EJC 8865 2022 Prelim

2022 JC2 H1 Mathematics Preliminary Examination [Turn over (modify this one, Update Field for the next 2 pages and the last page) EUNOIA JUNIOR COLLEGE JC2 Preliminary Examination 2022 General Certificate of Education Advanced Level Higher 1 CANDIDATE NAME CIVICS GROUP INDEX NO. MATHEMATICS Paper 1 [100 marks] 8865/01 13 September 2022 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name, civics group and index number 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, glue or correction fluid. DO NOT WRITE ON ANY BARCODES. Answer all 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. This document consists of 21 printed pages and 1 blank page. Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Total

2 2022 JC2 H1 Mathematics Preliminary Examination Section A: Pure Mathematics [40 marks] 1 Find the set of values of the real constant k for which the expression ( ) 2 1 2 1kx k x k+ − + + is always negative. [4] 2 Starluck Coffee is a distributor that brings in three types of coffee beans: Arabica, Robusta and Liberica. The selling price of Arabica coffee beans, Robusta coffee beans and Liberica coffee beans is $4.00 per kilogram, $1.50 per kilogram and $3.50 per kilogram respectively. On International Coffee Day, Starluck Coffee brought in a total of 550 kilograms of coffee beans to sell at a bazaar. At the end of the day, 4 5 of the Arabica coffee beans and 4 9 of the Robusta coffee beans brought in were sold. The amount of Liberica coffee beans sold was 70 kilograms less than the amount of Arabica coffee beans sold. The amount collected from selling these coffee beans was $1315. If all the coffee beans brought in were sold at the end of International Coffee Day, Starluck Coffee would have collected $1685. (i) By writing down three linear equations, find the mass of each type of coffee beans, in kilograms, brought in by Starluck Coffee on International Coffee Day. [4] (ii) Given that the cost of one kilogram of Arabica coffee beans, Robusta coffee beans and Liberica coffee beans is $3.70, $1.10 and $ 3.20 respectively, determine if Starluck Coffee made a profit or loss on International Coffee Day. Justify your answer. [2] 3 (a) Differentiate 52 3 23 −  x x with respect to x. [2] (b) Find the exact value of 1 0 1 d 32 x x−  . [3] 4 The diagram shows a sketch of the curve with equation ( )3 ln 2y px= − − , where p is a constant. The point A is where the curve cuts the x-axis. (a) Find the equation of the asymptote to the curve, giving your answer in terms of p. [1] y x A O

3 2022 JC2 H1 Mathematics Preliminary Examination (b) Show that the x-coordinate of A is 32e p − . [1] (c) Find the equation of the tangent to the curve at the point A, giving your answer in the form y mx c=+ where m is in terms of p and c is an exact constant. [4] 5 The curve C has equation 91y x=+ . (i) Sketch the graph of C, stating the exact coordinates of any points where the curve crosses the x-axis and the equation of any asymptote(s). [3] (ii) Find the area of the region bounded by C, the x-axis, the y-axis, and the lines 24yx=+ and xa= where 3 2a . Give your answer in terms of a. [4] 6 Kenrus, a young entreprenuer, sets up his business to sell tablets on 1 st January 2022. He tries to model his cost, C million dollars per year, at time t years. The model he uses is 0.5 1 12e 0.4 e −= − −tCt , for 0t . (i) Find the startup cost of his business. [1] (ii) Use differentiation to find the minimum value of C, justifying that this value is a minimum. [4] (iii) Sketch the graph of C against t, stating the coordinates of the minimum point and any intersections with the coordinate axes. [3] (iv) Use your calculat or to find the value of 3 0.5 1 1 12e 0.4 e d− −− t t t . In the context of the question, what does this value represent? [2] Kenrus also models his total profit, P million dollars, at time t years. He believes that 4 0.4532 e 1 tPt=− + . (v) Find the rate of increase of his total profit on 1st June 2022. [2]

4 2022 JC2 H1 Mathematics Preliminary Examination Section B: Probability and Statistics [60 marks] 7 A dance club of 18 students consist s of 4 from the School of Engineering, 6 from the School of Humanities and 8 from the School of Business. 10 students are selected at random to form a team to represent the club in a dance competition. (i) Find the number of different teams that can be formed by the club if the team consists of 2 students from the School of Engineering, 3 students from the School of Humanities and 5 students from the School of Business. [2] After the competition, all the 18 students decide to watch a movie in a cinema, and they are to sit in a row of 18 seats at random. It is known that one of the students from the School of Humanities and one of the students from the School of Business are sisters. (ii) Find the probability that the 2 sisters sit next to each other. [2] (iii) Find the probability that the 4 students from the School of Engineering are all separated given that the 2 sisters sit next to each other. [3] 8 Two events A and B are such that P( )Ap= , P( ) 2Bp= , ( )P '| 0.8AB = and P( ) 0.728AB= . (a) Explain what is meant by ( )P '|AB . [1] (b) Show that 0.28p= . [2] (c) Determine whether the events A and B are mutually exclusive. [2] (d) Complete the Venn diagram to show the probability in each of the four regions. [1] A B

5 2022 JC2 H1 Mathematics Preliminary Examination 9 A certain college has two year groups, Year 1 and Year 2. There are 600 students in Year 1 and 500 students in Year 2. Each student takes up exactly one co-curricular activity from Sports, Performing Arts or Clubs and Societies. The number of students in each activity are summarised in the table. Sports Performing Arts Clubs and Societies Year 1 320 146 134 Year 2 285 154 61 A student is chosen at random from the college. (a) Find the probability that the student is in Year 1 and takes up sports. [1] (b) Find the probability that the student is either in Year 2 or takes up sports. [1] (c) Find the probability that the student is in Year 2 given that the student does not take up performing arts. [1] On another occasion, 3 of these students are chosen at random, without replacement. (d) Find the probability that exactly 2 are in clubs and societies. [3] 10 In a large supply of avocados, %p are rotten. A shop sells avocados in boxes of 16 which are randomly chosen. (i) Given that the mean number of rotten avocados in a box is 3.52, show that the value of p is 22. [1] A box of avocados is rejected if there are more than 3 rotten avocados in it. (ii) Find the probability that a randomly selected box of avocados was rejected. [2] Fifteen boxes of avocados are selected at random. (iii) Find the probability that fewer than 5 boxes were rejected. [2] Six boxes of avocados are