2010 as Math & Stat Paper

8/8/2019 2010 as Math & Stat Paper

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2010-ASLM&SHONG KONG EXAMINATIONS AND ASSESSMENT AUTHORITY

HONG KONG ADVANCED LEVEL EXAMINATION 2O1O

MATHEMATICS AND STATFTrcS AS.LEVEL

8.30 am - 11.30 am (3 hours)This paper must be answered in English

l. This paper consists of Section A and Section B.2. Answer ALL questions in Section A, using the AL(E) answer book.3. Answer any FOUR questions in Section B, using the AL(C) answer book.4. Unless otherwise specified, all working must be clearly shown.5. Unless otherwise specified, numerical answers should be either exact or given to 4 decimal places.

o6ffi4;fr&i+t*.ffi {xffiHF{EHong Kong Examinations and Assessment AuthorityAII Rights Reserved 2010

Not to be taken away before theend of the examination session

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l.

Section A (40 marks)Answer ALL questions in this section.Write your answers in the AL(E) answer book.

It is given that/\r*2Ir*11 =t*3- * +terns involvinshigherpowers of x,\ a) 24 s76where r is a rational number and a is a non-zero integer.(a) Find the values of r and, a .I(b) Expand (8+x)3 inascendingpowersof x uptotheterminvolving 12 . Stut"therangeofx for which the expansion is valid. (6 marks)

2. Let f(x) = "2* .(a) Use trapezoidal rule with 2 intervals of equal width to find the approximate value of.lJnf(x)dx .(b) Evaluate the exact value of

uses trapezoidal rule with 2 trapeziums of unequal widths to approximateThe first trapezium has width n (Oclt


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3. An archaeologist models the presence of carbon-14 remaining in animal skulls fossil AV 4 = -fe'dtwhere I (in grams) is the amount of carbon-I4 present in the skull at time r (in years) and t is aconstant. Let As (ingrams) betheoriginalamountofcarbon-l4intheskull. Itisknownthathalfofthe carbon-I4 will disappear after 5 730 years.(a) By expressing I tn terms of I , or otherwise, find the value of & correct to 3 significant'dA

figures.(b) In an animal skull fossil,

carbon -I4 is still present.years.the archaeologist found that 30% of the original amount ofFind the approximate age of the skull correct to the nearest ten

(6 marks)

4. Let A and B be two exhaustive events of a certain sample space. Denote P(B) = b andP(Ar-:B)=c,where 0


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6. The coach of a girls school basketball team recruits new members from the Form One students, ofwhom I I .7% are taller than 152 cm . Assume that their heights are normally distributed with amean of p cm and a standard deviation of 5 cm.(a) Find the value of p(b) It is known that}Oo/o of the Form One students taller than 152 cm do not apply to join thebasketball team, while l0% of students shorter than 152 cm apply to join.If a Form One student is selected at random, find the probability that

(D the student applies to join the basketball team;(ii) the student is shortbr than 152 cm given that she does not apply to join the basketball

team. (7 marks)

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Section B (60 marks)Answer any FOUR questions in this section. Each question caries 15 marks.Write your answers in the AL(C) answer book

7. Define f(r)=I{ where a isaconstantand x +-a . Let C1 bethecurve y=f(x) withx+athe vertical asymptote x =2(a) (i) Find the value of a .

(ii) Find the equation(s) of the horizontal asymptote(s) of C1 .(iii) Find f'(x) and hence find the range of x for which f(x) is decreasing.(iv) Sketch the curve of C1 . Indicate its asymptote(s) and intercept(s).

(8 marks)

O) Let C2 bethecurve y=g(x),where g(x)=f(x-2)+2 .(i) Using the result in (aXrv), or otherwise, sketch the curve C2 . Indicate its asymptote(s)

and intercept(s).(ii) Let C3 be the curve y =lg(x)12 . Find the area of the region bounded by the curveC3 and the axes.

(7 marks)

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8. A company launches a campaign to increase the sales of a product. The monthly increase in sales(in thousand dollars) I months after the launch can be modelled by the functionf (t) = -25g"2ot + 3ooeot - 5o

where a is a non-zero constant.It is known that the monthly increase in sales attains the maximum 5 months after the launch.(a) Find the value of a . (3 marks)(b) After at least 7j months, the campaign will not increase the sales.

