TMJC 2026 H3 Math Prelim

Tampines Meridian Junior College 2026 JC2 Preliminary Examination H3 Mathematics CANDIDATE NAME: ___________________________________________________ CIVICS GROUP: _______________________________________________________ __________________________________________________________________________________ H3 MATHEMATICS 9820/01 Paper 1 18 SEPTEMBER 2026 3 hours Additional materials: Printed Answer Booklet List of Formulae and Results (MF27) ________________________________________________________________________________ READ THESE INSTRUCTIONS FIRST Answer all the questions. Write your answers in the spaces provided in the Printed Answer Booklet. Follow the instructions on the front cover of the Printed 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 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 6 printed pages and 0 blank pages. TAMPINES MERIDIAN JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION [Turn Over

2 1 (a) Prove that if 21n − is a prime number for any integer 1n , then n must be a prime number. [3] (b) Prove that for any integer 1n , the expression 221n − is not a prime number, and it is divisible by 3. [3] (c) Show that 221n − cannot be expressed as the sum of two consecutive prime numbers for any integer 1n . [4] 2 (a) Determine, with reason, if 2 21 cos dx xx converges or diverges. [4] (b) Evaluate 20 e de3 xx x + exactly. [4] (c) (i) Let x, y and z be positive real numbers. Prove that 2 2 2 2 x y z x y z y z x z x y +++ + +++ . [2] (ii) Hence solve the equation 22 313 1 1 2 xy xyy x x y+ + =+++ . [3] 3 (a) Prove that the differential equation d Fdyyxx =  can be transformed into the differential equation d F( ) d v v v xx −= by using the substitution y vx= . [2] (b) A solution curve of the differential equation 22d d y x yxy xx +=+ passes through the point ( )1, 0 . Using the result from part (a), find the equation of the curve in the form f ( )yx= . [6] (c) Hence, evaluate π 4e dlim dx yx→ . [4]

3 [Turn Over 4 (a) The distinct points A, Q and C lie on a straight line in the Argand diagram and represent the distinct complex numbers a, q and c respectively. Show that qa ca − − is real and deduce that ( )( ) ( )( )* * * * .c a q a c a q a− − = − − Given that ** 1,aa cc== show that * .q acq a c+ = + [6] (b) (i) The Cauchy -Schwarz inequality for complex numbe rs 12, ,..., nz z z and 12, ,..., nw w w , where kw is the conjugate of kw , states that 2 22 1 1 1 n n n k k k k k k k z w z w = = =             for some positive integers .n By considering 2 1 , where , n kk k z tw t = −  prove the inequality. Hence, obtain a relationship between kz and kw for equality to hold. [5] (ii) Suppose 222 1 2 3 12.z z z+ + = Find the largest possible value of 1 2 32 2 .z z z++ [2]

4 5 (a) An ice-cream parlour offers 10 flavours to choose from. A girl wishes to buy bowls of ice-cream for three friends such that all 10 flavours are chosen at least once, and that no flavour is chosen for all three bowls at the same time. She also wishes not to choose any flavour twice for the same bowl. (i) Let the sets A and B be defined as follows. A : Ways in which the girl can buy the three bowls of ice-cream. B : Table of numbers that have 10 rows and 3 columns such that each entry is either a ‘0’ or a ‘1’, no row is completely filled with ‘0’s or completely filled with ‘1’s, and no column is completely filled with ‘0’s. Explain how a bijection may be defined between the sets A and B . [3] (ii) Find the number of ways in which the girl may buy the bowls of ice-creams. [4] (b) (i) Calculate the number of ways of distributing 6 identical balls into 4 distinct boxes, A, B, C and D, in each of the following cases. (a) Boxes B and C each contain at least one ball. [2] (b) Each box contains an even number of balls. [2] (ii) A sandwich shop sells t types of sandwiches. Consider the problem of calculating the number of distinct purchases of s sandwiches. Describe a distribution problem equivalent to this problem and hence, or otherwise, find the number of distinct purchases. [3]

5 [Turn Over Use the information in the mathematical text below to answer Question 6 (next page). You should read the whole mathematical text before you start answering the questions. Bernoulli's Inequality and Its Applications Bernoulli's inequality states that for any real number 1x− and integer 1,n ( )1 1 , nx nx+  + with equality if and only if 0x= or 1.n= This inequality gives a useful lower bound for powers of expressions of the form ( )1 nx+ . One way to prove Bernoulli's inequality is to use the concavity of ( ) ( )f ln 1 ,xx=+ defined for 1x− . Since ( ) ( )" 21f 0, 1x x=− + the function is strictly concave. Geometrically, this means that the graph of the function lies above any chord joining two points on the graph. This property leads to Jensen's inequality: ( )( ) ( ) ( ) ( )f 1 1 f f , a b a b   − +  − + where 01  and .ab By choosing 1 n = , Bernoulli's inequality can be obtained . This shows how the concavity of the logarithm leads to an algebraic inequality involving powers. Together with the AM-GM inequality, these two inequalities are powerful tools for analysing the monotonicity of sequences. A sequence  na is monotone increasing if 1nnaa+  and monotone decreasing if 1nnaa+  for all n , and this is established by examining the ratio 1n n a a + or the difference 1 .nnaa+ − Together, Bernoulli's inequality, concavity, and the AM-GM inequality provide useful tools for investigating important sequences in mathematics.

6 6 (a) Using a sketch graph of the function ( ) ( )f ln 1xx=+ where x  , 1x− , prove that for any real numbers a, b and  where 0 ab and 01 , ( )( ) ( ) ( ) ( )f 1 1 f f a b a b− +  − + . [3] Hence, using 1 and 0an= = , prove Bernoulli’s inequality ( )11 n x nx+  + where n +   , 1x− . [2] (b) Using part (a), show that ( ) 1 2 111 1 1 1 n n n + + −   + . Hence prove that the sequence 11 n n + is a monotone increasing sequence. [4] (c) By applying the AM-GM inequality 1 2 3 22 1 2 3 2 2 nn n a a a aa a a a n ++ + + + + + +   such