XJC H2 Mathematics 9758 – Set II: Paper 2 (Questions)

X Junior College [Turn over This document consists of 26 printed pages and 2 blank page. X JUNIOR COLLEGE YEAR 6 PRELIMINARY EXAMINATION Mock Arrangement in preparation for Candidates’ Examination Higher (than) 2 CANDIDATE NAME MATHEMATICS Paper 2 Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) 9758/01 Set II 3 hours READ THESE INSTRUCTIONS FIRST Write your name on the work you hand in. 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 or correction fluid. Answer all questions. Write your answers in the space provided in the Question Paper. 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 of clear presentation in your answers. 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. (i) Sketch 𝑦=f(𝑥) for −2𝑘≤𝑥<4𝑘. Indicate clearly all axial intercepts and minimum points. [2] (ii) Find, in terms of 𝑘, the exact area bounded by the curve 𝑦=f(𝑥) and the 𝑥–axis for 0≤𝑥<2𝑘. [1]

© XJC 9758/02/II [Turn over 3 (iii) Deduce that ∫f(𝑥)∞−2𝑘d𝑥=𝑎𝑘2,for some exact real value 𝑎 to be determined. [2]

© XJC 9758/02/II 4 2 A sequence 𝑢1,𝑢2,𝑢3,… is given by 𝑢1=1, 𝑢2=2 and 𝑢𝑛+2=2𝑢𝑛+1−𝑢𝑛+2𝑛−1 for 𝑛≥1. Using the substitution 𝜈𝑛=𝑢𝑛+1−𝑢𝑛, show that the recurrent relation above can be reduced to 𝜈𝑛+1−𝜈𝑛=2𝑛−1. Use this result to find a general formula for 𝑢𝑛 in terms of 𝑛. [6]

© XJC 9758/02/II [Turn over 5 2 [Continued]

© XJC 9758/02/II 6 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] 𝐴 𝐵 𝐶 𝐷 𝐸 𝐹 𝑂 𝜃 𝑟

© XJC 9758/02/II [Turn over 7 3 [Continued]

© XJC 9758/02/II 8 4 (a) The complex number 𝑧 is such that 𝑧1+𝑧2 is real.If 𝑧 is not real,show that |𝑧|=1. [5]

© XJC 9758/02/II [Turn over 9 (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]

© XJC 9758/02/II 10 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. (i) Show that 𝑂𝐶⃖⃖⃖⃖⃑=(|𝐛||𝐚|−1)(𝐛−𝐚) and 𝐴𝐷⃖⃖⃖⃖⃖⃑=(|𝐛||𝐚|−1)𝐛. [5]