XJC H2 Mathematics 9758 – Set I: Paper 1 (Questions)

X Junior College [Turn over This document consists of 25 printed pages and 3 blank pages. X JUNIOR COLLEGE YEAR 6 PRELIMINARY EXAMINATION Mock Arrangement in preparation for Candidates’ Examination Higher (than) 2 CANDIDATE NAME MATHEMATICS Paper 1 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)

© XJC 9758/01/I 2 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]

© XJC 9758/01/I [Turn over 3 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]

© XJC 9758/01/I 4 3 An arithmetic sequence with first term 𝑎 and common difference 𝑑 is such that the sum of its first 𝑛 terms is 𝑚, and the sum its first 𝑚 terms is 𝑛. (i) Find 𝑎 and 𝑑 in terms of 𝑚 and 𝑛. [4]

© XJC 9758/01/I [Turn over 5 (ii) Hence, show that the sum of the first (𝑚+𝑛) terms of the sequence is −(𝑚+𝑛). [2]

© XJC 9758/01/I 6 4 A curve 𝐶 has equation 𝑥2+3𝑥𝑦−𝑦2+4𝑥=1. (i) Find d𝑦d𝑥 in terms of 𝑥 and 𝑦. [1] The tangents to 𝐶 at two distinct points meet at (6,−4). (ii) Show that these points satisfy the equation 2𝑥+13𝑦=11. [3]

© XJC 9758/01/I [Turn over 7 (iii) Hence, find the equation of these tangents, giving your answer in the form 𝑎𝑥+𝑏𝑦+𝑐=0, where 𝑎, 𝑏 and 𝑐 are integers to be determined. [3]

© XJC 9758/01/I 8 5 The complex numbers 𝑢 and 𝑣 have the same modulus 𝑟 and arguments 𝛼 and 𝛽 respectively, with 0<𝛽<𝛼<12𝜋. (i) Express 𝑢−𝑣𝑢+𝑣 in the form 𝑘tan(𝛼−𝛽2),where 𝑘 is a complex number to be found. [4]

© XJC 9758/01/I [Turn over 9 On an Argand diagram, the points 𝑍 and 𝑊 represent the complex numbers (𝑢−𝑣) and (𝑢+𝑣) respectively and angle 𝑂𝑊𝑍 = 𝜃. (ii) Use the result in (i) to show that triangle 𝑂𝑊𝑍 is a right triangle, stating the right angle. [1] (iii) By considering the ratio of the lengths 𝑂𝑍:𝑂𝑊, deduce an expression for the angle 𝜃 in terms of 𝛼 and 𝛽. Hence, express |𝑢−𝑣| and |𝑢+𝑣| in terms of 𝑟 and 𝜃. [3]

© XJC 9758/01/I 10 6 Relative to the origin 𝑂, the points 𝐴 and 𝐵 have position vectors 𝐚 and 𝐛 respectively such that 𝑂, 𝐴 and 𝐵 are not collinear. The point 𝐶 lies on the line segment 𝐴𝐵 such that 𝐴𝐶:𝐶𝐵=(1−𝜆):𝜆 and that 𝑂𝐶 bisects angle 𝐴𝑂𝐵. (i) Find 𝜆 in terms of |𝐚| and |𝐛|. [4]