XJC H2 Math - Set 1 - P2 (QP)

X Junior College [Turn over This document consists of 7 printed pages and 1 blank page. X JUNIOR COLLEGE YEAR 6 PRELIMINARY EXAMINATION Mock Arrangement in preparation for Candidates’ Examination Higher 2 MATHEMATICS Paper 2 Additional Materials: Answer Paper Graph paper List of Formulae and Results (MF27) 9758/02 Set I 3 hours READ THESE INSTRUCTIONS FIRST Write your name on the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. 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. You are expected to use an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. ABOUT THIS PAPER X Junior College (XJC) is an unofficial initiative aimed at preparing pre-university and/or junior college students in Singapore for school-level and/or national-level H2 Mathematics examinations through self-prepared mock papers. It has no affiliation with any existing institution in Singapore or worldwide. This mock paper follows closely the 9758 H2 Mathematics GCE Advanced Level syllabus, most suitable for preparation towards preliminary examinations and A–Levels. The paper intends to explore the unconventional ways and/or applications in which topics within the syllabus can be tested, which may affect the difficulty of this paper to varying degrees. While it is ideal to attempt this paper under examination constraints, prospective candidates are reminded not to use this potentially non-conforming paper as a definitive gauge for actual performance. (For enquiries, mail to: xjuniorcollege@gmail.com)

© XJC 9758/02/I 2 Section A: Pure Mathematics [40 marks] 1 The lines 𝐿1 and 𝐿2 on the 𝑥𝑦–plane has vector equations 𝐿1∶𝐫=(𝑘1)+𝜆(1𝑘), 𝜆∈ℝ,𝑘>1, 𝐿2∶𝐫=(1𝑘)+𝜇(𝑘1), 𝜇∈ℝ,𝑘>1. By considering cartesian equations, describe a pair of transformations which transforms 𝐿1 onto 𝐿2. [4] 2 It is given that 𝑥 satisfies the inequality 𝑎2−𝑥−2𝑥2−𝑥−2≥1,for some real constant 𝑎>0. Write down, in terms of 𝑎 where appropriate, the solution intervals of the above inequality for all possible 𝑎. [5] 3 [The volume of a pyramid is 13× base area × height.] A sphere with fixed radius 𝑟 is inscribed in an octahedron shape formed by joining two square-based right pyramids of height ℎ base-to-base (see diagram). Find, in terms of 𝑟, the value of ℎ which minimises the volume of the inscribing octahedron. Justify that the resulting volume is minimum and find its value exactly in terms of 𝑟. [8] 4 It is given that 𝑦=e2𝑥cos𝑎𝑥, where 𝑎 is a real constant. (a) Show that d2𝑦d𝑥2=4d𝑦d𝑥−(𝑎2+4)𝑦. [3] (b) Using the result in (i), find the first four terms of the Maclaurin expansion of 𝑦. [4] (c) Hence, find the Maclaurin expansion of e2𝑥sin𝑎𝑥 as far as the term in 𝑥2. [2] ℎ 𝑟

