 1)
2. ) :
2 + 3 = . 2 2 3
) y = 1 + 2 x . 1 dy y, , 2 2 x = x . 2 dx
3. 8. 14 log 8 = 2 . ) . )
= 6 ( + 4 )4. ( 3, 1) (3, 3) . : : 4 x 3 y + 9 = 0 .y = 2 x + 3 ,
) , . ) . ) . ) .
5. f(x) = x + x , . f = 6 f = 4 , 2 23/4
: ) , ) lim f ( x)x 0
6. . : =4cm, =8 cm, =5 cm =13 cm. . : ) ) .
A
B
. .. . . .
H . .
. .
4/4
2009/2010 /2010 :B :(): 1) 2) 3) 4) ( ) . 4 .
:26/5/2010 :2,5
15 12. 1 . 1. , f ( x ) x3 + x2 + x + 7 , -1 1. 2. 32cm 5cm. . 3. : x 2 + 3x 4 ) lim x 2x 2 + 1 3x 2x ) lim x 0 x 4. 50 3 m 2 30 . . 5. ( 2x 30 ) = x
[0,360] .
1
6. . V 1 , . 2 1 .. 2 i) V. ii) . 7. (-1,10), (7, 4) (-2,7) . i) . ii) . iii) .
8. 4m, 8m 15m. 4m. 10m. . 9. ) y = 2x + 1 + ( 4x )
dy : dx
) y = 2x + ln ( x )
10. :
2 = 1 + 2 1 + 2
11. x lo g
x
= 100
12. f ( x ) = 2e x +3 + 1. ) ) f f 1 ( x ) = x 13. 4 x + 4 x = , 4x = 4 3 14. = = 6 = 4 . . 15. log ( ) , lo g . i) =
(
2 , log
)
(
3 ,
)
0, 3, 4, 5 2
3 ii) 2
6 4. 2 .
1. y 2 + x 2 + 2x 4y = 5 . x=2 y 0. 12. 64m 3 . 4 m 3 m. , : i) . ii) .
13. (1,3), (-2,2) : 3++=0 , R. i) , . ii) =4 . 14. , R f, g f(x)=3x2+x+1 g(x)= 2x2+x+ , -2. 15. 3 cm. E . E AB. .
Z
2
:
6 4. 10 .
1.
(2,0), (4,5), (6,) R-{10}. i) , 8 . ii) =2 ) () . ) () () (). iii) () ()
2. f(x) =2
)
x x +x g(x)= 2 . fg x1 x 1 R f=g.
) f(x) = x + 1 + x 2i. ii.
(
)
2
. :
f(x) 1 + x2 ( 1 + 2 ) .f () + .f () = 4.f()
f (x) =
2
3. , R. ) ln 6 + f ln 5 = ln , : 2
f(x)=ln(x+-),
i. ii. e f ( x ) e f ( x ) =
: =
1 2 ) f (1,0), :
( )
( )
3
i. ii.
: -=04 f x 16 2 ( ) = 2 ln(2 e )
3
4. ) 0 x
2x x x = . , 2 2 ( 1 + 2x )( 1 + x )
) M i) 15 = 2 3 ii) : 1 ln ( 2x ) + ln ( x ) = ln ( 1 + 2x ) + ln ( 1 + x ) ln 3 2
0, 2
5. ) 70. 5 4 . .
+ 2 + 3 = ) , , , : 3
6. ABEZ . : i) ii)
A 6cm Z
8cm
16cm
6cm
E
8cm
4
5
:2009-10
- 2010 : : ( ) :26/5/2010 : 2:30
: . . (ippex). . ME : 15 12. 5/100 . 3 . 3 + 1 1. : . ( 3)( + 2)
2. , 20 . . 3. : ) lim 3 2 + 7 + 1 6 2 5 + 2 ) lim 0
5 3 + 3 5
+
4. ( 1 ) (1,4 ) (3,2 ) , ( 2 ) : 3 + 4 = 0 . 5. : 2 2 1 4 2 = 0 . 6. = = 90 0 , = = 3 cm = 6 cm . ( ). .
7. 12 m 384 m 3 . . 8. : 1 ) = ) = e ln (1 + 2 )
9. : log( ) + 2 log + 3 = 1 . 10. 32 cm 2 60 . . 11. : 2 ( )
(2 ) = 0 . 1
12. :
6 2 + 2 = 2 . , 2
13. : + = 4 R .
14. f ( ) = + 4 g ( ) = 4 . f f + g , f g , , f g ( g ).
2 15. ( ) 2 + 2 = 1 ( , )
) ( ) 1 : + = . ) ( 1 ) 2 : (1 2 ) 2 = 2010 .
