2020 RI Y5 H2 Math Prelims

BP-462 RAFFLES INSTITUTION 2O2O YEAR 6 PRELIMINARY EXAMINATION CANDIDATE NAME CLASS ')i MATHEMAT]CS PAPER 1 Candidates answer on the Question Paper Additional Materials: List of Formulae (MF26) 97s8/01 3 hours READ THESE INSTRUCTIONS FIRST Write your name and class on all the wo* 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 conection fluid. Answer all the questions. Write your answers in the spaces provided in the question paper. Give non-exact numerical answers conect 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 ftom 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 lo 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 is 100. FOR EXAMINER'S USE l4 l4 ls l6 l7 ls lto Total lLo Ito ILL It2 It? lLOo This document consists of 28 printed pages and 0 blank page. RAFFLES INSTITUTION Mathematics Dgpartment PartnerlnLeaming 624 o Rt2020 [Tum over ax q2 03 Q4 Q5 Q5 Q7 QlZQ8 q9 Ql0 Q11

BP-463 2 I A particle is moving along the curve with equation 16xz +9yz =144. 9=Z" s-' -d I is positive when x=.r6.&& Find the rate of increase ofx at this instant. l4l 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 2Yo and 5%o accounts and gave the i.nterest eamed to his son. The amount in the 3% account, including interest, was re-invesled in the same account for another year. At the end ofthe second year, Mr. Li withdrew all the money out ofthe 3% account and gave t}re interest eamed to his son. The total interest received by the son from the tkee accounts was $1423.50 Given that the amount invested in the 2o/o ascovnt was $1000 more than the amount invested in the 5%o accotnt, find the amounts invested in each ofthe three accounts. t4l (ii) Hence find the general solution ofthe differential equation (dv v\ Id,-;.,] t3l(v-") =y2+2xzforx>0 (D Sketch the curve with equation x2 + yz - 6x =7 . l2l (ii) The region R is bouoded by the curve x'+y'- 6x=7,for x>3,andthe line x=3. Given that J" a"-x'dx= 2 'o , frod the exact volume of the solid of revolution4 formed when R is rotated completely about the]r-axis. H2 MA 9758/2020 Rl Year 6 Preliminary B€mination Paper 1 ParinerlnLeaming 625 3 (i) Prove that for x>0, the substitution y=r.s reduces the differential equation . ( d, ,\U-r)lY-Ll=y2 +2x2 to \cx ''l t'.r--.r-)[gl)=,. tz]\u2 +2 u2 +z)lax ) 4 t4) 2

BP-464 3 5 (a) Find J x cos.r' dx. tll i t3lO) Use integration by pa(s to find rcos 2x dx. Hence or otherwise find t3l The diagram below shows the curve of y = f(x) . 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 x=l , x--4 and y = 3 are the asymptotes to the curve. y = f(x) l ' -rcos'x dx- 6 v Y=l (-2,2) (2,-3) 4 On separate diagrams, draw sketches ofthe following graphs, stating the exact coordinates ofany turning points and/or points of intersection with the axes, and the equations of any asymptotes, where possible. (a) y --f (1-x) t3l 0,r) I x x1JT (b) 1 ' f(,) v=f'(x). t3l H2 MA 9758/2020 Rl Yoar 6 Preliminary Exam,nation Paper 1 PartnerlnLeaming 626 (c) t3l

