NJC JC2 H2 Maths 2012 Question Paper 1

NJC 2012 9740/01/2012 MATHEMATICS 9740/01 Paper 1 14 September 2012 3 hours Additional Materials: Answer Paper List of Formulae (MF15) Cover Sheet 0815 – 1115 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 9 printed pages and 2 blank pages. National Junior College NATIONAL JUNIOR COLLEGE PRELIMINARY EXAMINATIONS Higher 2

NJC 2012 9740/01/2012 1 The diagram shows the graph of  f 'yx , where  f x is a cubic polyn omial, has a minimum point at (2.5, – 9). Given that  f 0 1 , find  f x . [4] State the range of values of x for which  f '' 0x  . [1] 2 Relative to the origin O, the position vectors of three points, A, B and P are, 32  i j k , 52ik and  1 2 ( 2) 2   i j k , where  is a real parameter, 1 . (i) Show that A, B and P are collinear. [2] (ii) Find the value of  such that P is on the line BA produced and area of triangle OAP is 162 5 square units. Give a reason for your choice. [4] x  f 'yx  f ' x 0 – 2 7 ( 2.5, – 9)

3 NJC 2012 9740/01/2012 [Turn Over 3 Use the standard series for e x and (1 ) nx to find the Maclaurin series for f ( ),x where   2 f ( ) e 1 2 , xxx up to and including the term in 3.x [3] (a) By substituting 1 ,3x find an approximation for   1 9135 e . [2] (b) Find the series for f ( )x up to and including the term in 2.x Hence or otherwise , find the Maclaurin series for 2 e , 12 x x up to and including the term in 2.x [3] 4 It is known that a particular type of bacteria grows very well under certain controlled conditions in a specially prepared Petri dish . The researcher believes that the growth rate of such bacteria can be modeled by 22d 2d xt xt xt   , where x milligrams is the amount of bacteria grown in the dish after t hours. (i) Using the substitution 2x ut , show that the differential equation can be reduced to 2d d u ut  . [2] (ii) Find x in terms of t, given that there was 0.2 milligrams of bacteria after 15 minutes. Hence find the amount of this particular type of bacteria after 4 hours. [4] (iii) Explain if this mathematical model is a realistic one. [2]

4 NJC 2012 9740/01/2012 [Turn Over 5 A curve C has parametric equations cos2xt , tanyt , for 22 t   . (i) Sketch the curve C, indicating clearly any asymptotes, and axial intercept(s). [2] (ii) The point P on the curve has parameter t = 3  . If the normal to the curve at P passes the through the point (b, 0), find the exact value of b. [3] (iii) Show that the area bounded by the curve C, y-axis and 4y , is 13 2 tan 42   . [4] 6 The planes 1p , 2p and 3p have equations 1x , 25x y az   and 2x y z b   , where a and b are real constants. Given that 1p and 2p intersect at the line l, show that the vector equation of l, in terms of a, is (3 ) a   r i j k , where  is a real constant. [2] (a) The acute angle between l and 3p is 60 . Find the possible values of a. [3] (b) Given that the shortest distance from origin to 3p is 6 3 and without solving for the value of b, determine the possible position vectors of the foot of perpendicular from the origin to 3p . [2] (c) What can be said about a and b if 1p , 2p and 3p do not have any points in common? [4]

5 NJC 2012 9740/01/2012 [Turn Over 7(a) A geometric series has common ratio r, and an arithmetic series has first term a and common difference d, where a and d are non -zero. The first three terms of the geometric series are equal to the ninth, fourth and second terms respectively of the arithmetic series. (i) Show that 3.da [2] (ii) Deduce that the geometric series is con vergent, and find in terms of a, the sum to infinity. [3] (b) A mountaineer climbs a mountain of height x metres. The amount of distance he climbs for the first hour is 300 metres. For each subsequent hour, he climbs 10 metres less than the previous hour. The number of whole hours that has passed just before he reaches the summit is n. (i) Write down an expression for the total distance climbed by the mountaineer and hence show that 2 ,pn qn x where p and q are constants to be determined. [2] (ii) Deduce the value of n if 2500.x [2]

6 NJC 2012 9740/01/2012 [Turn Over 8 A calculator is not to be used in answering this question. The polynomial  p z is