ACJC 9649 2023 Prelim P2

ANGLO-CHINESE JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 FURTHER MATHEMATICS 9649/02 Paper 2 14 September 2023 3 hours Additional Materials: Cover Sheet Answer Paper List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your index number, class and name on the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, 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 brackets [ ] at the end of each question or part question. __________________________________________________________________________________ This document consists of 7 printed pages and 1 blank page. [Turn Over

2 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 FURTHER MATHEMATICS 9649/02 Section A: Pure Mathematics [50 marks] 1 Using mathematical induction, prove that     1 3 1 1 22 1 2 2 2 N nN n n n n N     for all positive integers N. [5] Deduce the value of   1 3 2 1 2n n n nn     . [1] 2 (i) Let 2V  and 2, uv such that 1 2 u u  u and 1 2 v v  v . Define addition and scalar multiplication as: 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 when addition and scalar multiplication are defined in this way. [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 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 FURTHER MATHEMATICS 9649/02 [Turn Over 3 It is given that 2 2 3dxIx   . (i) Use the trapezium rule with five ordinates to find an approximation to I. [2] (ii) Explain whether the approximation to I found in (i) is an under -estimate or over - estimate of I. [2] (iii) Use Simpson’s rule with five ordinates to find an approximation to I. [2] (iv) Find the exact value of I. [2] (v) Explain whether the trapezium rule or Simpson’s rule , used with the same number of ordinates, wo