HCI_JC_2_H2_Maths_2012_Paper 1 Questions

HWA CHONG INSTITUTION 2012 PRELIMINARY EXAMINATION Higher 2 MATHEMATICS 9740/01 Paper 1 12 September 2012 3 hours Additional Materials: Answer Paper List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your Centre number, index number, 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 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 notatio ns and not calculator commands. You are reminded of the need for clear presentation in your answers. At the end of the examination, place the completed cover page on top of your answer scripts and 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.

[Turn over 2 1. The function f is defined for 2x by 2 53f: 2 kx xx x   , where k is a real constant. Find the range of values of k such that the function f is an increasing function for all real values of x, 2x . [5] 2. The point A represents a fixed complex number a such that arg 02 a   . The complex numbers i a and iaa are represented by the points B and C respectively. On a clearly labelled Argand diagram, show the points A, B, C and the set of points representing the complex numbers z satisfying the following: (i) iz a a , (ii) ( i )z z a a   . Write down the complex number z that gives the greatest value of ,z giving your answer in terms of a. [5] 3. It is given that xyyx , where 0x , 0y . Find d d y x in terms of x and y. Hence find the equation of the tangent to the curve xyyx which is parallel to the y-axis. [6] 4. (a) Find the comp lex number z in the form ixy , where x, y  ℝ such that 3i i1 2i z z  . [ Note: z is the conjugate of z .] [3] (b) The two roots 12, zz of the equation 2 0,z z p   where ,p are such that 1 iz a b where a, b  ℝ, 0b and 12 3.zz Find the value of .p [3]

[Turn over 3 5. (i) Use the stand ard series for ln(1 )x to find the first three ter ms of the Maclaurin's series for   1 1 ln 1 2 y x   . [3] (ii) Find the range of values of x for the above expansion to be valid. [3] (iii) Use your answer in part (i) to calculate an approx