BP-465 4 It is given that f {r) = !.Show that f (r+z) -f (r) = r(r-2)' (i) Showthat ) 3r -8 z'. 3r-8 2'. 7 12) 14) t4l /=3 r-2 n(n -1)( ) (ii) Hence hnd ir=l 3r -z)2' r(r +2) in the form lSnl!:)z'-': +c wherc A,B and c are (n+ B\(n+1) 8 integers to be determined. t4l Do not use a calculator in answering this question. (a) (r) Solve the equation z2:4i-3. t3l (ii) Solve the equation za +622 +25--O. i3l (b) Find the modulus and argument of the complex numb ", * =9:4. Hence find the 5+3i possible values ofthe positive integer z for which y' is real. t4I 9 A curve C has parametric equations x = tlnt, (iii) 4 Y=-+e',e for t>1. 2 (i) C meets the y-axis at point P and line Z is the normal to C at P. Show that the equation ofZ is e 4+e2 ' 4-e' e (ii) Skerch the curve C stating the coordinates of any turning points and points of intenection with the axes. t3l The finite region bounded by C, Z and the li". , = +hfl) is denored by.R. Find' 2\2) the area ofR. t3l H2 MA 9758/2020 Rl Year 6 Preliminary Examination Paper 1 PartnerlnLearning 627

BP-466 5 10 The plane z, contains the point A(7,2,-l) and the line -11- = ==, y = -1 . The plurc x, contains the pointB(-5.5,3,2) 2 3 '' line /. i with equation and meets n, in the t3l 0 (i) Find the equation of r, in scalar product form. tll 11 (ii) Show that the vector 8I- is , where F is the foot ofperpendicular from B to /. t3t (iii) Find the exact vaiue offhe shortest distance from B to 2,. 12) (iv) Hence or otherwise find the acute angle between r., atd r, giving your answer to the nearest 0.1". t3l Figure 1 shows an open container in the form of a trapezoidal pism ABCDEFGH with square base IBCD alo'd, AB = AE -- BF -- EH = a crn, 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 d radians, O < e . + ,to the borizontal surface, 2 ard faces ABFE ard DCGH are perpendicular to base ABCD. Figure 2 shows its cross- sectional view. Figure I H Figure 2 a d E a Ca e AaB Trapezoidal Prism Height of the container Horizontal surface Cross-Sectional View (i) Show that the volume / cm3 of the container is given by V = a3 sin0(7 + cos 0) .f2) H2 MA 9758/2020 Rl Year 6 Preliminary bGmination Paper 1 PartnerlnLeaming 628 (ii) Use differentiation to find, in terms of a, the maximum value of I/ in exact form, proving that it is a maximum. t5l

BP -467 6 (iii) A particular container is constructed with d = 1 and it is filled with water to half its J height. Find, in terms of a, the exact volume of water in this container. t3] t2 The container is then tilted in the direction of the face BCGF u:ntil face BCGF ar,d' base ABCD makes the same angle with the horizontal surface, Figure 3 shows its cross-sectional view. Figure 3 Base ABCD Fzce BCGF d Horizontal surface Cross-Sectional View Explain if it is possible to tilt the container to this position without any water flowing out from the container. l2l The von Bertalanffu growth model, introduced in 1938, is widely used in fisheries studies. It is used to predict the length, I mm of a fish over a period of time, I years. If I. is the maximum length for a species, then the model assumes that the rate of gro*'th in length of a fish is proportio nal to L--L. (D By setting up and solving a differential equation, show that the general solution of this differeutial equation is given by L= L*- Ae-b , where /r is the constant of proportionality and I is a positive constant t5l For the species of fish known as the Atlantic croaker, it has been deterrnined that tr- = 419mm and at one year of age, its length is 219 mm and the rate of growth in length is 55 mm per year. Using the above model, obtain an expression forZ in terms of/- t3] (ii) (iii) Find its age when the Atlantic croaker grows to a length of 300 mm. Sketch a graph ofl, against t. H2 MA 9758/2020 Rl Year 6 Preliminary E)Gmination Paper 't PartnerlnLeaming 629 t21 t2)

BP-468 RAFFLES INSTITUTION 2O2O YEAR 6 PRELIMINARY EXAMINATION 20CLASS MATHEMATICS PAPER 2 Candidates answer on the Question Paper Additional Materials: List of Formulae (MF26) 9758102 3 hours READ THESE INSTRUCTIONS FIRS