b lukeiou algebra theoria askiseis
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8/22/2019 B Lukeiou Algebra Theoria Askiseis
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:
[ , y= (x),
y=(x).
[ : x =,x=, x= .
[ - ,
-
.
[ .[ .
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10. -
: -
+
+
+
+
+
-
-
-
-
-
+
+
V
-
+
-
-
o o o o o o0 30 45 60 90 180 270 360
30 2
6 4 3 2 2
1 2 30 1 0 1 0
2 2 2
3 2 11 0 1 0 1
2 2 2
30 1 3 0 0
3
33 1 0 0
3
()
(rad)
2 2 1+ = R
= R
, 0
= R , 0
1 = R , 0
2
2
2
1 =
+ R , 0
22
1
1 =
+ R , 0
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11. -
x
x
x
x
x
x2 90 xx
x
x
x
x2 +90 x+x
x
x
x
x180 x
x
xx
x
x+180 x+
x
xx
x
3
x2 270 x
xx
x
x
3
x2 +270 x+
xx
x
x
2 x360 x
xx
x
x
2 x+360 x+
x
x
x
x
f , *T R+
x A :
i. x T A, x T A+ , ii. ( ) ( ) ( )f x T f x T f x+ = =O T f.
-
x (x rad)
- :
x = (x rad)
-
2 :
(2+x) = x, x R -
2.
[0,2] -
.
-
x (x
rad)
:
x = (x rad)
2., :
(2+x)=x x R
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12. -
-
2.
[0,2]
.
-
.
:x
f(x) x
x
= = A {x R : x 0}=
x : ( + x) = x,
x A . .
x () 2
x
2< x +
x2
= -
f f(x) = x.
f(x) = (x), , > 0 g(x) = (x),
, > 0
1 x 1 1 x(x 1 ) > 0 :
(x) - f (x) f - .
f :2
T
=
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13. -
x x 2 = = + x 2 = + ,
x x 2 = = ,
x x = = + ,
x x = = + ,
( + ) = - ( - ) = +
( + ) = + ( - ) = -
( )1
++ =
( )1
=
+
1
( )
+ =
+ 1
( )
+ =
1. 2 2=
2. 2 22 = 2. 22 2 1= 2. 22 1 2 =
3.2
22
1 =
4.
2 12
2
= 5.
2
22
1 =
+
6.
2
2
1 2
1
=
+7. 2
1 2
1 2
=
+8. 2
1 2
2
=
9. 21 2
2
+= 10. 33 3 4 = 11. 33 4 3=
2
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14. 1
[ ]
1 ( ) = +
2 ( ) =
(1) :
[ ( )] ( )
[ ( )] ( ) ( )
= + = + =
3 ( ) = +
(/ 2 x) x = (/ 2 x) x = :
( ) [/ 2 ( )] [(/ 2 ) ]( / 2 ) ( / 2 )
+ = + = == + = +
4 ( ) =
(3) :
[ ( )] ( )+ =
[ ( )] ( ) ( ) + = + =
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15. 1
5
( )
1
++ = , ( ) 0 0 0 .
( ) ( )
( )
+ ++ = =
+ = ( 0 )
1
++
= = +
6
( )1
= +
A (5) :
( ) [ ( )]
( ) [ ( )]
1 ( ) 1
= + +
+ = = +
[ 2]
7 2 2
2 ( ) 2= + = + =
8 2 2 2 22 2 1 1 2 = =
2 22 ( ) = + = =
2 2 2 2 2 2 2 (1 ) 1 2 1 = = + =
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16. 1
2 2 2 2 2 1 1 2 = =
9 22
21
=
2
22 ( )
1 1
+= + = =
102 1 2
2+=
(2) :
2 2 2 1 22 2 1 2 1 2
2
+= + = =
11
2 1 2
2
=
(2) :
2 2 2 1 22 1 2 2 1 2
2
= = =
122 1 2
1 2
= +
22
2
1 2 1 22
1 2 1 2
2
= = =+ +
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17. 2
.
2003. 1.1 : 5, 6, 7, 8
: 2, 3
1.2 : 5, 6, 8, 9, 10, 11, 12
: 1, 2, 3
1.3 : 6, 8, 10, 11
: 1, 2, 3, 4, 6, 8
1.4 : 6, 7, 8, 9, 10
: 2, 3, 4, 5, 6, 7, 8, 9
.
