KRSS AM Prelim P2 2024 Question Paper

KENT RIDGE SECONDARY SCHOOL PRELIMINARY EXAMINATION 2024 ADDITIONAL MATHEMATICS 4049/02 PAPER 2 SECONDARY 4 EXPRESS/ 5 NORMAL ACADEMIC Monday 26 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 19 printed pages, including this page. Setter: Vivian Tan [Turn over Calculator Model: For Examiner’s Use Total 90

Secondary 4 Express 2 4049 / 02 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 / 02 Additional Mathematics Kent Ridge Secondary School Preliminary Examination 2024 1 Express 4 (𝑥2+1)(𝑥+1) in partial fractions. [5]

Secondary 4 Express 4 4049 / 02 Additional Mathematics Kent Ridge Secondary School Preliminary Examination 2024 2 (a) Radiation intensity, R, varies inversely with the square of d, the distance from the source of radiation such that 𝑅 = 𝑘 𝑑2, where k is a constant. Values of R for different values of d have been collected and tabulated. Explain how a straight-lined graph can be drawn to determine the formula connecting R and d. [4] (b) The number of particles present in a room, t minutes after turning on the air filter is y. When corresponding values of lg y and t are plotted on a lg y against t axes, the points form a straight line that passes through (0,3) and (10,1.5) as drawn on the axes on the next page. (i) Find y in terms of t. [4]

Secondary 4 Express 5 4049 / 02 Additional Mathematics Kent Ridge Secondary School Preliminary Examination 2024 (ii) Use the graph to estimate the time taken for the number of particles in the room to be halved. [3]

Secondary 4 Express 6 4049 / 02 Additional Mathematics Kent Ridge Secondary School Preliminary Examination 2024 3 (a) The diagram below shows the graph of the curve 𝑦 = cos 𝑥 + 𝑥 2 for 𝑥 ≥ 0 radians. The tangent to the curve when 𝑥 = 0 at C, is drawn to D which is vertically above point B, the minimum point of the curve. Points A and B are the first two stationary points of the curve. Find 𝑥𝑏, the x coordinates of point B. You do not need to show that it is a minimum point. [4] (b) (i) Find the equation of CD. [2] x 𝑥𝑏 𝑥 𝑦 𝐶 𝐵 𝐷 𝐴 0 𝑦 = cos 𝑥 + 𝑥 2 x

Secondary 4 Express 7 4049 / 02 Additional Mathematics Kent Ridge Secondary School Preliminary Examination 2024 (ii) Find the area shaded that is bounded by the tangent to the curve 𝑦 = cos 𝑥 + 𝑥 2 at 𝑥 = 0, the curve and the line 𝑥 = 𝑥𝑏. [5]

Secondary 4 Express 8 4049 / 02 Additional Mathematics Kent Ridge Secondary School Preliminary Examination 2024 4 (a) 2𝑦 = 16𝑥 + 𝑘 is a tangent to the curve 𝑦 = 1 2𝑥 + 2𝑘𝑥. Find the value of constant k. [4]

Secondary 4 Express 9 4049 / 02 Additional Mathematics Kent Ridge Secondary School Preliminary Examination 2024 (b) Find the range of values of a such that 𝑎𝑥2 + √8𝑥 + (𝑎 − 1) < 0 for all values of x. [4]

Secondary 4 Express 10 4049 / 02 Additional Mathematics Kent Ridge Secondary School Preliminary Examination 2024 5 Given 𝑦 = 𝑒2𝑥 sin 3𝑥. (a) Find 𝑑𝑦 𝑑𝑥. [2] (b) Find 𝑑2𝑦 𝑑𝑥2. [2]