2021 NJC H3 Math Questions

NATIONAL JUNIOR COLLEGE SENIOR HIGH 2 PRELIMINARY EXAMINATION Higher 3 MATHEMATICS 9820/01 Paper 1 17 September 2021 3 hours Additional Materials: Answer Booklet List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name, registration number, subject tutorial group, on all the work you hand in. Write in dark blue or black pen on both sides of the paper. 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 6 printed pages and 0 blank pages.

2 [Turn Over 1 Let S be a set containing 2n integers. By considering the pigeonhole principle, show that there are two elements in the set S whose difference is a multiple of 2n or there are two elements in the set S whose sum is divisible by 2.n [5] 2 (i) Given integers a, b and m such that  gcd , 1bm  , show that the m numbers  , , 2 , ..., 1a a b a b a m b    are all incongruent modulo m. [2] (ii) Let 2n . Prove that if all the n terms of the arithmetic progression  , , 2 , ..., 1p p d p d p n d    are primes, then the common difference d is divisible by every prime smaller than n. [4] 3 Prove that 2 2 2a b c ab bc ca     for any real numbers a, b and c. [3] Hence prove that for any positive real numbers x, y and z satisfying xy yz zx x y z S      , (i) 3S  , and [2] (ii) 2 2 2 1 1 1 11 1 1x y y z z x        . [4] 4 (a) Show that the sequence 99, 999, 9999, 99999, … does not contain a number which is a sum of two squares. [2] (b) Dirichlet’s Theorem states the following: If a and b are coprime positive integers, then the arithmetic progression , , 2 , 3 , ...a a b a b a b   contains infinitely many primes. (i) Show that for any positive integer n, there exists a prime ending with n consecutive 9’s. [3] (ii) Prove that the arithmetic prog ression above contains infinitely many composite numbers for any pair of coprime positive integers a and b. [4]

3 [Turn Over 5 Use the substitution 2 tanux to find tan d ,xx showing all your working clea