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

X Junior College [Turn over This document consists of 23 printed pages and 1 blank page. 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 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 9758/01/II 2 1 Describe a sequence of transformations that transform the graph of 𝑦=sin𝑥 onto the graph of 𝑦=sin2𝑥. [4]

© XJC 9758/01/II [Turn over 3 2 (i) Using standard series from the List of Formulae (MF26), expand cos(𝑥+sin𝑥) as far as the term in 𝑥4. [3] (ii) By expanding cos(𝑥+sin𝑥)+cos(𝑥−sin𝑥) as far as the term in 𝑥4, or otherwise, evaluate ∫[cos(𝑥+sin𝑥)+cos(𝑥−sin𝑥)]𝜋60 d𝑥, giving your answer in exact form. [3]

© XJC 9758/01/II 4 3 An arithmetic series has an integer common difference 𝑑. The sum of the first 𝑝, 2𝑝 and 3𝑝 terms of the series are 185, 670, and 1455 respectively. Given that 0<𝑑<𝑝, find a general formula for the 𝑛th term of this series. [6]

© XJC 9758/01/II [Turn over 5 4 Given a continuous function f, explain, with the aid of a sketch, why the expression limℎ→0f(𝑥+ℎ)−f(𝑥)ℎ approximates to f′(𝑥). [2] Given that f(𝑥)=cos𝑏𝑥, for some real value 𝑏>0, use the expression above to show that f′(𝑥)=−𝑏sin𝑏𝑥. [4]

© XJC 9758/01/II 6 5 (i) Show algebraically that the inequality √2𝑥+3>1+√4𝑥−1 can be reduced to 14≤𝑥<12. [4]

© XJC 9758/01/II [Turn over 7 (ii) Find the range(s) of values of 𝑥, where 0≤𝑥≤𝜋, such that √cos2𝑥+4>1+√2cos2𝑥+1. [3]

© XJC 9758/01/II 8 6 A complex number 𝑧=cos𝜃+isin𝜃 is such that sin𝜃≠0. (i) Show that ∑𝑧2𝑟−1𝑛𝑟=1=1−𝑧2𝑛𝑧−1−𝑧. [1] (ii) De Moivre’s theorem states that for any complex number 𝜔 and any integer 𝑟, 𝜔𝑟=|𝜔|𝑟(cos𝑟𝜃+𝑖sin𝑟𝜃). Use this theorem and the result in (i) to show that ∑sin[(2𝑟−1)𝜃]𝑛𝑟=1 =sin2𝑛𝜃sin𝜃. [5]

© XJC 9758/01/II [Turn over 9 (iii) Hence,deduce that∑(2𝑟−1)cos[(2𝑟−1)𝜋2𝑛]𝑛𝑟=1 =−cosec(𝜋2𝑛)cot(𝜋2𝑛). [3]

© XJC 9758/01/II 10 7 The integral 𝐼𝑛 where 𝑛=1,2,3,… is given by 𝐼𝑛=∫1(1+𝑥2)𝑛𝑐0d𝑥, 𝑐>0. (i) Find 𝐼1 in terms of 𝑐. [1] (ii) Show that 𝐼𝑛+1=𝑐2𝑛(1+𝑐2)𝑛+(2𝑛−12𝑛)𝐼𝑛 for 𝑛>1. [5]