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2 Section A: Pure Mathematics [40 Marks] 1 Express 2cos sin 2 TT in the form sin sinabTT , where a and b are constants to be found. [2] Hence, find the exact value of D , where 0 DS , for which cos sin 2313cos cos e d 4 122 e S TT D TT T§· § · ¨¸ ¨ ¸©¹ © ¹ ´ µ¶ .[ 5] 2( i ) Show that 231 2 1 2 3 2 5 (2 1)(2 3)(2 5) Ar B rr r r r r , where A and B are constants to be found. [2] (ii) Hence find 1 29 (2 1)(2 3)(2 5) n r r rr r

¦ . (There is no need to express your answer as a single algebraic fraction.) [4] (iii) It is given that 1 29 (2 1)(2 3)(2 5) n r r rr r

¦ is within 0.01 of the sum to infinity. Write down an inequality in terms of n, and hence find the smallest possible value of n. [3] 3 The function f is defined by 2 51 1f: , , 2 2 xxxx x x z6\ . (i) Find the equations of the asymptotes of the curve f( )yx .[ 3] (ii) Determine whether f has an inverse, justifying your answer. [2] Given that the function g is defined by g : f( ), , 2 4xx x x 6\ - , find 1g x and state the domain of 1g .[ 4] Sketch the graph of 1ggyx .[ 2] www.KiasuExamPaper.com 775

3 [Turn over 4 A curve C has parametric equations 2 4xt , lnty t ,w h e r e 0t ! . (i) Show that 2 3 ln 1 4d d tty xt .[ 3] (ii) Find the exact coordinates of the turning point on C, and explain why it is a maximum. [4] (iii) Sketch C.[ 3] (iv) Show that the area bounded by C and the lines 13x and 5x is given by 3 2 1 ln d 4 t t t ´ µ¶ . Find the area, giving your answer to 4 decimal places. [3] Section B: Probability and Statistics [60 Marks] 5 Mr and Mrs Lee participate in a game show, together with 3 othe r men and 5 other women. In the first round, the 10 participants are grouped into 5 pairs. (i) Find the number of ways the pairings can be done if there is only 1 pair of the same gender. [3] After the first round, Mr and Mrs Lee are both eliminated. The remaining 8 participants are seated around a round table. Find the number of ways this can be done if (ii) there are no restrictions, [1] (iii) the 3 men are not all seated together. [3] www.KiasuExamPaper.com 776

4 6 As the use of email becomes more prevalent, the number of unsol icited email (also known as spam) received increases. Besides advertisements, spam can now b e cleverly disguised as business emails and contain malware. Hence, there is a need to use spam filter. The probability that Yip receives a spam email is p. He uses a spam filter, Spam Guard Plus, to filter his emails. He has the following information: P(an email is classified correctly) = 41 50 ; P(an email is classified correctly | it is classified as spam) = 38 45 ; P(an email is classified correctly | it is a spam email) = 19 20 . (i) Show that p = . Hence, complete the probability tree below. [5] Actual Classified Spam Spam Non-spam Spam Non-spam Non-spam (ii) Andy and Betty notices that, on average, 30% and 70% of their ema ils are spam respectively. State whether Spam Guard Plus would be (a) more appropriate for Andy, (b) more appropriate for Betty, or (c) just as appropriate for both Andy and Betty. Justify your answer. [2] 4 5 www.KiasuExamPaper.com 777

5 [Turn over 7 Seven members of the school cross country team undergo a new training programme to improve their fitness. During a particular session, each of them has to complete a 200 metre run and to achieve as many push-ups as possible in one minute. The times ta ken for the 200 metre run, t seconds, together with the number of push-ups each runner achieves, n, are shown in the table. Student A B C D E F G t 38.3 42.1 35.1 40.1 32.0 31.6 41.0 n 44 35 48 42 49 49 40 (i) Draw a scatter diagram to illustrate the data, labelling the axes. [1] (ii) Explain using your scatter diagram why the linear model n = at + b would not be appropriate. [1] It is thought that the relationship between n and t can be modelled by one of the formulae

2 30nc t d or 3 30ne t f where c, d, e and f are constants. (iii) The product moment correlation coefficient between n and 2 30t is 0.980 , correct to 3 decimal places. Determine, with a reason, which of the 2 models is more appropriate. [2] (iv) It is known that student H is able to do 48 push-ups in one minute. It is required to estimate student H’s timing for the 200 metre run. Find the equation of a suitable regression line, and use it to find the required estimate. Comment on the reliability of this estimate. [5] 8 A bag contains two balls numbered 3, n balls numbered 2 and three balls numbered 1. A player picks two balls at random from the bag at the same time. If the difference between the numbers on the two balls is 2, the player receives $6. If the difference between the num bers on the two balls is 1, the player does not receive or lose any money. If the numbers on the 2 balls are the same, the player loses $1. (i) Show that the largest value of n such that player is expected to receive money from this game is 8. [5] For the rest of this question, take the value of n to be 8. (ii) Show that the probability that a player loses money in a game is16 39 .[ 1] Victoria plays this game 50 times. (iii) Find the probability that she lost money for at least 20 games. [2] (iv) The probability that Victoria loses money in r games is more than 0.1. Find the set of values of r.[ 3] www.KiasuExamPaper.com 778

6 9 Exposure to Volatile Organic Com pounds (VOCs), which have been identified in indoor air, is suspected as a cause for headaches and respiratory symptoms. In door plants have not only a positive psychological effect on humans, but may also improve the air quality. Certain species of indoor plants were found to be effective removers of VOCs. A commonly known VOC is Benzene. The following data gives the benzene levels, x (in ppm) in 40 test chambers containing the indoor plant Epipremnum aureum. 40n , 26.0 30.1x ¦ , 2 26.0 214.61x ¦ . The initial mean Benzene level (in ppm) without Epipremnum aureum was found to be 26.0. (i) Test, at the 5% level of significance, the claim that the mean Benzene level, P (in ppm), has decreased as a result of the indoor plant Epipremnum aureum. You should state your hypotheses clearly. [5] (ii) S t a t e , g i v i n g a r e a s o n , w h e t h e r t h e r e i s a n e e d t o m a k e a n y a s sumptions about the population distribution of the Benzene level in order for the test to be valid. [2] The Benzene levels of another 50 test chambers containing the indoor plant Epipremnum aureum were recorded, The sample mean is x ppm and the sample variance is 8.33 ppm2. (iii) The acceptance region of a test of the null hypothesis 26.0P is 25.1x ! . State the alternative hypothesis and find the level of significance of the test. [4] (iv) If the null hypothesis is 0PP , where 0 26.0P ! , would the significance level of a test with the same acceptance region in part (iii) be larger or smaller than that found in part (iii)? Give a reason for your answer. [2] www.KiasuExamPaper.com 779

7 [Turn over 10 In this question, you should state clearly the values of the parameters of any distribution you use. A bus service plies from a point A in the city, through a point B, and then to its terminal station at point C. Journey times in minutes from A to B have the distribution 2N2 8 , 4. (i) Find the probability that a randomly selected bus journey from A to B is completed within 35 minutes. [1] The journey times in minutes from A to C, have the distribution 2N 46.2,4.8 . (ii) The journey times in minutes from B to C have the distribution 2N, PV . Given that the journey times from A to B are independent of the journey times from B to C, find the value of P and show that 2V = 7.04. [3] (iii) Find the set of values of k such that at least 90% of all journey times from A to C can be completed within k minutes. [2] The performance of the bus operation is d