2:
:
1, 2, 4, 5, 7, 8
3:
:
2, 4, 6, 9, 13, 14, 16, 18
4:
:
2, 3, 5, 8, 9, 10, 12
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18. 3
1. :
2x
f(x) x 0 , 3
= + > R Ri. f 3 C
f
y y (0,1), ,R .ii. -
.
iii. f xx
.
:
i. i H fC y y (0,1), ( ) fM 0,1 C f (0) 1 1 = =
i xR :2x
1 13
( 0> )
2x
3
( 1)
2x
1 1 13
+ + +
1 f (x) 1 + + f 1+ , 1 3 2+ = = :
2xf (x) 2 1 x
3
= +
R
ii. i H f 2
32
3
= = .
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19. 3
[ ]0,3 .
f :
f :
f :
iii. [ ]f (x) 0 0,3= :
2x2 1 0
3+ =
2x 1
3 2 6= =
2x 2
3 6
= 2x 7 7
2 x 3 x 3
3 6 4 4
= + = = +
:
0 3 34
7
0 3 34
+
3 3
4 4 +
7 73 3
4 4
1 13
12 12
7 5
12 12
1= 0=
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20. 3
f x x 11
,0
4
x 34
= 1= , 7
,04
7
x 34
= + 0= .
2. , -
f
: ( ) ( )f t t + , f(t) -
.
i. .
ii. -
f(t) 100.000 ;
iii. 100.000 ;:
i. O f :
( )f(t) t 0 t 8 0= + >
8 :2
8 4
= =
:
1 t 1
4
t
4
t 4
+ +
f (t) +
25 f ( t) 125 , : 25
125
= + =
2 150
125
= =
75
50
= =
f :
f (t) 50 t 75 0 t 84
= +
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21. 3
ii. :
f ( t) 100= 150 t 75 100 t4 4 2
+ = =
t 4 6
=
t 2
4 6 = +
t 2
4 6 = +
2
t 83
= + 10
t 8 3
= +
[ ]t 0,8 :
2 2 220 8 8 8
3 3 3
0
10 10 140 8 8 8
3 3 3
+
= +
:2
t3
= 10
t3
=
iii. : 1 t 5
f ( t) 100 t4 2 6 4 6
2 10 2 10t t ,3 3 3 3
> > < <
< <
3.i. : ( ) ( ) 2 2 x y x y x y 1 = + ii. :
1 x x
6 6 2
+ = iii. :
2 2 1 = + , .
:
i.: ( ) ( ) x y x y+ =
( ) ( )x y x y x y x y= + =2 2 2 2
x y x y= =
( ) ( )2 2 2 2 x y 1 x 1 y= =2 2 2 2 2 2
x y 1 y x x y= + + =2 2
x y 1= +
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22. 3
ii. :
1 x x6 6 2
+ =
2 2 1 x 16 2+ =
2 3 1 x 1
4 2+ = 2
3 x
4= 3x
2=
3x
2=
x
3=
2x
3=
x 2
3=
2x 2 ,
3=
iii. : 2 2 1 + = + 2 2
1 + =
( ) ( ) + =
( ) =
( ) 0+ =
( )[ ] 1 0+ =
0= ( ) 1 =
90= =, . 90= .
4. .
i. :
12 2 2 2 2 2 + + =
ii. 1
2 2
= 12 3= , .
:
i.: 180+ + =
902 2 2
+ + =
902 2 2
+ =
:
902 2 2
+ =
2 2
21
2 2
+ =
12 2
1 2 2 2
+=
1
2 2 2 2 2
+ =
1
2 2 2 2 2 2 + + =
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23. 3
ii. 1
2 2
= 1
2 3
= i. :
1 1 1 1
2 2 3 2 6 + + =
3 2 5
2 2 + =
1
2=
45
2=
90= .
5. 5 24 2 12< < = .
i. : 5 .
ii. ( )3 x 3 x + = , 3 225 3
= .:
i. :5
212
= 2
2 5
1 12=
25 5 24 = 25 24 5 0 =
y
= :
25y 24y 5 0 =
( ) 224 26y
25
=
24 26y
10
=
y 5= 1
y5
= . 5= 1
5
=
4 2
< < 0> , 5= .
ii. ( )
2
2 2
x 3 x 3 x 3 5 3 x
1 x
x 53 5 3 x 3 x 15 5 25 x 3 x 15 x
1 5 x
15 x 25 x 10 0 3 x 5 x 2 0
+ = + = +
+ = + + = +
+
+ = + =
y x= :
23y 5y 2 0 + = ( )5 1
y
2 3
=
5 1y
6
= y 1=
2y
3
=
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24. 3
x 1= 2
x
3
=
x
4
= + 3
x
25
= + Z .
