NJC_H2_MATHS_P1_Question

[Turn_over NATIONAL JUNIOR COLLEGE SENIOR HIGH 2 PRELIMINARY EXAMINATION Higher 2 MATHEMATICS 9740/01 Paper 1 27 August 2015 3 hours Additional Materials: Answer Paper List of Formulae (MF15) Cover Sheet READ THESE INSTRUCTIONS FIRST Write your name, registration number, subject tutorial 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 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 an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing 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, fasten all your work securely together. The number of marks is given in the brackets [ ] at the end of each question or part question. This document consists of 5 printed pages. National Junior College

[Turn_over 1 Prove by the method of mathematical induction that 11 0 3 2 1 1( 1)!3 ( 1)!3 N nN n n nN     for N ≥ 0. [5] 2 On separate diagrams, sketch the following curves, labelling clearly any asymptotes, axial intercepts and turning points. (i) 1 f ( )y x [3] (ii) f ( )yx  [3] 3 (i) Use the Maclaurin series for cos 2x to find the Maclaurin series for 2 cos 2f ( ) 1 xx x  , up to and including the term in 4x . [4] (ii) By substituting 1 3x in your answer in part (i), find an approximation for 2cos , 3  giving your answer as an exact fraction in its simplest form. [2] 4 It is given that two vectors a and b satisfy the equation 0.ab (i) What can be said about the vectors a and b? [2] Assume that a is a non-zero vector. (ii) Find, in terms of the vectors a and/or b, the projection vector of (a – b) onto a, simplifying your answer as far as possible. Show your working clearly. [3] (iii) Show that .a b a b [2]  f4yx y

[Turn_over 5 (i) Prove by the method of differences that      1 1 1 1 2 1 2 3 3 2 1 2 3 n r n r r n n     . [3] (ii) Hence, find (a) the exact value of   5 1 2 1 2 3r rr    , [3] (b)   0 1 2 1 2 5 n r rr  in terms of n . [2] 6 (a) On the same diagram, sketch the graphs of 11 1y x  and 2xy , labelling the equations of the asymptote(s), if any. [2] Hence, solve the inequality 112 1 x x  . [2] (b) K