2019 J2 NYJC H2 Math Prelim (with ans)

This document consists of 6 printed pages. NANYANG JUNIOR COLLEGE Internal Examinations [Turn Over NANYANG JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 Candidate Name CT Class 1 8 Centre Number/ Index Number / MATHEMATICS 9758/01 Paper 1 2nd September 2019 3 Hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS 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, highlighters, 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 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. The total number of marks for this paper is 100. For examiner’s use only Question number Mark 1 2 3 4 5 6 7 8 9 10 11 12 Total www.KiasuExamPaper.com 529

2 NYJC 2019 JC2 Preliminary Examination 9758/01 1 The curve with equation 22 9yx  is transformed by a stretch with scale factor 2 parallel to th e x-axis, followed by a translation of 4 units in the negative x-direction, followed by a translation of 1 2 units in the positive y-direction. Find the equation of the new curve and state the equations of a ny asymptote(s). Sketch the new curve, indicating the coordinates of any turning points. [6] 2 The diagram shows a mechanism for converting rotational motion into linear motion. The point P, on the circumference of a disc of radius r, rotates about a fixed point O. The point Q moves along the line OX, and P and Q are connected by a rod of fixed length 3 r. As the disc rotates, the point Q is made to slide backwards and forwards along OX. At time t, angle POQ is  , measured anticlockwise from OX, and the distance OQ is x. (i) Show that   2cos 9 sinxr   .[ 2] (ii) State the maximum value of x.[ 1] (iii) Express x as a polynomial in  if  is sufficiently small for 3 and higher powers of  are to be neglected. [3] 3 Without using a calculato r, solve the inequality 2 34 212 xx xx   . Hence solve the inequality 2 34 212 xx x x aa aa   where 2.a  [6] 4 (i) Using double angle formula, prove that  4 1 34 c o s 2 o 4si c s 8n    . [2] (ii) By using the substitution 2cosx  , find the exact value of  3 2 2 2 0 4d xx .[ 4] www.KiasuExamPaper.com 530

3 NYJC 2019 JC2 Preliminary Examination 9758/01 [Turn Over 5 Relative to the origin O, the points A, B, and C, have non-zero position vectors a , b , and 3a respectively. D lies on AB such that AD AB , where 01  . (i) Write down a vector equation of the line OD.[ 1] (ii) The point E is the midpoint of BC. Find the value of  if E lies on the line OD. Show that the area of BED' is given by k ab , where k is a constant to be determined. [5] 6 The function f is given by 2f: 2 4xxx k 6 for 5 xa  , where a and k are constants and k > 2. (i) State the largest value of a for the inverse of f to exist. [1] For the value of a found in (i), (ii) find 1f( )x and the domain of 1f  , leaving your answer in terms of k,[ 3] (iii) on the same diagram, sketch the graphs of  1ffyx  and  1ffyx  , labelling your graphs clearly. Determine the number of solutions to   11ff f fxx  .[ 4] 7 A spherical tank with negligible thickness and internal radius a cm contains water. At time t s, the water surface is at a height x cm above the lowest point of the tank and the volume of water in the tank, V cm3, is given by  21 33Vx a x . Water flows from the tank, through an outlet at its lowest point, at a rate kx cm3 s-1 , where k is a positive constant. (i) Show that  2 d2 d xax x k x t  .[ 2] (ii) Find the general solution for t in terms of x , a and k .[ 3] (iii) Find the ratio 12:TT , where 1T is the time taken to empty the tank when initially it is compl etely full, and 2T is the time taken to empty the tank when initially it is half full. [4] 8 A curve C has equation 2 4yx y , where 0y  . (i) Without using a calculator, find the coordinates of the point on C at which the gradient is 1 5 .[4] (ii) Variables z and y are related by the equation 22 10yz y , where 0z  . Given that x increases at a constant rate of 0.5 unit/s, find the rate of change of z when 3x  .[ 5] www.KiasuExamPaper.com 531

4 NYJC 2019 JC2 Preliminary Examination 9758/01 9( a) The complex numbers z and w satisfy the simultaneous equations * 32 izw   and * 5 1 20iwwz   , where *w is the complex conjugate of w . Find the value ofz and the corresponding value ofw . [4] (b) It is given that 8i is a root of the equation  32i 8 2i 40 0zz az    where a i s a c o m p l e x number. (i) Find a.[ 2] (ii) Hence, find the other roots of the equation, leaving your answe r in the form iab where a and b are real constants. [3] (iii) Deduce the number of real roots the equation  32 8 2i i 40 0zz a z    has. [1] 10 For this question, you may use the results   2 1 12 1 6 n r nn nr   and 3 22 1 4 1 n r nnr   . (i) Find  2 1 12 n r rr   in terms of n.[ 2] (ii) Find  2 1 1 n r rr   in terms of n. Hence find  1 2 2 1 n r rr    in terms of n.[ 5] (iii) Without using a graphing calculator, find the sum of the series    4 25 5 36 6(49) 7(64) 59 3600    .[ 3] www.KiasuExamPaper.com 532

5 NYJC 2019 JC2 Preliminary Examination 9758/01 [Turn Over 11 A boy is playing a ball game on a field. He arranges two cones A and B along the end of the field such that the cones are a and b metres respectively from one corner, O, of the field as shown in the diagram below. The boy stands along the edge of the field at x metres from O and kicks the ball between the two cones. The angle that the two cones subtends at the position of the boy is denoted by  . (i) Show that  2tan xba xa b   .[ 2] (ii) It is given that 15a  and 20b  . Find by differentiation, the value of x such that  is at a maximum. [3] (iii) It is given instead that the boy gets two friends to vary the p osition of both cones A and B along the end of the field such that 51 2a and 2ba , and the boy moves along the edge of the field such that his distance from cone A remains unchanged at 18 metres. Sketch a graph that shows how  varies with a and find the largest possible value of  .[ 4] (iv) The boy runs until he is at a distance k metres from the goal line that is formed by the two cones and kicks the ball toward the goal line. The path of the ball i s modelled by the equation 2 1 2610hk     , where k is the distance of the ball from the goal line and h its corresponding height above the ground respectiv ely. Find the angle that the p ath of the ball makes with the horizontal at the instant the ball crosses the goal line. [3] a A B O x . b www.KiasuExamPaper.com 533

6 NYJC 2019 JC2 Preliminary Examination 9758/01 12 In the study of force field, we are often interested in whether the work done in moving an object from point A to point B is independent of the path taken. If a force field is such that the work done is independent of the path taken, it is said to be a conservative field. A force field F can be regarded as a vector (, ) (, )Mxy NxyFi j where M and N are functions of x and y. The path that t