JJC_H2_MATHS_P2

JURONG JUNIOR COLLEGE Preliminary Examinations MATHEMATICS 9740/02 Higher 2 17 September 2014 Paper 2 3 hours Additional materials: Answer Paper Cover Page List of Formulae (MF 15) READ THESE INSTRUCTIONS FIRST Write your name and civics 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 a graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic 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. The number of marks is given in brackets [ ] at the end of each question or part question. At the end of the examination, fasten all your work securely together, with the cover page in front. This document consists of 6 printed pages. [Turn over

2 Section A: Pure Mathematics [40 marks] 1 (a) Given that e cosyx x  , show that d tan 1 0d y xx  . (i) By further differentiation of this result obtain the Maclaurin’s series for y in terms of x, up to and including the term in x2. [3] (ii) Let the result in (i) be h( )x . Find the set of values of x for which h( )x is within 0.2 of the value of y. [2] (b) Expand, in ascending powers of x, 3 n xa  where a and n is a non- positive integer, up to and including the term in x2. [2] It is given that the coefficient of x is four times the coefficient of x2 and the constant in the expansion is 1 4 . Find a and n. [3] 2 (a) Find the general solution of the differential equation 2 2 2 d ed xy ax  , where a is a constant. [2] (b) Mr Tan borrowed $10 000 from a bank and at time t, the sum of money he owed the bank is denoted by x thousand dollars. The su m of money he owed increases, due to interest, at a rate proportional to the sum of money owed. Money is also repaid at a constant rate p. When x = 12, the interest and repayment balance. Taking both x and t as continuous variables, show that for x > 0, d 12d1 2 xp xt  . [2] (i) Find x in terms of t and p. [4] (ii) Find the time T that it will take Mr Tan to repay the loan, leaving your answer in terms of p. [1] (iii) Sketch the graph of x against t. [1]

3 3 The plane 1 has equation, 30 0      r. , where  and  are positive constants, and contains the point A with coordinates ( 10, 0, 5) . (i) Given that