: 6 4. 10/100
1. (1,2 ) , : 3 + 11 = 0 : + 7 = 0 . : ) ) ) ) : ) ) 2. (..) (..) .. .. .. 1 .. .. ) . ) .. . 3. f ( ) = g ( ) = ln (ln x ) x (1,+ ) . ) f g . ) 1 f = 2 g = , 1 2 . 4. ) : f1 ( ) = log ( 4 2 + 9 ) + log 2 f 2 ( ) = log ( 2 2 + 1) + 1
f1 (1) = f 2 (1) . ) ) : 2 = 2 ) , : 2 = 2 [0,2 ] 2
5. ) ) f1 , f 2 , f 3 R f i ( ) 0 (i = 1,2,3) .g ' f1 ' f 2 ' f 3 ' = + + g f1 f2 f3 ) 2 k ( ) = ln e +1 3 2 + 5 1 3 2 + 1
g = f 1 f 2 f 3 , :
[ (+1
) (
)]
) f ( ) = e 5 R = f (5 ln 3) + f (5 ln 3) + f (5 ln 3) + f (5 ln 3) +
6. 2, 2 B = 60 o . .
:
:
3
2009 -2010
2010 : : :25- 5- 2010 : 2,5
() . () ( ) () . 4 . 15 12. 1 . 1) :2 x + 16 ( x + 3) ( x 2 )
2) 17 . : ) . ) 11 . 3) : () = 2e5 x + ln x x + 5 () = x 3 (3 x 2)
4) : () limx4
x 2 7 x + 12 x4
() limx 7
x 6 1 x7
5) 64 cm 2 . To 30 . .
1/4
6) ( 3,3) , ( 0, 6 ) ( 2, 2 ) . ) . ) . 7)
1 + 6 6 + = 2 3 3 . 1 6 6
8) K R = 12cm . . 15 cm , . 9) f ( x ) =
x+2 . 3 9 + 8 3x 3x
10) ( K , R ) = 4 .
( ). .11) ( 1 ) : 2 x 3 = 0 ( 2 ) : 3 x + 5 11 = 0 2 . 12) f ( x) = x 2 4 x + 7, x 2 . ) f 1 ( x) , . ) f 1 ( x) x = 4 . 13) = 2 x
1 d 2 y dy 2 + 4 x ln 2 = 0 ln 2 dx dx
14) . = 60, = 8 cm = 5 cm .15) : 3 log 5 5 log 5 5 + log 5 4 5 ...
(
)
log 2 x
= 16log x 2
2/4
6 4. 2 . 1) (1,3) , ( 4, 1) . H 2 x = 1 . : ) . ) . ) < 90 . ) . 2) ( 3, 2 ) , ( 3, 6 ) ( 5, 4 ) . ( , , ) ) . )
.3) > 0 . To 18 . eln ( 3 ) log 2 8 = 5 = 3x 7 x = lim . x o 2 x 8 x 4) ) + = 4 = 4 . 2 2 = R . 2 ) : ( x 5 ) ( x 5 ) ( x 5 ) + 4 + 5 + ... . lim 3 2 3 4 x 5 x 5x x 5x x 5x 5) ) f f ( x ) = ln ii) f 1 ( x ) = 17 . ) : 4 log 2 ( x ) + log 2 (1 + 2 x ) = 3 , x ( , 2 ) . 2
(
x 1 + 1 x 1 .
)
i) f 1 ( x ) .
3/4
6) ) f ( x ) = 8 x, x [ 15, 7 ]
g ( x ) = x 2 9 x 7, x [ 1,10] .
(f
g )( x ) .
) = x x , x R + =
4
ln ( ln 2 x ) , x (1, + ) .
x = .
2
4/4
20092010
2010
: () :
: 2:30 : 04/06/2010
: 1. . ( ) 2. . 3. . 4. . 5. 4 .
: 15 12. 5 . 1. 1 = 3 2 =7 . : ) (11) ) ( 11) . 2. o o :x 2 + 2x 3 x + x 1 lim
3. : = 300 , = 3 cm = 3 cm . 4. : log(x + 1) + 2log 5x = 2 5. : 6. (2, 1) (3, 4) . : ) ) (). 7. 24m. 4m , : ) ) .1 43x ( x + 1) ( x 2 )
8. 250 cm3 . . 9. d :dx
) = 2x + x + ln4 3
x
) = e3x 4x
10. x2 + 2 = 25 , 0 = 3 . 11. : (5x 300 ) =
2x 2x
12. (1 , 4) , (2, 1) (2 , 1). : ) ) A .2 x + 1 = eln ( x ) .
13. :
14. f(x) = ln(x + 3) . : ) ) ) f 1 ( ) . 15. f (x ) =1 x 3
g(x ) = x 2 .
f g g f . .
2 4
: 6 4 . 10 . 1. ) : 32x 4 3x + 3 = 0 x [ , ] .
) , , : + + = .
2. ( = = 900 ) 1 13 4 4 , (4,4) , = 450
,
x+ = 2. : ) ) ) ) ) .
3. xx. 450 xx () = 2 cm. ()=3cm, ()=4cm ()=5cm xx : .
