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Answer Marking Scheme F4 Add Math (DOC)

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					SULIT                                                                                                          3472/1/2




                                                                                                        FORM IV 2012




             SEKOLAH MENENGAH KEBANGSAAN TINGGI KAJANG
                                JALAN SEMENYIH, 43000 KAJANG, SELANGOR




                               PEPERIKSAAN PERTENGAHAN TAHUN 2012
                                           ADDITIONAL MATHEMATICS




                                            Answer and Marking Scheme



Prepared by :                                    Checked by:                                    Verified by:

........................................        ...........................................    ..…………………………
( NG KOK LYE )                               (NOORAIDA MOHD ZIN)                              ( SOH BOON CHUAN )

                              This question paper consists of 7 printed pages and 1 blank page




3472/1/2                                                                                                       SULIT
   For          SULIT                                                                                                      3472/1/2
examiner’s                                                         SECTION 1
 use only

                                                                    BAHAGIAN 1
                                                               [ 44 marks/markah ]
                                                     Answer all questions /Jawab semua soalan.


                1. Diagram 1 shows the relation between set A and set B.
                     Rajah 1 menunjukkan hubungan antara set A dan set B.
                                                                                       Set A                     Set B
                     (a) Express the relation in the form of ordered pairs.
                           Ungkapkan hubungan itu dalam bentuk pasangan tertib.          1                             p
                     (b) State the type of the relation.                                                               q
                         Nyatakan jenis hubungan itu          [ 3 marks/markah ]         2
                                                                                         3                             r
                     Answer / Jawapan:
                                                                                         4                             s
                     (a) {(1,p), (2,r), (3,s), (4,p)} √ 2

                           Any one pair correct √ B1                                                 DIAGRAM 1
                                                                                                      Rajah 1

        1
   3                  (b) many – to – one √ 1
            3



                2.    Diagram 2 shows the linear function f.
                      Rajah 2 menunjukkan fungsi linear f.                                       f
                                                                                   x                            f(x)
                     (a) State the value of w.
                            Nyatakan nilai bagi w.
                                                                                  0                            2

                      (b) Using the function notation,                            1                            3
                          express f in terms of x.
                                                                                  5                            w
                            Dengan menggunakan tatatanda
                            fungsi, ungkapkan f dalam sebutan x.
                                                                                  8                             10
                                            [ 2 marks/markah ]
                      Answer / Jawapan:                                                 DIAGRAM 2
                                                                                             Rajah 2

                      (a) 7 √ 1




                      (b) f(x) = x + 2     or    f:x          x+2 √1
        2
    2
            2


        5
                3472/1/2                                               2                                                    SULIT
SULIT                                                                                                     3472/1/2      For
                                                                                                                     examiner’s
                                                                                                                      use only
3.   Diagram 3 shows the graph of the function f(x) = |3x – 2|, for the domain 0 ≤ x ≤ 5.
     Rajah 3 menunjukkan graf bagi fungsi f(x) = |3x – 2| untuk domain 0 ≤ x ≤ 5.    [ 3 marks/markah ]
     State / Nyatakan
     (a) the value of m.                                                       y
             nilai m.

     (b) the range of f(x) corresponding to the given domain.                                      f(x) = |3x – 2|
           julat bagi f(x) sepadan dengan domain yang diberikan.

     Answer / Jawapan:
                                                                         2
               2
       (a)       √1
               3
                                                                          0        m               5
                                                                                                                         3
       (b)     0 ≤ f(x) ≤ 13 √ 2                                                      DIAGRAM 3
                                                                                         Rajah 3
                                                                                                                     3
               13 √ B1                                                                                                       3


4.   Two functions are defined by f : x 2x + 1 and g :x x2 + 2x – 6.
     Given that gf : x 4x2 + px + q, find the value of p and of q.
     Dua fungsi ditakrifkan sebagai f : x    2x + 1 dan g : x         x 2 + 2x – 6.
     Diberi gf : x   4x2 + px + q, cari nilai p dan nilai q.                                   [ 3 marks/markah ]

