HCI 9758 2023 Prelim P2

Section A: Pure Mathematics [40 marks] 1 A curve G has equation 1 coscosyx x= + . The region bounded by G , the line π 3x= , the x-axis and the y-axis is rotated through 4 right angles about the x-axis. Find the exact volume generated, giving your answer in terms of π . [4] 2 (a) Use the substitution 31ux= − to find 5 3 d 1 x x x−∫ . [4] (b) Hence find 8 3 d 1 x x x−∫ . [4] 3 A complex number 1z is given by 1 ( 3) izaa= +− , where a is a positive real constant. It is given that 1arg z θ= , where π 02 θ−<< . (a) Find, leaving your answers in terms of θ, (i) 1arg( 2 )z− , [2] (ii) 1arg( 2 )za− . [2] Another complex number 2z is given by 2 1 3iz = + . (b) Without using a calculator, find the range of values of a such that 2 12 12 10.Im ( ) zz zz ≤ [5] 4 A curve C has parametric equations 2 3,1x t yt= = + for t∈  . (a) Sketch the graph of C , giving the coordinates of any axial intercepts. [2] (b) Without using a calculator, find the equation of the tangent to C at the point P where 1t = . [2] (c) The tangent to C at P meets the curve again at the point Q . Find the exact coordinates of Q . [3] (d) Hence find the exact area of the region bounded by C and the tangent to C at P . [3]

2 © HCI 2023 5 The equations of two planes 1p and 2p are 2 3, 2 3 1, x sy z xyz ++= − −+= respectively, where s is a negative real constant. (a) Find a vector parallel to both 1p and 2p , giving your answer in terms of .s [2] (b) It is given that 1p and 2p intersect in a line l , and l meets the xz -plane at a point A . Find the coordinates of A . [3] (c) The shortest distance between the origin O and 1p is 3 6 . Find the value of s . [2] It is now given that 2,s=− with the normal vectors to 1p and 2p denoted as 2 2 1  = − 1n and 1 2 3  = − 2n respectively. It is given that 1p and 2p divide the real space into four regions 1R , 2R , 3R and 4R as shown in the following two -dimensional diagram (not drawn to scale). (d) Relative to the origin O , the point B has position vector k . Determine whether B lies in 1R , 2R , 3R or 4R , justifying your answer. [2]

3 © HCI 2023 Section B: Probability and Statistics [60 marks] 6 Eight students are shortlisted to be prize recipients for a competition. Two students are awarded the first prize and another two students are awarded the second prize. The eight shortlisted students are to stand in a single row for a photo- taking session with a guest of honour. Find the number of different possible arrangements if the two first prize awardees must stand together while the two second prize awardees also must stand together during the session. [2] After the photo-taking session, the eight students proceed for a tea reception and are to be seated at a round table with 10 identical seats. If 4 students are to be seated on each side of the guest of honour at the round table, find the total number of possible seating arrangements. [2] 7 In a school with a large number of students, p%, where p > 40, of its students study H2 Biology. In a random sample of 40 students, the probability of no less than nine but no more than twenty students studying H2 Biology is 0.25. Find the value of p. [3] It is given that p = 60. A group of 12 randomly chosen students are to attend an interview session at a meeting room, with only one interviewee being interviewed at a time. Find the probability that the third and tenth interviewee are the first and fourth students respectively among them studying H2 Biology. [3] 8 A biased 4-sided die gives the scores 1, 2, 3, and 4 with the probabilities shown in the table, where a and b are constants. The random variable S denotes the score on the die. It is given that the mean of the score is 2.56. (a) Show that 7 25b= . [2] (b) 1S and 2S are two independent observations of S. (i) Find the probability that the sum of the two independent observations of S is at least 6 given that one of the observed scores is 3, [3] Let 12YSS= − . Without finding the probability distribution table of Y , (ii) find ( )( )Var 2 ESY− . [3] (iii) state the value of ( )( )EEYY− . [1] Score ( )s 1 2 3 4 ( )P Ss= a a a b

4 © HCI 2023 9 Players A and B play against each other in a racquet match consisting of at most 3 sets. Each set is won by either Player A or B, and the match is won by the first person to win two sets. Player A has a probability of 8 ak− , where a is a real positive constant, of winning at the kth set where 1 , 2 , 3 .k = (a) (i) Show that the probability that player A will win the match is 32 18 59 50 256 aaa−+ − + . [3] (ii) Find all the values of a for the two players to have an equal chance of winning. Hence explain why only one value is suitable in this context. [3] (b) Given that 7a= and player A wins the match, find the probability that player A wins the second set. [2] 10 In this question you should, where applicable, state clearly all the distributions that you use, together with the values of the appropriate parameters. A supermarket produces its own packs of white sesame seeds and fennel seeds for sale under its house brand. Th e masses (in grams) of a randomly chosen pack of white sesame seeds and that of a randomly chosen pack of fennel seeds, denoted by W and F respectively, have independent normal distributions. The means and standard deviations of these distributions are shown in the following table. It is given that 5% of the packs of white sesame seeds have a mass of less than 245. (a) Show that the value of σ is 3.0398, correct to 5 significant figures. [2] (b) Find the probability that the total mass of six randomly chosen packs of white sesame seeds is less than five times the mass of a randomly chosen pack of fennel seeds by not more than 20 grams. [3] (c) n packs of white sesame seeds and n packs of fennel seeds are chosen at random. Find the smallest value of n such that the probability that the mean mass of these 2n packs being at least 278 grams is less than 0.015. [4] Mass of a pack of Mean (g) Standard deviation (g) white sesame seeds (W) 250 σ fennel seeds (F) 300 2.5

5 © HCI 2023 11 Researchers at University S claimed that Fungus A, a strain of fungi that is typically found in soil and plants, was able to break down polypropylene, a plastic that is hard to recycle, that had been pre-treated with either UV light or heat by at least 27 grams per 100 grams of polypropylene over 90 days. Researchers at University M believe that the researchers at University S overstated their claim and decided to perform a hypothesis test on a random sample of 120 batches of polypropylene, each of mass 100 grams. The researchers recorded the amount x grams of polypropylene for each batch that Fungus A was able to break down over a 90- day period. The results are summarised as follows: ( 27) 81x   , 2( 27) 2070.8x  (a) Calculate the unbiased estimates of the population mean and variance. [3] (b) Test, at 5% level of significance, the claim that the population mean mass of polypropylene per 100 grams that were broken down is at least 27 grams. You should state your hypotheses and define the symbols you use. [4] (c) State, in the context of the question, what you understand by the expression ‘at 5% level of significance’ found in part (b). [1] (d) Explain if it is necessary for University M to assume a normal distribution for the mass of polypropylene per 100 grams that Fungus A was able to break down for this test to be valid. [1] University S responded by experi