Κατατακτήριες Μαθηματικών ΑΠΘ_Άλγεβρα Β' Ενιαίου...
DESCRIPTION
Εδώ θα βρείτε την ύλη για τις κατατακτήριες Εξετάσεις Μαθηματικών ΑΠΘ από το βιβλίο της Άλγεβρας Β' Ενιαίου Λυκείου. Στο συγκεκριμένο pdf έχω συμπεριλάβει μόνο την ύλη και όχι τα περιττά κεφάλαια. Για περισσότερα αλλά και για να γνωριστούμε καλύτερα επισκεφθείτε το http://katataktiriesmathimatikon.blogspot.gr/TRANSCRIPT
-
1
1.1
x + y =
x + y = , 0 0,
, . : :
0, :
x y y x
y x
+ = = +
= + ,
-
= y'y
.
:
9 0, (. ),
x+y=, =0
O x
y
x+y=, 0
x
y
O
001-126_update.indd 9 10/5/2013 8:48:44 pi
-
10 1:
9 = 0 , y =
x'x y'y
(. ).
= 0 ( 0 ), .
- y'y
x'x
.
:
9 x y =2 2
=12
y'y 1.
9 y = 2 x'x - y'y 2 .
9 x = 2 y'y - x'x 2.
.
y x= 12
1
x x= =
001-126_update.indd 10 10/5/2013 8:48:44 pi
-
1.1 11
, (4, 1) x y =2 6, 4 2 1 4 2 6 = + =( ) . , , (16,5), ( , ) 10 8
.
2 x 2 x+y= x+y= , , , 2 x 2
x yx y+ =+ =
' ' '
. . - . . :
. () (') , .. (),
+ ( ) ' ( ') , () - 0, (') '.
+ ( ) ' ( ') () (').
- 2o :
9 0, : = =9 = = , + = +.
, , :
x yx y =+ =
2 6 13 4 8 2
( )( )
001-126_update.indd 11 10/5/2013 8:48:45 pi
-
12 1:
, ( )
, .. (1) x.
x
3 2 6 4 8 6 18 4 8
10 101
( )y y y y
yy
+ + = + + =
= =
y :
x = + =2 1 6 4( )
( , ).4 1
,
.
- , .
, x = 4 y = 1, :1 : 4 2(1) = 6
2 : 3 4 4 1 12 4 8 + = =( )
( ) , :
x yx y= ++ =
2 63 4 8
x yy= +=
2 61
001-126_update.indd 12 10/5/2013 8:48:46 pi
-
1.1 13
, , : . :
x x x = + = =2 1 6 2 6 4( ) . ( , )4 1 ( - ).
2 x 2
. , - .
,
, , . 2 x 2 . , 2 x 2 , , , .
+ = + =
3 6 183 4 8
x yx y
x yx y =+ =
2 63 4 8
+ + + = + = = 3 6 3 4 18 8 10 10 1x y x y y y
x yx y =+ =
2 63 4 8
x yx y+ =+ =
' ' '
A(4, 1)
3x+4y=8 x2y=6y
O x
. (3)
. 1
001-126_update.indd 13 10/5/2013 8:48:47 pi
-
14 1:
2 x 2 .
:
9 2 3 34 9 1x yx y+ = =
y x
y x
= +
=
23
1
49
19
,
, .
,
(1, 0,3) .
,
- 1
13,
.
9 x yx y+ =+ =
2 32 4 5
y x
y x
= +
=
12
32
12
54
, ,
.
9
y xx y+ = =
1 24 2 2
y xy x= =
2 12 1
, .
.
y
x
21
1
1
O
y
x
1
2O
x
y
2
1
O,
001-126_update.indd 14 10/5/2013 8:48:48 pi
-
1.1 15
, 2 x 2 :9 9 9 .
2 x 2
- 2 x 2 .
0 0. - :
1 2
1
= 2
= ''
.
'
', ' ' 0 , 1 2
,
+ = +
=
=
''
''
''
''
( ' ' ) '
x x x
x
'' '' '
=
x
:
x yx y+ =+ =
' ' '
y x
y x
= +
= +
( )
''
''
( )
1
2
y =
+ =
+ +
' '' '
' ' ' '( ' '
)
( ' ' )( ' ' )
=
001-126_update.indd 15 10/5/2013 8:48:49 pi
-
16 1:
y =
' '' '
, ,
=
'
', =0 , 1 2
, . . =0 ' =0.
:
' ' 0
' ' = 0 . ' ' ,
:
.
D, x , : .
, D, y , :
.
( , ) ' '' '
, ' '' '
x y =
x yx y+ =+ =
' ' '
( , ) ' '' '
, ' '' '
x y =
D = ' '
D = =
' '' '
Dy = =
' ' ''
001-126_update.indd 16 10/5/2013 8:48:50 pi
-
1.1 17
, , :
:
9
,
.
: .
, ( , ) ( , )x y = 4 1 .
9
. 2,
2 3 40x y = .