3 4
4. ) x log(x+1) , log (x 2 +9) ( x +1) , log7492
. ) x 2 x 3 x 4 x 5 + + + + ................ = 1 , 4 12 32 80
2 < x < 2 ,
x .5. ) f f(x) = 2A (lnx)2 lnx2 + 2 ( e , 1 2 ) . , , , . 2 ) : = 2 . 23 2 56. ) R , x 2 3x + 2 x 2 1 f (x) = g(x ) = . x2 x +1 x f , . ) h ( x ) = h x +3 x ) lim x 0 h ( x )
) g(x) + h(0) =
1 . 3
:
(..)
(..) (..)
4 4
. -
. 2009 - 10
E - 2010 : : : 25/05/10 : 7:30 -10:00 :1. . 2. ( ). 3. . 4. 3 .
( 2.30)
15 12. 5 .
1.
6 ( 1)( + 4)
.
2. =
1 . +2
3. = log ,
R + { } log + log( + 3) = 1 . 1
4. 4 (1005 ) 4 (1005 ) = (2010 ) . 5. ( , R). R 4 E 4 . 6. f ( ) =
2 + 3 1 g ( ) = , f = g . f g , 2 4 9 2 3 R f = g .
7. 6 3 15 360. 8. 3 =
3 + 3 . 2
9. (3,2), (3,4) (6,0). : ) , ) , ) .
1
10. 300 . 8m : ) ) . 11. : ) lim
3 3 4 81
) lim 1
1 + 3 22
12.
= . 2 2
13. f () = 2 2 ln , . ) = f ( ) = 1 . ) ; 14. :4 2 ) = ( e )
) = ln(
2 + 2 )
15. =90 , = 42cm = 56cm. . .
6 4. 10 .
A
B
1. 3 + + 3 = 0 . ) (1, -1). ) , .2
2. log 2 , log 4 , log16 ,... . ) .. = ) N
log 2 +log 4 +log16 + ... = 2 .
1 . 2
2
3. ) ( 2 + 2 2 ) 2 = 1 + 4 . ) ( 2 + 2 2) 2 = 3 + 4 4 2 2 4 . 4. (,R) (,) (,R) (,) . . 5. (6,4). = 4 . : ) , ) , ) .
6. 2
( = 90 ) . . : ) , ) .
3
2009 2010
2010 : : 26 2010 : 5 : : 2,5 . :117
: () . () . () . : 15 12. 5 . 1.
(
7 x + 1 x + 4 x - 5
)(
)
.
2. 5,15,45 ,... : ) ) . 3. 4. o o li mx +
= x2
+ 3 x + 5
- 5 x 4 x - 5 x 22
x
9 0 5. = ( 4, -2 ) , ( 6, 0 ) ( ,0 ) .
.6 c m 6. . 1 2 c m . :
) . ) .
1
7. x
lo g
x
=
1 0
.
8. 4 7 24 10 10 5 . ) . ) .
9
2 + 2 = R
.
10. V=972cm3 45. . = 2 x x '' + 2 ' + 4 x = 2 x
11.
12.
2 ( 2 + 2 ) + 2 ( 2 + 2 ) = 2
(1- )
2
1+2
13.
5
x
-5
2 x
+4=0
0 x
14. (,R) =3. . R.
2
lo g
2
15. f f x =
x
x3 -
2
-
x .2 3 ln
) f x . 2 9 2 lo g 2 1 2 ) f x x x
x+
3
=
x l+ 1 0 n
x
: 6 4. 2 . 1. ) i. . x > 2 ii. . ) :
y
x2
=
- 3 x + 2 x 2 - 2 x .
i. li m yx 2
ii. li mx 0
y
6 x
2.
=(+2)+(+3)+(+4)++(+25), 0
) . ) . ) =372, =2. ) (x-)=3 x (0, ).
3. : ) = 2x ) = 2x
A =
2 x + 2 x +
+ 2 x - + 2 x -
B =
6 x - 2 x 5 x 1 - 4 x
6 A B ) 1 - 2 x [ 2 - 2x(+x) + 2(+x) ] = 3
4. 1:
4 , 7 ( ) . : 3
2 x - + 5 = 0
2:
x - 2 + 4 = 0
) . ) . ) 2x + 5 = 0. ) .
5. x2+2=25 .5
) 5 5 3 , 3 2 x = 0 >0, ( 2 ) () ,
).1 -5 1 5
(). ) () xx , . -5
6. =2cm, =2 , =9cm =5cm. xx.
Z
x'
4
: 2009 - 2010
- 2010 : : 2 30 T 3 . : ) . ) ( ). ) . : 15 12. 5 . 1. x +1 . ( x 2)( x 1) e3x . x2 1 : : 2.6.2010
2. : () = x 3 ln(2 x) () = 3. : () limx 3
x2 x 6 3x 2 + 5 x 6 () lim 2 . x + x + 2 x + 1 x3
4. f ( x) =
x . x +1
: ) ) f-1(x). 5. 3, 7, 11, 15, . . . : ) ) 15 ) 20 .
6. :
1 2 = . 2
7. = 2 3 cm, = 2 cm = 120 .
