Juying 4E5N AM Prelims P2 2024 Solutions

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] 3 (𝑥2 − (2) ( 4 3) 𝑥 + ( 4 3) 2 − ( 4 3) 2 ) − 3 M1 = 3 (𝑥 − 4 3) 2 − 3 ( 16 9 ) − 3 = 3 (𝑥 − 4 3) 2 − 25 3 A1 𝑎 = 3 𝑏 = − 4 3 𝑐 = − 25 3 B1 (b) Hence, find the turning point and the value of 𝑥 where it occurs. [2] Turning Point = ( 4 3 , − 25 3 ) Value of 𝑥 = 4 3

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] (2 + 𝑥 2) 𝑛 = 2𝑛 + 𝑛(2)𝑛−1 ( 𝑥 2) + 𝑛(𝑛−1) 2 (2)𝑛−2 ( 𝑥 2) 2 + ⋯ M2 = 2𝑛 + 𝑥𝑛(2)𝑛 4 + 𝑛(𝑛−1)𝑥2(2)𝑛 32 + ⋯ A1 (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] ( 4 3 − 3𝑥) (2 + 𝑥 2) 𝑛 = ( 4 3 − 3𝑥) [2𝑛 + 𝑥𝑛(2)𝑛 4 + 𝑛(𝑛−1)𝑥2(2)𝑛 32 + ⋯ ] = 4 3 × 2𝑛 + 𝑥𝑛(2)𝑛 3 + 𝑛(𝑛−1)𝑥2(2)𝑛 24 − 3𝑥2𝑛 − 3𝑥2𝑛(2)𝑛 4 + ⋯ B1 No 𝑥 term: 𝑛(2)𝑛 3 − 3(2)𝑛 = 0 M1 𝑛 = 9 A1 (c) Find the term independent of 𝑥 in the expansion of (𝑥 − 1 2𝑥2) 9 . [3] 𝑻𝒓+𝟏 = (𝟗 𝒓) (𝒙)𝟗−𝒓 (− 𝟏 𝟐𝒙𝟐) 𝒓 Term independent of 𝒙: 𝟗 − 𝒓 − 𝟐𝒓 = 𝟎 M1 𝒓 = 𝟑 A1 Term independent of 𝒙 = (𝟗 𝟑) (𝒙)𝟗−𝟑 (− 𝟏 𝟐𝒙𝟐) 𝟑 = − 𝟐𝟏 𝟐 B1

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 tech