IJC_H2_MATH_P2_QP

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2    IJC/2015/JC2 9740/02/Sept/2015 Section A: Pure Mathematics [40 marks] 1 The parametric equations of a curve are 21e , ettxy  . (i) Find the equation of the normal to the curve at the point   21e, ep pP  . [3] (ii) This normal meets the x-axis at the point Q. Find the cartesian equation of the locus of the mid-point of PQ as p varies. [4] 2 (i) Find the general solution of the differential equation 2 3 2 d 2 d yx x x   . [3] (ii) It is given that 1y when 1x . On a single diagram, sketch three members of the family of solution curves for 0x  . [5] 3 (i) Given that sin 2f( ) cos 2 3 xx x   , where x is sufficiently small, find the series expansion of f( )x in ascending powers of x, up to and including the term in 2x . [4] (ii) Use your answer to part (i) to give an approximation for 0 f( ) d n x x in terms of n. Evaluate this approximation in the case where 0.5n , leaving your answer in 6 decimal places. [3] (iii) Use your calculator to fi nd an accurate value for 0.5 0 f( )