6.. :2
33 12 57 782
=. : 22 67 1 22 67+ = . : ( ) ( )2 2 45 45 2 =. :
3 3 2
8 8 8 8 4 + =
:
. :
33 57 90 + = i 57 33 =
12 78 90 + = i 78 12 =
33 12 57 78 = 33 12 33 12 =
( ) 33 12 + =2
452
=
. : 23 67 90 + = i 67 23 =
:
22 67 1 22 67 + = 22 23 1 22 23 + =
22 231
1 22 23
+ =
( ) 22 23 1 + = 45 1 = ,
.: ( ) ( ) ( )[ ]
( )
2 2 45 45 2 45
90 2 2
= =
=
. : 3 3 2 2
8 8 8 8 8 8 8 8
1 1 1 2 2 2
8 8 2 8 8 2 4 2 2 4
+ = + =
= = = =
7.i. : 8
x 3 x3 3 5
+ + + =
, :4
x
5
= .
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25. 3
ii.
0 x
2
< < 2x , 2x .iii. : ( ) y 2x 1= 17y
31= .
:
i. :
8 x 3 x
3 3 5
8x x 3 x x 3 3 3 3 5
1 3 3 3 8 8 4x x x x 2 x x
2 2 2 2 5 5 5
+ + + =
+ + =
+ + = = =
ii. :
2 2 x 1 x=
2 16 x 1
25= 2
9 x
25=
3x
5=
3x
5=
0 x 2< < x 0> . 3
x 5= .
:3 4 24
2x 2xx 25 5 25
= = =i
2 16 72x 2 x 1 2 1
25 25= = =i
2x 242x
2x 7= =i
iii. : ( ) y 2x 1+ = y 2x
11 y 2x
+ =
24 24y 1 y
7 7+ =
7 y 24 7 24 y + = 31 y 17 = 17
y31
=
8.i. : 2 2 x x 2
x8 2 8 2 2
+ =
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26. 3
ii. :
2
2 2 x x
4 2x8 2 8 2
+ =
:
i. :
2 2 x x 8 2 8 2
+ =
1 x 1 x
4 4
2 2
+ =
1 x 1 x
4 42
+ + =
x x
4 42
+ =
x x x x
4 4 4 4
2
+ + =
2x
24 x x2 4 2
= =
ii. :
2
2 2
2
2
x 4 2x
8 2 8 4
24 x 2x 2 x 2x
2
1 2x2 2x 1 2x 2x
2
1 1 2 2x 2x 2x
2 3
2x 2 x ,
3 6
+ =
= =
= =
= = =
= = Z
9.i.
4 : 2 4x x 0+ = , 2 .
ii. 0 , .
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27. 3
:
i. 4
:
2 0
4 + =
1 0
2 + = 2=
ii. :
2 4x 2 x 0 0 + =
24x 2 x 0+ = 21 2x
2 2x 1 2 02
+ =
22 2x 1 1 2x 0 + = 22 2x 2x 0 =
( )2x 2 2x 1 0 = 2x 0= 1
2x2
=
2x 2
= +
2x 3
=
x2 4
= +
2x 23
=
x2 4
= +
x , 6
= Z
10. 3 3f(x) x x x x x .i. :
1f(x) 4x
4=
ii. : 1
f(x) f x3 8 4
+ = iii. :
g(x) 8 f (x) 1 x :
i. x :
( )2 21
f(x) x x x x f (x) 2x x 2x2
1 1 1f(x) 2x 2x f (x) 2 2x 2x f (x) 4x
2 4 4
= =
= = =
ii. :
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28. 3
1 1 14x 4x
4 3 4 2 4
+ =
4x 4x 1
3+ =
4x 4x 1
3+ =
34x 4x 1
3
+ =
4x 4x
3 3 3 + =
4x
3 6
+ =
4x 23 6
+ = + 54x 23 6
+ = +
4x 2
6=
4x 2
2= + x
2 24=
x ,
2 8= + Z
iii. x : g(x) 8 f (x) 1 2 4x 1= =
x : 1 4x 1
2 2 4x 2
3 2 4x 1 1
3 g(x) 1 -3, 1 g.
