2019 Y6 RI H2 Math Prelim (with ans)

¤ RI2019 [Turn over CANDIDATE NAME CLASS 19 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 figur es, 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 unle ss a question specifically states otherwise. Where unsupported answers from a graphing calculator are not all owed in a question, you are required to present the mathematical steps using mathematical n otations 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 Q8 Q9 Q10 Q11 Total This document consists of 6 printed pages. RAFFLES INSTITUTION Mathematics Department RAFFLES INSTITUTION 2019 YEAR 6 PRELIMINARY EXAMINATION www.KiasuExamPaper.com 578

2 H2 MA 9758/2019 RI Year 6 Preliminary Examination Paper 1 1 A curve C has equation 3 ayb x cx  , where a, b and c are constants. It is given that C has a stationary point  1.2, 6.6 and it also passes through the point 2.1, 4.5 . (i) Find the values of , and ab c , giving your answers correct to 1 decimal place. [4] (ii) One asymptote of C is the line with equation x = 0. Write down the equation of the other asymptote of C. [1] 2 Two variables u and v are connected by the equation 11 1 20uv . Given that u and v both vary with time t, find an equation connecting dd , , and .dd uv uvtt Given also that u is decreasing at a rate of 2 units per second, calculate the rate of increase of v when u = 60 units. r [4] 3 Fig. 1 Fig. 2 Fig. 1 shows a square sheet of metal with side a cm. A square x cm by x cm is cut from each corner. The sides are then bent upwards to form an open box as shown in Fig. 2. Use differentiation to find, in terms of a, the maximum volume of the box, proving that it is a maximum. [6] a x www.KiasuExamPaper.com 579

3 H2 MA 9758/2019 RI Year 6 Preliminary Examination Paper 1 4 A curve C has parametric equations   22 1, 2 1.xt y t   (i) Find the coordinates of the point A where the tangent to C is parallel to the x-axis. [4] (ii) The line yx d  intersects C at the point A and another point B. Find the exact coordinates of B. [4] (iii) Find the area of the triangle formed by A, B and the origin. [1] 5 The diagram shows the graph of Folium of Descartes with cartesian equation 33 3,xy a x y where a is a positive constant. The curve passes through the origin, a nd has an oblique asymptote with equation yx a  . (i) Given that (0, 0) is a stationary point on the curve, find, in terms of a, the coordinates of the other stationary point. [5] (ii) Sketch the graph of 3 3 3,xy a x y including the equations of any as ymptotes, coordinates of the stationary points and the point where the graph crosses the x-axis. [3] (iii) yx a  O www.KiasuExamPaper.com 580

4 H2 MA 9758/2019 RI Year 6 Preliminary Examination Paper 1 6 (a) Given that   2 1 1 12 16 n r rn n n    , find an expression for  2 1 1 n rn rr   , simplifying your answer. [3] (b) (i) Use the method of differences to find  1 +1 e e n r r rr      . [3] (ii) Hence find 1 2 ee er n r rr       . [2] 7( a ) The complex numbers 32 i , z, 46 i are the first three terms in a geometric progression. Without using a calculator, find the two possible values of z.[ 4] (b) (i) The complex number w is such that iwa b , where a and b are non-zero real numbers. The complex conjugate of w is denoted by *w . Given that 2 * w w is a real number, find the possible values of w in terms of a only. [4] (ii) Hence, find the exact possible arguments of w if a is positive. [2] 8( a ) Find the exact value of m such that 1 3 2 2 220 0 11 d d9 1 m xxx mx       . [5] (b) (i) Use the substitution sin 2ux to show that  14 33 3 5 00 1sin 2 cos 2 d d 2xx x u u u     . [4] (ii) Hence find the exact value of 4 33 0 sin 2 cos 2 dxx x     . [2] www.KiasuExamPaper.com 581

5 H2 MA 9758/2019 RI Year 6 Preliminary Examination Paper 1 9( i) Write down 22 1 dvav     , where a is a positive constant. [1] (ii) In the motion of an object through a certain medium, the medium furnishes a resisting force proportional to the square o f the velocity of the moving object. Suppose that a body falls vertically through the medium, the model used to descr ibe the velocity, v ms–1 of the body at time t seconds after release from rest is given by the differential equation 2 22d ,10 d av avt  where a is a positive constant. (a) Show that 20 20 e1 e1 t a t a va      . [8] (b) The rate of change of the displacement, x metres, of the body from the point of release is the velocity of the body. Given that 2a  , find the value of x when t = 1, giving your answer correct to 3 decimal places. [3] 10 The curve 1C has equation 22 25, 0xy y  . The curve 2C has equation 243 3yx x   . (i) Verify that  3, 4 lies on both 1C and 2C . [1] (ii) Sketch 1C and 2C on the same diagram, stating the coordinates of any stationary points, points of intersection with the axes and the equations of any asymptotes. [4] The region bounded by 1C and 2C is R. (iii) Find the exact volume of solid obtained when R is rotated through 2 radians about the x-axis. [6] The region bounded by 1C , the x-axis and the vertical asymptote of 2C , where 3x  ,i s S. (iv) Write down the equation of the curve obtained when 1C is translated by 3 units in the negative x-direction. Hence, or otherwise, find the volume of solid obtained when S is rotated through 2 radians about the vertical asymptote of 2C . [4] www.KiasuExamPaper.com 582

6 H2 MA 9758/2019 RI Year 6 Preliminary Examination Paper 1 11 Path integration is a predominant mode of navigation strategy used by many animals to return home by the shortest possible route during a food fora ging journey. In path integrat ion, animals continuously compute a homebound global vector relative to their starting position by integrating the angles steered and distances travelled during t he entire foraging run. Once a food item has been found, the anim al commences its homing run b y using the homebound global vector, which was acquired during the outbound run. (a) A Honeybee’s hive is located at the origin O. The Honeybee travels 6 units in the direction 22 ij k before moving 15 units in the direction 34ik . The Honeybee is now at point A. (i) Show that the homebound global vector AO  is 741 6 ij k . Hence find the exact distance the Honeybee is from its hive. [3] (ii) Explain why path integration may fail. [1] A row of flowers is planted along the line 3 2, 25 x yz   . (iii) The Honeybee will take the shortest distance from point A to the row of flowers. Find the position vector of the point along the row of flowers which the Honeybee will fly to. [4] (b) To further improve their chances of returning home, apart from relying on the path integration technique, a