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 clearly. Express all values in exact form. [11] 6 (a) Show that     14 1 2 4 x x x x     can be expressed as   1 2 4 Ax B C x x x     , where A , B and C are constants to be determined. [2] (b) Let     px qx be a rational function such that the degree of the numerator  px is less than the degree of the denominator  qx . uurthermore, assume that       12 nq x x x x x x x    , where ix are all distinct. Show, by mathematical induction, that for any 1n ,     px qx can be written in the form 1 n i i i c xx  for suitable constants ic . [8]

4 [Turn Over 7 In basketball, any single shot is worth either one, two or three points. Mid-way through a basketball game, the scoreboard shows 6 points. (a) uind the number of ways t o score 6 points in basketball if we are interested in the order of each type of shot made. [2] Suppose that now we want to count the number of ways to score 6 points but we are only interested in the number of each type of shot made and not the order in which they were made. The following approach will help with the counting systematically . In scoring 6 points, the contribution from one-point shots to that score is 0 point  1 point  2 points  3 points  4 points  5 points  6 points, where  means the exclusive-or. We symbolise this algebraically as 0 1 2 3 4 5 6x x x x x x x      ---- (1), where  signs have been replaced by ordinary addition and where the total contribution appears in the exponents. Similarly, we can algebraically symbolise the contribution from two -point shots and three-point shots to the score of 6 points respectively as 20 46x x x x --- (2) and 30 6x x x --- (3) . Proceed to multiply the expressions in (1), (2) and (3) to each other. This product is called a generating function. Lastly, identify the required coefficient of rx , where 0 18r , which in this case, is the coefficient of 6x . You may consider using the above approach to answer the following questions. (b) Suppose that we are only interested in the number of each type of shot made and not the order, (i) find the number of ways to score 6 points. [1] (ii) write down the generating function to find the number of ways to score 10 points. [2] (c) (i) uind the coefficient of 24x in the product          3 5 7 11 13 17 1921 1 1 1 1 1 1 1x x x x x x x x        . [1] (ii) Interpret the answer in part (c) (i) in terms of the partitions of 24. [2] (d) (i) Given that        2 4 81 1 1 1 ... Px x x x x     where  P x is a power series, find  P x . [1] (ii) What does the equation in part (d) (i) say about all positive integers? [2]

5 [Turn Over 8 The uundamental Theorem of Algebra states that every polynomial function  p x of degree n can be expressed as a product of linear factors     12p( ) , nx x x x       where 12, , , n   are the roots of  p0x  in , not necessarily all distinct. Subject to certain conditions, this result can be extended to non-polynomial functions. In particular, , 0 sin 1 , k k k xxx       where k ’s are the roots of the equation sin 0,x and ku denotes the product of the terms 12, , , , nu u u of a sequence. (i) State an expression for k in exact form. [1] (ii) Express sin x x in the form 2 1 1, k k x       showing your working clearly. [2] (iii) By considering the coefficient of a suitable term in the expansion of sin ,x x find exactly the value of 2 1 1 . n n    [3] (iv) Obtain a formula for 2 1 1 m n n    in terms of m for any positive integer m. [3] Prove that if p  1, then the series 1 1 p n n    converges. [3]

6 [Turn Over 9 A cow produces one calf every year. Beginning in its fourth year, each calf produces one calf at the beginning of each year. Let ng be the number of calves born in the thn year. It is given that 1 2 3 1g g g   . (i) Explain why 13n n ng g g  for 4.n [3] (ii) uind 10.g [3] An ordered composition of a positive integer n is a sequence  1 2 3, , , ..., ka a a a of positive integers such that ian . uor example, all ordered compositions of 4 are                4 , 3,1 , 2, 2 , 1,3 , 2,1,1 , 1, 2,1 , 1,1, 2 and 1,1,1, 1 . Given that n  , we let nh be the number of ordered compositions of n, consisting of positive integers 3 or 1 (or both). It is given that 1 1h  . (iii) uind 2h and 3h . [2] (iv) Explain why 13n n nh h h  for 4n . [2] (v) Express 2nh as a sum of p q   , where p q   counts the number of ways to choose q objects from p distinct objects. [3] 10 A lattice