: ) , ) ) . 1/3
8. x 2 + 2 4 x + 5 11 = 0 . = 1 > 0. 9. 384 , 2 8.
10. = 2 3 :
d 3 = . dx 2
11. : .
+ = +
12. , >0 1 :
1 log 45 + 4 log 2 log 81 log 10 = 3 . 213. 2. . 4 2 cm 2 . 14. 2+=3 3-2=8 135 . , . 15. R = 4 cm. ,
=
. 2
.
2/3
: 6 4. 10 . 1. ) 1, , 2 . ) log 5 + log 5 = 4 . log 5 , log 5 , . 2. 10 cm. . 3. + 1 = 0 + 3 =0. (7, 2) : () . () . () . () . 4. ( // ) = 10 cm , = 16 cm 4 cm 2cm. . 5. ) = e 2 x : 4 + 5 = 0 . ) f ( ) = +1 + 2 1 . f ( ) = 1. 2 + 9 20 = 0 . 6. log ( ) , log 2 ( ) , log 4 ( ) , . . .
. 4 2 ) . ) = log 3 .
.
6. ) : = 0 - - +
(
)
(
)
) : 1 - 2
(
1 + 2
(
2
2
)
)=
4 2
3
A
: 2009 2010
- 2010 : : : 03/06/2010 : 2,5
4 . : . . . . ( ). . . . .
' (15) (12). (1) .
1.
.
2. 3, 7, 11, 15, : () () . 3. 6% 3 . .
4. : 5. : 8 .
. .
1
6.
.
7. 24 , 16% . () ; () 9 . N . 8. , , :
9. , , .
.
10.
.
30%. ... 15%; 11. >0, 20 15. . 12. : .
13. : ()
12
.
() .
14. : .
.
2
15. . ,
.
' (6) (4). (2) .
1. 18, . . 2. ( ) ,
3. () : () x : .
.
4. 1800 6 . , 4200 4 15 . ; 5. 25% . 20% 600, .3
6. , , .
, . () .
() .
4
: 2009 2010
2010 : 25 05 2010 : 7.30 .. : 2,5
3 . : . ( ). . .
(15) (12). (5) . 1. 2. 3. 3, 7, 11, 15, ... . . 3500 5% 4 ; : ) 22 1 = 8
) 64 = 3
4.
R = 6cm. : ) , ) 30 . : A = 60, B = 30 = 4cm. ) . ) . 8500 10% . ; : ) 20 + 4 8 = 1 ) 16 =2 4
5.
6. 7.
1/3
8.
x + 2, x, x 1 . x . (, R) 4 = 5 2 cm. 4 . : : ( 2 ) + = 8 . , 10 31. . (, R) , . : 3 4 = 4 6 . 2, 4, 8, 16, ... 126. = 17cm = 12cm. . . = . 2 2
9.
10. 11. 12. 13.
14. 15.
(6) (4). (10) .
1. 2.
10000 25% . , ... 15%; . 20, . R = 4cm. AB = 90, = 60 = 120 . .
3.
2/3
4.
) : 5 +1 + 5 6 = 0 . ) : 4 + 4 16 = 6. 90 2 3
5.
. : ) : 2 = 2 + 2 , , ) : = 2 , .
6.
: = 90, = 4cm, = 45 . AEB . .
3/3
2009-2010
: : : 26 /5/2010 : 2,5
: ( ). . . T 3 . : 15 12. 5 . 1. x x + 3 , 2 x + 1 , x + 9 . 2. , , 15000 . 3.5% , 8 ; 3. 60 4 cm . . 4. 820. 15%, ; 5. : A =
log 2 8 + log 3 81 + log10 log 5 5 + log107
6. =
3 cm, = 1 cm 2 =60. .
7. a2 = 108 a5 = 4 . . 8. :() log 2 ( 3 x + 5 ) = 3 () 9 x 8i3x + 15 = 0
9. 4 R 2 2 = 0 ,
1
10. 12 cm. , : () ()
11. log 2 x 2 log x + 1 = 0 . , a22 = 212 . 22 .. 12. 1 1 2 2 + 2 = 2 4R 2 2 2 13. 240.000. 1 2009 40% 5% 3%. 1 2010 249.000 ; 14. : 4log3 x 9 2log3 x + 8 = 0 15. , , 2 + 2 =60 , 2 = 2 : 6 4. 10 .
1. 52 210. . 2. : () log 2 log 2 ( 4 x + 2 x ) = 0 2+log x (log x3)
() 5
=1
2
3. 1 2009 (, , ) 120.000 , . 14 , 22 24 . 4.5%. 22% . 3 4% 4 3.5%. 1 2010..4. (,R) =4 3 cm . (,R). = 120 , .
5. ( =90 ) : = 4 R 2 ( 2 + 2 1)
6. :
log x log x log x log x log 36 + + + + ........ = . 2 4 8 log 2 log 2 log 2 log 2 log 2
3
20092010
2010
:25/05/10 /:_____________________________:________ :___________ :_______________ :2.30/:_______________
: )(). ). ). :
1215. .
