RI H2 2020 Prelim P2

 RI2020 [Turn over CANDIDATE NAME CLASS 20 MATHEMATICS 9758/02 PAPER 2 3 hours Candidates answer on the Question Paper Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name and class on all 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. 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. The use of an approved graphing calculator is expected, where appropriate. 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 ques tion, 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. The total number of marks for this paper is 100. FOR EXAMINER’S USE Q1 Q2 Q3 Q4 Q5 Sub-Total Total /5 /5 /8 /10 /12 /40 Q6 Q7 Q8 Q9 Q10 Q11 Sub-Total /100 /7 /8 /9 /12 /12 /12 /60 This document consists of 28 printed pages and 0 blank page. RAFFLES INSTITUTION Mathematics Department RAFFLES INSTITUTION 2020 YEAR 6 PRELIMINARY EXAMINATION

2 H2 MA 9758/2020 RI Year 6 Preliminary Examination Paper 2 Section A: Pure Mathematics [40 marks] 1 An arithmetic series has first term 3 2 and fourth term 4.u A geometric series also has first term 3 2 and fourth term 4.u Given that the common ratio of the geometric series is non -negative and the sum of its first 3 terms is 21,2 find the sum of the first 10 odd -numbered terms of the arithmetic series. [5] 2 In a geometric series, the ratio of the sum of the first 8 terms to the sum of the first 4 terms is 17:16. Find the 2 possible values of the common ratio, 12,rr where 12 .rr The sum to infinity of the geometric series with first term a and common ratio 1r is denoted by 1S and the sum to infinity of the geometric series with first term b and common ratio 2r is denoted by 2 .S Find 12:SS in terms of a and b. [5] 3 The functions f and g are defined by       f 1 3e , , 0, g 1 3 , , where is a real constant. xx x x x x x x x c c          (i) Given that 4,c determine if the composite functions fg and gf exist, justifying your answers. Find the range of the composite function that exists. [4] (ii) Given that 1g exists, state the largest possible value of .c Using this value of ,c find   1g x . [4]

3 H2 MA 9758/2020 RI Year 6 Preliminary Examination Paper 2 4 Given that 1tan 2e, x y    show that   2 d4 2 . d yxy x [2] (i) By repeated differentiation of the above result, find the Maclaurin series for 1tan 2e x   up to and including the term in 3.x [5] (ii) Hence find the Maclaurin series for   1 2 2 tan e 1 x x     up to and including the term in 2.x [3] 5 Referred to the origin O , points A and B have position vectors a and b respectively, where a and b are non-zero and non-parallel vectors. Point C lies on OA , between O and ,A such that : 2 :1OC CA  . Point D lies on OB produced such that : 3: 2OD BD  . (i) Find the position vectors OC   and ,OD   giving your answers in terms of a and b. [2] (ii) Show that the point E where the lines BC and AD meet has position vector 4 3 ab . [4] (iii) Show that the area of triangle CDE can be written as k ab , where k is a constant to be found. [3] (iv) It is given that the point F is on BO produced, and OE bisects the angle AOF . Find the ratio :OA OB . [3]

4 H2 MA 9758/2020 RI Year 6 Preliminary Examination Paper 2 Section B: Probability and Statistics [60 marks] 6 For events X and Y, it is given that 1P( ') , 2XY 3P( ') 4XY and 50P( ') . 63XY  Find (i) P( '),Y [2] (ii) P( ),X [2] (iii) P( ) and state with a reason whether and are independent events.X Y X Y [3] 7 In a factory, machines pack sugar into bags of 1 kg each on average, with variance 2 kg2. The manufacturer is concerned that the machines are putting too much sugar into the bags and decides to carry out a hypoth esis test. A random sample of 8 bags are selec ted and their total mass is 8.4 kg. (i) Stating a necessary assumption, carry out a test of the manufacturer’s concern at the 5% significance level if 0.08  . [5] (ii) Use an algebraic method to calculate the range of values of 2 for which the null hypothesis would not be rejected at the 5% significance level. [3] 8 In this question you should state clearly the values of the parameters of any normal distribution you use. In a supermarket, the masses in grams of apples have the distribution 2N(90,13 ) and the masses in grams of potatoes have the distribution 2N(170,30 ) . (i) Find the probability that the mass of a randomly chosen potato is more than twice the mass of a randomly chosen apple. [3] A certain salad recipe requires 5 apples and 6 potatoes. (ii) Find the probability that the total mass of 5 randomly chosen apples and 6 randomly chosen potatoes is between 1.2 and 1.5 kilograms. [3] The salad recipe requires the apples and potatoes to be prepared by peeling and slicing them. The process reduces the mass of each apple by 15% and the mass of each potato by 25%. (iii) Find the probability that the total mass, after preparation, of 5 randomly chosen apples and 6 randomly chosen potatoes is not more than 1.2 kilograms. [3]

5 H2 MA 9758/2020 RI Year 6 Preliminary Examination Paper 2 9 The continuous random variable X has the distribution 2N( , ). It is known that P( ) 0.2Xk and P( 7) 0.8.X  (i) Show that P( 7) 0.6kX and write down the value of P( 7). X [2] (ii) Express  in terms of k. [1] You may use 2 12  for the rest of the question. (iii) Show that 4.0845 correct to 4 decimal places. [3] (iv) Find P( ).Xk [2] It is given that 2P( ) 3P( )X r X r   for a certain constant, r. (v) Ten independent observations of X are randomly selected. Find the probability that there are more observations of X with values greater than r than observations of X with values less than r in the selection. [4] 10 A group of 13 people consists of 6 single men and 5 single women and a married couple. A committee of 7 is to be selected from the group. (i) Find the number of committees that can be formed if there is no restriction in the selection. [1] (ii) Show that there are 658 such committees with more women than men. [2] Given that a committee of 7 people is to be selecte