2021 CJC H3 Math Questions

9820/01/PRELIM/2021 [Turn over CATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 3 JC2 Preliminary Examination MATHEMATICS 9820/01 Paper 1 17 September 2021 3 hours Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST An answer booklet and a graph paper booklet will be provided with this question paper. You should follow the instruction on the front cover of both booklets. If you need additional answer paper or graph paper, ask the invigilator for a continuation booklet or graph paper 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 st ates 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 5 printed pages and 1 blank page.

2 9820/01/PRELIM/2021 Questions Q1 Topic : Numbers and Proofs Q2 Topic : Functions and Graphs Q3 Topic : Counting 1. Let  1f ( ) 13 6 7 1nnn    . By considering f ( 1) f ( )kk , where k  , prove using Mathematical Induction that f ( )n is divisible by 18 for every positive integer n. [6] 2. (a) For any real number x, the largest integer less than or equal to x is denoted by x . For example, 3.7 3 and 44 . (i) Use a sketch graph of yx for 05 x to evaluate 5 0 dxx . [2] (ii) Use a sketch graph of exy  for 0 lnxn , where n is an integer, to show that ln 0 e d ln ln( !) n x x n n n  . [3] (iii) Hence, show that 1!e nnnn  . [3] (b) Find the exact value of the integral 2 23 2 19 cos d 22 xx x x    . [4] 3. An n-digit number uses only digits 1, 2 and 3. It does not contain any occurrence of ‘12’ or ‘21’. Let there be nT such numbers, with nX of these having first digit 1 and nY having first digit 3. (i) Prove that, for 2n , (a) 11n n nX X Y  , (b) 112n n nY X Y  . [1] [2] (ii) Hence, find a recurrence relation for 1nY  in terms of nY and 1nY  for 2n . [2] (iii) Prove that     111 2 1 222 n n n Y     for n  . [5] (iv) Find and simplify an expression for nT for n  . [2]

3 9820/01/PRELIM/2021 [Turn over Q4 Topic : Functions and Graphs Q5 Topic : Counting Q6 Topic : Numbers and Proofs 4. The Bernoulli polynomials, B ( )n x , where n = 0, 1, 2, …, are defined by 0B ( ) 1x  and, for 1n , 1 dB B ( )d n nnxx  and 1 0 B ( )d 0n xx  . (i) Show that 22 4B ( ) ( 1)x x x A   , where A is a constant (that need not be evaluated). [4] (ii) Show that, for 2n , B (1) B (0) 0nn  . [2] (iii) Show that 1B ( 1) B ( ) n nn x x nx    for all positive integers n. [4] (iv) Hence, for any positive integer N, show that  1 1 1 B ( 1) B (1) N n nn m mN n      , and deduce that 2 3 1 ( 1) 2 N m NNm   . [3] 5. (a) New Chang Lee sells 5 types of puffs: curry, sardine, black-pepper chicken, tuna and yam. Mr. Ong wants to order a total of 26 puffs such that each type is included and there is an even number of puffs of each type. Find the number of ways he can make the order. [4] (b) 4 married couples are randomly seated at a round table with 8 chairs. By using the principle of inclusion and exclusion, find the probability that no wife sits next to her husband. [4] (c) Let 1 2 10, , ..., a a a be a sequence of 10 natural numbers. By considering the sums 1,a 12 ,aa … , 1 2 10a a a   , and using the pigeonhole principle, prove that there is a sequence of n consecutive term(s) whose sum is divisible by 10, for some 1 10n . [5] 6. (a) Let p be a prime number and ,rs  such that 0, r s p . (i) For any a , show that  modra sa p if and only if rs . [3] (ii) By considering the product  2 3 ... 1a a a p a     , prove that for any a but not divisible by p,  1 1 modpap  (Fermat’s Little Theorem). [2] (iii) Hence, show that for any integers a, b, p divides ppab a b . [2] (b) Let ,,m n k  and gcd  ,1mn  . Prove that 22n m n km if and only if 1n m k . [7]

4 9820/01/PRELIM/2021 Q7 Topic : Inequalities 7. (a) (i) Show that 1 2 for , 0xx x yyy   . [1] (ii) Hence, show that 50 1 2 49 50 2 3 50 1 1 1 1 1 ... 2x x x xx x x x                          , for 1 2 3 50, , , ... , 0x x x x  . [2] (iii) Deduce the positive solutions of the system of equations 1 2 2 3 3 4 49 50 50 1 1 8 11 2 1 8 1 8 11 .2 x x x x x x x x x x      [6] (b) (i) Let ,,x y z be positive real numbers satisfying 1xyz . Determine, with proof, the minimum value of 2 2 2 .xyz y z z x x y   [4] (ii) Hence, prove that if ,,abc are positive real numbers satisfying 1abc , then 3 3 3 1 1 1 3 .2( ) ( ) ( )a b c b c a c a b       [2]

5 9820/01/PRELIM/2021 [Turn over Q8 Topic : Sequences and Series 8. (a) For each positive integer r, let 23 1 1 1 ... ,1 ( 1)( 2) ( 1)( 2)( 3) 1 1 1 ... .1 ( 1) ( 1) r r a r r r r r r b r rr               (i) Find rb in terms of r. [2] (ii) Deduce that 10 ra r . [2] (iii) Show that !e !era r r  , where x denotes the integer part of x. [3] (iv) Hence show that e is irrational. [3] (b) Given that a sequence 1 2 3, , , ...y y y is defined by 11 2, 1 2 n n n y yy y  , for all n . Show that the sequence converges. Find the exact limit of the sequence. [5]