Juying (without plane geom) Prelim P2 Qn

CANDIDATE NAME CENTRE NUMBER S INDEX NUMBER ADDITIONAL MATHEMATICS 4049/02 Paper 2 31 August 2021 2 hour 15 minutes Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your name, index number and class on this cover page. 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 questions. The number of marks is given in brackets [ ] at the end of each question or part question. If working is needed for any question it must be shown with the answer. Omission of essential working will result in loss of marks. The total number of marks for this paper is 90. The use of an approved scientific calculator is expected, where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For 𝜋, use either your calculator value or 3.142. Setter: Mdm Mak Wai Han Vetter: Mdm Norhafiani A Majid This document consists of 15 printed pages General Certificate of Education Ordinary Level JUYING SECONDARY SCHOOL, SINGAPORE Secondary Four Express / Five Normal (Academic) Preliminary Examination

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3 1(a) The daily production cost, $𝑦, of making lead crystal bowls in a factory can be represented by the equation 𝑦 = 1 5 𝑥2 − 10𝑥 + 650, where x is the number of bowls made. (i) Explain the meaning of the constant term 650 in the equation. [1] (ii) Mr Tan, the manager, wants to reduce the cost of production to $500. Show, with clear working, how you would convince him that it is not possible. [3] (b) Find the range of values of 𝑎 for which −𝑎𝑥2 + 5𝑎𝑥 + (𝑎 − 4) is always negative for all real values of 𝑥. [5]

4 (c) Show that the line 𝑦 = −2𝑥 − 𝑘 intersects the curve 𝑦 = 𝑥2 + 𝑘𝑥 + 16 at two distinct points for all real values of 𝑘 except those in the interval of −2√15 ≤ 𝑘 ≤ 2√15. [5] 2 A rectangular box without the lid is made of thin cardboard of negligible thickness. The sides of the base are 2x cm and 3x cm and the height of the box is h cm. Given that the total surface area is 200 cm2, show that ℎ = 20 𝑥 − 3𝑥 5 . Hence find the dimensions of the box for which the volume is a maximum. [9]

5 3(a) In the expansion of (3 − 𝑝𝑥2) (1 + 𝑥 2) 7 , the coefficient of the 𝑥4 term is 525 16 . Find the value of p. [4] (b) In the expansion ( 𝑥3− 2 𝑥2 ) 𝑛 in ascending powers of x, the sixth term is independent of x. Find the value of 𝑛. [4]

6 4(a) The points with coordinates (27, 3) and (1, 𝑝) lie on the graph of 𝑦 = 𝑙𝑜𝑔𝑐𝑥 . (i) Find the value of each of the constants 𝑐 𝑎𝑛𝑑 𝑝. [2] (ii) Sketch the graph of 𝑦 = 𝑙𝑜𝑔𝑐𝑥. [2] (b) Find the equation of the straight line which must be drawn on the graph of 𝑦 = 3 − 𝑒2𝑥 to obtain a solution of the equation 𝑥 =𝑙𝑛 𝑙𝑛 √2 − 𝑥 . [3]

7 5(i) Using the law for the change of base of logarithm, or otherwise, show that 𝑙𝑜𝑔𝑏 𝑎 = 1 𝑙𝑜𝑔𝑎 𝑏 . [1] (ii) Hence, or otherwise, solve the equation 2 𝑙𝑜𝑔9 𝑥 + 1 = 2 𝑙𝑜𝑔𝑥 3 . [5] 6 The diagram below shows a hollow cone of semi-vertical angle of 45° held with its axis vertical and vertex downwards. The hollow cone has radius r cm and height h cm. At the beginning of an

8 experiment, the cone is filled with 390 cm3 of liquid. The liquid leaks through a small hole at the vertex at a rate of 2 cm3 per second. (i) Calculate, in cm3, the volume of water that was still in the cone 3 minutes after the start of the experiment. [2] (ii) Find the rate at which the depth of the liquid is decreasing 3 minutes after the start of the experiment. [4] 7 The number of bacteria, N, in a petri dish, is observed to increase with time, t hours. Measured values of N and t are given in the table given below. t 2 4 6 8 10 N 61 150 402 915 2250 It is known that N and t are related by the equation 𝑁 = 𝑁0𝑒𝑘𝑡 where 𝑁0 and 𝑘 are constants. (i) By expressing the equation 𝑁 = 𝑁0𝑒𝑘𝑡 in a form suitable for plotting a straight line graph, plot the graph on the grid given below. Label your axes clearly. [3]

9 (ii) Use your graph to estimate the value of 𝑘 and 𝑁0. [3]

10 8 (iii) Using your graph, estimate the number of hours required for the number of bacteria to increase to 1800. The diagram shows two trigonometric graphs labelled 𝑦1 and 𝑦2 in the interval 0° ≤ 𝑥 ≤ 360°. [2] (i) State the number of solutions for which 𝑦1 − 𝑦2 = 0 for 0° ≤ 𝑥 ≤ 360°. [1] (ii) Find the equations of the curves 𝑦1 and 𝑦2. [4]