.
x yx y =+ =
2 63 4 8
D =
= = + = 13 4
21 4 3 2 4 6 10 0( )
Dx =
= + =68 4
224 16 40
yDDy= =
=
1010
1x DDx= = =
4010
4
2 3 40 2 403
x y y x = =
D0, , (x,y)
D=0, .
x yx y+ =+ =
' ' '
yDDy=
2 3 404 6 80x yx y = =
2 3 402 3 40x yx y = =
,
x DDx=
001-126_update.indd 17 10/5/2013 8:48:51 pi
-
18 1:
9
.
.
1
- , . , . , . , .
D, Dx, Dy. :
k k k, ,2 403
3 119 3 6x yx y+ =+ =
3 113 2x yx y+ =+ =
x yx y =
=
122
D
D
D
x
y
=
= + =
= = + =
=
1
11
2
2
2 2
2 1 2
1
22
2
( )
( )
== = + = 2 2 2 21 1 1 2( ) ( ) ( )
001-126_update.indd 18 10/5/2013 8:48:52 pi
-
1.1 19
, D = 0 . :
:9 D 0, 0 2, (x, y), :
.
, 0 2 ,
.
9 D = 0 , = 0 = 2 , . : = 0 ,
. = 2 ,
. , ,
3 x 3 x+y+z=0 , - , , , . . 2x+3y+z=6 - ( , , )2 1 5 , 22+3(1)+5=6.
D = = = =0 2 0 0 2 ( )
x DDx= =
=
=
22
22
1
( )( )( )
yDD
y= =
=
=
2 22
22
2( )( )
( )( )
0 10 2 0
10
x yx y
yy
= =
==
2 14 2 2
2 12 1
x yx y
x yx y
= =
= =
.
1
,
2 1 2 1x y y x = =
001-126_update.indd 19 7/5/2014 4:36:51
-
20 1:
:
1 1 1 1x y z+ + = , 2 2 2 2x y z+ + = 3 3 3 3x y z+ + = , - , , 3 x 3
- 2 x 2 .
. , .. (3) z, - . : (4)
(1) (2) :
9 2 3 9 2 3 3 8 9 11 2 15x y z x y x y x y + = + + = + =( ) (5)9 (6)
(5), (6)
x=1 y=2 . x y (4) z = 3. ( , , ).1 2 3
x y z x y x y x y+ = + + = + =3 10 3 3 8 10 1( )
1 1 1 1
2 2 2 2
3 3 3 3
x y zx y zx y z
+ + =+ + =+ + =
2 33 10 2
3 8 3
x y zx y zx y z
+ = + =+ =
( )( )
3 8 3 8x y z z x y+ = = +
3 x 3, ,
2 x 2, 3 x 3
.
11 2 151
x yx y+ =
+ =
,
001-126_update.indd 20 10/5/2013 8:48:54 pi
-
1.1 21
1.
i) ii) .
2. i) ii)
3. :
i) ii)
4. : i) ii)
5. : i) ii)
6. , .
i) ii) iii)
7. :
i) ii)
x y
x y7 8
45
=
+ =
x yx y
=
=
3 3
32
2 212
1 0
y x
x y
= +
+ =
2 73 5 4x yx y+ = =
2 5 46 7 100x yx y =+ =
( )
( )
3 1 4 712
3 1 1
+ + =
+ =
x y
x y
2 3 404 6 80x yx y = =
x y
x y
=
+ =
13
24
4 3 8
x y
x y
+
++ =
+
=
52
2 17
2 0
63
62
8
2 3 83 1 0
y xx y
= + + =
x yx y =+ =
42
( )
( )
3 1 2 2
3 1 1 3
+ =
+ + =
x y
x y
3 119 3 2x yx y+ =
=
2 13
4 24
32
33
x y
x x y
=
+
+ =
001-126_update.indd 21 10/5/2013 8:48:55 pi
-
22 1:
8. :
i) ii) iii)
1. i) 1 2 .
ii) 1 2 ;
2. 26 , 68 . ;
3. 1,5 4. 2200 - 5050 . .
4. R T R =T + . 20C 0,4 80C 0,5 , .
5. , 50% - 80% . 100 ml 68% ;
6. 1 2 4 3: x y+ = 2 2: x y+ = , . i) 1 2 .
ii) -;
iii) ;
3 2 112 5 2 35 2 33
x yx yx y
= =+ =
5 3 43 2
3 2 2 2
x yx yx y
+ = + = + =
x y
x y
x y
+ =
+ + =
+ =
22 3
32
5
5 3 2 16
y
x1 41
1 22
O
001-126_update.indd 22 10/5/2013 8:48:56 pi
-
1.1 23
7. :
i) 12: x y+ = 2 1: x y+ = .
ii) 3 : x y = 4 1: x y+ = .
8. :
i) ii)
9. 1 2 6= , O O1 3 5=
O O2 3 7= . .
10. , . N = x , B = y = z , - , .
11. . 50% , 10% 30% . 52 lit 32% , - . .
