2014 H2 Maths Prelims 2 P2 Qn Paper

2   IJC/2014/JC2 9740/02/S/2014   Section A: Pure Mathematics [40 marks] 1 (i) Given that   21cosyx  , show that 2 2 2 dd(1 ) 2 dd yyxx xx  . [3] (ii) Hence find the Maclaurin’s series for   21cos x , up to and including the term in 3x . Give the coefficients in exact form. [3] (iii) Show that, if x is sufficiently small for powers of x above x2 to be neglected, then   22 1cos 24ee 1 x ax bx     , where a and b are constants to be determined exactly. [You may use standard results given in the List of Formulae (MF15).] [3] 2 A family of curves is given by the differential equation d d yx yx xyx  . By substituting yv x , show that the family of curves is given by  1e xyx A  . [6] Sketch, on a single diagram, three members of the family of curves corresponding to positive values of A. [2] Find a cartesian equation of the locus of the stationary points for the family of curves corresponding to positive values of A. [1] 3 The function f is defined by  2 5f : for , 4 4 xx x x     . (i) Find 1f( ) x and state the domain of 1f . [3] (ii) On a single clearly labelled diagram, sketch the graphs of f( )yx , 1f( )y x and 1ff ( )y x . [4] (iii) Show that the x-coordinates of the points of intersection of the curves f( )yx and 1f( )y x satisfy the equation 32 81 6 5 0xx x   . Hence find the values of these x-coordinates in exact form. [3]

3   IJC/2014/JC2 9740/02/S/2014 [Turn over   4 In the diagram, O is centre of the rectangular base ABCD of a right pyramid with vertex V. Perpendicular unit vectors i, j, k are parallel to AB, BC, OV respectively. The length of AB, BC and OV are 12 units, 6 units and 6 units respectively. The point M is the mid-point of CV and the point O is taken as the origin for position vectors. (i) Show that the equation of the line AM may be expressed as 66 33 , 02 t        r+ where t is a parameter. [3] (ii) Find the perpendicular distance from B to the line AM. [3] (iii) Find the acute angle between the line DV and the plane AMB. [4] The plane    has equation 1 44 a    r .      (iv) Given that the three planes AMB, AMD and  have no point in common, find the value of a. [2]          D    V C B M O            A

4   IJC/2014/JC2 9740/02/S/2014   Section B: Statistics [60 marks] 5 An automatic dispensing machine is set to dispense hot chocolate into cups. Based on observations over a long period of time, the cups were found to contain volumes of hot chocolate, in ml, with mean 55 and standard deviation 10. A large random sample of n cups of hot chocolate is taken. Gi ven that the probability that the sample mean exceeds 54 ml is more than 0.77, find the least value of n. [3] 6 The Ministry of Urban Development wish es to find out how the recent property cooling measures have affected Singaporeans. A survey is to be carried out on 5% of the households in a particular constituency. (i) Explain how a systematic sample might be carried out. [2] (ii) Describe an alternative sampling method which would be better in this case. [2] 7 A blogger observes that his blog receives an average of 3.2 visitors a day. Assuming a Poisson distribution, (i) find the probability that his blog receives at least 5 but fewer than 10 visitors over a two-day period, [2] (ii) use a suitable approximation to find the probab ility that his blog receives more than 30 visitors in a particular week. [3] Explain why the Poisson distribu tion may not be a good model for the number of visitors his blog receives in a year. [2] 8 A group of seven people consisting of two si ngle men, three single women and one married couple are arranged to be seated in a row. (i) Find the probability that the marri ed couple is seated together. [2] (ii) The married couple is seated together. Find the probability that all the women are seated next to one another. [2] (iii) Find the probability that the men a nd women are seated alternately. [2] The seven people are now seated at a round table. (iv) Find the probability that one particular woman is seated between two men. [3]

5   IJC/2014/JC2 9740/02/S/2014 [Turn over   9 In a standardized assessment test conducte d annually, it is found that , on average, 1 in 5 students who sit for the test are awarded distinction. (i) Out of 23 students who sit for the test, find the expected number of students who will be awarded distinction. [1] (ii) Find the minimum number of students who must sit for the test so that the probability of having at least 9 students being awarded distinction is more than 0.7. [3] It is further found that, on average, 93% of students who sit for the test pass the test. (iii) 60 students from a junior college sit for the test. Use a suitable approximation to find the probability that more than 55 students pass the test. [3] 10 ‘Famous Amas’ makes two types of cookies, almond cookie and chocolate chip cookie. The mass of an almond cookie and a chocolate chip cookie, in grams, are independent random variables with the distribution  2N3 2 , 3 . 0 and   2N2 8 , 1 . 8 respectively. (i) Find the probability that the total mass of four randomly chosen chocolate chip cookies is between 100 g and 120 g. [2] (ii) Find the probability that total mass of f our randomly chosen chocolate chip cookies differs from twice the mass of a randomly chosen almond cookie by less than 50 g. [3] Almond cookies are sold at $6 per 100 g and chocolate chip cookies at $5 per 100 g. John buys 4 almond cookies and 4 chocolate chip cookies. Cindy buys 10 chocolate chip cookies. (iii) Find the probability that John pays more than Cindy. [5] 11 The Department of Environmental Health did a research on how the number of mosquitoes, x, in a container varies with time, t. Observations on successive days give the data shown in the following table. t 1 2 3 4 5 6 7 x 32 47 65 92 132 190 275 (i) Draw a scatter diagram for the data. Comment on whether a linear model would be appropriate. [3] (ii) It is given that x can be modelled by e btx a , where a and b are constants. Find the equation of the least sq uares regression line of ln on x t . Hence calculate the estimates of a and b. [4] (iii) Calculate an estimate for the number of mos quitoes in the container if the experiment has lasted for 30 days. [1] (iv) Comment on the reliability of your answer in part (iii). [1]

6   IJC/2014/JC2 9740/02/S/2014   12 A manufacturer claims that the mean lifetime of the light bulbs he produces is at least 1200 hours. A random sample of 120 bulbs is taken and the lifetime per bulb, x hours, is measured. The results are summarised by   2 1200 60 and 1200 2014.xx    (i) Find unbiased estimates of the population mean and variance. [2] (ii) Test the manufacturer’s claim at the 10% significance level. [4] The same manufacturer makes a change in th e process which is in tended to improve the average lifetime. A random samp le of 20 bulbs is taken from the output of the new process. The mean and standard deviation of this sample are k hours and 9.8 hours respectively. A test at the 5% significance level indicates that the ma nufacturer’s claim is valid for this improved process. (iii) Stating a necessary assumption, find the least possible value of k, giving your answer correct to 3 decimal places. [5]