PJC P2

3 Section A : Pure Mathematics [40 marks] 1 (i) Given that    1 2 1 cos3h 2 sin     , and that  is a sufficiently small angle, show that   13h 2     . [2] (ii) Given that n is a positive integer, by expanding binomially, show that   h n ab  , where a and b are constants to be found in terms of n. [2] (iii) State the range of values of  for which the expansion is valid. [1] (iv) Verify your answers in (ii) by finding a suitable Maclaurin’s series. [2] 2 The complex number z satisfies the relations 4i 12 z z   , 2i 2z ,   1 3arg 2 4z  . (i) Illustrate these loci on a single Argand diagram, showing clearly the locus of z. [5] (ii) Find the complex number z such that z is the smallest, giving your answer in exact form. [3] [Turn over Name : Class :

4 3 The diagram shows the parts of the graph of 1 1y x  . The four rectangles each of equal width, as shown in diagram below, are rotated through  radians about the x-axis. Prove that the volume generated may be expressed as   4 2 1 4 8r r    . [2] n rectangles, each of equal width under the curve between 1x and 2x , are used to estimate the volume of solid obtained when the region bounded by the graph, the lines 1x and 2x , is rotated through  radians about the x-axis. Find the estimated volume in the form   1 f n r r   , where f(r) is to be determined in terms of r and n. [2] (i) Deduce that   21 11 62r n nnr   . [3] (ii) State the value of   21 2 n r n nr   as n . [1] x 1 1y x  O y x=1 x=2

5 4 (a) A, B and C are constants such that   2 22 4 1 xB A x Cx    for all values of x. Find the values of A, B and C. [1] State precisely a sequence of transformations by which the graph of 2 2 4 xy x   may be obtained from the graph of 2 1 1 y x   . [3] (b) The sketches above show the graphs of 2 f( )yx and f ( )'yx for a certain function f. (i) Sketch the graph of f ( )yx . [2] (ii) Sketch the graph of  f1yx . [2] [Turn over x y 0 f '( )yx 1 1 x y 0 2 f ( ) yx  1 ,3  1 , 3 2 2y 2y

6 5 The functions f and g are defined as follows f : , , , and xx x x x         , 2g : 3 6 2,x x x x    . (i) Show that f 1 exists. [2] (ii) Show that f 1(x) = f(x) if 1  . Deduce the value of  (2013)f  . [3] (iii) Given that the composite function fg does not exist, show that 1  . [2] (iv) Given that the composite function fg exists and that fg(0) 2 , find the range of values of  . [2] Section B : Statistics [60 marks] 6 A feedback manager of a mobile internet service provider, which has a large pool of subscribers, aims to determine the quality of his customers’ experience with its service. (i) Suggest the most informative sampling method that can be used to help the manager achieve his aim. [1] (ii) Describe precisely how the manager should carry out your suggestion. [3] 7 Find the number of 6-digit codes that can be formed using the digits 1 to 9 if (i) there are no restrictions, [1] (ii) there are three distinct pairs of identical digits, [2] (iii) there is more than one odd digit and all odd digits are separated. [3] 8 An event A occurs at random times, at an average rate of  times per day. (a) Fifty days are chosen at random, and the number of times A occurs in ea ch day is recorded. Given that 8 , find the probability that the average number of times A occurs is not more than 7. [3] (b) Given that 0.5m where m is a large positive integer, estimate the pr obability of A occurring at most m times in a randomly chosen day. [3]

7 9 In a university, a 3 year degree programme is offered to students. The proportion of students in the first, second and third year of study are in arithmetic progression. The pro babilities that a candidate did not complete the first year and third year of study are 0.1 and 0.12 respectively. (i) Show that the probability that a randomly chosen student from the degree programme will be in his second year of study is 1 3 . [1] (ii) Given that the probability that a randomly chosen s tudent is a second year student and has completed his second year of study is 14 45 , find the probability t hat a student will not complete his second year of study. [2] (iii) It was found that if a student did not complete his year of study, the probability that he is in his third year of study is 0.6. Find the proportion of third year students in the degree programme. [4] (iv) Are the 2 events “a randomly chosen student did not complete his year of study” and “a randomly chosen student is in his third year” independent? Justify your answer. [1] 10 A certain number of mobile phones purchased at a handphone shop is monitored and the numb er of those phones that are camera phones is denoted by C. (i) State, in context, one assumption needed for C to be well modelled by a Binomial distribution. [1] (ii) Given that 1 in 20 phones purchased are non-camera phones and the probability that there are more than 5 non -camera phones purchased is less than 0.1, find the largest number of mobile phones purchased that can be monitored. [3] (iii) The records kept by the hand phone shop suggested that 98% of the mobiles phones purc hased are camera phones. Using a suitable approximation, find the probability that less than 171 mobile phones purchased are camera phones out of 180 mobile phones purchased. [4] [Turn over

8 11 Ten randomly selected people were as ked to indicate the number of hours they spent listening to radio, s, and watching TV, t, during a 2-week period, as shown in the table below. s 3.8 2.4 3.0 7.2 9.4 3.6 8.2 2.6 2.0 4.8 t 47 19 30 78 84 38 76 23 10 64 (i) Calculate the product moment correlation coefficient for the data. [1] (ii) Give a sketch of the scatter diagram for the data using the model t a bs and comment on the value of the product moment correlation coefficient found in (i). [2] (iii) State, with reason, which of the following models is more appropriate. A : bta s B : 2t a bs C : lnt a b s [1] (iv) Explain how the product moment correlation coefficient can be used to justify that your choice in (iii) is a better model than in (ii). [1] (v) Using the appropriate regression line, calculate an estimate for the time spent on listening to radio, given that the time spent on watching TV is 5 hours. Comment on your answer. [3] 12 A promoter of a certain type of battery claimed that his battery can last for at least 2000 hours when used continuously in a graphic calculator. Tom heard this and is keen to recommend the battery to his classmates only if the battery’s life-span is as claimed. He bought less than 20 of this battery to test out on his own graphic calculator. In the process, however, he misplaced some readings and were left only with the following records of battery