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