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 radius and height of the silo respectively. Show that the volume of the silo can be expressed as 31 4 y y π   −    km 3. Find the exact maximum volume of the silo which can be constructed. [5] A power surge causes the dome to be enlarged (still retaining its hemispherical shape) such that its volume is increasing at a constant rate of 0.75 km 3s− 1. Find the exact rate of increase of surface area of the dome when 1d 3 kms d 4 r t π −= . [5] [The formula for the volume and surface area of a sphere are V = 34 3 rπ and A = 4 2rπ .] 4 (a) The curve C has equation ( )ln 2 y x = − . The region R is bounded by C, the lines ln 2 y = , 1x = and 1. x = − Find the exact area of R. [5] (b) The curve C is defined parametrically by ( ) ( ) 2 231 , ln for 1 x t y t t = + = ≤ − . Find the volume of the solid generated when the re gion enclosed by C, the lines x = 0 and x = 1 and the x-axis is rotated 2 π radians about the x-axis, giving your answer correct to 2 decimal places. [6] y x r

4 2012 NYJC 9740/02 [Turn Over Section B [60 marks] 5 The organisers of a conference wish to conduct a su rvey on 80 participants to gather feedback about their conference experience. The 800 participants who attended the conference come from various educational institutions, as shown in the table below: Primary school Secondary school Junior college Poly technic 350 250 150 50 (i) Describe briefly how a sample of 80 participants c an be selected by using an appropriate sampling method. [2] (ii) State one advantage and one disadvantage of the sa mpling method suggested in (i) . [2] 6 A committee consisting of six people is to be selec ted from five women and six men. Three of the women are sisters. Find the number of ways the committee can be formed if (i) the chosen committee will contain exactly two men, [1] (ii) at least one of the sisters are included. [2] The chosen committee consists of two particular sis ters together with 3 other men and one other woman. They are seated at a round table meant for six persons. Find the number of possible arrangements if (iii) one of the men is to be seated between the two sis ters, [2] (iv) the two sisters are sitting directly opposite each other. [2]

5 2012 NYJC 9740/02 [Turn Over 7 The average trade-in value, y thousand dollars, of a particular make of used car depreciates with the age of the car, x years, according to the following table. Age, x 2 3 4 5 6 7 8 Trade-in value, y 53.9 49.9 44.8 38.0 34.6 32.5 29.8 (i) Draw a scatter diagram to illustrate the data. [2] (ii) Comment on whether a linear model would be appropr iate, referring both to the scatter diagram and the context of the question. [2] (iii) Explain why in this context a quadratic model would probably not be appropriate for long-term predictions. [1] (iv) Fit a model of the form ln y ax b = + to the data, and use it to predict the trade-in va lue of a 15 2 -year-old car. [2] 8 A bag contains 10 red and 15 green beads which are indistinguishable apart from colour. Beads are drawn from the bag singly and at random, with replacement. (i) Calculate the probability that the first red bead is obtained on or before the 5 th draw. [2] (ii) Calculate the probability of first obtaining a gre en bead on the 8 th draw, given that no green bead has been obtained in the first 5 draws. [2] (iii) Find the probability that exactly r draws are required for beads of both colours to be obtained, leaving your answer in the form 2 2 ( ) r r a b c − − + , where a, b and c are of the form p q , ,p q +∈ /Zbb. [2] (iv) Calculate the probability of first obtaining beads of different colours after 5 or more draws. [3]

6 2012 NYJC 9740/02 [Turn Over 9 In a box of cheese rings from a certain manufacture r, the mass of sodium, X mg, is a continuous random variable with mean 0µ . Following a change in production chain, 15 rando mly chosen boxes of cheese rings were analysed. The masses of sodium (in mg) in the boxes are summarised by ( 1180) 17.5 x − = ∑ , 2( 1180) 190.5 x − = ∑ (i) Given that 0 1183 µ = , test at 5% significance level, whether the mean m ass of sodium in a box of cheese rings has changed. You should state cle arly any necessary assumption you need to make when carrying out the test. [6] (ii) Given that the standard deviation of mass of sodiu m in a box of cheese ring is 6.8 mg, find the range of values of 0µ if the test based on the same set of data leads to the conclusion that the mass of sodium has increased at 5% significance le vel. Explain briefly if the assumption made in part (i) is sti