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文憑試 數學 試卷一 答案及題解 by A.Ho

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  • by A.Ho

  • 2013DSE by A.Ho

    1

    ()

    1.

    12 3 6 72 18

    8 5 8 5

    80

    23

    ( )a b a ba b a b

    a

    b

    =

    =

    2. 3 55 2

    xyx

    +=

    5 2 3 53 2 5 5

    5 53 2

    y xy xx xy y

    yx

    y

    = +

    + =

    =

    +

    3. (a) 9a2 16b2 = (3a + 4b)(3a 4b) 2 2 2 2(b) 18 24 9 16 6(3 4 ) (9 16 )

    6(3 4 ) (3 4 )(3 4 )(3 4 )[6 (3 4 )](3 4 )(6 3 4 )

    a b a b a b a ba b a b a b

    a b a ba b a b

    + + = +

    = + +

    = +

    = + +

    4. (a) 25400 (1 25%) $20320 + =

    (b) 25400 (1 25%) $19050 =

    5. (a) : 2 : 3: 6 : 9

    x yx y

    =

    =

    : 3 : 4: 6 :8

    x z

    x z

    =

    =

    x : y : z = 6 : 9 : 8

    (b) 6k9k 8k

    9 38 3060 4

    36 24 9180765

    kk

    k kk

    =

    += +

    =

    (6 9 8) 76517595

    SS

    = + +

    =

  • 2013DSE by A.Ho

    2

    6. (a) P ( 8, 1) Q (8, 5)

    (b) K P PQ K (x, y) K

    2 2 2 2

    2 2

    [ ( 1)] ( 8) ( 8 8) [ 1 ( 5)]( 1) ( 8) 272

    x y

    x y

    + = +

    + + =

    7. (a) 4 1202 3 4 360x

    =

    + + +

    32 3 4 360

    yx

    =

    + + +

    12 93

    x

    x

    = +

    =

    32 3 4 3 360

    90

    y

    y

    =

    + + +=

    (b)

    1 1 1 1, , ,6 4 3 4

    16

    16

    8. (a) b = 150 b = 330 (b) b b = 150

    (4 3) 72

    2

    a

    a

    + =

    =

  • 2013DSE by A.Ho

    3

    9. PS RPS = 2kQPR = k QSP = 2k

    PTS 2 2 106 18018.5

    k kk

    + + =

    =

    PQT 10687.5

    PQS kPQS

    + = =

    () 10. (a) : x2 + x

    2

    3

    3

    1

    x x

    x x x Ax x

    A

    +

    (b)

    2 3

    3

    ( 1 )

    x

    x A x x Ax Ax

    A x A

    +

    A = 1

    11. (a) 0.5 cm 39.5 2.5 7.5 178.125 cm =

    (b) 310.5 3.5 8.5 312.375 cm =

    12. (a) 21 2kC kd

    = + k1 k2

    21 2

    1 2

    1234003

    1110600 9 ......(1)

    kk

    k k

    = +

    = +

    21 2

    1 2

    551506

    1985400 36 ......(2)

    kk

    k k

    = +

    = +

    (2) (1), 11

    874800 2732400

    kk

    =

    =

    k2 = 819000

    281900032400C

    d= +

  • 2013DSE by A.Ho

    4

    (b) 1 10d

    2

    2

    2

    1 11100

    1819000 8190

    1851400 40590

    d

    d

    d

    $40590 10 m

    13. (a) (i) 66.5 65.8 65 68

    (ii) 3

    3

    (b)

  • 2013DSE by A.Ho

    5

    14. (a) AB = BC (G)

    EBD = EBC (I ) EBC = BAD (BE // AD) EBD = ADB (BE // AD)

    BAD = ADB AB = BD ()

    AB = BD = BC B ADC B ACD

    (b)(i) DOC = 90 BDO CDO BO = CO = 4 BD = BC = 8

    BOD OD2 = 82 42 OD = 4 3

    D (0, 4 3 )

    (ii) BIO, tan 30

    44 4 3

    (or )33

    OI

    OI

    =

    =

    I (0, 4 33

    )

    BC D:

    2 2 2

    2 2 2 2

    4 3 4 3( 0) ( ) (4 3 )3 3

    4 3 64 8 3( ) ( 16 0)3 3 3

    x y

    x y x y y

    + =

    + = + =

  • 2013DSE by A.Ho

    6

    15. (a) 6410

    8

    4! 5! 142

    CP

    =

    (b) P(): 6 5

    4 410

    8

    142

    P PP

    =

    16. (a) (2 3 )

    2 3 4 92 313 13

    a a ii

    a a i

    +=

    +

    = +

    (2 3 )2 3 4 9

    2 313 13

    b b ii

    b b i

    =

    + +

    =

    (b)(i) 2 3 2 3 2 2 3 3( ) ( )13 13 13 13 13 13

    a a b b a b a bi i i+ + + = +

    22 3 2 3 4 6 6 9( ) ( )13 13 13 13 169 169 169 169

    13

    a a b b ab ab ab abi i i i i

    ab

    + = +

    =

    2 2 2 3 3( ) 013 13 13

    a b a b aby i y+ + + =

    (ii) 2 2 3 3( ) 813 13

    a b a b i+ + =

    2 2 813

    2 2 104......(1)

    a b

    a b

    + =

    + =

    3 3 013

    ......(2)

    a b

    a b

    =

    =

    (2) (1) 2 2 10426

    a a

    a

    + =

    =

    b = 26

  • 2013DSE by A.Ho

    7

    17. (a) (1 20%) 1.2x x + = 2 (1 20%) 2.4x x + =

    2(1 20%) 0.364x

    x + =

    (b) $0.2x

    18. (a) 2

    n n n nn

    B C C DS =

    1 2

    2

    2 2

    2

    2 1

    1[ ( ) ]22

    22

    2

    n

    n

    n

    x

    x

    x

    =

    =

    =

    (b) 2 1 2 3( ...)x S S S= + + +

    2 2 22

    3 5

    2

    2

    2

    ... 1082 2 2

    2( ) 108114

    1 1083

    18

    x x xx

    x

    x

    x

    x

    + + + =

    =

    =

    =

  • 2013DSE by A.Ho

    8

    19. (a) 6 3 5

    27

    s

    s

    + +=

    =

    27(7 6)(7 3)(7 5) 2 14 cm =

    (b) AM BC AMBC

    ABC 146 cm2

    2

    6 AM = 146

    AM = 142 cm

    2'6 MA

    = 142

    AM = 3142

    cm

    AMA, cos x = )

    3142)(142(2

    8)3142()142( 222 +

    x = 92.7294 x 92.4

    y

    sin sin8

    2 14 sin 92.7294sin

    869.1239

    y xAM

    y

    y

    =

    =

    =

    69.1

    (c) 2

    'MABC

    x 70 A A AM BC

    A HoMATH Paper 1_A Ho_A (Chi)