NJC H3 Math 2024 Prelim Questions

[Turn_over NATIONAL JUNIOR COLLEGE SENIOR HIGH 2 PRELIMINARY EXAMINATION Higher 3 MATHEMATICS 9820/01 Paper 1 20 September 2024 3 hours Additional Materials: Answer Booklet List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name and registration number on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for 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 the brackets [ ] at the end of each question or part question. This document consists of 8 printed pages.

2 1 Let a, b, c, d, e and f be positive integers such that the sum S a b c d e f= + + + + + divides both abc def+ and ab bc ca de ef fd+ + − − − . By considering the polynomial ( ) ( )( )( ) ( )( )( )p, x x a x b x c x d x e x f= + + + − − − − or otherwise, prove that S is composite. [6] 2 (a) Determine the number of solutions to the equation 1 2 3 28x x x+ + = where 1x , 2x and 3x are non-negative integers. [1] (b) Determine the number of solutions to the equation 1 2 3 28x x x+ + = where 1x , 2x and 3x are non-negative integers and 1 12.x  [2] (c) Determine the number of solutions to the equation 1 2 3 28x x x+ + = where 1x , 2x and 3x are non-negative integers less than 12. [2] (d) Let n, k and r be positive integers such that ( )1.k r n− By considering the number of solutions to the equation 12 kx x x n+ + + = where 12, , , kx x x are non -negative integers less than r, and using the Principle of Inclusion and Exclusion, evaluate ( ) 0 11 1 n r m m k n mr k mk   = − + −  −    −   . [6]

3 [Turn over 3 (i) Express 4 1 1u + in the form 22 11 Au B Au B u Cu u Cu +− −+ + − + , where A, B and C are positive constants to be determined. [5] (ii) By substituting 2 tanux= , or otherwise, evaluate 1 d. tan x x  [6] 4 (a) A sequence of terms is defined by ( ) 2 .2 1 ! n nx n= − (i) Prove that for all positive integers n, 1 2 .3 n n x x +  [2] (ii) Show that 1 n n x  =  converges. [