NJC 2025 H3 Mathematics Prelim Question Paper

* © NJC 2025 [Turn_over NATIONAL JUNIOR COLLEGE SENIOR HIGH 2 Higher 3 MATHEMATICS 9820/01 Preliminary Examination 25 September 2025 Paper 1 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Answer all questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. 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 must present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. You are reminded of the need for clear presentation in your answers. Up to 2 marks may be deducted for improper presentation. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 6 printed pages and 2 blank pages.

2 © NJC 2025 1 (a) Explain geometrically why for any positive real numbers a, b, c and d, if ,ac bd< then .a ac c b bd d +<< + [3] (b) Prove that if π0, 2x<< then sin 2tan tan .2 xxx<< [4] (c) Prove that if a and b are distinct positive real numbers, then 2 4 2. a b ab b a ba  <+ <+   [4] 2 (a) Given that f: → satisfies ( ) ( ) ( ) 2f 2 2f fm n mn+= + for all integer values of m and n, find all possible functions f. [6] (b) Given that x > y, find exactly the values of the real variables x and y such that 55 4, 464. xy xy += += [6] 3 (a) A soccer player is training for a competition over a period of 20 weeks, starting on a Sunday. Each day, he trains for at least 1 hour. In order not to overtrain and injure himself, the player decides not to train for more than 13 hours in any calendar week – a week beginning with Sunday and ending with Saturday. The number of hours trained each day is a positive integer. Show that there exists a succession of consecutive days during which the player will have trained for exactly 19 hours. [5] (b) (i) Explain how the integer solutions of the inequality 1234 50xxxx+++≤ can be related to the solutions of 12345 50.yyyyy++++= [1] (ii) Use the principle of inclusion and exclusion to find the number of integer solutions of 1234 50xxxx+++≤ such that 13 12,x≤≤ 20 7,x≤≤ 38 12x−≤ ≤ and 46 20.x≤≤ [7]

3 © NJC 2025 [Turn_over 4 (a) A curve C has the following property. At any point P on the curve C , the normal line cuts the x - and y-axes at A and B respectively, and B is always the midpoint of AP. Find an equation for C. [6] (b) The Weierstrass substitution suggests that 2 2sin 1 tx t= + and 2 2 1cos 1 tx t −= + , where tan2 xt = . You may use the above results without proof in solving the following question. A student forgot the Product Rule for differentiation and wrongly assumed ( )d dd d dd uvuvx xx   =     , where u and v are functions of x. However, he was lucky enough to get the correct answer. Given that ( ) 3sine xux = , show that 1 d 3cos d 3cos 1 vx vx x = − . [2] Hence find ( )vx . [5] 5 The nth Fermat number, ,nF is defined by 22 1, n nF = + for non-negative integers n. (i) Write down the values of 012 3, , and .FFF F [1] (ii) By considering 1, 2 and 3,kk k= = = conjecture a formula, in terms of ,kF for the product 012 1 ... .kFFF F − [2] (iii) Prove, by mathematical induction, that the conjecture in part (ii) holds for all positive integers k. [5] (iv) Deduce that any two distinct Fermat numbers are coprime. [3] (v) It is given that ( ) 44 7641 2 5 5 2 1.=+=× + (a) Find the remainder when 4 2852× is divided by 641. [2] (b) Hence, show, without the use of a calculator, that ( )5 0 modulo 641 .F ≡ [2]

4 © NJC 2025 Use the information in the mathematical text to answer Question 6. You should read the whole mathematical text before you start answering the questions. Investigating Catalan Numbers The Catalan numbers are a sequence of positive integers that are very useful in counting problems. The thn Catalan number can be expressed using binomial coefficients: 21 1 n nC nn = +  . nC counts the number of ways 1np + to associate – or to group – the product 112 ... nxx x + into products of two expressions using parenthesis. Hence, we can write 1nnCp += , for 1n≥ . 1A Recursive Formula for np + We define 1 1p = . For 1n= , we have a two-fold product, ab, which has one way to do the above grouping using parenthesis, namely ( )ab , so that 2 1p = . For 2n= , we can associate the three-fold product abc as ( )( ) ( )( ),ab c a bc . Hence, 3 2p = . For 3n= , we can associate the four-fold product abcd in five different groupings, namely ( )( )( ) ( )( )( ) ( )( )( ), , a b cd a bc d a bc d , ( )( )( ) ( )( )( ) , ab cd ab c d , so 4 5p = . Thus, the value of 1np + can be obtained by first grouping the product 112 ... nxx x + into two subproducts of lengths r and 1nr−+ for 1,2,...,rn= . As there are rp ways to group an r-fold product and 1nrp −+ ways to group an ( )1nr−+ -fold product, we see that 1 1 21 1 ...n nn np pp pp pp+− = + ++ , for 2n≥ . Relations between groupings The relations between the different ways to associate an ( )1n+ -fold product can be represented with a graph called the associahedron, of order 1.n+ This is a diagram whose vertices are the 1np + groupings of an ( )1n+ -fold product, and two vertices are joined by an edge if each can be transformed into the other by a single application of the associative law ( )( ) ( )( )x yz xy z= . The associahedron of order 5 has 14 vertices corresponding to the 14 groupings of a five -fold product. The diagram below shows this associahedron with 2 missing groupings, indicated as 1g and 2g .

5 © NJC 2025 [Turn_over 1A Formula for in terms o f n np + Here is another way to look at groupings of products. Each grouping of an ( )1n+ -fold product corresponds to a string of n O’s (for open parenthesis) and n C’s (for close parenthesis) in which the number of O ’s never falls below the number of C’s. For example, the string OOCOOCCC corresponds to the grouping ( ) ( )( )( )ab c de . Replacing each O by a 1 and each C by a (−1) yields a sequence 12 2, ,..., nyy y of n 1’s and n (−1)’s in which 12 0... kyy y++ ≥+ for 1 2.kn≤≤ We call such sequences good; thus, the good sequence 1, 1, −1, 1, 1, −1, −1, −1 corresponds to the grouping ( ) ( )( )( )ab c de . So, the number of good sequences of length 2𝑛𝑛 equal the number of groupings 1np + of an ( )1n+ -fold product. That number is the Catalan number nC .     ab c d e      ab cd e     ab c de      ab c de     ab cd e     a b cd e 1g 2g     a bc d e     a b cd e     a bc d e     abc d e     a debc     bc d ea

6 © NJC 2025 6 (i) Verify that the number of ways to associate a five -fold product into products of two expressions using parenthesis is 14. [1] (ii) Show that ( )21 0 modulo 2iP + ≡ for .i +∈  [1] (iii) (a) Fin