(D Find the value of fi .(iD Estimate the total amount of sales increased 7j months after the launch. (5 marks)

(c) The start up cost of the campaign is 100 thousand dollars and the running cost at time I isl0? tfrourund dollars. The campaign will be terminated after T2 months when the totalt+9expenditure reaches 200 thousand dollars.(D Express the total expenditure E (in thousand dollars) in terms of I .(iD Find the value of 12 .(iii) During the period of the campaiga, the manager of the company suggests replacing thecampaign by a less costly plan. The monthly increase in sales (in thousand dollars) dueto the plan can be modelled by the function

g(t)=-(t -a)(t -2a) , a 1t


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9 . The population of a kind of bacterium p(t) at time r (in days elapsed since 9 am on 1614/2010 ,and can be positive or negative) is modelled by

P(t)=--!- '+c ' -@(l(ob+e-'where a , b and c are positive constants. Define the primordial population be the population ofthe bacterium long time ago and the ultimate population be the population of the bacterium after along time.

(a) Find, in terms of a , b and c ,(i) the time when the growth rate attains the maximum value;(iD the primordial population;(iii) the ultimate population.

(5 marks)

(b) A scientist studies the population of the bacterium by plouing a linear graph of ln[p(t) - c]against ln(b + e-t ) and the graph shows the intercept on the vertical axis to be ln 8000 . Ifat 9 am on 16/4/2010 the population and the growth rate of the bacterium are 6000 and2000 per day respectively, find the values of a , b and c . (3 marks)

(c) Another scientist claims that the population of the bacterium at the time of maximum growthrate is the mean of the primordial population and ultimate population. Do you agree? Explainyour answer. (2 marks)

(d) By expressing e-t in terms of a , b ,iror ) - al[p1) - BJ, where a < BoHence express a and B in terms of a , bSketch p'(/) against p(t) for d,


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10. The marketing manager of a hotel studies the number of wedding banquets held per week in thehotel for the past 52 weeks. He proposes two Poisson models to explain the observed data in thefollowing table. He uses the sample mean of the observed data to estimate 21 .Number of weddingbanquets per week Observedfrequency fo

* Expected frequency f"Po( h) Po( 1z)0 7 a kI l8 b 12.682 t2 c t3.943 ll 8.07 10.234 4 d 5.62* Correct to 2 decimal places.

(a) (D Find the sample mean of the observed data.(iD Find 12 correct to 2 decimal places.(iii) Findthevaluesof a, b, c, d and t correctto 2 decimalplaces.(iv) The absolute value of the difference between the observed frequency and thecorresponding expected frequency is defined as error. The distribution with a smallertotal sum of errors is regarded as fitting the observed data better. Which distribution fitsthe observed data better? Explain your answer. (8 marks)

(b) The distribution that fits the observed data better in (a)(iv) is adopted. Assume that theexpense per table in a wedding banquet follows a normal distribution with a mean of $ 7 388and a standard deviation of $ I 200 .(D Find the probability that the expense per table is between $ 6 188 and $ 8 888 in acertain wedding banquet.(ii) Given that there are at most 3 wedding banquets in a certain week, find the probability

that there is exactly I banquet of which the expense per table is between $ 6 188 and$8888. (7 marks)

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I l. In a promotion period of an electronic shopping card with spending limit of $3 000 , cardholderswho spend over $400 in the maximum amount transaction are classified as VIPs and are eligiblefor entering an online "click-and-get-point" game once. The rules of the game are detailed in thefollowing table.Spending ($ x ) VIP Category Number of clicks allowed400


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12. A manufacturer produces a specific kind of tablets. He uses one machine to produce ingredient Iand ingredient B , and then one mixer to mix the ingredients to produce the tablets and pack themin bags. The bags of tablets are then delivered to a hospital.Past records indicate that 0.6Yo of ingredients A and B respectively are contaminated during theingredient production process, while 0.1% of the tablets are contaminated during the mixing andpacking process. A tablet is regarded as a contaminated tablet if

o the ingredient A in the tablet is contaminated, or. the ingredient B in the tablet is contaminated, oro the tablet is contaminated during the mixing and packing process.The pharmacist of the hospital draws a random sample of 20 tablets from each bag to test forcontamination. A bag is considered unsafe if it contains more than I tablet tested positive as acontamindted tdblet.(a) Find the probability that a randomly selected tablet from a certain bag is a contaminqted tablet.

