HCI JC 2 H2 Maths 2012 Paper 2 Questions

© Hwa Chong Institution 2012 9740/02/Prelim/12 [Turn over HWA CHONG INSTITUTION Preliminary Examination Higher 2 MATHEMATICS 9740/02 Paper 2 14 September 2012 3 hours Additional Materials: Answer Paper List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your name 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 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 mathematical 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 6 printed pages.

2 © Hwa Chong Institution 2012 9740/02/Prelim/12 Section A: Pure Mathematics [40 marks] 1 The functions f, g and h are defined as follows: f : sin( )xx  , , 0 2xx   , g: xxa , ,1xx  , where a is a constant and 01 a , h : f ( )xx , ,x m x n   . (i) Explain why 1f  does not exist. Find m and n so that 1h exists and the range of h is equal to the range of f. Find 1 1h 2    . [3] (ii) Show that the composite function gh exists, and find the range of gh in terms of a . [3] 2 A Hwa Chong student taking part in Project Day needs to design a rectangular poster to showcase his project. According to the rules set for the poster design, the poster should contain 1352 cm2 of printing with margins of 4 cm each at the top and bottom, and 2 cm each on the left and right side s. By using differentiation, determine the dimensions of the poster to minimise the amount of paper used. [6] 3 (i) Show that for any complex number ie,z  1 2cos .n nzn z  [2] (ii) By taking 1,n show that 33 3 1 3 1cos 3 . 8 zz zz     Deduce that 3 13cos cos3 cos .44   [4] (iii) Find 3cos 3 d . [2]

3 © Hwa Chong Institution 2012 9740/02/Prelim/12 [Turn over 4 (a) Referred to the origin O, the position vectors of three points A, B and P are ,ab and 5ab respectively. Given that b is a unit vector , the angle AOB is 60 and AB is perpendicular to OP, find a . [4] (b) In the triangle OAB where O is the origin, the position vectors of the points A and B are a and b respectively. The point C is the midpoint of OA, the point E on BC is such that : 3: 4CE EB  , and the line OE meets AB at D. Find the ratio :AD AB . [4] 5 A curve G has equation 2( 1) cy ax b x    , where , and a b c are positive integers. Let R be the region bounded by the curve G, the axes and the line 3x . (i) Explain why the graph of G lies above the x-axis for all 0x . [1] (ii) If the area of the region R is 42 units2, show that 93 + 3 + = 4224a b c . [2] (iii) It is also known that the curve G has a minimum turning point at ( 0, 5). Find the equation of the curve G. [3] (iv) Sketch the curve G, stating cle arly the equation of any asymptote (s), turning point(s) and axial intercept(s). [3] (v) The region S is bounded by the curve G, the y-axis, the lines 3x and 81yx . Find the volume of solid generated when r egion S is rotated through 2 radians about the x-axis. [3] O A C B D E

4 © Hwa Chong Institution 2012 9740/02/Prelim/12 Section B: Statistics [60 marks] 6 A manager claims that the time spent by each customer at his supermarket follows a normal distribution with mean 35 minutes and standard deviation 30 minute s. A statistician comments that the distribution 2N(35, 30 ) will not provide an adequate model. (i) Do you agree with the statistician’s comment? Give a reason to support your answer. [2] (ii) Suppose the time spent by e ach customer follows a normal distribution and its standard deviation is 10 minutes instead of 30 minutes , find the probability that the total time spent by 2 randomly chosen customers in the supermarket is more than 3 times that of another randomly chosen customer. [3] 7 In a particular year, a company awards scholarships to 4 applicants from a group of n applicants. If the order of selection is not taken into account, the number of ways in which the 4 scholarship recipients can be chosen is 7315. (i) Find the number of ways in which the 4 scholarship recipients can be chosen if the order of selection is taken into account. [1] (ii) Find the value of n. [2] In the year 2012, there are 13 scholarship applicants. The 13 applicants, which include the eventual 4 scholarship recipients Ann, Belle, Connie and Don, are required to attend a briefing. The applicants are to be seated in 2 rows as shown below: Find the number of ways in which the 13 applicants can be seated if all 4 scholarship recipients must sit together during the briefing. [3] 8 An eatery is giving out free burgers in a promotion to celebrate its anniversary. A total of 80 burgers, consisting of 8 different flavours, 10 from each flavour, are to be given out. The burgers are rand omly chosen and given out one at a time. Of the 8 burger flavours, the ‘Curry’ and ‘Spicy’ burgers are most popular among customers. In the first four burgers to be given out, the events A and B are defined as follows: A: At least 2 ‘Curry’ burgers. B: Exactly one ‘Spicy’ burger. (i) Find P( )AB . [3] (ii) Justify if 'A and 'B are independent events. [3] seat seat seat seat seat seat seat seat seat seat seat seat seat

5 © Hwa Chong Institution 2012 9740/02/Prelim/12 [Turn over 9 In each large batch of SIM cards, 15% of the cards are defectiv e. From each batch, a random sample of 10 cards is drawn for inspection. (i) State the distribution of the number of defective SIM cards in the sample. Write down an assumption in order for the distribution stated to be valid. [2] (ii) A batch is lab eled as ‘good’ if the sample drawn has less than 2 defective SIM cards. If 33 batches are produced daily, find using a suitable approximation, the probability that in a day, more than 80% produced are ‘good’ batches. [4] 10 (a) The supervisor of a petrol kiosk wants to monitor the service provided by his staff. He decides to choose a sample of 30 customers to take part in a survey at the payment counter before they leave the petrol kiosk. (i) Describe how a systematic sample of 30 customers can be chosen fro m the first 120 customers in a day. [2] (ii) Give a reason whether stratified sampling is appropriate in this situation. [1] (b) The waiting time , in minutes, for a customer to be served at a petrol kiosk is a random variable with mean  and standard deviation  . The following are observed based on a random sample of 100 customers:  for 95% of the time, the mean waiting time is between 7.5 minutes and 10 minutes;  the probability that the mean w aiting time is more tha