HCI 2026 H3 Math Prelim

1 © HCI 2026 9820/01/JC2 Preliminary Examination 2026 [Turn Over HWA CHONG INSTITUTION 2026 JC2 Preliminary Examination MATHEMATICS Higher 3 Paper 1 Wednesday 21 September 2026 3 hours Additional materials: 12-page Answer Booklet List of Formula (MF27) 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 (MF27). 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 2026 9820/01/JC2 Preliminary Examination 2026 [Turn Over 1. (a) Given that nmmn and knnk , where ,,m n k +  , prove that kmmk . [3] (b) Given that e axyx= , prove by mathematical induction that ( ) 1d ed n n n axny a x nax −=+ for every positive integer n. [4] 2. (a) Explain why 1 1 1 1! 2! 3! ( 1)! !n nn + + + + −  is a positive integer for all positive integers n. [1] It is given that ( )( ) ( )( )( ) 1 1 1 1 1 2 1 2 3 na n n n n n n= + + ++ + + + + +  for n +  . (b) Show that 10 na n . [4] The notation x denotes the greatest integer less than or equal to the real number x. For example, 3 = , 4.2 5− =− . (c) Deduce that !e !ena n n=−  [4] (d) Hence, show that e is an irrational number. [3] 3. The integral ( , )I m n is defined by ( ) 1 2 0 ( , ) 1 d nmI m n x x x=− for , and , 0m n m n  . (a) By applying integration by parts, show that for 1,nn  , 2( , ) ( , 1) 21nI m n I m nmn=− ++ [5] (b) Hence, show that ( )( ) 1(3, ) for 0, .2 1 2I n n n nn=  ++  [5] (c) Show that  (0, )In is a strictly decreasing sequence. [2]

3 © HCI 2026 9820/01/JC2 Preliminary Examination 2026 [Turn Over 4. (a) A teacher collects 1 book each from n students. No two book are alike. She randomly redistributes them so that each student receives exactly one book. Let nD denote the number of ways the books are distributed such that no student receives the book they submitted. (i) Use the principle of inclusion and exclusion to show that ( ) 0 1! ! kn n k Dn k= −=  [4] (ii) Using a combinatorial argument, prove that for 3n , ( )( )121n n nD n D D −−= − + [3] (b) Let  1, 2,...,36S = (i) Let X be a subset of S of size 19. Prove that X must contain two distinct integers a and b , where 37.ab+= [2] (ii) Determine the maximum possible size of a subset of S that contains no two distinct numbers whose sum is a multiple of 7. [3] (iii) Let Y be a subset of S of size 10. Prove that Y must contain two distinct non-empty subsets whose elements have the same sum. [3] 5. (a) Given that ,,x y z +  and 1x y z+ + = , find the minimum value of 1 4 9 x y z++ . [4] (b) Given that 1xy+= and ,xy +  , find the exact minimum value of 11xy xy+ + + . [6] (c) The real numbers 01, , ... nx x x satisfy 01 ... nx x x a+ + + = and 2 2 2 01 ... nx x x b+ + + = , where a and b are fixed real numbers and n  , 3n . (i) Show that 2 1 ab n + . [2] (ii) Show that 0 11 a M a M xnn−+ ++ , where M is an expression in terms of a, b and n. [4]

4 © HCI 2026 9820/01/JC2 Preliminary Examination 2026 [Turn Over 6. Sundaram’s Sieve and the Generation of Prime Numbers Primes are those numbers that are divisible only by themselves and 1. They are the atoms amongst the integers because every whole number can be written as a product of a unique set of primes. We have known since the time of Euclid that there are infinitely many primes — there is no largest prime — but there is no general formula that generates all of them. Their distribution among the other numbers is still a mystery and motivates some of the biggest open questions in mathematics As the eighteenth century m athematical genius Leonhard Euler put it, "Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate." However, we do have algorithms that can find all the prime numbers up to a given number. For example, Sundaram’s Sieve is an algorithm, in which the prime numbers pop out of a very simple, and highly regular, structure. It is named after an obscure East In dian mathematician, S.P. Sundaram, who discovered the algorithm in the 1930s. Sundaram's algorithm is not a real sieve, but a clever way of spotting numbers that are not prime, thus "sieving out" those that are. The Sundaram’s Sieve is shown as the following infinite array of positive integers: 4 7 10 13 16 ... 7 12 17 22 27 ... 10 17 24 31 38 ... 13 22 31 40 49 ... 16 27 38 49 60 ... .. .. .. .. .. ... In the 1st row and 1st column, we have arithmetic sequences with common difference of 3. In the 2nd row and 2nd column, we have arithmetic sequences with common difference of 5. The pattern continues. Let mna denote the number in the thm row and thn column. For example, 11 4a = and 24 22a = . When we pick any positive integer, N, from the array, 21N+ is not a prime number. On the other hand, for any positive integer, N, not in the array, 21N+ is always a prime

5 © HCI 2026 9820/01/JC2 Preliminary Examination 2026 [Turn Over number. Hence, making use of the positive integers not in the array, we can generate all the odd prime numbers up to 20. 1 2 3 4 5 6 7 8 9 10 3 5 7 11 13 17 19 In this way, Sundaram’s Sieve can generate all the odd prime numbers up to any given number. (a) Find 61a and 66a . [3] (b) Find mna in terms of m and n, giving your answer in the simplified form. [3] (c) It is observed that certain numbers appear in the a