ACSI 2022 Promo paper 2 ans

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Exam / Paper 2 / Solutions 1 FINAL EXAMINATION 2022 YEAR 5 IB DIPLOMA PROGRAMME MATHEMATICS HIGHER LEVEL PAPER 2 ANALYSIS AND APPROACHES SECTION A Qn Solution Marks 1*. In an infinite geometric sequence, 𝑢1 = 𝑎, 𝑢2 = 𝑎2 − 2. [6 marks] 1(a) Given that 𝑎 = 9 5 show that the sum to infinity of the sequence exists. Students found the sum to infinity, when what they are required to do is to find the common ratio r and show that -1<r<1. Common error to conclude that r < 1 for sum to infinity to exist. When 𝑎 = 9 5 , 𝑟 = 𝑎2−2 𝑎 = 31 45 Since −1 < 31 45 < 1, the sum to infinity exists 1(b) Find the least value of 𝑛 such that 𝑆∞ − 𝑆𝑛 < 1 100. There is mostly error in computing least value of n and handling of equations involving inequalities. 𝑆∞ − 𝑆𝑛 = 𝑎 1−𝑟 − 𝑎(1−𝑟𝑛) 1−𝑟 = 𝑎𝑟𝑛 1−𝑟 9 5(31 45) 𝑛 1−31 45 < 1 100 By GDC, 𝑛 > 17.1 The least value of 𝑛 is 18

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Exam / Paper 2 / Solutions 2 Qn Solution Marks 2*. The time 𝑇 hours taken for a machine to cool down to temperature 𝑥℃ after it has been turned off can be modelled by 𝑇 = 5𝑙𝑛 ( 13 𝑥−𝑘), where 𝑇 ≥ 0 and 𝑘 is a positive constant. The machine has an initial temperature of 36℃ when it is turned off. [4 marks] 2(a) Show that 𝑘 = 23. Done correctly by most students 0 = 5𝑙𝑛 ( 13 36−𝑘) 𝑘 = 23 (Shown) 2(b) Sketch the graph of the machine’s temperature for 𝑇 ≥ 0, clearly stating the coordinates of any axial intercept and the equation of any asymptote. Sketching x against T which leads to wrong graph. Note that the equation given has T as the y- axis variable by convention from the way it is expressed. x-intercept, x = 36 Vertical asymptote, x = 23 Qn Solution Marks 3*. On 1st January 2022, Sam invested $600 in an account that pays a nominal annual interest rate of 1.2%, compounded monthly. The amount of money in Sam’s account at the end of each year follows a geometric sequence with common ratio, 𝑟. [6 marks] 3(a) Find the value of 𝑟, giving your answer correct to 4 significant figures. Generally well done 𝑟 = (1 + 0.012 12 ) 12 = 1.012 3(b) Sam makes no further deposits or withdrawals from the account. Find the year in which the amount of money in Sam’s account becomes more than $1000. There are several ways of doing it, by

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Exam / Paper 2 / Solutions 3 Qn Solution Marks 600 × 1.012𝑛 > 1000 𝑛 > ln (1000 600 ) ln (1.012) = 42.8 Year 2064 letting the general term of a GP to be greater than 10