ASRJC 9758 2023 Prelim P2

[Turn over ANDERSON SERANGOON JUNIOR COLLEGE H2 MATHEMATICS JC2 Preliminary Examination Paper 2 (100 marks) 9758/02 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 different 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 question or part question. Question number Marks 1 2 3 4 5 6 7 8 9 10 11 Total This document consists of 21 printed and 3 blank pages.

2 © ASRJC 2023 y x R Section A: Pure Mathematics [40 marks] 1 Relative to the origin O, two fixed points, A and B, have position vectors a and b respectively. The point P has position vector p given by ( )p a1λλ= +− b where λ is a parameter such that 01 λ<< . (a) Show that for all real values of λ, the point P is collinear with A and B. [2] (b) If angle AOB is 090 , show that the position vector of the foot of perpendicular from O to AB is ( ) 2 22+− + aa ba ab . [3] 2 One of the roots of the equation 432 4 78 0z z az bz− + ++= , where a and b are real, is 3 2i+ . Find the values of a and b. Hence find the other roots of the equation, giving your answers in exact form. [6] 3 (a) Integrate by parts twice to show that 22 1e sin d e (2sin cos )5 xx xx x x c= −+∫ . [4] (b) The diagram below shows the shaded region R that is bounded by part of the curve 2 = e sin x yx π− , the line 4 ( 1)xy π= + and the x – axis. The line 4 ( 1)xy π= + cuts the curve and the x – axis at , 12 π  and , 04 π  respectively. Show that the volume generated when region R is rotated 2π radians about the axisx− is given by ( ) 2eAB C πππ −++ , where A, B and C are constants to be determined. [4]

3 © ASRJC 2023 [Turn over 4 The functions f and g are defined as follows: 3f : e 1 xx − − , 3x≤ 22g: , , 0xax x x −∈< , where 01 a<< . (a) Find f −1 in similar form. [3] (b) Sketch the graphs of ( )gyx= , 1g()yx −= and 1g g( )yx −= on the same diagram. [3] (c) Find the exact solution of ( ) 1g g ()xx −= , leaving your answer in terms of a. [2] (d) Show that the composite function fg exists and find the range of fg, leaving your answer in terms of a. [2] 5 The curve C is given by the equation 2 2 3y ax x ax −= + , 0x≠ , where a is a positive constant. (a) Use an algebraic method to determine the range of values for which x can take , leaving your answer in term of a. [3] For the rest of this question, take 1a= . (b) Find the coordinates of the point(s) at which the tangent is parallel to the x-axis. [4] (c) The normals to C at the points where 1x= intersect each other at the point N . Find the coordinates of N. [4]

4 © ASRJC 2023 Section B: Probability and Statistics [60 marks] 6 In a chemical reaction, a chemical mixture is left to cool down to room temperature. The temperature of the chemical mixture, θ oCat time t minutes is given in this table. (a) Sketch a scatter diagram for the data given in the table. [2] A student attempts to model the relationship between θ and t using two models: (A) a btθ = + or (B) dc tθ = + . (b) By calculating the product moment correlation coefficients and referring to the scatter diagram, explain clearly which of the two models is a more appropriate model for this set of data. [2] (c) Use the model you identified in ( b) to find the equation of a suitable regression line and use your equation to estimate to predict the temperature when the time is 45 minutes. Comment on the reliability of the estimate. [3] A temperature of T Kelvins is equivalent to a temperature of θ degrees Celsius, where 273.15T θ= + . (d) Re-write your equation in (c) so that it can be used when the temperature of the chemical, θ, is given in Kelvins. [1] 7 A circular target board is made up of 3 concentric circles as shown below. The circles have the same centre and their radii are 10 cm, 20 cm and 30 cm respectively. A player plays a game of darts using this target board. The probability of the player missing the target board completely is 1 5 and the player scores no point if he misses the target board. It can be assumed that he is equally likely to land his dart anywhere on the target board if he does not miss the target board. The player scores 5 points if the dart lands on the inner most circle, 3 points if the dart lands between the inner most and the middle circles, and 1 point if the dart lands between the middle and the outer most circles. It can be assumed that the dart will not land on the circumference of the circles. Time (t minutes) 1 6 12 18 24 30 Temperature (θ oC) 97 62 45 37 31 27 5 3 1

5 © ASRJC 2023 [Turn over (a) Show that the probabilities that the player scores 5 points, 3 points and 1 point with 1 throw are 4 45 , 4 15 and 4 9 respectively. [3] The player throws the dart twice and the random variable X denotes the higher score out of the 2 throws. (b) Tabulate the probability distribution of X. [4] (c) Find the exact value of E( )X . [2] 8 Some MRT trains have cars featur ing 12 seats each. Passengers take the trains to travel from station to station each day. Seats located nearest to the train doors are labelled as priority seats reserved for vulnerable commuters, namely those who are pregnant, elderly, with infants, or the differently-abled. Anyone can sit at the priority seats if there are sufficient seats for all vulnerable commuters in the car. For one trip, 12 passengers, of which 4 are vulnerable commuters, got onto Car B at the first station. (a) Given that all the vulnerable commuters are seated at priority seats, find the total number of ways all the 12 passengers can all sit in Car B. [2] (b) Find the number of ways the 12 passengers can be seated in Car B if (i) two particular girls are seated in the same row but not beside each other. [3] (ii) exactly 4 of the 5 girls are seated together. [4] Car B Train door P P P P Train door Train door Train door P: Priority seat Car A Car C

6 © ASRJC 2023 9 In this question you should state clearly the values of the parameters of any normal distributions that you use. The masses, in grams, of apples and pears have the independent distributions N(260, 152) and N(210, 102) respectively. (a) Find the probability that the mass of a randomly chosen apple is at least 250 grams. (b) Find the probability that the mass of a randomly chosen apple differs from the mass of a randomly chosen pear by more than 55 grams. [2] A certain recipe requires 5 apples and 6 pears. The recipe requires the apples and pears to be prepared by removing the cores. This process reduces the mass of each apple by 5% and the mass of each pear by 15%.