RVHS P1

Name ( ) Class RIVER VALLEY HIGH SCHOOL 2012 Year 6 Preliminary Examination Higher 2 MATHEMATICS Paper 1 Additional Materials: Answer Paper List of Formulae (MF15) Cover Page 9740/01 12 September 2012 3 hours READ THESE INSTRUCTIONS FIRST Do not open this booklet until you are told to do so. Write your name, class and index number in the space at the top of this page. 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 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. 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, place the cover page on top of your answer paper 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 5 printed pages.

2 1 One root of the equation 4 3 22 14 0z z z az b     , where a and b are real, is 1 2iz . Find the values of a and b and the other roots. [5] Deduce the roots of the equation 4 3 22 14 18 45 0z iz z iz     . [2] 2 Without the use of a calculator, solve the inequality 9x x . [3] Hence find 4 9 d n xx x in terms of n, where 03 n . [4] Describe the behaviour of the value of the integral as 0n . [1] 3 Relative to the origin O , the points A and B have position vectors a and b given by ( 1) 2pp   a i j k and 22  b i j k where 0p . The point Q is such that OAQB is a parallelogram. (i) Find the position vector of Q when 1p . [2] (ii) The vector c is a unit vector in the direction of OB   . Give the geometrical meaning of  a c c . [1] (iii) Find the range of values of p if |a| < |b|. [3] (iv) When 2p , determine whether ab and ab are perpendicular. Hence determine the geometrical meaning of | a b| . [3] 4 (For this question, leave all your answers in terms of  .) It is given that   21cos xy  . Show that yx yx 4d d)1( 2 2      . [2] By further differentiation of this result, find the Maclaurin’s series for y up to and including the term in 2x . [4] Deduce (i) the equat