12. -, y x x= + + 2 . .
( )( )
,
= + =
1 2 1
4 1 2x y
x y( )
( ),
+ =+ + =
2 5 5
2 5x y
x y
1 2
3
x+y=, 0
x
y
0
y
x1
K(1,4)
y=g(x)
O
y
x
y=h(x)
O 2 4
4
y
x
y=f(x)
OK(2, 1)
3
001-126_update.indd 23 10/5/2013 8:48:58 pi
-
24 1:
1.2
, -. , 13 89.
x, y , x y+ =13 x y2 2 89+ = .
, :
: (1), , .. x , (2).
x y y x+ = = 13 13 (3).
2 = 9 . :
(3), x=8 y=5 , x=5 y=8. (8, 5) (5, 8). 5 8. , , .
1o
x yx y+ =
+ =
1389 22 2 ( )
x x x x x
x xx x
2 2 2 2
2
2
13 89 169 26 89
2 26 80 013 40 0
+ = + + =
+ =
+ =
( )
.
x = 13 32
8
5
001-126_update.indd 24 10/5/2013 8:48:59 pi
-
(1) x (2). :
(3) x=2 y=3 , x=3 y=2 . (2, 3) (3, 2) .
- 5 6. , - Vieta
2 5 6 0 + = . 2 3 (2,3) (3,2).
x+y=5 , xy = 6
yx
=6
. -
.
x y y x+ = = 5 5
y
xO 11
AB
(3).xy x x
x xx x
= = = + =
6 5 65 6
5 6 0
2
2
( )
=x 2 x = 3.
1.2 25
001-126_update.indd 25 7/5/2014 4:37:35
-
26 1:
(2,3) (3,2) . (2,3) (3,2) .
2
(1) x (2).
(2) :
. x2 = , 2 13 36 0 + = , =9 =4 .
9 =9 x2= 9 x = 3 x = 3 (1) x=3 y=2 x = 3 y = 2 .9 = 4
.
(1) x=2 y=3 x = 2 y = 3. (3,2), ( , ), 3 2 (2,3) ( , ). 2 3
xy = 6 yx
=6
, -
x y2 2 13+ = (0,0) = 13 .
xy yx
= =6 6
x y xx
xx
x xx x
2 2 22
22
4 2
4 2
13 6 13
36 13
36 1313 36
+ = +
=
+ =
+ =
+ = 00
x x2 4 2= = x = 2
xyx y
=
+ =
6 113 22 2
( )( )
001-126_update.indd 26 10/5/2013 8:49:01 pi
-
1.2 27
1. :
2. :
i) ii) iii) .
3. S t t= + 021
2 = +0 t ,
S t= + 02
.
1. o - .
x y xyx y
2 2 31
+ + =+ =
.
y xx y= =
312 3 4
2 x yx y
2 2 90
+ = =
x yxy
2 2 52
+ ==
.
x yx y
2
2 2
2 1025
= +
+ =
y
x 1
AB
A B
001-126_update.indd 27 10/5/2013 8:49:02 pi
-
28 1:
2. :
3. 120cm2. - 3cm , 2cm , - . .
4. y x= 2 y x= + . k .
5. o 22y x
y x== +
.
1
I. :
, , ,
, , .
) , ) , ) .
II. , , .
1. , . A
2. D = 0 , .
3. xyx y
=+ =
10
.
4. x y2 2 1+ = y x= +2 1
. A
( ) :12 1
2 4 0x yx y+ =+ =
( ) :21
2 4x yx y+ =+ =
( ) :32 4 2
2 1x y
x y+ =+ =
( ) :4 21
1x yx y =
+ =
(1) (2) (3) (4)
2 5 04 3
2
2
xy y yy x x
=
= +
001-126_update.indd 28 10/5/2013 8:49:03 pi
-
( ) :4 21
1x yx y =
+ =
001-126_update.indd 29 10/5/2013 8:49:03 pi
-
4
-
4.1
x .
x x, .
x .
, : : 2, 3 , 0 x.
x :
x x x+ + + +
11
1 0................. , - .
- . 0 .
0, , . 0 -.
, 3 2 23 2x x x+ + , 0 5 12x x + ,
5 23
0 13
3 2x x x + + 2, 0 . x.
127-208_update.indd 128 10/5/2013 8:52:44 pi
-
4.1 129
:
x x+ + +................. 1 0
x x+ + +................. ,1 0 :
0 0 1 1= = =, ,....., + += = = =1 2 0.... 0 0 2 14 3 2x x x x+ + + 2 12x x +
. x x2 + + 2 3x + =3, =2 =0.
P(x), Q(x), .
P x x x x( ) .................= + + + +
1
11 0
, (x) 0 ( ). ,
(x) :
x x x+ + + +
11
1 0................. , 0 (x).
0. .
P x x x( ) = + 4 3 73 3 , Q(x) = 7 .
P x x x x( ) .................= + + + +
11
1 0. x , - P( ) .................
= + + + +
11
1 0 x=.
