2012 NYJC H2 Math Prelim_P2_Question

[Turn Over NANYANG JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 MATHEMATICS 9740/02 Paper 2 17 Sep 2012 3 hours Additional Materials: Answer Papers List of Formulae (MF15) Graph paper READ THESE INSTRUCTIONS FIRST Write your name and class on every script you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, 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 a graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic 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. This document consists of 7 printed pages.

2 2012 NYJC 9740/02 [Turn Over Section A [40 marks] 1 (a) The complex numbers z and w are such that z = 1 + i a, w = – b – i where a and b are real and positive. Given that zw = 6 – 11i, find the exact values of a and b. [4] (b) The conjugate of a complex number u has modulus 2 and argument 2 3 π and another complex number v has modulus 5 and argument 3 4 π . Find the exact values of the modulus and argument of 2 v u . [2] Hence, find the smallest positive integer n such that the point representing 2 n v u       lies on the negative imaginary axis. [2] 2 The function f is defined by f : x /arrowbarrightx2 – 4 x, x ∈ /Rbb, 1x < . (i) Show that 1f − exists and find 1f − in a similar form. [4] The function g is defined by g : x /arrowbarright 11 3x+ + , , 3 x x ∈ ≠ − /Rbb. (ii) Show that the composite function gf exists. [1] (iii) Find an expression for gf( x). [1] The function h is defined by h : x /arrowbarrightgf( x), x ∈ /Rbb, 1x < . (iv) Using differentiation, determine whether h is stric tly increasing or strictly decreasing. Hence or otherwise, find the range of h. [5]

3 2012 NYJC 9740/02 [Turn Over 3 In a futuristic movie, a hemispherical force field dome (an open hemisphere) is created to protect a cylindrical silo under construction to house deadly weapons. The circumference of the top of the silo lies just within the dome as shown in the diagram b elow. The power supply is such that the radius r of the dome generated is fixed at 0.5 km. Let x km and y km be the ra