NJC H2 MATHS P2 Question

[Turn_over NATIONAL JUNIOR COLLEGE SENIOR HIGH 2 PRELIMINARY EXAMINATION Higher 2 MATHEMATICS 9740/02 Paper 2 15 September 2015 3 hours Additional Materials: Answer Paper List of Formulae (MF15) Cover Sheet READ THESE INSTRUCTIONS FIRST Write your name, registration number, subject tutorial 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 diagrams or graphs. Do not use 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 an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing 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 the brackets [ ] at the end of each question or part question. This document consists of 6 printed pages. National Junior College

[Turn_over Section A: Pure Mathematics [40 marks] 1 (a) Use the substitution 3tanx  to find the exact value of 3 22 3 1 d 9 x xx    . [4] (b) Using integration by parts, find   2ln 4 dxx . [4] 2 (a) Joanne begins a monthly savings plan. In the first month, she puts $1000 into her savings, and for each subsequent month she puts 5% more than in the previous month. How many months would it take for Joanne’s total savings to first exceed $20000? [3] Jim starts a monthly savings plan two months later than Joanne with an initial savings of $2000, and for each subsequent month he puts $100 more than in the previous month. The table below shows the person whose total savings exceeds the other person’s total savings in the Nth month since Joanne started saving. Find the value of n. [4] (b) Suppose instead that Joanne puts a fixed amount of $1000 into her bank account on the first day of every month . The interest rate is r% per month, so that on the last day of each month the amount in the account on that day is increased by r%. At the end of 2 years, the total amount in her account will be at least $30000. Find the smallest value of r, correct to 1 decimal place. [3] N Person with more total savings in Nth month 1 Joanne 2 Joanne 3 Joanne 4 Joanne 5 Jim 6 Jim n – 1 Jim n Joanne

[Turn_over 3 The planes 1p and 2p have equations 1 10 1   r and 2 16 2     r respectively. (i) Find a vector equation of l, the line of intersection between 1p and 2.p [2] Another line, 1l , has equation 21 ,01 85                  r  where µ is a real parameter. (ii) Use the result in part (i) to show that 1p , 2p and 1l have exactly one common point of intersection. (You are not allowed to use any calculator for this part of the question.) [3] A third plane, 3,p has equation 1 .ad b   r If 1,p 2p and 3p have a common line of intersection, show that 1.b [1] (iii) If it is further given that 2p bisects the acute angle between 1p and 3,p find the equation for 3p completely, in scalar-product form. [5] 4 The complex number z satisfies both the relations 3 3i 5 2  z and   30 arg 6 4z    . (i) On an Argand diagram, shade the region in which the points representing z can lie. [3] (ii) Label the point(s) that correspond to the maximum value of 4 10iz on your diagram with the letter P. (You do not have to find the coordinates of the point(s).) [2] (iii) Express the smallest value of 4 10iz in the form 2,m where m is an integer - valued constant to be determined. Show your working clearly. [2] It is given that the complex number w satisfies the relation 3 3i 5 2w   only. (iv) Find the m inimum value of  arg 6 w , giving your answer in radians, correct to 3 decimal places. [4]

[Turn_over Section B: Statistics [60 marks] 5 A junior college comprises a total of 2500 students. A canteen operator conduct s a survey on the students’ preferences on the types of food. A sample of 500 students is to be selected to take part in the survey. Describe briefly how this sample can be obtained via (a) random sampling, [2] (b) systematic sampling. [2] 6 In a certain school, students are allowed to wear either their school uniforms, PE T -shirts or House T-shirts on Fridays. It was found that on a particular Friday, 78% of the students wore their school uniforms, 8% wore PE T -shirts and the rest of the students wore their House T - shirts. (You may assume that the school population size is very large.) (i) A random sample of 20 students was taken. Find the probability that exactly 3 students from this sample wore their house T-shirts on that day. [2] (ii) Another random sample of 60 students was taken. Using a suitable approximation, find the probability that more than 45 students wore their school uniforms on that day. [4] 7 A box contains 6 orange, 6 red, 6 green, 6 blue and 6 yellow balls. Balls of the same colour are considered to be indistinguishable. Find the number of ways to select (i) 4 distinct balls, [1] (ii) 4 balls, of which exactly 3 are identical, [2] (iii) 4 balls without any restriction. [3] 8 A hotel has 34 single-bed rooms and 60 double-bed rooms, each of which can be booked for a day at a time. (i) State, in context, two assumptions needed for the number of demands for a single -bed room for a particular day to be well modelled by a Poisson distribution. [2] Assume now that these assumptions in fact hold, and also that the number of demands per day for a single -bed room and the number of demands per day for a double -bed room follow independent Poisson distributions with means 5.8 and 37.1 respectively. (ii) On a particular day, 24 of the single -bed rooms are undergoing renovation and not available for booking. Find the probability that all the remaining single -bed rooms are booked for this day. [2] (iii) Show that there is a probability of 0.0927 that, for a given day, there are at least 3 6 demands for a double-bed room and at least 1 demand for a single-bed room, given that there is a total of 38 demands for that day. [3]

[Turn_over 9 In this question, you should state clearly the values of the parameters of any normal distributions you use. The weight, in grams, of a handphone is a random variable with the distribution   2N ,5 , where  is a constant. T he weight, in grams, of a tablet is a random variable with the independent distribution   2N 330,6 . (a) Suppose 130. Calculate the probability that the total weight of 5 randomly selected handphones is less than twice the weight of one randomly selected tablet. [3] (b) Suppose instead that  is unknown. A random sample of n handphones is taken. There is a probability of at most 0.04 that the mean weight of these handphones differs from the population mean weight of the handphones by more than 1 gram. Calculate the least value of n. [4] 10 In a particular country, cultured milk are manufactured, in bottles, by two companies, Yacoat and Vitergent. 60% of all cultured milk in that country are manufactured by Yacoat while the remaining 40% are manufactured by Vitergent. Both companies manufacture the cultured milk in three different flavours, “Apple”, “Gr