1. 1 = 2 = 3 . : ). ) 26 ( 26 ). 2. 6005%. ;
20092010
3. 36005%10; 4. 2 x4 = 8 x 5. 18+523=1 6. 10cm: ). )72. 211
20092010
7. x 3, 4, x + 3 . )x. ). 8. = , . 9. : 4 x 7 2 x + 12 = 0 311
20092010
10...:=
1 = 27 .(4). 3
11. 81m2. )N. )9m2, . 411
20092010
12. , = 5 3cm , = 10cm = 5cm .: ), < 90 . ). .
13. 6%26720; 511
20092010
1 14. x x, , ( x 3 ) 2 .
15. ,( = 90 )=20cm,
A = 30 .. B
A
611
20092010
:46. .
1. 31 . ). )10.
711
20092010
2. . 9 9 ,: 8 2 ) 1 (>0). ).
811
20092010
4.: ( 7 x+1 + 7 x1 ) = 1 + 5
3. 6000. 25%...10005 ...15%, (%);
911
20092010
5.) = , A = 90 .
)=2,.
1011
20092010
6. R = 6cm = 3cm .
, : ), ).
A
M
B
K
.
:: (...)
1111
2009-10 AIOY IOYNIOY 2010 M: : B : 31/05/10 : 7:45 . 10:15 .
: ) . ) ( ). ) . ) . : 15 12. 5/100. 1. 5,8,11, : ) , ) . 2. 6m. . 3. : ) 5-2 = 625 ) 32 = 5
4. A -1, +5, 2-3 . 5. 500. 20%; 6. : = 4+12+36++972.
7. : (+3)-2 = 8. 5000 2 400. 9. (.) . 10. :
2 8 + 10000 = 5 125 2
11. 81 m2. 60. 12.
= 30 0 .
+ = + 2
13. = 8 3 cm2 = 8cm = 4cm.1
14. 6 36. >0 . 15. 16 cm. N .
: 6 MONO 4. 10/100.
1. 20, 26. ) . ) . ) 725; 2. O 2 5% 6%. 5 3 11680. .
3. : ) )
4 x + 4 x = 23 x +1 = 4 x + 2
4. 90000. 20% 8000 . 14 18 . : ) ; ) ; 5. ) + =2R, .
2
) : 2 + 2 2 = 2
6. = = 20cm , = = 36 0 = 8cm . . .
: . . . . .
3
. 2009-2010
/ 2010 : : 28 /05/ 2010 : 2 30
: . , . , . , ., .
:
- . . - . - . 4
:
15 12. 5/100 .
1.
: 100, 96, 92, 88 : ) . ) .
2. 5 : 6, 12, 24, . . . 3. 5% 600. 4. x : 2 , x 1 , 3x + 1 . 5. :3x-2 = 27 x
6. : ) 36 = 2 ) = 1
7. : + 4 = ( + 6 ) .
8. : 9 , 3 , 1 , 1 , ... 3 9. :
4 x +1 - 4.2 x - 8 = 0
10. 500 460. (%) ; 11. E = 3 3 cm 2 = 3 cm = 4 cm. ( < 900 ) .12. :
+ + =
13. 20 cm. : () () .
14. = 7cm , = 7 3 cm , = 60 . .
15. , () = () = 8 cm , .
: 6 4. 10/100 . 1. : x.(x +2 ) 8 = 0 2. 3 7 34 6 1 .
3. 48 . () . () .
4. 5 :
3 5 4% 3.5%. 4% 3800.
5. 1000 m2 30% . , 15% . 120 m2 : () . () (%) 6. = 90 .
(
)
. = 4 2 cm , .
..
.
. - (.)
. - .. .
: 2009-2010
2010 : : B 3 : 1. . 2. ( ). 3. . : 15 12. 5 . : 31/5/2010
1. : 2, 5, 8, 11 , : () () (12) 2. 4000 3 5%; 3. 64 cm2. : () () () 900 4. : () 327=x () 2x=16
5. = 6cm2 =3cm, =8cm. (1 .. 14. 2= 15. , . 4cm 300.
B:
6 4. 10 .2 8% 9%. 5 3
1.
12750. 8 %0 2. 0 (==90 ), =4 2 cm, =12cm 45 . . .
3. : = 2 3cm, = 4 cm 450. 4. 30 . 30 . 30000. 20% . 1794. 15%. . 5. : 10, 15, 20 ... : () 105 () () 6. : (+3)=2+6
: 2009-2010
MAOY- 2010 : : : 25/ 5/ 2010 : 2.30
*** 3 *** : . . ( ).
15 12. (5) . 1. : 1 , 3 , 5 , 7 , ... . : () () 150 . : 34 x1 = 27 . 16 cm2 . . 5000 , 2 8% . : = 216 5 5 + 0,1 + 13 61 + 100 . : (3x + 5) 2 = x + 4 . : = 5 cm , = 5 3 cm , = 30 . :
2. 3. 4. 5. 6. 7. 8. 9.
() ( < 90 )
() .
