2012 TJC JC2 H2 Prelim Paper 2

1 © TJC 2012 TJC/MA 9740/Preliminary Exam 2012 TEMASEK JUNIOR COLLEGE, SINGAPORE Preliminary Examination Higher 2 MATHEMATICS 9740/02 Paper 2 19 September 2012 Additional Materials: Answer paper 3 hours List of Formula (MF15) READ THESE INSTRUCTIONS FIRST Write your Name and Civics 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 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. 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 mathematic al notations and not calculator commands. You are reminded of the need for clear presentation in your answers. At the end of the examination, 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 7 printed pages. [Turn Over

2 © TJC 2012 TJC/MA 9740/Preliminary Exam 2012 Section A: Pure Mathematics [40 marks] 1 A curve C is defined by the parametric equations 2 3 cos 3x  ,  2 3 1 siny  where 02 π. (i) Find d d y x in terms of  . [2] (ii) Show that the Cartesian equation of the curve C is 2 2 3 1 2 3 2 3 1 xy     . Hence or otherwise, sketch the graph of C, indicating clearly the x-intercepts in exact form. [3] (iii) The point P 32 3, 3 2   lies on curve C. The region R is bounded by the curve C for 3x , the x-axis and the line segment joining the points P and  3,0 . Show that the area of R is     3 2 0 3 2 3 1 2 6 3 sin d4      units2. [4] 2 The diagram above shows the curve of 2 ex xy . Two points A and B on the curve have coordinates (, 1 2 ) and (, 1 2 ) respectively. A sequence of real numbers 1 2 3, , ,...x x x satisfies the recurrence relation 1 1 e4 nx nx   for n  1. (i) Show algebraically that if the sequence converges, then it converges to either  or . [3] (ii) Show that 1nx   if nx <  . [2] (iii) Show that 1nnxx  if nx <  . [2] (iv) Explain briefly how the results in (ii) and (iii) may be used to deduce that the sequence converges to  when 1 0x  . [2]

3 © TJC 2012 TJC/MA 9740/Preliminary Exam 2012 3 The function f is defined by 1f : , , 0.x x x x x   (i) Sketch the graph of  fyx , showing clearly the coordinates of th e stationary points and the equations of asymptotes, if any. [2] (ii) Given that    g f 2x x b   where 2b , state a sequence of transformations which transform  fyx to  gyx . [2] Sketch the graph of  gyx , showing clearly the coordinates of the stationary points and the equations of asymptotes, if any. [2] On a separate diagram, sketch g ( )'yx and solve the inequality   g ( ) 1 1' x x b b x     . [5] 4 (i) Find the roots of the equation 3 8i 0z  , giving them in cartesian form a + ib, where a and b are exact real numbers. [3] (ii) The roots of the equation   3 3 2i 8i 0z    are 1 2 3, andz z z such that      1 2 3Re Re Rez z z . Hence find 1 2 3, andz z z in cartesian form a + ib, where a and b are exact real numbers. [2] (iii) Show 1 2 3, andz z z on an Argand diagram. [1] (iv) Explain why the locus of all points z such that 23z z z z   passes through the point representing 1z . Draw this locus on your Argand diagram and find the minimum value of z . [5] [Turn Over

4 © TJC 2012 TJC/MA 9740/Preliminary Exam 2012 Section B: Statistics [60 marks] 5 2 men and 5 women go to a restaurant for a meal. They choose an outdoor round table with 7 seats. Find the number of ways the group can be seated if (i) the two men are not seated next to each other, [2] (ii) one of the women, Mary, is to be seated between the two men. [2] Before their orders arrive, they request to shift to a table in the 'non -smoking' section of the restaurant. They are then given a round table with 10 seats. Find the number of ways they can be seated if (iii) the empty seats are adjacent to each other, [2] (iv) none of the empty seats are adjacent to each other and there must be more than 1 person between any two empty seats. [2] 6 Eighteen numbers are arranged in three groups of six as follows: Group A: 0, 2, 2, 2, 2, 9 Group B: 3, 3, 3, 7, 7, 16 Group C: 1, 1, 1, 1, 6, 6 One number is dr awn at random from each group. Let a, b and c denote the number drawn from groups A, B and C respectively. Event X is defined as "b is greater than the sum of a and c". Event Y is defined as "b is greater than both a and c". (i) Show that 23P( ) 54X  . [3] (ii) Find P( | )XY . [4] A game is played with a biased coin where the probability of getting a head is p . A player first flips a coin. If the coin shows a Head, the player draws a number from Group B and the score is the number dr awn. If the coin shows a Tail, the player draws a number from Group A and C each, and the score is the sum of the numbers drawn. If the probability of obtaining a score of 3 is 13 27 , find the value of p . [2]

5 © TJC 2012 TJC/MA 9740/Preliminary Exam 2012 7 Two friends, Bob an d Patrick, meet up each week at a swimming complex for a 200m freestyle friendly match. The time (in secon ds) taken by Bob to complete a 200m freestyle swim follows a normal distribution with mean 152 and standard deviation 2.2 while the corresponding time taken by Patrick is also normally distributed with mean 156 and standard deviation 3.0. (i) Show that the probability of Patrick beating Bob in a 200m freestyle match is 0.141, correct to 3 decimal places. [1] (ii) Find the probability that the total time taken by Bob to complete a 200m freestyle swim on two different occasions is less than twice the time taken by Patrick to complete a 200m freestyle swim on one occasion by less than 5 seconds. [4] Bob and Patrick maintained their weekly swimm ing matches for a total of k weeks, where 50k . Use a suitable approximation to find the least value of k such that the probability of Patrick beating Bob on fewer than four occasions is not larger than 5%. [5] 8 An online web sur vey company wishes to find out the number of hours spent per week, on average, by a typical teenager on Facebook. A survey was conducted on a random sample of 70 teenagers and the time spent per week, x hrs, was recorded and summarized:     2 18 208, 18 8967xx    (i) Find, corr ect to 1 decimal place, the unbiased estimates of the population mean and variance. [2] (ii) It is claimed that a typical teenager spends an average of 18 hours a week on Facebook. Test, at the 5% level of significance, whether the