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RAFFLES INSTITUTION 2012 Year 6 Preliminary Examination MATHEMATICS 9740/01 Paper 1 12 September 2012 3 hours Additional materials: Answer Paper List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your name and CT group on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. 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 a graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic 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. At the end of the test, fasten all your work securely together. This document consists of 5 printed pages. RAFFLES INSTITUTION  RI 2012 Math Department [Turn over

H2 MA 9740/ 2012 RI Year 6 Preliminary Examination [Turn Over 2 1 A study conducted by the Department of Health classified a population of size 21000 into three categories: underweight, normal and overweight. At the start of the study, it was found that x, y and z (in thousands) were the number of individuals in the population who were underweight, normal and overweight respect ively. At that time, the number of individuals who were overweight was three times the number of individuals who were underweight. Two years later, the Department found that 10% of the individuals who were underweight and 20% of the individuals who were overweight became normal but 5% of the individuals who were normal became overweight. The weight categories of the rest of the indivi duals remain unchanged. However, the number of individuals who were overweight was still three times the number of individuals who were underweight. Assuming that the population remains unchanged during the period of study, find the values of x, y and z. [3] 2 Expand 22 1 (1 2 )x as a series in ascending powers of x, up to and including the term in 6,x giving the coefficients in their simplest form. [2] Find the coefficient of 2rx and the range of values of x for the expansion to be valid. [3] 3 A graphic calculator is not to be used in answering this question. By writing i where , ,w a b a b   R solve the equation 2 5 12i.w  Hence find the roots of the equation  2 1 3i 0zz    . [6] 4 Find how many positive integers between 811 and 2013 are (i) even numbers; (ii) even numbers which are divisible by 7. Find the sum of the even numbers between 811 and 2013 which are not divisible by 7. [6] 5(a) Find 2 1 14ln(2e ) dx x  . [2] (b) Using the substitution ,ux find the exact value of 4 1 d1 12 x xx . [5]

H2 MA 9740/ 2012 RI Year 6 Preliminary Examination [Turn Over 3 6 In the diagram above, the curve f ( )yx cuts the x axis at 1 2x and 4x , has turning points at ( 2, 4) and (2, 2) , and has a horizontal asymptote 1y . Sketch, on separate diagrams, the graphs of (i) f (1 )yx , [3] (ii) 1 f ( )y x , [3] stating the equations of any asymptotes and the coordinates of any turning points and points of intersection with the axes. 7(a) The position vectors of the points A and B relative to an origin O are a and b respectively where a and b are non-zero, non-parallel vectors. The points P and Q have position vectors a  b and 3a 3b respectively. Find, in terms of a and b, the position vector of the point R on PQ such that 3PR RQ . What can you say about the points O, A and R? [3] (b) The planes 12 and pp , which meet in the line l, have equations 2 2 0 and 2 2 0x y z x y z      respectively. (i) Find an equation of l in Cartesian form. [2] The plane 3p has equation    2 2 2 2x y z c x y z d      . (ii) Given that 0d  , show that all 3 planes meet in the line l for any constant .c [2] (iii) Given instead that the 3 planes have no point in common, what can be said about the value of ?d [1] O x y (2, 2 ) ( 2 , 4) y = 1 1 2 4 f ( )yx

H2 MA 9740/ 2012 RI Year 6 Preliminary Examination [Turn Over 4 8 The curve C has equation 1tan ( )yx  and the line l has equation 22 4yx    . The region R is bounded by the curve C, the line l and the x-axis. (i) Verify that the curve C and the line l meet at the point where 1x and find the exact area of the region R. [5] (ii) Write down the equations of the curve C and the line l when each of them is translated by 8  units in the negative x-direction. [2] Hence find the volume of solid formed when R is rotated completely about the line ,8x  giving your answer correct to 2 decimal places. [2] 9 By using the substitution 2e xzy , find the gener al solution of the differential equation 2d 2 ( 1)ed xy yxx    , expressing your answer in the form f( ).yx [4] It is given that 1 when 0yx . (i) Find the particular solution. [1] (ii) By repeated differentiation of the given differential equation, find the Maclaurin expansion for y up to and including the term in 3.x [4] (iii) Without carrying out the calculation, d escribe briefly how you would use the answer in (i), to check the correctness of your answer in (ii). [2] 10(a) A paper drinking cup in the shape of a cone (as shown in the diagram below) is to hold 120 3cm of water. Use differentiation to find the height cmh and radius cmr of the cup that will require the least amount of paper. [7] [You do not need to verify that the amount of paper required is the least.] [Volume of cone 21 3V r h  ; Curved surface area of cone S rl ] (b) The curve C has parametric equations 5cos , 3sin ,0 2 .x t y t t     The normal to C at the point P (5cos , 3sin )tt is denoted by l. (i) Find an equation of l. [3] The normal l meets the x and y axes at the points A and B respectively, and M is the mid-point of AB. (ii) Find the Cartesian equation of the locus of M as P varies. Sketch this locus. [4] r h l

H2 MA 9740/ 2012 RI Year 6 Preliminary Examination [Turn Over 5 11(a) It is given that 1x for 1 1 ,x   h( )x  2 for 1 3,x 5x for 3 5.x and that h( ) h( 6)xx for all real values of x. (i) Sketch the graph of h( )yx for 1 11x   . [2] (ii) It is given that 0 63h( ) d 2 a xx  . Determine the value of a. [2] (b) Functions f and g are defined by   2 f : 2 3x a x   , x where a is a positive constant; g: 5 xx x  , , 5.xx (i) Sketch the graph of f ( )yx , indicating the coor dinates of the stationary point and intersections with the axes if any. [2] (ii) The function k, a restriction of the function f is defined by   2 k : 2 3, , 7.x a x x x     R Given that the composite function gk exists, find the range of values for a. [3] (iii) Given that 1 2a , s