XJC H2 Math - Set 1 - 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 I 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/I 2 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] 2 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. (a) Find 𝑎 and 𝑑 in terms of 𝑚 and 𝑛. [4] (b) Hence, show that the sum of the first (𝑚+𝑛) terms of the sequence is −(𝑚+𝑛). [2] 3 A complex number 𝑧=𝑥+i𝑦, where 𝑥 and 𝑦 are real, satisfies |𝑧+3|=2Re(𝑧). (a) Show that 𝑥 and 𝑦 are related by the equation of a hyperbola. State the equations of its asymptotes. [3] (b) 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] (c) Deduce the exact range of arg(𝑧−1). [2] 4 A curve 𝐶 has equation 𝑥2+3𝑥𝑦−𝑦2+4𝑥=1. (a) Find d𝑦d𝑥 in terms of 𝑥 and 𝑦. [1] The tangents to 𝐶 at two distinct points intersect at (6,−4). (b) Show that these points satisfy the equation 2𝑥+13𝑦=11. [3] (c) 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] 5 It is given that 𝑢=𝑟cos𝜃+i 𝑟sin𝜃, for some real values 𝑟 and 𝜃 such that 𝑟>0 and 0<𝜃<12𝜋. (a) Express 𝑢−𝑟𝑢+𝑟 in the form 𝑘tan12𝜃,where 𝑘 is a complex number to be found. [3] With respect to the origin 𝑂 on an Argand diagram, the points 𝑍 and 𝑊 represent the complex numbers (𝑢−𝑟) and (𝑢+𝑟) respectively. (b) Explain why triangle 𝑂𝑊𝑍 is a right triangle, stating the right angle. [1] (c) Deduce an expression for angle 𝑂𝑊𝑍 in terms of 𝜃 and the distance between 𝑍 and 𝑊 in terms of 𝑟. Hence, express |𝑢−𝑟| and |𝑢+𝑟| in terms of 𝑟 and 𝜃. [4]

© XJC 9758/01/I [Turn over 3 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. (a) Find 𝜆 in terms of |𝐚| and |𝐛|. [4] (b) Another point 𝐷 lies on the line segment 𝐴𝐵 such that 𝐴𝐶=𝐵𝐷. By considering suitable dot products, show that 𝑂𝐷2−𝑂𝐶2=(|𝐛|−|𝐚|)2. [4] 7 (a) The functions f and g are defined such that gf exists and f(𝑥)=𝑥+12𝑥−1, 𝑥∈ℝ,𝑥≠12, gf(𝑥)=2𝑥−7𝑥−2, 𝑥∈ℝ,𝑥≠2. (i) Explain why f has an inverse. Find its inverse in a similar form, stating its domain. [3] (ii) Find a simplified expression for g(𝑥). [2] (b) A function h is given such that h(𝑥)−2h(𝑥−1𝑥)+h(11−𝑥)=𝑥3+1𝑥2−𝑥,𝑥∈ℝ,𝑥≠0,1. (i) Find the value of h(2), h(0.5) and h(−1). [3] (ii) Deduce an expression for hhh(𝑥). Hence,given that hh(𝑥)=11−𝑥,find h(𝑥). [2] 8 The integral 𝐼𝑛, where 𝑛=0,1,2,3,…, is defined by 𝐼𝑛=∫𝑥𝑛√𝑎𝑥−𝑥2𝑎0 d𝑥, for some constant 𝑎>0. (a) By substituting 𝑥=𝑎sin2𝜃, find the exact value of 𝐼0 in terms of 𝑎. [4] (b) Show that 𝐼𝑛+1=(2𝑛+32𝑛+6)𝑎𝐼𝑛 for 𝑛≥0. [4] (c) Find the exact area of the region enclosed by the curve 𝑦=√(2𝑥−𝑥2)5 and the 𝑥–axis. [4]

© XJC 9758/01/I 4 9 A cardboard with negligible thickness 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). (a) Show that 𝑎=6.25 and find 𝑏. [2] (b) The outer surface area of the shape can be expressed as 𝑝|𝐷𝐺⃖⃖⃖⃖⃖⃗×𝐸𝐺⃖⃖⃖⃖⃖⃗|+𝑞|𝐷𝐺⃖⃖⃖⃖⃖⃗×𝐹𝐺⃖⃖⃖⃖⃗|+𝑟|𝐸𝐺⃖⃖⃖⃖⃖⃗×𝐹𝐺⃖⃖⃖⃖⃗|. Find the exact value of 𝑝, 𝑞 and 𝑟 and hence evaluate this area numerically. [6] [The area of a trapezium is 12× height × sum of parallel sides.] (c) The folded shape is detached along DG and flattened such that BCD remains attached to BDGE at BD and to CDGF at CD. Find the shortest distance between C and E on this flat surface. [5] 10 A free-hanging rope is held by two people standing at a distance of 2𝑎 units from one another. Due to its weight, the rope curves downwards in the shape of a catenary defined by parametric equations 𝑥=𝑎ln𝑡,𝑦=𝑎2(𝑡+1𝑡),for 𝑡1≤𝑡≤𝑡2, where parameters 𝑡1 and 𝑡2 correspond to the two ends of the rope at which it is being held. (a) Find d𝑦d𝑥 in terms of 𝑡. [1] (b) Given that the lowest point on the rope is horizontally in the middle of both people, find 𝑡1 and 𝑡2. [3] (c) Find, in degrees, the acute angle that the rope ends make with the horizontal as it curves downwards. [2] The rope is rotated about the line 𝑦=𝑘 at which height it is being held. The rope remains taut throughout its motion and the shape of the curve is maintained. (d) Show that 𝑦=𝑎2(e𝑥𝑎+e−𝑥𝑎). Hence find 𝑘 exactly in terms of 𝑎. [2] (e) Determine the exact volume of the space enclosed by the rotating rope, expressing your answer in the form 𝐴𝑎3(𝐵+𝐶e2+𝐷e−2), where 𝐴, 𝐵, 𝐶 and 𝐷 are exact constants to be determined. [5] 𝐶 𝐷(4,𝑎,𝑏) 𝐵(1,7,2) 𝐺(3,0,8) 𝐹(8,2,1) 𝐸(−1,1,−2) 𝐴

© XJC 9758/01/I 5 11 In Ecology, the Lotka-Volterra equations are often used to model the population dynamics of a predator-prey ecosystem. Under this model, the population of prey and predators are each defined by a differential equation: the prey population grows at a rate proportional to its own population and declines at a rate proportional to the product of its own population and the predator population, while the predator population declines at a rate proportional to its own populatio