NYJC TJC VJC 2026 H3 Math Prelim

1 TEMASEK JUNIOR COLLEGE 2026 JC2 PRELIMINARY EXAMINATIONS Higher 3 MATHEMATICS 9820 21 Sep 2026 3 hours Additional Materials: Printed Answer Booklet List of Formulae (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. 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. [Turn over

2 1 (a) In this question we define two functions: • x (known as the floor function) denotes the greatest integer less than or equal to x, • x (known as the ceiling function) denotes the smallest integer greater than or equal to x. For example, 2.9 2, 1.5 2, 2.1 3 and 1.5 1= − =− = − =−               . On separate diagrams draw the graphs, for −2 ≤ x ≤ 2, of: (i) yx= , [2] (ii) yx= , [2] (iii) y x x=+       . [3] In each case you should indicate clearly the points where the graph is discontinuous. (b) The function f is defined by f( x) = |x − p| − |x − q|, where 0 < p < q. Sketch the graph of f(x), giving f( x) in each of the regions x < p, p < x < q and x > q. Sketch on the same diagram the graph of g(x), where g(x) = |2x − p − q|. [4] 2 In this question, we consider the distributions of n identical balls into 1n+ labelled boxes. (a) For 1k , let ,nkT be the set of distributions in which exactly k boxes are non-empty. Show that , 11 1nk nnT kk+−  =   −   . [3] (b) Let nV be the set of distributions in which every non-empty box contains at least 2 balls. Show that for n > 2, 2 1 11 1 n n k n n kV kk= + − −  =    −   . [3] ( x (known as the floor function) denotes the greatest integer less than or equal to x.) (c) Let nE be the set of distributions such that every non-empty box contains an even number of balls. For the case when n is even, find a summation expression for nE . [2] (d) Let nW be the set of distributions such that each non-empty box contains at most 2 balls. Find a summation expression for nW . [4]

3 3 (a) Let 1x , 2x , 1y and 2y be positive real numbers. By considering ( ) 22 12 12 12 xx yyyy ++ and applying the Cauchy-Schwarz inequality, prove that ( ) 222 1212 1 2 1 2 xxxx y y y y ++ + . State the necessary and sufficient condition for which the equality holds . [3] (b) For each integer 2n and positive real numbers ix and iy , where 1 in , prove that 2 2 1 1 1 n in ii n i i i i xx y y = = =     . [3] Let a, b and c be positive real numbers. (c) (i) Prove that 2 2 2a b c ab bc ac+ +  + + . [2] (ii) Hence, show that 12 2 2a b cb c c a a b+ + + + + , stating the condition for which equality holds. [5] 4 The Riccati differential equation takes the general form ( ) ( ) ( ) 2d P Q Rdy x x y x yx = + + , where P, Q and R are continuous functions of x. (a) (i) Given that 1y is a particular solution, show that the substitution 1 1yy u=+ , where u is a non-zero differentiable function of x, transforms the differential equation to ( ) ( ) ( )1 d Q 2R R 0du x x y u xx + +  + = . [3] (ii) Given that 1yx= is a particular solution to the differential equation 22d 1 2 4 2dy x xy yx = − + − , solve for its general solution, expressing y in terms of x. [3] (b) (i) If 1y , 2y , 3y and 4y are unique particular solutions to the general Riccati differential equation, show that the cross-ratio given by ( )( ) ( )( ) 4 1 3 2 4 2 3 1 y y y y y y y y −−= −− Is independent of x. [3] (ii) It is obvious that 1yx= and 2yx=− are solutions to the differential equation ( ) ( )( ) 22 22 e1d d e x x xyy yxxx xx −= + − − . Verify that 3 exy = is also a particular solution. Hence, solve for its general solution. [3]

4 5 For any integer 0n , let 0 de sinxnn xIx  −= and 0 de sinxnn xJx  −= . (a) Show that ( ) ( ) 12d sin cos 1 sin sind n n nx x n x n xx −− = − − . [1] (b) Show that, for 2n , 22 ( 1) 1 nn nnII n −−= + . [4] (c) Find 21 21 lim n n n I I + → − . [1] By partitioning the interval  )0, into intervals of length  , nI can be rewritten as ( 1) 0 de sin r xn n rr xIx   + − = = . (d) Use the substitution u x r =− to show that ( ) 1 1 1 e nn nIJ −= −− . [3] Let 0 2k  . (e) Explain why sin cosxk for 0 2xk   − . Deduce that ( ) 2 0 de sin cos k nxn xxk− −  . [2] (f) It can be shown that  nJ converges to 0. Hence, show that lim 0nn I→ = . [1]

5 Use the information in the mathematical text to answer Question 6. You should read the whole mathematical text before you start answering the questions. Compositions, Partitions and Ferrers Diagrams A composition of a positive integer n is a way of writing n as a sum of positive integers where the order of the summands matters. For example, the four distinct compositions of 3n= are: 3 , 21+ , 12+ , 111++ . If the order of the summands does not matter, the representation is instead called a partition of n. To avoid duplicate arrangements, we conventionally list the summands (called parts) of a partition in a non-increasing order. For instance, the integer 4n= has exactly 5 unique partitions: 4 , 31+ , 22+ , 2 1 1++ , 1111+++ . We use the notation ( )12, ,..., k   = to denote a partition of n, where 1 k k i n = = and 12 ... 1 k      . Partitions can be visualised geometrically using a Ferrers diagram. A Ferrers diagram represents a partition  as a pattern of dots arranged in left -aligned rows, where the i-th row from the top contains exactly i dots. For example, the partition ( )4,2,1= of the integer 7n= is represented by the following layout: By reflecting a Ferrers diagram across its main diagonal (flipping rows into columns), we obtain a new Ferrers diagram. The partition represented by this flipped diagram is called the conjugate partition, denoted by * . Reflecting the diagram above transforms it into: Reading the rows of this flipped diagram reveals that the conjugate of ( )4,2,1= is ( ) * 3,2,1,1 = , which is also a valid partition of 7n= . A partition that satisfies *= is said to be self-conjugate. For positive integers n and k , ( )pn denotes the number of partitions of n , and ( ),p n k denotes the number of partitions of n where the largest part is at most k . Defining ( )0, : 1pk = , it is known that ( ) ( ) 1 , n k p n p n k k = =− .

6 6 Let n be a positive integer. (a) (i) List all distinct compositions of 4 and of 5. [3] (ii) State a formula for the number of distinct compositions of n , explaining your answer. [2] (b) Let ( )12, ,..., k   = be a pa rtition of n and let ( ) * * * * 12, ,..., m   = . Explain why * 1 k = and 1m = . [2] (c) Let ( )cn be the number of self -conjugate partitions of n , and let ( )dn be the number of partitions of n with parts that are all odd