()=0, o . , P x x x x( ) ,= + + +3 22 4 1 x=1 P( ) ,1 1 2 1 4 1 1 63 2= + + + = x = 1 P( ) ( ) ( ) ( ) = + + + =1 1 2 1 4 1 1 03 2 , 1 (x).
: c c x x(*)
(*) , : c x, c x, .
127-208_update.indd 129 7/5/2014 4:45:42
-
130 4: -
, , -, , :
1. i)
ii)
iii)
2.
3.
- : -, . .
-
1 i)
P x x x( ) ( ) ( )= + + + 2 3 21 3 2 1 .
ii) Q x x x( ) ( )= + + 2 3 22 3 R x x x( ) ( ) ( )= + + +5 6 4 13 2 2 .
i) To (x) , :
2 1 0 = , 2 3 2 0 + = =1 0
= 1. = 1 (x) .
[ 3 ]
[ 2 ]
( ) ( )2 1 2 2 3 2 1 2 2 33 2 3 3 2 3x x x x x x x x + + + = + +
= + x x2 2 2
( ) ( )x x x x x x x x3 2 3 2 3 2 3 23 1 3 1 3 1 3 1 0 + + + = + + = [ ]
[ 5 ]
2( 5 7 4 5 3 2 5 7 4 5 33 7
3 2 3 2 3 2 3 2
3 2) ( )x x x x x x x x x x
x x+ + + = + + +
= + +5 4x
( )( ) ( ) ( )x x x x x x x x x x2 3 2 3 35 2 3 1 2 3 1 5 2 3 1+ + = + + +
= +5 3 5 103 2x x x
= + + +
= + + +
2 3 10 15 52 10 3 14 5
5 3 2 4 2
5 4 3 2
x x x x x xx x x x x
( ) ( )( ) (
x x x x x x x x x xx
3 2 3 2 3 2 3 2
3
2 5 7 4 5 3 2 5 7 4 5 31 4 2
+ + + + = + + + +
= + + + +5 5 7 32) ( )x x
127-208_update.indd 130 10/5/2013 8:52:47 pi
-
4.1 131
ii) Q(x) R(x) - :
2 5 6= , = 2 42 3 1= +
=2. =2 - Q(x) R(x) .
2 P x x x( ) ,= + + 2 23 1 P( ) . =1 1
: 2, 2.
' 1. x: i) 1 3 x ii) 3 2 2 33 3 + x x x
iii) xx
+1
iv)
2. P x x x( ) = +2 5 2 Q x x x( ) .= + +3 3 1 :
i) P x Q x( ) ( )+ ii) 2 3P x Q x( ) ( ) iii) P x Q x( ) ( ) iv) [ ( )]P x 2
3. ,
P x x x( ) ( )= +
+4 4
14
2 13 3 2 .
4. Q x x x x( ) ( )= + + +2 1 13 2 2 3 .
5. , -, .
i) P x x x x( ) = + +2 3 2 73 2 x = 1, x=1 ii) Q x x( ) = +4 1 x = 1, x=1, x=3.
6. k 2
P x x kx x k( ) .= + +3 2 5
( ) ( ) ( ) = + + = =
1 1 1 3 1 1 14 0
2 2
2
= 2 = 2
+
P x x x( ) ( )= + + 2 3 23
127-208_update.indd 131 10/5/2013 8:52:50 pi
-
132 4: -
7. R, P x x x( ) = + + 5 3 22 2 x = 1 1.
'
1. , , , - f x x x( ) = +3 7 52 f x x x x( ) ( )= + + + 1
2. , , P x x x x( ) = + + 3 63 2 2 3.
3. , - P x x x x( ) = + + +2 63 2 1 P( ) . = 2 12
4. P x x x( ) ( ) ( )= + +9 4 9 4 3 23 3 2 .
5. (x), ( ) ( ) .2 1 2 9 3 13 2x P x x x x+ = +
. : 0 , -
, =+, 0
-
4.2 133
(x) (x) (x) (x), , . - x x x3 25 2 1 + x 3 .
1. .
2. x2 -
x3
x -.
3. x2 x 3
x3 3x
2
. 2x
2 + 2x 1.
4. 2 3 2x
2 + 2x 1. -
- 4 1x .
5. 2 3 4x 1. - 13 x
2 2x 4.
:
() = () () + ()
-.
- 4x
4 + x
2 3x 1
2x2 + x, :
x x x x3 25 2 1 3 +
x x x xx
3 225 2 13
+
x x xx x
x x
3 2
3 2
2
5 2 132 2 1
+
+ +
xx 32
x x xx x
x xx x
x
3 2
3 2
2
2
5 2 132 2 12 6
4 1
+
+ +
xx x322
x x xx x
x xx x
xx
3 2
3 2
2
2
5 2 132 2 12 6
4 14 12
13
+ +
+
xx x 32 42
4 0 3 14 2
2 3 12
2 3 1
4
4 3 2
4 3
3 2
3 2
2
x x x xx x
x x xx x
x x
x
+ +
+ +
1
22 1
2
2
x xx x
+ +
x3 5x2 + 2x 1 = (x3) . (x22x4) + (13)
127-208_update.indd 133 10/5/2013 8:52:53 pi
-
134 4: -
x3 . 2 2 13 2x x x+ 2 12x
, .
