NJC JC2 H2 Maths 2012 Question Paper 2

NJC 2012 9740/02/2012 MATHEMATICS 9740/02 Paper 2 17 September 2012 3 hours Additional Materials: Answer Paper List of Formulae (MF15) Cover Sheet 1400 – 1700 hours READ THESE INSTRUCTIONS FIRST Write your name, registration number, subject tutorial group, on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for diagrams or graphs. Do not use 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 the brackets [ ] at the end of each question or part question. This document consists of 8 printed pages and 0 blank page. National Junior College NATIONAL JUNIOR COLLEGE PRELIMINARY EXAMINATIONS Higher 2

2 NJC 2012 9740/02/2012 [Turn Over Section A: Pure Mathematics [40 marks] 1(a) Given that iez  ,where 0 and 0 2  , and i3wz , find (i) zw , in terms of  , [2] (ii)  arg zw , in terms of  . [2] (b) The complex number z satisfies the relations 1 i 3 3z   and  arg 2 i4 3 3z  . (i) Illustrate, on an Argand diagram, the locus of points representing the complex number z. [3] (ii) Find the exact value of z that gives the greatest possible value of arg(z). [3] 2(a) A car C is traveling at a bearing of 010 o towards intersection P at a rate of 60 km/h while a truck T is at rest, 48 km at a bearing of 070 o away from the intersection. By using Cosine Rule, show that the square of the distance, y2, between the car C and truck T is 22 48 2304y x x   , when the car C is x km from intersection P. [1] Find the rate of change of the distance between the car C and truck T when the car is 15 km from of the intersection. [3] Car C Truck T P 48 km 010 o 070 o N

3 NJC 2012 9740/02/2012 [Turn Over (b) A closed cylindrical container with a hemispherical base is constructed to store water. The cost of each unit area of the circular top is fixed at P dollars and the cost of each unit area of the curved surface and hemispherical base of the cylinder is fixed at 3P dollars. The total cost of the surface is a fixed amount, C dollars. Given that the radius of the hemisphere is r, show that the volume V of the structure can be expressed as V 3 62 Cr r P  . [3] Show that, as r varies, V is at its maximum value when the cost of circular top is 9 C dollars. [3] [Curved surface area of hemisphere = 22 r ; volume of hemisphere = 32 3 r ] 3(a) By expressing x + 3 = A 2x + 4( ) + B , where A and B are constants to be determined, evaluate x + 3 x2 + 4x + 9 dxò . [4] (b) A curve y = f x( ) , defined for x > - 1 3 , has gradient value of 2 at the point  0,1 . Given further that d2y dx 2 = 2 1+ 3x , find the equation of the curve. [6] r r r

4 NJC 2012 9740/02/2012 [Turn Over 4 (i) Without using integration, state the exact value of 2 2 2 4 d .   xx Justify your answer. [2] (ii) Let the region bounded by the curves whose equations are   22 24xy   and 12 xy be R. Show that the volume of solid generated when R is rotated completely about the x-axis can be expressed as 2 22 2 5 7 4 4 d .4 x x x x       Hence find this volume in its exact form. [8] Section B: Statistics [60 marks] 5 The organis er of the international YAYA m usical concert desires to sample 2% of the audience to find their opinions of the venue facilities. (i) Describe how a systematic sample could be obtained. [2] (ii) Give a reason why a stratified sample would not be easy to obtain. [1] R   22 24xy   x y O 12 xy 4 1

5 NJC 2012 9740/02/2012 [Turn Over 6 The table below gives the values of ten observations of bivariate data, x and y. (i) Sketch the scatter diagram and determine the value of the product moment correlation coefficient between y and x. [2] (ii) Determine which of the following is the best model for this set of data , justifying your choice clearly. (A) y ax b (B) 2y cx d (C) y e x f [2] (iii) Find the equation of the least -squares regression line of your selected best model in part (ii). Use your e quation to estimate the value of y when x = 3.8. Comment on the reliability of the estimation. [3] 7(a) Find the number of ways in which the letters of the word NATIONAL can be arranged if (i) there are no restrictions, [1] (ii) not all the identical letters are next to each other, [2] (iii) the vowels must not be next to one another. [2] (b) The Rational Junior College Symphonic Band comprises 4 trumpet players, 4 horn players, 3 trombone players and 4 saxophone players. A mini stage band of 9 players are to be chosen from these sections to perform for a segment of their annual concert. Find the number of ways the group can be chosen if it includes (i) exactly 3 trumpet players, 2 horn players, 2 trombone players and 2 saxophone players, [2] (ii) at least 3 players from each of the trumpet and saxophone sections. [3] x 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 y 13.1 12.9 12.5 11.7 10.9 9.9 8.9 7.7 6.0 4.1

6 NJC 2012 9740/02/2012 [Turn Over 8 In a particular school, all students are required to have a first Co