2024 VJC H2 JC2 Math Prelim P2 Qn (modified)

2024 Victoria Junior College Preliminary Examination H2 Mathematics Paper 2 (modified) Section A: Pure Mathematics (40 marks) 1 (a) Show that ( ) 22 222 11 11 211 xx xxx = + + − + − − . [1] (b) Hence use appropriate expansions from the List of Formulae (MF26) to find the first two non- zero terms in the series expansion of 2 2211 x xx+ − − , 0x in ascending powers of x. [3] (c) State the set of values of x for which the series expansion is valid. [1] (d) It is given that the two terms found in part (b) are equal to the first two terms in the series expansion of ( )cos bax . Find the possible value(s) of the constants a and b. [2] 2 Do not use a calculator in answering this question. The complex numbers 1z , 2z and 3z are such that 2πi 3 1 ez =− , 2 3iz =− + and 1 3 2 zz z= . (a) Express each of 1z , 2z and 3z in the form ier  , where 0r and ππ −   . [3] (b) Sketch an Argand diagram showing the points 1P , 2P and 3P where 1P , 2P and 3P represent the complex numbers 1z , 2z and 3z respectively. [2] (c) Find the area of triangle 12OPP . [2] (d) Find the smallest positive integer n for which ( ) * 2 n z is purely imaginary. [2] 3 The line 1l has equation ( )3 4 5 2 = − − + − −r i j k i j k , where  is a real parameter. The point A has position vector 2−+i j k . (a) The plane p contains the line 1l and the point A. Find a cartesian equation of the plane p. [3] (b) Find the position vector of the point 'A , the reflection of the point A in the line 1l . [4] (c) The plane q is such that q is parallel to p and passes through the point with position vector 3−+jk . Find a cartesian equation of q and the exact shortest distance between p and q. [3] (d) The line l2 has the equation 37 23 yz−− = , 2x= . Given that l2 intersects p at point S, find the area of the triangle OAS. [4]

4 The curve C is defined by the parametric equations 11xa t =+ and 2 1y a t t =−  where a is a positive constant and 0t . (a) Show that 3d2 d yt xt  +=−   . [3] (b) Find, in terms of a, the coordinates of the turning point on C, and explain why it is a maximum. [4] (c) Sketch C. [3] Section B: Statistics (60 marks) 5 Two married couples, two single adults and two children formed a team of 8 to take part in a series of games. (a) In the first game, the team sits in a circle. Find the number of arrangements that can be formed if each married couple must be seated together. [2] (b) A group of three people are to be selected from the team for the second game. Find the number of different groups that can be formed if there must not be a married couple in the group. [2] (c) In the third game, each team member selects a unique number from the set  1, 2, , 8 . Find the number of different ways this can be done if the numbers selected by the children are both greater than the numbers selected by the two single adults. [2] 6 A random variable X has the probability distribution given in the following table. x 1 4 6 8 ( )P Xx= a b c d Given that ( )E4X = , ( ) 19Var 4X = and ( ) ( )P 4 P 4XX =  , find the values of a, b, c and d. [5] 7 For events A, B and C, it is given that ( )P 0.7A = , ( )P 0.5B = , ( )P | 0.6CA  = and ( )P | 0.76AC  = . (a) Find the greatest and least possible values of ( )P AB . [2] (b) Find ( )P. CA  [1] (c) Find ( )P. CA [2] (d) Find ( )P. C [3]

8 A small company makes wine glasses. Each day, n randomly chosen wine glasses are checked and the number of wine glasses found to be cracked is denoted by X. (a) State, in context of the question, two assumptions needed for X to be well modelled by a binomial distribution. [2] Assume now that X has the distribution ( )B, np , where 3n . (b) Given that the mean of X and the variance of X are 1.8 and 1.773 respectively, find the value of n and the value of p. [2] (c) Given instead that the probability of finding 2 cracked wine glasses is thrice the probability of finding 3 cracked wine glasses, find p in terms of n. [2] 9 (a) S and T are independent random variables with the distributions ( ) 2N 18,3 and ( ) 2N, respectively. It is given that ( ) ( )P 4 P 9TT =  and ( )P 3 0.65ST= . Calculate the values of  and  . [4] (b) A fruit stal l sells grapes that are packed in packets with masses in grams that follow the distribution ( ) 2N 850,30 . The grapes are sold at $18 per kilogram. (i) Find the probability that a customer pays more than $30 for two packets of grapes. [2] (ii) The fruit stall accepts payment by c ash or PayNow. The number of customers who pay by PayNow in a day is a random variable with mean 12 and variance 4.8. In a month of 30 days, find the probability that the average number of customers per day who pay by PayNow is more than 12.3. [3] 10 The yield per hectare, y kg, of a crop is believed to depend on the average rainfall, x mm, in the month of June. For 10 regions, records are kept of the values of x and y , and these are shown in the table below. The yield from the tenth region was accidentally deleted from the records after the data was analysed, and this is indicated by the value p. Average rainfall (x mm) 149 110 188 135 156 140 168 118 122 174 Yield of crop (y kg) 13.8 6.5 15.2 12.2 14.4 12.2 14.7 9.5 9.9 p Given that the equation of the regression line of y on x is 2.5652 0.10168yx=− + , show that 14.4p= . [2] (a) Draw a scatter diagram for these values, labelling the axes clearly. Calculate the product moment correlation coefficient between x and y. [2] (b) It is thought that a model of the form lny a b x=+ may also be a suitable fit to the data. Calculate least square estimates of a and b , and find the value of the product moment correlation coefficient between y and lnx . [3] (c) Use your answers to parts (a) and (b) to explain which of 2.5652 0.10168yx=− + or lny a b x=+ is the better model. [2] (d) Using an appropriate regression line, estimate the yield for a region that experienced 200 mm of rainfall in June. Comment on the reliability of your estimate. [2] (e) In some regions, rainfall is measured in inches instead of in mm. Given that there are 25.4 mm in an inch, show how the regression line found in part (b) can be re-written so that it can be used when x, the average rainfall in June, is given in inches. [1]

11 In the swimming training school AquaV, the time taken to swim a lap of the pool by the trainees is found to have a mean of 35 seconds. The school adopted a new international training programme Breakthru for 3 months and wanted to analyse if Breakthru is effective in improving the timings of the trainees. A sample of 30 trainees is taken and the times taken, x seconds, to swim a lap of the pool by the trainees are summarised by ( 30) 94x−= , 2 ( 30) 758x−= . (a) Test, at the 5% significance level, whether there is any evidence that the mean time taken to swim a lap of the pool has improved after the trainees underwent 3 months of Breakthru , defining any parameters you use. [7] (b) State an assumption used in carrying out the test. [1] In another swimming training school AquaZ, the mean time taken to swim a lap of the pool by the trainees is 38 seconds. AquaZ similarly adopted Breakthru for 3 months and then also carried out a test at the 5% significance level to determine whether there is an improvement in the swimming times. A sample of 30 trainees was taken and their timings were measured. The sample standard deviation was found to be 4 seconds and the mean time was denoted by y . Assume that the times taken to swim a lap by trainees in this school follow a normal distribution. (c) Find the set of values of y for which the result of the test would be to reject the null hypothesis. [4] (d) If the times taken