0. .
, (x) = 0,
( ) ( ) ( )x x x= pi (x) (x) (x)
(x) (x) (x) (x)
(x). 2 12x 2 2 13 2x x x+ .
x.
(x) x . P x x x x( ) ( ) ( ) ( )= + pi x , . : P x x x( ) ( ) ( )= + pi , x = , P( ) ( ) ( ) pi = + = + =0
P x x x P( ) ( ) ( ) ( )= + pi
:
(x) x x=. = ()
2 2 12
+120
3 2
3
2
2
x x xx
xx
+
1
x 112x
x
+ +
2
2
127-208_update.indd 134 10/5/2013 8:52:54 pi
-
4.2 135
, P x x x x( ) = + 3 23 13 15 x 2 = = + = P( ) ,2 2 3 2 13 2 15 213 2 x+1 x ( ),1 = = + =P( ) ( ) ( ) ( ) .1 1 3 1 13 1 15 03 2 - : P( ) , =1 0 1 (x) P x x x x x( ) ( ) ( ) ( ) ( ),= + + = +1 0 1pi pi x+1 (x). :
x (x).
P x x x( ) ( ) ( )= pi x =
P( ) ( ) ( ) , pi = = 0 (x).
: (x), () = 0.
P x x x P( ) ( ) ( ) ( )= + pi
P x x x( ) ( ) ( ),= pi x (x).
-
1 x+2 x 1 - P x x x x( ) .= + +3 2 2
x+2 x ( ).2 P( ) ( ) ( ) ( ) , = + + =2 2 2 2 2 03 2 2 (x). , , x+2 (x). P( ) ,1 1 1 1 2 3 03 2= + + = 1 (x). x 1 (x).
(x) x (x), () = 0.
127-208_update.indd 135 10/5/2013 8:52:55 pi
-
136 4: -
2 :i) P x x x x( ) = + 3 23 3 1 x +
. ii) Q x x x( ) = + 2 4 23 3 x 1
1.
i) x x+ = ( ), (x) x+ = P( ). , = 0 :
ii) Q(x) x 1 = Q(1). , = 1 :
P( ) ( ) ( ) ( ) = + = =
+ + + =
0 3 3 1 03 3 1 0
3 3 1
3 2
3 2
3 2 001 01
3 + = =
( )
Horner () -, .. P x x x x( ) ,= + +3 8 7 23 2 x . (x) x :
Horner.
= 1 = 4
+ xx x
+ + +
3 3 8 3 8 72 ( ) ( )
Q( )1 1 1 3 1 3 13 4 0
2 4 2
2= + =
+ =
127-208_update.indd 136 10/5/2013 8:52:57 pi
-
4.2 137
:
(x) (x). :
.
- .
(x) ( ),x (x) x = . .
Horner P x x x x( ) = + + 3 3 6 135 4 x 2.
pi( )x x x x x= + + + +3 9 18 36 784 3 2 = (2) = 143
, Horner
, . , (2) x 2. Horner (x) . , x (), Horner.
[ ] 0 x .3 3 0 0 6 13 =26 18 36 72 1563 9 18 36 78 143
P(x)
3 8 7 2 3 (38) [(38)+7]
3 38 (38)+7 [(38)+7]+2
127-208_update.indd 137 10/5/2013 8:52:57 pi
-
138 4: -
1 0 0 ....... 0 - =
2 ....... -1
1 2 ....... -1 0
1 ( ) : ( )4 8 42 2x x x +
Horner 4 8 42 2x x + x :
(x) = 4x4 (x) = 0
2 , :x x x x x = + + + + ( )( ...... )1 2 3 2 1
Horner x x
( ) : ( )x x , , :
3 x+ - x + , 0. , x + (x+) (x).
P x x( ) ,= + P( ) ( ) . = + : , P( ) , = + = 2 0 (x). x+ x + . , P( ) = + = 0 (x). x+ x + .
, 2 - : x x x x x + = + + + ( )( ...... )1 2 3 2 1
4 8 42
4 42
4 4 0
pi ( ) ......x x x x= + + + + 1 2 2 3 1
x x x x x = + + + + ( )( ...... )1 2 3 2 1x x x pi = +( ) ( ) 0
127-208_update.indd 138 10/5/2013 8:52:59 pi
-
4.2 139
1.
