2024 Outram Prelim P2 QP

1 Subject : Mathematics Level (Stream) : Secondary Four Express & Five Normal Academic Paper : 4052/02 Date : 23 August 2024 Duration : 2 hours 15 mins Marks : 90 READ THESE INSTRUCTIONS FIRST Write your name, class and index 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 staples, paper clips, highlighters, glue or correction fluid. Answer all questions on the Question Paper. 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 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 , use either your calculator value or 3.142, unless the question requires the answer in terms of . The number of marks is given in brackets [ ] at the end of each question or part question. 90 This document consists of 22 printed pages, including this cover page. OUTRAM SECONDARY SCHOOL PRELIMINARY EXAMINATION 2024 Class Index No Name: __________________________________________________

2 Mathematical Formulae Compound interest Total amount = Mensuration Curved surface area of a cone = Surface area of a sphere = Volume of a cone = Volume of a sphere = Area of triangle ABC = Arc length = , where is in radians Sector area = , where is in radians Trigonometry Statistics Mean = Standard deviation = n rP      + 1001 rl 24 r hr 2 3 1 3 3 4 r Cabsin2 1 r  2 2 1 r  C c B b A a sinsinsin == Abccba cos2222 −+=   f fx 22       −     f fx f fx

3 Answer all the questions. 1 (a) It is given that 1 microgram = 10− 6 grams and 1 milligram = 10− 3 grams. (i) If 1 milligram = p micrograms, find the integer value of p. Answer p = …………..……………….. [1] (ii) The recommended daily amount of vitamin A intake for an adult man is 900 micrograms. The amount of vitamin A in half a cup of boiled spinach is about 0.573 milligrams. An adult man consumed one cup of boiled spinach in a day. Explain whether he has met the daily recommended intake of vitamin A. ………………..………………..……………………..…………………… ………………..………………..……………………..…………………… ………………..………………..……………………..…………………… [2] (b) A map has a scale of 1 : 30 000. (i) The distance on the map between Town A and Town B is 8.5 cm. Find the actual distance, in km, between Town A and Town B. Answer …………………..………… km [1] (ii) On the same map, the actual area of a pond is 0.36 km2. Find the area of the pond represented on the map. Answer ……………..…………… cm2 [2]

4 (iii) The same pond has an area of 0.5625 cm2 on another map with a scale of 1 : n. Find the value of n. Answer n = …………..……………….. [2] 2 (a) 242 3 yzxz z −= (i) Find y when x = 3 and z = – 1. Answer y = …………..……………….. [2] (ii) Express z in terms of x and y. Answer z = ………………..……………….. [3]

5 (b) Solve the inequality 7336 2 xx −−   . Answer …….………………..……………….. [2] (c) Solve the equation 25 11 2 5xx +=+− . Answer …….………………..……………….. [4]

6 3 The diagram shows an open funnel, consisting of a cylinder and a cone. The heights of the cylinder and cone are 1.5h cm each, and the radius of both the cylinder and cone is 0.8h cm. (a) Show that the curved surface area of the cone is 1.36πh2. Answer [2] (b) Given that the total surface area of the funnel is 1504π cm2, find the value of h. Answer h = …………..……………….. [2] 1.5h 1.5h 0.8h

7 (c) A glass rod is placed in the funnel so that the one end of the glass rod rests on the edge of the funnel as shown. When the glass rod is placed vertically from the bottom of the funnel, a part of the glass rod protrudes from the top of the funnel. Calculate this length of the part of the glass rod that is outside the funnel when placed vertically. Answer …………..………………. cm [3] rod

8 (d) Water is being poured into the funnel at a constant rate and it takes 16 seconds to fill the funnel completely. Sketch the change in height of the water level with respect to time, showing all critical values clearly. [2] height time 0 2 4 6 8 10 12 14 16 10 20 30 40 50 60

9 4 The cost, $y, of making a round plate of radius x cm is given by the function 2 210y ax x=+ , where a is a constant. (a) Given that x = 2, y = 107, show that a = 1 2 . Answer [1] (b) The table below shows some of the corresponding values of x and y. x 8 10 12 y 107 60.5 53 58.3 71 p Find p. Answer p = …………..……………….. [1] (c) On the grid, draw the graph for 2 210y ax x=+ for 2 ≤ x ≤ 12. [3] (d) Using your graph, (i) find the radius which gives the minimum cost, Answer ………………..…………… cm [1] (ii) the range of value of x such that the cost is below $70. Answer …....……………..……………. [1]

10 (e) (i) On the same grid as in part (b), draw the line 6y = 25x + 180. [2] (ii) Write down the x-coordinate of the points where the line intersects the curve. Answer x = ...………… and ...………… [2] (iii) These values of x are the solutions of the equation x3 + Ax2 + Bx + C = 0. Find the values of A, B and C. Answer A = ..……………… B = ………..……… C = ………..……… [3]