XJC H2 Math - Set 2 - P2 (QP)

X Junior College [Turn over This document consists of 6 printed pages and 2 blank pages. 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 II 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/II 2 Section A: Pure Mathematics [40 marks] 1 A piecewise function f is given such that f(𝑥)={−√𝑘2−(𝑥−𝑘)2,0≤𝑥<𝑘,𝑥−2𝑘, 𝑘≤𝑥<2𝑘, and f(𝑥+2𝑘)=12f(𝑥) for all real values of 𝑥,where 𝑘 is a positive constant. (a) Sketch 𝑦=f(𝑥) for −2𝑘≤𝑥<4𝑘. Indicate clearly all axial intercepts and minimum points. [2] (b) Find, in terms of 𝑘, the exact area bounded by the curve 𝑦=f(𝑥) and the 𝑥–axis for 0≤𝑥<2𝑘. [1] (c) Deduce that ∫f(𝑥)∞−2𝑘d𝑥=𝑎𝑘2,for some exact real value 𝑎 to be determined. [2] 2 A resort is built around a right conical mountain of height 85 metres and base radius 720 metres. Connecting the entrance at A, located on the mountain base, to the main resort at B, located on the mountain slope, is a gondola track of length 𝐷 metres. The gondola track is built along the slope such that it is the shortest possible track connecting A to B. A gondola cart travels from A to B at a constant speed of 1 metre per second. As the cart moves, its distance from the summit C is 𝑧 metres. It is known that BC =𝑀 metres (see diagram). (a) Given that the cart has travelled 𝑠 metres from A through track AB, show that 7252+𝐷2−𝑀2𝐷=7252+𝑠2−𝑧2𝑠. [2] (b) It is given further that 𝐷=900 and 𝑀=300. At a point 𝑃 on the track, the altitude of the gondola is decreasing at 4 metres per second. Show that 𝑧 increases at 14536 metres per second at this point. Find also the time taken for the cart to go from A to 𝑃, giving your answer to the nearest second. [4] A B C CART 𝑧 𝑀

© XJC 9758/02/II [Turn over 3 3 [The volume of a square-based pyramid is 13× base area × height.] A sphere with centre 𝑂 has a fixed radius 𝑟. A right pyramid with a square base is inscribed within it, with point 𝐴 as its apex and 𝐵𝐶𝐷𝐸 as its square base. A perpendicular dropped from 𝐴 passes through 𝑂 and intersects 𝐵𝐶𝐷𝐸 at point 𝐹. Point 𝐴 subtends an angle 𝜃 with the corners of the base at 𝑂, where 0°<𝜃<180° (see diagram). Find, in degrees, the angle 𝜃 which maximises the volume of the pyramid. Justify that the resulting volume is maximum and find its value exactly in terms of 𝑟. Show all your working clearly. [8] 4 (a) The complex number 𝑧 is such that 𝑧1+𝑧2 is real.If 𝑧 is not real,show that |𝑧|=1. [5] (b) A positive integer 𝑚 is the smallest possible such that 𝛾=(6+𝑝i)𝜔2+(−3𝑚+𝑞i)𝜔+2𝑚, where 𝑝 and 𝑞 are real values, is purely imaginary at two distinct 𝜔 values. Given that the two possible 𝛾 values are conjugate pairs, find these 𝛾 values in terms of 𝑞. [5] 5 With respect to an origin 𝑂, points 𝐴 and 𝐵 have position vectors 𝐚 and 𝐛 respectively such that |𝐚|<|𝐛| and the angle 𝜃 between 𝐚 and 𝐛 is acute. Points 𝐶 and 𝐷 are given such that 𝑂𝐴𝐵𝐶 and 𝑂𝐴𝐷𝐵 are isosceles trapeziums, and that 𝐶𝐷 is parallel to 𝐚 and equal in length to 𝑂𝐵. (a) Show that 𝑂𝐶⃖⃖⃖⃖⃑=(|𝐛||𝐚|−1)(𝐛−𝐚) and 𝐴𝐷⃖⃖⃖⃖⃖⃑=(|𝐛||𝐚|−1)𝐛. [5] (b) Express |𝐛−𝐚|2 in terms of 𝐚, 𝐛 and 𝜃. Hence, given that |𝑂𝐶⃖⃖⃖⃖⃑|=|𝐴𝐷⃖⃖⃖⃖⃖⃑|=|𝐚|, find 𝜃 in degrees. [5] (c) Name the shape of 𝑂𝐴𝐷𝐵𝐶. [1] 𝐴 𝐵 𝐶 𝐷 𝐸 𝐹 𝑂 𝜃 𝑟

© XJC 9758/02/II 4 Section B: Probability and Statistics [60 marks] 6 A 3×3×3 solid cube is white inside and has a black surface all around. It is then sliced into 27 pieces of 1×1×1 cubes. A machine shuffles the cube slices and then blindly reassembles them into a 3×3×3 cube. Find the probability that the resulting cube will appear completely black on the surface. [6] 7 Two players 𝐴 and 𝐵 are contestants of a gaming competition which consists of multiple rounds of duels. To win a round, a player must be the first to win three duels. It is known that the probability of 𝐴 winning a duel is 𝑝, and the outcomes of different duels are independent. Let 𝑋 be the number of duels played in one round. (a) Show that P(𝑋=3)=1−3𝑝+3𝑝2. [2] (b) Tabulate the probability distribution of 𝑋, expressing each probability in increasing powers of 𝑝. [3] In the middle of the competition, a commentator notices that none of the rounds played have ended at the third duel. He makes a statement that the competition will continue as such in the long run. (c) Find the range of values of 𝑝 for which the commentator’s statement is true. [3] 8 A confectionery company owns a machine which produces chocolate bars such that the mass 𝑀, in grams, of a chocolate bar has an average of 250 grams with standard deviation 150 grams. (a) Explain why the masses of the chocolate bars may not be approximated properly using a normal distribution with the given average and standard deviation. [2] The company then decides to upgrade its machineries. To evaluate the extent of upgrade, the mass of each chocolate bars is measured using a weighing scale. It is found that the measurements are normally distributed with mean 450 grams and standard deviation 𝜎 grams such that 3P(437.33<𝑀<462.67)=2P(462.67<𝑀<514.09) and P(437.33<𝑀<514.09)=P(𝑀<450). (b) Find 𝜎. [3] For the rest of the question, 𝜎=50. The weighing scale used is found to be faulty, and thus the recorded measurements need to be corrected. The old measurements are each scaled by a positive factor 𝑘 and then moderated by 𝑚 units. The new measurements now follow the normal distribution N(550,252). (c) Find the value of 𝑘 and 𝑚. Hence, calculate the probability that the new measurement of a randomly selected chocolate bar is now at least twice as heavy as its old measurement. [5]

© XJC 9758/02/II [Turn over 5 9 A city is experiencing an outbreak due to a virus known to infect 2% of the population. Medical experts are examining 5000 people and their blood samples to find out th