NYJC H2 2020 Prelim P2

This document consists of 6 printed pages and 0 blank page. NANYANG JUNIOR COLLEGE Internal Examinations © NYJC 2020 [Turn Over NANYANG JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME CT CLASS 1 9 Centre Number/ Index Number / MATHEMATICS 9758/02 Paper 2 15 September 2020 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name and class on the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Write your answers in the spaces provided in the Question Paper. 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. The number of marks is given in brackets [ ] at the end of each question or part question. For examiner’s use only Question number Marks 1 2 3 4 5 6 7 8 9 10 11 Total

2 NYJC 2020 JC2 Preliminary examination 9758/02 Section A: Pure Mathematics [40 marks] 1 In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. It provides a method of rendering graphs and indicating the positions of points on a two-dimensional surface. The polar coordinate system is employed in mathematics, physics, engineering, navigation, robotics, and other sciences. The area of a region enclosed by a curve with polar equation f ( )r  where    , is given by the integral   21 f ( ) d2    . A curve C has polar equation  cos2 sinra  , where a is a positive constant and 26     . By using the formulae from the List of Formulae (MF26), show that the area of the region enclosed by C is given by   62 2 1 cos4 cos2 4cos2 sin 2 d4 a            . [2] Hence find, in terms of a, the exact area enclosed by C. [4] 2 The function f is given by 2f : 2x x kx  , for ,x x k  . (i) Find   1f x and state the domain of 1f  in terms of k. [3] For the rest of this question, let 2k  . (ii) The function g is given by   2 2 1g : 5 7 1 4 xx  , for x , 35 x  . If fg(x) = 32, find the exact value(s) of x. [4] 3 A curve C has parametric equations   1 21 21 4 , cos (2 )xy     , where 1 02    . (i) Find d d y x in terms of  . Hence find the exact coordinates of the point on C such that the normal to the curve at this point is parallel to the line 2yx . [4] (ii) Q is a point moving on C such that its y-coordinate increases at a rate of 0.25 units per second. Find the value of  at the point Q when the rate of change of the gradient at Q is 1 unit per second. [4]

3 NYJC 2020 JC2 Preliminary examination 9758/02 [Turn Over 4 Referred to the origin O, points A and B have position vectors a and b respectively. Points X and Y are such that B is the mid-point of OX and B lies between A and Y with AB : BY = 2 : 1. The lines AX and OY meet at the point Z. (i) Find the position vectors OX , OY and OZ giving your answers in terms of a and b. [6] (ii) Given that 27OA , OB = 3 and the area of triangle OAZ is 1701 , find the acute angle between OA and OB, giving your answer exactly. [3] 5 (a) One of the roots of the equation 32 60x px x q    , where p and q are real, is 5 – i. Find the other roots of the equation and the values of p and q. [4] (b) Do not use a calculator in answering this part. (i) The complex number z has modulus 23 and argument 3  . Find 23z in the form ier  , where r > 0 and      . Give r and  in exact form. [3] (ii) Find the three smallest positive whole number values of n for which     * 23 23 n z z   is a real number. [3] Section B: Probability and Statistics [60 marks] 6 (a) Find the number of ways of arranging all the eight letters in the word ATTITUDE if the first and last letter must be a consonant and the T’s are separated. [3] (b) An organisation has 7 members in the Marketing team and 10 members in the Operations team. A committee consisting of 6 people is to be formed and it has been decided that it must include at least 2 people from the Marketing team and at least 3 people from the Operations team. (i) Find the number of ways in which the committee can be formed. [2] The committee of 6 people is then finalised and they join another 3 members from the Finance team to go for a bonding activity. During the activity, these 9 people are required to stand in a circle. (ii) Find the number of ways to arrange them. [1] (iii) Find the number of ways to arrange them such that all members of the Finance team are standing next to each other. [2] O B X A Y Z

4 NYJC 2020 JC2 Preliminary examination 9758/02 7 A jar contains five 20 -cent coins and n 50-cent coins, where 2n . In a game, Lee removes coins at random from the jar, one at a time, until he has at least 70 cents. He scores zero points if he removes only two coins, otherwise his score is half the number of coins he removed. The score Lee obtains is denoted by the random variable S. (i) Show that    ( 9)P( 0) 54 nnS nn   . [1] (ii) Find P(S = s) for all other possible values of s. [2] (iii) Show that    30E( ) 53S nn  and        2 22 15 3 40 113 Var( ) 5 4 3 n n n S n n n      . [5] 8 A manufacturer produces teacups, saucers and plates. Past records indicate that on average 5% of the teacups, 2% of the saucers and 1% of the plates produced are flawed. The quality of the teacups, saucers and plates are independent of one another. A box contains a teacup, a saucer and a plate. A box is considered imperfect if any of the three items are flawed. A consignment consists of 250 boxes. The number of boxes in a consignment that are imperfect is denoted by X. You may assume that X can be modelled by a binomial distribution. (i) Show that the probability that a randomly selected box is imperfect is 0.07831. [1] (ii) Find the probability that there are at le ast 10 but less than 30 boxes that are imperfect in a consignment. [2] (iii) Find the least value of r such that the probability that there are more than r boxes that are imperfect in a consignment is at most 0.125. [3] (iv) Using a suitable approximation, find the probability that the total number of boxes that are imperfect in a sample of 100 consignments is at most 2000. State an assumption that you have made in your calculations. [4] 9 A group of 240 residents who live in NY Gardens are surveyed on whether they like cats, hamsters or dogs as pets. The survey result are as follows:  80 residents like hamsters,  120 residents like either hamsters or cats (or both),  24 residents like both hamsters and dogs. There are some residents who like both cats and hamsters but n o r esident like both cats and dogs . Furthermore, no residents like all three animals. You may assume that a resident’s liking for cats and hamsters are independent. One resident from this group is chosen at random. The events where a resident likes cats, hamsters and do