KRSS AM Prelim P1 2024 Question Paper

KENT RIDGE SECONDARY SCHOOL PRELIMINARY EXAMINATION 2024 ADDITIONAL MATHEMATICS 4049/01 PAPER 1 SECONDARY 4 EXPRESS/ 5 NORMAL ACADEMIC Thursday 22 August 2024 2 hour 15 minutes KENT RIDGE SECONDARY SCHOOL KENT RIDGE SECONDARY SCHOOL KENT RIDGE SECONDARY SCHOOL KENT RIDGE SECONDARY SCHOOL KENT RIDGE SECONDARY SCHOOL KENT RIDGE KENT RIDGE SECONDARY SCHOOL KENT RIDGE SECONDARY SCHOOL KENT RIDGE SECONDARY SCHOOL KENT RIDGE SECONDARY SCHOOL KENT RIDGE SECONDARY SCHOOL KENT RIDGE KENT RIDGE SECONDARY SCHOOL KENT RIDGE SECONDARY SCHOOL KENT RIDGE SECONDARY SCHOOL KENT RIDGE SECONDARY SCHOOL KENT RIDGE SECONDARY SCHOOL KENT RIDGE KENT RIDGE SECONDARY SCHOOL KENT RIDGE SECONDARY SCHOOL KENT RIDGE SECONDARY SCHOOL KENT RIDGE SECONDARY SCHOOL KENT RIDGE SECONDARY SCHOOL KENT RIDGE Name: ____________________________________ ( ) Class: Sec ____________ Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your name, index number and class in the spaces at the top of this page. Do not open this question paper until you are told to do so. 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, correction fluid or correction tape. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an approved scientific calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 90. _________________________________________________________________________________ This Question Paper consists of 20 printed pages, including this page. Setter: Vivian Tan [Turn over Calculator Model: For Examiner’s Use Total 90

Secondary 4 Express 2 4049 / 01 Additional Mathematics Kent Ridge Secondary School Preliminary Examination 2024 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, 𝑥 = −𝑏 ± √𝑏2 − 4𝑎𝑐 2𝑎 Binomial Theorem (𝑎 + 𝑏)𝑛 = 𝑎𝑛 + (𝑛 1) 𝑎𝑛−1𝑏 + (𝑛 2) 𝑎𝑛−2𝑏2+. . . + (𝑛 𝑟) 𝑎𝑛−𝑟𝑏𝑟+. . . +𝑏𝑛, where n is a positive integer and (𝑛 𝑟) = 𝑛! 𝑟!(𝑛−𝑟)! = 𝑛(𝑛−1)...(𝑛−𝑟+1) 𝑟! . 2. TRIGONOMETRY Identities sin2 𝐴 + cos2 𝐴 = 1 sec2 𝐴 = 1 + tan2 𝐴 cosec2𝐴 = 1 + cot2 𝐴 sin( 𝐴 ± 𝐵) = sin 𝐴 cos 𝐵 ± cos 𝐴 sin 𝐵 cos( 𝐴 ± 𝐵) = cos 𝐴 cos 𝐵 ∓ sin 𝐴 sin 𝐵 tan( 𝐴 ± 𝐵) = tan 𝐴±tan 𝐵 1∓tan 𝐴 tan 𝐵 sin 2 𝐴 = 2 sin 𝐴 cos 𝐴 cos 2 𝐴 = cos2 𝐴 − sin2 𝐴 = 2 cos2 𝐴 − 1 = 1 − 2 sin2 𝐴 tan 2 𝐴 = 2 tan 𝐴 1−tan2 𝐴 Formulae for ABC 𝑎 sin 𝐴 = 𝑏 sin 𝐵 = 𝑐 sin 𝐶 𝑎2 = 𝑏2 + 𝑐2 − 2𝑏𝑐 cos 𝐴 ∆ = 1 2 𝑎𝑏 sin 𝐶