     Answer / Jawapan:

     p = 8, q = −3          √ 3 (both correct)

     4x2 + 8x – 3 √ B2                                                                                                   4
     (2x + 1)2 + 2(2x + 1) – 6 √ B1                                                                                  3
                                                                                                                             3


                                                    x5
5.   The function of f is defined as f (x)                  , x  m . Find,
                                                    3  2x
                                           x5
     Fungsi f ditakrifkan sebagai f(x) =            , x  m. Cari                              [ 3 marks/markah ]
                                           3  2x
     (a) the value of m,
           nilai m,

     (b) f 1( x ) .

     Answer / Jawapan:
              3
     (a)        √1
              2                                                                                                          5
                          5  3x      1                         y5
     (b)      f -1(x) =          ,x    √2              x            √ B1
                                                                                                                     3
                          2x  1      2                        3  2y                                                        3


                                                                                                                         9
3472/1/2                                                 3                                                  SULIT
   For          SULIT                                                                                          3472/1/2
examiner’s      6.                                                     2
                     Determine the roots of the quadratic equation 5x = 3x + 2.
 use only
                     Tentukan punca-punca bagi persamaan kuadratik 5x2 = 3x + 2.                     [ 3 marks/markah ]

                     Answer / Jawapan:
                                2
                     x=             or − 0.4 , x = 1 √ 3 (both correct)
                                5
                                                                   ( 3)  ( 3)2  4(5)( 2)
                     (5x + 2) (x – 1) = 0 √ B2        or    x                                 √ B2
                                                                              2(5)
        6
    3                5x2 – 3x – 2 = 0 √ B1
            3



                7.   Find the range of values of k if the following quadratic equation (k + 1)x2 + 6x + 3 = 0 which
                     has two different roots.
                     Cari julat bagi nilai k jika persamaan kuadratik berikut (k + 1)x2 + 6x + 3 = 0 mempunyai dua punca
                     yang berbeza.
                                                                                                       [ 3 marks/markah ]
                     Answer / Jawapan:

                     k<2 √3

                     6 2 – 4(k + 1)(3) > 0 √ B2

                      62 – 4(k + 1)(3)   or b2 – 4ac > 0     or    a = (k+1), b = 6, c = 3 √ B1

        7
    3
            3


                8. Given that  and  are the roots of the quadratic equation 2x2 – 5x + 3 = 0. Form the quadratic
                   equation whose roots are 2 and 2 .
                   Diberi  dan  adalah punca-punca bagi persamaan kuadratik 2x2 – 5x + 3 = 0. Bentuk satu persamaan
                   kuadratik yang mempunyai punca-punca 2  dan 2  .
                                                                                                  [ 3 marks/markah ]
                   Answer / Jawapan:

                     x2 – 5x + 12 = 0 √ 3

                     4αβ = 12
                     2(α + β) = 5    √ B2 (both)

        8                        5
                     α+β=
    3                            2   √ B1 (both correct)
            3           αβ = 3


        9

                3472/1/2                                           4                                            SULIT
SULIT                                                                                                           3472/1/2
                                                                                                                              For
9. Diagram 3 shows the graph of a curve y = a(x + p) ² + q that passes through the point (0, 3)                            examiner’s
   and has the minimum point (2, −1). Find the values of a, p and q.                                                        use only
   Rajah 3 menunjukkan geraf bagi satu lengkung y = a(x + p) 2 + q yang melalui satu titik (0.3) dan
   mempunyai titik minima (2 , −1). Cari nilai-nilai a, p dan q.
                                 y
                                                                                                  [ 3 marks/markah ]



                      (0, 3)
                                                                x              DIAGRAM 3
                             0
                                       (2, −1)                                   Rajah 3

    Answer / Jawapan:

    p = −2 √ 1
                                                                                                                               9
   q = −1    √1
                                                                                                                           3
   a=1       √1                                                                                                                     3



10. Find the range of values of x for which x(x − 3) ≤ 4.
    Cari julat bagi nilai x yang mana x(x − 3) ≤ 4.                                               [ 3 marks/markah ]

