XJC H2 Math - Set 3 - P1 (QP)

X Junior College [Turn over This document consists of 5 printed pages and 3 blank pages. X JUNIOR COLLEGE YEAR 6 PRELIMINARY EXAMINATION Mock Arrangement in preparation for Candidates’ Examination Higher 2 MATHEMATICS Paper 1 Additional Materials: Answer Paper Graph paper List of Formulae and Results (MF27) 9758/01 Set III 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 9758/01/III 2 1 Show that the 𝑦-coordinates of the turning points of 𝑦=e𝑥cos𝑥 in increasing 𝑥-coordinate values follow a geometric progression, stating its common ratio in exact form. [3] 2 A non-constant arithmetic sequence 𝑢1, 𝑢2, 𝑢3, … with first term 1 and common difference 𝑑 is such that the value of (𝑢𝑛+1+𝑢𝑛+2+⋯+𝑢2𝑛) is a constant multiple of the value of (𝑢1+𝑢2+⋯+𝑢𝑛), for all integers 𝑛≥1. (a) Show that 2+(3𝑛−1)𝑑2+(𝑛−1)𝑑 is constant. [2] (b) Hence, find the value of 𝑑. [2] 3 (a) On the same axes, sketch the curves 𝑦2=𝑎𝑥 and 𝑎𝑦=𝑥2−2𝑎𝑥, where 𝑎 is a positive constant. Indicate, for both curves, the point where 𝑦=−𝑎. [2] (b) Hence, using a suitable value for 𝑎, solve exactly the inequality √𝑥≤|𝑥2−2𝑥|. [3] 4 A curve has equation 𝑦=f(𝑥). It is known that the curve has asymptotes at 𝑥=0 and 𝑥=1, and that it has exactly three stationary points. (a) Given that f(𝑥)=f(1−𝑥), show that one of the stationary points has an 𝑥–coordinate of 12. [2] (b) Given further that f(𝑥)=f(1𝑥), find the 𝑥–coordinate of the other two stationary points. [4] 5 It is given that f(𝑥)=ln(ln𝑥) and g(𝑥)=1ln𝑥. Using a suitable integration by parts,find (a) ∫{f(𝑥)+g(𝑥)}d𝑥. [2] (b) ∫{f(𝑥)+(g(𝑥))2}e2 e d𝑥. Give your answer in exact form. [4] 6 A curve 𝐶 has an equation 𝑦2=𝑥𝑘f(𝑥), where 𝑘 is a constant and f is a function in 𝑥. It is given that 𝐶 passes through the point (1,1) and that it also satisfies the differential equation 2𝑥𝑦d𝑦d𝑥=𝑦2+𝑥2−2. (a) Using calculus, find 𝑘 and f(𝑥). [4] (b) Sketch 𝐶, labelling exactly the coordinates of any stationary point and any point where the curve meets the axes, as well as the equation of any asymptotes. [3]