11.i. : ( ) ( )2
1 2 1 2
=+ +ii. :
1 2
2
+ + =+iii. : ( ) ( )
2x x 1 2x x
1 2x 1 x 2x x
+ +=+ + +:
i. : ( ) ( ) ( ) ( )22 2
1 2 1 1 2 1 1
= =
+ + + +
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29. 3
( ) ( )
2
22
2
2 2 2 21 2 1 1 2 12
= = =+ + +
2
2
2 2 2 2
2 2 2
= =
ii. :
( )
( )
21 2 1 2 1
2 2
2 1
2 1
+ + + += =+ +
+= =
+
iii. :( ) ( )
2x x 1 2x x
1 2x 1 x 2x x
+ +=
+ + +
x x2 =
x 1 2 x=
xx 12 = 2
x2
x2 1x 2
1 2
=
2 2x x2 1
2 2 = 2
x3 1
2 = 2
x 1
2 3=
x 3
2 3=
x 3
2 3=
x
2 6=
x
2 6=
x
2 6= + x
2 6=
x 2 3= +
x 2 , 3= Z
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30. 4
1. :
i.x
f (x) 2 2= , ii. g(x) 23x , iii.
xh(x) 1 3
4= + : . ,
. , .
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2. t -
t :t
12
= 0 C, [ ]t 0, 24 .. 2
4 C.
.
.
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31. 4
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3. :
i. 1x2
= , ii. 32x2 3
= , iii. x3 3 04+ =
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4. :i. 3 3 x x 3xx(x x)= +ii. 2 2 2x 3x 1=iii.
1x x
x+ =
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32. 4
............................................................................................................................
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............................................................................................................................
............................................................................................................................
5. : f (x) 3 x4
= + . x f
[ ]0, 2 .. f
xx.
............................................................................................................................
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6. : 2 5 3 0+ =i. :
1
2=
ii. :3
2
: 2, 2 2.
...........................................................................................................................
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33. 4
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7. :
x x1 x + =
2
1 2 x 1 x 2 += + +
i. x.
ii. =
3, : 3 3= + .
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8. :
i. 1 2x 2( x)+ = ii. 2 4x 1 x2 2
=iii.
2 2 1 2x 2x2
=...........................................................................................................................
...........................................................................................................................
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34. 4
............................................................................................................................
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9. : 5 11 7 11 1 1 1 12 12 12 12 16
+ + + + = ............................................................................................................................
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10. :2
2
1 (135 )2
1 (135 )
= + ............................................................................................................................
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35. 4
11. ( )f x x x .i. f : ( ) 5f x 2 x
4
= ii. : x x 1=
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12. 1 21 1 2
+ =
12.
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36. 4
13. ( ) 4 f x x x3 3
= + .i. : ( ) 3 1f x 2x
4 2=
ii. : ( ) ( )f x f x= iii. [ )x 0, 2 f .
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37. 5
1
.
, ,
2
+ + , , : ( )
1
++ =
( 5)
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
.i.
6 2 2
1 x 2x
2+ = , .
( 5)
ii. . :2
22
=
( 5)
iii. :2 2 2
4 4
=
( 5)
iv. :
2
1 2 2
+ = +
( 5)
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38. 5
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2
:
.2
19 26 71 642
+ = ( 5)
.
2 2
2 2
5 38 2
1 5 3
=
( 10)
. 3 3 2
2 8 8 4
+ = ( 10)
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3
: .
x x 13 3
+ =
( 10)
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39. 5
. 2x
2x 2 0
2
+ = ( 15)
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4
( ) 4 4f x x x= + x R .
. ( )4x 3
f x4
+= , x R . ( 5)
. f. ( 10)
. ( )8f x 7=
0,2
. ( 10)
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:
[ - x, : ,
, .
[ - , -, , .
[ - , , -, - -
.
[ P(x):(x).[ Horner
P(x):(x).
[ : ( ) ( ) ( ) ( )= = P p 0 P x x x .[ 2 -
.
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42. -
: -
x x R ,* N x R.
x .
, ..
3 50x , 0x .
, .. o o7 7 x , 9 9 x= = .
x
1 1 1 0
x x ... x + + + + , 0 1 1 , , ..., , R , x
R.
x ( ) ( ) ( ) ( )P x , Q x , f x , x ....
: ( ) 1 1 1 0
P x x x ... x = + + + +
: ( ) 3 2Q x x 4x x 5= + + x.
( ) 00
x x= .
0.
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43. -
( ) 1 1 1 0
P x x x ... x , 0+ + + + -: 0 x ,...,
- : 0 x.
-: 1 1 0 , ,..., , .
- : .