    Answer / Jawapan:

    −1 ≤ x ≤ 4          or           x ≥ −1 , x ≤ 4         √3

    (x + 1)(x – 4) ≤ 0 √ B2              or                                     or
                                                                        √ B2             +    −        − √ B2
                                                                           x
                                                                                         +    +        − x
                                                 -1                 4                              4                           10
                                                                                     +
                                                                                         -1   −        +
    x2 – 3x – 4 ≤ 0 √ B1                                                                                                   3
                                                                                                                                    3



             81 x 1
11. Solve             243
               3x
                                                                                                  [ 4 marks markah ]
    Answer / Jawapan:
     x=3 √4
     4x – 4 = 5 + x      or            3x – 4 = 5 √ B3
     34x–4 = 3 5+x       or            33 x – 4 = 35 √ B2                                                                      11
      4       5
     3 or 3 √ B1                                                                                                           4
                                                                                                                                    3




                                                                                                                               10
3472/1/2                                                    5                                                    SULIT
   For        SULIT                                                                                        3472/1/2
examiner’s
 use only
              12.   Given that lg 2  0  3 and lg 5 = 0.7. Find, without using scientific calculator or mathematical
                    tables, the value of log 2 20 .
                    Diberi lg 2 = 0.3 dan lg 5 = 0.7. Cari, tanpa menggunakan kalkulator saintifik atau jadual sifir
                    matematk, nilai bagi log2 20.
                                                                                                 [ 4 marks/markah ]
                    Answer / Jawapan:

                    13
                     3 or 4.333 √ 4

                    2(0.3) 0.7
                                       √ B3
                        0.3
                    2lg2  lg5
    12                 lg2            √ B2

    4
          4         lg20
                               √ B1
                     lg2

              13.   Given that log5 x = k, find logx 125x2 in terms of k.
                    Diberi log5 x = k, find logx 125x2 dalam sebutan k.
                                                                                                 [ 4 marks/markah ]
                    Answer / Jawapan:

                    3  2k
                           √4
                       k
                    3log5 5  2log5 x
                                              √ B3
                         log5 x

                    log5 125  log5 x 2
                                        √ B2
    13                    log5 x
    4
                     log5 125x 2
          4                      √ B1
                        log5 x


              14.   Express 2n 3  2n  5(2n 1 ) in the simplest form.                          [ 3 marks/markah ]
                    Ungakapkan 2n + 3 −2n + 5(2n – 1 ) dalam sebutan paling ringkas.


                    2 n -1(19) √ 3
                                  5
                    2 n(8 – 1 +        ) √ B2
                                  2
                               2n              
    14              2 .2 –2 +5
                      n    3
                               2
                                  n              √ B1
                                                
    3                                          
          3


     11
              3472/1/2                                               6                                      SULIT
SULIT                                                                                                 3472/1/2
                                                    SECTION 2
                                                    BAHAGIAN 2
                                             [ 26 marks/markah ]
                                  Answer all questions / Jawab semua soalan.

1.   Solve the simultaneous equations. Give your answers correct to three decimal places.
     Selesaikan persamaan serentak. Berikan jawapan anda betul kepada tiga angka perpuluhan.

                        2x  y  1
                        x 2  2 y  xy  5 .                                              [ 5 marks/markah ]

     Answer / Jawapan:
                                                                          y1
     y = 2x – 1 √ P1                                      or         x=          √ P1
                                                                           2
     x2 + 2(2x – 1) + x(2x – 1) = 5 √ K1                                      2       y  1
                                                                      y  1
                                                                            + 2y +  2  y = 5 √ K1
     3x2 + 3x – 7 = 0                                                 2                  

                                                                     3y2 + 12y – 19 = 0
         3  3 2  4(3)( 7)                √ K1
     x
                2(3)
                                                                           12  12 2  4(3)( 19)
                                                                     y                            √ K1
     x = 1.107, - 2.107 √ N1                                                       3(3)

     y = 1.214, - 5.214           √ N1                              y = 1.214, - 5.214    √ N1
                                                5                                                           5
                                                                          x = 1.107, - 2.107      √
     N1


2.   Given that f : x       x2 − 2 and g : x              3x + 4.
                        2
     Diberi f : x    x − 2 dan g : x            3x + 4.

