PJC_H2_MATHS_P1

Candidate Name: ________________________ Class: _____________ JC2 PRELIMINARY EXAM Higher 2 MATHEMATICS 9740/01 Paper 1 14 Sept 2015 3 hours Additional Materials: Cover page Answer papers List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your full name and class on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 6 printed pages.

2 @PJC 2015 [Turn Over] 1 The volume of water V in a filtration tank at time t satisfies the differential equation d 5d V kVt  , where k is a positive constant. Find V in terms of k and t, given that the tank is initially empty. [5] State what happens to V for large values of t. [1] 2 (i) Given that 1sin (2 )e xy   , show that   2 22 d1 4 4 d yxy x   . [2] (ii) By further differentiation of this result, find the first three te rms of the Maclaurin series for y in ascending powers of x. [3] (iii) Deduce the first three terms of the Maclaurin series for 1sin (2 )e cos x x  in ascending powers of x. [3] 3 The curve C has equation 24xy xq  , where q is a non-zero constant. It is given that C has a stationary point at 4x and an asymptote 4y x r , where r is a non-zero constant. (i) Find the values of q and r. [3] (ii) Sketch C, stating clearly the equations of its asymptotes, stationary points and the coordinates of any point(s) of intersection with the axes. [3] (iii) State the set of values that y can take. [1] (iv) Using the graph in part (ii), find the range of values of a such that the equation   22 2 42 16 xxa xq      has a negative real root. [2]

3 @PJC 2015 [Turn Over] 4 (a) State a sequence of transformatio ns which transform the graph of lnyx to the graph of  ln 1 2yx . [3] (b) It is given that