TJC 9758 2023 Prelim P1

1 TEMASEK JUNIOR COLLEGE 2023 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME CENTRE NUMBER S INDEX NUMBER MATHEMATICS 9758/01 Paper 1 24 August 2023 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name, centre number and index number on 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 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 are required to present the mathematica l 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. This document consists of 19 printed pages and 1 blank page. [Turn over For Examiner’s Use Question Number Marks Obtained 1 2 3 4 5 6 7 8 9 10 11 Total Marks

2 DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN @TJC 2023 9758 1 (a) Find 1tan dx xx − ∫ . [3] (b) Find sin 2 cos5 dx xx∫ . [2] 2 A curve C has equation 4x2 − 24x + 9y2 = 0. (a) Sketch C, giving the exact coordinates of points where the curve meets the axes.[2] (b) Describe a sequence of transformations that transform the graph of C onto the graph of ( ) 22 1 194 yx α −+= where α is a positive constant. [3] (c) State the value of α for which the resulting curve takes on the shape of a circle. [1] 3 The function g is defined by ( )g 4 xx x = − , , 0 4.xx∈ << It is given that ( ) 23f x ax bx cx= ++ is the binomial expansion of ( )g 4 xx x = − in ascending powers of x up to and including terms in 3x . (a) Find the exact value of a, b and c. [4] (b) Find the range of values of x such that the percentage error in using ( )f x to approximate ( )g x is less than 4%. [2] 4 A curve C has equation ( ) 23 85 3 xxy kx − +−= − where k is a positive real constant and x ≠ 3. (a) Sketch C and give the equations of any asymptotes and the exact coordinates of the points where C crosses the axes. [4] (b) On the same axes, sketch the graph of 37yx kk= − . [1] (c) Hence solve the inequality 23 85 373 xx xx − +− ≥−− . [2]

3 @TJC 2023 9758 [Turn over DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN 5 (a) James puts $500 into an empty safe on 1 January 2023. On the first day of each subsequent month he puts $10 more than in the previous month, so that he puts $510 on 1 February 2023, $520 on 1 March 2023, and so on. On what date will he first have over $10,000 in total in his safe? [3] (b) Nora puts $x on 1 January 2023 into a bank account which pays a compound interest at a rate of 0.5% per month on the last day of each month. She puts a further $x into the account on the first day of each subsequent month. Given that she does not withdraw any money from her account, find the least integer value of x for which her account will accumulate at least $50,000 on 31 December 2027. [5] 6 The diagram shows the triangle PQR with 30QPR∠=  , ( 1)PQ x= + cm and 2(4 )PR x= − cm, where 14 x−< < . (a) Show that the area, 2 cmA , of the triangle is given by 321 ( 7 8 16)4 xxx− ++ . [2] (b) Using differentiation, find the length QR when A is a maximum. [6] 7 The curve C is defined by the parametric equations 2 14 2 , , where , 0x y t tttt= + = − ∈≠  . (a) Find the exact coordinates of the stationary point A. Hence write down the equation of the tangent of C at A. [4] (b) The tangent at A cuts C again at the point B. Find the equation of the normal of C at B. [4] (c) The normal of C at B cuts the y -axis at the point F . Find the area of the triangle ABF. [2] Q P R cm cm

4 DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN @TJC 2023 9758 8 Do not use a calculator in answering this question. (a) Solve the equation *( i) 2i 4zz+= + . [4] (b) The complex number z and w are such that * 2 i3 6 1 e16 z w π = and 3iw= − . (i) By expressing w in exponential form, find the value of z, giving your answer in the cartesian form a + ib, where a and b are real numbers. [5] (ii) Sketch an Argand diagram, with origin O , showing the points Z , W and Q representing the complex numbers z , w and z + w respectively. State the geometrical shape of OZQW. [2] 9 The curve C is defined by the parametric equations 2 tanx θ= , 2cosy θ= , where πθπ−<< . The finite region R is bounded by C , the x -axis and the lines xa=− and xa= , where 0a> . (a) Sketch the curve C and shade the region R. [2] (b) Show that the area of R can be expressed as ( )1 2tan 0 d a k θ − ∫ where k is an integer to be determined. Hence find the exact value of a if the area of R is 2 3 π units2 . [4] (c) Show that the cartesian equation of C is 2 4 4y x= + . [1] (d) For the case when a = 2, find the exact volume of the solid form when R is rotated π radians about the y-axis. [5]

5 @TJC 2023 9758 [Turn over DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN 10 Water is being pumped from reservoir into a filtration device at a constant rate of 8 gallons per hour. The filtration device processes the water and discharges it at a rate proportional to the volume of water currently in it. At time t hours, the volume of water in the device is v gallons. The filtration device is initially empty. After 10 hours, the volume of water in the device is 5 gallons. (a) Find an expression for v in terms of t. [7] (b) Hence find the volume of water in the filtration device in the long run. [1] A fault in the filtration device is discovered when there are 4.5 gallons of water in the device. The device now discharges water at a rate that is equal to the square of the volume of water currently in it. (c) Write down a differential equation to model the new situation. [1] (d) Find the time (to the nearest minute) that elapsed from the moment the fault is discovered to the moment when the water in the device drops to 3 gallons. [4]

6 DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN @TJC 2023 9758 11 The diagram shows the cross