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Program Latih Tubi SPM 2012 - Matematik Tambahan Kertas 2 - Download Now DOC

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 TOPIC: SIMULTANEOUS EQUATIONS

                                            1
1.   Solve the simultaneous equations x +       y = 1 and y2 – 10 = 2x.   [5 marks]
                                            2




2.   Solve the simultaneous equations x + 2y = 1 and x2 + 2y2 + xy = 5.
     Give your answers correct to three decimal places.                   [5 marks]




3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012
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 TOPIC: TRIGONOMETRIC FUNCTIONS


1. (a) Prove that cosec2 x – 2 sin2 x  cot2 x = cos 2x.                                       [2 marks]

   (b) (i) Sketch the graph of y = cos 2x for 0 ≤ x ≤ 2.
       (ii) Hence, using the same axes, draw a suitable straight line
                                                                                               x
           to find the number of solutions to the equation 3(cosec2 x  2 sin2 x – cot2 x) =     1
                                                                                               
           for 0 ≤ x ≤ 2.. State the number of solutions.
                                                                                               [6 marks]




2. (a) Sketch the graph of y =  2 sin 2x for 0 ≤ x ≤ 2.                                      [4 marks]

   (b) Hence, using the same axis, sketch a suitable graph to find the number of solutions to the
                  
       equation     + 2 sin 2x = 0 for 0 ≤ x ≤ 2.. State the number of solutions.             [3 marks]
                  x




3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012
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3. (a) Sketch the graph of y = | 3cos 2x | for 0 ≤ x ≤ 2.                                    [4 marks]

   (b) Hence, using the same axis, sketch a suitable graph to find the number of solutions to the
                                    x
       equation 2 - |3cos 2x | =      for 0 ≤ x ≤ 2. State the number of solutions.           [3 marks]
                                   2




4. (a) Sketch the graph of y = 1 + 3 sin x for 0 ≤ x ≤ 2.                                     [4 marks]

   (b) Hence, using the same axis, sketch a suitable straight line to find the number of solutions to the
       equation 6π sin x = 4π – 3x for 0 ≤ x ≤ 2. State the number of solutions.              [3 marks]




3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012
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     TOPICS: COORDINATE GEOMETRIC AND DIFFERENTIATION


1. A curve has a gradient function px2 – 4x, where p is a constant. The tangent to the curve at the point
   (1, 3) is parallel to the straight line y + x – 5 = 0. Find

     (a)   the value of p.                                                                     [3 marks]
     (b)   the equation of the curve.                                                         [3 marks]




                                             2
2.   A curve with a gradient function 2x –        has a turning point at (k, 8).
                                             x2

     (a) Find the value of k.                                                                 [3 marks]
     (b) Determine whether the turning point is a maximum or a minimum point.                 [2 marks]
     (c) Find the equation of the curve.                                                       [3 marks]




3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012
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  TOPICS: QUADRATIC FUNCTIONS

1. Diagram 2 shows the curve of a quadratic function f(x) = −x2 + kx – 5. The curve has a maximum
   point at B(2, p) and intersects the f(x)-axis at point A.

   (a) State the coordinates of A.                                                        [1 mark]
   (b) By using the method of completing the square, find the value of k and of p.        [4 marks]
   (c) Determine the range of values of x, If f(x) ≥ − 5.                                [2 marks]
                                                                   f(x)


                                                                                                x
                                                                0           B(2, p)

                                                               A



                                                                          Diagram 2




2. The gradient function of a curve is hx2 – kx, where h and k are constants. The curve has a turning
   point at (3, −4). The gradient of the tangent to the curve at the point x = −1 is 8.

   (a) the value of h and of k..                                                         [5 marks]
   (b) the equation of the curve.                                                        [3 marks]




3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012
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 TOPIC: COORDINATE GEOMETRIC


                                                                       3
1. Solution by scale drawing is not accepted.                 y   P(       , k)
                                                                       2
   Diagram 10 shows a triangle OPQ. Point S lies
   on the line PQ.

    (a)    A point W moves such that its distance                                 S(3, 1)
           from point S is always 2½       units.
           Find the equation of the locus of W.
                                         [3 marks]        O                                         x

                                                                                            Q

                                                                       Diagram 10




   (b) It is given that point P and point Q lie on the locus of W. Calculate
          (i) the value of k.
          (ii) the coordinate of Q.                                                             [5 marks]




   (c) Hence, find the area, in unit2, of triangle OPQ.                                         [2 marks]




3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012
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    2. Solution by scale drawing is not accepted.
        Diagram 3 shows the triangle AOB where O is the origin. Point C lies on the straight line AB
.
        (a) Calculate the area, in unit2 , of the triangle AOB.                                    [2 marks]
        (b) Given AC : CB = 3 : 2, find the coordinate of C.                                       [2 marks]
    .
                                                                                 y

                                                                  A(−3, 4)



                                                                                        C
                                                                                                       x
                                                                             O

                                                                                                 B(6,−2)

                                                                                     Diagram 3




    (c) A point P moves such that its distance from point A is always twice its distance from point B.
        (i)   Find the equation of the locus of P.
         (ii) Hence, determine whether this locus intersects the y-axis.                           [6 marks]




    3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012
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  TOPIC: VECTORS

                     5          2              k
1. Given that AB =  , OB =   and CD =   , find
                   7       3          5 
                                       

   (a) the coordinates of A.                                                                  [2 marks]
   (b) the unit vector in the direction of OA .                                               [2 marks]
   (c) the value of k, if CD is parallel to AB .                                              [2 marks]




2. In diagram 6, ABCD is a quadrilateral. The diagonals
   BD and AC intersect at point R. Point P lies on AD.                     D
                        1        1
   It is given that AP = AD, BR = BD ,
                        3        3
    AB = x and AP = y.                                                                      C
                                                                 P          R
   (a) Express in terms of x and y :
        (i) BD                                               A                    B
        (ii) AR                          [3 marks]                                      Diagram 6

   (b) Given that DC  k x  y and AR = h AC , where h and k are constant, find the value of h, and of k.
                                                                                                [4 marks]




   3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012

				
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