(3 marks)

(b) Find the probability that a bag oftablets is regarded unsafe. (2 marks)(c) In a certain week, 100 bags of such tablets are delivered to the hospital. The hospital will

suspend the supply of the tablets from the manufacturer if more than 4 bags are found unsafewithin a week.(i) Find the probability that the l0th bag will be the first one which is regarded unsafe.(ii) Find the probability that the supply from the manufacturer will be suspended in a certainweek.

(5 marks)

(d) The manufacturer wants to increase the production and requires the probability of a tabletbeing contaminated to be less than lYo. To achieve this, he plans to add r new machines forproducing the ingredients I and B which has contamination probability of 0.4%respectively. Suppose equal amount ofingredients I and B are produced by the originalmachine and each of the n new machines.(D Express the probability that the ingredient I is contaminated in terms of n .(iD What is the least value of n ? (5 marks)

END OFPAPER

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Table: Area under the Standard Normal Currez

0.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.5t.61.71.81.92.02.12.22.32.42.52.62.72.82.93.03.13.23.33.43.5

0000 .00400398 .043 80793 .0832tt79 .t217t554 . I 591l9l5 .19502257 .22912580 .26n288 I .29103r59 .31863413 .34383643 .36653849 .38694032 .40494192 .42074332 .43454452 .44634554 .45644641 .46494713 .47 t94772 .47784821 .48264861 .48644893 .489649t8 .49204938 .49404953 .495s4965 .4966497 4 .497 s4981 .49824987 .49874990 .49914993 .49934995 .499s4997 .49974998 .4998

0080 .01200478 .05170871 .09101255 .12931628 .1664r 985 .20192324 .23572642 .26732939 .29673212 .32383461 .34853686 .37083 888 .39074066 .40824222 .42364357 .43704474 .44844573 .45824656 .46644726 .47324783 .47884830 .48344868 .487 |4898 .49014922 .4925494r .49434956 .49574967 .49684976 .49774982 .49834987 .49884991 .499t4994 .4994499s .49964997 .49974998 .4998

0160 .01990557 .05960948 .0987133 l .13681700 .17362054 .20882389 .24222704 .2734299s .30233264 .32893508 .35313729 .3749392s .39444099 .41 l54251 .42654382 .43944495 .45054591 .45994671 .46784738 .47444793 .47984838 .4842487 s .48784904 .49064927 .49294945 .49464959 .49604969 .49704977 .49784984 .49844988 .49894992 .49924994 .49944996 .49964997 .49974998 .4998

0239 .02790636 .067 st026 .1064t406 .14431772 . l 8082t23 .21572454 .24862764 .27943051 .307833 l5 .33403554 .35773770 .37903962 .39804t3t .41474279 .42924406 .44184515 .4s254608 .46164686 .469347 50 .47 564803 .48084846 .4850488 t .48844909 .49n4931 .49324948 .49494961 .49624971 .49724979 .49794985 .49854989 .49894992 .49924994 .49954996 .49964997 .49974998 .4998

03 l9 .035907 t4 .07 53,I 103 .11411480 .t5t7t844 .18792190 .22242517 .25492823 .28523 106 .3t33336s .33 893599 36213810 .38303997 .40154162 .41774306 .43194429 .444t4535 .45454625 .46334699 .470647 6t .47 674812 .48174854 .48574887 .489049t3 .49164934 .49364951 .49524963 .49644973 .497 44980 .49814986 .49864990 .49904993 .4993499s .49954996 .49974997 .49984998 .4998

Note : An entry in the table is the proportion of thebetween z = 0 and a positive value of z .obtained by symmetry.area under the entire curve which isAreas for negative values of z are

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2010 as Math & Stat Paper