ASRJC 8865 2022 Prelim

1 ASRJC J2 H1 Mathematics Preliminary Examination ANDERSON SERANGOON JUNIOR COLLEGE Preliminary Examination H1 Mathematics Paper (100 marks) 8865/01 30 August 2022 3 hours Additional Material(s): List of Formulae (MF26) CANDIDATE NAME CLASS / READ THESE INSTRUCTIONS FIRST Write your name and class in the boxes above. Please write clearly and use capital letters. Write in dark blue or black pen. HB pencil may be used for graphs and diagrams only. Do not use staples, paper clips, glue or correction fluid. Answer all the questions and write your answers in this booklet. Do not tear out any part of this booklet. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a d ifferent level of accuracy is specified in the question. You are expected to use an approved graphing calculator. 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. All work must be handed in at the end of the examination. If you have used any additional paper, please insert them inside this booklet. The number of marks is given in brackets [ ] at the end of each que stion or part question. This document consists of 19 printed pages and 5 blank pages. Question number Marks 1 2 3 4 5 6 7 8 9 10 11 12 13 Total

2 ASRJC J2 H1 Mathematics Preliminary Examination Section A: Pure Mathematics [40 marks] 1 (i) Solve the inequality 22 3 2xx− . [2] (ii) Hence by means of the substitution exy= , find the range of values which will satisfy the inequality 22e 3e 2xx− . [2] 2 (a) Differentiate 2 2 23x x  − with respect to x. [2] (b) Find 3 2 d 54 x x− . [2] 3 The curve C has equation y = 8 – 2e1 – 4x . (a) Find the equation of the tangent to C at the point x = 1, giving your answer in the form y = mx + c, where m and c are exact. [4] (b) The finite area, A, between the curve C, the x – axis and the lines x = 1 4 and x = 1 is given by A = 3e qp+ where p and q are constants. Use integration to find the exact values of p and q. [3] 4 The curve C has equation y = x4 – 10x3 + 36x2 – 54x + 17 for 05 x . (i) Find d d y x . Hence find the coordinates of the stationary points on the curve. [4] (ii) Determine the nature of each of the stationary points. [2] (iii) Sketch the graph of C, stating the coordinates of any point (s) where the curve crosses the axes. [3] (iv) Determine the equation of a curve which, when used with the equation of C, will enable you to solve the equation x4 – 10x3 + 37x2 – 52x + 18 = 0. Sketch this curve on your diagram. [2] (v) Hence solve the equation x4 – 10x3 + 37 x2 – 52x + 18 = 0, giving your answers correct to 3 decimal places. [1]

3 ASRJC J2 H1 Mathematics Preliminary Examination 5 A particular company sells three brands of laptop: Brand A, Brand B and Brand C. Last week, the company sold 16 Brand A, 23 Brand B and 20 Brand C laptops and the total income from these sales was $109 741. The income from Brand C sales was $ 23 596 more than the income from Brand A sales. This week, the company has sold 2 less of each of the three brands of laptop than last week and the total income has decreased by $1 0 994. The prices of the laptops have not changed. (a) Using this information, form three linear equations to find the total price of two Brand A laptops and three Brand C laptops. [4] (b) The director of the company develops a model for predicting the annual profits. He believes that the annual profits, P million dollars, in the year up to time t years can be modelled by P = 1.8 + 0.6ln (3t – 2), for t  1. (i) Sketch the graph of P against t. [2] (ii) Find the value of P when t = 3 and explain in context of the question. [2] (iii) Find d d P t . [2] (iv) Explain in the context what the expression for d d P t indicates about the annual profits in the future. [2] (v) Hence find the rate at which P is increasing when t = 5. [1]

4 ASRJC J2 H1 Mathematics Preliminary Examination Section B: Probability and Statistics [60 marks] 6 An internet retailer has compiled a list of 10 000 regular customers and wishes to carry out a survey of customer opinions involving 10% of its customers. (i) Describe how the customers could be chosen to do this survey using random sampling. [2] (ii) Give one advantage and one disadvantage of random sampling in this context. [2] 7 A board of directors consist of Mr Koh, Mrs Ho and 7 other people. A committee consisting of 4 people is to be formed from this board of directors. (i) How many committees can be formed if it must include both Mr Koh and Mrs Ho? [1] (ii) Calculate the probability for the committee to include either Mr Koh or Mrs Ho but not both. [2] The committee is formed and consists of Mr Koh, Mrs Ho and 2 other people. For a photograph, the 4 people need to stand in a line. (iii) However, Mr Koh and Mrs Ho refused to stand next to each other. In how many ways can the committee members stand in a line? [2] 8 At a certain college, the number of male and female students studying Business, Engineering and Humanities are as shown in the following table. Each student studies exactly one of these three subjects. Business Engineering Humanities Male 23 80 35 Female 40 57 65 A student is chosen at random from these 300 students. Let B be the event that the student studies Business, E be the event that the student studies Engineering, H be the event that the student studies Humanities, F be the event that the student is female, M be the event that the student is male. (i) Find ( )P' FB . [1]

5 ASRJC J2 H1 Mathematics Preliminary Examination (ii) Find ( )P| BM . [1] (iii) Find ( )P ME . [1] (iv) Determine whether the events M and E are independent. [1] On another occasion, 3 of these students are chosen at random without replacement. (v) Find the probability that exactly 2 are studying Business, giving your answer correct to 3 decimal places. [2] 9 The events A and B are such that 1 1 4P( ) , P( | ') and P( | ') .2 2 9A B A A B= = = (i) Find the probability that at least one of A and B occurs, [2] (ii) Show that the probability that both A and B occurs is 0.3. [3] (iii) Draw a Venn diagram to represent this situation, showing the probability in each of the four regions. [2] 10 The probability that a pear tree, of one particular type, will produce pears in the first year after planting is 0.85. The owner of a garden centre sells 5 pear trees each time to his customers with a guarantee that if fewer than 3 trees produce pears in the first year, he will replace all the ones that fail. (i) Find the probability that no replacement will be required for a customer. [2] The owner has signed a deal to sell 5 trees to each of the 3000 customers. (ii) How many customers will be expected to receive replacement(s)? [1] (iii) Find the probability that more than 100 customers will require a replacement. [2] (iv) Find the probability that the average number of trees per customer that will produce pears in the first year is more than 4