RI H2 2020 Prelim P1

 RI2020 [Turn over CANDIDATE NAME CLASS 20 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 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. The use of an approved graphing calculator is expected, where appropriate. Unsupported answers from a graphing calculator are allowed unless a quest ion 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. FOR EXAMINER’S USE Q1 Q2 Q3 Q4 Q5 Q6 Q7 /4 /4 /5 /6 /7 /9 /10 Q8 Q9 Q10 Q11 Q12 Total /10 /10 /11 /12 /12 /100 This document consists of 28 printed pages and 0 blank page. RAFFLES INSTITUTION Mathematics Department RAFFLES INSTITUTION 2020 YEAR 6 PRELIMINARY EXAMINATION

2 H2 MA 9758/2020 RI Year 6 Preliminary Examination Paper 1 1 A particle is moving along the curve with equation 2216 9 144xy . 1d 2cm sd y t  and d d x t is positive when 5x . Find the rate of increase of x at this instant. [4] 2 Mr. Li invested a total of $30000 and divided this sum into three accounts, which paid 2%, 3% and 5% annual interest respectively. At the end of the first year, Mr . Li withdrew all the money out from the 2% and 5% accounts and gave the interest earned to his son. The amount in the 3% account, including interest, was re-invested in the same account for another year. At the end of the second year, Mr . Li withdrew all the money out of the 3% account and gave the interest earned to his son. The total interest received by the son from the three accounts was $1423.50. Given that the amount invested in the 2% account was $1000 more than the amount invested in the 5% account, find the amounts invested in each of the three accounts. [4] 3 (i) Prove that for 0x , the substitution y ux reduces the differential equation 22d( ) 2 d yyy x y x xx     to 22 1d 12 2 d uu u u x         . [2] (ii) Hence find the general solution of the differential equation 22d( ) 2 d yyy x y x xx     for 0x . [3] 4 (i) Sketch the curve with equation 22 6 7.x y x   [2] (ii) The region R is bounded by the curve 22 67x y x   , for 3x , and the line 3.x Given that 2 22 0 d 4 a aa x x  , find the exact volume of the solid of revolution formed when R is rotated completely about the y-axis. [4]

3 H2 MA 9758/2020 RI Year 6 Preliminary Examination Paper 1 5 (a) Find 2cos d .x x x [1] (b) Use integration by parts to find cos 2 d .x x x [3] Hence or otherwise find 22 4 cos d .x x x   [3] 6 The diagram below shows the curve of  fyx . The curve has a minimum point at  2, 2 , a maximum point at  2, 3 and cuts the y-axis at  0,3 . The lines 1x , 4x and 3y are the asymptotes to the curve. On separate diagrams, draw sketches of the following graphs, stating the exact coordinates of any turning points and/or points of intersection with the axes, and the equations of any asymptotes, where possible. (a) f (1 )yx . [3] (b)   1 fy x . [3] (c) f '( )yx . [3] x

4 H2 MA 9758/2020 RI Year 6 Preliminary Examination Paper 1 7 It is given that   2f. 2 r r r  Show that         3 8 2f 2 f 2 rrrr rr     . [2] (i) Show that     1 3 3 8 2 (3 2)2 162 ( 1) r nn r r n r r n n     . [4] (ii) Hence find    1 3 2 2 2 rn r r rr   in the form 1(3 )2 ( )( 1) nnA Cn B n   where A, B and C are integers to be determined. [4] 8 Do not use a calculator in answering this question. (a) (i) Solve the equation 2 4i 3.z  [3] (ii) Solve the equation 42 6 25 0.zz   [3] (b) Find the modulus and argument of the complex number 8 2i 5 3iw   . Hence find the possible values of the positive integer n for which nw is real. [4] 9 A curve C has parametric equations 4ln , e , e t tx t t y   for 1 2t . (i) C meets the y-axis at point P and line L is the normal to C at P. Show that the equation of L is 2 2 e 4 e .4 e eyx   [4] (ii) Sketch the curve C, stating the coordinates of any turning points and points of intersection with the axes. [3] (iii) The finite region bounded by C, L and the line 11ln22x   is denoted by R. Find the area of R. [3]

5 H2 MA 9758/2020 RI Year 6 Preliminary Examination Paper 1 10 The plane 1 contains the point (1,2, 1)A  and the line l with equation 12 ,123 xz y  . The plane 2 contains the point ( 5.5,3,2)B  and meets 1 in the line l. (i) Find the equation of 1 in scalar product form. [3] (ii) Show that the vector BF   is 4.5 4 3   , where F is the foot of perpendicular from B to l. [3] (iii) Find the exact value of the shortest distance from B to 1 . [2] (iv) Hence or otherwise find the acute angle between 1 and 2 , giving your answer to the nearest 0.1 . [3] 11 Figure 1 shows a n open container in the form of a trapezoidal prism ABCDEFGH with square base ABCD and cm,AB AE BF EH a    where a is a constant. The container is made of plastic of negligible thickness and is placed on a horizontal surface . The faces BCGF and ADHE are inclined at an angle  radians, 0 2  , to the horizontal surface, and faces ABFE and DCGH are perpendicular to base ABCD. Figure 2 shows its cross - sectional view. (i) Show that the volume V cm3 of the container is given by 3 sin (1 cos )Va  . [2] (ii) Use differentiation to find, in terms of a, the maximum value of V in exact form, proving that it is a maximum. [5] Trapezoidal Prism a a Height of the container a a a Cross-Sectional View a a Figure 1 E B F A D C G H Horizontal surface Figure 2

6 H2 MA 9758/2020 RI Year 6 Preliminary Examination Paper 1 (iii) A particular container is constructed with 3   and it is filled with water to half its height. Find, in terms of a, the exact volume of water in this container. [3] The container is then tilted in the direction of the face BCGF until face BCGF and base ABCD makes the same angle with the horizontal surface. Figure 3 shows its cross-sectional view. Explain if it is possible to tilt the container to this position without any water flowing out from the container. [2] 12 The von Bertalanffy growth model, introduced in 1938, is widely used in fisheries studies. It is used to predict the length, L mm of a fish over a period of time, t years.