2021 HCI H3 Math Prelim P1 Questions

HWA CHONG INSTITUTION 2021 JC2 Preliminary Examination MATHEMATICS Higher 3 Paper 1 Monday 20 September 2021 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 cor rect 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 7 printed pages and 1 blank page. 9820/01

2 HCI 2021 9820/01/JC2 Preliminary Examination 2021 [Turn Over 1. Let N be an integer greater than 1. (i) Show that for any positive integer k, ( ) ( ) 2 2 11 1 k kNN NN + − + +− is an integer. [4] (ii) Hence or otherwise, show that for any positive integer k, ( ) 2 1 k NN+− differs from the integer closest to it by less than 12 2 k N −  − . [4] 2. Let a, b and c be real numbers for all real x, ( )( )( ) 231 1 1 1 .ax bx cx qx rx+ + + = + + (i) Express q and r in terms of a, b and c, and show that 0.abc+ + = [3] (ii) Find, in terms of q and r, the first four nonzero terms in the series expansion of ( ) 23ln 1 .y qx rx= + + [2] (iii) Find the coefficient of nx , 1,n in the series expansion of ( ) 23ln 1 ,y qx rx= + + leaving your answer in the form ( ) 1 1 n nT + − , where nT is in terms of ,,abc and n. [2] (iv) Using the results in parts (ii) and (iii), show that ( )( ) 2 2 2 3 3 3 5 5 5 65 a b c a b c abc+ + + + ++= [2]

3 HCI 2021 9820/01/JC2 Preliminary Examination 2021 [Turn Over 3. Find functions ( )a x and ( )b x such that ux= and exu= both satisfy the differential equation ( ) ( ) 2 2 dd a b 0.dd uu x x uxx+ + = (1) [3] The general sol