NYJC 9758 2024 Promo Qns

This document consists of 4 printed pages and 0 blank page. NANYANG JUNIOR COLLEGE Internal Examinations © NYJC 2024 [Turn Over NANYANG JUNIOR COLLEGE JC1 END-OF-YEAR EXAMINATION Higher 2 CANDIDATE NAME CT CLASS 2 4 MATHEMATICS 9758/01 Paper 1 2 October 2024 2.5 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Answer all questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. 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 an approved graphing calculator. 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 must present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. 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 80.

2 NYJC 2024 JC1 End-of-Year Examination 9758/01 1 A curve C has equation 2 2 e xayb c x   , where a, b and c are constants. It is given that C passes through the point with coordinates  21, 2e 1 . When C undergoes a scaling of factor 2 parallel to the x-axis, the transformed curve passes through the point with coordinates 21, 8 e  . When C undergoes a translation by negativ e 1 unit in the direction of the y-axis, the transformed curve passes through the point with coordinates   233,2e 4  . Find the values of a, b and c. [4] 2 (a) Show that 2 49xx  is always positive for all real values of x. [2] (b) Hence, without using a calculator, solve   22 2 49 2 0 67 xx x xx     . [3] 3 The curve D has equation 2 25 23 xy xx   . (a) Sketch the graph of D, stating the equations of any asymptotes, the coordinates of the points where the curve crosses the axes and the stationary point(s). [3] (b) By considering a suitable curve, hence, find the exact range of values of k, where 0k  , such that the equation  2 2 2 2 253 23 xx k xx     has at least one positive real root. [3] 4 During a clinical trial, the concentration U, of a specific blood agent is measured at one-hour intervals following the initial ad ministration of the trial drug to a patient. The following readings were obtained 34 5 88, 76 and 70,UU U  where tU denotes the reading t hours after the drug was first administered. It is believed that U satisfies the relationship 1 , 0 ,ttUp q U t   where p and q are constants. (a) Find the values of p and q. [2] (b) Determine the initial conc entration of the blood agent when the drug was initially administered. [3] (c) Describe how the concentration behaves over a long period of time. [1]

3 NYJC 2024 JC1 End-of-Year Examination 9758/01 [Turn Over 5 The position vectors of points A, B and C, relative to the origin O are a, b and c respectively, where 32 , 432 , 3   a i jk b i j k c i jk . (a) Find ab and deduce the exact area of triangle OAB. [3] (b) Hence, find the volu me of the tetrahedron OABC. [The volume of a tetrahedron is 1 3  base area perpendicular height.] [3] (c) Find the perpendicular distance from B to line OA. [2] 6 (a) The sum, nS , of the first n terms of a series is given by 2 2nSn n . Show that this is an arithmetic series. Find the values of the first term and the common difference. [4] (b) In a geometric progression, the first term is 12 and the sixth term is 3 8 . Let the sum of the first n terms of the progression an d the sum to infinity be nS and S respectively. Find the least value of n for which the difference between nS and S is less than 0.001. [4] 7 Functions f and g are defined by g 20f, , ,, : : 0 . xx x xx x x x       (a) Explain why the composite function gf does not exist. [2] (b) Determine the range of values of  such that 1g exists. [2] For the rest of the qu estion, it is given that 4  and the domain of g is further restricted to x  . (c) Determine the least value of  for which 1g exists. Using this value of  , hence find  1g x and state its domain. [4] 8 (a) Given that ln eyx for 1ex  , show that 2d2e d yyy x  . Hence find the first three terms of the Maclaurin expansion of y. Give the coefficients as exact fractions in their simplest form. [4] (b) Using standard series from the List of Formulae (MF27), expand  ln ex as far as the term in 2x . Hence, verify the correctness of your expansion in (a), showing your working clearly. [4] (c) Use your expansion in part (a) to find an ap proximation to 11 0 eln 10   , leaving your answer in terms of e. [2]

4 NYJC 2024 JC1 End-of-Year Examination 9758/01 9 The plane 1 has equation 22 1 13 0 44 1                    r , where  and  are parameters. The plane 2 has equation 45 1 2xyz . (a) Find the acute angle between 1 and 2 . [3] (b) Find a vector equation of th e line of intersection between 1 and 2 . [2] (c) The line l passes through the point A with position vector 213mm ij k and is parallel to 33nni j k , where m and n are positive constants. Given that the perpendicular distance from A to the plane 1 is 15 6 and that the acute angle between l and 1 is 1 2sin 6  , find the values of m and n. [6] 10 A tiny robot moves in a way that traces out the curve with parametric equations given by 211, 2 s i nx ty t   from the point where 1t to the point where 1t . (a) Sketch the curve traced out by the robot, indicating clearly the exact coordinates of the endpoints. [2] (b) Find d d y x in terms of t. [2] The robot crosses the x-axis at the point P. (c) Find the cartesian equation of the tangent to the curve at P. [2] (d) Find the angle between the di rection in which the robot is moving and the positive x-axis at the instant it reaches the point where 3y  . [2] A tiny magnet is placed at the point  2, 2 . (e) Show that the distance s between the robot and the magnet is given by   2221 14 s i n 1st t    . [1] (f) The magnet will attract the robot if it comes within 0.25 unit of its position. By using differentiation to find the minimum distance between the magnet and the robot, determine whether the robot will be attracted by the magnet. [You need not show that the distance is a minimum.] [5]