2012_ VJC_ H2_MA_Prelim P1_ Questions

1 VICTORIA JUNIOR COLLEGE Preliminary Examination Higher 2 MATHEMATICS 9740/01 PAPER 1 September 2012 3 hours Additional materials: Answer paper Graph paper List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your name and CT group 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 calcula tor 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 5 printed pages © VJC 2012 VICTORIA JUNIOR COLLEGE [Turn over

2 1 A curve C is defined by the parametric equations 2 1, tan , 0 2xy        . Find the area of the region which is bounded by C, the y-axis and the x-axis for which 0x . Give your answer to 4 decimal places. [4] 2 The position vectors of A and B with respect to the origin O are a and b respectively. The point C is such that OACB is a parallelogram. The point P on AC is such that AP : PC = 1: 2 and the point Q on BC is such that BQ : QC = 1: 2. (i) Find QP in terms of a and b. [2] (ii) Show that OC QP can be written as 22  ab , where and  are constants to be found. [2] (iii) Given that 0OC QP , identify the shape of the parallelogram OACB, justifying your answer. [2] 3 (i) Given that the first three terms in the expansion of 1 a bx are 211 ,24 x cx find the values of a, b and c. [3] (ii) Use your answer in part (i) to find an approximation to the area of the region bounded by the curve 1y a bx  , the two axes and the line 1 2x . Give your answer in the form p q where p and q are integers. Hence find an estimate for ln 4 3 . [3] 4 (i) Prove by induction that, for all positive integers n and for , 0, 1x x x  R ,   2 1 1 1 1 1 ...11 nnx x x x x x     . [4] (ii) Find the range of values of x for which 1 n n u    exists given that 2 1 1 1 1 1 n nu x x x x     . [3]

3 [Turn over 5 (i) A hospital patient is given a pill containing 50 uni ts of antibiotic at 12pm.