CJC H2 MATH P2 QP

9740/02/Prelim/2015 [Turn over CATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 JC2 Preliminary Examination MATHEMATICS 9740/02 Paper 2 31 Aug 2015 3 hours Additional Materials: List of Formulae (MF15) Graph Paper Name: ___________________________ Class: ________________ READ THESE INSTRUCTIONS FIRST Write your name and 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 an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing 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, arrange your answers in NUMERICAL ORDER. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 6 printed pages, including the cover page.

2 9740/02/Prelim/2015 [Turn over Section A: Pure Mathematics [40 marks] 1 A Singaporean tourist is visiting three countries, Denmark, Engl and and Russia and is planning to buy souvenirs back for her loved ones. She pl ans to spend SGD$84 on cheese, SGD$85 on chocolate, and SGD$77 on candy. She can only buy in souvenir packs consisting of 1kg cheese, 1kg chocolate and 1kg candy. The exchange rates to the three countries are as follows: SGD$1 – 5 Danish Krone SGD$1 – 0.5 British Pound SGD$1 – 35 Russian Ruble The prices of the commodities in the various countries’ currency are given in the following table. Cheese/kg Chocolate/kg Candy/kg Price in Danish Krone 20 30 30 Price in British Pound 4 5 2 Price in Russian Ruble 280 175 245 Find the total number of souvenir packs she s hould buy so that she spends all her money. [4] 2 The diagram below shows the graph of f( )yx . The curve has a minimum point at 11, 2   and crosses the x–axis at  2, 0 , 0, 0 and  2, 0 . The lines 1, 3 and 3xx y   are the asymptotes of the curve. (i) Sketch, on separate diagrams, the graphs of (a) 2 f( )yx  , [4] (b) f '( )yx , [3] stating clearly in each case the equations of as ymptotes, the coordinates of turning points and axial intercepts whenever possible. (ii) State the number of distinct real roots of the equation 1f( ) 0 2x   . [1] (iii) Describe fully a sequence of transformati ons which would transform the graph of f( )yx to the graph of f( 2 1 ) 3yx  . [3] x y O  fyx

3 9740/02/Prelim/2015 [Turn over 3 A calculator is not to be used in answering this question. The complex numbers a and b are given by 1i 1i   and i 2 1   respectively. (i) Find the moduli and arguments of a and b . [3] (ii) In an Argand diagram, the points A , B and C represent the complex numbers a , b and ab respectively. The origin is denoted by O. By considering the quadrilateral OACB and the argument of ab , show that 3tan π 8 12     . [3] (iii) Using a single Argand diagram, sketch the loci (a) 2za , (b)  πarg 2zb . Find the exact complex number z , in the form ixy , that satisfies parts (a) and (b). [5] 4 In a research project, the population is modelled by the following logistic differential equation, d 0.64 1d1 0 PP Pt   , where P is the population function of time t . (i) Solve the differential equation by expressing P in terms of t , given that 1P when 0t . [5] Sketch the solution curve for 0t… . Comment on the population in the long run. [2] An alternative model for the population is the Gompertz function, which is the solution to the following differential equation, d 0.4 ln10 lnd P PPt  . (ii) By solving the differential equation, show that the general solution is 0.4e10e A t P  , where A is a constant. [3] Given the same initial condition that 1P when 0t , sketch the solution curve of the particular solution for 0t… on the same diagram in part (i). Comment on the similarity and difference between the two models. [4]

4 9740/02/Prelim/2015 [Turn over Section B: Statistics [60 marks] 5 A class consists of 15 female and 10 male studen ts. The form teacher needs to select 5 students from this class to attend a school function. The teacher wrote each student’s name (all the students’ names are distinct) on a small piece of paper of the same size, folded it into half and placed it in a large bowl. He then shook the bowl and took out five pieces of pa per one by one without replacement. (i) State the name of this method of sampling and explain a disadvantage of this method in the context of the question. [2] (ii) Find the probability that 3 female students and 2 male students are selected. [3] Amy is one of the 15 female students and Bertrand is one of the 10 male students. (iii) Find the probability that Amy and Bertrand are selected. [2] (iv) Given that 3 females and 2 males are selected, find the probability that Amy and Bertrand are selected. [4] The school management decided that the students selected must be representative of the class gender make-up. (v) Describe how the form teacher could select the 5 students. Write down the probability that Amy and Bertrand are selected using this method of sampling. [3] 6 On average 8% of cherries sold in supermarke ts are rotten. A customer randomly selected 26 cherries from a large number of cherries. (i) State, in context, two assumptions for the numbe r of rotten cherries in the sample to be well- modelled by a binomial distribution. [2] (ii) Find the most likely number of rotten cherries that the customer could have picked. [2] (iii) Another customer randomly selected n cherries such that the proba bility of having at most one rotten cherry is less than 0.1. Expre ss this information as an inequality in n , and hence find the smallest possible integer value of n. [3] (iv) The cherries are packed in boxes, each containing 60 cherries. Using a suitable approximation, find the probability that a rando mly chosen box contains at most one rotten cherry. [3] On average, 1 box of rotten cherries will be discarded every month. (v) Using a suitable approximation, find the probabi lity that in a year, th e number of boxes of rotten cherries that will be discarded is between 2 and 5 inclusive. [3]

5 9740/02/Prelim/2015 [Turn over 7 The masses, in grams, of carrots and onions ar e normally distributed w ith means and standard deviations as shown in the table below. Mean (g) Standard deviation (g) Carrot c 5 Onion 75 3 (i) The probability that twice the mass of a randoml y chosen carrot exceeds the total mass of 5 randomly chosen onions is more than 0.9. Find the range of values of c . State an assumption needed for your calculation. [5] It is given that 200gc . (ii) Find the probability that th e average mass of two carrots and three onions exceeds 130g. [3] (iii) Carrots are sold at $1.80 per kg and onions at $1.50 per kg. Find the probability that the price difference between 3 carrots and 4 onions is less than $0.60. [4] 8 A manufacturer claims that the mean mass of peanut butter in a jar is 0 g. A shopke