2024 RI H2 Math Prelim P2 (Qn)

This document consists of 21 printed pages and 3 blank pages. RAFFLES INSTITUTION  RI2024 Mathematics Department [Turn over CANDIDATE NAME CLASS 24 MATHEMATICS 9758/02 Paper 2 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name and class 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 P aper. You may use the blank page s on page 22, 23 and 24 if necessary and you are reminded to indicate the question number(s) clearly. 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. For Examiner’s Use Only TOTAL Section A: Pure Math Q1 Q2 Q3 Q4 Q5 / 8 / 5 / 7 / 10 / 10 _____ 100 Section B: Prob & Stats Q6 Q7 Q8 Q9 Q10 Q11 / 7 / 8 / 11 / 10 / 12 / 12 RAFFLES INSTITUTION 2024 YEAR 6 PRELIMINARY EXAMINATION CHairperson

2 H2 MA 9758/2024 RI Year 6 Preliminary Examination Paper 2 Section A: Pure Mathematics [40 marks] 1 The function f is defined by 2f: 2 xx x− , for x∈  , 2x≠ . (a) Sketch the graph of f and find its range. [3] Another function g is defined by g: 3 2xx ++ , for x∈  . (b) Show that the composite function fg exists. Find fg( )x and state the domain and range of fg. [5] y x CHairperson

3 H2 MA 9758/2024 RI Year 6 Preliminary Examination Paper 2 [Turn over 2 The function f is defined by 432f ( ) 45,z z Az Bz Cz=+ + ++ where A, B and C are real numbers. Given that 2i+ is a root of f( ) 0z = and 2()zk− is a factor of f( ),z where k is a positive real number, find the values of A, B, C and k . [5] CHairperson

4 H2 MA 9758/2024 RI Year 6 Preliminary Examination Paper 2 3 (a) The points A , B and C on the plane π have position vectors a , b and c respectively. Show that a vector perpendicular to π is parallel to ×+×+×bccaab . [3] (b) p and q are non-zero vectors such that ( ).=p pqq . (i) Find the relationship between p and q . [1] CHairperson

5 H2 MA 9758/2024 RI Year 6 Preliminary Examination Paper 2 [Turn over (ii) Find q . [1] (c) u is the position vector of a fixed point U relative to the origin O. A variable point V has position vector v relative to O. Given that ( ) 0. −=vvu , describe geometrically the set of all possible positions of the point V. [2] CHairperson

6 H2 MA 9758/2024 RI Year 6 Preliminary Examination Paper 2 4 (a) Given that 12e xy += , show that ( ) 2 2d12 d yxy x +=  and 2 2 dd(1 2 ) dd yyxy xx+ += . [3] (b) Hence, or otherwise, obtain the series expansion for y in terms of x up to and including the term in 3x . [3] CHairperson

7 H2 MA 9758/2024 RI Year 6 Preliminary Examination Paper 2 [Turn over (c) Verify that the same series expansion for y in part (b) is obtained if the standard series expansions for ex and (1 ) nx+ are used. [4] CHairperson

8 H2 MA 9758/2024 RI Year 6 Preliminary Examination Paper 2 5 The diagram shows the curve C with parametric equations ( ) 2 1xt= + , ( ) 2 3yt= − . The curve C meets the axes at ( )16, 0 and ( )0, 16 . (a) Show that the line 16x= meets C at the point P where 5t =− . [1] The normal to C at P is denoted by l . (b) Find the cartesian equation of l . [3] x y 16 16 O CHairperson

9 H2 MA 9758/2024 RI Year 6 Preliminary Examination Paper 2 [Turn over (c) The line l meets C again at the point Q where xb= . Show that the area of the region bounded by l , the lines 16x= , xb= and the x -axis i s 2266240 units .81 [3] (d) Show that the area 2(in units ) of the region bounded by C and l can be given by ( )266240 fd81 d c tt+ ∫ , where ( )f t , and the constants c and d are to be determined. Hence find the value of this area. [3] CHairperson

10 H2 MA 9758/2024 RI Year 6 Preliminary Examination Paper 2 Section B: Probability and Statistics [60 marks] 6 Eleven cards each bears a single letter and together they can be made to spell the word COFFEEHOUSE. The 11 cards are arranged in a row. (a) Find the number of different arrangements that can be made. [1] (b) Find the number of different arrangements in which the 2 F’s are next to each other and no E’s are next to each other. [3] CHairperson