JPJC 2024 J2 Prelim Math P2 Qn paper

Name:____________________________________ Class:_____________ JURONG PIONEER JUNIOR COLLEGE JC2 Preliminary Examination 2024 MATHEMATICS 9758/02 Higher 2 13 September 2024 Paper 2 3 hours Candidates answer on the Question Paper. Additional materials: List of Formulae (MF 26) READ THESE INSTRUCTIONS FIRST Write your name and civics class on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. This document consists of 6 printed pages. [Turn over For Candidate’s Use For Examiner’s Use Question Number Marks Obtained 1 2 3 4 5 6 7 8 9 10 11 Total Marks / 100 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 ans wers 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 by [ ] at the end of each question or part question.

2 Section A: Pure Mathematics [40 marks] 1 (a) Sketch, on the same axes, the graphs of y = 3x and y = |x2 – a2| where a > 0. [2] (b) Find the exact solution of | x2 – a2| = 3x for 0 < x < a. [2] 2 (a) Find sin cos dpx qx x , where p and q are constants such that p  q and p  q. [2] (b) Given that m  0, find sin dx mx x . [3] (c) Using the result in part (b), for all positive integers m, evaluate 0 sin dx mx x   , giving your answers in the form k m where the possible value(s) of k are to be determined. [2] 3 It is given that 1tanf ( ) e xx   , where 1tan x denotes the principal values. (i) Show that     2(1 )f ' fx x x  . [1] (ii) By further differentiation of the above result, find the Maclaurin series for f ( )x , up to and including the term in 3x . [5] (iii) Using the series in part (ii) find an approximate value of 0.5 0 f ( ) dx x , giving your answer to 4 significant figures. [1] (iv) Comment on the suitability of substituting 1x into the series in part (ii) to estimate the value of 4e  . [1] 4 (i) Show that    7 5 2 9 4 2 1 2 1 n n n n n n n       . [2] (ii) Hence find   3 9 4 2 1 N n n n n n    , giving your answer in the form f ( )k N , where k is a constant. [3] (iii) Show that   3 9 4 2 1 N n n n n n    is convergent and state the sum to infinity of this series. [2] (iv) Use your answer in (ii) to find 2 9 14 ( 1)( 2) N n n n n n    . [3]

3 5 A closed cylinder has a base in the shape of a circle with centre O. The coordinates of points A, B and C are  5, 0, 0 ,  4, 3, 0  and  5, 0, 6 respectively. Point E is directly above O with coordinates  0, 0, 10 . (i) Point D lies in the cylinder such that ABCD is a parallelogram. Find the position vector of D and determine the shape of ABCD, justifying your answer. [3] (ii) Find the cartesian equation of the plane ABC. [2] (iii) Find the acute angle between the plane ABC and the base of the cylinder. [2] (iv) Point F lies on AE such that : 1: 5AF AE  . Find the length of projection of AF onto the plane ABC. [4] Section B: Probability and Statistics [60 marks] 6 The individual letters of the word APPROPRIATE are printed on identical cards and arranged in a straight line. (a) Find the number of arrangements of all 11 letters of the word such that (i) the letters are not in alphabetical order, [2] (ii) all the vowels are together and only two of the Ps are together. [3] (b) The cards are now placed in a bag and 3 cards are drawn without replacement. Find the probability that there are at least two vowels drawn. [2] x y z O C 5, 0, 6 A  5, 0, 0 B  4, 3, 0  E  0, 0, 10

4 7 The number of years ( x) an employee has worked for the company and the corresponding salary increment, in dollars (y), received by the employee are given in the table. Years, x 6 7 9 11 13 15 18 Amount, y 155 170 211 230 248 260 265 (i) Draw the scatter diagram for these values, labelling the axes clearly. [1] It is thought that the salary increment, $y, can be modelled by one of the formulae y ax b  or lny c x d  where a, b, c and d are constants. (ii) Find, correct to 4 decimal places, the value of the product moment correlation coefficient between (a) x and y, (b) ln x and y. [2] (iii) Use your answers to parts (i) and (ii) to explain which of y ax b  or lny c x d  is the better model. [2] It is required to estimate the value of x for which y = 200. (iv) Explain why neither the regression line of x on y nor the regression line of ln x on y should be used. [1] (v) Find the equation of a suitable regression line and use it to find the required estimate, commenting on its reliability. [3] 8 A store owner receives a shipment of stationery items, including notebooks, pens, and correction tapes. Historical data indicates that 1% of the notebooks, 2% of the pens, and 4% of the correction tapes are defective. The quality of notebooks, pens and correction tapes is independent of one another. The store owner decides to sell the stationery in 180 packets, each containing one notebook, two pens and one correction tape. A packet is deemed unsatisfactory if any of the four items is defective. (i) Show that the probability that a randomly selected packet is unsatisfactory is 0.0872, correct to 3 significant figures. [1] The number of packets that are unsatisfactory is denoted by X. You may assume that X can be modelled by a binomial distribution. (ii) Find the probability that there are at least 5 but less than 10 packets that are unsatisfactory. [2] (iii) Find the least value of r such that the probability that there are more than r packets that are unsatisfactory is at most 0.12. [2] Before selling the packets of stationery, he decided to select a sample of 9 packets to check for unsatisfactory packets. (iv) How should the packets be selected? Give a reason for this method of selection. [2] (v) Find the probability that the ninth packet is the third unsatisfactory packet selected. [2]

5 9 Box A contains five cards numbered 1, 2, 2, 3 and 3. Box B contains three cards numbered 4, 5 and 5. Cards that are numbered 2, 3 and 5 are red, while cards that are numbered 1 and 4 are blue. A card is drawn from each of the two boxes. If both cards are of the same colour, then the score will be the sum of the numbers on the two cards. If both cards are of different colours, then the score will be the product of the numbers on the two cards. Let X be the score obtained. (i) Show that P( X = 8) = 2 5 . [2] (ii) Find the probability distribution of X. [3] (iii) Find E( X) and Var(X). [3] (iv) Find the probability that the mean score of 50 independent observations of X lies between 7.5 and 8.5. [2] 10 In a drinks factory, a machine is programmed