IJC H2 MATH P2 QP

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2    IJC/2015/JC2 9740/02/Sept/2015 Section A: Pure Mathematics [40 marks] 1 The parametric equations of a curve are 21e , ettxy  . (i) Find the equation of the normal to the curve at the point   21e, ep pP  . [3] (ii) This normal meets the x-axis at the point Q. Find the cartesian equation of the locus of the mid-point of PQ as p varies. [4] 2 (i) Find the general solution of the differential equation 2 3 2 d 2 d yx x x   . [3] (ii) It is given that 1y when 1x . On a single diagram, sketch three members of the family of solution curves for 0x  . [5] 3 (i) Given that sin 2f( ) cos 2 3 xx x   , where x is sufficiently small, find the series expansion of f( )x in ascending powers of x, up to and including the term in 2x . [4] (ii) Use your answer to part (i) to give an approximation for 0 f( ) d n x x in terms of n. Evaluate this approximation in the case where 0.5n , leaving your answer in 6 decimal places. [3] (iii) Use your calculator to fi nd an accurate value for 0.5 0 f( ) dx x , correct to 6 decimal places. Explain why this value is more than the approximation obtained in part (ii). [3]

3    IJC/2015/JC2 9740/02/Sept/2015 [Turn over 4 (a) The complex number z satisfies the relations 33z  and 33 izz  . (i) Illustrate both of these relations on a single Argand diagram. [3] (ii) Find exactly the maximum and minimum possible values of 2 z . [4] (b) The complex number w is given by 2 3i 2i      . Without using a calculator, find (i) w and the exact value of arg w , [4] (ii) the set of values of n, where n is a positive integer, for which *nww is a real number. [4] Section B: Statistics [60 marks] 5 A manufacturing company has three factorie s that produce packets of instant noodles. The manager wants to test whether the lead co ntent in its latest batch of instant noodles produced exceeds the legally permitted levels . The number of packets of instant noodles produced from each factory for the latest batch is shown in the table below. Factory Number of packets produced A 4000 B 2000 C 1000 To carry out the test, a sample of 100 packets will be chosen from this batch of instant noodles produced. (i) Describe how the sample could be c hosen using stratified sampling. [2] (ii) State one advantage of using strati fied sampling in this context. [1]

4    IJC/2015/JC2 9740/02/Sept/2015 6 A jackpot game machine at an arcade cont ains 4 slots where each of the first 2 slots displays any of the twelve zodiac signs and each of the next 2 slots display any of the twenty-six letters of the alphabets A ‒Z. The jackpot is won if the 4 slots display two identical zodiac signs and two identical letters. Find the probability that a random game played at the machine results in (i) two different zodiac signs and two different vowels, [2] (ii) winning the jackpot, [2] (iii) exactly two identical zodi ac signs or exactly two identical letters or both. [3] 7 Bernard is carrying out an experiment with a fair tetrahedral die, which has its four triangular faces numbered from ‘6’ to ‘9’, and a biased 10-sided die numbered from ‘1’ to ‘10’. (i) Bernard rolls the fair die 9 times. Find the probability that the die shows a ‘8’ between 3 and 7 times, inclusive. [2] (ii) Bernard now rolls the fair die 65 times. Use a suitable approximate distribution, which should be stated, to find the probabi lity that the die shows a ‘9’ more than 12 times. [3] The probability that the bi ased die shows a ‘9’ is 1 25 . (iii) Bernard rolls the biased die 65 times. Use a suitable approximate distribution, which should be stated, to find the probabi lity that the bias ed die shows a ‘9’ more than 5 times. [3] 8 A large office building is busy during the fi ve weekdays, Monday to Friday, and less busy during the two weekend days, Saturday and Sunday. The block is illuminated by fluorescent light tubes which frequently fail an d must be replaced with new tubes. It is assumed that the number of fluorescent tubes that fail on a particular weekday has the distribution Po(1.2) . The number of fluorescent tubes that fail on a particular weekend day is also assumed to be an independent random variable with distribution Po(0.5) . (i) Find the probability that at least 8 fluores cent light tubes fail in a period of five consecutive weekdays. [2] A week refers to a complete seven-day week. (ii) Given that a total of 10 fluorescent light tubes fail during a week, find the probability that at most 2 fluorescent light tubes fail during Saturday and Sunday. [3] (iii) Using a suitable approximation, find the probability that at most 30 fluorescent light tubes fail during a period of 4 weeks. [3]

5    IJC/2015/JC2 9740/02/Sept/2015 [Turn over 9 The continuous random variable X has the distribution 2N( , )  . It is known that  P2 0 . 1 0Xa and P0 . 3 0Xa . (i) Find E( )X and Var( )X in terms of a. [5] (ii) Given that 1X , 2X and 3X are three independent observations of ,X find  12 3P2X XX a  . [4] 10 A firm of solicitors claims that the average duration of the interviews with their clients is 45 minutes. (a) A random sample of 12 interviews is chosen, and the time taken for each interview, x minutes, is noted. The results are shown in the following data. 53 40 61 48 51 43 50 35 42 55 65 60 (i) Calculate unbiased estimates of the population mean and variance. [2] (ii) Stating a necessary assumption, carry out a test to determine, at the 5% significance level, whether the firm is understating the average interview time. You should define any symbols that you use. [5] (b) Another sample of 60 interviews is chos en and the time taken for each interview, in minutes, is noted. The sample mean is found to be m minutes and the sample standard deviation is 9 minutes. A test is carried out at the 10%