HCI_9820_2024_H3_Prelim_Qn

1 © HCI 2024 9820/01/JC2 Preliminary Examination 2024 [Turn Over HWA CHONG INSTITUTION 2024 JC2 Preliminary Examination MATHEMATICS Higher 3 Paper 1 Wednesday 18 September 2024 3 hours Additional materials: 12-page Answer Booklet List of Formula (MF26) 4-page Additional Answer Booklet (upon request) READ THESE INSTRUCTIONS FIRST Write your name and class on the 12-page Answer Booklet and any other additional 4 -page Answer Booklets you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use paper clips, highlighters, glue or correction fluid. Do not write anything on the List of Formula (MF26). 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 a 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. At the end of the examination, slot any additional 4-page Answer Booklets used in your 12-page Answer Booklet and indicate on the 12 -page Answer Booklet the number of additional 4 -page Answer Booklets used (if any). This question paper consists of 5 printed pages and 1 blank page. 9820/01

2 © HCI 2024 9820/01/JC2 Preliminary Examination 2024 [Turn Over 1. A sequence generated from the integer set  1, 2,3,...,n , is denoted as a special sequence if the sequence is strictly increasing and the first term is odd, the second term is even, the third term is odd, the fourth term is even, etc. For example, from the set  1, 2,3, 4,5,6 , some special sequences generated from the set are  1, 2,3,6 ,  1, 4,5 ,  3, 4 ,  5 . Let ()An be the total number of distinct special sequences obtained from  1, 2,3,..., .n (a) Form a recurrence relation involving ( ), ( 1), ( 2).A n A n A n−− [4] (b) Find the number of special sequences that can be formed from  1, 2,...,15 . [2] 2. Let f( )x be a differentiable function such that f( ) f '( ) 1xx+ for all x , and f(0) a= , where a is a constant with 1a . It is given that g( ) e f( )xxx= . (a) Show that g'( ) e xx  , for all x . [2] (b) Hence, or otherwise, find the largest possible va