- x = : ( ) 1 0
P ... = + + +
( )P x x .
-: R ( )P x , , ( )P 0= .
.
x ,
.
( ) 1 1 1 0
P x x x ... x = + + + +
( ) 1 1 1 0
Q x x x ... x = + + + +
x , :
( ) ( ) 0 0 1 1
1 2
, , ..., P x Q x
... 0+ +
= = == = = = =
1 ( )
(x) (x) ( ) x 0
(x) ( ) x : ( ) ( ) ( ) ( ) x x x x= +
( ) x -
( ) x .
( )
( )
( )
( )
x x
x
x ( ) ( ) ( ) ( ) x x x x= +
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44. -
( ) x ( ) x ( ) x
( ) x ( ) x , ( ) x 0= .
( ) ( ) x : x ( ) x 0= .
: ( ) ( ) ( ) x x x= ( ) x
( ) x .
:
- ( ) x ( ) x , ( ) x ( ) x , ( ) ( ) x : x , ( ) x : ( ) ( ) ( ) x x x=
2
( )P x x
( )P .
( )
( )
( )
x x
x
=
: ( ) ( ) ( ) ( )P x x x = +
3
( )P x x , ,
( )P x .
: ( ) ( ) ( ) ( )P x x x P 0= =
4 ( )
( ) v 1 0
P x x ... x = + + + , , ..., 1, 0 ,
, 0 0 .
0 .
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45. 1
1 (x) (x) ( ) x 0 - (x) ( ) x : ( ) ( ) ( ) ( ) x x x x+ (x)
(x).
( )
( )
( )( )
x x
x
x
: ( ) ( ) ( ) ( ) x x x x+ ( ) x ( ) x ( ) x ( ) x -
( ) x , ( ) x 0 .
2 ( )P x x x . -
( ) P .
( )P x x ( ) ( ) ( )P x x x = + ,
( ) x -
x .
A x = , ( ) ( ) ( ) ( )P 0 = + = + , ( )P = .
( )
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46. 1
3 ( )P x x , , ( )P x .
x ( )P x ( ) ( ) ( )P x x x= .
x = , ( ) ( ) ( )P 0= = , ( )P x . :
( )P x ( )P 0= ,
( )P x x : P() 0= =:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )P x x x P P x (x ) x 0 P x x x= + = + = , -
x ( )P x .
4 :
1 1 1 0
x x x 0 + + + = , . 0
.
0 , : 1 1 1 0 0
+ + + + =
1 2
1 2 1 0 0
( ) 0 ( )
+ ++ + = =
, 1 1 1 ...
= + + .
Z 0 .
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47. 2
.
2003. 2.1 : 3, 4, 6, 7
: 1, 2, 3, 4, 5
2.2 : 1, 2, 3, 4, 5, 7, 8, 9, 10
: 1, 2, 3, 4, 5
2.3 : 1, 2, 4, 5, 6, 7
: 1, 2, 3, 5
2.4 : 1, 2, 3
: 5
.
5:
:
2, 5, 6, 9, 10, 11, 12, 13
6:
:
4, 5, 6, 7, 8, 9, 10, 11
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48. 3
1. : ( ) ( )3 3 3 2 2 2q(x) 4 x 4 4 x 3 2 + + + +i. q(x) .
ii. x 1 q(x) .
iii.
q y 7x 8 + .:
i.
x.
3 4 0 = ( )2 4 0 = ( ) ( ) 2 2 0 + = 0= 2 0
= 2 0
+ = 0= 2= 2= :
1. 0 , 2 , 2 q(x)
2. 0= q(x) 2= q(x)
3. 2= q(x) 0= q(x) .
4. 2= 2q(x) 32x 12= + q(x) .
ii. x 1 q(x), , 1 q(x) :
( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
3 3 2 2
3 2 3
2
2 2
q(1) 0 4 4 4 3 2 0
2 3 3 2 0 2 1 3 1 0
2 1 1 3 1 0
1 2 2 2 3 0 1 2 5 2 0
= + + + + =
+ = + + =
+ + + =
+ + = + + =
1 0+ = 2
2 5 2 0 + = 1=
( )5 9
2 2
=
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49. 3
1= 5 3
4
= 1= 2=
1
2
=
iii. 1= : 3 2q(x) 3x 9x 6= +
q(x) y 7x 8= + :
3 2 3 2q(x) y 3x 9x 6 7x 8 3x 9x 7x 2 0= + = + + =
3 2G(x) 3x 9x 7x 2= + , - Horner 2 ( 2 -
-2 ) :
( ) ( )2G(x) x 2 3x 3x 1= +
: ( ) ( )2x 2 3x 3x 1 0 + =
x 2 0 = 23x 3x 1 0 + =
x 2= 0 >
< > +
9. :
( ) ( )3 2 2x 2 x 1 1 6x 16 x 10 x 2 0 = + + + =i. .
