EJC_9649_2023_Prelim_P2

2023 JC2 H2 Further Mathematics Preliminary Examination Paper 2 [Turn over EUNOIA JUNIOR COLLEGE JC2 Preliminary Examination 2023 General Certificate of Education Advanced Level Higher 2 CANDIDATE NAME CIVICS GROUP INDEX NO. FURTHER MATHEMATICS Paper 2 [100 marks] 9649/02 14 September 2023 3 hours Additional Materials: Answer Booklet List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name, civics group and index number on the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all 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 brackets [ ] at the end of each question or part question. This document consists of 6 printed pages.

2 2023 JC2 H2 Further Mathematics Preliminary Examination Paper 2 Section A: Pure Mathematics [50 Marks] 1 Using mathematical induction, prove that 1 3 1 1 2( 1)( 2)2 ( 2)2 N nN n n n n N     for all positive integers N. [5] Deduce the value of 1 3 ( 1)( 2)2 n n n nn     . [1] 2 (i) Let 2V  and , Vuv such that 1 2 u u  u and 1 2 v v  v . Define addition and scalar multiplication as follow: 11 22 uv uv   uv and 1 2 kuk ku  u where k is a real number. Explain why V is not a vector space with the stated operations. [2] (ii) Let M be the vector space of all real nn matrices with the standard addition and scalar multiplication for matrices. Determine if the following are subspaces of M. (a) All nn matrices A such that AB BA for a fixed nn matrix B. (b) All nn matrices A such that T A A I . [5] 3 It is given that 2 2 3dxIx   . (i) Use the trapezium rule with five ordinates to approximate I. [2] (ii) Explain whether the approximation to I found in part (i) is an under-estimate or over-estimate to I. [2] (iii) Use Simpson’s rule with five ordinates to approximate I. [2] (iv) Find the exact value of I. [2] (v) Explain clearly which of the two rules would give a more accurate approximation to I when used with the same number of ordinates. [2] (vi) Calculate the absolute percentage error in the approximation obtained in part (iii). [1]

3 2023 JC2