JJC H2 MATHS P2

JURONG JUNIOR COLLEGE Preliminary Examinations MATHEMATICS 9740/02 Higher 2 17 September 2014 Paper 2 3 hours Additional materials: Answer Paper Cover Page List of Formulae (MF 15) 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 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. At the end of the examination, fasten all your work securely together, with the cover page in front. This document consists of 6 printed pages. [Turn over

2 Section A: Pure Mathematics [40 marks] 1 (a) Given that e cosyx x  , show that d tan 1 0d y xx  . (i) By further differentiation of this result obtain the Maclaurin’s series for y in terms of x, up to and including the term in x2. [3] (ii) Let the result in (i) be h( )x . Find the set of values of x for which h( )x is within 0.2 of the value of y. [2] (b) Expand, in ascending powers of x, 3 n xa  where a and n is a non- positive integer, up to and including the term in x2. [2] It is given that the coefficient of x is four times the coefficient of x2 and the constant in the expansion is 1 4 . Find a and n. [3] 2 (a) Find the general solution of the differential equation 2 2 2 d ed xy ax  , where a is a constant. [2] (b) Mr Tan borrowed $10 000 from a bank and at time t, the sum of money he owed the bank is denoted by x thousand dollars. The su m of money he owed increases, due to interest, at a rate proportional to the sum of money owed. Money is also repaid at a constant rate p. When x = 12, the interest and repayment balance. Taking both x and t as continuous variables, show that for x > 0, d 12d1 2 xp xt  . [2] (i) Find x in terms of t and p. [4] (ii) Find the time T that it will take Mr Tan to repay the loan, leaving your answer in terms of p. [1] (iii) Sketch the graph of x against t. [1]

3 3 The plane 1 has equation, 30 0      r. , where  and  are positive constants, and contains the point A with coordinates ( 10, 0, 5) . (i) Given that the perpendicular distance from the origin O to the plane 1 is 6, find and   . [5] Another plane 2 has equation 1 14 2    r. . (ii) Find the acute angle between the line OA and the plane 2 . [2] (iii) Find a cartesian equation of the plane 3 which contains the line OA and is perpendicular to the plane 2 . [3] 4 The complex numbers andp q are given by 1 i and 1 3 ipq  . (i) Two other complex numbers andrs are given by irk 2 3and p rs q , where is a real number.k Given that 1 2s  and 2arg( ) 3s  , find the value of k . [5] (ii) Solve the equation 3 4zq . [2] Hence find the roots of the equation   3 43iw   . [3] [Express all answers in the form ier  , where 0r and   , giving r and  in exact form.] [Turn over

4 Section B: Statistics [60 marks] 5 A company wants to intervie w residents from a town about the brand of shoes that they prefer. An interviewer stationed himself at the town centre and interviewed 100 residents of different age groups. (i) Name the sampling method used by the interviewer and state one disadvantage of this method in the context of the question. [2] The company would now like to find out more about the preferences of the youths in a youth club with members aged 18 to 35. The table below shows the number of male and female members of each age group. 18 - 25 year old 26 - 35 year old Male 75 36 Female 99 40 (ii) Explain how a stratified sample of size 100 could be obtained. [2] 6 The random variable X has the distribution   2N,   . Given that 1X and 2X are two independent observations of X and    12 1 2P2 P 2 0 . 8XX XX a   , where a is a constant, find  in terms of a. [7]

5 7 For events A and B, it is given that P( ) 0.6A  ,  P ' 0.75BA  and P0 . 4AB  . Find (i) P' 'A B , [3] (ii) P( )B . [3] Justifying your conclusion, determine whether A and B are independent. [1] 8 Four letters of the word ABSENCE are se lected and arranged to form four-letter code words. Show that the number of four-letter code words that can be formed without restrictions is 480. [3] (i) Find the probability that a randomly chosen four-l etter code word contains distinct letters. [2] (ii) A four-letter code word containing distinct letters is randomly chosen. Find the probability that it does not contain any vowels (A, E). [3] 9 A random sample of nine pairs of values of x and y are given in the table. x 2.5 2.0 3.0 3.5 5.0 4.0 5.3 7.5 6.0 y 3.20 3.40 3.00 2.86 2.61 2.75 2.57 k 2.55 (i) The equation of the regression line of y on x is 0.175 3.57.yx  Show that 2.40k  . [3] (ii) Draw a scatter diagram for this set of data and obtain the product moment correlation coefficient. Comment on the suitability of the linear model. [3] (iii) State, with a reason, which of the follo wing would be an appropriate model to represent the above data. The letters a, b, c, d, e and f represent constants. 2A. y ab x , where a is positive and b is negative, B. lny cd x , where c is negative and d is positive, C. fye x , where e is positive and f is positive. [1] (iv) It is required to estimate the value of y for which x = 8.0. Find the equation of a suitable regression line, and use it to find the required estimate. Comment on the reliability of your estimation. [3] [Turn over

6 10 The number of packets of cereal sold in a supermarket is a normally distributed random variable. Over a long period it is know n that the mean number of packets of cereal sold per day is 124.5. Following a promotional campaign on television advertisement, the daily sales, x packets, is measured for a random sample of 12 days. The results are summarized as follows: 1620x  and 2 221175x  . (i) Find unbiased estimates of the population mean and variance. [2] (ii) Test, at the 1% level of significance, whether the mean number of packets of cereal sold per day has changed. [4] After some modifications to the promotiona l campaign, a new sample of 12 days is taken and the population standard deviati on is known to be 13. The owner of the supermarket claims that the mean daily sales has increased from 124.5. Find the range of values of the mean daily sales of the new sample if the owner’s claim is to be accepted at the 1% level of significance. [4] Explain what you understand by the expression ‘at the 1% level of significance’ in the context of this question. [1] 11 A small hospital receives emergency admissions randomly at an average rate of 3 per day. (i) Find the probability that there will be a total of 3 emergency admissions i