SRJC H2 MATH P2 STUDENT S

1 SERANGOON JUNIOR COLLEGE 2015 JC2 PRELIMINARY EXAMINATION MATHEMATICS Higher 2 9740/2 Tuesday 25 Aug 2015 Additional materials: Writing paper List of Formulae (MF15) TIME : 3 hours READ THESE INSTRUCTIONS FIRST Write your name and class on the cover page and 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. Total marks for this paper is 100 marks. This question paper consists of 6 printed pages (inclusive of this page) and 2 blank pages. [Turn Over

2 Section A: Pure Mathematics [40 marks] 1 Functions f and g are defined as below. 1f : , 2 1xx x  g : ln(2 ) , 2x x x x    (i) Sketch the graph of y = g(x) and state its exact range. [3] (ii) Determine whether the composite function s fg and g f exist, justifying your answer. Find the range of the composite function if it exists. [4] (iii) On the same diagram as part (i), sketch the graph of y = g −1(x), indicating on your sketch the line in which the graph of y = g(x) must be reflected in order to obtain the graph of y = g−1(x). [2] 2 A tank contains water which is heated by an electric water heater working under the action of a thermosta t. When the water heater is first switched on, the temperature of the water is 35 C . The heater causes the temperature to increase at a rate Cr per minute, where r is a constant, until the water temperature hits 75 C .The heater then switches off. (i) Write down, in terms of r, the time taken for the temperature to increase from 35 C to 75 C . [1] The temperature of the water then immediately starts to d ecrease. The temperature of the water at time t minutes after the heater is switched off is C . It is known that the temperature of the water decreases at a variable rate ( 25) Ck  per minute, where k is a positive constant, until 35  . (ii) Write down a differential equation involving  and t, to represent the situation as the temperature is decreasing. [1] (iii) Given that when 55  , the temperature is decreasing at a rate of 5C per minute, find the total length of time for the temperature to increase from 35 C to 75 C and then decrease to 35 C , leaving your answ er in exact form, in terms of r. [7]

3 3 [It is given that the volume of a sphere of radius r is 34 π3 r and that the volume of a right square pyramid with a square base of length x and height h is 21 3 xh .] In the di agram below, a hemisphere of fixed radius a cm lies on the base of a right pyramid such that its curved surface is in contact with all four faces of the pyramid. The pyramid has a square base of length x cm and height y cm. Given that the volume of region inside the pyramid that is not part of the hemisphere is denoted by V, (i) show that V = 3 3 22 12 π33 4 ax a xa   . [3] (ii) use differentiation to find, in terms of a, the minimum value of V exactly, proving that it is a minimum. [7] 4 (a) The complex number z is given by iezr  , where r > 0 and 0 1 2 . (i) Given that w = ( 3 – i) z, find |w| in terms of r and arg w in terms of  . [2] (ii) For a fixed value of r, draw on the same Argand diagram the the locus of z and w as  varies. [2] (iii) If r = 1.5 units, calculate (in terms of  ) the length of the locus of w for Im(w)  0 as  varies. [2] (b) Sketch on a single Argand diagram the set of points representing all complex numbers v satisfying both of the following inequalities: 5 8i 5v- - £ and 12 8i 12 10ivv- - ³ - - . Hence find (in radians) the least value of arg ( 5 3i)v-+ . [6] [Turn Over x x y a

4 Section B: Statistics [60 marks] 5 A company with eight hundred employees wishes to find out how much time its employees take to travel to work. It is given that the employees go to work either by car or by bus/train and that each of them takes the same form of transport to work every day. The following table shows the number of employees going to work by car and the number of employees going to work by bus/train. Car Bus/Train Men 165 260 Women 82 293 The company wants to take a random sample of 180 employees. (i) Explain what is meant, in this context, by the term “a random sample”. [1] (ii) Describe how a random stratified sample can be obtained. [2] (iii) Give a reason why quota sam pling is not as suitable in this context compared to stratified sampling. [1] 6 Salt is packed in bags to be sold. T he manufacturer claims each bag contains at least 300 g of salt. To test this claim, a random sample of 15 bags of salt is examined and the mass, x g, of the contents of each bag is determined. It is found that the sample has a mean of 299.1 g and variance of 3.864 g2. (i) Test at the 10% significance level whether the manufacturer’s claim is valid. [5] (ii) State an assumption necessary for the test in (i) to be valid. [1] 7 A bag contains w white balls and b black balls. One ball is selected at random from the bag, its colour noted and it is then returned to the bag along with n additional balls of the same colour. A second ball is then randomly selected from the bag. (i) Construct a probability tree showing this information. [2] (ii) Show that the probability that the second ball selected is black is independent of n. [2] It is now known that the second ball drawn is black. Show that the probabilit y that the first ball drawn is white is w w b n . [2]

5 8 The table below shows Singapore’s GDP per capita over the years from 1965. Year, x GDP per capita (in thousands), y 1965 1.580 1970 2.832 1975 6.607 1980 10.714 1985 14.921 1990 23.139 1995 35.346 2000 41.018 2005 49.715 2010 63.498 Source: Department of Statistics Singapore (a) Using the data available, (i) draw the scatter diagram, labelling the axes clearly, [2] (ii) find the least square regression line of y on x, [1] (iii) estimate the GDP per capita in the year 2015, correct to the nearest whole number. Comment on the reliability of the value obtained. [2] (b) It is suggested that the data from 1965 to 2010 can be modelled by   2 1965y a b x   instead of a linear model . Find th e value of the product moment correlation coefficient for each of the proposed models and determine which is the better model. [2] 9 In a factory manufacturing calculators, it is found that 1.5% of the calculators produced are defective. In a random sample of 90 calculators, find the probability that (i) there are exactly 2 defective calculators, [1] (ii) the 90th calculator is the second defective calculator given that there are exactly 2 defective calculators in that sample. [3] The calculators are packed in boxes of 90. (iii) Using a suitable approximation, find the probability that not more than 1 box of calculators, out of 60 boxes, contain more than 2 defective calculators. [5] [Turn Over

6 10 Seven men and seven women