NYJC 2025 Prelim

This document consists of 6 printed pages and 2 blank pages. NANYANG JUNIOR COLLEGE Internal Examinations © NYJC 2025 [Turn Over NANYANG JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 3 __________________________________________________________________________ ____________ MATHEMATICS 9820/01 Paper 1 19 September 2025 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) __________________________________________________________________________ ____________ READ THESE INSTRUCTIONS FIRST Answer all the questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. 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 must show all necessary working clearly. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 80.

2 NYJC 2025 JC2 Preliminary Examination 9820/01 1 The variables x and y are related by the differential equation 22d 2 4 4 3d y x y xy x yx = + − − + + . (a) Find the two solutions of the differential equation that have the form y mx c=+ . [3] Let f ( )yx= be another solution of the differential equation. (b) Show that any stationary point on the curve f ( )yx= must lie on one of two lines whose equations are 3yx=− and 1yx=− . [2] (c) Show that any stationary point on the curve f ( )yx= that lies on the line 3yx=− is a minimum point and that on the line 1yx=− is a maximum point. [3] The curve f ( )yx= has two stationary points and f (0) 2=− . It is given that the solution curves of this differential equation do not intersect each other. (d) On the same diagram, sketch the curve f ( )yx= , the two straight line solutions and the two lines which the stationary points lie on. [3] 2 In this question, the domains of the functions are the set of integers. (a) The functions f, g and h are defined by 2f ( ) 6 11n n n= + + , 2g( ) 4 6n n n= − + and 2h( ) 2 5n n n= − + . By completing the square, show that f and g have the same range. Show also that there are no common integers in the ranges of f and h. [5] (b) For any integers a and b, (i) show that 22 0a ab b+ +  . [1] (ii) show that 3 3 2 2( 1) ( 1 ) ( 1) ( 1)a b a b a a b b − − = − − − + − + . [1] (c) The functions p and q are defined by 32p( ) 3 7n n n n= − + and 3q( ) 4 6n n n= + − . By writing p( )n in the form 3()n a bn c+ + + , where a, b and c are real constants, find the integers that are common in the ranges of p and q. (Note that the integers common in the ranges may or may not correspond to the same n value.) [4]

3 NYJC 2025 JC2 Preliminary Examination 9820/01 [Turn Over 3 Let f be a continuous function on the interval 0 xa , where a is a constant. (a) (i) Use a substitution to prove that 00 f ( ) d f ( ) d aa x x a x x −= . [2] (ii) Hence find the exact value of π 2 0 cos d sin cos n nn x x xx+ , where n is a fixed positive integer. [3] It is further given that f ( ) f ( )x a x=− . (b) Show that 00 2 f ( ) d f ( ) d aa ax x x x x = . [2] (c) For non-negative integers n, let π 2 0 cos (2 ) dn n I xx= . (i) For each 2n , show that 2( 1)nnnI n I −=− . [3] (ii) Hence find the exact value of 8 π 0 cos (2 ) dx x x . [4] 4 (a) Show that for any complex numbers 1z and 2z , 1 2 1 2z z z z+  + . [2] Use mathematical induction to prove that for any complex numbers 12, , , nz z z  , 1 2 1 2 nnz z z z z z+ + +  + + +  . [3] (b) The real numbers 1u , 2u , …, 2023u satisfy 2 1 0rruu +− for 1, 2, 3, ..., 2022r = . If 2 2 2 2 1 2 2 3 2022 2023 2023 1( )( )...( )( )W u u u u u u u u= − − − − and 2 2023 1 0uu − , find the maximum value of 20254 W . [6]

4 NYJC 2025 JC2 Preliminary Examination 9820/01 5 A team is made up of 18 boys. (a) Each boy writes down two numbers on a piece of paper, which states the number of other boys having the same surname as him, and the number of other boys being born in the same month as he is. When all the pieces of paper were collected, all the integers from 0 to 7 inclusive were seen. Show that there must be at least two boys who share the same surname and the same birth month. [3] The team of 18 boys takes part in a competition where in the first stage, there are 4 different challenges. (b) (i) Find the number of ways in which the team can split themselves to compete in the challenges, if all students take part in exactly one challenge and each challenge must have at least 4 students. [3] (ii) Six boys take part in one of the challenges where they are tasked to distribute 80 identical dice amongst themselves where they had to then stack the allocated dice in the shortest time possible. Find the number of ways they can distribute 80 dice amongst themselves if each of them must receive at least 5 dice. [2] The team progresses to the second stage where three of them, Alex, Ben and Charlie, each completes a test of 50 multiple-choice questions. Each of the 50 questions was answered correctly by at least one of the three boys. Alex, Ben and Charlie answered 40, 30 and 20 questions correctly. Questions answered correctly by all three of them were considered easy, while questions answered correctly by only one of the three of them were considered difficult. (c) Find the difference in the number of easy questions and difficult questions, stating which type has more in number. [3] This competition is held monthly. Alex has a group of five friends from other teams. At the competition, he met each of them 1 1 times, every two of them 5 times, every three of them 3 times, every four of them 2 times, and all five of them only once. Alex has participated in the competition 3 times where he did not meet any of these friends. (d) Find the number of times Alex has taken part in this competition. [2]

5 NYJC 2025 JC2 Preliminary Examination 9820/01 [Turn Over Please turn over.

6 NYJC 2025 JC2 Preliminary Examination 9820/01 Use the information in the mathematical text to answer Question 6. You should read the whole mathematical text before you start answering the questions. Continued Fractions A continued fraction for a real number x is an expression of the form 0 1 2 1 1 1 xa a a =+ + +  , or simply expressed as  0 1 2; , ,x a a a=  , where 0a   and ia   for each 1i . A continued fraction with ia +  for each 1i is known as a simple continued fraction. It is known that the simple continued fraction for x is finite if and only if x is rational. For example, the number 91 1 [1;3,2]17 3 2 = + = + is rational while the number 1ln 3 1 [1;10,7,9, ] 110 17 9 = + = + + +   is irrational. Let 0 1 2, , , ...a a a be a sequence of integers, with all the terms positive except perhaps 0a . We define two sequences of integers as follows: 1 1p− = , , 12n n n np a p p −−=+ , 1n , 1 0q− = , 0 1q = , 12n n n nq a q q −−=+ , 1n . (*) For  0 1 2; , ,x a a a=  ,