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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 7