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

X Junior College [Turn over This document consists of 24 printed pages and 4 blank pages. 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 I 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)

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© XJC 9758/02/I [Turn over 3 Section A: Pure Mathematics [40 marks] 1 When the complex polynomial P(𝑧) is divided by (𝑧+i), (𝑧−i) and (𝑧2+1), the remainders are 1+i, 1−i and 𝐴𝑧+𝐵 respectively. Find 𝐴 and 𝐵. [3]

© XJC 9758/02/I 4 2 A complex number 𝑧 can be expressed as 𝑥+i𝑦, where 𝑥 and 𝑦 are real, and satisfies |𝑧+3|=2Re(𝑧). (i) Show that 𝑥 and 𝑦 are related by the equation of a hyperbola. State the equations of its asymptotes. [3] (ii) Sketch the part of the hyperbola on which any point (𝑥,𝑦) satisfies the above equation in 𝑧, including its asymptotes. On your sketch, indicate the values of the axial intercepts of these asymptotes. [2]

© XJC 9758/02/I [Turn over 5 (iii) Deduce the exact range of arg(𝑧−1). [2]

© XJC 9758/02/I 6 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] ℎ 𝑟

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© XJC 9758/02/I 8 4 It is given that 𝑦=e2𝑥cos𝑎𝑥, where 𝑎 is a real constant. (i) Show that d2𝑦d𝑥2=4d𝑦d𝑥−(𝑎2+4)𝑦. [3] (ii) Using the result in (i), find the first four terms of the Maclaurin expansion of 𝑦. [4]

© XJC 9758/02/I [Turn over 9 (iii) Hence, find the Maclaurin expansion of e2𝑥sin𝑎𝑥 as far as the term in 𝑥2. [2]

© XJC 9758/02/I 10 5 A cardboard is cut and folded into the shape following the surface of a truncated tetrahedron BCDEFG, which is formed by removing the tetrahedron ABCD from the larger tetrahedron AEFG. It is known that the faces BCFE and EFG are hollow, and that the plane BCD is parallel to plane EFG. The coordinates of B, D, E, F and G are (1,7,2), (4,𝑎,𝑏), (−1,1,2), (8,2,1) and (3,0,8) respectively (see diagram). (i) Show that 𝑎=6.25 and find 𝑏. [2] 𝐶 𝐷(4,𝑎,𝑏) 𝐵(1,7,2) 𝐺(3,0,8) 𝐹(8,2,1) 𝐸(−1,1,−2) 𝐴