Juying 4E5N AM Prelims P2 2024

1 CANDIDATE NAME CENTRE NUMBER S INDEX NUMBER ADDITIONAL MATHEMATICS 4049/02 Paper 2 27 August 2024 2 hours 15 minutes Candidates answer on the Question Paper. READ THESE INSTRUCTIONS FIRST Write your Centre number, index number and name on all the work you hand in. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, 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 in 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. This document consists of 18 printed pages. Set by: Mr Albert Lui Vetted by: Mdm Norhafiani Bte Abdul Majid General Certificate of Education Ordinary Level JUYING SECONDARY SCHOOL, SINGAPORE Secondary Four Express/Five Normal Academic Preliminary Examination

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3 Answer ALL the questions 1 (a) Express 3𝑥2 − 8𝑥 − 3 in the form 𝑎(𝑥 + 𝑏)2 + 𝑐, where 𝑎, 𝑏 and 𝑐 are constants to be determined. [3] (b) Hence, find the turning point and the value of 𝑥 where it occurs. [2]

4 2 (a) Write down and simplify the first three terms in the expansion, in ascending powers of x, of (2 + 𝑥 2) 𝑛 , where n is a positive integer. [3] (b) The first two terms in the expansion, in ascending powers of x, of ( 4 3 − 3𝑥) (2 + 𝑥 2) 𝑛 are a + bx2. Find the value of 𝑛. [3] (c) Find the term independent of 𝑥 in the expansion of (𝑥 − 1 2𝑥2) 9 . [3]

5 3 Radiocarbon dating, or carbon-14 dating, is a scientific method that can accurately determine the age of organic materials as old as approximately 60,000 years. The technique is based on the decay of the carbon-14 isotope. The time in which half of the original number of atom decay is defined as the half-life. It is modelled by the equation 𝑁 = 𝑁0𝑒−𝑘𝑡, where 𝑘 is a constant and 𝑡 is the time in years. 𝑁0 is the initial amount of material in an object. Carbon-14 has a half life of 5730 years, meaning that 5730 years after an organism dies, half of its carbon-14 atoms have decayed. (a) Find the value of 𝑘. [2] (b) A mummified body was found to have 8% of its original atoms left. How many years ago did the person die? Leave your answer to the nearest whole number. [2]

6 4 Solutions to this question by accurate drawing will not be accepted. OABC is a trapezium with right angle OAB. The coordinates of A and B are (−2k, 3k) and (0, t) respectively, where k > 0, and OA = √117 units. Find (a) the value of k and show that A is (−6, 9), [2] (b) the equation of AB, [3] x y O B (0, t) A (−2k, 3k) . C

7 (c) the coordinates of C if 𝑂𝐶 = 1 2 𝐴𝐵, [1] (d) the area of 𝑂𝐴𝐵𝐶. [2] Triangle 𝑂𝐴𝐵 lies in a circle 𝐶. (e) Explain if 𝑂𝐵 is the diameter of the circle. [1] (f) Find the equation of the circle 𝐶. [3]

8 5 The diagram shows two right-angled triangles, ABC and DEC. ∠𝐵𝐴𝐶 = ∠𝐶𝐷𝐸 = 𝜃 , AB = 3.5 cm and DE = 6 cm. (a) Show that BD = 6 cos 𝜃 + 3.5 sin 𝜃. [2] (b) Express BD in the form 𝑅 cos(𝜃 − 𝛼), where 𝑅 > 0 and 𝛼 is an acute angle. [3] (c) State the maximum length of BD and the corresponding value of 𝜃. [2] 3.5cm 6cm 𝐶 𝐷 𝐵 𝐴 𝐸 𝜃 𝜃

9 (d) Find the value of 𝜃 when 𝐵𝐷 = 6 cm. [2] 6 A conical vessel, whose vertical angle is as shown in the diagram, has water poured into it at such a rate that after 𝑡 seconds, the depth of the water is cm and the radius of the water surface is r cm. (a) Show that the volume of water, 𝑉 = 𝜋𝑥3 9 cm3. [3] 30 x

10 (b) Given further that the volume of water, 𝑉 = 8𝑡, find the rate of which the water level is increasing after 3 seconds. [5]