15% . 16000 ; = , . . 1 3
10. . = 8 . 5 . 11. ( , R ) = 90 , = 60 , = 90 . R . 12. 90 8 cm. 13. x = 4 x 3 2 x+1 16 = 0 , y = 3 y = 2 , , , . 14. ( , R) ( , ) , R > . = 3 = 8 3 cm ( , ) , : () R . () .
i
15. .
1 3 : 100 20 . 580 ,
.
: 6 4. (10) . 1. . 32 . : () () 10 . : () 4 3x +1 5 x = 4 5 x + 3x + 2 () 3 (4 x 10) = 2 + 3 (2 x 2) . 2 3
2.
3.
. 55000. 15% 2 4 % 5 6 %. 12155 ;
, :
4.
()
+ 1 = . 2
() = 2 cm , = 2 3 cm = 30 .
5.
= 8 cm = 6 . : () 1 2 = () = 8 3 cm2
1 2
6.
3000 85000 20% . : () () 20% .
. 3 3
. - : 2009-2010 : : : 27/ 6 / 2010 : 2.30
: 1) . 2) . ( ) 3) . 4) .
: 15 12 . 5 . 1. 25 cm2 . : () . () . 2. : x+1 () 5 = 125
() log 16 = 2
3. x , x+3 , 2x+4 . . 4. 780 . 15%. ;log 24 log 3 =2 log 2 + log 2
5. :
6. =3cm , = 3 3 cm = 600 . o . 7. :1 1 1 1 + + + + ... 2 6 18 54
1
8. 1600 3 . 160. 9. :
log( x 2) + log( x 3) = 1 log 5
10. 800 18 cm2 . . 11. . . =
12. - 64 8. 13. :
=
14. R= =4 . 15. : 22 x 1
2 x+2 + 6 = 0
: 6 4 . 10 . 1. () E = 2 3 cm 2 , = 4 cm = 2 3 cm . () 2 = 0 , . 2. :x x () 9 2.3 15 = 0
()
log(log(2 x + 2)) = 0
2
3. () : 3 + 7 + 11 + ... + 51 () .. 26 33 . . 4. : 5 35% 10%. , 6800.3
5. () 989 ... .. 15% , ... () .. 8 2. .
6. ==10cm = 1200 . . (: ) .
3
.
2009-2010
/ 2010 : : : 26/05/2010 : 2.30
: ) .) . ) ( ) ) .
: 15 12 . 5 . 1. : 5,8,11,14.. 20 . 2. 6 cm.( ). 3. , 1 , 2 . 4. : ) ) 4 = 3
5. 3700 6% , 20 . 6. , :=
48 6 12 6
7. 3, -9 , 27 , -81 , . . 8. = m, =1m B = 300 . : ) 900
(
)
) . 9. 600, 10 cm. 10. :
+ ( 2) = 3
1
11. :
+ 1 = 2
12. 25% 675 . . 13. :
4 6 2 16 = 0
14. 18 , 22. 15. : = = 7cm, = 60 .
: 6 4. 10 1. :
2 2 5 + 2 = 02. 68, 34. ) . ) . 3. 2 : 8%. 1088 ; 4. ) =300 . :
6%
+ = + 2
) = 2 . , .
2
5. 25000. 12% . 15% . ;
6. = 90 , = 60 =12 cm . , : ) , ) .
60 A B
.. ..
. .
.
3
2009/2010 / 2010 : B : ( ) : 26/5/2010 : 2, 5
: 1) ( ) 2) 3) . 4) 3 .
15 12. 1 . 1. x x+2, 3x-2, 4x-3 . 2. 16 cm, : () () 80 . 3. : (x-3)+2=(x+2)-3 .3 4. : 12+6+3+ + . . . 2
5. 4 x 9.2x + 8 = 0 .
6. 644. 15% .... ..; 7. 12000 4% 800; 1
8. 1:250 000 20 cm . Km . 2 9. . 3 18m, . . 10. 48 6. . 11. -=, . 12. : 4 16+0,04 =2 . 61+ 5 25
13. 600. 8% . , 1480. 14. : ( ) + (-) =(-). 15. , =10cm =6cm. .
6 4. 2 . 1. A 26 . : ) ) 100.
2
2. =4 3 cm2, =4 3 cm =4 cm ( 900 ).
3. 4 x + 6 = 5 + 2 4. 25% . 20% . 60 : ) ) .
(
)
5.
2 12% 10% 2 5 72 . ;
6. : = 8cm .
= 60 , .
: ( )
:
3
2009 2010
: _____________________ : _________________ : ___________________
2010 : : , 25.5.10 : 7.45 .. : 2,5
: ( ). . 3 . .
( 12 ) 15 12. 1. : () 2132 x = 8 , () 81=2 , () = 3
2. 20% 888. ; 3. 12, -6, 3, .. 4. 100 40 cm2. , . 5. 7% 3 4 980 ; 6. : ()5 15 = 6
()
3 2 1 = 7 9
()
4 = 25 3 12
7. 2+10 , , 3-25 , 4 = -2. 8. = 10 3 cm , = 10cm =60 . .
