AJC H2 MATH P2 Question

Page 1 of 6 AJC / 2015 Preliminary Examination / 9740 / P2 Anderson Junior College 2015 Preliminary Examination H2 Mathematics Paper 2 (9740/02) Section A: Pure Mathematics (40 marks) 1 (a) The variables w, x and y are connected by the following differential equations: 22 de d x w wx  and d d y wx  . (i) Verify that 2 2 e xw A  is the general solution of 22 de d x w wx  , where A is an arbitrary constant. [2] (ii) Hence find y in terms of x. [3] (b) A tank initially contains 50 grams of salt dissolved in 100 litres of water. Brine that contains 2 grams of salt per litre of brine fl ows into the tank at a rate of 5 litres per minute. The solution is kept thoroughly mixe d and flows out from the tank at a rate of 5 litres per minute. Given that the amount of salt in the tank at time t minutes is given by S , show that d 200 d2 0 SS t  . Hence find the time, in minutes, at whic h the concentration of salt in the tank reaches 1 gram per litre. [7] 2 Solve the equation 5 32 0z  , expressing your answers in the form ier  , where 0r  and   . [2] 1z , 2z and 3z are three of the roots of 5 32 0z   such that 1230 arg arg argzz z  . (i) Find the smallest positive integer n such that 1 2 n z z     is real and positive. [3] (ii) The points A and B represent the roots 1z and 3z respectively in the Argand diagram. The line segment 'BA is obtained by rota ting the line segment BA through 2  clockwise about the point B. Find the real part of the complex number represented by point 'A , giving your answer in exact trigonometric form. [4]

Page 2 of 6 AJC / 2015 Preliminary Examination / 9740 / P2 3 Referred to the origin O, the position vectors of points A, B and C are 6i , 424i j k and 326i j k respectively. The plane p1 is given by the equation 1x yz . (i) Find the equation of the plane ABC in scalar product form. [3] (ii) The perpendicular from the point D(3, 1, 4) to the plane p1 meets the plane ABC at the point S. Find the coordinates of S. [3] (iii) Find the equation of the plane p2 such that every point on p2 is equidistant from points S and D. [3] 4 Betty needs to decorate a wall of length 5 metres for a party. She attaches hooks, starting from the extreme left end of the wall, and numbers each hook “1”, “2”, “3” and so on. The spacing between the 1 st and 2nd hook is 50 cm and each subsequent spacing is 2 cm shorter than the previous spacing. This will c ontinue till she is unable to place the next hook due to insufficient space. (i) How many hooks can she attach in total? [4] Betty also cuts a 6 metre roll of ribbon into pieces of varying lengths. The first piece cut off is of length 80 cm and each s ubsequent piece cut off is 10 % shorter than the previous piece. (ii) If the length of the remaining roll of ribbon is less than 1 metre after n cuts, find the smallest value of n. [3] Betty wants to hang the ribbons between every 2 hooks such that the ribbons are of decreasing lengths starting from that between the 1 st and 2 nd hook. She starts by using the ribbon of length 80 cm. (Assume negligible length of ribbon is required to hang them on the hooks.) (iii) By considering the length of the ribbon to be hung between the nth and (n+1)th hook, write down an inequality to be satisfied by n. [1] (iv) Hence find the number of hooks which will have ribbons hung on them. [2]

Page 3 of 6 AJC / 2015 Preliminary Examination / 9740 / P2 Section B: Statistics (60 marks) 5 As part of SG50 celebrations, a local gras sroots organization invited the residents of the neighbourhood to the screening of a film on the 1964 racial riots. (i) A news reporter wants to interview 15 people at the event venue to find out how the film has impacted them. De scribe how a quota sample can be obtained in this context. [1] Before the screening of the film, reside nts were asked to re gister at a counter, providing their personal particulars such as names, contact numbers and their ages. Their age profiles were as follows: Age group below 21 years old 21 – 35 years old 36 – 50 years old above 50 years old Number of residents 130 195 260 65 The organiser intends to collect feedback on the event based on the various age groups by contacting some of them through telephone after the event. (ii) Describe in context, how a representa tive random sample of size 30 can be obtained by the organizer. State a disa dvantage of using this method as compared to using quota sampling (besid es the need for sampling frame and profile information). [3] 6 Three friends, Andy, Ben and Charlie, atte nd a graduation lunch with their parents. How many ways can the nine people be seated around a round table if each family must be seated together? [2] After the lunch, the nine pe ople stand in a single row to take a group photograph. How many ways can they be arranged if Andy and Ben are each standing between their respective parents and Charlie is standing either beside his father or his mother (or both)? [4]

Page 4 of 6 AJC / 2015 Preliminary Examination / 9740 / P2 7 A nursery supplies plants to Evergreen Pte. Ltd. The heights, in centimetres, of plants supplied may be taken to follow the distribution   2N,   . Over the years, Evergreen rejects 6% of th e plants supplied as too short and 4% as too tall. It is known that Evergreen only accepts plants of heights between 25 cm and 91 cm. Show that 20  correct to the nearest integer. Hence find the median height of the plants accepted by Evergreen. [5] Evergreen inspects the plants supplied in batches of 10. Find the probability that a total of at most 4 plants are rejected in two randomly chosen batches. [2] Three children, Anna, Bella and Cinderella , each chooses a plant at random from the nursery. Find the probability that the average height of the three plants is at least twice the average height of the plants chosen by Anna and Bella. [3] 8 It is believed that there is a correlat ion between the number of fogging sessions and the number of new dengue cases. To test th e effectiveness of fogging on controlling the spread of dengue, a particular nei ghbourhood is subjected to a number of fogging sessions ( x) spread out regularly in a mo nth and the corresponding number of new dengue cases ( y) is recorded. The results are summarised in the following table: No. of fogging sessions, x 2 4 6 8 10 12 14 No. of new dengue cases, y 20 14 9 6 7 2 1 (i) Calculate the product moment co rrelation coefficient between x and y, and explain whether your answer suggests that a linear model is appropriate. [2] (ii) Draw a scatter diagram for the data. [1] One of the data points appears to be incorrect. (iii) Indicate this point by labelling it as P, and explain why the scatter diagram for the remaining points may be cons istent with a model of the form ebxy A . [2] (iv) Omitting P, use an appropriate regression line to give the best estimate, to the nearest whole number, of the number of fogging sessions in a month when the number of new dengue cases is 7. [2]

Page 5 of 6 AJC / 2015 Preliminary Examination / 9740 / P2 9 Three red balls and two blue balls are placed in a bag. All the balls are indistinguishable except for their colours. Mr Red and Mr Blue take turns to dr aw a ball from the bag at random with replacement. The first player to draw th e ball whose colour matches his name wins the game and the game stops immediately. If Mr Red draws first, find the probability that (i) Mr Red wins the game at his third draw. [2] (ii) Mr Blue wins the game at his nth draw. [2] (iii) Mr Red wins the game given that the winner wins the game at his third draw. [2] (iv) Mr Blue wins the game. [2] 10 A farme