.
i) ( ) : ( )3 6 17 20 33 2x x x x+ + + ii) ( ) : ( )x x4 81 3
iii) ( ) : ( )24 20 16 15 6 55 3 2 2x x x x+ + iv)
v) x x4 31: ( ) vi)
2. ( ) : ( ).18 6 4 2 180 50 20x x x x + +
3. k, x1
g x k x kx( ) .= + 2 4 23 4
4. Horner :
i) ( ) : ( ) + +x x x3 75 250 10 ii) ( ) : ( )x x3 512 8+ +
iii) ( ) : ( )x x5 1 1+ iv) 3 24x x: ( )
v) ( ) :4 16 23 15 12
3 2x x x x+ +
5. P x x x x( ) = +2 2 24093 2 , (11).
6. x (x).
i) P x x x( ) ,= +4 225 144 x+3
ii) P x x x x x( ) ,= + + 16 8 9 14 44 3 2 x 14
iii) P x x x( ) ,= +3 23 2 x 1 3
7. , x+y x y .
8. x .
i) P x x x( ) = + +4 7 124 2 ii)
9. , x+1
x +1. ( ) : ( ).x x + +1 1
10. : i) ( ) : ( )3 2 8 22 2x x x ii) ( ) : ( )x x x x3 2 2 3+ +
( ) : ( )2 4 5 3 2 2 34 3 2 2x x x x x x+ + +
( ) : ( )x x5 37 1+
Q x x x( ) = 5 3 46 2
127-208_update.indd 139 10/5/2013 8:53:01 pi
-
140 4: -
1. , , x
x , (, ).
2. i) (x)
x+, 0
=
P .
ii) , x3 + - x+.
3. Horner , -
P x x x x x( ) = + +2 6 5 3 24 3 2 (x-1)(x-2) .
4. P x x x x( ) ( ) ,= + 2 2 2 33 2x x x+ + .
5. , , P x x x( ) = + ++ 1 1 ( ) .x 1 2
x x x+ = + + =0 02, x x4 2 0+ + = , 0
(x) = 0, (x) , .:
x x x+ + + + =
11
1 0 0...... , 0
, 2 5 2 03 2x x x + = + + =3 5 1 06 2x x - 3 6 .
P x x x x( ) ......= + + + +
11
1 0, , () = 0.
1 2 , - 3 4 . - . ,
4.3
127-208_update.indd 140 10/5/2013 8:53:03 pi
-
4.3 141
4 . , 2, -.
...
, (x) = 0, - (x) - . , . :
x , -
.
( )
x x x+ + + + =
11
1 0 0...... , -. 0 , 0.
0 ,
, 1, 2,...,
11
21......
. 0.
P x P x P x P xk1 2 10 0( ) ( ) ...... ( ) ( ( ) = = P xk ( ) )= 0
x x2 2 0+ =
P x2 0( ) =
=x 1
+ + + + = =
1
11 0 0 1
110...... ......
=
0
11
21( ...... )
x x x x xx x xx x x
3 3
2
3 2 0 2 2 01 2 1 0
1 1 2 0
+ = + = = + =
( ) ( )( )[ ( ) ](xx x x + =1 2 02)( )
=x 1 0
x = 2
127-208_update.indd 141 10/5/2013 8:53:05 pi
-
142 4: -
1 x x x3 23 2 0 + + =
, . Horner P x x x x( ) .= + +3 23 2:
1 (x)
1 (x)
2 (x)
x2 (x).
1 -3 1 2 =1
1 -2 -1
1 -2 -1 1
1 -3 1 2 =1
-1 4 -5
1 -4 5 -3
P x x x x( ) ( )( )= 2 12
(x 2)(x2 x 1) = 0
2, 1 5
2
1 5
2+ .
- . 0 , .. =1.
2 x x x x4 3 25 9 8 4 0+ + + + = .
4 : 1, 2, 4. - , 1, 2, 4 . - 1, 2, 4.
1 -3 1 2 =2
2 -2 -2
1 -1 -1 0
(1) = 1 0
(1) = 3 0
(2) = 0
127-208_update.indd 142 10/5/2013 8:53:05 pi
-
4.3 143
, P( ) = 1 1 0,
= 2 :
2 (x)
( )( )x x x x+ + + + =2 3 3 2 03 2
Q x x x x( ) = + + +3 23 3 2 = 2
- 2 Q(x)
x x x x x x3 2 23 3 2 2 1+ + + = + + +( )( )
( ) ( )x x x+ + + =2 1 02 2
2.
P x A x B x x( ) ( ) ( ) .... ( )= , A x B x x( ), ( ),...., ( ) x + () x x2 + + ().
- P(x) , .
x P x x x x x x( ) ( )( )( ).= + + +1 6 2 12 2
:
0
x x 1 0 1,
x 1 x >1 , x =1 x
-
144 4: -
0 x x x x x2 6 0 3 2 0 3+ + ( )( ) x 2, x x2 6+ x < 3 x > 2 , x = 3 x = 2 < 2
1 52
1 52
0
127-208_update.indd 144 10/5/2013 8:53:09 pi
-
4.3 145
. 1 52
2+
, (x) ,
, x 2, . , (x) .
x , 1 52
1 52
< 2.
, . , - , - :
: f (, f()) (,f()) x'x, .
,
-
146 4: -
f x x x( ) = +3 3 1
1o :
, , (1,2).
