PJC H2 MATH P2

PJC2011 [Turn over] Candidate Name : ____________________________ CT Group : ________ Index no : ________ PIONEER JUNIOR COLLEGE JC 2 Preliminary Examination MATHEMATICS Higher 2 Paper 2 ( 9740 / 2 ) Wednesday 21 Sept 2011 Additional material: Answer paper, List of Formulae MF15 TIME 3 hours INSTRUCTIONS TO CANDIDATES Do not open this booklet until you are told to do so. Write your full name, index number and CT group 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 correc t 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. The number of marks is given in brackets [ ] at the end of each question or part question. Attach this question paper with your answers, and arrange your answers in numerical order. For Examiner’s Use Qn Marks Qn Marks Qn Marks 1 7 13 2 8 14 3 9 4 10 5 11 6 12 Sub-total Sub-total Total _________________________________________________________________________________ This question paper consists of 6 printed pages and 2 blank pages.

PJC2011 [Turn Over] 2

PJC2011 [Turn Over] 3 Section A : Pure Mathematics [40 Marks] 1 The first four terms of a sequence are given by 1 2341, 2, 4, 8TT T T  . Given that rT is a cubic polynomial in r, find rT in terms of r. [4] 2 A sequence of positive real numbers 1x , 2x , 3x , ... satisfies the recurrence relation 1 3nnx x  for 1n . (i) If this sequence converges to , determine the exact value of . [2] (ii) By using a graphical method, prove that 1nnx x  if 0 x  . [2] 3 Without using a graphic calculator a nd showing your working clearly, solve the inequality 21 14 x x   . [ 3 ] Hence, solve the inequality 2e 114 e x x   , giving your answer in exact form. [3] 4 The diagram shows a pyramid with a square base OABC of side 4 cm. The vertex D is 6 cm vertically above M, the mid-point of AB. Taking O as the origin and unit vectors i, j and k as indicated, find (i) the position vector of L, the mid-point of CD, [2] (ii) the area of triangle OAL giving your answer in exact form, [2] (iii) the angle between the lines ML and OD. [3] O A B C D i j k

PJC2011 [Turn Over] 4 5 (i) Find 21d 44 s i nd2 xxxx     , giving your answer in its simplest f o r m . [ 3 ] (ii) Hence, or otherwise, show that 22 1 0 1 4d 4 s i n22 k kxxa k     where a is to be found in terms of k, 0 < k < 2. [2] (iii) Sketch the curve C with equation 2244yx  and show by shading, a region whose area is gi ven by the integral in (ii). [2] (iv) Hence, find the area of the region lying inside C and between the lines x = –1 and x = 1, giving your answer in exact form. [3] 6 (i) Illustrate, on an Argand diagram , the locus of a point Q representing the complex number z, where 3 11 z z   . [2] (ii) Illustrate, using the same Argand diagram, the locus of a point P representing the complex number z, where 3iarg66 z      . [ 3 ] (iii) Indicate, clearly, in your diagram the locus of z such that 3iarg66 z      and 3 11 z z   . [1] (iv) Hence find the range of  arg 2 3iz , giving your answer in exact f o r m . [ 3 ] Section B : Statistics [60 Marks] 7 A university faculty consists of 90 memb ers of whom 9 are professors, 18 are senior lecturers and the rest are lecturers. A random sample of 10 members is re quired. Explain how this may be obtained as (i) a systematic random sample, [2] (ii) a stratified random sample. [2] State one advantage of the sampling method used in (ii). [1]

PJC2011 [Turn Over] 5 8 A laboratory technician conducted an experiment and obtained the following observations of the variables x and y: (i) Draw a scatter diagram to illustrate the data and find the product moment correlation coefficient for the sample. [3] (ii) Comment on whether a linear model is appropriate. [1] (iii) Fit a model of the form ln y ab x  to the data, stating clearly the values of a and b. [ 1 ] (iv) It is given that x represents a certain chemical (in mg) added to a solution, and y represents the reaction time (in minutes) taken. Use an appropriate regression line to give the best estimate of the amount of chemical added when the reacti on time is 15 minutes. Explain your choice of regression line for this estimate. [2] 9 In a group of 100 students, 25 own an IPod, 40 own an IPhone, and 35 own either an IPod or an IPhone, but not both. Find the probability that a student chosen at random (i) owns both an IPod and an IPhone, [2] (ii) does not own an IPod or an IPhone, [2] (iii) owns an IPod, given that he owns an IPhone. [2] 10 Joshua tries to recall the 6-digit pin number for his ATM card. (Assuming that the number 000 000 is even and valid.) How many possible numbers can th ere be if he remembers that: (i) the number starts and ends with the digit 5? [1] (ii) the number is odd and the digits do not repeat? [2] (iii) the digits do not repeat and there are exactly 3 odd digits? [2] x 5 6 7 8 9 10 11 12 13 14 y 8 9 9 10 12 16 18 20 21 24

PJC2011 [Turn Over] 6 11 It is known that 1% of the pens produced by a company are defective. The company sells the pens in boxes of 10. A box can be rejected by customers if it contains more than 1 defective pen. (i) What is the probability that a randomly chosen box of pens is rejected? [ 2 ] (ii) A junior college bought 1000 boxes of pens. Using a suitable approximation, find the probability that at least 990 boxes are accepted. [ 3 ] (iii) A polytechnic bought 2000 boxes of pens. Estimate the probability that at most 12 boxes are rejected. [3] 12 Traffic police measured the speeds of vehicles travelling along a particular stretch of expressway to monitor the accident rate along the expressway. For a sample of 24 vehicles, the speeds, x km/h, were summarised by (x  30) = 1266 and (x  30) 2 = 78060.5 (i) Calculate unbiased estimates for the population mean and variance. [2] (ii) Test, at the 3% level of significance, the hypothesis that the mean speed is 80 km/h against the alternative that it is greater than this. [4] The supervisor of the traffic police f ound that there was an error in recording the results - the last digit 0 was missing in the record. So the c