© XJC 9758/02/I [Turn over 3 5 Mr Wong is considering investing money in a savings plan to purchase a car. At the first day of January 2023, he puts $100,000 into a bank account which pays compound interest at a rate of 2% per month on the last day of each month. He then puts $1,000 into the account on the first day of each subsequent month. The price of the car he wishes to buy is $300,000. (a) Find the month and year in which the total in the account will first exceed the price of the car. Explain whether this occurs on the first day or the last day of the month. [4] Instead of buying the car immediately after his account balance first exceeds the price of the car, Mr Wong decides to buy much later. Starting from the first month he pays for the car, he no longer puts additional $1,000 into the account on the first day of that month and subsequent months. The bank interest rate still applies. The car will be purchased under a hire purchase scheme. In this scheme, Mr Wong pays a deposit of $75,000 at the beginning of the first payment month. On subsequent months, the remaining amount will be paid by equal monthly instalments of $15,000 at the beginning of each month. Starting from the first payment month, interest rate is charged at the end of each month at 0.5% of the outstanding amount. All payments are deducted from Mr Wong’s account. (b) Show that, under the hire purchase scheme, Mr Wong would have completely paid for the car at the beginning of the seventeenth payment month, stating the amount to be paid on this month to the nearest cent. [5] (c) After paying the amount due on the last payment month, Mr Wong has $527,328.50 in his bank account. Find the month and year in which Mr Wong made the first payment. [5] Section B: Probability and Statistics [60 marks] 6 A continuous random variable 𝑋 has the distribution N(𝜇,𝜎2). It is known that P(𝑋≤6𝑎)=0.15866, and that the average of four independent observations of 𝑋 lies between 6𝑎 and 7𝑎 with a probability of 0.81859. (a) Find 𝜇 and 𝜎 in terms 𝑎, giving any numerical constants up to 3 decimal places. [5] (b) Sketch a graph of P(𝑋≤𝑥) against 𝑥, indicating any important features of the graph and the coordinates of the point where 𝑥=6𝑎 and 𝑥=𝜇. [2] 7 A computer simulates a repeated summation. Starting from an initial sum 0, the computer does either one of two actions: add 1 to the sum with probability 𝑝, where 0<𝑝<1, or subtract 1 from the sum with probability 𝑞, where 𝑞=1−𝑝. Each action is done independently of all other actions. (a) Show that the probability that the sum is 3 with 9 actions is 84𝑝6𝑞3. [1] (b) If it is equally likely that the sum is 3 with 9 actions as it is with 5 actions, find the possible values of 𝑝. [3] (c) Find the range of values of 𝑝 such that, for integers 𝑘>1, the computer is more likely to end up with a positive sum after (2𝑘+1) actions than it is after (2𝑘−1) actions. [5]

© XJC 9758/02/I 4 8 Box 𝐴 and Box 𝐵 each contain 𝑁 balls labelled 1, 2, 3, …, 𝑁. Each ball is equally likely to be picked from their respective box. A person picks a ball each from Box 𝐴 and Box 𝐵, and the label on each ball is denoted by 𝑋𝐴 and 𝑋𝐵 respectively. The random variable 𝑋 is defined by 𝑋={𝑋𝐴,if 𝑋𝐴≥𝑋𝐵,𝑋𝐵,if 𝑋𝐵≥𝑋𝐴. (a) Show that P(𝑋=𝑟)=2𝑟−1𝑁2 for 𝑟=1,2,3,…,𝑁. [2] (b) Find an expression for E(𝑋) in terms of 𝑁, leaving your answer as a simplified fraction. [2] [You may use the result ∑𝑟2𝑛 𝑟=1=16𝑛(𝑛+1)(2𝑛+1) without proof.] The 𝑚𝑒𝑑𝑖𝑎𝑛 of 𝑋 is defined to be the integer 𝑚 such that P(𝑋≥𝑚)≥12 and P(𝑋≤𝑚)≥12. (c) Show that √22𝑁≤𝑚≤√22𝑁+1. Hence find the median of 𝑋 when E(𝑋)=40299600. [5] 9 The events 𝐴 and 𝐵 are such that P(𝐴)=13, P(𝐴∪𝐵)=12 and P(𝐴|𝐵′)=16. (a) Find P(𝐵) and P(𝐴∩𝐵). [3] Another event 𝐶 is such that P(𝐶)=14. It is given that 𝐴 is independent of 𝐶, and 𝐵 is independent of 𝐶. (b) Using the result P(𝐴∩(𝐵∪𝐶))=P((𝐴∩𝐵)∪(𝐴∩𝐶)), or otherwise, show that P(𝐴∪𝐵∪𝐶)=P(𝐴)+P(𝐵)+P(𝐶)−P(𝐴∩𝐵)−P(𝐴∩𝐶)−P(𝐵∩𝐶)+P(𝐴∩𝐵∩𝐶). [1] (c) Given that P(𝐴′∩𝐵′∩𝐶′)=1130, find P(𝐴∩𝐵∩𝐶). [2] (d) Find exactly the maximum and minimum possible values of P(𝐴′∩𝐵′∩𝐶′). [4]

© XJC 9758/02/I [Turn over 5 10 An electronics company produces chips for devices. Since its establishment, the company has produced 100 000 chips, 1 000 of which are of the newest model. The electronics company takes a sample of chips and is then able to carry out a 𝑧-test to determine the effect of the ne