ii. :
( ) ( )3 2 26x 16 x 10 x 2 0+ + + >
4 3 24x 4x 7x 2x 1 + + + 24x 1 4 24x x + 2x x 2 + 3 24x 8x 2x 1 + + +
34x x
2
8x x 1+ + 28x 2 + x 3+
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57. 3
:
i. :
( ) ( )2 2
x 2 0
x 1 0x 2 x 1 1 x 2 x 1 1
x 2 x 1 1
+ + = + = +
+ = +
x 2x 1 x 1 x 1
x 1 x 21 x 1 x 21 x 1
x 2 x 1 1 2 x 1
= = == + = + + 2 , :
( ) ( )
( )
2 2
2 2
48 4 16 2 10 2 0 48 4 64 2 20 2 0
1 9 14 2 2 0 2 1 0 1
2 2 2
+ + + = + + =
+ + = = = = =
ii. =1 : 3 26x 17x 11x 2 0 + =
3 2q(x) 6x 17x 11x 2= +
Horner 2 :: ( ) ( )2q(x) x 2 6x 5x 1= +
: ( ) ( )2x 2 6x 5x 1 0 + =
x 2 0 = 26x 5x 1 0 + =
x 2= ( )5 1
x2 6
=
x 2= 5 1
x12
=
x 2= 1
x2
= 1
x3
=
q(x) :
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58. 3
: ( )3 21 1
6x 17x 11x 2 0 q(x) 0 x , 2 ,
3 2
+ > > +
10. mgr :
3 2q(t) t t 9t , t 0 + + + > .
1 16 mgr -
2 27 mgr:
i. , .
ii. ;
:
i. : ( )q 1 16= ( )q 2 27=
q(1) 16 1 9 16 8= + + + = + =
( )q 2 27 8 4 18 27 4 17= + + + = + =
: 8 8 3 9 3
4 17 4 17 8 5
+ = = = = + = + = + = =
: 3 2q(t) t 3t 9t 5, t 0= + + + >ii. , ( )q t 0= .
: q(t) 0= 3 2t 3t 9t 5 0 + + + = Horner 1 -
: ( ) ( )2q(t) t 1 t 4t 5= + + + :
( ) ( )2t 1 t 4t 5 0+ + + = t 1 0+ = 2t 4t 5 0 + + =
t 1= 4 36
t2
=
t 1=
4 6t
2
=
t 1= t 5= t 1= t 0> t 5= . 5 .
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59. 4
1. : ( ) ( )2 4P(x) 1 x 1 x 1 + + + . R , (x) ............................................................................................................................
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2. :
2f (x) 4x x 9+ + ( )( ) ( )h(x) x 4 x 3 x x 2 + + + + ,, ............................................................................................................................
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60. 4
3.
.i. ( ) ( )4 3 2 210x x 24x 19x 14 : 2x 3x 1 + +ii. ( ) ( )4 2x 4x x 2 : x 2 + iii. ( ) ( )3 2x x 5x 6 : x 1+ +iv. ( ) ( )3 2 2x x x 2 : x + + +...........................................................................................................................
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...........................................................................................................................
4. : ( )3 2 3P(x) x x 2 1+ + + , 0,2
, ( )x 1.i. .
ii. : P(x) 3 .
...........................................................................................................................
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...........................................................................................................................
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61. 4
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
5. 3 2P(x) x x 3x 2 + , - 2x 1 , (x) 3x 2 + . , ............................................................................................................................
...........................................................................................................................
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6. (x) ( )x 2 ( )x 3 1 10 . (x) 2x x 6 ............................................................................................................................
...........................................................................................................................
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...........................................................................................................................
...........................................................................................................................
7. : 3 2f (x) 2x 3x 5x 6 +i. f
xx yy .
ii. f (x) 0 .
...........................................................................................................................
...........................................................................................................................
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62. 4
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
8. : i. 5 4 3 2x 4x 6x 6x 5x 2 0+ + =ii. ( ) ( )8 4x 2 3 x 2 4 0 + =iii. 3 22 x 5 x 4x 2 0 + =
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9. : i.
