HCI 2022 FM Prelim P2 - modified

2022 HCI Prelim Paper 2_Modified [100 marks] Section A: Pure Mathematics [50 marks] 1 Two curves 1C and 2C are given by the equations 2 2 1 2 : cos , : 1 sin , 4 4 C y x C y x   =−  = − −  for 0 2x  . (i) Find the exact x-coordinates 1x and 2x of the points of intersection of 1C and 2C . [3] The region bounded by 1C and 2C between 1x and 2x is given by R . (ii) Find the exact volume when R is rotated by 2 radians about the y-axis. [3] 2 In this question, all matrices A are square matrices whose elements are real numbers. For each of the following statements, provide a proof if it is true or a counter example if it is false. (a) Every A that has zero as one of its eigenvalues is not invertible. [2] (b) Every A that is a diagonal matrix is diagonalisable. [2] (c) Every A that is diagonalisable is also a diagonal matrix. [2] (d) Every A that is diagonalisable has distinct eigenvalues. [2] 3 (i) Illustrate in an Argand diagram, the set of points z for which 2Re( ) 0zz+ . [4] (ii) Hence, or otherwise, determine the values of 1 2 1 2 22zz +−+ − + for z satisfying 2Re( ) 0zz+= . [3] 4 Let A and B be 22 matrices whose elements are real numbers. (i) Let T be the set of linear combinations of the columns of AB . Show that T is a subset of the column space of A . Hence show that T is a subspace. [4] (ii) Hence show that )rank ( r n) a k (AB A . [2] (iii) Show that if B is invertible, then )rank ( r n) a k (=AB A . [2]

3 5 Adnan received the following message. The viability of the offer is investigated. Assuming that the offer is taken up, let nu be the amount of money in the account including interest at the end of the nth month. (i) State the value of 1u and write down a recurrence relation relating nu and 1nu + for 1n in terms of k . [3] (ii) Find the general solution of the recurrence relation in part (i) in terms of k and n . [2] (iii) Suppose that the amount of money in the account is at least one million dollars after 10 years, find the minimum value of k . [2] (iv) Using the result from part (iii), advise Adnan if he should take up the offer. Give reason(s) to support your answer. [2] 6 By considering the substitution d d y px = , (a) solve the differential equation 2 2 dd 1 1dd yy xxx −= , where 0x . [5] (b) show that the differential equation 32 2 dd e dd yyy xx =  can be expressed as 2 d1 ed yp yp = . It is given that when 0x= , 0y= and d 1d y x =− . Solve the differential equation , leaving your answer in the form f ( )yx= , where 01 x . [7] Hey, you want make big money? We are registered financial firm and have unique offer just for you! Every month, you pay only $1000. An admin fee of 10% on your payment is charged and remainder goes into your account. End of every month, you earn a monthly k% COMPOUNDED interest and we know how fast compound interest grows! You confirm a millionaire in 10 years and trillionaire in no time! Don't wait no more, contact me ASAP at +65 XXXX XXXX before offer expires!

4 Section B: Probability and Statistics [50 marks] 7 In a straw poll, 7 students were asked to rate the quality of their learning experiences on a scale of 1 to 10 (1 being the lowest and 10 being the highest) before and after the implementation of Home Based Learning (HBL) . It is known that each student gave a different rating before and after the implementation of HBL. The results are shown in the table below, where a is a positive integer. Student A B C D E F G Before HBL 4 8 7 7 1 9 9 After HBL 6 9 3 4 a 1 2 (a) Given that there is no tied ranking among the students, carry out a Wilcoxon matched-pairs signed rank test, at the 5% significance level, to determine if the data supports the hypothesis that the quality of their learning experiences was lower after HBL was implemented. State an assumption for the Wilcoxon test to be valid in this context. [5] (b) Explain why Wilcoxon matched-pairs signed rank test is more appropriate than a paired-sample t-test in testing whether HBL lowers the quality of learning experiences. [1] 8 The continuous random variable X has cumulative distribution function F given by 2 0 , 0; 1 , 0 1;3F( ) 1 (2 1) (2 ) , 1 2;3 1 , 2; x x kx x x x k x x x    +  =  − + −     where k is a real constant. (i) Find the range of values that k can take. [3] (ii) Find in terms of k , (a) the first quartile q where 1P( ) 4Xq= . [3] (b) the exact value of P( 4)Y  given that 211 6Y X X=− . [3]

5 [Turn over 9 A beverage manufacturer produces instant coffee mix in sachets, and claims that the mass of sachets follows a normal distribution. A random sample of 100 sachets of the coffee mix is taken and the mass of each sachet measured. The data are summarised as follows. Mass (g) 16-18 18-19 19-20 20-21 21-22 22-25 Frequency 9 25 27 16 19 4 (i) Carry out a chi-squared goodness-of-fit test on the manufacturer’s claim. Discuss what the test indicates about the manufacturer’s claim. You should refer to the level of significance and the p-value for your test. [9] (ii) It is now given that the population mean mass of sachets of coffee mix is known, and the mean mass obtained from the sample above is exactly the same as the population mean mass of sachets of coffee mix. With the help of a suitable diagram or otherwise, comment on the p-value thus obtained with reference to the p-value found in your test. [2] 10 The duration of a written examination for a particular subject is 3 hours. Candidates write their answers in a single 12-page answer booklet, and may request for additional 4-page answer booklets if they run out of writing space in the 12-page answer booklet. In the first hour of one such examination, the number of additional 4 -page answer booklets requested by candidates from classes A , B and C are assumed to follow independent Poisson distributions with mean s 0.2, 0.3 and  respectively. The probability that no candidates from these 3 classes requested for additional 4-page answer booklets in the first hour is 1 2 . (i) Show that 1ln k k =− , where k is a constant to be determined. [2] For the remaining parts of the question, use the value of  found in part (i). (ii) Find the probability that at least one of the 3 classes has no candidates requesting for the additional 4-page answer booklets in the first hour of the examination. [2] (iii) Let M be the minimum number of additional 4 -page answer booklets requested among the 3 classes in the first hour. Find P( 1)M  . [2] (iv) Find the probability that the mean number of additional 4 -page answer booklets requested by the 3 classes in the first hour is 1, given that there are at least 2 classes with exactly 1 additional 4-page answer booklets requested by the 3 classes during the first hour. [3] (v) Let m denote the most likely total number of additional 4 -page answer booklets requested by the 3 classes in the first hour of the examination. Find m. Comment with justification if m is also the most likely total number of additional 4- page answer booklets requested by the 3 classes in the second hour of the examination. [3]

6 11 A medical student investigated the masses of new-born babies. A first baby is defined as the first child borne by a woman. A random sample of 20 new-born first babies was taken and their masses in kilograms (kg) measured. The 95% confidence interval of the average mass of new-born first babies constructed from these data is (3.056,3.384) . (i) Explain the significance of a 95% confidence interval in this context. [1] (ii) By stating clearly