Κατατακτήριες Μαθηματικών ΑΠΘ_Άλγεβρα Β' Ενιαίου...

55

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 of Κατατακτήριες Μαθηματικών ΑΠΘ_Άλγεβρα Β' Ενιαίου...

  • 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