MJC JC2 H2 Maths 2012 Paper 2

Write your name and civics 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. MERIDIAN JUNIOR COLLEGE JC2 Preliminary Examination Higher 2 ___________________________________________________________________ H2 Mathematics 9740/02 Paper 2 17 September 2012 3 Hours Additional Materials: Writing paper List of Formulae (MF 15) ___________________________________________________________________ READ THESE INSTRUCTIONS FIRST ___________________________________________________________________ This document consists of 7 printed pages and 1 blank page [Turn Over]

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3 MJC/2012 JC2 Preliminary Examination/9740/02 Section A: Pure Mathematics [40 marks] 1 A lake in Pangaea has a population of 600 000 adult fish at the start of 2012 and it is increasing at a rate of approximately 5% per year. It is proposed that at the end of every year 40 000 adult fish should be harvested. Let nu denote the size of the population (in thousands) at the start of n years after 2012. (i) Write down a recurrence relation for nu , and show that  800 200 1.05 n nu  . [4] (ii) Hence find the predicted population of the adult fish at the start of 2020. State what happens to the population of the adult fish for large values of n. [2] (iii) If the population of adult fish is to be maintained at 600 000, what should be the proposed number of adult fish to be harvested at the end of every year? [1] 2 The functions f and g are defined by      f : 2 1 2 9 , , g : ln 2 , 2. x x x x x x x      (i) Given that -1f exist when domain is restricted to [ a, b] where a < 0 and b > 0, find the least value of a and greatest value of b. [1] Using the restricted domain found in part (i), (ii) find the domain of -1f and the expression for   1f x , [5] (iii) find the range of the composite function gf, leaving your answer in exact form. [2] 3 A curve C is defined by parametric equations 3243x t t y t t    . (i) Find the equation of the tangent at 1t . [2] (ii) Given that the normal at another point P on the curve is perpendicular to the tangent found in part (i), find the value of t at point P. [2] (iii) Sketch the graph C for  1 ,3t , labeling the exact coordinates of the point where the gradient is undefined and the end points of C. [4] [Turn Over]

4 MJC/2012 JC2 Preliminary Examination/9740/01 4 The region R is bounded by the curve 2 2e1 x y   , the x-axis, the y-axis, and the line x = a, where a > 0. (i) Find the area of the region R in terms of a. [3] (ii) Hence find the exact value of   2 21 0 fd yy  , where    f 2ln 1xx , and give a geometrical interpretation of the value found. [4] 5 The line l1 passes through point A, whose position vector is 4 5 6 ,i j k and is parallel to the vector 2 3 .  i j k The plane p1 has equation 2 3 4.x y z   (i) Show that l1 is perpendicular to p1. Hence find the coordinates of the foot of perpendicular from A to p1. [3] The line l2 is given by 1 1.x y z    (ii) Given that the plane p2 is parallel to the line l1 and contains the line l2, show that the equation of p2 is 4 3 2.x y z   What is the relationship between the lines l1 and l2? [3] (iii) The planes p1 and p2 intersect in a line l3. Find the vector equation of the line l3. [1] (iv) The plane p3 has equation  2 7 2 2 1 0.x y z x y z         Given that the three planes p1, p2 and p3 have no point in common, what can be said about the values of and ? [3]

5 MJC/2012 JC2 Preliminary Examination/9740/02 Section B: Statistics [60 marks] 6 The table below shows the breakdown of the students in Tenaz Junior College. Males Females JC 1 480 560 JC 2 240 320 A sample of 100 students from different levels and gender is to be selected from the college to find out the amount of time they spent using the sports facilities. (i) Roarbert plans to survey 25 students from each stratum. State the name of this sampling method. [1] (ii) State a disadvantage in Roarbert’s sampling method. [1] (iii) Suggest another sampling method that would not have this di sadvantage, and describe how it can be carried out. [3] 7 (a) Describe the difference in the hypothesis between a one -tailed test and a two-tailed test. [1] (b) The mean weight of boys in a particular school is known to be m kg. A new weight gain programme was tried out on a random sample of 50 boys in this school. The weight of these 5 0 students, x, after the implementation of the new weight gain programme gave the following data: (x – 30) = 1279 and  x2 = 155233 A hypothesis test is carried out at the 5% significance level and it is found that the weight gain programme has been effective. Find the range of values of m. [5] [Turn Over]

6 MJC/2012 JC2 Preliminary Examination/9740/01 8 Three men, three women and a married couple are randomly seated at a round table with eight seats. Find the probability that (i) the married couple is seated together, [3] (ii) no two women are seated next to each other given that the married couple is seated together, [3] giving each of your answers as a fraction in its lowest terms. State, with a reason, whether or not the events ‘no two women are seated next to each other’ and ‘the couple is seated together’ are independent. [2] 9 A student wants to investigate the growth of mould on one side of a slice of expired bread measuring 8 cm by 8 cm. He placed the slice of bread in a sealed bag and measured the area of the bread covered by mould over 10 days. The following is the data recorded by the student. (i) Draw a scatter diagram to illustrate the data. [2] (ii) Calculate the product moment correlation coefficient and use a suitable regression line to estimate the area of the bread covered by mould on the 7th day. Comment on the reliability of your answer. [4] (iii) The student would like to use the regression line in part (ii) to predict the area of the bread covered by mould after 80 days. By calculating the predicted area of the mould, explain why a linear model may not be appropriate in the context of the question. [2] Time, t days 0 2 3 4 6 8 9 10 Area, y cm2 0.3 1.5 2.3 4.1 6.3 8.2 8.7 8.8

7 MJC/2012 JC2 Preliminary Examination/9740/02 10 The time taken for Miss Lau to wrap a large hamper is n ormally distributed with mean 18 minutes and standard deviation 4 minutes. The time taken for her to wrap a sm all hamper is n ormally distributed with mean 10 minutes and standa rd deviation  minutes. The time taken by her to wrap different hampers may be