2020 JC2 H2MATH Prelim P1 Questions

This document consists of 5 printed pages and 0 blank page. NANYANG JUNIOR COLLEGE Internal Examinations © NYJC 2020 [Turn Over NANYANG JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME CT CLASS 1 9 Centre Number/ Index Number / MATHEMATICS 9758/01 Paper 1 28 August 2020 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name and class 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, 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. 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 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. For examiner’s use only Question number Marks 1 2 3 4 5 6 7 8 9 10 11 12 Total

2 NYJC 2020 JC2 Preliminary examination  9758/01 1 A household's monthly utility bill is calculated by summing up the charges it incurs for its usage of electricity, gas and water based on the prevailing price rate per unit. The table below shows Mrs Ang’s household utilities usage for the months of April, May, June and July in a particular year. Monthly usage (units) April May June July Electricity 23 33 49 35 Gas 12 16 22 17 Water 16 21 33 26 The prevailing price rate per unit were unchanged for the period of April to June. However, in the month of July, the electricity rate was increased by 7%, the gas rate was reduced by 5% and the water rate remained unchanged. Given that Mrs Ang’s monthly utility bills for April, May and June were $103.00, $142.00 and $209.00 respectively, find her monthly utility bill for the month of July. [4] 2 (i) Sketch the graph of 2 4 7 1 2 3 xy x x  . Give the equations of the asymptotes and the coordinates of the turning points and the point(s) where the curve crosses the axes. [4] (ii) Find the range of values of k such that 2 4 72 31 x kx x   has only positive root(s). [1] 3 A curve C has parametric equations ,aaxt yt tt  , where t  , 0t  and a is a positive constant. (i) Find a cartesian equation of C. [2] (ii) The point P on C has parameter p. Show that the equation of the tangent at P can be expressed in the form   22 ,y pax pak a p  where k is a constant to be determined. [4] 4 (a) By writing  32sin as sin sinx xx , find 33sin cos dx xx . [3] (b) Find 2 56 d 53 2 x x xx   . [4] 5 (i) On the same axes, sketch the curves with equations 1 1 axy ax   and 1 1yx a , where 1a , giving the equations of the asymptotes and the coordinates of the points where the curves meet the axes. [3] (ii) Solve the inequality 11 11 ax xax a   , giving your answers in terms of a. [4]

3 NYJC 2020 JC2 Preliminary examination  9758/01 [Turn Over 6 The tangent at the point (, )x y on the curve with equation f( )yx passes through the point (,)ax , where a is a positive constant and x a . (i) Write down a differential equation relating d d y x , a, x and y. [1] (ii) Using the substitution ()ya x z , show that the differential eq uation can be transformed to  2 d d zx x ax   . By using partial fractions, find the general solution for y in terms of a and x. [5] (iii) Given that the curve f ( )yx passes through the origin, show that f( ) ( )l n axx ax x a    . [2] 7 A sequence of numbers 1u , 2u , 3u ,  has a sum nS where 1 n nr r Su   . It is given that 2 1nSA n n B n  , where A and B are non-zero constants. (i) Find a simplified expression for nu in terms of A, B and n. [2] (ii) It is also given that the first an d second term of the sequence are 1 and 9 respectively. Show that 1A and 1B . [3] (iii) Hence, find  2 1 31 n r rr    , giving your answer in the form 32an bn , where a and b are constants to be determined . [3] 8 You are given that 22 00 e cos d and e sin dxxCx x Sx x   . Use integration by parts to show 22 and 2 1 eCS S C     . Hence, find the exact values of C and S. [6] The finite region bounded by the axes and the curve 1 2ec o sxyx  , for 0 x  , is rotated through four right angles about the x-axis. Find the exact volume of the solid of revolution. [4] 9 (i) Using standard series from the List of Formula (MF26), expand ln( 2 ) nkx in ascending powers of x, as far as the term in 3x , where k and n are real constants, 0k  , 0n and 2 kx . [2] (ii) State the range of values of x for which the expansion in part (i) is valid. [1] (iii) Find the exact value of  1 2 0 ln 1 2 dx x   . [3] (iv) Use your series from part (i) to estimate  1 2 0 ln 1 2 dx x   . [2] (v) Calculate the percentage error of the estimate in part (iv), and comment on the accuracy of your approximation using your answer from part (ii). [2]

4 NYJC 2020 JC2 Preliminary examination  9758/01 10 Referred to the origin O, the points P and Q have position vectors 4i + 4j + 3k and 8i + 8j  k respectively. (i) Find the position vector of the point R where the line PQ meets the x-y plane. [3] (ii) The plane  contains R and is perpendicular to PQ. Show that a cartesian equation of  is x + y  z = c, where c is a constant to be determined. [2] (iii) Find the acute angle between  and the x-y plane. [2] (iv) Point A has position vector i + j. Show that AR  is perpendicular to the line of intersection of  and the x-y plane. [2] (v) Find the exact shortest distance from A to  . [2] 11 A farmer has a plot of the land, OAB, with adjacent sides OA and OB. With reference to O as the origin, OA and OB are parallel to the x-axis and y-axis respectively. The arc AB is decribed by the parametric equations 1cos , sin 2 , 0 .22xy      (i) Show that the area of the plot of land is 2 3 units2. [4] Due to a strong worldwide demand for latex, the farmer plans to use a portion of his land to grow rubber trees. He divides his land into three portions . One portion is a rectangle with vertices at O and along arc AB, and sides parallel to OA and OB. He intends to use only this rectangular plot of land which has an area of K units2 to plant rubber trees. (ii) By using differentiation, find the largest value of K. [4] In the midst of his planning, the futures contract for latex shot to historical high. To maximise his returns, the farmer decides to redistribute his plot of land to plant more rubber trees. He divides his plot of land into two portions using the line 20 . 1yx  and allocates the larger plot to plant rubber trees. (iii) Find the area of this larger plot of land meant for rubber trees. [4] 12 The reproduction number, 0R , of a disease is a mathematical term that indicates how contagious an infectious disease is. It gives the average number of people who will contract a contagious disease from a person with that disease. That replication will conti nue if no one has been vaccinated against the disease or is already immune to it in his or her community. The infectious period of a disease is the time period during which an infected pe