9.
= = 3 + 2 = 130 , , . 5 3 7
10. = . . 11. 2 2 7 + 3 = 0 12. (.) 5 (5) 162 8 (8) 4374. , 10 . 13. = 7 , = 9 3
= 2 ,
2 4 5
.
14. ( = 90 ) (=) = 16 2 cm . . , , . .
A
B
15. 6 (6) 20 12 (12) 258.2
( 8 ) 6 4. . 1. : () (3 2) + ( 1) = (2)
() 25 x 3 5 x 10 = 0 2. 2 8 20 = 0 (1) (.) () .. () . () 12 = 7 . 3. = 4cm , = 4 3 cm = 4 3 cm 2 . () , () .
4. 20700. 20% 15% ... : () ... () . () . 5. =26 m. 8 m = 24 m. .
6. () (.) 6 2. (i) 3 8 , (ii) 6+12+24++768. () 4500 3 5445. : M 3
. 2009 10
: : : 2.30 : . . . 3 . : 26/05/10
: 15 12. 5 . 1. 2, 8, 14, 20, .... . : ) . ) 31 . 2. : ) 2 5 x 1 = 16
) 5 5 = x
3. (, R) 64 cm. : ) . ) . 4. 12000 4 9%. 5. : x + ( x 1) = (7 6) 1
6. R = 5cm. : ) . ) . 7. 15 % 920. . 8. 9. .
1 2 3 36 + 6 2 =2 9. : 12 4
10. = 6cm, = 6 3 cm, = 60 . .11. 8 64. 10 .
12. : E = 2 R 2 13. 31,4 cm. .
14. 45 18 m. . 15. 1 3 8% 15400.
: 6 4. 10 . 1. 18 . () () 165;
2. : ) (9 x 1 + 7) = 2 2 + (3 x 1 + 1) ) 2 x +
1 =2 x
3. 34500 ... 15%. .., 3 : 7% 4 4%. 11250;
4 ) . =5cm , B = 600 1 cm ,
) + = 2 R .
5. :
x x x x 25 + + + + ... = 2 4 8 3 3 3 3 36. R. H AB , 30. R ,
.
30
:
.
:
2009 -2010
2010 : : :25- 5- 2010 : 2,5
: () . () ( ) () . 3 . 15 12. 1 . 1. 0, 3, 6, 9, . . . . : . . . 18o . 2. 5 cm . : . . . .
3. : . 33 x 7 = 9 . x 36 = 2
4. 3000, 4 6%. 5. +2, , -1 . 6. x ( x 3) = 1 . 7. : = 30, =4 3cm =4cm. : . . . . 8. 600 , 510. % .1
9. 4 cm , , .
10. : 4 2 + 5 8 = 60 6 11. :
32R
= 3 .
12. 27 3. : . . . . 13. 20000. 10% , , 8 10. ; 14. 36 cm 2 12 cm 2 . : . . . T . 15. 4 . 3 . 6 4. 2 .
1. :
9 x 2 3x +1 + 5 = 0
2. 20, 6 . : . 1 . . .
2
3. . , 23000. 15% 4 : 5% 5 6%. 3120. 4. . = , .. :
2 x 2 x 3 = 0 .
5.
( = 90 ) =3 cm =6 cm . . .
6. 255 : 1 , 2 , 4 ... . . . 2 , ;
- -
3
20092010
2010 : ( ) :1. 2. 3. 4. . ( ) . . .
: 2,5
:
: 04/06/2010
(60 ) 15 12. 5 . 1. 1 = 5 = 3. . 2. 7 3x 2 = 7 x + 4 . 3. R=4cm . : ) ) .
1 . . 2 5. (2x 1)+3=9 .4. .. 1 = 3 =
6. 2400 4%. 2 ; 7. 13 cm2. R=6cm, . 8. = 2 3 cm2 , = 60o =2cm. .
9. 4350. 650 4500. % ; 10. 3, 7, 11, . . . 20 .
1 3
11. R. 4 R = , .
12. 8cm. . .
25 x 6 5 x + 5 = 0 .
13. :
14. 2x 2 , x , x 3 , x .1 32 + 144 8 2 2 =1 15. : 2 + 3
(40 ) 6 4. 10 .
1. : ) 4x = 2 ) 2 x + 5x 6 = 0
2. 2 5 5, 3 8 17. ) ) 12
3. 6%, 2 4025 ;
1 2
2 3
4. ( = 90 )
= 60 =12cm. . .
5. 3000. 5% . ) ; ) 20% , ; ) 15% ... , ...
6. ) = 30 , = 8 3 cm =12cm. . ) =, .
