TMJC H2 Mathematics Prelims Paper 2 (Q)

Tampines Meridian Junior College 2024 JC2 Preliminary Examination H2 Mathematics CANDIDATE NAME: ___________________________________________________ CIVICS GROUP: _______________________________________________________ __________________________________________________________________________________ H2 MATHEMATICS 9758/01 Paper 1 10 SEPTEMBER 2024 3 hours Candidates answer on the question paper. Additional material: List of Formulae (MF26) ________________________________________________________________________________ READ THESE INSTRUCTIONS FIRST Write your name and Civics Group on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. Write your answers in the spaces provided in the question paper. 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 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. The total number of marks for this paper is 100. _________________________________________________________________________________ This document consists of 25 printed pages and 1 blank page. TAMPINES MERIDIAN JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION For Examiners’ Use 1 2 3 4 5 6 7 8 9 10 11 Total [Turn Over

2 1 (i) Express 21 4 xy x += − in the form 4 bya x=+ − where a and b are constants to be determined. [1] (ii) Hence, state a sequence of transformations that will transform the curve with equation 1y x= to the curve with equation 21 4 xy x += − . [3]

3 [Turn Over 2 A cylinder with height h cm and base radius r cm is inscribed within a sphere with fixed radius k cm. The circumference of the circular top and bottom of the cylinder is in contact with the sphere (see figure above). By using differentiation, find, in terms of k, the exact value of h for which the volume of the cylinder is maximum. [6] k h r

4 3 The function f is defined by ( ) for 0 2,f for π2sin 6 2 2 4 3, xx x x x  =       −  and that ( ) ( )f f 3xx=+ for all real values of .x (i) Find the value of ( )f 7 . [1]

5 [Turn Over (ii) Sketch the graph of ( )fyx= for 4 7.x−   [3] (iii) The region R is bounded by the curve ( )fyx= and the x-axis from 0x= to 5.x= Find the exact volume of solid generated when R is rotated 2π radians about x-axis. [4]

6 4 In the diagram above, OAB is an isosceles triangle where cm.OA OB r== It is given that the length 2 cmAB= and angle AOB = π 3 − radians. (i) Using cosine rule, show that 2 4 2 3 sin cos r  = −− . [3] O r 2 A r B

7 [Turn Over (ii) Given that  is a sufficiently small angle, show that 22r a b  + + where a and b are constants to be determined. [4]

8 5 An arithmetic progression A has first term a and common difference d, where a and d are non-zero. A geometric progression G has first term b and common ratio r. (i) The first, third and eleventh terms of A are equal to the fourth, third and second term of G respectively. Prove that the geometric series of G is convergent. [4]

9 [Turn Over (ii) It is given instead that 1 3r = and the terms of another sequence H is formed by squaring the terms of G. Find the range of values of b such that the sum to infinity of H exceeds the sum to infinity of G by more than 3 .2 [3]

10 6 It is given that ( ) 1f 21rr = + . (i) Show that ( ) ( ) ( )( ) 1 2f f 1 2 1 2 1 r rrrr +− + = ++ . [1] (ii) Using the result in part (i), find ( )( ) 1 1 2 2 1 2 1 rn rr r + = ++ . [3]