2o : 1,1, 1,2, ... 1,9 :
, (1,5 , 1,6).
3o : (1,5 , 1,6) :
, (1,53 , 1,54) 1,53 < < 1,54. = 1,5.
ff( , ) ,( , ) ,1 5 0 13 01 6 0 30 0
ff( , ) ,( , ) ,1 53 0 01 01 54 0 03 0
1. .
i) 5 64 2x x= ii) x x x3 22 9 18 0+ =
iii) 3 5 3 55 4 3 2x x x x+ = + iv) x6 64 0 = v) x x3 2 2 0+ = vi) x x3 7 6 0 + =vii) ( )x + + =1 1 03 viii) 7 3 2 1 3 2 1 0
2 2 3( ) ( ) ( )( )x x x x+ + =
ix) x x x x3 2 28 7 5 6 9 36+ = + + + ( ) x) x x x
4 33 6 4 0 + =
x x3 3 1 0 + = - 1 2. .
ff( )( )1 1 02 3 0=
1 2
B(2,3)
A(1,1)
1,5x
127-208_update.indd 146 10/5/2013 8:53:11 pi
-
4.3 147
2.
i) x x x3 23 2 0 + + = ii) 3 8 15 4 03 2x x x+ + =
iii) x x3 10 12 0 = iv) x x x3 22 7 6 0+ + + =
3. .
i) x x4 3 2 0+ = ii) 2 3 6 24 5 04 3 2x x x x + + =
4. , x.
i) P x x x x x x( ) ( )( )( )= +2 3 2 12 2 ii) Q x x x x x x( ) ( )( )( )= + + + +2 2 24 3 2 1
5. :
i) 2 162 05x x ii) ( )( )x x x x3 2 22 2 9 0 + > iii) 2 5 6 9 03 2x x x + > iv) x x x3 24 3 18 0 +
6. :
i) x x x3 22 3 6 0+ + + > ii) x x x x4 3 26 22 30 13 0 + + iii) x x3 3 2 0 + < iv) x x x x4 3 2 3 6 0 +
7. x'x :
i) f x x x x( ) = 3 3 5 23 2 ii) g x x x( ) = 4 3 13
8. , - f x x x x x( ) = + +4 3 25 3 x'x.
9. :
i) x x8 415 16 0 = ii) ( ) ( )x x + =1 9 1 8 06 3
iii) 61
51
6 02x
xx
x+
+ +
=
10. x x3 5 3 0+ = (0,1) .
127-208_update.indd 147 10/5/2013 8:53:13 pi
-
148 4: -
1. :
i) 110
12
15
45
03 2x x x+ + = , ii) x x x3 256
223
52
0 + =
2. , P x x x x x( ) = + + 4 3 2 16 12 x+1 x 2. (x) = 0.
3. k , x x kx3 2 3 0 + + = .
* 4. x x + =2 2 0, N, 2, .
5. P x x x x k( ) = +6 4 25 10 , k x 1 (x). k (x) = 0.
6. - 5dm 9dm, . , - dm 21dm3.
7. t
c t tt
=+
+3
50
2
3 . ,
3 2 300 200 04 3t t t+ = . t .
8. 36m3, x.
3 x 3
3
3
x
x
x
1m
xm
xmxm
127-208_update.indd 148 7/5/2014 4:46:42
-
4.3 149
9. . V,
.
10. t , mm - d = 15t(t2 6t 9). ;
11. , , 108cm ( ). - , 11664 cm3.
12. i) (1,2)
ii) y = x
3+x2 x - .
iii) .
V = + 5003
2000 100 20 2 3pi ( ),
B 12
12
, .
3 33x xy
x3 + x2 5x + 3 = 0
127-208_update.indd 149 10/5/2013 8:53:16 pi
-
150 4: -
13. . :
) 200ml , 1cm, - . x x x3 23 2 200 0+ + = . x mm.
) - 1lit 10cm . r r3 210 318 0+ = r
mm. ( 1000 318pi
).
) - , 250ml, 5cm .
h ( -).
4.4
, , . .
1 xx x x
2 22 1
12 1
0+
=( )
x x 0 x 12. -
:
xcm
x+2 cmx+1 cm
r+10 cm
r cm
h+5cmh+5cm
127-208_update.indd 150 10/5/2013 8:53:17 pi
-
4.4 151
3 2 7 2x x+ = .
x x 72. x :
( )
3 1. x=1 .
xx x x
x x x x xx
x xx x
2 222 1
12 1
0 2 1 2 1 22 1
2 1 12 1
+
= +
( )( ) ( ) ( )
( )==
+ =
+ =
+ =
0
2 2 1 02 1 2 1 0
2 1 1 0
4 3
3
3
x x xx x xx x( )
( )( )
, , . - - .
2 7 2x x+ = +
( ) ( )2 7 22 2x x+ = +
2 7 4 42x x x+ = + +
x x2 2 3 0+ =
12
1. - x = 1.
