MATHEMATICS Paper 2 22 nd September 20. Answer Papers List of Formulae (MF15)

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1 NANYANG JUNIOR COLLEGE JC PRELIMINARY EXAMINATION Higher MATHEMATICS 9740 Paper d September 0 3 Ho Additioal Materials: Cover Sheet Aswer Papers List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your ame ad class o all the work you had i. Write i dark blue or black pe o both sides of the paper. You may use a soft pecil for ay diagrams or graphs. Do ot use staples, paper clips, highlighters, glue or correctio fluid. Aswer all the questios. Give o-eact umerical aswers correct to 3 sigificat figures, or 1 decimal place i the case of agles i degrees, uless a differet level of accuracy is specified i the questio. You are epected to use a graphic calculator. Usupported aswers from a graphic calculator are allowed uless a questio specifically states otherwise. Where usupported aswers from a graphic calculator are ot allowed i a questio, you are required to preset the mathematical steps usig mathematical otatios ad ot calculator commads. You are remided of the eed for clear presetatio i your aswers. At the ed of the eamiatio, faste all your work securely together. The umber of marks is give i brackets [ ] at the ed of each questio or part questio. This documet cosists of 6 prited pages.

2 Sectio A: Pure Mathematics [40 marks] 1 The first four terms of a sequece of umbers are 10, 6, 5 ad 7. S is the sum of the first terms of this sequece. Give that S is a cubic polyomial i, fid S i terms of. [4] 3 17 Show that U = + 17, where U deotes the th term of the sequece. [] Fid the set of values of for which S < 3U. [] O separate diagrams, draw sketches of the graphs of (3 ) y =, 1+ (ii) y ( 3 ) = 1+, icludig the coordiates of the poits where the graphs cross the aes ad the equatios of ay asymptotes. You should show the features of the graphs at the poits where it crosses the -ais clearly. Show that the area of the regio eclosed by the graph i (ii) may be epressed i the form d. (4 ( 1) ) By usig the substitutio 1 = siθ, evaluate this area eactly. [10] z iz + 8 3i z 5+ i= 0, give that oe of the three roots is real. [5] 3 3 (a) Solve ( ) ( ) (b) The comple umber u is give by u = cosθ + isiθ, where 0 π < θ <. Show that 1 u = iusiθ ad hece fid the modulus ad argumet of 1 u i terms of θ. [4] (ii) Give that ( ) 10 1 u is real ad egative, fid the possible values of θ i terms of π. [3] H Math Prelim /0

3 4 [I this questio, you may use the result that for a circle with radius r, a sector with agle θ has arc legth rθ ad area (a) 1 r θ.] A circle of radius r is divided ito 16 sectors of decreasig arc legth. Let L ad A be the arc legth ad the area of the th sector respectively. Suppose L is a arithmetic sequece with first term r ad commo differece d. Show that π 8 d = r 60. [] (ii) Show that A is a arithmetic sequece. [3] (b) Let G be the area of a sector of a circle with radius a. Suppose that geometric sequece with first term a ad commo ratio r, where 0< r < 1. G is a If N sectors are eeded to form the circle, show that r satisfies the equatio N r πar+ ( πa 1) = 0. [3] (ii) If a ifiite umber of sectors are eeded to form the circle, fid r i terms of a. [ Sectio B: Statistics [60 marks] 5 A compay sells a certai brad of baby milk powder ad would like to gather feedback o their product. Eplai why quota samplig is appropriate i this situatio ad describe briefly how a sample of 50 could be chose usig quota samplig. [3] The compay wishes to radomly reward 5 customers with free milk vouchers through a lucky draw. Suppose that 000 customers qualify for the draw, show that there will be equal probability of a particular customer beig the first to be selected or the third to be selected for the free milk vouchers. [] H Math Prelim /0 3 [Tur Over]

