ASRJC 2023 JC1 Promos 9649

ASRJC 2023 [Turn over FURTHER MATHEMATICS 9649/01 Paper 1 28 September 2023 (Thursday) 8 – 11 a.m. 3 hours Additional Materials: Li st of Formulae (MF26) READ THESE INSTRUCTIONS FIRST An answer booklet will be provided with this quest ion paper. You should follow the instructions on the front cover of the booklet. If you need additional answer paper ask the invigi lator for a continuation booklet. Write your name, class, syllabus and paper number on the answer booklet you hand in. Write in dark blue or black pen on both sides of the answer booklet. 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. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 5 printed pages and 3 blank pages. Anderson Serangoon Junior College ANDERSON SERANGOON JUNIOR COLLEGE JC1 Promotional Examination 2023 Higher 2

2 ASRJC 2023 1 A conic has cartesian equation 22 22 ( 12) 113 5 xy  . (a) Show that one of its foci is at the origin. [2] (b) Obtain a polar equation of the conic. [3] 2 A dog bowl is produced by rotating the region bounded by the curve 4ln( 1)yx and the lines 0x , 1x and 1 5y fully about the y-axis. By using a substitution e1yu or otherwise, find the exact volume of the material required to produce the dog bowl. [7] 3 A polar curve has equation 1 cos , where 0r     . ๠e curve has two points where the gradient of the curve is 1 . (a) Show that cos cos 2 sin sin 2    at these two points. [3] (b) Without using a calculator, determine the value of  for each of these two points. [3] 4 (a) It is given that f is a function where f( ) 0x  for 0x . If 1 22f n n r rU nn   , where n is a positive integer and 2 0 f( ) dI xx , explain with the aid of a sketch, why  nUI  for all positive integers n,  UI  . [3] (b) If 1 0 22f n n r rV nn     , state an inequality relating nV and I for 1n . [1] (c) By choosing a suitable function f, determine the exact value of 111 1lim 246 3n nnn n      . [3]

3 ASRJC 2023 [Turn over 5 ๠e complex number z satisfies 44 izz and 1 1i