YJC P2

This paper consists of 6 printed pages. YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YSIHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YSIHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YSIHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YSIHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YSIHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE YSIHUN JUNIOR COLLEGE YISHUN JUNIOR COLLEGE 2012 JC2 PRELIMINARY EXAMINATION Higher 2 MATHEMATICS 9740/2 Paper 2 17 AUGUST 2012 FRIDAY 0800h – 1100h Additional materials: Answer paper Graph Paper List of Formulae (MF15) TIME 3 hours READ THESE INSTRUCTIONS FIRST Write your name and CTG in the spaces provided 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 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, write down the question number of the questions attempted, model of calculator used on the spaces provided on the cover page. Tie your cover page on top of the answer scripts before submission. The number of marks is given in brackets [ ] at the end of each question or part question.

Yishun Junior College  JC2 Preliminary Examination 2012  H2 Mathematics 9740/2 Page 2 of 6 Section A : Pure Mathematics [40 marks] 1. Two planes 1 and 2 have equations 22x y z   and 23x y z   respectively. The point A has coordinates (4, –1, 2). (i) Find the acute angle between the planes 1 and 2 . [2] (ii) Let B be the foot of the perpendicular from A to 1 . Find the coordinates of B. [3] (iii) If 3 contains the line AB and is perpendicular to 1 and 2 , find the position vector of the point of intersection of the 3 planes. [3] 2. The complex number z is given by iz x y , where x > 0 and y > 0. Sketch an Argand diagram, with the origin O, showing the points P, R and Q representing z, 3iz and  3izz respectively. [2] Describe the geometrical relationship between O, P, Q and R. [1] (a) Given that y = 3x, show that the point representing z2 is collinear with the origin and the point Q. [2] (b) Given that 3z  and   11 1tan arg tan 33 z  , illustrate both of these relations on a single Argand diagram. Find the area of the region in which the point P can lie. [3] 3. The functions f and g are defined by 2f : 4 4 , , where is a constant,x x x x     g : ln 1 , 1.x x x  (i) Explain why function f does not have an inverse. [1] (ii) The function f has an inverse if its domain is restricted to xk . Find the largest value of k in terms of  , and define 1f corresponding to this domain of f. [4] (iii) If 1 2  , show that the composite function 1gf  exists and find the range of 1gf  in terms of  . [3] 4. A sequence 1 2 3, , , ....u u u is such that 1 1u  and 1 13 3 2, for all 1.nn nnu u n     (i) Find the values of 2 3 4, and u u u , in the form of 3n k , where k and n are positive integers. [1] (ii) Give a conjecture for nu in the form f ( ) 3n n , where f ( )n is a linear expression in n. [1] (iii) Prove your conjecture by induction. [4]

Yishun Junior College  JC2 Preliminary Examination 2012  H2 Mathematics 9740/2 Page 3 of 6 5. (a) A cylindrical container has a height of 200 cm. The container was initially filled with a liquid chemical but there was a leak from a hole in the base. When the leak was noticed, the container was half-full and the level of the chemical was decreasing at a rate of 1 cm per minute. To model the situation, it is assumed that when the depth of the chemical remaining is x cm, the rate at which the level is decreasing is proportional to x . By setting up and solving a differential equation, show that 1 2 12 2 200 .10xt   Hence, find, to the nearest minute, the duration for which the container has been leaking. [6] (b) Using the substitution v = x – y, solve the equation   2d 1 ( ) cos 1 ,d y x y xx    expressing y in terms of x. [4]

Yishun Junior College  JC2 Preliminary Examination 2012  H2 Mathematics 9740/2 Page 4 of 6 Section B : Statistics [60 marks] 6. An employer has appointed an opinion polling company to survey 40 staff on the proposed changes in the shift work hours. The staff comprises 60 managers, 140 technicians and 300 factory workers. (i) Give a disadvantage of using simple random sampling in this case. [1] (ii) Describe how the sample could be chosen using stratified sampling and state one advantage of adopting this sampling method in this context. [3] 7. A manufacturer believes that its new coffee product will lower the low-density lipoprotein (LDL) level in the body. A group of 12 volunteers are randomly selected to consume this new coffee product every day, over a trial period of 10 weeks. The decrease in LDL level, denoted by X mg/dL, for each volunteer, is as follows: 32 44 52 59 37 15 18 21 28 51 40 23 The manufacturer will launch the new coffee product if the sample provides significant evidence that the mean decrease in LDL level is more than 25 mg/dL. (i) Test, at the 5% level of significance, whether the mean decrease in LDL level is more than 25 mg/dL. [4] (ii) State the assumption which is necessary fo