© XJC 9758/01/III [Turn over 3 7 A sequence {𝑢𝑛} for integers 𝑛≥1 is given by 𝑢𝑛=52(1+2i)𝑛+52(1−2i)𝑛. (a) Show that 𝑢𝑛+2=2𝑢𝑛+1−5𝑢𝑛, stating the values of the two base cases for this recursion. Hence, explain why 𝑢𝑛 is real for all 𝑛. [4] It is given that 𝑢𝑛+1𝑢𝑛→𝑟 as 𝑛→∞. (b) Show that 𝑟2+𝑎𝑟+𝑏=0, where 𝑎 and 𝑏 are constants to be determined. [2] (c) Let the roots of the equation in (b) be 𝛼 and 𝛽. Show that, for a positive integer 𝑚, 𝛼𝑚+𝛽𝑚 can be expressed in terms of 𝑢𝑚. Hence, or otherwise, find the least 𝑚 such that 𝛼𝑚+𝛽𝑚<0, stating this value. [2] 8 The roots of 𝑧5=1 in increasing arguments are 𝑧1, 𝑧2, 𝑧3, 𝑧4 and 𝑧5, which are represented in an Argand diagram by points 𝑍1, 𝑍2, 𝑍3, 𝑍4 and 𝑍5 respectively. It is known that the five points are vertices of a regular polygon. (a) Explain why you can expect the polygon to be symmetrical about the real axis. [1] (b) Determine which of the five roots is real and state its value. Hence, explain why arg(𝑧4)=25𝜋. [2] It is given that 𝑧4=14(−1+√5)+14√(10+2√5)i. (c) Find |𝑧4−𝑧3|, giving your answer in the form √𝑘, where 𝑘 is an exact constant to be determined. [2] (d) Find the exact area of the regular polygon. [2] (e) The points 𝑍1, 𝑍2, 𝑍3, 𝑍4 and 𝑍5 in the Argand diagram are transformed such that the complex numbers they represent are now roots of 𝑧5=−243i. Plot these transformed points on an Argand diagram, indicating the moduli of the transformed complex numbers. [2] 9 A curve 𝐶 has parametric equations 𝑥=2cos𝜃+cos2𝜃, 𝑦=2sin𝜃−sin2𝜃, for 0≤𝜃≤2𝜋. (a) Express d𝑦d𝑥 in the form cos𝜃+𝑎sin𝜃,where 𝑎 is a constant to be determined. [3] (b) Find the equation of the tangent to 𝐶 at the point with parameter 𝜃 and show that it can be written as 𝑦sin𝜃+cos2𝜃=𝑥(cos𝜃−1)+cos𝜃. [2] Points 𝑃 and 𝑄 on 𝐶 have parameters 𝑝 and 𝑞 respectively. The tangent at 𝑃 meets the tangent at 𝑄 at the point 𝑅. It is given that the two tangents are perpendicular to one another. (c) Verify that 𝑞=𝑝+𝜋. Hence, find the 𝑥-coordinate of 𝑅 in the form 𝑘cos2𝑝, where 𝑘 is a constant to be determined, and find similarly the 𝑦-coordinate of 𝑅 in terms of 𝑝. [4] (d) Show that 𝑅 traces a circle as 𝑝 varies from 0 to 2𝜋, stating its radius and centre, as well as the direction in which 𝑅 traces this circle (clockwise or counterclockwise). [2]

© XJC 9758/01/III 4 10 The variables 𝑥 and 𝑦 are given such that their rates of change with respect to time 𝑡 satisfies the equation d𝑥d𝑡=2𝑦−2𝑥 and d𝑦d𝑡=−d𝑥d𝑡−3𝑦. It is also given that initially 𝑥=5 and 𝑦=0. (a) Show that d2𝑥d𝑡2+𝑎d𝑥d𝑡+𝑏𝑥=0,where 𝑎 and 𝑏 are constants to be determined. [3] (b) Given that 𝑥=𝐴e𝑘𝑡+𝐵e𝑚𝑡, where 𝐴, 𝐵, 𝑘 and 𝑚 are constants and 𝑘≠𝑚, find 𝑥 and 𝑦 in terms of 𝑡. [6] (c) Sketch a graph of 𝑦 against 𝑥, indicating the coordinates of the endpoints and the turning point. [4] 11 Functions f and h are given by f(𝑥)=1−|2𝑥−1|, for 0≤𝑥≤1, h(𝑥)=2cos(𝜋𝑥), for 0≤𝑥≤1. (a) Explain why the following functions exist. (i) h−1 [1] (ii) fh−1 [1] (iii) hfh−1 [2] A function g is given such that f, g and h satisfy the relationship g(𝑥)=hfh−1(𝑥). Under this relationship, f and g is said to be topologically conjugate as there exists a function h which conjugates f to g, and vice versa. Scientists and mathematicians examine functions which exhibit topological conjugacy in the study of dynamical systems to predict the long-term behaviour of a system after multiple iterations. (b) By considering f(𝑥) as two non-modulus functions in the domain 0≤𝑥≤12 and 12≤𝑥≤1 respectively, show that g(𝑥)=𝑥2−2, for −2≤𝑥≤2, explaining how the domain of g is obtained. [5] (c) A function P is given by P(𝑥)=𝑥4−4𝑥2+2, where −2≤𝑥≤2. Another function Q is such that P and Q are topologically conjugate. Find Q, giving your answer in a similar form. [4]

© XJC 9758/01/III 5 12 Astronomers use satellite