4 3x 3x 6x 4 0+
ii.( )
22 5 x 1x
x 3 3 x
> ...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
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63. 4
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
10. : ( )3 2P(x) 2x 5x 1 x 2 3+ + . , R ( )2x 1 (x) .
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...........................................................................................................................
11. : ( )3 2P(x) x 2x 1 x 3 + , , Ri. (x) ( )2x 2 , 21
21 .ii. , i.:
. 1
(x) P( x) : ( x 2) .
. 2
(x) 2P( x) : ( x 2) .
. 1 2
(x) (x) 0> ............................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
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64. 4
...........................................................................................................................
...........................................................................................................................
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...........................................................................................................................
12. 4 3 2f(x) x 2x 2x x 6+ + , R . f xx M(2,0),
.
...........................................................................................................................
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...........................................................................................................................
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...........................................................................................................................
...........................................................................................................................
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...........................................................................................................................
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65. 5
1
. (x) 1
. :
x ( ) ( ) x x( 5)
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
.i. R ( ) 2P x 1= .( 5)
ii. ( ) 2004 2P x x x 1= + + . - (x) x 1.
( 5)
iii. : ( ) ( )2v 1 2v 1 2v 1P x x y x y+ + += + + + , *v N R . x y+ (x).
( 5)
iv. : ( ) 3 2P x x 2x x 1= + + + , . () .
( 5)
...........................................................................................................................
...........................................................................................................................
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66. 5
...........................................................................................................................
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...........................................................................................................................
...........................................................................................................................
2
: ( ) 3 2P x x 2x 6x
= + + +. x (x) 4 4= .
( 5)
. x 1+ (x) 6 .( 10)
.
,02
, : ( )P x 0=
( 10)
...........................................................................................................................
...........................................................................................................................
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...........................................................................................................................
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...........................................................................................................................
...........................................................................................................................
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67. 5
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
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...........................................................................................................................
...........................................................................................................................
3
: x 3 2x 7 1+ + =
( ) ( ) ( )4 3 2P x x 22 1 x 3x 12 8 x 4= + + . .
( 5)
. x (x) .
( 10)
.
0,2
( ) x 0= ( ) x 0< .
( 10)...........................................................................................................................
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...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
4
: ( ) 3 2 x x 2x x = + +i. x (x) 2, .
( 5)
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68. 5
ii. : ( ) x 0=
( 10)iii. 2x 22 4 + (x), R .
( 10)
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:
[ , , .
[ , , ,
.
[ : , , , S .[ -
( - ).
[ .
[ - +1
+ 1
.
[ .[ ,
.
[ -
.
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70. - -
- : -
*
.
.
1,
2,
3... , ,
... .
.
. :
.
.. 2 1= , 1 1= , 2 3= , 3 5= ,
.
.. 1 1= , 2 3= 2 1 + += +
: 3 2 1 1 1 2= + = + =
4 3 2 2 1 3= + = + =
5 4 3 3 2 5= + = + =
.. 100
99 .
(..)
(.) -
.
.
, , :*
1 1 , + += + =
.
1
: ( ) 1 1 = +
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71. - -
. , , ,
2
+=
. ,, .. , ,
2
+=
. .. :
1 2 S .... = + + + :
( ) 1
S 2
= + ( ) 1
S 2 1 2
= +
(..) -
.
.
1
0
0
0 , * .
( )
1
1
++ = = , .
. 1
: 1 1
=
. , 0 , ,
=
. , , .. , , 2 =. ..
1 2 S ... = + + +
1
1S
1
=
1
1S = 1= .
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72. - 1
1
1 : ( )
1 1 +
:
1 1
2 1
3 2
4 3 1
1 2
1
: ( 1)
= = + = += + = +
= += +
2 ( ). , ,
:
2 + 2
+= , , . :
=
2 2
+ = = + = + = + =
3 1
: 1 1
=
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73. 1 -
:
1 1
2 1
3 2
1
4 3 1
1 2
1
:
= = == ===
4 ( ). , ,
:
2
,, 0 . :
=
2
= = =
5 .
1 :
1
1S
1
= .
:2 2 1
1 1 1 1 1S = + + + + + (1)
(1) :
2 3 1 1 1 1 1 1S
= + + + + + (2)
: 1 1S S =
1S ( 1) ( 1) =
1 :
1
1S
1
=
:
= 1 1 1S = .
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74. - 2
.
