ACJC 9649 2023 Prelim P1

ANGLO-CHINESE JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 FURTHER MATHEMATICS 9649/01 Paper 1 12 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 9649/01 1 A curve has parametric equations 4 2 4ln , 4 x t t yt   where 0t  . The surface area generated when the arc of the curve from 1t  to tk , where 1k  , is rotated through one complete revolution about the x-axis is 384π square units. Find the value of k. [4] 2 The sequence  nx is given by 1 1x  and 1 1 1 14 1 2 n n nnnx x n n       for 2n . By multiplying the recurrence relation throughout by 1nn  , use a suitable substitution to determine nx as a function of n, simplifying your answer. [7] 3 The variables x and y are related by the differential equation 2 3 2 3d d yx y x x y yx    , (*) and it is given that 0xy . (a) Use the substitution y ux to find the general solution of (*). [5] A particular solution of (*) has 1y when 2x . (b) Use one step of the Euler method to calculate an approximate value of y when 2.5x . [2] (c) Given that the solution curve is concave downwards near the point  2,1 , determine with a sketch whether the approximate value of y found in (b) is an overestimate or underestimate. [1]

3 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 FURTHER MATHEMATICS 9649/01 [Turn Over 4 (i) A 33 square matrix A is said to be skew symmetric if T AA . Prove that for a given skew symmetric A,  det A is equal to zero. [3] (ii) Let T be the linear transformation such that 33T:  and T( ) x a x