2019 J2 EJC H2 Math Prelim (with ans)

EUNOIA JUNIOR COLLEGE JC2 Preliminary Examination 2019 General Certificate of Education Advanced Level Higher 2 CANDIDATE NAME CLASS INDEX NO. MATHEMATICS Paper 1 [100 marks] 9758/01 04 September2019 3 hours Candidates answer on the Question Paper Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name, civic s group and question number 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 paper clips, highlighters, glue or correction fluid. Answer all 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. 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 al lowed 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. This document consists of 27 printed pages (including this cover page) and1 blank page. For markers’ use: Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Total www.KiasuExamPaper.com 247

2019 JC2 H2 Mathematics Preliminary Examination 1 Electricity cost per household is calculated by multiplying th e electricity consumption (in kWh), by the tariff (in cents/kWh). The tariff is set by the government and reviewed every 4 months. The amount of electricity used by each household for each 4-mon th period, together with the total electricity cost for each household in the year, are given in the following table. Jan – April (in kWh) May – Aug (in kWh) Sept – Dec (in kWh) Total electricity cost in the year ($) Household 1 677 586 699 529.53 Household 2 1011 871 1048 790.63 Household 3 1349 1174 1417 1063.28 Write down and solve equations to find the tariff, in cents/kWh, to 2 decimal places, for each 4- month period. [4] 2 A string of fixed length l is cut into two pieces. The first piece is used to form a square of side s and the second piece is used to form a circle of radius r. Find the ratio of the length of the first piece to the second piece that gives the smallest possible combined area of the square and circle. [6] 3 A geometric progression has first term a and common ratio r, and an arithmetic progression has first term a and common difference d, where a and d are non-zero. The sums of the first 2 and 4 terms of the arithmetic progression are equal to the respective sums of the first 2 and 4 terms of the geometric progression. (i) By showing 32 53 0rr r , or otherwise, find the value of the common ratio. [5] (ii) Given that 0a and the nth term of the geometric progression is positive, find the smallest possible value of n such that the nth term of the geometric progression is more than 1000 times the nth term of the arithmetic progression. [3] 4 (i) Find the series expansion for 1 n ax in ascending powers of x, up to and including the term in 3x , where a is non-zero and 1a . [1] (ii) It is given that the coefficients of the terms in 23, , and xx x are three consecutive terms in a geometric progression. Show that 1n . [2] (iii) Show that the coefficients of the terms in the series expansion of 1 1 ax

form a geometric progression. [3] (iv) Evaluate the sum to infinity of the coefficients of the terms in x of odd powers. [2] www.KiasuExamPaper.com 248

5 (a) It is given that the equation f xa has three roots 123,,xxx where 12 30xx x , and a i s a constant. (i) How many roots does the equation f xa have? With the aid of a diagram, or otherwise, explain your answer briefly. [2] (ii) How many roots does the equation f xa a have? With the aid of a diagram, or otherwise, explain your answer briefly. [2] (b) Solve the inequality 2ln2 ln 1 sin ,3 xxS d where 02 x Sd . [4] 6 The curve C has equation 2 53 1 xxy x . (i) Show algebraically that the curve C has no stationary points. [2] (ii) Sketch the curve C, indicating the equations of any asymptotes, and the coordinat es of points where C intersects the axes. [4] (iii) Region S is bounded by C, the y-axis, and the line 9 2y . Find the volume of the solid formed when region S is rotated about the x-axis completely. [3] 7 In the Argand diagram, the points 1P and 2P represent the complex numbers z and 2z respectively, where 3i 3z . (i) Find the exact modulus and argument of .z [2] (ii) Mark the points 1P and 2P on an Argand diagram and find the area of the triangle 12OPP , where O represents the complex number 0. [3] Let i 32ew S§·¨¸ ©¹ . (iii) Find the set of integer values n such that 3arg 4 nwz S . [4] www.KiasuExamPaper.com 249

2019 JC2 H2 Mathematics Preliminary Examination 8 (a) G i v e n t h a t 22 s in 2 ,y x use repeated differentiation to find the Maclaurin series for y , up to and including the term in 2.x [5] (b) The points A, B, C, and D lie on a semi-circle with AC as its diameter. Furthermore, angle ,DAB T and angle 3ACB S . (i) Show that 1 . cos 3 sin BC DC TT

[3] (ii) Given that T is a sufficiently small angle, show that 21,BC abDC TT| for constants a and b to be determined. [3] 9 (a) (i) Find 2sin cos3 d .xx x³ [3] (ii) Hence, show that >@12 sin cos3 d 4 cos 4 8 cos 2 sin 4 4sin 2 ,16xx x x x xx x x x C ³ where C is an arbitrary constant. [3] (b) The curve C has parametric equations 2x T , sin cos3y TT , where 0. 2 STdd (i) Sketch the curve C, giving the exact coordinates of the points where it intersects the x-axis. [2] (ii) By using the result in (a)(ii), find the exact total area of the regions bounded by the curve C and the x-axis. [4] A B C D www.KiasuExamPaper.com 250

10 An object is heated up by placing it on a hotplate kept at a h igh temperature. A simple model for the temperature of the object over time is given by the differential equation d ,d H T kT Tt where T is the temperature of the object in degrees Celsius, HT is the temperature of the hotplate in degrees Celsius, t is time measured in seconds and k is a real constant. (i) State the sign of k and explain your answer. [1] (ii) It is given that the temperature of the object is 25 degrees C elsius at 0t , and the temperature of the hotplate is kept constant at 275 degrees Celsius. If the temperature of the object is 75 degrees Celsius at 100t , find T in terms of t , giving the value of k to 5 significant figures. [6] The model is now modified to account for heat lost by the object to its surroundings. The new model is given by the equation d ,d HS T kT T mT Tt where ST is the temperature of the surrounding environment in degrees C elsius and m is a positive real constant. (iii) It is given that the object eventually approaches an equilibri um temperature of 125 degrees Cel sius, and that the surrounding environment has a constant temperature which is lower than 125 degrees Celsius. One of the two curves (A and B) shown below is a possible graph of the object’s temperature over time. State which curve this is, and explain cle arly why the other curve cannot be a graph of the object’s temperature over time. [2] (iv) Using the same value of k as found in part (ii) and assuming 25ST , find the value of m . (You need not solve the revised differential equation.) [3] A B T t 125 25 www.KiasuExamPaper.com 251

2019 JC2 H2 Mathematics Preliminary Examination 11 Methane ( 4CH ) is an example of a chemical compound with a tetrahedral structure. The 4 hydrogen (H) atoms form a regu