2003. 3.2 : 3, 4, 5, 10, 12
: 1, 2, 4, 5, 6, 7, 8, 12, 13, 14, 16
3.3 : 3, 4, 5, 6, 7, 11, 12, 13
: 1, 2, 3, 4, 5, 6, 7, 9, 10, 13, 14
.
7:
:
2, 3, 4, 5, 6, 8, 12, 13, 14, 17,
18, 22, 25, 27
8: :
2, 3, 4, 5, 7, 9, 14, 15, 18, 24,
28, 29, 31
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75. 3 -
1.i. :3 2
2
3 2
1
+ += + + . ..* .
ii. 28.
iii. :1 4 7 61
+ + +iv. : ( )2 6 S 3 2= + + , .
:
i. :
( ) ( )2
2
3 2 1 3 2,
1
+ += = + +
: ( ) ( ) 1 3 1 2 3 2+ = +
1 3 3 2 3 2+ = + +
1 3+ = 1 3+ = +
.. : 1 1 3= =
ii.: 28= 3 2 28 = 3 30= 10=
28.
iii. 1 4 7 61 , , , , .. 1 1 1= =
3 3 3 9 = = = :
3 23 2+ + + 2 1+ + 3 23 3 3 3 2
22 2 2 + 22 2 2+ 0
*
.. .
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76. - 3
( ) ( ) ( )
( )
1 61 1 1 1 1 1 9 60 1 9
180 1 9 20 1 21
= + = + + = +
= = =
( )11 4 7 61 12 20
21 10 212
+ + + + + = = + =
( )1 90 21 91 21 1911= + = =iv. :
( )( )[ ] ( )1 2
2 1 6 S 6 3 2 1 3 3 3 1 9 3
2
+ = = + = = i
( ) ( )22 23 2 3 3 2 2 3 9 + + = + + = +i
: ( )2
6 S 3 2 = + + .
2.. .. 18 , S 88 2 . 1 .
. ..
16 15
:
. :
( ) ( ) 1 1 1
1 1
1 18 1 2 18 2 2
20 2 20 2
= + = + = + = + =
i( )
1
2 1 S
2
+ = ( )88 20 2 1 = + ( )88 19= +
2 19 88 0 + = ( )19 9
2
=
19 3
2
= 11= 8=
1
1
11 2
8 4
= =
= =
.: x 3 , x , x , x 3 + + , :
x 3 x x x 3 16 + + + + + = i 4x 16 x 4= =
( ) ( )x x 15 + = i 2 2x 15 = 216 15 = 2 1 1= =
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77. 3 -
: 1= , : 1, 3, 5, 7
1= , : 7, 5, 3, 1
3. :
1
3 1
2
= + =
i. .. ..*
ii. : ( ) ( ) ( ) ( ) 4 1 7 2 10 4 31 512+ + + + + + + +iii.
2
17 1 2 3 x x x ... x< < :
( ) 224 17 15 17 15
4y 17 0 4y 17y 4 0 y yy 2 4 8
+ = + = = =
y 4= 1
y4
=
2 4= 2
1
4= 2= 2=
1
2=
1
2=
: 2, 4, 8 2, 4,8 8, 4, 2 8, 4,2.
8.. ..
1
x 3
5
= .
:
2 x
3
x
+ .
2 x , , x
6 2 6
+ , .. x :
0 x
2<
x
6
=
x 2 ,
6
= Z . 0 x2
< < ,
x
6
=
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83. 3 -
9. 106 -
90% ..
.
. 4 .
. 4
100 .
106 .
:
. 60 10= .
1 : 51 0 0 0 0,9 0,1 10= = =
2 :2 1 0,1 =
..............................................................................................................
: 1 0,1 =
. 51 10= 0,1= .
. : ( )33 5
4 1 10 0,1 100= = = .
. 4 :0 100= .
1 :1 0 100 200= + = .
2 :2 1 100= + .
..............................................................................................................
: 1 100= + .
.. 1 200= 100= .
6 10= :
( ) ( )6 6 6 1
4 4
10 1 10 200 1 100 10
2 1 10 10 1 10.000 1 9.999
= + = + =
+ = = = =
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8/22/2019 B Lukeiou Algebra Theoria Askiseis
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84. - 4
1.i. (x) x 1 3
x 2 9. (x) 2x 3x 2+ , U(x) .
ii. U() .. -
57 2700.
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2. : 3 2q(x) x 2x x + + 2G(x) x 4 U(x) 5x 10 + .. , .. q(x) 0 .
. ..
q(x) 0 .
: i. 24.
ii. 15.
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