2014 H2 Maths Prelims 2 P2 Qn Paper

2   IJC/2014/JC2 9740/02/S/2014   Section A: Pure Mathematics [40 marks] 1 (i) Given that   21cosyx  , show that 2 2 2 dd(1 ) 2 dd yyxx xx  . [3] (ii) Hence find the Maclaurin’s series for   21cos x , up to and including the term in 3x . Give the coefficients in exact form. [3] (iii) Show that, if x is sufficiently small for powers of x above x2 to be neglected, then   22 1cos 24ee 1 x ax bx     , where a and b are constants to be determined exactly. [You may use standard results given in the List of Formulae (MF15).] [3] 2 A family of curves is given by the differential equation d d yx yx xyx  . By substituting yv x , show that the family of curves is given by  1e xyx A  . [6] Sketch, on a single diagram, three members of the family of curves corresponding to positive values of A. [2] Find a cartesian equation of the locus of the stationary points for the family of curves corresponding to positive values of A. [1] 3 The function f is defined by  2 5f : for , 4 4 xx x x     . (i) Find 1f( ) x and state the domain of 1f . [3] (ii) On a single clearly labelled diagram, sketch the graphs of f( )yx , 1f( )y x and 1ff ( )y x . [4] (iii) Show that the x-coordinates of the points of intersection of the curves f( )yx and 1f( )y x satisfy the equation 32 81 6 5 0xx x   . Hence find the values of these x-coordinates in exact form. [3]

3   IJC/2014/JC2 9740/02/S/2014 [Turn over   4 In the diagram, O is centre of the rectangular base ABCD of a right pyramid with vertex V. Perpendicular unit vectors i, j, k are parallel to AB, BC, OV respectively. The length of AB, BC and OV are 12 units, 6 units and 6 units respectively. The point M is the mid-point of CV and the point O is taken as the origin for position vectors. (i) Show that the equation of the line AM may be expressed as 66 33 , 02 t        r+ where t is a parameter. [3] (ii) Find the perpendicular distance from B to the line AM. [3] (iii) Find the acute angle between the line DV and the plane AMB. [4] The plane    has equation 1 44 a    r .      (iv) Given that the three planes AMB, AMD and  have no point in common, find the value of a. [2]          D    V C B M O            A

4   IJC/2014/JC2 9740/02/S/2014   Section B: Statistics [60 marks] 5 An automatic dispensing machine is set to dispense hot chocolate into cups. Based on observations over a long period of time, the cups were found to contain volumes of hot chocolate, in ml, with mean 55 and standard deviation 10. A large random sample of n cups of hot chocolate is taken. Gi ven that the probability that the sample mean exceeds 54 ml is more than 0.77, find the least value of n. [3] 6 The Ministry of Urban Development wish es to find out how the recent property cooling measures h