EJC 9758 2023 Prelim P2

2023 JC2 H2 Mathematics Preliminary Examination Paper 2 [Turn over EUNOIA JUNIOR COLLEGE JC2 Preliminary Examination 2023 General Certificate of Education Advanced Level Higher 2 CANDIDATE NAME CIVICS GROUP INDEX NO. MATHEMATICS Paper 2 [100 marks] 9758/02 21 September 2023 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 24 printed pages and 4 blank pages. Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Total

2 of 6 (the “of n” is hidden for our own reference) 2023 JC2 H2 Mathematics Preliminary Examination Paper 2 Section A: Pure Mathematics (40 marks) 1 The complex numbers z and w satisfy the following equations. 4 5i 7 (1 i) 8 30 zw zw += −+= Find z and w, giving your answers in the form iab+ where a and b are real numbers. [4] 2 (a) Find the exact roots of the equation 2 7 3 13xx x− +=− . [3] (b) On the same axes, sketch the curves with equations 2 73yx x=−+ and 13 ,yx= − indicating the value of the x-coordinates of any intersections. Hence solve exactly the inequality 2 7 3 13xx x− +<− . [5] 3 The variables x and y are related by the differential equation ( )dπ 3 π tan 0d y yxx +− = . (a) Using the substitution secyz x= , show that d3 d π zz x −= . Hence, show that the particular solution for which 2y= when π 3x= can be expressed in the form cos e a bxyx += where a and b are constants to be determined. [6] (b) For the graph of the solution found in part (a), find the equations of the two vertical asymptotes closest to the y-axis. [2] 4 A curve C has equation 33x y xy A+−= where A is a non-zero constant. (a) Show that any stationary points on C lie on 23yx= . [2] (b) Find, in terms of A, the x-coordinates of the stationary points, and hence determine the range of values of A for which C has two distinct stationary points. [4] (c) Suppose that C has two stationary points. Determine the nature of each of the stationary points found in part (b). [3]

3 of 6 (the “of n” is hidden for our own reference) 2023 JC2 H2 Mathematics Preliminary Examination Paper 2 [Turn over 5 Planes p and q are perpendicular. Plane p has equation 2 10xz−= . Plane q contains the line l with equation 2 5 3 4 2 1 λ  = −  +    −  r , where λ is a parameter. (a) Show that q has equation 227xyz++= . [2] (b) Find a vector equation of the line m where p and q meet. [2] The point A with coordinates ( )2,1, 6− lies on p and the shortest distance from A to q is k. (c) Find the position vector of the foot of perpendicular, F from A to q. Hence find the value of k. [4] (d) Find vector equations of the lines in p such that the shortest distance from each line to q is k. [3] Section B: Probability & Statistics (60 marks) 6 For the events A, B and C it is given that 1P( ) P( ) 5A AB= ∩= and P( ) 2P( )BA= . It is also given that 3P( ) 10BC∩= . (a) Find the greatest and least possible values of P( )AC∩ . [4] (b) Draw a Venn diagram showing all 3 events representing the case where the value of P( )AC∩ is the greatest. [1] It is now known that 2P 1( 1)ABC′′′∩∩ = . (c) If A and C are independent events, calculate the value of P( )AC∩ . [2] 7 X is a normal random variable with mean 1 and variance 1, and Y is also a normal random variable wit h mean µ and variance 2. (a) Given that ( )P 0 0.44 3XY >≤+< , find the range of possible values of µ. [3] (b) Given instead that 10µ = and the probability that the sum of n independent observations of X exceeds 2Y by at least 10 is less than 0.03, find the largest possible value of n. [4] (c) State an assumption needed for the above calculations to be valid. [1]

4 of 6 (the “of n” is hidden for our own reference) 2023 JC2 H2 Mathematics Preliminary Examination Paper 2 8 A company produces ceramic vases. A fixed number of randomly chosen vases are inspected each day and the number of defective vases found in a day is denoted by X. (a) State, in context, two assumptions needed for X to be well modelled by a binomial distribution. [2] Assume now that X has the distribution B(30, 0.04). (b) Find the probability that, on a randomly chosen day, more than 2 defective vases are found. [2] The number of defective vases found each day is independent of that on other days. (c) Find the probability that, in a randomly chosen 5- day working week, more than 2 defective vases are found on at most 1 day. [2] (d) In a particular 5-day working week, a total of 5 defective vases were found. Find the probability that all 5 of them were found on exactly 2 days that are consecutive. [4] 9 A personality test assesses 32 key traits. These 32 traits are grouped into 4 domains – Cognition, Influence, Rapport and Planning – with each domain consisting of 8 traits. The test result for a person consists of his/her top 5 traits in ascending order of strength. (a) Find the number of different results that the personality test can produce. [2] (b) Find the number of different results that can be produced with traits from at least 2 domains. [2] (c) Find the number of different results that can be produced with at least one trait from each domain and neither of the top 2 traits are from the Cognition domain. [3] (d) A result contains 2 particular traits from Rapport, 2 particular traits from Planning and 1 particular trait from Influence. Find the number of different possible results that can be produced with no consecutive traits from the same domain. [3]

5 of 6 (the “of n” is hidden for our own reference) 2023 JC2 H2 Mathematics Preliminary Examination Paper 2 [Turn over 10 A batch of plants is grown and then studied to examine how well they are growing. For each plant, data on two indicators are collected: the average root hair length x mm, and the amount of potassium found in the roots y mg. The data from a random sample of 8 plants are shown in the table below. Average root hair length (x mm) 3.3 3.9 7.5 8.1 13.3 22.1 32.2 36.1 Amount of potassium (y mg) 9 t 25 27 33 37 41 44 (a) (i) Calculate an unbiased estimate of the population variance of the average root hair lengths. [1] (ii) Find an expression for the unbiased estimate of the population variance of the amount of potassium in the roots, in the form ( ) ( ) 2 21 CtBtAD  ++−  , where A, B, C and D are positive integers to be determined. [2] You are now given that 17t = . (b) Sketch a scatter diagram of y against x. State the value of the product moment correlation coefficient between y and x. [2] (c) With reference to your answers in part (b), explain whether an equation of the form y mx k= + , where m and k are constants, provides an accurate model of the relationship between x and y. [1] (d) The data is instead modelled by the regression equation lny p xq= + . Find the values of p and q