2019 J2 Hwa Chong H2 Math Prelim (with ans)

1 The number of units, D(x) of a particular product that people are willing to purchase per week in city A at a price $ x is given by the function 40320() g( )Dx x , where g( x) is a quadratic polynomial in x. The following table shows the number of units people are willing to purchase at different prices. x 581 0 D(x) 384 224 168 Find the number of units of the p roduct that people are willing to purchase at a price of $18. [4] 2 It is given that 1 1ln 1 n n n xp x    , where 11 x  and n is a positive integer. (i) Find 1 N n n p   , giving your answer in terms of N and x . [3] (ii) Hence find the sum to infinity of the series in part (i) in terms of x. [2] 3 In the triangle ABC, 1AB  , 2AC  and angle 2ABC x   radians. Given that x is sufficiently small for 3x and higher powers of x to be ignored, show that 2BC p qx rx  , where p, q, r are constants to be determined in exact form. [5] 4 A part of a hyperbola has equation given by  2(5 )f1 36 xx  , xD , where D Ž  . (i) State the largest possible setD . [1] (ii) State the equations of the asymptotes of fyx . [1] (iii) Sketch the graph of fyx for D in part (i), showing clearly all the features of the curve. [2] (iv) On separate diagrams, sketch the graphs of  1 fy x and f 'yx , showing clearly all the features of the curves. [4] +0DWK3UHOLP+&, www.KiasuExamPaper.com 289

2 5 Referred to the origin O, points A and B have position vectors a and b respectively. Point C lies on OB produced such that OC OB     where 1.  Point D is such that OCDA is a parallelogram. Point M lies on AD, between A and D, such that :1 : 2AM MD  . Point N lies on OC, between O and C, such that :4 : 3ON NC  . (i) Find the position vectors of M and N, in terms of a, b and  .[ 2] (ii) Show that the area of triangle OMD is k ab , where k is a constant to be determined. [4] (iii) The vector p is a unit vector in the direction of OD   . Give a geometrical meaning of .pa .[ 1] 6 A curve C is defined by the parametric equations 225sinxt , 2c o syt where 0 t  . (i) Find d d y x .[ 2] (ii) The tangent to C at the point where 4t  cuts the x-axis at P and the y-axis at Q, find the exact area of the triangle OPQ, where O is the origin. [4] (iii) State the equation of the normal to C where the normal is parallel to the x-axis. [1] 7 (a) (i) Find cos 2d 2ed x x     .[ 1] (ii) Hence find cos 21 sin e d2 x xx .[ 4] (b) Using the substitution 1e xu  , find 1 d1e x x .[ 4] www.KiasuExamPaper.com 290

3 8 The function f is defined by 2 2 ()f: 4 xaxa 6 for x   , 3ax a  , where a is a positive constant. (i) Sketch the graph of fyx , giving the coordinates of any stationary points and the points where the graph meets the axes.