2024 RI H2 Math Prelim P1 (Qn)

This document consists of 21 printed pages and 3 blank pages. RAFFLES INSTITUTION  RI2024 Mathematics Department [Turn over CANDIDATE NAME CLASS 24 MATHEMATICS 9758/01 Paper 1 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 Q1 Q2 Q3 Q4 Q5 Q6 / 4 / 5 / 8 / 8 / 8 / 7 Q7 Q8 Q9 Q10 Q11 TOTAL / 10 / 12 / 12 / 14 / 12 / 100 RAFFLES INSTITUTION 2024 YEAR 6 PRELIMINARY EXAMINATION CHairperson

2 H2 MA 9758/2024 RI Year 6 Preliminary Examination Paper 1 1 A function f is defined by 32f( ) .x ax bx cx d= + ++ The graph of f( )yx= passes through the points ( 3,4)− and (1, 8). Given that the graph of 1 f( )y x= has a turning point at 1 4(2, ), find the values of , , and .abc d [4] CHairperson

3 H2 MA 9758/2024 RI Year 6 Preliminary Examination Paper 1 [Turn over 2 [The volume of a sphere with radius r is given by 34 3 rπ and the surface area of a sphere with radius r is given by 24 rπ .] (a) The volume of an expanding sphere is increasing at a constant rate of 315 cm s .− Show that, at any instant, the rate of increase of the surface area is 21cm s ,k r − where r is the radius of the sphere and k is a constant to be determined. [3] (b) Find the exact rate of change of surface area of the expanding sphere when the surface area is 220 cm . [2] CHairperson

4 H2 MA 9758/2024 RI Year 6 Preliminary Examination Paper 1 3 (a) Without using a calculator, solve exactly 2 1 1.1 x x x −− ≤+ [4] (b) Hence solve exactly 2 1 11 xx x − +≤− . [4] CHairperson

5 H2 MA 9758/2024 RI Year 6 Preliminary Examination Paper 1 [Turn over 4 (a) Find cos d.cos3 cos x xxx+ ⌠⌡ [3] (b) Find 12tan dx xx − ∫ . Hence find the exact value of 1 12 1 tan d .x xx − −∫ [5] CHairperson

6 H2 MA 9758/2024 RI Year 6 Preliminary Examination Paper 1 5 (a) Using the formulae for , prove that ( ) ( )sin 2 1 sin 2 1 2cos 2 sinrr r θ θ θθ+− −= . [1] (b) Hence find a formula for 1 cos 2 , where 0 , n r rθ θπ = <<∑ in terms of ( )sin 2 1n θ+ and sinθ. [3] ( )sin AB± CHairperson

7 H2 MA 9758/2024 RI Year 6 Preliminary Examination Paper 1 [Turn over (c) Using the formula found in part (b), show that the sum of the series 2 22 2sin 10 sin 11 sin 12 ... sin 20 , for 0θ θ θ θ θπ+ + ++ << is sin(41 ) sin(19 ) 4sink θθ θ −− , where k is a constant to be determined. [4] CHairperson

8 H2 MA 9758/2024 RI Year 6 Preliminary Examination Paper 1 6 (a) The diagram below shows the graph of f( )yx= . The graph cuts the x-axis at point A (5, 0). It has a turning point at (12, )Bk , where 0k < and asymptotes with equations 0x= and 0y= . On separate diagrams, sketch the graph of (i) 2f ( )y xk= + , stating the equations of any asymptotes and the coordinates of any turning point(s). [2] O y x x y CHairperson

9 H2 MA 9758/2024 RI Year 6 Preliminary Examination Paper 1 [Turn over (ii) f( )yx ′= , stating the equations of any asymptotes and the coordinates of any point(s) where the curve crosses the axes. [2] (b) The graph with equation g( )yx= , where 2g( ) ( 1)x xx= − undergoes a single transformation and the equation of the resultant graph is h( )yx= . Describe the transformation if (i) 2h ( ) ( 1) ,x xx= −+ [1] (ii) 21h( ) ( 2) .8x xx= − [2] x y CHairperson

10 H2 MA 9758/2024 RI Year 6 Preliminary Examination Paper 1 7 An arithmetic series has first term a and common difference d, where 0a> and 0d ≠ . The first, sixth and ninth terms of the arithmetic series are consecutive terms of a geometric series. (a) Show that 25 2da=− . [2] (b) The sum of the first n terms of the arithmetic series is denoted by S. Find the set of possible values of n for which S exceeds 6a. [3] CHairperson