TMJC H2 Mathematics Prelims Paper 2 (Q)

Tampines Meridian Junior College 2024 JC2 Preliminary Examination H2 Mathematics CANDIDATE NAME: ___________________________________________________ CIVICS GROUP: _______________________________________________________ __________________________________________________________________________________ H2 MATHEMATICS 9758/02 Paper 2 16 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 22 printed pages and 0 blank page. TAMPINES MERIDIAN JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION For Examiners’ Use 1 2 3 4 5 6 7 8 9 10 Total [Turn Over

2 Section A: Pure Mathematics [40 marks] 1 (i) Given that ( ) 2ln 1 3 2y x x= + + , show that ( ) ( ) 2 2 2 dd1 3 2 3 4 4. dd yyx x x xx+ + + + = By further differentiating the above result, find the Maclaurin series for ,y up to and including the term in 3x . [5]

3 [Turn Over (ii) Verify the correctness of your answer in part (i) by using the standard results given in the List of Formulae (MF26). [2] (iii) Hence, by using 1 2x= , estimate the value of ln 3. [2]

4 2 Referred to the origin O, the points A and B have position vectors a and b respectively. The point C is such that O divides the line segment AC in the ratio 2 : 3. The point D divides the line segment AB in the ratio 1: 4. (i) Show that the area of triangle OCD is given by , a b where  is a constant to be determined. [5]

5 [Turn Over (ii) It is now given that a is perpendicular to b. Given that AP is the projection vector of onto AO AB , show that ( ) ,AP =− ba where  is a constant to be determined in terms of a and .b [3]

6 3 Do not use a graphing calculator when answering this question. (a) The equation 32 8 23 0z z z k− + + = , where k is a real constant, has a root 2 3i.+ (i) Find the value of k and solve the equation. [5] (ii) Hence, solve the equation 32 8i 23 i 0z z z k+ − + = [2]

7 [Turn Over (b) The complex number w is given by cos isinw =+ , where 0 2  . Find, in terms of , the modulus and argument of 2 * 1 w w + , where w* is the complex conjugate of w. [4]

8 4 (a) (i) Find ( ) cosd e.d x x [1] (ii) Hence, show that ( )cos cossin 2 e d 2e 1 cos ,xxx x x C = − + where C is an arbitrary constant. [3] (iii) Find the exact value of π cos 0 sin 2 e d .xxx [3]

9 [Turn Over (b) Using the substitution sec ,x = where π0, 2 find 2 1 d . 1 x x − [5]

10 Section B: Probability and Statistics [60 marks] 5 For events A and B, it is given that ( ) 1P 8A = , ( ) 9P 16AB= and ( ) 1P' 16AB= . (i) Find ( )P B . [1] (ii) Determine, with a reason, whether the events A and B are independent. [2] For a third event C, it is given that ( ) 3P 8C = , ( ) 1P 64A B C  = and C is independent of A. (iii) Find ( )P' AC . [1]