PJC H2 MATH P1

PJC2011 [Turn over] Candidate Name : ____________________________ CT Group : ________ Index no : ________ PIONEER JUNIOR COLLEGE JC 2 Preliminary Examination MATHEMATICS Higher 2 Paper 1 ( 9740 / 1 ) Wednesday 14 Sept 2011 Additional material: Answer paper, List of Formulae MF15 TIME 3 hours INSTRUCTIONS TO CANDIDATES Do not open this booklet until you are told to do so. Write your full name, index number and CT 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 any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correc t 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. The number of marks is given in brackets [ ] at the end of each question or part question. Attach this question paper with your answers, and arrange your answers in numerical order. For Examiner’s Use Qn Marks Qn Marks Qn Marks 1 5 9 2 6 10 3 7 11 4 8 12 Sub-total Sub-total Total _________________________________________________________________________________ This question paper consists of 7 printed pages and 1 blank page.

PJC2011 [Turn over] 2

PJC2011 [Turn over] 3 1 (a) Find  12 tan 2 dx xx   . [ 3 ] (b) The region R is bounded by the curve 2 4y xx  , the line 29y x , and t h e y-axis. Find the volume of the solid when R is rotated completely about the x-axis, giving your answer correct to 2 decimal places. [3] 2 (i) Find the first three terms in the expansion of 2 1 4 x in ascending powers o f x . [ 3 ] (ii) Hence find the first four terms in the expansion of 2 1 4 x x   . [2] (iii) State the set of values of x for which this expansion is valid. [1] 3 Each time a ball falls vertically onto a horizontal floor, it rebounds to three - quarters of the height from which it fell. It is initially dropped from a point 4 m above the floor. (i) Show that the total distance the ball travels until it is about to touch the floor for the  1t hn time is given by 328 24 4 n   . [3] (ii) Find the least number of times the ball must bounce for it to travel m o r e t h a n 2 4 m . [ 2 ] (iii) Explain why the ball will not travel more than 28 m. [1] 4 (i) By writing   1 32rr in partial fractions, find  4 1 32 N r rr  . [ 4 ] (ii) Hence find   6 0 1 21 N r rr     . [ 2 ] (iii) Give a reason why the series  4 1 32r rr     converges, and write d o w n i t s v a l u e . [ 2 ]

PJC2011 [Turn over] 4 5 Given that 2ec o sxy x , prove that 2 2 dd 22 e c o s 2dd xyy y xxx    . [3] By further differentiation of this resu lt, obtain the series expansion of y in ascending powers of x up to and including the term in 3x . [3] Given that x is sufficiently small for 4x and higher powers of x to be neglected, deduce the series expansion of e cos sin 2x x x . [2] 6 The functions f and g are defined as follows:  12f: , 2 , 2 g: ln 3 , 3. xxx x xx x       (i) Sketch the graph of  fyx , showing clearly the asymptote(s) of the g r a p h . [ 2 ] (ii) Find the expression for   -1 f x and state its domain. [3] (iii) Only one of the composite functions fg and gf exists. Give a definition (including the domain) of the comp osite that exists, and explain why the other composite does not exist. [3] 7 A curve is defined by the parametric equations 3 ,aaxy tt  where a is a constant. (i) Find the equations of the ta ngent and the normal at point P where 1 2t  . [4] (ii) Find the coordinates of the point where the tangent cuts the curve a g a i n . [ 2 ] (iii) The tangent at P meets the x-axis at Q and the normal at P meets the x-axis at R. Show that the area of triangle PQR is 2145 6 a . [2]

PJC2011 [Turn over] 5 8 In a chemical plant, the amount of substance X in a chemical reaction is being observed. The amount of substance X , in kg at any time t minutes after the start of the chemical reaction is denoted by x and it satisfies the differential equation d 12d x kxt  , where k is a positive constant. It is known that at the start of the chemical reaction, the amount of substance X is 1 kg and is decreasing at a rate of 0.05 kg per minute. (i) Obtain an expression for x in terms of t. [ 4 ] (ii) Using a non-graphical method, show that the amount of substance X is a l w a y s d e c r e a s i n g . [ 2 ] (iii) State, with a reason whether substance X will be used up in the long r u n . [ 1 ] (iv) Sketch a graph to show how the amount of substance X varies with t i m e . [ 2 ] 9 The polynomial P z has real coefficients. The equation  P0z  has a root ier  , where 0r  and   . (i) Write down a second root in terms of r and , and hence show that a quadratic factor of  P z is 22 2c o szr z r   . [3] (ii) 1z and 2z are roots of the equation  P0z  , where i 3 1 2ez   and 21 izz . Write down the exact modulus and argument of 2z . State the geometrical relationship between 1z and 2z and illustrate this relationship clearly on an Argand diagram. [3] (iii) Given further that  P z is of degree four, express P z as a product of two quadratic factors with r eal coefficients, giving each factor in exact non - trigonometrical form. [3]

PJC2011 [Turn over] 6 y 0 1 2 2 ( 3 , 3 ) x 10 (a) The diagram below shows the graph of f ( ) y x . The graph crosses the x-axis at 0x , 2x and has a turning point at  3, 3 . The asymptotes of the graph are 1x and 2y . Sketch, on separate clearly labelled diagrams, the graphs of (i)  1 fy x , [ 3 ] (ii) f1yx . [ 3 ] (b) A graph with the equation  fy x undergoes, in succession, the following transformations: A: A translation of 1 unit in the direction of the x-axis. B: A stretch parallel to the x-axis by a scale factor 2 1 . C: A reflection in the y-axis. The equation of the resulting curve is 2 4 44 1y x x   . Determine the equation of the graph  fy x , giving your answer in the simplest form. [4]

PJC2011 [Turn over] 7 11 The curve C has equation 22 4 ,,xx ky xkxk   and k is a constant such that 0k  and 2k  . (i) Find the equations of the asymptotes of C. [ 2 ] (ii) Show that if C has 2 stationary points, then 0k  or 2k  . [3] (iii) Given that yx is an asymptote of C, find the value of k. With this value of k, sketch C, showing clearly the asymptotes and the stationary p o i n t s .