                             −1
     (a) (i) Determine f         (x).                                                      [ 2 marks/markah ]
                            −1
               Tentukan f     (x).
                                        −1
          (ii) State whether the f       (x) exist. Give a reason to your answer by showing the evidence of
               the reason given.
               Nyatakan sama ada f −1(x) itu wujud. Berikan sebab kepada jawapan anda dengan
               menunjukkan bukti kepada sebab yang anda berikan .            [ 3 marks/markah ]
     (b) Given gh(x) = 6x + 7, determine h(x).
          Diberi gh(x) = 6x + 7, tentukan h(x).                                            [ 2 marks/markah ]

          Answer / Jawapan:
                                                                          (b) Let h(x) = y
     (a) x = y2 - 2 √ K1                                                          g(y) = 6x + 7
           y2 = x + 2                                                             g(y) = 3y + 4
           f -1(x) =  x  2 √ N1                                               3y + 4 = 6x + 7 √ K1
                                                                                     y = 2x + 1
          Let x = any value,
          f -1(x) =  the value after substitution into x  2 , √ K1             h(x) = 2x + 1 √ N1
          an object has two images or
          it’s a one-to-many relation or the function undefined
                                                                                                        7
         Therefore the invest function does not exist. √ N1
3472/1/2                                     7                                                         SULIT
SULIT                                                                                                    3472/1/2


3.   (a) Simplify:
           Permudahkan:
                                 4 x + 2 – 2 2x + 3                                             [ 4 marks/markah ]
     (b) Hence, solve the equation:
           Seterusnya, selesaikan persamaan;

                                 4 x + 2 – 2 2x + 3 = 64                                        [ 2 marks/markah ]

           Answer / Jawapan:

     (a) 2 2(x + 2) – 2 2x + 3             √ P1                   (b) 2 2x + 3 = 2 6 √ P1
           2 2x . 2 4 − 2 2x . 2 3 √ K1                               2x + 3 = 6
           2 2x (2 4 – 2 3) √ K1                                                  3
                                                                             x=     √ N1           2
               2x                                                                 2
           2        (8)
               2x + 3                                  4                                           6
           2              √ N1

4.   The curve of a quadratic function f(x) = x2 + 2hx – 5 has minimum point of (2, k).
     Lengkung bagi satu fungsi kuadratik f(x) = x2 + 2hx – 5 mempunyai titik minimum (2, k).

     (a) State the equation of the axis of symmetry of the curve.
           Nyatakan pesamaan paksi simmetri bagi lengkung itu.                                  [ 1 mark/markah ]
     (b) By using the method of completing the square, determine the value of h and of k.
           Dengan menggunakan kaedah penyempurnaan kuasadua, tentukan nilai bagi h dan bagi k.
                                                                                   [ 4 marks/markah ]
     (c) Hence, sketch the graph of the curve.
           Seterusnya, lakarkan graf bagi lengkung itu.                                         [ 3 marks/markah ]

     Answer / Jawapan:

     (a) x = 2 √ N1                                                                         1
     (b) f(x) = x2 + 2hx + h2 – h2 – 5 √ K1
             = (x + h)2 – h2 – 5 √ K1
               h = −2 √ N1
               k = −h2 – 5
                    = −4 – 5
                                                                                            4
                    = −9 √ N1
                                            f(x)

     (c)                                                       √ P1
                                                                         x
                                           0          2
                                                                                            3
                                          -5
                                   √ P1
                                                      (2, −9) √ P1                          8


                                 END OF THE ANSWER & MARKING SCHME
3472/1/2                                                      8                                           SULIT
SULIT                                            3472/1/2
           SKEMA PEMARAKAHAN DAN JAWAPAN TAMAT




3472/1/2                    9                     SULIT

				
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