JJC H2 MATH P2

JURONG JUNIOR COLLEGE J2 Preliminary Examination MATHEMATICS 9740/02 Higher 2 24 August 2009 Paper 2 3 hours Additional materials: Answer Paper List of Formulae (MF15) Cover Page READ THESE INSTRUCTIONS FIRST Write your name and civics class 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 ar e 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. At the end of the examination, fasten all your work securely together, with the cover page in front. This document consists of 7 printed pages and 1 blank page. [Turn over

2 8 1 , Section A: Pure Mathematics [40 marks] 1 Given that lines and where 1 15 :0 31 l         r 2 13 :0 30 l         r   , meet at point A. B is the poin 4) and lies on 1l while C is a point on 2l with non-negative coordinates. t (6, 8, (i) Given that AB and AC are of equal lengths, show that  . [3] 10 3 3 OC      (ii) A point D is such that ABDC is a parallelogram. Find the position vector of D. [2] (iii) Find the sine of the angle BAC exactly and hence find the exact area of the parallelogram ABDC. [3] 0 11n n  6 n 5 n 4 n 2 n 3 n 1 n y2 x The graph of 2 xyx e , for 01 x is shown in the diagram. Rectangles, each of width 1 n are drawn under the curve as shown. (a) Show that A, the total area of the rectangles may be expressed as 1 3 1 1 f( ) , n r rn    where f( is to be found. [2] )r Find, using integration by parts, the limiting value of A when n , giving your answer in exact form. [4]  (b) A region R in the first quadrant is bounded completely by the curve 2 xy xe , the line y e and the y-axis. Find the volume of revolution formed when R is rotated completely about the x-axis, giving your answer correct to 2 decimal places. [2]

3 3 (a) Expand 13 2 n x x      where n  , in ascending powers of x , up to and including the term in 2x . State the range of values of x for which the expansion is valid. [4] (b) (i) Given that 1tanln 2 xy   , show that  2 2 2 d21 ( 1 4 ) dd dy yxx xx  . [2] (ii) Find the first four terms of the Maclaurin’s series of in ascending powers of x.[3] y (iii) Given that x is sufficiently small for 3x and higher powers of x to be neglected, deduce an expansion for 1tan 2 ln (sec ) e x x    in ascending powers of x. [2] 4 The curve C has equation 22 1 1x kxy x    , where k is a constant. (i) Show that C has no stationary point if k > 1. [2] (ii) Given that y = 2x + 1 is an asymptote of C, show that k = 3. [2] (iii) State a sequence of transformations which transform the graph of 1yx x to the graph of 223 1 1x xy x   . [3] (iv) Sketch on separate diagrams, the graphs of 223 1 1x xy x    and 2 2 23 1 1x xy x    , showing clearly the coordinates of the points where the graphs cross the axes and the equations of any asymptotes. [4] Hence deduce the number of real roots of the equation 43 223xx x x 1 .   [2] [Turn over

4 Section B: Statistics [60 marks] 5 A company has 2000 employees belonging to the following groups: Production 1200 Marketing 600 Management 100 Others 100 The company's president wants to obtain an es timate of the views of all employees about a pending executive decision. A sample of 100 employees is to be chosen to take part in a survey. (i) Describe one disadvantag e of obtaining the sample using quota sampling. [1] (ii) Name and briefly describe another method of sampling, in which each group is represented proportionately. [3] 6 Past records have shown that the mean reading speed per minute of Primary 2 school children is 163 words. The numbers of words, w, per minute read by a random sample of 80 Primary 2 school children are taken and summarised by 13560w 2 2388670.w  (i) Test, at the 5% level of significance, wh ether the mean reading speed of Primary 2 school children has increased. [5] (ii) Explain what do you understa nd by the phrase "a t the 5% level of significance" in the context of this question. [1] (iii) Explain whether the test is still valid if the 80 children were all chosen from the same school. [1]

5 7 (a) In an organisation of 10 workers, how ma ny ways can a leader and 3 other committee members be appointed? [1] After appointing the leader and the 3 committee members, in how many ways can these 10 people sit at a round table, if the lead er must have any tw o of his committee members seated beside him, when (i) the seats are not numbered? [2] (ii) the seats are numbered? [1] (b) Five cards each have a single digit written on them. The digits are 1, 2, 2, 4, 6 respectively. How many even numbers gr eater than 4000 can be formed by placing some or all of these cards side by side? [3] 8 A deck of 36 cards has four colours, red, blue, green and yellow. Each colour set consists of 9 cards numbered 1, 2, 3, …, 9. (a) One card is taken at random from the deck. Events A and B are defined as follows: A: The card taken is blue. B: The card taken is a 1. (i) Determine whether A and B are independent events. [2] (ii) Describe in words what the event A B represents and find the probab ility of this event. [2]  (iii) Find P' | 'A B . [2] (b) Three cards are taken from the deck , at random and without replacement. Find the probability that all three cards are of different colours with the same number. [2] (c) James takes 20 cards from this deck of 36 cards at random and without replacement. Find the probability that none of these cards are numbered ‘1’. [2] [Turn over

6 9 In a large canteen, a quarter of the customers drink coffee. (a) Find the probability that at least 3 out of a random sample of 7 customers drink coffee.[2] (b) If the probability that fewer than k out of a random sample of 500 customers drink coffee is at least 0.99, using a suitable approximation, find the least value of k. [5] (c) Assuming that 99.9% of the customers do not make a complaint and that complaints occur independently, using a suitable approxima tion, find the probabilit y that more than 498 out of a random sample of 500 customers will not make a complaint. [3] 10 Eric loves to solve online Sudoku puzzles. The puzzles are classified as “Beginner”, “Intermediate” and “Expert”. The times, in minut es, that Eric takes to solve the puzzles are independent and normally distri buted with the means and standa rd deviations shown in the following table: Type Mean (min.) Standard Deviation (min.) Beginner 3.2 0.5 Intermediate  2.1 Ex