2 x x= 2.
x 0. -, x x x= +2 4 4, x x2 5 4 0 + = x1 = 4 x2 = 1. x, - x 0 . , :
x = 4 : 4 4 2= x = 1 : 1 1 2= . x=4.
( )
127-208_update.indd 151 10/5/2013 8:53:19 pi
-
152 4: -
4 2 6 4 1x x+ + = .
x, 2 6 0x + x + 4 0, x 3. ' x :
3 5. x = 5 .
2 6 1 4x x+ = + +
( ) ( )2 6 1 42 2x x+ = + +
2 6 1 2 4 4x x x+ = + + + +
x x+ = +1 2 4
( ) ( )x x+ = +1 4 42
x x2 2 15 0 =
( )
( )
( )
(x)
> 0 (< 0)B(x)
A xB x
A x B x( )( )
( ) ( )> >0 0 A xB x
A x B x( )( )
( ) ( ) ,< 0 A x B x( ) ( ) < 0 (x).
, ( )( ) .x x x
x x + +
+ >
1 2 16
02
2
( )( )( )x x x x x + + + >1 6 2 1 02 2 , P(x)> 0 , , P(x)
x +( , ) ( , ).3 1 2
.:
2, 3 4, x , .
127-208_update.indd 152 10/5/2013 8:53:21 pi
-
4.4 153
x xx x
2
2
4 33 4
0 ++
. :
x xx x
x x x x2
22 24 3
3 40 4 3 3 4 0 +
+ + + ( )( ) x x2 3 4 0+ .
x x2 4 3 + 1 3 , x x2 3 4+ 1 4 . :
P x x x x x( ) ( )( )= + + 2 24 3 3 4
x - +x2- 4x + 3 + + - +x2 + 3x-4 + - + +
P(x) + - - +
-4
0
1
00
30
x +( , ) [ , ).4 3
1. :
i) 3 1
12 3 22
2
2xx x x
x xx
= +
ii)
xx x x
2
2121
41
+
=
2. : 2 2 2 03 2 x x x+ + =
3. :
i) x x3 4=
ii) 3 2 4x = iii) 5 1 4x =
iv) x x+ = +3 1 v) x x+ = +3 10 1 vi) x x+ =20 10
vii) x xx
=
+8
23
viii)
1 2 1+ = +x x
4. :
i)
xx+
>21
0
ii) 2 1
30x
x+
iii) x xx x
2
2
22
0 +
A xB x( )( )
0 -
x
A x B x( ) ( ) 0 B x( ) . 0
127-208_update.indd 153 10/5/2013 8:53:23 pi
-
154 4: -
( )
5. :
i) 2 3
14x
x+
>
ii) xx+
2
3 54 iii)
x xx
2 3 101
2 0
+
iv) xx x3 5
21
v) xx x2 1
32
+
6. : xx x x
2 22 1
12 1
+
( )
B
1. :
i) 2 3 1 3x x+ < ii) x x > 3 52. :
i) x x+ =3 10 0 ii) x x23 3 6 0+ =3. :
i) x x x x2 24 2+ = + ii)
4. :
i) x =1 ii) 4 1 22x x+ = 5. 2 3 3 3 4 04 3 2 x x x x + =
1. ( )( ) ,x x x x2 31 1 1+ + = - P x x x x( ) = + ++ +3 3 1 3 2 x
2+x+1 (, , ).
2. :
i) f x x x( ) ( )= + ++ 1 1 1 ( ) .x 12 -
.
ii) g x x x x( ) ( )= + + 2 21 ( ) .x 13
3. :
i) x x x4 32 2 1 0+ + + = ii) x x x x4 3 24 1 0+ + + =
( x x x x4 3 2 0+ + + + = , 0. ).
x x x + = +1 4 4
127-208_update.indd 154 10/5/2013 8:53:25 pi
-
(' ) 155
: x2
4.
i) x x x x4 3 216 2 4 0+ + = ii) x x x x4 3 28 13 8 1 0+ + + =5. ( ) ( )x x x x2 2 22 1 3 2 3 14 0+ + + + =
: x x y2 2 1+ = .6. , x x x5 23+ + + , -
x2 2 , 5x+8.
7. P x x x x x x( ) ... ,= + + + 17 16 15 1412 12 12 12 1 (11) (13).
8. - . - C , x = + 10 100 270 1803 2x x x .
i) ;
ii) ;
xx
y+ =1 .
127-208_update.indd 155 10/5/2013 8:53:27 pi
-
156 4: -
127-208_update.indd 156 10/5/2013 8:53:29 pi
-
157
* 3 : x x x3 2 0+ + + = - x y=
3 x Ax B3 + = .
127-208_update.indd 157 10/5/2013 8:53:31 pi
-
158 4: -
127-208_update.indd 158 10/5/2013 8:53:33 pi
-
127-208_update.indd 159 10/5/2013 8:53:33 pi
-
22-0207-0122-0207-020-22-0207-02-front.pdfPage 1
0-22-0207-02-back.pdfPage 1