:
(.) (.) (..) -
3 3
. - : : : 25/05/10 : 7:30 -10:00 :
. 2009- 10
E - 2010
( 2.30)
1. . 2. ( ). 3. . 4. 3 . 15 12. 5 . 1. 32x = 81 . 2. 3, 6, 12, ........ : () () . 3. =8m, =2m = 30 . 4. :12 - 2 1 = . 9 + 2 2 2
5. 100 5 cm. . 6. 4500 . . 7. : [ ( 2-6) ] =0. 12
2 km 2 1 km,
8. 75% . 144. : () () . 9. 4% 10 5600.
10. ..
+
=
, ,
11. 20m . 364 m. (%) ; 12. 3x +1 28 + 9 3 x = 0
=1 14. ( 01 ) 13. : 6 . 15. . 1 36cm, 2 33cm, 3 30cm ... () 8 ; () ; 6 4. 10 . 1. 640. () 20% . () ... 15%; 2. 20 8. ) . ) 120.
+
25 . 4
2
3. () = . () = 30 = 2cm . 4. K :
1 3 6% 8%. 8% 2400 6% , .
5.
2x + x
2 + x3 +...........
=2 2
1 x 1 .
6. 8cm , , . EZ . .
............................ ......................... .............................. .............................. ..............................
3
2009 2010
2010 : : : 2,5
: 26 2010 . : 176
: , , , , ,
. . . . . . To 3 .
15 12. 5/100.1. 2.
2 16 = x 2,6,10,14 : ) ) 51 3+6+12+.+384 6% 3 720 . 900 15% . . 49 cm2. : ) ) . +15,2+3,3 . .
3. 4.
5.
6.
7.
1
8.
=10cm. =5cm. = .
.
9.
4cm 4cm2. : ) ) : = 2 . 7 .. .
10.
11. 12.
13.
7
64
4 8.
14.
4 x +1 4 . 2 x 8 = 0
15.
=12m , . . : ) ) .
6 4. 10/100.1.
7 2 7 19. ) ) 30 . 6% 8% . 2
2.
1156 .
3.
) =12 cm, =6 3 cm = 600. ) 2 + 2 = 2 .
2
4.
=6 , =3 , =8cm. .
5.
150,000 15%. 20% 13,21,30. .
6.
)
)
3
2009-2010
-------------------------------------------------------------------------------------------
2010 : : B : 02- 2010 : 2 30
: ------------------------------------------------------------------ : ----------- ---------- 12 . : ) . ) ( ). ) . : 15 12. 5 . 1. 6 cm, : () () , 60 .
2. 2, 7, 12, 17,.... . : ) (30 ) ) ( 15 ) .
1 ... /12
3. : ) 4 = 3
) 63 1 = 1
4. 30% 5200. ;
5. 2-3 , 2+1 , 3+1 , .
2 ... /12
6. . 3 , . 2
7. : (3+2) + 2= (45+70) -1
3 ... /12
8. ( 900 ) : = 3cm,
= 3 3 cm B = 600 .
9. ( ) :
3 + 3 4 12 = 2. 2 2
4 ... /12
10. = 8cm. To . . .
11. = 10cm = 22cm = = 10cm. .
5 ... /12
12. : + = 2
13. 2 9 6 729, 4 ( ) 0 .
14. 600. 8% . , 1480.
6 ... /12
15. R, (24 + 8) cm. .
: 6 4. 10 . 1. :
() 16 -3 . 4 - 4=0
7 ... /12
() 2 2 - 5 + 2=0
2. ) , 112 13, 9 6 . ;
) 3, 9 x , 27 ..
8 ... /12
3. 19. 16. ( ): ) 51 ) 144.
9 ... /12
4. ) . = , .
) = 5 cm, = 8 cm = 10 3 cm2, ( < 90 ) .
10 ... /12
5. 8000 20% . 3 8 1776. ;
11 ... /12
6. = 6cm = 30 0 . . . .
:
12 ... /12
:20092010
2010 ::25/05/2010:7:45.. :() :2.30 : ) . ) . ) ,. ) 3. : 1512. 5100. 1) 3040. ...15%,...; 2) 3,7,11,15,.... : ) ) . 3) 3cm 3 cm2. . 4) 4%23 324. 5) : ) 5719
= 25
) ( + 5) = 1 .
6) , - 2, + 2, + 4 . 7) 15000 Km. 1:100000000, cm.
-1-
1 8) : = 3 9 - 2 + 20, 01 . 8 2 2 9) : 18, 2, 9 , 81 , . . . .
10) =20cm,=10cm = 60 . ) . ) .
11) 2 = 4 7 =:
1 . 8
). ).
12) 6% . 4794 . .
13) = 12 cm . = 45 , .
14) 2 8 38 14 3 . 1 . 15) + = . : 64. 10100. 1) : ) )4 - 7 2 +12 = 0 ( + 3) (1 2 ) = 1 .
2
2
2
-2-
2) 28.3cm,5cm, 7cm... : )28 )28. 3) , , = 4 cm
( ) = 2 ( ) .
.
4) : ) + = ) ( ) + ( ) = ( ) . 2R 5) , , 48000 3 5 . . 428%. . 6% 4 .4.
6) . 50 600 . 800 240 . , 20 800 . 25%.
-3-