4 6 The mass, i grams, of a ice-cube has the distributio N( µσ, ). The mea mass of a radom sample of ice-cubes is deoted by X. It is give that P( X < 35.0) = ad P( X 0.0) = Obtai a epressio for σ i terms of. [3] (ii) Fid P( X > 3). [] Assume ow that the mass of a ice-cube has the distributio N(5,50). A ice dispeser discharges 15 ice cubes each time ito a cup. State the distributio of the mass of a discharge of 15 ice cubes. [1] (iv) Fid the mass eceeded by 10% of these discharges, correct to 1 decimal place. Fid the probability that the mass of the first discharge of ice-cubes is more tha the secod discharge. [] [] 7 A team of 5 me ad 5 wome is to be picked from 8 me ad 9 wome such that two of the 9 wome, A ad Lucy, must both be selected or ot at all. Fid the umber of ways i which this ca be doe. [] Assume ow the team is selected ad A, Carrie ad Lucy are icluded. The selected team is to form a queue. Fid the umber of possible arragemets if A a (ii) O aother occasio, the selected team is required to be seated at a roud table with 10 H Math Prelim /0 4

5 8 Two teams, the Ramblers ad the Strollers, meet aually for a quiz which always has a wier. If the Ramblers wis the quiz, the probability of them wiig the followig year is 0.7. If the Strollers wis the quiz, the probability of them wiig the followig year is 0.5. The Ramblers wo the quiz i 015. Fid the probability that the Strollers will wi i 018. [] (ii) If the Strollers were to wi i 018, what is the probability that it will be their first wi for at least three years sice 015? [] Assumig that the Strollers wis i 018, fid the smallest value of such that the probability of the Ramblers wiig the quiz for cosecutive years after 018 is less tha 5%. [3] 9 It is believed that the probability p of a radomly chose pregat woma givig birth to a Dow Sydrome child is related to the woma s age, i years. The table gives observed values of p for 6 differet values of p Sketch the scatter diagram for the give data. [1] (ii) Fid, correct to 4 decimal places, the product momet correlatio coefficiet betwee (a) p ad, (b) l p ad, (iv) (c) p ad. [] Usig the most appropriate case from part (ii), fid the equatio which best models the probability of a pregat woma givig birth to a Dow Sydrome child at differet ages. [] Hece, estimate the epected umber of childre with Dow Sydrome that will be bor to H Math Prelim /0 5 [Tur Over]

6 10 At a early stage i aalysig the marks,, scored by a large umber of cadidates i a eamiatio paper, the Eamiatio Board takes the scores from a radom sample of 50 cadidates. The results are summarised as follows: =1187 ad = Calculate ubiased estimates of the populatio mea ad variace to 3 decimal places. [] (ii) I a 1-tail test of the ull hypothesis µ = 49.5, the alterative hypothesis is accepted. State the alterative hypothesis ad fid a iequality satisfied by the sigificace level of the test. [4] It is subsequetly foud that the populatio mea ad stadard deviatio for the eamiatio paper are 45.9 ad respectively. Fid the probability that i a radom sample of size 50, the sample mea is at least as high as the oe foud i the sample above. [] 11 O a typical weekday morig, customers arrive at the post office idepedetly ad at a rate State, i cotet, a coditio eeded for the umber of customers who arrived at the post o (ii) Fid the probability that o more tha 4 customers arrive betwee a.m. ad 11.3 The period from a.m. to a.m. o a Tuesday morig is divided ito 6 periods of 5 miutes each. Fid the probability that o customers arrive i at most oe of these periods. [] The post office opes for 3.5 hours each i the morig ad afteroo ad it is oted that o a typical weekday afteroo, customers arrive at the post office idepedetly ad at a rate of 1 per 10 miute period. Arrivals of customers take place idepedetly at radom times. (iv) Show that the probability that the umber of customers who arrived i the afteroo i (v) Fid the probability that more tha 38 customers arrived i a morig give that a total of 40 customers arrived i a day. [4] (vi) Usig a suitable approimatio, estimate the probability that more tha 100 customers arri H Math Prelim /0 6