RVHS P2

Name ( ) Class RIVER VALLEY HIGH SCHOOL 2012 Year 6 Preliminary Examination Higher 2 MATHEMATICS Paper 2 Additional Materials: Answer Paper List of Formulae (MF15) Cover Page 9740/02 14 September 2012 3 hours READ THESE INSTRUCTIONS FIRST Do not open this booklet until you are told to do so. Write your name, class and index number in the space at the top of this page. Write your name and class 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. 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, place the cover page on top of your answer paper and 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 6 printed pages.

Section A: Pure Mathematics [40 marks] 1 Use the substitution u xy , where u is a function of x, to reduce the differential equation 32d (1 ) 0d yx x y xx    to 2d ( 1)d u uxx  . Hence show that the general solution to the differential equation is of the form f ( )e1 xAy x  , where f is a function in x. [5] Sketch on separate diagrams a member of the family of curves corresponding to (i) 1A and (ii) 1A . [2] 2 (a) Use the substitution tan e xy to find 2 1 de (1 e )xx x . [4] (b) The region bounded by the x-axis, the curve 2 1exy  , the lines 0x and 1x is rotated about the y-axis to form a solid of revolution of volume V. Find the exact value of V. [6] 3 (a) The diagram shows the graph of  fyx . It has a maximum point at at  ,2a where 40 a   and passes through the points  6, 0 and  2, 0 . The curve has asymptotes at 0x and 4y  . On separate diagrams, sketch the graphs of (i)  f'yx , [3] (ii) 1 f ( )y x , [3] (iii)  f4yx [3] (b) Describe a s equence of transformations that maps the graph of 2 2 14 yx  onto the graph of   22 11xy   . [2] 6 O x y 2  ,2a 4y 

4 (i) Solve the equation 3 80z  , giving the roots in the form ier  , where 0r and ππ    . [3] (ii) The roots denoted by 1z , 2z and 3z are such that 1 2 3π arg( ) arg( ) arg( ) πz z z     . The complex numbers 1z , 2z and 3z are represented by the points 1Z , 2Z and 3Z in the Argand diagram respectively. Show and label the roots of the equation 3 80z  on the Argand diagram. [2] (iii) The set of points P in the Argand diagram represents the complex number z that satisfies 23| | | |z z z z   . Explain why the locus of P passes through the point 1Z . [1] (iv) The set o f points Q represents another complex number w that satisfies 1| | 2 3wz . Sketch the loci of P and Q on the same diagram, indicating clearly the positions of 1Z , 2Z and 3Z . [3] (v) The complex number v is represented by the point of intersection of the loci of P and Q in the first quadrant . Find v in the form ixy , giving the exact values o f x and y . [3] Section B: Statistics [60 marks] 5 It is desired to interview residents of a district to find out whether there is a need to set up a nursing home in their district. In particular it i s necessary to obtain a sample of 100 interviews representative of the various age groups. (i) Explain how a quota sample might be obtained. [2] (ii) Explain a disadvantage of quota sampling in the context of your answer to part (i). [1] (iii) Suggest an alternative method of sampling that would not have this disadvantage. [1]

6 In a particular restaurant, set lunch menu is offered to customers. Under this menu, a customer can choose 1 soup, 1 main course, 1 dessert and 1 drink from among the following choices: 2 soups, 3 main courses, 3 desserts and 4 drinks. Assuming every customer would choose 1 soup, 1 main course, 1 dessert and 1 drink from the menu, w rite down the number of combinations a customer can order for set lunch. [1] Two friends place two sets of order under the set lunch menu. Suppose they do not mind sharing food and drinks with each other, how many ways can they place their order such that they get to consume the maximum variety of food and drinks? [4] The restaurant also offers a special sandwich whereby customers can select the ingredients to be placed in a sandwich order. Suppose there are a total of 8 ingredients to choose from and a customer must select at least 1 ingredient, find the number of ways one can place a special sandwich order. [2] 7 (a) The equation of the estimated least square s regression line of y on x for a set of bivariate data is y a bx . Explain what do you under stand by the least squares regression line of y on x . [2] (b) The table below gives the values of seven observations of bivariate data, x and y. x 1.2 2.0 2.7 3.8 4.8 5.6 6.9 y 2.2 4.5 5.8 7.3 7.6 9.0 9.9 (i) Calculate the value of the product moment correlation coefficient, and explain why its value does not necessarily indicate that y = c + dx is a suitable model for the data. [2] (ii) Explain how t he product moment correlation coefficient can b e used to decide, for this data, whether lny a b x or y c dx is a better model. Hence determine the equation of the suitable regression line for the above data. [2] (iii) It is desired to use the given data to estimate the value of x when 6.4y . Use the regression line found in part (ii), to find the required estimate and comment on the reliability of your answer. [2]

8 A factory produces a particular brand of cooking oil in bottles of 2 sizes, namely Small and Regular. The amount of cooking oil in each type of bottle, in millilitres (ml), is assumed to be normally distributed with means and standard deviation s as shown in the following table. Mean Standard Deviation Small 545 20 Regular 1020 50 (i) A Regular bottle of cooking oil will be rejected if the amount of cooking oil is less than  ml. Find t he greatest value of  , to the nearest integer such that at least 97% of the Regular bottles of cooking oil will not be rejected. [2] (ii) Find the probability that the mean volume of three Small bottles and two Regular bottles of cooking oil is less than 730 ml. [3] (iii) Three Small bottles of cooking oil are randomly chosen. Find the probability that the bottle containing the least content holds more than 547 ml of cooking oil. [2] (iv) The factory found out that there was a change in the cooking oil dispensed for the Regular bottle. Assuming that the Regular bottle of co oking oil has a maximum capacity of 1080 ml, explain if the use of a normal distribution with the given mean and standard deviation is appropriate in modelling the amount of cooking oil dispensed. [1] 9 Singapore households are on the move to install fib re optic cables in their units. The continuous random variable X denotes the time, in minutes, spent by a technician of Open Network to install a fibre optic cable u