ASRJC H3 Math 2024 Prelim Questions

1 MATHEMATICS 9820/01 Paper 1 17 September 2024 3 hours Additional Materials: Answer Booklets List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST An answer booklet will be provided with this question paper. You should follow the instructions on the front cover of the booklet. If you need additional answer paper ask the invigilator for a continuation booklet. 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. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 6 printed pages and 0 blank page. ANDERSON SERANGOON JUNIOR COLLEGE JC2 Preliminary Examinations 2024 Higher 3

2 Answer all the questions. 1 (a) A function is concave on an interval if its second derivative is negative within the interval. Show that sin x is concave for 0 x π and ln x is concave for 0x . [1] If f is concave in an interval and 12, , , nx x x are elements in the interval, Jensen’s inequality states that 11 11 f ( ) f , nn ii ii xxnn and equality holds if and only if 12 .nx x x You may use this result without proof. (i) Given that A , B and C are angles of a triangle, prove that 33sin sin sin 2A B C+ +  . [2] (ii) By selecting a suitable function f, use Jensen’s inequality to derive the AM- GM inequality 12 12 n n n a a a a a an for any positive real numbers 12, , , na a a and any positive integer n. [2] (b) The product P( )n is defined, for any positive integer n, by 1 1 2 3 5 9 2 1( ) ...... .1 2 4 8 2 n nPn − − +=      (i) Use the AM-GM inequality to show that 2P( ) 1 n n n . (ii) Use binomial theorem to show that 2P( ) en for all positive integers n. [4] [3]

3 2 Given a sequence 0 1 2, , ,.....w w w , the sequence 1 2 3, , ,.....F F F is defined by 22 11 4.n n n n nF w w w w −−= + − Show that ( )( )1 2 2 1 4n n n n n n nF F w w w w w− − − −− = − + − for 2.n [2] (a) The sequence 0 1 2, , ,.....u u u has 0 1u = and 1 2u = and satisfies ( )124 2 .