CJC 9758 2023 Prelim P2

9758/01/J2Prelim/2023 [Turn Over CATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 JC2 Preliminary Examination CANDIDATE NAME CLASS INDEX NUMBER MATHEMATICS 9758/02 Paper 2 13 Sep 2023 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your class, index number and name on the work you hand in. Write in dark blue or black pen. You may use a HB pencil for any diagrams or graphs. Do not use staples, paper clips, 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 sp ecifically 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. Question 1 2 3 4 5 6 7 8 9 10 Total Marks Total 7 8 12 13 8 8 10 10 12 12 100 This document consists of 3 printed pages, including this cover page.

2 9758/02/J2PRELIM/2023 Section A: Pure Mathematics [40 marks] 1 Without using a calculator, solve the inequality 2 2 25 22 xx xx +− − . [4] Hence, solve the inequality 2 2 25 22 xx xx +− − . [3] 2 It is given that e sin 1xy x x −= + − . (a) Show that 2 2 d e cosd xy kxx −= , where k is a constant to be determined. [2] (b) By further differentiation of this result, find the Maclaurin series for y , up to and including the term in 3x . [3] (c) By using the result in part (b) and standard series from the List of Formulae (MF26), find the expansion of e sin 1 cos 2 x xx x − +− in ascending powers of x , up to and including the term in 3x , giving the coefficients in exact form. [3] 3 Following the popularity of the action role -playing game, Ginseng Im pact, three years ago, developers have developed a strategy game, Ginseng Impactful. The number of people who download Ginseng Impactful, P (in thousands), in a particular city, at time t months, can be modelled by the differential equation ( )d1 13 2d 26 P PPt =− . There were 2000 people who download Ginseng Impactful when it is launched. (a) Show that 1 2 26 9e 4 t P − = + . [6] (b) Determine, the time taken, in months, for the number of people who download Ginseng Impactful to double since the launch. [2] (c) Find the number of people that download Ginseng Impactful in the long run. [2] (d) Hence sketch the graph showing the number of people that download Ginseng Impactful against time. [2]

3 9758/02/J2PRELIM/2023 [Turn Over 4 The plane 1Π and the line l have equations 1 2 Π : 1 3 1  =− r and 24:3 32 xylz +− = = − respectively. (a) Find the acute angle between 1Π and l . [2] (b) Find the coordinates of the point of intersection between 1Π and l . [3] (c) Find the perpendicular distance from ( )10, 4,7B − to 1Π . [3] The plane 2Π contains l and is perpendicular to 1Π . (d) Find a cartesian equation of 2Π . [3] (e) Without using a calculator, find a vector equation of the line which lies in both 1Π and 2Π . [2] Section B: Probability and Statistics [60 marks] 5 During the Great Singapore Sale, a certain electronics store organises a lucky draw to attract more customers. The lucky draw is designed as follows: A circular board is divided into four sectors labelled with numbers 1, 2, 3, 4 and ha s angles 144°, 108°, 72 °, 36° respectively. The board has a spinner pivoted at the centre of the circular board. When a customer spins the spinner, the spinner comes to rest randomly in one of the four sectors. Every customer who visits the store is allowed to play one round of th e lucky draw. In each round of the lucky draw, the customer gets to spin the spinner twice. The score, X , of the customer is • the sum of the two numbers if the numbers from the two spins are different, • three times the number if the numbers from the two spins are the same. A customer wins a prize if the score, X , is more than 6. (a) Show that ( )P 6 0.15X == . [2] (b) Find the probability distribution of X . [3] (c) Find the probability that a customer scores less than 10, given that the customer wins a prize. [3]

4 9758/02/J2PRELIM/2023 6 The sales manager of a company that sells air condition ing systems presented the data of average daily temperature, t (°C) and sales, s (in hundred units) at a meeting. Average daily temperature, t (°C) 18 20 23 26 30 32 33 34 Sales, s (in hundred units) 307 366 497 523 565 580 588 596 (a) Draw a scatter diagram of these data and calculate the value of the product moment correlation coefficient between s and t . Comment on whether a linear model would be appropriate, referring both to the scatter diagram and the value of the product moment correlation coefficient found. [3] The marketing director proposes that the data should be modelled instead by the regression equation lns a t b=+ , where a and b are constants. (b) Find the values of a and b , giving your answers to 3 decimal places. [1] (c) Calculate the product moment correlation coefficient between s and lnt . [1] (d) Using parts (a) and (c), explain which is a better model. [1] (e) Use the model proposed by the marketing director to estimate the number of units sold when the average daily temperature is 38°C and comment on its reliability. [2] 7 In a class of 18 students, there are 12 girls and 6 boys. A chairperson, a vice -chairperson and a secretary are chosen from the 18 students. (a) Find the number of ways the chairperson, the vice-chairperson and the secretary can be chosen so that (i) they are all girls, [1] (ii) there are at least one girl and at least one boy. [3] The 18 students sit at random in a circle for a lesson. Find the probability that (b) the chairperson, the vice-chairperson and the secretary are all separated from one another, [3] (c) there are exactly 2 girls sitting between each boy. [3]

5 9758/02/J2PRELIM/2023 [Turn Over 8 The store manager at CJStore keeps a bin of large number of oranges. It is known that, on average, p % of the oranges are rotten. The oranges are packed into packets of 10 oranges each. The number of rotten oranges in each packet is denoted by the random variable X . (a) State, in context, two assumptions needed for X to be well modelled by a binomial distribution. [2] Assume now that X follows a binomial distribution. (b) It is given that 20p= . (i) Find the probability that there are at least two rotten oranges in a randomly chosen packet. [2] (ii) 100 packets of oranges are sold at a profit of $2 per packet. The store offers a discount of $ d for any packet of oranges that contains more than 1 rotten orange. By finding the expected number of packets of oranges that contains more than 1 rotten orange, find the range of values of d , correct to 2 decimal places, if the store manager is expecting a net profit. [3] (c) The store manager wants to ensure that 95% of the packets of oranges contain at most one rotten orange. Write down an equation satisfied by p . Hence find the value of p . [3] 9 In this question, you should state the parameters of any normal distributio