Additional Mathematics - Paper 1

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					ADDITIONAL MATHEMATICS
              SPM 2012
               PAPER 1
              ( PART TWO )


LAST KOPEK REVISION
      TRY TO ANSWER THESE QUESTIONS
AND CHECK YOUR ANSWERS FROM THE FOLLOWING
       SCHEME OF ANSWER AND MARKING
14.      The variables x and y are related by the equation
              p
         y = x . The diagram below shows the straight line graph
             3
         obtained by plotting log10 y against x.
                                     p
         (a) Express the equation x in its linear form used to obtain
                                    3
              the straight line graph as shown in the diagram.

          (b) Find the value of p.

      Answer:                                                log10 y

      (a) log10 y = log10 p – x log10 3   √1                         (0,2)

      (b) log10 p = 2     √ M1                                   0
                                                                             x


                p = 100    √2
15.   Given that y = 2x + 1 , x = 3
                    (x – 3)2
      (a) find the value of dy/dx, when x = 4 ,
      (b) the approximate change in y when x increases
          from 4 to 4.01

Answer:
(a) dy/dx = (2x + 1) (2) (x – 3) (1) – (x – 3) (2)
                                              2
                                                        √ M1
                             (x – 3)4
            [2(4) + 1] (2) (4 – 3) (1) – (4 – 3)2 (2)
          =
                            (4 – 3)4
          = 16 √ 2
                   15 (b) Sambungan




(b) x = 4
    ∂x = 0.01    √ M1

    dy/dx = 16

    ∂y = dy/dx x ∂x
    ∂y = 16 (0.01)      √ M1

    ∂y = 0.16    √2
16.   Given y = 5h i + (h – 2) j is a non-zero vector and is
      parallel to x-axis. Find the value of h.



      Answer:

      5h i + (h – 2) j = x i + 0 j
      ( parallel to x-axis, vertical component of the vector = 0 )


      h–2=0         √ M1

        h=2         √2
17.   Diagram shows the position of the point A, B and C relative
      to origin, O.
      Given B(1, 8), OA = 7i + 2j and OC = ¾ AB.
      Find in terms of i and j,
      (a) OC,
      (b) CB.


  Answer:                                            y

                                              C.
                                                         .B
  (a) AB = − OA + OB
                                                              .A
         = − (7 i + 2 j) + (i + 8 j) √ M1
                                                                   x
         =−6i+6j                                   O

      OC = −9/2 i + 9/2 j √ 2
  (b) CB = − OC + OB
         = − 9/2 i + 9/2 j + (i + 8 j) √ M1
         = − 11/2 i + 7/2 j √ 2
18.   Diagram shows a unit circle. T (−h, k) is a point on
      the rotating ray of an angle θ. Find in terms of h, the
      value of
      (a) cos θ.
      (b) sin 2θ.
                                                     y

                                       (-h, k)
      Answer:                    1
                            k
                                                         θ
      (a) cos θ = −h            θ
                                                             x
                   √1           −p               O

       (b) sin 2θ = 2 sinθ cosθ
                  = 2 (k) (−p) √ M1
                  = −2pk √ 2
19.   Given sin A = 3/5, cos B = 12/13. If both the angle A
      and B are at the same quarter, find the value of
      (a) sin (A + B),
      (b) tan (A − B)


                                                       5
      Answer:                                  3
                                                    A
      (a) sin A cos B + cos A sin B                4
           = (3/5) (12/13) + (4/5) (5/13) √ M1         13
                                               5
           = 56/65 √ 2                              B
           tan A − tan B                           12
      (b)
          1 + tan A tan B
          = (3/4) – (5/12) √ M1 = 7/16 √ 2
            1 + (3 /4) (5 /12)
20.     Point A lies on a curve y = 2x4 – x, find the
        coordinates of point A where the gradient of the
        normal at point A is -1/7.

Answer:

 dy/dx = mt = 8x3 − 1    √ M1

      8x3 – 1 = 7 √ M2
          8x3 = 8
            x=1
            y = 2(1)4 – 1
              =1
           A = ( 1, 1) √ 3
21. The standard deviation of a set of six numbers is √15.
    Given that the sum of square for the set of numbers is
    144. Find the new mean when a member 10 is added
    to this set.

      Answer:
     N=6                       x2 =9
     σ2 = 15                   x =3
    ∑ x2 = 144                  ∑x = x (N)
           144                     = 3 (6) = 18
      15 =     −x2    √ M1
            6                   ∑xnew = 18 + 10     √ M2

                                 x new = 28/7 = 4   3
22. A set of positive integers consists of 6, 7, k, 1, 8, 3, 3.
    (a) Find the value of k if the mean of the data is 5.
    (b) State the range of the values of k if the median of
        the data is k.


     Answer:
     (a)   28 + k       √ M1
                  =5
             7
                 k=7    √2

      (b) 1, 3, 3, k, 6, 7, 8

            3≤k≤6      √1
23.   A Proton Wira can accommodate 1 driver and 3 adult.
      Find the number of different ways the selection can
      be made from 3 men and 4 women if,
      (a) there are no restriction,
      (b) the driver must be a man.

      Answer:

      (a) 7C4
          = 35

      (b) 3C1 x 6C3
           = 60
         THE END
   Those who have worked hard
with the correct method of learning
       Additional Mathematics
 surely will receive their rewards

				
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