ACJC Question Paper 2

ANGLO-CHINESE JUNIOR COLLEGE MATHEMATICS DEPARTMENT MATHEMATICS Higher 2 Paper 2 27 August 2012 JC 2 PRELIMINARY EXAMINATION Time allowed : 3 hours Additional Materials: List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your Index number, Form Class, graphic and/or scientific calculator model/s on the cover page. Write your Index number and full name on all the work you hand in. Write in dark blue or black pen on your answer scripts. You may use a soft pencil for any 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 signi ficant 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 a llowed unless a question specifically states otherwise. Where unsupported answers from a graphic calculator are not allowed in the 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. This document consists of 6 printed pages. [Turn Over 9740 / 02

Anglo-Chinese Junior College H2 Mathematics 9740: 2012 JC 2 Preliminary Examination Paper 2 Page 2 of 6 ANGLO-CHINESE JUNIOR COLLEGE MATHEMATICS DEPARTMENT JC2 Preliminary Examination 2012 MATHEMATICS 9740 Higher 2 Paper 2 Calculator model : _____________________ Arrange your answers in the same numerical order. Place this cover sheet on top of them and tie them together with the string provided. Question No. Marks 1 /4 2 /9 3 /11 4 /16 5 /3 6 /7 7 /9 8 /7 9 /9 10 /7 11 /9 12 /9 / 100 Index No: Form Class: ___________ Name: _________________________

Anglo-Chinese Junior College H2 Mathematics 9740: 2012 JC 2 Preliminary Examination Paper 2 Page 3 of 6 Section A: Pure Mathematics [40 marks] 1 The lines 1l and 2l have equations 2.5 1 3 2 1 r k k k λ         = +             and 3 4 0.5 7 0.5 5 r k k µ         = +             , λ , µ ∈ /Rbb respectively, where k is a constant. (i) Given that 1l and 2l intersect at a point, find the value of k. [3] (ii) Hence find the coordinates of the point of interse ction. [1] 2 Given that 1sin 2 e xy − = , show that ( ) 2 2 2 d d 1 4 4 4 dd y y x y x xx − = + . [3] By differentiating this result, or otherwise, find the series expansion of y in ascending powers of x, up to and including the term in 3x . [4] Hence find an approximate value of 6e π as a fraction in its simplest form. [2] 3 (i) Use the substitution 1 sin x u = + to find 22 d x x x −∫ . [5] (ii) The finite region R, which includes the point (0,1), is bounded by the curves 9 y x x = + and 22 2 y x x x = + − − . Find (a) the exact area of R, [4] (b) the numerical value of the volume of revolution fo rmed when R is rotated completely about the x-axis, giving your answer to 3 decimal places. [2] 4 The equations of four planes 1p , 2p , 3p and 4p are 2x y z + + = , 1x ay z − + + = , 2 2 2 x y z b + + = , 2 3 cx y z d + + = , respectively, where a, b, c and d are constants. The planes 1p and 2p intersect in a line l whose equation is 11 2 3 y z x −− = = − . (i) Show that 2a = . [2] (ii) Given that point A has coordinates ( )0,0,1 , find the position vector of the foot N of the perpendicular from A to the plane 1p . [3] [Turn Over

Anglo-Chinese Junior College H2 Mathematics 9740: 2012 JC 2 Preliminary Examination Paper 2 Page 4 of 6 (iii) Find the acute angle between planes 1p and 2p . [2] (iv) Given that the distance between 1p and 3p is 3 , find the value of b where 0b > . [3] (v) Given that 1p , 2p and 4p meet in a line, find the values of c and d. [4] (vi) Given instead that the planes 1p , 2p and 4p have no point in common, what can be said about the values of c and d ? [2] Section B: Statistics [60 marks] 5 A company comprises 150 programmers, 15 secretarie s and 15 section heads. The owner of the company wants to gather feedback from the entire company. Describe, in this context, how a sample of 12 could be chosen using stratified sampling and state an advantage of stratified sampling. [3] 6 Four boys and three girls are at a playground. (a) In one of their games, all seven of them have to s tand in a straight line. (i) If all the girls are separated, find the number of ways in which this can be done. [2] (ii) If all the boys are to be together, find the numbe r of ways in which this can be done. [2] (b) In another game, only six of them can play at a ti me and the six players have to stand in a circle with all the girls separated. Fin d the number of ways in which this can be done. [3] 7 In an experiment involving two chemicals x and y, a researcher recorded observations of values of y for controlled values of x as shown in the table. x 6.3 12.3 8.1 7.5 4.4 5.1 10.6 11.7 9.4 y 6.3 4.5 5.3 5.6 3.2 2.7 4.6 4.5 4.8 (i) Draw a scatter diagram to illustrate the data and calculate the value of the correlation coefficient. [2] (ii) Explain whether your answers in (i) suggest that a linear model is appropriate. [2] The researcher realised that some of the observati ons came from contaminated materials. He then considered only the seven pairs of observations for which the values of x exceeded 6 and discarded the other observations. B y considering the product moment correlation coefficient for these se ven pairs of observations, identify which of the following is a better model: (a) y = ax 2 + b, or (b) y = a ln x + b, where a and b are constants, giving a reason for your choice. [2] Hence using the better model, estimate the value o f x when y = 6.1. For this model, comment on the validity of this estimated value. [3]

Anglo-Chinese Junior College H2 Mathematics 9740: 2012 JC 2 Preliminary Examination Paper 2 Page 5 of 6 8 Cars arrive at a jetty randomly at an average rate of one every 5 minutes. From the jetty, a ferry transports the cars to an island. Th e ferry can transport at most 3 cars at a time. The ferry departs the jetty when it is full or half an hour after its arrival whichever is sooner. (a) If the jetty is empty when the ferry arrives, fin d the probability that the ferry (i) departs with 2 cars [2] (ii) departs when it is still not full. [2] (b) The ferry takes 5 minutes to travel from the jet ty to the island and vice versa. The ferry leaves the jetty when it is empty and sta ys at the island for 10 minutes. The probability that more than k cars are waiting at the jetty when the ferry arrives is less than 0.1. Find the least value of k. [3] 9 (a) Each day a train travels from Town A to Town B and it is scheduled to arrive at Town B at 0800 hrs. During the first year of operation it leaves Town A at 0700 hrs and is found to be late arriving at Town B on an average of once every 4 days. In the second year it leaves Town A at 0650 hrs and is late once every 20 days. Assuming the time of journey has a n ormal distribution, find, to the nearest minute, the standard deviation of the time of journey. [4] (b) Each day another train travels from Town B to Town C. The train is scheduled to arrive at Town C at 1200 hrs. The time taken for this journey is a normal random variable with mean 80 mins and variance 9.