2021 ASRJC H3 Math Questions

1 ASRJC 2021 MATHEMATICS 9820/01 Paper 1 22 September 2021 3 hours Additional Materials: Answer Booklet List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST An answer booklet will be provided with this question paper. You should follow the instructions on the front cover of the booklet. If you need additional answer paper ask the invigilator for a continuation booklet. 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 brackets [ ] at the end of each question or part question. This document consists of 6 printed pages ANDERSON SERANGOON JUNIOR COLLEGE JC2 Preliminary Examinations 2021 Higher 3

2 ASRJC 2021 Answer all the questions. 1 Find the set of integer values of x that satisfy   43 1 0 mod5x x x    Express 43 1x x x   as a product (modulo 5) of four linear factors with integer coefficients. [6] 2 It is given that g is a function with domain  1, . Using the substitution tx  or otherwise , show that 00 g(sin )d g(sin )d2x x x x x   . Hence find the exact values of the following definite integrals, simplifying your answers. (a) › 2 0 sin d 8 sin xx x x     (b) › 0 1 sin dx x x   [9] 3 There are n identical balls to be distributed into k distinct boxes such that every box must be filled. (i) Find the number of ways if there are no restrictions. [1] (ii) It is given that n and k are both even and 2nk . (a) Find the number of ways if every box contains an even number of balls. [2] (b) Find the number of ways if every box contains an odd number of balls. [2] (iii) Use principle of inclusion and exclusion to find the number of ways to distribute the n objects if no box can have more than m objects. [4]

3 ASRJC 2021 4 It is given that a, b and c are the lengths of the sides of a triangle. (i)› Prove that it is always possible to form a triangle with sides of lengths 3 a , 3 b and 3 c . [2] (ii)› Prove that it is never possible to form a triangle with sides of lengths 3ab , 3 4bc and 4 ca . [2] (iii)› Determine with a proof whether it is always possible to form a triangle with sides of lengths 2 2a bc , 2 2b ca and 2 2c ab . [3] (iv)› Prove that if the triangle with sides of lengths a, b and c is a right -angled triangle, the triangle in part (i)›cannot be a right-angled triangle. [3] 5 It is given that 0 f ( ) n r r r x c x   , where 0 1ncc , rck  for all 1, 2, , 1rn , and k is a positive constant greater than or equal to 1. (i) By using triangle inequality, show that f ( ) 1 1 kxx x  if 11 x   . [3] (ii) Prove that if r is a root of f and 11 r   , then r must satisfy the inequality 1 11 rkk    . [2] (iii) Show that the inequality in (ii) also holds if r is a root of f and 1r  . [3] (iv) Consider the polynomial equation 5 4 3 2(2021 2 ) 2020 2019 2018 2017 2021 2 0n x x x x x n        . where n is a non-negative integer. Given that the above equation has integer solution(s), determine all possible values of n, justifying your answer clearly using the result in (iii). [3]

4 ASRJC 2021 6 (a) Show, by induction or otherwise, that if n is a positive integer , then any odd number a satisfies   22 1 mod 2 nn a  . [7] (b) Show that if a and k are positive integers then 1a divides 21 1ka   . [1] By considering the case 22 r a , or otherwise, show that if 21n is a prime then n is a power of 2. [4] 7 Show that, if n and r are positive integers, then    ,, n n r n r , where  ,nr denotes the greatest common divisor of n and r. [2] Show further that , if n is odd,       2 , for odd, 2, , for even. n r rn n r n r r    [5] For any positive integer n,  n is defined to be the number of positive integers not exceeding n which are coprime to n. For example,  42  since positive integers not exceeding 4 and coprime to 4 are 1 and 3. Using the above results or otherwise, show that (i) if n is odd,    2,nn  (ii) if n is even,    2 2 ,nn  (iii)   12 2 ,nn  where n is a positive integer. [7] [Note that any result involving  n should be proven before it can be used.]

5 ASRJC 2021 8 (i)› Verify that 42 2 4 6 4 22 1 111 n n tt t t ttt         . [2] Let 42 20 ( ) d 1 nx n tP x t t    for n  , 01 x . (ii)› Prove that 10 ( ) 43 nPx n  . Hence or otherwise, show that 1 2 1 0 ( 1)tan 21 n n n xx n      . [4] (iii)› D’Alembert’s ratio test states that a series of the form 0 r r a    converges if 1lim 1 n n n a a    , diverges if 1lim 1 n n n a a    , and is inconclusive if 1lim 1 n n n a a    . Given that the series expansion for 1tan x is convergent for 1x , prove that the series expansion in (ii)›is valid for 11 x   . [3] (iv)› Justify that 11 1tan tan 2t t   if t is a positive real number. Demonstrate how a numerical approximation of 1tan 2021 can be obtained using the first two terms of the series expansion in (ii). [2] One way to obtain an approximation of  is to sum sufficient initial terms of the series (iii)› By considering 1 1tan 2  and 1 1tan 3  , explain how the above result is obtained. [2] 1 1 3 3 5 5 7 7 1 1 1 1 1 1 1 14 1(2 ) 1(3 ) 3(2 ) 3(3 ) 5(2 ) 5(3 ) 7(2 ) 7(3 )         

6 ASRJC 2021 9 The Bell number nB gives the nu