Secondary 4 Express 3 4049 / 01 Additional Mathematics Kent Ridge Secondary School Preliminary Examination 2024 1 Do not use a calculator in this question. (a) Use the identity for tan 2𝐴 to find the value of tan 𝐴 in the form 𝑎 + 𝑏√2, where 𝑎 and 𝑏 are rational when tan 2𝐴 = 1. It is given that 𝐴 is acute. [4] (b) Hence find the exact value of sec2 𝐴. [2]

Secondary 4 Express 4 4049 / 01 Additional Mathematics Kent Ridge Secondary School Preliminary Examination 2024 2 Two perpendicular lines 𝐿1 and 𝐿2 meet at 𝐴 (2, −3). 𝐿1 has equation 2𝑥 + 𝑦 = 1 and 𝐿2 has equation 𝑦 = 𝑚𝑥 + 𝑐. 𝐿2 meets the y – axis at point B. (a) Find the coordinates of B. [3] The distance from A to a point 𝐶(4, 𝑘) is 5. (b) Find the exact value(s) of k. [3]

Secondary 4 Express 5 4049 / 01 Additional Mathematics Kent Ridge Secondary School Preliminary Examination 2024 3 It is given that f(𝑥) = 2 cos ( 𝑥 2) and g(𝑥) = 3 sin 𝑥 + 1. (a) State the minimum and maximum values of f(𝑥). [1] (b) State the minimum and maximum values of g(𝑥). [1] (c) Sketch, on the same axes, the graphs of 𝑦 = f(𝑥) and 𝑦 = g(𝑥) for 0° ≤ 𝑥 ≤ 360°. [4] (d) State the number of solutions to 3 sin 𝑥 + 1 = 2 cos ( 𝑥 2) for 0° ≤ 𝑥 ≤ 360°. [1]

Secondary 4 Express 6 4049 / 01 Additional Mathematics Kent Ridge Secondary School Preliminary Examination 2024 4 (a) Find, in simplest form, the first 3 terms in the expansion, in ascending powers of 𝑥 of (𝑎 − 𝑥2)6, where a is a non-zero constant. [3] (b) The first 2 non-zero terms in the expansion of ( 1 𝑥 + 𝑥) 2 (𝑎 − 𝑥2)6 in ascending powers of 𝑥 are 𝑎6 𝑥2 + 𝑏𝑥2. There is no term independent of x. Find the value of constant a and constant b. [5]

Secondary 4 Express 7 4049 / 01 Additional Mathematics Kent Ridge Secondary School Preliminary Examination 2024 5 (a) Explain why the function f(𝑥) = 𝑥+1 𝑥−3 , 𝑥 ≠ 3 is a decreasing function. [4]

Secondary 4 Express 8 4049 / 01 Additional Mathematics Kent Ridge Secondary School Preliminary Examination 2024 (b) Express f(𝑥) in the form 𝑎 + 𝑏 𝑥−3. Hence find ∫ 𝑥+1 𝑥−3 𝑑𝑥. [4]

Secondary 4 Express 9 4049 / 01 Additional Mathematics Kent Ridge Secondary School Preliminary Examination 2024 6 Roast chicken and fish baked in foil are taken out from different ovens. The temperature, 𝑇𝑐 ℃, of the chicken t minutes after it is removed from the oven is modelled by the formula 𝑇𝑐 = 75𝑒−0.02𝑡. (a) State the initial temperature of the chicken. [1] (b) Find the time taken for the temperature of the chicken to drop to 65 ℃. [3]

Secondary 4 Express 10 4049 / 01 Additional Mathematics Kent Ridge Secondary School Preliminary Examination 2024 The temperature, 𝑇𝑓 ℃, of the fish kept wrapped in foil t minutes after it is removed from the oven is modelled by the formula 𝑇𝑓 = 63𝑒𝑘𝑡. (c) After 1 4 of an hour, the temperature of the fish is 54.2 ℃. Find k, correct to 2 decimal places. [3] (d) Using the value of k corrected to 2 decimal places in part (c), find the time when the fish has the same temperature as the chicken. [3]