CJC SAJC JPJC 2026 H3 Math Prelim

Name:____________________________________ Class:_____________ JURONG PIONEER JUNIOR COLLEGE JC2 Preliminary Examination 2026 MATHEMATICS 9820/01 Higher 3 18 September 2026 Paper 1 3 hours Additional materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST This document consists of 7 printed pages and 1 blank page. [Turn over 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 co rrect 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 by [ ] at the end of each question or part question .

2 9820/01/PRELIM/2026 1 (a) For all real numbers x > 1, y > 1 and z > 1 such that 1 1 1 2x y z+ + = . (i) Show that 1 1 1 1x y zx y z− − −+ + = . [1] (ii) Prove that 1 1 1x y z x y z+ +  − + − + − . [2] (b) (i) By considering the function f ( ) lnx x x=− , show that ln 1xx− for all real values of x. [1] (ii) Hence, by letting 1 1 n i i Aa n ==  where n is a positive integer and ia are positive real numbers for all i, or otherwise, prove that 1 2 3 1 1 n n in i a a a a an =   and state the necessary condition for equality to hold. [3] (iii) For all positive real numbers a, b, c and d, they satisfy ( )( )a c b d ac bd+ + = + . Hence, find, with proof, the minimum possible value of a b c d b c d a+ + + . [5] 2 Consider a sequence of partial sums  nS for 1n , n  , defined by 3 1 1n n k S k== . (a) Prove that  nS is strictly increasing . [1] (b) By considering 3 1 k and 3 1 kk− , show that  nS is convergent. [5] (c) Show that ( ) 2 2 11 221 nSS nn  − + . Hence, evaluate ( ) 2lim nn n S S→ − , justifying your answer. [3] Consider an alternating series ( ) 1 3 1 1 k k A k − = −= . (d) Express A in terms of S . Hence, deduce a strict upper bound for A. [4]

3 9820/01/PRELIM/2026 3 Let f be a continuous function on the interval a x a−   , where a is a positive constant. (a) Use a suitable substitution to prove that  0f ( ) d f ( ) f ( ) d . aa a x x x x x − = + − [2] Suppose H is a continuous function where the graph of H( )yx= is symmetrical about the y -axis; and G is a positive continuous function where the graph of ln(G( ))yx= has rotational symmetry about the origin. (b) Using the result from part (a), show that 0 H( ) d H( ) d .1 G( )aaa x x x xx− =+ [3] Let 2 0 cos d mmI x x  = for integers 0m . (c) (i) Show that 2( 1)mmmI m I −=− for 2m . (ii) Hence, show that for 1n , 2 22 (2 )! .2 ( !) 2n n nI n = [3] [3] (d) Deduce the exact value of 6 2 sin 2 cos d . 1 xx xe− + [3] 4 Ten wooden planks are placed in a row to form a fence. Each plank is coloured either red, blue, green, yellow or black. (a) (i) Write down the total number of ways to form a row of 10 planks. [1] (ii) Find the number of ways this can be done if each of the colors is selected at least once. [3] (b) Let np be the total number of ways to place n planks if no two consecutive planks are red. (i) Write down the value of 1p and find 2p . [1] (ii) Show that 124( )n n np p p −−=+ for 3n . [2] (iii) Show that 32kp − is divisible by 5 for all integer 1k [3] (c) A collection of one hundred wooden planks is labelled with the integers 1 to 100. Prove that there must be two wooden planks out of fifty-five randomly chosen planks whose labels differ by 6. [4] [Turn over

4 9820/01/PRELIM/2026 5 In the following Argand diagram, a regular pentagon ABCDE is inscribed in a unit circle centred at the origin. B represents the complex number w . (a) Show that 2π 2πcos isin55w    =+        . [1] (b) Show that (i) 5 1w = ; (ii) 432 10w w w w+ + + + = . [1] [1] (c) Let 1xw w=+ . (i) Show that 432 10w w w w+ + + + = can be written as 2 10xx+ − = . (ii) Find the exact values of x and deduce the exact value of 2πcos 5 . [3] [3] (d) By finding the exact value of 2πsin 5 , find the area of the pentagon. [2]

5 9820/01/PRELIM/2026 Use the information in the mathematical text to answer Question 6. You should read the whole mathematical text before you start answering the questions. Derangements and Subfactorials In combinatorics, a permutation is an arrangement of a set of objects in a specific order. A derangement is a special type of permutation where none of the objects appear in their original ordered positions. Let { 1,2,3, , }nSn= be the set of the first n positive integers. A bijection : nnf S S → is a derangement if ()f i i  for all niS . The total number of unique derangements of a set of size n is denoted by !n , which is read as the subfactorial of n . For example, if 3n= , the set { 1,2,3} has 6 possible permutations. The permutations (3,1,2) and (2,3,1) are valid derangements because element 1 is not in position 1, element 2 is not in position 2, and element 3 is not in position 3. By listing all possibilities, it can be shown that !3 2= . For any integer 2n , the sequence of subfactorials satisfies the first-order recurrence relation: ! (!( 1)) ( where !1 01)nn n n= − + − = . This relation can also be rewritten as a second-order recurrence relation for 3n : ! ( 1)(!( 1) !( 2)) where !1 0 and !2 1n n n n= − − + − = = . A permutation is considered even if it can be obtained by performing an even number of transpositions (swaps) within a set while an odd permutation can be obtained by an odd number of transpositions. Let nE denote the number of even derangements of nS , and nO denote the number of odd derangements of nS . For example, if 4n= , the set { 1,2,3,4} has possible derangements of (2,1,4,3) and (2,4,1,3), (2,1,4,3) is an even derangement as it can be obtained by performing two transpositions within the set while (2,4,1,3) is an odd derangement obtained by 3 transpositions. Since every derangement is either even or odd, the total number of derangements of a set satisfies ! nnn E O=+ . [Turn over

6 9820/01/PRELIM/2026 6 (a) Show that ! (!( 1)) ( 1) nn n n= − + − can be rewritten as ! ( 1)(!( 1) !( 2))n n n n= − − + − . [3] (b) A student attempts to prove that for all 2n , the subfactorial !n is an even number if n is odd, and an odd number if n is even: Identify the logical flaw in the student’s proof in the fourth statement, and write down the amended proof. [4] (c) The table below shows the number of even derangements nE , the number of odd derangements nO , and their difference n n nD E O=− for small values of n : n !n nE nO nnnD OE= − 1 0 0 0 0 2 1 0 1 1− 3 2 2 0 2 4 9 3 6 3− 5 44 24 20 4 (i) Based on the pattern in the table, formulate a conjecture for the general expression o