MI_H2_MATH_P1

Class Adm No Candidate Name: This question paper consists of 5 printed pages. [Turn over 2015 Preliminary Examination II Pre-University 3 MATHEMATICS 9740/01 Paper 1 16 September 2015 3 hours Additional Materials: Answer Paper List of Formulae (MF 15) READ THESE INSTRUCTIONS FIRST Write your 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 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. At the end of the examination, arrange your answers in NUMERICAL ORDER and fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question.

2 Answer all the questions [100 marks] 1. It is given that   32f, x ax bx cx d  where , , abc and d are constants. The curve fyx passes through the points    1, 3 , 3, 13 and has a maximum point at 13 1, 32 7   . Find f. x [4] 2. Find (i) sin d,12 c o s x xx [2] (ii) 2 0 cos 2 d .xex x   [4] 3. (i) Expand 24 13 2 x x   in ascending powers of x, up to and including the term in 2x . [3] (ii) Find the set of values of x for which the expansion in part (i) is valid. [2] (iii) By substituting 1 4x and using your result in part (i), show that 4 19 p q , where p and q are integers to be determined. [2] 4. (i) The nth term of a sequence is 1ln 3Tn nx  where x is a constant. Show that the sequence is an arithmetic progression for all positive integers n. [2] (ii) When ln 3 is subtracted from the 19th, 7th and 3rd terms of the arithmetic progression in part (i), these terms become the first three terms of a geometric progression. (a) Find the common ratio of the geometric progression. [2] (b) Find the range of values of x for which the sum of the first 20 terms of the arithmetic progression exceeds the sum to infinity of the geometric progression. [3]

3 5. (i) Find the general solution of the differential equation 2d 1d y yx   . [3] (ii) Find the particular solution of the differential equation for which 1 3y  when 0.x [1] (iii) What can you